Abstract
Comparing probabilistic linguistic term sets (PLTSs) is quite essential in solving PLTS-expressed multi-attribute group decision-making problems (PLTS-MAGDM). Researchers have designed various comparison measures to obtain the rank of PLTSs. However, most of the existing PLTS comparison measures need additional tedious adjustments before conducting a specific computation. Besides, these measures do not adequately consider the effects of the semantics of the basic linguistic term set and the probabilistic distributions. This paper proposes a new preference degree for g-granularity probabilistic term sets (g-GPLTSs) to overcome the two shortcomings simultaneously by integrating the effect from basic linguistic terms and probabilistic distributions without any adjustment. Moreover, the g-GPLTS preference degree also shows the extended adaptability for comparing PLTSs with unbalanced semantics. Based on the newly proposed preference degree, we construct a useful min-conflict model to solve PLTS-MAGDM with a large number of experts expressing the three-way primary grading. Finally, an illustrative example concerning software supplier selections, followed by the comparative analysis, is presented to verify the feasibility and effectiveness of the proposed method.
Similar content being viewed by others
References
Li, C.C., Dong, Y.C., Herrera, F.: 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 Trans. Fuzzy Syst. 27, 221–233 (2019)
Sun, B.Z., Ma, W.M., Zhao, H.Y.: An approach to emergency decision-making based on decision-theoretic rough set over two universes. Soft Comput. 20, 3617–3628 (2016)
Alonso, S., Pérez, I. J., Cabrerizo, F. J., Herrera-Viedma, E.: A fuzzy group decision making model for large groups of individuals. In: Proceeding of IEEE international conference on fuzzy systems, pp. 643–648 Jeju Island, SouthKorea (2009)
Wu, Z.B., Xu, J.P.: A consensus model for large-scale group decision making with hesitant fuzzy information and changeable clusters. Inf. Fusion. 41, 217–231 (2018)
Liu, Y., Fan, Z.P., Zhang, X.: A method for large group decision-making based on evaluation information provided by participators from multiple groups. Inf. Fusion. 29, 132–141 (2016)
Bonissone, P. P., Decker, K. S.: Selecting uncertainty calculi and granularity: An experiment in trading off precision and complexity. In: Kanal, L. H., Lemmer, J. F. (eds) Proc. Uncertainty Artif. Intell, pp. 217–247. North-Holland, Amsterdam (1986)
Degani, R., Bortolan, G.: The problem of linguistic approximation in clinical decision making. Int. J. Approx. Reas. 2, 143–162 (1988)
Herrera, F., Martínez, L.: A 2-tuple linguistic representational model for computing with words. IEEE T. Fuzzy Syst. 8, 746–752 (2000)
Xu, Z.S.: Deviation measures of linguistic preference relations in group decision making. Omega 33, 249–254 (2005)
Zhang, C., Li, D.Y., Liang, J.Y.: Interval-valued hesitant fuzzy multi-granularity three-way decisions in consensus processes with applications to multi-attribute group decision making. Inf. Sci. 511, 192–211 (2020)
Dong, Y.C., Hong, W.C., Xu, Y.F., Yu, S.: Selecting the individual numerical scale and prioritization method in the analytic hierarchy process: a 2-tuple fuzzy linguistic approach. IEEE Trans. Fuzzy Syst. 19, 13–25 (2011)
Wang, B.L., Liang, J.Y., Pang, J.F.: Deviation degree: a perspective on score functions in hesitant fuzzy sets. Int. J. Fuzzy syst. 21, 2299–2316 (2019)
Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8, 199–49 (1975)
Wang, J.H., Hao, J.: A new version of 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 14, 435–445 (2006)
Rodríguez, R.M., Martínez, L., Herrera, F.: Hesitant fuzzy linguistic terms sets for decision making. IEEE Trans. Fuzzy Syst. 20, 109–119 (2012)
Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy systems, Jeju Island, Kerea, pp. 1378–1382 (2009)
Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25, 529–539 (2010)
Liang, D.C., Xu, Z.S.: The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets. Appl. Soft Comput. 60, 167–179 (2017)
Zhang, C., Li, D.Y., Liang, J.Y.: Multi-granularity three-way decisions with adjustable hesitant fuzzy linguistic multigranulation decision-theoretic rough sets over two universes. Inf. Sci. 507, 665–683 (2020)
Zhang, C., Li, D.Y., Liang, J.Y.: Hesitant fuzzy linguistic rough set over two universes model and its applications. Int. J. Machin. Learn. Cybern. 9, 577–588 (2018)
Pang, Q., Wang, H., Xu, Z.S.: Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci. 369, 128–143 (2016)
Wu, X.L., Liao, H.C., Xu, Z.S., Hafezalkotob, A., Herrera, F.: Probabilistic linguistic MULTIMOORA: a multicriteria decision making method based on the probabilistic linguistic expectation function and the Improved Borda Rule. IEEE Trans. Fuzzy Syst. 20, 3688–3702 (2018)
Wu, X.L., Liao, H.C.: A consensus-based probabilistic linguisitic gained and lost dominance score method. Eur. J. Oper. Res. 272, 1017–1027 (2019)
Zhang, Y.X., Xu, Z.S., Wang, H., Liao, H.C.: Consistency-based risk assessment with probabilistic linguistic preference relation. Appl. Soft. Compt. 49, 817–833 (2016)
Liu, P.D., Teng, F.: Some Muirhead mean operators for probabilistic linguistic term sets and their applications to multiple attribute decision-making. Appl. Soft Compt. 68, 396–431 (2018)
