Interval valued trapezoidal neutrosophic set: multi-attribute decision making for prioritization of non-functional requirements

Abstract

The paper discusses the trapezoidal fuzzy number (TrFN); interval-valued intuitionistic fuzzy number (IVIFN); neutrosophic set and its operational laws; and, trapezoidal neutrosophic set (TrNS) and its operational laws. Based on the combination of IVIFN and TrNS, an interval valued trapezoidal neutrosophic set (IVTrNS) is proposed followed by its operational laws. The paper also presents the score and accuracy functions for the proposed interval valued trapezoidal neutrosophic number (IVTrNN). Then, an interval valued trapezoidal neutrosophic weighted arithmetic averaging (IVTrNWAA) operator is introduced to combine the trapezoidal information which is neutrosophic and in the unit interval of real numbers. Finally, a method is developed to handle the problems in the multi attribute decision making (MADM) environment using IVTrNWAA operator followed by a numerical example of NFRs prioritization to illustrate the relevance of the developed method.

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References

  1. Akram M, Ilyas F, Garg H (2019) Multi-criteria group decision making based on ELECTRE I method in Pythagorean fuzzy information. Soft Comput 24(5):3425–3453. https://doi.org/10.1007/s00500-019-04105-0

    Article  Google Scholar 

  2. Anton A (1997) Goal identification and refinement in the specification of software-based information systems. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, USA

  3. Ashraf S, Abdullah S, Mahmood T (2019) Spherical fuzzy Dombi aggregation operators and their application in group decision-making problems. J Ambient Intell Hum Comput. https://doi.org/10.1007/s12652-019-01333-y

    Article  Google Scholar 

  4. Atanassov K (2017) Type-1 fuzzy sets and intuitionistic fuzzy sets. Algorithms 10(3):106. https://doi.org/10.3390/a10030106

    MathSciNet  Article  MATH  Google Scholar 

  5. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96. https://doi.org/10.1016/s0165-0114(86)80034-3

    Article  MATH  Google Scholar 

  6. Atanassov KT (1999) Interval valued intuitionistic fuzzy sets. Intuitionistic Fuzzy Sets Stud Fuzziness Soft Comput. https://doi.org/10.1007/978-3-7908-1870-3_2

    Article  MATH  Google Scholar 

  7. Atanassov KT (2000) Two theorems for intuitionistic fuzzy sets. Fuzzy Sets Syst 110(2):267–269. https://doi.org/10.1016/s0165-0114(99)00112-8

    MathSciNet  Article  MATH  Google Scholar 

  8. Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349. https://doi.org/10.1016/0165-0114(89)90205-4

    MathSciNet  Article  MATH  Google Scholar 

  9. Chen TY (2010) An outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy optimistic/pessimistic operators. Expert Syst Appl 37(12):7762–7774. https://doi.org/10.1016/j.eswa.2010.04.064

    Article  Google Scholar 

  10. Dey PP, Pramanik S, Giri BC (2016) TOPSIS for solving multi-attribute decision making problems under bi-polar neutrosophic environment. In: Smarandache F, Pramanik S (eds) New trends in neutrosophic theory and applications. Pons Editions, Brussels, pp 65–77

    Google Scholar 

  11. Dong JY, Wan SP (2015) Interval-valued trapezoidal intuitionistic fuzzy generalized aggregation operators and application to multi-attribute group decision making. Scientia Iranica 22(6):2702–2715

    Google Scholar 

  12. Glinz M (2007) On non-functional requirements. In: 15th IEEE International Requirements Engineering Conference, 21–26

  13. Gong Z, Hai S (2014) The interval-valued trapezoidal approximation of interval-valued fuzzy numbers and its application in fuzzy risk analysis. J Appl Math 2014:1–22. https://doi.org/10.1155/2014/254853

    Article  Google Scholar 

  14. Goguen J (1967) L-fuzzy sets. J Math Anal Appl 18:145–174

    MathSciNet  Article  Google Scholar 

  15. Jana C, Pal M, Wang J (2019) Bipolar fuzzy Dombi aggregation operators and its application in multiple-attribute decision-making process. J Ambient Intell Hum Comput 10:3533–3549. https://doi.org/10.1007/s12652-018-1076-9

    Article  Google Scholar 

  16. Joshi R (2020) A new multi-criteria decision-making method based on intuitionistic fuzzy information and its application to fault detection in a machine. J Ambient Intell Hum Comput 11:739–753. https://doi.org/10.1007/s12652-019-01322-1

