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)
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 , in interval quadratic function have the same sign, we can formulate the upper and lower bound submodels corresponding to and .
- (4)
- If the , in interval quadratic function have different signs, we should formulate and solve mid-values model first, have and or and whether 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
3.2. Problem Statement
3.3. Data Collection and Processing
3.4. Application of IQFDCP Model
4. Result Analysis
- (1)
- Compared with the actual situation, the interval of the system revenue of the triangular fuzzy number is too big, especially for obtaining r3. 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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Typical Year | Crop | Precipitation (P)/mm | Water Demand (ETmax)/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/×104 m3 | Ground Water/×104 m3 |
---|---|---|---|---|---|
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 |
---|---|
The credibility level of the event | |
The upper and lower limits of the parameters | |
System revenue (×109 Yuan) | |
Different irrigation areas | |
Different crop types | |
Acreage (hm2) | |
Purchase price (Yuan/kg) | |
The parameters of the crop water production function | |
The total amount of surface water in the whole growth period (mm), the decision variables | |
The total amount of groundwater in the whole growth period (mm), the decision variables | |
The availability factor of surface water | |
The availability factor of groundwater | |
The precipitation during the whole growth period (mm) | |
Maximum water demand (mm) | |
Minimum water demand (mm) | |
The amount of available surface water (m3) | |
The amount of available groundwater (m3) | |
The minimum grain demand of per capita (kg) | |
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 (×108 Yuan) | Actual System Revenue (×108 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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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
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 StyleWang, 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