Research on sports grounds’ management performance evaluation with triangular fuzzy information

Abstract

Sports ground for the construction of public sports service system, services, and improving the livelihood of the people plays an important role, which is the material base to carry out the national fitness activities and improve the competitive sports achievement, it plays an indispensable role of meeting people’s spiritual and cultural life as well as the growing fitness demand. There is the reference value to comprehensively understand the current situation of sports development of new technology development zone, and it’s important to carry out the national fitness activities, increase participate in regular physical exercise population, develop sports undertakings, enhance the level of sports services, improve the sports people’s livelihood. In this paper, we investigate the multiple attribute decision making problems for sports grounds’ management performance evaluation with triangular fuzzy information. Motivated by the ideal of geometric Bonferroni mean and dependent aggregating operators, we develop the dependent triangular fuzzy geometric Bonferroni mean (DTFGBM) operator for aggregating the triangular fuzzy information. Finally, a practical example for sports grounds’ management performance evaluation is given to verify the developed approach.

Keywords

Performance evaluation triangular fuzzy numbers geometric Bonferroni mean dependent triangular fuzzy geometric Bonferroni mean (DTFGBM) operator sports grounds management performance

1 Introduction

City civilization’s progress cannot leave the development of urban community sports, but the urban community sport’s development must take the powerful facility construction as a premise. Along with our country city scale’s expansion as well as the day-by-day growth of sports fitness entertainment demand for resident, the sports facility has received the community resident’s general reception by its unique value function, presents the irresistible rapid development situation. For understanding the present situation of urban community sports facility construction, tracking investigation “Urban Community Sports facilities Construction Land Target”, and defining urban community sports facility’s optimized construction mentality, this research is holding the literature material and develops in thoroughly investigation foundation, Using case analysis, comparison research and mathematical statistic research techniques and so on, conducted the research of urban community sports facility construction question. Many cities community sports facility’s construction presents the total quantity to be somewhat low, and indoor outside splits up a more obvious; The sports facility construction’s type coverage is broad, but the related management and the servicing facility are more deficient; The different sports items facility presents “heavily small light big, honors old abandons newly” characteristic; The urban development area’s sports facility type is more than the old area, the development area by the outdoor sports location facility primarily, the old area by indoor sports location facility primarily; The sports facility’s construction type and quantity are approximately related the house complete time, and the facility construction’s standard enhances year by year; The sports facility’s construction funds investment scale is small, urban development area take government allocation as the most important source channel, but in old area facility construction important source by property developer investment primarily; The community superintendents and the property developer don’t fully understand “Urban Community Sports facilities Construction Land Target” document, and the document execution with carries out the overall situation not to be good, the policy propaganda does not arrive, policy’s scientific nature to be short of as well as Government department’s supervision strength is not its execution not strength substantial clause; The resident is bad to the sports facility construction’s overall degree of satisfaction, the old area resident’s degree of satisfaction presents the dispersion, and the relatively average characteristic, the development area resident presents two to the sports location facility’s degree of satisfaction condition lowly and proportion distribution relative centralism characteristic; In indoor sports facility, the one who most receives the resident to welcome is the ping-pang, what the utilization ratio is lowest is the fitness instrument, but in outdoor sports facility, the one who most receives the resident to welcome is the fitness instrument, what the utilization ratio is lowest is the ping-pong; The resident most hoped increases badminton facility both indoor and outside; The community sports facility construction’s experience in overseas developed country inspires us should establishment perfect policies and regulations system, form the multiplication investment mechanism, take “school-community” development model and science reasonable plan to increases the work dynamics.

The sports grounds’ management performance evaluation with triangular fuzzy information is fuzzy multiple attribute decision making problems [1 –14]. In this paper, we investigate the multiple attribute decision making problems for sports grounds’ management performance evaluation with triangular fuzzy information. Motivated by the ideal of geometric Bonferroni mean [15 –20] and dependent aggregating operators [21, 22], we develop the dependent triangular fuzzy geometric Bonferroni mean (DTFGBM) operator for aggregating the triangular fuzzy information. Finally, a practical example for sports grounds’ management performance evaluation is given to verify the developed approach.

