Document Type : Article

**Authors**

^{1}
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan

^{2}
Department of Mathematics, Division of Science and Technology, University of Education, Lahore, Pakistan

^{3}
Department of Mathematics, Minhaj University, Lahore, Pakistan

**Abstract**

The q-rung orthopair fuzzy sets (q-ROFSs) are increasingly valuable to express fuzzy and vague

information, as the generalization of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets

(PFSs). In this paper, we propose complex $q$-rung orthopair fuzzy sets (C$q$-ROFSs) as a new tool to deal with vagueness, uncertainty and fuzziness by extending the range of membership and non-membership function of $q$-ROFS from real to complex number with the unit disc.

We develop some new complex $q$-rung orthopair fuzzy Hamacher operations and complex $q$-rung orthopair fuzzy Hamacher aggregation operators, i.e., the complex $q$-rung orthopair fuzzy Hamacher weighted average (C$q$-ROFHWA) operator, and the complex $q$-rung orthopair fuzzy Hamacher weighted geometric (C$q$-ROFHWG) operator. Subsequently, we introduce the innovative concept of a complex $q$-rung orthopair fuzzy graphs based on Hamacher operator called complex $q$-rung orthopair fuzzy Hamacher graphs (C$q$-ROFHGs) and determine its energy and Randi'{c} energy.

In particular, we present the energy of a splitting C$q$-ROFHG and shadow C$q$-ROFHG. Further, we describe the notions of complex $q$-rung orthopair fuzzy Hamacher digraphs (C$q$-ROFHDGs).

Finally, a numerical instance related to the facade clothing systems selection is presented to demonstrate the validity of the proposed concepts in decision making (DM).

information, as the generalization of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets

(PFSs). In this paper, we propose complex $q$-rung orthopair fuzzy sets (C$q$-ROFSs) as a new tool to deal with vagueness, uncertainty and fuzziness by extending the range of membership and non-membership function of $q$-ROFS from real to complex number with the unit disc.

We develop some new complex $q$-rung orthopair fuzzy Hamacher operations and complex $q$-rung orthopair fuzzy Hamacher aggregation operators, i.e., the complex $q$-rung orthopair fuzzy Hamacher weighted average (C$q$-ROFHWA) operator, and the complex $q$-rung orthopair fuzzy Hamacher weighted geometric (C$q$-ROFHWG) operator. Subsequently, we introduce the innovative concept of a complex $q$-rung orthopair fuzzy graphs based on Hamacher operator called complex $q$-rung orthopair fuzzy Hamacher graphs (C$q$-ROFHGs) and determine its energy and Randi'{c} energy.

In particular, we present the energy of a splitting C$q$-ROFHG and shadow C$q$-ROFHG. Further, we describe the notions of complex $q$-rung orthopair fuzzy Hamacher digraphs (C$q$-ROFHDGs).

Finally, a numerical instance related to the facade clothing systems selection is presented to demonstrate the validity of the proposed concepts in decision making (DM).

**Keywords**

References:

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9. Senapati, T. and Yager, R.R. "Fermatean fuzzy sets", Journal of Ambient Intelligence and Humanized Computing, 11(2), pp. 663-674 (2020).

10. Yager, R.R. "Generalized orthopair fuzzy sets", IEEE Transactions on Fuzzy Systems, 25(5), pp. 1222-1230 (2016).

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15. Rani, D. and H. Garg, "Distance measures between the complex intuitionistic fuzzy sets and their applications to the decision-making process", International Journal for Uncertainty Quantification, 7(5), pp. 423-439 (2017).

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17. Garg, H. and Rani, D. "Exponential, logarithmic and compensative generalized aggregation operators under complex intuitionistic fuzzy environment", Group Decision and Negotiation, 28(5), pp. 991-1050 (2019).

18. Rani, D. and Garg, H. "Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making", Expert Systems, 35(6), 12325 (2018).

19. Liu, P., Mahmood, T., and Ali, Z. "Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision-making", Information, 11(1), 5 (2020).

20. Garg, H., Gwak, J., Mahmood, T., et al. "Power aggregation operators and VIKOR methods for complex q-rung orthopair fuzzy sets and their applications", Mathematics, 8(4), p. 538 (2020).

