M.Sc Thesis
M.Sc Student Dahan Ella The Transshipment Problem with Partial Lost Sales Department of Industrial Engineering and Management Professor Yale Herer

Abstract

In this thesis, we consider two retailers between which transshipments can take place at the end of the period. The retailers differ in cost and demand distributions, operate in a single period, and cooperate to minimize joint costs. Our work differs from previous analyses as it considers the possibility that customers are not always willing to wait for transshipments. Instead, only some customers are willing to wait and return to the retailer for transshipments.

The objective of the research is to find the replenishment levels and transshipment quantities that minimize the total expected system cost. We consider two cases - a partially deterministic case, and a fully stochastic case. In the partially deterministic case, the number of returning customers is a known fraction of those that could not be satisfied off-the-shelf. The fully stochastic case treats the number of returning customers as a random variable whose probability density function is known and whose expected value is a fraction of the customers that could not be satisfied off-the-shelf.

In the partially deterministic case, we show that the transshipment decision has a form similar to complete pooling. Thereafter, we prove that the objective function is convex in the replenishment levels, and suggest numerical methods for finding the optimal replenishment levels.

In the fully stochastic case the number of returning customers is unknown. Thus, the transshipment decision is a stochastic planning problem. We have a newsvendor problem (the optimal transshipment quantity) nested within a larger newsvendor problem (the optimal replenishment levels). We show that the optimal transshipment quantity is found by solving a capacitated newsvendor problem. Thereafter, we analyze the convexity of the objective function with respect to the replenishment levels. We illustrate the analysis with a probability density function of returning customers which is normally distributed. We show that for this distribution, the objective function is not convex in the decision variables. Two approximations to the objective function are presented, and shown to be convex. We propose a solution methodology which utilizes numerical methods on the objective function and the two approximations.