# Inexact Mathematical Modeling for the Identification of Water Trading Policy under Uncertainty

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{h})x + W(ω

_{h}) = g(ω

_{h}), ∀h = 1, 2, …, H

_{h}) ≥ 0

^{T}x is the first-stage benefits, ω is the random events after the first-stage decisions are made, p

_{h}is the probability of an event, ω

_{h}, Q(y, ω

_{h}) is the recourse at the second-stage under the occurrence of the event, ω

_{h}, and is the expected value of the second-stage penalties [12]. However, the parameter of a model may fluctuate within a certain interval, and it is difficult to state a meaningful probability distribution for this variation. Interval-parameter programming (IPP) can deal with uncertainties in objective function and system constraints, which can be expressed as intervals without distribution information. An interval number, x

^{±}, can be defined as an interval with a known lower-bound and upper-bound, but unknown distribution information [34,35]. It can be expressed as [x

^{−}, x

^{+}], representing a number (or an interval), which can have a minimum value of x

^{−}and a maximum one of x

^{+}.

^{+}= [x

^{−}, x

^{+}] = {a ∊ x ǀ x

^{−}≤ a ≤ x

^{+ }}

^{−}and x

^{+}are the lower and upper bounds of x

^{±}, respectively. When x

^{−}= x

^{+}, x

^{+}becomes a deterministic number. When uncertainties presented as probabilities and intervals exist in water resource management systems, based on IPP and TSP techniques, an inexact TSP (ITSP) model can be formulated as follows [17]:

^{±}x

^{±}≤ b

^{±}

^{± }≥ 0

^{T }X

^{±}≥ 0

^{+}, which corresponds to the upper bound of the objective function value, it can be formulated as follows:

^{+}

_{j}(j = 1,2,…,q

_{1}) > 0; c

^{+}

_{j}(j = q

_{1}+ 1, q

_{1}+ 2,…,n) < 0; d

^{+}

_{k}(j = 1,2,…,q

_{2}) > 0; d

^{+}

_{k}(j = q

_{2}+ 1, q

_{2}+ 2,…,m) < 0; when , when and λ

^{-}= the lower bound of the credibility level value.

_{1,}for j = q

_{1}+ 1 to n, for k = 1 to q

_{2}and for k = q

_{2}+ 1 to m, can be obtained. Accordingly, the second submodel corresponding to the lower bound of the objective function value can be formulated as:

_{1}, for j = q

_{1}+ 1 to n, for k = 1 to q

_{2}and for k = q

_{2}+ 1 to m. Therefore, by integrating the solutions of the two submodels, the solution of the TICP model can be generated.

- Step 1: Formulate the TICP model.
- Step 2: Transform the TICP model into two submodels, where the submodel corresponding to ƒ
^{+}is desired first, since the objective is to maximize ƒ^{±}. - Step 3: Obtain the optimal solutions by solving the ƒ
^{+ }submodel under each λ. - Step 4: Formulate and solve the ƒ
^{-}submodel by importing optimal solutions from the ƒ^{+}submodel into the ƒ^{-}submodel under each λ. - Step 5: Obtain the optimal solution interval value under each λ.

## 3. Case Study

^{3}km

^{2}[36] (as shown in Figure 1). It is a typical arid region, due to an extremely dry climate and a low and uneven distribution of rainfall. For example, the climate in the basin is extremely dry, with the average rainfall being about 273 mm/year, which is more than 80% of the total annual precipitation fall from May to September and less than 20% of the total fall from November to the following April [37]. The basin includes six counties (i.e., Kuerle, Yanqi, Hejing, Heshuo, Bohu and Yuli) and has a population of more than one million [36]. It is suitable for the growth of crops, such as wheat, corn, sugar beet, tomato and fruit, which have provided high-speed growth in agricultural product processing and manufacturing. Moreover, the rich mineral and oil resources of the basin form an industrial structure dominated by mining, the chemical industry and the fossil oil industry, while textiles, electric power, papermaking and transportation are keeping pace with the development of the mainstay industries. The water demands of four users (e.g., municipal, agricultural, industrial and ecological) in six districts rely on the river’s streamflow, which is mainly derived from upstream flow, snow melt and rainfall. Due to the dry climate, low-rainfall and high evaporation, the water supply capacity of the river is quite low, which presents difficulties in satisfying the water demands from the six counties. Particularly in recent years, the demand for water has reached the limits of what the natural system can provide, so that water shortage could become a major obstacle to social and economic development for this region. Unfortunately, in the study of the basin, there is a lack of effective tools for facilitating efficient, equitable and sustainable water resource management. Therefore, population growth, the food security challenge, industrial sector development and the potential threat of climate change elevate the attention given to efficient and sustainable water management [38].

