3.1 Method Design of Multi-objective Optimization System Configuration of Rural Land

Land is a carrier of environmental elements and natural resources and is the basis for human survival and reproduction. The rational use of land resources affects the development of rural industries ^[16]. Under the background of the comprehensive implementation of the national rural revitalization strategy and the rapid development of the digital economy, how to optimize rural land resources to the greatest extent is the difficulty and focus that needs to be solved urgently ^[17]. This chapter combines the multi-objective optimization method to design the rural land multi-objective optimization system configuration. The trend of sustainable development is often predicted by the degree of land adaptability because the theory of sustainable development must be followed in the optimal allocation of rural land ^[18]. Therefore, the land adaptability and land agglomeration are optimized.

Multi-objective optimization is used mainly to find a vector composed mainly of decision variables that can satisfy all constraints and vector functions. The aim was to find one or more solutions so the decision-maker could accept the final target value ^[19]. The mathematical expressions of commonly used multi-objective optimization problems are expressed as Eq. (1).

(1)

where $F(x)$ is the objective function; $x$ is the decision vector; $n$ is the total number of sub-objective functions of the decision vector. $T$ is the matrix transpose. The relationship between the decision function and the constraints is expressed as Eq. (2).

(2)

where $G(x)$ is a constraint condition, and the control decision vector $(x)$ takes values within a feasible range. The limited relationship between the specific value range of decision vectors and the constraint condition is expressed as Eq. (3).

(3)

where the value ranges of the decision vector and constraints are given. The difference between multi-objective and single-objective optimization is that the latter has only a unique optimal solution. In contrast, finding the optimal unique solution to optimize each objective function is often impossible in multi-objective optimization. In most cases, the solution of multi-objective optimization is an optimal solution set consisting of multiple solutions called the Pareto optimal solution set. In practice, according to different situations, one or more suitable solutions are selected from the Pareto optimal solution to be the optimal solution or solution set of the optimal objective.

The evaluation process of sustainable land development was divided into two parts. First, a comprehensive evaluation of each index evaluation subsystem was carried out, and the comprehensive development level of the subsystem was calculated. The comprehensive development level was obtained from the calculation, and a coordination degree model was constructed for further calculation. The mathematical evaluation model of each subsystem is expressed as Eq. (4).

(4)

where $X_{ij}$ represents the i$^{\mathrm{th}}$ $j$ index of the i$^{\mathrm{th}}$ subsystem; $W_{ij}$ is the weight of the corresponding index value of each subsystem; $F_{i}$ is the comprehensive score of each subsystem that optimizes the allocation of land and evaluates the sustainable development level of each subsystem of the land. The level of sustainable development represents the suitability of land use in a region. Its mathematical expression is expressed as Eq. (5).

(5)

where $R$ is the coordination degree of sustainable use of rural land after optimization; $\gamma $ is the coordination coefficient; $F_{i}^{t}$ is the comprehensive score of the i$^{\mathrm{th}}$ subsystem after the first optimization $F_{i}$; $t$ is the initial score of the i$^{\mathrm{th}}$ subsystem, and there is one $m$ subsystem. Condition the above coordination model can be expressed as follows:

(6)

Eq. (6) expresses the constraint condition of the coordination degree model of Eq. (5), where $s$ is the minimum comprehensive score before and after optimization. The sustainable development of rural land can be optimized using this model. In addition, the spatial agglomeration of rural land needs to be optimized. Agglomeration can represent the spatial utilization and distribution concentration of the different land types. The cost of land management will be lower when the degree of land agglomeration is higher, and the economic benefits will be higher. A land use cluster is a spatial collective organization connected by the same land use type. A smaller number of clusters indicates a better connection. In contrast to land use clusters, land compactness encourages the same type of land to be located in close proximity without the need to be connected. Fig. 1 shows the distribution of a particular type of land in the region.

Fig. 1. Different distribution of a particular type of land.

Fig. 1 reveals three types of land distribution from left to right: scattered, connected, and adjacent. Determining whether the utilization rate of a particular piece of land is high requires a look at its compactness. Among them, Eq. (7) expresses the calculation formula of the land shape index, one of the measurement indicators.

(7)

where $\delta _{k}$ is the shape index; $k$ is the land type; $c$ is the land use cluster; $A_{kc}$ is the land area; $Y_{kc}$ is the land perimeter. The domain identity index (Eq. (8)) is another measure.

