Linear Diophantine fuzzy graphs with new decision-making approach

Muhammad Zeeshan Hanif; Naveed Yaqoob; Muhammad Riaz; Muhammad Aslam; Muhammad Zeeshan Hanif; Naveed Yaqoob; Muhammad Riaz; Muhammad Aslam

doi:10.3934/math.2022801

AIMS Mathematics

2022, Volume 7, Issue 8: 14532-14556. doi: 10.3934/math.2022801

Previous Article Next Article

Research article

Linear Diophantine fuzzy graphs with new decision-making approach

1.
Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad, Pakistan
2.
Department of Mathematics, University of the Punjab, Lahore, Pakistan
3.
Department of Mathematics, College of Sciences, King Khalid University, Abha, Saudi Arabia

Received: 14 February 2022 Revised: 08 May 2022 Accepted: 24 May 2022 Published: 07 June 2022
MSC : 03E72, 05C72

The concept of linear Diophantine fuzzy set (LDFS) is a new mathematical tool for optimization, soft computing, and decision analysis. The aim of this article is to extend the notion of graph theory towards LDFSs. We initiate the idea of linear Diophantine fuzzy graph (LDF-graph) as a generalization of certain theoretical concepts including, q-rung orthopair fuzzy graph, Pythagorean fuzzy graph, and intuitionistic fuzzy graph. We extend certain properties of crisp graph theory towards LDF-graph including, composition, join, and union of LDF-graphs. We elucidate these operations with various illustrations. We analyze some interesting results that the composition of two LDF-graphs is a LDF-graph, cartesian product of two LDF-graphs is a LDF-graph, and the join of two LDF-graphs is a LDF-graph. We describe the idea of homomorphisms for LDF-graphs. We observe the equivalence relation via an isomorphism between LDF-graphs. Some significant results related to complement of LDF-graph are also investigated. Lastly, an algorithm based on LDFSs and LDF-relations is proposed for decision-making problems. A numerical example of medical diagnosis application is presented based on proposed approach.
- LDFSs,
- LDF-graphs,
- properties of LDF-graph,
- LDF-relations,
- decision-making
Citation: Muhammad Zeeshan Hanif, Naveed Yaqoob, Muhammad Riaz, Muhammad Aslam. Linear Diophantine fuzzy graphs with new decision-making approach[J]. AIMS Mathematics, 2022, 7(8): 14532-14556. doi: 10.3934/math.2022801

Related Papers:

Abstract

The concept of linear Diophantine fuzzy set (LDFS) is a new mathematical tool for optimization, soft computing, and decision analysis. The aim of this article is to extend the notion of graph theory towards LDFSs. We initiate the idea of linear Diophantine fuzzy graph (LDF-graph) as a generalization of certain theoretical concepts including, q-rung orthopair fuzzy graph, Pythagorean fuzzy graph, and intuitionistic fuzzy graph. We extend certain properties of crisp graph theory towards LDF-graph including, composition, join, and union of LDF-graphs. We elucidate these operations with various illustrations. We analyze some interesting results that the composition of two LDF-graphs is a LDF-graph, cartesian product of two LDF-graphs is a LDF-graph, and the join of two LDF-graphs is a LDF-graph. We describe the idea of homomorphisms for LDF-graphs. We observe the equivalence relation via an isomorphism between LDF-graphs. Some significant results related to complement of LDF-graph are also investigated. Lastly, an algorithm based on LDFSs and LDF-relations is proposed for decision-making problems. A numerical example of medical diagnosis application is presented based on proposed approach.

