New generalization of fuzzy soft sets: $ (a, b) $-Fuzzy soft sets

Tareq M. Al-shami; José Carlos R. Alcantud; Abdelwaheb Mhemdi; Tareq M. Al-shami; José Carlos R. Alcantud; Abdelwaheb Mhemdi

doi:10.3934/math.2023155

AIMS Mathematics

2023, Volume 8, Issue 2: 2995-3025. doi: 10.3934/math.2023155

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New generalization of fuzzy soft sets: $ (a, b) $-Fuzzy soft sets

1.
Department of Mathematics, Sana'a University, Sana'a, Yemen
2.
BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, E37007 Salamanca, Spain
3.
Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia

Received: 01 October 2022 Revised: 03 November 2022 Accepted: 10 November 2022 Published: 14 November 2022
MSC : 03E72, 03E99, 08A72

Many models of uncertain knowledge have been designed that combine expanded views of fuzziness (expressions of partial memberships) with parameterization (multiple subsethood indexed by a parameter set). The standard orthopair fuzzy soft set is a very general example of this successful blend initiated by fuzzy soft sets. It is a mapping from a set of parameters to the family of all orthopair fuzzy sets (which allow for a very general view of acceptable membership and non-membership evaluations). To expand the scope of application of fuzzy soft set theory, the restriction of orthopair fuzzy sets that membership and non-membership must be calibrated with the same power should be removed. To this purpose we introduce the concept of $ (a, b) $-fuzzy soft set, shortened as $ (a, b) $-FSS. They enable us to address situations that impose evaluations with different importances for membership and non-membership degrees, a problem that cannot be modeled by the existing generalizations of intuitionistic fuzzy soft sets. We establish the fundamental set of arithmetic operations for $ (a, b) $-FSSs and explore their main characteristics. Then we define aggregation operators for $ (a, b) $-FSSs and discuss their main properties and the relationships between them. Finally, with the help of suitably defined scores and accuracies we design a multi-criteria decision-making strategy that operates in this novel framework. We also analyze a decision-making problem to endorse the validity of $ (a, b) $-FSSs for decision-making purposes.
- $ (a, b) $-fuzzy soft set,
- score and accuracy functions,
- aggregation operators,
- multi-criteria decision-making
Citation: Tareq M. Al-shami, José Carlos R. Alcantud, Abdelwaheb Mhemdi. New generalization of fuzzy soft sets: $ (a, b) $-Fuzzy soft sets[J]. AIMS Mathematics, 2023, 8(2): 2995-3025. doi: 10.3934/math.2023155

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Abstract

Many models of uncertain knowledge have been designed that combine expanded views of fuzziness (expressions of partial memberships) with parameterization (multiple subsethood indexed by a parameter set). The standard orthopair fuzzy soft set is a very general example of this successful blend initiated by fuzzy soft sets. It is a mapping from a set of parameters to the family of all orthopair fuzzy sets (which allow for a very general view of acceptable membership and non-membership evaluations). To expand the scope of application of fuzzy soft set theory, the restriction of orthopair fuzzy sets that membership and non-membership must be calibrated with the same power should be removed. To this purpose we introduce the concept of $ (a, b) $-fuzzy soft set, shortened as $ (a, b) $-FSS. They enable us to address situations that impose evaluations with different importances for membership and non-membership degrees, a problem that cannot be modeled by the existing generalizations of intuitionistic fuzzy soft sets. We establish the fundamental set of arithmetic operations for $ (a, b) $-FSSs and explore their main characteristics. Then we define aggregation operators for $ (a, b) $-FSSs and discuss their main properties and the relationships between them. Finally, with the help of suitably defined scores and accuracies we design a multi-criteria decision-making strategy that operates in this novel framework. We also analyze a decision-making problem to endorse the validity of $ (a, b) $-FSSs for decision-making purposes.

