Augustine O. Esogbue

Fuzzy dynamic programming: main developments and applications

Abstract We survey major developments and applications of fuzzy dynamic programming which is here advocated as a promising attempt at making dynamic programming models more realistic by a relaxation of often artificial assumptions of precision as to the constraints, goals, states and their transitions, termination time, etc. Issues related to numerical efficiency are considered. Applications in R&D planning, health care and medicine, socioeconomic regional development, energy systems, water and environmental systems, chemical engineering, etc. are surveyed.

On the application of fuzzy sets theory to the optimal flood control problem arising in water resources systems

Abstract We first survey the application of fuzzy sets theory to various problems occurring in water resources systems. The problem of optimal flood control planning by an appropriate integration of structural and non-structural measures with the objective of optimizing the flood damage reduction due to recurrent floods is modeled via fuzzy sets methodologies. A two level optimization model appropriate for planning decisions first on a regional or local level and then on a national level is proposed. A third phase involving coordination is appended. In particular, a fuzzy optimization model involving an adroit combination of fuzzy dynamic programming-type reasoning and a branch and bound-type search procedure is offered as a more utilitarian approach than its crisp equivalent. The performance of the algorithm is illustrated with an example.

Fuzzy sets and the modelling of physician decision processes, part II: fuzzy diagnosis decision models

Abstract Our concern is with the development of a more accurate mathematical model of physician diagnosis decision processes. Using the complete information net of matrices outputed from Part I, the various diagnoses processes—medical hypothesis, initial preliminary diagnoses, other preliminary diagnosis and final diagnoses, are modelled as fuzzy matrices. A general fuzzy decision model which could be particularized to each of the foregoing cases is then developed. Our general technique is based on fuzzy clustering theory with the criterion function being a modified form of the minimum Minkowski metric.

Measurement and valuation of a fuzzy mathematical model for medical diagnosis

Classical mathematical models for medical diagnosis which have been computerized are known to perform very poorly when compared to diagnoses made by the physician. Factors which contribute to their poor performance relate to the omission by these models of important information on the patient such as symptoms of past undiagnosed diseases which can only be vaguely recalled by the patient. Other deficiencies include failure to model the stage of development of the disease, and certain intrinsically fuzzy aspects of the pertinent information nets that are needed to develop a medical hypothesis. Models which attempted to remedy these shortcomings were developed and presented by the authors elsewhere. In this effort, we describe a study in which our fuzzy diagnosis models were computerized, validated and compared with a mock physician hypothesis as well as existing mathematical models. The example involved a medical hypothesis concerning a medical condition of valvular heart disease. The fuzzy sets together with their membership functions were measured using a distance criterion measure similar to the one employed by Kochen to quantify fuzzy adjectives in psychology. The results show that while there were discrepancies between the fuzzy models and the physicians hypotheses, the models performance was vastly superior to that of existing mathematical models. While the fuzzy models did well in naming the diseases, they performed less satisfactorily in the specification of the disease stage development, a factor completely ignored by classical models. Improvement in measurement aspects and the choice of diseases possessing discriminant attributes will greatly increase the accuracy of these models. It is however shown that a fuzzy systems approach is not only practical but results in models of greater validity than those based on classical set theoretic approaches.

Fuzzy dynamic programming

We provide a brief review of basic problem classes and developments of fuzzy dynamic programming which is a promising tool for dealing with multistage decision making and optimization problems under fuzziness (under fuzzy constraints on decisions made and fuzzy goals on states attained). We discuss cases of a deterministic, stochastic, and fuzzy state transitions, and of the fixed and specified, implicitly given, fuzzy, and infinite termination times.

Fuzzy sets and the modelling of physician decision processes, part I: The initial interview-information gathering session

Abstract Most mathematical models of physician decision processes offered to date, especially those relative to diagnosis and patient treatment, suffer from the inability to incorporate all useful data on the patient. Pertinent information so neglected or poorly modelled relate to variables that are intrinsically fuzzy but which describe the patients health status. We present mathematical models based on fuzzy set theory for physician aided evaluation of a complete representation of information emanating from the initial interview including patient past history, present symptoms, signs observed upon physical examination, and results of clinical and diagnostic tests.

The imbedded state space approach to reducing dimensionality in dynamic programs of higher dimensions

By exploiting discontinuity properties of the maximal convolution it is possible to drastically reduce dimensionality in finite dynamic programs. In fact, we show how the search over the usual M-dimensional state space can be reduced to a one-dimensional search over an imbedded state space. The versatility of our approach is illustrated on a number of example problems.

Optimal clustering of fuzzy data via fuzzy dynamic programming

Abstract The problem of clustering fuzzy data occurs in a variety of scenarios and numerous algorithms for their treatment abound in classical and fuzzy systems literature. We consider an extension of the conventional dynamic programming model introduced by Bellman to the fuzzy case. Two fuzzy dynamic programming models are developed and converted into algo r performance of these algorithms is compared to two others based on heuristics. Application to the evaluation of fuzzy data generated in connection with non-point source water pollution control strategies is reported.

A fuzzy adaptive controller using reinforcement learning neural networks

The authors describe an adaptive controller for complex processes which is capable of learning effective control using process data and improving its control through online adaptation. The controller is applicable to processes with multivariable states and with uncertain or nonlinear dynamics for which analytical models or standard control algorithms are either impractical or cannot be derived. This controller performs a fuzzy discretization of the process state and control variable spaces, and implements fuzzy logic control rules as a fuzzy relation. The membership functions of the fuzzy discretization are adjusted online and the fuzzy control rules are learned using a performance measure as feedback reinforcement. The fuzzy discretization procedure employs a data compression technique permitting multivariable state vector inputs. The controller is implemented with neural networks. Simulation results for the controller applied to a simple dynamical system demonstrate its effectiveness.<<ETX>>

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 inflation. 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 numbers. In particular, a fuzzy set approach to economic life of an asset calculation as well as a finite-horizon single asset replacement problem with multiple challengers is discussed. Because the use of triangular fuzzy numbers provides a compromise between computational efficiency 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 (DP) 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.