# Self-Adaptive Models for Water Distribution System Design Using Single-/Multi-Objective Optimization Approaches

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Addition of new operators to generate a new solution;
- Application of dynamic parameters (i.e., HMCR, PAR, and Bw) depending on number of iterations, harmony memory size, and decision variable (DV) size; and
- Consideration of the operation type in optimization process.

## 2. Optimal Design of Water Distribution Systems

#### 2.1. Minimizing Construction Cost

_{i}) is the cost function of the ith pipe per unit length (m) of each pipe diameter, L

_{i}is the length (m) of the ith pipe, D

_{i}is the pipe diameter (mm) of the ith pipe, and N is the total number of pipes.

#### 2.2. Maximizing System Resilience

_{j}is a head at node j, h

_{j}* is the minimum required head at node j, H

_{K}is the water level of reservoir K, Q

_{K}is water flow of reservoir K, and P

_{i}is the power of pump i.

#### 2.3. Hydraulic Constraints and the Penalty Function

_{i}is the pressure head at node i (m); v

_{j}is the water velocity at pipe j (m/s); h

_{min}and h

_{max}are the minimum and maximum pressure heads (m), respectively; v

_{min}and v

_{max}are the minimum and maximum water velocity (m/s), respectively, and α and β are the penalty constants.

## 3. Optimization Algorithms

#### 3.1. Harmony Search

_{i}

^{New}denotes a new decision variable, and x

_{i}

^{Lower}, x

_{i}

^{Upper}are the boundary conditions of the decision variables. Rnd is the uniform random value, and Bw is the bandwidth.

#### 3.2. Parameter-Setting-Free Harmony Search

_{n}denotes an operation type for the decision variable. Each y

_{n}represents one of three cases: (i) RS, (ii) MC, and (iii) PA. n is the count function of the number of operations (e.g., MC or PA) in HM. HMCR

_{i}and PAR

_{i}represent the HMCR and PAR of the ith decision variable, respectively.

_{i}and PAR

_{i}between 0 and 1, as indicated by Equation (9).

#### 3.3. Almost-Parameter-Free Harmony Search

_{i}

^{j}is ith decision variable in the jth iteration.

#### 3.4. Novel Self-Adaptive Harmony Search

_{std}is the standard deviation calculated by ${f}_{std}=std(f({x}_{1}),f({x}_{2}),\dots ,f({x}_{HMS}))$, Bw

_{i}

^{Lower}and Bw

_{i}

^{Upper}are the boundaries of the bandwidth at the ith decision variable in the jth iteration, and NI is the total number of iterations.

#### 3.5. Self-Adaptive Global-Based Harmony Search Algorithm

#### 3.6. Parameter Adaptive Harmony Search

## 4. Multi-Objective Optimization Formulation

_{1}

^{max,min}and f

_{2}

^{max,min}) and the distance between beside two Pareto optimal solutions (i.e., d

_{1}and d

_{2}).

## 5. Application and Results

#### 5.1. Performance Indices

^{−10}(DV = 2, 5, 10) and 10

^{−5}(DV = 30, 50). The parameter settings of all the approaches used in this study are tabulated in Table 2.

_{m}

^{Max}and F

_{m}

^{Min}are the maximum and minimum values of the Pareto front (PF), f

_{m}is the mth objective function value, and M is the number of objective functions.

_{pf}is the number of members in the generated PF, d

_{i}is the Euclidean distance between member i in PF and the nearest member in PF

_{optimal}, and the Euclidean distance (d) is calculated based on Equation (22).

_{q1}, f

_{q2}, …, f

_{qn}) is a point on PF, and p = (f

_{p1}, f

_{p2}, …, f

_{pn}) is the member nearest to q in PF

_{optimal}. The best possible value for the GD metric is 0, which corresponds to the PF

_{g}exactly covering PF

_{optimal}.

_{i}denotes ${\mathrm{min}}_{j}\left(\left|{f}_{1}^{i}\left(x\right)-{f}_{1}^{j}\left(x\right)\right|+\left|{f}_{2}^{i}\left(x\right)-{f}_{2}^{j}\left(x\right)\right|\right)$, i, j = 1, …, n; $\overline{d}$ is the mean value of all d

_{i}; and n is the number of non-dominated solutions. The values of SP are in the range of 0–1. Values closer to 0 indicate that the optimal solution is distributed more homogeneously.

