Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 3271969, 15 pages
https://doi.org/10.1155/2017/3271969

Research Article

Large-Scale Network Plan Optimization Using Improved Particle Swarm Optimization Algorithm

Houxian Zhang¹ and Zhaolan Yang²

¹School of Architecture and Civil Engineering, Nanjing Institute of Technology, Nanjing 211167, China
²Industrial Center, Nanjing Institute of Technology, Nanjing 211167, China

Correspondence should be addressed to Houxian Zhang; houxianzhang@sina.com

Received 21 October 2016; Revised 12 January 2017; Accepted 29 January 2017; Published 27 February 2017

Academic Editor: Shuming Wang

Copyright © 2017 Houxian Zhang and Zhaolan Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

No relevant reports have been reported on the optimization of a large-scale network plan with more than 200 works due to the complexity of the problem and the huge amount of computation. In this paper, an improved particle swarm optimization algorithm via optimization of initial particle swarm (OIPSO) is first explained by the stochastic processes theory. Then two optimization examples are solved using this method which are the optimization of resource-leveling with fixed duration and the optimization of resources constraints with shortest project duration in a large network plan with 223 works. Through these two examples, under the same number of iterations, it is proven that the improved algorithm (OIPSO) can accelerate the optimization speed and improve the optimization effect of particle swarm optimization (PSO).

1. Introduction

A large-scale network plan that is composed of more than 50 works has become an essential tool for managing large-scale engineering project [1, 2]. However, due to the rapidity of solution increase (called the combustion explosion) and the exponential growth of computing time with the complexity of the problem, which far exceeds the processing capacity of computing resources, the optimization of large-scale network plan becomes an unsolvable problem in the mathematics and computer science fields, also called the NP problem [3–7]. Among the existing optimization methods for network plan, accurate algorithm such as the dynamic planning [8], 0-1 planning [9, 10], and branch and bound method [11, 12] can solve the small network plan optimization; various heuristic algorithms [13–18] cannot solve large-scale network plan optimization. An effective way to solve the complex network plan is by using genetic algorithm (GA), but the works numbers of presented examples (86 and 122) are not large enough [19–23].

Proposed in 1995, PSO was applied to optimization, biomedicine, communication, control, plan, prediction, filter, and parameter estimation in rainfall-runoff modeling and so forth [24–30]. It was improved in selecting the parameter, the velocity equation of the particle, uncertainty stimulation, learning abilities, stability, convergence, and more [31–41]. Wang et al. and Chen et al. applied PSO to solve optimization of a nine-work network plan [42, 43]. The initial particle swarm was determined randomly by the improved and the initial PSO.

The Monte Carlo method can be applied to solve equations, integral equations, difference equations, integral, shielding radioactive particles, neutron fission security problems, the random service (queuing theory) of economic service problems, signal detection and system simulation, flow field simulation, life test, and more [44]. As it optimizes initial particle swarm to solve the optimization problem of large-scale network plans, it is PSO’s foundation. Zhang and Shi [45] adopted the Monte Carlo method to solve the optimization of the resource-leveling with fixed duration and the resources constraints with shortest project duration of a network plan. But the works number of the presented examples (9) is not large enough. To solve the optimization problem of the resource-leveling with fixed duration in a large network plan, Du et al. [46] proposed partition optimization based on the Monte Carlo method. However, there is not a large enough works number (61), and partition optimization combination may lose the global optimal solution in theory.

Without requiring any advanced knowledge of the reliability function, PSO combining with the Monte Carlo method was used to solve complex network reliability problems, while Monte Carlo method was used to evaluate system reliability, but its motivation is different from this paper [47]. In comparison to the random method, the Monte Carlo method in the Monte Carlo Enhanced PSO can calculate the probability of initial particles’ elements and form better initial particles, but there was no analyzation of the improvement mechanism to optimize the initial particle swarm [48]. To solve resource optimization and cost optimization of a large-scale network plan by using PSO, Zhang and Yang used the Monte Carlo method under limited conditions to optimize the initial particle swarm [49, 50]. However, there was no analyzation of the mechanism improvement to optimize the initial particle swarm, and the works number of presented examples (61) is not large enough.

In this paper, an improved particle swarm optimization algorithm via optimization of initial particle swarm (OIPSO) is first explained by the stochastic processes theory. Then two optimization examples of a large-scale network plan are solved using this method, which are the optimization of resource-leveling with fixed duration and the optimization of resources constraints with shortest project duration in a large network plan with 223 works. The optimization effect of the improved algorithm (OIPSO) is proven through these two examples.

This paper is organized as follows: Section 2 analyzes the improvement mechanism of OIPSO, Section 3 solves large-scale (223 works) network plans by OIPSO, Section 4 introduces the superiority of OIPSO compared with the original and existing PSOs, and Section 5 makes conclusions.

