Int. J. Production Economics 145 (2013) 790–798
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Newsvendor problem with random shortage cost under a risk criterion Meng Wu a,b, Stuart X. Zhu b, Ruud H. Teunter b,n a b
Business School, Sichuan University, Chengdu 610064, China Department of Operations, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
art ic l e i nf o
a b s t r a c t
Article history: Received 21 September 2012 Accepted 7 June 2013 Available online 3 July 2013
We study profit maximization vs risk approaches for the standard newsvendor problem with uncertainty in demand as well as a generalized version with uncertainty in the shortage cost (as often applies in practice). We consider two well-known risk approaches: Value-at-Risk (VaR) included as a constraint and Conditional Value-at-Risk (CVaR). We first derive the explicit expressions of the optimal solution with uncertainty of shortage cost under different risk measures and then perform a numerical analysis to quantify the effect of uncertainty in shortage cost and risk measures. For the standard newsvendor problem, we find that the optimal quantity under CVaR is always lower than that under the VaR constraint, which in turn is lower than the order quantity that maximizes the expected profit. Insightful explanations for this result are that: (a) a higher degree of risk aversion drives the newsvendor to order fewer products, increasing the likelihood that all will be sold; (b) this effect is stronger for the CVaR approach as this does not consider the expected profit at all. Another interesting and counter-intuitive observation for the CVaR approach is that a higher retail price may lead to a smaller order quantity, as fewer items need to be sold in order to attain a sufficient profit. These results show that one should be careful in employing the CVaR risk measure for newsvendor type problems. The results remain valid if the shortage cost becomes uncertain. However, increased uncertainty of this type does improve the relative profitability under the CVaR approach by increasing the order quantity under that criterion whilst there is no effect under the other criteria. & 2013 Elsevier B.V. All rights reserved.
Keywords: Conditional value-at-risk Value-at-risk Newsvendor problem Risk aversion Random shortage cost
1. Introduction As a fundamental problem in inventory control, the newsvendor problem has been studied for a long time and applied in a broad array of business settings. The classic newsvendor model assumes that managers select an order quantity to maximize the expected profit under the risk neutral assumption. However, Schweitzer and Cachon (2000) provide experimental evidence suggesting that inventory managers are risk averse for highvalue products. Thus, the assumption of risk neutrality is not always applicable. A number of authors have therefore considered risk related objectives, such as utility functions (e.g., Lau, 1980; Agrawal and Seshadri, 2000; Chen et al., 2007); the mean-variance objective function (e.g., Chen and Federgruen, 2000; Wu et al., 2009); VaR and downside risk (e.g., Özler et al., 2009; Gan et al., 2005), CVaR and coherent measures (e.g., Ahmed et al., 2007; Chen et al., 2009). An overview of the key results is provided in the next section.
n
Corresponding author. Tel.: +31 503638617; fax: +31 503632032. E-mail addresses:
[email protected] (M. Wu),
[email protected] (S.X. Zhu),
[email protected] (R.H. Teunter). 0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/http://dx.doi.org/10.1016/j.ijpe.2013.06.007
VaR and CVaR as financial risk measures have emerged and been widely used in recent years. In this paper, we focus on VaR and CVaR as the risk measures of the downside risk. The standard VaR criterion maximizes the minimum profit that is attained with a certain probability, e.g., with a probability of 95% the realized profit is at least an amount to be maximized. More formally, the standard VaR maximizes the lower end of a one-sided confidence interval (Artzner et al., 1999). A clear disadvantage of the standard VaR criterion is that it purely considers risk but not the expected profits. Although many inventory manager may be risk averse, they still strive to attain a high (expected) profit. To reflect this, VaR can be used as a constraint, e.g., with a probability of 95% the realized profit is at least some fixed amount. See also Gan et al. (2005). Another way to trade-off the expected profit and risk is by using the CVaR criterion. Loosely formulated, the CVaR maximizes the profit to be achieved after subtracting the expected amount by which the actual profit may fall short of that amount, multiplied by a risk factor. A formal definition is provided in Section 4. See also Rockafellar and Uryasev (2000, 2002) for more details. And we refer interest readers to Qin et al. (2011) for summaries of the ordering policies of newsvendors with various risk preferences. In this paper, we consider both the VaR constraint and the CVaR approach. Indeed, an important contribution of our research is to show that the choice of risk measures has an important effect on
M. Wu et al. / Int. J. Production Economics 145 (2013) 790–798
