## 3 Topisis  
MCDA is applied to tasks where the decisions are taken in order to fulfil often contradictory criteria, e.g., minimum cost and minimum required time to deliver a project. The decision is a product of a systematic approach that partially or fully satisfies the conditions or limitations that each criterion places. The criteria may refer to technical and economic constrains, risk related factors, environmental restrictions and others. Basic tools of MCDA are the Analytical Hierarchical Process (AHP) and TOPSIS. During the last years, MCDA has witnessed a vast variety of applications [96]. In TOPSIS method, the solutions refer to the available alternative approaches for addressing the problem. In the present paper, the problem is the selection of the clustering algorithm that optimally clusters a given set of load data. The solutions are the clustering algorithms themselves and the criteria that need to be taken into account are \"Criterion#1,\"..., \"Criterion#6.\" Also, 2 solutions need to be defined, namely the \"ideal\" and the \"anti-ideal.\" The distances of each solution from the ideal and the anti-ideal ones are calculated. The selected solution should have minimum distance from the ideal and maximum distance from the anti-ideal solutions. Let \(A_{i},i=1,...,r\) be the alternative solutions and \(z_{j},j=1,..,p\) the criteria. The steps that construct the TOPSIS method are [19; 97]:  
Step#1. Build the decision matrix \(D_{matrix}\) with \(i\) alternatives and \(j\) solutions: \[D_{matrix}=\begin{array}{c}A_{1}\\ A_{2}\\ \vdots\\ A_{i}\end{array}\left[\begin{array}{c}z_{1}\ z_{2}\ \cdots z_{j}\\ z_{21}\ z_{22}\cdots z_{2j}\\ \vdots\ \vdots\ \vdots\\ z_{i1}\ z_{i2}\cdots z_{ij}\end{array}\right]\] (33) Step#2. Construct the normalized \(D_{matrix}\) denoted as \(R\) with elements according to the following equation: \[r_{ij}=\frac{z_{ij}}{\sqrt{\sum\limits_{i=1}^{r}z_{ij}^{2}}}\] (34)Step#3. Construct the weighted matrix \(R\) denoted as \(V\) according to:  
\[V=\left[\begin{array}{c}v_{11}\,v_{12}\cdots v_{1j}\\ v_{21}\,v_{22}\cdots v_{2j}\\ \vdots\ \vdots\ \vdots\\ v_{11}\,v_{i2}\cdots v_{ij}\end{array}\right] \tag{35}\]  
where \(v_{ij}=w_{ij}r_{ij}\) and \(w_{ij}\) is the weight that solution \(A_{i}\) is connected with criterion \(z_{j}\). It should be noted that the weights are fixed by the decision maker. The weights are user-centric and their values influence the results of the decision making. This fact is an inherent characteristic of TOPSIS method. Thus, TOPSIS offers a framework for the decision maker to include its expertise on a decision problem by setting the weights and reach into a solution that is in accordance to his/hers needs.  
Step#4. Calculate the ideal \(V^{+}\) and the anti-ideal solution \(V^{-}\) according to:  
\[\begin{split} V^{+}&=\left\{(\underset{i}{\max}v_{ ij}|j\in J),(\underset{i}{\min}v_{ij}|j\in J)\right\}\\ V^{-}&=\left\{(\underset{i}{\min}v_{ij}|j\in J),( \underset{i}{\max}v_{ij}|j\in J)\right\}\end{split} \tag{36}\]  
where \(J\) and \(J\) are the positive and negative impact, respectively. More specifically, the ideal solution refers to is the maximum value for the positive impact and the minimum value for the negative impact in each column. Similarly, the anti-ideal solution, is the minimum and the maximum values for the positive and the negative impacts in each column, respectively.  
Step#5. Calculate the distances between each solution and the ideal and anti-ideal solutions:  
\[S_{i}^{+} =\sqrt{\sum_{i=1}^{r}\left(v_{ij}-V^{+}\right)^{2}} \tag{37}\] \[S_{i}^{-} =\sqrt{\sum_{i=1}^{r}\left(v_{ij}-V^{-}\right)^{2}} \tag{38}\]  
Step#6. Calculate the mean distance between each solution and anti-ideal solution as:  
\[B_{i}=\frac{S_{i}^{-}}{S_{i}^{-}+S_{i}^{+}} \tag{39}\]  
Step#7. Sort the solutions according to the \(B_{i}\) value.