A procurement model using capacity reservation

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European Journal of Operational Research 193 (2009) 303–316 www.elsevier.com/locate/ejor

O.R. Applications

A procurement model using capacity reservation Jishnu Hazra, B. Mahadevan

*

Indian Institute of Management Bangalore, Bannerghatta Road, Bangalore 560 076, India Received 27 March 2007; accepted 22 October 2007 Available online 30 October 2007

Abstract In this paper we model a scenario where a buyer reserves capacity from one or more suppliers in the presence of demand uncertainty. We explicitly derive suppliers’ capacity reservation price, which is a function of their capacity, amount of capacity reserved by the buyer and other parameters. The buyer operates in a ‘‘built-to-order’’ environment and needs to decide how much capacity to reserve and from how many suppliers. For a strategy of equal allocation of capacity among the selected suppliers we develop closed form solutions and show that the model is robust to the number of suppliers from whom capacity is procured through reservation. When the parameters of demand distribution changes the supply base is likely to remain more or less the same. Our analysis further shows that increasing the number of pre-qualified suppliers does not provide significant advantages to the buyer. On the other hand, a pre-qualified supply base with greater capacity heterogeneity will benefit the buyer. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Supplier selection; Capacity reservation

1. Introduction Firms in several sectors of industry such as apparels, toys, sports goods and electronics are witnessing supply chain problems arising out of increased product variety. As product variety increases, product life cycle shrinks, requiring firms to shorten lead times in procurement, manufacture and distribution. Moreover, firms also experience greater demand volatility (Jin and Wu, 2001) than before. In order to respond to these developments in the market place, firms either accumulate excess inventory or spend excessively to source capacity to meet the demand as it materialises. Often firms also face risks of shortages and lost sales. It is a common practice in several sectors of the industry to have contractual agreement with suppliers that enables buyers to reserve capacity in advance (Tsay, 1999). If manufacturers require more capacity to meet additional demand, they could procure it in a shorter time by tapping the ‘‘spot market’’ at a higher price. Capacity reservation offers several benefits to supply chain members such as mitigating the ‘‘bull-whip effect’’ (Lee et al., 1997), providing flexibility to handle uncertain demand and permitting better capacity planning (Serel et al., 2001). Additionally, suppliers derive benefits from better upstream procurement planning. Making appropriate choices in procurement of capacity during periods of greater demand volatility therefore becomes crucial. It has generally been the practice to reserve capacity with a single supplier. However, with the advent of the Internet, search costs have been falling and buying firms are able to locate several new suppliers who meet its requirements. Li and Fung, for example, reserves capacity at multiple mills for weaving and dying for undyed yarns in order to improve responsiveness (Magretta, 1998). German chipmaker Infineon Technologies has capacity reservation contracts with multiple vendors for the supply of memory chips (Dow Jones Newswire, 2002). *

Corresponding author. Tel.: +91 80 2699 3275; fax: +91 80 2658 4050. E-mail addresses: [email protected] (J. Hazra), [email protected] (B. Mahadevan).

0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.10.039

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In this paper we analyse a situation where a buyer facing uncertain demand could source capacity from either a set of suppliers by contracting in advance at a lower price or from the spot market later at a higher price. We derive a policy framework for an optimal split between advance reservation and spot procurement of capacity by the buyer and develop useful managerial insights to the problem. Under certain conditions we develop closed form solutions and show that the model is robust to the number of suppliers from whom capacity is procured through reservation. We also show how the utilisation of supplier capacity influences the pricing decisions. We develop several useful managerial insights based on the analysis of the model. There are multiple motivating factors for the suppliers to secure contracts with buyers. When the parameters of the demand distribution changes the supply base is likely to remain more or less the same. Our analysis further shows that increasing the number of pre-qualified suppliers does not provide significant advantages to the buyer. On the other hand, a pre-qualified supply base with greater capacity heterogeneity will benefit the buyer. We begin by briefly introducing the problem. Consider an item that a buyer would like to procure, for which there is a set of qualified suppliers available to supply as per specifications. The item requires a certain degree of customisation for the buyer and therefore represents medium asset specificity. In the case of such items, the suppliers will be able to sell their capacity through a contract agreed in advance or to a number of buyers in the spot market by making minimal modifications. Rangan (1999) reported that a number of items with some custom tooling fall under this category. The buyer faces uncertain demand for the item and would therefore like to reserve capacity through a contract entered with a set of suppliers. The decision to reserve capacity is made in the beginning of the season when the buyer has information only of demand distribution. After the demand is observed and if there is a shortfall the buyer procures the rest from the spot market. A qualified supplier has the option of selling her capacity to a specific buyer through a contract and also to several others through the spot market. While attempting to sell her capacity in the spot market, the supplier finds that the demand is stochastic. Therefore, she will consider selling some capacity to the buyer through a contract. The buyer faces two important decisions in this setting: (a) How much capacity should the buyer procure through capacity reservation? (b) Among how many suppliers should the buyer allocate the reserved capacity? We analyse the scenario in which the buyer procures capacity in advance through reservation using a bidding process. The salient features of the supplier selection process are as follows: The buyer pre-qualifies the number of suppliers to participate in the bidding process, through a request for quotation (RFQ), along with bidding rules and other terms of fulfillment. The buyer will a priori announce the number of suppliers that will be awarded the contract as well as the quantity allocated to selected suppliers. The rest of the paper is organized as follows. We begin with a review of relevant literature in the next section. In Section 3, we elaborate on the buyer’s problem, the costs and decisions. In Section 4 we model the pricing behavior of suppliers willing to offer capacity both through contracts and at the spot market and derive the total cost to the buyer. We derive expressions for optimal values of the number of suppliers selected and the amount of capacity reserved by the buyer in Section 5. In Section 6 we draw managerial insights and finally conclude the paper in Section 7. 2. Literature review Several studies address issues related to contractual agreements between the supply chain members for capacity reservation. Alternative contract mechanisms have been proposed in the past. These include quantity flexibility (Tsay, 1999), deductible reservation (Jin and Wu, 2001), pay-to-delay (Brown and Lee, 1997), and take-or-pay (Jin and Wu, 2001). We refer the readers to the respective papers for details. In a pay-to-delay capacity reservation (Brown and Lee, 1997), the buyer makes an obligation to buy a certain unit of capacity at a lower price. The reserved capacity is typically take-or-pay, i.e. even if the buyer does not fully utilise the reserved capacity she pays for it. Further, the buyer can later procure additional capacity beyond what she reserved at a higher price. The contract mechanism that we have employed in this paper is similar to this. We set the higher cost at the spot market price of the item and derive a basis for obtaining the capacity reservation price. Anupindi and Bassok (1998) discussed the general framework for supply contracts in a multi-period setting and proposed alternative models for quantitative commitments. They showed that contracts permit greater channel coordination using an example from the fashion industry. Henig et al. (1997) studied the contract mechanisms adopted by automobile manufacturers with trucking firms. The trucking capacity was contracted in advance for a lower-than-market cost. Serel et al. (2001) proposed a similar contract mechanism in which the supplier offers a portion of her capacity in advance at a cost lower than the market. For additional capacity, the buyer pays the market price. In both of these cases, the study focused on inventory control policies of the buyer in a periodic review setting. In contrast, our study focuses on the singleperiod, multiple-supplier setting. Moreover, we also make a similar assumption about price paid for reserving capacity in advance, but derive it explicitly. Cachon (2002) provides an excellent review of the supply chain contract literature.

