Modelo del Negocio

Experiments were conducted on a 2.50 GHz Intel Core i7-4710 processor with 16 GB of memory running under Windows 8. RStudio 1.1.453 with R 3.5.1 was used to implement and solve the heuristics. The instances of Beezão et al. (2017) were used but had been modified for multiple non-identical parallel machines by adding machine specific processing and tool

instances and 60 instances for the large (|J| ≥ 50) instances. A total of 1440 small instances and 1440 large instances had been tested.

Table 2 summarizes the results and shows the percentage of finding the best known solution per heuristic. Overall, the mean CPU time was 0.05 seconds for the SPT and LPT heuristics and 0.47 seconds for the GI-heuristic. Even large instances could be solved below 1 second.

Table 2. Average percentage (%) of finding the best known solution for TS, TFT and Fmax.

Problem Type

SPT-based heuristic LPT-based heuristic GI-heuristic

TS TFT Fmax TS TFT Fmax TS TFT Fmax

Small 10.8 84.7 58.8 9.6 0.6 3.3 88.5 14.8 38.1

Large 3.5 48.8 44.4 0.2 0.1 0.0 96.3 51.1 55.6

Note: Boldface numbers indicate the maximum percentage among the heuristics.

The results confirm that the SPT-based heuristic performs well for minimizing TFT or Fmax.

The LPT-based heuristic performs worst for all objectives and the GI-heuristic performs best for minimizing TS and well for minimizing time-related objectives especially for large instances.

Yet, the SPT-based heuristic outperforms the GI-based heuristic in terms of solution quality for small problem instances.

5. Conclusion

This paper presents simple and fast heuristics for the SSP-NPM with timetabling. Small and large instances were solved for different objective functions. The steadiest performance is achieved by the GI-heuristic for minimizing the number of tool switches while the SPT heuristic works rather well for minimizing total flow time or makespan. Further investigations and statistics tests are required in order to investigate the influence of the problem instance specifications on the solution quality. The quality of the solutions seems to differ significantly for variations in the relation between processing time and tool switching time but also variation of the job-tool matrix density. Future research may integrate the tool setup time as reducing the tool switches becomes especially important when the switching time is significant in regard to the processing time of a job. Overall, the heuristics seem to provide promising initial solutions for further local search procedures or metaheuristics.

Heuristics for the Job Sequencing and Tool Switching Problem with Non-Identical Parallel Machines

References

Bard, J. F. 1988. A Heuristic for Minimizing the Number of Tool Switches on a Flexible Machine. IIE Transactions 20(4) 382–391.

Beezão, A. C., J.-F. Cordeau, G. Laporte, H. H. Yanasse. 2017. Scheduling identical parallel machines with tooling constraints. European Journal of Operational Research 257(3) 834–844.

Crama, Y., A. W. J. Kolen, A. G. Oerlemans, F. C. R. Spieksma. 1994. Minimizing the number of tool switches on a flexible machine. International Journal of Flexible Manufacturing Systems 6(1) 33–54.

Fathi, Y., K. W. Barnette. 2002. Heuristic procedures for the parallel machine problem with tool switches. International Journal of Production Research 40(1) 151–164.

Ghrayeb, O. A., N. Phojanamongkolkij, P. R. Finch. 2003. A mathematical model and heuristic procedure to schedule printed circuit packs on sequencers. International Journal of Production Research 41(16) 3849–3860.

Gökgür, B., B. Hnich, S. Özpeynirci. 2018. Parallel machine scheduling with tool loading: A constraint programming approach. International Journal of Production Research 54 1–17.

Laporte, G., J. J. Salazar-González, F. Semet. 2004. Exact algorithms for the job sequencing and tool switching problem. IIE Transactions 36(1) 37–45.

Özpeynirci, S., B. Gökgür, B. Hnich. 2016. Parallel machine scheduling with tool loading.

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Sarmadi, H., S. Gholami. 2011. Modeling of Tool Switching Problem in a Flexible Manufacturing Cell: with two or More Machines. G. Lee, ed. International Conference on Mechanical and Electrical Technology, 3rd, (ICMET-London 2011), Volumes 1–3. ASME, Three Park Avenue New York, NY 10016-5990, 2345–2349.

