IIE Transactions (2006) 38, 765–775 C “IIE” Copyright ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/07408170600575447
Coordinating dyadic supply chains when production costs are disrupted MINGHUI XU1,∗ , XIANGTONG QI2 , GANG YU3 and HANQIN ZHANG4 1
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, P.R. China E-mail: xu
[email protected] 2 Department of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 3 Department of Information, Risk, and Operations Management, The University of Texas, Austin, TX 78712, USA 4 Academy of Mathematics and Systems Science, The Chinese Academy of Sciences, Beijing 100080, P.R. China Received November 2004 and accepted December 2005
This paper studies a supply chain coordination problem under production cost disruptions. When a production cost disruption occurs, the coordination scheme designed for the initially estimated production cost needs to be revised. To resolve this issue, it is necessary to explicitly consider the possible deviation costs caused by any changes in the original production plan. How to model the production cost disruptions and their impacts, and how to design coordination schemes under disruptions is discussed. Results obtained under various scenarios including the cases of a single retailer and multiple retailers, the cases of linear and nonlinear demand-price functions, and different wholesale policies are presented.
1. Introduction Supply chain management plays a significant role in modern manufacturing industries. The importance of effective supply chain coordination is widely recognized since it enables all parties in a supply chain to work together so as to maximize the profit of the entire system. The coordination schemes are generally designed based on various assumptions and estimates of the supply, production, and distribution levels. In contrast to most research on the design of supply chain coordination schemes, this paper will address another aspect of the problem: how to ensure an effective execution of a coordination scheme under disruptions. In particular, our focus is on production cost disruptions. Generally speaking, production cost disruptions refer to situations in which the production costs change from their estimated values used to design the coordination scheme. Cost disruptions may occur when raw material prices change, certain equipments fail, or interest rates fluctuate. Cost disruptions can occur in various forms at any stage of the production process with varying consequences. Thus, different solutions are needed for each situation. In this paper, we concentrate our analysis on the following scenario. Suppose that a supplier (called her) produces a single type of product that requires two major operations, Op∗
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C 2006 “IIE” 0740-817X
eration one (OpI) followed by Operation two (OpII). For example, OpI is to purchase raw materials with some preprocessing, and OpII is to convert the OpI product into the final product. Then she sells the final product to customers via a single retailer (called him), or equivalently, a group of homogeneous retailers. The demand for the final product is price sensitive. Therefore, the supply chain profit is determined by the demand-price relationship as well as the production costs for the two operations. Both the supplier and the retailer are independent decision makers that seek individual profit maximization. Following conventional assumptions, we model the problem as a Stackelberg game in which the supplier is the leader and the retailer is the follower. In such a situation, the retailer has a reservation profit, which is the condition for him to play the game. In other words, the retailer will not participate in the business if he cannot obtain the reservation profit. Knowing the reservation profit of the retailer, the supplier will try to maximize her own profit, which in turn is equivalent to maximizing the total supply chain profit. The supplier implements such a goal by designing an effective coordination scheme so that the total maximum supply chain profit can be achieved, the retailer can earn his reservation profit, and the supplier can maximize her profit. We have the following sequence of events. The supplier first makes a production plan in terms of a production quantity based on the information of production costs for the two operations. Such a production plan is part of the
766 supply chain coordination scheme that aims to maximize the total supply chain profit. Because of the leadtime between the two operations, OpI is conducted in advance. At this moment, the cost information for OpI is accurate, but the cost information for OpII is only an estimation. When OpII starts, its actual cost may be different from the anticipated cost, which results in the suboptimality of the original production plan. Observing the realized production cost for OpII, the supplier may be able to change the production plan so that the supply chain is re-coordinated in the new environment. In other words, we need an appropriate response to the production cost disruption. A typical example of such a scenario is the practice of production outsourcing. Suppose that the supplier outsources the production of OpI to an overseas contractor. Due to the long transportation leadtime, the production for OpI can only be made based on an estimation of the cost of OpII. If the realized cost of OpII is lower than the anticipated value, the supplier may be able to earn extra profit by selling more final product than that produced in OpI. In this case, she needs an additional supply of the OpI product which she obtains by either expediting an extra shipment or by acquisition from a local supplier, which is naturally more expensive than the cost of the original OpI production. On the other hand, if the realized cost of OpII is higher than the anticipated value, the supplier may want to sell less final product than the amount produced in OpI. Consequently, she has to handle the surplus of the OpI product by either a disposal or a sale on the secondary market. Some extra cost will be incurred for this case. Our interest in this research is to study how to make an optimal change for the above production cost disruption. There are essentially two questions to answer: (i) what is the optimal solution for a centralized decision maker who controls the entire supply chain? and (ii) what coordination mechanism is needed to achieve the centralized optimal solution when the supplier and retailer make decisions independently? To address these two questions, we study a problem in which the supply chain is coordinated by an all-unit wholesale quantity discount policy. Under a general demandprice relationship, we show how to optimally revise the production plan, and how to coordinate the supply chain using discount policies. Closed-form solutions are given for two special cases with linear and nonlinear price-demand functions. To show that our modeling and analysis approach is generally applicable to other situations, we also discuss two other cases, one with a different quantity discount policy, and one with multiple retailers. Supply chain coordination has been extensively studied in the literature. In the following, we only briefly review the work that is closely related to our model, specifically, supply chain coordination by wholesale quantity discount policies. The early work on supply chain coordination can be traced back to Jeuland and Shugan (1983) who considered a dyadic supply chain coordination problem by quantity discount.
