Analysis of a Quantity-Flexibility Supply Contract with Postponement Strategy

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1 Analysis of a Quantity-Flexibility Supply Contract with Postponement Strategy Zhen Li 1 Zhaotong Lian 1 Wenhui Zhou 2 1. Faculty of Business Administration, University of Macau, Macau SAR, China 2. School of Business Administration, South China University of Technology, China helenli@umac.mo lianzt@umac.mo whzhou@scut.edu.cn Abstract This paper studies a supply contract between a manufacturer and a retailer in the cosmetics industry under which a retailer receives discounts for committing to purchase products before the point of differentiation. The retailer can adjust the order quantities after the trial period based on updated demand forecast information and inventory status by paying a higher per-unit cost for the incremental units or giving up the deposit. Such contracts are popular in the cosmetics industry based on fast-moving consumer goods. Given the manufacturer s price structure, we first develop a two-period dynamic programming model to the optimal replenishment strategy for the retailer. We then further derive the quantity of the common material for the manufacturer satisfying a certain service level. Numerical results show difference in material quantities with and without using postponement strategy. Keywords Supply contract, inventory, postponement, material commonality, dynamic programming I. INTRODUCTION The characteristics of cosmetics are varieties with small order size, short delivery time and rapid renewal of products. Therefore, a company promotes new products from time to time, usually once or twice a year, and the promotion period is about one month. The amount of sales can be ignored or can be covered by their safety stock. The amount of sales fluctuate between promotion period and non-promotion period. Because of this fluctuation and the large amount of cosmetic products, a big challenge is to master production schedule. Usually, the company adopts Make-To-Forecast (MTF) policy in the production planning. In order to balance the tradeoff between service level and inventory level, it is necessary to re-organize the supply chain structure, which involves time postponement (TP) or form postponement (FP) labeled by Zinn and Bowersox [1]. To a cosmetic company, the FP is a more suitable strategy. There is usually a quantity-flexibility supply contract between the manufacturer and each retailer, in which, a retailer has a chance to revise the initial commitments based on the updated forecast demand obtained at a later stage. On the other hand, the company tries to reduce the uncertain risk by pooling inventory and adopting a postponement strategy. There is significant prior work in the general area of supply contracts. Bassok and Anupindi [2] provide an early study of supply contracts with quantity flexibility. They analyze a supply contract for a single product that specifies that the cumulative orders placed by a buyer, over a finite horizon, be at least as large as a (contracted upon) given quantity. They assume that the demand for the product is uncertain, and the buyer places orders periodically. They derive the optimal purchase policy for the buyer for a given total minimum quantity commitment and a discounted price. The focus of their work is on how the buyer makes the ordering decision in each period. In Bassok and Anupindi s [3] quantity flexibility contract, the demand forecast for each period of the entire contract horizon is given and the buyer needs to place an (initial) order for every period at the beginning of the first period. As time goes on, the buyer updates orders from the current period to the last period according to the current inventory level and the pre-fixed adjustment limits. Henig, Gerchak and Pyke [4] consider a different minimum ordering quantity contract under which the buyer decides whether to order the pre-fixed contract amount or order more than this amount at the beginning of each period. The buyer will be charged an incremental cost for the excess amount ordered. The authors explore the joint optimization of contract parameters and inventory control policy to show that the optimal policy is a modified order-up-to policy under the assumption that the demands are independent and identically distributed. Tsay [5] considers a supply chain consisting of two independent agents, a supplier and its buyer, the latter in turn serving an uncertain market demand. The author models the incentives of both the supplier and the buyer in a setting in which the buyer first estimates a