Optimal production and selling policies with fixed-price contracts and contingent-price offers

Int. J. Production Economics 137 (2012) 94–101

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Optimal production and selling policies with fixed-price contracts and contingent-price offers Cheng-Yuan Ku n, Yi-Wen Chang Department of Information Management, National Chung Cheng University, 168, University Rd., Min-Hsiung, Chia-Yi County, Taiwan, ROC

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 November 2009 Accepted 11 January 2012 Available online 28 January 2012

This paper investigates optimal production and selling decisions for a single supplier with two types of customers. Specifically, risk-averse buyers would rather pay higher fixed prices with guaranteed supply contracts whereas risk-prone buyers prefer to secure remaining stocks and pay lower contingent prices. This study formulized this problem with a dynamic programming model and analyzed it further using successive approximations. Theoretical results indicate that no controls are needed for fixed-price orders. However, thresholds exist for manufacturing and contingent-price ordering policies. These two types of threshold planes increased with the addition of waiting customers. Furthermore, a sensitivity analysis of seasonal factors revealed that the optimal threshold plane shifts upward during high demand periods. & 2012 Elsevier B.V. All rights reserved.

Keywords: Contingent prices Manufacturing and selling decisions Optimal policy Dynamic programming Sensitivity analysis

1. Introduction The tradeoff between risk and profit is always a difficult decision in business transactions and these decisions continue to be difficult to navigate despite substantial research undertaken in this arena. Contingent-pricing contracts provide possible alternatives for coping with this uncertainty by delaying difficult decisions until optimal solutions become clear (Bazerman and Gillespie, 1999). One well-known example of contingent contracting involved salary negotiations with the U.S. National Basketball Association (NBA) basketball player, Dennis Rodman. In 1997, while Dennis Rodman’s total possible salary was $10.5 million, the Chicago Bulls limited his salary guarantee to $4.5 million because of his unpredictable behavior and propensity to miss games. The remaining $6 million of his salary was contingent upon performance and behavior clauses (Bazerman and Gillespie, 1999). Compensation or price reductions for surplus stock are also preferentially applied during contingent-pricing contract negotiations. Thus, contingent-pricing contracts offer the advantage of avoiding negative tradeoffs and providing mutual benefits among buyers and sellers (Bazerman and Gillespie, 1999). However, this strategy is still overlooked by many industries because it appears to involve high risk that lacks a systematic approach to decision making.

n Corresponding author. Tel.: þ886 5 2720411x34603, 24608; fax: þ 886 5 2721501. E-mail addresses: [email protected] (C.-Y. Ku), [email protected] (Y.-W. Chang).

0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2012.01.019

Contingent pricing is often incorporated into contingent contracts and can be implemented by designing pricing mechanisms. In this way, the seller provides a fixed-price contract with a guaranteed supply. If an order cannot be fulfilled due to inadequate stocks, the seller becomes responsible for compensating the customer for any subsequent cancellations. In addition to fixed, high-price mechanisms, buyers can opt for lower prices with the stipulation that little or no compensation will be forthcoming for supply failures. The contingent-price strategy is currently applied in many trade sectors, including in the retail, fashion, aviation and hotel industries. For example, airline companies may offer two types of tickets whereby higher-priced reservations guarantee a seat and lower-priced reservations apply to vacancies. The fashion industry may sell guaranteed, in-season products at higher prices, but discounted sale prices could be offered during times of uncertainty. This mechanism is feasible due to the mutual benefits gained by both suppliers and customers (Biyalogorsky and Gerstner, 2004). Although the usefulness of the contingent-price policy has been demonstrated, few applications of this strategy have been observed in the production-oriented supply chain. The complexity and high uncertainty of many supply chains makes it difficult for managers to make pricing policy decisions. Suppliers and manufacturers face uncertainty surrounding customer demand and production processes. Downstream customers face tradeoffs between lower prices and guaranteed supplies. The bullwhip effect in a supply chain system makes planning and pricing even harder. However, the contingent-price mechanism is an attractive option because it provides suppliers with a strong motivation to be flexible and adjust their prices based on inventory levels.

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Customers would then have different options for placing orders based on their willingness-to-pay and whether they are riskprone or risk-averse (Biyalogorsky and Gerstner, 2004). This study discusses different kinds of contracts and manufacturing plans by considering a relatively unsophisticated supply chain of one supplier and two types of customers. Customers can select a fixed-price contract or a contingent-price contract and the supplier makes a decision about whether to accept or decline the offer and determines the production schedule. By constructing a mathematical model, this decision making problem is theoretically analyzed and useful properties of optimal policies are derived. The remainder of this paper is organized as follows. Section 2 reviews related pricing issues, including supply chain risks, price discrimination and contingent contracts. Section 3 presents a theoretical model of dynamic programming with explanations of assumptions and computational notations. Next, five essential theorems concerning optimal control policies are derived and proven. Section 4 outlines numerical analyses undertaken to demonstrate optimal threshold polices. The following sensitivity analyses describe variations in thresholds under different system parameters. Finally, we conclude the paper by offering insights gained in this study and possible directions for future research.

