Research Article
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Year 2020, Issue: 31, 1 - 19, 30.06.2020

Muhammed Riaz , Filiz Çıtak , Nabeela Wali , Amna Mushtaq

Abstract

References

  • W. D. Blizard, Multiset Theory, Notre Dame Journal of Formal Logic 30 (1989) 36-65.
  • L. A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338-353.
  • D. Molodtsov, Soft Set Theory-First Results, Computers and Mathematics with Applications 37(4-5) (1999) 19-31.
  • M. Abbas, M. I. Ali, S. Romaguerac, Generalized Operations in Soft Set Theory via Relaxed Conditions on Parameters, Filomat 31(19) (2017) 5955-5964. https://doi.org/10.2298/FIL1719955A
  • M. I. Ali, F. Feng, X. Y. Liu, W. K. Min, M. Shabir, On Some New Operations in Soft Set Theory, Computers and Mathematics with Applications 57 (2009) 1547-1553.
  • M. I. Ali, A Note on Soft Sets, Rough Soft Sets and Fuzzy Soft Sets, Applied Soft Computing 11 (2011) 3329-3332.
  • M. I. Ali, M. Shabir, F. Feng, Representation of Graphs Based on Neighborhoods and Soft Sets, International Journal of Machine Learning and Cybernetics 8(5) (2017) 1525-1534. https://doi.org/10.1007/s13042-016-0525-z
  • F. Feng, M. Irfan Ali, M. Shabir, Soft Relations Applied to Semigroups, Filomat 27(7) (2013) 1183-1196. https://doi.org/10.2298/FIL1307183F
  • F. Feng, M. Akram, B. Davvaz, V. L. Fotea, Attribute Analysis of Information Systems Based on Elementary Soft Implications, Knowledge-Based Systems 70 (2014) 281-292.
  • F. Feng, Y.B. Jun, X. Liu, L. Li, An Adjustable Approach to Fuzzy Soft Set Based Decision Making, Journal of Computational and Applied Mathematics 234(1) (2010) 10-20.
  • F. Feng, C. Li, B. Davvaz, M.I. Ali, Soft Sets Combined with Fuzzy Sets and Rough Sets, A Tentative Approach, Soft Computing 14(9) (2010) 899-911.
  • F. Feng, X. Y. Liu, V. Leoreanu-Fotea, Y. B. Jun, Soft Sets and Soft Rough Sets, Information Sciences 181(6) (2011) 1125-1137.
  • F. Feng, H. Fujita, M. I. Ali, R. R. Yager, X. Liu, Another View on Generalized Intuitionistic Fuzzy Soft Sets and Related Multiattribute Decision Making Methods, IEEE Transactions on Fuzzy Systems 27(3) (2019) 474-488.
  • K. Hayat, M. I. Ali, BY. Cao, F. Karaaslan, New Results on Type-2 Soft Sets, Hacettepe Journal of Mathematics and Statistics 47(4) (2018) 855-876.
  • P. K. Maji, A. R. Roy, R. Biswas, An Application of Soft Sets in A Decision Making Problem, Computers and Mathematics with Applications 44(8-9) (2002) 1077-1083.
  • P. K. Maji, A. R. Roy, R. Biswas, Soft Set Theory, Computers and Mathematics with Applications 45(4-5) (2003) 555-562.
  • H. Aktaş, N. Çağman, Soft Sets and Soft Group, Information Sciences 1(77) (2007) 2726-2735.
  • Z. Kong, L. Gaoa, L. Wang, S. Li, The Normal Parameter Reduction of Soft Sets and Its Algorithm, Computers and Mathematics with Applications 56 (2008) 3029-3037.
  • K. V. Babitha, J. J. Sunil, Soft Set Relations and Functions, Computers and Mathematics with Applications 60 (2010) 1840-1849.
  • K. V. Babitha, S. J. John, On Soft Multi Sets, Annals of Fuzzy Mathematics and Informatics 5(1) (2013) 35-44.
  • S. Alkhazaleh, A. R. Salleh, N. Hassan, Soft Multisets Theory, Applied Mathematical Sciences 5(72) (2011) 3561-3573.
  • A. Adeel, M. Akram, Ali N. A. Koam, Group Decision-Making Based on m-Polar Fuzzy Linguistic TOPSIS Method, Symmetry 11(735) (2019) 1-20. https://doi.org/10.3390/sym11060735
