Fuzzy SAW (Bonissone 1982) - L-R trapezoidal Simple Additive Weighting

Overview

Fuzzy SAW - L-R trapezoidal (a, b, alpha, beta). Output typically utility (higher value = preferred).

Strengths

• •Method-specific: Fuzzy SAW - L-R trapezoidal (a, b, alpha, beta)
• •Preserves fuzzy_LR_TrFN uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Assumes: Decision matrix entries are valid L-R 4-tuples (a <= b, alpha >= 0, beta >= 0)
• •Assumes: Compensation assumption holds: a strong score on one criterion can offset a weak score on another

Assumptions to verify

• •Decision matrix entries are valid L-R 4-tuples (a <= b, alpha >= 0, beta >= 0)
• •Compensation assumption holds: a strong score on one criterion can offset a weak score on another

When not to use

• •Crisp data sufficient - use base SAW
• •Multiplicative semantics required - use FUZZY-WPM

Edge cases

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

Common pitfalls

• •Hatalı: 'FUZZY-SAW bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Decision matrix entries are valid L-R 4-tuples (a <= b, alpha >= 0, beta >= 0)
• •Hatalı: 'FUZZY-SAW bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Compensation assumption holds: a strong score on one criterion can offset a weak score on another
• •Hatalı: FUZZY-SAW'yi 'Crisp data sufficient - use base SAW' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FUZZY-SAW'yi 'Multiplicative semantics required - use FUZZY-WPM' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Construct decision matrix X_tilde = [x_tilde_ij] and weight vector w_tilde, all L-R 4-tuples. Formül: \tilde{x}_{ij} = (a_{ij}, b_{ij}, \alpha_{ij}, \beta_{ij});\quad \tilde{w}_j = (a_j, b_j, \alpha_j, \beta_j) Anchor: Kahraman 2008 Ch.7 Eq.(7)-(8), p.192; Bonissone 1982 p.331
2. 2.Adım 2 (F2): Compute weighted contribution w_tilde_j (x) x_tilde_ij using Bonissone L-R product, Eq.(29). Formül: \tilde{M} \otimes \tilde{N} = (ac,\; bd,\; a\gamma + c\alpha - \alpha\gamma,\; b\delta + d\beta + \beta\delta) Anchor: Kahraman 2008 Ch.7 Eq.(29), p.200; Bonissone 1982
3. 3.Adım 3 (F3): Aggregate weighted contributions per alternative via Bonissone L-R sum, Eq.(27). Formül: \tilde{U}_i = \bigoplus_{j=1}^{n} (\tilde{w}_j \otimes \tilde{x}_{ij});\quad (a+c,\; b+d,\; \alpha+\gamma,\; \beta+\delta) Anchor: Kahraman 2008 Ch.7 Eq.(27), (31), p.200; Bonissone 1982
4. 4.Adım 4 (F4): Defuzzify each U_tilde_i via L-R trapezoidal centroid (left triangle + plateau + right triangle). Formül: C_i = \dfrac{(\alpha_i/2)(a_i - \alpha_i/3) + (b_i - a_i)\,(a_i+b_i)/2 + (\beta_i/2)(b_i + \beta_i/3)}{(b_i - a_i) + (\alpha_i + \beta_i)/2} Anchor: Bonissone 1982 (canonical L-R centroid); standard trapezoidal area centroid
5. 5.Adım 5 (F5): 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.201 (U_2 >- U_1 >- U_3)

Commonly paired with

• •FUZZY-AHP + FUZZY-SAW (common)

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