Optimal Irrigation Scheduling for Multi-Cropping Systems: A Chance-Constrained Multi-Objective Robust Programming Under Hybrid Uncertainty

Abstract

Drought and water scarcity are threatening the food security in irrigation districts around the world, urging water-saving and highly efficient irrigation scheduling. However, the increasing uncertainty jointly caused by changing environment and human activities makes it much more difficult and unreliable. In this study, a chance-constrained multi-objective robust programming framework was proposed to optimize irrigation scheduling of multi-cropping systems, while dealing with hybrid uncertainty and multiple objectives simultaneously. It integrates deficit irrigation theory, soil water movement and reservoir regulation and was applied in a seasonal drought region of Southwest China with multi-cropping systems. The results show that: (1) the optimal irrigation schemes can substantially mitigate the seasonal drought and reveal the influence of uncertainty from parameters on the objectives; (2) compared with the current practice, the optimized irrigation scheduling can help save water for wet-season crops and reflect the response of crops to various water demand and supply scenarios; (3) the risk preferences, goal preferences, the expected objectives interval, and the preferences for robust penalty of decision makers were investigated to show their interactive influence on the results. Although there are still limitations, the developed model can help improve drought resistance, save water and realize sustainable development of agriculture.

1. Introduction

With population growth, rapid economic development and environmental contamination, the world is experiencing unprecedented water and food deficit [1,2]. As a main water-intensive industry, agriculture is assumed to have great potential of saving water [3]. It is believed that proper water allocation can improve food supply and help maintain ecosystem balance [4]. However, the increasing complexity of agricultural systems and multiple uncertain factors from changing environment make it hard to obtain optimal water use strategies.

Multiple cropping is a practice of growing two or more crops in the same field during a year [5]. According to the intensification features of cropping in temporal and spatial dimensions, multiple cropping is divided into sequential cropping and intercropping. Sequential cropping means that a second crop is cultivated after the first one is harvested, while intercropping is growing two or more crops simultaneously in the same field [6]. There is evidence showing that multiple cropping accounts for 12% of global cropland, mostly in tropical and subtropical regions and over 90% of it is double cropping [7]. Multiple cropping has been proved to be an important way to increase crop productivity, reduce food crisis, and improve farmers’ income. In addition, multiple cropping is also considered as a potential way to reduce the environmental consequences from crop production [8]. Optimal water management has been proved to enhance agricultural sustainability for multiple cropping systems, such as replacing flood irrigation with precision irrigation, adjusting planting structure and optimizing crop calendar [9,10,11]. Many existing studies on multi-cropping systems have focused on the economic and environmental effects of multiple cropping systems, including crop yield, soil quality, carbon sequestration and greenhouse gas emission reduction, resource use efficiency, and soil microbial diversity [12,13,14]. However, the optimization of irrigation scheduling for multi-cropping systems under existing irrigation methods has been largely overlooked, and further research is needed to enhance agricultural sustainability.

Although the diversity of crop types and crop growth habits in multiple cropping systems makes the optimization of irrigation scheduling much more complicated than that in single cropping areas, the irrigation scheduling optimization of multiple crops in single cropping patterns still provides a good reference. Optimization models are widely used to generate optimal irrigation scheduling based on the relationship of crop yield and water consumption [15,16,17]. Although this idea is applicable for multi-cropping systems, there are some critical and specific problems to be solved. One is that crop combinations vary considerably in multi-cropping systems, and their growth habits and phenology can differ substantially. Planting dates, growth stages and growing periods vary across different crop groups, which increases the difficulty of model development. Previous irrigation scheduling studies in multi-cropping agriculture have tended to focus on one specific group, such as wheat–maize and rice–rice [18,19,20,21,22]. Another challenge is that the regional water cycle in multi-cropping systems is much more complicated than in monocropping. For example, the soil water content fluctuates more frequently in rotations of dryland crops and paddy field crops. However, irrigation-induced water exchange can have significant impacts on regional agriculture and should not be overlooked [23,24,25]. The evidence shows that irrigation expansion in many regions has substantial influence on the water cycle, further aggravating the existing drying trends caused by climate change [26]. Therefore, regional water cycle should also be concerned in irrigation scheduling of large-scale multi-cropping systems.

If the complexity of multi-cropping systems increases the difficulty of modelling on irrigation scheduling optimization, widespread uncertainty in nature and agricultural production challenges the solving of established models and the reliability of outputs [27,28,29]. Generally, uncertainty can be categorized into randomness, fuzziness, and interval [30]. Randomness describes stochastic characteristics, shown by the probabilities of events happening [31]. Fuzziness depicts the influence of human cognition, expressed as membership, which reflects variability of people’s attitudes [32,33]. Interval uncertainty is a description of the boundary, which can be shown as upper and lower limits [34,35]. The quantification of randomness and fuzziness needs larger sample data than interval, which means interval contains less information compared with randomness and fuzziness. On the other hand, interval is a convenient method to show the uncertainty when the available data is limited. Aimed at agricultural systems, the natural elements (e.g., evapotranspiration, precipitation, and runoff) and social–economic elements (e.g., price, cost, and policy) contain a huge amount of uncertainty [36,37]. Considering that natural and social-economic factors play a decisive role in agricultural production, uncertainty analysis and algorithm development are indispensable in optimal decision making under changing environments.

Therefore, the target of the study is to establish a highly efficient optimization system of irrigation scheduling for multi-cropping agricultural systems under the changing environment. In this paper, an irrigation scheduling optimization framework is constructed for multiple crop combinations in multi-cropping regions under hybrid uncertainty. The deficit irrigation theory, soil water exchange in the field and water balance of reservoirs are integrated within the optimization framework to reduce water stress risk from seasonal droughts, improve agricultural production and save water resources. Considering changing environments, uncertainty of factors in agricultural systems is quantified into different types based on their features and data volume, and a chance-constrained multi-objective robust programming method is proposed to solve the model and show the influence of uncertainty from natural and human factors on irrigation scheduling strategies.

2. Methodology

2.1. Modelling

The targets of the model are to generate optimal irrigation scheduling of multi-cropping systems to alleviate possible yield loss from seasonal droughts, increase profitability, and save water resources. Therefore, net benefit of crop production, water productivity and marginal yield were selected as objectives and irrigation scheduling of crops in different multi-cropping groups were decision variables of the optimization model. In order to achieve the objectives, deficit irrigation and reservoir regulation were combined. Meanwhile, water holding capacity of soil was also considered and water cycle of reservoir and soil was described. Constraints about soil water content, water demand of crops, operation security of reservoirs, and water availability were also involved as the most fundamental security guarantees of the optimization system. As introduced before, uncertainty exists in natural and social–economic factors. The possible sources of uncertainty can come from data noise of collected data, natural law, attitudes of decision groups, market fluctuation, etc. In this study, two categories of uncertainty were adopted considering different attributes of factors. Random theory was used to show the uncertainty of natural conditions like meteorological factors and hydrological factors. Since the mechanism of economy and society are always ambiguous and it is sometimes difficult to collect large samples, interval theory was adopted to show the fluctuation of markets and policy as well as some data noise. Therefore, the modelling framework is shown in Figure 1.

Before the optimization model was constructed, there are some assumptions to be clarified. Numerous crop groups would exist in multi-cropping systems and decision variables need to be categorized by the planting order of crops. Since double cropping shares over 90% of multi-cropping systems [7], a double cropping system was taken as an example for convenience. Crops are usually divided into winter crops and summer crops based on their sowing dates. Practical planting dates may differ for the same crop grown on different fields, as well as different types of winter or summer crops. For simplicity, we assumed that all crops sown within a given period were assigned the first day of that period as their sowing date. Meanwhile, considering the typical characteristics of some parameters, their uncertainty types have been assigned in the model. For example, randomness is suitable to express the uncertainty of water supply and demand factors, both of which are mainly influenced by natural factors and often have long-term time-series data. However, due to the timeliness of economic data such as prices and costs, a relatively short time-series historical data is recommended for uncertainty analysis.

