An ant colony optimization approach for the proportionate multiprocessor open shop

Abstract

Multiprocessor open shop makes a generalization to classical open shop by allowing parallel machines for the same task. Scheduling of this shop environment to minimize the makespan is a strongly NP-Hard problem. Despite its wide application areas in industry, the research in the field is still limited. In this paper, the proportionate case is considered where a task requires a fixed processing time independent of the job identity. A novel highly efficient solution representation is developed for the problem. An ant colony optimization model based on this representation is proposed with makespan minimization objective. It carries out a random exploration of the solution space and allows to search for good solution characteristics in a less time-consuming way. The algorithm performs full exploitation of search knowledge, and it successfully incorporates problem knowledge. To increase solution quality, a local exploration approach analogous to a local search, is further employed on the solution constructed. The proposed algorithm is tested over 100 benchmark instances from the literature. It outperforms the current state-of-the-art algorithm both in terms of solution quality and computational time.

Appendix: Explanation about the optimality of solutions for 2-stage problem set

The test results in this paper for the 2-stage problem instances of Matta (2009) revealed makespan values equal to the lower bound in 16 of the instances, hence provably optimal solutions. For the other 9 instances, the results showed a deviation from the lower bound. However, the results of these 9 instances are also claimed here to give optimum makespan values. The rationale behind this claim is given in this appendix through a schematic explanation and no technical proofs are provided. Past results for the 2-stage problem instances reported in literature by Matta (2009), Abdelmaguid et al. (2014) and Abdelmaguid (2020a) match exactly the findings of this paper. This one-to-one correspondence between 4 different algorithms also supports that these common results are indeed the optimum ones.

The statement is presented on a single instance, S2-P5, while the same reasoning holds for all the other 2-stage instances. Shop features for S2-P5 are given in Table

Table 10 Shop features for the benchmark problem S2-P5

10. There are 29 jobs to be processed in both stages. To complete processing of the 29 jobs in stage 1, where there are only 17 machines in parallel, at least two time-blocks of length 13 (remind the proportionate property of the shop) are required. A block is defined as a $p_{i}$-length time period in a stage. To process all the jobs in stage 2, on the other hand, at least 3 blocks are required as there are only 12 machines in parallel. These minimal requirements cause a lower bound of 26 for the latest job completion time in stage 1 and 27 in stage 2. Thus, the overall lower bound for the makespan is 27 for the problem. There is no way to complete all jobs in both stages in a time less than 27. Block representations for the lower bounds are shown in Fig. 

Fig. 9

Representation of minimum number of blocks in each stage

9. B1-1 and B1-2 are the first and second blocks of stage 1, while B2-1, B2-2 and B2-3 are the three blocks of stage 2.

A close analysis of Fig. 9 reveals that any job scheduled at time blocks B1-1 or B1-2 cannot be scheduled at B2-2 since a job is allowed to be processed on a single machine at a time and preemption is not allowed. However, in a final schedule all the jobs should be processed in either of the two blocks in stage 1 (otherwise a third block in stage 1 would render a makespan of 39), making it impossible to schedule any job at the time block B2-2 in stage 2. To create a feasible schedule, the intersection of B2-2 with at least one of the blocks of stage 1 should be eliminated. One possible solution is to shift the position of blocks (thus leaving machines idle) in stage 1, since there is a 1 time-unit flexibility that does not cause the makespan exceed the lower bound of 27. However, a 1 time-unit shift would not eliminate the intersection with B2-2 for neither of the blocks. At least a 5 time-unit shift to the right is required in B1-2 to allow for a feasible schedule. This shift results in a new makespan of 31. Again, instead of stage 1, one can shift B2-2 in stage 2 to prevent the intersection. The lowest possible length of shift is 4 time-units to the right, resulting in a new makespan of 31. Since, the least increase in the overall lower bound of 27 is by one of those shifts, the makespan of 31 should be the optimum. Figures 

Fig. 10

A block shift resulting in the optimum makespan

10 and

Fig. 11

An alternative block shift resulting in the optimum makespan

11 show these optimal solutions (another optimum solution can be generated by shifting B2-1 to the right by 4 time-units and starting the schedule in stage 2 from time 5). The makespan of 31 is consistent with the best makespan value reported for S2-P5 in the test results.