an integrated pricing and inventory model for...

Scientia Iranica E (2017) 24(1), 342{354

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

An integrated pricing and inventory model fordeteriorating products in a two-stage supply chainunder replacement and shortage

E. Teimoury� and S.M.M. Kazemi

Department of Industrial Engineering, Iran University of Science and Technology, Narmak Street, Tehran, Iran.

Received 17 April 2015; received in revised form 6 September 2015; accepted 7 December 2015

KEYWORDSOptimization;Supply chain;Inventory;Pricing;Deterioration;Replacement;Shortage.

Abstract. In this paper, a two-stage supply chain, including a wholesaler and a retailer,is investigated, in which the retailer acts as a vendor and sells the products to the�nal customers. This chain only produces a single deteriorating product with constantdeteriorating rate. In this chain, demand is deterministic, and lead time for replenishmentand replacement of retailer's stock is considered to be zero. The objective of this studyis to maximize total chain's pro�t by determining the optimal values of retailer's sellingprice (p) and order cycle length (T ). Since due to deterioration, a part of the initial stockis deteriorated, the retailer residuals recycle the deteriorated part and replace them withhealthy product as needed instead. Two scenarios with and without shortage assumptionsare developed. Finally, numerical examples are presented to show the applicability of theproposed models, and sensitivity analysis on some parameter's values is conducted.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Nowadays, a supply chain coordination has great im-portance in competitive business environments. Closecooperation between the members of the multi-stagesupply chain results in a noticeable increase in totalpro�t as well as faster response to the client's demand.Also, determining the optimal inventory control policyand selling price for di�erent products is one of themain issues of industrial and scienti�c studies, espe-cially when the product is perishable. So, in this paper,we concentrate on simultaneous pricing and inventorydecisions in a two-stage supply chain for deterioratingproducts with replacement and shortage assumptions.In this section, we will review some research piecesabout replacement, pricing, and inventory decision fordeteriorating product.

*. Corresponding author. Tel: +98 21 73225000E-mail addresses: [email protected] (E. Teimoury);sm [email protected] (S.M.M. Kazemi)

Since in the absence of product price is usuallythe main factor for customer's buying decision, jointdetermination of pricing and inventory decisions isan important problem. Yu et al. [1] determined thejoint optimal price-inventory decisions in a fuzzy price-dependent newsvendor framework. Huang et al. [2]modeled the coordination of suppliers and componentsselection, pricing and replenishment decisions in athree-level supply chain as a dynamic non-cooperativegame model. Ghoreishi et al. [3] dealt with aneconomic production quantity inventory model for non-instantaneous deteriorating items under in ationaryconditions, permissible delay in payments, customer re-turns, and price- and time-dependent demand. Zhu [4]formulated the combined pricing and inventory controlproblem in a random demand condition and �niteplanning horizon with return and expediting. Youet al. [5] studied a seasonal inventory model withtrial periods during which customers can return theirpurchases without paying any penalty and developed amathematical model. Lee et al. [6] formulated a novel

E. Teimoury and S.M.M. Kazemi/Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 342{354 343

fuzzy multi-level multi-objective production planningmodel for a supply chain under a fuzzy environment.Su and Geunes [7] examined the degree to whichthe bullwhip e�ect results from price uctuation ina two-stage supply chain with deterministic demand.Mutlu and C�etinkaya [8] concentrated on a carrier-retailer channel and compared the pro�tability of thecentralized and decentralized channels under a price-dependent demand. Hong and Lee [9] studied asingle-product inventory system and applied shipmentconsolidation (time-based policy) in order to maximizetotal pro�t and determine optimum values of sellingprice, replenishment quantity and dispatch cycle lengthsimultaneously. Abad [10] formulated a model to deter-mine selling price and order size for perishable productswith partial backordering. Adida and Perakis [11]presented a model for dynamic pricing and inventoryproblem for a make-to-stock manufacturing system.Dong et al. [12] studied the problem of product lineselection and pricing. G�um�us and G�uneri [13] reviewedthe literature, addressing multi-echelon inventory man-agement in supply chains from 1996 to 2005.