    Article  Google Scholar 

  17. Joshi BP, Kumar S (2012) Fuzzy time series model based on intuitionistic fuzzy sets for empirical research in stock market. Int J Appl Evol Comput 3(4):71–84. https://doi.org/10.4018/jaec.2012100105

    Article  Google Scholar 

  18. Khatter K, Kalia A (2012) Goal based analysis of non-functional requirements for webbased Systems. CSI J Comput 1(4):3–20 (ISSN: 2277–7091)

    Google Scholar 

  19. Khatter K, Kalia A (2013a) Impact of non-functional requirements on requirements evolution. Sixth international conference on emerging trends in engineering and technology-ICETET 2013, IEEE Computer Society 61–68. https://doi.org/10.1109/ICETET.2013.15

  20. Khatter K, Kalia A (2013b) Integration of non-functional requirements in model-driven architecture. Fifth International conference on advances in recent technologies in communication and computing (ARTCom 2013). https://doi.org/10.1049/cp.2013.2213

  21. Khatter K, Kalia A (2014) Quantification of non-functional requirements. Sixth international conference on contemporary computing-IC3 2014. 224–229, IEEE Computer Society, https://doi.org/10.1109/IC3.2014.6897177

  22. Kumar R, Khatter K, Kalia A (2011) Measuring software reliability: a fuzzy model. SIGSOFT Softw Eng Notes 36:1–6

    Google Scholar 

  23. Li J, Zeng W, Guo P (2014) Interval-valued intuitionistic trapezoidal fuzzy number and its application. 2014 IEEE International Conference on Systems, Man, and Cybernetics (SMC). https://doi.org/10.1109/smc.2014.6973997

  24. Liang Y, Qin J, Martínez L, Liu J (2020) A heterogeneous QUALIFLEX method with criteria interaction for multi-criteria group decision making. Inf Sci 512:1481–1502. https://doi.org/10.1016/j.ins.2019.10.044

    MathSciNet  Article  Google Scholar 

  25. Liu F, Yuan XH (2007) Fuzzy number intuitionistic fuzzy set. Fuzzy Syst Math 21(1):88–91

    MATH  Google Scholar 

  26. Liu H-W, Wang G-J (2007) Multi-criteria decision-making methods based on intuitionistic fuzzy sets. Eur J Oper Res 179(1):220–233. https://doi.org/10.1016/j.ejor.2006.04.009

    Article  MATH  Google Scholar 

  27. Meng F, Tan C, Zhang Q (2013) The induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging operator and its application in decision making. Knowl Based Syst 42:9–19. https://doi.org/10.1016/j.knosys.2012.12.006

    Article  Google Scholar 

  28. Meng F, Tang J, Cabrerizo FJ, Herrera-Viedma E (2020) A rational and consensual method for group decision making with interval-valued intuitionistic multiplicative preference relations. Eng Appl Artif Intell 90:103514. https://doi.org/10.1016/j.engappai.2020.103514

    Article  Google Scholar 

  29. Morente-Molinera J, Kou G, Pérez I, Samuylov K, Selamat A, Herrera-Viedma E (2018) A group decision making support system for the Web: how to work in environments with a high number of participants and alternatives. Appl Soft Comput 68:191–201. https://doi.org/10.1016/j.asoc.2018.03.047

    Article  Google Scholar 

  30. Morente-Molinera J, Wu X, Morfeq A, Al-Hmouz R, Herrera-Viedma E (2020) A novel multi-criteria group decision-making method for heterogeneous and dynamic contexts using multi-granular fuzzy linguistic modelling and consensus measures. Inf Fusion 53:240–250. https://doi.org/10.1016/j.inffus.2019.06.028

    Article  Google Scholar 

  31. Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11:341–356. https://doi.org/10.1007/BF01001956

    Article  MATH  Google Scholar 

  32. Perera I (2011) Impact of poor requirement engineering in software outsourcing: a study on software developers’ experience. Int J Comput Commun Control 6(2):337. https://doi.org/10.15837/ijccc.2011.2.2182

    Article  Google Scholar 

  33. Robertson S, Robertson J (1999) Mastering the requirements process, 2nd edn. Addison-Wesley Professional, USA

    Google Scholar 

  34. Shino TK, Sunil JJ (2012) Intuitionistic fuzzy multisets and its application in medical diagnosis. Int J Math Comput Sci 6:34–37

