obg

eong

gdo

ation

rsity

Newsvendor problem

wh

me

wsv

es

er t

rdi

ase

Approximation Method. A numerical example is discussed to show the optimal convergence of ordering

quantities and discuss the properties of the proposed algorithms.

inatedd (Arson auton be ca un

xists a

gura-ntric

nderpose

partial information sharing environment. The ADM is a general

Contents lists available at SciVerse ScienceDirect

journal homepage: www.e

Int. J. Productio

Int. J. Production Economics 135 (2012) 412419also guarantees the convergence to the global optimal solution.1 Tel.: 1 979 845 4993.methodology to solve a convex programming using the augmen-ted Lagrangian function (Bertsekas and Tsitsiklis, 1989; see Leeand Jeong, 2010 for details). The ADM guarantees the convergenceto the global optimal solution theoretically. The proposed method

0925-5273/$ - see front matter & 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.ijpe.2011.08.015

n Corresponding author at: Department of Technology Management, Hanyang

University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea.

Tel.: 82 2 2220 0412; fax: 82 2 2296 0471.E-mail addresses: ijeong@hanyang.ac.kr (I.-J. Jeong),

LEON@entc.tamu.edu (V. Jorge Leon).the ADM (Alternating Direction Method) and DQAM (DiagonalQuadratic Approximation Method) based SCC method under theoften there can be restrictions on how the information is sharedamong the members (i.e., partial information sharing). In this case,SCC cannot be achieved by a single member in the supply chain.Instead all members need to participate in the decision makingprocess.

this paper suggest that at least in some simple network contions and cost model, global SCC can be achieved in both uniceand polycentric supply chain system environments.

In this paper, we present the global optimal solution ucomplete information sharing among members. Also we prohas the power to request all required information (i.e., completeinformation) from the members. In turn, the focal companyunilaterally makes decisions for supply chain coordination (SCC)that all members accept without objection. The unicentric supplychain can be found where the market is monopolized or domi-nated by a member in supply chain. In a polycentric supply chainthe decision authority is distributed among the members and

rely less on all members having to transmit information to acentralized location, would allow each member to autonomouslynegotiate based on its local objectives; and would be more robust totime and system variations since there is no dependence on amonolithic formulation. From this perspective, it would be ideal toallow a SC to operate as a polycentric SC as long as it can warrantythe same performance as a unicentric SC. The models presented in1. Introduction

A supply chain is said to be coordthe supply chain (SC) is maximize2008). Based on the degree of decisisharing among the members, SCs caand polycentric supply chains. In(Stadtler and Kilger, 2005), there e& 2011 Elsevier B.V. All rights reserved.

if the entire benet ofhinder and Deshmukh,nomy and informationlassied into unicentricicentric supply chainfocal company, which

Most of the work in the literature seems to suggest that SCC iseasier to attain in a unicentric supply chains. At the same time, it iswidely accepted that global supply chain optimization remains oneof the main challenges in supply chain management due to networkcomplexity, members with conicting objectives, system dynamics,system variations over time and information uncertainty manage-ment (Simchi-Levi et al., 2003). Interestingly, many of these reasonswould be attenuated in a polycentric supply chain since they willA serial supply chain of newsvendor prcomplete and partial information sharin

In-Jae Jeong a,b,n, V. Jorge Leon c,d,1

a Department of Technology Management, Hanyang University, 17 Haengdang-dong, Sb Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seonc Department of Industrial and Systems Engineering, Texas A&M University, College Std Department of Engineering Technology and Industrial Distribution, Texas A&M Unive

a r t i c l e i n f o

Article history:

Received 1 February 2011

Accepted 8 August 2011Available online 26 August 2011

Keywords:

