Stock rationing under service level constraints in a vertically integrated distribution system

This paper addresses a rationing problem in a two-level, vertically integrated distribution system composed of one manufacturer and several retail points. The motivating case, developed in the vending machine sector and modeled as a newsvendor-like problem, is representative of many real settings where short-term changes in demand can be substantial while capacity modification is not a viable option. The paper provides an analytical discussion of the problem from two different standpoints: a pure profit-maximization perspective and a minimum service-level perspective, both subject to a product availability constraint that affects the service level the company can provide, and the related expected profit. By analyzing the Lagrangean formulation of the problem, we devise efficient computational procedures based on dichotomy search to find the optimal allotments to retailers, maximizing the expected profit and ensuring a minimum service level. Then, we extend the analysis to the evaluation of the highest service level that can be provided, under a product availability constraint. We identify conditions such that the proposed search procedures succeed in finding the optimal solutions, as well as bounds for the search domains. The proposed approach is legitimated under several demand distribution functions subject to a few commonly adopted restrictions that encompass many of the usually adopted continuous distributions. Finally, the paper presents a three-step decision-making framework using the proposed procedures, summarizing the decision paths the manufacturer might follow in order to optimize the allocation decision.