Interval neutrosophic hesitant fuzzy Einstein Choquet integral operator for multicriteria decision making

Abstract

Recently interval neutrosophic hesitant fuzzy sets are found to be more general and useful to express incomplete, indeterminate and inconsistent information. In this paper, we define some new Einstein operational rules on interval neutrosophic hesitant fuzzy elements, then we propose the interval neutrosophic hesitant fuzzy Einstein Choquet integral (INHFECI) operator and discuss its properties. Further, an approach for multicriteria decision making is developed to study the interaction between the input arguments under the interval neutrosophic hesitant fuzzy environment. The main advantage of the proposed operator is that, it can deal with the situations of the positive interaction, negative interaction or non-interaction among the criteria, during the decision making process. Also, the proposed operator can replace the weighted average to aggregate dependent criteria of interval neutrosophic hesistant fuzzy information for obtaining more accurate results. Moreover, some interval neutrosophic hesitant fuzzy weighted average operators are proposed as special cases of INHFECI operator. Finally, an illustrative example follows.

Appendices

Appendices

1.1 Useful definition

The following definition is used in Sect. 7.1

Liu and Shi (2015) proposed several weighted aggregation operators for decision making under interval neutrosophic hesitant fuzzy environment.

Definition 15

(Liu and Shi 2015) Let ${\tilde{n}}_j=\{{\tilde{t}}_j,{\tilde{i}}_j,{\tilde{f}}_j\}=\underset{[\gamma _j^L,\gamma _j^U]\in {\tilde{t}}_j,[\delta _j^L,\delta _j^U]\in {\tilde{i}}_j,[\eta _j^L,\eta _j^U]\in {\tilde{f}}_j}{\bigcup }\bigg \{[\gamma _j^L,\gamma _j^U],[\delta _j^L,\delta _j^U],[\eta _j^L,\eta _j^U]\bigg \} (j=1,2,\ldots ,n)$ be a collection of INHFEs with the weight vector $w=(w_1,w_2,\ldots ,w_n)^T$ such that $w_j>0$, ${\sum \nolimits _{j=1}^{n}}w_j=1$ and parameter $\lambda >0$, then an interval neutrosophic hesitant fuzzy generalized weighted average (INHFGWA) operator of dimension n is a mapping $\mathrm {INHFGWA}:\Omega ^n\rightarrow \Omega$ is given by,

$$\begin{aligned}&\mathrm {INHFGWA}({\tilde{n}}_1,{\tilde{n}}_2,\ldots ,{\tilde{n}}_n) =\left( \overset{n}{\underset{j=1}{\sum }}w_j{\tilde{n}}_{j}^{\lambda }\right) ^{1/\lambda }\nonumber \\&\quad =\left\{ \left[ \begin{array}{l} \left( \begin{array}{l} 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\gamma }}_j\in {\tilde{t}}_j}{\bigcup }\bigg (1-\big (\gamma _{j}^L\big )^{\lambda }\bigg )^{w_j}\end{array}\right) ^{1/\lambda }, \left( \begin{array}{l} 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\gamma }}_j\in {\tilde{t}}_j}{\bigcup }\bigg (1-\big (\gamma _{j}^U\big )^{\lambda }\bigg )^{w_j}\end{array}\right) ^{1/\lambda } \end{array}\right] \right. ,\nonumber \\&\qquad \left[ 1-\left( \begin{array}{l} 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\delta }}_j\in {\tilde{i}}_j}{\bigcup }\bigg (1-\big (1-\delta _{j}^L\big )^{\lambda }\bigg )^{w_j}\end{array}\right) ^{1/\lambda },\right. \nonumber \\&\qquad \left. 1-\left( \begin{array}{l} 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\delta }}_j\in {\tilde{i}}_j}{\bigcup }\bigg (1-\big (1-\delta _{j}^U\big )^{\lambda }\bigg )^{w_j}\end{array}\right) ^{1/\lambda }\right] ,\nonumber \\&\qquad \left. \left[ 1-\left( \begin{array}{l} 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\eta }}_j\in {\tilde{f}}_j}{\bigcup }\bigg (1-\big (1-\eta _{j}^L\big )^{\lambda }\bigg )^{w_j}\end{array}\right) ^{1/\lambda },\right. \right. \nonumber \\&\qquad \left. \left. 1-\left( \begin{array}{l} 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\eta }}_j\in {\tilde{f}}_j}{\bigcup }\bigg (1-\big (1-\eta _{j}^U\big )^{\lambda }\bigg )^{w_j}\end{array}\right) ^{1/\lambda }\right] \right\} \end{aligned}$$

(A.1)

In particular, if $\lambda =1$, then the INHFGWA (Liu and Shi 2015) operator reduces the interval neutrosophic hesitant fuzzy weighted averaging (INHFWA) (Liu and Shi 2015) operator.

