european journal of operational research · invited review the newsvendor problem: review and...

European Journal of Operational Research 213 (2011) 361–374

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Invited Review

The newsvendor problem: Review and directions for future research

Yan Qin a, Ruoxuan Wang a, Asoo J. Vakharia a,⇑, Yuwen Chen b, Michelle M.H. Seref c

a Department of Information Systems and Operations Management, Warrington College of Business Administration, University of Florida, Gainesville, FL 32611, United Statesb University of Rhode Island, Kingston, RI 02881, United Statesc Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0002, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 April 2009Accepted 20 November 2010Available online 16 December 2010

Keywords:InventoryLogisticsSupply chain management

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.11.024

⇑ Corresponding author. Tel.: +1 352 392 8571; faxE-mail addresses: [email protected] (Y. Qi

[email protected] (A.J. Vakharia), [email protected](M.M.H. Seref).

1 For obvious reasons, it is generally assumed that g

In this paper, we review the contributions to date for analyzing the newsvendor problem. Our focus is onexamining the specific extensions for analyzing this problem in the context of modeling customerdemand, supplier costs, and the buyer risk profile. More specifically, we analyze the impact of marketprice, marketing effort, and stocking quantity on customer demand; how supplier prices can serve as acoordination mechanism in a supply chain setting; integrating alternative supplier pricing policies withinthe newsvendor framework; and how the buyer’s risk profile moderates the newsvendor order quantitydecision. For each of these areas, we summarize the current literature and develop extensions. Finally, wealso propose directions for future research.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The newsvendor problem is one of the classical problems in theliterature on inventory management [7,43]. Key insights stemmingfrom an analysis of this problem have wide ranging implicationsfor managing inventory decisions for organizations in, for example,the hospitality, airline, and fashion goods industries. A generic con-temporary setting in which this problem could be framed is as fol-lows. Consider a three-node supply chain consisting of a supplier, abuyer, and customers. At the beginning of a single period, thebuyer is interested in determining an ‘‘optimal’’ stocking policyðQÞ to satisfy total customer demand for a single product. This cus-tomer demand is assumed to be stochastic and characterized by arandom variable x with the probability density function f ðxÞ andcumulative distribution function FðxÞ. The quantity Q is purchasedby the buyer from the supplier for a fixed price per unit v. The sup-plier is assumed to operate with no capacity restrictions and zerolead time of supply and thus, an order placed by the buyer withthe supplier at the beginning of a period is immediately filled. Salesof the product occur during (or at the end of) the single period and:(a) if Q P x, then Q � x units which are left over at the end of theperiod are salvaged by the buyer for a per unit revenue of g1; and(b) if Q < x, then x� Q units which represent ‘‘lost’’ sales cost thebuyer B per unit. Assuming a fixed market price of p, then the actualend of period profit for the buyer is:

ll rights reserved.

: +1 352 392 5438.n), [email protected] (R. Wang),(Y. Chen), [email protected]

< v .

PðQ ; xÞ ¼px� vQ þ gðQ � xÞ if Q P x;

pQ � vQ � Bðx� QÞ if Q < x:

�ð1Þ

Since the demand has not been realized at the beginning of the per-iod, the buyer cannot observe the actual profit. Hence, the tradi-tional approach to analyze the problem is based on assuming arisk neutral buyer who makes the optimal quantity decision atthe beginning of the period by maximizing total expected profits.These profits are:

E½PðQÞ� ¼Z Q

0½px� vQ þ gðQ � xÞ�f ðxÞdxþ

Z 1

Q½pQ � vQ

� Bðx� QÞ�f ðxÞdx

¼ ðp� gÞl� ðv � gÞQ � ðp� g þ BÞESðQÞ; ð2Þ

where E½�� is the expectation operator, l is the mean demand, andESðQÞ represents the expected units short assuming Q units arestocked and can be determined as

R1Q ðx� QÞf ðxÞdx. It is relatively

easy to show that Eq. (2) is strictly concave in Q [43] and hence,the first order conditions (FOCs) are necessary and sufficient todetermine the value of Q which maximizes Eq. (2). This optimal or-der quantity ðQ �Þ is set such that:

FðQ �Þ ¼ p� v þ Bp� g þ B

ð3Þ

and the corresponding profit for the buyer is:

E½PðQ �Þ� ¼ ðp� gÞl� ðp� g þ BÞZ 1

Q�xf ðxÞdx: ð4Þ

Prior researchers have sometimes adopted an ‘‘overage’’ ðcoÞ and‘‘underage’’ ðcuÞ cost perspective in analyzing the model from an

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362 Y. Qin et al. / European Journal of Operational Research 213 (2011) 361–374

expected cost perspective. In this case, FðQ �Þ ¼ cucoþcu where

cu ¼ p� v þ B and co ¼ v � g and E½PðQÞ� ¼ ðp� vÞl� coR Q

0 ðQ�xÞf ðxÞdx� cu

R1Q ðx� QÞf ðxÞdx.

In the last decade, Khouja [30] has also reviewed the extensivecontributions to the newsvendor problem. We extend this prior re-view by considering several specific extensions in our paper suchas integrating marketing effort, stock dependent demand, andbuyer risk profiles and how they all play a role in determiningthe optimal newsvendor order quantity. In addition, we also re-view 20 plus additional contributions since the review by Khouja[30]. We also refer the reader to a review of the newsvendor prob-lem in supply chain settings with a primary focus on contractualmechanisms which is available in Cachon [15]. The motivationfor our review, critique, and extensions stems from the fact thatkey parametric assumptions drive the relatively straightforwardanalysis for the single period newsvendor problem. In essence,our intent is on reviewing and examining how changes in theassumptions related to key parameters might moderate thecurrent set of results. More specifically, this paper focuses onreviewing prior work and developing extensions related to thefollowing:

1. Customer demand: This is typically treated as an exogenousparameter in the prior analysis. However, depending upon thecontext, it could be argued that:� customer demand x and price p are interrelated;� customer demand x could be influenced by marketing effort

e expended by the buyer; and� customer demand x could be influenced by the quantity

stocked Q by the buyer.

2. Supplier pricing policies: A fixed supplier price per unit (v)quoted by the supplier is the standard assumption in the news-vendor setting. However, current practice (see, for example,[11]) indicates suppliers tend to quote quantity discountschemes to generate larger order quantities from the buyer.Thus, it would be interesting to examine how alternative sup-plier price discounting schemes might moderate the buyer’soptimal quantity decision.

3. Buyer risk profile: The assumption underlying the basic struc-tural result for the newsvendor model is that the buyer is riskneutral and thus, would choose to optimize expected profits.Given that sourcing practice often indicates that buyers are riskaverse [13] and/or ‘‘risk’’ takers, we focus on how such risk pro-files would moderate the sourcing decision in a newsvendorsetting.

The remainder of this paper is organized as follows. In the nextsection, we start by analyzing the impact of alternative customerdemand related changes in a newsvendor setting; Section 3 focuseson examining the impact of alternative supplier pricing schemes;and Section 4 describes how the risk profile of the buyer impactsoptimal decision making in this setting. Finally, implications anddirections for future research are discussed in Section 5.

2. Customer demand

The analysis in this section focuses on how the optimal orderquantity in a newsvendor setting would be impacted if demandwas a function of: (a) market price; (b) marketing effort; and (c)stocking quantity.

2.1. Demand and market price

In a classic Newsvendor problem, selling price is considered asexogenous, over which the retailer has absolutely no control. This

is true in a perfectly competitive market where buyers are mereprice-takers. In practice, however, retailers may adjust the currentselling price in order to increase or reduce demand. Whitin [50],Mills [36], and Karlin and Carr [28] are three of the early publica-tions that investigated the effects of different demand processeson the seller’s pricing and ordering decisions.

Whitin [50] provided necessary optimality conditions for twomodels, an economic order quantity (EOQ) model and a style goodsmodel. In contrast to the well-known EOQ approach whichassumes an infinite planning horizon, a style goods model focuseson environments where there is a relatively short selling seasonwith a well defined beginning and end. Although Whitin [50] onlyprovided the closed-form solution for a style goods model with arectangular demand distribution, he believed that structuralsolutions are possible for other cases provided the price-dependentdemand distribution can be expressed as an explicit function of theexpected demand.

Mills [36] assumed that the price-dependent demand ðxÞ isaffected additively by a random term, �, independent of price,i.e., x ¼ lðpÞ þ � where lðpÞ is the mean demand as a function ofmarket price p. In contrast to the deterministic demand model thatoptimizes at the price which equates riskless marginal revenue andmarginal cost, the model under stochastic demand is optimized atthe price which equates riskless marginal revenue to marginal costplus an additional non-zero term. He concluded that the directionof change in the magnitude of the optimal price in the risky situa-tion depends on the shape of the marginal cost curve and showedthat this optimal price cannot be higher than it would be in theriskless case for a constant marginal cost. A contrary result was ob-tained by Karlin and Carr [28] for the case of a multiplicative modelin which the stochastic demand x ¼ lðpÞ�. The demand model em-ployed in Mills [36] is referred to as the additive demand form andthe one in Karlin and Carr [28] as the multiplicative demand form.In what follows, we will first derive some simple results for thesetwo special demand-price relationships, and then review the anal-ysis which leads to the results mentioned above.

