MAIRCA - Multi-Attributive Ideal-Real Comparative Analysis

Overview

Q_i ≥ 0. Lower Q means the alternative's actual preference is closer to the theoretical (equal-probability) preference - i.e., it performs well across all criteria. Q=0 would mean the alternative is ideal on every criterion. MAIRCA is robust because equal probability T_Aij avoids subjective preference assumptions at the starting point.

Output

utility, lower 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

   Define theoretical preference T_Aij: uniform probability that each alternative is chosen = 1/m. Multiply by weights.

   $$T_{Aij}=w_j\cdot \frac{1}{m}$$

   Pamučar et al. 2014, p.90 Eq.(1)

2. 2

   Normalise decision matrix using linear normalisation.

   $$n_{ij}=\frac{x_{ij}-{\min }_k{x}_{kj}}{{\max }_k{x}_{kj}-{\min }_k{x}_{kj}}\text{ (benefit)};n_{ij}=\frac{{\max }_k{x}_{kj}-x_{ij}}{{\max }_k{x}_{kj}-{\min }_k{x}_{kj}}\text{ (cost)}$$

   Pamučar et al. 2014, p.90 Eq.(2)

3. 3

   Compute actual preference R_Aij = T_Aij × n_ij.

   $$R_{Aij}=T_{Aij}\cdot n_{ij}$$

   Pamučar et al. 2014, p.90 Eq.(3)

4. 4

   Compute total gap G_ij = T_Aij - R_Aij and total gap per alternative Q_i. Rank in ascending order (lower gap = better).

   $$G_{ij}=T_{Aij}-R_{Aij};Q_i=\sum _{j=1}^n{G}_{ij}$$

   Pamučar et al. 2014, p.91 Eqs.(4)-(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

• Constant criterion column: normalisation denominator is zero - check E-4.

Works with

How to cite