# PRUNING SEARCH SPACE BY DOMINANCE RULES IN BEST FIRST SEARCH FOR THE JOB SHOP SCHEDULING PROBLEM

### María R. Sierra, Ramiro Varela

#### Abstract

Best-first graph search is a classic problem solving paradigm capable of obtaining exact solutions to optimization problems. As it usually requires a large amount of memory to store the effective search space, in practice it is only suitable for small instances. In this paper, we propose a pruning method, based on dominance relations among states, for reducing the search space. We apply this method to an A∗ algorithm that explores the space of active schedules for the Job Shop Scheduling Problem with makespan minimization. The A∗ algorithm is guided by a consistent heuristic and it is combined with a greedy algorithm to obtain upper bounds during the search process. We conducted an experimental study over a conventional benchmark. The results show that the proposed method is able to reduce both the space and the time in searching for optimal schedules so as it is able to solve instances with 20 jobs and 5 machines or 9 jobs and 9 machines. Also, the A∗ is exploited with heuristic weighting to obtain sub-optimal solutions for larger instances.

#### References

- Brucker, P. (2004). Scheduling Algorithms. Springer, 4th edition.
- Brucker, P., Jurisch, B., and Sievers, B. (1994). A branch and bound algorithm for the job-shop scheduling problem. Discrete Applied Mathematics, 49:107-127.
- Brucker, P. and Knust, S. (2006). Complex Scheduling. Springer.
- Carlier, J. and Pinson, E. (1989). An algorithm for solving the job-shop problem. Management Science, 35(2):164-176.
- Carlier, J. and Pinson, E. (1994). Adjustment of heads and tails for the job-shop problem. European Journal of Operational Research, 78:146-161.
- Dorndorf, U., Pesch, E., and Phan-Huy, T. (2000). Constraint propagation techniques for the disjunctive scheduling problem. Artificial Intelligence, 122:189- 240.
- Dorndorf, U., Pesch, E., and Phan-Huy, T. (2002). Constraint propagation and problem descomposition: A preprocessing procedure for the job shop problem.
- Annals of Operations Research, 115:125-142.
- Giffler, B. and Thomson, G. L. (1960). Algorithms for solving production scheduling problems. Operations Research, 8:487-503.
- Hart, P., Nilsson, N., and Raphael, B. (1968). A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. on Sys. Science and Cybernetics, 4(2):100-107.
- Korf, R. (2003). An improved algorithm for optimal binpacking. In Proceedings of the 13th International Conference on Artificial Intelligence (IJCAI03), pages 1252-1258.
- Korf, R. (2004). Optimal rectangle packing: New results. In Proceedings of the 14th International Conference on Automated Planning and Scheduling (ICAPS04), pages 132-141.
- Nazaret, T., Verma, S., Bhattacharya, S., and Bagchi, A. (1999). The multiple resource constrained project scheduling problem: A breadth-first approach. European Journal of Operational Research, 112:347-366.
- Nilsson, N. (1980). Principles of Artificial Intelligence. Tioga, Palo Alto, CA.
- Pearl, J. (1984). Heuristics: Intelligent Search strategies for Computer Problem Solving. Addison-Wesley.
- Pohl, I. (1973). The avoidance of relative catastrophe, heuristic competence, genuine dynamic weigting and computational issues in heuristic problem solving. In Proceedings of IJCAI73, pages 20-23.
- Sadeh, N. and Fox, M. S. (1996). Variable and value ordering heuristics for the job shop scheduling constraint satisfaction problem. Artificial Intelligence, 86:1-41.
- Sierra, M. and Varela, R. (2005). Optimal scheduling with heuristic best first search. Proceedings of AI*IA'2005, Lecture Notes in Computer Science, 3673:173-176.
- Varela, R. and Soto, E. (2002). Sheduling as heuristic search with state space reduction. Lecture Notes in Computer Science, 2527:815-824.

#### Paper Citation

#### in Harvard Style

R. Sierra M. and Varela R. (2008). **PRUNING SEARCH SPACE BY DOMINANCE RULES IN BEST FIRST SEARCH FOR THE JOB SHOP SCHEDULING PROBLEM** . In *Proceedings of the Third International Conference on Software and Data Technologies - Volume 1: ICSOFT,* ISBN 978-989-8111-51-7, pages 273-280. DOI: 10.5220/0001896102730280

#### in Bibtex Style

@conference{icsoft08,

author={María R. Sierra and Ramiro Varela},

title={PRUNING SEARCH SPACE BY DOMINANCE RULES IN BEST FIRST SEARCH FOR THE JOB SHOP SCHEDULING PROBLEM},

booktitle={Proceedings of the Third International Conference on Software and Data Technologies - Volume 1: ICSOFT,},

year={2008},

pages={273-280},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001896102730280},

isbn={978-989-8111-51-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Third International Conference on Software and Data Technologies - Volume 1: ICSOFT,

TI - PRUNING SEARCH SPACE BY DOMINANCE RULES IN BEST FIRST SEARCH FOR THE JOB SHOP SCHEDULING PROBLEM

SN - 978-989-8111-51-7

AU - R. Sierra M.

AU - Varela R.

PY - 2008

SP - 273

EP - 280

DO - 10.5220/0001896102730280