IF-MABAC - MABAC yönteminin Intuitionistic uzantısı

Formül adımları

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

1. Step 1 — Build each DM's intuitionistic fuzzy decision matrix Q^(k)=(q^k_ij)_{m×n}, q^k_ij=(μ^k_ij, ν^k_ij), μ+ν≤1; aggregate over l DMs via IFWA with weights h_k (Σh_k=1) to obtain Q=(q_ij)_{m×n}; then normalize per criterion type to Q^N: benefit ↦ (μ_ij, ν_ij), cost ↦ (ν_ij, μ_ij).

   $$Q^{(}k)=(q_i^{k}j{)}_{m\times n},q_i^{k}j=({\mu }_i^{k}j,{\nu }_i^{k}j),\mu +\nu \le 1Q=IFW{A}_h(Q^{(}1),...,Q^{(}l));q_{i}j=(1-{\prod }_{k=1}^l(1-{\mu }_i^{k}j{)}^{h_k},{\prod }_{k=1}^l({\nu }_i^{k}j{)}^{h_k})[Eq.(10)-(11)]q_i^{N}j=({\mu }_{i}j,{\nu }_{i}j)if{Z}_j\in benefit{q}_i^{N}j=({\nu }_{i}j,{\mu }_{i}j)if{Z}_j\in cost[Eq.(17)]$$
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   Q^(k) = (q^k_ij)_{m×n}, q^k_ij = (μ^k_ij, ν^k_ij), μ + ν ≤ 1 Q = IFWA_h(Q^(1),...,Q^(l)); q_ij = ( 1 − ∏_{k=1}^l (1 − μ^k_ij)^{h_k}, ∏_{k=1}^l (ν^k_ij)^{h_k} ) [Eq.(10)-(11)] q^N_ij = (μ_ij, ν_ij) if Z_j ∈ benefit q^N_ij = (ν_ij, μ_ij) if Z_j ∈ cost [Eq.(17)]

2. Step 2 — Determine attribute weights z_j (j=1,...,n) by the maximizing-deviation method (Wang 1998 style) using the IF distance d(·,·) of Eq.(8). Solve the M-1 program max Σ_j Σ_i Σ_t z_j · d(q^N_ij, q^N_tj) s.t. z_j ≥ 0, Σ z_j² = 1, then normalize Σ z_j = 1.

   $$d(I_1,I_2)=(1/6)({\ell }_1+{\ell }_2+{\ell }_3)[Eq.(8)]{\ell }_1=(|{\mu }_1-{\mu }_2|+|{\nu }_1-{\nu }_2|+|({\mu }_1+1-{\nu }_1)-({\mu }_2+1-{\nu }_2)|)/2{\ell }_2=({\pi }_1+{\pi }_2)/2{\ell }_3=max(|{\mu }_1-{\mu }_2|,|{\nu }_1-{\nu }_2|,|{\pi }_1-{\pi }_2|)/2z_j*=\surd ({\Sigma }_{i=1}^m{\Sigma }_{t=1}^{m}d(q_i^{N}j,q_t^{N}j)/{\Sigma }_{j=1}^n[{\Sigma }_i{\Sigma }_{t}d(q_i^{N}j,q_t^{N}j)]²)[Eq.(24)]z_j=({\Sigma }_i{\Sigma }_{t}d(q_i^{N}j,q_t^{N}j))/{\Sigma }_{j=1}^n{\Sigma }_i{\Sigma }_{t}d(q_i^{N}j,q_t^{N}j),{\Sigma }_j{z}_j=1[Eq.(25)]$$
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   d(I_1, I_2) = (1/6)(ℓ_1 + ℓ_2 + ℓ_3) [Eq.(8)] ℓ_1 = ( |μ_1 − μ_2| + |ν_1 − ν_2| + |(μ_1+1−ν_1) − (μ_2+1−ν_2)| ) / 2 ℓ_2 = (π_1 + π_2) / 2 ℓ_3 = max( |μ_1−μ_2|, |ν_1−ν_2|, |π_1−π_2| ) / 2 z_j* = √( Σ_{i=1}^m Σ_{t=1}^m d(q^N_ij, q^N_tj) / Σ_{j=1}^n [ Σ_i Σ_t d(q^N_ij, q^N_tj) ]² ) [Eq.(24)] z_j = ( Σ_i Σ_t d(q^N_ij, q^N_tj) ) / Σ_{j=1}^n Σ_i Σ_t d(q^N_ij, q^N_tj), Σ_j z_j = 1 [Eq.(25)]

3. Step 3 — Compute the IF weighted normalized matrix O=(o_ij)_{m×n} by applying the scalar multiplication λI = (1−(1−μ)^λ, ν^λ) to q^N_ij with λ = z_j.