2. Methodology

2.1. OIPSO

The process of the original PSO is as follows [30]:

Step 1. Determining the initial particle swarm

Step 2. Evolving the particle location

Step 3. Determining each particle’s best experiencing position and all particles doing

Step 4. Outputting the optimization results when the maximal number of iterations is reached; otherwise return to Step 2

The improved algorithm (OIPSO) in this paper is the same as PSO, except that Step 1 determines the initial particle swarm via optimization of initial particle swarm.

In the optimization of resource-leveling with fixed duration and the optimization of resources constraints with shortest project duration of a network plan, the following expression determines the initial particle swarm:

where is related to its start time of work , is the normal duration of the work , is the random function, and is the total float of the work . The limiting conditions are resource variance in the optimization of resource-leveling with fixed duration, as well as resources and project duration in the optimization of resources constraints with shortest project duration. In the Monte Carlo method, using the random function is the basic principle.

2.2. The Improvement Mechanism of OIPSO

Markov chains are constituted by the PSO M particles [51]. And then by randomly selecting initial particles and setting them out to a certain point, the stochastic optimization series of particles constitute the Markov chains. The probability for a particle to set out from is , and the probability for a particle to transfer to after an time transfer can be determined by the following formula ( represents the optimal position of a limited number of iterations conditions, it can also be the optimization solution, , and is the state space):

where represents the arrival state of a particle subsequent to time transfer; is called the probability distribution of ; and is the initial distribution which is matrix of the Markov chains, namely, the probability of the Markov chains starting from , , (state space), and is the state for a particle to set out from; is equal to the product of one-time transfer matrix ( order phalanx), and is also called matrix of transition probability, expressed as

where is the probability for a particle to transfer to in the next time setting out from [52].

Optimized and unoptimized initial particles comprise the two columns of Markov chains. The probability of particles starting from the location of the initial particle is equal to 100%, while that of particles starting from the locations of other particles is equal to 0. The excellent particle position close to the optimal solution is increased in the n particle positions, which is the position of the excellent initial particle, regardless of the subsequent excellent particle positions that were generated based on the initial particle. There is a higher probability of the excellent particle flying from the position near the optimal solution than it flying away from the optimal solution. Therefore, as shown in (4),    column of one-time transfer matrix of Markov chains to optimize initial particle is bigger.

As a result, column of and of Markov chains to optimize initial particle is bigger.

3. Solving Large-Scale Network Plans with 223 Works by OIPSO

As shown in Figure 1, a large-scale network plan has a works number of 223 and a calculated project duration of 135. Table 1 shows each work’s resources amount, duration, and earliest start time, corresponding to the resource variance of 37.51. The biggest quantity of resources at one period is 27. The optimization of resource-leveling with fixed duration can be unchanged project duration and resource demand equilibrium of each period. The resources supply capacity limit can be met by the optimization of resources constraints with shortest project duration, and it can have minimal extended project duration.

Table 1: The parameters and their optimization solution for the optimization example of the resource-leveling with fixed duration.