On the other hand, since in today's market, thenumber of deteriorating products shows a real increas-ing trend, so many researchers have concentrated oninventory control of these products. In this way, Chewet al. [14] studied an inventory system and determinedthe inventory allocation and the price for a perishableproduct with a predetermined lifetime. Taleizadehand Nematollahi [15] developed an inventory controlmodel that investigates the e�ects of time value ofmoney and in ation on optimal ordering policy ofdeteriorating products. Lee and Chung [16] used sys-tem dynamics thinking to propose a new order systemfor deteriorating products and prepare a systematicalsimulation. Maity and Maiti [17] presented an opti-mum production and advertising policy for a multi-product system with in ation and time discountingunder di�erent constraints. Yu et al. [18] studiedan inventory problem for a VMI system, where boththe raw material and �nished products are perishable.Mahata [19] formulated an EPQ-based inventory modelfor deteriorating products that investigates the optimalretailer's replenishment decisions under trade creditpolicy. Also, as we know, the pricing models ofdeteriorating items due to similarity to real businessenvironments, has a special place in scienti�c research.For example, Chung et al. [20] investigated the e�ectsof stock- and warranty- dependent demands on anintegrated two-stage production-inventory deteriorat-ing model. Yang [21] developed an optimal pricingand ordering policy for a deteriorating item using aquantity discount pricing strategy. Dye et al. [22]found the optimal inventory and pricing strategies fora single-product economic order quantity that maxi-mize total pro�t over an in�nite horizon. Tsao and

Sheen [23] studied the problem of dynamic pricing,promotion, and replenishment for a deteriorating itemwith supplier's trade credit and retailer's promotionale�orts. Huang [24] developed an integrated inventorymodel in a two-stage supply chain to determine theoptimal policy under order processing cost reductionand permissible delay in payments. Maihami andNakhai [25] considered a joint pricing and inventorycontrol model for non-instantaneously deterioratingitems with permissible delay in payments; shortage isallowed and partially backlogged in their model.

Furthermore, in case of deteriorating products,using a kind of replacement policies in relation withdeteriorated items can be useful. For example, Gomezet al. [26] investigated the simultaneous productionplanning and quality problem for an unreliable singlemachine manufacturing system responding to a single-product type demand. Nguyen et al. [27] studiedthe impact of spare parts inventory on equipmentmaintenance and replacement decisions under tech-nological change via a Markov process formulation.Jain and Gupta [28] studied an optimal replace-ment policy for a repairable system with multiplevacations and imperfect fault coverage. Berthautet al. [29] considered a joint preventive maintenanceand production/inventory control policy of an unreli-able single machine, mono-product manufacturing cell.Guchhait et al. [30] developed Economic ProductionQuality (EPQ) models for breakable items. Chengand Li [31] studied an optimal replacement policyfor a degenerative system with two types of failurestates. Chang [32] considered an optimal replacementpolicy for a degenerative system with two types offailure states. Sivazlian and Danusaputro [33] dis-cussed economic inventory and replacement manage-ment of a system in which components are subjectto failure. Teimoury and Fathi [34] studied OrderPenetration Point (OPP) strategic decision-making,which is the boundary between Make-To-Order (MTO)and Make-To-Stock (MTS) policies, considering two-echelon supply chain. Teimoury et al. [35] employedsome innovative methodologies of multi-agent systemsdevelopment to create an automation of the supplychain performance measurement based on multi-agentsystem. Teimoury and Fathi [36] studied An integratedoperations-marketing perspective for making decisionsabout order penetration point in multi-product supplychain using a queuing approach. Teimoury and Had-dad [37] studied a supply chain production schedulingwith deterioration and release date. Taleizadeh etal. [38] developed a VMI model for a two-level supplychain in which both the raw material and the �nishedproduct have di�erent deterioration rates. The marketdemand for the �nished product is deterministic andprice-sensitive. The Stackelberg approach is consideredbetween the chain partners, where the vendor is the

344 E. Teimoury and S.M.M. Kazemi/Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 342{354

leader, and the retailers are the followers. Rahdarand Nookabadi [39] considered a supply chain includinga manufacturer and several buyers and assumed thatthe inventory items deteriorate over time and itsinventory level decreases. In order to determine theorder policies, coordination over the supply chain isachieved by scheduling the buyers' delivery days andtheir coordination with the manufacturer's productioncycle. Sarkar [40] considered that a production-inventory model is developed for a deteriorating item ina two-echelon Supply Chain Management (SCM). Analgebraic approach is applied to �nd the minimum costrelated to this entire SCM. Three types of the continu-ous probabilistic deterioration function are consideredto �nd the associated cost.

