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International Journal of Production
Research
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Bicriteria scheduling in a two-machine
permutation flowshop
B. M. T. Lin a & J. M. Wu b a
Department of Information and Finance Management , Institute of Information Management National Chiao Tung University Hsinchu 300 , Taiwan
b
Department of Information Management , National Chi Nan University , Nantou 545, Taiwan
Published online: 22 Feb 2007.
To cite this article: B. M. T. Lin & J. M. Wu (2006) Bicriteria scheduling in a two-machine permutation flowshop, International Journal of Production Research, 44:12, 2299-2312, DOI: 10.1080/00207540500446394
To link to this article: http://dx.doi.org/10.1080/00207540500446394
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International Journal of Production Research, Vol. 44, No. 12, 15 June 2006, 2299–2312
Bicriteria scheduling in a two-machine permutation flowshop
B. M. T. LIN*y and J. M. WUz
yDepartment of Information and Finance Management, Institute of Information Management National Chiao Tung University Hsinchu 300, Taiwan
zDepartment of Information Management, National Chi Nan University, Nantou 545, Taiwan
(Revision received October 2005)
In this paper we consider a production scheduling problem in a two-machine flowshop. The bicriteria objective is a linear combination or weighted sum of the makespan and total completion time. This problem is computationally hard because the special case concerning the minimization of the total completion time is already known to be strongly NP-hard. To find an optimal schedule, we deploy the Johnson algorithm and a lower bound scheme that was previously developed for total completion time scheduling. Computational experiments are presented to study the relative performance of different lower bounds. While the best known bound for the bicriteria problem can successfully solve test cases of 10 jobs within a time limit of 30 min, under the same setting our branch-and-bound algorithm solely equipped with the new scheme can produce optimal schedules for most instances with 30 or less jobs. The results demonstrate the convincing capability of the lower bound scheme in curtailing unnecessary branching during problem-solving sessions. The computational experience also suggests the practical significance and potential implications of this scheme.
Keywords: Flowshop; Makespan; Total completion time; Lower bound; Branch-and-bound algorithm
1. Introduction
Research on production scheduling seeks to develop systematic ways for allocating limited resources to tasks subject to specified requirements or constraints. In the scheduling literature, due to their practical significance and theoretical challenges, flowshop problems are among the most well-studied topics (Dudek et al. 1992, Reisman et al. 1997). Flowshops are widely adopted to describe the organizational operations process as well as inter-organizational relationships in industrial networks. Flowshop scheduling research was inspired by Johnson’s (1954) seminal work that not only proposed the flowshop model but also provided an efficient algorithm that can produce a schedule with an optimal maximum completion time, or makespan, in a two-machine permutation flowshop. While makespan minimization can be optimally achieved by Johnson’s algorithm, to find a schedule that has the smallest total flow time is, however, computationally challenging (Garey et al. 1976, Garey and Johnson 1979). In this paper, we consider a two-machine permutation flowshop scheduling
*Corresponding author. Email: [email protected]
International Journal of Production Research
ISSN 0020–7543 print/ISSN 1366–588X onlineß 2006 Taylor & Francis
http://www.tandf.co.uk/journals DOI: 10.1080/00207540500446394
problem in which the objective function is a weighted sum of the makespan and total flow time.
The Makespan is commonly adopted as a measure for machine utilization. Total flow time, defined as the sum of the durations in which jobs stay in the system, is another important measure. This measure can be interpreted from aspects such as the average WIP level within an organization and the average waiting time of customers. Roughly speaking, the two measures are related to efficiency management and customer service, respectively. Because the two measures are crucial to the manage-ment of resources and service quality, a general decision practice might bring them into consideration simultaneously to measure the quality of a schedule with different criteria. Multiple criteria considerations provide flexibility to decision makers. Furthermore, Dudek et al. (1992) even suggest that the absence of multiple criteria from flowshop scheduling may be one of the reasons for the practical applications of flowshop scheduling problems. For such multiple and bicriteria scheduling problems, the reader is refereed to Nargar et al. (1995) for a comprehensive survey. Adopting the established three-field notation (Graham et al. 1979), we denote the two-machine flowshop scheduling problem by F2==CiþCmax: The first field indicates a
flowshop manufacturing system consisting of two machines. The third field specifies the objective function defined by weights and with þ ¼ 1 and 0 , 1: Note that the flow time of a job is equal to its completion time whenever the job is available for processing from time zero onwards. In this paper, we use the total completion time instead of the total flow time. As a sequel, Cidenotes the sum of
completion times in the objective function. It is easy to recognize the computational intractability of F2==CiþCmaxbecause the special case with ¼ 1 is known to be
strongly NP-hard (Garey et al. 1976). This negative result indicates that it is very unlikely to be able to devise polynomial or pseudo-polynomial time algorithms.