Number
Work
Duration
Resources quantity
ES
Optimized ES

1
1–3
2
2
0
0

2
1–4
2
1
0
0

3
3-4
2
1
2
2

4
1-2
1
2
0
0

5
4–6
4
1
4
4

6
5–8
4
1
7
7

7
5–9
5
1
7
7

8
6–8
5
3
13
13

9
7-8
8
3
15
15

10
3–6
3
2
2
2

11
2–4
3
2
1
1

12
3–5
5
1
2
2

13
2–8
3
2
1
1

14
5-6
6
2
7
7

15
2–6
1
3
1
1

16
6-7
2
0
13
13

17
8-9
7
3
23
23

18
8–11
2
2
23
23

19
8–10
2
1
23
23

20
7–10
2
1
15
15

21
9-10
1
2
30
30

22
9–12
4
1
30
30

23
18-19
8
3
57
58

24
17–20
2
1
51
51

25
16–19
4
1
49
49

26
18–21
5
1
57
71

27
17–19
5
3
51
51

28
16-17
2
2
49
49

29
15–17
2
1
46
46

30
14–18
2
1
42
42

31
14–17
1
2
42
42

32
13–15
2
1
44
44

33
10-11
4
1
31
31

34
12-13
5
1
39
39

35
10–13
5
3
31
31

36
10–12
8
3
31
31

37
17-18
6
0
51
51

38
15-16
3
2
46
46

39
11–15
5
1
35
35

40
14-15
3
2
42
42

41
11–17
5
1
35
35

42
12–14
3
2
39
39

43
11–13
6
2
35
35

44
12–15
2
0
39
39

45
21-22
2
2
70
86

46
22–24
2
1
75
88

47
20–22
2
1
73
74

48
20–24
1
2
73
74

49
20–26
2
1
73
74

50
19–21
5
1
65
81

51
19–22
5
3
65
66

52
19-20
8
3
65
66

53
24-25
3
2
77
96

54
24–26
5
1
77
96

55
26-27
3
2
85
104

56
25-26
5
1
80
99

57
23–27
3
2
72
88

58
23–26
6
2
72
88

59
23-24
1
3
72
88

60
21–23
2
0
70
86

61
21–24
7
3
70
89

62
3–28
1
2
2
2

63
3–30
3
2
2
2

64
3–31
1
3
2
2

65
28–30
2
1
3
3

66
28-29
2
2
3
3

67
29-30
2
1
5
5

68
29–31
3
2
5
5

69
29–34
3
2
5
5

70
30-31
4
1
7
7

71
5–31
6
2
7
7

72
5–34
5
3
7
7

73
31-32
2
0
13
13

74
31–34
4
1
13
13

75
32-33
2
1
15
15

76
32–34
8
3
15
15

77
9–34
7
3
30
30

78
9–33
2
1
30
30

79
34-35
2
2
44
44

80
33-34
1
2
43
43

81
33–35
8
3
43
43

82
33–36
5
3
43
43

83
12–33
4
1
39
39

84
12–36
6
2
39
39

85
12–37
5
1
39
39

86
35-36
5
1
51
51

87
35–37
2
0
51
51

88
35–39
5
1
51
51

89
36-37
2
1
56
56

90
14–37
3
2
42
42

91
14–39
2
1
42
42

92
37-38
3
2
58
58

93
37–39
1
2
58
58

94
38-39
2
2
61
76

95
18–39
6
0
57
57

96
18–40
5
3
57
57

97
39-40
8
3
63
76

98
39–41
2
1
63
63

99
40-41
5
1
78
98

100
40–42
5
3
78
98

101
21–40
8
3
70
90

102
21–42
2
1
70
86

103
21–43
1
2
70
86

104
41-42
2
2
83
103

105
41–43
7
3
83
103

106
41–45
2
1
83
103

107
42-43
2
1
85
105

108
23–43
1
3
72
88

109
23–45
5
1
72
88

110
43-44
3
2
90
110

111
43–45
6
2
90
110

112
44-45
5
1
93
113

113
27–45
3
2
88
107

114
38–40
4
1
61
61

115
47-48
2
2
8
8

116
48–50
2
1
10
10

117
47–50
2
1
8
8

118
48-49
1
2
10
10

119
50–53
4
1
14
14

120
52–57
4
1
49
49

121
52–58
5
1
49
49

122
53–57
5
3
55
55

123
56-57
8
3
57
57

124
47–53
3
2
8
8

125
49-50
3
2
11
11

126
47–52
5
1
8
8

127
49–57
3
2
11
11

128
52-53
6
2
49
49

129
49–53
1
3
11
11

130
53–56
2
0
55
55

131
57-58
7
3
65
65

132
57–61
2
2
65
65

133
57–59
2
1
65
65

134
56–59
2
1
57
57

135
58-59
1
2
72
72

136
58–60
4
1
72
72

137
71-72
8
3
99
99

138
69–75
2
1
93
93

139
68–72
4
1
91
91

140
71–73
5
1
99
99

141
69–72
5
3
93
93

142
68-69
2
2
91
91

143
67–69
2
1
88
88

144
65–71
2
1
84
84

145
65–69
1
2
84
84

146
64–67
2
1
86
86

147
59–61
4
1
73
73

148
60–64
5
1
81
81

149
59–64
5
3
73
73

150
59-60
8
3
73
73

151
69–71
6
0
93
93

152
67-68
3
2
88
88

153
61–67
5
1
77
77

154
65–67
3
2
84
84

155
61–69
5
1
77
77

156
60–65
3
2
81
81

157
61–64
6
2
77
77

158
60–67
2
0
81
81

159
73–76
2
2
112
112

160
76–79
2
1
117
117

161
75-76
2
1
115
115

162
75–79
1
2
115
115

163
75–83
2
1
115
115

164
72-73
5
1
107
107

165
72–76
5
3
107
107

166
72–75
8
3
107
107

167
79–82
3
2
119
124

168
79–83
5
1
119
124

169
83-84
3
2
127
132

170
82-83
5
1
122
127

171
78–84
3
2
114
123

172
78–83
6
2
114
123

173
78-79
1
3
114
123

174
73–78
2
0
112
112

175
73–79
7
3
112
112

176
46-47
1
2
7
7

177
47–51
3
2
8
8

178
47–54
1
3
8
8

179
46–51
2
1
7
7

180
29–46
2
2
5
5

181
29–51
2
1
5
5

182
29–54
3
2
5
5

183
51–54
4
1
11
11

184
52–54
6
2
49
49

185
34–52
5
3
44
44

186
54-55
2
0
55
55

187
34–54
4
1
44
44

188
55–62
2
1
57
57

189
34–55
8
3
44
44

190
34–58
7
3
44
44

191
58–62
2
1
72
72

192
34–62
1
2
44
44

193
35–62
8
3
51
51

194
62-63
5
3
85
85

195
60–62
4
1
81
81

196
60–63
6
2
81
81

197
60–66
2
1
81
81

198
35–63
5
1
51
51

199
35–66
2
0
51
51

200
63–66
2
1
90
90

201
65-66
3
2
84
84

202
39–65
2
1
63
63

203
66–70
3
2
92
92

204
39–66
1
2
63
63

205
39–70
2
2
63
63

206
39–71
6
0
63
63

207
71–74
5
3
99
99

208
39–74
8
3
63
63

209
41–74
5
1
83
103

210
74–77
5
3
120
120

211
73-74
8
3
112
112

212
73–77
2
1
112
112

213
73–80
1
2
112
112

214
41–77
2
2
83
103

215
41–80
7
3
83
103

216
77–80
2
1
125
125

217
78–80
1
3
114
123

218
45–78
5
1
98
118

219
80-81
3
2
127
127

220
45–80
6
2
98
118

221
81–84
5
1
130
130

222
45–84
3
2
98
118

223
70–74
4
1
95
95

“ES” is the early start time of each work. “Optimized ES” is the optimized start time.

