Research on Optimizing the Development of Sports and Leisure Industry Using Genetic Algorithm to Promote the Growth of Local Sports Economy

Since the emergence of leisure, sports have become the first choice for people’s leisure with its incomparable participation value. The combination of sports and leisure has attracted more and more people to participate in it with its unique charm and positive and healthy recreation, reflecting people’s pursuit of happiness and free spirit after a certain stage of economic development, and reflecting the real charm of sports as a culture [1–3]. Leisure sports can be summarized as a positive lifestyle based on sports activities in leisure time to achieve the goal of healthy physical and mental development. Leisure sports industry belongs to the field of sports industry, is to meet the increasing demand for people’s management of their own corresponding to the provision of fitness, entertainment, tourism, leisure in one of the sports products, which is mainly a service industry [4–6]. The development of sports and leisure requires the input of production materials such as people, money and materials, and can provide health products for the public, and such products have been partially or completely into the market, in line with the requirements of industrial economics and market economics, is the application of sports means to realize their own value and significance of the development, production and provision of leisure commodities industry [7–10]. The continuous development of sports and leisure industry has an important role in promoting the local sports economy, which can be summarized in four aspects: it can continuously improve the sports infrastructure and provide good conditions for people. It can provide people with more employment opportunities, alleviate the employment pressure in the region and improve people’s living standard. It can enrich people’s cultural life, improve the comprehensive quality of the population, and promote the continuous growth of local sports economy [11–14].

At the present stage, China’s sports and leisure market, with the sports sector as the mainstay, joint investment by a variety of economic constituents, and broad participation by all sectors of society, has taken initial shape to basically satisfy the needs of consumers of different grades, and is relying on the rapid development of the national economy and the vigorous emergence of the national fitness movement. The sports and leisure industry has become one of the most promising industries in China due to its extensive, complex and close industrial association, which has driven the development of sports tourism, sports finance and insurance, sports labor service, sports intermediary industry, sports goods industry and other related industries [15–18].

In this paper, a multi-objective optimal allocation model of sports and leisure resources based on improved genetic algorithm is constructed, and the traditional variation operator of genetic algorithm is improved to non-uniform variation operator, which realizes the fast and accurate optimization seeking effect on the multi-objective planning model of sports and leisure resources, and thus promotes the development of sports and leisure industry. In addition, in order to further verify the effect of the model on the development of local sports economy, vector autoregression (VAR) model is used to analyze the correlation between the development of sports and leisure industry and the development of local economy, in order to provide a direction guide to promote the design of synergistic development of sports and leisure industry and local sports economy.

2

Multi-objective optimization model of sports and leisure resources based on genetic algorithm

This paper first constructs a cross-regional multi-objective planning model for sports and leisure resources to achieve the integration decision of sports and leisure resources, and then uses an improved genetic algorithm to find the optimal model, so as to achieve the optimization of the development of sports and leisure industry and to promote the growth of local sports economy.

2.1

Cross-regional multi-objective integration model of sports and leisure resources

The study of multi-objective planning problems, which involves defining multi-objective functions with constraints and conflicting multi-objective functions to construct multi-objective optimization models and solve the optimal decision space. It is widely used to analyze and solve real problems in many fields such as engineering optimization, mathematical finance, etc. The general form of its optimal decision planning model is: 1 $(V M P) {V - \min f (x); x}; (s \in X)$ where x = (x₁,…,x₂)^T, f(x) = (f₁(x),…,f_n(x)^T) are vector multi-objective functions of the model.

Constraints: x <<, 0G(x) >> 0, G(x) are the values of the constraints.

2.1.1

Maximizing the benefits of each single subject’s objectives

The integration and sustainable development of sports and leisure resources need to seek a balance between conservation and development. First of all, conservation objectives include two major categories: natural resources and socio-cultural resources. The objective of conservation of natural resources is to maintain and restore the natural and ecological environments. The protection of sociocultural resources includes the uniqueness of local culture, the traditional characteristics of folkways and customs, and the “adaptability” of the lives and psychology of local residents. Secondly, the developmental goals are divided into three main categories: first, the growth of income and growth and expansion of sports and leisure tourism enterprises, the expansion of the structure, scale and market life of the tourism industry itself, and the maximization of the economic benefits of sports and leisure. Secondly, to maximize the sports and leisure experience and satisfaction of tourists. Third, to realize the coordinated development of sports and leisure resources and the sports economy. Regional sports and leisure resources have their own characteristics and complement each other. Based on the above single-objective analysis, a multi-objective model is constructed to achieve harmony between humans and nature and to coordinate the economic and social environments. 2 $M a x f (X) = {β (x), τ (x), ε (x), \dots \dots}$

Where x is a decision vector with non-negative value. 1)

Sports and Leisure Economic Development Objective β(x): It refers to the maximization of the total regional sports and leisure industry development revenue (YRTI) as the main economic objective. Namely: 3 $\max {Y R T I = \sum_{s = 1}^{m} \sum_{j = 1}^{n} T I (s, j)}$

Among them, TI(s, j) is the income of regional sports and leisure industry, j is the sub-district, j= 1,2,…n and S = 1,2,…m are the economic sectors involved in the development of sports and leisure industry.

2)

Tourist sports and leisure tour experience goal τ(x): mainly includes the level of sports and leisure tourism services, product participation and entertainment in the region. Comprehensive use of tourist experience, that is, the maximization of the minimum tourism experience benefits as the goal: 4 $M a x = {\min U (s, j)}$

Where, U(s, j) is the tourism experience effect function, i.e., tourist satisfaction.

3)

Sports and leisure ecological resources protection and restoration Objective ε(x): Including maintaining environmental quality up to standards and maintaining ecosystem balance. The maximum green equivalent area and the minimum emission of pollutant COD are comprehensively adopted: 5 $\max \sum_{s = 1}^{m} \sum_{j = 1}^{n} G R E E N (s, j)$ 6 $\min \sum_{S = 1}^{m} \sum_{j = 1}^{n} C O D (s, j)$

Where GREEN(s, j) is the integrated regional ecological evaluation indicator “green equivalent area”. COD(s, j) is the main pollutant factor contained in the discharged wastewater and exhaust gas.

