Keywords

1 Introduction

Fuzziness is the imperfection and skepticality, if the limits and margins of the set under the concern is not well defined, i.e., whether the element belongs to or does not belong to a set. The incipient approaches and theories treating imprecision and dubiousness were introduced by Zadeh [1]. Grzegorzewski [2] discussed the promising and indispensable inclusion of IFS followed [3] who introduced the concept of intuitionistic fuzzy set (IFS). Mishra et al. [4, 5] proposed exponential and trigonometric intuitionistic fuzzy information measures and applied these to assessment of service quality and rating of township development. Castineira [6] recognized the incompatibility measure in the framework of the IFS of Atanassov in the form of axioms.

While Liu and Wang [7] introduced incipient methods to solve multi-criteria decision-making quandaries in an intuitionistic fuzzy environment. An approach in group decision-making was developed by Xu [8, 9] on intuitionistic preference relations and also proposed clustering algorithm for IFS. The distance between IFS was investigated by Szmidt and Kacprzyk [10], Wang and Xin [11], and many other researchers further studied the distance measures in IFS and applied it to many real-life problems of pattern recognition.

The degree of homogeneous attribute or similarity of fuzzy sets is known as similarity measures. It is a consequential concept in the fuzzy set theory. It was introduced by Wang [12] who also provided computation formula. Thereafter it occupied the main focus of the various research findings. Many investigations have been done and extensively applied in fuzzy clustering, image processing, fuzzy reasoning, and fuzzy neural networks. Comparative study of similarity measures was conducted by Wang et al. [13]. Many studies fixate on the similarity measure and the information measure for IFS [11, 14] and the relationship between them, particularly, on the organized conversion of the information measure to the similarity measure and similarity measure to information measure for IFSs.

In today’s scenario, it is common to make decision in the presence of multiple conflicting criteria, which is known as multiple criteria (or attribute) decision-making (MCDM/MADM) quandaries. For example, if one has to buy a refrigerator, he has to look at many parameters like prize, capacity, safety, power consumption, technology, etc. A decision in multi-criteria decision-making quandaries can be taken by selecting the best suitable alternatives based on quantitative and qualitative assessment. Required option can be chosen by providing the preference set in the terms of numerical value or interval. But, in real-life situation the data set or preference values are not always precise, it can be vague or imprecise, which can be great dispute for the decision taking authorities and hence was a leading factor toward MCDM techniques in fuzzy scenarios.

The technique for ordering the preferable option by similarity to an ideal elucidation (TOPSIS) was developed by Hwang and Yoon [15]. In this technique, the concept is that the chosen option should be the closest from the positive ideal elucidation and the farthest from the negative ideal elucidation. Joshi and Kumar [16] proposed intuitionistic fuzzy information and distance measure-based TOPSIS technique for multi-criteria decision-making (MCDM). Likewise various authors introduced IFS with TOPSIS to give a hybrid technique for MCDM problems. Some methods were based on IFS using entropy weights and linear programming given by Liu and Wang [7, 17, 18].

The present communication is arranged as follows: Sect. 2 introduces the fundamental conceptions of fuzzy sets, intuitionistic fuzzy sets, intuitionistic fuzzy aggregation operator, and intuitionistic fuzzy information and similarity measures, followed by the study of new similarity and intuitionistic fuzzy information measures and their validity in Sect. 3. In Sect. 4, empirical illustration is listed, and comparison is tabulated between new and existing intuitionistic fuzzy information measures, followed by the application of the proposed intuitionistic fuzzy information measure in physical education teaching assessment in Sect. 5.

2 Prerequisite

In this section, authors discuss some fundamental conceptions of fuzzy sets and intuitionistic fuzzy sets, and information and similarity measures for IFSs. Throughout this paper, FSs and IFSs represent the fuzzy sets and intuitionistic fuzzy sets in \( X, \) respectively.

