Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

This paper investigates a wide range of generalized Maclaurin symmetric mean (MSM) aggregation operators, such as the generalized arithmetic MSM and the generalized geometric MSM, whose predominant characteristic is capturing the interrelationships among multi-input arguments. The single-valued neutrosophic linguistic set plays an essential role in decision making, which can serve as an extension of either a linguistic term set or a single-valued neutrosophic set. This study centers on multi-criteria decision-making (MCDM) issues in which criteria are weighed differently and criteria values are expressed as single-valued neutrosophic linguistic numbers. Based on this foundation, we extend a series of MSM aggregation techniques under single-valued neutrosophic linguistic environments and propose procedures for solving MCDM problems. We also explore the influence of parameters on aggregation results. Finally, we provide a practical example and conduct a comparison analysis between the proposed approach and other existing methods in order to verify the proposed approach and demonstrate its validity.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Smarandache F (1995) Neutrosophic logic and set, mss. http://fs.gallup.unm.edu/neutrosophy.htm

  2. Guo Y-H, Sengur A (2012) A novel color image segmentation approach based on neutrosophic set and modified fuzzy c-means. Circuits Syst Signal Process 32(4):1699–1723

    Article  MathSciNet  Google Scholar 

  3. Khoshnevisan M, Bhattacharya S (2003) Neutrosophic information fusion applied to financial market. In: Proceedings of the sixth international conference of information fusion, Cairns, Australia, pp 1252–1257

  4. Rivieccio U (2008) Neutrosophic logics: prospects and problems. Fuzzy Sets Syst 159(14):1860–1868

    Article  MathSciNet  MATH  Google Scholar 

  5. Salama AA, Alblowi SA (2012) Neutrosophic set and neutrosophic topological spaces. J Math 3(4):31–35

    Google Scholar 

  6. Bausys R, Zavadskas E, Kaklauskas A (2015) Application of neutrosophic set to multicriteria decision making by COPRAS. Econ Comput Econ Cybern Stud Res 49(1):91–106

    Google Scholar 

  7. Ye J (2014) A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst 26(5):2459–2466

    MathSciNet  MATH  Google Scholar 

  8. Tian Z-P, Wang J, Wang J-Q, Zhang H-Y (2016) An improved MULTIMOORA approach for multi-criteria decision-making based on interdependent inputs of simplified neutrosophic linguistic information. Neural Comput Appl. doi:10.1007/s00521-016-2378-5

    Google Scholar 

  9. Tian Z-P, Wang J, Wang J-Q, Zhang H-Y (2016) Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development. Group Decis Negot. doi:10.1007/s10726-016-9479-5

    Google Scholar 

  10. Smarandache F (1998) A unifying field in logics: neutrosophic logic. Neutrosophy, neutrosophic set, neutrosophic probability and statistics. American Research Press, Rehoboth

    MATH  Google Scholar 

  11. Smarandache F, Wang H-B, Zhang Y-Q, Sunderraman R (2005) Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Phoenix

    MATH  Google Scholar 

  12. Deli I, Şubaş Y (2016) A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems. Int J Mach Learn Cybern. doi:10.1007/s13042-016-0505-3

    Google Scholar 

  13. Biswas P, Pramanik S, Giri B (2016) TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput Appl 27(3):727–737

    Article  Google Scholar 

  14. Bausys R, Zavadskas E (2015) Multi criteria decision making approach by VIKOR under interval neutrosophic set environment. Econ Comput Econ Cybern Stud Res 49(4):33–48

    Google Scholar 

  15. Broumi S, Ye J, Smarandache F (2015) An extended TOPSIS method for multiple attribute decision making based on interval neutrosophic uncertain linguistic variables. Neutrosophic Sets Syst 8:22–31

    Google Scholar 

  16. Wang J-Q, Li X-E (2015) TODIM method with multi-valued neutrosophic sets. Control Decis 30(6):1139–1142

    Google Scholar 

  17. Ji P, Zhang H-Y, Wang J-Q (2016) A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput Appl. doi:10.1007/s00521-016-2436-z

    Google Scholar 

  18. Peng J-J, Wang J-Q, Wu X-H (2016) An extension of the ELECTRE approach with multi-valued neutrosophic information. Neural Comput Appl. doi:10.1007/s00521-016-2411-8

    Google Scholar 

  19. Ye J (2014) Multiple-attribute decision-making method under a single-valued neutrosophic hesitant fuzzy environment. J Intell Syst. doi:10.1515/jisys-2014-0001

    Google Scholar 

  20. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning. Inf Sci 8(3):199–249

    Article  MathSciNet  MATH  Google Scholar 

  21. Yu S-M, Wang J, Wang J-Q (2016) An extended TODIM approach with intuitionistic linguistic numbers. Int Trans Oper Res. doi:10.1111/itor.12363

