Optimal procurement strategies for online spot markets

European Journal of Operational Research 152 (2004) 781–799 www.elsevier.com/locate/dsw

Optimal procurement strategies for online spot markets Ralf W. Seifert a

a,*

, Ulrich W. Thonemann b, Warren H. Hausman

c

IMD, International Institute for Management Development, Chemin de Bellerive 23, P.O. Box 915, Lausanne CH-1001, Switzerland b ikb, Universit€at M€unster, Universit€atsstraße 14-16, M€unster D-48143, Germany c Department of Management Science and Engineering, Stanford University, Stanford, CA 94305-4026, USA Received 31 May 2001; accepted 16 September 2002

Abstract Spot markets have emerged for a broad range of commodities, and companies have started to use them in addition to their traditional, long-term procurement contracts (forward contracts). In comparison to forward contracts, spot markets offer products at essentially negligible lead time, but typically command a higher expected price for this added flexibility while also exhibiting substantial price uncertainty. In our research, we analyze the resulting procurement challenge and quantify the benefits of using spot markets from a supply chain perspective. We develop and solve mathematical models that determine the optimal order quantity to purchase via forward contracts and the optimal quantity to purchase via spot markets. We analyze the most general situation where commodities can be both bought and sold via a spot market and derive closed-form results for this case. We compare the obtained results to the reference scenario of pure contract sourcing and we include results for situations where the use of spot markets is restricted to either buying or selling only. Our approaches can be used by decision makers to determine optimal procurement strategies based on key parameters such as, demand and spot price volatilities, correlation between demand and spot prices, and risk aversion. The results of our analysis demonstrate that significant profit improvements can be achieved if a moderate fraction of the commodity demand is procured via spot markets. The results also show that companies who use spot markets can offer a higher expected service level, but that they might experience a higher variability in profits than companies who do not use spot markets. We illustrate our analytical results with numerical examples throughout the paper. Ó 2002 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Purchasing; Internet; Spot markets

1. Introduction The creation of online spot markets provides for significant opportunities in the procurement of *

Corresponding author. Tel.: +41-21-618-0111; fax: +41-21618-0707. E-mail address: [email protected] (R.W. Seifert).

supplies and has the potential to radically alter the business landscape [1,6,8,21]. While spot markets have long been used for grains, livestock, and oil [41], only recently have they been formed for memory chips, chemicals, energy, telecommunication bandwidth, etc. Online spot markets are proliferating relatively rapidly for a broad range of (near-) commodities with prominent players such

0377-2217/$ - see front matter Ó 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00754-3

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as ChemConnect, AltraNet, E-Steel, PaperExchange, and Converge serving as examples [8,21]. Many companies have already started to use spot markets (whether independent or consortiumbased) in addition to their traditional procurement via forward contracts. For example, spot market volumes as a percentage of total volume are estimated to account for 5% of chemicals and steel sales [2,9] and up to 30% of memory chips sales [13,18]. Yet, many companies appear to utilize spot markets in an ad hoc manner as opposed to viewing them as an integral part of their procurement strategy. The lack of quantitative decision tools contributes to the remaining hesitation of many buyers and sellers to participate in spot markets. As Kaplan and Sawhney [25] rightly note: ‘‘The matching mechanism works best when buyers and sellers are sophisticated enough to deal with dynamic pricing.’’ Likewise, McKinsey and Company and CAPS Research identified the need ‘‘to determine the appropriate contract-to-spot ratio for commodity purchases’’ in a broad survey on the impact of B2B e
marketplaces [29]. In this paper, we address this issue and analyze in detail optimal procurement strategies in situations where supplies can be bought both via forward contracts and via spot markets. In this context, a forward contract is an agreement to buy a commodity at a certain future time for a certain price. In contrast, a spot contract is an agreement to buy (or sell) a commodity today [23]. We develop mathematical models for using spot markets for both buying and selling and we include results for situations where the use of spot markets is restricted to either buying or selling only. These models enable us to analyze how spot markets affect the optimal order quantity via forward contract (subsequently simply referred to as contract quantity), to quantify the magnitude of the performance improvement that can be achieved by taking optimal advantage of spot markets, and to assess the effect of risk-averse decision making on the optimal procurement strategy. Our mathematical models incorporate the major parameters of supply chains where supplies can be bought via forward contracts and via spot markets. The results of our analysis can be used by decision makers to determine the effect of the key

spot market characteristics on optimal contract quantities and profits, to identify commodities for which the use of spot markets is attractive, and to decide how to take optimal advantage of spot markets. To foster additional managerial insights, we perform extensive sensitivity analyses and illustrate our results with numerical examples. This paper is organized as follows. In Section 2, we review the relevant literature. In Section 3, we describe our mathematical models in detail. We characterize the optimal procurement strategy and discuss the managerial implications of our analysis in detail. In Section 4, we analyze the benefits of using spot markets for buying and selling as opposed to not using them at all or using them for buying or for selling only. In Section 5, we show how the spot market parameters can be estimated in a way consistent with our model. In Section 6, we summarize our key findings. Mathematical references and derivations are contained in the Appendix.

2. Literature review Two major lines of literature are relevant to our research: the finance literature on commodity pricing and hedging as well as the supply chain management literature on dual sourcing and capacity-related supply contracting. In the finance literature, there exists a substantial amount of research on pricing and hedging of commodity-linked contingent claims [36]. The dominant approach is to devise equilibrium models of commodity spot prices and forward prices that assume one or multiple exogenously given stochastic input factors such as spot prices, convenience yields (defined in Section 3), or interest rates. Gibson and Schwartz [20] developed a twofactor model where the first factor is the spot price of the commodity and the second factor is the instantaneous convenience yield. Schwartz [38] extends this model by introducing a third stochastic factor, instantaneous interest rates. Miltersen and Schwartz [33] further distinguish between forward and future convenience yields when interest rates are stochastic. These models provide powerful tools for analyzing the term

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structure of futures prices. However, they do not explicitly consider demand uncertainty nor do they allow for inventories. A closely related stream of research that considers the role of inventories employs rational expectation models to analyze commodity price dynamics [11,14,45]. The general focus of this research is on the economic theory of storage. Most relevant to our work is Routledge et al. [36]. They developed a one-factor equilibrium model of forward prices for commodities with Markovian netdemand shocks as a stochastic input factor. In their model, the spot price and convenience yield are endogenous stochastic processes. This setup allows Routledge et al. to show an inherent correlation between spot prices and convenience yields. In addition, they explicitly incorporate a non-negativity constraint on inventory and demonstrate that convenience yields (and therefore spot prices) are high only if there is a shortage of the commodity. While this research generates valuable economic insights into the effect of inventories on spot prices, the effect of spot markets on supply chain operation is not explicitly considered. Within the supply chain management literature several authors have analyzed the effect of dual sourcing on the optimal control of inventory systems. Ramasesh et al. [35] analyzed an inventory system where companies can choose between two supply options with different lead times. In their model, lead times are uniformly or exponentially distributed and demand is constant. Fong et al. [19] extend this work and allow for normally distributed demand and Erlang distributed supplier lead times. However, Ramasesh et al. and Fong et al. assumed that orders for both supply options are placed concurrently. Other researchers have assumed sequential decision making. Rudi [37] uses a two-stage stochastic linear programming formulation to analyze the optimal order split between make-to-stock and assemble-to-order. Eeckhoudt et al. [16] consider risk aversion and analyze general comparative-static effects of changes in cost parameters when complementary units can be bought after demand has been realized. However, all cost parameters are assumed to be known with certainty.