Xu et al. Moorthy (1987) showed that a two-part tariff can also fully coordinate such a supply chain. Weng (1995) incorporated operational costs into the supply chain when devising a quantity discount policy to coordinate the supply chain. These results were extended to cases with multiple retailers by making various assumptions. For example, Ingene and Parry (1995a) addressed a linear quantity discount schedule for a system with one manufacturer and two competitive retailers; Ingene and Parry (1995b) presented a two-part tariff to coordinate a supply chain with multiple independent retailers; Boyaci and Gallego (2002) discussed the impact of transfer prices and ownership of retail-inventory; and Chen et al. (2002) extended Weng’s model to multiple retailers. In addition to quantity discounts and two-part tariffs, other coordination schemes have been proposed, for example, a price discount for orders at particular time points by Moinzadeh et al. (2002) and a revenue sharing contract by Cachon and Lariviere (2005). While the above research has resulted in many interesting designs for supply chain coordination, it is not clear how these coordination schemes work under disruptions. Our research is also related to the emerging field of disruption management (Yu and Qi, 2004). Generally speaking, disruption management studies the problem of how to dynamically revise an operation plan that has been made based on an in-advance prediction. In contrast to conventional research on handling uncertainty, disruption management emphasizes a new term, the deviation cost or penalty for changing the original plan, in the objective function. Disruption management has been successfully applied to the airline industry (Thengvall et al., 2000; Yu et al., 2003). Recently disruption management research has been extended to many other fields such as supply chain management (Xu et al., 2003; Qi et al., 2004; Xiao et al., 2005), machine scheduling (Qi et al., 2005), and production planning (Xia et al., 2004; Yang et al., 2005). The most closely related studies to our model are Xu et al. (2003) and Qi et al. (2004). They considered the impact of demand disruptions on supply chain coordination. We work on a similar supply chain coordination model but under production cost disruptions. The purpose of this paper is to understand the impact on the supply chain coordination of production cost disruptions and to propose coordination schemes when such disruptions occur. The rest of the paper is organized as follows. In Section 2, we build and analyze our model with an all-unit wholesale quantity discount policy. We then extend the results to the case with a continuous wholesale quantity discount policy in Section 3, and to the case with multiple retailers in Section 4. We draw conclusions in Section 5.
2. Coordinating the supply chains when production costs are disrupted In this section, we discuss in detail a supply chain coordination model under an all-unit wholesale quantity discount
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Coordinating supply chains with production cost disruptions policy. We start with some preliminaries for the normal case (no disruptions), then introduce our model for the production cost disruption, derive the centralized optimal solution for handling the disruption, and discuss how to coordinate the supply chain for the case of decentralized decision making. 2.1. Preliminaries Our basic model includes one supplier and one retailer. The supplier sells a single type of product to the retailer at a wholesale price, and the retailer sells the product to his consumers at a retail price. The relationship between the demand and the retail price is given by: Q = Q(p),
(1)
where p is the unit retail price, Q is the realized demand under retail price p, i.e., the product quantity the retailer orders from the supplier, and Q(p) is strictly decreasing in p. We will analyze our problem under such a general price-demand function, and also provide closed-form solutions for two special cases: (i) a linear demand function Q = D¯ − kp with D¯ > 0 and k > 0; and (ii) a nonlinˆ −2k with Dˆ > 0 and k > 1/2. ear demand function Q = Dp These two specific demand functions have been commonly used in the marketing and operations management literature (see, e.g., Weng (1995) and also Petruzzi and Dada (1999)). Note that we have slightly abused the notation k in these two models. However, because these two models will not be discussed in the same equation, the context makes it unambiguous when we consider a particular model. The supplier faces having to make both production and wholesale decisions, specifically, two production operations, OpI and OpII, followed by wholesale supply to the retailer. First of all she needs to decide the production quantity Q and then play a Stackelberg game with the retailer. Formally, the supplier declares a wholesale price and then the retailer decides how much to order from the supplier. The retailer takes part in the game only when he can earn at least his reservation profit. If the supplier has accurate information regarding the production costs and the demand-price relationship, then it is easy for her to make a perfect plan. Let ci be the unit production cost for operation i, i = 1, 2, and c = c1 + c2 . Then the supply chain profit is: fsc (p) = Q(p)(p − c).
(2)
Let Q−1 (·) be the inverse function of Q(·). The supply chain profit can also be formulated as a function of Q, fsc (Q) = Q(Q−1 (Q) − c). In order to make the following analysis meaningful, we assume that the profit function fsc (p) (or fsc (Q)) is strictly concave in p (or Q). It can be easily verified that the assumption is valid for the two specific demand functions. Let Q˜ be the optimal production quantity, then the optimal retail
˜ and the maximum supply chain profit price is p˜ = Q−1 (Q) ˜ −1 (Q) ˜ − c). Because of the concavity, Q˜ can be is f˜sc = Q(Q uniquely determined by the first-order condition: Q−1 (Q) − c + Q(Q−1 (Q)) = 0.
