Skip to main content
Log in

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

  • Published:
Artificial Intelligence Review Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    Article  MATH  Google Scholar 

  • Atanassov KT (1989) More on intuitionistic fuzzy sets. Fuzzy Sets Syst 33(1):37–45

    Article  MathSciNet  MATH  Google Scholar 

  • Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349

    Article  MathSciNet  MATH  Google Scholar 

  • Berrah L, Mauris G, Montmain J (2008) Monitoring the improvement of an overall industrial performance based on a Choquet integral aggregation. Omega 36(3):340–351

    Article  Google Scholar 

  • Broumi S, Smarandache F (2013) Correlation coefficient of interval neutrosophic set. Appl Mech Mater 436:511–517

    Article  Google Scholar 

  • Broumi S, Talea M, Smarandache F, Bakali A (2016a) Decision-making method based on the interval valued neutrosophic graph. In: Future technologies conference (FTC). IEEE, pp. 44–50

  • Broumi S, Bakali A, Talea M, Smarandache F, Vladareanu L (2016b) Computation of shortest path problem in a network with SV-trapezoidal neutrosophic numbers. In: International conference on advanced mechatronic systems (ICAMechS). IEEE, pp 417–422

  • Broumi S, Bakal A, Talea M, Smarandache F, Vladareanu L (2016c) Applying Dijkstra algorithm for solving neutrosophic shortest path problem. In: International conference on advanced mechatronic systems (ICAMechS). IEEE, pp 412–416

  • Broumi S, Bakali A, Talea M, Smarandache F, ALi M (2017) Shortest path problem under bipolar neutrosphic setting. Appl Mech Mater 859:59–66

    Article  Google Scholar 

  • Büyüközkan G, Feyzioğlu O, Ersoy MŞ (2009) Evaluation of 4PL operating models: a decision making approach based on 2-additive Choquet integral. Int J Prod Econ 121(1):112–120

    Article  Google Scholar 

  • Chao CT, Teng CC (1995) Implementation of a fuzzy inference system using a normalized fuzzy neural network. Fuzzy Sets Syst 75(1):17–31

    Article  MathSciNet  MATH  Google Scholar 

  • Chateauneuf A, Eichberger J, Grant S (2007) Choice under uncertainty with the best and worst in mind: neo-additive capacities. J Econ Theory 137(1):538–567

    Article  MathSciNet  MATH  Google Scholar 

  • Chen TY (2011) Bivariate models of optimism and pessimism in multi-criteria decision-making based on intuitionistic fuzzy sets. Inf Sci 181(11):2139–2165

    Article  MathSciNet  MATH  Google Scholar 

  • Chen YC, Teng CC (1995) A model reference control structure using a fuzzy neural network. Fuzzy Sets Syst 73(3):291–312

    Article  MathSciNet  MATH  Google Scholar 

  • Chen N, Xu Z, Xia M (2013) Interval-valued hesitant preference relations and their applications to group decision making. Knowl Based Syst 37:528–540

    Article  Google Scholar 

  • Chiang JH (1999) Choquet fuzzy integral-based hierarchical networks for decision analysis. IEEE Trans Fuzzy Syst 7(1):63–71

    Article  Google Scholar 

  • Choquet G (1954) Theory of capacities. Annales de l’institut Fourier 5:131–295

    Article  MathSciNet  MATH  Google Scholar 

  • Devaraj D, Selvabala B (2009) Real-coded genetic algorithm and fuzzy logic approach for real-time tuning of proportional-integral-derivative controller in automatic voltage regulator system. IET Gener Transm Distrib 3(7):641–649

    Article  Google Scholar 

  • Frayman Y, Wang L (2002) A dynamically-constructed fuzzy neural controller for direct model reference adaptive control of multi-input-multi-output nonlinear processes. Soft Comput 6(3–4):244–253

    Article  MATH  Google Scholar 

  • Garg H (2019) A novel divergence measure and its based TOPSIS method for multi criteria decision-making under single-valued neutrosophic environment. J Intell Fuzzy Syst 36(1):101–115

    Article  MathSciNet  Google Scholar 

  • Gorzałczany MB (1987) A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst 21(1):1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Grabisch M (1997) K-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst 92(2):167–189

    Article  MathSciNet  MATH  Google Scholar 

  • Haibin WANG, Smarandache F, Zhang Y, Sunderraman R (2010) Single valued neutrosophic sets. Rev Air Force Acad 17:4–10

    MATH  Google Scholar 

  • Iliadis LS, Spartalis S, Tachos S (2008) Application of fuzzy T-norms towards a new Artificial Neural Networks’ evaluation framework: A case from wood industry. Inf Sci 178(20):3828–3839

    Article  Google Scholar 

  • Jia W, Zhenyuan W (1997) Using neural networks to determine Sugeno measures by statistics. Neural Netw 10(1):183–195

    Article  Google Scholar 

  • Juan-juan P, Jian-qiang W, Jun-hua H (2018) Multi-criteria decision-making approach based on single-valued neutrosophic hesitant fuzzy geometric weighted Choquet integral heronian mean operator. J Intell Fuzzy Syst, 1–14 (preprint)

