# A Comparison of Preference Handling Techniques in Multi-Objective Optimisation for Water Distribution Systems

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## Abstract

**:**

## 1. Introduction

## 2. Review

## 3. Proposal and Experiment Description

- It uses spherical pruning [32] in order to promote diversity in the approximated Pareto front. Basically, the objective space is partitioned using spherical coordinates, and one solution is selected in each spherical sector, avoiding overcrowding areas.
- It uses physical programming (PP) [6] for pertinency improvement and as a mechanism for many-objectives optimisation. It states such preferences in aspiration levels in a matrix M as depicted in Table 2. This PP index is used as an additional mechanism to prune solutions, according to the preference index, in order to get a manageable size of the Pareto front approximation.

- It uses as many subplots as design objectives to depict trade-off information.
- Solutions are synchronised by the vertical axis, while the horizontal axis keeps their original units. That is, no normalisation deforming the units scale is used.
- Trade-off relationships might be propagated to design variables by synchronising the same vertical axis.

## 4. Test Cases

#### 4.1. Case Study 1: Dissolved Oxygen Control in a Waste-Water Treatment Process

- ${J}_{1}\left(\mathit{x}\right)$:
- Settling time (day) for a setpoint reference change (minimise).
- ${J}_{2}\left(\mathit{x}\right)$:
- Settling time (day) for an input disturbance in the sludge process (minimise).
- ${J}_{3}\left(\mathit{x}\right)$:
- Maximum deviation from setpoint ($gCOD$/m${}^{3}$) due to an input disturbance in the sludge process (minimise).
- ${J}_{4}\left(\mathit{x}\right)$:
- Total variation of oxygen mass transfer coefficient (day${}^{-1}$) due to the setpoint reference change and the input disturbance (minimise).
- ${J}_{5}\left(\mathit{x}\right)$:
- Aeration energy cost (kWh/day) due to the setpoint reference change and the input disturbance (minimise). Given a value of the control action for a given instant ${u}_{i}$, the instant aeration energy cost $A{E}_{i}$ is calculated as:$$A{E}_{i}=0.4032{u}_{i}^{2}+7.8408{u}_{i}$$

#### 4.2. Case Study 2: Pollution Management in Water Distribution Systems

- ${J}_{1}\left(\mathit{x}\right)$:
- DO level at Bowville (mg/L) (maximise).
- ${J}_{2}\left(\mathit{x}\right)$:
- DO level at Robin State Park (mg/L) (maximise).
- ${J}_{3}\left(\mathit{x}\right)$:
- DO level at Plymton (mg/L) (maximise).
- ${J}_{4}\left(\mathit{x}\right)$:
- Return on equity (%) (maximise).
- ${J}_{5}\left(\mathit{x}\right)$:
- Tax increment (Bowville) (minimise).
- ${J}_{6}\left(\mathit{x}\right)$:
- Tax increment (Plymton) (minimise).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

