Rough-WASPAS - WASPAS yönteminin Rough uzantısı

Formül adımları

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

1. Adım 1 — IRN alanında maksimum tabanlı lineer normalleştirme.

   $$\begin{gathered} j\in {\Omega }_{+}:n_{ij}^{IRN}=x_{ij}^{IRN}\oslash [{max}_k{l}_{kj},{max}_k{u}_{kj}]=[x_{ij}^L/{max}_k{u}_{kj},x_{ij}^U/{max}_k{l}_{kj}] \\ [\mathrm{4pt}]j\in {\Omega }_{-}:n_{ij}^{IRN}=[{min}_k{l}_{kj},{min}_k{u}_{kj}]\oslash x_{ij}^{IRN}=[{min}_k{l}_{kj}/x_{ij}^U,{min}_k{u}_{kj}/x_{ij}^L] \\ [\mathrm{4pt}]\text{(Eq.10--11; divisor is rough interval }[{max}_k{l}_{kj},{max}_k{u}_{kj}]\text{ for benefit; IRN division rule: }[a,b]\oslash [c,d]=[a/d,b/c]\text{)} \end{gathered}$$
   LaTeX

   j\in\Omega_{+}: n_{ij}^{IRN} = x_{ij}^{IRN}\oslash[\max_k l_{kj},\max_k u_{kj}] = \bigl[x_{ij}^{L}/\max_k u_{kj},\;x_{ij}^{U}/\max_k l_{kj}\bigr]\\[4pt]j\in\Omega_{-}: n_{ij}^{IRN} = [\min_k l_{kj},\min_k u_{kj}]\oslash x_{ij}^{IRN} = \bigl[\min_k l_{kj}/x_{ij}^{U},\;\min_k u_{kj}/x_{ij}^{L}\bigr]\\[4pt]\text{(Eq.10--11; divisor is rough interval }[\max_k l_{kj},\max_k u_{kj}]\text{ for benefit; IRN division rule: }[a,b]\oslash[c,d]=[a/d,b/c]\text{)}

2. Adım 2 — Ağırlıklı toplam kısmı (WSM): normalleştirilmiş değerleri ağırlıklandır ve topla.

   $$\begin{gathered} v_{ij}=[w_j^L·n_{ij}^L,w_j^U·n_{ij}^U]\text{(Eq.13)} \\ [\mathrm{4pt}]Q_i^{(1)IRN}=[\sum _j{w}_j^L{n}_{ij}^L,\sum _j{w}_j^U{n}_{ij}^U]\text{(Eq.14)} \end{gathered}$$
   LaTeX

   v_{ij}=\bigl[w_j^{L}\cdot n_{ij}^{L},\;w_j^{U}\cdot n_{ij}^{U}\bigr]\quad\text{(Eq.13)}\\[4pt]Q_i^{(1)IRN}=\Bigl[\sum_j w_j^{L}\,n_{ij}^{L},\;\sum_j w_j^{U}\,n_{ij}^{U}\Bigr]\quad\text{(Eq.14)}

3. Adım 3 — Ağırlıklı çarpım kısmı (WPM).

   $$v_{ij}=[w_{jL}{r}_{ijL},w_{jU}{r}_{ijU}];Q_i^{(2)IRN}=[\prod _j{v}_{ijL}^{w_{jL}},\prod _j{v}_{ijU}^{w_{jU}}]$$
   LaTeX

   v_{ij}=[w_{jL}r_{ijL},\,w_{jU}r_{ijU}];\quad Q_i^{(2)IRN} = \Bigl[\prod_j v_{ijL}^{w_{jL}},\,\prod_j v_{ijU}^{w_{jU}}\Bigr]

4. Adım 4 — Birleşik Rough aralık skoru A_i (Denklem 16); λ rough aralık (Denklem 17); azalan midpoint ile sıralama.

   $$\begin{gathered} \lambda =[{\lambda }^L,{\lambda }^U]=[0.5+\frac{{\sum }_i{Q}_i^{(2)L}}{{\sum }_i{Q}_i^{(1)L}+{\sum }_i{Q}_i^{(2)L}},0.5+\frac{{\sum }_i{Q}_i^{(2)U}}{{\sum }_i{Q}_i^{(1)U}+{\sum }_i{Q}_i^{(2)U}}]\text{(Eq.17)} \\ [\mathrm{4pt}]A_i^L={\lambda }^L·Q_i^{(1)L}+(1-{\lambda }^U)·Q_i^{(2)L} \\ {A}_i^U={\lambda }^U·Q_i^{(1)U}+(1-{\lambda }^L)·Q_i^{(2)U}\text{(Eq.16)} \\ [\mathrm{4pt}]{\overline{A}}_i=\frac{A_i^L+A_i^U}{2};\text{rank by }{\overline{A}}_i\downarrow \\ [\mathrm{2pt}]\text{(case study: }\lambda =[0.503,0.581]\text{)} \end{gathered}$$
   LaTeX

   \lambda=[\lambda^L,\lambda^U]=\left[0.5+\frac{\sum_i Q_i^{(2)L}}{\sum_i Q_i^{(1)L}+\sum_i Q_i^{(2)L}},\;0.5+\frac{\sum_i Q_i^{(2)U}}{\sum_i Q_i^{(1)U}+\sum_i Q_i^{(2)U}}\right]\quad\text{(Eq.17)}\\[4pt]A_i^{L}=\lambda^{L}\cdot Q_i^{(1)L}+(1-\lambda^{U})\cdot Q_i^{(2)L}\\A_i^{U}=\lambda^{U}\cdot Q_i^{(1)U}+(1-\lambda^{L})\cdot Q_i^{(2)U}\quad\text{(Eq.16)}\\[4pt]\bar{A}_i=\frac{A_i^{L}+A_i^{U}}{2};\quad\text{rank by }\bar{A}_i\downarrow\\[2pt]\text{(case study: }\lambda=[0.503,0.581]\text{)}

Sezgi

Sonucu okuma: ROUGH-WASPAS (Stojić et al. 2018) extends WASPAS to rough number intervals [L,U]. Normalization uses rough interval divisor [max_L, max_U] (not scalar max). WSM and WPM operate entirely in rough interval arithmetic with rough interval weights [w^L_j, w^U_j]. The combined score A_i is kept as a rough interval via data-driven λ (itself a rough interval computed from Eq.17); defuzzification to crisp midpoint (A^L+A^U)/2 occurs only at the final ranking step. Weights must be rough AHP intervals (or equivalent rough weight source). Sensitivity on λ: setting λ=[0.5, 0.5] gives slightly lower scores but identical ranking in published case study.

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 WASPAS 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.