# Non-Dominated Sorting Harmony Search Differential Evolution (NS-HS-DE): A Hybrid Algorithm for Multi-Objective Design of Water Distribution Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- DE and HS show high efficiency in solving numerous single-objective test problems, especially the least-cost design problems involving WDNs. Moreover, since the proposed algorithm is constituted of DE and HS operators, it is advantageous to include source algorithms in the comparison.
- (2)
- (3)
- NSGA2 and SPEA2 are widely used in representative MOEAs in water resource and environmental engineering. More importantly, NSGA2 was reported to be the superior method relative to several contenders.
- (4)
- MOEA/D is a new approach for solving multi-objective optimization problems (Zhang and Li, [26]), which has a specific characteristic based on the decomposition scheme to separate the original problem into several sub-problems that can be solved collaboratively and simultaneously.

## 2. Methods

#### 2.1. Proposed Algorithm, Non-Dominated Sorting Harmony Search Differential Evolution (NSHSDE)

- (I).
- A mutation vector ${V}_{i}=\left\{{v}_{i,1},{v}_{i,1},\dots ,{v}_{i,n}\right\}$ is computed according to the equation:$${V}_{i}={H}_{C1}+F\times \left({H}_{C2}-{H}_{C3}\right)$$
_{C}_{1}, H_{C}_{2}and H_{C}_{3}are three members randomly selected from HM and F is a scaling factor in (0,1], preferably within the range 0.3–0.7. - (II).
- A pitch adjustment is used to enhance the diversity of the perturbed harmony vector and is considered as new harmony:$${r}_{i,j}=\{\begin{array}{ll}{v}_{i,j}\text{\hspace{0.17em}}+\Delta \text{\hspace{0.17em}};& \mathrm{if}\text{\hspace{0.17em}}(ran{d}_{j}\le PAR)\\ {v}_{i,j}\text{\hspace{0.17em}};& \mathrm{if}\text{\hspace{0.17em}}(ran{d}_{j}>PAR)\end{array}$$
_{w}is the fret width (or band width) parameter with a value considered as a small percentage of the range of decision variable j, and N(0,1) is a normal random number with mean 0 and variance 1. r_{i,j}is the decision variable j of the new harmony, R_{i}.

_{1}, F

_{2}, …, F

_{l}according to their non-domination sort order, where l is the index of the last front. Harmonies in front 1 dominate the harmonies in higher fronts; similarly, harmonies in front 2 dominate the harmonies in fronts 3, 4, …, l, but are dominated by those in front 1, and so on.

_{1}, F

_{2}, …, F

_{l}are sorted according to the crowding distance [27].

_{l}is the number of non-dominated fronts in the current iteration.

_{1}) and proceeding with the subsequently ranked non-dominated fronts (F

_{2}, F

_{3}, …, F

_{l}) until the size exceeds the full capacity. It is necessary to reject some of the lower-ranked non-dominated solutions to reduce the total number of the non-dominated solutions to render it equal to the HMS. This is done by using the crowding distance comparison operator; using this procedure, the elements in the HM are updated.

_{min}and Fw

_{max}are the minimum and maximum values of “fret width” respectively, and MaxIt is the number of iterations (generations).

#### 2.2. Algorithms Implemented for Comparison

#### 2.2.1. Non-Dominated Sorting Genetic Algorithm 2 (NSGA2)

#### 2.2.2. Non-Dominated Sorting Harmony Search (NSHS) Algorithm

#### 2.2.3. Non-Dominated Sorting Differential Evolution (NSDE) Algorithm

#### 2.2.4. Improving the Strength Pareto Evolutionary Algorithm (SPEA2)

#### 2.2.5. Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D)

#### 2.3. Performance Measures

#### 2.3.1. Generational Distance (GD)

_{i}is the distance (measured in objective space) between each of these and the nearest member of the global Pareto-optimal set. The parameter P stands for the Pth norm of the distance which is assumed equal 2–i.e. Euclidean distance, in this research. An ideal value of GD = 0 indicates that all elements generated are in the global Pareto-optimal set. Thus, any other value indicates how “far” the generated elements are from the global Pareto front. The lower GD, the algorithm’s performance better in terms of convergence.

#### 2.3.2. Diversity (D)

_{j}is the vector of jth objective function values of the Pareto front solutions and n is the number of objective functions. The higher the value of D metric, the better the diversity of MOEA. There is no specific lower and upper bound for this metric and its value is problem-dependent.

