The prevailing concepts of intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PFSs) and q-rung orthopair fuzzy sets (q-ROFSs) have numerous applications in various fields from real life. Unfortunately, these theories have their own limitations related to the membership and non-membership grades. To eradicate these restrictions, we introduce the novel concept of linear Diophantine fuzzy set (LDFS) with the addition of reference parameters. This idea removes the restrictions of existing methodologies and the decision maker (DM) can freely choose the grades without any limitations. This structure also categorizes the problem by changing the physical sense of reference parameters. We present some fundamental operations on linear Diophantine fuzzy sets (LDFSs). We present geometrical interpretation for different operations of LDFSs. We also introduce the novel concepts of linear Diophantine fuzzy topological space (LDFTS) and linear Diophantine fuzzy weighted geometric aggregation (LDFWGA) operator. We discuss several properties of LDFTS with the help of examples. We introduce score functions and accuracy functions with different orders for the comparison of linear Diophantine fuzzy numbers (LDFNs). We propose two algorithms for solving multi-attribute decision-making (MADM) problem accompanied by an interesting application employing LDFTSs and LDFWGA operator.
ZadehL.A., Fuzzy sets, Information Control8 (1965), 338–353.
2.
ZadehL.A., The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences8(3) (1975), 199–249.
3.
AtanassovK.T., Intuitionistic fuzzy sets, in: V. Sgurev, Ed., VII ITKRs Session, Sofia, June, 1983 (Central Sci. and Techn. Library, Bulg. Academy of Sciences, 1984.
4.
AtanassovK.T. and StoevaS.S., Intuitionistic fuzzy sets, in: Polish Symp On Interval & Fuzzy Mathematics, Poznan, 1983, pp. 23–26.
5.
AtanassovK.T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems20(1) (1986), 87–96.
6.
AtanassovK.T., Geometrical interpretation of the elemets of the intuitionistic fuzzy objects, International Journal of Bio Automation20(S1) (2016), S27–S42.
7.
AtanassovK.T., Two variants of intuitionistic fuzzy prepositional calculus, Mathematical foundations of artificial intelligence seminar, Sofia, 1988, Reprintd: International Journal of Bio Automation20(S1) (2016), S17–S26.
8.
YagerR.P., Pythagorean fuzzy subsets, In Proceedings of the IFSA World Congress and NAFIPS Annual Meeting, Edmonton, AB, Canada, 2013, pp. 57–61.
9.
YagerR.P., Pythagorean membership grades in multi criteria decision-making, IEEE Transecrtions on Fuzzy Systems22 (2014), 958–965.
LiuP. and WangP., Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decisionmaking, International Journal of Intelligent Systems33 (1983), 259–280.
12.
AliM.I., Another view on q-rung orthopair fuzzy sets, International Journal of Intelligent Systems33 (2018), 2139–2153.
13.
MolodtsovD., Soft set theory-first results, Computers and Mathematics with Applications37 (1999), 19–31.
14.
AliM.I. and ShabirM., Logic connectives for soft sets and fuzzy soft sets, IEEE Transecrtions on Fuzzy Systems22 (2014), 1431–1442.
15.
AgarwalM., BiswasK.K. and HanmandluM., Generalized intuitionitic fuzzy soft sets with applications in decision-making, Applied Soft Computing20 (2013), 3552–3566.
16.
MahmoodT., UllahK., KhanQ. and JanN., An approach toward decision-making and medical diagnosis problems using the concept of spherical fuzzy sets, Neural Computing and Applications (2018). https://doi.org/10.1007/s00521-018-3521-2.
17.
ÇağmanN., EnginogluS. and ÇitakF., Fuzzy soft set theory and its applications, Iranian Journal of Fuzzy Systems8(8) (2011), 137–147.
18.
ChangC.L., Fuzzy topological spaces, Journal of Mathematical Analysis and Applications24 (1968), 182–190.
19.
CokerD., An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems88(1) (1997), 81–89.
20.
GargH., A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision-making, International Journal of Intelligent Systems31(9) (2016), 886–920.
21.
GargH., Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision-making, Computers & Industrial Engineering101 (2016), 53–69.
22.
GargH., A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multi-criteria decision-making problem, Journal of Intelligent and Fuzzy Systems31(1) (2016), 529–540.
23.
ChenS.M. and TanJ.M., Handling multicriteria fuzzy decision-makling problems based on vague set theory, Fuzzy Sets and Systems67 (1994), 163–172.
24.
