# Solution Approaches for the Management of the Water Resources in Irrigation Water Systems with Fuzzy Costs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Sustainability Concept in Water Systems

#### 1.2. Political Context

^{3}in some areas of Valencia (located in the mid-eastern part of Spain in the Mediterranean region) to 0.60 €/m

^{3}in Alicante (the southeast of Spain) (desalinated water) or Andalusian (the south of Spain) (groundwater source) [9]. It is common that the same irrigation community has different water sources, which usually have different extraction costs (e.g., an irrigation water system can receive water from groundwater that has a different piezometric head, water transferred between basins, or reused water that was treated in a wastewater treatment plant). For this reason, the decision about the most adequate water source from an economic viewpoint is critical. This selection significantly affects the final price that farmers have to pay [19], showing that there is a vital need for resilient and effective resource management strategies. This management has to be focused on societal, environmental, and economic aspects to promote sustainable development. Moreover, they also pointed to the need for decision-making mechanisms in order to address the question of resource utilization completely. To deal with this imperative need, different solution approaches have been developed.

#### 1.3. Mathematical Programming Modelling Applied to Water Systems: Initial Overview

- Multistage stochastic programming: Some studies use this method. For example, the work in [21] developed an interval multistage water allocation model to optimize water allocation between different growth stages to obtain the maximum food production in reservoir irrigation systems characterized by inputs’ uncertainties. The study developed by [22] considered a fuzzy probability distribution based multistage stochastic robust programming method. This model supported regional water supply management. The developed model was applied to a water resources management system with three water users.
- Stochastic dynamic programming: Among the references analyzed and in order to show their applications, the work in [23] used stochastic dynamic programming to model a farmer’s choice whether to invest in a sprinkler irrigation system or in a more water efficient drip irrigation system under uncertainty. The work in [24] developed a stochastic dynamic programming model, but in this case, in the context of hydro-economic models to maximize irrigation benefits while minimizing the costs of power generation within a power market. The work in [25] also developed a stochastic dynamic programming model with fuzzy state variables for irrigation of multiple crops. This model, in which the reservoir storage and soil moisture of the crops are considered as fuzzy numbers and the reservoir inflow is considered as a stochastic variable, has the main objective of minimizing crop yield deficits, resulting in optimal water allocations to the crops by maintaining storage continuity and soil moisture balance.
- Inexact programming including fuzzy and interval based programming: The work in [26] formulated a fuzzy mathematical programming model for a multi-reservoir system applied to a three reservoir system in the Upper Cauvery River basin, South India. The study tried to minimize the sum of deviations of the irrigation withdrawals from their target demands, on a monthly basis, over a year. Another study developed an interval-fuzzy two stage stochastic quadratic programming model. The goal was to allocate the limited irrigation water to different crops, maximizing the net benefit under uncertainty and to analyze how water allocation schemes change under different climate change scenarios [27].
- Nonlinear programming: The work in [28] used a non-linear programming model to estimate farmers’ willingness-to-pay for irrigation water that maximizes revenue from crop production under different shortage levels. In this case, Monte Carlo simulation was implemented considering model parameters’ uncertainty to assess the variation of farmers’ willingness-to-pay and avoid water shortage [28]. Other authors used nonlinear programing for the optimization of profitability and productivity in an irrigation command area with conjunctive water use options [29].
- Multiobjective fuzzy linear programming: The work in [30] proposed a multiobjective fuzzy linear programming irrigation planning model for the evaluation of the management strategy in the case study of the Jayakwadi irrigation project, Maharashtra, India. Three conflicting objectives, net benefits, agricultural production, and labor employment, were considered in the irrigation planning scenario. However, the objectives pursued in this study are far from the main aim of the present research. In the same line, the work in [31] proposed a model of multiobjective fuzzy linear programming based on fuzzy parametric programming to solve the problem of optimal cropping pattern in an irrigation system. The objective of the irrigation planning model is to find out an optimal cropping pattern that maximizes simultaneously the net benefits, crop production, employment generation, and manure utilization.

