MULTIMOORA - Multi-Objective Optimisation by Ratio Analysis plus Full Multiplicative Form

Overview

MULTIMOORA produces three sub-rankings (RS: higher y better; RP: lower d better; FMF: higher U better) and aggregates them via dominance theory - the alternative dominating in ≥2 sub-rankings wins its position. When all three sub-rankings agree, the result is robust. When they disagree, inspect each sub-ranking individually to understand the trade-off.

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.

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

   Brauers-Zavadskas 2010, p.7 Eq.(2)

2. 2

   Ratio System y_i = Σ benefit − Σ cost.

   $$y_i=\sum _{j\in J^{+}}{w}_j{r}_{ij}-\sum _{j\in J^{-}}{w}_j{r}_{ij}$$

   Brauers-Zavadskas 2010, p.7 Eq.(3)

3. 3

   Reference Point z_i = max(|w_j r_j* − w_j r_ij|).

   $$z_i=\underset{j}{\max }|w_j{r}_j^{\ast }-w_j{r}_{ij}|,r_j^{\ast }=\underset{i}{\max }{r}_{ij}(J^{+})\text{or}\underset{i}{\min }{r}_{ij}(J^{-})$$

   Brauers-Zavadskas 2010, p.8 Eq.(5)

4. 4

   Full Multiplicative Form U_i = Π benefit / Π cost.

   $$U_i=\frac{{\prod }_{j\in J^{+}}(x_{ij}{)}^{w_j}}{{\prod }_{j\in J^{-}}(x_{ij}{)}^{w_j}}$$

   Brauers-Zavadskas 2010, p.8 Eq.(7)

5. 5

   Dominance theory aggregation of three rankings.

   $$\text{Final rank}=\text{dominance}(\text{rank}(y_i),\text{rank}(z_i),\text{rank}(U_i))$$

   Brauers-Zavadskas 2010, p.9

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

• FMF requires all x_ij > 0 - zero or negative raw values invalidate the full multiplicative form.

Works with

How to cite