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This work introduces a comprehensive approach to the sensitivity analysis (SA) of risk-coherent inventory models. We address the issues posed by (i) the piecewise-defined nature of risk-coherent objective functions and (ii) by the need of multiple model evaluations. The solutions of these issues are found by introducing the extended finite change sensitivity indices (FCSI's). We obtain properties and invariance conditions for the sensitivity of risk-coherent optimization problems. An inventory management case study involving risk-neutral and conditional value-at-risk (CVaR) objective function illustrates our methodology. Three SA settings are formulated to obtain managerial insights. Numerical findings show that risk-neutral decision-makers are more exposed to variations in exogenous variables than CVaR decision-makers.

Recent works have demonstrated the use of coherent measures as a novel and effective way to manage risk in inventory problems Ahmed et al. (2007), Gotoh and Takano (2007), Borgonovo and Peccati (2009a). The convexity of the objective functions insures feasibility in a broad variety of applications. However, an explicit expression of the solution is generally not available. This prevents a direct interpretation of model results and a straightforward derivation of managerial insights. The need to explain “what it was about the inputs that made the outputs come out as they did Little (1970); p. B469” is underlined in Little's seminal paper on the creation and utilization of decision-support models for managers. Eschenbach (1992) underlines the need of identifying the “most critical factors” on which to focus “managerial attention during implementation ( Eschenbach, 1992; pp. 40–41).” Works as Rabitz and Alis (1999), Wallace (2000), Saltelli et al. (2000), Saltelli and Tarantola (2002), Saltelli et al. (2004) have established the awareness that these questions are answered only by a systematic application of sensitivity analysis (SA). Wallace (2000) and Higle and Wallace (2003) address the use of SA in examining management science model output. They underline the key-issue of establishing consistency between the managerial questions and the SA method selected for the analysis. In linear programming, Jansen et al. (1997), Koltai and Terlaky (2000), Koltay and Tatay (2008) discuss the differences in the mathematical and managerial interpretation of SA results. Saltelli et al. (2008) (p. 24) recognize that “a poor definition of the objectives of a sensitivity analysis can lead to confused or inconclusive results.” The works by Saltelli and Tarantola (2002), Saltelli et al. (2004) and Saltelli et al. (2008) demonstrate that these issues are solved by SA settings. A setting is “a way of framing the sensitivity quest in such a way that the answer can be confidently entrusted to a well-identified sensitivity measure Saltelli et al. (2008), p. 24.” Purpose of this work is to establish a comprehensive and consistent approach to the SA of risk-coherent inventory problems. To achieve this goal, we proceed as follows. We first address the specific (1) technical and (2) result communication issues. Technical issues are posed by the piecewise-defined character of risk-coherent objective functions Borgonovo and Peccati (2009b). This non-smoothness makes comparative statics and differential approaches not applicable. We show that the integral function decomposition at the basis of the finite change sensitivity indices (FCSI) provides the required generality and solves the technical issues. Result communication issues are posed by the multi-item nature of the problem and, more in general, by the presence of multiple outputs of interests to the decision-maker. We introduce two alternative ways for dealing with result communication. The utilization of the norm of the optimal policy and the technique of the Savage Score correlation coefficients Iman and Conover (1987). We highlight advantages and drawbacks of each approach. The second step is to enrich information further by enabling a deeper exploration of the exogenous variable space. In SA practice, decision-makers assess a set of efficient scenarios Tietje (2005). The model is tested at each scenario. Since, in previous inventory management works one [in perturbation approaches Bogataj and Cibej (1994), in comparative statics Borgonovo (2008) or two points (Borgonovo, 2010) were explored, we need to formalize the application of FCSI's in the presence of multiple scenarios. We show that this is achieved by applying the finite-change decomposition at each model jump. As a result, plentiful information is obtained on the behavior of the decision criteria and on the determinants of the problem. We synthesize this information in sensitivity measures called extended FCSI's. By the extended FCSI's one obtains insights on both the magnitude and direction of impact and on the importance of the exogenous variables. Flexibility in assessing the effect of individual variables and groups is offered by the approach. The third step is to derive general properties of extended FCSI's in risk-coherent problems. We show that, if the loss function of the system at hand (not necessarily an inventory system) is separable in a group of exogenous variables, then: (i) the optimal risk-coherent policy is insensitive on that group; (ii) the value of the risk-measure at the optimum is sensitive and responds additively to changes in the parameters of the group. We then discuss the SA settings that allow one to interpret numerical results and obtaining managerial insights consistence with Eschenbach's and Little's questions. We apply the proposed methodology to a stochastic inventory problem with risk-neutral and conditional value at risk (CVaR) objective functions. Numerical results confirm the theoretical expectations on the behavior of the sensitivity measures. We discuss managerial insights in the light of the SA settings. Comparison of the numerical findings for risk-neutral and CVaR decision-makers show that both the CVaR optimal policy and value-at-the-optimum are less sensitive to exogenous variable changes than the corresponding risk-neutral optimal policy and expected loss. The remainder of this work is organized as follows. Section 2 discusses technical aspects and the choice of the sensitivity measures. Section 3 formalizes the notion of extended FCSI's. Section 4 proves relevant properties of the sensitivity of risk-coherent problems under separability conditions of the loss function. Section 5 discusses the SA settings for gaining managerial insights. Section 6 presents the case study and illustrated numerical findings. Conclusions are offered in Section 7.

In this work, we have introduced a comprehensive approach to the SA of risk-coherent multi-item inventory problems. We have addressed specific issues raised by: (i) the piecewise-defined character of risk-coherent objective functions; (ii) the presence of simultaneous and discrete changes in the parameters when shifted across multiple scenarios. We solved these problems by decomposing of the endogenous variable changes across all scenarios according to the finite change decomposition of Borgonovo (2010). We have formalized the notion of extended FCSI's. We have seen that they retain all properties of FCSI's granting full flexibility in performing SA with groups of exogenous variables. We have obtained new results for the sensitivity analysis of risk-coherent optimization problems. We have seen that, if a loss function (not necessarily describing an inventory system) is separable in a group of exogenous variables, then the optimal policy is insensitive to that group. The risk-measure-at-the-optimum is sensitive, with an additive response. These results hold for any change and at all scenario jumps. We have discussed the interpretation of results in the light three SA settings. We have seen that one is endowed with the exact explanation of what has made the outputs come out as they did (Little's question) and with information about the factors on which to focus managerial attention during implementation (Eschenbach's question). To illustrate the methodology, we have applied it to an inventory management case study, involving a risk-neutral and CVaR problems. Numerical results confirm the theoretical findings. The extended FCSI's of fixed costs are null on the optimal policy, and equal to View the MathML source∑i=1IΔai˜ on both E[Z*]E[Z*] and View the MathML sourceCVaRα*, due to the separable structure of the loss function. The sensitivity measures allow us to identify the key drivers of the changes and unveil the detailed structure of the model response across scenarios — in the specific case, only the interaction between cc and rr is relevant — the values of the sensitivity measures reveal that a CVaR decision-maker is less sensitive to exogenous variable changes than a risk-neutral one.