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An optimization model for a crop deficitirrigation system under uncertaintyPing Guoab, Xiaohong Chenc, Ling Tonga, Jianbing Lib & Mo Liaa Centre for Agricultural Water Research in China, ChinaAgricultural University, Beijing, PR Chinab Environmental Science and Engineering Program, Universityof Northern British Columbia, Prince George, British Columbia,Canadac Centre for Water Resource and Environment, Sun Yat-senUniversity, Guangzhou, PR ChinaPublished online: 03 Dec 2012.

To cite this article: Ping Guo, Xiaohong Chen, Ling Tong, Jianbing Li & Mo Li (2014) An optimizationmodel for a crop deficit irrigation system under uncertainty, Engineering Optimization, 46:1, 1-14,DOI: 10.1080/0305215X.2012.737786

To link to this article: http://dx.doi.org/10.1080/0305215X.2012.737786

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Engineering Optimization, 2014Vol. 46, No. 1, 1–14, http://dx.doi.org/10.1080/0305215X.2012.737786

An optimization model for a crop deficit irrigation systemunder uncertainty

Ping Guoa,b, Xiaohong Chenc*, Ling Tonga, Jianbing Lib and Mo Lia

aCentre for Agricultural Water Research in China, China Agricultural University, Beijing, PR China;bEnvironmental Science and Engineering Program, University of Northern British Columbia,

Prince George, British Columbia, Canada; cCentre for Water Resource and Environment,Sun Yat-sen University, Guangzhou, PR China

(Received 17 July 2012; final version received 26 September 2012)

In this study, an inexact nonlinear programming model under uncertainty is developed by incorporatinga water production function into the crop irrigation system optimization framework. By introducing atime parameter, this model can address the uncertainty associated with the irrigation schedule for differentcrops and their planting stages. The developed model was applied to a case study of an agricultural waterresources management problem to demonstrate its applicability. Through scenario analysis under differentprecipitation levels, the key planting stage of crops and the amount of water for the irrigation schedulethat could significantly affect system benefits were identified. By using intervals to represent uncertainparameters, more reliable and practical decision alternatives were generated through the presented modelin typical hydrological years (i.e. wet, normal and dry years).

Keywords: crop irrigation system; uncertainty; water production function; scenario analysis

1. Introduction

Water shortages have become a major barrier to sustainable water resource management in recentyears (Lu, Huang, and He 2010). This is particularly true for the agricultural sector in arid andsemi-arid regions, where the optimal use of limited water resources is of critical importance formaximizing the benefits from crop production (Khare, Jat, and Ediwahyunan 2006). In orderto provide optimal agricultural irrigation schemes for maximizing the related benefits, variousmathematical programming models have been developed and utilized. However, uncertainty isalways an important factor that affects the management of an agricultural water resources systemowing to the existence of spatial and temporal variations. As a result, optimization techniques thatcan effectively address different uncertainty issues are desired. A number of methods have beenintroduced to address the uncertainties in the water resources management problem (Quinn et al.2004; Rees et al. 2006; Letcher, Croke, and Jakeman 2007; Pahl-Wostl 2007; Li et al. 2009; Zenget al. 2010; Huo et al. 2011). For example, Maqsood et al. (2005) developed an interval-parametertwo-stage optimization model for the stochastic planning of water resources systems. Li, Li et al.(2010) presented a fuzzy-boundary interval-stochastic programming method for planning water

*Corresponding author. Email: Eescxh@Mail.sysu.edu.cn

© 2013 Taylor & Francis

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2 P. Guo et al.

resources management systems. Lu, Huang, and He (2010) proposed an interval-valued fuzzylinear-programming method and applied it to water resources management. Guo, Huang, and Li(2010) developed an inexact fuzzy-stochastic programming model for agricultural water resourcesmanagement under multiple uncertainties. Among various methods for addressing uncertainties,the interval mathematical programming (IMP) model is considered effective when the availableinformation is insufficient for creating a probability distribution or a fuzzy membership function(Huang, 1996, 1998). By characterizing the uncertainty using an upper and a lower bound, theIMP model has been applied to a number of environmental management problems with linearfeatures (Li, Huang, and Nie 2010; Li and Huang 2011).

In general, agricultural water resources management is a nonlinear problem that can hardlybe addressed using conventional IMP models. Although other efforts have been made in waterresources management through interval nonlinear programming (Chen and Huang 2001; Guo,Huang, and Li 2008), fuzzy nonlinear programming and stochastic nonlinear programming (SNP)(Zhu et al. 2009; Pereira and Pinto 1991), few studies have simultaneously dealt with inter-val parameter programming and nonlinear programming, especially for the agricultural waterresources management problem. It is thus important to develop a new modelling method foraddressing this challenge.