References

[1]	B. Ahmad, A. Kharal, On fuzzy soft sets, Adv. Fuzzy Syst., 2009 (2009). https://doi.org/10.1155/2009/586507 doi: 10.1155/2009/586507
[2]	M. Akram, A. Adeel, J. C. R. Alcantud, Fuzzy $N$-soft sets: A novel model with applications, J. Intell. Fuzzy Syst., 35 (2018), 4757–4771. https://doi.org/10.3233/JIFS-18244 doi: 10.3233/JIFS-18244
[3]	J. C. R. Alcantud, Convex soft geometries, J. Comput. Cognitive Eng., 1 (2022), 2–12. https://doi.org/10.47852/bonviewJCCE597820 doi: 10.47852/bonviewJCCE597820
[4]	J. C. R. Alcantud, The semantics of $N$-soft sets, their applications, and a coda about three-way decision, Inform. Sci., 606 (2022), 837–852. https://doi.org/10.1016/j.ins.2022.05.084 doi: 10.1016/j.ins.2022.05.084
[5]	J. C. R. Alcantud, T. M. Al-shami, A. A. Azzam, Caliber and chain conditions in soft topologies, Mathematics, 9 (2021), 2349. https://doi.org/10.3390/math9192349 doi: 10.3390/math9192349
[6]	J. C. R. Alcantud, A. Z. Khameneh, A. Kilicman, Aggregation of infinite chains of intuitionistic fuzzy sets and their application to choices with temporal intuitionistic fuzzy information, Inform. Sci., 514 (2020), 106–117. https://doi.org/10.1016/j.ins.2019.12.008 doi: 10.1016/j.ins.2019.12.008
[7]	J. C. R. Alcantud, G. Varela, B. Santos-Buitrago, G. Santos-García, M. F. Jiménez, Analysis of survival for lung cancer resections cases with fuzzy and soft set theory in surgical decision making, Plos One, 14 (2019), e0218283. https://doi.org/10.1371/journal.pone.0218283 doi: 10.1371/journal.pone.0218283
[8]	T. M. Al-shami, (2, 1)-Fuzzy sets: Properties, weighted aggregated operators and their applications to multi-criteria decision-making methods, Complex Intell. Syst., 2022. https://doi.org/10.1007/s40747-022-00878-4 doi: 10.1007/s40747-022-00878-4
[9]	T. M. Al-shami, Generalized umbral for orthopair fuzzy sets: (m, n)-Fuzzy sets and their applications to multi-criteria decision-making methods, Submitted.
[10]	T. M. Al-shami, Soft somewhat open sets: Soft separation axioms and medical application to nutrition, Comput. Appl. Math., 41 (2022), 216. https://doi.org/10.1007/s40314-022-01919-x doi: 10.1007/s40314-022-01919-x
[11]	T. M. Al-shami, H. Z. Ibrahim, A. A. Azzam, A. I. EL-Maghrabi, SR-fuzzy sets and their applications to weighted aggregated operators in decision-making, J. Funct. Space., 2022 (2022). https://doi.org/10.1155/2022/3653225 doi: 10.1155/2022/3653225
[12]	T. M. Al-shami, L. D. R. Kočinac, Almost soft Menger and weakly soft Menger spaces, Appl. Comput. Math., 21 (2022), 35–51. https://doi.org/10.30546/1683-6154.21.1.2022.35 doi: 10.30546/1683-6154.21.1.2022.35
[13]	T. M. Al-shami, A. Mhemdi, R. Abu-Gdairi, M. E. El-Shafei, Compactness and connectedness via the class of soft somewhat open sets, AIMS Math., 8 (2023), 815–840. https://doi.org/10.3934/math.2023040 doi: 10.3934/math.2023040
[14]	T. M. Al-shami, A. Mhemdi, A. Rawshdeh, H. Al-jarrah, Soft version of compact and Lindelöf spaces using soft somewhere dense set, AIMS Math., 6 (2021), 8064–8077. https://doi.org/10.3934/math.2021468 doi: 10.3934/math.2021468
