Abstract: This paper considers a multi-product newsboy system that producesmultiple products for fulfilling independentlyuncertain demands, which share the same production capacity. To deal withpossible shortage of limited capacity, productions can be outsourced. We considertwo outsourcing strategies: zero lead time outsourcing, and nonzero lead timeoutsourcing. The structural properties and solution procedures for theprofit-maximization models are developed. Numerical results are provided forobtaining some managerial insights.
Keywords: Nonlinear programming;Newsboy problem; Capacity constraint; Outsourcing
1. Introduction
The multi-product newsboyproblem with capacity constraints is a classical production/inventorymanagement problem, which was firstly studied by Hadley and Whitin (1963).After their seminal work, the capacitated multi-product newsboy problem hasbeen extensively studied in production/inventoryresearch. Due to the difficulty for solving large-scale problemswithmultiple constraints, many works focus on the singlyconstrained problems, and develop exact solution methods for some special cases, and approximate solution methods for general problems.
Erlebacher(2000) studied multi-product newsboy problem with one capacity constraint. Heproved the optimality of the order quantities for two special cases, andproposed heuristics for a few specific probability distribution functions.Vairaktarakis (2000) developed some minimax regret formulations for singlyconstrained multi-product newsboy problem. Moon and Silver (2000) modeled fixedordering costs in the multi-product newsvendor problem with a budget constraint,and proposed heuristics for the assumed and free demand distribution situations. Abdel-Malek et al. (2004) developed Lagrangianbased methods, and their method generates the optimal solution for uniformdemand distribution, and near optimal solution for other continuousdistributions.
As Abdel-Malekand Montanari (2005) pointed out, some of the existing researches ignored the non-negativityconstraints on order quantities. This negligence, as observed by Lau and Lau (1996),could lead to negative order quantities for some products. Hadley and Whitin(1963), Kabak and Weinberg (1972), and Niederhoff (2007), Zhang (2009) haveconsidered the non-negativity constraints in some ways. For examples, Hadleyand Whitin (1963) presented a solution method to the constrained multi-product newsboyproblem with non-negativity constraints, which encounters difficulties,particularly for large-scale problems; Niederhoff (2007) providedan approximatingprogramming technique to the multi-productnewsvendor problemswith multiple constraints and non-negativityconstraints, and showed that a close approximation of the optimal solution can be found by using convex separable programming for any demand distribution.
To address non-negativityconstraints of order quantities, Abdel-Malek and Montanari (2005) proposed amodified Lagrangian based method by analyzing the solution space, and theirapproach can solve exact and approximate solutions for continuousdemand distributions. Recently,the non-negativity constraints are explicitly modeled in two relevant literatures. Zhanget al. (2009) developeda solution method for the singlyconstrained multi-product newsboyproblem with non-negativity constraints, which can provide exact solution for any continuous demanddistribution. Zhang and Hua (2008) proposeda unified algorithm for solving a class of convex separablenonlinear knapsack problems, which includes the singly constrainedmulti-product newsboy problem with non-negativity constraint.