In spite of the strong NP-hardness of F2//Ci, in the scheduling literature several
researchers still center on the development of exact algorithms that can solve the problem to a certain scale. To design implicit enumeration algorithms for deriving exact optimal schedules, several lower bounds and dominance properties have been proposed in research work such as Ahmadi and Bagchi (1990), Della Croce et al. (1996, 2002), Hoogeveen and Kawaguchi (1999), Hoogeveen and van de Velde (1995), Ignall and Schrage (1965), Lin and Wu (2005), and van de Velde (1990). Most of the proposed bounds are based upon Lagrangian relaxation techniques (Fisher 1981, 1985), which have been widely recognized and adopted to successfully cope with hard combinatorial optimization problems. Lin and Wu (2005) proposed a simple lower bound scheme for the total completion time problem. Their computational experiments show that this new scheme could solve optimally most of the test instances with 50 jobs and some instances with 65 jobs. For the bicriteria two-machine flowshop scheduling problem, Nagar et al. (1995a) first considered the weighted sum measure. Nagar et al. (1995b) proposed a lower bound for the development of branch-and-bound algorithms. Yeh (1999) developed some optimality properties and improved the lower bound by considering the inevitable idle time for the remaining unscheduled jobs. In a computational study, Yeh (1999) claimed the superiority of his algorithm over previous works. Later, Yeh (2001) improved the branch-and-bound algorithm by incorporating new properties and implementation skills. Their approaches, including initial incumbent values, lower bounds, dominance rules and reinforced implementa-tions, can solve instances with up to 20 jobs. In this paper, our goal is to apply our
results of the case ¼ 1 to the general F2==CiþCmaxproblem. Our technique will
not only lead to better results but also provide an easy-to-implement approach to exactly solve the hard problem.
The rest of this paper is organized as follows. In section 2, we shall introduce the notation that will be used throughout this paper. We shall also present some preliminary results. Section 3 is dedicated to the new scheme for establishing lower bounds. Examples will be given for illustration. The computational study and analysis are given in section 4. Section 5 contains some concluding remarks. 2. Notation and previous results
This section first defines the notation that will be used throughout this paper. Then, some previous results from the literature will be introduced. With the exception that the weights and are real numbers, all other variables are integers.
Notation
N ¼{1, 2, . . . , n} job set to be processed
pi processing time of machine-1 operation of job i
qi processing time of machine-2 operation of job i
p(i) the ith smallest processing times in {p1, p2, . . . , pn}
q(i) the ith smallest processing times in {q1, q2, . . . , qn}
S schedule of job set N
, weights associated with total completion time and makespan, 0 , 1 and þ ¼ 1
Cm
i completion time of job i on machine m, m 2 {1, 2}, in some
schedule
Z(S) objective value of schedule S Z(N) optimal objective value of job set N
To cope with hard combinatorial optimization problems, one may apply several approaches, such as heuristics for deriving the initial incumbent value and dominance rules or lower bounds for curtailing unnecessary branching, to boost the efficiency of the solution algorithms. Because this study is centered around the lower bounds, dominance properties and heuristic approaches are not included. The reader is referred to Yeh (1999, 2001) for details. The first lower bound, called the I-bound and denoted LBI, was proposed by Nagar et al. (1995b) for a given
partial schedule Si for the first i jobs. This bound directly calculates the objective
value of the partial schedule and the weighted sum of the machine-2 processing times, which are arranged in the shortest processing time (SPT) order:
LBIðSiÞ ¼ZðSiÞ þ Xn j¼iþ1 ðn j þ1Þ qðjÞþ Xn j¼iþ1 qðjÞ ¼ZðSiÞ þ Xn j¼iþ1 ðn jÞ qðjÞþ ð þ Þ Xn j¼iþ1 qðjÞ ¼ZðSiÞ þ Xn j¼iþ1 ððn jÞ þ1Þ qðjÞ:
In the above formula, Z(Si) is the cost already incurred by the first i jobs, and
Pnj¼iþ1ðn j þ1Þ qðjÞand Pnj¼iþ1qðjÞare the lower bounds of the weighted total
completion time and the weighted makespan, respectively. Yeh (2001) later improved the I-bound by including the potential idle times that might be incurred for unscheduled jobs. That is, we assume the remaining jobs are to be scheduled by Johnson’s algorithm and then add the total idle time in this schedule to the I-bound. The second bound, called the J-bound, is defined as
LBJðSiÞ ¼ZðSiÞ þ Xn j¼iþ1 ððn jÞ þ1Þ qðjÞþ#iðSiÞ, where #iðSiÞ ¼ Xn j¼iþ1 ðn jÞ maxð0, C1j t2j1Þ þ X n j¼iþ1 max 0,Cn 1j t2j1o
is the total idle time in the schedule derived by applying Johnson’s algorithm to the remaining unscheduled jobs. For implementation details and skills of the J-bound, the reader is referred to Yeh (2001).
