An inexact stochastic-fuzzy optimization model for agricultural water allocation and land resources utilization management under considering effective rainfall

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An inexact stochastic-fuzzy optimization model for agricultural water allocation and land resources utilization management under considering effective rainfall ⁎

Y.L. Xiea, , D.X. Xiaa, L. Jib, G.H. Huangc a b c

School of Energy and Environmental Engineering, University of Science Technology Beijing, Beijing, 100083, China School of Economics and Management, Beijing University of Technology, Beijing, 100124, China Environmental Systems Engineering Program, Faculty of Engineering, University of Regina, Regina, Sask, S4S 0A2, Canada

A R T I C L E I N F O

A B S T R A C T

Keywords: Land resources utilization Agricultural water management Inexact stochastic-fuzzy programming Imprecise crops water consumption Random effective rainfall

Agricultural water management faces challenges from ecological environment stress (e.g. water scarcity issues, land resources pressure, and climate conditions) and uncertainties exist among multifarious activities in agricultural water resources management systems. In this study, an inexact stochastic-fuzzy programming model was proposed for irrigation water resources allocation and land resources utilization management under considering multiple uncertainties. In the model, uncertainties can be directly integrated into the optimization process through reflecting parameters and coefficients as interval values, fuzzy sets, random variables, and their combinations. The developed method is applied to planning irrigation water resources allocation and cropland pattern under considering the limited surface water and groundwater, the random effective rainfall, and the imprecise crops water requirements in Jining City. A number of scenarios corresponding to different fuzzy probability of violating constraint are examined in order to obtain the best management program under various scenarios, and search reasonable tradeoffs between varied system benefit and system-failure risk. The results indicated that agricultural water allocation is explicitly affected by uncertainties expressed as randomness and fuzziness, and the results are valuable for supporting the adjustment or justification of the existing water resources management schemes and a desired land utilization plan for regions socioeconomic development under uncertainty.

1. Introduction Increased population, rapid urbanization process, and mindbending economic development have cased tremendous ecological environment stress and increasing water demand, and the water scarcity issues are being intensified that have directly influenced agricultural water resources and food security, and regional stable and prosperous development in many areas (Zhang et al., 2008; Singh and Panda, 2012; Zhang and Yang, 2014; Fan et al., 2015a). Moreover, in agricultural water resources system, various uncertainties that exist in many system parameters and their interrelationships, are multiplied by the mixed features of natural variations and human interference, such as random rainfall process, available water and cultivated land conditions, and the complexities in water transferring, irrigation, and crops water requirements (Li et al., 2010; Bryan et al., 2011; Lu et al., 2012a,b; Xie et al., 2013; Juwana et al., 2016). All of these uncertainties could intensify the conflict-laden issue of water resources among multiple

⁎

crops, and have concern with the effectiveness of agricultural water resources management (Geng et al., 2014). Therefore, effective planning for agricultural water resources management incorporating of uncertainties within a general regional irrigation and cropland utilization optimization framework is desired. Previously, a number of inexact optimization techniques were proposed for reflecting uncertainties and complexities in agricultural water resources and land utilization management, including interval parameter programming (IPP), fuzzy mathematical programming (FMP), and stochastic mathematical programming (SMP) (Cai and Rosegrant, 2004; Lacroix et al., 2005; Zhang et al., 2009, 2011; Tan et al., 2011; Lu et al., 2009, 2011; Koichi et al., 2013; Liu et al., 2013; Peña-Haro et al., 2011; Li et al., 2014a,b; Chen et al., 2015; Fan et al., 2015b; Dai et al., 2016). For example, Ghahraman and Sepaskhah (2002) proposed a nonlinear stochastic-dynamic programming to derive an optimal convergent reservoir operation schemes for the irrigation of predetermined multiple cropping patterns. Marques et al. (2005) developed a two-

Corresponding author. E-mail addresses: [email protected] (Y.L. Xie), [email protected] (D.X. Xia), [email protected] (L. Ji), [email protected] (G.H. Huang).

http://dx.doi.org/10.1016/j.ecolind.2017.09.026 Received 26 June 2017; Received in revised form 10 August 2017; Accepted 13 September 2017 1470-160X/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Xie, Y.L., Ecological Indicators (2017), http://dx.doi.org/10.1016/j.ecolind.2017.09.026