- (1)
- Constraints of water permit:
- (2)
- Constraints of water shortage:
- (3)
- Constraints of water surplus:
- (4)
- Constraints of water trading:
- (5)
- Constraints of technical:

^{3}m

^{3}), is the water demand target for user i in district j (10

^{6}m

^{3}), is the total water availability of the entire system under probability P

_{h }(10

^{6}m

^{3}), P

_{h }denotes the probability of random water availability under level h (%), is the economic loss to user i in district j per volume of water not being delivered (US$/10

^{3}m

^{3}), is the water deficiency for user i in district j when demand is not met (10

^{6}m

^{3}), is the allocated allowable water permit to user i in district j (10

^{6}m

^{3}), λ is the credibility level, which measures the degrees of satisfaction of the constraints, d is the percentage of the reduced total allowable water allocation, is the water released to user i in district j when the total water availability exceeds the allowable water reallocation with trading scheme (10

^{6}m

^{3}), is the trading fixed cost to user i in district j with trading scheme (US$/10

^{3}m

^{3}), is the variable trading cost to user i in district j with trading scheme (US$/10

^{3}m

^{3}), is the amount of water trading from other water sources to user i in district j with trading scheme (10

^{6}m

^{3}) and t

^{ʹ}

_{ijh}/ t

_{ijh}is the ratio of water trading from other water sources to user i in district j with the trading scheme.

^{3}m

^{3}in 2008, which was estimated by the gross amount of crops (i.e., wheat = 45.1 × 10

^{6}US$; oil plant = 11.3 × 10

^{6}US$; tomato = 15.93 × 10

^{6}US$; cotton t = 44.02 × 10

^{6}US$ and other crops = 47.33 × 10

^{6}US$) and the water demand (i.e., wheat = 24.1 × 10

^{6}m

^{3}; oil plant = 2.91 × 10

^{6}m

^{3}; tomato = 3.41 × 10

^{6}m

^{3}; cotton = 29.4 × 10

^{6}m

^{3}and other crops = 28.18 × 10

^{6}m

^{3}). Meanwhile, the lowest net benefit was 1530 US$/10

^{3}m

^{3}in 2005. Therefore, the interval value of the net benefit is acquired as [1530, 1860] US$/10

^{3}m

^{3}, which is on the basis of the highest and lowest value of the net benefit. The method calculated for the net benefit of the agricultural user in Hejing County can be applied to other users, districts and even other economic data. is a basic form of trading cost, which is estimated by the actual price of water exceed water permit in Kaidu-kongque Basin. is estimated according to the opportunity cost of water, which is affected by a number of factors, such as the scarcity of water resources, the relationship between supply and demand and the status of socio-economic development. Table 2 shows policy data , which are acquired from the water permits of the water authority of Uygur Autonomous Region from 2005 to 2010 directly. Additionally, water target is estimated by the users’ actual water usage in recent years, which takes the situation of economic development into consideration. The value of should be derived by conducting statistical analyses with the results of the annual streamflow of the Kaidu-kongque River (2005–2010). Due to the rainy seasons in the Kaidu-kongque River Basin, more than 80% of the total annual precipitation falls from May to September, and less than 20% of the total falls from November to the following April [37]. Therefore, the total water availability can be converted to several levels. Table 3 shows the total water availability of the Kaidu-kongque River Basin under several level probabilities.

^{3}m

^{3}in 2008, which was estimated by the gross amount of crops (i.e., wheat = 45.1 × 10

^{6}US$; oil plant = 11.3 × 10

^{6}US$; tomato = 15.93 × 10

^{6}US$; cotton t = 44.02 × 10

^{6}US$ and other crops = 47.33 × 10

^{6}US$) and the water demand (i.e., wheat = 24.1 × 10

^{6}m

^{3}; oil plant = 2.91 × 10

^{6}m

^{3}; tomato = 3.41 × 10

^{6}m

^{3}; cotton = 29.4 × 10

^{6}m

^{3}and other crops = 28.18 × 10

^{6}m

^{3}). Meanwhile, the lowest net benefit was 1530 US$/10

^{3}m

^{3}in 2005. Therefore, the interval value of the net benefit is acquired as [1530, 1860] US$/10