Moreover, different crops have different planting stages in an agricultural irrigation problem.However, the previously reported optimization models can only deal with various crops withoutconsidering their planting stages. When the planting stages were considered, those models dealtonly with a single crop. Few studies have addressed the water resources management problemby simultaneously considering various crops and their planting stages. The objective of thisstudy was to develop an inexact nonlinear programming (INP) model for agricultural irrigationmanagement under uncertainty through a case study in an arid region in north-western China.A time parameter was introduced into the optimization modelling framework to facilitate thecomprehensive consideration of various crops and their planting stages. The INP model can beused for a crop deficit irrigation system management that aims to obtain maximum economicbenefits under the condition of limited water resources. Through the modelling application, theoptimal allocation plan of irrigation water from different water supply sources for various studycrops in the study area was obtained, and a detailed analysis of the optimal solutions under differenthydrological conditions is presented. The obtained modelling results could provide a sound basisfor agricultural water resources management decision support.

2. Methodology

2.1. General modelling framework

The growing pressure on agricultural water resources calls for a more productive and efficientirrigation system under the limitation of its current water availability. Figure 1 illustrates thegeneral framework for developing an optimization model to effectively manage the crop irri-gation system associated with parameter uncertainties. It involves a number of procedures,including: (1) collection and examination of uncertain parameters through site investigation;(2) identification of a typical hydrological year based on the probability distribution obtainedfrom 30 years of precipitation data; (3) simulation of the main crops in the study area usinga yield production function whose coefficients were represented as intervals; (4) construc-tion of the INP model for crop irrigation by considering different crops and their plantingstages through time parameters; and (5) solution of the model to obtain optimal irrigation deci-sions. The components of the modelling framework are described in detail in the followingsections.

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Figure 1. Overall optimization modelling process for crop irrigation planning under uncertainty.

2.2. Optimization model formulation for crop irrigation

A crop water production function (CWPF) reflects the relationship between crop production andits water requirement, which is related to the irrigation water, precipitation and soil storage water,and practical evapotranspiration. Many different types of CWPF models have been developed,such as the Stewart model, the Blank model and the Jensen model (Wang, Rong, and Li 1997; Cui,Mou, and Li 1999; Liu, Li, and Li 2004). However, most of them generally present the relationshipbetween crop production and the total amount of water requirement during the entire plantingstages, without capturing the difference in water requirements during each planting stage. In fact,the irrigation frequency and water requirements during each planting stage can greatly affectcrop production. Crop production loss due to water scarcity in the key planting stage is hard tocompensate through more irrigation during other planting stages. In this study, the Jensen modelwas selected as the CWPF model since it can tackle the water requirement not only during theentire crop growth period but also during the individual planting stages. The model formulationis presented as follows:

Ya = max Ymax

n∏k=1

(ETak

ETmax k

)λk

k

(1)

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where Ya represents the real crop production during each planting stage, Ymax represents maximumcrop production, ETak indicates the real crop evapotranspiration during the kth crop planting stage(k = 1, 2, 3, . . . , n), n is the total number of crop planting stages, ETmax k represents the maximumcrop evapotranspiration during the kth crop planting stage, and λk is the sensitivity index of cropproduction to water scarcity during the kth planting stage. In general, ETak/ETmax k ≤ 1, andλk ≥ 0 (but λk can also be less than zero). The value of λk and Ya/Ymax can affect crop productionduring each planting stage, and vice versa.

Using the Jensen model, a system optimization model was developed for crop irrigation underthe condition of insufficient water. The irrigation quantities for the main crops were used as thedecision variables, and the objective function was to maximize the benefit from crop production.The corresponding nonlinear programming model is described as follows:

max f =J∑

j=1

⎛⎝BjAjYmax j

K∏k=1

(∑Ii=1 Xijk + pk

ETmax jk

)λjk⎞⎠

−I∑

i=1

J∑j=1

K∑k=1

(AjXijk/ηij)(CAi + CTij) (2a)

where f is the benefit from crop production, i is the index of water supply source (i = 1 for surfacewater, i= 2 for groundwater), j is the index of crops, k is the index of crop planting stage, Bj isthe price of crop j (RMB kg−1), Aj is the planting area of crop j (104 ha), Ymax j is the maximumproduction of crop j (kg ha−1), ETmax jk is the maximum water requirement of crop j during thekth planting stage (mm ha−1), λjk is the sensitivity index of crop production to water scarcity forcrop j during the kth planting stage, Xijk is the decision variable [i.e. the amount of water irrigationfrom the ith water supply source for crop j during the kth planting stage (m3)], ηij is the utilizationcoefficient from the ith water supply source to the jth crop, CAi represents the cost of water fromthe ith water supply source (RMB m−3), and CTij is the cost of delivering water from the ith watersupply source to the jth crop (RMB m−3).

