Modeling Dual-Uncertainty In Sustainable Supplier Selection Using Bipolar Complex Pythagorean Fuzzy Sets

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-17
Published: Jan 29, 2026
Cite as:
Murad M. Arar, Hariwan Z. Ibrahim, Modeling Dual-Uncertainty In Sustainable Supplier Selection Using Bipolar Complex Pythagorean Fuzzy Sets, Int. J. Anal. Appl., 24 (2026), 17.

Main Article Content

Murad M. Arar, Hariwan Z. Ibrahim

Abstract

Managing sustainability in modern supply chains requires decision-making tools that can accommodate conflicting criteria, uncertain data, and evaluations that involve both positive and negative impacts. To address these challenges, this study develops a new decision-making framework based on bipolar complex Pythagorean fuzzy sets (BCPFSs). The model integrates bipolarity, complex-valued membership degrees, and Pythagorean structures to capture the nuanced interplay of economic, environmental, and social considerations. On this foundation, two aggregation operators—the bipolar complex Pythagorean fuzzy weighted averaging (BCPFWA) and weighted geometric (BCPFWG) operators—are introduced to synthesize multidimensional information while preserving uncertainty and dual evaluations. The applicability of the framework is demonstrated through a case study on green supply chain management (GSCM). Six supplier strategies, ranging from cost-oriented to fully balanced sustainability-focused approaches, are assessed against eight attributes including cost efficiency, product quality, carbon emissions, waste management, technological integration, and social responsibility. The analysis reveals that the balanced sustainability supplier emerges as the most effective choice, consistently ranked highest by both operators. Comparative results with conventional fuzzy aggregation approaches show that the proposed operators provide richer, more stable, and more interpretable rankings, especially when trade-offs between cost and sustainability are present. This research contributes to both theory and practice: it extends the scope of fuzzy decision-making by unifying multiple existing models as special cases, and it offers a practical toolset for organizations seeking resilient and environmentally responsible supply chain solutions. The findings demonstrate that BCPF-based aggregation can enhance strategic decision-making in contexts where sustainability and uncertainty are inseparably linked.

