Current Situation and Optimization Strategy of Family Service Industry in Hebei Province Based on Fuzzy Mathematical Model Processing

Aiming at the current situation of the family service industry in Hebei Province and combining the characteristics of different cities, the construction of the province’s top-to-bottom, layer-by-layer decomposition of the family service industry development indicator system is shown in Table 1. Due to the different degree of development of the family service industry in different regions, the third-level indicators should be the characteristic indicators of different regions, which can support the unified second-level indicators of the province, but can be adapted to the local conditions, so as to reflect the characteristics of the system to build a perfect and flexible. On the basis of the standard indicator system, the tertiary indicators under the secondary indicators are analyzed, so as to provide support for the policy formulation of the relevant government departments, provide better guidance and assistance for the development of the relevant enterprises, and comprehensively promote the rapid, healthy and sustainable development of the family service industry in Hebei Province. Table 1.

Family service indicator system

This paper takes Hebei Province as the research object, considering the data accessibility, the panel data of 10 cities in Hebei Province, including Shijiazhuang City, from 2010 to 2020 are finally selected, and the data used are mainly from China Statistical Yearbook, China Social Statistical Yearbook, Hebei Provincial Statistical Bulletin, and statistical yearbooks and websites such as the Bureau of Statistics, and some of the missing data are obtained by interpolation method.

1)

Entropy value method

Entropy value method is an effective multi-indicator comprehensive evaluation method, which can measure the weight of each indicator according to the degree of variability of the indicator [25]. Compared to other similar methods, the entropy value method can effectively avoid the subjective nature of the assignment, thus making the results more scientific and accurate. In this study, the entropy value method is used to calculate the weight of each index, and the comprehensive score of Hebei Province’s family service industry is further quantified by the comprehensive index method, and its calculation formula is as follows: (1)Ai=Si+Si++Si−

where Si+ and Si− denote the degree of proximity and distance from the ideal solution, respectively. 2)

Kernel kernel density estimation

Kernel density estimation is a nonparametric estimation method that smoothes the data points by means of a probability density function, which has a better fit and continuity compared with other methods [26]. It does not depend on the length of time span and model selection, and the estimation results have stability. In this study, we analyze the shape, ductility and degree of polarization of the kernel density distribution curve of the development level of the household service industry in Hebei Province to reveal its time-series evolution law. Its calculation formula is: (2)f(x)=1Nh×∑i=1NK(Xi−x¯h) (3)K(x)=12π×exp(−x22)

Where: f(x) is the density function. K(x) is the probability density. N is the number of samples. h is the bandwidth. 3)

Spatial Markov chain

On the basis of the traditional Markov chain, the spatial Markov chain introduces spatial lag and fully considers the influence of the neighborhood type of the research object on its state transition [27], which can more profoundly reveal the spatiotemporal dynamic evolution characteristics of the research object. Based on the condition of the spatial lag of region i in the initial year, the conditional transition probability matrix of N and k × k is formed on the basis of decomposing the Markov matrix of traditional k × k. If the neighborhood of region i is j, then the spatial lag of region i is calculated as follows: (4)Lag=∑xiwij

Where: Lag denotes the spatial lag operator, x_i is the level of development of the household services industry in region i, and w_ij is the spatial lag weight of the observations in neighborhood j with respect to location i.

Based on the entropy value method to construct the evaluation system of the development level of family service industry in Guanzhong Plain City Cluster, after collecting and processing the data, we get the comprehensive level of the development of family service industry in 10 cities from 2010 to 2020, and the results are shown in Fig. 1, in which C1-C10 denote the 10 major cities in Hebei Province, such as Shijiazhuang, Tangshan, Handan, Xingtai, Baoding, Zhangjiakou, Chengde, Cangzhou, Langfang and Hengshui, etc., respectively. major cities. From the mean value, during the period of 2010-2020, the average value of the comprehensive level of family service industry development in 10 major cities in Hebei Province is 0.0739, 0.0733, 0.0819, 0.0953, 0.0971, 0.1068, 0.1141, 0.1207, 0.1215, 0.1269, and 0.1305, respectively.The comprehensive level of family service industry development shows a stable upward trend. The comprehensive level has shown a steady upward trend, from 0.0739 in 2010 to 0.1305 in 2020.The development of Shijiazhuang, Handan, Cangzhou and Langfang has been rapid during this period, of which Shijiazhuang has the most prominent development and has been in the first place, and Handan, Cangzhou and Langfang, although their comprehensive rankings are not outstanding, have developed at a fairly rapid pace, with the comprehensive development level of all three cities in 2020 compared with that in 2010 showing an increase in the level of development of the domestic service industry. All three cities have seen exponential growth in their comprehensive development levels in 2020 compared to 2010. From the viewpoint of each city, the average value of the development of household service industry in Hebei Province in 2010 was 0.0739, of which three cities, Shijiazhuang, Xingtai and Baoding, were higher than the average value of the Guanzhong Plain City Cluster, with the development level of 0.2585, 0.0774, 0.0819, respectively, and the comprehensive level of the development of Shijiazhuang’s household service industry was much higher than that of the other cities in Hebei Province. By 2020, the average value of development for the family service industry in Hebei Province will be 0.1305, which is a significant improvement.