EMO | Evolutionary Multi-Objective Optimisation |

LD | Level Diagrams |

MCDM | Multi-criteria Decision Making |

MOP | Multi-Objective Problem |

MOO | Multi-Objective Optimisation |

PP | Physical Programming |

WDS | Water Distribution System |

## Appendix A. Background

- Pareto optimality [3]: An objective vector $\mathit{J}\left({\mathit{x}}^{1}\right)$ is Pareto optimal if there is not another objective vector $\mathit{J}\left({\mathit{x}}^{2}\right)$ such that ${J}_{i}\left({\mathit{x}}^{2}\right)\le {J}_{i}\left({\mathit{x}}^{1}\right)$ for all $i\in [1,2,\cdots ,m]$ and ${J}_{j}\left({\mathit{x}}^{2}\right)<{J}_{j}\left({\mathit{x}}^{1}\right)$ for at least one j, $j\in [1,2,\cdots ,m]$.
- Dominance: An objective vector $\mathit{J}\left({\mathit{x}}^{1}\right)$ is dominated by another objective vector $\mathit{J}\left({\mathit{x}}^{2}\right)$ if ${J}_{i}\left({\mathit{x}}^{2}\right)\le {J}_{i}\left({\mathit{x}}^{1}\right)$ for all $i\in [1,2,\cdots ,m]$ and ${J}_{j}\left({\mathit{x}}^{2}\right)<{J}_{j}\left({\mathit{x}}^{1}\right)$ for at least one j, $j\in [1,2,\cdots ,m]$. This is denoted as $\mathit{J}\left({\mathit{x}}_{\mathbf{2}}\right)\u2aaf\mathit{J}\left({\mathit{x}}_{\mathbf{1}}\right)$.

**Figure A1.**Pareto optimality and dominance concepts for a min-min problem. Dark solutions is the subset of non-dominated solutions which approximates a Pareto front (

**right**) and a Pareto set (

**left**). Remainder solutios are dominated solutions, because it is possible to find at least one solution with better values in all design objectives (Source: [39]).

- Constrained optimisation. Results from the optimisation problem are not always feasible in a practical sense; therefore constraints must be incorporated in order to assure their feasibility.
- Many-objectives optimisation. If a MOP has more than 3 design objectives, it is considered a many-objectives optimisation problem. It is important to consider such a sub-classification, given that converge and diversity mechanisms might be in conflict.
- Computational expensive optimisation. Extensive or exhaustive simulations might be required in order to compute one or more design objectives requires.
- Multi-modal optimisation. It might happen that two or more decision vector points to the same objective vector.

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**Figure 3.**Pareto front approximation for MOP II with preference pruning: PP (red +); PROMETHEE II (blue *) and TOPSIS (black x).

**Figure 4.**Pareto set approximation for MOP II with preference pruning: PP (red +), PROMETHEE II (blue *) and TOPSIS (black x).

**Table 1.**Summary of MOOD procedures for WDS design concept. $J\left(\theta \right)$ refers to the number of objectives; $\theta $ to the number of sets of decision variables and $g\left(\theta \right)$, $h\left(\theta \right)$ to the number of sets of inequality and equality constraints respectively.

Reference | MOP | MOO | MCDM | ||||
---|---|---|---|---|---|---|---|

$\mathit{J}\left(\mathit{\theta}\right)$ | $\mathit{\theta}$ | $\left(\mathit{g}\right(\mathit{\theta}),\mathit{h}(\mathit{\theta}\left)\right)$ | Algorithm | Features | Plot | Insights | |

Savic et al. [10] | 2 | 1 | (2, 1) | Hybrid GA | Local search | - | - |

Sotelo and Baran [11] | 4 | 1 | (4, 0) | SPEA | - | Scatter plot | - |

Kelner and Leonard [12] | 2 | 3 | (2, 2) | GAPS | Penalised tournament selection scheme | Scatter plot | - |

Baran et al. [13] | 4 | 1 | (4, 0) | SPEA NSGA NSGA-II CNSGA NPGA MOGA | Comparison between algorithms | - | - |

Lopez-Ibanez et al. [14] | 2 | 1 | (3, 1) | SPEA2 | Comparison between initial population generation methods | Scatter Plot | Attainment surfaces |

Odan et al. [15] | 2 | 1 | (3, 1) | AMALGAM | Real-time | Scatter Plot | - |

Stokes et al. [16] | 2 | 1 | (2, 0) | NSGA-II | - | Scatter Plot | Minimum objective values |

Prasad et al. [17] | 2 | 2 | (3, 0) | NSGA-II | - | Scatter plot | - |

Kurek and Brdys [18] | 3 | 2 | (5, 0) | NSGA-II | Problem specific modification | Scatter Plot | - |

Ewald et al. [19] | 3 | 2 | (4, 0) | Distributed MOGA | Distributed application using grid computing | Scatter Plot | - |

