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A flexible flowshop problem with total flow time minimization
European Journal of Operational Research 132 (2001) 528±538
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Theory and Methodology
A ¯exible ¯owshop problem with total ¯ow time minimization Meral Azizo glu a
a,*
, Ergin C akmak b, Suna Kondakci
a
Industrial Engineering Department, Middle East Technical University, Ankara, Turkey b Industrial Engineering Department, Anadolu University, Eskis ehir, Turkey Accepted 12 May 1999
Abstract In this study, we consider total ¯ow time problem in a ¯exible ¯owshop environment. We develop a branch and bound algorithm to ®nd the optimal schedule. The eciency of the algorithm is enhanced by upper and lower bounds and a dominance criterion. Computational experience reveals that the algorithm solves moderate sized problems in reasonable solution times. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Flexible ¯owshop; Parallel machines; Total ¯ow time
1. Introduction This paper addresses the problem of scheduling n jobs on w serial stages, each stage including several parallel identical machines. A job should be processed on any one of the parallel machines at each stage. Such an environment is called a ¯exible ¯owshop. Our scheduling objective is to minimize total ¯ow time. Flexible ¯owshops are generalizations of ¯owshops. The literature for the ¯owshop problem has grown after JohnsonÕs (1954) well-known algorithm for the two stage maximum completion time, i.e. the makespan problem. Many of the studies on ¯owshops consider the minimization of makespan
* Corresponding author. Tel.: +90-312-210-2281; fax: +90312-210-1268. E-mail address: [email protected] (M. Azizoglu).
and total ¯ow time. The studies by Gupta and Dudek (1971) and Panwalker et al. (1972) have revealed that total ¯ow time problem is more representative of scheduling costs than makespan. The branch and bound algorithms by Ignall and Schrage (1965), Bansal (1977) and Ahmadi and Bagchi (1990) and heuristic approaches by Ho and Chang (1991), Miyazaki et al. (1978) and Rajendran and Chaudhuri (1991) are among various attempts to solve the total ¯ow time problem on ¯owshops. Recent studies have recognized the importance of ¯exible ¯owshops to reduce the delays caused by bottleneck stages. Flexible ¯owshop problems arise in a number of dierent settings including polymer, chemical, and petrochemical industries (Salvador, 1973). It has been encountered in certain manufacturing systems (Zijm and Nelissen, 1990) and in assembly lines with parallel machines at workstations (Brah and Hunsucker, 1991),
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 1 4 2 - 9
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
electronics industry (Guinet and Solomon, 1996) and textile industry (Guinet, 1991). Many of the studies on ¯exible ¯owshops consider makespan minimization. Some special cases of the makespan problem are studied by Arthanari and Ramamurthy (1971), Mittal and Bagga (1972), Murty (1974), Rajendran et al. (1986), Gupta (1988), Gupta and Tunc (1994) and Kim et al. (1998). The studies by Sriskandarajah and Sethi (1989) and Brah and Hunsucker (1991) consider the general ¯exible ¯owshop makespan problem. Sriskandarajah and Sethi (1989) study a number of heuristics in terms of their worst and average case performances, whereas Brah and Hunsucker (1991) propose a branch and bound algorithm to ®nd an optimal schedule. To the best of our knowledge, the only reported research on total ¯ow time problem in ¯exible ¯owshops is due to Rajendran and Chaudhuri (1992). The study proposes a branch and bound algorithm to obtain an optimal permutation schedule. In this study, we propose a branch and bound algorithm to ®nd an optimal solution for the total ¯ow time problem in ¯exible ¯owshops. This optimal schedule need not be a permutation schedule. The lower bounds and the dominance rule developed considerably reduce the size of the branch and bound tree. The rest of the paper is organized as follows. In the next section, we de®ne our notation and describe the problem. In Section 3, we present the branch and bound algorithm along with the lower and upper bounds and the dominance theorem. In Section 4, we give the results of our computational experience. We conclude in Section 5.
2. Problem de®nition We consider the ¯owshop scheduling problem. We assume that there are identical parallel machines that are continuously available at each stage. Processing of a job on a machine cannot be interrupted. Let w denote the number of stages, n the number of jobs, mj the number of machines at stage j, pij the processing time of job i at stage j and
529
Cij the completion time of job i at stage j. All jobs follow the same processing order, 1; 2; . . . ; w. The completion time of job i in schedule S is Ciw S and the objective is to ®nd S that minimizes total ¯ow time, i.e. ( ) X X Ciw S Min Ciw S ; i
S2p
i
where p is the set of all feasible schedules. Since a special case of our problem, where there is a single machine at each stage has been shown to be NP-hard (Garey et al., 1976) our problem is also NP-hard. When the problem has a single stage, it can be solved in O n log n complexity (Conway et al., 1967). Assigning jobs to the earliest available machine in the order they appear in the shortest processing time (SPT) list solves the problem. A common approach to ¯owshop problems is to use permutation schedules which preserve the same order of assignment at each stage. In our problem, this corresponds to using the same list from which the jobs are assigned to the earliest available machine at each stage. Permutation schedules are not guaranteed to produce optimal solutions except in some restrictive special cases. Note that permutation schedules constitute only n! of the (n!)w possible schedules. Our approach does not restrict itself with permutation schedules and is guaranteed to ®nd the optimal schedule. 3. The branch and bound algorithm We solve a branch and bound algorithm for the ®rst stage of our ¯exible ¯owshop problem. The end nodes correspond to complete solutions and are listed in nondecreasing order of their ¯ow time values. Starting from the ®rst node of the list, we proceed to the branch and bound procedure of the second stage. The completion times of the jobs at stage 1 are their ready times at stage 2. Similarly the end nodes of stage k provide ready times for stage k 1. A complete solution is obtained when the branch and bound solution of stage w is reached. We backtrack to a previous stage when
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all potential solutions at the current stage are examined. The procedure terminates when all nodes at the ®rst stage are explored. Brah and HunsuckerÕs (1991) branching scheme, developed for minimizing makespan in ¯exible ¯owshops, is employed in our study for minimizing total ¯ow time. The enumeration is accomplished by creating two types of nodes for each unscheduled job at each stage. If the path passes through a type 1 node, then an unscheduled job is assigned to the current machine, if it passes through a type 2 node then it is scheduled on a new machine. Brah and Hunsucker (1991) have shown that the number of possible branches under this enumeration scheme is
n mj
1 1
n! mj !
at stage j. Therefore, the total number of possible end nodes for a w-stage ¯exible ¯owshop becomes w Y n m j j1
1 1
n! ; mj!
some of which are duplications of each other.
