# Parameter Improvement of the Soccer League Competition Algorithm by Introducing Stubborn Players: Application to Water Distribution Network

Authors: Godfrey Chagwiza; Absalom Jaison; Tavengwa Masamha
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

A modified soccer league algorithm is presented in this paper. The effect of stubborn fixed players is investigated and the algorithm is implemented to three benchmark water distribution networks. The modified algorithm is compared to several algorithms. The results show that the modified algorithm performs better than the soccer league competition algorithm, in particular, on the average number of evaluations required to find the optimal cost. Computational results show that the utility benefit of both the individual player and team is essential. The algorithm becomes more reliable when utility benefits are high and as the number of fixed players increases.


---

## Body

## 1. Introduction

Optimization techniques have been in recent years regarded as important in line with technological advancement. The reliability of these optimization techniques is essential in real world applications [1–4]. Some techniques have been introduced to improve system reliability such as the techniques that increase reliability of system components and using redundancy components in many different subsystems [5]. Several hybrid algorithms have been introduced to improve performance of heuristics such as genetic algorithms [6], simulated annealing [7], ant colony optimization system [8], and particle swarm optimization [9].Metaheuristic algorithms have been in recent years used to solve water distribution systems such as Max-Min Ant System [10], soccer league competition (SLC) [11], harmony search optimization approach [12], and shuffled frog leaping algorithm [13]. This research is motivated by the success of the soccer league competition algorithm of Moosavian [11] to solve water distribution network problems.The remainder of the paper is organised as follows. In Section2, a review and modification of SLC algorithm is presented. Implementation of the modified SLC algorithm is done in Section 3 and conclusions are drawn in Section 4.


## 2. Review and Modification of Soccer League Competition (SLC) Algorithm

The successful implementation of the Soccer League Competition algorithm of Moosavian [11] to solve water distribution network motivates this research. The soccer league algorithm’s ideas are based on the soccer league and competition among players and teams and is relatively new but its promising results have motivated this research to explore the possibilities of expansion of the algorithm. The soccer league algorithm has the following steps.Step 1.
Initialisation of the problem and algorithm parameters: this involves defining the objective function and decision variables. The number of seasons, teams in the league, fixed players, and substitutes are determined in this step.Step 2.
Samples generation is carried out in this step of the algorithm. The total number of players in the league is calculated.Step 3.
Teams assessment is done by first arranging players according to their power. The power of each team is equal to the summation of power of each fixed player in the team.Step 4.
The league starts and competitions are initialised between all possible pairs of the teams that are in the league. Winners and losers in each match are determined. Imitation operator is introduced to fixed players in the winning team. The solution vector related to the fixed players in the team is regarded as moving toward the best solution vector of the league. A new fixed player is generated based on the solution vector of the winner team’s fixed players. Provocation operator is introduced to screen the substitutes. The substitutes need to prove that they are better than the fixed player in the winner team. A new solution vector of substitutes in the winner team moves toward the solution direction of the fixed players. If the new solution vector generated is better than the older solution vector, the old solution vector is replaced (see Moosavian [11] for details).Step 5.
League update is done after every season. The players are arranged taking into account their updated power. Best players are allocated to best teams, average performers to average performing teams, and weakest players to bottom teams in the league table.Step 6.
Steps3, 4, and 5 are repeated until the number of seasons required is achieved. Relegation of weakest teams and promotion of better performing teams take place.The third step of the SLC algorithm is modified in order to achieve a better assessment of the teams in the league. Since power of fixed players in the team is used to determine the power of the teams, an attempt to incorporate stubbornness among the fixed players is carried out. It is a fact that most super star players become stubborn due to praises and they are usually assured of their fixed player status in most teams. It is therefore imperative to include this fact in the metaheuristic so as to improve its performance. Definitions of terms that are used are presented in “Definitions of Terms” section.The team power is basically defined as the average power of its fixed players. The algorithm of Moosavian [11] did not incorporate the possibility of stubborn players. Stubborn players have an effect on the power of team which is an important aspect of the SLC algorithm. Team power is the most important part of winning the game. The power of player, as defined in the SLC algorithm, P
i
,
j, is(1)
P
i
,
j
=
1
F
i
,
j
,
i
∈
T
,
j
∈
J
.Incorporating the proposed effect of the stubborn players into (1) is presented as follows. We start by looking at the utility function of the team, T, which is given as follows:(2)
U
i
=
B
m
-
C
m
+
b
.The team’s performance can be affected by bribes given to either the couch or player or both. The utility function of the individual player is presented by (3) taking note of the effect of bribe. Bribes cannot be neglected when we are talking of soccer games:(3)
U
j
=
α
π
+
H
-
β
W
m
-
b
.Using (1) to (3), the power of an individual player can be formulated as in (4) since performance of an individual player is affected by the behavior of the couch toward each game:(4)
P
i
,
j
=
1
F
i
,
j
×
γ
U
i
+
U
j
.The total team power is based on the power of each fixed player. It is therefore important to find the probability that the stubborn player secures the fixed position in the team. The probability that a stubborn player secures a fixed position in the team can be calculated as follows:(5)
ϵ
j
=
η
~
σ
δ
x
j
R
+
1
-
δ
x
j
Q
+
1
-
η
~
σ
x
j
E
-
Q
.Now, the total team power can be calculated by the following:(6)
Y
i
=
1
N
i
∑
j
=
1
N
i
ϵ
j
P
i
,
j
.Figure 1 shows the flow chart of the modified soccer league algorithm.Figure 1
Modified soccer league algorithm flow chart.