Supply chain

Augmented Lagrangian function

a b s t r a c t

We consider a supply chain

the ordering quantity of a

model is similar to the ne

presents optimal approach

information sharing in ord

we develop an optimal coo

coordination algorithm blem with safety stocks under

dong-gu, Seoul 133-791, Republic of Korea

ng-gu, Seoul 133-791, Republic of Korea

, TX 77843-3131, USA

, College Station, TX 77843-3367, USA

ere multiple members are serially connected. The decision is to determine

mber to the next upstream member in the supply chain. The basic cost

endor problem with additional consideration to safety stock. This paper

for coordination of the supply chain under both complete and partial

o maximize the total expected benet. For complete information sharing

nation algorithm. For partial information sharing, we propose an optimal

d on the Alternating Direction Method and the Diagonal Quadratic

lsevier.com/locate/ijpe

n Economics

chain considered in this paper and the global optimal solution

The above mentioned researches focus on a simple two-

of the retailer and the manufacturer, respectively, must be known to

I.-J. Jeong, V. Jorge Leon / Int. J. Production Economics 135 (2012) 412419 413under compete information. Section 5 introduces the n-serialsupply chain model under partial information sharing environmentand optimal solution approaches. The solution approaches basedon ADM and DQAM are shown in Section 6. The numerical analysisfor 4-serial supply chain problem is shown in Section 7. Section 8concludes the paper with possible extension in the future.

2. Literature review

There are many tactics to achieve SCC under complete infor-mation sharing such as the return contracts, the revenue sharingcontracts, risk sharing contracts, quantity exibility contracts andcapacity allocation. In return contracts, the retailer can return theunsold inventory fully (i.e., full return contract) or to some xedamount (i.e., partial return contract) at agreed upon prices(Pasternack, 1985; Emmons and Gilbert, 1998; Marvel and Peck,1995; Kandel, 1996; Padmanabhan and Png, 1997; Lariviere,1999). These works have been extended to consider the attitudeof decision makers to the risk (Webster and Weng, 2000; Lee andLim, 2005). In the revenue sharing contracts, the manufactureroffers the retailer a low wholesale price, lower than the unitmarginal cost. The retailer shares the fraction of his revenue withthe supplier (Giannoccaro and Pontrandolfo, 2004; Gan et al.,2005). In quantity exibility contracts, the retailer can change theordered quantity after observing the actual customer demands(Tsay, 1999; Tsay and Lovejoy, 1999). The capacity allocation isanother way to coordinate the manufacturer and retailers whenretailers order more than the available capacity of the manufac-turer (Chachon and Lariviere, 1999, 2001).

The partial information sharing in SCC is a challenging issuebut a little effort has been reported in the literature. Karabati andSayin (2008) considered the information sharing issues among asingle supplier and multiple buyers where the supplier offersquantity discounts. The considering problem was the Stackelbergtype model. Chu and Leon (2008) proposed a distributed SCCIn addition, the local information of the safety stock policy of themanufacturer is not disclosed to others and local parameters arekept in private. It is important to note that the objectives ofoptimization under complete information in a unicentric SC andpartial information sharing environment in a polycentric SC arethe same, the maximization of the expected benet of theentire SC.

The ADM can only be applied to a convex programming wherewe have two parties of decision makers. One partys decisionmaking must be made on other partys decision. We will show inthe later part of the paper that ADM can be applied to a serialsupply chain when we separate members into odd numberedmembers and even numbered members. In addition, we proposeanother algorithm with different interaction type among mem-bers, which is simultaneous and independent to other memberswithout classifying members into parties which is the DQAM(Ruszczynski, 1995; see Lee and Jeong, 2010 for details) basedalgorithm. The DQAM utilizes a reference solution in order toseparate the augmented Lagrangian function into the indepen-dent problems of members. The DQAM based algorithm alsoguarantees the optimal convergence. The proposed algorithmsguarantee optimal convergences if and only if members acceptthe proposed coordination mechanism, which is an importantassumption. Also algorithms take innite number of iterations toconverge in worst case.