We propose an alternative branching scheme to generate all feasible sequences. In each stage, there are n nodes at the ®rst level of the tree, each node representing the assignment of a particular job to the earliest available machine. A node at the kth level of the tree corresponds to a partial sequence with k jobs scheduled. Each node at level k branches to n k nodes each representing the assignment of an unscheduled job to the earliest available machine. The number of possible branches under this enumeration scheme is n! at each stage. Therefore the total number of possible nodes for w stages is (n!)w . Figs. 1 and 2 give sample tree representations of a 4-job 2-machine problem for a particular stage, under the Brah and HunsuckerÕs and proposed branching schemes, respectively. A node at stage j represents a partial solution, where all jobs in stages 1 through j 1 are scheduled under both schemes. Our scheme generates only a subset of the schedules generated by Brah and HunsuckerÕs scheme. For example, in Fig. 1 the solution represented by according to Brah and HunsuckerÕs scheme corresponds to the schedule, where the ®rst machine processes job 1 and the
Fig. 1. Tree representation of four jobs on two parallel machines with Brah and Hunsucker's branching scheme.
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
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Fig. 2. Tree representation of four jobs on two parallel machines with proposed branching scheme.
second machine processes jobs 2, 3 and 4. Such a schedule would not be generated by the proposed scheme if p1j p2j p3j . Note that in such a case, job 4 could start earlier if processed by the ®rst machine. That is, the schedule represented by dominates the one represented by . We propose three heuristic procedures to ®nd an initial solution for the branch and bound algorithm. The procedures are given below. Heuristic 1. Stages are considered separately. Problem at each stage, i.e. minimizing total completion time on parallel machines, is solved optimally by the SPT rule. The sequences found are imposed over all stages. The sequence that gives the minimum total ¯ow time is chosen. Heuristic 2. This procedure is the same as Heuristic 1 except that the job sequence found is only used at the ®rst stage to assign the jobs. For the
other stages, among the available jobs the one having the shortest processing time is assigned to the earliest available machine. Heuristic 3. Rajendran and Chaudhuri (1991) propose a heuristic procedure for minimizing total ¯ow time in pure permutation ¯owshops. They form a sequence by ordering thePjobs according to w j 1pij their nondecreasing order of j1 w values. We extend the heuristic to our ¯exible ¯owshop problem by assigning the ®rst job of this sequence to the earliest available machine at each stage. We run each of the heuristics described above and use the best solution obtained as an initial upper bound in our branch and bound algorithm. We eliminate a partial solution from consideration by using a dominance theorem. We de®ne
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M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
r: the partial list rk: the partial list, where job k is appended to r Ej : the time at which machine j becomes available when the jobs in r are scheduled r: the earliest available machine when the jobs in r are scheduled, i.e. Er Minj Ej Theorem. If MaxfEr ; Cij 1 g pij 6 Ckj 1 , where i 6 k and i; k 62 r, then the partial list ri dominates rk. Proof. The partial list ri assigns job i to machine r at stage j. The completion time of job i at this stage is Cij MaxfEr ; Cij 1 g pij . Since job k becomes available for processing only after job i is completed, any optimal completion of the partial list rk, which will assign job k to machine r, will not lead to a better solution than that of the partial list ri. By the above theorem, we eliminate a partial sequence ri if there exists another partial sequence rj that satis®es the conditions of the theorem. Machines are allowed to stay idle for incoming jobs so long as the idle times are not long enough to process other jobs that are available. We calculate a lower bound for each partial solution that cannot be eliminated by the above theorem. Let Pj be the total completion time problem on parallel identical machines with processing times p1j ; p2j ; . . . ; pnj and ready times C1j 1 ;C2j 1 ; . . . ; Cnj 1P . Let P Lj be an optimal solution to Pj . Then n w Lj i1 qj1 piq is a lower bound on the optimal solution to the w-stage problem. Pj is NPhard as is its single machine version. Hence in place of optimal solutions, we use two dierent lower bounds on Lj each of which is explained below. Lower bound 1, LB1 . A special version of Pj with identical ready times is solvable in polynomial time by SPT list. Such a solution gives a lower bound on Lj when all ready times are set to mini fCi;j 1 g. LB1 is the total completion time of the SPT list, where the jobs start no earlier than mini fCi;j 1 g.
Lower bound 2, LB2 . Consider a relaxed version of Pj that allows job splitting. Such a relaxation converts Pj to a single machine total completion time problem, where the processing time of job i is pij =mj . The resulting problem is also NP-Hard (Lenstra et al., 1977), however its pre-emptive relaxation is solvable by the shortest remaining processing time (SRPT) rule (Schrage, 1968). The pre-emptive solution is a lower bound on the single machine problem, therefore on Lj . LB2 is the total completion time when jobs having processing times pij =mj on a single machine are scheduled by the SRPT rule. 4. Computational experience In this section, we discuss the results of our computational experiment. All algorithms are coded in Turbo C 3.0 and the computational experiments are conducted on a Pentium-166 MHz MMX under operating system Dos 7.0. We generate two problem sets. In set I, we consider small-sized problems to test the performances of lower bounds and the branching schemes proposed. In Set II, we consider largersized problems to test the performance of the branch and bound algorithm that gives the best results by the computational runs of Set I. In Set I, we generate problems with 6, 8, 9 and 10 jobs and 2, 3 and 4 stages. We generate the processing times using a discrete uniform distribution between 1; 25. The number of parallel machines at the stages are generated from a discrete uniform distribution between 1 and 4. For each problem combination, we solve 10 problem instances. We refer to proposed branching scheme as BB1 and Brah and HunsuckerÕs branching scheme as BB2. Each problem instance is solved by Lower Bound 1 (LB1) and Lower Bound 2 (LB2), and for each lower bound, we try both branching schemes, BB1 and BB2, resulting in a total of 480 runs. In Table 1, we report the average CPU times and the average number of nodes generated for each problem combination. We terminate the algorithm after 900 seconds of execution. The unsolved instances contribute to the CPU time
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
533
Table 1 Computational results of the branch and bound algorithms n
a b
w
mj
BB1
BB2
LB1 Average
LB1
LB2 Average
LB2
LB1 Average
LB1
LB2 Average
LB2
CPU
# of nodes
CPU
# of nodes
CPU
# of nodes
CPU
# of nodes
6
2 3 4
4; 3 3; 1; 2 1; 2; 4; 1
0.03 0.09 6.38
8
2 3 4
3; 4 2; 1; 2 1; 3; 2; 1
0.52 9.31 315.00
9
2 3
2; 2 2; 3; 3
10
2 3
2; 4 2; 2; 4
2001 8325 433838
0.56 0.16 9.48
3.35 0.39 13.63
0.01 0.15 16.27
34.00 14.62
213.00 54.06
0.68 24.22 510.00
352 11145 221322
0.10 5.78 60.32
3806 139409 4780000
34760 1341854 3805125 (3)
21.00 460.00 ±
660936 14653387 ± 3581316 2488000
30125 560128 3031902 (3)a
±b
±
14.58 22.66
800588 1192032
69.24 493.00
37000 174.5
85.84 105.50
3631075 (5) 4408900 (5)
545.00 501.00
1.24 109.00
56380 5351086
± ±
± ±
31.36 305.00
1165542 11322638
± ±
± ±
The numbers in parentheses show # of unsolved instances in 900 seconds. The empty entries show that no problem could be solved in 900 seconds.