The model was constructed based on Figure 1. The objectives include maximizing net benefit of crop production, maximizing irrigation water productivity and maximizing marginal yield.

• Maximizing net benefit

Net benefit covers the income from crop production and the cost during the process, including irrigation and other inputs.

$$\begin{array}{l}N{B}_{lh}={\displaystyle \sum _{i,j}(yield{w}_{ijlh}\cdot are{a}_{ij}\cdot price{w}_i^{\pm }+yield{s}_{ijlh}\cdot are{a}_{ij}\cdot price{s}_j^{\pm })}\\ -{\displaystyle \sum _{i,j,t}i{w}_{ijlht}\cdot are{a}_{ij}\cdot wpric{e}^{\pm }}-{\displaystyle \sum _{i,j}(cost{w}_i^{\pm }\cdot are{a}_{ij}+cost{s}_j^{\pm }\cdot are{a}_{ij})}\end{array}$$

(1)

where i and j are crops sowed in winter and summer seasons, l and h are discrete scenarios of different random events, t is time series, NB is net benefit of crop production (), yieldw and yields are yield of winter and summer crops (kg/ha), area is planting area of a double-cropping combination (ha), pricew and prices are price of winter and summer crops (/kg), iw is gross irrigation water, wprice is price of irrigation water (/m³), costw and costs are other cost of winter and summer crops excluding irrigation (/ha), (·)^± means interval number, with + denoting upper bound and − lower bound. The Jensen model was adopted to show the response of crop production to water, expressed as $yield=y_m\cdot {{\displaystyle \prod _t(\frac{et{a}_t}{et{c}_t})}}^{{\lambda }_t}$, where yield is the actual crop yield (kg/ha), y_m is the potential maximum yield (kg/ha), eta is actual evapotranspiration (m³/ha), etc is crop evapotranspiration (m³/ha), λ is crop sensitivity coefficient.

2.

Maximizing irrigation water productivity

High irrigation water productivity means high irrigation water use efficiency and could help save water and increase yield.

$$WU{E}_{lh}={\displaystyle \frac{{\displaystyle \sum _{i,j}(yield{w}_{ijlh}\cdot are{a}_{ij}+yield{s}_{ijlh}\cdot are{a}_{ij})}}{{\displaystyle \sum _{i,j,t}i{w}_{ijlht}\cdot are{a}_{ij}}}}$$

(2)

where WUE is irrigation water productivity (kg/m³).

3.

Maximize marginal yield

Different from the irrigation water productivity, the marginal yield emphasizes the yield increase from irrigation and excludes the contribution of rainfall.

$$M{Y}_{lh}={\displaystyle \frac{{\displaystyle \sum _{i,j}(yield{w}_{ijlh}-yieldr{w}_{il})\cdot are{a}_{ij}}+{\displaystyle \sum _{i,j}(yield{s}_{ijlh}-yieldr{s}_{il})\cdot are{a}_{ij}}}{{\displaystyle \sum _{i,j,t}i{w}_{ijlht}\cdot are{a}_{ij}}}}$$

(3)

where MY is marginal yield of irrigation (kg/m³), yieldrw and yieldrs are rain-fed yield of winter and summer crops, which are quantified by the Jensen model. For dry-land crops, the rain-fed actual evapotranspiration was calculated as $et{a}_t=\left\{\begin{array}{l}e{p}_t,ep<et{c}_t\\ et{c}_t,ep\ge et{c}_t\end{array}\right.$, where ep is effective precipitation (m³/ha). For paddy crops such as rice, the Jensen model is not applicable and the rain-fed yield should be determined based on the field survey or experiments.

In order to guarantee crop growth, operation security of reservoirs and water use of other sectors, basic constraints should be satisfied. The details are given below.

• Soil water constraints

For paddy field, soil water is saturated when there is water layer on the soil, while for dryland, soil water content should consider the porosity. In order to quantify water content in the soil and on the soil in a unified way, the soil surface was treated as zero, which means the water content on the soil is positive and the water in the soil is negative. The soil water at the initial stage is expressed by Equation (4) assuming the soil water unsaturated. Water exchange includes the recharge from irrigation, precipitation, groundwater evaporation and the increased depth of roots, as well as the discharge from evapotranspiration, deep seepage, and field drainage. The water content in or on the soil should meet the corresponding constraints for different crops.

$$s{w}_{0ijlh}=10,000\cdot r{d}_{0ij}\cdot ({\theta }_0-{\theta }_{fc})$$

(4)

$$s{w}_{ijlht}=s{w}_{ijlh(t-1)}+e{p}_{tl}+\sigma \cdot i{w}_{ijlht}+△r{w}_{ijt}-et{a}_{ijlht}-d{r}_{ijlht}-d{s}_{ijt}$$

(5)

$$△r{w}_{ijt}=10,000\cdot (r{d}_{ijt}-r{d}_{ij(t-1)})\cdot {\theta }_{(t-1)}+gw{r}_{ijt}$$

(6)

$$ls{w}_{ijt}\le s{w}_{ijlht}\le us{w}_{ijt}$$

(7)

where sw₀ is the soil water content in the initial stage (m³/ha), rd₀ is root depth in the initial stage (m), θ₀ is soil water content in the initial stage (m³/m³), θ_fc is field capacity (m³/m³), sw is soil water content or water layer on the soil (m³/ha), σ is efficiency coefficient of irrigation water delivery, Δrw is water recharge (m³/ha), dr is drained water (m³/ha), ds is deep seepage (m³/ha), rd is root depth (m), θ is soil water content (m³/m³), gwr is water recharge from groundwater evaporation (m³/ha), lsw and usw are lower and upper bounds of soil water (m³/ha).

2.

Crop water demand constraints

Actual evapotranspiration should not exceed the crop evapotranspiration. The minimal evapotranspiration should be guaranteed for crop survival.

$$e{t}_{\min ijlt}\le et{a}_{ijlht}\le et{c}_{ijlt}$$

(8)

where et_min is minimum evapotranspiration that keeps crops alive (m³/ha).

3.

Operation security of reservoirs

As an effective way to reduce water stress risk from seasonal droughts, reservoir regulation is considered. The water storage of reservoirs should be constrained to guarantee the operation security.

$$w{c}_{mlh}={\displaystyle \sum _{t\in m}i{w}_{ijlht}\cdot are{a}_{ij}}$$

(9)

$$r{s}_{mlh}=r{s}_{(m-1)lh}+inflo{w}_{ml}-w{c}_{mlh}-ow{u}_{mlh}-rd{r}_{mlh}$$

(10)

$$r{s}_{\min }\le r{s}_{mlh}\le r{s}_{\max }$$

(11)

where wc is irrigation water consumption (m³), m means month, rs is reservoir water storage (m³), inflow is the water recharging to the reservoir (m³), owu is water use for other sectors from the reservoir (m³), rdr is surplus water released from the reservoir (m³), rs_min and rs_max are the minimum and maximum reservoir storage (m³).

4.

Water availability constraints

Based on the water consumption in the previous year, along with the forecasts of water supply and demand for the current year, managers formulate water use plans for the current year in advance and require that these plans be followed as closely as possible. In practice, however, deviations often occur, resulting in either overuse or underuse. Managers generally have a certain level of tolerance for overuse, but their acceptance decreases as the actual water use approaches the upper limit of the planned range. Thus, the water availability constraints can be relaxed and a chance-constrained programming was adopted to reflect the tolerance.

$$ti{w}_{lh}={\displaystyle \sum _{t}i{w}_{ijlht}\cdot are{a}_{ij}}$$

(12)

$$Pos(ti{w}_{lh}\le w{a}_h^{\pm })\ge \alpha$$

(13)

where tiw is total water consumption in a year (m³), Pos( ) means chance measure of possibility chance constraints, wa is water availability set by policy makers (m³), and α is satisfaction degree.