## 3. Numerical Examples

Implementation of the modified soccer league algorithm is carried out in this section. Formalisation of steps required for comparing metaheuristics is important. In Chiarandini et al. [30], two main models were introduced and these are the univariate and multivariate models taking into account the solution cost and run time. There are good practices required to fairly compare metaheuristics. It is important to use benchmark problems [31]. It is also vital to present results in a way that allows fair comparisons of metaheuristics. These include showing execution time and the mean number of iterations to obtain the best result. The execution time and quality of solution are regarded as the main performance measures of primary interest. McGeoch [32] presents a detailed explanation of experimental analysis of algorithms.In this paper, these best practices are implemented for a fair and effective comparison of the proposed modified heuristic with the other heuristics available in the literature. Three benchmark problems, that is, the two-loop, Hanoi, and New York Tunnels water networks, are used to test the performance of the modified heuristic. Computations are executed in MATLAB environment on a PC with AMD E-300 APU with RadeonTM @1.30 GHz and 4.00 GB RAM. A total of 100 runs are performed for each problem recording the number of evaluations and the optimal cost.The control parameters areα, γ, and β. Sequential Parameter Optimization (SPO) introduced by Bartz-Beielstein et al. [27] is used to tune the parameters. The algorithm is allowed to perform a total of 1000 tests and is repeated 50 times. An initial population of 100 is used. Table 1 presents the initial values of the parameters used to perform the tests and the average values that are then used in the computational experiments. It is noticed that there is an inverse relationship between b and β; that is, as b
→
∞, then β
→
0.Table 1
Control parameters choice and average values.
Parameter
Initial/manual value
Average computational value
γ
0.1
0.821
β
30% ofb
51% ofb
α
2%
45.9%


