PIF-TOPSIS - Verilen ağırlıklarla resim benzerliği

Formül adımları

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

1. Yerel Resim matrisini doğrulayın

   $$R=[({\mu }_{ij},{\eta }_{ij},{\nu }_{ij}){]}_{m\times n},0\le {\mu }_{ij},{\eta }_{ij},{\nu }_{ij}\le 1,{\mu }_{ij}+{\eta }_{ij}+{\nu }_{ij}\le 1$$
   LaTeX

   R=[(\mu_{ij},\eta_{ij},\nu_{ij})]_{m\times n},\quad 0\le\mu_{ij},\eta_{ij},\nu_{ij}\le1,\quad\mu_{ij}+\eta_{ij}+\nu_{ij}\le1

2. Picture idealleri oluşturun ve doğrulayın

   $$P_j^{+}=\{\begin{array}{cc}({max}_i{\mu }_{ij},{min}_i{\eta }_{ij},{min}_i{\nu }_{ij}),\hfill & j\in J_1\hfill \\ ({max}_i{\mu }_{ij},{max}_i{\eta }_{ij},{min}_i{\nu }_{ij}),\hfill & j\in J_2\hfill \end{array}{P}_j^{-}=\{\begin{array}{cc}({min}_i{\mu }_{ij},{min}_i{\eta }_{ij},{max}_i{\nu }_{ij}),\hfill & j\in J_1\hfill \\ ({min}_i{\mu }_{ij},{max}_i{\eta }_{ij},{max}_i{\nu }_{ij}),\hfill & j\in J_2\hfill \end{array}{P}_j^{\pm }\in \{(\mu ,\eta ,\nu ):\mu +\eta +\nu \le 1\},J_1=\{j:{\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_j=\text{benefit}\},J_2=\{j:{\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_j=\text{cost}\}$$
   LaTeX

   P_j^+=\begin{cases}(\max_i\mu_{ij},\min_i\eta_{ij},\min_i\nu_{ij}),&j\in J_1\\(\max_i\mu_{ij},\max_i\eta_{ij},\min_i\nu_{ij}),&j\in J_2\end{cases}\quad P_j^-=\begin{cases}(\min_i\mu_{ij},\min_i\eta_{ij},\max_i\nu_{ij}),&j\in J_1\\(\min_i\mu_{ij},\max_i\eta_{ij},\max_i\nu_{ij}),&j\in J_2\end{cases}\quad P_j^\pm\in\{(\mu,\eta,\nu):\mu+\eta+\nu\le1\},\quad J_1=\{j:\mathrm{direction}_j=\text{benefit}\},\quad J_2=\{j:\mathrm{direction}_j=\text{cost}\}

3. Kriter ağırlıklarını uygulayın ve yerel benzerlikleri hesaplayın

   $$w_j=\frac{w_j^{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}}{{\sum }_k{w}_k^{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}},{\Delta }_{ij,t}^{\pm }=|R_{ij,t}-P_{j,t}^{\pm }|,S_i^{\pm }=1-\frac{1}{3}\sum _{j=1}^n{w}_j({\sum }_{t=1}^3{\Delta }_{ij,t}^{\pm }+{max}_{1\le t\le 3}{\Delta }_{ij,t}^{\pm }),w_j^{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}\ge 0,|{\sum }_k{w}_k^{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}-1|\le {10}^{-9}$$
   LaTeX

   w_j=\frac{w_j^{\mathrm{input}}}{\sum_k w_k^{\mathrm{input}}},\quad \Delta_{ij,t}^{\pm}=|R_{ij,t}-P_{j,t}^{\pm}|,\quad S_i^\pm=1-\frac13\sum_{j=1}^n w_j\left(\sum_{t=1}^3\Delta_{ij,t}^{\pm}+\max_{1\le t\le3}\Delta_{ij,t}^{\pm}\right),\quad w_j^{\mathrm{input}}\ge0,\quad \left|\sum_k w_k^{\mathrm{input}}-1\right|\le10^{-9}

4. Sağlanan ağırlık profilini kaydedin

   $$\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{\_}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}=\text{exogenous}$$
   LaTeX

   \mathrm{weight\_mode}=\text{exogenous}

5. Uygulanan kriter ağırlıklarını yayınlayın

   $$w_j^{\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}=w_j,w_j\ge 0,\sum _j{w}_j=1$$
   LaTeX

   w_j^{\mathrm{final}}=w_j,\quad w_j\ge0,\quad \sum_j w_j=1

6. Nihai benzerlikleri yayınlayın

   $$S_{i,\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}^{+}=S_i^{+},S_{i,\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}}^{-}=S_i^{-}$$
   LaTeX

   S_{i,\mathrm{final}}^+=S_i^+,\qquad S_{i,\mathrm{final}}^-=S_i^-

7. Yakınlığı ve eşit puanlı sıralamaları hesaplayın

   $$C{R}_i=\frac{S_i^{+}}{S_i^{+}+S_i^{-}},S_i^{+}+S_i^{-}>0,rank(A_i)=1+\#\{k:C{R}_k>C{R}_i\}$$
   LaTeX

   CR_i=\frac{S_i^+}{S_i^++S_i^-},\quad S_i^++S_i^->0,\qquad\operatorname{rank}(A_i)=1+\#\{k:CR_k>CR_i\}

Sezgi

Sonucu okuma: This implementation uses supplied real criterion weights and native Picture triples. It evaluates the published extrema and similarity rules, then ranks by CR. LP weight derivation and native expert aggregation are unsupported. If the published cost extrema leave the Picture domain, calculation stops without a ranking.

Varsayımlar

• Complete native Picture triples and valid Picture ideals.
• Supplied finite nonnegative real criterion weights summing to1.
• Explicit criterion directions and defined positive similarity denominator.

Ne zaman kullanılmaz

• Classical data sufficient - use base TOPSIS directly.
• Linguistic terms - use Zeng 2019 (LPF-TOPSIS) or fuzzy linguistic methods instead.
• Higher-rung membership constraint needed (μ + η + ν > 1 but μ^q + η^q + ν^q ≤ 1 for some q > 1) - use Ali 2024 q-ROPFS-TOPSIS variant.

Sınırlılıklar

• Full LP pipeline and group aggregation are unsupported.
• Source arithmetic and cost ideal defects prevent full scientific approval.

Sık yapılan hatalar

• Nötrlük ve reddetme farklıdır. Yerel eta kaynak gamma’yı (nötrlük) belirtir; reddetme 1-mu-eta-nu’dur.
• Geçerli kaynak derecelendirmeleri, basılı maliyet-ideal kuralının geçerli Resim idealleri üreteceğini garanti etmez; bu tür durumlar açıkça reddedilir.
• Kaynak benzerliğinde maksimum mutlak fark gereklidir. Hiçbir yerel üçlü bileşen ortalaması ile değiştirilmez.
• Yayınlanmış Örnek3 aritmetiği basılı denklemlerle tutarsızdır; tarihsel J düzeni ile eşleşmek makale çoğaltımı sağlamaz.