Shipment Consolidation in Discrete Time and Discrete Quantity: Matrix-Analytic Methods

Abstract

Shipment consolidation is a logistics strategy whereby many small shipments are combined into a few larger loads. The economies of scale achieved by shipment consolidation help in reducing the transportation costs and improving the utilization of logistics resources. The fundamental questions about shipment consolidation are i) to how large a size should the consolidated loads be allowed to accumulate? And ii) when is the best time to dispatch such loads? The answers to these questions lie in the set of decision rules known as shipment consolidation policies. A number of studies have been done in an attempt to find the optimal consolidation policy. However, these studies are restricted to only a few types of consolidation policies and are constrained by the input parameters, mainly the order arrival process and the order weight distribution. Some results on the optimal policy parameters have been obtained, but they are limited to a couple of specific types of policies. No comprehensive method has yet been developed which allows the evaluation of different types of consolidation policies in general, and permits a comparison of their performance levels. Our goal in this thesis is to develop such a method and use it to evaluate a variety of instances of shipment consolidation problem and policies. In order to achieve that goal, we will venture to use matrix-analytic methods to model and solve the shipment consolidation problem. The main advantage of applying such methods is that they can help us create a more versatile and accurate model while keeping the difficulties of computational procedures in check. More specifically, we employ a discrete batch Markovian arrival process (BMAP) to model the weight-arrival process, and for some special cases, we use phase-type (PH) distributions to represent order weights. Then we model a dispatch policy by a discrete monotonic function, and construct a discrete time Markov chain for the shipment consolidation process. Borrowing an idea from matrix-analytic methods, we develop an efficient algorithm for computing the steady state distribution of the Markov chain and various performance measures such as i) the mean accumulated weight per load, ii) the average dispatch interval and iii) the average delay per order. Lastly, after specifying the cost structures, we will compute the expected long-run cost per unit time for both the private carriage and common carriage cases.