In the additive demand case, assume that demand x ¼ lðpÞ þ �,where lðpÞ ¼ a� bp. In this case, a and b are both positive con-stants, p is the market price, and � is a random variable with aprobability density function hð�Þ and cumulative distribution func-tion Hð�Þwith a mean of zero. The expected profit function is now abivariate function with price and order quantity as the decisionvariables and can be expressed in terms of overage and underagecosts as:

E½Pðp;QÞ� ¼ ðp� vÞlðpÞ � co � E½Q � x�þ � cu � E½x� Q �þ; ð5Þ

where E½Q � x�þ is the expected number of units overstocked andE½x� Q �þ is the expected number of units understocked. By observ-ing that x ¼ aþ bpþ �, it is easy to show that [39]:

E½Q � x�þ ¼Z Q�aþbp

0Hð�Þd�;

E½x� Q �þ ¼Z þ1

Q�aþbp½1�Hð�Þ�d�:

This expected profit can be interpreted as being equal to risklessprofit minus expected cost. Next, let’s consider the partial deriva-tives of the expected profit function with respect to Q and p. Differ-entiating E½Pðp;QÞ� with respect to Q, we obtain

@E½Pðp;QÞ�@Q

¼ cu � ðco þ cuÞHðQ � aþ bpÞ;

@2E½Pðp;QÞ�@Q 2 ¼ �ðco þ cuÞhðQ � aþ bpÞ:

Clearly, @2E½Pðp;QÞ�@Q2 6 0, which indicates that the expected profit func-

tion is concave in Q for a given p. We can then solve for the optimal

Y. Qin et al. / European Journal of Operational Research 213 (2011) 361–374 363

order quantity for a given p by setting the FOC equal to zero. Thisresults in:

HðQ �ðpÞ � aþ bpÞ ¼ cu

co þ cu; ð6Þ

where Q �ðpÞ is the optimal order quantity2 for a given price p. Aftersubstituting Q �ðpÞ ¼ H�1½ cu

coþcu� þ a� bp (obtained from Eq. (6)) into

the expected profit function (Eq. (5)), we can reduce the bivariateobjective function into a univariate function of p as follows:

E½PðQ �ðpÞÞ� ¼ ðp� vÞlðpÞ � co � E½Q �ðpÞ � x�þ � cu � E½x� Q �ðpÞ�þ

¼ ðp� vÞða� bpÞ � co

Z Q�ðpÞ�aþbp

0Hð�Þd�

" #

� cu

Z þ1

Q�ðpÞ�aþbp½1� Hð�Þ�d�

" #;

where Q �ðpÞ ¼ H�1½ cucoþcu� þ a� bp. Differentiating E½PðQ �ðpÞÞ� with

respect to p gives us the following FOC and second order condition:

dE½PðQ �ðpÞÞ�dp

¼ a� 2bpþ bv � co � HðQ �ðpÞ � aþ bpÞ½

� bþ dQ �ðpÞdp

� �� E½x� Q �ðpÞ�þ

þ cu � ½1� HðQ �ðpÞ � aþ bpÞ� � bþ dQ �ðpÞdp

� �� ¼ a� 2bp� ES½Q �ðpÞ� þ bv;

d2E½PðQ �ðpÞÞ�dp2 ¼ �2bþ 1

hðQ �ðpÞ � aþ bpÞ �ðv � gÞ2

ðp� g þ BÞ3;

where ES½Q �ðpÞ� ¼ E½x� Q �ðpÞ�þ. The sign of the second-order deriv-

ative d2E½PðQ�ðpÞÞ�dp2 depends on the demand distribution and parameter

values assumed. If d2E½PðQ�ðpÞÞ�dp2 < 0, by setting the first-order condition

to zero, we can get

p� ¼ 12bðaþ bv � ES½Q �ðp�Þ�Þ: ð7Þ

It is worth noting that in the additive deterministic demand case(i.e., x ¼ aþ bp), the optimal price is given by the expression aþbv

2b .Since we know that ES½Q �ðp�Þ�P 0, we can conclude that the opti-mal price in the stochastic setting is always lower than the optimalprice in the deterministic setting. Finally, it is also worth noting thatPetruzzi and Dada [39] approached the same problem by expressingexpected profit as a univariate function of quantity Q. They pointout that a unique Q � exists for an optimal price p� if the distributionof the random term � satisfies the inequality 2r2ð�Þ þ @rð�Þ

@� > 0 for� 2 ½A;M�, where rð�Þ � hð�Þ=½1� Hð�Þ� and a� bðv � 2MÞ þ A > 0,where AðMÞ is the minimum (maximum) possible value of �. Thefirst inequality generalizes the conditions for which optimal solu-tions to single period newsvendor pricing problems can be identi-fied analytically. Zhan and Shen [51] achieved the same results asabove under the linear additive demand form and developed aniterative algorithm and a simulation based algorithm to find theoptimal solution. In contrast to the previous papers discussed sofar, the authors treated this problem as a nonlinear system withtwo variables, instead of reducing the bivariate objective functionto a univariate function.

2 The concavity result guarantees the existence of Q �ðpÞ. However, the existence ofa unique value of Q �ðpÞ requires the assumption that Hð�Þ is also continuous anddifferentiable.

In the multiplicative demand case, assume that x ¼ lðpÞ�,where lðpÞ ¼ ap�b. Similar to the additive demand setting, a andb are both positive constants with the additional restriction thatb > 1; p is the market price, and � is a random variable with acumulative distribution function H1ð�Þ with a mean of 1. The ex-pected profit function is again a bivariate function of price and or-der quantity as given by Eq. (5). The only difference is that theterms E½Q � x�þ and E½x� Q �þ are redefined as follows [28]:

E½Q � x�þ ¼Z Qa�1pb

0H1ð�Þap�b d�;

E½x� Q �þ ¼Z þ1

Qa�1pb½1� H1ð�Þ�ap�b d�:

By applying the same procedure as in the additive case, we are ableto show that the expected profit function is again concave in Q for agiven p and that:

H1ðQ �ðpÞa�1pbÞ ¼ cu

cu þ co: ð8Þ

After substituting Q �ðpÞ ¼ H�11

cucuþco

h iap�b (obtained from Eq. (8))

into the expected profit function Eq. (5) with the terms E½Q � x�þ

and E½x� Q �þ redefined as above, we can reduce the bivariate objec-tive function into a univariate function of p. DifferentiatingE½PðQ �ðpÞÞ� with respect to p gives us the following FOC3:

dE½PðQ �ðpÞÞ�dp

¼ ½1� bð1� p�1vÞ�ap�bl� þ bp�1ðv � gÞ � EL½Q �ðpÞ�

� ½1� bp�1ðp� v þ BÞ� � ES½Q�ðpÞ�;

where EL½Q �ðpÞ� ¼R Q�ðpÞa�1pb

0 H1ð�Þap�bd� and ES½Q �ðpÞ� ¼Rþ1Q�ðpÞa�1pb ½1� H1ð�Þ�ap�bd�. If d2E½PðQ�ðpÞÞ�

dp2 6 0, then p� ¼ bvb�1þ D,

where

D ¼ a�1bp�b½B � ESðQ �ðp�ÞÞ þ ðv � gÞ � ELðQ �ðp�ÞÞ�ðb� 1Þð1� a�1p�b � ESðQ �ðp�ÞÞÞ : ð9Þ

In the multiplicative deterministic demand case (i.e., x ¼ ap�b), theoptimal price is given by the expression p� ¼ bv

b�1. Since D P 0, wecan conclude that the optimal price in the stochastic setting is al-ways greater than the optimal price in the deterministic setting.This is in direct contrast to the result for the additive price depen-dent demand setting. Finally, it is also worth noting that Petruzziand Dada [39] approached the same problem by expressing ex-pected profit as a univariate function of quantity Q. They pointedout that a unique Q � exists provided the distribution of the randomterm satisfies the inequality 2r2ð�Þ þ drð�Þ

d� > 0 for � 2 ½A;M� andb > 2. A possible explanation is also given in their paper regardingthe relative magnitudes of riskless and risky optimal prices underdifferent demand forms in terms of demand variance and demandcoefficient of variation.

Lau and Lau [32] examined the pricing and ordering policies of anewsvendor facing with a random price-dependent demand undertwo different objectives, (1) the objective of maximizing the ex-pected profit and (2) the objective of maximizing the probabilityof achieving a certain profit level. Two price-demand forms areconsidered in this paper, an additive form as assumed in our anal-ysis above with a homoscedastic and normally distributed randomterm (Model A), and a relationship fitted with a four-parameterbeta distribution based on subjective estimates (Model B). Inscenario (1A), where Model A is considered under objective (1), anumerical example is used to illustrate the effects of price

3 As before, it is necessary to assume that H1ð�Þ is continuous and differentiable.

4 For example, if demand is normally distributed with mean lðe�Þ and standarddeviation r, then Q� ¼ lðe�Þ þ zr where z is the unit normal deviate such thatFðzÞ ¼ p�vþB

p�gþB.


sensitivity and demand variation on the pricing and ordering deci-sions. The authors showed that lower price sensitivity and demandvariation lead to higher optimal price but lower optimal orderquantity. In scenario (2A) with zero shortage cost, a counter-intu-itive result is obtained that more uncertainty in demand may actu-ally increase the probability of achieving a certain profit level. Inthis paper, numerical solution procedures are provided in eachscenario, (1A), (1B), (2A) and (2B), while analytical results arerestricted to scenarios (1A) and (2A) with zero shortage cost.

Polatog�lu [40] investigated the joint pricing and ordering deci-sions under general demand uncertainty, aimed to reveal the fun-damental properties independent of demand pattern. Unimodalityof the expected profit function that traces the best price trajectoryover the order-up-to decision was proved under the assumptionsthat the mean demand is a monotone decreasing function of priceand that the riskless revenue function, which is equal to pricetimes mean demand, is uni-modal. Polatog�lu [40] also approachedthe same results proven by Whitin [50], Mills [36] and Karlin andCarr [28], regarding the relative magnitudes of the optimal pricesin additive and multiplicative models.

