IVIF-COPRAS: Interval-Valued Intuitionistic Fuzzy COPRAS (Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019)

Overview

Compound-proportional ranking MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ⟨[μ⁻,μ⁺],[ν⁻,ν⁺]⟩; μ⁺+ν⁺ ≤ 1): Xu normalization + IVIF profit/cost sums + Garg score (GIS) compound Q_i + utility degree D_i (%). Output typically utility_degree_percent (higher value = preferred).

Strengths

• •Method-specific: Compound-proportional ranking MCDM under Interval-Valued Intuitionistic Fuzzy uncertainty (IVIFN: ⟨[μ⁻,μ⁺],[ν⁻,ν⁺]⟩; μ⁺+ν⁺ ≤ 1): Xu normalization + IVIF profit/cost sums + Garg score (GIS) compound Q_i + utility degree D_i (%)
• •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 COPRAS (Zavadskas-Kaklauskas 1996; cf. Belton-Gear 1983, Wang-Luo 2009). The relative-weight Q_i formula depends on the cost-side denominator sum_GIS_C × (1/GIS_C_i) × Σ(1/GIS_C_k) which couples all alternatives; adding/removing an alternative re-scales this denominator and can shift the ranking.)
• •Assumes: Decision matrix entries are valid IVIFNs (b+d ≤ 1)
• •Assumes: Weights are non-negative and sum to 1 (paper Table 4 sums to 0.997 due to rounding: lenient O-3 tolerance 1e-3)
• •Assumes: At least one profit criterion exists (NB ≠ ∅)
• •Assumes: λ ∈ [0,1] (default 0.6 from Davoudabadi 2019 §4.1)

Assumptions to verify

• •Decision matrix entries are valid IVIFNs (b+d ≤ 1)
• •Weights are non-negative and sum to 1 (paper Table 4 sums to 0.997 due to rounding: lenient O-3 tolerance 1e-3)
• •At least one profit criterion exists (NB ≠ ∅)
• •λ ∈ [0,1] (default 0.6 from Davoudabadi 2019 §4.1)
• •Decision-maker accepts compound-proportional-ranking framing (vs. distance-to-ideal TOPSIS or compromise VIKOR)

When not to use

• •Crisp data sufficient: use base COPRAS (Zavadskas-Kaklauskas 1996) directly
• •Single-valued IFS already provides enough granularity: use IF-COPRAS
• •Decision-maker wants prospect-theory loss aversion: use IVIF-TODIM instead
• •Decision-maker wants compromise-stability conditions (C1/C2): use IVIF-VIKOR instead
• •All criteria are cost-direction (pure-all-cost configuration unsupported per Eq. 31)

Edge cases

• •IF scalar multiplication. Davoudabadi 2019 §3 Eq. 28.
• •IF sum B_i (Eq. 29) over benefit criteria j ∈ NB (max-direction), cost IVIF sum C_i (Eq. 30) over cost criteria j ∈ NC (min-direction). Both use Eq. 1 IVIF ⊕ operation.
• •Default λ = 0.6 (Davoudabadi 2019 §4.1).

