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Journal of Systems Engineering and Electronics Vol. 22, No. 2, April 2011, pp. 247–256

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Collaborative supply chain planning under dynamic lotsizing costs with capacity decision

Hongyan Li1, 2, Tianhui You1,∗, and Xiaoyi Luo1

1. School of Business Administration, Northeastern University, Shenyang 110004, P. R. China;2. Aarhus School of Business, Aarhus University, Aarhus V 8210, Denmark

Abstract: Studies show that supply chain cooperation improvessupply chain performance. However, it remains a challenge todevelop and implement the realistic supply chain cooperationscheme. We investigate a two-echelon supply chain planningproblem with capacity acquisition decision under asymmetric costand demand information. A simple negotiation-based coordinationmechanism is developed to synchronize production/order strate-gies of a supplier and a buyer. The coordination scheme showshow the supplier and the buyer modify their production and orderpolicy in order to find a joint economic lot sizing plan, which savesthe overall supply chain cost. The allocation of the cooperationbenefit is determined by negotiation. Due to the complexity of themultiple periods, multiple level supply chain lot sizing with capacitydecision, a heuristic algorithm is developed to find coordinationsolutions. Finally, the results of the numerical study indicate theperformance of supply chain coordination scheme.

Keywords: supply chain coordination, joint economic lot siz-ing, negotiation, supply chain management, capacity acquisition,heuristics.

DOI: 10.3969/j.issn.1004-4132.2011.02.010

1. Introduction and literature review

Centralized supply chain decision provides optimal so-lution, however, it is often unrealistic to centralize theproduction or investment decisions of a supply chain con-sisting of independent firms. In the decentralized frame-work, the supply chain partners hold the cost and demandinformation privately, make capacity investment decision,and plan production independently. For example, in theautomobile industry, the supply chain consists of a mainassembler and multiple component manufacturers. Theyoften operate in decentralized way. However, the decen-tralized decisions often cause high inventory or productioncosts or excess and shortage of capacity or production.In order to mitigate or avoid this mismatch, supply chain

Manuscript received May 9, 2009.*Corresponding author.This work was partly supported by the National Natural Science Foun-

dation of China (70701008).

coordination mechanisms are studied intensively. It hasbeen common sense that coordination can improve perfor-mance of a decentralized supply chain system [1,2].

The simplest coordination solution is based on sequen-tial supply chain planning. The retailer decides his optimalorder policy first according to the external demand, andthen places orders to the supplier. The supplier decides hercapacity and production policy to satisfy the order. How-ever, the sequential solution only provides an upper boundof the supply chain coordination cost. Since the retaileronly considers his own costs, his order policy might causesupplier’s cost high, and thereby cause high supply chainoverall cost. In this paper, we provide a simple supplychain coordination scheme to improve the overall supplychain performance.

Collaborative supply chain planning is a broad topic,and has been studied extensively by scholars in economics,operations research and marketing areas. Here, we onlyreview the literature which is closely relevant with ourstudy in the field of operations management. Many sup-ply chain coordination mechanisms based on informationsharing have been developed [3,4]. However, in reality, thesupply chain partners are often reluctant to reveal their fulldemand or cost information in practice. Thus, it is nec-essary to develop more realistic supply chain coordinationschemes based on partial information sharing or withoutinformation sharing.

Some researchers also intend to develop complex co-ordination mechanisms, and for example, [5] addresses apricing and inventory replenishment policy for a distribu-tion system with one single supplier and multiple retailers.Reference [6] presents analytical models to coordinatinginvestment, production and subcontracting applying gametheory. However, they restrict the constant demand andcost parameters over the entire planning horizon. Whilesupply chain collaboration has been studied increasinglylately, the existed studies are often based on strong as-sumptions, and focus on finding structural properties of the

248 Journal of Systems Engineering and Electronics Vol. 22, No. 2, April 2011

solutions. It remains a challenge to implement the resultsin a realistic business environment.

In addition, a stream of studies on capacity commit-ment contract is also relevant with our research. Refer-ence [7] addresses a capacity game in an assembly sys-tem consisting of a single assembler and multiple suppli-ers with uncertain demands. Reference [8] also proposesa game model where a downstream buyer offers a contractto induce a single supplier to construct production capacity.References [9,10] enrich the study [8] by allowing buyer topurchase capacity options. While the papers make great ef-forts to accommodate the capacity decision in supply chaincoordination, they all assume very simple cost structureand focus on the optimality of the solution.

Although most studies addressed above provide the sys-tematic optimal solution, they all ignored how to imple-ment a coordination scheme and achieve the solution. Infact, negotiation is a common business practice to rendercoordination. From the theoretical perspective, negotiationprocesses are often analyzed and solved based on bargain-ing theory. References [11, 12] address general two-partybargaining problems with incomplete information. Refer-ence [13] proposes a distributor lead negotiation processto induce supply chain coordination. From the implemen-tation perspective, [14] develops a negotiation-based co-ordination scheme to synchronize supply chain partners’inventory/production plan decisions. Although the overallcost of the collaborative plan is not better than the tradi-tional centralized master planning method, their approachsolves supply chain planning problems effectively underlimited information exchange.

