IF-MABAC - Intuitionistic extension of MABAC

Overview

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.

Output

utility, higher is better

Data

Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Intuitionistic Fuzzy MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 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 = 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)]

   $$Q^{(}k)=(q_i^{k}j{)}_{m\times n},q_i^{k}j=({\mu }_i^{k}j,{\nu }_i^{k}j),\mu +\nu \le 1$$

   Li 2021 (J. Math., DOI 10.1155/2021/5536751), Steps 1-2, p.3 Eqs.(10)-(11),(17)

2. 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. 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)]

   $$d(I_1,I_2)=(1/6)({\ell }_1+{\ell }_2+{\ell }_3)[Eq.(8)]$$

   Li 2021, Step 3, p.4-5 Eqs.(8),(24)-(25)

3. 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.

   $$o_{i}j=z_j\cdot q_i^{N}j=(1-(1-{\mu }_{q_i^{N}j}{)}^{z_j},({\nu }_{q_i^{N}j}{)}^{z_j})[Eq.(4)applied;Eq.(26)]$$

   Li 2021, Step 4, p.5 Eq.(26)

4. 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.

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

   Li 2021, Step 5, p.5 Eq.(27)

5. 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. 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.

   $$S(I)={\mu }_I+{\mu }_I\cdot (1-{\mu }_I-{\nu }_I)[Eq.(6)]$$

   Li 2021, Steps 6-8, p.5-6 Eqs.(6),(28),(29)

Fits when / Look elsewhere when

Assumptions to verify

• 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

Edge cases and pitfalls

• •IF-MABAC için kritik edge case'ler: (1) IFS geçerlilik: μ + ν ≤ 1 zorunlu - Atanassov 1986 aksiyomu; engine input validation bu kısıtı kontrol etmeli, ihlal → fail-fast. (2) Cost kriter normalize: (μ,ν) → (ν,μ) swap Eq.17 - μ membership iyi-derecesidir, cost'ta ters çevrilir; ν non-membership de swap olur. (3) Maximizing deviation feasibility - Σ z_j² = 1 normalize sabit nokta; tüm alternatifler tüm kriterlerde identik (zero deviation) ise model dejenere → equal-weight fallback önerilir (paper'da explicit değil). (4) BAA g_j = Π^(1/m) - geometric mean; o_ij'lerden herhangi biri (μ=0 olduğunda) sıfırlanırsa g_j = 0 olur ve d_ij sonsuz log-ratio'ya yol açabilir; engine ε-perturbation veya pre-normalize gerektirir. (5) Prospect parametreler θ=ς=0.88 (sıkı concave gain + convex loss), ρ=2.25 (loss aversion 2.25× ağırlık) Tversky-Kahneman kanonik; sosyokültürel context'e göre değiştirilebilir ama paper'da değiştirilmez.

• Value-space violation: ensure all entries satisfy IFN: μ ∈ [0,1], ν ∈ [0,1], μ+ν ≤ 1; π = 1−μ−ν ≥ 0 before computation.

• Defuzzification method affects ranking: score function S = μ − ν is the canonical choice but alternatives exist.

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