Chen et al. [17] investigated the price-dependent newsvendormodel in a competitive environment. The general demand is sto-chastic and price-dependent and the firm uses price to competefor the demand. They proved and showed the conditions for theexistence of the pure-strategy Nash equilibrium and its unique-ness. Raising individual prices at any equilibrium of the gamecould increase the total system profit, but no single firm would fol-low this strategy since it has already optimized its own profit con-ditioned on the other firms’ pricing schemes. At the cooperativeequilibrium, each self-interested firm has an incentive to lowerits price to grab more demand. They concluded that competitionleads to a lower equilibrium price and higher stock levels in allfirms.

Arcelus et al. [5] studied the pricing and ordering policies of aprofit-maximizing newsvendor that faces trade incentives offeredby the supplier (i.e., wholesale price discount to the newsvendorand direct rebate to the end customers) and stochastic price-dependent demand. Let ðv ;RÞ denote the supplier’s pricing/rebatepolicy, where v is the wholesale price and R is the direct rebateto end customers. Suppose the original wholesale price is v0. Thenewsvendor receives an offer ðv0 � d; 0Þ under the price discountpolicy and ðv0;RÞ under the direct rebate policy. No specific func-tional form is assumed for the deterministic component of the de-mand and both additive-error and multiplicative-error demandsare considered in this paper. The authors derived the demand con-ditions under which the results obtained in Petruzzi and Dada [39]regarding the relative magnitude of the retail prices in riskless andrisky cases still hold in this setting. The effectiveness of the twotrade incentives on the newsvendor’s decisions is then measuredand contrasted by the pass-through ratio, defined as d ¼ dp

dv, underthe price-discount policy, and the claw-back ratio, defined ask ¼ dp

dR, under the direct-rebate policy. They showed analyticallythat the pass-through ratio is always greater than the claw-backratio for any deterministic demand and that the claw-back ratioin the risky case is smaller than that in the riskless case for demandforms with additive-error and linear deterministic component. It isworth noting that negative claw-back ratio is possible, that is, thenewsvendor may offer a lower retail price to the end customers inspite of the rebate the consumers have already received from thesupplier. However, no further analytical results are obtained con-cerning the impact of demand uncertainty on the newsvendor’soptimal order quantity in this setting. Numerical analysis is con-ducted under the assumption of uniformly distributed error term.Observations include the dominance of the pass-through ratio overthe claw-back ratio in the risky case and the improved newsven-dor’s expected profits under either trade incentive.

2.2. Demand and marketing effort

Customer demand can often be influenced by a number of mar-keting activities (e.g., advertising, sales calls, and in-store displays).Kraiseburd et al. [31] studied a model that incorporates such an as-pect in the newsvendor scenario. Their setting is as follows. Let erepresent the marketing ‘‘effort’’ expended by the buyer. Mean cus-tomer demand is modeled as a non-decreasing function of e (i.e.,dlðeÞ

de P 0 and d2lðeÞde2 < 0) while the variance in customer demand

(i.e., r2) is assumed to be independent of the marketing effort e.cðeÞ represents the cost per unit of effort and is assumed to be con-vex in e. In this case, the optimal marketing effort for the firm (i.e.,e�) can be determined by solving the following FOC:

ðp� vÞdlðeÞde� dcðeÞ

de¼ 0: ð10Þ

The optimal order quantity Q � can then be found by settingFðQ �Þ ¼ p�vþB

p�gþB where the cumulative distribution function of de-

mand4 has a mean lðe�Þ.The analysis presented above could also be easily extended to a

situation where marketing effort affects demand in a way that de-mand variance decreases as more effort is made in the selling sea-son. Gerchak and Mossman [22] examined the effects of demandrandomness on optimal order quantities and the associated ex-pected costs by applying mean-preserving transformations to thedemand variable. They considered the family of random variablesxa representing demand as follows:

xa ¼ axþ ð1� aÞl;

where a P 0 is an arbitrary constant. Obviously, EðxaÞ ¼ l for all a.We note that if for some constants a1 > a2, then xa1 is more variablethan xa2 and using this property the authors are able to show that:

Q �a ¼ aQ � þ ð1� aÞl;CaðQ�aÞ ¼ aCðQ �Þ;

where CðQ �Þ represents the expected cost associated with an opti-mal order quantity of Q � under demand x, and CaðQ �aÞ representsthe expected cost associated with an optimal order quantity Q �a un-der demand xa.

To analyze the case in which demand uncertainty is affected bymarketing effort, replace a with aðeÞ in the above three equations.Assume that aðeÞ is a monotone decreasing function of e and thatað0Þ ¼ 1, and aðþ1Þ ¼ 0. The equation aðþ1Þ ¼ 0 simply says thatdemand uncertainty may be eliminated by tremendous marketingeffort, which is impossible in practice due to the unbearable cost ofthose marketing activities. Also assume that cð0Þ ¼ 0.

After substituting and rearranging, we obtain

Q �aðe�Þ ¼ aðe�ÞQ � þ ð1� aðe�ÞÞl: ð11Þ

Consider the relative magnitudes of Q �aðe�Þ and Q�. As

dQ�aðeÞde

¼ daðeÞdeðQ � � lÞ; ð12Þ

the optimal order quantity increases in marketing effort e if andonly if Q � < l, i.e., FðlÞ > p�vþB

p�gþB.The maximum expected profit for a specific marketing effort e

can be expressed as

E½PðQ �aðeÞ; eÞ� ¼ ðp� vÞl� CaðeÞðQ �aðeÞ; eÞ � cðeÞ¼ ðp� vÞl� aðeÞCðQ �Þ � cðeÞ: ð13Þ

5 The assumption of linearity is made for tractability but the results hold as long asdemand is non-decreasing in quantity.

6 The mean of the random term � is not required to be zero for the linear additivedemand form.


Differentiating E½PðQ �aðeÞÞ� with respect to e, we get

dE½PðQ �aðeÞÞ�de

¼ � daðeÞde

CðQ�Þ � dcðeÞde

; ð14Þ

d2E½PðQ �aðeÞÞ�de2 ¼ � d2aðeÞ

de2 CðQ �Þ � d2cðeÞde2 : ð15Þ

Ifd2E½PðQ�aðeÞÞ�

de2 < 0, we can derive the optimal marketing effort e� by

setting the partial derivativedE½PðQ�aðeÞÞ�

de equal to zero, which deter-mines the optimal order quantity of the buyer. Clearly, the sec-ond-order condition holds for a concavely decreasing aðeÞ and aconvexly increasing cðeÞ. In this case, e� can be characterized bythe following equation:

daðeÞde

CðQ �Þ þ dcðeÞde¼ 0: ð16Þ

For this setting (i.e., demand is influenced by marketing effort),our analysis indicates that an increase (decrease) in mean demand(with demand variability unchanged) due to marketing effort leadsto an increase (decrease) in the optimal stocking quantity Q �. Onthe other hand, the impact of a decrease in demand variability(with mean demand unchanged) due to marketing effort is not asclear. However, we can identify parametric conditions under whicha decrease in demand variability due to marketing effort will leadto increases in the optimal stocking quantity Q �.

2.3. Demand and stocking quantity

Marketing researchers often argue that quantity displayed canhave a positive impact on demand and thus, ultimate sales. In es-sence two types of stimuli that impact demand positively due tothe quantity displayed are: (a) a selective effect; and (b) an adver-tising effect. There is a significant body of literature on inventorysystems where demand is positively impacted by stock level. Thesestock dependent inventory models developed so far can be classi-fied into two categories. Models in the first category dealt with de-mand that is a function of the initial stock level, while the othersassumed that demand is dependent on the instantaneous stock le-vel and may thus vary over time. We will focus on newsvendormodels with stock-dependent demand. Readers who are also inter-ested in other inventory systems may refer to Urban [45] for acomprehensive review of the stock-level-dependent periodic-re-view models.

Baker and Urban [8] studied a deterministic single-period mod-el in which demand at time t; xðtÞ, is a polynomial function of theinstantaneous stock level, iðtÞ, and solved for the optimal orderquantity ðQÞ, which is also the initial stock level, under the objec-tive of maximizing the total profit. Specifically, they assumed a de-mand function xðtÞ ¼ a½iðtÞ�1�b, where a > 0; 1 > b > 0. Theconstant parameter b is constrained to be some value between zeroand one to ensure that the marginal increase in demand decreasesas the stock level increases. Sensitivity analysis conducted in thispaper showed that failing to consider the stimulating effect ofstock on demand when it exists may lead to miserable optimiza-tion results, since the model is quite sensitive to changes in b.Urban and Baker [46] investigated a similar deterministic news-vendor problem with stock-dependent, price sensitive and timeproportional demand. They expressed the demand rate asxðp; t; iÞ ¼ ap�ktr�1i1�b, where a > 0; k > 0; r > 0 and 1 > b > 0.Although no structural results were obtained for the general case,sensitivity analysis performed regarding the effects of price, timeand stock level on demand reassured the importance of recogniz-ing the correct demand pattern.

Urban [44] also examined a stock-dependent newsvendor mod-el. Demand in this paper takes the same polynomial function as inBaker and Urban [8], but with the total stock level iðtÞ replaced by

the observed stock level uðtÞ assuming that customers are onlyinfluenced by the displayed products. That is,xðuðtÞÞ ¼ a½uðtÞ�1�b

; a > 0; 1 > b > 0, where uðtÞ ¼minðs; iðtÞÞand s denotes the shelf space allocated to that product. With thefurther assumption of the quasi-concavity of the profit function,Urban derived the closed form expressions for the optimal orderquantity Q �, which is equal to i�ð0Þ, and the optimal shelf spaces� and investigated the interdependency between the two deci-sions. The relationship between Q � and s� can be characterizedby the equation s�

Q� ¼ð1�bÞðp�cÞð1�bÞðp�cÞþv, where v is the unit shelf-space cost.