purchase quantity for a given selling season, the supplier then commits to production, and finally the buyer makes the actual purchase based on the updated information of an uncertain demand. Tsay and Lovejoy [6] consider the quantity commitment contract in a multi-echelon setting, allowing for non-stationary demand with information updating. Ho, Ouyang and Su [7] develop an integrated supplier-buyer inventory model with the assumption that the market demand is sensitive to the retail price and the supplier adopts a trade credit policy. The trade credit policy discussed in their paper is a two-part strategy: cash discount and delayed payment. Their objective is to determine the optimal pricing, ordering, shipping, and payment policy to maximize the joint expected total profit per unit time. Lian and Deshmukh [8] study a supply contract with quantity flexibility based on a rolling horizon. They explore a class

2 of supply contracts under which a buyer receives discounts for committing to purchases in advance. Other literature dealing with quantity commitment contracts can be found in Anupindi and Bassok [9], Lariviere [1], Corbett [11], and the review paper by Tsay [12]. The postponement is an approach that helps to deliver more responsive supply chains (see Skipworth and Harrison [13]). It involves the delay of final manufacturing until a customer order is received and is commonly regarded as an approach to mass customization. In a push-pull strategy, some stages of the supply chain, typically the initial stages, are operated in a push-based manner, while the remaining stages employ a pull-based strategy. The interface between the pushbased stages and the pull-based stages is known as the pushpull boundary. In postponement, the firm designs the product and the manufacturing process so that decisions about which specific products are being manufactured can be delayed as long as possible (see Simchi-Levi, Kaminsky and Simchi-Levi [14]). Both supply contract and postponement strategy are very hot topics in the supply chain management. However people usually study the supply contract without considering the postponement strategy, or study the postponement strategy without considering the supply contract between the supplier and the buyer. We contribute to the literature by studying the combination of these two situations, which is actually very popular in reality. The rest of the paper is organized as follows. We first describe the model in Section II. We develop a stochastic dynamic program for the retailer and derive the optimal order policy for each product in Section III. We give the formula to calculate the quantity of the common material satisfying a service level in Section IV. We analyze the numerical results and give some managerial insights in Section V and we conclude the paper in the final section. II. MODEL DESCRIPTION Consider a quantity flexibility contract that has been negotiated between a manufacturer and a retailer for a serial similar products. Without loss of generality, we assume that there are only two products, named product A 1 and product A 2. The contract works in a two-period horizon as follows: the retailer has commitment orders, S i = R 1,i R 2,i of product A i for the whole promotion season. At the beginning of the first period, R 1,i units of product A i are delivered to the retailer at a price a i, i = 1, 2. The retailer pays h j,i of holding cost per unit item and p j,i of shortage cost per unit item of unsatisfied demand at the end of period j, j = 1, 2. After observing the early demand, the retailer can order up to the rest of the committed quantity with the original purchase price a i and receive quick delivery, but will pay with a higher price c i for the extra amount, or get back a refund η i per unit item for the backup units it does not buy. It is reasonable to assume that c i a i η i. Further, the retailer will get s i of salvage revenue per unit item for product A i, i = 1, 2 at the end of the life cycle if there are some items left. Denote by D i,j the demands for product A i at period j which have cumulative distribution function H i,j (x), i, j = 1, 2. The objective of this paper is two folds. On the retailer side, given the price structure by the manufacturer, the optimal order quantities for product A i are needed to derive so as to minimize the retailer s total cost. On the manufacturer side, rather than do the optimization, we will figure out the quantity of the common material A that we need to prepare, so as to satisfy the retailer requirement with a certain service level. III. STOCHASTIC DYNAMIC PROGRAM FOR