(3)
For the two specific demand models that we are interested in, closed-form solutions for Equation (3) can be de¯ − kp, the optirived. For the linear demand model Q = D mal production quantity, the optimal retail price, and the corresponding maximum profit of the supply chain are, respectively: (D¯ − kc)2 D¯ + kc , and f¯sc = . (4) 2k 4k ˆ −2k , the optimal For the nonlinear demand model Q = Dp production quantity, the optimal retail price, and the corresponding maximum profit of the supply chain are, respectively: 2ck 2k − 1 2k , pˆ = , and Qˆ = Dˆ 2ck 2k − 1 ˆ 2k − 1 2k ˆf sc = Dc . (5) 2k − 1 2ck D¯ − kc Q¯ = , 2
p¯ =
Such a supply chain can be coordinated by an all-unit wholesale quantity discount policy, denoted by AWQD(w1 , w2 , q0 ) with w1 > w2 (Xu et al., 2003; Qi et al., 2004). In this setting, if the retailer orders less than q0 , the unit wholesale price is w1 ; otherwise, the unit wholesale price is w2 . Knowing the reservation profit of the retailer, the supplier can easily calculate w1 and w2 , and ˜ Then the claim the wholesale policy AWQD(w1 , w2 , Q). retailer, when trying to maximize his profit, will order exactly Q˜ units and get his reservation profit. At the same time both the supply chain profit and the supplier’s profit are maximized. Further details can be found in Qi et al. (2004). 2.2. Cost disruption and optimal centralized solution Now we are ready to model the production cost disruption and solve the centralized optimization problem. When OpI production starts, the supplier needs to decide the production quantity Q˜ based on c1 , the accurate value for the OpI production cost, and c2 , the estimate of the production cost for OpII. Note that the optimal Q˜ is given by one of Equations (3), (4) or (5). When OpII starts, the production cost may be different from its estimated value if a disruption occurs. We denote the actual production cost for OpII as c2 + c. Due to the change in the production cost, the supply chain profit from the Q˜ units of the final product becomes: ˜ − c − c). ˜ −1 (Q) (6) Q(Q Observing the production cost disruption c, however, the supplier may decide to produce Q units of the final
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product, rather than the original Q˜ units, to maximize her profit. This will cause a production quantity deviation of ˜ If Q < 0, there are −Q units of the OpI Q = Q − Q. product that will not be further converted into the final product. They have to be disposed of or sold on a secondary market at a very low price. If Q > 0, Q units of extra OpI product have to be acquired, at a higher unit cost than the original c1 . If either Q > 0 or Q < 0, then the production quantity change may result in a deviation cost that has to be considered when the optimal Q is to be determined. Formally, we assume that λ1 > 0 is the unit extra cost that has to be paid for any Q units of the final product when Q > 0, and λ2 > 0 is the unit cost caused by handling the unused −Q units of the OpI product when Q < 0. Then the supply chain profit should be calculated as: ˜ + fscd (Q) = Q(Q−1 (Q) − c − c) − λ1 (Q − Q) + − λ2 (Q˜ − Q) , (7) where (x)+ = max{x, 0}. If there is a centralized decision maker who tries to maximize the total supply chain profit, then he will find an optimal Q∗ that maximizes Equation (7). Intuitively, when the production cost decreases (increases), the optimal production quantity should increase (decrease). This is stated in the following lemma. Lemma 1. Suppose Q∗ is the optimal production quantity under the production cost disruption described by Equation (7), then: (i) Q∗ ≥ Q˜ if c < 0; and (ii) Q∗ ≤ Q˜ if c > 0. Proof. Recall that Q˜ is the original optimal pro˜ −1 (Q) ˜ − c) ≥ duction quantity, then the inequality Q(Q −1 ˜ = Q(Q (Q) − c) holds for any Q > 0. Note that fscd (Q) −1 ˜ ˜ Q(Q (Q) − c − c). Suppose that Q∗ < Q˜ when c < 0. Then we have fscd (Q∗ ) = Q∗ (Q−1 (Q∗ ) − c) − c × Q∗ − λ2 (Q˜ − Q∗ ) ≤ ˜ − c) − c × Q˜ − λ2 (Q˜ − Q∗ ) < fscd (Q). ˜ ˜ −1 (Q) This Q(Q is a contradiction to the assumption that Q∗ maximizes Equation (7). Therefore, when c < 0, Q∗ ≥ Q˜ must hold. Similarly, we must have Q∗ ≤ Q˜ when c > 0. Next we analyze how to find the optimal production quantity Q∗ , as well as the corresponding optimal retail price and the supply chain profit. Based on Lemma 1, when c < 0, the supply chain profit Equation (7) is simplified to: 1 ˜ (Q) = Q(Q−1 (Q) − c − c) − λ1 (Q − Q), (8) fscd ˜ This is a concave function in Q. Let Q1 be with Q ≥ Q. the production quantity satisfying the following first-order condition: −1
−1
Q (Q1 ) − (c + c + λ1 ) + Q1 (Q (Q1 )) = 0. −1
−1
(9)
Note that c + λ1 = Q (Q1 ) − c + Q1 (Q (Q1 )) = (fsc (Q)) |Q=Q1 . When c ≤ −λ1 , we have (fsc (Q)) |Q=Q1 ≤ 0. Due to the concavity of fsc (Q) and the fact that Q˜
˜ which implies maximizes fsc (Q), we must have Q1 ≥ Q, that Q1 maximizes Equation (8) under the constraint Q ≥ ˜ Thus, when c ≤ −λ1 , the optimal production quanQ. tity is Q1 , and the corresponding optimal retail price is Q−1 (Q1 ). When −λ1 < c < 0, the above analysis shows that we must have Q1 < Q˜ which violates the constraint of Q ≥ ˜ The concavity of f 1 (Q) implies that Equation (8) is Q. scd ˜ ∞). Thus the optimal maximized at Q˜ on the interval [Q, ˜ production quantity is Q, the corresponding optimal retail ˜ the same as the original p˜ , and the supply price is Q−1 (Q), ˜ chain profit is f˜sc − c × Q. Similarly, when c > 0, the supply chain profit Equation (7) is simplified to: 2 (Q) = Q(Q−1 (Q) − c − c) − λ2 (Q˜ − Q) fscd
(10)
with the constraint that Q˜ ≥ Q. Following similar arguments to those of the case where c < 0, we know that when 0 < c < λ2 , the optimal production quantity, the optimal retail price and the optimal supply chain profit ˜ and f˜sc − c × Q, ˜ respectively; and when ˜ Q−1 (Q) are Q, λ2 ≤ c, the optimal production quantity, the optimal retail price and the optimal supply chain profit are Q2 , 2 (Q2 ), respectively, where Q2 is determined Q−1 (Q2 ) and fscd by: Q−1 (Q2 ) − (c + c − λ2 ) + Q2 (Q−1 (Q2 )) = 0.
