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for each problem is limited to 200 seconds. In Table 2, AVE, MIN, and STD
indicate the values of average, minimum, and standard deviation, respectively.
T presents the average computation time (given in seconds) that the solution
converges to the final solution.
Tabl e 2.
Comparison results on Liao's benchmark problems
AIS
PSO
IDABC
Problem
AVE MIN STD T(s) AVE MIN STD T(s) AVE
MIN STD T(s)
j
30
c
5
e
1 485
.
35 479 2
.
58 99
.
44 474
.
70 471
1
.
42
96
.
16
465
.
15 463
1
.
50
56
.
81
j
30
c
5
e
2 620
.
70 619 1
.
63 80
.
24 616
.
25
616
0
.
44 55
.
28
616
51
j
30
c
5
e
3 625
.
70 614 4
.
81 116
.
70 610
.
25 602 4
.
70 64
.
56
596
.
4 593 1
.
70 49
.
14
j
30
c
5
e
4 588
.
55 582 3
.
38 108
.
63 577
.
10 575 1
.
52 86
.
98
566
.
2 565 1
.
20 39
.
29
j
30
c
5
e
5 618
.
75 610 3
.
42 101
.
19 606
.
80 605
1
616
0
1
.
.
11
79
.
84
602
600
1
.
56
57
.
67
02
j
30
c
5
e
7 641
.
30 635 4
.
67 93
.
56 630
.
60 629 0
.
75 87
.
18
626 626 0 18
.
68
j
30
c
5
e
8 697
.
50 686 5
.
14 100
.
68 684
.
20 678 2
.
50 97
.
67
674
.
65 674 0
.
88 55
.
18
j
30
c
5
e
9 670
.
20 662 3
.
85 100
.
75 654
.
65 651 1
.
87 83
.
80
643
.
65 642 1
.
04 67
.
49
j
30
c
5
e
10 613
.
45 604 5
.
33 89
.
29 599
.
75 594 5
.
28 77
.
46
576
j
30
c
5
e
6 625
.
75 620 3
.
01 100
.
47 612
.
50 605 3
.
49 67
.
99
603
.
05 601 1
.
47 55
.
05
Average 618
.
73 611
.
10 3
.
78 99
.
09 606
.
68 602
.
60 2
.
31 79
.
69
596
.
94 595
.
31
.
08 47
.
69
.
25 573 1
.
52 76
.
In Table 2, the smallest values of AVE, MIN, STD, and T in the rows are
shown in bold, respectively. It can be noted that the overall mean values of
AVE, MIN, and STD yielded by the IDABC algorithm are equal to 596.94,
595.3, and 1.08, respectively, which are much better than those generated by AIS
and PSO. Besides, the average computation time T of IDABC: 47.69 seconds is
much shorter than that generated by AIS and PSO. From these observations, it
is shown that the IDABC algorithm can obtain a better solution than AIS and
PSO in an obviously shorter computational time. This means that the IDABC
algorithm can converge to the good solutions faster than AIS and PSO. Also, it
can be seen that the IDABC algorithm is more robust than both AIS and PSO
for Liao's benchmark problems.
5 Conclusions
This paper establishes the model of the HFS problem by employing the vector
representation and presents an improved discrete artificial bee colony (IDABC)
algorithm for the HFS problem to minimize the makespan. Our future work is
to extend the IDABC algorithm to other kinds of scheduling problems such as
stochastic scheduling and multi-objective scheduling.
Acknowledgments.
The authors are grateful to Carlier, Neron, and Liao for
making the benchmark set available and to the anonymous reviewers for giving
us helpful suggestions. This work was supported by National Natural Science
Foundation of China (Grant no. 61174040, 61104178), the Fundamental Research
Funds for the Central Universities.
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