2.1.2

Conflict of Interest and Decision Making Attributes of the Relevant Subjects

Decision-making attributes, i.e. constraints, need to construct constraint terms according to the industrial economy, natural environment, regional cooperation level and other factors in the development stage of local sports and leisure, and the study presupposes the following decisionmaking attributes (but not limited to): 1)

Sports and leisure industry structure coordination: 7 $Y_{\min} ≪ (1 - a) Q P ≪ Y_{\max}$ where Y_min, Y_max are the upper and lower constraints on i industrial development, respectively. a is the production technology coefficient matrix. QP is the amount of resources to ensure the development of each sports and leisure industry.

2)

The regional natural environment reaches the standard: 8 $\sum_{k = 1}^{K} \sum_{j = 1}^{J (k)} 0 .001 * (s, j) d_{j}^{k} p_{j}^{k} [\sum_{i = 1}^{I (K)} x_{i j}^{K}] ≪ C_{0}$

Where C₀ is the regional natural environment tourism capacity. $d_{j}^{k}$ is the content of the main pollutant factors in the emissions of wastewater and exhaust gas of k subarea j user units. $p_{j}^{k}$ is the pollutant emission factor for k subarea j users.

3)

Analysis of factors for decision-making attributes

In the process of integrating local sports and leisure resources, “coordination of industrial structure” and “regional natural environment” are measurable hard indicators, while empirical analysis needs to transform soft conclusions into measurable data through interviews and surveys, and the formation of constraints involves the following major conflicts of interest. The formation of the constraints involves the following major conflicts of interest:

(1)

Conflict of interest in objectives between administrative regions

Although the cross-regional sports and leisure tourism cooperation has broken the administrative boundaries between regions, it is largely subject to the constraints of the administrative power of the local governments, and the constraints of the “region-oriented” concept and the “administrative region economy” have led to the asymmetry of talents, capital, technology, information, resource endowments and the lack of smooth communication and coordination channels, and the cross-regional flow of production factors is often subject to the rigid constraints of administrative divisions. Problems such as homogeneous competition of sports and leisure tourism products, inconsistent marketing actions of sports and leisure tourism, and different standards and policies of sports and leisure tourism. The management model of “sports dismemberment” hinders the process of cross-regional integration and development of sports and leisure resources, which not only restricts the strategic cooperation between provinces and cities, but also affects the overall planning and resource allocation between different urban areas of the same province and city.

(2)

Conflict of interest among government departments

Due to the arrangement of administrative affiliation and functional management mechanisms, the development, utilization, and management of sports and leisure resources involve many functional departments. In the face of the development, utilization and management of sports and leisure resources, under the premise that each administrative department is responsible for its own duties, the concept of overall layout is generally ignored, which hinders the transformation of sports and leisure resources from general leisure resources to high-quality sports and leisure resources, and affects the enhancement of the grade of sports and leisure resources as well as their efficient development and utilization.

There are two main reasons for this situation: first, the planning between departments is not unified, and there is a serious conflict of objectives. The second is that departmental “overlapping management boundaries” have resulted in the misalignment of management boundaries and functional boundaries, hindering the overall integration and rational allocation of sports and leisure resources.

(3)

Conflict of interest between governmental enterprises’ goals

The cooperative development of sports and leisure tourism resources is often led by the local government, while regular and specific cooperation must be realized through the behavior of enterprises, and the two paths lead to an easy conflict of goals between the government and enterprises. For example, enterprises need high-quality talents, while the government wants to arrange employment for local residents as much as possible. Enterprises prioritize short-term economic gains, while resource management departments prioritize sustainable development. In addition, the improper behavior and management omission of government departments can also cause the enthusiasm and initiative of enterprises, which is the main body of sports and leisure tourism development, resulting in the slow development of sports and leisure tourism, ineffective market cultivation strategy, low level of marketization and service standards, and low level of enterprise specialization.

(4)

Conflict of interest among enterprise subjects

The cooperation of enterprises upstream and downstream of the value chain of the sports and leisure industry is conducive to extending the scope of the existing business, but too much focus on the competition for their own interests has led to the competitive behavior of maximizing local interests, which ultimately stalls the process of market-oriented operation of sports and leisure tourism cooperation. Enterprises in a certain link in the same value chain are bound to undergo some degree of differentiation in the cooperative development of resources.

2.1.3

Decision-making methods

1)

Decision-making threshold and target satisfaction function

Set the interval of indicators acceptable for decision-making as [L_i0,H_i0]. For the benefittype indicators (e.g., economic benefits, degree of social development, etc.) that are better as the indicators get bigger, L_i0 is the tolerance value of the indicators, and H_i0 is the ideal value. Construct the satisfaction affiliation function: 9 $u_{i} (x) = {\begin{array}{l} \frac{r_{i} - L_{i 0}}{H_{i 0} - L_{i 0}} & r_{i} ≫ L_{i 0} \\ < 0 & r_{i} < 0 Re f u s e d \end{array}$ where r_i is the attribute value of the decision indicator. The satisfaction level of the decision maker’s goal is judged by constructing the satisfaction affiliation function. At the same time, the corresponding interval distribution by u_i(x) is used to illustrate the situation such as whether the decision-making program is accepted or the critical state of non-acceptance.

2)

Integration of regional sports and leisure total goal satisfaction function

The total satisfaction function of the regional decision maker is a linear integration of the coordinated coupling of the three major goal satisfaction in the region, which can be further derived by the integration of the satisfaction function between the decision indicators of each political region and sector. If the satisfaction level of each specific region or sector K and goal i on the allocation scheme is u_ki(x), the satisfaction affiliation function of the regional total goal can be constructed through linear integration: 10 $\begin{array}{l} \max = \sum_{i = 1}^{3} λ_{i} U_{l} (x) \\ \max U = \sum_{i = 1}^{n} \sum_{k = 1}^{m} λ_{k i} U_{k i} (x) \end{array}$ where U is the total satisfaction function, U_l(x)(l = 1,2,3) is the satisfaction of the three main goals, and λ_ki is the weight of the satisfaction of goal i to the total goal satisfaction contribution affiliation function. $\sum_{i = 1}^{3} = 1$ or $\sum_{i = 1}^{n} \sum_{i = 1}^{m} λ_{k j} = 1$ , denotes a non-conflicting alignment of interests.

