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Soergel Distance Measures for q-Rung Orthopair Fuzzy Sets and Their Applications

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q-Rung Orthopair Fuzzy Sets

Abstract

The virtue of the q-rung orthopair fuzzy set inherits those of the intuitionistic fuzzy set and the Pythagorean fuzzy set in loosening the constraint on support and counter-support. The very lax requirement gives the evaluators great freedom in expressing their beliefs about membership degrees and non-membership degrees, which makes q-rung orthopair fuzzy sets having a wide scope of application in practice. A distance measure is an important mathematical tool for distinguishing the difference between q-rung orthopair fuzzy sets and allows to deal with problems such as multi-criteria decision-making, medical diagnosis, and pattern recognition under a q-rung orthopair fuzzy environment. Unfortunately, many of the existing q-rung orthopair fuzzy distance measures have their limitations. To eliminate such restrictions, in this chapter, the Soergel-type distances of q-rung orthopair fuzzy sets are introduced and the basis on which the orthopairs can be ranked is established. The weighted types of the proposed Soergel distances and their corresponding similarity coefficients are derived. In addition, the validity of the emerging distance measures is shown by comparing them with the distance measures described in some recent research studies through numerical examples. Some charts are provided to visually display the various characteristics and to analyze the properties of the proposed distance measures. The outputs verify that these Soergel distance measures of q-rung orthopair fuzzy sets outperform other existing metrics in measuring uncertainty and avoiding counterintuitive cases. Some illustrative examples of decision-making in real life are presented, demonstrating the strong discrimination capability and effectiveness of the proposed Soergel distance measures.

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References

  1. P. Albert, The algebra of fuzzy logic. Fuzzy Sets Syst. 1(3), 203–230 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. K.T. Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)

    Article  MATH  Google Scholar 

  3. K.T. Atanassov, Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Aygün, H. Kamacı, Some generalized operations in soft set theory and their role in similarity and decision making. J. Intell. Fuzzy Syst. 36(6), 6537–6547 (2019)

    Article  Google Scholar 

  5. E. Aygün, H. Kamacı, Some new algebraic structures of soft sets. Soft Comput. 25(13), 8609–8626 (2021)

    Article  Google Scholar 

  6. H. Bashir, S. Inayatullah, A. Alsanad, R. Anjum, M. Mosleh, P. Ashraf, Some improved correlation coefficients for q-rung orthopair fuzzy sets and their applications in cluster analysis. Math. Probl. Eng. 2021, 11 (2021)

    Google Scholar 

  7. K. Bhattacharya, S.K. De, Decision making under intuitionistic fuzzy metric distances. Ann. Optim. Theory Pract. 3(2), 49–64 (2020)

    Google Scholar 

  8. F.E. Boran, D.A. Akay, A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition. Inf. Sci. 255, 45–57 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Broumi, F. Smarandache, New operations over interval valued intuitionistic hesitant fuzzy set. Math. Stat. 2(2), 62–71 (2014)

    Article  Google Scholar 

  10. S.-M. Chen, Similarity measures between vague sets and between elements. IEEE Trans. Syst. Man Cybern. 27, 153–158 (1997)

    Article  Google Scholar 

  11. S.-M. Chen, C.-H. Chang, A novel similarity measure between Atanssov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf. Sci. 291, 96–114 (2015)

    Article  Google Scholar 

  12. I. Couso, L. Garrido, L. Sánchez, Similarity and dissimilarity measures between fuzzy sets: a formal relational study. Inf. Sci. 229, 122–141 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Dehghan, M. Ghatee, B. Hashemi, Inverse of a fuzzy matrix of fuzzy numbers. Int. J. Comput. Math. 86(8), 1433–1452 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. W.S. Du, Minkowski-type distance measures for generalized orthopair fuzzy sets. Int. J. Intell. Syst. 33, 802–817 (2018)

