Because our objective is minimum makespan, total setup time and the distribution of jobs to parallel machines both are related to the qualities of testing solutions. In order to find the setting parameters of improving heuristics for reducing total setup time, we put these WPSP algorithms into a lot of pre-tests. The following settings of WPSP algorithms are efficient in solving identical parallel-machine scheduling problems. The parameters A and B of modified sequential savings algorithm (MSSA) are set to be 0.975 and 0.55. The parameter λ of parallel insertion with the slackness (PIA II) is set to be 0.7. And the param ter e ϕ of parallel insertion with
e (PIA IV) is set to be 0.8. The improving heuristics are encoded in Visual Basic 6.0, which are implemented in the compiled form on a PC with
And the average of testing solutions solved by PIA III in 16 problems is the smallest, which means PIA III is most efficient with minimum makespan. By considering one experimental factor once, the computational results of 16 problems can be tran
all kinds of situations, PIA III having 8 and 7 smallest ones individually also shows the variance of regret measur
AMD 1150 MHz CPU and 512 MB RAM.
6.1 ANOVA Analysis of Improving Heuristics
The CPU times cost by saving algorithms is 1137.396 seconds in average, and the CPU time cost by insertion algorithms is 73.629 seconds in average. The computational results of WPSP algorithms in 16 problems are expressed in Table 3.
The testing results show that SSA and SIA are not robust in our testing problems because SSA and SIA would generate unfeasible solutions in some cases. Therefore, we look for best solutions exclusive of SSA and SIA and find that PIA III has the largest number of best solutions via Table 3.
in solving the 16 testing problems of WPSP
sformed into the performance comparison of all situations as shown in Table 4. From the opinion of comparing mean and standard deviation of solutions in
that it outperforms other WPSP algorithms except for the situation, where total processing time is low.
Table 3. Computational results of WPSP algorithms in 16 test problems.
SSA MSSA SIA PIA PIA I PIA II PIA III PIA IV
1 2730 2965 2724 2837 2660 2599* 2600 2639 2653
2 3297 3709 3240 3194 3218 3157* 3291
3 2730 2661 2611* 2611 2703 2616 2632 2611* 2669
4 3297 3186 3083* 3116 3197 3115 3158 3105 3140
5 3196 3152 2795 2837 2879 2728* 2757 2730 2778
6 3763 4308 3461 3265 3357 3248* 3955
7 3196 2682 2739 2611 2805 2683 2758 2659* 2770
8 3763 3253 3223 3116 3376 3201 3223 3194* 3242
9 2687 2692 2697 2664 2635 2566* 2569 2594 2644
10 3253 3648 3476 3193 3140 3111* 3125 3225
11 2687 2548 2597 2604 2627 2577 2624 2565* 2624
12 3253 3113 3127 3063 3094 3081 3066* 3083 3116
13 3153 3600 2766 2977 2782 2732 2714* 2731 2930
14 3719 3596 3447 3284 3274* 3274* 3675
15 3153 2721
16 3719 3201
Problem
No. Cmax Cmax Cmax Cmax Cmax Cmax Cmax Cmax
2713 2696 2728 2662 2669 2654* 2729
3235 3245 3295 3223* 3228 3223* 3269
Mean 3087.438 3007.625 2916.625 2934.875 2912 3044.375
No. of * 2 4 4 9
Result with grey bcakground indicates a unfeasible solution in scena Label * means the best of all exclusive of SSA and SIA.
Expected Capacity
Table 4. Computational results of improving heuristics under all kinds of situations.