2. Related work 2.1. Supply chain risks A supply chain faces risks from internal dynamics and the external environment (Goh et al., 2007). While considerable research has examined supply chain uncertainties, Zsidisin et al. (1999) and Harland et al. (2003) made especially important contributions by providing precise definitions of supply chain risks. Zsidisin et al. (1999) defined supply risk as the transpiration of significant and/or disappointing failures with inbound goods and services. Harland et al. (2003) claimed that supply risks adversely affect the inward flow of any type of resource needed for operations to take place. More recently, Goh et al. (2007) categorized supply chain risks into two categories; namely, internal risks, like supply, demand and trade credit risks and external risks arising from environmental threats, such as natural disasters and terrorism. Regardless of the risk type, they can make it impossible for suppliers to meet customer demands (Zsidisin, 2002). Industries and scholarly institutions have examined this topic ¨ previously. For example, Juttner (2005) outlined requirements for managing supply chain risk from the perspective of a practitioner. Norrman and Jansson (2004) studied the approach applied by Ericsson to supply chain risk management. Manuj and Mentzer (2008) used an interview approach to gain insights into the applicability of six risk management strategies. Sinha et al. (2004) purposed a methodology to mitigate supplier risk in an aerospace supply chain. Paulsson (2003), Tang (2006) and Peck (2006) reviewed and discussed related research results of supply chain risk management and Ritchie and Brindley (2007) suggested avenues for its future development. The recent research which identifies risk issues and research advancements in supply chain risk management appeared in Tang and Musa (2011). Organizations require practical methods to properly manage supply chain risks so as to maintain a competitive advantage. Risk-sharing involves a series of activities, such as vendormanaged inventory (VMI) and risk-pooling and forecasting (Peck, 2006) to reduce risks. Jin and Wu (2007) have also suggested deductible reservation contracts for high-tech supply chains to avoid supply risks and Goh et al. (2007) proposed an

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algorithm that uses a multi-stage supply chain to minimize risks and maximize profits. Giri (2011) reduced inventory risk by souring two types of suppliers with different costs. 2.2. Price discrimination Price discrimination is one of the most popular marketing strategies applied across a broad range of industries. According to the conventional definition, price discrimination means that the same commodity is sold to different consumers at different prices (Philips, 1983). However, three conditions are necessary for a firm to consider price discrimination. First, this firm must have the power to influence the market. Second, the firm must be able to clearly identify different customer groups and their purchasing characteristics. Third, the firm must be able to prevent arbitrage from resale (Varian, 1989). In practice, several different forms of price discrimination have developed. For example, nonlinear pricing adjusts marginal prices according to the quantity requested in purchase orders. Bundling strategies determine the marginal price by products grouped in a sale. Customer poaching strategies reduce prices for firsttime buyers (Armstrong, 2006). The use of coupons as a pricing strategy has also been analyzed by Narasimhan (1984) and several studies have examined price discrimination across different fields. Moorthy (1984) attempted to segment customers into groups to facilitate marketing and Langenfeld et al. (2003) studied how price discrimination affects social welfare and individual profits. Clerides (2002) examined price discrimination in the book publication industry. The results showed that pricing practices had no association with cost but rather could be affected by quality. Iyer and Seetharaman (2003) investigated price discrimination in the retail gasoline industry and concluded that heterogeneity of local markets influenced pricing decisions. Comprehensive microeconomic surveys can be found in Philips (1983), Varian (1989) and Langenfeld et al. (2003). 2.3. Contingent contracts Many contract models for supply chain have been proposed to improve the performance under uncertainty (Wang et al., 2011) and the contingent contract is one of them. Bazerman and Gillespie (1999) have published an important paper that discusses contingent contracts. This paper explained benefits of contingent contracts and concluded that organizations should confront uncertainties and parlay them into advantages rather than conflicts to be avoided. Early strategies of contingent-price contracts included money-back guarantees, warranties and trial periods (Lutz and Padmanabhan, 1998; Bhargava and Sundaresan, 2003; Bhargava and Sun, 2005). However, these approaches focused on use and involved extra transaction costs, such as returning merchandise, refunding payments and repairing merchandise. Price and quality guarantees are similar to these early ideas but provide a mechanism for reducing risks to buyers. There are two main areas of research on contingent contracts and strategies. The first area focuses on uncertainty of quality. Bhargava and Sundaresan (2003) have used contingent-price strategies to reduce uncertainties surrounding quality in IT services and their effects are discussed under private and public information disclosures. Bhargava and Sun (2005) have examined the applicability of performance-based contingent-price schemes for broadband services. The second area of research has focused on reducing seller risks. Biyalogorsky and Gerstner (2004) have demonstrated contingent-price structures, economic efficiencies and comparisons with high-pricing and low-pricing strategies. Contingent-price contracts are also related to concepts of clearance sales (Sallstrom, 2001; Lee and Rhee, 2007; Nocke and