(11)
We can give the closed-form solutions for the above Q1 and Q2 for the two specific demand models. The results are summarized as follows. Theorem 1. Suppose the price-demand relationship is Q = ¯ − kp. When there is a production cost disruption c, the D total supply chain profit is maximized at the optimal production quantity: ¯ D − ck c + λ1 − k 2 2 ¯ D − ck Q¯ ∗ = 2 ¯ − ck λ2 − c D + k 2 2
if
c ≤ −λ1 ,
if
−λ1 < c < λ2 , (12)
if
λ2 ≤ c,
with the corresponding optimal retail price being: ¯ D + ck λ1 + c + 2k 2 ¯ D + ck p¯ ∗ = 2k D¯ + ck c − λ2 + 2k 2
if c ≤ −λ1 , if −λ1 < c < λ2 , if λ2 ≤ c,
(13)
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Coordinating supply chains with production cost disruptions and a corresponding supply chain profit of: f¯∗scd = fscd (Q∗ ) (D¯ − kc)2 − c · 4k (D¯ − kc)2 − c · = 4k (D¯ − kc)2 − c · 4k
D¯ − ck k(λ1 + c)2 + 2 4 if c ≤ −λ1 , D¯ − ck 2 if − λ1 < c < λ2 , D¯ − ck k(λ2 − c)2 + 2 4 if λ2 ≤ c. (14)
ˆ −2k Theorem 2. If the price-demand relationship is Q = Dp and the production cost disruption is c, then the optimal production quantity is: 2k 2k − 1 ˆ if c ≤ −λ1 , D 2(c + c + λ1 )k 2k ˆ ∗ = Dˆ 2k − 1 Q if −λ1 < c < λ2 , (15) 2ck 2k 2k − 1 Dˆ if λ2 ≤ c, 2(c + c − λ2 )k the corresponding optimal retail price is: 2(c + c + λ1 )k if c ≤ −λ1 , 2k − 1 2ck pˆ ∗ = if −λ1 < c < λ2 , 2k − 1 2(c + c − λ2 )k if λ2 ≤ c, 2k − 1
a small production cost disruption. The range [−λ1 , λ2 ], referred to as the cost disruption robust region, can be intuitively explained as follows. First, when the production cost decreases, c < 0, there is some opportunity to increase the production quantity for a higher profit, which will also cause a deviation cost λ1 . If −λ1 < c, the deviation cost exceeds the saving created by the production cost decrease. Thus, it is optimal to keep the original plan rather than taking the opportunity to produce more. Second, when the production cost increases, c > 0, we may want to decrease the production quantity, which will also cause a deviation cost λ2 . If c < λ2 , the production cost increase is less than the deviation cost. So it is optimal to keep the original plan rather than to revise it with a deviation cost being incurred. Comparing the two price-demand relationships, we have the following observations. We have the same cost disruption robust region [−λ1 , λ2 ] for the two models. This indicates that the cost disruption robust region is a general concept that does not depend on a specific assumption on the price-demand relationship. On the other hand, we have different forms for the optimal response when the cost disruption is beyond the robust region. For example, the optimal retail price does not depend on the price sensitivity coefficient k for the case of the linear price-demand relationship, but it does for the case of the nonlinear price-demand relationship. This implies the case of the nonlinear pricedemand relationship is more complicated. 2.3. Supply chain coordination under cost disruptions
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and the corresponding optimal supply chain profit is: fˆ∗scd = 2k ˆ + c + λ1 ) 2k − 1 2k − 1 2k D(c ˆ + λ1 D 2k − 1 2(c + c + λ1 )k 2ck if c ≤ −λ1 , Dc 2k − 1 2k 2k − 1 2k ˆ − c × Dˆ (17) 2k − 1 2ck 2ck if − λ1 < c < λ2 , 2k ˆ + c − λ2 ) 2k − 1 2k − 1 2k D(c ˆ − λ2 D 2k − 1 2(c + c − λ2 )k 2ck if λ2 ≤ c. From the above analysis, we can see that when the production cost disruption is not large, specifically, −λ1 < c < λ2 , it is optimal to keep the original production plan unchanged, regardless of the specific demand models. This implies that the original production plan Q˜ is robust under
Up to now, we have known the optimal centralized solution for a production cost change, but it is not clear whether such a centralized solution can be achieved when the supplier and the retailer make decisions independently. In the following, we propose the coordination schemes and show that the supply chain can still be coordinated under some wholesale quantity discount policies. Let w be the wholesale price for the transaction. Then the supplier’s profit is: ˜ + − λ2 (Q˜ − Q)+ , fsd (w, Q) = Q(w − c − c) − λ1 (Q − Q) (18) and the retailer’s profit is: frd (w, Q) = Q(Q−1 (Q) − w).
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Recall that the supplier and the retailer play a Stackelberg game in which the supplier is the leader and knows the reservation profit of the retailer. Denote the maximum ∗ . Let profit of the supply chain under cost disruption as fscd frd∗ be the reservation profit of the retailer. If the supply chain is coordinated, then the supplier would project her ∗ − frd∗ . maximum profit to be fsd∗ = fscd In designing a coordination scheme by using a wholesale quantity discount, we need to make sure that the retailer can obtain his reservation profit frd∗ when he tries to maximize his profit, and at the same time, the maximum supply chain
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∗ profit fscd is also achieved. That is, to achieve the maximum ∗ , the coordination scheme should supply chain profit fscd induce the retailer to purchase Q∗ from the supplier at the wholesale price w. In the game, we assume that frd∗ is known, which implies that the supplier’s projected maximum profit fsd∗ is also known when the supply chain is coordinated. In other words, the profit allocation is given in a coordinated supply chain. The problem we need to solve is how to implement the coordination. For this reason, we may assume that both fsd∗ and frd∗ are known to allow the design of the supply chain coordination scheme. In deriving the form and parameters of a specific coordination scheme, we may use either fsd∗ or frd∗ , depending on which may lead to a simple presentation. We discuss the problem for the following cases.
Case 1: c ≤ −λ1 . In this case, in view of Equations (8) and (9), we can ∗ as: rewrite fscd ∗ 1 ˜ fscd = fscd (Q∗ ) = Q∗ (Q−1 (Q∗ ) − c − c) − λ1 (Q∗ − Q), ∗
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where Q is the unique solution of Q (Q) − (c + c + λ1 ) + Q(Q−1 (Q)) = 0. ˜ If the supplier’s projected profit fsd∗ is larger than λ1 Q, we can show that an all-unit wholesale quantity discount policy can coordinate the supply chain. The exact policy parameters can be derived for the two specific demand models. Firstly, we consider the linear price-demand relationship Q = D¯ − kp. Theorem 3. Suppose the price-demand relationship is Q = ¯ with δ ≥ 1, the D¯ − kp. When c ≤ −λ1 and fsd∗ = δλ1 Q supply chain can be coordinated by an all-unit wholesale quantity discount policy AWQD(w1 , w2 , Q¯ ∗ ), where: 1 ¯ w1 > [D k − −2(δ − 1)λ1 (D¯ − ck)k + (D¯ − ck − ck − λ1 k)2 ], and w2 = c + c + λ1 + (δ − 1)
¯ − ck) λ1 (D , ¯ − ck − ck − λ1 k D
and Q¯ ∗ is given by Theorem 1.
