Abstract
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.
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- 1.
Instances ft06, ft10, abz5, abz6, la01–la05, la16–la20, orb01–orb10.
- 2.
Instances in which the MIP solver was unable to prove optimality within the 30 min time limit: abz7–abz9, ft20, la06–la15, la21–la40, swv01–swv20, yn1–yn4.
- 3.
This table also repeats the MIP results in order to facilitate the comparison instance by instance.
- 4.
I.e., there existed a delta on the objective function value given by \(\varDelta = (\text {ObjValue}_{CP}-\text {ObjValue}_{MIP})/ \text {ObjValue}_{MIP}\).
- 5.
Since the number of variables and constraints is different in the two models, size is herein considered as the number of jobs multiplied by the number of machines.
- 6.
MIP model: proven by optimality gap. CP Model: the only way for a CP program to prove optimality is by exploring all possible solutions; therefore, if the solver stops before the time limit, it means that optimality was proven.
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Acknowledgements
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.
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Oliveira, B.B., Carravilla, M.A. (2018). Understanding Complexity in a Practical Combinatorial Problem Using Mathematical Programming and Constraint Programming. In: Vaz, A., Almeida, J., Oliveira, J., Pinto, A. (eds) Operational Research. APDIO 2017. Springer Proceedings in Mathematics & Statistics, vol 223. Springer, Cham. https://doi.org/10.1007/978-3-319-71583-4_19
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