Research on the management performance evaluation of the sports sites with intuitionistic fuzzy information

Abstract

Sports field is important material foundation which popularizes 〈〈National Fitness Program〉〉 and 〈〈the Olympics Glory Program〉〉 in our country. It is an importance carrier of the development of our country’s sports undertaking. In the meantime, the development scale and level of the sports field is one of importance sign of the national economies development level and social civilization degree. There exists difference in the society, economy, culture...etc. in our country, which causes the “out of line” of sports field construction and development in each region.And the “out of line” of the structure of regional sports field in our country is the key problem that restricts the development of the region sports and is also the key problem that restricts the development of our country’s sports undertaking. In this paper, we first introduce some operations on the intuitionistic fuzzy sets, such as Hamacher sum, Hamacher product, Hamacher exponentiation, etc., and further develop the intuitionistic fuzzy Hamacher correlated averaging (IFHCA) operator. The prominent characteristic of the operators is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have applied the IFHCA operators to multiple attribute decision making with intuitionistic fuzzy information. Finally, a practical example for evaluating the management performance of the sports sites is used to illustrate the developed procedures.

Keywords

Intuitionistic fuzzy numbers operational laws intuitionistic fuzzy Hamacher correlated averaging (IFHCA) operator management performance evaluation sports sites

1 Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set(IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set [3] whose basic component is only a membership function. The intuitionistic fuzzy set has received more and more attention since its appearance [4 –20]. Xu [21] developed some arithmetic aggregation operators, such as the intuitionistic fuzzy arithmetic averaging (IFAA) operator and the intuitionistic fuzzy weighted averaging (IFWA) operator. Furthermore, Xu [22] developed the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator, and the intuitionistic fuzzy hybrid aggregation (IFHA) operator. Xu [23] developed some geometric aggregation operators with intuitionistic fuzzy information. Xu and Yager [24] investigated the dynamic intuitionistic fuzzy multiple attribute decision making problems and developed some aggregation operators such as the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator and uncertain dynamic intuitionistic fuzzy weighted averaging (UDIFWA) operator to aggregate dynamic or uncertain dynamicintuitionistic fuzzy information. Wei [25] proposed some geometric aggregation operators such as the dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator to aggregate dynamic or uncertain dynamic intuitionistic fuzzy information. Wei [26] proposed two new aggregation operators: induced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced Intuitionistic fuzzy ordered weighted geometric (I-IIFOWG) operator. Wei and Zhao [27] developed two new aggregation operators: induced intuitionistic fuzzy correlated averaging (I-IFCA) operator and induced intuitionistic fuzzy correlated geometric (I-IFCG) operator.

In university art education research field, the music teaching and dance teaching to promote each other and influence each other between the relationships has become one of the hot topics today. Based on the previous research on the basis of results, the research object will be further defined the scope of normal music education major, for the professional dance course teaching content on the specific research and analysis. Dance is the earliest human the art that is produced. In ancient human language has not been produced before, people use body movement, posture, facial expression to transfer information, thoughts and feelings in communication. Dance is a national and an important part of national culture. China since ancient times has put great emphasis on dance education. Chinese dynasties have court music education institutions. But due to the dance practice, visual characteristics and the history of the written records can provide people with infinite daydream space, but can’t restore once brilliant real style. The establishment of the new China, especially since the reform and opening up, China’s dance art in a new platform has started a new round of revolution. In this paper, we first introduce some operations on the intuitionistic fuzzy sets, such as Hamacher sum, Hamacher product, Hamacher exponentiation, etc., and further develop the intuitionistic fuzzy Hamacher correlated averaging (IFHCA) operator. The prominent characteristic of the operators is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have applied the IFHCA operators to multiple attribute decision making with intuitionistic fuzzy information. Finally, a practical example for evaluating the management performance of the sports sites is used to illustrate the developed procedures.

2 Preliminaries

In the following, we introduce some basic concepts related to intuitionistic fuzzy sets.

Definition 1. Let X be a universe of discourse, then a fuzzy set is defined as: $A = {〈 x, μ_{A} (x) 〉 | x \in X}$ (1)

Which is characterized by a membership function μ_A : X → [0, 1], where μ_A (x) denotes the degree of membership of the element x to the set A [3].

Definition 2. An IFS A in X is given by $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 | x \in X}$ (2)

Where μ_A : X → [0, 1] and ν_A : X → [0, 1], with the condition 0 ≤ μ_A (x) + ν_A (x) ≤ 1, ∀ x ∈ X . The numbers μ_A (x) and ν_A (x) represent, respectively, the membership degree and non- membership degree of the element x to the set A [1, 2].

