BeCoMe method description¶

Contents¶

Overview
Mathematical foundation
The BeCoMe algorithm, step by step
A complete worked example
Why combine mean and median?
Limitations
Reference

Overview¶

BeCoMe (Best Compromise Mean) is a method for aggregating expert opinions expressed as fuzzy triangular numbers. I. Vrana, J. Tyrychtr, and M. Pelikán developed it 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. An outlier still moves the result, but by half of what it moves the mean alone.

Mathematical foundation¶

1. Fuzzy triangular numbers¶

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

Visual representation¶

  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

Notation¶

A fuzzy triangular number is denoted as: FTN(A, C, B) or (A, C, B)

Example¶

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)

2. Centroid calculation¶

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.

Example¶

For FTN(5, 8, 12):

Gx = (5 + 8 + 12) / 3 = 25 / 3 ≈ 8.33

The BeCoMe algorithm, step by step¶

The BeCoMe method aggregates M expert opinions into a single best compromise result through five steps.

Input¶

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)

Step 1: calculate the arithmetic mean (Γ)¶

Averaging each component separately gives the arithmetic mean Γ(α, γ, β).

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)

Example (3 experts)¶

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)

Step 2: calculate the median (Ω)¶

To find the median Ω(ρ, ω, σ), sort the opinions by centroid and take the middle value. With an even count, take the average of the two middle values.

Step 2.1: calculate centroids¶

For each expert opinion, calculate the centroid:

Gxₖ = (Aₖ + Cₖ + Bₖ) / 3

Step 2.2: sort by centroid¶

Sort all expert opinions in ascending order by their centroid values.

Step 2.3: find the median¶

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)

Example (3 experts - odd)¶

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)

Example (4 experts - even)¶

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)

Step 3: calculate the best compromise (ΓΩMean)¶

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)

Example (3 experts)¶

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)

Step 4: calculate the maximum error (Δmax)¶

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

Example (3 experts)¶

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

Step 5: summary of results¶

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

A complete worked example¶

Scenario: estimating a project budget¶

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)

Solution¶

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).

Why combine mean and 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. An outlier barely moves it, but it throws away what the non-central experts said.

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, which is strong consensus. Large Δmax signals polarization or outliers, suggesting the group should discuss further before deciding.

Limitations¶

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.

Reference¶

Vrana, I., Tyrychtr, J., & Pelikán, M. (2021). BeCoMe: Easy-to-implement optimized method for best-compromise group decision making: Flood-prevention and COVID-19 case studies. Environmental Modelling & Software, 136, 104953. https://doi.org/10.1016/j.envsoft.2020.104953