Figure 1: A large-scale network plan (223 works).

3.1. Solving the Optimization of Resource-Leveling with Fixed Duration

The variance method can be applied to evaluate the resource leveling, and the calculation formula of the variance is

where the total number of the samples is ; the arithmetic average of is .

The evolution equation is [49]

where is the number of iterations, is the -dimensional space coordinates of the particle at times of iterations, is the -dimensional space coordinates of the particle at times of iterations, is inertia weight (its general value is 1), is the -dimensional flight velocity of particle , and are the acceleration constant with a general value of 0–2, and are the random function with the value in the range of (0, 1), is the best position of particle experienced, is the best position of all particles, and is the integral function.

Table 1 also shows the start time of each work for resource-leveling optimization with fixed duration solution, corresponding to the resource variance of 22.41. The parameters applied in the OIPSO are the following: the inertia weight (empirical value), the acceleration constant (empirical value), (empirical value), the particle number (experimental value), the initial particle variance <30 (experimental value), and the number of iterations (experimental value). It is obvious that the optimized resource variance (22.41) is much less than the original one (37.51).

3.2. Solving the Optimization of Resources Constraints with Shortest Project Duration

As shown above, the methods of the optimization of resources constraints with shortest project duration in a large-scale network plan and the optimization of resource-leveling with fixed duration are similar. Besides, the initial particle swarm constraint is applied with resources constraints and as small as possible duration. The optimal solutions criteria are different; the constraint range of the feasible solution is also different, where the spatial coordinates of particles are the -dimension variables related to the work start time; their initial values are the coordinates of the initial particle swarm ; is the flight speed of particles.

The example is as shown in Figure 1, and Table 2 lists the start time of each work for resources constraints with shortest project duration optimization, where 25 is the resource constraint, 199 is the project duration, and 22 is the biggest resources quantity. The applied parameters of the resources constraints with shortest construction period optimization of the OIPSO are the following: the inertia weight (empirical value), the acceleration constant (empirical value), (empirical value), the quantity of particle (experimental value), the number of iterations (experimental value), the resource constraint of the initial particle swarm is 25 (experimental value), and the constrained duration is 300 (experimental value). The range of is and (the meaning of , , and is the same as the previous). The resource constraint (25) is met, and the corresponding project duration (199) is not too long after optimization.

Table 2: The parameters and their optimal solution of resources constraints with shortest project duration.

