SF-ARAS

SF-ARAS - ARAS yönteminin Spherical uzantısı

Spherical üstünlük/sıralama - Küresel Bulanık Küme (KBK: μ, ν, π; μ²+ν²+π² ≤ 1)

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

normalization

${\tilde{a}}_{i j}^{c} = (ν_{i j}, μ_{i j}, π_{i j}) for cost criteria$

LaTeX
\tilde{a}_{ij}^{c}=(\nu_{ij},\mu_{ij},\pi_{ij})\quad\text{for cost criteria}
ranking_computation

$A_{0} = ({max}_{i} μ_{i j}, {min}_{i} ν_{i j}, {min}_{i} π_{i j})$

LaTeX
A_0=\bigl(\max_i\mu_{ij},\;\min_i\nu_{ij},\;\min_i\pi_{ij}\bigr)
ranking_computation

$s ({\tilde{a}}_{i j}) = \frac{1}{3} (2 + μ_{i j} - ν_{i j} - π_{i j}), {\hat{s}}_{i j} = \frac{s ({\tilde{a}}_{i j})}{\sum_{i = 0}^{m} s ({\tilde{a}}_{i j})}$

LaTeX
s(\tilde{a}_{ij})=\frac{1}{3}(2+\mu_{ij}-\nu_{ij}-\pi_{ij}),\quad \hat{s}_{ij}=\frac{s(\tilde{a}_{ij})}{\sum_{i=0}^{m}s(\tilde{a}_{ij})}
aggregation

${\tilde{S}}_{i} = ⨁_{j = 1}^{n} w_{j} {\tilde{v}}_{i j} = (\sqrt{1 - \prod_{j = 1}^{n} (1 - μ_{i j}^{2})^{w_{j}}}, \prod_{j = 1}^{n} ν_{i j}^{w_{j}}, \sqrt{\prod_{j = 1}^{n} (1 - μ_{i j}^{2})^{w_{j}} - \prod_{j = 1}^{n} (1 - μ_{i j}^{2} - π_{i j}^{2})^{w_{j}}})$

LaTeX
\tilde{S}_i=\bigoplus_{j=1}^{n}w_j\tilde{v}_{ij}=\Bigl(\sqrt{1-\prod_{j=1}^n(1-\mu_{ij}^2)^{w_j}},\;\prod_{j=1}^n\nu_{ij}^{w_j},\;\sqrt{\prod_{j=1}^n(1-\mu_{ij}^2)^{w_j}-\prod_{j=1}^n(1-\mu_{ij}^2-\pi_{ij}^2)^{w_j}}\Bigr)
ranking_computation

$K_{i} = \frac{s ({\tilde{S}}_{i})}{s ({\tilde{S}}_{0})}, rank descending$

LaTeX
K_i=\frac{s(\tilde{S}_i)}{s(\tilde{S}_0)},\quad\text{rank descending}

Yöntem ayrıntıları kaynak kütüphanedeki özgün (İngilizce) metindir.

Sezgi

Sonucu okuma: sf-aras extends ARAS to handle Spherical uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Spherical Fuzzy Set (SFS: μ, ν, π; μ²+ν²+π² ≤ 1) algebra. The final scores are defuzzified via score function S = μ² − ν² before ranking.

Varsayımlar

Decision matrix entries are valid Spherical Fuzzy numbers/tuples
Underlying crisp method's compensation assumption holds in uncertain space
All decision-maker(s) and experts use the same linguistic/uncertainty scale

Ne zaman kullanılmaz

Classical data sufficient - use base ARAS directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sık yapılan hatalar

Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin SFS: μ,ν,π ∈ [0,1]; μ²+ν²+π² ≤ 1 koşulunu sağladığından emin olun.
Defuzzifikasyon yöntemi sıralamayı etkiler: score function S = μ² − ν² kanonik seçimdir.