IVIF-VIKOR: Interval-Valued Intuitionistic Fuzzy VIKOR (Park, Cho & Kwun 2011)

Overview

Compromise-ranking MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ⟨[μ⁻,μ⁺],[ν⁻,ν⁺]⟩; μ⁺+ν⁺ ≤ 1): Lp-metric distance to interval-valued intuitionistic PIS. Output typically compromise_index (lower value = preferred).

Strengths

• •Method-specific: Compromise-ranking MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ⟨[μ⁻,μ⁺],[ν⁻,ν⁺]⟩; μ⁺+ν⁺ ≤ 1): Lp-metric distance to interval-valued intuitionistic PIS
• •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 VIKOR (Opricovic 1998). Compromise solution depends on PIS/NIS bounds: alternative addition/removal can shift PIS/NIS and trigger rank reversal. Acceptable-advantage condition C1 (Q̃_{[2]}-Q̃_{[1]} ≥ 1/(m-1)) acts as a stability gate but does not eliminate reversal.)
• •Assumes: Decision matrix entries are valid IVIFNs (b+d ≤ 1)
• •Assumes: Weights are positive and sum to 1
• •Assumes: Strategy weight v ∈ [0,1] (default 0.5 = consensus)
• •Assumes: PIS and NIS are distinct for every criterion (d(r̃_i+, r̃_i-) > 0)

Assumptions to verify

• •Decision matrix entries are valid IVIFNs (b+d ≤ 1)
• •Weights are positive and sum to 1
• •Strategy weight v ∈ [0,1] (default 0.5 = consensus)
• •PIS and NIS are distinct for every criterion (d(r̃_i+, r̃_i-) > 0)
• •Decision-maker accepts compromise-ranking framing (vs. distance-to-ideal TOPSIS or prospect-theory TODIM)

When not to use

• •Crisp data sufficient: use base VIKOR (Opricovic 1998) directly
• •Single-valued IFS already provides enough granularity: use IF-VIKOR (Devi 2011)
• •Decision-maker wants prospect-theory loss aversion: use IVIF-TODIM instead
• •Decision-maker wants strict distance-to-ideal ranking without compromise stability conditions: use IVIF-TOPSIS instead

Edge cases

• •default 0.5 = consensus). v > 0.5 = voting by majority (S̃ dominant), v < 0.5 = veto (R̃ dominant).
• •if both C1 (acceptable advantage) and C2 (stability) hold.

