MULTIMOORA Method

Multi-Objective Optimization by Ratio Analysis Plus Full Multiplicative Form

2010 Willem K. M. Brauers and Edmundas K. Zavadskas 7 Steps

MULTIMOORA (Multi-Objective Optimization by Ratio Analysis Plus Full Multiplicative Form) is a multi-criteria decision-making (MCDM) method that ranks alternatives by combining three subordinate approaches: Ratio System, Reference Point Approach and Full Multiplicative Form. The final ranking is obtained by aggregating the results of these three methods using Dominance Theory.

Video Demonstration

Step-by-Step Algorithm

Calculated cell

Source cells in formula

Form the decision matrix

Create the decision matrix with $m$ alternatives and $n$ criteria :

\[ X=\begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1n}\\ x_{21} & x_{22} & \cdots & x_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ x_{m1} & x_{m2} & \cdots & x_{mn} \end{bmatrix} \]

$x_{ij}$ presents the performance value of ith alternative on jth criterion, m and n are the numbers of alternatives and criteria respectively.

Alternative	$C_1$	$C_2$	$C_3$
$A_1$	10	20	30
$A_2$	15	25	35
$A_3$	25	15	35

Criterion type: C₁ Benefit C₂ Cost C₃ Benefit

Weights: $w_1=0.7279$ · $w_2=0.2009$ · $w_3=0.0712$ (Entropy method)

Normalize the decision matrix

Formula

Normalize the decision matrix using vector normalization :

$$ x_{ij}^{*}=\frac{x_{ij}} {\sqrt{\displaystyle\sum_{i=1}^{m}x_{ij}^{2}}} $$

The normalized values are dimensionless and therefore comparable between criteria.

Alternative	$C_1$	$C_2$	$C_3$
$A_1$	0.3244	0.5657	0.5183
$A_2$	0.4867	0.7071	0.6047
$A_3$	0.8111	0.4243	0.6047

Calculate the weighted normalized decision matrix

Formula

When criterion weights are used, multiply each normalized value by its criterion weight :

$$ v_{ij}=w_jx_{ij}^{*} $$

Where $w_j$ is the weight of criterion $C_j$.

$$ \sum_{j=1}^{n}w_j=1 $$

Using weights from the Entropy method: $w_1=0.7279$, $w_2=0.2009$, $w_3=0.0712$

Alternative	$C_1$	$C_2$	$C_3$
$A_1$	0.2361	0.1136	0.0369
$A_2$	0.3543	0.1421	0.0431
$A_3$	0.5904	0.0852	0.0431

Ratio system

Formula

Add the weighted normalized values of benefit criteria and subtract the values of cost criteria :

$$ Y_i=\sum_{j=1}^{g}v_{ij} - \sum_{j=g+1}^{n}v_{ij} $$

Where $g$ is the number of benefit criteria.

A larger $Y_i$ indicates a better alternative.

$$ Y_i>Y_k \quad\Longrightarrow\quad A_i>A_k $$

Benefit criteria: $C_1, C_3$. Cost criterion: $C_2$ ($g=2$).

Alternative	Ratio System ($Y_i$)
$A_1$	0.1594
$A_2$	0.2553
$A_3$	0.5483

Ratio system ranking (descending $Y_i$): $A_3 \succ A_2 \succ A_1$

Determine the reference point

Formula

The weighted normalized decision matrix is :

$$ V=[v_{ij}]_{m\times n} $$

For a benefit criterion , select the maximum weighted normalized value :

$$ r_j=\max_i v_{ij} $$

For a cost criterion , select the minimum weighted normalized value :

$$ r_j=\min_i v_{ij} $$

Therefore, the weighted ideal reference point is :

$$ r=(r_1,r_2,\ldots,r_n) $$

Calculate the absolute deviation between each weighted normalized value and the weighted ideal reference point :

$$ d_{ij}=\left|r_j-v_{ij}\right| $$

Determine the largest weighted deviation for each alternative :

$$ P_i=\max_j \left|r_j-v_{ij}\right| $$

$P_i$ represents the maximum weighted deviation of alternative $A_i$ from the ideal reference point.

Alternatives are ranked in ascending order of their reference-point assessment values :

$$ P_i < P_k \quad\Longrightarrow\quad A_i>A_k $$

Weighted Ideal Reference Point

Metric	$C_1$ (Benefit)	$C_2$ (Cost)	$C_3$ (Benefit)
$r_j$	0.5904	0.0852	0.0431

Absolute Deviation Matrix ($d_{ij}$)

Alternative	$C_1$	$C_2$	$C_3$
$A_1$	0.3543	0.0284	0.0062
$A_2$	0.2361	0.0569	0.0000
$A_3$	0.0000	0.0000	0.0000

Maximum Deviation ($P_i$)

Alternative	$P_i$
$A_1$	0.3543
$A_2$	0.2361
$A_3$	0.0000

Reference point ranking (ascending $P_i$): $A_3 \succ A_2 \succ A_1$

Full multiplicative form

Formula

Calculate the product of the normalized values of the benefit criteria :

$$ A_i=\prod_{j=1}^{g}x_{ij} $$

Calculate the product of the normalized values of the cost criteria :

$$ B_i=\prod_{j=g+1}^{n}x_{ij} $$

The overall multiplicative utility is :

$$ U_i=\frac{A_i}{B_i}=\frac{ \displaystyle\prod_{j=1}^{g}x_{ij}}{ \displaystyle\prod_{j=g+1}^{n}x_{ij}} $$

A higher $U_i$ indicates a better alternative.

$$ U_i>U_k \quad\Longrightarrow\quad A_i >A_k $$

Uses raw values from Step 1 (not normalized or weighted). Benefit: $C_1, C_3$. Cost: $C_2$.

Alternative	$A_i$ (Benefit)	$B_i$ (Cost)	$U_i=A_i / B_i$
$A_1$	300	20	15.0000
$A_2$	525	25	21.0000
$A_3$	875	15	58.3333

Multiplicative ranking (descending $U_i$): $A_3 \succ A_2 \succ A_1$

Ranking

Formula

The three subordinate rankings are compared :

$$ R_i^{(1)}=\text{Ratio System Rank} $$

$$ R_i^{(2)}=\text{Reference Point Rank} $$

$$ R_i^{(3)}=\text{Full Multiplicative Form Rank} $$

Dominance theory is then used to integrate the three rankings and obtain the final MULTIMOORA ranking.

An alternative that dominates another alternative in two or more of the three subordinate rankings receives the stronger final position.

Alternative	$R_i^{(1)}$	$R_i^{(2)}$	$R_i^{(3)}$	Final Rank
$A_1$	3	3	3	3
$A_2$	2	2	2	2
$A_3$	1	1	1	1

Subordinate rankings: All three methods yield $A_3 \succ A_2 \succ A_1$
Dominance result: $A_3$ dominates $A_2$ and $A_1$ in all three rankings; $A_2$ dominates $A_1$ in all three rankings.
Final MULTIMOORA ranking: $A_3 \succ A_2 \succ A_1$

MULTIMOORA Algorithm Flowchart

START

Form the decision matrix

Normalize the decision matrix

Calculate the weighted normalized decision Matrix

Ratio system

Determine the reference point

Full multiplicative form

Ranking

END