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Peitz, 2005), advance selling (Weatherford and Pfeifer, 1994; Xie and Shugan, 2001; Geng et al., 2007) and overbooking (Sharbel and Muhammad, 2004; Takeshi and Hiroaki, 2005; Popescu et al., 2006; LaGanga and Lawrence, 2007). For clearance sales, the discount, which is similar to a contingent-price contract, attracts potential buyers to purchase goods before or near expiration. Advance selling uses discounts to allow sellers to push purchase times forward, while overbooking mitigates risks associated with no-shows after making reservations. Yield management (Bodily and Weatherford, 1995) and revenue management (Mitra, 2007) are also other types of contingent-price contracts. For a manufacturer with fixed production capacities and uncertain demand, yield management strategies assign a higher priority to buyers paying higher prices. Other related reviews can be found in papers by Takeshi and Hiroaki (2005), Parijat et al. (2005) and Walczak and Erumelle (2007). Despite a growing interest in supply chain research, to the best of our knowledge, little study is being done to improve the integrated aspects of supply chain risk controls, price discrimination advantages and contingent-price contracts. Therefore, this study provides a win–win choice for customers and suppliers with contingent-price and fixed-price contracts. We investigated the optimal production and selling policies for a supplier with two types of customers (i.e., risk-averse and risk-prone buyers). Although two-customer models have been examined previously by Nocke and Peitz (2005) in the context of clearance sale strategies, this study explored more complex problems involving production and order fulfillment based on cost structures and risk attitudes. Furthermore, inventory clearance was the research goal of Nocke and Peitz (2005) but not ours in this study. A detailed description of our problem formulation and approach is described in the next section.

(1) a1(X,Y,Z)¼1 (0) indicates that the control action fulfills (rejects) the fixed-price order when the system is in state (X,Y,Z). (2) a2(X,Y,Z)¼1 (0) indicates that the control action fulfills (rejects) the contingent-price order when the system is in state (X,Y,Z). (3) a3(X,Y,Z)¼1 (0) indicates that the control action continues (stops) the manufacturing process when the system is in state (X,Y,Z). For the analysis of tradeoff conditions, several model assumptions were made and are summarized below. Assumption 1. There are two categories of customers. The riskaverse buyer chooses to place orders at a fixed price (i.e., higher price) with guaranteed supplies. The risk-prone buyer prefers to take remaining stocks but pay contingent prices (i.e., lower price). This categorization assumes that the arrival process of the riskaverse buyer is a Poisson stream with an arrival rate l1, while the arrival process of the risk-prone buyers is a Poisson stream with an arrival rate l2. Assumption 2. Waiting times, including buffer time, advanced reservation and delayed decision making, prior to purchase order fulfillment are assumed to be i.i.d. Waiting times for risk-averse and risk-prone customers are also exponentially distributed at rates m1 and m2, respectively. Assumption 3. The order quantity from either risk-averse or riskprone customers is assumed to be one unit. Assumption 4. The payment for each fixed-price sale is Pf and the payment for each contingent-price sale is Pc; therefore, Pf 4Pc Z0.

3. Problem formulation The research problem was formulated as a two-stage stochastic model, which contains two upstream parallel queues for two different customer types and one manufacturer with a warehouse to hold stock. This system can be modeled as a three-dimensional, continuous time Markov chain with state (X,Y,Z)AO defined as the stock level, the quantity of unfulfilled fixed-price orders and the quantity of unfulfilled contingent-price orders. The value O  f0,1,. . .,Xgxf0,1,. . .,Ygxf0,1,. . .,Zg is the state space of (X,Y,Z), where the maximum capacity of inventory is X, the maximum quantity of unfulfilled fixed-price orders is Y and the maximum quantity of unfulfilled contingent-price orders is Z due to processing capacity limitations. For computational convenience, discounted dynamic programming was chosen to optimize control policies. In addition, uniformization was used to produce the equivalent, discrete time Markov chain by allowing fictitious transitions from a state to itself. As provided by the following optimality equation, optimal control policies were chosen in each state to maximize expected discounted profits. RðX,Y,ZÞ ¼ max afHðXÞ þ p1 RðX,Y þ1,ZÞ þ p2 RðX,Y,Z þ 1Þ AðX,Y,ZÞ

þ p3 ½a3 RðX þ 1,Y,ZÞ þ ð1a3 ÞRðX,Y,ZÞ þ q1 Y½a1 ðP f þ RðX1,Y1,ZÞÞ þ ð1a1 ÞðRðX,Y1,ZÞC 1 Þ þ q2 Z½a2 ðP c þ RðX1,Y,Z1ÞÞ þ ð1a2 ÞðRðX,Y,Z1ÞC 2 Þ þ ð1p1 p2 p3 Yq1 Zq2 ÞRðX,Y,ZÞg

mapping function to A(X,Y,Z) ¼[a1(X,Y,Z),a2(X,Y,Z),a3(X,Y,Z)] with ai(X,Y,Z):O-{0,1}3, where i¼1, 2, 3.