5.

Non-negative constraints

$$i{w}_{ijlht}\ge 0$$

(14)

Each crop planted in a given period has its own characteristic values for deep seepage, crop evapotranspiration and soil water content requirement, which can be obtained from experiments and practice. However, different crop combinations within multi-cropping systems can lead to different outcomes in a single year. To provide a general mathematical formulation for the free combinations of winter and summer crops, the following matrix transformation was adopted, where dsw and dss are deep seepage of winter and summer crops (m³/ha), Δrww and Δrws are water recharge of winter and summer crops (m³/ha), etcw and etcs are crop evapotranspiration of winter and summer crops (m³/ha), lsww and lsws are lower bounds of soil water of winter and summer crops (m³/ha), usww and usws are upper bounds of soil water of winter and summer crops (m³/ha).

$$d{s}_{ijt}=ds{w}_{it}+ds{s}_{jt}$$

(15)

$$△r{w}_{ijt}=△rw{w}_{it}+△rw{s}_{jt}$$

(16)

$$et{c}_{ijlt}=etc{w}_{ilt}+etc{s}_{jlt}$$

(17)

$$ls{w}_{ijt}=lsw{w}_{it}+lsw{s}_{jt}$$

(18)

$$us{w}_{ijt}=usw{w}_{it}+usw{s}_{jt}$$

(19)

2.2. Chance-Constrained Multi-Objective Robust Programming

The proposed optimization model is characterized as nonlinear, multi-objective, multi-uncertainty chance-constrained programming. The key to the solution is to transform the prime programming model into a series of determined single-objective sub-models [36]. The established model can be generalized as follows.

$$\left\{\begin{array}{l}\min C_{RV}^{\pm }x\\ s.t.\\ Pos(A^{\pm }x\le d^{\pm })\ge \gamma \\ x\ge 0\end{array}\right.$$

(20)

where $C_{RV}^{\pm }$ is the parameter in objective, characterized as random and interval; $A^{\pm }$ and $d^{\pm }$ are interval numbers in constraints; Pos(·) denotes the chance constraint with a minimum confidence level of γ.

In this study, a multi-objective robust programming method (MRP) was proposed to solve the complicated uncertain model. The method integrates robust stochastic programming [38,39,40], robust interval programming [41,42,43], and chance-constrained programming [31] to handle uncertainties in objectives and constraints. The modified determined single-objective sub-model can be expressed as below.

$$\begin{array}{l}\min F_i={\displaystyle {\displaystyle \sum _{h=1}^H}{p}_h\cdot \frac{c_{ih}^{+}+c_{ih}^{-}}{2}\cdot x_h}+\alpha {\displaystyle {\displaystyle \sum _{h=1}^H}{p}_h[\frac{c_{ih}^{+}+c_{ih}^{-}}{2}\cdot x_h-{\displaystyle \sum _{h=1}^H{p}_h\cdot \frac{c_{ih}^{+}+c_{ih}^{-}}{2}\cdot x_h}+2{\delta }_{ih}]}\\ \hspace{1em}\hspace{1em} +\beta {\displaystyle \sum _{h=1}^H{p}_h\cdot \frac{c_{ih}^{+}-c_{ih}^{-}}{2}\cdot x_h}\end{array}$$

(21)

s.t.

$${\displaystyle \frac{c_{ih}^{+}+c_{ih}^{-}}{2}}\cdot x_h-{\displaystyle \sum _{h=1}^H{p}_h\cdot \frac{c_{ih}^{+}+c_{ih}^{-}}{2}\cdot x_h}+{\delta }_{ih}\ge 0$$

(22)

$$\gamma A^{+}{x}_h+(1-\gamma )A^{-}{x}_h\le (1-\gamma )d^{+}+\gamma d^{-}$$

(23)

$$x_h,{\delta }_{ih}\ge 0$$

(24)

where F_i is single objective of sub-model i; h denotes discrete scenario from a random event; p_h is the associated probability of random scenario; x_h is decision variable under the hth scenario, $c_h^{+}$ and $c_h^{-}$ are upper and lower bounds of parameters related to random events in objectives; α and β are weight coefficients, δ_h is auxiliary variable and Equation (22) is associated auxiliary constraint. The purpose of auxiliary variables and constraints is to calculate the absolute deviation between values in various scenarios and the average. The details of the derivation can refer to the reference studies [38,39,40].

The modified objective seen in Equation (21) can be divided into three parts. The first part ${\displaystyle \sum _{h=1}^H{p}_h\cdot \frac{c_{ih}^{+}+c_{ih}^{-}}{2}\cdot x_h}$ is the expectation of original objective. The second one $\alpha {\displaystyle \sum _{h=1}^H{p}_h[\frac{c_{ih}^{+}+c_{ih}^{-}}{2}\cdot x_h-{\displaystyle \sum _{h=1}^H{p}_h\cdot \frac{c_{ih}^{+}+c_{ih}^{-}}{2}\cdot x_h}+2{\delta }_{ih}]}$ is the expectation of absolute deviation caused by random scenarios. The third one $\beta {\displaystyle \sum _{h=1}^H{p}_h\cdot \frac{c_{ih}^{+}-c_{ih}^{-}}{2}\cdot x_h}$ is the expectation of deviation from interval numbers. Note that the modified programming can only deal with a single objective. In order to handle multiple objectives, the fuzzy goal programming [44] was introduced. The solving method can be expressed as follows.

Step 1: Transform the prime problem into the modified multi-objective robust programming as Equations (21)–(24).

Step 2: Solve single-objective sub-models separately and record the results for all objectives.

Step 3: Select the best value of each objective as the theoretically optimum value (TOV) and the worst result as the theoretically worst value (TWV). The TOVs and TWVs are the boundaries of multi-objective programming. Note that, the TOVs and TWVs can also be based on the suggestion from decision makers.

Step 4: Construct the membership function for each objective as ${\mu }_i=\left\{\begin{array}{ll}1& F_i(x)\le TO{V}_i\\ {\displaystyle \frac{TW{V}_i-F_i(x)}{TW{V}_i-TO{V}_i}}& TO{V}_i<F_i(x)<TW{V}_i\\ 0& F_i(x)\ge TW{V}_i\end{array}\right.$. The meaning of the membership function is shown in Figure 2. The final objective of the problem is rewritten as $\max \left[\min \left({\mu }_i\right)\right]$ with constraints in Equations (22)–(24). The multiple objectives are transformed into a single objective.

Step 5: Solve the modified model in Step 4 under different confidence levels.

3. Study Area and Data Preparation

3.1. Overview of the Study Area

Dongfeng Reservoir Irrigation District (DRID), located in Danlin County of Meishan City, Sichuan Province, China (103°14′24.3″–103°35′24.2″ E, 29°52′19.4″–30°08′15.6″ N) was selected as study area (Figure 3). It has four distinct seasons and a moderate climate, typical of a humid subtropical region. The average annual temperature is 16.5 °C, the average relative humidity is 83%, the average cumulative sunshine hours are 1079.2 h, and the average annual evaporation and precipitation are 968.4 mm and 1250.3 mm, respectively. Figure 4 shows that the rainfall is distributed unevenly during the year. In practice, May to September is considered the wet season and October to April the dry season. The study area faces serious seasonal drought risk in dry seasons and the additional irrigation water is transferred from Dongfeng Reservoir. Wheat, rapeseed, rice, maize, and citrus are the main crops in the study area. Double cropping is widely applied in this area with crop groups of wheat–rice, wheat–maize, rapeseed–rice, and rapeseed–maize. Besides multiple cropping, monocropping may exist in the same region. Therefore, citrus is also considered in this study, and as a perennial crop, its vegetative growth and fruit development follow an annual cycle.