  • M. Akram, M. Arshad, A Novel Trapezoidal Bipolar Fuzzy TOPSIS Method for Group Decision-Making, Group Decision and Negotiation 28 (2018) 565-584. https://doi.org/10.1007/s10726-018-9606-6
  • N. Çağman, S. Karataş, S. Enginoglu, Soft Topology, Computers and Mathematics with Applications 62 (2011) 351-358.
  • K. Chakrabarty, R. Biswas, S. Nanda, Fuzziness in Rough Sets, Fuzzy Sets and Systems 110(2) (2000) 247-251.
  • K. P. Girsh, S. J. John, General Relations between Partially Ordered Multisets and Their Chains and Antichains, Mathematical Communications14(2) (2009) 193-206.
  • M. B. Gorzalzany, A Method of Inference in Approximate Reasoning Based on Interval-Valued Fuzzy Sets, Fuzzy Sets and Systems 21 (1987) 1-17.
  • D. Pie, D. Miao, From Soft Sets to Information System, IEEE International Conference on Granular Computing (2005) 617-621 Beijing, China.
  • M. Shabir, M. I. Ali, Soft Ideals and Generalized Fuzzy Ideals in Semigroups, New Mathematics and Natural Computation 5(3) (2009) 599-615.
  • X. Liu, Y. Ju, S. Yang, Hesitant Intuitionistic Fuzzy Linguistic Aggregation Operators and Their Applications to Multi Attribute Decision Making, Journal of Intelligent and Fuzzy Systems 26(3) (2014) 1187-1201.
  • M. R. Hashmi, M. Riaz, F. Smarandache, m-polar Neutrosophic Topology with Applications to Multi-Criteria Decision-Making in Medical Diagnosis and Clustering Analysis, International Journal of Fuzzy Systems 22(1) (2020), 273-292. https://doi.org/10.1007/s40815-019-00763-2
  • M. R. Hashmi, M. Riaz, A Novel Approach to Censuses Process by using Pythagorean m-polar Fuzzy Dombi’s Aggregation Operators, Journal of Intelligent and Fuzzy Systems 38(2) (2020) 1977-1995. https://doi.org/10.3233/JIFS-190613
  • K. Naeem, M. Riaz, X. D. Peng, D. Afzal, Pythagorean Fuzzy Soft MCGDM Methods Based on TOPSIS, VIKOR and Aggregation Operators, Journal of Intelligent and Fuzzy Systems 37(5) (2019) 6937-6957. https://doi.org/10.3233/JIFS-190905
  • K. Naeem, M. Riaz, D. Afzal, Pythagorean m-polar Fuzzy Sets and TOPSIS method for the Selection of Advertisement Mode, Journal of Intelligent and Fuzzy Systems 37(6) (2019) 8441-8458. https://doi.org/10.3233/JIFS-191087
  • M. Riaz, N. Çağman, I. Zareef, M. Aslam, N-Soft Topology and its Applications to Multi-Criteria Group Decision Making, Journal of Intelligent and Fuzzy Systems 36(6) (2019) 6521-6536. https://doi.org/10.3233/JIFS-182919
  • M. Riaz, F. Samrandache, A. Firdous, A. Fakhar, On Soft Rough Topology with Multi-Attribute Group Decision Making, Mathematics 7(67) (2019) 1-18. https://doi.org/10.3390/math7010067
  • M. Riaz, B. Davvaz, A. Firdous, A. Fakhar, Novel Concepts of Soft Rough Set Topology with Applications, Journal of Intelligent and Fuzzy Systems 36(4) (2019) 3579-3590.
  • M. Riaz, M. R. Hashmi, MAGDM for Agribusiness in the Environment of Various Cubic m-polar Fuzzy Averaging Aggregation Operators, Journal of Intelligent and Fuzzy Systems 37(3) (2019) 3671-3691. https://doi.org/10.3233/JIFS-182809
  • M. Riaz, M. R. Hashmi, Linear Diophantine Fuzzy Set and Its Applications Towards Multi-Attribute Decision Making Problems, Journal of Intelligent and Fuzzy Systems 37(4) (2019) 5417-5439. https://doi.org/10.3233/JIFS-190550
  • M. Riaz and M. R. Hashmi, Soft Rough Pythagorean m-Polar Fuzzy Sets and Pythagorean m-Polar Fuzzy Soft Rough Sets with Application to Decision-Making, Computational and Applied Mathematics (2019). https://doi.org/10.1007/s40314-019-0989-z
