Novel distance measures for cubic intuitionistic fuzzy sets and their applications to pattern recognitions and medical diagnosis

Abstract

Cubic intuitionistic fuzzy set (IFS), handles the uncertainties by characterizing them into membership and non-membership interval in form of interval-valued IFS and further the degree of agreement, as well as disagreement corresponding to these intervals, is given in the form of an IFS. Under this environment, some series of distance measures based on Hamming, Euclidean, and Hausdorff metrics are proposed. Various relations among them are derived. The practical relevance of our work is justified by giving two real-life examples, one on medical diagnosis and other on pattern recognition. Further, comparison analysis has been done with the existing decision-making approaches and the advantages of the proposed approach are highlighted.

1 Introduction

Fuzzy set (FS) theory, introduced by Zadeh (1965), is one of the most successful theories to represent the uncertainty in the data. Since then, many scholars have utilized the FS to handle the uncertainties in the real fields (Chen and Tanuwijaya 2011; Chen et al. 2001, 2012a; Garg 2018b; Garg and Ansha 2018; Wang and Chen 2008). However, FS considers only the degree of agreement (also named as membership) of each object which is lying between [0,1] and considers that degree of disagreement (also named as non-membership) is the complement of the agreement. But mere considering of the degree of agreement was not fetching appropriate precise results. Later, Atanassov (1986) extended the theory of FSs to an IFS which is characterized by a degree of membership and non-membership functions. Subsequently, Atanassov and Gargov (1989) presented the concept of interval-valued IFS (IVIFS), which is a further extension of IFSs, where the membership degrees are represented by interval-valued intuitionistic fuzzy numbers (IVIFNs).

With economic and social developments, decision-making (DM) has been widely applied in various fields such as aggregation operators (AOs), pattern recognition, medical diagnosis, and clustering analysis, etc. However, in the past few decades, researchers have applied IFSs and IVIFSs to such fields. For instance, Garg (2016a, 2017b) presented some interactive AOs for different intuitionistic fuzzy numbers (IFNs). Garg (2018c) presented some geometric AOs for IVIFNs. Liu and Wang (2018) presented some ordered weighted AOs for solving DM problems. Mahmood et al. (2018) presented some hybrid AOs for solving DM problems under the triangular IFNs. Apart from these, several other AOs are developed by the scholars in the literature to solve the DM problems by using IFNs and/or IVIFNs information (Chen and Chang 2015; Chen et al. 2012b, 2016; Liu et al. 2017).

However, in the field of an information measure, the concept of distance, similarity, correlation, etc., are paying more attention by the scholars to solve the DM problems. Under it, Chen (1997) presented the similarity measure between the vague sets. Based on Hausdorff distance, Hung and Yang (2004) presented some similarity measures for IFSs. Garg (2016b) presented a generalized score function to rank the different IVIFNs. Szmidt and Kacprzyk (2000) presented the distance and similarity measures between the IFSs. Singh and Garg (2017) developed the distance measures between the type-2 IFSs. Garg and Kumar (2018a) presented new similarity measures for IFSs using set pair analysis. Xu (2007) presented similarity measures between the two IVIFSs and applied them to solve the pattern recognition problems. Dugenci (2016) presented some distance measures for IVIFSs and applied them to solve the group DM problems. Park et al. (2009), Wei et al. (2011) presented the correlation coefficients for IVIFSs. Garg (2017a) presented the distance and similarity measures for an intuitionistic multiplicative preference relation. Zhang et al. (2014) presented some entropy measures based on the distance measures for IVIFSs to solve the DM problems. Rani and Garg (2017) presented the family of the distance measures for the complex IFS, which is an extension of IFSs. Kumar and Garg (2018a, b) presented a technique for order preference with respect to the similarity to the ideal solution (TOPSIS) approach for solving the DM problems by using connection number of the set pair analysis theory. However, apart from them, some other theories for solving the DM problems by using distance or similarity measures are presented in the literature. For them, we refer to Arora and Garg (2018b), Garg (2018b), Huang and Li (2018), Jamkhaneh and Garg (2018) and Mitchell (2003).

Since all these facilitate the uncertainties to a great extent, but still they cannot withstand the situations where the decision-maker has to consider the falsity corresponding to the truth value ranging over an interval. Jun et al. (2012) combined the theory of the interval-valued FS with the FS and presented the concept of a cubic fuzzy set (CFS). After it, Mahmood et al. (2016) extended the CFS into the cubic hesitant FS by combining interval-valued hesitant FS and hesitant FS to solve the DM problems. Since CFSs considered only acceptance region into the analysis, but it is quite clear that the degree of rejection/non-membership plays an equivalent role during the performance analysis of any DM problems. So, by keeping these facts in mind, Kaur and Garg (2018a, c) introduced the idea of the cubic intuitionistic fuzzy set (CIFS), which is a hybrid set and formed by combining the features of IVIFSs and IFSs. Clearly, CIFS contains more information to represent the data in terms of IVIFNs and IFNs simultaneously. Furthermore, it allows us to consider the non-membership corresponding to the membership interval expressed in the form of IVIFS. Also, they have a fundamental characteristic of clubbing the information varying over different time spans. For example, suppose at the start of a financial year, a stock market analyst estimates the return on his investment to be 60–70% tending towards profit state and 10–15% tending towards the loss state. However, at the end of the financial year, the analyst found his returns to be 55% agreeing to the profit estimate and 35% disagreeing towards the loss estimate. Thus, a P-order CIFS is formed as $(\langle [0.60, 0.70],[0.10,0.15]\rangle ,$$\langle 0.55,0.35\rangle )$ . On the other hand, if at end of the financial year, the analyst found the returns to be 30% disagreeing to the profit estimate and 20% agreeing to the loss state then an R-order CIFS is formed as $\big ( \langle [0.60,0.70],[0.10,0.15]\rangle$, $\langle 0.30,0.20 \rangle \big )$. Thus, CIFSs have a powerful ability to express the uncertainty and fuzzy decision process more precisely and objectively during the DM process. Keeping the advantages of this set, recently, Kaur and Garg (2018b) presented some generalized AOs for cubic IFNs to solve the DM problems. Garg and Kaur (2018) presented a TOPSIS method based on the distance measures under CIFS environment to solve the group DM problems. However, to the best of authors knowledge, no work has been conducted so far on the information measures for the different CIFSs. Thus, motivated by the aforementioned studies and the advantages of CIFSs, in this manuscript, we present a novel series of non-weighted as well as weighted distance measures to process the available information. In the existing DM theories, the evaluation is done by processing the information confined to one-time frame only whereas using the distances on CIFSs the analysis can be done varying over two time spans simultaneously. Based on all these features, the objectives of this work are:

• 1. To extend the theory of the cubic set to CIFS, by considering the degree of the rejection into the analysis and hence develops some series of distance measures for them.

• 2. To discuss some desirable properties on the proposed distance measures.

• 3. To propose an efficient decision-making approach based on the developed measures and illustrate them with real-life cases associated with medical diagnosis and pattern recognition.

• 4. To compare the outcomes of the proposed approach with the existing approaches and to justify the proposed approach’s adaptability to the real-life cases.

To achieve the first objective, in this article, we represent the information towards the different objects under the CIFSs where the preferences related to each element are represented in the form of cubic intuitionistic fuzzy numbers (CIFNs). Then, we present a family of the distance measures based on Hamming, Euclidean and Hausdorff metrics. The desirable characteristics and relations between the various proposed measures are investigated to achieve the second objective. The third objective is achieved by establishing a DM approach based on the proposed measures to solve the problems associated with the medical diagnosis and pattern recognition under CIFS environment. The obtained results are compared with the existing approaches under IVIFSs for fulfilling the objective 4.

The rest of the manuscript is summarized as follows. Section 2 presents some basic concepts of IVIFSs, CFSs, and CIFSs. In Sect. 3, we present a family of distance measures between the two CIFSs and their desirable relations. Section 4 describes an approach based on the proposed measures to solve the DM problems followed by an illustrative example. Finally, a concrete conclusion is drawn in Sect. 5.

2 Preliminaries

Some basic concepts on IVIFS, CFS and CIFSs are summarized here over the universal set U.

Definition 1

(Atanassov 1986) An IFS A in a set U is given as:

$$\begin{aligned} A=\{(x, \zeta _A(x), \vartheta _A(x))\mid x\in U\}, \end{aligned}$$

(1)

where $\zeta _A$ and $\vartheta _A$ are the mappings from U to [0, 1] such that $0 \le \zeta _A(x) \le 1$ and $0\le \vartheta _A(x) \le 1$ and $0\le \zeta _A(x) + \vartheta _A(x) \le 1$. We denote this pair as $A=\langle \zeta _A, \vartheta _A\rangle$ and called as IFN.

Definition 2

(Atanassov and Gargov 1989) An IVIFS A in U is given as

$$\begin{aligned} A=\{\langle x, [\zeta _A^-(x), \zeta _A^+(x)], [\vartheta _A^-(x), \vartheta _A^+(x)]\rangle \mid x \in U \}, \end{aligned}$$

(2)

where $0 \le \zeta _A^-(x) \le \zeta _A^+(x) \le 1$, $0 \le \vartheta _A^-(x) \le \vartheta _A^+(x) \le 1$ and $\zeta _A^+(x) + \vartheta _A^+(x) \le 1$. This pair is often called as IVIFN.