3.2. Data Acquisition and Processing

The data of the established model, including economic data, crop features, the natural factors and other data like reservoir were obtained from various sources, including statistical yearbooks, field experiments, farmer survey, hydrometric stations, governmental reports and published studies. The economic data includes crop price, irrigation water price and the cost of crop production, which were obtained from market and farmer survey. A 90% confidence level was adopted to analyze boundaries of these data. The price, cost, and maximum yield of crops are shown in

Table 1. Price, cost, and maximum yield of crops considered in this study.

Crops	Price (/kg)	Cost (/ha)	Max Yield (kg/ha)
Wheat	[2.6, 3.18]	[3498, 4276]	7650
Rapeseed	[5.81, 7.1]	[3432, 4195]	2625
Rice	[2.46, 3.01]	[5064, 6189]	7995
Maize	[2.33, 2.85]	[3939, 4815]	11,955
Citrus	[4.95, 6.05]	[16,217, 19,821]	42,750

. The price of water varied within the interval of [0.315, 0.385] /m³. Meteorological data during 1960–2019 were downloaded from China Meteorological Data Network (http://data.cma.cn/), and the daily reference evapotranspiration (ET₀) was computed using the FAO-56 Penman–Monteith equation. Crop evapotranspiration (ET_c) was derived by multiplying ET₀ with crop coefficients, provided by a previous study [45]. The planting structure in 2022 was adopted and the associated planting area of the double cropping system is shown in

Table 2. Planting area of double cropping system considered in this study (ha).

	Rice	Maize
Wheat	370	101.5
Rapeseed	370	142.1

. The cultivated area of citrus is 1084 ha.

A ten-day timestep was adopted considering the irrigation habits of local farmers and a monthly timestep for the reservoir regulation due to the statistical data characteristics. Thus, a month is divided into three periods, i.e., the first ten days (FTD), the middle ten days (MTD) and the last days (LTD). The Jensen model was used to describe the response of crop yield to water supply and the associated water sensitivity index was obtained from previous studies in the same study area, shown in

Table 3. Water sensitivity index in the Jensen model, where FTD, MTD and LTD denotes the first, middle and last ten days in a month. Note that the growth period of citrus begins on 1 January.

		Wheat	Rapeseed	Rice	Maize	Citrus
October	FTD	0.016	0.003	0	0	0.008
	MTD	0.021	0.006	0	0	0.010
	LTD	0.025	0.008	0	0	0.014
November	FTD	0.029	0.013	0	0	0.016
	MTD	0.034	0.019	0	0	0.020
	LTD	0.039	0.028	0	0	0.024
December	FTD	0.045	0.040	0	0	0.029
	MTD	0.051	0.057	0	0	0.035
	LTD	0.064	0.090	0	0	0.046
January	FTD	0.066	0.111	0	0	0.049
	MTD	0.073	0.141	0	0	0.055
	LTD	0.089	0.187	0	0	0.068
February	FTD	0.089	0.189	0	0	0.067
	MTD	0.095	0.192	0	0	0.070
	LTD	0.080	0.145	0	0	0.058
March	FTD	0.103	0.160	0	0	0.072
	MTD	0.106	0.131	0	0	0.070
	LTD	0.118	0.109	0	0	0.073
April	FTD	0.107	0.072	0	0	0.060
	MTD	0.104	0.051	0	0	0.054
	LTD	0.101	0.035	0	0	0.048
May	FTD	0.095	0.024	0	0	0.041
	MTD	0	0	0	0	0.035
	LTD	0	0	0.002	0.027	0.032
June	FTD	0	0	0.006	0.046	0.024
	MTD	0	0	0.018	0.075	0.019
	LTD	0	0	0.051	0.117	0.016
July	FTD	0	0	0.135	0.172	0.013
	MTD	0	0	0.303	0.233	0.010
	LTD	0	0	0.539	0.310	0.009
August	FTD	0	0	0.461	0.294	0.006
	MTD	0	0	0.266	0.263	0.005
	LTD	0	0	0.120	0.225	0.004
September	FTD	0	0	0.038	0.141	0.003
	MTD	0	0	0.012	0.077	0.002
	LTD	0	0	0	0	0.002

[46,47]. The 10-day effective precipitation (EP) was calculated using Equation (25) [47]. It is assumed that the growth period is 1 October to 10 May for wheat and rapeseed, 21 May to 20 September for maize and rice, and 1 January to 31 December for citrus. There are two fallow periods in a year, happening from 11 to 20 May and 21 to 30 September. 1 October was chosen as the beginning of the growth period for multi-cropping systems and 1 January for citrus. The root depth of crops and water layer depth of rice were obtained from observation of field experiments, shown in

Table 4. The root depth and water depth of crops, where soil surface is treated as zero and the negative means the root depth while the positive means the depth of water layer of paddy field (m). Note that the growth period of citrus begins on 1 January.

		Root Depth				Water Layer Depth of Rice
		Wheat	Rapeseed	Maize	Citrus	Lower Bound	Upper Bound
October	FTD	−0.3	−0.3		−0.7
	MTD	−0.3	−0.3		−0.7
	LTD	−0.3	−0.3		−0.7
November	FTD	−0.3	−0.3		−0.7
	MTD	−0.3	−0.3		−0.7
	LTD	−0.3	−0.3		−0.7
December	FTD	−0.3	−0.3		−0.7
	MTD	−0.4	−0.3		−0.7
	LTD	−0.4	−0.3		−0.7
January	FTD	−0.4	−0.3		−0.7
	MTD	−0.4	−0.4		−0.7
	LTD	−0.4	−0.4		−0.7
February	FTD	−0.5	−0.4		−0.7
	MTD	−0.5	−0.5		−0.7
	LTD	−0.5	−0.5		−0.7
March	FTD	−0.6	−0.5		−0.7
	MTD	−0.6	−0.5		−0.7
	LTD	−0.6	−0.5		−0.7
April	FTD	−0.6	−0.6		−0.7
	MTD	−0.6	−0.6		−0.7
	LTD	−0.6	−0.6		−0.7
May	FTD	−0.6	−0.6		−0.7
	MTD				−0.7
	LTD			−0.3	−0.7	0.03	0.05
June	FTD			−0.3	−0.7	0.03	0.05
	MTD			−0.3	−0.7	0.015	0.03
	LTD			−0.3	−0.7	0.015	0.03
July	FTD			−0.4	−0.7	−0.2	0.03
	MTD			−0.4	−0.7	0.03	0.05
	LTD			−0.4	−0.7	0.03	0.05
August	FTD			−0.4	−0.7	0.03	0.05
	MTD			−0.6	−0.7	0.015	0.03
	LTD			−0.6	−0.7	0.015	0.03
September	FTD			−0.6	−0.7	−0.4	0
	MTD			−0.6	−0.7	−0.5	0
	LTD				−0.7

. The wilting point and field capacity of the soil were 0.16 and 0.48 cm³/cm³.

$$E{P}_{10\text{-}\mathrm{d}\mathrm{a}\mathrm{y}}=\left\{\begin{array}{ll}{P}_{10\text{-}\mathrm{d}\mathrm{a}\mathrm{y}},& P_{10\text{-}\mathrm{d}\mathrm{a}\mathrm{y}}\le 50 \mathrm{mm}\\ 0.75\cdot P_{10\text{-}\mathrm{d}\mathrm{a}\mathrm{y}},& 50 \mathrm{mm}<P_{10\text{-}\mathrm{d}\mathrm{a}\mathrm{y}}\le 150 \mathrm{mm}\\ 0.7\cdot P_{10\text{-}\mathrm{d}\mathrm{a}\mathrm{y}},& 150 \mathrm{mm}<P_{10\text{-}\mathrm{d}\mathrm{a}\mathrm{y}}\end{array}\right.$$

(25)

where EP_10-day is 10-day effective precipitation, P_10-day is precipitation in a 10-day timescale.

