1 Introduction

In real life, most of the mathematical problems do not contain complete or exact information about the given problem and hence there is a big task for the decision maker to handle it before analyzing the problem. This inexact information has been handled by a theory of fuzzy set (FS) [40] and their corresponding extensions such as an intuitionistic fuzzy set (IFS) [1], type-2 fuzzy set (T2FS) [41] and so on. In the literature, various researchers [10, 11, 14, 20, 21, 36, 38] have analyzed the decision making problem under FS as well as IFS environment. Xu [36] and Xu and Yager [38] presented an averaging and geometric aggregation operators for aggregating the different IFSs information. Xu and Chen [35] and Xu [37] developed some averaging and geometric aggregation operators for aggregating the interval-valued intuitionistic fuzzy information. Garg [8] presented a generalized intuitionistic fuzzy interactive geometric aggregation operators using Einstein t-norm and t-conorm operations. Garg [9, 11] extended the theory of IFS to the Pythagorean fuzzy sets (PFSs) and presented a generalized geometric as well as averaging aggregation operators for solving the decision-making problems. Also, Garg [12, 13] presented an novel accuracy function as well as the correlation coefficient measures for the PFSs, respectively. As the above work has been considered under the FS and/or IFS environment but they have some sort of drawbacks. For instance, it is difficult, in some circumstances, for the decision maker to determine an exact membership function of a fuzzy set corresponding to its element. To overcome it, an extension of FS named as T2FS has been utilized which is characterized of the primary, secondary functions, and a footprint of uncertainty (FOU). But due to the high complexity of the T2FS, it is difficult for the decision makers to apply in the real situation. For this, an interval type-2 fuzzy set (IT2FS) [26] has been considered which contains a membership value from zero to one. After their pioneer work, various authors have applied an IT2FS theory to the field of decision making problems by using linguistic weighted average [32], ranking and arithmetic operation [4, 19], interval type-2 TOPSIS method [3, 18], ranking interval type-2 fuzzy set [5], OWA operator [45, 46], signed distance [6], triangular interval type-2 [27], distance and similarity measures [17, 33, 34] and many others [7, 15, 22, 24, 25, 30, 44].

It has been observed from the above analysis that they have conducted an analysis by considering the degree of acceptance of an element only. But, in the real world, it is not possible for a decision-maker to give their preferences toward the object under the different parameters in terms of only acceptance region (membership degree). Thus, for handling this, there is a need of the degree of non-membership degree (rejection degree) such that the sum of its membership and non-membership degree is less than or equal to one. Therefore to overcome it, a degree of membership, non-membership and their corresponding FOU have been considered during the present analysis and called a theory as an type-2 intuitionistic fuzzy set (T2IFS). To the best of our knowledge, no work has been done on T2IFS and hence no one makes a general comment about the distance measure for T2IFS.

Considering the fact the T2IFS has the great and powerful ability to model the imprecise and ambiguous information in real-world applications, this paper presents the notion of the T2IFS and hence their corresponding definition of the distance measure. Based on it, a series of some distance measures based on Hamming, Euclidean and the utmost metrics have been proposed. Various desirable properties between the proposed measures have been investigated in detail. Finally, based on these measures, a ranking method has been proposed for ordering the T2IFS. A procedure for selecting the best alternative has been explained through an illustrative example, as a group decision-making problem. The rest of the article is structured as follows: Section 2 reviews some basic definitions about T2FS, T2IFS, distance measures etc. Section 3 proposes the new normalized, weighted normalized and utmost distance measures along with their desirable properties for two T2IFSs. In Section 4, we presented a ranking method for group decision-making problems based on these measures and illustrated by numerical examples. A conclusion has been drawn in Section 5.

2 Basic concepts

2.1 Type 2 fuzzy set

Definition 1

(Type 2 fuzzy set) [23]. Let X be a fixed universe. A type-2 fuzzy set \(A \subseteq X\), is characterized by the membership function

$$\begin{array}{@{}rcl@{}} A = \{((x,u_{A}),\mu_{A}(x,u_{A}))|x\in X, u_{A}\in j_{x} \subseteq [0,1]\} \end{array} $$

in which 0≤μ A (x,u A )≤1. Another expression for A is

$$\begin{array}{@{}rcl@{}} A = {\int}_{x\in X} \mu_{A}(x)/x = {\int}_{x\in X} [ {\int}_{u_{A}\in j_{x}} f_{x}(u_{A})/u_{A}]/x, \end{array} $$

where \(\mu _{A}(x)={\int }_{x\in j_{x}}f_{x}(u_{A})/u_{A}\) is the grade of the membership, f x (u A )=μ A (x,u A ) is named as a secondary membership function (SMF) where u A denotes the primary membership function (PMF) of A and j x is named as the PMF of x.

Definition 2

(Footprint of uncertainty) [23] Uncertainty in the primary memberships of a type-2 fuzzy set consists of a boundary region that we call the “footprint of the uncertainty” (FOU). Mathematically, it is the union of all primary membership functions, i.e., \(FOU(A)=\cup _{x\in X} j_{x}\).

Definition 3

The variance margin function of T2FS is defined as the difference between PMF and SMF and is denoted by ξ. For T2FS A, ξ A =|u A (x)−f x (u A )| for all x.

Definition 4

(Distance measure for T2FSs) [28] Let F 2(X) be set of all T2FSs. A real function d:F 2(X)×F 2(X)→[0,1] is called distance measure, where d satisfies the following axioms:

  1. (P1)

    0≤d(A 1,A 2)≤1,∀A 1,A 2∈F 2(X)

  2. (P2)

    d(A 1,A 2)=0, if A 1=A 2

  3. (P3)

    d(A 1,A 2)=d(A 2,A 1)

  4. (P4)

    If d(A 1,A 2)=0,d(A 1,A 3)=0,A 3∈F 2(X) then d(A 2,A 3)=0

For convenience, two T2FSs A 1 and A 2 in X are denoted by A 1=〈x(u,f x (u A 1))∣x∈X〉 and A 2=〈x(u,f x (u A 2))∣x∈X〉. Based on these notations, [28] has proposed some distance measures for two T2FSs A 1 and A 2:

  1. ∙

    The normalized Hamming distance,

    $$\begin{array}{@{}rcl@{}} h_{2}(A_{1},A_{2}) &=& \frac{1}{2n}\sum\limits_{i=1}^{n}\bigg(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i}) \\ &&| + |f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) | + |\xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i})| \bigg) \end{array} $$
    (1)
  2. ∙