3. Our approach
In this section, we introduce Lin and Wu’s lower bound scheme for the total completion time problem. The new approach is based upon a data rearrangement mechanism, developed by Cheng et al. (2000), that transforms an instance of a strongly NP-hard problem into an ideal form that exhibits polynomial solvability and provides a lower bound for the original hard problem. In Lin and Wu’s computational study, under the same settings, branch-and-bound algorithms equipped with this bound can solve most of the F2//Ci problems with 55 jobs,
whereas the best lower bound (Della Croce et al. 2002) known in the literature can solve problems with data set of 25 or 30 jobs.
To underestimate the bicriteria CiþCmax, one may find a schedule whose
objective value is smaller than or equal to the optimum value. It is also viable to instead find lower bounds for Cmaxand Ci, respectively. The weighted sum of the
two bounds will also be a lower bound. In our study, the latter approach is employed. Because an optimal solution to F2//Cmax is attainable in O(n log n)
time using Johnson’s algorithm, we use the optimal makespan as a lower bound for the Cmaxpart of CiþCmax. As for the derivation of the lower bounds of Ci, the
data rearrangement methodology newly developed by Lin and Wu (2005) is applied. In the following, we introduce this method and give an example for illustration.
Given job set N ¼ {1, 2, . . . , n}, we create a new job set N0¼{10, 20, . . . , n0} in
which the processing times of job i0 are defined as p0
i, the ith smallest element in
{p1, p2, . . . , pn}, and qi0, the ith smallest element in {q1, q2, . . . , qn}. That is, each job i0
of N0 is defined by p
(i)and q(i). Tables 1 and 2 contain the original data set and the
data set derived through the rearrangement process, respectively. Although set N0
has two ideal SPT sequences on both machines, it remains NP-hard to find an optimal schedule for it (Hoogeveen and Kawaguchi 1999). Furthermore, it is not guaranteed that an optimal solution value of N0 would be smaller than that of the
original set N. Therefore, a second phase for further refinement is needed to find a lower bound in polynomial time.
With the derived job set N0, we exploit the following procedure, called Truncation
that schedules the jobs of N0 in ascending order of their indices and truncates some
machine-2 processing times when necessary. PROCEDURETRUNCATION
INPUT: Derived job set N0;
OUTPUT: Lower bound for the total completion time of N0;
Step 1: t1¼t2¼0; TCT ¼ 0; / Initialize the completion times and the total
completion time; Step2: flag ¼ 0; i ¼ 1; Step3: Do loop { t1¼t1þq0i; IF(t1t2)THENt2t2q0i; TCT ¼ TCT þ t2;
ELSEflag ¼1; t2¼t1þmin{q0i, p0iþ1}; TCT ¼ TCT þ t2;
i ¼ i þ1 }
WHILE(i < n 1)AND( flag ¼ 0);
Step4: For j ¼ i þ 1 to n 1 do { t1¼t1þp0j; t2¼min{max{t1, t2} þ q0j, t1þp0jþ1}; TCT ¼ TCT þ t2; } Step5: TCT ¼ TCT þ t2; Step6: Return TCT.
The time complexity of this procedure is O(n log n)for sorting the jobs. It could be reduced to O(n) if we deploy a preparatory procedure that arranges the processing times on machine 1 and machine 2 in SPT order before the solution procedure is activated. Lin and Wu have shown that the derived total completion time is no greater than the optimal solution value for the original data set N. The key operation
Table 1. Original data set N.
Job 1 2 3 4 5
pi 8 20 18 10 8
qi 5 16 11 20 17
Table 2. Data set N0derived from rearrangement of the processing times.
Job 1 2 3 4 5
p0
i 8 8 10 18 20
q0
i 5 11 16 17 20
in PROCEDURETRUNCATIONis locating the first job, in front of which an idle time is
incurred on machine 2. Such a non-zero idle time is likely to drive the remaining jobs to have longer completion times and consequently results in a total completion time larger than the optimal one. Therefore, to ensure the existence of a lower bound, truncation is required whenever the machine-2 completion time of a job is strictly later than the machine-1 completion time of its immediate successor in the sequence. Such a mechanism corresponds to the Else part of Step 3. When some non-trivial idle time exists to trigger the truncation mechanism, for any remaining job with a machine-2 completion time longer than the machine-1 completion time of its immediate successor, we trim its machine-2 operation such that its machine-2 completion time is the same as the machine-1 completion time of its immediate successor. Statement t2 ¼minfmaxft1,t2g þq0j, t1þp0jþ1gof Step 4 specifies when and
how the truncation operation is performed.