The modelling constraints are described below:

J∑j=1

K∑k=1

AjXijk/ηij ≤ Qi ∀i (2b)

K∑k=1

AjXijk/ηij ≥ DjQi ∀i, j (2c)

ETmin jk ≤(

I∑i=1

X±ijk + pk

)≤ ETmax jk ∀j, k (2d)

X±ijk ≥ 0 ∀i, j, k (2e)

where constraint (2b) illustrates that the total water irrigation for all the crops during the entireplanting stage from the ith water supply source cannot exceed its total available water (Qi), andQi corresponds to a typical hydrological year (e.g. wet, normal and dry). The available waterfrom groundwater can be at a high level during the dry year, but a low level in the wet year.Constraint (2c) indicates that the total water irrigation for a certain crop j should be greater thana given proportion (Dj) of the total available water from the ith water supply source (Qi), whileDj represents the percentage of the planting area of crop j in the total planting area of the maincrops. Constraint (2d) indicates that the sum of Xijk for crop j from all the water supply sources

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and the effective rainfall (pk) at the kth planting stage should be less than the maximum cropwater requirement but greater than the minimum crop water requirement. Constraint (2e) statesthe non-negativity condition.

Using intervals to represent the uncertainties associated with crop irrigation management, anINP model can then be formulated as:

max f ± =J∑

j=1

⎛⎝B±

j AjYmax j

K∏k=1

(∑Ii=1 X±

ijk + pk

ET±max jk

)λjk⎞⎠

−I∑

i=1

J∑j=1

K∑k=1

(AjX±ijk/ηij)(CA±

i + CTij) (3a)

subject to:

J∑j=1

K∑k=1

AjX±ijk/ηij ≤ Q±

i ∀i (3b)

K∑k=1

AjX±ijk/ηij ≥ DjQ

±i ∀i, j (3c)

ETmin jk ≤ (

I∑i=1

X±ijk + pk) ≤ ETmax jk ∀j, k (3d)

X±ijk ≥ 0, ∀i, j, k (3e)

where the ‘+’ and ‘−’ signs represent the upper and lower bounds of the modelling parameters ordecision variables, respectively. Model (3) can be solved by transforming into two deterministicsubmodels, with one each corresponding to the lower and upper bound solution.

The first submodel is described as follows:

max f + =J∑

j=1

⎛⎝B+

j AjYmax j

K∏k=1

(∑Ii=1 X+

ijk + pk

ET+max jk

)λjk⎞⎠

−I∑

i=1

J∑j=1

K∑k=1

(AjX+ijk/ηij)(CA+

i + CTij) (4a)

Subject to :J∑

j=1

K∑k=1

AjX+ijk/ηij ≤ Q+

i ∀i (4b)

K∑k=1

AjX+ijk/ηij ≥ DjQ

+i ∀i, j (4c)

ETmin jk ≤(

I∑i=1

X+ijk + pk

)≤ ETmax jk ∀j, k (4d)

X+ijk ≥ 0 ∀i, j, k (4e)

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The second submodel is formulated as follows:

max f − =J∑

j=1

(B−

j AjYmax j

K∏k=1

(∑Ii=1 X−

ijk + pk

ET−max jk

))

−I∑

i=1

J∑j=1

K∑k=1

(AjX−ijk/ηij)(CA−

i + CTij) (5a)

Subject to :J∑

j=1

K∑k=1

AjX−ijk/ηij ≤ Q−

i ∀i (5b)

K∑k=1

AjX+ijk/ηij ≥ DjQ

+i ∀i, j (5c)

ETmin jk ≤(

I∑i=1

X−ijk + pk

)≤ ETmax jk ∀j, k (5d)

X−ijk ≥ 0 ∀i, j, k (5e)

By solving the first submodel, the upper bound of the optimal solution of the decision variablesand objective function can be obtained as X+

ijk opt and f +. By solving the second submodel, thelower bound of the optimal solution of the decision variables and objective function can beobtained as X−

ijk opt and f −. Consequently, the optimal interval solutions for crop irrigation canbe found.

