WEBIRA - WEighted Bi-directional Ideal Ratio Analysis

Overview

B_i can be any real number. Higher B means the alternative is closer to the ideal and farther from the anti-ideal, summed across all criteria weighted. B_i = 0 means equal proximity to ideal and anti-ideal on a weighted average.

Output

utility, higher is better

Data

Crisp, complete numeric matrix

Weights

Needs a weight source

Size

2+ alternatives, 3-12 criteria works best

Used for

Alternative selection, Supplier evaluation

How it works

1. 1

   Identify ideal x*_j and anti-ideal x°_j for each criterion.

   $$x_j^{\ast }=\underset{i}{\max }{x}_{ij}\text{ (benefit)},\underset{i}{\min }{x}_{ij}\text{ (cost)};x_j^{\circ }=\underset{i}{\min }{x}_{ij}\text{ (benefit)},\underset{i}{\max }{x}_{ij}\text{ (cost)}$$

   Krstović et al. 2022, p.42 Eqs.(1)-(2)

2. 2

   Compute positive ratio R⁺_ij (closeness to ideal) and negative ratio R⁻_ij (closeness to anti-ideal).

   $$R_{ij}^{+}=\frac{x_{ij}}{x_j^{\ast }},R_{ij}^{-}=\frac{x_j^{\circ }}{x_{ij}}$$

   Krstović et al. 2022, p.43 Eqs.(3)-(4)

3. 3

   Compute WEBIRA score B_i = weighted sum of (R⁺_ij − R⁻_ij). Rank in descending order.

   $$B_i=\sum _{j=1}^n{w}_j(R_{ij}^{+}-R_{ij}^{-})$$

   Krstović et al. 2022, p.43 Eq.(5)

Look elsewhere when

Assumptions to verify

• Criteria preferences are independent (no synergistic interactions)
• Compensation is acceptable: high score on one criterion can offset low on another
• Decision matrix is complete (no missing values)

Edge cases and pitfalls

• All alternatives identical on a criterion: ideal = anti-ideal, R⁺ = R⁻ = 1 for all, so that criterion contributes 0 to score - acceptable but check for degenerate columns.

Works with

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