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A recent exception that admits price and demand uncertainty is an inventory control model by Cohen and Agrawal [12]. Cohen and Agrawal formulate and numerically solve a stochastic dynamic program to analyze the trade-off between long- and short-term contracts, but do not allow for a combined use of long- and short-term contracts. This combined use, however, is explicitly considered and shown to be favorable by Araman et al. [4] in a model developed subsequent to our research. While Araman et al. consider a setting similar to ours, they employ a very different set of underlining assumptions. In particular, Araman et al. assume that the decision-maker is strictly risk-neutral, that the manufacturer is restricted to buying only via the spot market (despite having contracted additional units at a potentially much lower price than the current spot market price), and that spot markets are not liquid, thus, they exhibit a pronounced (yet independent) pricequantity effect. Closed-form results cannot be obtained. Finally, we would like to reference a substantial stream of supply chain management literature that focuses on optimal capacity-related design of supply chain contracts and analyzes the resulting operational effects in terms of mitigating supply risk under pre-negotiated terms. An excellent classification of this literature is provided in the recent review papers by Anupindi and Bassok [3] and Tsay et al. [42]. In this paper, we go beyond the existing literature by explicitly analyzing the effect of spot markets on optimal inventory control. Our model takes into account spot price uncertainty, correlation between demand and spot prices, and risk aversion.

3. Mathematical models The objective of our research is to analyze and quantify the benefits of utilizing online spot markets in supply chain operations. In this section, we develop and solve mathematical models to explicitly characterize the optimal usage of spot markets in the procurement of consumption commodities. We consider supply chains with two

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supply options: under option 1, the buyer uses a forward contract with known lead time T and fixed unit price c. Under option 2, the buyer uses a spot market with essentially negligible lead time but stochastic spot price s. Stochastic demand n occurs at time T , and units purchased via both options can be used to fill this demand. The economics of the supply chain are as follows: sales (consumption or usage in assembly) of the commodity translate into revenue r > c per unit. If the company can use spot markets for buying, demand in excess of on-hand inventory is met at spot price s. If the company does not use spot markets for buying, unmet demand is lost and an additional stockout penalty cost p P 0 per unit might be incurred. Similarly, if the company can use spot markets for selling, excess inventory is ‘‘salvaged’’ at spot price s. Alternatively, if the company does not use spot markets for selling, excess inventory is salvaged at unit price v (0 6 v < c). The expected spot price for both buying or selling at time T is ls with standard deviation rs . The actual spot price s at time T does not depend on the quantity traded by the firm on the spot market. The buyerÕs decision problem is as follows: at time 0, the buyer must decide on the contract quantity q. The buyer has an unbiased forecast of the demand at time T and knows the cumulative distribution function (c.d.f.) Fd ð
Þ with mean demand ld and standard deviation rd . At time T , the contract quantity q is delivered and demand n occurs. If the contract quantity q is less (greater) than the demand n, the buyer will buy n  q (salvage q  n) units. An extension to our research may assume more complex decision making instances but the above description captures the essential trade-offs for an initial analysis. The buyer seeks to maximize expected profits EPðqÞ but is generally averse to a variance of profits VarPðqÞ. We reflect this in our model by maximizing ZðqÞ ¼ EPðqÞ  kVarPðqÞ, with k P 0. This form of ZðqÞ is widely used in portfolio theory and yields mean–variance efficient outcomes [28]. Similarly, the inclusion of higher moments into the objective function has been widely advocated in the inventory control literature, e.g., Lau [26]. An in-depth discussion of the justification and

limitations of this approach is provided by BarShira and Finkelshtain [7], Meyer [32], and Tsiang [43]. An alternative approach is to maximize expected utility. This approach is often used in economics to analyze the direction of change induced by a hypothetical shift in parameters but is less suitable in our setting because our objective is to characterize the optimal solution explicitly [43]. Finally, we should note that our managerial discussion will be focused on cases where buying via spot markets involves a price premium in expectation, i.e., ls P c. Such a price premium is reflective of a positive convenience yield [24,46]. Positive convenience yields are considered a fundamental characteristic of commodity markets for consumption goods [27,36] and reflect the marketÕs expectation concerning the future availability of the commodity. The greater the possibility that general shortages may arise during the lead time of the forward contract, the higher the convenience yield [23]. Thus, the convenience yield may be thought of as the value of being able to profit from temporary local shortages of the commodity through ownership of the commodity [10]. As a procurement manager at Hewlett–Packard noted: ‘‘I was delighted to find a much needed component for assembly of one of our products at only three times the regular market price when faced with a supply shortage.’’ Subsequently, we will discuss a spot market model where commodities can be both bought and sold (Section 3.1) and we review a pure contract sourcing model (Section 3.2) to provide for a performance benchmark. 3.1. Buy and sell model 3.1.1. Model description If supplies can be both bought and sold (BS) via spot markets, the profit function is þ

PBS ðqÞ ¼ rn  cq  sðn  qÞ þ sðq  nÞ

þ

¼ rn  cq  sðn  qÞ; where s denotes the spot price at time T . When demand n exceeds the contract quantity q, we use a spot market to procure n  q units to prevent

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stockouts. When demand n is less than the contract quantity q, we salvage q  n units of excess inventory by selling them on spot markets at spot price s. Demand n and spot price s are typically positively correlated [22,27]. When spot markets are liquid, this correlation will be due to the realized company demand being correlated with the overall industry demand. The industry demand governs the spot market price while the much smaller company-specific demand should not be subject to a price-quantity effect. Thus in stochastic terms, when demand is high, the spot price is more likely to be high than when demand is low. We explicitly analyze such positive correlation between demand n and spot price s by modelling ðn; sÞ as a Bivariate Normal (BN) distribution with correlation q P 0: ðn; sÞ  BN ½ld ; ls ; r2d ; r2s ; q . While the Normal demand assumption is commonplace in the supply chain literature, the implied Normal assumption on the spot market price is reconciled in Section 5. (In Section 5, we show how the spot market parameters can be efficiently estimated based on a mean-reverting Ornstein–Uhlenbeck process, which is consistent with Normally distributed spot prices.) The expected value and variance of PBS ðqÞ can be computed as (Appendix A.1) EPBS ðqÞ ¼ ðr  ls Þld  qrs rd þ ðls  cÞq and

