An interval multi-objective programming model for irrigation water allocation under uncertainty

Agricultural Water Management 196 (2018) 24–36

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Agricultural Water Management journal homepage: www.elsevier.com/locate/agwat

Research paper

An interval multi-objective programming model for irrigation water allocation under uncertainty Mo Li a , Qiang Fu a,b,∗ , Vijay P. Singh c , Dong Liu a,b a

School of Water Conservancy & Civil Engineering, Northeast Agricultural University, Harbin, Heilongjiang 150030, China Key Laboratory of Efficient Utilization of Agricultural Water Resources of Ministry of Agriculture, Northeast Agricultural University, Harbin, Heilongjiang 150030, China c Department of Biological and Agricultural Engineering & Zachry Department of Civil Engineering, Texas A & M University, 201 Scoates Hall, 2117 TAMU, College Station, TX 77843-2117, USA b

a r t i c l e

i n f o

Article history: Received 28 July 2017 Received in revised form 16 October 2017 Accepted 16 October 2017 Available online 23 October 2017 Keywords: Irrigation water allocation Multi-objective programming Interval number Bootstrap Scenario analysis

a b s t r a c t An interval linear multi-objective programming (ILMP) model for irrigation water allocation was developed, considering conflicting objectives and uncertainties. Based on the generation of interval numbers through statistical simulation, the ILMP model was solved using a fuzzy programming method. The model balances contradictions among economic net benefit, crop yield and water-saving in irrigation systems incorporating uncertainties in both objective functions and constraints that are based on the conjunctive use of surface water and groundwater. The model was applied to Hulan River irrigation district, northeast China. Tradeoffs between various crops in different subareas under different frequencies were analyzed, and scenarios with different objectives were considered to evaluate the changing trend of irrigation water allocation.Results indicated that the ILMP model provided effective linkages between revenue/output promotion and water saving, and offers insights into tradeoffs for irrigation water management under uncertainty. © 2017 Elsevier B.V. All rights reserved.

1. Introduction These days water crisis is occurring too frequently and at too many places, underscoring the importance of sustainable and efficient water resources management. Among all the water users, agriculture is the dominant user and irrigated agriculture consumes more than 70% of available water resources in the world (GalánMartín et al., 2017). Due to rapid socio-economic development and continuing population growth, future irrigated agriculture will face challenges to meet the growing food demand, while the water available for agriculture will simultaneously be decreasing (Wang et al., 2017). Therefore, optimal allocation of water availability for agricultural irrigation in an efficient manner is a critical issue for agricultural water management. Optimal allocation of agricultural irrigation water can be determined using optimization techniques. A number of optimization techniques, such as linear programming, dynamic programming, nonlinear programming, and stochastic programming, have been

∗ Corresponding author at: School of Water Conservancy & Civil Engineering, Northeast Agricultural University, Harbin, Heilongjiang, 150030, China. E-mail addresses: [email protected], [email protected] (Q. Fu). https://doi.org/10.1016/j.agwat.2017.10.016 0378-3774/© 2017 Elsevier B.V. All rights reserved.

employed to drive optimal irrigation patterns subject to the maximization or minimization of certain objectives (Singh, 2012). Among these optimization techniques, linear programming for irrigation water allocation has been most popular (Das et al., 2015). However, irrigation water allocation systems depend on various independent aspects, such as economic, social and natural, that often conflict with each other, and are therefore too complex for linear programming. Such a water allocation problem can be handled by multi-objective programming (MOP) that is capable of incorporating multiple conflicting objectives functions, such as maximizing net return/crop yield versus reducing water consumption. Some investigators have used MOP for allocating irrigation water with a different emphasis. For example, Su et al. (2014) developed a multi-objective optimal allocation model for agricultural water resources considering three objectives of the maximum net benefit from agriculture, the minimum fairness difference in the utilization of water, and the maximum proportion of green water utilization. Galán-Martín et al. (2017) formulated a multi-objective linear programming model that simultaneously accounts for the maximization of crop production and the minimization of environmental impact caused by water consumption. However, optimal allocation of irrigation water in real world is complicated by various uncertainties that arise in the interac-

M. Li et al. / Agricultural Water Management 196 (2018) 24–36

tions among many system components. Examples include temporal and spatial variations of hydrological elements, fluctuations of economic parameters, and errors in estimating other related parameters. Therefore, it is essential to optimize irrigation water allocation in the framework of MOP considering uncertainties in order to better represent the complexity of irrigation systems. Generally, the widely used uncertainty methods are stochastic mathematical programming (SMP), fuzzy mathematical programming (FMP), and interval mathematical programming (IMP). SMP is an optimization model wherein parameters in the objective functions or constraints can be represented by probability distributions. FMP addresses vagueness in decision maker’ aspirations (or preference) and ambiguity in knowledge or information in an optimization model. IMP deals with uncertainties that are approximated by only the lower and upper boundaries. These uncertainty methods can handle different types of uncertainties, but some deficiencies also exist. For example, although SMP is capable for adequately tackling uncertainties but the high computations data requirement for specifying probability distributions may affect its practical application. For the FMP, there exists subjectivity in the membership function determination and results generation. Recently, some investigators focused on fuzzy uncertainties in the MOP model for irrigation water allocation (Regulwar and Gurav, 2011; Mirajkar and Patel, 2013; Li and Guo, 2014; Morankar et al., 2016; Li et al., 2016a,b). In practice, specification of fuzzy sets or probability distributions is more difficult than obtaining interval numbers, especially in the absence of data. For an irrigation system, many basic data are obtained by field experiments and monitoring, resulting in the time series of these primary data are fairly short. Therefore, considering the availability of data and computational efficiency, the use of interval numbers is particularly appealing for an irrigation system compared with SMP and FMP although the IMP may encounter difficulties in tackling higher uncertain parameters. Thus far, few investigations have considered interval uncertainties in MOP for irrigation water allocation. Because of the capability to consider uncertainties with known lower and upper bounds in both objective functions and constraints, IMP has been successfully integrated in several single objective programming models for irrigation water allocation (Lu et al., 2011; Li et al., 2014; Guo et al., 2014; Yang et al., 2015, 2016). However, few studies provide details on methods for acquisition of interval numbers in the optimization of irrigation water allocation. Hydrological elements, such as runoff, precipitation, and evapotranspiration, directly affect optimal irrigation water allocation. These elements spatially and temporally change and can essentially be regarded as random parameters. Considerable differences occur in the quantities of these hydrological elements for different frequencies. Determining the fluctuation values of these elements for each frequency in advance can not only benefit irrigation optimization but also help ameliorate natural disasters, such as flooding and drought. Therefore, generation of interval numbers of hydrological elements for different frequencies that are incorporated in the MOP optimization model can be valuable not only for avoiding sophisticated calculations but also for precision irrigation. The primary objective of this study therefore is to develop an interval linear multi-objective programming (ILMP) model for optimal irrigation water allocation under uncertainty. The model incorporates the optimization technique of interval parameters into linear multi-objective programming to handle uncertainties of irrigation systems and achieves a balance among net revenue, output, and water-saving by optimally allocating available water. The study entails several elements. First, interval numbers for socialeconomic data and hydrological elements for different frequencies are generated; second, the ILMP model is developed for irrigation water allocation, based on the generation of interval parameters; third, results are analyzed for different hydrological frequencies;

