# Multi-Criteria Decision-Making Techniques for Optimal In-House Storage Location Selection
## 3 Methodology  
### Additive Ratio Assessment (ARAS) Method  
The Additive Ratio Assessment (ARAS) method is created by [PERSON] and [PERSON] [16]. This method can be presented through the five steps [16]:  
Step 1. Define a decision-making matrix \(Z\)  
\[Z= \begin{bmatrix}z_{01}&\cdots&z_{0j}&\cdots&z_{0n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ z_{i1}&\cdots&z_{ij}&\cdots&z_{in}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ z_{m1}&\cdots&z_{mj}&\cdots&z_{mn}\end{bmatrix} ;\;i=\overline{0,m},\;j= \overline{1,n}. \tag{1}\]  
where: \(m\)--number of alternatives, \(n\)--number of criteria describing each alternative, \(z_{ij}\)--a value representing the performance value of the \(i\)-th alternative in terms of the \(j\)-th criterion, \(z_{0j}\)--the optimal value of \(j\)-th criterion. If the optimal value of the \(j\)-th criterion is unknown, then:  
\[z_{0j}=max_{i}z_{ij},\;\text{if}\;max_{i}z_{ij}\;\text{is preferable}. \tag{2}\]  
\[z_{0j}=min_{i}z_{ij}^{*},\;\text{if}\;min_{i}z_{ij}^{*}\;\text{is preferable}.\]  
Usually, the performance values \(z_{ij}\) and the criteria weights \(W_{j}\) are considered entries in a DMM. The system of criteria, as well as values and initial weights of criteria, is determined by experts. The interested parties can correct the information by considering their goals and opportunities. Step 2. Normalize the data form Step 1. In this step, the initial values of all the criteria are normalized--defining values \(\overline{z_{ij}}\) of normalized decision-making matrix \(\overline{Z}\). \[\overline{Z}= \begin{bmatrix}\overline{z}_{01}&\cdots&\overline{z}_{0j}&\cdots& \overline{z}_{0n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \overline{z}_{i1}&\cdots&\overline{z}_{ij}&\cdots&\overline{z}_{in}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \overline{z}_{m1}&\cdots&\overline{z}_{mj}&\cdots&\overline{z}_{mn}\\ \end{bmatrix} ;\;i=\overline{0,\;m,\;j=\overline{1,n}}; \tag{3}\]  
For the criteria with the maximal preferable numbers, the normalization is calculated by the following equation:  
\[\overline{z_{ij}}=\frac{z_{ij}}{\sum_{i=0}^{m}z_{ij}}; \tag{4}\]  
For minimal type of criteria, the normalization is performed through two steps, by the following equation:  
\[z_{ij}=\frac{1}{z_{ij}^{*}},\;\;\;\overline{z_{ij}}=\frac{z_{ij}}{\sum_{i=0}^ {m}z_{ij}}. \tag{5}\]  
Step 3. Formulate normalized-weighted matrix--\(\underline{Z}\). The criteria with weights are structured between intervals \(0<W_{j}<1\). The values of weight \(W_{j}\) are usually computed by the experts' assessment. The sum of weights \(W_{j}\) is to be equal to 1, as follows:  
\[\sum_{j=1}^{n}w_{j}=1; \tag{6}\]  
\[\hat{Z}= \begin{bmatrix}z_{01}&\cdots&z_{0j}&\cdots&\hat{z}_{0n}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ z_{i1}&\cdots&z_{ij}&\cdots&z_{in}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \hat{z}_{m1}&\cdots&\hat{z}_{mj}&\cdots&z_{mn}\\ \end{bmatrix} ;\;i=\overline{0,m,\;j=\overline{1,n}}; \tag{7}\]  
Normalized-weighted values are found:  
\[\hat{z}_{ij}=\overline{z}_{ij}\cdot\mathsf{W}_{j};\;i=\overline{0,m}; \tag{8}\]  
where \(\mathsf{W}_{j}\) is the significance of the \(j\)-th criterion, and \(\overline{z}_{ij}\) is the normalized rating of the \(j\)-th criterion. Step 4. Find the value of the optimality function. \[S_{i}=\sum_{j=1}^{n}\hat{z}_{ij},i=\overline{0,m}. \tag{9}\]  
where: \(S_{i}\)--the value of the optimality function of \(i\)-th alternative. The higher value of \(S_{i}\) is the best one, while the lowest is the worst. Therefore, the greater the value of the optimality function \(S_{i}\), the more influential the alternative. The priorities of other alternatives can be determined according to the value \(S_{i}\). Step 5. Calculate the degree of the alternative utility. To calculate the degree of the alternative utility, it is necessary to compare the variants with the ideally best one subscript base, one \(S_{0}\).