# Optimal Configuration of Modular Strongrooms Using Multi-Attribute Decision Making
## 3 Method to Support Decision-Making  
Given the nature of the problem, a solution was found by integrating the Analytical Hierarchy Process (AHP) methods and Simple Additive Weighting (SAW) methods from the MADM group. To avoid the subjective assignment of weighting factors by the decision-makers, the AHP method was used. On the other hand, the problem often contains many alternatives whose performance can be quantified (numerically); thus, it was expedient to choose the SAW method to select the optimal alternative. The algorithm for applying the integration of these two methods (Figure 4) consists of 10 steps which will be explained below. Due to the specifics of the MSR, the proposed selection procedure using the SAW method (steps 7 to 10) was carried out twice, i.e., for walls and separately for floor and ceiling MSR. Step 1: Defining the problem and objective of decision-making. The first step is to clearly define the problem to be solved and the corresponding objective to be achieved. Step 2: Structuring decision-making and graphical representation. Structuring the decision-making problem involves decomposing the problem according to a hierarchy of at least three levels [13; 14; 15]. Step 3: Construct a comparison matrix. To determine the priorities of the criteria, i.e., importance of each criterion relative to the main objective, the AHP method uses pairwise comparison [14; 15; 16]. For this purpose, a comparison matrix is formed (square matrix \(n\times n\)) in which each criterion in a row is compared with each criterion in the column at each level of the hierarchical structure. The Saaty scale for the importance of value was used for comparison [16; 17]. Step 4: Normalizing the compared matrix data. Normalization of the comparison matrix data is performed by dividing each entry in the column by the sum of the corresponding column to obtain the following normalized result [14; 15; 18]:  
Figure 4: Integration of the AHP and SAW methods in a flow graph. \[a_{ij}^{*}=\frac{a_{ij}}{\sum\limits_{i=1}^{n}a_{ij}}\ ;\ \ \ \ \ \ j=1,2,\ldots n \tag{9}\]  
Step 5: Consistency analysis. In order to notice possible contradictions between the compared pairs, it is necessary to check the consistency. The Consistency Ratio (\(CR\)) is calculated as the quotient of the Consistency Index (\(CI\)) and the Random Index (\(RI\)):  
\[CR=\frac{CI}{RI} \tag{10}\]  
The AHP method allows an inconsistency of 10% (\(CR\leq\) 0.10) in relation to 500 randomly filled matrices [14; 16]. Step 6: Adopting the Criteria Significance Level. If the inconsistency is less than 10%, the assessment of the relative importance of the criteria is considered acceptable; otherwise, it is necessary to revise the comparison of the pairs of criteria. The SAW Method was chosen to select the optimal alternative. The SAW method is the best known, most popular, and most frequently used method of multi-attribute decision making [19; 20; 21]. Step 7: Construct decision matrix. The problem of multi-attribute decision-making can be expressed concisely in matrix form [21] as follows:  
\[ \begin{array}{l}A=\left\{A_{i}|i=1,2,3,\ldots,m\right\}\\ C=\left\{C_{j}|j=1,2,3,\ldots,n\right\}\\ W=\left\{W_{1},W_{2},W_{3},\ldots,W_{n}\right\}\\ X=\left[\begin{array}{cccc}x_{11}&\ldots&x_{1j}&\ldots&x_{1n}\\ \ldots&&\ldots&&\ldots\\ x_{i1}&\ldots&x_{ij}&\ldots&x_{in}\\ \ldots&&\ldots&&\ldots\\ x_{m1}&\ldots&x_{mj}&\ldots&x_{mn}\end{array}\right]\end{array}  \tag{11}\]  
where:  
\(A_{1}\), \(A_{2}\), \(\ldots\), \(A_{m}\)--feasible alternatives;  
\(C_{1}\), \(C_{2}\), \(\ldots\), \(C_{n}\)--criteria;  
\(x_{ij}\)--value of the \(i\)-th alternative relative to the \(j\)-th criterion;  
\(W_{j}\)--weighting coefficient (significance) of the \(j\)-th criterion. The matrix \(X\) is a (\(m\times n\)) matrix and is called a Decision matrix. Criteria (\(C_{j}\)) can be of maximization (max) or minimization (min) types. Step 8: Normalize decision matrix. To rank alternatives, the proposed SAW method requires data normalization in the decision matrix.