    The normalized weighted Hamming distance,

    $$\begin{array}{@{}rcl@{}} h_{2w}(A_{1},A_{2}) &=& \frac{1}{2n}\sum\limits_{i=1}^{n}\omega_{i} \bigg(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})| + |f_{x_{i}}(u_{A_{1}})\\ &&-f_{x_{i}}(u_{A_{2}}) | + |\xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i})| \bigg) \end{array} $$
    (2)
  3. ∙

    The normalized Euclidean distance

    $$\begin{array}{@{}rcl@{}} e_{2}(A_{1},A_{2}) &=& \bigg\{\frac{1}{2n}\sum\limits_{i=1}^{n}\bigg(\mid u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i}) \mid^{2} + \mid f_{x_{i}}(u_{A_{1}})\\ &&-f_{x_{i}}(u_{A_{2}}) \mid^{2} + \mid \xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i}) \mid^{2} \bigg)\bigg\}^{\frac{1}{2}} \end{array} $$
    (3)
  4. ∙

    The normalized weighted Euclidean distance

    $$\begin{array}{@{}rcl@{}} e_{2w}(A_{1},A_{2}) &=& \bigg\{\frac{1}{2n}\sum\limits_{i=1}^{n}\omega_{i} \bigg(\mid u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i}) \mid^{2} + \mid \\ && f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) \mid^{2} + \mid \\ &&\xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i}) \mid^{2} \bigg)\bigg\}^{\frac{1}{2}} \end{array} $$
    (4)

2.2 Type-2 intuitionistic fuzzy set(T2IFS)

Definition 5

A type-2 intuitionistic fuzzy set A in the universe of discourse X is set of pairs {x,μ A (x),ν A (x)} where x is the element of T2IFS, μ A (x) and ν A (x) are the grades of the membership and non-membership respectively which are defined in the interval [0,1] as

$$\begin{array}{@{}rcl@{}} \mu_{A}(x) = \int\limits_{x\in {j_{x}^{1}}} f_{x}(u_{A})/u_{A} \quad; \quad \nu_{A}(x)=\int\limits_{x\in {j_{x}^{2}}} t_{x}(v_{A})/v_{A} \end{array} $$

where f x (u A ) and t x (v A ) are named as secondary membership function (SMF) and secondary non-membership functions (SNMF). In addition, u A ,v A denotes the primary membership function (PMF) and primary non-membership functions (PNMF) and \({j_{x}^{1}}, {j_{x}^{2}}\) are named as the PMF and PNMF of x, respectively.

In other words, T2IFS A in the universe of discourse is defined as

$$\begin{array}{@{}rcl@{}} A=\bigg\{\bigg\langle(x,u_{A},v_{A}), f_{x}(u_{A}), t_{x}(v_{A}) \bigg\rangle \\ \mid x \in X, u_{A} \in {j_{x}^{1}}, v_{A} \in {j_{x}^{2}} \bigg\} \end{array} $$

where the element of the domain (x,u A ,v A ) called as PMF (u A ) and PNMF (v A ) of x∈X while f x (u A ) and t x (v A ) be the memberships of the PMF and PNMF called as the SMF and SNMF respectively where \(u_{A} \in {j_{x}^{1}} \subseteq [0,1]\), \(v_{A} \in {j_{x}^{2}} \subseteq [0,1]\). For convenience, we denote this pair to be A=〈x(u A ,f x (u A ),v A ,t x (v A ))〉 and called as type-2 intuitionistic fuzzy number (T2IFN).

Definition 6

Let A 1,A 2 be two T2IFSs then \(A_{1} \subseteq A_{2} \) if and only if \( 0 \leq f_{x}(u_{A_{1}}) \leq f_{x}(u_{A_{2}}) \leq 1, \forall u_{A_{1}}, u_{A_{2}} \in {j_{x}^{1}} \subseteq [0,1] \) and \( 0 \leq t_{x}(v_{A_{1}}) \leq t_{x}(v_{A_{2}}) \leq 1, \forall v_{A_{1}}, v_{A_{2}} \in {j_{x}^{2}} \subseteq [0,1]\).

Definition 7

The variance margin function (VMF) of T2IFS is defined as the difference between PMF and PNMF, and SMF and SNMF. It is denoted by ξ and η i.e., for T2IFS A, variance margin functions are ξ A =∣u A (x i )−f x i (u A )∣ and η A =∣v A (x i )−t x i (v A )∣∀i.

3 Distance measures between T2IFS

In this section, we present the Hamming and the Euclidean distances between T2IFNs which can be used in real scientific and engineering applications. Let \({F_{2}^{I}}(X)\) be the class of T2IFSs over the universal set X.

Definition 8

A real function \(d:{F_{2}^{I}}(X)\times {F_{2}^{I}}(X) \to [0,1]\) is called distance measure, where d satisfies the following axioms:

  1. (P1)

    \( 0 \leq d(A_{1},A_{2}) \leq 1, \forall A_{1},A_{2} \in {F_{2}^{I}}(X) \)

  2. (P2)

    d(A 1,A 2)=0, if A 1=A 2

  3. (P3)

    d(A 1,A 2)=d(A 2,A 1)

  4. (P4)

    If \( d(A_{1},A_{2}) = 0, d(A_{1},A_{3}) = 0, A_{3} \in {F_{2}^{I}}(X) \) then d(A 2,A 3)=0

For convenience, two T2IFSs A 1 and A 2 in X are denoted by A 1=〈x(u,f x (u A 1),v, t x (v A 1))∣x∈X〉 and A 2=〈x(u,f x (u A 2),v,t x (v A 2))∣x∈X〉. Then we define the following distances for A 1 and A 2 by considering the PMF, PNMF, SMF, SNMF, FOU and VMF.