### 3.1. Two-Loop Water Network

A two-loop network problem of Alperovits and Shamir [28] is considered. The problem has 1 reservoir, 7 nodes, and 8 pipes. Figure 2 is the diagrammatic presentation of the network. The pipes in the network, all of them, are 1000 m long. Table 3 shows the cost data of available pipe diameters in both inches and millimeters. Table 2 shows the node information of the network. The results are compared to those of genetic algorithm (GA) [14], simulated annealing algorithm (SA) [15], shuffled leapfrog algorithm (SLA) [13], shuffled complex algorithm (SCA) [16], modified genetic algorithm [17], particle swarm optimization (PSO) [18], differential evolution (DE) [19], harmony search (HS) [12], scatter search (SS) [21], PSO + DE [18], particle swarm harmony search (PSHS) [20], and SLC [11].Table 2
Two-loop network node information.
Nodes
Demand (m3/h)
Elevation (m)
1
-
1120
210
2
100
150
3
100
160
4
120
155
5
270
150
6
330
165
7
200
160Table 3
Cost and diameter information for the two-loop water network.
Diameter (in)
Diameter (mm)
Cost ($/m)
1
25.4
2
2
50.8
5
3
76.2
8
4
101.6
11
6
152.4
16
8
203.2
23
10
254
32
12
304.8
50
14
355.6
60
16
406.4
90
18
457.2
130
20
508
170
22
558.8
300
24
609.6
550Figure 2
Two-loop water network.The computational results of the two-loop network are shown in Table4. The modified soccer league algorithm that uses stubborn players performs better in terms of average number of evaluations (1031) as compared to other heuristics, especially, the soccer league competition algorithm. As reported in Moosavian [11], the optimal costs ($419 000) are exactly the same for all the algorithms.Table 4
Results obtained by different metaheuristics for two-loop network.
Algorithm
Mean evaluations
Cost ($)
GA [14]
65 000
419 000
SA [15]
25 000
419 000
SLA [13]
11 155
419 000
SCA [16]
11 019
419 000
Modified genetic algorithm [17]
2440
419 000
PSO [18]
5138
419 000
DE [19]
4750
419 000
HS [20]
2891
419 000
SS [21]
3215
419 000
PSO + DE [18]
3080
419 000
PSHS [20]
233
419 000
SLC [11]
2051
419 000
Modified soccer league algorithm
1031
419 000


### 3.2. Hanoi Water Network

The Hanoi network layout is shown by Figure3. The network has 34 pipes, 32 nodes, of which one is a reservoir, and 3 loops. Table 5 presents the node and pipe data and Table 6 presents the cost information as described by Fujiwara and Khang [29]. There are 2.87
×
1
0
26 possible designs and 6 commercially available diameters. The modified soccer league algorithm is compared to Max-Min Ant System (MMAS) [22], PSO [18], hybrid discrete dynamically dimensioned search (HD-DDS) [23], genetic algorithm pipe network optimization model (GENOME) [24], genetic heritage evolution by stochastic transmission (GHEST) [25], SS [21], DE [19], self-adaptive differential evolution (SADE) [26], and SLC [11].Table 5
The Hanoi network pipe and node data.
Pipe
Length (m)
Pipe
Length (m)
Node
Demand (m3/h)
Node
Demand (m3/h)
1
100
18
800
1
-
19940
18
1345
2
1350
19
400
2
890
19
60
3
900
20
2200
3
850
20
1275
4
1150
21
1500
4
130
21
930
5
1450
22
500
5
725
22
485
6
450
23
2650
6
1005
23
1045
7
850
24
1230
7
1350
24
820
8
850
25
1300
8
550
25
170
9
800
26
850
9
525
26
900
10
950
27
300
10
525
27
370
11
1200
28
750
11
500
28
290
12
3500
29
1500
12
560
29
360
13
800
30
2000
13
940
30
360
14
500
31
1600
14
615
31
105
15
550
32
150
15
280
32
805
16
2730
33
860
16
310
—
—
17
1750
34
950
17
865
—
—Table 6
The cost data of the Hanoi network.
Diameter (inches)
Cost per unit length (unit)
12
45.726
16
70.400
20
98.387
24
129.333
30
180.748
40
278.280Figure 3
Hanoi water networkt.The computational results are shown in Table7. The results show that the optimal cost is the same for all algorithms. The average evaluations required by the modified algorithm to find the optimal solution are 65 443, fewer than that of MMAS, HD-DDS, and SLC. It is important to note that the modified algorithm has a success rate of 100% for the 100 runs. The algorithm performs better than the SLC algorithm in particular.Table 7
Computational results of the Hanoi network.
Algorithm
Number of runs
Optimal cost ($)
Mean evaluations
MMAS [22]
20
6 134 087
85 600
PSO [18]
2000
6 081 087
NA
HD-DDS [23]
50
6 081 087
100 000
GENOME [24]
10
6 081 087
NA
GHEST [25]
60
6 081 087
50 134
SS [21]
100
6 081 087
43 149
DE [19]
300
6 081 087
48 724
SADE [26]
50
6 081 087
60 532
SLC [11]
50
6 081 087
71 789
Modified soccer league algorithm
100
6 081 087
68 443


