Skip to main content

An approach for solving fuzzy multi-criteria decision problem under linguistic information

Fuzzy Optimization and Decision Making volume 21, pages 45–69 (2022)Cite this article

Abstract

Linguistic information processing exists in multi-criteria decision making, and linguistic truth-valued lattice implication algebra (LTV-LIA) has definite advantages in handling comparable and incomparable linguistic values. To deal with the preference relations with linguistic evaluation information, we establish a novel approach for solving fuzzy multi-criteria decision problem under linguistic information based on LTV-LIA. In this paper, we propose linguistic lattice-valued preference relation (LLVPR). LLVPR positive and negative matrixes are introduced to evaluate the advantages and disadvantages of alternatives respectively. In order to get a reasonable result, we introduce a new algorithm to check and repair the consistency of a LLVPR. A linguistic lattice-valued 2-tuple representation model (LLV2-tuple) and some new aggregation operations are presented to get the comprehensive linguistic information without information loss. Considering different decision makers have different preferences, a multiple preferences implication operation of LLV2-tuple is introduced. Finally, we propose a novel linguistic analytic hierarchy process embedded in aggregation layer and implication layer, introducing algorithm and numerical examples. A comparative analysis is adopted to illustrate the rationality.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

Fig. 1
Fig. 2
Fig. 3

References

  • Cabrerizo, F. J., Morente-Molinera, J. A., Pedrycz, W., Taghavi, A., & Herrera-Viedma, E. (2018). Granulating linguistic information in decision making under consensus and consistency. Expert Systems with Applications, 99, 83–92.

    Article  Google Scholar 

  • Callejas, E. A., Cerrada, J. A., Cerrada, C., & Cabrerizo, F. J. (2019). Group decision making based on a framework of granular computing for multi-criteria and linguistic contexts. IEEE Access, 7, 54670–54681.

    Article  Google Scholar 

  • Dong, Y., & Herrera-Viedma, E. (2017). Consistency-driven automatic methodology to set interval numerical scales of 2-tuple linguistic term sets and its use in the linguistic gdm with preference relation. IEEE Transactions on Cybernetics, 45(4), 780–792.

    Article  Google Scholar 

  • Dong, Y., Hong, W. C., Xu, Y., & Yu, S. (2011). Selecting the individual numerical scale and prioritization method in the analytic hierarchy process: A 2-tuple fuzzy linguistic approach. IEEE Transactions on Fuzzy Systems, 19(1), 13–25.

    Article  Google Scholar 

  • Dong, Y., Liu, W., Chiclana, F., Kou, G., & Herrera-Viedma, E. (2019). Are incomplete and self-confident preference relations better in multicriteria decision making? A simulation-based investigation. Information Sciences, 492, 40–57.

    MathSciNet  Article  Google Scholar 

  • Dong, Y., Xu, Y., & Yu, S. (2009). Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model. IEEE Transactions on Fuzzy Systems, 17(6), 1366–1378.

    Article  Google Scholar 

  • Dong, Y., Zhang, G., Hong, W. C., & Yu, S. (2013). Linguistic computational model based on 2-tuples and intervals. IEEE Transactions on Fuzzy Systems, 21(6), 1006–1018.

    Article  Google Scholar 

  • Herrera, F., & Martinez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Transactions on Fuzzy Systems, 8(6), 746–752.

    Article  Google Scholar 

  • Li, C. C., Dong, Y., & Herrera, F. (2019). A consensus model for large-scale linguistic group decision making with a feedback recommendation based on clustered personalized individual semantics and opposing consensus groups. IEEE Transactions on Fuzzy Systems, 27(2), 221–233.

    Article  Google Scholar 

  • Li, C. C., Dong, Y., Herrera, F., Herrera-Viedma, E., & Martłnez, L. (2017). Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching. Information Fusion, 33, 29–40.

    Article  Google Scholar 

  • Li, C. C., Rodriguez, R. M., Martinez, L., Dong, Y., & Herrera, F. (2018). Personalized individual semantics based on consistency in hesitant linguistic group decision making with comparative linguistic expressions. Knowledge-Based Systems, 145, 156–165.

    Article  Google Scholar 

  • Liu, P., Cui, H., Luo, S., Diao, H., & Zou, L. (2018). Linguistic-valued lattice-valued 2-tuple representation model. In 2018 5th IEEE international conference on cloud computing and intelligence systems (CCIS) (pp. 620–624). IEEE.

  • Liu, P., Diao, H., Zou, L., & Deng, A. (2020). Uncertain multi-attribute group decision making based on linguistic-valued intuitionistic fuzzy preference relations. Information Sciences, 508, 293–308.