    Article  Google Scholar 

  15. P.A. Ejegwa, New operations on intuitionistic fuzzy multisets. J. Math. Inf. 3, 17–23 (2015)

    Google Scholar 

  16. P.A. Ejegwa, I.C. Onyeke, V. Adah, An algorithm for an improved intuitionistic fuzzy correlation measure with medical diagnostic application. Ann. Optim. Theory Pract. 3(3), 51–68 (2020)

    Google Scholar 

  17. B. Farhadinia, S. Effati, F. Chiclana, A family of similarity measures for q-rung orthopair fuzzy sets and their applications to multiple criteria decision making. Int. J. Intell. Syst. 36(4), 1535–1559 (2021)

    Article  Google Scholar 

  18. H. Garg, A new possibility degree measure for interval-valued q-rung orthopair fuzzy sets in decision-making. Int. J. Intell. Syst. 36(1), 526–557 (2021)

    Article  Google Scholar 

  19. H. Garg, A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem. J. Intell. Fuzzy Syst. 31(1), 529–540 (2016)

    Article  MATH  Google Scholar 

  20. H. Garg, A novel trigonometric operation-based q-rung orthopair fuzzy aggregation operator and its fundamental properties. Neural Comput. Appl. 32(18), 15077–15099 (2020)

    Article  Google Scholar 

  21. H. Garg, Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process. Comput. Math. Organ. Theory 23(4), 546–571 (2017)

    Article  MathSciNet  Google Scholar 

  22. H. Garg, New exponential operation laws and operators for interval-valued q-rung orthopair fuzzy sets in group decision making process. Neural Comput. Appl. 33(20), 13937–13963 (2021). https://doi.org/10.1007/s00521-021-06036-0

    Article  Google Scholar 

  23. H. Garg, CN-q-ROFS: connection number-based q-rung orthopair fuzzy set and their application to decision-making process. Int. J. Intell. Syst. 36(7), 3106–3143 (2021)

    Article  Google Scholar 

  24. H. Garg, S.-M. Chen, Multiattribute group decision making based on neutrality aggregation operators of q-rung orthopair fuzzy sets. Inf. Sci. 517, 427–447 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Garg, K. Kumar, A novel possibility measure to interval-valued intuitionistic fuzzy set using connection number of set pair analysis and its applications. Neural Comput. Appl. 32, 3337–3348 (2020)

    Article  Google Scholar 

  26. D.H. Hong, C. Kim, A note on similarity measures between vague sets and between elements. Inf. Sci. 115, 83–96 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. W.-L. Hung, M.-S. Yang, Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recogn. Lett. 25, 1603–1611 (2004)

    Article  Google Scholar 

  28. M. Imtiaz, M. Saqlain, M. Saeed, TOPSIS for multi criteria decision making in octagonal intuitionistic fuzzy environment by using accuracy function. J. New Theory 31, 32–40 (2020)

    Google Scholar 

  29. N. Jan, L. Zedam, T. Mahmood, E. Rak, Z. Ali, Generalized dice similarity measures for q-rung orthopair fuzzy sets with applications. Complex Intell. Syst. 6, 545–558 (2020)

    Article  Google Scholar 

  30. H. Kamacı, Interval-valued fuzzy parameterized intuitionistic fuzzy soft sets and their applications. Cumhur. Sci. J. 40(2), 317–331 (2019)

    Article  Google Scholar 

  31. H. Kamacı, A novel approach to similarity of soft sets. Adıyaman Univ. J. Sci. 9(1), 23–35 (2019)

    Google Scholar 

  32. H. Kamacı, Similarity measure for soft matrices and its applications. J. Intell. Fuzzy Syst. 36(4), 3061–3072 (2019)

    Article  Google Scholar 

  33. H. Kamacı, Simplified neutrosophic multiplicative refined sets and their correlation coefficients with application in medical pattern recognition. Neutrosophic Sets Syst. 41, 270–285 (2021)

    Google Scholar 

  34. A. Kandel, W.J. Byatt, Fuzzy sets, fuzzy algebra, and fuzzy statistics. Proc. IEEE 66(12), 1619–1639 (1978)

    Article  Google Scholar 

  35. M.J. Khan, P. Kumam, M. Shutaywi, Knowledge measure for the q-rung orthopair fuzzy sets. Int. J. Intell. Syst. 36, 628–655 (2021)

