IVIF-MABAC - Aralık-Değerli Sezgisel Bulanık MABAC (Xue, You, Lai, Liu 2016)

Formül adımları

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

Step 1 — Build each DM's IVIF decision matrix X̃^(k) = (x̃^k_ij)_{m×n} with x̃^k_ij = ([a^k_ij, b^k_ij], [c^k_ij, d^k_ij]); aggregate the l DM matrices via IVIFWG operator with DM weights λ_k (Σ λ_k = 1) into the group IVIF decision matrix X̃ = [x̃_ij]_{m×n}. Then normalize entries per criterion type using Eqs. 12-15: benefit, cost, fixation, deviation — yielding IVIFN normalized matrix.

$X ̃^{(} k) = (x ̃_{i}^{k} j)_{m \times n}, x ̃_{i}^{k} j = ([a_{i}^{k} j, b_{i}^{k} j], [c_{i}^{k} j, d_{i}^{k} j]), μ ⁺ + ν ⁺ \leq 1 x ̃_{i} j = I V I F W G_{λ} (x ̃_{i}^{1} j, . . ., x ̃_{i}^{l} j) = \prod_{k = 1}^{l} (x ̃_{i}^{k} j)^{λ_{k}} [E q . (11)] \Rightarrow a_{i} j = \prod_{k} (a_{i}^{k} j)^{λ_{k}}, b_{i} j = \prod_{k} (b_{i}^{k} j)^{λ_{k}} \Rightarrow c_{i} j = 1 - \prod_{k} (1 - c_{i}^{k} j)^{λ_{k}}, d_{i} j = 1 - \prod_{k} (1 - d_{i}^{k} j)^{λ_{k}} B e n e f i t n o r m a l i z a t i o n : x ̃_{i} j \mapsto (x_{i} j - m i n_{j}) / (m a x_{j} - m i n_{j}) [E q . (12)] C o s t n o r m a l i z a t i o n : x ̃_{i} j \mapsto (m a x_{j} - x_{i} j) / (m a x_{j} - m i n_{j}) [E q . (13)] F i x a t i o n : x ̃_{i} j \mapsto (m a x_{x ⁺_{j}} - | x_{i} j - x ⁺_{j} |) / m a x_{x ⁺_{j}} [E q . (14)] D e v i a t i o n : x ̃_{i} j \mapsto (| x_{i} j - x ⁻_{j} | - m i n_{x ⁻_{j}}) / (m a x_{x ⁻_{j}} - m i n_{x ⁻_{j}}) [E q . (15)]$

LaTeX
X̃^(k) = (x̃^k_ij)_{m×n}, x̃^k_ij = ([a^k_ij, b^k_ij], [c^k_ij, d^k_ij]), μ⁺ + ν⁺ ≤ 1 x̃_ij = IVIFWG_λ(x̃^1_ij,...,x̃^l_ij) = ∏_{k=1}^l (x̃^k_ij)^{λ_k} [Eq.(11)] ⇒ a_ij = ∏_k (a^k_ij)^{λ_k}, b_ij = ∏_k (b^k_ij)^{λ_k} ⇒ c_ij = 1 − ∏_k (1 − c^k_ij)^{λ_k}, d_ij = 1 − ∏_k (1 − d^k_ij)^{λ_k} Benefit normalization: x̃_ij ↦ (x_ij − min_j) / (max_j − min_j) [Eq.(12)] Cost normalization: x̃_ij ↦ (max_j − x_ij) / (max_j − min_j) [Eq.(13)] Fixation: x̃_ij ↦ (max_{x⁺_j} − |x_ij − x⁺_j|) / max_{x⁺_j} [Eq.(14)] Deviation: x̃_ij ↦ (|x_ij − x⁻_j| − min_{x⁻_j}) / (max_{x⁻_j} − min_{x⁻_j}) [Eq.(15)]
Step 2 — Determine optimal criterion weights w* under incomplete weight information H = H_1 ∪ H_2 ∪ H_3 ∪ H_4 ∪ H_5 (Xue 2016 §4.2). If H is partial, solve linear-programming model M-1 (Eq. 19) maximizing total IVIF Hamming distance D(w). If H is completely unknown, solve M-2 (Eq. 20) via Lagrange method giving closed-form Eq. 21. When weights are externally supplied (e.g., from chained AHP/ENTROPY), skip this step.