^{3}m

^{3}, which is on the basis of the highest and lowest value of the net benefit. The method calculated for the net benefit of the agricultural user in Hejing County can be applied to other users, districts and even other economic data. is a basic form of trading cost, which is estimated by the actual price of water exceed water permit in Kaidu-kongque Basin. is estimated according to the opportunity cost of water, which is affected by a number of factors, such as the scarcity of water resources, the relationship between supply and demand and the status of socio-economic development. Table 2 shows policy data , which are acquired from the water permits of the water authority of Uygur Autonomous Region from 2005 to 2010 directly. Additionally, water target is estimated by the users’ actual water usage in recent years, which takes the situation of economic development into consideration. The value of should be derived by conducting statistical analyses with the results of the annual streamflow of the Kaidu-kongque River (2005–2010). Due to the rainy seasons in the Kaidu-kongque River Basin, more than 80% of the total annual precipitation falls from May to September, and less than 20% of the total falls from November to the following April [37]. Therefore, the total water availability can be converted to several levels. Table 3 shows the total water availability of the Kaidu-kongque River Basin under several level probabilities.

District | User | |||
---|---|---|---|---|

i = 1 | i = 2 | i = 3 | i = 4 | |

Municipality | Agriculture | Industry | Ecology | |

Net benefits (unit: US$/10^{3 }m^{3}) | ||||

j = 1 Kuerle county | [6030, 6670] | [2320, 2520] | [4530, 4670] | [1960, 2120] |

j = 2 Yanqi county | [5500, 6040] | [1420, 1560] | [2600, 2930] | [1680, 1930] |

j = 3 Hejing county | [4670, 4800] | [1530, 1860] | [3730, 3810] | [1540, 1780] |

j = 4 Heshuo county | [5300, 5530] | [2010, 2340] | [3440, 3620] | [1660, 1940] |

j = 5 Bohu county | [4910, 5100] | [1780, 2010] | [3620, 3740] | [1530, 1840] |

j = 6 Yuli county | [4600, 5260] | [2230, 2460] | [3220, 3440] | [1690, 1990] |

Penalties (unit: US$/10^{3 }m^{3}) | ||||

j = 1 Kuerle county | [9045, 10005] | [3480, 3765] | [6795, 7005] | [2940, 3180] |

j = 2 Yanqi county | [8250, 9060] | [2130, 2340] | [3900, 4395] | [2520, 2895] |

j = 3 Hejing county | [7005, 7200] | [2295, 2790] | [5595, 5725] | [2310, 2670] |

j = 4 Heshuo county | [7950, 8295] | [3015, 3510] | [5160, 5430] | [2190, 2490] |

j = 5 Bohu county | [7365, 7650] | [2670, 3015] | [5430, 5610] | [2295, 2760] |

j = 6 Yuli county | [6900, 7890] | [3345, 3690] | [4830, 5160] | [2535, 2985] |

Trading fix cost (unit: US$/10^{3 }m^{3}) | ||||

j = 1 to 6 | [3050, 3150] | [550, 650] | [2400, 2600] | [280, 350] |

Trading variable cost (unit: US$/10^{3 }m^{3}) | ||||

j = 1 to 6 | [1200, 1350] | [700, 800] | [150, 200] | [100, 150] |

District | User | |||
---|---|---|---|---|

i = 1 | i = 2 | i = 3 | i = 4 | |

Municipality | Agriculture | Industry | Ecology | |

Water target (unit: 10^{6} m^{3}) | ||||

j = 1 Kuerle County | [8.80, 14.00] | [258.00, 275.00] | [53.30, 620.00] | [56.00, 76.00] |

j = 2 Yanqi County | [6.00, 8.20] | [158.00, 165.00] | [28.00, 39.00] | [31.00, 47.00] |

j = 3 Hejing County | [2.40, 4.30] | [81.00, 88.00] | [16.00, 20.10] | [14.70, 23.00] |

j = 4 Heshuo County | [0.24, 0.50] | [9.70, 10.10] | [1.78, 2.25] | [1.28, 2.60] |

j = 5 Bohu County | [2.20, 4.30] | [75.00, 85.00] | [15.60, 19.00] | [13.70, 23.00] |

j = 6 Yuli County | [4.60, 6.00] | [110.00, 120.00] | [21.60, 27.00] | [24.00, 33.00] |