  • M. Riaz, S. T. Tehrim, Cubic Bipolar Fuzzy Ordered Weighted Geometric Aggregation Operators and Their Application Using Internal and External Cubic Bipolar Fuzzy Data, Computational and Applied Mathematics, 38(2) (2019) 1-25.
  • M. Riaz, S. T. Tehrim, Multi-Attribute Group Decision Making Based Cubic Bipolar Fuzzy Information Using Averaging Aggregation Operators, Journal of Intelligent and Fuzzy Systems 37(2) (2019) 2473-2494.
  • M. Riaz, S. T. Tehrim, Bipolar Fuzzy Soft Mappings with Application to Bipolar Disorders, International Journal of Biomathematics 12(7) (2019). https://doi.org/10.1142/S1793524519500803
  • R. Roy, P. K. Maji, A Fuzzy Soft Set Theoretic Approach to Decision Making Problems, Journal of Computational and Applied Mathematics 203(2) (2007) 412-418.
  • G. Şenel, The Relation Between Soft Topological Space and Soft Ditopological Space, Communications Series A1: Mathematics and Statistics 67(2) (2018) 209-219. https://doi.org/10.1501/Commua1-0000000875
  • G. Şenel, Characterization of Soft Sets by delta-Soft Operations, Journal of Polytechnic 21(3) (2018) 707-713. https://doi.org/10.2339/politeknik.389631
  • A. Sezgin, A. O. Atagün, On Operations of Soft Sets, Computers and Mathematics with Applications 61(5) (2011) 1457-1467.
  • A. Sezgin, A. O. Atagün, N. Çağman, Soft Intersection Near-Rings with Applications, Neural Computing and Applications 21(1) (2012) 221-229.
  • M. Shabir, M. Naz, On Soft Topological Spaces, Computers and Mathematics with Applications 61 (2011) 1786-1799.
  • S. T. Tehrim, M. Riaz, A Novel Extension of TOPSIS to MCGDM with Bipolar Neutrosophic Soft Topology, Journal of Intelligent and Fuzzy Systems 37(4) (2019) 5531-5549. https://doi.org/10.3233/JIFS-190668
  • G. Wei, H. Wang, X. Zhao, R. Lin, Hesitant Triangular Fuzzy Information Aggregation In Multiple Attribute Decision Making, Journal of Intelligence and Fuzzy Systems 26(3) (2014) 1201-1209.
  • Xueling Ma, Q. Liu, J. Zhan, A Survey of Decision Making Methods Based on Certain Hybrid Soft Set Models, Artificial Intelligence Review 47 (2017) 507-530.
  • Z. S. Xu, X. L. Zhang, Hesitant Fuzzy Multi-Attribute Decision-Making Based on TOPSIS with Incomplete Weight Information, Knowledge-Based Systems 52 (2013) 53-64.
  • Z. S. Xu, Intuitionistic Fuzzy Aggregation Operators, IEEE Transactions on Fuzzy Systems 15(6) (2007) 1179-1187.
  • Z. S. Xu, X.Q. Cai, Intuitionistic Fuzzy Information Aggregation: Theory and Applications, Science Press Beijing and Springer-Verlag Berlin Heidelberg, 2012.
  • Z. S. Xu, Studies in Fuzziness and Soft Computing: Hesitant Fuzzy Sets Theory, Springer International Publishing Switzerland, 2014.
  • J. Zhan, Q. Liu, B. Davvaz, A New Rough Set Theory: Rough Soft Hemirings, Journal of Intelligence and Fuzzy Systems 28(4) (2015) 1687-1697.
  • J. Zhan, J. C. R. Alcantud, A Novel Type of Soft Rough Covering and Its Application to Multicriteria Group Decision Making, Artificial Intelligence Review 52 (2019) 2381-2410. https://doi.org/10.1007/s10462-018-9617-3
  • X. L. Zhang, Z. S. Xu, Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets, International Journal of Intelligent Systems 29(12) (2014) 1061-1078.
  • A. Syropoulos, Mathematics of Multisets, Multiset Processing (2001) 347–358.
  • Z. Pawlak, Rough Sets, International Journal of Computer Science 11(1982) 341-356.