n Mean Stdev Mean Stdev Mean Stdev Mean Stdev
Total 16 3087.44 485.11 3007.63 309.04
R=2 8 3149.00 589.21 3040.13 314.85
R=6 8 3025.88 385.29 2975.13 321.12
Te=Yes 8 3171.88 558.31 3096.63 325.18
Te=No 8 3003.00 419.85 2918.63 284.27
T_Due=Stable 8 3258.88 600.47 3037.13 339.57
T_Due=Increase 8 2920.63 292.66 2916.00 276.65 2882.75* 275.89 2978.13 295.58 Total_PT=Low 8 2877.63 350.02 2705.25 69.66 2729.63 138.16 2727.38 89.24
Total_PT=High 8 3469.63 406.89 3287.88 131.07
Number of * 1
n Mean Stdev Mean Stdev Mean Stdev Mean Stdev Total 16 2916.63 287.85 2934.88 289.49 2912.00* 279.90 3044.38 392.77 R=2 8 2925.13 292.62 2962.88 305.12 2917.87* 280.73 3062.25 443.37 R=6 8 2908.13 302.93 2906.88 291.02 2906.13* 298.32 3026.50 365.06 Te=Yes 8 2972.25 291.64 2997.50 295.96 2964.13* 291.52 3168.50 455.44 Te=No 8 2861.00 292.28 2872.25 288.14 2859.88* 276.92 2920.25 296.54 T_Due=Stable 8 2938.50 310.03 2950.00 322.71 2937.25* 289.28 3143.88 483.84 T_Due=Increase 8 2894.75 283.45 2919.75 273.71 2886.75 287.65 2944.88 271.75 Total_PT=Low 8 2645.37* 65.48 2665.38 71.45 2647.88 59.79 2724.63 101.39 Total_PT=High 8 3187.88 71.37 3204.38 92.00 3176.13* 69.67 3364.13 294.29
Number of * 1 7
Result with grey bcakground means there are unfeasible solutions in scenario
Label * indicates the best among algorithms in situations considering single factor and whole conditions.
PIA III PIAIV
PIAI PIAII
SSA MSSA SIA PIA
In order to find the effects of improving heuristics and experimental factors on the problem design, we use statistical analysis by applying statistical software, SAS.
First of all, we check the satisfaction of normality assumption for 96 data of Table 3 except for SSA and SIA. The check of normality assumption is expressed in Table 5 and the solutions are normally distributed. Then use ANOVA to check for the significances of all experimental factors and interactions. The summary of ANOVA table shown in Table 6 shows that five single factors, product family ratio, temperature changing consideration, tightness of due date, total processing time level,
and algorithms, would significantly affect the solutions of WPSP with minimum makespan under 99% confidential intervals. Besides, p values of interactions less than 0.01 also have significant effect on the performance of testing problems.
Through Duncan’s multiple comparison as shown in Table 7, the statistical results show that the multiple comparisons divide WPSP algorithms into two groups, A and B.
The same letter of Duncan’s groups indicates that there is no significant difference between WPSP algorithms. So the first group is MSSA, PIA IV, and PIA, of which the performance of solutions is inferior to the second group of PIA II, PIA I, and PIA III.
Table 5. Check of normality assumption for 96 solutions in 16 test problems.
Statistic p Value Test
Shapiro-Wilk W 0.886611 Pr < W <0.0001 Kolmogorov-Smirnov D 0.17839 Pr > D <0.0100 Cramer-von Mises W-Sq 0.58319 Pr > W-Sq <0.0050 Anderson-Darling A-Sq 3.464467 Pr > A-Sq <0.0050
Table 6. The summary of ANOVA table under 99% confidential intervals.
Factor SS d.f. MS F value p-value
R 63500 1 63500 10.548 <0.01
Te 583908 1 583908 96.991 <0.01
Total_PT 8516246 1 8516246 1414.611 <0.01
Algorithm 432625 5 86525 14.372 <0.01
R*Te 184 1 184 0.031
R*T_Due 11726 1 11726 1.948
Te*T_Due 47126 1 47126 7.828 <0.01
R*Total_PT 13325 1 13325 2.213
Te*Total_PT 32893 1 32893 5.464
T_Due*Total_PT 170775 1 170775 28.367 <0.01
R*Algorithm 33404 5 6681 1.110
Te*Algorithm 59144 5 11829 1.965
T_Due*Algorithm 313319 5 62664 10.409 <0.01 Total_PT*Algorithm 168813 5 33763 5.608 <0.01
Error 156525.3 26 6020
T_Due 350779 1 350779 58.267 <0.01
72 .
26
7
, 1 , 01 .
0
=
F F0.01,5,26
= 3 . 82
Table 7. Duncan’s multiple comparisons for the performance of WPSP algorithms.
