Journal of Service Science and Management, 2011, 4, 2734 doi:10.4236/jssm.2011.41005 Published Online March 2011 (http://www.SciRP.org/journal/jssm) Copyright © 2011 SciRes. JSSM 27 Optimal Capacity Expansion Policy with a Deductible Reservation Contract Jianbin Li1, Minghui Xu2*, Ruina Yang3 1School of Management, Huazhong University of Science and Technology, Wuhan, China; 2* School of Economics and Management, Wuhan University, Wuhan, China; 3Department of Industrial Engineering and Logistics Management, The Hong Kong University of Science and Technology, Hong Kong, China. Email: jimlee@amss.ac.cn, xu_mingh@yahoo.com.cn, rnyang@ust.hk Received October 18th, 2010; revised November 27th, 2010; accepted December 2nd, 2010. ABSTRACT This paper investigates an optimal capacity expansion policy for innovative product in a context of one supplier and one retailer. With a fully deductible contract, we employ the Stackelberg game model to examine the negotiation proc ess of capacity expansion in a single period. We first derive the retailer’s optimal reservation strategy and then char acterize the optimal capa city expansion policy for the supplier. We also investigate the impacts of reservation pr ice on the optimal strateg y of capacity reservation and expansion as well as th e supplier’s expected profits. Keywords: Supply Chain, Capacity Expansion, Deductible Reservation Contract, Stackelberg Game 1. Introduction This paper is concerned with capacity expansion policy for innovative product in a setting of one supplier and one retailer. Evidently, capacity management is an im portant issue for innovative product, which is often char acterized by volatile demand, short life cycle and long lead time. In fact, due to the highly volatile demand, the supplier often suffers from capacity shortage with the adoption of exact capacity expansion policy. Therefore, the retailer also loses revenues and his market reputation is damaged. Despite the need for higher revenue and im proved service levels, the supplier may not be ready to expand capacity proactively because of financial risks due to higher capacity cost, long (capacity) lead time and high demand volatility. However, if the retailer agrees to share the financial risks by forward reservation, then the supplier may be motivated to expand capacity more ag gressively. In this paper, the retailer reserves a capacity prior to demand realization, and in exchange, the supplier commits to have the “excess” capacity in addition to the reservation amount. This kind of capacity expansion pol icy provides a winwin situation for both the supplier and the retailer. In the paper, we employ the fully deductible contract: the retailer pays a fee upfront for each unit of capacity reserved. When the retailer actually utilizes the reserved capacity (i.e., placing a firm order), the reservation fee is deductible from the order payment. However, if the re served capacity is not fully utilized within the specified time period, the reservation fee associated with unused capacity is not refundable. Interestingly, supplier’s an nouncement of excess capacity is a unique feature of the deductible reservation (DR) contract. We consider a twolevel supply chain in which a sup plier offers an innovative product to one retailer facing a stochastic demand. Throughout the paper, we assume that the reservation price of the DR contract is exoge nously determined. Obviously, the negotiation process for capacity expansion policy can be described as a Stackelberg game in which the supplier is the leader and the retailer is the follower. The objective of the current paper is to design an appropriate capacity expansion pol icy that allows both the supplier and the retailer to opti mize their expected profits. Specifically, with an exoge nously given reservation contract, we firstly analyze the retailer’s optimal strategy, and then study the supplier’s optimal capacity expansion policy. Finally, we illustrate the impacts of reservation price on the optimal capacity expansion policy and provide with some managerial in sights. The literature on capacity reservation is fairly abun dant. There are some earlier literature related to capacity reservation mainly discuss the retailer’s optimal strategy.