by 900 seconds, and the number of nodes generated before termination is also considered in ®nding the average number of nodes. Notice that, only 21 of the 480 runs are not completed in 900 seconds. As can be observed from the table, the computational times increase signi®cantly with the increase in the number of jobs and the number of stages. Comparing LB1 and LB2, we see that under both branching schemes LB1 results in significantly smaller CPU times. We also observe that BB1 is consistently better than BB2 when either of the bounding schemes is used. Therefore, the best algorithm for all problem combinations is the one using LB1 under branching scheme BB1. We refer to this algorithm as BBBEST . To test the performance of BBBEST on problems of larger sizes, we generate Set II. The combinations of n, w and mj Õs are taken from Rajendran and Chaudhuri (1992). For each combination, we generate 10 problem instances. We generate the processing times again using a discrete uniform distribution between 1; 25. We terminate the execution of the algorithm after 3600 seconds. The results are tabulated in Tables 2 and 3 for w 2 and w P 3, respectively. The tables report the average and maximum CPU times and
the number of nodes created. The unsolved instances also contribute to these statistics. As can be observed from the tables, CPU times increase signi®cantly with an increase in the number of jobs and the number of stages. Notice that the number of machines in stages plays an important role in problem diculties. The problems having relatively small number of machines at earlier stages are easier to solve. This is because of the fact that the completion times estimated by lower bounds take only processing times into account. Hence, the lower bounds are close to the actual completion times when there are fewer number of machines at earlier stages due to the short waiting times between the stages. The tables also include the percentage of nodes searched until the optimal solution is found. We can observe that the optimum solutions are found at the very early stages of branching and most of the eort is spent on proving the optimality of the solution. Extending these results to the problems of larger sizes, one can expect that terminating the branch and bound algorithm after a certain time limit may produce near optimal solutions and can be used as a heuristic. We run a set of experiments to test the eectiveness of LB1 and LB2 . Proximity of lower
534
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
Table 2 Computational experience with BBBEST for w 2 N
w
m1
m2
CPU time (s) Average
# of nodes created Max.
Average
Average % of nodes created until optimality
Max.
8
2
1
2 3 4 5
0.02 0.00 0.00 0.00
0.16 0.00 0.00 0.00
1026 8 8 8
8918 8 8 8
29.63 0 0 0
10
2
1
2 3 4 5
0.08 0.00 0.00 0.00
0.33 0.00 0.00 0.00
3897 10 10 10
14983 10 10 10
48.70 0 0 0
8
2
2
1 2 3 4 5
8.59 0.97 0.02 0.00 0.00
10.00 7.69 0.11 0.00 0.00
511739 59826 1532 8 8
599576 478334 8829 8 8
15.59 9.40 0.52 0 0
10
2
2
1 2 3 4 5
1244.08 215.71 2.03 0.00 0.00
1562.00 1589.00 19.67 0.00 0.00
48733072 8702226 93093 10 10
59585536 63886749 899571 10 10
9.77 4.77 18.92 0 0
8
2
3
1 2 3 4 5
9.18 6.31 0.80 0.42 0.00
12.53 14.72 15.60 3.35 0.00
557353 391696 113351 25503 23
784960 914545 990775 202193 155
0.73 20.01 32.06 0.03 30.43
10
2
3
1 2 3 4 5
1276.33 809.21 79.41 52.07 0.00
1586.37 3147.64 710.99 413.79 0.00
60881165 32945293 3948090 2100029 10
71971729 128977697 35707598 16459241 10
6.20 13.71 3.27 7.62 0
8
2
4
1 2 3 4 5
9.92 16.56 20.32 2.44 0.00
15.11 42.53 53.63 23.68 0.00
611362 1020742 1230568 122853 23
1002038 2689728 351521 1186907 159
0.19 1.31 1.50 2.23 43.47
10
2
4
1 2 3 4 5
1789.26 2528.13 850.18 273.26 0.00
3600.00 3600.00 3600.00 2590.00 0.05
71571104 103084209 33868970 10100669 54
148753081 152013000 146566000 95583002 447
0.40 2.35 7.42 25.16 0
12
2
1
2 3 4 5
12.05 0.13 6.90 0.03
93.41 10.80 68.59 0.27
472187 42556 251193 1064
3676060 408602 2490183 10529
43.12 26.03 0 65.22
15
2
2
2 3 4 5
640.36 1800.66 132.34 0.00
3600.00 3600.00 1320.01 0.00
15505302 30605853 3663880 15
86329254 68466201 36502524 15
65.47 3.59 0.17 0
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535
Table 2 (Continued)
a
N
w
m1
m2
Average
Max.
12
2
2
1 2 3 4 5
3600.00 2340.00 18.01 0.15 0.00
a
12
2
3
1 2 3 4 5
a
a
3600.00 1635.56 30.20 0.00
a
CPU time (s)
# of nodes created Average
Max.
120772000 64012000 555206 4695 10
3600.00 113.34 0.54 0.00
3600.00 287.53 0.00
Average % of nodes created until optimality a
29.02 20.77 18.72 1.70 0
119172100 3428739 46841 10
a
a
142738000 60928000 987210 12
a
a
11.41 34.23 23.52 0
137241000 9420027 12
The problems are not solved in 3600 seconds.
Table 3 Computational experience with BBBEST for w > 2 N
a
w
m1
m2
m3
m4
m5
CPU time (s) Average
Max.