785

3.1.2. Closed-form results For k > 0, the resulting optimal contract quantity q BS can be computed as l c rd ð1Þ q BS ¼ ld þ s 2  q ðr  ls Þ: 2krs rs Using Eq. (1), we can further compute the optimal expected profits and the optimal variances of profits as EP BS ¼ ðr  cÞld þ q

ðls  cÞ 2kr2s

2

rd ½ðr  ls Þðls  cÞ þ r2s rs

ð2Þ

and 2

VarP BS ¼ ð1  q2 Þðr  ls Þ r2d þ r2s r2d þ

ðls  cÞ2  q2 r2s ð3l2d  r2d Þ: 4k 2 r2s

ð3Þ

Based on Eqs. (1)–(3), we can conduct full sensitivity analysis with respect to all major parameters of the given procurement problem. In what follows, we analyze the effects of risk-averse decision making, spot-price premium, spot-price volatility, and the correlation between demand and spot prices on the optimal contract quantity, the optimal expected profits, and the optimal variances of profits in detail. In the numerical experiments, we used c ¼ 10, r ¼ 30, ld ¼ 100, rd ¼ 25, ls ¼ 12, rs ¼ 2:5, q ¼ 0:2; and k ¼ 0:005 to exemplify our results (unless specified otherwise).

2

VarPBS ðqÞ ¼ ½ðr  ls Þ þ r2s r2d 2

þ r2s ðld  qÞ  qrs ½qrs ð3l2d  r2d Þ þ 2rd ðr  ls Þðld  qÞ : The expected profits are a linear function of the contract quantity q with coefficient ðls  cÞ. Thus, a risk-neutral buyer (indifferent about the variance in profits) would order an infinite quantity via forward contract whenever the expected spot price ls at time T is higher than the contract price c. Of course, such a procurement strategy would be unrealistic and thus it is imperative to take into account the variance of profits as we maximize ZBS ðqÞ ¼ EPBS ðqÞ  kVarPBS ðqÞ, i.e., to require k > 0.

3.1.3. Risk-averse decision making Risk aversion in the buyerÕs decision making towards demand and spot price variability is reflected in our objective function by means of factor k. The more risk-averse the buyer, the larger the value of k. Thus, we can analyze how a more risk-averse buyer would optimally use spot markets by analyzing the effect of an increase in k on the optimal solution. As k increases, the optimal contract quantity decreases (see Eq. (1)), which in turn results in lower optimal expected profits and lower optimal variances of profits (see Eqs. (2) and (3)). From Eqs. (1)–(3), we can furthermore see that q BS , EP BS , and VarP BS are convex decreasing in k.

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pend on k for q ¼ 0. Each value of k corresponds to a tangent point with slope k to the efficient frontier of the BS-model, i.e., to a mean-variance efficient combination of expected profits EPBS ð
Þ and variances of profits VarPBS ð
Þ. For example, for k ¼ 0:005 we have q BS ¼ 132, EP BS ¼ 2; 064, and VarP BS ¼ 212; 806. Finally, we can see from Eqs. (1)–(3) that the sensitivity in k exclusively depends on the spotprice premium ðls  cÞ and the spot-price volatility rs . Demand variability rd does not affect the sensitivity in k as long as all demand can be filled and all excess inventory can be salvaged via spot markets.

This implies that for very small values of k, the optimal order quantity q BS and the optimal expected profits EP BS are very large. Thus, an essentially risk-neutral buyer would act predominately as a speculator by ordering well in excess of the mean demand via forward contract and betting on selling the remaining excess supplies to spot markets at a spot-price premium at time T . However, as k ! 0 the optimal variance of profits VarP BS increases even faster than the optimal expected profits because EP BS contains k in the denominator (Eq. (2)), whereas VarP BS contains k 2 in the denominator (Eq. (3)). On the other hand, as k grows large, q BS , EP BS , and VarP BS approach finite limits. Without correlation between demand and spot prices, i.e., q ¼ 0, the optimal contract quantity q BS approaches the mean demand ld because the buyer seeks to minimize the use of spot markets (both for buying and for selling) due to the inherent spot price uncertainty. The optimal expected profits approach ðr  cÞld , which is the maximal expected revenue from commodity consumption or baseline profits, because the buyer no longer acts as a speculator. Similarly, the optimal variances of profits approach their minimum given by ½ðr  2 ls Þ þ r2s r2d for q ¼ 0. For q > 0, q BS , EP BS , and VarP BS still approach finite limits as k grows large, but the limits are lower than for q ¼ 0 (see Eqs. (1)–(3)). To illustrate the above, Fig. 1 shows how the optimal contract quantity, the optimal expected profits, and the optimal variances of profits de-

E

3.1.4. Spot-price premium Here, we analyze how the expected spot-price premium ðls  cÞ, or equivalently, the expected spot price ls affects the optimal solution. Without a price premium ðls ¼ cÞ, the buyer has no incentive to speculate in the spot markets––resulting in an optimal solution which is identical to the one obtained for k ! 1. With a price-premium ðls > cÞ, the optimal contract quantity q BS , the optimal expected profits EP BS , and the optimal variances of profits VarP BS directly depend on ls . The optimal contract quantity q BS increases in ls and exceeds the mean demand because the buyer can sell excess inventory via spot markets at an increasing marginal contribution ðls  cÞ. The increase of q BS is linear in ls and more pronounced the less risk-averse the buyer is (see Eq. (1)). The increase of q BS is linear in ls because the buyer is

q*BS 250

BS 2,200

k =0.0025 q*BS =164

200

k =0.005 q*BS =132

2,100

150

µ d= 100

2,000

k= ∞ q*BS = µ d =100 1,900 200,000

50

0 220,000

240,000

260,000

Var

0

0.0025

BS

Fig. 1. Effect of k on EP BS , VarP BS and q BS for q ¼ 0.