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 28 Sample references are [13]. Moreover, the work in this filed can be divided into two main categories in terms of retailer’s motivations to reserve capacity. The first cate gory considers the case in which the retailer reserves a certain portion of the future capacity to achieve potential cost reduction, for the sample references we refer readers to [410]. The second category, including [1113], studies the problem that the retailer is motivated to offer early com mitment on the future capacity so as to ensure a certain level of production availability. Moreover, Cachon and Lariviere [14] investigate capacity contracting in the context of supplierbuyer forecast coordination. Murat and Wu [15] show that, by fully deductible reservation contracts, the supplier has the incentive to expand the capacity proactively. They conclude that as the buyer’s revenue margin decreases, the supplier faces a sequence of four profit scenarios with decreasing desirability. Jin and Wu [16] propose a capacity expansion policy that the supplier will have excess capacity in addition to reserva tion amount from the buyer. With a deductible reserva tion contract, they show that supply chain coordination can be achieved and both players benefit from supply chain coordination. Evidently, our work on capacity expansion policy for innovative product mainly differs from earlier work in four aspects. First of all, the papers reviewed above mainly discuss the retailer’s decisionmaking behavior, with little concern on supplier’s perspective. We investi gate the supplier’s optimal strategy on capacity expan sion policy in addition to the retailer’s optimal strategy. Secondly, most existing papers consider endogenous wholesale price. However, in this paper we assume that the wholesale price is determined exogenously by the market or by earlier negotiations. Thirdly, the papers re viewed above assume that the supplier does not build any capacity without retailer’s upfront commitment. How ever, with knowledge of market demand information, the supplier has the incentive to build capacity even without retailer's commitment. Finally, different from the per spective of supply chain coordination, we pay our atten tion on the players’ interactions through modeling the process as a Stackelberg Game to derive optimal capacity expansion policy. The rest of this paper is organized as follows. Section 2 presents the model. Section 3 discusses the retailer’s optimal reservation strategy with an exogenously given reservation contract, and then describes the optimal ca pacity expansion policy for the supplier. Section 4 inves tigates the impacts of reservation price on the optimal capacity reservation and expansion policy. Finally, Sec tion 5 concludes the paper. 2. Model Description We consider a twoechelon supply chain in a single pe riod, in which a supplier (called her) sells an innovative product to a retailer (called him). The retailer faces a stochastic demand D, with the probability density func tion and cumulative distribution function . Assume that the two parties hold symmetrical informa tion about the market demand and the cost structure. The initial capacity level of the supplier is assumed to be zero. In the model, we propose a capacity reservation contract with fully deductible payment and assume that the reser vation price r is an exogenously given constant parame ter. To encourage the retailer to reserve capacity more readily, we let rw , where w is the unit purchasing price charged by the supplier. The sequence of the events is as follows: 1) At stage 0, the supplier announces the excess capac ity E, which is the amount of capacity the supplier pre pares to have in addition to (and regardless of) the re tailer’s reservation amount R. 2) Based on the excess capacity E and the demand forecasting information, the retailer decides the reserva tion amount R and pays rR to the supplier. 