Figure 1.

Integrated level of development of family service industry

In general, although the comprehensive level of development of the family service industry in Hebei Province has been rising year by year, it is still at a low level and is vulnerable to interference by other factors. Shijiazhuang’s comprehensive level of family service industry development is higher than that of other cities in Hebei Province, and has been in a leading position, while the comprehensive level of family service industry development in other cities is lower, and the trend of differentiation in the level of development of the family service industry in these cities is relatively small, and the problem of imbalance in the open development of different regions still exists.

In order to further explore the industry development situation, the analysis will also be conducted at the level of city clusters to explore the evolutionary dynamics of the development level of the household service industry in city clusters. 1)

Kernel density estimation

Based on the fact that Kernel kernel density estimation method can reflect the absolute change of the development level of family service industry in different regions, Kernel kernel density method is used to analyze the characteristics of the extension and development trend of the development level of the family service industry in Hebei Province from 2010 to 2020.Kernel kernel density estimation of the development level of the family service industry in Hebei Province from 2010 to 2020 is shown in Fig. 2. From the point of view of distribution location, the overall distribution of kernel density curve in Hebei Province shows a trend of rightward shift, indicating that the development level of family service industry in Hebei Province is constantly improving. In terms of the shape of the distribution, the right tail is shortened and the extension of the distribution is contracting, implying that the spatial differences in the level of development of the household service industry are shrinking. From the position of the main peak, there are obviously three peaks, but there is no significant change in the spacing of the three peaks, indicating that there is a differentiation in the level of development of the family service industry in Hebei Province.

Figure 2.

Kernel density estimation of the development level of family service industry

Let X be the set consisting of the m design indicator factors for all: (5)X={x1,x2,⋯,xm}

x_i are the indicator factors in the set, i = 1, 2, 3, ⋯, m set the weights of the m indicator factors and express them as weight vectors: (6)W={w1,w2,w3,⋯,wm}

A for the set consisting of all optimization schemes: (7)A={A1,A2,A3,⋯,An}

G is the optimal subset derived from the fuzzy comparison on x, then: (8)dw(Aj,G)=11+[1−1∑i=1mwjyij]2

Scenario A_j is the optimal scenario resulting from the phase comparison, and G takes the scenario superiority at the Hemming distance.

There are many factors that affect the preferred development strategy of Hebei Province’s family service industry, and this paper selects nine evaluation index factors in the preferred selection process. Among the nine evaluation index factors, the development foundation, the degree of marketization, the degree of socialization, and professionalism are qualitative indexes. Per capita GDP, per capita disposable income of urban residents, the proportion of added value of the service industry, the urbanization rate, the proportion of investment in the service industry, and the number of loss of talents in the service industry are quantitative indicators. These two main indicators are constructed as a system of preferred indicators for program evaluation. Among them, quantitative and qualitative indicators are calculated in a very different way. 1)

Quantitative factor affiliation matrix calculation

GDP per capita, the share of value added in the service sector, the urbanization rate, and other quantitative evaluation of these indicative factors have different units, so there is no comparability. In order to make these indicators can be comparable, it is necessary to construct the affiliation matrix in this concept. Generally, quantitative indicative factors are determined using a linear affiliation function. Linear affiliation function matrix is the matrix of eigenvector functions obtained by using the m indicator factors of n optimization program for the development of family service industry: (9)Y=[ y11 y12 ⋯ y1n y21 y22 ⋯ y2n ⋯ ⋯ ⋯ ⋯ ym1 ym2 ⋯ ymn]

where i = 1, 2, ⋯, m; j = 1, 2, ⋯, n.