    MathSciNet  Article  Google Scholar 

  • Liu, W., Dong, Y., Chiclana, F., Cabrerizo, F. J., & Herrera-Viedma, E. (2017). Group decision-making based on heterogeneous preference relations with self-confidence. Fuzzy Optimization and Decision Making, 16(4), 429–447.

    MathSciNet  Article  Google Scholar 

  • Maghsoodi, A. I., Rasoulipanah, H., López, L. M., Liao, H., & Zavadskas, E. K. (2020). Integrating interval-valued multi-granular 2-tuple linguistic bwm-codas approach with target-based attributes: Site selection for a construction project. Computers & Industrial Engineering, 139, 106147.

    Article  Google Scholar 

  • Martłnez, L., Liu, J., & Xu, Y. (2010). Information fusion and logic-based reasoning approaches for decision making under uncertainty. Journal of Universal Computerence, 16(16), 1–2.

    Google Scholar 

  • Meng, F., & Chen, X. (2018). A robust additive consistency-based method for decision making with triangular fuzzy reciprocal preference relations. Fuzzy Optimization and Decision Making, 17(1), 49–73.

    MathSciNet  Article  Google Scholar 

  • Meng, F. Y., Tang, J., & Fujita, H. (2019). Consistency-based algorithms for decision-making with interval fuzzy preference relations. IEEE Transactions on Fuzzy Systems, 27(10), 2052–2066.

    Article  Google Scholar 

  • Tang, J., Meng, F., Cabrerizo, F. J., & Herrera-Viedma, E. (2019). A procedure for group decision making with interval-valued intuitionistic linguistic fuzzy preference relations. Fuzzy Optimization and Decision Making, 18(4), 493–527.

    MathSciNet  Article  Google Scholar 

  • Xu, Y., Liu, J., Martłnez, L., & Ruan, D. (2010). Some views on information fusion and logic based approaches in decision making under uncertainty. Journal of Universal Computerence, 16(1), 3–19.

    MathSciNet  Google Scholar 

  • Xu, Y., Ruan, D., Qin, K., & Liu, J. (2003). Lattice-valued logic. In Studies in fuzziness and soft computing (Vol. 132).

  • Xu, Z., & Liao, H. (2014). Intuitionistic fuzzy analytic hierarchy process. IEEE Transactions on Fuzzy Systems, 22(4), 749–761.

    Article  Google Scholar 

  • Xu, Z., & Zhou, W. (2017). Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optimization and Decision Making, 16(4), 481–503.

    MathSciNet  Article  Google Scholar 

  • Zhang, Z., & Pedrycz, W. (2018). Goal programming approaches to managing consistency and consensus for intuitionistic multiplicative preference relations in group decision making. IEEE Transactions on Fuzzy Systems, 26(6), 3261–3275.

    Article  Google Scholar 

  • Zou, L., Liu, X., Ruan, D., & Xu, Y. (2012). Linguistic truth-valued intuitionistic fuzzy algebra. Journal of Multiple Valued Logic & Soft Computing, 18(5–6), 445–456.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is partially supported by the National Natural Science Foundation of P.R. China (Nos. 61502068, 61772250) and the Scientific Research Project of Department of Education of Liaoning Province (No. LJ2020007).

Author information

Authors and Affiliations

  1. Information Science and Technology College, Dalian Maritime University, Dalian, China

    Hongyue Diao, Ansheng Deng & Hui Cui

  2. School of Mathematics, Liaoning Normal University, Dalian, China

    Xin Liu

  3. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, China

    Li Zou

Authors
  1. Hongyue Diao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ansheng Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hui Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Li Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ansheng Deng or Xin Liu.

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.

Appendix

Appendix

In this appendix, detailed calculation process of numerical examples is introduced.

Determine the LLVPR according to the linguistic fundamental scale by pair-wise comparisons respectively.