ð1Þ

Control policies were capable of filling or rejecting product deliveries under each purchase order and these policies could continue or stop the manufacturing process. The primary objective was to maximize the total discounted profit R(X,Y,Z) over an infinite time horizon. Control policies were defined by a X  Y  Z

Assumption 5. H(X) is the holding cost function (e.g., deterioration risk, storage cost and management cost) for stock level X. It is reasonable to consider this holding cost function to be increasing and a convex function of X. Assumption 6. The supplier will pay compensation C1 per unit to risk-averse buyers if their orders cannot be fulfilled. Similarly, the supplier will pay compensation C2 per unit to risk-prone buyers if their orders cannot be fulfilled. Because risk-averse buyers pay more, they will receive higher compensations for lost sales, i.e., C1 4C2 Z0. Assumption 7. The production process is a Poisson stream with arrival rate l3 if the supplier decides to manufacture the product. For uniformization, an arbitrary QAR is chosen s.t. Q 4 l1 þ l2 þ l3 þ X m1 þY m2 , where pi ¼(li/Q) for i¼1, 2, 3 and qj ¼(mj/Q) for j¼1, 2 (Ross, 1995). Thus, the equivalent discrete time system has corresponding parameters p1,p2,p3,q1,q2 and 0o a o1 is selected to be the discount factor for dynamic programming. The expected profit for the optimality equation includes the following: H(X) (i.e., negative holding cost function), profit from two types of purchase orders p1R(X,Yþ 1,Z)þp2R(X,Y,Z þ1), profit from production p3[a3R(Xþ1,Y,Z) þ(1 a3)R(X,Y,Z)], profit from the fulfillment of two purchase types q1Y[a1(Pf þR(X 1, Y  1,Z))þ(1  a1)(R(X,Y  1,Z) C1)]þ q2Z[a2(Pc þR(X 1,Y,Z  1)) þ (1 a2)(R(X,Y,Z  1)  C2)] and profit from no action (1 p1  p2  p3 Yq1 Zq2)R(X,Y,Z). Preferred policies based on the optimality equation can be determined using successive approximations. Specifically, this approximation was completed in this study by choosing an arbitrary step 0 profit R0 and defining the step h profit, Rh(X,Y,Z),

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to be the expected discounted profit starting in state (X,Y,Z) associated with choosing actions to maximize future expected discounted profits. This approach assumed that transitions of more than one time step into the future generated an average profit according to the step h 1 profit. Thus, successive approximations give: Rh ðX,Y,ZÞ ¼ max

AðX,Y,ZÞ

(

a

X

) P X,Y,Z-X 0 ,Y 0 ,Z 0 ðAÞfrðX,Y,Z,X 0 ,Y 0 ,Z 0 Þ þ Rh1 ðX 0 ,Y 0 ,Z 0 Þg

X 0 ,Y 0 ,Z 0

ð2Þ

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Theorem 2. D(X,Y,Z) Z  (Pf þC1) for all (X,Y,Z)AO. Consequently, it is optimal to always fulfill the fixed-price order, i.e., Xf(Y,Z)¼0 for 0 r Y r Y and 0 rZ r Z . It is also equivalent to a1(X,Y,Z) ¼1 for all states. Although Theorem 1 demonstrates that the optimal control policy for fixed-price orders operates under a threshold, coincidentally D(X,Y,Z)Z (Pf þ C1) for all (X,Y,Z)AO means that the requirement of fixed-price orders should be satisfied if there is any remaining stock. Hence, the control threshold for fixed-price orders is the bottom YZ plane.

where PX,Y,Z-X0 ,Y0 ,Z0 (A) is a one-step transition probability and r(X,Y,Z,X0 ,Y0 ,Z0 ) is the earned profit for state transitions from (X,Y,Z) to (X0 ,Y0 ,Z0 ) (Ross, 1983). The optimal profit function obeys the following rule (Ross, 1983):

Theorem 3. D(X,Y,Z) is monotonically decreasing for Y under fixed X and Z values, (X,Y,Z)AO. Consequently, optimal threshold planes for manufacturing and contingent-prices are monotonically increasing in the Y axis.

