A Multi-Product Risk-Averse Newsvendor
with Exponential Utility Function
Sungyong Choi ∗
Andrzej Ruszczyński †
Received January 14, 2009; revised February 11 and December 19, 2010;
accepted at European Journal of Operational Research in April 5, 2011
Abstract
We consider a multi-product newsvendor using an exponential utility function. We
first establish a few basic properties for the newsvendor regarding the convexity of the
model and monotonicity of the impact of risk aversion on the solution.
When the product demands are independent and the ratio of the degree of risk aversion to the number of products is sufficiently small, we obtain closed-form approximations of the optimal order quantities. The approximations are as easy to compute as the
risk-neutral solution. We prove that when this ratio approaches zero, the risk-averse
solution converges to the corresponding risk-neutral solution. When the product demands are positively (negatively) correlated, we show that risk aversion leads to lower
(higher) optimal order quantities than the solution with independent demands.
Using a numerical study, we examine convergence rates of the approximations and
thoroughly study the interplay of demand correlation and risk aversion. The numerical
study confirms our analytical results and further shows that an increased risk aversion
does not always lead to lower order quantities, when demands are strongly negatively
correlated.
Keywords: Supply chain management, risk analysis, expected utility theory
∗
Nanyang Technological University, Division of Systems and Engineering Management, 50 Nanyang Avenue, Singapore, 639798, E-mail: [email protected]
†
Rutgers University, Department of Management Science and Information Systems, 94 Rockefeller Road,
Piscataway, New Jersey 08854, E-mail: [email protected]
1
Introduction
The multi-product newsvendor problem is a classical model in the inventory management
literature. In this model, we consider multiple products to be procured and sold in a single selling season. We only know the demand for each product as a form of probability
distribution at the time of making orders and the demand realization is determined after
some time during the season. When we order less than the actual demand realization for
any product, the excessive demand is lost. On the other hand, when we order more than
the actual demand realization, the excessive inventory is sold at a loss. The objective of a
company is to determine its ordering quantity for each product to optimize some profit or
cost function.
The literature of the multi-product newsvendor model has mainly used risk-neutral performance measures as an objective function. For example, the company optimizes the expected average profit or average cost per product. Under these objective functions, the
model is decomposable and we can consider each product separately as multiple singleproduct newsvendor models, unless resource constraints are imposed or demand substitution is allowed. Under risk-averse objective functions, however, the model is generally not
decomposable. One needs to consider all products simultaneously, as a portfolio.
Below, we first review the literature of risk-neutral multi-product inventory models by
ways products interact. Then, we review the literature of risk models and its recent applications in supply chain inventory management.
Hadley and Whitin (1962) consider a multi-product newsvendor model with storage capacity or budget constraints, and provide the solution methods based on Lagrangian multiplier. Porteus (1990) presents a thorough review of various newsvendor models. Veinott
(1965) considers the dynamic version of the multi-product inventory models in a multiperiod setting, with general assumptions in demand process, cost parameters and lead times.
Conditions under which myopic policy is optimal are identified. Ignall and Veinott (1969)
and Heyman and Sobel (1984) extend the work by identifying new conditions for the myopic policy in models with risk-neutral assumption, see Evans (1967), Federgruen and Zipkin (1984), DeCroix and Arreola-Risa (1998) and Aviv and Federgruen (2001) for exact
analysis and approximations. Other than resource constraints, multi-product newsvendor
models are also studied under demand substitution, where unsatisfied demand of one product can be satisfied by on-hand inventory of another product. We refer to van Ryzin and
Mahazan (1999) for a review on multi-item inventory systems with substitution.
All the aforementioned studies focus on the risk-neutral inventory manager, that is, they
provide the best decision on average. This may be justified by the Law of Large Numbers.
However, the outcomes actually observed are random. The first few outcomes may turn
out to be very bad and cause unacceptable losses. Schweitzer and Cachon (2000) provide
experimental evidence suggesting that inventory manager may be risk-averse for high-value
1
products.
Attempts to overcome the drawbacks of the expected value optimization have a long
history. Chen, Sim, Simchi-Levi and Sun (2007) provide an excellent review and a summary
of the results for this literature. Choi, Ruszczyński and Zhao (2009) review the recent
literature after Chen et al. (2007) and categorize it by the four typical approaches to model
decision making under risk. They are expected utility theory, stochastic dominance, chance
constraints and mean-risk analysis. Each of the approaches has its own characteristics and
advantages although they are closely related and consistent.
In this study, we focus on the exponential utility function of the profit to model risk aversion in the multi-product newsvendor problem. Exponential utility function is a particular
form of a nondecreasing and concave utility function. It is also the unique function to satisfy constant absolute risk aversion (CARA) property. For those reasons, exponential utility
