IVAIF-CODAS - Aralık Değerli Atanassov Sezgisel Bulanık CODAS

Formül adımları

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

Adım 1 — Her DM için IVAISF karar matrisi X̃^l oluştur. Her giriş x̃_ijl = ([μL_ijl, μU_ijl], [vL_ijl, vU_ijl]) dilsel ölçekten atanır. DM l, kriter önem değerlendirmesi de yapar.

${\tilde{x}}_{i j l} = ([μ_{i j l}^{L}, μ_{i j l}^{U}], [v_{i j l}^{L}, v_{i j l}^{U}]); 0 \leq μ_{i j l}^{U} + v_{i j l}^{U} \leq 1$

LaTeX
\tilde{x}_{ijl} = \left([\mu^L_{ijl}, \mu^U_{ijl}], [v^L_{ijl}, v^U_{ijl}]\right); \quad 0 \le \mu^U_{ijl} + v^U_{ijl} \le 1
Adım 2 — IIFWA operatörü ile bireysel DM matrislerini tek IVAISF karar matrisi X̃ ve kriter ağırlıkları W̃ ile topla.

${IIFWA}_{w} ({\tilde{A}}_{1}, \dots, {\tilde{A}}_{k}) = ([1 - \prod_{l = 1}^{k} (1 - a_{l})^{w_{l}}, 1 - \prod_{l = 1}^{k} (1 - b_{l})^{w_{l}}], [\prod_{l = 1}^{k} c_{l}^{w_{l}}, \prod_{l = 1}^{k} d_{l}^{w_{l}}])$

LaTeX
\text{IIFWA}_w(\tilde{A}_1,\ldots,\tilde{A}_k) = \left(\left[1-\prod_{l=1}^k(1-a_l)^{w_l},\; 1-\prod_{l=1}^k(1-b_l)^{w_l}\right],\; \left[\prod_{l=1}^k c_l^{w_l},\; \prod_{l=1}^k d_l^{w_l}\right]\right)
Adım 3 — IVAISF ağırlıklı karar matrisi R̃ hesapla. Her ağırlıklı giriş r̃_ij = w̃_j ⊗ x̃_ij, IVAIFN çarpma operasyonunu kullanır.

${\tilde{r}}_{i j} = {\tilde{w}}_{j} \otimes {\tilde{x}}_{i j} = ([μ_{w j}^{L} μ_{i j}^{L}, μ_{w j}^{U} μ_{i j}^{U}], [v_{w j}^{L} + v_{i j}^{L} - v_{w j}^{L} v_{i j}^{L}, v_{w j}^{U} + v_{i j}^{U} - v_{w j}^{U} v_{i j}^{U}])$

LaTeX
\tilde{r}_{ij} = \tilde{w}_j \otimes \tilde{x}_{ij} = \left([\mu^L_{wj}\mu^L_{ij},\; \mu^U_{wj}\mu^U_{ij}],\; [v^L_{wj}+v^L_{ij}-v^L_{wj}v^L_{ij},\; v^U_{wj}+v^U_{ij}-v^U_{wj}v^U_{ij}]\right)
Adım 4 — IVAISF negatif ideal çözüm NS̃ belirle. Her kriter j için negatif ideal; tüm alternatifler arasında minimum üyelik, maksimum üye-olmama değerlerini taşıyan IVAIFN'dir.

${\tilde{n}}_{j} = {min}_{i} {\tilde{r}}_{i j} = ([{min}_{i} μ_{i j}^{L}, {min}_{i} μ_{i j}^{U}], [{max}_{i} v_{i j}^{L}, {max}_{i} v_{i j}^{U}])$

LaTeX
\tilde{n}_j = \min_i \tilde{r}_{ij} = \left([\min_i \mu^L_{ij},\; \min_i \mu^U_{ij}],\; [\max_i v^L_{ij},\; \max_i v^U_{ij}]\right)
Adım 5 — Her alternatif için NS̃'den normalize Öklid mesafesi (ED_i) ve normalize Hamming mesafesi (HD_i) hesapla (Park ve ark. 2008 IVAISF mesafe formülleri).

$E D_{i} = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} [(μ_{i j}^{L} - μ_{n j}^{L})^{2} + (μ_{i j}^{U} - μ_{n j}^{U})^{2} + (v_{i j}^{L} - v_{n j}^{L})^{2} + (v_{i j}^{U} - v_{n j}^{U})^{2}]}; H D_{i} = \frac{1}{n} \sum_{j = 1}^{n} [| μ_{i j}^{L} - μ_{n j}^{L} | + | μ_{i j}^{U} - μ_{n j}^{U} | + | v_{i j}^{L} - v_{n j}^{L} | + | v_{i j}^{U} - v_{n j}^{U} |] N o r m a l i z a t i o n n o t e : d i v i s o r i s n (n u m b e r o f c r i t e r i a), N O T 4 n . Y e n i - Ö z ç e l i k 2018 T a b l e 9 n u m e r i c s c o n f i r m t h i s; P a r k e t a l . 2008 u s e 1 / (2 n) o v e r 8 c o m p o n e n t s (μ L, μ U, ν L, ν U, π L, π U) w i t h h e s i t a n c y b o u n d s, w h i c h Y e n i - Ö z ç e l i k a d a p t t o 1 / n o v e r t h e 4 s t a n d a r d c o m p o n e n t s (μ L, μ U, ν L, ν U) . R e p r o d u c e r m a t c h e s T a b l e 9 E D / H D e n t r i e s t o \leq 5 e - 5 a b s o l u t e .$

