Optimization of the irrigation water resources for Shijin irrigation district in north China

Agricultural Water Management 158 (2015) 82–98

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

Optimization of the irrigation water resources for Shijin irrigation district in north China Gaiqiang Yang a,b , Ping Guo a,∗ , Lijuan Huo b,c , Chongfeng Ren a a b c

Centre for Agricultural Water Research in China, China Agricultural University, Beijing 100083, China Institute of Environmental Science, Taiyuan University of Science and Technology, Taiyuan, Shanxi 030024, China Institute of Environment and Sustainable Development in Agriculture (IEDA), Chinese Academy of Agricultural Sciences, Beijing 100081, China

a r t i c l e

i n f o

Article history: Received 10 January 2014 Accepted 20 April 2015 Keywords: Fuzzy Uncertainty Agricultural irrigation Optimal allocation Irrigation strategy

a b s t r a c t In this study, a nonlinear fuzzy interval irrigation optimization model is developed for optimal agricultural water allocation in Shijin irrigation district, Hebei province, China. The developed method can deal with interval values and fuzzy numbers in a nonlinear objective and a set of constraints including water availability, evapotranspiration, reservoir water priority, fairness, etc. Three different scenarios (under dry, normal, and wet years) were considered. As the result, the optimal water allocation schemes, both from reservoir water and ground water, were obtained. Under fuzzy degree of membership level that equals to 1.0, the system benefits are 1.53 × 109 RMB for dry years, 1.61 × 109 RMB for normal years and 1.61 × 109 RMB for wet years. The solutions can not only provide an effective evaluation under preset scenarios, but also reveal the associated economic implications. Three aspects have been improved in this study compared with previous studies: (1) the model has capability of incorporating multiple water resources, multiple subareas, multiple crops, and multiple growth stages of water resources management within the optimization framework; (2) in order to address the uncertain parameters which are not sufficient or imprecise in the practical problems, a nonlinear fuzzy interval irrigation optimization model (NFIIOM) is developed by incorporating interval parameters programming and fuzzy programming; (3) after applying the developed model to a real case study, the reasonable solutions has been gotten and the results can be used to help the manager to establish effective water allocation plan for agricultural irrigation systems under uncertainty, and thus maximize the benefit of local farmers. © 2015 Elsevier B.V. All rights reserved.

1. Introduction New incentives and policies are very important for ensuring the sustainability of agriculture (Tilman et al., 2002). Agriculture consumption accounts for about 70% of the total water withdrawals in the world (Schlager, 2005). Distribution of agricultural water is intrinsically related to food production for the largest water user among all sectors (Yang et al., 2003). With continuously rapid population growth and socio-economic development, conflicts between decreased available water supply and increased water demand have escalated to a higher level of intensity (Lu et al., 2008). Moreover, drying-up of rivers, falling of groundwater levels, degradation of lakes and wetlands, and water pollution have led to water resources overcommitted which is one of the major issues facing North China. For example, 4000 km of the lower reaches of Hai

∗ Corresponding author at: Centre for Agricultural Water Research in China, China Agricultural University, Beijing 100083, China. Tel.: +86 1062738496. E-mail address: [email protected] (P. Guo). http://dx.doi.org/10.1016/j.agwat.2015.04.006 0378-3774/© 2015 Elsevier B.V. All rights reserved.

River, about 40% of its length, has experienced zero flows since 1998 (Xia et al., 2007). The renewable internal freshwater resources per capita (internal river flows and groundwater from rainfall in the country for one person) in China was 2093 m3 in 2011, according to the World Bank: Databank Indicator (Turner et al., 2013). To improve management of irrigation water demand through better water management systems, some methods can be utilized, such as reducing irrigation water distribution losses, improving irrigation scheduling, changing cropping patterns (Hamdy et al., 1995). Thus, sound optimization strategies are necessary for better allocating agriculture water resources in China. Under the guidance of sound optimization, maximum yield, more saving water or maximum economic benefit can be achieved. There have been some successful applications in optimization of the irrigation water resources. The objective of most studies was to maximize economic benefits, based on developing for optimal land and water resources, maintaining the existing cropping patterns or expressing farmers’ willingness to pay for irrigation water (Salman and Al-Karablieh, 2004; Brown et al., 2010; Singh and Panda, 2012; Nazer et al., 2010). The objective of some studies

G. Yang et al. / Agricultural Water Management 158 (2015) 82–98

was to maximize the sum of relative yields of crops over a year, with three sets of constraints: mass balance at the reservoir, soil moisture balance for individual crops, and governing equations for ground water flow (Vedula et al., 2005). The objective of other studies was to distribute the water resources according to the water demand (Takahashi et al., 2013; Galelli et al., 2010). It was concluded that these problems can be solved through mathematical programming methods. In these problems, model parameterization needs to be accurate. However, system parameter information is often not available, or some available information may be inaccurate. The deterministic parameters directly obtained are only a fraction of all parameters. The rest may be fuzzy information and difficult to quantify, such as the amount of water available, costs, capacities, water demand and time (Hernandes et al., 2007). The deterministic planning methods are often inadequate for supporting water resources management under uncertainty of model parameters (Li and Huang, 2007). To deal with these uncertainties, the approach of fuzzy programming, interval programming and stochastic programming can be adopted (Qin et al., 2007). Interval programming approach is used to describe the uncertain parameters for the available data when they are not sufficient to generate a distribution or membership function. But the solutions might be unreasonable for oversized interval solutions when the right hand sides of the programming model are highly uncertain. In addition, the interval solutions cannot describe the information between the upper bounds and the lower bounds (Guo et al., 2008). Therefore, the deficiencies limit their application to solve practical problems for decision makers. Fuzzy programming approach offers a means of handling optimization problems with fuzzy parameters. It is used to describe the uncertain parameters by fuzzy membership functions. It is applicable to describe the amount of water available with uncertain or imprecise characteristics. The fuzzy parameter is more rigorous than interval parameter, because it needs more information to be described as a fuzzy membership function. The approach is used to support subjective decisions for managers, and cannot accurately reflect the specific parameter information. Moreover, the stochastic approach can accurately describe the uncertain information of parameters by probability density functions. But it requires a lot of objective data. In this study, the precipitation data is limited, because a complete statistics just started in recent years. Therefore, the stochastic approach was not suitable for this study, so a method with interval and fuzzy parameters was developed to solve optimal water allocation for an irrigation district. In the uncertain model, some fuzzy decision parameters in the programming problems can be described as fuzzy numbers (Maleki and Mashinchi, 2004). Some multi-objective fuzzy linear programming models were formulated to solve the problem of optimal cropping pattern in an irrigation system and to evaluate management strategy of irrigation planning, by dealing with the fuzziness in resources and decision variables (Raju and Duckstein, 2003; Gurav and Regulwar, 2012; Regulwar and Gurav, 2012). A two-step infinite ˛-cuts fuzzy linear programming method was developed, and was applied in an agricultural irrigation system (Lu et al., 2009a,b). Although some fuzzy programming models have been developed for solving water allocation problems, there are still some inadequacies that limit their application. For example, fuzzy membership functions are difficult to determine in an accurate and simple way in a programming model which has multiple uncertain parameters and nonlinear structure. As a parameter has two or more different uncertain features, it is difficult to describe it through a general member function. If it has a nonlinear structure, solving is more difficult. In addition, most models are targeted for a specific system, and cannot be directly applied to other systems, so the specific scope limits its application. Considering multiple uncertainties, some models have been developed for water allocation or water transfer problems, such as