0.005

0.0075

k

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variances of profits depend on ls for q ¼ 0. For ls ¼ c, the efficient frontier collapses to a single efficient point because the buyer can not increase expected profits by speculating in the spot markets.

torn between being risk averse both towards buying and towards selling via spot markets at time T . The optimal expected profits EP BS are decreasing–increasing in the expected spot price ls . For q > 0, we can see from Eq. (2) that EP BS is decreasing in ls when ls 6 ðc þ qkrs rd ðr þ cÞÞ= ð1 þ 2qkrs rd Þ and increasing in ls otherwise. EP BS initially decreases in ls if q > 0 because buying complementary units via spot markets to prevent stockouts becomes more costly. After the initial decrease, the optimal expected profits increase in ls and exceed the base-line profits ðr  cÞld because additional revenues can be generated by selling excess inventory to spot markets at an increasing marginal contribution ðls  cÞ. Note that for q ¼ 0 the optimal expected profits are strictly increasing in ls (Eq. (2)). The optimal variance of profits VarP BS is also decreasing–increasing in the expected spot price ls . From Eq. (3), it can be shown that the optimal variances of profits are decreasing in ls when ls 6 ðc þ 4k 2 ð1  q2 Þr2d r2s rÞ=ð1 þ 4k 2 ð1  q2 Þr2d r2s Þ and strictly increasing in ls otherwise. The optimal variances of profits initially decrease in ls because an increasing optimal contract quantity provides better protection against demand variability. However, as ls increases further, VarP BS increases because the buyer is already sufficiently protected against demand variability but becomes increasingly exposed to the variability in spot prices when selling excess inventory. Fig. 2 shows how the optimal contract quantity, the optimal expected profits, and the optimal

E

BS

787

3.1.5. Spot-price volatility The optimal contract quantity q BS is strictly decreasing in spot price volatility rs when rs < ðls  cÞ=ðqkrd ðr  ls ÞÞ and increasing in rs otherwise (see Eq. (1)). When the variability in spot prices is low, selling excess inventory via spot markets at a spot-price premium is attractive. In addition, a high contract quantity provides good protection against demand variability. For larger values of rs , however, the buyer limits the use of spot markets (both buying and selling) and, thus, the optimal contract quantity q BS approaches the mean demand ld as rs ! 1 (Fig. 3). Fig. 3 also shows how the optimal expected profits and the optimal variances of profits depend on rs . From Fig. 3, we can see that a reduced spot market volatility rs improves the efficient frontier by allowing for higher optimal expected profits and lower optimal variances of profits. 3.1.6. Correlation between demand and spot prices The optimal contract quantity q BS is strictly decreasing in correlation q if ls < r (see Eq. (1)). As q increases, the contribution from selling excess inventory via spot markets decreases because when excess inventory is high, the spot price is more likely to be low than when excess inventory is low

q*BS 250

2,500

200 2,250

µ s=12 µ s=11 2,000

1,750 200,000

k =0.005

150

µ d = 100 µ s=10

50

0 250,000

300,000

Var

10

11

BS

Fig. 2. Effect of ls on EP BS , VarP BS and q BS for q ¼ 0.

12

13

14

µs

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BS

2,500

q*BS 250 200

σs=1.25 2,250

σ s=2.5

ρ =0.2

µ d = 100

σ s=3.75

2,000

ρ =0.0

150

ρ =0.4

50 0

1,750 200,000

250,000

0

300,000

Var

1,25

2,5

3,75

5 σs

BS

Fig. 3. Effect of rs on EP BS , VarP BS , and q BS .

and vice versa. Furthermore, we can see from Eqs. (2) and (3) that the optimal expected profits

EP BS and the optimal variances of profits VarP pffiffiffi BS are strictly decreasing in q given ðrd =ld Þ < 3, a condition we would expect to hold in practise. Fig. 4 illustrates the effect of q on the optimal solution. Note that in Fig. 4, the optimal contract quantity q BS would be negative for large values of q because of the Normal demand assumption. In these cases, we would set q BS equal zero.

inventory via spot markets is reduced. Thus, the buyer will procure more from spot markets as rd increases in order to reduce the likelihood of excess inventory. For q ¼ 0, however, the optimal contract quantity q BS and the optimal expected profits EP BS are unaffected by the demand variability rd (see Eqs. (1) and (2)) because the optimal balance between buying and selling via spot markets is not affected by rd since all demand will be filled. Finally, we can see from Eq. (3) that an increase in demand variability rd increases the optimal variances of profits VarP BS , as we would expect.

3.1.7. Demand distribution The optimal contract quantity q BS and the optimal expected profits EP BS are linearly increasing in the mean demand ld because the company will fill all demand. For q > 0, q BS and EP BS are decreasing in demand variability rd because the expected profit contribution from selling excess

E

BS

3.1.8. Comments on modelling assumptions The original stockout penalty cost p and salvage value outside the spot market v do not enter the BS-model because of our implicit assumption of using the spot market exclusively. This assumption

2,500

q*BS 250 200

ρ =0.0

2,250

150

k =0.005 2,000

µ d = 100

ρ =0.4 ρ =0.2

1,750 100,000

150,000

50

200,000

250,000

300,000

Var

BS

0 0

0.2

Fig. 4. Effect of q and k on EP BS , VarP BS , and q BS .

0.4

0.6

0.8

1

ρ

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799

is conservative in that it leads to a lower bound on expected profits and to an upper bound on variance of profits. For completeness, we include results relaxing this assumption in Section 4. 3.2. Pure contract sourcing Following our detailed analysis of the BSmodel, we now analyze a pure contract (PC) sourcing model where spot markets are not used. The PC-model is then used as a reference point for the evaluation of models that take advantage of spot markets in Section 4. The profit function for PC sourcing is þ

þ

PPC ðqÞ ¼ ðr  cÞq  ðr  vÞðq  nÞ  pðn  qÞ : The expected value and variance of PPC ðqÞ can be computed as (Appendix A.2) EPPC ðqÞ ¼ ðr  cÞq  ðr  vÞV1 ðqÞ  pP1 ðqÞ and VarPPC ðqÞ ¼ ðr  vÞ2 V2 ðqÞ þ p2 P2 ðqÞ 2

 ½ðr  vÞV1 ðqÞ þ pP1 ðqÞ : convenience, we define Vn ðqÞ ¼ R q For notational R1 n n ðq  nÞ dFd ðnÞ and Pn ðqÞ ¼ n¼q ðn  qÞ  n¼1 dFd ðnÞ. V1 ðqÞ denotes the expected excess inventory and P1 ðqÞ denotes the expected number of shortages if q units are at hand to satisfy demand n (Appendix A.3). Analogous to the BS-model, the buyer maximizes ZPC ðqÞ ¼ EPPC ðqÞ  kVarPPC ðqÞ. Fig. 5 shows how the optimal solution depends on k for differ-