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Elmaghraby (2000) reviews the trends in the sourcing literature and identifies a central issue of multiple sourcing: the number of vendors to be selected. Very few papers in the supply contracting literature have addressed the issue of allocation (Tsay et al., 1998). In a multiple-supplier, single-buyer setting, the presence of multiple suppliers will permit the buyer to setup a competitive mechanism such as a bidding process for capacity allocation among the selected suppliers. Several studies utilised such a process as we also do in this paper. However, other models of supplier competition in a queueing context were also proposed (see Cachon and Zhang, 2006 and Bernstein and De Vericourt, to be published for details). Martinez de Albeniz and Simchi-Levi (2003) consider a multiple-supplier, one-buyer single period model where each supplier offers an option price and an execution price and buyer has to decide on the optimal number of suppliers. Although our model is similar to Martinez de Albeniz and Simchi-Levi (2003) there are important differences. First, in our setting both the buyer and the suppliers have access to the spot market and the suppliers have a risk of unsold capacity, which affects pricing. Secondly, Martinez and Simchi-Levi assume that all parameters are known to the participating players whereas we assume capacity to be a private information. As we shall see, this has important implications to price determination and the winning suppliers. Finally, we utilise a different contract structure than theirs. The differences in structuring the contract and the participant profiles in the capacity reservation context provide an interesting perspective to analyse the problem and develop useful managerial insights. Araman et al. (2003) determine a class of contracts that allows two sources of supply, spot market and long term contract. We model a similar scenario, however, while Araman et al. have addressed a multi-period, single supplier scenario we, on the other hand, have studied a single-period, multiple supplier scenario. Golovachkina and Bradley (2002) have modelled the supplier–manufacturer relationship similar to Araman et al. They model a single period, one supplier–one manufacturer scenario where the manufacturer has the option of sourcing from the spot market. While we study a very similar problem, our model deals with multiple suppliers. Furthermore, our study differs from the earlier studies on the issue of capacity reservation price. Earlier studies in capacity contracts often assumed the capacity reservation price to be exogenous. We however, explicitly model this situation and derive price capacity curves for suppliers. Although suppliers have an option to sell capacity in both the spot market and through pre-negotiated contract with buyers, they face different sets of costs (Grey et al., 2002). Consequently, suppliers will deploy alternative strategies for pricing their capacity in these markets. We believe that business environment aspects such as search cost differentials and risks associated with selling capacity in the open market would influence suppliers’ pricing strategies and capacity allocation through contractual agreement with the buyers. We therefore derive an expression for pricing of supplier’s capacity. 3. The capacity reservation problem Consider an item that a buyer would like to procure from the market, for which there are n qualified suppliers available to supply as per specifications. The item procured is typically built-to-order. By that, we mean, a certain level of customisation is required before the component could be supplied. This assumption motivates reservation of capacity due to demand uncertainty, as procuring the item in the short term may be expensive due to finite lead time between placing the order and obtaining the item. The total amount of capacity the buyer is seeking is a stochastic variable D, sampled from a distribution F(x). The decision to reserve capacity is made in the beginning of the season when it has information only of demand distribution. The buyer utilises a pay-to-delay contract to reserve capacity from a set of suppliers. Let us denote the capacity procured through contract by X, and the number of suppliers among whom X is allocated as m, (1 6 m < n). All m suppliers, finally selected, will get an equal allocation of the capacity at the price of the first rejected supplier. This is akin to the second price sealed bid auction format widely discussed in the literature (see for details McAfee and McMillan, 1987). Therefore, the fraction of the capacity awarded to each supplier is Xm. Equal allocation has been adopted by earlier researchers (see review paper by Seshadri, 2005; Elmaghraby, 2000; Seshadri et al., 1991 and Anton and Yao, 1992 for details). We are aware of other types of allocation (prevalent in practice) notably unequal allocation at the supplier’s bid price, but, in most cases it is not truth inducing (Krishna, 2002) and therefore the analysis within our framework becomes difficult. Each supplier has a finite capacity denoted by l and possibly different from each other (we have suppressed the supplier’s subscript for notational convenience). The supplier’s capacity is private information and not known to the buyer or other suppliers. However, the buyer assumes that the suppliers’ capacities are sampled from a uniform distribution U(ll, lh). The private information assumption stems from the fact that capacity has a strategic dimension. However, at the industry level, capacity availability is known at aggregate levels in terms of the range of capacity available among the existing players. In this paper, we have assumed that the unit production cost for the item under consideration is known to the buyer. This may not be true in general. However, we note that in situations where similar process and production technologies are employed, such an assumption is not unreasonable. A case in point is the reported cost estimation of electronic components by research firms such as iSupply (Hesseldahl, 2005).