3.3. Scenario Design

Regional water demand is primarily governed by evapotranspiration and rainfall, while water supply is mainly influenced by water inflow and reservoir regulation for the reservoir-control area. In order to optimize irrigation and reservoir water use, the random features cannot be neglected. The net irrigation water demand was represented by ET₀ minus EP, regardless of crop types. The water inflow mainly denotes the upstream runoff. In this study, four levels were defined for water demand and supply, and the Pearson type III (P-III) distribution was employed to estimate the characteristic values and discrete marginal probabilities. The independence between water demand and supply was assumed, and their joint probabilities were calculated by $p{(A,B)}_{ij}=p{(A)}_i\cdot p{(B)}_j$ (A and B are random events), presented in

Table 5. The joint probability of coupled random scenarios, where H1–4 represent high to very low levels of water inflow and L1–4 represent low to very high levels of water demand.

	Water Inflow	H1	H2	H3	H4	Cumulative Probability
Water Demand
L1		0.0625	0.125	0.0375	0.025	0.25
L2		0.125	0.25	0.075	0.05	0.5
L3		0.0375	0.075	0.0225	0.015	0.15
L4		0.025	0.05	0.015	0.01	0.1
Cumulative probability		0.25	0.5	0.15	0.1	1

. For water demand, levels L1 to L4 denote low, medium, high, and very high demand, respectively. For water inflow, levels H1 to H4 denote high, medium, low, and very low, respectively. Based on these scenario classifications, the corresponding representative years were selected to provide the basis for estimating crop evapotranspiration, effective precipitation, and water inflow. The crop evapotranspiration of different crops and the corresponding effective precipitation under different water-demand scenarios are summarized in

Table 6. Crop evapotranspiration (ET_c) under different levels of water demand (m³/ha), where the values in red denotes the evaporation of bare soil during fallow periods, and L1–4 represent low to very high levels of water demand. Note that the growth period of citrus begins on 1 January.

		Wheat				Rape				Citrus
		L1	L2	L3	L4	L1	L2	L3	L4	L1	L2	L3	L4
October	FTD	196.2	117.6	132.1	172.6	198.1	118.8	133.4	174.2	180.8	108.4	121.7	159.0
	MTD	145.0	107.4	104.4	130.4	146.4	108.4	105.4	131.7	133.6	99.0	96.2	120.2
	LTD	126.5	173.5	109.1	146.7	127.8	175.2	110.2	148.1	116.6	159.9	100.6	135.2
November	FTD	109.7	211.0	127.8	203.6	110.5	212.6	128.8	205.1	75.6	145.5	88.1	140.4
	MTD	165.1	89.5	97.9	89.7	166.3	90.2	98.6	90.4	113.8	61.7	67.5	61.9
	LTD	100.9	99.3	93.3	239.5	101.7	100.0	94.1	241.3	69.6	68.4	64.4	165.1
December	FTD	165.1	122.9	108.5	112.4	179.9	133.9	118.2	122.4	134.3	99.9	88.2	91.3
	MTD	60.1	90.4	160.4	101.0	65.5	98.5	174.7	110.0	48.9	73.5	130.4	82.1
	LTD	178.2	114.5	129.6	338.8	194.1	124.7	141.1	369.1	144.8	93.1	105.3	275.4
January	FTD	112.8	280.6	225.3	260.2	120.7	300.4	241.2	278.6	61.4	152.9	122.8	141.8
	MTD	250.6	189.8	245.4	145.9	268.3	203.2	262.7	156.1	136.6	103.4	133.7	79.5
	LTD	181.1	241.2	226.2	130.4	193.8	258.2	242.2	139.6	98.7	131.4	123.3	71.0
February	FTD	121.5	130.0	331.8	169.4	123.4	132.0	336.9	172.0	71.0	76.0	194.0	99.0
	MTD	120.6	215.5	133.6	225.3	122.4	218.8	135.7	228.8	70.5	126.0	78.1	131.7
	LTD	137.9	155.3	139.3	126.5	140.0	157.7	141.4	128.5	80.6	90.8	81.4	74.0
March	FTD	124.5	186.2	257.0	221.0	159.1	238.1	328.6	282.5	74.7	111.7	154.2	132.6
	MTD	90.9	103.4	296.0	248.7	116.2	132.2	378.4	317.9	54.6	62.1	177.6	149.2
	LTD	225.1	303.9	166.3	256.5	287.8	388.5	212.6	327.8	135.1	182.3	99.8	153.9
April	FTD	176.9	153.9	174.4	178.4	191.2	166.3	188.5	192.8	116.2	101.0	114.5	117.1
	MTD	155.5	276.2	149.9	237.2	168.1	298.5	162.0	256.4	102.1	181.3	98.4	155.7
	LTD	245.8	255.8	268.8	274.8	265.7	276.5	290.5	297.0	161.4	168.0	176.5	180.4
May	FTD	284.7	216.6	322.2	200.2	376.4	286.4	425.9	264.7	229.2	174.4	259.4	161.2
	MTD	198.9	137.9	142.2	182.5	198.9	137.9	142.2	182.5	209.9	145.5	150.1	189.3
		Rice				Maize				Citrus
		L1	L2	L3	L4	L1	L2	L3	L4	L1	L2	L3	L4
May	LTD	238.5	238.5	196.8	321.6	269.6	269.6	222.4	363.5	197.0	197.0	162.6	265.6
June	FTD	277.4	315.4	312.4	340.6	292.3	332.4	329.2	358.9	247.5	281.4	278.7	303.9
	MTD	218.2	333.2	378.0	232.5	229.9	351.2	398.4	245.0	194.7	297.4	337.3	207.4
	LTD	232.2	308.6	381.6	262.1	244.7	325.2	402.1	276.3	207.2	275.4	340.5	233.9
July	FTD	280.9	325.7	229.3	301.9	271.5	314.8	221.7	291.9	202.2	234.5	165.1	217.4
	MTD	414.7	354.6	396.9	278.7	400.9	342.8	383.7	269.4	298.6	255.3	285.8	200.7
	LTD	342.2	316.3	447.5	491.5	330.8	305.7	432.6	475.1	246.4	227.7	322.2	353.9
August	FTD	367.8	403.0	310.6	327.1	291.7	319.6	246.3	259.4	276.5	302.9	233.5	245.9
	MTD	356.1	354.3	382.5	342.3	282.4	281.0	303.4	271.5	267.7	266.3	287.5	257.3
	LTD	279.1	255.3	221.0	349.3	221.4	202.5	175.3	277.0	209.8	191.9	166.1	262.6
September	FTD	212.2	304.9	267.9	246.6	166.7	239.5	210.5	193.7	189.5	272.2	239.2	220.2
	MTD	195.8	229.0	188.1	237.1	153.8	180.0	147.8	186.3	174.8	204.5	167.9	211.7
	LTD	130.5	129.6	108.1	124.8	130.5	129.6	108.1	124.8	163.1	162.0	135.1	161.6

and

Table 7. Effective precipitation under different levels of water demand (m³/ha), where L1–4 represent low to very high levels of water demand.

