Multi-Objective Crop Planting Structure Optimisation Based on Game Theory

: To realise the ecological protection and high-quality development of the Yellow River Basin and transition from extensive utilisation to intensive conservation of agricultural water, a multi-objective crop planting structure optimisation model was established. The model enabled highly efﬁcient crop planting in terms of net income with high yield and low consumption of water. Thereafter, the game algorithm was used to balance different requirements of each objective function under each constraint, both competitively and cooperatively, to obtain an optimal crop planting structure. Finally, the proposed model and analysis method were demonstrated and veriﬁed using the Xiaolangdi south bank irrigation area as an example. The results indicated that using the competitive game algorithm produced a superior crop planting structure in terms of high net income, high yield, and low water usage, suggesting that the relationships between game players and objective functions should be considered in designing the optimisation model. Thus, the proposed approach provides a theoretical basis for the sustainable development of the agricultural industry by realising the intensive utilisation of water resources in a particular irrigation area.


Introduction
Water scarcity is a common global issue owing to rapid population growth and accelerating economic development. In particular, it is an urgent problem in agricultural countries. China is one among many countries with severe water shortage issues, and agriculture is a major water consumer. Notably, agriculture in the Yellow River basin guarantees national ecological security and regional food security.
The efficient, economic, and intensive utilisation of water resources in the Yellow River basin requires rational planning of population, urban, and industrial development. Such planning is necessary to develop water-saving industries and technologies, promote agricultural water conservation, and transform the utilisation of water resources from the extensive mode to the economical and intensive mode [1]. From 1956 to 2000, the average quantity of water resources in the Yellow River water supply area of Henan province was 19.198 billion m 3 ; the per capita consumption of water was only 275 m 3 ; and the average consumption of water by cultivated land was 4185 m 3 /hm 2 , only 13.7% of the national average during this period. Therefore, the problem of insufficient water resources was prominent [2]. Within the Yellow River basin, agriculture is a large consumer of water resources, accounting for 55% of all water consumption. In particular, the 52 large and medium-sized irrigation areas fed by the Yellow River have not yet been modernised, the benefits of water resource irrigation have not been fully realised, and the water conservation transformation has yet to be fully implemented [3]. To achieve high-quality development, conservation, and intensive utilisation of water resources in harmony with agricultural development in the Yellow River basin, it is necessary to consider factors such as social and economic development and ecological protection, optimisation of crop planting structure, development of a system for the reasonable allocation of limited water resources, and optimal allocation of limited water and soil resources among different crops to obtain the most comprehensive benefits.
Various technologies have been used to realise agricultural water conservation, including water-saving irrigation techniques [4], crop breeding [5], soil management [6], and crop planting structure optimisation (CPSO) [7]. Currently, CPSO methods are mostly multi-objective optimisation models, which are marked by their simultaneous consideration of economic, social, and ecological objectives and constraints to determine an optimal crop planting structure. Pearson and McRoberts [8] used linear programming to conduct a multi-objective optimisation of crop production in Germany based on trade balance maximisation, self-sufficiency, and virtual water footprint minimisation objectives. Qian [7] conducted a multi-objective fuzzy-robust programming method to obtain the optimal use of land and water resources in agriculture. However, major challenges remain when conducting the multi-objective planning of water resources in agricultural systems. In the CPSO of an irrigation area, the demand imposed by each objective on the crop planting area is different. For example, the water conservation objective imposes a higher demand for crops with smaller irrigation quotas, whereas meeting the economic objective requires more high-yield crops. Therefore, the process of meeting each target objective involves balancing various conflicting interests.
Generally, the multi-objective model is often transformed into a single objective model to find a solution. The transformation methods include the objective weight method, entropy weight coefficient method, and fuzzy optimisation method. However, selection of the weight coefficient or preference function of each objective is subjective and more empirical.
New multi-objective optimisation methods have been proposed to overcome this problem, including optimisation based on multi-objective particle swarm optimisation algorithm [9] and ant colony algorithm [10]. However, these methods must solve many number of Pareto non-inferior solutions with numerous calculations and slow convergence and the problem worsens with the increase of optimisation objectives and variables.
In recent years, game theory has been applied to solve multi-objective optimisation problems because it is similar to multi-objective optimisation design. B Zhang applied game theory to highlight the problem of water in the Heihe River basin [11]. Based on the game theory, Zhang introduced both external driving forces and internal equilibrium measures to explore a cooperation mechanism to mitigate the conflicts and encourage cooperation between upstream and downstream regions [12]. Currently, this method is mostly aimed at water resource management in water resource systems, and research on its application to CPSO is limited.
From this perspective, a multi-objective CPSO model of the Yellow River irrigation area was established in this study based on the intensive utilisation of water resources; the model was then solved using the game algorithm [13,14], considering cooperation or competition among the different demands of the target objectives, thereby obtaining an optimal strategy for crop planting in the irrigation area. To solve the model, each target is considered a game participant. The multi-objective model is standardised and multiplied to be solved as a cooperative game model. For a competitive game, the variable set is decomposed into the strategy set of each player. Each game player aims to maximise their benefit; thus, they combine the optimisation goals of the crop planting structure to find an optimum solution in their strategy set, form a round of game strategy combinations from all the optimisation results, and obtain the equilibrium solution through multiple rounds of games. Finally, the optimal strategy of the crop planting structure in the irrigation area is obtained by comparison.

