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Frontiers of Environmental Science & Engineering

ISSN 2095-2201

ISSN 2095-221X(Online)

CN 10-1013/X

Postal Subscription Code 80-973

2018 Impact Factor: 3.883

Front. Environ. Sci. Eng.    2016, Vol. 10 Issue (2) : 341-351    https://doi.org/10.1007/s11783-015-0776-z
RESEARCH ARTICLE
Estimation of distribution algorithm enhanced particle swarm optimization for water distribution network optimization
Xuewei QI1, Ke LI2(), Walter D. POTTER3
1. Department of Electrical and Computer Engineering, University of California, Riverside, CA 92507, USA
2. College of Engineering, University of Georgia, Athens, GA 30605, USA
3. Institute of Artificial Intelligence, University of Georgia, Athens, GA 30605, USA
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Abstract

The optimization of a water distribution network (WDN) is a highly nonlinear, multi-modal, and constrained combinatorial problem. Particle swarm optimization (PSO) has been shown to be a fast converging algorithm for WDN optimization. An improved estimation of distribution algorithm (EDA) using historic best positions to construct a sample space is hybridized with PSO both in sequential and in parallel to improve population diversity control and avoid premature convergence. Two water distribution network benchmark examples from the literature are adopted to evaluate the performance of the proposed hybrid algorithms. The experimental results indicate that the proposed algorithms achieved the literature record minimum (6.081 M$) for the small size Hanoi network. For the large size Balerma network, the parallel hybrid achieved a slightly lower minimum (1.921M€) than the current literature reported best minimum (1.923M€). The average number of evaluations needed to achieve the minimum is one order smaller than most existing algorithms. With a fixed, small number of evaluations, the sequential hybrid outperforms the parallel hybrid showing its capability for fast convergence. The fitness and diversity of the populations were tracked for the proposed algorithms. The track record suggests that constructing an EDA sample space with historic best positions can improve diversity control significantly. Parallel hybridization also helps to improve diversity control yet its effect is relatively less significant.

Keywords particle swarm optimization (PSO)      diversity control      estimation of distribution algorithm (EDA)      water distribution network (WDN)      premature convergence      hybrid strategy     
Corresponding Author(s): Ke LI   
Online First Date: 11 February 2015    Issue Date: 01 February 2016
 Cite this article:   
Xuewei QI,Ke LI,Walter D. POTTER. Estimation of distribution algorithm enhanced particle swarm optimization for water distribution network optimization[J]. Front. Environ. Sci. Eng., 2016, 10(2): 341-351.
 URL:  
https://academic.hep.com.cn/fese/EN/10.1007/s11783-015-0776-z
https://academic.hep.com.cn/fese/EN/Y2016/V10/I2/341
generation t generation t+1
Xmn(t) X^mn(t) Ωmn(t) Hmn(t) Xmn(t +1) X^mn(t+1) Ωmn(t+1) Hmn(t +1)
fitness 2 3 6 6 4 3 6 6
5 6 6 3 4 6 6 5
1 2 6 2 3 2 6 4
3 1 6 2 3 3 6 3
4 2 6 1 5 4 6 2
Tab.1  Explanation of historical best positions
Fig.1  Flow chart of parallel hybridization of EDA and PSO(PEDPSO)
Fig.2  Experimental results of the Hanoi (a) and the Balerma network (b)
Fig.3  Fitness track of different algorithms (Balerma network)
Fig.4  Diversity track of different algorithms (Balerma network)
algorithms min cost/M$ average success rate average no. of evaluations
GA1[28] 6.173 n/a n/a 26,457
ACO2 [29] 6.134 n/a n/a 35,433
HS3 [30] 6.081 n/a 1/81 27,721
PSHS4 [31] 6.081 n/a 1/81 17,980
GHEST5[32]
PSO6 [8]
HD-DDS7 [33]
NLP-DE8 [34]
6.081
6.133
6.081
6.081
n/a
n/a
6.252
n/a
6/10
3/100
8/100
98/100
16,600
n/a
5000,000
48,724
ISEDPSO (Present study) 6.081 6.102 28/30 17,600
PEDPSO (Present study) 6.081 6.103 27/30 23,400
Tab.2  Comparison of the best cost achieved by different algorithms (Hanoi network)
algorithms min cost
/M€
average success rate average no. of evaluations
GA [28] 2.302,000 n/a n/a 10,000,000
HS [30]
HD-DDS [33]
2.018,000
1.940,923
n/a
2.165,000
1/81
n/a
10,000,000
30,000,000
GHEST [32]
NLP-DE [34]
2.002,387
1.923,000
n/a
1.927,000
1/10
n/a
254,400
1,427,850
ISEDPSO(Present study) 1.933,407 1.976,672 16/30 201,400
PEDPSO(Present study) 1.921,428 1.942,231 17/30 217,400
Tab.3  Comparison of the best cost achieved by different algorithms (Balerma network)
algorithms min cost/M€ No. of evaluations
GA [28] 3,738 45,400
SA1 [27] 3.476 45,400
MSATS2 [27] 3.298 45,400
HS [30] 2.601 45,400
PSHS [31] 2.633 45,400
MA3 [35] 3.120 45,400
GHEST [32] 2.178 45,400
ISEDPSO (Present study) 2.1083 45,400
PEDPSO (Present study) 2.3781 45,400
Tab.4  Comparison with the same number of evaluations (Balerma network)
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