  1. (i)

    The Hamming distance,

    $$\begin{array}{@{}rcl@{}} d_{1}(A_{1},A_{2}) &=& \frac{1}{4}\sum\limits_{i=1}^{n}\bigg(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})| \\ && + |f_{x_{i}}(u_{A_{1}}\!)\,-\,\!f_{x_{i}}(u_{A_{2}}\!) | \!+ |\xi_{A_{1}}(x_{i})\\&&-\xi_{A_{2}}(x_{i})| \\ && + |v_{A_{1}}(x_{i})\,-\,v_{A_{2}}(x_{i}) |+ |t_{x_{i}}(v_{A_{1}})\\&&-t_{x_{i}}(v_{A_{2}}) | + | \eta_{A_{1}} \\ && (x_{i})-\eta_{A_{2}}(x_{i}) | \bigg) \end{array} $$
    (5)
  2. (ii)

    The normalized Hamming distance,

    $$\begin{array}{@{}rcl@{}} d_{2}(A_{1},A_{2}) &=& \frac{1}{4n}\sum\limits_{i=1}^{n}\bigg(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})| \\&&+ |f_{x_{i}}(u_{A_{1}})\\&&-f_{x_{i}}(u_{A_{2}})| \\ && + |\xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i})| \\ && + |v_{A_{1}}\!(x_{i})\!-v_{A_{2}}\!(x_{i}) |\,+\, |t_{x_{i}}(v_{A_{1}})\\&&-t_{x_{i}}(v_{A_{2}}) | \\ && + | \eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i}) | \bigg) \end{array} $$
    (6)
  3. (iii)

    The Euclidean distance

    $$\begin{array}{@{}rcl@{}} d_{3}(A_{1},A_{2}) &=& \bigg\{\frac{1}{4}\sum\limits_{i=1}^{n}\bigg(\mid u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i}) \mid^{2}\\ &&+ \mid f_{x_{i}}(u_{A_{1}}) \\ &&-f_{x_{i}}(u_{A_{2}}) \mid^{2} + \mid \xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i}) \mid^{2} \\ && + \mid v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) \mid^{2} + \mid t_{x_{i}}(v_{A_{1}}) \\ && -t_{x_{i}}(v_{A_{2}}) \mid^{2} + \mid \eta_{A_{1}}(x_{i})\\&&-\eta_{A_{2}}(x_{i}) \mid^{2}\bigg)\bigg\}^{\frac{1}{2}} \end{array} $$
    (7)
  4. (iv)

    The normalized Euclidean distance

    $$\begin{array}{@{}rcl@{}} d_{4}(A_{1},A_{2}) &=& \bigg\{\frac{1}{4n}\sum\limits_{i=1}^{n}\bigg(\mid u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i}) \mid^{2} + \\ && \mid f_{x_{i}}(u_{A_{1}}) -f_{x_{i}}(u_{A_{2}}) \mid^{2} + \mid \xi_{A_{1}}(x_{i})\\&&-\xi_{A_{2}}(x_{i}) \mid^{2} \\ && + \mid v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) \mid^{2} + \mid t_{x_{i}}(v_{A_{1}}) \\ &&-t_{x_{i}}(v_{A_{2}}) \mid^{2} + \mid \eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i}) \mid^{2}\bigg)\bigg\}^{\frac{1}{2}} \end{array} $$
    (8)

Then based on the distance properties as defined in Definition 8, we can obtain the following propositions:

Proposition 1

The above defined distance d k (A 1 ,A 2 ), (k=2,4) between T2IFSs A 1 and A 2 satisfies the following properties (P1)-(P4):

  1. (P1)

    \( 0 \leq d_{k}(A_{1},A_{2}) \leq 1, \forall A_{1},A_{2} \in {F_{2}^{I}}(X) \)

  2. (P2)

    d k (A 1 ,A 2 )=0, if A 1 =A 2

  3. (P3)

    d k (A 1 ,A 2 )=d k (A 2 ,A 1)

  4. (P4)

    If \( d_{k}(A_{1},A_{2}) = 0, d_{k}(A_{1},A_{3}) = 0, A_{3} \in {F_{2}^{I}}(X) \) then d k (A 2 ,A 3) = 0

Proof

For p=1,2, we have

  1. (P1)

    Since A 1 and A 2 are T2IFSs, we have,\(| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p} \geq 0\), \(| f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |^{p} \geq 0\), \(| \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p} \geq 0\), \(| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} \geq 0\), \(| t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} \geq 0\) and \(| \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \geq 0\). Thus, \(| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p} + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p} + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p} + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} + | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \geq 0\) which implies that d k (A 1,A 2)≥0. Further, \(| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p} \leq 1\), \(| f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p} \leq 1\), \(| \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p} \leq \) 1, \(| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} \leq 1\), \(| t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} \leq 1\) and \(| \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \leq 1\). Therefore, \({\sum }_{i=1}^{n} |u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p} + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p} + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p} + |v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} + |t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} + | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \leq 4\) \(\Rightarrow \) d k (A 1,A 2)≤1. Thus 0≤d k (A 1,A 2)≤1.

  2. (P2)

    Let d k (A 1,A 2)=0, \(\Leftrightarrow | u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p} + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p} + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p} + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} + | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} = 0\) \(\Leftrightarrow \) \(u_{A_{1}}(x_{i})=u_{A_{2}}(x_{i})\), \(f_{x_{i}}(u_{A_{1}}) = f_{x_{i}}(u_{A_{2}})\), \(\xi _{A_{1}}(x_{i})=\xi _{A_{2}}(x_{i})\), \(v_{A_{1}}(x_{i})=v_{A_{2}}(x_{i})\), \(t_{x_{i}}(v_{A_{1}})=t_{x_{i}}(v_{A_{2}})\), \(\eta _{A_{1}}(x_{i})=\eta _{A_{2}}(x_{i})\). Therefore A 1=A 2.

  3. (P3)

    \(d_{k}(A_{1},A_{2}) = \frac {1}{4n} {\sum }_{i=1}^{n} \big [ |u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p}+ | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p} + |\xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p}\) + \(| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} + |t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} + | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \big ]\) = \(\frac {1}{4n} {\sum }_{i=1}^{n} \big [ |u_{A_{2}}(x_{i})-u_{A_{1}}(x_{i})|^{p} + |f_{x_{i}}(u_{A_{2}})-f_{x_{i}}(u_{A_{1}})|^{p} + | \xi _{A_{2}}(x_{i})-\xi _{A_{1}}(x_{i})|^{p}\) + \(| v_{A_{2}}(x_{i})-v_{A_{1}}(x_{i})|^{p} + |t_{x_{i}}(v_{A_{2}})-t_{x_{i}}(v_{A_{1}})|^{p} + | \eta _{A_{2}}(x_{i})-\eta _{A_{1}}(x_{i})|^{p} \big ]\) = d k (A 2,A 1)

  4. (P4)