As a result, although many outstanding researchpieces about pricing and inventory control model fordeteriorating product have been separately developedup to now, none of them has considered replacement,inventory, and pricing policies in a two-stage supplychain together. As for more clari�cation, the sig-ni�cance of our research comes up whenever one isstudying the pro�tability of food industries by focusingon a special dairy product factory in which a largenumber of products are deteriorating every minute andthe inventory of healthy products are becoming smaller;since there is a direct relationship between stock'shealthy inventory and pro�t, a smaller pro�t will beobtained. Now, here a replacement policy can help usto replace the deteriorated items with healthy ones,and in this way to compensate for a part of our loss.The above-mentioned hints are the noticeable gaps inthis context that motivate us for this study. In thisresearch, we develop an integrated pricing-inventorysystem for a deteriorating item in a two-stage supplychain, in which a kind of replacement policy is used.

The rest of the paper is organized as follows.The problem is de�ned in Section 2. In Section3, we develop a mathematical model for productswith constant deteriorating rate and price-dependentdemand, and then, we determine the optimum valuesof product price and order cycle length simultaneously,and a solution method is presented. In Section 4, wepresent our �rst model by adding shortage assumption.Numerical examples are provided in Section 5. Finally,paper is concluded in Section 6.

2. Problem de�nition

Consider a two-stage supply chain consisting of awholesaler and a retailer, that is, a single item deterio-rating system with constant deteriorating rate. In thischain, the wholesaler buys the products from the upperlevel, the manufacturer (that is not considered in thisstudy), and sells them to the retailer. Then, the retailersells the products to the �nal customers. The main

purpose of this paper is to study the inventory systemof this chain and extend models to determine theoptimum values of retailer's selling price (P ) and ordercycle length (T ) simultaneously in order to maximizetotal chain's pro�t. In this paper, the retailer at thebeginning of the order cycle purchases (orders) as cy-cle's demand, but since the products are deteriorating,a part of the initial stock will deteriorate during theorder cycle. So, in order to respond to the customer'sdemand, the retailer replaces deteriorated quantitieswith healthy products. In other words, retailer recycles(residuals) the deteriorated products, and then replacesthe deteriorated products with the healthy products.Replacement is instantaneous and repurchasing costis considered as equal as the initial purchasing price.Also, lead time for stock's replenishment is consideredto be zero. Moreover, in order to determine theretailer's optimal selling price, it is assumed that cus-tomer's demand is deterministic and a linear functionof product price D(p) = (a � bp), where a and b arepositive and constant values a; b > 0. Two scenarios aredeveloped with and without shortage assumptions. Inthe �rst scenario, shortage is not permitted and all de-mands must be satis�ed; in the second scenario, short-age is permitted and will be completely backlogged.

In this section, we present mathematical modelsfor integrated pricing and inventory decisions for de-teriorating products in a two-stage supply chain withreplacement and shortage assumption. In order tomodel the problem, the following notations are used.

ParametersI(t) The inventory level at time t in interval

[0; T ]�(t) The inventory level at time t in interval

[T; T 0]� The constant deteriorating rateD(p) The price-sensitive demandFD The �xed cost of a dispatchCD The dispatching cost per unit productFp The �xed cost of a ordering

(purchasing)Cp The ordering (purchasing) cost per

unit productCH The recycling cost (salvage value) of

per unit deteriorated productQp The amount of deteriorated producth The holding cost per unit product per

unit of timeS The maximum backlogged shortage� The backlogged shortage cost per unit

productT 0 The ordering time


C 0 The wholesaler's purchasing price

Decision variablesP The retailer's selling priceT The ordering (purchasing) cycle length

3. Mathematical modelling

3.1. The �rst scenario: without shortageIn this scenario, shortage is not permitted and alldemands must be satis�ed. In this part, we present amathematical model for integrated pricing and inven-tory decisions for deteriorating products in a two-stagesupply chain in which shortage is not permitted. First,we model the retailer and wholesaler's pro�t functionsseparately, and then form an integrated pro�t function.