the inventory decision. We find that the CVaR typically (and always with a constant shortage cost) leads to much smaller order quantities and lower profits than the VaR constraint approach. This result, which to the best of our knowledge has gone unnoticed in the literature, shows that risk criteria should be carefully selected for inventory decisions. Another main contribution is that we consider uncertainty in both demand volume and shortage cost. The classical newsvendor problem, reflecting the majority of the inventory control literature, only considers the first type of uncertainty. In practice, however, shortage costs are often variable and unknown. For example, Corsten and Gruen (2003) conduct a worldwide study of consumer responses to out-of-stocks in fast-moving consumer goods industry. They categorize five primary responses to out-of-stock, i.e., 31% of customers choose to buy the same item at another retailer, 15% of customers choose to buy later at the same retailer, 19% of customers choose a substitute with same brand for a different size or type, 26% of customers choose a substitute with a different brand, 9% of customers do not purchase the item. Since all these five responses result in different amounts of profit losses for a retailer, the shortage cost is uncertain. Another cause of randomness in shortage cost, especially in a service logistics environment, is that the cost of dealing with a shortage may vary considerably. For instance, if some spare part is needed but not available for completing a repair, then that part can sometimes be cannibalized from a partially manufactured product or a product that is collected/returned after use. If this is indeed possible, then the shortage cost is relatively low. If, however, the repair cannot be completed and the customer is offered a new (comparable) product, then the shortage cost is much higher. We remark that even if the shortage cost is fairly constant, it may be difficult to estimate, which also creates uncertainty. The uncertainty of the shortage cost in inventory problems has received research attention in recent years. Some authors defined shortage costs as a fuzzy variable. For example, Ishii and Konno (1998) introduced fuzzy shortage cost into traditional discrete random newsvendor problem; Wee et al. (2009) constructed a multi-objective joint replenishment deteriorating items inventory model, where the demand and shortage cost were assumed to be fuzzy variables; Petrović et al. (1996) investigated the newsvendor problem with fuzzy demand and fuzzy costs. However, this paper presents the first analysis of the newsvendor problem with a random shortage cost under different risk measures. We find that uncertainty of the shortage cost does not affect the order quantity under the traditional criterion and under the VaR constraint. However, it results in an increase of the order quantity under the CVaR criterion. The remainder of this paper is organized as follows. In the next section, we review the literature on the risk-averse newsvendor problem. Then, in Section 3, we derive the optimal order decision for the classic newsvendor model with random shortage cost. In Section 4, we describe the risk measure CVaR and get the optimal order quantity under the CVaR criterion. Section 5 considers the newsvendor problem with a VaR constraint and derives the optimal order quantity for a special case. Section 6 presents numerical results and discusses the effects of different risk measures and uncertainty in the shortage on order quantity. Section 7 concludes with a general discussion and directions for future research. All proofs can be found in the online appendix.
2. Literature review Three main streams of modelling risk-averse newsvendor problems can be identified in the literature: utility functions, mean variance approach, VaR/CVaR. Next, we review key contributions for
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each stream separately, after which we draw some general conclusion. The first stream is related to utility function. Eeckhoudt et al. (1995) considered the newsvendor model with a risk-averse decision maker where risk is measured by a utility function. They examined the effects of risk aversion on a prudent newsvendor without shortage costs. Schweitzer and Cachon (2000) provided a complete investigation of the relationship between the newsvendor's profit-maximizing order quantity and optimal order quantities under various alternative objectives. They showed that a loss-averse newsvendor would order strictly less than a riskneutral newsvendor. Wang and Webster (2009) continued their research by including a shortage cost. They found that a lossaverse newsvendor may order more than a risk-neutral newsvendor if shortage cost is not negligible, and that the optimal order quantity may increase in the wholesale price and decrease in the retail price. The second is related to mean-variance objective function. Chen and Federgruen (2000) studied the newsvendor problem using the mean-variance framework. They showed that the variance of the stochastic profit is a monotone increasing function of the order quantity. Without shortage cost, the optimal riskaverse order quantity is shown to be smaller than the risk-neutral order quantity and to decrease in the degree of risk aversion. Wu et al. (2009) showed that this result does not extend in