Quantitative indicators include positive and negative indicators, for positive indicators, the larger the value the better, such as GDP per capita. For negative indicators, the smaller the value, the better, such as the number of brain drain in the service industry.

The formula for positive indicators is: (10)rij=yijmaxyijDoing the specification

The negative indicator formula is: (11)rij=minyijyijDoing the specification

Get the affiliation matrix: (12)R=[ r11 r12 ⋯ r1n r21 r22 ⋯ r2n ⋯ ⋯ ⋯ ⋯ rm1 rm2 ⋯ rmn]=(rij)

In the formula, i = 1, 2, ⋯, m; j = 1, 2, ⋯, n. 2)

Qualitative factors affiliation matrix calculation

Non-quantitative (qualitative) indicators are determined using binary comparison primary ranking. The binary comparison primary ranking is to have a target set P={P1,P2,P3,⋯Pm} of non-quantitative factors, and use the binary comparison method to compare the importance of each factor and rank them accordingly. The relative P_k and P_l in the target set are compared bivarially, and if P_k is more important than P_l, then the ranking scale is e_kl = 1, e_lk = 0. If P_k is as important as P_l, then e_kl = 0.5, e_lk = 0.5. If P_l is more important than P_k, then e_lk = 1, e_kl = 0. k = 1, 2, ⋯, m; l = 1, 2, ⋯n, and then the matrix E is obtained: (13)E=[ e11 e12 ⋯ e1n e21 e22 ⋯ e2n ⋯ ⋯ ⋯ ⋯ em1 em2 ⋯ emn]

Sorting matrix E in order of importance yields: (14)β=∑j=1mβij(i≠j,i=1,2,⋯,m)

According to the matrix E importance ordering obtained by row ordering β control relative affiliation relationship is shown in Table 3, where T₁-T₁₁ represent the same, slightly, slightly, more, obviously, significantly, very, very, extremely, extremely, incomparably, and other 11 tone operators, respectively. Using the data in the table can be derived from the affiliation of non-quantitative (qualitative) evaluation indicator factors.

The weight is the relative importance of the indicator values in the whole evaluation according to the different factors influencing the development of service industry of Hebei Province families, which are assigned different values for the indicators. And use the square root method to calculate the weights, and ask the relevant experts to get the assignment matrix according to the assignment. The steps are as follows: 1)

Make D=(d1,d2,⋯,dm)

where: (15)di=∏j=1maijn(i=1,2,⋯,n) 2)

Normalize the vector to get the weight vector: (16)W=(W1,W2,⋯,Wn)

Among them: (17)Wi=di∑j=1ndj

The weight vector W and the fuzzy affiliation matrix R are utilized B = W × R to obtain the final judgment vector B. The optimal solution is selected based on the size of the final judgment vector B.

The main influencing factors of each family service industry optimization scheme and its judging indexes are shown in Table 4. F1-F9 indicate 9 influencing factors respectively. The optimization scheme affiliation matrix selects six quantitative indicators of per capita GDP, per capita disposable income of urban residents, the proportion of value-added of service industry, the urbanization rate, the proportion of investment in service industry, and the number of loss of talents in service industry, and three qualitative indicators of the development foundation, the degree of marketization, and the degree of socialization and specialization for the preference according to the principle of the optimization scheme selection.

From Table 4, the eigenvector matrix is given as Y1−6=[ 300 600 500 1000 400 4573 5624 4482 5036 4975 26.79 32.83 34.96 23.54 28.97 72.38 72.46 72.38 75.49 77.67 34.78 35.94 33.26 41.05 37.86 538 694 372 409 533]

Applying Eqs. (10) and (11) the specification Y₁₋₆ matrix is obtained: R1−6=[ 0.300 0.600 0.500 1.000 0.400 0.813 1.000 0.797 0.895 0.885 0.766 0.939 1.000 0.673 0.828 0.937 0.933 0.937 0.972 1.000 0.847 0.876 0.810 1.000 0.922 0.691 0.536 1.000 0.910 0.698]