3.1.1. Retailer's pro�t functionAccording to Figure 1, di�erential equations, shown inEq. (1), present the changes of inventory level of theretailer during the ordering cycle, interval [0; T ]:

dI(t)dt

= ��I(t)�D(p); t 2 [0; T ]: (1)

Using the boundary condition I(T ) = 0, we have:

I(t) =D(p)�

�e�(T�t) � 1

�: (2)

Moreover, the maximum inventory level during theinterval [0; T ] will be:

I(0) =D(p)�

�e�T � 1

�: (3)

In order to determine the optimal value of decisionvariables, �rstly, we need to model the pro�t function,and we know that pro�t is the di�erence betweenincome and cost. In this study, T is considered asordering cycle and demand during this cycle is D(p).So, income will be:

Figure 1. Inventory diagram without shortage.

D(p)Tp = (a� bp)Tp: (4)

Moreover, the cost function of retailer includes purchas-ing, dispatch, holding, and replacement costs which arecalculated as below.

Purchasing cost:

In this model, the purchasing quantity is equal to thecycle's demand D(p)T . By having Fp as the �xed costof purchasing and Cp as purchasing cost of per unitproduct, for purchasing cost, we have:

Fp + CpD(p)T: (5)

Dispatch cost:

Dispatch quantity is equal to D(p)T . So, by having FDas the �xed cost of dispatching and CD as dispatchingcost of per unit product, the dispatch cost during theordering cycle will be:

FD + CDD(p)T: (6)

Holding cost:

According to Eq. (2), we know that the inventory levelat time t is I(t) = D(p)

�

��1 + e�(T�t)�. Utilizing a

truncated Taylor series expansion for the exponentialterm, e�T = 1 + �T + �2T 2

2 , the inventory holding costin each ordering cycle is:

hZ T

0I(t)dt = h

Z T

0

�D(p)�

(�1 + e�(T�t))�dt

=hD(p)T 3

2=h(a� bp)T 2

2: (7)

Replacement cost:

Replacement cost is de�ned as the di�erence betweenthe purchasing cost of per unit deteriorated productand its salvage value (recycling cost). Now, thedeteriorated quantity is equal to:

Qp = I(0)� I(T )�D(p)T; (8)

where I(0) = D(p)� (e�T � 1), I(T ) = 0 and demand

during the ordering cost is D(p)T . So, deterioratedquantity is:

Qp = I(0)� I(T )�D(p)T

=D(p)�

(e�T � 1)�D(p)T: (9)

Utilizing a truncated Taylor series expansion for theexponential term, e�T = 1 + �T + �2T 2

2 , replacementcost in each ordering cycle is:


Cp(Qp)�CH(Qp)=(CP�CH)Qp

=(Cp�CH)�D(p)�T 2

2

�; (10)

where Cp is the purchasing cost of per unit deterio-rated product, which is considered equal to the initialpurchasing price. And, CH is recycling cost (residualvalue), which means that if the deteriorated productis repairable, then CH is the recycling cost; otherwise,CH is the residual value. Using Eqs. (4)-(7) and (10),the retailer's total pro�t function is:

TPr(p; T ) =D(p)pT ��FP + CPD(p)T

+hD(p)T 2

2+ FD + CDD(p)T

+ (CP � CH)�D(p)�T 2

2

��: (11)

3.1.2. Wholesaler's pro�t functionWholesaler's pro�t function is also the di�erence be-tween its income and cost. Here, wholesaler's costfunction only includes purchasing cost, and income isthe obtained from selling the products to the retailer:

TPw(p; T ) = CpI(0)� C 0I(0) = (Cp � C 0)I(0)

= (Cp � C 0)D(p)T

+(Cp � C 0)D(p)�T 2

2; (12)

where I(0) is the product quantity that wholesaler sellsto the retailer, Cp is the selling price from wholesalerto retailer, and C 0 is the wholesaler's purchasing cost(the wholesaler buys the products with purchasing costC 0 from manufacturer; manufacturer is not consideredin this study).