general to the case with a positive shortage cost. The third is to use the risk measure of VaR and/or CVaR. Gan et al. (2005) incorporated the VaR concept to a newsvendor problem with a downside risk constraint for a single product. They studied the channel coordination without shortage cost where the supplier is risk-neutral and the retailer is constrained by a downside risk. Özler et al. (2009) considered the single period newsvendor problem with VaR constraints in the multi-product case. They derived the exact profit distribution function for the two-product newsvendor problem and developed an approximation method for the profit distribution of multi-product case. However, they did not incorporate shortage cost into the models. Ahmed et al. (2007) studied the multi-period newsvendor problem without shortage cost where the objective function is a coherent risk measure. They showed that the structure of the optimal solution of the risk-averse model is similar to that of the risk-neutral case. Without shortage cost, Jammernegg and Kischka (2007) compared the ordering policy and its corresponding performance measures, such as the cycle service level, under the CVaR and the expected profit approach. They found that a riskaverse (risk-seeking) newsvendor orders less (more) than a riskneutral newsvendor does. Xu and Li (2010) discussed the newsvendor problem under a hybrid model with a combination of the CVaR and the expected profit. The authors showed that the optimal order quantity under the CVaR criterion is increasing in the shortage cost; incorporating a substantial shortage cost, the optimal order quantity of the risk-averse newsvendor may be larger or less than that of the risk-neutral newsvendor. Choi et al. (2011) considered a multiproduct risk-averse newsvendor under the law-invariant coherent measures of risk. For identical products with independent and generally distributed demands, they showed that increased risk aversion leads to decreased orders. Jammernegg and Kischka (2012) gave a comparative analysis of CVaR, VaR, and the mean-CVaR rule in the newsvendor problem without shortage cost. They analyzed the impact of these risk measures on the distribution functions of the profit, the optimal order quantity, and the optimal expected profit. In particular, they showed that the CVaR newsvendor orders less than the VaR newsvendor. Katariya et al. (2013) gave a comparative analysis of expected utility, mean-variance and CVaR in the newsvendor problem with shortage cost. They showed that the optimal order
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quantities of risk neutral and risk averse newsvendors depend on the demand distribution and the cost parameters. A common conclusion of these above literature is that the inclusion of a shortage cost alters the direction of the results: a risk-averse newsvendor may order more than a risk-neutral newsvendor if shortage cost is not negligible. An interesting and unexplored question is what happens in an, arguably even more realistic, situation with a random shortage cost. This is the main focus of our research, where we consider both the VaR and the CVaR approach to gain insights of the effects of these two types of risk modelling on the optimal ordering quantity and the corresponding expected profit.
3. Classic newsvendor model with random shortage cost Let Q be the newsvendor's order quantity and D be the stochastic demand during the selling season. The purchasing cost of the product is c per unit, the selling price is r per unit, the salvage value is s per unit and the shortage cost is P where P is a random variable and independent of D. Let Fð Þ and Gð Þ be the cumulative distribution functions, and f ð Þ and gð Þ be the probability density functions of demand and shortage cost, respectively. To avoid unrealistic and trivial cases, we assume that r 4 c 4s 4 0 and P 40. Without loss of generality, we assume that Fð0Þ ¼ Gð0Þ ¼ 0. If Q units are ordered before the selling season, the retailer's profit is πðQ ; D; PÞ ¼ ðP þ r−cÞQ −ðP þ r−sÞðQ −DÞþ −PD
ð1Þ
and the expected value of πðQ Þ is Z Q EðπðQ ; D; PÞÞ ¼ −ðr þ EðPÞ−sÞ FðxÞ dx 0
þðr þ EðPÞ−cÞQ −EðPÞEðDÞ: By noting that this function is concave and setting the derivative to zero, we obtain the following theorem. Theorem 1. Suppose that the random shortage cost P is independent of stochastic demand D. The optimal order quantity of the classic newsvendor problem is r þ EðPÞ−c : ð2Þ Q nE ¼ F −1 r þ EðPÞ−s
Note from this theorem that the optimal order quantity formula is similar to that of the classic newsvendor problem, but with the constant shortage cost replaced by the expected shortage cost.
4. Newsvendor model with random shortage cost under CVaR criterion Given order quantity Q and the corresponding distribution of the profit πðQ ; DÞ, CVaR is defined as (see Rockafellar and Uryasev, 2000, 2002) 1 CVaRη ðπðQ ; D; PÞÞ ¼ max qðQ ; ϕÞ≔ϕ− Eðϕ−πðQ ; D; PÞÞþ ; ð3Þ ϕ∈R η where η∈ð0; 1 reflects the degree of risk aversion, i.e., a lower value implies a higher degree of risk aversion and a value of 1 implies risk neutrality. CVaR can be treated as the conditional expectation of the profit when the probability that the realized profit is not more than the target profit ϕ is less than η, i.e., Prob:ðπðQ ; D; PÞ ≤ϕÞ ≤η.