### 3.3. New York Tunnels Network

The New York Tunnels city network was first presented by Neelakantan and Suribabu [17]. The network consists of 1 loop, 21 pipes, and 20 nodes as shown in Figure 4. It is fed by gravity from a reservoir. Table 8 presents the pipe length and node data of the network. The search space of this optimization problem has 1
6
21
=
1.93
×
1
0
25 possible designs [33]. Available commercial diameters of the network and the respective costs are shown in Table 9.Table 8
Node and pipe information of the New York Tunnels network.
Node
Demand (m
3)
Minimum head (ft)
Pipe
Length (ft)
Existing diameter (ft)
1
-
22018
300
1
11 600
180
2
92.4
255
2
19 800
180
3
92.4
255
3
7 300
180
4
88.2
255
4
8 300
180
5
88.2
255
5
8 600
180
6
88.2
255
6
19 100
180
7
88.2
255
7
9 600
132
8
88.2
255
8
12 500
132
9
170
255
9
9 600
180
10
1
255
10
11 200
204
11
170
255
11
14 500
204
12
117.1
255
12
12 200
204
13
117.1
255
13
24 100
204
14
92.47
255
14
21 100
204
15
92.4
255
15
15 500
204
16
170
260
16
26 400
72
17
57.5
272.8
17
31 200
72
18
117.1
255
18
24 000
60
19
117.1
255
19
14 400
60
20
170
255
20
38 400
60
21
26 400
72Table 9
New York Tunnels commercially available pipe diameters and their respective costs.
Diameter (in)
Cost (US$/ft)
36
93.5
48
134
60
176
72
221
84
267
96
316
108
365
120
417
132
469
144
522
156
577
168
632
180
689
192
746
204
804Figure 4
New York Tunnels water network.The results are shown in Table10 and 15 311 average evaluations, lower than those of SLC, SS, and HD-DDS, produced the optimal cost of $38.64 million. The best cost is found by the modified soccer league algorithm with an average success probability of 100% of 100 runs. A significant reduction in the mean evaluations from 15764 of SLC to 15 311 is worth noting.Table 10
Computational results of the New York Tunnels network.
Algorithm
Number of runs
Best cost ($)
Mean evaluations
MMAS [22]
20
38.64
30 700
PSO [18]
2000
38.64
NA
HD-DDS [23]
50
38.64
47 000
GHEST [25]
60
38.64
11 464
SS [21]
100
38.64
57 583
DE [19]
50
38.64
5 494
SADE [26]
50
38.64
6 598
SLC [11]
100
38.64
15 764
Modified soccer league algorithm
100
38.64
15 311Statistical analysis of the experimental results is performed in SPSS. Table11 presents results of the Analysis of Variance (ANOVA). The results show that the results of the algorithms used are significantly different from each other. Nonparametric tests, as suggested by García et al. [34], are performed to analyse the behavior of the modified soccer league algorithm as compared to the other algorithms used in each experimental problem. Table 12 presents results of the Wilcoxon test. Best 30 results among the runs are selected and used to avoid the challenge of using high runs that may result in the statistical test to detect insignificant difference as significant. The results show that the modified algorithm outperformed all the other algorithms in all experimental problems.Table 11
ANOVA results of the three experimental problems.
Problem
Definition
Degrees of freedom
p value
Two-loop
Between group
12
0.002
Within groups
1288
Total
1300
Hanoi
Between group
9
0.037
Within groups