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ent values of the stockout penalty cost p, with salvage value v ¼ 8 and n  Normal½ld ; r2d . The optimal contract quantity q PC must satisfy (Appendix A.2) pþrc  2k½ðr  vÞ½1  Fd ðqÞ V1 ðqÞ Fd ðqÞ ¼ pþrv  pFd ðqÞP1 ðqÞ : ð4Þ For k ¼ 0, Eq. (4) reduces to a standard newsvendor problem [34], while for k > 0, q PC is also affected by the variance of profits. We can see from Eq. (4) that q PC is decreasing in k for p 6 p, p can be computed by solving    1 p þ r  c V1 Fd pþrv    pðp þ r  cÞ 1 p þ r  c P1 Fd ¼ ðr  vÞðc  vÞ pþrv and increasing in k for p > p. For p 6 p, q PC decreases in k because the buyer is risk averse towards excess inventory. For p > p, q PC increases in k because the buyerÕs stockout aversion dominates over the buyerÕs aversion towards excess inventory [39]. In all subsequent numerical experiments, we will use p ¼ 20, which corresponds to the profit margin, to exemplify our results. We note p ¼ 146 as a reference point.

4. Model comparison In this section, we analyze the performance improvement over the PC-model that can be

Fig. 5. Effect of k and p on EP PC , VarP PC , and q PC .

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achieved by taking optimal advantage of spot markets. We base our comparison on the optimal contract quantity, the optimal expected profits and the optimal variance of profits. Furthermore, we consider the implied expected fill rate (EFR) because some demands are lost in the PC-model, while all demands can be filled in the BS-model. To foster additional structural insights, we also include results for similar models where spot markets are either used for buying only (BOmodel) or selling only (SO-model). A full development and sensitivity analysis of the BO-model and the SO-model are provided by Seifert [40] (Appendices A.4 and A.5 provide a brief reference of the respective results). Table 1 summarizes the optimal solution and the corresponding EFR for k ¼ 0:005 and q ¼ 0:2. Table 1 shows that significant profit gains can be achieved by taking full advantage of spot markets. Furthermore, the EFR is improved when spot markets are used. The resulting optimal variance of profits when buying via spot markets, however, can be higher than in the PC-model be-

Table 1 Performance summary for k ¼ 0:005 and q ¼ 0:2 Model

q

EP

VarP

EFR (%)

BS PC BO SO

96.0 106.8 63.1 112.4

1,980 1,695 1,911 1,826

197,363 113,697 202,963 104,945

100 93 100 95

cause all demands are filled in the BS-model and in the BO-model. Since both the BS-model and the BO-model share this assumption, we first compare these two models. Then we compare the PC-model and the SO-model. From Fig. 6, we can see that the BS-model strictly dominates the BO-model. The optimal expected profits in the BO-model are bounded above by the base-line profits ðr  cÞld (represented by the horizontal line in Fig. 6), while in the BSmodel, additional revenues can be generated by selling excess inventory to spot markets. Both procurement strategies yield an EFR of 100%. However, the optimal contract quantities (shown in brackets in Fig. 6) are lower in the BO-model than in the BS-model (for k constant) because under the BO-model excess inventory cannot be salvaged at a profit. Thus, on average the buyer will buy more via forward contract (and thus, less via spot markets) under the BS-model than under the BO-model. Similarly, we can see from Fig. 6 that the SOmodel strictly dominates the PC-model. The optimal expected profits in the PC-model are bounded above by the base-line profits while in the SOmodel, additional profits can be generated by selling excess inventory to spot markets. Both procurement strategies yield an EFR less than 100% and the optimal variance of profits is significantly lower when the EFR is low. The optimal contract quantities and the EFR (shown in brackets in Fig. 6) are lower in the PC-model than

Fig. 6. Comparison of spot market models.

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799

in the SO-model (for k constant) because under the PC-model excess inventory cannot be salvaged profitably. Thus, on average, the buyer will buy more via forward contracts under the SO-model than under the PC-model. To gain additional insights into the optimal use of spot markets, we can complement our model comparison by allowing the service level to be variable in all four models. Specifically, we may relax the assumption that all demand be filled when buying via spot markets and assume instead that units will be bought via spot markets only if the spot price s is less than the corresponding lost sales cost. These modified models no longer imply a 100% expected fill rate and are subsequently rec and d ferred to as BS BSBO-model. The profit funcBO c tion for the BS BS-model is P b ðqÞ ¼ rn  cq  min½s; r þ p ðn  qÞþ BS

þ

þ sðq  nÞ

d and the profit function for the BO-model BO is þ

Pc ðqÞ ¼ rn  cq  min½s; r þ p ðn  qÞ BO BO

þ

þ vðq  nÞ : The expected profit and the variance of profc and BO d its for the BS BSBO-model can be readily computed following standard definitions. As in our previous models, the buyer maximizes ZðqÞ ¼ EPðqÞ  kVarPðqÞ but the effective service level (or the EFR) will now depend on the given shortage penalty cost p in all four models. To il-

791

lustrate the model behavior we employ separate numerical experiments where the shortage penalty cost p is varied between 0 to 10 given k ¼ 0:005 and q ¼ 0:2. Furthermore, we use c ¼ 10; r ¼ 14; v ¼ 6; ld ¼ 100, rd ¼ 25, ls ¼ 11, and rs ¼ 2:5 (i.e., product data with a lower revenue figure than originally) to allow for more pronounced sensitivity of the model performance and the implied service level in the shortage penalty cost p than in our original numerical experiment. The resulting c d model performances for the BS BS-, BO-, SO-, and BO PC-models are shown in Fig. 7. c From Fig. 7, we can see that the BS BS-model dominates all other models both in terms of the attained EFR and the attained objective function value Z. Furthermore, we can see that being able to buy complementary units via spot markets is most valuable when the shortage penalty cost p is high, while the SO-model can be seen to perform equally c well to the BS BS-model and better than the d BOBO model in terms of Z for lower values of p. We next analyze how the optimal usages of spot markets in terms of buying and selling is affected by the spot-price premium and the spot-price volatility. We define EPB ¼ ðP1 ðq Þ=ld Þ  100% and EPS ¼ ðV1 ðq Þ=ld Þ  100%. EPB denotes the expected percentage bought via spot markets and EPS denotes the expected percentage sold via spot markets if q units were ordered via forward contract to satisfy demand n. Fig. 8 shows how EPB (left bar) and EPS (right bar) are affected by the spot-price premium and the spot-price volatility

Fig. 7. Effect of p on EFR and Z.