In order to minimize the downside risk of the profit, the objective of the newsvendor is to choose the order quantity that maximizes the CVaR, i.e., max CVaRη ðπðQ ; D; PÞÞ: Q
The following theorem can be used to determine the optimal order quantity under the CVaR criterion. Theorem 2. Suppose that the random shortage cost P is independent of stochastic demand D. Under the CVaR criterion, the optimal order n quantity Qn and the optimal target profit ϕ^ ¼ arg maxϕ∈R ^ CVaRη ðπðQ ; D; PÞÞ are the solutions of the following equations: !! ! 8 n n R > ϕ^ −ðs−cÞQ n ðy þ r−cÞQ n −ϕ^ > > þ∞ y 1−F dGðyÞ ¼ ηðc−rÞ þ ðr−sÞF ; > > y r−s < 0 ! ! n > R þ∞ ðy þ r−cÞQ n −ϕ^ > > > ðy þ r−sÞ 1−F dGðyÞ ¼ ηðc−sÞ: > : 0 y
ð4Þ Proof. See Appendix A. Remark 1. When the shortage cost is a constant, from (4) we obtain Theorem 2 in Xu and Li (2010). The complexity of the derived optimality conditions obviously depends on the distributions of demand and shortage cost. In order to obtain structural insights into the optimal order decision, we select the uniform demand and a binary (high-low) shortage cost since it will make the problem tractable. This assumption of the uniform demand has also been used by Pasternack (2005) and Burke et al. (2009). Moreover, managers often find it easier to specify lower and upper bounds for demand rather than a suitable distribution (or moments). For the estimation of the shortage cost, Oral et al. (1972) propose a modified version of a decision tree model that considers a set of possible cost outcomes caused by a stock out. Then, by multiplying each cost with the corresponding probability, the expected value is treated as the shortage cost. Further, Ishii and Konno (1998) have used a fuzzy number with a lower bound and an upper bound to describe the shortage cost. We follow their approach and, for simplicity, only consider the best and the worst consequence of a stock out. In other words, we use the binary distribution to model the shortage cost. We remark that this high-low shortage cost is especially representative of situations where managers may or may not be able to find low cost solutions for dealing with a shortage, as also discussed in the introduction. 4.1. Special case: uniform distributed demand and binary distributed shortage costs Without loss of generality, we use 0 and 1 as the minimum and maximum demand, respectively. So, demand is uniform distributed with parameter 1, i.e., D∼U½0; 1, and shortage cost is binary distributed, i.e., ProbðP ¼ pL Þ ¼ α and ProbðP ¼ pH Þ ¼ 1−α where pH ≥pL and α∈ð0; 1Þ. Let the mean of random shortage cost be p, i.e., EðPÞ ¼ p. Then, the mean absolute deviation of random shortage cost EjP−pj is 2ð1−αÞðpH −pÞ or 2αðp−pL Þ. We define Δ ¼ EjP−pj=2, which is an indicator of the variability of shortage cost randomness. Thus, pL and pH can be written as pL ¼ p−Δ=α and pH ¼ p þ Δ=ð1−αÞ. Further, differentiating Δ with respect to α gives dΔ=dα ¼ ð1−2αÞðpH −pL Þ, which implies that, if α ≤1=2, then the variability is increasing in α, otherwise the variability is decreasing in α. Theorem 3. Under the CVaR criterion, for uniform distributed demand and binary distributed shortage costs, the optimal ordering
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quantity is 8 EðPÞ þ ηðr−cÞ; > > > < EðPÞ þ r−s Q nCVaR ¼ pH ð1−αÞðr−sÞðpH þ r−cÞ−pH ðc−sÞ > > > : p þ r−s þ η ð1−αÞðpH þ r−sÞ2 H
if ηn ≤η o 1; if 0 o η o ηn ;
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shown in Appendix C, (5) is equivalent to Z Q max −ðr þ EðPÞ−sÞ FðxÞ dx þ ðr þ EðPÞ−cÞQ −EðPÞEðDÞ: Q
s:t:
0
R ðπ 0 þðc−sÞQ Þ=ðr−sÞ 0
f ðxÞ dx þ
R þ∞ R þ∞ 0
ððyþr−cÞQ −π 0 Þ=y
f ðxÞgðyÞ dx dy ≤η;
π 0 −ðr−cÞQ ≤0: ð6Þ
where ð1−αÞðpH þ r−sÞðr−sÞðpH −pL Þ ηn ¼ : ð1−αÞðr−sÞðpH þ r−cÞðpH −pL Þ þ pH ðc−sÞðpL þ r−sÞ
5.1. Special case: uniform distributed demand and binary distributed shortage costs