25

fourth, a scenarios analysis is done as a complementary method to develop more decision-making plans under different policy conditions; and the model is tested by applying to a real-world study in northeast China. 2. Methodology This section develops an interval linear multi-objective programming (ILMP) framework, with emphasis on the (1) generation of interval parameters using bootstrap method; (2) development of the ILMP model based on the generation of interval parameters; (3) solution of the ILMP model based on fuzzy programming (FP) method; and (4) development of a framework for multi-objective programming under uncertainty. Each of these elements is now discussed. 2.1. Bootstrap method The bootstrap method is a resampling technique. This method only needs resampling from the original sample series without making assumptions for the overall distribution, then continually estimates parameter values of the drawn samples, and finally deduces the parameter characteristics of the unknown overall samples and quantitatively describes the uncertainty of parameter estimation. Thus, it is an effective way to estimate and generate interval numbers based on long-term data. Assuming X = (x1 , x2 , · · ·, xn ) is an original sample and  is the unknown parameter of the overall distribution. From the lowest value to the highest value, the original sample can be written as x(1) ≤ x(2) ≤ · · · ≤ x(n) . The empirical distribution Fn for the sample can be described as follows (Hu et al., 2015):

Fn =

⎧ 0, x  x(1) ⎪ ⎪ ⎪ ⎪ ⎨ k

(1)

, x(k) ≤ x  x(k+1) , k = 1, 2, · · ·, n − 1 ⎪ n ⎪ ⎪ ⎪ ⎩ 1, x ≥ x(n)

∗ from the distribution Fn , the same size sample X = By∗resampling ∗ ∗

x1 , x2 , · · ·, xn can be obtained. Based on the bootstrap sample X*, the estimation * of parameter  of the distribution function can be calculated by a proper parameter estimation method. Repeating the bootstrap sampling N times, the N groups of  bootstrap samples can be obtained and can be described as X ∗(j) =

∗(j)

∗(j)

∗(j)

x1 , x2 , · · ·, xn

(j = 1, 2, · · ·, N). Then N parameter estimates  ∗(j) (j = 1, 2, · · ·, N) of parameter  can be derived. Taking  ∗(j) (j = 1, 2, · · ·, N) as the sample of the unknown parameter , the distribution of parameter  can thus be obtained. Based on this distribution, interval estimation of the values under each frequency can be derived based on the confidence intervals under predetermined confidence levels. 2.2. Interval linear multi-objective programming 2.2.1. Property of the interval numbers Before formulating the ILMP model, it is appropriate to first discuss the properties of interval numbers. Property 1: Let A denote a closed and bounded set of real numbers, and A± define an interval number with known upper and lower bounds of A. Then A± can be expressed asA± = [A− , A+ ] =

A− + z (A+ − A− ) |0 ≤ z ≤ 1 , with A− and A+ representing the lower and upper bounds of interval number A± , respectively, and z representing an auxiliary variable that can be used to transform the interval parameter into a determination one. Property 2: For A± , the following relationships hold: (1) A± ≥ 0 if A− ≥ 0 and A+ > 0; and (2) A±  0 if A− ≤ 0 and A+ ≤ 0. Further, for

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M. Li et al. / Agricultural Water Management 196 (2018) 24–36

A± and B± , the other relations are as follows: (1) A± ≤ B± if A− ≤ B− and A+ ≤ B+ ; and (2) A± ≥ B± if A− ≥ B− and A+ ≥ B+ . Property 3: Let ∗ ∈ {+,−, ×, ÷} be a binary operation on interval numbers. For A± and B± , we have: ±



±







A ∗ B = min A ∗ B , max A ∗ B



−

+

−

,A ≤ A ≤ A ,B ≤ B ≤ B

+

In the case of division, it is assumed that B± does not contain a zero. Hence, we have:



A± +B± = A− + B− , A+ + B+ ,



A± − B± = A− − B− , A+ − B+ ,





















A± × B± = min A × B , max A × B ±

±

A ÷ B = min A ÷ B , max A ÷ B

, A− ≤ A ≤ A+ , B− ≤ B ≤ B+ ,

xj ≥ 0

(4d)

zkj , sij , ti ∈ [0, 1]

(4e)

where i = 1, 2, · · · , m, j = 1, 2, · · · , n, k = 1, 2, · · · , p, and they are adapted for all the models in this section; zkj , sij , ti are auxiliary ± variables to transform the interval numbers of ckj , a± , and b± into ij i the corresponding deterministic expressions. Combined with the max-min operator (Bellmanr and Zaden, 1970), the ILMP model can be transformed into the following two forms corresponding to the linear and non-linear membership functions, respectively (Hulsurkar et al., 1997). (a) Equivalence model for linear membership function: max 

(5a)

subject to −

+

−

+

,A ≤A≤A ,B ≤B≤B .

 (fk (X)) ≥ 





n

2.2.2. Model development The ILMP model can be formulated as follows: min fk± (X) = Ck± X

(2a)

A± X ≤ B±

(2b)

X≥0

(2c)

where, fk± (X) (k = 1, 2, · · ·, p) is the linear objective function;

(x1 , x2 , · · ·, xn )T is the independent variable vector; Ck± = X =  ± ± ±

ck1 , ck2 , · · ·, ckn (k = 1, 2, · · ·, p) is the coefficient of the objective



a± , a± , · · ·, a± 1j 2j mj

T

(j = 1, 2, · · ·, n) is the coefficient

B±

matrix of the constraints; and = on the right hand of constraints.