    Now, \(d_{k}(A_{1},A_{2})=0 \Rightarrow u_{A_{1}}(x_{i})=u_{A_{2}}(x_{i})\), \(f_{x_{i}}(u_{A_{1}}) = f_{x_{i}}(u_{A_{2}})\), \(\xi _{A_{1}}(x_{i})=\xi _{A_{2}}(x_{i})\), \(v_{A_{1}}(x_{i})=v_{A_{2}}(x_{i})\), \(t_{x_{i}}(v_{A_{1}})=t_{x_{i}}(v_{A_{2}})\), \(\eta _{A_{1}}(x_{i})=\eta _{A_{2}}(x_{i})\) and \(d_{k}(A_{1},A_{3})=0 \Rightarrow u_{A_{1}}(x_{i})=u_{A_{3}}(x_{i})\), \(f_{x_{i}}(u_{A_{1}}) = f_{x_{i}}(u_{A_{3}})\), \(\xi _{A_{1}}(x_{i})=\xi _{A_{3}}(x_{i})\), \(v_{A_{1}}(x_{i})=v_{A_{3}}(x_{i})\), \(t_{x_{i}}(v_{A_{1}})=t_{x_{i}}(v_{A_{3}})\), \(\eta _{A_{1}}(x_{i})=\eta _{A_{3}}(x_{i})\), therefore \(u_{A_{2}}(x_{i})=u_{A_{3}}(x_{i})\), \(f_{x_{i}}(u_{A_{2}}) = f_{x_{i}}(u_{A_{3}})\), \(\xi _{A_{2}}(x_{i})=\xi _{A_{3}}(x_{i})\), \(v_{A_{2}}(x_{i})=v_{A_{3}}(x_{i})\), \(t_{x_{i}}(v_{A_{2}})=t_{x_{i}}(v_{A_{3}})\), \(\eta _{A_{2}}(x_{i})=\eta _{A_{3}}(x_{i})\) \(\Rightarrow d_{k}(A_{2},A_{3})=0\).

Hence, d k (A 1,A 2), (k=2,4) are distance measures. □

Proposition 2

Distance measures d 1 and d 3 satisfies the following properties

  1. 1.

    0≤d 1 ≤n

  2. 2.

    0≤d 3 ≤n 1/2

However, in many practical situations, the different set may have taken different weights and thus, weight ω i (i= 1,2,…,n) with ω i ≥0, \(\sum \limits _{i=1}^{n} \omega _{i} = 1\), of the element x i ∈X should be taken into account. In the following, we develop a normalized weighted Hamming and the normalized weighted Euclidean distance measure between T2IFSs.

  1. (i)

    The normalized weighted Hamming distance,

    $$\begin{array}{@{}rcl@{}} d_{5}(A_{1},A_{2}) &=& \frac{1}{4n}\sum\limits_{i=1}^{n} \omega_{i}\bigg(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i}) |\\&&+ | f_{x_{i}}(u_{A_{1}}) \\ &&-f_{x_{i}}(u_{A_{2}}) | + | \xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i}) | \\ &&+ |v_{A_{1}}(x_{i})\,-\,v_{A_{2}}(x_{i}) | \,+\, | t_{x_{i}}(v_{A_{1}})\\&&-t_{x_{i}}(v_{A_{2}}) | \\ &&+ | \eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i})| \bigg) \end{array} $$
    (9)
  2. (ii)

    The normalized weighted Euclidean distance,

    $$\begin{array}{@{}rcl@{}} d_{6}(A_{1},A_{2}) \!&=&\! \bigg\{\frac{1}{4n}\sum\limits_{i=1}^{n}\omega_{i}\bigg(| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{2} \\ && +| f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |^{2} + | \xi_{A_{1}}(x_{i}) \\ &&-\xi_{A_{2}}(x_{i}) |^{2} + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{2} \\ && + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{2} \\ && + | \eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i}) |^{2}\bigg)\bigg\}^{\frac{1}{2}} \end{array} $$
    (10)

Proposition 3

Let ω i is the weight vector of the element x i ∈X, then the weighted distance measure d k (A 1 ,A 2 ), (k=5,6) also satisfies the following properties (P1)-(P4):

  1. (P1)

    \( 0 \leq d_{k}(A_{1},A_{2}) \leq 1, \forall A_{1},A_{2} \in {F_{2}^{I}}(X) \)

  2. (P2)

    d k (A 1 ,A 2 )=0, if A 1 =A 2

  3. (P3)

    d k (A 1 ,A 2 )=d k (A 2 ,A 1)

  4. (P4)

    If \( d_{k}(A_{1},A_{2}) = 0, d_{k}(A_{1},A_{3}) = 0, A_{3} \in {F_{2}^{I}}(X) \) then d k (A 2 ,A 3)=0

Proof

Since ω i ∈[0,1] and \(\sum \limits _{i=1}^{n} \omega _{i} = 1\) then we can easily obtain 0≤d 5(A 1,A 2)≤d 2(A 1,A 2). Thus, d 5(A 1,A 2) satisfies (P1). The proofs of (P2)−(P4) are similar to those of Proposition 1. Similarly for d 6. □

Further, it can be easily verify that when ω i =1/n for all i=1,2,…,n, then Eqs. (9) and (10) reduce to Eqs. (6) and (9) respectively.

Proposition 4

Distance measures d 2 and d 5 satisfies the relation d 5 ≤d 2.

