**2. Preliminary**

In this section, we review the basic concepts related to IVIFSs that will be used in this paper.

**Definition 1** ([1])**.** *A fuzzy set A in the unverse of discourse X* = {*<sup>x</sup>*1, *x*2,..., *xn*} *is defined as follows:*

$$A = \{<\mathfrak{x}, \mu\_A(\mathfrak{x})>|\mathfrak{x} \in X\},$$

*where μA*(*x*) : *X* → [0, 1] *is the membership degree.* **Definition 2** ([2])**.** *An intuitionistic fuzzy set A in a universe of discourse X* = {*<sup>x</sup>*1, *x*2, ... , *xn*} *is defined as follows:*

$$A = \{ <\ x\_\prime \mu\_A(\mathfrak{x}), \nu\_A(\mathfrak{x}) > |\mathfrak{x} \in X \},$$

*where μA*(*x*) : *X* → [0, 1] *and <sup>ν</sup>A*(*x*) : *X* → [0, 1] *are membership and non-membership degree, respectively, such that:* 0 ≤ *μA*(*x*) + *<sup>ν</sup>A*(*x*) ≤ 1*.*

*The third parameter of intuitionistic fuzzy set A is: <sup>π</sup>A*(*x*) = 1 − *μA*(*x*) − *<sup>ν</sup>A*(*x*)*, which is known as the intuitionistic fuzzy index or the hesitation degree of whether x belongs to A or not. It is obviously seen that* 0 ≤ *<sup>π</sup>A*(*x*) ≤ 1*. If <sup>π</sup>A*(*x*) *is small; then, knowledge about x is more certain; if <sup>π</sup>A*(*x*) *is great, then knowledge about x is more uncertain.*

**Definition 3** ([22])**.** *An interval-valued intuitionistic fuzzy set A in a universe of discourse X* = {*<sup>x</sup>*1, *x*2,..., *xn*} *is defined as follows:*

$$A = \{ <\mathbf{x}, \mu\_A(\mathbf{x}), \nu\_A(\mathbf{x}) > |\mathbf{x} \in X \} = \{ <\mathbf{x}, [\mu\_A^-(\mathbf{x}), \mu\_A^+(\mathbf{x})], [\nu\_A^-(\mathbf{x}), \nu\_A^+(\mathbf{x})] > |\mathbf{x} \in X \},$$

*where μA*(*x*) ⊆ [0, 1]*, <sup>ν</sup>A*(*x*) ⊆ [0, 1]*, which satisfies* 0 ≤ *<sup>μ</sup>*<sup>+</sup>*A*(*x*) + *ν*+*A* (*x*) ≤ 1*.*

*The intervals μA*(*x*) *and <sup>ν</sup>A*(*x*) *denote the membership degree and non-membership degree, respectively. Furthermore, for each x* ∈ *X, we can compute the hesitance degree <sup>π</sup>A*(*x*)=[*π*<sup>−</sup>*A* (*xi*), *<sup>π</sup>*+*A* (*xi*)] = [1 − *<sup>μ</sup>*<sup>+</sup>*A*(*x*) − *ν*+*A* (*x*), 1 − *<sup>μ</sup>*<sup>−</sup>*A*(*x*) − *<sup>ν</sup>*<sup>−</sup>*A* (*x*)]*.*

**Definition 4** ([29])**.** *For every two IVIFSs A and B in the universe of discourse X, we have the following relations:*


.

*(3): A* ∩ *B* = 8*x*, [min(*μ*<sup>−</sup>*A*(*x*), *μ*<sup>−</sup>*B*(*x*)), min(*μ*<sup>+</sup>*A*(*x*), *μ*+*B* (*x*))], [max(*ν*<sup>−</sup>*A*(*x*), *<sup>ν</sup>*<sup>−</sup>*B*(*x*)), max(*ν*+*A*(*x*), *ν*+*B*(*x*))]9 .

*(4): A* = *B iff* (∀*x* ∈ *<sup>X</sup>*)*μ*<sup>−</sup>*A*(*x*) = *μ*<sup>−</sup>*B* (*x*) *and <sup>μ</sup>*<sup>+</sup>*A*(*x*) = *μ*+*B* (*x*) *and <sup>ν</sup>*<sup>−</sup>*A* (*x*) = *<sup>ν</sup>*<sup>−</sup>*B* (*x*) *and ν*+*A* (*x*) = *ν*+*B* (*x*)*.*

$$(5) \colon \quad A^{\mathfrak{c}} = \langle \mathfrak{x}, [\nu\_A^{-}(\mathfrak{x}), \nu\_A^{+}(\mathfrak{x})], [\mu\_A^{-}(\mathfrak{x}), \mu\_A^{+}(\mathfrak{x})] \rangle$$

**Definition 5** ([18])**.** *Let A and B be interval-valued intuitionistic fuzzy sets in the unverse of discourse X* = {*<sup>x</sup>*1, *x*2, ... , *xn*}*, a mapping S* : *IVIFS*(*X*) × *IVIFS*(*X*) → [0, 1]*, <sup>S</sup>*(*<sup>A</sup>*, *B*) *is called to be a similarity measure between A and B, if <sup>S</sup>*(*<sup>A</sup>*, *B*) *satisfies the following properties:*

*(S1):* 0 ≤ *<sup>S</sup>*(*<sup>A</sup>*, *B*) ≤ 1,


#### **3. Some Existing Similarity Measures**

In this section, we review some existing similarity measures.

