MAKESPAN DISTRIBUTIONS IN FLOW SHOPS
WITH MULTIPLE PROCESSORS
Wei Wang1 and John L. Hunsucker
Department of Industrial Engineering
University of Houston, Houston, Texas 77204-4812, USA
Abstract
The makespan distributions in a flow shop with multiple processors (FSMP) are investigated in
this study. The FSMP problem is characterized as the processing of n jobs through an m stage
flow shop, where there exists one or more processors at each stage. Since it is known that the
minimum makespan FSMP problem is computationally explosive, it was necessary to investigate
the distribution of makespans in FSMP in order to provide guidance to the FSMP research.
Results of this study may help gain a better understanding of the FSMP makespan problem.
Keywords: Scheduling, Flow shop, Makespan, Distribution, Heuristic
1. Introduction
This paper is concerned with the makespan distributions in flow shops with multiple processors
(FSMP). Scheduling in an FSMP involves the sequencing of n jobs through m stages where there
can be one or more identical processors at each stage. The makespan is the total elapsed time
required to process all of a set of given jobs. It is an important characteristic of a schedule. The
optimization objective of many studies is to minimize the makespan. It is known that the FSMP
makespan problems are computationally explosive. This has caused researchers to turn to
heuristic methods or computer simulation programs to hopefully find near-optimal solutions to
the FSMP problems. Without knowing the optimal solutions, a tool is then needed to estimate the
optimal solution and to evaluate the quality of various heuristics. Determination of a strong global
lower bound has been shown empirically to be an effective tool to achieve this objective.
Nevertheless, it is not the only method to evaluate the effectiveness of various heuristics.
Investigation of the makespan distributions in FSMP can also help gain a better understanding of
the makespan problem. Hence, this study examines the makespan distributions in the FSMP
environment and the result of this study reveals the statistical nature of the FSMP makespan
problems.
2. Background Review
The pure flow shop scheduling problem with the makespan objective has been the focus of a lot
of research for last several decades. In the pure flow shop environment, there is only one
processor available at each stage throughout the entire flow shop. Johnson’s (1954) constructive
algorithm for the two-stage pure flow shop makespan problem is a very important contribution to
the makespan study because it can optimally solve the problem. Under the tenet of Johnson’s
algorithm, roughly defined as “start quick and finish quick”, other important heuristics were
developed to solve the makespan problem in the pure flow shop with more than two stages. These
significant heuristics are those developed by Palmer (1965), Campbell et al. (1970), Gupta (1971)
1
Corresponding Author
1
and Dannenbring (1977). These four heuristics can quickly produce a sequence for the multiplestage scheduling problem but will not always produce an optimal solution.
The FSMP environment is a more complex class of the flow shop than the pure flow shop. The
FSMP makespan problem has been the focus of recent research efforts. Some significant work
done on FSMP is reviewed because they formed the foundation for this research. Starting from
two-stage FSMP’s, work has been done by Deal and Hunsucker (1991), Gupta and Tunc (1991),
Lee and Vairaktarakis (1994), and Deal et al. (1994). Other research has been conducted on
FSMP’s with more than two stages. For instance, Brah et al. (1991a) performed studies on the
mathematical modeling of FSMP’s. Brah and Hunsucker (1991b) developed a branch and bound
model for FSMP’s. Hunsucker and Shah (1992) evaluated the priority rules in scheduling
FSMP’s. Since the optimal solutions are unknown to researchers, a good makespan lower bound
is needed in order to test a heuristic. Santos (1993) developed a lower bound, the SHD lower
bound, to evaluate the quality of various heuristics when the optimal makespan is unknown. It
was shown by Santos’ study that the lower bound is a strong indicator of the optimal makespan in
the FSMP environment. Moreover, Santos et al. (1995b) developed a computer heuristic called
FLOWMULT which can help solve the makespan problem in a timely manner.
3. Experimental Design
In order to better understand the FSMP makespan problems it is necessary to investigate the
distribution of the makespans in FSMP. Hence, the primary objective of this study is to examine
the statistical nature of the makespan distributions in the FSMP environment. The investigation
analyzes the results of all n! multiple-permutation (MP) schedules for each job configuration. A
multiple-permutation schedule is a non-delay schedule in which the start order of jobs on the
current stage is determined by the finish order of jobs in the previous stage. Ties are handled by
using the start order on the previous stage. This new term, MP, applies only to the FSMP
environment. Considering only MP schedules effectively reduces the solution space to n!
different sequences. A MP schedule is generally close to an optimal schedule even if it is not
optimal. Therefore, examining only the MP schedules does not significantly affect the quality of
the study. To fulfill the primary objective of this study, the following specific tasks are
performed:
1.
