MOORA - Multi-Objective Optimisation by Ratio Analysis

Overview

y_i can be any real number. Higher y_i (Ratio System) means better. Lower d_i (Reference Point) means better. MOORA produces two rankings - Ratio System (RS) and Reference Point (RP). They may disagree; the full MULTIMOORA method resolves this via dominance theory.

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

   Vector normalisation r_ij = x_ij / √(Σ x_ij²).

   $$r_{ij}=\frac{x_{ij}}{\sqrt{{\sum }_{i=1}^m{x}_{ij}^2}}$$

   Brauers-Zavadskas 2006, p.453 Eq.(2)

2. 2

   Weighted normalised matrix v_ij = w_j · r_ij.

   $$v_{ij}=w_j{r}_{ij}$$

   Brauers-Zavadskas 2006, p.453 Eq.(3)

3. 3

   Ratio system y_i = Σ_{j∈J+} v_ij − Σ_{j∈J−} v_ij.

   $$y_i=\sum _{j\in J^{+}}{v}_{ij}-\sum _{j\in J^{-}}{v}_{ij}$$

   Brauers-Zavadskas 2006, p.453 Eq.(4)

4. 4

   Descending ranking by y_i.

   $$\text{Rank descending: }{y}_{(1)}\ge y_{(2)}\ge \cdots \ge y_{(m)}$$

   Brauers-Zavadskas 2006, p.453

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

• Using unweighted MOORA: the seminal method uses weights; some implementations omit them, yielding a different ranking.

Works with

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