2.2

Optimal allocation of resources based on improved genetic algorithm

2.2.1

Basic principles of genetic algorithms

Genetic Algorithm (GA) is a search algorithm based on the principles of natural selection and genetics, which abstracts the problem space as a population of individuals and explores the most adapted individuals by repeatedly iterating to produce offspring [19]. Genetic algorithms evolve the initial population into a high-quality population where each individual represents a solution to its problem. The quality of each individual is measured by a fitness function, which is a quantitative representation of each individual’s adaptation to a particular environment. The process starts with an initial population of randomly generated individuals. For example let the function optimization problem be: 11 ${\begin{matrix} \max f (x) = x \\ a \leq x \leq b \end{matrix}$

Where x is regarded as the chromosome composed of genes, f(x) is the individual fitness function, and f(x) and the constraints together constitute the survival environment of the individual.

The flow of the genetic algorithm is shown in Fig. 1, and its solution steps are as follows: 1)

Coding

Traditional genetic algorithms generally use binary coding, set the length of the code as e, each binary bit is called a gene, the range of values of the variable is divided into 2e – 1 subintervals, that is: 12 $x = a + (b - a) * \frac{\sum_{k = 1}^{e} g_{k} * 2^{e - k}}{2^{e - 1}}$

In the formula g_k is the knd value (gene value) in the binary digit, so a certain binary digit string can be represented as S_i = g₁g₂g₃ ⋯ g_e.

The range of variable values [a, b] is discretized into 2e grid points by coding, and each grid point corresponds to the binary digit string and individual one by one.

2)

Random generation of initial parent population

Set the population size as n, and randomly select n strings of length e from the range of values of the encoded variables, and this set containing n individuals is the initial parent population.

3)

Setting the fitness function

Decode the binary into variable x, define the fitness function as F(x) = f (x), and substitute the ith individual, the larger F_i is, the higher the fitness is.

In fact genetic algorithms consist of two main processes. The first process is the selection of individuals to reproduce the next generation, and the second process is the manipulation of the selected individuals to form the next generation through hybridization and mutation techniques. The mechanism that determines which individuals are selected for reproduction also determines how many offspring each selected individual produces. The main principle of the selection strategy is that the better the individual, the more likely they are to become a parent.

4)

Selection

The selection operation is to choose elite individuals as parents in the current population that can produce offspring. The fitness value is used as a criterion to determine whether an individual is elite or not. There are many methods to select the best individual, such as roulette selection, Boltzmann selection, tournament selection, rank selection, steady-state selection, elite selection, etc. Roulette selection is commonly used, i.e., the selection probability of the i st individual is made to be: 13 $P_{i} = \frac{F_{i}}{\sum_{j = 1}^{n} F_{j}}$

Calculate the probability of selecting each individual, draw a pie chart of the probabilities, and imagine spinning a roulette wheel and the hands of the wheel randomly selecting one. Think of this as selecting a ird individual from n with probability P_i, for a total of n individuals.

5)

Crossbreeding of parent individuals

The generation of offspring in a genetic algorithm is determined by a set of operators that recombine and mutate selected members of the current population. The principle of hybridization operator is to generate two new offspring from two parent strings by copying the genes selected with specified hybridization rules from both parent strings. The gene selected at position I in each offspring is copied from the gene at position I in one of the two parents. Which parent string node is selected as the I replication position is determined by an additional string called the hybridization mask.

In single-point hybridization, the construction of the hybridization mask always starts with a string containing n consecutive 1’s, supplemented by (e–n) 0’s in the remaining positions. This means that in the offspring, the first n bits of the gene are copied from the first parent, and the remaining bits are copied from the second parent. That is, the single-point hybridization operator consists of only two steps, selecting any hybridization point n and then generating and applying a hybridization mask.

Two-point hybridization, on the other hand, reproduces offspring by replacing the middle segment of one parent string with the middle of a second parent string. The hybridization mask is a string starting with n₀ 0, with a continuous string of n₁ 1s in the middle, and ending with the required number of 0s to complete the string.Each time the two-point hybridization operator is applied, the mask is generated by randomly selecting the integers n₁ and n₁. The selected n individuals are paired two by two to reproduce their offspring by selecting some pair of two-parent number strings with hybridization probability P_c.

6)

Mutation

The control parameter for variation is P_m, which means that each individual in the population has a P_m probability of randomly varying the value in the binary number string, changing 1 to 0 or 0 to 1. This operation is performed to maintain genetic diversity, allowing the genetic algorithm to perform a global search, and is a key step in preventing a locally optimal solution from being found.

7)

Evolutionary Iteration

The n children obtained from the mutation operation in turn become the parents for the next iteration of evolution, go to step 3), and iterate repeatedly until a satisfactory individual is obtained or a predetermined number of evolutionary iterations is reached and the algorithm terminates.

2.2.2

Improved genetic algorithm based on non-uniform variation operator

The genetic algorithm includes three operators: selection, hybridization, and mutation. However, this paper only improves the mutation operator. The selection operator still uses the classic roulette wheel, where the higher the fitness of an individual, the higher the probability that its genes will be retained, and replaces the contemporary worst-fit individual with the selected historically optimal individual. Then all individuals mutate with a certain probability to produce the next generation of individuals. The advantage of the mutation operator is that it improves the global search ability of the algorithm, but it may slow down the convergence time and thus have poor local optimization ability. In order to improve the disadvantage of the traditional mutation operator, this paper proposes to replace the traditional genetic algorithm with a non-uniform mutation operator [20]. The specific idea is: select the knd component x_k in individual X to carry out the mutation operation according to equation (14) to obtain the new individual $X^{'} = {x_{1} \dots {x^{'}}_{k} \dots x_{m}}$ : 14 ${x^{'}}_{k} = {\begin{cases} x_{k} + Δ_{k} (g, r i g h t (k) - x_{k}), i f ξ_{k} = 0 \\ x_{k} - Δ_{k} (g, x_{k} - l e f f (k)), i f ξ_{k} = 1 \end{cases}$