#### 2.3.3. Hypervolume (HV)

_{R}), was used to reduce the bias arising out of the magnitudes of different objective functions and to evaluate the quality of solutions found by each MOEA. HV

_{R}varies between zero and one with the ideal value of one.

#### 2.3.4. Coverage Set (CS)

#### 2.4. Multi-Objective Design of WDNs

#### 2.4.1. Mathematical Formulation

_{i}= unit cost of pipe i of diameter D

_{i}, L

_{i}= length of pipe i, I

_{n}= network resilience, nn = number of demand nodes, C

_{j}, Q

_{j}, H

_{j}, and ${H}_{j}^{req}$ are uniformity, demand, actual head and minimum required head respectively, of node j; nr = number of reservoirs, Q

_{k}and H

_{k}are discharge and actual head respectively, of reservoir k, npu = number of pumps, P

_{i}= power of pump i, γ = specific weight of water, npj = number of pipes connected to node j, and D

_{i}= diameter of pipe i connected to demand node j. The constraints of the optimization problem are as follows:

_{in}and Q

_{out}are in-flow and out-flow of the node, respectively, and Q

_{e}is the external flow rate or demand at the node.

_{k}is the head loss in pipe k and Nl is the total number of loops in the system. The head loss in each pipe is the difference in head between connected nodes, and is a function of discharge, pipe diameter, and roughness coefficient of the pipe. Head loss is usually calculated using empirical equations, such as the Darcy-Weisbach or the Hazen-Williams equation.

_{j}is the pressure head at node j, ${H}_{j}^{l}$ is the minimum required pressure head at node j, ${H}_{j}^{u}$ is the maximum allowed pressure head at node j, and nn is the number of demand nodes in the network.

_{i}is the diameter of pipe i, {A} denotes the set of commercially available pipe diameters, and np is the number of pipes.

_{i}Is velocity in pipe i, ${v}_{i}^{l}$ and ${v}_{i}^{u}$ are the minimum and maximum allowed velocity in pipe i, respectively.

^{6}in this study. Cp

_{1}is the summation of the penalties of all nodes with pressure violation, Cp

_{2}is the summation of the penalties of all pipes with velocity violation and Cp is the total penalty. Therefore, the total cost of the network is the sum of network cost C and penalty cost Cp in nodes and pipes with pressure and velocity violation, respectively.

#### 2.4.2. Experimental Tests on WDNs

## 3. Results and Discussions

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Layout of two-loop network, TLN [21].

**Figure 2.**Layout of Hanoi network (HAN) [20].

**Figure 3.**Layout of Fossolo network (FOS) [21].

**Figure 4.**Layout of Balerma network (BIN) [21].

**Figure 5.**Pareto front (PF) of benchmark problems, (

**a**) Two-loop network (TLN) problem; (

**b**) Hanoi network (HAN) problem.

**Figure 6.**Pareto front (PF) of benchmark problems, (

**a**) Fossolo network (FOS) problem; (

**b**) Balerma network (BIN) problem.

**Figure 7.**A close-up view of Pareto fronts (PFs) generated by Non-Dominated Sorting Harmony Search (NSHS) and nondominated sorting genetic algorithm 2 (NSGA2), (

**a**) FOS problem; (

**b**) BIN problem.

Diameter (in.) | Unit Cost ($/m) | Diameter (in.) | Unit Cost ($/m) |
---|---|---|---|

1 | 2 | 12 | 50 |

2 | 5 | 14 | 60 |

3 | 8 | 16 | 90 |

4 | 11 | 18 | 130 |

6 | 16 | 20 | 170 |

8 | 23 | 22 | 300 |

10 | 32 | 24 | 550 |

Diameter (in.) | Unit Cost ($/m) | Diameter (in.) | Unit Cost ($/m) | Diameter (in.) | Unit Cost ($/m) |
---|---|---|---|---|---|

12 | 45.73 | 20 | 98.39 | 30 | 180.75 |

16 | 70.40 | 24 | 129.33 | 40 | 278.28 |

NI | Pmax (m) | NI | Pmax (m) | NI | Pmax (m) | NI | Pmax (m) | NI | Pmax (m) | NI | Pmax (m) |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 55.85 | 7 | 53.1 | 13 | 59.1 | 19 | 58.1 | 25 | 56.6 | 31 | 56.6 |