Definition 3

(Jun et al. 2012) A CFS ${{\mathcal {A}}}$ in U is given as

$$\begin{aligned} {{\mathcal {A}}}=\{ (x,A_F(x),\lambda _F(x)) \mid x \in U \}, \end{aligned}$$

(3)

where $A_F(x)=[A^-(x), A^+(x)]$ and $\lambda _F(x)$, respectively, represents the interval-valued FS and FS in $x\in U$. We denote these pairs as ${{\mathcal {A}}}=\langle A_F,\lambda _F\rangle$.

Definition 4

(Jun et al. 2012) The complement of the CFS ${{\mathcal {A}}}$ is defined to be the cubic fuzzy set ${{\mathcal {A}}}^c =\{x,\langle A^c(x), 1-\lambda (x)\rangle \mid x\in U\}$.

Definition 5

(Kaur and Garg 2018a, c) A CIFS ${{\mathcal {A}}}$ is an ordered pair given by

$$\begin{aligned} {{\mathcal {A}}}=\{\langle x, A(x), \lambda (x)\rangle \mid x \in U\}, \end{aligned}$$

(4)

where $A= \{x, \langle [\zeta ^-(x),\zeta ^+(x)], [\vartheta ^-(x), \vartheta ^+(x)] \rangle \mid x\in U\}$ represents the IVIFS over U while $\lambda (x)=\{x, \langle \zeta (x), \vartheta (x) \rangle \mid x \in U\}$ represents an IFS. We denote this pair as ${\mathcal {\alpha }}=\big (A,\lambda \big )$, where $A=\langle [\zeta ^-, \zeta ^+], [\vartheta ^-, \vartheta ^+]\rangle$ and $\lambda =\langle \zeta , \vartheta \rangle$ and called as CIFN.

Remark 1

The special cases are to be considered from Eq. (4), which are summarized as follows:

• (i) A CIFS ${{\mathcal {A}}}$ in which $A(x)=\langle [0,0], [1,1]\rangle$ and $\lambda (x) = \langle 1,0 \rangle$ for all $x\in U$ is denoted by $\ddot{0}$.

• (ii) A CIFS ${{\mathcal {A}}}$ in which $A(x)=\langle [1,1], [0,0]\rangle$ and $\lambda (x) = \langle 0, 1\rangle$ for all $x \in U$ is denoted by $\ddot{1}$.

• (iii) A CIFS ${{\mathcal {A}}}$ in which $A(x)=\langle [0,0], [1,1]\rangle$ and $\lambda (x)= \langle 0,1\rangle$ for all $x \in U$ is denoted by $\hat{0}$.

• (iv) A CIFS ${{\mathcal {A}}}$ in which $A(x)=\langle [1,1], [0,0]\rangle$ and $\lambda (x)=\langle 1,0\rangle$ for all $x\in U$ is denoted by $\hat{1}$.

Definition 6

(Kaur and Garg 2018a, c) For any CIFNs $\alpha = \big (\big \langle [\zeta ^-, \zeta ^+], [\vartheta ^-, \vartheta ^+]\big \rangle , \big \langle \zeta , \vartheta \rangle \big )$ and $\alpha _i = \big ( \big \langle [\zeta _i^-, \zeta _i^+],$$[\vartheta _i^-, \vartheta _i^+]\big \rangle , \big \langle \zeta _i, \vartheta _i\rangle \big )$, $i\in \varLambda$, the following operations have been defined as follows:

• (i) $\alpha ^c = \big ( \langle [\vartheta ^-, \vartheta ^+], [\zeta ^-, \zeta ^+]\rangle , \langle \vartheta ,\zeta \rangle \big )$;

• (ii) (Equality) $\alpha _1=\alpha _2 \Leftrightarrow$$[\zeta _1^-,\zeta _1^+]=[\zeta _2^-,\zeta _2^+]$, $[\vartheta _1^-,\vartheta _1^+]=[\vartheta _2^-,\vartheta _2^+]$, $\zeta _1$ = $\zeta _2$ and $\vartheta _1$ = $\vartheta _2$;

• (iii) (R-order) $\alpha _1 \subseteq \alpha _2$ if $[\zeta _1^-,\zeta _1^+]\subseteq [\zeta _2^-,\zeta _2^+]$,$[\vartheta _1^-,\vartheta _1^+]\supseteq [\vartheta _2^-,\vartheta _2^+]$, $\zeta _1 \ge \zeta _2$ and $\vartheta _1 \le \vartheta _2$;

• (iv) (R-union): $\bigcup \alpha _i = \left( \left\langle \left[ \sup \limits _{i\in \varLambda }\zeta _i^-, \sup \limits _{i\in \varLambda }\zeta _i^+\right] , \left[ \inf \limits _{i \in \varLambda }\vartheta _i^-, \inf \limits _{i \in \varLambda }\vartheta _i^+ \right] \right\rangle , \left\langle \inf \limits _{i\in \varLambda } \zeta _i,\sup \limits _{i\in \varLambda } \vartheta _i\right\rangle \right)$;

• (v) (R-intersection): $\bigcap \alpha _i = \left( \left\langle \left[ \inf \limits _{i\in \varLambda }\zeta _i^-, \inf \limits _{i\in \varLambda }\zeta _i^+\right] , \left[ \sup \limits _{i \in \varLambda }\vartheta _i^-, \sup \limits _{i \in \varLambda }\vartheta _i^+\right] \right\rangle , \left\langle \sup \limits _{i\in \varLambda } \zeta _i,\inf \limits _{i\in \varLambda } \vartheta _i\right\rangle \right)$.

3 Proposed distance measures between CIFSs

Let $\varPhi (U)$ denote the set of all CIFSs defined on U.

Definition 7

For CIFSs ${{\mathcal {A}}}$, ${{\mathcal {B}}}$ and ${{\mathcal {C}}}$, distance measure is a real-valued function $d : \varPhi (U) \times \varPhi (U) \rightarrow [0,1]$ satisfying the following properties:

• (P1) $0 \le d({{\mathcal {A}}},{{\mathcal {B}}}) \le 1$;

• (P2) $d({{\mathcal {A}}},{{\mathcal {B}}})=0$ if and only if ${{\mathcal {A}}}={{\mathcal {B}}}$;

• (P3) $d({{\mathcal {A}}},{{\mathcal {B}}})=d({{\mathcal {B}}}{{\mathcal {A}}})$;

• (P4) If ${{\mathcal {A}}}\subseteq {{\mathcal {B}}}\subseteq {{\mathcal {C}}}$, then $d({{\mathcal {A}}},{{\mathcal {B}}}) \le d({{\mathcal {A}}},{{\mathcal {C}}})$ and $d({{\mathcal {B}}},{{\mathcal {C}}}) \le d({{\mathcal {A}}},{{\mathcal {C}}})$.

For CIFSs ${{\mathcal {A}}}=(\langle [\zeta _A^-(x),\zeta _A^+(x) ], [\vartheta _A^-(x),\vartheta _A^+(x) ] \rangle , \langle [\zeta _A(x),\vartheta _A(x) ] \rangle )$ and ${{\mathcal {B}}}= ( \langle [\zeta _B^-(x),\zeta _B^+(x) ]$, $[\vartheta _B^-(x), \vartheta _B^+(x) ] \rangle$, $\langle [\zeta _B(x),\vartheta _B(x) ] \rangle )$ over the universal set $U=\{x_1,x_2,\ldots ,x_n\}$, we define some series of the distance measures as follows:

• (i) Hamming distance measure:

  $$\begin{aligned} d_1({{\mathcal {A}}},{\mathcal {B}}) & = \frac{1}{6}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert +\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \hfill \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert +\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned}\right) ; \end{aligned}$$

  (5)

• (ii) Normalized Hamming distance measure:

  $$\begin{aligned} d_2({{\mathcal {A}}},{\mathcal {B}}) &= \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert +\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \hfill \\ + \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert +\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned}\right) ; \end{aligned}$$

  (6)

• (iii) Euclidean distance measure:

  $$\begin{aligned} d_3({{\mathcal {A}}},{\mathcal {B}})&= \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n}\left( \begin{aligned} {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2+{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 +{\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2 \hfill \\ +{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2+{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2+{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned} \right) ; \end{aligned}\right) ^{1/2} \qquad \end{aligned}$$

  (7)

• (iv) Normalized Euclidean distance measure:

  $$\begin{aligned} d_4({{\mathcal {A}}},{\mathcal {B}})&= \left( \begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2+{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 +{\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2 \hfill \\ +{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2+{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2+{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned} \right) . \end{aligned}\right) ^{1/2} \qquad \end{aligned}$$

  (8)

From them, the following results are obtained.

Theorem 1

The distance$d_2$, between two CIFSs${{\mathcal {A}}}$and${\mathcal {B}}$satisfies the following properties (P1)–(P4):

• (P1) $0 \le d_2({{\mathcal {A}}},{\mathcal {B}}) \le 1$;

• (P2) $d_2({{\mathcal {A}}},{\mathcal {B}})=0$if and only if${{\mathcal {A}}} = {\mathcal {B}}$;

• (P3) $d_2({{\mathcal {A}}},{\mathcal {B}}) = d_2({\mathcal {B}}, {{\mathcal {A}}})$;

• (P4) If${{\mathcal {A}}}\subseteq {\mathcal {B}}\subseteq {\mathcal {C}}$, then$d_2({{\mathcal {A}}},{\mathcal {B}})\le d_2({{\mathcal {A}}},{\mathcal {C}})$and$d_2({\mathcal {B}},{\mathcal {C}})\le d_2({{\mathcal {A}}},{\mathcal {C}})$, where${\mathcal {C}}\in \varPhi (U)$.

