DISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORDINATION WITH EXPONENTIAL DEMAND FUNCTION

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1 Acta Mathematica Scientia 2006,26B(4): ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION WITH EXPONENTIAL EMAN FUNCTION Huang Chongchao ( ) School of Mathematics and Statistics, Wuhan University, Wuhan , China Yu Gang ( ) McCombs School of Business, The University of Texas at Austin, Austin, Texas, U.S.A Wang Song ( ) School of Mathematics and Statistics, University of Western Australia, Perth, WA, Australia Wang Xianjia ( ) Institute of Systems Engineering, Wuhan University, Wuhan , China Abstract The coordination problem of a supply chain comprising one supplier and one retailer under maret demand disruption is studied in this article. A novel exponential demand function is adopted, and the penalty cost is introduced explicitly to capture the deviation production cost caused by the maret demand disruption. The optimal strategies are obtained for different disruption scale under the centralized mode. For the decentralized mode, it is proved that the supply chain can be fully coordinated by adjusting the price discount policy appropriately when disruption occurs. Furthermore, the authors point out that similar results can be established for more general demand functions that represent different maret circumstances if certain assumptions are satisfied. Key words Supply chain coordination, disruption management, real-time optimization, production and pricing 2000 MR Subject Classification 90B30 1 Introduction In this article, we study a supply chain coordination problem with demand disruption. The supply chain comprises one supplier and one retailer, where the supplier produces one type of product and sells it to the retailer at wholesale price and then the retailer sells the product at retail price. Both the supplier and the retailer want to maximize their individual profits. Usually, there are two inds of decision-maing modes for the supply chain: the centralized decision- maing mode and the decentralized decision-maing mode. In the centralized decisionmaing mode, there is a central decision-maer who sees to maximize the total profit of the Received September 15, 2005; revised March 21, This research was supported by National Science Foundation of China ( )

2 656 ACTA MATHEMATICA SCIENTIA Vol.26 Ser.B supply chain. In the decentralized mode, both the retailer and the supplier mae decisions independently to maximize their own profits. The sum of their individual profits is usually less than the maximum total profit of the supply chain in the centralized decision-maing mode due to the well-nown double marginalization phenomenon. Therefore, a coordination mechanism is needed to induce the retailer and the supplier to mae their decisions in a cooperative way so that the total profit of the supply chain can be maximized. Supply chain coordination has emerged as an attractive problem in recently years. Weng [1] analyzed a one-supplier one-retailer model in which the maret demand was sensitive to the retail price. In the scenario investigated, it was possible for the supplier to design a coordination scheme that induced the retailer to mae the right ordering decision and to set the retail price so that the maximum joint profit could be achieved. Weng and Zeng [2] extended the model to one-supplier two-retailer case, and Chen et al. [3] generalized the model to one-supplier manyretailer case. Zhao and Wang [4] studied the situation in which the price demand relationship varied from period to period, but was nown in each period. All the studies mentioned above depend on the assumptions that demand is deterministic and that the supplier has perfect information of the price demand relationship. But perfect maret information is rarely available in practice. Thus, i et al. [5] propose a two-period supply chain model to study the production and pricing problem for a product with a short life cycle. They mae use of the methods of disruption management to this problem and focus on how to adjust the production plan in the second period and how to coordinate the disrupted supply chain by a modified wholesale price policy. isruption management is an upcoming field, on which the main character is concerned with analyzing the costs incurred when an organization is forced to deviate from a predetermined plan or schedule. Successful applications include airline operations [6, 7], telecommunications [7], project scheduling [8, 9], and machine scheduling [10]. Xiao et al. [11] Studied another supply chain coordination model with demand disruptions in which there were one supplier and two competing retailers. They made use of the price subside rate contract to coordinate the investments of the competing retailers with sales promotion opportunities and demand disruption. They found that an appropriate contractual arrangement can fully coordinate the supply chain and the manufacturer can achieve a desired allocation of the total channel profit by varying the unit wholesale price and the subside rate. In this article, we study a supply chain coordination problem with demand disruptions as follow. The supplier maes an initial production plan before the selling season based on a maret forecast. When the selling season arrives, the real maret demand may be found to be different from the forecasted one. Thus, the supplier has to adjust the production plan to respond to the actual maret demand. But the adjustment to the initial production plan will usually cause deviation costs. So the supplier must decide how to adjust his production plan and how to design a new wholesale price mechanism by which he can induce the retailer to order the right quantity of product and maximize the profit of the whole supply chain. This problem was originally studied by i et al. [5]. In their model, the maret demand function is assumed to be a linear function of the retail price, = p, where is the maximum maret scale, p is the retail price, is a coefficient of price sensitivity, and is the real demand under retail price p. Xu et al. [12] studied the nonlinear demand function = p 2 for the similar problem. Note that is the maret scale and is the real demand,