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Stage 1

Stage 2

Buyer pre-approves n suppliers

Buyer discovers her demand D

Buyer announces m and Ω

If D > Ω buyer sources balance from spot market

Suppliers announces their bid price Suppliers sell their capacity in spot market

m lowest bid suppliers selected for award of contract Fig. 1. Time line of events.

The salient features of the supplier selection process are as follows: The buyer pre-qualifies the number of suppliers (denoted by n > 1) to participate in the bidding process, through a RFQ, along with bidding rules and other terms of fulfillment. A preliminary qualification for the supplier to be enlisted for participating in the auction is that the supplier’s capacity is at least as large as the capacity sought to be reserved through the contract. The assumption is necessary to avoid infeasibility of the solution in the event the number of suppliers selected for award of contract is just one. Similar assumptions have been made in Bernstein and De Vericourt, to be published. Moreover, such pre-qualification conditions are not uncommon in practice (Steinfield, 2002; Jap, 2002; Emiliani, 2000). The buyer will also a priori announce the number of suppliers, m (1 6 m 6 n  1), that will be selected. The buyer obtains quotes from the participating suppliers in the form of a price–capacity curve and uses it as the basis for allocating the order among the suppliers so as to minimize the total expected cost of the items procured. After the demand is observed and if there is a shortfall the buyer procures the rest from the spot market at an average unit price of P0. The time line of events is shown in Fig. 1. 4. Total cost for the buyer The cost equation for the buyer is given by    X CðX; mÞ ¼ XE P ½mþ1 ð1Þ þ P 0 E maxð0; D  XÞ: m    In Eq. (1), E P ½mþ1 Xm is the expected unit capacity reservation price of the first rejected supplier. This will be the unit price that the buyer will pay to the m selected suppliers to reserve capacity. In order to analyse the total cost equation, we need the capacity reservation price for the participating suppliers. We first derive an expression for this below. Using the price curve of the suppliers, we also derive an expression for capacity reservation price of the (m + 1)th supplier before we proceed with our analysis of the buyer’s problem. 4.1. Supplier’s price curve Consider a supplier who has the option of selling her capacity to a specific buyer through a contract and also through the spot market. While attempting to sell her capacity in the spot market, the supplier finds that the demand is stochastic. Hence, the capacity the supplier will be able to sell in the spot market is a random variable X. The demand distribution for suppliers’ capacity, in the spot market, is assumed to be independently and identically distributed. The random nature of demand imposes a financial burden on suppliers because of the risk of unsold capacity. From the suppliers’ perspective, she will sell her capacity in the spot market at an average unit price of P0. However, since she will face uncertain demand in the spot market she will consider selling some capacity to the buyer through a contract at a lower price, as long as her total expected profit is at least as large as the case when she sells her entire capacity in the spot market.

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The suppliers’ production cost is linear in number of units produced and is given by vx, where v(v > 0) is the unit production cost and x is number of units produced and v is the same for all pre-qualified suppliers. Here we have assumed that cost drivers are primarily independent of scale within a certain range. In these cases the assumption of uniform cost across the suppliers with differing capacity is reasonable. However, if capacity variation across suppliers is large then this assumption becomes restrictive. Ideally, in such situations economies of scale must reflect in the cost structure and it is important to model the capacity–cost relationships more fully than what we have currently attempted. Based on these considerations, we derive the price–capacity curve for a supplier who would like to sell her capacity to the buyer through contract. Her expected profit if she offers her entire capacity in the spot market is given by P 0 EfminðX ; lÞg  vðEfminðX ; lÞgÞ:

ð2Þ

The first term in the above expression is the expected revenue earned and the second term is the expected production cost. The E in the equation is the expectation operator. Suppose the supplier offers x(where x 6 l) units of capacity to the buyer through a capacity reservation contract. She will have to sell l  x units of capacity in the spot market. Her expected profit, if she wins the contract, will then be P ðxÞx þ P 0 EfminðX ; l  xÞg  vðx þ EfminðX ; l  xÞgÞ:

ð3Þ

In (3), P(x) is the price the supplier quotes to the buyer, which is a function of units of capacity, the buyer procures from this supplier through capacity reservation contract. The second term is the expected revenue the supplier earns from the spot market and last term is the expected production cost. The supplier will commit x units of capacity provided (3) is at least as large as (2). Therefore the price–capacity curve is given by ðP 0  vÞfEðminðX ; lÞÞ  EðminðX ; l  xÞÞg þ v: ð4Þ x Similar to Araman et al. (2003), we assume that the spot market is not capacity constrained. Therefore, the contract between the buyer and the participating suppliers will not have any significant impact in the spot market. P ðxÞ ¼

