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Multi-Criteria Decision-Making Method Based on Distance Measure and Choquet Integral for Linguistic Z-Numbers

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Abstract

Z-numbers are a new concept considering both the description of cognitive information and the reliability of information. Linguistic terms are useful tools to adequately and effectively model real-life cognitive information, as well as to characterize the randomness of events. However, a form of Z-numbers, in which their two components are in the form of linguistic terms, is rarely studied, although it is common in decision-making problems. In terms of Z-numbers and linguistic term sets, we provided the definition of linguistic Z-numbers as a form of Z-numbers or a subclass of Z-numbers. Then, we defined some operations of linguistic Z-numbers and proposed a comparison method based on the score and accuracy functions of linguistic Z-numbers. We also presented the distance measure of linguistic Z-numbers. Next, we developed an extended TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method based on the Choquet integral for multi-criteria decision-making (MCDM) problems with linguistic Z-numbers. Finally, we provided an example concerning the selection of medical inquiry applications to demonstrate the feasibility of our proposed approach. We then verified the applicability and superiority of our approach through comparative analyses with other existing methods. Illustrative and comparative analyses indicated that the proposed approach was valid and feasible for different decision-makers and cognitive environments. Furthermore, the final ranking results of the proposed approach were closer to real decision-making processes. Linguistic Z-numbers can flexibly characterize real cognitive information as well as describe the reliability of information. This method not only is a more comprehensive reflection of the decision-makers’ cognition but also is more in line with expression habits. The proposed method inherited the merits of the classical TODIM method and considers the interactivity of criteria; therefore, the proposed method was effective for dealing with real-life MCDM problems. Consideration about bounded rational and the interactivity of criteria made final outcomes convincing and consistent with real decision-making.

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References

  1. Liu Y, Zhang L, Deng PL, He Z. Common subspace learning via cross-domain extreme learning machine. Cogn Comput. 2017; Doi:https://doi.org/10.1007/s12559-017-9473-5.

  2. Thanh ND, Ali M. A novel clustering algorithm in a Neutrosophic recommender system for medical diagnosis. Cogn Comput. 2017; Doi:https://doi.org/10.1007/s12559-017-9462-8.

  3. Wootton AJ, Taylor SL, Day CR, Haycock PW. Optimizing echo state networks for static pattern recognition. Cogn Comput. 2017;9:391-399. 

  4. Yao Y. Three-way decisions and cognitive computing. Cogn Comput. 2016;8:543–54.

    Article  Google Scholar 

  5. Oliva J, Serrano JI, Castillo MDD, Ángel I. Cross-linguistic cognitive modeling of verbal morphology acquisition. Cogn Comput. 2017;9:237-258.

  6. Peng HY, Cambria E, Hussain A. A review of sentiment analysis research in Chinese language. Cogn Comput. 2017; Doi:https://doi.org/10.1007/s12559-017-9470-8.

  7. Guo T, Zhang L, Tan XH. Neuron pruning-based discriminative extreme learning machine for pattern classification. Cogn Comput. 2017; Doi:https://doi.org/10.1007/s12559-017-9474-4.

  8. Messaoud MAB, Bouzid A, Ellouze N. A new biologically inspired fuzzy expert system-based voiced/unvoiced decision algorithm for speech enhancement. Cogn Comput. 2016;8:478–93.

    Article  Google Scholar 

  9. Tian ZP, Wang J, Wang JQ, Zhang HY. Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development. Group Decis Negot. 2017;26:597–627.

    Article  Google Scholar 

  10. Zhang H, Ji P, Wang J, et al. A novel decision support model for satisfactory restaurants utilizing social information: a case study of TripAdvisor. Com. Tour Manag 2017; 59: 281–297.

  11. Zadeh LA. Fuzzy sets. Inf Control. 1965;8:338–53.

    Article  Google Scholar 

  12. Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986;20:87–96.

    Article  Google Scholar 

  13. Torra V. Hesitant fuzzy sets. Int J Intell Syst. 2010;25:529–39.

    Google Scholar 

  14. Liu P, Tang G. Multi-criteria group decision-making based on interval neutrosophic uncertain linguistic variables and Choquet integral. Cogn Comput. 2016;8:1036–56.

    Article  Google Scholar 

  15. Meng F, Wang C, Chen X. Linguistic interval hesitant fuzzy sets and their application in decision making. Cogn Comput. 2016;8:52–68.

    Article  Google Scholar 

  16. Wang JQ, Kuang JJ, Wang J, Zhang H. An extended outranking approach to rough stochastic multi-criteria decision-making problems. Cogn Comput. 2016;8:1144–60.

    Article  Google Scholar 

  17. 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;248:183–91.