Proof

For a collection of CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$, we have

• (P1) By definition of $d_2$, we have $d_2({{\mathcal {A}}},{\mathcal {B}})\ge 0$, so for arbitrary CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$, it is enough to show that $d_2({{\mathcal {A}}},{\mathcal {B}})\le 1$ . For two CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$, we get $0 \le \zeta _A^-(x_i), \zeta _A^+(x_i), \vartheta _A^-(x_i), \vartheta _A^+(x_i) \le 1$, $0 \le \zeta _A(x_i), \vartheta _A(x_i) \le 1$ and $0 \le \zeta _B^-(x_i), \zeta _B^+(x_i)\le 1$, $0\le \vartheta _B^-(x_i), \vartheta _B^+(x_i) \le 1$ and $0 \le \zeta _B(x_i), \vartheta _B(x_i) \le 1$. This implies that $0 \le \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert \le 1$, $0 \le \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert \le 1$, $0 \le \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \le 1$, $0 \le \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert \le 1$, and $0 \le \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert \le 1$ , $0 \le \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \le 1$. Therefore,

  $$\begin{aligned} 0 \le \left( \begin{aligned} &\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert + \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ &\quad +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert + \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \le 6 \end{aligned}$$

  which implies that $d_2({{\mathcal {A}}},{\mathcal {B}})\le 1$.

• (P2) For two CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$, assume that $d_2({{\mathcal {A}}},{\mathcal {B}})=0$ which implies that

  $$\begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} & \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ &\quad +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) = 0 \end{aligned}$$

  if and only if, for all i,

  $$\begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert =0, \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert =0, \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert =0,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert =0 , \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert =0, \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert =0 \end{aligned}$$

  which is equivalent to

  $$\begin{aligned} \zeta _A^-(x_i) &= \zeta _B^-(x_i), \zeta _A^+(x_i)=\zeta _B^+(x_i), \vartheta _A^-(x_i)=\vartheta _B^-(x_i), \\ \vartheta _A^+(x_i) &= \vartheta _B^+(x_i), \zeta _A(x_i)=\zeta _B(x_i), \vartheta _A(x_i)=\vartheta _B(x_i) \end{aligned}$$

  Thus, $d_2({{\mathcal {A}}},{\mathcal {B}})=0$ implies that ${{\mathcal {A}}}={\mathcal {B}}$.

• (P3) For two CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$,

  $$\begin{aligned} d_2({{\mathcal {A}}},{\mathcal {B}}) &= \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned}\right) \\ &= \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _B^-(x_i)-\zeta _A^-(x_i)\big \vert + \big \vert \zeta _B^+(x_i)-\zeta _A^+(x_i)\big \vert +\big \vert \vartheta _B^-(x_i)-\vartheta _A^-(x_i)\big \vert \\ + \big \vert \vartheta _B^+(x_i)-\vartheta _A^+(x_i)\big \vert + \big \vert \zeta _B(x_i)-\zeta _A(x_i)\big \vert + \big \vert \vartheta _B(x_i)-\vartheta _A(x_i)\big \vert \end{aligned} \right) \\ &= d_2({\mathcal {B}}, {{\mathcal {A}}}). \end{aligned}$$

  Hence, $d_2({{\mathcal {A}}}, {\mathcal {B}})=d_2({\mathcal {B}}, {{\mathcal {A}}})$.

• (P4) If ${{\mathcal {A}}} \subseteq {\mathcal {B}} \subseteq {\mathcal {C}}$, then $[\zeta _A^-(x_i),\zeta _A^+(x_i)] \subseteq [\zeta _B^-(x_i),\zeta _B^+(x_i)] \subseteq [\zeta _C^-(x_i),\zeta _C^+(x_i)],$$[\vartheta _A^-(x_i),\vartheta _A^+(x_i)] \supseteq [\vartheta _B^-(x_i),\vartheta _B^+(x_i)] \supseteq [\vartheta _C^-(x_i),\vartheta _C^+(x_i)]$. Also, $\zeta _A(x_i) \ge \zeta _B(x_i) \ge \zeta _C(x_i)$ and $\vartheta _A(x_i) \le \vartheta _B(x_i) \le \vartheta _C(x_i).$ Therefore, $\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert \le \big \vert \zeta _A^-(x_i)-\zeta _C^-(x_i)\big \vert ,$$\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert \le \big \vert \zeta _A^+(x_i)-\zeta _C^+(x_i)\big \vert ,$$\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \le \big \vert \vartheta _A^-(x_i)-\vartheta _C^-(x_i)\big \vert ,$$\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert \le \big \vert \vartheta _A^+(x_i)-\vartheta _C^+(x_i)\big \vert ,$$\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert \le \big \vert \zeta _A(x_i) - \zeta _C(x_i)\big \vert ,$ and $\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \le \big \vert \vartheta _A(x_i) - \vartheta _C(x_i)\big \vert$. Thus,

  $$\begin{aligned} d_2({{\mathcal {A}}},{\mathcal {C}}) & = \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _C^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _C^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _C^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _C^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _C(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _C(x_i)\big \vert \end{aligned}\right) \\ & \ge \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert +\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned}\right) \\ & \ge d_2({{\mathcal {A}}}, {\mathcal {B}}). \end{aligned}$$

  Similarly, $d_2({{\mathcal {A}}},{\mathcal {C}}) \ge d_2({\mathcal {B}},{\mathcal {C}})$.

Hence, $d_2$ is a valid distance measure.

Theorem 2

For two CIFSs${{\mathcal {A}}}$and${\mathcal {B}}$, the distance measure$d_4({{\mathcal {A}}}, {\mathcal {B}})$satisfies the properties (P1)–(P4) as described in Definition 7.

Proof

For CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$ , we have

• (P1) Since $0 \le \zeta _A^-(x_i), \zeta _A^+(x_i), \vartheta _A^-(x_i), \vartheta _A^+(x_i) \le 1$, $0 \le \zeta _A(x_i), \vartheta _A(x_i) \le 1$ , $0 \le \zeta _B^-(x_i)$, $\zeta _B^+(x_i)$, $\vartheta _B^-(x_i)$, $\vartheta _B^+(x_i) \le 1$ and $0 \le \zeta _B(x_i), \vartheta _B(x_i) \le 1$. This implies that $0 \le {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2 \le 1$, $0 \le {\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 \le 1$, $0 \le {\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2$$\le 1$, $0 \le {\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2$$\le 1$, $0 \le {\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2 \le 1$, $0 \le {\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2$$\le 1$. Thus, it follows that $0 \le d_4({{\mathcal {A}}}, {\mathcal {B}}) \le 1$.

• (P2) For two CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$ , assume that $d_4({{\mathcal {A}}}, {\mathcal {B}})=0$ which implies that

  $$\begin{aligned}&\left( \begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2 +{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2+{\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2\\ +{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2 +{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2+{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned}\right) \end{aligned} \right) ^{1/2} = 0 \\&\quad \Leftrightarrow \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert =0, \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert =0, \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert =0,\\&\qquad \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert =0, \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert =0, \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert =0 \\&\quad \Leftrightarrow \zeta _A^-(x_i)=\zeta _B^-(x_i), \zeta _A^+(x_i)=\zeta _B^+(x_i), \vartheta _A^-(x_i)=\vartheta _B^-(x_i),\\&\qquad \vartheta _A^+(x_i)=\vartheta _B^+(x_i), \zeta _A(x_i)=\zeta _B(x_i), \vartheta _A(x_i)=\vartheta _B(x_i) \quad {\text {for all }} \ i\\&\quad \Leftrightarrow A=B \end{aligned}$$

• (P3) For any two numbers a and b, we have $\big \vert a-b\big \vert =\big \vert b-a\big \vert$. Thus, we have $d_4({{\mathcal {A}}}, {\mathcal {B}}) = d_4({\mathcal {B}}, {{\mathcal {A}}})$.