#### 1.4. Research Goals

## 2. Methodology

#### 2.1. Formulation Model

- i ∈ I
- Procurement water sources of the water network
- m ∈ M
- Procurement methods
- t ∈ T
- Time periods; in this case, 8760 periods were considered (one year)
- k ∈ K
- Months in the year

- ${M}_{t}^{K}$
- Set of time periods in month k (720 periods for months that have 30 days)

- d
_{t} - Required demand in period t (in m
^{3}); it includes the evaporation, leakages, and non-measured volume of the water network; the data were obtained through the irrigation water manager - CM
_{it} - Maximum flow for source i in period t (in m
^{3}/h) - CMT
_{i} - Monthly maximum volume for source i (in m
^{3}) - CH
_{i}_{,m} - Monthly available time for the procurement from source i with method m (in hours); CH is used when the water source requires grid consumption to procure it
- SMIN
_{t} - Safety stock of stored volume in period t (in m
^{3}) - SMAX
_{t} - Maximum stored volume in period t (in m
^{3}) - ${\tilde{cpv}}_{imt}$
- Variable cost for source i with method m in period t (in €/m
^{3}) - ${\tilde{cpf}}_{imt}$
- Fixed cost for source i with method m in period t (in €/m
^{3}) - ${\tilde{ci}}_{t}$
- Storage cost in period t (in €/m
^{3}) - ${\tilde{cf}}_{im}$
- Fixed cost for source i with method m over the planning horizon (in €/m
^{3})

- S
_{t} - Storage in period t (in m
^{3}) - Q
_{imt} - Flow from source i with method m in period t (in m
^{3}/h) - Y
_{imt} - 1 if any amount of water is required from source i with method m in period t, and 0 otherwise
- F
_{im} - 1 if any procurement from source i with method m is placed over the planning horizon, and 0 otherwise

_{imt}when decision variable Q

_{imt}is higher than 0.

#### 2.2. Solution Approaches

_{ij}, b

_{i}$\in $M, j$\in $N, and ${\tilde{c}}_{j}\in F(\mathbb{R})$ where $F(\mathbb{R})$ is the set of fuzzy numbers whose membership function ${\mu}_{j}$ represents the lack of precision of the values in objective function costs. The membership function is defined as follows:

_{j}, c

_{j}, and R

_{j}correspond to the left (optimistic), center, and right (pessimistic) values, the membership function is given in the following way:

_{j}≥ 0, then the membership function of $\tilde{y}$ can be formulated as follows:

#### 2.2.1. First Index of Yager

#### 2.2.2. Third Index of Yager

#### 2.2.3. Lai and Hwang’s Approach

_{1}), as well as the difference between pessimistic and average costs (z

_{3}). Besides, simultaneously, the model maximized the difference between normal and optimistic costs (z

_{2}). Thus, minimization of the original fuzzy objective can be obtained by pushing these three critical points in the direction of the left-hand side. Therefore, the following auxiliary problem is obtained taking into account the previous nomenclature:

#### 2.3. Application of the Solution Approaches

#### 2.3.1. First Index of Yager

#### 2.3.2. Third Index of Yager

#### 2.3.3. Lai and Hwang’s Approach

_{1}, μ

_{2}, μ

_{3}) for each objective function (z

_{1}, z

_{2}, z

_{3}) are formulated according to [50] as follows:

#### 2.3.4. The Zimmerman Solution Method

#### 2.3.5. The Werners Solution Method

_{0}has an equivalent meaning to Zimmermann’s solution method. In contrast, γ corresponds to the compensation coefficient among the objectives.

#### 2.3.6. Selim and Ozkarahan’s Solution Method

#### 2.3.7. Torabi and Hassini’s Solution Method

## 3. Results and Discussion

#### 3.1. Case Study

^{3}). The storage is defined as a function of water level using Equation (38):

^{2}+ 43,556 WL

^{3}and WL is the water level of the reservoir in m.