TverskyA. and KahnemanD., Advances in prospect theory: Cumulative representation of uncertainity, Journal of Risk and Uncertainity5 (1992), 297–323.
25.
DombiJ., A general class of fuzzy operators, the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators, Fuzzy Sets and Systems8 (1982), 149–163.
26.
FengF., JunY.B., LiuX. and LiL., An adjustable approach to fuzzy soft set based decision making, Journal of Computational and Applied Mathematics234(1) (2010), 10–20.
27.
FengF., FujitaH., AliM.I., YagerR.R. and LiuX., Another view on generalized intuitionistic fuzzy soft sets and related multiattribute decision making methods, IEEE Transecrtions on Fuzzy Systems27(3) (2019), 474–488.
28.
JoseS. and KuriaskoseS., Aggregation operators, score function and accuracy function for multi-criteria decision-making in intuitionistic fuzzy context, Notes on Intuitionistic Fuzzy Sets20(1) (2014), 40–44.
29.
KaurG. and GargH., Cubic intuitionistic fuzzy aggregation operators, International Journal of Uncertainity Quantification8(5) (2018), 405–427.
30.
MahmoodT., MehmoodF. and KhanQ., Some generalized aggregation operators for cubic hesitant fuzzy sets and their application to multi-criteria decision-making, Punjab University Journal of Mathematics49(1) (2017), 31–49.
31.
RiazM. and HashmiM.R., Certain applications of fuzzy parameterized fuzzy soft sets in decision-making problems, International Journal of Algebra and Statistics5(2) (2016), 135–146.
32.
RiazM. and HashmiM.R., Fuzzy parameterized fuzzy soft topology with applications, Annals of Fuzzy Mathematics and Informatics13(5) (2017), 593–613.
33.
RiazM. and HashmiM.R., Fuzzy parameterized fuzzy soft compact spaces with decision-making, Punjab University Journal of Mathematics50(2) (2018), 131–145.
34.
RiazM. and HashmiM.R., Fixed points of fuzzy neutrosophic soft mapping with decision-making, Fixed Point Theory and Applications7 (2018), 1–10.
35.
RiazM., HashmiM.R. and FarooqA., Fuzzy parameterized fuzzy soft metric spaces, Journal of Mathematical Analysis9(2) (2018), 25–36.
36.
RiazM., SamrandacheF., FirdousA. and FakharA., On soft rough topology with multi-attribute group decision making, Mathematics7(67) (2019). doi: 10.3390/math7010067
37.
RiazM., DavvazB., FirdousA. and FakharF., Novel concepts of soft rough set topology with applications, Journal of Intelligent and Fuzzy Systems36(4) (2019), 3579–3590. doi: 10.3233/JIFS-181648
38.
RiazM., ÇağmanN., ZareefI. and AslamM., N-soft topology and its applications to multi-criteria group decision making, Journal of Intelligent and Fuzzy Systems36(6) (2019), 6521–6536.
39.
RiazM. and TehrimS.T., Cubic bipolar fuzzy ordered weighted geometric aggregation operators and their application using internal and external cubic bipolar fuzzy data, Computational & Applied Mathematics38(87) (2019), 1–25. doi: doi.org/10.1007/s40314-019-0843-3
40.
RiazM. and TehrimS.T., Multi-attribute group decisionmaking based cubic bipolar fuzzy information using averaging aggregation operators, Journal of Intelligent and Fuzzy Systems (2019), (In Press). DOI: 10.3233/JIFS-182751
41.
RiazM. and TehrimS.T., A novel extension of TOPSIS to MCGDM with bipolar neutrosophic soft topology, Journal of Intelligent and Fuzzy Systems (2019), (In Press). 10.3233/JIFS-190668.
42.
ZhanJ., LiuQ. and DavvazB., A new rough set theory: Rough soft hemirings, Journal of Intelligent and Fuzzy Systems28(4) (2015), 1687–1697.
43.
ZhanJ. and AlcantudJ.C.R., A novel type of soft rough covering and its application to multi-criteria group decision-making, Artificial Intelligence Review (2018). https://doi.org/10.1007/s10462-018-9617-3.
44.
ZhangL. and ZhanJ., Fuzzy soft β-covering based fuzzy rough sets and corresponding decision-making applications, International Journal of Machine Learning and Cybernetics10(6) (2018), 1487–1502. DOI: 10.1007/s13042-018-0828-3
45.
