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

Overview

Distance-based (fuzzy, vertex metric). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Distance-based (fuzzy, vertex metric)
• •Preserves fuzzy_TFN uncertainty through the pipeline rather than premature crispification at elicitation
• •Native group-decision support (multi-DM aggregation built into the pipeline)

Limitations

• •Rank reversal known on alternative-set changes (ref: inherits from TOPSIS (Belton-Gear-1983))
• •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

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

When not to use

• •Crisp data sufficient: use base TOPSIS-CHEN2000 directly (avoid unnecessary uncertainty layer)
• •Aggregation operator (PFWA/PFOWA/etc.) not specified: output ambiguous

Edge cases

• •See F.steps and D.parameters for FUZZY-TOPSIS-CHEN2000-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'FUZZY-TOPSIS-CHEN2000 bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• •Hatalı: 'FUZZY-TOPSIS-CHEN2000 bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'FUZZY-TOPSIS-CHEN2000 bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: FUZZY-TOPSIS-CHEN2000'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FUZZY-TOPSIS-CHEN2000'yi 'Aggregation operator (PFWA/PFOWA/etc.) not specified' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Construct the fuzzy decision matrix X̃ from linguistic ratings (or numeric TFNs) and the fuzzy weight vector W̃. Formül: \tilde{X} = [\tilde{x}_{ij}]_{m \times n};\quad \tilde{x}_{ij} = (a_{ij}, b_{ij}, c_{ij});\quad \tilde{W} = [\tilde{w}_1, \ldots, \tilde{w}_n];\ \tilde{w}_j = (w_{j1}, w_{j2}, w_{j3}) Anchor: Chen 2000, p.3 Eqs.(2)-(3)
2. 2.Adım 2 (F2): Step 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. Formül: \text{Benefit: } \tilde{r}_{ij} = \left( \frac{a_{ij}}{c_j^{*}},\ \frac{b_{ij}}{c_j^{*}},\ \frac{c_{ij}}{c_j^{*}} \right),\ c_j^{*} = \max_i c_{ij};\quad \text{Cost: } \tilde{r}_{ij} = \left( \frac{a_j^{-}}{c_{ij}},\ \frac{a_j^{-}}{b_{ij}},\ \frac{a_j^{-}}{a_{ij}} \right),\ a_j^{-} = \min_i a_{ij} Anchor: Chen 2000, p.3 Eq.(4)
3. 3.Adım 3 (F3): Step 3: Weighted normalised fuzzy matrix: ṽ_ij = r̃_ij ⊗ w̃_j (component-wise TFN multiplication). Formül: \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})\quad \text{where } \tilde{r}_{ij} = (a_{ij}^{r}, b_{ij}^{r}, c_{ij}^{r}) Anchor: Chen 2000, p.4 Eq.(7)
4. 4.Adım 4 (F4): Step 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). Formül: A^{+} = (\tilde{v}_1^{*}, \ldots, \tilde{v}_n^{*}),\ \tilde{v}_j^{*} = (1, 1, 1)\ \forall j;\quad A^{-} = (\tilde{v}_1^{-}, \ldots, \tilde{v}_n^{-}),\ \tilde{v}_j^{-} = (0, 0, 0)\ \forall j Anchor: Chen 2000, p.4 Eqs.(8)-(9)
5. 5.Adım 5 (F5): Step 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. Formül: 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]};\\[4pt] d_i^{+} = \sum_{j=1}^{n} d(\tilde{v}_{ij}, \tilde{v}_j^{*}),\quad d_i^{-} = \sum_{j=1}^{n} d(\tilde{v}_{ij}, \tilde{v}_j^{-}) Anchor: Chen 2000, p.4 Eqs.(10)-(11)
6. 6.Adım 6 (F6): Step 6: Closeness coefficient CC_i = d_i⁻ / (d_i⁺ + d_i⁻); rank alternatives by CC descending. Formül: CC_i = \frac{d_i^{-}}{d_i^{+} + d_i^{-}};\quad \text{rank by descending } CC_i Anchor: Chen 2000, p.4 Eq.(12)

Commonly paired with

• •FUZZY-AHP + FUZZY-TOPSIS-CHEN2000 (common)

How to cite