1. Introduction
The World Health Organization (WHO) advocates that humans can promote physical health and develop a healthy lifestyle through appropriate physical exercise, but progress is slow at present [
1]. With rapid urbanization, China faces problems of obesity and a lack of physical exercise. Therefore, how sports are promoted in China is gaining people’s attention more and more [
2]. One of the core aspects of China’s sports policy is to build a viable environment to promote people’s participation in sports activities in a way that is consistent with the health-promotion strategy advocated by the World Health Organization [
3]. Compared with developed countries, the average level of sports facilities in China is relatively backward. Compared with the city, the average level of rural sports facilities has an obvious lag, and the number is small, unable to meet the needs of the majority of residents for physical exercise, which has become one of the factors restricting the physical fitness training of villagers. The above situation can be improved by promoting the process of rural sports development, and the construction of rural sports facilities is the key to promoting the development of rural sports. The state promotes the implementation of the rural revitalization strategy by increasing the capital investment in the construction of rural sports facilities. In the process of the construction of rural sports facilities, how to choose the right location for the construction of rural sports facilities is one of the key problems in the construction of rural sports facilities. The construction of sports facilities can promote the physical exercise of rural residents, improve the physical fitness of residents, contribute to the sustainable development of rural areas, and promote the implementation of rural revitalization strategies. Based on the classical continuous location model, this paper constructs a multi-objective location model of rural sports facilities from the perspectives of fairness and efficiency.
Over the past few decades, different meta-heuristic algorithms have been developed to handle these complex optimization problems, such as differential evolution (DE) [
4,
5,
6,
7], particle swarm optimization (PSO) [
8,
9,
10,
11], flower pollination algorithm (FA) [
12], ant colony optimization (ACO) [
13,
14], sine cosine algorithm (SCA) [
15,
16], gray wolf optimization (GWO) [
17,
18,
19], Tianji’s horse racing optimization (THRO) [
20], the Divine Religions Algorithm (DRA) [
21], and Stellar oscillation optimizer (SOO) [
22], etc. It is proven that the metaheuristic algorithm has important application value. In academia and industry, these algorithms are widely used to solve various types of optimization problems. Because of their simple principle and easy implementation, these algorithms have attracted more and more attention. Unlike traditional methods to solve optimization problems, metaheuristic algorithms do not need to analyze objective functions and constraints, which is the most significant advantage of metaheuristic algorithms. In general, a metaheuristic algorithm starts with a random set and then searches globally for the best solution according to the set rules. These rules often mimic the laws of natural evolution, the laws of human behavior [
23], the social behavior of birds, insects, and other organisms [
24,
25], the laws of plant growth [
26], mathematical planning [
27], and the laws of physics [
28,
29], etc. The literature review is summarized in
Table 1.
Multi-objective optimization is a common problem in all fields. In multi-objective problems, each objective will not reach the optimal solution at the same time. It is important to find multiple sets of solutions for multi-objective problems because different solutions can be applied to decision-makers with different needs. In recent years, many meta-heuristic algorithms have been successfully applied to multi-objective optimization problems. One of the very classic and popular multi-objective algorithms is multi-objective particle swarm optimization (MOPSO) [
30]. In MOPSO, the optimal Pareto solution is selected by external archiving and the adaptive grid mechanism. K. Deb improved NSGA by introducing fast elite non-dominated sorting to obtain NSGA-II [
31,
32]. The algorithm proposes a fast non-dominated ranking to improve the population level. In addition, crowding ranking and elite selection strategies are introduced. There are other classical multi-objective algorithms, such as SPEA2 [
33] and Omni-optimizer [
34], etc. In addition, there are some excellent multi-objective algorithms, such as DN-NSGAII, MO_Ring_ MO_Ring_PSO_SCD, MOHHO, MOFOA, MaODA, DPDCA, and so on [
35,
36,
37,
38,
39]. These multi-objective algorithms can solve multi-objective optimization problems well. A new swarm intelligence optimization algorithm called white shark optimizer (WSO) was proposed by Malik B. et al. in 2022 [
40]. Inspired by the great white shark’s search for food and tracking of prey in the deep sea, it combines the great white shark’s keen hearing and smell abilities with fish behavior. Based on the mathematical modeling of great-white-shark hunting behavior, the great-white-shark population can search through random search, locate the behavior of prey and fish, and search in the search space, so as to update position and finally achieve the purpose of searching for the best. At present, the white shark optimizer algorithm has been applied to some optimization problems and has achieved good results [
41,
42,
43,
44,
45,
46]. In this paper, the white shark optimizer is improved to the multi-objective white shark optimizer (MOWSO) to solve the multi-objective problem.