#### 3.2. Results

_{1}), most optimistic (z

_{2}), and most pessimistic (z

_{3}) total costs. Therefore, the decision-maker could develop a water management plan with a triangular possibility distribution for total storage and procurements costs (z

_{1}, z

_{2}, z

_{3}) = (278,450, 236,683, 348,063).

_{1}) when Lai and Hwang’s approach was used was 97.4% comparing it to FY. The different approaches varied between 278,450 (Selim and Ozkarahan’s approach) and 278,579 € (Zimmerman’s approach). However, the variation between the maximum and minimum costs was 0.05%. Therefore, the use of the different approaches was indifferent with respect to obtaining the optimal solution. When the most optimistic and pessimistic costs were analyzed (Table 6), a reduction of 17.8% was obtained, while the pessimistic costs showed an increase of 21.86 %. The annual volume used of each source as a function of the considered solution approach is shown in Table 7. Besides, Table 7 shows the percentage used of each considered water source for each solution approach.

^{3}. This trend was stable over the planning horizon once the water system reached continuous time. Therefore, the planning demonstrated that it was not necessary to reach the maximum water level in the reservoir in the winter season since the water sources were enough to meet the annual water demand, minimizing the exploitation costs. Accordingly, the proposed model showed a significant improvement in comparison with the results given by the water manager, enhancing the current management [26]. In this sense, the proposed model allowed distributing the energy, storage, and management costs over the planning horizon simultaneously with incomes provided from selling water to farmers. Otherwise, the current planning method used by water manager concentrates most of the amounts of required water in winter months, and consequently, it is necessary to have large flows of cash available or, in some cases, to apply for loans from credit institutions.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Data of the water demand according to [30].

**Figure 4.**Example of demanded sources in the third week of June by applying Selim and Ozkarahan’s approach.

**Table 1.**Number of publications according to the different searches performed in the literature review.

Keywords | And “Water” | And “Irrigation” | Number of Publications Per Year and “Irrigation” Limited to Related Areas | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Total | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |||

“mathematical model” | 238,864 | 14,810 | 403 | 11 | 6 | 9 | 12 | 13 | 10 | 10 | 8 | 11 | 15 | 6 |

“mathematical modelling” | 18,655 | 1862 | 60 | 1 | 1 | 2 | 3 | 5 | 1 | 3 | 2 | 1 | ||

“mathematical programming” | 11,092 | 643 | 150 | 4 | 8 | 10 | 6 | 8 | 6 | 4 | 10 | 9 | 10 | 9 |

“mathematical optimisation” | 2630 | 276 | 10 | 2 | 1 | 1 | ||||||||

“fuzzy mathematical programming” | 362 | 12 | 4 | 1 |

**Table 2.**Water prices as a function of the different sources according to [39].

Source | Method | Price |
---|---|---|

(€/m^{3}) | ||

Source 1 | Fixed | 0.25 |

Source 2 | Fixed | 0.35 |

Source 3 | Fixed | 0.60 |

Source 4 | Variable | − |

P1 | 0.56 | |

P2 | 0.50 | |

P3 | 0.35 | |

P4 | 0.30 | |

P5 | 0.20 | |

P6 | 0.12 | |

Source 5 | Variable | − |

P1 | 0.70 | |

P2 | 0.65 | |

P3 | 0.49 | |

P4 | 0.42 | |

P5 | 0.35 | |

P6 | 0.25 |

Hour | Month | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | 1–15 June | 16–30 June | July | August | September | October | November | December | |