Definition 2.1

([1]) Let \( X = \left\{ {x_{1} ,{\kern 1pt} x_{2} , \ldots ,x_{n} } \right\} \) be a finite universe of discourse and \( \tilde{A} \subset X. \) Then \( \tilde{A} \) is a fuzzy set defined by

$$ {\kern 1pt} {\kern 1pt} {\kern 1pt} \tilde{A} = \left\{ {\left( {x_{i} ,\mu_{{\tilde{A}}} \left( {x_{i} } \right)} \right):\mu_{{\tilde{A}}} \left( {x_{i} } \right) \in \left[ {0,1} \right];\forall \,x_{i} \in X} \right\}, $$
(1)

where \( \mu_{{\tilde{A}}} (x_{i} ) \) is the membership function \( \mu_{{\tilde{A}}} (x_{i} ):X \to [0,1] \) such that

$$ \mu_{{\tilde{A}}} \left( {x_{i} } \right) = \left\{ {\begin{array}{*{20}l} {0,\;{\text{if}}\,x_{i} \notin \tilde{A}\,{\text{and}}\,{\text{there}}\,{\text{is}}\,{\text{no}}\,{\text{ambiguity}}} \hfill \\ {1,\;{\text{if}}\,x_{i} \in \tilde{A}\,{\text{and}}\,{\text{there}}\,{\text{is}}\,{\text{no}}\,{\text{ambiguity}}} \hfill \\ {0.5,\;{\text{there}}\,{\text{is}}\,{\text{maximum}}{\kern 1pt} \,{\text{ambiguity}}\,{\text{whether}}\,x_{i} \notin \tilde{A}\,{\text{or}}\,x_{i} \in \tilde{A}.} \hfill \\ \end{array} } \right.{\kern 1pt} $$

Definition 2.2

([3]) The intuitionistic fuzzy set \( A \subset X = \left\{ {x_{1} ,x_{2} , \ldots ,x_{n} } \right\} \) is defined as

$$ A = \left\{ {\left\langle {x,{\kern 1pt} \mu_{A} (x),\nu_{A} (x)} \right\rangle :x \in X} \right\}, $$
(2)

where \( \mu_{A} :X \to [0,1] \) is the degree of membership and \( \nu_{A} :X \to [0,1] \) is the degree of nonmembership of \( x \in X \) in \( A, \) respectively, such that

$$ 0 \le \mu_{A} (x) + \nu_{A} (x) \le 1,\forall x \in X. $$

The intuitionistic index (or hesitancy degree) of an element \( x \in X \) in \( A \) is as

$$ \pi_{A} (x) = 1 - \mu_{A} (x) - \nu_{A} (x). $$

It implies \( 0 \le \pi_{A} (x) \le 1,\forall x \in X \) [3].

If \( \pi_{A} (x) = 0 \) then IFSs can formed FSs, i.e., \( A = \left\{ {\left\langle {x,\mu_{A} (x),1 - \mu_{A} (x)} \right\rangle {\kern 1pt} :x \in X} \right\}, \) with \( \pi_{A} (x) = 0,\forall x \in X. \)

The complement set of \( A \) is \( A^{c} \) and defined as

$$ A^{c} = \left\{ {\left\langle {x,\nu_{A} (x),\mu_{A} (x)} \right\rangle :x \in X} \right\}. $$
(3)

[19] represents \( \alpha = \left\langle {a,{\kern 1pt} {\kern 1pt} b} \right\rangle \) an intuitionistic fuzzy number (IFN), if \( 0 \le a + b \le 1. \)

Definition 2.3

Let \( \alpha_{1} = \left\langle {a_{1} ,{\kern 1pt} {\kern 1pt} b_{1} } \right\rangle \) and \( \alpha_{2} = \left\langle {a_{2} ,b_{2} } \right\rangle \) be two IFNs, three intuitionistic fuzzy aggregation operators are defined by

$$ \begin{aligned} \alpha_{1} \oplus \alpha_{2} & = \left\langle {a_{1} + a_{2} - a_{1} a_{2} ,{\kern 1pt} {\kern 1pt} b_{1} b_{2} } \right\rangle ,\alpha_{1} \otimes \alpha_{2} = \left\langle {a_{1} a_{2} ,b_{1} + b_{2} - {\kern 1pt} b_{1} b_{2} } \right\rangle \;{\text{and}} \\ w\alpha_{1} & = \left\langle {1 - \left( {1 - a_{1} } \right)^{w} ,b_{1}^{w} } \right\rangle ,w > 0. \\ \end{aligned} $$
(4)