CPSO Model Based on Intensive Utilisation of Water Resources
The concept of intensive utilisation of water resources represents an economic evaluation in which, by considering a relatively unchanged level of production technology, adjusting water resource use efficiency or reorganising factors to reduce the input cost, income can be increased relative to the utilised water resources [15]. Considering the water demand in and agricultural output of the irrigation area, the intensive use of water resources within the irrigation area can be reflected by three objectives: high efficiency, high output, and low consumption. A multi-objective optimisation model was accordingly established considering the net income generated by the crops grown in the irrigation area as the high-efficiency objective, the crop yield as the high output objective, and the irrigation water consumption as the low consumption objective. The crop planting area X Maj was the decision variable.

High-Efficiency Objective
The high-efficiency objective is defined to reflect the efficient use of agricultural water in the irrigation area and can be captured by the net income generated by the crops, which includes the degree of resource utilisation and production efficiency as follows: where Y j is the yield per hectare of crop j, P j is the net income generated by crop j, and n is the total number of different crops.

High Output Objective
The high output objective is defined as the maximisation of the total crop output, which can be calculated as follows:

Low Consumption Objective
The low consumption objective is defined as the minimisation of water consumed by crop irrigation and is expressed as follows: where IQ j is the irrigation quota for crop j.

Constraint Conditions
The planting industry is restricted by both natural and social conditions. Comprehensively considering the agricultural and water supply planning in the irrigation area, the crop planting constraints comprise the crop planting proportion, irrigation area, strictest restriction of the water resources management system, and non-negative constraints.

Crop Planting Proportion Constraint
The overall planting area is limited by regional land planning. The protection of cultivated land and basic farmland is determined by adhering to the strictest cultivated land protection system using the following constraint: where X MA is the total area of cultivated land, p jmin is the planning-allowed minimum crop planting ratio, and p jmax is the planning-allowed maximum crop planting ratio.

Irrigation Area Constraint
The total planting area of crops shall not exceed the arable land; the constraint is as follows: where I mc is the multiple cropping index in irrigation area, and A g is the maximum available irrigation area.

Strictest Restriction of the Water Resources Management System
The water available for irrigation in the irrigation area must be less than the planned water supply from each water source in the irrigation area as follows: where Q A is the total planned water supply for irrigation.

Multi-Objective Optimisation Solution Based on the Game Algorithm
Game theory represents a process of balancing multiple objectives to obtain an optimal solution through a game-based strategy and has considerable advantages in multi-objective and standard optimisation decision-making processes [14]. Because the high efficiency, high output, and low consumption objectives are contradictory in the multi-objective CPSO model developed in this study, the game algorithm is well-suited to obtaining an optimal balance among them. There are three elements in the game algorithm: the decision maker, strategy set, and benefit function. For transformation of the multi-objective model into a game model G = {P, S, F}, the decision maker is i ∈ P, P = (1, 2, 3), the policy set for each decision maker is S = (s 1 , s 2 , s 3 ), and the profit function of the decision maker is F = (f 1 , f 2 , f 3 ). When solving for the desired goal, cooperative and competitive games are used to solve the problem. The key to transforming the multi-objective model into a game is to divide the variable set into the strategy set owned by the game player, and the process is as follows: 1.
According to the multi-objective CPSO model, a single objective optimisation of each of three objectives is conducted, and the optimisation values The single objective functions of the CPSO model are divided into game parties, each of which strives to optimise its objective through competition. Considering the different objective targets, each objective function is processed into a dimensionless function as follows: 3.
If the decision maker considers the decision as a cooperative game, the multi-objective model is transformed into the income function for a cooperative game by where k is the number of objects. If the decision maker assumes the decision as a comparative game, the priority of each player is determined, and then the initial policy set is obtained according to the objective function.