Urban also pointed out that the holding cost, which can be consid-ered as the negative salvage cost in our setting, has an equal impacton the optimal values of the two decision variables.

Most of the inventory models where demand is assumed to be afunction of instantaneous stock level seldom incorporate theuncertainty of demand due to technical complexity. Similarly,when stochastic stock-dependent demand is considered in aninventory system, researchers usually interpret the word ‘‘stock’’as ‘‘initial stock’’, which can be viewed as ‘‘order quantity’’ in a sin-gle period newsvendor model. In the rest of the section, some sim-ple results will first be derived for a special case of the stochasticnewsvendor model that uses initial stock level as an independentvariable, followed by a brief literature review of stochastic stock-dependent newsvendor models.

Assume that customer demand is linearly increasing in thebuyer’s stocks as follows5: xðQÞ ¼ aþ bQ þ �; a > 0; 0 < b < 1,where � is a random variable with probability density functionh2ð�Þ and cumulative distribution function H2ð�Þ.6 Assuming thatthe buyer is risk neutral and would like to maximize the expectedprofit, the objective function for this setting is as follows:

E½Pðp;QÞ� ¼ ðp� vÞðaþ bQ þ l�Þ � co � E½Q � xðQÞ�þ

� cu � E½xðQÞ � Q �þ; ð17Þ

where

E½Q � xðQÞ�þ ¼Z ð1�bÞQ�a

0H2ð�Þd�;

E½xðQÞ � Q �þ ¼Z þ1

ð1�bÞQ�a½1� H2ð�Þ�d�:

It is quite straightforward to show that the expected profit functionis concave in Q. By equating the first order derivative of the objec-tive function with respect to Q to zero, we obtain

H2ðð1� bÞQ � � aÞ ¼p�v1�b þ B

p� g þ B: ð18Þ

Note that H2ðð1� bÞQ � aÞ ¼ Pðð1� bÞQ � a 6 �Þ ¼ PðQ 6 xðQÞÞ.Since 0 < b < 1, the fractile value given by

p�v1�bþBp�gþB is greater than

the critical fractile value p�vþBp�gþB for the classic newsvendor model.

Hence, we can conclude that the stock-dependent demand in anadditive form causes an increase in the order quantity as comparedto the classic newsvendor model.

Dana and Petruzzi [19] investigated an extension of the classicnewsvendor model where each customer is interested in maximiz-ing her own expected utility and chooses whether to visit the firmor go for an outside option, u, based on the firm’s price and initialstock level she observes as well as her own valuations of the prod-uct and the outside option. Given these assumptions of consumerbehavior, the firm faces uncertain aggregate demand which canbe modeled as a multiplicative function of both its price and its ini-tial stock level. The authors considered two cases: the exogenous


price case for a firm that is merely a price taker and the endoge-nous price case where price can be chosen optimally. Both thebenchmark policy, which ignores the effect of stock on demand,and the optimal policy are determined in each case. The unique-ness of the optimal policies again requires specific restrictions onthe distribution of consumers’ outside options and/or the distribu-tion of aggregate demand as mentioned in the previous section.The result of a comparison between the optimal policy and its cor-responding benchmark policy is that if a firm internalizes the effectof stock on demand, it usually holds more stock, provides higherservice level and earns higher expected profit no matter it is a pricetaker or price setter. It is also worth mentioning that sequentialoptimization of price and stocking quantity can be applied in theendogenous price case.

Balakrishnan et al. [9] generalized the newsvendor model toincorporate the stochastic and initial-stock-level dependent de-mand. The authors proposed a general stochastic demand modelvia an inverse fractile function to capture the stimulating effectof stock on demand. Let f ðQ ; xÞ denote the probability density func-tion (pdf) of xðQÞ and lðQÞ denote the stock-dependent lower sup-port of the pdf. The corresponding inverse fractile function isdefined as /ðQ ;qÞ ¼ F�1ðQ ;qÞ, where q is some given fractile valuebetween 0 and 1. To ensure that demand increases at a decreasingrate with the initial stock level, /ðQ ;qÞ is assumed to be a con-cavely increasing function of Q, which is later shown to be the suf-ficient condition for the concavity of the profit function. Applying avariable transformation x ¼ /ðQ ;qÞ, the expected sales for a givenorder quantity Q can be expressed as SðQÞ ¼

R FðQ ;QÞ0 /ðQ ;qÞdqþ

Qð1� FðQ ;QÞÞ. The marginal expected sales S0ðQÞ was found toconsist of two parts, 1� FðQ ;QÞ and

R FðQ ;QÞ0 /0ðQ ;qÞdq. The former

was interpreted as the availability effect of stocking higher quanti-ties and the latter as the promotional effect of stocking more.When the stock stimulating effect is ignored, the term representingthe promotional effect drops out in the optimality condition. Byobserving so, the authors extended, from a multiplicative demandform to a general demand form, the conclusion reached in Danaand Petruzzi [19] that the consideration of the demand-stimulatingeffect of stock leads to higher initial stock levels and higher fillrates.

7 A two-part tariff can be considered a special case of a quantity discount where forany quantity purchased, the retailer is required to pay a fixed fee plus a per unit fee.

3. Supplier pricing policies and stocking quantity

Quantity discounts are a wildly adopted pricing scheme in prac-tice. Basically, the notion is that the larger the order quantity, thelower the unit procurement cost and the justification for its popu-larity includes, but not limited to, costs transfer, price discrimina-tion and channel coordination. When a supplier has a high fixedcost, s/he may choose to offer quantity discounts to encourage lar-ger orders, which helps achieve economies of scale. Given that therisk of larger orders is mainly carried by the buyer, suppliers are of-ten more than willing to practice quantity discounts when inven-tory costs are high. As for price discrimination, quantitydiscounts helps suppliers to identify price-insensitive retailersfrom price-sensitive ones and charge higher prices (if possible) toretailers who are willing to pay more. In this area, there are twointerrelated streams of literature which are of relevance: (a) usingdifferent supplier pricing policies to coordinate a supply chain; and(b) modifying the traditional newsvendor analysis to handle differ-ent types of supplier pricing policies. Research contributions ineach area are discussed in this section.

3.1. Coordination in a supply chain through supplier pricing policies

Consider a supply chain (SC) with a single supplier (S) and a sin-gle newsvendor-type retailer (R) facing uncertain demand. Assume

that the two channel members have full information about de-mand and each other’s cost structure and that the retailer placesan order at the beginning of the selling season. Then, the retailer’sexpected profit ðE½PRðQÞ�Þ can be expressed as

E½PRðQÞ� ¼ ðp� vÞl� ðp� v þ BÞE½x� Q �þ � ðv � gÞE½Q � x�þ

ð19Þ

and based on the order placed by the retailer ðQÞ and a supplier var-iable production cost ðcÞ, the supplier’s expected profit ðE½PS�Þ is:

E½PS� ¼ ðv � cÞQ ; ð20Þ

respectively. Since the supplier will not operate with negative orzero profit, it is reasonable to assume v > c. The expected supplychain profit ðE½PSC �Þ, which is the sum of the retailer’s and supplier’sprofits, is

E½PSC � ¼ ðp� cÞl� ðp� c þ BÞE½x� Q �þ � ðc � gÞE½Q � x�þ: ð21Þ

When the channel members are only interested in maximizing theirown profits, the retailer will place an order of F�1ðp�vþB

p�gþBÞ units,

which is smaller than the order quantity F�1ðp�cþBp�gþBÞ that maximizes

the channel profit. Hence, offering some form of a quantity discountto the retailer, which results in a lower wholesale price, can be afeasible solution to achieving channel coordination. However, itmust be made clear that not all forms of quantity discounts can helpobtain channel coordination.

According to Jeuland and Shugan [26], among various quantitydiscounts, only those that make channel members’ profit maximiz-ing incentives consistent with overall channel profit maximizationcan coordinate the supply chain. This implies that channel mem-bers’ profit functions should be some fixed linear function of thechannel profit under the channel-coordinating quantity discount.Define xðpÞ as a price-dependent demand function such that de-mand is decreasing in price (thus, pðxÞ is the inverse of xðpÞ),k1ð0 < k1 < 1Þ as some fraction of total supply chain profits re-ceived by the supplier (hence, the retailer receives 1� k1), k2 assome fixed amount of total supply chain profit allocated to the sup-plier, and cðCÞ as the variable per unit cost for the supplier (retai-ler). Based on this, one possible form of a quantity discounted pricevðxÞ which can coordinate the supply chain is vðxÞ ¼ k1ðpðxÞ�C � cÞ þ ðk2

x Þ þ C.Ingene and Parry [24] showed the existence of a two-part tariff

wholesale pricing policy that coordinates a supply chain with a sin-gle supplier and multiple independent retailers. It is assumed thatthe deterministic demand faced by each retailer is unique and con-cavely decreasing in retail price. The number of retailers whochoose to participate (by obtaining non-negative profits) can alsobe considered as a decision in this problem. Let vðQiÞ denote theunit payment by retailer i for an order of size Qi and c denotethe supplier’s marginal cost. Utilizing marginal analysis tech-niques, the authors showed that a two-part tariff7 in whichvðQiÞ ¼ c þ /c=Q i coordinates the supply chain, where /c is a fixedfee paid by each participating retailer to the supplier. The numberof participating retailers under this type of pricing policy, denotedby N�c , should be no more than the number of participating retailersin the integrated system since each retailer in the decentralized sys-tem has a non-negative /c to cover. Finally, it is shown that this typeof coordinating two-part tariff policy is different from the two-parttariff policy that maximizes the supplier’s profit except in thefollowing two cases: the supply chain consists of a single retailer,or the sales of the retailer that achieves exactly zero profit from


participation is equal to the total sales divided by the number of par-ticipating retailers.