3 No.4 Huang et al: ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION 657 so must satisfy that 0 for any value of p. But unfortunately, it is not satisfied for the above-mentioned demand functions. Hence, we introduce another ind of nonlinear demand function = e p in this article. The rest of the article is organized as follows. In Section 2, the coordination of supply chain without demand disruption is studied with the exponential demand function; then the centralized mode of the supply chain with demand disruptions is analyzed in Section 3; in Section 4, the coordination of the decentralized supply chain with disruptions is investigated in detail; Section 5 illustrates some results derived in this article by several numerical examples; Section 6 points out a possible extension; and Section 7 concludes this article. 2 Supply Chain Coordination Without isruption We first consider the coordination problem of supply chain without disruptions. In this case, the maret scale is assumed to be nown exactly during the planning stage and that it will not change later. The supplier manufactures a product that is purchased by the retailer who then sells it to consumer. The situation can be viewed as a Stacelberg game in which supplier plays first by declaring a wholesale price policy and then the retailer reacts to that policy by deciding how much to order and what retailer price to set. We assume the demand function is = e p, where, p, and represent the maret scale, retailer price, and real demand, respectively, and > 0 is the price sensitivity coefficient. Suppose the unit production cost of the supplier is c. Then, the total profit of the supply chain can be written as f() = (p c) = ( 1 c). (1) It is easy to verify that f() is strictly concave with respect to, so there must be a unique optimal point maximizing the supply chain s total profit. From the first-order optimality condition, we now that the supply chain profit is maximized at = e (1+c), (2) the optimal retail price is and the maximum supply chain profit is p = c + 1, (3) f SC max = ( p c) = 1 e (1+c). (4) We now from the above equations that only when the retailer decides to order and set the retail price at p can the maximum profit of the supply chain be achieved. But both the supplier and the retailer are independent decision-maers, so the supplier should devise a wholesale price mechanism to induce the retailer to order and set the retail price at p. As in [5], for the case of linear demand function, it can be proved that the following wholesale quantity discount policy can be used to coordinate the supply chain with the exponential demand function.

4 658 ACTA MATHEMATICA SCIENTIA Vol.26 Ser.B The all-unit quantity discount policy, denoted by AP(w 1, w 2, q 0 ) with w 1 > w 2, wors as follows. If the retailer orders < q 0, the unit wholesale price is w 1. If the retailer orders q 0, the unit wholesale price becomes w 2. Suppose f S is the profit that the supplier wishes to achieve. Then, fs can be written as f S SC = η f max, with 0 < η < 1. The following theorem indicates that the supplier can use the all-unit quantity discount policy to induce the retailer to order quantity of and set the retailer price to p. As a consequence, the supplier s goal and the maximum supply chain profit can be achieved. Theorem 1 For f S SC = η f max, 0 < η < 1, the supply chain can be coordinated under the quantity discount policy AP( w 1, w 2, ), where w 2 = c + η and w 1 > c 1 (1 η). Proof If the retailer wants to tae the wholesale price w 2, then he needs to order no less than from the supplier. The retailer s profit can be expressed as f r 2() = (p w 2 ) = ( 1 w 2), subject to, which is maximized at 2 = e (1+ w2). It can be seen that 2 = e (1+ w2) < e (1+c) = because w 2 > c. But the retailer cannot order 2 if he wants to tae the price w 2. By the strict concavity of f r 2() as well as the constraint, the retailer should order to maximize his profit if he wants to tae the price w 2. The maximal profit with price w 2 is f r 2( ) = e (1+c) ( 1 + c w 2 ). If the retailer orders less than and taes the wholesale price w 1, his profit can be expressed as f r 1 () = (1 w 1), subject to <, which is maximized at 1 = e (1+ w1). It is easy to see that So the maximal profit with price w 1 is From w 1 > c 1 (1 η), we have f r 1 ( 1) = e (1+ w1) ( 1 + w 1 1 = e (1+ w1) < e (1+c) =. f r 1( 1 ) = e (1+ w1) ( 1 + w 1 w 1 ). w 1 ) < e (1+c) ( 1 + c w 2 ) = f r 2 ( ). So the retailer would order to maximize his profit. The corresponding retailer s profit would be f 2( r ) = e (1+c) ( 1 + c w 2 ) = 1 η e (1+c) sc = (1 η) f max, which implies the supplier s desired profit is achieved and the maximum supply chain profit is also obtained.

