# A Fuzzy Max–Min Decision Bi-Level Fuzzy Programming Model for Water Resources Optimization Allocation under Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Bi-Level Programming

- (The upper-level)$$\begin{array}{ll}\underset{x\in X,y}{\mathrm{max}}& F(x,y)\\ \mathrm{s}.\mathrm{t}.& G(x,y)\le 0,\end{array}$$
- (The lower-level)$$\begin{array}{ll}\underset{y}{\mathrm{max}}& f(x,y)\\ \mathrm{s}.\mathrm{t}.& g(x,y)\le 0,\end{array}$$

#### 2.2. Fuzzy Set Theory

#### 2.3. Fuzzy Max–Min Decision Bi-Level Programming

#### 2.4. Fuzzy Max–Min Decision Bi-Level Fuzzy Programming (FMDBLFP)

- Upper level$$\begin{array}{l}\mathrm{max}\hspace{1em}F(x)=Ax+B\\ \mathrm{s}.\mathrm{t}.\\ \hspace{1em}\hspace{1em}\hspace{1em}Ex+H\le 0\end{array}$$
- Lower level$$\begin{array}{l}\mathrm{max}\hspace{1em}f(x)=Cx+D\\ \mathrm{s}.\mathrm{t}.\\ \hspace{1em}\hspace{1em}\hspace{1em}Ex+H\le 0\end{array}$$
- Build original FMDBLFP model [Equations (14) and (15)];
- Convert the fuzzy coefficients of [Equations (14) and (15)] into the closed intervals as [Equation (6)] by [Equation (5)];
- Preset the value of $\alpha $ and solve the FMDBLFP model by the solving method [Equations (2) and (3)];
- Change the value of $\alpha $ and repeat steps 2 and 3;
- Get the optimal solution under different $\alpha $ levels.

## 3. Application

#### 3.1. Study Area

#### 3.2. Model Building

- $M{A}_{i}$, ${A}_{i}$: Maximum irrigation areas, irrigation areas of region i (10
^{4}mŭ); (mŭ is equal to 614.4 m^{2}); - $E{P}_{i}$: Population of region i (10
^{4}p); - $L{F}_{i}$: Food demand per capita of region i (t/p);
- ${I}_{i}$: Irrigation water of per unit irrigation areas in region i (m
^{3}/ mŭ); - $S{W}_{i}$, $T{W}_{i}$: Water supply for the secondary and tertiary industry of region i (10
^{4}m^{3}); - $W{D}_{i}$, $W{E}_{i}$: Domestic water and ecological water in region i;
- ${O}_{i}$, ${S}_{i}$, ${T}_{i}$: Per unit of water resources benefit of planting, secondary and tertiary industry in region i (yuan/m
^{3}); - ${B}_{i}$: Yield per unit of region i (t/ mŭ);
- $W$: Water supply for Wuwei City, which is fuzzy sets;
- $A{W}_{i\mathrm{min}}$, $A{W}_{i\mathrm{max}}$: Minimum and maximum water supply for the planting industry of region i (10
^{4}m^{3}); - $S{W}_{i\mathrm{min}}$, $S{W}_{i\mathrm{max}}$: Minimum and maximum water supply for the secondary industry of region i (10
^{4}m^{3}); - $T{W}_{i\mathrm{min}}$, $T{W}_{i\mathrm{max}}$: Minimum and maximum water supply for the tertiary industry of region i (10
^{4}m^{3});

_{i}10

^{4}mŭ), yield per unit area (B

_{i}t/mŭ), population (EP

_{i}10

^{4}P), food demand per capita (LF

_{i}t/p), irrigation water per unit (I

_{i}m

^{3}), per unit of water resources benefit of the planting (O

_{i}yuan/m

^{3}), secondary and tertiary industry (S

_{i}yuan/m

^{3}), domestic water demand (WD

_{i}10

^{4}m

^{3}), and ecological water demand (WE

_{i}10

^{4}m

^{3}) of the four regions. Table 2 represents the maximum and minimum water supply of the planting industry, the secondary industry, and the tertiary industry. In addition, the water supply of Wuwei City has the characteristic of fuzzy uncertainty. Moreover, it can be represented as trapezoidal fuzzy sets, which are presented as [15.49 × 10

^{8}, 16.14 × 10

^{8}, 16.84 × 10

^{8}, 17.97 × 10

^{8}] m

^{3}.