Weng [48] studied the coordination issues of a single-supplier-single-distributor system under price-dependent stochasticdemand. A two-part tariff policy, a fixed payment plus the whole-sale price, was shown to be sufficient to achieve the coordination.The author assumed that all demand must be satisfied and a sec-ond order is allowed at a higher cost when demand exceeds theinitial order quantity. Decisions to be made include the distribu-tor’s selling price and initial order quantity as well as the supplier’sunit profit margin. The supplier’s selling price is equal to the sumof the unit profit margin and the unit production cost. In the ab-sence of coordination, it is assumed that the supplier may notknow the distributor’s actual response to its wholesale price andthus determines the unit profit margin based on its estimatedorder quantity, which may deviate from the distributor’s actualorder quantity. When coordination is enabled, the supplier hascomplete information about the distributor’s response. The price-dependent random demand is modeled by a generalized phase-

type distribution, i.e., FðxjpÞ ¼ 1� e�x=lðpÞPþ1n¼1an

Pn�1i¼0

ðx=lðpÞÞii! , where

an; n ¼ 1;2; . . . ;1 is some appropriately-chosen discrete probabil-

ity distribution and lðpÞ ¼ dp�k; d > 0; k > 1. The parameter k

represents the sensitivity of demand to price. The optimal pricingand ordering policies with and without coordination are then char-acterized and constrasted under previous assumptions. It is shownanalytically that the optimal initial order quantity with coordina-tion is always greater than the optimal initial order quantity with-out coordination and the gap increases in k. Two effects ofcoordination are found to contribute to the above result, the reduc-tion in the unit overage and underage costs and the reduction inthe supplier’s selling price. Numerical analysis demonstrated thenecessity of coordination for both parties when demand is highlysensitive to the distributor’s selling price. The author also devel-oped a two-part tariff policy to coordinate the system in whichthe supplier sets its selling price equal to its unit production costand the distributor makes a fixed payment to the supplier.

In Weng [49], a generalized newsvendor model was developedto examine the effect of coordination for a single-supplier-single-retailer supply chain under stochastic demand and an all-unitsquantity discount was designed to coordinate the system. The onlydecision included in this model is the buyer’s order size. No specificdistribution is assumed for the demand. Unfilled demand may bebackordered at a cost and a second order may be placed to the sup-plier depending on whether the additional order generates positiveprofits for both parties. Let b denote the unit backorder cost, t2 de-note the ordering cost for the second order and s2 be the supplier’ssetup cost for the second run. The buyer is willing to place a secondorder if the unfilled demand, q, satisfies the inequalityðp� v � bÞq� ðt2 þ s2Þ þ Bq P 0, where the sum of the first twoterms is the profit to be obtained from placing a second orderand the last term is the shortage cost incurred if a second orderis not placed (recall that B is the per unit shortage/lost sales cost).This modeling of unfilled demand can be used to cover situationsin which all demand must be satisfied by setting B! þ1 or nore-supply is available from the supplier by letting b ¼ p� v þ B.Analytically, the author showed that, for an increasing concavedemand distribution function, the supply chain’s optimal orderquantity ðQ �SCÞ is always larger than the retailer’s optimal orderquantity, Q �R when coordination is not implemented. LetcminðQ �SC ;qÞ (where q is some non-negative constant) denote thesupplier’s lowest acceptable per unit price v satisfyingE½PSðQ �SCÞ�P ð1þ qÞE½PSðQ �RÞ�, where E½PSð�Þ� refers to the sup-plier’s expected profit with a particular order quantity. LetcmaxðQ �R; cÞ (where c is some non-negative constant) denote theretailer’s highest acceptable per unit price v satisfying

E½PRðQ �SCÞ�P ð1þ cÞE½PRðQ �RÞ�. If cmaxðQ �R; cÞP cminðQ �SC ;qÞ, theauthor argued that an all-units quantity discount in which the sin-gle price break quantity and the corresponding unit price be set toQ �SC and cmaxðQ �R; cÞ, respectively, can coordinate the system sinceboth parties are no worse off. The two parameters q and c can beinterpreted as specifying the allocation of the increased systemprofit due to coordination. For example, if q ¼ 0, then all increasesin supply chain profits are allocated to the buyer, while if c ¼ 0,then all increases in supply chain profits are allocated to thesupplier.

Cachon and Lariviere [16] focused on the use of revenue-sharingcontracts in supply chain coordination and offered comparisonsbetween revenue-sharing and other contracts including quantitydiscounts. Again, consider a single-supplier–single-retailer supplychain. The profit-maximizing retailer makes two decisions, orderquantity and retail price (shortage cost is not included). LetSðQ ; pÞ be the expected sales for the quantity-price pair ðQ ; pÞ, cbe the unit production cost of the supplier and C be the retailer’sunit operating cost. The retailer’s expected revenue can then beexpressed as RðQ ; pÞ þ gQ , where RðQ ; pÞ ¼ ðp� gÞSðQ ; pÞ, and theexpected system profit E½PSCðQ ; pÞ� ¼ RðQ ; pÞ � ðc þ C � gÞQ . LetðQ �; p�Þ denote the quantity-price pair that maximizes theexpected system profit. The authors demonstrated that, for a fixedretail price p�, the quantity discount scheme vðQÞ ¼ ð1� aÞ RðQ ;p�Þ

Q þaðc þ CÞ þ ð1� aÞg � C can also coordinate the supply chain.However, this specific quantity discount may fail to coordinatethe system when the retail price deviates from p�, while the linearrelationship between the retailer’s expected profit and the sys-tem’s expected profit maintains under revenue sharing. Further-more, as RðQ ; pÞ is included in vðQÞ, quantity discounts may onlyapply to a multi-retailer situation where the retailers have thesame revenue function.

3.2. Alternative pricing policies and the newsvendor analysis

Three common types of quantity discount supplier pricing are:

� Linear quantity discount (LD). A linear discount schemeassumes that the supplier offers a per unit price vLDðQÞ decreas-ing in the quantity Q ordered by the retailer. The specific lineardiscount scheme we consider is vLDðQÞ ¼ v f � tQ where t issome per unit constant discount rate, and v f is the base price.� All-units (AU) quantity discount. As with the linear discount

scheme, under the all-units quantity discount policy the sup-plier also offers a per unit price vAUðQÞ which is declining inthe quantity Q ordered by the retailer. The key difference is thatthe price per unit decline is a step function and is specified asfollows. Define wiði ¼ 1; . . . ;nÞ as alternative per unit pricessuch that w1 > w2 > � � � > wn and corresponding maximumquantities to each per unit price as qiþ1ði ¼ 1; . . . ;nÞ such thatq1 < q2 < � � � < qn (in general, q1 is assumed to be 0). Then underthe all-units discount scheme, the per unit price charged by thesupplier is: wi for qi < Q 6 qiþ1ði ¼ 1; . . . ;nÞ where Q is thequantity ordered by the retailer.� Incremental-units (IU) quantity discount. In comparison to the

all-units quantity discount scheme, under the incremental-units policy the price quoted by the supplier only applies toincremental units ordered by the retailer. Hence, under thispricing scheme, the per unit average price charged by the sup-plier is: w1 for q1 < Q 6 q2ðw1q2 þw3ðQ � q2ÞÞ=Q for q2 < Q 6q3ðw1q2 þw3ðq3 � q2Þ þw3ðQ � q3ÞÞ=Q for q3 < Q 6 q4, and soon.

The cost per unit (vLDðQÞ;vAUðQÞ, and v IUðQÞ) and the total costof each policy is illustrated in Fig. 1. Assuming the retailer operates

Fig. 1. Graphical illustration of alternative pricing schemes.


as a newsvendor, the impact of each of these schemes on the opti-mal order quantity for each of these schemes is as follows.

Assuming a risk-neutral retailer facing with a linear quantitydiscount, the retailer’s expected profit function can be attainedby simply replacing v with vðQÞ ¼ v f � tQ in its expected profitfunction in the basic case

E½PR� ¼ ðp� vðQÞÞl� ðp� vðQÞ þ BÞE½x� Q �þ � ðvðQÞ � gÞE½Q � x�þ:ð22Þ

By imposing certain parametric restrictions,8 the optimal orderquantity Q �LD is characterized as follows:

FðQ�LDÞ ¼p� v0 þ Bp� g þ B

þ 2tQp� g þ B

: ð23Þ

Given that 2tQp�gþB > 0, it is obvious that the buyer would be induced

to order more under linear discounting scheme as compared to thefixed price scheme. While a closed form expression for Q �LD cannotbe derived for an arbitrary demand distribution, the optimal valueof Q �LD can be found via a simple uni-dimension search algorithm.

To solve the buyer’s optimal order quantity decision ðQ �AUÞassuming that the supplier offers an all-units quantity discount,the algorithm in Fig. 2 can be used. To solve the buyer’s optimal or-der quantity decision ðQ �IUÞ assuming that the supplier offers anincremental quantity discount, the algorithm shown in Fig. 3 canbe used.9

Jucker and Rosenblatt [27] studied the implications of quantitydiscounts for both the retailer and supplier in the context of thenewsvendor problem. Consider a profit maximizing retailer thatfaces deterministic demand and an all-units quantity discountoffered by the supplier. The authors showed that the retailer orders

8 These restrictions are: 0 < t < min½0:5f ðQÞðp� g þ BÞ; ðv � gÞQ�1� for all possiblevalues of Q.

9 The process is similar to that of the all-units quantity discount except that in thiscase, we only compute the expected profit when order quantities are in the feasiblerange.

up to di units if the amount of units she must purchase, denoted byQ 0, falls into the interval ½di; qiþ1Þ, and orders exactly Q 0 units if Q 0 isin ½qi; diÞ, where di ¼ qiþ1

wiþ1þgwiþg . di can be interpreted as the break-

even point at which the marginal cost remains the same betweenordering di and qiþ1 units. Since this kind of behavior reflects zeromarginal cost of units in the interval ½di; qiþ1Þ, the all-units quantitydiscount can be characterized by a marginal profit function whichequals wi for qi 6 Q < di and 0 for di 6 Q < qiþ1. Incremental quan-tity discounts can be modeled as a special case where di ¼ qiþ1.Using marginal analysis, the authors then proposed a new solutionprocedure for newsvendor problems with all-units quantity dis-count. Traditionally, the global maximum is found through thealgorithm outlined earlier. However, the authors suggested that,for cases in which the marginal profit of an additional unit equalswi cannot be solved analytically, it be better to first compare thevalues of the marginal cost function and marginal revenue functionfrom the lowest price break quantity to locate the intersections ifexist. While price break quantities can be candidates for the globaloptimum in the case of all-units quantity discount, the paper men-tioned that we only need to pay attention to the intersectionswhen an incremental quantity discount is considered.

