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

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.

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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).

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Correspondence to Jian-qiang Wang.

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

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

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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

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Keywords

  • Single-valued neutrosophic linguistic
  • Maclaurin symmetric mean
  • Multi-criteria decision-making