Exploiting structural similarities for solving inventory and replacement problems

Abstract

This dissertation is concerned with the study of the Markov process for two-product joint setup cost inventory problems with continuous review, instantaneous replenishment and compound Poisson demands for each product. The inventory position for each product at each successive demand epoch forms the transition states of the Markov chain. By the special structure of the Markov rate matrix (G) using (pi)G = 0 and (pi)1 = 1, steady state probabilities can be obtained and the average expected cost per unit time for the inventory system can be evaluated. By applying search techniques, optimal solutions for the independent and the dependent joint setup cost inventory models can be obtained. Curry's algorithm using the iterative-optimization solution technique is demonstrated as an efficient technique for the dependent joint setup cost inventory problem. Using this technique, a near optimal can-order policy is obtained. Further improvement over the approximate optimal solution using search techniques is insignificant. The Markov process model is also used to study machine replacement problems for a single machine under an observable sequence of random shocks with compound Poisson damage. Using the Markov process, an optimal replacement level is obtained through a one dimensional search technique. For two machine joint replacement problems using the Markov process model, a similar structural transition rate matrix to the transition rate matrix of the joint order inventory problem can be revealed. Because of the structural similarities of the problems, the same approach in modeling the system, determining the steady state probabilities and determining the optimal policy can be used to solve the two machine joint replacement problems with compound Poisson shocks.