3. Overview of the case study system

The developed INP model was applied to a case study in the Wuwei area, north-western China, forsupporting agricultural water resources management. This area is situated within the eastern partof Hexi Corridor and the northern part of the Qilian Mountains. The elevation of its agriculturalareas is 1400–1700 m (Gong 1995). As a region with a typical continental climate, Wuwei has anannual average temperature of 7.8◦C, annual average sunshine of 2968.2 hours and an annual frost-free period of 158 days (Lv 2007). Its annual rainfall and evaporation are 161 mm and 2020 mm,respectively. The distribution of its rainfall is uneven, with 59.4% of annual precipitation occurringfrom July to September. Wuwei has a land area of 3,324,900 ha, with a population of 1,942,400.It is one of the regions of China, and even the world, with the most serious shortage of waterresources. Its water resources total less than 600 m3 per capita and less than 3300 m3 per ha ofland, and there is also serious wastage of water resources owing to ineffective irrigation. Thiscould significantly restrict its economic and social development (Dong, Hu, and Luan 2009).The total area of Wuwei that is irrigated is 18.2 × 104 ha, and its soil texture is a medium loam.Figure 2 shows a map of the case study area. In total, there are15 irrigation zones in Wuwei, witheight of them in LiangZhou section, three in GuLang County, three in Minqin County and one inTianZhu County (Lv 2007).

The main crops in Wuwei include spring wheat, maize, oil flax and seed watermelon. Table 1lists the information of the INP modelling parameters. Two water supply sources (i.e. surface waterand groundwater) were considered. The modelling data were mainly obtained from governmentalreports, field investigations and public survey. Each main crop has different planting stages. Springwheat has four planting stages (seeding stage, jointing stage, heading stage and maturity stage),

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Engineering Optimization 7

Figure 2. Overview of the study area in Wuwei.

Table 1. Information on modelling parameters for the case study area.

Modelling parameters Main crops

Oil SeedWheat Maize flax watermelon

Total planting area of thecrops (104 ha)

11.52

Proportion of each crop’splanting area (%)

26.51 16.33 1.15 1.25

Maximum crop yield(kg ha−1)

Wet year 8227.4 12,509.69 3385.5 3,106.07Normal yearDry year

Irrigation utilizationcoefficient

Surface water Wet year 0.4613Normal year 0.4587Dry year 0.4687

Groundwater – 0.60Water supply from each

source (104 m3)Surface water Wet year [24,889.04, 33,098.35]

Normal year [26,798.25, 36,320.80]Dry year [32,950.82, 34,331.03]

Groundwater Wet year [38,133.62, 45,199.40]Normal year [32,429.96, 46,103.55]Dry year [33,654.50, 39,302.05]

Unit price of each crop(RMB kg−1)

[1.5,2] [1.3,1.8] [3,4] [6,8]

Water price (RMB m−3) Surface water [0.11,0.15]Groundwater [0.2,0.24]

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8 P. Guo et al.

usually spanning from 22 March to 26 July every year. Maize has four planting stages (seeding,shooting, heading, and filling and maturity) from 14 April to 26 July. Oil flax has six plantingstages (seeding, stemming, squaring, blooming, blossoming and maturity) from 17 April to 27August. Seed watermelon has five planting stages (seeding, blooming, expanding, stereotyping

(a)

(c)

(b)

(d)

Figure 3. Sensitivity index of different crops in different hydrological years: (a) spring wheat; (b) maize; (c) oil flax;(d) seed watermelon.

(a) (b)

(c) (d)

Figure 4. Effective rainfall for different crops in different hydrological years: (a) spring wheat; (b) maize; (c) oil flax;(d) seed watermelon.

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Engineering Optimization 9

(a)

(c)

(b)

Figure 5. Upper and lower bounds of the maximum evapotranspiration of spring wheat in different hydrological years:(a) wet year; (b) normal year; (c) dry year.

and maturity) from 1 May to 17 September. In this study, the entire growth period (from 22 Marchto 17 September) for all of the crops was divided into 18 periods (i.e., k = 1, 2, . . . , 18). Eachcrop’s planting stage is within these periods. For example, the filling and maturity stage of maizecorresponds to a k value from 14 to 18. By introducing such a time parameter to the model, thewater deficit sensitivity index (λi), precipitation (Pi) and evapotranspiration (ETmi) for differentcrops during different planting stages in different hydrological years can be obtained, as shownin Figures 3–5.