   $$o_{i}j=z_j·q_i^{N}j=(1-(1-{\mu }_{q_i^{N}j}{)}^{z_j},({\nu }_{q_i^{N}j}{)}^{z_j})[Eq.(4)applied;Eq.(26)]$$
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   o_ij = z_j · q^N_ij = ( 1 − (1 − μ_{q^N_ij})^{z_j}, (ν_{q^N_ij})^{z_j} ) [Eq.(4) applied; Eq.(26)]

4. Step 4 — Build the border approximation area (BAA) row G=(g_j)_{1×n} by taking the geometric mean of each column of O across the m alternatives.

   $$g_j=({\prod }_{i=1}^m{o}_{i}j{)}^{1/m}=(({\prod }_{i=1}^m{\mu }_{o_{i}j}{)}^{1/m},1-({\prod }_{i=1}^m(1-{\nu }_{o_{i}j}){)}^{1/m})[Eq.(27)]$$
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   g_j = ( ∏_{i=1}^m o_ij )^{1/m} = ( ( ∏_{i=1}^m μ_{o_ij} )^{1/m}, 1 − ( ∏_{i=1}^m (1 − ν_{o_ij}) )^{1/m} ) [Eq.(27)]

5. Step 5 — Compute the prospect-theory weighted IF distance from BAA, sum across criteria, and rank alternatives in descending order of F_i. Tversky-Kahneman parameters: ϑ = ς = 0.88, ρ = 2.25; the score function S(I) = μ + μ·(1−μ−ν) determines whether an alternative lies above/below the BAA per criterion.

   $$S(I)={\mu }_I+{\mu }_I·(1-{\mu }_I-{\nu }_I)[Eq.(6)]d_{i}j=(d(o_{i}j,g_j){)}^{\vartheta }ifS(o_{i}j)\ge S(g_j)[Eq.(28)]d_{i}j=-\rho ·(d(o_{i}j,g_j){)}^{\varsigma }ifS(o_{i}j)<S(g_j)with\vartheta =0.88,\varsigma =0.88,\rho =2.25(Tversky-Kahneman1992)F_i={\Sigma }_{j=1}^n{d}_{i}j,i=1,...,m[Eq.(29)]Ranking:sortalternativesbydescending{F}_i;thelarger{F}_i,thebetterthealternative.$$
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   S(I) = μ_I + μ_I · (1 − μ_I − ν_I) [Eq.(6)] d_ij = ( d(o_ij, g_j) )^ϑ if S(o_ij) ≥ S(g_j) [Eq.(28)] d_ij = − ρ · ( d(o_ij, g_j) )^ς if S(o_ij) < S(g_j) with ϑ = 0.88, ς = 0.88, ρ = 2.25 (Tversky-Kahneman 1992) F_i = Σ_{j=1}^n d_ij, i = 1,...,m [Eq.(29)] Ranking: sort alternatives by descending F_i; the larger F_i, the better the alternative.

Sezgi

IF-MABAC, Pamucar-Cirovic (2015) tarafından geliştirilen MABAC (Multi-Attributive Border Approximation area Comparison) yönteminin Y. Li (2021, J.Mathematics) tarafından intuitionistic fuzzy ortama uyarlanmış halidir. MABAC'ın özgün sezgisi diğer MCDM yöntemlerinden temel olarak ayrılır: PIS/NIS gibi uç-nokta referansları değil, BORDER APPROXIMATION AREA (BAA) g_j = (Π_{i=1}^m o_ij)^(1/m) - GEOMETRIK ORTALAMA - referans alır ve alternatifin BAA'dan uzaklığını ±işaretli ölçer. Pozitif uzaklık (G^+) iyi, negatif (G^-) kötü. Y. Li'nin paper'a iki ÖZGÜN katkısı vardır: (i) Novel IF Hamming distance IFHD Def.4 (Eq.8-9) - üç-bileşenli (l_1: μ ve ν farkı; l_2: hesitation π toplamı; l_3: max-component); klasik Euclidean'den daha 'comprehensive' olduğu iddiası; (ii) Maximizing deviation method ile objektif ağırlık (Wang 1997) - Lagrange multiplier ile kapalı-form z_j Eq.25. Distance d_ij Eq.28 PROSPECT THEORY (Tversky-Kahneman 1992) parametre seti içerir: θ=0.88 (gain power), ς=0.88 (loss power), ρ=2.25 (loss aversion). Yani BAA'ya göre alternatif iyi ise d^θ ile yumuşatılır; kötü ise -ρ·d^ς ile AĞIRLAŞTIRILIR (kayıp aversiyonu).

Sonucu okuma: if-mabac extends MABAC to handle Intuitionistic uncertainty. All arithmetic operations (normalisation, weighting, distance computation) are performed using Intuitionistic Fuzzy Number (IFN: μ, ν; μ+ν ≤ 1) algebra. The final scores are defuzzified via score function S = μ − ν before ranking.

Varsayımlar

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

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: inherited from crisp base; cf. Belton-Gear 1983, Wang-Luo 2009)
• Assumes: Decision matrix entries are valid Intuitionistic Fuzzy 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

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