RETAILERS We assume that the demands for product A 1 and product A 2 are independent. Therefore, we can derive the order policy for different products individually. Without loss of generality, we consider a product A and we will ignore the subscriptions of the parameters for different products. For example, we denote the unit costs by a, c and η, rather than a i, c i and η i. The holding cost per unit item in period j is h j, and the shortage cost per unit item is p j of unsatisfied demand at the end of period j, j = 1, 2. Denote by D j the demands for the product at period j which have cumulative distribution function H j (x), j = 1, 2. Denote by C(R 1, S) the total cost function of the retailer for this product in the whole promotion period with respect to the committed amount of R 1 in the first period and the total committed amount of S in the whole promotion period. Then C(R 1, S) = as EL[R 1 D 1 ] E D1 {min F (Z I)} (1) Z where I = R 1 D 1 is the inventory level at the end of period 1, Z is the inventory level at the beginning of period 2 after received the order, and EL(R 1 D 1 ) = h 1 E(R 1 D 1 ) p 1 E(R 1 D 1 ) R1 = h 1 dh 1 (x) p 1 R 1 dh 1 (x) (2) F (Z I) = c(z I R 2 ) η(z I R 2 ) G(Z), = c(z S D 1 ) η(z S D 1 ) G(Z) (3) where G(Z) = se(z D 2 ) p 2 E(Z D 2 ). Given the price structure, the retailer s objective is to derive the optimal commitment quantities of R 1 and S. A. The optimal solution for period 2 After more updated information that the demand is obtained at the beginning of period 2, the retailer may revise the amounts of the order quantities. In this section, we first explore the structural properties in period 2. Let G c (Z) = cz G(Z), (4) G η (Z) = ηz G(Z). (5)

3 Lemma 1: Let S L and S U respectively minimize G c (Z) and G η (Z), then S L = H2 1 ), p s (6) S U = H2 1 p s ) (7) and S L S U. Proof: By taking the differentiation to (4) and (5), it is easy to obtain (6) and (7). Further, cs L G(S L ) cs U G(S U ) = (c η)s U ηs U G(S U ) (c η)s U ηs L G(S L ). (8) So, (c η)s L (c η)s U, it follows S L S U because c η. Theorem 1: If G(Z) is convex then the optimal policy for problem min{f (Z I)} is determined by Z I, if I > S U, Z S = U, if S U R 2 < I S U, I R 2, if S L R 2 < I S U (9) R 2, S L, if I S L R 2. Proof: Let M(I) = η(i R 2 ) min {ηz I Z IR 2 G(Z)} and N(I) = c(ir 2 ) min {czg(z)}. By the Z IR 2 convexity of ηz G n (Z) and cz G n (Z) and the definitions of S L and S U, we have the following: First, if I R 2 S L, then I R 2 S U. N(I) = c(i R 2 ) cs L G(S L ) c(i R 2 ) c(i R 2 ) G(I R 2 ) = G(I R 2 ) = M(I). (1) Namely, it is cost-efficient to increase order quantities, which yields Z = S L. Secondly, If S L I R 2 < S U, it is straightforward to show N(I) = G(I R 2 ) = M(I). Consequently, the buyer has no incentive to modify the previous order, i.e., Z = I R 2. Next, if I S U I R 2, then S L I R 2. Then, M(I) = η(i R 2 ) ηs U G(S U ) η(i R 2 ) η(i R 2 ) G(I R 2 ) = G(I R 2 ) = N(I). (11) Therefore, it is cost-efficient to decrease order quantities, which yields Z = S U. Finally, if S U I, we have M(I) = η(i R 2 ) di G(I) = ηr 2 G(I) G(I R 2 ) = N(I), (12) which means that the buyer should fully cancel the previous order, i.e., Z = I. B. The optimal order quantities of R 1 and S After we obtain the order policy of the retailer at the second period, we now go back to the dynamic program (1) to derive R 1 and S, the optimal order quantities of R 1 and S for a product in the first period. Then R 2, the optimal value of R 2, equals to S R 1. Theorem 2: The optimal order quantity R 1 is a solution of the equation: (h p)h 1 (R 1 ) p (η p)h 1 (R 1 S U ) (p s) R S U H 2 (R x)dh 1 (x) =. (13) The optimal order quantity S = max{s, R 1} where S is a solution of the equation: a c (p η)h 1 (S S U ) (c p)h 1 (S S L ) (p s) S S L S S U H 2 (S x)dh 1 (x) = (14) respectively. Proof: Because I = R 1 D 1, the threshold in Theorem 1 can be written as where Z = I, if D 1 U 1, S U, if D 1 U 2, I R 2, if D 1 U 3, S L, if D 1 U 4 (15) U 1 = (, R 1 S U ], (16) U 2 = (R 1 S U, S S U ], (17) U 3 = (S S U, S S L ], (18) U 4 = (S S L, ). (19) Therefore, the dynamic function (1) can be rewritten as a non-linear function: C(R 1, S) = as R1 R 1 R1 S U S S U h(r 1 x)dh 1 (x) p(x R 1 )dh 1 (x) [η(r 1 S) G(R 1 x)]dh 1 (x) R 1 S U [ η(s S U x) G(S U )]dh 1 (x) S S L S S U G(S x)dh 1 (x) S S L [c(x S L S) G(S L )]dh 1 (x). (2) Because G(x) is a convex function of x, it is easy to see that C(R 1, S) is also a convex function of R 1 and S. By taking the partial derivatives of C(R 1, S) with respect to R 1 and S, and letting the derivatives be zero, we have (13) and (14).