The following is a summary of the principal achievements:
- (1)
A novel multi-objective white shark optimizer (MOWSO) is proposed;
- (2)
The archived mechanism is introduced in MOWSO to store the current Pareto optimal solution, and it is screened by calculating the true distance of the Pareto optimal solution;
- (3)
The performance is tested and compared to MOWSO on CEC2020 benchmark functions;
- (4)
A novel rural sports-facilities location problem model is proposed;
- (5)
We used MOWSO to solve the rural sports-facilities location problem and obtained the best result.
The rest of this article is divided into four parts.
Section 2 is the introduction of the white shark optimizer algorithm and location problem. In
Section 3, the multi-objective problem is introduced, and the multi-objective white shark optimizer algorithm is designed.
Section 4 evaluates the performance of the multi-objective white shark optimizer algorithm on the test set.
Section 5 introduces the rural sports-facility location model and uses the white shark algorithm to solve it.
Section 6 is the summary of this paper and future work.
2. Related Work
2.1. White Shark Optimizer
The white shark is one of the most ferocious and dangerous predators in the ocean. It has powerful muscles, keen eyesight, and a keen sense of smell. When searching for prey, great white sharks first use their hearing to scout large spaces. Second, they can find prey by using their keen sense of smell. These features help them explore nearby areas. Great white sharks can use two lines on their sides to detect changes in water pressure and thus find the direction of prey movement. Great white sharks sense changes in pressure in the ocean and move toward prey. They can also sense the weak electromagnetic field generated by the movement of prey. When locating prey, the great white shark moves toward it in a fluctuating motion. The algorithm is divided into three parts: speed of movement towards prey, movement towards optimal prey, and clustering behavior of fish.
2.1.1. Movement Speed Towards Prey
White sharks spend most of their time hunting and tracking prey. They track their prey in a variety of ways, using their superior sensory abilities like hearing, sight, and smell. When white sharks sense the location of their prey based on the hesitation of the waves they hear as the prey moves, they move toward the prey in an undulating manner. This mode of motion can be defined as shown in Equation (1).
where
represents the speed of the
-th white shark to step
in the white-shark population.
represent the contraction factor, which is used to analyze the convergence behavior of great white sharks, and its definition is shown in Equation (5).
is the
-th indicator vector when the white shark searches for the best prey, and its definition is shown in Equation (2).
and
are used to control the influence coefficients of
and
on
, and their calculation methods are shown in Equations (3) and (4), respectively.
is the global optimal position obtained by the white shark in the first
k searches so far in the search process,
is the optimal position corresponding to the
-th white shark in the population in the
k iterations, and
is the position of the
-th white shark in the population at the
k searches.
and
are randomly generated numbers between 0 and 1.
represents the
-th white shark in the white-shark population.
where
is a uniformly distributed random number generated in the range of 0 to 1.
where
k and
K, respectively, represent the number of iterations and the maximum number of iterations of the white-shark population so far.
and
are two fixed values set empirically, 0.5 and 1.5, respectively, in order to control the
and
values in combination with the number of iterations.
where
represents the coefficient of acceleration, thorough analysis results in a value of 4.125.
