CHAPTER 6 A PARTICLE SWARM OPTIMIZATION FOR THE OPEN SHOP
6.4 Decoding Operators
6.6.2 Comparison with Other Metaheuristics
We compared our PSOs with the GA proposed by Liaw (2000) (denoted by GA-Liaw), the GA proposed by Prins (2000) (denoted by GA-Prins), and the ACO hybridized with beam search proposed by Blum (2005) (denoted by Beam-ACO). Our results were obtained by 20 runs on each problem, and the time limit of each run is half of Beam-ACO. The program only stops at the CPU time limits even if the lower bound reached. It is important to remember that both our PSO and Beam-ACO performed 20 runs, but the two GAs, GA-liaw and GA-Prins, performed only one run.
Moreover, in our PSO and Beam-ACO, the computation time of each run is substantially longer than GA-liaw and GA-Prins. Therefore, we will mainly compare our computational results with Beam-ACO.
Table 6.2 shows the results of the test problems proposed by Taillard (1993). The term ‘t’ is the average time needed by one run to get its best makespan value.
Compared with other test problem sets, Taillard test problems are easier to solve.
Therefore, Beam-ACO and PSO-mP-ASG2+BS obtained all of the optimal solutions, but PSO-mP-ASG2+BS is slightly better than Beam-ACO on average gaps. The t of PSO-mP-ASG2 and PSO-mP-ASG2+BS are quite similar on the problem set tai_4×
4_*. The reason for this is that the search area of these two PSOs is restricted by the parameter delayweight . In the decoding operator, the delayweight increases linearly form 0 to 1 during a run. This means that the search area of PSO is near to the non-delay set in earlier iterations, and then enlarges after delayweight increases. If the problem size is quite small but the t is rather large, it means that the optimal solution is far from the non-delay set, and the PSO can only obtain the optimal solution when delayweight increases.
Table 6.3 shows the results of the test problems proposed by Brucker et al.
(1997). Although there are two best solutions obtained by PSO-mP-ASG2+BS worse than Beam-ACO on test problem j7-per0-0 and j7-per20-1, all the average gaps obtained by PSO-mP-ASG2+BS are better than Beam-ACO. Furthermore, PSO-mP-ASG2+BS obtained new best-known solutions on test problem j8-per10-1 and j8-per10-2.
Table 6.2 Results of the test problems proposed by Taillard (1993)
Beam-ACO PSO-mP-ASG2 PSO-mP-ASG2+BS Time Problem BKS GA-Liaw GA-Prins
Best Average Best Average t Best Average t limit(s) tai_4×4_1 193 193 193 193 193.0 193 193.0 4.7 193 193.0 4.7 8
Table 6.2 (continued)
Beam-ACO PSO-mP-ASG2 PSO-mP-ASG2+BS Time Problem BKS GA-Liaw GA-Prins
Best Average Best Average t Best Average t limit(s) tai_15×15_1 937 937 937 937 937.0 937 937.0 4.6 937 937.0 4.3 112.5 tai_15×15_2 918 918 918 918 918.0 918 918.9 20.5 918 918.0 9.1 112.5 tai_15×15_3 871 871 871 871 871.0 871 871.0 5.0 871 871.0 4.3 112.5 tai_15×15_4 934 934 934 934 934.0 934 934.0 3.2 934 934.0 3.9 112.5 tai_15×15_5 946 946 946 946 946.0 946 946.4 16.1 946 946.0 5.7 112.5 tai_15×15_6 933 933 933 933 933.0 933 933.0 10.0 933 933.0 4.7 112.5 tai_15×15_7 891 891 891 891 891.0 891 891.6 22.3 891 891.0 10.4 112.5 tai_15×15_8 893 893 893 893 893.0 893 893.0 1.2 893 893.0 17.3 112.5 tai_15×15_9 899 899 899 899 899.7 899 903.3 46.6 899 899.2 26.6 112.5 tai_15×15_10 902 902 902 902 902.0 902 903.0 12.7 902 902.0 6.9 112.5 Average gap 0.000% 0.000% 0.000% 0.007% 0.000% 0.079% 0.000% 0.002%
tai_20×20_1 1155 1155 1155 1155 1155.0 1155 1155.8 24.9 1155 1155.0 16.6 200 tai_20×20_2 1241 1241 1241 1241 1241.0 1242 1245.2 56.8 1241 1241.0 23.5 200 tai_20×20_3 1257 1257 1257 1257 1257.0 1257 1257.0 5.5 1257 1257.0 19.6 200 tai_20×20_4 1248 1248 1248 1248 1248.0 1248 1248.0 4.0 1248 1248.0 19.6 200 tai_20×20_5 1256 1256 1256 1256 1256.0 1256 1256.0 7.9 1256 1256.0 19.6 200 tai_20×20_6 1204 1204 1204 1204 1204.0 1204 1204.1 12.0 1204 1204.0 19.6 200 tai_20×20_7 1294 1294 1294 1294 1294.0 1294 1296.6 48.5 1294 1294.0 25.4 200 tai_20×20_8 1169 1177 1171 1169 1170.3 1173 1177.6 80.9 1169 1170.0 50.9 200 tai_20×20_9 1289 1289 1289 1289 1289.0 1289 1289.0 2.7 1289 1289.0 78.2 200 tai_20×20_10 1241 1241 1241 1241 1241.0 1241 1241.0 1.2 1241 1241.0 78.2 200 Average gap 0.068% 0.017% 0.000% 0.011% 0.042% 0.134% 0.000% 0.008%