We consider job set N0 derived above as an illustration for the procedure.
The Gantt chart of the running session is shown in figure 1. Idle time occurs when job 20is scheduled. From this position on, the processing of any machine-2 operation
that is completed later than the processing of its successor on machine 1 must be trimmed. As a sequel, the completion time of job 20 is 26 instead of 27. Such a
trimming operation is conducted when the machine-1 completion time of the newly inserted job is greater than the machine-2 completion time of the current job, i.e. idle time occurs on machine 2 for the newly inserted job. The total completion time reported when the procedure stops is (13 þ 26 þ 42 þ 61 þ 84) ¼ 226. In this example, only one machine-2 operation is trimmed. Combining the optimal makespan 77
produced by Johnson’s algorithm, we have the objective value
0.3 226 þ 0.7 77 ¼ 121.7. The optimal schedule has an objective value of 127.8. Given the same data set, the two previous bounds can be derived as shown in the following: LBI¼ Xn j¼iþ1 ½ðn jÞ þ1 q½j ¼ ð0:3 4 þ 1Þ 5 þ ð0:3 3 þ 1Þ 11 þ ð0:3 2 þ 1Þ 16 þ ð0:3 1 þ 1Þ 17 þ ð0:3 0 þ 1Þ 2 ¼99:6, and LBJ¼LBIþ#ðSiÞ ¼99:6 þ ð0:3 4 þ 1Þ 8 ¼117:2: 8 8 10 18 20 5 16 10 17 20 13 26 42 61 84 M1 M2
Figure 1. Example of Procedure Truncation.
In the derivation of #ðSiÞ, an idle time of 8 units is incurred for the first job in
Johnson’s sequence for the unscheduled jobs. Therefore, an increment of 17.6 is incorporated. Considering the three lower bounds for the data set given in table 1, we readily realize that the bound derived by Johnson’s algorithm and the data rearrangement scheme is tighter than the other two.
In addition to the capability of composing optimal schedules for middle-scale problems, the lower bound based upon data rearrangement also demonstrates several advantages. First of all, the fundamentals of the scheme are free from mathematical skills such as Lagrangian relaxation. Second, the implementation is straightforward and simple. Furthermore, verification of the correctness of the scheme can be conducted through combinatorial arguments instead of complicated mathematical derivations.
Although Lin and Wu’s computational study has indicated that the data rearrangement scheme is effective in reducing the effort demanded for probing the solution space of the total completion time problem, whether or not it works well for the bicriteria problem is still unknown. In the next section, we shall design and conduct computational experiments to study the joint effectiveness of Johnson’s algorithm and the data rearrangement scheme for the F2kCiþCmax problem.
4. Computational experiments
Because there is no theoretical analysis concerning the relative performances of the existing bounds and the new scheme, we circumvent this problem and use computational experiments to investigate the efficiency issue. We wrote the codes in C language and performed the experiments under the Linux Red Hat 7.0 operating system running on a personal computer with a Pentium III 1.6 GHz CPU, 256 MB RAM and a 40 GB hard disk. In the experiments, two branch-and-bound algorithms were implemented. The first algorithm is based upon the data rearrangement approach. The second algorithm first uses the I-bound for each partial schedule. If a partial solution is not pruned by the I-bound, the second part of the J-bound will be calculated to derive the value of the J-bound. The two algorithms are denoted LBTR-Jand LBI-J. The solution trees of both algorithms are explored in
a depth-first fashion. Such a choice was made to keep the implementations simple in order to avoid excess memory requirement and programming skills that might be needed by strategies such as breadth-first search or best-first search. Such simplicity also avoids potential bias that might result from the implementation details, such as sophisticated data structures. Also note that we did not use any heuristic to derive initial upper bounds or incumbent values. Initially, the incumbent is a large number that would gradually decrease as better solutions are encountered.
The job instances were randomly generated in three different modes: (1) pi2 ½0,100, qi2 ½0,100; (2) pi2 ½0,100, qi2 ½0,50; and (3) pi2 ½0,50, qi2 ½0,100.