Liang, D.C., Kobina, A., quan, W.: Grey relation analysis method for probabilistic linguistic multi-criteria group decision-making based on geometric Bonferroni mean. Int. J. Fuzzy Syst 20, 2234–2244 (2018)
Gao, J., Xu, Z.S., Liang, Z.L., Liao, H.C.: Expected consistency-based emergency decision making with incomplete probabilistic linguisitic preference relations. Knowl. Based Syst. 176, 15–28 (2019)
Zhang, Y.X., Xu, Z.S., Liao, H.C.: A consensus process for group decision making with probabilistic linguistic preference relations. Inf. Sci. 414, 260–275 (2017)
Bai, C.Z., Zhang, R., Qian, L.X., Wu, Y.N.: Comparisons of probabilistic linguistic term sets for multi-criteria decision making. Knowl. Based Syst. 119, 284–291 (2017)
Xian, S.D., Chai, J.H., Yin, Y.B.: A visual comparison method and similarity measure for probabilistic linguistic term sets and their applications in multi-criteria decision making. Int. J. Fuzzy Syst. 21, 1154–1169 (2019)
Mao, X.B., Wu, M., Dong, J.Y., Wan, S.P., Jin, Z.: A new method for probabilistic linguisitic multi-attribute group decision making: Application to the selection of financial technologies. Appl. Soft Comput. 77, 155–175 (2019)
Herrera, F., Herrera-Viedma, E., Martínez, L.: A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst. 114, 13–58 (2000)
Herrera-Viedma, E., Cordón, O., Luque, M., López, A.G., Munoz, A.M.: A model of fuzzy linguistic IRS based on multi-granular linguistic information. Int. J. Approx. Reason. 34, 221–239 (2003)
Li, C.C., Rodríguez, R.M., Martínez, L., Dong, Y.C., Herrera, F.: Personalized individual semantics based on consistency in hesitant linguistic group decision making with comparative linguistic expressions. Knowl. Based Syst. 145, 156–165 (2018)
Wang, B.L., Liang, J.Y., Qian, Y.H., Dang, C.Y.: A normalized numerical scaling method for the unbalanced multi-granular linguistic sets. Int. J. Uncertain. Fuzz. 23, 221–243 (2015)
Ma, J., Fan, Z.P., Huang, L.H.: A subjective and objective integrated approach to determine attribute weights. Eur. J. Oper. Res. 112, 397–404 (1999)
Wang, Y.M., Luo, Y.: Integration of correlations with standard deviations for determining attribute weights in multiple attribute decision making. Math. Comput. Model. 51, 1–12 (2010)
Zavadskas, E.K., Podvezko, V.: Integrated determination of objective criteria weights in MCDM. Int. J. Inf. Tech. Decis. 15, 267–283 (2016)
Herrera, F., Herrera-Viedma, E., Martínez, L.: A fuzzy linguistic methodology to deal with unbalanced linguistic term sets. IEEE Trans. Fuzzy Syst. 16, 354–370 (2019)
Dong, Y.C., Xu, Y.F., Yu, S.: Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model. IEEE Trans. Fuzzy Syst. 17, 1366–1378 (2009)
Xu, Z.S.: A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf. Sci. 166, 19–30 (2004)
Yager, R.R.: Quantifier guided aggregation using OWA operators. Int. J. Intell. Syst. 11, 49–73 (1996)
Xu, Z.S.: Study on the prioritizing method for fuzzy complementary judgement matrices. J. Syst. Eng. Electron. 24, 74–75 (2002)
Yao, Y.Y.: Three-way decisions and cognitive computing. Cogn. Comput. 8, 543–554 (2016)
Yao, Y.Y.: The superiority of three-way decisions in probabilistic rough set models. Inf. Sci. 181, 1080–1096 (2011)
Acknowledgements
We would like to thank the reviewers for their insightful suggestions. The authors also thank assistant professor Yarong Hu for her help in completing the theoretical proof. This work was supported by the National Natural Science Foundation of China (Nos. 61703363, 61876103), the Applied Basic Research Program of Shanxi Province (Nos. 201901D211462, 201801D121148), the Key R&D program of Shanxi Province (International Cooperation, 201903D421041), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province (No. 2019L0864), and the Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (No. CICIP2018008).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
None.
Appendices
Appendix A
Proof of the transitivity of the GLTS preference degree
Let \(p^{1}=(p_1^1,p_2^1,\ldots ,p_n^1),\)\(p^2=(p_1^2,p_2^2,\ldots ,p_n^2)\) and \(p^3=(p_1^3,p_2^3,\ldots ,p_n^3)\) be three probability distributions. n discrete numbers \(s_i\ (i=1,2,\ldots ,n)\) satisfy that \(0\le s_1<s_2<\ldots <s_n\le 1.\) Let
be the joint distribution of \(p^i\) and \(p^j.\) Let
Define
If \(\parallel P^{12}\otimes S\parallel \ge 0\) and \(\parallel P^{23}\otimes S\parallel \ge 0,\) then \(\parallel P^{13}\otimes S\parallel \ge 0.\)
Proof
Compute that
Similarly, we have
and
Since \(\parallel P^{12}\otimes S\parallel \ge 0\) and \(\parallel P^{23}\otimes S\parallel \ge 0,\) we can easily obtain that
Therefore, we have \(\parallel P^{13}\otimes S\parallel \ge 0.\)\(\square \)
Appendix B
Definition of Hadamard product
Let \(A=\big (a_{ij}\big )_{m\times n}\) and \(B=\big (b_{ij}\big )_{m\times n}\) be two matrices.
is called the Hadamard Product of A and B.
Rights and permissions
About this article
Cite this article
Wang, B., Liang, J. A Novel Preference Measure for Multi-Granularity Probabilistic Linguistic Term Sets and its Applications in Large-Scale Group Decision-Making. Int. J. Fuzzy Syst. 22, 2350–2368 (2020). https://doi.org/10.1007/s40815-020-00887-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40815-020-00887-w