M♮-Convexity and Its Applications in Operations

Abstract

A new approach for structural analysis of operations models with substitutability structures. In many operations models with substitutability structures, one often ends up with parametric optimization models that maximize submodular objective functions, and it is desirable to derive structural properties including monotone comparative statics of the optimal solutions or preservation of submodularity under the optimization operations. Yet, this task is challenging because the classical and commonly used results in lattice programming, applicable to optimization models with supermodular objective function maximization, do not apply. Using a key concept in discrete convex analysis, M♮-convexity, Chen and Li establish conditions under which the optimal solutions are nonincreasing in the parameters and the preservation property holds for parametric maximization models with submodular objectives, together with the development of several new fundamental properties of M♮-convexity. Their approach is powerful as demonstrated by applications in a classical multiproduct stochastic inventory model and a portfolio contract model.