This paper is organized as follows: Relevant literature survey isshown in Section 2. Section 3 describes the model development.Section 4 presents the problem description of the n-serial supplyprocedure for single warehouse multiple retailers problem undera unique decision maker. In addition, the CDF of demand which isprivate information of the retailer must be known to the decisionmaker. This is called an optimization under complete information ina unicentric SC. This optimal decision may not possible in apolycentric SC where only partial information sharing is availableamong members. This is an optimization under partial informationsharing in a polycentric SC. This paper presents optimal solutionapproaches for problems under complete information and partialinformation sharing.

4. Optimization under complete information

We have a serial supply chain where nZ3 members are seriallyconnected from the upstream to the downstream of the supplychain as shown in Fig. 1. The member i orders quantity Qi to themember i1 in the upstream with unit purchasing price pi1.The downmost member has information of ultimate customerechelon supply chain which is a supplierretailer or a warehouseretailer structure. In this research, we study a serial supply chain,which can be found in distribution channels where goods owfrom the upstream to the downstream of the supply chain and theordering information ow in the opposite direction.

Chu and Leon (2009) studied the dynamic lot sizing model of aconvergent supply chain under private information environment.However, the model considered in this paper is the Newsvendor-type problem where the customer demand is probabilistic.

3. Model development

Consider a supply chain of a manufacturer and a retailer (i.e.,MR problem) with a Newsvendor-type problem. Let Y be thecustomer demand with f(y) and F(y), the probability densityfunction (PDF) and cumulative density function (CDF) of demand,respectively. The retailer sells items with the retail price, p perunit and orders Q quantity to the manufacturer with the wholesaleprice, c per unit. The manufacturer produces items with m unitcost. Without loss of generality, we assume that moc and cop.For the problem setting of the traditional Newsvendor problem,the expected prot of the retailer is

GRQ pcQpZ Q0

Fydy

where R represents retailer. The expected prot of the manufac-ture is GM cmQ .

The expected system prot for the entire supply chain is asfollows:

GQ pcQpZ Q0

FydycmQ pmQpZ Q0

Fydy:

Then the optimal ordering quantity of the retailer to themanufacturer is F1pm=p in order to maximize the expectedbenet of the retailer and the manufacturer.

In order to determine the optimal order quantity, the selling priceto customer and the manufacturing cost that are private informationpower-of-two policy where the warehouse and the retailerscommunicate with minimal private information. Lee and Jeong(2010) considered the same problem as Chu and Leon (2008) andproposed a distributed SCC method which possesses a heterarch-ical communication structure among the warehouse and theretailers. The model considered in Chu and Leon (2008) and Leeand Jeong (2010) was Roundys model (1985).demand, which is probabilistic.

For the member 1, there is no reason to maintain safety stocks

multipliers at iteration t, respectively.

ial s

I.-J. Jeong, V. Jorge Leon / Int. J. Production Economics 135 (2012) 412419414without selling them to customers. For the last member n, weassume that there exists innite number of inventory such thatwe do not have shortages in the considering problem. For theintermediate members i 2,. . .,n1, they may serve as membersfor other supply chains. In order to reduce the variability of theordering from different supply chain, they may maintain safetystocks. Note that the safety stock constraint also includes zero-safety stock policy if ai 0. Also the safety stock policy is privateinformation of the member which may not be disclosed to otherdecision makers in the supply chain.

4.1. Global optimal solution

Theorem 1. The global optimal solution under complete informationenvironment can be found by the following equations:

@G

@Q1 p1pn1

Yn1k 2

1akp1FQn1 p2a2Fa2Qn1

Xn1i 3

piaiYi1k 2

1akF aiYi1k 2

1akQn1 !( )

0 3

Qni 1aiQni1, i 2,. . .,n1:

Proof. Consider the rst derivate of G on Qi in Eq. (2) as follows:

@G

@QipiFQiQi1pi1FQi1Qio0, i 2,. . .,n2

@G

@Qipn1pn1FQn1Qn2o0, i n1 4

Since the rst derivative is negative, Qni 1aiQni1, i 2,. . .,The expected benets of members are as follows:

Gi pipi1Qipi

RQi0 Fydy, i 1

piQi1pi1QipiR QiQi10 Fydy, i 2,. . .,n1

pipi1Qi1, i n:

8>>>: 1

The decision making problem of the serial supply chain undercomplete information can be dened as follows:

maxGXni 1

Gi p1Q1pn1Qn1p1Z Q10

Fydy

Xn1i 2

pi

Z QiQi10

Fydy 2

st: QiZ1aiQi1, i 2,. . .,n1The objective function is the maximization of the expected

system benet. The constraint represents the safety stock policyof intermediate members. aiZ0 is a real value, i 2,. . .,n1.

pi+1

n Qi

Pn+1i

Qi-1

pi

Fig. 1. A general n-sern1 or equivalently

Qni Yik 2

1akQn1 , i 2,. . .,n1: 5

By replacing Qi of G with Eq. (5) and taking the rst derivative

on Q1, Eq. (3) can be derived. &5.1. Downmost member problemNote that local information of members such as purchasingprices, demand distribution and safety stock policies must beknown to a single decision maker.

5. Optimization under partial information sharing

In this section, we propose an optimal solution approach forSCC of the supply chain under partial information sharing envir-onment. In this case the safety stock level determined by a givenai is only known by member i; furthermore, the prices are onlyshared between adjacent members. Because of these informationrestrictions the problem can no longer be solved as in theprevious section. We refer to this case as the partial informationsharing case. Let Qi,j be the ordering quantity proposed by themember j for Qi where j i1 or i, i 1,2, . . .,n1. The equivalentrepresentation of the problem (3) is as follows:

maxpnpn1Qn1,nXn1i 2

piQi1,ipi1Qi,ipiZ Qi,iQi 1,i0

Fydy

p1p2Q1,1p1Z Q1,10

Fydy 6

st: Qi,i1 Qi,i, i 1,. . .,n1 7

Qi,iZ 1aiQi1,i, i 2,. . .,n1: 8The objective function can be divided into independent mem-

bers problems. However the constraints cannot be separated bymembers due to the coupling constraint (7). The constraint (7)implies that two adjacent members must agree on the same orderquantity. The constraint (8) is a local constraint of safety stocksfor member i.

To solve the problem (6)(8), it is decomposed into subpro-blems requiring only the information available to the correspond-ing decision maker. We introduce the Lagrangian multiplier uiassociated to the ith constraint of constraint (7). The augmentedLagrangian function can be constructed as follows:

L pnpn1Qn1,nXn1i 2

piQi1,ipi1Qi,ipiZ Qi,iQi 1,i0

Fydy

p1p2Q1,1p1Z Q1,10

Fydy

Xn1i 1

uiQi,i1Qi,iXn1i 1

1

2Qi,i1Qi,i2 9

Let Qi,i1t, Qi1,i1t and uit be the optimal solutions givenfrom member i1, i1 and the current settings of the Lagrangian

1 2Y, f(y),F(y)Q1

p2 p1

Q2

p3

upply chain problem.The decision making problem of the downmost member 1 insupply chain is as follows:

max LQ1,1t19 Q1,2t,p2

,fg, Fy p1p2Q1,1t1p1

Z Q1,1t10

Fydyu1tQ1,1t1

12Q1,2tQ1,1t12: 10

If the safety stock constraint is not satised by Eqs. (16) and (17),

I.-J. Jeong, V. Jorge Leon / Int. J. Production Economics 135 (2012) 412419 415As shown in the augmented Lagrangian function in Eq. (10),data given to solve the problem at iteration t1 consists of threeelements: (i) information that must be received from theupstream member to member 1, (ii) information that must bereceived from the downstream member to member 1 and (iii)global information that must be known to all members. Eq. (10),implies that Q1,2t must be received from member 2 and p2 mustbe shared with the member 2. Since the member 1 does not haveits downstream member, the second element is null set. Finallythe CDF is the information that must be known to all members inthe SC. It will be shown in Section 6 that the Lagrangian multi-pliers can be calculated locally. The optimal Q1,1t1 can becalculated as follows:

@L

@Q1,1t1 p1p2p1FQ1,1t1u1t

Q1,2tQ1,1t1 0: 11The above solution is global optimal since

@2L

@Q21,1t1p1f Q1,1t11o0:

5.2. Uppermost member problem

The decision making problem of the uppermost member n inthe supply chain is

max LQn1,nt19fg, Qn1,n1t,un1t,pn

,fg pnpn1

Qn1,nt1un1tQn1,nt11

2Qn1,nt1Qn1,n1t2:

12Once Qn1,n1t is given from the member n1 and pn are

shared with the optimal solution can be found as follows:

@L

@Qn1,nt1 pnpn1un1tQn1,nt1Qn1,n1t 0

13Also the above solution is global optimal since @2L=@Q2n1,n

t1 1o0.

5.3. Intermediate member problem

The decision making problem of the intermediate member i,i 2,. . .,n1 in the supply chain is as follows:max LQi1,it1,Qi,it19 Qi1,i1t,pi

, Qi,i1t,pi1

, Fy piQi1,it1pi1Qi,it1

piZ Qi,it1Qi1,it10

Fydyui1tQi1,it1uitQi,it1

12Qi1,it1Qi1,i1t2

1

2Qi,i1tQi,it12 14

st: Qi,it1Z1aiQi1,it1: 15Once the constraint Qi,it1Z1aiQi1,it1 is satised,

the optimal solution can be found as follows:

@L

@Qi,it1 pi1piFQi,it1Qi1,it1

uiQi,i1tQi,it1 0 16

@L

@Qi1,it1 pipiFQi,it1Qi1,it1ui1tQi1,it1Qi1,i1t 0 17Qi1,it1 and Qi,it1 are determined by the following theorem:

Theorem 3. The solution Qi1,it1 and Qi,it1 satisfyingEqs. (16) and (17) that does not satisfy Qi,it1Z 1aiQi1,it1, the optimal solution Qni1,it1 and Qni,it1 must satisfythe following equations:

piui1tQi1,i1t1aiQi,i1tpi1uit11ai2Qni1,it1piaiFaiQni1,it1 0 19

Qni,it1 1aiQni1,it1

Proof. If the solution satisfying Eqs. (14) and (15) does not satisfyQi,it1Z1aiQi1,it1, the optimal solution exists alongQi,it1 1aiQi1,it1. Let Q Qi1,it1 andQi,it1 1aiQ then the objective function is

max LQ piQpi11aiQpiZ aiQ0

Fydyui1tQuit1aiQ12QQi1,i1t2

1

2Qi,i1t1aiQ 2: 20

The rst derivative of L(Q) on Q gives the result of Eq. (19) andthe equation gives the global optimal solution since

@2L

@Q211ai2piai2f aiQ o0: &

In the following section, we propose two optimal algorithmsunder partial information sharing, which converge to the globaloptimal solution. The algorithms iteratively solve the abovementioned members problems and communicate the optimalsolutions to the adjacent members in the supply chain.

6. Optimal algorithms under partial information sharing

First, we propose ADM based algorithm. For n3 example, wehave the downmost member 1, uppermost member 3 and oneintermediate member 2 in the supply chain. Assume that we startwith an arbitrary solution of member 2 at iteration 0 as Q1,20and Q2,20. Then the optimal solution of the downmost and theuppermost member at iteration 1, Q2,11 and Q2,31 can becalculated from the Eqs. (11) and (13). Then the intermediatemember 2 can nd its own optimal solution, Q1,21 and Q2,21 atiteration 1 using Eqs. (16) and (17) and Theorem 1. If the solutionsare not feasible (i.e., violate the coupling constraints), the Lagran-gian multipliers updated as follows:

u11 u10Q1,21Q1,11Theorem 2. The solution Qi1,it1 and Qi,it1 satisfying Eqs. (16)and (17) are global optimal.

Proof.