Proof. See Appendix B. From Theorem 3, we find that Q nCVaR is increasing in η. Note that η measures the degree of risk aversion and a lower value of η implies a higher degree of risk aversion. Therefore, this finding indicates that the optimal ordering quantity is decreasing in the degree of risk aversion. The reason is that the newsvendor focuses on the reduction of the downside risk of the profit. When the degree of risk aversion increases, the newsvendor intends to order less so that it is more likely that the order can be fully sold and a sufficient profit can be achieved. Corollary 1. For a constant shortage cost p, the optimal order quantity for uniform distributed demand is Q nCVaRc ¼
p þ ηðr−cÞ : p þ r−s
In the next corollary, we reformulate the result in Theorem 3 to facilitate a comparison with the case of a constant shortage cost (which is a special case of that corollary with Δ equal to 0). Corollary 2. Under the CVaR criterion, the optimal order quantity for uniform distributed demand and binary distributed shortage costs is
For uniform distributed demand and binary distributed shortage costs, (6) can be simplified to Q2 EðPÞ : þ ðr þ EðPÞ−cÞQ − −ðr þ EðPÞ−sÞ Q 2 2 R∞ ðy þ r−cÞQ −π 0 π 0 þ ðc−sÞQ ≥1−η; dGðyÞ−F F 0 r−s y s:t: π 0 −ðr−cÞQ ≤0:
max
To solve (7), we have to consider several cases, depending on the value of η, as shown in Appendix D where the solution is derived in Theorem 7 for all those cases. The last case of 0 o η o η3 in Theorem 7 where η3 is defined in Theorem 7 is most likely to occur in practice as we will argue next. It is not hard to verify that ðr−cÞ=ðr−sÞ o η3 . A sufficient condition for this case to hold is that ðr−cÞ=ðr−sÞ is larger than η, i.e., the (salvage value corrected) profit margin is larger than η. Since values of 1, 5, or 10 percent are typically used for η, we usually end up in this case. Thus, we restrict ourselves to that case in the main text. The following results give the optimal ordering quantity for uniform distributed demand and binary distributed shortage costs under the VaR constraint.
8 p þ ηðr−cÞ > > > < p þ r−s ; Q nCVaR ¼ ð1−αÞp þ Δ ð1−αÞðr−sÞðð1−αÞðp þ r−cÞ þ ΔÞ−ðð1−αÞp þ ΔÞðc−sÞ > > > : ð1−αÞðp þ r−sÞ þ Δ þ η ðð1−αÞðp þ r−sÞ þ ΔÞ2
where ðð1−αÞðp þ r−sÞ þ ΔÞðr−sÞΔ η ¼ : ðð1−αÞðp þ r−cÞ þ ΔÞðr−sÞΔ þ ðð1−αÞp þ ΔÞðc−sÞðαðp þ r−sÞ−ΔÞ n
5. Newsvendor model with random shortage cost under the VaR constraint In this section, we consider the newsvendor problem with the VaR constraints. Let π 0 be the VaR or target profit. The newsvendor wants to choose an order quantity Q in order to maximize his/her expected profit EðπðQ ; D; PÞÞ, while specifying that his/her actual profit should not fall below the VaR (target profit level) with a probability exceeding a specified η. So the decision problem is given by max Q
s:t:
EðπðQ ; D; PÞÞ ProbfπðQ ; D; PÞ ≤π 0 g ≤η;
ð5Þ
where π 0 is a given constant and η∈ð0; 1. Note that, for η ¼ 1, the VaR constraint always holds and (2) is the optimal solution. As
ð7Þ
if ηn ≤η o1; if 0 o η oηn ;
Theorem 4. Under the VaR constraint, if 0 ≤η oðr−cÞ=ðr−sÞ, then the optimal order quantity for uniform distributed demand and binary distributed shortage costs is 8 < ηðr−sÞ−π 0 if π n o π 0 ≤π~ ; n c−s Q VaR ¼ : Qn if π ≤π n ; E
0
n
where Q E ¼ ðEðPÞ þ r−cÞ=ðEðPÞ þ r−sÞ, π~ ¼ ηðr−cÞ−
pL pH ðc−sÞð1−ηÞ pL pH þ ðr−sÞðαpH þ ð1−αÞpL Þ
π n ¼ ηðr−sÞ−
and
ðEðPÞ þ r−cÞðc−sÞ : EðPÞ þ r−s
Proof. This follows directly from the last case of Theorem 7 in Appendix D. Theorem 4 shows that, as under the CVaR approach (see Theorem 3), Q nVaR is increasing in η under the VaR constraint approach. So, although the VaR constraint does consider the expected profit next to the downside risk, an increased amount of downside risk still leads to a lower order quantity, thereby
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increasing the probability that all items are sold and a sufficient profit is achieved. In the following corollary, we reformulate Theorem 4 to facilitate a comparison with the case of a constant shortage cost (which is a special case of that corollary with Δ equal to 0). Corollary 3. Under the VaR constraint, if 0 ≤η o ðr−cÞ=ðr−sÞ, then the optimal order quantity for uniform distributed demand and binary distributed shortage costs is 8 > < ηðr−sÞ−π 0 if π n oπ 0 ≤π~ ; n c−s Q VaR ¼ > : Q nE if π 0 ≤π n ; c where Q nEc ¼ ðp þ r−cÞ=ðp þ r−sÞ, π~ ¼ ηðr−cÞ−
ðc−sÞð1−ηÞ α 1−α þ pþΔ=ð1−αÞ Þ 1 þ ðr−sÞð p−Δ=α