#### 5.2. Comparison of Algorithm Performance in Mathematical Benchmark Problems

#### 5.2.1. Single-Objective Optimization Problems

^{−10}, and 10

^{−5}if it is not) and examined the initial convergence ability of each algorithm.

^{−10}(DV = 2, 5, 10) and 10

^{−5}(DV = 30, 50), depending on their decision variables. In all cases, SGHSA has the best success ratio and NSHS shows the lowest NFEs-Fs. However, as the number of decision variables is increased, the success ratio of NSHS decreases drastically.

#### 5.2.2. Multi-Objective Optimization Problems

#### 5.3. Comparison of Self-Adaptive Technique Performance in WDS Design Problems

#### 5.3.1. Single-Objective Optimal Design of WDSs

#### 5.3.2. Multi-Objective Optimal Design of WDSs

#### 5.4. Comparison of Algorithm Characteristics (i.e., Operator and Parameter)

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Comparison of results in the multi-objective optimal design of water distribution systems: (

**a**,

**c**,

**e**) Pareto optimal solutions of each network; (

**b**,

**d**,

**f**) Performance measures of the best solution of each network.

**Figure 6.**Distribution of non-dominated solutions from each self-adaptive approach in the Pareto optimal solutions: (

**a**) Hanoi network, (

**b**) Saemangeum network, (

**c**) P-city network.

**Table 1.**Category of self-adaptive harmony search (HS). PAR = pitch adjustment rate; HMCR = harmony memory consideration rate; Bw = bandwidth.

Algorithm Category | Application Field | Improvement, [Reference] |
---|---|---|

Single-objective Self-adaptive HS | Water resource engineering | HMCR, PAR, [29] |

Water resource engineering | HMCR, PAR, [30] | |

Water resource engineering, mathematics | HMCR, PAR, [31] | |

Economic dispatch | HMCR, PAR, [32] | |

Water quality engineering | HMCR, PAR, Bw, [33] | |

Mathematics | Bw, [25] | |

Mathematics | Bw, [26] | |

Mathematics | HMCR, Bw, [36] | |

Mathematics | HMCR, PAR, [37] | |

Structure engineering | Bw, [27] | |

Mathematics | PAR, Bw, [38] | |

Mathematics | HMCR, PAR, Bw, [28] | |

Traffic engineering | PAR, Bw, [39] | |

Mathematics | HMCR, [40] | |

Data mining | PAR, [41] | |

Economic dispatch problem | PAR, [42] | |

Electricity system | PAR, Bw, [43] | |

Electricity system | HMCR, PAR, [44] | |

Mathematics | HMCR, PAR, [45] | |

Mathematics | PAR, Bw, [46] | |

Mathematics | HMCR, PAR, Bw, [47] | |

Multi-objective self-adaptive HS | Mathematics | HMCR, PAR, Bw, [34] |

Mechanical engineering | HMCR, PAR, Bw, [35] |

Algorithms | HMS | HMCR | PAR | Bw | Noise Value | |||
---|---|---|---|---|---|---|---|---|

LB | UB | LB | UB | LB | UB | |||

HS | 5/10 (if DV ≤10, HMS = 5 otherwise, HMS = 10) | 0.95 | 0.1 | 10^{−4} | - | |||

PSF-HS first | 0.05 | 0.1 | 10^{−5} | 10^{−4} | 10^{−4} | 10^{−3} | ||

PSF-HS second | ||||||||

APF-HS | 10^{−5} | 10^{−4} | ||||||

NSHS | - | |||||||

SGHSA | 0.95 | 0.05 | 0.1 | 10^{−5} | 10^{−4} | - | ||

PAHS | 0.5 | 0.95 |

Name (Function Shape) | Formulation | Search Domain | |
---|---|---|---|

Unconstrained problems | Sphere function (Bowl-shaped) | $Min\text{}f(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}$ | [−∞, ∞]n |