5 No.4 Huang et al: ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION Centralized Supply Chain with isruptions Now we consider the situation with demand disruptions. Suppose that the maret scale changes from to +. The demand function becomes = ( + )e p with + > 0. (5) Assume that there is a central decision-maer in the supply chain who sees to maximize the total profit of the supply chain. Let be the real demand under the new maret and = be the corresponding deviation. When < 0, there will be excess inventory that has to be sold in a secondary maret at a low price. When > 0, production must be increased in order to meet new maret demand. Usually the unit production cost for the additional products will be higher than the normal unit production cost c because it has to get some extra production resources at a higher price. In both cases, the demand disruption will cause adjustment to the original production plan and finally affect the whole supply chain. The supply chain profit with production deviation cost can be written as f() = ( 1 + c) λ 1 ( ) + λ 2 ( ) +, (6) where λ 1 is the unit extra cost for producing additional products, and λ 2 is the unit cost for disposing the excess inventory and (x) + = max{x, 0}. Lemma 1 Suppose f() in (6) is maximized at. Then, if > 0 and if < 0. Proof Note that maximizes f() when = 0. This implies that for any > 0 ( 1 c) ( 1 c). Suppose > 0 but <, then f( ) = ( 1 + c) λ 2 ( ) = ( 1 + c) + λ 2 ( ) ( 1 c) + + λ 2 ( ) < ( 1 c) + + = ( 1 + c) = f( ). This contradicts the assumption that maximizes f() in (6). Therefore, when > 0. Similarly, it can be proved that when < 0. From Lemma 1, when > 0, the problem of maximizing f() reduces to maximizing the following strictly concave function f 1 () = ( 1 + c) λ 1 ( ), (7)

6 660 ACTA MATHEMATICA SCIENTIA Vol.26 Ser.B subject to. From the first-order optimality condition f 1 () = 0, we have 1 = ( + )e (1+(λ1+c)). (8) Therefore, if 1 or equivalently (e λ1 1), f 1 () is maximized at 1. If 1 < or equivalently 0 < < (e λ1 1), f 1 () is maximized at due to the concavity of f 1 (). For these two different cases of, the optimal production quantities are 1 = ( + )e (1+(λ1+c)) (e λ1 1) (case 1), = 2 = e (1+c) 0 < < (e λ1 1) (case 2). The corresponding retailer prices are p p 1 = c λ 1 = p + λ 1 (case 1), = p 2 = c = p (case 2). The maximum total profits of the supply chain are f f 1 = 1 ( + )e (1+(λ1+c)) + λ 1 e (1+c) (case 1), = f2 = 1 e (1+c) (1 + + ) (case 2). When < 0, the problem of maximizing f() is reduced to maximizing the following strictly concave function f 2 () = ( 1 + c) λ 2 ( ), (9) subject to. Again from the first-order optimality condition f 2 () = 0, we have 2 = ( + )e (1+(c λ2)). (10) Similar to the analysis for > 0, we have case 3 with (e λ2 1) < < 0 and case 4 with (e λ2 1). The optimal production quantities are 3 = = e (1+c) (e λ2 1) < < 0 (case 3), 4 = ( + )e (1+(c λ2)) < (e λ2 1) (case 4). The corresponding retailer prices are p p 3 = c = p (case 3), = p 4 = c + 1 λ 2 = p λ 2 (case 4). The maximum total profits of the supply chain for case 3 and case 4 are f f 3 = 1 e (1+c) (1 + + ) (case 3), = f4 = 1 ( + )e (1+(c λ2)) λ 2 e (1+c) (case 4).