T b± , b± , · · ·, b± m 1 2

 (fk (X)) =

⎪ ⎪ ⎪ ⎩

f k,min ≤ fk (X) ≤ fk,max , 1 ≤ k ≤ p

fk,max − fk,min

(3)

where  (fk (X)) is the membership function of the kth objective function; fk,min and fk,max are the lower and upper bounds of fk (X), respectively, and it is assumed that fk,min is not equal to fk,max . The models for calculating fk,min and fk,max share the same constraints with different objective functions and are described as follows: Objective function for calculating fk,min : min fk (X) =

n  



− + − ckj + zkj ckj − ckj



xj

(4a)



− + − ckj + zkj ckj − ckj



xj

(4b)

Both (4a) and (4b) are subject to

j=1



Where  refers to the overall satisfactory degree. (b) Equivalence model for non-linear membership function: max

p 

k

(6a)

k=1

 (fk (X)) ≥ k n  



a− + sij a+ − a− ij ij ij

(6b)





xj ≤ b− + ti b+ − b−

a− + sij a+ − a− ij ij ij





xj ≤ b− + ti b+ − b−





(6c)

xj ≥ 0

(6d)

k , sij , ti ∈ [0, 1]

(6e)

Where k refers to the satisfactory degree of the kth objective function. From the equivalence model for non-linear membership function, the importance of each objective can be quantitatively expressed by endowing with weights. Thus, the objective function p 

k based on the weighted geoω k

k=1

metric method, where ωk refers to the weight of the kth objective function (Guo et al., 2015). Models (5) and (6) are single objective programming models and the best-worst models are available to obtain the interval solutions of the ILMP model. For linear membership function, the best and worst models are expressed as follows: Best model: (7a)





+ ckj xj +  fk,max − fk,min ≤ fk,max

(7b)

a− x ≤ b+ ij j i

(7c)

j=1 n 

j=1

n  

(5e)

n 

Objective for calculating fk,max : max fk (X) =

, sij , ti ∈ [0, 1]

max 

j=1

n  

(5c) (5d)

of (6a) can be described as max

f k (X) ≥ fk,max

0



j=1

f k (X) ≤ fk,min

fk,max − fk (X)



xj ≤ b− + ti b+ − b−

xj ≥ 0

is the vector

2.2.3. Solution method The solution for the ILMP model is based on the FP method the essence of which is the membership function. The membership function for a vector-minimum problem can usually be written as

⎧ 1 ⎪ ⎪ ⎪ ⎨



j=1

k = 1, 2, . . ., p

subject to:

function; A± =

a− + sij a+ − a− ij ij ij

(5b)

j=1

(4c)

xj ≥ 0

(7d)

0≤≤1

(7e)

M. Li et al. / Agricultural Water Management 196 (2018) 24–36

27

Worst model: max 



(8a)

n





+ ckj xj +  fk,max − fk,min ≤ fk,max

(8b)

a+ij xj ≤ b− i

(8c)

j=1 n  j=1

xj ≥ 0

(8d)

0≤≤1

(8e)

For nonlinear membership function, the best and worst models are expressed as follows: Best model: max

p 

k ω k

(9a)

k=1 n 





− ckj xj + k fk,max − fk,min ≤ fk,max

(9b)

a− x ≤ b+ ij j i

(9c)

j=1 n  j=1

xj ≥ 0

(9d)

0 ≤ k ≤ 1

(9e)

Worst model: max

p 

k ω k

(10a)

k=1 n 





+ ckj xj + k fk,max − fk,min ≤ fk,max

(10b)

a+ij xj ≤ b− i

(10c)

j=1 n  j=1

xj ≥ 0

(10d)

0 ≤ k ≤ 1

(10e)

Models (7) ∼ (10) are the ultimate solution models for the generation of interval results of the ILMP model. 2.2.4. Framework for multi-objective programming under uncertainty The framework for multi-objective programming under uncertainty contains mainly two parts. The first part focuses on the generation of interval parameters, in which the bootstrap method is introduced. For water resources management, the bootstrap method is capable of generating interval numbers for hydrological elements, such as runoff, precipitation, and evapotranspiration under different frequencies based on long-terms data. However, there are still short-term data for some parameters associated with economy and society, such as market price, crop planting area, grain production efficiency. For such parameters with short-term data, a CI, which is derived by ordering all potential values and later identifying the upper and lower thresholds that act as good estimates of unknown data sets (Shen et al., 2011), and a probable error range for which each simulated or measured data point is determined with an error range (Chen et al., 2014), are effective ways for the generation of interval numbers. Based on the solution of the first

Fig. 1. Flow chart of the developed framework.

part, the second part based on the ILMP model will be undertaken. The interval numbers generated in the first part was regarded as the inputs of the ILMP model in the second part and this is the connection of the two parts. The basic properties of interval numbers, the model establishment, and the solution method have been discussed and the flow chart of the developed framework is shown in Fig. 1. 3. Application 3.1. Area description Hulan River irrigation district is located in Qing’an County, Suihua City, the middle of Heilongjiang province in northeast China. It ranges from 125◦ 55 to 128◦ 43 E and 45◦ 52 to 48◦ 03 N, with a total area of 1.89 × 105 ha, among which, the actual irrigation area is 2.03 × 104 ha. The irrigation district is in temperate semi-arid and semi-humid continental monsoon climate, with characteristics of seasonal weather. Annual average precipitation, evaporation, temperature, and frost-free days are 545 mm, 665 mm, 1.7 ◦ C, and 128 d, respectively. Hulan River irrigation district faces water shortage problems. Only surface water cannot satisfy the water demand,

28

M. Li et al. / Agricultural Water Management 196 (2018) 24–36

Fig. 2. Schematic diagram of the study area.

leading to the necessity of the conjunctive use of surface water and groundwater. The district is divided by Hulan River into two subdistricts, where the southern part includes Heping, Jianye and Liuhe areas, while the northern part includes Lanhe, Fengtian and Laomo areas. Heping area contains three headworks: Lishantun which brings water from Hulan River, Anbanghe which brings water from Anbang River, and Zhengwenju which brings water from Lalinqing River. Jianye area has one headwork bringing water from Lalinqing River; Liuhe area brings water from Liuhe reservoir. The headworks of Lanhe district and Laomo district are, respectively, Fucheng and Begai and they both bring water from Ougen River. The Fengtian area contains two headworks, among which Fengtian headwork brings water from Yijimi River and Wanghai headwork brings water from Hulan River. Anbang River, Lalinqing River, Yijimi River, and Ougen River are all branches of Hulan River. The location of these partitions can be seen in Fig. 2. Food crops including rice, maize and soybean occupy most of the planting area of Hulan River irrigation district, among which rice is the main crop to plant because of higher net benefits. Most water consumption in Hulan River irrigation district is for agricultural irrigation. Therefore, optimally allocating available water resources for food crops is significant for the commend area. Because of natural and artificial factors, the elements that affect agricultural irrigation water allocation present uncertainties, especially hydrological elements, such as runoff, precipitation and crop evapotranspiration. In this study, many rivers are involved and they present different values under different flow levels. These provoke managers to ask as to (1) how to consider the uncertainties of both hydrological elements and non-hydrological elements; (2) for hydrological elements, how to quantitatively express the uncertainties for different frequencies of different water supplies; and (3) how to generate desired water allocation (both surface water and groundwater) schemes to achieve the goal of yield and/or benefit promotion and blue water saving. Based on the above considerations, a multi-objective programming model with interval parameters can be developed, the optimal results of which will provide a certain guide in the development of water-saving and efficient agriculture. 3.2. Model development This part develops the ILMP model for optimally allocating available irrigation water for different food crops in different subareas

of Hulan River irrigation district for different frequencies of different water supplies. The model consists of three linear objective functions and a set of constraints.