Allocated allowable water permit (unit: 10^{6} m^{3}) | ||||

j = 1 Kuerle County | [9.72, 13.75] | [261.60, 275.08] | [55.44, 61.89] | [59.56, 75.65] |

j = 2 Yanqi County | [[6.64, 8.14] | [158.33, 162.87] | [30.31, 36.65] | [32.26, 44.79] |

j = 3 Hejing County | [2.89, 4.23] | [82.22, 84.50] | [15.70, 19.01] | [17.08, 23.24] |

j = 4 Heshuo County | [0.36, 0.53] | [9.11, 9.38] | [1.89, 2.11] | [1.69, 2.70] |

j = 5 Bohu County | [2.56, 4.09] | [78.89, 81.84] | [15.78, 18.41] | [15.58, 22.58] |

j = 6 Yuli County | [4.94, 5.87] | [110.56, 117.41] | [22.33, 26.42] | [25.56, 32.29] |

District | Level | Probability | User | |||
---|---|---|---|---|---|---|

i = 1 | i = 2 | i = 3 | i = 4 | |||

Municipality | Agriculture | Industry | Ecology | |||

Total water availability (unit: 10^{6} m^{3}) | ||||||

j = 1 Kuerle County | h = 1 (low) | 0.6 | [6.57, 10.67] | [202.32, 216.00] | [43.47, 49.96] | [42.30, 58.50] |

h = 2 (medium) | 0.3 | [7.30, 11.85] | [224.80, 240.00] | [48.30, 55.51] | [47.00, 65.00] | |

h = 3 (high) | 0.1 | [8.03, 13.04] | [247.28, 264.00] | [53.13, 61.06] | [51.70, 71.50] | |

j = 2 Yanqi County | h = 1 (low) | 0.6 | [4.77, 6.38] | [124.02, 128.70] | [22.86, 29.41] | [22.95, 35.55] |

h = 2 (medium) | 0.3 | [5.30, 7.09] | [137.80, 143.00] | [25.40, 32.68] | [25.50, 39.50] | |

h = 3 (high) | 0.1 | [5.83, 7.80] | [151.58, 157.30] | [27.94, 35.95] | [28.05, 43.45] | |

j = 3 Hejing County | h = 1 (low) | 0.6 | [1.89, 3.24] | [66.11, 68.40] | [12.96, 16.29] | [11.43, 17.82] |

h = 2 (medium) | 0.3 | [2.10, 3.60] | [73.45, 76.00] | [14.40, 18.10] | [12.70, 19.80] | |

h = 3 (high) | 0.1 | [2.31, 3.96] | [80.80, 83.60] | [15.84, 19.91] | [13.97, 21.78] | |

j = 4 Heshuo County | h = 1 (low) | 0.6 | [0.19, 0.38] | [7.90, 8.19] | [1.40, 1.62] | [0.88, 1.90] |

h = 2 (medium) | 0.3 | [0.21, 0.42] | [8.78, 9.10] | [1.55, 1.80] | [0.98, 2.11] | |

h = 3 (high) | 0.1 | [0.23, 0.46] | [9.66, 10.01] | [1.71, 1.98] | [1.08, 2.32] | |

j = 5 Bohu County | h = 1 (low) | 0.6 | [1.79, 3.33] | [58.50, 65.70] | [12.69, 15.38] | [11.16, 18.20] |

h = 2 (medium) | 0.3 | [1.99, 3.70] | [65.00, 73.00] | [14.10, 17.09] | [12.40, 20.23] | |

h = 3 (high) | 0.1 | [2.19, 4.07] | [71.50, 80.30] | [15.51, 18.80] | [13.64, 22.25] | |

j = 6 Yuli County | h = 1 (low) | 0.6 | [3.69, 4.75] | [86.85, 93.82] | [17.64, 21.79] | [19.35, 26.10] |

h = 2 (medium) | 0.3 | [4.10, 5.28] | [96.50, 104.25] | [19.60, 24.21] | [21.50, 29.00] | |

h = 3 (high) | 0.1 | [4.51, 5.81] | [106.15, 114.67] | [21.56, 26.63] | [23.65, 31.90] |