		L1	L2	L3	L4
October	FTD	61	1176	388	231
	MTD	52	728	429	277
	LTD	674	57	417	165
November	FTD	260	374	139	34
	MTD	209	143	110	239
	LTD	142	54	72	54
December	FTD	21	76	16	57
	MTD	39	99	5	122
	LTD	36	107	81	0
January	FTD	86	1	9	11
	MTD	26	39	26	33
	LTD	164	108	122	31
February	FTD	196	123	1	250
	MTD	155	77	134	24
	LTD	55	28	33	132
March	FTD	253	109	25	41
	MTD	208	246	102	210
	LTD	213	272	248	132
April	FTD	133	125	238	222
	MTD	988	151	306	204
	LTD	366	464	470	143
May	FTD	181	346	269	457
	MTD	1163	561	373	644
	LTD	446	333	675	479
June	FTD	504	694	241	558
	MTD	626	210	721	320
	LTD	648	910	958	1096
July	FTD	606	800	1068	562
	MTD	814	2108	583	727
	LTD	2039	3117	2396	1597
August	FTD	1034	1933	2385	225
	MTD	937	2553	1450	504
	LTD	3703	1638	2428	1556
September	FTD	702	458	469	573
	MTD	888	644	597	486
	LTD	762	428	851	409

. The bare-soil evaporation during the first and second fallow periods (in May MTD and September LTD, respectively) is also presented in these tables. The four levels of reservoir inflow are described in

Table 8. Water inflow under different levels of water supply and the other uses of reservoir water (10⁴ m³), where H1–4 represent high to very low levels of water inflow.

	H1	H2	H3	H4	Other Uses
October	109.58	60.96	13.41	39.16	11.16
November	24.98	15.53	13.50	17.94	10.38
December	2.89	3.09	2.80	12.73	10.66
January	5.40	4.92	3.28	5.50	10.36
February	4.24	6.08	2.60	7.04	9.33
March	11.58	13.50	13.89	9.45	10.06
April	25.47	24.98	25.47	12.93	9.88
May	103.89	30.39	34.73	11.77	10.36
June	335.69	98.01	109.00	50.45	10.58
July	273.57	388.84	155.31	65.21	10.16
August	158.68	109.97	220.90	157.04	11.06
September	86.33	155.98	97.24	129.36	10.88
Sum	1142.32	912.26	692.13	518.59	124.87

. The irrigation water use efficiency in the study area is 0.6. The reservoir has a minimum storage of 27 × 10⁴ m³ and a maximum storage of 480 × 10⁴ m³. The initial storage was set to its maximum capacity (480 × 10⁴ m³). To ensure sufficient water supply for the following year, the reservoir is expected to be refilled to its maximum capacity as much as possible by the end of the wet season. Lingo v11.0 was used to solve the model.

In this study, tolerance of decision makers for water overuse was considered and fuzzy chance-constrained programming was applied to represent their risk preferences. Three confidence levels were adopted to show the risk attitude of decision makers, i.e., γ = (0, 0.5, 1), and three risk scenarios were generated. Higher confidence degree means lower risk. In particular, the constraints will be strictly satisfied when γ = 1. The weighting coefficients α and β show the preference of decision makers for random and interval robustness penalty, which are assumed as 1 for convenience.

4. Results and Discussion

4.1. Rain-Fed Yield

Before solving the model, rain-fed yield should be estimated first. Although rice can tolerate some variation in water, a shallow water layer is essential during key growth stages, such as the booting and heading stages. Therefore, the Jensen model is not applicable to rain-fed rice and the rain-fed rice yield was assumed to be 0 under all scenarios according to field survey. The results are shown in

Table 9. The rain-fed yield (kg/ha) and yield reduction ratio of crops under different water demand scenarios, where the values in brackets are yield reduction and these out of brackets are rain-fed yield, L1–L4 denote water demand levels from low to very high.

	L1	L2	L3	L4	Potential Maximum Yield
Wheat	4178 (0.45)	2730 (0.64)	1347 (0.82)	0 (1.00)	7650
Rapeseed	1081 (0.59)	445 (0.83)	122 (0.95)	0 (1.00)	2625
Rice	0 (1.00)	0 (1.00)	0 (1.00)	0 (1.00)	7995
Maize	11,955 (0.00)	11,505 (0.04)	11,785 (0.01)	11,465 (0.04)	11,955
Citrus	32,734 (0.23)	26,921 (0.37)	16,154 (0.62)	0 (1.00)	42,750

. Among five crops, maize performed best and yield reduction can be neglected, largely because of its inherent drought tolerance and the fact that its growth period aligns with the wet seasons. In contrast, wheat and rapeseed suffered substantial yield loss as they grow in dry seasons, and their yield declined rapidly with intensifying drought. Citrus had moderate yield reduction during normal and wet years, while considerable yield reduction happened under severe drought. Therefore, except for maize, irrigation is essential for yield stabilization and serves as an effective measure to mitigate drought.

4.2. Objective Analysis Under Typical Risk Scenario

In practice, the water use targets set by decision makers should be met as much as possible. Therefore, the scenario characterized by actual water use being equal or lower than the targets, i.e., the confidence level γ = 1, was considered as a typical scenario in this study. The results of various objectives and penalty caused by uncertainty are shown in

Table 10. The optimal results of different objective indicators and penalty of uncertainty under the typical risk scenario, where expectation value equals robust value plus penalty.

Objective	Net Benefit	Irrigation Water Productivity	Marginal Yield
Unit	106	kg/m3	kg/m3
Expectation value	174.34	22.46	10.10
Robust value	138.89	18.67	8.47
Penalty from uncertainty	35.45	3.79	1.63
Membership of robust value	0.85	0.85	0.95
Ratio of penalty to expectation	0.20	0.17	0.16

. The results show that the smallest membership is 0.85 and both memberships of the objective of net benefit and irrigation water productivity reach the bottom, denoting the trade off of these two objectives limits the further improvement in model performance. Meanwhile, the results also imply that 0.85 is the best compromise that the trade off among three objectives can reach by means of optimizing water resource allocation under current situation. Further potential would lie in other non-water-management measures, like adopting more effective irrigation measures or reducing irrigation loss during delivery. However, these methods also have some flaws, such as high costs. In addition, the uncertainty penalty reflects the influence of uncertainty on the robustness of optimized objectives. Among the three objectives, the penalty ratio of net benefit is largest, denoting that it is most difficult to keep the objective stable and reliable. The reason might be the coexistence of interval and randomness, while irrigation water productivity and marginal yield are influenced only by randomness. Furthermore, the penalty from randomness and interval of net benefit was 0.58 and 34.87 × 10⁶ , accounting for 1.64% and 98.36% of the total penalty, respectively. This implies that the influence from interval is predominant in determining robustness of net benefit. Since interval uncertainty is mainly from economic parameters like prices and costs, market fluctuation may be the reason for bringing large uncertainty to net benefit. In other words, the objective of net benefit is more sensitive to the price fluctuation than to water supply and demand. For irrigation water productivity and marginal yield, the randomness from water supply and demand will influence the most. Additionally, the error during data collection may enlarge the uncertainty. Therefore, the sample error should be controlled as much as possible to avoid unnecessary uncertainty.

In order to analyze the impact of random uncertainty, the objectives under different random scenarios are shown in

Table 11. The objectives under different water supply and demand scenarios, where L1–4 denotes low to high crop water demand and H1–4 denotes high to low water inflow.