Pantumsinchai and Knowles [37] presented solution algorithmsto the standard container size discounts in the newsvendor envi-ronment. The newsvendor determines the optimal order quantityand the number of different standard containers to make up the or-der under random demand. Suppose that there are m standard con-tainer sizes available with q1 > q2 > � � � > qm. The unit purchasecosts associated with container sizes satisfy the inequality0 < p1 6 p2 6 � � � 6 pm. Let I denote the newsvendor’s initial inven-tory level and yi denote the optimal order-up-to level that mini-mizes Gðyi; IÞ, the purchase cost if applicable plus the expectedoverage and underage cost when only the ith container size isavailable. Define ni ¼ ðyi � IÞ=qi; i ¼ 1;2; . . . ;m. The authorsshowed that the newsvendor should not order when I P y1. Inthe case where I < y1, the optimal order quantity Q � is boundedby nuqu and ðnu þ 1Þqu, where nu ¼ ðyu � IÞ=qu and u is chosen such

Fig. 2. Algorithm for the all-units discount scheme.

Fig. 3. Algorithm for the incremental-units discount scheme.


that ni ¼ 0; i ¼ 1; . . . ;u� 1 and G1ððn1 þ 1Þq1 þ I; IÞP G2ððn2þ1Þq2 þ I; IÞP � � �P Guððnu þ 1Þqu þ I; IÞ. Solution algorithms arethen proposed for solving the general case and the restricted casewhere the newsvendor is required to use successively smaller con-tainer sizes. In addition, when a fixed cost K is incurred for placingan order, the authors suggested comparing Gðy�; IÞ þ K with GðI; IÞand ordering if GðI; IÞ is larger.

Lin and Kroll [35] considered both all-units quantity discountand incremental discount in the newsvendor problem with dual

performance measures: ‘‘maximize the expected profit subject toa constraint that the probability of achieving a target profit levelis no less than a predetermined risk level’’. The retailer’s orderingdecision was analyzed under four models: Model 1: all-units dis-count with zero shortage cost; Model 2: incremental discount withzero shortage cost; Model 3: all-units discount with positive short-age cost; and Model 4: incremental discount with positive shortagecost. For Model 1, using results from Lau [34], the authors showedthat the constraint ProbðP P TÞP h, where P is the actual profit,


T is the target profit level, and h is the predetermined risk level,actually imposes another feasible condition on Q associated withwi, i.e., T=ðp�wiÞ 6 Q 6 ððp� gÞF�1ð1� hÞ � TÞ=ðwi � gÞ. There-fore, the feasible interval for Q corresponding to wi can be obtainedby combining this interval with the interval ½qi; qiþ1Þ in this case.The optimal order quantity can be found by solving the newsven-dor model with an all-units quantity discount scheme using thealgorithm described earlier. For Model 2, the decision variable Qhas the same feasible interval as for Model 1 for each wi andclosed-form expression for the optimal solution can be derived.For the other two models with positive shortage cost, again usingthe results from Lau [34], the constraint on the risk level can bereplaced by the inequality FðD2i � FðD1iÞÞP h, where D2i ¼½T þ ðwi� gÞQi�=ðp� gÞ and D1i ¼ ½ðp� v i þ BÞQi � T�=B where Qi

is the feasible order quantity in the interval ½qi; qiþ1�. Since the con-straint is nonlinear, no closed form solution can be obtained inthese two cases.

Burnetas et al. [14] addressed a pricing-policy design problemof a profit-maximizing supplier that has less demand informationthan the newsvendor-type buyer does. The optimal all-units dis-count and the optimal incremental discount were also contrastedin terms of the effectiveness of increasing the supplier’s expectedprofit in this paper. The authors modeled information asymmetryby considering n buyer types that have different demand distribu-tion FiðxÞ; i ¼ 1;2; . . . ;n, but the same cost structure. For the sup-plier, each type i has a probability of pi to be the real type of thebuyer. And it is assumed that FjðxÞ stochastically dominates FiðxÞfor j > i.10 The buyer has an outside supply source to purchase theproducts at a price of u, c < u < p. In such an environment, followingthe revelation principle, the pricing mechanism design problem canbe formulated as maximizing the supplier’s expected profit subjectto constraints that each buyer type prefers the quantity-price optionintended for him and accepting the intended option should yield atleast as much expected profit as resorting to the outside supplysource. The solution to this problem is referred to as the fixed-pack-age pricing policy (i.e., a pricing policy that specifies a quantity-pricepair). The optimal all-units discount and the optimal incrementaldiscount can be obtained by solving a similar problem. The authorswere able to show that, both analytically and numerically, the fixed-package pricing policy dominates the optimal all-units discount,which in term outperforms the optimal incremental discount, whenexpected profit is concerned.

Altintas et al. [3] addressed the problem of designing the opti-mal all-units quantity discount with a single price break fromthe perspective of a supplier that faces a newsvendor-type buyerin a multiple-period setting. It is assumed that, for every periodin the planning horizon except the last one, leftover units at theend of one period will be carried over to the next period and un-filled demand will be backlogged at a cost. At the end of the plan-ning horizon, leftover units must be disposed and unfilled demandmust be satisfied from an outside supply source. The supplier isconcerned with maximizing its own profit, which is the amountof money coming from the retailer less the trucking cost. Thetrucking cost for each order of size Q can be expressed as KdQCe,where K is the fixed operating cost per truck and C is the capacityof each truck. The buyer seeks to minimize her average cost. Theauthors first considered a single period model that allows positiveinitial inventory level y. Let ki denote the critical fractile valuecorresponding to the whole sale price v i; i ¼ 0;1. DefineSi ¼ F�1ðkiÞ; i ¼ 0;1, and S01 as some value that equates the result-ing expected costs of ordering q1 or ordering up to S0 if necessary.The authors demonstrated that a three-index policy with indices

10 Stochastic dominance implies that the cumulative density function of demand forbuyer i is strictly greater than the cumulative density function for buyer j for allpossible values of x ([38]).

ðS0; S1; S01Þ can be the buyer’s optimal ordering policy. To be spe-cific, order maxfS1 � y; q1g if the initial inventory level y is lessthan S01 and order maxfS0 � y;0g otherwise. However, this three-index policy does not apply to a multi-period model according tothe numerical study. One interesting finding is that demand vari-ance and the variance of orders not including the orders of size 0may move in opposite directions, since the buyer may follow po-lices with fewer indices under more variant demand. For the sup-plier’s problem, fixing v1 and treating v0 and q1 as variables, theauthors were able to show numerically that the supplier should of-fer quantity discounts that can act as minimum order quantityschemes to reduce the order variance when the fixed cost K is high.

Lau et al. [33] investigated the design of a supplier’s wholesalepricing policy in the form of an all-units discount with a singleprice-break point. They considered a supply chain with a singleStackelberg-leader supplier and numerous heterogeneous news-vendor-type retailers. Each retailer i faces a normally distributeddemand with mean li and standard deviation ri, where ri ¼ kli

(where k is a common scale parameter across all retailers) andall li’s follow another stochastic distribution H3ð�Þ by assumption.According to the authors, all demand characteristics are to be esti-mated from empirical data and they assumed that the parametersk; p (market price) and c (retailer specific cost) are identical for allretailers. Based on a numerical analysis, they concluded that for aStackelberg-leader supplier whose only objective is profit maximi-zation, the optimal all-units quantity discount dominates theoptimal single wholesale price scheme and that retailers’ heteroge-neity, indicated by the standard deviation of H3ð�Þ, adversely affectsthe effectiveness of the quantity discount scheme. In addition, for aprofit-maximizing supplier that is also concerned with allocating acertain amount of profits to the retailers, the optimal quantity dis-count again performs better than the optimal single wholesaleprice scheme in terms of the supplier’s expected profit.

4. Buyer risk profile

This part of the paper is mainly devoted to the analysis of theordering policies of newsvendors with various risk preferences,including, but not limited to, risk-averse and risk-seeking prefer-ences. Motivated by empirical and experimental observations,alternative risk preferences (such as loss-aversion, and stock-outaversion) have been analyzed in the context of the newsvendorsetting. General utility functions, mean–variance, and coherent riskmeasures are the three most popular approaches to date. In whatfollows, we will introduce some general results for a risk-aversenewsvendor derived and provide a detailed review of other contri-butions in this domain.

