TIFN-CODAS: Triangular Intuitionistic Fuzzy Number CODAS (Daami Remadi & Frikha 2023)

Overview

Combinative Distance-based Assessment under Triangular Intuitionistic Fuzzy uncertainty (TIFN: {(a1,a2,a3); (a'1,a2,a'3)}; a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3): MCGDM with linguistic-to-TIFN translation. Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Combinative Distance-based Assessment under Triangular Intuitionistic Fuzzy uncertainty (TIFN: {(a1,a2,a3); (a'1,a2,a'3)}; a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3): MCGDM with linguistic-to-TIFN translation
• •Preserves triangular_intuitionistic uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: Inherited from crisp CODAS (Keshavarz Ghorabaee 2016); empirical comparison with IVIF-VIKOR/TOPSIS in Daami Remadi & Frikha 2023 Table 10 shows close rankings across methods.)
• •Assumes: Decision matrix entries are valid TIFNs (ordering a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3)
• •Assumes: DM weights λ_l sum to 1 in MAGDM group setting
• •Assumes: All decision-makers use the same linguistic-to-TIFN translation table (paper Tables 1-2)
• •Assumes: τ threshold reflects domain-appropriate Euclidean equality tolerance

Assumptions to verify

• •Decision matrix entries are valid TIFNs (ordering a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3)
• •DM weights λ_l sum to 1 in MAGDM group setting
• •All decision-makers use the same linguistic-to-TIFN translation table (paper Tables 1-2)
• •τ threshold reflects domain-appropriate Euclidean equality tolerance

When not to use

• •Crisp data sufficient: use base CODAS directly
• •Single-valued IFS already provides enough granularity: use IF-CODAS (Ren 2018)
• •Interval bounds on μ/ν needed: use IVIF-CODAS (Boltürk-Kahraman 2018)
• •Linguistic granularity coarser than 7 points: consider crisp methods

Edge cases

• •When w_j is itself a TIFN (linguistic weight, paper canon), use full TIFN multiplication; when w_j is crisp, the scalar form (Eq. 8) applies component-wise.
• •default τ = 0.02) per Eq. 19; sum row-wise to obtain individual assessment score H_i (Eq. 20).