7 No.4 Huang et al: ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION 661 Theorem 2 For a given disruption in maret demand, if the corresponding demand function is = ( + )e p, then the supply chain profit is maximized at ( + )e (1+(c+λ1)) (e λ1 1), = e (1+c) (e λ2 1) < < (e λ1 1), ( + )e (1+(c λ2)) (e λ2 1), the corresponding retailer price is p + λ 1 (e λ1 1), p = p (e λ2 1) < < (e λ1 1), p λ 2 (e λ2 1), and the maximum total profits of the supply chain is 1 ( + )e (1+(c+λ1)) + λ 1 e (1+c) (e λ1 1), f 1 = e (1+c) (1 + + ) 1) < < (e (e λ2 λ1 1), 1 ( + )e (1+(c λ2)) λ 2 e (1+c) (e λ2 1). From Theorem 2, we could draw the following conclusions. (I) When the maret scale change is small, eep the original production plan and adjust the retail price to achieve the maximum profit of the supply chain. This shows that the original production plan has certain robustness under the uncertain maret scale. (II) When the maret scale changes significantly, adjusting both production quantity and the retail price becomes necessary. However, although the production quantity change is proportional to the maret scale change, the retail price change is a constant independent of. 4 Coordination of ecentralized Supply Chain with isruptions We now study the decentralized supply chain and investigate how the supplier could devise a new mechanism to achieve the maximum supply chain profit and coordinate the disrupted supply chain. In correspondence with the analysis of Section 3, we investigate in four different cases, respectively. 4.1 Case 1: (e λ1 1) Recall that the maximum supply chain profit in this case is f 1 = 1 ( + )e (1+(λ1+c)) + λ 1 e (1+c). (11) Suppose the profit the supplier wishes to achieve is f s. We consider the following two cases. Sub-case 1.1 f s λ 1 e (1+c)

8 662 ACTA MATHEMATICA SCIENTIA Vol.26 Ser.B In this case, f s can be expressed as f s = η 1 ( + )e (1+(λ1+c)) + λ 1 e (1+c) (0 η < 1). (12) Theorem 3 When (e λ1 1) and f s λ 1 e (1+c), the supply chain can be coordinated by AP(w 1, w 2, 1 ), where w 2 = c + λ 1 + η and w 1 > c + λ 1 1 (1 η). Proof If the retailer taes the wholesale price w 2, he should order more than 1 from the supplier and his profit is f r () = ( 1 + w 2 ), which is maximized at 1 = ( + )e (1+w2). It can be shown that 1 < 1. But the retailer should not order quantity 1 if he chooses the price w 2. From the concavity of f r (), the retailer should order 1 to maximize his profit with the price w 2. His profit is f r ( 1 ) = 1 (1 + 1 w 2 ) = ( + )e (1+(λ1+c)) ( 1 + (λ 1 + c) w 2 ) = 1 η ( + )e (1+(λ1+c)). (13) If the retailer orders less than 1, he has to tae the wholesale price w 1 and his profit can be written as f r () = ( 1 + w 1 ), subject to <, which is maximized at 2 = ( + )e (1+w1). From we have w 1 > c + λ 1 1 (1 η), f r ( 2 ) = 1 ( + )e (1+w1) < 1 η ( + )e (1+(λ1+c)) = f r ( 1 ). (14) So the retailer maximizes his profit by ordering 1. As a result, both the supplier s desired profit and the maximum channel profit are achieved. From the proof of Theorem 1 and Theorem 3, it can be seen that we may always choose a sufficiently large w 1 to satisfy the inequality f r ( 2 ) < f r ( 1 ). For the sae of simplicity, we will no longer give out the lower bound for w 1 explicitly in what follows. Sub-case 1.2 f s < λ 1 e (1+c) Lemma 2 When (e λ1 1) and f s < λ 1 e (1+c), the supply chain cannot be coordinated by any AP(w 1, w 2, q 0 ). Proof Suppose the supply chain can be coordinated by an AP(w 1, w 2, q 0 ). Then the retailer should order = ( + )e (1+(λ1+c)) and the profit of supplier should be 1 f s = 1 (w 2 c) λ 1 ( 1 ) = ( + )e (1+(λ1+c)) (w 2 c) λ 1 (( + )e (1+(λ1+c)) e (1+c) ).

