Research ArticleAParticle SwarmOptimization Algorithm for Solving Pricing andLead Time Quotation in a Dual-Channel Supply Chain withMultiple Customer Classes

Mahboobeh Honarvar 1 Majid Alimohammadi Ardakani2 and Mohammad Modarres3

1Department of Industrial Engineering Yazd University Yazd Iran2Department of Industrial Engineering Ardakan University Yazd Iran3Department of Industrial Engineering Sharif University of Technology PO Box 14588-89694 Tehran Iran

Correspondence should be addressed to Mahboobeh Honarvar mhonarvaryazdacir

Received 29 September 2019 Revised 5 January 2020 Accepted 16 March 2020 Published 22 April 2020

Academic Editor Viliam Makis

Copyright copy 2020Mahboobeh Honarvar et alis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

e combination of traditional retail channel with direct channel adds a new dimension of competition to manufacturersrsquodistribution system In this paper we consider a make-to-order manufacturer with two channels of sale sale through retailers andonline direct sale e customers are classified into different classes based on their sensitivity to price and due date e orders oftraditional retail channel customers are fulfilled in the same period of ordering However price and due date are quoted to theonline customers based on the available capacity as well as the other orders in the pipeline We develop two different structures ofthe supply chain centralized and decentralized dual-channel supply chain which are formulated as bilevel binary nonlinearmodelse Particle SwarmOptimization algorithm is also developed to obtain a satisfactory near-optimal solution and comparedto a genetic algorithm rough various numerical analyses we investigate the effects of the customersrsquo preference of a directchannel on the modelrsquos variables

1 Introduction

e rapidly expanding Internet provides an opportunity fororganizations to distribute their products via both directchannel and traditional retail channel Some personalcomputer manufacturers (like Dell company) apparel re-tailers and automotive industries (like General Motors) areexamples of companies that use hybrid of both direct andretailer channels

In the direct channel the firm interacts with consumersdirectly through Internet ere are a number of benefitsfrom direct channel distribution such as controlling thedistribution and pricing directly providing a broaderproduct selection and improving firmsrsquo visibility [1]

Despite hybrid channelrsquos benefits such as capturing alarger share of the market combining the retail distributionchannel with direct channel may pose some challengesincluding pricing policies distribution strategies and

conflicting demands placed on internal company resourcessuch as capital personal products and technology bymultiple channels [2]

To overcome these challenges supply chainrsquos membersnegotiate the retail as well as wholesale price to cooperatewith each other In addition to the product price there areother factors such as product availability and service qualitythat contribute to consumer preference of the direct channel[3]

Delivery lead time is one of the important factors thatcan affect customersrsquo demand [3] is is the reason manye-retailers such as Amazoncom BestBuycom Walmartcom and FYEcom try to offer competitively quoted leadtimes [4]

is paper is focused on competitive pricing strategy aswell lead time decisions in a supply chain for a manufacturerthat sells the products through two channels One channel isthe traditional one in which the firm uses an intermediary to

HindawiAdvances in Operations ResearchVolume 2020 Article ID 5917126 21 pageshttpsdoiorg10115520205917126

reach final consumers while the other is a direct channel inwhich the customer places direct orders through the In-ternet It is assumed that there are multiple classes of cus-tomers in the market e demand of each class depends onthe price and lead time A finite planning horizon is con-sidered where the production capacity in each period isfinite but varies e manufacturer has to respond to thecustomers quickly based on the available capacity and theorders in pipeline Due to the finite production capacity andthe demand sensitivity to price and lead time changes themanufacturer needs to decide on the selling price and thelead time of direct customers the contract terms with theretailer and the wholesale price as well as the productionschedule to maximize his own profit

We consider two different cases centralized anddecentralized dual-channel supply chain We propose amodel to determine suitable lead time and price simulta-neously for each case In a decentralized dual-channel supplychain the two members interact within the framework ofStackelberg game Under this framework the manufactureras the leader determines the wholesale price for retailer andalso a price and lead time for direct sale in each period eintermediary then reacts by choosing a retail price tomaximize its own profit On the other hand for a centralizeddual-channel supply chain the direct sale price the tradi-tional retail price and the quoted lead time in the directchannel are determined by a vertically integratedmanufacturer

e resulting models happen to be binary nonlinearprograms ese kinds of problems with large dimensionsare not usually solved by exact algorithms erefore analternative method for solving the models is to use meta-heuristic algorithms to obtain a near-optimal solution withreasonable computational time us an algorithm ofParticle SwarmOptimization (PSO) is developed to solve themodels e results will be compared in terms of solutionquality and computational time with genetic algorithm(GA)

e remainder of this paper is structured as followsSection 2 reviews the more relevant literature We formulatethe models in Section 3 e principles of the PSO meta-heuristic and our algorithm are explained in Section 4Section 5 presents some numerical examples of our model toexamine the effects of customer preference of a directchannel on the pricing strategies and lead time decisionsOur conclusions are summarized in Section 6

2 Literature Review

is paper focuses on competition in a dual-channel supplychain A comprehensive review of multichannel models canbe found in [1 5]

Several researchers and practitioners have focused ondual-channel supply chains during the last decade Differentaspects of this chain are investigated Most of the papers thatformulated the dual-channel supply chain focused on thecompetition context in the supply chain and pricing opti-mization issues in each channel [6ndash9] e researcherscombined the pricing issues with other aspects of supply

chain In this regard we can refer to sales effort determiningand service management in each channel [10ndash13] contractoptimization [6 14] disruption management [15ndash17]product variety in supply chain [18 19] and multiperiodmodel [20]

Besides parameters such as price services and qualitythat can affect the demand process the lead time (or duedate) is also an important parameter Some researchers haveconsidered that the quoted lead times (or due dates) alsoaffect customersrsquo decisions on placing an order (eg Due-nyas and Hopp [21] So and Song [22] Easton and Moodie[23] Keskinocak et al [24] Webster [25] Watanapa andTechanitisawad [26] Charnsirisakskul et al [27] Mustafaet al [28] Chaharsooghi et al [29] and Chaharsooghi et al[30]) However the above papers have not addressed thedual-channel distribution issue e papers by Chen et al[31] Hua et al [3] Xu et al [32] Batarfi et al [33] Yang et al[4] and Modak and Kelle [34] are some examples thatinvestigated lead time optimization in the dual-channelsupply chain

Xu et al [32] considered the unit delivery cost (mt) withmgt 0 if the product is to be delivered with lead time t Batarfiet al [33] considered the combination of two productionapproaches make-to-order and make-to-stock in directonline channel and indirect offline channel respectivelyedemand depends on the prices the quoted delivery leadtime and the product differentiation

Yang et al [4] modeled delivery lead time optimizationfor perishable products in a dual-channel supply chainModak and Kelle [34] considered that the demand not onlyis dependent on price and delivery time but also is stochastic

e lead times in the above papers are determined ex-ogenously (determined by the sales department withoutknowing the actual production schedule) In addition theproduction capacity is unlimited Table 1 illustrates themajor literature review with our paper included

Based on the above literature this paper investigates thejoint decision on production pricing and lead time in adual-channel distribution system where the lead times areassigned internally by the scheduling model Aside fromconsidering the above literature we address a new model inwhich the production capacity in each period is limited Wealso consider multiple customer classes that differ in theirarrival (commitment) times quantities demanded andsensitivity to price and lead time

3 Problem Description

We consider a dual-channel supply chain in which amanufacturer sells to retailers as well as directly to endcustomers e planning horizon is assumed to be finite anddivided into periods of equal length e capacity of themanufacturer is limited but may vary in different periodsCustomers can be classified with respect to their sensitivityto price and lead time Furthermore the attributes of cus-tomers such as arrival (commitment) times quantitiesdemanded unit production and holding costs are different

In each period the manufacturer must set a wholesaleprice for the traditional retail channel as well as setting both

2 Advances in Operations Research

price and lead time (or due date) for the customers of directchannel For the customers of traditional retail channel theproduction must be scheduled at the same period at whichthe order arrives However the manufacturer can quote alead time (due date) to the customers of direct channel eproduction for these customers is scheduled within anyperiod between the arrival time of order and the quoted duedate A holding cost occurs for any order of direct channelwhich is completed before the quoted due date

As mentioned before in order to evaluate the effects ofthe delivery lead time and customerrsquos preference of the directchannel on the pricing decisions of the manufacturer andretailer we consider two different dual-channel supplychains centralized and decentralized systems

31 Notation We use the following notation

311 Sets

Ψ 1 N set of customer classes based onsensitivity to price and due dateT 1 T set of planning periods

312 Parameters

e(i) arrival time of customer of class i isin ΨChi third party holding cost per time period per unit ofcustomer of class i isin ΨCp1

it production cost of customer of class i isin Ψ in thedirect channel in period t isin T

Cp2it production cost of customer of class i isin Ψ in the

retail channel in period t isin T

Cri operational cost in the retail channel for customerof class i isin Ψ

Cei operational cost in the direct channel for customerof class i isin Ψ

Kt production capacity available in period t isin T

Drij demand (in terms of production capacity units) for

customer of class i isin Ψ in the retail channel quoted duedate j isin [e(i) T]

Dsij demand (in terms of production capacity units) for

customer of class i isin Ψ in the direct channel quoteddue date j isin [e(i) T]

313 Decision Variables

Zij 1 if the due date j j isin [e(i) T] is selected(quoted) for customer of class i isin Ψ in direct channel 0otherwisePr

i price in the retail channel for customer of class i isin ΨPs

i price in the direct channel for customer of classi isin ΨWi wholesale price charged for customer of class i isin Ψin the retail channelYit total production (in units of capacity) for customerof class i isin Ψ in the direct channel in period t isin T

Hi total inventory of customer of class i isin Ψ

32 Demand Function In line with Kurata et al [35] Caiet al [6] and Hua et al [3] we assume that the demand is alinear function of the self- and cross-price and lead time asfollows

Table 1 Related literature

Research paperProduction type

Pricingdecision

Due date quotationCapacityconstraint

Productionplanning

Customerclasses

CompetitionapproachMake-

to-stockMake-to-order Internally Exogenously

Tsay and Agrawal[13]

lowast lowast Nash

Chiang et al [7] lowast lowast Nash

Yao and Liu [9] lowast lowast Bertrand andStackelberg

Cai et al [6] lowast lowast StackelbergNash

Dan et al [10] lowast lowast StackelbergHuang et al [15] lowast lowast lowast StackelbergSoleimani et al [17] lowast lowast NashRoy et al [12] lowast lowast StackelbargHua et al [3] lowast lowast lowast NashXu et al [32] lowast lowast lowast NashBatarfi et al [33] lowast lowast lowast lowast -Yang et al [4] lowast lowast lowast NashRofin and Mahanty[18]

lowast lowast Nash

Pi et al [16] lowast lowast StackelbergModak and Kelle [34] lowast lowast lowast Stackelbergis paper lowast lowast lowast lowast lowast lowast Stackelberg

Advances in Operations Research 3

Dsi ai middot θi minus b

si middot P

si + αr

i middot Pri minus c

si middot L

si (1)

Dri ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri L

si (2)

where Dsi and Dr

i denote the consumer demand to themanufacturer and the consumer demand to the retailerrespectively ai denotes the base level of industry demandor the demand rate for customer of class i andθi(0lt θi lt 1) represents the initial portion of customers ofclass i that prefer the direct channel if the price and leadtime are zero us 1 minus θi represents the portion ofcustomers of class i that prefer purchasing from the re-tailer e coefficients bs

i and bri are the coefficients of the

price elasticity in the direct channel and retail channeldemand functions respectively e cross-price sensi-tivities αr

i and αsi reflect the substitutionrsquos degree of the

two channels and csi is the lead time sensitivity of the

demand in the direct channel If the lead time Lsi increases

by one unit cri units of demand will transfer to the retail

channel and csi units of direct channelrsquos demand will be

decreased e total demand of the two channels shouldhave a negative slope with respect to the retailerrsquos pricethe direct sale price and the quoted lead time us wehave αr

i lt bri α

si lt bs

i and csi gt cr

i Following Hua et al [3] it is assumed that the man-

ufacturer uses dual channels to sell their goods and thebase level of industry demand or demand rate in both

channels is very large ie θi should not be unreasonablysmall or large

33 Model of the Decentralized Dual-Channel Supply ChainIn this section we study a decentralized dual-channel supplychain In this model both the manufacturer and the retailermake their own decisions separately to maximize theirprofits e manufacturer as the Stackelberg leader firstdetermines the wholesale price Wi the direct sale price Ps

i and the direct sale quoted lead time Ls

i en the retailerchooses his own optimal retail price Pr

i based on themanufacturerrsquos decisions

331 Manufacturerrsquos Best Response e goal of the man-ufacturer as a leader is to maximize his profit consideringthe capacity constraints and demand constraints in thesystems and constraints determined by the retailer opti-mization problem e problem is formulated within theframework of bilevel programming (BLP) first level (themanufacturer model called the leader) and second level (theretailer model called a follower) In BLP model each de-cision maker tries to optimize its own objective functionwithout considering the objective of the other partyHowever the decision of each party affects the objectivevalue of the other one as well as the decision space

332 First-Level Model Manufacturer Model

Max Πs 1113944N

i11113944

T

je(i)

psi times D

sij times Zij1113872 1113873 minus 1113944

N

i11113944

T

te(i)

Yit times Cp1it minus Cei1113872 11138731113872 1113873 + 1113944

N

i11113944

T

je(i)

Wi minus Cp2ie(i)1113872 1113873 D

rij times Zij1113872 1113873 minus 1113944

N

i1Chi times Hi

⎛⎝ ⎞⎠

(3)

STWi lePs

i foralli isin Ψ (4)

1113944

T

je(i)

Zij 1 foralli isin Ψ (5)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (6)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)t

1113944

T

je(i)

DrijZij leKt t 1 T (7)

1113944

T

tj

Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T (8)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (9)

4 Advances in Operations Research

Dsij ai middot θi minus bs

i lePsi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) infin T (10)

333 Second-Level Model Retailer Model

MaxPr

i

Πr 1113936iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (11)

ST

1113936i|te(i)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij leKt t 1 T (12)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij ge 0 foralli isin Ψ (13)

Yit Hi Dsij Pr

i Psi Wi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(14)

e first and second terms in the first-level objectiverepresent the total revenue and production cost of directsales e third term represents the total profit obtained bysales to the retailer and the fourth one is the carrying cost

Constraint (4) indicates that the wholesale price cannotbe higher than the direct channel price otherwise the re-tailer may purchase from the direct channel at a lower priceConstraints (5) ensure that only one due date is chosen foreach direct channel customer order Constraint (6) ensuresthat if due date j is selected for customer of class i in thedirect channel then exact Ds

ij units must be produced anddelivered where Ds

ij depends on the selected price and duedate Constraint (7) is a capacity constraint that ensures thatthe production capacity in each period is not exceeded Anorder of a retail channel customer is produced in the cus-tomerrsquos arrival time period and delivered instantaneously inthe same period whereas an order of a direct channelcustomer can be produced in any period between its arrivalperiod and the quoted due date e first term in the lefthand side of constraint (7) indicates the total production fordirect channel customers and the second term is the totalproduction for retail channel customers Constraint (8)indicates that an order of a direct channel customer can beproduced in any period between its arrival period and thequoted due date

e required inventory for orders is scheduled in anyperiod prior to its commitment and the negotiated due dateis controlled by constraint (9) where M is a sufficiently large

number Constraint (10) is demand functions for customerorders in the direct channel e term j minus e(i) + 1 is a timeinterval between the arrival time of an order and the quoteddue date which is called the lead time e objectivefunction of second-level optimization problem for the re-tailer channel is represented by (11) Constraints (12) and(13) control feasibility as well as demand restrictions

Without considering constraints (12)ndash(14) for retailer itsbest response to the wholesale price can be defined inProposition 1

Proposition 1 e retailerrsquos best response to the wholesaleprice Wi the direct sale price Ps

i and the direct quoted leadtime Ls

i set by the manufacturer is as follows

Pri

ai middot 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

foralli isin Ψ

(15)

e Proof of Proposition 1 as well as the other propositionsis given in the appendix

e BLP model (4)ndash(14) can be formulated as a singlelevel mixed binary problemis is achieved by replacing thelower level problem (12)ndash(14) with its KuhnndashTucker con-ditions which we name Model I as follows

334 Model I

Max Πs 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

Wi minus Cp1ie(i)1113872 1113873 Dr

ij times Zij1113872 1113873 minus 1113936N

i1Chi times Hi

⎛⎝ ⎞⎠ (16)

ST

Advances in Operations Research 5

Wi lePsi (17)

1113936T

te(i)

Yit 1113936T

je(i)

DsijZij foralli isin Ψ (18)

1113936T

je(i)

Zij 1 foralli isin Ψ (19)

1113936iisinΨ|e(i)let

Yit + 1113936iisinΨ|e(i)t

1113936T

je(i)

DrijZij leKt t 1 T (20)

1113936T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (21)

Hi ge 1113936j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j 1 T (22)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (23)

ai 1 minus θi( 1113857 minus 2Pri middot br

i + αsi P

si + cs

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij + Wi middot bri + Cri middot br

i + λe(i) middot bri minus λiprime middot b

ri + ]i 0 foralli isin Ψ (24)

1113936i|e(i)t

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ + St Kt t 1 T (25)

ai 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij minus Siprime 0 (26)

λt middot St + λiprime middot Siprime + ]i middot P

ri 0 (27)

Yit Hi Dsij Pr

i Psi Wi λi λi

prime Siprime Si ]i ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(28)

Constraints (24)ndash(27) indicate the optimality conditionfor the retailer to convert the bilevel problem into a singlelevel one