Proposition 1. The unit price quoted by the supplier increases with capacity committed to the buyer. The proof is available in the Appendix. The above proposition is valid for any probability distribution of demand for supplier’s capacity in the spot market. At the outset, the result appears counter-intuitive. As x increases, the capacity available for sale in the spot market at a higher unit price of P0 is reduced. Therefore, the supplier will increase P(x) to compensate for this. We also provide further analysis later in the paper (in Proposition 5) to formally explain this observed pattern. Proposition 2. Assume that the demand for capacity in the spot market for the supplier is uniformly distributed (0, b), b > 0. When l 6 b then the price quoted by the supplier is given by   P0  v P ðxÞ ¼ P 0  ð2l  xÞ: ð5Þ 2b The proof is available in the Appendix. We present a few observations pertinent to the price quoted by the supplier: (a) The price quoted by the supplier is always less than the average spot market price. Furthermore, this price decreases as the supplier’s capacity increases. This is evident due to the fact that with an increase in x, the risk of unsold capacity is reduced, as the supplier may not be able to sell (X  x) in the spot market always. Hence, the supplier would be satisfied with a lower capacity reservation price than P0.   (b) The maximum price the supplier will charge to the buyer is P ðlÞ ¼ P 0  l P 02bv . The second term in the RHS of the above equation denotes the minimum discount the buyer gets from the supplier over the spot market price. The result is intuitive because we find that the discount increases with supplier capacity and contribution margin P0  v. (c) Furthermore, if the supplier’s mean demand (which is 0.5b) increases then the discount reduces. This happens because the supplier will face lower risk of unsold capacity in the spot market and will consequently charge a higher price for capacity reservation.

4.2. Expected capacity reservation price of (m + 1)th supplier Let us now consider the multiple suppliers setting. Once the buyer makes an appropriate choice of X, she will allocate this equally among the m selected suppliers. However, in order to determine the price at which the buyer reserves

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the capacity, she needs to know the price of the first rejected supplier. In (5), we assume that the buyer knows the value of all parameters except each supplier’s capacity. Further, as stated earlier, the suppliers’ capacity is sampled from U(ll, lh). We proceed the expected  to determine    price of the  first rejected supplier. Let P ½1 Xm 6 P ½2 Xm 6    6 P ½mþ1 Xm    6 P ½n Xm denote the order statistics of the unit capacity reservation price (bids) received and P ½mþ1 Xm is the lowest bid amongst the rejected suppliers. Similarly, let l[1] 6 l[2] 6    6 l[m] 6    l[n 1] 6 l[n] denote the order statistics of suppliers’ capacities.      From (5), it is clear that E P ½mþ1 Xm ¼ P 0  P 02bv 2Eðl½nm Þ  Xm . As li’s are uniformly distributed between lh and ll, one can get a closed form expression for the expected value of the order statistics. Substituting Eðl½nm Þ ¼ ll þ nm ðlh  ll Þ (David, 1981) in the above equation and simplifying we get an nþ1 expression for the expected unit price to be paid for reserving the capacity of the selected suppliers      

 X P0  v nm X ðl  ll Þ  E P ½mþ1 ¼ P0  2 ll þ : ð6Þ m 2b nþ1 h m Substituting (6) into (1), we can re-write the cost equation, for the buyer, as   

 P0  v nm X2 ðlh  ll Þ  CðX; mÞ ¼ XP 0  2X ll þ þ P 0 E maxð0; D  XÞ: 2b nþ1 m

ð7Þ

5. Optimal capacity to reserve The buyer would like to minimize the cost of procurement given the stochastic nature of the demand and the existence of n qualified suppliers to choose from. The buyer’s problem boils down to an optimal choice of X and m. We consider two variations of this problem and derive expressions and insights into the problem. 5.1. Buyer’s demand is deterministic We first consider the case of deterministic demand. Assume that the demand is known in advance to be K. Since the demand is known and the reservation price is less than the spot market price, the buyer would like to reserve the entire capacity. The problem reduces to identifying the optimal value of m, among which the reserved capacity will be equally split. Moreover, since, there is no demand uncertainty, the second part of the RHS of Eq. (7) is irrelevant. Replacing X with K in (7) and dropping X from the LHS and the second part on the RHS, we obtain the cost equation as   

 P0  v nm K2 CðmÞ ¼ KP 0  ðl  ll Þ  2K ll þ : ð8Þ 2b nþ1 h m The buyer’s optimization problem can therefore be stated as MinmC(m) and solved by simple calculus to obtain the optimum value of m. Proposition 3. If the buyer’s demand is deterministic and equal to K then the optimal number of suppliers that the buyer will select is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Kðn þ 1Þ  m ¼ : ð9Þ 2ðlh  ll Þ See Appendix for proof. The expression for optimal number of suppliers exhibits several properties of interest to the buyer. It is dependent only on the total capacity requirement of the buyer, the number of participating suppliers and the buyer’s knowledge of the supplier’s capacity distribution parameters. Interestingly, all these parameters are known to the buyer and are independent of supplier specific parameters such as manufacturing cost and demand distribution parameters. The only error that the buyer can introduce in the computation of m* is in the estimation of lh and ll. Since m* is less sensitive to these two parameters (because of the square root factor), it is expected to be robust to errors in estimating the model parameters. 5.2. Buyer’s demand is stochastic We now analyse the case of stochastic demand. We assume that D (the buyer’s demand) follows a uniform distribution U(b1, b2), 0 6 b1 < b2. This assumption enables us to obtain closed form solutions and derive some useful insights. The supplier selection process remains the same as in the deterministic case. However, suppliers are enlisted to participate in the auction only if their capacity is at least b2. As D is uniformly distributed we have

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2

E maxð0; D  XÞ ¼

ðX  b1 Þ b þ b2 þ 1  X: 2ðb2  b1 Þ 2

ð10Þ

Substituting (12) in (7), the cost equation is expressed as !   

