Ranking

Fuzzy WPM (Kahraman-Birgun-Yenen 2008) - Triangular Fuzzy Weighted Product

Kahraman, C., Birgun, S., Yenen, V. Z. · 2008

Overview

Fuzzy WP / Multiplicative Weighting - Triangular Fuzzy Number (l, m, u). Output typically utility (product) (higher value = preferred).

Strengths

•Method-specific: Fuzzy WP / Multiplicative Weighting - Triangular Fuzzy Number (l, m, u)
•Preserves fuzzy_TFN uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

•Assumes: All ratings strictly positive
•Assumes: Partial compensation acceptable: a very weak rating dominates the product via low exponent

Method assistant

Grounded explanations: it explains the method, it does not compute.

Assumptions to verify

•All ratings strictly positive
•Partial compensation acceptable: a very weak rating dominates the product via low exponent

When not to use

•Any zero or negative rating present - use FUZZY-SAW
•Full compensation desired - use FUZZY-SAW

Edge cases

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

Common pitfalls

•Hatalı: 'FUZZY-WPM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All ratings strictly positive
•Hatalı: 'FUZZY-WPM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Partial compensation acceptable: a very weak rating dominates the product via low exponent
•Hatalı: FUZZY-WPM'yi 'Any zero or negative rating present - use FUZZY-SAW' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
•Hatalı: FUZZY-WPM'yi 'Full compensation desired - use FUZZY-SAW' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1.Adım 1 (F1): Construct decision matrix X_tilde = [x_tilde_ij] and weight vector w_tilde, all TFNs (l, m, u). Formül: \tilde{x}_{ij} = (l_{ij}, m_{ij}, u_{ij});\quad \tilde{w}_j = (l_j, m_j, u_j) Anchor: Kahraman 2008 Ch.7 Sec.3.2 p.205
2.Adım 2 (F2): Compute fuzzy weighted product per alternative componentwise per Eq.(38): each TFN parameter is raised to its matching weight TFN parameter, then multiplied across criteria. Formül: \tilde{V}(A_i) = \left( \prod_{j=1}^{n} l_{ij}^{l_j},\; \prod_{j=1}^{n} m_{ij}^{m_j},\; \prod_{j=1}^{n} u_{ij}^{u_j} \right) Anchor: Kahraman 2008 Ch.7 Eq.(38), p.205
3.Adım 3 (F3): Defuzzify each V_tilde(A_i) via TFN centroid (l + m + u) / 3. Formül: C_i = (l_i + m_i + u_i) / 3 Anchor: Kahraman 2008 Ch.7 p.206 (book uses fuzzy comparison via Fig.6; centroid is standard equivalent)
4.Adım 4 (F4): Rank alternatives by defuzzified centroid C_i in descending order. Formül: \text{rank}(A_i) = \text{argsort}_{\downarrow}(C_i) Anchor: Kahraman 2008 Ch.7 p.206 (FMS_1 selected over FMS_2)

Commonly paired with

•FUZZY-AHP + FUZZY-WPM (moderate)

How to cite

Kahraman, C.; Birgun, S.; Yenen, V. Z. (2008). Fuzzy Multi-Attribute Scoring Methods with Applications (Ch.7 Section 3.2 - Fuzzy Multiplicative Weighting Method). Fuzzy Multi-Criteria Decision Making (Kahraman, C., Ed.). https://doi.org/10.1007/978-0-387-76813-7_7

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Common pitfalls

•Hatalı: 'FUZZY-WPM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: All ratings strictly positive

•Hatalı: 'FUZZY-WPM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Partial compensation acceptable: a very weak rating dominates the product via low exponent

•Hatalı: FUZZY-WPM'yi 'Any zero or negative rating present - use FUZZY-SAW' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

•Hatalı: FUZZY-WPM'yi 'Full compensation desired - use FUZZY-SAW' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1.Adım 1 (F1): Construct decision matrix X_tilde = [x_tilde_ij] and weight vector w_tilde, all TFNs (l, m, u). Formül: \tilde{x}_{ij} = (l_{ij}, m_{ij}, u_{ij});\quad \tilde{w}_j = (l_j, m_j, u_j) Anchor: Kahraman 2008 Ch.7 Sec.3.2 p.205

2.Adım 2 (F2): Compute fuzzy weighted product per alternative componentwise per Eq.(38): each TFN parameter is raised to its matching weight TFN parameter, then multiplied across criteria. Formül: \tilde{V}(A_i) = \left( \prod_{j=1}^{n} l_{ij}^{l_j},\; \prod_{j=1}^{n} m_{ij}^{m_j},\; \prod_{j=1}^{n} u_{ij}^{u_j} \right) Anchor: Kahraman 2008 Ch.7 Eq.(38), p.205

3.Adım 3 (F3): Defuzzify each V_tilde(A_i) via TFN centroid (l + m + u) / 3. Formül: C_i = (l_i + m_i + u_i) / 3 Anchor: Kahraman 2008 Ch.7 p.206 (book uses fuzzy comparison via Fig.6; centroid is standard equivalent)

4.Adım 4 (F4): Rank alternatives by defuzzified centroid C_i in descending order. Formül: \text{rank}(A_i) = \text{argsort}_{\downarrow}(C_i) Anchor: Kahraman 2008 Ch.7 p.206 (FMS_1 selected over FMS_2)