106
CHAPTER 8
FIREFLY ALGORITHM
8.1
INTRODUCTION
The Firefly Algorithm (FA) is a nature - inspired algorithm which is
based on the social flashing behavior of fireflies. A significant advantage of
the algorithm is the fact that it uses mainly real random numbers, and it is
based on the global communication among the swarming particles i.e., the
fireflies, and as a result, it seems more effective multi objective optimization.
In this algorithm, the flashing light helps fireflies for finding mates, attracting
their potential prey and protecting themselves from their predators. The
swarm of fireflies will move to brighter and more attractive locations by the
flashing light intensity that associated with the objective function of problem
considered in order to obtain efficient optimal solutions.
The development of firefly-inspired algorithm was based on three
idealized rules:
i)
Artificial fireflies are unisex so that sex is not an issue for
attraction.
ii)
Attractiveness is proportional to their flashing brightness
which decreases as the distance from the other firefly
increases due to the fact that the air absorbs light. Since the
most attractive firefly is the brightest one, to which it
convinces neighbours moving toward. In case of no brighter
one, it freely moves any direction
107
iii) The brightness of the flashing light can be considered as
objective function to be optimized.
The main steps of the FA start from initializing a swarm of fireflies,
each of which is determined the flashing light intensity. During the loop of
pair wise comparison of light intensity, the firefly with lower light intensity
will move toward the higher one. The moving distance depends on the
attractiveness. After moving, the new firefly is evaluated and updated for the
light intensity. During pair wise comparison loop, the best-so-far solution is
iteratively updated. The pair wise comparison process is repeated until
termination criteria are satisfied. Finally, the best-so-far solution is visualized.
In this work, the evaluation on the goodness of schedules is
measured by the makespan, which can be calculated using equation (8.1),
where Ck is completed time of job k.
Minimize: Cmax = max (C1, C2…Ck)
8.2
(8.1)
IMPORTANT DEFINITONS IN FIREFLY ALGORITHM
There are some definitions in FA namely, Distance, Attractiveness
and Movement.
(a)
Distance:
The distance between any two fireflies i and j at positions xi and xj,
respectively, can be defined as a Cartesian or Euclidean distance (rij) using
equation (8.2), where xi,k is the kth component of the spatial coordinate xi of the
ith firefly and d is the number of dimensions.
rij =
i
- xj =
(
,
,
)
(8.2)
108
(b)
Attractiveness:
The calculation of attractiveness function of a firefly is shown in
equation (8.3),
(r)
(c)
0
× exp (-
m
), with m
1
(8.3)
Movement:
The movement of a firefly i which is attracted by a more attractive
(i.e., brighter) firefly j is given by the following equation (8.4),
xi = xi +
0
× exp (-
2
ij )
× (xj – xi
- )
(8.4)
Where the first term is the current position of a firefly, the second
term is used for considering a firefly’s attractiveness to light intensity seen by
adjacent fireflies, and the third term is used for the random movement of a
firefly in case there are not any brighter ones.
The settings of FA parameters: Light absorption coefficient
( ) = 1.0, Randomization parameter ( ) = 0.3, Attractiveness value ( 0) = 1.0
and random number (rand) generator uniformly distributed in the space
[0, 1] = 0.2.
109
8.3
GENERAL SCHEMA OF FA
Objective function makespan, z = (C1, C2...Cn)
Generate machines (i = 1, 2... m)
Generate job sequences (j = 1, 2... n)
Evaluate makespan (C1, C2, C3….Cn) for all population
While Gen. < Max Gen.
For each job sequence j = (1, 2, 3…, n)
For each lot sizes (Xij)
Move firefly in d-dimensional space
Determine the attractiveness based on distance rij
Evaluate makespan
End For
End For
Assess light intensity
Select job sequences and lot sizes for Gen. +1
End While
Rank and choose the best job sequences and Sub lot sizes
8.4
FLOW CHART OF FA
The frame work of the proposed FA is illustrated in Figure 8.1. The
various steps involved in FA is explained in this section.
110
Start
Initialize a population of fireflies
xi (i = 1,2,…n)
Define light absorption co- efficient ( )
Move firefly i towards j in all d dimensions
Yes
Is there is any
good location?
No
Find location attractiveness
Location
found
Evaluate new solutions and update
light intensity
Rank the fireflies and find the current best
Stop
Figure 8.1 Flow chart of FA
Go to that location
111
8.5
NUMERICAL ILLUSTRTION OF FA (EQUAL SUB LOTS)
The algorithm is illustrated with an example problem (5 jobs- 2
machines) in this section. Table 4.1 provides the data of the example
problem.