$$\begin{aligned}&\mathrm {INHFOWA}({\tilde{n}}_1,{\tilde{n}}_2,\ldots ,{\tilde{n}}_n) =\overset{n}{\underset{j=1}{\sum }}w_j{\tilde{n}}_{j}\nonumber&\\&\quad =\left\{ \left[ \begin{array}{l} 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\gamma }}_j\in {\tilde{t}}_j}{\bigcup }\bigg (1-\gamma _{j}^L\bigg )^{w_j}, 1-{\prod \limits _{j=1}^{n}}\underset{{\tilde{\gamma }}_j\in {\tilde{t}}_j}{\bigcup }\bigg (1-\gamma _{j}^U\bigg )^{w_j} \end{array}\right] \right. ,\nonumber \\&\qquad \left[ \begin{array}{l} {\prod \limits _{j=1}^{n}}\underset{{\tilde{\delta }}_j\in {\tilde{i}}_j}{\bigcup }\big (\delta _{j}^L\big )^{w_j}, {\prod \limits _{j=1}^{n}}\underset{{\tilde{\delta }}_j\in {\tilde{i}}_j}{\bigcup }\big (\delta _{j}^U\big )^{w_j}\end{array}\right] , \left. \left[ \begin{array}{l} {\prod \limits _{j=1}^{n}}\underset{{\tilde{\eta }}_j\in {\tilde{f}}_j}{\bigcup }\big (\eta _{j}^L\big )^{w_j}, {\prod \limits _{j=1}^{n}}\underset{{\tilde{\eta }}_j\in {\tilde{f}}_j}{\bigcup }\big (\eta _{j}^U\big )^{w_j}\end{array}\right] \right\} \end{aligned}$$

(A.2)

1.2 Definition and equations moved to appendices

Continuing from Definition 11, the remaining Einstein operations of INHFEs are as follows.

Let ${\tilde{n}}_1=\{{\tilde{t}}_1,{\tilde{i}}_1,{\tilde{f}}_1\} , {\tilde{n}}_2=\{{\tilde{t}}_2,{\tilde{i}}_2,{\tilde{f}}_2\}$ be two INHFEs and $k>0$ be a scalar ,then the INHFEs has the following Einstein operations:

$$\begin{aligned}&\mathrm{(iii)}~~kn_1=\underset{{{\tilde{\gamma }}_1\in {\tilde{t}}_1,{\tilde{\delta }}_1\in {\tilde{i}}_1,\tilde{\eta _1}\in {\tilde{f}}_1}}{\bigcup } \Bigg \{\bigg [\frac{(1+\gamma _1^L)^k-(1-\gamma _1^L)^k}{(1+\gamma _1^L)^k+(1-\gamma _1^L)^k},\frac{(1+\gamma _1^U)^k-(1-\gamma _1^U)^k}{(1+\gamma _1^U)^k+(1-\gamma _1^U)^k}\bigg ],\\&\quad \bigg [\frac{2\cdot (\delta _1^L)^k}{(2-\delta _1^L)^k+(\delta _1^L)^k},\frac{2\cdot (\delta _1^U)^k}{(2-\delta _1^U)^k+(\delta _1^U)^k}\bigg ], \bigg [\frac{2\cdot (\eta _1^L)^k}{(2-\eta _1^L)^k+(\eta _1^L)^k},\frac{2\cdot (\eta _1^U)^k}{(2-\eta _1^U)^k+(\eta _1^U)^k} \bigg ]\Bigg \}\\&\mathrm{(iv)}~~{\tilde{n}}_1^k =\underset{{{\tilde{\gamma }}_1\in {\tilde{t}}_1,{\tilde{\delta }}_1\in {\tilde{i}}_1,\tilde{\eta _1}\in {\tilde{f}}_1}}{\bigcup } \Bigg \{\bigg [\frac{2\cdot (\gamma _1^L)^k}{(2-\gamma _1^L)^k+(\gamma _1^L)^k},\frac{2\cdot (\gamma _1^U)^k}{(2-\gamma _1^U)^k+(\gamma _1^U)^k}\bigg ],\\&\quad \bigg [\frac{(1+\delta _1^L)^k-(1-\delta _1^L)^k}{(1+\delta _1^L)^k+(1-\delta _1^L)^k},\frac{(1+\delta _1^U)^k-(1-\delta _1^U)^k}{(1+\delta _1^U)^k+(1-\delta _1^U)^k}\bigg ],\\&\quad \bigg [\frac{(1+\eta _1^L)^k-(1-\eta _1^L)^k}{(1+\eta _1^L)^k+(1-\eta _1^L)^k},\frac{(1+\eta _1^U)^k-(1-\eta _1^U)^k}{(1+\eta _1^U)^k+(1-\eta _1^U)^k}\bigg ]\Bigg \}. \end{aligned}$$