Application of hybrid algorithms in an optimal allocation model of water and land resources

Irrigated agriculture accounts for more than 70% of freshwater withdrawals from rivers, lakes, and aquifers, and the development of countries worldwide is directly or indirectly affected by the restriction of water resources (Fan et al. 2020). Agricultural irrigation water and crop planting layout planning affect each other. To achieve the dual goals of food security and sustainable economic development, we need to focus on the synchronous and optimal allocation of water and land resources.

Since the 1970s, models have been developed for the optimal allocation of water and land resources. In the literature, such models are divided into three categories according to different decision variables: (1) optimizing the planting area of different crops when the irrigation water volume is sufficient to maximize the yield per unit area (Maji & Heady 1978; Morales et al. 1987; Paudyal & Gupta 1990). (2) When irrigation water volume is limited but the crop planting area is determined, the limited irrigation water is allocated at different growth periods of the crops, to obtain the maximum yield and benefit throughout the entire growth period of crops (Hiessl & Plate 1990; Paudyal & Manguerra 1990; Akhand et al. 1995). (3) When irrigation water volume and crop planting area are limited, water resources and land resources need to be optimized simultaneously to obtain the maximum yield and benefits of different crops.

Hitherto, many scholars have conducted in-depth research on the third type of model. Khandelwal & Dhiman (2018) proposed a deterministic linear programming (DLP) model and a chance constrained linear programming (CCLP) model to jointly dispatch surface water and groundwater and formulate a sustainable land and water resource management plan. Ren et al. (2017) proposed a multi-objective fuzzy optimization method for water and land resources based on the objectives of administrative, economic, and ecological benefits, and solved it at the level of fuzzy parameters to obtain the optimal irrigation scheme. Li et al. (2022) quantified the changing environmental model and established a multi-dimensional model for the synchronous optimal allocation of water and land resources in combination with climate and socio-economic changes. However, the water source projects of the irrigation system involved in the above research were mostly single reservoirs and multiple reservoirs, or the joint operation of surface water and groundwater. There are relatively few studies on irrigation systems with special hydraulic connections. In China, with the gradual modernization of irrigation areas to meet the water demand of irrigation areas, a large number of pumping stations have been built in water-deficient irrigation areas to reduce the diversion of water resources outside the area, thus forming an irrigation system comprising a joint operation ‘reservoir and pump station’. Concurrently, with the gradual improvement of the water rights distribution system (Molle 2004; Heikkila 2015), the annual diversion volume of the water diversion project is strictly limited; as such, when optimizing the allocation of water and land resources in water-deficient irrigation areas, the impact of regional water rights restrictions must be considered.

There are many decision variables and engineering constraints that must be considered in the optimal allocation model of water and land resources. Furthermore, classical optimization algorithms, such as linear/nonlinear programming and dynamic programming (DP), struggle to solve such high-dimensional and nonlinear problems (Nagesh Kumar & Janga Reddy 2007; Tsoukalas & Makropoulos 2015). However, with the rapid development of computer technology, modern heuristic algorithms (such as genetic algorithm (Nagesh Kumar et al. 2006), simulated annealing algorithm (Georgiou et al. 2006), and ant colony algorithm (Nguyen et al. 2016)) have been developed to solve corresponding problems with satisfactory results. However, intelligent algorithms are limited owing to their tendency for premature convergence or convergence to local optimal solutions. On this basis, some scholars have integrated the characteristics of different algorithms and proposed a new hybrid algorithm to overcome the defects of classical and intelligent algorithms. For example, Dahmani & Yebdri (2020) proposed a new hybrid algorithm (HPSOGWO) based on the particle swarm optimization algorithm and the gray wolf algorithm and compared it with the real-coded genetic algorithm (RGA) and the gravitational search algorithm (GSA). The results showed that the combined algorithm can obtain higher-quality solutions than the other algorithms. Rasoulzadeh-Gharibdousti et al. (2011) proposed a new hybrid algorithm, the nonlinear-programming-genetic-algorithm (NLP-GA), based on nonlinear programming and a genetic algorithm. The solution results show that the hybrid algorithm could not only improve the convergence speed of the algorithm but also avoid local optimization. Masoud et al. (2021) proposed a new algorithm combining the Pareto evolutionary algorithm and multi-objective particle swarm optimization (MOPSO) for the optimization of pipe networks. The results of the case study showed that the hybrid algorithm was more stable in obtaining a global optimal solution.