Proposition 2 For any given due date j the wholesale priceWi and the direct sale price Ps

i for customer class i theoptimal retail price Pr

i is always positiveAs a result of Proposition 2 it can be concluded that ]i is

always equal to zero because ]i middot Pri 0 in Model I

Proposition 3 Consider an unlimited production capacityin each period for a decentralized dual-channel supply chain

If 8bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 6αs

i middot αri gt 0 the dual channel Πs is

strictly jointly concave in Wi and Psi

From Proposition 3 if αsi αr

i then 8bri middot bs

i minus (αsi )2 minus

(αri )2 minus 6αs

i middot αri is always greater than zerous in this case

the manufacturer profit Πs is always strictly jointly concavein Wi and Ps

i us for the sake of analytical tractability wewill assume that the cross-price effects are symmetric andαs

i αri

Proposition 4 Let the production capacity be unlimited ineach period for a decentralized dual-channel supply chain andsuppose there is a L0

i isin [Lli Lu

i ] with

6 Advances in Operations Research

L0i

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 (29)

en the optimal lead time Lsi isin [Ll

i Lui ] can be found

from Lli Lu

i and L0i by comparing their Πs values the one with

the largest Πs value is the optimal lead time

34 Model of the Centralized Dual-Channel Supply Chain

In a centralized dual-channel supply chain a vertically in-tegrated manufacturer controls the wholesale price Wi thedirect sale price Ps

i the direct sale quoted lead time Lsi and the

retailer sale price Pri e model called Model II is as follows

341 Model II

Max Πc Πs +Πr 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

pri minus Cp2

ie(i) minus Cri1113872 1113873 Drij times Zij1113872 1113873 minus 1113936

N

i1Chi times Hi

⎛⎝ ⎞⎠ (30)

ST

1113944

T

je(i)

Zij 1 foralli isin Ψ (31)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (32)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)let

1113944

T

je(i)

DrijZij leKt t 1 T (33)

1113944

T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (34)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (35)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (36)

Drij ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (37)

Yit Hi Dsij Pr

i Psi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(38)

e objective function (30) maximizes the total profit ofthe centralized supply chain which is the sum of manu-facturerrsquos profit and retailerrsquos profit e remaining con-straints are the same as those in Model I

Proposition 5 Consider an unlimited production capacityin each period for a centralized dual-channel supply chain If4br

i middot bsi minus (αs

i )2 minus (αr

i )2 minus 2αs

i middot αri gt 0 then the optimal lead

time Lsi isin [Ll

i Lui ] can be found from Ll

i and Lui by comparing

their Πc values the largest Πc value is the optimal lead time

4 Solution Approach

Models I and II are 0-1 nonlinear programs with continuousprice and production decision variables and binary variables

for due date selection It is well known that the problem ofnonlinear 0-1 programming is NP-hard [36] us theseproblems cannot be solved easily and exact algorithms canbe time consuming when the number of 0-1 variables islarge An alternative for solving these models is to use ap-proximate algorithms or heuristic methods with reasonablecomputational times

In this paper PSO metaheuristic is used and comparedto the GA for modeling and optimization of the pricingand due date setting problems in a dual-channel supplychain

PSO and GA can handle the whole MINLP easily andnaturally and it is easy to apply it to various problems forcomparison with conventional methods

Advances in Operations Research 7

If we fix the set of binary variables Z in the modelsthen both models are converted to quadratic programsthat can be solved by the nonlinear programming solverLINGO

41 Particle SwarmOptimization (PSO) e PSO algorithmwas first proposed by Kennedy and Eberhart [37] ismethod is based on the information shared among membersof a species and then used for evolution

Update the local best position Pkd global best

Send values of Zij to the Lingo solve the model andCalculate every particlersquos fitness value

Satisfy stopcriterion

Initialize a population of particles with random positionsand velocities

Fix the set of variables Zij in problems I and II according to thevalues for the lead times

Set up the searching range for L according to proposition 6

Initialize the parameters population size the inertial weight (ω) and two learning factors (c1 and c2)

No

Yes

End

Update the velocity and position for every particle

Lkd(new) = INT(Lkd

(old) +Vkd(new))

Vkd(new) = ωVkd

(old) + c1r1(Pkd(old) ndash Lkd

(old)) + c2r2(Pgd(old)ndashLkd

(old))

Figure 1 PSO flowchart for pricing and due date quotation

8 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

price and lead time (or due date) for the customers of directchannel For the customers of traditional retail channel theproduction must be scheduled at the same period at whichthe order arrives However the manufacturer can quote alead time (due date) to the customers of direct channel eproduction for these customers is scheduled within anyperiod between the arrival time of order and the quoted duedate A holding cost occurs for any order of direct channelwhich is completed before the quoted due date

As mentioned before in order to evaluate the effects ofthe delivery lead time and customerrsquos preference of the directchannel on the pricing decisions of the manufacturer andretailer we consider two different dual-channel supplychains centralized and decentralized systems

31 Notation We use the following notation

311 Sets

Ψ 1 N set of customer classes based onsensitivity to price and due dateT 1 T set of planning periods

312 Parameters

e(i) arrival time of customer of class i isin ΨChi third party holding cost per time period per unit ofcustomer of class i isin ΨCp1

it production cost of customer of class i isin Ψ in thedirect channel in period t isin T

Cp2it production cost of customer of class i isin Ψ in the

retail channel in period t isin T

Cri operational cost in the retail channel for customerof class i isin Ψ

Cei operational cost in the direct channel for customerof class i isin Ψ

Kt production capacity available in period t isin T

Drij demand (in terms of production capacity units) for

customer of class i isin Ψ in the retail channel quoted duedate j isin [e(i) T]

Dsij demand (in terms of production capacity units) for

customer of class i isin Ψ in the direct channel quoteddue date j isin [e(i) T]

313 Decision Variables

Zij 1 if the due date j j isin [e(i) T] is selected(quoted) for customer of class i isin Ψ in direct channel 0otherwisePr

i price in the retail channel for customer of class i isin ΨPs

i price in the direct channel for customer of classi isin ΨWi wholesale price charged for customer of class i isin Ψin the retail channelYit total production (in units of capacity) for customerof class i isin Ψ in the direct channel in period t isin T

Hi total inventory of customer of class i isin Ψ

32 Demand Function In line with Kurata et al [35] Caiet al [6] and Hua et al [3] we assume that the demand is alinear function of the self- and cross-price and lead time asfollows

Table 1 Related literature

Research paperProduction type

Pricingdecision

Due date quotationCapacityconstraint

Productionplanning

Customerclasses

CompetitionapproachMake-

to-stockMake-to-order Internally Exogenously

Tsay and Agrawal[13]

lowast lowast Nash

Chiang et al [7] lowast lowast Nash

Yao and Liu [9] lowast lowast Bertrand andStackelberg

Cai et al [6] lowast lowast StackelbergNash

Dan et al [10] lowast lowast StackelbergHuang et al [15] lowast lowast lowast StackelbergSoleimani et al [17] lowast lowast NashRoy et al [12] lowast lowast StackelbargHua et al [3] lowast lowast lowast NashXu et al [32] lowast lowast lowast NashBatarfi et al [33] lowast lowast lowast lowast -Yang et al [4] lowast lowast lowast NashRofin and Mahanty[18]

lowast lowast Nash

Pi et al [16] lowast lowast StackelbergModak and Kelle [34] lowast lowast lowast Stackelbergis paper lowast lowast lowast lowast lowast lowast Stackelberg

Advances in Operations Research 3

Dsi ai middot θi minus b

si middot P

si + αr

i middot Pri minus c

si middot L

si (1)

Dri ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri L

si (2)

where Dsi and Dr

i denote the consumer demand to themanufacturer and the consumer demand to the retailerrespectively ai denotes the base level of industry demandor the demand rate for customer of class i andθi(0lt θi lt 1) represents the initial portion of customers ofclass i that prefer the direct channel if the price and leadtime are zero us 1 minus θi represents the portion ofcustomers of class i that prefer purchasing from the re-tailer e coefficients bs

i and bri are the coefficients of the

price elasticity in the direct channel and retail channeldemand functions respectively e cross-price sensi-tivities αr

i and αsi reflect the substitutionrsquos degree of the

two channels and csi is the lead time sensitivity of the

demand in the direct channel If the lead time Lsi increases

by one unit cri units of demand will transfer to the retail

channel and csi units of direct channelrsquos demand will be

decreased e total demand of the two channels shouldhave a negative slope with respect to the retailerrsquos pricethe direct sale price and the quoted lead time us wehave αr

i lt bri α

si lt bs

i and csi gt cr

i Following Hua et al [3] it is assumed that the man-

ufacturer uses dual channels to sell their goods and thebase level of industry demand or demand rate in both

channels is very large ie θi should not be unreasonablysmall or large

33 Model of the Decentralized Dual-Channel Supply ChainIn this section we study a decentralized dual-channel supplychain In this model both the manufacturer and the retailermake their own decisions separately to maximize theirprofits e manufacturer as the Stackelberg leader firstdetermines the wholesale price Wi the direct sale price Ps

i and the direct sale quoted lead time Ls

i en the retailerchooses his own optimal retail price Pr

i based on themanufacturerrsquos decisions

331 Manufacturerrsquos Best Response e goal of the man-ufacturer as a leader is to maximize his profit consideringthe capacity constraints and demand constraints in thesystems and constraints determined by the retailer opti-mization problem e problem is formulated within theframework of bilevel programming (BLP) first level (themanufacturer model called the leader) and second level (theretailer model called a follower) In BLP model each de-cision maker tries to optimize its own objective functionwithout considering the objective of the other partyHowever the decision of each party affects the objectivevalue of the other one as well as the decision space

332 First-Level Model Manufacturer Model

Max Πs 1113944N

i11113944

T

je(i)

psi times D

sij times Zij1113872 1113873 minus 1113944

N

i11113944

T

te(i)

Yit times Cp1it minus Cei1113872 11138731113872 1113873 + 1113944

N

i11113944

T

je(i)

Wi minus Cp2ie(i)1113872 1113873 D

rij times Zij1113872 1113873 minus 1113944

N

i1Chi times Hi

⎛⎝ ⎞⎠

(3)

STWi lePs

i foralli isin Ψ (4)

1113944

T

je(i)

Zij 1 foralli isin Ψ (5)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (6)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)t

1113944

T

je(i)

DrijZij leKt t 1 T (7)

1113944

T

tj

Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T (8)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (9)

4 Advances in Operations Research

Dsij ai middot θi minus bs

i lePsi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) infin T (10)

333 Second-Level Model Retailer Model

MaxPr

i

Πr 1113936iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (11)

ST

1113936i|te(i)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij leKt t 1 T (12)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij ge 0 foralli isin Ψ (13)

Yit Hi Dsij Pr

i Psi Wi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(14)

e first and second terms in the first-level objectiverepresent the total revenue and production cost of directsales e third term represents the total profit obtained bysales to the retailer and the fourth one is the carrying cost

Constraint (4) indicates that the wholesale price cannotbe higher than the direct channel price otherwise the re-tailer may purchase from the direct channel at a lower priceConstraints (5) ensure that only one due date is chosen foreach direct channel customer order Constraint (6) ensuresthat if due date j is selected for customer of class i in thedirect channel then exact Ds

ij units must be produced anddelivered where Ds

ij depends on the selected price and duedate Constraint (7) is a capacity constraint that ensures thatthe production capacity in each period is not exceeded Anorder of a retail channel customer is produced in the cus-tomerrsquos arrival time period and delivered instantaneously inthe same period whereas an order of a direct channelcustomer can be produced in any period between its arrivalperiod and the quoted due date e first term in the lefthand side of constraint (7) indicates the total production fordirect channel customers and the second term is the totalproduction for retail channel customers Constraint (8)indicates that an order of a direct channel customer can beproduced in any period between its arrival period and thequoted due date

e required inventory for orders is scheduled in anyperiod prior to its commitment and the negotiated due dateis controlled by constraint (9) where M is a sufficiently large

number Constraint (10) is demand functions for customerorders in the direct channel e term j minus e(i) + 1 is a timeinterval between the arrival time of an order and the quoteddue date which is called the lead time e objectivefunction of second-level optimization problem for the re-tailer channel is represented by (11) Constraints (12) and(13) control feasibility as well as demand restrictions

Without considering constraints (12)ndash(14) for retailer itsbest response to the wholesale price can be defined inProposition 1

Proposition 1 e retailerrsquos best response to the wholesaleprice Wi the direct sale price Ps

i and the direct quoted leadtime Ls

i set by the manufacturer is as follows

Pri

ai middot 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

foralli isin Ψ

(15)

e Proof of Proposition 1 as well as the other propositionsis given in the appendix

e BLP model (4)ndash(14) can be formulated as a singlelevel mixed binary problemis is achieved by replacing thelower level problem (12)ndash(14) with its KuhnndashTucker con-ditions which we name Model I as follows

334 Model I

Max Πs 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

Wi minus Cp1ie(i)1113872 1113873 Dr

ij times Zij1113872 1113873 minus 1113936N

i1Chi times Hi

⎛⎝ ⎞⎠ (16)

ST

Advances in Operations Research 5

Wi lePsi (17)

1113936T

te(i)

Yit 1113936T

je(i)

DsijZij foralli isin Ψ (18)

1113936T

je(i)

Zij 1 foralli isin Ψ (19)

1113936iisinΨ|e(i)let

Yit + 1113936iisinΨ|e(i)t

1113936T

je(i)

DrijZij leKt t 1 T (20)

1113936T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (21)

Hi ge 1113936j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j 1 T (22)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (23)

ai 1 minus θi( 1113857 minus 2Pri middot br

i + αsi P

si + cs

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij + Wi middot bri + Cri middot br

i + λe(i) middot bri minus λiprime middot b

ri + ]i 0 foralli isin Ψ (24)

1113936i|e(i)t

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ + St Kt t 1 T (25)

ai 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij minus Siprime 0 (26)

λt middot St + λiprime middot Siprime + ]i middot P

ri 0 (27)

Yit Hi Dsij Pr

i Psi Wi λi λi

prime Siprime Si ]i ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(28)

Constraints (24)ndash(27) indicate the optimality conditionfor the retailer to convert the bilevel problem into a singlelevel one

Proposition 2 For any given due date j the wholesale priceWi and the direct sale price Ps

i for customer class i theoptimal retail price Pr

i is always positiveAs a result of Proposition 2 it can be concluded that ]i is

always equal to zero because ]i middot Pri 0 in Model I

Proposition 3 Consider an unlimited production capacityin each period for a decentralized dual-channel supply chain

If 8bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 6αs

i middot αri gt 0 the dual channel Πs is

strictly jointly concave in Wi and Psi

From Proposition 3 if αsi αr

i then 8bri middot bs

i minus (αsi )2 minus

(αri )2 minus 6αs

i middot αri is always greater than zerous in this case

the manufacturer profit Πs is always strictly jointly concavein Wi and Ps

i us for the sake of analytical tractability wewill assume that the cross-price effects are symmetric andαs

i αri

Proposition 4 Let the production capacity be unlimited ineach period for a decentralized dual-channel supply chain andsuppose there is a L0

i isin [Lli Lu

i ] with

6 Advances in Operations Research

L0i

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 (29)

en the optimal lead time Lsi isin [Ll

i Lui ] can be found

from Lli Lu

i and L0i by comparing their Πs values the one with

the largest Πs value is the optimal lead time

34 Model of the Centralized Dual-Channel Supply Chain

In a centralized dual-channel supply chain a vertically in-tegrated manufacturer controls the wholesale price Wi thedirect sale price Ps

i the direct sale quoted lead time Lsi and the

retailer sale price Pri e model called Model II is as follows

341 Model II

Max Πc Πs +Πr 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

pri minus Cp2

ie(i) minus Cri1113872 1113873 Drij times Zij1113872 1113873 minus 1113936

N

i1Chi times Hi

⎛⎝ ⎞⎠ (30)

ST

1113944

T

je(i)

Zij 1 foralli isin Ψ (31)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (32)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)let

1113944

T

je(i)

DrijZij leKt t 1 T (33)

1113944

T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (34)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (35)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (36)

Drij ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (37)

Yit Hi Dsij Pr

i Psi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(38)

e objective function (30) maximizes the total profit ofthe centralized supply chain which is the sum of manu-facturerrsquos profit and retailerrsquos profit e remaining con-straints are the same as those in Model I

Proposition 5 Consider an unlimited production capacityin each period for a centralized dual-channel supply chain If4br

i middot bsi minus (αs

i )2 minus (αr

i )2 minus 2αs

i middot αri gt 0 then the optimal lead

time Lsi isin [Ll

i Lui ] can be found from Ll

i and Lui by comparing

their Πc values the largest Πc value is the optimal lead time

4 Solution Approach

Models I and II are 0-1 nonlinear programs with continuousprice and production decision variables and binary variables

for due date selection It is well known that the problem ofnonlinear 0-1 programming is NP-hard [36] us theseproblems cannot be solved easily and exact algorithms canbe time consuming when the number of 0-1 variables islarge An alternative for solving these models is to use ap-proximate algorithms or heuristic methods with reasonablecomputational times

In this paper PSO metaheuristic is used and comparedto the GA for modeling and optimization of the pricingand due date setting problems in a dual-channel supplychain

PSO and GA can handle the whole MINLP easily andnaturally and it is easy to apply it to various problems forcomparison with conventional methods

Advances in Operations Research 7

If we fix the set of binary variables Z in the modelsthen both models are converted to quadratic programsthat can be solved by the nonlinear programming solverLINGO

41 Particle SwarmOptimization (PSO) e PSO algorithmwas first proposed by Kennedy and Eberhart [37] ismethod is based on the information shared among membersof a species and then used for evolution

Update the local best position Pkd global best

Send values of Zij to the Lingo solve the model andCalculate every particlersquos fitness value

Satisfy stopcriterion

Initialize a population of particles with random positionsand velocities

Fix the set of variables Zij in problems I and II according to thevalues for the lead times