3) After receiving R, the supplier expands her capacity to R + E with marginal cost c. 4) At stage1, the demand D is realized, then the re tailer places an order min ,DR E, with the unit pur chasing cost w. The selling price for each product is p and any unmet demand will be lost. 5) The supplier deducts the amount of min ,rDR from the retailer’s purchasing cost, but keeps the amount max, 0rRD. 6) The supplier salvages the residual capacity with unit salvage value s. Obviously, the above negotiation process for capacity expansion policy can be modeled as a Stackelberg game, in which the supplier is the leader and the retailer is the follower. The supplier has complete visibility to the re tailer’s decisionmaking process. Suppose the two parties in the supply chain are riskneutral. The aim of this paper is to characterize the optimal capacity expansion policy that allows both the supplier and the retailer to maximize their respective expected profits. In order to avoid trivial cases, we assume that cwp . As salvaging re sidual capacity will incur additional logistical and proc essing costs, we assume that the salvage value s is strictly less than the capacity expansion cost c. 3. The Optimal Capacity Expansion Policy 3.1. The Optimal Strategy for the Retailer As reservation price r of the fully deductible reservation Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 29 contract is an exogenous given constant parameter, the retailer offers an early commitment on a certain portion of future capacity just before the supplier expands capac ity. When the stochastic demand is realized, the retailer places an order and the reservation cost can be deducted from the purchasing cost. In this situation, at stage 0 with the excess capacity E offered by the supplier, the retailer determines the optimal reservation amount ˆ RE that maximizes his expected profit. 0 0 ˆmin ,min , d d. M R RE REp wDR ErR rDR pwRERx Fx pwREx Fx (1) The first term on the righthand side of Equation (1) denotes the retailer’s profit from selling the innovative product, the second and the third term represents the re tailer’s effective reservation cost paid to the supplier so as to ensure a certain level of availability. Evidently, the more the effective reservation cost is, the higher the available capacity level will be in future. Lemma 1 Given the supplier’s excess capacity E, there exists a unique optimal reservation amount ˆ RE to maximize the retailer’s expected profit, which is deter mined by ˆˆ 1rF REpwF REE . (2) Proof. For any given E, taking the first and second de rivatives of with respect to R, we get that ˆMR 2 2 ˆ 1 ˆ 0. M M Rp wFRErFR R RpwfRE rfR R , This implies that is strictly concave in R, and hence the optimal reservation amount ˆMR ˆ RE is uni quely determined by the first order condition given in Equation (2). Lemma 1 shows that, for a given E, the retailer’s op timal strategy is to reserve , which is uniquely determined by Equation (2). It follows from Lemma 1 that the optimal reservation amount ˆ RE ˆ RE is mono tonically decreasing in E. To see this, differentiating Equation (2) on both sides with respect to E and rear ranging items, we get that ˆˆˆ rfR EpwfR EER EEp w. , which implies that ˆ0fRE E ˆ0 RE E. Therefore, for a certain level of future demand, the larger the excess capacity is, the smaller the possibility of disruptions in future supply will be. Since the supplier prepares to set a higher level of excess ca pacity (i.e., increasing E), the retailer will reserve less. In this scenario, the supplier undertakes more financial risks in contrast with the retailer. Furthermore, we can see that ˆ RE 0 (and hence ) as ; and ˆ0 RE0E FRE ˆ pw pwr as . E 3.2. The Optimal Capacity Expansion Policy for the Supplier In anticipation of the retailer’s optimal response behavior for any given E, we proceed to investigate the supplier’s optimal capacity expansion policy that maximizes her expected profit. By taking the retailer’s response function ˆ RE into account, the supplier’s expected profit can be expressed as ˆ ˆmin , ˆˆ ˆˆ min,. SEEw DREE sREE DrRE rDREcREE (3) where ˆ RE is an implicit function of E given in Equa tion (2). On the right hand side of Equation (3), the first term is the supplier’s revenues from delivering the inno vative product to the retailer; the second term denotes the supplier’s revenues from salvaging the residual capacity; the third and the forth terms represent the retailer’s effec tive reservation cost paid to the supplier and the last term is the cost of expanding capacity. To derive an explicit expression of and make future analysis easier, throughout the paper we mainly discuss the scenario that the customer demand is uni formly distributed (other distributions can be analyzed similarly). ˆ RE We assume that the customer demand D is uniformly distributed over the interval 0, with 0 . Note that the assumption of uniform distribution is a