 2 P0  v nm X2 ðX  b1 Þ b1 þ b2 ðl  ll Þ  þ X : CðX; mÞ ¼ XP 0  2X ll þ þ P0 2b nþ1 h m 2ðb2  b1 Þ 2

ð11Þ

Proposition 4. When the buyer’s demand is uniformly distributed U(b1, b2) then (a) The optimal number of suppliers is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi All ðnþ1Þ P 0 b1 ðnþ1Þ 4P 0 2 3A þ 9A þ b2 b1 lh ll þ An þ 2ðlh ll Þðb2 b1 Þ X ðn þ 1Þ  ¼ m ¼ ; 2P 0 2ðlh  ll Þ

where A ¼

b2 b1

P0  v ; 2b

and (b) The total amount of capacity reserved is given by h i  b1 2A ll þ nm ðl  l Þ þ bP20b h l nþ1 1 X ¼ : P0 2 A þ m b2 b1 The proof is in the Appendix. The expressions for m* and X* show the relationship between these two decision variables and the impact of other model parameters on them. For instance, when the range of demand distribution parameters (b1  b2) increases, both m* and X* will decrease. On the contrary, when the range of suppliers’ capacity distribution parameters (lh  ll) increases, m* decreases while X* increases. Finally, when n increases both m* and X* will increase. However, as in the deterministic case, m* is likely to be robust to changes in model parameters. We investigate these relationships in some detail in the next section to provide some insights on the proposed model. 6. Managerial insights The expressions for the decision variables and their relationships with other model parameters provide a broad understanding of the relevance of the proposed capacity reservation model to managers. However, to sharpen the understanding and provide additional insights we perform more analysis of the model behavior. We performed a series of ‘‘what if’’ analyses using a base case scenario with the following values for the model parameters: P 0 ¼ 120;

v ¼ 60;

b ¼ 500;

lh ¼ 400;

ll ¼ 200;

b1 ¼ 100;

b2 ¼ 200;

n ¼ 8:

6.1. Supplier motivation issues In general, contractual agreements with buyers provide numerous benefits arising out of stable capacity and production planning. These translate into better up-stream planning and material and cost control. However, the proposed model indicates additional motivations for the suppliers to participate in the contract. We see three significant aspects from the suppliers’ perspective. First the supplier’s capacity utilization increases when she commits x > 0 units of capacity to the buyer. The expected capacity utilization is denoted by q. Proposition 5. When the supplier commits x units of capacity to the buyer her expected capacity utilization is given by 2

q¼

x þ E½minðX ; l  xÞ ðl  xÞ ¼1 : l 2bl

ð12Þ

The proof is omitted as it follows from simple algebra. l If the supplier sells her capacity only in the spot market then her expected utilization is given by 1  2b . Her capacity utilization will increase if she sells part of her capacity to the buyer through a capacity reservation contract. Second is the ability of the supplier to charge a higher price. As the capacity utilization increases, the opportunity cost of unsold capacity also increases resulting in a higher price quoted by the supplier. Using Eqs. (5) and (12) we can show that the expected bid price quoted by the supplier is given by

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 P ðqÞ ¼ P 0 

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P0  v  l þ 2blð1  qÞ : b

ð13Þ

pffiffiffiffiffiffiffiffiffiffiffi In other words, price increases with the capacity utilization and is proportional to 1  q. Greer and Liao (1986) provide empirical evidence for this phenomenon in the aerospace industry. They reported that suppliers were charging higher prices at higher utilization. Finally, it can be seen in this model that winning suppliers are rewarded with additional profit. The benefit will be the  maximum for the supplier who quoted the lowest price, quantified as fEðP ½mþ1 Þ  EðP ½1 Þg Xm . This is the additional profit that the supplier with the lowest price will make by entering into a contract with the buyer. Similarly the remaining m  1 selected suppliers will also benefit. 6.2. Variance in demand distribution parameters Buyers would like to understand the impact of demand variance while engaging in contracts with the selected suppliers. Let us denote the mean of the demand distribution by a and the standard deviation by r. To analyse the effect of variability of the demand distribution, we keep the mean of the demand distribution fixed and change the standard deviation. We 2 ffiffiffiffi2 . Replacing b1 and b2 with a and r we can rewrite know that in the case of a uniform distribution a ¼ b1 þb and r ¼ bp1 b 2 12 * * the expressions for m and X . 2 1 b ffiffiffiffi2 Proposition 6. When the mean and the standard deviation of buyer’s demand distribution are a ¼ b1 þb and r ¼ bp 2 12 respectively, the optimal number of suppliers and the optimal amount of capacity reserved decrease as variance of buyer’s demand increases.