Definition 2.4

([20]) Let \( h:IFS(X) \to [0,1] \) be an information measure defined on IFSs. Then it is valid, if it holds the given postulate:

  • (P1). \( h(A) = \) (minimum) \( \Leftrightarrow \) \( A \) is a crisp set;

  • (P2). \( h(A) = 1 \) (maximum) \( \Leftrightarrow \) \( \mu_{A} (x_{i} ) = \nu_{A} (x_{i} ) \) for any \( x_{i} \in X; \)

  • (P3). \( h(A) \le h(B) \) and if \( A \) is less fuzzy than \( B \), i.e., \( \mu_{A} (x_{i} ) \le \mu_{B} (x_{i} ) \) and \( \nu_{A} (x_{i} ) \ge \nu_{B} (x_{i} ) \) for \( \mu_{B} (x_{i} ) \le \nu_{B} (x_{i} ) \) or \( \mu_{A} (x_{i} ) \ge \mu_{B} (x_{i} ) \) and \( \nu_{A} (x_{i} ) \le \nu_{B} (x_{i} ) \) for \( \mu_{B} (x_{i} ) \ge \nu_{B} (x_{i} ) \) for any \( x_{i} \in X; \)

  • (P4). \( h(A) = h(A^{c} ) \).

Definition 2.5

The function \( Sim:IFS(X) \times IFS(X) \to [0,1] \) is said to be a valid measure of similarity on \( IFS(X), \) if \( Sim \) holds

  • (S1). \( Sim{\kern 1pt} (A,A^{c} ) = 0 \) if \( A \) is a crisp set;

  • (S2). \( Sim{\kern 1pt} (A,B) = 1, \) iff \( A = B; \)

  • (S3). \( Sim{\kern 1pt} (A,B) = Sim{\kern 1pt} (B,A); \)

  • (S4). \( Sim{\kern 1pt} (A,C) \le Sim{\kern 1pt} (A,B) \) and \( Sim{\kern 1pt} (A,C) \le Sim{\kern 1pt} (A,B), \) For all \( A,B,C \in IFS{\kern 1pt} (X) \) and \( A \, \subseteq \, B \, \subseteq \, C. \)

3 Similarity and Information Measures for IFSs

Here, similarity and information measures for IFSs are proposed and their validity are proved.

Let \( A = \left\{ {\left( {x,\mu_{A} (x),\nu_{A} (x)} \right):x \in X} \right\} \) and \( B = \left\{ {\left( {x,\mu_{B} (x),\nu_{B} (x)} \right):x \in X} \right\} \) be two IFSs in \( X \) defines

$$ Sim{\kern 1pt} (A,B) = 1 - \frac{1}{n}\sum\limits_{i = 1}^{n} \sin \left[ {\frac{{\left\{ {\hbox{max} \left( {\left| {\mu_{A} (x_{i} ) - \mu_{B} (x_{i} )} \right|,{\kern 1pt} \left| {\nu_{A} (x_{i} ) - \nu_{B} (x_{i} )} \right|} \right)} \right\}\pi }}{2}} \right]. $$
(5)

Theorem 1

The mapping \( Sim, \) defined by (5) is a similarity measure for IFSs.

Proof

It is obvious that \( Sim{\kern 1pt} (A,{\kern 1pt} B) \) holds the postulate (S1)–(S3). Hence only (S4) has to be proved. For \( A \subset B \subset C, \) we have \( \mu_{A} (x_{i} ) \le \mu_{B} (x_{i} ) \le \mu_{C} (x_{i} ), \) and \( \nu_{A} (x_{i} ) \ge \nu_{B} (x_{i} ) \ge \nu_{C} (x_{i} ),\forall {\kern 1pt} x_{i} \in X. \) It implies