Rosenbrock function (Valley-shaped) | $Min\text{}f(x)={\displaystyle \sum _{i=1}^{n-1}\left[100{({x}_{i+1}^{}-{x}_{i}^{2})}^{2}+{({x}_{i}^{}-1)}^{2}\right]}$ | [−30, 30]n | |

Rastrigin function (Many local optima) | $Min\text{}f(x)=10n+{\displaystyle \sum _{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}(2\pi {x}_{i}^{})\right]}$ | [−5.12, 5.12]n | |

Griewank function (Many local optima) | $Min\text{}f(x)={\displaystyle \sum _{i=1}^{n}\frac{{x}_{i}^{2}}{4000}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}}\left(\frac{{x}_{i}^{}}{\sqrt{i}}\right)+1}$ | [−600, 600]n | |

Ackley function (Many local optima) | $\begin{array}{l}Min\text{}f(x)=-20\mathrm{exp}\left(-0.2\sqrt{(1/n){\displaystyle \sum _{i=1}^{n}({x}_{i}^{2})}}\right)\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}}-\mathrm{exp}\left((1/n){\displaystyle \sum _{i=1}^{n}cos(2\pi {x}_{i}^{})}\right)+20+\mathrm{exp}(1)\end{array}$ | [−32.768, 32.768]n | |

Constrained problem 1 | $\begin{array}{l}Min\text{}f(x)={({x}_{1}-10)}^{2}+{({x}_{2}-20)}^{3}\\ h(x)=-{({x}_{1}-5)}^{2}-{({x}_{2}-5)}^{2}+100\le 0\\ g(x)=-{({x}_{1}-6)}^{2}-{({x}_{1}-5)}^{2}-82.81\le 0\end{array}$ | 13 < x_{1} < 1000 < x _{2} < 100 | |

Constrained problem 2 | $\begin{array}{l}Min\text{}f(x)={({x}_{1}-10)}^{2}+5{({x}_{2}-12)}^{2}+{x}_{3}^{4}+3{({x}_{4}-11)}^{2}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}+10{x}_{5}^{6}+7{x}_{6}^{2}+{x}_{7}^{4}-4{x}_{6}^{}{x}_{7}^{}-10{x}_{6}^{}-8{x}_{7}^{}\\ {g}_{1}(x)=127-2{x}_{1}^{2}-3{x}_{2}^{4}-{x}_{3}^{}-4{x}_{4}^{2}-5{x}_{5}^{}\ge 0\\ {g}_{2}(x)=282-7{x}_{1}^{}-3{x}_{2}^{}-10{x}_{3}^{2}-{x}_{4}^{}+{x}_{5}^{}\ge 0\\ {g}_{3}(x)=196-23{x}_{1}^{}-{x}_{2}^{}-6{x}_{6}^{2}+8{x}_{8}^{}\ge 0\\ {g}_{4}(x)=-4{x}_{1}^{2}-{x}_{2}^{2}-3{x}_{1}^{}{x}_{2}^{}-2{x}_{3}^{2}-5{x}_{6}^{}+11{x}_{7}^{}\ge 0\end{array}$ | [−10, 10]n n = 1,2,3…,6,7 |

Problem Name | Algorithm | Worst Solution | Mean Solution | Best Solution | SD |
---|---|---|---|---|---|

Constrained problem 1 | SHS | −6473.900 | −6642.600 | −6952.100 | 2.43 × 10^{2} |