Eeckhoudt et al. [20] provided a detailed investigation of the ef-fects of risk, risk aversion and changes in various price and costparameters for a risk-averse retailer. One important assumptionthat simplifies the analysis is that all demand must be satisfied.To be specific, when the initial order is insufficient to fulfill the de-mand, the retailer purchases additional products at a cost of v̂ ,which is higher than the initial order cost v. The profit functioncan then be reformulated as

Pðx;QÞ ¼P�ðx;QÞ ¼ w0 þ px� vQ þ gðQ � xÞ; x 6 Q ;

Pþðx;QÞ ¼ w0 þ pQ � vQ � v̂ðx� QÞ; x > Q ;

�

where w0 is the newsvendor’s initial wealth. Obviously, for a givenQ, profit increases in demand x, which is defined in the interval½0; T�. For a risk-averse newsvendor, the objective is to maximizethe expected utility of the random profit. Let uð�Þ denote theincreasing and concave utility function. The problem can be statedas max E½uðPðx;QÞÞ�, where


E½uðPðx;QÞÞ� ¼Z Q

0u½P�ðx;QÞ�f ðxÞdxþ

Z T

Qu½Pþðx;QÞ�f ðxÞdx:

ð24Þ

The first and second derivatives of the objective function with re-spect to Q are as follows:

@E½uðPðx;QÞÞ�@Q

¼ ðg � vÞZ Q

0u0½P�ðx;QÞ�f ðxÞdx

þ ðv̂ � vÞZ T

Qu0½Pþðx;QÞ�f ðxÞdx

@2E½uðPðx;QÞÞ�@2Q

¼ ðg � vÞ2Z Q

0u00½P�ðx;QÞ�f ðxÞdx� ðv̂ � gÞu0½pQ

� vQ �f ðQÞ þ ðv̂ � vÞ2Z T

Qu00½Pþðx;QÞ�f ðxÞdx:

It is relatively easy to show that the objective function E½uðPðx;QÞÞ�is concave in Q and thus the optimal order quantity can bedetermined by first-order condition. Let Q �N and Q �A denote therisk-neutral and risk-averse optimal order quantities, respectively.Since

u0½P�ðx;QÞ�P u0½pQ � vQ �; x 6 Q ;

u0½Pþðx;QÞ� < u0½pQ � vQ �; x > Q ;

we get:

@E½uðPðx;Q �AÞÞ�@Q

¼ ðg�vÞZ Q�N

0u0½P�ðx;Q �NÞ�f ðxÞdx

þðv̂ �vÞZ T

Q�A

u0½Pþðx;Q �AÞ�f ðxÞdx< u0ðpQ �N �vQ �NÞ

� ðg�vÞZ Q�A

0f ðxÞdxþðv̂ �vÞ

Z T

Q�N

f ðxÞdx

" #¼ 0

This indicates that a risk-averse newsvendor orders less than a risk-neutral newsvendor in this setting. Keren and Pliskin [29] alsoshowed that, under uniformly distributed demand, the optimal or-der quantity decreases in the degree of risk aversion even if penaltyis incurred for lost sales. As an increase in risk aversion can be mod-eled by a concave transformation of the current utility function, it isnot so difficult to show that Q �A decreases in the degree of risk aver-sion. Eeckhoudt et al. [20] showed that increases in the salvage va-lue and the re-order cost both leads to a larger initial order, whilethe effect of changes in the initial order cost v on Q �A is still ambig-uous. As for the changes in retail price, the authors showed that Q �Aincreases in p if the retailer’s preference displays decreasing partialrelative risk aversion. The effects of increases in two types of riskwere examined in this paper, risk associated with the initial wealthand demand risk. It is shown that the optimal size of the initial or-der is unaffected by the additional risk if the utility is quadratic orexhibits constant absolute risk aversion, decreases if the utility dis-plays decreasing absolute risk aversion. Riskier demand is describedby a probability distribution G which is a mean-preserving with anincrease in risk (MIR) of F. Let Q �AF and Q �AG denote the optimal orderquantity associated with demand distribution F and G, respectively.The authors were able to prove that (1) Q �AF > Q �AG if an MIR is re-stricted to ½0;Q �AF � or G and F satisfies the inequality½GðxÞ � FðxÞ�½Q � x�P 0 and (2) Q �AF < Q �AG if MIR is restricted to½Q �AF ; T�.

Lau [34] provided solution procedures for solving the newsven-dor problems under two alternative optimization objectives, max-imizing expected utility and maximizing the probability ofachieving a target profit level. Let A denote the actual sales of thenewsvendor. The profit can be expressed as Pðx;QÞ ¼ ðp� gþBÞA� Bx� ðv � gÞQ , where A ¼minðx;QÞ. When only the

tradeoff between mean and standard deviation of the profit is con-sidered, the optimization problem becomes max E½Pðx;QÞ��kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivarðPðx;QÞÞ

p, where k is some constant that reflects the decision

maker’s degree of risk aversion. The author demonstrated that theoptimal solution to the classic newsvendor problem always servesas an upper bound on the optimal order quantity in this scenario.Besides, he suggested determining the optimal solution valuenumerically due to the complexity of the first-order condition.For the case in which the decision maker is concerned withmaximizing the profit’s expected utility, the author considered autility function approximated by an nth degree polynomial,uðPðx;QÞÞ ¼

Pni¼0aiP

iðx;QÞ and derived the corresponding first-or-der condition. Under the objective of maximizing the probability ofachieving a target profit level T, i.e., max probðPðx;QÞP TÞ, theoptimal order quantity is equal to T=ðp� vÞ when B ¼ 0, which isindependent of demand distribution. When B > 0, the first ordercondition can be expressed as f ðD1Þ

f ðD2Þ¼ ðp�vþBÞðp�gÞ

Bðv�gÞ , where D1 and D2

are the two demand realizations that generates the target profit le-vel T. However, no conclusion can be reached regarding thebehavior of the objective function for an arbitrary demanddistribution.

Bouakiz and Sobel [10] showed the optimality of a base-stockpolicy for a multi-period newsvendor problem in which thenewsvendor’s risk attitude can be described by an exponentialutility function. Specifically, they assumed a utility functionuðCðNÞÞ ¼ �e�sCðNÞ, where s is some positive constant measuringthe risk-aversion degree of the retailer and CðNÞ is the present va-lue of the total cost incurred in the N-period planning horizon. Let rdenote the discount factor. With the infinite-horizon model, theauthors proved the existence of a functions yð�Þ that gives the opti-mal base stock level for period i in the form of yðri�1sÞ, which even-tually approaches the optimal base stock level for a risk-neutralretailer.

Agrawal and Seshadri [1] considered a similar problem as Eec-khoudt et al. [20] except that the demand now depends on the re-tail price. Assuming that the mean demand is equal to aðpÞ, whichis a decreasing and concave function of p, the authors modeled theeffect of price on demand in two ways: (1) x ¼ aðpÞ�1, where �1 is arandom variable with unity mean; (2) x ¼ aðpÞ þ �2, where �2 is arandom variable with zero mean. The authors first reduce thebivariate objective function to a univariate function through agraphical analysis of price p versus all possible demand realiza-tions, and then examine the impact of risk aversion on the retailer’spricing and ordering decisions. With the multiplicative demandmodel (Model 1), it is demonstrated that the optimal retail priceincreases in the degree of risk aversion, while the optimal size ofthe initial order decreases. With the additive demand model (Mod-el 2), a more risk-averse retailer will choose a lower retailer price,but the effect of risk aversion on the optimal initial order quantitycannot be precisely identified.

Schweitzer and Cachon [42] provided a complete investigationof the relationship between the newsvendor’s profit-maximizingorder quantity and optimal order quantities under various alterna-tive objectives. A symmetric demand distribution and zero short-age cost are assumed in the analysis. Let w0 denote the initialwealth. For a loss-averse newsvendor, the authors considered autility function in which uðwÞ is equal to w for w P w0 and kwotherwise, where k is a measure of the newsvendor’s degree of lossaversion. It is proved that the optimal order quantity of a loss-averse newsvendor is smaller than the profit-maximizing orderquantity and decreases in k. For a newsvendor with preferencefor waste-aversion (dislike of excess inventory), stock-out aver-sion, or minimizing ex-post inventory error (minimizing absolutedifferences between Q and x), the authors modeled the respectiveutilities as expected profit less the cost incurred by the correspond-ing deviation. They showed that a waste-averse newsvendor orders


less than a risk-neutral newsvendor, while a stockout-aversenewsvendor orders more. By defining products with the criticalfractile greater than 1=2 as high-profit products and as low-profitproducts otherwise, then to minimize ex-post inventory error,the newsvendor tends to order less than profit-maximizing quan-tity for high-profit products and more for low-profit products.The same ordering policy applies to newsvendors adopting ananchoring policy in which the decision maker anchors on mean de-mand and adjusts the profit-maximizing order quantity. The re-sults derived from the last two situations are exactly what theauthors observed in their experiments.

Arcelus et al. [6] investigated the pricing, ordering and promo-tion policies of a risk-sensitive (risk-averse or risk-seeking) news-vendor under price-dependent and stochastic demand. Threepromotion schemes are considered in this paper: rebate, advertis-ing and the simultaneous use of the previous two schemes. It is as-sumed that the price and promotion decisions (i.e., rebate’s facevalue and advertising expenditure) only affect the deterministicpart of demand. To be specific, price and rebate decisions havean exponential relationship with demand, while advertising entersin logarithmic form. Both additive and multiplicative demand er-rors are allowed in this model. It is also worth mentioning thatin contrast to most of the previous papers we have reviewed, riskpreferences are included in this model by introducing a risk-adjustment factor k for the stochastic part of the expected profit.k > 1 for risk-averse newsvendors and k < 1 for risk-seeking deci-sion makers. Necessary optimality conditions are derived for all thepossible decisions to be made for both additive error and multipli-cative error. It is demonstrated that both the optimal order quan-tity and the optimal price are independent of the advertisingexpenditure. Since no closed form solution can be achieved dueto technical complexity, numerical analysis is conducted for anexample with uniformly distributed demand. One observation isthat if no promotion scheme is available, price and expected profitincreases in the value of k, but order quantity falls.