9 No.4 Huang et al: ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION 663 On the other hand, we now f s < λ 1 e (1+c). Thus, f s can be expressed as f s = ηλ 1 e (1+c) (0 < η < 1). From ηλ 1 e (1+c) = ( + )e (1+(λ1+c)) (w 2 c) λ 1 (( + )e (1+(λ1+c)) e (1+c) ), we have w 2 = c + λ 1 (1 η)λ 1 + eλ1 < c + λ 1. For this value of w 2, the corresponding profit of the retailer is f r () = ( 1 + w 2 ), which is maximized at 1 = ( + )e (1+w2). Because w 2 < c + λ 1, we have 1 = ( + )e (1+w2) > ( + )e (1+(λ1+c)) = 1. This implies that the retailer will order 1 rather than 1. Thus, the supply chain cannot be coordinated. In order to coordinate the supply chain in this case, we use the capacity limited pricing policy CLPP(w, q), which wors as follow. The unit wholesale price is w, but the retailer s order is not permitted to excess q. Theorem 4 When (e λ1 1) and f1 s = ηλ 1e (1+c) (0 < η < 1), the supply chain can be coordinated by CLPP(w, 1), where w = c + λ 1 (1 η)λ1 +. eλ1 Proof By Lemma 2, the profit of the retailer is maximized at 1 = ( + )e (1+w2), which is greater than 1. By the definition of CLPP(w, 1) and the strict concavity of f r (), the retailer should order 1 in order to maximize his profit. This maes the supply chain coordinated. 4.2 Case 2: 0 < < (e λ1 1) Recall that in this case the supply chain s maximum profit is f2 = 1 e (1+c) (1 + + ). (15) To coordinate the supply chain, the retailer should order 2 = = e (1+c). Suppose the profit that the supplier wishes to achieve is On the other hand, f s can be expressed as From (16) and (17) we obtain that f s = ηf2 = η e (1+c) (1 + + ). (16) f s = 2 (w 2 c) = e (1+c) (w 2 c). (17) w 2 = c + η + (1 + ). Theorem 5 Suppose 0 < < (e λ1 1) and f s is defined as in (16). (i) When η , AP(w 1, w 2, ) can be used to coordinate the supply chain;

10 664 ACTA MATHEMATICA SCIENTIA Vol.26 Ser.B (ii) When 0 < η < +, CLPP(w , ) can be used to coordinate the supply chain. ) and w 1 is sufficiently large. Where w 2 = c + η + (1 + Proof A sufficiently large w 1 will always prevent the retailer from ordering products at that price. This means the retailer definitely taes the wholesale price w 2. Thus, his profit can be written as f r () = ( 1 + w 2 ), which is maximized at 1 = ( + )e (1+w2). It can be shown that the inequality 1 is equivalent to w 2 c + 1 +, (18) which is equivalent to η From the analysis above, we have that (i) If η , then 1. Note that the retailer s order cannot be less than if he wishes to tae price w 2. By the strict concavity of f r (), the retailer should order in order to maximize his profit. + (ii) If η <, then >. By the concavity of f r ()and the rule of CLPP(w 2, ), the retailer has to order in order to maximize his profit. In both cases, the retailer s profits are f r ( ) = (p 2 w 2 ) = 1 η e (1+c) (1 + + ). (19) From (16) and (19), it can be seen that the maximum profit of the supply chain is achieved and the system is coordinated. 4.3 Case 3: (e λ2 1) < < 0 For this case, the total supply chain profit f 3 = 1 e (1+c) (1 + + ) (20) is positive only when > 0, which is equivalent to > (1 e 1 ). If this is true, we have the following result similar to Theorem 5. Theorem 6 Suppose (e λ2 1) < < 0 and > (1 e 1 ). If the profit that the supplier wishes to achieve is f s = ηf3 = η e (1+c) (1+ + ) for 0 < η < 1, then (i) when η (ii) when η < , AP(w 1, w 2, ) can be used to coordinate the supply chain; +, CLPP(w , ) can be used to coordinate the supply chain. Here ) and w 1 is large enough. w 2 = c + η + (1 + The proof is similar to that of Theorem 5 and is thus omitted. Let us now consider the situation with (1 e λ2 ) < < (1 e 1 ). In this case, the total supply chain f 3 will be negative. The retailer would refuse to order any quantity of product if he cannot earn a certain profit. This will force the supplier to dispose his entire production in a secondary maret with the loss of λ 2. To reduce his loss, the supplier

