Mean-Risk Analysis of Multiperiod Inventory Problems

In this chapter, we carry out mean-risk analysis of multiperiod inventory problems. We select the well-known (R, nQ) multiperiod inventory replenishment model (see Chen and Zheng 1994, 1998; Larsen and Kiesmüller 2007; Li and Sridharan 2008; Shang and Zhou 2010; Lagodimos et al. 2012 and the references therein for more details of the recent developments and extensions of this model) as an example to demonstrate how to perform a mean-risk analysis for multiperiod inventory problems. As shown later on in this chapter, the mean-risk analysis of multiperiod inventory problems is very different from the mean-risk analysis of single-period analysis, in terms of problem formulations and methodology applied. In particular, the (R, nQ) model considers an infinite-horizon replenishment problem under which the total profit/cost is infinite, too. Therefore, the expected (total) profit and the variance of (total) profit cannot be used directly as the “mean” and the “risk,” respectively, in the mean-risk analysis of the (R, nQ) model. To perform the mean-risk analysis, we take the long-run average profit as the “mean,” and propose the variance of on-hand inventory and the variance of one-period profit as “risk” of the (R, nQ) model. We first derive the closed form expressions of the long-run average profit, the variance of on-hand inventory, and the variance of one-period profit. Then, we apply the numerical analysis to demonstrate how to construct the efficient frontier, in the mean-risk sense.