Rough-TODIM - TODIM yönteminin Rough uzantısı

Formül adımları

Analiz motorunun yöntem bildirimindeki (manifest F.steps) adımlar; raporlardaki formüllerle aynı kaynaktır.

1. Adım 1 — Kaba karar matrisini global maksimuma göre normalleştir. Tüm kriterler aynı şekilde işlenir (fayda/maliyet ayrımı yok): p_ij = [l_ij / max_global, u_ij / max_global]; max_global = tüm u_ij üzerindeki maksimum. Eq.5'ten doğrulandı: tüm bölenler = 6.5.

   $$p_{ij}=[\frac{l_{ij}}{{max}_{\text{global}}},\frac{u_{ij}}{{max}_{\text{global}}}],{max}_{\text{global}}={max}_{i,j}\{u_{ij}\}$$
   LaTeX

   p_{ij} = \left[\frac{l_{ij}}{\max_{\text{global}}},\; \frac{u_{ij}}{\max_{\text{global}}}\right],\quad \max_{\text{global}} = \max_{i,j}\{u_{ij}\}

2. Adım 2 — Referans kriter r: en büyük ağırlık; göreli ağırlıklar w_jr = w_j/w_r.

   $$r=\mathrm{\backslash arg}{max}_j{w}_j;w_{jr}=\frac{w_j}{w_r},j=1,\dots ,n$$
   LaTeX

   r = \arg\max_j w_j;\quad w_{jr} = \frac{w_j}{w_r},\quad j=1,\ldots,n

3. Adım 3 — Kriter c başına önem tabanlı ölçekleme ile TODIM değer fonksiyonu (θ=1). Mesafe (Denklem 2): d_c = |p_ic^L − p_jc^L| + |p_ic^U − p_jc^U|. Önem kuralları: C1/C2 tam ölçek (α=β=1); C3 (Orta Önemli) her iki terime 0.5 çarpanı; C4 (Düşük Önemli) Kural 3.

   $$\begin{gathered} d_c(A_i,A_j)=|p_{ic}^L-p_{jc}^L|+|p_{ic}^U-p_{jc}^U|(\text{Eq.2}) \\ [\mathrm{6pt}]{\phi }_c(A_i,A_j)=\{\begin{array}{cc}{\alpha }_c\sqrt{{\displaystyle \frac{W_{rc}^{(c)}}{{\sum }_c{W}_{rc}}}·d_c(A_i,A_j)}\hfill & p_{ij}>p_{jc}\hfill \\ 0\hfill & p_{ij}=p_{jc}\hfill \\ -{\displaystyle \frac{{\beta }_c}{\theta }}\sqrt{{\displaystyle \frac{W_{rc}^{(c)}}{{\sum }_c{W}_{rc}}}·d_c(A_i,A_j)}\hfill & p_{ij}<p_{jc}\hfill \end{array} \\ [\mathrm{6pt}]{\alpha }_c={\beta }_c=1(\text{C1 Important, C2 Most Important});{\alpha }_c={\beta }_c=0.5(\text{C3 Averagely Important});\text{C4: Rule 3 (Low: loss if low preference, else gain)};\theta =1 \end{gathered}$$
   LaTeX

   d_c(A_i,A_j)=|p_{ic}^{L}-p_{jc}^{L}|+|p_{ic}^{U}-p_{jc}^{U}|\quad(\text{Eq.2})\\[6pt]\varphi_c(A_i,A_j)=\begin{cases}\alpha_c\sqrt{\dfrac{W_{rc}^{(c)}}{\sum_c W_{rc}}\cdot d_c(A_i,A_j)}&p_{ij}>p_{jc}\\0&p_{ij}=p_{jc}\\-\dfrac{\beta_c}{\theta}\sqrt{\dfrac{W_{rc}^{(c)}}{\sum_c W_{rc}}\cdot d_c(A_i,A_j)}&p_{ij}<p_{jc}\end{cases}\\[6pt]\alpha_c=\beta_c=1\;(\text{C1 Important, C2 Most Important});\;\alpha_c=\beta_c=0.5\;(\text{C3 Averagely Important});\;\text{C4: Rule 3 (Low: loss if low preference, else gain)};\;\theta=1

4. Adım 4 — Toplam baskınlık δ(A_i,A_k) = Σ_j Φ_j(A_i,A_k).

   $$\delta (A_i,A_k)=\sum _{j=1}^n{\Phi }_j(A_i,A_k)$$
   LaTeX

   \delta(A_i,A_k) = \sum_{j=1}^{n} \Phi_j(A_i,A_k)

5. Adım 5 — [0,1]'e normalize edilmiş global beklenti ξ_i; azalan sıralama.

   $${\xi }_i=\frac{{\sum }_k\delta (A_i,A_k)-{min}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)}{{max}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)-{min}_{i^{\prime }}{\sum }_k\delta (A_{i^{\prime }},A_k)};\text{rank by }{\xi }_i\downarrow$$
   LaTeX

   \xi_i = \frac{\sum_k\delta(A_i,A_k)-\min_{i'}\sum_k\delta(A_{i'},A_k)}{\max_{i'}\sum_k\delta(A_{i'},A_k)-\min_{i'}\sum_k\delta(A_{i'},A_k)};\quad\text{rank by }\xi_i\downarrow

Sezgi

Sonucu okuma: rough-todim extends TODIM to handle Rough uncertainty. Decision matrix entries are rough numbers [L,U] representing customer preference scores. Normalization uses global-max (all bounds divided by max upper bound across matrix). The TODIM φ_c function computes crisp scalar distances d_c = |p^L_ic - p^L_jc| + |p^U_ic - p^U_jc| and applies importance-based scaling: full scale (α=β=1) for Important/Most-Important criteria, ×0.5 for Averagely Important, Rule 3 for Low Important. All dominance computations (δ, Σδ) are crisp. The final ξ scores are crisp [0,1] values computed directly from crisp dominance sums - NOT from midpoint defuzzification. CAUTION: paper does not specify how to compare rough intervals [L,U] for gain/loss determination; paper's Table 8 δ values cannot be reproduced from the described formula alone.

Varsayımlar

• Decision matrix entries are valid Rough numbers/tuples
• Underlying crisp method's compensation assumption holds in uncertain space
• All decision-maker(s) and experts use the same linguistic/uncertainty scale

Ne zaman kullanılmaz

• Classical data sufficient - use base TODIM directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sık yapılan hatalar

• Değer-uzayı ihlali: hesaplamadan önce tüm girişlerin Rough: L ≤ U; approximations defined by equivalence classes koşulunu sağladığından emin olun.
• ROUGH-TODIM nihai sıralama için midpoint defuzzifikasyon kullanmaz. Mesafeler d_c crisp skaler; tüm baskınlık toplamları crisp. Çözülmemiş sorun: φ_c'de kazanç/kayıp tespiti için rough aralıkların nasıl karşılaştırılacağı belirtilmemiş.