2 | 56.6 | 8 | 54.5 | 14 | 58.4 | 20 | 58.17 | 26 | 57.6 | 32 | 56.8 |

3 | 57.65 | 9 | 55.0 | 15 | 57.5 | 21 | 58.2 | 27 | 57.1 | 33 | 56.4 |

4 | 58.5 | 10 | 56.83 | 16 | 56.7 | 22 | 57.1 | 28 | 55.35 | 34 | 56.3 |

5 | 59.76 | 11 | 57.3 | 17 | 55.5 | 23 | 56.8 | 29 | 56.5 | 35 | 55.57 |

6 | 55.60 | 12 | 58.36 | 18 | 56.9 | 24 | 53.5 | 30 | 56.9 | 36 | 55.1 |

Diameter (mm) | Unit Cost (€/m) | Diameter (mm) | Unit Cost (€/m) | Diameter (mm) | Unit Cost (€/m) | Diameter (mm) | Unit Cost (€/m) |
---|---|---|---|---|---|---|---|

16 | 0.38 | 61.4 | 4.44 | 147.20 | 24.78 | 290.6 | 99.58 |

20.4 | 0.56 | 73.6 | 6.45 | 163.6 | 30.55 | 327.4 | 126.48 |

26 | 0.88 | 90 | 9.59 | 184.00 | 38.71 | 368.2 | 160.29 |

32.6 | 1.35 | 102.2 | 11.98 | 204.6 | 47.63 | 409.2 | 197.71 |

40.8 | 2.02 | 114.6 | 14.93 | 229.2 | 59.7 | ||

51.4 | 3.21 | 130.80 | 19.61 | 257.8 | 75.61 |

Diameter (mm) | Unit Cost (€/m) | Diameter (mm) | Unit Cost (€/m) |
---|---|---|---|

113 | 7.22 | 226.2 | 28.6 |

126.6 | 9.10 | 285 | 45.39 |

144.6 | 11.92 | 361.8 | 76.32 |

162.8 | 14.84 | 452.2 | 124.64 |

180.8 | 18.38 | 581.8 | 215.85 |

Problem | NFE ^{a} | Population Size | DV ^{b} | PD ^{c} | Search Space Size |
---|---|---|---|---|---|

Two-loop Network | 20,000 | 40 | 8 | 14 | 1.48 × 10^{9} |

Hanoi Network | 50,000 | 60 | 34 | 6 | 2.87 × 10^{26} |

Fossolo Network | 200,000 | 100 | 58 | 22 | 7.25 × 10^{77} |

Balerma Irrigation Network | 1,000,000 | 400 | 454 | 10 | 1.0 × 10^{454} |

^{a}NFE = number of function evaluations,

^{b}DV = number of decision variables,

^{c}PD = number of pipe diameter options.

Algorithm | Parameter | Value |
---|---|---|

NSGA2 | Mutation rate | 1/(no. variables) |

Crossover prob. | 0.9 | |

Tournament size | 2 | |

Mutation step size | 0.1 × Variable range | |

SPEA2 | Mutation rate | 1/(no. variables) |

Crossover prob. | 0.9 | |

Tournament size | 2 | |

Mutation step size | 0.1 × Variable range | |

NSHS | HMCR | 0.98 |

PAR | 0.4 | |

Fw | (0.05–0.005) × Variable range | |

NSDE | F (scaling factor) | 0.5 |

Crossover prob. | 0.7 | |

NSHSDE | F (scaling factor) | 0.5 |

PAR | 0.4 | |

Fw | (0.05–0.005) × Variable range | |

MOEA/D | Mutation prob. | 0.3 |

Mutation rate. | 0.1 | |

T (number of neighbors) | 0.2 × Pop size | |

Z * (Goal point) | The ideal values for objective functions, zero for the cost and one for the resiliency index |

**Table 8.**Comparison of multi-objective algorithms using the average values of generational distance (GD), relative diversity (D), and hypervolume (HV) metrics.

