Markov Chains: Models, Algorithms and Applications
Description
Markov chains are a particularly powerful and widely used tool for analyzing a variety of stochastic (probabilistic) systems over time. This monograph will present a series of Markov models, starting from the basic models and then building up to higher-order models. Included in the higher-order discussions are multivariate models, higher-order multivariate models, and higher-order hidden models. In each case, the focus is on the important kinds of applications that can be made with the class of models being considered in the current chapter. Special attention is given to numerical algorithms that can efficiently solve the models.
Therefore, Markov Chains: Models, Algorithms and Applications outlines recent developments of Markov chain models for modeling queueing sequences, Internet, re-manufacturing systems, reverse logistics, inventory systems, bio-informatics, DNA sequences, genetic networks, data mining, and many other practical systems.
Reviews / Votes
From the reviews:
"The authors outline recent developments of Markov chain models . . This book is aimed at students, professionals, practitioners, and researchers in scientific computing and operational research, who are interested in the formulation and computation of queuing and manufacturing systems. It gives a number of useful tools for researchers in real applications . ." (Alexander I. Zejfman, Zentralblatt MATH, Vol. 1089 (15), 2006)
"In this book's . essential notions on Markov chains, hidden Markov models, and Markov decision processes are covered, with special emphasis on iterative methods for solving linear systems. . Each chapter finishes with a short summary and sometimes a selection of open problems. . This book is intended for students and researchers in applied mathematics, scientific computing, and operations research . . Overall, this book offers much interesting and up-to-date material on a wide variety of topics, dealing with finite-space Markov processes." (Jozef L. Teugels, Journal of the American Statistical Association, Vol. 103 (483), September, 2008)
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3 Re-manufacturing Systems (p. 61-62)
3.1 Introduction
In this chapter, the inventory controls of demands and returns of single-item inventory systems is discussed. In fact, there are many research papers on inventory control of repairable items and returns, most of them describe the system as a closed-loop queueing network with constant number of items inside [78, 158, 201]. Disposal of returns [127, 200] is allowed in the models presented here. The justi.cation for disposal is that accepting all returns will lead to extremely high inventory level and hence very high inventory cost. Sometimes transshipment of returns is allowed among the inventory systems to reduce the rejection rate of returns. Other re-manufacturing models can be found in [117, 200, 196] and good reviews and current advances of the related topics can be found in [23, 84, 92, 132, 157].
As a modern marketing strategy to encourage the customers to buy products, the customers are allowed to return the bought product with full refund within a period of one week. As a result, many customers may take advantage of this policy and the manufacturers have to handle a lot of such returns. Very often, the returns are still in good condition, and can be put back to the market after checking and packaging. The .rst model we introduce here attempt to model this situation. The model is a single-item inventory system for handling returns is captured by using a queueing network. In this model, the demands and the returns are assumed to follow two independent Poisson processes. The returns are tested and repaired with the standard requirements. Repaired returns will be put into the serviceable inventory and non-repairable returns will be disposed. The repairing time is assumed to be negligible. A similar inventory model with returns has been discussed in [110]. However, the model in [110] includes neither the replenishment costs nor the transshipment of returns. In this model, the inventory system is controlled by a popular (r,Q) continuous review policy. The inventory level of the serviceable product is modelled as an irreducible continuous time Markov chain.
The generator matrix for the model is given and a closed form solution for the system steady state probability distribution is also derived. Next, two independent identical inventory systems are considered and transshipment of returns from one inventory system to another is allowed. The joint inventory levels of the serviceable product is modelled as a twodimensional irreducible continuous time Markov chain. The generator matrix for this advanced model is given and a closed form approximation of the solution of the system steady state probability distribution is derived. Analysis of the average running cost of the joint inventory system can be carried out by using the approximated probability distribution. The focus is on the inventory cost and the replenishment cost of the system because the replenishment lead time is assumed to be zero and there is no backlog or loss of demands. It is shown that in the transshipment model, the rejection rate of the returns is extremely small and decreases signi.cantly when the re-order size (Q + 1) is large. The model is then extended to multiple inventory/return systems with a single depot. This kind of model is of particular interest when the remanufacturer has several re-cycling locations. Since the locations can be easily connected by an information network, excessive returns can be forwarded to the nearby locations or to the main depot directly. This will greatly cut down the disposal rate. The handling of used machines in IBM (a big recovery network) serves as a good example for the application of this model [92]. More examples and related models can be found in [92, pp. 106-131].
Finally, a hybrid system consists of a re-manufacturing process and a manufacturing process is discussed. The hybrid system captures the remanufacturing process and the system can produce serviceable product when the return rate is zero.
The remainder of this chapter is organized as follows. In Section 3.2, a single-item inventory model for handling returns is presented. In Section 3.3, the model is extended to the case that lateral transshipment of returns is allowed among the inventory systems. In Section 3.4, we discuss a hybrid remanufacturing system. Finally, concluding remarks are given in Section 3.5.