Proof

Consider two T2IFSs A 1 and A 2. Since \(\omega _{i}\geq 0, {\sum }_{i=1}^{n} \omega _{i} = 1\) then we have, \(d_{5}(A_{1},A_{2}) = \frac {1}{4n} {\sum }_{i=1}^{n} \omega _{i} \big [ | u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})| + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})| + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})| + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})| + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})| + | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})| \big ]\) \(=\frac {1}{4}\big (\omega _{1} \big [ | u_{A_{1}}(x_{1})-u_{A_{2}}(x_{1})| + | f_{x_{1}}(u_{A_{1}})-t_{x_{1}}(u_{A_{2}})| + | \xi _{A_{1}}(x_{1})-\xi _{A_{2}}(x_{1})| + | v_{A_{1}}(x_{1})-v_{A_{2}}(x_{1})| + | t_{x_{1}}(v_{A_{1}})-t_{x_{1}}(v_{A_{2}})| + | \eta _{A_{1}}(x_{1})-\eta _{A_{2}}(x_{1})| \big ]+ \omega _{2} \big [ | u_{A_{1}}(x_{2})-u_{A_{2}}(x_{2})| + | f_{x_{2}}(u_{A_{1}})-f_{x_{2}}(u_{A_{2}})| + | \xi _{A_{1}}(x_{2})-\xi _{A_{2}}(x_{2})| + | v_{A_{1}}(x_{2})-v_{A_{2}}(x_{2})| + | t_{x_{2}}(v_{A_{1}})-t_{x_{2}}(v_{A_{2}})| + | \eta _{A_{1}}(x_{2})-\eta _{A_{2}}(x_{2})|\big ] + {\ldots } + \omega _{n} \big [ | u_{A_{1}}(x_{n})-u_{A_{2}}(x_{n})| + | f_{x_{n}}(u_{A_{1}})-f_{x_{n}}(u_{A_{2}})| + | \xi _{A_{1}}(x_{n})-\xi _{A_{2}}(x_{n})| + | v_{A_{1}}(x_{n})-v_{A_{2}}(x_{n})| + | t_{x_{n}}(v_{A_{1}})-t_{x_{n}}(v_{A_{2}})| + | \eta _{A_{1}}(x_{n})-\eta _{A_{2}}(x_{n})|\big ] \big )\). As ω i ∈[0,1] thus, \(d_{5}(A_{1},A_{2})\leq \frac {1}{4n}\big (\big [ | u_{A_{1}}(x_{1})-u_{A_{2}}(x_{1})| + | f_{x_{1}}(u_{A_{1}})-f_{x_{1}}(u_{A_{2}})| + | \xi _{A_{1}}(x_{1})-\xi _{A_{2}}(x_{1})| + | v_{A_{1}}(x_{1})-v_{A_{2}}(x_{1})| + | t_{x_{1}}(v_{A_{1}})-t_{x_{1}}(v_{A_{2}})| + | \eta _{A_{1}}(x_{1})-\eta _{A_{2}}(x_{1})| \big ]+ \big [ | u_{A_{1}}(x_{2})-u_{A_{2}}(x_{2})| + | f_{x_{2}}(u_{A_{1}})-f_{x_{2}}(u_{A_{2}})| + | \xi _{A_{1}}(x_{2})-\xi _{A_{2}}(x_{2})| + | v_{A_{1}}(x_{2})-v_{A_{2}}(x_{2})| + | t_{x_{2}}(v_{A_{1}})-t_{x_{2}}(v_{A_{2}})| + | \eta _{A_{1}}(x_{2})-\eta _{A_{2}}(x_{2})|\big ] + {\ldots } + \big [ | u_{A_{1}}(x_{n})-u_{A_{2}}(x_{n})| + | f_{x_{n}}(u_{A_{1}})-f_{x_{n}}(u_{A_{2}})| + | \xi _{A_{1}}(x_{n})-\xi _{A_{2}}(x_{n})| + | v_{A_{1}}(x_{n})-v_{A_{2}}(x_{n})| + | t_{x_{n}}(v_{A_{1}})-t_{x_{n}}(v_{A_{2}})| + | \eta _{A_{1}}(x_{n})-\eta _{A_{2}}(x_{n})|\big ] \big ) \leq \frac {1}{4n}{\sum }_{i=1}^{n} \big (\mid u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i}) \mid + \mid f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) \mid + \mid \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i}) \mid + |v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})| + |t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})| + |\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|\big ) = d_{2}(A_{1},A_{2})\). Hence, d 5(A 1,A 2)≤d 2(A 1,A 2). Since A 1 and A 2 are arbitrary T2IFSs and therefore d 5≤d 2. □

Proposition 5

Distance measures d 4 and d 6 satisfies the relation d 6 ≤d 4.

Proof

Follows from Proposition 4. □

Proposition 6

Distance measures d 2 and d 4 satisfies the inequalities \(\sqrt {d_{2}} \geq d_{4}\).

Proof

For any two T2IFSs A 1 and A 2, we have \(0\leq u_{A_{1}}, u_{A_{2}} \leq 1\), \(0\leq v_{A_{1}}, v_{A_{2}} \leq 1\), \(0\leq f_{x_{i}}(u_{A_{1}}), f_{x_{i}}(u_{A_{2}}) \leq 1\), \(0\leq \xi _{A_{1}}, \xi _{A_{2}} \leq 1\), \(0\leq t_{x_{i}}(v_{A_{1}}), t_{x_{i}}(v_{A_{2}}) \leq 1\) and \(0\leq \eta _{A_{1}}, \eta _{A_{2}} \leq 1\). Thus, \(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{2} \leq |u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|\), \(|f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |^{2}\leq |f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |\), \(| \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{2}\leq | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|\), \(| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |^{2}\leq | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |\), \( | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) |^{2} \leq | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) |\), \(|\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) |^{2} \leq |\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) |\). Now, \(d_{4}(A_{1},A_{2}) = \bigg \{\frac {1}{4n}{\sum }_{i=1}^{n}\bigg (| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{2} + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |^{2} + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{2} + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |^{2} + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) |^{2} + |\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) |^{2}\bigg )\bigg \}^{\frac {1}{2}}\) which implies that \(d_{4}(A_{1},A_{2}) \leq \bigg \{\frac {1}{4n}{\sum }_{i=1}^{n}\bigg (| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})| + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) | + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})| + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) | + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) | + |\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) |\bigg )\bigg \}^{\frac {1}{2}} = (d_{2}(A_{1},A_{2}))^{1/2}\). Hence, \(d_{4} \leq \sqrt {d_{2}}\). □

Proposition 7

Distance measures d 5 and d 6 satisfies the inequalities \(d_{6} \leq \sqrt {d_{5}}\)

Proof

Follows from Proposition 6. □

Proposition 8

Distance measures d 2 and d 6 satisfies the inequalities \(d_{6} \leq \sqrt {d_{2}}\)

Proof

For any two arbitrary T2IFSs A 1 and A 2, we have \(d_{6}(A_{1},A_{2})= \bigg \{\frac {1}{4n}{\sum }_{i=1}^{n}\omega _{i} \bigg (| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{2} + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |^{2} + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{2} + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |^{2} + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) |^{2} + |\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) |^{2}\bigg )\bigg \}^{1/2} \leq \bigg \{\frac {1}{4n}{\sum }_{i=1}^{n}\bigg (| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{2} + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |^{2} + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{2} + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |^{2} + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) |^{2} + |\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) |^{2}\bigg )\bigg \}^{1/2} \leq \bigg \{ \frac {1}{4n}{\sum }_{i=1}^{n}\bigg (| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})| + | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) | + | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})| + | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) | + | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) | + |\eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) |\bigg )\bigg \}^{1/2}=(d_{2}(A_{1},A_{2}))^{1/2}\). Hence, \(d_{6} \leq \sqrt {d_{2}}\). □