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2. Methodology

stage economic production model to examine the effects of hydrologic uncertainty and water prices on agricultural production, cropping patterns, and water and irrigation technology use. Lu et al. (2012a,b) proposed a simulation-based inexact rough-interval programming approach for agricultural irrigation management in a China’s rural region, where the conjunctive use of multiple water sources is examined under a set of land-area, water-availability, environmental standard, capital, and technical constraints. Li et al. (2013) advanced an inexact twostage stochastic programming for irrigation water allocation optimization in Minqin County of China, which was derived from incorporation of interval parameter programming within a two-stage stochastic programming framework. Dong et al. (2014) developed an inexact-stochastic programming approach for agricultural water resources planning and land use management, where the inexact modeling approach was integrated with interval parameter programming and chance-constraint programming methods. Li et al. (2014a,b) proposed an inexact stochastic programming model for agricultural irrigation management in Yongxin County of China, where functional intervals were first introduced to deal with water management problems, along with probability distribution functions and conventional intervals to express a variety of parameter uncertainties. Tan et al. (2016) developed a robust interval fuzzy programming approach with superiority-inferiority and risk-aversion analyses for identifying sustainable agricultural and industrial production strategies at the watershed scale in a highly uncertain environment. Cai et al. (2016) proposed an integrated approach through incorporating operational research and uncertainty analysis methods within a general life cycle assessment framework for water resources allocation management. In general, the above methods have advantages in their effectiveness for dealing with uncertainties in water resources system. However, in many agricultural water resources management problems, several types of uncertainties may exist as multiple levels in a complex system, and the multiple uncertainties characteristics exist among many of irrigation water resources management components, such as such as the vagueness and/or impreciseness in the outcomes of a random rainfall sample, and the randomness and/or fuzziness in the lower and upper bounds of an interval (Liu et al., 2003; Li and Huang, 2009). In addition, the parameters of linear programs such as the right-hand-sides and coefficients of the objective and constraints could be fuzzy random variables due to the fact that they depend on many factors (Nguyen, 2007; Li and Zhang, 2010). Thus, it is difficult to determine exactly the values of these parameters. When applying the above approaches to real agricultural irrigation management problems, difficulties in formulating and solving the resulting agricultural water resources allocation optimization problems under uncertainty arise due to the system multiple uncertainties, and these complexities have placed water resources management programs beyond the conventional systems analysis methods (Huang and Loucks, 2000). Therefore, as an extension of the previous study, the aim of this study is to develop an inexact stochastic-fuzzy programming (ISFP) for water resources allocation and cropland structure management under multiple uncertainties. In the ISFP, uncertainties can be directly integrated into the optimization process through reflecting the uncertain parameters and coefficients as interval values, fuzzy sets, random variables, and their combinations. Moreover, the developed model will be applied to planning irrigation water resources allocation and cropland pattern under considering surface water, groundwater, effective rainfall, and crops water requirements in Jining City. The City, as a major base of grain-production in China with a comparatively nice natural condition, faces serious ecological environment stress due to increasing demand and decreasing availability, and water shortage is crucial for restricting agricultural development. The generated management strategies from the ISFP model will be used for facilitating decision makers in regulating sustainable development plans.

An inexact stochastic-fuzzy programming model for regional agricultural water resources management and planning was based on interval-parameter programming (IPP), and stochastic-fuzzy programming. In the ISFP, each technique has its unique contribution in enhancing the capacities for reflecting multiple uncertainties and system risk. For example, the uncertainties presented as discrete intervals (e.g. technical-economic parameters) were reflected through IPP, and the relationship between fuzzy parameter (e.g. crops water requirements) and stochastic information (e.g. effective rainfall) can be reflected by the stochastic-fuzzy programming. The modeling framework would offer feasible and reliable solutions under different fuzzy tolerance levels, which are helpful for decision makers in the future agricultural system development. 2.1. Interval-parameter programming In interval-parameter programming (ILP), interval values are allowed to be directly communicated into the optimization process (Huang et al., 1992). An IPP model can be defined as follows (Huang, 1996): n

Maxf ± =

∑

c j±x j± (1a)

j=1

subject to n

∑

aij±x j± ≤ bi±, ∀ i = 1, 2, ..., m (1b)

j=1

x j±

≥0

(1c)

aij±

{R±}m × n ,

bi±

{R±}m × 1,

c j±

{R±}1 × n ,

±

where and R denotes a set ∈ ∈ ∈ of interval numbers. Let x ± be a set of intervals with crisp lower bound (e.g., x−) and upper bounds (i.e., x+), but unknown distribution information. Let x be a set of closed and bounded interval numbers x ± , x ± = [x−, x+] = {t|x− ≤ t ≤ x+}. The IPP can directly handle uncertainties presented as interval numbers. However, it has difficulties in reflecting uncertainties expressed as probabilistic distributions; moreover, there is a lack of linkage to the economic consequences of violated polices predefined by the authorities (Huang, 1998). 2.2. Inexact stochastic- fuzzy programming IPP can tackle uncertainties expressed as intervals, but has difficulties in handling stochastic variables with known probability density distributions. In real-world practical problems, the incompleteness or impreciseness of observed information would lead to dual uncertainties of randomness and fuzziness due to the fact that decision makers express different subjective judgments upon a same problem. When uncertainties of some elements in aij± can be expressed as fuzzy sets and those in bi± can be expressed as probability distributions, the inexact stochastic- fuzzy programming (ISFP) problem can be formulated as follows: n

Maxf ± =

∑

c j±x j±

j=1

(2a)

subject to n

∑

a͠ ij x j± ≤ bi , ∀ i = 1,2,..., is (2b)

j=1 n

∑

2

aij±x j± ≤ bi±, ∀ i = is + 1, is + 2, ..., m

j=1

(2c)

x j ≥ 0, ∀ j = 1,2,..., n

(2d)

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Here x j±, j = 1, 2, ..., n are non-negative decision variables; c j±,j = 1, 2, ..., nare the coefficients in the objective function; bi ,i = 1, 2, ..., is are independent random variables with known distribution functions; a͠ ij ,i = 1, 2, ..., is represents the fuzzy coefficient of the jth decision variable in the ith stochastic constraint; bi± and aij± ,i = is + 1, is + 2, ..., m represent the interval variables and interval coefficient, respectively. Constraints of model (2b) contain stochastic variables in the righthand and fuzzy coefficient in the left-hand, and can be violated with a low tolerance level. Thus, through incorporating fuzzy tolerance measuresδ͠ i , 0 ≤ δ͠ i ≤ 1,i = 1, 2, ..., is, and chance-constraint approach, then the inexact stochastic fuzzy constraints (2b) can be converted to (Iskander, 2005): n

Pr

⎞ ⎛ a͠ x ± ≤ bi ≥ δ͠ i, ∀ i = 1,2,..., is ⎟ ⎜ ∑ ij j ⎠ ⎝ j=1

(3a)

then, n

a͠ ij x j± ≤ Fi−1 (β͠ i ), ∀ i = 1,2,..., is

∑

(3b)

j=1

and ͠ bi (βi) = Fi−1 (β͠ i ), ∀ i = 1,2,..., is

(3c)

Fig. 1. Framework of the ISFP model.

