Shu-Cherng Fang

Fuzzy data envelopment analysis (DEA): a possibility approach

Abstract Evaluating the performance of activities or organizations by traditional data envelopment analysis (DEA) models requires crisp input/output data. However, in real-world problems inputs and outputs are often imprecise. This paper develops DEA models using imprecise data represented by fuzzy sets (i.e., “fuzzy DEA” models). It is shown that fuzzy DEA models take the form of fuzzy linear programming which typically are solved with the aid of some methods to rank fuzzy sets. As an alternative, a possibility approach is introduced in which constraints are treated as fuzzy events. The approach transforms fuzzy DEA models into possibility DEA models by using possibility measures of fuzzy events (fuzzy constraints). We show that for the special case, in which fuzzy membership functions of fuzzy data are of trapezoidal types, possibility DEA models become linear programming models. A numerical experiment is used to illustrate the approach and compare the results with those obtained with alternative approaches.

Entropy Optimization and Mathematical Programming

Preface. 1. Introduction to Entropy and Entropy Optimization Principles. 2. Entropy Optimization Models. 3. Entropy Optimization Methods: Linear Case. 4. Entropy Optimization Methods: General Convex Case. 5. Entropic Perturbation Approach to Mathematical Programming. 6. Lp-Norm Perturbation Approach: A Generalization of Entropic Perturbation. 7. Extensions and Related Results. Bibliography. Index.

Generalized variational inequalities

This paper introduces and analyzes generalized variational inequalities. The most general existence theory is established, traditional coercivity conditions are extended, properties of solution sets under various monotonicity conditions are investigated, and a computational scheme is considered. Similar results can be obtained for generalized complementarity and fixed-point problems.

On the Convergence of a Population-Based Global Optimization Algorithm

In global optimization, a typical population-based stochastic search method works on a set of sample points from the feasible region. In this paper, we study a recently proposed method of this sort. The method utilizes an attraction-repulsion mechanism to move sample points toward optimality and is thus referred to as electromagnetism-like method (EM). The computational results showed that EM is robust in practice, so we further investigate the theoretical structure. After reviewing the original method, we present some necessary modifications for the convergence proof. We show that in the limit, the modified method converges to the vicinity of global optimum with probability one.

Solving fuzzy relation equations with a linear objective function

Abstract An optimization model with a linear objective function subject to a system of fuzzy relation equations is presented. Due to the non-convexity of its feasible domain defined by fuzzy relation equations, designing an efficient solution procedure for solving such problems is not a trivial job. In this paper, we first characterize the feasible domain and then convert the problem to an equivalent problem involving 0–1 integer programming with a branch-and-bound solution technique. After presenting our solution procedure, a concrete example is included for illustration purpose.

Solving nonlinear optimization problems with fuzzy relation equation constraints

Abstract An optimization model with a nonlinear objective function subject to a system of fuzzy relation equations is presented. Since the solution set of the fuzzy relation equations is in general a non-convex set, when it is not empty, conventional nonlinear programming methods are not ideal for solving such a problem. In this paper, a genetic algorithm (GA) is proposed. This GA is designed to be domain specific by taking advantage of the structure of the solution set of fuzzy relation equations. The individuals from the initial population are chosen from the feasible solution set and are kept within the feasible region during the mutation and crossover operations. The construction of test problems is also developed to evaluate the performance of the proposed algorithm.

Optimization of fuzzy relation equations with max-product composition

Abstract An optimization problem with a linear objective function subject to a system of fuzzy relation equations using max-product composition is considered. Since the feasible domain is non-convex, traditional linear programming methods cannot be applied. We study this problem and capture some special characteristics of its feasible domain and the optimal solutions. Some procedures for reducing the original problem are presented. The problem is transformed into a 0–1 integer program which is then solved by the branch-and-bound method. For illustration purpose, an example of the procedures is provided.

Linear programming with fuzzy coefficients in constraints

Abstract This paper presents a new method for solving linear programming problems with fuzzy coefficients in constraints. It is shown that such problems can be reduced to a linear semi-infinite programming problem. The relations between optimal solutions and extreme points of the linear semi-infinite program are established. A cutting plane algorithm is introduced with a convergence proof, and a numerical example is included to illustrate the solution procedure.

On the entropic regularization method for solving min-max problems with applications

Consider a min-max problem in the form of minxεXmax1≤i≤m{fi(x)}. It is well-known that the non-differentiability of the max functionF(x) ≡ max1≤i≤m{fi(x)} presents difficulty in finding an optimal solution. An entropic regularization procedure provides a smooth approximationFp(x) that uniformly converges toF(x) overX with a difference bounded by ln(m)/p, forp > 0. In this way, withp being sufficiently large, minimizing the smooth functionFp(x) overX provides a very accurate solution to the min-max problem. The same procedure can be applied to solve systems of inequalities, linear programming problems, and constrained min-max problems.

Fuzzy BCC Model for Data Envelopment Analysis

Fuzzy Data Envelopment Analysis (FDEA) is a tool for comparing the performance of a set of activities or organizations under uncertainty environment. Imprecise data in FDEA models is represented by fuzzy sets and FDEA models take the form of fuzzy linear programming models. Previous research focused on solving the FDEA model of the CCR (named after Charnes, Cooper, and Rhodes) type (FCCR). In this paper, the FDEA model of the BCC (named after Banker, Charnes, and Cooper) type (FBCC) is studied. Possibility and Credibility approaches are provided and compared with an α-level based approach for solving the FDEA models. Using the possibility approach, the relationship between the primal and dual models of FBCC models is revealed and fuzzy efficiency can be constructed. Using the credibility approach, an efficiency value for each DMU (Decision Making Unit) is obtained as a representative of its possible range. A numerical example is given to illustrate the proposed approaches and results are compared with those obtained with the α-level based approach.