Fuzzy BWM - Triangular Fuzzy Best-Worst Method

Overview

Fuzzy BWM returns crisp criterion weights (Σ w_j = 1, w_j ≥ 0) and a defuzzified consistency R(ξ̃). It needs only 2n−3 fuzzy pairwise comparisons (the BO and OW vectors). Compared with crisp BWM, Fuzzy BWM lets experts express uncertainty in each ratio using linguistic terms (EI/WI/FI/VI/AI). R(ξ̃) = 0 means perfectly consistent comparisons; large R(ξ̃) indicates inconsistency.

Output

Weight, higher is better

Data

Fuzzy (TFN), linguistic expert assessments

Weights

Derived internally, no weight source needed

Size

0+ alternatives, 3-10 criteria works best

Used for

Criterion-importance elicitation under uncertainty, expert-driven group decision making, Supplier selection

How it works

1. 1

   Identify Best (B) and Worst (W) criteria from the expert; B ≠ W.

   $$B\in \{c_j\},W\in \{c_j\},B\ne W$$

   Guo-Zhao 2017 Step 1; Özdağoğlu 2025 Bölüm 10 Adım 1

2. 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).

   $${\tilde{A}}_B=({\tilde{a}}_{B1},\dots ,{\tilde{a}}_{Bn}),{\tilde{A}}_W=({\tilde{a}}_{1W},\dots ,{\tilde{a}}_{nW})$$

   Guo-Zhao 2017 Eqs.(1)-(2); Özdağoğlu 2025 Bölüm 10 Eşitlik (5)-(8)

3. 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.

   $$\min R(\tilde{\xi })\text{s.t.}\left|\frac{{\tilde{w}}_B}{{\tilde{w}}_j}-{\tilde{a}}_{Bj}\right|\le \tilde{\xi },\left|\frac{{\tilde{w}}_j}{{\tilde{w}}_W}-{\tilde{a}}_{jW}\right|\le \tilde{\xi },\sum _{j}R({\tilde{w}}_j)=1,l_j^w\le m_j^w\le u_j^w,l_j^w\ge 0$$

   Guo-Zhao 2017 Eqs.(3)-(4); Özdağoğlu 2025 Bölüm 10 Eşitlik (12)-(18)

4. 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).

   $$w_j=\frac{l_j^w+4m_j^w+u_j^w}{6}$$

   Guo-Zhao 2017 Step 5; Özdağoğlu 2025 Bölüm 10 Eşitlik (19)-(20)

Fits when / Look elsewhere when

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

Edge cases and pitfalls

• Best ≠ worst is required; identical B and W yields a degenerate NLP.

• n=2: the four ratio constraints reduce to a single non-trivial constraint w̃_B/w̃_W = ã_BW; the NLP has a closed-form solution with ξ̃ = 0.

Works with

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