H@2L

@Q2i,it1

@2L@Qi,it1Qi1,it1

@2L@Qi1,it1Qi,it1

@2L@Q2

i1,i1t1

pif Qi,it1Qi1,it11 pif Qi,it1Qi1,it1

pif Qi,it1Qi1,it1 pif Qi,it1Qi1,it11

18

The rst minor determinant of the Hessian matrix is pif Qi,it1Qi1,it11o0 and the second minor determinant is2pif Qi,it1Qi1,it1140. Thus it is global optimal. &u21 u20Q2,31Q2,21

where u10 and u20 are the arbitrary determined initial Lagrangianmultipliers. The iteration among one party of member 1 and 3 andanother party of member 2 continues until the solution converges toa solution, which is the global optimal. This approach is called ADMand it guarantees the global optimal convergence theoretically. Thisconcept can be extended to more general cases. Let one party be oddnumbered members, f1,3,. . ., n1=2 21g and another be evennumbered members, f2,4,. . ., n=2 2g, where y xb c is the largestinteger such that yrx. Once solutions of odd numberedmembers aregiven, the solutions of even numbered members can be determinedby Eqs. (16) and (17) and vice versa.

The ADM based optimal algorithm is summarized as follows:

given, each member can solve the individual problem usingEqs. (11), (13), (16) and (17) and Theorem 3 independently. Theresulting solution is an updated reference point and the proce-dure continues until the reference point converges to a solution. Ifthe converged solution is feasible (i.e., the converged solutionsatises the coupling constraint), the solution is the globaloptimal. Otherwise, we update the Lagrangian multipliers andthe updating the reference point continues. This procedure iscalled DQAM, which guarantees the global optimal convergencetheoretically. The proposed DQAM based algorithm is as follows:

6.2. DQAM based algorithm for the odd and even number of member

problem

I.-J. Jeong, V. Jorge Leon / Int. J. Production Economics 135 (2012) 412419416Sharing the information of customer demand is a well knownstrategy to alleviate the so called Bullwhip Effect in a supplychain. The Bullwhip effect is the phenomenon that the amplica-tion of order quantity and inventory level increases from thedownstream to the upstream of a supply chain.

The DQAM based algorithm starts with an arbitrary solution ofall members in the problem. Since all the adjacent solutions areadjacent members. Also the demand information (i.e., CDF ofcustomer demand) must be shared among all the members in thesupply chain in order to determine optimal ordering quantities.problem can be separated as independent member problems.In terms of information sharing, note that stock inventory ratesare kept locally and the selling price are shared between the{2,4}. Once the order quantities for {1,3,5} are given to {2,4},members of {2,4} can make decision on their own order quantitiesindependently. In general, if we group odd numbered members asone group, even numbered members as another group, the6.1. ADM based algorithm for the odd number of member problem

Step 0. Set t0.Arbitrary determine uit, i 1,2, :::,n1.Arbitrary determine the solutions of odd numberedmembersQ2,1t, Qn1,nt, Qi1,it and Qi1,it.

Step 1. Obtain new solutions for even numbered members,Qi1,it1 and

Qi1,it1using Eqs. (11), (13), (16) and (17) and Theorem 3.Step 2. Obtain new solutions for odd numbered members,Qi1,it1 and

Qi1,it1using Eqs. (11), (13), (16) and (17) and Theorem 3.Step 3. If Qi,i1t1 Qi,it1, i 1,2,. . .,n1 then stop.

Otherwise set uit1 uitQi,i1t1Qi,it1,i 1,2,. . .,n1and increase t by 1 and go to Step 1.

The information ows of ADM is shown in Fig. 2.For example, when n5, the decision for each members can be

separated when the decision of members are grouped {1,3,5} andFig. 2. Informationupdating the reference point and t represents the number ofiteration of updating the Lagrangian multipliers.

