Fuzzy BWM - Triangular Fuzzy Best-Worst Method

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

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

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).

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

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.

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

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).

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