792

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799 Spot-Price Volatility Low (σs=1.25) EPB

BO Low (µs=11)

EPS

17 5

SO Spot-Price Premium

High (µs=12)

BS SO

EPS

EPB

3

22

16

EPS

44

EPB

EPS

56

18

3

15

EPB

38

0

EPS

31

25

17

EPB

BO

EPB

High (σs=3.75)

41

51

BS

Medium (σ s=2.5)

EPS

31

12 8

17

16 6

16

Fig. 8. Effect of ls and rs on EPB and EPS.

for the BO-, BS-, and SO-model. As the spot-price premium increases, emphasis on using spot markets shifts from buying to selling because when expected spot prices are low, the buyer benefits from buying via spot markets ‘‘as needed’’, while when expected spot prices are high, the buyer gains by selling excess inventory to spot markets at a price premium. Furthermore, Fig. 8 shows that a risk averse buyer under the BO-model and under the SO-model will generally use spot markets less (reflected by lower EPB and EPS) when the spotprice volatility is higher. Under the BS-model, however, an increased spot-price volatility may increase the use of spot markets while simultaneously shifting emphasis on using spot markets from selling to buying. The above model comparison and sensitivity analysis exemplify the results that can be obtained with the presented mathematical models for taking advantage of spot markets. By enabling decisionmakers to quantify the corresponding magnitude of the performance improvements, one may indirectly also assess the relative importance of additional cost factors such as membership fees, trade commissions, etc. not currently accounted for in our model but typically associated with sourcing via spot markets.

5. Parameter estimation Throughout Sections 3 and 4, we regarded the expected spot price ls and its standard deviation rs to be exogenously given. In this section, we show how the expected spot price ls and its standard deviation rs can be obtained from spot market data in a way consistent with the presented model. We decompose spot prices for a specific commodity into two elements: a general price pattern gt that is common to a family of commodities and a random price pattern ~st that is unique to a specific commodity. The spot price can then be expressed as st ¼ gt þ ~st . The general price pattern gt is due to technology diffusion and can be determined for an entire family of commodities (e.g., DRAMs, CPUs, etc.) over time. For example, gt could reflect an approximately linear or geometric price decline over time, i.e., gt ¼ g0  at or gt ¼ g0 eat , with a P 0. a can be estimated via least square linear regression on st or lnðst Þ using historical spot prices of a family of commodities. Fig. 9 shows an example for the general price patterns of DRAMs over time. The random price pattern ~st is due to commodity specific supply and demand shocks and

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799

793

US $ 1000 64M 4M

64K 100

256K

16M

1M

16K

logarithmic scale

4K

10

1

0.1 1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

Time

Fig. 9. General price patterns for DRAM components.

often exhibits a mean-reverting pattern where prices fluctuate around an equilibrium level. A price process with mean-reversion is considered a natural choice for many commodities [10,36,38]. In particular, commodity spot prices are frequently represented as a mean-reverting Ornstein–Uhlenbeck (OU) process [15], which we adopt for our representation of ~st . To arrive at an estimate for the expected spot price ls and its standard deviation rs at time t þ T (present time plus lead time), we proceed as follows: 1. Compute the trend coefficient a using a regression on historic spot prices for commodities from the same family of commodities. 2. Compute st ¼ st  gt using trend coefficient a and g0 ¼ s0 , where s0 denotes the first available data point for the particular commodity. 3. Run a regression on st  st1 ¼ a þ bst1 þ et , where et is Gaussian noise with mean zero and variance r2s ðT ¼ 1Þ. The result of this regression allows us to estimate the parameters of mean ¼ ða=bÞ, reversion: the equilibrium level m the speed of reversion g ¼  lnð1 þ bÞ, and

2

the variance r2 ¼ r2e ðlnð1 þ bÞÞ=ðð1 þ bÞ  1Þ, where re is the standard error of the regression analysis [15].  þ gt , i.e., adjust gt for hav4. Compute gt ¼ m ing initially used g0 ¼ s0 by the equilibrium . level m 5. The distribution of the spot price stþT conditioned on st is Normal distributed with mean ls ðt þ T j st Þ ¼ gtþT þ ðst  gt ÞegT and variance r2s ðt þ T j st Þ ¼ ð1  e2gT Þðr2 =2gÞ [15]. Note that the speed of reversion g reflects a restoring force directed towards gt and is of a magnitude proportional to the distance. As a result, the variance r2s grows only initially but stabilizes after some time, i.e., r2s ! ðr2 =2gÞ as T ! 1 [15]. Let us consider a specific example to illustrate our approach. Fig. 10 shows spot prices for a 64M (8M  8) SDRAM PC66 component starting April 1998. The lead time for forward contracts of this component is T ¼ 3 months (¼63 business days). First, we estimate a ¼ 0:007 using gt;i ¼ s0;i  at across multiple similar DRAM components i. Then, we use the actual spot prices st¼0

794

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799

US$ 14

12

10

8

6 4/1/98

6/1/98

8/1/98

10/1/98

12/1/98

2/1/99

4/1/99

Fig. 10. Daily spot prices and forecast with 90% confidence interval (Source: AICE).

through st¼200 for the 64M (8M  8) SDRAM PC66 component (with s0 ¼ 12:35 and s200 ¼ 10:29) to determine the parameters of mean reversion: m ¼ 1:82, g ¼ 0:12, and r2 ¼ 0:19 as well as g200 ¼ 9:15 and gtþT ¼263 ¼ 8:7. Finally, we compute ls ð263js200 Þ ¼ 8:7 and rs ð263 j s200 Þ ¼ 0:9. The resulting forecast, along with its 90% confidence interval, is shown in Fig. 10 for t ¼ 201 to 263 (i.e., to the right of the vertical line). Finally, we would like to point out that our representation of spot prices is but one of many alternatives [15,38,44]. In particular, more elaborate models such as a geometric Ornstein– Uhlenbeck model with drift term [31] or a Poisson–Gaussian model of mean-reversion with jumps [30] could be used instead. However, the limited spot price history for most commodities is often not sufficient to estimate the parameter for these more complete models.

6. Conclusion Spot markets have recently emerged for a broad range of commodities and many companies have started to utilize them in addition to their traditional procurement via forward contracts. In this paper, we analyzed this development from a supply chain perspective. We developed and solved

mathematical models that determine the optimal order quantity to purchase via traditional forward contracts and the optimal quantity to purchase via spot markets. We analyzed the most general situation where commodities can be both bought and sold via a spot market. We obtained closed-form results that enabled us to provide full sensitivity analysis on the optimal procurement strategies. To gain additional structural insights, we complemented our research by considering situations where we restrict the use of spot markets to buying or selling only. These additions allowed us to further analyze the merits of leveraging online spot markets and to enrich the performance comparison versus a pure contract sourcing via traditional forward contracts. The approaches presented in our paper can be used by decision makers to identify commodities for which the use of spot markets is attractive as well as to determine optimal procurement strategies for those commodities based on some key parameters, such as demand and spot price volatilities, correlation between demand and spot prices, and risk aversion. Our analysis allows to explicitly quantify the potential performance gains that companies may realize by making optimal use of online spot markets and shows that significant profit improvements can indeed be achieved if a moderate fraction of the commodity demand is

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799

procured via spot markets. Our results also show that companies who use spot markets can offer a higher expected service level, but that they might experience a higher variability in profits than companies who do not use spot markets. Finally, the presented model provides valuable insights into the required tools and capabilities that companies will need to acquire to more fully leverage spot markets.