Using Eqs. (11) and (12), the integrated totalpro�t function is the sum of total pro�t functions ofretailer and wholesaler, and it is obtained as below:

TP (p; T ) = TPw(p; T ) + TPr(p; T )

= (Cp � C 0)D(p)T +(Cp � C 0)D(p)�T 2

2

+D(p)pT ��FP + CPD(p)T

+hD(p)T 2

2+ FD+CDD(p)T

+(CP�CH)�D(p)�T 2

2

��: (13)

3.2. Solution methodTheorem1. TP (p; T ) is strictly concave if and onlyif:

CD < 3p� C 0� ab

+T2

(�(C 0�CH)+h)�abp�3�:

Proof. TP (p; T ) is strictly concave if and only if:

X:H:XT = [p T ]�H � [p T ]T < 0

where:

H=

"@2TP@p2

@2TP@p@T

@2TP@T@p

@2TP@T 2

#=� �2bTb�(C 0 + CD)+T (�(C 0�CH+h)

�+(a�2bp)

b�(C 0+CD)+T (�(C 0�CH)+h)

�+(a�2bp)

�D(p)(�(C 0�CH)+h)

�:(14)

So, we have:

X�H �XT = �6bTp2 + 3bpT 2�C 0 + 3bpT 2h

� 3bp�CHT 2 + 2bpTC 0 + 2apT + 2bpTCD

� a�C 0T 2 � ahT 2 + a�CHt2: (15)

We should show that:

X�H �XT =��6bTp2 + 3bpT 2�C 0 + 3bpT 2h

� 3bp�CHT 2 + 2bpTC 0 + 2apT + 2bpTCD

� a�C 0T 2 � ahT 2 + a�CHT 2�< 0:

(16)

So, the pro�t function TP (p; T ) is strictly concave ifand only if:

CD < 3p�C 0� ab + T

2 (�(C 0 � CH) + h)�abp � 3

�.

The proof is completed �.Now, taking the �rst derivative of Eq. (13) with

respect to p, we have:

@TP (p; T )@p

=aT � 2bpT + bTC 0 + btCD +bC 0�T 2

2

� bCH�T 2

2+bhT 2

2: (17)

On the other hand, the �rst derivatives of Eq. (13) withrespect to T yield:@TP (p; T )

@T=��C 0 � C 0�T � hT � CD+ CH�T

�D(p) +D(p)p: (18)

By setting Eq. (17) equal to zero, we have:


p� =a2b

+C 02

+CD2

+C 0�T

4� CH�T

4+hT4: (19)

By substituting Eq. (19) into Eq. (18) and setting itequal to zero, Eq. (18) changes to Eq. (20) as follows:

AT 2 +BT + C = 0; (20)

where:

A =3b16

(�(C 0 � CH) + h)2 ; (21)

B =12

(b(C 0 � CD)� a) (�(C 0 � CH) + h) ; (22)

C =14

�pb(C 0 + CD)� ap

b

�2

: (23)

By solving the above equation, T � is obtained asfollows:

T � =2�ab � (C 0 + CD)

�[�(C 0 � CH) + h]

or

T � = 2�ab � (C 0 + CD)

�3 [�(C 0 � CH) + h]

: (24)

4. The second scenario: with shortage

4.1. Mathematical modelRetailer's pro�t function:

According to Figure 2, di�erential equations, shown inEq. (1), present the changes of inventory level of theretailer during the �rst interval.

In second interval, shortage will happen and willbe completely backlogged. So, di�erential Eq. (25)presents the changes of inventory level during thesecond interval:

Figure 2. Inventory diagram under shortage.

d�(t)dt

= D(p); t 2 [T; T 0]: (25)

Using the boundary condition, �(T ) = 0, for interval[T; T 0], we have:

�(t) = D(p)(t� T ): (26)

The maximum level of backlogged shortage (S) will be:

S = �(T 0) = D(p)(T 0 � T ): (27)

In the following model, as we know, income will bethe sum of which obtained from selling of productsin the �rst interval and income which obtained fromselling of backlogged shortages in the second interval.According to Eq. (4), income in the �rst interval isD(p)Tp; income in the second interval is:

p� S=p� (D(p)(T 0�T ))=pD(p)T 0�pD(p)T: (28)

So, the total income of replenishment cycle will be:

pD(p)T + pD(p)T 0 � pD(p)T = pD(p)T 0: (29)

Moreover, the cost function of retailer includes pur-chasing, dispatch, holding, replacement, and shortagecosts which are calculated as below.