Taking the water-deficient irrigation area in the northern hilly area of Jiangsu Province as a case study, this study constructed an optimal allocation model of water and land resources for a ‘reservoir and pumping station’ irrigation system supplying a water-deficient area under crop rotation with consideration of water rights restrictions.

When the area decision variables in the large-scale system model are determined, the model can be transformed into a high-dimensional DP model. When decision variables such as water supply and spill of the reservoir and water replenishment of the pumping station are determined in the large-scale system model, the model can be transformed into a linear model. Therefore, this study proposes two-hybrid optimization algorithms to solve the above model. The first-hybrid algorithm is a large-scale system successive approximation method [(DADP–LP)SA]. The algorithm uses DADP (Gong & Cheng 2018; Gong et al. 2019) to solve the high-dimensional DP model, linear programming (LP) to solve the linear model and obtains the global optimal solution of the large-scale system model through the successive approximation of DADP and LP. The second-hybrid algorithm (OD–DADP) is based on the theory of experimental optimization and solves the large-scale system model by mixing the orthogonal design (OD) method and DADP. In addition, these two-hybrid algorithms were compared with a real-coded genetic algorithm (RGA) and PSO, and the optimality and applicability of the various algorithms were analyzed and compared.

• (1)

  Water supply constraints:

• • (a)
    Total annual water supply constraint of the system:

    

    (2)
  • (b)
    Water supply restriction during the entire growth period of a single crop:

    

    (3)

    where SK is the total annual water supply of the annual regulation reservoir, MCM; BZ is the water rights of the pumping station, that is, the maximum allowable water replenishment volume of the pump station in a year, MCM; IW_i is the upper limit of the water supply during the entire growth period of a single crop, MCM.

• (2)

  Crop planting area constraints:

• • (a)
    Constraints on the total planting area of multiple crops:

    

    (4)
  • (b)
    Single crop planting area constraints:

    

    (5)

    where TA is the total planned planting area of various crops, hm²; A_i^min is the minimum planting area of i crops, hm²; A_i^max is the maximum planting area of i crops, hm².

• (3)
  Scheduling criteria constraints:

  

  (6)

  where V_i,j is determined according to the water balance equation:

  

  (7)

In the formula, V_i,j is the reservoir storage capacity at the end of stage j of crop i, MCM.

The (DADP–LP)SA algorithm is a hybrid algorithm for the successive approximation of large-scale systems, mainly aimed at the optimal allocation model of water and land resources. When the area decision variables in the large-scale system model are determined, the model can be solved by using the DADP method. When the decision variables such as water supply and spill of the reservoir and water replenishment of the pumping station are determined, the model can be solved by using the LP method. The global optimal solution of the large-scale system model can be obtained by successive approximations of DADP and LP.

Among these, DADP is based on the decomposition aggregation method. The large-scale system model is decomposed into a series of subsystem models. The subsystem model can be solved using the one-dimensional DP method to ensure optimal solution results are obtained for the subsystem model. Then, based on the corresponding relationship between each subsystem and the large-scale system, a series of subsystem models are aggregated according to the recurrence principle of DP. The aggregation model can also be solved by DP, and the global optimal solution of the large-scale system model can be obtained by checking the optimal solution of each subsystem model back from the optimal solution of the aggregation model. The essence of the algorithm is to nest multiple new DP processes in a recursive process of DP.

In addition, the two subsystem models must meet the constraints of scheduling criteria, water replenishment of the pumping station, crop planting area, and crop water demand.

• Step 2.2: Subsystem solution. The subsystems are obtained by the system decomposition belonging to the DP model, which can be solved using the classical DP method, where the state variable λ_i,j is the total water supply of the reservoir to the entire growth period of a single crop, which is discretized in a fixed step d in [0, IW_i]. For each state variable λ_i,j, the decision variable X_i,j in its feasible region [0, λ_i,j] is discretized in step d. Using sequential recurrence, the state transition equation λ_i,j−1 = λ_i,,j−X_i,j, the optimal water supply process X_i,j(i = 1,2;j = 1,2,…,M_i), and the objective function value F of the reservoir for each growth stage of dry crops in the subsystem can be obtained. Based on the optimal water supply process of the reservoir, the optimal spill process of the reservoir PS_i,j(i = 1,2;j = 1,2,…,M_i) and the optimal water replenishment process Y_i,j(i = 1,2;j = 1,2,…,M_i) of the pumping station can be determined.