795

The variance of profits can be computed as VarPBS ðqÞ ¼ Var½rn  sðn  qÞ ¼ E½Var½rn  sðn  qÞjn þ Var½E½rn  sðn  qÞjn 2

¼ E½ðn  qÞ Var½s j n þ Var½rn  E½s j n ðn  qÞ

Acknowledgements The authors thank personnel at HP TradingHubs (now part of Converge) for their cooperation and feedback in conducting this research. The authors would also like to thank Professor H. L. Lee, Stanford University, as well as the anonymous referees for their constructive feedback that improved this paper.

¼ ð1  q2 Þr2s E½n2  2qn þ q2



rs þ Var rn  ls þ q ðn  ld Þ ðn  qÞ rd ¼ ð1  q2 Þr2s ½r2d þ ðld  qÞ2  

rs rs þ Var r  ls þ q ðq þ ld Þ n  q n2 rd rd 2

2

¼ r2d ðr  ls Þ þ r2s ½r2d þ ðld  qÞ 2

Appendix A

 q2 r2s ½r2d þ ðld  qÞ þ 2q2 r2s ðr2d  l2d Þ

A.1. Buy and sell model

 2qrd rs ld ðr  ls Þ

The profit function for buying and selling (BS) via spot markets is þ

PBS ðqÞ ¼ rn  cq  sðn  qÞ þ sðq  nÞ

þ

 2qrs ðqrs ld  rd ðr  ls ÞÞq þ q2 r2s q2 ¼ ðr  ls Þ2 r2d þ r2s ½r2d þ ðld  qÞ2  qrs  ½qrs ð3l2d  r2d Þ þ 2rd ðr  ls Þðld  qÞ

¼ rn  cq  sðn  qÞ: The expected profit and its derivative can be computed as EPBS ðqÞ ¼ cq þ

Z

1

n¼0

Z

1

½rn  ðn  qÞs fd;s ðn; sÞ ds dn

s¼0

  rs ¼ cq þ rn  ðn  qÞ½ls þ q ðn  ld Þ dFd ðnÞ rd n¼0 Z

1

¼ ðr  ls Þld  qrs rd þ ðls  cÞq

and

using

rs rs Var ðr  ls þ q ðq þ ld ÞÞn  q n2 rd rd ¼ Var½aX þ bX 2   rs with a ¼ r  ls  bðq þ ld Þ and b ¼  q rd ¼ r2d ½a2 þ 4abld þ b2 l2d þ 2b2 r2d ¼ r2d ½ðr  ls Þ2 þ 2b2 ðr2d  l2d Þ þ 2bld ðr  ls Þ  2bðbld þ r  ls Þq þ b2 q2 2

o EPBS ðqÞ ¼ ls  c: oq

¼ r2d ðr  ls Þ þ 2q2 r2s ðr2d  l2d Þ  2qrd rs ld  ðr  ls Þ  2qrs ðqrs ld  rd ðr  ls ÞÞq þ q2 r2s q2

796

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799

and

and 2

o VarPPC ðqÞ ¼ 2ðr  vÞ2 V1 ðqÞ  2p2 P1 ðqÞ oq

Var½aX þ bX 2 2

2

¼ E½ðaX þ bX Þ  ðE½aX þ bX Þ

2

 2½ðr  vÞV1 ðqÞ þ pP1 ðqÞ

2

¼ a2 E½X 2 þ 2abE½X 3 þ b2 E½X 4  a2 ðE½X Þ  2abE½X E½X 2  b2 ðE½X 2 Þ

 ½ðv  rÞFd ðqÞ þ p½1  Fd ðqÞ

2

¼ 2ðp þ r  vÞ½ðr  vÞ½1  Fd ðqÞ V1 ðqÞ

¼ r2d ½a2 þ 4abld þ b2 l2d þ 2b2 r2d

 pFd ðqÞP1 ðqÞ :

given E½X ¼ ld , E½X 2 ¼ l2d þ r2d , E½X 3 ¼ l3d þ 3ld r2d , and E½X 4 ¼ l4d þ 3l2d r2d þ 3r4d for X  Normal½ld ; r2d . The first derivative of VarPBS ðqÞ with respect of q is o VarPBS ðqÞ ¼ 2qrs rd ðr  ls Þ  2r2s ðld  qÞ: oq The objective function is ZBS ðqÞ ¼ EPBS ðqÞ  kVarPBS ðqÞ and is strictly concave in q for k > 0 2 and rs > 0 because ðoo2 qÞZBS ðqÞ ¼ 2kr2s < 0. Setting ðoqo ÞZBS ðqÞ ¼ 0, we arrive at Eq. (1) which must be satisfied by the unique maximum of ZBS ðqÞ, the optimal contract quantity q BS . A.2. Pure contract sourcing The profit function for pure contract (PC) sourcing is þ

PPC ðqÞ ¼ r min½n; q  cq þ vðq  nÞ  pðn  qÞ

þ

¼ ðr  cÞq  ðr  vÞðq  nÞþ  pðn  qÞþ : The expected profit and its derivative can be computed as EPPC ðqÞ ¼ ðr  cÞq  ðr  vÞV1 ðqÞ  pP1 ðqÞ and o EPPC ðqÞ ¼ p þ r  c  ðp þ r  vÞFd ðqÞ: oq The variance of profits and its derivative can be computed as VarPPC ðqÞ ¼ E½ððr  vÞðq  nÞþ þ pðn  qÞþ Þ2  ðE½ðr  vÞðq  nÞþ þ pðn  qÞþ Þ2 2

