FUCOM-F - Fuzzy Full Consistency Method (TFN)

Overview

FUCOM-F returns crisp criterion weights (Σ w_j = 1, w_j ≥ 0) and a defuzzified consistency deviation S(χ̃). It needs only n−1 fuzzy pairwise priority ratios on the ranked criterion chain; the NLP additionally enforces transitivity. S(χ̃) = 0 means perfect fuzzy transitivity. Compared with FUCOM (crisp), FUCOM-F lets experts express uncertainty in each priority ratio.

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

   Rank criteria by importance C_(1) ≻ C_(2) ≻ … ≻ C_(n).

   $$C_{(1)}⪰C_{(2)}⪰\cdots ⪰C_{(n)}$$

   Pamucar-Ecer 2020 Step 1; Keleş 2025 Bölüm 4 Adım 1

2. 2

   For each consecutive pair, elicit the fuzzy comparative significance φ̃_{j,j+1} = (l_j, m_j, u_j) from the linguistic scale (n−1 TFNs).

   $${\tilde{\phi }}_{j,j+1}=(l_j,m_j,u_j),j=1,\dots ,n-1$$

   Pamucar-Ecer 2020 Eq.(8); Keleş 2025 Bölüm 4 Adım 2

3. 3

   Solve the FUCOM-F nonlinear program for fuzzy weights w̃_j = (l_j^w, m_j^w, u_j^w) and deviation χ̃ = (l_χ, m_χ, u_χ): minimise S(χ̃) subject to (a) ratio constraints |w̃_(j)/w̃_(j+1) − φ̃_{j,j+1}| ≤ χ̃ componentwise; (b) transitivity |w̃_(j)/w̃_(j+2) − φ̃_{j,j+1}⊗φ̃_{j+1,j+2}| ≤ χ̃; (c) Σ_j S(w̃_j) = 1; (d) l_j^w ≤ m_j^w ≤ u_j^w; (e) l_j^w ≥ 0.

   $$\min S(\tilde{\chi })\text{s.t.}\left|\frac{{\tilde{w}}_{(j)}}{{\tilde{w}}_{(j+1)}}-{\tilde{\phi }}_{j,j+1}\right|\le \tilde{\chi },\left|\frac{{\tilde{w}}_{(j)}}{{\tilde{w}}_{(j+2)}}-{\tilde{\phi }}_{j,j+1}\otimes {\tilde{\phi }}_{j+1,j+2}\right|\le \tilde{\chi },\sum _{j}S({\tilde{w}}_j)=1,l_j^w\le m_j^w\le u_j^w,l_j^w\ge 0$$

   Pamucar-Ecer 2020 Eq.(9); Keleş 2025 Bölüm 4 Adım 3

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 deviation S(χ̃).

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

   Pamucar-Ecer 2020 Step 4; Keleş 2025 Bölüm 4 Adım 4

Fits when / Look elsewhere when

Assumptions to verify

• Experts can rank criteria
• Each consecutive ratio φ̃_{j,j+1} expressible on the linguistic scale
• TFN shape (l ≤ m ≤ u) preserved

Edge cases and pitfalls

• Each φ̃_{j,j+1} must satisfy l ≥ 1: the ranked side cannot be 'less important' than the next.

• n=2: only the ratio constraint applies (no transitivity). The NLP becomes trivial: w̃_{(1)} = φ̃_{12} ⊗ w̃_{(2)} and χ̃ = 0.

Works with

How to cite