    Article  Google Scholar 

  36. M.J. Khan, P. Kumam, M. Shutaywi, W. Kumam, Improved knowledge measures for q-rung orthopair fuzzy sets. Int. J. Comput. Intell. Syst. 14(1), 1700–1713 (2021)

    Article  Google Scholar 

  37. M.J. Khan, P. Kumam, N.A. Alreshidi, W. Kumam, Improved cosine and cotangent function-based similarity measures for q-rung orthopair fuzzy sets and TOPSIS method. Complex Intell. Syst. (2021), in press. https://doi.org/10.1007/s40747-021-00425-7

  38. K.H. Kim, F.W. Roush, Generalized fuzzy matrices. Fuzzy Sets Syst. 4(3), 293–315 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Kon, Fuzzy distance and fuzzy norm. J. Oper. Res. Soc. Jpn. 60(2), 66–77 (2017)

    MathSciNet  MATH  Google Scholar 

  40. D. Li, C. Cheng, New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recogn. Lett. 23, 221–225 (2002)

    Article  MATH  Google Scholar 

  41. D. Li, W. Zeng, Distance measure of Pythagorean fuzzy sets. Int. J. Intell. Syst. 33, 348–361 (2018)

    Article  Google Scholar 

  42. Y. Li, D.L. Olson, Z. Qin, Similarity measures between intuitionistic fuzzy (vague) sets a comparative analysis. Pattern Recogn. Lett. 28, 278–285 (2007)

    Article  Google Scholar 

  43. Z. Li, M. Lu, Some novel similarity and distance measures of Pythagorean fuzzy sets and their applications. J. Intell. Fuzzy Syst. 37(2), 1781–1799 (2019)

    Article  Google Scholar 

  44. Z. Liang, P. Shi, Similarity measures on intuitionistic fuzzy sets. Pattern Recogn. Lett. 24, 2687–2693 (2003)

    Article  MATH  Google Scholar 

  45. P. Liu, P. Wang, Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making. Int. J. Intell. Syst. 33, 259–280 (2018)

    Article  Google Scholar 

  46. P. Liu, T. Mahmood, Z. Ali, Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision making. Information 11(1), 1–27 (2020)

    Google Scholar 

  47. H.B. Mitchell, On the Dengfeng-Chuntian similarity measure and its application to pattern recognition. Pattern Recogn. Lett. 24, 3101–3104 (2003)

    Article  Google Scholar 

  48. A. Pal, B. Mondal, N. Bhattacharyya, S. Raha, Similarity in fuzzy systems. J. Uncertain. Anal. Appl. 2, 18 (2014)

    Article  Google Scholar 

  49. Z. Pawlak, Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)

    Article  MATH  Google Scholar 

  50. Z. Pawlak, Rough sets and intelligent data analysis. Inf. Sci. 147, 1–12 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning. Inf. Sci. 177, 41–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Z. Pawlak, A. Skowron, Rudiments of rough sets. Inf. Sci. 177, 3–27 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. X. Peng, J. Dai, H. Garg, Exponential operation and aggregation operator for q-rung orthopair fuzzy set and their decision-making method with a new score function. Int. J. Intell. Syst. 33(11), 2255–2282 (2018)

    Article  Google Scholar 

  54. X. Peng, L. Liu, Information measures for q-rung orthopair fuzzy sets. Int. J. Intell. Syst. 34, 1795–1834 (2019)

    Article  Google Scholar 

  55. X. Peng, H. Yuan, Y. Yang, Pythagorean fuzzy information measures and their applications. Int. J. Intell. Syst. 32, 991–1029 (2017)

    Google Scholar 

  56. S. Petchimuthu, H. Garg, H. Kamacı, A.O. Atagün, The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM. Comput. Appl. Math. 39(2), 1–32 (2020). Article number 68

    Google Scholar 

  57. S. Petchimuthu, H. Kamacı, The adjustable approaches to multi-criteria group decision making based on inverse fuzzy soft matrices. Sci. Iran. (in press). https://doi.org/10.24200/sci.2020.54294.3686