LaTeX
ED_i = \sqrt{\frac{1}{n}\sum_{j=1}^n \left[(\mu^L_{ij}-\mu^L_{nj})^2 + (\mu^U_{ij}-\mu^U_{nj})^2 + (v^L_{ij}-v^L_{nj})^2 + (v^U_{ij}-v^U_{nj})^2\right]}; \quad HD_i = \frac{1}{n}\sum_{j=1}^n \left[|\mu^L_{ij}-\mu^L_{nj}| + |\mu^U_{ij}-\mu^U_{nj}| + |v^L_{ij}-v^L_{nj}| + |v^U_{ij}-v^U_{nj}|\right] Normalization note: divisor is n (number of criteria), NOT 4n. Yeni-Özçelik 2018 Table 9 numerics confirm this; Park et al. 2008 use 1/(2n) over 8 components (μL, μU, νL, νU, πL, πU) with hesitancy bounds, which Yeni-Özçelik adapt to 1/n over the 4 standard components (μL, μU, νL, νU). Reproducer matches Table 9 ED/HD entries to ≤ 5e-5 absolute.
Adım 6 — Göreli Değerlendirme matrisi (RA) oluştur. p_ik = (ED_i − ED_k) + t(ED_i − ED_k) × (HD_i − HD_k); t(x) = 1 eğer |x| ≥ θ, aksi hâlde 0.

$p_{i k} = (E D_{i} - E D_{k}) + t (E D_{i} - E D_{k}) \cdot (H D_{i} - H D_{k}); t (x) = {\begin{matrix} 1 & | x | \geq θ \\ 0 & | x | < θ \end{matrix}$

LaTeX
p_{ik} = (ED_i - ED_k) + t(ED_i - ED_k) \cdot (HD_i - HD_k); \quad t(x) = \begin{cases}1 & |x| \ge \theta \\ 0 & |x| < \theta \end{cases}
Adım 7 — Her alternatif için Değerlendirme Skoru (AS_i), RA matrisinin satır toplamı olarak hesaplanır.

$A S_{i} = \sum_{k = 1}^{n} p_{i k}$

LaTeX
AS_i = \sum_{k=1}^{n} p_{ik}
Adım 8 — Alternatifleri AS_i azalan sırasına göre sırala. En yüksek AS_i'ye sahip alternatif en iyisidir.

$rank (A_{i}) = rank by descending A S_{i}$

LaTeX
\text{rank}(A_i) = \text{rank by descending } AS_i

Sezgi

Interval-valued intuitionistic fuzzy outranking - IVAIFS. Output typically utility (higher value = preferred).

Sonucu okuma: IVAIF-CODAS extends CODAS to handle Interval-Valued Atanassov Intuitionistic Fuzzy (IVAIF) data. DM assessments are expressed as IVAIFNs ([μL,μU],[vL,vU]) using a linguistic scale, then aggregated via IIFWA. The method uses IVAIF Euclidean and Hamming distances from the negative ideal solution. Higher AS score = better alternative. Threshold θ (default 0.02) controls when the Hamming distance refinement is activated.

Varsayımlar

Decision matrix entries are valid Interval 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 CODAS directly (avoid unnecessary uncertainty layer)
Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

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

Sık yapılan hatalar

IVAIFN geçerliliği: tüm girişlerde μU + vU ≤ 1 koşulunu sağla; ihlali durumunda IVAIFN tanımsız olur.
Eşik θ hassasiyeti: farklı θ değerleri için sıralama değişebilir (makalede 0.01-0.05 test edilmiş, bu örnekte stabil).
IIFWA vs IIFWG: makale agregasyon için IIFWA kullanır; geometrik ortalama operatörü sonuçları değiştirir.

Hesap adımları ve dayanakları

Construct the IVAIF decision matrix X̃^l for each DM l. Each entry x̃_ijl = ([μL_ijl, μU_ijl], [vL_ijl, vU_ijl]) is assigned from a linguistic scale (Table 2). DM l also rates criterion importance using Table 3, giving weight vector W̃^l.

Dayanak: Yeni & Özçelik 2018, p.437 Eq.(13)-(14)
Aggregate individual DM matrices into a single IVAIF decision matrix X̃ and aggregate criterion weights W̃ using IIFWA operator (Eq. 12). DM weights w_l are given (w1=0.40, w2=0.35, w3=0.25 in paper example).

Dayanak: Yeni & Özçelik 2018, p.438 Eq.(12), Eq.(15)-(16)
Calculate the IVAIF weighted decision matrix R̃. Each weighted entry r̃_ij = w̃_j ⊗ x̃_ij uses the IVAIFN multiplication operation (Eq. 6).

Dayanak: Yeni & Özçelik 2018, p.439 Eq.(17), based on Eq.(6)
Determine IVAIF negative ideal solution (NS̃). For each criterion j, the negative ideal is the IVAIFN with minimum membership and maximum non-membership across all alternatives.

Dayanak: Yeni & Özçelik 2018, p.439 Eq.(18)
Calculate normalized Euclidean distance (ED_i) and normalized Hamming distance (HD_i) of each alternative from NS̃, using IVAIF distance formulas (Park et al. 2008).

Dayanak: Yeni & Özçelik 2018, p.440 Eq.(19)-(20)
Construct the Relative Assessment matrix (RA). Entry p_ik = (ED_i − ED_k) + t(ED_i − ED_k) × (HD_i − HD_k), where t(x) = 1 if |x| ≥ θ, else 0. Threshold θ ∈ [0.01, 0.05]; default θ = 0.02.

Dayanak: Yeni & Özçelik 2018, p.440 Eq.(21)-(22)
Calculate the Assessment Score (AS_i) for each alternative as the row sum of RA.

Dayanak: Yeni & Özçelik 2018, p.440 Eq.(23)
Rank alternatives in decreasing order of AS_i. The alternative with the highest AS_i is the best.

Dayanak: Yeni & Özçelik 2018, p.440 Step 8