Average
# of nodes created Max.
# of nodes created untiloptimal solution
10592492 18 532482 204114
2052808 103 999194 2020747
4769113 0 56513 77125
a
a
a
25702 60777830 482944
165153 8378698 4565639
9756 3353454 69169
8
3
4 1 3 2
2 3 1 3
2 4 4 4
1747.78 0.00 10.67 4.69
3600.00 0.00 19.12 46.48
10
3
4 1 3 2
3 3 1 3
2 4 4 4
a
a
0.56 1563.79 12.63
3.74 2068.02 119.40
12
3
4 1 3 4
3 3 1 2
2 4 4 3
a
a
a
6.67
a
23.24
256747
908311
105051
433.36
3600.00
14345093
1175330
4556788
a
a
a
a
a
a
8
5
1
2
3
3
4
0.61
5.22
29312
250952
4724
10
5
1
2
3
3
4
53.25
526.00
2097269
2073834
249337
12
5
1
2
3
3
4
383.01
3600.00
11954945
1450230
2529047
The problems are not be solved in 3600 seconds.
bound values to the optimal solutions are reported in Table 4. We observe that, both lower bounds at the root node are very close to the optimal solutions. Hence it is not surprising that they perform quite well in a branch and bound algorithm. We also investigate the performances of the heuristic procedures. We calculate the percentage deviation from the optimal solution for each upper bounding heuristic as Hi
OPT=OPT;
where Hi : total ¯ow time value of heuristic i OPT: optimal ¯ow time found by the branch and bound algorithm Table 5 reports the results of such evaluations on various heuristic solutions for n P 12 or m P 3, i.e. relatively large size problems. As can be observed from the table all heuristics consistently produce solutions that are very close to optimum.
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M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
Table 4 Performances of lower bounds n
w
m1
m2
2 3 4
4 3 1
3 1 2
2 4
2 3 4
3 2 1
4 1 3
2 2
9
2 3
2 2
2 3
10
2 3
2 2
4 2
6
8
m3
m4
LB1 /OPT
LB2 /OPT
1
0.99 0.72 0.92
0.95 0.69 0.92
1
0.99 0.80 0.95
0.85 0.70 0.95
3
0.97 0.98
0.81 0.87
4
0.99 0.97
0.84 0.86
Table 5 Performances of heuristic procedures m4
H1
m5
H2
H3
w
m1
m2
m3
#Best
Dev.(%)
#Best
8
3
4 1 3 2
3 3 1 3
2 4 4 4
9.45 0.00 9.68 1.21
0 10 6 7
8.92 0.00 9.97 1.21
2 10 3 7
5.26 0.35 7.11 2.13
8 3 5 3
10
3
1 3 2
3 1 3
4 4 4
0.57 12.91 0.48
10 3 9
0.60 11.62 0.43
8 3 9
0.65 7.10 1.13
2 6 1
12
3
1 3 4
3 1 2
4 4 3
0.79 12.84 1.11
6 2 6
0.78 12.77 0.53
6 1 10
0.90 7.51 1.70
4 7 0
8
5
1
2
3
3
4
0.80
8
0.57
9
1.21
3
10
5
1
2
3
3
4
0.65
8
0.49
9
1.39
1
12
5
1
2
3
3
4
0.95
8
0.69
8
1.66
2
n
Dev.(%)
# Best
Dev.(%)
Table 6 The performance of the elimination rule n
w 6
m1
m2
m3
m4
Average CPU time (s)
Average # of nodes created
BBBEST
BB0 BEST
0.061 0.632 1.890
0.061 1.181 17.555
4165 47605 129298
4446 99341 1461376
BBBEST
BB0 BEST
2 3 4
4 3 1
3 2 2
2 4
8
2 3
3 2
4 1
2
0.011 10.252
0.011 11.115
578 630476
578 640312
9
2 3
2 2
2 3
3
47.555 48.148
54.621 48.785
2607480 2587109
3175835 2759597
10
2
2
4
0.027
0.027
1236
1241
1
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
We tested the eect of the elimination rule on algorithm performance. Table 6 compares the performance of BBBEST with a modi®ed variation of BB0 BEST . BB0 BEST is the same, except that the elimination rule is not checked. The results show that the improvement achieved by elimination rule overweighs the extra eort to test it.
5. Conclusions We considered the problem of minimizing total ¯ow time in a ¯exible ¯owshop problem. To our knowledge, there is no other published work that considers ®nding an optimal schedule for this problem. We proposed lower and upper bounding schemes and incorporated them into a branch and bound algorithm. We used two branching schemes. Our computational experiments showed that the algorithm is capable of generating optimal solutions for medium-sized problems. The branch and bound algorithm produces the optimal solution at very early stages and the majority of the time is spent to show that the solution is optimal. A branch and bound algorithm terminated after a prespeci®ed time limit may be an attractive heuristic solution for larger size problems if proving the optimality of the solution is not crucial. The model we studied can be generalized to include due-date related performance measures and/or ¯exible job shops.
References Ahmadi, R.H., Bagchi, U., 1990. Improved lower bounds for minimizing the sum of completion times of n jobs over m machines in a ¯owshop. European Journal of Operational Research 44, 331±336. Arthanari, T.S., Ramamurthy, K.S., 1971. An extension of two machines sequencing problem. Opsearch 8, 10±22. Bansal, S.R., 1977. Minimizing the sum of completion times of n-jobs over m-machines in a ¯owshop ± a branch and bound approach. AIIE Transactions 9, 306±311. Brah, S.A., Hunsucker, J.L., 1991. Branch and bound algorithm for the ¯owshop with multiple processor. European Journal of Operational Research 51, 88±99. Conway, R.W., Maxwell, W.L., Miller, L.W., 1967. Theory of Scheduling. Addison-Wesley, Reading, MA.
