Penalty method for fuzzy linear programming with trapezoidal numbers

In this paper we shall propose an algorithm for solving fuzzy linear programming problems with trapezoidal numbers using a penalty method. We will transform the problem of maximizing a function having trapezoidal fuzzy number values under some constraints into a deterministic multi-objective programming problem by penalizing the objective function for possible constraint violation. Furthermore, the obtained deterministic problem will have only unavoidable inequalities between trapezoidal fuzzy numbers parameters as constraints

Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics

On fuzzy input data and the worst scenario method

summary:In practice, input data entering a state problem are almost always uncertain to some extent. Thus it is natural to consider a set $\mathcal U_{\mathrm ad}$ of admissible input data instead of a fixed and unique input. The worst scenario method takes into account all states generated by $\mathcal U_{\mathrm ad}$ and maximizes a functional criterion reflecting a particular feature of the state solution, as local stress, displacement, or temperature, for instance. An increase in the criterion value indicates a deterioration in the featured quantity. The method takes all the elements of $\mathcal U_{\mathrm ad}$ as equally important though this can be unrealistic and can lead to too pessimistic conclusions. Often, however, additional information expressed through a membership function of $\mathcal U_{\mathrm ad}$ is available, i.e., $\mathcal U_{\mathrm ad}$ becomes a fuzzy set. In the article, infinite-dimensional $\mathcal U_{\mathrm ad}$ are considered, two ways of introducing fuzziness into $\mathcal U_{\mathrm ad}$ are suggested, and the worst scenario method operating on fuzzy admissible sets is proposed to obtain a fuzzy set of outputs

Reducing the number of membership functions in linguistic variables

Dissertation presented at Universidade Nova de Lisboa, Faculdade de Ciências e Tecnologia in fulfilment of the requirements for the Masters degree in Mathematics and Applications, specialization in Actuarial Sciences, Statistics and Operations ResearchThe purpose of this thesis was to develop algorithms to reduce the number of membership functions in a fuzzy linguistic variable. Groups of similar membership functions to be merged were found using clustering algorithms. By “summarizing” the information given by a similar group of membership functions into a new membership function we obtain a smaller set of membership functions representing the same concept as the initial linguistic variable. The complexity of clustering problems makes it difficult for exact methods to solve them in practical time. Heuristic methods were therefore used to find good quality solutions. A Scatter Search clustering algorithm was implemented in Matlab and compared to a variation of the K-Means algorithm. Computational results on two data sets are discussed. A case study with linguistic variables belonging to a fuzzy inference system automatically constructed from data collected by sensors while drilling in different scenarios is also studied. With these systems already constructed, the task was to reduce the number of membership functions in its linguistic variables without losing performance. A hierarchical clustering algorithm relying on performance measures for the inference system was implemented in Matlab. It was possible not only to simplify the inference system by reducing the number of membership functions in each linguistic variable but also to improve its performance

Approximate Membership Function Shapes of Solutions to Intuitionistic Fuzzy Transportation Problems

In this paper, proposing a mathematical model with disjunctive constraint system, and providing approximate membership function shapes to the optimal values of the decision variables, we improve the solution approach to transportation problems with trapezoidal fuzzy parameters. We further extend the approach to solving transportation problems with intuitionistic fuzzy parameters; and compare the membership function shapes of the fuzzy solutions obtained by our approach to the fuzzy solutions to full fuzzy transportation problems yielded by approaches found in the literature