$$ \begin{aligned} \hbox{max} & \left( {\left| {\mu_{A} (x_{i} ) - \mu_{B} (x_{i} )} \right|,\left| {\nu_{A} (x_{i} ) - \nu_{B} (x_{i} )} \right|} \right) \\ & \le \hbox{max} \left( {\left| {\mu_{A} (x_{i} ) - \mu_{C} (x_{i} )} \right|,{\kern 1pt} \left| {\nu_{A} (x_{i} ) - \nu_{C} (x_{i} )} \right|} \right),\forall {\kern 1pt} x_{i} \in X. \\ \end{aligned} $$
$$ \begin{aligned} \sin & \left[ {\frac{{\left\{ {\hbox{max} \left( {\left| {\mu_{A} (x_{i} ) - \mu_{B} (x_{i} )} \right|,\left| {\nu_{A} (x_{i} ) - \nu_{B} (x_{i} )} \right|} \right)} \right\}\pi }}{2}} \right]{\kern 1pt} \\ & {\kern 1pt} \le \,\sin \left[ {\frac{{\left\{ {\hbox{max} \left( {\left| {\mu_{A} (x_{i} ) - \mu_{C} (x_{i} )} \right|,{\kern 1pt} \left| {\nu_{A} (x_{i} ) - \nu_{C} (x_{i} )} \right|} \right)} \right\}\pi }}{2}} \right],\forall \,x_{i} \in X. \\ \end{aligned} $$

These imply

$$ \begin{aligned} 1 & - \frac{1}{n}\sum\limits_{i = 1}^{n} \sin \left[ {\frac{{\left\{ {\hbox{max} \left( {\left| {\mu_{A} (x_{i} ) - \mu_{B} (x_{i} )} \right|,{\kern 1pt} \left| {\nu_{A} (x_{i} ) - \nu_{B} (x_{i} )} \right|} \right)} \right\}\pi }}{2}} \right] \\ {\kern 1pt} & \ge 1 - \frac{1}{n}\sum\limits_{i = 1}^{n} \sin \left[ {\frac{{\left\{ {\hbox{max} \left( {\left| {\mu_{A} (x_{i} ) - \mu_{C} (x_{i} )} \right|,{\kern 1pt} \left| {\nu_{A} (x_{i} ) - \nu_{C} (x_{i} )} \right|} \right)} \right\}\pi }}{2}} \right],\forall x_{i} \in X. \\ \end{aligned} $$

This means that

$$ Sim{\kern 1pt} (A,B) \ge Sim(A,C). $$
(6)

Similarly,

$$ Sim{\kern 1pt} (A,B) \ge Sim{\kern 1pt} (A,C). $$
(7)

Let \( A = \left\{ {\left( {x,\mu_{A} (x),\nu_{A} (x)} \right):x \in X} \right\} \) be an IFS in \( X \) defines

$$ h(A) = \frac{1}{n}\sum\limits_{i = 1}^{n} \sin \left\{ {\left( {\frac{{\mu_{A} (x_{i} ) \wedge \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \vee \nu_{A} (x_{i} )}}} \right)\frac{\pi }{2}} \right\}. $$
(8)

Theorem 2

The mapping \( h, \) defined by (8) is information measure on IFSs.

Proof

Let \( h_{i} (A) = \sin \left\{ {\left( {\frac{{\mu_{A} (x_{i} ) \, \wedge \, \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \, \vee \, \nu_{A} (x_{i} )}}} \right)\frac{\pi }{2}} \right\}, \) for \( i = 1,2, \ldots ,n. \) From \( 0 \le \mu_{A} (x_{i} ) \le 1, \) \( 0 \le \nu_{A} (x_{i} ) \le 1, \) we have \( 0 \le \left( {\frac{{\mu_{A} (x_{i} ) \, \wedge \, \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \, \vee \, \nu_{A} (x_{i} )}}} \right)\frac{\pi }{2} \le \frac{\pi }{2}. \) Thus \( 0 \le h_{i} (A) \le 1. \)

(P1). For \( \mu_{A} (x_{i} ) = 0,{\kern 1pt} {\kern 1pt} \nu_{A} (x_{i} ) = 1 \) or \( \mu_{A} (x_{i} ) = 1,{\kern 1pt} {\kern 1pt} \nu_{A} (x_{i} ) = 0 \) for any \( x_{i} \in X. \) It implies \( h_{i} (A) = 0. \) Hence \( h{\kern 1pt} (A) = 0. \)

