The Single Period Inventory and Pricing Problem with Spectral Risk Measures

Inventory management decisions based on quantitative models both in industrial practice and academic works often rely on minimizing expected cost or maximizing expected revenues or profits, which refers to the concept of risk-neutrality of the decision maker. Although many useful insights in operational problems can be obtained by such an approach, it is well understood that incorporating attitudes toward risk is an important lever for building new theories in other fields such as economics and finance. The level of risk associated with an investment might be as important as the expected gain from the investment. Hence, it is necessary to find appropriate measures of risk and the appropriate objectives related to or including these risk measures for inventory control problems. After the axiomatic foundation of coherent risk measures the application of risk measures to inventory models such as Conditional Value-at-Risk (CVaR) or convex combinations of mean and CVaR became popular. In our work we apply spectral risk measures to the single-period, single-item, linear cost inventory control problem (also known as newsvendor problem) and derive optimal policies. By doing so, we are able to unify results obtained so far in the literature under the common concept of spectral risk measures for the case of zero and non-zero shortage penalty cost. In particular, we show convexity results and structural properties for the inventory control problem. An extensive numerical analysis illustrates the findings