Let *A* = {< *xi*, [*μ*<sup>−</sup>*A*(*xi*), *<sup>μ</sup>*<sup>+</sup>*A*(*xi*)], [*ν*<sup>−</sup>*A* (*xi*), *ν*+*A* (*xi*)] > |*xi* ∈ *<sup>X</sup>*}, *B* = {< *xi*, [*μ*<sup>−</sup>*B* (*xi*), *μ*+*B* (*xi*)], [*ν*<sup>−</sup>*B* (*xi*), *ν*+*B* (*xi*)] > |*xi* ∈ *X*} be IVIFSs defined on a universe of discourse *X* = {*<sup>x</sup>*1, *x*2, ... , *xn*}. The following Formulas (1)–(4) are similarity measures based on IVIFSs:

Xu's similarity measure([24]):

$$S\_1(A,B) = 1 - \sqrt{\frac{1}{4n} \sum\_{i=1}^{n} (\left| \mu\_A^-(\mathbf{x}\_i) - \mu\_B^-(\mathbf{x}\_i) \right|^p + \left| \mu\_A^+(\mathbf{x}\_i) - \mu\_B^+(\mathbf{x}\_i) \right|^p + \left| \nu\_A^-(\mathbf{x}\_i) - \nu\_B^-(\mathbf{x}\_i) \right|^p + \left| \nu\_A^+(\mathbf{x}\_i) - \nu\_B^+(\mathbf{x}\_i) \right|^p}, \quad \{1, \ldots, n\}$$
 
$$\sqrt{\frac{1}{4n} \sum\_{i=1}^{n} (\left| \mu\_A^-(\mathbf{x}\_i) - \mu\_B^-(\mathbf{x}\_i) \right|^p + \left| \mu\_A^+(\mathbf{x}\_i) - \mu\_B^-(\mathbf{x}\_i) \right|^p}$$

$$S\_{2}(A,B) = 1 - \sqrt[n]{\frac{1}{n} \sum\_{i=1}^{n} \max(\left|\mu\_{A}^{-}(\mathbf{x}\_{i}) - \mu\_{B}^{-}(\mathbf{x}\_{i})\right|^{p}, \left|\mu\_{A}^{+}(\mathbf{x}\_{i}) - \mu\_{B}^{+}(\mathbf{x}\_{i})\right|^{p}, \left|\nu\_{A}^{-}(\mathbf{x}\_{i}) - \nu\_{B}^{-}(\mathbf{x}\_{i})\right|^{p}, \left|\nu\_{A}^{+}(\mathbf{x}\_{i}) - \nu\_{B}^{+}(\mathbf{x}\_{i})\right|^{p}).\tag{2}$$

> Wei's similarity measure ([25]):

$$S\_W(A,B) = \frac{1}{n} \sum\_{i=1}^n \frac{2 - \min(\mu\_i^-, \nu\_i^-) - \min(\mu\_i^+, \nu\_i^+)}{2 + \max(\mu\_i^-, \nu\_i^-) + \max(\mu\_i^+, \nu\_i^+)},\tag{3}$$

,

where

$$\begin{aligned} \mu\_i^- &= \left| \mu\_A^-(\mathbf{x}\_i) - \mu\_B^-(\mathbf{x}\_i) \right| \; \mu\_i^+ = \left| \mu\_A^+(\mathbf{x}\_i) - \mu\_B^+(\mathbf{x}\_i) \right| \; \mu\_i^- \\\nu\_i^- &= \left| \nu\_A^-(\mathbf{x}\_i) - \nu\_B^-(\mathbf{x}\_i) \right| \; \nu\_i^+ = \left| \nu\_A^+(\mathbf{x}\_i) - \nu\_B^+(\mathbf{x}\_i) \right| \; \end{aligned}$$

Dhivya's similarity measure ([28]):

$$\begin{array}{rcl} S\_D(A, B) &=& 1 - \frac{1}{n} \sum\_{i=1}^n \left( \frac{1}{2} (|\psi\_A^-(\mathbf{x}\_i) - \psi\_B^-(\mathbf{x}\_i)| + |\psi\_A^+(\mathbf{x}\_i) - \psi\_B^+(\mathbf{x}\_i)|) \cdot (1 - \frac{\sigma\_A(\mathbf{x}\_i) + \sigma\_B(\mathbf{x}\_i)}{2}) + \\ & & |\sigma\_A(\mathbf{x}\_i) - \sigma\_B(\mathbf{x}\_i)| \cdot (\frac{\sigma\_A(\mathbf{x}\_i) + \sigma\_B(\mathbf{x}\_i)}{2}) \right), \end{array} \tag{4}$$

where

$$\begin{split} \psi\_{A}^{-} &= \frac{\mu\_{A}^{-}(\mathbf{x}\_{i}) + 1 - \nu\_{A}^{-}(\mathbf{x}\_{i})}{2}, \; \psi\_{A}^{+} = \frac{\mu\_{A}^{+}(\mathbf{x}\_{i}) + 1 - \nu\_{A}^{+}(\mathbf{x}\_{i})}{2}, \\ \psi\_{B}^{-} &= \frac{\mu\_{B}^{-}(\mathbf{x}\_{i}) + 1 - \nu\_{B}^{-}(\mathbf{x}\_{i})}{2}, \; \psi\_{B}^{+} = \frac{\mu\_{B}^{+}\upsilon + 1 - \nu\_{B}^{+}(\mathbf{x}\_{i})}{2}, \\ \sigma\_{A}(\mathbf{x}\_{i}) &= 1 - \frac{1}{2}(\mu\_{A}^{-}(\mathbf{x}\_{i}) + \mu\_{A}^{+}(\mathbf{x}\_{i}) + \nu\_{A}^{-}(\mathbf{x}\_{i}) + \nu\_{A}^{+}(\mathbf{x}\_{i})), \\ \sigma\_{B}(\mathbf{x}\_{i}) &= 1 - \frac{1}{2}(\mu\_{B}^{-}(\mathbf{x}\_{i}) + \mu\_{B}^{+}(\mathbf{x}\_{i}) + \nu\_{B}^{-}(\mathbf{x}\_{i}) + \nu\_{B}^{+}(\mathbf{x}\_{i})). \end{split}$$

