Periodic review stochastic inventory problem with forecast updates: Worst-case bounds for the myopic solution

Muth first considered the linear cost periodic review inventory problem in which the mean demand in a period undergoes a non-observed random walk. Assuming the random walk variance and the within period demand variance to be known, and stationary, he showed that the Best Linear Unbiased Estimate for the mean is given by exponential smoothing and derived the formula for the optimal steady state smoothing constant in terms of the variances. We first show that the corresponding Bayesian analysis is useful under transient conditions, and converges to the Muth results under steady state. For the steady state solution, we prove that the myopic policy is near-optimal, using the concepts of P-myopic and D-myopic introduced in this paper, in the sense that the worst case bounds on policy errors in using it may be given analytically, and are small for reasonable values of parameters. As further validation, we use dynamic programming to compute optimal policies and compare them with myopic policies; policy errors are very small. By considering analogous results in recent literature, it is conjectured that additions of such factors as non-stationarity, lead-times, perishability, setup costs would produce near-myopic results.