Hung and Yang [17] proposed a Hausdorff metric for two intervals A=[a 1,a 2] and B=[b 1,b 2], as follows:

$$\begin{array}{@{}rcl@{}} d(A,B) = \max\{|a_{1}-a_{2}|, |b_{1}-b_{2}|\} \end{array} $$

Now, for any two T2IFSs A 1 and A 2 over X={x 1,x 2,…,x n }, we propose the following utmost distance measures:

  1. (i)

    utmost normalized Hamming distance

    $$\begin{array}{@{}rcl@{}} {d_{1}^{U}}(A_{1},A_{2})&=&\frac{1}{4n}\sum\limits_{i=1}^{n} \max\bigg\{|u_{A_{1}}(x_{i})\\&&-u_{A_{2}}(x_{i})|, | f_{x_{i}}(u_{A_{1}})\\ &&-f_{x_{i}}(u_{A_{2}}) |, |\xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i})|,\\ &&| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |, | t_{x_{i}}(v_{A_{1}}) \\ &&-t_{x_{i}}(v_{A_{2}}) |, |\eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i}) |\bigg\} \end{array} $$
  2. (ii)

    utmost normalized weighted Hamming distance

    $$\begin{array}{@{}rcl@{}} {d_{2}^{U}}(A_{1},A_{2})\!&=&\!\frac{1}{4n}\sum\limits_{i=1}^{n} \omega_{i}\max\!\bigg\{\!|u_{A_{1}}(x_{i})\,-\,u_{A_{2}}(x_{i})|,\\ &&| f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |, |\xi_{A_{1}}\\&&-\xi_{A_{2}}(x_{i})|, \\ &&| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |, | t_{x_{i}}(v_{A_{1}})\\&&-t_{x_{i}}(v_{A_{2}}) |,\\ &&|\eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i}) |\bigg\} \end{array} $$
  3. (iii)

    utmost normalized Euclidean distance

    $$\begin{array}{@{}rcl@{}} {d_{3}^{U}}(A_{1},A_{2})\!&=&\!\bigg\{\frac{1}{4n}\sum\limits_{i=1}^{n} \max\!\bigg(|u_{A_{1}}(x_{i})\,-\,u_{A_{2}}(x_{i})|^{2}\!, \\ &&| f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{2}, |\xi_{A_{1}}(x_{i})\\&&-\xi_{A_{2}}(x_{i})|^{2}, \\ &&| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |^{2}, | t_{x_{i}}(v_{A_{1}})\\&&-t_{x_{i}}(v_{A_{2}}) |^{2}, \\ &&|\eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i}) |^{2} \bigg) \bigg\}^{1/2} \end{array} $$
  4. (iv)

    utmost normalized weighted Euclidean distance

    $$\begin{array}{@{}rcl@{}} {d_{4}^{U}}(A_{1},A_{2})\!&=&\!\!\bigg\{\frac{1}{4n}\sum\limits_{i=1}^{n} \omega_{i}\max\bigg(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{2}, \\ &&| f_{x_{i}}(u_{A_{1}})\,-\,f_{x_{i}}(u_{A_{2}}) |^{2}, |\xi_{A_{1}}(x_{i})-\xi_{A_{2}}(x_{i})|^{2}, \\ &&| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |^{2}, | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) |^{2}, \\ &&|\eta_{A_{1}}(x_{i})-\eta_{A_{2}}(x_{i}) |^{2} \bigg) \bigg\}^{1/2} \end{array} $$

Proposition 9

For \(A_{1},A_{2}\in {F_{2}^{I}}(X)\) , \({d_{k}^{U}}\) (k=1,3) is the distance measure.

Proof

For p=1,2 and \(A_{1},A_{2}\in {F_{2}^{I}}(X)\) with n attribute. Then we have to prove that \({d_{k}^{U}}\), (k=1,3) satisfies the axioms (P1)-(P4) as described in Proposition 1.

  1. (P1)

    As A 1 and A 2 be T2IFSs so \(|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p}\geq 0\), \(| f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |^{p} \geq 0\), \(| \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p} \geq 0\), \(| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} \geq 0\), \(| t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} \geq 0\) and \(| \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \geq 0\). Thus, \(\max \big \{| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p}, |f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p}, |\xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p}, | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p}, | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p}, | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p}\big \} \geq 0\) \(\Rightarrow {d_{k}^{U}}(A_{1},A_{2}) \geq 0\). Further, \(| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p} \leq 1\), \(| f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p} \leq 1\), \(| \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p} \leq 1\), \(| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p} \leq 1\), \(| t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p} \leq 1\) and \(| \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \leq 1\). Therefore, \(\frac {1}{4n}{\sum }_{i=1}^{n} \max \big \{|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p}, | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p}, | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p}, |v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p}, |t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p}, | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p}\big \} \leq 1\) \(\Rightarrow \) \({d_{k}^{U}}(A_{1},A_{2})\leq 1\). Thus \(0\leq {d_{k}^{U}}(A_{1},A_{2})\leq 1\).

  2. (P2)

    Let \({d_{k}^{U}}(A_{1},A_{2})= 0\), \(\Leftrightarrow \max \big \{| u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p}, | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p}, | \xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p}, | v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p}, | t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p}, | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \big \} = 0\) \(\Leftrightarrow \) \(u_{A_{1}}(x_{i})=u_{A_{2}}(x_{i})\), \(f_{x_{i}}(u_{A_{1}}) = f_{x_{i}}(u_{A_{2}})\), \(\xi _{A_{1}}(x_{i})=\xi _{A_{2}}(x_{i})\), \(v_{A_{1}}(x_{i})=v_{A_{2}}(x_{i})\), \(t_{x_{i}}(v_{A_{1}})=t_{x_{i}}(v_{A_{2}})\), \(\eta _{A_{1}}(x_{i})=\eta _{A_{2}}(x_{i})\). Therefore A 1=A 2.