simplifi cation of reality, but it is sufficient to capture the main features of capacity reservation policy and derive mana gerial insights in practice. Specifically, from Lemma 1 we get that, ˆ, pw E RE pwr (4) and the total capacity of the supplier after capacity ex pansion is ˆ. pw rE RE Epwr (5) Different values of E represent different capacity ex Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 30 pansion strategies. means exact capacity expan sion policy; represents aggressive capacity ex pansion policy with potentially higher gains and 0E 0E 0E represents overbooking. However, credibility with the retailer is crucial for the supplier in the industry, so overbooking is not considered as an acceptable business. We will only consider the case where . Obviously, the total capacity should be no more than the maximum possible demand 0E , i.e., ˆ RE E , which turns out to be E . Therefore, we will confine our analysis on 0,E in the rest of the paper. By Equation (2), the supplier’s objective function can be reformulated as: 22 ˆ ˆ ˆˆ . 22 SEwcREE RE ERE ws r (6) Now, we characterize the properties of supplier’s ex pected profit function with an exogenously given reser vation price r ( 0,rw), which are stated in the fol lowing lemma. Lemma 2 Let 12pps , 1pw ws , then we have the following results. 1) If 1 wp , then is convex in E for any ˆSE 0,rw; 2) If 1 wp , then is convex in E for any ˆE S ˆE p w w 1 0,r , and is concave in E for any S 1,rpw . Proof. Since ˆ REEpwp wr and ˆ REEErp wr by Equation (4), we get from Equation (6) that the first and the second deriva tives of with respect to E are ˆSE 2 2 ˆˆˆ ˆ ˆ ˆ SRE E ERER wc r EEE RE E RE E ws E rs crE pwr pwr pw wsr E (7) and 2 2 22 ˆ . Srw sp ww sr E Epwr Obviously, we know that if 2 rpwws 1pw , then we have 2 ˆ0 SEE 2 ; and if 1 rp w , then 22 ˆ0 SEE . Since 0rw p, we know that at 1pw w 1 w . Therefore, we get that 1) 1 pw w when 1 wp , and hence ˆSE is convex in E for any 0,rw; 2) pw w 1 when 1 wp , and hence ˆSE is convex in E for any given 1 0,rp w and ˆSE is concave in E for any given 1,rpw w . Lemma 2 indicates that if the purchasing price is no more than 1p , then with any exogenously given r, the supplier’s expected profit is decreasing in E as the excess capacity is smaller than a critical point while increasing in E as the excess capacity exceeds the critical point. On the other hand, if the purchasing price is larger than 1p , then when r is not greater than a threshold w 1, the supplier’s expected profit is decreasing in E as the excess capacity is smaller than a critical point while in creasing in E as excess capacity exceeds the critical point; when the reservation price exceeds the threshold, the supplier’s expected profit is increasing in E as the excess capacity is smaller than a critical point while decreasing in E as excess capacity exceeds the critical point. p Following from Lemma 2, we can obtain the supplier’s optimal level of excess capacity with any given reserva tion price 0,rw. Proposition 1 Let 22cs ps and 2)pwcswc . 1) If pw )( 21 , then the supplier will set the optimal excess capacity ˆ0E ; 2) If wp 12 , then 2 2 2 1 0, if 0,, ˆ ,if ,. () rpw Ewcr pwrpww wsr pw Proof. From Equation (7), we have that 2 0 2 2 2 ˆ , S E Ersc r Epwr pwr pw wsr rw crpw pwr ˆ 0. S E Ersc Epwr We consider the following cases by noting that 12 . Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 31 1) 1 wp . For this case, we get that 1 wpw and hence is convex in for any given ˆSEE 0,rw p (by Lemma 2). Moreover, 1 w implies that 2 pw 0wsr. Then, it follows from Equation (7) and c that ˆ 0 EE S in 0,E ˆ E . Therefore, the supplier obtains her maximum profit at . 0 2) . 11 p 2 For this case, we get that pw 12 pw wpw ˆEE . From Lemma 2, we know that is convex in S with a given . Similar to case (a), we 1 0,rp w get that the optimal excess capacity is . ˆ0 E ˆ For any given , is con 1,rpw w SE cave in E (Lemma 2(2)). Since , we 2 rw pw know that 0 ˆ0 SE EE , which implies that the optimal excess capacity is ˆ0 E . 3) . 