991
Total
1000
New York
Between group
8
0.041
Within groups
892
Total
1000Table 12
Wilcoxon test of the experimental problems.
Proposed algorithm versus
Two-loop
Hanoi
New York
R
+
R
-
p
R
+
R
-
p
R
+
R
-
p
GA [27]
210
88
0.001
—
—
—
—
—
—
SA [28]
120
36
0.012
—
—
—
—
—
—
SLA [13]
123
37
0.033
—
—
—
—
—
—
SCA [14]
144
41
0.002
—
—
—
—
—
—
Modified genetic algorithm [15]
101
29
0.011
—
—
—
—
—
—
PSO [16]
111
33
0.012
93
24
0.002
119
89
0.003
DE [17]
213
56
0.03
117
34
0.021
144
47
0.034
HS [19]
122
37
0.041
—
—
—
—
—
—
SS [18]
144
54
0.032
127
51
0.043
154
71
0.011
PSO + DE [16]
123
77
0.037
—
—
—
—
—
—
PSHS [19]
142
56
0.000
—
—
—
—
—
—
SLC [11]
13
6
0.049
43
6
0.39
33
8
0.042
MMAS [20]
—
—
—
134
67
0.001
89
23
0.003
HD-DDS [29]
—
—
—
129
66
0.037
65
29
0.016
GENOME [22]
—
—
—
178
50
0.045
—
—
—
GHEST [23]
—
—
—
280
67
0.001
90
34
0.019
SADE [24]
—
—
—
128
44
0.029
138
79
0.022An experiment has been carried out to understand the behavior of the modified algorithm as the input parameters change. Unlike the SLC algorithm which uses the number of teams, fixed players, and substitutes as input parameters only, the modified algorithm uses the utility benefit of individual player, as a percentage,U
j, the utility function of the team, as a percentage, U
i, and the number of fixed players as additional input parameters. Table 13 shows the summary of computational results in terms of number of evaluations (Evalu), optimal cost (cost), and the computational time (run time).Table 13
Sensitivity analysis of the input parameters.
Parameter
Two-loop
Hanoi
New York
U
j (%)
U
i (%)
N
j
Evalu
Cost
Run time
Evalu
Cost
Run time
Evalu
Cost
Run time
0
0
1
1900
419 000
2.64
68500
6 081 087
4.02
15800
38.64
3.18
5
5
2
1502
419 000
2.33
68495
6 081 087
3.99
15792
38.64
3.04
10
10
3
1465
419 000
2.27
68490
6 081 087
3.56
15788
38.64
2.67
20
20
4
1392
419 000
2.12
68485
6 081 087
3.42
15781
38.64
2.55
30
30
5
1109
419 000
2.09
68480
6 081 087
3.37
15779
38.64
2.33
40
40
6
1001
419 000
1.73
68475
6 081 087
3.35
15768
38.64
2.21
50
50
7
954
419 000
1.65
68445
6 081 087
3.31
15765
38.64
2.15
60
60
8
869
419 000
1.62
68440
6 081 087
3.29
15742
38.64
2.08
70
70
9
701
419 000
1.45
68435
6 081 087
3.25
15703
38.64
2.01
80
80
10
602
419 000
1.37
68433
6 081 087
3.17
15596
38.64
2.00
90
90
11
503
419 000
1.28
68302
6 081 087
3.15
15584
38.64
1.98
100
100
12
375
419 000
1.24
68219
6 081 087
3.09
15347
38.64
1.85
Mean
1031
419 000
1.82
68433
6 081 087
3.41
15 311
38.64
2.34It is shown that if both percentage benefits of the fixed player and team are high and the number of fixed players is high, the number of evaluations is reduced significantly. The computational time is as well reduced significantly. This might be as a result of the fact that the set of fixed players constructed by the algorithm eliminate those that are stubborn and concentrate only on players that are motivated by benefits they will get for winning the match. Fixed players that cost the team are penalised and eliminated from calculating the team power. The modified algorithm becomes more reliable and efficient by increasing the number of fixed players.