(β͠ i)

whereβ͠ i = 1 − δ͠ i ,andbi = Fi−1 (β͠ i ) represents the corresponding values given the cumulative distribution function of bi and the fuzzy probability of violating constraint (β͠ i ). It is apparent that this transformation requires the independent random variables to be continuous (Iskander, 2003; Ben Abdelaziz et al., 2007). In addition, the deterministic fuzzy constraints (3b) can be represented by its crisp equivalent, and the suggested approach, for formulating the crisp constraints, depends on utilizing the αcut approach, for the membership functions of a͠ ij andδ͠ i . Assuming that a͠ ij andδ͠ i are presented as triangular fuzzy numbers for a͠ ij andβ͠ i , i.e., a͠ ij = ( aij , a0ij , aij ) andβ͠ i = ( β i , β0i , βi ) = (1 − δi, 1 − δ0i, 1 − δ i ) . Then the α-cut for each of the membership functions of a͠ ij and β͠ should, re-

k1

((1 − α ) aij − sign ( aij ) + α a′0ij sign (a′0ij )) x+j

∑ j=1

n

∑

+

((1 − α) aij + sign(aij ) + α a′0ij Sign(a′0ij )) x −j

j = k1+ 1 ((1 − α) βi + αβ0i)

≤ bi

, ∀ α, i = 1,2,..., is

k1

n

∑

aij− − sign(aij−) x+j +

j=1

∑

aij+ + sign(aij+) x −j ≤ bi+, ∀ i = is + 1, is

j = k1+ 1

i

+ 2, ..., m

spectively, derivate the following two closed crisp intervals, [(1 − α) aij + αa0ij , (1 − α) aij + αa0ij]and[(1 − α ) β i + αβ0i , (1 − α ) βi + αβ0i]. When incorporating the transformation within inexact stochasticfuzzy program, the ISFP can be inverted as follows:

x+j

=

∑

(4a)

subject to

[(1 − α ) aij + αa 0ij , (1 − α ) aij + αa 0ij] x j±

k1

Maxf − =

j=1

≤

[(1 − α) β i + αβ0i,(−α) βi + αβ0i] bi ,

∀ i = 1,2,..., is

(5d) (5e)

where j = 1, 2,…, k1, are interval variables with positive coefficients in the objective function; x j± , j = k1 + 1, k1 + 2,…, n are interval variables with negative coefficients. Solutions of x+jopt (j = 1, 2,…, k1) and x −jopt (j = k1 + 1, k1 + 2,…, n) can be obtained through submodel (5). Based on the above solutions, the second submodel for f− can be formulated as follows:

n

∑

(5c)

x j± ,

c j±x j±

j=1

≥ 0, j = 1,2,..., k1

x −j ≥ 0, j = k1 + 1, k1 + 2,..., n

n

Maxf ±

(5b)

∑

n

c −j x −j +

j=1

(4b)

∑

c −j x+j (6a)

j = k1+ 1

subject to n

∑

aij±x j±

≤

bi±,

k1

∀ i = is + 1, is + 2, ..., m

((1 − α ) aij + sign (aij ) + α a0ij sign (a0ij )) x −j

∑

(4c)

j=1

j=1

x j ≥ 0, ∀ j = 1,2,..., n

n

(4d)

Maxf + =

∑ j=1

((1 − α ) aij − sign ( aij ) + α a 0ij Sign (a 0ij )) x+j

j = k1+ 1

According to Huang and Loucks (2000), based on an interactive algorithm, Model (4) can be solved by being converted into two submodels. Since the objective is to maximize the net system benefit, the submodel corresponding tof+is desired firstly. The submodel corresponding to f ± can be formulated as follows (assume b ± ≥ 0 andf ± ≥ 0): k1

∑

+

((1 − α ) β i + αβ0i)

≤ bi

, ∀ α, i = 1,2,..., is

k1

∑

(6b)

n

aij+ + sign(aij+) x −j +

j=1

∑

aij− − sign(aij−) x+j ≤ bi−, ∀ i = is + 1, is

j = k1+ 1

+ 2, ..., m

(6c)

n

c+j x+j

+

∑ j = k1+ 1

c+j x −j

x+jopt

x −j

≥

0≤

x −jopt

≥ 0, j = 1,2,..., k1

(5a)

subject to 3

≤

x+j ,

j = k1 + 1, k1 + 2,..., n

(6d) (6e)

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Solutions of x −jopt (j = 1, 2,…, k1), and x+jopt (j = k1 + 1, k1 + 2,…, n) can be obtained through submodel (11). Through integrating solutions of submodels (5) and (6), interval solution for model (4) can be obtained. ± f opt = [x −jopt , x+jopt ]

(7a)

± x jopt = [x −jopt , x+jopt ]