Where: right(k) and left(k) are the upper and lower bounds of the k rd component x_k, {ξ_k}_1≤k≤m is an independent identically distributed random variable obeying a distribution P(ξ = 0) = P(ξ_k = 1) = 0.5, the function Δ_k (g, y) returns a number in the range [0, y] between and: 15 $\lim_{g \to \infty} Δ_{k} (g, y) = 0$

Among them: 16 $Δ_{k} (g, y) = y \cdot (1 - r_{k}^{{(1 - \frac{g}{G})}^{b}})$ where: {r_k}_1≤k≤m is an independent identically distributed random variable obeying a uniform distribution over [0,1] intervals, g is the current number of generations, G is the maximum number of evolutionary generations, and b is a system parameter determining the degree of nonuniformity, which determines how much the stochastic perturbation depends on the number of evolutionary generations g.

From Eq. (16), the changeable range of Δ_k (g, y) is large when the number of iterations g is small, enabling the non-uniform variation operator to perform a globally uniform search. Whereas, in the later stages of evolution, the changeable range of Δ_k (g, y) shrinks as g increases, i.e., the search step size becomes shorter, allowing the algorithm to perform optimization in a small range. The larger the maximum number of evolutionary generations G is, the longer the pre-search step is, the more the search range can be enlarged, avoiding the situation that the population, which is not conducive to finding the globally optimal solution, gathers into a certain locally optimal domain. With the increase of g, the population gradually converges to the value domain with the highest probability of existence of the optimal solution, at this time, the search step length becomes shorter, which is conducive to local optimization until the termination condition of the algorithm is met, and the globally optimal solution is found.

2.2.3

Solution Process of the Optimization Problem of Sports and Leisure Resource Allocation

The parameter population size Num, maximum number of evolutionary generations G, and system parameter b are controlled, then the specific flow of the genetic algorithm based on non-uniform variation operator to solve the problem of optimal allocation of sports and leisure resources is as follows: Step1:

Real number coding. The objective of the optimal allocation model of sports and leisure resources is to find n·T decision variables Q_1,1, Q_1,2,…Q_n,T in order to maximize the total regional output value supported by the corresponding amount of sports and leisure resources. By dividing the target region into one, two and three circles and analyzing the amount of sport and leisure resources in each circle, the problem is essentially to solve for n·T sport and leisure resource volume decision values Q_1,1,Q_1,2,…Q_n,T. Where n is the number of circles and T is the number of sectors, there are a total of 12 decision volumes. In this paper, we use real number coding, each chromosome corresponds to one kind of sport and leisure resource allocation scheme, and each gene on each chromosome represents the amount of water allocated to a certain sector in a circle, and there are 12 genes on a chromosome.

Step2:

Select a reasonable fitness function. In this paper, we directly take the maximum total value of sports economic output as the evaluation function of individual f(X). In practice, the fitness value of each individual is calculated by the decision value of the amount of sports and leisure resources Q_1,1, Q_1,2,…Q_n,T based on the value of economic output supported by the use of sports and leisure resources of each type of unit in each circle by substituting it into the function, and the advantages and disadvantages of the individual are decided according to the size of the fitness value, and individuals are selected by using roulette wheeler method, and the better the individual, the higher the probability that it is selected. In the calculation process, boundary constraints such as the size of available sports and leisure resources are considered.

Step3:

Generate initialized population. Randomly generate the initial population with a population size of Num and all individuals $X_{i}^{0}$ are n·T -dimensional variables, $X_{i}^{0} = (x_{1} \dots x_{k} \dots x_{n * T})$ , at which point the number of iterations g = 1, and the k th component x_k, rand is a random number on the interval [0,1], as calculated by the following equation: 17 $x_{k} = l e f t (k) + r a n d \cdot (r i g h t (k) - l e f t (k))$

Step4:

Individual evaluation. The fitness values of all individuals in the population are calculated by the fitness function $f (X_{i}^{0})$ , the larger the fitness value, the higher the evaluation of the individual, the optimal individual will be saved as $X^{*} = (x_{1}^{*} \dots x_{k}^{*} \dots x_{n \cdot T}^{*})$ .

Step5:

Individual crossbreeding. Adopt the gambling wheel method to select individual $X_{i}^{g} = (x_{1}^{g} \dots x_{k}^{g} \dots x_{n \cdot T}^{g})$ in the population for crossbreeding to produce offspring individuals, and calculate the fitness function of the offspring individuals.

Step6:

Mutation operation. Generate $x_{k}^{g'}$ by utilizing each one-dimensional component $x_{k}^{g}$ of the offspring individual $X_{i}^{g} = (x_{1}^{g} \dots x_{k}^{g} \dots x_{n \cdot T}^{g})$ in the population, such that $X_{i}^{g'} = (x_{1}^{g'} \dots x_{k}^{g^{'} \dots} \dots x_{n \cdot T}^{g'})$ , a total of Num offspring individuals are generated by the non-uniform variation operator operation.

Step7:

Evaluate the child individuals. For the Num offspring generated by Step6, the operation of Step4 is performed again, evaluating each new offspring individual with the fitness function, and comparing the best evaluated individual with the fitness value of the individual saved in X*, always ensuring that the optimal solution is retained in X*.

Step8:

Check the termination condition. When the number of iterations of the algorithm g is equal to the preset maximum number of evolutionary generations G or enough excellent individuals are obtained, it is considered that the termination condition of the algorithm is satisfied, and the result X* is output, and the algorithm ends. Otherwise, g = g + 1, return to Step5.