    MATH  Google Scholar 

  22. Wang J, Wang J-Q, Zhang H-Y (2016) A likelihood-based TODIM approach based on multi-hesitant fuzzy linguistic information for evaluation in logistics outsourcing. Comput Ind Eng 99:287–299

    Article  Google Scholar 

  23. Moharrer M, Tahayori H, Livi L (2015) Interval type-2 fuzzy sets to model linguistic label perception in online services satisfaction. Soft Comp 19(1):237–250

    Article  Google Scholar 

  24. Ye J (2015) An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers. J Intell Fuzzy Syst 28(1):247–255

    MathSciNet  Google Scholar 

  25. Ye J (2014) Some aggregation operators of interval neutrosophic linguistic numbers for multiple attribute decision making. J Intell Fuzzy Syst 27(5):2231–2241

    MathSciNet  MATH  Google Scholar 

  26. Ma Y-X, Wang J-Q, Wang J, Wu X-H (2016) An interval neutrosophic linguistic multi-criteria group decision-making method and its application in selecting medical treatment options. Neural Comput Appl. doi:10.1007/s00521-016-2203-1

    Google Scholar 

  27. Tian Z-P, Wang J, Zhang H-Y, Wang J-Q (2016) Multi-criteria decision-making based on generalized prioritized aggregation operators under simplified neutrosophic uncertain linguistic environment. Int J Mach Learn Cybern. doi:10.1007/s13042-016-0552-9

    Google Scholar 

  28. Liu P, Li Y, Antuchevičienė J (2016) Multi-criteria decision-making method based on intuitionistic trapezoidal fuzzy prioritised owa operator. Technol Econ Dev Econ 22(3):453–469

    Article  Google Scholar 

  29. Liang R-X, Wang J-Q, Li L (2016) Multi-criteria group decision making method based on interdependent inputs of single valued trapezoidal neutrosophic information. Neural Comput Appl. doi:10.1007/s00521-016-2672-2

    Google Scholar 

  30. Ji P, Wang J-Q, Zhang H-Y (2016) Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third party logistics. Neural Comput Appl. doi:10.1007/s00521-016-2660-6

    Google Scholar 

  31. Liu P-D, Shi L-L (2015) Some neutrosophic uncertain linguistic number Heronian mean operators and their application to multi-attribute group decision making. Neural Comput Appl. doi:10.1007/s00521-015-2122-6

    Google Scholar 

  32. Maclaurin C (1729) A second letter to Martin Folkes, Esq.: concerning the roots of equations, with the demonstration of other rules of algebra. Philos Trans R Soc Lond Ser A 36(1729):59–96

    Google Scholar 

  33. Detemple D, Robertson J (1979) On generalized symmetric means of two variables. Angew Chem 47(25):4638–4660

    MATH  Google Scholar 

  34. Aydoğdu A (2015) On similarity and entropy of single valued neutrosophic sets. Gen Math Notes 29(1):67–74

    Google Scholar 

  35. Broumi S, Smarandache F, Talea M, Bakali A (2016) An introduction to bipolar single valued neutrosophic graph theory. Appl Mech Mater 841:184–191

    Article  Google Scholar 

  36. Ye J (2014) Single valued neutrosophic minimum spanning tree and its clustering method. J Intell Syst 23(3):311–324

    Google Scholar 

  37. Karaaslan F (2016) Correlation coefficients of single valued neutrosophic refined soft sets and their applications in clustering analysis. Neural Comput Appl. doi:10.1007/s00521-016-2209-8

    Google Scholar 

  38. Broumi S, Smarandache F (2014) Single valued neutrosophic trapezoid linguistic aggregation operators based multi-attribute decision making. Bull Pure Appl Sci Math Stat 33e(2):135–155

    Article  Google Scholar 

  39. Ju Y-B, Liu X-Y, Ju D-W (2015) Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making. Soft Comput. doi:10.1007/s00500-015-1761-y

    MATH  Google Scholar 

  40. Qin J-D, Liu X-W (2015) Approaches to uncertain linguistic multiple attribute decision making based on dual Maclaurin symmetric mean. Intell Fuzzy Syst 29:171–186

    Article  MathSciNet  MATH  Google Scholar 

  41. Qin J-D, Liu X-W, Pedrycz W (2015) Hesitant fuzzy Maclaurin symmetric mean operators and its application to multiple-attribute decision making. Int J Fuzzy Syst 17(4):509–520