$D_{i} j = (1 / (m - 1)) Σ_{g = 1, g \neq i}^{m} d_{H} (x ̃_{i} j, x ̃_{g} j), i = 1, . . ., m; j = 1, . . ., n [E q . (16)] D_{j} = (1 / (m - 1)) Σ_{i} Σ_{g \neq i} d_{H} (x ̃_{i} j, x ̃_{g} j), j = 1, . . ., n [E q . (17)] D (w) = Σ_{j} D_{j} w_{j} = (1 / (m - 1)) Σ_{j} Σ_{i} Σ_{g \neq i} d_{H} (x ̃_{i} j, x ̃_{g} j) w_{j} [E q . (18)] M - 1 : m a x D (w) s . t . w \in H, Σ_{j} w_{j} = 1, w_{j} \geq 0 [E q . (19)] M - 2 (c o m p l e t e l y u n k n o w n H) : [E q . (20)] w_{j} = (Σ_{i} Σ_{g \neq i} d_{H} (x ̃_{i} j, x ̃_{g} j)) / Σ_{j} Σ_{i} Σ_{g \neq i} d_{H} (x ̃_{i} j, x ̃_{g} j) [E q . (21)] d_{H} (α ̃_{1}, α ̃_{2}) = (1 / 4) (| a_{1} - a_{2} | + | b_{1} - b_{2} | + | c_{1} - c_{2} | + | d_{1} - d_{2} |) [E q . (6)]$

LaTeX
D_ij = (1/(m−1)) Σ_{g=1, g≠i}^m d_H(x̃_ij, x̃_gj), i=1,...,m; j=1,...,n [Eq.(16)] D_j = (1/(m−1)) Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj), j=1,...,n [Eq.(17)] D(w) = Σ_j D_j w_j = (1/(m−1)) Σ_j Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj) w_j [Eq.(18)] M-1: max D(w) s.t. w ∈ H, Σ_j w_j = 1, w_j ≥ 0 [Eq.(19)] M-2 (completely unknown H): [Eq.(20)] w_j = (Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj)) / Σ_j Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj) [Eq.(21)] d_H(α̃_1, α̃_2) = (1/4)(|a_1−a_2| + |b_1−b_2| + |c_1−c_2| + |d_1−d_2|) [Eq.(6)]
Step 3 — Compute the weighted group IVIF decision matrix R̃ = (r̃_ij)_{m×n} by applying scalar-IVIFN multiplication (Eq. 22) to each entry x̃_ij of the normalized group matrix with λ = w_j.

$r ̃_{i} j = w_{j} \cdot x ̃_{i} j = ([1 - (1 - a_{i} j)^{w_{j}}, 1 - (1 - b_{i} j)^{w_{j}}], [c_{i} j^{w_{j}}, d_{i} j^{w_{j}}]) [E q . (22)]$

LaTeX
r̃_ij = w_j · x̃_ij = ([1 − (1 − a_ij)^{w_j}, 1 − (1 − b_ij)^{w_j}], [c_ij^{w_j}, d_ij^{w_j}]) [Eq.(22)]
Step 4 — Construct the Border Approximation Area (BAA) vector G̃ = [g̃_1, g̃_2, ..., g̃_n] by applying the IVIFG operator column-wise across the m alternatives (geometric mean of R̃).