8.5.1
(i)
Calculation of Distance, Attractiveness and Movement
For sequence 2-3-5-1-4
rij =
i
=
(r)
0
(
- xj =
× exp (-
m
,
), with m
,
)
= (0
1 ; (here,
0.6) = 0.6
0=1,
= 1, m = 1)
= 1× exp [- 1× (0.6)1] = 0.55
xi = xi +
× exp (-
0
2
ij )
- ); (here,
× (xj – xi
0.3, rand = 0.2)
= 0.6 + 1× exp [- 1× (0.6)2] × (0.6 –0) + 0.3(0.2 -0.5) = 0.9986
(ii)
For sequence 3-2-5-1-4
ri j =
(r)
=
i
0
(
- xj =
× exp (-
m
,
), with m
,
)
= (0.6
1 ; (here,
0=1,
1.3) = 0.7
= 1, m = 1)
= 1× exp [- 1× (0.7)1] = 0.4966
xi = xi +
0
× exp (-
2
ij )
× (xj – xi
- ); (here,
= 1.3 + 1× exp [- 1× (0.7)2] × (1.3 –0.6) + 0.3(0.2 -0.5)
= 1.6388
0.3, rand = 0.2)
112
(iii)
For sequence 2-4-3-5-1
rij =
(r)
i
=
0
(
- xj =
m
× exp (-
,
), with m
)
,
= (1.3
1 ; (here,
0=1,
0.9) = 0.4
= 1, m = 1)
= 1× exp [- 1× (0.4)1] = 0.6703
xi = xi +
0
× exp (-
2
ij )
- ); (here,
× (xj – xi
0.3, rand = 0.2)
= 0.9 + 1× exp [- 1× (0.4)2] × (0.9 –1.3) + 0.3(0.2 -0.5)
= 0.4692
(iv)
For sequence 1-5-2-3-4
rij =
(r)
i
0
(
- xj =
× exp (-
m
,
), with m
,
)
= (0.9
1 ; (here,
0=1,
2.1) = 1.2
= 1, m = 1)
= 1× exp [- 1× (1.2)1] = 0.3012
xi = xi +
0
× exp (-
2
ij )
× (xj – xi
- ) ; (here,
0.3, rand = 0.2)
= 2.1 + 1× exp [- 1× (1.2)2] × (2.1 –0.9) + 0.3(0.2 -0.5)
= 2.2943
8.5.2
FA for makespan criterion
The illustration of FA (for equal sub lots) is shown in Table 8.1
113
Table 8.1 FA Illustration (equal sub lots)
Seed
Distance Attractiveness Movement
Sequence
for next
(xi)
makespan
Sequence
(rij)
r)
23514
0.6
0.55
0.9986
23514
48
32514
0.7
0.497
1.6388
32514
48
24351
0.4
0.6703
0.4692
24351
45
15234
1.2
0.3012
2.2943
12534
45*
generation
The final job sequence is 1- 2 -5 -3- 4, the corresponding makespan
time is 45 and it is shown in Figure 7.2.
8.5.3
FA for total flow time criterion
The method to find total flow time is illustrated in this section. The
final result is 2-5-4-1-3 and the corresponding total flow time is 144. It is
shown in Figure 8.2.Total flow time calculation for the sequence 2-5-4-1-3 =
14+22+27+35+46 = 144. The following Figure 8.2 shows the time schedule
of the optimal solution.
The following fig. shows the time schedule of the optimal solution.
114
Figure 8.2
Schedule for the sequence 2-5-4-1-3 in FA (total flow time
objective for equal sub lots)
8.6
NUMERICAL ILLUSTRTION OF FA (VARIABLE SUB
LOTS)
The algorithm is illustrated with an example problem (5 jobs- 2
machines) in this section. Table 4.4 provides the data of the example problem.
8.6.1
FA for makespan criterion
The illustration of FA (for variable sub lots) is shown in Table 8.2.
Table 8.2 FA Illustration (variable sub lots)
Seed
Distance Attractiveness
r)
Movement Sequence
(xi)
for next makespan
Sub lot size
Sequence
(rij)
54231
1.2
0.3012
2.2943
53124
85*
{132}{543}{46}{3212}{1233}
31452
0.4
0.6703
1.2692
31452
88
{453}{64}{2133}{231}{2132}
45132
0.2
0.8187
1.2178
45132
89
{2133}{231}{64}{453}{2132}
34125
0.3
0.7408
0.8358
34125
89
{453}{2133}{64}{2132}{231}
generation
115
The final result is 5-3-1-2-4, the corresponding makespan time is 85,
sub lot size is {132}{543}{46}{3212}{1233} and it is shown in Figure 6.7.
8.6.2
FA for total flow time criterion
The method to find total flow time is illustrated in this section. The
final result is 3-1-4-5-2 and the corresponding total flow time is 299, sub lot
size is {453}{64}{2133}{231}{2132}. Total flow time calculation for the
sequence 3-1-4-5-2 = 22+44+64+79+90 = 299. The following Figure 8.3
shows the time schedule of the optimal solution.