## 4. Results and Analysis

#### 4.1. Solution of the FMDBLFP

^{4}t ($\alpha =0$) to 110.24 × 10

^{4}t ($\alpha =1$). The lower-bound yields would vary from 98.34 × 10

^{4}t ($\alpha =0$) to 104.07 × 10

^{4}t ($\alpha =1$). In this paper, trapezoidal function was selected as fuzzy membership functions, which has the characteristic as the larger $\alpha -\mathrm{cut}$ levels, the larger the possibility of the occurrence of events shown in Figure 1. As $\alpha -\mathrm{cut}$ levels increased, the fuzzification weakens. Therefore, the optimized yields and net economic benefit gap between the upper bound and the lower bound was wide when $\alpha =0$, and narrow when $\alpha =1$. Therefore, different policies might be adopted for water resources allocation and water saving targets under different water resources shortage conditions, thus lead to broad options of varied system objects and system-failure risks.

#### 4.2. Comparison of FMDBLFP with ULDM and LLDM

^{4}t ($\alpha =0$) to 108.32 × 10

^{4}t ($\alpha =1$), while the lower-bound yields of ULDM would vary from 95.32 × 10

^{4}t ($\alpha =0$) to 101.41 × 10

^{4}t ($\alpha =1$). However, there were great differences in the optimized yields and net benefits obtained by the three models. From Figure 9, the optimized yield of ULDM was largest under each $\alpha -\mathrm{cut}$ level. Furthermore, Table 5 shows that the optimized economic benefit of LLDM was largest under each $\alpha -\mathrm{cut}$ level. In addition, regardless of yields or economic benefits, the optimized result of FMDBLFP was always between ULDM and LLDM. For example, the optimized yields of LLDM, FMDBLFP, and ULDM were [114.30 × 10

^{4}t, 126.75 × 10

^{4}t], [101.78 × 10

^{4}t, 114.21 × 10

^{4}t], and [98.85 × 10

^{4}t, 112.76 × 10

^{4}t], respectively, when $\alpha =0.6$. When $\alpha =1$, the optimized economic benefits of ULDM, FMDBLFP, and LLDM were [384.98 × 10