Each interval is discrete and uniformly distributed. The arrangement will depict different situations concerning the relative length of processing for each individual job on different machines, and will provide more extensive observations on the behavior of the implemented algorithms. For each different problem size (n), 10 job sets were generated and run by the branch-and-bound algorithms equipped with different bounds. For each job set, a limit of 30 min was given to confine the
execution of the algorithms. That is, if an algorithm cannot optimally solve a job set in 30 min, it will abort and report a failure. The statistics of the experiments shown were averaged over successful running sessions of all 10 instances. Throughout the experiments, we recorded (1) #_opt: the number of instances successfully solved; (2) avg_time: the average run time for the successfully solved instances; (3) max_time: the longest time elapsed of the successfully solved instances; (4) avg_node: the average number of nodes visited for the successfully solved instances; and (5) max_node: the largest number of nodes visited of the successfully solved instances.
Table 3 contains the results for two algorithms solving instances with 10 jobs. The statistics clearly demonstrate the superiority of our new approach over the existing one. The elapsed computation time and the number of visited nodes of our algorithm are almost negligible in comparison with those of the algorithm with LBIJ. The ratio is around 1000. Our test continued with an increment of five jobs.
Because the algorithm with LBIJ could not solve any instance with 15 or more
jobs, the results are not shown. Table 4 lists the results of our algorithm for solving instances of 15, 20, 30 and 35 jobs. For 15 job problems, the number of candidate sequences is 15!, which is on the order of 1012. On average, our algorithm visited only about 105or 106nodes, which is 107or 106times less than the size of the solution space. A very important observation to address is the relationship between the performance of our algorithm and the modes by which the data were generated. The most difficult cases were encountered when the processing times on both machines were taken from the same interval [0,100]. The algorithm performed well for the other two data modes, pi2 ½0,100, qi2 ½0,50 and pi2 ½0,50, qi2 ½0,100.
This is reflected in all terms including time, nodes and number of instances solved. Figure 2 compares the total number of instances solved for different problem sizes and data modes. Further examination suggests that our algorithm demonstrates the best problem-solving capability when the data size is relatively large and generated using the mode pi2 ½0,50, qi2 ½0,100. This might be due to the fact that when piis
relatively smaller than qi, the potential idle time on machine 2 could be reduced to a
certain degree and the lower bounds were much closer to the optimal solution values. We further scrutinized the dimensions of the weights and . When the value of is large, the objective value is anticipated to be more dependent on the total completion time. This might then imply that the role the lower bound of the total completion time plays will become more crucial. However, the statistics do not reflect this.