It is important to note here both the ADM and DQAM basedalgorithm guarantee the optimal convergence. Two algorithmsare different in terms of the frequency of updating the Lagrangianmultipliers. ADM based algorithms update the Lagrangian multi-pliers for each iteration; however, in DQAM based algorithms, theto a solution. Step 3 checks whether the converged solutionsatises the coupling constraints. If not all the coupling con-straints are satised, the Lagrangian multipliers are updated andthe procedure is repeated. L implies the number of iteration ofQ^ i,i1l1 Q^ i,i1lQi,i1l1Q^ i,i1l, fori 1,2, :::,n1 andincrease l by 1 and go to Step 1.Step 3. If Qi,i1l1 Qi,il1, i 1,2,. . .,n1 then stop.

Otherwise set uit1 uitrQi,i1l1Qi,il1and increase t by 1 and go to Step 1.

Step 0 initializes the corresponding parameters and set anarbitrarily solution as a reference solution. In Steps 1 and 2, theinitial reference point is consecutively updated until it convergesows foStep 0. Initialization: Set e, L, t0 and l1.Arbitrarily determine Qi,it, Qi,i1ti 1,2,. . .,n1 and uit.

Set Q^ i,il Qi,it and Q^ i,i1l Qi,i1t i 1,2,. . .,n1:

Step 1. For i 1,2,. . .,n, obtain a new solution Q1,1l1,Qi,il1, Qi1,il1

and Qn1,nl1 using Eqs. (11), (13), (16) and (17) andTheorem 3.

Step 2. If 9Qi,i1l1Qi,il1Qi,i1lQi,il9re,i 1,2,. . .,n1or lL then go to Step 3. Otherwise setQ^ i,il1 Q^ i,ilQi,il1Q^ i,il,r ADM.

Qn3 1a3Qn2 117:94

a general n-serial supply chain problem.

0

20

40

60

80

100

120

i

Q

RetailerCenter

1 2927252321191715131193 5 7

Fig. 4. Ordering quantities of the retailer and the center of ADM for example.

20

40

60

80

100

120

Q

Center

I.-J. Jeong, V. Jorge Leon / Int. J. Production Economics 135 (2012) 412419 417In order to apply algorithms, we set the initial solution as follows:Q1,10100, Q1,2010, Q2,2050, Q2,30100, Q3,3070,Q3,40 30, u10 20, u20 20, u30 20. For DQAM, we sete 0:001 and L1000.

Figs. 46 are the results of ADM based algorithm for theexample. Fig. 4 shows the behavior of order quantities betweenthe center and the retailer. Even though the center and theretailer start the iteration with 100 and 10, respectively, theyLagrangian multipliers are updated if and only if the referencepoint converges to a certain solution. The different behavior ofalgorithms will be explained by numerical analysis in Section 7.

The information ow for the proposed DQAM based algorithmis shown in Fig. 3. Note that members share selling price onlywith their neighborhood. Also note that the demand informationmust be shared among all the members in order to calculateoptimal ordering quantities.

7. Numerical example

In this section, we examine the example of n4 serial supplychain problem. We apply both the ADM and DQAM basedalgorithms and compare the behavior of the optimal convergenceof the algorithms.

Example, consider a SupplierManufacturerCenterRetailerproblem (SMCR problem), which is n4 serial supply chainproblem. Parameter settings are as follows: p1$100, p2$75,p3$50, p4$25, p5$1, f(y)U(0,100), a2 0:1 and a3 0:1.

Considering Eq. (5), the global optimal solution of the retailercan be calculated as follows:

Qn1 p1p51a21a3

p1=100p2a22=100p3a231a22=100 97:47,

Qn2 1a2Qn1 107:22,

Fig. 3. Information ows of DQAM forconverge to the global optimal quantity, 97.47. Similarly Fig. 5shows the order quantities between the manufacture and theretailer and the result also shows that their order quantitiesconverge to the global optimal quantity, 107.22. The manufac-turer and the supplier also agreed on the global optimal orderingquantity, 117.94 as shown in Fig. 6.

