A Note on Mean-variance Analysis of the Newsvendor
Model with Stockout Cost
Jun Wu
1
, Jian Li
2,4
, Shouyang Wang
2
and T.C.E Cheng
3∗
1
School of Economics and Management
Beijing University of Posts and Telecommunications, Beijing 100876, CHINA
2
Institute of Systems Science, Academy of Mathematics and Systems Science
Chinese Academy of Sciences, Beijing 100080, CHINA
3
Department of Logistics, The Hong Kong Polytechnic University
Hung Hom, Kowloon, Hong Kong SAR, CHINA
4
College of Economics and Management
Beijing University of Chemical Technology, Beijing 100029, CHINA
Abstract
We apply the mean-variance approach to analyze the risk-averse newsvendor problem
with stockout cost. We first derive an explicit form and some new properties of the
variance of the profit function. Then, under the assumption that demand follows the
power distribution, and its special case the uniform distribution, we obtain the set of
optimal ordering quantities. We also give a counterexample to one result presented in
the literature. Contrary to the traditional result in the literature that the risk-averse
newsvendor always orders less than the risk-neutral newsvendor, our findings show that
this may not be the case when stockout cost is considered because the newsvendor may
order more than the risk-neutral newsvendor order quantity under a stockout situation
with mean-variance tradeoff.
Keywords: supply chain, newsvendor problem, mean-variance analysis, stockout cost
1 Introduction
As a fundamental problem in stochastic inventory control, the newsvendor problem has been
studied for a long time and applied in a broad array of business settings with the objective
of expected profit maximization or expected cost minimization. However, modern supply
chains are very complex and increasingly becoming more vulnerable to uncertainties. What
∗
Corresponding author. Tel.: (852)-2766-5216; Fax: (852)-2364-5245. E-mail address:
LGTcheng@polyu.edu.hk.
1
This is the Pre-Published Version.
supply chain managers concern most is not only profit, but also risk or loss to their firms.
The assumption of risk-neutrality seems to be inadequate for contemporary supply chain
management. In view of this, a number of papers have been devoted to risk analysis of supply
chain models. Recent studies include, but are not limited to, those by Lau [13], Bouakiz and
Sobel [5], Choi et al. [9], Eeckhoudt et al. [10], Lau and Lau [14], Agrawal and Seshadri
[1, 2], Chen and Federgruen [7], Buzacott et al.[6], Chen et al. [8], Wang and Webster [18],
Wu et al. [19], Bogataj and Bogataj [4], He and Zhang [11], Sounderpandian, Prasad and
Madan [16], and Agrawal and Ganeshan [3]. For an extensive review of the literature on
supply chain risk management or extension of different objectives on newsvendor problem,
the reader is referred to Khouja [12], Tang [17] and Wu et al. [20].
In the newsvendor problem, if there is not enough stock to satisfy all the demand occur-
ring in the selling season, the newsvendor may incur a stockout cost. Besides making a loss
in marginal profit, the stockout cost may include such adverse effects on a firm as tarnishing
the firm’s reputation and jeopardizing the loyalty of the firm’s customers, which can greatly
impair the firm’s performance and profitability. For example, the Wall Street Journal [21]
reported that IBM, as a result of under-producing its Aptiva PC line, lost more than $100
million in potential revenue in 1994.
Although stockout cost plays an important role in the practice of supply chain man-
agement, it is often ignored or has not been studied in depth in risk analysis of supply
chain models. Chen and Federgruen [7] studied the newsvendor problem using the mean-
variance framework. Without stockout cost, the variance function of the stochastic profit
is a monotone increasing function of order quantity, so the mean-variance tradeoff can be
carried out efficiently. However, if stockout cost is considered, the variance function will lose
this monotonicity property and the mean-variance tradeoff becomes much more complicated.