^{8}yuan, 381.60 × 10

^{8}yuan], [384.67 × 10

^{8}yuan, 381.12 × 10

^{8}yuan], and [268.14 × 10

^{8}yuan, 264.80 × 10

^{8}yuan], respectively. Based on the above analysis, the ULDM just satisfied the requirement of upper level but caused great loss to the lower-level, while the LLDM also just satisfied the requirement of lower-level but caused great loss to the upper-level. However, the optimized yields and economic benefits of FMDBLFP were always between ULDM and LLDM, which meant that it took the requirements of upper and lower levels into account. Moreover, it demonstrated that it has the ability to deal with water competing conflict among water competing sectors from different levels.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Wang, S.; Huang, G.H. A multi-level Taguchi-factorial two-stage stochastic programming approach for characterization of parameter uncertainties and their interactions: An application to water resources to water resources management. Eur. J. Oper. Res.
**2015**, 240, 572–581. [Google Scholar] [CrossRef] - García-Garizábal, I.; Causapé, J.; Abrahao, R. Application of the irrigation land environmental evaluation tool for flood irrigation management and evaluation of water use. Catena
**2011**, 87, 260–267. [Google Scholar] [CrossRef] - Dai, Z.Y.; Li, Y.P. A multistage irrigation water allocation model for agricultural land-use planning under uncertainty. Agric. Water Manag.
**2013**, 129, 69–79. [Google Scholar] [CrossRef] - Huang, Y.; Li, Y.P.; Chen, X.; Ma, Y.G. Optimization of the irrigation water resources for agricultural sustainability in Tarim river basin, China. Agric. Water Manag.
**2012**, 107, 74–85. [Google Scholar] [CrossRef] - Salman, A.Z.; AI-Karablieh, E.K.; Fisher, F.M. An inter-seasonal agricultural water allocation system (SAWAS). Agric. Syst.
**2001**, 68, 233–252. [Google Scholar] [CrossRef] - Sethi, L.N.; Panda, S.N.; Nayak, M.K. Optimal crop planning and water resources allocation in a coastal groundwater basin, Orissa, India. Agric. Water Manag.
**2006**, 83, 209–220. [Google Scholar] [CrossRef] - Bravo, M.; Gonzalez, I. Applying stochastic goal programming: A case study on water use planning. Eur. J. Oper. Res.
**2009**, 196, 1123–1129. [Google Scholar] [CrossRef] - Gu, J.J.; Huang, G.H.; Guo, P.; Shen, N. Interval multistage joint-probabilistic integer programming approach for water resources allocation and management. J. Environ. Manag.
**2013**, 128, 615–624. [Google Scholar] [CrossRef] [PubMed] - Ren, C.H.; Guo, P.; Li, M.; Gu, J.J. Optimization of industrial structure considering the uncertainty of water resources. Water Resour. Manag.
**2013**, 27, 3885–3989. [Google Scholar] [CrossRef] - Guo, P.; Chen, X.H.; Li, M.; Li, J.B. Fuzzy chance-constrained linear fractional programming approach for optimal water allocation. Stoch. Environ. Res. Risk Assess.
**2014**, 28, 1601–1612. [Google Scholar] [CrossRef] - Li, M.; Guo, P. A multi-objective optimal allocation model for irrigation water resources under multiple uncertainties. Appl. Math. Model.
**2014**, 38, 4897–4911. [Google Scholar] [CrossRef] - Fan, Y.R.; Huang, G.H.; Huang, K.; Baetz, B.W. Planning water resources allcation under multiple uncertainties through a generalized fuzzy two-stage stochastic programming method. IEEE Trans. Fuzzy Syst.
**2015**, 23, 1488–1504. [Google Scholar] [CrossRef] - Wang, S.; Huang, G.H.; Zhou, Y. A fractional-factorial probabilistic-possibilistic optimization framework for planning water resources management systems with multi-level parametric interactions. J. Environ. Manag.
**2016**, 172, 97–106. [Google Scholar] [CrossRef] [PubMed] - Guo, P.; Huang, G.H.; Zhu, H.; Wang, X.L. A two-stage programming approach for water resources management under randomness and fuzziness. Environ. Model. Softw.
**2010**, 25, 1573–1581. [Google Scholar] [CrossRef] - Gu, J.J.; Guo, P.; Huang, G.H. Inexact stochastic dynamic programming method and application to water resources management in Shandong China under uncertainty. Stoch. Environ. Res. Risk Assess.
**2013**, 27, 1207–1219. [Google Scholar] [CrossRef] - Li, M.; Guo, P.; Ren, C.H. Water resources management models based on two-level linear fractional programming method under uncertainty. J. Water Resour. Plan. Manag.
**2015**, 141, 05015001. [Google Scholar] [CrossRef] - Davijani, M.H.; Banihabib, M.E.; Anvar, A.N.; Hashemi, S.R. Multi-objective optimization model for the allocation of water resources in arid regions based on the maximization of socioeconomic efficiency. Water Resour. Manag.
**2016**, 30, 927–946. [Google Scholar] [CrossRef] - Ren, C.H.; Li, R.H.; Zhang, L.D.; Guo, P. Multiobjective stochastic fractional goal programming model for water resources optimal allocation among industries. J. Water Resour. Plan. Manag.
**2016**, 142, 04016036. [Google Scholar] [CrossRef] - Maher, M.J.; Zhang, X.Y.; Vliet, D.V. A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows. Transp. Res. Part B Methodol.
**2001**, 35, 23–40. [Google Scholar] [CrossRef] - Emam, O.E. A fuzzy approach for bi-level integer non-linear programming problem. Appl. Math. Comput.
**2006**, 172, 62–71. [Google Scholar] [CrossRef] - Li, Y.P.; Huang, G.H.; Nie, S.L. An interval-parameter multi-stage stochastic programming model for water resources management under uncertainty. Adv. Water Resour.
**2006**, 29, 776–789. [Google Scholar] [CrossRef] - Guo, P.; Huang, G.H.; Li, Y.P. Inexact fuzzy-stochastic programming for water resources management under multiple uncertainties. Environ. Model. Assess.
**2010**, 15, 111–124. [Google Scholar] [CrossRef] - Zhu, H.; Haung, G.H.; Guo, P.; Qin, X.S. A fuzzy robust nonlinear programming model for stream water quality management. Water Resour. Manag.
**2009**, 23, 2913–2940. [Google Scholar] [CrossRef] - Yan, X.P.; Ma, X.F.; Huang, G.H.; Wu, C.Z. An inexact transportation planning model for supporting vehicle emissions management. J. Environ. Inf.
**2010**, 15, 97–98. [Google Scholar] [CrossRef] - Colson, B.; Marcotte, P.; Savard, G. Bilevel programming: A survey. 4OR
**2005**, 3, 87–107. [Google Scholar] [CrossRef] - Li, M.; Guo, P.; Fang, S.Q.; Zhang, L.D. An inexact fuzzy parameter two-stage stochastic programming model for irrigation water allocation under uncertainty. Stoch. Environ. Res. Risk Assess.
**2013**, 27, 1441–1452. [Google Scholar] [CrossRef] - Sakawa, M. Fuzzy Sets and Interactive Multi-Objective Optimization; Plenum Press: New York, NY, USA, 1993. [Google Scholar]