2 Preliminaries

In this section, we shall introduce some basic concepts and basic operational laws of triangular fuzzy numbers.

Definition 1. [23] A triangular fuzzy numbers $\tilde{a}$ can be defined by a triplet (a^L, a^M, a^U). The membership function $μ_{\tilde{a}} (x)$ is defined as: $μ_{\tilde{a}} (x) = {\begin{matrix} 0, & x < a^{L}, \\ \frac{x - a^{L}}{a^{M} - a^{L}}, & a^{L} \leq x \leq a^{M}, \\ \frac{x - a^{U}}{a^{M} - a^{U}}, & a^{M} \leq x \leq a^{U}, \\ 0, & x \geq a^{U} . \end{matrix}$ (1) where 0 < a^L ≤ a^M ≤ a^U, a^L and a^U stand for the lower and upper values of the support of $\tilde{a}$ , respectively, and a^M for the modal value.

Definition 2. [1, 2] Let $\tilde{b} = [b^{L}, b^{M}, b^{U}]$ and $\tilde{a} = [a^{L}, a^{M}, a^{U}]$ be two triangular fuzzy numbers, then the degree of possibility of a ≥ b is defined as

$\begin{matrix} p (a \geq b) & = & λ max {1 - max [\frac{b^{M} - a^{L}}{a^{M} - a^{L} + b^{M} - b^{L}}, 0], 0} \\ + (1 - λ) max {1 - max [\frac{b^{U} - a^{M}}{a^{U} - a^{M} + b^{U} - b^{M}}, 0], 0} \end{matrix}$ (2) where the value λ refers to an index of rating attitude, which can reflect the decision maker’s risk-bearing attitude. If λ is larger than 0.5, the decision maker is risk lover. If λ is equal to 0.5, the decision maker may be neutral to risk. If the value of parameter λ is smaller than 0.5, the decision maker is risk avertor.

Definition 3. [23] Let $\tilde{a} = [a^{L}, a^{M}, a^{U}]$ , $\tilde{b} = [b^{L}, b^{M}, b^{U}]$ , the distance between $\tilde{a}$ and $\tilde{b}$ is defined as follows: $d (\tilde{a}, \tilde{b}) = \frac{| a^{L} - b^{L} + | + | a^{M} - b^{M} | + | a^{U} - b^{U} |}{3}$ (3)

Definition 4. [24, 25] Let ${\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]$ be a group of triangular fuzzy numbers, the average value of the of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ is computed as $\tilde{a} = \frac{1}{n} \otimes_{j = 1}^{n} ({\tilde{a}}_{j}) = (\frac{1}{n} \sum_{j = 1}^{n} a_{j}^{L}, \frac{1}{n} \sum_{j = 1}^{n} a_{j}^{M}, \frac{1}{n} \sum_{j = 1}^{n} a_{j}^{U})$ (4)

Zhu et al. [15] studied on the geometric Bonferroni mean (GBM) which utilizing both the BM [26] and the geometric mean (GM) [27 –30].

Definition 5. [15] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of non-negative real numbers. Then the aggregation functions:

$\begin{matrix} {GBM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({pa}_{i} + {qa}_{j}))}^{\frac{1}{n (n - 1)}} \end{matrix}$ (5) is named the geometric Bonferroni mean (GBM) operator.

3 Dependent triangular fuzzy geometric Bonferroni mean operators

Based on the geometric Bonferroni mean (GBM) operator [15], Guo et al. [31] proposed the triangular fuzzy geometric Bonferroni mean (TFGBM) operator and triangular fuzzy weighted Bonferroni mean (TFWBM) operator.