Article Details

References

  1. L. Zadeh, Fuzzy Sets, Inf. Control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65)90241-x.
  2. K.T. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets Syst. 20 (1986), 87–96. https://doi.org/10.1016/s0165-0114(86)80034-3.
  3. R.R. Yager, Pythagorean Fuzzy Subsets, in: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, 2013, pp. 57–61. https://doi.org/10.1109/ifsa-nafips.2013.6608375.
  4. D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex Fuzzy Sets, IEEE Trans. Fuzzy Syst. 10 (2002), 171–186. https://doi.org/10.1109/91.995119.
  5. A.S. Alkouri, A.R. Salleh, Complex Intuitionistic Fuzzy Sets, AIP Conf. Proc. 1482 (2012), 464–470. https://doi.org/10.1063/1.4757515.
  6. K. Ullah, T. Mahmood, Z. Ali, N. Jan, On Some Distance Measures of Complex Pythagorean Fuzzy Sets and Their Applications in Pattern Recognition, Complex Intell. Syst. 6 (2019), 15–27. https://doi.org/10.1007/s40747-019-0103-6.
  7. W.R. Zhang, Bipolar Fuzzy Sets and Relations: A Computational Framework for Cognitive Modeling and Multiagent Decision Analysis, in: Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, IEEE, pp. 305–309. https://doi.org/10.1109/ijcf.1994.375115.
  8. W.R. Zhang, (Yin) (Yang) Bipolar Fuzzy Sets, in: 1998 IEEE International Conference on Fuzzy Systems Proceedings, IEEE, pp. 835–840. https://doi.org/10.1109/fuzzy.1998.687599.
  9. D. Ezhilmaran, K. Sankar, Morphism of Bipolar Intuitionistic Fuzzy Graphs, J. Discret. Math. Sci. Cryptogr. 18 (2015), 605–621. https://doi.org/10.1080/09720529.2015.1013673.
  10. K. Mohana, R. Jansi, Bipolar Pythagorean Fuzzy Sets and Their Application Based on Multi-Criteria Decision Making Problems, Int. J. Res. Advent Technol. 6 (2018), 3754–3764.
  11. H.Z. Ibrahim, Multi-Attribute Group Decision-Making Based on Bipolar N,m-Rung Orthopair Fuzzy Sets, Granul. Comput. 8 (2023), 1819–1836. https://doi.org/10.1007/s41066-023-00405-x.
  12. A.U.M.J. Alkouri, M.O. Massa’deh, M. Ali, On Bipolar Complex Fuzzy Sets and Its Application, J. Intell. Fuzzy Syst. 39 (2020), 383–397. https://doi.org/10.3233/jifs-191350.
  13. A. Al-Husban, A. Amourah, J.J. Jaber, Bipolar Complex Fuzzy Sets and Their Properties, Ital. J. Pure Appl. Math. 43 (2020), 754–761.
  14. M. Gulistan, N. Yaqoob, A. Elmoasry, J. Alebraheem, Complex Bipolar Fuzzy Sets: An Application in a Transport’s Company, J. Intell. Fuzzy Syst. 40 (2021), 3981–3997. https://doi.org/10.3233/jifs-200234.
  15. T. Mahmood, U. Rehman, A Novel Approach Towards Bipolar Complex Fuzzy Sets and Their Applications in Generalized Similarity Measures, Int. J. Intell. Syst. 37 (2021), 535–567. https://doi.org/10.1002/int.22639.
  16. A. Al-Husban, Bipolar Complex Intuitionistic Fuzzy Sets, Earthline J. Math. Sci. 8 (2022), 273–280. https://doi.org/10.34198/ejms.8222.273280.
  17. A.U.M.J.S. Alkouri, Z.A. Alshboul, Environmental Impact Assessment Using Bipolar Complex Intuitionistic Fuzzy Sets, Soft Comput. 29 (2025), 3795–3809. https://doi.org/10.1007/s00500-025-10608-w.
  18. R. Nandhinii, D. Amsaveni, On Bipolar Complex Intuitionistic Fuzzy Graphs, TWMS J. Appl. Eng. Math. 12 (2022), 92–106.
  19. R. Nandhini, D. Amsaveni, Bipolar Complex Pythagorean Fuzzy Graphs, in: Studies in Fuzziness and Soft Computing, Springer Singapore, 2022: pp. 1–31. https://doi.org/10.1007/978-981-19-0471-4_1.
  20. M.H. Alqahtani, D. Lou, F. Sikander, Y. Saber, C. Lee, Novel Fuzzy Ostrowski Integral Inequalities for Convex Fuzzy-Valued Mappings Over a Harmonic Convex Set: Extending Real-Valued Intervals Without the Sugeno Integrals, Mathematics 12 (2024), 3495. https://doi.org/10.3390/math12223495.