  58. S. Petchimuthu, H. Kamacı, The row-products of inverse soft matrices in multicriteria decision making. J. Intell. Fuzzy Syst. 36(6), 6425–6441 (2019)

    Article  Google Scholar 

  59. A.U. Rahman, M.R. Ahmad, M. Saeed, M. Ahsan, M. Arshad, M. Ihsan, A study on fundamentals of refined intuitionistic fuzzy set with some properties. J. Fuzzy Ext. Appl. 1(4), 300–314 (2020)

    Google Scholar 

  60. A.U. Rahman, M. Saeed, M. Arshad, M. Ihsan, M.R. Ahmad, (m, n)-convexity-cum-concavity on fuzzy soft set with applications in first and second sense. Punjab Univ. J. Math. 53(1), 19–33 (2021)

    Article  MathSciNet  Google Scholar 

  61. M. Riaz, A. Razzaq, H. Kalsoom, D. Pamučar, H.M. A. Farid, Y.-M. Chu, q-rung orthopair fuzzy geometric aggregation operators based on generalized and group-generalized parameters with application to water loss management. Symmetry 12(8), 1236 (2020). https://doi.org/10.3390/sym12081236

  62. M. Riaz, H. Garg, H.M.A. Farid, M. Aslam, Novel q-rung orthopair fuzzy interaction aggregation operators and their application to low-carbon green supply chain management. J. Intell. Fuzzy Syst. 41(2), 4109–4126 (2021). https://doi.org/10.3233/JIFS-210506

    Article  Google Scholar 

  63. M. Riaz, H.M.A. Farid, F. Karaaslan, M.R. Hashmi, Some q-rung orthopair fuzzy hybrid aggregation operators and TOPSIS method for multi-attribute decision-making. J. Intell. Fuzzy Syst. 39(1), 1227–1241 (2020)

    Article  Google Scholar 

  64. M. Riaz, W. Salabun, H.M.A. Farid, N. Ali, J. Watróbski, A robust q-rung orthopair fuzzy information aggregation using Einstein operations with application to sustainable energy planning decision management. Energies 13(9), 2155 (2020). https://doi.org/10.3390/en13092155

  65. S. Samanta, B. Sarkar, Representation of competitions by generalized fuzzy graphs. Int. J. Comput. Intell. Syst. 11(1), 1005–1015 (2018)

    Article  Google Scholar 

  66. A.K. Shaw, T.K. Roy, Trapezoidal intuitionistic fuzzy number with some arithmetic operations and its application on reliability evaluation. Int. J. Math. Oper. Res. 5(1), 55–73 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  67. T.K. Shinoj, S.J. John, Intuitionistic fuzzy multisets and its application in medical diagnosis. World Acad. Sci. Eng. Technol. 6, 1–28 (2012)

    Google Scholar 

  68. M. Sitara, M. Akram, M.Y. Bhatti, Fuzzy graph structures with application. Mathematics 7(1), 63 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  69. R. Taghaodi, A novel solution approach for solving intuitionistic fuzzy transportation problem of type-2. Ann. Optim. Theory Pract. 2(2), 11–24 (2019)

    Google Scholar 

  70. P. Talukdar, S. Goala, P. Dutta, B. Limboo, Fuzzy multicriteria decision making in medical diagnosis using an advanced distance measure on linguistic Pythagorean fuzzy sets. Ann. Optim. Theory Pract. 3(4), 113–131 (2020)

    Google Scholar 

  71. V. Uluçay, I. Deli, M. Şahin, Intuitionistic trapezoidal fuzzy multi-numbers and its application to multi-criteria decision-making problems. Complex Intell. Syst. 5, 65–78 (2019)

    Article  Google Scholar 

  72. P. Wang, J. Wang, G. Wei, C. Wei, Similarity measures of q-rung orthopair fuzzy sets based on cosine function and their applications. Mathematics 7(4), 340 (2019)