¼ ðr  vÞ V2 ðqÞ þ p2 P2 ðqÞ  ½ðr  vÞV1 ðqÞ þ pP1 ðqÞ

2

The objective function is ZPC ðqÞ ¼ EPPC ðqÞ  kVarPPC ðqÞ; setting ðoqo ÞZPC ðqÞ ¼ 0, we arrive at Eq. (4) which must be satisfied by the optimal contract quantity q PC . For p ¼ 0, the extreme point q PC must be a unique maximum of ZPC ðqÞ because Zðq j p ¼ 0Þ is concave-convex in q and limq!1 ZPC ðqÞ ¼ 1. To show that Zðq j p ¼ 0Þ is concave–convex in q consider EPPC ðqÞ and kVarPPC ðqÞ separately. EPPC ðqÞ is strictly 2 concave in q because ðoo2 qÞEPPC ðqÞ < 0, and kVarPPC  ðqÞ is concave–convex in q because limq!1 VarPPC ðq j p ¼ 0Þ ¼ 0, limq!1 VarPPC  ðq j p ¼ 0Þ ¼ ðr  vÞ2 r2d > 0, and ðoqo ÞVarPPC  ðq j p ¼ 0Þ > 0. To show that limq!1 ZPC ðqÞ ¼ 1, it suffices to show that limq!1 EPPC ðqÞ ¼ 1 because kVarPPC ðqÞ 6 0; 8q. This can be shown directly, i.e., limq!1 EPPC ðqÞ ¼ ðr  vÞ  ld  ðc  vÞ limq!1 q ¼ 1 and limq!1 EPPC  ðqÞ ¼ pld þ ðp þ r  cÞ limq!1 q ¼ 1 given v < c < r and p P 0. For p > 0, limq!1 ZðqÞ ¼ 1 continues to hold, but we have to rely on numerical analysis to support our conjecture that ZPC ðqÞ is unimode. A.3. Mathematical references Rq n R 1Let Vn ðqÞn ¼ n¼1 ðq  nÞ dFd ðnÞ and Pn ðqÞ ¼ ðn  qÞ dFd ðnÞ. Notice that ðo=oqVn ðqÞÞ ¼ n¼q nVn1 ðqÞ and ðo=oqPn ðqÞÞ ¼ nPn1 ðqÞ while V0 ðqÞ ¼ Fd ðqÞ and P0 ðqÞ ¼ 1  Fd ðqÞ. For Fd ð
Þ  Normal½ld ; r2dR , we have fd0 ðqÞ ¼ q ððq  ld Þ=r2d Þfd ðqÞ and n dFd ðnÞ ¼ ld  n¼1 2 Fd ðqÞ  rd fd ðqÞ (adapted from [17]). Thus, V1 ðqÞ and P1 ðqÞ can be computed as V1 ðqÞ ¼ ðq  ld Þ Fd ðqÞ þ r2d fd ðqÞ and P1 ðqÞ ¼ ðld  qÞ½1  Fd ðqÞ þ r2d fd ðqÞ. Finally, for ðn; sÞ  BN½ld ; ls ; r2d ; r2s ; q with correlation q, we repeatedly use E½s j n ¼ ls þ qðrs = rd Þðn  ld Þ and Var½s j n ¼ ð1  q2 Þr2s [5].

R.W. Seifert et al. / European Journal of Operational Research 152 (2004) 781–799

A.4. Spot markets used for buying only

þ

PSO ðqÞ ¼ rn  rðn  qÞ  cq þ sðq  nÞ  pðn  qÞ

If the use of spot markets is restricted to buying only (BO), the profit function is þ

PBO ðqÞ ¼ rn  cq  sðn  qÞ þ vðq  nÞ ¼ ðr  cÞq þ ðr  sÞðn  qÞ

797

þ

þ

þ

þ ðv  rÞðq  nÞ : When demand n exceeds the contract quantity q, we use a spot market to procure n  q units to prevent stockouts. When demand n is less than the contract quantity q, we salvage q  n units of excess inventory by selling them outside the spot market at unit price v. As in the BS-model, demand n and spot price s are Bivariate Normal distributed: ðn; sÞ  BN ½ld ; ls ; r2d ; r2s ; q . The expected value and variance of PBO ðqÞ can be computed as [40] EPBO ðqÞ ¼ ðr  cÞq þ ðv  rÞV1 ðqÞ þ uP1 ðqÞ

þ

þ þ

þ

¼ ðr  cÞq þ ðs  rÞðq  nÞ  pðn  qÞ : When demand n exceeds the contract quantity q, the buyer incurs a stockout penalty cost p P 0 per unit. When demand n is less than the contract quantity q, we salvage q  n units of excess inventory by selling them on spot markets at spot price s. As in the BS-model, demand n and spot price s are Bivariate Normal distributed: ðn; sÞ  BN½ld ; ls ; r2d ; r2s ; q . The expected value and variance of PSO ðqÞ can be computed as [40] EPSO ðqÞ ¼ ðr  cÞq  uV1 ðqÞ  pP1 ðqÞ þ

 wE½ðq  nÞ n and VarPSO ðqÞ ¼ ðr2s  w2 r2d þ u2 ÞV2 ðqÞ þ p2 P2 ðqÞ þ wE½ð2un þ wn2 Þ½ðq  nÞþ 2

þ

þ wE½ðn  qÞ n

 ðuV1 ðqÞ þ pP1 ðqÞ

and

þ wE½ðq  nÞþ n Þ2

VarPBO ðqÞ ¼ ðr2s  w2 r2d þ u2 ÞP2 ðqÞ

where

2

þ ðv  rÞ V2 ðqÞ

u ¼ r  ls þ qðrs =rd Þld and w ¼ qðrs =rd Þ: þ 2

þ wE½ð2un þ wn2 Þ½ðn  qÞ  ðuP1 ðqÞ þ ðv  rÞV1 ðqÞ þ

2

þ wE½ðn  qÞ n Þ ; where u ¼ r  ls þ qðrs =rd Þld > 0 and w ¼ qðrs =rd Þ: As in our previous models, the buyer maximizes ZBO ðqÞ ¼ EPBO ðqÞ  kVarPBO ðqÞ. The optimal contract quantity q BO , however, cannot be derived in closed-form and numerical analysis is needed to conduct sensitivity analysis as detailed in Seifert [40]. A.5. Spot markets used for selling only If the use of spot markets is restricted to selling only (SO), the profit function is

As in our previous models, the buyer maximizes ZSO ðqÞ ¼ EPSO ðqÞ  kVarPSO ðqÞ. The optimal contract quantity q SO , however, cannot be derived in closed-form and numerical analysis is needed to conduct sensitivity analysis as detailed in Seifert [40]. References [1] AMR Research, Inc. Independent Trading Exchanges–– The Next Wave of B2B e-commerce, 1999. [2] E. Andren, Steel marketplaces are strengthening their mettle, Gartner Group, Inc. January 4, 2000. [3] R. Anupindi, Y. Bassok, Supply Contracts with Quantity Commitments and Stochastic Demands (Chapter 7), in: S. Tayur, R. Ganeshan, M. Magazine (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, Dordrecht, 1998. [4] V. Araman, J. Kleinknecht, R. Akella, Supplier and procurement risk management in e
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