An analysis of the multi-product newsboy problem with a budget constraint

This is a sequel of an earlier paper entitled “Exact, approximate, and generic iterative models for the multi-product newsboy problem with budget constraint'' (Abdel-Malek et al., 2004) that appeared in this journal. Motivated by Lau and Lau's (Eur. J. Oper. Res., 94 (1996) 29) observation where infeasible ordering quantities (negative) were obtained when applying existing methods, the extension here examines the solution space of the problem in order to provide the necessary insight into this phenomenon. The resulting insight shows that the solution space can be divided into three regions that are marked by two distinct thresholds. The first region is where the budget is large and the solution is the same as the unconstrained problem. The second region is where the budget is medium and the constraint is binding, however the newsboy can order all the products on the list. The third region is where the budget is very tight and if the non-negativity constraints are relaxed negative order quantities may be obtained, and therefore some products have to be deleted from the original list. We show how the values of the thresholds that divide the regions are computed and extend the previous methods, when necessary, to cover each of the three-solution's domains in order to determine the optimum order quantity for the various products. Numerical examples are given to illustrate the application of the developed procedures.