    Article  Google Scholar 

  18. Zadeh LA. A note on Z-numbers. Inf Sci. 2011;181:2923–32.

    Article  Google Scholar 

  19. Aliev RA, Alizadeh AV, Huseynov OH. The arithmetic of discrete Z-numbers. Inf Sci. 2015;290:134–55.

    Article  Google Scholar 

  20. Bhanu MS, Velammal G. Operations on Zadeh’s Z-number. IOSR Journal of Mathematics. 2015;11:88–94.

    Google Scholar 

  21. Kang B, Wei D, Li Y. A method of converting Z-number to classical fuzzy number. J Inf Comput Sci. 2012;9:703–9.

    Google Scholar 

  22. Akhbari M, Sadi-Nezhad S. Equilibrium solution of non-cooperative bimatrix game of Z-numbers. Bull Geor Natl Acad Sci. 2015;9:33–47.

    Google Scholar 

  23. Azadeh A, Saberi M, Atashbar NZ. Z-AHP: a Z-number extension of fuzzy analytical hierarchy process. Digital Ecosystems and Technologies (DEST), 2013 7th IEEE International Conference on. IEEE. 2013; 141–147.

  24. Banerjee R, Pal SK. Z*-numbers: augmented Z-numbers for machine-subjectivity representation. Inf Sci. 2015;323:143–78.

    Article  Google Scholar 

  25. Peng H, Wang J. Hesitant uncertain linguistic Z-numbers and their application in multi-criteria group decision-making problems. Int J Fuzzy Syst. 2016; Doi:https://doi.org/10.1007/s40815-016-0257-y.

  26. Velammal G, Bhanu MS. Intuitionistic Z-Numbers. American International Journal of Research in Science, Technology, Engineering & Mathematics. 2015.

  27. Yager RR. On Z-valuations using Zadeh’s Z-numbers. Int J Intell Syst. 2012;27:259–78.

    Article  Google Scholar 

  28. Gardashova LA. Application of operational approaches to solving decision making problem using Z-numbers. Appl Math. 2014;5:1323–34.

    Article  Google Scholar 

  29. Kang B, Wei D, Li Y. Decision making using Z-numbers under uncertain environment. J Comput Inf Syst. 2012;8:2807–14.

    Google Scholar 

  30. Kang B, Hu Y, Deng Y. A new methodology of multi-criteria decision-making in supplier selection based on Z-numbers. Mathematical Problems in Engineering. 2016.

  31. Yaakob AM, Gegov A. Interactive TOPSIS based group decision making methodology using Z-numbers. Int J Comput Intell Syst. 2016;9:311–24.

    Article  Google Scholar 

  32. Zeinalova LM. Choquet aggregation decision making under Z-information. ICTACT J Soft Comput. 2014;4:819–24.

    Article  Google Scholar 

  33. Banerjee R, Pal SK. On Z-numbers and the machine-mind for natural language comprehension. Fifty Years of Fuzzy Logic and its Applications. Springer International Publishing. 2015; 415–457.

  34. Banerjee R, Pal SK. The Z-number enigma: a study through an experiment. Soft computing: state of the art theory and novel applications. Springer Berlin Heidelberg. 2013; 71–88.

  35. Pal SK, Banerjee R, Dutta S. An insight into the Z-number approach to CWW. Fundamental Information. 2013;124:197–229.

    Google Scholar 

  36. Patel P, Khorasani ES, Rahimi S. Modeling and implementation of Z-number. Soft Comput. 2015:1–24.

  37. Li D, Liu C, Gan W. A new cognitive model: cloud model. Int J Intell Syst. 2009;24:357–75.

    Article  CAS  Google Scholar 

  38. Wang JQ, Lu P, Zhang HY. Method of multi-criteria group decision-making based on cloud aggregation operators with linguistic information. Inf Sci. 2014;274:177–91.

    Article  Google Scholar 

  39. Wang JQ, Peng JJ, Zhang HY, Liu T, Chen XT. An uncertain linguistic multi-criteria group decision-making method based on a cloud model. Group Decis Negot. 2015;24:171–92.

    Article  Google Scholar 

  40. Sugeno M Theory of fuzzy integral and its application. Doctorial Dissertation, Tokyo Institute of Technology. 1974.

  41. Liu P. Some generalized dependent aggregation operators with intuitionistic linguistic numbers and their application to group decision making. J Comput Syst Sci. 2013;79:131–43.

    Article  Google Scholar 

  42. Gomes LFAM, Lima MMPP. TODIM: basics and application to multi-criteria ranking of projects with environmental impacts. Found Comput Decis Sci. 1992;16:113–27.

    Google Scholar 

  43. Gomes LFAM, Lima MMPP. From modeling individual preferences to multi-criteria ranking of discrete alternatives: a look at prospect theory and the additive difference model. Found Comput Decis Sci. 1992;17:171–84.

    Google Scholar 

  44. Kahneman D, Tversky A. Prospect theory: an analysis of decision under risk. Econometrica. 1979;47:263–92.

    Article  Google Scholar 

  45. Zhou H, Wang JQ, Zhang HY. Grey stochastic multi-criteria decision-making approach based on prospect theory and distance measures. J Grey Syst. 2017;29:15–33.