• (P4) If ${{\mathcal {A}}} \subseteq {\mathcal {B}} \subseteq {\mathcal {C}}$, then $[\zeta _A^-(x_i),\zeta _A^+(x_i)] \subseteq [\zeta _B^-(x_i),\zeta _B^+(x_i)] \subseteq [\zeta _C^-(x_i),\zeta _C^+(x_i)],$$[\vartheta _A^-(x_i),\vartheta _A^+(x_i)] \supseteq [\vartheta _B^-(x_i),\vartheta _B^+(x_i)] \supseteq [\vartheta _C^-(x_i),\vartheta _C^+(x_i)].$ Also, $\zeta _A(x_i) \ge \zeta _B(x_i) \ge \zeta _C(x_i)$ and $\vartheta _A(x_i) \le \vartheta _B(x_i) \le \vartheta _C(x_i).$ Therefore,

  $$\begin{aligned} {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2\le & {} {\big \vert \zeta _A^-(x_i)-\zeta _C^-(x_i)\big \vert }^2, {\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 \le {\big \vert \zeta _A^+(x_i)-\zeta _C^+(x_i)\big \vert }^2,\\ {\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2\le & {} {\big \vert \vartheta _A^-(x_i)-\vartheta _C^-(x_i)\big \vert }^2, {\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2 \le {\big \vert \vartheta _A^+(x_i)-\vartheta _C^+(x_i)\big \vert }^2, \\ {\big \vert \zeta _A(x_i) - \zeta _B(x_i)\big \vert }^2\le & {} {\big \vert \zeta _A(x_i) - \zeta _C(x_i)\big \vert }^2, {\big \vert \vartheta _A(x_i) - \vartheta _B(x_i)\big \vert }^2 \le {\big \vert \vartheta _A(x_i) - \vartheta _C(x_i)\big \vert }^2. \end{aligned}$$

  Thus,

  $$\begin{aligned} d_4({{\mathcal {A}}},{\mathcal {C}}) & = \left( \begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)- \zeta _C^-(x_i)\big \vert ^2 + \big \vert \zeta _A^+(x_i) - \zeta _C^+(x_i)\big \vert ^2 +\big \vert \vartheta _A^-(x_i)-\vartheta _C^-(x_i)\big \vert ^2 \\ +{\big \vert \vartheta _A^+(x_i)-\vartheta _C^+(x_i)\big \vert }^2 +{\big \vert \zeta _A(x_i)-\zeta _C(x_i)\big \vert }^2+{\big \vert \vartheta _A(x_i)-\vartheta _C(x_i)\big \vert }^2 \end{aligned} \right) \end{aligned} \right) ^{1/2} \\ & \ge \left( \begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n} \left( \begin{aligned} {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2+{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 +{\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2 \\ +{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2 +{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2+{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned} \right) \end{aligned} \right) ^{1/2} \\ & = d_4({{\mathcal {A}}}, {\mathcal {B}}). \end{aligned}$$

  Similarly, $d_4({{\mathcal {A}}},{\mathcal {C}}) \ge d_4({\mathcal {B}},{\mathcal {C}})$. Hence, $d_4$ is a valid distance measure.

Example 1

Let ${\mathcal {A}}$ and ${\mathcal {B}}$ be two known pattern expressed as ${{\mathcal {A}}}=\big \{\big ( x_1,\big (\langle [0.2,0.4]$, $[0.3,0.5]\rangle$, $\langle 0.3,0.4\rangle \big )\big )$, $\big ( x_2,\big ( \langle [0.3,0.4]$, $[0.2,0.6]\rangle$, $\langle 0.1,0.3\rangle \big )\big )\big \}$ and ${\mathcal {B}}=\big \{\big ( x_1,\big (\langle [0.3,0.4],[0.1,0.2]\rangle , \langle 0.25,0.40\rangle \big )\big ) ,$$\big ( x_2,\big (\langle [0.15,0.25]$, $[0.30,0.45]\rangle$, $\langle 0.4,0.2\rangle \big )\big )\big \}$, then

$$\begin{aligned} d_2({{\mathcal {A}}}, {\mathcal {B}})& = \frac{1}{6\times 2}\left( \begin{aligned} \big \vert 0.20-0.30\big \vert +\big \vert 0.40-0.40\big \vert +\big \vert 0.30-0.10\big \vert +\big \vert 0.50-0.20\big \vert \\ +\big \vert 0.30-0.25\big \vert +\big \vert 0.40-0.40\big \vert +\big \vert 0.30-0.15\big \vert +\big \vert 0.40-0.25\big \vert \\ +\big \vert 0.20-0.30\big \vert +\big \vert 0.60-0.45\big \vert +\big \vert 0.10-0.40\big \vert +\big \vert 0.30-0.20\big \vert \end{aligned}\right) \\ & = 0.1333 \end{aligned}$$

and

$$\begin{aligned} d_4({{\mathcal {A}}}, {\mathcal {B}})& = \left( \begin{aligned} \frac{1}{6\times 2} \left( \begin{aligned} {\big \vert 0.20-0.30\big \vert }^2+\big \vert {0.40-0.40\big \vert }^2+{\big \vert 0.30-0.10\big \vert }^2+{\big \vert 0.50-0.20\big \vert }^2\\ +{\big \vert 0.30-0.25\big \vert }^2+{\big \vert 0.40-0.40\big \vert }^2 +{\big \vert 0.30-0.15\big \vert }^2+{\big \vert 0.40-0.25\big \vert }^2\\ +{\big \vert 0.20-0.30\big \vert }^2 +{\big \vert 0.60-0.45\big \vert }^2+{\big \vert 0.10-0.40\big \vert }^2+{\big \vert 0.30-0.20\big \vert }^2 \end{aligned} \right) \end{aligned} \right) ^{1/2} \\ & = 0.1633 \end{aligned}$$

Theorem 3

The measures $d_1$ and $d_3$ satisfies the following inequalities:

• (i) $0 \le d_1 \le n$,

• (ii) $0 \le d_3 \le \sqrt{n}$.

Proof

Let ${{\mathcal {A}}}$ and ${\mathcal {B}}$ be any two CIFSs. Clearly, $d_1({{\mathcal {A}}}, {\mathcal {B}})=nd_2({{\mathcal {A}}}, {\mathcal {B}})$. By Theorem 1, we have $0 \le d_2({{\mathcal {A}}}, {\mathcal {B}}) \le 1$. It implies that $0 \le \frac{d_1({{\mathcal {A}}}, {\mathcal {B}})}{n} \le 1$ and hence $0 \le d_1({{\mathcal {A}}}, {\mathcal {B}}) \le n$. Similarly, we can prove that $0 \le d_3({{\mathcal {A}}}, {\mathcal {B}}) \le \sqrt{n}$. As ${{\mathcal {A}}}$ and ${\mathcal {B}}$ are arbitrary, so we get $0 \le d_1 \le n$ and $0 \le d_3 \le \sqrt{n} .$

Theorem 4

The measures $d_{1},d_{3}$ and $d_{2}, d_{4}$ satisfy the following inequalities:

• (i) $d_{3} \le \sqrt{d_{1}}$

• (ii) $d_{4} \le \sqrt{d_{2}}$

Proof

Let ${{\mathcal {A}}}$ and ${\mathcal {B}}$ be two CIFSs.

• (i) Since $0 \le \zeta _A^-(x_i), \zeta _A^+(x_i), \vartheta _A^-(x_i), \vartheta _A^+(x_i) \le 1$, $0 \le \zeta _A(x_i), \vartheta _A(x_i) \le 1$ , $0 \le \zeta _B^-(x_i)$, $\zeta _B^+(x_i)$, $\vartheta _B^-(x_i)$, $\vartheta _B^+(x_i) \le 1$ and $0 \le \zeta _B(x_i), \vartheta _B(x_i) \le 1$. Then, by using the property, ${\eta }^2 < \eta$, for any $\eta \in [0,1]$, we have $\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ^2 \le \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert$, $\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ^2 \le \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert$, $\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ^2 \le \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert$, $\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ^2 \le \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert$ ,$\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^2 \le \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert$ and $\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert ^2 \le \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert$, for all i. This implies that,

  $$\begin{aligned}&\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ^2 +\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ^2 +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ^2\\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ^2 +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^2+\big \vert \vartheta _A(x_i) -\vartheta _B(x_i)\big \vert ^2 \end{aligned} \right) \\&\quad \le \left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert +\big \vert \zeta _A^+(x_i) -\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert +\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) . \end{aligned}$$

  Therefore, for all i, we have

  $$\begin{aligned}&\frac{1}{6} \sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ^2+\big \vert \zeta _A^+(x_i) -\zeta _B^+(x_i)\big \vert ^2+\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ^2\\ + \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ^2 +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^2+\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert ^2 \end{aligned} \right) \\&\quad \le \frac{1}{6}\sum \limits _{i=1}^{n} \left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert + \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) , \end{aligned}$$

  which implies that $d_3({{\mathcal {A}}}, {\mathcal {B}}) \le \sqrt{d_1({{\mathcal {A}}}, {\mathcal {B}})}$. Since, ${{\mathcal {A}}}$ and ${\mathcal {B}}$ are arbitrary CIFSs thus, $d_3 \le \sqrt{d_1}$ is true for all CIFSs.

• (ii) This can be proceeded in the similar manner as above.

As in practical situations, many times we have to deal with such situations in which various CIFSs may have weights assigned to them. So, taking into account weights $\kappa _i(i=1,2,\ldots ,n)$, where each $\kappa _i> 0$ and $\sum \nolimits _{i=1}^{n}\kappa _i=1$, we define weighted Hamming as well as weighted Euclidean distances between two CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$ as follows:

• (i) Weighted Hamming distance measure:

  $$\begin{aligned} d_5({{\mathcal {A}}}, {\mathcal {B}})& = \frac{1}{6}\sum \limits _{i=1}^{n}\kappa _i \left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert + \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert +\big \vert \vartheta _A(x_i) -\vartheta _B(x_i)\big \vert \end{aligned} \right) \end{aligned}$$

  (9)

• (ii) Weighted Euclidean distance measure:

  $$\begin{aligned} d_6({{\mathcal {A}}}, {\mathcal {B}})& = \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n} \kappa _i \left( \begin{aligned} {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2+{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 +{\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2 \\ +{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2+{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2+{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned} \right) \end{aligned} \right) ^{1/2} \qquad \end{aligned}$$

  (10)

Especially, when $\kappa _{i}=1/n$, for $i=1,2,\ldots ,n$, then distances $d_5$ and $d_6$ reduces to $d_2$ and $d_4$, respectively.

