IVIF-TODIM - Aralık-Değerli Sezgisel Bulanık TODIM (Krohling & Pacheco 2014)

Formül adımları

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

Step 1 — Normalize the IVIF decision matrix Ã = [x̃_ij]_{m×n} into R̃ = [r̃_ij]_{m×n} where r̃_ij = ([μ^L_ij, μ^U_ij], [ν^L_ij, ν^U_ij]) using Xu-Yager (2008) vector normalization (Eqs. 4-5). Cost criteria use IVIFN complement (a,b,c,d) ↦ (c,d,a,b) before normalization.

$μ_{i}^{L} j = a_{i}^{L} j / (Σ_{k = 1}^{m} ((a_{k}^{L} j)^{2} + (a_{k}^{U} j)^{2}))^{1 / 2} [E q . (4)] μ_{i}^{U} j = a_{i}^{U} j / (Σ_{k = 1}^{m} ((a_{k}^{L} j)^{2} + (a_{k}^{U} j)^{2}))^{1 / 2} ν_{i}^{L} j = b_{i}^{L} j / (Σ_{k = 1}^{m} ((b_{k}^{L} j)^{2} + (b_{k}^{U} j)^{2}))^{1 / 2} [E q . (5)] ν_{i}^{U} j = b_{i}^{U} j / (Σ_{k = 1}^{m} ((b_{k}^{L} j)^{2} + (b_{k}^{U} j)^{2}))^{1 / 2} F o r c o s t c r i t e r i a : a p p l y I V I F N c o m p l e m e n t (a, b, c, d) \mapsto (c, d, a, b) b e f o r e a p p l y i n g E q s . (4) - (5) .$

LaTeX
μ^L_ij = a^L_ij / ( Σ_{k=1}^m ((a^L_kj)^2 + (a^U_kj)^2) )^{1/2} [Eq.(4)] μ^U_ij = a^U_ij / ( Σ_{k=1}^m ((a^L_kj)^2 + (a^U_kj)^2) )^{1/2} ν^L_ij = b^L_ij / ( Σ_{k=1}^m ((b^L_kj)^2 + (b^U_kj)^2) )^{1/2} [Eq.(5)] ν^U_ij = b^U_ij / ( Σ_{k=1}^m ((b^L_kj)^2 + (b^U_kj)^2) )^{1/2} For cost criteria: apply IVIFN complement (a,b,c,d) ↦ (c,d,a,b) before applying Eqs. (4)-(5).
Step 2 — Calculate the dominance δ(R̃_i, R̃_j) of each alternative R̃_i over each alternative R̃_j by summing the partial dominance φ_c across all criteria. Partial dominance φ_c uses prospect-theory branching: gain (r̃_ic > r̃_jc) yields positive contribution scaled by sqrt(w_rc / Σw_rc); loss (r̃_ic < r̃_jc) yields negative contribution scaled by -(1/θ)·sqrt(Σw_rc / w_rc); nil (r̃_ic = r̃_jc) yields zero. Comparison uses score function S(α̃) = (a1-a3+a2-a4)/2 with accuracy H(α̃) = (a1+a2+a3+a4)/2 as tiebreaker (Def 5). Distance uses IVIFN Xu-Yager Eq. 3. The reference weight w_r = max_c(w_c); ratio w_rc = w_c / w_r.

$δ (R ̃_{i}, R ̃_{j}) = Σ_{c = 1}^{n} φ_{c} (R ̃_{i}, R ̃_{j}), \forall (i, j) [E q . (6)] φ_{c} (R ̃_{i}, R ̃_{j}) = s q r t (w_{r} c / Σ_{c = 1}^{n} w_{r} c) \cdot d (r ̃_{i} c, r ̃_{j} c) i f r ̃_{i} c > r ̃_{j} c [E q . (7) g a i n] φ_{c} (R ̃_{i}, R ̃_{j}) = 0 i f r ̃_{i} c = r ̃_{j} c [E q . (7) n i l] φ_{c} (R ̃_{i}, R ̃_{j}) = - (1 / θ) \cdot s q r t ((Σ_{c = 1}^{n} w_{r} c) / w_{r} c) \cdot d (r ̃_{i} c, r ̃_{j} c) i f r ̃_{i} c < r ̃_{j} c [E q . (7) l o s s] w_{r} = m a x_{c} (w_{c}), w_{r} c = w_{c} / w_{r} (r e f e r e n c e c r i t e r i o n h a s g r e a t e s t w e i g h t) d (ã, b ̃) = ((1 / 4) \cdot (| a 1 - b 1 | + | a 2 - b 2 | + | a 3 - b 3 | + | a 4 - b 4 |))^{1 / 2} [E q . (3)] S (α ̃) = (a 1 - a 3 + a 2 - a 4) / 2 \in [- 1, 1] [E q . (1)] H (α ̃) = (a 1 + a 2 + a 3 + a 4) / 2 \in [0, 1] [E q . (2)] O r d e r (D e f 5) : S (ã) > S (b ̃) \Rightarrow ã > b ̃; S (ã) = S (b ̃) \land H (ã) > H (b ̃) \Rightarrow ã > b ̃; S (ã) = S (b ̃) \land H (ã) = H (b ̃) \Rightarrow ã = b ̃ .$

