Improving Water and Energy Resource Management: A Comparative Study of Solution Representations for the Pump Scheduling Optimization Problem
Abstract
:1. Introduction
- Minimizing pumping operation costs.
- Enhancing water quality.
- Execution time.
- Convergence.
- Pareto front coverage.
- Diversity.
- Sensitivity.
2. Model Outline and Encoding Decision Variables in the Pump Scheduling Problem
2.1. Model Outline and Mathematical Notations
2.2. Encoding Decision Variables
- Binary Representation (bin) [24,25,26,31]: This strategy is used to represent the pump states observed in each time interval using accepted values , where . Here, 0 and 1 are representative of the off and on states, respectively. The size of the solution vector is determined by the number of time intervals () and the number of pumps (), and it is calculated as . Additionally, the search space for this representation type equals .
- Integer Representation (int) [36]: The operations of pumps are represented by integers, which are in the range , where NP is the number of pumps. Once the valid values are defined, a conversion from each integer to its binary equivalent is performed to represent the state of each pump. For example, if we have a time interval and obtain the corresponding integer value, we convert it to a binary number. Each bit of the binary number is used to define the state of each pump in the interval . In this representation, the size of the solution space is the same as that in the binary representation, i.e., , and the size of the solution vector is equal to .
- Restricted Formulation (int_r) [18,37]: In this variant, the decision variables represent the start and end times of pump operations, and they are bounded between 0 (pump off) and Δt (the duration of the time interval). To determine the number of decision variables, the ranges of the time intervals () are defined. For example, if we consider a total of 24 h and define as 4, we have a total of 6 decision variables for each set of pumps (). In general, the formula for calculating the total number of decision variables is , and the total search space is , where .
- Absolute Time-Controlled Triggers (int_at) [15]: In this representation strategy, the decision variables are absolute times, meaning that each decision variable represents the time elapsed from the start of the scheduling period until the point at which the status of a pump changes. A pair of decision variables represents the operating interval during which the associated pump is active. This representation approach allows for scheduling the turning on and turning off of pumps at specific times, and a maximum change limit (SW) must be defined. The total number of decision variables is , and the size of the search space is .
- Relative Time-Controlled Triggers (int_rt) [15]: For decision variables that represent relative time intervals, each pair signifies the duration from the beginning of the scheduling period to the first state change exhibited by the corresponding pump. In other words, they denote the periods of inactivity and activity for a pump, respectively. The number of decision variables is , and must be satisfied, where represents each decision variable in the vector.
3. Materials and Methods
3.1. Solution Representation Comparison Methodology
3.2. Computational Environment
3.3. Case Studies
4. Results and Discussion
4.1. Overview of the Simulation Results
4.2. Performance Metrics
4.3. Benefits and Disadvantages of the Suggested Solution Codification
- Enhanced Convergence: The int_r method achieves superior convergence towards the Pareto front, as indicated by the lower Epsilon (EP) and IGD+ values across all networks (Table 2). The int_r representation consistently shows the closest proximity to the reference Pareto front, ensuring high-quality solutions that balance energy costs and water quality objectives.
- Improved Solution Diversity: The int_r representation excels in generating diverse solutions, as evidenced by the higher Hypervolume (HV) values in all three case studies (Table 2). This indicates extensive coverage in the objective space, which enhances the robustness and applicability of the solutions.
- Problem Simplification: The int_r representation significantly simplifies the problem by reducing the number of decision variables. While this leads to faster convergence, it also means that the method might not explore the entire search space thoroughly. Consequently, it could potentially miss some solutions that are closer to the optimal, which might be found using more detailed representations.
- Resource Consumption: The int_r method, while efficient in some scenarios, can be resource-intensive, particularly for larger and more complex networks like Curicó. As observed in Table 4, the execution time for the Curicó network is significantly higher (36,730.12 s) compared to the other representations. This indicates that the computational demands increase with the complexity and size of the network, potentially requiring substantial computational resources and longer processing times.
5. Conclusions
- -
- The binary representation strategy excels at generating a high number of unique solutions, implying robust exploration of the solution space. However, this advantage does not necessarily translate into higher solution quality, as indicated by the lower feasibility levels observed, especially in complex networks like Curicó. In contrast, integer representations demonstrate higher feasibility rates across most experiments, showcasing their effectiveness in meeting problem constraints. Furthermore, the analysis of nondominated solutions reveals comparable quantities across representations, suggesting that each strategy yields solutions with similar dominance characteristics.
- -
- However, a closer examination shows that certain representations contribute more significantly to shaping the approximate Pareto front, particularly the int_r representation. The visualizations provided in Figure 5, Figure 6 and Figure 7 highlight the unique contributions of specific representations to the Pareto fronts, emphasizing differences near the theoretical optimum.
- -
- To quantitatively evaluate solution diversity and convergence, metrics such as HV, Epsilon, and IGD+ were calculated. The results consistently favor the int_r representation, indicating its superior performance in terms of solution diversity, proximity to the Pareto front, and convergence.
- -
- Statistical analysis using the Wilcoxon test confirms the significance of these differences, establishing int_r as the most suitable representation strategy for the optimization problem studied. Conversely, the binary representation strategy demonstrates inferior numerical performance across comparisons, suggesting its inadequacy for this type of optimization problem.