**Figure 9.**Yields of the upper-level decision-maker (ULDM), fuzzy max–min decision bi-level programming (FMDBLFP) model, and lower-level decision-maker (LLDM) under different $\alpha -\mathrm{cut}$ levels.

Region | MA_{i} (10^{4} mŭ) | B_{i} (t/ mŭ) | I_{i} (m^{3}) | EP_{i} (10^{4} P) | LF_{i} (t/p) | O_{i} (yuan/m^{3}) | S_{i} (yuan/m^{3}) | T_{i} (yuan/m^{3}) | WD_{i} (10^{4} m^{3}) | WE_{i} (10^{4} m^{3}) |
---|---|---|---|---|---|---|---|---|---|---|

Liangzhou | 167.12 | 0.41 | 465 | 101.43 | 0.3 | 5.06 | 68.86 | 664.50 | 6303.40 | 9288.40 |

Minqin | 72.65 | 0.45 | 456 | 24.16 | 0.3 | 4.84 | 131.75 | 689.53 | 1869.30 | 4357.50 |

Gulang | 91.16 | 0.46 | 425 | 38.95 | 0.3 | 4.82 | 97.21 | 686.07 | 1145.30 | 1603.20 |

Tianzhu | 33.22 | 0.42 | 475 | 17.62 | 0.3 | 5.04 | 111.05 | 710.48 | 1196.20 | 295.70 |

**Table 2.**The maximum and minimum water supply of planting industry, the secondary industry and the tertiary industry.

Region Districts | Planting Industry (10^{4} m^{3}) | Secondary Industry (10^{4} m^{3}) | Tertiary Industry (10^{4} m^{3}) | |||
---|---|---|---|---|---|---|

AW_{i}_{min} | AW_{i}_{max} | SW_{i}_{min} | SW_{i}_{max} | TW_{i}_{min} | TW_{i}_{max} | |

Liangzhou | 31,393.82 | 73,958.20 | 10,951 | 16,425.50 | 921 | 1555.57 |

Minqin | 6845.33 | 33,128.40 | 907 | 1360.50 | 172 | 290.51 |

Gulang | 9449.61 | 38,743.00 | 1253 | 1879.50 | 122 | 206.06 |

Tianzhu | 5348.93 | 15,779.50 | 1448 | 2172.00 | 124 | 209.44 |

**Table 3.**The optimized irrigation areas of different irrigation districts under different $\alpha -\mathrm{cut}$ levels.

