Abstract
Conventional trapezoidal intuitionistic fuzzy numbers (CTrIFNs) are used in the literature to handle many real-life problems with imprecise information. However, the CTrIFNs are not the real generalization of interval-valued intuitionistic fuzzy numbers (IVIFNs) and triangular intuitionistic fuzzy numbers (TIFNs). This study discusses the non-conventional trapezoidal intuitionistic fuzzy numbers called trapezoidal-valued intuitionistic fuzzy numbers (TrVIFNs) that generalize (contain) all real-valued intuitionistic fuzzy numbers, IVIFNs and TIFNs. Various researchers worldwide are looking for a complete ranking principle on the set of TrVIFNs. However, none of them yields a complete ranking. The paper’s main aim is to propose a new complete ranking principle on the class of TrVIFNs. The complete ranking principle on the set of TrVIFNs makes the decision-making algorithm more robust. To achieve the main aim, firstly, we introduce eight different score functions on the set of TrVIFNs and study their mathematical properties. Secondly, we present a new ranking principle by considering all eight score functions in a linear order. Further, we prove (mathematically) that the introduced ranking principle defines complete ranking on the set of TrVIFNs. Thirdly, we show the significance of the proposed method compared with the existing methods on various classes of fuzzy and intuitionistic fuzzy numbers. Finally, we implement a case application of the proposed complete ranking principle for evaluating the social sustainability performance of Indian firms.
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References
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Atanassov KT (1994) Operators over interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 64(2):159–174
Atanassov KT, Gargov G (1989) Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349
Chen SM, Tan JM (1994) Handling multi-criteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 67:163–172
Dejian Yu, Yingyu W, Ting L (2012) Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowl Based Syst 30:57–66
Li D-F (2010) A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems. Comput Math Appl 60:1557–1570
Geetha S, Vishnukumar P, EdwinAntonyRaj M (2020) MCDM based on new membership and non-membership accuracy functions on trapezoidal-valued intuitionistic fuzzy numbers. Soft Comput 24:4283–4293
Gong R, Ma LY (2019) A new score function and accuracy function of interval-valued intuitionistic fuzzy number and its application. Syst Eng Theor Pract 39(2):463–475
Hammer J, Pivo G (2017) The triple bottom line and sustainable economic development theory and practice. Econ Dev Q 31(1):25–36
Heinzelmann N (2018) Deontology defended. Synthese 195(12):5197–5216
Hennig M, Hutter M (2020) Revisiting the divide between deontology and utilitarianism in moral dilemma judgment: a multinomial modeling approach. J Pers Soc Psychol 118(1):22
Hong DH, Choi CH (2000) Multi-criteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 114:103–113
Hussain N, Rigoni U, Orij RP (2018) Corporate governance and sustainability performance: Analysis of triple bottom line performance. J Bus Ethics 149(2):411–432
Ishmat B, Rashid T (2014) Multi-criteria trapezoidal valued intuitionistic fuzzy decision making with Choquet integral based TOPSIS. Opsearch 51(1):98–129
Jain R (1977) A procedure for multi-aspect decision making using fuzzy sets. Int J Syst Sci 8:1–7
Ye J (2009) Multi-criteria fuzzy decision making method based on a novel accuracy score function under interval valued intuitionistic fuzzy environment. Expert Syst Appl 36:6899–6902
Jeevaraj S (2021) Similarity measure on interval valued intuitionistic fuzzy numbers based on non-hesitance score and its application to pattern recognition. Comput Appl Math 39(3):1–15
Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall, Hoboken
Lakshmana Gomathi Nayagam V, Sivaraman G (2011) Ranking of interval valued intutionistic fuzzy sets. Appl Soft Comput 11(4):3368–3372
Lakshmana Gomathi Nayagam V, Jeevaraj S, Sivaraman G (2017) Ranking of incomplete trapezoidal information. Soft Comput 21(23):7125–7140
Lakshmana Gomathi Nayagam V, Jeevaraj S, Sivaraman G (2016) Total ordering defined on the set of all intuitionistic fuzzy numbers. J Intell Fuzzy Syst 30(4):2015–2028
Lin LG, Xu LZ, Wang JY (2005) Multi-criteria fusion decision-making method based on vague set. Comput Eng 31:11–13 ((in Chinese))
Lin L, Yuan XH, Xia ZQ (2007) Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets. J Comput Syst Sci 73:84–88
Liu HW (2004) Vague set methods of multi-criteria fuzzy decision making. Syst Eng Theory Pract 24:103–109 ((in Chinese))
Mani V, Gunasekaran A, Delgado C (2018) Enhancing supply chain performance through supplier social sustainability: an emerging economy perspective. Int J Prod Econ 195:259–272
Mihailov E (2016) Is deontology a moral confabulation? Neuroethics 9(1):1–13
Missimer M, Robèrt KH, Broman G (2017) A strategic approach to social sustainability-Part 1: exploring the social system. J Clean Prod 140:32–41
Pope J, Bond A, Huge J, Morrison-Saunders A (2017) Reconceptualising sustainability assessment. Environ Impact Assess Rev 62:205–215
Quintelier K, Van Speybroeck L, Braeckman J (2011) Normative ethics does not need a foundation: it needs more science. Acta Biotheor 59(1):29–51
Rajesh R, Rajeev A, Rajendran C (2022) Corporate social performances of firms in select developed economies: a comparative study. Socioecon Plann Sci 81:101194
Sachin N, Rajesh R (2022) An empirical study of supply chain sustainability with financial performances of Indian firms. Environ Dev Sustain 24(5):6577–6601
Sacco DF, Brown M, Lustgraaf CJ, Hugenberg K (2017) The adaptive utility of deontology: deontological moral decision-making fosters perceptions of trust and likeability. Evol Psychol Sci 3(2):125–132
Sandberg J (2015) Moral economy and normative ethics. J Global Ethics 11(2):176–187
Thomson Reuters ESG Scores (2019) https://www.refinitiv.com/content/dam/marketing/en_us/documents/methodology/esg-scores-methodology.pdf
Wang J, Zhang J, Liu SY (2006) A new score function for fuzzy MCDM based on vague set theory. Int J Comput Cogn 4:44–48
Wang JQ, Zhang Z (2009) Multi-criteria decision making method with incomplete certain information based on intuitionistic fuzzy number. Control Decis 24:226–230 ((in Chinese))
Xu ZS (2007) Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control Decis 22(2):215–219
Xu ZS, Chen J (2007) An approach to group decision making based on intervalvalued intuitionistic fuzzy judgement matrices. Syst Eng Theory Pract 27(4):126–133
Ye J (2010) Using an improved measure function of vague sets for multicriteria fuzzy decision-making. Expert Syst Appl 37:4706–4709
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–356
Zhang H, Yu L (2012) MADM method based on cross-entropy and extended TOPSIS with interval-valued intuitionistic fuzzy sets. Knowl Based Syst 30:115–120
Zhou Z, Wu QZ (2005) Multicriteria fuzzy decision making method based on vague set. Mini-Micro Syst 26:1350–1353 ((in Chinese))
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Jeevaraj, S., Rajesh, R. & Lakshmana Gomathi Nayagam, V. A complete ranking of trapezoidal-valued intuitionistic fuzzy number: an application in evaluating social sustainability. Neural Comput & Applic 35, 5939–5962 (2023). https://doi.org/10.1007/s00521-022-07983-y
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DOI: https://doi.org/10.1007/s00521-022-07983-y