تجزیه و تحلیل حساسیت جهانی در مدیریت موجودی

This paper deals with the sensitivity analysis (SA) of inventory management models when uncertainty in the input parameters is given full consideration. We make use of Sobol’ function and variance decomposition method for determining the most influential parameters on the model output. We first illustrate the method by means of an analytical example. We provide the expression of the global importance of demand, holding costs, order costs of the Harris economic order quantity (EOQ) formula. We then present the global SA of the inventory management model developed by Luciano and Peccati [1999. Capital structure and inventory management: the temporary sale problem. International Journal of Production Economics 59, 169–178] for the economic order quantity estimation in the context of the temporary sale problem. We show that by performing global SA in parallel to the modeling process an analyst derives insights not only on the EOQ structure when its expression is not analytically known, but also on the relevance of modeling choices, as the inclusion of financing policies and special orders.

Uncertainty in inventory policy making stems from a variety of factors. Just as a simple example, consider a firm that uses the Harris economic order quantity (EOQ) formula as a support to its inventory policies (Erlenkotter, 1990, Harris, 1915 and Piasecki, 2001). In order to come to a final decision on the EOQ, the firm must estimate demand, unit order costs and holding costs. Demand is seldom steady, and its value cannot be determined with certainty in most of the cases (Alstrom, 2001, Ray and Chaudhuri, 1997, Grubbstrõm, 1996, Teng and Yang, 2004, Matheus and Gelders, 2000 and Boylan and Johnston, 1996). Costs can be a further source of uncertainty (see Piasecki, 2001): on the one hand the criteria of cost classification are not always be sharply set and, on the other hand, even once the criteria are set, variability characterizes the costs themselves (Piasecki, 2001). Hence, rarely one can predict the behavior of an inventory system with the inputs fixed at a certain value; more likely, the decision-maker will be able to assign parameters within ranges determined by the analysis (Piasecki, 2001, Bogataj, 1998 and Bogataj and Hvalica, 2003). To cope with the corresponding uncertainty in model predictions, usually a sensitivity analysis (SA) exercise is performed. The more direct SA scheme is the testing of the change in model output that follows a change in the parameters when they are shifted within the limits of their variation ranges: this type of one-variable-at-a-time SA is performed in Ray and Sahu (1992), Arcelus and Rowcroft (1993), Ray and Chaudhuri (1997), and Powell (2000). Ganeshan et al. (2001) study the sensitivity of supply chain performance to three inventory parameters. Perturbation analysis has been developed and employed in the works of Glasserman and Tayur (1995), Bogataj and Bogataj (1992), Cibej and Bogataj (1994), Bogataj and Cibej (1994), and Bogataj and Bogataj (2004). The previous approaches belong to the family of local SA approaches. Local SA techniques are the set of methods that study the behavior of a mathematical model around a point of the input parameter space for finite or small changes in the input parameters (Borgonovo et al., 2003). To the family of local SA methods belong the technique of comparative statics (Samuelson, 1947) and the differential importance measure (Borgonovo and Apostolakis, 2001 and Borgonovo and Peccati, 2004). An SA method is global if it tests the sensitivity of the model in consideration of the uncertainty distribution reflecting the decision-maker state of knowledge in the parameters. Global SA techniques can be divided in the categories of non-parametric techniques (Saltelli and Marivoet, 1990), screening methods (Morris, 1991), response surface methodology (Downing et al., 1985), and variance-based methods (Saltelli et al., 1999, Saltelli et al., 2000b and Sobol’, 1993). Target of variance-based global SA methods is the model output variance, which is decomposed in a series of summands of increasing dimensionality (Saltelli et al., 1999, Sobol’, 1993, Sobol’, 2003 and Saltelli et al., 2000b). In this work, we focus on variance decomposition through the Sobol’ (Saltelli et al., 2000b and Sobol’, 1993) and the extended FAST methods (Saltelli et al., 1999). Our first application is the determination of the global importance of the parameters in the classical Harris inventory management model (Harris, 1915 and Erlenkotter, 1990). We derive analytically Sobol’ function and variance decomposition of the Harris EOQ formula and provide the expression of the input parameter global importance (GI). By means of numerical results, we illustrate that parameters associated with the highest value of GI are the most effective in reducing the variance of the EOQ. We then apply the techniques to the inventory management model introduced by Luciano and Peccati (1999) in the context of the temporary sale problem. Starting point of the model is the loss function corresponding to the consideration of the cost of capital which is obtained by making use of the adjusted present value (APV) technique (Grubbstrõm and Ashcroft, 1991, Grubbstrõm and Thorstenson, 1986, Peccati, 1989, Peccati, 1996 and Thorstenson, 1988). Luciano and Peccati (1999) then formulate the cost functions that include third party financing, and the presence of special orders. None of the cost functions allows for the analytical expression of the EOQ. We show that performing a step-by-step global SA an analyst gains insights on both the modeling aspects and the EOQ structure. As far as modeling aspects are concerned, the use of global SA allows to ascertain whether the inclusion/exclusion of a certain assumption has a significant or negligible impact on the EOQ. As far as the EOQ structure is concerned, one gains a quantitative indication on the type of dependency of the EOQ on the parameters. This information would not be gained without a global SA when one does not possess the analytical expression of the EOQ. Results show that, for the uncertainty ranges at hand, the cost of debt, followed by the cost of capital and the special order discount are the most relevant parameters. Not only, but their inclusion impacts the EOQ structure in a significant way, shifting the EOQ dependence on the parameters from additive to non-additive. Section 2 describes the principles and theorems at the basis of global SA. Section 3 illustrates the application and provides analytical results for the Harris EOQ formula. Section 4 presents the global SA of the Luciano–Peccati model, comparing the different cost functions and the corresponding SA results. Conclusions are offered in Section 5.

