On the multi-product newsvendor with bounded demand distributions

Abstract

We consider a multi-item newsvendor problem with side constraints and common continuous demand distributions that are bounded implying that the items’ profit functions are nondifferentiable. In particular we focus on the cases of uniform and triangular distributions. These distributions naturally describe demand that is guaranteed to exceed a certain threshold – for example, consumption of basic food products or electric power consumption over any given day. Moreover, in practice, it is often difficult to estimate the demand distribution. Accordingly, the uniform and triangular distributions become default modeling choices when only information regarding the bounds and possibly the mode of the distribution is known. We generalize a previous quadratic programming model for uniformly distributed demand on [a,b] to allow a to be nonzero and to allow the order quantity to be smaller than a. We study the corrected model and propose an efficient algorithm for determining an optimal solution. The algorithm is motivated by a structural result of an upper bound on the number of guaranteed shortage products, which typically appear in multiproduct settings with a positive demand distribution lower bound. The performance of our specialized algorithm is compared to that achieved when solving our formulation with a piecewise quadratic objective using a state-of-the-art standard solver. We also extend the modeling technique to propose a nonlinear programming formulation for triangular demand distributions. A similar approach can be adopted to approximate other demand distributions with a possibly non-finite support, such as truncated normal with strictly positive lower bounds.