  3. (P3)

    \({d_{k}^{U}}(A_{1},A_{2}) = \frac {1}{4n} {\sum }_{i=1}^{n} \max \big [ |u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|^{p}, | f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}})|^{p}, |\xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|^{p}\), \(| v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i})|^{p}, |t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}})|^{p}, | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i})|^{p} \big ]\) = \(\frac {1}{4n} {\sum }_{i=1}^{n} \max \big [ |u_{A_{2}}(x_{i})-u_{A_{1}}(x_{i})|^{p}, |f_{x_{i}}(u_{A_{2}})-f_{x_{i}}(u_{A_{1}})|^{p}\), \(| \xi _{A_{2}}(x_{i})-\xi _{A_{1}}(x_{i})|^{p}\), \(| v_{A_{2}}(x_{i})-v_{A_{1}}(x_{i})|^{p}, |t_{x_{i}}(v_{A_{2}})-t_{x_{i}}(v_{A_{1}})|^{p}, | \eta _{A_{2}}(x_{i})-\eta _{A_{1}}(x_{i})|^{p}\big ]\) = \({d_{k}^{U}}(A_{2},A_{1})\)

  4. (P4)

    Now, \({d_{k}^{U}}(A_{1},A_{2})=0 \Rightarrow u_{A_{1}}(x_{i})=u_{A_{2}}(x_{i})\), \(f_{x_{i}}(u_{A_{1}}) = f_{x_{i}}(u_{A_{2}})\), \(\xi _{A_{1}}(x_{i})=\xi _{A_{2}}(x_{i})\), \(v_{A_{1}}(x_{i})=v_{A_{2}}(x_{i})\), \(t_{x_{i}}(v_{A_{1}})=t_{x_{i}}(v_{A_{2}})\), \(\eta _{A_{1}}(x_{i})=\eta _{A_{2}}(x_{i})\) and \({d_{k}^{U}}(A_{1},A_{3})=0 \Rightarrow u_{A_{1}}(x_{i})=u_{A_{3}}(x_{i})\), \(f_{x_{i}}(u_{A_{1}}) = f_{x_{i}}(u_{A_{3}})\), \(\xi _{A_{1}}(x_{i})=\xi _{A_{3}}(x_{i})\), \(v_{A_{1}}(x_{i})=v_{A_{3}}(x_{i})\), \(t_{x_{i}}(v_{A_{1}})=t_{x_{i}}(v_{A_{3}})\), \(\eta _{A_{1}}(x_{i})=\eta _{A_{3}}(x_{i})\), therefore \(u_{A_{2}}(x_{i})=u_{A_{3}}(x_{i})\), \(f_{x_{i}}(u_{A_{2}}) = f_{x_{i}}(u_{A_{3}})\), \(\xi _{A_{2}}(x_{i})=\xi _{A_{3}}(x_{i})\), \(v_{A_{2}}(x_{i})=v_{A_{3}}(x_{i})\), \(t_{x_{i}}(v_{A_{2}})=t_{x_{i}}(v_{A_{3}})\), \(\eta _{A_{2}}(x_{i})=\eta _{A_{3}}(x_{i})\) \(\Rightarrow {d_{k}^{U}}(A_{2},A_{3})=0\).

□

Proposition 10

For \(A_{1},A_{2}\in {F_{2}^{I}}(X)\) , \({d_{2}^{U}}\) and \({d_{4}^{U}}\) are the distance measures.

Proof

Follows from above proposition. □

Proposition 11

For \(A_{1},A_{2}\in {F_{2}^{I}}(X)\) , \({d_{2}^{U}}\) , \({d_{1}^{U}}\) and \({d_{3}^{U}}\) , \({d_{4}^{U}}\) respectively, satisfies the following inequalities

  1. (i)

    \({d_{2}^{U}} \leq {d_{1}^{U}}\)

  2. (ii)

    \({d_{4}^{U}} \leq {d_{3}^{U}}\)

Proof

Proof follow from the definitions of utmost measures as ω i ∈[0,1] for all i. □

Proposition 12

Measures \({d_{1}^{U}}\) and d 2 have the following inequality

$$d_{2}\geq {d_{1}^{U}}.$$

Proof

Since for any positive numbers a i ,i=1,2,…,n, \({\sum }_{i=1}^{n} a_{i} \geq \underset {i}\max \{a_{i}\}\). Thus, for any T2IFSs A 1 and A 2, we have \(d_{2}(A_{1},A_{2})= \frac {1}{4n}{\sum }_{i=1}^{n}\big (|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})| + |f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) | + |\xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})| + |v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |+ |t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) | + | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) | \big ) \geq \frac {1}{4n}{\sum }_{i=1}^{n}\max \big (|u_{A_{1}}(x_{i})-u_{A_{2}}(x_{i})|, |f_{x_{i}}(u_{A_{1}})-f_{x_{i}}(u_{A_{2}}) |, |\xi _{A_{1}}(x_{i})-\xi _{A_{2}}(x_{i})|, |v_{A_{1}}(x_{i})-v_{A_{2}}(x_{i}) |, |t_{x_{i}}(v_{A_{1}})-t_{x_{i}}(v_{A_{2}}) |, | \eta _{A_{1}}(x_{i})-\eta _{A_{2}}(x_{i}) | \big )\) \(\Rightarrow d_{2}(A_{1},A_{2})\geq {d_{1}^{U}}(A_{1},A_{2})\). Hence, \(d_{2}\geq {d_{1}^{U}}\). □

Proposition 13

The measures \({d_{3}^{U}}\) and d 4 satisfies the following inequality

$$d_{4}\geq {d_{3}^{U}}.$$

Proof is similar as that of above.

Proposition 14

Measures d 2 , d 5 and \({d_{1}^{U}}\) satisfies the following inequalities

  1. (i)

    \(d_{2} \geq \frac {d_{5} + {d_{1}^{U}}}{2}\)

  2. (ii)

    \(d_{2} \geq \sqrt {d_{5}\cdot {d_{1}^{U}}}\)

Proof

Since d 2≥d 5 and \(d_{2}\geq {d_{1}^{U}}\). So by adding these inequalities, we get \(d_{2} \geq \frac {d_{5} + {d_{1}^{U}}}{2}\). On the other hand, by multiplying these, we get \(d_{2} \geq \sqrt {d_{5}\cdot {d_{1}^{U}}}\). □

4 Group decision making with T2IFSs based on distance measures

In this section, we propose a method for ranking the different T2IFSs based on the proposed distance measures for group decision-making problems.

