Fuzzy TOPSIS (Chen-Hwang 1992) - Trapezoidal fuzzy TOPSIS with Zadeh sup-min similarity distance

Overview

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.

Output

utility, higher is better

Data

Fuzzy Tr FN, uncertainty tuples complete

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy (Triangular) MCDM, MAGDM under epistemic uncertainty, expert-driven evaluation with linguistic terms

How it works

1. 1

   Linear scale transformation: benefit criteria normalised by componentwise column ideal x_j*; cost criteria normalised reciprocally by column anti-ideal x_j-.

   $${\tilde{r}}_{ij}=\{\begin{array}{ll}\left(\frac{a_{ij}}{d_j^{\ast }},\frac{b_{ij}}{c_j^{\ast }},\frac{c_{ij}}{b_j^{\ast }},\frac{d_{ij}}{a_j^{\ast }}\right)& j\in J(\text{benefit})\\ \left(\frac{a_j^{-}}{d_{ij}},\frac{b_j^{-}}{c_{ij}},\frac{c_j^{-}}{b_{ij}},\frac{d_j^{-}}{a_{ij}}\right)& j\in J^{\prime }(\text{cost})\end{array}$$

   Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(5)-(7)

2. 2

   Weighted normalised fuzzy matrix: multiply each normalised TrFN by the fuzzy weight component-by-component (Chen-Hwang 1992 fuzzy product).

   $$\begin{gathered} {\tilde{v}}_{ij}={\tilde{r}}_{ij}(\cdot ){\tilde{w}}_j;{\tilde{w}}_j=({\alpha }_j,{\beta }_j,{\chi }_j,{\delta }_j); \\ \text{Benefit: }{\tilde{v}}_{ij}=\left(\frac{a_{ij}}{d_j^{\ast }}{\alpha }_j,\frac{b_{ij}}{c_j^{\ast }}{\beta }_j,\frac{c_{ij}}{b_j^{\ast }}{\chi }_j,\frac{d_{ij}}{a_j^{\ast }}{\delta }_j\right); \\ \text{Cost: }{\tilde{v}}_{ij}=\left(\frac{a_j^{-}}{d_{ij}}{\alpha }_j,\frac{b_j^{-}}{c_{ij}}{\beta }_j,\frac{c_j^{-}}{b_{ij}}{\chi }_j,\frac{d_j^{-}}{a_{ij}}{\delta }_j\right) \end{gathered}$$

   Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(8)-(11)

3. 3

   PIS A* and NIS A^- selected per column by Chen-Hwang generalized mean ranking M(v_ij). v_j* is the TrFN with largest M; v_j^- with smallest M.

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

   Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(12)-(14)

4. 4

   Separation measures S_i* and S_i^- via Zadeh max-min similarity. For each (i,j) compute D_ij = 1 - sup_x[mu_v_ij(x) AND mu_v_j*(x)] = 1 - L_ij; sum across criteria.

   $$\begin{gathered} S_i^{\ast }=\sum _{j=1}^n{D}_{ij}^{\ast },S_i^{-}=\sum _{j=1}^n{D}_{ij}^{-}; \\ {D}_{ij}^{\ast }=1-\underset{x}{\sup }\left[{\mu }_{{\tilde{v}}_{ij}}(x)\wedge {\mu }_{{\tilde{v}}_j^{\ast }}(x)\right]=1-L_{ij}^{\ast }; \\ {D}_{ij}^{-}=1-\underset{x}{\sup }\left[{\mu }_{{\tilde{v}}_{ij}}(x)\wedge {\mu }_{{\tilde{v}}_j^{-}}(x)\right]=1-L_{ij}^{-} \end{gathered}$$

   Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(15)-(20)

5. 5

   Relative closeness C_i = S_i^- / (S_i* + S_i^-). Rank alternatives by C_i in descending order.

   $$C_i=\frac{S_i^{-}}{S_i^{\ast }+S_i^{-}}$$

   Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eq.(21)

Fits when / Look elsewhere when

Assumptions to verify

• 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

Edge cases and pitfalls

• Confusing this Chen-Hwang 1992 trapezoidal method with Chen 2000 triangular vertex-method (see FUZZY-TOPSIS-CHEN2000 manifest). Different normalisation and different distance metric.

• Defuzzifying TrFN entries before Step 4 destroys the fuzzy information and reduces the algorithm to crisp TOPSIS.

Works with

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