Hans-Jürgen Zimmermann

Fuzzy programming and linear programming with several objective functions

Abstract In the recent past numerous models and methods have been suggested to solve the vectormaximum problem. Most of these approaches center their attention on linear programming problems with several objective functions. Apart from these approaches the theory of fuzzy sets has been employed to formulate and solve fuzzy linear programming problems. This paper presents the application of fuzzy linear programming approaches to the linear vectormaximum problem. It shows that solutions obtained by fuzzy linear programming are always efficient solutions. It also shows the consequences of using different ways of combining individual objective functions in order to determine an “optimal” compromise solution.

Latent connectives in human decision making

Abstract The interpretation of a decision as the intersection of fuzzy sets, computed by applying either the minimum or the product operator to the membership functions of the fuzzy sets concerned implies that there is no compensation between low and high degrees of membership. If, on the other hand, a decision is defined to be the union of fuzzy sets, represented by the maximum or algebraic sum of the degrees of membership, full compensation is assumed. Managerial decisions hardly even represent any of these extremes. The aggregation of subjective categories in the framework of human decisions or evaluations almost always shows some degree of compensation. This indicates that human beings partially are using non-verbal aggregation procedures which do not correspond to the verbal and logical connectives ‘and’ and ‘or’. The results of our experiments support the hypothesis that people often use compensatory procedures. Several well-known operators are tested. However, they do not predict our data very well. Therefore a new class of operators is suggested which varies with respect to a parameter of compensation. Our data do confirm this concept.

Fuzzy Mathematical Programming

Abstract Fuzzy linear programming (FLP) was originally suggested to solve problems which could be formulated as LP-models, the parameters of which, however, were fuzzy rather than crisp numbers. It has turned out in the meantime that FLP is also well suited to solve LP-problems with several objective functions. FLP belongs to goal programming in the sense that implicitly or explicitly aspiration levels have to be defined at which the membership functions of the fuzzy sets reach their maximum or minimum. Main advantages of FLP are, that the models used are numerically very efficient and that they can in many ways be well adopted to different decision behaviors and contexts.

An application-oriented view of modeling uncertainty

Abstract Uncertainty is involved in many real phenomena. Whether one considers uncertainty explicitly when modeling such a phenomenon is one of the modeling decisions, the result of which will depend on the context. If, however, the modeler decides to consider uncertainty, he or she will have to select the method for modeling it. Some scientists claim that one theory, e.g. probability theory, is sufficient to model all kinds of uncertainty. Here it is claimed, however, that the choice of the appropriate method is context dependent and an approach is suggested to determine context-dependently a suitable method to model uncertainty.

Applications of fuzzy set theory to mathematical programming

Mathematical programming is one of the areas to which fuzzy set theory has been applied extensively. Primarily based on Bellman and Zadehs model of decision in fuzzy environments, models have been suggested which allow flexibility in constraints and fuzziness in the objective function in traditional linear and nonlinear programming, in integer and fractional programming, and in dynamic programming. These models in turn have been used to offer computationally efficient approaches for solving vector maximum problems. This paper surveys major models and theories in this area and offers some indication on future developments which can be expected.

On the suitability of minimum and product operators for the intersection of fuzzy sets

Abstract In many papers concerning fuzzy set theory it is assumed that the membership or an element in the intersection of two or more fuzzy sets is given by the minimum of product of the corresponding membership values. To use these operators in modelling aspects of the real world, such as decision making, however, it is necessary to prove their appropriateness empirically. The main question of this study is whether people rating the membership of objects in the intersection of two fuzzy sets behave in accordance with one of these models. An important problem in answering this question is how to measure membership which seems to have the characteristics of an absolute scale. No measurement structure is available at present, but a practical method for scaling is suggested. The results of our experiments indicate that neither the product nor the minimum fit the data sufficiently well, but the latter seems to be preferable.

Fuzzy sets theory and applications

1: Some theoretical Aspects.- 1.1 Mathematics and fuzziness.- 1.2 Radon-Nikodym Theorem for fuzzy set-valued measures.- 1.3 Construction of a probability distribution from a fuzzy information.- 1.4 Convolution of fuzzyness and probability.- 1.5 Fuzzy sets and subobjects.- 2: From theory to applications.- 2.1 Outline of a theory of usuality based on fuzzy logic.- 2.2 Fuzzy sets theory and mathematical programming.- 2.3 Decisions with usual values.- 2.4 Support logic programming.- 2.5 Hybrid data - various associations between fuzzy subsets and random variables.- 2.6 Fuzzy relation equations : methodology and applications.- 3: Various particular applications.- 3.1 Multi criteria decision making in crisp and fuzzy environments.- 3.2 Fuzzy subsets applications in O.R. and management.- 3.3 Character recognition by means of fuzzy set reasoning.- 3.4 Computerized electrocardiography and fuzzy sets.- 3.5 Medical applications with fuzzy sets.- 3.6 Fuzzy subsets in didactic processes.

Sensitivity analysis in fuzzy linear programming

Abstract In this paper we propose a certain interpretation of a partially fuzzy LP-Problem. This LP-is dualized and the corresponding variables of the Dual are analyzed and an economic interpretation is given.

Fuzzy global optimization of complex system reliability

The problem of optimizing the reliability of complex systems has been modeled as a fuzzy multi-objective optimization problem. Three different kinds of optimization problems: reliability optimization of a complex system with constraints on cost and weight; optimal redundancy allocation in a multistage mixed system with constraints on cost and weight; and optimal reliability allocation in a multistage mixed system with constraints on cost, weight, and volume have been solved. Four numerical examples have been solved to demonstrate the effectiveness of the present methodology. The influence of various kinds of aggregators is also studied. The inefficiency of the noncompensatory min operator has been demonstrated. One of the well-known global optimization meta-heuristics, the threshold accepting, has been invoked to take care of the optimization part of the model. A software has been developed to implement the above model. The results obtained are encouraging.

Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems

Abstract In this paper, several kinds of possibility distributions of fuzzy variables are studied in possibilistic linear programming problems to reflect the inherent fuzziness in fuzzy decision problems. Interval and triangular possibility distributions are used to express the non-interactive cases between the fuzzy decision variables, and exponential possibility distributions are used to represent the interrelated cases. Possibilistic linear programming problems based on exponential possibility distributions become non-linear optimization problems. In order to solve optimization problems easily, algorithms for obtaining center vectors and distribution matrices in sequence are proposed. By the proposed algorithms, the possibility distribution of fuzzy decision variables can be obtained.