Abstract
Two new mixed-integer linear programming (MILP) models for the regular permutation flowshop problem, called TBA and TS3, are derived using a combination of JAML (job-adjacency, machine-linkage) diagrams and variable substitution techniques. These new models are then compared to the incumbent best MILP models (Wilson, WST2, and TS2) for this problem found in the flowshop sequencing literature. We define the term best to mean that a particular model or set of models can solve a common set of test flowshop problems in significantly less time than other competing models. In other words, the two new MILP models (TBA and TS3) become the challengers to the current incumbent best models (Wilson, WST2, TS2.). Both new models are shown to require less time, on average, than the current best models for solving this set of problems; and the TS3 model is shown to solve these problems in statistically significantly less time than the other four models combined.
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Acknowledgements
The authors appreciated the useful comments from anonymous referees. Any remaining errors are the authors’ responsibility. Further, this research was partially supported by the awarding of a C. David Billings Faculty Fellowship, University of Alabama in Huntsville to each of the authors: Stafford (2003–2004), and Tseng (2004–2005).
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†Sadly, Ted Stafford passed away on 9 June 2007. This paper is dedicated to his memory.
Appendix
Appendix
The three Wagner family MILP models are presented here because they are used for comparative evaluations of the new models in this paper, and because the new models may be derived from the Wilson model. All three existing MILP models use Equations (2) and (3) in the main text above to control the assignment of jobs to positions in the production sequence. They differ in the description of JAML constraints. The reader is referred to Stafford (1988), Wilson (1989), and Stafford et al (2005) for a complete explanation of these three models.
The WST2 model
The WST2 model presented here is different than the earlier developed WST model of Stafford and Tseng (2002), The WST model did not enforce the initial condition that all jobs are processed on machine 1 without any in-sequence machine idleness. WST2 is thus equivalent to the Wilson model which does enforce this initial condition.
The JAML constraints of the WST2 model are as follows:
The objective function for makespan for this model may be stated as:
The following additional equation facilitates the development of an expression for mean flowtime as the optimizing criterion for the WST2 model:
The objective function for minimizing mean flowtime is then:
The WST2 model may then be summarized as:
Minimize (A.6) or (A.8), subject to: (2), (3), (A.1), (A.2), (A.3), (A.4), (A.5), and (A.7)
Equation (A.7) may be omitted if makespan is the sole performance measure under consideration, or if the second version of Equation (A.8) is used for minimizing mean flowtime.
The Wilson model
Instead of the JAML constraints of the WST2 model, Wilson (1989) used sets of equations based on the start times, B rj 's, of the N jobs on the M machines. This eliminates the need for the X rj (machine idle time) and Y rj (job idle time) variables that were used in the WST2 model. The JAML constraints of the Wilson model are as follows:
In the Wilson model, the flowtime for each job is simply the start time of that job on the last machine plus that job's processing time on the last machine. Thus for the job in position j in the sequence, the flowtime may be expressed as:
Makespan, the flowtime of the last job in the processing sequence, then may be expressed as:
Substituting Equation (A.14) into Equation (5) yields the following expression for mean flowtime for the Wilson model:
The Wilson model may be summarized as:
Minimize (A.15) or (A.16), subject to (2), (3), (A.9), (A.10), (A.11), (A.12), and (A.13).
The TS2 model
The TS2 MILP model for the regular flowshop is based on an earlier model with the same name that was developed by Tseng and Stafford (2001) for the SDST flowshop problem. This model uses the job ending or completion time approaches of Srikar and Ghosh (1986) and Stafford and Tseng (1990), which also eliminates the need for the X rj and Y rj variables that were used in the WST2 model. The JAML constraints of the TS2 model may be stated as follows:
For the TS2 model, makespan may be expressed as:
Likewise, mean flowtime may be expressed as:
The TS2 model may then be summarized as:
Minimize (A.22) or (A.23), subject to (2), (3), (A.17), (A.18), (A.19), (A.20), and (A.21).
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Tseng, F., Stafford, E. New MILP models for the permutation flowshop problem. J Oper Res Soc 59, 1373–1386 (2008). https://doi.org/10.1057/palgrave.jors.2602455
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DOI: https://doi.org/10.1057/palgrave.jors.2602455