Fuzzy TOPSIS (Chen 2000) - TOPSIS extension for group decision-making under fuzzy environment with triangular fuzzy numbers

Overview

CC ∈ [0, 1]. Higher CC means the alternative is closer to the FPIS and farther from the FNIS under the fuzzy vertex distance. Rank by descending CC.

Output

utility, higher is better

Data

Fuzzy (TFN), 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

   Construct the fuzzy decision matrix X̃ from linguistic ratings (or numeric TFNs) and the fuzzy weight vector W̃.

   $$\tilde{X}=[{\tilde{x}}_{ij}{]}_{m\times n};{\tilde{x}}_{ij}=(a_{ij},b_{ij},c_{ij});\tilde{W}=[{\tilde{w}}_1,\dots ,{\tilde{w}}_n];{\tilde{w}}_j=(w_{j1},w_{j2},w_{j3})$$

   Chen 2000, p.3 Eqs.(2)-(3)

2. 2

   Chen's normalisation: for benefit criteria divide each TFN component by c_j* = max_i c_ij; for cost criteria use the reciprocal form with a_j^- = min_i a_ij.

   $$\text{Benefit: }{\tilde{r}}_{ij}=\left(\frac{a_{ij}}{c_j^{\ast }},\frac{b_{ij}}{c_j^{\ast }},\frac{c_{ij}}{c_j^{\ast }}\right),c_j^{\ast }=\underset{i}{\max }{c}_{ij};\text{Cost: }{\tilde{r}}_{ij}=\left(\frac{a_j^{-}}{c_{ij}},\frac{a_j^{-}}{b_{ij}},\frac{a_j^{-}}{a_{ij}}\right),a_j^{-}=\underset{i}{\min }{a}_{ij}$$

   Chen 2000, p.3 Eq.(4)

3. 3

   Weighted normalised fuzzy matrix: ṽ_ij = r̃_ij ⊗ w̃_j (component-wise TFN multiplication).

   $${\tilde{v}}_{ij}={\tilde{r}}_{ij}\otimes {\tilde{w}}_j=(a_{ij}^r{w}_{j1},b_{ij}^r{w}_{j2},c_{ij}^r{w}_{j3})\text{where }{\tilde{r}}_{ij}=(a_{ij}^r,b_{ij}^r,c_{ij}^r)$$

   Chen 2000, p.4 Eq.(7)

4. 4

   Define the Fuzzy Positive-Ideal Solution (FPIS) A⁺ = (ṽ_1*, …, ṽ_n*) with ṽ_j* = (1, 1, 1) and the Fuzzy Negative-Ideal Solution (FNIS) A⁻ with ṽ_j⁻ = (0, 0, 0).

   $$A^{+}=({\tilde{v}}_1^{\ast },\dots ,{\tilde{v}}_n^{\ast }),{\tilde{v}}_j^{\ast }=(1,1,1)\forall j;A^{-}=({\tilde{v}}_1^{-},\dots ,{\tilde{v}}_n^{-}),{\tilde{v}}_j^{-}=(0,0,0)\forall j$$

   Chen 2000, p.4 Eqs.(8)-(9)

5. 5

   Compute fuzzy distances from FPIS and FNIS using the vertex method: d(ã, b̃) = √((1/3)·[(a₁−b₁)² + (a₂−b₂)² + (a₃−b₃)²]); sum over all criteria.

   $$\begin{gathered} d(\tilde{a},\tilde{b})=\sqrt{\frac{1}{3}\left[(a_1-b_1{)}^2+(a_2-b_2{)}^2+(a_3-b_3{)}^2\right]}; \\ {d}_i^{+}=\sum _{j=1}^{n}d({\tilde{v}}_{ij},{\tilde{v}}_j^{\ast }),d_i^{-}=\sum _{j=1}^{n}d({\tilde{v}}_{ij},{\tilde{v}}_j^{-}) \end{gathered}$$

   Chen 2000, p.4 Eqs.(10)-(11)

6. 6

   Closeness coefficient CC_i = d_i⁻ / (d_i⁺ + d_i⁻); rank alternatives by CC descending.

   $$C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}};\text{rank by descending }C{C}_i$$

   Chen 2000, p.4 Eq.(12)

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

• Using Hwang-Yoon vector normalisation instead of Chen's max-c* normalisation produces different CC values - Chen's normalisation maps the FPIS exactly to (1,1,1) per criterion, which is essential for Step 4.

• Defuzzifying TFNs before Step 5 collapses fuzzy uncertainty and reduces Fuzzy TOPSIS to crisp TOPSIS - defer defuzzification (if any) to reporting only.

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