11 No.4 Huang et al: ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION 665 needs to set a wholesale price policy that induces the retailer to order at least some amount of product. Suppose the least profit that the retailer must achieve is The supplier s profit becomes f r = µf3 = µ e (1+c) (1 + + ). (21) f s = f 3 fr = (1 + µ)f 3 = 1 + µ e (1+c) (1 + + ). (22) Clearly, the Supplier hopes f s > λ 2. Otherwise, he will simply dispose the entire production in the secondary maret with the loss of λ 2. Theorem 7 If (1 e λ2 ) < < (1 e 1 ) and f r is as in (21), then 1 (i) when 0 < µ, AP(w , w 2, ) can be used to coordinate the supply chain; 1 (ii) when 1+ + the supply chain; (iii) when µ λ < µ < production in the secondary maret. Where w 2 = c + 1+µ Proof Note that f s = 1+µ λ When µ < loss. From λ , CLPP(w 2, ) can be used to coordinate 1, the best choice for the supplier is disposing of the entire + (1 + ) and w 1 is sufficiently large. e (1+c) (1 + + ) λ 2 is equivalent to µ 1. This completes the third part of the theorem. λ we obtain w 2 = c + 1+µ 1, the supplier can devise a wholesale price policy to reduce his f s = 1 + µ e (1+c) (1 + + ) = 3 (w 2 c) = (w 2 c) = e (1+c) (w 2 c), + (1 + ). For this value of w 2, the profit of retailer is f r () = ( 1 + w 2 ), which is maximized at 1 = ( + )e (1+w2) 1. When 0 < µ, it can be seen 1+ + that 1. Remember that the retailer s order quantity should be no less than if he wants to tae price w 2. By the concavity of f r, the retailer should order in order to maximize his profit if he wants to tae the price w 2. Because w 1 is very large, the retailer cannot gain more profit by taing price w 1. So the retailer s ordering is exactly, which implies the supply chain is coordinated by AP(w 1, w 2, ). When < µ < λ , it can be shown that 1 >. From the definition of CLPP(w 2, ), the retailer cannot order more than. Again from the concavity of f r, the retailer should order in order to maximize his profit. This means the supply chain is coordinated by CLPP(w 2, ) policy. 4.4 Case 4: (e λ2 1) Recall that the maximum supply chain profit for this case is f 4 = 1 ( + )e (1+(c λ2)) λ 2 e (1+c). (23)

12 666 ACTA MATHEMATICA SCIENTIA Vol.26 Ser.B Suppose the desired profit of the retailer is Thus the supplier s profit is f r = 1 ( + )e (1+(c λ2)) µλ 2 e (1+c). (24) f s = (µ 1)λ 2 e (1+c). (25) In order that the retailer participates in the game, f r should satisfy f r > 0 or equivalently µ < + λ 2 eλ2. On the other hand, f s should satisfy f s > λ 2 = λ2 e (1+c) or equivalently µ > 0. When µ + λ 2 eλ2, the retailer cannot earn a positive profit. As a result, he will refuse to order any quantity of product and the supplier has to dispose all his production at the secondary maret. Similarly, when µ 0, the supplier will also dispose all his production at the secondary maret to avoid more loss. Except for the above two situations, the supplier can use a proper wholesale price policy to coordinate the supply chain. Theorem 8 For the case (e λ2 1), suppose f r is defined as in (24), then (i) when 0 < µ < + λ 2, the supply chain can be coordinated by AP(w eλ2 1, w 2, 4 ); (ii) when µ + λ 2 or µ 0, the supplier will sell his entire product at the eλ2 secondary maret. Where w 2 = c λ 2 + µλ2 + and w e λ2 1 is sufficiently large. Proof Note that On the other hand, f s can be expressed as f s = (µ 1)λ 2 e (1+c). (26) f s = 4(w 2 c) λ 2 ( 4) = 4(w 2 c + λ 2 ) λ 2. (27) From (26) and (27), it follows that w 2 = c λ 2 + µλ 2 + e λ2. For this value of w 2, the profit of retailer is f r () = ( 1 which is maximized at 1 = ( + )e (1+w2). + w 2 ), When µ > 0, we have w 2 > c λ 2. It follows that 1 < 4. By the strict concavity of f r (), the retailer should order 4 in order to maximize his profit. That maes the supply chain coordinated. The second part of the theorem follows directly from the analysis before this theorem. 4.5 Effects Analysis of isruption Management Now we analyze the effect of the disruption management studied above. If the supplier fails to react to the disruption and eeps the given policy AP( w 1, w 2, ), then the retailer s profit for wholesale price w 2 is f r () = ( 1 + w 2 ),

