Computational results for the flowshop tardiness problem

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Abstract

This paper reports on computational experiments involving optimal solutions to the flowshop tardiness problem. Of primary interest was a generic approach: solutions were obtained using a spreadsheet-based, mixed-integer programming code. However, the results compare favorably with those from a specially-tailored branch and bound algorithm. The main implication is that hardware and software have developed to the point that generic tools may offer the best way to solve combinatorial problems in scheduling.

Highlights

► The flowshop tardiness problem was solved using a spreadsheet-based optimization approach. ► Solution performance was better than results for the best branch and bound procedure. ► The implication is that generic tools may often be competitive with specialized tools in solving complex sequencing problems.

Introduction

In this paper we address the minimization of total tardiness in the permutation flowshop. This problem has been the subject of research in the past, but progress on finding optimal solutions has been limited. The computational results summarized here are arguably the best yet reported. However, the more important point is that they have been achieved without a tailored solution algorithm. Instead, the optimization problem was formulated as a mixed-integer program and solved with spreadsheet-based optimization software. The implication is that hardware and software have advanced to the point that generic tools are becoming competitive when it comes to solving some difficult scheduling problems.

This trend has important consequences for scheduling practitioners. The opportunity to use generic tools to solve complex scheduling problems is a major advantage. For practical scheduling problems, it may not be necessary to search the literature for a highly specialized, state-of-the-art solution algorithm when a generic solution may work just fine. A generic solution approach (that is, a spreadsheet-based formulation together with a publicly available mixed-integer programming code) is usually more accessible than a highly specialized algorithm. Moreover, it may not be easy to determine what problem sizes are amenable to solution by state-of-the-art algorithms if published results are out of date. (They are typically not updated as hardware tools develop.) For a generic approach, however, it is not difficult to keep a sample integer programming formulation on hand and test its solution with each new generation of hardware or software. Thus, the main implication is that spreadsheet-based optimization is becoming a viable solution tool for scheduling applications.

This result also has implications for scheduling researchers. When specialized algorithms are developed for combinatorial scheduling problems, it makes sense to use mixed-integer programming as a benchmark solution procedure, especially for cases in which algorithm development has been limited. In an earlier era, scheduling problems were sometimes solved using integer programming in order to gain insight into integer programming methods. Now, finally, the opposite seems to be true: integer programming methods can reveal our ability to solve the scheduling problems themselves.

Section snippets

Minimizing total tardiness in the flowshop model

The flowshop model contains n jobs and m  2 machines (referred to as an n × m problem). Job j has a given processing time, pij, on machine i and a given due date, dj. Each job must visit the machines in the same machine order, and the machines can process at most one job at a time. As a result of scheduling decisions, job j achieves a completion time, Cj. Its tardiness is defined as Tj = max{0, Cj  dj}. The total tardiness in the schedule is ∑Tj, which is the objective to be minimized.

We consider

Progress in finding optimal solutions

Even the two-machine version of the flowshop tardiness problem is NP-hard, and for that reason, some research studies have addressed only that case. Other studies have investigated versions with more than two machines, where we would expect that solutions would be computationally even more difficult to obtain. The article by Vallada, Ruiz, and Minella (2008), which primarily covers heuristic methods, provides a careful review of previous research on the flowshop tardiness problem. Part of this

Parameters of the test data

Most computational studies of tardiness problems have used the tardiness factor and the due-date range as parameters to guide the generation of random test problems. This pair of parameters was first used in combination by Baker and Martin (1974) for the single-machine problem and has been used in various studies of tardiness problems ever since. In the flowshop tardiness problem, these same two factors were used in the studies cited above, although sometimes with different interpretations. We

Computational results

We followed the design of (TF, DDR) pairs used for example by Potts and Van Wassenhove (1985) and Szwarc, Grasso, and Della Croce (2001) by taking DDR = {0.2, 0.4, 0.6, 0.8, 1.0} and TF = {0.2, 0.4, 0.6, 0.8}. For a given problem size and for each possible combination of DDR and TF, we drew four random instances. We then found optimal solutions to each of those 80 instances using RSP and BB. We repeated that design for several problem sizes from 10 × 4 to 16 × 8. Our summary statistics are the average,

Summary and conclusions

This paper examined computational aspects of solving the flowshop tardiness problem. We compared the best specially-tailored algorithm in the literature with a generic, mixed-integer programming approach, using a spreadsheet-based platform in both cases. Our main observation is that the spreadsheet-based approach is at least competitive with (and usually better than) a specialized branch-and-bound approach. For the practitioner who faces a combinatorial scheduling problem, it may not be the

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    Chung, Flynn, and Kirca (2006) was able to solve problems up to 15 jobs and 2 machines, and 20 jobs and 8 machines. Most recently, Baker (2013) reports the results of a comparison with Chung et al.’s branch-and-bound algorithm and a commercial Excel add-in (RSP), where RSP was able to solve all the generated problems having 16 jobs with 4 and 8 machines, all of the 8-machine problems with 18 jobs except for one, and all of the 4-machine problems with 22 jobs except for two; within an hour of CPU time on a computer that has an Intel Core2 2.7 GHz processor with 8 GB of RAM. Heuristic algorithms construct solutions from scratch, based on priority rules such as earliest due date (EDD), minimum slack (SLACK), shortest processing times (SPT) and modified due date (MDD) (Vallada, Ruiz, & Minella, 2008).

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