3.2.1. Objective functions (1) Economic objective function With consideration of crop planting cost and water use cost, the target for economic objective function is to maximize the net return from different crops grown:. J  I  

max fE =

MPij±

i=1 j=1

−CGW

J I  

· YA± ij

− Cij±



A± ij



− CSW

J I  

SWij

i=1 j=1

GWij

(11a)

i=1 j=1

(2) Yield objective function The yield objective function considers maximizing the total crop yield as the target through precipitation and irrigation with both surface water and groundwater:

max fY =

J I  



YWij± SWij · cwei · fwei + GWij · fwei

i=1 j=1

+EPij± · A± ij



(11b)

(3) Water saving objective function Minimizing the total renewable blue water including streamflow and shallow groundwater is the target of water saving objective function. This is because that irrigated agriculture uses both green water and blue water and both of them are important for food production. Besides its contribution to crop production, blue water can also be directly used by socio-economic sectors while green water cannot. Therefore, green water should be well utilized and thus the blue water can be saved to increase irrigation efficiency in the premise of ensuring the minimum water demand of

M. Li et al. / Agricultural Water Management 196 (2018) 24–36

crops. Therefore, in this study, the blue water including both water from rivers and groundwater should be minimized. min fW =

J I   

SWij + GWij



(11c)

in area i (Yuan/kg); YA± is the yield per unit area of crop j in area ij

i (kg/ha); Cij± is the planting cost per unit area of crop j in area i

is the irrigated area of crop j in area i (104 ha); CSW (Yuan/ha); A± ij and GSW are the supply cost of surface water and groundwater, respectively (Yuan/m3 ); SWij and GWij are respectively the gross amounts of surface water and groundwater and they are decision variables (104 m3 ); YWij± is the water utilization efficiency of crop j in area i (kg/m3 ); cwei and fwei are water use coefficients for canal and field, respectively; and EPij± is the effective precipitation of crop j in area i (m3 /ha).

3.2.2. Constraints (1) Surface water availability constraint The availability of canal water for irrigation is limited. So, canal water allocation must not exceed the available canal water: SWij ≤ TSWi± ∀i

(11d)

j=1

Specifically, for this study we have: =

± pHP1 · QHL

± + pHP2 · QAB

± + pHP3 · QLLQ

(11d-1)

± TSW2± = pJY · QLLQ

(11d-2)

± TSW3± = pLHR · QLLQ

(11d-3)

± ± TSW4± = pFT 1 · QHL +pFT 2 · QYJM

(11d-4)

± TSW5± = pLM · QOG

(11d-5)

± TSW6± = pLH · QOG

(11d-6)

(11g)

GWij ≤ i · TGWi ∀i

(11e)

j=1

(3) Irrigation requirement constraint The irrigation requirement during the period of growth of all crops should be satisfied from precipitation, available canal water, and groundwater resources: SWij · cwei · fwei + GWij · fwei + EP ± · A± ≥ IR± · A± ∀i, j ij ij ij ij (4) Maximum irrigation constraint

(5) Hydrologic balance of aquifer Hydrological balance of the groundwater aquifer will help keep the water table at a predetermined level: J I  





GWij − 1 SWij + 2 SWij · cwei + GWij

i=1 j=1

+3 · EPij± · A± ij





≤ MA

(11h)

(6) Non-negativity constraint The allocated irrigation amount of both surface water and groundwater for various crops in each subarea should not be negative: SW ij ≥ 0∀ i,j GW ij ≥ 0∀ i,j where TSWi± is the surface water availability in area i ± ± ± ± ± QHL , QAB , QLLQ ,QYJM , QOG are, respectively, the runoffs

(11i-1) (11i-2) (104 m3 );

of Hulan River, Anbang River, Lalinqing River, Yijimi River and Ougen River (104 m3 ); pHP 1 , pHP 2 , pHP 3 are, respectively, the irrigation proportions of Hulan River, Anbang River, and Lalinqing River for Heping area; pJY is the irrigation proportion of Lalinqing River for Jianye area; pLHR is the irrigation proportion of Lalinqing River for Liuhe reservoir; pFT are, respectively, the irrigation proportions of Hulan River and Yijimi River for Fengtian area; pLM is the irrigation proportion of Ougen River for Laomo area; pLH is the irrigation proportion of Ougen River for Liuhe area; TGWi± is the groundwater availability in area i (104 m3 ); ␩i is the groundwater utilization proportion for irrigation in area i; IRij± is the crop irrigation requirement of crop j in area i (m3 /ha); WMi± is the maximum irrigation amount in area i (104 m3 ); ␪1 , ␪2 , ␪3 are the coefficients for conveyance loss of surface water, field water application loss, rainfall infiltration, respectively; and MA is the permissible annual mining allowance of the aquifer (104 m3 ).