function has been used frequently in finance and also in the supply chain management literature such as Bouakiz and Sobel (1992) and Chen et al. (2007).
This paper contributes to literature in the following ways. In §3.1 we establish some basic
analytical results for the model when the product demands are independent: the convexity
of the model and monotonicity of the impact of risk aversion on the solution. We then
consider limiting properties of the solution with the degree of risk aversion. We analyze
asymptotic behavior of the solution when the degree of risk aversion coefficient converges
to zero or infinity. When this coefficient is sufficiently small but not zero, we also develop
closed-form approximations which are as easy to compute as the risk-neutral solution.
The analysis of the model with dependent demands is substantially more challenging.
By employing the concept of associated random variables, we prove that the risk-averse solution with negative (positive) correlation is always higher (lower) than the risk-averse solution with independent demands. We then conduct a numerical study in §5. For independent
demands case, our numerical study shows that the approximation converges quickly to the
optimal solution as the risk aversion coefficient decreases. For the dependent demands case,
the numerical study confirms our analytical results and also provides additional insights into
the interplay between demand correlation and risk aversion. Finally, we summarize the paper in §6.
2
Problem Formulation
Given products j = 1, . . . , n, let x = (x1 , . . . , xn ) be the vector of ordering quantities and
let D = (D1 , . . . , Dn ) be the demand vector. We also define r = (r1 , . . . , rn ) to be the price
vector, c = (c1 , . . . , cn ) to be the cost vector, and s = (s1 , . . . , sn ) to be the vector of salvage
values. Finally, let fD j (·) and F D j (·) be the marginal probability density function (pdf), if it
exists, and the marginal cumulative distribution function (cdf) of D j , respectively. Denote
2
F̄ D j (·) = 1 − F D j (·).
Setting c̄ j = c j − s j and r̄ j = r j − s j with r j > c j > s j ≥ 0 for all j = 1, . . . , n, we can
write the profit function as follows:
Π(x, D) =
n
X
Π j (x j , D j ).
(2.1)
j=1
where
Π j (x j , D j ) = −c̄ j x j + r̄ j min{x j , D j }
= (r j − c j )x j − (r j − s j )(x j − D j )+ ,
j = 1, . . . , n.
(2.2)
Here (z)+ = max(0, z). The demand vector D is random and nonnegative. Thus, for every
x ≥ 0 the profit Π(x, D) is a real-valued and bounded random variable.
The risk-neutral multi-product newsvendor optimization problem is to maximize the expected profit:
max E[Π(x, D)].
(2.3)
x≥0
This problem can be decomposed into independent problems, one for each product. Thus,
under risk-neutrality, a multi-product newsvendor problem is equivalent to multiple singleproduct newsvendor problems with the well-known solutions:
−1
x̂RN
j = F̄ D j (α j ),
α j = (c j − s j )/(r j − s j ),
j = 1, . . . , n.
(2.4)
However, as we have mentioned it in the introduction, this formulation is inappropriate, if
we are concerned with few (or just one) realizations and the Law of Large Numbers cannot
be invoked.
The exponential utility function of a profit z ∈ R is defined as follows:
u1 (z) = −e−λz .
It is nondecreasing and concave. Here, λ > 0 is a given degree of risk aversion. The
expected utility of a random profit Z is defined as follows:
h
i
U1 (Z) = E − e−λZ .
Setting Z = Π(x, D), we obtain the expected utility in the newsvendor problem,
h
i
U1 (Π(x, D)) = E − e−λΠ(x,D) .
Thus, the problem to maximize the expected utility can be represented equivalently to a
minimization problem as follows:
h
i
min E e−λΠ(x,D) .
(2.5)
x≥0
3
Alternatively, we can apply a “certainty equivalent” operator to the exponential utility
function, which leads to the following entropic utility function:
h
i
1
U2 (Z) = − ln E e−λZ .
λ
(2.6)
Setting Z = Π(x, D), we can formulate the problem by the entropic utility function as
follows:
h
i
1
min ln E e−λΠ(x,D) .
(2.7)
x≥0 λ
We point out that problems (2.5) and (2.7) cannot be decomposed into separate problems
for each product, unless product demands are independent.
As U2 (Π(x, D)) = −λ−1 ln − U1 (Π(x, D)) and the function ln(·) is increasing, problems
(2.5) and (2.7) are equivalent. The reader is referred to Chen and Sun (2007) for applications
of the entropic utility function of form (2.7) in inventory models.
Our analysis of problem (2.5) applies, with minor adjustments, to the problem of optimizing the expected utility of the average profit per product:
h λ
i
min E e− n Π(x,D) .
(2.8)
x≥0
The only difference is that the parameter λ in (2.5) is replaced by λ/n in (2.8). In particular,
our asymptotic results for λ → 0 in model (2.5) can also be interpreted as asymptotic results
for a constant λ and n → ∞ in model (2.8).
3
Analytical Results in the Independent Case
In the following subsections, we provide several analytical results for the multi-product
newsvendor model under the exponential utility function. These results lay the theoretical
foundation for the paper.
3.1
Basic Analytical Results
In this subsection, we provide a few analytical results for the multi-product newsvendor
model under the exponential utility function.
h
i
Lemma 1. E e−λΠ(x,D) is a convex function of x.
P
Proof. We first note that the function x 7→ Π(x, D) = nj=1 Π j (x j , D j ) is concave, and
thus x 7→ −λΠ(x, D) is convex. Because the function et is increasing and convex of t, the
composition is convex as well.
4
Proposition 1. Assume that all products have independent demands. Let x̂RA1 be the solution of problem (2.5) for λ = λ1 > 0. If λ2 ≥ λ1 then there exists a solution x̂RA2 of problem
2
1
(2.5) for λ = λ2 such that x̂RA
≤ x̂RA
j
j , j = 1, . . . , n.
Proof. Because all products have independent demands, problem (2.5) is separable into
individual problems for each product:



n
n

 X

Y
h
i
min E exp −λ
Π j (x j , D j ) = min
E exp −λΠ j (x j , D j )
x≥0
x≥0
j=1
j=1
(3.9)
n
Y
h
i
=
min E exp −λΠ j (x j , D j ) .
j=1
x j ≥0
Consider the function
h
i
Φ j (x j , λ) = E exp −λΠ j (x j , D j ) .
(3.10)
Observe that the function Π j (·, D j ) is piecewise linear and its derivative is uniformly bounded.
Thus, to calculate the right-hand side derivative of Φ j (·, λ) we can interchange integration
and differentiation. We obtain
h
i
Φ0j (x j , λ) = −λE exp −λΠ j (x j , D j ) Π 0j (x j , D j ) ,
with Π 0j (x j , D j ) denoting the right-hand side derivative of the profit. Substituting formula
(2.2) we have



r j − c j if x j < D j ,
Π 0j (x j , D j ) = 

 s j − c j if x j ≥ D j .
Thus, after straightforward calculations
Φ0j (x j , λ) = −λ exp −λ(r j − c j )x j × G j (x j , λ),
(3.11)
with
G j (x j , λ) = (s j − c j )E exp λ(r j − s j )(x j − D j ) 1{D j ≤x j } + (r j − c j )F̄ D j (x j )
= (s j − c j )
Zx j
exp λ(r j − s j )(x j − ξ) dF D j (ξ) + (r j − c j )F̄ D j (x j ).
(3.12)
0
Let us denote by H1 j (x j , λ) and H2 j (x j ) as the first and second terms on the right hand side
of (3.12):
Zx j
H1 j (x j , λ) = (s j − c j ) exp λ(r j − s j )(x j − ξ) dF D j (ξ),
0
H2 j (x j ) = (r j − c j )F̄ D j (x j ).
5
1
For λ = λ1 and x j = x̂RA
j , the right-hand side derivative is nonnegative. Therefore,
RA
RA 1
1
G j x̂RA
,
λ
=
H
x̂
,
λ
+
H
x̂ j 1 ≤ 0.
1
1
j
1
2
j
j
j
(3.13)
RA 1
As s j ≤ c j , r j ≥ s j , and x̂RA
≥
ξ,
H
x̂ j 1 , λ is nonincreasing with respect to λ, while
1
j
j
RA 1
H2 j x̂ j 1 does not depend on λ at all. Therefore, G j x̂RA
j , λ remains non-positive when
λ = λ1 is replaced by λ = λ2 . Hence,
1
Φ0j x̂RA
,
λ
≥ 0.
2
j
2
1
The function Φ j (·, λ2 ) is convex, and thus it has a minimizer x̂RA
≤ x̂RA
j
j .
3.2
Asymptotic Results and Closed-Form Approximations
In this section, we first consider asymptotic results with respect to the degree of risk aversion. Then we develop closed-form approximations to the solution. We assume that each
demand D j has a continuous probability distribution and that the demands of the products
are independent. Moreover, with no loss of generality we assume that each demand D j is
supported on a possibly infinite interval [0, ∆ j ), where we allow ∆ j = ∞. Indeed, if the
support of some demand D j was an interval [δ j , ∆ j ), with δ j > 0, we could eliminate the
sure part δ j of the demand from the problem entirely, and consider only the excess demand
D j − δ j.
As discussed in the proof of Proposition 1, due to the independence of demands, the
problem is decomposable into individual problems for the products:
min Φ j (x j , λ),
j = 1, . . . , n,
x j ≥0
with Φ j (·, ·) given by (3.10). If the demand distribution is continuous, the function Φ j (·, λ)
is differentiable, with its derivative given by the equations (3.11) and (3.12). This can be
easily verified by calculating the left-hand side derivative, similarly to (3.11) and (3.12),
and noticing that the right and left-hand side derivatives are equal, if P[D j = x j ] = 0. The
derivative of the equation (3.10), Φ0j (x j , λ), is zero at x j = x̂RA
j if
G j x j , λ = 0.
We consider three cases.
Case 1: λ ↓ 0.
6
(3.14)
Proposition 2. Suppose λk ↓ 0, as k → ∞, and let x̂RA (λk ) be the corresponding solutions
of problem (2.5). Then every accumulation point of the sequence x̂RA (λk ) is a solution of
problem (2.3).
Proof. Again, by the certainty equivalent operation at the equation (2.6), problems (2.5)
and (2.7) are equivalent. We first show that
lim U2 (Π) = E[Π].
λ↓0
h
i
where U2 (Π) = − λ1 ln E e−λΠ , as defined in the equation (2.6). Then, using de l’Hôpital
rule and interchanging differentiation and integration we obtain the following chain of equations:
h
i
h
i