    Google Scholar 

  35. Smarandache F (1998) Neutrosophy/neutrosophic probability, set, and logic. American Research Press, Rehoboth

    Google Scholar 

  36. Smarandache F (2019) Neutrosophic set is a generalization of intuitionistic fuzzy set, inconsistent intuitionistic fuzzy set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set (Atanassov's Intuitionistic Fuzzy Set of second type), q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, and n-HyperSpherical Fuzzy Set, while Neutrosophication is a Generalization of Regret Theory, Grey System Theory, and Three-Ways Decision (revisited), arXiv, Cornell University, New York City, NY, USA, (Submitted on 17 Nov 2019 (v1), last revised 29 Nov 2019 (this version, v2)), https://arxiv.org/ ftp/ arxiv/ papers/1911/1911.07333.pdf

  37. Tan C (2011) A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS. Expert Syst Appl 38(4):3023–3033. https://doi.org/10.1016/j.eswa.2010.08.092

    Article  Google Scholar 

  38. Tan CQ, Ma BJ, Wu DSD, Chen XH (2014) Multi-criteria decision making methods based on interval-valued intuitionistic fuzzy sets. Int J Uncertainty Fuzziness Knowl Based Syst 22(3):475–494

    MathSciNet  Article  Google Scholar 

  39. Turksen I (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 20(2):191–210. https://doi.org/10.1016/0165-0114(86)90077-1

    MathSciNet  Article  MATH  Google Scholar 

  40. Wang XF (2008) Fuzzy number intuitionistic fuzzy geometric aggregation operators and their application to decision making. Control Decis 23(6):607–612

    MathSciNet  MATH  Google Scholar 

  41. Wang H, Smarandache F, Zhang YQ, Sunderraman R (2005) Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Arizona

    Google Scholar 

  42. Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single valued neutrosophic sets. Multisp Multistruct 4:410–413

    MATH  Google Scholar 

  43. Wei G, Zhao X, Lin R (2010) Some induced aggregating operators with fuzzy intuitionistic number information and their applications for group decision making. Int J Comput Intell Syst 3(1):84. https://doi.org/10.2991/ijcis.2010.3.1.8

    Article  Google Scholar 

  44. Weinberg GM (1972) The psychology of improved programmer performance. Datamation 82–85

  45. Wiegers KE, Forte S, Suojanen N (2003) Software requirements, 2nd edn. Microsoft Press, Redmond

    Google Scholar 

  46. Wu J, Liu Y (2013) An approach for multiple attribute group decision making problems with interval-valued intuitionistic trapezoidal fuzzy numbers. Comput Ind Eng 66(2):311–324. https://doi.org/10.1016/j.cie.2013.07.001

    Article  Google Scholar 

  47. Xu Z (2012) Intuitionistic fuzzy multiattribute decision making: an interactive method. IEEE Trans Fuzzy Syst 20(3):514–525. https://doi.org/10.1109/tfuzz.2011.2177466

    Article  Google Scholar 

  48. Ye J (2014a) Prioritized aggregation operators of trapezoidal intuitionistic fuzzy sets and their application to multicriteria decision-making. Neural Comput Appl 25(6):1447–1454. https://doi.org/10.1007/s00521-014-1635-8

    Article  Google Scholar 

  49. Ye J (2014b) Trapezoidal neutrosophic set and its application to multiple attribute decision-making. Neural Comput Appl 26(5):1157–1166. https://doi.org/10.1007/s00521-014-1787-6

    Article  Google Scholar 

  50. Ye J (2017) Some weighted aggregation operators of trapezoidal neutrosophic numbers and their multiple attribute decision making method. Informatica 28(2):387–402. https://doi.org/10.15388/informatica.2017.108

    Article  Google Scholar 

  51. Zadeh LA (1965) Fuzzy sets. Inf Control 8(5):338–353

    Article  Google Scholar 

  52. Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoning—II. Inf Sci 8(4):301–357. https://doi.org/10.1016/0020-0255(75)90046-8

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Kiran Khatter.

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Khatter, K. Interval valued trapezoidal neutrosophic set: multi-attribute decision making for prioritization of non-functional requirements. J Ambient Intell Human Comput (2020). https://doi.org/10.1007/s12652-020-02130-8

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Keywords

  • Non-functional requirements (NFRs)
  • Multi criteria decision making (MCDM)
  • Multi attribute decision making (MADM)
  • Neutrosophic set
  • Interval valued fuzzy set (IVFS)
  • Interval valued intuitionistic fuzzy set (IVIFS)
  • Interval valued neutrosophic set
  • Trapezoidal neutrosophic set
  • Interval valued trapezoidal neutrosophic set (IVTrNS)
  • Interval valued trapezoidal neutrosophic number (IVTrNN)
  • Interval valued trapezoidal neutrosophic weighted arithmetic averaging operator (IVTrNWAA)