On the complexity of some fuzzy integer programs

TL;DR

This paper explores the computational complexity of fuzzy integer programming, extending classical results to cases where problem elements are imprecise and modeled with fuzzy logic, revealing new complexity insights.

Contribution

It provides novel complexity results for fuzzy integer programs, utilizing short generating functions to analyze problems with fuzzy elements.

Findings

New complexity classifications for fuzzy integer programming

Application of short generating functions to fuzzy optimization

Insights into the computational difficulty of fuzzy constraints

Abstract

Fuzzy optimization deals with the problem of determining 'optimal'solutions of an optimization problem when some of the elements that appear in the problem are not precise. In real situations it is usual to have information, in systems under consideration, that is not exact. Thisimprecision can be modeled in a fuzzy environment. Zadeh (1965) analyzed systems of logic that permit truth values between zero and one instead of the classical binary true-false logic. In this framework, satisfying a certain condition means to evaluate how close are the elements involved to the complete satisfaction. Then, each element in a 'fuzzy set' is coupled with a value in [0,1] that represents the membership level to the set. In linear programming some or all the elements that describe a problem may be considered fuzzy: objective function, right-hand side vector or constraint matrix, the notion of optimality (ordering over the feasible solutions), the level of satisfaction of the constraints, etc. Moreover, in integer programs, the integrality constraints may be seen as fuzzy constraints. Here, we present new complexity results about linear integer programming where some of its elements are considered fuzzy using previous results on short generating functions for solving multiobjective integer programs.