Definition 3. Let $\tilde{a} = (μ, ν)$ be an intuitionistic fuzzy number, a score function S of an intuitionistic fuzzy value can be represented as follows [28]: $S (\tilde{a}) = μ - ν, S (\tilde{a}) \in [- 1, 1] .$ (3)

Definition 4. Let $\tilde{a} = (μ, ν)$ be an intuitionistic fuzzy number, an accuracy function H of an intuitionistic fuzzy value can be represented as follows [29]: $H (\tilde{a}) = μ + ν, H (\tilde{a}) \in [0, 1] .$ (4) to evaluate the degree of accuracy of the intuitionistic fuzzy value $\tilde{a} = (μ, ν)$ , where $H (\tilde{a}) \in [0, 1]$ .

Based on the score function S and the accracy function H, in the following, Xu [23] give an order relation between two intuitionistic fuzzy values.

T-norm and t-conorm are an important notion in fuzzy set theory, which are used to define a generalized union and intersection of fuzzy sets [30]. Roychowdhury and Wang [31] gave the definition and conditions of t-norm and t-conorm. Based on a t-norm (T) and t-conorm (T^*), a generalized union and a generalized intersection of intuitionistic fuzzy sets were introduced by Deschrijver and Kerre [32]. Further, Hamacher [33] proposed a more generalized t-norm and t-conorm. They are defined as follows:

Hamacher product ⊗ is a t-norm and Hamacher sum ⊕ is a t-conorm, where $T (a, b) = a \otimes b = \frac{ab}{γ + (1 - γ) (a + b - ab)}$ (5) $T * (a, b) = a \oplus b = \frac{a + b - ab - (1 - γ) ab}{1 - (1 - γ) ab}$ (6)

Then, we shall introduce the Hamacher operations on intuitionistic fuzzy sets and analyze some desirable properties of these operations. Motivated by (5-6), let the t-norm T and t-conorm S be Hamacher product T” and Hamacher sum S” respectively, then the generalised intersection and union on two IFSs A and B become the Hamacher product (denoted by ${\tilde{a}}_{1} \otimes {\tilde{a}}_{2}$ ) and Hamacher sum (denoted by ${\tilde{a}}_{1} \oplus {\tilde{a}}_{2}$ ) on two IFSs ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ , respectively, asfollows. $\begin{matrix} {\tilde{a}}_{1} \oplus {\tilde{a}}_{2} = (\frac{μ_{1} + μ_{2} - μ_{1} μ_{2} - (1 - γ) μ_{1} μ_{2}}{1 - (1 - γ) μ_{1} μ_{2}}, \\ \frac{ν_{1} ν_{2}}{γ + (1 - γ) (ν_{1} + ν_{2} - ν_{1} ν_{2})}) \end{matrix}$ $\begin{matrix} λ {\tilde{a}}_{1} = (\frac{{(1 + (γ - 1) μ_{1})}^{λ} - {(1 - μ_{1})}^{λ}}{{(1 + (γ - 1) μ_{1})}^{λ} + (γ - 1) {(1 - μ_{1})}^{λ}}, \\ \frac{γ {(ν_{1})}^{λ}}{{(1 + (γ - 1) (1 - ν_{1}))}^{λ} + (γ - 1) {(ν_{1})}^{λ}}) λ > 0 . \end{matrix}$

3 Intuitionistic fuzzy hamacher correlated operators

In the following, we shall propose the intuitionistic fuzzy correlated aggregation operators with the help of the Hamacher operations.

Definition 6. [34] Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2,$ ⋯ , n) (j = 1, 2, ⋯ , n) be a collection of intuitionistic fuzzy values, and let IFHWA: Qⁿ → Q, if $\begin{matrix} {IFHWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} (ω_{j} {\tilde{a}}_{j}) \\ = (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - μ_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - μ_{j})}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} ν_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - ν_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} ν_{j}^{ω_{j}}}) \end{matrix}$ (7) where ω = (ω₁, ω₂, ⋯ , ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ ,then IFHWA is called the Intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator.