[15]	Z. A. Ameen, T. M. Al-shami, A. A. Azzam, A. Mhemdi, A novel fuzzy structure: Infra-fuzzy topological spaces, J. Funct. Space., 2022 (2022) https://doi.org/10.1155/2022/9778069 doi: 10.1155/2022/9778069
[16]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96.
[17]	M. Atef, M. I. Ali, T. M. Al-shami, Fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications, Comput. Appl. Math., 40 (2021), 115. https://doi.org/10.1007/s40314-021-01501-x doi: 10.1007/s40314-021-01501-x
[18]	N. Cağman, S. Enginoğlu, F. Çitak, Fuzzy soft set theory and its application, Iran. J. Fuzzy Syst., 8 (2011), 137–147.
[19]	S. K. De, R. B. Akhil, R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Set. Syst., 117 (2001), 209–213. https://doi.org/10.1016/S0165-0114(98)00235-8 doi: 10.1016/S0165-0114(98)00235-8
[20]	F. Fatimah, D. Rosadi, R. B. F. Hakim, J. C. R. Alcantud, $N$-soft sets and their decision-making algorithms, Soft Comput., 22 (2018), 3829–3842. https://doi.org/10.1007/s00500-017-2838-6 doi: 10.1007/s00500-017-2838-6
[21]	F. Fatimah, J. C. R. Alcantud, The multi-fuzzy $N$-soft set and its applications to decision-making, Neural Comput. Appl., 33 (2021), 11437–11446. https://doi.org/10.1007/s00521-020-05647-3 doi: 10.1007/s00521-020-05647-3
[22]	M. Gulzar, D. Alghazzawi, M. H. Mateen, N. Kausar, A certain class of t-intuitionistic fuzzy subgroups, IEEE Access, 8 (2020), 163260–163268. https://doi.org/10.1109/ACCESS.2020.3020366 doi: 10.1109/ACCESS.2020.3020366
[23]	M. T. Hamida, M. Riaz, D. Afzal, Novel MCGDM with q-rung orthopair fuzzy soft sets and TOPSIS approach under q-Rung orthopair fuzzy soft topology, J. Intell. Fuzzy Syst., 39 (2020), 3853–3871. https://doi.org/10.3233/JIFS-192195 doi: 10.3233/JIFS-192195
[24]	H. Z. Ibrahim, T. M. Al-shami, O. G. Elbarbary, (3, 2)-Fuzzy sets and their applications to topology and optimal choices, Comput. Intell. Neurosci., 2021 (2021). https://doi.org/10.1155/2021/1272266 doi: 10.1155/2021/1272266
[25]	S. J. John, Soft sets: Theory and applications, Springer Cham, 2021.
[26]	H. Liao, Z. Xu, Intuitionistic fuzzy hybrid weighted aggregation operators, Int. J. Intell. Syst., 29 (2014), 971–993. https://doi.org/10.1002/int.21672 doi: 10.1002/int.21672
[27]	A. A. Khan, S. Ashraf, S. Abdullah, M. Qiyas, J. Luo, S. U. Khan, Pythagorean fuzzy Dombi aggregation operators and their application in decision support system, Symmetry, 11 (2019), 383. https://doi.org/10.3390/sym11030383 doi: 10.3390/sym11030383
[28]	M. Kirişci, A case study for medical decision making with the fuzzy soft sets, Afr. Mat., 31 (2020), 557–564.
[29]	M. Kirişci, N. Şimşek, Decision making method related to Pythagorean fuzzy soft sets with infectious diseases application, J. King Saud Univ.-Com., 34 (2022), 5968–5978. https://doi.org/10.1016/j.jksuci.2021.08.010 doi: 10.1016/j.jksuci.2021.08.010
[30]	P. K. Maji, R. Biswas, A. R. Roy, Fuzzy soft sets, J. Fuzzy Math., 9 (2001), 589–602.
[31]	A. R. Roy, P. K. Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math., 203 (2007), 412–418. https://doi.org/10.1016/j.cam.2006.04.008 doi: 10.1016/j.cam.2006.04.008
[32]	D. Molodtsov, Soft set theory–-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