13 No.4 Huang et al: ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION 667 which is maximized at 1 = ( + )e (1+c+η). When > (e η 1), it can be seen that 1 >, thus retailer will order 1 to maximize his profit. If w 2 < c + λ 1, the supplier s projected profit f s would be reduced by c + λ 1 w 2 for each unit product above. However, ordering more than is always beneficial for the retailer because the wholesale price remains unchanged. Next, we consider the coordination of the supply chain. Note that only if > (e λ1 1) does 1 > hold and we have obtained in the above that when > (e η 1), 1 >. This means that only when 1 = 1, or equivalently, η = λ 1, the supply chain is coordinated. Note that even in this case, the supplier would miss a opportunity for profit increase by failing to adjust the wholesale price accordingly. If (e η 1), we have 1. Supposing w 1 is too high for the retailer to tae, the the retailer has two choices. When 1 + > w 2 = c+ η, or equivalently, > (eη 1 1), he will order to maximize his profit. In this case, the supplier s target profit is achieved exactly because this is what he planed to sell before the disruption. If > 0, some extra profit will go to the retailer. If < 0, the retailer s profit is less than what was projected. When (e η 1 1), 1 w 2. This means the retail price is even lower than the wholesale price and retailer would withdraw from the game and leave the supplier with a huge loss λ 2. If the supplier could react to the disruption properly, he would adjust his wholesale policy to induce the retailer to order some amount of product and reduce his loss to a certain extent. + 5 Numerical Example In this section, we give Several numerical examples to illustrate some of the results derived throughout the paper. First, consider the situation without disruption. Let c = 1, = 100, and = 0.2. In this case, the optimal retail price is p = 6, the production quantity = 30.12, sc and the total profit of the whole supply chain f max = Then, consider how the supplier can coordinate the supply chain. Suppose the supplier wishes to tae 60% of the total profit, that is η = 0.6. From Theorem 1, we now that w 2 = 4 and w 1 > 5.58, so the supplier can use the quantity discount policy AP(6, 4, 30.12) to coordinate the supply chain. It can be verified that the best choice for the retailer is taing wholesale price w 2 = 4, order quantity = 30.12, and set the retail price to p = 6. His profit is under such decision, which is exactly 40% of the total profit. This means the supply chain has been coordinated. Next, consider the centralized supply chain with demand disruption. Let = 40 and λ 1 = λ 2 = 1. From Theorem 2, we now the optimal retail price is p = 7, the production quantity is = 34.52, and the total profit of the supply chain is f = If the disruption was not responded to correctly and the original retail price p = 6 remained unchanged, then a total product would be sold, giving a total profit f sc = It is inferior to the optimal solution. Now, consider the case of decreased maret scale, where = 40. Again, from Theorem 2, we now the optimal retail price is p = 5, the production quantity is = 22.07, and the total profit of the supply chain is f = If the original retail price p = 6 were ept unchanged, then a total product would be sold, giving a total profit f sc = It is less than the value of the optimal solution.