PSF-HS first | −6577.123 | −6740.288 | −6956.251 | 2.70 × 10^{2} | |

PSF-HS second | −6901.721 | −6961.814 | −6961.814 | 7.30 × 10^{−4} | |

APF-HS | −6960.415 | −6961.814 | −6961.814 | 1.70 × 10^{−2} | |

NSHS | −6961.284 | −6961.813 | −6961.814 | 6.00 × 10^{−3} | |

SGHSA | −6961.813 | −6961.813 | −6961.814 | 5.77 × 10^{−4} | |

PAHS | −6350.262 | −6875.94 | −6961.814 | 1.60 × 10^{−1} | |

GA | −6821.511 | −6861.196 | −6961.813 | 7.23 × 10^{−4} | |

PSO | −6791.813 | −6921.133 | −6961.813 | 8.88 × 10^{−4} | |

DE | −6960.813 | −6961.412 | −6961.814 | 5.04 × 10^{−5} | |

Constrained problem 2 | SHS | 683.181 | 681.160 | 680.911 | 4.11 × 10^{−2} |

PSF-HS first | 680.721 | 680.681 | 680.631 | 1.00 × 10^{−5} | |

PSF-HS second | 682.965 | 681.347 | 680.631 | 5.70 × 10^{−1} | |

APF-HS | 682.651 | 681.642 | 680.426 | 2.70 × 10^{−2} | |

NSHS | 682.081 | 681.246 | 680.426 | 3.22 × 10^{−3} | |

SGHSA | 680.763 | 680.656 | 680.426 | 3.40 × 10^{−4} | |

PAHS | 680.719 | 680.643 | 680.632 | 1.55 × 10^{−2} | |

GA | 680.653 | 680.638 | 680.631 | 6.61 × 10^{−3} | |

PSO | 684.528 | 680.971 | 680.634 | 5.10 × 10^{−1} | |

DE | 681.144 | 680.503 | 680.426 | 6.71 × 10^{−1} |

Number of Decision Variables | Algorithm | Success Ratio (%) | Best NFEs-Fs | Average NFEs-Fs | Average NIS |
---|---|---|---|---|---|

2 | SHS | 14 | 3453 | 36,727.3 | 53.4 |

PSF-HS first | 74 | 1291 | 9770.6 | 75.7 | |

PSF-HS second | 64 | 1258 | 10,235.3 | 76.4 | |

APF-HS | 96 | 104 | 7709.4 | 89.8 | |

NSHS | 100 | 51 | 190.1 | 168.3 | |

SGHSA | 100 | 176 | 529.0 | 173.7 | |

PAHS | 96 | 200 | 3000.4 | 111.3 | |

5 | SHS | 0 | - | - | 19.2 |

PSF-HS first | 46 | 12,218 | 21,572.1 | 38.1 | |

PSF-HS second | 6 | 38,247 | 39,332.3 | 37.1 | |

APF-HS | 94 | 7086 | 15,585.0 | 48.3 | |

NSHS | 36 | 5762 | 9863.4 | 41.2 | |

SGHSA | 100 | 417 | 886.7 | 162.5 | |

PAHS | 56 | 18,588 | 26,523.3 | 35.3 | |

10 | SHS | 0 | - | - | 22.2 |

PSF-HS first | 32 | 32,522 | 44,321.3 | 70.7 | |

PSF-HS second | 4 | 42,776 | 48,721.1 | 62.9 | |

APF-HS | 96 | 9177 | 21,627.1 | 88.8 | |

NSHS | 32 | 36,251 | 45,126.2 | 69.2 | |

SGHSA | 100 | 1263 | 1974.3 | 270.2 | |

PAHS | 24 | 39,853 | 47,953,3 | 36.8 | |

30 | SHS | 15 | - | - | 28.0 |

PSF-HS first | 100 | 94 | 233.4 | 117.6 | |

PSF-HS second | 100 | 101 | 214.2 | 58.7 | |

APF-HS | 100 | 63 | 112.3 | 281.9 | |

NSHS | 100 | 18 | 30.1 | 399.3 | |

SGHSA | 100 | 57 | 116.9 | 425.1 | |

PAHS | 100 | 5062 | 9677.9 | 209.3 | |

50 | SHS | 0 | - | - | 33.3 |

PSF-HS first | 50 | 369 | 929.3 | 49.1 | |

PSF-HS second | 100 | 447 | 808.2 | 56.9 | |

APF-HS | 100 | 97 | 223.8 | 381.1 | |

NSHS | 100 | 30 | 39.8 | 682.1 | |

SGHSA | 100 | 148 | 302.9 | 752.5 | |

PAHS | 80 | 37,905 | 42,695.4 | 390.3 |

Name | Formulation | Search Domain |
---|---|---|

Zitzler Deb Thiele’s function No. 1 | $Min=\{\begin{array}{l}{f}_{1}(x)={x}_{1}\\ {f}_{2}(x)=g(x)h\left({f}_{1}(x),g(x)\right)\\ g(x)=1+\frac{9}{n-1}\left({\displaystyle \sum _{i=2}^{n}{x}_{i}}\right)\\ h({f}_{1}(x),g(x))=1-\sqrt{\frac{{f}_{1}(x)}{g(x)}}\end{array}$ | 0 ≤ x_{i} ≤ 11 ≤ i ≤ 30 |

