Bulanık TOPSIS (Chen-Hwang 1992) - Yamuksel bulanık sayılarla TOPSIS ve Zadeh sup-min benzerlik uzaklığı

Formül adımları

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

1. Adim 1 - Dogrusal olcek donusumu: fayda kriterleri bilesen-bilesen sutun ideali x_j* ile, maliyet kriterleri ise sutun anti-ideali x_j- ile resiprokal normalize edilir.

   $${\tilde{r}}_{ij}=\{\begin{array}{cc}({\displaystyle \frac{a_{ij}}{d_j^{*}}},{\displaystyle \frac{b_{ij}}{c_j^{*}}},{\displaystyle \frac{c_{ij}}{b_j^{*}}},{\displaystyle \frac{d_{ij}}{a_j^{*}}})\hfill & j\in J(\text{benefit})\hfill \\ [\mathrm{8pt}]({\displaystyle \frac{a_j^{-}}{d_{ij}}},{\displaystyle \frac{b_j^{-}}{c_{ij}}},{\displaystyle \frac{c_j^{-}}{b_{ij}}},{\displaystyle \frac{d_j^{-}}{a_{ij}}})\hfill & j\in J^{\prime }(\text{cost})\hfill \end{array}$$
   LaTeX

   \tilde{r}_{ij} = \begin{cases} \left( \dfrac{a_{ij}}{d_j^{*}},\ \dfrac{b_{ij}}{c_j^{*}},\ \dfrac{c_{ij}}{b_j^{*}},\ \dfrac{d_{ij}}{a_j^{*}} \right) & j \in J\ (\text{benefit}) \\[8pt] \left( \dfrac{a_j^{-}}{d_{ij}},\ \dfrac{b_j^{-}}{c_{ij}},\ \dfrac{c_j^{-}}{b_{ij}},\ \dfrac{d_j^{-}}{a_{ij}} \right) & j \in J'\ (\text{cost}) \end{cases}

2. Adim 2 - Agirliklandirilmis normallestirilmis bulanik matris: her TrFN'yi bulanik agirlikla bilesen-bilesen carp.

   $$\begin{gathered} {\tilde{v}}_{ij}={\tilde{r}}_{ij}(·){\tilde{w}}_j;{\tilde{w}}_j=({\alpha }_j,{\beta }_j,{\chi }_j,{\delta }_j); \\ [\mathrm{4pt}]\text{Benefit: }{\tilde{v}}_{ij}=({\displaystyle \frac{a_{ij}}{d_j^{*}}}{\alpha }_j,{\displaystyle \frac{b_{ij}}{c_j^{*}}}{\beta }_j,{\displaystyle \frac{c_{ij}}{b_j^{*}}}{\chi }_j,{\displaystyle \frac{d_{ij}}{a_j^{*}}}{\delta }_j); \\ [\mathrm{4pt}]\text{Cost: }{\tilde{v}}_{ij}=({\displaystyle \frac{a_j^{-}}{d_{ij}}}{\alpha }_j,{\displaystyle \frac{b_j^{-}}{c_{ij}}}{\beta }_j,{\displaystyle \frac{c_j^{-}}{b_{ij}}}{\chi }_j,{\displaystyle \frac{d_j^{-}}{a_{ij}}}{\delta }_j) \end{gathered}$$
   LaTeX

   \tilde{v}_{ij} = \tilde{r}_{ij}\,(\cdot)\,\tilde{w}_j;\quad \tilde{w}_j = (\alpha_j, \beta_j, \chi_j, \delta_j);\\[4pt] \text{Benefit: } \tilde{v}_{ij} = \left( \dfrac{a_{ij}}{d_j^{*}}\alpha_j,\ \dfrac{b_{ij}}{c_j^{*}}\beta_j,\ \dfrac{c_{ij}}{b_j^{*}}\chi_j,\ \dfrac{d_{ij}}{a_j^{*}}\delta_j \right);\\[4pt] \text{Cost: } \tilde{v}_{ij} = \left( \dfrac{a_j^{-}}{d_{ij}}\alpha_j,\ \dfrac{b_j^{-}}{c_{ij}}\beta_j,\ \dfrac{c_j^{-}}{b_{ij}}\chi_j,\ \dfrac{d_j^{-}}{a_{ij}}\delta_j \right)

3. Adim 3 - PIS A* ve NIS A^- her sutunda Chen-Hwang genellestirilmis ortalama M(v_ij) ile secilir. v_j* en buyuk M'ye sahip TrFN; v_j^- en kucuk M'ye sahip.

