Analytic hierarchy process based on interval type-2 intuitionistic fuzzy sets with their application to multicriteria decision making

Abstract

Based on the concepts of interval type-2 fuzzy sets and intuitionistic fuzzy sets, this paper introduces the concept of interval type-2 intuitionistic fuzzy set that provides an extra dimension to intuitionistic fuzzy sets for the first time. Different arithmetic operations on interval type-2 intuitionistic fuzzy set are defined and a defuzzification method of interval type-2 intuitionistic fuzzy sets is developed for ranking purpose. Also, existing analytic hierarchy processes are further generalized with the introduction of interval type-2 intuitionistic fuzzy analytic hierarchy process to assess the relative importance among the alternatives in a convenient way under multi criteria decision-making environment. This process is efficient enough to evaluate the differences among the alternatives more prominently than the existing method that plays an important role in multi criteria decision-making contexts. Finally, a case example on supplier selection problem is considered and is solved to illustrate the applicability and validity of the proposed methodology.

Keywords

Interval type-2 intuitionistic fuzzy set operations on fuzzy sets defuzzification processes fuzzy analytic hierarchy process multicriteria decision making

1. Introduction

Analytic hierarchy process (AHP) [30] is one of the most efficient methodology for solving different multicriteria decision making (MCDM) [16] problems due to its simple structure which can create a chance of searching and evaluating different cause and effect relationships between goals, factors, sub-factors and alternatives using hierarchical structure of the problems [44]. Most of the times, AHP [15] experiences difficulties in capturing uncertain parameters and imprecise judgment of experts. In this context, ordinary i.e., type-1 fuzzy set (T1 FS) [43] based fuzzy AHP (FAHP) [33, 38] is developed to handle these difficulties.

In 1975, Zadeh generalised the concept of T1 FS [4, 9] by introducing type-2 fuzzy set (T2 FS) [25]. But due to the computational complexities, the generalized T2 FSs are simplified using interval T2 FSs (IT2 FSs) [12, 26]. Further, FAHP method is extended by Abdullah and Najib [1] with the introduction of IT2 FAHP to solve complex decision making problems having various conflicting criteria. Kahraman et al. [21] developed IT2 FAHP method by defining a new defuzzification process to solve various MCDM problems.

Grattan-Guinness [19] pointed out that the presentation of a linguistic expression in the form of fuzzy set is not enough as it is difficult for decision makers (DMs) to determine a unique number as the membership degree of each element of the set. In fact, when a membership degree is determined for each element of a FS [7], it is assumed that there is no hesitancy for the DM in his or her preferences. However, this hesitancy is natural. Considering these phenomena, Atanassov [2, 3] introduced the notion of intuitionistic fuzzy (IF) set (IFS) [36] as a generalization of T1 FSs [43]. In addition to membership degree of each element in T1FSs [6], IFS [35] assigns a degree of non-membership to each element. The hesitancy of decision maker is considered in IFS as the difference between membership degree and non-membership degree of each element. Szmidt and Kacprzyk [37] proposed a method to aggregate the individual IF preference relations into a social fuzzy preference relation on the basis of a fuzzy linguistic quantifier. Silavi et al. [34] demonstrated the possibility of extending AHP using IFS. In recent years, the applications of IF AHP [22, 31, 40, 41] and interval valued IFS [20, 29] in the context of decision making are increased rapidly.

Again, the defuzzification of IFS [42] plays an important role in ranking of IFSs [28]. In 2011, Saneifard and Saneifard [32] introduced a method for defuzzification of fuzzy sets using Mellin’s transformation [17]. Nowadays, various defuzzification methods [18, 45] are available in the literature; but most of them failed to provide effective output in a justified way. In such circumstances, Biswas and De [5] introduced a method using Mellin’s transform. In the present study, IFS is defuzzified using the concept of probability density function.

In MCDM environments, DMs frequently suffer to describe exact situation using suitable IF linguistic variables [23]. Although, previous theories [1, 26] already established that IT2 FSs can be used as an effective tool to deal such types of inconsistencies. The main purpose of this paper is to propose a decision support system which can handle possible vagueness and uncertainties in a justified way in order to prioritize the best alternative in MCDM contexts. In this perspective, the present study makes a fusion among IT2 FS and IFS to define a new fuzzy set viz., interval type-2 IFS (IT2 IFS). Based on the newly defined arithmetic operations on triangular type IT2 IFSs, this article proposes, for the first time in the literature, FAHP method viz., interval type-2 IF (IT2 IF) AHP (IT2 IF-AHP) which is characterised by IT2 IFS to solve different MCDM [8] problems. The developed IT2 IFS with its inherent flexibility for capturing possibilistic uncertainties provides extra freedom while determining equivalent linguistic variable corresponding to criteria of MCDM problems under vague situation. Finally, the proposed IT2 IF-AHP method is applied to solve a real life decision making problem and the achieved results are compared with the FAHP and IT2 FAHP methods to investigate the validation of the proposed methodology.

2. Preliminaries

To introduce the concepts of IT2 IFS and arithmetic operations on triangular IT2 FSs some basic concepts on the variants of FSs are presented in this section.

2.1 T1 FS

In fuzzy set theory, a T1 FS [11, 14, 24] $\tilde{A}$ defined on the universe of discourse, $X$ is described as follows:

$\displaystyle\tilde{A}=\left\{\left(x,\mu_{\tilde{A}}(x)\right):x\in X,0% \leqslant\mu_{\tilde{A}}(x)\leqslant 1\right\}$

where $\mu_{\tilde{A}}:X\to[0,1]$ represents the membership function of the set $\tilde{A}$ . For any element $x$ of universe of discourse, $X$ , membership value $\mu{}_{\bar{A}}(x)$ represents the degree of belongingness of $x$ to the set $\tilde{A}.$

2.2 T2 FS

T2 FS [25] is a fuzzy set whose membership functions are itself T1 FSs. A T2 FS, $\tilde{\tilde{A}}$ , in the universe of discourse $X$ can be represented by a type-2 membership function $\mu{}_{\tilde{\tilde{A}}}(x)$ shown as follows:

$\displaystyle\tilde{\tilde{A}}=\{(\left(x,u\right)\mu_{\tilde{\tilde{A}}}\left% (x,u\right)):x\!\in\!X,u\!\in\!J_{x}\subseteq[0,1],0\leqslant\mu_{\tilde{% \tilde{A}}}\left(x,u\right)\leqslant 1\}$

where $J_{x}$ denotes asubinterval in [0, 1].

A T2 FS, $\tilde{\tilde{A}}$ , can also be represented as

$\displaystyle\tilde{\tilde{A}}=\int_{x\in X}\int_{u\in J_{x}}{\mu_{\tilde{% \tilde{A}}}\left(x,u\right)/(x,u)}$

where $J_{x}\subseteq[0,1]$ .

2.3 IT2 FS

Let $\tilde{\tilde{A}}$ be a T2 FS, [39] defined on the universe of discourse $X$ , represented by the type-2 membership function $\mu_{\tilde{\tilde{A}}}.$ If $\mu_{\tilde{\tilde{A}}}\left(x,u\right)=1$ is considered for each pair $\left(x,u\right)$ , then $\tilde{\tilde{A}}$ is called an IT2 FS. It is represented by

$\displaystyle\tilde{\tilde{A}}=\int_{x\in X}\int_{u\in J_{x}}{1/(x,u)}$

where $J_{x}\subseteq[0,1].$

Thus in an IT2 FS $\tilde{\tilde{A}}$ , each point $x\in X$ corresponds to a membership interval, $[\mu_{A}^{L}(x),\mu_{A}^{U}(x)]$ , where $\mu_{A}^{L}:X\to[0,1]$ and $\mu_{A}^{U}:X\to[0,1]$ are respective upper and lower T1 membership functions of $\tilde{\tilde{A}}$ .

