Solution approaches to the three-index assignment problem

Abstract

This paper delves into the axial Three-index Assignment Problem (3IAP), alternatively known as the Multi-dimensional Assignment Problem, defined as an extension of the classical two-dimensional assignment problem. The 3IAP entails allocating $n$  tasks to $n$ machines in $n$  factories, ensuring one task is completed by one machine in one factory at a minimum total cost. This combinatorial optimization problem is classified as \emph{NP}-hard due to its inherent complexity and being the subject of much scholarly research and investigation.                      The study employs an algorithmic approach to devise rapid and effective solutions for the 3IAP. A new heuristic Greedy-style Procedure (GSP) is introduced for solving the 3IAP, achieving feasible solutions within polynomial time. Particular configurations of cost matrices enable us to reach quality solutions. Examining tie-cases and matrix ordering unveiled innovative variants. Further investigation of cost matrix attributes facilitates the development of two new heuristic categories, offering optimal or nearly optimal solutions for the 3IAP. Extensive numerical experiments validate the effectiveness of the heuristics, generating quality solutions in a short computational time. Furthermore, we implement two potent methods using optimization solvers, achieving optimal solutions for the 3IAP within competitive CPU times.