As a general summary, our approach demonstrates its capability in dealing with middle-scale instances. Most of the test cases with 30 or less jobs were solved successfully. Compared with the existing method, our algorithm represents a significant improvement. Furthermore, the simplicity of implementation makes its practical use much more viable.
5. Conclusions
In this paper, we have considered a two-machine flowshop scheduling problem to minimize the weighted sum of the makespan and the total completion time. To optimally solve the problem, a branch-and-bound algorithm equipped with two lower bounds has been addressed. The lower bound of the total completion time
Table 3. Numerical results for bounds LB TR – J and LB I– J . LB TR-J LB I-J Data mode #_opt Avg_time Max_time Avg_node Max_node #_opt Avg_time Max_time Avg_node Max_node pi 2 [0, 100], qi 2 [0, 100] 0.1 10 0.011 0.02 4342.2 9694 10 5.896 9.57 3 630 226 5 797 565 0.2 10 0.008 0.01 3585.4 5830 10 8.103 13.08 5 066 051 8 243 532 0.3 10 0.011 0.03 4981.5 12 860 10 10.265 14.38 6 313 707 8 897 275 0.4 10 0.007 0.01 2616.3 6327 10 8.214 11.88 5 106 893 7 300 271 0.5 10 0.007 0.01 3104.6 4644 10 8.642 11.38 5 360 178 7 174 942 0.6 10 0.007 0.01 3635.9 6495 10 9.870 13.67 6 113 064 8 423 607 0.7 10 0.009 0.02 3716.0 10 996 10 9.683 11.43 6 033 337 7 186 903 0.8 10 0.006 0.02 2736.1 5855 10 8.747 11.02 5 472 960 6 918 081 0.9 10 0.010 0.04 3969.0 16 826 10 10.132 13.94 6 309 212 8 657 193 pi 2 [0, 100], qi 2 [0, 50] 0.1 10 0.010 0.03 5685.3 15 041 10 7.407 10.89 4 459 487 6 649 113 0.2 10 0.011 0.02 5685.6 13 599 10 9.532 13.51 5 834 734 8 413 749 0.3 10 0.007 0.04 3965.0 19 666 10 8.353 13.39 5 130 353 8 331 656 0.4 10 0.006 0.01 2398.2 6523 10 9.305 12.79 5 716 079 7 928 900 0.5 10 0.008 0.02 2772.9 9601 10 10.329 14.15 6 415 446 8 886 679 0.6 10 0.006 0.01 1589.3 2938 10 8.550 11.04 5 272 271 6 833 493 0.7 10 0.005 0.01 1940.7 3243 10 8.768 11.85 5 413 319 7 370 102 0.8 10 0.003 0.01 1302.0 2593 10 9.862 13.92 6 107 623 8 703 403 0.9 10 0.003 0.01 1411.2 3291 10 10.834 12.81 6 734 367 8 025 638 pi 2 [0, 50], qi 2 [0, 100] 0.1 10 0.004 0.01 2091.0 6014 10 6.843 9.91 4 301 645 6 234 171 0.2 10 0.003 0.01 1581.9 4346 10 7.290 9.09 4 578 888 5 716 712 0.3 10 0.005 0.01 2141.9 4931 10 9.197 13.54 5 783 879 8 517 798 0.4 10 0.007 0.04 3092.4 19 125 10 9.436 12.28 5 927 541 7 725 276 0.5 10 0.008 0.04 3421.2 17 412 10 7.979 9.72 5 008 025 6 105 567 0.6 10 0.004 0.02 1770.5 7467 10 8.396 11.01 5 277 513 6 905 758 0.7 10 0.003 0.01 1479.8 3138 10 9.131 13.75 5 746 402 8 670 108 0.8 10 0.004 0.02 1773.8 5635 10 8.826 11.42 5 549 160 7 194 248 0.9 10 0.002 0.01 1822.9 3287 10 8.905 11.13 5 602 277 6 997 718
Table 4(a). Numerical results for bound LB TR – J . n ¼ 15 n ¼ 20 Data mode #_opt Avg_time Max_time Avg_node Max_node #_opt Avg_time Max_time Avg_node Max_node pi 2 [0, 100], qi 2 [0, 100] 0.1 10 3.419 20.36 1.45E þ 06 8.59E þ 06 6 91.613 239.90 3.48E þ 07 9.08E þ 07 0.2 10 0.717 3.85 3.09E þ 05 1.66E þ 06 6 89.083 240.11 3.43E þ 07 9.49E þ 07 0.3 10 0.512 3.06 2.17E þ 05 1.28E þ 06 8 255.731 719.18 9.71E þ 07 2.69E þ 08 0.4 10 0.874 3.34 3.77E þ 05 1.42E þ 06 10 97.489 356.73 3.66E þ 07 1.33E þ 08 0.5 10 0.831 3.88 3.55E þ 05 1.64E þ 06 9 164.307 859.06 6.23E þ 07 3.22E þ 08 0.6 10 1.969 14.25 8.46E þ 05 6.12E þ 06 9 269.424 669.52 1.02E þ 08 2.56E þ 