Our study will extend existing studies to include sup-plier’s capacity acquisition decision and not to require di-rect information exchange. We assume that supplier ac-quires capacity at the beginning of the planning horizon,and the capacity will remain constant until end of the plan-ning horizon. The supplier and the retailer face their capac-ity and production decisions under the dynamic lot-sizingcost and demand structure. In a coordination situation, twopartners have common interests to cooperate, but have con-flicting interests over how to sharing benefits. We developa simple negotiation-based coordination scheme that canalign the objectives of the supply chain partners in a finiteplanning horizon.

2. Formulations

In a supply chain, the buyer determines the produc-tion/order plan to satisfy a deterministic demand stream{dt} over a finite planning horizon which is partitionedinto periods {t = 1, 2, . . . , T}. A supplier needs to se-cure a production capacity with a lot-sizing plan to meet a

buyer’s orders on a major component. A typical examplefor the problem might be the automobile assembly supplychain.

It is assumed that variable and fixed, production and or-der costs, and inventory holding costs occur in each pe-riod at both the supplier’s and the buyer’s site. Costs aretime-varying and there is no speculative inventory for ei-ther supply chain partner. Neither partner knows the otherparty’s cost structure. Additionally, the supplier purchasesher capacity at a unit and fixed cost. The cost parametersare defined below:

ast : The variable unit-production cost of the supplier in

period t;

fs : Fixed setup cost of the supplier for each batch ofproduction;

hst : The unit inventory-holding cost of the supplier in

period t;

art : The variable unit-order cost of the retailer in period

t;

f r : Fixed setup cost of the retailer for each order;

hrt : The unit inventory-holding cost of the retailer in

period t;

Λ : The unit production-capacity-purchasing cost of thesupplier;

ρ : The fixed-capacity-acquisition cost of the supplier.

We assume that there is a fixed unit-price contract be-tween the supplier and the buyer for the product. Giventhe buyer’s demand has to be satisfied fully, it is just as ef-fective to consider a cost minimization objective as a profitmaximization objective because the end product demand isdeterministic. The objectives of the decisions in the rangesof supply chain system, supplier and buyer are thereforedefined respectively as:

C = The supply chain system total cost;

Cr = The retailer’s total order cost;

Cs = The supplier’s production and capacity-acqui-sition cost.

The supplier’s capacity and the production amount aswell as the retailer’s order amount are all described in unitsof the end product demand. The decision variables include

xrt = The order amount of the retailer in period t;

yrt =

{1, xr

t > 00, otherwise

, ∀t. The retailer’s setup vari-

able;

Irt = The inventory amount of the retailer at the end of

the period t;

xst = The production amount of the supplier in period t;

yst =

{1, xs

t > 00, otherwise

, ∀t. The supplier’s setup vari-

able;

Hongyan Li et al.: Collaborative supply chain planning under dynamic lot sizing costs with capacity decision 249

Ist = The inventory amount of the supplier at the end of

the period t;K = The supplier’s capacity to acquire.Based on the cost and demand parameters above and rel-

evant assumptions, a centralized and a decentralized modelare formulated in the following sections.

2.1 Centralized model

In a centralized supply chain, a central decision maker hasaccess to the cost and demand information of the supplychain partners and be able to minimize the system cost.Denote the problem as CM, and it is formulated as

CM : C =min(Cs + Cr)=T∑

t=1

(astx

st + hs

tIst + fs

t yst )+

T∑t=1

(art x

rt + hr

tIrt + f r

t yrt ) + A(K) (1)

s.t.xr

t + Ir(t−1) = dt + Ir

t , ∀t (2a)

Ir0 = Ir

T = 0 (2b)

yrt =

{1, xr

t > 00, otherwise,

∀t (2c)

xst + Is

(t−1) = xrt + Is

t , ∀t (2d)

Is0 = Is

T = 0 (2e)

xst � Kys

t , ∀t (2f)

yst =

{1, xs

t > 00, otherwise

, ∀t (2g)

A(K) ={

0, K = 0ΛK + ρ, otherwise

(2h)

xst , I

st , xr

t , Irt � 0, ∀t (2i)

Constraint (2a) keeps the buyer’s production, inventory,and demand balance. The retailer’s inventory is zero atthe beginning of the first period and the end of the plan-ning horizon according to (2b). The retailer’s productionsetup variable (2c) is restricted to be binary. The sup-plier’s production, inventory and demand balance is keptby (2d). (2e) sets the supplier’s initial and final inventoryto zero, analogous to the retailer’s constraint in (2b). Con-straint (2f) restricts the supplier’s production under capac-ity level. The supplier’s production-setup variable is set tobe binary by (2g). (2h) defines the supplier’s capacity ac-quisition cost function. Finally, the decision variables areconstrained to be non-negative by (2i).