Roughness and Fuzziness Associated with Soft Multi-Sets and their Application to MADM

Year 2020, Issue: 31, 1 - 19, 30.06.2020

Muhammed Riaz , Filiz Çıtak , Nabeela Wali , Amna Mushtaq

Abstract

In this paper, we tried to hybrid soft sets with multisets. We define some basic properties of soft multiset and present some important results. We define some binary relations, equivalence relations and an indiscernibility relation on soft multiset with examples. The concept of an approximation space associated with each parameter in a soft multiset is discussed and an approximation space associated with the soft multiset is defined. We introduce the novel concepts of roughness and fuzziness associated with soft multiset. We use soft multiset in multi-valued information system. Furthermore, we present an algorithm to cope with uncertainties in multi-attribute decision making (MADM) problems by utilizing soft multisets and related concepts. The efficiency of the algorithm is verified by a case study to find the optimal choice of the real-world problems having uncertainties.

Keywords

Soft multi-set roughness and fuzziness multiu-valued information systems MADM

References

  • W. D. Blizard, Multiset Theory, Notre Dame Journal of Formal Logic 30 (1989) 36-65.
  • L. A. Zadeh, Fuzzy Sets, Information and Control 8 (1965) 338-353.
  • D. Molodtsov, Soft Set Theory-First Results, Computers and Mathematics with Applications 37(4-5) (1999) 19-31.
  • M. Abbas, M. I. Ali, S. Romaguerac, Generalized Operations in Soft Set Theory via Relaxed Conditions on Parameters, Filomat 31(19) (2017) 5955-5964. https://doi.org/10.2298/FIL1719955A
  • M. I. Ali, F. Feng, X. Y. Liu, W. K. Min, M. Shabir, On Some New Operations in Soft Set Theory, Computers and Mathematics with Applications 57 (2009) 1547-1553.
  • M. I. Ali, A Note on Soft Sets, Rough Soft Sets and Fuzzy Soft Sets, Applied Soft Computing 11 (2011) 3329-3332.
  • M. I. Ali, M. Shabir, F. Feng, Representation of Graphs Based on Neighborhoods and Soft Sets, International Journal of Machine Learning and Cybernetics 8(5) (2017) 1525-1534. https://doi.org/10.1007/s13042-016-0525-z
  • F. Feng, M. Irfan Ali, M. Shabir, Soft Relations Applied to Semigroups, Filomat 27(7) (2013) 1183-1196. https://doi.org/10.2298/FIL1307183F
  • F. Feng, M. Akram, B. Davvaz, V. L. Fotea, Attribute Analysis of Information Systems Based on Elementary Soft Implications, Knowledge-Based Systems 70 (2014) 281-292.
  • F. Feng, Y.B. Jun, X. Liu, L. Li, An Adjustable Approach to Fuzzy Soft Set Based Decision Making, Journal of Computational and Applied Mathematics 234(1) (2010) 10-20.
  • F. Feng, C. Li, B. Davvaz, M.I. Ali, Soft Sets Combined with Fuzzy Sets and Rough Sets, A Tentative Approach, Soft Computing 14(9) (2010) 899-911.
  • F. Feng, X. Y. Liu, V. Leoreanu-Fotea, Y. B. Jun, Soft Sets and Soft Rough Sets, Information Sciences 181(6) (2011) 1125-1137.
  • F. Feng, H. Fujita, M. I. Ali, R. R. Yager, X. Liu, Another View on Generalized Intuitionistic Fuzzy Soft Sets and Related Multiattribute Decision Making Methods, IEEE Transactions on Fuzzy Systems 27(3) (2019) 474-488.
  • K. Hayat, M. I. Ali, BY. Cao, F. Karaaslan, New Results on Type-2 Soft Sets, Hacettepe Journal of Mathematics and Statistics 47(4) (2018) 855-876.
  • P. K. Maji, A. R. Roy, R. Biswas, An Application of Soft Sets in A Decision Making Problem, Computers and Mathematics with Applications 44(8-9) (2002) 1077-1083.
  • P. K. Maji, A. R. Roy, R. Biswas, Soft Set Theory, Computers and Mathematics with Applications 45(4-5) (2003) 555-562.
  • H. Aktaş, N. Çağman, Soft Sets and Soft Group, Information Sciences 1(77) (2007) 2726-2735.
  • Z. Kong, L. Gaoa, L. Wang, S. Li, The Normal Parameter Reduction of Soft Sets and Its Algorithm, Computers and Mathematics with Applications 56 (2008) 3029-3037.