#### **4. A New Similarity Measure between Interval-Valued Intuitionistic Fuzzy Sets**

**Definition 6.** *Let A, B be IVIFSs defined in universe of discourse X* = {*<sup>x</sup>*1, *x*2, ... , *xn*}*, and A* = {< *xi*, [*μ*<sup>−</sup>*A*(*xi*), *<sup>μ</sup>*<sup>+</sup>*A*(*xi*)], [*ν*<sup>−</sup>*A* (*xi*), *ν*+*A* (*xi*)] > |*xi* ∈ *<sup>X</sup>*}*, B* = {< *xi*, [*μ*<sup>−</sup>*B* (*xi*), *μ*+*B* (*xi*)], [*ν*<sup>−</sup>*B* (*xi*), *ν*+*B* (*xi*)] > |*xi* ∈ *<sup>X</sup>*}*. We call*

$$S^{p}(A,B) = 1 - \left\{ \begin{array}{c} \frac{1}{2n} \sum\_{i=1}^{n} \left| \frac{t\_1[(\mu\_A^-(\mathbf{x}) - \mu\_B^-(\mathbf{x})) + (\mu\_A^+(\mathbf{x}) - \mu\_B^+(\mathbf{x}))] - [(\nu\_A^-(\mathbf{x}) - \nu\_B^-(\mathbf{x})) + (\nu\_A^+(\mathbf{x}) - \nu\_B^+(\mathbf{x}))]}{2(t\_1+1)} \right|^p \right|^{\frac{1}{p}} \\ + \left| \frac{t\_2[(\nu\_A^-(\mathbf{x}) - \nu\_B^-(\mathbf{x})) + (\nu\_A^+(\mathbf{x}) - \nu\_B^+(\mathbf{x}))] - [(\mu\_A^-(\mathbf{x}) - \mu\_B^-(\mathbf{x})) + (\mu\_A^+(\mathbf{x}) - \mu\_B^+(\mathbf{x}))]}{2(t\_2+1)} \right|^p \end{array} \right\}^{\frac{1}{p}} \tag{5}$$

*a similarity measure between A and B. t*1, *t*2, *p* ∈ [1, <sup>+</sup>∞)*. Here, three parameters: p is the Lp-norm and t*1, *t*2 *identifies the level of uncertainty.*

**Theorem 1.** *Sp*(*<sup>A</sup>*, *B*) *is a similarity measure between IVIFSs A and B.*

**Proof.** Let *A*, *B*, *C* be IVIFSs defined on a universe of discourse *X* = {*<sup>x</sup>*1, *x*2, ... , *xn*}, and *A* = {< *xi*, [*μ*<sup>−</sup>*A*(*xi*), *<sup>μ</sup>*<sup>+</sup>*A*(*xi*)], [*ν*<sup>−</sup>*A* (*xi*), *ν*+*A* (*xi*)] > |*xi* ∈ *<sup>X</sup>*}, *B* = {< *xi*, [*μ*<sup>−</sup>*B* (*xi*), *μ*+*B* (*xi*)], [*ν*<sup>−</sup>*B* (*xi*), *ν*+*B* (*xi*)] > |*xi* ∈ *<sup>X</sup>*}, and *C* = {< *xi*, [*μ*<sup>−</sup>*C* (*xi*), *μ*+*C* (*xi*)], [*ν*<sup>−</sup>*C* (*xi*), *ν*+*C* (*xi*)] > |*xi* ∈ *<sup>X</sup>*}.

(1) Firstly, we know that, for arbitrary *xi* ∈ *X*:

$$\begin{split} & \quad t\_{1} [ (\mu\_{A}^{-}(\mathbf{x}\_{i}) - \mu\_{B}^{-}(\mathbf{x}\_{i})) + (\mu\_{A}^{+}(\mathbf{x}\_{i}) - \mu\_{B}^{+}(\mathbf{x}\_{i}))] - [ (\nu\_{A}^{-}(\mathbf{x}\_{i}) - \nu\_{B}^{-}(\mathbf{x}\_{i})) + (\nu\_{A}^{+}(\mathbf{x}\_{i}) - \nu\_{B}^{+}(\mathbf{x}\_{i})) ] \\ & = & \left[ t\_{1} (\mu\_{A}^{-}(\mathbf{x}\_{i}) - \mu\_{B}^{-}(\mathbf{x}\_{i})) - (\nu\_{A}^{-}(\mathbf{x}\_{i}) - \nu\_{B}^{-}(\mathbf{x}\_{i})) \right] + \left[ t\_{1} (\mu\_{A}^{+}(\mathbf{x}\_{i}) - \mu\_{B}^{+}(\mathbf{x}\_{i})) - (\nu\_{A}^{+}(\mathbf{x}\_{i}) - \nu\_{B}^{+}(\mathbf{x}\_{i})) \right]. \end{split}$$

For *<sup>μ</sup>*<sup>−</sup>*A*(*xi*), *μ*<sup>−</sup>*B* (*xi*), *<sup>ν</sup>*<sup>−</sup>*A* (*xi*), *<sup>ν</sup>*<sup>−</sup>*B* (*xi*) ∈ [0, 1], then we have −*t*<sup>1</sup> ≤ *<sup>t</sup>*1(*μ*<sup>−</sup>*A*(*xi*) − *μ*<sup>−</sup>*B* (*xi*)) ≤ *t*1, −1 ≤ *<sup>ν</sup>*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*B* (*xi*) ≤ 1. Thus, we obtain that

$$-(t\_1 + 1) \le t\_1(\mu\_A^-(\mathbf{x}\_i) - \mu\_B^-(\mathbf{x}\_i)) - (\nu\_A^-(\mathbf{x}\_i) - \nu\_B^-(\mathbf{x}\_i)) \le t\_1 + 1.$$

