# An Interval Quadratic Fuzzy Dependent-Chance Programming Model for Optimal Irrigation Water Allocation under Uncertainty

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Interval Quadratic Programming (IQP)

#### 2.2. Fuzzy Dependent-Chance Programming Theory (FDCP)

_{1}, r

_{2}, r

_{3}), $Cr$ is a membership function, then the credibility level of $x\ge \epsilon $ can be expressed as:

#### 2.3. Interval Quadratic Fuzzy Dependent-Chance Programming (IQFDCP) Model

- (1)
- Determine the input parameters of the model and its fuzzy membership functions.
- (2)
- Analyze the credibility level of the model. According to the triangular fuzzy membership functions, the developed model can be transformed into its equivalent model.
- (3)
- If the ${c}_{j}{}^{\pm}$, ${d}_{j}{}^{\pm}$ in interval quadratic function have the same sign, we can formulate the upper and lower bound submodels corresponding to ${f}^{+}$ and ${f}^{-}$.
- (4)
- If the ${c}_{j}{}^{\pm}$, ${d}_{j}{}^{\pm}$ in interval quadratic function have different signs, we should formulate and solve mid-values model first, have ${f}^{+}\left({x}_{j}{}^{+}\right)$ and ${f}^{-}\left({x}_{j}{}^{-}\right)$ or ${f}^{+}\left({x}_{j}{}^{-}\right)$ and ${f}^{-}\left({x}_{j}{}^{+}\right)$ whether ${f}^{\prime}{\{{\left({x}_{j}^{}\right)}_{mvopt}\}}^{+}$ is more than 0.
- (5)
- Solve the above submodels and obtain the optimal objective function and the decision variables.

## 3. Case Study

#### 3.1. Study Area

^{2}and lies between 102°45′–103°55′ E and 38°20′–39°10′ N in the contiguous zone and between the Badain Jaran desert and Tengger desert [34]. The annual average precipitation is 113 mm, while the annual evapotranspiration reaches 2644 mm [35]. Minqin Oasis is mainly focused on agricultural production, and agricultural water is the largest water use section of Minqin Oasis, accounting for 70–80% of the total water consumption. The surface water resource is delivered only from the Shiyang River, which originates from the Qilian mountains [36]. There are three irrigation districts in study area: Changning irrigation district (CN), Huanhe irrigation district (HH), and Hongyashan irrigation district (HYS). CN and HH are fully irrigated by groundwater, and HYS is irrigated by both groundwater and surface water. The study crops consist of grain crops (e.g., spring wheat, maize) and economic crops (e.g., cotton). The main irrigation method is furrow irrigation.

#### 3.2. Problem Statement

^{8}m

^{3}in 1956 to only 1.79 × 10

^{8}m

^{3}in 2006 [32]. Moreover, the amount of extracted groundwater accounts for over 85% of annual water consumption of the oasis [34]. Owing to excessive extraction, the groundwater table has continuously declined. The desertification and salinization of the region are serious and have become the growing concerns. Therefore, irrigation water resources are increasingly limited, and the disparity between supply and demand of agricultural water resources is expanding. Therefore, it is necessary to optimize irrigation water resources so as to alleviate the contradiction between supply and demand.

#### 3.3. Data Collection and Processing

_{1}, r

_{2}, r

_{3}) of the system revenue can be obtained by using the planting area data of the past 5 years (2012–2017) and the corresponding economic parameters. In this case, r

_{1}, r

_{2}, r

_{3}in the triangular fuzzy number of evaluation values were “poor”, “general” and “good”. Therefore, the acreage and economic parameters are selected from 2012–2017 as the lowest, average, the highest value calculation, and r

_{1}= 2.89 × 10

^{9}Yuan, r

_{2}= 3.21 × 10

^{9}Yuan, r

_{3}= 4.32 × 10

^{9}Yuan.