Table 6.3 Results of the test problems proposed by Brucker et al. (1997)
Beam-ACO PSO-mP-ASG2 PSO-mP-ASG2+BS Time Problem BKS GA-Liaw GA-Prins
Best Average Best Average t Best Average t limit(s) j5-per0-0 1042 1042 1050 1042 1042.0 1042 1042.0 6.4 1042 1042.0 6.3 125
Table 6.4 shows the results of the test problems proposed by Guéret and Prins (1999). In this problem set, PSO-mP-ASG2 performs better than other algorithms and obtained 16 new best-known solutions. The PSO hybridized with beam search does not perform well, because the instances from (Guéret & Prins, 1999) are designed to maximize a new lower bound greater than the traditional one. Therefore, the traditional lower bound calculated by equations (5.6), (5.7), and (5.8) is hard to discriminate in the good and bad partial solutions in this problem set (Guéret & Prins, 1999).
Consequently, better solutions can be obtained when the PSO performs much more iterations instead of spending computation time on an inefficient beam search.
However, it is evident that both the PSOs outperform Beam-ACO in this problem set.
Table 6.4 Results of the test problems proposed by Guéret and Prins (1999)
Beam-ACO PSO-mP-ASG2 PSO-mP-ASG2+BS Time Problem BKS GA-Prins
Best Average Best Average t Best Average t limit(s) gp03-01 1168 1168 1168 1168.0 1168 1168.0 0.0 1168 1168.0 0.0 45
Table 6.4 (continued)
Beam-ACO PSO-mP-ASG2 PSO-mP-ASG2+BS Time Problem BKS GA-Prins
Best Average Best Average t Best Average t limit(s) gp07-01 (1159) 1159 1159 1159.0 1159 1159.0 180.5 1159 1159.3 223.7 245 gp07-02 (1185) 1185 1185 1185.0 1185 1185.0 0.3 1185 1185.0 1.2 245 gp07-03 1237 1237 1237 1237.0 1237 1237.0 3.9 1237 1237.0 9.5 245 gp07-04 (1167) 1167 1167 1167.0 1167 1167.0 188.2 1167 1167.0 160.4 245 gp07-05 1157 1157 1157 1157.0 1157 1157.0 130.3 1157 1157.0 139.1 245 gp07-06 (1193) 1193 1193 1193.9 1193 1193.0 153.1 1193 1193.1 198.6 245
6.7 Concluding Remarks
We have presented a PSO for open shop scheduling problems in this chapter. We modified the representation of particle position, particle movement, and particle velocity to better suit it for OSSP. We also proposed a new decoding operator (mP-ASG), which decodes particle positions into parameterized active schedules.
Furthermore, we added fathoming constraints to mP-ASG and then hybridized it with beam search. The computational results show that our PSO can obtain many new best-known solutions of the test problems.
For further research, we will try to apply our PSO to other shop scheduling problems. In addition, further research topics include how to modify the particle position representation, particle movement, and particle velocity to better suit them to the problem. Table 6.5 shows the summary of the PSO for OSSP.
Table 6.5 Summary of the PSO for OSSP
Components The concept of this components
1 Particle Position
Representation Priority weights
We represent the preference-list by priorities, which can save computation time when we implement insert operator.
Particle Velocity Inertia
2
Particle Movement Insert operator
Because there is no preference constraint between the operations of a job, if we swap two operations at the same time, the new solution will be much different than the original one and lose the correlation property. Therefore, we implement the insert operator, just move one operation at a time, and earn more correlation property.
3 Decoding Operator
G&T algorithm mP-ASG Beam search
The mP-ASG can much restrict the search area but not exclude the optimal solution.
4 Other Strategies Diversification
The diversification strategy can prevent particles rapped in local optima.
We do not implement the local search strategy, because the neighborhood size of OSSP is huge and the local search strategy is in efficient.