Zitzler Deb Thiele’s function No. 2 | $Min=\{\begin{array}{l}{f}_{1}(x)={x}_{1}\\ {f}_{2}(x)=g(x)h\left({f}_{1}(x),g(x)\right)\\ g(x)=1+\frac{9}{n-1}\left({\displaystyle \sum _{i=2}^{n}{x}_{i}}\right)\\ h({f}_{1}(x),g(x))=1-{\left(\frac{{f}_{1}(x)}{g(x)}\right)}^{2}\end{array}$ | 0 ≤ x_{i} ≤ 11 ≤ i ≤ 30 |

Zitzler Deb Thiele’s function No. 3 | $Min=\{\begin{array}{l}{f}_{1}(x)={x}_{1}\\ {f}_{2}(x)=g(x)h\left({f}_{1}(x),g(x)\right)\\ g(x)=1+\frac{9}{n-1}\left({\displaystyle \sum _{i=2}^{n}{x}_{i}}\right)\\ h({f}_{1}(x),g(x))=1-\sqrt{\frac{{f}_{1}(x)}{g(x)}}-\left(\frac{{f}_{1}(x)}{g(x)}\right)\mathrm{sin}(10\pi {f}_{1}(x))\end{array}$ | 0 ≤ x_{i} ≤ 11 ≤ i ≤ 30 |

Zitzler Deb Thiele’s function No. 4 | $Min=\{\begin{array}{l}{f}_{1}(x)={x}_{1}\\ {f}_{2}(x)=g(x)h\left({f}_{1}(x),g(x)\right)\\ g(x)=1+10(n-1)+{\displaystyle \sum _{i=2}^{n}({x}_{i}^{2}-10\mathrm{cos}(4\pi {x}_{i}^{})})\\ h({f}_{1}(x),g(x))=1-\sqrt{\frac{{f}_{1}(x)}{g(x)}}\end{array}$ | 0 ≤ x_{1} ≤ 1−5 ≤ x _{i} ≤ 52 ≤ i ≤ 10 |

Zitzler Deb Thiele’s function No. 6 | $Min=\{\begin{array}{l}{f}_{1}(x)=1-\mathrm{exp}(-4{x}_{1}){\mathrm{sin}}^{6}(6\pi {x}_{1})\\ {f}_{2}(x)=g(x)h\left({f}_{1}(x),g(x)\right)\\ g(x)=1+9{\left(\left({\displaystyle \sum _{i=2}^{n}{x}_{i}}\right)/(n-1)\right)}^{0.25}\\ h({f}_{1}(x),g(x))=1-{\left(\frac{{f}_{1}(x)}{g(x)}\right)}^{2}\end{array}$ | 0 ≤ x_{i} ≤ 11 ≤ i ≤ 10 |

Multi-Objective Optimization Problems | Algorithms | Convergence | Diversity | ||
---|---|---|---|---|---|