Purchasing cost:

In this model, we purchase according (as) to thedemand of the �rst interval D(p)T and the maximumlevel of backlogged shortage S. So, for purchasing cost,we have:FP + CP (D(p)T + S) = FP + CP

�D(p)T �D(p)T

+D(p)T 0�

= FP + CPD(p)T 0: (30)

Dispatch cost:

As mentioned, the dispatch cost in the �rst interval isFD + CDD(p)T ; in second interval, in order to reducecustomer's waiting time, it is assumed that all thebacklogged shortages will be dispatched at once. So,the total dispatch costs in each ordering cycle is:

FD +CDD(p)T+FD+CDS=FD+CDD(p)T+FD

+CDf�D(p)T+D(p)T 0g=2FD+CDD(p)T 0:(31)

Holding cost:

Since inventory exists in the �rst interval, the holdingcost is de�ned only in the �rst interval. And, accordingto Eq. (7), it is hD(p)T 2

2 .

Backlogged shortage cost:

Backlogged shortage cost is only de�ned in the secondinterval and is calculated as below:


��S=� (D(p)(T 0�T ))=��(Dp)T+�D(p)T 0: (32)

Replacement cost:

Accordingly, Eq. (10) is equal to (Cp�CH)�D(p)�T 2

2

�.

Using Eqs. (7), (10) and (29) to (32), the retailer'stotal pro�t function is:

TPr(p; T ) =D(p)pT 0 ��FP + CPD(p)T 0

+hD(p)T 2

2+ 2FD + CDD(p)T 0

+ (CP � CH)�D(p)�T 2

2

�+ �D(p)T 0

� �D(p)T�: (33)

Wholesaler's pro�t function:

As mentioned before, wholesaler's pro�t function is:

TPw(p; T ) = Cp (I(0) + S)� C 0(I(0) + S)

= (Cp�C 0)(I(0)+S)=(Cp�C 0)D(p)T 0

+(Cp � C 0)D(p)�T 2

2; (34)

in which (I(0) + S) is the product quantity thatwholesaler sells to the retailer.

Using Eqs. (33) and (34), the integrated totalpro�t function is the sum of retailer and wholesaler'stotal pro�t functions and is obtained as below:

TP (p; T ) = TPw(p; T ) + TPr(p; T )

= (Cp � C 0)D(p)T 0 + (Cp � C 0)D(p)�T 2

2

+D(p)pT 0��FP +CPD(p)T 0+ hD(p)T 2

2

+ 2FD + CDD(p)T 0 + (CP � CH)�D(p)�T 2

2

�+ �D(p)T 0 � �D(p)T

�:

(35)

4.2. Solution methodTheorem 2. TP (p; T ) is strictly concave if and onlyif a > 3bp.

Proof. TP (p; T ) is strictly concave if and only if:

X:H:XT = [p T ]�H � [p T ]T < 0

where:

H =

"@2TP@p2

@2TP@p@T

@2TP@T@p

@2TP@T 2

#=� �2bT 0�bT (�(CH � C 0)� h)� b��bT (�(CH � C 0)� h)� b�D(p) (�(CH � C 0)� h)

�: (36)

So, we have:

X �H �XT =� T 2 (�(C 0 � CH) + h) (a� 3bp)

� 2bp(T 0p+ T�): (37)

We should show that:

X �H �XT =� T 2 (�(C 0 � CH) + h) (a� 3bp)

� 2bp(T 0p+ T�) < 0: (38)

By having this assumption that C 0 � CH , the pro�tfunction, TP (p; T ), is strictly concave if and only ifa > 3bp. The proof is completed �.

Considering Eq. (35) with respect to p, we have:

@TP (p; T )@p

=aT 0 � 2bpT 0 + bT 0C 0 + bT 0CD

+bC 0�T 2

2� bCH�T 2

2+ b�T 0

� b�T +bhT 2

2: (39)

On the other hand, the �rst derivatives of Eq. (35) withrespect to T yield:

@TP (p; T )@T

=(�C 0�T � hT + CH�T + �)D(p): (40)

By setting Eq. (39) equal to zero, we have:

p�=a2b

+C 02

+CD2

+�2� �T

2T 0+C 0�T 2

4T 0 +hT 2

4T 0 : (41)

By substituting Eq. (41) into Eq. (40) and setting itequal to zero, Eq. (40) changes to Eq. (42) as follows:

AT 3 +BT 2 + CT +D = 0; (42)

where:

A =b

4T 0 (�(CH � C 0)� h)2 ; (43)

B =3�b4T 0 (�(CH � C 0)� h) ; (44)

C=12

(�(CH�C 0)�h) (a�b(C 0+CD+�))+b�2

2T 0 ; (45)

D =�2

(a� b(C 0 + CD + �)) : (46)


Finally, the optimal values of period length shouldbe determined by solving Eq. (42).

