A risk-averse newsvendor with law invariant coherent measures of risk

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Abstract

For general law invariant coherent measures of risk, we derive an equivalent representation of a risk-averse newsvendor problem as a mean-risk model. We prove that the higher the weight of the risk functional, the smaller the order quantity. Our theoretical results are confirmed by sample-based optimization.

Introduction

The classical newsvendor problem, a stochastic inventory replenishment problem with random demand, can be described as follows. There is a perishable single product in a single period of time; backlog is not allowed. The vendor has to order the item before the demand is realized. The demand is only known in the form of its probability distribution function. If the vendor orders too much, the leftovers have only a small salvage value; if the vendor orders too little, he faces lost sales. The problem is to maximize the expected profit. It has a simple analytical solution and many applications, such as plant capacity and overbooking problems.

Due to its simplicity and versatility, many variants of the classical problem have been studied. However, much less attention has been paid to the effect of risk aversion in the newsvendor problem.

In our note, we study a risk-averse newsvendor problem with coherent measures of risk. We first reformulate the problem as a mean-risk model and consider the risk-averse newsvendor solutions with different risk functionals. Then we extend our analysis to general law invariant coherent measures of risk. Finally, we present a numerical illustration.

Our study is motivated by the recent paper [2], where the authors (under a different set of assumptions involving a large backlog penalty) conclude that, for a specific measure of risk, the order of the risk-averse newsvendor should be higher. By using the standard definition of the problem and employing modern theory of law invariant measures of risk, we arrive, in the general case without the backlog penalty, to the opposite conclusion: the more risk-averse the newsvendor is, the smaller his order. We also confirm these findings experimentally. This is in harmony with our intuition and with the results of [4], where a utility model was considered, with [3], for a mean-variance model, and with [6], for a Conditional Value at Risk model.

Section snippets

The mean-risk newsvendor problem

We introduce the following constants: unit resale price p, unit ordering cost c, and unit salvage value s. We assume that p>c>s. Let x denote the amount ordered by the newsvendor and let D be the random demand. The net profit can be calculated as follows: Z(x,D)=-cx+pmin(D,x)+smax(0,x-D).Simple manipulation yields an equivalent formula:Z(x,D)=-cx+smin(D,x)+smax(0,x-D)+(p-s)min(D,x)=-cx+sx+(p-s)min(D,x)=-(c-s)x+(p-s)min(D,x).It is thus sufficient to consider the newsvendor problem with resale

Analysis of basic mean-risk models

In this section we briefly analyze two mean–risk models with risk functionals rβ[Z] and σ1[Z]. There are three reasons for that: practical importance, ease of analysis, and the fundamental role of rβ[Z] in the theory of law invariant coherent measures of risk.

Lemma 1

For every β(0,1) the function xrβ[Z(x)] is nondecreasing on R+.

Proof

The result can be proved (for a continuous distribution of D) by differentiating rβ[Z(x)] with respect to x and changing the order of integration (see [2] and [6] for

Extension to law invariant coherent measures of risk

Problem (3) can be equivalently written as follows:minx0ρ[Z(x,D)],where ρ[·] is a certain measure of riskρ[Z]=-E[Z]+λr[Z].The paper [1] initiated a general theory of measures of risk, by specifying a number of axioms that these measures should satisfy. For further generalization and overview of other developments, see [5], [13] and [14]. Here we provide a brief information sufficient for our purposes.

Let (Ω,F) be a certain measurable space. In our case, Ω is the probability space on which D is

Sample-based optimization

With a general probability distribution of demand, it is impossible to solve our problem in a closed form. We resort to sample-based optimization. That is, we randomly generate a large sample from the demand distribution: Dk, k=1,...,N. Then we solve the mean-risk model with the empirical distribution of D, by treating the sampled values Dk as equally likely scenarios (with probabilities pk=1/N). For an introduction to sample-based optimization, see [16].

In the case of the semideviation σ1[·],

Numerical example

In order to illustrate the results of this note, we consider the problem with c=10, p=15, and s=7. We consider two distributions of the demand: uniform in [0,100] and lognormal. The parameters of the lognormal distribution were chosen so as to match the mean and the variance of the uniform distribution. The risk-neutral solution x^N equals 62.5 for the uniform distribution and 51.37 for the lognormal distribution.

To find the risk-averse solutions, we generate a sample of size N=1000, and we

Acknowledgements

This research was supported by the NSF awards DMS-0603728 and DMI-0354678.

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