Abstract: Netessine and Rudi(2003) consider a consumer-driven substitution problem with an arbitrary number of products under both centralized management and competition.They obtain analytically tractable solutions,establish concavity of the objective function,i.e.,the expected profit function generated by each product and uniqueness of the equilibrium for the decentralize case.For the centralized case,they indicate that the objective function,i.e., the expected profit function,might not be concave and not even quasiconcave.In this note we show,for the centralize case,that the objective function is submodular,and partially verify Netessine and Rudi’s observation that the objective function be unimodal in each of the decision variables for some case. 

Key words: operation research, inventory, substitution, submodular