Theorem 5

The weighted measures$d_{k}({{\mathcal {A}}}, {\mathcal {B}}), (k=5,6)$satisfy the following properties for${{\mathcal {A}}},{\mathcal {B}}, {\mathcal {C}} \in \varPhi (U)$:

• (P1) $0\le d_{k}({{\mathcal {A}}}, {\mathcal {B}})\le 1$;

• (P2) $d_{k}({{\mathcal {A}}}, {\mathcal {B}})= 0$if and only if${{\mathcal {A}}} = {\mathcal {B}}$;

• (P3) $d_{k}({{\mathcal {A}}}, {\mathcal {B}})=d_{k}({\mathcal {B}}, {{\mathcal {A}}})$.

• (P4) If${{\mathcal {A}}}\subseteq {\mathcal {B}} \subseteq {\mathcal {C}}$then$d_{k}({{\mathcal {A}}}, {\mathcal {B}}) \le d_{k}({{\mathcal {A}}},{\mathcal {C}})$and$d_{k}({\mathcal {B}},{\mathcal {C}}) \le d_{k}({{\mathcal {A}}},{\mathcal {C}})$.

Proof

Since $\kappa _{i}>0$ and $\sum \nolimits _{i=1}^{n}\kappa _{i}=1$, then we can easily get the proof of above theorem. Hence, we omit it here.

Theorem 6

The measures$d_5$and$d_1$satisfy the inequality$d_5 \le d_1$.

Proof

Since $\kappa _i>0$ and $\sum \nolimits _{i=1}^{n} \kappa _i = 1$, then for any two CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$ , we have

$$\begin{aligned} d_5({{\mathcal {A}}}, {\mathcal {B}}) & = \frac{1}{6}\sum \limits _{i=1}^{n} \kappa _i\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ + \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert + \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned}\right) \\ & \le \frac{1}{6}\sum \limits _{i=1}^{n} \left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert +\big \vert \zeta _A^+(x_i) -\zeta _B^+(x_i)\big \vert + \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \\ &= d_1({{\mathcal {A}}}, {\mathcal {B}}). \end{aligned}$$

Since, ${{\mathcal {A}}}$ and ${\mathcal {B}}$ are arbitrary, therefore we obtain $d_5\le d_1$.

Theorem 7

The measures$d_6$and$d_3$satisfy the inequality$d_6 \le d_3$.

Proof

Similar to Theorem 6, so we omit here.

Theorem 8

The measures$d_6$and$d_1$satisfy the inequality$d_6 \le \sqrt{d_1}$.

Proof

Since, $0\le \zeta _A^-(x_i),\zeta _A^+(x_i),\vartheta _A^-(x_i),\vartheta _A^+(x_i) \le 1$ and $0\le \zeta _B^-(x_i),$$\zeta _B^+(x_i),$$\vartheta _B^-(x_i),$$\vartheta _B^+(x_i) \le 1$. and hence $0 \le \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert \le 1$. Now for any real number $\eta \in [0,1]$, we know that $\big \vert \eta \big \vert ^2 \le \big \vert \eta \big \vert$. Thus, it follows that ${\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2 \le \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert$. Similarly, ${\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 \le \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert$, ${\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2 \le \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert$, ${\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2 \le \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert$ and ${\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \le \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert$. Now, for $\kappa _i>0$ such that $\sum _{i=1}^n \kappa _i =1$, we have

$$\begin{aligned} d_6({{\mathcal {A}}}, {\mathcal {B}}) & = \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n} \kappa _i \left( \begin{aligned} {\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2+{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 +{\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2 \\ +{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2+ \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^2+{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned} \right) \end{aligned} \right) ^{1/2} \\\le & {} \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n} \left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert + \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert + \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert + \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert + \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned}\right) \end{aligned} \right) ^{1/2} \\\le & {} \sqrt{d_1({{\mathcal {A}}}, {\mathcal {B}})} \end{aligned}$$

As ${{\mathcal {A}}}$ and ${\mathcal {B}}$ are arbitrary, so we get $d_6 \le \sqrt{d_1}$.

Theorem 9

The measures$d_6$and$d_5$have the inequality$d_6 \le \sqrt{d_5}$.

Proof

Similar to Theorem 8.

Next, we propose Hausdorff measures for two CIFSs ${{\mathcal {A}}}$ and ${\mathcal {B}}$ as follows:

• (i) Hausdorff hamming distance measure:

  $$\begin{aligned} d_1^H({{\mathcal {A}}}, {\mathcal {B}})& = \frac{1}{6}\sum \limits _{i=1}^{n} \left( \begin{aligned} \max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \end{aligned} \right) ; \qquad \end{aligned}$$

  (11)

• (ii) Hausdorff normalized hamming distance measure:

  $$\begin{aligned} d_2^H({{\mathcal {A}}}, {\mathcal {B}})& = \frac{1}{6n}\sum \limits _{i=1}^{n} \left( \begin{aligned} \max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \end{aligned} \right) ; \qquad \end{aligned}$$

  (12)

• (iii) Hausdorff Euclidean distance measure:

  $$\begin{aligned} d_3^H({\mathcal {A}}, {\mathcal {B}})& = \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n} \left( \begin{aligned} \max \left( \begin{aligned}&{\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2,{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2, {\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2,\\&{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2,{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2,{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned} \right) \end{aligned} \right) \end{aligned} \right) ^{1/2}; \qquad \end{aligned}$$

  (13)

• (iv) Hausdorff normalized Euclidean distance measure:

  $$\begin{aligned} d_4^H({\mathcal {A}}, {\mathcal {B}})& = \left( \begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \max \left( \begin{aligned}&{\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2,{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2, {\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2,\\&{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2,{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2,{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned} \right) \end{aligned} \right) \end{aligned} \right) ^{1/2}; \qquad \end{aligned}$$

  (14)

• (v) Hausdorff weighted hamming distance measure:

  $$\begin{aligned} d_5^H({\mathcal {A}}, {\mathcal {B}})& = \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n} \kappa _i \max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \end{aligned} \right) ; \qquad \end{aligned}$$

  (15)

• (vi) Hausdorff weighted Euclidean distance measure:

  $$\begin{aligned} d_6^H({\mathcal {A}}, {\mathcal {B}})& = \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n}\kappa _i \max \left( \begin{aligned}&{\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2, {\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2, {\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2,\\&{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2,{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2,{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \end{aligned}\right) \end{aligned}\right) ^{1/2}. \qquad \end{aligned}$$

  (16)

Theorem 10

For${\mathcal {A}},{\mathcal {B}},{\mathcal {C}} \in \varPhi (U)$, the measures$d_{k}^{H}({\mathcal {A}}, {\mathcal {B}}), (k=2,4)$satisfies the following properties.

• (P1) $0\le d_{k}^{H}({\mathcal {A}}, {\mathcal {B}})\le 1$;

• (P2) $d_{k}^{H}({\mathcal {A}}, {\mathcal {B}})= 0$if and only if$A = B$;

• (P3) $d_{k}^{H}({\mathcal {A}}, {\mathcal {B}})=d_{k}^{H}({\mathcal {B}}, {\mathcal {A}})$;

• (P4) If${\mathcal {A}}\subseteq {\mathcal {B}} \subseteq {\mathcal {C}}$then$d_{k}^{H}({\mathcal {A}}, {\mathcal {B}}) \le d_{k}^{H}({\mathcal {A}},{\mathcal {C}})$and$d_{k}^{H}({\mathcal {B}},{\mathcal {C}}) \le d_{k}^{H}({\mathcal {A}},{\mathcal {C}})$.

Proof

Let ${\mathcal {A}},{\mathcal {B}},{\mathcal {C}} \in \varPhi (U)$ be any three CIFSs. Then, for $q=1,2$, we have

• (P1) Since ${\mathcal {A}}$ and ${\mathcal {B}}$ are two CIFSs, so by the definitions of CIFSs, we have $0 \le \mid \zeta _A^-(x_i)-\zeta _B^-(x_i)\mid ^q\le 1$, $0\le \mid \zeta _A^+(x_i)-\zeta _B^+(x_i)\mid ^q\le 1$, $0\le \mid \vartheta _{A}^-(x_i)-\vartheta _{B}^-(x_i)\mid ^q\le 1$, $0\le \mid \vartheta _{A}^+(x_i)-\vartheta _{B}^+(x_i)\mid ^q\le 1$, $0\le \mid \zeta _A(x_i)-\zeta _B(x_i)\mid ^q \le 1$ and $0\le \mid \vartheta _A(x_i)-\vartheta _B(x_i)\mid ^q\le 1$ which implies that $0\le \max \Big (\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ^q, \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ^q, \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ^q, \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ^q, \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^q, \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert ^q\Big )\le 1$. Thus, $0\le d_{k}^{H}({\mathcal {A}}, {\mathcal {B}})\le 1$.