  21. S.Y. Musa, B.A. Asaad, H. Alohali, Z.A. Ameen, M.H. Alqahtani, Fuzzy N-Bipolar Soft Sets for Multi-Criteria Decision-Making: Theory and Application, Comput. Model. Eng. Sci. 143 (2025), 911–943. https://doi.org/10.32604/cmes.2025.062524.
  22. M. Rizwan khan, K. Ullah, A. Raza, T. Senapati, S. Moslem, Multi-Attribute Decision-Making Method Based on Complex T-Spherical Fuzzy Frank Prioritized Aggregation Operators, Heliyon 10 (2024), e25368. https://doi.org/10.1016/j.heliyon.2024.e25368.
  23. H. Garg, T. Mahmood, U. ur Rehman, G.N. Nguyen, Multi-Attribute Decision-Making Approach Based on Aczel-Alsina Power Aggregation Operators Under Bipolar Fuzzy Information & Its Application to Quantum Computing, Alex. Eng. J. 82 (2023), 248–259. https://doi.org/10.1016/j.aej.2023.09.073.
  24. T. Mahmood, U. ur Rehman, Multi-Attribute Decision-Making Method Based on Bipolar Complex Fuzzy Maclaurin Symmetric Mean Operators, Comput. Appl. Math. 41 (2022), 331. https://doi.org/10.1007/s40314-022-02016-9.
  25. Z. Zhao, A. Hussain, N. Zhang, K. Ullah, S. Yin, et al., Decision Support System Based on Bipolar Complex Fuzzy Hamy Mean Operators, Heliyon 10 (2024), e36461. https://doi.org/10.1016/j.heliyon.2024.e36461.
  26. T. Mahmood, Z. Ali, Multi-Attribute Decision-Making Methods Based on Aczel–Alsina Power Aggregation Operators for Managing Complex Intuitionistic Fuzzy Sets, Comput. Appl. Math. 42 (2023), 87. https://doi.org/10.1007/s40314-023-02204-1.
  27. E. Natarajan, F. Augustin, R. Saraswathy, S. Narayanamoorthy, S. Salahshour, et al., A Bipolar Intuitionistic Fuzzy Decision-Making Model for Selection of Effective Diagnosis Method of Tuberculosis, Acta Trop. 252 (2024), 107132. https://doi.org/10.1016/j.actatropica.2024.107132.
  28. T. Mahmood, U.U. Rehman, S. Shahab, Z. Ali, M. Anjum, Decision-Making by Using TOPSIS Techniques in the Framework of Bipolar Complex Intuitionistic Fuzzy N-Soft Sets, IEEE Access 11 (2023), 105677–105697. https://doi.org/10.1109/access.2023.3316879.
  29. H. Jin, A. Hussain, K. Ullah, A. Javed, Novel Complex Pythagorean Fuzzy Sets Under Aczel–Alsina Operators and Their Application in Multi-Attribute Decision Making, Symmetry 15 (2022), 68. https://doi.org/10.3390/sym15010068.
  30. W.A. Mandal, Bipolar Pythagorean Fuzzy Sets and Their Application in Multi-Attribute Decision Making Problems, Ann. Data Sci. 10 (2021), 555–587. https://doi.org/10.1007/s40745-020-00315-8.
  31. S. Yin, Y. Wang, Q. Zhang, Mechanisms and Implementation Pathways for Distributed Photovoltaic Grid Integration in Rural Power Systems: A Study Based on Multi-Agent Game Theory Approach, Energy Strat. Rev. 60 (2025), 101801. https://doi.org/10.1016/j.esr.2025.101801.
  32. Z. Xu, Intuitionistic Fuzzy Aggregation Operators, IEEE Trans. Fuzzy Syst. 15 (2007), 1179–1187. https://doi.org/10.1109/tfuzz.2006.890678.
  33. Z. Xu, R.R. Yager, Some Geometric Aggregation Operators Based on Intuitionistic Fuzzy Sets, Int. J. Gen. Syst. 35 (2006), 417–433. https://doi.org/10.1080/03081070600574353.
  34. K. Rahman, A. Ali, M. Shakeel, M.S. A. Khan, M. Ullah, Pythagorean Fuzzy Weighted Averaging Aggregation Operator and Its Application to Decision Making Theory, Nucleus 54 (2017), 190–196. https://doi.org/10.71330/thenucleus.2017.184.
  35. K. Rahman, M.A. Khan, M. Ullah, A. Fahmi, Multiple Attribute Group Decision Making for Plant Location Selection with Pythagorean Fuzzy Weighted Geometric Aggregation Operator, Nucleus 54 (2017), 66–74. https://doi.org/10.71330/thenucleus.2017.107.
  36. T. Mahmood, U.U. Rehman, Z. Ali, M. Aslam, R. Chinram, Identification and Classification of Aggregation Operators Using Bipolar Complex Fuzzy Settings and Their Application in Decision Support Systems, Mathematics 10 (2022), 1726. https://doi.org/10.3390/math10101726.