Multiple criteria risk averse model for multi-product newsvendor problem using conditional value at risk constraints

Highlights

• Developed a new risk averse model for multi-product newsvendor problem.

• Multiple performance measures are used to assess the risk.

• Conditional Value at risk is used as the risk measure on scalarized loss function.

• A cut-generation algorithm is used to solve the problem and performance is compared.

• Application of model on illustrative example validates intuitive characteristics.

Abstract

Contemporary decision makers exhibit greater risk aversion than before and often have multiple performance criteria for comparing order quantities. Moreover, there is uncertainty in both demand and manager's risk preference. Uncertainties are also associated with suppliers, suppliers’ suppliers and transporters. Optimizing only profit does not consider risk in any way. To capture the risk associated with the decisions taken in multiproduct newsvendor scenario, there is a need to incorporate coherent risk measures in the classical newsvendor problems’ formulation. To address these issues, risk averse model for order quantity decision in a multi-product newsvendor scenario with multiple performance criteria using polyhedral-scalarized Conditional Value at Risk constraints is developed to establish quantified dominance of solution over the classical newsvendor solution. “Improvement Factor” is introduced to specify the degree of preference of the decision maker. We formulate the master problem and adopt a cut-generation algorithm to solve the problem and present a simulated numerical study to illustrate the application of the model. Building upon the results of the model, we draw significant inferences crucial to the development of a risk averse order policy framework. This is shown for newsvendor scenario but can be extended for multi stage inventory problems that future research can address.

Introduction

Characterized by rapid development of products and short selling seasons, the present business environment experiences highly volatile demand of products such as electronic items, fast moving consumer goods (FMCG), software packages and fashion apparel. This uncertainty in demand reflects in the sales and losses, thereby increasing the associated risk. As a result, the executives are increasingly seeking to incorporate risk aversion in developing an order policy framework.

In particular, our research focuses on providing a single period order policy for multiple products with stochastic demand with the objective of maximizing the expected profit. We target multi-criteria risk aversion in a newsvendor scenario with uncertain preferences for each of the criteria. The solution is required to be risk preferable to a specified benchmark by a predefined ‘Improvement Factor (IF). In this paper, we consider the classical newsvendor order policy as the benchmark to gauge the reduction in the corresponding risk. We incorporate this factor to stress on the quantification of the risk preference of the decision maker.

To improve practicality of the model, we introduce a budget constraint on the order policy. The multiple criteria, represented as newsvendor sensitive losses, are mapped using Conditional Value-at-Risk (CVaR) preference relations. The formulation takes a non-linear form, which is computationally expensive to solve. Hence, we approximately linearize the model using pseudo variables and discrete demand scenarios. Therefore, we further extend the cut generation algorithm used by Homem-De-Mello and Mehrotra [7] and later tailored by Noyan and Rudolf [9] to solve the model. The problem is outlined as an optimization model in close relation with the multi-product newsvendor problem (MPNP) of the inventory theory (Khouja [8]; Ozler et al. [10]).

Variance and value-at-risk are some widely used risk measures that have been found inadequate because now properties such as monotonicity, convexity, translation invariance, positive homogeneity, and law invariance need to be considered. Unfortunately, the challenge remains that it is unclear how to choose a risk measure that faithfully represents a decision maker's true risk attitude. As described by Delage, and Li [5] in their work, one can account precisely for (neither more nor less than) what we know of the risk preferences of an investor/policy maker when comparing and optimizing financial positions.

Widely used risk measures such as semi-derivatives, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are functions that assess the risk associated with a random variable through a scalar quantity that allows convenient comparison between random outcomes. In order to capture risk, our model employs CVaR (Rockafellar and Uryasev [13], [14]; Rockafellar [11]), which is a law invariant and coherent measure of risk. These mathematical properties make it a preferred choice for risk measurement. Moreover, CVaR based dominance constraints can account for both risk neutral and worst case based decision rules. Although, CVaR finds extensive application in financial optimization models as univariate risk constraints, its application in the inventory theory is a recent area of research. Zhou et al. [17] worked upon the application of CVaR in Multi-Products Newsvendor Model for only single criteria. Our research takes into consideration multiple performance criteria. Fessi et al. [6] have deployed weighted linear combination (WLC) approach to scalarize the criteria. In nutshell, this multivariate risk-averse problem is proposed in order to address the uncertainty in decision maker's preferences and to ensure a convergent solution algorithm which is addressed using polyhedral scalarization on the multiple performance criteria.

The contributions of the study are summarized as follows:

• Application of the multivariate risk-averse preference relation based on CVaR and linear scalarization on the newsvendor scenario for better decision making and to reduce the risk associated with the optimal solution by introducing loss functions. The two losses due to overage and underage are captured using polyhedral scalarization and Cut Generation algorithm is used to find the solution.

• Introduction of “Improvement Factor” to specify the degree of preference of the decision maker. Major motivation behind introducing a new factor is to quantify the preference between two random variables under certain coherent risk measure. Finally, we show how the improvement factor affects the solution space and optimal solution based on numerical results.

Further attempts have been made to incorporate risk aversion in multi-product newsvendor environment by employing VaR considerations by Ozler et al. [10]. Rockafellar and Royset [12] presented second order superquantile/CVaR as a risk measure for risk averse optimization. The problem of optimal pricing and order quantity decision of a risk-averse newsvendor faced with price-dependent demand has been addressed by Chen et al. [3] using CVaR as risk measure. Choi et al. [4] studied the multiproduct risk averse newsvendor problem under law invariant coherent risk measure as objective function while considering the case for dependent and independent product demands. Arıkan and Fichtinger [1] defined the objective function using a spectral risk measure to address the risk-averse newsvendor problem. Another dimension, from which such single period, multi-product inventory models are studied, is under multi-objective decision making with fuzzy rough coefficients (Xu and Zhao [16]). Chen and Ho [2] proposed the optimal inventory policy with consideration of complex quantity discounts and fuzzy uncertainties in newsvendor scenario. However, uncertainties in stochastic single period inventory environments can be fuzzy, rough or random, our focus is more inclined towards random uncertainties.

While modeling decision-making problems under uncertainty, it is crucial to compare random outcomes based on decision maker's risk preference. Such multi-criteria, preference based comparisons have yet not been addressed in a multi-product newsvendor scenario. In order to fill this research gap, our work broadens the idea of Zhou et al. [17] by considering multiple stochastic performance measures and establishing dominance of our solution over a specified benchmark.

Employing the solution algorithm, we present a numerical analysis on a simulated inventory system of a newsvendor, selling multiple products with stochastic demands. The data for the example has been inspired from the work of Zhou et al. [17] with suitable alterations and extensions to fit the model regime. The results are tested for different risk averse scenarios by varying the confidence levels, the scalarization regions and the Improvement Factor, wherever feasible. Based on the analysis of the results, significant insights are drawn to guide the executives while making the order for the products.

The paper is organized as follows. In Section 2, we briefly explore the basic concepts related to Conditional Value At Risk in the context of our research. Subsequently, in Section 3 we provide a linear formulation of the risk averse model in detail. In Section 4, we discuss a cut-generation algorithm to solve the model. Section 5 contains a numerical analysis to illustrate the application of the model using a sample dataset. Thereafter, we draw inferences and develop managerial insights from the results obtained in this section. Section 6 concludes the methodology and results presented in the paper, as well as hints at the possibilities of future scope of research in this field.