Definition 7. [34] Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of Intuitionistic fuzzy numbers. An Intuitionistic fuzzy Hamacher ordered weighted averaging (IFHOWA) operator of dimension n is a mapping IFHOWA: Qⁿ → Q, that has an associated vector w = (w₁, w₂,..., _wn)^T such that _wj >0 and $\sum_{j = 1}^{n} w_{j} = 1$ . Furthermore, $\begin{matrix} {IFHOWA}_{w} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} (w_{j} {\tilde{a}}_{σ (j)}) \\ = (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{σ (j)})}^{w_{j}} - \prod_{j = 1}^{n} {(1 - μ_{σ (j)})}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{σ (j)})}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - μ_{σ (j)})}^{w_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} ν_{σ (j)}^{w_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - ν_{σ (j)}))}^{w_{j}} + (γ - 1) \prod_{j = 1}^{n} ν_{σ (j)}^{w_{j}}}) \end{matrix}$ (8) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{α}}_{σ (j - 1)} \geq {\tilde{α}}_{σ (j)}$ for all j = 2, ⋯ , n.

For real decision making problems, there is always some degree of inter-dependent characteristics between attributes. Usually, there is interaction among attributes of decision makers. However, this assumption is too strong to match decision behaviors in the real world. The independence axiom generally can’t be satisfied. Thus, it is necessary to consider this issue.

Definition 8. [35] Let f be a positive real-valued function on X, and μ be a fuzzy measure on X. The discrete Choquet integral of f with respective to μ is defined by $C_{μ} (f) = \sum_{i = 1}^{n} f_{σ (i)} [μ (A_{σ (i)}) - μ (A_{σ (i - 1)})]$ (9) where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that f_σ(i-1) ≥ f_σ(i) for all j = 2, ⋯ , n, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

Based on the aggregation principle for IFs and Choquet integral, in the following, we shall develop some Hamacher correlated aggregation operators with intuitionistic fuzzy information.

Definition 9. Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of intuitionistic fuzzy values on X, and μ be a fuzzy measure on X, then we call $\begin{matrix} IFHCA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{a}}_{σ (j)}) \end{matrix}$ (10) the intuitionistic fuzzy Hamacher correlated averaging (IFHCA) operator, where (σ (1) , σ (2) , ⋯ , σ (n)) is a permutation of (1, 2, ⋯ , n), such that ${\tilde{a}}_{σ (j - 1)} \geq {\tilde{a}}_{σ (j)}$ for all j = 2, ⋯ , n, A_σ(k) ={ x_σ(j)|j ≤ k }, for k ≥ 1, and A_σ(0) = φ.

With the operation of intuitionistic fuzzy numbers, the IFHCA operator can be transformed into the following from by induction on n:

$\begin{matrix} IFHCA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{a}}_{σ (j)}) \\ = (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - μ_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - μ_{σ (j)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} ν_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - ν_{σ (j)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} ν_{σ (j)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}) \end{matrix}$ (11) whose aggregated value is also an intuitionistic fuzzy numbers.

It can be easily proved that the IFHCA operator has the following properties.

Theorem 1.(Idempotency) If all (μ_j, ν_j) are equal, i.e. ${\tilde{a}}_{j} = \tilde{a}$ for allj, then $IFHCA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = \tilde{a}$ (12)

Theorem 2. (Boundedness) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ be a collection of I F VN s, and let ${\tilde{a}}^{-} = min_{j} {\tilde{a}}_{j}, {\tilde{a}}^{+} = max_{j} {\tilde{a}}_{j}$

Then ${\tilde{a}}^{-} \leq IFHCA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \leq {\tilde{a}}^{+}$ (13)

Theorem 3.(Monotonicity) Let ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ and ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ be two set of I F VN s, if ${\tilde{a}}_{j} \leq {\tilde{a}}_{j}^{'}$ , for allj, then $\begin{matrix} IFHCA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ \leq IFHCA ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (14)

Theorem 4.(Commutativity) Let ${\tilde{a}}_{j} (j = 1, 2,$ ⋯ , n) and ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ be two set of I F VN s, then $\begin{matrix} IFHCA ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = IFHCA ({\tilde{a}}_{1}^{'}, {\tilde{a}}_{2}^{'}, \dots, {\tilde{a}}_{n}^{'}) \end{matrix}$ (15) where ${\tilde{a}}_{j}^{'} (j = 1, 2, \dots, n)$ is any permutation of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ .

4 Research on the management performance evaluation of the sports sites with intuitionistic fuzzy information

In this section, we shall investigate the multiple attribute decision making problems based on the IFHCA operator with intuitionistic fuzzy information. Let A ={ A₁, A₂, ⋯ , A_m } be a discrete set of alternatives, and G ={ G₁, G₂, ⋯ , G_n } be the set of attributes. The information about attribute weights is completely known. Let w = (w₁, w₂, ⋯ , w_n) ∈ H be the weight vector of attributes, where w_j ≥ 0, j = 1, 2, ⋯ , n, $\sum_{j = 1}^{n} w_{j} = 1$ . Suppose that $\tilde{R} = {({\tilde{r}}_{ij})}_{m \times n} = {(μ_{ij}, ν_{ij})}_{m \times n}$ is the intuitionistic fuzzy decision matrix, where μ_ij ⊂ [0, 1], ν_ij ⊂ [0, 1], μ_ij + ν_ij ≤ 1, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n.

In the following, we apply the IFHCA operator to multiple attribute decision making with intuitionistic fuzzy information. The method involves the following steps:

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the IFHCA operator

$\begin{matrix} {\tilde{r}}_{i} = (μ_{i}, ν_{i}) = IFHCA ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) = \oplus_{j = 1}^{n} ((μ (A_{σ (j)}) - μ (A_{σ (j - 1)})) {\tilde{r}}_{σ (ij)}) \\ = (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} - \prod_{j = 1}^{n} {(1 - μ_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} {(1 - μ_{σ (ij)})}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}, \\ \frac{γ \prod_{j = 1}^{n} ν_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - ν_{σ (ij)}))}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})} + (γ - 1) \prod_{j = 1}^{n} ν_{σ (ij)}^{μ (A_{σ (j)}) - μ (A_{σ (j - 1)})}}), i = 1, 2, \dots, m . \end{matrix}$ (16) to derive the collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i, where ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weighting vector of the attributes and w = (w₁, w₂, ⋯ , w_n) ^T is the associated vector of the IFHHA operator, such that w_j > 0 and $\sum_{j = 1}^{n} w_{j} = 1$ .

Step 2. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2,$ ⋯ , m) of the collective overall values ${\tilde{r}}_{i} (i = 1,$ 2, ⋯ , m) to rank all the alternatives A_i (i = 1, 2, ⋯ , m) and then to select the best one(s).

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ .

Step 4. End.

5 Numerical example

Scientific and technological progress not only led to the rapid development of all aspects of human society, also sharpen the conflicts between human beings and their living environments, in particular, as the escalation of the conflicts, a new way of think, ecology and ecologic philosophy was born, from which the research touches the education, pedagogic ecology was gestated and derived. It supplies a new eye-view for school education and theories and practices of school physical education. The balance of pedagogic ecology decides the healthy development in school physical education. In recent years, the physical constitution of students in elementary and middle schools declines, school physical education is being edged out, etc. It shows that school physical education ecology is out of balance or even a serious imbalance in the evolution process. Teaching is the main task in universities. The management of teaching effectiveness must be strengthened in order to effectively train person with ability, while it is indispensable to teaching effectiveness evaluation in process of teaching effectiveness management. Teaching effectiveness evaluation index system has been mainly researched in the paper, in order to provide a more scientific and reasonable teaching effectiveness evaluation system for universities. The author analyzed and defined some important concepts correlating with teaching effectiveness and teaching effectiveness evaluation. The teaching effectiveness was defined as increment in knowledge, ability, values, and as the degree to meet the customer and relative party’s demand. It is a result of synthesizing function of whole teaching system of the university. At the same time, the author pointed out that teaching effectiveness is composing of process quality, outcome quality, and conditional quality. Among them, the conditional quality is the basic of teaching effectiveness; process quality is that teachers effectively instruct student with knowledge, while the outcome quality of teaching process is named as outcome quality of teaching. With the new period of developing the socialist modernization, higher education with high quality meets a challenge in the new century. The core task of higher education is teaching. Improving the teaching effectiveness is the key element for improving the reform and developing, establishing the scientific evaluation system for teaching is the important way to strengthen the management and get the higher quality. The teaching effectiveness is the base for the existing and developing of higher education, it is also the inevitable request of internationalism. The teaching effectiveness is reflected by the quality of respective departments whose quality is reflected by respective courses whose quality is reflected by the teachers. In this section, we present an empirical case study of evaluating the management performance of the sports sites. The project’s aim is to evaluate the best sports sites from the different management performance of the sports sites. The management performance of five possible sports sites A_i (i = 1, 2, 3, 4, 5) is evaluated according to the following four attributes: G₁ is the number of sports sites; G₂ is the distribution of the sports sites; G₃ is the investment and financing situation of sports sites; G₄ is the use of the sports sites. The five possible sports sites A_i (i = 1, 2, 3, 4, 5) are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above four attributes, as listed in the following matrix. $\tilde{R} = [\begin{matrix} (0.2, 0.7) & (0.1, 0.8) & (0.4, 0.5) & (0.5, 0.4) \\ (0.6, 0.3) & (0.3, 0.6) & (0.6, 0.4) & (0.6, 0.3) \\ (0.4, 0.4) & (0.5, 0.4) & (0.5, 0.5) & (0.4, 0.5) \\ (0.2, 0.5) & (0.1, 0.7) & (0.7, 0.2) & (0.5, 0.4) \\ (0.6, 0.2) & (0.4, 0.4) & (0.5, 0.3) & (0.3, 0.4) \end{matrix}]$

