Skip to main content

An ant colony optimization approach for the proportionate multiprocessor open shop


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Availability of data and materials

The data that support the findings of this study are available from the corresponding author on reasonable request.


  1. Abdelmaguid TF (2014) A hybrid PSO-TS approach for proportionate multiprocessor open shop scheduling. In: 2014 IEEE international conference on industrial engineering and engineering management. pp 107–111

  2. Abdelmaguid TF (2020a) Scatter search with path relinking for multiprocessor open shop scheduling. Comput Ind Eng 141:106292

    Article  Google Scholar 

  3. Abdelmaguid TF (2020b) Bi-objective dynamic multiprocessor open shop scheduling: an exact algorithm. Algorithms 13(3):74

    MathSciNet  Article  Google Scholar 

  4. Abdelmaguid TF, Shalaby MA, Awwad MA (2014) A tabu search approach for proportionate multiprocessor open shop scheduling. Comput Optim Appl 58(1):187–203

    MathSciNet  Article  Google Scholar 

  5. Adak Z, Arıoğlu Akan MÖ, Bulkan S (2020) Multiprocessor open shop problem: literature review and future directions. J Comb Optim 40(2):547–569

    MathSciNet  Article  Google Scholar 

  6. Azadeh A, Farahani MH, Torabzadeh S, Baghersad M (2014) Scheduling prioritized patients in emergency department laboratories. Comput Methods Programs Biomed 117(2):61–70

    Article  Google Scholar 

  7. Azadeh A, Goldansaz S, Zahedi-Anaraki A (2016) Solving and optimizing a bi-objective open shop scheduling problem by a modified genetic algorithm. Int J Adv Manuf Technol 85:1603–1613

    Article  Google Scholar 

  8. Bai D, Zhang ZH, Zhang Q (2016) Flexible open shop scheduling problem to minimize makespan. Comput Oper Res 67:207–215

    MathSciNet  Article  Google Scholar 

  9. Behnamian J, Memar Dezfooli S, Asgari H (2021) A scatter search algorithm with a novel solution representation for flexible open shop scheduling: a multi-objective optimization. J Supercomput.

    Article  Google Scholar 

  10. Blum C, Dorigo M (2004) The hyper-cube framework for ant colony optimization. IEEE Trans Syst Man Cybern Part B Cybern 34(2):1161–1172

    Article  Google Scholar 

  11. Den Besten M, Stützle T, Dorigo M (2000) Ant colony optimization for the total weighted tardiness problem. In: International conference on parallel problem solving from nature. Springer, Berlin, pp 611–620

  12. Dorigo M, Gambardella LM (1997) Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput 1(1):53–66

    Article  Google Scholar 

  13. Dorigo M, Maniezzo V, Colorni A (1996) The ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B 26:29–41

    Article  Google Scholar 

  14. Dorigo M, Stützle T (2004) Ant colony optimization. MIT Press, Cambridge

    Book  Google Scholar 

  15. Goldansaz SM, Jolai F, Anaraki AHZ (2013) A hybrid imperialist competitive algorithm for minimizing makespan in a multi-processor open shop. Appl Math Model 37(23):9603–9616

    MathSciNet  Article  Google Scholar 

  16. Gonzalez T, Sahni S (1976) Open shop scheduling to minimize finish time. J ACM JACM 23(4):665–679

    MathSciNet  Article  Google Scholar 

  17. Graham RL, Lawler EL, Lenstra JK, Kan AR (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 5:287–326

    MathSciNet  Article  Google Scholar 

  18. Matta ME (2009) A genetic algorithm for the proportionate multiprocessor open shop. Comput Oper Res 36(9):2601–2618

    MathSciNet  Article  Google Scholar 

  19. Matta ME, Elmaghraby SE (2010) Polynomial time algorithms for two special classes of the proportionate multiprocessor open shop. Eur J Oper Res 201(3):720–728

    MathSciNet  Article  Google Scholar 

  20. Mao W (1995) Multi-operation multi-machine scheduling. In: International conference on high-performance computing and networking. Springer, Berlin, pp 33–38

  21. Merkle D, Middendorf M (2003) Ant colony optimization with global pheromone evaluation for scheduling a single machine. Appl Intell 18(1):105–111

    Article  Google Scholar 

  22. Merkle D, Middendorf M, Schmeck H (2000) Pheromone evaluation in ant colony optimization. In: 2000 26th annual conference of the IEEE industrial electronics society. IECON 2000. 2000 IEEE international conference on industrial electronics, control and instrumentation. 21st century technologies, vol 4. IEEE, pp 2726–2731

  23. Merkle D, Middendorf M, Schmeck H (2002) Ant colony optimization for resource-constrained project scheduling. IEEE Trans Evol Comput 6(4):333–346

    Article  Google Scholar 

  24. Naderi B, Ghomi SF, Aminnayeri M, Zandieh M (2011) Scheduling open shops with parallel machines to minimize total completion time. J Comput Appl Math 235(5):1275–1287

    MathSciNet  Article  Google Scholar 

  25. Stützle T, Hoos HH (2000) MAX–MIN ant system. Futur Gener Comput Syst 16(8):889–914

    Article  Google Scholar 

  26. Zhang J, Wang L, Xing L (2019) Large-scale medical examination scheduling technology based on intelligent optimization. J Comb Optim 37(1):385–404

    MathSciNet  Article  Google Scholar 

Download references


No funds, grants, or other support were received.

Author information




All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by ZA, MÖA and SB. The first draft of the manuscript was written by ZA and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Zeynep Adak.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Adak, Z., Arıoğlu, M.Ö. & Bulkan, S. An ant colony optimization approach for the proportionate multiprocessor open shop. J Comb Optim (2021).

Download citation


  • Proportionate multiprocessor open shop
  • Ant colony optimization
  • Scheduling
  • Makespan
  • Implicit stage permutation representation