According to the high or low degree of the development base of the household service industry, the matrix of eigenvectors is obtained with reference to equation (13): R7=[ 0 1 0 0 0 1 1 0 0.5 1 0.5 0 1 0 0 1 0.5 1 0.5 1 0 1 1 0.5 0]

According to the ordering of equation (14), according to its order in Table 3 kinds of appropriate selection of parameters, to obtain the relative superiority vector of the development base is: R7={0.317,0.824,0.976,0.528,0.836}

A matrix of eigenvectors is obtained based on the high or low increase in the degree of marketization of the household services industry by each scenario: R8=[ 1 0.5 0.5 0 1 1 1 0.5 1 0 1 1 0 0 0 0.5 0.5 0 0.5 1 0 0.5 0.5 0.5 0]

The relative superiority vector of the degree of marketization is obtained from Table 3: R8={0.642,0.657,0.894,0.903,0.721}

According to the effect of each home service industry optimization scheme on the degree of socialization and professionalism, the eigenvector matrix is obtained: R9=[ 0.5 0.5 1 0 0.5 1 0.5 0 1 1 0 0.5 1 0 0.5 0.5 1 0 1 0.5 0 0 0.5 1 1]

From Table 3, the relative superiority vector for the degree of socialization expertise of each program is: R9={0.712,0.638,0.509,0.836,0.471}

Summarizing the above, the integrated affiliation matrix is obtained: R=[ 0.300 0.600 0.500 1.000 0.400 0.813 1.000 0.797 0.895 0.885 0.766 0.939 1.000 0.673 0.828 0.937 0.933 0.937 0.972 1.000 0.847 0.876 0.810 1.000 0.922 0.691 0.536 1.000 0.910 0.698 0.317 0.824 0.976 0.528 0.836 0.642 0.657 0.894 0.903 0.721 0.712 0.638 0.509 0.836 0.471]

The weights of the above nine factors on the impact of the optimization plan of Hebei Province’s household service industry are requested to be assigned by experts according to the fuzzy quantitative method, from which the judgment matrix B is constructed as follows: B=[ b11 b12 ⋯ b19 b21 b22 ⋯ b29 ⋯ ⋯ ⋯ ⋯ b91 b92 ⋯ b99]=[ 1 0.25 1 0.33 3 4 4 0.5 2 4 1 0.5 0.25 0.33 1 3 5 4 1 2 1 0.33 2 0.25 3 0.33 0.2 3 4 3 1 0.2 1 0.2 4 0.25 0.33 3 0.5 5 1 0.5 0.33 1 5 0.25 1 4 1 2 1 0.33 1 0.33 0.25 0.33 0.33 5 3 3 1 0.2 0.25 2 0.2 3 0.25 1 1 5 1 2 0.5 0.25 5 4 2 3 4 0.5 1]

Calculate the value of each factor weight according to equation (15): d1=0.165,d2=1.290,d3=0.738,d4=1.041,d5=1.083, d6=0.844,d7=0.733,d8=1.130,d9=1.459

Gained after normalization: d1=0.0195,d2=0.1521,d3=0.0870,d4=0.1226,d5=0.1277, d6=0.0995,d7=0.0864,d8=0.1332,d9=0.1720

The weight vector is: W={0.0195,0.1521,0.0870,0.1226,0.1277,0.0995,0.0864,0.1332,0.1720}

The final judgment is obtained from the weight vector and the combined affiliation matrix: W×R={0.0804,0.0882,0.0919,0.0957,0.0859}

From the evaluation results, we can conclude that the optimization programs suitable for the development of the domestic service industry in Hebei Province are, in order: building a model domestic helper training base (Program 4), developing the employee-based domestic service industry (Program 3), encouraging employment and entrepreneurship and talent cultivation (Program 2), strengthening the construction of the information platform and the regional docking (Program 5), and strengthening vocational skills training and appraisal (Program 1).

Comprehensive fuzzy mathematical model judging results, Hebei Province can take the optimal family service industry development program for the construction of demonstrative domestic helper training bases, under this program can be taken to optimize the strategy of family services also as follows: (1) Establish demonstrative domestic helper training bases in the cities of the districts and key counties and cities. (2) Increase financial support and equipment investment in the bases to improve training output capacity. (3) Evaluate and dynamically manage the bases on a regular basis to ensure the quality and effectiveness of training. This will effectively enhance the construction and optimization of the domestic service industry in Hebei Province.