CS | GD | DI | SP | ||

ZDT1 | SHS | 0.112 | 8.41 × 10^{−16} | 1.314 | 4.21 × 10^{−3} |

PSF-HS first | 0.129 | 2.34 × 10^{−17} | 1.414 | 2.53 × 10^{−3} | |

PSF-HS second | 0.133 | 8.09 × 10^{−18} | 1.247 | 2.05 × 10^{−2} | |

APF-HS | 0.135 | 2.66 × 10^{−18} | 1.288 | 4.02 × 10^{−2} | |

SGHSA | 0.145 | 0 | 1.207 | 3.71 × 10^{−2} | |

NSHS | 0.098 | 0 | 0.786 | 6.73 × 10^{−5} | |

PAHS | 0.139 | 0 | 1.414 | 3.16 × 10^{−3} | |

ZDT2 | SHS | 0.143 | 1.96 × 10^{−9} | 1.236 | 4.87 × 10^{−3} |

PSF-HS first | 0.151 | 1.93 × 10^{−9} | 1.401 | 4.15 × 10^{−3} | |

PSF-HS second | 0.159 | 1.92 × 10^{−9} | 0.915 | 7.38 × 10^{−3} | |

APF-HS | 0.161 | 1.85 × 10^{−9} | 0.889 | 6.19 × 10^{−3} | |

SGHSA | 0.168 | 1.88 × 10^{−9} | 1.414 | 9.91 × 10^{−3} | |

NSHS | 0.062 | 1.82 × 10^{−9} | 0.781 | 1.23 × 10^{−6} | |

PAHS | 0.136 | 1.81 × 10^{−9} | 1.414 | 4.06 × 10^{−3} | |

ZDT3 | SHS | 0.144 | 1.53 × 10^{−9} | 1.106 | 6.13 × 10^{−3} |

PSF-HS first | 0.120 | 3.72 × 10^{−17} | 1.108 | 6.01 × 10^{−3} | |

PSF-HS second | 0.131 | 1.41 × 10^{−17} | 0.856 | 3.55 × 10^{−2} | |

APF-HS | 0.135 | 7.85 × 10^{−18} | 1.236 | 9.36 × 10^{−3} | |

SGHSA | 0.149 | 3.97 × 10^{−18} | 1.399 | 4.69 × 10^{−2} | |

NSHS | 0.104 | 3.38 × 10^{−18} | 1.108 | 8.28 × 10^{−3} | |

PAHS | 0.127 | 2.49 × 10^{−18} | 1.088 | 5.75 × 10^{−3} | |

ZDT4 | SHS | 0.151 | 2.75 × 10^{−9} | 1.414 | 3.73 × 10^{−3} |

PSF-HS first | 0.151 | 2.74 × 10^{−9} | 1.314 | 3.12 × 10^{−3} | |

PSF-HS second | 0.159 | 2.76 × 10^{−9} | 1.160 | 1.21 × 10^{−2} | |

APF-HS | 0.161 | 2.73 × 10^{−9} | 1.265 | 5.44 × 10^{−3} | |

SGHSA | 0.168 | 2.72 × 10^{−9} | 1.414 | 5.57 × 10^{−3} | |

NSHS | 0 | 2.71 × 10^{−9} | 1.003 | 1.18 × 10^{−6} | |

PAHS | 0.147 | 2.75 × 10^{−9} | 1.414 | 3.39 × 10^{−3} | |

ZDT6 | SHS | 0.126 | 3.18 × 10^{−9} | 1.216 | 4.12 × 10^{−3} |

PSF-HS first | 0.127 | 3.17 × 10^{−9} | 1.361 | 2.93 × 10^{−3} | |

PSF-HS second | 0.134 | 3.15 × 10^{−9} | 0.848 | 2.71 × 10^{−2} | |

APF-HS | 0.141 | 3.16 × 10^{−9} | 1.361 | 6.64 × 10^{−3} | |

SGHSA | 0.148 | 3.13 × 10^{−9} | 1.361 | 6.68 × 10^{−3} | |

NSHS | 0.121 | 3.12 × 10^{−9} | 1.361 | 7.55 × 10^{−3} | |

PAHS | 0.141 | 3.11 × 10^{−9} | 1.361 | 3.06 × 10^{−3} |

Problem | NP | NN | PD | PCD | SS | NFEs | KS | |
---|---|---|---|---|---|---|---|---|

SOOD | MOOD | |||||||

Hanoi network (HAN) | 34 | 32 | 304.8, 406.4, 508.0, 609.6, 762.0, 1016 | 45.72, 70.40, 98.37, 129.33, 180.74, 278.28 | 2.87 × 10^{26} | 50,000 | 100,000 | USD 6.081 million |