Set up the searching range for L according to proposition 6

Initialize the parameters population size the inertial weight (ω) and two learning factors (c1 and c2)

No

Yes

End

Update the velocity and position for every particle

Lkd(new) = INT(Lkd

(old) +Vkd(new))

Vkd(new) = ωVkd

(old) + c1r1(Pkd(old) ndash Lkd

(old)) + c2r2(Pgd(old)ndashLkd

(old))

Figure 1 PSO flowchart for pricing and due date quotation

8 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

Dsi ai middot θi minus b

si middot P

si + αr

i middot Pri minus c

si middot L

si (1)

Dri ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri L

si (2)

where Dsi and Dr

i denote the consumer demand to themanufacturer and the consumer demand to the retailerrespectively ai denotes the base level of industry demandor the demand rate for customer of class i andθi(0lt θi lt 1) represents the initial portion of customers ofclass i that prefer the direct channel if the price and leadtime are zero us 1 minus θi represents the portion ofcustomers of class i that prefer purchasing from the re-tailer e coefficients bs

i and bri are the coefficients of the

price elasticity in the direct channel and retail channeldemand functions respectively e cross-price sensi-tivities αr

i and αsi reflect the substitutionrsquos degree of the

two channels and csi is the lead time sensitivity of the

demand in the direct channel If the lead time Lsi increases

by one unit cri units of demand will transfer to the retail

channel and csi units of direct channelrsquos demand will be

decreased e total demand of the two channels shouldhave a negative slope with respect to the retailerrsquos pricethe direct sale price and the quoted lead time us wehave αr

i lt bri α

si lt bs

i and csi gt cr

i Following Hua et al [3] it is assumed that the man-

ufacturer uses dual channels to sell their goods and thebase level of industry demand or demand rate in both

channels is very large ie θi should not be unreasonablysmall or large

33 Model of the Decentralized Dual-Channel Supply ChainIn this section we study a decentralized dual-channel supplychain In this model both the manufacturer and the retailermake their own decisions separately to maximize theirprofits e manufacturer as the Stackelberg leader firstdetermines the wholesale price Wi the direct sale price Ps

i and the direct sale quoted lead time Ls

i en the retailerchooses his own optimal retail price Pr

i based on themanufacturerrsquos decisions

331 Manufacturerrsquos Best Response e goal of the man-ufacturer as a leader is to maximize his profit consideringthe capacity constraints and demand constraints in thesystems and constraints determined by the retailer opti-mization problem e problem is formulated within theframework of bilevel programming (BLP) first level (themanufacturer model called the leader) and second level (theretailer model called a follower) In BLP model each de-cision maker tries to optimize its own objective functionwithout considering the objective of the other partyHowever the decision of each party affects the objectivevalue of the other one as well as the decision space

332 First-Level Model Manufacturer Model

Max Πs 1113944N

i11113944

T

je(i)

psi times D

sij times Zij1113872 1113873 minus 1113944

N

i11113944

T

te(i)

Yit times Cp1it minus Cei1113872 11138731113872 1113873 + 1113944

N

i11113944

T

je(i)

Wi minus Cp2ie(i)1113872 1113873 D

rij times Zij1113872 1113873 minus 1113944

N

i1Chi times Hi

⎛⎝ ⎞⎠

(3)

STWi lePs

i foralli isin Ψ (4)

1113944

T

je(i)

Zij 1 foralli isin Ψ (5)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (6)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)t

1113944

T

je(i)

DrijZij leKt t 1 T (7)

1113944

T

tj

Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T (8)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (9)

4 Advances in Operations Research

Dsij ai middot θi minus bs

i lePsi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) infin T (10)

333 Second-Level Model Retailer Model

MaxPr

i

Πr 1113936iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (11)

ST

1113936i|te(i)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij leKt t 1 T (12)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij ge 0 foralli isin Ψ (13)

Yit Hi Dsij Pr

i Psi Wi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(14)

e first and second terms in the first-level objectiverepresent the total revenue and production cost of directsales e third term represents the total profit obtained bysales to the retailer and the fourth one is the carrying cost

Constraint (4) indicates that the wholesale price cannotbe higher than the direct channel price otherwise the re-tailer may purchase from the direct channel at a lower priceConstraints (5) ensure that only one due date is chosen foreach direct channel customer order Constraint (6) ensuresthat if due date j is selected for customer of class i in thedirect channel then exact Ds

ij units must be produced anddelivered where Ds

ij depends on the selected price and duedate Constraint (7) is a capacity constraint that ensures thatthe production capacity in each period is not exceeded Anorder of a retail channel customer is produced in the cus-tomerrsquos arrival time period and delivered instantaneously inthe same period whereas an order of a direct channelcustomer can be produced in any period between its arrivalperiod and the quoted due date e first term in the lefthand side of constraint (7) indicates the total production fordirect channel customers and the second term is the totalproduction for retail channel customers Constraint (8)indicates that an order of a direct channel customer can beproduced in any period between its arrival period and thequoted due date

e required inventory for orders is scheduled in anyperiod prior to its commitment and the negotiated due dateis controlled by constraint (9) where M is a sufficiently large

number Constraint (10) is demand functions for customerorders in the direct channel e term j minus e(i) + 1 is a timeinterval between the arrival time of an order and the quoteddue date which is called the lead time e objectivefunction of second-level optimization problem for the re-tailer channel is represented by (11) Constraints (12) and(13) control feasibility as well as demand restrictions

Without considering constraints (12)ndash(14) for retailer itsbest response to the wholesale price can be defined inProposition 1

Proposition 1 e retailerrsquos best response to the wholesaleprice Wi the direct sale price Ps

i and the direct quoted leadtime Ls

i set by the manufacturer is as follows

Pri

ai middot 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

foralli isin Ψ

(15)

e Proof of Proposition 1 as well as the other propositionsis given in the appendix

e BLP model (4)ndash(14) can be formulated as a singlelevel mixed binary problemis is achieved by replacing thelower level problem (12)ndash(14) with its KuhnndashTucker con-ditions which we name Model I as follows

334 Model I

Max Πs 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

Wi minus Cp1ie(i)1113872 1113873 Dr

ij times Zij1113872 1113873 minus 1113936N

i1Chi times Hi

⎛⎝ ⎞⎠ (16)

ST

Advances in Operations Research 5

Wi lePsi (17)

1113936T

te(i)

Yit 1113936T

je(i)

DsijZij foralli isin Ψ (18)

1113936T

je(i)

Zij 1 foralli isin Ψ (19)

1113936iisinΨ|e(i)let

Yit + 1113936iisinΨ|e(i)t

1113936T

je(i)

DrijZij leKt t 1 T (20)

1113936T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (21)

Hi ge 1113936j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j 1 T (22)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (23)

ai 1 minus θi( 1113857 minus 2Pri middot br

i + αsi P

si + cs

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij + Wi middot bri + Cri middot br

i + λe(i) middot bri minus λiprime middot b

ri + ]i 0 foralli isin Ψ (24)

1113936i|e(i)t

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ + St Kt t 1 T (25)

ai 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij minus Siprime 0 (26)

λt middot St + λiprime middot Siprime + ]i middot P

ri 0 (27)

Yit Hi Dsij Pr

i Psi Wi λi λi

prime Siprime Si ]i ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(28)

Constraints (24)ndash(27) indicate the optimality conditionfor the retailer to convert the bilevel problem into a singlelevel one

Proposition 2 For any given due date j the wholesale priceWi and the direct sale price Ps

i for customer class i theoptimal retail price Pr

i is always positiveAs a result of Proposition 2 it can be concluded that ]i is

always equal to zero because ]i middot Pri 0 in Model I

Proposition 3 Consider an unlimited production capacityin each period for a decentralized dual-channel supply chain

If 8bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 6αs

i middot αri gt 0 the dual channel Πs is

strictly jointly concave in Wi and Psi

From Proposition 3 if αsi αr

i then 8bri middot bs

i minus (αsi )2 minus

(αri )2 minus 6αs

i middot αri is always greater than zerous in this case

the manufacturer profit Πs is always strictly jointly concavein Wi and Ps

i us for the sake of analytical tractability wewill assume that the cross-price effects are symmetric andαs

i αri

Proposition 4 Let the production capacity be unlimited ineach period for a decentralized dual-channel supply chain andsuppose there is a L0

i isin [Lli Lu

i ] with

6 Advances in Operations Research

L0i

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 (29)

en the optimal lead time Lsi isin [Ll

i Lui ] can be found

from Lli Lu

i and L0i by comparing their Πs values the one with

the largest Πs value is the optimal lead time

34 Model of the Centralized Dual-Channel Supply Chain

In a centralized dual-channel supply chain a vertically in-tegrated manufacturer controls the wholesale price Wi thedirect sale price Ps

i the direct sale quoted lead time Lsi and the

retailer sale price Pri e model called Model II is as follows

341 Model II

Max Πc Πs +Πr 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

pri minus Cp2

ie(i) minus Cri1113872 1113873 Drij times Zij1113872 1113873 minus 1113936

N

i1Chi times Hi

⎛⎝ ⎞⎠ (30)

ST

1113944

T

je(i)

Zij 1 foralli isin Ψ (31)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (32)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)let

1113944

T

je(i)

DrijZij leKt t 1 T (33)

1113944

T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (34)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (35)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (36)

Drij ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (37)

Yit Hi Dsij Pr

i Psi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(38)

e objective function (30) maximizes the total profit ofthe centralized supply chain which is the sum of manu-facturerrsquos profit and retailerrsquos profit e remaining con-straints are the same as those in Model I

Proposition 5 Consider an unlimited production capacityin each period for a centralized dual-channel supply chain If4br

i middot bsi minus (αs

i )2 minus (αr

i )2 minus 2αs

i middot αri gt 0 then the optimal lead

time Lsi isin [Ll

i Lui ] can be found from Ll

i and Lui by comparing

their Πc values the largest Πc value is the optimal lead time

4 Solution Approach

Models I and II are 0-1 nonlinear programs with continuousprice and production decision variables and binary variables

for due date selection It is well known that the problem ofnonlinear 0-1 programming is NP-hard [36] us theseproblems cannot be solved easily and exact algorithms canbe time consuming when the number of 0-1 variables islarge An alternative for solving these models is to use ap-proximate algorithms or heuristic methods with reasonablecomputational times

In this paper PSO metaheuristic is used and comparedto the GA for modeling and optimization of the pricingand due date setting problems in a dual-channel supplychain

PSO and GA can handle the whole MINLP easily andnaturally and it is easy to apply it to various problems forcomparison with conventional methods

Advances in Operations Research 7

If we fix the set of binary variables Z in the modelsthen both models are converted to quadratic programsthat can be solved by the nonlinear programming solverLINGO

41 Particle SwarmOptimization (PSO) e PSO algorithmwas first proposed by Kennedy and Eberhart [37] ismethod is based on the information shared among membersof a species and then used for evolution

Update the local best position Pkd global best

Send values of Zij to the Lingo solve the model andCalculate every particlersquos fitness value

Satisfy stopcriterion

Initialize a population of particles with random positionsand velocities

Fix the set of variables Zij in problems I and II according to thevalues for the lead times

Set up the searching range for L according to proposition 6

Initialize the parameters population size the inertial weight (ω) and two learning factors (c1 and c2)

No

Yes

End

Update the velocity and position for every particle

Lkd(new) = INT(Lkd

(old) +Vkd(new))

Vkd(new) = ωVkd

(old) + c1r1(Pkd(old) ndash Lkd

(old)) + c2r2(Pgd(old)ndashLkd

(old))

Figure 1 PSO flowchart for pricing and due date quotation

8 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

Dsij ai middot θi minus bs

i lePsi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) infin T (10)

333 Second-Level Model Retailer Model

MaxPr

i

Πr 1113936iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (11)

ST

1113936i|te(i)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij leKt t 1 T (12)

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij ge 0 foralli isin Ψ (13)

Yit Hi Dsij Pr

i Psi Wi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(14)

e first and second terms in the first-level objectiverepresent the total revenue and production cost of directsales e third term represents the total profit obtained bysales to the retailer and the fourth one is the carrying cost

Constraint (4) indicates that the wholesale price cannotbe higher than the direct channel price otherwise the re-tailer may purchase from the direct channel at a lower priceConstraints (5) ensure that only one due date is chosen foreach direct channel customer order Constraint (6) ensuresthat if due date j is selected for customer of class i in thedirect channel then exact Ds

ij units must be produced anddelivered where Ds

ij depends on the selected price and duedate Constraint (7) is a capacity constraint that ensures thatthe production capacity in each period is not exceeded Anorder of a retail channel customer is produced in the cus-tomerrsquos arrival time period and delivered instantaneously inthe same period whereas an order of a direct channelcustomer can be produced in any period between its arrivalperiod and the quoted due date e first term in the lefthand side of constraint (7) indicates the total production fordirect channel customers and the second term is the totalproduction for retail channel customers Constraint (8)indicates that an order of a direct channel customer can beproduced in any period between its arrival period and thequoted due date

e required inventory for orders is scheduled in anyperiod prior to its commitment and the negotiated due dateis controlled by constraint (9) where M is a sufficiently large

number Constraint (10) is demand functions for customerorders in the direct channel e term j minus e(i) + 1 is a timeinterval between the arrival time of an order and the quoteddue date which is called the lead time e objectivefunction of second-level optimization problem for the re-tailer channel is represented by (11) Constraints (12) and(13) control feasibility as well as demand restrictions

Without considering constraints (12)ndash(14) for retailer itsbest response to the wholesale price can be defined inProposition 1

Proposition 1 e retailerrsquos best response to the wholesaleprice Wi the direct sale price Ps

i and the direct quoted leadtime Ls

i set by the manufacturer is as follows

Pri

ai middot 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

foralli isin Ψ

(15)

e Proof of Proposition 1 as well as the other propositionsis given in the appendix

e BLP model (4)ndash(14) can be formulated as a singlelevel mixed binary problemis is achieved by replacing thelower level problem (12)ndash(14) with its KuhnndashTucker con-ditions which we name Model I as follows

334 Model I

Max Πs 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

Wi minus Cp1ie(i)1113872 1113873 Dr

ij times Zij1113872 1113873 minus 1113936N

i1Chi times Hi

⎛⎝ ⎞⎠ (16)

ST

Advances in Operations Research 5

Wi lePsi (17)

1113936T

te(i)

Yit 1113936T

je(i)

DsijZij foralli isin Ψ (18)

1113936T

je(i)

Zij 1 foralli isin Ψ (19)

1113936iisinΨ|e(i)let

Yit + 1113936iisinΨ|e(i)t

1113936T

je(i)

DrijZij leKt t 1 T (20)

1113936T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (21)

Hi ge 1113936j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j 1 T (22)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (23)

ai 1 minus θi( 1113857 minus 2Pri middot br

i + αsi P

si + cs

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij + Wi middot bri + Cri middot br

i + λe(i) middot bri minus λiprime middot b

ri + ]i 0 foralli isin Ψ (24)

1113936i|e(i)t

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ + St Kt t 1 T (25)

ai 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij minus Siprime 0 (26)

λt middot St + λiprime middot Siprime + ]i middot P

ri 0 (27)

Yit Hi Dsij Pr

i Psi Wi λi λi

prime Siprime Si ]i ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(28)

Constraints (24)ndash(27) indicate the optimality conditionfor the retailer to convert the bilevel problem into a singlelevel one

Proposition 2 For any given due date j the wholesale priceWi and the direct sale price Ps

i for customer class i theoptimal retail price Pr

i is always positiveAs a result of Proposition 2 it can be concluded that ]i is

always equal to zero because ]i middot Pri 0 in Model I

Proposition 3 Consider an unlimited production capacityin each period for a decentralized dual-channel supply chain

If 8bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 6αs

i middot αri gt 0 the dual channel Πs is

strictly jointly concave in Wi and Psi

From Proposition 3 if αsi αr

i then 8bri middot bs

i minus (αsi )2 minus

(αri )2 minus 6αs

i middot αri is always greater than zerous in this case

the manufacturer profit Πs is always strictly jointly concavein Wi and Ps

i us for the sake of analytical tractability wewill assume that the cross-price effects are symmetric andαs

i αri

Proposition 4 Let the production capacity be unlimited ineach period for a decentralized dual-channel supply chain andsuppose there is a L0

i isin [Lli Lu

i ] with

6 Advances in Operations Research

L0i

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 (29)

en the optimal lead time Lsi isin [Ll

i Lui ] can be found

from Lli Lu

i and L0i by comparing their Πs values the one with

the largest Πs value is the optimal lead time

34 Model of the Centralized Dual-Channel Supply Chain

In a centralized dual-channel supply chain a vertically in-tegrated manufacturer controls the wholesale price Wi thedirect sale price Ps

i the direct sale quoted lead time Lsi and the

retailer sale price Pri e model called Model II is as follows

341 Model II

Max Πc Πs +Πr 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

pri minus Cp2

ie(i) minus Cri1113872 1113873 Drij times Zij1113872 1113873 minus 1113936

N

i1Chi times Hi

⎛⎝ ⎞⎠ (30)

ST

1113944

T

je(i)

Zij 1 foralli isin Ψ (31)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (32)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)let

1113944

T

je(i)

DrijZij leKt t 1 T (33)

1113944

T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (34)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (35)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (36)

Drij ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (37)

Yit Hi Dsij Pr

i Psi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(38)

e objective function (30) maximizes the total profit ofthe centralized supply chain which is the sum of manu-facturerrsquos profit and retailerrsquos profit e remaining con-straints are the same as those in Model I