Similarly,

$$-(t\_1 + 1) \le t\_1(\mu\_A^+(\mathbf{x}\_i) - \mu\_B^+(\mathbf{x}\_i)) - (\nu\_A^+(\mathbf{x}\_i) - \nu\_B^+(\mathbf{x}\_i)) \le t\_1 + 1.$$

Thus,

$$0 \le \left| \frac{t\_1[(\mu\_A^-(\mathbf{x}\_i) - \mu\_B^-(\mathbf{x}\_i)) + (\mu\_A^+(\mathbf{x}\_i) - \mu\_B^+(\mathbf{x}\_i))] - [(\nu\_A^-(\mathbf{x}\_i) - \nu\_B^-(\mathbf{x}\_i)) + (\nu\_A^+(\mathbf{x}\_i) - \nu\_B^+(\mathbf{x}\_i))]}{2(t\_1 + 1)} \right|^p \le 1.$$

By the same way, we have

$$0 \le \left| \frac{t\_2[(\boldsymbol{\nu}\_A^-(\mathbf{x}\_i) - \boldsymbol{\nu}\_B^-(\mathbf{x}\_i)) + (\boldsymbol{\nu}\_A^+(\mathbf{x}\_i) - \boldsymbol{\nu}\_B^+(\mathbf{x}\_i))] - [(\boldsymbol{\mu}\_A^-(\mathbf{x}\_i) - \boldsymbol{\mu}\_B^-(\mathbf{x}\_i)) + (\boldsymbol{\mu}\_A^+(\mathbf{x}\_i) - \boldsymbol{\mu}\_B^+(\mathbf{x}\_i))]}{2(t\_2 + 1)} \right|^p \le 1.$$

Therefore,

$$\begin{split} 0 \leq \left\{ \begin{array}{l} \frac{1}{2\pi} \sum\_{i=1}^{n} \left| \frac{t\_{1}\left[ (\mu\_{A}^{-}(\mathbf{x}\_{i}) - \mu\_{B}^{-}(\mathbf{x}\_{i})) + (\mu\_{A}^{+}(\mathbf{x}\_{i}) - \mu\_{B}^{+}(\mathbf{x}\_{i})) \right] - \left[ (\boldsymbol{\nu}\_{A}^{-}(\mathbf{x}\_{i}) - \boldsymbol{\nu}\_{B}^{-}(\mathbf{x}\_{i})) + (\boldsymbol{\nu}\_{A}^{+}(\mathbf{x}\_{i}) - \boldsymbol{\nu}\_{B}^{+}(\mathbf{x}\_{i})) \right] \right|^{p} \\ + \left| \frac{t\_{2}\left[ (\boldsymbol{\nu}\_{A}^{-}(\mathbf{x}\_{i}) - \boldsymbol{\nu}\_{B}^{-}(\mathbf{x}\_{i})) + (\boldsymbol{\nu}\_{A}^{+}(\mathbf{x}\_{i}) - \boldsymbol{\nu}\_{B}^{+}(\mathbf{x}\_{i})) \right] - \left[ (\boldsymbol{\mu}\_{A}^{-}(\mathbf{x}\_{i}) - \boldsymbol{\mu}\_{B}^{-}(\mathbf{x}\_{i})) + (\boldsymbol{\mu}\_{A}^{+}(\mathbf{x}\_{i}) - \boldsymbol{\mu}\_{B}^{+}(\mathbf{x}\_{i})) \right] \end{array} \right\} \leq 1. \end{split}$$

That is, 0 ≤ *Sp*(*<sup>A</sup>*, *B*) ≤ 1.

(2) *A* = *B*, if and only if for arbitrary *xi* ∈ *X*, we have *<sup>μ</sup>*<sup>−</sup>*A*(*xi*) = *μ*<sup>−</sup>*B* (*xi*), *<sup>μ</sup>*<sup>+</sup>*A*(*xi*) = *μ*+*B* (*xi*), *ν* − *A* (*xi*) = *<sup>ν</sup>*<sup>−</sup>*B* (*xi*), *ν*+*A* (*xi*) = *ν*+*B* (*xi*). It is obvious that *Sp*(*<sup>A</sup>*, *B*) = 1. (3) For *Sp*(*<sup>A</sup>*, *<sup>B</sup>*),wehave

HH*<sup>t</sup>*1[(*μ*<sup>−</sup>*A*(*xi*) − *μ*<sup>−</sup>*B* (*xi*)) + (*μ*<sup>+</sup>*A*(*xi*) − *μ*+*B* (*xi*))] − [(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*B* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*B* (*xi*))]HH*<sup>p</sup>* = HH−*t*1[(*μ*<sup>−</sup>*A*(*xi*) − *μ*<sup>−</sup>*B* (*xi*)) + (*μ*<sup>+</sup>*A*(*xi*) − *μ*+*B* (*xi*))] + [(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*B* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*B v*)]HH*<sup>p</sup>* = HH*<sup>t</sup>*1[(*μ*<sup>−</sup>*B* (*xi*) − *<sup>μ</sup>*<sup>−</sup>*A*(*xi*)) + (*μ*<sup>+</sup>*B* (*xi*) − *<sup>μ</sup>*<sup>+</sup>*A*(*xi*))] − [(*ν*<sup>−</sup>*B* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*A* (*xi*)) − (*ν*+*B* (*xi*) − *ν*+*A* (*xi*))]HH*<sup>p</sup>* .