  • K. V. Babitha, J. J. Sunil, Soft Set Relations and Functions, Computers and Mathematics with Applications 60 (2010) 1840-1849.
  • K. V. Babitha, S. J. John, On Soft Multi Sets, Annals of Fuzzy Mathematics and Informatics 5(1) (2013) 35-44.
  • S. Alkhazaleh, A. R. Salleh, N. Hassan, Soft Multisets Theory, Applied Mathematical Sciences 5(72) (2011) 3561-3573.
  • A. Adeel, M. Akram, Ali N. A. Koam, Group Decision-Making Based on m-Polar Fuzzy Linguistic TOPSIS Method, Symmetry 11(735) (2019) 1-20. https://doi.org/10.3390/sym11060735
  • M. Akram, M. Arshad, A Novel Trapezoidal Bipolar Fuzzy TOPSIS Method for Group Decision-Making, Group Decision and Negotiation 28 (2018) 565-584. https://doi.org/10.1007/s10726-018-9606-6
  • N. Çağman, S. Karataş, S. Enginoglu, Soft Topology, Computers and Mathematics with Applications 62 (2011) 351-358.
  • K. Chakrabarty, R. Biswas, S. Nanda, Fuzziness in Rough Sets, Fuzzy Sets and Systems 110(2) (2000) 247-251.
  • K. P. Girsh, S. J. John, General Relations between Partially Ordered Multisets and Their Chains and Antichains, Mathematical Communications14(2) (2009) 193-206.
  • M. B. Gorzalzany, A Method of Inference in Approximate Reasoning Based on Interval-Valued Fuzzy Sets, Fuzzy Sets and Systems 21 (1987) 1-17.
  • D. Pie, D. Miao, From Soft Sets to Information System, IEEE International Conference on Granular Computing (2005) 617-621 Beijing, China.
  • M. Shabir, M. I. Ali, Soft Ideals and Generalized Fuzzy Ideals in Semigroups, New Mathematics and Natural Computation 5(3) (2009) 599-615.
  • X. Liu, Y. Ju, S. Yang, Hesitant Intuitionistic Fuzzy Linguistic Aggregation Operators and Their Applications to Multi Attribute Decision Making, Journal of Intelligent and Fuzzy Systems 26(3) (2014) 1187-1201.
  • M. R. Hashmi, M. Riaz, F. Smarandache, m-polar Neutrosophic Topology with Applications to Multi-Criteria Decision-Making in Medical Diagnosis and Clustering Analysis, International Journal of Fuzzy Systems 22(1) (2020), 273-292. https://doi.org/10.1007/s40815-019-00763-2
  • M. R. Hashmi, M. Riaz, A Novel Approach to Censuses Process by using Pythagorean m-polar Fuzzy Dombi’s Aggregation Operators, Journal of Intelligent and Fuzzy Systems 38(2) (2020) 1977-1995. https://doi.org/10.3233/JIFS-190613
  • K. Naeem, M. Riaz, X. D. Peng, D. Afzal, Pythagorean Fuzzy Soft MCGDM Methods Based on TOPSIS, VIKOR and Aggregation Operators, Journal of Intelligent and Fuzzy Systems 37(5) (2019) 6937-6957. https://doi.org/10.3233/JIFS-190905
  • K. Naeem, M. Riaz, D. Afzal, Pythagorean m-polar Fuzzy Sets and TOPSIS method for the Selection of Advertisement Mode, Journal of Intelligent and Fuzzy Systems 37(6) (2019) 8441-8458. https://doi.org/10.3233/JIFS-191087
  • M. Riaz, N. Çağman, I. Zareef, M. Aslam, N-Soft Topology and its Applications to Multi-Criteria Group Decision Making, Journal of Intelligent and Fuzzy Systems 36(6) (2019) 6521-6536. https://doi.org/10.3233/JIFS-182919
  • M. Riaz, F. Samrandache, A. Firdous, A. Fakhar, On Soft Rough Topology with Multi-Attribute Group Decision Making, Mathematics 7(67) (2019) 1-18. https://doi.org/10.3390/math7010067
  • M. Riaz, B. Davvaz, A. Firdous, A. Fakhar, Novel Concepts of Soft Rough Set Topology with Applications, Journal of Intelligent and Fuzzy Systems 36(4) (2019) 3579-3590.
  • M. Riaz, M. R. Hashmi, MAGDM for Agribusiness in the Environment of Various Cubic m-polar Fuzzy Averaging Aggregation Operators, Journal of Intelligent and Fuzzy Systems 37(3) (2019) 3671-3691. https://doi.org/10.3233/JIFS-182809
  • M. Riaz, M. R. Hashmi, Linear Diophantine Fuzzy Set and Its Applications Towards Multi-Attribute Decision Making Problems, Journal of Intelligent and Fuzzy Systems 37(4) (2019) 5417-5439. https://doi.org/10.3233/JIFS-190550
