Rough-MARCOS - MARCOS yönteminin Rough uzantısı

Formül adımları

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

1. Adım 1 — Defuzz karşılaştırmaları kullanarak IRN ideal (AI) ve anti-ideal (AAI) referans satırlarını tanımla.

   LaTeX

   ID_j^{IRN}=\begin{cases}[\max_i v^L_{ij},\,\max_i v^U_{ij}] & j\in\Omega_{+}\[\min_i v^L_{ij},\,\min_i v^U_{ij}] & j\in\Omega_{-}\end{cases}\\[4pt]AID_j^{IRN}=\begin{cases}[\min_i v^L_{ij},\,\min_i v^U_{ij}] & j\in\Omega_{+}\[\max_i v^L_{ij},\,\max_i v^U_{ij}] & j\in\Omega_{-}\end{cases}\quad\text{(Eqs.8--9)}

2. Adım 2 — IRN alanında AI'ye göre normalleştir.

   $$\begin{gathered} j\in {\Omega }_{+}:[t_{ij}^L,t_{ij}^U]=[\frac{v_{ij}^L}{v_{id}^U},\frac{v_{ij}^U}{v_{id}^L}]\text{(Eq.11)} \\ [\mathrm{4pt}]j\in {\Omega }_{-}:[t_{ij}^L,t_{ij}^U]=[\frac{v_{id}^L}{v_{ij}^U},\frac{v_{id}^U}{v_{ij}^L}]\text{(Eq.12)} \end{gathered}$$
   LaTeX

   j\in\Omega_{+}:\;[t^L_{ij},t^U_{ij}]=\Bigl[\tfrac{v^L_{ij}}{v^U_{id}},\tfrac{v^U_{ij}}{v^L_{id}}\Bigr]\quad\text{(Eq.11)}\\[4pt]j\in\Omega_{-}:\;[t^L_{ij},t^U_{ij}]=\Bigl[\tfrac{v^L_{id}}{v^U_{ij}},\tfrac{v^U_{id}}{v^L_{ij}}\Bigr]\quad\text{(Eq.12)}

3. Adım 3 — Ağırlıklı normalleştirilmiş matris.

   $$\begin{gathered} [e_{ij}^L,e_{ij}^U]=[t_{ij}^L\times w_j^L,t_{ij}^U\times w_j^U]\text{(Eq.13)} \\ [\mathrm{4pt}]\text{Note: this paper uses crisp }{w}_j\text{ (}{w}_j^L=w_j^U=w_j\text{) from IMF D-SWARA (Eq.6)} \end{gathered}$$
   LaTeX

   [e^L_{ij},e^U_{ij}]=[t^L_{ij}\times w^L_j,\;t^U_{ij}\times w^U_j]\quad\text{(Eq.13)}\\[4pt]\text{Note: this paper uses crisp }w_j\text{ (}w^L_j=w^U_j=w_j\text{) from IMF D-SWARA (Eq.6)}

4. Adım 4 — Toplam skor S_i^IRN ve defuzzify; S_AI ve S_AAI hesapla.

   $$\begin{gathered} RN(Z{)}_i=[z_i^L,z_i^U]=\sum _{j=1}^n[e_{ij}^L,e_{ij}^U]\text{(Eq.14)} \\ [\mathrm{4pt}]\text{(Row sum kept as rough interval; defuzz occurs in F5 after IRN division)} \end{gathered}$$
   LaTeX

   RN(Z)_i=\bigl[z^L_i,\,z^U_i\bigr]=\sum_{j=1}^{n}[e^L_{ij},\,e^U_{ij}]\quad\text{(Eq.14)}\\[4pt]\text{(Row sum kept as rough interval; defuzz occurs in F5 after IRN division)}

5. Adım 5 — AI ve AAI'ye göre fayda dereceleri K+_i ve K−_i.

   $$\begin{gathered} RN(Y_i^{-})=[y_i^{-L},y_i^{-U}]=[\frac{z_i^L}{z_{aid}^U},\frac{z_i^U}{z_{aid}^L}]\text{(Eq.15)} \\ [\mathrm{4pt}]RN(Y_i^{+})=[y_i^{+L},y_i^{+U}]=[\frac{z_i^L}{z_{id}^U},\frac{z_i^U}{z_{id}^L}]\text{(Eq.16)} \\ [\mathrm{4pt}]Y_i^{-}={\displaystyle \frac{y_i^{-L}+y_i^{-U}}{2}},Y_i^{+}={\displaystyle \frac{y_i^{+L}+y_i^{+U}}{2}}\text{(Eqs.17--18)} \end{gathered}$$
   LaTeX

   RN(Y^-_i)=\bigl[y^{-L}_i,y^{-U}_i\bigr]=\Bigl[\tfrac{z^L_i}{z^U_{aid}},\tfrac{z^U_i}{z^L_{aid}}\Bigr]\quad\text{(Eq.15)}\\[4pt]RN(Y^+_i)=\bigl[y^{+L}_i,y^{+U}_i\bigr]=\Bigl[\tfrac{z^L_i}{z^U_{id}},\tfrac{z^U_i}{z^L_{id}}\Bigr]\quad\text{(Eq.16)}\\[4pt]Y^-_i=\dfrac{y^{-L}_i+y^{-U}_i}{2},\quad Y^+_i=\dfrac{y^{+L}_i+y^{+U}_i}{2}\quad\text{(Eqs.17--18)}

6. Adım 6 — MARCOS fayda fonksiyonları f(K+) ve f(K−).

   $$f(Y_i^{+})=\frac{Y_i^{-}}{Y_i^{+}+Y_i^{-}},f(Y_i^{-})=\frac{Y_i^{+}}{Y_i^{+}+Y_i^{-}}\text{(Eqs.20--21)}$$
   LaTeX

   f(Y^+_i)=\frac{Y^-_i}{Y^+_i+Y^-_i},\quad f(Y^-_i)=\frac{Y^+_i}{Y^+_i+Y^-_i}\quad\text{(Eqs.20--21)}

7. Adım 7 — Bileşik fayda f(K_i) ve azalan sıralama.

   $$f(Y_i)=\frac{Y_i^{+}+Y_i^{-}}{1+{\displaystyle \frac{1-f(Y_i^{+})}{f(Y_i^{+})}}+{\displaystyle \frac{1-f(Y_i^{-})}{f(Y_i^{-})}}};\text{rank by }f(Y_i)\downarrow \text{(Eq.19)}$$
   LaTeX

   f(Y_i)=\frac{Y^+_i+Y^-_i}{1+\dfrac{1-f(Y^+_i)}{f(Y^+_i)}+\dfrac{1-f(Y^-_i)}{f(Y^-_i)}};\quad\text{rank by }f(Y_i)\downarrow\quad\text{(Eq.19)}

Sezgi

Sonucu okuma: rough-marcos extends MARCOS to handle Rough uncertainty. All arithmetic operations (normalisation, weighting, row summation, utility ratio computation) are performed using Rough number (lower approximation L, upper approximation U) algebra. Utility degrees Y+/Y- are computed via IRN division before defuzzification. The final composite score f(Y_i) is obtained via MARCOS utility functions and ranked in descending order.

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 MARCOS 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.
• Defuzzifikasyon yöntemi sıralamayı etkiler: midpoint (L+U)/2 kanonik seçimdir.