IVIF-ARAS - Interval-Valued Intuitionistic Fuzzy ARAS (Büyüközkan & Göçer 2018)

Overview

IVIF-ARAS extends Zavadskas-Turskis 2010 Additive Ratio ASsessment (ARAS) to handle Interval-Valued Intuitionistic Fuzzy uncertainty (Atanassov-Gargov 1989). Each alternative's overall performance is summarized by an optimality function P̃_i (IVIF sum of weighted-normalized criterion ratings) and benchmarked against an optimal reference row x̃_0 (componentwise max of memberships and min of non-memberships for benefit criteria, reversed for cost). The utility degree Q_i = P_i/P_0 expresses how close each alternative is to the optimal benchmark; values lie in (0, 1] for non-degenerate inputs and the highest Q_i identifies the best alternative.

Output

utility degree, higher is better

Data

Interval Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-20 criteria works best

Used for

Supplier selection in digital supply chains, Logistics service provider evaluation, Renewable energy alternative ranking, Strategic decision-making with utility-degree benchmarking, MAGDM under epistemic uncertainty (with IVIF AHP weight elicitation extension)

How it works

1. 1

   Step 10 - Build the IVIF decision matrix x̃_ij and prepend an optimal performance row x̃_0 (Eq. 18). For benefit criterion j ∈ J1: μ^L(x̃_0j) = max_i μ^L(x̃_ij), μ^U(x̃_0j) = max_i μ^U(x̃_ij), ν^L(x̃_0j) = min_i ν^L(x̃_ij), ν^U(x̃_0j) = min_i ν^U(x̃_ij). For cost criterion j ∈ J2: reverse (min on μ, max on ν). Büyüközkan-Göçer 2018 Eqs. 19-20. For j ∈ J1 (benefit): μ^L(x̃_0j) = max_i μ^L(x̃_ij), μ^U(x̃_0j) = max_i μ^U(x̃_ij) ν^L(x̃_0j) = min_i ν^L(x̃_ij), ν^U(x̃_0j) = min_i ν^U(x̃_ij) [Eq.(19)] For j ∈ J2 (cost): μ^L(x̃_0j) = min_i μ^L(x̃_ij), μ^U(x̃_0j) = min_i μ^U(x̃_ij) ν^L(x̃_0j) = max_i ν^L(x̃_ij), ν^U(x̃_0j) = max_i ν^U(x̃_ij) [Eq.(20)]

   $$\tilde{X}=[{\tilde{x}}_{0}j;{\tilde{x}}_{i}j](m+1)\times nmatrixwithoptimalrowprepended[Eq.(18)]$$

   Büyüközkan-Göçer 2018 §3.2 Steps 10 + Eqs.(18)-(20)

2. 2

   Step 11 - Normalize each criterion column j across alternatives (including the optimal row x̃_0) using the Xu vector normalization (Eqs. 21-22). Each component of the IVIFN is divided by the L2 norm of its column. a_ij^U = μ_ij^U / sqrt(Σ_{l=0}^m ((μ_lj^L)^2 + (μ_lj^U)^2)) b_ij^L = ν_ij^L / sqrt(Σ_{l=0}^m ((ν_lj^L)^2 + (ν_lj^U)^2)) [Eq.(22)] b_ij^U = ν_ij^U / sqrt(Σ_{l=0}^m ((ν_lj^L)^2 + (ν_lj^U)^2)) Result: r̃_ij = ⟨[a_ij^L, a_ij^U], [b_ij^L, b_ij^U]⟩, i ∈ {0,1,...,m}, j ∈ {1,...,n}.

   $$a_i{j}^L={\mu }_i{j}^L/sqrt({\Sigma }_{l=0}^m(({\mu }_l{j}^L{)}^2+({\mu }_l{j}^U{)}^2))[Eq.(21)]$$

   Büyüközkan-Göçer 2018 §3.2 Step 11 + Eqs.(21)-(22)