Buzacott et al. [6] studied a class of commitment-option supply contracts under the mean-
variance framework. They used the mean-variance criterion as the objective function, which
is a newsvendor type of problem without sto ckout cost, and obtained a similar monotone
increasing property of the variance function. They further emphasized that monotonicity is
a fundamental result in this type of stochastic planning models. Choi et al. [9] investigated
the issues of channel coordination in a supply chain when individual supply chain decision
makers take the mean-variance objective. Eeckhoudt et al. [10] examined the effects of risk
and risk aversion on a risk-averse and prudent newsvendor without considering the stockout
cost. They pointed out that risk aversion will lead to a reduced initial newspaper order. Lau
[13] considered the risk-averse newsvendor problem with mean-standard deviation tradeoff
under two cases: one is without stockout cost; the other is with stockout cost. He proved
that the risk-neutral newsvendor order quantity is an upper bound on the optimal order
quantity of the risk-averse newsvendor without stockout cost. He also stated without proof
that a similar result still holds when stockout cost is considered.
Our study is most related to [18], but with several major differences. Both studies
consider the newsvendor problem with stockout cost based upon objectives different from
profit maximization and find some results different from the risk-neutral newsvendor model.
The significant differences are: (1) Wang and Webster [18] used loss aversion to mo del
the newsvendor problem, while we use mean-variance tradeoff. Loss aversion belongs to a
2
class of utility function, while mean-variance tradeoff belongs to the return-risk framework.
Generally speaking, utility maximization is mainly used in theoretical study, while mean-
variance tradeoff is widely applied both in theoretical study and in practice. We select
mean-variance tradeoff mainly based on the following two considerations. First, since there
are various utility functions, it is not easy to construct a proper one convenient for analysis.
Second, return-risk models usually have a much more intuitive explanation than the utility
maximization approach. Here we must point out that, unlike many other utility functions,
the loss aversion used in [18] is also intuitively appealing. (2) Wang and Webster [18] studied
the risk-averse newsvendor problem within the loss aversion framework, while our paper is
motivated mainly from previous studies. Thus, we also carry out comparisons with previous
risk-averse newsvendor problems and present a counterexample to one result presented in
the literature. (3) Besides studying the impact of stockout cost on the ordering quantity, we
also study its impact on the newsvendor’s profit.
If stockout cost is considered in the newsvendor mo del, the properties of the variance
function and the mean-variance tradeoff may be very different from those of the model
without stockout cost. Moreover, some results obtained in the above literature may no longer
be valid. Motivated by this observation, we study in this paper the risk-averse newsvendor
model presented in Chen and Federgruen [7] but with stockout cost consideration. We derive
an explicit form of the variance of the profit function and obtain its properties. We show that
the variance of the profit function is no longer a monotone increasing function. Furthermore,
under the assumption that the demand function follows the power distribution, we work out
the set of optimal ordering quantities. Contrary to the traditional result in the literature
that the risk-averse newsvendor (without stockout cost) always orders less than the risk-
neutral newsvendor order quantity, our findings show that this may not be the case when
stockout cost is considered because the newsvendor may order more than the risk-neutral
newsvendor order quantity under a stockout situation with mean-variance tradeoff. We also
give a counterexample to one result presented in Lau [13].
The rest of this paper is organized as follows. In Section 2 we analyse the newsvendor
problem under study using the mean-variance approach. In Section 3, under the assumption
that demand follows the power distribution, we derive the new properties and results due to
the stockout cost. We give conclusions in Section 4.
2 Mean-variance analysis with stockout cost
Let Q be the newsvendor’s order quantity. Let D be the future stochastic demand during the
selling season. Let F be the cumulative distribution function and f the probability density
function of demand, respectively. We assume that F is a continuous and strictly increasing
function and f is a nonnegative function.
The purchasing cost of the product is c per unit, the selling price is r per unit, the salvage
value of any unsold product is s per unit, and the stockout cost of unsatisfied demand is
p per unit. To avoid unrealistic and trivial cases, we assume that 0 < s < c < r and
0 < p. Throughout the paper, we use the following notation: for any numbers a and b,
3
a
+
= max{a, 0}, and a ∧ b = min{a, b}.