 d h −λΠ i

d −λΠ 
−λΠ ] 
 ln E e−λΠ 

 dλ E e
/
E
[e
e
E



dλ


 = lim −

lim −

 = lim −



−λΠ

λ↓0
λ↓0
λ↓0
λ
1
E[e ] 
!
E[−Πe−λΠ ]
= lim −
= E[Π].
λ↓0
E[e−λΠ ]
This implies our assertion.
Case 2: λ → ∞.
Proposition 3. If λk → ∞, as k → ∞, then limk→∞ x̂RA (λk ) = 0.
Proof. With no loss of generality we can assume that the points x̂RA (λk ) are the largest
solutions of problem (2.5) with λ = λk . By Proposition 1, they form a nonincreasing
sequence. As they are nonnegative, they have a limit x∗ .
Passing to the limit in the equation (3.14), we obtain:
k
k
∗
RA k
k
lim G j x̂RA
j (λ ), λ = H2 j x + lim H1 j x̂ j (λ ), λ = 0.
k→∞
k→∞
As r j > s j , for the second limit to be finite it is necessary that F D j (x∗j ) = 0. Thus x∗j = 0.
Case 3: λ is small, but finite.
In this case, we can use the following approximation:
exp λ(r j − s j )(x j − D j ) ' 1 + λ(r j − s j )(x j − D j ).
(3.15)
Its error is of order λ2 . Then, substituting (3.15) into (3.14), we obtain the equation:
(c j − s j )F D j (x j ) + λ(r j − s j )(c j − s j )E (x j − D j )+ = (r j − c j )F̄ D j (x j ).
(3.16)
7
RN
As x̂RA
j ≤ x̂ j , we have
h
h
h
+ i
+ i
+ i
RA
RN
RA
RN
E x̂RN
−
D
+
x̂
−
x̂
≤
E
x̂
−
D
≤
E
x̂
−
D
.
j
j
j
j
j
j
j
j
RN
By virtue of Case 1, the risk-averse solution x̂RA
j is very close to x̂ j . Therefore, we use in
(3.16) the following approximation:
h
h
+ i
+ i
RN
x̂
−
D
.
(3.17)
−
D
'
E
E x̂RA
j
j
j
j
RA , which is infinitely
The error introduced into equation (3.16) is of order λ x̂RN
j − x̂ j
smaller than λ. Substituting equation (3.17) into (3.16), we get the following closed-form
approximation of the risk-averse solution:
!
h
+ i
−1 c j − s j
RN
x̂RA
'
F̄
+
λ(c
−
s
)E
x̂
−
D
, j = 1, . . . , n.
(3.18)
j
j
j
j
Dj
j
rj − sj
If the density fD j (·) is bounded away from zero in the neighborhood of the risk-averse
solution, the error of our approximation is infinitely smaller than λ.
3.3
Iterative Procedures for Approximations
We cannot expect the approximation obtained in equation (3.18) to be very accurate unless
λ is sufficiently small. In order to make the error rates smaller, we generate a sequence of
(0)
RN
approximations x̂(ν)
j , ν = 0, 1, 2, . . ., by an iterative method. At first, x̂ j = x̂ j . Then we
(ν−1)
calculate x̂(1)
instead of
j by applying equation (3.18). In iteration ν = 1, 2, . . ., we use x̂ j
RN
x̂ j in our approximation, calculating
x̂(ν)
j
'
−1
F̄ D
j
!
h (ν−1)
cj − sj
+ i
− Dj
+ λ(c j − s j )E x̂ j
,
rj − sj
j = 1, . . . , n.
(3.19)
The iterative method is efficient, if the initial approximation x̂(0) is sufficiently close to
the risk-averse solution. This is true if the risk aversion coefficient λ is close to zero. For
a moderately small λ, the risk-neutral solution may not be a good starting point for the
iterative method. An alternative and more accurate method is then the continuation method.
In this approach, we apply the iterative method for a small value of λ, starting from the riskneutral solution. Then we increase λ a little, and we apply the iterative method again, but
starting from the best solution found for the previous value of λ. In this way, we gradually
increase λ, until we recover optimal solutions for all values of the risk aversion coefficient
which are of interest for us.
8
4
Impact of Dependent Demands
Under risk-averse performance measures, dependence of product demands can greatly affect the optimal order decisions for the newsvendor, because the objective function is no
longer decomposable. However, an intuitive and appealing property is that positively (or
negatively) dependent demands generate larger (or smaller) variability and thus pose a
larger (or smaller) risk than independent demands. Thus, one tends to decrease (or increase) the order quantity in case of positively (or negatively) dependent demands relative
to the case of independent demands.
To characterize the impact of demand dependence on the optimal order quantity, we use
the concept of “associated” random variables. Consider a random vector D = (D1 , D2 , . . . , Dn ).