Similarly,

HH*<sup>t</sup>*2[(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*B* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*B* (*xi*))] − [(*μ*<sup>−</sup>*A*(*xi*) − *μ*<sup>−</sup>*B* (*xi*)) + (*μ*<sup>+</sup>*A*(*xi*) − *μ*+*B* (*xi*))]HH*<sup>p</sup>* = HH−*t*2[(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*B* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*B* (*xi*))] + [(*μ*<sup>−</sup>*A*(*xi*) − *μ*<sup>−</sup>*B* (*xi*)) + (*μ*<sup>+</sup>*A*(*xi*) − *μ*+*B* (*xi*))]HH*<sup>p</sup>* = HH*<sup>t</sup>*2[(*ν*<sup>−</sup>*B* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*A* (*xi*)) − (*ν*+*B* (*xi*) − *ν*+*A* (*xi*))] − [(*μ*<sup>−</sup>*B* (*xi*) − *<sup>μ</sup>*<sup>−</sup>*A*(*xi*)) − (*μ*<sup>+</sup>*B* (*xi*) − *<sup>μ</sup>*<sup>+</sup>*A*(*xi*))]HH*<sup>p</sup>* .

Thus, *Sp*(*<sup>A</sup>*, *B*) = *Sp*(*<sup>B</sup>*, *<sup>A</sup>*). (4) For *A*, *B*, *C* be IVIFSs, the similarity measure *A* and *B*, and *A* and *C* are the following:

$$S^{p}(A,B) = 1 - \left\{ \begin{array}{l} \frac{1}{2\pi} \sum\_{i=1}^{\mathsf{H}} \left| \frac{t\_{1}[(\mu\_{A}^{-}(\mathbf{z}\_{i}) - \mu\_{B}^{-}(\mathbf{z}\_{i})) + (\mu\_{A}^{+}(\mathbf{z}\_{i}) - \mu\_{B}^{+}(\mathbf{z}\_{i}))] - [(\nu\_{A}^{-}(\mathbf{z}\_{i}) - \nu\_{B}^{-}(\mathbf{z}\_{i})) + (\nu\_{A}^{+}(\mathbf{z}\_{i}) - \nu\_{B}^{+}(\mathbf{z}\_{i}))] \right|^{p}}{2(t\_{1}+1)} \right|^{p} \right\}^{\frac{1}{p}}{\mathsf{H}} $$

*<sup>S</sup><sup>p</sup>*(*<sup>A</sup>*, *C*) = 1 − ⎧⎪⎪⎨⎪⎪⎩ 12*n n*∑*i*=1 HHHH*t*1[(*μ*−*A* (*xi*)−*μ*<sup>−</sup>*C* (*xi*))+(*μ*+*A* (*xi*)−*μ*+*C* (*xi*))]−[(*ν*<sup>−</sup>*A* (*xi*)−*ν*<sup>−</sup>*C* (*xi*))+(*ν*+*A* (*xi*)−*ν*+*C* (*xi*))] <sup>2</sup>(*<sup>t</sup>*1+<sup>1</sup>) HHHH*p* + HHHH*t*2[(*ν*−*A* (*xi*)−*ν*<sup>−</sup>*C* (*xi*))+(*ν*+*A* (*xi*)−*ν*+*C* (*xi*))]−[(*μ*<sup>−</sup>*A* (*xi*)−*μ*<sup>−</sup>*C* (*xi*))+(*μ*+*A* (*xi*)−*μ*+*C* (*xi*))] <sup>2</sup>(*<sup>t</sup>*2+<sup>1</sup>) HHHH*p* ⎫⎪⎪⎬⎪⎪⎭ 1 *p* .

If *A* ⊆ *B* ⊆ *C*, then *<sup>μ</sup>*<sup>−</sup>*A*(*xi*) ≤ *μ*<sup>−</sup>*B* (*xi*) ≤ *μ*<sup>−</sup>*C* (*xi*), *<sup>μ</sup>*<sup>+</sup>*A*(*xi*) ≤ *μ*+*B* (*xi*) ≤ *μ*+*C* (*xi*), *<sup>ν</sup>*<sup>−</sup>*C* (*xi*) ≤ *<sup>ν</sup>*<sup>−</sup>*B* (*xi*) ≤ *ν* − *A* (*xi*), and *ν*+*C* (*xi*) ≤ *ν*+*B* (*xi*) ≤ *ν*+*A* (*xi*). Then, we have

HH*<sup>t</sup>*1[(*μ*<sup>−</sup>*A*(*xi*) − *μ*<sup>−</sup>*B* (*xi*)) + (*μ*<sup>+</sup>*A*(*xi*) − *μ*+*B* (*xi*))] − [(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*B* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*B* (*xi*))]HH = *<sup>t</sup>*1[(*μ*<sup>−</sup>*B* (*xi*) − *<sup>μ</sup>*<sup>−</sup>*A*(*xi*)) + (*μ*<sup>+</sup>*B* (*xi*) − *<sup>μ</sup>*<sup>+</sup>*A*(*xi*))] + [(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*B* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*B* (*xi*))] ≤ *<sup>t</sup>*1[(*μ*<sup>−</sup>*C* (*xi*) − *<sup>μ</sup>*<sup>−</sup>*A*(*xi*)) + (*μ*<sup>+</sup>*C* (*xi*) − *<sup>μ</sup>*<sup>+</sup>*A*(*xi*))] + [(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*C* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*C* (*xi*))] = HH*<sup>t</sup>*1[(*μ*<sup>−</sup>*A*(*xi*) − *μ*<sup>−</sup>*C* (*xi*)) + (*μ*<sup>+</sup>*A*(*xi*) − *μ*+*C* (*xi*))] − [(*ν*<sup>−</sup>*A* (*xi*) − *<sup>ν</sup>*<sup>−</sup>*C* (*xi*)) + (*ν*+*A* (*xi*) − *ν*+*C* (*xi*))]HH .

