BeCoMe (Best Compromise Mean) is a method for aggregating expert opinions expressed as fuzzy triangular numbers. The method was developed by I. Vrana, J. Tyrychtr, and M. Pelikan at the Czech University of Life Sciences Prague.
BeCoMe combines two classical aggregation approaches:
- Arithmetic mean (Γ) - represents the average opinion
- Statistical median (Ω) - represents the central tendency, resistant to outliers
The final result is the best compromise (ΓΩMean) - the average of these two measures, providing a robust consensus estimate.
A fuzzy triangular number is a special type of fuzzy set that represents uncertain or imprecise information. It is defined by three characteristic values:
- A (
lower_bound) - minimum possible value - C (
peak) - most likely value (mode) - B (
upper_bound) - maximum possible value
Constraint: A ≤ C ≤ B
Membership
μ(x)
1 | /\
| / \
| / \
| / \
0 |____/________\____
A C B x
The triangular shape represents the membership function:
- At point C (peak), membership = 1 (fully certain)
- At points A and B, membership = 0 (boundary values)
- Between A-C and C-B, membership decreases linearly
A fuzzy triangular number is denoted as: FTN(A, C, B) or (A, C, B)
An expert estimates project duration:
- Optimistic: 5 days (minimum)
- Most likely: 8 days (peak)
- Pessimistic: 12 days (maximum)
This is represented as: FTN(5, 8, 12)
The centroid (center of gravity) is the x-coordinate of the center of mass of the triangular fuzzy number.
Formula:
Gx = (A + C + B) / 3
Geometric interpretation: The centroid represents the "balance point" of the triangle.
Purpose: Used for sorting expert opinions in median calculation.
For FTN(5, 8, 12):
Gx = (5 + 8 + 12) / 3 = 25 / 3 ≈ 8.33
The BeCoMe method aggregates M expert opinions into a single best compromise result through five steps.
A set of M expert opinions, each represented as a fuzzy triangular number:
E₁ = (A₁, C₁, B₁)
E₂ = (A₂, C₂, B₂)
...
Eₘ = (Aₘ, Cₘ, Bₘ)
The arithmetic mean Γ(α, γ, β) is calculated by averaging each component separately.
Formulas:
α = (1/M) × Σ(Aₖ) for k = 1 to M (average of lower bounds)
γ = (1/M) × Σ(Cₖ) for k = 1 to M (average of peaks)
β = (1/M) × Σ(Bₖ) for k = 1 to M (average of upper bounds)
Result: Γ = (α, γ, β)
Three experts estimate project cost (in thousands):
Expert 1: E₁ = (10, 15, 20)
Expert 2: E₂ = (12, 18, 25)
Expert 3: E₃ = (8, 14, 22)
Calculation:
α = (10 + 12 + 8) / 3 = 30 / 3 = 10.00
γ = (15 + 18 + 14) / 3 = 47 / 3 ≈ 15.67
β = (20 + 25 + 22) / 3 = 67 / 3 ≈ 22.33
Result: Γ = (10.00, 15.67, 22.33)
The median Ω(ρ, ω, σ) is found by sorting opinions by their centroids and taking the middle value(s).
For each expert opinion, calculate the centroid:
Gxₖ = (Aₖ + Cₖ + Bₖ) / 3
Sort all expert opinions in ascending order by their centroid values.
Case A: Odd number of experts (M = 2n + 1)
Take the middle element at position n + 1:
Ω = E_{middle}
ρ = A_{middle}
ω = C_{middle}
σ = B_{middle}
Case B: Even number of experts (M = 2n)
Average the two middle elements at positions n and n + 1:
ρ = (A_n + A_{n+1}) / 2
ω = (C_n + C_{n+1}) / 2
σ = (B_n + B_{n+1}) / 2
Result: Ω = (ρ, ω, σ)
Using the same data from Step 1:
Expert 1: E₁ = (10, 15, 20) → Gx₁ = 45/3 = 15.00
Expert 2: E₂ = (12, 18, 25) → Gx₂ = 55/3 ≈ 18.33
Expert 3: E₃ = (8, 14, 22) → Gx₃ = 44/3 ≈ 14.67
Sorted by centroid:
1. Expert 3: (8, 14, 22) → Gx = 14.67
2. Expert 1: (10, 15, 20) → Gx = 15.00 ← Middle element (n=1, position 2)
3. Expert 2: (12, 18, 25) → Gx = 18.33
Result: Ω = (10, 15, 20) (the middle element)
Adding a fourth expert:
Expert 4: E₄ = (11, 16, 23) → Gx₄ = 50/3 ≈ 16.67
Sorted by centroid:
1. Expert 3: (8, 14, 22) → Gx = 14.67
2. Expert 1: (10, 15, 20) → Gx = 15.00 ← Position n (2)
3. Expert 4: (11, 16, 23) → Gx = 16.67 ← Position n+1 (3)
4. Expert 2: (12, 18, 25) → Gx = 18.33
Calculation:
ρ = (10 + 11) / 2 = 10.5
ω = (15 + 16) / 2 = 15.5
σ = (20 + 23) / 2 = 21.5
Result: Ω = (10.5, 15.5, 21.5)
The best compromise ΓΩMean(π, φ, ξ) is the average of the arithmetic mean and median.