Definition 6. [31] Let ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers, and let p, q > 0. If

$\begin{matrix} {TFGBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {\tilde{a}}_{i} + q {\tilde{a}}_{j}))}^{\frac{1}{n (n - 1)}} \\ = [\frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({pa}_{i}^{L} + {qa}_{j}^{L}))}^{\frac{1}{n (n - 1)}}, \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({pa}_{i}^{M} + {qa}_{j}^{M}))}^{\frac{1}{n (n - 1)}}, \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({pa}_{i}^{U} + {qa}_{j}^{U}))}^{\frac{1}{n (n - 1)}}] ∥ \end{matrix}$ (6) then TFGBM^p,q is called the triangular fuzzy geometric Bonferroni mean (TFGBM) operator.

Furthermore, Guo et al. [31] proposed the triangular fuzzy weighted geometric Bonferroni mean (TFWGBM) operator.

Definition 7. [31] ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ be a set of triangular fuzzy numbers and p, q > 0, w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of ${\tilde{a}}_{i} = [a_{i}^{L}, a_{i}^{M}, a_{i}^{U}] (i = 1, 2, \dots, n)$ , where w_i indicates the importance degree of ${\tilde{a}}_{i}$ , satisfying w_i > 0 (i = 1, 2, ⋯ , n), and $\sum_{i = 1}^{n} w_{i} = 1$ . If

$\begin{matrix} {TFWGBM}_{w}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {({\tilde{a}}_{i})}^{w_{i}} + q {({\tilde{a}}_{j})}^{w_{j}}))}^{\frac{1}{n (n - 1)}} \\ = [\frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {(a_{i}^{L})}^{w_{i}} + q (a_{j}^{L}))}^{w_{j}})}^{\frac{1}{n (n - 1)}}, \\ \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {(a_{i}^{M})}^{w_{i}} + q (a_{j}^{M}))}^{w_{j}})}^{\frac{1}{n (n - 1)}}, \\ \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {(a_{i}^{U})}^{w_{i}} + q (a_{j}^{U}))}^{w_{j}})}^{\frac{1}{n (n - 1)}}] \end{matrix}$ (7) then ${TFWGBM}_{w}^{p, q}$ is called the triangular fuzzy weighted geometric Bonferroni mean (TFWGBM) operator.

Definition 8. Let ${\tilde{a}}_{j} = [a_{j}^{L}, a_{j}^{M}, a_{j}^{U}]$ be a group of triangular fuzzy numbers, then call $s ({\tilde{a}}_{j}, \tilde{a}) = 1 - \frac{d ({\tilde{a}}_{j}, \tilde{a})}{\sum_{j = 1}^{n} d ({\tilde{a}}_{j}, \tilde{a})}, j = 1, 2, \dots, n .$ (8) the degree of similarity between the triangular fuzzy numbers ${\tilde{a}}_{j}$ and the mean $\tilde{a}$ .

In real-life situations, some individuals may assign unduly high or unduly low preference values to their preferred or repugnant objects. In such a case, we shall assign very low weights to these “false” or “biased” opinions, that is to say, the closer a preference value (argument) is to the mid one(s), the more the weight. As a result, based on (7), we define the triangular fuzzy geometric Bonferroni mean (TFGBM) operator weights as follows. $w_{j} = \frac{s ({\tilde{a}}_{j}, \tilde{a})}{\sum_{j = 1}^{n} s ({\tilde{a}}_{j}, \tilde{a})}, j = 1, 2, \dots, n .$ (9)

Obviously, w_j ≥ 0, j = 1, 2, ⋯ , n and $\sum_{j = 1}^{n} w_{j} = 1$ .