The following definition is due to Esary, Proschan and Walkup (1967); see Tong (1980) for
a review.
Definition 1. The random variables D1 , D2 , . . . , Dn are positively associated, if
E f (D)g(D) ≥ E f (D) E g(D) ,
for all non-decreasing real functions f :
E[g(D)], and E[ f (D)g(D)] exist.
(4.20)
Rn → R and g : Rn → R for which E[ f (D)],
The following definition is due to Joag-Dev and Proschan (1983).
Definition 2. The random variables D1 , D2 , . . . , Dn are negatively associated, if for every
pair of disjoint subsets I1 and I2 of {1, . . . , n} and all nondecreasing functions f : R|I1 | → R
and g : R|I2 | → R
E f (Di , i ∈ I1 )g(Di , i ∈ I2 ) ≤ E f (Di , i ∈ I1 ) E g(Di , i ∈ I2 ) ,
(4.21)
whenever these expected values are finite.
The lemma below summarizes the properties of associated random variables that we use
in this paper. We refer to Tong (1980) and Joag-Dev and Proschan (1983) for proofs.
Lemma 2. (i) Any subset of a set of positively (negatively) associated random variables
is positively (negatively) associated.
(ii) If two sets of positively (negatively) associated random variables are independent of
each other, their union is a set of positively (negatively) associated random variables.
(iii) Non-decreasing (or non-increasing) functions of positively (negatively) associated random variables are positively (negatively) associated.
(iv) If D1 , D2 , . . . , Dn are positively associated, then for all (y1 , y2 , . . . , yn ) ∈ Rn
P{D1 ≤ y1 , D2 ≤ y2 , . . . , Dn ≤ yn } ≥
n
Y
k=1
9
P{Dk ≤ yk }.
(v) If D1 , D2 , . . . , Dn are negatively associated, then for all (y1 , y2 , . . . , yn ) ∈ Rn
P{D1 ≤ y1 , D2 ≤ y2 , . . . , Dn ≤ yn } ≤
n
Y
P{Dk ≤ yk }.
k=1
Association is closely related to correlation. By Tong (1980, pg. 99), a set of multivariate normal random variables is positively associated if their correlation matrix has the
structure l (Tong 1980, pg. 13) in which the correlation coefficient ρi j = γi γ j for all i , j and
0 ≤ γi < 1 for all i. This means that we can represent the random demands (Di , i = 1, . . . , n)
as having one common random factor (D0 ) with error terms (∆i , i = 1, . . . , n):
Di = γi D0 + ∆i ,
i = 1, . . . , n,
where D0 and ∆i , i = 1, . . . , n, are independent. A special case is the bi-variate normal
random variable with a positive correlation coefficient.
Due to section 3.4 in Joag-Dev and Proschan (1983), a set of multi-variate normal random variables with non-positive covariances σi j for all pairs (i, j) with i , j, is negatively
associated.
By Lemma 2 (iii), the exponents of positively (negatively) associated normal variables,
which follow lognormal distributions, are positively (negatively) associated as well.
To provide some analytical insight on the impact of dependent demand under exponential
utility function we consider three cases of (D1 , D2 , . . . , Dn ). In Case 1, (D1 , D2 , . . . , Dn )
are positively associated random variables and x̂P,λ
j is the optimal solution for product j =
1, 2, . . . , n; In Case 2, (D1 , D2 , . . . , Dn ) are independent and x̂I,λ
j denotes the optimal solution
for product j = 1, 2, . . . , n; In Case 3, (D1 , D2 , . . . , Dn ) are negatively associated random
variables and x̂N,λ
is the optimal solution for product j = 1, 2, . . . , n. We assume that all
j
these solutions are unique. In each case λ denotes the positive degree of risk aversion.
We assume that the demand vector has a continuous distribution.
I,λ
N,λ
Proposition 4. x̂P,λ
j ≤ x̂ j ≤ x̂ j ,
j = 1, 2, . . . , n.
Proof. Let’s start from positively associated random variables (D1 , D2 , . . . , Dn ). Differen
tiating E exp(−λΠ) with respect to x1 , we obtain:
∂
∂x1
∂
E exp (−λΠ1 ) · exp (−λ(Π2 + · · · + Πn ))
∂x1
#
"
∂
=E
exp (−λΠ1 ) · exp (−λ(Π2 + · · · + Πn )) .
∂x1
E exp(−λΠ) =
10
We could interchange differentiation and integration because the derivative ∂x∂1 exp − λΠ1
is uniformly bounded. The factors under the expectation operator have the following forms:


n n
 X
o

+
exp (−λ(Π2 + · · · + Πn )) = exp −λ
(r j − c j )x j − (r j − s j )(x j − D j )  ,
j=2
and
∂
exp (−λΠ1 )
∂x1



λ(r1 − c1 )x1 + λ(r1 − s1 )(x1 − D1 ) , if D1 < x1 ,
λ(c1 − s1 ) exp −
=

−λ(r1 − c1 ) exp − λ(r1 − c1 )x1 ,
if D1 ≥ x1 .
Thus, exp − λ(Π2 + · · · + Πn ) is a nonincreasing and positive function of D2 , . . . , Dn .
Also, ∂x∂1 exp − λΠ1 is a nonincreasing and positive function of D1 . As D1 , D2 , . . . , Dn
are positively associated, we obtain the inequality:
∂
h
i
i
∂ h
E exp − λΠ ≥ E
exp (−λΠ1 ) E exp (−λ(Π2 + · · · + Πn )) .
∂x1
∂x1
It is true at every x. If x1 = x̂1I,λ then
∂
(−λΠ
)
E
exp
= 0.
1
∂x1
Thus,
i
∂ h
E exp(−λΠ) n x = x̂I,λ o ≥ 0.
1
∂x1
j
This inequality is true for all values of x2 , . . . , xn , in particular, for x j = x̂P,λ
j , j = 2, . . . , n.
h
i
As the function x1 7→ E exp − λΠ is convex, it has a minimum at a point x̂1P,λ ≤ x̂1I,λ .
I,λ
Similarly, x̂P,λ
j ≤ x̂ j , j = 2, . . . , n.
Consider now negatively associated random variables (D1 , D2 , · · · , Dn ). Arguing as before, we obtain the inequality
i
∂ h
∂ E exp(−λΠ) =
E exp (−λΠ1 ) exp (−λ(Π2 + · · · + Πn ))
∂x1
∂x1
∂
(−λΠ
)
(−λ(Π
=E
exp
1 exp
2 + · · · + Πn ))
∂x1
∂
h
i
exp (−λΠ1 ) E exp (−λ(Π2 + · · · + Πn )) .
≤E
∂x1
11
Thus, the inequality direction is reversed as follows:
i
∂ h
E exp(−λΠ) n x = x̂I,λ o ≤ 0.
1
∂x1
j
I,λ
Consequently, x̂1N,λ ≥ x̂1I,λ . Similarly, x̂N,λ
j ≥ x̂ j , j = 2, . . . , n.
5
Numerical Examples
The objective of this section is two-fold. First, we demonstrate the accuracy and the convergence rate of the approximations. Second, we illustrate Proposition 4 and provide additional
insights on the interplay between demand dependence and risk aversion.
5.1
Sample-based Optimization
In all examples considered we apply sample-based optimization to solve the resulting stochastic programming problems. We generate a sample D1 , D2 , · · · , DT of the demand vectors
where
(5.22)
Dt = (dt1 , dt2 , . . . , dtn ), t = 1, . . . , T.
Then we replace the original demand distribution by the empirical distribution based on
the sample, that is, we assign to each of the sample points the probability pt = 1/T . It
is known that when T → ∞, the optimal value of the sample problem converges to the
optimal value of the original problem (see Shapiro, Dentcheva and Ruszczyński (2009)).
In all our examples we used T = 1, 000. For the empirical distribution, the corresponding
optimization problem (2.5) has an equivalent nonlinear programming formulation. For each
j = 1, . . . , n and t = 1, . . . , T , we introduce the variable u jt to represent the salvaged number
of product j in scenario t. We obtain the following formulation:


n
T
 X

1X
min
(5.23)
exp −λ
Π jt 
T t=1
j=1
subject to Π jt = (r j − c j )x j − (r j − s j )u jt ,
j = 1, . . . , n,
x j − d jt ≤ u jt ,
t = 1, . . . , T,
j = 1, . . . , n,
u jt ≥ 0,
j = 1, . . . , n,
x j ≥ 0,
j = 1, . . . , n.
12
t = 1, . . . , T,
t = 1, . . . , T,
120
Continuation, average
Continuation, worst
100
Risk-neutral, average
Risk-neutral, worst
Error Rate [%]
80
60
40
20
0
0.005
0.01
0.015
Ratio (λ/n)
0.02
0.025
Figure 1: Independent products – The average and maximum percentage errors of the approximate
solutions and risk-neutral solutions.
5.2
Accuracy of Approximation
In this section, we test the accuracy of the closed-form approximations of §3.2 on ten randomly selected problems. For each problem, we calculated sample-based nonlinear programming solution by CPLEX and an approximation solution by the continuation method
from §3.3. At each value of λ = 0.005, 0.01, 0.015, 0.02, 0.025, we made just one step of
the continuation method. We must point out that our approximation given by the equation
(3.18) does not guarantee a feasible solution because the term
h
cj − sj
+ i
+ λ(c j − s j )E x̂RN
j − Dj
rj − sj
might be negative or greater than 1 (due to approximation). When this occurs less frequently, we say that the approximation method is more stable. Generally, the approximation
method is more stable for smaller values of λ. The reason is that the risk-averse solution
with smaller λ is closer to the risk-neutral solution and this implies the risk-neutral solution is a better starting point than in the case with larger λ. Our numerical study shows
13
that the continuation method was much more stable and accurate than the straightforward
approximation method starting from the risk-neutral solution.
For each instance, we compute the average and maximum percentage errors of the approximate solution relative to the sample-based nonlinear programming solution for each
value of λ. Then, for comparison, we also compute the average and maximum percentage
errors of the risk-neutral solutions relative to the sample-based nonlinear programming solution. These errors are displayed in Figure 1. First, we see that our approximation cuts the
error rates of the risk-neutral solution 3 to 5 times in all cases. Second, we observe that the
average and maximum errors of risk-neutral solution and our approximation are decreasing
when λ → 0.
5.3
Impact of Dependent Demands under Risk Aversion
44
40
36
32
Optimal
Ordering
Quantity
for
Product 1
28
24
20
λ = 0.02
λ = 0.04
16
λ = 0.06
12
λ = 0.08
λ = 0.1
8
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Correlation coefficient of the generating bivariate normal
Figure 2: Less variable and profitable products – The impact of demand correlation under risk
aversion.
The objective of this section is to study the impact of demand dependence on the op14
1
20
18
16
14
Optimal
Ordering
Quantity
for
Product 2
12
10
λ = 0.02
8
λ = 0.04
λ = 0.06
6
λ = 0.08
λ = 0.1
4
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Correlation coefficient of the generating bivariate normal
Figure 3: More variable and profitable products – The impact of demand correlation under risk
aversion.
timal ordering amount under risk aversion. For this purpose, we consider a two-product
system and the numerical results are obtained by the sample nonlinear programming problem. We choose the following parameters for the system: r1 = 15, c1 = 10, s1 = 7 and
r2 = 30, c2 = 10, s2 = 4 with λ = 0.02, 0.04, 0.06, 0.08, 0.1. We also assume that the
demand vector follows a bivariate lognormal distribution, which is generated by exponentiating a bivariate normal with the parameters µ1 = µ2 = 3 and σ1 = 0.4724, σ2 = 1.2684 to
achieve the assumed coefficients of variance (CV) of 0.5 and 2. From this setting, product
1 (product 2) represents less (more) profitable and less (more) risky. The numerical results
are summarized in Figures 2 and 3.
First, consistent with our analysis, risk aversion reduces the optimal order quantity for
independent demands. This is also true for positively correlated demands. But surprisingly,
this may not be true for strongly negatively correlated demands. To explain the intuition behind these counterexamples, let’s consider two identical products with perfectly negatively
correlated demands, D1 and D2 . A larger quantity, x, increases negative correlation between
15
1
the sales min(D1 , x) and min(D2 , x), and thus leads to smaller variability of the total sales
min(D1 , x) + min(D2 , x). A special case of perfectly correlated bivariate uniform demand is
analyzed in Choi (2009).
Second, consistent with our analysis, negatively correlated demands result in higher optimal order quantities than independent demands under risk aversion, while positively correlated demands lead to lower optimal order quantities under risk aversion. Finally, the