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inexact rough-interval two-stage stochastic programming model (Lu et al., 2009a,b; 2011), inexact full-infinite two-stage stochastic programming model (Zhu et al., 2013), interval two-stage stochastic programming model (Lu et al., 2009a,b; Xiaoyan et al., 2011; Li et al., 2013), interval fuzzy linear programming model based on infinite ˛-cuts (Lu et al., 2010). Generally speaking, the above interval methods were based on the assumption that the objective function was linear, resulting in difficulties in dealing with such issues wherein yield of crop is calculated based on the growth stages and thus make the relevant objective function nonlinear. The methods combining fuzzy with interval can describe more accurately the objective systems by establishing inexact programming models, but relevant cases applied to agricultural water resources management are rather rare. The objective systems are the systems which can reflect and describe the actual problems. Therefore, it was desired that an optimization method be developed for handling uncertainties and complexities in optimization of agricultural irrigation water allocation. This study aimed to optimize the irrigation water resources for Shijin irrigation district in north China by developing a nonlinear fuzzy interval irrigation optimization model (NFIIOM) incorporating approaches of interval parameters programming and fuzzy programming. The developed NFIIOM can not only handle uncertainties expressed as interval values and fuzzy numbers, but also tackle nonlinearity in the objective function. The method is demonstrated in Shijin irrigation district in north China for optimal agricultural water allocation. The results can be used to help the manager to establish effective water allocation plans for agricultural irrigation systems, and thus maximize the benefit of local farmers.

2. Study system Shijin irrigation district is located in central south of Hebei plain, China, near Hutuo River and Fuyang River. It is the largest agricultural irrigation district in Hebei province. The water in the district is used mainly for agricultural irrigation, power generation and city industrial water supply. The designed irrigation area is 162.7 × 103 ha. The climate is semi-arid, continental monsoon climate. The annual average temperatures are around 13 ◦ C(55 ◦ F). January is the coldest month of the year with an average temperature of 0.6 ◦ C(33 ◦ F), July is the hottest month of the year with an average temperature of 18 ◦ C(64 ◦ F). The annual amount of precipitation is about 480 mm. The distribution of precipitation throughout the year is much uneven for it is concentrated in June, July and August. The precipitation rainfall in these three months accounts for 70% of the annual precipitation. The average annual evaporation is about 1100 mm. The study area is shown in Fig. 1. In Shijin irrigation district, multiple water suppliers (Huangbizhuang Reservoir, Gangnan Reservoir and groundwater aquifers) are available for irrigation. Huangbizhuang Reservoir and Gangnan Reservoir are connected with the irrigation district. Reservoir water irrigates all eight subareas through water supply channels. Ground water merely irrigates subareas 2, 3, 4, 6, 7 and 8, through respective pipelines distributed in the subareas. The irrigation data comes from local investigations. Irrigation water comes from upstream reservoir, and local well exploitation is only permitted in some faraway region, where farms cannot afford high water transportation cost for long distance from the water supply canal. Number of irrigations also has a relationship with the water availability within a year. For example, there is only one irrigation during spring in a dry year (when the probability of occurrence of the flows is higher than 75%), but there might be two or three times irrigation during spring and autumn in a wet year (when the probability of occurrence of the flows is lower than 25%). The classification of different hydrological

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Fig. 1. The study area, Shijin irrigation district, is located in central south of Hebei plain, China. The distribution of reservoir, waste water canals and rivers are exhibited (a). The distribution of water supply canals is exhibited in (b). The irrigation district is composed of two independent areas, for distribution of cities and villages. According to the water supply canals, eight subareas are divided, delineated in (c).

years depends on local frequency distribution of annual precipitation which is shown in Fig. 2. Three principal crops, i.e. winter wheat, maize, and cotton, are produced in Shijin irrigation district at present. A double cropping system, which combined two crops, winter wheat and summer maize, in a single year rotation is applied in most of the irrigation district (Binder et al., 2007). The double cropping system includes winter wheat and after harvest, maize is immediately sown without tillage, and is harvested at the end of September (He et al., 2009). Some subareas have a small area of cotton, owing to higher costs of water transport and utilization for longer channels, higher terrain,

Fig. 2. Frequency distribution of annual precipitation. The statistical data is collected from 1963 to 2012.

or sandy soil suitable for cotton. The cotton producing subareas are shown in Fig. 3. Due to large spatial and temporal variations of the available water resources, the optimal agricultural irrigation water allocation plans vary from year to year correspondingly. According to the statistical data from 2007 to 2012, there are mainly three irrigation targets (annual volume of water used for irrigation), which were draw up under limitation of water availability for irrigation and adopted by the managers of Shijin irrigation district. The irrigation target was 250 × 106 m3 in dry years, 350 × 106 m3 in normal years, and 400 × 106 m3 in wet years. Some input data of different subareas are presented in Table 1. Moreover, utilization coefficient of ground water (the ratio of the net water utilized by crops and the total pumping capacity in an irrigation period) is 0.850. Market price of winter wheat is [2.16, 2.4] RMB/kg, the values of maize and cotton are [1.96, 2.05] RMB/kg and [3.9, 4.3] RMB/kg respectively. Jensen model (Jensen, 1968) can describe the relationship between crop yield and ET under growth stages quite well (Li, 1998). Parameters in Jensen model for all growth stages of the three crops come from Farmland Irrigation Research Institute, Chinese Academy of Agricultural Sciences (Xiao et al., 2008) as shown in Table 2. Since the establishment of Shijin irrigation district, the irrigation area is expanding gradually, and crop production is increasing year after year. With the rapid population growth and economic development, water resource has become the primary factor restricting the sustainable social and economic development of the irrigation district. Domestic and industrial water consumption will continue to limit the water available to agriculture. Additionally, it is