Conversely, suppose that \( h{\kern 1pt} (A) = 0 \) for \( i = 1,2, \ldots ,n. \) Since \( 0 \le h_{i} (A) \le 1, \) therefore, it follows that \( h_{i} (A) = 0. \) Also since \( 0 \le \left( {\frac{{\mu_{A} (x_{i} ) \, \wedge \, \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \, \vee \, \nu_{A} (x_{i} )}}} \right)\frac{\pi }{2} \le \frac{\pi }{2}, \) we have

$$ \left( {\frac{{\mu_{A} (x_{i} ) \wedge \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \vee \nu_{A} (x_{i} )}}} \right)\frac{\pi }{2} = 0\;{\text{or}}\;\left( {\frac{{\mu_{A} (x_{i} ) \wedge \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \vee \nu_{A} (x_{i} )}}} \right)\frac{\pi }{2} = \frac{\pi }{2}. $$

Hence \( A \) is a crisp set.

(P2). Let \( \mu_{A} (x_{i} ) = \nu_{A} (x_{i} ) \) for each \( x_{i} \in X. \) Applying this condition to (8), we easily obtain \( h{\kern 1pt} (A) = 1. \)

Conversely, suppose that \( h{\kern 1pt} (A) = 1. \) From (8) and \( 0 \le h_{i} (A) \le 1, \) we obtain that \( h_{i} (A) = 1 \) for each \( x_{i} \in X. \) Also from \( 0 \le \left( {\frac{{\mu_{A} (x_{i} ) \, \wedge \, \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \, \, \vee \,\nu_{A} (x_{i} )}}} \right)\frac{\pi }{2} \le \frac{\pi }{2}, \) we have \( \mu_{A} (x_{i} ) = \nu_{A} (x_{i} ) \) for each \( x_{i} \in X. \)

(P3). Let \( \mu_{A} (x_{i} ) \ge \mu_{B} (x_{i} ) \) and \( \nu_{B} (x_{i} ) \ge \nu_{A} (x_{i} ) \) for \( \mu_{B} (x_{i} ) \ge \nu_{B} (x_{i} ), \) i.e., \( \mu_{A} (x_{i} ) \ge \mu_{B} (x_{i} ) \ge \nu_{B} (x_{i} ) \ge \nu_{A} (x_{i} ). \) Thus

$$ \left( {\frac{{\mu_{A} (x_{i} ) \wedge \nu_{A} (x_{i} )}}{{\mu_{A} (x_{i} ) \vee \nu_{A} (x_{i} )}}} \right)\frac{\pi }{2} \le \left( {\frac{{\mu_{B} (x_{i} ) \wedge \nu_{B} (x_{i} )}}{{\mu_{B} (x_{i} ) \vee \nu_{B} (x_{i} )}}} \right)\frac{\pi }{2}. $$
(9)

It follows that \( h_{i} (A) \le h_{i} (B). \)

Similarly, when \( \mu_{A} (x_{i} ) \le \mu_{B} (x_{i} ) \) and \( \nu_{B} (x_{i} ) \le \nu_{A} (x_{i} ) \) for \( \mu_{B} (x_{i} ) \le \nu_{B} (x_{i} ),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{i.e.}}, \) \( \mu_{A} (x_{i} ) \le \mu_{B} (x_{i} ) \le \nu_{B} (x_{i} ) \le \nu_{A} (x_{i} ). \) we can also prove that \( h_{i} (A) \le h_{i} (B). \)

Hence we have

$$ h(A) \le h(B). $$
(10)

(P4). For \( A^{c} = \left\{ {\left( {x_{i} ,\nu_{A} (x_{i} ),\mu_{A} (x_{i} )} \right):x_{i} \in X} \right\}, \) we can easily get that

$$ h(A^{c} ) = h(A). $$
(11)

Hence \( h(A) \) is valid intuitionistic fuzzy information measure.

4 Numerical Comparisons

In present section, the efficiency of information measure through comparisons with some existing information measures for IFSs is demonstrated.

Example 1

Let

$$ \begin{aligned} A_{1} & = \left\{ {\left( {x,0.2,0.4} \right):x \in X} \right\},A_{2} = \left\{ {\left( {x,0.3,0.4} \right):x \in X} \right\}, \\ A_{3} & = \left\{ {\left( {x,0.4,0.5} \right):x \in X} \right\},A_{4} = \left\{ {\left( {x,0.5,0.5} \right):x \in X} \right\}. \\ \end{aligned} $$

We first recall these information measures.

