Binary fuzzy linear programming problems: a new solution

Abstract

This paper presents a novel solution approach for binary fuzzy linear programming problems (BFLPP) where both the objective function and constraints are defined using triangular fuzzy numbers. The core contribution is a method that preserves the fuzzy nature of the problem throughout the entire solution process, in contrast to conventional defuzzification or fuzzy ranking techniques that often convert fuzzy parameters into crisp values prematurely. By maintaining fuzziness, the proposed approach yields solutions that are more consistent with the inherent uncertainty of real-world decision-making scenarios, offering a more realistic and robust framework.

The methodology integrates Kerre’s adapted method with a fuzzy branch-and-bound algorithm. At each node of the branch-and-bound tree, a linear relaxation of the fuzzy problem is solved using a newly developed fuzzy simplex method, also based on Kerre’s adapted method. This allows the fuzzy constraints to be handled directly without eliminating their imprecise nature. The approach effectively transforms fuzzy inequality constraints into a set of crisp mixed-integer constraints through systematic modeling of fuzzy relations, and then linearizes the resulting model to make it tractable for the fuzzy branch-and-bound procedure.

Key achievements of the research include the development of a fully fuzzy solution framework that avoids early defuzzification, thus better capturing uncertainty. The proposed fuzzy simplex and branch-and-bound algorithms are shown to be effective in solving BFLPP while retaining fuzzy information throughout the optimization process. Numerical examples—including a fuzzy investment selection problem and a constrained shortest path problem—demonstrate that the method outperforms existing fuzzy ranking and credibility programming approaches, yielding superior objective function values and more reliable solutions.

In conclusion, the paper provides a significant advancement in fuzzy integer programming by offering a direct, fuzzy-preserving solution technique for binary fuzzy linear programs. It successfully bridges a gap in the literature where most methods rely on defuzzification, thereby enhancing both the theoretical and practical applicability of fuzzy optimization in areas such as capital budgeting, logistics, and resource allocation under uncertainty. Future research may explore the integration of heuristic and metaheuristic algorithms to address larger-scale instances and extend the approach to other types of fuzzy integer and multi-objective problems.