## 3.1. Two-Loop Water Network

A two-loop network problem of Alperovits and Shamir [28] is considered. The problem has 1 reservoir, 7 nodes, and 8 pipes. Figure 2 is the diagrammatic presentation of the network. The pipes in the network, all of them, are 1000 m long. Table 3 shows the cost data of available pipe diameters in both inches and millimeters. Table 2 shows the node information of the network. The results are compared to those of genetic algorithm (GA) [14], simulated annealing algorithm (SA) [15], shuffled leapfrog algorithm (SLA) [13], shuffled complex algorithm (SCA) [16], modified genetic algorithm [17], particle swarm optimization (PSO) [18], differential evolution (DE) [19], harmony search (HS) [12], scatter search (SS) [21], PSO + DE [18], particle swarm harmony search (PSHS) [20], and SLC [11].Table 2
Two-loop network node information.
Nodes
Demand (m3/h)
Elevation (m)
1
-
1120
210
2
100
150
3
100
160
4
120
155
5
270
150
6
330
165
7
200
160Table 3
Cost and diameter information for the two-loop water network.
Diameter (in)
Diameter (mm)
Cost ($/m)
1
25.4
2
2
50.8
5
3
76.2
8
4
101.6
11
6
152.4
16
8
203.2
23
10
254
32
12
304.8
50
14
355.6
60
16
406.4
90
18
457.2
130
20
508
170
22
558.8
300
24
609.6
550Figure 2
Two-loop water network.The computational results of the two-loop network are shown in Table4. The modified soccer league algorithm that uses stubborn players performs better in terms of average number of evaluations (1031) as compared to other heuristics, especially, the soccer league competition algorithm. As reported in Moosavian [11], the optimal costs ($419 000) are exactly the same for all the algorithms.Table 4
Results obtained by different metaheuristics for two-loop network.
Algorithm
Mean evaluations
Cost ($)
GA [14]
65 000
419 000
SA [15]
25 000
419 000
SLA [13]
11 155
419 000
SCA [16]
11 019
419 000
Modified genetic algorithm [17]
2440
419 000
PSO [18]
5138
419 000
DE [19]
4750
419 000
HS [20]
2891
419 000
SS [21]
3215
419 000
PSO + DE [18]
3080
419 000
PSHS [20]
233
419 000
SLC [11]
2051
419 000
Modified soccer league algorithm
1031
419 000


## 3.2. Hanoi Water Network

The Hanoi network layout is shown by Figure3. The network has 34 pipes, 32 nodes, of which one is a reservoir, and 3 loops. Table 5 presents the node and pipe data and Table 6 presents the cost information as described by Fujiwara and Khang [29]. There are 2.87
×
1
0
26 possible designs and 6 commercially available diameters. The modified soccer league algorithm is compared to Max-Min Ant System (MMAS) [22], PSO [18], hybrid discrete dynamically dimensioned search (HD-DDS) [23], genetic algorithm pipe network optimization model (GENOME) [24], genetic heritage evolution by stochastic transmission (GHEST) [25], SS [21], DE [19], self-adaptive differential evolution (SADE) [26], and SLC [11].Table 5
The Hanoi network pipe and node data.
Pipe
Length (m)
Pipe
Length (m)
Node
Demand (m3/h)
Node
Demand (m3/h)
1
100
18
800
1
-
19940
18
1345
2
1350
19
400
2
890
19
60
3
900
20
2200
3
850
20
1275
4
1150
21
1500
4
130
21
930
5
1450
22
500
5
725
22
485
6
450
23
2650
6
1005
23
1045
7
850
24
1230
7
1350
24
820
8
850
25
1300
8
550
25
170
9
800
26
850
9
525
26
900
10
950
27
300
10
525
27
370
11
1200
28
750
11
500
28
290
12
3500
29
1500
12
560
29
360
13
800
30
2000
13
940
30
360
14
500
31
1600
14
615
31
105
15
550
32
150
15
280
32
805
16
2730
33
860
16
310
—
—
17
1750
34
950
17
865
—
—Table 6
The cost data of the Hanoi network.
Diameter (inches)
Cost per unit length (unit)
12
45.726
16
70.400
20
98.387
24
129.333
30
180.748
40
278.280Figure 3
Hanoi water networkt.The computational results are shown in Table7. The results show that the optimal cost is the same for all algorithms. The average evaluations required by the modified algorithm to find the optimal solution are 65 443, fewer than that of MMAS, HD-DDS, and SLC. It is important to note that the modified algorithm has a success rate of 100% for the 100 runs. The algorithm performs better than the SLC algorithm in particular.Table 7
Computational results of the Hanoi network.
Algorithm
Number of runs
Optimal cost ($)
Mean evaluations
MMAS [22]
20
6 134 087
85 600
PSO [18]
2000
6 081 087
NA
HD-DDS [23]
50
6 081 087
100 000
GENOME [24]
10
6 081 087
NA
GHEST [25]
60
6 081 087
50 134
SS [21]
100
6 081 087
43 149
DE [19]
300
6 081 087
48 724
SADE [26]
50
6 081 087
60 532
SLC [11]
50
6 081 087
71 789
Modified soccer league algorithm
100
6 081 087
68 443


