On the Complexity of Some Fuzzy Integer Programs

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. This imprecision can be modeled in a fuzzy environment. Zadeh [20] 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.