Üçgen sezgisel bulanık CODAS (Daami Remadi ve Frikha, 2023)

Temel Yöntemden Ne Değişir?

Çıktı Nasıl Yorumlanır?

Ne Zaman Temel Yöntem Yerine Bu?

Bu Uzantıya Özgü Hatalar

Vakalar

Manifestteki J örneği literatür sayfasından değil, DecisionMind'ın kendi doğrulama örneğinden gelir; üç seçenek üç kriterde, Daami Remadi ve Frikha'nın (2023) yedi terimli dilsel ölçeğinden seçilmiş TIFN'lerle kurulmuştur. Birinci vaka bu örnektir. İkinci vaka öğretici kurgudur.

Formüller

Hesabın adımları ve her adımın formülü. Adım açıklamaları İngilizcedir.

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)}.

   Linguistic-to-TIFN map (Daami Remadi & Frikha 2023 Table 1, 7-point): VL = {(0.5,0.5,0.5); (0.5,0.5,0.5)} 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.

2. 2. For each decision maker, compute the TIFN normalised decision matrix n_ij. Benefit criteria divide every component by x_j+ = max_i x'3_ij (Eq. 13); cost criteria use x_j- over the components with the order reversed so the TIFN stays ordered (Eq. 14, x_j- = min_i x'1_ij; the paper text says the TIFN with the low x'3, see L.deviations_from_seminal). The five TIFN parameters (a1, a2, a3, a'1, a'3) are normalised together.

   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

3. 3. 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.

   $$ScalarTIFNmultiplication(k>0):k\ast A_{T}IFN=(k\ast a1,k\ast a2,k\ast a3);(k\ast a^{\prime }1,k\ast a2,k\ast a^{\prime }3)[Eq.(8)]TIFN-TIFNmultiplication(A\ast B):A\ast B=(a1\ast b1,a2\ast b2,a3\ast b3);(a^{\prime }1\ast b^{\prime }1,a2\ast b2,a^{\prime }3\ast b^{\prime }3)[Eq.(7)]Weightedmatrix:{\hat{r}}_{i}j=w_j\otimes {\hat{n}}_{i}j\forall i=\mathrm{1..}n,j=\mathrm{1..}m[Eq.(15)]$$

4. 4. 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 = [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.

5. 5. The paper defuzzifies the TIFN weighted normalised matrix and the TIFN negative-ideal solution to crisp scalars with the Gani-Abbas (2014) average (Eq. 10, 17-18) before the CODAS distances; this is the point where the source leaves the TIFN algebra. Each TIFN {(a1,a2,a3);(a'1,a2,a'3)} maps to a real number.

   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)]

6. 6. 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).

   $$Euclideandistance:E_i=sqrt({\Sigma }_{j=1}^m(r_i^{d}j-n{s}_j^d{)}^2)[Eq.(1)]Taxicabdistance:T_i={\Sigma }_{j=1}^m|r_i^{d}j-n{s}_j^d|[Eq.(2)]Relativeassessmentmatrix:h_{i}k=(E_i-E_k)+\psi (E_i-E_k)\times (T_i-T_k)[Eq.(19)]\psi (x)=1if|x|\ge \tau ;0otherwise(\tau default=0.02perKeshavarzGhorabaee2016)Assessmentscore:H_i={\Sigma }_{k=1}^n{h}_{i}k[Eq.(20)]$$

7. 7. 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.

   $$H{G}_i={\Sigma }_{l=1}^y{\lambda }_l\times H_i^{(}l)[Eq.(21)]Where{H}_i^{(}l)istheindividualassessmentscoreofalternativeiunderdecision-maker{d}_l,and{\lambda }_{l}isthecrispDMweight.$$

8. 8. 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).

   $$Ranking:sortalternativesbyH{G}_{i}descending.A_{(}{i}_1)\succ A_{(}{i}_2)\succ ...\succ A_{(}{i}_n)\iff H{G}_{i_1}>H{G}_{i_2}>...>H{G}_{i_n}$$

Kaynaklar