Number
Work
Duration
Resources quantity
ES
Optimized ES

1
1–3
2
2
0
1

2
1–4
2
1
0
7

3
3-4
2
1
2
4

4
1-2
1
2
0
0

5
4–6
4
1
4
10

6
5–8
4
1
7
9

7
5–9
5
1
7
15

8
6–8
5
3
13
18

9
7-8
8
3
15
20

10
3–6
3
2
2
5

11
2–4
3
2
1
2

12
3–5
5
1
2
3

13
2–8
3
2
1
2

14
5-6
6
2
7
9

15
2–6
1
3
1
1

16
6-7
2
0
13
16

17
8-9
7
3
23
33

18
8–11
2
2
23
28

19
8–10
2
1
23
28

20
7–10
2
1
15
19

21
9-10
1
2
30
40

22
9–12
4
1
30
43

23
18-19
8
3
57
84

24
17–20
2
1
51
82

25
16–19
4
1
49
81

26
18–21
5
1
57
97

27
17–19
5
3
51
77

28
16-17
2
2
49
72

29
15–17
2
1
46
69

30
14–18
2
1
42
76

31
14–17
1
2
42
61

32
13–15
2
1
44
64

33
10-11
4
1
31
44

34
12-13
5
1
39
58

35
10–13
5
3
31
41

36
10–12
8
3
31
45

37
17-18
6
0
51
74

38
15-16
3
2
46
68

39
11–15
5
1
35
49

40
14-15
3
2
42
63

41
11–17
5
1
35
51

42
12–14
3
2
39
58

43
11–13
6
2
35
48

44
12–15
2
0
39
53

45
21-22
2
2
70
104

46
22–24
2
1
75
126

47
20–22
2
1
73
107

48
20–24
1
2
73
112

49
20–26
2
1
73
122

50
19–21
5
1
65
96

51
19–22
5
3
65
95

52
19-20
8
3
65
99

53
24-25
3
2
77
129

54
24–26
5
1
77
131

55
26-27
3
2
85
148

56
25-26
5
1
80
142

57
23–27
3
2
72
108

58
23–26
6
2
72
109

59
23-24
1
3
72
112

60
21–23
2
0
70
102

61
21–24
7
3
70
107

62
3–28
1
2
2
3

63
3–30
3
2
2
10

64
3–31
1
3
2
29

65
28–30
2
1
3
30

66
28-29
2
2
3
5

67
29-30
2
1
5
7

68
29–31
3
2
5
8

69
29–34
3
2
5
9

70
30-31
4
1
7
35

71
5–31
6
2
7
12

72
5–34
5
3
7
43

73
31-32
2
0
13
42

74
31–34
4
1
13
44

75
32-33
2
1
15
61

76
32–34
8
3
15
49

77
9–34
7
3
30
46

78
9–33
2
1
30
41

79
34-35
2
2
44
76

80
33-34
1
2
43
63

81
33–35
8
3
43
67

82
33–36
5
3
43
79

83
12–33
4
1
39
53

84
12–36
6
2
39
57

85
12–37
5
1
39
57

86
35-36
5
1
51
81

87
35–37
2
0
51
79

88
35–39
5
1
51
88

89
36-37
2
1
56
87

90
14–37
3
2
42
63

91
14–39
2
1
42
62

92
37-38
3
2
58
91

93
37–39
1
2
58
94

94
38-39
2
2
61
94

95
18–39
6
0
57
82

96
18–40
5
3
57
83

97
39-40
8
3
63
97

98
39–41
2
1
63
97

99
40-41
5
1
78
118

100
40–42
5
3
78
116

101
21–40
8
3
70
107

102
21–42
2
1
70
121

103
21–43
1
2
70
120

104
41-42
2
2
83
123

105
41–43
7
3
83
127

106
41–45
2
1
83
130

107
42-43
2
1
85
126

108
23–43
1
3
72
129

109
23–45
5
1
72
108

110
43-44
3
2
90
134

111
43–45
6
2
90
139

112
44-45
5
1
93
141

113
27–45
3
2
88
151

114
38–40
4
1
61
96

115
47-48
2
2
8
31

116
48–50
2
1
10
48

117
47–50
2
1
8
32

118
48-49
1
2
10
43

119
50–53
4
1
14
70

120
52–57
4
1
49
74

121
52–58
5
1
49
73

122
53–57
5
3
55
87

123
56-57
8
3
57
86

124
47–53
3
2
8
32

125
49-50
3
2
11
67

126
47–52
5
1
8
33

127
49–57
3
2
11
49

128
52-53
6
2
49
69

129
49–53
1
3
11
82

130
53–56
2
0
55
84

131
57-58
7
3
65
94

132
57–61
2
2
65
103

133
57–59
2
1
65
95

134
56–59
2
1
57
100

135
58-59
1
2
72
101

136
58–60
4
1
72
104

137
71-72
8
3
99
138

138
69–75
2
1
93
130

139
68–72
4
1
91
127

140
71–73
5
1
99
138

141
69–72
5
3
93
131

142
68-69
2
2
91
126

143
67–69
2
1
88
123

144
65–71
2
1
84
118

145
65–69
1
2
84
128

146
64–67
2
1
86
119

147
59–61
4
1
73
104

148
60–64
5
1
81
113

149
59–64
5
3
73
106

150
59-60
8
3
73
105

151
69–71
6
0
93
129

152
67-68
3
2
88
122

153
61–67
5
1
77
111

154
65–67
3
2
84
119

155
61–69
5
1
77
112

156
60–65
3
2
81
114

157
61–64
6
2
77
112

158
60–67
2
0
81
114

159
73–76
2
2
112
152

160
76–79
2
1
117
162

161
75-76
2
1
115
159

162
75–79
1
2
115
168

163
75–83
2
1
115
160

164
72-73
5
1
107
146

165
72–76
5
3
107
150

166
72–75
8
3
107
151

167
79–82
3
2
119
171

168
79–83
5
1
119
174

169
83-84
3
2
127
181

170
82-83
5
1
122
175

171
78–84
3
2
114
163

172
78–83
6
2
114
162

173
78-79
1
3
114
161

174
73–78
2
0
112
151

175
73–79
7
3
112
155

176
46-47
1
2
7
30

177
47–51
3
2
8
33

178
47–54
1
3
8
101

179
46–51
2
1
7
37

180
29–46
2
2
5
25

181
29–51
2
1
5
9

182
29–54
3
2
5
98

183
51–54
4
1
11
130

184
52–54
6
2
49
69

185
34–52
5
3
44
64

186
54-55
2
0
55
134

187
34–54
4
1
44
110

188
55–62
2
1
57
138

189
34–55
8
3
44
71

190
34–58
7
3
44
70

191
58–62
2
1
72
132

192
34–62
1
2
44
123

193
35–62
8
3
51
83

194
62-63
5
3
85
142

195
60–62
4
1
81
115

196
60–63
6
2
81
118

197
60–66
2
1
81
118

198
35–63
5
1
51
81

199
35–66
2
0
51
83

200
63–66
2
1
90
148

201
65-66
3
2
84
130

202
39–65
2
1
63
98

203
66–70
3
2
92
152

204
39–66
1
2
63
100

205
39–70
2
2
63
129

206
39–71
6
0
63
101

207
71–74
5
3
99
148

208
39–74
8
3
63
105

209
41–74
5
1
83
140

210
74–77
5
3
120
164

211
73-74
8
3
112
155

212
73–77
2
1
112
154

213
73–80
1
2
112
168

214
41–77
2
2
83
123

215
41–80
7
3
83
123

216
77–80
2
1
125
170

217
78–80
1
3
114
161

218
45–78
5
1
98
156

219
80-81
3
2
127
174

220
45–80
6
2
98
159

221
81–84
5
1
130
177

222
45–84
3
2
98
155

223