G. Yang et al. / Agricultural Water Management 158 (2015) 82–98

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Fig. 3. The study system includes water sources, subareas and crops. Surface water from Huangbizhuang Reservoir and Gangnan Reservoir can be used in all subareas. Ground water irrigates subareas 2, 3, 4, 6, 7 and 8. Winter wheat and maize are planted in all subareas. The cotton producing subareas are subarea 3, 4, 5, 6, 7, and 8.

difficult to obtain accurate information because of the lack of realtime online monitoring system in Shijin irrigation district. Most monitoring information, such as precipitation and the amount of irrigation water determined by instruments, is imprecise, insufficient or untimely, so the managers of Shijin irrigation district cannot make more effective decisions without a comprehensive understanding of the irrigation district situation. The current water utilization coefficient of irrigation canals is still low for obsolete agricultural irrigation engineering. A large amount of irrigation water is needed for a non-advanced irrigation mode. Unsound water-resources management planning leads to unreasonable allocation and utilization of water resources. These factors limit the irrigation district benefit. Therefore, in current conditions, it is necessary to optimize the irrigation water resources in Shijin irrigation district for not only improving local crop production but also increasing income of local farmers. An advanced optimization model can be used for allocating agriculture water in a more efficient way.

3. Methodology In irrigation systems, multiple parameters exist with respective special characteristics. Such parameters can hardly be reflected in mathematical models as determined values due to insufficient monitoring practices, poor data availability or limitations of practical conditions in obtaining exact data (Lu et al., 2012). These uncertainty parameters widely exist in many real-world systems, so the relevant water allocation optimization analysis and the associated decision may be affected. It is difficult to apply limited data and extensive uncertainties by conventional programming approaches. To handle such problems, a mathematical tool (i.e. interval analysis) can be applied for decision makers. These uncertain parameters can be described as interval parameters which represent the fluctuation range of the data when the distribution within the range can hardly be clarified. The results of interval mathematical programming can cope with complex uncertainty parameters and provide more convenience to managers (Huang

Table 1 Input data by different subarea. Subarea

Irrigation area of the crops (ha) Winter wheat

1 2 3 4 5 6 7 8

326.13 39.00 13,893.33 8379.33 18.00 13,549.53 9427.33 9660.07

Maize 326.13 39.00 14,043.20 8339.33 18.00 13,549.53 9946.67 9660.07

Groundwater available (106 m3 )

Water supply cost (RMB/m3 )

Irrigation scheme efficiency

0 0 (19.29, 20.79, 21.44) (14.61, 18.39, 23.40) 0 (5.97, 6.75, 7.70) (6.86, 7.08, 7.14) (12.06, 16.69, 21.42)

0.205 0.205 0.184 0.190 0.205 0.218 0.204 0.198

0.800 0.800 0.611 0.592 0.800 0.627 0.645 0.623

Cotton 0 0 1255.13 2633.33 6.20 7718.93 8614.60 2350.67

Note: In the groundwater available column, the numbers in the brackets indicate the minimum, mean and maximum available groundwater respectively.

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Table 2 Parameters in Jensen model for all growth stages. Crop

Winter wheat

Crop

Maize

Crop

Cotton

Parameters

Growth stages

Name

Seedling

Overwintering

Reviving

Booting

Heading

Ripening

 T (day) ETcm (mm) Ym (kg/ha)

0.1721 55 66.00 7569

0.0411 50 80.40

0.0591 50 115.96

0.1694 30 134.00

0.3108 20 67.00

0.1895 20 127.82

Parameters

Growth stages

Name

Seedling

Booting

Tasseling

Ripening

 T (day) ETcm (mm) Ym (kg/ha)

0.0557 25 28.50 6900

0.1106 30 62.80

0.3197 15 133.00

0.2113 35 201.00

Parameters

Growth stages

Name

Seedling

Budding

Boll-forming

Boll-opening

 T (day) ETcm (mm) Ym (kg/ha)

0.0730 40 171.80 1320

0.3166 30 100.20

0.1320 50 256.60

0.0422 70 61.20

Note: ETcm is the maximum evapotranspiration obtained under non-limiting water conditions;  is the sensitivity to water stress during the growth stage; T is the duration of each growth stage; Ym is the maximum yield under given management conditions.

et al., 1993; Bass et al., 1997). The introduction of interval parameters into irrigation systems makes it possible to generate optimal strategic plans in consideration of parameter uncertainty.

Table 3 Symbols and their meanings for objective function (1a). Symbol

Maximal value

Meaning

i

iCorp = 3

j k

iDistrict = 8 iSource = 2

Winter wheat (i = 1) Maize (i = 2) Spring cotton (i = 3) Eight subareas Water from reservoir (k = 1) Water from wells (k = 2) Six growth stages of winter wheat, including seedling, overwintering, reviving, booting, heading and ripening (i = 1) Four growth stages of maize, including seedling, booting, tasseling and ripening (i = 2) Four growth stages of cotton, including seedling, budding, boll-forming and boll-opening (i = 3)

3.1. Modeling formulation Considering the local development plan, available water sources and other related restrictions, decision makers have to face two challenges: (1) how to plan multiple available sources in a most economical manner, and (2) how to allocate agricultural water during crop growth periods to pursue higher system benefit. Thus, in response to the above concerns, a nonlinear fuzzy interval irrigation optimization model (NFIIOM) for Shijin irrigation district is formulated as follows. 3.1.1. System objective For most agricultural systems, the objective is relevant to economic concern, such as maximization of the system benefit, and minimization of the system cost (Tilman et al., 2002). In this study, the optimal system benefit arises from the water allocated to farmers with an extended consideration of uncertain parameters. Consequently, the study problem can be formulated as follows:

 

iCropiSubarea

Max f1± =

i=1

j=1

± CC,i Aij Ym,i

  ETc,±

iStagei

l=1

ETcm,

ijl il

il

iStage1 = 6 l iStage2 = 4 iStage3 = 4

stage l. The symbols of the index and their meanings are listed in Table 3. In this model, the objective is the maximization of the system benefits from local crop production, and the decision variables are

  

iCropiSubareaiSourceiStage

−

i=1

j=1

k=1

where f± , (RMB), benefit of local farmers; i, index for crops (i = 1, 2, . . ., iCorp); j, index for subareas (j = 1, 2, . . ., iSubarea); k, index for two water resources (k = 1, . . ., iSource); l, index for growth stage of crop i (l = 1, 2, . . ., iStagei ); iCorp, the number of crop species; iSubarea, the number of subareas; iSource, the number of water sources; ± iStagei , the number of growth stages of crop i; CC,i , (RMB/kg), market price of crop i; Aij , (ha), planting area of crop i; Ym,i , (kg/ha), the max± imum yield of crop i under given management conditions; ETc, , ijl (mm), the actual evapotranspiration; ETcm,ijl , (mm), the maximum evapotranspiration obtained under non-limiting water conditions; il , the sensitivity of crop i to water stress during the growth stage ± , (m3 ), water l; CW,j , (RMB/m3 ), water supply cost in district j; Wijkl supply for crop i in district j from water source k during the growth

± CW,j Wijkl

(1a)

l=1 ± Wijkl . In the objective, the yields of crops are calculated by Jensen Model which expresses a mathematical relationship between rela± tive yield (Ya,i /Ym,i ) and relative evapotranspiration (ETc,ijl /ETcm,il ), where Ya,i means the actual yield of crop i.