2.4 IFS [2,10,27]

An IFS is represented as

$\displaystyle A=\left\{\left(x,\mu_{A}(x),v_{A}(x)\right)\thinspace|\thinspace% {x\in X}\right\}$

where the functions

$\displaystyle\mu_{A}:X\to\left[0,1\right]\mbox{ and }v_{A}:X\to\left[0,1\right]$

are defined, respectively, the degree of membership and non-membership of the element $x\in X$ to the set $A$ , and for every element $x\in X$ satisfying the condition

$\displaystyle 0\leqslant\mu_{A}(x)+v_{A}(x)\leqslant 1.$

Furthermore, the degree of indeterminacy of $x\in X$ in $A$ is defined as

$\displaystyle\pi_{A}=1-\mu_{A}-v_{A};\quad\pi_{A}:X\to\left[0,1\right]$

3. Methodological development of IT2 IFS

In this section, the concepts of IT2 IFS with arithmetic operations on triangular IT2 IFSs are introduced.

An IFS is said to be an IT2 IFS if the degree of membership and non-membership are determined by IT2 FSs instead of T1 FSs. An IT2 IFS $\bar{\bar{A}}$ is expressed as follows:

$\displaystyle\bar{\bar{A}}=\left\{\left(x,\widetilde{\widetilde{\mu_{\bar{\bar% {A}}}(x)}},\widetilde{\widetilde{v_{\bar{\bar{A}}}(x)}}\right)\Bigg{|}{x\in X}\right\}$

where $\widetilde{\widetilde{{\mu_{\bar{\bar{A}}}}:}}X\to D\left[0,1\right]$ and $\widetilde{\widetilde{{v_{\bar{\bar{A}}}}:}}X\to D\left[0,1\right]$ represent the membership and non-membership functions, respectively. $D\left[0,1\right]$ denotes the set of all closed subintervals in [0, 1]. Also, for every element $x\in X$ , the degree of membership $\widetilde{\widetilde{\mu_{\bar{\bar{A}}}(x)}}=[\mu_{A}^{L}(x),\mu_{A}^{U}(x)]$ and the degree of non-membership $\widetilde{\widetilde{v_{\bar{\bar{A}}}(x)}}=[v_{A}^{L}(x),v_{A}^{U}(x)]$ satisfying the condition

$\displaystyle 0<sup_{x}\widetilde{\widetilde{{\mu_{\mathop{A}\limits^{=}}}(x)}% }+sup_{x}\widetilde{\widetilde{{v_{\mathop{A}\limits^{=}}}(x)}}\leqslant 1,i.e.$ $\displaystyle 0<\mu_{A}^{U}(x)+v_{A}^{U}(x)\leqslant 1.$

As like other variants of FSs, triangular FSs are of special interests, the concept of triangular IT2 IFSs is introduced in the next sub section.

Figure 1.

Triangular IT2 IFS.

3.1 Triangular IT2 IFS

A triangular IT2 IFS is represented as follows:

$\displaystyle\bar{\bar{A}}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}[{[{({a_{11}^{U},a,a_{12}^{U}% });\mu_{A}^{U}(a)}],[{({a_{21}^{L},a,a_{22}^{L}});\mu_{A}^{L}(a)}]}],\\ [{[{({a_{31}^{U},a,a_{32}^{U}});v_{A}^{U}(a)}],[{({a_{41}^{L},a,a_{42}^{L}});v% _{A}^{L}(a)}]}]\end{array}\right\rangle$

where $\mu_{A}^{U}\mbox{ and }\mu_{A}^{L}$ are, respectively, the upper and lower triangular type membershipfunctions; and $v_{A}^{U}$ and $\mu_{A}^{L}$ are, respectively, the upper and lower triangular type non-membership functions, $\mu_{A}^{U}(a)$ , $\mu_{A}^{L}(a)$ , $v_{A}^{U}(a),v_{A}^{L}(a)\in[0,1]$ are, respectively, the membership and non-membership grades of $a$ , and $a_{1i}^{U}$ , $a_{2i}^{L}$ , $a_{3i}^{U}$ and $a_{4i}^{L}(i=1,2,3)$ are the parameters of functions $\mu_{A}^{U},\mu_{A}^{L},v_{A}^{U}$ and $v_{A}^{L}$ respectively, and for every $x$ in the universe of discourse satisfying the condition $0<\mu_{A}^{U}(x)+v_{A}^{U}(x)\leqslant 1$ . A triangular IT2 IFS is presented in Fig. 1.

3.1.1 Arithmetic operations on triangular IT2 IFSs

$\displaystyle{\rm Let}∼{}\bar{\bar{A}}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\left[{[{({a_{11}^{U},a,a_{12% }^{U}});\mu_{A}^{U}(a)}],[{({a_{21}^{L},a,a_{22}^{L}});\mu_{A}^{L}(a)}]}\right% ],\\ \left[{[{({a_{31}^{U},a,a_{32}^{U}});v_{A}^{U}(a)}],[{({a_{41}^{L},a,a_{42}^{L% }});v_{A}^{L}(a)}]}\right]\end{array}\right\rangle$

and

$\displaystyle\bar{\bar{B}}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\left[{[{({b_{11}^{U},b,b_{12% }^{U}});\mu_{A}^{U}(b)}],[{({b_{21}^{L},b,b_{22}^{L}});\mu_{A}^{L}(b)}]}\right% ],\\ \left[{[{({b_{31}^{U},b,b_{32}^{U}});v_{A}^{U}(b)}],[{({b_{41}^{L},b,b_{42}^{L% }});v_{A}^{L}(b)}]}\right]\end{array}\right\rangle$

be two triangular IT2 IFSs.

The arithmetic operations on triangular IT2 IFSs are described as follows.

3.1.1.1 Additon

The addition of two triangular IT2 IFSs is denoted by, $\bar{\bar{A}}\oplus\bar{\bar{B}}$ , and is defined as follows:

$\displaystyle\bar{\bar{A}}\oplus\bar{\bar{B}}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\left[\begin{array}[]{@{}l@{}% }\scalebox{.93}{${{[{({a_{11}^{U}{+}b_{11}^{U},a{+}b,a_{12}^{U}{+}b_{12}^{U}})% ;\min(\mu_{A}^{U}(a),\mu_{B}^{U}(b))}]},}$}\\ \scalebox{.93}{${[{({a_{11}^{L}{+}b_{11}^{L},a{+}b,a_{12}^{L}{+}b_{12}^{L}});% \min(\mu_{A}^{L}(a),\mu_{B}^{L}(b))}]}$}\end{array}\right],\\ \left[\begin{array}[]{@{}l@{}}\scalebox{.93}{${{[{({a_{31}^{U}{+}b_{31}^{U},a{% +}b,a_{32}^{U}{+}b_{32}^{U}});\max(v_{A}^{U}(a),v_{B}^{U}(b))}]},}$}\\ \scalebox{.93}{${[{({a_{41}^{L}{+}b_{41}^{L},a{+}b,a_{42}^{L}{+}b_{42}^{L}});% \max(v_{A}^{L}(a),v_{B}^{L}(b))}]}$}\end{array}\right]\end{array}\right\rangle$

3.1.1.2 Subtraction

The subtraction of the IT2 IFS, $\bar{\bar{B}}$ , from the IT2 IFS, $\bar{\bar{A}}$ , is denoted by $\bar{\bar{A}}\ominus\bar{\bar{B}}$ and is defined as

$\displaystyle\bar{\bar{A}}\ominus\bar{\bar{B}}=$ $\displaystyle\left\langle\begin{array}[]{l}{\left[\begin{array}[]{@{}l@{}}% \scalebox{.93}{${{[{({a_{11}^{U}{-}b_{11}^{U},a{-}b,a_{12}^{U}{-}b_{12}^{U}});% \min(\mu_{A}^{U}(a),\mu_{B}^{U}(b))}]},}$}\\ \scalebox{.93}{${[{({a_{21}^{L}{-}b_{21}^{L},a{-}b,a_{22}^{L}{-}b_{22}^{L}});% \min(\mu_{A}^{L}(a),\mu_{B}^{L}(b))}]}$}\end{array}\right]},\\ \left[\begin{array}[]{@{}l@{}}\scalebox{.93}{${{[{({a_{31}^{U}{-}b_{31}^{U},a{% -}b,a_{32}^{U}{-}b_{32}^{U}});\max(v_{A}^{U}(a),v_{B}^{U}(b))}]},}$}\\ \scalebox{.93}{${[{({a_{41}^{L}{-}b_{41}^{L},a{-}b,a_{42}^{L}{-}b_{42}^{L}});% \max(v_{A}^{L}(a),v_{B}^{L}(b))}]}$}\end{array}\right]\end{array}\right\rangle$