12 wp w 12 p implies that 21 wpw p . w Following from Lemma 2, we know that ˆES is convex in E when 1 0,rp ,wpw w and concave in E when . For both cases, 12 rp since is decreasing over ˆSE 0,E by noting that 0 ˆS 0 E EE , we can conclude that the op timal excess capacity is . ˆ0 E When , is concave in E, 2,rpw w ˆSE and 0 ˆ0 SE EE . Therefore, the optimal excess capacity can be derived from the first order condi tion determined by Equation (7), which is ˆ E 2 1 ˆ. wcr pw Ewsr pw In conclusion, 1) when , the opti mal excess capacity is for any exogenously given 12 0wp 0 ˆ E 0,rw; and 2) when 12 wp , we have 2 2 2 1 0,if 0,, ˆ ,if ,. rpw Ewcrpwrpw wsrpw w The proof is completed. Obviously, Proposition 1 clearly implies that the opti mal capacity expansion policy for the supplier is to adopt the exact capacity expansion policy if 12 w p . Moreover, the intuition underlying Proposition 1 is clear. If the purchasing price is no more than a threshold , which means that the retailer’s marginal profit by selling one unit of innovative product is larger than a critical value 12 p 12 1p ˆ0E , then the retailer has an incentive to reserve a larger amount capacity be cause the revenue loss due to capacity shortage is very big, and the retailer is willing to undertake more financial risk for capacity expansion to ensure a higher level of capacity availability. By observing this, the supplier be lieves that the retailer’s reservation amount is large enough to meet the future demand and thus take the exact capacity expansion policy with . On the other hand, if the purchasing price exceeds the threshold 12 p , which means the retailer’s mar ginal profit is smaller than the critical value, then the retailer is encouraged to reserve more to ensure a higher level of capacity availability in future with a smaller res ervation price pw 2. In this situation, the sup plier will also adopt exact capacity expansion policy with r ˆ0E . However, when the reservation price is larger than pw 2 , the retailer will reserve less. To avoid future capacity shortage, the supplier will expand the capacity aggressively with . Therefore, the sup plier and the retailer’s optimal strategies with any ex ogenous constant ˆ0E 0, wr can be summarized as the following proposition. Proposition 2 1) If 12 wp , then ˆ0E and ˆˆˆ RREpw pwr . 2) If 12 wp , then 2 2 0, , ˆ , . pw rpw pw Rpw rpww wsr 2 ,if ,if r cs p w 2 2 2 1 0, if ,if 0,, ˆ ,. rpw Ewcrpwrpww wsrpw Hence, the supplier’s total capacity is ˆˆ RE p wpwr if p w 2 0,r ; and 22 r pw ˆˆ RE wcs r pww if 2,rpw w . Proof. The results follow directly from Proposition 1 and Equation (4). 4. The Impact of Reservation Price In this subsection, we investigate the impacts of reserva tion price r on the optimal capacity reservation policy. Specially, we would like to show how r affects the opti mal excess capacity , the retailer’s optimal reservation amount and the supplier’s capacity level ˆ E ˆ R Rˆˆ . Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 32 4.1. The Case 12 wp The following proposition presents the results of com parative statics of reservation price. Proposition 3 If , then 1) is de 12 wp 0,rp w ˆ R ˆ R creasing in r over ; and 2) is de 2 ˆˆˆ creasing in r and and ERE are both increasing in r over . 2,rpw w Proof. For part 1), we know that and it is easy to see that ˆ0E ˆˆˆ RREpw pwr w is decreasing in r. We now consider part 2). For , 2,rpw we know that 2 ˆ Rpwcswsrpw 11 cs rpw (Proposition 2). Then, 1 2 1 ˆ d0. d cs R rrpw (8) By noticing that 21 , it follows from Proposition 2 that for 2,rpw w 21 2 1 ˆ d0. d wc pw E rws rpw w (9) which indicates that is increasing in r over . Combining the above two equations and rearranging items, we get that ˆ E 2,rpw 2 2 2 1 ˆˆ d 0. d RE pw cs rws rpw (10) Hence, is creasing in r for the case ˆˆ RE 1 w and . 2p 2,rpw w Proposition 3 shows that, under the condition that the purchasing price w is larger than and the reservation price r exceeds 2, the supplier’s optimal excess capacity increases while the retailer’s optimal reservation amount decreases as r increases. This is because the retailer takes more risks of over reserva tion and the supplier benefits more from the capacity reservation. Even though, the supplier’s optimal capacity level is still increasing in r since the decreasing rate of is smaller than the increasing rate of . 12 p pw ˆˆ RE ˆ Rˆ E Corollary 1 If , then for 12 wp w 2,rpw , the supplier’s optimal expected pro fit is decreasing in r. ˆ SE Proof It follows from Proposition 2 and Equation (10) that 2 2 2 2 2 1 22 2 2 11 42 3 3 1 ˆˆ ˆˆ d d . wsRE RE wc r wcrpw wc wswsrpw pw cs ws rpw pw cspw cs wsrpw ws rpw pw cs ws rpw From Equation (8) and Proposition 2, we have 1 22 1 22 3 2 1 ˆˆ d d . rp wc sc s rR R rwsrpwrpw rp wc s ws rpw Therefore, we get that 2 42 3 3 1 22 3 2 1 22 2 2 1 ˆˆˆ dd dd ˆˆ ˆ d d2 2 SEwsRE wc rr rR RR r pw cs ws rpw rp wc s ws rpw pw cs ws rpw ˆˆ RE 22 2 2 1 0. 