Duncan
Grouping Mean No. of
problems Algorithm
A 3087.44 16 MSSA
A 3044.38 16 PIA IV
A 3007.63 16 PIA
B 2934.88 16 PIA II
B 2916.63 16 PIA I
B 2912 16 PIA III
6.2 Computational Results of GA with Initial Population from WPSP algorithms
We use WPSP algorithms except for SSA and SIA for generating initial population
of GA these
initial solutions to enlarge our population size for larger species of chromosomes. In our problem design, we set the population size to be 30. Other genetic factors, mutation rate and generation size, are considered in testing problems because they would affect genetic combinations of chromosomes. We select problems No.7 and No. 8 of Table 2 for testing the performance and solution time of GA. One is the situation that product family ratio is 2, temperature changing is considered, total processing time level is low, and tightness of due dates is increasing. The other selected is the situation that product family ratio is 2, temperature changing is considered, total processing time level is high, and tightness of due dates is increasing.
The mutation rate (denoted as pm) is divided into five levels, 0, 0.25, 0.5, 0.75, and 1.
A s
equal to 1500. Each problem is solved by hybrid GA with different mutation rates and repeated four times for checking if the mutation rate is significantly effective.
The statistical results of hybrid GA in problem No. 7 and No. 8 are shown in Table 8. It shows that hybrid GA would improve the initial solutions while the mutation rate is larger than 0. And we know that the generation number is proportional to the
running times of GA in 250, 500, and 750 generations are about 191 seconds, 401 se
. Because the initial solutions are not sufficient enough, we make use of
nd the hybrid GA is proceeding until the number of generation (denoted as gen) i
performance of scheduling solutions of WPSP with minimum makespan. The
conds, and 563 seconds individually. They are apparently larger than the running time of improving heuristics. Because the roulette wheel method is selecting chromosomes based on fitness values randomly, there is not a definite mutation rate used for finding the best solution of hybrid GA. So we consider all kinds of mutation rates in generations and plot the flowcharts of solutions solved by hybrid GA as shown in Figure 7 to Figure 10 in the appendix. They show the trend of mean performance solved by hybrid GA in problem No. 7 and No. 8. Observing the tendency of solutions solved in Figure 7 and Figure 9, it reveals that the value of
average performance via hybrid GA would drop very fast before initial generations (about 100 generations). Based on statistical data of hybrid GA repeated four times in 1500 generations, we find that the best solution of hybrid GA is tending to be improved after later generations (about 40 generations) when the total processing time level is low. In contrast with the situation while the total processing time level is high, we can see that the improving time point of hybrid GA in problem No. 7 is significantly later. So hybrid GA may have a faster speed of feedback for improving the best solution unde
8 1 250 3675 0.25 200.91
m No.
2659* 75.64*
r tough situations.
Table 8. The comparison of hybrid GA with WPSP algorithms and improving heuristics in testing problems.
The best sol. by
Label * means the better between hybrid GA and improving heuristics.
Proble
6.3 Further Improvement of Hybrid GA with Initial Population from Improving Heuristics
We apply scheduling so s in initial population of
hybrid GA for further improvement in 16 testing problems. We run 1500 generations of hybrid GA with 0.5, 0.75, and 1 of mutation rate for confirming
lutions of improving heuristic
whether the scheduling solutions of improving heuristics can be improved or not.
The computational results are shown in Table 9 and expresses that we may not use hybrid GA for improving solutions of improving heuristics in the situation where the tightness of due date is stable. And hybrid GA can improve solutions generated by improving heuristics in problem No. 8 and No. 16, where the temperature change consideration is yes, tightness of due date is increasing, and total processing time level is high. The trends of solutions generated by hybrid GA in problem No. 8 and No.
16 are shown in Figure 11 and Figure 12 individually. The evidential results show that hybrid GA can improve in earlier generations in problem No. 8 (R=2, Te=yes, T_Due=increase, and Total_PT=high) than hybrid GA in problem No. 16 (R=6, Te=yes, T_Due=increase, and Total_PT=high). Through solutions of hybrid GA with initial population generated by WPSP algorithms and improving heuristics, we find that the performance of hybrid GA would stop improving after latter periods of generations (about 1500 generations).
Table 9. Computation results of GA with initialization of improving heuristics.