3.1.1.3 Multiplication

The multiplication of two triangular IT2 IFSs is denoted by $\bar{\bar{A}}\otimes\bar{\bar{B}}$ and is defined as follows:

$\displaystyle\bar{\bar{A}}\otimes\bar{\bar{B}}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\left[\begin{array}[]{@{}l@{}% }\scalebox{.93}{$[{({a_{11}^{U}{\times}b_{11}^{U},a{\times}b,a_{12}^{U}{\times% }b_{12}^{U}});\min(\mu_{A}^{U}(a),\mu_{B}^{U}(b))}],$}\\ \scalebox{.93}{$[({a_{21}^{L}{\times}b_{21}^{L},a{\times}b,a_{22}^{L}{\times}b% _{22}^{L}});\min(\mu_{A}^{L}(a),\mu_{B}^{L}(b))]$}\end{array}\right],\\ \left[\begin{array}[]{@{}l@{}}\scalebox{.93}{${{[{({a_{31}^{U}{\times}b_{31}^{% U},a{\times}b,a_{32}^{U}{\times}b_{32}^{U}});\max(v_{A}^{U}(a),v_{B}^{U}(b))}]% },}$}\\ \scalebox{.93}{${[{({a_{41}^{L}{\times}b_{41}^{L},a{\times}b,a_{42}^{L}{\times% }b_{42}^{L}});\max(v_{A}^{L}(a),v_{B}^{L}(b))}]}$}\end{array}\right]\end{array% }\right\rangle$

3.1.1.4 Multiplication by a crisp value

The multiplication of a triangular IT2 IFS $\tilde{\tilde{A}}$ with a crisp value $k(>0)$ is denoted by $k\tilde{\tilde{A}}$ and is described as

$\displaystyle k\mathop{A}\limits^{=}=\left\langle\begin{array}[]{@{}l@{}}{% \left[{\begin{array}[]{@{}l@{}}{\left[{\left({k\times a_{11}^{U},k\times a,k% \times a_{12}^{U}}\right);\mu_{A}^{U}(a)}\right],}\\ {\left[{\left({k\times a_{21}^{L},k\times a,k\times a_{22}^{L}}\right);\mu_{A}% ^{L}(a)}\right]}\end{array}}\right],}\\ {\left[{\begin{array}[]{@{}l@{}}{\left[{\left({k\times a_{31}^{U},k\times a,k% \times a_{32}^{U}}\right);v_{A}^{U}(a)}\right],}\\ {\left[{\left({k\times a_{41}^{L},k\times a,k\times a_{42}^{L}}\right);v_{A}^{% L}(a)}\right]}\end{array}}\right]}\end{array}\right\rangle$

3.1.1.5 Division

Let $\bar{\bar{A}}$ and $\bar{\bar{B}}$ be two IT2 IFSs. The division of $\bar{\bar{A}}$ by $\bar{\bar{B}}$ is denoted by $\bar{\bar{A}}/\bar{\bar{B}}$ , and is defined by

$\displaystyle{\frac{{\mathop{A}\limits^{=}}}{{\mathop{B}\limits^{=}}}}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\left[{\begin{array}[]{@{}l@{% }}{\left[{\left({{\displaystyle\frac{{a_{11}^{U}}}{{b_{12}^{U}}}},{% \displaystyle\frac{a}{b}},{\displaystyle\frac{{a_{12}^{U}}}{{b_{11}^{U}}}}}% \right);\min\left({\mu_{A}^{U}(a),\mu_{B}^{U}(b)}\right)}\right],}\\ {\left[{\left({{\displaystyle\frac{{a_{21}^{L}}}{{b_{22}^{L}}}},{\displaystyle% \frac{a}{b}},{\displaystyle\frac{{a_{22}^{L}}}{{b_{21}^{L}}}}}\right);\min% \left({\mu_{A}^{L}(a),\mu_{B}^{L}(b)}\right)}\right]}\end{array}}\right],\\ \left[{\begin{array}[]{@{}l@{}}{\left[{\left({{\displaystyle\frac{{a_{31}^{U}}% }{{b_{32}^{U}}}},{\displaystyle\frac{a}{b}},{\displaystyle\frac{{a_{32}^{U}}}{% {b_{31}^{U}}}}}\right);{\rm{max}}\left({v_{A}^{U}(a),v_{B}^{U}(b)}\right)}% \right],}\\ {\left[{\left({{\displaystyle\frac{{a_{41}^{L}}}{{b_{42}^{L}}}},{\displaystyle% \frac{a}{b}},{\displaystyle\frac{{a_{42}^{L}}}{{b_{41}^{L}}}}}\right);{\rm{max% }}\left({v_{A}^{L}(a),v_{B}^{L}(b)}\right)}\right]}\end{array}}\right]\end{% array}\right\rangle$

provided $b_{11}^{U},b_{21}^{L},b_{31}^{U},b_{41}^{U}$ and $b$ are positive scalars.

3.1.1.6 Integral power of a triangular IT2 IFS

Let $\bar{\bar{A}}$ be a IT2 IFS. Then $\bar{\bar{A}}^{\lambda}$ is described as

$\displaystyle\bar{\bar{A}}^{\lambda}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\left[{\begin{array}[]{@{}l@{% }}{\left[{({{{({a_{11}^{U}})}^{\lambda}},{a^{\lambda}},{{({a_{12}^{U}})}^{% \lambda}}});\mu_{A}^{U}(a)}\right],}\\ {\left[{({{{({a_{21}^{L}})}^{\lambda}},{a^{\lambda}},{{({a_{22}^{L}})}^{% \lambda}}});\mu_{A}^{L}(a)}\right]}\end{array}}\right],\\ \left[{\begin{array}[]{@{}l@{}}{\left[{({{{({a_{11}^{U}})}^{\lambda}},{a^{% \lambda}},{{({a_{12}^{U}})}^{\lambda}}});\mu_{A}^{U}(a)}\right],}\\ {\left[{({{{({a_{21}^{L}})}^{\lambda}},{a^{\lambda}},{{({a_{22}^{L}})}^{% \lambda}}});\mu_{A}^{L}(a)}\right]}\end{array}}\right]\end{array}\right\rangle$ (3)

where $\lambda$ is a positive real number.

3.1.2 Defuzzification method of triangular IT2 IFS

Any two IT2 IFSs can be compared on a basis of their quantitative values which are extracted by the process of defuzzification. In this section, a defuzzification method for triangular IT2 IFSs has been introduced:

Let

$\displaystyle\bar{\bar{A}}=\left\langle\begin{array}[]{@{}l@{}}\scalebox{.94}{% $\displaystyle{[{[{({a_{3}^{U},a,a_{6}^{U}});\mu_{A}^{U}(a)}],[{({a_{4}^{L},a,% a_{5}^{L}});\mu_{A}^{L}(a)}]}]}$}\\ [{[{({a_{2}^{U},a,a_{7}^{U}});v_{A}^{U}(a)}],[{({a_{1}^{L},a,a_{8}^{L}});v_{A}% ^{L}(a)}]}]\end{array}\right\rangle$

be an IT2 IFS, and the upper and lower membership and non-membership functions of $\bar{\bar{A}}$ are represented as follows:

