# A Linguistic Neutrosophic Multi-Criteria Group Decision-Making Method to University Human Resource Management

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## Abstract

**:**

## 1. Introduction

## 2. New Operations and Distance Measure for LNNs

#### 2.1. Linguistic Neutrosophic Set

**Definition**

**1**

**.**Let $X$ be a universe of discourse and $\overline{H}=\left\{{h}_{\alpha}|{h}_{0}\le {h}_{\alpha}\le {h}_{2t},\alpha \in \left[0,2t\right]\right\}$, and the LNSs can be defined as follows:

#### 2.2. New Operations for LNNs

**Definition**

**2.**

- (1)
- $\tilde{a}\oplus \tilde{b}=\langle {f}^{*-1}\left(\frac{{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)+{f}^{*}\left({h}_{{T}_{\tilde{b}}}\right)}{1+{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right){f}^{*}\left({h}_{{T}_{\tilde{b}}}\right)}\right),{f}^{*-1}\left(\frac{{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)+{f}^{*}\left({h}_{{I}_{\tilde{b}}}\right)}{1+\left(1-{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)\left(1-{f}^{*}\left({h}_{{I}_{\tilde{b}}}\right)\right)}\right),$${f}^{*-1}\left(\frac{{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)+{f}^{*}\left({h}_{{F}_{\tilde{b}}}\right)}{1+\left(1-{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)\left(1-{f}^{*}\left({h}_{{F}_{\tilde{b}}}\right)\right)}\right)\rangle $;
- (2)
- $\tilde{a}\otimes \tilde{b}=\langle {f}^{*-1}\left(\frac{{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)+{f}^{*}\left({h}_{{T}_{\tilde{b}}}\right)}{1+\left(1-{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)\left(1-{f}^{*}\left({h}_{{T}_{\tilde{b}}}\right)\right)}\right),{f}^{*-1}\left(\frac{{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)+{f}^{*}\left({h}_{{I}_{\tilde{b}}}\right)}{1+{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right){f}^{*}\left({h}_{{I}_{\tilde{b}}}\right)}\right),$${f}^{*-1}\left(\frac{{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)+{f}^{*}\left({h}_{{F}_{\tilde{b}}}\right)}{1+{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right){f}^{*}\left({h}_{{F}_{\tilde{b}}}\right)}\right)\rangle $;
- (3)
- $\zeta \tilde{a}=\langle {f}^{*-1}\left(\frac{{\left(1+{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)}^{\zeta}-{\left(1-{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)}^{\zeta}}{{\left(1+{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)}^{\zeta}+{\left(1-{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)}^{\zeta}}\right),{f}^{*-1}\left(\frac{2{\left({f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)}^{\zeta}}{{\left(2-{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)}^{\zeta}+{\left({f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)}^{\zeta}}\right),$${f}^{*-1}\left(\frac{2{\left({f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)}^{\zeta}}{{\left(2-{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)}^{\zeta}+{\left({f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)}^{\zeta}}\right)\rangle $;
- (4)
- ${\tilde{a}}^{\zeta}=\langle {f}^{*-1}\left(\frac{2{\left({f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)}^{\zeta}}{{\left(2-{f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)}^{\zeta}+{\left({f}^{*}\left({h}_{{T}_{\tilde{a}}}\right)\right)}^{\zeta}}\right),{f}^{*-1}\left(\frac{{\left(1+{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)}^{\zeta}-{\left(1-{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)}^{\zeta}}{{\left(1+{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)}^{\zeta}+{\left(1-{f}^{*}\left({h}_{{I}_{\tilde{a}}}\right)\right)}^{\zeta}}\right),$${f}^{*-1}\left(\frac{{\left(1+{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)}^{\zeta}-{\left(1-{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)}^{\zeta}}{{\left(1+{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)}^{\zeta}+{\left(1-{f}^{*}\left({h}_{{F}_{\tilde{a}}}\right)\right)}^{\zeta}}\right)\rangle $; and
- (5)
- $neg\left(\tilde{a}\right)=\langle {h}_{{F}_{\tilde{a}}},1-{h}_{{I}_{\tilde{a}}},{h}_{{T}_{\tilde{a}}}\rangle $.

