Novel Distance Measure for Linear Diophantine Fuzzy Sets with CODAS-Based Decision-Making

Abstract

Decision-making under uncertainty requires mathematical tools capable of capturing the complexity of real-world expert evaluations. Linear Diophantine fuzzy sets address this need by incorporating reference parameters alongside membership and non-membership degrees, offering greater flexibility than earlier fuzzy frameworks. However, existing distance measures for this setting rely entirely on component-wise differences and can fail to distinguish between two elements whenever those differences are equal. This paper proposes a new distance measure that extends the standard component-wise sum by introducing cross-interaction terms between the membership degrees and between the reference parameters of the two elements under consideration. Axiomatic validity is established through a formal proof, and the measure is compared against nine existing distance measures, revealing cases where existing measures lose discriminating power while the proposed one does not. Building on this distance, a CODAS-based multi-criteria decision-making procedure is developed for the linear Diophantine fuzzy environment and applied to a logistics specialist selection problem. Results are benchmarked against existing methods, and the consistency of the proposed approach is discussed alongside its limitations and directions for future work.

Keywords

• linear Diophantine fuzzy sets
• distance measure
• CODAS
• multi-criteria decision-making

CODAS Tabanlı Karar Verme ile Doğrusal Diophantine Bulanık Kümeler için Yeni Bir Mesafe Ölçüsü

Abstract

Belirsizlik altında karar verme, gerçek dünyadaki uzman değerlendirmelerinin karmaşıklığını yakalayabilen matematiksel araçlar gerektirir. Doğrusal Diophantine bulanık kümeler, üyelik ve üye olmama derecelerinin yanı sıra referans parametrelerini de dahil ederek bu ihtiyacı karşılar ve önceki bulanık çerçevelere göre daha fazla esneklik sunar. Bununla birlikte, bu ortam için mevcut mesafe ölçüleri tamamen bileşen bazındaki farklara dayanır ve bu farklar eşit olduğunda iki öğeyi ayırt edemeyebilir. Bu makale, üyelik dereceleri ve incelenen iki öğenin referans parametreleri arasında çapraz etkileşim terimleri ekleyerek standart bileşen bazındaki toplamı genişleten yeni bir mesafe ölçüsü önermektedir. Aksiyomatik geçerlilik, biçimsel bir ispatla ortaya konulmuş ve ölçü, dokuz mevcut mesafe ölçüsüyle karşılaştırılmıştır; bu karşılaştırma, mevcut ölçülerin ayırt edici gücünü kaybettiği, önerilen ölçünün ise kaybetmediği durumları ortaya koymaktadır. Bu mesafeye dayanarak, doğrusal Diophantine bulanık ortam için CODAS tabanlı çok kriterli bir karar verme prosedürü geliştirilmiş ve bir lojistik uzmanı seçim problemine uygulanmıştır. Sonuçlar mevcut yöntemlerle karşılaştırılmış ve önerilen yaklaşımın tutarlılığı, sınırlamaları ve gelecekteki çalışmalar için yönleriyle birlikte tartışılmıştır.

Keywords

• Doğrusal Diophantine bulanık kümeler
• mesafe ölçüsü
• CODAS
• çok kriterli karar verme

References

1. Zadeh LA. Fuzzy sets. Inf Control. 1965; 8(3): 338-353.
2. Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986; 20(1): 87-96.
3. Yager RR. Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst. 2014; 22(4): 958-965.
4. Yager RR. Generalized orthopair fuzzy sets. IEEE Trans Fuzzy Syst. 2017; 25(5): 1222-1230.
5. Aydoğdu E, Gül A. A novel entropy proposition for spherical fuzzy sets and its application in multiple attribute decision-making. Int J Intell Syst. 2020; 35(9): 1354-1374.
6. Aydoğdu A, Gül S. New entropy propositions for interval-valued spherical fuzzy sets and their usage in an extension of ARAS (ARAS-IVSFS). Expert Syst. 2022; 39(4): e12898.
7. Aydoğdu E, Güner E, Aldemir B, Aygün H. Complex spherical fuzzy TOPSIS based on entropy. Expert Syst Appl. 2023; 215: 119331.
8. Aydoğdu E, Aldemir B, Güner E, Aygün H. A novel entropy measure with its application to the COPRAS method in complex spherical fuzzy environment. Informatica. 2023; 34(4): 679-711.