## 3.3. New York Tunnels Network

The New York Tunnels city network was first presented by Neelakantan and Suribabu [17]. The network consists of 1 loop, 21 pipes, and 20 nodes as shown in Figure 4. It is fed by gravity from a reservoir. Table 8 presents the pipe length and node data of the network. The search space of this optimization problem has 1
6
21
=
1.93
×
1
0
25 possible designs [33]. Available commercial diameters of the network and the respective costs are shown in Table 9.Table 8
Node and pipe information of the New York Tunnels network.
Node
Demand (m
3)
Minimum head (ft)
Pipe
Length (ft)
Existing diameter (ft)
1
-
22018
300
1
11 600
180
2
92.4
255
2
19 800
180
3
92.4
255
3
7 300
180
4
88.2
255
4
8 300
180
5
88.2
255
5
8 600
180
6
88.2
255
6
19 100
180
7
88.2
255
7
9 600
132
8
88.2
255
8
12 500
132
9
170
255
9
9 600
180
10
1
255
10
11 200
204
11
170
255
11
14 500
204
12
117.1
255
12
12 200
204
13
117.1
255
13
24 100
204
14
92.47
255
14
21 100
204
15
92.4
255
15
15 500
204
16
170
260
16
26 400
72
17
57.5
272.8
17
31 200
72
18
117.1
255
18
24 000
60
19
117.1
255
19
14 400
60
20
170
255
20
38 400
60
21
26 400
72Table 9
New York Tunnels commercially available pipe diameters and their respective costs.
Diameter (in)
Cost (US$/ft)
36
93.5
48
134
60
176
72
221
84
267
96
316
108
365
120
417
132
469
144
522
156
577
168
632
180
689
192
746
204
804Figure 4
New York Tunnels water network.The results are shown in Table10 and 15 311 average evaluations, lower than those of SLC, SS, and HD-DDS, produced the optimal cost of $38.64 million. The best cost is found by the modified soccer league algorithm with an average success probability of 100% of 100 runs. A significant reduction in the mean evaluations from 15764 of SLC to 15 311 is worth noting.Table 10
Computational results of the New York Tunnels network.
Algorithm
Number of runs
Best cost ($)
Mean evaluations
MMAS [22]
20
38.64
30 700
PSO [18]
2000
38.64
NA
HD-DDS [23]
50
38.64
47 000
GHEST [25]
60
38.64
11 464
SS [21]
100
38.64
57 583
DE [19]
50
38.64
5 494
SADE [26]
50
38.64
6 598
SLC [11]
100
38.64
15 764
Modified soccer league algorithm
100
38.64
15 311Statistical analysis of the experimental results is performed in SPSS. Table11 presents results of the Analysis of Variance (ANOVA). The results show that the results of the algorithms used are significantly different from each other. Nonparametric tests, as suggested by García et al. [34], are performed to analyse the behavior of the modified soccer league algorithm as compared to the other algorithms used in each experimental problem. Table 12 presents results of the Wilcoxon test. Best 30 results among the runs are selected and used to avoid the challenge of using high runs that may result in the statistical test to detect insignificant difference as significant. The results show that the modified algorithm outperformed all the other algorithms in all experimental problems.Table 11
ANOVA results of the three experimental problems.
Problem
Definition
Degrees of freedom
p value
Two-loop
Between group
12
0.002
Within groups
1288
Total
1300
Hanoi
Between group
9
0.037
Within groups
991
Total
1000
New York
Between group
8
0.041
Within groups
892
Total
1000Table 12
Wilcoxon test of the experimental problems.
Proposed algorithm versus
Two-loop
Hanoi
New York
R
+
R
-
p
R
+
R
-
p
R
+
R
-
p
GA [27]
210
88
0.001
—
—
—
—
—
—
SA [28]
120
36
0.012
—
—
—
—
—
—
SLA [13]
123
37
0.033
—
—
—
—
—
—
SCA [14]
144
41
0.002
—
—
—
—
—
—
Modified genetic algorithm [15]
101
29
0.011
—
—
—
—
—
—
PSO [16]
111
33
0.012
93
24
0.002
119
89
0.003
DE [17]
213
56
0.03
117
34
0.021
144
47
0.034
HS [19]
122
37
0.041