RðX,Y,ZÞ ¼ lim Rh ðXÞ ¼ lim Rh1 ðX,Y,ZÞ

Theorem 4. D(X,Y,Z) is monotonically decreasing for Z under fixed X and Y values, (X,Y,Z)AO. Consequently, optimal threshold planes for manufacturing and contingent-prices are monotonically increasing in the Z axis.

h-1

h-1

ð3Þ

For the purpose of maximization, the optimal policy fulfills any purchase order if revenues exceed expected losses caused by future blocking by the customer. Therefore, difference functions are defined as follows for (X,Y,Z)AO, X4K, Y o Y and Z oZ:

Dh ðX,Y,ZÞ ¼ Rh ðX,Y,ZÞRh ðX þ 1,Y,ZÞ

ð4Þ

DðX,Y,ZÞ ¼ RðX,Y,ZÞRðX þ 1,Y,ZÞ

ð5Þ

Thus, the optimal policy in state (X,Y,Z) involves the following:

 Continuing to manufacture the product, i.e., a3(X,Y,Z) ¼1 iff D(X,Y,Z)r0

 Fulfilling a fixed-price order, i.e., a1(X,Y,Z)¼1 iff D(X  1, Y  1,Z)Z  (Pf þC1)

 Fulfilling a contingent-price order, i.e., a2(X,Y,Z)¼1 iff D(X 1, Y,Z 1)Z  (Pc þC2) Based on difference functions, the following theorems are derived. Theorem 1. D(X,Y,Z) is monotonically increasing for X under fixed Y and Z values; (X,Y,Z)AO. Consequently, a threshold plane exists whereby the optimal policy entails product manufacturing if the state is below (not including) this plane. There is a threshold plane for the optimal policy in fulfilling fixed-price orders if the state is above (not including) this plane. Likewise, there is a threshold plane for the optimal policy in contingent-price fulfillment if the state is above (not including) this plane. Because the production manager will only continue to manufacture products if D(X,Y,Z)r0 and D(X,Y,Z) is monotonically increasing in the X axis, there may exist an inventory level Xm that satisfies the condition of D(Xm,Y,Z)40 and D(Xm  1,Y,Z) r0. Therefore, production should be stopped while the system is in this state or above this threshold level. Similarly, because D(X,Y,Z) is monotonically increasing in X, an inventory level Xf may exist, such that D(X  1,Y 1,Z)o (Pf þC1) for X rXf. Thus, the supplier should forego the fixed-price order while the system is in this state or below this threshold level. The same relationship will also hold true for contingent-price orders. Assumption 8. The optimal control plane policy for manufacturing is denoted by Xm(Y,Z), such that manufacturing will continue if the state is below (not including) this plane. The optimal control plane policy for accepting fixed-price orders is denoted by Xf(Y,Z), such that fixed-price orders will be fulfilled if the state is above (not including) this plane. Similarly, the optimal control plane policy for accepting contingent-price orders is denoted by Xc(Y,Z), such that contingent-price orders will be fulfilled if the state is above (not including) this plane.

Because D(X,Y,Z) is monotonically decreasing under Y or Z axes, one more unit of fixed-price or contingent-price orders will reduce net profits due to competition between customers. Hence, the manufacturer should produce more product (i.e., raise the manufacturing threshold plane) to increase net revenues. Furthermore, the optimal policy applied should reserve more stock if there are higher units of waiting fixed-price orders. This relationship explains why the threshold plane for contingent-price customers is monotonically increasing in the Y axis. The optimal policy applied would also hold more stock in circumstances of increased pending contingent-price orders due to the enhanced probability that contingent-price customers will want their orders fulfilled. Thus, the threshold plane for contingent-price customers is monotonically increasing in the Z axis. Theorem 5. Xc(Y,Z) r Xm(Y,Z) for 0 r Y r Y and 0 r Z r Z . That is, the upper bound of Xc(Y,Z) is Xm(Y,Z). In general, the threshold plane of manufacturing will be higher than the threshold plane of contingent pricing. However, if Pc þ C2 approaches zero, then Xc(Y,Z) will approach Xm(Y,Z). This condition means that the system neither achieves profits from fulfilled sales nor suffers punishment from lost sales for contingent-price orders. Making matters worse, the system needs to pay holding costs for existing stocks. The supplier should stop production in this situation to fulfill any contingent-price orders and maximize benefits. Therefore, contingent-price thresholds are at most the same as manufacturing thresholds and the supplier continues manufacturing for contingent-price orders only if Pc þC2 is larger enough to cover holding costs.