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Garey, M.S., Johnson, D.S., Sethi, R., 1976. The complexity of ¯owshop and jobshop scheduling. Mathematics of Operations Research 1, 117±129. Guinet, A.G.P., 1991. Textile production systems: A succession of non-identical parallel processor shops. Journal of Operations Research Society 42, 655±671. Guinet, A.G.P., Solomon, M.M., 1996. Scheduling hybrid ¯owshops to minimize maximum tardiness or maximum completion time. International Journal of Production Research 34, 1643±1654. Gupta, J.N.D., 1988. Two stage hybrid ¯owshop sequencing problem. Journal of Operations Research Society 39, 359± 364. Gupta, J.N.D., Dudek, R.A., 1971. Optimality criteria for ¯ow shop schedules. International Journal of Production Research 29, 1489±1502. Gupta, J.N.D., Tunc, E., 1994. Scheduling a two-stage hybrid ¯owshop with separable setup and removal times. European Journal of Operational Research 77, 415±428. Ho, J.C., Chang, Y.L., 1991. A new heuristic for the n-job mmachine ¯ow shop problem. European Journal of Operational Research 52, 194±202. Ignall, E., Schrage, L.E., 1965. Application of branch and bound technique to some ¯ow shop problem. Operations Research 13, 400±412. Johnson, S.M., 1954. Optimal two and three stage production schedules with setup times included. Naval Research Logistics Quarterly 1, 61±68. Kim, J.S., Kang, S.H., Lee, S.M., 1998. Transfer batch scheduling for a two-stage ¯owshop with identical parallel machines at each stage. Omega 25, 547±555. Lenstra, J.K., Rinnooy Kan, A.H.G., Brucker, P., 1977. Complexity of machine scheduling problems. Annals of Discrete Mathematics 1, 343±362. Mittal, B.S., Bagga, P.C., 1972. Two machine sequencing problem with parallel machines. Opsearch 9, 50±59. Miyazaki, S., Nishiyama, N., Hastimoto, F., 1978. An adjacent pairwise approach to the mean ¯owtime scheduling problem. Journal of the Operations Research Society of Japan 21, 287±299. Murty, R.N., 1974. On two machine sequencing problem with parallel machines. Opsearch 11, 42±44. Panwalkar, S.S., Dudek, R.A., Smith, M.L., 1972. Sequencing research and the industrial scheduling problem. In: Proceedings of the Symposium on Theory of Scheduling and Its Applications, Berlin. Rajendran, C., Chaudhuri, D., 1991. An ecient heuristic approach to the scheduling of jobs in a ¯owshop. European Journal of Operational Research 61, 318±325. Rajendran, C., Chaudhuri, D., 1992. A multi-stage parallelprocessor ¯owshop problem with minimum ¯owtime. European Journal of Operational Research 57, 111±112. Rajendran, C., Narendan, T.T., Chaudhuri, D., 1986. A branch and bound algorithm for scheduling in a ¯ow shop. In: Proceedings of the XIX Annual Convention of the Operational Society of India, vol. 57, New Delhi, pp. 472±484.
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Salvador, M.S., 1973. A solution of a special class of ¯owshop scheduling problems. In: Proceedings of the Symposium on the Theory of Scheduling and its Applications. Springer, Berlin, pp. 83±91. Schrage, L., 1968. A proof of the optimality of shortest remaining processing time discipline. Operations Research 16, 687±690.
Sriskandarajah, C., Sethi, S.P., 1989. Scheduling algorithms for ¯exible ¯owshops: Worst and average case performance. European Journal of Operational Research 43, 143±160. Zijm, W.H.M., Nelissen, E.H.L.B., 1990. Scheduling a ¯exible machining centre. Engineering Costs and Production Economics 19, 249±258.
www.elsevier.com/locate/dsw
Theory and Methodology
A ¯exible ¯owshop problem with total ¯ow time minimization Meral Azizo glu a
a,*
, Ergin C akmak b, Suna Kondakci
a
Industrial Engineering Department, Middle East Technical University, Ankara, Turkey b Industrial Engineering Department, Anadolu University, Eskis ehir, Turkey Accepted 12 May 1999
Abstract In this study, we consider total ¯ow time problem in a ¯exible ¯owshop environment. We develop a branch and bound algorithm to ®nd the optimal schedule. The eciency of the algorithm is enhanced by upper and lower bounds and a dominance criterion. Computational experience reveals that the algorithm solves moderate sized problems in reasonable solution times. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Flexible ¯owshop; Parallel machines; Total ¯ow time
1. Introduction This paper addresses the problem of scheduling n jobs on w serial stages, each stage including several parallel identical machines. A job should be processed on any one of the parallel machines at each stage. Such an environment is called a ¯exible ¯owshop. Our scheduling objective is to minimize total ¯ow time. Flexible ¯owshops are generalizations of ¯owshops. The literature for the ¯owshop problem has grown after JohnsonÕs (1954) well-known algorithm for the two stage maximum completion time, i.e. the makespan problem. Many of the studies on ¯owshops consider the minimization of makespan
* Corresponding author. Tel.: +90-312-210-2281; fax: +90312-210-1268. E-mail address: [email protected] (M. Azizoglu).
and total ¯ow time. The studies by Gupta and Dudek (1971) and Panwalker et al. (1972) have revealed that total ¯ow time problem is more representative of scheduling costs than makespan. The branch and bound algorithms by Ignall and Schrage (1965), Bansal (1977) and Ahmadi and Bagchi (1990) and heuristic approaches by Ho and Chang (1991), Miyazaki et al. (1978) and Rajendran and Chaudhuri (1991) are among various attempts to solve the total ¯ow time problem on ¯owshops. Recent studies have recognized the importance of ¯exible ¯owshops to reduce the delays caused by bottleneck stages. Flexible ¯owshop problems arise in a number of dierent settings including polymer, chemical, and petrochemical industries (Salvador, 1973). It has been encountered in certain manufacturing systems (Zijm and Nelissen, 1990) and in assembly lines with parallel machines at workstations (Brah and Hunsucker, 1991),
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 1 4 2 - 9
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
electronics industry (Guinet and Solomon, 1996) and textile industry (Guinet, 1991). Many of the studies on ¯exible ¯owshops consider makespan minimization. Some special cases of the makespan problem are studied by Arthanari and Ramamurthy (1971), Mittal and Bagga (1972), Murty (1974), Rajendran et al. (1986), Gupta (1988), Gupta and Tunc (1994) and Kim et al. (1998). The studies by Sriskandarajah and Sethi (1989) and Brah and Hunsucker (1991) consider the general ¯exible ¯owshop makespan problem. Sriskandarajah and Sethi (1989) study a number of heuristics in terms of their worst and average case performances, whereas Brah and Hunsucker (1991) propose a branch and bound algorithm to ®nd an optimal schedule. To the best of our knowledge, the only reported research on total ¯ow time problem in ¯exible ¯owshops is due to Rajendran and Chaudhuri (1992). The study proposes a branch and bound algorithm to obtain an optimal permutation schedule. In this study, we propose a branch and bound algorithm to ®nd an optimal solution for the total ¯ow time problem in ¯exible ¯owshops. This optimal schedule need not be a permutation schedule. The lower bounds and the dominance rule developed considerably reduce the size of the branch and bound tree. The rest of the paper is organized as follows. In the next section, we de®ne our notation and describe the problem. In Section 3, we present the branch and bound algorithm along with the lower and upper bounds and the dominance theorem. In Section 4, we give the results of our computational experience. We conclude in Section 5.