2.2 Decentralized model

In the decentralized supply chain, the retailer has knowl-edge of the external demand and places orders to satisfy

the demand; the supplier does not have accurate externaldemand information and usually determines a capacity-acquisition level and production pattern according to herown forecasts of the retailer’s orders. In this case, eachfirm has the objective to minimize its own cost individu-ally without taking into consideration the other party’s ex-penses. Therefore, the retailer’s decision and the supplier’sdecision are modeled separately below.

Retailer’s domain (BDM)

Cr = minT∑

t=1

(art x

rt + hr

t Irt + f r

t yrt ) (3)

subject to (2a)–(2c) and (2i).In order to distinguish the supplier’s demand forecast

anticipating the buyer’s order from the external demandstream, we denote the supplier’s forecast as {ds

t , t =1, 2, . . . , T}. Retailer’s problem BDM is a classic unca-pacitated lot-sizing model. There are numerous polyno-mial time algorithms to solve the model to optimal, and weuse the classic W–W lot-sizing algorithm [15].

Supplier’s domain (SDM)

Cs = maxT∑

t=1

(astx

st + hs

tIst + fs

t yst ) + A(K) (4)

subject to (2e)–(2i) and

xst + Is

t−1 = dst + Is

t , ∀t (5)

The supplier’s problem SDM cannot be solved to opti-mal in a polynomial time. In this paper, we use a heuris-tic algorithm developed by [16]. SDM can also be solvedto optimality by discretizing capacity levels and usingCPLEX.

The upper bound solution of the supply chain is ob-tained when retailer places order to supplier based on itsoptimal lot size plan, and the supplier arranges productionby letting. We denote the solution as sequential supplychain plan.

With the progress of the negotiation, the supplier maysuggest a delivery plan which might be deviate from thebuyer’s optimal lot size plan. Without loss of generality,the final agreed supply chain plan will not allow shortage inany stage, and aims at improving the overall supply chainoperation performance such that a close approximation ofoptimal solution.

3. Coordination mechanism

The supplier’s capacity investment and production and in-ventory cost can be saved dramatically if the buyer acceptsa modified delivery plan which is different from its localoptimal plan. Applying a pragmatic contract framework to


examine cost-saving allocations in a decentralized supplychain is common and natural. While coordination can im-prove decentralized supply chain operation performance,the real challenge is how to realize the benefits of coordi-nation. We construct a coordination scheme including analgorithm to coordinate the decisions and activities of sup-ply chain members and an incentive scheme to allocate thebenefits of the coordination.

3.1 Construction of coordination solutions

Based on the demand information and their own cost struc-ture, the supplier and buyer negotiate on a common jointdelivery plan and amount of the price discount. Thereby,the supplier and buyer can share the benefit of coordinationand they both are better off. Since the supplier initiates thedeviation, so the modified delivery plan will always savethe supplier’s total cost comparing with that of the sequen-tial plan. The outcome of negotiation relies on first if thedelivery proposal is feasible to meet the buyer’s end prod-uct demands, and save the buyer’s total costs. Therefore,the only possibility to achieve the coordination is that theoverall supply chain operation cost is reduced.

In the coordination scheme, the supplier initiates a de-livery proposal to the buyer, but it probably differs fromthe buyer’s local optimal plan attribute to the supplier’scost structure or production conjecture situation. In orderto compensate the deviation of the buyer’s order plan, thesupplier offers a direct transfer payment with the deliveryproposal. To facilitate our analysis, we further define fol-lowing notations.

Φ = {φ1, φ2, . . . , φT }, the delivery proposal of the sup-plier;

ω(Φ) = the transfer payment the supplier offers to theretailer.

The retailer evaluates the delivery proposal Φ and trans-fer payment ω(Φ), and Cr(Φ) denotes the retailer’s costgiven a delivery proposal Φ. The coordination model canbe characterized by a supply chain cost function (6)

C(Φ∗) =

min(astx

st + hs

tIst + fs

t yst + A(K) + Cr(Φ)) (6)

Considering the supply chain’s external demand, the ob-jective function (6) is subject to (2e)−(2h) and

t∑i=1

φi �t∑

i=1

di, t = 1, 2, . . . , T (7a)

T∑i=1

φi =T∑

i=1

di (7b)

xst + Is

(t−1) = φt + Ist , t = 1, 2, . . . , T (7c)

xst , I

st , φt, K � 0, t = 1, 2, . . . , T (7d)

Where the delivery proposal is kept feasible by (7a), andthe retailer’s beginning and ending inventory are set to zeroby (7b). (7c) is the subrogation of the (5d). The supplierarranges her production according to the delivery proposalin (7d).

