Mean-Variance Assortment and Inventory Optimization for a Newsvendor

Abstract

The single-period newsvendor model is a widely used application in Operations Management. Usually, fashion retailers encounter a problem in deciding the size of their orders before the start of the season. Most of the times they incur overage and underage costs while ordering too much or too little. The newsvendor model typically helps avoiding these costs by setting an order quantity that maximizes the expected profit of the retailer. Recent literature on the single-product case has shown, however, that the expected profit-maximizing (risk-neutral) newsvendor is prone to a high risk level reflected in a high profit variance. This literature also observes adopting a slightly smaller order quantity that the one utilized by the risk-neutral newsvendor caries significant variance reductions. Motivated by the single-product observation on the high variance bared by the risk-neutral vendor, we consider the case of a fashion retailer managing an assortment of substitutable products under logit demand. We develop a model inspired by the classic mean-variance portfolio optimization problem in Finance, whereby the retailer sets the inventory levels of products in the assortment in a way that minimizes the profit variance while achieving a minimum targeted expected profit level. We develop useful analytical properties of this mean-variance assortment planning model. For example, we show that the ordered quantities are always below those of the risk-neutral newsvendor and that the expected profit target constraint is always binding. Numerical results indicate that our model is well-behaved in the sense that an optimal solution is reached quickly with a reasonable choice of the starting order quantities solution. In addition, we observe ample opportunities to reduce the profit variance involving small sacrifices in the expected profit. That is, multiple product management seems to allow better harnessing the risk-reward tradeoff than the single product one. Numerical results are also developed on the structure of the optimal assortment in the mean-variance setting. We observe some deviations from the common risk-neutral results. For example, for horizontally differentiated products, the optimal assortment among those having the same cardinality is not necessarily a popular set.