2. Problem de®nition We consider the ¯owshop scheduling problem. We assume that there are identical parallel machines that are continuously available at each stage. Processing of a job on a machine cannot be interrupted. Let w denote the number of stages, n the number of jobs, mj the number of machines at stage j, pij the processing time of job i at stage j and
529
Cij the completion time of job i at stage j. All jobs follow the same processing order, 1; 2; . . . ; w. The completion time of job i in schedule S is Ciw S and the objective is to ®nd S that minimizes total ¯ow time, i.e. ( ) X X Ciw S Min Ciw S ; i
S2p
i
where p is the set of all feasible schedules. Since a special case of our problem, where there is a single machine at each stage has been shown to be NP-hard (Garey et al., 1976) our problem is also NP-hard. When the problem has a single stage, it can be solved in O n log n complexity (Conway et al., 1967). Assigning jobs to the earliest available machine in the order they appear in the shortest processing time (SPT) list solves the problem. A common approach to ¯owshop problems is to use permutation schedules which preserve the same order of assignment at each stage. In our problem, this corresponds to using the same list from which the jobs are assigned to the earliest available machine at each stage. Permutation schedules are not guaranteed to produce optimal solutions except in some restrictive special cases. Note that permutation schedules constitute only n! of the (n!)w possible schedules. Our approach does not restrict itself with permutation schedules and is guaranteed to ®nd the optimal schedule. 3. The branch and bound algorithm We solve a branch and bound algorithm for the ®rst stage of our ¯exible ¯owshop problem. The end nodes correspond to complete solutions and are listed in nondecreasing order of their ¯ow time values. Starting from the ®rst node of the list, we proceed to the branch and bound procedure of the second stage. The completion times of the jobs at stage 1 are their ready times at stage 2. Similarly the end nodes of stage k provide ready times for stage k 1. A complete solution is obtained when the branch and bound solution of stage w is reached. We backtrack to a previous stage when
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M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
all potential solutions at the current stage are examined. The procedure terminates when all nodes at the ®rst stage are explored. Brah and HunsuckerÕs (1991) branching scheme, developed for minimizing makespan in ¯exible ¯owshops, is employed in our study for minimizing total ¯ow time. The enumeration is accomplished by creating two types of nodes for each unscheduled job at each stage. If the path passes through a type 1 node, then an unscheduled job is assigned to the current machine, if it passes through a type 2 node then it is scheduled on a new machine. Brah and Hunsucker (1991) have shown that the number of possible branches under this enumeration scheme is
n mj
1 1
n! mj !
at stage j. Therefore, the total number of possible end nodes for a w-stage ¯exible ¯owshop becomes w Y n m j j1
1 1
n! ; mj!
some of which are duplications of each other.
We propose an alternative branching scheme to generate all feasible sequences. In each stage, there are n nodes at the ®rst level of the tree, each node representing the assignment of a particular job to the earliest available machine. A node at the kth level of the tree corresponds to a partial sequence with k jobs scheduled. Each node at level k branches to n k nodes each representing the assignment of an unscheduled job to the earliest available machine. The number of possible branches under this enumeration scheme is n! at each stage. Therefore the total number of possible nodes for w stages is (n!)w . Figs. 1 and 2 give sample tree representations of a 4-job 2-machine problem for a particular stage, under the Brah and HunsuckerÕs and proposed branching schemes, respectively. A node at stage j represents a partial solution, where all jobs in stages 1 through j 1 are scheduled under both schemes. Our scheme generates only a subset of the schedules generated by Brah and HunsuckerÕs scheme. For example, in Fig. 1 the solution represented by according to Brah and HunsuckerÕs scheme corresponds to the schedule, where the ®rst machine processes job 1 and the
Fig. 1. Tree representation of four jobs on two parallel machines with Brah and Hunsucker's branching scheme.
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
531
Fig. 2. Tree representation of four jobs on two parallel machines with proposed branching scheme.
second machine processes jobs 2, 3 and 4. Such a schedule would not be generated by the proposed scheme if p1j p2j p3j . Note that in such a case, job 4 could start earlier if processed by the ®rst machine. That is, the schedule represented by dominates the one represented by . We propose three heuristic procedures to ®nd an initial solution for the branch and bound algorithm. The procedures are given below. Heuristic 1. Stages are considered separately. Problem at each stage, i.e. minimizing total completion time on parallel machines, is solved optimally by the SPT rule. The sequences found are imposed over all stages. The sequence that gives the minimum total ¯ow time is chosen. Heuristic 2. This procedure is the same as Heuristic 1 except that the job sequence found is only used at the ®rst stage to assign the jobs. For the
other stages, among the available jobs the one having the shortest processing time is assigned to the earliest available machine. Heuristic 3. Rajendran and Chaudhuri (1991) propose a heuristic procedure for minimizing total ¯ow time in pure permutation ¯owshops. They form a sequence by ordering thePjobs according to w j 1pij their nondecreasing order of j1 w values. We extend the heuristic to our ¯exible ¯owshop problem by assigning the ®rst job of this sequence to the earliest available machine at each stage. We run each of the heuristics described above and use the best solution obtained as an initial upper bound in our branch and bound algorithm. We eliminate a partial solution from consideration by using a dominance theorem. We de®ne
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M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
r: the partial list rk: the partial list, where job k is appended to r Ej : the time at which machine j becomes available when the jobs in r are scheduled r: the earliest available machine when the jobs in r are scheduled, i.e. Er Minj Ej Theorem. If MaxfEr ; Cij 1 g pij 6 Ckj 1 , where i 6 k and i; k 62 r, then the partial list ri dominates rk. Proof. The partial list ri assigns job i to machine r at stage j. The completion time of job i at this stage is Cij MaxfEr ; Cij 1 g pij . Since job k becomes available for processing only after job i is completed, any optimal completion of the partial list rk, which will assign job k to machine r, will not lead to a better solution than that of the partial list ri. By the above theorem, we eliminate a partial sequence ri if there exists another partial sequence rj that satis®es the conditions of the theorem. Machines are allowed to stay idle for incoming jobs so long as the idle times are not long enough to process other jobs that are available. We calculate a lower bound for each partial solution that cannot be eliminated by the above theorem. Let Pj be the total completion time problem on parallel identical machines with processing times p1j ; p2j ; . . . ; pnj and ready times C1j 1 ;C2j 1 ; . . . ; Cnj 1P . Let P Lj be an optimal solution to Pj . Then n w Lj i1 qj1 piq is a lower bound on the optimal solution to the w-stage problem. Pj is NPhard as is its single machine version. Hence in place of optimal solutions, we use two dierent lower bounds on Lj each of which is explained below. Lower bound 1, LB1 . A special version of Pj with identical ready times is solvable in polynomial time by SPT list. Such a solution gives a lower bound on Lj when all ready times are set to mini fCi;j 1 g. LB1 is the total completion time of the SPT list, where the jobs start no earlier than mini fCi;j 1 g.