$\displaystyle\mu_{A}^{U}(x)=\begin{cases}{\displaystyle\frac{x-a_{3}^{U}}{a-a_% {3}^{U}}}&if∼{}a_{3}^{U}\leqslant x\leqslant a\\ {\displaystyle\frac{a_{6}^{U}-x}{a_{6}^{U}-a}}&if∼{}a\leqslant x\leqslant a_{6% }^{U}\\ 0&if∼{}x<a_{3}^{U}∼{}or∼{}x>a_{6}^{U}\\ \end{cases}$ $\displaystyle\mu_{A}^{L}(x)=\begin{cases}{\displaystyle\frac{x-a_{4}^{L}}{a-a_% {4}^{L}}}&if∼{}a_{4}^{L}\leqslant x\leqslant a\\ {\displaystyle\frac{a_{5}^{L}-x}{a_{5}^{L}-a}}&if∼{}a\leqslant x\leqslant a_{5% }^{L}\\ 0&if∼{}x<a_{4}^{L}∼{}or∼{}x>a_{5}^{L}\\ \end{cases}$ $\displaystyle v_{A}^{U}(x)=\begin{cases}{\displaystyle\frac{a-x}{a-a_{2}^{U}}}% &if∼{}a_{2}^{U}\leqslant x\leqslant a\\ {\displaystyle\frac{x-a}{a_{7}^{U}-a}}&if∼{}a\leqslant x\leqslant a_{7}^{U}\\ 1&if∼{}x<a_{2}^{U}∼{}or∼{}x>a_{7}^{U}\\ \end{cases}$ $\displaystyle v_{A}^{L}(x)=\begin{cases}{\displaystyle\frac{a-x}{a-a_{1}^{L}}}% &if∼{}a_{1}^{L}\leqslant x\leqslant a\\ {\displaystyle\frac{x-a}{a_{8}^{L}-a}}&if∼{}a\leqslant x\leqslant a_{8}^{L}\\ 1&if∼{}x<a_{1}^{L}∼{}or∼{}x>a_{8}^{L}\\ \end{cases}$

Now, considering the upper and lower membership functions and the standard complement of the upper and lower non-membership functions, an alternative representation of IT2 IFSs can be found which is represented in the following figure.

Figure 2.

IT2 IFS with complement of non-membership function.

The mathematical form of the upper and lower non-membership functions of $\bar{\bar{A}}$ is presented as follows.

$\displaystyle 1-v_{A}^{U}(x)=\begin{cases}{\displaystyle\frac{x-a_{2}^{U}}{a-a% _{2}^{U}}}&if∼{}a_{2}^{U}\leqslant x\leqslant a\\ {\displaystyle\frac{a_{7}^{U}-x}{a_{7}^{U}-a}}&if∼{}a\leqslant x\leqslant a_{7% }^{U}\\ 0&if∼{}∼{}x<a_{2}^{U}∼{}or∼{}x>a_{7}^{U}\\ \end{cases}$ $\displaystyle 1-v_{A}^{L}(x)=\begin{cases}{\displaystyle\frac{x-a^{L}_{1}}{a-a% _{1}^{L}}}&if∼{}a_{1}^{L}\leqslant x\leqslant a\\ {\displaystyle\frac{a_{8}^{L}-x}{a_{8}^{L}-a}}&if∼{}a\leqslant x\leqslant a_{8% }^{L}\\ 0&if∼{}x<a_{1}^{L}∼{}or∼{}x>a_{8}^{L}\\ \end{cases}$

It is to be noted here that membership function of any T1 FS can be represented in terms of some random variable [13] with a probability density function. Let $f_{1}(x)$ , $f_{2}(x)$ , $f_{3}(x)$ and $f_{4}(x)$ denote, respectively, the probability density function of the random variables corresponding to the upper and lower membership and complement of the upper and lower non-membership functions of $\bar{\bar{A}}$ .

Thus using the membership and complement of non-membership functions, the probability density functions are expressed as

$\displaystyle f_{1}(x)=c_{1}\mu_{A}^{U},f_{2}(x)=c_{2}\mu_{A}^{L},$ $\displaystyle f_{3}(x)=c_{3}(1-v_{A}^{U})$

and

$\displaystyle f_{4}(x)=c_{4}(1-v_{A}^{L})$

where the values of $c_{i}$ are calculated by using the equation $\int\nolimits_{-\infty}^{\infty}{f_{i}(x)}dx=1$ for all $i=1,2,3,4$ .

Executing the above integral, the value of $c_{i}(i=1,2,3,4)$ are obtained as

$\displaystyle c_{1}=\frac{2}{(a_{6}^{U}-a_{3}^{U})},∼{}c_{2}=\frac{2}{(a_{5}^{% L}-a_{4}^{L})},$ $\displaystyle c_{3}=\frac{2}{(a_{7}^{U}-a_{2}^{U})}∼{}∼{}\text{and}∼{}∼{}c_{4}% =\frac{2}{(a_{8}^{L}-a_{1}^{L})}.$

Thus the probability density functions of all the random variables corresponding to the membership and non-membership functions are described as follows:

$\displaystyle f_{1}(x)=$ $\displaystyle\begin{cases}{\displaystyle\frac{2(x-a_{3}^{U})}{(a_{6}^{U}-a_{3}% ^{U})(a-a_{3}^{U})}}&if∼{}a_{3}^{L}\leqslant x\leqslant a\\ {\displaystyle\frac{2(a_{6}^{U}-x)}{(a_{6}^{U}-a_{3}^{U})(a_{6}^{U}-a)}}&if∼{}% a_{3}^{L}\leqslant x\leqslant a\\ 0&if∼{}a_{3}^{L}\leqslant x\leqslant a\\ \end{cases}$ (4) $\displaystyle f_{2}(x)=$ $\displaystyle\begin{cases}{\displaystyle\frac{2(x-a_{4}^{L})}{\left(a_{5}^{L}-% a_{4}^{L}\right)(a-a_{4}^{L})}}&if∼{}a_{4}^{U}\leqslant x\leqslant a\\ {\displaystyle\frac{2(a_{5}^{L}-x)}{(a_{5}^{L}-a_{4}^{L})(a_{5}^{L}-a)}}&if∼{}% a\leqslant x\leqslant a_{5}^{L}\\ 0&if∼{}x<a_{4}^{L}∼{}or∼{}x>a_{5}^{L}\\ \end{cases}$ (5) $\displaystyle f_{3}(x)=$ $\displaystyle\begin{cases}{\displaystyle\frac{2(x-a_{2}^{U})}{(a_{7}^{U}-a_{2}% ^{U})(a-a_{2}^{U})}}&if∼{}a_{2}^{U}\leqslant x\leqslant a\\ {\displaystyle\frac{2(a_{7}^{U}-x)}{(a_{7}^{U}-a_{2}^{U})(a_{7}^{U}-a)}}&if∼{}% a\leqslant x\leqslant a_{7}^{U}\\ 0&if∼{}x<a_{2}^{U}∼{}or∼{}x>a_{7}^{U}\\ \end{cases}$ (6) $\displaystyle f_{4}(x)=$ $\displaystyle\begin{cases}{\displaystyle\frac{2(x-a_{1}^{L})}{\left(a_{8}^{L}-% a_{1}^{L}\right)(a-a_{1}^{L})}}&if∼{}a_{1}^{L}\leqslant x\leqslant a\\ {\displaystyle\frac{2(a_{8}^{L}-x)}{(a_{8}^{L}-a_{1}^{L})(a_{8}^{L}-a)}}&if∼{}% a\leqslant x\leqslant a_{8}^{L}\\ 0&if∼{}x<a_{1}^{L}∼{}or∼{}x>a_{8}^{L}\\ \end{cases}$ (7)

Now, since the IT2 IFS $\bar{\bar{A}}$ can be expressed using two membership and two complement of non-membership functions, and each of those functions are described using four probability density functions of the random variables $X$ as presented in Eqs (3.1.2)–(3.1.2), to achieve a combined probability density function corresponding to the random variable representing the IT2 IFS $\bar{\bar{A}}$ , the convex combination of the four probability density functions, defined earlier, is taken into account.

Let $f(x)$ be the probability density function of the random variable corresponding to the IT2 IFS $\bar{\bar{A}}$ which is the convex combination of $f_{1}(x)$ , $f_{2}(x)$ , $f_{3}(x)$ and $f_{4}(x)$ .

