Optimal myopic policy for a stochastic inventory problem with fixed and proportional backorder costs

In this paper, we consider a single product, periodic review, stochastic demand inventory problem where backorders are allowed and penalized via fixed and proportional backorder costs simultaneously. Fixed backorder cost associates a one-shot penalty with stockout situations whereas proportional backorder cost corresponds to a penalty for each demanded but yet waiting to be satisfied item. We discuss the optimality of a myopic base-stock policy for the infinite horizon case. Critical number of the infinite horizon myopic policy, i.e., the base-stock level, is denoted by S. If the initial inventory is below S then the optimal policy is myopic in general, i.e., regardless of the values of model parameters and demand density. Otherwise, the sufficient condition for a myopic optimum requires some restrictions on demand density or parameter values. However, this sufficient condition is not very restrictive, in the sense that it holds immediately for Erlang demand density family. We also show that the value of S can be computed easily for the case of Erlang demand. This special case is important since most real-life demand densities with coefficient of variation not exceeding unity can well be represented by an Erlang density. Thus, the myopic policy may be considered as an approximate solution, if the exact policy is intractable. Finally, we comment on a generalization of this study for the case of phase-type demands, and identify some related research problems which utilize the results presented here.