(8)

where $\beta _{k}$ is the neighborhood homogeneity index. $B_{ijk}$ is the number of all types of cells in the domain. $X_{ijk}$ is the number of cells of the type $k$ in the domain. $N$ and $M$ are the range of values of $i$ and $j$, respectively. The range of values for $i$ was $i \in(1,2, \cdots, N)$. The range of values for $j$ was $j \in(1,2, \cdots, N)$. The general domain was categorized into four and eight domains (Fig. 2).

Fig. 2. Common field types.

Among the two neighborhood types in Fig. 2, the left is a four-neighborhood, and the right is an eight-neighborhood. If the value range is 0-4 or 0-8, it is a four- or eight-neighborhood, respectively. If $\beta _{k}=0$, it means that there is no land of the same type in its neighborhood, and $\beta _{k}$ has the largest value when all land units are of the same type $k$. In this study, the $\beta _{k}$ objective function was used to reflect the compactness of the land. The mathematical expression of the objective function of spatial agglomeration is expressed as Eq. (9).

(9)

3.2 Construction of Multi-objective Optimization Model of Rural Land based on Improved Ant Colony Algorithm

As a swarm intelligence algorithm, the ant colony algorithm has been used widely in more fields because of its excellent searchability and global convergence ability ^[20]. The double bridge experiment analyzed the foraging behavior of ants. The ant colony could choose the shortest path from the ant nest to the food. Fig. 3 presents the foraging process.

Fig. 3. Foraging path diagram of ant colony.

The ability of the ant colony to find the optimal route was not high when the number of ant colonies was small (Fig. 3). On the other hand, it can quickly find the optimal route for foraging when the ant colony increases to a certain number. At the same time, the ant colony algorithm also has specific problems. For example, the movement of multiple individuals in the ant colony is random, which will slow down the evolution speed of the algorithm considerably; it is easy to fall into local optimal solutions in the later optimization stage. Therefore, this chapter improves the traditional ant colony algorithm, combined with the construction of sustainable land development and agglomeration function in the previous chapter. In addition, it proposes a multi-objective optimization model of rural land based on the improved ant colony algorithm. Fig. 4 presents a flow chart of the traditional ant colony algorithm.

Fig. 4. Flow chart of the traditional ant colony algorithm.

Generally, common optimal allocation problems can be solved directly using mathematical models. On the other hand, using conventional multi-objective optimization solutions to handle huge data sets is difficult because the problem of rural land resource utilization is a multi-objective optimization problem, and the land space is large. With the ant colony algorithm, as a general global optimization algorithm, its inherent operating mechanism and characteristics of global optimization are suitable for solving multi-objective optimization problems. Therefore, the content of this chapter is based on the two mathematical expressions of land optimization in the previous chapter, combined with the ant colony algorithm to establish the optimal allocation model of land use to find its Pareto optimal solution. Fig. 5 presents the distribution of Pareto optimal solutions for common multi-objective optimization models.

Fig. 5. Distribution of Pareto optimal solutions.

For a complex land optimization system, it is difficult to obtain the Pareto optimal set of the entire system accurately and obtain the Pareto optimal set as much as possible. The multi-objective optimization search is required to meet the following conditions: reduce the distance between the Pareto optimal frontier and the actual optimal frontier as much as possible. The distribution of each approximate solution obtained should be as uniform as possible on the Pareto frontier. The distribution of optimal fronts should be as wide as possible so each objective can be covered by the optimal solution. The traditional ant colony algorithm will make the population converge to a single optimal solution, which cannot meet the requirements of the Pareto optimal solution set. Therefore, a multi-objective optimization mechanism was introduced to improve the traditional ant colony algorithm. Fig. 6 outlines the basic process of multi-objective optimization of rural land combined with the ant colony algorithm.

Fig. 6. Multi-objective optimization process of rural land combined with ant colony algorithm.

The existence of the pheromone volatile factor, $\beta $, will allow the unsearched information to continue to decrease or even approach 0, reducing the global searchability of the algorithm. The search capability is improved by changing the value adaptively. Let $\beta (t_{0})$ be the initial value of $\beta $, and the calculation formula of $\beta $ is as follows (10).

(10)

where $\beta $ is the pheromone volatilization factor. $\beta _{\min }$ is the minimum value of $\beta $, which prevents the computational efficiency of the algorithm from being reduced because $\beta $ is too small. $t$ is the number of iterations. The behavior of land type allocation for a series of land use units is recorded as $V(l,\,\,j)$, and its contribution to the objective function is shown in Eq. (11).

