Abstract
In multiattribute decision-making (MADM), more and more attention is paid to the interaction between attributes when considering the actual decision environment. As a result, interactive MADM has become an emerging and challenging area of research whose success will greatly facilitate the development of decision-making. This paper models the interactive MADM problem and its contribution is multifaceted. First, the concept of probabilistic linguistic term set (PLTS) is extended by Dempster–Shafer theory (DST), which helps to express more uncertain information, followed by some basic operations, such as score function and aggregation operator. In virtue of evidential best-worst method and the principle of maximum entropy, then a novel nonadditive measure determination method is developed based on the 2-order additive measure to better model the interaction between attributes. Further, the generalized PLTS-based Choquet integral is defined by which generalized PLTSs on a nonadditive measure can be reasonably aggregated. Finally, an interactive MADM model is constructed and the technical details are described. The proposed approach is implemented to select the supplier for medical devices, and its effectiveness is emphasized by comparison with other methods.
Similar content being viewed by others
References
Liao, H., Wu, X.: DNMA: a double normalization-based multiple aggregation method for multi-expert multi-criteria decision making. Omega 94, 102058 (2020)
Tian, Y., Liu, L., Mi, X., Kang, B.: Zslf: a new soft likelihood function based on z-numbers and its application in expert decision system. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2997328
Fei, L., Xia, J., Feng, Y., Liu, L.: An electre-based multiple criteria decision making method for supplier selection using dempster-shafer theory. IEEE Access 7, 84701–84716 (2019)
Xiao, F.: GIQ: A generalized intelligent quality-based approach for fusing multi-source information. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2991296
Xiao, F.: Evidence combination based on prospect theory for multi-sensor data fusion. ISA Trans. (2020). https://doi.org/10.1016/j.isatra.2020.06.024
Xiao, F., Cao, Z., Jolfaei, A.: A novel conflict measurement in decision making and its application in fault diagnosis. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.3002431
Xiao, F.: A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems. IEEE Trans. Syst Man Cyber. (2019). https://doi.org/10.1109/TSMC.2019.2958635
Fei, L., Feng, Y., Liu, L.: On pythagorean fuzzy decision making using soft likelihood functions. Int. J. Intell. Syst. 34(12), 3317–3335 (2019)
Liao, H., Gou, X., Xu, Z., Zeng, X.-J., Herrera, F.: Hesitancy degree-based correlation measures for hesitant fuzzy linguistic term sets and their applications in multiple criteria decision making. Inf. Sci. 508, 275–292 (2020)
Xingli, W., Liao, H.: Utility-based hybrid fuzzy axiomatic design and its application in supply chain finance decision making with credit risk assessments. Comput. Ind. 114, 103144 (2020)
Xiao, F.: A new divergence measure for belief functions in D-S evidence theory for multisensor data fusion. Inf. Sci. 514, 462–483 (2020)
Xiao, F.: EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy. IEEE Trans. Fuzzy Syst. 28(7), 1477–1491 (2020)
Wang, Y.-M., Yang, J.-B., Xu, D.-L.: Environmental impact assessment using the evidential reasoning approach. Eur. J. Oper. Res. 174(3), 1885–1913 (2006)
Zhou, M., Liu, X.-B., Yang, J.-B., Chen, Y.-W., Wu, J.: Evidential reasoning approach with multiple kinds of attributes and entropy-based weight assignment. Knowl.-Based Syst. 163, 358–375 (2019)
Chen, Y., Chen, Y.-W., Xu, X.-B., Pan, C.-C., Yang, J.-B., Yang, G.-K.: A data-driven approximate causal inference model using the evidential reasoning rule. Knowl.-Based Syst. 88, 264–272 (2015)
Chen, S.-M., Cheng, S.-H., Chiou, C.-H.: Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology. Inf. Fusion 27, 215–227 (2016)
Pang, Q., Wang, H., Xu, Z.: Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci. 369, 128–143 (2016)
Yang, J.-B., Xu, D.-L.: Evidential reasoning rule for evidence combination. Artif. Intell. 205, 1–29 (2013)
Yang, J.-B., Xu, D.-L.: On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty. IEEE Trans. Syst. Man. Cyber. A 32(3), 289–304 (2002)