2.3

Model solution results and analysis

According to the improved genetic algorithm to solve the problem of optimal allocation of sports and leisure industry resources, Matlab R2022a is used to write a solution implementation program for the structural model of sports and leisure resources in Province H. The range of the distribution of each variable is set slightly, the population size is taken to be 60, and the crossover rate is 0.4, of which the number of single-point crossover individuals is 4 pairs, the number of arithmetic crossover individuals is 6 pairs, and the number of heuristic crossover individuals is 14 pairs. The variation rate was 0.3, with 2 individuals with boundary variation, 4 individuals with uniform variation, 4 individuals with non-uniform variation, and 8 individuals with multi-point non-uniform amount of variation, and a series of 50 pareto solutions (valid solutions) of the model were obtained after several iterations. Figure 2 depicts the spatial distribution of the resulting solutions.

Considering the sequence of pareto solutions, comprehensively measuring each variable, analyzing the results of the objective function and consulting relevant experts, the 30th effective solution is chosen as the “optimal solution” for the optimization of the resource allocation structure of the whole sports and leisure industry, as shown in Table 1. In Table 1, the unit of E(X) is billion yuan.

Table 1.

Pareto solution of sports leisure resource allocation optimization

Variables	X₁₁	X₁₂	X₁₃	X₁₄	X₁₅
Optimization value	162.37	41.28	201.45	305.74	9.159
Variables	X₁₆	X₁₇	X₁₈	X₁₈	X₁₁₀
Optimization value	13.412	29.508	69.631	24.259	43.083
Variables	X₁₁₁	X₁₁₂	X₁₁₃	X₁₁₄	X₁₁₅
Optimization value	26.564	58.317	28.243	21.267	30.524
Variables	X₂₁	X₂₂	X₂₃	X₂₄	X₂₅
Optimization value	501.21	324.52	2908.74	1224.32	30071
Variables	X₃	X₄	E(X)	δ(X)	D_δ(X)
Optimization value	2406.32	51.718	611.45	0.0009	849.62

3

VAR model analysis of sports and leisure industry development and local economic growth

In order to better assess the effectiveness of the constructed multi-objective optimal allocation model of sports and leisure resources, i.e., the efficacy of the improved genetic algorithm in optimizing the development of the sports and leisure industry and thus promoting the growth of the local sports economy, this paper explores the relationship between the development of the sports and leisure industry and the growth of the local sports economy by using the VAR model.

3.1

VAR modeling

The vector autoregressive (VAR) model employs an unstructured approach aimed at exploring the interactions between variables in an economic system and analyzing the dynamic shocks of random disturbance terms on economic variables [21].The core idea of the VAR model is that each endogenous variable is regressed on the lagged values of all the endogenous variables of the model to estimate the dynamics of the full set of endogenous variables.

VAR (p) The mathematical expression of the model is: 18 $y_{t} = β_{0} + β_{1} y_{t - 1} + \dots + β_{p} y_{t - p} + ε_{t}$

In Eq. (18), y_t denotes the k -dimensional endogenous variables, respectively, β is the coefficient matrix, p is the lag order, the explanatory variables (y_t−1,y_t−2,⋯) depend on (ε_t−1,ε_t−2,⋯), ε_t are uncorrelated with their lagged values (ε_t−1,ε_t−2,⋯), so that all the explanatory variables of the antecedent variables are uncorrelated with the current disturbance term ε_t, and so the equation can be regressed using OLS.

3.2

Stability test

The variables analyzed in this study are “Value Added of Sports and Leisure Industry (TYC)” and “Regional GDP (hereinafter referred to as GDP)”, and since both of them are time-series variables, there may be the influence of heteroskedasticity. Therefore, in this study, we firstly take the natural logarithm of the variables as InTYC and InGDP, and then take InGDP and InTYC as the explanatory variables and the explanatory variables respectively, and conduct OLS regression on both of them by defining the unit root variable. It is found that the regression coefficients of InGDP and InTYC are significant at the 5% level, and R² is as high as 0.49, which indicates that although GDP and TYC are independent unit root variables, there is the phenomenon of “pseudo-regression”, so it is necessary to carry out the smoothness test of the unit root for the two sets of series.

In this study, we use ADF (Augmented Dickey-Fuller) method to carry out the unit root test, and the mathematical expression of ADF test method is: 19 $Δ y_{t} = β_{0} + δ y_{t - 1} + γ_{1} Δ y_{t - 1} + \dots + γ_{p - 1} Δ y_{t - p + 1} + γ t + ε_{t}$

Eq. (19) where (Δy_t−1,⋯,Δy_t−p+1) is called the lagged difference term, i.e., the first to (p–1) order lag term of the first order difference Δy_t. t is the period trend. β₀ is the displacement term. γ is the coefficient of period t. γ_i is the unknown parameter. p is the lag term. ε_t is the random disturbance term. ADF test for the left one-sided test, its rejection region is only distributed in the leftmost, so its hypothesis is, H₀ : δ = 0, H₁ : δ < 0. That is, the value of the ADF test is greater than the critical value, accept the original hypothesis, means that there is a unit root of the time series, for the non-smooth series, and vice versa for the smooth series.

The results of the smoothness test of the value added of sports and leisure industry and gross regional product are shown in Table 2. The results show that the ADF values of InTYC and InGDP are greater than the 10% critical value, which are non-stationary series. After the first-order difference treatment, the ADF value of dInTYC is greater than the 10% critical value, and the ADF value of dInGDP is less than the 5% critical value but greater than the 1% critical value, and both of them are still non-stationary series. After second-order differencing again, the ADF test values of d2InTYC and d2InGDP are both less than the 1% critical value, which means that InTYC and InGDP are smooth under the second-order single integer I (2) and the VAR model can be constructed.

Table 2.

Stability test of sports industry added value and GDP

Variables	ADF test value	t statistic			Stability
Variables	ADF test value	1% threshold	5% threshold	10% threshold	Stability
InTYC	5.042	-2.704	-1.941	-1.598	Non-stationary
InGDP	7.641	-2.704	-1.941	-1.598	Non-stationary
dInTYC	-1.185	-2.704	-1.941	-1.598	Non-stationary
dInGDP	-2.235	-2.704	-1.941	-1.598	Non-stationary
d2InTYC	-2.934	-2.704	-1.941	-1.598	Smooth
d2InGDP	-2.738	-2.704	-1.941	-1.598	Smooth

3.3

Cointegration test

According to the results of ADF unit root test, it can be seen that there may be a smooth long-term equilibrium relationship between the two series of data, i.e. “co-integration relationship”. In order to further clarify whether there is a long-term equilibrium relationship between the value-added of sports and leisure industry and the gross regional product, this paper carries out the cointegration test.