• (P2) Assume that $d_k^H({\mathcal {A}}, {\mathcal {B}})=0$ if and only if

  $$\begin{aligned}&\max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ^q, \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ^q, \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ^q,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ^q, \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^q, \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert ^q \end{aligned} \right) =0 \\&\quad \Leftrightarrow \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ^q=0, \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ^q=0, \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ^q=0, \\&\qquad \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ^q=0, \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^q=0, \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert ^q=0 \qquad \text { for all } i \\&\quad \Leftrightarrow \zeta _A^-(x_i)=\zeta _B^-(x_i), \zeta _A^+(x_i)=\zeta _B^+(x_i), \vartheta _A^-(x_i)=\vartheta _B^-(x_i), \\&\qquad \vartheta _A^+(x_i)=\vartheta _B^+(x_i), \zeta _A(x_i)=\zeta _B(x_i), \vartheta _A(x_i)=\vartheta _B(x_i) \qquad \text { for all } i, \\&\quad \Leftrightarrow {\mathcal {A}} = {\mathcal {B}} \end{aligned}$$

• (P3) For two CIFSs ${\mathcal {A}}$ and ${\mathcal {B}}$ , we have

  $$\begin{aligned} d_k^H({\mathcal {A}}, {\mathcal {B}}) &= \left( \begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n} \max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ^q,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ^q, \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ^q,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ^q, \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ^q, \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert ^q \end{aligned} \right) \end{aligned} \right) ^{1/q} \\ &= \left( \begin{aligned} \frac{1}{6n}\sum \limits _{i=1}^{n} \max \left( \begin{aligned}&\big \vert \zeta _B^-(x_i)-\zeta _A^-(x_i)\big \vert ^q,\big \vert \zeta _B^+(x_i)-\zeta _A^+(x_i)\big \vert ^q, \big \vert \vartheta _B^-(x_i)-\vartheta _A^-(x_i)\big \vert ^q, \\&\big \vert \vartheta _B^+(x_i)-\vartheta _A^+(x_i)\big \vert ^q, \big \vert \zeta _B(x_i)-\zeta _A(x_i)\big \vert ^q, \big \vert \vartheta _B(x_i)-\vartheta _A(x_i)\big \vert ^q \end{aligned} \right) \end{aligned} \right) ^{1/q} \\&= d_k^H({\mathcal {B}}, {\mathcal {A}}) \end{aligned}$$

• (P4) If ${\mathcal {A}} \subseteq {\mathcal {B}} \subseteq {\mathcal {C}}$, then $[\zeta _A^-(x_i),\zeta _A^+(x_i)] \subseteq [\zeta _B^-(x_i),\zeta _B^+(x_i)] \subseteq [\zeta _C^-(x_i),\zeta _C^+(x_i)],$$[\vartheta _A^-(x_i),\vartheta _A^+(x_i)] \supseteq [\vartheta _B^-(x_i),\vartheta _B^+(x_i)] \supseteq [\vartheta _C^-(x_i),\vartheta _C^+(x_i)].$ Also, $\zeta _A(x_i) \ge \zeta _B(x_i) \ge \zeta _C(x_i)$ and $\vartheta _A(x_i) \le \vartheta _B(x_i) \le \vartheta _C(x_i).$ Therefore, $\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert \le \big \vert \zeta _A^-(x_i)-\zeta _C^-(x_i)\big \vert ,$$\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert \le \big \vert \zeta _A^+(x_i)-\zeta _C^+(x_i)\big \vert ,$$\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \le \big \vert \vartheta _A^-(x_i)-\vartheta _C^-(x_i)\big \vert ,$$\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert \le \big \vert \vartheta _A^+(x_i)-\vartheta _C^+(x_i)\big \vert ,$$\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert \le \big \vert \zeta _A(x_i)-\zeta _C(x_i)\big \vert ,$$\big \vert \vartheta _A(x_i) - \vartheta _B(x_i)\big \vert \le \big \vert \vartheta _A(x_i)-\vartheta _C(x_i)\big \vert .$ Thus,

  $$\begin{aligned} d_2^H({\mathcal {A}},{\mathcal {C}})&= \frac{1}{6n}\sum \limits _{i=1}^{n} \max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _C^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _C^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _C^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _C^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _C(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _C(x_i)\big \vert \end{aligned} \right) \\ &\ge \frac{1}{6n}\sum \limits _{i=1}^{n}\max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \\ &= d_2^H({\mathcal {A}}, {\mathcal {B}}). \end{aligned}$$

  Hence, $d_2^H({\mathcal {A}},{\mathcal {C}}) \ge d_2^H({\mathcal {A}}, {\mathcal {B}})$. Similarly, $d_2^H({\mathcal {A}},{\mathcal {C}}) \ge d_2^H({\mathcal {B}},{\mathcal {C}})$.

Theorem 11

The measures $d_5^H$ and $d_6^H$ are also the valid distance measures.

Proof

Similar to Theorem 10.

Theorem 12

The Hausdorff measures$d_1^H$and$d_5^H$satisfy the inequality$d_5^H \le d_1^H$.

Proof

For two CIFSs ${\mathcal {A}}$ and ${\mathcal {B}}$ and $\kappa _i>0$ be such that $\sum _{i=1}^n \kappa _i=1$. Then, we have

$$\begin{aligned} d_5^H({\mathcal {A}}, {\mathcal {B}}) &= \frac{1}{6}\sum \limits _{i=1}^{n}\kappa _i \max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \\ &\le \frac{1}{6}\sum \limits _{i=1}^{n}\max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert , \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \\ &= d_1^H({\mathcal {A}}, {\mathcal {B}}). \end{aligned}$$

As ${\mathcal {A}}$ and ${\mathcal {B}}$ are arbitrary CIFSs, so we get $d_5^H \le d_1^H$.

Theorem 13

The Hausdorff measures$d_3^H$and$d_6^H$satisfy the inequality$d_6^H \le d_3^H$.

Proof

Similar to Theorem 12.

Theorem 14

The measure$d_1^H$, $d_2^H$, $d_3^H$and$d_4^H$satisfy the following relations:

• (i) $d_3^H\le \sqrt{d_1^H}$;

• (ii) $d_4^H\le \sqrt{d_2^H}$.

Proof

We will prove part (i) only, while other can be proven similarly.

Let ${\mathcal {A}}$ and ${\mathcal {B}}$ be two CIFSs and for any number $\eta \in [0,1]$, we know that $\mid \eta \mid ^2 \le \mid \eta \mid$. Thus, by the definition of $d_3^H$, we have ${\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2 \le \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert$, ${\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2 \le \big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert$, ${\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2 \le \big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert$, ${\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2 \le \big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert$, ${\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2 \le \big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert$ and ${\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2 \le \big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert$. Thus,

$$\begin{aligned} d_3^H({\mathcal {A}}, {\mathcal {B}}) & = \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n}\max \left( \begin{aligned}&{\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert }^2,{\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert }^2, {\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert }^2,\\&{\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert }^2,{\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert }^2,{\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert }^2\end{aligned} \right) \end{aligned} \right) ^{1/2} \\\le & {} \left( \begin{aligned} \frac{1}{6}\sum \limits _{i=1}^{n}\max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \end{aligned} \right) ^{1/2} \\& = \sqrt{d_1^H({\mathcal {A}}, {\mathcal {B}})}. \end{aligned}$$

Since, ${\mathcal {A}}$ and ${\mathcal {B}}$ are arbitrary, so we get $d_3^H \le \sqrt{d_1^H}$.

Theorem 15

The measures$d_2^H$and$d_2$satisfy the inequality$d_2^H \le d_2$.

Proof

Since for any positive numbers $a_i(i=1,2,\ldots ,n)$, we know that $\max \nolimits _i \{a_i\}\le \sum \nolimits _{i=1}^n a_i$ and hence by the definition of $d_2^H$, we get

$$\begin{aligned} d_2^H({\mathcal {A}}, {\mathcal {B}})& = \frac{1}{6n}\sum \limits _{i=1}^{n}\max \left( \begin{aligned}&\big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert ,\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert ,\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert ,\\&\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert ,\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert ,\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \\\le & {} \frac{1}{6n}\sum \limits _{i=1}^{n}\left( \begin{aligned} \big \vert \zeta _A^-(x_i)-\zeta _B^-(x_i)\big \vert +\big \vert \zeta _A^+(x_i)-\zeta _B^+(x_i)\big \vert +\big \vert \vartheta _A^-(x_i)-\vartheta _B^-(x_i)\big \vert \\ +\big \vert \vartheta _A^+(x_i)-\vartheta _B^+(x_i)\big \vert +\big \vert \zeta _A(x_i)-\zeta _B(x_i)\big \vert +\big \vert \vartheta _A(x_i)-\vartheta _B(x_i)\big \vert \end{aligned} \right) \\& = d_2({\mathcal {A}}, {\mathcal {B}}). \end{aligned}$$

Since, ${\mathcal {A}}$ and ${\mathcal {B}}$ are arbitrary, therefore, $d_2^H \le d_2$.

Theorem 16

The measures$d_k^H$and$d_k$, $(k=1,2,\ldots ,6)$satisfy the inequality$d_k^H \le d_k$.

Proof

Similar to Theorem 15.

Theorem 17

The measures$d_5^H$and$d_1$satisfy the inequality$d_5^H \le d_1$.

Proof

Since from the above theorems, we get $d_5^H\le d_5$ and $d_5\le d_1$. Therefore, $d_5^H\le d_1$.

Theorem 18

The measures$d_6^H$and$d_3$satisfy the inequality$d_6^H \le d_3$.

Proof

From Theorem 16, we get $d_6^H \le d_6$ while from Theorem 7, we get $d_6 \le d_3$. Hence, $d_6^H \le d_3$.

Theorem 19

Distance measures$d_5$, $d_1$, $d_1^H$and$d_3$, $d_6$, $d_3^H$satisfy the following relations:

• (i) $d_1 \ge \frac{d_5+d_1^H}{2}$and$d_1 \ge \sqrt{d_5 \cdot d_1^H}$;

• (ii) $d_3 \ge \frac{d_{6}+ d_{3}^{H}}{2}$and$d_3\ge \sqrt{d_{6}\cdot d_{3}^{H}}$.