$g ̃_{j} = \prod_{i = 1}^{m} (r ̃_{i} j)^{1 / m} [E q . (23)] \Rightarrow a_{g_{j}} = \prod_{i} (a_{r_{i} j})^{1 / m}, b_{g_{j}} = \prod_{i} (b_{r_{i} j})^{1 / m} \Rightarrow c_{g_{j}} = 1 - \prod_{i} (1 - c_{r_{i} j})^{1 / m}, d_{g_{j}} = 1 - \prod_{i} (1 - d_{r_{i} j})^{1 / m} G ̃ = [g ̃_{1}, g ̃_{2}, . . ., g ̃_{n}] [E q . (24)]$

LaTeX
g̃_j = ∏_{i=1}^m (r̃_ij)^{1/m} [Eq.(23)] ⇒ a_{g_j} = ∏_i (a_{r_ij})^{1/m}, b_{g_j} = ∏_i (b_{r_ij})^{1/m} ⇒ c_{g_j} = 1 − ∏_i (1 − c_{r_ij})^{1/m}, d_{g_j} = 1 − ∏_i (1 − d_{r_ij})^{1/m} G̃ = [g̃_1, g̃_2, ..., g̃_n] [Eq.(24)]
Step 5 — Compute the signed distance matrix D = (d_ij)_{m×n} between r̃_ij and the BAA element g̃_j using the IVIF Euclidean distance (Eq. 26). Sign is positive if r̃_ij ≥ g̃_j by the comparison rule (score then accuracy, Def 5), otherwise negative — interpreting whether the alternative lies above (upper area G⁺) or below (lower area G⁻) the BAA.

$d_{E} (α ̃_{1}, α ̃_{2}) = \sqrt ((1 / 4) \cdot ((a_{1} - a_{2}) ² + (b_{1} - b_{2}) ² + (c_{1} - c_{2}) ² + (d_{1} - d_{2}) ²)) [E q . (26)] d_{i} j = d_{E} (r ̃_{i} j, g ̃_{j}) i f r ̃_{i} j \geq g ̃_{j} (i . e ., s c o r e (r ̃_{i} j) \geq s c o r e (g ̃_{j})) [E q . (25)] d_{i} j = - d_{E} (r ̃_{i} j, g ̃_{j}) i f r ̃_{i} j < g ̃_{j} S c o r e (D e f 4 E q .3) : S (α ̃) = (1 / 4) (2 + a - c + b - d) \in [0, 1] A c c u r a c y (D e f 4 E q .4) : H (α ̃) = (a + b - 1) + (c + d) / 2 \in [- 1, 1] (u s e d a s t i e - b r e a k e r)$

LaTeX
d_E(α̃_1, α̃_2) = √( (1/4) · ((a_1−a_2)² + (b_1−b_2)² + (c_1−c_2)² + (d_1−d_2)²) ) [Eq.(26)] d_ij = d_E(r̃_ij, g̃_j) if r̃_ij ≥ g̃_j (i.e., score(r̃_ij) ≥ score(g̃_j)) [Eq.(25)] d_ij = −d_E(r̃_ij, g̃_j) if r̃_ij < g̃_j Score (Def 4 Eq.3): S(α̃) = (1/4)(2 + a − c + b − d) ∈ [0, 1] Accuracy (Def 4 Eq.4): H(α̃) = (a + b − 1) + (c + d)/2 ∈ [−1, 1] (used as tie-breaker)
Step 6 — Compute the closeness coefficient CC_i = Σ_j d_ij for each alternative i, and rank alternatives in descending order of CC_i. Higher CC_i means the alternative lies closer to (or above) the BAA, indicating better material/option. If CC_i > 0, A_i is in the upper area G⁺ (near ideal A⁺); if CC_i < 0, A_i is in the lower area G⁻ (near anti-ideal A⁻).

$C C_{i} = Σ_{j = 1}^{n} d_{i} j, i = 1, . . ., m [E q . (27)] R a n k i n g : s o r t a l t e r n a t i v e s b y d e s c e n d i n g C C_{i} .$

LaTeX
CC_i = Σ_{j=1}^n d_ij, i = 1,...,m [Eq.(27)] Ranking: sort alternatives by descending CC_i.

Sezgi

Border-approximation-area MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ([μ⁻,μ⁺],[ν⁻,ν⁺]); μ⁺+ν⁺ ≤ 1). Output typically utility (higher value = preferred).