Determining the number of lags is a prerequisite for the cointegration test. According to the criteria of LL, LR, AIC, HQIC, SBIC, etc., the optimal lag period is found to be 2. In addition, through the parameter estimation of the cointegration test of the two series of data, it can be seen that the trace test statistic of the hypothesis that “there is no cointegration relationship between InTYC and InGDP” is 24.2751, which is greater than the 5% critical value of 16.27, and the trace test statistic of the hypothesis that The trace test statistic for the hypothesis “at least one cointegration relationship exists” is 2.4176, which is less than the 5% critical value of 3.84. Therefore, only the first hypothesis is accepted, i.e., there is a long-run equilibrium relationship between InTYC and InGDP.

3.4

Vector error correction model

The vector correction error model (VCE) is built on the basis that two time series variables have a long-term cointegration relationship, which is a measure of short-term fluctuations between the time series variables. Based on the cointegration relationship between InTYC and InGDP, a vector correction error model can be established for the value added of sports and leisure industry and gross regional product: 20 $Δ y_{t} = β_{0} + δ e c m_{t - 1} + \sum_{i = 1}^{p - 1} γ_{i} Δ y_{t - 1} + Δ_{t}$

In equation (20), ecm_t−1 is the correction error term, which reflects the change in short-term fluctuations of the explanatory variables, and δ is the coefficient of the error correction term, which reflects the speed of adjustment to equilibrium when deviation from the long-term equilibrium occurs among the time series variables. γ_i is the coefficient of the lagged difference term of the explanatory variables, which indicates the effect of short-term fluctuations on the explanatory variables. p is the optimal lag order of the differential explanatory variables. ε_t is the residual.

The fit index R² = 0.8159,chi² = 21.6325, P = 0.000 of the equation with the value added of the sports and leisure industry as the explanatory variable and the fit index R² = 0.9392,chi² = 85.7468, P = 0.000 of the equation with the Gross Regional Product as the explanatory variable indicate that the two equations have a good fit, which can better explain the impact of short-term fluctuations of the two time-series variables on the long-run equilibrium.

The results of the vector error correction model test are shown in Table 3. As can be seen from Table 3, the coefficient estimate of the error correction model with the explanatory variable being the value added of sports and leisure industry is -1.3026, and the corresponding companion probability P=0.179>0.05, indicating that the adjustment strength of regional GDP to the deviation of the value added of sports and leisure industry from the long-run equilibrium relationship is not significant. On the contrary, the adjustment of the value added of sports and leisure industry to the deviation of GDP from the long-run equilibrium relationship is significant (the estimated value of the model coefficient is -0.2526, and the probability of companionship is P=0.000<0.05), which indicates that the value added of the sports and leisure industry in the current period is able to adjust the deviation of the previous period with the strength of -0.2526, and pull it back to the long-run equilibrium state.

Table 3.

Test results of vector corrected error model

Explained variable	Interpretation variable	Coefficient estimate	Standard deviation	Z statistic	Interaction probability (P value)
TYC	ΔInGDP	2.2658	0.0758	-37.92	0.000
	ΔInGDP_t−1	1.3984	1.4934	1.03	0.328
	ΔInTYC_t−1	0.9243	0.7732	1.18	0.274
	ecm_t−1	-1.3026	0.9463	-1.52	0.179
	cons	0.0065	0.1842	0.04	0.993
GDP	ΔInTYC	0.4793	0.0187	-37.65	0.000
	ΔInTYC_t−1	0.0542	0.2075	0.20	0.829
	ΔInGDP_t−1	0.2567	0.3793	0.65	0.537
	ecm_t−1	-0.2526	0.5324	-0.53	0.000
	cons	0.0460	0.0459	0.88	0.418

The results of the vector correction error model test are shown in Table 3.

3.5

VAR model parameter estimation

According to the existing analysis, it is known that there is a smooth long-term positive equilibrium relationship between the value added of the sports and leisure industry and the gross regional product, and the optimal lag period is 2, so the model parameters of VAR(2) can be estimated. The results of parameter estimation for the VAR model for the series of value added in sports and leisure industry and gross regional product are shown in Table 4.

Table 4.

Parameter estimation of VAR model

Variable	InTYC	InGDP
InTYC(–1)	0.6648	0.1759
	(0.7458)	(0.1926)
	[0.94]	[0.96]
InTYC(–2)	-0.9642	-0.0674
	(0.8135)	(0.2082)
	[-1.22]	[-3.38]
InGDP(–1)	3.4278	0.7869
	(3.4275)	(0.8523)
	[1.07]	[0.98]
InGDP(–2)	-0.7025	-0.0456
	(1.8051)	(0.4373)
	[-0.39]	[-0.14]
Constant term	-25.0564	3.0226
	(21.3515)	(5.2713)
	[-1.19]	[0.62]
R–sq	0.9843	0.9964
F statistic	41.0548	139.8421
P value	0.0019	0.0002
AIC	-6.4927	-3.1954
HQIC	-7.2149	-3.8516
SBIC	-6.4208	-3.1454

The results in Table 4 show that the goodness of fit of mnTYC and hGDP are 0.9758 and 0.9931, respectively, and the two fit indices are good. Meanwhile, the characteristic polynomial inverse roots of the VAR(2) model are shown in Fig. 3, which shows that the inverse of the characteristic polynomial roots are all within the unit circle range, indicating that the stability of the two serial data is good. This indicates that the economic system between the development of sports and leisure industry and local economic growth has stability, and can gradually recover to a stable equilibrium state with the passage of time when it is hit by external disturbing factors.