  • Kakati P, Borkotokey S, Mesiar R, Rahman S (2018) Interval neutrosophic hesitant fuzzy choquet integral in multicriteria decision making. J Intell Fuzzy Syst, 1–19 (preprint)

  • Kalyanmoy D (2001) Multi objective optimization using evolutionary algorithms. Wiley, New York, pp 124–124

    MATH  Google Scholar 

  • Kecman V (2001) Learning and soft computing: support vector machines, neural networks, and fuzzy logic models. MIT Press, Cambridge

    MATH  Google Scholar 

  • Keller JM, Gader PD, Hocaoglu AK (2000) Fuzzy integrals in image processing and recognition. In: Grabisch M, Murofushi T, Sugeno M (eds) Fuzzy measures and integrals: theory and applications. Springer, Physica, pp 435–466

    MATH  Google Scholar 

  • Lamata MT (2004) Ranking of alternatives with ordered weighted averaging operators. Int J Intell Syst 19(5):473–482

    Article  MATH  Google Scholar 

  • Li X, Zhang X (2018) Single-valued neutrosophic hesitant fuzzy Choquet aggregation operators for multi-attribute decision making. Symmetry 10(2):50

    Article  MATH  Google Scholar 

  • Li L, Wang L, Liao B (2016) Einstein Choquet integral operators for promethee II group decision making method with triangular intuitionistic fuzzy numbers. International conference on oriental thinking and fuzzy logic. Springer, Cham, pp 137–149

    Chapter  Google Scholar 

  • Lin CM, Hsu CF (2004) Supervisory recurrent fuzzy neural network control of wing rock for slender delta wings. IEEE Trans Fuzzy Syst 12(5):733–742

    Article  Google Scholar 

  • Liu P, Shi L (2015) The generalized hybrid weighted average operator based on interval neutrosophic hesitant set and its application to multiple attribute decision making. Neural Comput Appl 26(2):457–471

    Article  Google Scholar 

  • Liu P, Tang G (2016) Multi-criteria group decision-making based on interval neutrosophic uncertain linguistic variables and Choquet integral. Cognit Comput 8(6):1036–1056

    Article  Google Scholar 

  • Lucca G, Sanz JA, Dimuro GP, Bedregal B, Asiain MJ, Elkano M, Bustince H (2017) CC-integrals: Choquet-like copula-based aggregation functions and its application in fuzzy rule-based classification systems. Knowl Based Syst 119:32–43

    Article  Google Scholar 

  • Marichal JL (2000) An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. IEEE Trans Fuzzy Syst 8(6):800–807

    Article  MathSciNet  Google Scholar 

  • Meng F, Chen X (2014) An approach to interval-valued hesitant fuzzy multi-attribute decision making with incomplete weight information based on hybrid Shapley operators. Informatica 25(4):617–642

    Article  MATH  Google Scholar 

  • Murofushi T, Sugeno M (1989) An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure. Fuzzy Sets Syst 29(2):201–227

    Article  MathSciNet  MATH  Google Scholar 

  • Öztürk Ş, Akdemir B (2018) Fuzzy logic-based segmentation of manufacturing defects on reflective surfaces. Neural Comput Appl 29(8):107–116

    Article  Google Scholar 

  • Peng HG, Zhang HY, Wang JQ (2018) Probability multi-valued neutrosophic sets and its application in multi-criteria group decision-making problems. Neural Comput Appl 30(2):563–583

    Article  Google Scholar 

  • Shahzadi G, Akram M, Saeid AB (2018) An application of single-valued neutrosophic sets in medical diagnosis. Neutrosophic Sets Syst 18:80–88

    Google Scholar 

  • Smarandache F (1999) A unifying field in logics: Neutrosophic logic. In: Perez M (ed) Philosophy. American Research Press, Rehoboth, pp 1–141

  • Sodenkamp MA, Tavana M, Di Caprio D (2018) An aggregation method for solving group multi-criteria decision-making problems with single-valued neutrosophic sets. Appl Soft Comput 71:715–727

    Article  Google Scholar 

  • Soria-Frisch A, Köppen M, Sy T (2003) Is she gonna like it? Automated inspection system using fuzzy aggregation. In: Intelligent systems for information processing. Elsevier Science, pp 465–476

  • Srivastava S, Bansal A, Chopra D, Goel G (2006) Implementation of a Choquet fuzzy integral based controller on a real time system. In: Proceedings of the 7th WSEAS international conference on neural networks. World Scientific and Engineering Academy and Society (WSEAS), pp 34–40

  • Sugeno M (1974) Theory of fuzzy integrals and its applications, Doctorial Thesis. Doctoral Thesis, Tokyo Institute of Technology

  • Sun HX, Yang HX, Wu JZ, Ouyang Y (2015) Interval neutrosophic numbers Choquet integral operator for multi-criteria decision making. J Intell Fuzzy Syst 28(6):2443–2455