By the same reason, we have

$$\begin{split} & \left| t\_2 \big[ \left( \nu\_A^- (\mathbf{x}\_i) - \nu\_B^- (\mathbf{x}\_i) \right) + \left( \nu\_A^+ (\mathbf{x}\_i) - \nu\_B^+ (\mathbf{x}\_i) \right) \big] - \left[ \left( \mu\_A^- (\mathbf{x}\_i) - \mu\_B^- (\mathbf{x}\_i) \right) + \left( \mu\_A^+ (\mathbf{x}\_i) - \mu\_B^+ (\mathbf{x}\_i) \right) \right] \right| \\ & \leq \quad \left| t\_2 \big[ \left( \nu\_A^- (\mathbf{x}\_i) - \nu\_C^- (\mathbf{x}\_i) \right) + \left( \nu\_A^+ (\mathbf{x}\_i) - \nu\_C^+ (\mathbf{x}\_i) \right) \big] - \left[ \left( \mu\_A^- (\mathbf{x}\_i) - \mu\_C^- (\mathbf{x}\_i) \right) + \left( \mu\_A^+ (\mathbf{x}\_i) - \mu\_C^+ (\mathbf{x}\_i) \right) \right] \right|. \end{split}$$

Therefore, *Sp*(*<sup>A</sup>*, *B*) ≥ *Sp*(*<sup>A</sup>*, *<sup>C</sup>*), and *Sp*(*<sup>B</sup>*, *C*) ≥ *Sp*(*<sup>A</sup>*, *<sup>C</sup>*). In conclusion, *Sp*(*<sup>A</sup>*, *B*) is a similarity measure between IVIFSs *A* and *B*.

**Remark 1.** *If interval-valued intuitionistic fuzzy sets A and B degenerates to intuitionistic fuzzy set, i.e., μ* − *A* = *<sup>μ</sup>*<sup>+</sup>*A, <sup>ν</sup>*<sup>−</sup>*A* = *ν*+*A , and μ*<sup>−</sup>*B* = *μ*+*B , <sup>ν</sup>*<sup>−</sup>*B* = *ν*+*B , then*

$$S^{p}(A,B) = 1 - \left\{ \frac{1}{n} \sum\_{i=1}^{n} \left| \frac{t\_1(\mu\_A - \mu\_B) - (\nu\_A - \nu\_B)}{2(t\_1 + 1)} \right|^p + \left| \frac{t\_2(\nu\_A - \nu\_B) - (\mu\_A - \mu\_B)}{2(t\_2 + 1)} \right|^p \right\}^{\frac{1}{p}} \tag{6}$$

*is a new similarity measure between intuitionistic fuzzy sets A and B.*

**Remark 2.** *In the environment of IFSs, and when t*1 = *t*2 = *t, the proposed similarity measure*

$$S^p(A,B) = 1 - \left\{ \frac{1}{2n(t+1)^p} \sum\_{i=1}^n \left( |t(\mu\_A - \mu\_B) - (\nu\_A - \nu\_B)|^p + |t(\nu\_A - \nu\_B) - (\mu\_A - \mu\_B)|^p \right) \right\}^{\frac{1}{p}} \tag{7}$$

*is the similarity measure between intuitionistic fuzzy sets A and B in the literature ([19]).*

**Example 1.** *Supposing that Ai and Bi are two IVIFSs, we can compute the similarity measures between Ai and Biby different similarity measures listed in Table 1.*

**Table 1.** Comparison of similarity measures in the environment of IVIFSs (interval-valued intuitionistic fuzzy set) (counter-intuitive cases are in bold type; *p* = 1 in *S*1 and *S*2; *p* = 1, *t*1 = 2, *t*2 = 3 in *Sp*).


*In Table 1, by comparing the first column and the second column, we can find that Si*(*<sup>A</sup>*1, *<sup>B</sup>*1) = *Si*(*<sup>A</sup>*2, *<sup>B</sup>*2)(*<sup>i</sup>* = 1, 2) *when A*1 = *A*2*, B*<sup>1</sup> <sup>=</sup>*B*2*. Similarly, by comparing the third column and the fourth column, we can find <sup>S</sup>*2(*<sup>A</sup>*3, *<sup>B</sup>*3) = *<sup>S</sup>*2(*<sup>A</sup>*4, *<sup>B</sup>*4) *when A*3 = *A*4*, B*<sup>3</sup> <sup>=</sup>*B*4*. Therefore, we can determine that the similarity measure S*1 *and S*2 *is not reasonable. Meanwhile, we find that SD*(*<sup>A</sup>*1, *<sup>B</sup>*1) = 1 *when A*<sup>1</sup> <sup>=</sup>*B*1*, which is not satisfy the second axiom of the definition for similarity measure. Most importantly, we can observe that the proposed similarity measure Sp can overcome these drawbacks. Therefore, our novel similarity measure for IVIFSs is more reasonable than others.*

#### **5. Geometric Interpretation of the Novel Similarity Measure**

In this section, we briefly interpret the proposed similarity measure and explain the functionality of parameters *t*1, *t*2 and *p* defined in the proposed similarity measure.