—
—
—
—
—
—
SS [18]
144
54
0.032
127
51
0.043
154
71
0.011
PSO + DE [16]
123
77
0.037
—
—
—
—
—
—
PSHS [19]
142
56
0.000
—
—
—
—
—
—
SLC [11]
13
6
0.049
43
6
0.39
33
8
0.042
MMAS [20]
—
—
—
134
67
0.001
89
23
0.003
HD-DDS [29]
—
—
—
129
66
0.037
65
29
0.016
GENOME [22]
—
—
—
178
50
0.045
—
—
—
GHEST [23]
—
—
—
280
67
0.001
90
34
0.019
SADE [24]
—
—
—
128
44
0.029
138
79
0.022An experiment has been carried out to understand the behavior of the modified algorithm as the input parameters change. Unlike the SLC algorithm which uses the number of teams, fixed players, and substitutes as input parameters only, the modified algorithm uses the utility benefit of individual player, as a percentage,U
j, the utility function of the team, as a percentage, U
i, and the number of fixed players as additional input parameters. Table 13 shows the summary of computational results in terms of number of evaluations (Evalu), optimal cost (cost), and the computational time (run time).Table 13
Sensitivity analysis of the input parameters.
Parameter
Two-loop
Hanoi
New York
U
j (%)
U
i (%)
N
j
Evalu
Cost
Run time
Evalu
Cost
Run time
Evalu
Cost
Run time
0
0
1
1900
419 000
2.64
68500
6 081 087
4.02
15800
38.64
3.18
5
5
2
1502
419 000
2.33
68495
6 081 087
3.99
15792
38.64
3.04
10
10
3
1465
419 000
2.27
68490
6 081 087
3.56
15788
38.64
2.67
20
20
4
1392
419 000
2.12
68485
6 081 087
3.42
15781
38.64
2.55
30
30
5
1109
419 000
2.09
68480
6 081 087
3.37
15779
38.64
2.33
40
40
6
1001
419 000
1.73
68475
6 081 087
3.35
15768
38.64
2.21
50
50
7
954
419 000
1.65
68445
6 081 087
3.31
15765
38.64
2.15
60
60
8
869
419 000
1.62
68440
6 081 087
3.29
15742
38.64
2.08
70
70
9
701
419 000
1.45
68435
6 081 087
3.25
15703
38.64
2.01
80
80
10
602
419 000
1.37
68433
6 081 087
3.17
15596
38.64
2.00
90
90
11
503
419 000
1.28
68302
6 081 087
3.15
15584
38.64
1.98
100
100
12
375
419 000
1.24
68219
6 081 087
3.09
15347
38.64
1.85
Mean
1031
419 000
1.82
68433
6 081 087
3.41
15 311
38.64
2.34It is shown that if both percentage benefits of the fixed player and team are high and the number of fixed players is high, the number of evaluations is reduced significantly. The computational time is as well reduced significantly. This might be as a result of the fact that the set of fixed players constructed by the algorithm eliminate those that are stubborn and concentrate only on players that are motivated by benefits they will get for winning the match. Fixed players that cost the team are penalised and eliminated from calculating the team power. The modified algorithm becomes more reliable and efficient by increasing the number of fixed players.


## 4. Conclusion

This paper presented a modified soccer league algorithm by introducing stubborn fixed players. Each stubborn player is taken as a cost to the algorithm. Utility benefit of each individual fixed player and the team is used to calculate power of each fixed player. Probability of including a fixed player who has not reformed is also used to calculate the total team power. The algorithm is implemented to three benchmark problems and compared to the other algorithms available in the literature. The computational results show that the modified algorithm performs better than the SLC algorithm in particular. It is also shown that the number of evaluations and computational time are reduced significantly as percentage benefits of both the player and team increase and the number of fixed players increases as well. It is recommended to increase the number of fixed players for the algorithm to be more reliable.



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*Source: 3425374-2016-03-14.xml*