6.8 Appendix
A pseudo code of the PSO for OSSP is given below:
//↓initializing
Initialize a population of particles with random positions.
for each particle k do
Decode X (the position of particle k) into a schedule k Sk. Set the kth pbest solution ( pbest ) equal to k Sk, pbest ←k Sk. end for
Set gbest solution equal to the best pbest . k //↑initializing
repeat
Update velocities according to Figure 6.1.
for each particle k do
Move particle k according to Figure 6.2.
Decode X into k Sk.
Update pbest solutions and gbest solution according to Figure 5.6.
end for
until maximum CPU time limit is reached.
CHAPTER 7
CONCLUSION AND FUTURE WORK
7.1 Conclusions
The original PSO is used to solve continuous optimization problems. Due to solution spaces of discrete optimization problems are discrete, we have to develop new PSO designs to better suit it for discrete optimization problems. The contribution of this research is that we proposed several PSO designs for discrete optimization problems. The new PSO designs are better suit for discrete optimization problems, and differ from the original PSO. In this research, we separated a PSO design into five parts: particle position representation, particle velocity, particle movement, decoding operator, and other search strategies. We can develop a new PSO design by redesign these five parts.
In chapter 4, we presented a binary PSO for the multidimensional 0-1 knapsack problem (MKP). This PSO design focuses on the concept of building blocks, which is the basis of another evolution computation algorithm—genetic algorithm. The particle velocity is represented by blocks. Particles obtain new blocks from gbest and pbest solutions. Moreover, the selection strategy can recognize superior blocks and and accumulate the superior blocks in the swarm. The computational results show that the concept of building blocks works in PSO design. Therefore, we can design other new PSOs based on the concept of building blocks.
In chapter 5, we presented a PSO for the job shop scheduling problem (JSSP).
This PSO design focuses on the particle position representation. In this PSO, the particle position is represented by preference list-based representation, which has
also tested and compared these two particle position representations. The computational results show that the preference list-based representation we proposed outperforms the priority based representation. It also demonstrated that the particle position representation with more Lamarckian performs better. Therefore, we can design new particle position representation based on Lamarckian property in further researches.
In chapter 6, we presented a PSO for the open shop scheduling problem (OSSP).
This PSO design focuses on comparing decoding operators. In this PSO, we implemented a modified parameterized active schedule generation algorithm (mP-ASG), which decodes particle positions into parameterized active schedules. In mP-ASG, we can reduce or increase the search area between non-delay schedules and active schedules by controlling the maximum delay time allowed. Furthermore, we added fathoming constraints to mP-ASG and then hybridized it with beam search. The computational results show that a decoding operator, which can map the positions to the solution space in a smaller region but not excluding the optimal solution, is performs better.
7.2 Future Works
There are two aspects for further research: (1) new PSO designs, and (2) other applications. We described some principles for new PSO designs in chapter 3. In the further research, we can develop new PSO designs by these principles and find out new design principles at the same time.
On the other hand, we can also implement the new PSO designs to other combinatorial optimization problems for example: assignment problems, network problems, multiobject combinatorial optimization problems…etc.
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作者簡介 (Biography)
姓名(Name)：徐誠佑 (Cheng-Yu Hsu) 學歷：
專士 明志工專 工業工程與管理科 (80.9~85.7)
學士 台灣科技大學 工業管理系 (88.9~90.7)
碩士 清華大學 工業工程與工程管理學系 (90.9~92.7)
博士 交通大學 工業工程與管理學系 (92.9~96.6)
著作：
一、 已接受之期刊論文
1. D.Y. Sha, C.-Y. Hsu, 2007, “A new particle swarm optimization for the open shop scheduling problem,” Computers and Operations Research (Accepted) (SCI).
2. D.Y. Sha, C.-Y. Hsu, 2006, “A hybrid particle swarm optimization for job shop scheduling problem,” Computers & Industrial Engineering, Vol. 51, No. 4, pp.
791-808 (SCI).
二、 審查中期刊論文
1. D.Y. Sha, C.-Y. Hsu, “A new discrete binary particle swarm optimization for the multidimensional 0-1 knapsack problem,” (submitted to European Journal of Operational Research) (SCI).
三、 研討會論文
1. D.Y. Sha, C.-Y. Hsu, 2006, “A particle swarm optimization with parameterized active schedules for the open shop scheduling problem,” Proceedings of the 2006 CIIE Annual Conference.
2. D.Y. Sha, C.-Y. Hsu, 2006, “A modified parameterized active schedule generation algorithm for the job shop scheduling problem,” Proceedings of the 36th International Conference on Computers and Industrial Engineering (ICCIE 2006), pp. 702-712, Taiwan, ROC.