2.
3.
4.
Investigate the distribution of makespans over all experimental problems.
Investigate the distribution of makespans in terms of the number of processors per stage.
Investigate the distribution of makespans in terms of the number of stages.
Investigate the distribution of makespans in terms of the number of jobs.
A computer program called AUTOMSCT is developed in this study. AUTOMSCT basically
enumerates the makespans of all MP schedules and counts the number of all possible makespans
generated by each FSMP problem. AUTOMSCT is coded in Microsoft Quick Basic. The
processing times of the jobs are generated from a uniform distribution over the integers from 1 to
99, exclusively. A random number generator in Quick Basic is used to produce these random
numbers. The test was performed on a Dell Latitude C600 laptop computer. It has a Pentium III
750MHZ processor and 256MB memory.
The FSMP problem size examined in this study is relatively small because the computational time
grows exponentially when the number of jobs increases. This limits the study to a maximum
number of jobs of seven. For each job configuration, there are at least two stages and at most four
stages. The number of identical processors at each stage ranges from 2 to N-1, where N is the
2
Table 1: Problem Breakdown of the Uniform FSMP Configuration
Jobs
3
4
Stages
2
3
4
2
Processors/Stage
2
2
2
2
3
2
3
2
3
2
3
4
2
3
4
2
3
4
2
3
4
5
2
3
4
5
2
3
4
5
2
3
4
5
6
2
3
4
5
6
2
3
4
5
6
3
4
5
2
3
4
6
2
3
4
7
2
3
4
3
No. of Problems
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
number of jobs. Table 1 is the breakdown of all job configurations tested in the experiment. The
table contains the number of jobs, the number of stages, the number of identical processors per
stage and the number of test problems. For each test configuration, 30 test problems are run.
Therefore, the test for the uniform FSMP environment contains 45 test configurations and 1350
problems.
To better evaluate the makespan distributions, the makespan values are standardized using the
following formula:
Relative Deviation =
Makespan  Optimal MP
Makespan
 100%.
Optimal MP
Makespan
(1)
The optimal makespans of these experimental problems are unknown. Therefore, the optimal
multiple-permutation (MP) makespan for each problem is used for calculating the relative
deviation. The optimal MP makespan is the minimum makespan of all n! makespans produced by
the AUTOMSCT program.
4. Experimental Results
4.1 Overall Results
The histogram in Figure 1 illustrates the makespan distributions of each problem configuration
and the overall distribution. It is obvious that the makespan distribution of the 7-job configuration
dominates the makespan distribution of all experimental problems because even the sum of the
numbers of schedules (296460) generated by all other job configurations is still too small when
comparing to the number of schedules (2268000) generated by the 7-job configuration. Table 2
shows a comparison of the overall performance of the 5 different job configurations. As shown by
Table 2, the number of schedules increases factorially as the number of jobs (N) increases. The
number of test problems for each job configuration is different from one configuration to another.
This is because the increase of number of jobs allows more number of processors to be added to
each stage and therefore increases the number of test problems for a certain job configuration. For
each problem, there are n! schedules generated. The overall number of schedules tested for each
job configuration varies significantly simply because the size of jobs and the number of problems
vary significantly. For example, the 3-job configuration generates 540 schedules over all test
problems and the 7-job configuration generates 2268000 schedules for all test problems. This
again explains why the overall makespan distribution shows the same performance as the 7-job
configuration. According to Table 2, it seems that the number of optimal schedules decreases
when the number of jobs increases. This needs more investigation because the comparison in
Table 2 is too generic to reflect the differences between the job configurations tested. Table 2 also
shows that the worst relative deviation is more than doubled when the number of jobs increases
from 3 to 7 jobs. The worst case happens mostly when there are 2 processors at each stage. This
means that increasing the number of processors at each stage will help improve the makespan
performance. More investigation needs to be conducted in order to better understand the influence
on the makespan when more processors are added to the FSMP. In any job configuration, more
than 50% of the schedules produce makespans that are within 10% of the optimal MP solution.