4. Numerical and sensitivity analysis From theoretical results presented here, optimal policies to maximize expected discounted profits over an infinite time horizon are given by threshold planes in three-dimensional space. These results are demonstrated by a numerical analysis with nominal system parameters set as follows: X ¼ Y ¼ Z ¼ 6, l1 ¼ l2 ¼ l3 ¼1, m1 ¼ m2 ¼1, Pf ¼30, Pc ¼10, C1 ¼50, C2 ¼15 and the holding cost H¼(X2/100). Numerical values are obtained by successive approximations and optimal policies are inferred from the optimal value function. Corresponding optimal thresholds are shown in Figs. 1 and 2. As demonstrated in Theorem 2, it is optimal to always fulfill fixed-price orders. The manufacturing and contingent-price ordering policies comprise threshold-type planes in

C.-Y. Ku, Y.-W. Chang / Int. J. Production Economics 137 (2012) 94–101

6

6

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Fig. 2. Threshold for optimal contingent order policy.

three-dimensional space based on Theorem 1. Both of these policy planes are increasing with fixed-price and contingent-price orders. These characteristics were proven in Theorems 3 and 4. As indicated by Theorem 5, the threshold of contingent-price ordering is lower than the threshold of manufacturing decision making. Seasonal factors may also influence supply chain operations. To consider these effects, arrival rates of the two customer types were adjusted to conduct a sensitivity analysis of optimal policies. This analysis retained all previously used parameters, except l1 and l2. Rather, l1 ¼ l2 ¼2 was considered a season of average demand and l1 ¼ l2 ¼3 was considered the season of high demand. The threshold plane for optimal manufacturing policy and contingent-price ordering under these two scenarios are shown in Figs. 3 and 4, respectively.

6

3 5

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1

.o

ff

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de 2 r

1 1 0 0

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0 6 5 r 4 orde t 3 n ge 2 tin on c of

Fig. 3. Thresholds for optimal manufacturing policy (a) average season, l1 ¼ l2 ¼ 2 (b) a season with high demand, l1 ¼ l2 ¼ 3.

From these results, it is clear that both the manufacturing and contingent-price ordering policies moved upwards as customer orders increased. For high-demand seasons, more customer requests will result in substantial consumption of inventory in the near term. Therefore, the manufacturing threshold policy should be increased to avoid stock shortages. The threshold for optimal contingent-price orders should also be increased to avoid blocking future fixed-price orders. For production capacity expansions, l3 ¼2, l3 ¼3 and l3 ¼4 were another three additional cases that were considered. Fig. 5 shows threshold planes of optimal manufacturing policies under these three cases, while Fig. 6 shows threshold planes of optimal contingent-price ordering policies. If numerical results from Figs. 1, 2, 5 and 6 are combined, it is apparent that optimal planes for manufacturing and contingent-price policies decreased with production capacities. As production capacity increases, the manufacturer can handle customer demand much more easily. Therefore, retaining high stock levels is unnecessary and the

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6 5 4 3 2 1 0 5 6 4 t 3 gen 2 ntin o 1 c of r 00 No. orde

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Fig. 4. Thresholds for optimal contingent-price order policy (a) average season, l1 ¼ l2 ¼ 2 (b) a season with high demand, l1 ¼ l2 ¼3.

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optimal manufacturing threshold decreases. Under this same reasoning, more stocks can also be allocated to fulfill contingent-price orders, causing the optimal threshold for contingentprice orders to shift downward.

5. Conclusions and future research This study demonstrates the usefulness of contingent-price strategies for managing a supply chain and concludes that optimal control policies are of a threshold and monotonic nature. Relative to fixed-price strategies, the contingent-price design provides suppliers with the flexibility to adjust their pricing based on inventory levels. Moreover, customers could have different choices for placing orders based on their willingnessto-pay and risk attitudes. Although this basic supply-demand risk model, with its numerous assumptions and simplifications, may not fully describe real-world complexities, theoretical results

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Fig. 5. The optimal manufacturing policy with different production capacity (a) l3 ¼ 2 (b) l3 ¼3 (c) l3 ¼4.

presented here will provide important insights into managing supply chains. Until now, the implementation of contingent-price strategies was undertaken by trial and error rather than using theoretical estimation techniques (Biyalogorsky and Gerstner, 2004), probably due to capricious customer preferences. Meanwhile, costsensitive customers may search for alternate sellers to avoid unnecessary risk rather than choose contingent pricing options. Strong communication between customers and suppliers is critical for contingent-pricing strategies to work effectively. Customers are more apt to choose contingent pricing if they have a full understanding of their associated risks. Carefully designed

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C.-Y. Ku, Y.-W. Chang / Int. J. Production Economics 137 (2012) 94–101

Capacity

100

6 45 2 3 ent g 0 1 n i t con f o No. order

Fig. 6. The optimal contingent policy with different production capacity (a) l3 ¼ 2 (b) l3 ¼3 (c) l3 ¼4.

contingent-price contracts offer multiple pricing strategies that can satisfy the needs of a range of customers. Future research is needed to examine the following: suitable time frames for contingent-price contracts; problematic models to consider variable customer preferences and characteristics of contingent-price contracts in competitive markets. Sociological research is also needed to better understand customer reactions and acceptance levels of contingent-price arrangements and additional data collection efforts would elucidate the efficiency of contingent-price strategies. Furthermore, managers need to better understand how to adjust prices and compensation schemes to stimulate or restrain demand under conditions of limited production capacity. Based on additional numerical analyses, we have determined that managers should be apt to accept fixed-price contracts when the difference of prices and compensations between fixed-price contracts and contingent-price orders, i.e., Pf þC1 and Pc þ C2 increases. Therefore, the threshold of the contingent-price order moves upward. It seems that this intuitive property could be proven mathematically.