(7b)

including autumn crops (May to October) and summer crops (November to next April). Wheat, garlic, onion, and melon are the major crop in during November to April, and rice, cotton, maize, peanut, beans, millet, sesame, sorghum, and tuber crops are the main summer crops in the study area (JNSB, 2012). In the study system, land, light, and heat resources provide favorable conditions for the development of agriculture. The amount of agricultural irrigation water resources accounts for nearly 80% of the total water resources consumption. Previously, surface water is the major water supplier supporting the agricultural water demand due to easy-to-use feature. As regional social-economic development in Jining City, the current watersupply pattern can hardly satisfy the water demand of agricultural activities, and the groundwater and water shortage for irrigation simultaneously increased year by year. In 2015, the surface and groundwater amount for irrigation had reached to 203.56 × 106 m3 and 67.21 × 106 m3 in Jining City (JNWCB, 2016; JNSB, 2016). With the rapid development in Jining city, agricultural activities must have sufficient water resources, and water resource becomes more and more significant. The main task is to formulate reasonable crops irrigation schemes under considering rainfall performance with the aim of reducing the water demand, and to obtain optimized planting structure adjustment program under available water resources and land constraints. Besides, there are many complexities and uncertainties (e.g. available water resources and land area, irrigation efficiency, and economic parameters) that should be considered when planning, developing, managing, and operating regional agricultural development strategies under considering agricultural irrigation problems. Especially, the random variations of available precipitation and the fuzzy characteristics of water requirement during crops growth period in irrigation system, that would correspondingly effect the optimal schemes for effective irrigation (i.e. optimal crops planting area and agricultural water-allocation schemes). Moreover, many impact factors and their interrelationships are uncertain in nature. These lead to a number of challenging questions such as (a) how to identify the optimal policy which could result in the highest benefit under the limited available water resources and land area constraints; (b) how much water should be allocated to each crop under different levels of stochastic precipitation and vague demand during crops growth periods; (c) how to adapt/adjust the crops planting structure under different probabilities of system-constraint violation under demanding conditions for water requirement and rainfall performance availabilities under uncertain system conditions. From the above analysis, the uncertain information for a variety of system components in such an agricultural water management system may exist in terms of intervals, fuzzy, and probability distributions; in addition, linkage to the relationship between the fuzzy water demand and the random rainfall performance in the planning horizon. The inexact stochastic-fuzzy programming is thus considered to be suitable for tackling this type of agricultural irrigation management problem.

The general solution algorithm of the ISFP model is illustrated in Fig. 1. 3. Case study 3.1. Overview of the study area The study area focuses on Jining City, situated at the lower Nansihu Lake Basin in the southwest of Shandong province, China. The area ranges in longitude from 115°52′ to 117°36′E and in latitude from 34°26′ to 35°57′ N, and covers an area of 10684.9 km2 (Fig. 2). The available area of cultivated land, garden, forest land, and grassland is 6113.2 km2, 97.5 km2, 625.4 km2 and 74.8 km2, respectively (JNSB, 2012). The study area features semi-humid climatic conditions with an average annual rainfall of 597 mm to 820 mm, approximately 72.3% of which is received during June and September. The area is a temperate continental monsoon climate, and the annual average temperature was 13.6 °C, with a maximum temperature of 26.8 °C and minimum temperature of −1.7 °C. Due to the advantage on climate, topographic and geomorphic conditions, agriculture is the main industry in the study area, and the productivity of agricultural activities is relatively greater. It has an agricultural acreage of 10649.3 km2, and an effective irrigation area of about 8093.5 km2, with a variety of crops being planted (JNSB, 2013). Traditionally, the year is divided into two principal crop seasons,

3.2. Model development Consider a one-year planning horizon, the planning horizon is subdivided into two periods corresponding to the crops growing times, the first crops growth period is from May to October, and November to April next year is the second crops growth period. Surface water and groundwater are the main irrigation water sources. Policies in terms of the related agricultural activities and irrigation areas in each period are critical for ensuring maximized system benefits, and controlling over the total amount of irrigation water resources under considering the random available precipitation and different satisfaction degrees for crop fuzzy water requirement. Therefore, the ISFP model for the agricultural water management can be formulated as follows: Fig. 2. The study system.

4

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Y.L. Xie et al. K

Maxf ± =

Ik

∑∑

K

NBki±⋅Y ki±⋅Aki± −

k=1 i=1 K

Ik

∑∑

−

K

k=1 i=1

Ik

∑∑

SWki±⋅CSWki ±

(GWki± + SWki±) ≤ AVW ±

(8j)

k=1 i=1

k=1 i=1 Ik R

∑∑∑

GWki±⋅CGWki± −

K

Ik

∑∑

± CFkr± ⋅Fkir

where ASWk±, available surface water in periodk (ha-m); and AGW j± , available groundwater in periodk (ha-m); AVW ± , the total amount of allowable irrigation water resources (ha-m). (5) Water requirement constraint. Based on the government notice of Shandong agriculture department, groundwater depth of the irrigation areas are 13 ∼ 49 m, and the groundwater cannot be directly utilized by crops during their growth period. The net irrigation requirement of crops in different periods must be fully satisfied with the available water resources, including surface water, groundwater, and rainfall.