Moreover, more than 99% of the time, a schedule will produce a makespan that is within 50% of
the optimal MP value. This suggests that the probability that the makespan obtained from a
random schedule is within 10% of the optimal is more than 50%; and the probability that the
makespan obtained from a random schedule is within 50% of the optimal is more than 99%. This
4
Makespan Distributions
50%
45%
40%
Frequency (%)
35%
3 job
30%
4 job
5 job
25%
6 job
20%
7 job
15%
Over All
10%
5%
0%
0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90%
Relative Deviation from Optimal MP Makepsan
Figure 1 Makespan Distributions for FSMP
Table 2: Comparison of Makespan Distributions
Schedule Performance
3 job
4 job
5 job
6 job
7 job
1. N!
6
24
120
720
5040
2. No. of test problems
90
180
270
360
450
3. No. of schedules
540
4320
32400
259200
2268000
4. Percent of optimal schedules
43.33%
36.44%
30.69%
32.36%
29.21%
5. Worst relative deviation
40.53%
50.29%
64.63%
87.71%
85.71%
6. No. of stages (worst case)
2
3
4
2
2
7. No. of processors (worst case)
2
2
2
2
3
8. No. of worst cases
2
2
2
2
6
9. Within 10% of optimal
76.29%
67.04%
58.67%
57.61%
52.39%
10. Within 50% of optimal
100%
99.95%
99.68%
99.54%
99.06%
5
information indicates that it is quite difficult to get extremely bad makespan values (i.e. makespan
values that are not within 50% of the optimal).
4.2 Makespan Distributions vs. Number of Processors
It is found from previous section that the number of processors per stage may play a pivotal role
in the makespan distributions. Hence, the relationship between the number of processors at each
stage and the makespan distributions is further investigated. As shown by Table 1, there are three
different stage configurations, namely the two-stage, three-stage and four-stage configurations.
Theoretically the stage configuration will not affect the makespan distributions. Thus the twostage configuration is chosen to study the relationship between the number of processors and the
makespan distribution. Since the number of jobs ranges from 2 to N – 1 (N is the number of jobs),
the 7-job configuration is selected for better representation. Figure 2 and Table 3 show the
comparison of relative deviations between different processor configurations. It is noticed that the
number of optimal MP schedules is about 75% when there are 6 (N-1) processors. The worst
relative deviation is only about 40% when there are 6 (N-1) processors. It is obvious that
distributions of the relative deviations are getting less normal when the number of processors per
stage increases.
Figure 3 shows that the percentage of the optimal MP makespans may fit into an exponential
distribution when the number of processors increases from 2 to 6 (N-1). Hence, a hypothesis is
used in order to test the observations. The hypothesis is constructed as follows:
H0: For a 7-job, 2-stage FSMP problem, the number of optimal MP schedules in
percentage follows the exponential distribution with the mean equal to 0.8 when the
number of processors per stage increases from 2 to N-1 processors.
H1: For a 7-job, 2-stage FSMP problem, the number of optimal MP schedules in
percentage doesn’t follow the exponential distribution with the mean equals to 0.8
when the number of processors per stage increases from 2 to N-1 processors.
The Kolmogorov-Smirnov (K-S) Goodness of Fit Test is carried out in order to test the above
hypothesis. The Kolmogorov-Smirnov test is used to decide if a sample of data comes from a
specific distribution. Since the test statistic D (=0.417) is less than the critical value D0.1, 5 (=
0.510), the K-S test failed to reject the hypothesis H0 at the 0.1 significance level. Therefore, it
can be concluded that the number of optimal MP schedules in percentage follows exponential
distribution with mean equals to 0.8 with the number of processors per stage ranging from 2 to 6
(N-1). Details of this test may be found in Wang (2001).
Similarly, the K-S test was carried out to test the observations of 2-stage FSMP problems that
have 4, 5 and 6 jobs. For the K-S test of the 6-job, 2-stage FSMP problem, the null hypothesis
assumes that the mean of the exponential distribution equals to 0.9. The test statistic D equals to
0.435, which is less than the critical value D, n at 0.1 significance level (D0.1, 4 = 0.564). Hence,
the K-S test failed to reject the null hypothesis and the number of the best MP schedules in
percentage was concluded to follow the exponential distribution with mean = 0.9 with the number
of processors per stage ranges from 2 to 5.
