MULTI-OBJECTIVE PROBLEM WITH MULTIPLE JOBS ASSIGNED TO A SINGLE MACHINE WITHIN AVAILABLE COST UNDER UNCERTAIN ENVIRONMENT

The assignment problem is a key challenge in optimization and operations research, finding applications in diverse real-world scenarios. The Hungarian method is a widely employed algorithm for solving this problem, especially in its balanced form. However, for unbalanced assignment problems, where tasks outnumber resources (or vice versa), an extension is necessary. One common approach introduces a dummy resource, but this may leave tasks unassigned. The Modified Hungarian method improves upon the standard algorithm for unbalanced problems, ensuring that all tasks are assigned to real resources. This is achieved by modifying the cost matrix and algorithm steps to accommodate additional tasks and resources. Triangular fuzzy numbers are discussed when exact parameter information is undefined, and fuzzy programming is applied to determine a compromise result. Incorporating cost and profit per resource, the Modified Hungarian algorithm addresses the problem of unspecified job allocations to a single machine by introducing a cost parameter for each machine. The methodology is demonstrated on a numerical example for better comprehension.