The following solution method mentioned in Fig-ure 3 has been presented to simplify the solutionprocedure for both of the proposed models as follows:

- Step 1. Computing the coe�cients of polynomialsshown in Eqs. (20) and (42);

- Step 2. Calculate all positive real roots of Eqs. (20)and (42) using MATLAB software. Then, check thefeasibility. If the results are feasible, then go to Step3, terminate the solution process;

- Step 3. Determine p for all feasible values of (T )resulted from Step 2;

- Step 4. For all combinations of (p; T ), computethe total pro�t function and choose the higher one.Also, check the concavity condition. If the resultsshow the concavity, then the results are the optimal

values and terminate the procedure, or else go toStep 5;

- Step 5. Using a nonlinear solver toolbox solves theproblem.

5. Numerical examples

5.1. The �rst exampleFor the �rst model, consider the parameters values asFp = 40, FD = 40, h = 14, CH = 10, � = 0:04,C 0 = 20, b = 0:3, a = 10, CD = 3, Cp = 30,and CR = 40. According to the above-mentionedsolution procedure, the highest pro�t obtains TP =78:2973 which corresponds to the optimum value ofp� = 29:8889 as retailer's selling price and optimumvalue of T � = 0:4784 as order cycle length.

5.2. The second exampleFor the second model, consider the parameters valuesas Fp = 40, FD = 40, h = 14, CH = 10, � = 0:04, C 0 =

Figure 3. Flowchart of solution method.


Table 1. E�ects of parameter changes on optimal values of the �rst scenario.

Changes in parameter's values% Changes in parameter T p TP

h

-0.5 +94.58 -0.0003 -2.057-0.25 +32.10 0 -0.698+0.25 -19.54 +0.0006 +0.425+0.5 -32.71 -0.0003 +0.711

�

-0.5 +1.400 -0.0003 -0.03-0.25 +0.689 -0.0006 -0.015+0.25 -0.689 0 9.583+0.5 -1.379 -0.006 0.0297

CH

-0.5 -1.379 -0.0006 +0.029-0.25 -0.689 0 +0.014+0.25 +0.689 -0.0006 -0.015+0.5 +1.400 -0.0003 -0.030

C0-0.5 +102.4038 -11.151 -14.875-0.25 +50.480 -5.575 -5.030+0.25 -49.101 +5.575 +1.879+0.5 -96.86 +11.152 +2.1745

Table 2. E�ects of parameter changes on optimal values of the second scenario.

Changes in parameter's values% Changes in parameter T p TP

h

-0.5 +16.711 -5.45 +5.679-0.25 +8.755 -2.639 +3.175+0.25 -9.173 +2.4116 -3.604+0.5 -18.064 +4.528 -7.322

�

-0.5 -0.356 +0.001 -0.002-0.25 -0.172 -0.0002 +0.0003+0.25 +0.184 -0.001 +0.002+0.5 +0.368 -0.002 +0.003

CH

-0.5 +0.368 -0.002 +0.003-0.25 +0.184 -0.001 +0.002+0.25 -0.172 -0.0002 +0.0003+0.5 -0.356 +0.001 -0.002

C0-0.5 -32.870 -7.343 +6.840-0.25 -14.96 -4.066 +4.557+0.25 +11.85 +4.770 -7.78+0.5 +20.67 +10.133 -19.66

20, b = 0:3, a = 10, CD = 3, Cp = 30, T 0 = 1:5, � = 35,and CR = 40. According to the above-mentionedsolution procedure, the highest pro�t obtains TP =111:1546 which corresponds to the optimum value ofp� = 37:7664 as retailer's selling price and optimumvalue of T � = 0:8132 as order cycle length.