21. Liu, P., Ali, Z., and Mahmood, T., "A method to multi-attribute group decision-making problem with complex q-rung orthopair linguistic information based on Heronian mean operators", International Journal of Computational Intelligence Systems, 12(2), pp. 1465- 1496 (2019).

22. Gutman, I. "The energy of a graph: old and new results", In Algebraic Combinatorics and Applications, Springer Berlin Heidelberg, pp. 196-211 (2001).

23. Vaidya, S.K. and Popat, K.M. "Some new results on energy of graphs", MATCH Commun. Math. Comput. Chem, 77, pp. 589-594 (2017) .

24. Rosenfeld, A. "Fuzzy graphs, fuzzy sets and their applications" (L.A. Zadeh, K.S. Fu, M. Shimura, Eds.) Academic Press, New York, pp. 77-95 (1975).

25. Anjali, N. and Mathew, S. "Energy of a fuzzy graph", Annals of Fuzzy Mathematics and Informatics, 6, pp. 455-465 (2013).

26. Akram, M. and Naz, S. "Energy of Pythagorean fuzzy graphs with applications", Mathematics, 6(8), p. 136 (2018).

27. Akram, M., Naz, S., and Davvaz, B. "Simplified interval-valued Pythagorean fuzzy graphs with application", Complex and Intelligent Systems, 5(2), pp. 229-253 (2019).

28. Naz, S. and Akram, M. "Novel decision-making approach based on hesitant fuzzy sets and graph theory", Computational and Applied Mathematics, 38(1), p. 7 (2019).

29. Thirunavukarasu, P., Suresh, R., and Viswanathan, K.K. "Energy of a complex fuzzy graph", International Journal of Mathematical Sciences and Engineering Applications, 10(1), pp. 243-248 (2016).

30. Luqman, A., Akram, M., Al-Kenani, A.N., et al. "A study on hypergraph representations of complex fuzzy information", Symmetry, 11(11), p. 1381 (2019).

31. Luqman, A., Akram, M., and Smarandache, F. "Complex neutrosophic hypergraphs: new social network models", Algorithms, 12(11), pp. 234-261 (2019).

32. Naz, S., Ashraf, S., and Akram, M. "A novel approach to decision-making with Pythagorean fuzzy information", Mathematics, 6(6), pp. 95-122 (2018).

33. Akram, M. and Naz, S. "A novel decision-making approach under complex pythagorean fuzzy environment", Mathematical and Computational Applications, 24(3), pp. 73-105 (2019).

34. Habib, A., Akram, M., and Farooq, A. "q-Rung orthopair fuzzy competition graphs with application in the soil ecosystem", Mathematics, 7(1), p. 91 (2019).

35. Yin, S., Li, H., and Yang, Y. "Product operations on q-rung orthopair fuzzy graphs", Symmetry, 11(4), pp. 588-610 (2019).

36. Akram, M., Alsulami, S., Karaaslan, F., et al. "q-rung orthopair fuzzy graphs under Hamacher operators", Journal of Intelligent and Fuzzy Systems, 40(1), pp. 1367-1390 (2021).

37. Guleria, A. and Bajaj, R.K. "T-spherical fuzzy graphs: operations and applications in various selection processes", Arabian Journal for Science and Engineering, 45(3), pp. 2177-2193 (2020).

38. Akram, M., Naz, S., Shahzadi, S., et al. Geometricarithmetic energy and atom bond connectivity energy of dual hesitant q-rung orthopair fuzzy graphs", Journal of Intelligent and Fuzzy Systems, 40, pp. 1287- 1307 (2021).

39. Naz, S., Akram, M., Alsulami, S., et al. "Decisionmaking analysis under interval-valued q-rung orthopair dual hesitant fuzzy environment", International Journal of Computational Intelligence Systems, 14(1), pp. 332-357 (2021).

40. Hamacher, H. "Uber logische verknupfungen unscharfer aussagen und deren zugehrige bewertungsfunktionen", Progress in Cybernetics and Systems Research, 3, pp. 276-288 (1978).

41. Liu, P., Mahmood, T., and Ali, Z. "Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision-making", Information, 11(1), pp. 5-31 (2020).