References

[1]	S. Ashraf, S. Abdullah, Emergency decision support modeling for COVID-19 based on spherical fuzzy information, Int. J. Intell. Syst., 35 (2020), 1601–1645. http://dx.doi.org/10.1002/int.22262 doi: 10.1002/int.22262
[2]	H. Garg, R. Arora, TOPSIS method based on correlation coefficient for solving decision-making problems with intuitionistic fuzzy soft set information, AIMS Mathematics, 5 (2020), 2944–2966. http://dx.doi.org/10.3934/math.2020190 doi: 10.3934/math.2020190
[3]	F. Kutlu Gündoğdu, C. Kahraman, Spherical fuzzy sets and spherical fuzzy TOPSIS method, J. Intell. Fuzzy Syst., 36 (2019), 337–352.
[4]	F. Karaaslan, Correlation coefficient of neutrosophic sets and its applications in decision-making, In: Fuzzy multi-criteria decision-making using neutrosophic sets, Cham: Springer, 2019,327–360. http://dx.doi.org/10.1007/978-3-030-00045-5_13
[5]	L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
[6]	C. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182–190. http://dx.doi.org/10.1016/0022-247X(68)90057-7
[7]	R. Smithson, Topologies generated by relations, B. Aust. Math. Soc., 1 (1969), 297–306. http://dx.doi.org/10.1017/S0004972700042167 doi: 10.1017/S0004972700042167
[8]	J. Fang, Y. Qiu, Fuzzy orders and fuzzifying topologies, Int. J. Approx. Reason., 48 (2008), 98–109. http://dx.doi.org/10.1016/j.ijar.2007.06.001 doi: 10.1016/j.ijar.2007.06.001
[9]	K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. http://dx.doi.org/10.1016/S0165-0114(86)80034-3
[10]	K. Atanssov, More on intuintionistic fuzzy sets, Fuzzy Set. Syst., 33 (1989), 37–45. http://dx.doi.org/10.1016/0165-0114(89)90215-7 doi: 10.1016/0165-0114(89)90215-7
[11]	N. Palaniappan, R. Srinivasan, Applications of intuitionistic fuzzy sets of root type in image processing, Proceddings of Annual Meeting of the North American Fuzzy Information Processing Society, 2009, 1–5. http://dx.doi.org/10.1109/NAFIPS.2009.5156480
[12]	E. Szmidt, J. Kacprzyk, An application of intuitionistic fuzzy set similarity measures to a multi-criteria decision making problem, In: Artificial intelligence and soft computing, Berlin: Springer, 2006,314–323. http://dx.doi.org/10.1007/11785231_34
[13]	I. Vlachos, G. Sergiadis, Intuitionistic fuzzy information: applications to pattern recognition, Pattern Recogn. Lett., 28 (2007), 197–206. http://dx.doi.org/10.1016/j.patrec.2006.07.004 doi: 10.1016/j.patrec.2006.07.004
[14]	A. Kaufmann, Introduction a la thiorie des sous-ensemble flous, Paris: Masson, 1973.
[15]	J. Mordeson, P. Nair, Fuzzy graphs and fuzzy hypergraphs, Heidelberg: Physica, 1998. http://dx.doi.org/10.1007/978-3-7908-1854-3
[16]	P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recogn. Lett., 6 (1987), 297–302. http://dx.doi.org/10.1016/0167-8655(87)90012-2
[17]	P. Thirunavukarasu, R. Suresh, K. Viswanathan, Energy of a complex fuzzy graph, Int. J. Math. Sci. Eng. Appl., 10 (2016), 243–248.
[18]	A. Shannon, K. Atanassov, A first step to a theory of the intuitionistic fuzzy graphs, Proceedings of the First Workshop on Fuzzy Based Expert Systems, 1994, 59–61.
[19]	M. Akram, B. Davvaz, Strong intuitionistic fuzzy graphs, Filomat, 26 (2012), 177–196.