LaTeX
δ(R̃_i, R̃_j) = Σ_{c=1}^n φ_c(R̃_i, R̃_j), ∀(i,j) [Eq.(6)] φ_c(R̃_i, R̃_j) = sqrt( w_rc / Σ_{c=1}^n w_rc ) · d(r̃_ic, r̃_jc) if r̃_ic > r̃_jc [Eq.(7) gain] φ_c(R̃_i, R̃_j) = 0 if r̃_ic = r̃_jc [Eq.(7) nil] φ_c(R̃_i, R̃_j) = -(1/θ) · sqrt( (Σ_{c=1}^n w_rc) / w_rc ) · d(r̃_ic, r̃_jc) if r̃_ic < r̃_jc [Eq.(7) loss] w_r = max_c(w_c), w_rc = w_c / w_r (reference criterion has greatest weight) d(ã, b̃) = ( (1/4) · (|a1-b1| + |a2-b2| + |a3-b3| + |a4-b4|) )^{1/2} [Eq.(3)] S(α̃) = (a1 - a3 + a2 - a4) / 2 ∈ [-1, 1] [Eq.(1)] H(α̃) = (a1 + a2 + a3 + a4) / 2 ∈ [ 0, 1] [Eq.(2)] Order (Def 5): S(ã) > S(b̃) ⇒ ã > b̃; S(ã) = S(b̃) ∧ H(ã) > H(b̃) ⇒ ã > b̃; S(ã) = S(b̃) ∧ H(ã) = H(b̃) ⇒ ã = b̃.
Step 3 — Compute the global value ξ_i of each alternative by normalizing Σ_j δ(i,j) to [0,1] via min-max scaling. Rank alternatives in descending order of ξ_i (higher = better).

$ξ_{i} = (Σ_{j} δ (i, j) - m i n_{i} Σ_{j} δ (i, j)) / (m a x_{i} Σ_{j} δ (i, j) - m i n_{i} Σ_{j} δ (i, j)) [E q . (8)] R a n k i n g : s o r t a l t e r n a t i v e s b y d e s c e n d i n g ξ_{i} .$

LaTeX
ξ_i = ( Σ_j δ(i,j) - min_i Σ_j δ(i,j) ) / ( max_i Σ_j δ(i,j) - min_i Σ_j δ(i,j) ) [Eq.(8)] Ranking: sort alternatives by descending ξ_i.

Sezgi

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

Sonucu okuma: IVIF-TODIM extends Gomes-Lima 1992 prospect-theory TODIM to handle Interval-Valued Intuitionistic Fuzzy uncertainty (Atanassov-Gargov 1989). All operations use IVIFN arithmetic. Each pairwise alternative comparison decomposes into gain (positive φ) or loss (negative φ scaled by 1/θ) per criterion via score-based ordering. Dominance δ(i,j) sums all partial contributions; total Σ_j δ(i,j) is normalized to [0,1] yielding global value ξ_i. Higher ξ = better. Loss-aversion parameter θ > 1 amplifies losses (prospect-theory loss aversion); θ = 1 is neutral.

Varsayımlar

Decision matrix entries are valid IVIFNs (a2+a4 ≤ 1)
Weights are positive and sum to 1
θ > 0 (loss attenuation factor)
Decision-maker exhibits prospect-theory loss aversion behavior (otherwise prefer additive methods like TOPSIS, MABAC)

Ne zaman kullanılmaz

Classical data sufficient - use base TODIM directly
Single-valued IFS already provides enough granularity - use IF-TODIM (Krohling-Pacheco-Siviero 2013)
Decision-maker is risk-neutral (no loss aversion) - use IVIF-TOPSIS or IVIF-MABAC instead
Need group decision aggregation - IVIF-TODIM is single-DM; for MAGDM use IVIF-MABAC (Xue 2016) or wrap with IVIFWG/IVIFWA pre-aggregation

Sınırlılıklar

Rank reversal known on alternative-set changes (ref: Inherited from crisp TODIM (Gomes-Lima 1992); θ-attenuation factor mitigates but does not eliminate rank reversal under alternative addition.)
Assumes: Decision matrix entries are valid IVIFNs (a2+a4 ≤ 1)
Assumes: Weights are positive and sum to 1
Assumes: θ > 0 (loss attenuation factor)
Assumes: Decision-maker exhibits prospect-theory loss aversion behavior (otherwise prefer additive methods like TOPSIS, MABAC)

