Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques
Abstract
:1. Introduction
2. Materials and Methods
2.1. Adopted Multi-Objective Optimization Approach
- In a minimization problem, a vector u = (u1, . . . , uM)T is said to dominate another vector v = (v1, . . . , vM)T if ui ≤ vi for i = 1, . . . , M and u ≠ v. This property may be denoted as u ≺v.
- A feasible solution x∈ X is called a Pareto-optimal solution, if there is no alternative solution y∈ X such that F(y) ≺ F(x).
- The Pareto-optimal set, PS, is the union of all Pareto-optimal solutions, and may be defined as PS = {x ∈ X :y ∈ X, F(y) ≺ F(x)}.
- The Pareto-optimal front, PF, is the set comprising the Pareto-optimal solutions in the objective space. It may be expressed as PF = {F(x)|x ∈ PS}.
2.2. Details of Epsilon-Dominance-Driven Self-Adaptive Evolutionary Algorithm (ε-DSEA) Optimization Algorithm
- Diversity expansion to increase decision variables’ search space exploitation
- Self-adaptive operators’ parameters for parameters in process tuning
- Exploration extension for algorithm revival and stagnation coping
- Virtual dominance archive to improve diversity and convergence.
2.2.1. Diversity Expansion
2.2.2. Self-Adaptive Mechanism and Formulae
2.2.3. Exploration Extension Mechanism
2.2.4. Virtual Dominance Archive
2.2.5. Constraint Handling Strategy
2.3. Comparative Paradigms
- If there is no change in the archive size for a certain number of evaluations;
- If there is no improvement indicated by the -progress indicator; and
- If the current population to archive ratio exceeds 1.25×γ
2.4. Identification of a Real-World Experimental Test Problem
2.4.1. Objectives Functions Formulae
2.4.2. Reservoir System Constraints
2.5. Computational Properties
3. Results
3.1. Performance Achievement
3.1.1. Algorithms’ Reliability
3.1.2. Algorithms’ Robustness and Efficiency
3.1.3. Algorithms’ Effectiveness
3.2. Strategic Achievement
4. Discussion
4.1. Algorithms’ Optimization Techniques
4.2. Water Resources Management Case Study
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Operator | Parameters | Domain | Adaptation Functions | Comments |
---|---|---|---|---|
SBX 1 | [0, 100] | Distribution index | ||
DE 2 | CR F | [0.1,1.0] [0.5, 1.0] | Crossover probability Step size | |
SPX 3 | [2.5, 3.5] | Expansion rate | ||
PCX 4 | [0.1, 0.3] | These parameters (standard deviations) control the spatial distribution of the offspring for PCX and UNDX | ||
UNDX 4 | [0.4, 0.6] [0.1, 0.35] |
Parameters | Borg | ε-DSEAa | Parameters | Borg | ε-DSEA |
---|---|---|---|---|---|
Initial population size | 100 | 100 | SPX parents | 10 | 3 |
Tournament selection size | 2 | 2 | SPX offspring | 2 | 2 |
SBX crossover rate | 1.0 | 1.0 | SPX expansion rate λ | 3 | [2.5, 3.5] |
SBX distribution index η | 15.0 | [0, 100] | UNDX parents | 10 | 10 |
DE crossover rate CR | 0.1 | [0.1, 1.0] | UNDX offspring | 2 | 2 |
DE step size F | 0.5 | [0.5, 1.0] | UNDX σζ | 0.5 | [0.4, 0.6] |
PCX parents | 10 | 10 | UNDX ση | 0.35/ | [0.1, 0.35]/ |
PCX offspring | 2 | 2 | UM mutation rate | 1/L | 1/L |
PCX ση | 0.1 | [0.1, 0.3] | PM mutation rate | 1/L | 1/L |
PCX σζ | 0.1 | [0.1, 0.3] | PM distribution index ηm | 20 | 20 |
Borg MOEA | ε-DSEA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Area 1 | Head 2 | Power 3 | Storage 4 | Releases 5 | Area | Head | Power | Storage | Releases | |
3 Objective problem | ||||||||||
Min. | 19.18 | 437.79 | 24.50 | 433.55 | 129.94 | 17.32 | 434.73 | 24.83 | 373.82 | 129.75 |
Max. | 122.79 | 485.97 | 249.00 | 2565.84 | 866.25 | 121.80 | 485.86 | 246.37 | 2551.05 | 877.20 |
Mean | 74.37 | 474.71 | 94.76 | 1732.33 | 336.06 | 76.39 | 475.53 | 94.77 | 1775.22 | 336.17 |
Median | 72.03 | 477.12 | 83.00 | 1743.17 | 297.19 | 78.92 | 478.95 | 84.37 | 1867.41 | 316.10 |
St.7 | 27.19 | 10.11 | 50.88 | 523.86 | 174.84 | 24.99 | 10.42 | 51.88 | 496.30 | 183.22 |
Gross6 | 37.52 | 686.00 | 133.08 | 37.53 | 702.99 | 133.12 | ||||
5 Objective problem | ||||||||||
Min. | 16.94 | 434.09 | 23.44 | 361.60 | 130.72 | 19.10 | 437.67 | 24.09 | 431.16 | 130.45 |
Max. | 122.97 | 485.98 | 249.00 | 2568.50 | 866.08 | 123.14 | 486.00 | 249.00 | 2570.98 | 797.97 |
Mean | 66.09 | 470.67 | 90.46 | 1555.39 | 337.72 | 71.90 | 472.95 | 91.77 | 1672.36 | 334.80 |
Median | 61.55 | 473.63 | 82.14 | 1540.19 | 316.81 | 71.78 | 477.05 | 82.00 | 1738.53 | 298.79 |
St. | 29.89 | 12.83 | 45.64 | 597.36 | 162.74 | 28.94 | 12.58 | 47.18 | 583.53 | 169.11 |
Gross | 35.82 | 615.94 | 133.74 | 36.34 | 662.25 | 132.58 |
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Y. Al-Jawad, J.; M. Kalin, R. Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques. Water 2019, 11, 2021. https://doi.org/10.3390/w11102021
Y. Al-Jawad J, M. Kalin R. Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques. Water. 2019; 11(10):2021. https://doi.org/10.3390/w11102021
Chicago/Turabian StyleY. Al-Jawad, Jafar, and Robert M. Kalin. 2019. "Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques" Water 11, no. 10: 2021. https://doi.org/10.3390/w11102021
APA StyleY. Al-Jawad, J., & M. Kalin, R. (2019). Assessment of Water Resources Management Strategy Under Different Evolutionary Optimization Techniques. Water, 11(10), 2021. https://doi.org/10.3390/w11102021