Chang, L., Zhou, Y., Jiang, J., Li, M., Zhang, X.: Structure learning for belief rule base expert system: a comparative study. Knowl.-Based Syst. 39, 159–172 (2013)
Fu, C., Yang, J.-B., Yang, S.-L.: A group evidential reasoning approach based on expert reliability. Eur. J. Oper. Res. 246(3), 886–893 (2015)
Yang, Jian-Bo, Liu, Jun, Wang, Jin, Sii, How-Sing, Wang, Hong-Wei: Belief rule-base inference methodology using the evidential reasoning approach-rimer. IEEE Trans. Syst. Man. Cyber. Part A 36(2), 266–285 (2006)
Zhou, Z.-J., Hu, C.-H., Xu, D.-L., Chen, M.-Y., Zhou, D.-H.: A model for real-time failure prognosis based on hidden Markov model and belief rule base. Eur. J. Oper. Res. 207(1), 269–283 (2010)
Liao, H., Peng, X., Gou, X.: Medical supplier selection with a group decision-making method based on incomplete probabilistic linguistic preference relations. Int. J. Fuzzy Syst. (2020). https://doi.org/10.1007/s40815-020-00885-y
Li, B., Zhang, Y., Xu, Z.: The medical treatment service matching based on the probabilistic linguistic term sets with unknown attribute weights. Int. J. Fuzzy Syst. (2020). https://doi.org/10.1007/s40815-020-00844-7
Liu, P., Li, Y., Teng, F.: Bidirectional projection method for probabilistic linguistic multi-criteria group decision-making based on power average operator. Int. J. Fuzzy Syst. 21(8), 2340–2353 (2019)
Lei, F., Wei, G., Gao, H., Wu, J., Wei, C.: Topsis method for developing supplier selection with probabilistic linguistic information. Int. J. Fuzzy Syst. 1–11 (2020)
Zhang, X., Gou, X., Xu, Z., Liao, H.: A projection method for multiple attribute group decision making with probabilistic linguistic term sets. Int. J. Mach. Learn. Cybern. 10(9), 2515–2528 (2019)
Wang, X., Xu, Z., Gou, X., Trajkovic, L.: Tracking a maneuvering target by multiple sensors using extended kalman filter with nested probabilistic-numerical linguistic information. IEEE Trans. Fuzzy Syst. 28(2), 346–360 (2020)
Li, Y., Zhang, Y., Xu, Z.: A decision-making model under probabilistic linguistic circumstances with unknown criteria weights for online customer reviews. Int. J. Fuzzy Syst. 1–13 (2020)
Qin, M., Tang, Y., Wen, J.: An improved total uncertainty measure in the evidence theory and its application in decision making. Entropy 22(4), 487 (2020)
Xue, Y., Deng, Y.: Entailment for intuitionistic fuzzy sets based on generalized belief structures. Int. J. Intell. Syst. 35(6), 963–982 (2020)
Fei, L., Feng, Y., Liu, L.: Evidence combination using owa-based soft likelihood functions. Int. J. Intell. Syst. 34(9), 2269–2290 (2019)
Mao, S., Han, Y., Deng, Y., Pelusi, D.: A hybrid DEMATEL-FRACTAL method of handling dependent evidences. Eng. Appl. Artif. Intell. 91, 103543 (2020)
Xiao, F.: Generalization of Dempster–Shafer theory: A complex mass function. Appl. Intell. (2019). https://doi.org/10.1007/s10489-019-01617-y
Liu, P., Wang, P.: Multiple attribute group decision making method based on intuitionistic fuzzy einstein interactive operations. Int. J. Fuzzy Syst. (2020) https://doi.org/10.1007/s40815-020-00809-w
Fei, L., Feng, Y.: A novel retrieval strategy for case-based reasoning based on attitudinal choquet integral. Eng. Appl. Artif. Intell. 94, 103791 (2020)
Zhang, D., Li, Y., Wu, C.: An extended todim method to rank products with online reviews under intuitionistic fuzzy environment. J. Oper. Res. Soc. 71(2), 322–334 (2020)
Aggarwal, M., Tehrani, A.F.: Modelling human decision behaviour with preference learning. Inf. J. Comput. 31(2), 318–334 (2019)
Beliakov, G.: Construction of aggregation functions from data using linear programming. Fuzzy Sets Syst. 160(1), 65–75 (2009)
Grabisch, M., Kojadinovic, I., Meyer, P.: A review of methods for capacity identification in choquet integral based multi-attribute utility theory: applications of the kappalab r package. Eur. J. Oper. Res. 186(2), 766–785 (2008)
Marichal, J.-L., Roubens, M.: Determination of weights of interacting criteria from a reference set. Eur. J. Oper. Res. 124(3), 641–650 (2000)
Murillo, J., Guillaume, S., Bulacio, P.: k-maxitive fuzzy measures: a scalable approach to model interactions. Fuzzy Sets Syst. 324, 33–48 (2017)
Tehrani, A.F., Cheng, W., Dembczyński, K., Hüllermeier, E.: Learning monotone nonlinear models using the choquet integral. Mach. Learn. 89(1–2), 183–211 (2012)
Wu, J.-Z., Pap, E., Szakal, A.: Two kinds of explicit preference information oriented capacity identification methods in the context of multicriteria decision analysis. Int. Trans. Oper. Res. 25(3), 807–830 (2018)