3.1.2. Constraints The objective is subjected to a set of constraints in water availability, evapotranspiration, reservoir water priority, fairness, and a number of technical concerns. The constraints define the reasonable range of some variables, such as nonnegative constraints. (1) Evapotranspiration constraints ± ETc,ijl ≥ ETc min,il

∀i, j, l

(1b)

G. Yang et al. / Agricultural Water Management 158 (2015) 82–98 ± ETc,ijl ≤ ETcm,il

∀i, j, l

⎧ = ETcm,il if (P˜ ijl + Rijl > ETcm,il ) ⎪ ⎪ ⎪ ⎪ ⎨ iSource  ± ± ETc,ijl Wij1l ⎪ ⎪ ⎪ ⎪ ⎩ = 0.1 k=1 + P˜ ijl + Rijl otherwise

(1c)

˜ W,j ) in each subarea, and shall meet the requirements irrigation (W of the local groundwater development and utilization plan. (4) Reservoir water priority constraints

(1d)

± Wijkl



∀i, j, l

Aij

where ETcmin,il , (mm), the minimum evapotranspiration for survival of crop; P˜ ijl , (mm), precipitation for crop i in district j during the growth stage l; Rijl , (mm), other water estimated value for crop i in district j during the growth stage l, which is decided by water recharge (such as groundwater recharge) and water losses (such as seepage losses and surface evaporation); 0.1 is a unit conversion factor. ± ) must be larger than the miniCrop evapotranspiration (ETc,ijl mum evapotranspiration (ETcmin,il ) for survival. Meanwhile, if too much water is supplied by precipitation and groundwater recharge, excess water will be discharged through drainage canals, ensuring ± that (ETc,ijl ) is not more than (ETcm,il ). (2) Reservoir water availability constraints



± Wijkl

= 0 if (P˜ ijl + Rijl > ETcm,il or ˇil = 0) ≤ Qmax,jk Til

otherwise

87

k = 1, ∀i, j, l (1e)

=0

± if (Wij1l ≤ Qmax,jk Til and ˇil = 1)

≤ Qmax,jk Til

otherwise

±    Wijkl i=1

j=1

l=1

jk

˜R ≤W

k=1

(1f)

where ˇil , dimensionless index for irrigating crop i in district j from reservoir; Qmax,jk , (m3 /d), maximum flow in district j from water source k; Til , (d), the number of days for crop i during the growth stage l; jk , irrigation scheme efficiency in district j from water ˜ R , (m3 ), the amount of water available for irrigation source k; W from reservoirs. Index ˇil = 0 means that water supply from reservoirs for crop i during the growth stage l is not allowed. Contrarily, ˇil = 1 means that water supply from reservoirs for crop i during the growth ± stage l is allowed. Water supply from reservoirs (Wijkl , k = 1) is equal to zero outside spring irrigation period or autumn irrigation period (˛il = 0). When precipitation and groundwater recharge is P˜ ijl + Rijl > ETcm,il , water supply from reservoirs is equal to zero ± too. For every subarea, water supply from reservoirs (Wijkl ) has to be less than the maximum water available (Qmax,kj Til ) during the corresponding growth stage through the respective canal. Moreover, the planning of the total amount of water supply from reservoirs to each subarea shall not exceed approved water drawings for ˜ R ). irrigation (W (3) Ground water availability constraints



± Wijkl

= 0 if (P˜ ijl + Rijl > ETcm,il ) ˜ W,j ≤ jk W

±   Wijkl

otherwise

iStagei     ± ± ETc,ij l − ETc,ij l 1 2 l=1 j1 j2 l=1

iSubareaiSubarea iStagei

(1g)

i=1

l=1

jk

˜ W,j ≤W

k = 2, ∀j

(1h)

˜ W,j , (m3 ), the amount of groundwater available in district where W j for irrigation. ± Water supply from wells (Wijkl , k = 2) is equal to zero when precipitation and groundwater recharge is P˜ ijl + Rijl > ETcm,il . The planning of the total amount of groundwater exploited shall not exceed the total amount of permitted groundwater drawings for

∀i

(1j)

1

bound of the Gini coefficient. Without considering the principle of fairness, the farmers near the water sources can be allocated more water resources for bigger irrigation scheme efficiency, while the farmers far away the water sources have to obtain lower crop yield with less water resources allocated. To achieve a fair allocation of water from a resource, the Gini coefficient is introduced. The Gini coefficient is known as the Gini index or Gini ratio, which is a widely used measure of economic inequality (Lambert and Aronson, 1993). The Gini coefficient can range from 0 to 1, with higher value indicating more unequal distribution. It means complete equality when the Gini coefficient is 0, and means completely inequality when the Gini coefficient is 1. According to the provisions of the relevant United Nations organizations, less than 0.2 represents the absolute average; 0.2–0.3 represents relatively average; 0.3 to 0.4 represents a relatively reasonable; 0.4–0.5 represents a big gap; more than 0.5 represents the disparity (Sun, 2013). Absolute average means that every farmer earn exactly the same amount of money; relatively average means that every farmer earn almost the same amount of money; a relatively reasonable means that an acceptable gap exist. In the system, let G0 = 0.4 in constraint (1j), owing to the interand Xu, 2011). In addition, national

warning level of 0.4 (Zhang 

iSubarea iStagei j

j

± ETc,ijl /N

∀i.

