# An Inexact Optimization Model for Crop Area Under Multiple Uncertainties

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

#### 2.1. Type-2 Fuzzy Interval Programming

#### 2.2. The Chance Constrained Programming Model

#### 2.3. Type-2 Fuzzy Interval Chance Constrained Programming

#### 2.4. Solution Process

- Step 1: Establish the T2FICCP model.
- Step 2: Convert the chance constraint into the deterministic constraint with a feasible constraint set based on the CCP model.
- Step 3: Transform the model, which has converted the chance constraint into the deterministic constraint, into two submodels with T2FS based on the interactive algorithm.
- Step 4: Each submodel is converted into its deterministic counterpart by using a type reduction technique.
- Step 5: The submodel corresponding to ${\tilde{f}}^{+}$ is formulated and solved first based on the solution steps presented in Section 2.1.
- Step 6: Based on the solution of the first submodel, the submodel corresponding to ${\tilde{f}}^{-}$ is formulated and solved.
- Step 7: Based on the solutions of two submodels, the solution of the established T2FICCP model can be obtained, which is expressed as a set of intervals: ${x}_{jopt}^{+}=\left[{x}_{jopt}^{-},{x}_{jopt}^{+}\right]$ and ${f}_{opt}^{\pm}=\left[{f}_{opt}^{-},{f}_{opt}^{+}\right]$.
- Step 8: Repeat steps 3–7 corresponding to different ${P}_{i}$ levels.

## 3. Case Study

^{2}) that can be used for planting is very small, most of which is desert and mountainous land [37,40].

^{3}, which accounts for approximately 34.82% of the national level (2260 m

^{3}) [2].

#### Model Building

- ${P}_{i1}^{\pm}$: Price of crop i under the conventional irrigation mode (yuan/t) (Interval parameter);
- ${P}_{i2}^{\pm}$: Price of crop i under the water-saving irrigation mode (yuan/t) (Interval parameter);
- ${Y}_{i1}^{\pm}$: Yield of crop i under the conventional irrigation mode (t/ha) (Interval parameter);
- ${Y}_{i2}^{\pm}$: Yield of crop i under the water saving irrigation mode (t/ha) (Interval parameter);
- ${C}_{i1}$: The cost of crop i under the conventional irrigation mode (yuan/ha);
- ${C}_{i2}$: The cost of crop i under the water saving irrigation mode (yuan/ha);
- ${A}_{i}^{\pm}$: Irrigation area of crop i (10
^{4}ha); - ${A}_{imin}$: Minimized irrigation area of crop i (10
^{4}ha); - ${A}_{imax}$: Maximized irrigation area of crop i (10
^{4}ha); - TA: Total irrigation area of the study area (10
^{4}ha); - $I{W}_{i1}^{\pm}$: Irrigation quota of crop i under the conventional irrigation mode (m
^{3}/ha) (Interval number); - $I{W}_{i2}^{\pm}$: Irrigation quota of crop i under the water saving irrigation mode (m
^{3}/ha) (Interval number); - IC: Irrigation water use efficiency of the study area;
- SW: Water demand of the secondary industry (10
^{4}m^{3}); - TW: Water demand of the tertiary industry (10
^{4}m^{3}); - DW: Domestic water consumption (10
^{4}m^{3}); - EW: Ecological water consumption (10
^{4}m^{3}); - $\tilde{G}{W}^{\pm}$: Groundwater exploration (10
^{4}m^{3}) (Type-2 fuzzy interval parameter); - SFW: Maximized surface water supply (10
^{4}m^{3}) (Random parameter); - $E{T}_{i1}^{\pm}$: The ET of the ith crop under the conventional irrigation mode (m
^{3}/ha) (Interval parameter); - $E{T}_{i2}^{\pm}$: The ET of the ith crop under the water saving irrigation mode (m
^{3}/ha) (Interval parameter); - TET: The control indicator of the total water consumption (m
^{3}); - FDP: Food demand per capita (t/p);
- TPR: Population of the study area (10
^{4}p).

^{8}m

^{3}, 9.89 × 10

^{8}m

^{3}, 10.36 × 10

^{8}m

^{3}and 10.64 × 10

^{8}m

^{3}, were obtained based on the constructed P-III; hydrographic curve (Figure 3).

