# Efficiency Criteria as a Solution to the Uncertainty in the Choice of Population Size in Population-Based Algorithms Applied to Water Network Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Problem Formulation and Selected Algorithms

_{D,j}(X

^{i}) and the flow of each pump Q

_{D,j}(X

^{i}) (if it exists on the network) are represented. F

_{e,j}is the energy cost in each of these points. The capital cost of the pipes is represented in the second term of the equation, where C

_{j}is the unit cost of the decision variable contained in link j of solution i, and L

_{j}is the length of pipe j. The last three terms in the equation represent the constraints of the problem, related with minimum pressure height in each node (H

_{min,k}), maximum velocity (V

_{max,k}) and minimum velocity (V

_{min,k}).

_{i}are large enough (10

^{7}) to reject all solutions that violate the constraints. More details about the methodology, the objective function and the penalty terms can be found in [20].

_{c}) and Mutation frequency (P

_{m}). This work uses a modified version of classical GA, PGA, whose complete description can be found in [20]

_{lim}) and the learning factors C

_{1}and C

_{2}. The original PSO algorithm is often trapped into local optima, causing early convergences. Therefore, the authors include a new parameter, creating a modified version of the algorithm. This parameter, named Confusion Probability (P

_{c}), determines the number of particles that do not follow the social behavior in each iteration [32].

_{s}) and a search acceleration factor (C).

#### 2.2. Specific Operators and Calibration

#### 2.3. Efficiency Criteria

_{quality}is associated with the quality of the simulations, representing the number of successful solutions N

_{successful}divided by the total number of simulations performed N

_{sim}. Note that N

_{successful}is the number of “known global lowest cost” or the number of “good” solutions, depending on the optimization objective, i.e., this rate can be defined according to the requirement of the solutions. For example, sometimes it may be preferable to find a set of solutions close to the lowest cost but that can be obtained with a smaller computational effort. In this study, a “good solution” can be defined as a solution with a fitness function that does not exceed a certain threshold above the known global lowest cost.

_{convergence}is related to the computational effort required by the algorithm to reach the final solution. η

_{convergence}refers to the number of objective function evaluations performed by the algorithm before finding the final solution to the problem. An evaluation of the OF represents a call to the hydraulic solver, so that only the chains that changed from the previous generation are re-evaluated. Therefore, the number of calls to a hydraulic package does not necessarily have to match the number of generations in the algorithm multiplied by the population size P. Optimization software was programmed to compute this data, which is obtained directly in each simulation.

#### 2.4. Programming Environment

## 3. Test Problems and Results

## 4. Analysis of Results

_{quality}, the speed of convergence η

_{convergence}of the algorithm and the relationship between the previous two, which is defined as the efficiency E of the algorithm.

_{quality}in Equation (3). Considering the overall performance of each algorithm in obtaining low-cost solutions, the results clearly show that the SFL algorithm is the most powerful of those tested because it produces more successful solutions in all networks. Conversely, the HS algorithm has great difficulty finding low-cost solutions, except in the small problems and in the New York network, which are problems with less complexity.

_{convergence}). Thus, the previous calibration of specific parameters and the differences in the search processes of each algorithm play key roles in the speed of convergence [32]. Figure 2 shows the average number of OF evaluations required to reach the final solution at each tested population size and for each algorithm and network.

_{quality}and η

_{convergence}, according to Equation (3). As noted above, E describes the performance of each algorithm while considering both the ability to obtain the desired solution and the computational effort required to reach this solution. This paper analyses E for all algorithms, considering both the possibility of obtaining the global minimum and obtaining a “good solution” to the problem (3% above the minimum). Thereby, the efficiency E of obtaining low-cost solutions for all tested networks is given in Figure 3.

## 5. Conclusions

- In terms of efficiency E and considering the complexity of the pipe-sizing problem, the PGA and SFLA are better for complex networks, such as Hanoi or Joao Pessoa, when the goal is to obtain the global minimum. These algorithms are more likely to identify the lowest-cost solution. Meanwhile, the HS and PSO algorithms hardly found global minimums, which severely decrease their efficiency.
- The HS algorithm exhibited improved performance in less complex water networks such as small problems, in which all algorithms are able to find minimal solutions easily. In addition, if we extend the range of solutions that is considered successful to include good solutions (3% above the global minimum), the HS algorithm is the best in the New York network and the second most efficient for the rest of networks (except in the Joao Pessoa network). In this case, the algorithms that require fewer OF evaluations to reach the final solution benefit, and HS is the fastest algorithm.
- The PGA is the most robust technique for finding low-cost solutions because it has a high value of E in all networks, which is similar to the values obtained by the best algorithm in each case.
- SFL algorithm efficiency E is close to the PGA in complex networks (Hanoi and Joao Pessoa) when the goal is to obtain the low-cost solution. However, if the optimization goal is based on obtaining “good solutions”, other techniques are better because the algorithm SFL is penalized due to the high number of evaluations of the objective function performed in the optimization process.
- Regarding the initial random population size, normally the efficiency improves as P increases to a certain limit, beyond which the performance no longer improves or worsens. This size “limit” depends on the complexity of the problem and the problem goal, but larger populations are generally less efficient than small populations when finding a global known.
- In addition, expanding the optimization goal to include a set of “good solutions” close to the global minimum (3% above) confirms the hypothesis, and in the range considered, the smallest population is the most efficient in almost all algorithms and networks.
- Numerically, all algorithms exhibit their highest values of E when finding good solutions near the minimum for 25 < P < 50. To obtain low-cost solutions, the tested algorithms need larger populations (except in the simplest problems), with the highest values of E observed for approximately 75 < P < 125. These values vary slightly for each algorithm and network, but they are a good starting point, regardless of the complexity of the network in the studied cases.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

PbA | Population-based algorithm |

WDN | water distribution network |

OF | Objective Function |

PGA | Pseudo-Genetic Algorithm |

PSO | Particle Swarm Optimization |

SFLA | Shuffled Frog Leaping Algorithm |

HS | Harmony Search |

HAWaNet | Heuristic Algorithms Water Networks |

NP | Nondeterministic, polynomial time |

BMP | Best Management Practice |

GA | Genetic Algorithm |

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**Figure 3.**Efficiency of Population-based algorithms in obtaining low-cost solutions in the four selected networks.