(6) Nonnegative constraints ± Wijkl ≥ 0 ∀i, j, k, l

iCropiStagei

≤ G0

where j1 and j2 , index for subareas (j = 1, 2. . ., iSubarea); N, the num± ber of subareas; i , (mm), average ETc,ij for crop i; G0 , the upper l

i =

k = 2, ∀i, j, l

(1i)

The ground subsidence is associated with the continuously increasing groundwater exploitation in the deep, confined aquifer (Chen et al., 2003). Over-exploitation of ground water can lead to the fall of the water table, even a large surface subsidence (Zhang et al., 2010). To prevent over-exploitation of ground water, control surface subsidence, and protect ground water from other environmental problems, decision makers give priority to the development and utilization of surface water, and limit the exploitation of ground water. In the system, if current water supply from reser± voirs (Wijkl , k = 1) is less than maximum water supply through canal (Qmax,j Til ) and water supply from reservoirs for crop i during ± the growth stage l is allowed (ˇil = 1), then Wijkl , k = 2 is equal to zero. That is to say, water supply from wells is not allowed when more water from reservoirs can be utilized. (5) Fairness constraints

2N 2 i iCropiSubareaiStagei

k = 2, ∀i, j, l

(1k)

Most of the decision variables are nonnegative in real-world problems. Water supply for crop i in district j from water source ± k during the growth stage l (Wijkl ) is nonnegative too. Consequently, the problem is formulated as: objective function (1a) subject to formulae (1b)–(1k). In this problem, decision ± variables are Wijkl , as the main control parameters in irrigation ± scheduling; intermediate variables are ETc,ijl , also decided by the model; while the others parameters are entered into the solving program as constants.

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G. Yang et al. / Agricultural Water Management 158 (2015) 82–98

precipitation value, the lower bound is defined as the minimum precipitation value, and the most probable value is defined as the main precipitation value. Thus, model (1) can be transformed into submodel (3) and submodel (4). Submodel (3) corresponding to f+ can be formulated as model A listed in Appendix. A solution for f+ provides the extreme + , f + be the upper bound of system under uncertainty. Let Wijkl opt opt solution of the submodel. Then submodel (4) corresponding to the lower bound of the objective value (f− ) can be formulated as model − B listed in Appendix. Let Wijkl , f − be the solution of the subopt opt model. Then, we have solutions for NFIIOM as follows: ± − Wijkl = [Wijkl



opt

− + f ± = fopt , fopt

+ , Wijkl

opt

] ∀i, j, k, l



Fig. 4. Triangular membership function and ˛-cut interval.

4. Results analysis

3.2. Method of solution The above model can be solved using a two-step method (TSM) (Huang and Cao, 2011). TSM has been widely used to deal with such programs with uncertain parameters. According to TSM, the original interval programming model is firstly transformed into two deterministic submodels based on an interactive algorithm. The first submodel is to obtain the upper bound of system benefit, and the second is to obtain the lower bound conditions. By solving them sequentially, and then integrating the solutions of the two models, interval solutions for decision variables and the objective value can be acquired. Some parameters in the system are fuzzy numbers. There are many approaches that can be adopted to deal with fuzzy numbers. One approach is triangular possibility distribution (YoungJou and ChingLai, 1992), and the conventional approach to solve fuzzy programming is defuzzifing and derandomizing fuzzy numbers, such as via ˛-cut levels. ˛-cut levels are also called level sets which perfectly describe the fuzzy degree of membership level (Chen, 1985). From a practical point of view, ˛-cut levels are important for a quantification of fuzzy event, particularly with the maximal ˛. Fig. 4 shows the triangular membership function and ˛-cut interval for ˛-cut level. A fuzzy number X can be illustrated as a triangular fuzzy number X˜ = (X, X − Xmin , Xmax − X) (Zwick, 1993), with the following membership function (2a). ˛-cut interval is [X˛− , X˛+ ] at ˛ level. Different ˛-cuts can reflect the changing trend of the optimal solutions under different degrees of uncertainty.

(x) =

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ x − Xmin ⎪ ⎨

X − Xmin

if (x ≤ Xmin ) if (Xmin < x ≤ X)

Xmax − x ⎪ ⎪ if (X < x ≤ Xmax ) ⎪ ⎪ X ⎪ max − X ⎪ ⎩ 0

(2)

if (Xmax < x)

where (x), membership function; x, the variable; X, the mode ˜ Xmin , the which is the most probable value; the lower bound of X; ˜ Xmax , the upper bound of X. ˜ lower bound of X; For this case, Shijin irrigation district experienced construction and water-saving reform in recent years, the statistics data of precipitation in all subareas is also recorded for a few years. The precipitation over the years fluctuated in interval with discrete form. For limited statistics data, the fuzzy parameter method is more suitable to describe them. Hence, in the system model, precipitation is considered as a fuzzy parameter (P˜ ijl ) and described by a triangular membership function. According to the statistical data, the upper + − bound (Pijl ), the lower bound (Pijl ) and the most probable value (Pijl ) are determined. The upper bound is defined as the maximum

In this study, three different scenarios are defined based on different water availabilities for irrigation from reservoirs. The three levels of water availability could reflect local decision makers’ attitudes to agricultural crop and the abundance of available water resources. Scenario 1 is under dry years, the annual precipitation is less than 414 mm; scenario 2 is under normal years, the annual precipitation is more than 414 mm and less than 646 mm; and scenario 3 is under wet years, the annual precipitation is more than 646 mm. Six ˛-cut levels (˛ = 0.0, 0.2, 0.4, 0.6, 0.8, or 1.0) are set, which makes the information contained relatively uniform. The solutions obtained can not only provide an effective evaluation under preset scenarios, but also reveal the associated economic implications. Different scenarios would result in varied system benefits. As shown in Fig. 5, the figure gives the system benefits of Shijin irrigation district under different ˛-cut levels. As ˛-cut level increases under one scenario, the upper bound of local farmers’ benefit decreases, while the lower bound increases. When ˛-cut level is closer to 1, the probability of the results would be lower. That means the lower-upper bound is closer to the higher-lower bound. In other words, the interval of the solution is narrow. The results show that the system benefit could reach 0.99 × 109 RMB or more in dry years, while the corresponding value could reach 1.08 × 109 RMB or more for more amount of water available for irrigation from reservoirs. From Fig. 5, it is obvious that the optimal system benefit under scenario 2 is higher than that under scenario 1, and lower than that under scenario 3. According to the optimization results, the lowest upper bound and the highest lower bound are 1.53 × 109 RMB under dry years, 1.61 × 109 RMB under normal years, and 1.62 × 109 RMB under wet years, respectively. The less the water availability, the greater the influence on economic yield. The gap of system benefit is 0.08 × 109 RMB between under dry years and normal years, but the gap is 0.01 × 109 RMB between under normal years and wet years. The gap between normal years and wet years is not obvious. This suggests that the system could achieve higher benefits in normal years or wet years. In order to facilitate the comparison and analysis, set ˛ = 0.4 to optimize water allocation in all subareas under each scenario. The optimal interval results in dry years are shown in Fig. 6, the optimal interval results in normal years are shown in Fig. 7, and the optimal interval results in wet years are shown in Fig. 8. Comparing Fig. 6(a) with (d) and (g), it can be found that the optimal reservoirs water allocation to winter wheat is greater than the optimal reservoir water allocation to maize and cotton. For example, under scenario 1, reservoir water allocation to winter wheat is [29.03, 35.61] × 106 m3 in subarea 3, [14.47, 18.35] × 106 m3 in subarea 4, [21.30, 28.67] × 106 m3 in subarea 6,

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Fig. 5. System benefit under different scenarios and ˛-cut level.