$$\begin{aligned} C= & {} \left[ {\begin{array}{*{20}{c}} - &{}{\left( {{h_1},t} \right) }&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_2},t} \right) }&{}{\left( {{h_2},t} \right) }\\ {\left( {{h_1},f} \right) }&{} - &{}{\left( {{h_1},t} \right) }&{}{\left( {{h_2},t} \right) }&{}{\left( {{h_2},t} \right) }\\ {\left( {{h_1},f} \right) }&{}{\left( {{h_1},f} \right) }&{} - &{}{\left( {{h_1},t} \right) }&{}{\left( {{h_1},t} \right) }\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{} - &{}{\left( {{h_1},t} \right) }\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{}{\left( {{h_1},f} \right) }&{} - \end{array}} \right] ;\\ {A^{\left( 1 \right) }}= & {} \left[ {\begin{array}{*{20}{c}} - &{}{\left( {{h_1},t} \right) }&{}{\left( {{h_0},t} \right) }\\ {\left( {{h_1},f} \right) }&{} - &{}{\left( {{h_1},f} \right) }\\ {\left( {{h_0},f} \right) }&{}{\left( {{h_1},t} \right) }&{} - \end{array}} \right] ; {A^{\left( 2 \right) }} = \left[ {\begin{array}{*{20}{c}} - &{}{\left( {{h_1},t} \right) }&{}{\left( {{h_0},t} \right) }\\ {\left( {{h_1},f} \right) }&{} - &{}{\left( {{h_1},f} \right) }\\ {\left( {{h_0},f} \right) }&{}{\left( {{h_1},t} \right) }&{} - \end{array}} \right] ;\\ {A^{\left( 3 \right) }}= & {} \left[ {\begin{array}{*{20}{c}} - &{}{\left( {{h_1},t} \right) }&{}{\left( {{h_2},t} \right) }\\ {\left( {{h_1},f} \right) }&{} - &{}{\left( {{h_1},t} \right) }\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{} - \end{array}} \right] ; {A^{\left( 4 \right) }} = \left[ {\begin{array}{*{20}{c}} - &{}{\left( {{h_3},t} \right) }&{}{\left( {{h_2},t} \right) }\\ {\left( {{h_3},f} \right) }&{} - &{}{\left( {{h_1},t} \right) }\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{} - \end{array}} \right] ;\\ {A^{\left( 5 \right) }}= & {} \left[ {\begin{array}{*{20}{c}} - &{}{\left( {{h_2},t} \right) }&{}{\left( {{h_2},t} \right) }\\ {\left( {{h_2},f} \right) }&{} - &{}{\left( {{h_1},t} \right) }\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{} - \end{array}} \right] . \end{aligned}$$

Compute the LLVPR positive matrix and negative matrix of every preference relation matrix.

$$\begin{aligned} {C^ + }= & {} \left[ {\begin{array}{*{20}{c}} *&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_2},t} \right) }&{}{\left( {{h_2},t} \right) }\\ *&{}*&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_2},t} \right) }&{}{\left( {{h_2},t} \right) }\\ *&{}*&{}*&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_1},t} \right) } \\ *&{}*&{}*&{}*&{}{\left( {{h_1},t} \right) }\\ *&{}*&{}*&{}*&{}* \end{array}} \right] , {C^ - } = \left[ {\begin{array}{*{20}{c}} *&{}*&{}*&{}*&{}* \\ {\left( {{h_1},f} \right) }&{}*&{}*&{}*&{}*\! \\ {\left( {{h_1},f} \right) }&{}{\left( {{h_1},f} \right) }&{}*&{}*&{}*\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{}*&{}*\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{}{\left( {{h_1},f} \right) }&{}* \end{array}} \right] ;\\ {A^{\left( 1 \right) , + }}= & {} \left[ {\begin{array}{*{20}{c}} *&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_0},t} \right) }\\ *&{}*&{}*\\ *&{}{\left( {{h_1},t} \right) }&{}* \end{array}} \right] , {A^{\left( 1 \right) , - }} = \left[ {\begin{array}{*{20}{c}} *&{}*&{}*\\ {\left( {{h_1},f} \right) }&{}*&{}{\left( {{h_1},f} \right) }\\ {\left( {{h_0},f} \right) }&{}*&{}* \end{array}} \right] ;\\ {A^{\left( 2 \right) , + }}= & {} \left[ {\begin{array}{*{20}{c}} *&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_0},t} \right) }\\ *&{}*&{}*\\ *&{}{\left( {{h_1},t} \right) }&{}* \end{array}} \right] , {A^{\left( 2 \right) , - }} = \left[ {\begin{array}{*{20}{c}} *&{}*&{}*\\ {\left( {{h_1},f} \right) }&{}*&{}{\left( {{h_1},f} \right) }\\ {\left( {{h_0},f} \right) }&{}*&{}* \end{array}} \right] ;\\ {A^{\left( 3 \right) , + }}= & {} \left[ {\begin{array}{*{20}{c}} *&{}{\left( {{h_1},t} \right) }&{}{\left( {{h_2},t} \right) }\\ *&{}*&{}{\left( {{h_1},t} \right) }\\ *&{}*&{}* \end{array}} \right] , {A^{\left( 3 \right) , - }} = \left[ {\begin{array}{*{20}{c}} *&{}*&{}*\\ {\left( {{h_1},f} \right) }&{}*&{}*\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{}* \end{array}} \right] ;\\ {A^{\left( 4 \right) , + }}= & {} \left[ {\begin{array}{*{20}{c}} *&{}{\left( {{h_3},t} \right) }&{}{\left( {{h_2},t} \right) }\\ *&{}*&{}{\left( {{h_1},t} \right) }\\ *&{}*&{}* \end{array}} \right] , {A^{\left( 4 \right) , - }} = \left[ {\begin{array}{*{20}{c}} *&{}*&{}*\\ {\left( {{h_3},f} \right) }&{}*&{}*\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{}* \end{array}} \right] ;\\ {A^{\left( 5 \right) , + }}= & {} \left[ {\begin{array}{*{20}{c}} *&{}{\left( {{h_2},t} \right) }&{}{\left( {{h_2},t} \right) }\\ *&{}*&{}{\left( {{h_1},t} \right) }\\ *&{}*&{}* \end{array}} \right] , {A^{\left( 5 \right) , - }} = \left[ {\begin{array}{*{20}{c}} *&{}*&{}*\\ {\left( {{h_2},f} \right) }&{}*&{}*\\ {\left( {{h_2},f} \right) }&{}{\left( {{h_1},f} \right) }&{}* \end{array}} \right] . \end{aligned}$$