Vlachos and Sergiagis [21] proposed an entropy measure \( E_{VS} \) according to cross entropy measure:

$$ \begin{aligned} E_{VS} = & \, - \frac{1}{n\,\ln 2}\sum\limits_{i = 1}^{n} \left[ {\mu_{A} (x_{i} )\ln \mu_{A} (x_{i} ) + \nu_{A} (x_{i} )\ln \nu_{A} (x_{i} )} \right.{\kern 1pt} \\ & \left. { - \,(1 - \pi_{A} (x_{i} ))\ln (1 - \pi_{A} (x_{i} )) - \pi_{A} (x_{i} )\ln 2} \right]. \\ \end{aligned} $$
(12)

Wei and Zhang [22] developed information measure based on cosine function for IFSs:

$$ E_{WZ} (A) = \frac{1}{n}\sum\limits_{i = 1}^{n} \cos \left\{ {\frac{{\left( {\mu_{A} (x_{i} ) - \nu_{A} (x_{i} )} \right)\pi }}{{2\left( {1 + \pi_{A} (x_{i} )} \right)}}} \right\}. $$
(13)

These two information measures satisfy the set of requirements in Definition 2.4.

From Table 1, it can be seen that \( h \) and \( E_{VS} , \) the nearer the degree of membership and nonmembership, the higher the information measure. Particularly, when \( \mu_{A} (x_{i} ) = \nu_{A} (x_{i} ) \), the information measure reaches its maximum value. The results obtained by using the information measures \( h \) and \( E_{VS} \) are in accordance with the intuition.

Table 1 Comparisons with existing entropies
Full size table

Table 1 shows that \( h(A_{1} ) < h(A_{2} ) < h(A_{3} ) < h(A_{4} ) \) and \( E_{VS} (A_{1} ) < E_{VS} (A_{2} ) < E_{VS} (A_{3} ) < E_{VS} (A_{4} ), \) which are consistent with the intuition. But, from the above table it is visible that \( E_{WZ} (A_{1} ) < E_{WZ} (A_{2} ) > E_{WZ} (A_{3} ) < E_{WZ} (A_{4} ). \) It implies that the information measure \( E_{WZ} \) is not in accordance with the intuition. And hence our measure is more efficient and authentic than the information measure \( E_{WZ} . \)

5 Physical Education Teaching Quality Assessment

So far, researchers concentrated on the empirical as well as theoretical evidences for the reform and development of the teaching quality assessment to deal with confounding quandaries of expansion and quality improvement of physical education. The association of teaching quality system and the proposal for the fundamental supervision is favorable to the growth and the suitable progress of the undergraduate physical education majors [23, 24].

Xu [19, 25] dealt with triangular intuitionistic fuzzy information measure in the multi-criteria attribute decision-making quandaries, while solving the problems of physical education teaching quality in the universities and colleges. Here, in this section, we develop an approach on intuitionistic fuzzy information measure to evaluate the physical education teaching quality in the universities and colleges. The developed IFMWAO method is as follows:

  1. Step 1.

    Let \( C = \left\{ {C_{1} ,{\kern 1pt} C_{2} , \ldots ,C_{m} } \right\} \) be a discrete set of colleges of a district region, and \( L = \left\{ {L_{1} ,{\kern 1pt} L_{2} , \ldots ,L_{n} } \right\} \) be the set of options. Let \( \tilde{D} = \left( {\tilde{r}_{i\,j} } \right)_{m \times n} = \left( {\mu_{{i{\kern 1pt} j}} ,\nu_{{i{\kern 1pt} j}} } \right)_{{m{\kern 1pt} \times n}} \) be the intuitionistic fuzzy judgment matrix, where \( \mu_{{i{\kern 1pt} j}} \) represents the extend up to which \( C_{i} \) holds \( L_{j} \) and \( \nu_{{i{\kern 1pt} j}} \) reflects \( C_{i} \) does not support the option \( L_{j} . \)

  2. Step 2.

    By using information measure formula (8), we can easily compute the information of each intuitionistic fuzzy value in the judgment matrix and obtain the information matrix of this judgment matrix as \( D = \left( {h_{i\,j} } \right)_{m \times n} , \) where \( h_{i\,j} = E\left( {\tilde{r}_{{i{\kern 1pt} j}} } \right). \)