Alfonso et al. [20] | 3 | 1 | (2, 1) | NSGA-II | - | Scatter Plot | - |

Giustolisi et al. [21] | 2 | 1 | (4, 1) | OPTIMOGA | - | Table | - |

Kougias and Theodossiou [22] | 3 | 1 | (4, 0) | MO-HSA | - | Scatter Plot | - |

Kurek and Ostfeld [23] | 3 | 3 | (3, 1) | SPEA2 | - | Scatter Plot | Utopian solution mechanism |

Kurek and Ostfeld [24] | 2 | 2 | (3, 1) | SPEA2 | - | Scatter Plot | Most “balanced” solution |

Mala-Jetmarova et al. [25] | 2 | 1 | (4, 0) | NSGA-II | - | Scatter Plot | - |

Mala-Jetmarova et al. [26] | 3 | 1 | (4, 0) | NSGA-II | - | Scatter Plot | - |

**Table 2.**Preference matrix M. Five preference ranges have been defined: highly desirable (HD), desirable (D), tolerable (T) undesirable (U) and highly undesirable (HU).

Preference Matrix | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

← | HD | →← | D | →← | T | →← | U | →← | HU | → | |

Objective | ${\mathit{J}}_{\mathit{i}}^{\mathbf{0}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{1}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{2}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{3}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{4}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{5}}$ | |||||

${J}_{1}\left(\mathit{x}\right)$ [-] | . | . | . | . | . | . | |||||

… | |||||||||||

${J}_{n}\left(\mathit{x}\right)$ [-] | . | . | . | . | . | . |

**Table 3.**Matrix with (in)significant differences. Significant (S) and Insignificant (I) differences for each design objectives are defined.

I/S Differences Matrix | ||
---|---|---|

Objective | I | S |

${J}_{1}\left(\mathit{x}\right)$ [-] | . | . |

… | ||

${J}_{n}\left(\mathit{x}\right)$ [-] | . | . |

Method | Input |
---|---|

TOPSIS | Pareto front approximation. |

PROMETHEE II | Pareto front approximation. |

Significant/Insignificant differences. | |

Physical Programming | Solution from the Pareto front approximation. |

Information about the desirability limits. |

**Table 5.**Preference matrix $\mathfrak{m}$ for MOP statement I. Five preference ranges have been defined: highly desirable (HD), desirable (D), tolerable (T) undesirable (U) and highly undesirable (HU).

Preference Matrix | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Objective | ← | HD | →← | D | →← | T | →← | U | →← | HU | → |

${\mathit{J}}_{\mathit{i}}^{\mathbf{0}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{1}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{2}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{3}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{4}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{5}}$ | ||||||

${J}_{1}\left(\mathit{x}\right)$ (day) | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | |||||

${J}_{2}\left(\mathit{x}\right)$ (day) | 0.005 | 0.010 | 0.015 | 0.020 | 0.025 | 0.030 | |||||

${J}_{3}\left(\mathit{x}\right)$ ($gCOD$/m${}^{3}$ day) | 0.000 | 0.005 | 0.010 | 0.0150 | 0.020 | 0.025 | |||||

${J}_{4}\left(\mathit{x}\right)$ (day${}^{-1}$) | 20.00 | 30.00 | 40.00 | 50.00 | 60.00 | 70.00 | |||||

${J}_{5}\left(\mathit{x}\right)$ (kWh/day) | 1.00 | 2.00 | 3.00 | 4.00 | 5.00 | 6.00 |

**Table 6.**Matrix with (in) significant differences for MOP statement I. Significant (S) and Insignificant (I) differences for each design objectives are defined.