3.3. Data collection and processing

(2) Groundwater availability constraint Groundwater allocation must not exceed the available groundwater supply to protect groundwater environment: J 



SWij · cwei · fwei + GWij · fwei ≤ WMi± ∀i

j=1

where fE is the objective function for economic benefit (104 Yuan, and Yuan is the monetary unit in China); fY is the objective function for yield promotion (104 kg); fW is the objective function for watersaving (104 m3 ); i is the index for subareas of Hulan River irrigation district, and I is the total number of subareas, in which, i = 1 for Heping area, i = 2 for Jianye area, i = 3 for Liuhe area, i = 4 for Fengtian area, i = 5 for Laomo area, and i = 6 for Lanhe area; j is the index for crops and J is the total number of crops, in which, j = 1 for rice, j = 2 for maize and j = 3 for soybean; MPij± is the market price of crop j

TSW1±

The irrigation allocation for each crop should not exceed a maximum value to avoid a waste. J  

i=1 j=1

J 

29

(11f)

The data for the ILMP model can be divided into four parts in this study. They are hydrological parameters, social-economic parameters, other parameters, and weight scenario for each objective function. Rice, maize, and soybean are the studied crops and their growth periods are from May to September. 3.3.1. Hydrological parameters Hydrological parameters contain runoff of each related river, effective precipitation, evapotranspiration of each crop, and groundwater availability. These hydrological parameters are essentially random, and the values of most of them have significant variations for different frequencies. Accordingly, it is necessary to estimate the interval values for different frequencies for these hydrological elements to reflect the randomness. Hydrological parameters usually have longer data series (more than 30 years) relative to non-hydrological parameters, thus, the interval number generation of these hydrological elements can be obtained using the bootstrap method. The time series for all these hydrological parameters are 55 years, and they were collected from hydrological stations and meteorological network.

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M. Li et al. / Agricultural Water Management 196 (2018) 24–36

Fig. 3. Annual variation of ET0 and effective precipitation.

For runoff, the bootstrap method was directly applied to generate interval numbers based on the time series data of each river. ± In this study, the interval numbers of runoff of Hulan River (QHL ), ± ± ± Anbang River (QAB ), Lalinqing River (QLLQ ), Yijimi River (QYJM ) and

± Ougen River (QOG ) were all generated by bootstrap method. First, the empirical distribution was calculated, and based on which a certain theoretical frequency curve was determined by comparing the fit with the empirical frequency curve. Then, using Monte Carlo method resampling from the original sample was done 1000 times and parameters were estimated for each new sample that yielded 1000 groups of parameter estimates. Thereafter, the probability distribution of a certain frequency was obtained. Finally, the distribution of each frequency was obtained and the corresponding interval numbers were generated. In this study, the frequencies were chosen 25%, 50%, 75%, and 95%, corresponding to the wet condition, normal condition, dry condition and extreme dry condition, respectively, according to the dividing standard of wet and dry conditions of China and the Planning and Design Report of Hulan River Irrigation District. The effective precipitation was calculated by multiplying precipitation by an effective coefficient. According to Kang and Cai (1996), if precipitation was less than 50 mm, then the coefficient was 0.9; if precipitation was greater than 50 mm but less than 150 mm, then the coefficient was 0.75; and if precipitation was greater than 150, the coefficient was 0.7. In this manner, the effective precipitation was obtained and the corresponding interval numbers for the selected frequencies were obtained with the same way as runoff. For crop evapotranspiration (ETc ), the FAO56 Penman-Monteith method (Allen et al., 1998) was used to estimate daily reference evapotranspiration (ET0 ). The FAO56 Penman-Monteith method was adopted because it was regarded as the basic method for calculating reference ET0 by Food and Agriculture Organization of United Nations in 1998, and it has strong theoretical property and computational accuracy. The FAO56 Penman-Monteith method was widely used throughout the world and climate factors, including average temperature, lowest temperature, highest temperature, average wind speed, sunshine duration, average relative humidity were the basic input data for the method. Monthly and annual values of ET0 were obtained by summing daily values. In this study, the coefficient of variation (Cv ) of ET0 was only 0.06, much less than the Cv values of the five rivers (Cv = 0.53 for Hulan River, Cv = 0.57 for Anbang River, Cv = 0.77 for Lalinqing River, Cv = 0.50 for Yijimin River, and Cv = 0.55 for Ougen River), indicating yearly ET0 changed little. The yearly trend can also be seen from Fig. 3 which suggests that it makes little sense to estimate the interval numbers of ET0 under each frequency which will otherwise add to the complexity of the optimization solution.

The stochastic simulation method (Li et al., 2016a,b) was employed to estimate the interval number of ET0 based on the known probability distribution function. The difference between bootstrap method and stochastic simulation method for the generation of interval number of hydrological elements is that the former is capable of conveniently generating interval numbers under different frequencies without the assumption of the overall distribution. Compared with the results of generation using bootstrap method, the interval number of ET0 was expressed by only one value while the interval numbers of runoff of each river and precipitation were expressed by four values corresponding to the frequencies of 25%, 50%, 75%, and 95%. Based on ET0 , the ETc was obtained using the formula ETc = Kc ET0 (Pereira et al., 2015), where Kc is the crop coefficient which changes with growth periods of each crop. The Kc value for each crop of each month is listed in Table 1. In this study, the value of water requirement equaled the value of ETc for each crop. The calculated annual variations of ET0 and effective precipitation are shown in Fig. 3, and their monthly variation in Fig. 4. Because of the lack of observation data, availability of groundwater was expressed as the average value according to the planning report. Groundwater available for different subareas and the corresponding irrigation proportions are given in Table 2. Fengtian area is not designed for well irrigation. The product of available groundwater and the irrigation proportion of groundwater is the value that was adopted for optimization in this study. 3.3.2. Socio-economic parameters Socio-economic parameters in this study included market price, planting area, yield per unit area, maximum irrigation amount, planting cost, water use productivity, and water utilization cost. These parameters were obtained mainly from yearbook, report, website, field survey, and previous studies. The time series of these socio-economic parameters were shorter than hydrological parameters (less than 10 years), so the bootstrap method was not suitable for the generation of interval numbers for these parameters. For market price, the maximum and minimum values during the whole growth period of each crop were collected from Agricultural Product Price Net. Crop planting area data and yield per unit area were obtained from planning report and yearbook (2010 ∼ 2015), and the corresponding interval numbers were calculated from the method of  ± x%, where  is the mean and x% is the deviation. Then, interval numbers of yield per unit area were obtained by dividing the yield of each crop of each irrigation area by the interval planting area. The maximum irrigation amount was the upper threshold in the right hand of the irrigation constraint, indicating that the optimal water allocation amount for different regions cannot exceed the maximum irrigation amount. In this study, the maximum irrigation amount was obtained by multiplying the irrigation quota under higher design dependability with the interval planting area. As the planting area was expressed as interval numbers, leading to that the maximum irrigation amount was expressed as interval numbers as well. Water use productivities for the three crops were obtained from data analysis of field experiments. Planting cost and water utilization cost were obtained from previous works (Li and Zhang, 2016; Fu et al., 2016). Tables 1–3 provide detailed information. 3.3.3. Other parameters Other parameters mainly included irrigation proportion of groundwater, irrigation proportions for each irrigation area from the corresponding rivers, canal and field water utilization coefficient, and hydraulic parameters. All these parameters were deterministic in this study that came from Planning and Design Report of Hulan River Irrigation District. Detailed information can be seen in Table 2. The values of pHP 1 , pHP 2 , pHP 3 , pJY , pLHR , pFT 1 ,