0 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

1 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

2 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

3 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

4 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

5 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

7 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 | P6 |

8 | P2 | P2 | P4 | P5 | P5 | P4 | P2 | P2 | P6 | P4 | P5 | P4 | P2 |

9 | P2 | P2 | P4 | P5 | P5 | P3 | P2 | P2 | P6 | P4 | P5 | P4 | P2 |

10 | P1 | P1 | P4 | P5 | P5 | P3 | P2 | P2 | P6 | P4 | P5 | P4 | P1 |

11 | P1 | P1 | P4 | P5 | P5 | P3 | P1 | P1 | P6 | P4 | P5 | P4 | P1 |

12 | P1 | P1 | P4 | P5 | P5 | P3 | P1 | P1 | P6 | P4 | P5 | P4 | P1 |

13 | P2 | P2 | P4 | P5 | P5 | P3 | P1 | P1 | P6 | P4 | P5 | P4 | P2 |

14 | P2 | P2 | P4 | P5 | P5 | P3 | P1 | P1 | P6 | P4 | P5 | P4 | P2 |

15 | P2 | P2 | P4 | P5 | P5 | P4 | P1 | P1 | P6 | P4 | P5 | P4 | P2 |

16 | P2 | P2 | P3 | P5 | P5 | P4 | P1 | P1 | P6 | P3 | P5 | P3 | P2 |

17 | P2 | P2 | P3 | P5 | P5 | P4 | P1 | P1 | P6 | P3 | P5 | P3 | P2 |

18 | P1 | P1 | P3 | P5 | P5 | P4 | P1 | P1 | P6 | P3 | P5 | P3 | P1 |

19 | P1 | P1 | P3 | P5 | P5 | P4 | P2 | P2 | P6 | P3 | P5 | P3 | P1 |

20 | P1 | P1 | P3 | P5 | P5 | P4 | P2 | P2 | P6 | P3 | P5 | P3 | P1 |

21 | P2 | P2 | P3 | P5 | P5 | P4 | P2 | P2 | P6 | P3 | P5 | P3 | P2 |

22 | P2 | P2 | P4 | P5 | P5 | P4 | P2 | P2 | P6 | P4 | P5 | P4 | P2 |

23 | P2 | P2 | P4 | P5 | P5 | P4 | P2 | P2 | P6 | P4 | P5 | P4 | P2 |

Source | Hourly Capacity (m^{3}) CM _{it} | Monthly Capacity (m^{3}) CMT _{it} |
---|---|---|

1 | 120 | 40,000 |

2 | 306 | 500,000 |

3 | 324 | 75,000 |

4 | 360 | 75,000 |

5 | 450 | 125,000 |

Hours of Water Service Per Month | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Source | Method | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

4 | 2 | 132 | 120 | 0 | 0 | 0 | 0 | 286 | 0 | 0 | 0 | 0 | 126 |

4 | 3 | 220 | 200 | 0 | 0 | 0 | 0 | 66 | 0 | 0 | 0 | 0 | 210 |

4 | 4 | 0 | 0 | 126 | 0 | 0 | 126 | 0 | 0 | 120 | 0 | 126 | 0 |

4 | 5 | 0 | 0 | 210 | 0 | 0 | 210 | 0 | 0 | 200 | 0 | 210 | 0 |

4 | 6 | 0 | 0 | 0 | 336 | 336 | 0 | 0 | 0 | 0 | 352 | 0 | 0 |

4 | 7 | 392 | 352 | 408 | 384 | 408 | 384 | 392 | 744 | 400 | 392 | 384 | 456 |

5 | 2 | 132 | 120 | 0 | 0 | 0 | 0 | 286 | 0 | 0 | 0 | 0 | 126 |

5 | 3 | 220 | 200 | 0 | 0 | 0 | 0 | 66 | 0 | 0 | 0 | 0 | 210 |

5 | 4 | 0 | 0 | 126 | 0 | 0 | 126 | 0 | 0 | 120 | 0 | 126 | 0 |

5 | 5 | 0 | 0 | 210 | 0 | 0 | 210 | 0 | 0 | 200 | 0 | 210 | 0 |

5 | 6 | 0 | 0 | 0 | 336 | 336 | 0 | 0 | 0 | 0 | 352 | 0 | 0 |

5 | 7 | 392 | 352 | 408 | 384 | 408 | 384 | 392 | 744 | 400 | 392 | 384 | 456 |

First Index of Yager | Third Index of Yager | Lai and Hwang Approach | ||||
---|---|---|---|---|---|---|