**Example**

**1.**

- (1)
- $\tilde{a}\oplus \tilde{b}=\langle {h}_{4.29},{h}_{3.75},{h}_{3.75}\rangle $;
- (2)
- $\tilde{a}\otimes \tilde{b}=\langle {h}_{3.75},{h}_{4.29},{h}_{4.29}\rangle $;
- (3)
- $2\tilde{a}=\langle {h}_{4.8},{h}_{0.46},{h}_{0.46}\rangle $; and
- (4)
- ${\tilde{a}}^{2}=\langle {h}_{1.2},{h}_{3.6},{h}_{3.6}\rangle $.

**Theorem**

**1.**

- (1)
- $\tilde{a}\oplus \tilde{b}=\tilde{b}\oplus \tilde{a}$;
- (2)
- $\left(\tilde{a}\oplus \tilde{b}\right)\oplus \tilde{c}=\tilde{a}\oplus \left(\tilde{b}\oplus \tilde{c}\right)$;
- (3)
- $\tilde{a}\otimes \tilde{b}=\tilde{b}\otimes \tilde{a}$;
- (4)
- $\left(\tilde{a}\otimes \tilde{b}\right)\otimes \tilde{c}=\tilde{a}\otimes \left(\tilde{b}\otimes \tilde{c}\right)$;
- (5)
- $\zeta \tilde{a}\oplus \zeta \tilde{b}=\zeta \left(\tilde{b}\oplus \tilde{a}\right)$; and
- (6)
- ${\left(\tilde{a}\otimes \tilde{b}\right)}^{\zeta}={\tilde{a}}^{\zeta}\otimes {\tilde{b}}^{\zeta}$.

#### 2.3. Distance between Two LNNs

**Definition**

**3.**

**Theorem**

**2.**

- (1)
- $d(\tilde{a},\tilde{b})\ge 0$;
- (2)
- $d(\tilde{a},\tilde{a})=0$;
- (3)
- $d(\tilde{a},\tilde{b})=d(\tilde{b},\tilde{a})$; and
- (4)
- $d(\tilde{a},\tilde{c})\le d(\tilde{a},\tilde{b})+d(\tilde{b},\tilde{c})$.

## 3. Linguistic Neutrosophic Aggregation Operators

**Definition**

**4**

**.**Let ${a}_{j}(j=1,2,\cdots ,n)$ be a collection of positive values and $\mathsf{\Omega}$ be the set of all given values; then the PA operator is the mapping $PA:{\mathsf{\Omega}}^{n}\to \mathsf{\Omega}$, which can be defined as follows:

- (1)
- $Sup({a}_{i},{a}_{j})\in [0,1]$;
- (2)
- $Sup({a}_{i},{a}_{j})=Sup({a}_{j},{a}_{i})$; and
- (3)
- $Sup({a}_{i},{a}_{j})\ge Sup({a}_{l},{a}_{r})$, when$d({a}_{i},{a}_{j})<d({a}_{l},{a}_{r})$, and$d({a}_{i},{a}_{j})$is the distance between${a}_{i}$and${a}_{j}$.

