IVIF-TODIM - Interval-Valued Intuitionistic Fuzzy TODIM (Krohling & Pacheco 2014)

Overview

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.

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

Investment decision under uncertainty, Project selection with environmental impact, Risk-aware alternative ranking, Behavioral economics applications, MCDM under epistemic uncertainty

How it works

1. 1

   Normalize the IVIF decision matrix à = [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).

   $${\mu }_i^{L}j=a_i^{L}j/({\Sigma }_{k=1}^m((a_k^{L}j{)}^2+(a_k^{U}j{)}^2){)}^{1/2}[Eq.(4)]$$

   Krohling-Pacheco 2014 §3 Step 1, p.239 Eqs.(4)-(5)

2. 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. 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.

   $$\delta ({\tilde{R}}_i,{\tilde{R}}_j)={\Sigma }_{c=1}^n{\phi }_c({\tilde{R}}_i,{\tilde{R}}_j),\forall (i,j)[Eq.(6)]$$

   Krohling-Pacheco 2014 §3 Step 2, p.239 Eqs.(6)-(7) + Eqs.(1)-(3) Def 3-6

3. 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). Ranking: sort alternatives by descending ξ_i.

   $${\xi }_i=({\Sigma }_j\delta (i,j)-mi{n}_i{\Sigma }_j\delta (i,j))/(ma{x}_i{\Sigma }_j\delta (i,j)-mi{n}_i{\Sigma }_j\delta (i,j))[Eq.(8)]$$

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

Fits when / Look elsewhere when

Assumptions to verify

• 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)

Edge cases and pitfalls

• •IF decision matrix à = [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 complemen
• •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); rat

• Value-space violation: ensure all entries satisfy IVIFN constraints 0 ≤ a1 ≤ a2 ≤ 1, 0 ≤ a3 ≤ a4 ≤ 1, a2+a4 ≤ 1.

• θ must be strictly positive. θ < 1 amplifies losses superlinearly; θ > 1 attenuates losses. θ = 1 (default) is prospect-theory neutral.

• Reference criterion: w_r = max_c(w_c) is the criterion with greatest weight; w_rc = w_c/w_r ∈ (0, 1]. Do not normalize weights to sum-1 before computing w_rc - TODIM uses the ratio relative to the reference, not absolute weight.

• Cost criteria handling: apply IVIFN complement (a,b,c,d) ↦ (c,d,a,b) before Xu normalization Eqs. 4-5; do NOT apply max-min inversion on the raw a/b/c/d components.

• Rank reversal: TODIM family is known to exhibit rank reversal under alternative addition/removal because ξ is min-max normalized to [0,1] across the current alternative set. Document the alternative set explicitly when reporting results.

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