Sonucu okuma: IVIF-MABAC extends MABAC to handle Interval-Valued Intuitionistic Fuzzy uncertainty (Atanassov-Gargov 1989). All operations use IVIFN arithmetic (Xue 2016 Defs 1-8). Each alternative is compared to the Border Approximation Area (BAA, geometric mean across alternatives) per criterion via signed IVIF Euclidean distance - positive if above BAA (upper area G⁺, near ideal), negative if below (lower area G⁻, near anti-ideal). The closeness coefficient CC_i sums these signed distances; higher CC_i = better. Supports incomplete weight information via optimization model M-1 (Eq. 19) or M-2/Eq. 21 closed-form when weights are completely unknown.

Varsayımlar

Decision matrix entries are valid IVIFNs (μ⁺+ν⁺ ≤ 1)
DM weights λ_k sum to 1 in MAGDM group setting
Underlying MABAC compensation assumption holds in uncertain interval space
All decision-makers use the same linguistic-to-IVIFN translation table

Ne zaman kullanılmaz

Classical data sufficient - use base MABAC directly
Single-valued IFS already provides enough granularity - use IF-MABAC (Li 2021)
Decision criteria are correlated - use IF-MABAC with Choquet integral (Liang 2019)

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: Inherited from crisp MABAC (Pamučar-Ćirović 2015); generally insensitive in IVIF setting per Xue 2016 §3 sensitivity analysis.)
Assumes: Decision matrix entries are valid IVIFNs (μ⁺+ν⁺ ≤ 1)
Assumes: DM weights λ_k sum to 1 in MAGDM group setting
Assumes: Underlying MABAC compensation assumption holds in uncertain interval space
Assumes: All decision-makers use the same linguistic-to-IVIFN translation table

Sık yapılan hatalar

Değer-uzayı ihlali: tüm girişlerin IVIFN kısıtlarını sağladığından emin olun.
Skor fonksiyonu S = (1/4)(2+a-c+b-d) BAA-tarafı işaretini belirler - accuracy H sadece tie-breaker; rollerini değiştirmeyin.
Ağırlıklar dışarıdan verilmişse (AHP/ENTROPY zincirinden) F2 (optimizasyon) adımını atla; w'yi doğrudan F3'e geçir.
IVIFWG (Eqs. 11, 23) çarpımsaldır - kesirli kuvvet alırken giriş bileşenleri kesin pozitif olmalı; herhangi a_ij = 0 ise 0^x = 0 çöküşünden kaçınmak için log-uzay formülasyonu kullanın.

Hesap adımları ve dayanakları

Build each DM's IVIF decision matrix X̃^(k) = (x̃^k_ij)_{m×n} with x̃^k_ij = ([a^k_ij, b^k_ij], [c^k_ij, d^k_ij]); aggregate the l DM matrices via IVIFWG operator with DM weights λ_k (Σ λ_k = 1) into the group IVIF decision matrix X̃ = [x̃_ij]_{m×n}. Then normalize entries per criterion type using Eqs. 12-15: benefit, cost, fixation, deviation - yielding IVIFN normalized matrix. x̃_ij = IVIFWG_λ(x̃^1_ij,...,x̃^l_ij) = ∏_{k=1}^l (x̃^k_ij)^{λ_k} [Eq.(11)] ⇒ a_ij = ∏_k (a^k_ij)^{λ_k}, b_ij = ∏_k (b^k_ij)^{λ_k} ⇒ c_ij = 1 − ∏_k (1 − c^k_ij)^{λ_k}, d_ij = 1 − ∏_k (1 − d^k_ij)^{λ_k} Benefit normalization: x̃_ij ↦ (x_ij − min_j) / (max_j − min_j) [Eq.(12)] Cost normalization: x̃_ij ↦ (max_j − x_ij) / (max_j − min_j) [Eq.(13)] Fixation: x̃_ij ↦ (max_{x⁺_j} − |x_ij − x⁺_j|) / max_{x⁺_j} [Eq.(14)] Deviation: x̃_ij ↦ (|x_ij − x⁻_j| − min_{x⁻_j}) / (max_{x⁻_j} − min_{x⁻_j}) [Eq.(15)]

