The Ant Colony Optimization For Solving Multi-Objective Assignment Problems Under Uncertainty

Abstract

This study investigates the effectiveness of the Triangular Distribution (TD) and Ant Colony Optimization Algorithm (ACOA) in solving Multi-Objective Assignment Problems (MOAPs) under uncertainty. ACOA is a well-known biologically inspired metaheuristic algorithm which is widely used for solving large scale optimization problems. Its extension, the Multi-Objective Ant Colony Optimization Algorithm (MOACOA), provides a probabilistic framework for identifying near-optimal solutions in MOAPs. In real-world assignment problems, Assignment Costs (ACs) are often uncertain and cannot be precisely determined due to limited or incomplete information. To address the uncertainty, this study assumes that uncertain ACs follow a TD characterized by optimistic, pessimistic, and most probable values. Monte Carlo simulation is employed to estimate the expected assignment costs, allowing the uncertain problem to be deterministic equivalent. The resulting deterministic model is solved considering each objective separately using ACOA, yielding the positive ideal solution and the negative ideal solution for each objective. To handle multiple objectives simultaneously, a Fuzzy Exponential Membership Function (FEMF) is defined for each objective. The overall performance of each assignment is evaluated by aggregating the FEMF values across all objectives. The results demonstrate that variations in the shape parameter of the FEMF significantly influence the satisfaction levels of individual objectives. The obtained solutions consistently align with the preferences of the Decision Maker (DM). When a particular assignment does not meet expectations, alternative solutions can be generated by adjusting the shape parameters. The combined TD and ACOA framework effectively produce multiple reliable assignment alternatives and illustrates how different shape parameters impact objective trade-offs and convergence behavior. Overall, the proposed approach successfully identifies optimal assignments that reflect the DM’s shape parameter under uncertainty. Without loss of generality, the algorithm is executed 100 times to compare the performance of the proposed ACOA with that of the Genetic Algorithm for a range of 1 to 10 shape parameter values. In general, the findings validate the capability of the proposed ACOA algorithm with triangular random parameters and exponential membership functions to reach near optimal solutions to multi-objective uncertain assignment problems. Different from prior studies that largely concentrate on either deterministic or fuzzy assignment problems, the proposed study focuses on uncertain assignment costs following TD, ACOA, and FEMFs to optimize multiple conflicting objectives, thereby addressing the limited research available with uncertain MOAPs along with decision preferences based on shape parameters. Future research can focus on metaheuristic hybrid approach to solve this problem to reach more stable solution with faster convergence.