Proposition 5 Consider an unlimited production capacityin each period for a centralized dual-channel supply chain If4br

i middot bsi minus (αs

i )2 minus (αr

i )2 minus 2αs

i middot αri gt 0 then the optimal lead

time Lsi isin [Ll

i Lui ] can be found from Ll

i and Lui by comparing

their Πc values the largest Πc value is the optimal lead time

4 Solution Approach

Models I and II are 0-1 nonlinear programs with continuousprice and production decision variables and binary variables

for due date selection It is well known that the problem ofnonlinear 0-1 programming is NP-hard [36] us theseproblems cannot be solved easily and exact algorithms canbe time consuming when the number of 0-1 variables islarge An alternative for solving these models is to use ap-proximate algorithms or heuristic methods with reasonablecomputational times

In this paper PSO metaheuristic is used and comparedto the GA for modeling and optimization of the pricingand due date setting problems in a dual-channel supplychain

PSO and GA can handle the whole MINLP easily andnaturally and it is easy to apply it to various problems forcomparison with conventional methods

Advances in Operations Research 7

If we fix the set of binary variables Z in the modelsthen both models are converted to quadratic programsthat can be solved by the nonlinear programming solverLINGO

41 Particle SwarmOptimization (PSO) e PSO algorithmwas first proposed by Kennedy and Eberhart [37] ismethod is based on the information shared among membersof a species and then used for evolution

Update the local best position Pkd global best

Send values of Zij to the Lingo solve the model andCalculate every particlersquos fitness value

Satisfy stopcriterion

Initialize a population of particles with random positionsand velocities

Fix the set of variables Zij in problems I and II according to thevalues for the lead times

Set up the searching range for L according to proposition 6

Initialize the parameters population size the inertial weight (ω) and two learning factors (c1 and c2)

No

Yes

End

Update the velocity and position for every particle

Lkd(new) = INT(Lkd

(old) +Vkd(new))

Vkd(new) = ωVkd

(old) + c1r1(Pkd(old) ndash Lkd

(old)) + c2r2(Pgd(old)ndashLkd

(old))

Figure 1 PSO flowchart for pricing and due date quotation

8 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

Wi lePsi (17)

1113936T

te(i)

Yit 1113936T

je(i)

DsijZij foralli isin Ψ (18)

1113936T

je(i)

Zij 1 foralli isin Ψ (19)

1113936iisinΨ|e(i)let

Yit + 1113936iisinΨ|e(i)t

1113936T

je(i)

DrijZij leKt t 1 T (20)

1113936T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (21)

Hi ge 1113936j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j 1 T (22)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (23)

ai 1 minus θi( 1113857 minus 2Pri middot br

i + αsi P

si + cs

i 1113936T

je(i)

(j minus e(i) + 1) middot Zij + Wi middot bri + Cri middot br

i + λe(i) middot bri minus λiprime middot b

ri + ]i 0 foralli isin Ψ (24)

1113936i|e(i)t

ai middot 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ + St Kt t 1 T (25)

ai 1 minus θi( 1113857 minus bri middot Pr

i + αsi middot Ps

i + cri 1113936

T

je(i)

(j minus e(i) + 1) middot Zij minus Siprime 0 (26)

λt middot St + λiprime middot Siprime + ]i middot P

ri 0 (27)

Yit Hi Dsij Pr

i Psi Wi λi λi

prime Siprime Si ]i ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(28)

Constraints (24)ndash(27) indicate the optimality conditionfor the retailer to convert the bilevel problem into a singlelevel one

Proposition 2 For any given due date j the wholesale priceWi and the direct sale price Ps

i for customer class i theoptimal retail price Pr

i is always positiveAs a result of Proposition 2 it can be concluded that ]i is

always equal to zero because ]i middot Pri 0 in Model I

Proposition 3 Consider an unlimited production capacityin each period for a decentralized dual-channel supply chain

If 8bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 6αs

i middot αri gt 0 the dual channel Πs is

strictly jointly concave in Wi and Psi

From Proposition 3 if αsi αr

i then 8bri middot bs

i minus (αsi )2 minus

(αri )2 minus 6αs

i middot αri is always greater than zerous in this case

the manufacturer profit Πs is always strictly jointly concavein Wi and Ps

i us for the sake of analytical tractability wewill assume that the cross-price effects are symmetric andαs

i αri

Proposition 4 Let the production capacity be unlimited ineach period for a decentralized dual-channel supply chain andsuppose there is a L0

i isin [Lli Lu

i ] with

6 Advances in Operations Research

L0i

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 (29)

en the optimal lead time Lsi isin [Ll

i Lui ] can be found

from Lli Lu

i and L0i by comparing their Πs values the one with

the largest Πs value is the optimal lead time

34 Model of the Centralized Dual-Channel Supply Chain

In a centralized dual-channel supply chain a vertically in-tegrated manufacturer controls the wholesale price Wi thedirect sale price Ps

i the direct sale quoted lead time Lsi and the

retailer sale price Pri e model called Model II is as follows

341 Model II

Max Πc Πs +Πr 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

pri minus Cp2

ie(i) minus Cri1113872 1113873 Drij times Zij1113872 1113873 minus 1113936

N

i1Chi times Hi

⎛⎝ ⎞⎠ (30)

ST

1113944

T

je(i)

Zij 1 foralli isin Ψ (31)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (32)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)let

1113944

T

je(i)

DrijZij leKt t 1 T (33)

1113944

T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (34)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (35)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (36)

Drij ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (37)

Yit Hi Dsij Pr

i Psi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(38)

e objective function (30) maximizes the total profit ofthe centralized supply chain which is the sum of manu-facturerrsquos profit and retailerrsquos profit e remaining con-straints are the same as those in Model I

Proposition 5 Consider an unlimited production capacityin each period for a centralized dual-channel supply chain If4br

i middot bsi minus (αs

i )2 minus (αr

i )2 minus 2αs

i middot αri gt 0 then the optimal lead

time Lsi isin [Ll

i Lui ] can be found from Ll

i and Lui by comparing

their Πc values the largest Πc value is the optimal lead time

4 Solution Approach

Models I and II are 0-1 nonlinear programs with continuousprice and production decision variables and binary variables

for due date selection It is well known that the problem ofnonlinear 0-1 programming is NP-hard [36] us theseproblems cannot be solved easily and exact algorithms canbe time consuming when the number of 0-1 variables islarge An alternative for solving these models is to use ap-proximate algorithms or heuristic methods with reasonablecomputational times

In this paper PSO metaheuristic is used and comparedto the GA for modeling and optimization of the pricingand due date setting problems in a dual-channel supplychain

PSO and GA can handle the whole MINLP easily andnaturally and it is easy to apply it to various problems forcomparison with conventional methods

Advances in Operations Research 7

If we fix the set of binary variables Z in the modelsthen both models are converted to quadratic programsthat can be solved by the nonlinear programming solverLINGO

41 Particle SwarmOptimization (PSO) e PSO algorithmwas first proposed by Kennedy and Eberhart [37] ismethod is based on the information shared among membersof a species and then used for evolution

Update the local best position Pkd global best

Send values of Zij to the Lingo solve the model andCalculate every particlersquos fitness value

Satisfy stopcriterion

Initialize a population of particles with random positionsand velocities

Fix the set of variables Zij in problems I and II according to thevalues for the lead times

Set up the searching range for L according to proposition 6

Initialize the parameters population size the inertial weight (ω) and two learning factors (c1 and c2)

No

Yes

End

Update the velocity and position for every particle

Lkd(new) = INT(Lkd

(old) +Vkd(new))

Vkd(new) = ωVkd

(old) + c1r1(Pkd(old) ndash Lkd

(old)) + c2r2(Pgd(old)ndashLkd

(old))

Figure 1 PSO flowchart for pricing and due date quotation

8 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

L0i

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 (29)

en the optimal lead time Lsi isin [Ll

i Lui ] can be found

from Lli Lu

i and L0i by comparing their Πs values the one with

the largest Πs value is the optimal lead time

34 Model of the Centralized Dual-Channel Supply Chain

In a centralized dual-channel supply chain a vertically in-tegrated manufacturer controls the wholesale price Wi thedirect sale price Ps

i the direct sale quoted lead time Lsi and the

retailer sale price Pri e model called Model II is as follows

341 Model II

Max Πc Πs +Πr 1113936N

i11113936T

je(i)

psi times Ds

ij times Zij1113872 1113873 minus 1113936N

i11113936T

te(i)

Yit times Cp1it1113872 1113873 + 1113936

N

i11113936T

je(i)

pri minus Cp2

ie(i) minus Cri1113872 1113873 Drij times Zij1113872 1113873 minus 1113936

N

i1Chi times Hi

⎛⎝ ⎞⎠ (30)

ST

1113944

T

je(i)

Zij 1 foralli isin Ψ (31)

1113944

T

te(i)

Yit 1113944T

je(i)

DsijZij foralli isin Ψ (32)

1113944iisinΨ|e(i)let

Yit + 1113944iisinΨ|e(i)let

1113944

T

je(i)

DrijZij leKt t 1 T (33)

1113944

T

tj+1Yit leM 1 minus Zij1113872 1113873 foralli isin Ψ j e(i) T minus 1 (34)

Hi ge 1113944

j

te(i)

(j minus t) Yit1113872 1113873 + M Zij minus 11113872 1113873 foralli isin Ψ j e(i) T (35)

Dsij ai middot θi minus bs

i middot Psi + αr

i middot Pri minus cs

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (36)

Drij ai middot 1 minus θi( 1113857 minus br

i middot Pri + αs

i middot Psi + cr

i middot (j minus e(i) + 1) foralli isin Ψ j e(i) T (37)

Yit Hi Dsij Pr

i Psi ge 0 foralli isin Ψ t e(i) T j e(i) T

Zij isin 0 1 foralli isin Ψ j e(i) T(38)

e objective function (30) maximizes the total profit ofthe centralized supply chain which is the sum of manu-facturerrsquos profit and retailerrsquos profit e remaining con-straints are the same as those in Model I

Proposition 5 Consider an unlimited production capacityin each period for a centralized dual-channel supply chain If4br

i middot bsi minus (αs

i )2 minus (αr

i )2 minus 2αs

i middot αri gt 0 then the optimal lead

time Lsi isin [Ll

i Lui ] can be found from Ll

i and Lui by comparing

their Πc values the largest Πc value is the optimal lead time

4 Solution Approach

Models I and II are 0-1 nonlinear programs with continuousprice and production decision variables and binary variables

for due date selection It is well known that the problem ofnonlinear 0-1 programming is NP-hard [36] us theseproblems cannot be solved easily and exact algorithms canbe time consuming when the number of 0-1 variables islarge An alternative for solving these models is to use ap-proximate algorithms or heuristic methods with reasonablecomputational times

In this paper PSO metaheuristic is used and comparedto the GA for modeling and optimization of the pricingand due date setting problems in a dual-channel supplychain

PSO and GA can handle the whole MINLP easily andnaturally and it is easy to apply it to various problems forcomparison with conventional methods

Advances in Operations Research 7

If we fix the set of binary variables Z in the modelsthen both models are converted to quadratic programsthat can be solved by the nonlinear programming solverLINGO

41 Particle SwarmOptimization (PSO) e PSO algorithmwas first proposed by Kennedy and Eberhart [37] ismethod is based on the information shared among membersof a species and then used for evolution

Update the local best position Pkd global best

Send values of Zij to the Lingo solve the model andCalculate every particlersquos fitness value

Satisfy stopcriterion

Initialize a population of particles with random positionsand velocities

Fix the set of variables Zij in problems I and II according to thevalues for the lead times

Set up the searching range for L according to proposition 6

Initialize the parameters population size the inertial weight (ω) and two learning factors (c1 and c2)

No

Yes

End

Update the velocity and position for every particle

Lkd(new) = INT(Lkd

(old) +Vkd(new))

Vkd(new) = ωVkd

(old) + c1r1(Pkd(old) ndash Lkd

(old)) + c2r2(Pgd(old)ndashLkd

(old))

Figure 1 PSO flowchart for pricing and due date quotation

8 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

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015

025 0

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750

850

95

015

025 0

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750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

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850

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015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

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850

95

015

025 0

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850

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015

025 0

50

750

850

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015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

If we fix the set of binary variables Z in the modelsthen both models are converted to quadratic programsthat can be solved by the nonlinear programming solverLINGO

41 Particle SwarmOptimization (PSO) e PSO algorithmwas first proposed by Kennedy and Eberhart [37] ismethod is based on the information shared among membersof a species and then used for evolution

Update the local best position Pkd global best

Send values of Zij to the Lingo solve the model andCalculate every particlersquos fitness value

Satisfy stopcriterion

Initialize a population of particles with random positionsand velocities

Fix the set of variables Zij in problems I and II according to thevalues for the lead times

Set up the searching range for L according to proposition 6

Initialize the parameters population size the inertial weight (ω) and two learning factors (c1 and c2)

No

Yes

End

Update the velocity and position for every particle

Lkd(new) = INT(Lkd

(old) +Vkd(new))

Vkd(new) = ωVkd

(old) + c1r1(Pkd(old) ndash Lkd

(old)) + c2r2(Pgd(old)ndashLkd

(old))

Figure 1 PSO flowchart for pricing and due date quotation

8 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

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015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

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015

025 0

50

750

850

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e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

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95

015

025 0

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750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

Simple structure ease of implementation speed of ac-quiring solutions and robustness are the advantages of thePSO that persuade us to use it in solving the models

Recently PSO algorithms were successfully applied to awide range of applications A comprehensive survey of PSOalgorithms and applications can be found in the paper byKennedy et al [38]

In the proposed PSO algorithm particle k is representedas Lk Lk1 Lk2 LkN1113966 1113967 which denotes the due datesquoted to N customers For each Lki in N-dimensionalspace the ZiLki

is set to one in Models I and II and the othervariables Zij j e(i) T and jne Lki are fixed to zero forcustomer order i isin Ψ

is section will present the application of the PSOalgorithm forModels I and IIe PSO algorithm for solvingthese models is illustrated in Figure 1

e key point in a constrained optimization process is todeal with the constraints We must modify the original PSOmethod for constrained optimization Many methods wereproposed for handling constraints such as methods basedon preserving the feasibility of solutions methods based onpenalty functions methods that make a clear distinctionbetween feasible and infeasible solutions and other hybridmethods [39] e most straightforward method is the onebased on preserving the feasibility of solutions In thismethod each particle searches the whole space but onlykeeps track of the feasible solutions To accelerate theprocess we derive an upper bound on the search range byfinding the largest lead times such that both demand in retailand direct channel will be nonnegative ie Ds

ij ge 0Drij ge 0

e upper bounds on lead times are given in the followingproposition

Proposition 6 e best due date for each customer i isin Ψ isrestricted to a range Ll

i leLi le Lui e lower bound of the range

is the arrival time of the customer ie e(i) and the upperbound of the range is

Luj

ai θi middot bri minus θi middot αr

i + αri( 1113857

csi middot br

i minus cri middot αr

i

+ e(i) (39)

42 Genetic Algorithm (GA) GA is a well-known meta-heuristic optimization technique based on Darwinrsquos theoryof the ldquosurvival of the fittestrdquo proposed by Holland [40] eGA starts with a group of individuals created randomly eindividuals in the population are then evaluated using afitness value Two individuals are then selected based ontheir fitness ese individuals ldquoreproducerdquo to create one ormore offspring by crossover and mutation operators eone-point crossover and swap mutation are used in thisarticle According to the description provided in PSO al-gorithm each solution can be represented by an integerstring with length N (number of customers) Each generepresents the due date quoted to each customer

e GA algorithm for solving these models is illustratedin Figure 2

5 Numerical Studies

In this section we investigate the relation between thewholesale price the direct sale price and the retail price inthe decentralized and centralized supply chain e pricingstrategies and quoted lead time decisions are compared forthe two models using numerical experiments

e parameters of the proposed PSO algorithm areselected based on parameter tuning With testing differentvalues for PSOrsquo parameters the ones selected are those thatgave the best results e best values of the computationalexperiments are as follows (1) the values of a population of30 individuals are used for both GA and PSO (2) the initialinertia weight is set to 09 and (3) the values of the accel-eration constants c1 and c2 are fixed to 09 e maximumvelocity is set as the difference between the upper and thelower bounds which ensures that the particles are able to flyacross the problem-specific constraints region e otherparameters used in GA are crossover rate of 080 andmutation rate of 03

Each instance was run for 50 iterations and 30 repli-cations were conducted for each instance e performanceof PSO algorithm was compared with MINLP solver GAMSand GA algorithm by testing on a set of 8 small size instancesin the decentralized (Model I) and centralized (Model II)supply chain Table 2 reports the comparison between the

Initialize the parameters the population size Crossoverprobability and Mutation probability

Randomly generate solutions or chromosomes according tothe ranges in the previous step

Generate offspring with crossover technique

Fix the set of variables Zij in problems I and II accordingto the values for the lead times

Set up the searching range for L according to proposition 6

Send values of Zij to the Lingo solve the model andcalculate every particlersquos fitness value

Perform mutation operation on the new offspring

Satisfy stopcriterion

NoYes

End

Figure 2 GA flowchart for pricing and due date quotation

Advances in Operations Research 9

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

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015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

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025 0

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025 0

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025 0

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850

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015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

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015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

GAMS solutions and the presented PSO algorithme sumofmanufacturerrsquos and retailerrsquos profit is shown in Table 2 Allthe results show that PSO algorithm can achieve betterresults than those obtained by GAMS solver

Comparing the results of the GA and PSO methods wecan see that there is no conclusive winner In some cases theGAmethod results in a better solution and in some cases thePSO method does Due to the simplicity of the PSOstructure we use it for our sensitivity analysis

To analyze the effects of modelsrsquo parameters on pricesand profits we considered 18 different problem groups tocarry out this numerical study For each group of problemsthe production capacity of each period (K) is categorized ashigh and low capacity e ratio of operational costs indirect channel and retailer (CeCr) is categorized as highmedium and low We can have single class customerproblems where the price and lead time sensitivities are thesame across the customers and multiclass customerproblems based on the variability of these sensitivitiesWith this assumption the additional categories are basedon price sensitivity variability (PV) and lead time sensi-tivity variability (LV) Table 3 presents the characteristics ofdifferent groups of problems

e other parameters are considered as follows

Table 2 A comparison between the GAMS solutions the presented PSO algorithm and the GA algorithm