Formulas:
π = (α + ρ) / 2 (average of lower bounds)
φ = (γ + ω) / 2 (average of peaks)
ξ = (β + σ) / 2 (average of upper bounds)
Result: ΓΩMean = (π, φ, ξ)
Using results from Steps 1 and 2:
Γ = (10.00, 15.67, 22.33)
Ω = (10.00, 15.00, 20.00)
Calculation:
π = (10.00 + 10.00) / 2 = 10.00
φ = (15.67 + 15.00) / 2 = 15.33
ξ = (22.33 + 20.00) / 2 = 21.17
Result: ΓΩMean = (10.00, 15.33, 21.17)
The maximum error Δmax is a precision indicator that measures the distance between the arithmetic mean and median.
Formula:
Δmax = |centroid(Γ) - centroid(Ω)| / 2
Expanded:
Δmax = |Gx(Γ) - Gx(Ω)| / 2
= |(α + γ + β)/3 - (ρ + ω + σ)/3| / 2
Interpretation:
- Lower Δmax → Higher agreement among experts
- Higher Δmax → Greater disagreement or presence of outliers
Using results from previous steps:
Γ = (10.00, 15.67, 22.33)
Ω = (10.00, 15.00, 20.00)
Calculation:
Gx(Γ) = (10.00 + 15.67 + 22.33) / 3 = 48.00 / 3 = 16.00
Gx(Ω) = (10.00 + 15.00 + 20.00) / 3 = 45.00 / 3 = 15.00
Δmax = |16.00 - 15.00| / 2 = 1.00 / 2 = 0.50
Result: Δmax = 0.50
The complete BeCoMe result includes:
| Component | Value | Description |
|---|---|---|
| Best Compromise (ΓΩMean) | (10.00, 15.33, 21.17) | Final aggregated opinion |
| Arithmetic Mean (Γ) | (10.00, 15.67, 22.33) | Average of all opinions |
| Median (Ω) | (10.00, 15.00, 20.00) | Central tendency |
| Maximum Error (Δmax) | 0.50 | Precision indicator |
| Number of Experts (M) | 3 | Count of opinions |
| Is Even? | False | Affects median calculation |
Five project managers estimate the required budget (in millions):
Manager 1: (5.0, 8.0, 12.0)
Manager 2: (6.0, 9.0, 14.0)
Manager 3: (4.0, 7.0, 11.0)
Manager 4: (7.0, 10.0, 15.0)
Manager 5: (5.5, 8.5, 13.0)
Step 1: Arithmetic Mean
α = (5.0 + 6.0 + 4.0 + 7.0 + 5.5) / 5 = 27.5 / 5 = 5.50
γ = (8.0 + 9.0 + 7.0 + 10.0 + 8.5) / 5 = 42.5 / 5 = 8.50
β = (12.0 + 14.0 + 11.0 + 15.0 + 13.0) / 5 = 65.0 / 5 = 13.00
Γ = (5.50, 8.50, 13.00)
Step 2: Median (M = 5, odd)
Calculate centroids:
Manager 1: Gx = (5.0 + 8.0 + 12.0) / 3 = 8.33
Manager 2: Gx = (6.0 + 9.0 + 14.0) / 3 = 9.67
Manager 3: Gx = (4.0 + 7.0 + 11.0) / 3 = 7.33
Manager 4: Gx = (7.0 + 10.0 + 15.0) / 3 = 10.67
Manager 5: Gx = (5.5 + 8.5 + 13.0) / 3 = 9.00
Sort by centroid:
1. Manager 3: (4.0, 7.0, 11.0) → Gx = 7.33
2. Manager 1: (5.0, 8.0, 12.0) → Gx = 8.33
3. Manager 5: (5.5, 8.5, 13.0) → Gx = 9.00 ← Middle (position 3)
4. Manager 2: (6.0, 9.0, 14.0) → Gx = 9.67
5. Manager 4: (7.0, 10.0, 15.0) → Gx = 10.67
Middle element (position 3):
Ω = (5.5, 8.5, 13.0)
Step 3: Best Compromise
π = (5.50 + 5.5) / 2 = 5.50
φ = (8.50 + 8.5) / 2 = 8.50
ξ = (13.00 + 13.0) / 2 = 13.00
ΓΩMean = (5.50, 8.50, 13.00)
Step 4: Maximum Error
Gx(Γ) = (5.50 + 8.50 + 13.00) / 3 = 9.00
Gx(Ω) = (5.5 + 8.5 + 13.0) / 3 = 9.00
Δmax = |9.00 - 9.00| / 2 = 0.00
Interpretation: Δmax = 0 means perfect agreement (arithmetic mean equals median).
Arithmetic mean uses all data points but gets skewed by outliers. One extreme opinion can pull the result away from the group consensus. Median ignores everything except the central value — robust to outliers, but throws away information from non-central experts.
BeCoMe splits the difference. The mean component ensures every opinion contributes; the median component prevents extremes from dominating. When experts largely agree, Γ and Ω are close, and ΓΩMean lands near both. When opinions diverge, the compromise falls between the pulled-mean and the stable-median.
The error metric Δmax quantifies this divergence. Near-zero Δmax means the mean and median nearly coincide — strong consensus. Large Δmax signals polarization or outliers, suggesting the group should discuss further before deciding.
BeCoMe assumes opinions fit the triangular fuzzy format. Not all expert judgments naturally decompose into (min, likely, max). Converting verbal assessments like "probably around 50, maybe up to 80" requires interpretation. The centroid-based sorting can also produce different orderings than peak-based sorting would, though in practice this rarely changes the median significantly.
Vrana, I., Tyrychtr, J., & Pelikan, M. (2021). BeCoMe – A new approach for fuzzy group decision making. Expert Systems with Applications, 177, 114936.