## 4. Result Analysis

#### 4.1. Results under Different Credibility Levels

^{6}m

^{3}, 97.75 × 10

^{6}m

^{3}, 21.85 × 10

^{6}m

^{3}and 28.75 ×10

^{6}m

^{3}, respectively. When inflow is high, shortages would be [0.19, 1.28] × 10

^{6}m

^{3}, [3.82, 12.11] × 10

^{6}m

^{3}, [2.55, 2.55] × 10

^{6}m

^{3}and [6.59, 7.06] × 10

^{6}m

^{3}; correspondingly, the actual allocations would be [3.66, 4.75] × 10

^{6}m

^{3}, [85.65, 93.93] × 10

^{6}m

^{3}, [19.30, 19.30] × 10

^{6}m

^{3}and [18.81, 19.29] × 10

^{6}m

^{3}. The total amount of allocated water to Bohu County would be from 127.42 × 10

^{6}m

^{3}to 137.27 × 10

^{6}m

^{3}; however, its total water demand would be 153.3 × 10

^{6}m

^{3}, indicating that there would be a shortage, even though the inflow is high. Due to more than 80% of the total annual precipitation falls from May to September in the study region, falls in other months were much less. Therefore, when inflows are medium or low, the shortage would be strengthened, whereas each user would have to obtain water from other sources to satisfy its essential demands.

^{6}m

^{3}. When inflow is high, shortages would be [0.19, 1.28] × 10

^{6}m

^{3}; correspondingly, the actual allocations would be [3.66, 4.75] × 10

^{6}m

^{3}. Meanwhile, when λ = 0.9, a lower water resource permit and water availability would correspond to the higher credibility satisfaction levels. The optimized municipal target for Bohu County (j = 5) was 4.95 × 10

^{6}m

^{3}. When inflow is high, shortages would be [0.21, 1.32] × 10

^{6}m

^{3}; correspondingly, the actual allocations would be [1.32, 4.75] × 10

^{6}m

^{3}. This implies that different λ levels lead to different credibility satisfaction levels and violation risks in the water planning system, corresponding to different water availabilities, which lead to different shortages and allocations.

^{6}m

^{3}. When inflow is high, shortages would be [0.19, 1.28] × 10

^{6}m

^{3}; correspondingly, the actual allocations would be [3.66, 4.75] × 10

^{6}m

^{3}. Meanwhile λ = 0.9, lower water resource permit and water availability would correspond to the higher credibility satisfaction levels. Optimized targets of municipal in Bohu County (j = 5) was 4.95 × 10

^{6}m

^{3}. When inflow is high, shortages would be [0.21, 1.32] × 10

^{6}m

^{3}; correspondingly, the actual allocations would be [1.32, 4.75] × 10

^{6}m

^{3}. It replied that different λ levels led to different credibility satisfactions and violation risks in water planning system, corresponding to different water availabilities, which lead to different shortages, and allocations.

^{6}m

^{3}at the low level, [0.62, 5.86] × 10

^{6}m

^{3}at the medium level and [0.66, 4.99] × 10

^{6}m

^{3}at the high level under Case 1 (λ = 0.6), while it would be [1.31, 6.39] × 10

^{6}m

^{3}at the low level, [0.62, 5.96] × 10

^{6}m

^{3}at the medium level and [1.54, 5.11] × 10

^{6}m

^{3}at the high level under Case 2 (λ = 0.6). Moreover, shortages are affected by the randomness of water availabilities. Due to a special climate situation, when the flow is high in the wet season, the shortages may be relatively low under advantageous conditions and would be increased when the flow is low in the dry season.

^{6}m

^{3}at the low level, [5.35, 6.64] × 10

^{6}m

^{3}at the medium level and [5.82, 6.22] × 10

^{6}m

^{3}at the high level under case 1 (λ = 0.6), while it would be [5.12, 6.20] × 10

^{6}m

^{3}at the low level, [5.35, 6.50] × 10

^{6}m

^{3}at the medium level and [5.82, 5.91] × 10

^{6}m

^{3}at the high level under Case 2 (λ = 0.6). The results of water allocations under the two cases indicated that different water allocations would be achieved, due to different shortages, based on different policies. It is implied that water allocations were sensitive to water permitting, since different water permits led to different shortages, resulting in different water allocations. Figure 5 presents the amount of water trading under different water permit and trading ratios. In the study region, due to the extremely dry climate, low rainfall and high evaporation rate, the losses from a water shortage are serious. Thus, trading was introduced to reduce the losses from, shortages and to obtain greater benefits. When the losses from water shortages are generated, each user would need to obtain water from released water and other sources to satisfy its essential demands. For example, the amount of water trading for agricultural users in Heshuo County (j = 4) would be [1.48, 1.62] × 10