4.1 Distance measure based approach

For this, consider a finite set of ‘n’ criteria C = {C 1,C 2,…,C n } and a set of ‘t’ alternatives A={A 1,A 2,…,A t } which are evaluated by the set of decision makers D M={D M 1,D M 2,…,D M k } whose weight vector is ω=(ω 1,ω 2,…,ω n )T where ω i ≥0,i=1,2,…,n and \(\sum \limits _{i=1}^{n}\omega _{i} =1\). Assume that the decision-maker provides the rating values corresponding to each alternative in terms of a PMF, SMF, PNMF, SNMF. Then the following steps have been described for finding the best alternative(s).

  1. (Step 1:)

    Arrange the collective information for each alternative with respect to criteria in terms of PMF, SMF, PNMF and SNMF.

  2. (Step 2:)

    Compute the distance measure between the decision makers D M k and the null decision N, i.e. d(D M k ,N), where N is a decision of the decision maker, having zero PMF and SMF while one PNMF and SNMF for each alternative A j corresponding to each criteria C i .

  3. (Step 3:)

    Find the maximum value of the distance measures corresponding to DM and hence construct the type-2 intuitionistic fuzzy alternative A j , j=1,2,…,t by considering all the alternative, criteria C i and their corresponding maximum value of distance measure d.

  4. (Step 4:)

    Calculate the distance measure between the alternative A j and the null decision N, i.e. d(P j ,N).

  5. (Step 5:)

    Rank the alternatives A j ,j=1,2,…,t and hence get the best one.

  6. (Step 6:)

    End.

4.2 Numerical example

Consider a decision-making problem in which a person wants to invest some money in to the market. For it, they bought a certain panel of experts (D M 1,D M 2 and D M 3) having weight vector is (0.40,0.35,0.25)T for the five alternatives such as (i) A 1 is a car company, (ii) A 2 is food company, (iii) A 3 is a computer company, (iv) A 4 is arms company, (v) A 5 is tire company. The investor takes a decision under the different criteria, namely, C 1 is the risk analysis, C 2 is the growth analysis, C 3 is the environmental impact analysis and C 4 is the available space whose weight vector is (0.35,0.30,0.20,0.15)T under the T2IFS environment. For this, the linguistic grades of PMF, PNMF and SMF, SNMF are shown in Tables 1 and 2 respectively.

  1. (Step 1:)

    Based on the decision makers’ knowledge and experience, the collective information of each alternative corresponding to each criteria are arranged in terms of the linguistic grades and are given in Table 3.

  2. (Step 2:)

    Calculate d(D M k ,N), (k=1,2,3) for each alterative. For the sake of calculation, we use d 2(D M k ,N) and their values are summarized in Table 4.

  3. (Step 3:)

    Find the maximum value of d 2(D M k ,N) from Table 4 for all alternatives A j , (j=1,2,…,5) corresponding to each criteria C i ,(i=1,2,3,4) and hence construct the T2IFS alternative A j ={C i (u A j ,f C i (A j ), \(v_{A_{j}},t_{C_{i}}(A_{j}))\}\) as

    $$\begin{array}{@{}rcl@{}} A_{1} &=& \{C_{1} (1,0.7,0,0.2), C_{2} (1,0.9,0,0.1),\\ && C_{3} (1,0.7,0,0.2), C_{4} (0.9,1,0.1,0)\}; \\ A_{2} &=& \{C_{1} (0.9,0.7,0,0.1), C_{2} (0.9,0.7,0.1,0.2), \\ && C_{3} (1,0.7,0,0.2), C_{4} (0.9,1,0.1,0)\}; \\ A_{3} &=& \{C_{1} (0.9,0.5,0.1,0.4), C_{2} (0.9,1,0.1,0), \\ && C_{3} (1.0,0.9,0,0.1), C_{4} (1,0.4,0,0.5)\}; \\ A_{4} &=& \{C_{1} (0.9,0.5,0.1,0.4), C_{2} (0.9,0.7,0.1,0.2), \\ && C_{3} (1,0.9,0,0.1), C_{4} (0.9,0.5,0.1,0.4)\}; \\ A_{5} &=& \{C_{1} (0.9,0.7,0,0.1), C_{2} (1,0.9,0,0.1), \\ && C_{3} (0.9,0.4,0.1,0.5), C_{4} (0.9,0.7,0.1,0.2)\} \end{array} $$
  4. (Step 4:)

    Now, the proposed distance measures, d 2 have been computed from N to A j (j=1,2,…,5) and their corresponding results are given as follows: d 2(A 1,N)=1.0000, d 2(A 2,N)=0.9625, d 2(A 3,N)=0.9500, d 2(A 4,N)=0.9250 and d 2(A 5,N)=0.9375

  5. (Step 5:)

    Thus, we conclude that A 1 is the best company for investing the money than the others.

Table 1 Linguistic grade and corresponding PMF and PNMF value
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Table 2 Linguistic grade and corresponding SMF and SNMF value
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Table 3 Graded values of the alternative corresponding to each attribute (criteria)
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Table 4 Distance measure between d 2 and N
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On the other hand, if we apply the other proposed distance measures such as \(d_{1},d_{3},\ldots ,{d_{3}^{U}}, {d_{4}^{U}}\) for computing the best alternative(s), then step 2 of the proposed approach has been executed for each distance measures and hence their corresponding T2IFS set has been constructed. Finally, based on these sets, the rating value of the distance measures corresponding to each alternative is computed and ranking has been done which are summarized in Table 5. From this table, it has been concluded that the best alternative is still A 1 in all cases.

Table 5 Final ranking
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4.3 Comparative study

In order to compare the performance of the proposed methods with some existing methods, a comparative studies based on interval-valued and type-2 fuzzy set as proposed by the authors [2, 16, 28, 29, 31, 39, 42, 43] have been taken and their corresponding results are summarized in Table 6. From this table, it has been seen that the best company for investing the money is A 1 than others and this result has been overlapped with the proposed results. Thus, the proposed technique can be suitably utilized to solve the problem of decision-making problem than the other existing measures.

Table 6 Comparative analysis
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5 Conclusion

In this manuscript, a family of Hamming, Euclidean and utmost distance measures for T2IFSs has been proposed by considering the PMF, SMF, PNMF, SNMF, FOU and VMF. Several desirable properties of these measures have been investigated in detail. Further, a ranking method based on these measures has been proposed for solving group decision making problems and illustrated with a numerical example. The proposed method has more fuzziness and uncertainties due to the fact that it uses type-2 intuitionistic fuzzy sets rather than existed fuzzy sets. From the studies it has been concluded that the proposed results coincide with the ones, shown in existing approaches and hence place an alternative way for solving the decision-making problems.