Fuzzy BWM: Triangular Fuzzy Best-Worst Method

Overview

Triangular-fuzzy Best-to-Others and Others-to-Worst pairwise comparison with nonlinearly-constrained programming. Output typically weight (higher value = preferred).

Strengths

• •Method-specific: Triangular-fuzzy Best-to-Others and Others-to-Worst pairwise comparison with nonlinearly-constrained programming
• •Preserves triangular_fuzzy uncertainty through the pipeline rather than premature crispification at elicitation

Limitations

• •Assumes: Experts can identify Best and Worst criteria
• •Assumes: Each ã_Bj and ã_jW expressible on the linguistic scale
• •Assumes: TFN shape (l ≤ m ≤ u) preserved

Assumptions to verify

• •Experts can identify Best and Worst criteria
• •Each ã_Bj and ã_jW expressible on the linguistic scale
• •TFN shape (l ≤ m ≤ u) preserved

When not to use

• •Only crisp ratio available → use crisp BWM
• •Probabilistic group disagreement modelling needed → use Bayesian BWM

Edge cases

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

Common pitfalls

• •Hatalı: 'FUZZY-BWM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Experts can identify Best and Worst criteria
• •Hatalı: 'FUZZY-BWM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: Each ã_Bj and ã_jW expressible on the linguistic scale
• •Hatalı: 'FUZZY-BWM bu varsayımı kontrol etmeden uygulamak'. Doğrusu: TFN shape (l ≤ m ≤ u) preserved
• •Hatalı: FUZZY-BWM'yi 'Only crisp ratio available → use crisp BWM' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.
• •Hatalı: FUZZY-BWM'yi 'Probabilistic group disagreement modelling needed → use Bayesian BWM' durumunda kullanmak: recommendation_metadata.not_recommended_when alternatif öneriyor.

Worked example

1. 1.Adım 1 (F1): Step 1: Identify Best (B) and Worst (W) criteria from the expert; B ≠ W. Formül: B \in \{c_{j}\},\ W \in \{c_{j}\},\ B\neq W Anchor: Guo-Zhao 2017 Step 1; Özdağoğlu 2025 Bölüm 10 Adım 1
2. 2.Adım 2 (F2): Step 2: Build fuzzy BO vector Ã_B = (ã_B1,…,ã_Bn) and fuzzy OW vector Ã_W = (ã_1W,…,ã_nW) from the linguistic scale (TFN). Formül: \tilde{A}_{B} = (\tilde{a}_{B1},\ldots,\tilde{a}_{Bn}),\quad \tilde{A}_{W} = (\tilde{a}_{1W},\ldots,\tilde{a}_{nW}) Anchor: Guo-Zhao 2017 Eqs.(1)-(2); Özdağoğlu 2025 Bölüm 10 Eşitlik (5)-(8)
3. 3.Adım 3 (F3): Step 3: Solve the fuzzy-BWM nonlinear program for fuzzy weights w̃_j = (l_j^w, m_j^w, u_j^w) and fuzzy consistency ξ̃ = (l_ξ, m_ξ, u_ξ): minimise R(ξ̃) subject to (a) |w̃_B/w̃_j − ã_Bj| ≤ ξ̃ componentwise; (b) |w̃_j/w̃_W − ã_jW| ≤ ξ̃ componentwise; (c) Σ_j R(w̃_j) = 1; (d) l_j^w ≤ m_j^w ≤ u_j^w; (e) l_j^w ≥ 0. Formül: \min\ R(\tilde{\xi})\quad \text{s.t.}\quad \left|\dfrac{\tilde{w}_{B}}{\tilde{w}_{j}} - \tilde{a}_{Bj}\right| \le \tilde{\xi},\ \left|\dfrac{\tilde{w}_{j}}{\tilde{w}_{W}} - \tilde{a}_{jW}\right| \le \tilde{\xi},\ \sum_{j} R(\tilde{w}_{j}) = 1,\ l_{j}^{w} \le m_{j}^{w} \le u_{j}^{w},\ l_{j}^{w} \ge 0 Anchor: Guo-Zhao 2017 Eqs.(3)-(4); Özdağoğlu 2025 Bölüm 10 Eşitlik (12)-(18)
4. 4.Adım 4 (F4): Step 4: Defuzzify w̃_j → w_j with graded mean: w_j = (l_j^w + 4 m_j^w + u_j^w)/6. Final output: crisp weights and crisp consistency R(ξ̃). Formül: w_{j} = \dfrac{l_{j}^{w} + 4 m_{j}^{w} + u_{j}^{w}}{6} Anchor: Guo-Zhao 2017 Step 5; Özdağoğlu 2025 Bölüm 10 Eşitlik (19)-(20)

Commonly paired with

• •FUZZY-BWM + FUZZY-TOPSIS (high)
• •FUZZY-BWM + FUZZY-MARCOS (high)
• •FUZZY-BWM + FUZZY-WASPAS (medium)
• •FUZZY-BWM + FUZZY-EDAS (medium)

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