The proof is in the Appendix. As demand uncertainty increases, the buyer will reduce the supply base. Moreover, the quantity allocated per supplier will also reduce. The optimal reserve capacity is less than the buyer’s mean demand, 0.5(b1 + b2). An intuitive explanation to this follows from the fact that the buyer has to take-or-pay for the reserved capacity. With an increase in demand uncertainty the buyer becomes conservative and reserves less capacity and would prefer to source most of his requirement from the spot market. Further as r ! 0, X* ! 0.5(b1 + b2). In this limiting case the optimal number of suppliers that will be selected is given by Eq. (9), which is to be expected. Fig. 2 is a plot of the optimal quantity to be contracted and the optimal number of suppliers for the base case with alternative values of co-efficient of variation. As evident from Fig. 2, the number of suppliers does not change significantly with change in buyer’s coefficient of variation of demand; only the contracted capacity changes. In other words, the supply base is likely to remain more or less the same with increase in demand variation. The problem could be analysed for a general distribution for buyer’s demand using certain properties as stated below. As C(X, m) is simultaneously convex with respect to both m and X for any general demand distribution for the buyer, the first order condition is sufficient to obtain the optimal solution (this has been shown in the proof of Proposition 4). Furthermore, it can be shown that maxð0; D  XÞ ¼ D  minðD; XÞ; oE minðD; XÞ ¼ PrðD > XÞ: oX Using these, we obtain an expression for m* and X* from (1) as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðn þ 1Þ ; m ¼ 2ðlh  ll Þ   P0  v n  m X ll þ P 0 F ðX Þ ¼ ðlh  ll Þ   : b nþ1 m where F(Æ) is the buyer’s demand distribution. We need to resort to numerical evaluation of the above expressions to obtain the optimal values. In order to compare the performance of uniform distribution, we have performed a numerical evaluation using a normal distribution for buyer’s demand. The mean of the normal distribution was kept the same as that of uniform and all other parameter values were kept at the base case value. The comparison has been made for alternative values of the variance of the demand. Fig. 3 illustrates the comparative picture. As we have already observed, m* is robust to a change in the buyer’s demand distribution. Furthermore, at lower levels of variability, uniform distribution and normal distribution estimate the optimal values very closely. However, the difference in the optimal capacity to contract between normal and uniform distribution increases with the variability of demand.

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2.0

150

1.8

140

1.6

130 1.4

120 110

1.2

100

1.0

90

0.8

80

0.6

70 0.4

60

0.0 47 % 51 % 54 % 57 %

40 11 % 14 % 17 % 21 % 24 % 27 % 31 % 34 % 37 % 41 % 44 %

0.2

4% 7%

50

Optimal number of suppliers

Optimal Number of suppliers

160

0% 1%

Optimal Capacity to contract

Optimal Capacity to contract

311

Coefficient of variation Fig. 2. Effect of buyer’s demand distribution on

X*

and

m*.

#Base case values: P0 = 120; v = 60; b = 500;lh = 400; ll = 200; b1 = 100; b2 = 200; n = 8.

No. of suppliers (Uniform)

No. of suppliers (Normal)

Capacity Contracted (Uniform)

Capacity Contracted (Normal) 160

2.00

1.80

1.60 120 1.40

100

1.20

1.00 80 0.80

Optimal capacity contracted

Optimal Number of suppliers

140

60 0.60

0.40

40 10%

20%

30%

33%

40%

50%

60%

70%

Coefficent of variation of demand Fig. 3. Comparison of normal and uniform distributions for buyer’s demand. #Base case values: P0 = 120; v = 60; b = 500; lh = 400; ll = 200; n = 8.

6.3. Distribution of supplier capacities The distribution of supplier’s capacity has several effects on the proposed contracting model. If the capacity variation among the suppliers (lh  ll) is large, then it indicates a heterogeneous supply base in terms of capacity available for contract. It ensures that suppliers with large capacity (relative to others) will quote a lower price than others and the buyer is likely to make use of the lower price offered by the large capacity suppliers. Hence the number of suppliers selected for the contract will be fewer. On the other hand, when suppliers are almost homogenous in terms of capacity, they do not differentiate from one another on the basis of price. Since in our model there are no transaction costs of dealing with multiple

J. Hazra, B. Mahadevan / European Journal of Operational Research 193 (2009) 303–316 Optimal capacity to contract

Optimal number of suppliers

160

8

Optimal capacity to contract

150

7

140 130

6

120

5

110 100

4

90

3

80 70

2

60

1

50 8% 11 % 14 % 18 % 21 % 24 % 28 % 31 % 34 % 38 % 41 % 44 % 48 % 51 % 54 % 58 %

0 4%

1%

40

Optimal number of suppliers

312

Coefficient of variation Fig. 4. Effect of suppliers’ capacity distribution on

X*

and

m*.

#Base case values: P0 = 120; v = 60; b = 500;lh = 400; ll = 200; b1 = 100; b2 = 200; n = 8.

suppliers, m* will tend to be larger in the latter case. Fig. 4 illustrates this phenomenon for the base case with alternative values for the coefficient of variation for suppliers’ capacities, defined similar to demand distribution. Clearly, it shows that all other things being equal, buyers are better off by pre-selecting a heterogeneous mix of suppliers. 6.4. Number of suppliers pre-qualified However, managers would like to know if there are realistic limits to the number of suppliers to be pre-qualified. Clearly, our model indicates that the optimal number of suppliers (m*) and the optimal capacity to contract (X*) increase with n. However, as shown in Fig. 5, buyers do not benefit much from arbitrarily increasing n. As n increases, the rate at which X* increases drop significantly. Furthermore, the price that they obtain through reservation does not decrease significantly. Hence, limiting the number of suppliers to be pre-qualified on the basis of other practical considerations may not introduce significant errors in the decision making process. Moreover, if we introduce transaction costs associated with the pre-selection process, the effect will be more pronounced.