  3. Step 3.

    Normalize the information values in the above decision matrix by the following equation

    $$ \overline{h}_{i\,j} = \frac{{h_{i\,j} }}{{\hbox{max} \,h_{i\,j} }},j = 1(1)n;i = 1(1)m. $$
    (14)

    And expressed it as \( \overline{D} = \left( {\overline{h}_{{i{\kern 1pt} j}} } \right)_{{m{\kern 1pt} \times {\kern 1pt} n}} . \)

  4. Step 4.

    Compute weight vector by applying the given formula

    $$ w_{i} = \frac{{1 - \sum\nolimits_{j = 1}^{n} {\overline{h}_{{i{\kern 1pt} j}} } }}{{m - \sum\nolimits_{i = 1}^{m} {\sum\nolimits_{j = 1}^{n} {\overline{h}_{{i{\kern 1pt} j}} } } }},i = 1(1)m. $$
    (15)
  5. Step 5.

    Chen and Tan [26] defines the score function \( \delta \) of x as

    $$ \delta (x) = \mu_{A} (x) - \nu_{A} (x), $$
    (16)

    where \( \delta (x) \in [ - 1,{\kern 1pt} 1]. \) For higher values of deviation \( \delta (x) \) the membership value \( \mu_{A} (x) \) should be high, while the nonmembership value should be low. The accuracy function h defined by Hong and Choi [27] to compute the degree of precision of the intuitionistic fuzzy value is

    $$ \eta (x) = \mu_{A} (x) + \nu_{A} (x), $$
    (17)

    where \( \eta (x) \in [0,1]. \) The greater value of \( \eta (x) \) results in the superior degree of membership of IFS. Let \( \alpha_{1} = (\mu_{1} ,\nu_{1} ),{\kern 1pt} \) \( \alpha_{2} = (\mu_{2} ,\nu_{2} ) \) be two intuitionistic fuzzy values, then (Fig. 1)

    Fig. 1
    figure 1

    Implementation flowchart of IFMWAO method

    Full size image
    1. 1.

      for \( \delta (\alpha_{1} ) < \delta (\alpha_{2} ), \) we obtain \( \alpha_{1} < \alpha_{2} \);

    2. 2.

      for \( \delta (\alpha_{1} ) = \delta (\alpha_{2} ), \) we get if \( \eta (\alpha_{1} ) < \eta (\alpha_{2} ), \) then \( \alpha_{1} < \alpha_{2} \); otherwise if \( \eta (\alpha_{1} ) = \eta (\alpha_{2} ), \) then \( \alpha_{1} = \alpha_{2} . \)

  6. Step 6.

    Wu and Zhang [28] proposed the intuitionistic fuzzy weighted averaging operator as

    $$ IFWA_{\omega } (\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} ) = \left( {1 - \prod\limits_{i = 1}^{n} \left( {1 - \mu_{i} } \right)^{{\omega_{i} }} ,{\kern 1pt} \prod\limits_{i = 1}^{n} \nu_{i}^{{\omega_{i} }} } \right), $$
    (18)

    where \( \omega_{i} \) is the weight of \( \alpha_{i} , \) \( \omega_{i} \in [0,1] \) and \( \sum\nolimits_{i = 1}^{n} \omega_{i} = 1. \)

5.1 A Real Case Study

The above-defined method is illustrated mathematically in this subsection. Let there be a panel of five colleges of a district region \( C_{j} \left( {j = 1,2, \ldots ,5} \right) \). The education institution has to take a decision based on the following four options: \( L_{1} \) is the sports basic theoretical knowledge evaluation; \( L_{2} \) is the physical fitness and sports skill evaluation; \( L_{3} \) is the learning attitude evaluation; \( L_{4} \) is the effective expression. The ranking of five colleges of a district region \( C_{j} \left( {j = 1,2, \ldots ,5} \right) \) are calculated by the triangular intuitionistic fuzzy set on the above four options, and the decision matrix is constructed as (Table 2)

  1. Step 1.

    The information measure is computed using formula (8) in the above judgment matrix and get the following information matrix:

    $$ D = \left( {h_{i\,j} } \right)_{{4{\kern 1pt} \times 5}} = \left[ {\begin{array}{*{20}c} {0.7071} & {0.8090} & {0.7071} & {0.4343} & {0.6228} \\ {0.9511} & {0.7071} & {0.5090} & {0.8090} & {0.8655} \\ {0.6228} & {0.9511} & {0.5090} & {1.0000} & {0.8655} \\ {0.8090} & {1.0000} & {0.9511} & {0.7071} & {0.8090} \\ \end{array} } \right]. $$
  2. Step 2.