2.1.2. Movement Towards Optimal Prey
White sharks are constantly shifting their position, and when they hear waves from prey movement or smell prey, they move closer to their prey. The location of the prey changes as it searches for food or the white shark moves towards it. However, the prey will leave some scent in the previous location. In this case, white sharks can search for prey based on the scent they left behind. This behavior of the white shark can be described in Equation (6).
where
represent the latest position of the
-th white shark in the population after the (
k + 1) iteration,
indicates a logical non-operation,
and
are binary vectors whose formulas are Equations (7) and (8), and
and
represent the upper and lower bounds of the entire search space, respectively.
represents a logical vector calculated by Equation (9), and
represents the frequency of the wave motion of the white shark, and the formula is represented by Equation (10).
represents a random number between 0 and 1, and
represents the motion force of a white shark approaching prey, which increases with the number of iterations and which is calculated by Equation (11).
where
stands for the XOR operation.
where
and
represent the maximum and minimum frequency of the great white shark’s fluctuating movement, respectively. By conducting a large number of experimental tests,
and
were finally selected as 0.75 and 0.07, respectively.
where
and
are constants used to manage exploration and exploitation capabilities. Typically, the values are 6.25 and 100.
2.1.3. The Swarming Behavior of White Sharks
When great white sharks hunt, they cooperate and move toward the white shark closest to their prey. This behavior is represented by Equation (12).
The
is the
-th update of the position of the white shark relative to prey. The
value is 1 or −1, indicating the search direction.
,
, and
represent random numbers between 0 and 1.
is the distance between the white shark and its prey, calculated by Equation (13).
represents the olfactory- and visual-intensity parameters of a great white shark following other great whites approaching prey, as calculated by Equation (14).
where
is a random number between 0 and 1.
where
is a constant, and through extensive experimental analysis, the value has been taken as 0.0005.
White sharks are highly social animals that prefer to hunt in groups. The fish behavior of white sharks is represented by Equation (15)
where
is a random number between 0 and 1. White sharks can update their location based on the nearest white shark to their prey.
2.2. Continuous Facility Location
There are many kinds of location problems, which exist in different disciplines, and different disciplines have different descriptions of location problems. However, according to the description of a location problem from different disciplines, the most essential definition of a location problem is given a metric space and the location of demand points in the space, and the location of new facilities in the space, a certain goal between facility points and demand points (generally called customers) can be optimized. This problem has an important position in real life. For more than half a century, with the development of mathematical tools, such as the introduction of optimization techniques to the problem of facility location, the field has entered a period of flourishing. People have conducted in-depth research into and analysis of facility-location problems in different fields, established optimization models and studied theoretical properties, proposed effective numerical algorithms to solve the models, and used the research results to solve practical problems.
The study of location problems spans numerous research fields such as management, engineering, geography, economics, computer science, and mathematics, etc. Its typical application is the location of factories, warehouses, hospitals, retail stores, and in addition, it is also used in fire centers, fire stations, gas stations, exploration oil wells, missile warehouses, and so on. From the perspective of decision makers, site selection is of great significance. Facility location has been accompanied by the development of human history; it has an important and profound impact on human life and production activities.
The rural location problem can be classified in different ways. According to the topology of the metric space, the location problem can be divided into discrete location, continuous location, and network location. Discrete location is the location of a given series of discrete points in the metric space. Continuous siting has a long history. New facilities in continuous siting can be located in a continuous region of the metric space. Continuous optimization, such as linear and nonlinear programming, can be used to solve the continuous siting problem. Network location is the location of the vertex or edge of a given graph or network, which is solved by combinatorial optimization and integer programming. Discrete location, continuous location, and network location have important applications in real life, which can be used to solve practical problems in different situations. The following are several key elements in continuous facility siting: the number of new facilities, the distance metric function, and the objective function, etc.
2.2.1. Number of Facilities
In the facility-location problem, the number of new facilities can be one facility point or multiple facility points. According to the different number of facility points, the problem of facility location can be divided into single-facility location and multi-facility location. In general, the need to locate multiple facilities is more difficult than the need to locate only one facility. This is because in the multi-facility location problem, it is necessary to consider to which facility point the customer goes to obtain the service, or the fact that, in some cases, a particular amount of service is required by the customer, but the amount of service provided by the facility cannot meet the conditions.
2.2.2. Distance Metric Function
There are various distance measurement functions in the field of location selection. One of the important distance measurement functions is the norm. It satisfies positive definiteness, homogeneity, and subadditivity, and its measured distance reflects the actual distance more truly. The LP-norm is a distance metric commonly used in continuous siting. When p = 1, this norm measures the L1-distance, which is often used to represent the distance between two points in a city. When p = 2, this norm measures the Euclidean distance, which represents the straight-line distance between two points. The distance measured by this norm when p = ∞ is often referred to as the Chebyshev distance. Some generalized LP-norms are also commonly used to measure the distance between a facility and a customer. When customers in the location of continuous facilities occupy a large range in the metric space, they should not be simply regarded as points but as regions. The distance between the facility and the customer becomes the distance between the point and the area.
