IVIF-MABAC - Interval-Valued Intuitionistic Fuzzy MABAC (Xue, You, Lai, Liu 2016)

Overview

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.

Output

utility, higher is better

Data

Interval Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Material selection, Supplier selection, MAGDM with l decision makers, Incomplete weight information problems, Engineering design alternative ranking

How it works

1. 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. 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)]

   $${\tilde{X}}^{(}k)=({\tilde{x}}_i^{k}j{)}_{m\times n},{\tilde{x}}_i^{k}j=([a_i^{k}j,b_i^{k}j],[c_i^{k}j,d_i^{k}j]),{\mu }^{+}+{\nu }^{+}\le 1$$

   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)

2. 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_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)]

   $$D_{i}j=(1/(m-1)){\Sigma }_{g=1,g\ne i}^m{d}_H({\tilde{x}}_{i}j,{\tilde{x}}_{g}j),i=1,...,m;j=1,...,n[Eq.(16)]$$

   Xue 2016, Stage 2 Step 2, p.707 Eqs.(16)-(21) + Eq.(6) Hamming distance

3. 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.

   $${\tilde{r}}_{i}j=w_j\cdot {\tilde{x}}_{i}j=([1-(1-a_{i}j{)}^{w_j},1-(1-b_{i}j{)}^{w_j}],[c_i{j}^{w_j},d_i{j}^{w_j}])[Eq.(22)]$$

   Xue 2016, Stage 3 Step 3, p.708 Eq.(22)

4. 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). ⇒ 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)]

   $${\tilde{g}}_j=\prod _{i=1}^m({\tilde{r}}_{i}j{)}^{1/m}[Eq.(23)]$$

   Xue 2016, Stage 3 Step 4, p.708 Eqs.(23)-(24)

5. 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_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)

   $$d_E({\tilde{\alpha }}_1,{\tilde{\alpha }}_2)=\surd ((1/4)\cdot ((a_1-a_2{)}^2+(b_1-b_2{)}^2+(c_1-c_2{)}^2+(d_1-d_2{)}^2))[Eq.(26)]$$

   Xue 2016, Stage 3 Step 5, p.708 Eqs.(25)-(26) + Def 4 Eq.(3)-(4)

6. 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⁻). Ranking: sort alternatives by descending CC_i.

   $$C{C}_i={\Sigma }_{j=1}^n{d}_{i}j,i=1,...,m[Eq.(27)]$$

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

Fits when / Look elsewhere when

Assumptions to verify

• 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

Edge cases and pitfalls

• •IF 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 dec
• •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.
• •IF 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.
• •IF 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
• •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⁻).

• Value-space violation: ensure all entries satisfy IVIFN constraints 0 ≤ μ⁻ ≤ μ⁺ ≤ 1, 0 ≤ ν⁻ ≤ ν⁺ ≤ 1, μ⁺+ν⁺ ≤ 1.

• Score function S = (1/4)(2+a-c+b-d) determines BAA-side sign - accuracy H is only a tiebreaker; do not swap their roles.

• When weights are externally supplied (chained from AHP/ENTROPY), skip F2 (optimization); pass w directly to F3.

• IVIFWG (Eqs. 11, 23) is multiplicative - input components must be strictly positive when raising to fractional powers; if any a_ij = 0, use the equivalent log-space formulation to avoid 0^x = 0 collapse.

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