Region Districts | A_{i} (10^{4} mŭ) | |||||

$\alpha =0$ | $\alpha =0.2$ | $\alpha =0.4$ | $\alpha =0.6$ | $\alpha =0.8$ | $\alpha =1$ | |

Liangzhou | [105.75, 159.05] | [108.55, 154.14] | [111.34, 149.41] | [114.14, 144.46] | [116.93, 139.73] | [119.73, 134.78] |

Minqin | [60.65, 60.65] | [60.65, 60.65] | [60.65, 60.65] | [60.65, 60.65] | [60.65, 60.65] | [60.65, 60.65] |

Gulang | [35.77, 35.77] | [35.77, 35.77] | [35.77, 35.77] | [35.77, 35.77] | [35.77, 35.77] | [35.77, 35.77] |

Tianzhu | [26.75, 26.55] | [26.75, 26.55] | [26.75, 26.55] | [26.75, 26.55] | [26.75, 26.55] | [26.75, 26.55] |

**Table 4.**The optimization allocation of water resources for SW

_{i}and TW

_{i}under different $\alpha -\mathrm{cut}$ levels.

Region Districts | $\alpha \text{}\mathrm{from}\text{}0\text{}\mathrm{to}\text{}1$ | |

SW_{i} (10^{4} m^{3}) | TW_{i} (10^{4} m^{3}) | |

Liangzhou | 16426.50 | 1555.57 |

Minqin | 1360.50 | 290.51 |

Gulang | 1879.50 | 206.06 |

Tianzhu | 2172.00 | 209.44 |

α-Cut Levels | UBUB (10^{8} yuan) | UBLB (10^{8} yuan) | FBUB (10^{8} yuan) | FBLB (10^{8} yuan) | LBUB (10^{8} yuan) | LBLB (10^{8} yuan) |
---|---|---|---|---|---|---|

α = 0 | 390.45 | 378.39 | 390.38 | 377.83 | 273.85 | 261.33 |

α = 0.2 | 389.34 | 379.05 | 389.22 | 378.49 | 272.69 | 261.98 |

α = 0.4 | 388.27 | 379.70 | 388.11 | 379.15 | 271.58 | 262.64 |

α = 0.6 | 387.16 | 380.34 | 386.94 | 379.81 | 270.41 | 263.29 |

α = 0.8 | 386.10 | 380.97 | 385.83 | 380.47 | 269.30 | 263.95 |

α = 1 | 384.98 | 381.60 | 384.67 | 381.12 | 268.14 | 264.60 |

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**MDPI and ACS Style**

Ren, C.; Zhang, H.
A Fuzzy Max–Min Decision Bi-Level Fuzzy Programming Model for Water Resources Optimization Allocation under Uncertainty. *Water* **2018**, *10*, 488.
https://doi.org/10.3390/w10040488

**AMA Style**

Ren C, Zhang H.
A Fuzzy Max–Min Decision Bi-Level Fuzzy Programming Model for Water Resources Optimization Allocation under Uncertainty. *Water*. 2018; 10(4):488.
https://doi.org/10.3390/w10040488

**Chicago/Turabian Style**

Ren, Chongfeng, and Hongbo Zhang.
2018. "A Fuzzy Max–Min Decision Bi-Level Fuzzy Programming Model for Water Resources Optimization Allocation under Uncertainty" *Water* 10, no. 4: 488.
https://doi.org/10.3390/w10040488