    Article  MathSciNet  MATH  Google Scholar 

  • Sundarabalan CK, Selvi K (2017) Real coded GA optimized fuzzy logic controlled PEMFC based dynamic voltage restorer for reparation of voltage disturbances in distribution system. Int J Hydr Energy 42(1):603–613

    Article  Google Scholar 

  • Tiwari R, Ramesh Babu N, Arunkrishna R, Sanjeevikumar P (2018) Comparison between PI controller and fuzzy logic-based control strategies for harmonic reduction in grid-integrated wind energy conversion system. In: SenGupta S, Zobaa A, Sherpa K, Bhoi A(eds) Advances in smart grid and renewable energy. Lecture notes in electrical engineering, vol 435. Springer, Singapore, pp 297–306

  • Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539

    MATH  Google Scholar 

  • Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: 2009 IEEE international conference on fuzzy systems. IEEE, pp 1378–1382

  • Wan S, Dong J (2014) A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making. J Comput Syst Sci 80(1):237–256

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Z, Klir G (1992) Fuzzy measure theory. Plenum press, New York

    Book  MATH  Google Scholar 

  • Wang W, Liu X (2011) Intuitionistic fuzzy geometric aggregation operators based on Einstein operations. Int J Intell Syst 26(11):1049–1075

    Article  Google Scholar 

  • Wang W, Liu X (2012) Intuitionistic fuzzy information aggregation using Einstein operations. IEEE Trans Fuzzy Syst 20(5):923–938

    Article  Google Scholar 

  • Wang H, Smarandache F, Sunderraman R, Zhang YQ (2005) Interval neutrosophic sets and logic: theory and applications in computing: theory and applications in computing, vol 5. Infinite Study

  • Xu Y, Wang H, Merigó JM (2014) Intuitionistic fuzzy Einstein Choquet integral operators for multiple attribute decision making. Technol Econ Dev Econ 20(2):227–253

    Article  Google Scholar 

  • Yager RR (1995) An approach to ordinal decision making. Int J Approx Reason 12(3–4):237–261

    Article  MathSciNet  MATH  Google Scholar 

  • Yager RR (2002) On the cardinality index and attitudinal character of fuzzy measures. Int J Gener Syst 31(3):303–329

    Article  MathSciNet  MATH  Google Scholar 

  • Ye J (2014) Single valued neutrosophic cross-entropy for multicriteria decision making problems. Appl Math Model 38(3):1170–1175

    Article  MathSciNet  MATH  Google Scholar 

  • Ye J (2015a) Multiple-attribute decision-making method under a single-valued neutrosophic hesitant fuzzy environment. J Intell Syst 24(1):23–36

    Google Scholar 

  • Ye J (2015b) Multiple attribute decision-making method based on the possibility degree ranking method and ordered weighted aggregation operators of interval neutrosophic numbers. J Intell Fuzzy Syst 28(3):1307–1317

    Article  MathSciNet  Google Scholar 

  • Ye J (2017) Projection and bidirectional projection measures of single-valued neutrosophic sets and their decision-making method for mechanical design schemes. J Exper Theor Artif Intell 29(4):731–740

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  • Zeshui X (2001) Algorithm for priority of fuzzy complementary judgment matrix. J Syst Eng 16(4):311–314

    Google Scholar 

  • Zeshui X (2005) On method for uncertain multiple attribute decision making problems with uncertain multiplicative preference information on alternatives. Fuzzy Optim Decis Mak 4(2):131–139

    Article  MathSciNet  MATH  Google Scholar 

  • Zeshui X (2008) Dependent uncertain ordered weighted aggregation operators. Inf Fus 9(2):310–316

    Article  Google Scholar 

  • ZeShui X, QingLi D (2002) The uncertain OWA operator. Int J Intell Syst 17(6):569–575

    Article  Google Scholar 

  • Zhang HY, Wang JQ, Chen XH (2014) Interval neutrosophic sets and their application in multicriteria decision making problems. Sci World J 2014:1–15

    Google Scholar 

  • Zhang Z (2017) Multi-criteria decision-making using interval-valued hesitant fuzzy QUALIFLEX methods based on a likelihood-based comparison approach. Neural Comput Appl 28(7):1835–1854

    Article  Google Scholar 

  • Zhang S, Yu D (2014) Some geometric Choquet aggregation operators using Einstein operations under intuitionistic fuzzy environment. J Intell Fuzzy Syst 26(1):491–500

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao S, Wang D, Changyong L, Lu W (2019) Induced Choquet integral aggregation operators with single-valued neutrosophic uncertain linguistic numbers and their application in multiple attribute group decision-making. Math Prob Eng 2019:1–14

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The first two authors acknowledge the grant by UKIERI No. 184-15/2017(IC). The authors also acknowledge the anonymous reviewers for their thoughtful, critical but constructive comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Saifur Rahman.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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:

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kakati, P., Borkotokey, S., Rahman, S. et al. Interval neutrosophic hesitant fuzzy Einstein Choquet integral operator for multicriteria decision making. Artif Intell Rev 53, 2171–2206 (2020). https://doi.org/10.1007/s10462-019-09730-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10462-019-09730-7

Keywords

Mathematics Subject Classification

Navigation