Since the buyer and supplier’s production plans interactwith each other, and no cost and demand information arerevealed, the supplier and buyer have to reach an agree-ment on delivery plan by an iterative negotiation process.In the each round of negotiation, the buyer and supplierevaluate the delivery plan offered by counter player respec-tively, and make the decision whether to accept it or not.

We assume that both supplier and buyer are rational, andtrusting. If the transfer payment is sufficient to compensatethe buyer’s cost increase, he will accept the supplier’s pro-posal. On the other hand, if the buyer asks more compen-sation, the supplier will pay for it as long as she can affordit. In addition, if the negotiation terminates without agree-ment struck, the supplier will accept the initial sequentialsolution.

Basically, the negotiation process includes two majorevaluation activities. First, the supplier initiates the negoti-ation by offering a different delivery pattern which wouldresult in lower cost on her side, and the buyer will evaluatethe plan on feasibility. Follows to the evaluations, the sup-plier and retailer negotiate to split the benefits. In details,the negotiation procedure consists of five steps. For conve-nience of description, the following notations are applied:

(X̃s, X̃r) = The coordinated supply chain plan;(C̃s, C̃r) = The coordinated costs of the supply chain

partners.Negotiation procedure:Step 1 Initiate parameters and variables of the sup-

plier’s and retailer’s models;Step 2 Find the initial decentralized solution (Xs,

Xr∗) (see Section 3.3 for the method), and denote thesupply chain partners’ costs as (Cs, Cr∗) and produc-tion/order plans as (Xs, Xr∗);

Step 3 The supplier proposes a modified delivery planΦ = {φ1, φ2, . . . , φT } and a transfer payment ω(Φ) to theretailer, and denote the corresponding supplier’s produc-tion plan and cost as {Xs∗, Cs∗}, where Cs∗ < Cs;

Step 4 The retailer evaluates the supplier’s deliveryproposal Φ, and denote the corresponding cost as Cr;

Step 5 The retailer compares the modified cost Cr andhis local optimal cost Cr∗. If Cr−ω(Φ) < Cr∗, go to Step5.1. Otherwise, go to Step 5.2.

Step 5.1 The retailer accepts the offer and modifies hisorder policy into Xr = Φ. Therefore, the coordinated so-lution is (X̃s, X̃r) = (Xs∗,Φ), and the coordinated cost


will be (C̃s, C̃r) = (Cs∗ + ω(Φ), Cr − ω(Φ)) The nego-tiation terminates!

Step 5.2 If Cr − ω(Φ) � Cr∗, the retailer rejects theoffer, go to Step 3.

In the coordination scheme above, the determination ofthe transfer payment is an important issue affecting the ob-taining of the coordinated solution. Two alternatives areinvestigated based on two information environments as dis-cussed below.

3.2 The determination of the transfer payment

Given the supply chain overall cost reduction by coordi-nating, it is reasonable to expect the supplier to offer com-pensation to the retailer and the retailer to accept a modi-fied solution. Reference [17] concludes that a direct cost-sharing negotiation based on total cost probably dominatesthe transfer payment approach because the direct cost shar-ing is more transparent, robust, and easier to implement.

If we assume, as in the coordination scheme of [14],that the collaboration partners exchange the effects on localcost incurred by the modification of the delivery proposal(note that the supplier and the retailer only exchange theirtotal cost variations instead of the full cost structure infor-mation), the transfer payment would be determined by (8)which is the sum of the retailer’s cost increase and half ofthe supply chain cost savings. This is straightforward andthe algorithm can converge to the maximum supply chaincost savings.

ω = (C̃r − C̃r∗) + (Cs + Cr∗ − C̃r − C̃s)/2 =(C̃r − C̃r∗ + Cs − C̃s)/2 (8)

On the other hand, the supply chain performance canalso be improved without the assumption that the supplychain partners share cost-variation information. We as-sume that the supplier offers the product to the retailer fora price that consists of own cost plus a fixed mark-up rate.The supplier calculates her own cost according to the re-tailer’s local optimal order pattern and offers the productat the price of this cost plus the margin. In this case, thesupplier retains the absolute profit, yet still has an incentiveto decrease costs to minimize capital investment. Reduc-ing costs but maintaining the same absolute profit results inincreasing the profit margin, is one of the key performanceindicators for managers.

The supplier initiates the negotiation by offering a dif-ferent delivery pattern, which would result in lower coston her side, but higher cost to the retailer. The initial com-pensation offered to the retailer equals the difference of thesupplier’s own production cost before and after the change.The process terminates when the supplier cannot find pro-duction plans that save her any cost or the retailer rejectsall of the supplier’s delivery proposals.