[20]	M. Akram, R. Akmal, Operations on intuitionistic fuzzy graph structures, Fuzzy Information and Engineering, 8 (2016), 389–410. http://dx.doi.org/10.1016/j.fiae.2017.01.001 doi: 10.1016/j.fiae.2017.01.001
[21]	N. Alshehri, M. Akram, Intuitionistic fuzzy planar graphs, Discrete Dyn. Nat. Soc., 2014 (2014), 397823. http://dx.doi.org/10.1155/2014/397823 doi: 10.1155/2014/397823
[22]	M. Karunambigai, M. Akram, R. Buvaneswari, Strong and superstrong vertices in intuitionistic fuzzy graphs, J. Intell. Fuzzy Syst., 30 (2016), 671–678. http://dx.doi.org/10.3233/IFS-151786 doi: 10.3233/IFS-151786
[23]	K. Myithili, R. Parvathi, M. Akram, Certain types of intuitionistic fuzzy directed hypergraphs, Int. J. Mach. Learn. Cyber., 7 (2016), 287–295. http://dx.doi.org/10.1007/s13042-014-0253-1 doi: 10.1007/s13042-014-0253-1
[24]	A. Nagoorgani, M. Akram, S. Anupriya, Double domination on intuitionistic fuzzy graphs, J. Appl. Math. Comput., 52 (2016), 515–528. http://dx.doi.org/10.1007/s12190-015-0952-0 doi: 10.1007/s12190-015-0952-0
[25]	R. Parvathi, M. Karunambigai, K. Atanassov, Operations on intuitionistic fuzzy graphs, Proceedings of IEEE International Conference on Fuzzy Systems, 2009, 1396–1401. http://dx.doi.org/10.1109/FUZZY.2009.5277067
[26]	P. Burillo, H. Bustince, Intuitionistic fuzzy relations (Part-I), Mathware and Soft Computing, 2 (1995), 5–38.
[27]	H. Bustince, P. Burillo, Structures on intuitionistic fuzzy relations, Fuzzy Set. Syst., 78 (1996), 293–303. http://dx.doi.org/10.1016/0165-0114(96)84610-0 doi: 10.1016/0165-0114(96)84610-0
[28]	G. Deschrijver, E. Kerre, On the composition of intuitionistic fuzzy relations, Fuzzy Set. Syst., 136 (2003), 333–361. http://dx.doi.org/10.1016/S0165-0114(02)00269-5 doi: 10.1016/S0165-0114(02)00269-5
[29]	K. Hur, S. Jang, Y. Ahn, Intuitionistic fuzzy equivalence relations, Honam Math. J., 27 (2005), 163–181.
[30]	R. Borzooei, H. Rashmanlou, S. Samanta, M. Pal, A study on fuzzy labeling graphs, J. Intell. Fuzzy Syst., 30 (2016), 3349–3355. http://dx.doi.org/10.3233/IFS-152082 doi: 10.3233/IFS-152082
[31]	R. Borzooei, H. Rashmanlou, Ring sum in product intuitionistic fuzzy graphs, International Journal of Advanced Research, 7 (2015), 16–31. http://dx.doi.org/10.5373/jarpm.1971.021614 doi: 10.5373/jarpm.1971.021614
[32]	H. Rashmanlou, S. Samanta, M. Pal, R. Borzooei, A study on bipolar fuzzy graphs, J. Intell. Fuzzy Syst., 28 (2015), 571–580.
[33]	H. Rashmanlou, Y. Jun, R. Borzooei, More results on highly irregular bipolar fuzzy graphs, Ann. Fuzzy Math. Inform., 8 (2014), 149–168.
[34]	S. Samanta, M. Pal, H. Rashmanlou, R. Borzooei, Vague graphs and strengths, J. Intell. Fuzzy Syst., 30 (2016), 3675–3680. http://dx.doi.org/10.3233/IFS-162113
[35]	R. Yager, Pythagorean fuzzy subsets, Proceedings of Joint IFSA World Congress and NAFIPS Annual Meeting, 2013, 57–61. http://dx.doi.org/10.1109/IFSA-NAFIPS.2013.6608375
[36]	R. Yager, A. Abbasov, Pythagorean membership grades, complex numbers, and decision making, Int. J. Intell. Syst., 28 (2013), 436–452. http://dx.doi.org/10.1002/int.21584 doi: 10.1002/int.21584
[37]	R. Yager, Generalized orthopair fuzzy sets, IEEE T. Fuzzy Syst., 25 (2017), 1222–1230. http://dx.doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