Acknowledgment This research was supported by project NSC 95-2416-H-194-028 of National Science Council, Taiwan, ROC. Moreover, we are also grateful to the anonymous referees for their useful suggestions.

Appendix A Outline of Proofs of Theorems 1–4. In order to prove D(X,Y,Z) is monotonically increasing in the X axis, two states (X1,Y,Z) and (X2,Y,Z) with X1 oX2 are considered and it remains necessary to prove that Dh þ 1(X2,Y,Z)  Dh þ 1(X1,Y,Z)40. The proof proceeds by substituting (2) into each term on the right hand side and by collecting and comparing similar terms. The key to proving the theorem involves demonstrating that terms generated by threshold or boundaries of the state space can be bounded by others. Results of individual comparisons show that Dh þ 1(X2,Y,Z)  Dh þ 1

(X1,Y,Z)40 is true. Theorem 1 is proven by applying mathematical inductions and Eq. (2). The computational procedure is similar for proving Theorems 2, 3 and 4. The detailed proofs are long and involved. They can be provided upon request. & Proof of Theorem 5. While in state (X,Y,Z), it is optimal to fulfill a contingent-price order if D(X  1,Y,Z 1) Z  (Pc þC2). Because D(X,Y,Z) is monotonically increasing in the X axis, Xc(Y,Z) will shift upward while Pc and C2 decrease. As Pc and C2 approach zero, the necessary condition for fulfilling a contingent-price order will become D(X 1,Y,Z  1)Z0. Because D(X,Y,Z) is monotonically decreasing in the Z axis, D(Xm(Y,Z),Y,Z  1)4 D(Xm(Y,Z),Y,Z)40. This condition means that the optimal policy will be to accept the contingent-price order for state (Xm(Y,Z) þ1,Y,Z). Thus, the optimal threshold plane for contingent-price order is lower than (Xm(Y,Z)þ1,Y,Z) and the upper bound of Xc(Y,Z) is Xm(Y,Z). &

References Armstrong, M., 2006. Recent Developments in the Economics of Price Discrimination. available on/http://www.econ.ucl.ac.uk/downloads/armstrong/pd.pdfS. Bazerman, M.H., Gillespie, J.J., 1999. Betting on the future: the virtues of contingent contracts. Harvard Business Review 77, 3–8. Bhargava, H.K., Sun, D., 2005. Performance-contingent pricing for broadband service. Proceeding of the 38th Hawaii International Conference on System Sciences 8, p. 221b. Bhargava, H.K., Sundaresan, S., 2003. Contingency pricing for information goods and services under industrywide performance standard. Journal of Management Information System 20, 113–136. Biyalogorsky, E., Gerstner, E., 2004. Contingent pricing to reduce price risks. Marketing Science 23, 146–155. Bodily, S.E., Weatherford, L.R., 1995. Perishable-asset revenue management: generic and multiple-price yield management with diversion. Omega 23, 173–185. Clerides, S.K., 2002. Book value: intertemporal pricing and quality discrimination in the US market for books. International Journal of Industrial Organization 20, 1385–1408. Geng, X.J., Wu, R.H., Andrew, B.W., 2007. Profiting from partial allowance of ticket resale. Journal of Marketing 71, 184–195. Giri, B.C., 2011. Managing inventory with two suppliers under yield uncertainty and risk aversion. International Journal of Production Economics 133, 80–85. Goh, M., Lim, J.Y.S., Meng, F., 2007. A stochastic model for risk management in global supply chain networks. European Journal of Operational Research 182, 164–173.