π n ¼ ηðr−sÞ−
and
ðp þ r−cÞðc−sÞ : p þ r−s
For uniform distributed demand and constant shortage cost p, (6) can be simplified to −ðr þ p−sÞ
max Q
s:t:
Q2 p þ ðr þ p−cÞQ − : 2 2
ð8Þ
ðp þ r−cÞQ −π 0 π 0 þ ðc−sÞQ ≥1−η; −F r−s p π 0 −ðr−cÞQ ≤0: F
Under the VaR constraint, if 0 ≤η o ðr−cÞ=ðr−sÞ, the optimal order quantity for uniform distributed demand and constant shortage cost is 8 > < ηðr−sÞ−π 0 if π n oπ 0 ≤π ; c−s Q nVaRc ¼ ð9Þ > : Q nE if π 0 ≤π n ; c where Q nEc ¼ ðp þ r−cÞ=ðp þ r−sÞ, π ¼ ðr−cÞ−ðr−sÞð1−ηÞðp þ r−cÞ=ðp þ r−sÞ and π n ¼ ηðr−sÞ−ðp þ r−cÞðc−sÞ=ðp þ r−sÞ. This result holds for generally distributed demand, as shown in Gan et al. (2005) and Jammernegg and Kischka (2012).
6. Discussion
(1) Uncertainty in the shortage cost does not affect the order quantity under the traditional expected profit criterion, i.e., Q nEc ¼ Q nE . (2) For uniform distributed demand and binary distributed shortage costs, if the optimal order quantity exists, uncertainty in the shortage cost does not affect the order quantity under the VaR constraint, i.e., Q nVaRc ¼ Q nVaR . (3) For uniform distributed demand and binary distributed shortage costs, uncertainty in the shortage cost increases the order quantity under the CVaR criterion, i.e., Q nCVaRc ≤Q nCVaR . Proof. See Appendix E.
Note from Theorem 5 that uncertainty of the shortage cost does not affect the order quantity under the traditional criterion and under the VaR constraint. However, it results in an increase of the order quantity under the CVaR criterion. This is explained as follows. A higher value of Δ implies more uncertainty in the shortage cost and therefore a higher maximum shortage cost pH. This, in turn, implies a higher profit loss from unmet demand for the worst case that pH is realised. CVaR focuses on the lower end of profits associated with this worst case, and therefore reacts by ordering more. The VaR constraint and standard approach considered the expected profit over the whole domain and do not react to a change in Δ. To further illustrate the effects of uncertainty in shortage cost, we perform a numerical experiment. We consider the following setting of the parameters: r¼5, EðPÞ ¼ p ¼ 5, c¼1, η ¼ 5%, π 0 ¼ −0:2, s¼0.5, α ¼ 0:5 and Δ∈½0:1; 2:4 where Δ represents the variability of shortage cost. Moreover, the uncertainty in the shortage cost is increasing with respect to Δ. Since ðr−cÞ=ðr−sÞ ¼ 88:8%, the condition η ≤ðr−cÞ =ðr−sÞ always holds. We define percentage profit decrease that results from using the CVaR criterion instead of maximizing expected profit, is ðΠ E −Π CVaR Þ=Π E , where Π E and Π CVaR are the expected profit at optimal order quantity, i.e., Π E ¼ πðQ E ; D; PÞ and Π CVaR ¼ πðQ CVaR ; D; PÞ. We use similar definitions of percentage profit decrease between other risk measures. Figs. 1 and 2 illustrate that the optimal order quantity under the CVaR criterion increases as Δ increases, and that percentage profit decrease as Δ increases. This result contrasts with the newsvendor under the traditional criterion and under the VaR constraint, whose optimal order quantities are independent of Δ. Note that the differences in order quantity and associated profit can be considerable, up to 28% and 47% respectively for the considered settings.
By comparing the results in Sections 3–5, we investigate the effects of using a certain risk measure and of uncertainty in the shortage cost in this section. We will discuss these effects separately. Besides discussing the comparative exact results for the effect on the order quantity, we also perform a numerical investigation to explore the significance of the effects on the ordering quantity as well as the profits. In the following discussion, we assume that the expected value of the random shortage cost is equal to the constant shortage cost, i.e., EðPÞ ¼ p, and the subscript c indicates the optimal order quantity with constant shortage cost. 6.1. Effects of uncertainty in the shortage cost We summarize the effects of uncertainty in the shortage cost in the following theorem. Theorem 5. Assume that demand is uniform distributed and shortage cost is binary distributed.
