On replacement models via a fuzzy set theoretic framework

| Uncertainty is present in virtually all replacement decisions due to unknown future events, such as revenue streams, maintenance costs, and in°ation. Fuzzy sets provide a mathematical framework for explicitly incorporating imprecision into the decision making model, especially when the system involves human subjectivity. This paper illustrates the use of fuzzy sets and possibility theory to explicitly model uncertainty in replacement decisions via fuzzy variables and fuzzy numbers. In particular, a fuzzy set approach to economic life of an asset calculation as well as a ̄nite horizon single asset replacement problem with multiple challengers is discussed. Because the use of triangular fuzzy numbers provides a compromise between computational ef̄ciency and realistic modeling of the uncertainty, this discussion emphasizes fuzzy numbers. The algorithms used to determine the optimal replacement policy incorporate fuzzy arithmetic, dynamic programming with fuzzy rewards, the vertex method, and various ranking methods for fuzzy numbers. A brief history of replacement analysis, current conventional techniques, the basic concepts of fuzzy sets and possibility theory, and the advantages of the fuzzy generalization are also discussed. Keywords| Replacement analysis, fuzzy sets, possibility theory, fuzzy numbers, decision making under uncertainty. I. Economic Decision Analysis Economic decision analysis is a useful tool, o®ering individuals and organizations the techniques to model economic decision making problems, such as maintenance and replacement decisions, and determine an optimal decision. However, the accuracy of the model determines the validity of the conclusion. In many cases, the assumption of certainty in many models is made not so much for validity but the need to obtain simpler and more readily solvable formulations. Essentially, the tradeo® is between an inaccurate but solvable model and a more accurate but potentially unsolvable one. In most real-world systems, however, there are elements of uncertainty in the process or its parameters, which may lack precise de ̄nition or precise measurement, especially when the system involves human subjectivity. When developing a model of a system with uncertainty, the decision maker can either ignore the uncertainty, implicitly acknowledge it, or explicitly model it. Ignoring the uncertainty usually results in a deterministic model of the process with precise values for all parameters. Implicitly acknowledging the uncertainty may still result in a deterministic model in which sensitivity analysis or discount factors can be used to get an idea of how this uncertainty a®ects the outcome. Lastly, the decision maker can explicitly model the uncertainty using speci ̄c paradigms such as interval analysis, possibility theory, probability theory, or evidence theory [3]. The proper paradigm depends on the nature of the uncertainty. When the probabilities are speci ̄ed for the outcomes, then the theory of Von Neumann and Morgenstern [40] provides the tools necessary to determine the optimal decision. However, in many cases these probabilities are neither de ̄ned nor directly attainable. Under these circumstances, other theories are needed. The most common choice is the use of subjective probability distributions and the theory of choice due to Savage [34]. However, considerable debate on the use of subjective probabilities exists and is well documented in the literature [6], [16], [23], [25], [27]. From a psychological standpoint, the methods used to elicit these subjective probabilities and the validity of the subjective probabilities themselves have been the focus of research led by Tversky and Kahneman [37], [39], [38]. Their studies show that the heuristics employed to assess probabilities and predict values can sometimes lead to \severe and systematic errors" [38]. Because humans do not think naturally in probabilistic terms, they tend to ̄nd the notions of fuzzy sets and their linguistic based approaches more user-friendly and appealing. We may view fuzzy set theory as a generalization of classical set theory since it provides us with a mathematical tool for describing sets that have no sharp transition from membership to nonmembership. Membership in a fuzzy set is de ̄ned by a generalized version of the classical indicator function called a membership function. Fuzzy sets allow the de ̄nition of vague or imprecise concepts such as \approximately 1000" where, for example, 1000 would have a membership of 1.0 and 975 a membership of 0.5 (see Figure 1). This theory has been developed and successfully applied to numerous areas such as control and decision making, engineering, and medicine. Its application to economic analysis is natural due to the uncertainty inherent in many ̄nancial and investment decisions. As noted earlier, it provides a precise mathematical language to model uncertainty due to vagueness and imprecision in events or statements describing a system. More information on fuzzy set theory, particularly fundamental concepts such as fuzzy numbers which are invoked in our presentation, is included in the Appendix. II. Replacement Analysis One of the most practical and topical areas of engineering economics is replacement analysis. Mathematical models and analysis methods are used to determine the sequence of replacement decisions that provides a required service for a speci ̄ed time horizon in an optimal manner. It is assumed that maintenance and replacement decisions occur on a periodic basis. The decision maker chooses from various options, such as to keep, overhaul, or perform preventive maintenance on the existing asset or replace it with