In the results of parameter estimation of VAR(2) model, as far as the development of sports and leisure industry is concerned, the coefficients of the influence of regional GDP on the value added of sports and leisure industry in lag 1 and 2 periods are 3.4278 and -0.7025, respectively, but this influence is statistically insignificant, which indicates that the regional economic growth has little influence on the development of sports and leisure industry in the short term. From the perspective of regional economic growth, the coefficients of the impact of the value added of sports and leisure industry on regional GDP in lag 1 and 2 periods are 0.1759 and -0.0674 respectively, and the coefficients of the lag 2 periods are significantly affected at the 1% level, which means that the development of sports and leisure industry is able to promote the growth of the local sports economy, and that this positive impact gradually slows down with the passage of time.

3.6

VAR model analysis

In order to further explore the specific correlation between the two variables of sports and leisure industry development and local sports economy, this paper explores the detailed influence mechanism through Granger causality analysis, impulse response function (IRF) analysis and variance decomposition analysis.

3.6.1

Granger causal analysis

In order to verify the existence of causal relationship between the added value of sports and leisure industry and local economic development, this paper selects Granger causality test for analysis on the basis of cointegration analysis. The test model is: 21 $Y_{t} = \sum_{i = 1}^{m} β_{i} X_{t - i} + \sum_{j = 1}^{n} γ_{j} Y_{t - j} + c + μ_{t}$

Where X_t and Y_t are variables, X_t−i is the lagged value of X_t, Y_t−j is the lagged value of Y_t, c is the constant term, β_i and γ_j are regression coefficients and μ_t is the random error. The results of Granger causality test between sequence InTYC and InGDP show that InTYC is the Granger cause of InGDP and InGDP is not the Granger cause of InTYC. It can be seen that the value added of sports and leisure industry can promote the growth of local sports economy, but the growth of local economy does not promote the development of sports and leisure industry. The reason for this is that, on the one hand, with the development of the sports and leisure industry, its importance in the local economy is becoming more and more prominent, and it can provide a driving force for the development of the local economy. On the other hand, although the local sports economy is growing and the living standard of the residents is improving day by day, the awareness of sports and fitness of the general public is still not strong, and the level of sports consumption is relatively low, which affects the development of sports and leisure industry to a certain extent, and also reflects that there is still a large room for improvement of sports and leisure industry, which can become a new growth point for the development of the local economy.

3.6.2

Impulse Response Function (IRF) Analysis

In a VAR model, a random disturbance shock to an endogenous variable not only affects itself directly, but can also affect other variables industry by passing this disturbance to other endogenous variables through the dynamic structure of the model. Impulse response function (IRF) analysis is mainly used to measure the impact of a one standard deviation shock to the randomly perturbed term on the current and future values of the endogenous variables. By analyzing the shock response function, it is possible to identify the dynamic response of each variable to the perturbation term’s shock.

The impulse response and cumulative impulse response of InGDP to one standard deviation information from itself are shown in Fig. 4(a)~(b). As can be seen in Figure 4, InGDP does not respond immediately to the shock of one standard deviation information from itself, and this response is zero in the first period InGDP, then shows a decreasing trend, and then goes to zero again in the 6th period, and the trend of the response from the 7th to the 12th period is the same as that of the previous 5 periods. As for the cumulative effect, the volatility shock itself has a negative impact on the long-term development of the local economy, and to a certain extent, stagnates the growth of the local economy. It can be seen that the fluctuation of the local economic system will affect its own development and produce a certain negative effect, and the impact of this negative effect on the stability of the local economic system shows a certain cyclicality. Therefore, in order to promote the development of local sports economy, we should try to create superior conditions, reduce the interference of external influences, and maintain the development and stability of the local economic system.

The impulse response and cumulative impulse response of InGDP to one standard deviation information from InTYC are shown in Fig. 5(a)~(b). As can be seen from Fig. 5, the impulse response of 4 shows an upward trend in response to the shock from InTYC. This upward trend gradually decreases after period InGDP and goes to zero in period 6. The response trend from period 7 to period 12 is similar to the first 6 periods. In addition, it can be seen that its cumulative response trend has been showing an upward trend. It can be seen that the development of sports and leisure industries can promote the growth of the local economy, and this promotion can show a certain cyclicality. Therefore, at this stage, vigorously developing sports and leisure industry is undoubtedly an important path to promote the rapid development of the local sports economy, and this conclusion is completely consistent with the previous cointegration analysis and Granger causality analysis.

The impulse response and cumulative impulse response of InTYC to one standard deviation information from InGDP are shown in Fig. 6(a)~(b). From the impulse response trend in Fig. 6, it can be seen that InTYC has a negative effect on the shock from InGDP, both in the short-term response and the long-term response. It can be seen that the development of the local economy has not been able to promote the development of the sports and leisure industry, but has instead had a stagnant effect on the development of the sports and leisure industry. This phenomenon, which seems to violate the economic routine, is not unrelated to the residents’ awareness of sports and fitness, the concept of sports consumption, and the starting state of the development of the sports and leisure industry. Therefore, further publicity and guidance should be provided to enhance the residents’ awareness of physical fitness, change the residents’ concept of sports consumption, increase the level of sports consumption expenditure, release the huge potential of sports consumption, and promote the optimization and upgrading of the structure of the sports and leisure industry, so as to help the development of local sports economy.

The impulse response and cumulative impulse response of InTYC to one standard deviation information from itself are shown in Fig. 7(a)~(b). From Figure 7, it is easy to see that the impulse response of InTYC to its own shock shows a positive correlation trend, and the cumulative impulse response shows a growing trend. Thus, in order to promote the development of sports and leisure industry, we need to design the system around the “sports and leisure industry itself”, and this conclusion is corroborated with the results of the VAR(2) model parameters estimation.

3.6.3

Analysis of variance decomposition

While the impulse response function describes the effect of a shock to one endogenous variable on the other endogenous variables, the variance decomposition provides another way of describing the dynamics of the system by analyzing the degree of contribution of each structural shock to the change in the endogenous variable and further evaluating the importance of the different structural shocks. By decomposing the mean square error of a variable shock into the contribution made by the random shocks of each variable in the system, and then calculating the relative importance of each variable shock, i.e., the ratio of the contribution of the variable shocks to the total contribution is the result of the equation decomposition. The results of the variance decomposition of the VAR model are shown in Table 5.