  • M. Riaz and M. R. Hashmi, Soft Rough Pythagorean m-Polar Fuzzy Sets and Pythagorean m-Polar Fuzzy Soft Rough Sets with Application to Decision-Making, Computational and Applied Mathematics (2019). https://doi.org/10.1007/s40314-019-0989-z
  • M. Riaz, S. T. Tehrim, Cubic Bipolar Fuzzy Ordered Weighted Geometric Aggregation Operators and Their Application Using Internal and External Cubic Bipolar Fuzzy Data, Computational and Applied Mathematics, 38(2) (2019) 1-25.
  • M. Riaz, S. T. Tehrim, Multi-Attribute Group Decision Making Based Cubic Bipolar Fuzzy Information Using Averaging Aggregation Operators, Journal of Intelligent and Fuzzy Systems 37(2) (2019) 2473-2494.
  • M. Riaz, S. T. Tehrim, Bipolar Fuzzy Soft Mappings with Application to Bipolar Disorders, International Journal of Biomathematics 12(7) (2019). https://doi.org/10.1142/S1793524519500803
  • R. Roy, P. K. Maji, A Fuzzy Soft Set Theoretic Approach to Decision Making Problems, Journal of Computational and Applied Mathematics 203(2) (2007) 412-418.
  • G. Şenel, The Relation Between Soft Topological Space and Soft Ditopological Space, Communications Series A1: Mathematics and Statistics 67(2) (2018) 209-219. https://doi.org/10.1501/Commua1-0000000875
  • G. Şenel, Characterization of Soft Sets by delta-Soft Operations, Journal of Polytechnic 21(3) (2018) 707-713. https://doi.org/10.2339/politeknik.389631
  • A. Sezgin, A. O. Atagün, On Operations of Soft Sets, Computers and Mathematics with Applications 61(5) (2011) 1457-1467.
  • A. Sezgin, A. O. Atagün, N. Çağman, Soft Intersection Near-Rings with Applications, Neural Computing and Applications 21(1) (2012) 221-229.
  • M. Shabir, M. Naz, On Soft Topological Spaces, Computers and Mathematics with Applications 61 (2011) 1786-1799.
  • S. T. Tehrim, M. Riaz, A Novel Extension of TOPSIS to MCGDM with Bipolar Neutrosophic Soft Topology, Journal of Intelligent and Fuzzy Systems 37(4) (2019) 5531-5549. https://doi.org/10.3233/JIFS-190668
  • G. Wei, H. Wang, X. Zhao, R. Lin, Hesitant Triangular Fuzzy Information Aggregation In Multiple Attribute Decision Making, Journal of Intelligence and Fuzzy Systems 26(3) (2014) 1201-1209.
  • Xueling Ma, Q. Liu, J. Zhan, A Survey of Decision Making Methods Based on Certain Hybrid Soft Set Models, Artificial Intelligence Review 47 (2017) 507-530.
  • Z. S. Xu, X. L. Zhang, Hesitant Fuzzy Multi-Attribute Decision-Making Based on TOPSIS with Incomplete Weight Information, Knowledge-Based Systems 52 (2013) 53-64.
  • Z. S. Xu, Intuitionistic Fuzzy Aggregation Operators, IEEE Transactions on Fuzzy Systems 15(6) (2007) 1179-1187.
  • Z. S. Xu, X.Q. Cai, Intuitionistic Fuzzy Information Aggregation: Theory and Applications, Science Press Beijing and Springer-Verlag Berlin Heidelberg, 2012.
  • Z. S. Xu, Studies in Fuzziness and Soft Computing: Hesitant Fuzzy Sets Theory, Springer International Publishing Switzerland, 2014.
  • J. Zhan, Q. Liu, B. Davvaz, A New Rough Set Theory: Rough Soft Hemirings, Journal of Intelligence and Fuzzy Systems 28(4) (2015) 1687-1697.
  • J. Zhan, J. C. R. Alcantud, A Novel Type of Soft Rough Covering and Its Application to Multicriteria Group Decision Making, Artificial Intelligence Review 52 (2019) 2381-2410. https://doi.org/10.1007/s10462-018-9617-3
  • X. L. Zhang, Z. S. Xu, Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets, International Journal of Intelligent Systems 29(12) (2014) 1061-1078.
  • A. Syropoulos, Mathematics of Multisets, Multiset Processing (2001) 347–358.
  • Z. Pawlak, Rough Sets, International Journal of Computer Science 11(1982) 341-356.
There are 61 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Muhammed Riaz