4 Remark 1: Based on Theorem 2, the optimal value of R 1 is independent to the total committed amount S, and the value of S is also independent to the first-period committed amount R 1. IV. ORDERING POLICY OF THE MANUFACTURER In this section, we will derive the quantity of the common material that the manufacturer needs to prepare. Theoretically, the manufacture may consider the profit maximizing as their objective, while it is complicated and time consuming to collect the precise data, specially, it is hard to precisely evaluate the inventory related costs including the setup cost, the holding cost and the shortage cost. Another easier way is that we derive the quantity of the material they need to prepare so as to satisfy the retailers with a certain service level, say, 99%. Denote by Q R the total quantity of the material that the manufacturer needs to prepare. In the following formula, we recover the subscriptions of all variables for different products. The justification in the second period can be written as α i = Z i R 1,i R 2,i D 1,i where the subscription i represents the product i. So the total order amount of product i is Q i = R 1,i R 2,i α i = Z i D 1,i. Denote by Z i,j the values of Z i and denote by Q i,j the total order amount of product i when D 1,i U i,j where U i,j represent the intervals showing in (15). We show the detail in the following table: Z i,1 = I i Q i,1 = R 1,i D 1,i U i,1 Z i,2 = Si U Q i,2 = Si U D 1,i D 1,i U i,2 Z i,3 = I i R 2,i Q i,3 = R 1,i R 2,i D 1,i U i,3 Z i,4 = Si L Q i,4 = Si L D 1,i D 1,i U i,4 For i = 1, 2, we denote Ψ i,1 ˆ=P {D 1,i U i,1 } = H 1,i (R 1,i S U i ), (21) Ψ i,2 ˆ=P {D 1,i U i,2 } = H 1,i (R 1,i R 2,i S U i ) H 1,i (R 1,i S U i ), (22) Ψ i,3 ˆ=P {D 1,i U i,3 } = H 1,i (R 1,i R 2,i S L i ) H 1,i (R 1,i R 2,i S U i ), (23) Ψ i,4 ˆ=P {D 1,i U i,4 } = 1 H 1,i (R 1,i R 2,i S L i ) (24) and we assume that the demands for different products are independent. Denote ξ B = { 1, B is true,, B is false. A. Material quantity without postponement strategy (25) If the postponement strategy is not considered, the product A 1 and products A 2 are needed to prepared to satisfy the demands independently. Denote by Q R,i the quantity of the product A i. Then u i ˆ=P {Q R,i > Q i } 4 = P {Q R,i > Q i,j D 1,i U i,j }P {D 1,i U i,j } j=1 = ξ {QR,i >R 1,i}Ψ i,1 H 1,i (Q R,i S U i )Ψ i,2 ξ {QR,i >R i}ψ i,3 H 1,i (Q R,i S L i )Ψ i,4. (26) Denote by v the service level which is defined as the probability that the total material quantity satisfy the both demands of product A 1 and A 2. Then v = u 1 u 2. We can derive Q R,i by considering u 1 = u 2 v. Obviously, u i is an increasing function of Q R,i. The smallest amount of Q R,i satisfies u i v, i = 1, 2: Q R,i = arg min Q R,i {u i = P {Q R,i > Q i }}. (27) We can adopt an effective algorithm to calculate Q R,i effectively. For example, to calculate Q R,1 : Algorithm 1: (Bisection Method) Initialize Q R,1 = and a small positive number ɛ; Initialize step; //the