impact of demand correlation is monotone. Small deviations from monotonicity are due to
sampling errors.
Economically, these observations imply that if the firm is risk averse, then demand dependence can have a significant impact on its optimal order quantities and this impact can
be product-specific. These observations confirm the intuition that stronger positively (negatively) correlated demands indicate higher (lower) risk, and therefore lead to lower (higher)
order quantities. More interestingly, if the firm becomes more risk averse, it should always
order less of the more risky and more profitable products. However, for the less risky and
less profitable products, while it should order less when demands are positively correlated,
it may order more when demands are strongly negatively correlated.
6
Conclusion
The multi-product newsvendor with exponential utility function is not decomposable, unless the product demands are independent. The main contributions are the closed-form
approximation in the equations (3.18) and (3.19), asymptotic behavior of the solution with
respect to the ratio of the degree of risk aversion to the number of products, and the impact
of demand correlation under demand dependence. Our numerical examples confirm the
accuracy of the approximations and illustrate the interplay of demand dependence and risk
aversion.
Acknowledgement
The research was partially supported by the NSF Award CMMI-0965689. The authors are
very grateful to three anonymous Referees for their comments and suggestions that lead to
significant improvements of the manuscript.
References
[1] Aviv, Y. and A. Federgruen (2001). Capacitated Multi-Item Inventory Systems with
Random and Fluctuating Demands: Implications for Postponement Strategies. Man16
agement Science, 47(4), 512-531.
[2] Bouakiz, M. and M. Sobel (1992). Inventory Control with an Exponential Utility Criterion. Operations Research, 40(3), 603-608.
[3] Chen, X., M. Sim, D. Simchi-Levi and P. Sun (2007). Risk Aversion in Inventory
Management. Operations Research, 55, 828-842.
[4] Chen, X. and P. Sun (2007). Optimal Structural Policies for Ambiguity and Risk
Averse Inventory and Pricing Models. Working Paper, University of Illinois.
[5] Choi, S., A. Ruszczyński and Y. Zhao (2009). A Multi-Product Risk-Averse Newsvendor with Law Invariant Coherent Measures of Risk. Fothcoming in Operations Research.
[6] Choi, S. (2009). Risk-Averse Newsvendor Models. PhD Dissertation, Department of
Management Science and Information Systems, Rutgers University, New Jersey.
[7] Decroix, G. and A. Arreola-Risa (1998). Optimal Production and Inventory Policy for
Multiple Products Under Resource Constraints. Management Science, 44, 950-961.
[8] Eeckhoudt, L., C. Gollier and H. Schlesinger (1995). The Risk-averse (and Prudent)
Newsboy. Management Science, 41(5), 786-794.
[9] Esary, J. D., F. Proschan & D. W. Walkup (1967). Association of Random Variables,
with Applications. Annals of Mathematical Statistics, 38, 1466-1474.
[10] Evans, R. (1967). Inventory Control of a Multiproduct System with a Limited Production Resource. Naval Research Logistics, 14(2), 173-184.
[11] Federgruen, A. and P. Zipkin (1984). Computational issues in an infinite-horizon,
multi-echelon inventory model. Operations Research, 32, 818-836.
[12] Hadley, G. and T. M. Whitin (1963). Analysis of Inventory Systems, Prentice-Hall,
Englewood Cliffs, New Jersey.
[13] Heyman, D.P. and M.J. Sobel (1984). Stochastic Models in Operations Research: Volume 2. Stochastic Optimization, McGraw Hill Company.
[14] Ignall, E. and D.P. Veinott Jr. (1969). Optimality of Myopic Inventory Policies for
several Substitute Products. Management Science, 15, 284-304.
[15] Porteus, E.L. (1990). Stochastic Inventory Theory, in: D.P. Heyman and M.J. Sobel
(Eds.). Handbook in OR and MS, Vol. 2, Elsevier Science Publishers B.V., Chapter
12, 605-652.
17
[16] Schweitzer, M. and G. Cachon (2000). Decision Bias in the Newsvendor Problem with
a Known Demand Distribution: Experimental Evidence. Management Science, 46(3),
404-420.
[17] Shapiro, A., D. Dentcheva and A. Ruszczyński (2009), Lectures on Stochastic Programming, SIAM Publications, Philadelphia 2009.
[18] Tong, Y. L. (1980). Probability Inequalities in Multivariate Distributions. Academic
Press, New York.
[19] van Ryzin, G.J. and S. Mahajan (1999). On the Relationship Between Inventory Cost
and Variety Benefits in Retail Assortments. Management Science, 45, 1496-1509.
[20] Veinott, A. (1965). Optimal policy for a multi-product, dynamic, nonstationary inventory problem. Management Science, 12, 206-222.
[21] Yoag-Dev, K. and F. Proschan (1983). Negative association of random variables, with
applications. The Annals of Statistics, 11, 286-295.
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