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Table 1 Basic data for different crops. Crop

Rice Maize Soybean

Kc

Market price

Planting cost

Water use productivity

May

Jun

Jul

Aug

Sep

(Yuan/kg)

(Yuan/ha)

(kg/m3 )

0.38 0.30 0.40

0.78 1.20 1.15

1.34 1.20 1.15

1.06 1.20 1.15

0.45 0.80 0.70

[3.16, 3.27] [2.25, 2.36] [5.40, 5.44]

9526 5010 4980

1.61 1.82 0.72

Fig. 4. Monthly variation of ET0 and precipitation. Table 2 Basic data for different areas. Area

Groundwater availability (104 m3 )

Irrigation proportion of groundwater

Maximum irrigation (104 m3 )

Heping Jianye Liuhe Fengtian Laomo Lanhe

1552 378 293 – 2432 1324

0.87 0.79 0.82 – 0.91 0.95

[4380,4651] [646,686] [680,722] [963,1022] [4124,4379] [2254,2394]

Table 3 Basic data for different crops in different areas. Subarea

Heping Jianye Liuhe Fengtian Laomo Lanhe

Planting area (102 ha)

Per unit yield (kg/ha)

Rice

Maize

Soybean

Rice

Maize

Soybean

[32.77, 34.13] [4.83, 5.03] [5.09, 5.30] [7.20, 7.50] [30.85, 32.13] [16.86, 17.57]

[20.48, 21.85] [3.02, 3.22] [3.18, 3.39] [4.50, 4.80] [19.28, 20.57] [10.54, 11.24]

[12.97, 15.02] [1.91, 2.21] [2.01, 2.33] [2.85, 3.30] [12.21, 14.14] [6.68, 7.73]

[8668,9204] [8151,8655] [8110,8612] [8243,8752] [8558,9088] [8252,8762]

[9286,9861] [8935,9488] [8887,9437] [8964,9519] [9158,9724] [9326,8762]

[2538,2695] [2453,2604] [2439,2590] [2656,2821] [2554,2712] [2560,2718]

pFT , pLH were 0.84, 9.8, 18.1, 7.5, 11.1, 0.84, 4.1, 6.8, 3.4 (%), respectively. The values of canal and field irrigation utilization coefficients were 0.5 and 0.8, respectively for each area. The coefficients for conveyance loss of surface water, field water application loss, rainfall infiltration were, respectively, 0.49, 0.10, 0.08.

3.3.4. Weight scenarios for objective functions The weighting coefficient method was used in the solution of the ILMP model when the membership function was non-linear. As a typical meta-synthesis engineering method from quality analysis to quantity analysis, analytical hierarchy process (AHP) has been widely used in the field of water resources management (Srdjevic and Medeiros, 2008). Therefore, in this study, the weights of importance for the three objectives of the developed ILMP model were determined by the AHP method. Four scenarios were set: Scenario 1 referred to the equal importance of the three objectives; Scenario 2 referred to the case where the objective of economic benefit was paramount; Scenario 3 referred to the case where the objective of yield increase was paramount; and Scenario 4 referred to the case where the objective of water-saving was paramount. Results of weights determined from the AHP method are shown in Table 4.

Table 4 Weighting coefficients of different scenarios. Scenario

Scenario 1 Scenario 2 Scenario 3 Scenario 4

Weighting coefficient Economic objective

Yield objective

Water saving objective

0.33 0.43 0.31 0.31

0.33 0.30 0.41 0.31

0.33 0.27 0.28 0.37

4. Analysis of results and discussion 4.1. Generation of interval numbers for hydrological parameters The interval numbers of hydrological parameters were obtained using the bootstrap method. Through hydrological curve fitting, the probability distribution of each hydrological parameter was determined and parameters were estimated. The Pearson typeIII distribution was fitted to the values of runoff of Hulan River, Lalinqing River, and Yijimi River and of precipitation and the distribution parameters were estimated using the method of moments. The runoff of Anbang River followed a 2- parameter gamma dis-

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M. Li et al. / Agricultural Water Management 196 (2018) 24–36

Fig. 5. Probability distribution and Mann-Kendall test for the runoff of Lalinqing River. Note: UF, UB represent the positive and the negative series of M-K test, respectively

Table 5 Interval numbers for hydrological parameters. Hydrological parameters

Runoff (108 m3 )

Frequency

Hulan River Anbang River Lalinqing River Yijimi River Ougen River

Precipitation (mm) Evapotranspiration (mm)

25%

50%

75%

95%

[8.17, 10.12] [1.32, 1.71] [0.63, 0.88] [3.91, 4.83] [3.31, 4.29] [558.41, 610.01] [780.32, 954.46]

[5.70, 7.35] [0.90, 1.17] [0.37, 0.54] [2.76, 3.50] [2.36, 3.08] [492.05, 536.30]

[3.68, 5.16] [0.56, 0.77] [0.18, 0.32] [1.92, 2.51] [1.69, 2.21] [431.35, 472.23]

[1.76, 2.89] [0.23, 0.40] [0.03, 0.14] [1.14, 1.61] [0.92, 1.39] [345.11, 393.20]

tribution, and the runoff of Ougen River followed a generalized extreme value distribution. The parameters of these hydrological parameters were estimated using the maximum likelihood estimation method. The runoff of Lalinqing River, which supplied water for most areas (including Heping area, Jianye area, and Liuhe area), was used as an example to demonstrate the generation of interval numbers by the bootstrap method. The probability distribution and the Mann-Kendall (M-K) test (Hamed and Rao, 1998) of the runoff of Lalinqing River are shown in Fig. 5. The M-K test showed a low temporal variability of the runoff time series of Lalinqin River, indicating that the sample could be considered continuous. Fig. 6 presents the frequency histogram and the normal probability plot of the runoff values of Lalinqing River under the frequencies of 25%, 50%, 75%, and 95%. The figure shows that the normal distribution function fitted the frequency histogram well under each frequency. The scatters were evenly distributed around the 45◦ line, which showed the distribution of runoff values of Lalinqing River under each frequency wa approximately a normal distribution. Therefore, using the normal distribution, the interval number of each frequency was obtained for the 95% confidence interval. Similarly, the interval numbers of other hydrological parameters are listed in Table 5. The Pearson type-III distribution was fitted to the ET0 values and the acceptance/rejection method (Gu et al., 2008) was used for the stochastic simulation to generate interval numbers with the 95% confidence interval. The results of generation of these interval numbers were then input in the ILMP model. 4.2. Agricultural irrigation water allocation The ILMP model was then solved, based on interval inputs. A series of water allocation results for different crops in different areas under different frequencies, membership functions, and weight scenarios were obtained. 4.2.1. Results of linear membership function For linear membership function, the results of total irrigation amount, expressed as intervals of different areas under different