#### 3.1. Linguistic Neutrosophic Power Weighted Averaging Operator

**Definition**

**5.**

**Theorem**

**3.**

**Theorem**

**4.**

#### 3.2. Linguistic Neutrosophic Power Weighted Geometric Operator

**Definition**

**6.**

**Theorem**

**5.**

## 4. MCGDM Method Based on the LNPWA and LNPWG Operators

**Step 1:**Normalize the decision matrices.

**Step 2:**Obtain the weighted decision matrices.

**Step 3:**Calculate the supports.

**Step 4:**Calculate the weights associated with ${r}_{ij}^{{k}_{1}}({k}_{1}=1,2,\cdots ,s)$.

**Step 5:**Obtain the comprehensive evaluation information.

**Step 6:**Determine the ideal decision vectors of all alternative decisions.

**Step 7:**Calculate the separations of each alternative decision vector from the ideal decision vector.

**Step 8:**Calculate the relative closeness of each alternative decision.

**Step 9:**Rank all the alternatives.

## 5. A Case of Human Resource Management Problem

#### 5.1. Problem Definition

#### 5.2. Evaluation Steps of the Proposed Method

**Step 1:**Normalize the decision matrices.

**Step 2:**Obtain the weighted decision matrices.

**Step 3:**Calculate the supports.

**Step 4:**Calculate the weights associated with ${r}_{ij}^{{k}_{1}}({k}_{1}=1,2,\cdots ,s)$.

**Step 5:**Obtain the comprehensive evaluation information.

**Step 6:**Determine the ideal decision vectors of all alternative decisions.

**Step 7:**Calculate the separations of each alternative decision vector from the ideal decision vector.

**Step 8:**Calculate the relative closeness of each alternative decision.

**Step 9:**Rank all the alternatives.

#### 5.3. Sensitivity Analysis and Discussion

#### 5.4. Comparison Analysis and Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Linguistic Scale Function

## Appendix B. The Archimedean T-norm and T-conorm

## Appendix C. The Proof of Theorem 2

**Proof.**

## Appendix D. The Proof of Theorem 3

## References

- Abdullah, L.; Zulkifli, N. Integration of fuzzy AHP and interval type-2 fuzzy DEMATEL: An application to human resource management. Expert Syst. Appl.
**2015**, 42, 4397–4409. [Google Scholar] [CrossRef] - Filho, C.F.F.C.; Rocha, D.A.R.; Costa, M.G.F. Using constraint satisfaction problem approach to solve human resource allocation problems in cooperative health services. Expert Syst. Appl.
**2012**, 39, 385–394. [Google Scholar] [CrossRef] - Marcolajara, B.; ÚbedaGarcía, M. Human resource management approaches in Spanish hotels: An introductory analysis. Int. J. Hosp. Manag.
**2013**, 35, 339–347. [Google Scholar] [CrossRef] - Bohlouli, M.; Mittas, N.; Kakarontzas, G.; Theodosiou, T.; Angelis, L.; Fathi, M. Competence assessment as an expert system for human resource management: A mathematical approach. Expert Syst. Appl.
**2017**, 70, 83–102. [Google Scholar] [CrossRef] - Zhang, X.; Wang, J.; Zhang, H.; Hu, J. A heterogeneous linguistic MAGDM framework to classroom teaching quality evaluation. Eurasia J. Math. Sci. Technol. Educ.
**2017**, 13, 4929–4956. [Google Scholar] [CrossRef] - Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic: Neutrosophy, Neutrosophic Set, Neutrosophic Probability; American Research Press: Rehoboth, DE, USA, 1999; pp. 1–141. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Wu, X.H.; Wang, J.Q.; Peng, J.J.; Qian, J. A novel group decision-making method with probability hesitant interval neutrosphic set and its application in middle level manager’ selection. Int. J. Uncertain. Quantif.
**2018**, 8, 291–319. [Google Scholar] [CrossRef] - Ji, P.; Wang, J.Q.; Zhang, H.Y. Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers. Neural Comput. Appl.
**2016**, 30, 799–823. [Google Scholar] [CrossRef] - Liang, R.; Wang, J.; Zhang, H. Evaluation of e-commerce websites: An integrated approach under a single-valued trapezoidal neutrosophic environment. Knowl.-Based Syst.
**2017**, 135, 44–59. [Google Scholar] [CrossRef] - Liang, R.X.; Wang, J.Q.; Zhang, H.Y. A multi-criteria decision-making method based on single-valued trapezoidal neutrosophic preference relations with complete weight information. Neural Comput. Appl.
**2017**. [Google Scholar] [CrossRef] - Liang, R.X.; Wang, J.Q.; Li, L. Multi-criteria group decision making method based on interdependent inputs of single valued trapezoidal neutrosophic information. Neural Comput. Appl.
**2018**, 30, 241–260. [Google Scholar] [CrossRef] - Tian, Z.P.; Wang, J.; Wang, J.Q.; Zhang, H.Y. Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development. Group Decis. Negot.
**2017**, 26, 597–627. [Google Scholar] [CrossRef] - Ji, P.; Zhang, H.Y.; Wang, J.Q. Selecting an outsourcing provider based on the combined MABAC–ELECTRE method using single-valued neutrosophic linguistic sets. Comput. Ind. Eng.
**2018**, 120, 429–441. [Google Scholar] [CrossRef] - Karaaslan, F. Correlation coefficients of single-valued neutrosophic refined soft sets and their applications in clustering analysis. Neural Comput. Appl.
**2017**, 28, 2781–2793. [Google Scholar] [CrossRef] - Ye, J. Single-valued neutrosophic clustering algorithms based on similarity measures. J. Classif.
**2017**, 34, 148–162. [Google Scholar] [CrossRef] - Li, Y.Y.; Wang, J.Q.; Wang, T.L. A linguistic neutrosophic multi-criteria group decision-making approach with EDAS method. Arab. J. Sci. Eng.
**2018**. [Google Scholar] [CrossRef] - Chen, Z.S.; Chin, K.S.; Li, Y.L.; Yang, Y. Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making. Inf. Sci.
**2016**, 357, 61–87. [Google Scholar] [CrossRef] - Rodríguez, R.M.; Martínez, L.; Herrera, F. Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Wang, H. Extended hesitant fuzzy linguistic term sets and their aggregation in group decision making. Int. J. Comput. Intell. Syst.
**2014**, 8, 14–33. [Google Scholar] [CrossRef] - Wang, X.K.; Peng, H.G.; Wang, J.Q. Hesitant linguistic intuitionistic fuzzy sets and their application in multi-criteria decision-making problems. Int. J. Uncertain. Quantif.
**2018**, 8, 321–341. [Google Scholar] [CrossRef] - Tian, Z.P.; Wang, J.Q.; Zhang, H.Y.; Wang, T.L. Signed distance-based consensus in multi-criteria group decision-making with multi-granular hesitant unbalanced linguistic information. Comput. Ind. Eng.
**2018**, 124, 125–138. [Google Scholar] [CrossRef] - Zhang, H.M. Linguistic intuitionistic fuzzy sets and application in MAGDM. J. Appl. Math.
**2014**, 2014. [Google Scholar] [CrossRef] - Chen, Z.C.; Liu, P.H.; Pei, Z. An approach to multiple attribute group decision making based on linguistic intuitionistic fuzzy numbers. Int. J. Comput. Intell. Syst.
**2015**, 8, 747–760. [Google Scholar] [CrossRef] [Green Version] - Peng, H.G.; Wang, J.Q. A multicriteria group decision-making method based on the normal cloud model with Zadeh’s Z-numbers. IEEE Trans. Fuzzy Syst.
**2018**. [Google Scholar] [CrossRef] - Peng, H.G.; Zhang, H.Y.; Wang, J.Q. Cloud decision support model for selecting hotels on TripAdvisor.com with probabilistic linguistic information. Int. J. Hosp. Manag.
**2018**, 68, 124–138. [Google Scholar] [CrossRef] - Luo, S.Z.; Zhang, H.Y.; Wang, J.Q.; Li, L. Group decision-making approach for evaluating the sustainability of constructed wetlands with probabilistic linguistic preference relations. J. Oper. Res. Soc.
**2018**. [Google Scholar] [CrossRef] - Fang, Z.B.; Ye, J. Multiple attribute group decision-making method based on linguistic neutrosophic numbers. Symmetry
**2017**, 9, 111. [Google Scholar] [CrossRef] - Li, Y.Y.; Zhang, H.Y.; Wang, J.Q. Linguistic neutrosophic sets and their application in multicriteria decision-making problems. Int. J. Uncertain. Quantif.
**2017**, 7, 135–154. [Google Scholar] [CrossRef] - Fan, C.X.; Ye, J.; Hu, K.L.; Fan, E. Bonferroni mean operators of linguistic neutrosophic numbers and their multiple attribute group decision-making methods. Information
**2017**, 8, 107. [Google Scholar] [CrossRef] - Shi, L.L.; Ye, J. Cosine measures of linguistic neutrosophic numbers and their application in multiple attribute group decision-making. Information
**2017**, 8, 117. [Google Scholar] - Liang, W.Z.; Zhao, G.Y.; Wu, H. Evaluating investment risks of metallic mines using an extended TOPSIS method with linguistic neutrosophic numbers. Symmetry
**2017**, 9, 149. [Google Scholar] [CrossRef] - Herrera, F.; Martínez, L. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2000**, 8, 746–752. [Google Scholar] - Delgado, M.; Verdegay, J.L.; Vila, M.A. Linguistic decision-making models. Int. J. Intell. Syst.
**1992**, 7, 479–492. [Google Scholar] [CrossRef] - Hu, J.; Zhang, X.; Yang, Y.; Liu, Y.; Chen, X. New doctors ranking system based on VIKOR method. Int. Trans. Oper. Res.
**2018**. [Google Scholar] [CrossRef] - Li, D.Y.; Meng, H.J.; Shi, X.M. Membership clouds and membership cloud generators. Comput. Res. Dev.