4.
For any design variable x k , the definition domain is divided into n segments according to the step length ∆x k . The influence factor ∆ ji of the variable x k on game f i (x) can be expressed as

5.
Defining ∆ j = (∆ j1 , ∆ j1 , . . . , ∆ jk ) as the set of influence factors of the jth variable on the kth objective function, the degree of similarity between influence factors can be determined according to the Euclidean distance as follows: where p and q define the solution points between which the distances are measured, as in 1, 2, . . . , n. Thereafter, similarity matrix D is established based on Equation (10) as follows: 6. The similarity matrix D is fuzzy clustered to obtain the classification results of ∆. Because ∆ = (∆ 1 , ∆ 2 , . . . , ∆ l ) and the policy set S = {x 1 , x 2 , . . . , x l } correspond to each other, the clustering results of ∆ are the same as those of S. The design variable strategy set S 1 , S 2 , . . . , S k is obtained according to the clustering results and can be selected by each player according to the sum of their respective objectives and influencing factors. 7.
If the policy set S * i is the optimal policy set of the ith crop planting structure optimisation objective, take this strategy is taken as the initial strategy S (0) to find the optimal strategy of other objectives. The new strategy combination is expressed as S (1) = {S * 1 , S * 2 , S * 3 }. It is then verified if the distance between two adjacent policy combinations S (0) and S (1) satisfies the inequality S (1) −S (0) ≤ ε, where ε is the game convergence criterion. If it meets the inequality criterion, the competitive game ends. Otherwise, it continues on the basis of S (1) until the optimal strategies of all the players are stable and satisfied, and the game equilibrium solution is obtained, that is, the optimal strategy of the competitive game for the optimisation problem. 8.
Comparing the optimal strategy of the cooperative game with that of the competitive game, the most suitable strategy is selected as the optimal strategy of the model.

Introduction to the Evaluated Irrigation Area
The Xiaolangdi south bank irrigation area is located in the North Mangling of Luoyang city and represents an important commercial grain production base in Henan Province, adjacent to the Yellow River to the north and the Luohe River to the south. Limited by natural conditions, the Xiaolangdi south bank irrigation area experiences severe droughts and water shortages; annual precipitation is scarce, and the available surface water in the irrigation area is limited. The industrial, agricultural, and domestic water supply depends on the exploitation of groundwater and the Xiaolangdi Reservoir on the Yellow River. Therefore, intensive utilisation of water resources and maximum social and economic benefits must be urgently realised in this area.
According to [16], agriculture in the Xiaolangdi south bank irrigation area is devoted to grain production, followed by cash crops, and then the forestry and fruit industries. Grain crops include cereals (wheat, corn), beans, and sweet potatoes (autumn miscellaneous); cash crops include oil (peanut, rape, sesame), cotton, vegetables, melons, tobacco leaves, medicinal materials, flowers, etc. As the planting area of each of such crops is relatively small, they are collectively referred to as cash crops in this study. In 2018, the total economic output of Xiaolangdi south bank irrigation area was 65.84 billion yuan; the controlled cultivated land area was 50,647 hm 2 ; the designed irrigation area was 35,787 hm 2 ; the multiple cropping index was 1.8; the total water supply of the irrigation area was 82.63 million m 3 ; and the net water supply of surface water, groundwater, and Yellow River water were 9.16 million m 3 , 25.65 million m 3 , and 47.82 million m 3 , respectively.