frequencies, are shown in Fig. 7. The figure shows that the amount of water allocated to Heping area was the highest, then was Laomo area, and the last one was Jianye area. It can also be seen from the figure that the total water allocation amount under different frequencies changed distinctly, from [1.01, 1.60] × 108 m3 under the frequency of 25% to [0.98, 1.33] × 108 m3 under the frequency of 95%, indicating the importance to prepare alternative water allocation plans to deal with the unknown water supply for the efficient use of limited water resources. In fact, when solving for the worst case of the ILMP model under the frequency of 95%, which corresponded to the condition of extreme dry, there was no feasible solution because both surface water and groundwater supplies were not able to meet crops’ highest irrigation demand. In such a case, one way was to transfer 2791 × 104 m3 from areas with relative abundant water supply or pumping of more groundwater within the allowable range which would increase the accompanying cost and the impact on groundwater. The other way was to decrease irrigation requirement which would reduce production and would affect benefit. This is worth a further study for seeking a satisfactory water allocation solution under extreme conditions. On the other hand, groundwater allocation under the frequency of 25% was the least, because the optimization model tended to prefer allocating surface water. This was attributed to two reasons: one was that the cost of groundwater was higher than that of surface water, and the other was that the surplus of groundwater was considered for transfer to other water use sectors, for example industry, to avoid wasting extra water for irrigation and meanwhile increasing profit. Comparison among water utilization, water requirement (WR) intervals, and maximum irrigation for each area under different frequencies is shown in Fig. 8, among which, the total water allocation contained both the net water allocation amount and effective precipitation. The net water allocation amount equaled the gross water allocation amount (optimal results) multiplied by the water use coefficients for canal and field. The figure shows that under all frequencies, the total water allocation was in the interval of maximum and minimum water requirement. For frequency of 25%, the total

M. Li et al. / Agricultural Water Management 196 (2018) 24–36

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Fig. 6. Normality of the runoff value of Lalinqing River under different frequencies.

water allocation amounts of Heping area, Jianye area, Liuhe area, and Fengtian area even exceeded their maximum water requirements, and nearly reached their maximum irrigation thresholds,

indicating that water supply was abundant under the frequency of 25%. However, for the 95% frequency, the total water allocation could only satisfy the minimum water requirement for most areas

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M. Li et al. / Agricultural Water Management 196 (2018) 24–36

Fig. 10. Values of objective functions under different scenarios.

Fig. 7. Total irrigation amount of different areas under different frequencies.

Fig. 8. Comparison among water utilization, water requirement (WR) intervals and maximum irrigation for each area under different frequencies.

Fig. 9. Irrigation water allocation for each crop of each area under the frequency of 50%.

except for Fengtian area. For the 50% and 75% frequencies, the total water allocation in most of the areas was in the intervals of minimum water requirement and maximum irrigation threshold. The water allocation for smaller areas, such Jianye, Liuhe, and Fentian, was reaching the maximum irrigation threshold in order to obtain the optimized comprehensive benefit. Taking the 50% frequency which represented the normal condition as an example, the water allocation for each crop in each area is shown in Fig. 9. The total water allocation amount for rice wa larger than that for maize and soybean because of the relative higher market price and water production efficiency. The total average water allocation for rice, maize and soybean was 5798 × 104 m3 , 2518 × 104 m3 , and 4174 × 104 m3 , respectively. Since maize is a water-consuming crop with lower

benefit, it is inadvisable to plant it in large quantities, especially in Heping irrigation area. 4.2.2. Results of non-linear membership function For nonlinear membership function of the ILMP model, the objective weight scenarios were taken into account. Different scenarios indicated different weights of importance for different objectives, and information on the weights is given in Table 4. Taking the 50% frequency as an example, Fig. 10 shows different values of the three objectives under different scenarios. In Fig. 10, OF1 means the first objective, i.e. the maximum net benefit (104 Yuan), OF2 means the second objective, i.e. the maximum crop yield (104 kg), and OF3 means the third objective, i.e. the minimum blue water use (104 m3 ). In this figure, both the upper bound and the lower bound of the values of objectives were in the range of maximum and minimum objectives. For each scenario, the first and third objectives were always given a priority, with the upper bounds nearly achieving the corresponding maximum values. Scenario 1 yielded optimal results with equal importance of the three objectives. Scenario 2 focused more on the system net benefit, scenario 3 focused more on crop yield, and scenario 4 focused more on water-saving. For scenario 2 and scenario 4, the changing trends of upper and lower bounds of all the three objectives were the same. This was attributed to the same properties of objective 1 and objective 3, whose aim was to decrease the water utilization cost and achieve water-saving. Specifically, the net benefit was [31348,38524] × 104 Yuan, with 430 × 104 Yuan lower than the maximum net benefit and the lower bound of which was equal to the minimum net benefit. The crop yield was [17772,20630] × 104 kg, with 5691 × 104 kg lower than the maximum net benefit and 3309 × 104 kg higher than the minimum net benefit. The water allocation was [12435,14763] × 104 m3 , with 7458 × 104 m3 higher than the minimum water allocation and the upper bound of which was equal to the maximum water allocation amount. The changing trend of the three objectives for scenario 3, which emphasized allocating more water to get more yield, was contrary to that of scenario 2 and scenario 4. The crop yield of scenario 3 was 167 × 104 kg higher than that of scenario 2 and scenario 4. For scenario 1, as the importance of all the three objectives was equal, the values of net benefit, crop yield and water allocation were in the range of that of scenarios 2–4. For the area that water was insufficient, the water use productivity was an important index to reflect the water use efficiency and it was expressed as the yield per unit water use. Calculating it, the water use productivity under the four scenarios was [1.34, 1.50] kg/m3 , [1.40, 1.74] kg/m3 , [1.34, 1.71] kg/m3 , [1.40, 1.74] kg/m3 , respectively. It was clear that water use productivities of scenario 2 and 4 were higher than those under other scenarios, attributing to the smaller water allocation amount under those two scenarios. The objective of transformations of the original ILMP model was to maximize the

M. Li et al. / Agricultural Water Management 196 (2018) 24–36

Fig. 11. Satisfaction degrees under different frequencies and different scenarios.

