ELSEVIER European Journal of Operational Research 83 (1995) 581-593 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology Three models of fuzzy integer linear programming F. Herrera *, J.L. Verdegay Department of Computer Science and Artificial Intelligence, University of Granada, 18071 Granada, Spain Received May 1992; revised June 1993 Abstract In this paper we study some models for dealing with Fuzzy Integer Linear Programming problems which have a certain lack of precision of a vague nature in their formulation. We present methods to solve them with either fuzzy constraints, or fuzzy numbers in the objective function or fuzzy numbers defining the set of constraints. These methods are based on the representation theorem and on fuzzy number ranking methods: Keywords: Integer Programming; Fuzzy constraints; Fuzzy numbers 1. Introduction Integer Linear Programming (ILP) problems have an outstanding relevance in many fields, such as those related to artificial intelligence, operations research, etc. They are especially important for representing and reasoning with propositional knowledge. Thus, the use of Mathematical Programming (MP) techniques for treating propositional logic is useful. In particular, several research efforts have involved the use of MP as a tool for modeling and performing deductive reasoning. An arbitrary system of rules can be represented and solved as an Integer Linear Program. The applications of Integer Programming to logic lead to new algorithms for inference in Knowledge-Based-Systems [13,15,20,23]. A classical ILP problem can be written as follows: max z=cx (1) s.t. ~ aijx j <_bi, i ~ M = { 1 .... , m}, yeN xi>_O , j~N={1 ..... n}, xj~N, j~N, where N is the set of integer numbers, c e ~n and aij, b i ~ ~, i ~ M, j ~ N. * This research has been supported by DGICYT PB92-933. * Corresponding author. 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0377-2217(93)E0338-X