• Step 2.3: Subsystem aggregation. If the two subsystem models are aggregated, the aggregation model can be solved using a DP method. The objective functions and constraints of the aggregation model are as follows:

The OD method is a scientific test method for the optimization of multifactor tests (Wang et al. 2017). A standardized orthogonal table was used to reasonably arrange the test. Using this method, we only need to perform a few experiments to determine the best scheme; then, through orthogonal analysis, we can obtain more comprehensive and systematic experimental results and find the optimal solution.

For the optimal allocation model of water and land resources established in this study, the OD could be used to solve the crop area, and the DADP could be used to solve the high-dimensional DP model. Therefore, the specific solution steps of OD–DADP are as follows:

• Step 1: Using crop type i (i = 1,2) as the test factor and the discrete value A_i of the crop area as the test level, an orthogonal table was constructed, and the test combination was determined.

• Step 2: Substituting the different crop area test combinations into Equation (1), the large-scale system model could be transformed into a high-dimensional DP model that could be solved, which can be solved using DADP. The solution process of the DADP can be referred to as the solution process of the (DADP–LP)SA, and the objective function value G was obtained at the end of the solution.

• Step 3: The best area scheme A_i* for all test combinations using orthogonal analysis was determined.

• Step 4: The best function value G* of the large-scale system model was obtained by substituting the best area scheme A_i* into Equation (1). Then, we obtained the optimal water supply process X_i,,j*(i = 1,2;j = 1,2,…,M_i), the optimal spill process PS_i,j*(i = 1,2;j = 1,2,…,M_i) of the reservoir, and the optimal water replenishment Y_i,j*(i = 1,2;j = 1,2,…,M_i) of the pumping stations.

Wright (1991) proposed a RGA and showed that the RGA has better convergence and optimality compared with the earlier binary-coded genetic algorithm. In the optimization process, the RGA usually randomly generates N possible solution sets X_i = (x_i¹,x_i²,…,x_i^d). In the iteration process, the population evolves through three operations: selection, crossover, and mutation, and finally converges to the optimal solution.

Similarly, in the PSO solution process, the penalty term in Equations (18)–(21) are constructed to deal with the corresponding constraints.

The two main crops in the study area showed the characteristics of rotation cultivation. The entire growth period of corn is from June to September and that of wheat is from October to May of the following year. The total growth period of the two rotation crops corresponds to the regulation and storage cycle of the entire hydrological year of the annual regulation reservoir (from October to September of the following year). Therefore, the regulation and storage periods of the GT reservoir are divided according to the growth stage of crops. The crop parameters of the irrigation area are shown in Tables 1 and 2.

The study area is in a hilly and mountainous area with high terrain. Artesian water diversion and reservoir storage cannot meet the water demand of the irrigation area. The newly built Shengli (SL) pumping station has transformed and modernized the irrigation area and water from the Shu River to supplement the GT reservoir to alleviate the water deficiency in the irrigation area. The characteristic parameters of the pump station are presented in Table 3.

Before comparing the performance of the algorithms, the OD–DADP using the orthogonal design method was analyzed. Range analysis can effectively clarify the interaction between various factors and the corresponding relationship between local and overall tests. In this study, two crops are mainly planted in the study area, so the factor number is 2. Then, nine values of the planting area of the two crops are discretized in the feasible area, that is, the level number is 9, and an orthogonal table L₈₁(9²) at 2 factors and 9 levels per factor is constructed. As shown in Table 5, a total of 81 test schemes will be generated, and corresponding objective function values can be obtained by substituting different combination schemes of area decision variables into the model. Then, orthogonal analysis was carried out on the test results to obtain the optimal combination scheme corresponding to the global optimal solution, as shown in Table 6. When factors 1 and 2 were both taken as the first discrete value, the global optimal solution of the model was (3.05 × 10⁷) rmb.

To compare the performances of (DADP–LP)SA, OD–DADP, RGA, and PSO, this study compared and analyzed the optimality of the objective function, stability of the algorithm, convergence of the algorithm, and speed of the solution.

First, the sensitivities of the four algorithms were analyzed. It can be seen from the range analysis in Table 6 that when the test combination was (1,1), it was the optimal combination, and the optimal value of the objective function of the OD–DADP was (3.05 × 10⁷) rmb. As shown in Table 7, the optimal solution obtained by the (DADP–LP)SA algorithm was (3.05 × 10⁷) rmb, and the convergence accuracy was 100,000. For the RGA, the optimal value of the objective function was (2.94 × 10⁷) rmb. At this time, the population size was 100, the crossover probability was 0.6, and the mutation probability was 0.2. For PSO, the optimal value of the objective function was (2.87 × 10⁷) rmb. At this time, the population size was 70, the inertia weight was 0.4, and the acceleration factor (c1 = c2) was 1.4.