For the K-S test of the 5-job, 2-stage FSMP problem, the null hypothesis assumes that the mean
of the exponential distribution equals to 1.0. The test statistic D equals to 0.517, which is less
than the critical value D, n at 0.1 significance level (D0.1, 3 = 0.642). Hence, the K-S test failed to
reject the null hypothesis and the number of the best MP schedules in percentage was concluded
6
Makespan Distribution - 7-job, 2-stage, 2-6 processor
80%
70%
60%
Percentage (%)
2processor
3processor
4processor
5processor
6processor
50%
40%
30%
20%
10%
0%
0%
20%
40%
60%
80%
100%
Relative Deviation
Figure 2 Makespan Distributions for 7-job, 2-stage and 2 to 6 processor FSMP
Table 3: Makespan Distributions vs. No. of Processors – 7-job, 2-stage FSMP
Schedule
Performance
7-job,
2-processor
5040
7-job,
3-processor
5040
7-job,
4-processor
5040
7-job,
5-processor
5040
7-job,
6-processor
5040
30
30
30
30
30
3. No. of schedules
151200
151200
151200
151200
151200
4. Percent of
optimal schedules
0.30%
3.21%
16.14%
43.41%
74.76%
5. Worst Deviation
67.41%
85.71%
61.33%
44.58%
39.29%
4
6
192
240
720
11.78%
21.97%
46.52%
71.74%
92.38%
1. N!
2. No. of test
problems
6. No. of worst
cases
7. Within 10% of
optimal
7
Optimality vs. Number of Processors
80%
74.76%
70%
60%
Percentage (%)
50%
43.41%
40%
30%
20%
16.14%
10%
3.21%
0.30%
0%
2-processor
3-processor
4-processor
5-processor
6-processor
Number of Processors
Figure 3 The MP Optimal Schedules – 7-job, 2-stage, 2 to 6 processors
to follow the exponential distribution with mean = 1.0 with the number of processors per stage
ranges from 2 to 4.
For the K-S test of the 4-job, 2-stage FSMP problem, the null hypothesis assumes that the mean
of the exponential distribution equals to 1.1. The test statistic D equals to 0.611, which is less
than the critical value D, n at 0.1 significance level (D0.1, 2 = 0.776). Hence, the K-S test failed to
reject the null hypothesis and the number of the best MP schedules in percentage was concluded
to follow the exponential distribution with mean = 1.1 with the number of processors per stage
ranges from 2 to 3.
Table 4 shows the means of the exponential distributions from 4- to 7-job FSMP problems, in
which there are 2 stages for each problem. A plot of the exponential distribution mean versus the
number of jobs is shown in Figure 4. As can be seen, a straight line fits the data very well, which
suggests that a direct linear relationship exists. A linear regression analysis was performed on the
data. Table 5 shows the results of the linear regression analysis. The correlation coefficient R^2 is
1, which means there is a direct linear relationship which exists between the number of jobs and
the exponential distribution mean. Therefore, if the number of jobs is known, it might be possible
to predict the exponential distribution mean. The equation of the fitted line is:
Y = 1.5 - 0.1X1,
where X1 is the number of jobs.
8
Table 4: The Mean of Exponential Distribution
Mean
4-job
5-job
6-job
7-job
1.1
1.0
0.9
0.8
Exponential Distribution Mean vs. Number of Jobs
1.2
1
0.8
0.6
0.4
0.2
0
4-job
5-job
6-job
7-job
Number of Jobs
Figure 4 Plot of the exponential distribution mean vs. the number of jobs
Table 5: Linear Regression Analysis on the Exponential Distribution Mean
Model is from : Y = B0 + B1*X1
COEFFICIENT
SE
T
B0 = 1.5
7.242792E-08
2.071024E+07
B1 = -0.1
1.290478E-08
-7749064
N=4
R^2 = 1
S = 2.885598E-08
9
In summary, this section investigated the relationship between the number of processor per stage
and the makespan distributions. It may be concluded from the study that the number of optimal
MP schedules increases exponentially as the number of processors per stage increases from 2 to
N-1 (N is the number of jobs). This means that increasing the number of processors per stage will
significantly improve the makespan performance. Moreover, the worst relative deviation from the
optimal MP makespan decreases when the number of processors increases. This implies that it is
more difficult to get extremely bad solutions when the number of processors per stage increases.
4.3 Makespan Distributions vs. Number of Stages
This section investigates the relationship between the makespan distributions and the number of
stages. The 7-job, 2-processor FSMP problems were tested in this study. The number of stages
ranges from 2 to 4. Figure 5 shows that the three different stage configurations follow similar
makespan distributions. To better compare the makespan distributions of these three different
stage configurations, a hypothesis is used and tested. The hypothesis is:
H0: There is no difference in the makespan distributions of the 7-job, 2-stage, 2-processor
FSMP problem, the 7-job, 3-stage, 2-processor FSMP problem and the 7-job, 4-stage, 2processor FSMP problem.