[38]	K. Naeem, M. Riaz, F. Karaaslan, Some novel features of Pythagorean m-polar fuzzy sets with applications, Complex Intell. Syst., 7 (2021), 459–475. http://dx.doi.org/10.1007/s40747-020-00219-3 doi: 10.1007/s40747-020-00219-3
[39]	M. Akram, J. Dar, S. Naz, Certain graphs under Pythagorean fuzzy environment, Complex Intell. Syst., 5 (2019), 127–144. http://dx.doi.org/10.1007/s40747-018-0089-5 doi: 10.1007/s40747-018-0089-5
[40]	M. Akram, S. Alsulami, F. Karaaslan, A. Khan, q-Rung orthopair fuzzy graphs under Hamacher operators, J. Intell. Fuzzy Syst., 40 (2021), 1367–1390. http://dx.doi.org/10.3233/JIFS-201700 doi: 10.3233/JIFS-201700
[41]	S. Yin, H. Li, Y. Yang, Product operations on q-rung orthopair fuzzy graphs, Symmetry, 11 (2019), 588. http://dx.doi.org/10.3390/sym11040588 doi: 10.3390/sym11040588
[42]	M. Sitara, M. Akram, M. Riaz, Decision-making analysis based on q-rung picture fuzzy graph structures, J. Appl. Math. Comput., 67 (2021), 541–577. http://dx.doi.org/10.1007/s12190-020-01471-z doi: 10.1007/s12190-020-01471-z
[43]	M. Riaz, M. Hashmi, Linear Diophantine fuzzy set and its applications towards multi-attribute decision-making problems, J. Intell. Fuzzy Syst., 37 (2019), 5417–5439. http://dx.doi.org/10.3233/JIFS-190550 doi: 10.3233/JIFS-190550
[44]	M. Riaz, M. Hashmi, H. Kulsoom, D. Pamucar, Y. Chu, Linear Diophantine fuzzy soft rough sets for the selection of sustainable material handling equipment, Symmetry, 12 (2020), 1215. http://dx.doi.org/10.3390/sym12081215 doi: 10.3390/sym12081215
[45]	H. Kamaci, Linear Diophantine fuzzy algebraic structures, J. Ambient Intell. Human. Comput., 12 (2021), 10353–10373. http://dx.doi.org/10.1007/s12652-020-02826-x doi: 10.1007/s12652-020-02826-x
[46]	S. Ayub, M. Shabir, M. Riaz, M. Aslam, R. Chinram, Linear Diophantine fuzzy relations and their algebraic properties with decision making, Symmetry, 13 (2021), 945. http://dx.doi.org/10.3390/sym13060945 doi: 10.3390/sym13060945
[47]	A. Almagrabi, S. Abdullah, M. Shams, Y. Al-Otaibi, S. Ashraf, A new approach to q-linear Diophantine fuzzy emergency decision support system for COVID-19, J. Ambient Intell. Human. Comput., 13 (2022), 1687–1713. http://dx.doi.org/10.1007/s12652-021-03130-y doi: 10.1007/s12652-021-03130-y
[48]	E. Klement, F. Kouchakinejad, D. Guha, R. Mesiar, Generalizing expected values to the case of L-fuzzy events, Int. J. Gen. Syst.*, 50 (2021), 36–62. http://dx.doi.org/10.1080/03081079.2020.1870459 doi: 10.1080/03081079.2020.1870459
[49]	E. Klement, R. Mesiar, L-fuzzy sets and isomorphic lattices: are all the new results really new? Mathematics, 6 (2018), 146. http://dx.doi.org/10.3390/math6090146
[50]	P. Liu, Z. Ali, T. Mahmood, Generalized complex q-rung orthopair fuzzy Einstein averaging aggregation operators and their application in multi-attribute decision making, Complex Intell. Syst., 7 (2021), 511–538. http://dx.doi.org/10.1007/s40747-020-00197-6 doi: 10.1007/s40747-020-00197-6
[51]	N. Yaqoob, M. Gulistan, S. Kadry, H. Wahab, Complex intuitionistic fuzzy graphs with application in cellular network provider companies, Mathematics, 7 (2019), 35. http://dx.doi.org/10.3390/math7010035 doi: 10.3390/math7010035

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)