Δ Fig. 1. Optimal ordering quantity with respect to Δ.
M. Wu et al. / Int. J. Production Economics 145 (2013) 790–798
795
Δ Fig. 2. Percentage profit decease with respect to Δ.
σ Fig. 4. Percentage profit decease with respect to s for normal distributed demand and shortage cost.
1
Order Quantity
0.9
0.8
0.7
0.6
0.5
σ Fig. 3. Optimal ordering quantity with respect to s for normal distributed demand and shortage cost.
Next, we consider the normal distribution that has been widely studied and applied in inventory management. Since there are no closed-form solutions, we numerically investigate the validity of our results for normal distributed demand and shortage cost. The parameters are set as follows: r ¼ 6, c¼ 2.5, η ¼ 10%, s¼ 0.5, π 0 ¼ 5:6, D∼Nð4; 1Þ, and P∼Nðp; s2 Þ where EðPÞ ¼ p. Figs. 3 and 4 show the impact of the standard deviation (s) of the shortage cost on the optimal order quantity and percentage profit decrease under different risk measures. The optimal order quantity under the CVaR criterion is increasing with respect to s, and percentage profit decrease reduces as s increases. The optimal order quantity under the traditional criterion and the VaR constraint are independent of the standard deviation. These results are consistent with those for uniform distributed demand and binary distributed shortage costs.
6.2. Effects of risk measure In this section, we investigate the effects of risk measure on the newsvendor's optimal order quantity. Theorem 6. Assume that demand is uniform distributed and shortage cost is binary distributed.
0.4
3
3.5
4
4.5
5
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6
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8
Retail Price Fig. 5. Optimal ordering quantity with respect to r.
(1) For uncertain shortage costs, the risk-averse newsvendor under the CVaR criterion orders less than risk-neutral newsvendor, i.e., Q nCVaR ≤Q nE . (2) For constant shortage cost, if 0 ≤η ≤ðr−cÞ=ðr−sÞ then the riskaverse newsvendor under the VaR constraint orders less than risk-neutral newsvendor, i.e., Q nVaRc ≤Q nEc ; otherwise, the riskaverse newsvendor under the VaR constraint orders more than the risk-neutral newsvendor, i.e., Q nVaRc ≥Q nEc . (3) For uncertain shortage cost, if 0 o η ≤ðr−cÞ=ðr−sÞ then the riskaverse newsvendor under VaR constraint orders less than the risk-neutral newsvendor, i.e., Q nVaR ≤Q nE . (4) For constant shortage cost, if ðr−cÞ=ðr−sÞ ≤η o1, then Q nVaRc ≥Q nEc ≥Q nCVaRc ; if 0 ≤η ≤ðr−cÞ=ðr−sÞ, then Q nEc ≥Q nVaRc ≥Q nCVaRc . Proof. See Appendix F. Remark 2. For a general demand distribution, Xu and Li (2010) prove that Q nCVaR may not be monotonic in η which means that Q nCVaR may be less or greater than Q nE . For uniform distributed demand, Part (4) of Theorem 6 shows that Q nEc is always at least Q nCVaRc . Note from Theorem 6 that, for 0 ≤η ≤ðr−cÞ=ðr−sÞ, using either risk criterion results in a decrease of the order quantity compared with
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Percentage Profit Decrease
the classical profit optimization. Selecting CVaR as the risk criterion leads to a larger drop in the ordering quantity and also a larger profit decrease. Now, we perform a numerical experiment to quantify the effects of the risk measures. We consider the following setting of the parameters: c ¼1, η ¼ 5%, π 0 ¼ −0:2, s ¼0.5, α ¼ 0:5, Δ ¼ 1, r∈½3:2; 7:8 and E(P) is a element of ½2:2; 8:8. Note that for all considered combinations of parameters values, it holds that ðr−cÞ=ðr−sÞ≥30%. Thus, the condition η ≤ðr−cÞ=ðr−sÞ always holds. Figs. 5 and 7 compare the optimal order quantity under different risk measures with respect to the retail price and the shortage cost, respectively. It is not surprising that the optimal order quantity under the VaR constraint and the CVaR criterion is non-decreasing with respect to the shortage cost, and that the optimal order quantity under the VaR constraint is increasing with respect to retail price. These results also hold for the risk-neutral newsvendor. An interesting and counter-intuitive result is that the optimal order quantity under the CVaR criterion is decreasing in the retail price. This is explained as follows. A higher retail price also improves the lower end of the profit, and therefore fewer items need to be sold to obtain a sufficient profit. If only the lower end of profit is considered, as under the CVaR criterion, then the risk averse newsvendor orders fewer items although they can be sold at a higher profit margin. The VaR approach does consider the whole profit range by maximizing profit under a risk constraint. As a result, it does lead to a larger order quantity if the retail price increases, as expected. Further, we observe that Q nE ≥Q nVaRc ≥Q nCVaRc ,
Q nE ≥Q VaR and Q nE ≥Q nCVaR , which has been shown to hold in general in Theorem 6. Figs. 6 and 8 present percentage profit decrease after comparing the optimal profit under the two risk measures with respect to retail price and shortage cost, respectively. Note that in line with the above discussions, the profit difference between the CVaR and
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Percentage Profit Decrease
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50% 40% 30% 20% E vs VaR E vs VaRC E vs CVaR E vs CVaRC
10% 0
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(Expected) Shortage Cost Fig. 8. Percentage profit decease with respect to p.