Saemangeum network (SAN) | 356 | 334 | 80, 100, 150, 200, 250, 300, 350, 400, 450, 500, 600, 700, 800 | 86,500, 100,182, 124,737, 153,347, 186,909, 219,089, 250,307, 288,313, 305,397, 344,394, 400,586, 506,082, 678,144 | 7.53 × 10^{446} | 100,000 | 500,000 | KRW 11.200 billion |

P-city network (PCN) | 1339 | 1297 | 25, 50, 80, 100, 150, 200, 250, 300, 500 | 43.80, 56.85, 72.51,82.95, 109.05,135.15, 161.25,187.35, 291.7 | 5.37 × 10^{1277} | 100,000 | 500,000 | KRW 34.946 billion |

Network | Method | KS | Best Cost | Average Cost | Worst Cost | Average NFEs-Fs | Reduction Rate between KS and Average Cost (%) |
---|---|---|---|---|---|---|---|

HAN | Simple HS | 6.081 million (Unit: USD) | 6.081 | 6.319 | 6.632 | 43.149 | - |

PSF-HS first | 6.081 | 6.252 | 6.508 | 40.200 | - | ||

PSF-HS second | 6.081 | 6.213 | 6.623 | 38.721 | - | ||

APF-HS | 6.081 | 6.223 | 6.782 | 39.842 | - | ||

SGHSA | 6.081 | 6.150 | 6.423 | 27.980 | - | ||

NSHS | 6.081 | 6.145 | 6.531 | 28.400 | - | ||

PAHS | 6.081 | 6.152 | 6.592 | 32.450 | - | ||

SAN | Simple HS | 11.200 billion (Unit: KRW) | 10.016 | 10.072 | 11.006 | - | 10.07 |

PSF-HS first | 10.017 | 10.068 | 10.982 | - | 10.11 | ||

PSF-HS second | 9.982 | 10.066 | 10.832 | - | 10.12 | ||

APF-HS | 9.991 | 10.006 | 10.320 | - | 10.65 | ||

SGHSA | 9.943 | 9.970 | 10.218 | - | 10.97 | ||

NSHS | 9.956 | 9.972 | 10.466 | - | 10.96 | ||

PAHS | 9.970 | 9.986 | 10.312 | - | 10.84 | ||

PCN | Simple HS | 34.946 billion (Unit: KRW) | 25.494 | 28.981 | 33.494 | - | 17.07 |

PSF-HS first | 25.545 | 28.321 | 32.545 | - | 18.96 | ||

PSF-HS second | 25.601 | 27.651 | 32.601 | - | 20.88 | ||

APF-HS | 25.512 | 27.654 | 31.512 | - | 20.87 | ||

SGHSA | 25.194 | 27.416 | 30.187 | - | 21.55 | ||

NSHS | 25.240 | 27.553 | 30.100 | - | 21.16 | ||

PAHS | 25.287 | 27.953 | 31.024 | - | 20.01 |

Algorithm | Additional Operators | Additional Parameters | Improvement |
---|---|---|---|

PSF-HS first | - | - | HMCR, PAR |

PSF-HS second | - | - | HMCR, PAR |

APF-HS | - | Max/Min Bw | HMCR, PAR, Bw |

NSHS | PA | Max/Min Bw, f_{std}, Adj^{min} | HMCR, Bw |

SGHSA | PA | Max/min Bw | Bw |

PAHS | - | Max/Min HMCR, PAR, Bw | HMCR, PAR, Bw |

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## Share and Cite

**MDPI and ACS Style**

Choi, Y.H.; Kim, J.H.
Self-Adaptive Models for Water Distribution System Design Using Single-/Multi-Objective Optimization Approaches. *Water* **2019**, *11*, 1293.
https://doi.org/10.3390/w11061293

**AMA Style**

Choi YH, Kim JH.
Self-Adaptive Models for Water Distribution System Design Using Single-/Multi-Objective Optimization Approaches. *Water*. 2019; 11(6):1293.
https://doi.org/10.3390/w11061293

**Chicago/Turabian Style**

Choi, Young Hwan, and Joong Hoon Kim.
2019. "Self-Adaptive Models for Water Distribution System Design Using Single-/Multi-Objective Optimization Approaches" *Water* 11, no. 6: 1293.
https://doi.org/10.3390/w11061293