Aggregate every row of every matrix and calculate the comprehensive aggregate value. We take the \({G_{LAAO}}\) as an example.

$$\begin{aligned} {C^ + }= & {} \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0.2} \right) }\\ {\left( {\left( {{h_1},t} \right) ,0} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.4} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.2} \right) }\\ * \end{array}} \right) , {C^ - } = \left( {\begin{array}{*{20}{c}} * \\ {\left( {\left( {{h_0},f} \right) ,0.2} \right) }\\ {\left( {\left( {{h_0},f} \right) ,0.4} \right) }\\ {\left( {\left( {{h_1},f} \right) ,0} \right) }\\ {\left( {\left( {{h_1},f} \right) ,0.2} \right) } \end{array}} \right) , {\tilde{C}} = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0.2} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.8} \right) }\\ - \\ {\left( {\left( {{h_0},f} \right) ,0.8} \right) }\\ {\left( {\left( {{h_1},f} \right) ,0.2} \right) } \end{array}} \right) ;\\ {A^{\left( 1 \right) , + }}= & {} \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_0},t} \right) ,0.33} \right) }\\ * \\ {\left( {\left( {{h_0},t} \right) ,0.33} \right) } \end{array}} \right) , {A^{\left( 1 \right) , - }} = \left( {\begin{array}{*{20}{c}} * \\ {\left( {\left( {{h_0},f} \right) ,0.67} \right) }\\ {\left( {\left( {{h_0},f} \right) ,0} \right) } \end{array}} \right) , {{\tilde{A}}^{\left( 1 \right) }} = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_0},t} \right) ,0.33} \right) }\\ {\left( {\left( {{h_0},f} \right) ,0.67} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.17} \right) } \end{array}} \right) ;\\ {A^{\left( 2 \right) , + }}= & {} \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_0},t} \right) ,0.33} \right) }\\ * \\ {\left( {\left( {{h_0},t} \right) ,0.33} \right) } \end{array}} \right) , {A^{\left( 2 \right) , - }} = \left( {\begin{array}{*{20}{c}} * \\ {\left( {\left( {{h_0},f} \right) ,0.67} \right) }\\ {\left( {\left( {{h_0},f} \right) ,0} \right) } \end{array}} \right) , {{\tilde{A}}^{\left( 2 \right) }} = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_0},t} \right) ,0.33} \right) }\\ {\left( {\left( {{h_0},f} \right) ,0.67} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.17} \right) } \end{array}} \right) ;\\ {A^{\left( 3 \right) , + }}= & {} \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.33} \right) }\\ * \end{array}} \right) , {A^{\left( 3 \right) , - }} = \left( {\begin{array}{*{20}{c}} * \\ {\left( {\left( {{h_0},f} \right) ,0.33} \right) }\\ {\left( {\left( {{h_1},f} \right) ,0} \right) } \end{array}} \right) , {{\tilde{A}}^{\left( 3 \right) }} = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0} \right) }\\ - \\ {\left( {\left( {{h_1},f} \right) ,0} \right) } \end{array}} \right) ;\\ {A^{\left( 4 \right) , + }}= & {} \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0.67} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.33} \right) }\\ * \end{array}} \right) , {A^{\left( 4 \right) , - }} = \left( {\begin{array}{*{20}{c}} * \\ {\left( {\left( {{h_1},f} \right) ,0} \right) }\\ {\left( {\left( {{h_1},f} \right) ,0} \right) } \end{array}} \right) , {{\tilde{A}}^{\left( 4 \right) }} = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0.67} \right) }\\ {\left( {\left( {{h_0},f} \right) ,0.67} \right) }\\ {\left( {\left( {{h_1},f} \right) ,0} \right) } \end{array}} \right) ;\\ {A^{\left( 5 \right) , + }}= & {} \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0.33} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.33} \right) }\\ * \end{array}} \right) , {A^{\left( 5 \right) , - }} = \left( {\begin{array}{*{20}{c}} * \\ {\left( {\left( {{h_0},f} \right) ,0.67} \right) }\\ {\left( {\left( {{h_{\mathrm{1}}},f} \right) ,0} \right) } \end{array}} \right) , {{\tilde{A}}^{\left( 5 \right) }} = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0.33} \right) }\\ {\left( {\left( {{h_0},f} \right) ,0.17} \right) }\\ {\left( {\left( {{h_{\mathrm{1}}},f} \right) ,0} \right) } \end{array}} \right) . \end{aligned}$$