Problem number N TModel I Model II

PSO GAMS solver GA PSO GAMS solver GATest1 6 12 10420290 9965964 10396030 131059 12618257 13119062Test2 6 12 19102460 17988790 18674310 20461450 19218688 20376016Test3 12 12 70010710 65649040 68780042 78671060 74520709 78461942Test4 12 12 12570530 11669220 11831432 13509720 12581600 13526622Test5 12 12 16089030 13923450 16142060 18506860 16468183 18634364Test6 12 20 12368550 10787850 11965970 12919940 11998329 1299349Test7 20 20 13841910 No feasible 13952010 14673770 8284983 14778033Test8 30 20 1499656 No feasible 14883240 15678881 No feasible 15761304N number of customer classes T planning periods

Table 3 Specifications of different groups of problems

Problem number K (CeCr) PV LV1 M M No No2 H M No No3 M M No Yes4 H M No Yes5 M M Yes No6 H M Yes No7 M H No No8 H H No No9 M H Yes No10 H H Yes No11 M H No Yes12 H H No Yes13 M L No No14 H L No No15 M L Yes No16 H L Yes No17 M L No Yes18 H L No Yes

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast s

Figure 3 Manufacturerrsquo profit in the decentralized dual-channelsupply chain for different instances

ndash10000

0

10000

20000

30000

40000

50000

60000

70000

015 025 05 075 085 095θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

πlowast r

Figure 4 Retailerrsquo profit in the decentralized dual-channel supplychain for different instances

10 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

0102030405060

Pric

e

Instance 1

PsPr

Cw

020406080

100120

Pric

e

Instance 5

01020304050607080

Pric

e

Instance 9

010203040506070

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 16

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

PsPr

Cw

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

θ

θ

Figure 5 Comparison of the retail price direct sale price and wholesale price in decentralized dual-channel supply chain for differentinstances

Advances in Operations Research 11

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

e holding costs per unit of time for each customer ofclass i are Chi 5 and the production cost in each period t

for each customer of class i is Cpit 10 e demand rate for

customer of class i ai is generated randomly fromai isin [500 3000] We considered a planning horizon with 12time periods and 6 customer classes arriving during theplanning horizon for all problems Two types of arrival timedistributions are considered the arrival times near the be-ginning of planning horizon (B) (e(i) sim uniform [1 4]) andthe uniform distribution along planning horizon (U) (e(i) simuniform [1 12]) [29 33]

In all instances we investigated the effect of customerpreference of the direct channel θ on pricing and lead timedecisions e results of some carried instances are sum-marized in Figures 3ndash9 and Tables 4 and 5 Tables 4 and 5show the differences between average values of retailerprices (P

r

II minus Pr

I ) and direct prices (Ps

II minus Ps

I) obtained forcustomer classes and the profits (ΠII minus ΠI) under bothdecentralized (Model I) and centralized (Model II) supplychains In the decentralized supply chain as illustrated in

Figures 3 and 4 in some examples the manufacturerrsquosprofit increases as the customer preference of the directchannel increases whereas the profit decreases for othersAlmost all the examples with decreasing profit functionsare consequence of high operational cost in the directchannel e retailerrsquos profit is decreasing with customerpreference of the direct channel

As we can see from Figures 5 and 8 in both the cen-tralized and decentralized dual-channel supply chain whencustomer preference of the direct channel θ is below a certainlevel the retail price is higher than the direct sale priceconversely when θ exceeds that level the direct price be-comes higher than the retail sale priceis result shows thatif the base level of demand or demand rate in one channel isrelatively higher than a certain threshold the sale price inthat channel should be set higher than the one in the otherchannel

From Figure 5 in some examples when θ is relativelylow the direct sale price should be set equal to thewholesale price and when θ is relatively high the direct sale

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 6 Comparison of the optimal quoted lead time in decentralized dual-channel supply chain for different instances

020000400006000080000

100000120000140000160000180000

015 025 05 075 085 095

πlowast

θ

instance 1instance 2instance 3instance 4instance 5instance 6

instance 7instance 8instance 9instance 10instance 11instance 12

instance 13instance 14instance 15instance 16instance 17instance 18

Figure 7 Supply chain profit in the centralized dual-channel supply chain for different instances

12 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

price should be set higher than the wholesale price In otherwords when θ is lower than a certain threshold the baselevel of demand in the retail channel is highus the retailprice can also be set to be high but according to the lowbase demand in the direct channel the direct sale price is

set to be lowe wholesale price must be less than or equalto the direct sale price erefore the direct sale price isequal to the wholesale price and the retail price is higherthan the wholesale price conversely when θ exceeds thethreshold the base level of demand in the retail channel is

Instance 9

0102030405060

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Pric

e

Instance 1

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6

01020304050607080

Pric

e

Instance 5

020406080

100120

Pric

e

010203040506070

Pric

e

Instance 16

θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Pr

Ps

Pr

Ps

Pr

Ps

Pr

Ps

Figure 8 Comparison of the retail price and direct sale price in centralized dual-channel supply chain for different instances

Advances in Operations Research 13

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

02468

101214

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

015

025 0

50

750

850

95

Lead

tim

e

Instance 1

Instance 2Instance 9

Instance 16

Instance 3

Instance 5Instance 17

Customer class 1 Customer class 2 Customer class 3 Customer class 4 Customer class 5 Customer class 6θ

Figure 9 Comparison of the optimal quoted lead time in centralized dual-channel supply chain for different instances

Table 4 Comparison of retailer and direct prices under both decentralized and centralized supply chain

Problem numberP

r

II minus Pr

I Ps

II minus Ps

I

θ 015 θ 025 θ 05 θ 075 θ 085 θ 095 θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 minus 520 minus 599 minus 023 088 015 000 006 minus 036 minus 194 minus 093 minus 006 0002 minus 524 minus 574 015 065 037 002 minus 019 minus 424 minus 518 minus 139 minus 036 minus 0013 minus 045 minus 044 minus 097 005 000 minus 005 minus 005 minus 008 minus 012 minus 005 000 minus 0014 minus 564 minus 452 019 minus 002 037 001 minus 026 minus 464 minus 492 minus 216 minus 036 minus 0015 minus 082 minus 111 minus 086 044 001 minus 349 003 minus 002 minus 007 minus 014 000 minus 0626 minus 447 minus 376 minus 034 minus 001 027 minus 381 minus 153 minus 506 minus 506 minus 218 minus 061 minus 0697 minus 422 minus 422 minus 153 000 000 000 128 128 047 000 000 0008 minus 524 minus 621 minus 132 minus 079 minus 058 minus 037 minus 019 minus 531 minus 699 minus 420 minus 308 minus 1979 minus 078 minus 142 minus 107 068 129 minus 567 067 055 046 minus 028 minus 029 minus 12510 minus 531 minus 460 minus 182 minus 074 minus 142 minus 664 minus 174 minus 544 minus 701 minus 446 minus 320 minus 28911 minus 499 minus 499 minus 263 000 000 minus 076 114 114 029 000 000 minus 01112 minus 524 minus 596 minus 132 minus 079 minus 058 minus 141 minus 019 minus 534 minus 699 minus 420 minus 308 minus 19413 000 000 000 000 000 045 000 000 000 000 000 minus 01014 000 000 000 000 000 002 000 000 000 000 000 minus 00115 000 000 007 009 minus 001 minus 396 000 000 001 002 000 minus 06916 000 000 000 000 000 minus 381 000 000 000 000 000 minus 06917 000 000 minus 001 005 000 028 000 000 003 001 000 minus 01418 000 000 000 000 000 001 000 000 000 000 000 minus 001

Table 5 Comparison of obtained profits under both decentralized and centralized supply chain

ΠII minus ΠI

Problem number θ 015 θ 025 θ 05 θ 075 θ 085 θ 095

1 388436 565756 309133 13657 9142 16272 658972 905578 649573 83946 7591 0003 7540 9832 33652 866 000 minus 167174 671326 996570 706269 90876 7591 0005 44087 58224 50225 27285 minus 43803 609736 881277 1071997 563536 96510 12380 0007 38500 38500 14000 000 000 67578 658972 1527077 1045274 377387 203484 971909 56632 61244 91249 10192 9621 1080210 1148149 1601967 1081605 440786 244925 10226011 35453 35453 8187 320 320 540612 660300 1360109 1045594 377707 203804 10623613 000 000 000 000 000 00014 000 000 000 000 000 00015 000 000 11102 13035 14726 minus 1856216 000 000 000 000 000 00017 000 000 minus 4752 15625 minus 1091 1666718 000 000 000 000 000 000

14 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

low and the retail price is generally also set to be low Weknow that the wholesale price must be lower than the retailprice erefore with increasing θ the retail price andconsequently the wholesale price decrease As we can seefrom Figures 5 and 8 for some values of θ both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set higher than those in the cen-tralized one while the total profit in the decentralized dual-channel supply chain is lower than that in the centralizedonee negative value of gap in price differences in Table 4and the positive value of gap for profits in Table 5 show thisfactis shows that when each partner in the decentralizeddual-channel supply chain maximizes his own profit thisleads to double marginalization Double marginalizationmeans that the retailer and the manufacturer indepen-dently set their price to maximize their profit margins as aresult the price is higher and the sales volume and profitsare lower than those of a vertically integrated channel [3]

In instances with low (CeCr) or with high Cr the directsale price in the centralized setting tends to equal the directsale price in decentralized settings is trend is also true forretail prices in the two settings Moreover when Cr is verylarge the retail channel has a small impact on the manu-facturerrsquos profits and in both settings the manufacturer hasa main role in determining prices erefore the pricesdefined in the two settings become close

As illustrated in Figure 7 in the centralized dual-channelsupply chain when customer preference of the direct channelθ is lower than a certain threshold the total profit decreases

with increasing θ conversely when θ exceeds the thresholdthe total profit increases with increasing θ is thresholddiffers from one instance to another In examples with highdirect channel operational cost the threshold is near 1meaning the function seems as decreasing function Tocompare the profitability changes of the firm according todifferent direct and retail channel operational costs wecarried somemore examples by varying these two parametersFigure 10 shows the results As we can see from Figure 10under high value of direct channel operational cost the profitdecreases with increasing θ erefore it is more profitable toencourage customers buying from retail channel

As we can see from Figures 6 and 9 the high lead timechanges depending on θ in two centralized and decentralizedsetting show that the delivery lead time has a strong effect onthe manufacturerrsquos and the retailerrsquos pricing strategies andprofits We can see the results of Propositions 4 and 5 ininstances 2 and 16 where the unlimited capacity is con-siderede quoted lead times for the customers are equal tothe lower bound or the upper bound of the lead time

We perform a numerical study to compare how the leadtime flexibility affects the profitability of the firm

We consider two policies of lead time flexibility P1 (leadtime flexibility) P2 (no lead time flexibility) With lead timeflexibility we quote different lead times to different cus-tomers When there is no lead time flexibility a single (fixed)lead time is quoted to all customers

We have generated the problems in three situations thecustomersrsquo sensitivity to price in direct channel is 1- greater

015 025 05 075 085 095

CeCr = 70CeCr = 70CeCr= 70

CeCr= 70CeCr = 70CeCr = 70

CeCr = 70CeCr = 20CeCr = 10

0

20000

40000

60000

80000

100000

120000

πlowast

θ

(a)

015 025 05 075 085 095

CeCr = 110CeCr = 120CeCr = 170

CeCr = 170CeCr = 170CeCr= 170

CeCr = 170CeCr = 170CeCr = 170

020000400006000080000

100000120000140000160000180000

πlowast

θ

(b)

Figure 10 Comparison of the profitability of the firm according to different direct and retail channel operational costs

Table 6 e percentage of profit increases over P2 policy with no lead time flexibility

Problem number Capacity (K) Lead time sensitivity variability (LV)Gap ((P1 minus P2)P2)lowast 100

bsi gt br

i bsi br

i bsi lt br

i

1 M Yes 1974 1981 22862 M No 1720 1598 18343 H Yes 108 164 0114 H No 0 080 0

Advances in Operations Research 15

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

2- equal and 3- lower than customersrsquo sensitivity to price inretail channel We have considered the same price sensi-tivities for all customers e categorization for productioncapacity and variability of customersrsquo lead time sensitivity isshown in Table 3 e percentage of profit increases over P2policy with no lead time flexibility under each policy andcategory is represented in Table 6e main conclusions onecan draw from Table 6 are as follows

(1) e effect of production capacity comparing in-stances 2 and 4 (instance 4 with high productioncapacity) we can see that more capacity makes themanufacturer able to charge equal lead times to allcustomers and manufacturers usually determine alead time equal to that of the first period for allcustomers

(2) e effect of variability of customersrsquo sensitivity tolead time comparing instances 1 and 2 (instance 1with high variability in customersrsquo sensitivity) wecan see that if there is high variability in customersrsquolead time sensitivities the manufacturer can obtainmore profit from lead time flexibility e manu-facturer can charge the high lead time for customerwith lower sensitivity to lead time and reserve thecapacity of the first period for customers with highersensitivity to lead time

6 Conclusion

In this paper we presented a pricing and due date settingmodel for a manufacturer with a dual sale channel an onlinedirect channel and a traditional retail channel We see that alarge number of e-retailers such as Amazoncom BestBuycom and Walmartcom attempt to offer a proper deliverylead time (Yang et al [4]) Also the model can be developedfor rsquos auto industry which is the second largest industry inIran after oil and gas Currently Iranrsquos auto industry has anonline sale and the proposed list price is sometimes based ondelivery time

We assumed a finite planning horizon divided intoperiods of equal length We considered the limit productioncapacity in each period and multiple classes of customersarriving during these periods We developed a Stackelberggame to model the decentralized dual-channel supply chainUnder this game framework the supplier as the leaderannounces the wholesale price to an intermediary in ad-dition to the direct channel sales price and lead time eintermediary then reacts by choosing a retail price tomaximize its own profit Bilevel programming was devel-oped to model this situation For a centralized dual-channelsupply chain the traditional retail price the direct sale priceand the quoted lead time in the direct channel are deter-mined in an integrated manner Considering the lead time

selection for each customer order mixed binary integernonlinear programming models for the design of the cen-tralized and decentralized dual-channel supply chains wereused

Because exact algorithms for solving the proposedmodels can be expensive and time consuming for instanceswith large numbers of 0-1 variables the PSO algorithmsolving model is adopted to get a satisfactory near-optimalsolution efficiently

rough numerical analyses we have examined theeffects of the customer preference of a direct channel on themanufacturersrsquo and retailersrsquo pricing behaviors We alsocompared the optimal lead times prices and profits in thetwo settings of a centralized and decentralized dual-channelsupply chain

Numerical examples have shown that the retail price isdecreasing with increasing customer preference of the directchannel and that the direct sale price is increasing withincreasing customer preference of the direct channelerefore when the customer preference of the directchannel is low the retail price is higher than the direct saleprice conversely when the customer preference of the directchannel is high the retail price is lower than the direct saleprice Comparing the two settings of dual-channel supplychains we found that when θ is relatively low and thus thebase level demand in the retail channel is high both the retailprice and the direct price in the decentralized dual-channelsupply chain should be set to be higher than those in thecentralized supply chain while the total profit in thedecentralized dual-channel supply chain is lower than that inthe centralized supply chain is shows double marginal-ization in the decentralized dual-channel supply chain whicheach member maximizes his own profit

e high changes in the lead time as a function of θ in thecentralized and decentralized settings show that the deliverylead time strongly influences the manufacturersrsquo and theretailersrsquo pricing strategies and profits erefore consid-ering a constant value for the lead time in all scenarios leadsto a decrease in profits

is research can be extended in several directions infuture work First our model is deterministic and we canstudy a model with stochastic demand Second we canconsider competition among several manufacturers andretailers ird it is worth investigating the coordination ofthe dual-channel supply chain by contracts when the priceand lead time are both considered in the models

Appendix

Proof of Proposition 1 For given Wi Psi and Ls

i the retailerrsquosprofit is determined by

MaxPr

i

Πr 1113944iisinψ

Pri minus Wi minus Cri( 1113857 middot ai middot 1 minus θi( 1113857 minus b

ri middot P

ri + αs

i middot Psi + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij⎛⎝ ⎞⎠ (A1)

16 Advances in Operations Research

Obviously Πr is a concave quadratic function of Pri

Using the partial first-order condition we get

Pri

ai 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

foralli isin Ψ

(A2)

Proof of Proposition 2 If Pri 0 then we can simplify

(24)ndash(26) as follows

ari 1 minus θi( 1113857 + αs

i Psi + c

si 1113944

T

je(i)

(j minus e(i) + 1) middot Zij + Wi middot bri + Cri middot b

ri + λi middot b

ri minus λiprime middot br

i + ]i 0 foralli isin Ψ (A3)

ai middot 1 minus θi( 1113857 + αsi middot P

si + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij + St Ktt e(i) foralli isin Ψ j 1 T (A4)

ai middot 1 minus θi( 1113857 + αsi middot P

si + c

ri 1113944

T

je(i)

(j minus e(i) + 1) middot Zij Siprime (A5)

Hence Siprime gt 0 Based on complementary conditions (27)

the λiprime 0 us the constraint (A3) can be simplified as

follows

minus ari 1 minus θi( 1113857 + αs

i Psi + c

si 1113944

T

je(i)

(j minus e(i) + 1) middot Zij + Wi middot bri + Cri middot b

ri + λi middot b

ri

⎛⎝ ⎞⎠ ]i foralli isin Ψ (A6)

erefore the ]i is always negative which is acontradiction

Proof of Proposition 3 e retailerrsquos best response to thewholesale price Wi the direct sale price Ps

i and the directsale quoted lead time Ls

i set by the manufacturer is given by(3)

