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

Overview

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

Strengths

• •Method-specific: Distance-based ranking: Trapezoidal Fuzzy Number (TrFN: a, b, c, d) with Zadeh max-min similarity
• •Preserves fuzzy_TrFN 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: 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

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 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-specific edge handling. recommendation_metadata.assumptions_to_verify also lists boundary conditions.

Common pitfalls

• •Hatalı: 'FUZZY-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid Fuzzy (Triangular) numbers/tuples
• •Hatalı: 'FUZZY-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Underlying crisp method's compensation assumption holds in uncertain space
• •Hatalı: 'FUZZY-TOPSIS bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All decision-maker(s) and experts use the same linguistic/uncertainty scale
• •Hatalı: FUZZY-TOPSIS'yi 'Crisp data sufficient' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FUZZY-TOPSIS'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 - Linear scale transformation: benefit criteria normalised by componentwise column ideal x_j*; cost criteria normalised reciprocally by column anti-ideal x_j-. Formül: \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} Anchor: Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(5)-(7)
2. 2.Adım 2 (F2): Step 2 - Weighted normalised fuzzy matrix: multiply each normalised TrFN by the fuzzy weight component-by-component (Chen-Hwang 1992 fuzzy product). Formül: \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) Anchor: Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(8)-(11)
3. 3.Adım 3 (F3): Step 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. Formül: 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})} Anchor: Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(12)-(14)
4. 4.Adım 4 (F4): Step 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. Formül: 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}^{-} Anchor: Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eqs.(15)-(20)
5. 5.Adım 5 (F5): Step 5 - Relative closeness C_i = S_i^- / (S_i* + S_i^-). Rank alternatives by C_i in descending order. Formül: C_i = \dfrac{S_i^{-}}{S_i^{*} + S_i^{-}} Anchor: Chen-Hwang 1992; cf. Kahraman 2008 Ch6 sec.3.2 Eq.(21)

Commonly paired with

• •FUZZY-AHP + FUZZY-TOPSIS (common)

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