Optimal capacity to contract

Capacity reservation price

160

120

150 110

130 120

100

110 100

90

90 80

80

70 60

Capacity reservation price

Optimal capacity to contract

140

70

50 40

60 2

5

8

11

14

17

20

23

26

29

32

35

Number of pre-selected suppliers Fig. 5. Effect of the number of pre-qualified suppliers on X* and capacity reservation price. #Base case values: P0 = 120; v = 60; b = 500;lh = 400; ll = 200; b1 = 100; b2 = 200; n = 8.

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313

7. Conclusions This paper addresses the issue of capacity contracts in a manufacturing setting involving multiple suppliers in detail and analyses alternative strategies that a buying firm would like to adopt. Earlier studies have exogenously assumed the reservation price of the supplier. In contrast, we have explicitly derived the capacity reservation price. Suppliers participating in such contracts are likely to quote a higher price at higher levels of utilisation and will have several motivational elements to participate in the contract. The optimal number of suppliers from which to contract capacity remains robust to changes in model parameters. However, the variability of demand distribution has a significant effect on the optimal capacity to contract. The proposed model points to several useful suggestions for practice. These include, limiting the number of suppliers at the pre-selection stage itself based on some practical considerations and working with the same set of suppliers and adjusting the capacity contracted in response to changing prices at the spot market. The assumption of a uniform distribution for spot market demand and suppliers’ capacity has, on the one hand enabled greater analytical treatment of the model, and on the other hand makes the problem less general. However, we utilise this trade-off to develop greater understanding of the problem and derive some managerial insights. In this paper we have not modelled the transaction costs of dealing with the m suppliers. For a coordination cost function of the form Cma, where a P 1 and C is a positive parameter, the cost structure will continue to be convex. Therefore, the structural properties of our model will not change, although one can expect the value of m to be lower than what our model has currently estimated. However, modelling transaction costs related to the pre-selection process and dealing with multiple suppliers in a post-selection setting in a more general form will sharpen the insights. Another interesting question to study is the structure of the contract and the policy framework in cases where cost and capacity are correlated. Acknowledgement Both authors gratefully acknowledge the funding support received from the EADS-SMI Endowed Chair for Sourcing and Supply Management for a part of this research. Appendix A Proof of Proposition 1. If X is a non-negative random with density function f(x) and distribution function F(x), and n is a constant then it can be shown that Z n EfminðX ; nÞg ¼ nð1  F ðnÞÞ þ xf ðxÞdx: ðA:1Þ 0

To prove Proposition 1 we need to show that P ðxÞ ¼

oP ðxÞ ox

P 0, where P(x) is given by Eq. (4), which is reproduced below:

ðP 0  vÞfEðminðX ; lÞÞ  EðminðX ; l  xÞÞg þ v: x

Ignoring the term (P0  v), which is positive (otherwise the supplier will go out of business), it suffices to show that the ;lxÞ function GðxÞ ¼ EfminðX ;lÞminðX is an increasing function of x. Substituting (A.1) in G(x) we get x Rl R lx lf1  F ðlÞg þ 0 xf ðxÞdx  ðl  xÞf1  F ðl  xÞg  0 xf ðxÞdx ; ðA:2Þ GðxÞ ¼ x Rl xf ðxÞdx oGðxÞ lf1  F ðlÞg ¼  0 x2 ox x2 xf1  F ðl  xÞg  xðl  xÞf ðl  xÞ þ ðl  xÞf1  F ðl  xÞ þ x2 R lx xf ðxÞdx þ xðl  xÞf ðl  xÞ þ 0 : x2 After algebraic simplification we get the expression below: Rl R lx R lx Rl ðl  xÞf ðxÞdx  0 ðl  xÞf ðxÞdx oGðxÞ  0 xf ðxÞdx þ lF ðlÞ þ 0 xf ðxÞdx  lF ðl  xÞ 0 ¼ ¼ P 0: x2 x2 ox As G(x) is increasing, therefore P(x) is also increasing with x. h Proof of Proposition 2. Let us assume that f(x) is a Uniform (0, b) where b > 0. Then Eq. (A.1) after simplification is written as

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J. Hazra, B. Mahadevan / European Journal of Operational Research 193 (2009) 303–316

2bn  n2 ; 06n6b 2b b ¼ ; n > b: 2

EfminðX ; nÞg ¼

Substituting (A.3) in (4) we get Eq. (5).

ðA:3Þ

h

Proof of Proposition 3. The buyer minimizes his cost by minimizing the function C(m) given by Eq. (8). C(m) is convex in m therefore solving for the first order condition (FOC) will give the optimal solution for m. The FOC is given by oCðmÞ 2AKðlh  ll Þ AK2 P0  v ¼  2 ¼ 0; where A ¼ : om nþ1 2b m qffiffiffiffiffiffiffiffiffiffiffiffiffi Kðnþ1Þ Solving for optimal m we get m ¼ 2ðl . h h ll Þ Proof of Proposition 4. The first order condition (FOC) for optimality is given by oCðX; mÞ ¼0 oX

and

oCðX; mÞ ¼ 0; om

ðA:4Þ

where C(X, m) is given by Eq. (11). Solving (A.4) we get the following expressions:

X ¼

2

P

0 v

2b

i h  b1 ðl  l Þ þ bP20b ll þ nm h l nþ1 1   P0 2 P 0 v þ m 2b b2 b1

ðA:5Þ

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðn þ 1Þ m ¼ : 2ðlh  ll Þ 

ðA:6Þ

Substituting (A.5) in (A.6) we get (A.7) which is quadratic in m.   P0 All ðn þ 1Þ P 0 b1 ðn þ 1Þ 2 þ An ¼ 0; ðm Þ þ 3Am  þ lh  ll 2ðlh  ll Þðb2  b1 Þ b2  b1 P0  v : where A ¼ 2b