2 pw cs ws rpw Hence, when 12 w p ˆ SE, is decreasing in r for ,rpww 2 . From Corollary 1 we get that the retailer’s optimal expected profit ˆ MR is increasing in r by noting that the total supply chain profit is unrelated to r. In this situation, the high reservation price results in much low reservation amount of capacity. Consequently, the sup plier needs to build excess capacity to match the demand Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract 33 in the future. However, the increasing costs of building up the capacity have a negative effect on the supplier’s optimal expected profit, that is, the supplier’s optimal expected profit is decreasing in r. Although, the expected profit for the entire supply chain can be increased since it will alleviate the effects of double marginalization. In view of this point, it is advised that the supplier should not to choose a higher reservation price in a decentralized supply chain. If we choose the reservation price r in the interval 2 0, pw ˆ , then it follows from Proposition 2 that 0E. Taking derivative of with respect to r, we get that ˆ SE 2 2 22 3 ˆˆ dˆˆ d dd 2 2 222. SEwsrR RR wc rr wsr pw wc pwr pw pw pwr pwr pw pwr pw crpwpw cs 2 (11) Let 322 2pw cs pw c . It can be ve rified that 23 , therefore, for this case, 0 S is increasing for smaller r () and decreasing for larger r (32 ). For smaller r, the supplier can benefit from the reservation since the retailer decreases his amount of reservation capacity and undertakes some risk of higher demand as r increases. However, for larger r, the supplier may lose some profit when r is increased since the reservation capacity is de creased too much. 3 rp r pw w pw In summary, the optimal profit for the supplier is uni modal in r when 12 w 3 r p w and the optimal res ervation price can be set to p . 4.2. The Case 12 wp When 12 w ˆ0Ep , the supplier will not expand her capacity, i.e., . For the retailer, the optimal reser vation amount is ˆ Rpwpwr . If the res ervation price approaches zero, the retailer would set the reservation amount at the highest possible demand λ. However, as r increases, the retailer’s optimal reservation amount will decrease. Proposition 4 If 12 wp , then (and ) is decreasing in r. And the supplier’s expected profit ˆ R ˆˆ RE )0( S is unimodal. Proof. Since ˆ0E , it is easy to see that ˆˆˆ E RR pwrpw is decreasing in r. The derivative of 0 S with respect to r is the same as Equation (11). By noting that 23 and 2pw w , it follows from Equation (11) that, if 3pw w , then 0 S is increasing in r over 0, w; and if pw 2 3 wp w, then 0 S is increasing in r over pw 3 0, and decreasing in r over pw 3,w . For the case 12 w p 0 S , is increasing for smaller r ( 3 rpw ) and decreasing for larger r ( 3p r). For smaller r, the supplier can benefit from the reservation since the retailer decreases his amount of reservation capacity and undertakes some risk of higher demand as r increases. However, for larger r, the supplier may lose some profit when r is increased since the reservation capacity is decreased too much. w In the previous analysis, we implicitly assume that the supplier always accept the retailer’s capacity reservation. However, the deductible reservation contract can be conducted only if the supplier could earn some profits. Now we identify the condition under which the supplier has an incentive to accept the retailer’s capacity reserva tion. For this case, if the supplier accepts the retailer’s reservation, then the supplier’s optimal expected profit is 2 ˆ ˆ 2 ˆ 2 ˆ 2 2. R wc R wsr wsr pw Rwc pwr R pwr pw crwcspw 0 2 S Therefore, the supplier accepts the retailer’s reserva tion only if 22pw crwcspw 0 , i.e. , 22rcwspwpw w c c . This condition holds if the purchasing cost is high relative to the capacity building cost . Then the supplier has an incentive to accept the retailer’s capacity reservation; otherwise, the supplier will raise the reservation price r or unit purchase price w so that she can obtain some profits. 5. Concluding Remarks Capacity management plays a significant role on innova tive product. In the paper, the capacity expansion policy not only provides with a risksharing mechanism for both the supplier and the retailer, but also improves the re tailer’s potential revenue. Specifically, we propose a fully deductible contract where the retailer reserves fu ture capacity with a fee that cab be deducted from the Copyright © 2011 SciRes. JSSM