   $$\begin{gathered} A^{*}=[{\tilde{v}}_1^{*},\dots ,{\tilde{v}}_n^{*}],{\tilde{v}}_j^{*}=\mathrm{\backslash arg}{max}_{i}M({\tilde{v}}_{ij});A^{-}=[{\tilde{v}}_1^{-},\dots ,{\tilde{v}}_n^{-}],{\tilde{v}}_j^{-}=\mathrm{\backslash arg}{min}_{i}M({\tilde{v}}_{ij}); \\ [\mathrm{6pt}]M({\tilde{v}}_{ij})={\displaystyle \frac{-a_{ij}^2-b_{ij}^2+c_{ij}^2+d_{ij}^2-a_{ij}{b}_{ij}+c_{ij}{d}_{ij}}{3(-a_{ij}-b_{ij}+c_{ij}+d_{ij})}} \end{gathered}$$
   LaTeX

   A^{*} = [\tilde{v}_1^{*}, \ldots, \tilde{v}_n^{*}],\ \tilde{v}_j^{*} = \arg\max_i M(\tilde{v}_{ij});\quad A^{-} = [\tilde{v}_1^{-}, \ldots, \tilde{v}_n^{-}],\ \tilde{v}_j^{-} = \arg\min_i M(\tilde{v}_{ij});\\[6pt] M(\tilde{v}_{ij}) = \dfrac{-a_{ij}^{2} - b_{ij}^{2} + c_{ij}^{2} + d_{ij}^{2} - a_{ij}b_{ij} + c_{ij}d_{ij}}{3\,(-a_{ij} - b_{ij} + c_{ij} + d_{ij})}

4. Adim 4 - Zadeh max-min benzerligi ile ayrim olculeri S_i* ve S_i^-. Her (i,j) icin D_ij = 1 - sup_x[mu_v_ij(x) ^ mu_v_j*(x)] = 1 - L_ij; kriterler boyunca topla.

   $$\begin{gathered} S_i^{*}=\sum _{j=1}^n{D}_{ij}^{*},S_i^{-}=\sum _{j=1}^n{D}_{ij}^{-}; \\ [\mathrm{6pt}]D_{ij}^{*}=1-{sup}_x[{\mu }_{{\tilde{v}}_{ij}}(x)\wedge {\mu }_{{\tilde{v}}_j^{*}}(x)]=1-L_{ij}^{*}; \\ [\mathrm{4pt}]D_{ij}^{-}=1-{sup}_x[{\mu }_{{\tilde{v}}_{ij}}(x)\wedge {\mu }_{{\tilde{v}}_j^{-}}(x)]=1-L_{ij}^{-} \end{gathered}$$
   LaTeX

   S_i^{*} = \sum_{j=1}^{n} D_{ij}^{*},\quad S_i^{-} = \sum_{j=1}^{n} D_{ij}^{-};\\[6pt] D_{ij}^{*} = 1 - \sup_{x}\left[ \mu_{\tilde{v}_{ij}}(x) \wedge \mu_{\tilde{v}_j^{*}}(x) \right] = 1 - L_{ij}^{*};\\[4pt] D_{ij}^{-} = 1 - \sup_{x}\left[ \mu_{\tilde{v}_{ij}}(x) \wedge \mu_{\tilde{v}_j^{-}}(x) \right] = 1 - L_{ij}^{-}

5. Adim 5 - Goreli yakinlik C_i = S_i^- / (S_i* + S_i^-). Alternatifleri C_i'ye gore azalan sirada sirala.

   $$C_i={\displaystyle \frac{S_i^{-}}{S_i^{*}+S_i^{-}}}$$
   LaTeX

   C_i = \dfrac{S_i^{-}}{S_i^{*} + S_i^{-}}

Sezgi

Distance-based ranking - Trapezoidal Fuzzy Number (TrFN: a, b, c, d) with Zadeh max-min similarity. Output typically utility (higher value = preferred).

Sonucu okuma: C_i in [0, 1]. Higher C_i means the alternative is closer to the fuzzy PIS A* and farther from the fuzzy NIS A^- under Zadeh's max-min similarity. Rank by descending C_i. Chen-Hwang 1992 trapezoidal arithmetic with linear scale transformation normalisation and Chen-Hwang generalized mean ranking for PIS/NIS extraction.

Varsayımlar

• Decision matrix entries are valid Fuzzy (Triangular) 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 TOPSIS directly (avoid unnecessary uncertainty layer)
• Aggregation operator (PFWA/PFOWA/etc.) not specified - output ambiguous

Sınırlılıklar

• Rank reversal known on alternative-set changes (ref: inherited from crisp TOPSIS; cf. Belton-Gear 1983, Wang-Luo 2009)
• Assumes: Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• Assumes: Underlying crisp method's compensation assumption holds in uncertain space
• Assumes: All decision-maker(s) and experts use the same linguistic/uncertainty scale

Sık yapılan hatalar

• Bu Chen-Hwang 1992 yamuksel yontemini Chen 2000 ucgen vertex-yontemi ile karistirmak (bkz FUZZY-TOPSIS-CHEN2000). Farkli normallestirme ve farkli uzaklik metrigi.
• TrFN girislerini Adim 4'ten once durulamak bulanik bilgiyi yok eder ve algoritmayi kesin TOPSIS'e indirir.