Table 5.

Variance decomposition of VAR model

Period	InGDP variance decomposition			InTYC variance decomposition
Period	S.E.	InGDP	InTYC	S.E.	InGDP	InTYC
1	0.2356	100.0000	0.0000	0.0881	51.6388	48.3612
2	0.0975	52.9653	47.0347	0.1117	52.9158	47.0842
3	0.1545	62.3246	37.6754	0.1332	55.2703	44.7297
4	0.1693	62.4273	37.5727	0.1567	57.0939	42.9061
5	0.1887	64.2145	35.7855	0.1745	58.2465	41.7535
6	0.1896	64.4615	35.5385	0.1796	57.9066	42.0934
7	0.1905	64.0687	35.9313	0.1922	57.1062	42.8938
8	0.2182	62.1418	37.8582	0.2109	57.1469	42.8531
9	0.2514	63.0612	36.9388	0.2198	57.9134	42.0866
10	0.2492	63.8279	36.1721	0.2295	58.0631	41.9369
11	0.2542	64.1278	35.8722	0.2413	58.3265	41.6735
12	0.2545	64.4675	35.5325	0.2442	58.0489	41.9511

InGDP The variance decomposition shows that in the first period, the prediction variance of InGDP is completely caused by the disturbance of InGDP itself, that is to say, its development is completely influenced by itself. By the 2nd period, 52.9653% of the forecast variance of InGDP is caused by its own perturbation, and 47.0347% is caused by the perturbation of InTYC, which means that the role of sports and leisure industry in the development of the local sports economy is rapidly appearing, and it has more impacts. Subsequently, in the process of period 3 to 12, the prediction variance of InTYC is basically stabilized at 35.5325%~37.8582%, and shows a certain periodicity, this result is basically consistent with the impulse result analysis of InTYC. In summary, it can be seen that InTYC can quickly promote the development of InGDP in the short term, and with the development of time this effect is relatively weakened, but still can inject vitality into the development of the local sports economy.

The variance decomposition of InTYC shows that in period 1, 51.6388% of the predicted variance of InTYC is caused by InGDP and 48.3612% by its own perturbation, and in period 2, the effect of InGDP increases to 52.9158% and its own effect decreases to 47.0842%. Subsequently, from the 3rd to the 12th period, the influence of InGDP shows a gradual increase, and its own influence keeps shrinking. It can be seen that the development of sports and leisure industry is gradually influenced by the development of local sports economy and its own influence is gradually reduced. This result reflects that the local sports economy occupies an important position in the development of sports and leisure industry. Although InTYC impulse analysis shows that InGDP fails to promote the development of InTYC, it does not negate the role of local sports economy as the economic foundation in the development of sports and leisure industry, and the results of the variance decomposition show that this influence will continue to increase with the development of time. Therefore, further consolidating the foundation of the local sports economy is one of the important paths to promote the development of the sports and leisure industry.

4

Conclusion

This paper constructs a cross-regional multi-objective integration model of sports and leisure resources, and realizes the further optimization of the model by using the improved genetic algorithm based on the non-uniform variation operator, and explores the relationship between the development of the sports and leisure industry and the development of the local economy based on the VAR model, and designing a practical path for the synergistic development of the two.

The multi-objective optimization model of sports and leisure resource allocation solves a total of 50 pareto solution sequences, and the 30th effective solution is selected as the “optimal solution” for the optimization of the resource allocation structure of the whole sports and leisure industry by comprehensively measuring the variables and analyzing the results of the objective function and consulting with relevant experts.

The optimal lag period of the cointegration test between the value added (TYC) of the sports and leisure industry and the series data of gross domestic product (GDP) is 2, and the trace test statistic of “InTYC and InGDP without cointegration relationship” is 24.2751, which is greater than the 5% cut-off value of 16.27, so the hypothesis is accepted, that is, there is a long-term equilibrium relationship between InTYC and InGDP. And the strength of the adjustment of the value added of sports and leisure industry on the deviation of regional GDP from the long-run equilibrium relationship is significant (the coefficient estimate is -0.2526, and the probability of companionship is P=0.000<0.05), i.e., the value added of the sports and leisure industry in the current period is capable of adjusting the deviation between the two variables in the previous period with the strength of -0.2526 times and pulling it back to the long-run equilibrium state. In addition, the coefficients of the value added of sports and leisure industry on regional GDP are 0.1759 and -0.0674 in lag 1 and 2 respectively, and the coefficients of lag 2 are significantly affected at 1% level, which implies that the development of sports and leisure industry is able to promote the growth of the local sports economy, and this positive effect is gradually slowed down with the passage of time.

The variance decomposition shows that in period 1 the predicted variance of InGDP and InTYC is mainly due to self perturbations. In period 2, the effect of self-perturbation decreases. Then, from period 3 to 12, the mutual influence between InGDP and InTYC is gradually increasing, and the self-influence is decreasing. It shows that InTYC can quickly promote the development of InGDP in the short term, and this influence is relatively weakened with the development of time, but it can still inject vitality into the development of local sports economy. At the same time, local sports economy occupies an important position in the development of sports and leisure industry. Therefore, the foundation of the local sports economy should be further strengthened to promote the synergy development between the sports and leisure industries and the local sports economy.

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Research on Optimizing the Development of Sports and Leisure Industry Using Genetic Algorithm to Promote the Growth of Local Sports Economy

Yanhua Jiao

Bing Cao

Publié en ligne: 21 mars 2025

Reçu: 07 oct. 2024

Accepté: 01 févr. 2025

DOI: https://doi.org/10.2478/amns-2025-0564

Mots clésMulti-objective optimization, Genetic algorithm, Non-uniform variation operator, Vector autoregressive model, Sports and leisure industry

© 2025 Yanhua Jiao et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Mots clés
Multi-objective optimization, Genetic algorithm, Non-uniform variation operator, Vector autoregressive model, Sports and leisure industry