08 0.7 10 2.122 15.70 9.09E þ 05 6.67E þ 06 10 319.935 775.67 1.21E þ 08 2.95E þ 08 0.8 10 2.078 17.76 8.88E þ 05 7.57E þ 06 9 171.083 912.00 6.48E þ 07 3.47E þ 08 0.9 10 0.410 1.00 1.74E þ 05 4.20E þ 05 10 105.745 274.95 3.96E þ 07 1.03E þ 08 pi 2 [0, 100], qi 2 [0, 50] 0.1 10 0.711 2.79 3.30E þ 05 1.32E þ 06 10 99.247 688.87 4.17E þ 07 2.91E þ 08 0.2 10 2.039 6.32 9.16E þ 05 2.85E þ 06 10 49.186 339.70 1.97E þ 07 1.33E þ 08 0.3 10 0.173 0.87 8.51E þ 04 4.31E þ 05 10 11.816 46.23 4.94E þ 06 2.02E þ 07 0.4 10 0.145 0.58 6.72E þ 04 2.67E þ 05 10 13.339 93.07 5.22E þ 06 3.55E þ 07 0.5 10 0.096 0.53 4.38E þ 04 2.27E þ 05 10 3.720 28.44 1.63E þ 06 1.26E þ 07 0.6 10 0.120 0.57 5.48E þ 04 2.46E þ 05 10 10.746 48.02 4.07E þ 06 1.81E þ 07 0.7 10 0.049 0.14 2.36E þ 04 6.40E þ 04 10 2.320 12.14 8.98E þ 05 4.41E þ 06 0.8 10 0.172 0.91 7.56E þ 04 3.82E þ 05 10 1.864 6.54 7.12E þ 05 2.47E þ 06 0.9 10 0.114 0.76 4.99E þ 04 3.14E þ 05 10 5.401 48.87 2.03E þ 06 1.83E þ 07 pi 2 [0, 50], qi 2 [0, 100] 0.1 10 0.929 5.27 3.92E þ 05 2.22E þ 06 8 10.109 74.06 3.77E þ 06 2.76E þ 07 0.2 10 0.860 6.91 3.69E þ 05 2.96E þ 06 10 1.269 9.69 4.76E þ 05 3.61E þ 06 0.3 10 5.231 35.38 2.28E þ 06 1.53E þ 07 10 12.253 111.14 4.60E þ 06 4.18E þ 07 0.4 10 0.182 1.00 7.85E þ 04 4.22E þ 05 10 0.964 6.57 3.59E þ 05 2.40E þ 06 0.5 10 0.045 0.21 2.13E þ 04 8.99E þ 04 10 0.295 1.47 1.13E þ 05 5.52E þ 05 0.6 10 2.279 16.36 9.99E þ 05 7.29E þ 06 10 1.864 14.35 6.99E þ 05 5.38E þ 06 0.7 10 3.514 18.13 1.52E þ 06 7.69E þ 06 8 0.994 2.94 3.68E þ 05 1.07E þ 06 0.8 10 0.084 0.59 3.77E þ 04 2.52E þ 05 10 1.515 10.14 5.60E þ 05 3.73E þ 06 0.9 10 35.190 351.47 1.58E þ 07 1.58E þ 08 10 0.266 1.12 1.02E þ 05 4.14E þ 05
Table 4(b). Numerical results for bound LB TR – J . n ¼ 25 n ¼ 40 Data mode #_opt Avg_time Max_time Avg_node Max_node #_opt Avg_time Max_time Avg_node Max_node pi 2 [0, 100], qi 2 [0, 100] 0.1 3 403.883 1167.90 1.35E þ 08 3.91E þ 08 0 – – – – 0.2 2 189.045 213.48 6.41E þ 07 7.26E þ 07 0 – – – – 0.3 2 363.750 501.65 1.21E þ 08 1.67E þ 08 1 598.5 598.5 1.75E þ 08 1.75E þ 08 0.4 1 634.040 634.04 2.11E þ 08 2.11E þ 08 1 17.55 17.55 4.95E þ 06 4.95E þ 06 0.5 3 230.687 518.97 7.70E þ 07 1.73E þ 08 0 – – – – 0.6 4 80.073 164.34 2.62E þ 07 5.33E þ 07 0 – – – – 0.7 2 4.585 8.41 1.58E þ 06 2.89E þ 06 1 1.37 1.37 4.70E þ 05 4.70E þ 05 0.8 6 635.328 1456.56 2.15E þ 08 4.85E þ 08 1 111.22 111.22 3.20E þ 07 3.20E þ 07 0.9 3 599.283 1267.27 2.06E þ 08 4.27E þ 08 0 – – – – pi 2 [0, 100], qi 2 [0, 50] 0.1 7 122.869 435.65 4.64E þ 07 1.59E þ 08 4 529.228 1323.78 1.63E þ 08 3.89E þ 08 0.2 8 53.781 257.71 2.07E þ 07 1.01E þ 08 4 306.526 406.35 1.01E þ 08 1.39E þ 08 0.3 8 508.314 1626.28 1.95E þ 08 6.33E þ 08 7 107.053 208.24 3.60E þ 07 7.26E þ 07 0.4 9 177.861 623.56 6.68E þ 07 2.51E þ 08 4 59.590 125.44 2.03E þ 07 4.45E þ 07 0.5 10 128.361 535.38 4.86E þ 07 2.21E þ 08 6 186.197 636.07 5.75E þ 07 1.88E þ 08 0.6 9 28.629 77.59 9.83E þ 06 2.59E þ 07 7 209.546 509.54 7.68E þ 07 1.90E þ 08 0.7 9 255.621 1723.42 8.47E þ 07 5.67E þ 08 7 81.684 200.45 2.66E þ 07 6.21E þ 07 0.8 10 330.733 1349.27 1.13E þ 08 4.51E þ 08 10 86.389 378.44 2.59E þ 07 1.11E þ 08 0.9 9 37.921 152.62 1.26E þ 07 5.08E þ 07 9 72.968 236.88 2.19E þ 07 6.92E þ 07 pi 2 [0, 50], qi 2 [0, 100] 0.1 9 8.954 58.42 2.91E þ 06 1.88E þ 07 9 65.390 367.99 1.86E þ 07 