**Figure 4.**Efficiency of Population-based algorithms in obtaining “good solutions” for the four selected networks.

Algorithm/Network | Two Loops | Bak-Ryan | Hanoi | Joao Pessoa | New York | GoYang |
---|---|---|---|---|---|---|

PGA | ||||||

P_{c} | 10%–90% | 10%–90% | 10%–90% | 10%–90% | >60% | 10%–90% |

* P_{m} | 4% | 3% | 3% | 2% | 4% | 3% |

PSO | ||||||

* V_{lim} | 10% | 10% | 20% | 20% | 30% | 10% |

* P_{conf} | 10%–20% | 10%–20% | 10%–20% | 10%–20% | 10% | 10%–20% |

C_{1} | 2 | 2 | 2 | 2 | 2 | 2 |

C_{2} | 2 | 2 | 2 | 2 | 2 | 2 |

HS | ||||||

* HMCR | 90% | 90% | 95% | 95% | 90% | 90% |

* PAR | 15% | 15% | 10% | 10% | 15% | 15% |

SFLA | ||||||

* C | 2 | 2 | 2 | 2 | 2 | 1.5 |

Q | 50% | 50% | 50% | 50% | 50% | 50% |

N_{s} | 30 | 30 | 30 | 30 | 30 | 30 |

Network | Number of Pipes | Number of Possible Diameters | Search Space | Minimum Pressure Requirements (mca) | Best Known Solution ($) | Number of Different Solutions |
---|---|---|---|---|---|---|

Two Loops | 8 | 14 | 1.48 × 10^{9} | 30 | 419,000 | 60 |

BakRyan | 9 | 11 | 2.36 × 10^{9} | 15 | 903,620 | 10 |

New York | 21 | 16 | 1.93 × 10^{25} | 30 | 38.642 × 10^{6} | 2163 |

GoYang | 30 | 8 | 1.24 × 10^{27} | 15 | 177.010 × 10^{6} ^{a} | 303 |

Hanoi | 34 | 14 | 2.87 × 10^{26} | 30 | 6.081 × 10^{6} | 5553 |

Joao Pessoa | 72 | 10 | 1 × 10^{72} | 15 | 192.366 × 10^{6} | 16,101 |

^{a}Cost in won (1,000 won ≈ 1 US$).

Algorithm/Network | Two Loops | BakRyan | Hanoi | Joao Pessoa | New York | GoYang |
---|---|---|---|---|---|---|

Pseudo-Genetic algorithm (PGA) | ||||||

Low cost solution | 25 | 25 | P = 50 | P = 100 | P = 100 | P = 25 |

Good solution (3% above) | 25 | 25 | P = 25 | P = 25–50 | P = 50 | P = 25 |

Particle Swarm Optimization algorithm (PSO) | ||||||

Low cost solution | 125 | 25 | P = 125 | P = 75 | P = 50 | P = 75 |

Good solution (3% above) | 25 | 25 | P = 50 | P = 75 | P = 25 | P = 25 |

Harmony Search algorithm (HS) | ||||||

Low cost solution | 75 | 25 | P = 175 | P = 100 | P = 125 | P = 25 |

Good solution (3% above) | 25 | 25 | P = 25–50 | P = 50 | P = 75 | P = 25 |

Shuffled Frog Leaping algorithm (SFLA) | ||||||

Low cost solution | 25 | 25 | P = 100 | P = 150 | P = 75 | P = 25 |

Good solution (3% above) | 25 | 25 | P = 25 | P = 25 | P = 75 | P = 25 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Mora-Melià, D.; Gutiérrez-Bahamondes, J.H.; Iglesias-Rey, P.L.; Martínez-Solano, F.J. Efficiency Criteria as a Solution to the Uncertainty in the Choice of Population Size in Population-Based Algorithms Applied to Water Network Optimization. *Water* **2016**, *8*, 583.
https://doi.org/10.3390/w8120583

**AMA Style**

Mora-Melià D, Gutiérrez-Bahamondes JH, Iglesias-Rey PL, Martínez-Solano FJ. Efficiency Criteria as a Solution to the Uncertainty in the Choice of Population Size in Population-Based Algorithms Applied to Water Network Optimization. *Water*. 2016; 8(12):583.
https://doi.org/10.3390/w8120583

**Chicago/Turabian Style**

Mora-Melià, Daniel, Jimmy H. Gutiérrez-Bahamondes, Pedro L. Iglesias-Rey, and F. Javier Martínez-Solano. 2016. "Efficiency Criteria as a Solution to the Uncertainty in the Choice of Population Size in Population-Based Algorithms Applied to Water Network Optimization" *Water* 8, no. 12: 583.
https://doi.org/10.3390/w8120583