3.3 Algorithms

In this section, we present a detailed computer algorithmon the modification of the solution and negotiation pro-cedure in the coordination scheme. The algorithm startsfrom the initial sequential supply chain plan which arethe retailer’s local optimal solution and the supplier’s best-response production plan. The basic idea of the algorithmis that the supplier investigates the buyer’s current orderpattern, modifying the delivery plan based on the setupnumbers.

The modification of the supplier’s delivery proposal isonly based on the increase or decrease of her delivery num-ber. We define Ns to be the supplier’s total setup numberin the decentralized production pattern and N r to be theretailer’s total setup number in the decentralized order pat-tern. Three types of decentralized solutions are possibleand can be identified by the relationships between the sup-plier’s and the retailer’s setup numbers:

(i) Ns < N r. In this scenario, the supplier prefers tolump some of the retailer’s orders together, to produce ear-lier and hold inventory to satisfy future demand. Increasingsetup is more expensive than building higher capacity andholding more inventories.

(ii) Ns > N r. Here, the supplier prefers to set upmore frequently and hold the inventory to satisfy the re-tailer’s one order. The capacity-acquisition cost dominatesthe setup and inventory costs.

(iii) Ns = N r. The supplier and the retailer have equalsetup numbers in the decentralized solution, but the sup-plier needs not produce exactly as much as the retailer or-ders in each order period since the supplier’s productionis limited by capacity. In this case, the setup and capaci-ty acquisition costs have comparable effects on the setupnumbers.

Whatever the type of decentralized solution, the supplierexpects to synchronize delivery and production to reduceher inventory cost. Therefore, the algorithm focuses on thecoordination between the supply chain partners on the tim-ing of the delivery and order.

Given the cost structure of the supply chain and the ini-tial decentralized solution, the specific cost variation char-acteristics of modifying the delivery plan are:

If Ns < N r, increasing the retailer’s setup number willincrease the supplier’s cost. Thus, the algorithm searchesfor possible modifications by reducing the delivery times.

If Ns > N r, decreasing the retailer’s setup number willincrease the supplier’s cost. So the algorithm searches forpossible modification by increasing the delivery times.

The algorithm analyzes and compares only existing or-ders in the retailer’s order plan, modifies one individual


order to provide maximum cost savings, then proposes themodification to the retailer. The retailer then evaluates thedelivery proposal and decides whether to accept it. If thedelivery proposal is feasible to the retailer, they start tonegotiate on the transfer payment. This process howeverrepresents only one round of the negotiation. Regardlessof the results of the negotiation, the algorithm continues tosearch for further modifications until savings possibilitieshave been exhausted. This forms an iterative modificationand evaluation procedure. The main routine of the proce-dure is given as Algorithm 1.

Algorithm 1 Construction of coordination solutionMain routine

D(t) =t∑

i=1

di;

DM(i, j) =j∑

t=i

dt;

L =Emptyset;for t=1:1:T do

if xr∗t > 0 then

L← L∪{t}end if

end forR = N r; Xr = N r∗; C̃s = Cs; C̃r = Cs∗; ω = 0if N s < N r then

while R � 1 do[SupplierCostSaving, Φ]← Find the maximum sav-

ings by reducing one order;if SupplierCostSaving > 0 then{X̃r, X̃s, ω}← Determining the coordinated so-

lution (Φ);else

break;end ifR= size(L)

end whileelseif N s < N r then

while R � T do[SupplierCostSaving, Φ]← Find the maximum saving

by increasing one order;if SupplierCostSaving > 0 then{X̃r, X̃s, ω}←Determining the coordinated solu-

tion (Φ);else

break;end ifR= size(L)

end whileelse

[SupplierCostSaving, Φ]← Bi-directional modifica-tion by subroutines 1 and 2;

if SupplierCostSaving > 0 then{X̃r, X̃s, ω} ← Determining the coordinated solu-

tion (Φ);end if

end ifend ifIf the setup numbers are not equal, either N s < N r or

Ns > N r, the supplier modifies her delivery pattern us-ing two specific subroutines. For the sake of brevity, wepresent the two routines in the Algorithm 2. Subroutine1 determines whether a large delivery should be split intotwo while subroutine 2 determines whether to merge twodeliveries fully or partially. Both routines check all the or-ders in each round and increase (reduce) one of them toobtain the supplier’s maximum cost savings.