Thus $f(x)$ can be expressed as

$\displaystyle f(x)=\omega_{1}f_{1}(x)+\omega_{2}f_{2}(x)+\omega_{3}f_{3}(x)+% \omega_{4}f_{4}(x)$

where $\omega_{i}(i=1,2,3,4)$ are non-negative scalars satisfying $\sum\nolimits_{i=1}^{4}\omega_{i}=1$ .

For computational simplicity, the values of $\omega_{i}(i=1,2,3,4)$ are chosen as $\frac{1}{4}$ .

It is to be noted here that other values of $\omega_{i}$ may be considered based on the decision making contexts, and would not affect the proposed technique.

Table 1

Scale of relative importance for pair-wise comparison

Triangular IT2 IFS	Linguistic variable
$<$ [[(1, 1, 2); 0.8], [(1, 1, 1.5); 0.7]], [[(1, 1, 2.5); 0.1], [(1, 1, 3); 0]] $>$	Just equal (JE)
$<$ [[(2, 3, 4); 0.8], [(2.5, 3, 3.5); 0.7]], [[(1.5, 3, 4.5); 0.1], [(1, 3, 5); 0]] $>$	Weakly important (WI)
$<$ [[(4, 5, 6); 0.8], [(4.5, 5, 5.5); 0.7]], [[(3.5, 5, 6.5); 0.1], [(3, 5, 7); 0]] $>$	Slightly important (SI)
$<$ [[(6, 7, 8); 0.8], [(6.5, 7, 7.5); 0.7]], [[(5.5, 7, 8.5); 0.1], [(5, 7, 9); 0]] $>$	Reasonably important (RI)
$<$ [[(8, 9, 9); 0.8], [(8.5, 9, 9); 0.7]], [[(7.5, 9, 9); 0.1], [(7, 9, 9); 0]] $>$	Absolutely important (AI)

Now, the expected defuzzified value of $\bar{\bar{A}}$ is obtained as

$\displaystyle D_{\bar{\bar{A}}}=\frac{1}{4}\int\nolimits_{-\infty}^{\infty}% \sum\nolimits_{i=1}^{4}{{xf}_{i}(x)}dx=\frac{1}{4}\bigg{[}\left\{\frac{\left(a% _{6}^{U}\right)^{2}+a\left(a_{6}^{U}-a_{3}^{U}\right)-\left(a_{3}^{U}\right)^{% 2}}{3\left(a_{6}^{U}-a_{3}^{U}\right)}\right\}+\left\{\frac{\left(a_{5}^{L}% \right)^{2}+a\left(a_{5}^{L}-a_{4}^{L}\right)-\left(a_{4}^{L}\right)^{2}}{3% \left(a_{5}^{L}-a_{4}^{L}\right)}\right\}+\left\{\frac{\left(a_{7}^{U}\right)^% {2}+a\left(a_{7}^{U}-a_{2}^{U}\right)-\left(a_{2}^{U}\right)^{2}}{3\left(a_{7}% ^{U}-a_{2}^{U}\right)}\right\}+\left\{\frac{\left(a_{8}^{L}\right)^{2}+a\left(% a_{8}^{L}-a_{1}^{L}\right)-\left(a_{1}^{L}\right)^{2}}{3\left(a_{8}^{L}-a_{1}^% {L}\right)}\right\}\bigg{]}$ (8)

Thus, the defuzzified value of an IT2 IFS can be derived.

3.1.3 Development of IT2 IF-AHP

In this section the concept of IT2 IFS is introduced in the context of AHP to achieve a systematic process which can capture the inconsistencies present in different MCDM problems in more precise way.

As like AHP, a hierarchical structure that consists of goals, criteria and sub-criteria (if any) is constructed. In this formulation, the pair wise comparisons of the alternatives and criteria are made using a comparison scale described in terms of using triangular IT2 IFSs. The comparison scale is formulated in Table 1.

Using the scale as presented in Table 1, an IT2 IF pair wise comparison matrix is constructed as follows:

$\displaystyle\bar{\bar{C}}\!=\!\left[\bar{\bar{C}}_{ij}\right]\!=\!\left[{% \begin{array}[]{l}{\begin{array}[]{*{20}c}\bar{\bar{C}}_{11}&\bar{\bar{C}}_{12% }\\ \bar{\bar{C}}_{21}&\bar{\bar{C}}_{22}\\ \end{array}}{\begin{array}[]{*{20}c}\ldots&\bar{\bar{C}}_{1n}\\ \ldots&\bar{\bar{C}}_{2n}\\ \end{array}}\\ {\begin{array}[]{*{20}c}\vdots&\vdots\\ \bar{\bar{C}}_{n1}&\bar{\bar{C}}_{n2}\\ \end{array}}{\begin{array}[]{*{20}c}\ddots&\vdots\\ \mathellipsis&\bar{\bar{C}}_{nn}\\ \end{array}}\\ \end{array}}\right],i,j=1,2,\ldots,n$

where each $\bar{\bar{C}}_{ij}$ denotes an IT2 IFS representing the quantified judgment on $c_{i}$ to $c_{j}$ .

Also, $\bar{\bar{C}}_{ji}=\frac{1}{\bar{\bar{C}}_{ij}}$ represents the quantified judgment on $c_{j}$ to $c_{i}$ ( $i,j=1,2,\ldots,n$ ). The IT2 IFS $\bar{\bar{C}}_{ji}$ can be calculated using (14). It is to be noted here that for an IT2 IFS

$\displaystyle\mathop{A}\limits^{=}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\left[{[{({a_{11}^{U},a,a_{12% }^{U}});\mu_{A}^{U}(a)}],[{({a_{21}^{L},a,a_{22}^{L}});\mu_{A}^{L}(a)}]}\right% ],\\ \left[{[{({a_{11}^{U},a,a_{12}^{U}});\nu_{A}^{U}(a)}],[{({a_{21}^{L},a,a_{22}^% {L}});\nu_{A}^{L}(a)}]}\right]\end{array}\right\rangle,$

$\frac{1}{{\mathop{A}\limits^{=}}}$ is given by

$\displaystyle\frac{1}{{\mathop{A}\limits^{=}}}=$ $\displaystyle\left\langle\begin{array}[]{@{}l@{}}\scalebox{.87}{$\displaystyle% {\left[{\left[{\left({\frac{1}{{a_{12}^{U}}},\frac{1}{a},\frac{1}{{a_{11}^{U}}% }}\right);\mu_{A}^{U}(a)}\right],\left[{\left({\frac{1}{{a_{22}^{L}}},\frac{1}% {a},\frac{1}{{a_{21}^{L}}}}\right);\mu_{A}^{L}(a)}\right]}\right],}$}\\ \scalebox{.87}{$\displaystyle{\left[{\left[{\left({\frac{1}{{a_{12}^{U}}},% \frac{1}{a},\frac{1}{{a_{11}^{U}}}}\right);\nu_{A}^{U}(a)}\right],\left[{\left% ({\frac{1}{{a_{22}^{L}}},\frac{1}{a},\frac{1}{{a_{21}^{L}}}}\right);\nu_{A}^{L% }(a)}\right]}\right]}$}\end{array}\right\rangle$

After formulating the comparison matrices, the consistency of each matrices are checked. In this perspective, the IT2 IFSs are to be converted first into its equivalent defuzzified values using the defuzzification process as described in the previous section.

Based on the defuzzified values, the comparison matrix $\bar{\bar{C}}$ is converted into the following form as

$\displaystyle C=\left[C_{ij}\right]=\left[{\begin{array}[]{l}{\begin{array}[]{% *{20}c}C_{11}&C_{12}\\ C_{21}&C_{22}\\ \end{array}}{\begin{array}[]{*{20}c}\ldots&C_{1n}\\ \ldots&C_{2n}\\ \end{array}}\\ {\begin{array}[]{*{20}c}\vdots&\vdots\\ C_{n1}&C_{n2}\\ \end{array}}{\begin{array}[]{*{20}c}\ddots&\vdots\\ \mathellipsis&C_{nn}\\ \end{array}}\\ \end{array}}\right]$

It is worthy to mention here that a fuzzy comparison matrix $\bar{\bar{C}}$ is consistent only whenthe corresponding traditional comparison matrix $C$ is consistent. The consistency of the comparison matrix $C$ is checked in usual manner. In this perspective, it is to be mentioned here that if CR $<$ 0.10, the consistency of the matrix is considered as acceptable, otherwise the pair-wise comparisons should be revised.