According to unlimited capacity in each period themanufacturerrsquos profit is determined by

Πs 1113944N

i1p

si minus Cp

2i1113872 1113873 times D

si + 1113944

N

i1Wi minus Cp

1i1113872 1113873 times D

ri (A7)

Substituting (15) into (A7) and simplifying it we get

Πs 1113944N

i1p

si minus Cp

2i1113872 1113873 times ai middot θi minus b

si middot P

si + αr

i middotai middot 1 minus θi( 1113857 + Wi middot br

i + αsi P

si + cr

i Lsi + Crib

ri

2bri

1113888 1113889 minus csi middot L

si1113888 1113889

+ 1113944N

i1Wi minus Cp

1i1113872 1113873 times ai middot 1 minus θi( 1113857 minus b

ri middot

ai middot 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

1113888 1113889 + αsi middot P

si + c

ri L

si1113888 1113889

(A8)

Taking the second-order partial derivatives of Πs withrespect to Ps

i and Wi we have the Hessian matrix as followsH

z2Πs

z psi( 11138572

z2Πs

z psi( 1113857z Wi( 1113857

z2Πs

z Wi( 1113857z psi( 1113857

z2Πs

z Wi( 11138572

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2bsi +

αri αs

i

bri

αri + αs

i

2

αri + αs

i

2minus br

i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A9)

Advances in Operations Research 17

From assumption 8bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 6αs

i αri gt 0

Πs is strictly jointly concave in Psi and Wi since

(z2Πsz(psi )2) minus 2bs

i + (αri α

si b

ri )lt 0 and

minus 2bsi +

αriαs

i

bri

αri + αs

i

2

αri + αs

i

2minus br

i

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

8br

i bsi minus αs

i( 11138572

minus αri( 1113857

2minus 6αs

i middot αri

4gt 0 (A10)

Proof of Proposition 4 As expressed in Proposition 3 underunlimited production capacity the manufacturerrsquos best priceis obtained from

Max Πs 1113936N

i1ps

i minus Cp1i( 1113857 times ai middot θi minus bs

i middot Psi + αr

i middotai middot 1 minus θi( 1113857 + Wi middot br

i + αsi P

si + cr

i Lsi + Crib

ri

2bri

1113888 1113889 minus csi middot Ls

i1113888 1113889 +

1113936N

i1Wi minus Cp2

i( 1113857 times ai middot 1 minus θi( 1113857 minus bri middot

ai middot 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

1113888 1113889 + αsi middot Ps

i + cri L

si1113888 1113889

ST Wi lepsi

(A11)

and the KKT conditions for this model are

2aiθibri + aiαi 1 minus θi( 1113857 minus p

si 4b

ri b

si minus 2 αi( 1113857

21113872 1113873 + αib

ri Cri + L

si αic

ri minus 2c

si b

ri( 1113857 + 2αib

ri Wi + Cp

2i 2b

ri b

si minus αi( 1113857

21113872 1113873 minus Cp

1i αib

ri + 2ηib

ri 0 (A12)

2αipsi minus αiCp

2i + ai 1 minus θi( 1113857 minus b

ri Cri + c

ri L

si minus 2b

ri Wi + b

ri Cp

1i minus 2ηi 0 (A13)

ηi Wi minus psi( 1113857 0 (A14)

Wi lepsi (A15)

ηi ge 0 (A16)

Case 1 When ηi 0 we obtain Wlowasti (Lsi ) and pslowast

i (Lsi ) from

((A12) and (A13)) Notice that Wlowasti (Lsi )lepslowast

i (Lsi ) ere-

fore we have

Wilowast

Lsi( 1113857 minus p

slowasti L

si( 1113857

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873 + cr

i Lsi bs

i minus αi( 1113857 + csi L

si br

i minus αi( 1113857

2 bri b

si minus αi( 1113857

21113872 1113873

le 0

or Lsi ge

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 L

0i

(A17)

e second-order partial derivative of Πs with respect toLs

i is (z2Πsz(Lsi )2) 0 which indicates that theΠs is a linear

function of Lsi erefore in this case the optimal value of Ls

i

will be equal to its upper bound (Lui ) or lower bound (L0

i )

18 Advances in Operations Research

Case 2 When ηi gt 0 thenWlowasti (Lsi ) pslowast

i (Lsi ) When Ls

i lt L0i

we have ηi gt 0 In this case the second-order partial de-rivative ofΠs with respect to Ls

i is also (z2Πsz(Lsi )2) 0 and

we only need to check the upper bound (L0i ) and lower

bound (LLi )

Overall the optimal value Lsi can be found from Ll

i Lui

and L0i by comparing the Πs values at them and the one at

which Πs is the largest is the optimal

Proof of Proposition 5 According to unlimited capacity ineach period the profit of the centralized supply chain is

Πc Πs + Πr 1113944N

i1p

si minus Cp

1i1113872 1113873 times ai middot θi minus b

si middot P

si + αr

i middot pri minus c

si middot L

si(⎛⎝ ⎞⎠ + 1113944

N

i1p

ri minus Cp

2i minus Cri1113872 1113873 times ai middot 1 minus θi( 1113857 minus b

ri middot p

ri + αs

i middot Psi + c

ri L

si( 1113857

(A18)

Taking the second-order partial derivatives of Πc withrespect to Pr

i Psi and Ls

i we have the Hessian matrix asfollows

H

z2Πc

z psi( 11138572

z2Πc

z psi( 1113857z pr

i( 1113857

z2Πc

z psi( 1113857z Ls

i( 1113857

z2Πc

z pri( 1113857z ps

i( 1113857

z2Πc

z pri( 1113857

2z2Πc

z pri( 1113857z Ls

i( 1113857

z2Πc

z Lsi( 1113857z ps

i( 1113857

z2Πc

z Lsi( 1113857z pr

i( 1113857

z2Πc

z Lsi( 11138572

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2bsi αr

i + αsi minus cs

i

αri + αs

i minus 2bri cr

i

minus csi cr

i 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A19)

From assumption 4bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 2αs

i middot αri gt 0

Πc is strictly jointly concave in Psi and Pr

i since(z2Πcz(ps

i )2) minus 2bs

i lt 0 andminus 2bs

i αri + αs

i

αri + αs

i minus 2bri

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 4b

ri middot b

si minus αs

i( 11138572

minus αri( 1113857

2minus 2αs

i middot αri gt 0

(A20)

e second-order partial derivative of Πc with respect toLs

i is (z2Πcz(Lsi )2) 0 which indicates that theΠc is a linear

function of Lsi According to the above properties the op-

timal value of Lsi will be equal to its upper bound or lower

bound

Proof of Proposition 6 From Dsi ge 0 Dr

i ge 0 and constraints(1) and (2) we obtain

Pri le

ai 1 minus θi( 1113857 + αsi middot Ps

i + cri middot Ls

i

bri

foralli isin Ψ

Pri ge

minus ai middot θi + bsi middot Ps

i + csi middot Ls

i

αri

foralli isin Ψ

(A21)

To have a feasible space for Psi and Pr

i the upper boundobtained for Pr

i must be greater than or equal to the lowerbound Consequently we have

ai 1 minus θi( 1113857le + αsi middot Ps

i + cri middot Ls

i

bri

geminus ai middot θi + bs

i middot Psi + cs

i middot Lsi

αri

foralli isin Ψ

Lsi le

ai θi middot bri minus θi middot αr

i + αri( 1113857

bri middot cs

i minus αri middot cr

i

minus Psi

bsi middot br

i minus αsi middot αr

i

bri middot cs

i minus αri middot cr

i

1113888 1113889 foralli isin Ψ

(A22)

Advances in Operations Research 19

Substituting Psi 0 into the above inequality we can

derive (39) which is upper bound for Lsi

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] A A Tsay and N Agrawal ldquoModeling conflict and coordi-nation in multi-channel distribution systems a reviewrdquo inHandbook of Quantitative Supply Chain Analysis Modeling inthe eBusiness Era 557ndash606) S D Simchi-Levi Wu andZ M Shen Eds Kluwer Academic Publishers Norwell MAUSA 2004

[2] K L Webb and J E Hogan ldquoHybrid channel conflict causesand effects on channel performancerdquo Journal of Business ampIndustrial Marketing vol 17 no 5 pp 338ndash356 2002

[3] G Hua S Wang and T C E Cheng ldquoPrice and lead timedecisions in dual-channel supply chainsrdquo European Journal ofOperational Research vol 205 no 1 pp 113ndash126 2010

[4] J Q Yang X M Zhang H Y Fu and C Liu ldquoInventorycompetition in a dual-channel supply chain with delivery leadtime considerationrdquo Applied Mathematical Modelling vol 42pp 675ndash692 2017

[5] K Cattani W G Gilland and J M Swaminathan ldquoCoordi-nating traditional and Internet supply chainsrdquo in Handbook ofQuantitative Supply Chain Analysis Modeling in the eBusinessEra D Simchi-Levi amp M Wu and Shen Eds pp 643ndash680Kluwer Academic Publishers Norwell MA USA 2004

[6] G Cai Z G Zhang and M Zhang ldquoGame theoretical per-spectives on dual-channel supply chain competition withprice discounts and pricing schemesrdquo International Journal ofProduction Economics vol 117 no 1 pp 80ndash96 2009

[7] W-y K Chiang D Chhajed and J D Hess ldquoDirect mar-keting indirect profits a strategic analysis of dual-channelsupply-chain designrdquo Management Science vol 49 no 1pp 1ndash20 2003

[8] S Panda N M Modak S S Sana and M Basu ldquoPricing andreplenishment policies in dual-channel supply chain undercontinuous unit cost decreaserdquo Applied Mathematics andComputation vol 256 pp 913ndash929 2015

[9] D Q Yao and J J Liu ldquoCompetitive pricing of mixed retailand e-tail distribution channelsrdquoOmega vol 33 pp 235ndash2472005

[10] B Dan G Xu and C Liu ldquoPricing policies in a dual-channelsupply chain with retail servicesrdquo International Journal ofProduction Economics vol 139 no 1 pp 312ndash320 2012

[11] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling vol 40 no 15-16pp 7349ndash7367 2016

[12] A Roy S S Sana and K Chaudhuri ldquoJoint decision on EOQand pricing strategy of a dual channel of mixed retail and e-tailcomprising of single manufacturer and retailer under sto-chastic demandrdquo Computers amp Industrial Engineeringvol 102 pp 423ndash434 2016

[13] A A Tsay and N Agrawal ldquoChannel conflict and coordi-nation in the e-commerce agerdquo Production and OperationsManagement vol 13 pp 93ndash110 2004

[14] Z Zhang S Liu and B Niu ldquoCoordination mechanism ofdual-channel closed-loop supply chains considering productquality and returnrdquo Journal of Cleaner Production vol 248Article ID 119273 2020

[15] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chain when productioncosts are disruptedrdquo EconomicModelling vol 30 pp 521ndash5382013

[16] Z Pi W Fang and B Zhang ldquoService and pricing strategieswith competition and cooperation in a dual-channel supplychain with demand disruptionrdquo Computers amp IndustrialEngineering vol 138 2019

[17] F Soleimani A Arshadi Khamseh and B Naderi ldquoOptimaldecisions in a dual-channel supply chain under simultaneousdemand and production cost disruptionsrdquo Annals of Oper-ations Research vol 243 no 1-2 pp 301ndash321 2016

[18] T M Rofin and B Mahanty ldquoOptimal dual-channel supplychain configuration for product categories with differentcustomer preference of online channelrdquo Electronic CommerceResearch vol 18 no 3 pp 507ndash536 2018

[19] T Xiao T-M Choi and T C E Cheng ldquoProduct variety andchannel structure strategy for a retailer-Stackelberg supplychainrdquo European Journal of Operational Research vol 233no 1 pp 114ndash124 2014

[20] J Zhou R Zhao and B Wang ldquoBehavior-based price dis-crimination in a dual-channel supply chain with retailerrsquosinformation disclosurerdquo Electronic Commerce Research andApplications vol 39 2020

[21] I Duenyas and W J Hopp ldquoQuoting customer lead timesrdquoManagement Science vol 41 no 1 pp 43ndash57 1995

[22] K C So and J-S Song ldquoPrice delivery time guarantees andcapacity selectionrdquo European Journal of Operational Researchvol 111 no 1 pp 28ndash49 1998

[23] F F Easton and D R Moodie ldquoPricing and lead time de-cisions for make-to-order firms with contingent ordersrdquoEuropean Journal of Operational Research vol 116 no 2pp 305ndash318 1999

[24] P Keskinocak R Ravi and S Tayur ldquoScheduling and reliablelead-time quotation for orders with availability intervals andlead-time sensitive revenuesrdquo Management Science vol 47no 2 pp 264ndash279 2001

[25] S Webster ldquoDynamic pricing and lead-time policies formake-to-order systemsrdquo Decision Sciences vol 33 pp 579ndash599 2002

[26] BWatanapa andA Techanitisawad ldquoSimultaneous price andduedate settings for multiple customer classesrdquo European Journal ofOperational Research vol 166 no 2 pp 351ndash368 2005

[27] K Charnsirisakskul P M Griffin and P KeskinocakldquoPricing and scheduling decisions with leadtime flexibilityrdquoEuropean Journal of Operational Research vol 171 no 1pp 153ndash169 2006

[28] A Mustafa B Ata and T Olsen ldquoCongestion based lead timequotation for heterogenous customers with convex-concavedelay costs optimality of a cost balancing policy based onconvex hull functionsrdquo Operations Research vol 57 no 3pp 753ndash768 2009

[29] S K Chaharsooghi M Honarvar M Modarres andI N Kamalabadi ldquoDeveloping a two stage stochastic pro-gramming model of the price and lead-time decision problemin the multi-class make-to-order firmrdquo Computers amp In-dustrial Engineering vol 61 no 4 pp 1086ndash1097 2011

20 Advances in Operations Research

[30] S K Chaharsooghi M Honarvar and M Modarres ldquoAmulti-stage stochastic programming model for dynamicpricing and lead time decisions in multi-class make-to-orderfirmrdquo Scientia Iranica vol 18 no 3 pp 711ndash721 2011

[31] K-Y Chen M Kaya and O Ozer ldquoDual sales channelmanagement with service competitionrdquo Manufacturing ampService Operations Management vol 10 no 4 pp 654ndash6752008

[32] H Xu Z Z Liu and S H Zhang ldquoA strategic analysis of dual-channel supply chain design with price and delivery lead timeconsiderationsrdquo International Journal of Production Eco-nomics vol 139 no 2 pp 654ndash663 2012

[33] R Batarfi M Y Jaber and S Zanoni ldquoDual-channel supplychain a strategy to maximize profitrdquo Applied MathematicalModelling vol 40 no 21-22 pp 9454ndash9473 2016

[34] N M Modak and P Kelle ldquoManaging a dual-channel supplychain under price and delivery-time dependent stochasticdemandrdquo European Journal of Operational Research vol 272no 1 pp 147ndash161 2019

[35] H Kurata D-Q Yao and J J Liu ldquoPricing policies underdirect vs indirect channel competition and national vs storebrand competitionrdquo European Journal of Operational Re-search vol 180 no 1 pp 262ndash281 2007

[36] G L Nemhauser and L AWolsey Integer and CombinatorialOptimization Wiley-Interscience New York NY USA 1988

[37] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of 1995 IEEE International Conference on NeuralNetworks pp 1942ndash1948 IEEE Perth Australia November1995

[38] J Kennedy RC Eberhart and Y Shi Swarm IntelligenceMorgan Kaufmann San Mateo CA USA 2001

[39] S Koziel and Z Michalewicz ldquoEvolutionary algorithmshomomorphous mappings and constrained parameter opti-mizationrdquo Evolutionary Computation vol 7 pp 19ndash44 1999

[40] J H Holland Adaptation in Natural and Artificial SystemsAn Introductory Analysis with Applications to Biology Con-trol and Artificial Intelligence University of Michigan PressAnn Arbor MI USA 1975

Advances in Operations Research 21

From assumption 8bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 6αs

i αri gt 0

Πs is strictly jointly concave in Psi and Wi since

(z2Πsz(psi )2) minus 2bs

i + (αri α

si b

ri )lt 0 and

minus 2bsi +

αriαs

i

bri

αri + αs

i

2

αri + αs

i

2minus br

i

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

8br

i bsi minus αs

i( 11138572

minus αri( 1113857

2minus 6αs

i middot αri

4gt 0 (A10)

Proof of Proposition 4 As expressed in Proposition 3 underunlimited production capacity the manufacturerrsquos best priceis obtained from

Max Πs 1113936N

i1ps

i minus Cp1i( 1113857 times ai middot θi minus bs

i middot Psi + αr

i middotai middot 1 minus θi( 1113857 + Wi middot br

i + αsi P

si + cr

i Lsi + Crib

ri

2bri

1113888 1113889 minus csi middot Ls

i1113888 1113889 +

1113936N

i1Wi minus Cp2

i( 1113857 times ai middot 1 minus θi( 1113857 minus bri middot

ai middot 1 minus θi( 1113857 + Wi middot bri + αs

i Psi + cr

i Lsi + Crib

ri

2bri

1113888 1113889 + αsi middot Ps

i + cri L

si1113888 1113889

ST Wi lepsi

(A11)

and the KKT conditions for this model are

2aiθibri + aiαi 1 minus θi( 1113857 minus p

si 4b

ri b

si minus 2 αi( 1113857

21113872 1113873 + αib

ri Cri + L

si αic

ri minus 2c

si b

ri( 1113857 + 2αib

ri Wi + Cp

2i 2b

ri b

si minus αi( 1113857

21113872 1113873 minus Cp

1i αib

ri + 2ηib

ri 0 (A12)

2αipsi minus αiCp

2i + ai 1 minus θi( 1113857 minus b

ri Cri + c

ri L

si minus 2b

ri Wi + b

ri Cp

1i minus 2ηi 0 (A13)