4. Analysis and discussion of results

Table 2 presents the optimal interval solutions of water irrigation for each crop during its var-ious planting stages for the scenarios of three typical hydrological years (wet, normal anddry). The solutions indicated that the allocation amount of surface water for spring wheatirrigation for a normal year would be [559.22, 838.96], [127.98, 210.79], [32.61, 51.50] and[101.93, 226.13] m3 ha−1 during its seeding, jointing, heading and maturity stages, respectively.It was found that the water irrigation during the seeding stage of spring wheat is much higher thanthat during its other planting stages. This is because the seeding stage is associated with a higherwater deficit sensitivity index. On the other hand, the optimal water irrigation during spring wheat’sheading stage is the lowest because of the negative water deficit sensitivity index at this stage. Interms of the oil flax during its blossoming stage, the modelling results show that the optimal alloca-tion amount of surface water was [202.83, 272.83], [252.55, 306.68] and [215.67, 274.98] m3 ha−1

in the wet year, normal year and dry year, respectively, and the corresponding allocation of ground-water was [442.92, 478.09], [407.90, 480.52] and [298.12, 384.72] m3 ha−1. This is due to thefact that the available water from surface water and groundwater in the dry year is less than thatin the normal year.

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uoetal.

Table 2. Optimal interval solutions of water irrigation to each crop during its various planting stages from different water supply sources in different hydrological years(m3ha−1).

Wet year Normal year Dry year

Crop Planting stage Surface water Groundwater Surface water Groundwater Surface water Groundwater

Springwheat Seeding [660.34, 901.91] [1203.31, 1295.76] [559.22, 838.96] [875.33, 1337.07] [804.16, 818.89] [660.43, 869.84]Jointing [174.25, 184.04] [303.12, 378.37] [127.98, 210.79] [285.51, 440.54] [254.84, 268.59] [332.01, 376.27]Heading [3.20, 3.25] [54.84, 63.38] [32.61, 51.50] 0 75.09 119.86Maturity [74.62, 78.33] [64.38, 95.18] [101.93, 226.13] 77.58 41.04 48.31

Maize Seeding [137.63, 236.51] [489.95, 556.75] [207.52, 252.90] [476.24, 582.81] [187.36, 207.05] [276.40, 372.67]Shooting [276.16, 314.73] [500.39, 662.81] [251.13, 310.07] [570.21, 732.41] [258.54, 264.25] [418.78, 492.34]Heading [381.30, 570.42] [940.30, 1144.50] [565.63, 608.10] [720.44, 1159.25] [465.54, 479.11] [516.08, 937.02]Filling and maturity [325.80, 441.97] [674.11, 864.77] [436.21, 468.96] [665.86, 852.42] [660.55, 676.56] [922.78, 1260.82]

Oil flax Seeding [136.46, 170.66] [168.99, 202.68] [149.53, 181.49] [145.14, 178.67] [65.31, 79.94] [63.34, 77.30]Stemming [371.83, 471.09] [433.57, 579.53] [372.87, 447.86] [522.97, 670.34] [365.27, 493.16] [363.88, 495.31]Squaring [138.97, 170.56] [137.06, 180.48] [85.36, 113.72] [219.68, 270.44] [103.54, 111.99] [125.24, 130.89]Blooming [281.97, 370.38] [354.59, 454.08] [308.94, 399.43] [419.81, 493.98] [138.00, 195.15] [185.12, 254.60]Blossoming [202.83, 272.83] [442.92, 478.09] [252.55, 306.68] [407.90, 480.52] [215.67, 274.98] [298.12, 384.72]Maturity [300.43, 426.08] [492.64, 596.54] [208.52, 313.58] [186.79, 304.71] [601.73, 668.05] [484.12, 656.74]

Seed watermelon Seeding [105.64, 112.80] [115.76, 133.10] [8.42, 88.32] [13.83, 109.68] 0 0Blooming [114.92, 146.82] [165.38, 207.07] [114.34, 138.67] [220.49, 270.55] [17.97, 25.31] [13.14, 28.61]Expanding [284.10, 388.68] [746.12, 913.45] [302.78, 371.75] [561.03, 734.17] [206.43, 237.57] [654.81, 796.84]Stereotyping [60.67, 87.07] [99.31, 121.52] [120.73, 140.36] [125.30, 184.58] [348.99, 402.91] [97.66, 167.31]Maturity [0, 55.98] [0, 30.45] [90.84, 124.40] [87.85, 134.73] [89.54, 113.96] [98.51, 169.10]

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Engineering Optimization 11

(a)

(b)

Figure 6. Upper and lower bounds of optimal water irrigation from different water supply sources in differenthydrological years: (a) surface water; (b)groundwater.