C.-Y. Ku, Y.-W. Chang / Int. J. Production Economics 137 (2012) 94–101

Harland, C., Brenchley, R., Walker, H., 2003. Risk in supply networks. Journal of Purchasing and Supply Management 9, 51–62. Iyer, G., Seetharaman, P.B., 2003. To price discrimination or not: product choice and the selection bias problem. Quantitative Marketing and Economics 1, 155–178. Jin, M., Wu, S.D., 2007. Capacity reservation contracts for high-tech industry. European Journal of Operational Research 176, 1659–1677. ¨ Juttner, U., 2005. Supply chain risk management: understanding the business requirements from a practitioner perspective. International Journal of Logistics Management 16, 120–141. LaGanga, L.R., Lawrence, S.R., 2007. Clinic overbooking to improve patient access and increase provider productivity. Decision Sciences 28, 251–276. Langenfeld, J., Li, W., Schink, G., 2003. Economic literature on price discrimination and its application to the uniform pricing of gasoline. International Journal of the Economics of Business 10, 179–193. Lee, C.H., Rhee, B.D., 2007. Channel coordination using product returns for a supply chain with stochastic salvage capacity. European Journal of Operational Research 177, 214–238. Lutz, N.A., Padmanabhan, V., 1998. Warranties, extended warranties, and product quality. International Journal of Industrial Organization 16, 463–493. Manuj, I., Mentzer, J.T., 2008. Global supply chain risk management strategies. International Journal of Physical Distribution & Logistics Management 38, 192–223. Mitra, S., 2007. Revenue management for remanufactured products. Omega 35, 553–562. Moorthy, K.S., 1984. Marketing segmentation, self-selection, and product line design. Marketing Science 3, 288–307. Narasimhan, C., 1984. A price discrimination theory of coupons. Marketing Science 3, 128–147. Nocke V., Peitz M., 2005. Monopoly pricing under demand uncertainty: the optimality of clearance sales. Working paper, available on /http://www.i-u. de/fileadmin/be_user/PDF_Docs/BA_School/Working_Papers/wp34_2005.pdfS. Norrman, A., Jansson, U., 2004. Ericsson’s proactive supply chain risk management approach after a serious sub-supplier accident. International Journal of Physical Distribution & Logistics Management 34, 434–456. Parijat, D., Yezekael, H., Laura, W., 2005. Yield management for IT resources on demand: analysis and validation of a new paradigm for managing computing centres. Journal of Revenue and Pricing Management 4, 24–38. Paulsson U., 2003. Managing risks in supply chain-an article review. Proceeding of OFOMA 2003, Oulu, Finland. Peck, H., 2006. Reconciling supply chain vulnerability, risk and supply chain management. International Journal of Logistics: Research and Applications 9, 127–142.

101

Philips, L., 1983. The Economics of Price Discrimination. Cambridge University Press. Popescu, A., Keskinocak, P., Johnson, E., LaDue, M., Kasilingam, R., 2006. Estimating air-cargo overbooking based on a discrete show-up-rate distribution. Interfaces 36, 248–258. Ritchie, B., Brindley, C., 2007. Supply chain risk management and performance: a guiding framework for future development. International Journal of Operations & Production Management 27, 303–322. Ross, S.M., 1983. Introduction to Stochastic Dynamic Programming. Academic Press. Ross, S.M., 1995. Stochastic Process. John Wiley & Sons. Sallstrom, S., 2001. Fashion and sales. International Journal of Industrial Organization 13, 1363–1385. Sharbel, E.H., Muhammad, E.T., 2004. Dynamic two-leg airline seat inventory control with overbooking, cancellations and no-shows. Journal of Revenue and Pricing Management 3, 143–170. Sinha, P.R., Whitman, L.E., Malzahn, D., 2004. Methodology to mitigate supplier risk in an aerospace supply chain. Supply Chain Management: An International Journal 9, 154–168. Takeshi, K., Hiroaki, I., 2005. The hotel yield management with two types of room prices, overbooking and cancellations. International Journal of Production Economics 93, 417–428. Tang, C.S., 2006. Perspectives in supply chain risk management. International Journal of Production Economics 103, 451–488. Tang, O., Musa, N., 2011. Identifying risk issues and research advancements in supply chain risk management. International Journal of Production Economics 133, 25–34. Varian, H.R., 1989. Price discrimination. In: Schmalensee, R., Willig, R. (Eds.), Handbook of Industrial Organization. Elsevier Science Publisher, pp. 597–654. Walczak, D., Erumelle, S., 2007. Semi-Markov information model for revenue management and dynamic pricing. Operations Research Spectrum 29, 61–83. Wang, Q., Chu, B., Wang, J., Kumakiri, Y., 2011. Risk analysis of supply contract with call options for buyers. International Journal of Production Economics. doi:10.1016/j.ijpe.2011.01.013. Weatherford, L.R., Pfeifer, P.E., 1994. The economic value of using advance booking of orders. Omega 22, 105–111. Xie, J., Shugan, S.M., 2001. Electronic tickets, smart cards, and online prepayments: when and how to advance sell. Marketing Science 20, 219–243. Zsidisin G.A., 2002. Defining supply risk: a grounded theory approach. Proceeding of the Decision Sciences Institute Annual Meeting, San Diego, CA. Zsidisin, G.A., Panelli, A., Upton, R., 1999. Purchasing organization involvement in risk assessments, contingency plans, and risk management: an exploratory study. Supply Chain Management: An International Journal 5, 187–197.