ηi Wi minus psi( 1113857 0 (A14)

Wi lepsi (A15)

ηi ge 0 (A16)

Case 1 When ηi 0 we obtain Wlowasti (Lsi ) and pslowast

i (Lsi ) from

((A12) and (A13)) Notice that Wlowasti (Lsi )lepslowast

i (Lsi ) ere-

fore we have

Wilowast

Lsi( 1113857 minus p

slowasti L

si( 1113857

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873 + cr

i Lsi bs

i minus αi( 1113857 + csi L

si br

i minus αi( 1113857

2 bri b

si minus αi( 1113857

21113872 1113873

le 0

or Lsi ge

minus aiαi 1 minus θi( 1113857 minus aiθi bri minus αi( 1113857 + aib

si 1 minus θi( 1113857 minus Cri br

i bsi minus αi( 1113857

21113872 1113873

cri bs

i minus αi( 1113857 + csi br

i minus αi( 1113857 L

0i

(A17)

e second-order partial derivative of Πs with respect toLs

i is (z2Πsz(Lsi )2) 0 which indicates that theΠs is a linear

function of Lsi erefore in this case the optimal value of Ls

i

will be equal to its upper bound (Lui ) or lower bound (L0

i )

18 Advances in Operations Research

Case 2 When ηi gt 0 thenWlowasti (Lsi ) pslowast

i (Lsi ) When Ls

i lt L0i

we have ηi gt 0 In this case the second-order partial de-rivative ofΠs with respect to Ls

i is also (z2Πsz(Lsi )2) 0 and

we only need to check the upper bound (L0i ) and lower

bound (LLi )

Overall the optimal value Lsi can be found from Ll

i Lui

and L0i by comparing the Πs values at them and the one at

which Πs is the largest is the optimal

Proof of Proposition 5 According to unlimited capacity ineach period the profit of the centralized supply chain is

Πc Πs + Πr 1113944N

i1p

si minus Cp

1i1113872 1113873 times ai middot θi minus b

si middot P

si + αr

i middot pri minus c

si middot L

si(⎛⎝ ⎞⎠ + 1113944

N

i1p

ri minus Cp

2i minus Cri1113872 1113873 times ai middot 1 minus θi( 1113857 minus b

ri middot p

ri + αs

i middot Psi + c

ri L

si( 1113857

(A18)

Taking the second-order partial derivatives of Πc withrespect to Pr

i Psi and Ls

i we have the Hessian matrix asfollows

H

z2Πc

z psi( 11138572

z2Πc

z psi( 1113857z pr

i( 1113857

z2Πc

z psi( 1113857z Ls

i( 1113857

z2Πc

z pri( 1113857z ps

i( 1113857

z2Πc

z pri( 1113857

2z2Πc

z pri( 1113857z Ls

i( 1113857

z2Πc

z Lsi( 1113857z ps

i( 1113857

z2Πc

z Lsi( 1113857z pr

i( 1113857

z2Πc

z Lsi( 11138572

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2bsi αr

i + αsi minus cs

i

αri + αs

i minus 2bri cr

i

minus csi cr

i 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A19)

From assumption 4bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 2αs

i middot αri gt 0

Πc is strictly jointly concave in Psi and Pr

i since(z2Πcz(ps

i )2) minus 2bs

i lt 0 andminus 2bs

i αri + αs

i

αri + αs

i minus 2bri

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 4b

ri middot b

si minus αs

i( 11138572

minus αri( 1113857

2minus 2αs

i middot αri gt 0

(A20)

e second-order partial derivative of Πc with respect toLs

i is (z2Πcz(Lsi )2) 0 which indicates that theΠc is a linear

function of Lsi According to the above properties the op-

timal value of Lsi will be equal to its upper bound or lower

bound

Proof of Proposition 6 From Dsi ge 0 Dr

i ge 0 and constraints(1) and (2) we obtain

Pri le

ai 1 minus θi( 1113857 + αsi middot Ps

i + cri middot Ls

i

bri

foralli isin Ψ

Pri ge

minus ai middot θi + bsi middot Ps

i + csi middot Ls

i

αri

foralli isin Ψ

(A21)

To have a feasible space for Psi and Pr

i the upper boundobtained for Pr

i must be greater than or equal to the lowerbound Consequently we have

ai 1 minus θi( 1113857le + αsi middot Ps

i + cri middot Ls

i

bri

geminus ai middot θi + bs

i middot Psi + cs

i middot Lsi

αri

foralli isin Ψ

Lsi le

ai θi middot bri minus θi middot αr

i + αri( 1113857

bri middot cs

i minus αri middot cr

i

minus Psi

bsi middot br

i minus αsi middot αr

i

bri middot cs

i minus αri middot cr

i

1113888 1113889 foralli isin Ψ

(A22)

Advances in Operations Research 19

Substituting Psi 0 into the above inequality we can

derive (39) which is upper bound for Lsi

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] A A Tsay and N Agrawal ldquoModeling conflict and coordi-nation in multi-channel distribution systems a reviewrdquo inHandbook of Quantitative Supply Chain Analysis Modeling inthe eBusiness Era 557ndash606) S D Simchi-Levi Wu andZ M Shen Eds Kluwer Academic Publishers Norwell MAUSA 2004

[2] K L Webb and J E Hogan ldquoHybrid channel conflict causesand effects on channel performancerdquo Journal of Business ampIndustrial Marketing vol 17 no 5 pp 338ndash356 2002

[3] G Hua S Wang and T C E Cheng ldquoPrice and lead timedecisions in dual-channel supply chainsrdquo European Journal ofOperational Research vol 205 no 1 pp 113ndash126 2010

[4] J Q Yang X M Zhang H Y Fu and C Liu ldquoInventorycompetition in a dual-channel supply chain with delivery leadtime considerationrdquo Applied Mathematical Modelling vol 42pp 675ndash692 2017

[5] K Cattani W G Gilland and J M Swaminathan ldquoCoordi-nating traditional and Internet supply chainsrdquo in Handbook ofQuantitative Supply Chain Analysis Modeling in the eBusinessEra D Simchi-Levi amp M Wu and Shen Eds pp 643ndash680Kluwer Academic Publishers Norwell MA USA 2004

[6] G Cai Z G Zhang and M Zhang ldquoGame theoretical per-spectives on dual-channel supply chain competition withprice discounts and pricing schemesrdquo International Journal ofProduction Economics vol 117 no 1 pp 80ndash96 2009

[7] W-y K Chiang D Chhajed and J D Hess ldquoDirect mar-keting indirect profits a strategic analysis of dual-channelsupply-chain designrdquo Management Science vol 49 no 1pp 1ndash20 2003

[8] S Panda N M Modak S S Sana and M Basu ldquoPricing andreplenishment policies in dual-channel supply chain undercontinuous unit cost decreaserdquo Applied Mathematics andComputation vol 256 pp 913ndash929 2015

[9] D Q Yao and J J Liu ldquoCompetitive pricing of mixed retailand e-tail distribution channelsrdquoOmega vol 33 pp 235ndash2472005

[10] B Dan G Xu and C Liu ldquoPricing policies in a dual-channelsupply chain with retail servicesrdquo International Journal ofProduction Economics vol 139 no 1 pp 312ndash320 2012

[11] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling vol 40 no 15-16pp 7349ndash7367 2016

[12] A Roy S S Sana and K Chaudhuri ldquoJoint decision on EOQand pricing strategy of a dual channel of mixed retail and e-tailcomprising of single manufacturer and retailer under sto-chastic demandrdquo Computers amp Industrial Engineeringvol 102 pp 423ndash434 2016

[13] A A Tsay and N Agrawal ldquoChannel conflict and coordi-nation in the e-commerce agerdquo Production and OperationsManagement vol 13 pp 93ndash110 2004

[14] Z Zhang S Liu and B Niu ldquoCoordination mechanism ofdual-channel closed-loop supply chains considering productquality and returnrdquo Journal of Cleaner Production vol 248Article ID 119273 2020

[15] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chain when productioncosts are disruptedrdquo EconomicModelling vol 30 pp 521ndash5382013

[16] Z Pi W Fang and B Zhang ldquoService and pricing strategieswith competition and cooperation in a dual-channel supplychain with demand disruptionrdquo Computers amp IndustrialEngineering vol 138 2019

[17] F Soleimani A Arshadi Khamseh and B Naderi ldquoOptimaldecisions in a dual-channel supply chain under simultaneousdemand and production cost disruptionsrdquo Annals of Oper-ations Research vol 243 no 1-2 pp 301ndash321 2016

[18] T M Rofin and B Mahanty ldquoOptimal dual-channel supplychain configuration for product categories with differentcustomer preference of online channelrdquo Electronic CommerceResearch vol 18 no 3 pp 507ndash536 2018

[19] T Xiao T-M Choi and T C E Cheng ldquoProduct variety andchannel structure strategy for a retailer-Stackelberg supplychainrdquo European Journal of Operational Research vol 233no 1 pp 114ndash124 2014

[20] J Zhou R Zhao and B Wang ldquoBehavior-based price dis-crimination in a dual-channel supply chain with retailerrsquosinformation disclosurerdquo Electronic Commerce Research andApplications vol 39 2020

[21] I Duenyas and W J Hopp ldquoQuoting customer lead timesrdquoManagement Science vol 41 no 1 pp 43ndash57 1995

[22] K C So and J-S Song ldquoPrice delivery time guarantees andcapacity selectionrdquo European Journal of Operational Researchvol 111 no 1 pp 28ndash49 1998

[23] F F Easton and D R Moodie ldquoPricing and lead time de-cisions for make-to-order firms with contingent ordersrdquoEuropean Journal of Operational Research vol 116 no 2pp 305ndash318 1999

[24] P Keskinocak R Ravi and S Tayur ldquoScheduling and reliablelead-time quotation for orders with availability intervals andlead-time sensitive revenuesrdquo Management Science vol 47no 2 pp 264ndash279 2001

[25] S Webster ldquoDynamic pricing and lead-time policies formake-to-order systemsrdquo Decision Sciences vol 33 pp 579ndash599 2002

[26] BWatanapa andA Techanitisawad ldquoSimultaneous price andduedate settings for multiple customer classesrdquo European Journal ofOperational Research vol 166 no 2 pp 351ndash368 2005

[27] K Charnsirisakskul P M Griffin and P KeskinocakldquoPricing and scheduling decisions with leadtime flexibilityrdquoEuropean Journal of Operational Research vol 171 no 1pp 153ndash169 2006

[28] A Mustafa B Ata and T Olsen ldquoCongestion based lead timequotation for heterogenous customers with convex-concavedelay costs optimality of a cost balancing policy based onconvex hull functionsrdquo Operations Research vol 57 no 3pp 753ndash768 2009

[29] S K Chaharsooghi M Honarvar M Modarres andI N Kamalabadi ldquoDeveloping a two stage stochastic pro-gramming model of the price and lead-time decision problemin the multi-class make-to-order firmrdquo Computers amp In-dustrial Engineering vol 61 no 4 pp 1086ndash1097 2011

20 Advances in Operations Research

[30] S K Chaharsooghi M Honarvar and M Modarres ldquoAmulti-stage stochastic programming model for dynamicpricing and lead time decisions in multi-class make-to-orderfirmrdquo Scientia Iranica vol 18 no 3 pp 711ndash721 2011

[31] K-Y Chen M Kaya and O Ozer ldquoDual sales channelmanagement with service competitionrdquo Manufacturing ampService Operations Management vol 10 no 4 pp 654ndash6752008

[32] H Xu Z Z Liu and S H Zhang ldquoA strategic analysis of dual-channel supply chain design with price and delivery lead timeconsiderationsrdquo International Journal of Production Eco-nomics vol 139 no 2 pp 654ndash663 2012

[33] R Batarfi M Y Jaber and S Zanoni ldquoDual-channel supplychain a strategy to maximize profitrdquo Applied MathematicalModelling vol 40 no 21-22 pp 9454ndash9473 2016

[34] N M Modak and P Kelle ldquoManaging a dual-channel supplychain under price and delivery-time dependent stochasticdemandrdquo European Journal of Operational Research vol 272no 1 pp 147ndash161 2019

[35] H Kurata D-Q Yao and J J Liu ldquoPricing policies underdirect vs indirect channel competition and national vs storebrand competitionrdquo European Journal of Operational Re-search vol 180 no 1 pp 262ndash281 2007

[36] G L Nemhauser and L AWolsey Integer and CombinatorialOptimization Wiley-Interscience New York NY USA 1988

[37] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of 1995 IEEE International Conference on NeuralNetworks pp 1942ndash1948 IEEE Perth Australia November1995

[38] J Kennedy RC Eberhart and Y Shi Swarm IntelligenceMorgan Kaufmann San Mateo CA USA 2001

[39] S Koziel and Z Michalewicz ldquoEvolutionary algorithmshomomorphous mappings and constrained parameter opti-mizationrdquo Evolutionary Computation vol 7 pp 19ndash44 1999

[40] J H Holland Adaptation in Natural and Artificial SystemsAn Introductory Analysis with Applications to Biology Con-trol and Artificial Intelligence University of Michigan PressAnn Arbor MI USA 1975

Advances in Operations Research 21

Case 2 When ηi gt 0 thenWlowasti (Lsi ) pslowast

i (Lsi ) When Ls

i lt L0i

we have ηi gt 0 In this case the second-order partial de-rivative ofΠs with respect to Ls

i is also (z2Πsz(Lsi )2) 0 and

we only need to check the upper bound (L0i ) and lower

bound (LLi )

Overall the optimal value Lsi can be found from Ll

i Lui

and L0i by comparing the Πs values at them and the one at

which Πs is the largest is the optimal

Proof of Proposition 5 According to unlimited capacity ineach period the profit of the centralized supply chain is

Πc Πs + Πr 1113944N

i1p

si minus Cp

1i1113872 1113873 times ai middot θi minus b

si middot P

si + αr

i middot pri minus c

si middot L

si(⎛⎝ ⎞⎠ + 1113944

N

i1p

ri minus Cp

2i minus Cri1113872 1113873 times ai middot 1 minus θi( 1113857 minus b

ri middot p

ri + αs

i middot Psi + c

ri L

si( 1113857

(A18)

Taking the second-order partial derivatives of Πc withrespect to Pr

i Psi and Ls

i we have the Hessian matrix asfollows

H

z2Πc

z psi( 11138572

z2Πc

z psi( 1113857z pr

i( 1113857

z2Πc

z psi( 1113857z Ls

i( 1113857

z2Πc

z pri( 1113857z ps

i( 1113857

z2Πc

z pri( 1113857

2z2Πc

z pri( 1113857z Ls

i( 1113857

z2Πc

z Lsi( 1113857z ps

i( 1113857

z2Πc

z Lsi( 1113857z pr

i( 1113857

z2Πc

z Lsi( 11138572

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2bsi αr

i + αsi minus cs

i

αri + αs

i minus 2bri cr

i

minus csi cr

i 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A19)

From assumption 4bri middot bs

i minus (αsi )2 minus (αr

i )2 minus 2αs

i middot αri gt 0

Πc is strictly jointly concave in Psi and Pr

i since(z2Πcz(ps

i )2) minus 2bs

i lt 0 andminus 2bs

i αri + αs

i

αri + αs

i minus 2bri

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 4b

ri middot b

si minus αs

i( 11138572

minus αri( 1113857

2minus 2αs

i middot αri gt 0

(A20)

e second-order partial derivative of Πc with respect toLs

i is (z2Πcz(Lsi )2) 0 which indicates that theΠc is a linear

function of Lsi According to the above properties the op-

timal value of Lsi will be equal to its upper bound or lower

bound

Proof of Proposition 6 From Dsi ge 0 Dr

i ge 0 and constraints(1) and (2) we obtain

Pri le

ai 1 minus θi( 1113857 + αsi middot Ps

i + cri middot Ls

i

bri

foralli isin Ψ

Pri ge

minus ai middot θi + bsi middot Ps

i + csi middot Ls

i

αri

foralli isin Ψ

(A21)

To have a feasible space for Psi and Pr

i the upper boundobtained for Pr

i must be greater than or equal to the lowerbound Consequently we have

ai 1 minus θi( 1113857le + αsi middot Ps

i + cri middot Ls

i

bri

geminus ai middot θi + bs

i middot Psi + cs

i middot Lsi

αri

foralli isin Ψ

Lsi le

ai θi middot bri minus θi middot αr

i + αri( 1113857

bri middot cs

i minus αri middot cr

i

minus Psi

bsi middot br

i minus αsi middot αr

i

bri middot cs

i minus αri middot cr

i

1113888 1113889 foralli isin Ψ

(A22)

Advances in Operations Research 19

Substituting Psi 0 into the above inequality we can

derive (39) which is upper bound for Lsi

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] A A Tsay and N Agrawal ldquoModeling conflict and coordi-nation in multi-channel distribution systems a reviewrdquo inHandbook of Quantitative Supply Chain Analysis Modeling inthe eBusiness Era 557ndash606) S D Simchi-Levi Wu andZ M Shen Eds Kluwer Academic Publishers Norwell MAUSA 2004

[2] K L Webb and J E Hogan ldquoHybrid channel conflict causesand effects on channel performancerdquo Journal of Business ampIndustrial Marketing vol 17 no 5 pp 338ndash356 2002

[3] G Hua S Wang and T C E Cheng ldquoPrice and lead timedecisions in dual-channel supply chainsrdquo European Journal ofOperational Research vol 205 no 1 pp 113ndash126 2010

[4] J Q Yang X M Zhang H Y Fu and C Liu ldquoInventorycompetition in a dual-channel supply chain with delivery leadtime considerationrdquo Applied Mathematical Modelling vol 42pp 675ndash692 2017