Lower bound 2, LB2 . Consider a relaxed version of Pj that allows job splitting. Such a relaxation converts Pj to a single machine total completion time problem, where the processing time of job i is pij =mj . The resulting problem is also NP-Hard (Lenstra et al., 1977), however its pre-emptive relaxation is solvable by the shortest remaining processing time (SRPT) rule (Schrage, 1968). The pre-emptive solution is a lower bound on the single machine problem, therefore on Lj . LB2 is the total completion time when jobs having processing times pij =mj on a single machine are scheduled by the SRPT rule. 4. Computational experience In this section, we discuss the results of our computational experiment. All algorithms are coded in Turbo C 3.0 and the computational experiments are conducted on a Pentium-166 MHz MMX under operating system Dos 7.0. We generate two problem sets. In set I, we consider small-sized problems to test the performances of lower bounds and the branching schemes proposed. In Set II, we consider largersized problems to test the performance of the branch and bound algorithm that gives the best results by the computational runs of Set I. In Set I, we generate problems with 6, 8, 9 and 10 jobs and 2, 3 and 4 stages. We generate the processing times using a discrete uniform distribution between 1; 25. The number of parallel machines at the stages are generated from a discrete uniform distribution between 1 and 4. For each problem combination, we solve 10 problem instances. We refer to proposed branching scheme as BB1 and Brah and HunsuckerÕs branching scheme as BB2. Each problem instance is solved by Lower Bound 1 (LB1) and Lower Bound 2 (LB2), and for each lower bound, we try both branching schemes, BB1 and BB2, resulting in a total of 480 runs. In Table 1, we report the average CPU times and the average number of nodes generated for each problem combination. We terminate the algorithm after 900 seconds of execution. The unsolved instances contribute to the CPU time
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
533
Table 1 Computational results of the branch and bound algorithms n
a b
w
mj
BB1
BB2
LB1 Average
LB1
LB2 Average
LB2
LB1 Average
LB1
LB2 Average
LB2
CPU
# of nodes
CPU
# of nodes
CPU
# of nodes
CPU
# of nodes
6
2 3 4
4; 3 3; 1; 2 1; 2; 4; 1
0.03 0.09 6.38
8
2 3 4
3; 4 2; 1; 2 1; 3; 2; 1
0.52 9.31 315.00
9
2 3
2; 2 2; 3; 3
10
2 3
2; 4 2; 2; 4
2001 8325 433838
0.56 0.16 9.48
3.35 0.39 13.63
0.01 0.15 16.27
34.00 14.62
213.00 54.06
0.68 24.22 510.00
352 11145 221322
0.10 5.78 60.32
3806 139409 4780000
34760 1341854 3805125 (3)
21.00 460.00 ±
660936 14653387 ± 3581316 2488000
30125 560128 3031902 (3)a
±b
±
14.58 22.66
800588 1192032
69.24 493.00
37000 174.5
85.84 105.50
3631075 (5) 4408900 (5)
545.00 501.00
1.24 109.00
56380 5351086
± ±
± ±
31.36 305.00
1165542 11322638
± ±
± ±
The numbers in parentheses show # of unsolved instances in 900 seconds. The empty entries show that no problem could be solved in 900 seconds.
by 900 seconds, and the number of nodes generated before termination is also considered in ®nding the average number of nodes. Notice that, only 21 of the 480 runs are not completed in 900 seconds. As can be observed from the table, the computational times increase signi®cantly with the increase in the number of jobs and the number of stages. Comparing LB1 and LB2, we see that under both branching schemes LB1 results in significantly smaller CPU times. We also observe that BB1 is consistently better than BB2 when either of the bounding schemes is used. Therefore, the best algorithm for all problem combinations is the one using LB1 under branching scheme BB1. We refer to this algorithm as BBBEST . To test the performance of BBBEST on problems of larger sizes, we generate Set II. The combinations of n, w and mj Õs are taken from Rajendran and Chaudhuri (1992). For each combination, we generate 10 problem instances. We generate the processing times again using a discrete uniform distribution between 1; 25. We terminate the execution of the algorithm after 3600 seconds. The results are tabulated in Tables 2 and 3 for w 2 and w P 3, respectively. The tables report the average and maximum CPU times and
the number of nodes created. The unsolved instances also contribute to these statistics. As can be observed from the tables, CPU times increase signi®cantly with an increase in the number of jobs and the number of stages. Notice that the number of machines in stages plays an important role in problem diculties. The problems having relatively small number of machines at earlier stages are easier to solve. This is because of the fact that the completion times estimated by lower bounds take only processing times into account. Hence, the lower bounds are close to the actual completion times when there are fewer number of machines at earlier stages due to the short waiting times between the stages. The tables also include the percentage of nodes searched until the optimal solution is found. We can observe that the optimum solutions are found at the very early stages of branching and most of the eort is spent on proving the optimality of the solution. Extending these results to the problems of larger sizes, one can expect that terminating the branch and bound algorithm after a certain time limit may produce near optimal solutions and can be used as a heuristic. We run a set of experiments to test the eectiveness of LB1 and LB2 . Proximity of lower
534
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
Table 2 Computational experience with BBBEST for w 2 N
w
m1
m2
CPU time (s) Average
# of nodes created Max.
Average
Average % of nodes created until optimality
Max.
8
2
1
2 3 4 5
0.02 0.00 0.00 0.00
0.16 0.00 0.00 0.00
1026 8 8 8
8918 8 8 8
29.63 0 0 0
10
2
1
2 3 4 5
0.08 0.00 0.00 0.00
0.33 0.00 0.00 0.00
3897 10 10 10
14983 10 10 10
48.70 0 0 0
8
2
2
1 2 3 4 5
8.59 0.97 0.02 0.00 0.00
10.00 7.69 0.11 0.00 0.00
511739 59826 1532 8 8
599576 478334 8829 8 8
15.59 9.40 0.52 0 0
10
2
2
1 2 3 4 5
1244.08 215.71 2.03 0.00 0.00
1562.00 1589.00 19.67 0.00 0.00
48733072 8702226 93093 10 10
59585536 63886749 899571 10 10
9.77 4.77 18.92 0 0
8
2
3
1 2 3 4 5
9.18 6.31 0.80 0.42 0.00
12.53 14.72 15.60 3.35 0.00
557353 391696 113351 25503 23
784960 914545 990775 202193 155
0.73 20.01 32.06 0.03 30.43
10
2
3
1 2 3 4 5
1276.33 809.21 79.41 52.07 0.00
1586.37 3147.64 710.99 413.79 0.00
60881165 32945293 3948090 2100029 10
71971729 128977697 35707598 16459241 10
6.20 13.71 3.27 7.62 0
8
2
4
1 2 3 4 5
9.92 16.56 20.32 2.44 0.00
15.11 42.53 53.63 23.68 0.00
611362 1020742 1230568 122853 23
1002038 2689728 351521 1186907 159
0.19 1.31 1.50 2.23 43.47
10
2
4
1 2 3 4 5
1789.26 2528.13 850.18 273.26 0.00
3600.00 3600.00 3600.00 2590.00 0.05
71571104 103084209 33868970 10100669 54
148753081 152013000 146566000 95583002 447
0.40 2.35 7.42 25.16 0
12
2
1
2 3 4 5
12.05 0.13 6.90 0.03
93.41 10.80 68.59 0.27
472187 42556 251193 1064
3676060 408602 2490183 10529
43.12 26.03 0 65.22
15
2
2
2 3 4 5
640.36 1800.66 132.34 0.00
3600.00 3600.00 1320.01 0.00
15505302 30605853 3663880 15
86329254 68466201 36502524 15
65.47 3.59 0.17 0
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
535
Table 2 (Continued)
a
N
w
m1
m2
Average
Max.