Common pitfalls

• •Hatalı: 'IVIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid IVIFNs (b+d ≤ 1)
• •Hatalı: 'IVIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Weights are positive and sum to 1
• •Hatalı: 'IVIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Strategy weight v ∈ [0,1] (default 0.5 = consensus)
• •Hatalı: 'IVIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: PIS and NIS are distinct for every criterion (d(r̃_i+, r̃_i-) > 0)
• •Hatalı: 'IVIF-VIKOR bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-maker accepts compromise-ranking framing (vs. distance-to-ideal TOPSIS or prospect-theory TODIM)
• •Hatalı: IVIF-VIKOR'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IVIF-VIKOR'yi 'Single-valued IFS already provides enough granularity' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IVIF-VIKOR'yi 'Decision-maker wants prospect-theory loss aversion' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Determine the interval-valued intuitionistic PIS O* and NIS O- by componentwise max (for benefit) / min (for cost) over alternatives. For benefit criterion i ∈ J1: r̃_i+ = ⟨[max_j a_ij, max_j b_ij], [min_j c_ij, min_j d_ij]⟩; r̃_i- = ⟨[min_j a_ij, min_j b_ij], [max_j c_ij, max_j d_ij]⟩. For cost criterion i ∈ J2: swap max ↔ min on a,b and c,d. Park 2011 Eqs. 16-17. Formül: O* = {⟨u_i, (max_j r̃_ij | i∈J1), (min_j r̃_ij | i∈J2)⟩ | i=1,...,m}^T = (r̃_1+, r̃_2+, ..., r̃_m+)^T [Eq.(16)] O- = {⟨u_i, (min_j r̃_ij | i∈J1), (max_j r̃_ij | i∈J2)⟩ | i=1,...,m}^T = (r̃_1-, r̃_2-, ..., r̃_m-)^T [Eq.(17)] For i ∈ J1 (benefit): r̃_i+ = ⟨[max_j a_ij, max_j b_ij], [min_j c_ij, min_j d_ij]⟩; r̃_i- = ⟨[min_j a_ij, min_j b_ij], [max_j c_ij, max_j d_ij]⟩ For i ∈ J2 (cost): r̃_i+ = ⟨[min_j a_ij, min_j b_ij], [max_j c_ij, max_j d_ij]⟩; r̃_i- = ⟨[max_j a_ij, max_j b_ij], [min_j c_ij, min_j d_ij]⟩ Anchor: Park-Cho-Kwun 2011 §4 p.241 Eqs.(16)-(17)
2. 2.Adım 2 (F2): Step 2: Compute the average score S̃_j (Eq. 18) and worst group score R̃_j (Eq. 21) for each alternative O_j using the Burillo-Bustince d1 distance (default). The ratio w_i·d1(r̃_i+, r̃_ij)/d1(r̃_i+, r̃_i-) measures the relative closeness of alternative j to the PIS under criterion i; S̃_j sums these over criteria (sum-aggregator), R̃_j takes the max (worst-case aggregator). Eqs. 19-20 / 22-23 give the d2 (modified Burillo-Bustince) and dH (Grzegorzewski) variants. Formül: d1(⟨[a1,b1],[c1,d1]⟩, ⟨[a2,b2],[c2,d2]⟩) = |a1-a2| + |b1-b2| + |c1-c2| + |d1-d2| S̃_j^{d1} = Σ_{i=1}^m w_i · d1(r̃_i+, r̃_ij) / d1(r̃_i+, r̃_i-) [Eq.(18)] R̃_j^{d1} = max_{1 ≤ i ≤ m} [ w_i · d1(r̃_i+, r̃_ij) / d1(r̃_i+, r̃_i-) ] [Eq.(21)] Alternative distance variants (selectable via parameter distance_measure): d2 modified Burillo-Bustince: Eqs.(19),(22) dH Grzegorzewski Hamming : Eqs.(20),(23) Anchor: Park-Cho-Kwun 2011 §4 p.241-242 Eqs.(18)-(23)
3. 3.Adım 3 (F3): Step 3: Compute the VIKOR compromise index Q̃_j blending the sum-aggregator S̃_j and the worst-case-aggregator R̃_j by the strategy weight v (default 0.5 = consensus). v > 0.5 = voting by majority (S̃ dominant), v < 0.5 = veto (R̃ dominant). Formül: Q̃_j = v · (S̃_j - S̃*) / (S̃- - S̃*) + (1 - v) · (R̃_j - R̃*) / (R̃- - R̃*) [Eq.(24)] where S̃* = min_j S̃_j, S̃- = max_j S̃_j [Eq.(25)] R̃* = min_j R̃_j, R̃- = max_j R̃_j [Eq.(26)] Default v = 0.5 (consensus). v > 0.5 = voting by majority; v < 0.5 = veto. Anchor: Park-Cho-Kwun 2011 §4 p.243 Eqs.(24)-(26)
4. 4.Adım 4 (F4): Step 4: Rank alternatives by S̃, R̃ and Q̃ values in ascending order. Three ranking lists S̃_[·], R̃_[·], Q̃_[·] are obtained. The alternative O_{j1} with minimum Q̃ is proposed as compromise solution if both C1 (acceptable advantage) and C2 (stability) hold. Formül: Rank O_j by ascending Q̃_j → list Q̃_[1] ≤ Q̃_[2] ≤ ... ≤ Q̃_[m] Also compute ascending lists for S̃ and R̃. Compromise conditions (Park 2011 §4 p.243 from Opricovic 1998): C1 (acceptable advantage): Q̃_[2] - Q̃_[1] ≥ DQ where DQ = 1 / (m - 1) C2 (stability in decision-making process): O_{j1} is also ranked best in S̃_[·] or R̃_[·] If C1 and C2 hold → unique compromise solution = O_{j1}. If only C2 fails → compromise set = {O_{j1}, O_{j2}} where Q̃_{j2} = Q̃_[2]. If C1 fails → compromise set = {O_{j1}, O_{j2}, ..., O_{jk}} for the maximum k with Q̃_[k] - Q̃_[1] < DQ. Anchor: Park-Cho-Kwun 2011 §4 p.243 (C1, C2) + Opricovic 1998 §2 Step 5

Commonly paired with

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

How to cite