    Article  MathSciNet  Google Scholar 

  42. Wen J-J, Shi H-N (2000) Optimizing sharpening for Maclaurin inequality. J Chengdu Univ 19(3):1–8

    Google Scholar 

  43. Pečarić J, Wen J, W-l Wang TLu (2005) A generalization of Maclaurin’s inequalities and its applications. Math Inequalities Appl 8(4):583–598

    MathSciNet  MATH  Google Scholar 

  44. Krnić M, Pečarić J (2006) A Hilbert inequality and an Euler-Maclaurin summation formula. Anziam J 48(3):419–431

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang X-M (2007) S-Geometric convexity of a function involving Maclaurin’s elementary symmetric mean. J Inequalities Pure Appl Math 8(2):156-165

    MathSciNet  Google Scholar 

  46. Herrera F, Martinez L (2000) An approach for combining numerical and linguistic information based on the 2-tuple fuzzy linguistic representation model in decision-making. Int J Uncertain Fuzziness Knowl Based Syst 8(5):539–562

    Article  MATH  Google Scholar 

  47. Xu Z-S (2006) A note on linguistic hybrid arithmetic averaging operator in multiple attribute group decision making with linguistic information. Group Decis Negot 15(6):593–604

    Article  Google Scholar 

  48. Tian Z-P, Wang J, Zhang H-Y, Chen X-H, Wang J-Q (2015) Simplified neutrosophic linguistic normalized weighted Bonferroni mean operator and its application to multi-criteria decision-making problems. Filomat. doi:10.2298/FIL1508576F

    MATH  Google Scholar 

  49. Wang J-Q, Wu J-T, Wang J, Zhang H-Y, Chen X-H (2014) Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf Sci 288(1):55–72

    Article  MathSciNet  MATH  Google Scholar 

  50. Yu S-M, Zhou H, Chen X-H, Wang J-Q (2015) A multi-criteria decision-making method based on Heronian mean operators under a linguistic hesitant fuzzy environment. Asia Pac J Oper Res 32(5):1–35

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang H-B, Smarandache F, Zhang Y-Q, Sunderraman R (2010) Single valued neutrosophic sets. Multispace Multistruct 4:410–413

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editors and anonymous reviewers for their helpful comments that improved the paper. This work was supported by the National Natural Science Foundation of China (No. 71571193).

Author information

Authors and Affiliations

  1. School of Business, Central South University, Changsha, 410083, People’s Republic of China

    Jian-qiang Wang & Yu Yang

  2. School of Business, Hunan University, Changsha, 410082, People’s Republic of China

    Lin Li

Authors
  1. Jian-qiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-qiang Wang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interests regarding the publication of this paper.

Appendix

Appendix

Proof of Theorem 1

$$\begin{gathered} a_{{i_{j}^{{(k)}} }} = s_{{\theta _{{i_{j}^{{(k)}} }} }} ,\left( {T_{{i_{j}^{{(k)}} }} ,I_{{i_{j}^{{(k)}} }} ,F_{{i_{j}^{{(k)}} }} } \right),(j = 1,2, \ldots ,m) \hfill \\ \Rightarrow \mathop \otimes \limits_{{j = 1}}^{m} a_{{i_{j}^{{(k)}} }} = \left\langle {f^{{* - 1}} \left( {\prod\limits_{{j = 1}}^{m} {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} } \right),\left( {\prod\limits_{{j = 1}}^{m} {T_{{i_{j}^{{(k)}} }} ,1 - \prod\limits_{{j = 1}}^{m} {\left( {1 - I_{{i_{j}^{{(k)}} }} } \right)} ,1 - \prod\limits_{{j = 1}}^{m} {\left( {1 - F_{{i_{j}^{{(k)}} }} } \right)} } } \right)} \right\rangle . \hfill \\ \Rightarrow \mathop \oplus \limits_{{1 \le i_{1} < \cdots < i_{{m \le n}} }} \left( {\mathop \otimes \limits_{{j = 1}}^{m} a_{{i_{j} }} } \right) = \left\langle {f^{{* - 1}} \left( {\sum\limits_{{k = 1}}^{{C_{n}^{m} }} {\prod\limits_{{j = 1}}^{m} {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} } } \right),\left( {\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right) \cdot \prod\limits_{{j = 1}}^{m} {\left( {T_{{i_{j}^{{(k)}} }} } \right)} } \right)} } }}{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} \right)} } }},\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right) \cdot \left( {1 - \prod\limits_{{j = 1}}^{m} {\left( {1 - I_{{i_{j}^{{(k)}} }} } \right)} } \right)} \right)} } }}{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} \right)} } }}} \right)} \right. \hfill \\ \left. {\left. {\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right) \cdot \left( {1 - \prod\limits_{{j = 1}}^{m} {\left( {1 - I_{{i_{j}^{{(k)}} }} } \right)} } \right)} \right)} } }}{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} \right)} } }},\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right) \cdot \left( {1 - \prod\limits_{{j = 1}}^{m} {\left( {1 - F_{{i_{j}^{{(k)}} }} } \right)} } \right)} \right)} } }}{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} \right)} } }}} \right)} \right\rangle . \hfill \\ \Rightarrow \left( {\frac{{ \oplus _{{1 \le i_{1} < \cdots < i_{m} \le n}} \left( { \otimes _{{j = 1}}^{m} a_{{i_{j} }} } \right)}}{{C_{n}^{m} }}} \right)^{{\frac{1}{m}}} = \left\langle {f^{{* - 1}} \left( {\left( {\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\prod\nolimits_{{j = 1}}^{m} {f^{*} } \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} }}{{C_{n}^{m} }}} \right)} \right)} \right.,\left( {\left( {\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right) \cdot \prod\nolimits_{{j = 1}}^{m} {T_{{i_{j}^{{(k)}} }} } } \right)} }}{{C_{n}^{m} \prod\nolimits_{{j = 1}}^{m} {f^{*} } \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)}}} \right)} \right., \hfill \\ \left. {\left. {1 - \left( {1 - \frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right) \cdot \left( {1 - \prod\nolimits_{{j = 1}}^{m} {\left( {1 - I_{{i_{j}^{{(k)}} }} } \right)} } \right)} \right)} }}{{C_{n}^{m} \prod\nolimits_{{j = 1}}^{m} {f^{*} } \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)}}} \right)^{{\frac{1}{m}}} } \right),1 - \left( {1 - \frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\left( {f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right) \cdot \left( {1 - \prod\nolimits_{{j = 1}}^{m} {\left( {1 - F_{{i_{j}^{{(k)}} }} } \right)} } \right)} \right)} }}{{C_{n}^{m} \prod\nolimits_{{j = 1}}^{m} {f^{*} } \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)}}} \right)^{{\frac{1}{m}}} } \right\rangle . \hfill \\ \end{gathered}$$