12
2
2
1 2 3 4 5
3600.00 2340.00 18.01 0.15 0.00
a
12
2
3
1 2 3 4 5
a
a
3600.00 1635.56 30.20 0.00
a
CPU time (s)
# of nodes created Average
Max.
120772000 64012000 555206 4695 10
3600.00 113.34 0.54 0.00
3600.00 287.53 0.00
Average % of nodes created until optimality a
29.02 20.77 18.72 1.70 0
119172100 3428739 46841 10
a
a
142738000 60928000 987210 12
a
a
11.41 34.23 23.52 0
137241000 9420027 12
The problems are not solved in 3600 seconds.
Table 3 Computational experience with BBBEST for w > 2 N
a
w
m1
m2
m3
m4
m5
CPU time (s) Average
Max.
Average
# of nodes created Max.
# of nodes created untiloptimal solution
10592492 18 532482 204114
2052808 103 999194 2020747
4769113 0 56513 77125
a
a
a
25702 60777830 482944
165153 8378698 4565639
9756 3353454 69169
8
3
4 1 3 2
2 3 1 3
2 4 4 4
1747.78 0.00 10.67 4.69
3600.00 0.00 19.12 46.48
10
3
4 1 3 2
3 3 1 3
2 4 4 4
a
a
0.56 1563.79 12.63
3.74 2068.02 119.40
12
3
4 1 3 4
3 3 1 2
2 4 4 3
a
a
a
6.67
a
23.24
256747
908311
105051
433.36
3600.00
14345093
1175330
4556788
a
a
a
a
a
a
8
5
1
2
3
3
4
0.61
5.22
29312
250952
4724
10
5
1
2
3
3
4
53.25
526.00
2097269
2073834
249337
12
5
1
2
3
3
4
383.01
3600.00
11954945
1450230
2529047
The problems are not be solved in 3600 seconds.
bound values to the optimal solutions are reported in Table 4. We observe that, both lower bounds at the root node are very close to the optimal solutions. Hence it is not surprising that they perform quite well in a branch and bound algorithm. We also investigate the performances of the heuristic procedures. We calculate the percentage deviation from the optimal solution for each upper bounding heuristic as Hi
OPT=OPT;
where Hi : total ¯ow time value of heuristic i OPT: optimal ¯ow time found by the branch and bound algorithm Table 5 reports the results of such evaluations on various heuristic solutions for n P 12 or m P 3, i.e. relatively large size problems. As can be observed from the table all heuristics consistently produce solutions that are very close to optimum.
536
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
Table 4 Performances of lower bounds n
w
m1
m2
2 3 4
4 3 1
3 1 2
2 4
2 3 4
3 2 1
4 1 3
2 2
9
2 3
2 2
2 3
10
2 3
2 2
4 2
6
8
m3
m4
LB1 /OPT
LB2 /OPT
1
0.99 0.72 0.92
0.95 0.69 0.92
1
0.99 0.80 0.95
0.85 0.70 0.95
3
0.97 0.98
0.81 0.87
4
0.99 0.97
0.84 0.86
Table 5 Performances of heuristic procedures m4
H1
m5
H2
H3
w
m1
m2
m3
#Best
Dev.(%)
#Best
8
3
4 1 3 2
3 3 1 3
2 4 4 4
9.45 0.00 9.68 1.21
0 10 6 7
8.92 0.00 9.97 1.21
2 10 3 7
5.26 0.35 7.11 2.13
8 3 5 3
10
3
1 3 2
3 1 3
4 4 4
0.57 12.91 0.48
10 3 9
0.60 11.62 0.43
8 3 9
0.65 7.10 1.13
2 6 1
12
3
1 3 4
3 1 2
4 4 3
0.79 12.84 1.11
6 2 6
0.78 12.77 0.53
6 1 10
0.90 7.51 1.70
4 7 0
8
5
1
2
3
3
4
0.80
8
0.57
9
1.21
3
10
5
1
2
3
3
4
0.65
8
0.49
9
1.39
1
12
5
1
2
3
3
4
0.95
8
0.69
8
1.66
2
n
Dev.(%)
# Best
Dev.(%)
Table 6 The performance of the elimination rule n
w 6
m1
m2
m3
m4
Average CPU time (s)
Average # of nodes created
BBBEST
BB0 BEST
0.061 0.632 1.890
0.061 1.181 17.555
4165 47605 129298
4446 99341 1461376
BBBEST
BB0 BEST
2 3 4
4 3 1
3 2 2
2 4
8
2 3
3 2
4 1
2
0.011 10.252
0.011 11.115
578 630476
578 640312
9
2 3
2 2
2 3
3
47.555 48.148
54.621 48.785
2607480 2587109
3175835 2759597
10
2
2
4
0.027
0.027
1236
1241
1
M. Azizo glu et al. / European Journal of Operational Research 132 (2001) 528±538
We tested the eect of the elimination rule on algorithm performance. Table 6 compares the performance of BBBEST with a modi®ed variation of BB0 BEST . BB0 BEST is the same, except that the elimination rule is not checked. The results show that the improvement achieved by elimination rule overweighs the extra eort to test it.
5. Conclusions We considered the problem of minimizing total ¯ow time in a ¯exible ¯owshop problem. To our knowledge, there is no other published work that considers ®nding an optimal schedule for this problem. We proposed lower and upper bounding schemes and incorporated them into a branch and bound algorithm. We used two branching schemes. Our computational experiments showed that the algorithm is capable of generating optimal solutions for medium-sized problems. The branch and bound algorithm produces the optimal solution at very early stages and the majority of the time is spent to show that the solution is optimal. A branch and bound algorithm terminated after a prespeci®ed time limit may be an attractive heuristic solution for larger size problems if proving the optimality of the solution is not crucial. The model we studied can be generalized to include due-date related performance measures and/or ¯exible job shops.
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