    Using (14), the above information matrix is transformed to the normalized information matrix as

    $$ \overline{D} = \left( {\overline{h}_{{i{\kern 1pt} j}} } \right)_{{4{\kern 1pt} \times 5}} = \left[ {\begin{array}{*{20}c} {0.7840} & {1.0000} & {0.8740} & {0.5386} & {0.7698} \\ {1.0000} & {0.7435} & {0.5352} & {0.8506} & {0.9100} \\ {0.6228} & {0.9511} & {0.5090} & {1.0000} & {0.8655} \\ {0.8090} & {1.0000} & {0.9511} & {0.7071} & {0.8090} \\ \end{array} } \right]. $$
  3. Step 3.

    Compute weight vector by utilizing the normalized information matrix and by the formula (14), we have

    $$ w_{i} = \frac{{1 - \sum\nolimits_{j = 1}^{n} {\overline{h}_{{i{\kern 1pt} j}} } }}{{m - \sum\nolimits_{i = 1}^{m} {\sum\nolimits_{j = 1}^{n} {\overline{h}_{{i{\kern 1pt} j}} } } }},i = 1(1)m. $$

    Hence, obtained as \( W = \left( {0.2455,{\kern 1pt} 0.2475,0.2401,0.2669} \right). \)

  4. Step 4.

    Obtain score function

    By applying \( IFWA_{\omega } \) operator (18), the set of intuitionistic fuzzy values \( \alpha_{j} = (\mu_{j} ,\nu_{j} ) \), where \( j = 1(1)n, \) is obtained as

    $$ \begin{aligned} \alpha_{1} & = \left( {0.5812,0.3222} \right),{\kern 1pt} \alpha_{2} = \left( {0.4770,0.3684} \right),\alpha_{3} = \left( {0.5458,0.3345} \right), \\ {\kern 1pt} \alpha_{4} & = \left( {0.4596,0.3142} \right),\alpha_{5} = \left( {0.4941,0.3816} \right). \\ \end{aligned} $$

    By (16), score function \( \delta \left( {\alpha_{j} } \right),\left( {j = 1,2,3,4,5} \right) \) is as

    $$ \delta (\alpha_{1} ) = 0.2590,\delta (\alpha_{2} ) = 0.1086,\delta (\alpha_{3} ) = 0.2113,\delta (\alpha_{4} ) = 0.1454,\delta (\alpha_{5} ) = 0.1125. $$
  5. Step 5.

    Desirable ranking

Table 2 Weighted intuitionistic fuzzy decision matrix
Full size table

The options are ranked in the descending order by comparing the score values. Here, we obtained the following ordering as \( C_{1} \succ C_{3} \succ C_{4} \succ C_{5} \succ C_{2} . \) Thus the most desirable college is \( C_{1} . \)

The ranking of these five alternatives are also shown in the following Table 3.

Table 3 Ranking order alternatives for different methods
Full size table

It is found that there is no conflict in the preference ordering of all the alternatives by intuitionistic fuzzy TOPSIS method and developed technique. There is only one conflict found in deciding the preference ordering of C 3 and C 4.

6 Conclusions

In this work, new intuitionistic fuzzy similarity and information measures are developed and analyzed the features of developed measures. Numerical results illustrate the proposed similarity and information measures are sensible and effective supplement of measures of IFSs. An intuitionistic fuzzy information method for MCDM problems has been proposed. A real case study to the rank five colleges of a district region based on four attributes using the proposed method is done. Moreover, the results of the developed technique and the already existing method are compared. Consequently, it is concluded that our method reduces the complexity, while evaluating the results and hence it is more effective and efficient.