Fig. 6. Optimal water allocation in different subareas (dry year, ˛ = 0.4). (a), (d), and (g) are optimal water allocation to winter wheat; (b), (e), and (h) are optimal water allocation to maize; (c), (f), and (i) are optimal water allocation to cotton. (a), (b), and (c) are optimal water allocation of total water resources; (d), (e), and (f) are optimal water allocation of reservoirs water; (g), (h), and (i) are optimal water allocation of ground water.

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Fig. 7. Optimal water allocation in different subareas (normal year, ˛ = 0.4). (a), (d), and (g) are optimal water allocation to winter wheat; (b), (e), and (h) are optimal water allocation to maize; (c), (f), and (i) are optimal water allocation to cotton. (a), (b), and (c) are optimal water allocation of total water resources; (d), (e), and (f) are optimal water allocation of reservoirs water; (g), (h), and (i) are optimal water allocation of ground water.

[17.35, 22.54] × 106 m3 in subarea 7, and [19.17, 22.91] × 106 m3 in subarea 8. However, the maximum reservoir water allocation to maize is [5.14, 11.51] × 106 m3 in subarea 3, and reservoirs water allocation to cotton is 0 in all subareas. Meanwhile, the optimal ground water allocation to winter wheat is also greater than the optimal ground water allocation to maize and cotton. For example, under scenario 1, ground water allocation to winter wheat is [9.72, 10.82] × 106 m3 in subarea 3, [9.71, 13.27] × 106 m3 in subarea 4, [9.09, 13.86] × 106 m3 in subarea 8. However, the maximum ground water allocation to maize is 7.16 × 106 m3 in subarea 3, and the maximum ground water allocation to cotton is [1.01, 1.93] × 106 m3 in subarea 4. Thus, the total water allocation to winter wheat is largest, and the total allocation to cotton is smallest. The results indicate that most of the decision variables, except for ground water, are intervals. Generally, solutions presented as intervals illustrate that the related decisions are sensitive to the input parameters of uncertain models (Huang et al., 2012). In case of dry years, water allocation to winter wheat should be first decreased on the premise of guaranteeing its minimum promised target. The maize should be first guaranteed since it brings the highest benefit.

Comparing Fig. 7(a) with (b) and (c), or comparing (d) with (e) and (f), it is obvious that winter wheat has greater interval than maize and cotton. For example, total water allocation to winter wheat is [30.81, 41.17] × 106 m3 in subarea 8, with an interval of 10.36 × 106 m3 . Total water allocation to maize is a number, 7.99 × 106 m3 in subarea 8. There is no water allocation to cotton in the same subarea. Winter wheat has larger degree of uncertainty than maize as well as cotton. One reason is that water production efficiency of maize is higher during the irrigation period. Maize will offer higher and more stable yields, so water supply should meet water demand of maize at first. Another reason is that the planting area of cotton is so small that cotton does not require large quantities of irrigation water. Thus, the system benefit is mainly affected by economic benefits of winter wheat in Shijin irrigation district, owing to large variability of optimal water allocation. To compare these subareas’ dependence on irrigation water, ring graphs are drawn in Fig. 9 (under scenario 1, ˛ = 0.4), Fig. 10 (under scenario 2, ˛ = 0.4), and Fig. 11 (under scenario 3, ˛ = 0.4). The inner rings refer to lower bounds of optimal water allocation, and the outer rings refer to upper bounds. The eight colors represent the eight subareas.

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Fig. 8. Optimal water allocation in different subareas (wet year, ˛ = 0.4). (a), (d), and (g) are optimal water allocation to winter wheat; (b), (e), and (h) are optimal water allocation to maize; (c), (f), and (i) are optimal water allocation to cotton. (a), (b), and (c) are optimal water allocation of total water resources; (d), (e), and (f) are optimal water allocation of reservoirs water; (g), (h), and (i) are optimal water allocation of ground water.

As shown in Fig. 9(a) and (d), Fig. 10(a) and (d), Fig. 11(a) and (d), the main subareas, which plant winter wheat and maize and consume reservoirs water are subareas 3, 4, 6, 7, and 8. Among these, irrigation water consumption in subarea 3 is the largest for its largest planting area. From Fig. 9(b), Fig. 10(b) and Fig. 11(b), the main subareas, which plant winter wheat and maize and consume ground water are subareas 3, 4, and 8. The sum of the three percentages accounts for three-quarters of total ground water supply. From Fig. 9(g), Fig. 10(g) and Fig. 11(g), reservoirs supply cotton mostly in subareas 3, 4, 5, 6, 7, and 8. According to similarities between Fig. 9(a) and (c), Fig. 10(a) and (c), and Fig. 11(a) and (c), it can be inferred that optimization of local irrigation water resources for winter wheat mainly relies on reservoirs water supply. Similarly, maize mainly relies on reservoirs water supply too. From Fig. 9(h), Fig. 10(h) and Fig. 11(h), most of groundwater irrigated to cotton occurs in subarea 4. According to similarities between Fig. 9(h) and (i), Fig. 10(h) and (i), and Fig. 11(h) and (i), it can be inferred that optimal local cotton mainly rely on ground water supply. Because maize and winter wheat are major water consumers, the total optimal water allocation percentages mainly depend on their share of water resources. Hence, optimal water allocation percentage to all crops is similar to optimal water allocation percentage to winter wheat. Comparing the results, it is obvious that optimal water allocation percentages, except for cotton, are basically affected by the scenarios, due to the similarity of graphs. However, irrigation water of local cotton does not play a critical role in total water supply between of the smallest planting area.