I/S Differences Matrix | ||
---|---|---|

Objective | I | S |

${J}_{1}\left(\mathit{x}\right)$ (day) | 0.01 | 0.03 |

${J}_{2}\left(\mathit{x}\right)$ (day) | 0.05 | 0.05 |

${J}_{3}\left(\mathit{x}\right)$ ($gCOD$/m${}^{3}$ day) | 0.025 | 0.05 |

${J}_{4}\left(\mathit{x}\right)$ (day${}^{-1}$) | 5.0 | 10.00 |

${J}_{5}\left(\mathit{x}\right)$ (kWh/day) | 0.5 | 1.00 |

**Table 7.**Matrix with (in)significant differences for MOP II statement. Significant (S) and Insignificant (I) differences for each design objectives are defined.

I/S Differences Matrix | ||
---|---|---|

Objective | I | S |

${J}_{1}\left(\mathit{x}\right)$ (mg/L) | 2.0 | 5.0 |

${J}_{2}\left(\mathit{x}\right)$ (mg/L) | 2.0 | 5.0 |

${J}_{3}\left(\mathit{x}\right)$ (mg/L) | 2.0 | 5.0 |

${J}_{4}\left(\mathit{x}\right)$ ($) | 1.0 | 3.0 |

${J}_{5}\left(\mathit{x}\right)$ (%) | 1.0 | 3.0 |

${J}_{6}\left(\mathit{x}\right)$ (%) | 1.0 | 3.0 |

${G}_{1}\left(\mathit{x}\right)$ (mg/L) | 0.5 | 1.5 |

**Table 8.**Preference matrix $\mathfrak{m}$ for MOP II statement. Five preference ranges have been defined: highly desirable (HD), desirable (D), tolerable (T) undesirable (U) and highly undesirable (HU).

Preference Matrix | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Objective | ← | HD | →← | D | →← | T | →← | U | →← | HU | → |

${\mathit{J}}_{\mathit{i}}^{\mathbf{0}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{1}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{2}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{3}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{4}}$ | ${\mathit{J}}_{\mathit{i}}^{\mathbf{5}}$ | ||||||

$-{J}_{1}\left(\mathit{x}\right)$ (mg/L) | −9 | −8 | −6 | −4 | −2 | 0.0 | |||||

$-{J}_{2}\left(\mathit{x}\right)$ (mg/L) | −9 | −8 | −6 | −4 | −2 | 0.0 | |||||

$-{J}_{3}\left(\mathit{x}\right)$ (mg/L) | −9 | −8 | −6 | −4 | −2 | 0.0 | |||||

$-{J}_{4}\left(\mathit{x}\right)$ ($) | −8 | −7 | −6 | −5 | −4 | −3 | |||||

${J}_{5}\left(\mathit{x}\right)$ (%) | 0 | 1 | 2 | 3 | 4 | 5 | |||||

${J}_{6}\left(\mathit{x}\right)$ (%) | 0 | 1 | 2 | 3 | 4 | 5 | |||||

$-{G}_{1}\left(\mathit{x}\right)$ (mg/L) | −5 | −4.5 | −4 | −3.5 | −3.0 | −2.5 |

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**MDPI and ACS Style**

Reynoso-Meza, G.; Alves Ribeiro, V.H.; Carreño-Alvarado, E.P.
A Comparison of Preference Handling Techniques in Multi-Objective Optimisation for Water Distribution Systems. *Water* **2017**, *9*, 996.
https://doi.org/10.3390/w9120996

**AMA Style**

Reynoso-Meza G, Alves Ribeiro VH, Carreño-Alvarado EP.
A Comparison of Preference Handling Techniques in Multi-Objective Optimisation for Water Distribution Systems. *Water*. 2017; 9(12):996.
https://doi.org/10.3390/w9120996

**Chicago/Turabian Style**

Reynoso-Meza, Gilberto, Victor Henrique Alves Ribeiro, and Elizabeth Pauline Carreño-Alvarado.
2017. "A Comparison of Preference Handling Techniques in Multi-Objective Optimisation for Water Distribution Systems" *Water* 9, no. 12: 996.
https://doi.org/10.3390/w9120996