This work has concerned the application of global sensitivity analysis (SA) to inventory management models. Global SA techniques are the most appropriate for performing the SA of model output when full consideration is given to the uncertainty in the parameters. After presenting the mathematical definitions and theorems at the bases of variance-based global SA, we have dealt with its application to inventory models in two ways: analytically, through the application of Sobol’ function and variance decomposition method to the classical Harris inventory model and numerically, through the application of the extended FAST method to the global SA of the Luciano–Peccati model. We have seen that variance-based global SA results have enabled us to gain insights on: Parameter influence : Through the computation of parameter global importance, one is able to obtain a quantitative measure of the influence of parameters on the model output. For the Harris EOQ, we have seen that the three parameters had the same importance in case 1, while holding cost (x3x3) resulted the most important parameter in case 2 (increased uncertainty). For the Luciano–Peccati model, the cost of debt, δδ, has turned out to be the most influential input, followed by the cost of capital ρρ and the special order discount, d. EOQ structure : Global SA enables one to understand the type of model structure even when the model is not an explicit function of the parameters. In the case of the Harris EOQ, we have seen that order cost (x1x1) and demand (x2x2) shared the same global importance, reflecting the symmetry in the EOQ formula. In the case of the Luciano–Peccati model we have seen that the EOQ dependence on the parameters changed from additive to non-additive as third party financing and special orders were included. Model construction : Global SA results provide an analyst with direct and quantitative information on whether the inclusion/exclusion of certain aspects in/from the model would correspond to consider/neglect relevant aspects. In the Luciano–Peccati model, for example, not to model third party financing (δδ), the cost of equity (ρρ) or the presence of a special orders would have resulted in neglecting aspects with a significant impact on the EOQ. Uncertainty management insights : By definition, parameters associated with high values of S1lS1l are the ones that are capable of reducing variance in the output the most. In the presence of limited time and resources, there can result a precious information the knowledge of the parameters on which to orient resource and data collection to reduce the variance of model output the most.