[5] K Cattani W G Gilland and J M Swaminathan ldquoCoordi-nating traditional and Internet supply chainsrdquo in Handbook ofQuantitative Supply Chain Analysis Modeling in the eBusinessEra D Simchi-Levi amp M Wu and Shen Eds pp 643ndash680Kluwer Academic Publishers Norwell MA USA 2004

[6] G Cai Z G Zhang and M Zhang ldquoGame theoretical per-spectives on dual-channel supply chain competition withprice discounts and pricing schemesrdquo International Journal ofProduction Economics vol 117 no 1 pp 80ndash96 2009

[7] W-y K Chiang D Chhajed and J D Hess ldquoDirect mar-keting indirect profits a strategic analysis of dual-channelsupply-chain designrdquo Management Science vol 49 no 1pp 1ndash20 2003

[8] S Panda N M Modak S S Sana and M Basu ldquoPricing andreplenishment policies in dual-channel supply chain undercontinuous unit cost decreaserdquo Applied Mathematics andComputation vol 256 pp 913ndash929 2015

[9] D Q Yao and J J Liu ldquoCompetitive pricing of mixed retailand e-tail distribution channelsrdquoOmega vol 33 pp 235ndash2472005

[10] B Dan G Xu and C Liu ldquoPricing policies in a dual-channelsupply chain with retail servicesrdquo International Journal ofProduction Economics vol 139 no 1 pp 312ndash320 2012

[11] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling vol 40 no 15-16pp 7349ndash7367 2016

[12] A Roy S S Sana and K Chaudhuri ldquoJoint decision on EOQand pricing strategy of a dual channel of mixed retail and e-tailcomprising of single manufacturer and retailer under sto-chastic demandrdquo Computers amp Industrial Engineeringvol 102 pp 423ndash434 2016

[13] A A Tsay and N Agrawal ldquoChannel conflict and coordi-nation in the e-commerce agerdquo Production and OperationsManagement vol 13 pp 93ndash110 2004

[14] Z Zhang S Liu and B Niu ldquoCoordination mechanism ofdual-channel closed-loop supply chains considering productquality and returnrdquo Journal of Cleaner Production vol 248Article ID 119273 2020

[15] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chain when productioncosts are disruptedrdquo EconomicModelling vol 30 pp 521ndash5382013

[16] Z Pi W Fang and B Zhang ldquoService and pricing strategieswith competition and cooperation in a dual-channel supplychain with demand disruptionrdquo Computers amp IndustrialEngineering vol 138 2019

[17] F Soleimani A Arshadi Khamseh and B Naderi ldquoOptimaldecisions in a dual-channel supply chain under simultaneousdemand and production cost disruptionsrdquo Annals of Oper-ations Research vol 243 no 1-2 pp 301ndash321 2016

[18] T M Rofin and B Mahanty ldquoOptimal dual-channel supplychain configuration for product categories with differentcustomer preference of online channelrdquo Electronic CommerceResearch vol 18 no 3 pp 507ndash536 2018

[19] T Xiao T-M Choi and T C E Cheng ldquoProduct variety andchannel structure strategy for a retailer-Stackelberg supplychainrdquo European Journal of Operational Research vol 233no 1 pp 114ndash124 2014

[20] J Zhou R Zhao and B Wang ldquoBehavior-based price dis-crimination in a dual-channel supply chain with retailerrsquosinformation disclosurerdquo Electronic Commerce Research andApplications vol 39 2020

[21] I Duenyas and W J Hopp ldquoQuoting customer lead timesrdquoManagement Science vol 41 no 1 pp 43ndash57 1995

[22] K C So and J-S Song ldquoPrice delivery time guarantees andcapacity selectionrdquo European Journal of Operational Researchvol 111 no 1 pp 28ndash49 1998

[23] F F Easton and D R Moodie ldquoPricing and lead time de-cisions for make-to-order firms with contingent ordersrdquoEuropean Journal of Operational Research vol 116 no 2pp 305ndash318 1999

[24] P Keskinocak R Ravi and S Tayur ldquoScheduling and reliablelead-time quotation for orders with availability intervals andlead-time sensitive revenuesrdquo Management Science vol 47no 2 pp 264ndash279 2001

[25] S Webster ldquoDynamic pricing and lead-time policies formake-to-order systemsrdquo Decision Sciences vol 33 pp 579ndash599 2002

[26] BWatanapa andA Techanitisawad ldquoSimultaneous price andduedate settings for multiple customer classesrdquo European Journal ofOperational Research vol 166 no 2 pp 351ndash368 2005

[27] K Charnsirisakskul P M Griffin and P KeskinocakldquoPricing and scheduling decisions with leadtime flexibilityrdquoEuropean Journal of Operational Research vol 171 no 1pp 153ndash169 2006

[28] A Mustafa B Ata and T Olsen ldquoCongestion based lead timequotation for heterogenous customers with convex-concavedelay costs optimality of a cost balancing policy based onconvex hull functionsrdquo Operations Research vol 57 no 3pp 753ndash768 2009

[29] S K Chaharsooghi M Honarvar M Modarres andI N Kamalabadi ldquoDeveloping a two stage stochastic pro-gramming model of the price and lead-time decision problemin the multi-class make-to-order firmrdquo Computers amp In-dustrial Engineering vol 61 no 4 pp 1086ndash1097 2011

20 Advances in Operations Research

[30] S K Chaharsooghi M Honarvar and M Modarres ldquoAmulti-stage stochastic programming model for dynamicpricing and lead time decisions in multi-class make-to-orderfirmrdquo Scientia Iranica vol 18 no 3 pp 711ndash721 2011

[31] K-Y Chen M Kaya and O Ozer ldquoDual sales channelmanagement with service competitionrdquo Manufacturing ampService Operations Management vol 10 no 4 pp 654ndash6752008

[32] H Xu Z Z Liu and S H Zhang ldquoA strategic analysis of dual-channel supply chain design with price and delivery lead timeconsiderationsrdquo International Journal of Production Eco-nomics vol 139 no 2 pp 654ndash663 2012

[33] R Batarfi M Y Jaber and S Zanoni ldquoDual-channel supplychain a strategy to maximize profitrdquo Applied MathematicalModelling vol 40 no 21-22 pp 9454ndash9473 2016

[34] N M Modak and P Kelle ldquoManaging a dual-channel supplychain under price and delivery-time dependent stochasticdemandrdquo European Journal of Operational Research vol 272no 1 pp 147ndash161 2019

[35] H Kurata D-Q Yao and J J Liu ldquoPricing policies underdirect vs indirect channel competition and national vs storebrand competitionrdquo European Journal of Operational Re-search vol 180 no 1 pp 262ndash281 2007

[36] G L Nemhauser and L AWolsey Integer and CombinatorialOptimization Wiley-Interscience New York NY USA 1988

[37] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of 1995 IEEE International Conference on NeuralNetworks pp 1942ndash1948 IEEE Perth Australia November1995

[38] J Kennedy RC Eberhart and Y Shi Swarm IntelligenceMorgan Kaufmann San Mateo CA USA 2001

[39] S Koziel and Z Michalewicz ldquoEvolutionary algorithmshomomorphous mappings and constrained parameter opti-mizationrdquo Evolutionary Computation vol 7 pp 19ndash44 1999

[40] J H Holland Adaptation in Natural and Artificial SystemsAn Introductory Analysis with Applications to Biology Con-trol and Artificial Intelligence University of Michigan PressAnn Arbor MI USA 1975

Advances in Operations Research 21

Substituting Psi 0 into the above inequality we can

derive (39) which is upper bound for Lsi

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] A A Tsay and N Agrawal ldquoModeling conflict and coordi-nation in multi-channel distribution systems a reviewrdquo inHandbook of Quantitative Supply Chain Analysis Modeling inthe eBusiness Era 557ndash606) S D Simchi-Levi Wu andZ M Shen Eds Kluwer Academic Publishers Norwell MAUSA 2004

[2] K L Webb and J E Hogan ldquoHybrid channel conflict causesand effects on channel performancerdquo Journal of Business ampIndustrial Marketing vol 17 no 5 pp 338ndash356 2002

[3] G Hua S Wang and T C E Cheng ldquoPrice and lead timedecisions in dual-channel supply chainsrdquo European Journal ofOperational Research vol 205 no 1 pp 113ndash126 2010

[4] J Q Yang X M Zhang H Y Fu and C Liu ldquoInventorycompetition in a dual-channel supply chain with delivery leadtime considerationrdquo Applied Mathematical Modelling vol 42pp 675ndash692 2017

[5] K Cattani W G Gilland and J M Swaminathan ldquoCoordi-nating traditional and Internet supply chainsrdquo in Handbook ofQuantitative Supply Chain Analysis Modeling in the eBusinessEra D Simchi-Levi amp M Wu and Shen Eds pp 643ndash680Kluwer Academic Publishers Norwell MA USA 2004

[6] G Cai Z G Zhang and M Zhang ldquoGame theoretical per-spectives on dual-channel supply chain competition withprice discounts and pricing schemesrdquo International Journal ofProduction Economics vol 117 no 1 pp 80ndash96 2009

[7] W-y K Chiang D Chhajed and J D Hess ldquoDirect mar-keting indirect profits a strategic analysis of dual-channelsupply-chain designrdquo Management Science vol 49 no 1pp 1ndash20 2003

[8] S Panda N M Modak S S Sana and M Basu ldquoPricing andreplenishment policies in dual-channel supply chain undercontinuous unit cost decreaserdquo Applied Mathematics andComputation vol 256 pp 913ndash929 2015

[9] D Q Yao and J J Liu ldquoCompetitive pricing of mixed retailand e-tail distribution channelsrdquoOmega vol 33 pp 235ndash2472005

[10] B Dan G Xu and C Liu ldquoPricing policies in a dual-channelsupply chain with retail servicesrdquo International Journal ofProduction Economics vol 139 no 1 pp 312ndash320 2012

[11] Q-H Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling vol 40 no 15-16pp 7349ndash7367 2016

[12] A Roy S S Sana and K Chaudhuri ldquoJoint decision on EOQand pricing strategy of a dual channel of mixed retail and e-tailcomprising of single manufacturer and retailer under sto-chastic demandrdquo Computers amp Industrial Engineeringvol 102 pp 423ndash434 2016

[13] A A Tsay and N Agrawal ldquoChannel conflict and coordi-nation in the e-commerce agerdquo Production and OperationsManagement vol 13 pp 93ndash110 2004

[14] Z Zhang S Liu and B Niu ldquoCoordination mechanism ofdual-channel closed-loop supply chains considering productquality and returnrdquo Journal of Cleaner Production vol 248Article ID 119273 2020

[15] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chain when productioncosts are disruptedrdquo EconomicModelling vol 30 pp 521ndash5382013

[16] Z Pi W Fang and B Zhang ldquoService and pricing strategieswith competition and cooperation in a dual-channel supplychain with demand disruptionrdquo Computers amp IndustrialEngineering vol 138 2019

[17] F Soleimani A Arshadi Khamseh and B Naderi ldquoOptimaldecisions in a dual-channel supply chain under simultaneousdemand and production cost disruptionsrdquo Annals of Oper-ations Research vol 243 no 1-2 pp 301ndash321 2016

[18] T M Rofin and B Mahanty ldquoOptimal dual-channel supplychain configuration for product categories with differentcustomer preference of online channelrdquo Electronic CommerceResearch vol 18 no 3 pp 507ndash536 2018

[19] T Xiao T-M Choi and T C E Cheng ldquoProduct variety andchannel structure strategy for a retailer-Stackelberg supplychainrdquo European Journal of Operational Research vol 233no 1 pp 114ndash124 2014

[20] J Zhou R Zhao and B Wang ldquoBehavior-based price dis-crimination in a dual-channel supply chain with retailerrsquosinformation disclosurerdquo Electronic Commerce Research andApplications vol 39 2020

[21] I Duenyas and W J Hopp ldquoQuoting customer lead timesrdquoManagement Science vol 41 no 1 pp 43ndash57 1995

[22] K C So and J-S Song ldquoPrice delivery time guarantees andcapacity selectionrdquo European Journal of Operational Researchvol 111 no 1 pp 28ndash49 1998

[23] F F Easton and D R Moodie ldquoPricing and lead time de-cisions for make-to-order firms with contingent ordersrdquoEuropean Journal of Operational Research vol 116 no 2pp 305ndash318 1999

[24] P Keskinocak R Ravi and S Tayur ldquoScheduling and reliablelead-time quotation for orders with availability intervals andlead-time sensitive revenuesrdquo Management Science vol 47no 2 pp 264ndash279 2001

[25] S Webster ldquoDynamic pricing and lead-time policies formake-to-order systemsrdquo Decision Sciences vol 33 pp 579ndash599 2002

[26] BWatanapa andA Techanitisawad ldquoSimultaneous price andduedate settings for multiple customer classesrdquo European Journal ofOperational Research vol 166 no 2 pp 351ndash368 2005

[27] K Charnsirisakskul P M Griffin and P KeskinocakldquoPricing and scheduling decisions with leadtime flexibilityrdquoEuropean Journal of Operational Research vol 171 no 1pp 153ndash169 2006

[28] A Mustafa B Ata and T Olsen ldquoCongestion based lead timequotation for heterogenous customers with convex-concavedelay costs optimality of a cost balancing policy based onconvex hull functionsrdquo Operations Research vol 57 no 3pp 753ndash768 2009

[29] S K Chaharsooghi M Honarvar M Modarres andI N Kamalabadi ldquoDeveloping a two stage stochastic pro-gramming model of the price and lead-time decision problemin the multi-class make-to-order firmrdquo Computers amp In-dustrial Engineering vol 61 no 4 pp 1086ndash1097 2011

20 Advances in Operations Research

[30] S K Chaharsooghi M Honarvar and M Modarres ldquoAmulti-stage stochastic programming model for dynamicpricing and lead time decisions in multi-class make-to-orderfirmrdquo Scientia Iranica vol 18 no 3 pp 711ndash721 2011

[31] K-Y Chen M Kaya and O Ozer ldquoDual sales channelmanagement with service competitionrdquo Manufacturing ampService Operations Management vol 10 no 4 pp 654ndash6752008

[32] H Xu Z Z Liu and S H Zhang ldquoA strategic analysis of dual-channel supply chain design with price and delivery lead timeconsiderationsrdquo International Journal of Production Eco-nomics vol 139 no 2 pp 654ndash663 2012

[33] R Batarfi M Y Jaber and S Zanoni ldquoDual-channel supplychain a strategy to maximize profitrdquo Applied MathematicalModelling vol 40 no 21-22 pp 9454ndash9473 2016

[34] N M Modak and P Kelle ldquoManaging a dual-channel supplychain under price and delivery-time dependent stochasticdemandrdquo European Journal of Operational Research vol 272no 1 pp 147ndash161 2019

[35] H Kurata D-Q Yao and J J Liu ldquoPricing policies underdirect vs indirect channel competition and national vs storebrand competitionrdquo European Journal of Operational Re-search vol 180 no 1 pp 262ndash281 2007

[36] G L Nemhauser and L AWolsey Integer and CombinatorialOptimization Wiley-Interscience New York NY USA 1988

[37] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of 1995 IEEE International Conference on NeuralNetworks pp 1942ndash1948 IEEE Perth Australia November1995

[38] J Kennedy RC Eberhart and Y Shi Swarm IntelligenceMorgan Kaufmann San Mateo CA USA 2001

[39] S Koziel and Z Michalewicz ldquoEvolutionary algorithmshomomorphous mappings and constrained parameter opti-mizationrdquo Evolutionary Computation vol 7 pp 19ndash44 1999

[40] J H Holland Adaptation in Natural and Artificial SystemsAn Introductory Analysis with Applications to Biology Con-trol and Artificial Intelligence University of Michigan PressAnn Arbor MI USA 1975

Advances in Operations Research 21

[30] S K Chaharsooghi M Honarvar and M Modarres ldquoAmulti-stage stochastic programming model for dynamicpricing and lead time decisions in multi-class make-to-orderfirmrdquo Scientia Iranica vol 18 no 3 pp 711ndash721 2011

[31] K-Y Chen M Kaya and O Ozer ldquoDual sales channelmanagement with service competitionrdquo Manufacturing ampService Operations Management vol 10 no 4 pp 654ndash6752008

[32] H Xu Z Z Liu and S H Zhang ldquoA strategic analysis of dual-channel supply chain design with price and delivery lead timeconsiderationsrdquo International Journal of Production Eco-nomics vol 139 no 2 pp 654ndash663 2012

[33] R Batarfi M Y Jaber and S Zanoni ldquoDual-channel supplychain a strategy to maximize profitrdquo Applied MathematicalModelling vol 40 no 21-22 pp 9454ndash9473 2016

[34] N M Modak and P Kelle ldquoManaging a dual-channel supplychain under price and delivery-time dependent stochasticdemandrdquo European Journal of Operational Research vol 272no 1 pp 147ndash161 2019

[35] H Kurata D-Q Yao and J J Liu ldquoPricing policies underdirect vs indirect channel competition and national vs storebrand competitionrdquo European Journal of Operational Re-search vol 180 no 1 pp 262ndash281 2007

[36] G L Nemhauser and L AWolsey Integer and CombinatorialOptimization Wiley-Interscience New York NY USA 1988

[37] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of 1995 IEEE International Conference on NeuralNetworks pp 1942ndash1948 IEEE Perth Australia November1995

[38] J Kennedy RC Eberhart and Y Shi Swarm IntelligenceMorgan Kaufmann San Mateo CA USA 2001

[39] S Koziel and Z Michalewicz ldquoEvolutionary algorithmshomomorphous mappings and constrained parameter opti-mizationrdquo Evolutionary Computation vol 7 pp 19ndash44 1999

[40] J H Holland Adaptation in Natural and Artificial SystemsAn Introductory Analysis with Applications to Biology Con-trol and Artificial Intelligence University of Michigan PressAnn Arbor MI USA 1975

Advances in Operations Research 21