IDOCRIW Method
Integrated Determination of Objective Criteria Weights
IDOCRIW (Integrated Determination of Objective Criteria Weights) is an objective MCDM weighting method that combines the information provided by different objective weighting techniques to obtain more reliable criterion weights. IDOCRIW integrates the Entropy Weight Method and the CILOS method. It considers both the variation/information content of the criteria and their relative impact/loss.
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.
| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 10 | 20 | 30 |
| $A_2$ | 15 | 25 | 35 |
| $A_3$ | 25 | 15 | 35 |
Normalize the decision matrix
The normalized value is calculated as :
\(r_{ij}\) indicates the normalized value of the decision matrix of ith alternative in jth criteria.
| 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 |
Calculate the Entropy value
The entropy value of criterion \(j\) is calculated as :
Where :
If \(r_{ij}=0\), the term \(r_{ij}\ln(r_{ij})\) is considered equal to \(0\).
The entropy value generally satisfies :
With $m=3$: $k=1/\ln(3)=0.9102$
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Entropy ($e_j$) | 0.9372 | 0.9808 | 0.9976 |
Weights of criteria (ENTROPY)
The degree of diversification of criterion \(j\) is :
The entropy weight of criterion \(j\) is calculated as :
The resulting weights satisfy :
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Entropy ($e_j$) | 0.9372 | 0.9808 | 0.9976 |
| Diversity ($d_j$) | 0.0628 | 0.0192 | 0.0024 |
| Weight ($w_j^{E}$) | 0.7441 | 0.2275 | 0.0284 |
Construct the transformed matrix
Convert all cost criteria into benefit criteria .
$$ x'_{ij}=\begin{cases} x_{ij}, & \text{if criterion}j\text{ is a benefit criterion}, \\[10pt] \dfrac{\min_i(x_{ij})}{x_{ij}}, & \text{if criterion}j\text{ is a cost criterion.} \end{cases} $$The transformed matrix is :
$$ X'=\begin{bmatrix} x'_{11}&x'_{12}&\cdots&x'_{1n}\\ x'_{21}&x'_{22}&\cdots&x'_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x'_{m1}&x'_{m2}&\cdots&x'_{mn} \end{bmatrix} $$C₁ and C₃ are benefit (unchanged). C₂ is cost: $\min(C_2)=15$, so $x'_{i2}=15/x_{i2}$.
| Alternative | $C_1$ (Benefit) | $C_2$ (Cost→Benefit) | $C_3$ (Benefit) |
|---|---|---|---|
| $A_1$ | 10 | 0.7500 | 30 |
| $A_2$ | 15 | 0.6000 | 35 |
| $A_3$ | 25 | 1.0000 | 35 |
Normalize the transformed matrix
Normalize every column using the column sum.
$$ r_{ij}=\frac{x'_{ij}} {\displaystyle\sum_{i=1}^{m}x'_{ij}} $$The normalized matrix is :
$$ R=[r_{ij}]=\begin{bmatrix} r_{11}&r_{12}&\cdots&r_{1n}\\ r_{21}&r_{22}&\cdots&r_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ r_{m1}&r_{m2}&\cdots&r_{mn} \end{bmatrix} $$| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.2000 | 0.3191 | 0.3000 |
| $A_2$ | 0.3000 | 0.2553 | 0.3500 |
| $A_3$ | 0.5000 | 0.4255 | 0.3500 |
Square matrix
Find the maximum value for every criterion.
If alternative \(k_j\) contains the maximum value of criterion \(j\), select that entire row to build the square matrix.
The diagonal elements satisfy :
| Metric | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| Max ($a_j$) | 0.5000 | 0.4255 | 0.3500 |
| Source ($k_j$) | $A_3$ | $A_3$ | $A_2$ |
The square matrix is formed by taking row $A_3$ for $C_1$ and $C_2$, and row $A_2$ for $C_3$.
| Row (criterion) | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ (from $A_3$) | 0.5000 | 0.4255 | 0.3500 |
| $C_2$ (from $A_3$) | 0.5000 | 0.4255 | 0.3500 |
| $C_3$ (from $A_2$) | 0.3000 | 0.2553 | 0.3500 |
Relative impact loss matrix
Where :
$$ p_{ij}=\frac{a_{jj}-a_{ij}} {a_{jj}} $$The diagonal elements are :
$$ p_{ii}=0 $$Thus, the relative impact loss matrix is :
$$ P=\begin{bmatrix} 0 & p_{12} & \cdots & p_{1n}\\ p_{21} & 0 & \cdots & p_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ p_{n1} & p_{n2} & \cdots & 0 \end{bmatrix} $$| $P$ | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | 0 | 0 | 0 |
| $C_2$ | 0 | 0 | 0 |
| $C_3$ | 0.4000 | 0.4000 | 0 |
Weights system matrix
Construct the weights system matrix.
where :
and
Hence,
| $F$ | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | -0.4000 | 0 | 0 |
| $C_2$ | 0 | -0.4000 | 0 |
| $C_3$ | 0.4000 | 0.4000 | 0 |
Weights of criteria (CILOS)
The objective weights are obtained by solving the following homogeneous system of equations :
$$ Fq=0 $$subject to the normalization constraint :
$$ \sum_{j=1}^{n} q_j=1 $$and the non-negativity constraint :
$$ q_j \ge 0, \qquad j=1,2,\ldots,n. $$The resulting objective weight vector is :
$$ q=\begin{bmatrix} q_1\\ q_2\\ \vdots\\ q_n \end{bmatrix} $$where each \(q_j\) is the objective weight of criterion \(j\).
| Criterion | CILOS Weight ($q_j$) |
|---|---|
| $C_1$ | 0 |
| $C_2$ | 0 |
| $C_3$ | 1.0000 |
Aggregate weights
The Entropy and CILOS weights are combined to obtain the final integrated criterion weights :
where \(w_j^{E}\) is the Entropy weight and \(q_j\) is the CILOS weight.
The final IDOCRIW weights satisfy :
| Criterion | IDOCRIW Weight ($w_j$) |
|---|---|
| $C_1$ | 0 |
| $C_2$ | 0 |
| $C_3$ | 1.0000 |
Check: $\sum_{j=1}^{3}w_j=0 + 0 + 1.0000=1.0000\;\checkmark$