Algorithm 2 Subroutines 1 & 2Find maximum savings by increasing or reducing one

orderSupplierCostSaving = Q

Xr ← Φif N r < Ns then

for i= 1 : 1 : R dofor j = �i : 1 : �i+1 − 1 do

V = xr�i

;xr

�j= DM(j, �i+1 − 1);

xr�i

= xr�i− xr

�j;

[Cs, Xs]← CapAcqLotSizing(X);if Cs < C̃s then

if SupplierCostSaving < C̃s − Cs thenSupplierCostSaving = C̃s − Cs;Φ = Xr; Cs∗ = Cs;Xs∗ = Xs;

end ifend ifxr

�i= V ; xr

�j= 0;

end forend for

elseif N r > Ns then

for i= 1 : 1 : R−1 dofor j = �i+1 : 1 : �i+2−1 do

V = xr�i

; F = xr�i+1

;xr

�i= xr

�i+ DM(�i+1, j);

xr�i+1

= xr�i+1−DM(�i+1, j);

[Cs, Xs]←CapAcqLotSizing(Xr);if Cs < C̃s then

if SupplierCostSaving < C̃s − Cs thenSupplierCostSaving= C̃s − Cs;Φ = Xr; Cs∗ = Cs;Xs∗ = Xs;

end if


end ifxr

�i= V ; xr

�i+1= F ;

end forend for

end ifend ifFor equal setup numbers, Ns = N r, the algorithm

searches the modified solutions using each subroutine.Given the cost structure, only one of the subroutines canresult in savings for the supplier; the other will break outin the first round of the modification. Therefore, the bi-directional searching will not incur significantly highercomputational cost than the other situations.

In Algorithm 2, Q is the minimum cost savings to thesupplier required to make the modification valid. CapAc-qLotSizing is a sub-procedure used to evaluate and deter-mine the supplier’s best solution based on the retailer’s or-der pattern Xr (see [16]).

With the new delivery proposal generated by Algorithm2, the supplier offers a transfer payment to the retailer toinduce his cooperation. If the supply chain partners ex-change information on the cost effects caused by the mod-ification, they share the supply chain cost reduction fairlyand reach a coordination solution. If, however, the partnersdo not share their cost-variation information, the supplieroffers all her cost reduction to the retailer in each roundof negotiation to raise her profit margin. Algorithm 3 ad-dresses either situation.

Algorithm 3 Subroutine 3Find the coordinated solution and transfer paymentcostInformation =1if costInformation =1 then

Cr = Cr(Φ)if Cs∗ + Cr < C̃s + C̃r then

ω = ω + (Cr − C̃r) + (C̃s + C̃r − Cs∗ − Cr)/2X̃s = Xs; X̃s = ΦC̃r = Cr; C̃s = Cs∗

end ifelse

Retailer evaluates Xr to have Cr privatelyω0 =SupplierCostSavingif retailer accepts the offer thenC̃r = Cr; X̃s = Φ;C̃s = Cs∗; X̃s = Xs

end ifend if

4. A numerical example

In this section, we use numerical instances to illustrate howto implement the coordination mechanism and test the ef-fectiveness of our approach. Multiple test instances are

applied. The instance is generated based on different costand demand structures. A time-varying demand stream isconsidered

dt = βt × (d̄)× U [0.5, 1.5], t = 1, 2, ..., T (9)

where {βt: t =1,2,. . . ,54} is the seasonality pattern. Weconsider six different seasonality patterns:

(i) Time-invariant demand functions

βt = 1, t = 1, . . . , 54

(ii) Linear growth

βt = 0.25 + 1.5(t− 1)

53, t = 1, . . . , 54

(iii) Linear decline

βt = 1.75− 1.5(t− 1)

53, t = 1, . . . , 54

(iv) Holiday season at the beginning of the planninghorizon

βt =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

54114

+540570

(t− 1), t = 1, . . . , 6

594114− 540

570(t− 7), t = 7, . . . , 12

54114

, t = 13, . . . , 54

(10)

(v) Holiday season at the end of the planning horizon

βt =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

54114

, t = 1, 2, . . . , 42

54114

+540570

(t− 43), t = 43, . . . , 48

594114− 540

570(t− 49), t = 49, . . . , 54

(11)

(vi) Cyclical pattern

βt =

⎧⎪⎪⎨⎪⎪⎩

0.25 + 0.75(t− 1), t = 1, . . . , 3

1.75− 0.75(t− 4), t = 4, . . . , 6

βt mod6, t = 7, . . . , 54

(12)

The first pattern reflects a situation where demand func-tions are time-invariant and the second and third patternsshow linear growth (decline) demand. The fourth and fifthpatterns represent a planning horizon with a single seasonof peak demands either at the beginning or at the end of theplanning horizon. The last pattern is cyclical with a cyclelength of six periods, such that demands in the two middleperiods of each cycle are 7 times their value in the first and


last periods, while βt =1 in the remaining two periods ofthe cycle.

d̄ is the average demand in each period and U [0.5,1.5]is a uniform distribution function. For both the supplierand the retailer, we pick variable production cost αt =15and inventory holding cost ht =5 based on the assumptionof no speculative inventory. The fixed setup cost is de-termined indirectly by first choosing time-between-orders(TBO)=

√2f/hd, EOQ-cycle time, and from this identity

determining the value of the fixed setup cost f . The TBOvalue is generated from a uniform distribution on the in-terval [1,3] when considering low TBO values, the interval[4,7] when considering medium TBO values, and [8,10]for the case of high TBO values.