Li, X., Zhang, X.: Sugeno integral of set-valued functions with respect to multi-submeasures and its application in madm. Int. J. Fuzzy Syst. 20(8), 2534–2544 (2018)
Liao, Z., Liao, H., Tang, M., Al-Barakati, A., Llopis-Albert, C.: A Choquet integral-based hesitant fuzzy gained and lost dominance score method for multi-criteria group decision making considering the risk preferences of experts: Case study of higher business education evaluation. Inf. Fusion 62, 121–133 (2020)
Liu, P., Zhang, X.: A multicriteria decision-making approach with linguistic d numbers based on the choquet integral. Cogn. Comput. 11(4), 560–575 (2019)
Sirbiladze, G.: Associated probabilities’ aggregations in interactive multiattribute decision making for q-rung orthopair fuzzy discrimination environment. Int. J. Intell. Syst. 35(3), 335–372 (2020)
Fei, L., Feng, Y.: An attitudinal nonlinear integral and applications in decision making (2020) https://doi.org/10.1007/s40815-020-00862-5
Mu, Z., Zeng, S.: Some novel intuitionistic fuzzy information fusion methods in decision making with interaction among attributes. Soft. Comput. 23(20), 10439–10448 (2019)
Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38(2), 325–339 (1967)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Smets, P., Kennes, R.: The transferable belief model. Artif. Intell. 66(2), 191–234 (1994)
Sugeno, M.: Theory of fuzzy integrals and its applications, Doct. Thesis, Tokyo Institute of technology
Choquet, G.: Theory of capacities, In: Annales de l’institut Fourier, Vol. 5, pp. 131–295 (1954)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)
Hwang, C.-L., Yoon, K.: Methods for multiple attribute decision making, In: Multiple attribute decision making, Springer, Berlin, pp. 58–191 (1981)
Wang, Z., Leung, K.sak, Wang, J.: A genetic algorithm for determining nonadditive set functions in information fusion. Fuzzy Sets Syst. 102(3), 463–469 (1999)
Aggarwal, M.: Attitudinal choquet integrals and applications in decision making. Int. J. Intell. Syst. 33(4), 879–898 (2018)
Wu, J.-Z., Zhang, Q.: 2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle. Fuzzy Optim. Decis. Making 9(4), 435–453 (2010)
Wu, J.-Z., Zhang, Q.: 2-order additive fuzzy measures identification method based on maximum entropy principle. Syst. Eng. Electr. 32(11), 2346–2351 (2010)
Wu, J.-Z., Beliakov, G.: k-order representative capacity. J. Intell. Fuzzy Syst. 38(3), 3105–3115 (2020)
Beliakov, G., James, S., Wu, J.-Z.: k-order fuzzy measures and k-order aggregation functions. Discrete Fuzzy Meas. 382, 193–203 (2019)
Wu, J., Zhang, Q.: Nonadditive Measure Theory and Multi-criteria Decision Making. Science Press, Beijing (2013)
Grabisch, M.: K-order additive discrete fuzzy measures and their representation. Fuzzy Sets Syst. 92(2), 167–189 (1997)
Fei, L., Lu, J., Feng, Y.: An extended best-worst multi-criteria decision-making method by belief functions and its applications in hospital service evaluation. Comput. Ind. Eng. 142, 106355 (2020)
Rezaei, J.: Best-worst multi-criteria decision-making method. Omega 53, 49–57 (2015)
Grabisch, M., Sugeno, M., Murofushi, T.: Fuzzy measures and integrals: theory and applications, p. 2010. Physica, Heidelberg (2000)
Rota, G.-C.: On the foundations of combinatorial theory i. theory of möbius functions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 2(4), 340–368 (1964)
Jia, F., Liu, Y., Wang, X.: An extended mabac method for multi-criteria group decision making based on intuitionistic fuzzy rough numbers. Expert Syst. Appl. 127, 241–255 (2019)
Corrente, S., Greco, S., Ishizaka, A.: Combining analytical hierarchy process and choquet integral within non-additive robust ordinal regression. Omega 61, 2–18 (2016)
Yager, R.R., Alajlan, N.: Sugeno integral with possibilistic inputs with application to multi-criteria decision making. Int. J. Intell. Syst. 31(8), 813–826 (2016)
Acknowledgements
This research was funded by the grants from the National Natural Science Foundation of China (#71472053, #91646105).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fei, L., Feng, Y. Modeling Interactive Multiattribute Decision-Making via Probabilistic Linguistic Term Set Extended by Dempster–Shafer Theory. Int. J. Fuzzy Syst. 23, 599–613 (2021). https://doi.org/10.1007/s40815-020-01019-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40815-020-01019-0