2.2.3. Correlation Function of Location Problem
According to different objectives, the functions of facility location can be roughly divided into pull objectives, push objectives, and push-and-pull objectives [
42,
43]. When the facility is of the type desired by the demand point (such as a supermarket, bank, fire station, etc.), the demand point always wants the facility as close to it as possible. Therefore, pull goals are often used, such as minimizing the distance between the demand point and the facility and maximizing market share. When the facility belongs to the type that is excluded by the demand point (such as polluting factories, garbage disposal stations, etc.), the demand point wants the facility to be as far away from itself as possible. Push goals are often used, such as maximizing the sum of distances between customers and facility points, minimizing the number of customers covered by the facility, and so on. In some cases, the type of facilities is between expectation and exclusion, and there will be two contradictory or opposite factors in the problem of attraction and exclusion. This kind of location problem often adopts push–pull goals.
The minisum function is a commonly used pull target to minimize the distance between the facility and the customer. The corresponding location problem is called the minisum problem. The Weber problem is the minisum problem, where the number of facilities is one. The problem with more than one facility is called the multi-facility minisum problem, which includes two kinds of important problems: the multi-facility Weber problem and the multi-source Weber problem. The minimax function is another commonly used pull goal that minimizes the maximum distance between the client and the setup, and the corresponding model is called the minimax problem. Because the model focuses on the farthest customers, it reflects fairness to some extent. The coverage problem is also an important problem in the use of pull targets, and the coverage problem includes two types of problems: maximum coverage and minimum coverage. The former is to make facilities cover as many customers as possible when the number of facilities is given, while the latter is to cover all customers with the least facilities when the number of facilities is variable. Problems such as locating a point on the network to maximize its weighted distance to the node are called maxisum problems [
47]. Like pull targets, the maxisum and maximin functions are two important types of push targets. The maxisum problem seeks to maximize the distance between the customer and the facility, while maximin seeks to maximize the closest distance between the customer and the facility. When the number of facilities is more than one, the multi-facility model of the push target can be divided into max–min–min, max–min–sum, max–sum–min, and max–sum–sum, etc., in which min/sum of the third layer represents the minimum distance or distance sum between a given customer and all facilities. Because of the attraction and repulsion of push–pull targets, the location problem of push–pull targets can be modeled as two-objective optimization. Therefore, the method and technology of double objective optimization can be effectively used to solve the location problem of push–pull targets.
6. Conclusions and Future Work
In this paper, we improve the white shark optimizer algorithm by adding two mechanisms. The first mechanism is to introduce the non-dominated Pareto solution obtained by archiving. The second mechanism is to calculate the true distance between Pareto solutions to filter the best Pareto solutions in the archive. The algorithm is tested on CEC2020 benchmark test function and compared with other classical multi-objective algorithms to verify the performance of MOWSO. The experimental results show that the Pareto solution obtained by MOWSO algorithm is closer to the real Pareto solution and has certain reliability. Then, MOWSO was applied to solve the location problem of rural sports facilities. Through the experiment, it was found that MOWSO can obtain multiple sets of solutions, corresponding to different layouts of sports facilities, and can provide reference for the location problem of rural sports facilities. In real life, there are many similar problems, such as the location of health stations, market sites, and so on. Although these problems have different constraints, similar ideas can be used to solve them. In this paper, the location of sports facilities is taken as a variable, and the location of sports facilities under different target conditions is studied carefully, but the terrain limitation is not considered. If the obtained sports facilities are located on non-construction land, such as paddy fields and agricultural land, etc., it is necessary to avoid these addresses manually. In real life, faced with these complex conditions, the constraints on the location of sports facilities will increase. Due to the high complexity of the specific problem, this experiment may not be considered enough. Therefore, it is very important to propose a new model and improve MOWSO to solve problems in this field under the condition of considering many factors comprehensively.