ZhangL. and ZhanJ., Novel classes of fuzzy soft β-coveringsbased fuzzy rough sets with applications to multi-criteria fuzzy group decision-making, Soft Computing23(14) (2018), 5327–5351. doi.org/10.1007/s00500-018-3470-9
46.
ZhangL., ZhanJ. and XuZ.S., Covering-based generalized IF rough sets with applications to multi-attribute decisionmaking, Information Sciences478 (2019), 275–302.
XuZ.S. and CaiX.Q., Intuitionistic fuzzy information aggregation: Theory and applications, Science Press Beijing and Springer-VerlagBerlin Heidelberg (2012).
49.
XuZ.S., Studies in fuzziness and soft computing: Hesitant fuzzy sets theory, Springer International PublishingSwitzerland (2014).
50.
YeJ., Interval-valued hesitant fuzzy prioritized weighted aggregation operators for multi attribute decision-making, Journal of Algorithms and Computational Technology8(2) (2013), 179–192.
51.
YeJ., Linguistic neutrosophic cubic numbers and their multiple attribute decision-making method, Information8 (2017), 1–11.
52.
GuoK., Amount of information and attitudinal-based method for ranking Atanassov’s intuitionistic fuzzy values, IEEE Transections on Fuzzy Systems22 (2014), 177–188.
53.
HongD.H. and ChoiC.H., Multi-criteria fuzzy decisionmaking problems based on vauge set theory, Fuzzy Sets and Systems114 (2000), 103–113.
54.
DemirciM., Fuzzy functions and their fundamental properties, Fuzzy Sets and Systems106(2) (1999), 239–246.
55.
MajiP.K., BiswasR. and RoyA.R., Soft set theory, Computers and Mathematics with Applications45 (2003), 555–562.
56.
SmarandacheF., Neutrosophy Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, USA, 1998.
57.
ShenY.H., WangF.X. and ChenW., A note on intuitionistic fuzzy mappings, Iranian Journal of Fuzzy Systems9(5) (2012), 63–76.
58.
WangH., SmarandacheF., ZhangY.Q. and SunderramanR., Single valued neutrosophic sets, Multispace and Multistructure4 (2010), 410–413.
59.
ChenJ., LiS., MaS. and WangX., m-polar fuzzy sets: An extension of bipolar fuzzy sets, The Scientific World Journal (2014).
60.
AkramM., WaseemN. and LiuP., Novel approach in decision making with m-polar fuzzy ELECTRE-I, International Journal of Fuzzy Systems (2019). doi.org/10.1007/s40815-019-00608-y
61.
GuleriaA. and BajajR.K., On pythagorean fuzzy soft matrices, operations and their applications in decision-making and medical diagnosis, Soft Computing (2018).
62.
PengandX. and YangY., Some results for pythagorean fuzzy sets, International Journal of Intelligent Systems30 (2015), 1133–1160.
63.
LiuY., ZhangH., WuY. and DongY., Ranking range based approach to MADM under incomplete context and its application in venture investment evaluation, Technological and Economic Development of Economy (2019), (in press).
64.
ZhangH., DongY., CarrascosaI.P. and ZhouH., Failure mode and effect analysis in a linguistic context: A consensus-based multi-attribute group decision-making approach, IEEE Transections on Reliability68(2) (2019), 566–582.
65.
ZhangH., DongY., ChiclanaF. and YuS., Consensus efficiency in group decision making: A comprehensive comparative study and its optimal design, European Journal of Operational Research275(2) (2019), 580–598.
66.
YuW., ZhangZ., ZhongQ. and SunL., Extended TODIM for multi-criteria group decision-making based on unbalanced hesitant fuzzy linguistic term sets, Computers and Industrial Engineering114 (2017), 316–328.
67.
ZhangZ., KouX., YuW. and GuoC., On priority weights and consistency for incomplete hesitant fuzzy preference relations, Knowledge Based Systems143 (2018), 115–126.
68.
ZhangZ. and GuoC., Deriving priority weights from intuitionistic multiplicative preference relations under group decisionmaking settings, Journal of the Operational Research Society68(12) (2017), 1582–1599.
69.
ZhangZ., GuoC. and MartineL., Managing multigranular linguistic distribution assessments in large-scale multiattribute group decision-making, IEEE Transactions on Systems, Man, and Cybernetics: Systems47(11) (2017), 3063–3076.
70.
ZhangB., DongY. and ViedmaE.H., Group decision making with heterogeneous preference structures: An automatic mechanism to support consensus reaching, Group Decision and Negotiation28(3) (2019), 585–617.