Proof. If the retailer takes the wholesale price w2 by ordering no less than Q¯ ∗ , then his profit function can be writ¯ − Q)/k − w2 ), which is maximized ten as: frd (Q) = Q((D ¯ − w2 k)/2. Note that we have Qr 1 ≤ Q¯ ∗ = (D ¯ − at Qr 1 = (D ck − ck − λ1 k)/2. By the concavity of the profit function, the retailer would like to order Q¯ ∗ units of the product to maximize his own profit, where frd (Q¯ ∗ ) = Q¯ ∗ (¯p ∗ − w2 ) = ¯ − ck − ck − λ1 k)2 /(4k). (1 − δ)λ1 Q¯ + (D If the retailer orders less than Q¯ ∗ , he has to take the wholesale price w1 , and his profit function is frd (Q) = ¯ − ¯ − Q)/k − w1 ), which is maximized at Qr 2 = (D Q((D w1 k)/2. It can be verified that frd (Qr 2 ) < frd (Q¯ ∗ ). Thus,
the retailer would like to order exactly Q¯ ∗ units and take the wholesale price w2 . Therefore, the supply chain is coordinated. ∗ Remark 1. In view of fsd∗ ≤ fscd , we know δ ≤ 1 + 2 ¯ − ck − c · k − λ1 k) /(2λ1 k(D¯ − ck)). When devising (D an AWQD(w1 , w2 , Q¯ ∗ ), once w2 is given, we can always find a large w1 such that w1 > w2 and the retailer’s profit with wholesale price w1 is less than the one whose wholesale price is w2 . For simplicity, we will only say that w1 is large enough instead of explicitly giving the lower bound of w1 in the following discussions.
Now we consider the case in which the supplier’s pro¯ In this case, the coordinajected profit fsd∗ is less than λ1 Q. tion policy is given by the following theorem where a capacitated linear pricing policy denoted by C LP(w, Q) works as follows. There is a unit wholesale price w, but the retailer’s order quantity cannot exceed Q, i.e., it is as if the supplier has a production capacity Q. Theorem 4. Suppose the price-demand relationship is Q = ¯ with 0 < δ < D¯ − kp. When c ≤ −λ1 and fsd∗ = δλ1 Q 1: (i) the supply chain can not be coordinated by an AWQD(w1 , w2 , q); and (ii) the supply chain can be coordinated by a capacitated linear pricing policy C LP(w, Q¯ ∗ ) with: w = c + c + λ1 + (δ − 1)
¯ − ck) λ1 (D . ¯ − ck − ck − λ1 k D
The proof is similar to the proof of Lemma 4 and Theorem 3 of Qi et al. (2004). We omit the details here. Similar to the case of a linear price-demand relationship, when the price-demand relationship is nonlinear, we have the following theorem. The proof is omitted due to its similarity to the proofs of Theorems 3 and 4. Theorem 5. Suppose the price-demand relationship is Q = ˆ with: ˆ −2k . If c ≤ −λ1 and f ∗ = δλ1 Q Dp sd 2k−1 c c , 0 c > 0. In this case, the maximum supply chain profit is: ∗ fscd
˜ −1 (Q) ˜ − c) − cQ˜ = f˜sc − cQ. ˜ = Q(Q
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Note that the maximum supply chain profit is positive if ˜ − c − c > 0. In other words, it is posand only if Q−1 (Q) sible that the total supply chain has a negative profit if the disruption is too large. Even for a negative profit, the supplier has to do something because she has made OpI first. What she can do is to try to reduce her loss by either continuing some of her OpII production so as to obtain some revenue from the final product or terminating production immediately by disposing of all her OpI product, depending on which may lead to a lower loss. The latter option of ˜ Therefore, the supplier entire disposal causes a loss of λ2 Q.
prefers to proceed with the OpII production if her profit, ˜ even if negative, is larger than −λ2 Q. No matter whether the profit is positive or negative, we have the coordination schemes for the supplier to enable her to achieve the best benefit. For ease of presentation, the coordination scheme is designed from the perspective of the retailer’s reservation profit. Suppose that the retailer’s reservation profit is frd∗ = δλ2 Q˜ with δ > 0, then the supplier’s projected profit will be: ˜ −1 (Q) ˜ − c) − cQ˜ − δλ2 Q. ˜ fsd∗ = Q(Q As discussed previously, if the supplier continues her pro˜ which turns duction, then fsd∗ should be larger than −λ2 Q, out to be: ˜ − c + λ2 − c]/λ2 . δ < [Q−1 (Q)
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The following theorem gives the coordination policy for the case where 0 < c < λ2 . The proof is similar to Theorem 6 and is thus omitted. ˜ Theorem 7. For the case where 0 < c < λ2 with frd∗ = δλ2 Q: ˜ − c)/λ2 , the policy AWQD (i) when 0 < δ ≤ (Q−1 (Q) ˜ can coordinate the supply chain; (w1 , w2 , Q) ˜ − c)/λ2 < δ < [Q−1 (Q) ˜ − c + λ2 − (ii) when (Q−1 (Q) ˜ can coordinate the c]/λ2 , the policy CLP(w2 , Q) supply chain; ˜ − c + λ2 − c]/λ2 , the supplier (iii) when δ ≥ [Q−1 (Q) would not continue any OpII production. ˜ − δλ2 . In the above, w1 is large enough and w2 = Q−1 (Q) Case 4: c ≥ λ2 . In this case, the maximum supply chain profit is: ∗ ˜ fscd = Q∗ (Q−1 (Q∗ ) − c − c + λ2 ) − λ2 Q,
where Q∗ is determined by Equation (11). Suppose the retailer’s reservation profit is frd∗ = δλ2 Q˜ with δ > 0. Then the supplier’s projected profit is fsd∗ = ˜ Similar to case Q∗ (Q−1 (Q∗ ) − c − c + λ2 ) − λ2 Q˜ − δλ2 Q. 3, the condition for the supplier to continue the OpII pro¯ i.e., duction is fsd∗ > −λ2 Q, δ
0 then: (i) when δ ≥ A(δ), it is optimal for the supplier to cancel the OpII production; (ii) when 0 < δ < A(δ), the policy AWQD(w1 , w2 , Q∗ ) can coordinate the supply chain where w1 is large enough, ˜ ∗ , and Q∗ is determined by w2 = Q−1 (Q∗ ) − δλ2 Q/Q Equation (11).