By (6), we have

$\begin{matrix} {DTFGBM}^{p, q} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \frac{1}{p + q} (\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {({\tilde{a}}_{i})}^{s ({\tilde{a}}_{i}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})} + q {({\tilde{a}}_{j})}^{s ({\tilde{a}}_{j}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})})) \\ = [\frac{1}{p + q} (\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {(a_{i}^{L})}^{s ({\tilde{a}}_{i}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})} + q (a_{j}^{L}))}^{s ({\tilde{a}}_{j}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})}), \\ \frac{1}{p + q} (\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {(a_{i}^{M})}^{s ({\tilde{a}}_{i}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})} + q (a_{j}^{M}))}^{s ({\tilde{a}}_{j}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})}), \\ \frac{1}{p + q} (\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(p {(a_{i}^{U})}^{s ({\tilde{a}}_{i}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})} + q (a_{j}^{U}))}^{s ({\tilde{a}}_{j}, \tilde{a}) / \sum_{k = 1}^{n} s ({\tilde{a}}_{k}, \tilde{a})})] \end{matrix}$ (10)

We call (9) a dependent triangular fuzzy geometric Bonferroni mean (DTFGBM) operator.

4 Research on sports grounds’ management performance evaluation with triangular fuzzy information

In this section, we shall use the dependent triangular fuzzy geometric Bonferroni mean (DTFGBM) operator for sports grounds’ management performance evaluation with triangular fuzzy information. Let A ={ A₁, A₂, ⋯ , A_m } be a discrete set of alternatives, G ={ G₁, G₂, ⋯ , G_n } be the set of attributes. Supposing that $A = {({\tilde{a}}_{ij})}_{m \times n} = {[a_{ij}^{L}, a_{ij}^{M}, a_{ij}^{U}]}_{m \times n}$ is the decision making matrix, where ${\tilde{a}}_{ij}$ are preference values, which take the form of triangular fuzzy numbers, given by the decision maker, for the alternative A_i ∈ A with respect to the attribute G_j ∈ G.

Then, we use the DTFGBM operator to solve multiple attribute decision making problems for sports grounds’ management performance evaluation with triangular fuzzy information, which can be described as following:

Step 1. Utilize the information given in matrix $\tilde{R}$ , and the DTFGBM operator (in general, we can take p = q = 1)

$\begin{matrix} {\tilde{r}}_{i} & = & (r_{i}^{L}, r_{i}^{M}, r_{i}^{U}) = {DTFGBM}^{p, q} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = & \frac{1}{p + q} (\prod_{\begin{matrix} j, k = 1 \\ j \neq k \end{matrix}}^{n} (p {({\tilde{r}}_{ij})}^{s ({\tilde{r}}_{ij}, {\tilde{r}}_{i}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{i})} + q {({\tilde{r}}_{ij})}^{s ({\tilde{r}}_{kj}, {\tilde{r}}_{j}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{j})})) \\ = & [\frac{1}{p + q} (\prod_{\begin{matrix} j, k = 1 \\ j \neq k \end{matrix}}^{n} {(p {(a_{ij}^{L})}^{s ({\tilde{r}}_{ij}, {\tilde{r}}_{i}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{i})} + q (a_{kj}^{L}))}^{s ({\tilde{r}}_{kj}, {\tilde{r}}_{j}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{j})}), \\ \frac{1}{p + q} (\prod_{\begin{matrix} j, k = 1 \\ j \neq k \end{matrix}}^{n} {(p {(a_{ij}^{M})}^{s ({\tilde{r}}_{ij}, {\tilde{r}}_{i}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{i})} + q (a_{kj}^{M}))}^{s ({\tilde{r}}_{kj}, {\tilde{r}}_{j}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{j})}), \\ \frac{1}{p + q} (\prod_{\begin{matrix} j, k = 1 \\ j \neq k \end{matrix}}^{n} {(p {(a_{ij}^{U})}^{s ({\tilde{r}}_{ij}, {\tilde{r}}_{i}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{i})} + q (a_{kj}^{U}))}^{s ({\tilde{r}}_{kj}, {\tilde{r}}_{j}) / \sum_{k = 1}^{n} s ({\tilde{r}}_{kj}, {\tilde{r}}_{j})})] \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix}$ (11)

Step 2. Let $p_{ij} = p ({\tilde{r}}_{i} \geq {\tilde{r}}_{j})$ , then develop a complementary matrix as P = (p_ij) _m×m, where p_ij ≥ 0, p_ij + p_ji = 1, p_ii = 0.5, i, j = 1, 2, ⋯ , n.