degree of overall satisfaction. Fig. 11 shows the satisfaction degrees under different frequencies and different scenarios. It can be seen from the figure that the satisfaction degrees vary as the frequencies and scenarios change. The changing trends of different frequencies under the four scenarios were nearly the same with different changing amplitudes. The lower bounds of satisfaction degrees were in the vicinity of 0.3, while the upper bounds of satisfaction degrees were in the vicinity of 0.8. In addition, for the area that water was insufficient, decision makers usually paid more attention to water allocation schemes with higher frequency to reduce the water shortage risk because of the randomness of water supply. Thus, taking the 75% frequency as an example, whose frequency was higher but not the extreme condition, the irrigation water allocation amount of both surface water and groundwater for each crop under different scenarios are shown in Fig. 12. It is clear that surface water was the primary water source for all the areas in every scenario. The percentages of surface water and groundwater were about 60% and 40% and the total surface water allocation amount was the same, while the groundwater allocation amount was changing under different scenarios. This was because surface water was fully utilized, and the groundwater utilization percentage was large because water supply from different rivers was lower under the 75% frequency, leading to more groundwater use to guarantee the growth of crops. 4.3. Discussion In summary, based on the generation of interval parameters, the ILMP model was developed with three objective functions. Based on the fuzzy programming method, the ILMP model was solved with a series of results associated with water allocation and objective values generated under different frequencies and scenarios. In this study, there were in total six parameters varying with different frequencies. Five of them were the runoffs of Hulan River, Anbang River, Lalinqing River, Ougen River and Yijimi River, and

35

the remaining one was precipitation. The yearly changing trends of all the five rivers were nearly the same, based on historical data. However, there was a difference of yearly changing trend between runoffs of the five rivers and precipitation. Taking Huanlan River as an example, the occurrence probabilities of runoff and precipitation under the same frequency were the highest, except for the condition when the frequencies of runoff and precipitation were both 75%. Therefore, the corresponding values of runoff and precipitation with combinations of the same frequency were used to decrease computation. For the condition when the frequencies of runoff and precipitation were both 75%, the runoffs under the 75% frequency and precipitation under 95% frequency were combined. However, such combinations would lead to the loss of information and thus affect results. Taking runoff under the 25% frequency as an example, there also existed occurrence probabilities when the frequencies of precipitation were 50%, 75% and 95%, and such conditions were not considered in this study. This suggests further researches on how to develop a joint probability function to reflect all the combinations of related elements. The aim of the ILMP model was to balance the contradictions among economic benefit, crop yield, and water-saving in irrigation systems. The solutions contained a combination of interval and deterministic information, and thus reflected uncertainty, better reflecting real conditions. The interval solutions can help managers obtain multiple decision alternatives, as well as provide a basis for further analyses of tradeoffs between different contradictories in an irrigation system. 5. Conclusion Considering uncertain factors in the course of water resources allocation, interval parameters were generated. The interval numbers of Hulan River, Anbang River, Lalinqing River, Ougen River, Yijimi River and precipitation were generated by the bootstrap method. The interval numbers of evapotranspiration were generated by stochastic simulation and those of other parameters by the upper and lower thresholds or the probable error range. Thus, an interval linear multi-objective programming (ILMP) model was developed for irrigation water resources management considering the contradictory objectives of system net benefit, crop yield, and water-saving. The interval parameter generation and the developed model constitute a framework which has advantages in: (1) generating relatively accurate interval ranges of different hydrological elements under different frequencies for water resources management; (2) allowing uncertainties presented as interval values to be incorporated within a general multi-objective optimization framework, in which tradeoffs between the aspects of economy, society and resources were considered.

Fig. 12. Irrigation allocation for each area for different scenarios under frequency of 75%.

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To generate solutions, the model was transformed into two deterministic sub-models, based on fuzzy programming with the consideration of both linear and non-linear membership functions. The developed framework was applied to a real case study for allocating limited water resources to different crops in different subareas in Hulan River irrigation district, northeast of China. Results suggest that the methodology is applicable for reflecting the complexities of multi-objective and uncertainties of agricultural irrigation management systems. This study attempts to provide a framework for comprehensive allocation of agricultural irrigation water to reflect uncertainties and multi-objective more precisely. The framework can also been applied to management of other resources where hydrological elements are involved, for example, regional water resources management. In this study, the impacts of the changes of different hydrological elements such as runoff and precipitation on the water resources allocation schemes were analyzed as independent scenarios, without considering the joint distributions of these hydrological elements. This may limit the allocative effect of the developed framework and requires further research. Acknowledgments This research was supported by the National Key R&D Plan of China (No: 2016YFC0400107), and National Natural Science Foundation of China (No: 51479032). References Bellmanr, E., Zaden, L.A., 1970. Decision making in fuzzy environment. Manage. Sci. 17, 141–164. Chen, L., Shen, Z.Y., Yang, X.H., Liao, Q., Yu, S.L., 2014. An interval-deviation approach for hydrology and water quality model evaluation within an uncertainty framework. J. Hydrol. 509, 207–214. Das, B., Singh, A., Panda, S.N., Yasuda, H., 2015. Optimal land and water resources allocation policies for sustainable irrigated agriculture. Land Use Policy 42, 527–537. Fu, Q., Liu, Y.F., Liu, D., Li, T.X., Liu, W., Amgad, O., 2016. Optimal allocation of multi-water resources in irrigation area based on interval-parameter multi-stage stochastic programming model. Trans. Chin. Soc. Agric. Eng. 32 (1), 132–139 (in Chinese with English abstract). Galán-Martín, Á, Vaskan, P., Antón, A., Esteller, L.J., Gonzalo, G., 2017. Multi-objective optimization of rained and irrigated agricultural areas considering production and environmental criteria: a case study of wheat production in Spain. J. Clean. Prod. 140 (Pt. 2), 816–830. Gu, W.Q., Shao, D.G., Huang, X.F., Dai, T., 2008. Multi-objective risk assessment on water resources optimal deployment. J. Hydraul. Eng. 39 (3), 339–345. Guo, P., Wang, X.L., Zhu, H., Li, M., 2014. Inexact fuzzy chance-constrained nonlinear programming approach for crop water allocation under precipitation variation and sustainable development. J. Water Resour. Plan. Manage. 140 (9), 05014003. Guo, Z.X., Zheng, Y.M., Li, S.S., 2015. Interval multi-objective programming problem based on fuzzy geometric weighting method. J. Heibei Univ. (Nat. Sci. Ed.) 35 (3), 230–235 (in Chinese with English abstract).

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