(8a)

k=1 i=1 r=1

where, k denotes the planning period, k = 1 for the first period (November to April) with summer crops, k = 2 for the second period (May to October) with autumn crops. i denotes the main crops, including summer crops (i = 1 for wheat, i = 2 for garlic, i = 3 for onion, and i = 4 for melon) and autumn crops (i = 5 for rice, i = 6 for cotton, i = 7 for maize, i = 8 for beans, i = 9 for peanut, i = 10 for millet, i = 11 for sesame, i = 12 for sorghum, and i = 13 for tuber crops). r denotes the fertilizer types, r = 1 for nitrogen fertilize, r = 2 for phosphate fertilizer, and r = 3 for potash fertilizer.NRki±, market price of product of cropi(RMB¥/100 kg); Y ki± , yield of cropi(100 kg/ha); Aki± , irrigation target for cropi(ha); SWki± , quantity of river water allocated to cropi in periodk((ha-m)); CSWki ± , cost of pumping the river water in periodk (RMB¥/m3); GWki±, quantity of groundwater pumped in periodkfor cropi(ha-m); CGWki± , cost of pumping the groundwater in period k (RMB¥/ha-m); CFkr± , variable cost of per fertilizer utilization amount ± (RMB¥/tonne); Fkir , the utilization amount of fertilizerrfor cropiin periodk(tonne). subject to: (1) Land availability constraints. Land allocated to various crops must not exceed total available cultivable area during the planning periods.

Ik

∼

Ik

±

⎧∑ NIRki ⋅Aki± − λ 2±⋅ ∑ ⎪ i=1 Pr i = 1 ⎨− γ⋅RFE ±⋅ Ik A ± ≤ 0 k ∑ ki ⎪ i=1 ⎩

(GWki± + μ1±⋅SWki±) ⎫

⎪ ≥ ⎬ ⎪ ⎭

͠ ∀k − β, (8k)

conveyance efficiency of surface water system (fraction); λ 2± , field water application efficiency (fraction); β͠ , fuzzy probability of

where, μ 1± ,

∼

±

violating constraint; NIRki (fuzzy parameters), net irrigation requirement for cropi grown in period k(ha-m); RFEk± (stochastic parameters), effective precipitation in period k(m). (6) Fertilizer demand and pollutant emission constraints ± ± (1 − ρir±)⋅Fkir ≥ Aki± ⋅NRkir , ∀ k, i, r , h

(8)

Ik

∑

Aki± ≤ TAk± , ∀ k , h

K

k = 1 i =1

TAk±

is the total cultivated area available (ha). where (2) Allowable area constraint. Management considerations restrict the minimum or maximum land acreages under certain crops to meet the local food production in the study area. (a) Lower bound limit

Aki± ≤ ζki±⋅TAk± , ∀ k , i

Ik

∑∑

(8b)

i=1

K ± Fkir −

Ik

∑∑

K ± Aki± ⋅NRkir ≤ αr±⋅ ∑

k=1 i=1

Ik

∑

Aki± , ∀ r , h

k=1 i=1

(8)

± where, ρir± , fertilizer loss coefficient of cropi in period k (fraction); NRkir , fertilizer demand of cropi in period k(tonne/ha); αr± , the available fertilizer r discharge amount of per planting area for cropi in period k (tonne/ha).

(8c)

3.3. Data acquisition and analysis

(b) Upper bound limit

Aki± ≥ ψki±⋅TAk± , ∀ k , i

The data on crops, rainfall, surface water and groundwater, and technical and economic were acquired from various central and government departments and district administrations, located in and around the study area and from personal contact, such as Jining Water Conservancy Bureau, Agricultural Bureau, Jining Meteorological Department, Water Conservancy Administration Authority of Nansihu Lake. Table 1 presents the water requirement for each crop from extensive field research in the study area in recent years (Liu et al., 2009; Huang, 2011; Wen et al., 2015). Table 2 presents the unit net benefit and the grain yield of each crop sector during the planning horizon. The random information about rainfall was obtained from the meteorological data in 1990–2015. The permeating coefficient of precipitation is correlated with precipitation amount, rainfall intensity, soil properties, and other relevant factors. It is generally recognized that the precipitation is less than 5 mm, the effective precipitation is equal to 0; the precipitation is 5–50 mm, the effective precipitation is about [0.8, 1.0] of the rainfall amount; and the amount of rainfall is greater than 50 mm, the effective precipitation is about [0.7, 0.8] of the

(8d)

whereζki± and ψki± , fraction of cultivable area can be allocated to cropi. (4) Water availability constraint. Water resources like surface water and groundwater must not exceed the corresponding available water resources during the cropping seasons. The total water amount used for irrigating crops should be less than the water consumption amount control for agricultural production. (a) Surface water Ik

∑

SWki± ≤ ASWk±, ∀ k

(8 h)

i=1

(b) Groundwater Ik

∑

GWki± ≤ AGWk±, ∀ k

(8i)

i=1

(c) Total amount of agricultural water consumption control Table 1 Crops irrigation water requirement. Crop

Water requirement (103m3/ha)

Crop

Water requirement (103m3/ha)

Crop

Water requirement (103m3/ha)

wheat garlic onion melon tuber crops

(3.05, (1.90, (2.10, (3.20, (4.61,

rice cotton maize peanut

(7.60, (3.20, (1.25, (2.15,

beans millet sesame sorghum

(2.62, (2.95, (1.86, (1.53,

3.33, 2.30, 2.50, 3.42, 5.35,

3.68) 2.60) 2.76) 3.76) 5.77)

5

8.00, 3.60, 1.40, 2.50,

8.60) 4.10) 1.65) 2.95)

3.30, 3.42, 2.42, 1.94,

3.75) 3.92) 2.72) 2.42)

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Table 2 The unit net benefit and the grain yield of each crop. Net benefit of each crop (103RMB¥/103kg) wheat rice beans tuber crops