H1: At least one of the makespan distributions is significantly different.
A one-way analysis variance (ANOVA) was carried out in order to test the above hypothesis. The
details of the ANOVA may be found in Wang (2001). The ANOVA test cannot reject the
hypothesis H0. Therefore, it can be concluded from the test that there are no significant
differences between the makespan distributions of FSMP problems that have the same number of
jobs and processors but with different number of stages. This result implies that the makespan
performance of an FSMP problem may be predicted using the makespan distributions of another
FSMP problem with the same number of jobs and processors but with different number of stages.
Makespan Distributions vs. Number of Stages
25%
20%
Percentage (%)
2-stage
15%
3-stage
4-stage
10%
5%
0%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
Relative Deviation
Figure 5 Makespan Distributions – 7-job, 2-processor and 2, 3, and 4 stages
10
4.4 Makespan Distributions vs. Number of Jobs
The relationship between the makespan distributions and the number of jobs is examined in this
section. Figure 6 illustrates the makespan distributions when the number of jobs increases. It can
be seen from Figure 6 that makespan distribution are more spread out when the number of jobs
increases. Also, more schedules are getting worse relative deviations when the number of jobs
increases. Figure 7 shows the change of the number of optimal MP schedules when the number of
jobs increases. A K-S test is carried out in order to test if the change of the number of MP optimal
schedules follows the exponential distribution. The hypothesis below is constructed as follows:
H0: For a 2-stage FSMP problem with 2 processors per stage, the change of the number of
optimal MP schedules in percentage follows the exponential distribution with the mean
equals to 0.7 when the number of jobs increases from 3 to 7.
H1: For a 2-stage FSMP problem with 2 processors per stage, the change of the number of
optimal MP schedules in percentage doesn’t follow the exponential distribution with the
mean equals to 0.7 when the number of jobs increases from 3 to 7.
The K-S test failed to reject the null hypothesis H0 at the 0.01 significance level and the number
of optimal MP schedules in percentage was concluded to follow the exponential distribution with
mean = 0.7. Details of this test may be found in Wang (2001). This means that the probability
that a bad makespan is generated by a random schedule increases significantly when the
number of jobs increases.
Relative Deviation vs. Number of jobs
45%
40%
35%
Percentage (%)
3 job
30%
4 job
25%
5 job
20%
6 job
15%
7 job
10%
5%
0%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
Relative Deviation
Figure 6 Makespan Distributions vs. Number of jobs – 2-stage, 2-processor FSMP
11
Optimality vs. Number of jobs
45%
42.22%
40%
35%
Percentage (%)
30%
25%
20%
16.11%
15%
10%
3.50%
5%
0.95%
0.30%
0%
3 job
4 job
5 job
6 job
7 job
Number of Jobs
Figure 7 Optimal MP Schedules vs. Number of Jobs – 2-stage, 2-processor FSMP
5. Conclusions
This paper reveals the statistical nature of the makespan distributions in FSMP. The results of this
research can be used as guidance for other FSMP research. There are several important findings
regarding the makespan distributions. First, the number of processors per stage is the determinant
of the makespan distributions in FSMP. As the number of processors increases, the number of
optimal MP schedules increases exponentially. Moreover, the makespan distributions are less
spread out when there are more processors. This implies that the probability that a fairly good
makespan is generated by a random schedule increases significantly as more processors are added
to the FSMP environment. The result about the number of processors also suggests that it is more
meaningful to specify the number of processors at each stage when performing statistical studies
in the FSMP environment. Second, the number of stages in the FSMP problem doesn’t affect the
makespan distributions significantly. This result suggests that the number of stages is an
insignificant variable in the FSMP statistical studies. The results of statistical studies on a specific
stage configuration may also apply to configurations with different number of stages (with the
same number of jobs and processors). Third, when the number of processors per stage is the
same, the number of optimal schedules decreases exponentially when the number of jobs
increases. This means that the probability that a bad makespan is generated by a random schedule
increases significantly when the number of jobs increases. The np-hard nature of the FSMP
problem forces researchers towards the development of heuristic procedures, which hopefully can
generate a good makespan solution in a timely manner. In reality, the optimality of the FSMP
problem’s solution is normally sacrificed for less time computational times. When dealing with
medium or large sized FSMP problems, heuristics that generate near-optimal makespan solutions
with less computational times will be more effective.
12
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13