Proof. If the retailer takes wholesale price w2 by ordering no less than Q∗ , then his profit function can be written as Q(Q−1 (Q) − w2 ), which is maximized at Qrd where Qrd is determined by Qrd (Q−1 (Qrd )) + Q−1 (Qrd ) − w2 = 0. ˜ When 0 < δ < A(δ) = Q∗ [Q−1 (Q∗ ) − c − c + λ2 ]/(λ2 Q), it can be shown that w2 > c + c − λ2 . Therefore, we have that Qrd (Q−1 (Qrd )) + Q−1 (Qrd ) − (c + c − λ2 ) > 0. Note that Q∗ is determined by Equation (11), we must have Qrd < Q∗ by the concavity of the supply chain profit function. Thus, the retailer has to order Q∗ to maximize his own profit, and it is easy to show that: ∗
−1
∗
˜ Q (Q (Q ) − w2 ) = δλ2 Q, which is just the retailer’s reservation profit. When δ ≥ A(δ), the retailer asks too much for his reservation profit, and the supplier’s profit will be less than −λ2 Q¯ if she continues any OpII production.
3. Continuous wholesale pricing schedule In the previous section, we have shown that an all-unit wholesale quantity discount policy AWQD(w1 , w2 , q) combined with a capacitated linear pricing policy can fully coordinate the supply chain when the production cost is disrupted. The policy AWQD(w1 , w2 , q) is discontinuous at the breakpoint q. In this section, we introduce a continuous quantity discount schedule, i.e., the wholesale price is a continuous function of the order quantity Q, denoted by w(Q), where the quantity discount means dw(Q)/dQ < 0. For the sake of simplicity, we only discuss the general demand form Q = Q(p), and omit the details for each specific demand function. Let f˜s (p), f˜r (p) and f˜sc (p) be the supplier’s profit, the retailer’s profit and the supply chain profit with the retail price p, respectively. Then: f˜s (p) = Q(p)[w(Q(p)) − c], f˜r (p) = Q(p)[p − w(Q(p))], f˜sc (p) = Q(p)(p − c). (24) Based on these expressions, we can design the coordination policy for a continuous quantity discount schedule for the case with perfect information. Let f˜sc be the maximum supply chain profit, and Q˜ the optimal production quantity as before. Lemma 2. When the supplier’s projected profit is δ f˜sc with 0 ≤ δ ≤ 1, the supply chain can be coordinated under the con-
tinuous wholesale pricing quantity discount schedule w(Q) = c + δ[Q−1 (Q) − c].
Proof. If the wholesale price is w(Q) = c + δ[Q−1 (Q) − c], then Equation (24) implies that the supplier’s profit function is Q(c + δ[Q−1 (Q) − c] − c) = δQ[Q−1 (Q) − c] = δ f˜sc (p), and the retailer’s profit function is Q(p − c − δ[Q−1 (Q) − c]) = (1 − δ)Q[Q−1 (Q) − c] = (1 − δ)f˜sc (p). Therefore, the whole price w(Q) given by the lemma simultaneously maximizes the supplier’s profit, the retailer’s profit and the supply chain profit. Hence, w(Q) = c + δ[Q−1 (Q) − c] can coordinate the supply chain. Furthermore, δdQ−1 (Q) d(c + δ[Q−1 (Q) − c]) = =δ dQ dQ
dQ(p) < 0. dp
Thus, the wholesale price schedule is a quantity discount. This lemma says that, no matter whether the demand function is linear or nonlinear, the continuous quantity discount schedule w(Q) = c + δ[Q−1 (Q) − c] can fully coordinate the supply chain. Consequently, we expect that when the production cost is disrupted, a similar continuous quantity discount schedule can coordinate the disrupted supply chain. Under production cost disruption, the supply chain profit is given by: ˜ + f˜scd (Q) = Q(Q−1 (Q) − c − c) − λ1 (Q − Q) + − λ2 (Q˜ − Q) .
(25)
Suppose that Equation (25) is optimized at Q˜ ∗ , and the corresponding retail price is p˜ ∗ . Let f˜∗rd be the retailer’s reservation profit and f˜∗scd be the maximum supply chain profit. The following two theorems give coordination policies for the cases where c < 0 and c > 0, respectively. The proofs are similar to Lemma 2. Theorem 9. If c < 0 and f˜∗rd = δ f˜∗scd with 0 ≤ δ ≤ 1, then the supply chain can be coordinated by the wholesale price schedule: −1 Q (Q) − δ(Q−1 (Q) − c − c), −λ1 < c < 0, ˜
Q wc1 (Q) = −1 −1 Q (Q) − δ Q (Q) − c − c − λ1 1 − , Q c ≤ −λ1 . Theorem 10. For the case where c > 0, the retailer’s reservation profit is f˜∗rd = δ f˜∗scd . (i) when f˜∗scd > 0 and δ > 0, if the supplier’s projected profit ˜ Then the supply chain can be coordinated f˜∗sd > −λ2 Q.