Figure 8.3
Schedule for the sequence 3-1-4-5-2 in FA (Total Flow Time
objective for variable sub lots)
116
8.7
PERFORMANCE EVALUATION
8.7.1
Proposed methodology illustrated with makespan criterion
The makespan results of FA algorithm for equal sub lots (Table 8.3 -
Table 8.6) and for variable sub lots (Table 8.10 - Table 8.12) is given in this
section.
8.7.2
Proposed methodology illustrated with total flow time criterion
The total flow time results of FA algorithm for equal sub lots
(Table 8.7 - Table 8.9) and for variable sub lots (Table 8.13 - Table 8.15) is
given in this section.
o Nt es mel bor P
.
118
Table 8.4
Makespan value for three – machine cases in FA (equal sub
lots)
Problem set
number
5
160
168
154
164
1
2
3
4
Table 8.5
10
380
388
378
384
Makespan
Number of jobs
15
20
410
570
412
568
408
564
412
570
25
632
648
642
640
30
794
688
620
618
Makespan value for five – machine cases in FA (equal sub
lots)
Problem set
number
5
180
188
180
174
1
2
3
4
Table 8.6
10
382
388
380
378
Makespan
Number of jobs
15
20
410
650
428
670
420
668
415
664
25
700
698
700
702
30
842
744
622
674
Makespan value for seven – machine cases in FA (equal sub
lots)
Problem set
number
1
2
3
4
5
200
202
198
200
10
202
204
208
198
Makespan
Number of jobs
15
20
460
590
504
600
508
608
498
594
25
709
708
714
722
30
854
745
698
676
119
Table 8.7
Total flow time value for three – machine cases in FA (equal
sub lots)
Problem set
number
1
2
3
4
Table 8.8
5
432
448
425
428
10
528
532
530
524
Total flow time
Number of jobs
15
20
3012
5712
3152
5845
3098
5523
3124
5512
25
7498
7523
7618
7609
30
10878
8545
8417
8098
Total flow time value for five – machine cases in FA (equal
sub lots)
Problem set
number
1
2
3
4
Table 8.9
5
478
460
500
502
10
2250
2276
2000
2068
Total flow time
Number of jobs
15
20
3142
11285
3196
9512
3109
9004
3032
9325
25
11252
11318
11316
11312
30
12644
9618
9434
9518
Total flow time value for seven – machine cases in FA (equal
sub lots)
Problem set
number
1
2
3
4
5
568
598
592
588
10
2200
2509
2404
2208
Total flow time
Number of jobs
15
20
4027
9126
4008
9377
4002
9318
4012
9315
25
9812
9989
9787
9957
30
12756
10518
10275
10164
120
Table 8.10
Makespan value for three – machine cases in FA (variable
sub lots)
Problem set
number
1
2
3
4
Table 8.11
5
608
1025
1002
975
10
708
3276
3289
3298
Makespan
Number of jobs
15
20
818
918
3512
4012
3398
3998
3494
3889
25
1007
4098
4078
4099
30
1017
4909
4989
5004
Makespan value for five – machine cases in FA (variable sub
lots)
Problem set
number
1
2
3
4
Table 8.12
5
704
3475
4004
4012
10
829
4004
4098
4095
Makespan
Number of jobs
15
20
1121
1305
4712
7178
5008
7004
5012
7314
25
1508
7812
7409
7676
30
1699
8287
8256
8102
Makespan value for seven – machine cases in FA (variable
sub lots)
Problem set
number
1
2
3
4
5
790
4134
4212
4018
10
1009
4927
4909
4927
Makespan
Number of jobs
15
20
1212
1321
6334
7298
6289
7389
6209
7042
25
1908
9823
10245
10432
30
1878
9989
9993
9945
121
Table 8.13
Total flow time value for three – machine cases in FA
(variable sub lots)
Problem set
number
1
2
3
4
Table 8.14
5
3342
15100
15212
15316
10
3972
15150
16356
15432
Total flow time
Number of jobs
15
20
8324
9871
26547
69562
26542
73108
26321
72002
25
23102
91175
140004
130028
30
27564
142875
172121
166523
Total flow time value for five – machine cases in FA
(variable sub lots)
Problem set
number
1
2
3
4
Table 8.15
5
3398
14323
14112
15002
10
4212
16024
18512
18316
Total flow time
Number of jobs
15
20
8108
12457
24417
70034
26012
72389
27023
73451
25
24032
95089
154132
140997
30
30012
174623
192978
166421
Total flow time value for seven – machine cases in FA
(variable sub lots)
Problem set
number
1
2
3
4
5
4189
14108
14235
14323
10
5027
20012
21000
19764
Total flow time
Number of jobs
15
20
9285
14012
29112
72025
30089
73112
30004
73997
25
25188
96118
157225
163123
30
28098
165113
165580
167772
122
8.8
SUMMARY
The numerical illustration and computational results of proposed FA
for equal size sub lots (from Table 8.3 – Table 8.9) and variable size sub lots
(from Table 8.10 - Table 8.15) are discussed.