Once the optimal comparison matrix is determined, the geometric mean $\bar{\bar{C}}_{i}$ of the $i^{\rm th}$ row in the comparison matrix is calculated using Eqs (3.1) and (3.1).

$\displaystyle\bar{\bar{C}}_{i}{=\left[\bar{\bar{C}}_{i1}\otimes\ldots\otimes% \bar{\bar{C}}_{in}\right]}^{1/n}$ (9)

where for an IT2 IFS $\bar{\bar{A}}$ ,

$\displaystyle\sqrt[n]{{\bar{\bar{A}}}}=\left\langle\begin{array}[]{@{}l@{}}% \Big{[}\Big{[}{\Big{(}{\sqrt[n]{{a_{11}^{U}}},\sqrt[n]{a},\sqrt[n]{{a_{12}^{U}% }}}\Big{)};\mu_{A}^{U}(a)}\Big{]},\\ \Big{[}{\Big{(}{\sqrt[n]{{a_{21}^{L}}},\sqrt[n]{a},\sqrt[n]{{a_{22}^{L}}}}\Big% {)};\mu_{A}^{L}(a)}\Big{]}\Big{]},\\ \Big{[}\Big{[}{\Big{(}{\sqrt[n]{{a_{31}^{U}}},\sqrt[n]{a},\sqrt[n]{{a_{32}^{U}% }}}\Big{)};v_{A}^{U}(a)}\Big{]},\\ \Big{[}{\Big{(}{\sqrt[n]{{a_{41}^{L}}},\sqrt[n]{a},\sqrt[n]{{a_{42}^{L}}}}\Big% {)};v_{A}^{L}(a)}\Big{]}\Big{]}\end{array}\right\rangle$

The IT2 IF local weight of the $i^{\rm th}$ criteria in the comparison matrix is evaluated using Eqs (3.1) and (3.1) as

$\displaystyle\bar{\bar{w}}_{i}^{L}{=\bar{\bar{C}}_{i}\otimes[\bar{\bar{C}}_{1}% \oplus\mathellipsis\oplus\bar{\bar{C}}_{i}\oplus\mathellipsis.\oplus\bar{\bar{% C}}_{n}]}^{-1}.$ (10)

After calculating the local weights, the IT2 IF global weight of the $i^{\rm th}$ alternative with respect to each criterion is determined based on Eq. (3.1) as

$\displaystyle\bar{\bar{w}}_{i}^{\textit{GA}}=\bar{\bar{w}}_{i}^{LC}\otimes\bar% {\bar{w}}_{i}^{\textit{{LA}}}$ (11)

where $\bar{\bar{w}}_{i}^{\textit{LC}}$ and $\bar{\bar{w}}_{i}^{\textit{LA}}$ represent the IT2 IF local weights of the criteria and alternatives, respectively.

Finally, the overall weights of each alternative are calculated as follows:

$\displaystyle\bar{\bar{w}}_{i}^{O}=\sum\nolimits_{i=1}^{n}\bar{\bar{w}}_{i}^{GA}$ (12)

IT2 IF overall weights ( $\bar{\bar{w}}_{i}^{O}$ ) are then converted into defuzzified crisp weights ( $w_{i}^{*}$ ) using Eq. (8).

Crisp weights are normalized to rank all the alternatives using the following equation as

$\displaystyle w_{i}=\frac{w_{i}^{*}}{\sum\nolimits_{i=1}^{n}w_{i}^{*}}$ (13)

where $w_{i}$ is the weight of $c_{i}$ after normalization process.

Thus the weight of the alternatives can be obtained by capturing uncertainties through IT2 IF-AHP in a justified manner.

3.2 The algorithm

The whole procedure of IT2 IF-AHP is summarized in the following algorithm.

Step 1: Step 1:
Determine criteria and alternatives to construct of hierarchical structure.
Step 2:
Based on the expert’s judgment establish initial IT2 IF comparison matrix.
Step 3:
Calculate CR values of each initial comparison matrices and modify those matrices where CR values are greater than 0.1.
Step 4:
Compute the geometric mean of each row of the comparison matrix using Eq. (9).
Step 5:
Evaluate the IF local weight of each criteriaof the comparison matrix using Eq. (10).
Step 6:
Determine the IF global weights of each alternative corresponding to each criterion using Eq. (11).
Step 7:
Evaluate the IF overall weights of each alternative as given in Eq. (12).
Step 8:
Calculate the equivalent crisp weights of the alternatives using the defuzzification process presented in Eq. (8).
Step 9:
Normalize the crisp weights in order to prioritize the alternatives.

The process may also be presented in the following flow chart (Fig. 3).
4. An illustrative example

In this section, the developed method of IT2 IF-AHP is applied to resolve a supplier selection process. Suppose, $S_{1}$ and $S_{2}$ are two suppliers of a renowned company. The administration of the company was in a need for selection of its best supplier from which they would like to purchase materials. The criteria that affect the company’s supplier selection process are price (P), quality (Q), delivery (D) and service (S). In evaluating criteria corresponding to each supplier, the comparison scales are used through IT2 IFNs for capturing uncertainties in a better way. The preferences among the criteria are presented using IT2 IF information as follows:

Table 2
Pairwise comparison of preferences among the criteria

Criteria $\to$	$P$	$Q$	$D$	$S$
$\downarrow$
$P$	$J E$	$\frac{1}{SI}$	$S I$	$R I$
$Q$	$S I$	$J E$	$R I$	$A I$
$D$	$\frac{1}{SI}$	$\frac{1}{RI}$	$J E$	$W I$
$S$	$\frac{1}{RI}$	$\frac{1}{AI}$	$\frac{1}{WI}$	$J E$

Figure 3.

Flowchart of the proposed method.

Figure 4.

Hierarchical structure a supplier selection problem.

The IT2 IF pairwise comparisons between the suppliers with respect to each criterion are presented in the following table:

Table 3

Pairwise comparison of suppliers

$P$	$S_{1}$	$S_{2}$	$Q$	$S_{1}$	$S_{2}$
$S_{1}$	$J E$	$W I$	$S_{1}$	$J E$	$\frac{1}{SI}$
$S_{2}$	$\frac{1}{WI}$	$J E$	$S_{2}$	$S I$	$J E$
$D$	$S_{1}$	$S_{2}$	$S$	$S_{1}$	$S_{2}$
$S_{1}$	$J E$	$W I$	$S_{1}$	$J E$	$\frac{1}{AI}$
$S_{2}$	$\frac{1}{WI}$	$J E$	$S_{2}$	$A I$	$J E$

Table 4

Geometric mean of the criteria

$P$	$<$ [[(1.42, 1.63, 2.21); 0.8], [(1.51, 1.63, 1.92); 0.7]], [[(1.3, 1.63, 2.49); 0.1], [(1.26, 1.63, 2.81); 0]] $>$
$Q$	$<$ [[(3.72, 4.21, 5.42); 0.8], [(3.97, 4.21, 4.86); 0.7]], [[(3.47, 4.21, 5.94); 0.1], [(3.2, 4.21, 6.42); 0]] $>$
$D$	$<$ [[(0.45, 0.54, 0.76); 0.8], [0.49, 0.54, 0.64); 0.7]], [[(0.41, 0.54, 0.87); 0.1], [(0.37, 0.54, 1); 0]] $>$
$S$	$<$ [[(0.24, 0.27, 0.38); 0.8], [0.25, 0.27, 0.32); 0.7]], [[(0.23, 0.27, 0.45); 0.1], [(0.22, 0.27, 0.54); 0]] $>$