^{6}m

^{3}at the low level, [0.70, 1.16] × 10

^{6}m

^{3}at the medium level and [0, 1.36] × 10

^{6}m

^{3}at the high level under Case 1 (λ = 0.9), while it would be [1.05, 1.46] × 10

^{6}m

^{3}at the low level, [0.25, 0.65] × 10

^{6}m

^{3}at the medium level and [0, 0] × 10

^{6}m

^{3}at the high level under Case 2 (λ = 0.9). The solutions indicated that the amount of trading based on shortages was relatively low under advantageous conditions and rose in the dry season. Meanwhile, by decreasing the water permit and trading ratios, the more the water surplus remedies the water shortage, the less is the amount of water trading from other sources. Moreover, the amount of water trading is sensitive to trading cost, particular in assuming different water polices. Under the situation of water permit and trading ratio change, the amount of water trading changes dissimilarly, due to the law of value. More released water permits lead to a greater difference in the amount of water trading, while a smaller trading ratio obtains a tremendous change in water trading. For example, in Figure 5, the amount of water trading for agricultural users in Yanqi County (j = 2) would be [7.54, 7.54] × 10

^{6}m

^{3}at the low level under Case 1 (λ = 0.9), while it would be [0, 6.65] × 10

^{6}m

^{3}at the low level under Case 2 (λ = 0.9). The difference of water trading between Case 1 and Case 2 was caused by the lowest trading cost and limited trading resources.

#### 4.2. Comparison of Trading and Non-Trading Schemes

^{9}US$ (λ = 0.6) and [0.68, 2.20] × 10

^{9}US$ (λ = 0.9) under Case 1. By decreasing the water permit, the benefits for the system would be [0.51, 1.89] × 10

^{9}US$ (λ = 0.6) and [0.47, 1.82] × 10

^{9}US$ (λ = 0.9) under Case 2, which indicates that the net benefit for the system would decrease by decreasing the water permit and trading ratio. However, the net benefit for the system under the two cases with the trading scheme was much higher than that with the non-trading scheme (i.e., [0.47, 1.81] × 10

^{9}US$ when λ = 0.6 and [0.31, 1.77] × 10

^{9}US$ when λ = 0.9). Comparing the net benefit for the system under trading and that under non-trading, the efficiency of trading and non-trading would be acquired. This implies that trading through water markets is likely to increase and improve economic efficiency overall.

**Figure 6.**Benefits for the system under Case 1 and Case 2 (λ = 0.6 to 0.9) with the trading scheme and non-trading scheme.

^{6}m

^{3}at the low level, [1.78, 5.74] × 10

^{6}m

^{3}at the medium level and [2.73, 6.50] × 10

^{6}m

^{3}at the high level under Case 1 (λ = 0.6), while with the non-trading scheme, they would be [1.96, 6.55] × 10

^{6}m

^{3}at the low level, [1.78, 8.05] × 10

^{6}m

^{3}at the medium level and [2.73, 8.81] × 10

^{6}m

^{3}at the high level. This implies that markets can provide incentives for adopting water saving practices, since market prices make the opportunity cost of water explicit to users. Therefore, water trading was considered as an effective way for not only reducing the shortages of water systems, but also for gaining a higher net benefit for the system in an arid region.

**Figure 7.**Solutions shortages under Case 1 (λ = 0.6) with the trading scheme and non-trading scheme.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Zeng, X.; Li, Y.; Huang, G.; Yu, L.
Inexact Mathematical Modeling for the Identification of Water Trading Policy under Uncertainty. *Water* **2014**, *6*, 229-252.
https://doi.org/10.3390/w6020229

**AMA Style**

Zeng X, Li Y, Huang G, Yu L.
Inexact Mathematical Modeling for the Identification of Water Trading Policy under Uncertainty. *Water*. 2014; 6(2):229-252.
https://doi.org/10.3390/w6020229

**Chicago/Turabian Style**

Zeng, Xueting, Yongping Li, Guohe Huang, and Liyang Yu.
2014. "Inexact Mathematical Modeling for the Identification of Water Trading Policy under Uncertainty" *Water* 6, no. 2: 229-252.
https://doi.org/10.3390/w6020229