In the following, we apply the IFHCA operator to multiple attribute decision making for evaluating the management performance of the sports sites with intuitionistic fuzzy information. The method involves the following steps:

Step 1. Suppose the fuzzy measure of attribute of G_j (j = 1, 2, ⋯ , n) and attribute sets of G as follows: $\begin{matrix} μ (G_{1}) = 0.20, μ (G_{2}) = 0.25 \\ μ (G_{3}) = 0.30, μ (G_{4}) = 0.28 \\ μ (G_{1}, G_{2}) = 0.60, μ (G_{1}, G_{3}) = 0.55 \\ μ (G_{1}, G_{4}) = 0.65, μ (G_{2}, G_{3}) = 0.60 \\ μ (G_{2}, G_{4}) = 0.55, μ (G_{3}, G_{4}) = 0.62 \\ μ (G_{1}, G_{2}, G_{3}) = 0.78, μ (G_{1}, G_{2}, G_{4}) = 0.80 \\ μ (G_{1}, G_{3}, G_{4}) = 0.85, μ (G_{2}, G_{3}, G_{4}) = 0.75 \\ μ (G_{1}, G_{2}, G_{3}, G_{4}) = 1.00 \end{matrix}$

Step 2. Utilize the decision information given in matrix $\tilde{R}$ , and the IFHCA operator to derive the collective overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the sports sites A_i. $\begin{matrix} {\tilde{r}}_{1} = (0.3525, 0.5445), {\tilde{r}}_{2} = (0.5725, 0.3494) \\ {\tilde{r}}_{3} = (0.4384, 0.4547), {\tilde{r}}_{4} = (0.4530, 0.3868) \\ {\tilde{r}}_{5} = (0.4666, 0.3015) \end{matrix}$

Step 3. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2,$ ⋯ , 5) of the overall intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ $\begin{matrix} S ({\tilde{r}}_{1}) = - 0.1920, S ({\tilde{r}}_{2}) = 0.2230 \\ S ({\tilde{r}}_{3}) = - 0.0163, S ({\tilde{r}}_{4}) = 0.0662 \\ S ({\tilde{r}}_{5}) = 0.1651 \end{matrix}$ (17)Step 4. Rank all the sports sites A_i (i = 1, 2, 3, 4, 5) in accordance with the scores $S ({\tilde{r}}_{i})$ (i = 1, 2, ⋯ , 5) of the overall preference values ${\tilde{r}}_{i} (i =$ 1, 2, ⋯ , 5): A₂ ≻ A₅ ≻ A₃ ≻ A₄ ≻ A₁, and thus the most desirable sports site is A₂.

6 Conclusion

Our country’s education got big development in these days. For its future’s more durative and healthy development, the research on all educational resources in our country’s education is recognized. The sports sites of National Education System which is one of the most important educational resources are the key ensurance and material basis for the smooth development of our country’s school sport. The data of Fifth National Sport-Site Investigation show the sports sites of National Education System take percentage of 65.6 in all of our country’s sports sites. As a result, the development situation of sports sites in National Education System not only has direct effect on our country’s school sport, but also on our country’s other sports career. In this paper, we first introduce some operations on the intuitionistic fuzzy sets, such as Hamacher sum, Hamacher product, Hamacher exponentiation, etc., and further develop the intuitionistic fuzzy Hamacher correlated averaging (IFHCA) operator. The prominent characteristic of the operators is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have applied the IFHCA operators to multiple attribute decision making with intuitionistic fuzzy information. Finally, a practical example for evaluating the management performance of the sports sites is used to illustrate the developed procedures. In the future, we shall continue working in the extension and application of the developed operators to other domains[36 –55].

Footnotes

Acknowledgment

The work was supported by the Humanities and Social Sciences Foundation Project of Anhui Education Commission of the People’s Republic of China (No. SK2015ZD001).

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