[33]	X. Peng, H. Yuan, Fundamental properties of Pythagorean fuzzy aggregation operators, Fund. Inform., 147 (2016), 415–446. https://doi.org/10.3233/FI-2016-1415 doi: 10.3233/FI-2016-1415
[34]	X. Peng, Y. Yang, J. Song, Y. Jiang, Pythagorean fuzzy soft set and its application, Comput. Eng., 41 (2015), 224–229.
[35]	K. Rahman, S. Abdullah, M. Jamil, M. Y. Khan, Some generalized intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute group decision making, Int. J. Fuzzy Syst., 20 (2018), 1567–1575. https://doi.org/10.1007/s40815-018-0452-0 doi: 10.1007/s40815-018-0452-0
[36]	K. Rahman, S. Abdullah, M. S. A. Khan, M. Shakeel, Pythagorean fuzzy hybrid geometric aggregation operator and their applications to multiple attribute decision making, Int. J. Comput. Sci. Inform. Sec., 14 (2016), 837–854.
[37]	A. Sivadas, S. J. John, Fermatean fuzzy soft sets and its applications, Communications in Computer and Information Science, Springer, Singapore, 1345 (2021).
[38]	S. Saleh, R. Abu-Gdairi, T. M. Al-shami, Mohammed S. Abdo, On categorical property of fuzzy soft topological spaces, Appl. Math. Inform. Sci., 16 (2022), 635–641. http://dx.doi.org/10.18576/amis/160417 doi: 10.18576/amis/160417
[39]	T. Senapati, R. R. Yager, Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision-making methods, Eng. Appl. Artif. Intell., 85 (2019), 112–121. https://doi.org/10.1016/j.engappai.2019.05.012 doi: 10.1016/j.engappai.2019.05.012
[40]	T. Senapati, R. R. Yager, Fermatean fuzzy sets, J. Amb. Intell. Hum. Comput., 11 (2020), 663–674. https://doi.org/10.1007/s12652-019-01377-0 doi: 10.1007/s12652-019-01377-0
[41]	Y. J. Xu, Y. K. Sun, D. F. Li, Intuitionistic fuzzy soft set, 2010 2nd International Workshop on Intelligent Systems and Applications, 2010, 1–4. https://doi.org/10.1109/IWISA.2010.5473444
[42]	J. Wan, H. Chen, T. Li, Z. Yuan, J. Liu, W. Huang, Interactive and complementary feature selection via fuzzy multigranularity uncertainty measures, IEEE T. Cybern., 2021. https://doi.org/10.1109/TCYB.2021.3112203 doi: 10.1109/TCYB.2021.3112203
[43]	R. R. Yager, Pythagorean membership grades in multicriteria decision making, IEEE T. Fuzzy Syst., 22 (2013), 958–965. https://doi.org/10.1109/TFUZZ.2013.2278989 doi: 10.1109/TFUZZ.2013.2278989
[44]	R. R. Yager, Generalized orthopair fuzzy sets, IEEE T. Fuzzy Syst., 25 (2017), 1222–1230. https://doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
[45]	B. Yang, Fuzzy covering-based rough set on two different universes and its application, Artif. Intell. Rev., 55 (2022), 4717–4753. https://doi.org/10.1007/s10462-021-10115-y doi: 10.1007/s10462-021-10115-y
[46]	B. Yang, B. Q. Hu, A fuzzy covering-based rough set model and its generalization over fuzzy lattice, Inform. Sci., 367–368 (2016), 463–486. https://doi.org/10.1016/j.ins.2016.05.053 doi: 10.1016/j.ins.2016.05.053
[47]	J. Yang, Y. Yao, Semantics of soft sets and three-way decision with soft sets, Inform. Sci., 194 (2020), 105538. https://doi.org/10.1016/j.knosys.2020.105538 doi: 10.1016/j.knosys.2020.105538
[48]	L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353.

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