In Shijin irrigation district, irrigation usually occurs in spring and autumn, which has great effect on growth of winter wheat. Three or four growth stages of winter wheat can be affected by reservoirs water availability. Comparatively speaking, the water supply situation for maize and cotton is more simple due to only one growth stage, during which reservoirs water can be abstracted. As shown in Fig. 12, the upper bound and lower bound are exhibited with six growth stages under scenario 2 (˛ = 0.4). As can be seen from Fig. 12, the distributions of irrigation water with growth stages have higher level of consistency in eight subareas. The upper bound of optimal water allocation remains high during reviving and booting stage, and remains low in seeding and heading stage. The large interval mainly occurs during reviving stage, such as [6.36, 14.61] × 106 m3 with interval 8.25 × 106 m3 in subarea 3, and [4.85, 13.84] × 106 m3 with interval 8.99 × 106 m3 in subarea 6. Contrarily, all interval values are zero in seedling, booting, and heading stages. From Fig. 12, it can be clearly seen that variabilities are mainly concentrated during overwintering and reviving stages. Therefore, willingness to accept a low system benefit could guarantee meeting the small irrigation water supply during overwintering and reviving stages, while a strong desire to acquire a high system benefit could run into a high risk of irrigation water shortage during the later stages. In scenario 1, variability of total optimal water allocation to winter wheat and maize is large, as shown in Fig. 6(a) and (b); in scenario 2, variability of total optimal water allocation to winter wheat is large, as shown in Fig. 7(a), but small to maize, as shown in Fig. 7(b); in scenario 3, variability of total optimal water allocation is

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Fig. 9. Optimal water allocation percentage (dry years, ˛ = 0.4). (a), (d), (g), and (j) are optimal water allocation percentages of reservoirs water; (b), (e), (h), and (k) are optimal water allocation percentages of ground water; (c), (f), (i), and (l) are optimal water allocation percentages of total water resources; (a), (b), and (c) are optimal water allocation percentages to winter wheat; (d), (e), and (f) are optimal water allocation percentages to maize; (g), (h), and (i) are optimal water allocation percentages to cotton; (j), (k) and (l) are optimal water allocation percentages to all crops.

small, as shown in Fig. 8(a) and (b). The results indicate that interactions exist among supplies for multiple competing crops when water resources available are limited, due to benefit effects on the water-allocation patterns. Under scenario 1, the water allocation to three crops could be decreased in case of insufficient water. However, under scenario 2, optimal water allocation to maize should be satisfied firstly, due to its highest benefit, and water allocation

to winter wheat and cotton should be decreased in case of insufficient water. Under scenario 3, it is easy to obtain higher system benefit due to the abundant reservoir water. Under this condition, these crops can be irrigated in accordance with the upper bound of optimal water allocation. Fig. 13 illustrates the comparison between actual system benefit and optimal system benefit. The system benefits were calculated

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Fig. 10. Optimal water allocation percentage (normal years, ˛ = 0.4). (a), (d), (g), and (j) are optimal water allocation percentages of reservoirs water; (b), (e), (h), and (k) are optimal water allocation percentages of ground water; (c), (f), (i), and (l) are optimal water allocation percentages of total water resources; (a), (b), and (c) are optimal water allocation percentages to winter wheat; (d), (e), and (f) are optimal water allocation percentages to maize; (g), (h), and (i) are optimal water allocation percentages to cotton; (j), (k) and (l) are optimal water allocation percentages to all crops.

at ˛ = 1. From the figure, it is obvious that the actual system benefits are lower than the optimal system benefit. In 2011, the actual system benefit was 1.60 × 109 RMB, while the optimal system benefit was 1.61 × 109 RMB. On the other hand, there are larger gaps between actual and optimal system benefits, such as 0.26 × 109 RMB in 2010. Thus the current water resource allocation could be improved and reach higher system benefit. The optimal water allocation could lead to higher benefit than the present allocation.

In each year, the system could reach higher level of management by allocating water resource among subareas based on the model. For example, comparison between actual and optimal yield in year 2009 is shown in Fig. 14. Yield of winter wheat changes little between actual situation and optimal situation. In subarea 2, the yield changes from 7200 kg/ha to 5759 kg/ha with optimization. This is the biggest change for winter wheat in 2009. Yield of maize would be reduced in all subareas. In addition, yield of cotton

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Fig. 11. Optimal water allocation percentage (wet years, ˛ = 0.4). (a), (d), (g), and (j) are optimal water allocation percentages of reservoirs water; (b), (e), (h), and (k) are optimal water allocation percentages of ground water; (c), (f), (i), and (l) are optimal water allocation percentages of total water resources; (a), (b), and (c) are optimal water allocation percentages to winter wheat; (d), (e), and (f) are optimal water allocation percentages to maize; (g), (h), and (i) are optimal water allocation percentages to cotton; (j), (k) and (l) are optimal water allocation percentages to all crops.

would be increased substantially. In subarea 4 and subarea 8, yield of cotton would reach 1320 kg/ha. Overall, optimal system could be more equal for each subarea, and the system benefit could be higher than the actual situation. In the case study, the solutions for optimal water allocation to achieve maximum economic gain have produced different results for three scenarios. Managers can set ˛-cut levels according to their own willingness, and get solutions from NFIIOM. The model takes

into account Gini coefficient to achieve a fair allocation of water, so the solutions would be receptive for the vast majority of farmers in the irrigation district. To acquire a high system benefit, the region closer to surface water, such as subarea 1, should be allocated more water than the region far away from surface water, such as subareas 7 and 8. In a larger sense, the water distribution is planned by Management Bureau of Shijin Irrigation Area in Hebei. Therefore, the local government can easily distribute water for every

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Fig. 12. Irrigation water allocation to winter wheat during its growth stages (normal years, ˛ = 0.4). (a), (b), . . ., (h) represent subareas 1, 2, . . ., 8. Solid lines represent upper bounds of total optimal water allocation, and dash lines represent lower bounds of total optimal water allocation.

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Fig. 13. Comparison between actual and optimal system benefits.

subarea in accordance with the optimal water allocation schemes. As usual, the water distribution is still concentrated in several specific growth stages in which the precipitation is not sufficient to meet the crop growth needs. The priority water supply to crops is surface water in the several stages, while ground water can be used in the other periods for supplementary irrigation. Through allocating water with the method developed in this study, more accurate water allocation schemes can be provided for all periods, crops and subareas. This has not been achieved in the past rough planning mode. NFIIOM takes into account two water sources, surface water and ground water, so the model can be directly used for optimal water allocation of other irrigation districts without any modification.