Therefore, the objective function values of (DADP–LP)SA and OD–DADP were the same, and better objective function values could be obtained than those of the RGA and PSO. It can be observed that in the solution process of (DADP–LP)SA and OD–DADP, the additional parameters of the algorithm do does not need calibrating. By adjusting the convergence accuracy of the (DADP–LP)SA or the level number discretization accuracy of the OD–DADP, the optimal value of the objective function could be obtained, which has better optimality of the objective function and operability of the algorithm.

According to the algorithm parameters in Table 7, the four algorithms were run randomly 10 times to verify algorithm stability. As shown in Table 8, the 10 running results indicate that the objective function values obtained by (DADP–LP)SA and OD–DADP were optimal. Moreover, the worst solutions, the averages and standard deviations of (DADP–LP)SA and OD–DADP were still superior to those of RGA and PSO. Therefore, the above results show that the (DADP–LP)SA and OD–DADP algorithms proposed in this study exhibit better stability.

It is known that the running time of OD–DADP is independent of the number of iterations; therefore, it is impossible to directly compare its convergence and running time with those of the other three algorithms. However, the running time of OD–DADP was much longer than that of (DADP–LP)SA.

Notably, the initial solution sets of the RGA and PSO were randomly generated, and it is difficult for a series of random numbers to simultaneously meet the judgment constraints (Equation (8)) and equality constraints (Equations (7) and (12)) involved in the model. Therefore, the intelligent algorithms, RGA and PSO, are prone to fall into a dead cycle. Therefore, in this study, the constraints in Equations (8) and (12) are not considered in the solution processes of RGA and PSO.

Overall, despite their longer running times, in terms of the optimality of the objective function, the stability of the algorithm, and the convergence of the algorithm, the two-hybrid algorithms were better than RGA and PSO and had better algorithm optimality.

In addition, (DADP–LP)SA and OD–DADP do not need to spill in the entire regulation and storage cycle of the reservoir, whereas the spill of the reservoir of RGA and PSO is obtained by random generation. Therefore, the phenomenon of spill occurs in individual periods. This means that the pumping station water replenishment phenomenon and the spill of the reservoir coexist in uncertain periods, which violate common sense reservoir regulation. It is also the existence of spill that makes the four algorithms slightly different from the total amount of water supply.

Table 9 lists the water supply assurance rate and the water shortage index of the four algorithms. The irrigation assurance rates of (DADP–LP)SA and OD–DADP were higher than those of RGA and PSO, and the water deficit index was lower than those of RGA and PSO. Therefore, (DADP–LP)SA and OD–DADP can ensure the optimal operation of the irrigation system.

Overall, in terms of the dispatching process and results of the ‘reservoir and pumping station’ irrigation system, (DADP–LP)SA and OD–DADP were more reasonable and applicable than RGA and PSO.

In this study, an optimal allocation model of water and land resources for the combined irrigation of reservoirs and pumping stations is established. The model considers the irrigation water supply and spill of the reservoir and water replenishment of the pumping station as related variables, suggests the constraints of operation criteria, and synchronously solves the optimal water supply and spill process of the reservoir, the optimal water replenishment process of the pumping station, and the optimal planting area of different crops.

Although the (DADP–LP)SA and OD–DADP proposed in this study are two complex hybrid algorithms, their basic algorithms are classical function optimization methods such as DP, decomposition aggregation, linear programming, and orthogonal design. To ensure the optimal value of the objective function of the mathematical model, the hybrid algorithm can also reasonably deal with equality and judgment constraints. Compared to modern heuristic algorithms, the hybrid algorithm was more effective advantages in dealing with complex constraints.

The simulation results show that the operation time of (DADP–LP)SA and OD–DADP algorithms is longer than that of RGA and PSO algorithms. However, this is outweighed by the superior optimization and applicability of the (DADP–LP)SA and OD–DADP algorithms. This means that the hybrid algorithms developed here can effectively improve the annual output value of the irrigation area, improve the utilization rate of limited water and land resources, and provide a reference for the planning and management of similar water-deficient irrigation areas. This study also provides new ideas for the planning and solving of complex nonlinear models.

This work was supported by the National Natural Science Foundation of China (NSFC) (grant no. 52079119).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.