3. 3

   Step 12 - Apply criterion weights via IVIF scalar multiplication (Eq. 10 with λ = w_j) to obtain the weighted-normalized matrix R̂ (Eq. 23). Step 13 - Sum the weighted-normalized IVIFN cells across criteria (IVIF sum Eq. 6) to get the optimality function P̃_i (Eq. 24). Defuzzify P̃_i → P_i using the score function (μ^L + μ^U + (1-ν^L) + (1-ν^U))/4 ∈ [0,1] (default), or paper Eq. 25 variant. λã = ⟨[1-(1-μ^L)^λ, 1-(1-μ^U)^λ], [(ν^L)^λ, (ν^U)^λ]⟩ [Eq.(10)] R̂_ij = w_j ⊗_λ r̃_ij [Eq.(23)] Step 13 - Optimality function (IVIF sum Eq. 6): ã ⊕ b̃ = ⟨[μ_a^L+μ_b^L - μ_a^L·μ_b^L, μ_a^U+μ_b^U - μ_a^U·μ_b^U], [ν_a^L·ν_b^L, ν_a^U·ν_b^U]⟩ [Eq.(6)] P̃_i = ⊕_{j=1}^n r̂_ij [Eq.(24)] Defuzzification (default - Hwang-Lin-style score, [0,1]-bounded, monotonic, best=1): P_i = (μ^L(P̃_i) + μ^U(P̃_i) + (1-ν^L(P̃_i)) + (1-ν^U(P̃_i))) / 4 Paper Eq. 25 variant available via parameter defuzzification_method='paper_eq25'.

   $$Step12-Weighted-normalizedmatrix(IVIFscalarmultEq.10):$$

   Büyüközkan-Göçer 2018 §3.2 Steps 12-13 + Eqs.(6),(10),(23)-(25)

4. 4

   Step 14 - Compute the utility degree Q_i = P_i / P_0 (Eq. 26), the ratio of each alternative's defuzzified optimality to that of the optimal reference row. Step 15 - Rank alternatives by descending Q_i; the alternative with the highest Q_i is the best. where P_0 is the defuzzified optimality of the optimal reference row x̃_0. Rank: Q_{[1]} ≥ Q_{[2]} ≥ ... ≥ Q_{[m]} (descending). Best alternative = A_{[1]}, worst = A_{[m]}.

   $$Q_i=P_i/P_0(i=1,...,m)[Eq.(26)]$$

   Büyüközkan-Göçer 2018 §3.2 Steps 14-15 + Eq.(26)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid IVIFNs (μ⁺+ν⁺ ≤ 1)
• Weights are positive and sum to 1
• Defuzzification choice documented (score default vs paper_eq25)
• Optimal reference row x̃_0 has at least one non-trivial criterion
• Decision-maker accepts utility-degree framing (vs. compromise-ranking VIKOR or distance-to-ideal TOPSIS)

Edge cases and pitfalls

• •IF decision matrix x̃_ij and prepend an optimal performance row x̃_0 (Eq. 18). For benefit criterion j ∈ J1: μ^L(x̃_0j) = max_i μ^L(x̃_ij), μ^U(x̃_0j) = max_i μ^U(x̃_ij), ν^L(x̃_0j) = min_i ν^L(x̃_ij)
• •IF scalar multiplication (Eq. 10 with λ = w_j) to obtain the weighted-normalized matrix R̂ (Eq. 23). Step 13 - Sum the weighted-normalized IVIFN cells across criteria (IVIF sum Eq. 6) to get the optim

• Value-space violation: ensure all entries satisfy IVIFN constraints 0 ≤ μ⁻ ≤ μ⁺ ≤ 1, 0 ≤ ν⁻ ≤ ν⁺ ≤ 1, μ⁺+ν⁺ ≤ 1 (Atanassov-Gargov 1989).

• Cost criteria handling: reverse the max ↔ min direction when constructing x̃_0 for cost criteria (Eq. 20) - use min for μ-components and max for ν-components. Do NOT invert IVIFN components themselves.

• Xu normalization includes x̃_0: the normalization sum (Eqs. 21-22) must include the optimal reference row alongside the m alternatives, not just the m alternatives. This is the canonical Büyüközkan-Göçer 2018 procedure.

• Defuzzification polarity: the score-based defuzzifier (μ^L+μ^U+(1-ν^L)+(1-ν^U))/4 maps best IVIFN to 1, worst to 0 - monotonic ascending in 'goodness'. Paper Eq. 25 has known polarity issues; use the score-based default unless specifically reproducing paper results.

• Rank reversal: ARAS uses an alternative-derived reference row x̃_0 (Eqs. 19-20) and Xu normalization across all rows. Adding or removing alternatives changes both x̃_0 and the normalization denominator, which can re-order surviving alternatives. Apply S-block sensitivity analysis to characterize stability.

• IVIF AHP weight elicitation (Steps 1-9, paper §3.2) is out of F.steps scope. F.steps single-DM kernel assumes weights given as a crisp simplex. To replicate the full Büyüközkan-Göçer 2018 case study, run an IVIF-AHP pipeline first (IIFWA aggregation + C.R. ≤ 0.10 + Eq. 17 weight extraction) and feed the result to IVIF-ARAS.

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