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

Overview

Prospect-theory dominance MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ([μ⁻,μ⁺],[ν⁻,ν⁺]); μ⁺+ν⁺ ≤ 1). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Prospect-theory dominance MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ([μ⁻,μ⁺],[ν⁻,ν⁺]); μ⁺+ν⁺ ≤ 1)
• •Preserves interval_intuitionistic uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from crisp TODIM (Gomes-Lima 1992); θ-attenuation factor mitigates but does not eliminate rank reversal under alternative addition.)
• •Assumes: Decision matrix entries are valid IVIFNs (a2+a4 ≤ 1)
• •Assumes: Weights are positive and sum to 1
• •Assumes: θ > 0 (loss attenuation factor)
• •Assumes: Decision-maker exhibits prospect-theory loss aversion behavior (otherwise prefer additive methods like TOPSIS, MABAC)

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)

When not to use

• •Crisp data sufficient: use base TODIM directly
• •Single-valued IFS already provides enough granularity: use IF-TODIM (Krohling-Pacheco-Siviero 2013)
• •Decision-maker is risk-neutral (no loss aversion): use IVIF-TOPSIS or IVIF-MABAC instead
• •Need group decision aggregation: IVIF-TODIM is single-DM; for MAGDM use IVIF-MABAC (Xue 2016) or wrap with IVIFWG/IVIFWA pre-aggregation

Edge cases

• •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

Common pitfalls

• •Hatalı: 'IVIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid IVIFNs (a2+a4 ≤ 1)
• •Hatalı: 'IVIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Weights are positive and sum to 1
• •Hatalı: 'IVIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: θ > 0 (loss attenuation factor)
• •Hatalı: 'IVIF-TODIM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-maker exhibits prospect-theory loss aversion behavior (otherwise prefer additive methods like TOPSIS, MABAC)
• •Hatalı: IVIF-TODIM'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IVIF-TODIM'yi 'Single-valued IFS already provides enough granularity' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IVIF-TODIM'yi 'Decision-maker is risk-neutral (no loss aversion)' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 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. Formül: μ^L_ij = a^L_ij / ( Σ_{k=1}^m ((a^L_kj)^2 + (a^U_kj)^2) )^{1/2} [Eq.(4)] 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). Anchor: Krohling-Pacheco 2014 §3 Step 1, p.239 Eqs.(4)-(5)
2. 2.Adım 2 (F2): Step 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. Formül: δ(R̃_i, R̃_j) = Σ_{c=1}^n φ_c(R̃_i, R̃_j), ∀(i,j) [Eq.(6)] 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. Anchor: Krohling-Pacheco 2014 §3 Step 2, p.239 Eqs.(6)-(7) + Eqs.(1)-(3) Def 3-6
3. 3.Adım 3 (F3): Step 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). Formül: ξ_i = ( Σ_j δ(i,j) - min_i Σ_j δ(i,j) ) / ( max_i Σ_j δ(i,j) - min_i Σ_j δ(i,j) ) [Eq.(8)] Ranking: sort alternatives by descending ξ_i. Anchor: Krohling-Pacheco 2014 §3 Step 3, p.240 Eq.(8)

Commonly paired with

• •IF-ENTROPY + IVIF-TODIM (common)
• •AHP + IVIF-TODIM (occasional)

How to cite