Figs. 79 show the behavior of ordering quantities betweenthe retailer and center, center and manufacturer, manufacturerand supplier respectively when we apply DQAM for the exampleproblem. Fig. 6 shows the behavior of ordering quantity, Q1 fromthe retailer to the center proposed by the retailer, Q1,1 and thecenter Q1,2 for the example problem. The gure shows that theinitial reference point converged to a solution about l105thiteration. However the converged solution is not feasible becausetwo ordering quantities are not the same. Thus at t2, theLagrangian multiplier is updated and the new reference pointconverged to the global optimal solution, 97.47 about 200thiteration.

Same reasoning can be applied to Figs. 8 and 9.It seems that the compromise gap (i.e., the difference between

ordering quantities) between two members in the downstream ofthe supply chain is larger than that in the upperstream. Howeverthe optimality gap (i.e., the deviation from the optimal orderingquantity) in the downstream is smaller than that in the

0

i

Manufacturer

1 2927252321191715131193 5 7

Fig. 5. Ordering quantities of the center and the manufacturer of ADM forexample.

I.-J. Jeong, V. Jorge Leon / Int. J. Production Economics 135 (2012) 412419418120

140upperstream of the supply chain. We believe that this phenom-enon is closely related to the Bullwhip effect. The maintaining ofthe safety stock in downstream of the supply chain amplies theordering quantities placed to upstream, which causes the Bullwhipeffect.

0

20

40

60

80

100

1i

Q

ManufacturerSupplier

2927252321191715131193 5 7

Fig. 6. Ordering quantities of the manufacturer and the supplier of ADM forexample.

Fig. 7. Ordering quantities of the retailer and the center of DQAM for example.

0

50

100

150

200

250

1l

Q

CenterManufacturer

23922521119718316915514112711399857157432915

Fig. 8. Ordering quantities of the center and the manufacturer of DQAM forexample.

1508. Conclusion

This paper presents optimal approaches for the coordination ofa serial supply chain in order to maximize the total expectedbenet under complete and partial information sharing environ-ments. We provided an optimal solution for the optimizationunder complete information. For supply chain coordination underpartial information sharing, we proposed ADM and DQAM basedalgorithms. ADM and DQAM are the iterative interaction meth-ods, which guarantee the optimal convergence of the solutionsproposed by individual decision makers for a special structure ofconvex programming. The numerical analysis indicates that ADMCompared to ADM, DQAM requires more iterations to con-verge and lesser updates of the Lagrangian multipliers. Howeverthe interaction method among members of ADM is more compli-cated than DQAM. In ADM, members must be partitioned into twogroups and interactions are performed group by group iteratively.Meanwhile, in DQAM, members interact with adjacent memberssimultaneously and independently.

In terms of information sharing, ADM and DQAM methods aresimilar. In ADM and DQAM, only the selling price is sharedbetween two adjacent members the ordering quantities are com-municating. However, members in DQAM must maintain andupdate the reference point locally during the solution procedure.

0

50

100

l

Q

1 23922521119718316915514112711399857157432915

Fig. 9. Ordering quantities of the manufacturer and the supplier of DQAM for example.200

250ManufacturerSupplieris better than DQAM in terms of convergence speed.The proposed decision making procedure can be proceeded

automatically in a computerized system without the interventionof human, if the single item considered in this paper is a B or Cclass item in ABC inventory management system. Thus, eventhough the proposed algorithms require many iterations, it willtake less than a second for computation. The research can beextended to a more complex supply chain structure rather than aserial supply chain such as a convergent or divergent supplychain. Also we do not consider the delivery lead time in thispaper. However, in real world scenario, it may cause a stochasticdelivery lead time from a member to the adjacent member.

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I.-J. Jeong, V. Jorge Leon / Int. J. Production Economics 135 (2012) 412419 419

A serial supply chain of newsvendor problem with safety stocks under complete and partial information sharingIntroductionLiterature reviewModel developmentOptimization under complete informationGlobal optimal solution

Optimization under partial information sharingDownmost member problemUppermost member problemIntermediate member problem

Optimal algorithms under partial information sharingADM based algorithm for the odd number of member problemDQAM based algorithm for the odd and even number of member problem

Numerical exampleConclusionReferences