70–74
4
1
95
158

“ES” is the early start time of each work. “Optimized ES” is the optimized start time.

4. The Superiority of OIPSO Compared with the Original and the Existing PSOs

After changing optimization parameters, the superiority of OIPSO in optimization of resource-leveling with fixed duration and resources constraints with shortest project duration of a large-scale network plan in Figure 1 is shown in Tables 3 and 4.

Table 3: The superiority of OIPSO in optimization of resource-leveling with fixed duration.

Optimization parameters
Case  1
Case  2
Case  3
Case  4

1
1
1
1

3.5
3.5
3.5
3.5

0.4
0.4
0.4
0.4

50
50
50
50

100
100
100
100

The project duration of the network plan
135
135
135
135

The project duration of the initial particle
135
135
135
135

The resource variance corresponding to the initial particle
30
50
—
25

The resource variance corresponding to the optimization results
22.41
22.99
22.99
—

“—” denotes “no restrictions” or “no solution in a limited amount of computing time.” The other symbols’ meanings are the same as the previous.

Table 4: The superiority of OIPSO in optimization of resources constraints with shortest project duration.

Optimization parameters
Case  5
Case  6
Case  7
Case  8
Case  9

1
1
1
1
1

3.5
3.5
3.5
3.5
3.5

0.4
0.4
0.4
0.4
0.4

10
10
10
10
10

1000
1000
1000
1000
1000

The constrained resources of the network plan
25
25
25
25
25

The constrained resources of the initial particle
25
25
—
—
25

The constrained project duration of the initial particle
300
—
300
—
200

The biggest resources quantity after optimization
22
21
22
19
—

The calculated project duration after optimization
199
206
199
206
—

Annotation is the same as Table 3.

Case  3 in Table 3 and case 8 in Table 4 are obtained by the original and existing PSOs in which the initial particles are randomly decided. Case  1 in Table 3 and case 5 in Table 4 are obtained by the improved algorithm (OIPSO) in which optimization is used to decide the initial particles. For the optimization of resource-leveling with fixed duration or resources constraints with shortest project duration of a large-scale network, it can be found that the improved algorithm (OIPSO) can accelerate the optimization speed and improve the optimization effect of particle swarm optimization under the same number of iterations by adding proper optimization constraints of the initial particle swarm, such as the variance restriction or resources limitation and project duration constraint. The optimization constraints of the initial particle swarm are decided gradually through the experiment such as case 2 (the resource variance corresponding to the initial particle is 50) and case 4 (the resource variance corresponding to the initial particle is 25) in Table 3 and case 6 (the constrained resources of the initial particle are 25 and there is no constrained project duration of the initial particle), case 7 (there are no constrained resources of the initial particle and the constrained project duration of the initial particle is 300), and case 9 (the constrained resources and the constrained project duration of the initial particle are, respectively, 25 and 200) in Table 4.

5. Conclusion

In this paper, the improved particle swarm optimization algorithm via optimization of initial particle swarm (OIPSO) has been proven to improve the solution probability of optimal solution or by the theory of Markov chains in random process and the optimization examples accelerates the optimization speed and improves optimization effect of particle swarm optimization under the same number of iterations. In existing publications on the larger-scale network plan optimization, the optimization examples of the resource-leveling with fixed duration and the resources constraints with shortest project duration on the large-scale network plan with 223 works have the largest work quantity.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is a part of the results of Methods Research on Resource Optimization of Large-scale Network Plan (2014-K3-039), which is the Science and Technology Plan Project from Housing and Urban-Rural Development of China.

References

S. E. Elmaghraby, “Activity nets: a guided tour through some recent developments,” European Journal of Operational Research, vol. 82, no. 3, pp. 383–408, 1995.

E. L. Demeulemeester and W. S. Herroelen, Project Scheduling, Kluwer Academic, Boston, Mass, USA, 2002.

J. Blazewicz, J. K. Lenstra, and A. H. G. R. Kan, “Scheduling subject to resource constraints: classification and complexity,” Discrete Applied Mathematics, vol. 5, no. 1, pp. 11–24, 1983.