Dayanak: Xue 2016 (ASC 38:703-713, DOI 10.1016/j.asoc.2015.10.010), Stage 1 Step 1, p.706 Eq.(11) + Eqs.(12)-(15)
Determine optimal criterion weights w* under incomplete weight information H = H_1 ∪ H_2 ∪ H_3 ∪ H_4 ∪ H_5 (Xue 2016 §4.2). If H is partial, solve linear-programming model M-1 (Eq. 19) maximizing total IVIF Hamming distance D(w). If H is completely unknown, solve M-2 (Eq. 20) via Lagrange method giving closed-form Eq. 21. When weights are externally supplied (e.g., from chained AHP/ENTROPY), skip this step. D_j = (1/(m−1)) Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj), j=1,...,n [Eq.(17)] D(w) = Σ_j D_j w_j = (1/(m−1)) Σ_j Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj) w_j [Eq.(18)] M-1: max D(w) s.t. w ∈ H, Σ_j w_j = 1, w_j ≥ 0 [Eq.(19)] M-2 (completely unknown H): [Eq.(20)] w_j = (Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj)) / Σ_j Σ_i Σ_{g≠i} d_H(x̃_ij, x̃_gj) [Eq.(21)] d_H(α̃_1, α̃_2) = (1/4)(|a_1−a_2| + |b_1−b_2| + |c_1−c_2| + |d_1−d_2|) [Eq.(6)]

Dayanak: Xue 2016, Stage 2 Step 2, p.707 Eqs.(16)-(21) + Eq.(6) Hamming distance
Compute the weighted group IVIF decision matrix R̃ = (r̃_ij)_{m×n} by applying scalar-IVIFN multiplication (Eq. 22) to each entry x̃_ij of the normalized group matrix with λ = w_j.

Dayanak: Xue 2016, Stage 3 Step 3, p.708 Eq.(22)
Construct the Border Approximation Area (BAA) vector G̃ = [g̃_1, g̃_2, ..., g̃_n] by applying the IVIFG operator column-wise across the m alternatives (geometric mean of R̃). ⇒ a_{g_j} = ∏_i (a_{r_ij})^{1/m}, b_{g_j} = ∏_i (b_{r_ij})^{1/m} ⇒ c_{g_j} = 1 − ∏_i (1 − c_{r_ij})^{1/m}, d_{g_j} = 1 − ∏_i (1 − d_{r_ij})^{1/m} G̃ = [g̃_1, g̃_2, ..., g̃_n] [Eq.(24)]

Dayanak: Xue 2016, Stage 3 Step 4, p.708 Eqs.(23)-(24)
Compute the signed distance matrix D = (d_ij)_{m×n} between r̃_ij and the BAA element g̃_j using the IVIF Euclidean distance (Eq. 26). Sign is positive if r̃_ij ≥ g̃_j by the comparison rule (score then accuracy, Def 5), otherwise negative - interpreting whether the alternative lies above (upper area G⁺) or below (lower area G⁻) the BAA. d_ij = d_E(r̃_ij, g̃_j) if r̃_ij ≥ g̃_j (i.e., score(r̃_ij) ≥ score(g̃_j)) [Eq.(25)] d_ij = −d_E(r̃_ij, g̃_j) if r̃_ij < g̃_j Score (Def 4 Eq.3): S(α̃) = (1/4)(2 + a − c + b − d) ∈ [0, 1] Accuracy (Def 4 Eq.4): H(α̃) = (a + b − 1) + (c + d)/2 ∈ [−1, 1] (used as tie-breaker)

Dayanak: Xue 2016, Stage 3 Step 5, p.708 Eqs.(25)-(26) + Def 4 Eq.(3)-(4)
Compute the closeness coefficient CC_i = Σ_j d_ij for each alternative i, and rank alternatives in descending order of CC_i. Higher CC_i means the alternative lies closer to (or above) the BAA, indicating better material/option. If CC_i > 0, A_i is in the upper area G⁺ (near ideal A⁺); if CC_i < 0, A_i is in the lower area G⁻ (near anti-ideal A⁻). Ranking: sort alternatives by descending CC_i.

Dayanak: Xue 2016, Stage 3 Step 6, p.708 Eq.(27)