## 4. Analysis of the Results and Discussion

^{8}m

^{3}, 9.89 × 10

^{8}m

^{3}, 10.36 × 10

^{8}m

^{3}and 10.64 × 10

^{8}m

^{3}, respectively. Through solving the established model (10), the optimal crop area schemes were obtained under the different violation probabilities, P

_{i}, of each water-saving scenario. The results, which are generally expressed as intervals, can reflect more sensitivity. Figure 4, Figure 5 and Figure 6 present the optimized allocation of the crop area, yield and economic benefit under different violation probabilities of each water-saving scenario, respectively. From the figures, in general, the total irrigation area would vary under different water-saving scenarios and multiple uncertainties. For example, the lower bound of the total irrigation area would increase from 12.15 × 10

^{4}ha (λ

_{i}= 10%) to 17.30 × 10

^{4}ha (λ

_{i}= 90%) when the violation probability P

_{i}is 0.05. In addition, no matter the total irrigation area, yield or economic benefit, they have the same characteristic that the lower and upper bound increase as the water-saving levels and violation probabilities increase. When the result of the developed model is compared with the result of the model with conventional fuzzy uncertainty, there is a great difference. The biggest difference is that the lower bound and the upper bound of the result with the conventional fuzzy uncertainty have different trends, which represented that the upper bound value is decreasing while the lower bound value is increasing as the α-cut level increases. This is because of the triangular or trapezoidal membership function, which has the characteristic that the fuzzy extent weakens as the α-cut level increases, was usually selected as the membership function to deal with fuzzy uncertainty problems. However, in this study, in order to deal with the complex uncertainty expressed as type-2 fuzzy intervals, type-2 fuzzy interval programming was introduced into the established model. Therefore, based on the interactive algorithm and type reduction algorithm, the results are as shown in Figure 4, Figure 5 and Figure 6.

^{4}ha (λ

_{i}= 10%) to (17.89 × 10

^{4}, 21.63 × 10

^{4}) ×10

^{4}ha (λ

_{i}= 90%) when violation probability P

_{i}is 0.1. The result accords with the actual conditions. This is because the available water resources would increase as the water-saving levels increase. Thus, the crop area would increase as the available water resources increase, correspondingly. In addition, the total yield would increase from (807.69, 885.94) × 10

^{4}t (λ

_{i}= 10%) to (917.57, 969.72) × 10

^{4}t (λ

_{i}= 90%) when the violation probability, P

_{i}

_{,}is 0.1. Furthermore, the economic benefit would increase from (69.61, 92.38) × 10

^{8}¥ (λ

_{i}= 10%) to (76.51, 100.48) × 10

^{8}¥ (λ

_{i}= 90%) when the violation probability P

_{i}is 0.1. This is because that there are positive relationships between the available water resources and the irrigation area, yield and economic benefit. As the water-saving levels increase, the available water resources would increase. Thus, the irrigation area, yield and economic benefit increase correspondingly.

^{4}ha (P

_{i}= 0.05) to (15.28, 18.24) ×10

^{4}ha (P

_{i}= 0.25) when the water-saving level is λ

_{i}= 30%. Moreover, the total yield would increase from (813.50, 889.38) × 10

^{4}t (P

_{i}= 0.05) to (855.58, 926.55) × 10

^{4}t (P

_{i}= 0.25) when the water-saving level is λ

_{i}= 30%. In addition, the economic benefit would increase from (72.63, 96.12) × 10

^{8}¥ (P

_{i}= 0.05) to (75.14, 99.19) × 10

^{8}¥ (P

_{i}= 0.25) when the water-saving level is λ

_{i}= 30%.

_{i}= 30% under P

_{i}= 0.1 was selected. From Figure 7, it shows that there are great differences in the irrigation areas of different crops. This is because some crops have a higher price or could produce a higher yield by using the same amount of water resources when compared with the other crops. For example, the yield per unit of maize under the conventional irrigation mode and water-saving mode are (10,470, 10,695) kg/ha and (12,750, 12,750) kg/ha, respectively, while the irrigation quota of maize under the conventional and water-saving mode are (5400, 5520) m

^{3}/ha and (3000, 3600) m

^{3}/ha, respectively. Furthermore, the price per unit and cost per unit under both irrigation modes are (1.3, 1.5) ¥/kg and 7788 ¥/ha, respectively. Therefore, when optimizing the irrigation water among the 11 crops, the minimum water resources constraint should be satisfied first. In addition, then the rest of the water resources would be assigned to the crops with a higher yield, higher prices, lower irrigation quota or lower cost.

_{i}is 0.1. As the figure shows, as water-saving levels increases, the total area would increase, while the consumption of irrigation water resources would not vary. Figure 8 and Figure 9 show that the of water resource shortage is very serious and Wuwei’s water demand cannot be met, even if the violation probability reaches 0.25. Moreover, it also shows that the crop area, yield and economic benefit would increase as water-saving levels increases when the available water resources do not change. Therefore, it also suggests that attention should not only be paid to optimizing the allocation of crop areas but also focus on improving the water-saving levels which to make full use of the limited water resources.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The total irrigation area of different water-saving modes under different violation probabilities, P

_{i}.