Proof

Since $d_{1}^{H}\le d_{1}$ and $d_{5}\le d_{1}$. So, by adding these inequalities, we get $\frac{d_{5}+ d_{1}^{H}}{2}\le d_{1}$ while by multiplying we get $\sqrt{d_{5}\cdot d_{1}^{H}} \le d_{1}$.

Also, $d_{3}^{H}\le d_3$ and $d_{6}\le d_{3}$. So, by adding these inequalities, we get $\frac{d_{6}+ d_3^{H}}{2}\le d_{3}$ and by multiplying, we get $\sqrt{d_{6}\cdot d_{3}^{H}} \le d_3 .$

4 Decision-making based on the proposed distance measure of CIFSs

In this section, a decision-making method by using the above defined distance measures for CIFSs has been presented followed by an illustrative example for demonstrating the approach.

4.1 Decision-making approach

For this, assume that there are m alternatives, denoted by $A_1,A_2,\ldots ,A_m$, which are evaluated by an expert under the set of n criteria, denoted by $C_1,C_2,\ldots ,C_n$, and gave his preferences in the form of the CIFNs $\alpha _{pq} = \big ( \langle [\zeta _{pq}^-, \zeta _{pq}^+]$, $[\vartheta _{pq}^-, \vartheta _{pq}^+]\rangle , \langle \zeta _{pq}, \vartheta _{pq} \rangle \big );\ p=1,2,\ldots ,m; \,q=1,2,\ldots ,n$. Thus, the rating values corresponding to each alternative are summarized in terms of CIFNs as follows:

$$\begin{aligned} A_p = \bigg\{ \big (x_q, \langle [\zeta _{pq}^-(x_q), \zeta _{pq}^+(x_q)], [\vartheta _{pq}^-(x_q), \vartheta _{pq}^+(x_q)]\rangle , \langle \zeta _{pq}(x_q), \vartheta _{pq}(x_q) \rangle \big )\mid x_q \in U \bigg\} . \end{aligned}$$

Let $\kappa _q (q=1,2,\ldots ,n)$ be the weight of the criteria $C_q$ such that $\kappa _q>0$ and $\sum \nolimits _{q=1}^n \kappa _q = 1$. Then, the following steps are proposed to solve the DM problems using proposed measures:

• Step 1 Collect all the information corresponding to each alternative in terms of CIFNs and hence an overall decision matrix D is expressed as

  

  (17)

• Step 2 Utilize the proposed distance measures ‘d’ to compute the measurement value of each alternative from its reference set.

• Step 3 Find the index value of $r_p=\arg \min \nolimits _{1\le p\le m}\{d\}$ and hence select the best one(s) accordingly.

4.2 Numerical examples

Two examples, related to pattern recognition and medical diagnosis are taken to demonstrate the approach.

Example 2

Consider a set of diseases ${\mathcal {Q}}= \{{\mathcal {Q}}_1 \text {(Viral fever)}$, ${\mathcal {Q}}_2 \text {(Malaria)}$, ${\mathcal {Q}}_3 \text {(Typhoid)}$, ${\mathcal {Q}}_4$ (Stomach problem)$\}$ and a set of symptoms $S=\{s_1 \text {(Temperature)}$, $s_2 \text {(Headache)}$, $s_3$ (Stomach-ache), $s_4 \text {(Cough)}\}$ which are represented in the form of CIFSs (where the IVIFS set represents the probable disease during initial stage, i.e., before diagnosis and IFS represents the corresponding measure after diagnosis) as below:

$$\begin{aligned} {\mathcal {Q}}_1& = \big \{\big (s_1,\big (\langle [0.10,0.20],[0.30,0.60]\rangle ,\langle 0.40,0.20\rangle \big )\big ), \big (s_2,\big (\langle [0.25,0.30],[0.45,0.50]\rangle ,\langle 0.60,0.30\rangle \big )\big ),\\&\big (s_3,\big (\langle [0.30,0.45],[0.20,0.25]\rangle ,\langle 0.10,0.80\rangle \big )\big ), \big (s_4,\big (\langle [0.40,0.50],[0.10,0.30]\rangle ,\langle 0.30,0.70\rangle \big )\big )\big \}\\ {\mathcal {Q}}_2& = \big \{\big (s_1,\big (\langle [0.15,0.45],[0.25,0.30]\rangle ,\langle 0.40,0.60\rangle \big )\big ), \big (s_2,\big (\langle [0.20,0.25],[0.30,0.35]\rangle ,\langle 0.15,0.20\rangle \big )\big ),\\&\big (s_3,\big (\langle [0.45,0.60],[0.20,0.25]\rangle ,\langle 0.29,0.63\rangle \big )\big ), \big (s_4,\big (\langle [0.16,0.20],[0.25,0.30]\rangle ,\langle 0.15,0.30\rangle \big )\big )\big \} \\ {\mathcal {Q}}_3& = \big \{\big (s_1,\big (\langle [0.20,0.30],[0.25,0.35]\rangle ,\langle 0.15,0.25\rangle \big )\big ), \big (s_2, \big (\langle [0.30,0.40],[0.35,0.45]\rangle ,\langle 0.35,0.40\rangle \big )\big ), \\&\big (s_3, \big (\langle [0.15,0.25],[0.30,0.35]\rangle ,\langle 0.18,0.35\rangle \big )\big ), \big (s_4, \big (\langle [0.19,0.36],[0.20,0.40]\rangle ,\langle 0.30,0.25\rangle \big )\big )\big \} \\ {\mathcal {Q}}_4& = \big \{\big (s_1,\big (\langle [0.14,0.28],[0.15,0.30]\rangle ,\langle 0.20,0.30\rangle \big )\big ), \big (s_2, \big (\langle [0.16,0.32],[0.17,0.34]\rangle ,\langle 0.18,0.20\rangle \big )\big ), \\&\big (s_3, \big (\langle [0.50,0.55],[0.20,0.30]\rangle ,\langle 0.40,0.30\rangle \big )\big ), \big (s_4,\big (\langle [0.30,0.40],[0.50,0.55]\rangle ,\langle 0.35,0.55\rangle \big ) \big )\big \}. \end{aligned}$$

Suppose a patient ${\mathcal {P}}$, has been evaluated by an expert in order to find that his symptoms has maximum compliance with that of diseases ${\mathcal {Q}}_1,{\mathcal {Q}}_2, {\mathcal {Q}}_3$ or ${\mathcal {Q}}_4$. For it, they have recorded the preferences of patient ${\mathcal {P}}$ with respect to all the symptoms in terms of CIFN represented by the following set:

$$\begin{aligned} {\mathcal {P}}& = \bigg \{\Big (s_1,\big (\langle [0.20,0.30],[0.40,0.50]\rangle ,\langle 0.10,0.40\rangle \big )\Big ), \Big (s_2,\big (\langle [0.30,0.40],[0.10,0.60]\rangle ,\langle 0.20,0.40\rangle \big )\Big ),\\&\Big (s_3,\big (\langle [0.40,0.50],[0.20,0.30]\rangle ,\langle 0.60,0.30\rangle \big )\Big ), \Big (s_4,\big (\langle [0.10,0.50],[0.20,0.30]\rangle ,\langle 0.40,0.30\rangle \big )\Big )\bigg \}. \end{aligned}$$

Now, the aim of the problem is to identify the disease of patient ${\mathcal {P}}$ among ${\mathcal {Q}}_1,{\mathcal {Q}}_2,{\mathcal {Q}}_3,{\mathcal {Q}}_4$. For it, the proposed non-weighted measures are utilized to compute the measurement values of ${\mathcal {Q}}_k(k = 1, 2, 3, 4)$ from ${\mathcal {P}}$. The results corresponding to these measures values are summarized in Table 1 and conclude that the ${\mathcal {Q}}_3$ is the disease by which patient ${\mathcal {P}}$ suffers.

If we assign weights 0.10, 0.30, 0.45 and 0.15 to set $s_k(k = 1, 2, 3 ,4)$ and utilize weighted measures $d_5$, $d_6$, $d_5^H$ and $d_6^H$, we get their measurement values and summarize in Table 2. Based on step 3 of proposed approach, we get that the patient ${\mathcal {P}}$ suffers from the disease ${\mathcal {Q}}_4$.

Table 1 Computed distance measures values for medical diagnosis problem

Table 2 Distance measurement values using weighted distance measures

To compare the proposed results with the existing approaches (Dugenci 2016; Park et al. 2009; Wei et al. 2011; Xu 2007; Zhang et al. 2014) under IVIFS environment, we first convert the preferences of the decision-maker from CIFNs to IVIFNs. For it, we have taken the fuzzy judgment as zero in CIFS so that it gets reduced to IVIFS and then the various existing approaches have been utilized to compute the most desirable diseases. The result corresponding to their approaches is summarized in Table 3. From it, the relativity of CIFS to the real-life scenario can be noticed. Since it facilitates us to study the symptoms for a prolonged time period and relating it to the existing IVIFS environment, it has been found that under it, the patient is found to be suffering from ${\mathcal {Q}}_1$ disease, i.e., Viral fever, but on analyzing the symptoms and its effect after diagnosis (under CIFS environment), the patient is found to be suffering from disease ${\mathcal {Q}}_3$, i.e., Typhoid whereas under the weighted criteria, the patient is diagnosed with disease ${\mathcal {Q}}_4$, i.e., stomach problem.

