A Multi-Objective Ant-Colony Algorithm for Permutation Flowshop Scheduling to Minimize the Makespan and Total Flowtime of Jobs

Abstract

The problem of scheduling in permutation flowshops is considered with the objectives of minimizing the makespan and total flowtime of jobs. A multi-objective ant-colony algorithm (MOACA) is proposed. The salient features of the proposed multi-objective ant-colony algorithm include the consideration of two ants (corresponding to the number of objectives considered) that make use of the same pheromone values in a given iteration; use of a compromise objective function that incorporates a heuristic solution’s makespan and total flowtime of jobs as well as an up per bound on the makespan and an upper bound on total flowtime of jobs, coupled with weights that vary uniformly in the range [0, 1]; increase in pheromone intensity of trails by reckoning with the best solution with respect to the compromise objective function; and updating of pheromone trail intensities being done only when the ant-sequence’s compromise objective function value is within a dynamically updated threshold level with respect to the best-known compromise objective function value obtained in the search process. In addition, every generated ant sequence is subjected to a concatenation of improvement schemes that act as local search schemes so that the resultant compromise objective function is improved upon. A sequence generated in the course of the ant-search process is con sidered for updating the set of heuristically non-dominated solutions. We consider the benchmark flowshop scheduling problems proposed by Taillard (1993), and solve them by using twenty variants of the MOACA. These variants of the MOACA are obtained by varying the values of parameters in the MOACA and also by changing the concatenation of improvement schemes. In order to benchmark the proposed MOACA, we rely on two recent research reports: one by Minella et al. (2008) that re ported an extensive computational evaluation of more than twenty existing multi-objective algorithms available up to 2007; and a study by Framinan and Leisten (2007) involving a multi-objective iterated greedy search algorithm, called MOIGS, for flowshop scheduling. The work by Minella concluded that the multi-objective simulated annealing algorithm by Varadharajan and Rajendran (2005), called MOSA, is the best performing multi-objective algorithm for permutation flowshop scheduling. Framinan and Leisten found that their MOIGS performed better than the MOSA in terms of generating more heuristically non-dominated solutions. They also obtained a set of heuristically non-dominated solutions for every benchmark problem instance provided by Taillard (1993) by consolidating the solutions obtained by them and the solutions reported by Varadharajan and Rajendran. This set of heuristically non-dominated solutions (for every problem instance, up to 100 jobs, of Taillard’s benchmark flowshop scheduling problems) forms the reference or benchmark for the present study. By considering this set of heuristically non-dominated solutions with the solutions given by the twenty variants of the MOACA, we form the net heuristically non-dominated solutions. It is found that most of the non-dominated solutions on the net non-dominated front are yielded by the variants of the MOACA, and that in most problem instances (especially in problem instances exceeding 20 jobs), the variants of the MOACA con tribute more solutions to the net non-dominated front than the corresponding solutions evolved as benchmark solutions by Framinan and Leisten, thereby proving the effectiveness of the MOACA. We also pro vide the complete set of heuristically non-dominated solutions for the ninety problem instances of Taillard (by consolidating the solutions obtained by us and the solutions obtained by Framinan and Leisten) so that researchers can use them as benchmarks for such research attempts.