#### 3.4. Application of IQFDCP Model

## 4. Result Analysis

^{9}Yuan). The normal year level of system revenue is the highest, followed by wet year level, dry year level. R in different scenarios is higher than the general value r

_{2}in the triangular fuzzy number, but less than r

_{3}. It can be concluded that the system revenue in different scenarios is higher than the “normal” level, but it is difficult to achieve the “good” level. The range of the interval decreased gradually, that is, 0.05, 0.04, and 0.03. The fluctuation of system revenue is small, and the risk space of irrigation water will decrease accordingly.

^{4}Yuan. Table 5 shows that the interval quadratic fuzzy dependent-chance programming model for optimal irrigation water allocation has achieved obvious water-saving effect and improved the economic benefit. Based on the fuzzy dependent-chance programming, it is possible to obtain the maximum system revenue and its credibility level. When considering the credibility level, there is a certain risk level for obtaining the theoretical maximum system revenue.

- (1)
- Compared with the actual situation, the interval of the system revenue of the triangular fuzzy number is too big, especially for obtaining r
_{3}. The planting area and the price take a more optimistic value. It is difficult to achieve in the actual situation, causing the lower level of credibility of system revenue. - (2)
- For three irrigation districts in Minqin Oasis, the CN and HH districts are irrigated by only groundwater. In order to decrease groundwater extraction, irrigation water is reduced, resulting in lower crop yield under the low amount of groundwater availability compared to HYS. Thus, low system revenue is obtained in Minqin Oasis and the credibility level is decreased.
- (3)
- There is no cotton in CN and HH. The main crops are spring wheat and maize. However, cotton is a major source of income for system, resulting in low system revenue, causing the low credibility level.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The framework of the interval quadratic fuzzy dependent-chance programming (IQFDCP) model.

**Table 1.**The whole growth stage water demand of different crops and precipitation statistics of Minqin Oasis under typical years.

Typical Year | Crop | Precipitation (P)/mm | Water Demand (ET_{max})/mm |
---|---|---|---|

Wet year (2002) | Spring wheat | (98.6, 120.6) | 658.0 |

Maize | (124.1, 151.7) | 561.0 | |

Cotton | (135.8, 166.0) | 375.3 | |

Normal year (1998) | Spring wheat | (77.0, 94.2) | 677.4 |

Maize | (79.8, 97.6) | 590.6 | |

Cotton | (90.5, 110.6) | 396.8 | |

Dry year (2013) | Spring wheat | (34.1, 41.7) | 695.9 |

Maize | (68.9, 84.3) | 589.8 | |

Cotton | (76.0, 92.8) | 398.5 |

Irrigation Area | Crop | Acreage/ha | Price/Yuan | Surface Water/×10^{4} m^{3} | Ground Water/×10^{4} m^{3} |
---|---|---|---|---|---|

CN | Spring wheat | 153.4 | (2.4, 2.7) | 0 | (260, 289) |

Maize | 530.3 | (2.5, 2.8) | |||

Cotton | 0 | (7.0, 8.0) | |||

HH | Spring wheat | 439.7 | (2.4, 2.7) | 0 | (530, 589) |

Maize | 1214.7 | (2.5, 2.8) | |||

Cotton | 0 | (7.0, 8.0) | |||

HYS | Spring wheat | 3764.9 | (2.4, 2.7) | (20180, 22422) | (3231, 3590) |

Maize | 6868.2 | (2.5, 2.8) | |||

Cotton | 709.1 | (7.0, 8.0) |

Parameters and Variables | Meaning and Description |
---|---|

$Cr$ | The credibility level of the event |

$\pm $ | The upper and lower limits of the parameters |

$\tilde{F}$ | System revenue (×10^{9} Yuan) |

$i$ | Different irrigation areas |

$j$ | Different crop types |

${A}_{ij}$ | Acreage (hm^{2}) |

${B}_{ij}$ | Purchase price (Yuan/kg) |

$a,b,c$ | The parameters of the crop water production function |

$S{W}_{ij}$ | The total amount of surface water in the whole growth period (mm), the decision variables |

$G{W}_{ij}$ | The total amount of groundwater in the whole growth period (mm), the decision variables |