Optimal Capacity Expansion Policy with a Deductible Reservation Contract Copyright © 2011 SciRes. JSSM 34 purchasing price. Additionally, the supplier’s ex ante an nouncement of “excess” capacity is a unique feature of the deductible reservation contract. Given the reservation contract, we figure out the optimal capacity expansion policy and also study the effects of reservation price on the optimal strategy as well as the supplier’s optimal profit. Finally, we address the issues of how to set the reservation price from the perspective of the supplier in different situations. 6. Acknowledgements This research was partially supported by the NSFC Grant Nos. 70901029 and 70901059, and the Fundamental Re search Funds for the Central Universities (Grant No. 105275171). REFERENCES [1] K. Jain and E. A. Silver, “The Single Period Procurement Problem Where Dedicated Supplier Capacity Can be Re served,” Naval Research Logistics, Vol. 42, No. 6, 1995, pp. 915934. doi:10.1002/15206750(199509)42:6<915::AIDNAV322 0420605>3.0.CO;2M [2] A. Brown and H. Lee, “Optimal PaytoDelay Capacity Reservation with Application to the Semiconductor In dustry,” Working Paper, Vanderbilt University, Nashville, Tennessee, 1998. [3] G. P. Cachon, “Supply Chain Coordination with Con tracts,” S. Graves and T. de Kok Eds., The Handbook of Operations Research and Management Science ， Supply Chain Management, Amsterdam, The Netherlands, 2003. [4] J. S. Bonser and S. D. Wu, “Procurement Planning to Maintain Both ShortTerm Adaptiveness and LongTerm Perspective,” Management Science, Vol. 47, No. 6, 2001, pp. 769786.doi:10.1287/mnsc.47.6.769.9814 [5] D. A. Serel, M. Dada and H. Moskowitz, “Sourcing De cisions with Capacity Reservation Contracts,” European Journal of Operational Research, Vol. 131, No. 3, 2001, pp. 635648. doi:10.1016/S03772217(00)001065 [6] R. W. Seifert, U. W. Thonemann and W. H. Hausman, “Optimal Procurement Strategies for Online Spot Mar kets,” European Journal of Operational Research, Vol. 152, No. 3, 2004, pp.781799. doi:10.1016/S03772217(02)007543 [7] D. J. Wu, P. R. Kleindorfer and J. E. Zhang, “Optimal Bidding and Contracting Strategies for CapitalIntensive Goods,” European Journal of Operational Research, Vol. 137, No. 3, 2002, pp. 657676. doi:10.1016/S03772217(01)000935 [8] V. MartinezdeAlbeniz and D. SimchiLevi, “A Portfolio Approach to Procurement Contracts,” Production and Operations Management, Vol. 14, No. 1, 2005, pp. 90 114. doi:10.1111/j.19375956.2005.tb00012.x [9] O. Ozer and W. Wei, “Strategic Commitments for an Optimal Capacity Decision under Asymmetric Forecast Information,” Management Science, Vol. 52, No. 8, 2006, pp. 12381257. doi:10.1287/mnsc.1060.0521 [10] Q. Fu, C. Y. Lee and C. P. Teo, “Procurement Manage ment Using Option Contracts: Random Spot Price and the Portfolio Effect,” IIE Transactions, Vol. 42, No. 11, 2010, pp. 793811. doi:10.1080/07408171003670983 [11] G. Eppen and A. Iyer, “Backup Agreements in Fashion Buying  the Value of Upstream Flexibility,” Manage ment Science, Vol. 43, No. 11, 1997, pp. 14691484. doi:10.1287/mnsc.43.11.1469 [12] D. BarnesSchuster, Y. Bassok and R. Anupindi, “Coor dination and Flexibility in Supply Contracts with Op tions,” Manufacturing & Service Operations Manage ment, Vol. 4, No. 3, 2002, pp. 171207. doi:10.1287/msom.4.3.171.7754 [13] X. Gan, S. P. Sethi and H. Yan, “Coordination of Supply Chains with RiskAverse Agents,” Production and Op erations Management, Vol. 13, No. 2, 2004, pp.135149. doi:10.1111/j.19375956.2004.tb00150.x [14] G. P. Cachon and M. A. Lariviere, “Supply Chain Coor dination with Revenue Sharing Contracts: Strengths and Limitations,” Management Science, Vol. 51, No. 1, 2005, pp. 3044.doi:10.1287/mnsc.1040.0215 [15] M. Erkoc and S. D. Wu, “Managing HighTech Capacity Expansion via Reservation Contracts,” Production and Operations Management, Vol. 14, No. 2, 2005, pp. 232251.doi:10.1111/j.19375956.2005.tb00021.x [16] M. Jin and S. D. Wu, “Capacity Reservation Contracts for HighTech Industry,” European Journal of Operational Research, Vol. 176, No. 3, 2007, pp. 16591677. doi:10.1016/j.ejor.2005.11.008