Let *A* =< [*μ*<sup>−</sup>*A*, *<sup>μ</sup>*<sup>+</sup>*A*], [*ν*<sup>−</sup>*A* , *ν*+*A* ] >, *B* =< [*μ*<sup>−</sup>*B* , *μ*+*B* ], [*ν*<sup>−</sup>*B* , *ν*+*B* ] > be interval-valued intuitionistic fuzzy numbers. We can split *A* into two intuitionistic fuzzy numbers, i.e., *A*− =< *μ*<sup>−</sup>*A*, *<sup>ν</sup>*<sup>−</sup>*A* > and *A*<sup>+</sup> =< *<sup>μ</sup>*<sup>+</sup>*A*, *ν*+*A* >. For intuitionistic fuzzy set *A*<sup>−</sup>, *μ*<sup>−</sup>*A* can be equal to any value in [*μ*<sup>−</sup>*A*, *μ*<sup>−</sup>*A* + *<sup>π</sup>*+*A* ] and *ν* − *A* can be equal to any value in [*ν*<sup>−</sup>*A* , *<sup>ν</sup>*<sup>−</sup>*A* + *<sup>π</sup>*+*A* ], where *<sup>π</sup>*+*A* = 1 − *μ*<sup>−</sup>*A* − *<sup>ν</sup>*<sup>−</sup>*A* . Similarly, *μ*+*A* can be equal to any value in [*μ*<sup>+</sup>*A*, *μ*+*A* + *<sup>π</sup>*<sup>−</sup>*A* ] and *ν*+*A* can be equal to any value in [*ν*+*A* , *ν*+*A* + *<sup>π</sup>*<sup>−</sup>*A* ] for intuitionistic fuzzy set *A*+, where *<sup>π</sup>*<sup>−</sup>*A* = 1 − *μ*+*A* − *ν*+*A* . Then, the possible values for *A*− and *A*<sup>+</sup> illustrated in Figure 1 as the two triangles. As the center of gravity, *D*− and *D*<sup>+</sup> are the most informative points in the triangle *A*− and *A*+, respectively.

However, < *μ*<sup>−</sup>*A* + *π*+*A t*1+1 , *<sup>ν</sup>*<sup>−</sup>*A* + *π*+*A t*2+1 > (*<sup>t</sup>*1, *<sup>t</sup>*2<sup>∈</sup>[1, <sup>+</sup>∞)) can represent any point in the triangle *A*<sup>−</sup>. Especially when *t*1 = *t*2 = *t*, < *μ*<sup>−</sup>*A* + *π*+*A t*+1 , *<sup>ν</sup>*<sup>−</sup>*A* + *π*+*A t*+1 > can denote the point of middle line of the triangle bevel. In the same way, < *μ*+*A* + *π* − *A t*1+1 , *ν*+*A* + *π* − *A t*2+1 > (*<sup>t</sup>*1, *<sup>t</sup>*2<sup>∈</sup>[1, <sup>+</sup>∞)) represents any point in the triangle *A*+.

The following is the calculation process:

Firstly, *A* − = *μ*−*A* + *π*+*A t*1+1 , *<sup>ν</sup>*<sup>−</sup>*A* + *π*+*A t*2+1denotes possible points of triangle *A*<sup>−</sup>. By the same token, *A* + = *μ*+*A* + *<sup>π</sup>*<sup>−</sup>*A t*1+1 , *ν*+*A* + *<sup>π</sup>*<sup>−</sup>*A t*2+1 denotes possible points of triangle *A*+. Similarly, we can obtain that *B* − = *μ*−*B* + *π*+*B t*1+1 , *<sup>ν</sup>*<sup>−</sup>*B* + *π*+*B t*1+1 and *B* + = *μ*+*B* + *<sup>π</sup>*<sup>−</sup>*B t*1+1 , *ν*+*B* + *<sup>π</sup>*<sup>−</sup>*B t*1+1 denote any points in triangles *B*− and *B*+, respectively.

Secondly, the average of *A* − and *A* + can be computed as follows:

$$A'' = <\mu\_{A'}'' \nu\_A'' > = \left\langle \frac{2 + t\_1(\mu\_A^- + \mu\_A^+) - (\nu\_A^- + \nu\_A^+)}{2(t\_1 + 1)}, \frac{2 + t\_2(\mu\_A^- + \mu\_A^+) - (\nu\_A^- + \nu\_A^+)}{2(t\_2 + 1)} \right\rangle.$$

We can also ge<sup>t</sup> the mean value of *B* − and *B* +:

$$B'' = <\mu\_{B'}'' \nu\_B'' > = \left\langle \frac{2 + t\_1(\mu\_B^- + \mu\_B^+) - (\nu\_B^- + \nu\_B^+)}{2(t\_1 + 1)}, \frac{2 + t\_2(\mu\_B^- + \mu\_B^+) - (\nu\_B^- + \nu\_B^+)}{2(t\_2 + 1)} \right\rangle \dots$$

The absolute difference between *A* and *B* is calculated as follows:

$$\left| \mu\_A'' - \mu\_B'' \right| = \left| \frac{t\_1[(\mu\_A^- - \mu\_B^-) + (\mu\_A^+ - \mu\_B^+)] - [(\nu\_A^- - \nu\_B^-) + (\nu\_A^+ - \nu\_B^+)]}{2(t\_1 + 1)} \right|,$$

$$\left| \nu\_A'' - \nu\_B'' \right| = \left| \frac{t\_2[(\nu\_A^- - \nu\_B^-) + (\nu\_A^+ - \nu\_B^+)] - [(\mu\_A^- - \mu\_B^-) + (\mu\_A^+ - \mu\_B^+)]}{2(t\_2 + 1)} \right|.$$

