Analysis of two substitute products newsvendor problem with a budget constraint

Abstract

This paper presents a substitute product newsvendor problem with a budget constraint: the newsvendor sells two substitute products and determines the optimal order quantity and selling price for each product. We present a nonlinear programming model with a budget constraint and solve the model mainly according to the Karush-Kuhn-Tucker (KKT) theorem. The resulting insights are as follows: (1) We prove the existence of unique optimal solutions in a stochastic demand scenario with a general demand probability density function. (2) When the substitute products’ information is symmetric, the demand variability, store-level factor and competitive factor have no effect on optimal order quantities, and the optimal prices are the same as long as the value of store-level factor minus competitive factor remains the same, when other factors are fixed. However, this result is not observed when the substitute products’ information is asymmetric. (3) If the budget constraint is tight, releasing the non-negative constraints of the decision variables will lead to negative order quantities, which mean the retailer has to cancel the corresponding order from the purchasing list. In addition, the retailer should decrease the optimal retail price in order to obtain more expected revenue if the budget increases.