CILOS Method
Criterion Impact Loss
The Criterion Impact Loss (CILOS) method is an objective criterion weighting technique for Multi-Criteria Decision-Making (MCDM), introduced by Zavadskas and Podvezko in 2016. It estimates criterion weights by evaluating the relative impact loss associated with each criterion, using only the decision matrix and without requiring expert judgments.
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 | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 30 | 40 | 15 |
| $A_2$ | 10 | 10 | 20 |
| $A_3$ | 20 | 25 | 35 |
Construct the transformed matrix
Since the CILOS method requires all criteria to be benefit criteria, each cost criterion is transformed into a benefit criterion.
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
$$ X'=\begin{bmatrix} x'_{11}&x'_{12}&\cdots&x'_{1n}\\ x'_{21}&x'_{22}&\cdots&x'_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ x'_{m1}&x'_{m2}&\cdots&x'_{mn} \end{bmatrix} $$| Alternative | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 30 | 0.2500 | 15 |
| $A_2$ | 10 | 1.0000 | 20 |
| $A_3$ | 20 | 0.4000 | 35 |
Normalize the decision 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 | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $A_1$ | 0.5000 | 0.1515 | 0.2143 |
| $A_2$ | 0.1667 | 0.6061 | 0.2857 |
| $A_3$ | 0.3333 | 0.2424 | 0.5000 |
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 value ($a_j$) | 0.5000 | 0.6061 | 0.5000 |
| Alternative ($k_j$) | $A_1$ | $A_2$ | $A_3$ |
Square matrix $A$: row $k_j$ is the entire row of alternative $k_j$ from the normalized matrix.
| Row | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $k_1$ ($A_1$) | 0.5000 | 0.1515 | 0.2143 |
| $k_2$ ($A_2$) | 0.1667 | 0.6061 | 0.2857 |
| $k_3$ ($A_3$) | 0.3333 | 0.2424 | 0.5000 |
Relative impact loss matrix
Where
The diagonal elements are
Thus, the relative impact loss matrix is
| $P$ | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | 0.0000 | 0.7500 | 0.5714 |
| $C_2$ | 0.6667 | 0.0000 | 0.4286 |
| $C_3$ | 0.3333 | 0.6000 | 0.0000 |
Weights system matrix
Construct the weights system matrix
Where
and
Hence,
| $F$ | $C_1$ | $C_2$ | $C_3$ |
|---|---|---|---|
| $C_1$ | -1.0000 | 0.7500 | 0.5714 |
| $C_2$ | 0.6667 | -1.3500 | 0.4286 |
| $C_3$ | 0.3333 | 0.6000 | -1.0000 |
Weights of criteria
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 | Weight ($q_j$) |
|---|---|
| $C_1$ | 0.3971 |
| $C_2$ | 0.2941 |
| $C_3$ | 0.3088 |
Verification: $q_1 + q_2 + q_3=0.3971 + 0.2941 + 0.3088=1.0000$ ā