Many researchers have considered the use of financial pricingand risk measures in inventory problems. Anvari [4] argued thatthe mean–variance approach proposed in Lau [34] may not applyto a decision maker facing with multiple investment options sincethe effect of covariance cannot be captured by the mean–varianceapproach. The author hence suggested the use of the capital assetpricing model (CAPM) in the inventory models and thus optimizingfrom the perspective of the shareholders, instead of the managers,to avoid agency problem. Consider a newsvendor with two invest-ment options, the inventory investment option (purchase Q unitsat a price of v) and the option representing all the other investmentopportunities other than the inventory option with a random re-turn rate. It is assumed that B ¼ 0 and that investment is madeat the beginning of the single period while returns will be collectedat the end of the period. Let C denote the total capital available forinvestment. The objective is now to determine Q maximizingthe shareholder’s wealth represented by ½E½DðQÞ� �XCovðDðQÞ;MÞ�=ð1þ rf Þ � C, where DðQÞ is the random liquidating dividend,X is the market price per unit of risk, M is the value of the marketportfolio at the end of period and rf is the risk-free rate. The behav-ior of CovðDðQÞ;MÞ was examined in this paper under the assump-tion that the joint distribution of demand x and M is bivariatenormal. (A solution procedure was developed in Chung [18] for thisspecial case.) Moreover, the author showed that the proposedmodel can be reduced to the classic newsvendor model by: (1) set-ting rf ¼ 0; (2) dropping out the term CovðDðQÞ;MÞ since risk is notconsidered in the classic setting; and (3) ignoring the return fromthe other investment option.

Gaur and Seshadri [21] investigated the problem of hedgingoperational risk using financial instruments in the newsvendorenvironment. Demand is assumed to be correlated with the price

of a financial asset in a complete financial market with a uniquerisk-neutral pricing measure. No penalty is considered for lostsales. The decision maker makes the inventory decision at time 0and sells the products at time T. Let ST denote the price of the finan-cial asset at time T. Demand is modeled as x ¼ aþ bST when de-mand is perfectly correlated with the asset price and asx ¼ aþ bST þ � when demand is only partially correlated. � is thedecision maker’s subjective forecast error. The authors were ableto show that, for a risk-neutral decision maker, the optimal orderquantity is not affected by the hedging decision in both cases.However, hedging transactions enable greater reduction in riskwhen demand is more correlated with the price of the asset. Undercertain conditions on the structure of the hedging portfolio and thedecision maker’s utility function, financial hedging increases theoptimal order quantity if a risk-averse decision maker is willingto invest in inventory rather than put all the money in the financialmarket.

Ahmed et al. [2] suggested the use of coherent risk measures ininventory problems to obtain convex objective function. Theyanalyzed a multi-period inventory problem with linear costs.The total cost for the single-period problem is expressed asCðQÞ ¼ vQ � cgðQ � xÞ þ B½x� Q �þ þ h½Q � x�þ, where c is a dis-count parameter within the interval ð0;1� and h is the holding costfor each unit left. Let qð�Þ denote a coherent risk measure. Whenrisk aversion is considered, the problem can be formulated asminqðCðQÞÞ. For an arbitrary coherent risk measure, the authorsproved its one-to-one correspondence with the min–max formula-tion. Using a mean-absolute deviation risk measure in the form ofqðzÞ ¼ E½z� þ kE½jz� E½z�j�; k 2 ½0;1=2�, the structure of the optimalsolution is found to be similar to that of the risk-neutral case forthe multi-period problem.

Gotoh and Takano [23] investigated the risk-sensitive newsven-dor problem under the objective of minimizing the conditional va-lue-at-risk (CVaR), which is a coherent risk measure. Let Dðx;QÞdenote a particular loss function of interest and a denote somepre-specified target level of the loss function. The CVaR minimiza-tion problem formulated in this paper is to find the minimizing Qand a values of the function aþ 1

1�b E½½Dðx;QÞ � a�þ� for a given b.The objective function is introduced in Rockafellar and Uryasev[41]. The parameter b refers to the probability of the loss functionexceeding the infimum of all the a values that satisfy the inequalityUðajQÞP b, where Uð�Þ is defined as the distribution function ofDðx;QÞ. Two loss functions are considered in this paper, the net loss�PðQÞ and the total cost CðQÞ. The authors proved the convexity ofthe objective function and derived closed form expressions for Qand a in both cases. The two CVaR minimization problems areshown to be generalizations of the classic newsvendor problem.However, unlike what happens in the classic newsvendor casewhere maximizing profit is equivalent to minimizing cost, thetwo sets of ðQ �;a�Þ can be very different from each other. Analyti-cal results are also obtained with the mean-risk criterion whererisk is measured by the net loss CVaR. A linear programming for-mulation is also developed for scenarios where the decision makerhas multiple products to handle.

Jammernegg and Kischka [25] analyzed the ordering policy andits corresponding performance measures, such as cycle service le-vel (CSL) and probability of loss, of a risk-sensitive newsvendor un-der the mean-deviation criterion. No penalty cost is considered.The problem is formulated as maxðQ ;aÞ k�a

1�a CVaRaðQÞ þ 1�k1�a E½PðQÞ�,

where CVaRaðQÞ ¼ E½PðQÞjPðQÞ 6 F�1ðaÞ�; 0 < a 6 1; 0 6 k 6 1.This objective applies to a risk-neutral newsvendor if a ¼ k, arisk-averse newsvendor if a < k, and to a risk-seeking newsvendorif a > k. The closed form expression for the optimal orderquantity is also derived under this criterion. Specifically, Q � ¼F�1 p�v

p�g þ a�k1�k �

v�gp�g

� for k 6 p�v

p�g, and F�1 p�vp�g � ak�

, otherwise. The


conclusion is reached that a risk-averse (risk-seeking) newsvendororders less (more) than a risk-neutral newsvendor does. The cycleservice level, probability of loss, defined as PðPðQ �Þ 6 0Þ, and ex-pected loss, defined as E½PðQ �ÞjPðQ �Þ 6 0�, increase in a and de-crease in k. In addition to the conditional expected valueapproach discussed above, the authors also suggested using perfor-mance measures to model the decision maker’s risk preference. Forexample, a decision maker is risk averse (risk-seeing) if his/herself-specified CSL is less (greater) than the optimal CSL in the clas-sic newsvendor scenario.

Wang and Webster [47] examined the ordering policy of a loss-averse newsvendor. They used the same utility function to describethe loss-aversion preference as in Schweitzer and Cachon [42] butassumed a positive shortage cost. Let q1ðQÞ ¼ ðv � gÞQ andq2ðQÞ ¼ p�vþB

B Q . The objective function can be written as

E½uðPðQ ; xÞÞ� ¼ E½PðQ ; xÞ� þ ðk � 1ÞR q1ðQÞ

0 P�ðx; QÞdFðxÞ þR supI

q2ðQÞPþ

hðx; QÞdFðxÞ�, where I is the feasible interval for demand x and k P 1is the degree of loss aversion. The two integrals in the objectivefunction represent the expected underage and overage losses andhave been explicitly defined earlier in the discussion of Eeckhoudtet al. [20]. Define ðp � v þ BÞð1 � Fðq2ðQÞÞÞ as the marginal under-age loss and ðv � gÞFðq1ðQÞÞ as the marginal overage cost. Theauthors showed that, for k > 1, whether a loss-averse newsvendororders more or less than a risk-neutral newsvendor depends on therelative magnitude of the marginal underage and overage losses,which in terms depends on the demand distribution and the valuesof the price and cost parameters. If the marginal underage loss ex-ceeds the marginal overage loss, the loss-averse newsvendor tendsto order more than a risk-neutral decision maker. However, whenthe shortage cost is lower than some threshold value, the loss-averse optimal order quantity is always less than the profit-maxi-mizing order quantity and decreases in the degree of loss-aversion.And it is not surprising to know that the loss-averse optimal orderquantity increases in the shortage cost.

5. Conclusions and directions for future research

In this paper, we have summarized the existing contributionson the newsvendor problem for managing inventories. Over andabove synthesizing, summarizing, and extending the current con-tributions, we also find that there are several opportunities for fu-ture research. These are as follows:

� Supplier capacity constraints are generally not considered in thenewsvendor setting. More recently, Burke et al. [11] have exam-ined how a fixed capacity constraint would moderate the news-vendor’s ordering decision. However, what has not beenaddressed is how the supplier could moderate the capacitydecision (i.e., increase or decrease capacity) in this setting.� One extension which has of late been receiving greater interest

is that of integrating both stochastic demand and stochasticsupply in newsvendor ordering decisions. A first attempt toaddress this issue is the research of Burke et al. [12] wherethe authors show that under assumptions of uniformly distrib-uted demand, closed form solutions could be obtained regard-less of the underlying distribution of supplier reliabilities. Itwould be interesting to examine whether their analysis holdsfor other demand distributions and also whether, it can beextended to multiple stochastic demand sources.� The idea of zero supplier lead time of supply is another assump-

tion in the newsvendor setting. Given that stochastic lead timesare quite common in certain settings (e.g., sourcing of seasonalgoods), how these lead times impact the buyer’s ordering deci-sion is another area for future research.

� There has been some prior research on the multi-period news-vendor’s ordering decision (see [43] for a review of this prob-lem). However, this has not addressed the dynamics of pricingdecisions, marketing effort, and/or buyer penalties in changingordering quantities which are obviously relevant in an appliedsetting.

In addition to the suggestions noted above, it is obvious that fu-ture research opportunities which integrate these distinct areas arealso relevant for future research. For example, practical settingswhere capacity and supplier price are related would obviouslymoderate the ordering policy for a newsvendor. Further, it mightalso be worthwhile to investigate settings where buyers arecharged a penalty for accepting quantities which are lower thanthose previously agreed upon since this would tend to increasesupplier prices.

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