Summing all the elements in each line of matrix P, we have $p_{i} = \sum_{j = 1}^{m} p_{ij}, i = 1, 2, \dots, m .$ (12)

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with the values p_i (i = 1, 2, ⋯ , m).

5 Numerical example

With the rapid development of the social economy, people’s living standard is increasing day by day, so great changes have taken place in their ways of lives. More and more people spend more time on physical exercise, and a healthy lifestyle has become a fashionable pursuit. The public fitness fever then grows rapidly. At the same time, it also leads to the contradiction between growing public fitness demand and the lack of public sports facilities. So, how to make full use of the sports facilities especially all kinds of schools sports facilities, to create a better fitness condition for the public, and to ease the contradiction between the large number of physical exercise and the lack of sports facilities have become a problem which has strong practical significance. In this section, we utilize a practical example for sports grounds’ management performance evaluation to illustrate the application of the developed approaches. The expert team selects four attributes for sports grounds’ management performance evaluation: G₁ is the number of sports grounds; G₂ is the distribution of the sports grounds; G₃ is the investment and financing situation of sports grounds; G₄ is the use of the sports grounds. The five possible sports grounds A_i (i = 1, 2, ⋯ , 5) are to be evaluated by using the triangular fuzzy numbers by the decision makers under the above four attributes (whose weighting vector is ω = (0.2, 0.4, 0.1, 0.3)), and construct the following matrix $A = {({\tilde{a}}_{ij})}_{5 \times 4}$ is shown in Table 1.

In the following, in order to select the most desirable sports grounds, we utilize the DTFGBM operator for sports grounds’ management performance evaluation with triangular fuzzy information, which can be described as following:

Step 1. Aggregate all triangular fuzzy value ${\tilde{r}}_{ij} (j = 1, 2, \dots, n)$ by using the DTFGBM to get the overall triangular fuzzy values ${\tilde{r}}_{i} (i = 1, 2, 3, 4, 5)$ of the sports grounds A_i. The aggregating results are shown in Table 2.

Step 2. According to the aggregating results shown in Table 2 and the formula of degree of possibility, the ordering of the sports grounds are shown in Table 3. As we can see, the best sports ground is A₄.

6 Conclusion

Sports grounds and sports activities are carried out, the development of sports undertakings to protect the most important one of the conditions for the material, the people are for fitness activities and competitive sports grounds for training and match the material carriers. Sixty years since the founding of new China in all aspects of government and with the support of our country has made sports more obvious results. The fifth national census data show that the sports grounds, sports grounds in our country at the number and size than the past, significant progress had been made. However, the achievements at the same time can not underestimate the current stadium to our country a number of problems, especially the problem of wastage of sports At present, China’s sports facilities and grounds are still unable to meet the broad masses of the people’s growing demand for cultural and sports life and sports career development of sports facilities, The loss of sports at a certain extent hindered the development of sports in our country, our country will have on the masses of sports, competitive sports and the sports industry in the development of much negative impact. In this paper, we investigate the multiple attribute decision making problems for sports grounds’ management performance evaluation with triangular fuzzy information. Motivated by the ideal of geometric Bonferroni mean and dependent aggregating operators, we develop the dependent triangular fuzzy geometric Bonferroni mean (DTFGBM) operator for aggregating the triangular fuzzy information. Finally, a practical example for sports grounds’ management performance evaluation is given to verify the developed approach. In the future, we shall continue working in the extension and application of the other decision analysis models [32 –42] to sports grounds’ management performance evaluation.

Footnotes

Acknowledgments

The work was supported by the Anhui province sports equalization of public service regulation research (SK2015ZD001).

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