S. F. Li, K. J. Zhu, and D. Y. Wang, “Complexity study of the application of network plan technique to large project,” Journal of China University of Geosciences (Social Science Edition), no. 9, pp. 90–94, 2010 (Chinese).

X. H. Zhao, CPM Network Plan Optimization Based on the Characteristic of Time Float, North China Electric Power University, Baoding, China, 2012 (Chinese).

D. D. Sun, Research oil Time-Cost Tradeoff Problem Based on Properties of Network, North China Electric Power University, Baoding, China, 2013 (Chinese).

X. F. Liu, Application Research of Network Plan Technique Optimization Methods to Building Construction Management, Tianjin University, Tianjin, China, 2013 (Chinese).

C. E. Bell and K. Park, “Solving resource-constrained project scheduling problems by a∗ search,” Naval Research Logistics, vol. 37, no. 1, pp. 61–84, 1990.

J. H. Patterson and W. D. Huber, “A horizon-varying, zero-one approach to project scheduling,” Management Science, vol. 20, no. 6, pp. 990–998, 1974.

R. Klein, Scheduling of Resource-Constrained Projects, Kluwer Academic, Boston, Mass, USA, 2000.

T. J. R. Johnson, An algorithm for the resource-con-strained project scheduling problem [dissertation], Massachusetts Institute of Technology (MIT), Cambridge, Mass, USA, 1967.

P. Brucker, S. Knust, A. Schoo, and O. Thiele, “A branch and bound algorithm for the resource-constrained project scheduling problem,” European Journal of Operational Research, vol. 107, no. 2, pp. 272–288, 1998.

A. Lova and R. Tormo, “Analysis of scheduling schemes and heuristic rules performance in resource-constrained multi-project scheduling,” Annals of Operations Research, vol. 102, no. 1, pp. 263–276, 2001.

J. D. Brand, W. L. Meyer, and L. R. Shaffer, “The resource scheduling method in construction,” Civil Engineering Studies Report 5, University of Illinois, Champaign, Ill, USA, 1964.

L. Özdamar and G. Ulusoy, “An iterative local constraints based analysis for solving the resource constrained project scheduling problem,” Journal of Operations Management, vol. 14, no. 3, pp. 193–208, 1996.

S. Hartmann and R. Kolisch, “Experimental evaluation of state-of-the-art heuristics for the resource-constrained project scheduling problem,” European Journal of Operational Research, vol. 127, no. 2, pp. 394–407, 2000.

P. Tormos and A. Lova, “A competitive heuristic solution technique for resource-constrained project scheduling,” Annals of Operations Research, vol. 102, pp. 65–81, 2001.

J. Alcaraz and C. Maroto, “A robust genetic algorithm for resource allocation in project scheduling,” Annals of Operations Research, vol. 102, no. 3, pp. 83–109, 2001.

W.-T. Chan, D. K. H. Chua, and G. Kannan, “Construction resource scheduling with genetic algorithms,” Journal of Construction Engineering and Management, vol. 122, no. 2, pp. 125–132, 1996.

S.-S. Leu and C.-H. Yang, “GA-based multicriteria optimal model for construction scheduling,” Journal of Construction Engineering and Management, vol. 125, no. 6, pp. 420–427, 1999.

S.-S. Leu and T.-H. Hung, “A genetic algorithm-based optimal resource-constrained scheduling simulation model,” Construction Management and Economics, vol. 20, no. 2, pp. 131–141, 2002.

J.-W. Huang, X.-X. Wang, and R. Chen, “Genetic algorithms for optimization of resource Allocation in Large Scale Construction Project Management,” Journal of Computers, vol. 5, no. 12, pp. 1916–1924, 2010.

D.-H. Zhong, M.-C. Li, W.-B. Zhang, and C.-S. Hu, “Resource optimization models and application for complex engineering construction system,” Tianjin: Journal of Tianjin University Science and Technology, vol. 37, no. 7, pp. 589–594, 2004.

J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948, IEEE, Perth, Australia, 1995.

Y. Liu, X. Zhang, X. Guan, and D. Delahaye, “Potential odor intensity grid based UAV path planning algorithm with particle swarm optimization approach,” Mathematical Problems in Engineering, vol. 2016, Article ID 7802798, 16 pages, 2016.

M. Wang and Q. Tian, “Dynamic heat supply prediction using support vector regression optimized by particle swarm optimization algorithm,” Mathematical Problems in Engineering, vol. 2016, Article ID 3968324, 10 pages, 2016.

J. Szklarski and M. Wikło, “Designing of elastoplastic adaptive truss structures with the use of particle swarm optimization,” Mathematical Problems in Engineering, vol. 2015, Article ID 652824, 14 pages, 2015.

K. Hu, A. Song, M. Xia et al., “An image filter based on shearlet transformation and particle swarm optimization algorithm,” Mathematical Problems in Engineering, vol. 2015, Article ID 414561, 9 pages, 2015.