(11)

where $s$ is the objective function; $l$ is the land use type; $M$ is the number of land use units. The calculation formula for realizing a specific behavior in the optimal land use allocation is as follows. $V(l,\,\,j)$

(12)

where $\eta _{lj1}$ is the heuristic function for the coordination degree of sustainable land use; $\eta _{lj2}$ is the heuristic function for the spatial aggregation function; $\eta _{lj3}$ is the heuristic function for the minimum planning cost function; $\eta _{lj}$ indicates that a specific land use unit allocates a certain type of land. In the single-objective optimization problem, the basic ant colony algorithm updates the pheromone on each path according to Eq. (13).

(13)

where $\rho $ is the pheromone volatilization factor; $\tau _{ij}(t)$ means from ant status $i$ to the ant status $j$ (ed note: What state are you referring to?); $j$ is the concentration of pheromone on the $\tau _{ij}(t-1)$ path after the pheromone update; $ij$ means the last pheromone concentration of the path; $\Delta \tau (t-1)$ represents the pheromone left by the ants on the path during the current iteration process. Its calculation formula is as shown in Eq. (14).

(14)

where $x_{t}$ is the candidate solution set of the ant colony algorithm. Therefore, further optimized $\Delta \tau (t-1)$ calculation methods are needed to solve the complex problem of land use optimization configuration. This research establishes a new rule (Eq. (15)) for the calculation.

(15)

where $\Delta \tau (t+1)$ is the pheromone left by the ant on path $ij$ in the $(t+1)$ iteration. $\delta $ is a positive number. $t$ is the number of iterations. After each ant completes the search of the entire space, the global pheromone needs to be updated to obtain the optimal solution, and the update rules are as follows.

(16)

Finally, combining the ant colony algorithm and multi-objective optimization of land use allowed optimized land use in quantitative structure and spatial layout. Fig. 7 presents the basic process.

Fig. 7. Improved ant colony algorithm with the multi-objective optimization problem.

4.1 Analysis of the Multi-objective Optimization Results of Rural Land Under Different Optimization Objectives

For a complete land optimization configuration, each objective function should be optimized to the greatest extent in their respective directions. Therefore, ensuring that each objective function has a gradually increasing trend in the optimization process is necessary. When each objective function tends to converge, it indicates that the entire optimization system is near the optimal solution. The evolutionary optimization of rural land use allocation under its system was carried out according to the objective function proposed in the previous article ^[21].

As shown in Fig. 8, there are two optimization objectives: land adaptability and spatial agglomeration. The iterations of the two optimization objectives are obtained under the premise that the land adaptability optimization is the main optimization objective (hereinafter referred to as optimization model 1). Fig. 8(a) shows the convergence of the land adaptability under optimization model 1. The objective function starts to converge after iterating approximately 45 times, suggesting that the adaptability value of the land reaches the vicinity of the optimal solution at this time. Fig. 8(b) presents the convergence of the land spatial agglomeration degree under the optimization model 1. The objective function iterates approximately 43 times and begins to converge. The spatial agglomeration degree value converges to the vicinity of the optimal solution.

Fig. 8. Optimal results of adaptability and agglomeration degree under optimization model 1.

Fig. 9 shows the iterative situation of obtaining the two optimization objectives of land adaptability and spatial agglomeration under the premise that the optimization of spatial agglomeration is the main optimization objective (hereinafter called optimization model 2). Fig. 9(a) presents the convergence of the land adaptability under optimization model 2. The objective function started to converge after iterating approximately 50 times, suggesting that the adaptability value of the land is near the optimal solution. Fig. 9(b) shows the convergence of the degree of spatial agglomeration of the land under optimization model 2. The objective function converged after iterating approximately 51 times, suggesting that the spatial agglomeration degree value converges to the vicinity of the optimal solution at this time.

Fig. 9. Optimal results of adaptability and agglomeration degree under optimization model 2.

Two optimization objectives of carbon emission and minimum planning cost were introduced to obtain a more comprehensive and accurate optimal solution set. Carbon emissions and land adaptability can be trend predictors for sustainable development, while spatial agglomeration determines the minimum planning cost of land. The iterative situation of the four optimization objectives was obtained by taking the adaptive value and the spatial agglomeration degree as the optimization objectives (hereinafter called optimization model 3). Fig. 10(a) shows the convergence of land adaptability. The objective function started to converge after 45 iterations. Fig. 10(b) shows the convergence of spatial agglomeration. The objective function was also iterative; it started to converge at 45 times. Fig. 10(c) shows the convergence of the minimum planning cost. The objective function iterated approximately 45 times and started to converge. Fig. 10(a) shows the convergence of carbon emissions. The objective function still iterated to convergence at 45 times. Therefore, all four optimization objectives could reach a stable convergence state at approximately 45 times under this optimization model. Under this optimization model, the number of iterations for each objective function to achieve stable convergence was faster than other models. Moreover, the number of iterations for each objective function to converge was relatively close. Hence, this optimization model was selected for multi-objective optimization. The best solution near the optimal solution can be considered the Pareto optimal solution.