**1995**, 32, 16–21. [Google Scholar] - Bordogna, G.; Fedrizzi, M.; Pasi, G. A linguistic modeling of consensus in group decision making based on OWA operators. IEEE Trans. Syst. Man Cybern.-Part A Syst. Hum.
**1997**, 27, 126–133. [Google Scholar] [CrossRef] - Doukas, H.; Karakosta, C.; Psarras, J. Computing with words to assess the sustainability of renewable energy options. Expert Syst. Appl.
**2010**, 37, 5491–5497. [Google Scholar] [CrossRef] - Wang, J.Q.; Wu, J.T.; Wang, J.; Zhang, H.Y.; Chen, X.H. Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf. Sci.
**2014**, 288, 55–72. [Google Scholar] [CrossRef] - Yager, R.R. The power average operator. IEEE Trans. Syst. Man Cybern.-Part A Syst. Hum.
**2001**, 31, 724–731. [Google Scholar] [CrossRef] - Jiang, W.; Wei, B.; Zhan, J.; Xie, C.; Zhou, D. A visibility graph power averaging aggregation operator: A methodology based on network analysis. Comput. Ind. Eng.
**2016**, 101, 260–268. [Google Scholar] [CrossRef] - Gong, Z.; Xu, X.; Zhang, H.; Aytun Ozturk, U.; Herrera-Viedma, E.; Xu, C. The consensus models with interval preference opinions and their economic interpretation. Omega
**2015**, 55, 81–90. [Google Scholar] [CrossRef] - Liu, P.D.; Qin, X.Y. Power average operators of linguistic intuitionistic fuzzy numbers and their application to multiple-attribute decision making. J. Intell. Fuzzy Syst.
**2017**, 32, 1029–1043. [Google Scholar] [CrossRef] - Yager, R.R. Applications and extensions of OWA aggregations. Int. J. Man-Mach. Stud.
**1992**, 37, 103–132. [Google Scholar] [CrossRef] - Yager, R.R. On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Syst. Man Cybern.
**1988**, 18, 183–190. [Google Scholar] [CrossRef] - Yager, R.R. Families of OWA operators. Fuzzy Sets Syst.
**1993**, 59, 125–148. [Google Scholar] [CrossRef] - Huang, J.J.; Yoon, K. Multiple Attribute Decision Making: Methods and Applications; Chapman and Hall/CRC: Boca Raton, FL, USA, 2011. [Google Scholar]
- Baykasoğlu, A.; Gölcük, İ. Development of an interval type-2 fuzzy sets based hierarchical MADM model by combining DEMATEL and TOPSIS. Expert Syst. Appl.
**2017**, 70, 37–51. [Google Scholar] [CrossRef] - Joshi, D.; Kumar, S. Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making. Eur. J. Oper. Res.
**2016**, 248, 183–191. [Google Scholar] [CrossRef] - Mehrdad, A.M.A.K.; Aghdas, B.; Alireza, A.; Mahdi, G.; Hamed, K. Introducing a procedure for developing a novel centrality measure (Sociability Centrality) for social networks using TOPSIS method and genetic algorithm. Comput. Hum. Behav.
**2016**, 56, 295–305. [Google Scholar] - Afsordegan, A.; Sánchez, M.; Agell, N.; Zahedi, S.; Cremades, L.V. Decision making under uncertainty using a qualitative TOPSIS method for selecting sustainable energy alternatives. Int. J. Environ. Sci. Technol.
**2016**, 13, 1419–1432. [Google Scholar] [CrossRef] [Green Version] - Xu, Z.S. A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf. Sci.
**2004**, 166, 19–30. [Google Scholar] [CrossRef] - Chou, Y.C.; Sun, C.C.; Yen, H.Y. Evaluating the criteria for human resource for science and technology (HRST) based on an integrated fuzzy AHP and fuzzy DEMATEL approach. Appl. Soft Comput.
**2012**, 12, 64–71. [Google Scholar] [CrossRef] - Yu, D.; Wu, Y.; Lu, T. Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowl.-Based Syst.
**2012**, 30, 57–66. [Google Scholar] [CrossRef] - Yu, S.M.; Wang, J.; Wang, J.Q.; Li, L. A multi-criteria decision-making model for hotel selection with linguistic distribution assessments. Appl. Soft Comput.
**2018**, 67, 741–755. [Google Scholar] [CrossRef] - Qiu, J.; Wang, T.; Yin, S.; Gao, H. Data-based optimal control for networked double-layer industrial processes. IEEE Trans. Ind. Electron.
**2017**, 64, 4179–4186. [Google Scholar] [CrossRef] - Qiu, J.; Wei, Y.; Karimi, H.R.; Gao, H. Reliable control of discrete-time piecewise-affine time-delay systems via output feedback. IEEE Trans. Reliab.
**2017**, 67, 79–91. [Google Scholar] [CrossRef] - Liu, A.Y.; Liu, F.J. Research on method of analyzing the posterior weight of experts based on new evaluation scale of linguistic information. Chin. J. Manag. Sci.
**2011**, 19, 149–155. [Google Scholar] - Klement, E.P.; Mesiar, R. Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms; Elsevier: New York, NY, USA, 2005. [Google Scholar]
- Beliakov, G.; Pradera, A.; Calvo, T. Aggregation Functions: A Guide for Practitioners; Springer: Berlin, Germany, 2007; Volume 12, pp. 139–141. [Google Scholar]