step length while(p {Q R,1 > Q 1 Q 2 } < v){ Q R,1 = step; } x = Q R,1 step; y = Q R,1 ; while(y x > epsilon){ if(p {(y x)/2 > Q 1 Q 2 } < v) {x = (y x)/2;} else{y = (y x)/2;} } Q R,1 = x; return(q R,1 ); } B. Material quantity with postponement strategy As the material is common for product A 1 and product A 2, with the postponement strategy, we can prepare enough materials to satisfy the demand of product A 1 and product A 2 with the service level v. According (15) and the total probability formula, where v P {Q R > Q 1 Q 2 } = 4 j=1 k=1 4 v j,k (28) v j,k = P {Q R > Q 1,j Q 2,k D 1,1 U 1,j, D 1,2 U 2,k } P {D 1,1 U 1,j, D 1,2 U 2,k }, (29) There are totally 16 terms in (28).

5 We can write these terms v j,k in detail below: v 1,1 = ξ {QR >R 1,1R 1,2}Ψ 1,1 Ψ 2,1, (3) v 1,2 = P {Q R > R 1,1 S U 2 D 1,2 }Ψ 1,1 Ψ 2,2 = H 1,2 (Q R R 1,1 S U 2 )Ψ 1,1 Ψ 2,2, (31) v 1,3 = P {Q R > R 1,1 R 2 }Ψ 1,1 Ψ 2,3 = ξ {QR >R 1,1R 2}Ψ 1,1 Ψ 2,3, (32) v 1,4 = P {Q R > R 1,1 S L 2 D 1,2 }Ψ 1,1 Ψ 2,4 = H 1,2 (Q R R 1,1 S L 2 )Ψ 1,1 Ψ 2,4, (33) v 2,1 = P {Q R > S U 1 D 1,1 R 1,2 }Ψ 1,2 Ψ 2,1 = H 1,1 (Q R S U 1 R 1,2 )Ψ 1,2 Ψ 2,1, (34) v 2,2 = P {Q R > S U 1 S U 2 D 1,1 D 1,2 }Ψ 1,2 Ψ 2,2 = H 1,1 H 1,2 (Q R S U 1 S U 2 )Ψ 1,2 Ψ 2,2, (35) v 2,3 = P {Q R > S U 1 D 1,1 R 2 }Ψ 1,2 Ψ 2,3 = H 1,1 (Q R S U 1 R 2 )Ψ 1,2 Ψ 2,3, (36) v 2,4 = P {Q R > S U 1 D 1,1 S L 2 D 1,2 }Ψ 1,2 Ψ 2,4 = H 1,1 H 1,2 (Q R S U 1 S L 2 )Ψ 1,2 Ψ 2,4, (37) v 3,1 = P {Q R > R 1 R 1,2 }Ψ 1,3 Ψ 2,1 = ξ {QR >R 1R 1,2}Ψ 1,3 Ψ 2,1, (38) v 3,2 = P {Q R > R 1 S U 2 D 1,2 }Ψ 1,3 Ψ 2,2 = H 1,2 (Q R R 1 S U 2 )Ψ 1,3 Ψ 2,2, (39) v 3,3 = P {Q R > R 1 R 2 }Ψ 1,3 Ψ 2,3 = ξ {QR >R 1R 2}Ψ 1,3 Ψ 2,3, (4) v 3,4 = P {Q R > R 1 S L 2 D 1,2 }Ψ 1,3 Ψ 2,4 = H 1,2 (Q R R 1 S L 2 )Ψ 1,3 Ψ 2,4, (41) v 4,1 = P {Q R > S L 1 D 1,1 R 1,2 }Ψ 1,4 Ψ 2,1 = H 1,1 (Q R S L 1 R 1,2 )Ψ 1,4 Ψ 2,1, (42) v 4,2 = P {Q R > S L 1 D 1,1 S U 2 D 1,2 }Ψ 1,4 Ψ 2,2 = H 1,1 H 1,2 (Q R S L 1 S U 2 )Ψ 1,4 Ψ 2,2, (43) v 4,3 = P {Q R > S L 1 D 1,1 R 2 }Ψ 1,4 Ψ 2,3 = H 1,1 (Q R S L 1 R 2 )Ψ 1,4 Ψ 2,3, (44) v 4,4 = P {Q R > S L 1 D 1,1 S L 2 D 1,2 }Ψ 1,4 Ψ 2,4 = H 1,1 H 1,2 (Q R S L 1 S L 2 )Ψ 1,4 Ψ 2,4. (45) P {Q R > Q 1 Q 2 } is also an increasing function of Q R. The smallest amount of Q R satisfies P {Q R > Q 1 Q 2 } v: Q R = arg min Q R {P {Q R > Q 1 Q 2 } v}. (46) Similarly, Q R can be calculated by using Algorithm 1. V. COMPUTATIONAL ANALYSIS Given the formulae in the previous section, we can calculate the amounts of the original material that the manufacturer needs to order so as to satisfy a committed service level. We now summarize the steps to calculate the order quantities of product A i for the retailer and the material quantity for the manufacturer, i = 1, 2: TABLE I DEMAND INFORMATION Product A 1 Product A 2 Period 1 Period 2 Period 1 Period 2 N(3, 75 2 ) N(7, ) N(45, ) N(15, ) TABLE II RETAILER S ORDER QUANTITIES Product A 1 Product A 2 Discount R 1,1 R 2,1 R 1,2 R 2, a) Input the values a i, p i, h i, and s i, and the production demand