Let π(Q) be the newsvendor’s random profit, namely
π(Q) = r(Q ∧ D) + s(Q − D)
+
− p(D − Q)
+
− cQ. (1)
Let Π(Q) be the mean profit, namely
Π(Q) = E[π(Q)] = −(r + p − s)
Z
Q
0
F (x)dx + (r + p − c)Q − pE[D], (2)
where E[D] is the mean of the random demand D.
The risk-neutral newsvendor problem is given by
max
Q≥0
{E[π(Q)]}. (3)
The optimal solution Q
∗
for problem (3) is called the newsvendor order quantity. It is
straightforward to verify that the expected profit function is a concave function of Q. By
using the first-order optimality conditions, we obtain the newsvendor order quantity Q
∗
as
follows
F (Q
∗
) =
r + p − c
r + p − s
. (4)
The mean-variance analysis was first proposed by Markowitz [15] to measure the risk
associated with the return of assets. It uses a parameter α (α ≥ 0) to characterize a decision
maker’s risk averseness, which is a quantitative balance between the mean profit and the
risk associated with its variance. α = 0 denotes the special case of maximizing the mean
profit function only. An increase in α indicates the decision maker’s increasing willingness
to sacrifice the mean profit to avoid the risk of its variance. Note that, for any given α,
a solution is optimal in the sense that we cannot improve the mean profit without bearing
more risk, or reduce the risk without decreasing the mean profit.
Under the mean-variance framework, the objective function of the newsvendor problem
is given by
max
Q≥0
{E[π(Q)] − αV ar[π(Q)]}, (5)
where Q is the order quantity, α is the parameter denoting the decision maker’s risk attitude,
π(Q) is the random profit given by Eq. (1), E[π(Q)] is the mean of the random profit given
by Eq. (2), and V ar[π(Q)] is the variance of the random profit given by Eq. (7).
Note that the variance of the random profit is given by
V ar[π(Q)] = E
h
(π(Q))
2
i
− (E[π(Q)])
2
. (6)
Substituting Eqs. (1) and (2) into Eq. (6), the variance function of the newsvendor’s
profit can be written as
V ar[π(Q)]
4
= −(r + p − s)
2
Ã
Z
Q
0
F (x)dx
!
2
+
n
2Q(r − s)(r + p − s) − 2p(r + p − s)E[D]
o
Z
Q
0
F (x)dx
−2(r + p − s)(r − p − s)
Z
Q
0
xF (x)dx + p
2
V ar[D], (7)
where V ar[D] is the variance of the random demand D.
The first-order derivative of the variance function of Eq. (7) with respect to Q is given
by
dV ar [π(Q)]
dQ
= −2(r + p − s)
2
F (Q)
Z
Q
0
F (x)dx + 2p(r + p − s)QF (Q)
+2(r + p − s)(r − s)
Z
Q
0
F (x)dx − 2p(r + p − s)E[D]F (Q). (8)
The second-order derivative of the variance function of Eq. (7) with respect to Q is given
by
d
2
V ar[π(Q)]
dQ
2
= 2(r + p − s)
(
f(Q)
h
p(Q −
Z
Q
0
F (x)dx − E[D]) − (r − s)
Z
Q
0
F (x)dx
i
+(r + p − s)F (Q)(1 − F (Q))
)
. (9)
Remark 2.1 From Eqs. ( 7), (8) and (9), we see that the variance function with stockout
cost is more complicated than that without stockout cost. Compared with the results without
stockout cost presented in Chen and Federgruen [7], where the variance function is an in-
creasing function of the order quantity, such results may no longer be valid when stockout
cost is considered.
1
Theorem 2.2 The expected profit function is a concave function of Q and asymptotically
linear with a slope (s − c) < 0.
Proof: See Appendix A1.
Remark 2.3 When Q → +∞, no stockout could happen; so the expected profit function
preserves the same property as that presented in Chen and Federgruen [7] without stockout
cost. And one unit of overstocking will result in a unit loss of |s − c|.
1
We give an example in the next section to show that the variance function is not an increasing function
of the order quantity when stockout cost is considered.
5