- -
- Regarding computational efficiency, int_r maintains competitive execution times across benchmarking networks, indicating that its enhanced solution quality does not compromise computational speed. However, in more complex scenarios like the Curicó network, where computational costs vary significantly among representations, the choice depends on specific optimization objectives.
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Network | Representation | Total Solutions | Average Unique Solutions per Experiment (%) | ||||
---|---|---|---|---|---|---|---|
Unique | Feasible | Nondominated | Feasibility (%) | Contribution to the Reference Front (%) | |||
Anytown | bin | 7743 | 7173 | 11 | 93.7 | 93.0 | 0.0 |
int | 2372 | 2099 | 13 | 27.0 | 88.6 | 58.8 | |
int_r | 253 | 245 | 9 | 2.8 | 96.8 | 41.2 | |
int_at | 476 | 465 | 16 | 5.4 | 97.7 | 0.0 | |
int_rt | 297 | 297 | 12 | 3.5 | 100.0 | 0.0 | |
Anytown Modified | bin | 5961 | 5961 | 13 | 67.3 | 100.0 | 0.0 |
int | 538 | 538 | 23 | 6.3 | 100.0 | 9.7 | |
int_r | 424 | 424 | 33 | 6.7 | 100.0 | 90.3 | |
int_at | 245 | 245 | 19 | 3.3 | 100.0 | 0.0 | |
int_rt | 69 | 69 | 12 | 2.8 | 100.0 | 0.0 | |
Curicó | bin | 8125 | 470 | 7 | 97.3 | 5.7 | 0.0 |
int | 153 | 153 | 6 | 1.7 | 100.0 | 20.0 | |
int_r | 214 | 214 | 9 | 2.4 | 100.0 | 60.0 | |
int_at | 235 | 229 | 11 | 2.6 | 97.4 | 0.0 | |
int_rt | 131 | 131 | 8 | 1.5 | 100.0 | 20.0 |
Network | Quality Indicator | bin | int | int_r | int_at | int_rt |
---|---|---|---|---|---|---|
Anytown | HV | 0.65 | 0.70 | 0.73 | 0.70 | 0.70 |
EP | 0.09 | 0.05 | 0.01 | 0.06 | 0.05 | |
IGD+ | 0.05 | 0.02 | 0.01 | 0.02 | 0.02 | |
Anytown Modified | HV | 0.29 | 0.30 | 0.35 | 0.28 | 0.24 |
EP | 0.07 | 0.07 | 0.01 | 0.09 | 0.13 | |
IGD+ | 0.05 | 0.04 | 0.00 | 0.06 | 0.09 | |
Curicó | HV | 0.15 | 0.19 | 0.22 | 0.16 | 0.16 |
EP | 0.10 | 0.06 | 0.03 | 0.09 | 0.09 | |
IGD+ | 0.05 | 0.02 | 0.01 | 0.04 | 0.04 |
bin | int | int_r | int_at | int_rt | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
HV | EP | IGD+ | HV | EP | IGD+ | HV | EP | IGD+ | HV | EP | IGD+ | HV | EP | IGD+ | |
+ | 2 | 2 | 2 | 5 | 5 | 5 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 |
− | 8 | 8 | 8 | 6 | 6 | 6 | 0 | 0 | 0 | 8 | 8 | 8 | 7 | 7 | 7 |
= | 2 | 2 | 2 | 1 | 1 | 1 | 0 | 0 | 0 | 2 | 2 | 2 | 3 | 3 | 3 |
Network | Average Times (s) | ||||
---|---|---|---|---|---|
bin | int | int_r | int_at | int_rt | |
Anytown | 216.46 | 212.58 | 209.72 | 218.08 | 212.88 |
Anytown Modified | 203.15 | 206.00 | 215.18 | 214.52 | 205.32 |
Curicó | 28,729.87 | 26,842.75 | 36,730.12 | 20,756.05 | 15,807.77 |
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Silva-Rubio, S.A.; Salgueiro, Y.; Mora-Meliá, D.; Gutiérrez-Bahamondes, J.H. Improving Water and Energy Resource Management: A Comparative Study of Solution Representations for the Pump Scheduling Optimization Problem. Mathematics 2024, 12, 1994. https://doi.org/10.3390/math12131994
Silva-Rubio SA, Salgueiro Y, Mora-Meliá D, Gutiérrez-Bahamondes JH. Improving Water and Energy Resource Management: A Comparative Study of Solution Representations for the Pump Scheduling Optimization Problem. Mathematics. 2024; 12(13):1994. https://doi.org/10.3390/math12131994
Chicago/Turabian StyleSilva-Rubio, Sergio A., Yamisleydi Salgueiro, Daniel Mora-Meliá, and Jimmy H. Gutiérrez-Bahamondes. 2024. "Improving Water and Energy Resource Management: A Comparative Study of Solution Representations for the Pump Scheduling Optimization Problem" Mathematics 12, no. 13: 1994. https://doi.org/10.3390/math12131994
APA StyleSilva-Rubio, S. A., Salgueiro, Y., Mora-Meliá, D., & Gutiérrez-Bahamondes, J. H. (2024). Improving Water and Energy Resource Management: A Comparative Study of Solution Representations for the Pump Scheduling Optimization Problem. Mathematics, 12(13), 1994. https://doi.org/10.3390/math12131994