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Coordinating supply chains with production cost disruptions by the wholesale price schedule: −1 Q (Q) − δ(Q−1 (Q) − c − c), 0 < c < λ2 , −1 wc2 (Q) = Q (Q) − δ Q−1 (Q) − c − c
˜ Q − λ2 − 1 , c ≥ λ2 , Q otherwise, the supplier would not continue OpII production; (ii) when f˜∗scd < 0 and δ < 0, if the supplier’s projected profit ˜ then the supply chain can be coordinated f˜∗sd > −λ2 Q, by the wholesale price schedule: −1 Q (Q) − (1 − δ)(Q−1 (Q) − c − c) (2δ − 1)f˜∗scd , 0 < c < λ2 , − Q −1 wc2 (Q) = Q (Q) − (1 − δ) Q−1 (Q) − c − c
˜ (2δ − 1)f˜∗scd Q − λ2 −1 − , Q Q c ≥ λ2 , otherwise the supplier would not continue OpII production. Remark 2. Although the continuous wholesale price schedule wci (Q) (i = 1, 2) can fully coordinate the supply chain, it may not be a quantity discount for some cases. For example, when −λ1 < c < 0, it can be shown that dwc1 (Q)/dQ = δdQ−1 (Q)/dQ < 0; but for c ≤ −λ1 , we have: Q˜ dwc1 (Q) dQ−1 (Q) = (1 − δ)λ1 2 + δ . dQ Q dQ The first term on the right-hand side of the above equation is positive, while the second term is negative. Therefore, the sign of dwc1 (Q)/dQ depends on the values of Q and δ. For the case where c > 0, the wholesale price schedule is a quantity discount policy when δ < 1 and f˜∗scd > 0. For other cases, we cannot determine whether or not the wholesale price schedule is a quantity discount.
As documented by Chen et al. (2001), even when the cost structure is uniform, a uniform order-quantity discount scheme cannot be guaranteed to coordinate the supply chain with multiple nonidentical retailers. So other coordination mechanisms are needed to coordinate a supply chain. Suppose that there are N retailers in the supply chain. The retailers are nonidentical, i.e., they have different demand functions and market parameters. The pricedemand relationship faced by retailer i is Qi = Qi (pi ) which may have either the linear form Qi = D¯ i − ki pi or nonlinear form Qi = Dˆ i pi−2ki . First, we consider the centralized solution. The profit for the supply chain is: ft (pi ; i = 1, . . . , N) =
(pi − c)Qi (pi ).
(26)
i=1
The problem is to determine a price vector p˜ ∗ = ∗ ) that maximizes ft (pi ; i = 1, . . . , N). Since the (˜p1∗ , . . . , p˜ N retailers’ demand-price functions are independent of each other, and (pi − c)Qi (pi ) is a concave function, it is easy to obtain the retailers’ optimal retail price p˜ ∗ , and the optimal ˜ ∗ = (Q˜ ∗ , . . . , Q˜ ∗ ) with Q˜ i∗ = Qi (˜pi∗ ). order quantity Q N 1 Given the supplier’s wholesale price w, the profit for retailer i is given by (pi − w)Qi (pi ). If the supplier sets the wholesale price to w = c, the retailers will take all profit from the supply chain. Now, we consider a revenue sharing plan under which the supplier receives a predetermined fraction η (0 < η < 1) of each retailer’s revenue. Given a wholesale price w, under the revenue sharing scheme, the profit function for retailer i is now given by: (1 − η)pi Qi (pi ) − wQi (pi ).
(27)
Choose (w, η) such that (1 − η)pi Qi (pi ) − wQi (pi ) = (1 − η)(pi − c)Qi (pi ). This leads to w = (1 − η)c. Therefore, the profit for retailer i is maximized at p˜ i∗ . The supplier will re N ceive revenue N shares η i=1 pi Qi (pi ) and transfer payment cQi (pi ) under the revenue sharing scheme. (1 − η) i=1 Thus, the supplier’s profit is: η
N
pi Qi (pi ) + (1 − η)
i=1
4. Extension to multiple retailers
N
=η
N i=1
N (pi − c)Qi (pi ).
cQi (pi ) − c
N
Qi (pi )
i=1
(28)
i=1
In this section we consider the case with one supplier and multiple retailers. The purpose is to illustrate how the idea of modeling and handling production cost disruptions can be applied to the case of multiple retails. We assume that a single supplier distributes a single product to multiple retailers who in turn sell to consumers. The retailers serve geographically dispersed, heterogeneous markets. The supplier charges a wholesale price and each of the retailers sets his retail price and order quantity to maximize his own profit.
Such a revenue sharing scheme is a fixed, predetermined and increasing function of the retailers’ profit function. Therefore, the maximum supply chain profit, the supplier’s maximum profit and all retailers’ maximum profits are achieved simultaneously, and then the supply chain is coordinated under the perfect information. When the production cost is disrupted, i.e., the production cost increases from c to c + c, a production deviation may be caused as a response to the disruption. Denote Qi = Qi − Q˜ i∗ as the production deviation for retailer i.
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The production cost disruption does not change retailer i’s profit function. The supply chain profit becomes: ftd (pi ; i = 1, . . . , N)
+ N N N ∗ (pi − c − c)Qi (pi ) − λ1 Qi (pi ) − Q˜ i = i=1
− λ2
N i=1
Q˜ i∗ −
N
i=1
+ Qi (pi )
i=1
.