5. Conclusions A nonlinear fuzzy interval programming model was developed for agricultural irrigation water resources planning and management. NFIIOM is based on nonlinear fuzzy interval programming. It can deal with nonlinearities in the benefit objective and uncertainties expressed as discrete intervals and fuzzy numbers. Solutions for the optimal water allocation under different scenarios were obtained. The main advantage of NFIIOM is its capability of incorporating multiple water resources, multiple subareas, multiple crops, and multiple crop growth stages within the optimization framework. The NFIIOM model represents a new effort to obtain optimal water allocation schemes through maximizing local farmers’ economic benefits, and it is subjected to a set of constraints in water availability, evapotranspiration, reservoir water priority, fairness, and a number of technical constraints. The method was applied to a real case of planning agricultural water management in Shijin irrigation district, Hebei province, China, where a number of scenarios based on different water resources management were analyzed. Reasonable solutions have been generated and have provided information of discrete intervals for optimal water allocation. Under fuzzy degree of membership level equal to 1.0, the system benefits are 1.53 × 109 RMB for dry years, 1.61 × 109 RMB for normal years and 1.62 × 109 RMB for wet years. Moreover, different scenarios for water resources planning would lead to varied allocation targets and system benefits. The results can be used to help managers of Shijin irrigation district to establish effective water allocation schemes, and maximize the benefit of local farmers. In addition, there was insufficient precipitation data of continuous years in all subareas, because some observation stations were established within the last dozen years, and channels and wells were constructed, repaired, reconstructed, or abandoned from year to year. To deal with such information, a fuzzy membership function was more suitable than stochastic methods, because stochastic methods requires a lot of objective data, and the study did not have such information. Consequently, future work will continue to focus on refining the model and collect precipitation data in all subareas for many years. For example, multistage or stochastic programming methods can be applied to improve the system model.

Acknowledgements

Fig. 14. Comparison between actual and optimal yield for year 2009.

This research was supported by the National Natural Science Foundation of China (No. 51321001), the Government Public Research Funds for Projects of Ministry of Agriculture (No. 201203077), and International Science & Technology Cooperation Program of China (2013DFG70990).

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Appendix. Model A:

 

iCropiSubarea

f1+

Max

=

i=1

j=1

+ ETc,ijl ≥ ETc min,il + ETc,ijl ≤ ETcm,

il

+ il   ETc,ijl

ETcm,il

l=1

  

iCropiSubareaiSourceiStage

iStagei

+ CC,i Aij Ym,i

−

i=1

j=1

k=1

+ CW,j Wijkl

∀i, j, l

(3b)

∀i, j, l

(3c)

⎧ + = ETcm,il if ((1 − ˛)Pijl + ˛Pijl + Rijl > ETcm,il ) ⎪ ⎨  + iSource + ETc,ijl Wijkl ⎪ + ⎩ = 0.1 k=1 + (1 − ˛)Pijl + ˛Pijl + Rijl otherwise

+ Wijkl

(3a)

l=1

∀i, j, l

(3d)

Aij

=0

+ if ((1 − ˛)Pijl + ˛Pijl + Rijl > ETcm,il or ˇil = 0)

≤ Qmax,jk Til

otherwise

+    Wijkl

k = 1, ∀i, j, l

(3e)

iCropiSubareaiStagei

i=1

j=1



+ Wijkl

jk

l=1

≤ (1 − ˛)WR+ + ˛WR

k=1

(3f)

=0

+ if ((1 − ˛)Pijl + ˛Pijl + Rijl > ETcm,il )

+ + ˛WW,j ) ≤ jk ((1 − ˛)WW,j

otherwise

+   Wijkl

k = 2, ∀i, j, l

(3g)

iCropiStagei

i=1

l=1



+ Wijkl

jk

+ ≤ (1 − ˛)WW,j + ˛WW,j

k = 2, ∀j

(3h)

=0

+ if (Wij1l ≤ Qmax,jk Til and ˇil = 1)

≤ Qmax,jk Til

otherwise

k = 2, ∀i, j, l

(3i)

iStage iSubarea     iStage i  + + + ET − ET ≤ 2NG ETc,ijl ∀i 0 c,ij1 l c,ij2 l

iSubareaiSubarea iStagei

j1

j2

l=1

l=1

j

(3j)

l

+ Wijkl ≥ 0 ∀i, j, k, l

(3k)

+ + where Wijkl are decision variables, ETc,ijl are intermediate variables, and ˇil are binary ones. Model B:

 

iCropiSubarea

f1−

Max

=

i=1 − ETc,ijl ≥ ETc min,il − ETc,ijl

≤ ETcm,il

− CC,i Aij Ym,i

j=1

− il   ETc,ijl l=1

ETcm,il

  

iCropiSubareaiSourceiStage

iStagei

−

i=1

j=1

k=1

− CW,j Wijkl

∀i, j, l

(4b)

∀i, j, l

⎧ − = ETcm,il if ((1 − ˛)Pijl + ˛Pijl + Rijl > ETcm,il ) ⎪ ⎨  − iSource − ETc,ijl Wij1l ⎪ − ⎩ = 0.1 k=1 + (1 − ˛)Pijl + ˛Pijl + Rijl otherwise

− Wijkl

(4a)

l=1

(4c)

∀i, j, l

(4d)

Aij

=0

− if ((1 − ˛)Pijl + ˛Pijl + Rijl > ETcm,il or ˇil = 0)

≤ Qmax,jk Til

otherwise

−    Wijkl

k = 1, ∀i, j, l

(4e)

iCropiSubareaiStagei

i=1 − Wijkl

j=1



l=1

jk

≤ (1 − ˛)WR− + ˛WR

k=1

=0

− if ((1 − ˛)Pijl + ˛Pijl + Rijl > ETcm,il )

− + ˛WW,j ) ≤ jk ((1 − ˛)WW,j

otherwise

(4f)

k = 2, ∀i, j, l

(4g)

98

G. Yang et al. / Agricultural Water Management 158 (2015) 82–98

−   Wijkl

iCropiStagei

i=1

l=1



− Wijkl

jk

− ≤ (1 − ˛)WW,j + ˛WW,j

k = 2, ∀j

=0

− if (Wij1l ≤ Qmax,jk Til and ˇil = 1)

≤ Qmax,jk Til

otherwise

(4h)

k = 2, ∀i, j, l

iStage iStagei iSubarea iStagei i      − − − ETc,ij l ≤ 2NG0 ETc,ijl ETc,ij1 l − 2

(4i)

iSubareaiSubarea

j1

j2

l=1

− + Wijkl ≤ Wijkl − Wijkl

opt

l=1

∀i, j, k, l

≥ 0 ∀i, j, k, l

j

∀i

(4j)

l

(4k) (4l)

− − where Wijkl are decision variables, ETc,ijl are intermediate variables,

and ˇil are binary ones.

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