${D}_{1}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{3}$ | ${C}_{4}$ |

${A}_{1}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ |

${A}_{2}$ | $\langle {h}_{5},{h}_{3},{h}_{1}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{0},{h}_{3},{h}_{0}\rangle $ |

${A}_{3}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ |

${A}_{4}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ |

${A}_{5}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{0},{h}_{3},{h}_{2}\rangle $ |

${A}_{6}$ | $\langle {h}_{6},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{0},{h}_{3},{h}_{2}\rangle $ |

${D}_{2}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{3}$ | ${C}_{4}$ |

${A}_{1}$ | $\langle {h}_{6},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ |

${A}_{2}$ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ |

${A}_{3}$ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{0},{h}_{0}\rangle $ |

${A}_{4}$ | $\langle {h}_{6},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{6},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ |

${A}_{5}$ | $\langle {h}_{5},{h}_{5},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{6},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{0},{h}_{3},{h}_{2}\rangle $ |

${A}_{6}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{6},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{1}\rangle $ |

${D}_{3}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{3}$ | ${C}_{4}$ |

${A}_{1}$ | $\langle {h}_{6},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{6},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ |

${A}_{2}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ |

${A}_{3}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{6},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ |

${A}_{4}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{6},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{0},{h}_{3},{h}_{2}\rangle $ |

${A}_{5}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{6},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ |

${A}_{6}$ | $\langle {h}_{5},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{0},{h}_{3},{h}_{2}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ | $\langle {h}_{5},{h}_{3},{h}_{0}\rangle $ |

${D}_{1}$ | ${C}_{1}$ | ${C}_{2}$ | ${C}_{3}$ | ${C}_{4}$ |

${A}_{1}$ | $\langle {h}_{2.0696},{h}_{5.0201},{h}_{4.579}\rangle $ | $\langle {h}_{0.8573},{h}_{5.6051},{h}_{0}\rangle $ | $\langle {h}_{2.1327},{h}_{4.9881},{h}_{0}\rangle $ | $\langle {h}_{1.8772},{h}_{5.1166},{h}_{4.7165}\rangle $ |

${A}_{2}$ | $\langle {h}_{2.0696},{h}_{5.0201},{h}_{3.9304}\rangle $ | $\langle {h}_{0.8573},{h}_{5.6051},{h}_{0}\rangle $ | $\langle {h}_{2.1327},{h}_{4.9881},{h}_{0}\rangle $ | $\langle {h}_{0},{h}_{5.1166},{h}_{0}\rangle $ |

${A}_{3}$ | $\langle {h}_{2.0696},{h}_{5.0201},{h}_{4.579}\rangle $ | $\langle {h}_{0.8573},{h}_{5.6051},{h}_{0}\rangle $ | $\langle {h}_{2.1327},{h}_{4.9881},{h}_{0}\rangle $ | $\langle {h}_{0},{h}_{5.1166},{h}_{0}\rangle $ |

${A}_{4}$ | $\langle {h}_{2.0696},{h}_{5.0201},{h}_{4.579}\rangle $ | $\langle {h}_{0.8573},{h}_{5.6051},{h}_{0}\rangle $ | $\langle {h}_{2.1327},{h}_{4.9881},{h}_{4.5335}\rangle $ | $\langle {h}_{0},{h}_{5.1166},{h}_{0}\rangle $ |

${A}_{5}$ | $\langle {h}_{}$ |