Sık yapılan hatalar

Değer-uzayı ihlali: tüm girişlerin IVIFN kısıtlarını sağladığından emin olun.
θ kesin pozitif olmalı. θ < 1 kayıpları süper-doğrusal büyütür; θ > 1 kayıpları zayıflatır. θ = 1 (varsayılan) prospect-teori nötr.
Referans kriter: w_r = max_c(w_c) en yüksek ağırlıklı kriter; w_rc = w_c/w_r ∈ (0, 1]. w_rc hesabından önce ağırlıkları toplam-1'e normalleştirmeyin - TODIM referansa göre oran kullanır, mutlak ağırlık değil.
Maliyet kriter işleme: Xu normalleştirme Eqs. 4-5'ten ÖNCE IVIFN komplementi (a,b,c,d) ↦ (c,d,a,b) uygula; ham a/b/c/d bileşenlerine max-min ters çevirme YAPMA.
Sıra tersine dönmesi: TODIM ailesi alternatif ekleme/çıkarma altında sıra tersine dönmesi gösterir çünkü ξ mevcut alternatif kümesinde min-max [0,1]'e normalleştirilir. Sonuç raporlanırken alternatif kümesini açıkça belgele.

Hesap adımları ve dayanakları

Normalize the IVIF decision matrix Ã = [x̃_ij]_{m×n} into R̃ = [r̃_ij]_{m×n} where r̃_ij = ([μ^L_ij, μ^U_ij], [ν^L_ij, ν^U_ij]) using Xu-Yager (2008) vector normalization (Eqs. 4-5). Cost criteria use IVIFN complement (a,b,c,d) ↦ (c,d,a,b) before normalization. μ^U_ij = a^U_ij / ( Σ_{k=1}^m ((a^L_kj)^2 + (a^U_kj)^2) )^{1/2} ν^L_ij = b^L_ij / ( Σ_{k=1}^m ((b^L_kj)^2 + (b^U_kj)^2) )^{1/2} [Eq.(5)] ν^U_ij = b^U_ij / ( Σ_{k=1}^m ((b^L_kj)^2 + (b^U_kj)^2) )^{1/2} For cost criteria: apply IVIFN complement (a,b,c,d) ↦ (c,d,a,b) before applying Eqs. (4)-(5).

Dayanak: Krohling-Pacheco 2014 §3 Step 1, p.239 Eqs.(4)-(5)
Calculate the dominance δ(R̃_i, R̃_j) of each alternative R̃_i over each alternative R̃_j by summing the partial dominance φ_c across all criteria. Partial dominance φ_c uses prospect-theory branching: gain (r̃_ic > r̃_jc) yields positive contribution scaled by sqrt(w_rc / Σw_rc); loss (r̃_ic < r̃_jc) yields negative contribution scaled by -(1/θ)·sqrt(Σw_rc / w_rc); nil (r̃_ic = r̃_jc) yields zero. Comparison uses score function S(α̃) = (a1-a3+a2-a4)/2 with accuracy H(α̃) = (a1+a2+a3+a4)/2 as tiebreaker (Def 5). Distance uses IVIFN Xu-Yager Eq. 3. The reference weight w_r = max_c(w_c); ratio w_rc = w_c / w_r. φ_c(R̃_i, R̃_j) = sqrt( w_rc / Σ_{c=1}^n w_rc ) · d(r̃_ic, r̃_jc) if r̃_ic > r̃_jc [Eq.(7) gain] φ_c(R̃_i, R̃_j) = 0 if r̃_ic = r̃_jc [Eq.(7) nil] φ_c(R̃_i, R̃_j) = -(1/θ) · sqrt( (Σ_{c=1}^n w_rc) / w_rc ) · d(r̃_ic, r̃_jc) if r̃_ic < r̃_jc [Eq.(7) loss] w_r = max_c(w_c), w_rc = w_c / w_r (reference criterion has greatest weight) d(ã, b̃) = ( (1/4) · (|a1-b1| + |a2-b2| + |a3-b3| + |a4-b4|) )^{1/2} [Eq.(3)] S(α̃) = (a1 - a3 + a2 - a4) / 2 ∈ [-1, 1] [Eq.(1)] H(α̃) = (a1 + a2 + a3 + a4) / 2 ∈ [ 0, 1] [Eq.(2)] Order (Def 5): S(ã) > S(b̃) ⇒ ã > b̃; S(ã) = S(b̃) ∧ H(ã) > H(b̃) ⇒ ã > b̃; S(ã) = S(b̃) ∧ H(ã) = H(b̃) ⇒ ã = b̃.

Dayanak: Krohling-Pacheco 2014 §3 Step 2, p.239 Eqs.(6)-(7) + Eqs.(1)-(3) Def 3-6
Compute the global value ξ_i of each alternative by normalizing Σ_j δ(i,j) to [0,1] via min-max scaling. Rank alternatives in descending order of ξ_i (higher = better). Ranking: sort alternatives by descending ξ_i.

Dayanak: Krohling-Pacheco 2014 §3 Step 3, p.240 Eq.(8)