[2.20, [2.70, [3.80, [1.20,

2.60] 3.20] 4.20] 1.80]

garlic cotton millet

[6.00, 6.50] [7.50, 8.30] [3.50, 3.80]

onion maize sesame

[0.68, 0.86] [1.50, 1.80] [5.80, 6.20]

melon peanut sorghum

[3.40, 4.20] [3.60, 3.90] [3.00, 3.40]

Grain yield of each crop (103kg/ha) wheat rice beans tuber crops

[5.99, [8.38, [2.61, [8.47,

6.19] 8.97] 2.96] 9.25]

garlic cotton millet

[18.75, 19.05] [1.27, 1.44] [4.00, 4.24]

onion maize sesame

4. Result analysis and discussion Feasible solutions for crops planting areas, irrigation water resources allocation from surface water and groundwater, and the total system net benefit were obtained by solving the ISFP model through the interactive algorithm. The solutions of most decision variables are intervals, indicating that the related decisions are sensitive to the uncertain modeling input (Huang and Loucks, 2000). The model was solved using the LINGO 11.0 software package. The model results are described below. 4.1. Optimal area allocation Thirteen crops (wheat, garlic, onion, melon, rice, cotton, maize, beans, peanut, millet, sesame, sorghum, and tuber crops) were considered for the model, and these crops already occupy a considerable part of the available cultivable area. Table 3 provides the optimal solutions of planting areas for various crops in the two planting periods under different α-cut levels that correspond to different crops water Table 3 The optimal solutions of planting areas for various crops. Planting area (103ha)

Summer crops wheat [134.55, 221.33] garlic [98.67, 109.45] onion [4.49, 24.88] melon [21.94, 39.80] Autumn crops rice [22.43, 24.88] cotton [44.85, 114.43] maize [224.25, 248.75] peanut [13.46, 50.75] beans [4.49, 24.88] millet [0.09, 0.40] sesame [0.05, 0.50] sorghum [0.45, 0.50] tuber crops [4.49, 5.97]

melon peanut sorghum

[38.50, 40.50] [4.28, 4.76] [3.27, 3.78]

requirement levels and different risk of violating available precipitation amounts. In general, wheat, maize, garlic, and cotton would be the main crops, and the planting area would account for over 70% of the total cultivated area in Jining City during the planning periods. For example, under α = 0.2, the optimized planting area of wheat, maize, garlic, and cotton would be [134.55, 221.33] × 103 ha, [224.25, 248.75] × 103 ha, [98.67, 109.45] × 103 ha, and [44.85, 114.43] × 103 ha, respectively. In addition, the garlic, maize, melon, and cotton would reach to their upper bound of limited planting area. It indicates that garlic, maize, melon, cotton are the most beneficial crops in the study area in their respective seasons, as compared with the other crops. As a major base of grain-production in China, although the main food crops area (e.g. wheat, rice, and tuber crops) would reach to their lower bound of limited planting area, and it still be controlled within a reasonable range in order to ensure the food normal supply. For example, the planting area for wheat, rice, and tuber crops would be [134.55, 149.25] × 103 ha, [22.43, 24.88] × 103 ha, and [4.49, 5.97] × 103 ha under α = 0.5, respectively. Being a high value crop, a higher limit would be imposed for the garlic, melon, peanut, and sesame crop; otherwise, it would have occupied a bigger proportion of the whole area during their respective seasons. Moreover, the optimized crops planting area would vary as α level increasing. Under the optimal cropping pattern, the wheat, cotton, peanut, beans crops area would be decreased significantly, the rice and tuber crops area would increase considerably, whereas garlic, maize, sorghum crops area would remain unchanged. For example, under α fixed with the value of 0.2, 0.5, 0.8 and 1.0, the cotton crop area would be [44.85, 114.43] × 103 ha, [75.08, 110.38] × 103 ha, 59.70 × 103 ha, 59.70 × 103 ha, and the tuber crops area would be [4.49, 5.97], [4.49, 5.97] × 103 ha, 5.97 × 103 ha, 5.97 × 103 ha, respectively.

precipitation amount. If the effective precipitation is more than the water requirement crop in any period, then the effective precipitation can take the place of than water requirement and make sure the water requirement is not negative. The fuzzy probability of violating constraintβ͠ was (0.05, 0.10, 0.15), and the α-cut level would be considered as 0.2, 0.5, 0.8, and 1.0 for crops irrigation and planting structure adjustment management.

α = 0.2

[72.50, 74.50] [7.43, 7.96] [1.34, 1.80]

4.2. Optimal water resources allocation

α = 0.5

α = 0.8

α = 1.0

[134.55, 149.25] [98.67, 109.45] [22.43, 24.88] [35.88, 39.80]

149.25

149.25

[98.67, 109.45] [22.43, 24.88] [35.88, 39.80]

[98.67, 109.45] 7.46 20.41

[22.43, 24.88] [75.08, 110.38] [224.25, 248.75] [45.75, 50.75] [22.43, 24.88] [0.36, 0.40] [0.45, 0.50] [0.45, 0.50] [4.49, 5.97]