Model Parameters
The four main crop types-wheat (x 1 ), corn (x 2 ), autumn miscellaneous (x 3 ), and cash crops (x 4 )-were considered as the decision variables in this study; the total planned irrigation water volume included surface water, groundwater, and Yellow River water. A multi-objective CPSO model was established using these parameters and optimised using the procedure provided in Section 2. The economic benefit is expressed by the income per unit area, and the social benefit is expressed by the output per unit area. The income per unit area and yield per unit area of wheat and maize are obtained from the Henan Statistical Yearbook and Henan Survey Yearbook. The income per unit area and yield per unit area of autumn miscellaneous and cash crops are weighted according to the planned planting area of the irrigation areas and calculated in combination with the Henan Statistical Yearbook and Henan Survey Yearbook. According to the irrigation area planning, the irrigation assurance rate is 50%. The irrigation quota of wheat and maize is obtained by selecting the irrigation quota of each crop in western Henan according to the local standard of Henan Province,-the 2020 agricultural and rural domestic water quota, and combining the autumn miscellaneous and cash crops with the proportion of crops in the irrigation area planning (Table 1). As wheat and corn are the major commercial grains in the Xiaolangdi south bank irrigation area, the irrigation area development plan states that the planting proportions of wheat, corn, autumn miscellaneous, and cash crops in the irrigation area shall not be less than 0.7, 0.4, 0.2, and 0.05, respectively, and shall not be greater than 0.9, 0.7, 0.4, and 0.2, respectively. Therefore, the design planting ratio for these crops was set as 0.8, 0.6, 0.3, and 0.1, respectively. Meanwhile, the total irrigation area of crops in the irrigation area was set to 35,787 hm 2 , and the irrigation water was limited to 82.63 million m 3 .

Model Solution
According to the multi-objective model and constraints established in Section 2 and defined in Section 3.2, the game algorithm was applied to first calculate the single-objective optimisation values and their corresponding strategies. For the solution of the game model, the strategy set is initially "fuzzy clustered"; that is, from the perspective of cluster analysis, f 1 (x) is entirely determined by (x 1 , x 4 ), f 2 (x) is determined by (x 1 , x 2 ), and f 3 (x) is determined by (x 2 , x 3 ). The income function of the game model can be defined according to the results of the strategy set division. Because the objective functions f 1 (x) and f 2 (x) are the maximisation objectives while f 3 (x) is the minimisation objective, Equation (7a) was selected to evaluate f 1 (x) and f 2 (x), while Equation (7b) was selected to evaluate f 3 (x). Thus, the income function was defined for the cooperative game model as follows: where f * i (x) and f − i (x) are the maximum and minimum values of the single objective targets, respectively, as defined in Table 2. The planting area of each crop in the singleobjective optimisation strategy is shown in Figure 1. The minimum value of the objective function provided by Equation (12) was obtained by the game algorithm, and the result represents the optimal crop planting structure under the cooperative game, as shown in Table 3. defined in Section 3.2, the game algorithm was applied to first calculate the single-objective optimisation values and their corresponding strategies. For the solution of the game model, the strategy set is initially "fuzzy clustered"; that is, from the perspective of cluster analysis, 1 ( ) is entirely determined by ( 1 , 4 ), 2 ( ) is determined by ( 1 , 2 ), and 3 ( ) is determined by ( 2 , 3 ). The income function of the game model can be defined according to the results of the strategy set division. Because the objective functions 1 ( ) and 2 ( ) are the maximisation objectives while 3 ( ) is the minimisation objective, Equation (7a) was selected to evaluate 1 ( ) and 2 ( ), while Equation (7b) was selected to evaluate 3 ( ). Thus, the income function was defined for the cooperative game model as follows: where + ( ) and − ( ) are the maximum and minimum values of the single objective targets, respectively, as defined in Table 2. The planting area of each crop in the singleobjective optimisation strategy is shown in Figure 1. The minimum value of the objective function provided by Equation (13) was obtained by the game algorithm, and the result represents the optimal crop planting structure under the cooperative game, as shown in Table 3.  To solve the competitive game, the optimisation value of 3 ( 3 ) is considered as the known quantity, the single objective optimisation of 1 ( ) in the variable set { 1 , 2 , 4 } is conducted, and the result represents the optimal scheme for the three variables except { 3 }. Similarly, only { 1 } is used to optimise 1 ( ) and 2 ( ), and the result is the optimal decision-making scheme for { 3 }. The union of these two schemes represents the optimal  To solve the competitive game, the optimisation value of f 3 (x 3 ) is considered as the known quantity, the single objective optimisation of f 1 (x) in the variable set {x 1 , x 2 , x 4 } is conducted, and the result represents the optimal scheme for the three variables except {x 3 }. Similarly, only {x 1 } is used to optimise f 1 (x) and f 2 (x), and the result is the optimal decision-making scheme for {x 3 }. The union of these two schemes represents the optimal decision scheme for all objective functions; the results are shown in Table 3, Figures 2 and 3. decision scheme for all objective functions; the results are shown in Table 3, Figure 2 and Figure 3.   decision scheme for all objective functions; the results are shown in Table 3, Figure 2 and Figure 3.    The comparison between the cooperative and competitive game optimisation strategies in Table 3 and Figure 3 shows that, under the cooperative game optimisation strategy,