Common pitfalls

• •Hatalı: 'TIFN-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid TIFNs (ordering a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3)
• •Hatalı: 'TIFN-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: DM weights λ_l sum to 1 in MAGDM group setting
• •Hatalı: 'TIFN-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-makers use the same linguistic-to-TIFN translation table (paper Tables 1-2)
• •Hatalı: 'TIFN-CODAS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: τ threshold reflects domain-appropriate Euclidean equality tolerance
• •Hatalı: TIFN-CODAS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: TIFN-CODAS'yi 'Single-valued IFS already provides enough granularity' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: TIFN-CODAS'yi 'Interval bounds on μ/ν needed' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Steps 1-2 (paper Part i): Build the linguistic decision matrix X^(l) for each DM d_l (l=1..y) over n alternatives × m criteria; transform linguistic ratings + linguistic criterion weights + DM weights to Triangular Intuitionistic Fuzzy Numbers (TIFN) via fixed 7-point scale (paper Tables 1-2). TIFN representation: x̂_ij^(l) = {(x_ij^1, x_ij^2, x_ij^3); (x'_ij^1, x_ij^2, x'_ij^3)}. Formül: Linguistic-to-TIFN map (Daami Remadi & Frikha 2023 Table 1, 7-point): VL = {(0,0,0.5); (0,0,0.5)} or {(0.5,0.5,0.5); (0.5,0.5,0.5)} (paper variant: anchor at scale boundary) L = {(0,1,3); (0,1,4)} ML = {(1,3,5); (0.5,3,5.5)} M = {(3,5,7); (2,5,8)} MH = {(5,7,9); (4.5,7,9.5)} H = {(7,9,10); (6,9,10)} VH = {(9,10,10); (8,10,10)} TIFN ordering invariant: a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3. Anchor: Daami Remadi & Frikha 2023, Part (i) S1-S2, p.5 + Tables 1-2
2. 2.Adım 2 (F2): Step 3 (paper Part ii): For each DM, compute the TIFN normalized decision matrix n̂_ij. Benefit criteria divide by x_j+ = max_i x_ij^'3 (Eq. 13); cost criteria use complementary form with x_j- = min_i x_ij^'1 (Eq. 14). Normalization is component-wise on all 6 TIFN parameters. Formül: Benefit (j ∈ Nb): n̂_ij = {(x_ij^1/x_j+, x_ij^2/x_j+, x_ij^3/x_j+); (x'_ij^1/x_j+, x_ij^2/x_j+, x'_ij^3/x_j+)} [Eq.(13)] where x_j+ = max_i x_ij^'3 Cost (j ∈ Nc): n̂_ij = {(x_j-/x_ij^3, x_j-/x_ij^2, x_j-/x_ij^1); (x_j-/x'_ij^3, x_j-/x_ij^2, x_j-/x'_ij^1)} [Eq.(14)] where x_j- = min_i x_ij^'1 Anchor: Daami Remadi & Frikha 2023, Part (ii) S3, p.6 Eqs.(13)-(14)
3. 3.Adım 3 (F3): Step 4: Compute the weighted normalized matrix r̂_ij = w_j ⊗ n̂_ij via TIFN scalar multiplication (Eq. 7-8). When w_j is itself a TIFN (linguistic weight, paper canon), use full TIFN multiplication; when w_j is crisp, the scalar form (Eq. 8) applies component-wise. Formül: Scalar TIFN multiplication (k > 0): k * A_TIFN = {(k*a1, k*a2, k*a3); (k*a'1, k*a2, k*a'3)} [Eq.(8)] TIFN-TIFN multiplication (A * B): A * B = {(a1*b1, a2*b2, a3*b3); (a'1*b'1, a2*b2, a'3*b'3)} [Eq.(7)] Weighted matrix: r̂_ij = w_j ⊗ n̂_ij ∀ i=1..n, j=1..m [Eq.(15)] Anchor: Daami Remadi & Frikha 2023, Part (ii) S4, p.6 Eq.(15) + Eqs.(7)-(8)
4. 4.Adım 4 (F4): Step 5: Determine the TIFN negative-ideal solution tns_j for each criterion j. Selection rule: tns_j = r̂_(i*)j where i* = argmin_i r'_ij^1 (smallest left-of-ν boundary). Vector tn̂s = [tns_1, ..., tns_m]. Formül: tn̂s = [tn̂s_j]_{1×m} [Eq.(16)] tn̂s_j = {(ns_j^1, ns_j^2, ns_j^3); (ns'_j^1, ns_j^2, ns'_j^3)} = min_i r̂_ij where min_i r̂_ij selects the r̂_ij with the lowest r'_ij^1. Anchor: Daami Remadi & Frikha 2023, Part (ii) S5, p.6 Eq.(16)
5. 5.Adım 5 (F5): Step 6: Defuzzify the TIFN weighted normalized matrix and the TIFN negative-ideal solution to crisp scalars via the Gani-Abbas (2014) weighted-average defuzzifier (Eq. 10). Each TIFN element A = {(a1,a2,a3);(a'1,a2,a'3)} maps to A_d ∈ ℝ. Formül: Defuzzifier (Gani-Abbas 2014): A_d = ((a1 + 2a2 + a3) + (a'1 + 2a2 + a'3)) / 8 [Eq.(10)] Weighted matrix defuzzification: r^d_ij = ((r_ij^1 + 2 r_ij^2 + r_ij^3) + (r'_ij^1 + 2 r_ij^2 + r'_ij^3)) / 8 [Eq.(17)] NIS defuzzification: ns^d_j = ((ns_j^1 + 2 ns_j^2 + ns_j^3) + (ns'_j^1 + 2 ns_j^2 + ns'_j^3)) / 8 [Eq.(18)] Anchor: Daami Remadi & Frikha 2023, Part (ii) S6, p.7 Eqs.(17)-(18) + Eq.(10)
6. 6.Adım 6 (F6): Step 7: Apply classical CODAS (Keshavarz Ghorabaee 2016) on the defuzzified matrix r^d_ij and NIS ns^d_j. Compute the Euclidean distance E_i (Eq. 1, L2 norm) and Taxicab distance T_i (Eq. 2, L1 norm) of each alternative from the negative-ideal. Build the n×n relative assessment matrix R_a using threshold function ψ (default τ = 0.02) per Eq. 19; sum row-wise to obtain individual assessment score H_i (Eq. 20). Formül: Euclidean distance: E_i = sqrt( Σ_{j=1}^m (r^d_ij − ns^d_j)^2 ) [Eq.(1)] Taxicab distance: T_i = Σ_{j=1}^m | r^d_ij − ns^d_j | [Eq.(2)] Relative assessment matrix: h_ik = (E_i − E_k) + ψ(E_i − E_k) × (T_i − T_k) [Eq.(19)] ψ(x) = 1 if |x| ≥ τ ; 0 otherwise (τ default = 0.02 per Keshavarz Ghorabaee 2016) Assessment score: H_i = Σ_{k=1}^n h_ik [Eq.(20)] Anchor: Daami Remadi & Frikha 2023, Part (ii) S7, p.7 Eqs.(19)-(20) + Eqs.(1)-(2)
7. 7.Adım 7 (F7): Step 8 (paper Part iii): Aggregate individual assessment scores across l decision makers using DM weights λ_l (Σ_l λ_l = 1) into the group assessment score HG_i. In single-DM mode (y=1), HG_i = H_i trivially. Formül: HG_i = Σ_{l=1}^y λ_l × H_i^(l) [Eq.(21)] Where H_i^(l) is the individual assessment score of alternative i under decision-maker d_l, and λ_l is the crisp DM weight. Anchor: Daami Remadi & Frikha 2023, Part (iii) S8, p.7 Eq.(21)
8. 8.Adım 8 (F8): Step 9: Rank alternatives in descending order of group assessment score HG_i. Higher HG_i indicates greater desirability (the alternative is farther from the negative-ideal in the L2/L1 combined metric). Formül: Ranking: sort alternatives by HG_i descending. A_(i_1) ≻ A_(i_2) ≻ ... ≻ A_(i_n) ⇔ HG_{i_1} > HG_{i_2} > ... > HG_{i_n} Anchor: Daami Remadi & Frikha 2023, Part (iii) S9, p.7

Commonly paired with

• •AHP + TIFN-CODAS (common)
• •ENTROPY + TIFN-CODAS (occasional)

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