9. Güner E, Aldemir B, Aydoğdu E, Aygün H. Novel Hamacher aggregation operators with applications to the AHP-COPRAS method for spherical fuzzy environment. Neural Comput Appl. 2025; 37(18): 12941-12990.
10. Aydoğdu E. A Spherical fuzzy entropy–driven TOPSIS framework with hybrid weighting for group decision-making. AIMS Mathematics, 2026 ;11(3): 7015-7046.
11. Güner E, Aygün H. A comparative study on the generalized spherical fuzzy Einstein aggregation operators. Gazi University Journal of Science, 2023; 37(1): 393-413.
12. Güner E, Aldemir B, Aygün H. A literature review on the theory and applications of spherical fuzzy sets. Expert Systems with Applications, 2026; 332C: 133610.
13. Aldemir B, Aygün H, Valero O. On inducing similarity measures on spherical fuzzy sets and application on colored edge detection. Measurement, 2026; 271: 120963.
14. Baydaş M, Ersoy N. Artificial intelligence-assisted multi-criteria decision-making methodology: from research trends to the future roadmap. Turkish Journal of Nature and Science, 2025; 14(1): 180-191.
15. Riaz M, Hashmi MR. Linear Diophantine fuzzy set and its applications towards multi-attribute decision-making problems. J Intell Fuzzy Syst. 2019; 37(4): 5417-5439.
16. Riaz M, Farid HMA, Karaaslan F. Linear Diophantine fuzzy aggregation operators with multi-criteria decision-making. J Comput Cogn Eng. 2025; 4(1): 24-35.
17. Gül S, Aydoğdu A. Novel distance and entropy definitions for linear Diophantine fuzzy sets and an extension of TOPSIS (LDF-TOPSIS). Expert Syst. 2023; 40(1): e13104.
18. Aydoğdu A, Gül S, Alniak T. New information measures for linear Diophantine fuzzy sets and their applications with LDF-ARAS on data storage system selection problem. Expert Syst Appl. 2024; 252(Part A): 124135.
19. Aydoğdu A. Novel linear Diophantine fuzzy information measures based decision making approach using extended VIKOR method. IEEE Access. 2023; 11: 95526-95544.
20. Aldemir B, Aydoğdu E, Güner E, Aygün H. Information measures on linear Diophantine fuzzy soft sets with their applications to the medical diagnosis. J Mult-Valued Logic Soft Comput. 2024; 44(1-2): 1-47.
21. Mohammad MMS, Abdullah S, Al-Shomrani MM. Some linear Diophantine fuzzy similarity measures and their application in decision making problem. IEEE Access. 2022; 10: 29859-29877.
22. Aydoğdu A, Özkan E. Novel distance measure for linear Diophantine fuzzy sets. Res Rev Discrete Math Struct. 2022; 9(2): 38-45.
23. Aldemir B, Güner E, Aygün H. Linear Diophantine fuzzy sets: Image edge detection techniques based on similarity measures. WSEAS Trans Signal Process. 2023; 19: 205-214.
24. Abirami KM, Srikanth R, Dhanasekaran P. Linear Diophantine fuzzy similarity measure with applications to TOPSIS and clustering analysis. Int J Fuzzy Syst. 2025.
25. Kannan J, Jayakumar V, Pethaperumal M. Advanced fuzzy-based decision-making: the linear Diophantine fuzzy CODAS method for logistic specialist selection. Spectr Oper Res. 2025; 2(1): 41-60.
26. Aldemir B, Güner E, Aydoğdu E, Aygün H. Some Fixed Point Theorems in Partial Fuzzy Metric Spaces. Journal of the Institute of Science and Technology, 2020; 10(4): 2889-2900.
27. Gül S, Aydoğdu A. Novel entropy measure definitions and their uses in a modified combinative distance-based assessment (CODAS) method under picture fuzzy environment. Informatica. 2021; 32(4): 759-794.
28. Keshavarz Ghorabaee M, Zavadskas EK, Turskis Z, Antucheviciene J. A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Econ Comput Econ Cybern Stud Res. 2016; 50: 25-44.
29. Karsak EE. Personnel selection using a fuzzy MCDM approach based on ideal and antiideal solutions. In: Köksalan M, Zionts S, editors. Multiple Criteria Decision Making in the New Millennium. Lecture Notes in Economics and Mathematical Systems, vol 507. Berlin:Springer; 2001.
30. Boran FE, Genç S, Akay D. Personnel selection based on intuitionistic fuzzy sets. Hum Factors Ergon Manuf Serv Ind. 2011; 21(5): 493–503.
31. Zhang SF, Liu SY. A GRA-based intuitionistic fuzzy multi-criteria group decision making method for personnel selection. Expert Systems with Applications, 2011; 38(9): 11401-11405.
32. Altuntas G, Yildirim BF. Logistics specialist selection with intuitionistic fuzzy TOPSIS method. International Journal of Logistics Systems and Management, 2022; 42(1): 1-34.