M. Jakubcová, P. Máca, and P. Pech, “Parameter estimation in rainfall-runoff modelling using distributed versions of particle swarm optimization algorithm,” Mathematical Problems in Engineering, vol. 2015, Article ID 968067, 13 pages, 2015.

F. Pan, W. X. Li, and Q. Gao, Particle Swarm Optimization and Multi-Objective Optimization, Beijing Institute of Technology Press, 2013 (Chinese).

C.-F. Wang and K. Liu, “A novel particle swarm optimization algorithm for global optimization,” Computational Intelligence and Neuroscience, vol. 2016, Article ID 9482073, 9 pages, 2016.

C. Pornsing, M. S. Sodhi, and B. F. Lamond, “Novel self-adaptive particle swarm optimization methods,” Soft Computing, vol. 20, no. 9, pp. 3579–3593, 2016.

A. Meng, Z. Li, H. Yin, S. Chen, and Z. Guo, “Accelerating particle swarm optimization using crisscross search,” Information Sciences, vol. 329, pp. 52–72, 2016.

S. Taghiyeh and J. Xu, “A new particle swarm optimization algorithm for noisy optimization problems,” Swarm Intelligence, vol. 10, no. 3, pp. 161–192, 2016.

M. Basu, “Improved particle swarm optimization for global optimization of unimodal and multimodal functions,” Journal of The Institution of Engineers (India): Series B, vol. 97, no. 4, pp. 525–535, 2016.

Y. Fu, Z. L. Xu, and J. L. Cao, “Application of heuristic particle swarm optimization method in power network planning,” Power System Technology, no. 15, pp. 31–35, 2008.

B. Q. Xu, L. C. Zhang, and B. Z. Xu, “Cloud adapt particle Swarm Optimization algorithm for distribution network planning with distributed generation,” Applied Mechanics and Materials, vol. 427–429, pp. 1136–1140, 2013.

A. Nawawi, K. Hasnan, and S. Ahmad Bareduan, “Correlation between RFID network planning (RNP) parameters and particle swarm optimization (PSO) solutions,” Applied Mechanics and Materials, vol. 465-466, pp. 1245–1249, 2014.

J. Sun, X. Wu, V. Palade, W. Fang, and Y. Shi, “Random drift particle swarm optimization algorithm: convergence analysis and parameter selection,” Machine Learning, vol. 101, no. 1–3, pp. 345–376, 2015.

B. Wang, S. Wang, X. Zhou, and J. Watada, “Multi-objective unit commitment with wind penetration and emission concerns under stochastic and fuzzy uncertainties,” Energy, vol. 111, pp. 18–31, 2016.

R. Cheng and Y. Jin, “A social learning particle swarm optimization algorithm for scalable optimization,” Information Sciences. An International Journal, vol. 291, pp. 43–60, 2015.

X. Wang, Q. Guo, Q. Li, and J. Zhang, “High-order cumulant-based adaptive filter using particle swarm optimization,” in Proceedings of the Chinese Control and Decision Conference (CCDC '08), Shandong, China, July 2008.

Z. Y. Chen, Z. D. Du, and H. Zhou, “Research on the unlimited resource leveling optimization with PSO,” China Civil Engineering Journal, vol. 40, no. 2, pp. 93–96, 2007 (Chinese).

Z. J. Xu, Monte Carlo Method, Shanghai Science and Technology Press, 1985 (Chinese).

H. X. Zhang and B. N. Shi, “Resource optimization for network plan based on Monte Carlo method,” Mathematics in Practice and Theory, no. 6, pp. 120–128, 2015 (Chinese).

H.-Z. Du, N. Xia, J.-G. Jiang, L.-N. Xu, and R. Zheng, “A monte carlo enhanced PSO algorithm for optimal QoM in multi-channel wireless networks,” Journal of Computer Science and Technology, vol. 28, no. 3, pp. 553–563, 2013.

W.-C. Yeh, Y.-C. Lin, Y. Y. Chung, and M. Chih, “A particle swarm optimization approach based on monte carlo simulation for solving the complex network reliability problem,” IEEE Transactions on Reliability, vol. 59, no. 1, pp. 212–221, 2010.

H. X. Zhang, “Resource-leveling optimization with fixed duration for a large network plan based on the Monte Carlo method,” Construction Technology, no. 18, pp. 81–85, 2015 (Chinese).

H. X. Zhang and Z. L. Yang, “Resource optimization for a large network plan on particle swarm optimization,” Mathematics in Practice and Theory, no. 12, pp. 125–132, 2015 (Chinese).

H. X. Zhang and Z. L. Yang, “Cost optimization for a large network plan based on particle swarm optimization,” Mathematics in Practice and Theory, no. 11, pp. 142–148, 2015 (Chinese).

F. Pan, Q. Zhou, W. X. Li, and Q. Gao, “Analysis of the standard particle swarm optimization algorithm based on Markov chain,” Acta Automatica Sinica, vol. 39, no. 4, pp. 381–389, 2013.

S. Y. He, Randomized Procedure, Peking University Press, Beijing, China, 2008 (Chinese).