Table 5

Geometric mean of the suppliers with respect to the criteria

$P$
$S_{1}$	$<$ [[(1.41, 1.73, 2.83); 0.8], [(1.58, 1.73, 2.29); 0.7]], [[(1.22, 1.73, 3.35); 0.1], [(1, 1.73, 3.87); 0]] $>$
$S_{2}$	$<$ [[(0.50, 0.57, 1); 0.8], [(0.53, 0.57, 0.77); 0.7]], [[(0.47, 0.57, 1.29); 0.1], [(0.45, 0.57, 1.73); 0]] $>$
$Q$
$S_{1}$	$<$ [[(0.41, 0.45, 0.71); 0.8], [(0.42, 0.45, 0.57); 0.7]], [[(0.39, 0.45, 0.84); 0.1], [(0.40, 0.45, 0.99); 0]] $>$
$S_{2}$	$<$ [[(2, 2.24, 3.46); 0.8], [(2.12, 2.24, 2.87); 0.7]], [[(1.87, 2.24, 4.03); 0.1], [(1.73, 2.24, 4.58); 0]] $>$
$D$
$S_{1}$	$<$ [[(1.41, 1.73, 2.83); 0.8], [(1.58, 1.73, 2.29); 0.7]], [[(1.22, 1.73, 3.35); 0.1], [(1, 1.73, 3.87); 0]] $>$
$S_{2}$	$<$ [[(0.50, 0.57, 1); 0.8], [(0.53, 0.57, 0.77); 0.7]], [[(0.47, 0.57, 1.29); 0.1], [(0.45, 0.57, 1.73); 0]] $>$
$S$
$S_{1}$	$<$ [[(0.33, 0.33, 0.49); 0.8], [(0.33, 0.33, 0.42); 0.7]], [[(0.33, 0.33, 0.57); 0.1], [(0.33, 0.33, 0.65); 0]] $>$
$S_{2}$	$<$ [[(2.83, 3, 4.24); 0.8], [(2.92, 3, 3.67); 0.7]], [[(2.74, 3, 4.74); 0.1], [(2.65, 3, 5.20); 0]] $>$

Table 6

IT2 IF local weight of the criteria and Suppliers with respect to each criterion

$P$	$<$ [[(0.16, 0.25, 0.38); 0.8], [(0.20, 0.25, 0.31); 0.7]], [[(0.13, 0.25, 0.46); 0.1], [(0.12, 0.25, 0.56); 0]] $>$
$Q$	$<$ [[(0.42, 0.63, 0.93); 0.8], [(0.51, 0.63, 0.78); 0.7]], [[(0.36, 0.63, 1.10); 0.1], [(0.30, 0.63, 1.27); 0]] $>$
$D$	$<$ [[(0.05, 0.08, 0.13); 0.8], [(0.06, 0.08, 0.10); 0.7]], [[(0.04, 0.08, 0.16); 0.1], [(0.03, 0.08, 0.20); 0]] $>$
$S$	$<$ [[(0.03, 0.04, 0.07); 0.8], [(0.03, 0.04, 0.05); 0.7]], [[(0.02, 0.04, 0.08); 0.1], [(0.02, 0.04, 0.11); 0]] $>$
$P$
$S_{1}$	$<$ [[(0.37, 0.75, 1.48); 0.8], [(1.58, 0.75, 1.09); 0.7]], [[(1.22, 0.75, 1.98); 0.1], [(1, 0.75, 2.68); 0]] $>$
$S_{2}$	$<$ [[(0.13, 0.25, 0.52); 0.8], [(0.17, 0.25, 0.37); 0.7]], [[(0.10, 0.25, 0.76); 0.1], [(0.08, 0.25, 1.20); 0]] $>$
$Q$
$S_{1}$	$<$ [[(0.10, 0.17, 0.29); 0.8], [(0.12, 0.17, 0.23); 0.7]], [[(0.08, 0.17, 0.37); 0.1], [(0.07, 0.17, 0.46); 0]] $>$
$S_{2}$	$<$ [[(0.48, 0.83, 1.44); 0.8], [(0.62, 0.83, 1.13); 0.7]], [[(0.38, 0.83, 1.79); 0.1], [(0.31, 0.83, 2.14); 0]] $>$
$D$
$S_{1}$	$<$ [[(0.37, 0.75, 1.48); 0.8], [(1.58, 0.75, 1.09); 0.7]], [[(1.22, 0.75, 1.98); 0.1], [(1, 0.75, 2.68); 0]] $>$
$S_{2}$	$<$ [[(0.13, 0.25, 0.52); 0.8], [(0.17, 0.25, 0.37); 0.7]], [[(0.10, 0.25, 0.76); 0.1], [(0.08, 0.25, 1.20); 0]] $>$
$S$
$S_{1}$	$<$ [[(0.07, 0.10, 0.17); 0.8], [(0.08, 0.10, 0.15); 0.7]], [[(0.06, 0.10, 0.21); 0.1], [(0.06, 0.10, 0.24); 0]] $>$
$S_{2}$	$<$ [[(0.60, 0.90, 1.34); 0.8], [(0.71, 0.90, 1.13); 0.7]], [[(0.52, 0.90, 1.54); 0.1], [(0.45, 0.90, 1.75); 0]] $>$

Table 7

IT2 IF global weights of the Suppliers with respect to each criterion

$P$
$S_{1}$	$<$ [[(0.06, 0.19, 0.56); 0.8], [(0.10, 0.19, 0.34); 0.7]], [[(0.03, 0.19, 0.91); 0.1], [(0.02, 0.19, 1.50); 0]] $>$
$S_{2}$	$<$ [[(0.02, 0.06, 0.20); 0.8], [(0.03, 0.06, 0.11); 0.7]], [[(0.01, 0.06, 0.35); 0.1], [(0.01, 0.06, 0.67); 0]] $>$
$Q$
$S_{1}$	$<$ [[(0.04, 0.11, 0.27); 0.8], [(0.06, 0.11, 0.18); 0.7]], [[(0.03, 0.11, 0.41); 0.1], [(0.02, 0.11, 0.58); 0]] $>$
$S_{2}$	$<$ [[(0.20, 0.52, 1.34); 0.8], [(0.32, 0.52, 0.88); 0.7]], [[(0.14, 0.52, 2); 0.1], [(0.09, 0.52, 2.72); 0]] $>$
$D$
$S_{1}$	$<$ [[(0.02, 0.06, 0.19); 0.8], [(0.03, 0.06, 0.11); 0.7]], [[(0.01, 0.06, 0.32); 0.1], [(0.005, 0.06, 0.54); 0]] $>$
$S_{2}$	$<$ [[(0.01, 0.02, 0.07); 0.8], [(0.01, 0.02, 0.04); 0.7]], [[(0.004, 0.02, 0.12); 0.1], [(0.002, 0.02, 0.24); 0]] $>$
$S$
$S_{1}$	$<$ [[(0.002, 0.004, 0.012); 0.8], [(0.02, 0.004, 0.008); 0.7]], [[(0.001, 0.004, 0.02); 0.1], [(0.001, 0.004, 0.03); 0]] $>$
$S_{2}$	$<$ [[(0.02, 0.04, 0.06); 0.8], [(0.02, 0.04, 0.07); 0.7]], [[(0.01, 0.04, 0.12); 0.1], [(0.009, 0.04, 0.19); 0]] $>$

Now, the problem is to select the suitable supplier for the company using the proposed IT2 IF-AHP method. The step by step evaluation is presented below.