HHH*μ A* − *μ B*HHH and HHH*ν A* − *ν B*HHH to the power of *p* is equal to the following:

$$\left|\mu\_A'' - \mu\_B''\right|^p = \frac{\left|t\_1\left[\left(\mu\_A^- - \mu\_B^-\right) + \left(\mu\_A^+ - \mu\_B^+\right)\right] - \left[\left(\nu\_A^- - \nu\_B^-\right) + \left(\nu\_A^+ - \nu\_B^+\right)\right]\right|^p}{2^p (t\_1 + 1)^p}.$$

$$\left|\nu\_A'' - \nu\_B''\right|^p = \frac{\left|t\_2\left[\left(\nu\_A^- - \nu\_B^-\right) + \left(\nu\_A^+ - \nu\_B^+\right)\right] - \left[\left(\mu\_A^- - \mu\_B^-\right) + \left(\mu\_A^+ - \mu\_B^+\right)\right]\right|^p}{2^p (t\_2 + 1)^p}.$$

The average value of HHH*μ A* − *μ B*HHH*p* and HHH*ν A* − *ν A*HHH*p* is calculated as follows:

$$\begin{split} &\frac{1}{2}\left(\left|\mu\_{A}^{''}-\mu\_{B}^{''}\right|^{p}+\left|\nu\_{A}^{''}-\nu\_{A}^{''}\right|^{p}\right) \\ &=\left.\frac{1}{2}\left|\frac{t\_{1}[(\mu\_{A}^{''}-\mu\_{B}^{'})+(\mu\_{A}^{+}-\mu\_{B}^{+})]-[(\nu\_{A}^{'}-\nu\_{B}^{'})+(\nu\_{A}^{+}-\nu\_{B}^{+})]}{2(t\_{1}+1)}\right|^{p}\right|^{p} \\ &+\frac{1}{2}\left|\frac{t\_{2}[(\nu\_{A}^{''}-\nu\_{B}^{-})+(\nu\_{A}^{+}-\nu\_{B}^{+})]-[(\mu\_{A}^{'}-\mu\_{B}^{-})+(\mu\_{A}^{+}-\mu\_{B}^{+})]}{2(t\_{2}+1)}\right|^{p} \end{split}$$

.

**Figure 1.** Possible value for *A*− and *A*+.

The *p* root of the average value of HHH*μ A* − *μ B*HHH*p* and HHH*ν A* − *ν A*HHH*p* is calculated as:

$$\left\{\frac{1}{2}\left(\left|\boldsymbol{\mu}\_{A}^{\prime\prime}-\boldsymbol{\mu}\_{B}^{\prime\prime}\right|^{p}+\left|\boldsymbol{\nu}\_{A}^{\prime}-\boldsymbol{\nu}\_{B}^{\prime\prime}\right|^{p}\right)\right\}^{\frac{1}{p}}=\left\{\left|\frac{t\_{1}(\left|\boldsymbol{\mu}\_{A}^{-}-\boldsymbol{\mu}\_{\mathcal{C}}^{\prime}\right|+\left|\boldsymbol{\mu}\_{A}^{+}-\boldsymbol{\mu}\_{\mathcal{C}}^{\prime}\right|)-\left[\left(\boldsymbol{\nu}\_{A}^{-}-\boldsymbol{\nu}\_{\mathcal{C}}^{-}\right)+\left(\boldsymbol{\nu}\_{A}^{+}-\boldsymbol{\nu}\_{\mathcal{C}}^{\prime}\right)\right]}{2\left(t\_{1}+1\right)}\right|^{p}\right\}^{\frac{1}{p}}.$$

For an interval-valued intuitionistic fuzzy set instead of interval-valued intuitionistic fuzzy number, i.e., there is more than one feature in the discourse of universe, such as *X* = {*<sup>x</sup>*1, *x*2,..., *xn*}:

*<sup>S</sup><sup>p</sup>*(*<sup>A</sup>*, *B*) = 1 − ⎧⎪⎪⎨⎪⎪⎩ 12*n n*∑*i*=1 HHHH*t*1[(*μ*−*A* (*xi*)−*μ*<sup>−</sup>*B* (*xi*))+(*μ*+*A* (*xi*)−*μ*+*B* (*xi*))]−[(*ν*<sup>−</sup>*A* (*xi*)−*ν*<sup>−</sup>*B* (*xi*))+(*ν*+*A* (*xi*)−*ν*+*B* (*xi*))] <sup>2</sup>(*<sup>t</sup>*1+<sup>1</sup>) HHHH*p* + HHHH*t*2[(*ν*−*A* (*xi*)−*ν*<sup>−</sup>*B* (*xi*))+(*ν*+*A* (*xi*)−*ν*+*B* (*xi*))]−[(*μ*<sup>−</sup>*A* (*xi*)−*μ*<sup>−</sup>*B* (*xi*))+(*μ*+*A* (*xi*)−*μ*+*B* (*xi*))] <sup>2</sup>(*<sup>t</sup>*2+<sup>1</sup>) HHHH*p* ⎫⎪⎪⎬⎪⎪⎭ 1 *p* .

In particular, *A* − = *D*− = *μ*−*A* + <sup>1</sup>−*μ*<sup>−</sup>*A*<sup>−</sup>*ν*<sup>−</sup>*A* 3 , *<sup>ν</sup>*<sup>−</sup>*A* + <sup>1</sup>−*μ*<sup>−</sup>*A*<sup>−</sup>*ν*<sup>−</sup>*A* 3 and *A* + = *D*<sup>+</sup> = *μ*+*A* + <sup>1</sup>−*μ*+*A*<sup>−</sup>*ν*+*A* 3 , *ν*+*A* + <sup>1</sup>−*μ*+*A*<sup>−</sup>*ν*+*A* 3 when *t*1 = *t*2 = 2. Without a doubt, *D*− and *D*<sup>+</sup> are the most concentrated points of information in triangle *A*− and *A*+, respectively; therefore, they are also the most significant points in all possible meaningful points.