24.88 59.70

24.88 59.70

[224.25, 248.75] 23.50 5.97 0.10 0.07 [0.45, 0.50] 5.97

[224.25, 248.75] 14.93 5.97 0.10 0.07 [0.45, 0.50] 5.97

Figs. 3 and 4 describe the optimized solutions of the total crops irrigation water amount (including surface water and groundwater) under different α levels during the planning period 1 and 2. Water resources for supporting regional agricultural development and food security would be mainly allocated to wheat, garlic, rice, and cotton crops, that corresponds to the crops planting area and crops water requirement. As shown in Fig. 3, in period 1, the water resources allocated to wheat crop would account for over 50% of the total water consumption. For example, under α = 0.5, the total water resources allocated to wheat, garlic, onion, and melon crops would be [584.21, 598.34] × 106 m3, [129.46, 158.43] × 106 m3, [36.53, 6 3 41.77] × 10 m , and [115.87, 116.05] × 106 m3, respectively. In period 2, the irrigation water amount for rice, cotton, tuber crops, beans, and peanut crops would be far more than that allocated to sorghum, maize, millet, and sesame crops, as shown in Fig. 4. For example, under α = 0.2, the total water resources allocated to sorghum, maize, millet, and sesame crops would be [0.166.59] × 103 m3, 0, [307.18, 6

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Fig. 3. Irrigation water amount for summer crops under different α levels.

732.64] × 103 m3, and [0.33.19] × 103 m3, respectively (Fig. 4 (a)); and the irrigation water amount for rice, cotton, tuber crops, beans, and peanut crops would be [281.14, 290.06] × 106 m3, [122.75, 186.02] × 106 m3, [22.84, 32.44] × 106 m3, [13.89, 31.17] × 106 m3, and [5.65, 13.80] × 106 m3, respectively (Fig. 4 (b)). In addition, the total water resources allocated to maize crop would be 0, and the maize crop area would be the largest among all crops under different α levels. From that point on, the water requirement for maize crop during its growth cycle is small, and the rainfall would meet the water demand in Jining City. Moreover, as α level increasing, the

lower bound of the crops water requirement and the risk of violating available precipitation amounts will increase, and the corresponding upper bound values will be decreased. Through interactive algorithm for solving Model (8), the lower and upper bound of crops water resources allocation amount would vary with different trends as α level increasing. For example, the total water resources allocated to melon crop would be [83.21, 105.30] × 106 m3, [115.87, 116.05] × 106 m3, [98.18, 125.77] × 106 m3, and [56.36, 69.12] × 106 m3 under α fixed with the value of 0.2, 0.5, 0.8 and 1.0, respectively (Fig. 3); and the corresponding values for cotton would be [122.75, 186.02] × 106 m3, Fig. 4. Irrigation water amount for summer crops under different α levels.

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Fig. 5. Surface water allocation schemes under different α levels [the amount for other crops is equal to 0].

[190.84, 257.57] × 106 m3, [98.07, 165.73] × 106 m3, and [93.17, 159.60] × 106 m3,respectively (Fig. 4 (a)). Fig. 5 and 6 present the optimal water allocation from the surface water and groundwater sources for different crops under different α levels. It can be found that most of the surface water would be allocated to the wheat and rice crops, the irrigation water for beans, sorghum, and tuber crops would be smaller, and the surface water amount

allocated to garlic, onion, melon, maize, peanut, sesame would be 0. For example, the irrigation water amount for wheat, rice, beans, sorghum, and tuber crops from surface water would be [584.21, 598.34] × 106 m3, [272.10, 294.24] × 106 m3, [48.53, 53.57] × 106 m3, 0, and [29.34, 36.26] × 106 m3 under α = 0.5, respectively (Fig. 5). In addition, the lower bound of rice irrigation water amount would be decreased, and the corresponding upper bond value

Fig. 6. Groundwater allocation schemes under different α levels [the amount for other crops is equal to 0].

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Fig. 7. System net benefit in each periods under different α levels.

would increase as α level increasing. For example, the rice irrigation water amount from surface water would be [281.14, 290.06] × 106 m3, [272.10, 294.24] × 106 m3, [189.31, 307.33] × 106 m3, and [91.83, 315.35] × 106 m3 under α fixed with the value of 0.2, 0.5, 0.8 and 1.0, respectively (Fig. 5). It indicated that as the largest water-consumption crop, with α level increasing, the lower of water requirement would increase, and rice crop area would decrease; the upper bounds of rice water-consumption would be decreased, and the planting area would increase. That would lead to sequenced changes of rice irrigation water amount from surface water. In general, surface water would be mainly used for irritating the crops with a large planting area, including wheat, rice, and cotton. Similarly, the groundwater would be the mainly water sources for the garlic, onion, melon, cotton, peanut, and sesame crops irrigation that with a smaller planting area, and the groundwater amount allocated to wheat, maize, beans, millet, and sorghum would be 0. For example, under α = 0.2, the groundwater allocated to garlic, onion, melon, cotton, peanut, and sesame crops would be [91.94, 211.86] × 106 m3, [10.71, 28.00] × 106 m3, [83.21, 105.30] × 106 m3, [110.39, 186.02] × 106 m3, [5.65, 6 3 6 3 13.80] × 10 m , and [0,0.03] × 10 m , respectively (Fig. 6). For rice and tuber crops, the groundwater would be a supplemental source for irrigation. For example, the groundwater amount for rice and tuber crops irrigation would be [0.65.42] × 106 m3, and [0.24.60] × 106 m3 under α = 0.8, respectively. As α level increasing, the groundwater allocation schemes