Filiz Çıtak

Nabeela Wali

Amna Mushtaq

Publication Date June 30, 2020
Submission Date January 1, 2020
Published in Issue Year 2020 Issue: 31

Cite

APA Riaz, M., Çıtak, F., Wali, N., Mushtaq, A. (2020). Roughness and Fuzziness Associated with Soft Multi-Sets and their Application to MADM. Journal of New Theory(31), 1-19.
AMA Riaz M, Çıtak F, Wali N, Mushtaq A. Roughness and Fuzziness Associated with Soft Multi-Sets and their Application to MADM. JNT. June 2020;(31):1-19.
Chicago Riaz, Muhammed, Filiz Çıtak, Nabeela Wali, and Amna Mushtaq. “Roughness and Fuzziness Associated With Soft Multi-Sets and Their Application to MADM”. Journal of New Theory, no. 31 (June 2020): 1-19.
EndNote Riaz M, Çıtak F, Wali N, Mushtaq A (June 1, 2020) Roughness and Fuzziness Associated with Soft Multi-Sets and their Application to MADM. Journal of New Theory 31 1–19.
IEEE M. Riaz, F. Çıtak, N. Wali, and A. Mushtaq, “Roughness and Fuzziness Associated with Soft Multi-Sets and their Application to MADM”, JNT, no. 31, pp. 1–19, June 2020.
ISNAD Riaz, Muhammed et al. “Roughness and Fuzziness Associated With Soft Multi-Sets and Their Application to MADM”. Journal of New Theory 31 (June 2020), 1-19.
JAMA Riaz M, Çıtak F, Wali N, Mushtaq A. Roughness and Fuzziness Associated with Soft Multi-Sets and their Application to MADM. JNT. 2020;:1–19.
MLA Riaz, Muhammed et al. “Roughness and Fuzziness Associated With Soft Multi-Sets and Their Application to MADM”. Journal of New Theory, no. 31, 2020, pp. 1-19.
Vancouver Riaz M, Çıtak F, Wali N, Mushtaq A. Roughness and Fuzziness Associated with Soft Multi-Sets and their Application to MADM. JNT. 2020(31):1-19.
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