Fig. 10. Results of the multi-objective optimization of rural land under optimization model 3.

4.2 Analysis of Multi-objective Optimization Results of Rural Land Under Different Algorithms

The difference in parameters will lead to differences in the calculation results and calculation efficiency of the algorithm. Therefore, the model optimization in this chapter selected the determined parameters according to previous research results and compared the genetic algorithm, the traditional ant colony algorithm, and the combination proposed in this study. The objective was to improve the convergence of three genetic algorithms. The relevant parameters were set as follows. Population size $Q=20$, pheromone volatility factor $\rho =0.1$, information heuristic factor $\alpha =0.1$, and expectation heuristic factor $\beta =1-\alpha $.

Fig. 11 presents the iterative situations of the three algorithms under the adaptive objective function and the agglomeration objective function. Fig. 11(a) shows the iterative situations of the three algorithms under the adaptive objective function. A comparison of the iteration times of the three algorithms showed that the improved ant colony algorithm first converges to a stable state and begins to converge after 50 iterations. The ant colony algorithm gradually began to converge after 80 times, and its convergence speed was much lower than that of the improved ant colony algorithm. Fig. 11(b) shows the iterations of the three algorithms under the agglomeration objective function. The ant colony algorithm converged to a stable state first, and began to converge when the number of iterations reached 40. The traditional and ant colony algorithms gradually began to converge after 60 iterations, and the convergence speed was slow.

Fig. 11. Iterative situations of the three algorithms under different objective functions.

The running times of the three algorithms under different objective functions were compared to demonstrate the superiority of the improved multi-objective ant colony algorithm. The improved multi-objective ant colony algorithm was significantly faster than the traditional and genetic algorithms. Combining the number of iterations and running time resulted in the improved multi-objective ant colony algorithm with higher convergence effect and computational efficiency. Hence, the feasibility of the improved ant colony algorithm proposed in this study in land use optimization and its space utilization was verified. The advantages in layout can optimize the utilization of land resources in rural industries in the context of the current digital economy.

A piece of rural land in southwest China was selected as the actual experimental object, and carbon emission, minimum planning cost, land adaptability value, and spatial agglomeration were used as evaluation indices to compare the actual optimization of rural land under the three algorithms, as shown in Fig. 12.

Fig. 12 shows the optimization of carbon emission, minimum planning cost, land adaptability value, and spatial agglomeration degree of a piece of rural land in Southwest China under three algorithms. The value range of 0-1.0 was used to describe the optimization strength of the three algorithms for the four indicators of carbon emission, minimum planning cost, land adaptability value, and spatial agglomeration. The effect worsens as the value increases. In Fig. 12(a), the optimization degree of the ant colony algorithm (denoted as ACO), genetic algorithm (denoted as GA), and improved multi-objective ant colony algorithm (denoted as IMO-ACO) for the carbon emission of this piece of land was 0.73, 0.56, and 0.11, respectively. The optimization time spent was 5.5 min, 4.2 min, and 1.8 min, respectively. The minimum planning cost for the land of ACO, GA, and IMO-ACO for this piece of land was 1.8 min. The ACO degree of optimization for the minimum planning cost of the land was 0.75, 0.81, and 0.08, respectively, and the optimization time spent was 3.7min, 4.3min, and 1.5min, respectively. The degree of ACO, GA, and IMO-ACO optimization for the land adaptability value was 0.73, 0.89, and 0.06, respectively; the optimization time spent was 3.5. The optimization degree of ACO, GA, and IMO-ACO for land spatial agglomeration was 0.66, 0.92, and 0.10, respectively, and the optimization time spent was 3.7 min, 5.8 min, and 1.9 min, respectively. In summary, the improved multi-objective ACO has a better optimization effect in the actual multi-objective optimization of land, and the optimization time spent was 3.7 min, 5.8 min, and 1.9 min, respectively.

Fig. 12. Actual optimization of rural land under the three algorithms.