information µ i,j and σ i,j, i, j = 1, 2; b) Calculate Si L and Si U by (6) and (7); c) Calculate the optimal commitment order quantities R j,i by solving (13) and (14); d) Given a service level v, derive the minimum material quantities of Q R,1, Q R,2 and Q R depending on whether the postponement strategy is using or not. We set the market price c 1 = 4., c 2 = 8., the holding cost h i =.5c i, p i =.4c i and the salvage value s i =.1c i, i = 1, 2. And we assume that the prices of the two products have the same discount in the supply contract and the committed prices a i = dc i and η i = da i. We set the discount d = We assume that the demands for product A i in period j follows normal distribution N(µ i,j, σi,j 2 ), i, j = 1, 2. The values of the parameters are shown in Table I. Table II shows the optimal order quantities of Product A 1 and Product A 2 for different shortage costs. We can see that the retailer orders much less than the expected demands for the second period at the beginning. Actually, more quantity will be ordered after the demand information is updated at the end of the first period. Table III shows the optimal quantities of the common material in the non-postponement case and the postponement case. In Table IV and Figure 1, we compare the quantities of the common material between the postponement strategy and non-postponement strategy for the different service levels and different discounts. By calculating the relative different value Diff ˆ= Q R,1Q R,2 Q R Q R,1 Q R,2 1%, we can see that the difference in using postponement strategy are not using postponement strategy is very significant, which are between 6.58% 15.14%, and the higher the service level, the bigger the difference. VI. CONCLUDING REMARKS We study a supply contract with quantity flexibility and postponement strategy. Given the cost structure and the demand information, we first conclude that the optimal order policy of the retailer for a product in the second period is a

6 TABLE III MATERIAL QUANTITIES WITH AND WITHOUT POSTPONEMENT (DISCOUNT=.9) Non-Postponement Postponement SL Q R,1 Q R,2 Q R,1 Q R,2 Q R Difference Rate (%) Discount =.85 Discount =.75 Discount =.95 threshold policy. We then derive the optimal order quantities of the retailer in the first period. Given the service level, we derive the quantities of the common material with or without postponement strategy. Numerical results shows that the optimal order quantities are very different for different values of the holding cost and shortage cost. We also found that, given a service level, the quantity of the common material under postponement strategy is significantly less than the quantity of the common material without using postponement strategy. On the manufacturer s side, we only show how to set the safety stock to satisfy the given service level without considering the cost structure. That can avoid lots of data collection work, but may not give the manufacturer an optimal solution. In the