مدیریت مالی در مشکلات موجودی: سیاست های گریز از ریسک در مقابل سیاست های خنثی ریسک

In this work, we discuss the effect of risk measure selection in the determination of inventory policies. We consider an inventory system characterized by the loss function of Luciano et al. [2003. VaR as a risk measure for multi-period static inventory models. International Journal of Production Economics 81–82, 375–384]. We derive the optimization problems faced by risk neutral, quadratic utility, mean-absolute and CVaR decision makers. Results show that while the global nature of the optimal policy is assured for risk coherent and risk neutral decision makers, the convexity of the quadratic utility problem depends on the stochastic properties of demand. We investigate the economic and stochastic determinants of the different policies. This allows us to establish the conditions under which each type of decision maker is indifferent to imprecision in the distribution families. Finally, we discuss the numerical impact of the choice of the risk measure by means of a multi-item inventory. The introduction of an approach based on Savage Scores allows us to offer a quantitative measurement of the similarity/discrepancy of policies reflecting different risk attitudes.

The purpose of this work is to investigate the quantitative implications of the risk measure choice on optimal inventory policies. We introduce a structured approach to allow the determination of the extent of the discrepancies in the policies selected by decision makers with different risk attitudes—in particular, we compare risk neutral policies to the policies of decision makers selecting variance, absolute deviation (MAD) and conditional value at risk (CVaR) as risk measures.1 Relevant literature in the last 20 years has evidenced the importance of financial and decision theoretical aspects in inventory management. The works of Grubbström and Thorstenson (1986), Thorstenson (1988), Luciano (1998), Luciano and Peccati (1999), Luciano et al. (2003) and Koltai (2006) focus on applications of the discounted cash flow methodology to inventory policies. Bogataj and Hvalica (2003) propose to utilize, besides an expected value criterion, the maximin approach. The maximin approach to the newsvendor problem is discussed in Gallego and Moon (1993) and Gallego et al. (2001). Earlier, alternative optimization criteria for the newsvendor problem have been studied in Lau (1980). This lines of thought can be seen as leading to the recent formulation of inventory management problems in terms of coherent risk measures (Chen et al., 2005; Ahmed et al., 2007; Gotoh and Takano, 2007). Chen et al. (2005) analyze risk aversion in inventory problems comparing risk measures and expected utility optimization. Ahmed et al. (2007) derive the structure of the solution of coherent risk measure optimization for the newsvendor loss function. Gotoh and Takano (2007) discuss the solution of the newsvendor problem with CVaR. Common feature in these works is the utilization of the newsvendor loss function. Distinctive features are, in Gotoh and Takano (2007) CVaR minimization with the extension of the loss function to a multi-item single-period problem, and, in Ahmed et al. (2007), the treatment of the single-item multi-period (finite horizon) problem. In this work, we consider a multi-item inventory system whose financial characteristics are described by the profit function of Luciano et al. (2003)—“LCP model” from now on. In order to assess the effect of alternative risk aversion representations, we are faced with formulating and studying the optimization problems of the four decision makers in the presence of the LCP model. Results show that while a risk neutral and a coherent risk averse optimal policy is always a global one, the conditions under which a mean-variance decision maker's optimal policy is globally optimal are determined by the stochastic properties of demand. We then investigate the determinants of the optimal policies. By deriving the analytical expressions of the optimization problems, we identify and discuss the stochastic properties that are needed by the four types of inventory managers to identify the optimal policies. This allows us to determine the conditions under which the decision makers are insensitive to imprecision in the demand distributions. As far as economic aspects are concerned, findings show that while a risk neutral policy can be determined based on the sole knowledge of revenues and variable costs, risk averse decision makers need the additional knowledge of the system fixed costs. We then carry out a numerical discussion aimed at highlighting the quantitative differences among the optimal policies selected by the different decision makers. To compare the policy structures we introduce a methodology based on Savage's score correlation coefficients (Iman and Conover, 1987). The numerical impact of imprecision in the demand distribution is assessed by confronting numerical results obtained with finite support distributions (Beta) to the results obtained via an infinite support distribution (Gamma). The remainder of the paper is organized as follows. Section 2 illustrates the problem settings in the presence of a generic loss function. Section 3 presents the problem settings for the LPC profit function. In particular, Section 3.1 discusses the optimization problem for a risk neutral decision maker. Section 3.2 derives the optimization problem for a quadratic utility risk averse decision maker. Section 3.3 discusses the problem for an inventory manager utilizing MAD. Section 3.4 presents the optimization problem for a CVaR decision maker. Section 4 compares the different problems, discusses numerical results and evaluates the effect of imprecision in the demand distributions. Section 5 offers conclusions.

In this work, we proposed an approach to quantify the differences among inventory policies determined by alternative risk aversion attitudes. We have dealt with the determination of inventory policies in the presence of alternative risk measures for a multi-item inventory system described by the (extended) LCP loss function. We have considered the policies of risk neutral decision makers, and of three risk averse inventory managers selecting variance, MAD and CVaR as risk measures. The optimization problem for each decision maker has been derived. This has enabled us to observe that decision makers choosing any coherent risk measure always solve a convex program, thus being assured of the global nature of the optimal policy. The same happens to a risk neutral decision maker. However, the nature of the optimization problem for decision maker selecting variance depends on the stochastic properties of demand. We have then studied the economic and stochastic determinants of the inventory policies to identify what information is relevant to the different types of managers in order to come to an inventory management decision. Findings show that per unit revenues and variable costs concur in the determination of all four policies. However, while a risk neutral decision makers would not need to measure fixed costs to identify the optimal policy, such knowledge is required to all risk averse decision makers. We have identified the stochastic properties needed to come to an optimal policy vary across the different risk measure selections. This has also enabled us to state conclusions about the sensitivity of the policies to imprecision in the demand distribution. More in detail: (i) risk neutral policies require the knowledge of mm (Table 1) and do not change in the presence of distributions leading to the same mm (Table 1); (ii) quadratic risk averse policies require the knowledge of M (Table 1) and are unaltered by distributions leading to the same M (Table 1); (iii) MAD policies require the knowledge of mm, m++m-m++m- and are invariant for distributions leading to the same mm, m++m-m++m- (Table 1) and finally (iv) CVaR policies require the knowledge of View the MathML sourceF(ΘDζ+) and mζ+mζ+ (Table 1) and are unaffected if imprecision in the distributions leads to the same View the MathML sourceF(ΘDζ+) and mζ+mζ+. We have finally addressed the quantitative comparison of optimal policies induced on the same system by different risk measures. The approach to compare the structure of inventory policies has been based on the statistical technique of Savage Scores (Iman and Conover, 1987). Results for the 10 item inventory system analyzed in this exercise are as follows. Risk aversion leads to a reduction in the optimal order quantity, with MAD decision makers ordering less than risk neutral ones but more than CVaR, and quadratic utility decision makers ordering the lowest number of items. However, risk aversion has not altered the structure of the policies in a significant way. Namely, items that were ordered the most by a risk neutral decision maker have remained the most ordered also across the three risk averse policies. The same has happened to the least ordered items. This work opens future research directions by the authors. The first direction is the study of optimal inventory policies in the presence of a non-additive loss function, so as to evidence the effects of synergies and discounts. The second direction is the implementation of the methodology in the reverse direction, to infer the risk measure which the decision maker is selecting from the actually chosen policy in the context of a case study.