**Figure 6.**Net economic benefit of different water-saving modes under different violation probabilities, P

_{i}.

**Figure 7.**Optimal allocation area distribution for 11 crops (λi = 30%, P

_{i}= 0.1) and the maximum and minimum crop area constraints.

**Figure 8.**Irrigation water resources of different water-saving modes under different violation probabilities, P

_{i.}

**Figure 9.**Irrigation water resource consumption and total crop area of different water-saving modes at violation probability P

_{i}= 0.1.

Crops | Yield (kg/ha) | Irrigation Quota (m^{3}/ha) | ET (m^{3}/ha) | P (¥/ha) | C (¥/ha) | A_{min} (ha) | A_{max} (ha) | |||
---|---|---|---|---|---|---|---|---|---|---|

Mode 1 | Mode 2 | Mode 1 | Mode 2 | Mode 1 | Mode 2 | |||||

Wheat | (5447, 5745) | (5175, 5250) | (5150, 5250) | (4150, 4450) | (5410, 6490) | (4050, 4500) | (1.8, 2) | 7179.5 | 0.9396 | 4.2282 |

Maize | (10,470, 10,695) | (12,750, 12,750) | (5400, 5520) | (3000, 3600) | (5400, 5520) | (3000, 3600) | (1.3, 1.5) | 7788 | 1.8782 | 8.4519 |

Bean | (2482, 2884) | (2482, 2884) | (3950, 4450) | (3950, 4450) | (5715, 5715) | (5715, 5715) | 4 | 5500 | 0.0866 | 0.3897 |

Potato | (27,000, 30,000) | 33,000 | (4370, 4850) | (3500, 3640) | (5200, 5705) | (4213, 4270) | (0.5, 0.6) | 5250 | 0.7768 | 3.4956 |

Maize seed | (14,041, 14,428) | (13,835, 15,418) | (2250, 3250) | (2250, 2250) | (3621, 4248) | (3617, 3500) | 2.2 | 16,075.5 | 0.2386 | 1.0737 |

Cotton | (1800, 1845) | (1980, 2030) | (3950, 4450) | (1950, 3150) | (3440, 4300) | (3600, 4695) | (5, 5.5) | 5235 | 0.1500 | 0.6750 |

Oilseed | (3559, 3674) | (3559, 3674) | (4800, 5250) | (4800, 5250) | (3750, 4695) | (3750, 4695) | (3, 3.4) | 3650 | 0.8018 | 3.6351 |

Vegetable | 11,5500 | 11,3250 | (4500, 5300) | (2680, 3100) | (5400, 6450) | (2200, 2900) | (1, 1.3) | 18,000 | 1.3540 | 6.0930 |

Cucurbit | 60,882 | 63,196 | (4200, 4800) | (2700, 3150) | (3900, 4190) | (3300, 3670) | 1.8 | 9125 | 0.1212 | 0.5454 |

Apple | 11,545 | 11,545 | (4780, 5290) | (4780, 5290) | (5825, 6742) | (5825, 6742) | (2, 2.2) | 4950 | 0.2050 | 0.9225 |

Grape | (13,500, 15,000) | 12,000 | 4130 | 2130 | (4296, 5000) | (3017, 3073) | (3, 3.5) | 8000 | 0.2670 | 1.2015 |

TA (10^{4} ha) | TPR (10^{4} P) | FDP (kg/P) | SW (10^{4} m^{3}) | EW (10^{4} m^{3}) | DW (10^{4} m^{3}) | TW (10^{4} m^{3}) | IC |
---|---|---|---|---|---|---|---|

24.28 | 186.14 | 300 | 21,838.5 | 15,544.8 | 5832.67 | 2261.57 | 0.59 |

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**MDPI and ACS Style**

Ren, C.; Zhang, H.
An Inexact Optimization Model for Crop Area Under Multiple Uncertainties. *Int. J. Environ. Res. Public Health* **2019**, *16*, 2610.
https://doi.org/10.3390/ijerph16142610

**AMA Style**

Ren C, Zhang H.
An Inexact Optimization Model for Crop Area Under Multiple Uncertainties. *International Journal of Environmental Research and Public Health*. 2019; 16(14):2610.
https://doi.org/10.3390/ijerph16142610

**Chicago/Turabian Style**

Ren, Chongfeng, and Hongbo Zhang.
2019. "An Inexact Optimization Model for Crop Area Under Multiple Uncertainties" *International Journal of Environmental Research and Public Health* 16, no. 14: 2610.
https://doi.org/10.3390/ijerph16142610