Common pitfalls

• •Hatalı: 'IVIF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid IVIFNs (b+d ≤ 1)
• •Hatalı: 'IVIF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Weights are non-negative and sum to 1 (paper Table 4 sums to 0.997 due to rounding: lenient O-3 tolerance 1e-3)
• •Hatalı: 'IVIF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: At least one profit criterion exists (NB ≠ ∅)
• •Hatalı: 'IVIF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: λ ∈ [0,1] (default 0.6 from Davoudabadi 2019 §4.1)
• •Hatalı: 'IVIF-COPRAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision-maker accepts compound-proportional-ranking framing (vs. distance-to-ideal TOPSIS or compromise VIKOR)
• •Hatalı: IVIF-COPRAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IVIF-COPRAS'yi 'Single-valued IFS already provides enough granularity' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: IVIF-COPRAS'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 IV: Xu normalization: for each criterion j, normalize μ-components by Σ_i sqrt(a_ij²+b_ij²) and ν-components by Σ_i sqrt(c_ij²+d_ij²). This preserves IVIFN value-space (μ⁺+ν⁺ ≤ 1) under columnwise rescaling. Davoudabadi 2019 §3 Eqs. 14-19. Formül: For each criterion j = 1, ..., n: denom_mu_j = sqrt( Σ_{i=1}^{m} (a_ij² + b_ij²) ) [Eq. (14)-(15)] denom_nu_j = sqrt( Σ_{i=1}^{m} (c_ij² + d_ij²) ) [Eq. (16)-(17)] Then for each alternative i: ñ_ij = ⟨[ a_ij/denom_mu_j , b_ij/denom_mu_j ], [ c_ij/denom_nu_j , d_ij/denom_nu_j ]⟩ [Eq. (18)-(19)] Notation note: Davoudabadi 2019 §3 uses (μ^L, μ^U, ν^L, ν^U) = (a, b, c, d) for the IVIFN ⟨[a,b],[c,d]⟩ components. Anchor: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step IV, Eqs.(14)-(19)
2. 2.Adım 2 (F2): Step VII: Weighted normalized matrix n̂_ij = w_j ⊙ ñ_ij via Eq. 3 IVIF scalar multiplication. Davoudabadi 2019 §3 Eq. 28. Formül: n̂_ij = w_j ⊙ ñ_ij = ⟨[ 1 - (1-a_ij)^{w_j} , 1 - (1-b_ij)^{w_j} ], [ c_ij^{w_j} , d_ij^{w_j} ]⟩ [Eq. (28) via Eq. (3)] Anchor: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step VII, Eq.(28) via Eq.(3)
3. 3.Adım 3 (F3): Step IX: Profit IVIF sum B_i (Eq. 29) over benefit criteria j ∈ NB (max-direction), cost IVIF sum C_i (Eq. 30) over cost criteria j ∈ NC (min-direction). Both use Eq. 1 IVIF ⊕ operation. Formül: IVIF ⊕ operation (Eq. 1): ⟨[a,b],[c,d]⟩ ⊕ ⟨[a',b'],[c',d']⟩ = ⟨[a+a'-aa', b+b'-bb'], [cc', dd']⟩ B_i = ⊕_{j ∈ NB} n̂_ij (max-direction criteria) [Eq. (29)] C_i = ⊕_{j ∈ NC} n̂_ij (min-direction criteria) [Eq. (30)] where NB = {j : criteria_types[j] = 'max'} (profit), NC = {j : criteria_types[j] = 'min'} (cost). Anchor: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step IX, Eqs.(29)-(30) via Eq.(1)
4. 4.Adım 4 (F4): Step X: Relative weight Q_i (Eq. 31). Uses Garg's IVIF score function GIS (Eq. 7) on B_i and C_i. GIS(α; λ) = λ · (a+b)/2 + (1-λ) · (1 - (c+d)/2) ∈ [0,1] balances membership average and non-membership complement. Default λ = 0.6 (Davoudabadi 2019 §4.1). Formül: GIS score function (Eq. 7, λ-parametrized): GIS(⟨[a,b],[c,d]⟩; λ) = λ · (a+b)/2 + (1-λ) · (1 - (c+d)/2) [Eq. (7)] Compound relative weight (Eq. 31): Q_i = GIS(B_i; λ) + ( Σ_k GIS(C_k; λ) ) / ( GIS(C_i; λ) · Σ_k (1/GIS(C_k; λ)) ) [Eq. (31)] Note: Sum_{k=1..m} runs over all m alternatives. The cost-side fraction is the classical Zavadskas-Kaklauskas 1996 COPRAS denominator transposed into IVIF score-space via GIS. Anchor: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step X, Eq.(31) using GIS Eq.(7)
5. 5.Adım 5 (F5): Step XI: Utility degree D_i (Eq. 32) as percentage of the maximum Q_i across alternatives. The alternative with the highest Q achieves D = 100%. Rank by descending D (higher is better). Formül: Q_max = max_{i=1..m} Q_i D_i = ( Q_i / Q_max ) × 100% [Eq. (32)] Ranking: descending D_i. Best alternative = argmax_i D_i (= argmax_i Q_i). Anchor: Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step XI, Eq.(32)

Commonly paired with

• •IF-ENTROPY + IVIF-COPRAS (common)
• •AHP + IVIF-COPRAS (occasional)
• •BWM + IVIF-COPRAS (occasional)

How to cite