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

Öz

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.

Anahtar Kelimeler

• 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

Öz

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.

Anahtar Kelimeler

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

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