Table 3. Statistical Analysis for Multiple Problems
Measurement
N
Average fitness
共makespan兲
Average CPU run
time 共s兲
Average unique
schedules
Serial schedule
generation
scheme
Parallel schedule
generation
scheme
t score
共p value兲
30
30
—
52.5
53.3
2.65共0.0065兲
14.40
19.88
5.03共0.0000兲
1,747
1,748
0.29共0.3900兲
Fig. 7. Comparison of the average fitness values
when possible 共Meyer and Krueger 1998兲. The input parameter
values for the Elitist GA were set as follows: the population size,
crossover rate, and mutation rate were set to 50, 0.5, and 0.03,
respectively. The Elitist GA was terminated when it met 50 generations.
T-statistic was used to make an inference on the mean of the
population differences with the assumption that the distribution of
differences is approximately normal. The point estimate of the
mean is the mean of the sample difference. Fig. 7 compares the
average fitness values obtained from the Elitist GA according to
the scheme. The result indicates that, for most pairs of the average
fitness values, those obtained from the parallel scheme had similar values to those obtained from the serial scheme. Only two
pairs resulted in lower values obtained from the parallel scheme.
Also, the result presents a lot of variation in the average fitness
values of the problems. This variation in the values is accounted
for by the matched pairs design; each pair is matched by factors
that might affect the number of unique schedules. A 95% confidence interval for the difference between mean fitness values was
constructed. The point estimate of the true mean difference in the
average fitness values is 0.833. With 95% accuracy, the mean
difference in the values is between 0.190 and 1.477. The mean
difference means that the values obtained from the serial scheme
have the average values from 0.190 to 1.477, which are lower
than the values obtained from the parallel scheme.
The hypotheses to test whether the average fitness values 共␮1兲
obtained from the serial scheme exceed those 共␮2兲 obtained from
the parallel scheme are Ho: ␮1 – ␮2 = 0 and Ha: ␮1 – ␮2 ⬎ 0.
Table 3 tabulates the statistical results for the average fitness values, total algorithm run time, and the number of unique schedules
as a result of scheduling 30-problem instances with 30 activities.
The Elitist GA found 52.5 and 53.5 average fitness values for the
serial scheme and the parallel scheme, respectively, as they converge to a single point across the number of generations. For the
test of the average fitness values, we reject the null hypothesis
because the observed significance level or p value of 0.006 5 is
less than ␣ = 0.05. Therefore, we have sufficient evidence to conclude that the mean difference is greater than 0 or the mean values
for the average fitness values obtained from the parallel scheme
exceeds the values obtained from the serial scheme. In other
words, the serial scheme produces smaller fitness values than the
parallel scheme.
The total CPU run times are 8 min. 35 s. and 10 min. 34 s. for
the serial scheme and the parallel scheme to solve the problem
instances, respectively. For the test of the average CPU run time
in seconds, we reject the null hypothesis since the observed significance level or p value of 0.000 0 is less than ␣ = 0.05. There-
fore, we have sufficient evidence to conclude that the mean
difference is greater than 0 or the mean values for the average
CPU run time obtained from the parallel scheme exceeds those
obtained from the serial scheme.
Finally, the Elitist GA generated 1,747 and 1,748 unique
schedules, which amounts to 69.88 and 69.92% of the total schedules of 2,500, respectively. For the test of the number of unique
schedules, we do not reject the null hypothesis since the observed
significance level or p value of 0.390 0 is greater than ␣ = 0.05.
Therefore, we do not have sufficient evidence to conclude that the
mean difference is greater than 0 or the mean values for the number of unique schedules obtained from the parallel scheme exceeds those obtained from the serial scheme. The number of
unique schedules generated from either the serial scheme or the
parallel scheme does not have any difference for the Elitist GA.
These findings coincide with several experimental studies which
supported that the serial scheme yields better results when a large
number of schedules were computed for one project instance
共Kolisch 1996; Hartmann and Kolisch 2000兲.
Application Two Schemes into Construction Project
A coke bunker construction project 共Willis and Hastings 1976兲
was used to verify the performance and robustness of the Elitist
GA, varying two different schedule generation schemes. The
warehouse project is to be completed for the purpose of storing
coke and loading and dispatching trucks. The main features include a block of bunker, interconnecting conveyors, and weighing
facilities. An area of hardstand and an access road have to be laid
to the site around the bunkers. The project consists of 29 nondummy activities and each activity has six multiple resources,
which include laborers, steel men, concrete men, bricklayers,
cranes, and dumpers. The project duration calculated from the
critical path method 共CPM兲 with the assumption of the unlimited
resources is 33 weeks while the optimal project duration obtained
from the branch-and-bound method of Willis and Hastings 共1976兲
is 35 weeks. Fig. 8 shows the performance of the Elitist GA with
two different schedule generation schemes for the coke bunker
project. The project duration obtained from the Elitist GA is 35
weeks, which is the same as the optimal project duration for the
bunker project. The performance result indicates that the series
scheme performs better than the parallel scheme in that the serial
scheme started to reach the best fitness for the first time relative to
the parallel scheme and the fitness values did converge constantly
to the optimal solution. The Elitist GA with the serial scheme is a
better combination for solving the RCPSP than that with the parallel scheme within a reasonable amount of time.
JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / FEBRUARY 2010 / 167
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