Table 3 Comparative study of Example 1

The comparison analysis clearly shows that from the information available during the initial stage (recorded in form of IVIFS) shows the patient to be suffering from disease ${\mathcal {Q}}_1$, i.e., Viral Fever. However, if we consider the information of post-diagnosis period clubbed together with the initial symptoms, i.e., in form of CIFS, then the patient is diagnosed with the disease ${\mathcal {Q}}_3$, i.e., Typhoid and from ${\mathcal {Q}}_4$, i.e., stomach problem under the suitable weighted criteria. This shows the effect of the time frame during which the information is recorded from the patient and CIFS allows us to handle the complete opinion of a patient of both pre-diagnosis and post-diagnosis stages. Thus, the results obtained from our approach show a significant difference from the existing ones because, in the prevailing approaches, the decision is made only on the basis of initial symptoms assessed before the diagnosis while our approach takes into account both the pre-diagnosis as well as the post-diagnosis stages.

Example 3

Consider four known patterns ${\mathcal {C}}_1$, ${\mathcal {C}}_2$, ${\mathcal {C}}_3$ and ${\mathcal {C}}_4$ whose rating values are given as

$$\begin{aligned} {\mathcal {C}}_1& = \big \{\big (x_1,\big (\langle [0.15,0.30],[0.35,0.40]\rangle ,\langle 0.20,0.65\rangle \big )\big ), \big (x_2,\big (\langle [0.13,0.25],[0.40,0.45]\rangle ,\langle 0.30,0.60\rangle \big )\big ),\\&\big (x_3,\big (\langle [0.30,0.45],[0.25,0.30]\rangle ,\langle 0.55,0.33\rangle \big )\big ), \big (x_4,\big (\langle [0.10,0.30],[0.25,0.35]\rangle ,\langle 0.11,0.20\rangle \big )\big )\big \}\\ {\mathcal {C}}_2& = \big \{\big (x_1,\big (\langle [0.10,0.15],[0.35,0.40]\rangle ,\langle 0.40,0.20\rangle \big )\big ), \big (x_2,\big (\langle [0.15,0.22],[0.27,0.30]\rangle ,\langle 0.15,0.60\rangle \big )\big ),\\&\big (x_3,\big (\langle [0.40,0.45],[0.21,0.33]\rangle ,\langle 0.16,0.40\rangle \big )\big ), \big (x_4,\big (\langle [0.50,0.60],[0.15,0.20]\rangle ,\langle 0.35,0.28\rangle \big )\big )\big \},\\ {\mathcal {C}}_3& = \big \{\big (x_1,\big (\langle [0.14,0.25],[0.35,0.65]\rangle ,\langle 0.10,0.40\rangle \big )\big ), \big (x_2,\big (\langle [0.35,0.45],[0.15,0.20]\rangle ,\langle 0.30,0.50\rangle \big )\big ),\\&\big (x_3,\big (\langle [0.45,0.55],[0.15,0.25]\rangle ,\langle 0.20,0.80\rangle \big )\big ), \big (x_4,\big (\langle [0.30,0.50],[0.10,0.30]\rangle ,\langle 0.20,0.35\rangle \big )\big )\big \},\\ {\mathcal {C}}_4& = \big \{\big (x_1,\big (\langle [0.30,0.35],[0.25,0.45]\rangle ,\langle 0.20,0.30\rangle \big )\big ), \big (x_2,\big (\langle [0.20,0.55],[0.40,0.45]\rangle ,\langle 0.20,0.45\rangle \big )\big ),\\&\big (x_3,\big (\langle [0.15,0.25],[0.20,0.35]\rangle ,\langle 0.60,0.20\rangle \big )\big ), \big (x_4,\big (\langle [0.10,0.29],[0.40,0.50]\rangle ,\langle 0.30,0.40\rangle \big )\big )\big \} \end{aligned}$$

Consider an unknown pattern ${\mathcal {P}}$, which will be recognized as:

$$\begin{aligned} {\mathcal {P}}& = \big \{\big (x_1,\big (\langle [0.10,0.30],[0.35,0.45]\rangle ,\langle 0.60,0.10\rangle \big )\big ), \big (x_2,\big (\langle [0.15,0.20],[0.25,0.29]\rangle ,\langle 0.18,0.66\rangle \big )\big ),\\&\big (x_3,\big (\langle [0.44,0.50],[0.20,0.30]\rangle ,\langle 0.18,0.35\rangle \big )\big ), \big (x_4,\big (\langle [0.60,0.70],[0.20,0.30]\rangle ,\langle 0.40,0.25\rangle \big )\big )\big \}. \end{aligned}$$

In these preference values, the IVIFN in each CIFN shows the probable interval of belongingness as well as non-belongingness of the pattern P to the classes ${\mathcal {C}}_1,{\mathcal {C}}_2,{\mathcal {C}}_3$ and ${\mathcal {C}}_4$ using Pattern-classifier 1. However, the IFS portion, shows the extent of agree-ness and disagree-ness shown by the Pattern-classifier 2 to that of Classifier 1. Now, the aim of the problem is to classify the pattern ${\mathcal {P}}$ with ${\mathcal {C}}_1$, ${\mathcal {C}}_2$, ${\mathcal {C}}_3$ and ${\mathcal {C}}_4$.

Initially, by using non-weighted measures $d_1$, $d_2$, $d_3$, $d_4$, $d_1^H$, $d_2^H$ and $d_3^H$, the measurement values of ${\mathcal {P}}$ from ${\mathcal {C}}_k$, $(k=1,2,3,4)$ are computed and their results are summarized in Table 4. Thus, based on the maximum recognition principle, it has been concluded that unknown pattern ${\mathcal {P}}$ belongs to the pattern ${\mathcal {C}}_2$. Later on, if we assign weight 0.33, 0.23, 0.24 and 0.20 to $x_1,x_2,x_3$ and $x_4$, then by using weighted measures $d_5$, $d_6$, $d_5^H$ and $d_6^H$, the measurement values of each pattern ${\mathcal {C}}_k (k=1,2,3,4)$ from the unknown pattern ${\mathcal {P}}$ are computed and summarized in Table 5. From these tabulated values, we see that the pattern ${\mathcal {P}}$ can be classified to the class of pattern ${\mathcal {C}}_2$.

Table 4 Proposed measures results for pattern recognition

Table 5 Measurement values corresponding to the weighted distance measures for pattern recognition

To compare the proposed results with the existing approaches (Dugenci 2016; Park et al. 2009; Wei et al. 2011; Xu 2007; Zhang et al. 2014) under IVIFS environment, we have taken the fuzzy judgments as zero in CIFS so that it gets reduced to IVIFS and hence their corresponding results are summarized in Table 6. It is clearly seen from their measurement values that the unknown pattern ${\mathcal {P}}$ recognized with the known pattern ${\mathcal {C}}_2$ and their results coincide with the proposed approach’s result.

Noticeably, although the proposed approach’s results coincide with the results of existing approaches, still the results obtained from the proposed method are superior because they bring the surety of the pattern ${\mathcal {P}}$, to be classified under Class ${\mathcal {C}}_2$, verified by two pattern-classifiers, whereas the existing approaches only give an outlook of the pattern recognized by the classifier 1 only. Moreover, by employing the proposed approach it is also found that both the Pattern-classifiers 1 and 2 recognizes the pattern ${\mathcal {P}}$ to be similar to class ${\mathcal {C}}_2$.

Table 6 Comparative analysis of pattern recognition example

5 Conclusion

In the present manuscript, we have extended the theory of the IFS to the cubic IFS which is an extension of the cubic set. In this set, preference corresponding to an element is expressed by means of the IVIFS and IFS which shows the importance of IVIFS to get the more appropriate results through IFS. Then, based on it, some family of distance measures based on Hamming, Euclidean, and Hausdorff measures, have been proposed along with their desired relations. These measures are advantageous over distance measures associated with CFS in the manner that these enhance our scope of analysis beyond the membership degrees. Instead of remaining confined to perform all the analysis sticking to only the membership portion, distance measures on CIFSs provide us with a wide perspective of handling the non-membership degrees too. Unlike all the existing and developed distance measures, measures in CIFSs allow us to consider the degree of disagreement (in form of IFS values) corresponding to the agreed interval region (in form of IVIFS) which clearly incline our results more towards the practically feasible values. To demonstrate the efficiency of the proposed operators, two illustrative examples have been taken into account. From the studies, it has been concluded that these distance measures employed together with the decision-making approach can model the uncertainties in a much better way than that of the existing approaches and provide us with a deep analysis of the real-life situations. As in the medical recognition example, the proposed approach not only provides us with the disease to which the patient is prone to, but also analyzes the patient’s state both during the pre-diagnosis and post-diagnosis states and in the pattern recognition example, it classifies the pattern to its most similar class and meanwhile provides us with the verification of uniform working of both the classifiers 1 and 2. Thus, it is clear that using the proposed method, we reach the desired results along with some additional information about the real-life scenarios. In the future, the result of this paper can be extended to a linguistic environment under the Pythagorean, neutrosophic and other uncertain and fuzzy environment (Arora and Garg 2018a; Chen et al. 2012a, 2016; Garg 2018a; Garg and Kumar 2018b; Garg and Nancy 2018; Liu et al. 2018, 2017; Rani and Garg 2018; Singh and Garg 2018; Wang and Chen 2008).