ðA:7Þ

It can be shown that one of the solutions for m* will be positive and the other will be negative. pWe, of course, take the positive root as solution for m*. This is given by Eq. (A.8). rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  P 0 b1 ðnþ1Þ 3A þ 9A2 þ b24Pb0 1 Alllhðnþ1Þ þ An þ ll 2ðlh ll Þðb2 b1 Þ  m ¼ : ðA:8Þ 2P 0 b2 b1

We now show that the optimal values of X* and m* are globally optimal. Consider Eq. (1), which is reproduced below:    X þ P 0 E maxð0; D  XÞ: CðX; mÞ ¼ XE P ½mþ1 m The second part of the right-hand side of Eq. (1) is convex (Bean, 2001) Consider now the first part of the right-hand side of Eq. (1):      X nm X2 Let R ¼ XE P ½mþ1 ðl  ll ÞÞ  ¼ P 0 X  A 2Xðll þ : m nþ1 h m Then   oR nm 2X ¼ P 0  A 2ðll þ ðl  ll ÞÞ  ; oX nþ1 h m and

o2 R 2A >0 ¼ m oX2

J. Hazra, B. Mahadevan / European Journal of Operational Research 193 (2009) 303–316

    oR lh  ll X2 ¼ A 2X þ 2 ; om nþ1 m

o2 R 2AX2 ¼ > 0; om2 m3

315

  o2 R o2 R 2ðlh  ll Þ 2X ¼ ¼A  2 : omoX oXom nþ1 m

To show that the Hessian is positive semi-definite we analyse the eigenvalues of the matrix H, where H is the Hessian matrix. We denote k as the eigenvalue of H and I as the identity matrix, then 0  1 2ðlh ll Þ 2A 2X  k A  2 m nþ1 m B C ðA:9Þ H  kI ¼ @  A: 2 2ðlh ll Þ 2AX A nþ1  2X k m3 m2 The eigenvalues are given by the solution of the quadratic Eq. (A.10)    2 2A 2AX2 4A2 X2 2X 2 2 2ðlh  ll Þ þ  2 A ¼ 0: kþ k  m nþ1 m m3 m4 Let k1 and k2 be the two roots of (A.10). Then from Eq. (A.10) we have k1 + k2 > 0. The product of the eigenvalues, k1k2, is greater than 0 if the following condition holds:  2 4A2 X2 2X 2 2ðlh  ll Þ   A >0 nþ1 m2 m4    4X 2ðlh  ll Þ 2ðlh  ll Þ  ) A2 >0 m2 nþ1 nþ1 2ðn þ 1ÞX : ) m2 < ðlh  ll Þ

ðA:10Þ

ðA:11Þ

From Eq. (A.6) it is readily seen that (A.11) is true. We have shown that C(X, m), which is given by Eq. (1), is a linear combination of two convex functions with positive weights. Thus C(X, m) is convex in X and m. Therefore the solution to the first order conditions will yield the global optimum solution. h Proof of Proposition 6. To show that the amount of capacity reserved in advance decreases with uncertainty (that is standard deviation of buyer’s demand) we keep the mean of buyer’s distribution fixed and change the standard deviation. Let b2 = 2a  b1, where a is the buyer’s mean demand. Let r be the standard deviation of buyer’s demand, then we have ffiffiffiffi1 . Replacing b1 and b2 by a and r in (A.5) we get r ¼ bp2 b 12 h i  2A ll þ nm ðl  l Þ þ P 0 ða1:732rÞ h l nþ1 6:93r  X ¼ ; ðA:12Þ P0 2A þ m 6:93r      6:93ða1:732rÞþ6:931:73 P0 P0 a1:732r  P 0 m2A þ 6:93r 6:93r2 oX  2Al½nm þ P 0 6:93r ð6:93rÞ2 ¼ ; 2A  P0 2 or  þ m

6:93r

nm nþ1

where l½nm ¼ ll þ ðlh  ll Þ. As b1 = a  1.732r P 0, therefore the above expression is always negative. Thus the optimal capacity that is reserved decreases with r. In other words, with increase uncertainty in the buyer’s demand the buyer will reserve less capacity. From  (A.6) it is clear that m* will also decrease with r. The amount allocated to each supplier is X m , which again from equation 2m ðlh ll Þ (A.6) is expressed as nþ1 . This again decreases with buyer’s demand uncertainty. h References Anton, J., Yao, D., 1992. Coordination in split award auctions. Quarterly Journal of Economics 20 (4), 681–707. Anupindi, A., Bassok, Y., 1998. Supply contracts with quantity commitments and stochastic demand. In: Tayur, S., Magazine, M., Ganesan, R. (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, pp. 197–232 (Chapter 7). Araman, V., Kleinknecht, J., Akella, R. 2003. Coordination and Risk-sharing in E-Business. Working Paper, Department of Management Science and Engineering, Stanford University. Bean, M.A. 2001. Probability: The Science of Uncertainty, Brooks/Cole (Thomson Learning) Pacific Grove, CA, USA (Chapter 7). Bernstein F., De Vericourt F., to be published. Allocation of supply contracts with service guarantees. Operations Research. Brown, A.O., Lee, H.L., 1997. Optimal pay to delay capacity reservation with applications to the semiconductor industry. Stanford University Working Paper. Cachon, G.P., 2002. Supply Chain Coordination with Contracts. The Wharton School of Business. University of Pennsylvania, Philadelphia, PA. Cachon, G., Zhang, F., 2006. Obtaining fast service in a queuing system via performance-based allocation of demand. Management Science 52 (6), 881– 896.

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