    Google Scholar 

  46. Tversky A, Kahneman D. Advances in prospect theory: cumulative representation of uncertainty. J Risk Uncertain. 1992;5:297–323.

    Article  Google Scholar 

  47. Ji P, Zhang HY, Wang JQ. A projection-based TODIM method under multi-valued neutrosophic environments and its application in personnel selection. Neural Comput & Applic. 2016; Doi:https://doi.org/10.1007/s00521-016-2436-z.

  48. Wang J, Wang J, Zhang H. A likelihood-based TODIM approach based on multi-hesitant fuzzy linguistic information for evaluation in logistics outsourcing. Comput Ind Eng. 2016;99:287–99.

    Article  Google Scholar 

  49. Zhou H, Wang JQ, Zhang HY. Multi-criteria decision-making approaches based on distance measures for linguistic hesitant fuzzy sets. J Oper Res Soc. 2016; Doi:https://doi.org/10.1057/jors.2016.41.

  50. Tan CQ, Jiang ZZ, Chen ZH. An extended TODIM method for hesitant fuzzy interactive multicriteria decision making based on generalized Choquet integral. J Intell Fuzzy Syst. 2015;29:293–305.

    Article  Google Scholar 

  51. Delgado M, Verdegay JL, Vila MA. Linguistic decision making models. Int J Intell Syst. 1992;7:479–92.

    Article  Google Scholar 

  52. Xu ZS. A note on linguistic hybrid arithmetic averaging operator in multiple attribute decision-making with linguistic information. Group Decis Negot. 2006;15:593–604.

    Article  Google Scholar 

  53. Xu ZS. Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Inf Sci. 2014;168:171–84.

    Article  Google Scholar 

  54. Wang JQ, Wu JT, Wang J. Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems. Inf Sci. 2014;288:55–72.

    Article  Google Scholar 

  55. Bao GY, Lian XL, Ming HE. Improved two-tuple linguistic representation model based on new linguistic evaluation scale. Control and Decision. 2010;25:780–4.

    Google Scholar 

  56. Simon HA. Administrative behavior-a study of decision making processes in administrative organization. New York: Macmillan Publishing Co, lnc; 1971.

    Google Scholar 

  57. Grabisch M, Murofushi T, Sugeno M. Fuzzy measure and integrals. New York: Physica-Verlag; 2000.

    Google Scholar 

  58. Pawlak Z. Rough sets and fuzzy sets. Fuzzy Sets Syst. 1985;17:99–102.

    Article  Google Scholar 

  59. Pawlak Z, Skowron A. Rough membership function: a tool for reasoning with uncertainty. Algebraic Meth Logic Comput Sci. 1993;28:135–50.

    Google Scholar 

  60. Liu P, Jin F. Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making. Inf Sci. 2012;205:58–71.

    Article  Google Scholar 

  61. Nie R, Wang J, Li L. 2-tuple linguistic intuitionistic preference relation and its application in sustainable location planning voting system. J Intell Fuzzy Syst. 2017;33:885-899.

  62. Yu SM, Wang J, Wang JQ. An extended TODIM approach with intuitionistic linguistic numbers. Int Trans Oper Res. 2016; Doi:https://doi.org/10.1111/itor.12363.

  63. Holt CA, Laury SK. Risk aversion and incentive effects. Am Econ Rev. 2002;92:1644–55.

    Article  Google Scholar 

  64. Aungst TD, Clauson KA, Misra S, et al. How to identify, assess and utilise mobile medical applications in clinical practice. Int J Clin Pract. 2014;68:155–62.

    Article  CAS  PubMed  Google Scholar 

  65. Gomes LFAM. An application of the TODIM method to the multicriteria rental evaluation of residential properties. Eur J Oper Res. 2009;193:204–11.

    Article  Google Scholar 

  66. Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput. 2002;6:182–97.

    Article  Google Scholar 

  67. Paelinck JHP. Qualitative multiple criteria analysis, environmental protection and multiregional development. Pap Reg Sci. 1976;36:59–76.

    Article  Google Scholar 

  68. Tian ZP, Wang J, Wang JQ, et al. A likelihood-based qualitative flexible approach with hesitant fuzzy linguistic information. Cogn Comput. 2016;8:670–83.

    Article  CAS  Google Scholar 

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Acknowledgments

The author would like to thank the editors and the anonymous referees for their valuable and constructive comments and suggestions that greatly help the improvement of this paper. This work was supported by the National Natural Science Foundation of China (No. 71571193).

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  1. School of Business, Central South University, Changsha, 410083, China

    Jian-qiang Wang, Yong-xi Cao & Hong-yu Zhang

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  1. Jian-qiang Wang
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Wang, Jq., Cao, Yx. & Zhang, Hy. Multi-Criteria Decision-Making Method Based on Distance Measure and Choquet Integral for Linguistic Z-Numbers. Cogn Comput 9, 827–842 (2017). https://doi.org/10.1007/s12559-017-9493-1

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