Multi-product Newsvendor with Submodular Demand Restrictions

We study a novel subclass of the multi-product newsvendor model where sales is restricted by a maximal demand that is submodular and random. This setting is prevalent in the inventory and revenue management literature and arises from (i) product indifference, where customers perceive no difference in net utility among a set of alternatives, and (ii) fulfillment deference, where due to indifference, choice of the specific product to fulfill customer demand is deferred to the retailer. We characterize the exact optimal solution, showing that optimal inventory levels should be decided greedily by prioritizing products in decreasing sequence of selling price, while cost only affects the specific inventory levels. Furthermore, these inventory levels are the critical quantiles of some demand distribution, akin to the traditional newsvendor problem. However, this distribution also captures the demand of other products that the relevant product could fulfill due to product indifference. Our work calls for deeper examination into stochastic optimization over submodular random variables.