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

Overview

IVIF-COPRAS extends Zavadskas-Kaklauskas 1996 crisp COPRAS (Complex Proportional Assessment) to handle Interval-Valued Intuitionistic Fuzzy uncertainty (Atanassov-Gargov 1989). Each alternative is summarized by two IVIF aggregates: B_i (profit IVIF sum over benefit criteria) and C_i (cost IVIF sum over cost criteria). These are projected to crisp scores via Garg's λ-parametrized score function GIS (Eq. 7). The compound relative weight Q_i adds GIS(B_i) to a cost-side fraction that rewards alternatives with LOW cost-side scores. Utility degree D_i = Q_i / Q_max × 100% gives the percentage-of-best ranking signal - higher D is better. The best alternative achieves D = 100%.

Output

utility degree percent, higher is better

Data

Interval Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Resilient supplier selection, Supply chain risk management, Procurement decision-making under epistemic uncertainty, Multi-criteria evaluation with linguistic expert judgments, MAGDM under interval-valued intuitionistic fuzzy information

How it works

1. 1

   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. 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.

   $$Foreachcriterionj=1,...,n:$$

   Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step IV, Eqs.(14)-(19)

2. 2

   Step VII - Weighted normalized matrix n̂_ij = w_j ⊙ ñ_ij via Eq. 3 IVIF scalar multiplication. Davoudabadi 2019 §3 Eq. 28.

   $${\hat{n}}_{i}j=w_j\odot {\tilde{n}}_{i}j=⟨[1-(1-a_{i}j{)}^{w_j},1-(1-b_{i}j{)}^{w_j}],[c_i{j}^{w_j},d_i{j}^{w_j}]⟩[Eq.(28)viaEq.(3)]$$

   Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step VII, Eq.(28) via Eq.(3)

3. 3

   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. ⟨[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).

   $$IVIF\oplus operation(Eq.1):$$

   Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step IX, Eqs.(29)-(30) via Eq.(1)

4. 4

   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). 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.

   $$GISscorefunction(Eq.7,\lambda -parametrized):$$

   Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step X, Eq.(31) using GIS Eq.(7)

5. 5

   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). D_i = ( Q_i / Q_max ) × 100% [Eq. (32)] Ranking: descending D_i. Best alternative = argmax_i D_i (= argmax_i Q_i).

   $$Q_{m}ax=ma{x}_{i=\mathrm{1..}m}{Q}_i$$

   Davoudabadi-Mousavi-Mohagheghi-Vahdani 2019 §3 Step XI, Eq.(32)

Fits when / Look elsewhere when

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)

Edge cases and pitfalls

• •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).

• Value-space violation: ensure all entries satisfy IVIFN constraints 0 ≤ a ≤ b ≤ 1, 0 ≤ c ≤ d ≤ 1, b+d ≤ 1 (Atanassov-Gargov 1989).

• Pure-all-cost configuration not supported: Davoudabadi 2019 Eq. 31 requires NB ≠ ∅ (at least one profit criterion) - the cost-side fraction is added to GIS(B_i). E-4 input check enforces this.

• λ parameter semantics: λ ∈ [0,1] is the GIS membership-vs-complement balance, NOT a probability. λ = 0 weights only the non-membership complement (1 - ν_avg); λ = 1 weights only membership (μ_avg); λ = 0.5 is symmetric Xu-style; λ = 0.6 is the Davoudabadi 2019 §4.1 default.

• GIS denominator zero: if GIS(C_i; λ) = 0 for any alternative (only possible at degenerate IVIFN extremes), the Eq. 31 cost-side fraction is undefined. Engine should emit a warning and either treat that alternative's cost-side contribution as 0 or flag the configuration.

• Cost criteria handling: cost vs benefit partition is applied at Step IX (B/C sum split) - NOT at normalization. Do NOT swap (μ,ν) ↔ (ν,μ) for cost columns before Xu normalization; the cost-direction effect is encoded inside Eq. 31's cost-side denominator fraction.

• Rank reversal: COPRAS is known to exhibit rank reversal (Belton-Gear 1983, Wang-Luo 2009) when alternatives are added or removed, since Eq. 31's cost-side denominator Σ_k(1/GIS(C_k)) couples all alternatives. Use sensitivity analysis on alternative_removal when stability is critical.

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