(29)
i=1
Suppose that Equation (29) is maximized at p∗ = quantity vector Qi∗ ≥ Q˜ i∗ when 1, . . . , N. Thus, when c < 0, Equation (29) reduces to:
∗ ) and the corresponding order (p1∗ , . . . , pN ∗ is Q = (Q1 , . . . , Q∗N ). It is obvious that c < 0 and Qi∗ ≤ Q˜ i∗ when c > 0 for i =
ftd1 (pi ; i = 1, . . . , N) =
N (pi − c − c)Qi (pi )
i=1
− λ1
N
Qi (pi ) −
N
i=1
Q˜ i∗
, (30)
i=1
and when c > 0 Equation (29) reduces to: N (pi − c − c)Qi (pi ) ftd2 (pi ; i = 1, . . . , N) = i=1
− λ2
N i=1
Q˜ i∗
−
N
Qi (pi ) . (31)
i=1
Similar to the analysis of the single-retailer case, we can derive the optimal order quantity and optimal retail price. When −λ1 ≤ c ≤ λ2 , the optimal order quantity and retail price stay unchanged, i.e., we still have the same disruption robust region. When c < −λ1 or c > λ2 , the optimal order quantity for retailer i with a linear demand price function or a nonlinear demand-price function has the same form as the single-retailer case. We can still take a (modified) revenue sharing scheme to coordinate the disrupted supply chain. When −λ1 ≤ c ≤ λ2 , the coordination parameters can be taken as (w, η) = ((1 − η)(c + c), η) with 0 ≤ η ≤ 1. When c < −λ1 or c > λ2 , a revenue-sharing scheme with a retailer-specific incentive can coordinate the disrupted supply chain. When c < −λ1 , we can take the revenue sharing parameters to be (w, η) = ((1 − η)(c + c + λ1 ), η), and if retailer i takes part in the supply chain, the supplier pays the retailer (1 − η)λ1 Q˜ i∗ as an award for participation. Thus, the profit function for retailer i will be: (1 − η)pi Qi (pi ) − (1 − η)(c + c + λ1 )Qi (pi ) + (1 − η)λ1 Q˜ i∗ , = (1 − η)[pi − (c + c + λ1 )]Qi (pi ) + (1 − η)λ1 Q˜ i∗ , and the profit that the supplier can obtain from retailer i is: η[pi∗ − (c + c + λ1 )]Qi∗ + ηλ1 Q˜ i∗ . When c > λ2 , the revenue sharing parameters can be taken to be (w, η) = ((1 − η)(c + c − λ2 ), η), and if re-
tailer i takes part in the supply chain he should pay the supplier (1 − η)λ2 Q˜ i∗ as a franchise fee. Under such a coordination mechanism, the profit for retailer i can be obtained similarly. For the case where c > 0, we have the following arguments. Let f˜i∗ scd be the maximum channel profit containing retailer i, and f˜i∗ rd be retailer i’s reservation profit. If f˜i∗ > 0, retailer i’s reservation profit can be written scd ˜i∗ with η < 1. When ηf˜i∗ > −λ2 Q˜ ∗ , we = (1 − η) f as f˜i∗ i rd scd scd can still take a revenue sharing policy with parameters of (w, η) = ((1 − η)(c + c − λ2 ), η), and the supplier charges retailer i a franchise fee of (1 − η)λ2 Q˜ i∗ and obtains a rev˜∗ enue share of (1 − η)pi Q(pi ). When ηf˜i∗ scd ≤ −λ2 Qi , the supplier would not produce anything for retailer i in the second production stage. It can be shown that such a revenue sharing plan can achieve the maximum channel profit. If f˜i∗ scd < 0, the previous revenue sharing scheme does not work.
5. Conclusions We have discussed a supply chain coordination model under production cost disruptions. We have built a model for a product with two major operations. When the production cost changes for the second operation, the revision of the production plan has to take into account the impact on the intermediate product that has been made by the first operation. Such a change to the original plan may cause a deviation cost, a term that is often neglected in the literature. We started our analysis with a simple supply chain model with one supplier and one retailer coordinated by an allunit wholesale quantity discount policy. We then extended the results to other cases, including a continuous wholesale pricing quantity discount and multiple retailers. The results show that our modeling and analysis approaches are applicable to many different scenarios rather than limited to specific models. Disruption management in supply chain coordination is a new and challenging field. In this paper, we only consider a single production cost disruption. Extending our work to more complicated and practical problems and exploring other types of disruptions are possible further research directions. For example, it would be interesting to investigate cases in which the production cost is disrupted several times, or the production cost is disrupted simultaneously with other disruptions such as the demand.
Acknowledgements We thank the associated editor and the two anonymous referees for their excellent and constructive comments which led to significant improvements in this paper. This research is supported by the Distinguished Young Investigator Grants from the National Natural Sciences
Coordinating supply chains with production cost disruptions Foundation of China, the Distinguished Oversea Chinese Scholarship, and a grant from Hundred Talents Program of the Chinese Academy of Sciences.
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Yu, G., Arguello, M. Song, G. McCowan, S.M. and White, A. (2003) A new era for crew recovery at Continental Airlines. Interfaces, 33, 5–22. Yu, G. and Qi, X. (2004) Disruption Management: Framework, Models and Applications, World Scientific Publishing Co, Singapore.
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Biographies Minghui Xu is currently a postdoctoral fellow in the Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong. Before that, he studied at Wuhan University in China, and received his B.S. in 1998, M.S. in 2001 and Ph.D. in 2005 from School of Mathematics and Statistics Sciences, respectively. His research interests include supply chain management, inventory control, risk management, disruption management and other related managerial areas. Xiangtong Qi is an assistant professor at the Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology. He obtained his Ph.D. degree in August 2003 from McCombs School of Business, The University of Texas at Austin. Before that, he studied at Nankai University in China, and obtained his bachelor, master and doctorate degrees from the Department of Computer and System Sciences. His main research interests include production scheduling, inventory control, supply chain management, and manpower planning. Gang Yu received M.S. from Cornell University and Ph.D. from the Wharton School, University of Pennsylvania. He joined Amazon.com in 2004 as the Vice President, Worldwide Supply Chain Operations. Before that, he was the Jack G. Taylor Professor in Business at the McCombs School of Business of The University of Texas at Austin. He is the Founder and former Chairman/CEO of CALEB Technologies Corp. He is the author of over 70 journal articles and 4 books. He won the 2002 Franz Edelman Management Science Achievement Award from INFORMS, the 2002 IIE Transaction Award for Best Application Paper, and the 2003 Outstanding IIE Publication Award from the Institute of Industrial Engineers in addition to many other awards. Hanqin Zhang is a professor in Operations Research Division, Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, Beijing, China. He received his Ph.D. in operations research from the Chinese Academy of Sciences in 1991. His research interests are in queueing networks, stochastic manufacturing systems, inventory models and supply chain management. He has published more than 60 papers in refereed journals such as Operations Research, Manufacturing & Service Operations Management, Mathematics of Operations Research, SIAM Journal on Applied Mathematics, Queueing Systems, and Advances in Applied Probability. He is a co-author of two monographs, Average-Cost Control of Stochastic Manufacturing Systems (with S. Sethi and Q. Zhang, Springer-Verlag, 2004), and Inventory and Supply Chain Management with Forecast Updates (with S. Sethi and H. Yan, Springer-Verlag, 2005).
