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

Overview

TIFN-CODAS extends classical CODAS (Keshavarz Ghorabaee 2016) to handle Triangular Intuitionistic Fuzzy uncertainty (Li-Nan-Zhang 2010 TIFN over Atanassov 1986 IFS). Linguistic ratings are mapped to TIFNs via a fixed 7-point scale (paper Table 1). TIFN arithmetic (normalize → weight) is performed in fuzzy space, then defuzzified via Gani-Abbas (2014) weighted-average to crisp scalars before applying CODAS distances (Euclidean L2 + Taxicab L1) from the TIFN negative-ideal. Combined assessment uses threshold ψ (default τ = 0.02) to recognize equivalent Euclidean distances. For MCGDM (l > 1), individual H_i scores are aggregated by DM weights λ. Higher HG_i = better alternative.

Output

utility, higher is better

Data

Triangular Intuitionistic Fuzzy, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Green material selection, Sustainable supplier evaluation, MAGDM with linguistic ratings, Environmental/economic dual-criterion problems, Construction material selection

How it works

1. 1

   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)}. 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.

   Linguistic-to-TIFN map (Daami Remadi & Frikha 2023 Table 1, 7-point):

   Daami Remadi & Frikha 2023, Part (i) S1-S2, p.5 + Tables 1-2

2. 2

   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. 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

   $$Benefit(j\in Nb):$$

   Daami Remadi & Frikha 2023, Part (ii) S3, p.6 Eqs.(13)-(14)

3. 3

   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. 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)]

   $$ScalarTIFNmultiplication(k>0):$$

   Daami Remadi & Frikha 2023, Part (ii) S4, p.6 Eq.(15) + Eqs.(7)-(8)

4. 4

   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]. 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.

   $$t\hat{n}s=[t\hat{n}{s}_j{]}_{1\times m}[Eq.(16)]$$

   Daami Remadi & Frikha 2023, Part (ii) S5, p.6 Eq.(16)

5. 5

   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. 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)]

   $$Defuzzifier(Gani-Abbas2014):$$

   Daami Remadi & Frikha 2023, Part (ii) S6, p.7 Eqs.(17)-(18) + Eq.(10)

6. 6

   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). 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)]

   $$Euclideandistance:$$

   Daami Remadi & Frikha 2023, Part (ii) S7, p.7 Eqs.(19)-(20) + Eqs.(1)-(2)

7. 7

   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. 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.

   $$H{G}_i={\Sigma }_{l=1}^y{\lambda }_l\times H_i^{(}l)[Eq.(21)]$$

   Daami Remadi & Frikha 2023, Part (iii) S8, p.7 Eq.(21)

8. 8

   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). A_(i_1) ≻ A_(i_2) ≻ ... ≻ A_(i_n) ⇔ HG_{i_1} > HG_{i_2} > ... > HG_{i_n}

   $$Ranking:sortalternativesbyH{G}_{i}descending.$$

   Daami Remadi & Frikha 2023, Part (iii) S9, p.7

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• •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).

• Value-space violation: ensure all TIFN entries satisfy ordering a'1 ≤ a1 ≤ a2 ≤ a3 ≤ a'3; the membership μ-support [a1,a3] must be inside the non-membership ν-support [a'1,a'3].

• ψ threshold sensitivity: τ value materially affects ranking when alternatives' Euclidean distances are close. Paper default τ = 0.02 (Keshavarz Ghorabaee 2016 recommendation); user can override via D.parameters.psi_threshold.

• Defuzzification order: defuzzify ONLY after weighted normalized matrix is built - never defuzzify the raw decision matrix first (loses TIFN-weighted scaling information).

• Cost criterion normalization: use Eq. 14 with x_j- = min_i x_ij^'1 - do NOT apply TIFN complement (μ↔ν swap); paper uses ratio-form min/max normalization, not complement.

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