42. Garg, H., Gwak, J., Mahmood, T., et al. "Power aggregation operators and VIKOR methods for complex q-rung orthopair fuzzy sets and their applications", Mathematics, 8(4), pp. 538-571 (2020).

2. Ye, J., Zhan, J., Ding, W., et al. "A novel fuzzy rough set model with fuzzy neighborhood operators", Information Sciences, 544, pp. 266-297 (2021).

3. Wang, J., Ma, X., Xu, Z., et al. "Three-way multiattribute decision making under hesitant fuzzy environments", Information Sciences, 552, pp. 328-351 (2021).

4. Zhan, J., Jiang, H., and Yao, Y. "Three-way multiattribute decision-making based on outranking relations", IEEE Transactions on Fuzzy Systems, (2020). DOI: 10.1109/TFUZZ.2020.3007423.

5. Ye, J., Zhan, J., and Xu, Z. "A novel decisionmaking approach based on three-way decisions in fuzzy information systems", Information Sciences , 541, pp. 362-390 (2020).

6. Zadeh, L.A. "Fuzzy sets", Information and Control, 8(3), pp. 338-353 (1965).

7. Atanassov, K.T. "Intuitionistic fuzzy sets", Fuzzy Sets and Systems, 20(1), pp. 87-96 (1986).

8. Yager, R.R. "Pythagorean membership grades in multi-criteria decision making", IEEE Transactions on Fuzzy Systems, 22(4), pp. 958-965 (2014).

9. Senapati, T. and Yager, R.R. "Fermatean fuzzy sets", Journal of Ambient Intelligence and Humanized Computing, 11(2), pp. 663-674 (2020).

10. Yager, R.R. "Generalized orthopair fuzzy sets", IEEE Transactions on Fuzzy Systems, 25(5), pp. 1222-1230 (2016).

11. Ramot, D., Milo, R., Friedman, M., et al. "Complex fuzzy sets", IEEE Transactions on Fuzzy Systems, 10(2), pp. 171-186 (2002).

12. Alkouri, A. and Salleh, A. "Complex atanassov's intuitionistic fuzzy relation", In Abstract and Applied Analysis, pp. 1-18 (2013).

13. Alkouri, A. and Salleh, A.R. "Complex intuitionistic fuzzy sets", In AIP Conference Proceedings, 1482(1), pp. 464-470 (2012).

14. Garg, H. and Rani, D. "Robust averaging-geometric aggregation operators for complex intuitionistic fuzzy sets and their applications to MCDM process", Arabian Journal for Science and Engineering, 45(3), pp. 2017-2033 (2020).

15. Rani, D. and H. Garg, "Distance measures between the complex intuitionistic fuzzy sets and their applications to the decision-making process", International Journal for Uncertainty Quantification, 7(5), pp. 423-439 (2017).

16. Garg, H. and Rani, D., "New generalised Bonferroni mean aggregation operators of complex intuitionistic fuzzy information based on Archimedean t-norm and t-conorm", Journal of Experimental & Theoretical Artificial Intelligence, 32(1), pp. 81-109 (2020).

17. Garg, H. and Rani, D. "Exponential, logarithmic and compensative generalized aggregation operators under complex intuitionistic fuzzy environment", Group Decision and Negotiation, 28(5), pp. 991-1050 (2019).

18. Rani, D. and Garg, H. "Complex intuitionistic fuzzy power aggregation operators and their applications in multi-criteria decision-making", Expert Systems, 35(6), 12325 (2018).

19. Liu, P., Mahmood, T., and Ali, Z. "Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision-making", Information, 11(1), 5 (2020).

20. Garg, H., Gwak, J., Mahmood, T., et al. "Power aggregation operators and VIKOR methods for complex q-rung orthopair fuzzy sets and their applications", Mathematics, 8(4), p. 538 (2020).

21. Liu, P., Ali, Z., and Mahmood, T., "A method to multi-attribute group decision-making problem with complex q-rung orthopair linguistic information based on Heronian mean operators", International Journal of Computational Intelligence Systems, 12(2), pp. 1465- 1496 (2019).

22. Gutman, I. "The energy of a graph: old and new results", In Algebraic Combinatorics and Applications, Springer Berlin Heidelberg, pp. 196-211 (2001).