Before look into the coordination mechanism, we firstanalyze the gap between the centralized supply chain so-lution and the initial sequential supply chain plan. Theresults demonstrate the opportunity of supply chain per-formance improvement by coordination.

4.1 Gap analysis

Although the centralized model has quadratic constraints,it can be solved using standard solver CPLEX 11.0 to opti-mality by discretizing the supplier’s capacity levels. Withdifferent combinations of each partner’s inventory cost lev-els and demand patterns, the instances of the test problems

are generated hypothetically and solved iteratively to ob-tain the solutions for costs, production plans, and capa-city for the centralized and decentralized situations. Here,the centralized solution is regarded as the optimal solution,lower bound of the total supply chain cost. The sequen-tial supply chain plan serves as the decentralized solution(upper bound).

We address the gap between the centralized objectivevalue and the optimal decentralized value. The results areshown in Table 1.

The overall average Gap 1 of 3.60 % (see Column 6 inTable 1) reveals the cost margin between the optimal de-centralized solutions and the centralized solutions that canbe narrowed by coordination.

4.2 Coordination example

In this section, we further test the coordination scheme.Nine classes of test problems are constructed with differ-ent combinations of the supplier’s fixed-setup and retailer’sfixed-setup cost levels for each of the problems. The algo-rithm is coded using MatLab7.5 and CPLEX 11.0 and runfor 10 iterative trials in a PC with 512 RAM and Pentium 4processor. The benchmarks for the coordination solutionsare the centralized and the exact decentralized costs. Theyallow us to analyze our coordination solutions from the

Table 1 The difference between the centralized and decentralized solutions

Supplier’s Retailer’s setup costsetup Demand Low Medium High Averagecost Pattern Gap 1/(%) Gap 1/(%) Gap 1/(%) Gap 1/(%)

(1) (2) (3) (5) (7) (9)

DP1 1.59 1.86 1.71DP2 2.67 0.61 1.52

LowDP3 1.30 0.53 4.63DP4 2.20 1.98 1.90DP5 1.63 1.17 2.76DP6 1.54 0.90 1.84

Sub-average 1.82 1.17 2.39 1.80

DP1 5.48 4.87 1.68DP2 5.69 3.36 1.93

MediumDP3 5.37 4.71 2.74DP4 4.54 4.27 2.91DP5 4.52 4.45 1.53DP6 5.38 1.99 1.60

Sub-average 5.16 3.94 2.06 3.72

High

DP1 6.25 4.53 4.70DP2 6.01 6.66 5.15DP3 6.35 7.00 4.89DP4 5.36 6.24 4.84DP5 4.38 5.95 3.17DP6 5.87 3.79 3.83

Sub-average 5.70 5.69 4.43 5.28

Average 4.23 3.60 2.96 3.60

Note: Gap 1 = (The decentralized cost−the centralized cost)/The centralized cost.


two perspectives of individual cost analysis and averagegap analysis. We first present an example of the individ-ual cost variations with nine instances from one trial of thesmall-scale test problems in Table 2.

The results in Table 3 present the cost variation betweenthe supplier and retailer as well as the amount of the trans-fer payment by coordination. The results indicate thatour coordination scheme provides a clear cost reductionand effective benefit-sharing opportunities for each supplychain partner. The retailer’s final cost is actually lower thanhis local optimal cost.

Gap 2 and Gap 3 are the differences between the co-ordination costs and the centralized and the decentralizedcosts, respectively. The gaps grow slightly from the small-scale problems to the large-scale problems, but the coor-dination solutions are quite close to the centralized solu-tion such as the Gap 3 of 0.92% for small-scale problems,0.96% for medium-scale problems and 1.18% for large-scale problems. Compared with an average gap of up to

4.81% between the centralized and decentralized costs, ourcoordination scheme and algorithm represent a substan-tial improvement over the decentralized supply chain. Theaverage gaps between the coordination solutions and thebenchmarks are calculated based on the results of the tentrials.

In addition, the coordination solution comes very closeto the centralized solution in smaller cases (see Column 4for the high supplier’s setup and retailer’s medium setupcase). In the cases when the heuristic solution does little toimprove upon the decentralized, the gap between the cen-tralized solution and the decentralized solution is alreadyvery small (see Column 7 for the example of supplier’shigh setup and retailer’s high setup).

