ITARA Method
Indifference Threshold-Based Attribute Ratio Analysis
ITARA (Indifference Threshold-Based Attribute Ratio Analysis) is an objective criteria-weighting method for MCDM. It determines the importance of criteria by analyzing the relative differences between alternatives and identifying an indifference threshold below which differences in criterion performance are considered insignificant.
Video Demonstration
Step-by-Step Algorithm
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.
For each criterion $C_j$, determine the indifference threshold $IT_j$. $IT_j$ should be less than the standard deviation of criterion $j$ ($\sigma_j$):
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 10 | 20 | 30 |
| $A_2$ | 15 | 25 | 35 |
| $A_3$ | 25 | 15 | 35 |
Normalized decision matrix
Normalize the criterion values using :
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2000 | 0.3333 | 0.3000 |
| $A_2$ | 0.3000 | 0.4167 | 0.3500 |
| $A_3$ | 0.5000 | 0.2500 | 0.3500 |
Normalized indifference threshold NIT
The normalized indifference threshold is :
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $IT_j$ | 5 | 3 | 2 |
| $NIT_j$ | 0.1000 | 0.0500 | 0.0200 |
Sorted normalized decision matrix
For each criterion \(j\), sort the normalized values in ascending order :
where \(\beta_{ij}\) represents the ordered normalized value.
| Rank | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| 1st | 0.2000 | 0.2500 | 0.3000 |
| 2nd | 0.3000 | 0.3333 | 0.3500 |
| 3rd | 0.5000 | 0.4167 | 0.3500 |
C₃ has a tie at rank 2nd–3rd: both $A_2$ and $A_3$ have $\alpha_{ij}=0.3500$.
Calculate the distance between adjacent values
Calculate the distance between each pair of adjacent ordered values :
for :
The value \(\gamma_{ij}\) represents the dispersion between two consecutive alternatives for criterion \(j\).
| Gap | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $\gamma_{1j}$ | 0.1000 | 0.0833 | 0.0500 |
| $\gamma_{2j}$ | 0.2000 | 0.0834 | 0.0000 |
Calculate the distance between $\gamma_{ij}$ and NIT
Compare each adjacent distance \(\gamma_{ij}\) with the normalized indifference threshold \(NIT_j\) :
Only differences greater than the indifference threshold contribute to the criterion weight.
Therefore, differences that are considered insignificant by the decision maker are assigned a value of zero.
$NIT_1=0.1000$, $NIT_2=0.0500$, $NIT_3=0.0200$
| Gap | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $\delta_{1j}$ | 0 | 0.0333 | 0.0300 |
| $\delta_{2j}$ | 0.1000 | 0.0334 | 0 |
Calculate the attribute adjustment value $v_j$
Sum all meaningful differences for each criterion :
A larger \(v_j\) indicates greater meaningful dispersion in criterion \(j\).
Using $p=1$ (Manhattan distance): $v_j=\sum_{i=1}^{m-1}\delta_{ij}$
| Criterion | Adjustment Value ($v_j$) |
|---|---|
| $C_1$ | 0.1000 |
| $C_2$ | 0.0667 |
| $C_3$ | 0.0300 |
Weights of criteria
Normalize the attribute adjustment values to obtain the final ITARA weights :
The weights satisfy
and
| Criterion | ITARA Weight ($w_j$) |
|---|---|
| $C_1$ | 0.5084 |
| $C_2$ | 0.3391 |
| $C_3$ | 0.1525 |
Check: $\sum_{j=1}^{3}w_j=0.5084 + 0.3391 + 0.1525=1.0000\;\checkmark$
ITARA Algorithm Flowchart
Sorted normalized decision matrix
Calculate the distance between Yij and NIT
Calculate the Attribute Adjustment Value Vj