Problem | Algorithm | GD | D | HV_{R} |
---|---|---|---|---|

TLN | NSHSDE | 193.96 | 0.79 | 1.00 |

NSDE | 1317.76 | 0.50 | 0.62 | |

NSGA2 | 3690.13 | 0.43 | 0.49 | |

NSHS | 3362.55 | 0.19 | 0.26 | |

SPEA2 | 3136.40 | 0.51 | 0.72 | |

MOEA/D | 4007.28 | 0.41 | 0.49 | |

HAN | NSHSDE | 1992.61 | 0.80 | 0.98 |

NSDE | 2420.65 | 0.66 | 0.96 | |

NSGA2 | 1987.79 | 0.22 | 0.30 | |

NSHS | 1670.78 | 0.17 | 0.24 | |

SPEA2 | 1866.20 | 0.12 | 0.18 | |

MOEA/D | 5130.38 | 0.81 | 0.86 | |

FOS | NSHSDE | 2253.75 | 0.5 | 0.96 |

NSDE | 10529.41 | 0.67 | 0.98 | |

NSGA2 | 213.13 | 0.02 | 0.03 | |

NSHS | 103.00 | 0.00 | 0.01 | |

SPEA2 | 310.30 | 0.00 | 0.00 | |

MOEA/D | 36463.37 | 0.33 | 0.37 | |

BIN | NSHSDE | 7032.75 | 0.32 | 0.36 |

NSDE | 28295.29 | 0.48 | 0.69 | |

NSGA2 | 2144.96 | 0.06 | 0.06 | |

NSHS | 1697.33 | 0.03 | 0.03 | |

SPEA2 | 1762.07 | 0.01 | 0.01 | |

MOEA/D | 1729.88 | 0.09 | 0.08 |

Problem | Algorithm | NSHSDE | NSDE | NSGA2 | NSHS | SPEA2 | MOEA/D |
---|---|---|---|---|---|---|---|

TLN | NSHSDE | - | 0.87 | 0.85 | 0.92 | 0.82 | 0.94 |

NSDE | 0.84 | - | 0.84 | 0.93 | 0.85 | 0.95 | |

NSGA2 | 0.74 | 0.76 | - | 0.89 | 0.78 | 0.86 | |

NSHS | 0.31 | 0.32 | 0.35 | - | 0.32 | 0.37 | |

SPEA2 | 0.66 | 0.67 | 0.7 | 0.83 | - | 0.85 | |

MOEA/D | 0.58 | 0.61 | 0.62 | 0.87 | 0.63 | - | |

HAN | NSHSDE | - | 0.6 | 0.72 | 0.53 | 0.99 | 0.95 |

NSDE | 0.54 | - | 0.71 | 0.51 | 0.99 | 0.93 | |

NSGA2 | 0.28 | 0.29 | - | 0.34 | 0.92 | 0.78 | |

NSHS | 0.47 | 0.49 | 0.66 | - | 0.92 | 0.85 | |

SPEA2 | 0.01 | 0.01 | 0.08 | 0.08 | - | 0.48 | |

MOEA/D | 0.06 | 0.07 | 0.22 | 0.15 | 0.52 | - | |

FOS | NSHSDE | - | 0.82 | 0.93 | 0.75 | 1 | 1 |

NSDE | 0.21 | - | 0.98 | 0.55 | 0.83 | 1 | |

NSGA2 | 0.07 | 0.02 | - | 0.17 | 0.44 | 0.71 | |

NSHS | 0.25 | 0.45 | 0.83 | - | 0.17 | 0.67 | |

SPEA2 | 0 | 0.17 | 0.566 | 0.83 | - | 0.92 | |

MOEA/D | 0 | 0 | 0.29 | 0.33 | 0.08 | - | |

BIN | NSHSDE | - | 0.59 | 1 | 0.64 | 1 | 0.53 |

NSDE | 0.41 | - | 1 | 0.62 | 0.55 | 0.38 | |

NSGA2 | 0 | 0 | - | 0.48 | 0.4 | 0 | |

NSHS | 0.36 | 0.38 | 0.52 | - | 1 | 0.46 | |

SPEA2 | 0 | 0.45 | 0.59 | 0 | - | 0 | |

MOEA/D | 0.48 | 0.61 | 1 | 0.54 | 1 | - |

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**MDPI and ACS Style**

Yazdi, J.; Choi, Y.H.; Kim, J.H.
Non-Dominated Sorting Harmony Search Differential Evolution (NS-HS-DE): A Hybrid Algorithm for Multi-Objective Design of Water Distribution Networks. *Water* **2017**, *9*, 587.
https://doi.org/10.3390/w9080587

**AMA Style**

Yazdi J, Choi YH, Kim JH.
Non-Dominated Sorting Harmony Search Differential Evolution (NS-HS-DE): A Hybrid Algorithm for Multi-Objective Design of Water Distribution Networks. *Water*. 2017; 9(8):587.
https://doi.org/10.3390/w9080587

**Chicago/Turabian Style**

Yazdi, Jafar, Young Hwan Choi, and Joong Hoon Kim.
2017. "Non-Dominated Sorting Harmony Search Differential Evolution (NS-HS-DE): A Hybrid Algorithm for Multi-Objective Design of Water Distribution Networks" *Water* 9, no. 8: 587.
https://doi.org/10.3390/w9080587