		H1	H2	H3	H4
Net benefit (106 )	L1	175.43	175.43	175.43	175.43
	L2	174.38	174.38	174.38	174.38
	L3	172.91	172.91	172.91	172.91
	L4	173.71	173.71	173.71	172.27
Irrigation water productivity (kg/m3)	L1	23.12	23.12	23.12	23.12
	L2	25.91	25.91	25.91	25.91
	L3	15.34	15.34	15.34	15.34
	L4	14.14	14.14	14.14	14.61
Marginal yield (kg/m3)	L1	7.04	7.04	7.04	7.04
	L2	11.04	11.04	11.04	11.04
	L3	9.76	9.76	9.76	9.76
	L4	13.46	13.46	13.46	13.90

. The table shows that with the increase in water demand from L1 to L4, the net benefit presents a declining trend, while irrigation water productivity first grows then drops significantly. Compared with water demand, the change in water supply had little influence on the objectives, which means that the model is more sensitive to water demand than water supply. Given the frequent seasonal droughts in the study area, the stable objectives under different water supply scenarios indicate that optimal irrigation and reservoir regulation can effectively reduce the sensitivity of crop growth to water inflow and mitigate most of seasonal drought, except for the extreme scenario (L4, H4). Statistically, the probability of extremely high water demand and extremely low water inflow is 1%, which means that there is theoretically 1% probability of inevitable drought loss. However, based on historical records for the Dongfeng Irrigation District (1971–2020), moderate to severe yield-reducing droughts occur in about 15–20% of the years. Therefore, there exists substantial potential to reduce drought-induced loss through optimized irrigation and reservoir regulation.

4.3. Optimal Strategy Analysis Under Typical Risk Scenario

Figure 5 presents: (a) the total irrigation quota of crops over the entire growth period, (b) the satisfaction degree of water supply at different growth stages except for citrus (value = 1 in all cases), and (c) irrigation quota of crops excluding citrus and effective rainfall under different water demand scenarios. The optimized irrigation quota for each crop in Figure 5a varies significantly across water demand scenarios, but generally remains insensitive to water supply scenarios, except for rice under the extreme dry condition (L4, H4). These findings are consistent with the objective results in Table 11 across all random scenarios, indicating that the available water is sufficient to achieve the optimal objectives through appropriate irrigation scheduling and reservoir regulation. Among the five crops, maize requires the least irrigation, as its water requirement can be largely met by the rainfall (see Table 9). However, although the reference crop water demand (i.e., ET₀−EP, if it is bigger than 0, else 0) was used to define four levels of water demand from low to very high, the optimal irrigation quota for a given crop does not follow a monotonic increasing trend with the water demand. The irrigation quotas under different water demand scenarios exhibit considerable variability, with ranges of wheat [773, 2073], rapeseed [689, 2210], rice [1141, 2171], maize [0, 57], and citrus [143, 1022] m³/ha, respectively. Two possible reasons may account for this pattern. First, deficit irrigation is widely applied. Except for citrus, whose water demand is fully satisfied across all scenarios, the water stress coefficients of the other crops are shown in Figure 5b. Notably, the dry-season crops (rapeseed and wheat) suffer more severe water stress than wet-season crops (maize and rice). Second, the sensitivity indices of crops and effective rainfall vary across growth stages under different water demand scenarios. Taking Scenario L2 for instance, sufficient rainfall occurring in the first ten days of October provides adequate soil moisture, which substantially reduce the need for irrigation. This also explains why the water productivity in Scenarios L2 is higher than that in Scenario L1 and why the marginal yield in Scenario L2 exceeds that in Scenario L3, as shown in Table 11.

Figure 5 shows substantial variation in the optimal irrigation strategies under the combined effects of crop types, natural conditions, and agricultural practices, even though the water demand and supply scenarios were predefined to reduce the uncertainty. Therefore, the optimization model is necessary and highly efficient for irrigation scheduling in complex irrigation systems.

The optimal results of irrigation water allocation of different crops were compared with the local irrigation quota from field survey and “Water Quota in Sichuan Province” issued by the Sichuan Provincial People’s Government in 2021, as shown in

Table 12. Comparison of irrigation quota from the model, local survey and Sichuan Provincial People’s Government, where p is irrigation design guarantee rate.

m3/ha	Optimized Irrigation Quota	Irrigation Quota from Survey	Irrigation Water Quota
			p = 0.5	p = 0.75	p = 0.9
Wheat	[773, 2073]	2237	1425	2175	2400
Rapeseed	[689, 2210]	2637	1575	1875	2100
Rice	[1141, 2171]	3417	4800	5400	6000
Maize	[0, 57]	146	960	1275	1500
Citrus	[143, 1022]	964	1425	1725	2700

. The results show that the irrigation water of wet-season crops was saved under optimal irrigation schemes, and the government-proposed irrigation quota of rapeseed should be increased when extreme dry situation happens. Great water-saving potential exists in wet-season crops for current study area, which aligns with the points of previous study [46].

Table 13. The crop yield under optimized irrigation schemes, the yield increased by irrigation compared with rain-fed yield, and the yield reduction rate compared with maximum yield, where L1–4 means low to very high water demand.

	Crop Yield Under Optimized Irrigation Schemes (kg/ha)				Yield Increased by Irrigation (kg/ha)				Yield Reduction Rate
	L1	L2	L3	L4	L1	L2	L3	L4
Wheat	5898	5374	5086	4949	1720	2644	3738	4949	[0.23, 0.35]
Rapeseed	1830	1625	1358	1588	749	1180	1236	1588	[0.3, 0.48]
Rice	7934	7928	7652	7857	7934	7928	7652	7857	[0.01, 0.04]
Maize	11,955	11,508	11,785	11,955	0	3	0	490	[0, 0.04]
Citrus	42,750	42,750	42,750	42,750	10,016	15,829	26,596	42,750	0

shows the crop yield under optimized irrigation schemes, the yield increased by irrigation compared with rain-fed yield, and the yield reduction rate compared with maximum yield. With the increase in crop water demand, the role played by irrigation in maintaining crop yield becomes significant. Among all crops, maize experiences the least water stress and the contribution of irrigation is small. However, although planted in wet seasons, rice highly relies on irrigation since water layer is necessary for some key growth stages. The expected yield of wheat, rapeseed, rice, maize, and citrus is 5419, 1633, 7881, 11,706, and 42,750 kg/ha, respectively. Compared with potential maximum yield, the average yield reduction rates are 0.29, 0.38, 0.01, 0.02, and 0 for wheat, rapeseed, rice, maize, and citrus. Even though water supply is ample, dry-season crops also experience water stress, especially for rapeseed. The reason might be the low benefit of wheat and rapeseed compared with their high water demand. The average yield of wheat, rapeseed, rice, maize, and citrus in the study area is 5400, 2026, 6288, 8784, and 32,724 kg/ha, respectively. Compared with the actual average yield, the yield of wheat, rapeseed, rice, maize, and citrus increased by 0.4%, −19.4%, 25.3%, 33.2%, and 30.6%, following optimal irrigation schemes. The average income after optimization increased by 27.9% compared with the actual practice. According to local survey, a net loss of RMB 192 to 4692 per hectare was reported to grow rapeseed alone under traditional farming practices, which explains why the water supply was restricted for rapeseed in this study. Despite this, farmers and local governments continue to promote the rice-rapeseed rotation system. The reasons are not confined to direct cash returns from rapeseed but lie in a broader set of agronomic, economic, and policy benefits, such as soil health, reduced pests, diseases, and agrochemical use, government subsidies, additional income from “flower economy”, and strategic national food security, which could be considered in further study.

The monthly water consumption, and total water consumption and supply under different water demand and inflow scenarios are respectively shown in Figure 6a,b. From Figure 6a, the main irrigation happens in October, January, and May, where October and January are dry seasons and the larger amount of water irrigated in May was used for paddy field soaking. With the increasing water demand from Scenario L3 and L4, irrigation is required throughout dry seasons. From Figure 6b, the total water demand can be satisfied under most scenarios, except for the extreme dry scenario (L4, H4). The irrigated water for rice decreased by 10% in Scenario (L4, H4).