future study, we will study the pricing strategy for the manufacturer by setting its cost structure. Given any price, we first derive the optimal order policy of the retailer by minimizing its total cost, then we obtain the optimal pricing strategy by maximizing the manufacturer s total profit. Acknowledgements This research was funded in part by the Grant of the University of Macau under RG6/9-1S/LZT/FBA and the National Natural Science Foundation of China under No We thank the referees for their detailed comments that improved the presentation of the paper. TABLE IV COMPARISON BETWEEN POSTPONEMENT AND NON-POSTPONEMENT Discount SL Service Level Fig. 1. Difference between Two Strategies REFERENCES [1] W. Zinn and D. Bowersox, Planning physical distribution with the principle of postponement, Journal of Business Logistics, vol. 9, pp , [2] Y. Bassok and R. Anupindi, Analysis of supply contracts with total minimum commitment, IIE Transactions, vol. 29, pp , [3] Y. Bassok, R. Srinivasan, A. Bixby, and H. Wiesei, Design of component supply contracts with commitment revision flexibility, IBM Journal of Research and Development, vol. 41, pp , [4] M. Henig, Y. Gerchak, and D. F. Pyke, An inventory model embedded in designing a supply contract, Management Science, vol. 43, pp , [5] A. A. Tsay, The quantity flexibility contract and supplier-customer incentives, Management Science, vol. 45, pp , [6] A. A. Tsay and W. S. Lovejoy, Quantity flexibility contracts and supply chain performance, Manufacturing and Service Operations Management, vol. 1, pp , [7] C.-H. Ho, L.-Y. Ouyang, and C.-H. Su, Optimal pricing, shipment and payment policy for an integrated supplier-buyer inventory model with two-part trade credit, European Journal of Operational Research, vol. 187, pp , June 28. [8] Z. Lian and A. Deshmukh, Analysis of supply contracts with quantity flexibility, European Journal of Operational Research, vol. 196, pp , 29. [9] R. Anupindi and Y. Bassok, Supply contracts with quantity commitments and stochastic demand, in Quantitative Models for Supply Chain Management, ch. 7, Kluwer Academic Publishers, [1] M. A. Lariviere, Supply chain contracting and coordination with stochastic demand, in Quantitative Models for Supply Chain Management, ch. 8, Kluwer Academic Publishers, [11] C. J. Corbett and C. S. Tang, Designing supply contracts: Contract type and information asymmetry, in Quantitative Models For Supply Chain Management, ch. 9, Kluwer Academic Publishers, [12] A. A. Tsay, S. Nahmias, and N. Agrawal, Modeling supply chain contracts: A review, in Quantitative Models For Supply Chain Management, ch. 1, Kluwer Academic Publishers, [13] H. Skipworth and A. Harrison, Implications of form postponement to manufacturing: a case study, International Journal of Production Research, vol. 42, no. 1, pp , 24. [14] D. Simchi-Levi, P. Kaminsky, and E. Simchi-Levi, Designing and Managing the Supply Chain: Concepts, Strategies and Case Studies. Third ed, McGRAW-Hill, 28.