would vary with different change trends. For example, irrigation water amount for melon from groundwater would be [83.21, 105.30] × 106 m3, [115.87, 116.05] × 106 m3, [98.18, 125.77] × 106 m3, and [56.36, 69.12] × 106 m3 under α fixed with the value of 0.2, 0.5, 0.8 and 1.0, respectively (Fig. 6). 4.3. Net annual return Fig. 7 shows the system net benefit under different α levels in the whole planning period. The α level would determine the value ofβ͠ that represent a set of probabilities at which the constraints will be violated (i.e., the admissible risk of violating the constraints). Thus, the relation between the system benefits and the α level demonstrates a tradeoff between benefit and constraint violation risk. There is an obvious decrease of the upper bound of the total system benefit, and a growth trend of the lower bound of the benefit from low levels to high ones. For example, the total system benefit would be RMB¥ [17.74, 30.26] × 109, RMB¥ [21.23, 29.18] × 109, RMB¥ [21.10, 27.92] × 109, and RMB¥ [19.79, 25.27] × 109 under α fixed with the value of 0.2, 0.5, 0.8 and 1.0, respectively. It indicates that an increased α means an increased strictness for the upper bound constraints, and a decreased strictness for the lower bound constraints, which may then result in a decreased and increased system benefit. In addition, under different α levels, the net benefit of summer crops in period 1 would be 9

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Acknowledgements

more than that of autumn crops in period 2. For example, under α = 0.5, the net benefit from the summer crops and autumn crops would be RMB¥ [17.57, 23.12] × 109, and RMB¥ [3.66, 6.06] × 109. The main reason is that the agricultural productions are the main economic crops (such as garlic, onion, and melon) with a higher income in period 1, and the main grain crops (such as rice, maize, millet, sorghum, and tuber crops) with a lower income and bigger planting area in period 2. Moreover, as α level increasing, the benefits from summer and autumn crops in period 1 and 2 would be similar with the total system benefit. For example, the net benefit would be RMB¥ [14.86, 24.09] × 109, RMB¥ [17.57, 23.12] × 109, RMB¥ [17.83, 23.06] × 109, and RMB¥ [16.63, 20.57] × 109 from the summer crops in period 1under α fixed with the value of 0.2, 0.5, 0.8 and 1.0, respectively; and RMB¥ [2.88, 6.17] × 109, RMB¥ [3.66, 6.06] × 109, RMB¥ [3.27, 4.86] × 109, and RMB¥ [3.16, 4.71] × 109 from the autumn crops in period 2under α fixed with the value of 0.2, 0.5, 0.8 and 1.0, respectively.

This research was supported by the National Natural Science Foundation of China (51609003 and 71603016), the Fundamental Research Funds for the Central Universities (FRF-TP-15-083A1), and the China Postdoctoral Science Foundation funded project (Grand No.2015M580046 and Grand No.2015M580034). The authors are extremely grateful to the editor and the anonymous editors and reviewers for their insightful comments and suggestions. References Ben Abdelaziz, F., Aouni, B., El Fayedh, R., 2007. Multi-objective stochastic programming for portfolio selection. Eur. J. Oper. Res. 177, 1811–1823. Bryan, B.A., King, D., Ward, J.R., 2011. Modelling and mapping agricultural opportunity costs to guide landscape planning for natural resource management. Ecol. Indic. 11 (1), 199–208. Cai, X.M., Rosegrant, M.W., 2004. Irrigation technology choices under hydrologic uncertainty: a case study from Maipo river basin. Chile.Water Resour. Res. 40, W04103. 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5. Conclusions In this study, an inexact stochastic-fuzzy programming model was proposed for irrigation water resources allocation management and cropland structure planning under uncertainty. Multiple uncertainties expressed as interval values, fuzzy sets, random variables, and their combinations could be directly communicated into the optimization process. Through incorporating fuzzy tolerance measures chance-constraint approach, the model can be converted and lead to enhanced system robustness for uncertainty reflection. The developed method is applied to planning irrigation water resources allocation and land utilization under considering the limited surface water and groundwater, the available land resources, the random effective rainfall, and the imprecise crops water requirements in Jining City. A number of scenarios corresponding to different fuzzy probability of violating constraint (e.g. tolerance levels) are examined, which indicates that different tolerance levels correspond to different water allocation schemes and cropland patterns, and thus lead to varied system benefit and system-failure risk. It indicated that agricultural water allocation is explicitly affected by uncertainties expressed as randomness and fuzziness, which challenges to make reasonable optimization decisions under multiple uncertainties. Jining City should make efforts to improve its irrigation policy and crops structure adjustment in the following aspects: (i) irrigation quota, coupled with considering effective rainfall, is significant for decreasing water consumption and increase crops production in agricultural system; (ii) it is necessary to improve the statistical accuracy of regional rainfall in order to make an reasonable water resources allocation schemes, and impose restrictions on groundwater and total water consumption control; (iii) system risk assessment for pollutants discharge is necessary to assist the best nutrient management mode. Moreover, the obtained results indicated that the model is valuable for supporting the adjustment or justification of the existing irrigation water allocation and crops planting schemes of the complicated agricultural water resources system, and identify a desired water-allocation plan for regions socioeconomic development under uncertainty. However, there are still many improvements and developments can be fulfilled. These include the improvement of algorithms and the exploration of application area. The main research topics of further study are shown as blow: 1) ISFP would have difficulties in dealing with the uncertainties in the model’s right-hand-side coefficients; 2) the selection of a suitable alternative under different scenarios and λ values should be presented for water resources managers; and 3) the proposed method can be applied in many energy and environmental system, such as energy structure adjustment, water quality planning, and air environmental management. Further studies are desired to deal with these limitations. 10

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