Zimmerman’s Approach | Werners’ Approach | Selim and Ozkarahan’s Approach | Torabi and Hassini’s Approach | |||

Total costs (z) | 285,753 | 284,076 | - | - | - | - |

Most possible total costs (z1) | - | - | 278,579 | 278,567 | 278,450 | 278,531 |

Most optimistic total costs (z2) | - | - | 236,792 | 236,782 | 236,683 | 236,752 |

Most pessimistic total costs (z3) | - | - | 348,224 | 348,209 | 348,063 | 348,164 |

Computational Time | - | 30% | 294% | 350% | 322% | 229% |

_{1}= 0.6; w

_{2}= 0.2; w

_{3}= 0.2.

Source | Method | Annual Volume (m^{3}/Year) | % Use with Respect to Annual Capacity of Each Water Source Used |
---|---|---|---|

First Index of Yager | |||

4 | 7 | 347,743 | 38.64% |

Third Index of Yager | |||

4 | 7 | 347,743 | 38.64% |

Zimmerman’s approach | |||

1 | 1 | 36 | 0.01% |

4 | 7 | 347,743 | 38.64% |

Werners’ approach | |||

1 | 1 | 2565 | 0.53% |

2 | 1 | 343,111 | 5.72% |

3 | 1 | 1797 | 0.20% |

Selim and Ozkarahan’s approach | |||

1 | 1 | 4489 | 0.94% |

2 | 1 | 951 | 0.02% |

4 | 7 | 346,457 | 38% |

Torabi and Hassini’s approach | |||

1 | 1 | 1970 | 0.41% |

2 | 1 | 5454 | 0.09% |

4 | 7 | 340,049 | 37.78% |

First Index of Yager | Third Index of Yager | Lai and Hwang Approach | ||||
---|---|---|---|---|---|---|

Zimmerman’s Approach | Werners’ Approach | Selim and Ozkarahan’s Approach | Torabi and Hassini’s Approach | |||

Constraints | 455,654 | 455,654 | 455,661 | 455,661 | 455,661 | 455,661 |

Variables | 96,372 | 96,372 | 96,376 | 96,379 | 96,379 | 96,376 |

Integers | 43,812 | 43,812 | 43,812 | 43,812 | 43,812 | 43,812 |

Nonzeros | 2,417,737 | 2,417,737 | 2,803,234 | 2,803,237 | 2,803,237 | 2,803,234 |

Density | 0.006% | 0.006% | 0.006% | 0.006% | 0.006% | 0.006% |

Iterations | 19,875 | 19,858 | 68,620 | 36,858 | 35,569 | 34,532 |

Solution time (s) | 22 | 28 | 85 | 97 | 91 | 76 |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Sanchis, R.; Díaz-Madroñero, M.; López-Jiménez, P.A.; Pérez-Sánchez, M. Solution Approaches for the Management of the Water Resources in Irrigation Water Systems with Fuzzy Costs. *Water* **2019**, *11*, 2432.
https://doi.org/10.3390/w11122432

**AMA Style**

Sanchis R, Díaz-Madroñero M, López-Jiménez PA, Pérez-Sánchez M. Solution Approaches for the Management of the Water Resources in Irrigation Water Systems with Fuzzy Costs. *Water*. 2019; 11(12):2432.
https://doi.org/10.3390/w11122432

**Chicago/Turabian Style**

Sanchis, Raquel, Manuel Díaz-Madroñero, P. Amparo López-Jiménez, and Modesto Pérez-Sánchez. 2019. "Solution Approaches for the Management of the Water Resources in Irrigation Water Systems with Fuzzy Costs" *Water* 11, no. 12: 2432.
https://doi.org/10.3390/w11122432