Figure 6(a,b) presents the comparison of the upper and lower bounds of the optimal waterirrigation amount for all of the crops from two different water supply sources (surface water andgroundwater) in different hydrological years. It can be seen that the upper bound of water irrigationfrom surface water was only slightly different, but the lower bound illustrated an increasing trendin the wet, normal and dry years. On the other hand, both the upper and the lower bounds of theoptimal water irrigation from groundwater in the dry year were less than in the wet year and thenormal year. Figure 7 shows the comparison of the upper and the lower bounds of the optimalirrigation amount from surface water for spring wheat, maize, oil flax and seed watermelon overtheir entire planning periods in the normal year. Figure 8 presents the upper and the lower boundsof the optimal total irrigation from two water supply sources for oil flax during its various plantingstages in a normal hydrological year.

The optimal water irrigation patterns with maximum crop production benefit can be obtainedfrom the INP modelling solutions. The variations in these optimal solutions (Table 2) could reflectdifferent irrigation schedules for agricultural water resources management under different precip-itation scenarios. As a result of limited water supply and increased water demand, complexitiescould arise for effective water irrigation in the study system. Considering water sustainability,more surface water should be used for irrigation in the wet year than in the normal year, and lessgroundwater should be used to maintain the groundwater levels. In the dry year, more ground-water should be used than in the normal year to maintain the productivity of crops. In the caseof insufficient water, irrigation should be first guaranteed to the crop planting stages with higherwater deficit sensitivity, such as oil flax’s stemming and maturity stages, and maize’s filling and

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Figure 7. Upper and lower bounds of the optimal water irrigation from surface water for different crops.

Figure 8. Upper and lower bounds of the optimal total water irrigation for spring wheat during its various plantingstages in a normal year.

maturity stage. This is because the highest benefit could be gained when water demand duringthese planting stages is satisfied, and the highest penalty could result if insufficient water isdelivered.

The main modelling parameters related to the optimal irrigation amount include theprecipitation, the available water for each typical hydrological year and the water deficit sen-sitivity index. The model could generate solutions for the objective function and the decisionvariables for different crops during different planting stages under the scenarios of differenthydrological years. Figure 9 compares the upper and lower bounds of benefit from optimal crop

Figure 9. Upper and lower bounds of the optimal benefits of crop irrigation during different hydrological years.

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irrigation under the scenarios of different hydrological years. An optimistic policy correspond-ing to the upper bound of the system benefit may be subject to a high constraint-violation risk,while a conservative policy may lead to water resources being wasted. The modelling solutionscould be interpreted for generating multiple decision alternatives. It was noted that the intervalof optimal water irrigation for seed watermelon during its seeding stage in the normal year wastoo wide to be used by the decision maker. As a result, a more effective approach for dealing withuncertainty would be needed to address such complexity. The INP model developed in this studycan simultaneously deal with crop irrigation and different planting stages, and has the potentialto be applied to other irrigation areas.

5. Conclusions

In this study, an INP optimization model was developed for crop irrigation planning under uncer-tainty. By introducing a time parameter into the model, the optimal allocation of irrigation waterfrom different water supply sources for different crops during their various planting stages can beaddressed. Using intervals to represent parameter uncertainty, the INP model can generate resultsthat reflect a practical problem more reliably than deterministic methods. The developed modelwas applied to a case study of agricultural water resources management in an arid region of north-western China to demonstrate its applicability. Four main crops were examined,each with differentplanting stages. The results indicated that a comprehensive solution for addressing the crop deficitirrigation problem was obtained. By analysing three different scenarios of precipitation levels (i.e.wet, normal and dry), the key planting stages of crops and the corresponding irrigation amountthat could significantly affect the system were identified. The modelling solutions could providea sound basis for decision makers to develop effective agricultural water resources managementplans when faced with serious water shortage problems.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (no. 41271536, 71071154and 91125017), the National High Technology Research and Development Program of China (863 Program) (no.2011AA100502), the Governmental Public Research Funds for Projects of the Ministry of Agriculture (no. 201203077)and the Ministry of Water Resources (no. 200901083, 201001060 and 201001061).

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