6. Sensitivity analysis

In this part, we have conducted several numericalexperiments to investigate the sensitivity of optimalsolutions to changes in parameter's values, and the

results are shown in Tables 1 and 2 and Figures 4 to 7.Also, it must be said that in both �gures and tables, thepercent of changes in decision variables with respect topercentage of parameter's values changes is shown. Inthis part, the sensitivity analysis is only conducted onthe most e�ective parameters such as CH , C 0, h, and �.

Figure 4 represent the e�ects of change in holdingcost (h) on optimal values of retailer's selling priceand order cycle length in both scenarios (with andwithout shortage). And, as is seen that there is adirect relationship between changes in holding cost andchanges in optimal values of retailer's selling price and


Figure 4. E�ects of changes in holding cost on decision variables.

Figure 5. E�ects of changes in wholesaler's purchasing price on decision variables.

Figure 6. E�ects of changes in deterioration rates on decision variables.

a reverse relationship between changes in holding costand changes in optimal values of order cycle length.The optimal values of retailer's selling price increase asper unit holding cost increases and the optimal valuesof order cycle length decrease as the per unit holdingcost increases.

Figure 5 represents the e�ects of changes inwholesaler's purchasing price (C 0) on optimal values

of retailer's selling price and order cycle length in bothscenarios (with and without shortage). And, as is seen,there is a direct relationship between changes in whole-saler's purchasing price and changes in optimal valueof retailer's selling price (in both scenarios) and ordercycle length (in the scenario with shortage). The opti-mal values of retailer's selling price (in both scenarios)and order cycle length (in the scenario with shortage)


Figure 7. E�ects of changes in recycling cost (residual value) on decision variables.

increase as the wholesaler's purchasing price increases.But, the optimal value of order cycle length (in thescenario without shortage) �rst increases, and thendecreases as the wholesaler's purchasing price increases.

Figure 6 represents the e�ects of changes indeterioration rate (�) on optimal values of retailer'sselling price and order cycle length in both scenarios(with and without shortage). And, as is seen, there isan alternative and also a reverse relationship betweenchanges in deterioration rate and changes in retailer'sselling price (in both scenarios), a reverse relationshipbetween changes in deterioration rate and changes inorder cycle length (in the scenario without shortage),and �nally, a direct relationship between changes indeterioration rate and changes in order cycle length (inthe scenario with shortage).

Figure 7 represents the e�ects of change in recy-cling cost (CH) on optimal values of retailer's sellingprice and order cycle length in both scenarios (withand without shortage). And, as is seen, there isan alternative and also a direct relationship betweenchanges in recycling cost and changes in optimal valuesof retailer's selling price (in both scenarios), a reverserelationship between changes in recycling cost andchanges in order cycle length (in the scenario withshortage), and �nally, a direct relationship betweenchanges in recycling cost and changes in order cyclelength (in the scenario without shortage).

7. Conclusion

In this paper, a two-stage supply chain with a singledeteriorating product system is considered, and twomathematical models for jointly determining the op-timum values of retailer's selling price p and ordercycle length T under two di�erent scenarios (with andwithout shortage assumption) are presented. Also,in this study, since products are deteriorating, a re-

placement cost for deteriorating items is considered,in which the deteriorated products are recycled andhealthy products are replaced instead. Then, usingsome theorems, the concavity of pro�t functions of thetwo proposed models is proved. Finally, a solutionmethod (algorithm) is proposed to solve the models. Atthe end, numerical examples are provided to show theapplicability of the proposed policies. The presentedmodels in this paper are comprehensive and considerednearly all costs of an inventory system. For the futurestudies, permissible delay in payments, promotions,and stochastic and fuzzy demands can be considered.

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Biographies

Ebrahim Teimoury is an Associate Professor ofUniversity of Science and Technology (Iran). Hegraduated from Amirkabir University of Technology in1990 and was awarded an MSc and a PhD degrees fromUniversity of Science and Technology, respectively. hehas published many papers in international journals.Most of his research papers have been cited in manyarticles.

Seyed Mohammad Mehdi Kazemi is currently aPhD Candidate in University of Science and Tech-nology since 2011. He graduated from University ofScience and Technology and was awarded an MScdegree from Amirkabir University of Technology. Also,he has inventions which have patents.

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