4.4. Model Performance

In order to recognize the influence of multiple objectives and robustness penalty, model comparison was conducted among the single-objective expectation models (ME), the single-objective robust models (MR), and the proposed multi-objective robust models (MRP) in this study. Meanwhile, the influence of other sources of uncertainty, such as the risk tolerance of constraint violation and the expected objective intervals (i.e., TOVs and TWVs) from decision makers was also considered. As constraint violation usually happens when water availability is limited, the scenarios were simplified and the dry scenario groups of (L3, H3), (L3, H4), (L4, H3) and (L4, H4) were selected with modified probabilities of (0.36, 0.24, 0.24, 0.16), respectively. The optimized results of expectation and robustness of net benefit, irrigation water productivity, and marginal yield are shown in

Table 14. The results of objective indicators of single-objective and multi-objective robust models under different risk levels and two groups of TOVs and TWVs, where ME1–3 denote single-objective expectation models with net benefit, irrigation water productivity, and marginal yield as the objectives; MR1–3 denote single-objective robust models with net benefit, irrigation water productivity, and marginal yield as the objectives; MRP1–2 denote multi-objective model with robust and expectation values of TOVs and TWVs; γ = 0, 0.5, 1 denote high, moderate, and low risk levels.

	Model	Confidence Level	Robust Values			Expectation Values
			Net Benefit	Irrigation Water Productivity	Marginal Yield	Net Benefit	Irrigation Water Productivity	Marginal Yield
Single-objective expectation model	ME1	γ = 1	138.53	10.50	7.74	176.04	12.38	9.49
	ME2	γ = 1	110.66	15.25	9.05	152.13	15.91	11.40
	ME3	γ = 1	132.78	15.18	9.62	168.63	15.53	11.71
Single-objective robust model	MR1	γ = 1	138.53	10.62	7.38	173.76	12.45	9.58
		γ = 0.5	140.96	10.32	7.02	176.33	11.31	8.69
		γ = 0	142.19	10.71	6.74	177.76	10.79	8.31
	MR2	γ = 1	110.66	15.25	9.05	152.13	15.91	11.40
		γ = 0.5	110.66	15.25	9.05	152.13	15.91	11.40
		γ = 0	110.66	15.25	9.05	152.13	15.91	11.40
	MR3	γ = 1	130.45	12.30	9.86	167.26	14.25	10.46
		γ = 0.5	130.64	11.30	9.95	167.50	13.79	10.03
		γ = 0	135.63	11.12	9.96	170.37	13.68	9.96
TOV			142.19	15.25	9.96	177.76	15.91	11.40
TWV			110.66	10.32	6.74	110.66	10.32	6.74
Multi-objective robust model	MRP1	γ = 1	136.09	14.85	9.64	170.25	15.38	11.57
		γ = 0.5	136.09	14.85	9.64	170.25	15.38	11.57
		γ = 0	136.09	14.85	9.64	170.25	15.38	11.57
	MRP2	γ = 1	138.51	12.79	8.67	173.76	13.65	10.37
		γ = 0.5	139.73	12.74	8.76	174.66	13.44	10.18
		γ = 0	139.73	12.74	8.76	174.66	13.44	10.18

.

The results of single-objective models reveal a clear conflict among net benefit, irrigation water productivity, and marginal yield—an increase in any one of these indicators is typically accompanied by a decline in the other two. The robust penalty associated with the single-objective expectation models is larger than that of single-objective robust models with γ = 1, indicating that the objectives from expectation models exhibit greater variability, although they generally generate better expected objectives. In contrast, the water productivity remains largely unaffected.

The TOVs generated from the robust results are more conservative than those from expectation values, whereas the same TWVs were assigned since the TWVs from expectation values could not be feasibly solved. The results of multi-objective models fall within the ranges of TOVs and TWVs, suggesting that the multi-objective robust models can effectively mitigate the conflict among three objectives and provide decision makers with compromise irrigation strategies. Furthermore, the performance of MR1, MR3 and MRP2 is sensitive to risk preferences, while MR2 and MRP2 remain relatively stable, implying that both goal preferences and expected objective intervals would influence the response of optimal decisions to risks. In turn, the risk attitudes would also influence the model performance under different objectives. However, risk-preferring decision makers should exercise caution when selecting the risk levels, as for some risk-insensitive models, taking on additional risk does not necessarily lead to improved performance. In summary, the interactions among risk preferences, goal preferences, and expected objective intervals should be carefully examined before a final decision is made.

The preferences of decision makers for different types of uncertainty also affect the performance of the robust models. Taking MR1 under γ = 1 as an example, the robust and expectation values of the net benefit, as well as associated penalty from randomness and interval are shown in

Table 15. The robust and expectation values of the net benefit, as well as the associated penalty from randomness and interval under different weighing coefficients α and β from single-objective robust model MR1.

α	β	Robust Value	Expectation Value	Weighed Penalty	Random Penalty	Interval Penalty
1	1	138.53	173.76	35.23	0.48	34.75
1.5	0.5	155.67	173.76	18.09	0.48	34.75
0.5	1.5	122.07	176.01	53.94	2.28	35.20
0.5	0.5	157.27	176.01	18.74	2.28	35.20
0	1	140.81	176.01	35.20	2.28	35.20
1	0	173.74	176.01	2.28	2.28	35.20
0.5	1	139.67	176.01	36.34	2.28	35.20
1	0.5	156.13	176.01	19.88	2.28	35.20

. The results show that different weighting coefficients of various types of uncertainty will influence the model performance.

Compared with the previous studies that focus on one specific group of multi-cropping systems [18,19,20], the research in this study can not only generate optimal irrigation strategies for multiple groups simultaneously, but also deal with multiple uncertainty. The crop growth and water movement were expressed with popular and simple methods. The objectives selected in this study were also representative for most irrigation regions. The constraints were applied to guarantee the secure operation of the system. All these efforts make the model framework capable of being applied to other similar regions, even for monocropping systems. According to the proposed chance-constrained multi-objective robust programming, ample strategies of irrigation and reservoir regulation can be obtained. Specially, the influence of uncertainty can be revealed based on the optimal results, which reduces the work of parameter sensitivity analysis. However, there are still some limitations in this study. First, the model assumptions are inevitable for all models and the default values of parameters, such as planting date, initial soil moisture and various scenarios will influence the model performance and optimal results, which should be investigated during practical application. Second, the uncertainty from economic and natural factors, as well as three kinds of human preferences has been investigated in this study. However, the uncertainty is widespread, and the involvement of a large quantity of uncertainty would make the difficulty in model solving increase considerably. Therefore, appropriate simplification and sensitivity analysis after optimization are necessary and helpful for high-efficient and reliable decision making. Third, with the development of technology and cognition, new methods should be considered. For example, with the improvement of sensor-based crop water-status monitoring, associated irrigation could also realize automatic and real-time management [48].

5. Conclusions

A hybrid chance-constrained multi-objective robust programming framework was developed for irrigation scheduling of multi-cropping systems to help reduce seasonal drought risk. The deficit irrigation theory, soil water balance and reservoir operation were integrated within the model. In order to satisfy different decision makers, net benefit, irrigation water productivity and marginal yield of irrigation were selected as objectives to trade off water use and crop production. Meanwhile, hybrid uncertainty was handled, for example, coupled random scenarios, interval economic parameters and chance constraints.

The proposed method was applied in the Dongfeng Reservoir Irrigation District and ample irrigation schemes were generated for 16 coupled scenarios. The results show the optimal irrigation schemes can substantially mitigate the seasonal drought and reveal the influence of uncertainty from parameters on the objectives. The optimized irrigation scheduling can help save water for wet-season crops compared with the practice and reflect the response of crops to various water demand and supply scenarios. Finally, the uncertainty from human factors, i.e., the risk and goal preferences, as well as the optimistic attitudes of decision makers was investigated to show their interactive influence on the results. Although there are still limitations in the framework, the developed model is capable of optimizing irrigation scheduling of multi-cropping systems under hybrid uncertainty, and help mitigate the seasonal drought, save water, and increase sustainability of agrosystems.