50%
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Fig. 9. Optimal ordering quantity with respect to r for normal distributed demand and shortage cost when p ¼10.
Retail Price Fig. 6. Percentage profit decease with respect to r.
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Fig. 10. Optimal ordering quantity with respect to p for normal distributed demand and shortage cost when r ¼6.
M. Wu et al. / Int. J. Production Economics 145 (2013) 790–798
the expected profit criterion increases as the retail price increases and decreases as the shortage cost increases. Meanwhile, the profit difference between the CVaR and the expected profit criterion decreases as the retail price increases and increases as the shortage cost increases. Also note that uncertainty in the shortage cost narrows the gap in profit between CVaR and VaR. Next, we consider the normal distributed demand and shortage cost. The parameters are set as follows: c ¼2.5, η ¼ 10%, s ¼0.5, π 0 ¼ 4, D∼Nð4; 1Þ, and P∼Nðp; 1Þ where EðPÞ ¼ p (Figs. 9 and 10). Figs. 9 and 10 compare the optimal order quantity for normal distributed demand and shortage costs under different risk measures. The optimal order quantity under the VaR constraint and the CVaR criterion is non-decreasing with respect to the expected shortage cost, and the optimal order quantity under the VaR constraint is increasing with respect to the retail price. Further, uncertainty in the shortage cost increases the order quantity under the CVaR criterion. These results are consistent with those for uniform distributed demand and binary distributed shortage costs. The only difference between two different distributions is that the risk averse newsvendor under the CVaR criterion may order more than the risk neutral newsvendor. This result is consistent with Xu and Li (2010)'s result (see Remark 2).
7. Conclusion In this paper, we study the effect of the objective (profit maximization, VaR, CVaR) and uncertainty in the shortage cost for the newsvendor problem. The VaR variant that we considered is to optimize the expected profit under a risk constraint rather than to change the objective function, which is the other common VaR variant. The CVaR approach does alter the objective function. An insightful result is that for constant shortage cost (as in the standard newsvendor problem), the order quantity under CVaR is less than that under the VaR constraint, which in turn is less than the order quantity that maximizes the expected profit. As a result, the expected profit under CVaR is always lower than under the VaR constraint. Moreover, a numerical investigation shows that the profit gap can be considerable. Furthermore, CVaR leads to some counter-intuitive results, for instance that the order quantity decreases with the retail price. All these results are related to the fact that CVaR only considers the lower end of the profit range, whilst the other approaches (also) focus on the expected profit. In practice, even risk-averse logistics managers still consider the expected profit, and therefore the constraint variant of the VaR approach seems more balanced and suitable than the CVaR approach that creates a tension for the decision maker between profit and risk. Because introducing uncertainty in the shortage cost complicates the analysis considerably, we consider a special case with uniform demand and a high/low shortage cost. For this case, it turns out that the added uncertainty to shortage cost does not affect the order quantity when the expected profit is maximized (with or without a VaR constraint), but the CVaR approach reacts with an increased order quantity. Further, the numerical experiments show that the results also hold for the normal distributed demand and shortage cost. Combined with the above discussed findings for a constant shortage cost, this implies that uncertainty in the shortage cost decreases the profit gap between CVaR and VaR constraint. An interesting direction for further research is to compare different risk approaches for different, more complex inventory systems. Indeed, although our analysis and findings are restricted to the newsvendor problem, the explanations also apply to other systems, suggesting that we may obtain similar results for such systems. Establishing whether this is indeed the case is certainly
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worthwhile, as the implication might be that the standard CVaR approach is generally less suitable for inventory systems, and modifications are worthwhile to be developed.
Acknowledgment The authors are very grateful to the referees and the editor for patiently giving critical and insightful comments and suggestions, which helped to significantly improve the models, results, and presentation. The research was supported in part by National Natural Sciences Foundation of China under Grants 71101099, 71171088, 71271092 and by Sichuan University SKQY201330.
Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org.10.1016/j.ijpe.2013.06.007.
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