14 668 ACTA MATHEMATICA SCIENTIA Vol.26 Ser.B Finally, we show how the supply chain can be coordinated under demand disruption. Continue the above example with = 40 and suppose that the supplier still wishes to tae 60% of the total profit. According to Theorem 3, we obtain that w 2 = 4.65 and w 1 > 5.78, thus the quantity discount policy AP(6, 4.65, 34.52) can coordinate the supply chain. The best choice for the retailer is to tae the wholesale price w 2 = 4.65, order quantity = 34.52, and set the retail price to p = 7. His profit is f r = Now, consider again the case where = 40. The maximum profit of the supply chain is f = Suppose the retailer wishes to achieve a profit of 50, from Theorem 8 we now the supplier can coordinate the supply chain by using quantity discount policy AP(6, 2.73, 22.07). Under such policy, the retailer s best decision is taing the wholesale price w 2 = 2.73, order quantity = 22.07, and set the retail price to p = 5. If the retailer is so greedy that he is seeing a profit greater than , the best decision for the supplier is disposing all his inventory in the secondary maret to avoid a greater loss. 6 Further Extensions In the previous sections, we have studied the coordination problem of supply chain with demand disruption, in which the exponential demand function is used. For different maret environments and different products, the demand functions may be in different forms. Fortunately, AP(w 1, w 2, q 0 ) and CLPP(w, q) wholesale pricing policies could also be used to coordinate those supply chains with other demand functions if the demand functions = g(p, ) satisfy the following assumptions. Here p,, and represent retailer price, demand, and maret scale, respectively. Assumption (1) = g(p, ) is a strictly decreasing function with respect to p ; (2) The profit function of the supply chain f() = (p c) = (g 1 (, ) c) is unimodal where p = g 1 (, ) represents the inverse of = g(p, ) and c the unit production cost. From assumption (1) and (2), similar results can be derived as in this article. Now we establish a sufficient condition for the unimodality of f(). Note that is a parameter, to avoid confusion, f() is rewritten as f() = (p c) = (g 1 () c). We now that and f () = 2 g (p) f () = (g 1 () c) + 1 g (p), (g (p)) 3 g (p) = 2 g (p) g(p) (g (p)) 3 g (p). If g (p) < 2 g(p) (g (p)) 2, we can easily obtainf () < 0. It follows that f() is strictly concave, which implies that f() is unimodal. Proposition Let = g(p) be the demand function that satisfies assumption (1). If it also satisfies the inequality g (p) < 2 g(p) (g (p)) 2, then the profit function of the supply chain is strictly concave. Analysis above implies that AP(w 1, w 2, q 0 ) and CLPP(w, q) quantity discount policy might be used to coordinate the supply chains with a variety of demand functions which correspond to different maret situations and even in the disrupted circumstances. We would lie

15 No.4 Huang et al: ISRUPTION MANAGEMENT FOR SUPPLY CHAIN COORINATION 669 to point out that the function e p used in this paper and p, p 2 used in [5] and [12] satisfy both Assumption (1) and (2). 7 Conclusions In this article, we introduced a new ind of demand function and investigate the corresponding supply chain coordination problem with demand disruptions. From the analysis above, we could draw a conclusion that if the supplier is able to adjust his wholesale policy according to the maret demand disruption in a timely and properly manner, he will have more opportunities to increase his profit and more chances to coordinate the supply chain. Furthermore, we found that the same wholesale price discount policies can also be used to coordinate general supply chains with disruptions as long as the corresponding demand functions satisfy certain assumptions. References 1 Weng K. Channel coordination and quantity discounts. Management Science, 1995, 41: Weng K, Zeng A-Z. The role of uantity discounts in the presence of heterogeneous buyers. Annals of Operations Research, 2001, 107: Chen F, Federgruen A, Zheng Y-S. Coordination mechanisms for a distribution system with one supplier and multiple retailers. Management Science, 2001, 47: Zhao W, Wang Y. Coordination of joint pricing-production decision in a supply chain. IIE Transactions, 2002, 34: i X, Bard J F, Yu G. Supply chain coordination with demand disruptions. Omega, 2004, 32: Thengvall B, Bard J F, Yu G. Balancing user preferences for aircraft schedule recovery during irregular operations. IIE Transactions, 2000, 32: Clausen J, Hansen J, Larsen J, Larsen A. isruption Management. ORMS Today, 2001, 28: Howic S, Eden C. The impact of disruption and delay when compressing large projects: going for incentives? Journal of the Operational Research Society, 2001, 52: Williams T, Acermann F, Eden C. Structuring a delay and disruption claim: an application of causemapping and system dynamics. European Journal of Operational Research, 2003, 148: i X, Bard J F, Yu G. isruption management for machine scheduling: the case of SPT schedules. Woring paper, epartment of Management Science and Information Systems, McCombs School of Business, The University of Texas, Austin, TX, Xiao T, Yu G, Sheng Z, Xia Y. Coordination of a supply chain with one manufacturer and two retailers under demand promotion and and disruption management decision. Annals of Operations Research, 2005, 135: Xu M, i X, Yu G, Zhang H, Gao C. The demand disruption management problem for a supply chain system with nonlinear demand function. Journal of Systems Science and Systems Engineering, 2003, 12: 82 97