${\eta}_{1}$ | The availability factor of surface water |

${\eta}_{2}$ | The availability factor of groundwater |

${P}_{ij}$ | The precipitation during the whole growth period (mm) |

${ET}_{\mathrm{max}ji}$ | Maximum water demand (mm) |

${ET}_{\mathrm{min}ji}$ | Minimum water demand (mm) |

${Q}_{i}$ | The amount of available surface water (m^{3}) |

${G}_{i}$ | The amount of available groundwater (m^{3}) |

$\overline{Y}$ | The minimum grain demand of per capita (kg) |

$S$ | The total population of research area |

Irrigation District | Typical Year | Irrigation Sources | Irrigation Water Allocation for Different Crops | ||
---|---|---|---|---|---|

Spring Wheat | Maize | Cotton | |||

CN | Wet year | Surface water | 0 | 0 | 0 |

Groundwater | (267.2, 279.4) | (293.3, 289.8) | 0 | ||

Normal year | Surface water | 0 | 0 | 0 | |

Groundwater | (275.9, 286.5) | (290.7, 287.7) | 0 | ||

Dry year | Surface water | 0 | 0 | 0 | |

Groundwater | (320.2, 322.8) | (278.0, 277.2) | 0 | ||

HH | Wet year | Surface water | 0 | 0 | 0 |

Groundwater | (250.3, 262.2) | (239.1, 234.8) | 0 | ||

Normal year | Surface water | 0 | 0 | 0 | |

Groundwater | (258.8, 259.2) | (236.1, 232.3) | 0 | ||

Dry year | Surface water | 0 | 0 | 0 | |

Groundwater | (302.2, 304.7) | (220.3, 219.4) | 0 | ||

HYS | Wet year | Surface water | (459.9, 481.9) | (278.4, 306.4) | (186.4, 212.8) |

Groundwater | 0 | (130.9, 130.5) | (22.9, 26.7) | ||

Normal year | Surface water | (486.3, 503.4) | (364.4, 380.0) | (242.1, 282.4) | |

Groundwater | 0 | (128.7, 130.8) | (44.2, 24.0) | ||

Dry year | Surface water | (538.8, 546.4) | (374.1, 388.3) | (287.3, 315.7) | |

Groundwater | 0 | (131.4, 132.6) | (18.4, 6.9) |

Irrigation District | Crop | Irrigation Water Amount | Water Saving Amount | Optimized System Revenue (×10^{8} Yuan) | Actual System Revenue (×10^{8} Yuan) | |
---|---|---|---|---|---|---|

Optimization | Actual | |||||

CN | Spring wheat | (275.9, 286.5) | 410 | (134.1, 123.5) | (3.58, 3.62) | 3.2 |

Maize | (290.7, 287.7) | 405 | (114.3, 117.3) | |||

Cotton | 0 | 0 | 0 | |||

HH | Spring wheat | (258.8, 259.2) | 410 | (151.2, 150.8) | ||

Maize | (236.1, 232.3) | 405 | (168.9, 172.7) | |||

Cotton | 0 | 0 | 0 | |||

HYS | Spring wheat | (486.3, 503.4) | 536 | (49.7, 32.6) | ||

Maize | (493.1, 510.8) | 563 | (69.9, 52.2) | |||

Cotton | (286.3, 306.4) | 323 | (36.7, 16.6) |

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## Share and Cite

**MDPI and ACS Style**

Wang, H.; Zhang, C.; Guo, P.
An Interval Quadratic Fuzzy Dependent-Chance Programming Model for Optimal Irrigation Water Allocation under Uncertainty. *Water* **2018**, *10*, 684.
https://doi.org/10.3390/w10060684

**AMA Style**

Wang H, Zhang C, Guo P.
An Interval Quadratic Fuzzy Dependent-Chance Programming Model for Optimal Irrigation Water Allocation under Uncertainty. *Water*. 2018; 10(6):684.
https://doi.org/10.3390/w10060684

**Chicago/Turabian Style**

Wang, Hang, Chenglong Zhang, and Ping Guo.
2018. "An Interval Quadratic Fuzzy Dependent-Chance Programming Model for Optimal Irrigation Water Allocation under Uncertainty" *Water* 10, no. 6: 684.
https://doi.org/10.3390/w10060684