For the small and medium problems, our coordinationalgorithm is very efficient, and all the instances are solvedwith very little computation time. For the large problems,however, due to the high complexity of the iterative

Table 2 The examples of cost savings and transfer payments

Supplier RetailerCentral Coordination solutions Decentralized solutions Supplier Retailer SC Transfer

TBO TBOSC Retailer Supplier SC Retailer Supplier SC cost cost cost paymentcost cost cost cost cost cost cost saving increase saving

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

LowLow 54 647 23 546 31 335 54 881 23 359 32 533 55 892 11 98 187 1 011 693

Medium 92 619 47 979 45 320 93 299 44 573 56 245 10 0818 10 925 3 406 7 519 7 166High 122 790 79 073 43 713 122 786 67 967 77 996 145 963 34 283 11 106 23 177 22 695

MediumLow 76 044 25 510 50 534 76 044 23 359 55 871 79 230 5 337 2 151 3 186 3 744

Medium 108 730 47 489 62 154 109 643 44 573 66 714 111 287 4 560 2 916 1 644 3 738High 135 560 79 073 56 485 135 558 67 967 82 253 150 220 25 768 11 106 14 662 18 437

HighLow 122 820 31 888 90 937 122 825 23 359 102 450 125 809 11 513 8 529 2 984 10 021

Medium 127 250 44 573 86 225 130 798 44 573 86 225 130 798 0 0 0 0High 165 080 76 188 88 893 165 081 67 967 97 269 165 236 8 376 8 221 155 8 299

Table 3 The average performance of the coordination solutions

Supplier Retailer T=12 T=30 T=54

TBO TBO Gap 2/(%) Gap 3/(%) Gap 1/(%) Gap 2/(%) Gap 3/(%) Gap 1/(%) Gap 2/(%) Gap 3/(%) Gap 1/(%)(Co-Cen) (Co-Decen) (Cen-Decen) (Co-Cen) (Co-Decen) (Cen-Decen) (Co-Cen) (Co-Decen) (Cen-Decen)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

Low 0.65 0.91 1.56 0.87 0.87 1.74 0.49 0.71 1.20Low Medium 1.30 5.78 7.08 0.47 2.03 2.51 0.97 1.65 2.61

High 1.45 11.09 12.54 0.78 2.26 3.05 1.92 2.77 4.69

Average 1.13 5.93 7.06 0.71 1.72 2.43 1.12 1.71 2.83

Low 1.54 2.55 4.09 0.23 3.45 3.68 0.14 3.25 3.39Medium Medium 1.47 0.66 2.13 1.34 1.10 2.44 0.76 2.55 3.31

High 0.27 7.06 7.33 1.04 0.87 1.91 0.80 0.00 0.80

Average 1.09 3.42 4.52 0.87 1.81 2.67 0.56 1.93 2.50

Low 0.00 4.08 4.08 0.39 3.50 3.90 1.85 4.88 6.73High Medium 1.23 0.00 1.23 2.80 2.13 4.93 0.84 2.59 3.43

High 0.39 2.84 3.23 0.76 0.00 0.76 2.90 1.08 3.98

Average 0.54 2.31 2.85 1.32 1.88 3.19 1.86 2.85 4.71

Total average/(%) 0.92 3.89 4.81 0.96 1.80 2.77 1.18 2.17 3.35

Note: Gap 2=(Coordination cost−Centralized cost)/Centralized cost;Gap 3=(Decentralized cost−Coordination cost)/Decentralized cost.


modifications and evaluations, the coordination algorithmtakes excessive computation time to search for possiblecost saving opportunities. We also test some instances(more than 54-period) for which an exact solution couldnot be found within an acceptable computation time. Re-placing the optimal capacity-acquisition and lot sizing al-gorithm with our heuristic algorithm, we find that, al-though the solution is degraded to some extent, it does pro-vide an alternative solution to solve the large-scale collab-orative supply chain planning problems.

5. Conclusion

We develop a simple supply chain coordination scheme toreduce the overall cost of the two-echelon decentralizedsupply chain. The supplier’s capacity-acquisition, produc-tion plan, and the retailer’s order plan are decided simulta-neously. The approach, based on limited information ex-change and benefit sharing, advances the traditional supplychain coordination model by considering time-varying de-mand and dynamic costs, common conditions in a realisticoperational environment. Computationally, the heuristicalgorithm provided solves the coordination model effec-tively. A future research possibility would be the investi-gation of a negotiation-based coordination scheme whereeither supply chain partner could be the initiator of themodified supply chain plan. In addition, demand uncer-tainty should also be considered in the further study.

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BiographiesHongyan Li was born in 1974. She is a Ph.D. andassociate professor in Northeastern University. Herresearch interests include supply chain management,revenue management and operations research etc.E-mail: [email protected]; [email protected]

Tianhui You was born in 1967. She is a Ph.D. andassociate professor in Northeastern University. Herresearch interests include decision analysis, knowl-edge management and supply chain management etcE-mail: [email protected].

Xiaoyi Luo was born in 1986. She is a postgrad-uate student in Northeastern University. Her currentresearch interests include supply chain managementand knowledge management etc.E-mail: [email protected]