According to the above calculation, the criterion aggregation value and alternative aggregation value can be obtained as follows

$$\begin{aligned} {\left( {{\tilde{C}}} \right) ^T}= & {} \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_1},t} \right) ,0.2} \right) }&{\left( {\left( {{h_0},t} \right) ,0.8} \right) }- & {} {\left( {\left( {{h_0},f} \right) ,0.8} \right) }&{\left( {\left( {{h_1},f} \right) ,0.2} \right) } \end{array}} \right) \\ {\tilde{A}}= & {} \left( {\begin{array}{*{20}{c}} {{{\left( {{{{\tilde{A}}}^{\left( 1 \right) }}} \right) }^T}}\\ {{{\left( {{{{\tilde{A}}}^{\left( 2 \right) }}} \right) }^T}}\\ {\begin{array}{*{20}{c}} {{{\left( {{{{\tilde{A}}}^{\left( 3 \right) }}} \right) }^T}}\\ {{{\left( {{{{\tilde{A}}}^{\left( 4 \right) }}} \right) }^T}} \end{array}}\\ {{{\left( {{{{\tilde{A}}}^{\left( 5 \right) }}} \right) }^T}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_0},t} \right) ,0.33} \right) }&{}{\left( {\left( {{h_0},f} \right) ,0.67} \right) }&{}{\left( {\left( {{h_0},t} \right) ,0.17} \right) }\\ {\left( {\left( {{h_0},t} \right) ,0.33} \right) }&{}{\left( {\left( {{h_0},f} \right) ,0.67} \right) }&{}{\left( {\left( {{h_0},t} \right) ,0.17} \right) }\\ {\left( {\left( {{h_1},t} \right) ,0} \right) }&{} - &{}{\left( {\left( {{h_1},f} \right) ,0} \right) }\\ {\left( {\left( {{h_1},t} \right) ,0.67} \right) }&{}{\left( {\left( {{h_0},f} \right) ,0.67} \right) }&{}{\left( {\left( {{h_1},f} \right) ,0} \right) }\\ {\left( {\left( {{h_1},t} \right) ,0.33} \right) }&{}{\left( {\left( {{h_0},f} \right) ,0.17} \right) }&{}{\left( {\left( {{h_{\mathrm{1}}},f} \right) ,0} \right) } \end{array}} \right) \\ R= & {} {\left( {{\tilde{C}}} \right) ^T} \rightarrow {\tilde{A}} = \left( {\begin{array}{*{20}{c}} {\left( {\left( {{h_2},t}\right) ,0.28}\right) }&{\left( {\left( {{h_1},t}\right) ,0.942}\right) }&{\left( {\left( {{h_2},t}\right) ,0.868}\right) } \end{array}}\right) \end{aligned}$$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Diao, H., Deng, A., Cui, H. et al. An approach for solving fuzzy multi-criteria decision problem under linguistic information. Fuzzy Optim Decis Making 21, 45–69 (2022). https://doi.org/10.1007/s10700-021-09356-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10700-021-09356-x

Keywords

  • 2-tuple linguistic representation model
  • Lattice implication algebra
  • Linguistic information processing
  • Multi-criteria decision problem
  • Preference relation