23. Vaidya, S.K. and Popat, K.M. "Some new results on energy of graphs", MATCH Commun. Math. Comput. Chem, 77, pp. 589-594 (2017) .

24. Rosenfeld, A. "Fuzzy graphs, fuzzy sets and their applications" (L.A. Zadeh, K.S. Fu, M. Shimura, Eds.) Academic Press, New York, pp. 77-95 (1975).

25. Anjali, N. and Mathew, S. "Energy of a fuzzy graph", Annals of Fuzzy Mathematics and Informatics, 6, pp. 455-465 (2013).

26. Akram, M. and Naz, S. "Energy of Pythagorean fuzzy graphs with applications", Mathematics, 6(8), p. 136 (2018).

27. Akram, M., Naz, S., and Davvaz, B. "Simplified interval-valued Pythagorean fuzzy graphs with application", Complex and Intelligent Systems, 5(2), pp. 229-253 (2019).

28. Naz, S. and Akram, M. "Novel decision-making approach based on hesitant fuzzy sets and graph theory", Computational and Applied Mathematics, 38(1), p. 7 (2019).

29. Thirunavukarasu, P., Suresh, R., and Viswanathan, K.K. "Energy of a complex fuzzy graph", International Journal of Mathematical Sciences and Engineering Applications, 10(1), pp. 243-248 (2016).

30. Luqman, A., Akram, M., Al-Kenani, A.N., et al. "A study on hypergraph representations of complex fuzzy information", Symmetry, 11(11), p. 1381 (2019).

31. Luqman, A., Akram, M., and Smarandache, F. "Complex neutrosophic hypergraphs: new social network models", Algorithms, 12(11), pp. 234-261 (2019).

32. Naz, S., Ashraf, S., and Akram, M. "A novel approach to decision-making with Pythagorean fuzzy information", Mathematics, 6(6), pp. 95-122 (2018).

33. Akram, M. and Naz, S. "A novel decision-making approach under complex pythagorean fuzzy environment", Mathematical and Computational Applications, 24(3), pp. 73-105 (2019).

34. Habib, A., Akram, M., and Farooq, A. "q-Rung orthopair fuzzy competition graphs with application in the soil ecosystem", Mathematics, 7(1), p. 91 (2019).

35. Yin, S., Li, H., and Yang, Y. "Product operations on q-rung orthopair fuzzy graphs", Symmetry, 11(4), pp. 588-610 (2019).

36. Akram, M., Alsulami, S., Karaaslan, F., et al. "q-rung orthopair fuzzy graphs under Hamacher operators", Journal of Intelligent and Fuzzy Systems, 40(1), pp. 1367-1390 (2021).

37. Guleria, A. and Bajaj, R.K. "T-spherical fuzzy graphs: operations and applications in various selection processes", Arabian Journal for Science and Engineering, 45(3), pp. 2177-2193 (2020).

38. Akram, M., Naz, S., Shahzadi, S., et al. Geometricarithmetic energy and atom bond connectivity energy of dual hesitant q-rung orthopair fuzzy graphs", Journal of Intelligent and Fuzzy Systems, 40, pp. 1287- 1307 (2021).

39. Naz, S., Akram, M., Alsulami, S., et al. "Decisionmaking analysis under interval-valued q-rung orthopair dual hesitant fuzzy environment", International Journal of Computational Intelligence Systems, 14(1), pp. 332-357 (2021).

40. Hamacher, H. "Uber logische verknupfungen unscharfer aussagen und deren zugehrige bewertungsfunktionen", Progress in Cybernetics and Systems Research, 3, pp. 276-288 (1978).

41. Liu, P., Mahmood, T., and Ali, Z. "Complex q-rung orthopair fuzzy aggregation operators and their applications in multi-attribute group decision-making", Information, 11(1), pp. 5-31 (2020).

42. Garg, H., Gwak, J., Mahmood, T., et al. "Power aggregation operators and VIKOR methods for complex q-rung orthopair fuzzy sets and their applications", Mathematics, 8(4), pp. 538-571 (2020).

Transactions on Industrial Engineering (E)

July and August 2023Pages 1450-1479