1.05E þ 08 0.2 9 32.761 99.00 1.08E þ 07 3.26E þ 07 8 14.948 95.19 4.29E þ 06 2.69E þ 07 0.3 9 2.108 8.70 6.98E þ 05 2.82E þ 06 4 64.585 157.03 1.86E þ 07 4.51E þ 07 0.4 10 7.742 56.19 2.57E þ 06 1.85E þ 07 8 190.838 1476.57 5.49E þ 07 4.24E þ 08 0.5 8 90.881 717.77 3.02E þ 07 2.39E þ 08 6 215.667 1259.46 6.20E þ 07 3.62E þ 08 0.6 9 11.711 67.65 3.84E þ 06 2.21E þ 07 8 424.113 1211.43 1.23E þ 08 3.49E þ 08 0.7 8 12.675 35.99 4.11E þ 06 1.16E þ 07 8 352.265 1130.42 1.02E þ 08 3.26E þ 08 0.8 6 25.378 144.36 8.35E þ 06 4.74E þ 07 9 234.053 850.15 6.73E þ 07 2.46E þ 08 0.9 8 71.446 283.35 2.33E þ 07 9.22E þ 07 8 249.576 1447.67 7.23E þ 07 4.19E þ 08
Table 4(c). Numerical results for bound LB TR – J . n ¼ 35 n ¼ 30 Data mode #_opt Avg_time Max_time Avg_node Max_node #_opt Avg_time Max_time Avg_node Max_node pi 2 [0, 100], qi 2 [0, 100] 0.1 0 – – – – 0 – – – – 0.2 0 – – – – 0 – – – – 0.3 0 – – – – 0 – – – – 0.4 0 – – – – 0 – – – – 0.5 0 – – – – 0 – – – – 0.6 0 – – – – 0 – – – – 0.7 0 – – – – 0 – – – – 0.8 0 – – – – 0 – – – – 0.9 0 – – – – 0 – – – – pi 2 [0, 100], qi 2 [0, 50] 0.1 0 – – – – 0 – – – – 0.2 2 481.705 716.85 1.6E þ 08 2.42E þ 08 0 – – – – 0.3 0 – – – – 4 513.455 1468.48 1.42E þ 08 4.11E þ 08 0.4 2 104.530 195.99 2.78E þ 07 5.16E þ 07 1 21.590 21.59 5.93E þ 06 5.93E þ 06 0.5 3 766.973 1776.81 2.80E þ 08 6.77E þ 08 1 304.770 304.77 7.49E þ 07 7.49E þ 07 0.6 3 554.173 1294.92 1.88E þ 08 4.38E þ 08 2 202.220 230.31 4.94E þ 07 5.63E þ 07 0.7 7 461.350 1440.32 1.46E þ 08 4.94E þ 08 2 975.170 1324.85 2.58E þ 08 3.16E þ 08 0.8 5 379.582 158.84 1.04E þ 08 3.33E þ 08 4 365.250 653.82 1.16E þ 08 1.96E þ 08 0.9 5 234.832 747.52 6.51E þ 07 1.94E þ 08 1 42.020 42.02 1.00E þ 07 1.00E þ 07 pi 2 [0, 50], qi 2 [0, 100] 0.1 4 18.550 57.25 4.98E þ 06 1.48E þ 07 6 681.730 1564.29 1.63E þ 08 3.70E þ 08 0.2 5 97.782 251.36 2.54E þ 07 6.57E þ 07 6 191.063 996.82 4.61E þ 07 2.40E þ 08 0.3 3 91.000 263.79 2.35E þ 07 6.79E þ 07 6 252.192 1201.54 6.13E þ 07 2.91E þ 08 0.4 3 134.037 298.63 3.53E þ 07 7.83E þ 07 3 141.737 411.55 3.38E þ 07 9.75E þ 07 0.5 7 28.894 176.34 7.76E þ 06 4.65E þ 07 5 112.400 489.16 2.66E þ 07 1.15E þ 08 0.6 6 251.925 976.01 6.49E þ 07 2.51E þ 08 5 118.022 357.19 2.82E þ 07 8.45E þ 07 0.7 5 23.734 52.24 6.28E þ 06 1.37E þ 07 5 101.100 372.93 2.45E þ 07 8.95E þ 07 0.8 7 271.600 976.20 6.99E þ 07 2.49E þ 08 3 114.773 310.02 2.77E þ 07 7.45E þ 07 0.9 6 458.083 1354.96 1.22E þ 08 3.63E þ 08 2 707.140 1405.87 1.68E þ 08 3.35E þ 08
is obtained by applying a data rearrangement scheme that was previously developed for the scheduling problem with a single objective function. The statistics obtained from computational experiments suggest that a strategy comprising Johnson’s algorithm and the data rearrangement scheme is a powerful tool for determining unnecessary branches in the solution tree. The success shown in this study also supports the significance of the data rearrangement scheme for flowshop-related problems. Adapting this approach to other flowshop problems or even other optimization problems could be a worthy direction for future research.
Acknowledgements
The authors are supported, in part, by the NSC of ROC and the NRC of Canada under grant number NSC 91-2213-E-002-111 and project grant NSC-91-2416-H-260–001.
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