Table 8

Overall, defuzzified and normalized weights of the suppliers

	Overall IT2 IF weights $(\bar{\bar{w}}_{i}^{O})$	Defuzzified weights $(w_{i}^{*})$	Normalized weights $(w_{i})$
$S_{1}$	$<$ [[(0.12, 0.36, 1.04); 0.8], [(0.20, 0.36, 0.63); 0.7]], [[(0.07, 0.36, 1.65); 0.1], [(0.05, 0.36, 2.65); 0]] $>$	0.654	0.382
$S_{2}$	$<$ [[(0.25, 0.64, 1.70); 0.8], [(0.38, 0.64, 1.09); 0.7]], [[(0.16, 0.64, 2.56); 0.1], [(0.11, 0.64, 3.82); 0]] $>$	1.054	0.617

Table 9

Overall, defuzzified and normalized weights of the suppliers

Criteria	T1 FAHP		IT2 FAHP		IT2 IF-AHP
	Normalized weights	Rank	Normalized weights	Rank	Normalized weights	Rank
$P$	0.282	2	0.256	2	0.250	2
$Q$	0.418	1	0.607	1	0.623	1
$D$	0.168	3	0.087	3	0.084	3
$S$	0.132	4	0.049	4	0.043	4

Alternatives		Weights	Normalized weights	Weights	Normalized weights	Weights	Normalized weights
$S_{1}$	With respect to $P$	0.59	0.440	0.72	0.384	0.73	0.382
	With respect to $Q$	0.34		0.17		0.17
	With respect to $D$	0.59		0.72		0.73
	With respect to $S$	0.25		0.11		0.11
$S_{2}$	With respect to $P$	0.41	0.560	0.28	0.616	0.27	0.617
	With respect to $Q$	0.66		0.83		0.83
	With respect to $D$	0.41		0.28		0.27
	With respect to $S$	0.75		0.89		0.89

Step 1: Step 1:

The hierarchical structure of the problem is given in Fig. 4.

Step 2:

Comparison matrices corresponding to the criteria, $P, Q, D$ and $S$ (as described in Table 2), and alternatives, $S_{1}$ and $S_{2}$ (as described in Table 2), are evaluated based on the preference scale defined in Table 1. It is to be mentioned here that the elements of each matrices are represented using IT2 IFS.

Step 3:

Defuzzified pairwise comparison matrices of each criteria and alternatives are evaluated from the corresponding IT2 IF comparison matrices in Tables 2 and 3 using Eq. (8). The consistency ratio (CR) of each matrix is checked. The consistency ratio of all matrices is found less than 0.10.

Step 4:

The geometric mean $\bar{\bar{C}}_{i}$ of the $i^{th}$ row in the comparison matrix of the criteria and suppliers areevaluated using Eq. (9) and presented in Tables 4 and 5 below.

Step 5:

The IT2 IF local weight ( $\bar{\bar{w}}_{i}^{L}$ ) of the $i^{\rm th}$ criteria in the comparison matrix is calculated using Eq. (10) and is presented in Table 6.

Step 6:

The global weight ( $\bar{\bar{w}}_{i}^{GA}$ ) of each supplier with respect to each criterion is evaluated using Eq. (11) and are presented in Table 7.

Step 7:

After calculating the global weights of each supplier, the IT2 IF overall weights ( $\bar{\bar{w}}_{i}^{O}$ ) of each job are obtained by using Eq. (12) and are given in Table 8.

Figure 5.

Graphical comparison of the final weights of the suppliers obtained using T1 FAHP, IT2 FAHP and IT2 IF-AHP method.

Step 8:

The IT2 IF overall weights ( $\bar{\bar{w}}_{i}^{O}$ ) of the suppliers are defuzzified $(w_{i}^{*})$ (Table 8) as described in Eq. (8).

Step 9:

Finally, the defuzzified weights $(w_{i}^{*})$ are normalised $(w_{i})$ using Eq. (13) and given in Table 8. Based the defuzzified value the relative importance of the suppliers are observed.

5. Results and discussions

Considering all the four criteria ( $P, Q, D$ and $S$ ), it is evident from the final results in Table 9 that the second supplier ( $S_{2}$ ) is more suitable than the first supplier ( $S_{1}$ ). Again for the validation of the proposed approach, a comparison is done by solving the same problem with T1 FAHP [38] and IT2 FAHP [1, 21]. The results obtained at different stages by these two techniques along with the proposed method are summarized in Table 9.

Figure 6.

Graphical comparison of the individual weights of the suppliers with respect to each criteria obtained using TI FAHP, IT2 FAHP and IT2 IF-AHP method.

From Table 9, it is observed that all the three techniques declare the second supplier ( $S_{2}$ ) as the best supplier. But if the individual weights and final weights of each criteria and alternatives corresponding to each method is compared the methodological improvement can be observed. For example, if the final weights of each alternative are considered the last method seems to be more accurate in assessing the relative importance (Fig. 5). Again, if the individual weights of each criteria and alternative are considered, the proposed IT2 IF-AHP appears as most suitable method which increases the difference of crisp weights by capturing the small relative importance (Fig. 6) due to its flexible membership and non-membership values. This flexibility is reflected to the results of weights with larger distinction. In Table 9, it is observed that the distinction between the final weights of the first supplier ( $S_{1}$ ) and the second supplier ( $S_{2}$ ) using TI FAHP method is 0.120 and IT2 FAHP method is 0.232 but applying the proposed IT2 IF-AHP method the distinction is found as 0.235. Similarly, the distinction between the final weights of first supplier ( $S_{1}$ ) and the second supplier ( $S_{2}$ ) with respect to the criteria $P$ and $D$ has increased (Fig. 6) significantly using the proposed method. Also, the superiority of the proposed method is reflected by calculating the differences between the weights of suppliers ( $S_{1}$ and $S_{2}$ ) with respect to the criteria $Q$ and $S$ . Thus, based on the achieved results, the efficiency of the developed method in MCDM environment is justified.

6. Conclusions

Most of the time it is difficult to define membership and non-membership functions using IFSs since it is not possible to model uncertainty and imprecision sufficiently. The IT2 FS capture this problem by incorporating footprint of uncertainty into IFSs. In the present study, an IFS based on IT2 FS concept is developed to introduce IT2 IFS for the first time in the literature. Also, the present article proposed a defuzzify technique for ranking the IT2 IFSs.

Based on the proposed IT2 IFS, the widely used AHP method is further generalised using a new IT2 IF preference scale and the IT2 IF-AHP method is thus introduced to solve decision making problems which are surrounded by higher degrees of possibilistic uncertainties. In decision making process, DMs very often face difficulties to assign a definite IFNs corresponding to linguistic terms which are used for comparison of criteria. These kinds of difficulties can be effectively resolved by the concepts of the developed IT2 IFS which with its inherent flexibility provides extra freedom while determining the equivalent linguistic variable corresponding to a vague situation. Also, IT2 IFSs provide an extra dimension in the form of IT2 upper and lower membership and non-membership functions which make it more flexible than normal IFSs. Again, the developed method can be appeared as an effective tool when it becomes difficult for the DMs to assign exact membership and non-membership values of the parameters. Due to the flexibility, the developed IT2 IF-AHP method out performs the traditional FAHP method [33] in handling any uncertainties inherent in decision making situation more precisely. The main benefits of this approach are that it converts a complex MCDM problem into simpler hierarchy as well as reduces the inconsistency of the DMs judgements.

In this modern era, selecting an appropriate supplier plays a vital role in order to achieve goals and success of overall performances of any industry. In such situation, the present model can be used for ranking the suppliers by prioritize each criterion according to their importance. Further, due to the computational simplicity of the developed model, this approach will also help the administrators in dividing the complex decision making problem into simpler hierarchy so that they can select the best alternative.

Further the concepts of IT2 IFS can be used in the context of T1 fuzzy inference system to provide an extra degree of freedom in a justified manner. Although, the IT2 IF-AHP can be improved using T2 FSs as linguistic variables. However, it is desired that proposed methodology may open up a new vistas in the environment of multi criteria decision making.

Footnotes

Acknowledgements

The authors remain grateful to the reviewers for their comments and suggestions for improving the quality of the manuscript.

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Analytic hierarchy process based on interval type-2 intuitionistic fuzzy sets with their application to multicriteria decision making

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 T1 FS

2.2 T2 FS

2.3 IT2 FS

2.4 IFS [2,10,27]

3. Methodological development of IT2 IFS

3.1.1 Arithmetic operations on triangular IT2 IFSs

3.1.1.1 Additon

3.1.1.2 Subtraction

3.1.1.3 Multiplication

3.1.1.4 Multiplication by a crisp value

3.1.1.5 Division

3.1.1.6 Integral power of a triangular IT2 IFS

Table 2 Pairwise comparison of preferences among the criteria

Footnotes

Acknowledgements

References

Table 2
Pairwise comparison of preferences among the criteria