Understanding Complexity in a Practical Combinatorial Problem Using Mathematical Programming and Constraint Programming
Optimization problems that are motivated by real-world settings are often complex to solve. Bridging the gap between theory and practice in this field starts by understanding the causes of complexity of each problem and measuring its impact in order to make better decisions on approaches and methods. The Job-Shop Scheduling Problem (JSSP) is a well-known complex combinatorial problem with several industrial applications. This problem is used to analyse what makes some instances difficult to solve for a commonly used solution approach – Mathematical Integer Programming (MIP) – and to compare the power of an alternative approach: Constraint Programming (CP). The causes of complexity are analysed and compared for both approaches and a measure of MIP complexity is proposed, based on the concept of load per machine. Also, the impact of problem-specific global constraints in CP modelling is analysed, making proof of the industrial practical interest of commercially available CP models for the JSSP.
KeywordsJob-shop scheduling problem Mathematical programming Constraint programming Global constraints Complexity
The first author was supported by grant SFRH/BD/103362/2014 from FCT - Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology). This work was also partially financed by the ERDF - European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme within project “POCI-01-0145-FEDER-006961”, and by National Funds through the FCT - Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) as part of project UID/EEA/50014/2013.
- 1.4COutreachProgram. CSP tutorial (2005), http://4c.ucc.ie/web/outreach/tutorial.html
- 6.H. Fisher, G.L. Thompson, Probabilistic learning combinations of local job-shop scheduling rules, in Industrial Scheduling, ed. by J.F. Muth, G.L. Thompson (Prentice-Hall, Englewood Cliffs, 1963), pp. 225–251Google Scholar
- 7.S. Lawrence, Resource constrained project scheduling: an experimental investigation of heuristic scheduling techniques (supplement), in Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, Pennsylvania (1984)Google Scholar
- 13.W.J. van Hoeve, Introduction to constraint programming, in ACP Summer School on Theory and Practice of Constraint Programming, September 24–28, 2012, Wrocław, Poland (2012)Google Scholar
- 14.T. Yamada, R. Nakano, A genetic algorithm applicable to large-scale job-shop instances, in PPSN’2 Proceedings of the 2nd International Workshop on Parallel Problem Solving from Nature, ed. by R. Manner, B. Manderick (1992), pp. 281–290Google Scholar