    Article  Google Scholar 

  73. G. Wei, C. Wei, J. Wang, H. Gao, Y. Wei, Some q-rung orthopair fuzzy Maclaurin symmetric mean operators and their applications to potential evaluation of emerging technology commercialization. Int. J. Intell. Syst. 34, 50–81 (2019)

    Article  Google Scholar 

  74. G. Wei, Y. Wei, Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications. Int. J. Intell. Syst. 33, 634–652 (2018)

    Article  Google Scholar 

  75. L. Xu, Y. Liu, H. Liu, Some improved q-rung orthopair fuzzy aggregation operators and their applications to multiattribute group decision-making. Math. Probl. Eng. 2019, 2019, 18 pages

    Google Scholar 

  76. R.P. Yager, Pythagorean fuzzy subsets, in Proceedings of the IFSA World Congress and NAFIPS Annual Meeting, Edmonton, Canada (2013), pp. 57–61

    Google Scholar 

  77. R.P. Yager, Generalized orthopair fuzzy sets. IEEE Trans. Fuzzy Syst. 25, 1222–1230 (2017)

    Article  Google Scholar 

  78. Z. Yang, H. Garg, Interaction power partitioned Maclaurin symmetric mean operators under q-rung orthopair uncertain linguistic information. Int. J. Fuzzy Syst. 1–19 (2021). https://doi.org/10.1007/s40815-021-01062-5

  79. Y.Y. Yao, A comparative study of fuzzy sets and rough sets. Inf. Sci. 109, 227–242 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  80. J. Ye, Cosine similarity measures for intuitionistic fuzzy sets and their applications. Math. Comput. Modell. 53, 91–97 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  81. L.A. Zadeh, Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  MATH  Google Scholar 

  82. S. Zeng, Y. Hu, X. Xie, Q-rung orthopair fuzzy weighted induced logarithmic distance measures and their application in multiple attribute decision making. Eng. Appl. Artif. Intell. 100, 104–167 (2021)

    Google Scholar 

  83. X. Zhang, A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int. J. Intell. Syst. 31, 593–611 (2016)

    Article  Google Scholar 

  84. F. Zhou, T.-Y. Chen, A novel distance measure for Pythagorean fuzzy sets and its applications to the technique for order preference by similarity to ideal solutions. Int. J. Comput. Intell. Syst. 12(2), 955–969 (2019)

    Article  MathSciNet  Google Scholar 

  85. R.M. Zulqarnain, X.L. Xin, H. Garg, W.A. Khan, Aggregation operators of Pythagorean fuzzy soft sets with their application for green supplier chain management. J. Intell. Fuzzy Syst. 40(3), 5545–5563 (2021)

    Article  Google Scholar 

  86. R.M. Zulqarnain, X.L. Xin, H. Garg, R. Ali, Interaction aggregation operators to solve multi criteria decision making problem under Pythagorean fuzzy soft environment. J. Intelli. Fuzzy Syst. 41(1), 1151–1171 (2021)

    Article  Google Scholar 

  87. R.M. Zulqarnain, X.L. Xin, M. Saqlain, W.A. Khan, TOPSIS method based on the correlation coefficient of interval-valued intuitionistic fuzzy soft sets and aggregation operators with their application in decision-making. J. Math. 12 pages (2021)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Science and Arts, Department of Mathematics, Yozgat Bozok University, 66100, Yozgat, Turkey

    Hüseyin Kamacı

  2. Department of Science and Humanities (Mathematics), University College of Engineering, Nagercoil, 629004, Tamil Nadu, India

    Subramanian Petchimuthu

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  1. Hüseyin Kamacı
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Correspondence to Hüseyin Kamacı .

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  1. Thapar Institute of Engineering and Technology, Patiala, Punjab, India

    Harish Garg

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Kamacı, H., Petchimuthu, S. (2022). Soergel Distance Measures for q-Rung Orthopair Fuzzy Sets and Their Applications. In: Garg, H. (eds) q-Rung Orthopair Fuzzy Sets. Springer, Singapore. https://doi.org/10.1007/978-981-19-1449-2_4

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