Denote \(\prod\nolimits_{j = 1}^{m} {\mathop f\nolimits^{*} (\mathop s\nolimits_{{\mathop \theta \nolimits_{{\mathop i\nolimits_{j}^{(k)} }} }} )}\) as A (k), and we can finally obtain a result that matches Theorem 1.

The proof of Property 6

Since \({\mathop A\limits_{ \cdot }}^{(k)} = \prod\nolimits_{j = 1}^{m} {\left( {n \cdot \frac{1}{n}} \right)f^{*} \left( {s_{{\theta_{{i_{j}^{(k)} }} }} } \right)} = A^{(k)} ,\)

$$\begin{gathered} WSVNLMSM^{{(m)}} (a_{1} , \ldots ,a_{n} ) = \left\langle {f^{{* - 1}} \left( {\left( {\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\left( {\prod\nolimits_{{j = 1}}^{m} {\left( {n \cdot \frac{1}{n}} \right)f^{*} \left( {s_{{\theta _{{i_{j}^{{(k)}} }} }} } \right)} } \right)} }}{{C_{n}^{m} }}} \right)^{{\frac{1}{m}}} } \right)} \right.,\left( {\left( {\frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\left\{ {A^{{(k)}} \cdot \prod\nolimits_{{j = 1}}^{m} {T_{{i_{j}^{{(k)}} }} } } \right\}} }}{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {A^{{(k)}} } }}} \right)} \right., \hfill \\ \left. {\left. {1 - \left( {1 - \frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\left\{ {A^{{(k)}} \cdot \left( {1 - \prod\nolimits_{{j = 1}}^{m} {\left( {1 - T_{{i_{j}^{{(k)}} }} } \right)} } \right)} \right\}} }}{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {A^{{(k)}} } }}} \right)^{{\frac{1}{m}}} ,1 - \left( {1 - \frac{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {\left\{ {A^{{(k)}} \cdot \left( {1 - \prod\nolimits_{{j = 1}}^{m} {\left( {1 - F_{{i_{j}^{{(k)}} }} } \right)} } \right)} \right\}} }}{{\sum\nolimits_{{k = 1}}^{{C_{n}^{m} }} {A^{{(k)}} } }}} \right)^{{\frac{1}{m}}} } \right)} \right\rangle \hfill \\ = SVNLMSM^{{(m)}} (a_{1} , \ldots ,a_{n} ).\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \square \hfill \\ \end{gathered}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Jq., Yang, Y. & Li, L. Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators. Neural Comput & Applic 30, 1529–1547 (2018). https://doi.org/10.1007/s00521-016-2747-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-016-2747-0

Keywords

Search

Navigation