Tang, C. S., E. V. Denardo. 1988. Models Arising from a Flexible Manufacturing Machine, Part I: Minimization of the Number of Tool Switches. Operations Research 36(5) 767–777.

Van Hop, N., N. N. Nagarur. 2004. The scheduling problem of PCBs for multiple non-identical parallel machines. European Journal of Operational Research 158(3) 577–594.

An Iterated Local Search Procedure for the Job Sequencing and Tool Switching

Problem with Non-Identical Parallel Machines

Author Dorothea Calmels, University of Passau, Germany.

Under Review in European Journal of Operations Research.

Abstract

In this paper, a new generalisation of the uniform job sequencing and tool switching problem is presented by considering non-identical parallel machines, which differ in tool magazine capacity and setup times. Applications of the problem can be found in the metal working or semiconductor manufacturing industries when the jobs require different tool sets for processing. The paper provides a Mixed Integer Programming formulation for the job sequencing and tool switching problem with machine-dependent processing and tool switching times under the consideration of three conflicting objectives (minimizing the total flowtime, minimizing the makespan and minimizing the total number of tool switches). Different efficient construction heuristics and Iterated Local Search methods are proposed and evaluated, using 640 new and publicly available instances. The results of the construction heuristics, the Iterated Local Search schemes and a Mixed Integer Programming Solver are discussed.

The extensive computational experiments show the merit of the perturbation strategy in order to overcome local optima. It is shown that certain methods are more or less suitable depending on the objective function.

Reprinted as part of the Author Copyright Agreement as defined by Elsevier B.V.

(https://www.elsevier.com/about/policies/copyright).

Acknowledgement: This is a pre-copyedited version of a contribution sent to the European Journal of Operational Research 2020, published by Elsevier B.V. The definitive authenticated version may later be available online via https://www.journals.elsevier.com/european-journal-of-operational-research.

An Iterated Local Search Procedure for the Job Sequencing and Tool Switching Problem withNon-Identical Parallel Machines

1. Introduction

Modern manufacturing systems have to allow for an adjustable machine structure in order to respond to product changes and complex product requirements with the upcoming of ‘smart factory’ and ‘industry 4.0’. A manufacturing machine may be equipped with a tool magazine of limited capacity and may provide different functions such as milling, turning, or tool handling operations. In general, the number of tools that is required to process a variety of jobs on a machine is so large that it exceeds the tool magazine capacity, and hence tool switches - defined by removing a tool from its slot and inserting another tool in the free slot Tang and Denardo (1988) - become necessary. If the tools cannot be interchanged during job processing, then the switching time is particularly significant if it is relatively high compared to the processing time of a job. In most real manufacturing systems, component or tool switching is the most time-consuming process (Van Hop and Nagarur 2004), and must be avoided. The objective of the standard uniform job sequencing and tool switching problem (SSP) is, therefore, to find the best sequence of jobs for a given set of jobs that minimizes the number of tool switches on a single machine. This problem has been addressed in production and operations research literature for more than 30 years, with a sharp increase in the last 5 years (Calmels 2019).

In the following, the job sequencing and tool switching problem with non-identical parallel machines (SSP-NPM) is addressed which is better adapted to challenges in the modern production environment. More precisely, the SSP-NPM is defined by three interdependent problems (see Figure 1): (1) assigning a set of jobs to machines, (2) sequencing the allocated jobs on the machines, and (3) arranging the tool loading. In contrast to the single machine problem, time-related objectives such as minimizing the makespan or total flowtime become more important when considering multi-machine problems. As the machines have a limited tool capacity, the number of tools to switch and hence the tool setup time of a job are influenced by the job sequence. Non-identical machines imply that all the machines can execute the same operations but may have different magazine capacities and different tools, and the processing time of a job and the tool switching time need not be equal for all machines.

The main contributions of this work include the formulation of a mixed integer linear (MIP) model that allows for time-related objectives, and the development of several heuristic solution methods. The mathematical formulation presents a more realistic and general approach to existing SSP research. Moreover, the paper provides two new data sets for the SSP-NPM and examines the solution quality of different problem groups. The remainder of this article is as follows. Section 2 presents an overview of related work. A problem description and the mathematical formulation are given in Section 3, followed by presenting the heuristic approaches in section 4. The computational experiments and the analysis are presented in section 5. The final section 6 concludes the study.

Figure 1. Schematic layout of the problem environment.