## 3 Materials and Methods
#### 3.3.1 Introduction of the TOPSIS Technique as One of the MCDM Approaches to Assess Effectiveness  
There are many different MCDM (Multi-Criteria Decision-Making) methods. The technique in [39] is one of the most frequently used MCDM methods. The origins of this method can be credited to [40; 41], and its usage is illustrated in Figure 1[42]. The TOPSIS technique is attractive because limited subjective input is needed from decision makers. The only subjective input needed is weights [43], which is another reason for the selection of this method, which was developed as an alternative to the ELECTRE method (ELimination Et Choix Traduisant la REalite). The authors in [44] describe the results of this method as a solution with the shortest distance to the Positive Ideal Solution (PIS) calculated using Euclidean distances. The TOPSIS technique provides a solution that is the closest under the specific conditions to the aforementioned PIS and also the farthest from the Negative Ideal Solution (NIS) [45]. The first step of processing is to build a matrix according to the pre-identified criteria (characteristics):\[D\ =\ \begin{pmatrix}&X_{1}&X_{2\cdots}&X_{j\cdots}&X_{n}\\ A_{1}&x_{11}&x_{12\cdots}&x_{1\cdots}&x_{1n}\\ A_{2}&x_{21}&x_{22\cdots}&x_{2\cdots}&x_{2n}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ A_{i}&x_{i1}&x_{i2\cdots}&x_{ij\cdots}&x_{in}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ A_{m}&x_{m1}&x_{m2\cdots}&x_{mj\cdots}&x_{mm}\\ \end{pmatrix} \]  
where \(A_{i}\) is the \(i\)-th variant, and \(x_{ij}\) is the value of the \(j\)-th criterion reached by the \(i\)-th variant. In the next step, this matrix is normalised using the relationship  
\[r_{ij}\ =\ x_{ij}/\sqrt{\sum_{j\ =\ 1}^{j}x_{ij}^{2}}\]  
where \(r_{ij}\) is the normalised value of the \(j\)-th criterion reached by the \(i\)-th variant, and \(x_{ij}\) is the value of the \(j\)-th criterion reached by the \(i\)-th variant. The matrix of data obtained is multiplied by the weights of the relevant criteria through that relationship:  
\[v_{ij}\ =\ w_{ij}*r_{ij}\]  
where \(v_{ij}\) is the weighted normalised value, \(w_{ij}\) is the criterion weight, and \(r_{ij}\) is the normalized value. Thus, the normalised matrix obtained contains values from which the PIS and NIS can be identified. These variants can be both real alternatives and hypothetical alternatives (out of the best or worst achieved results). The identification of the PIS and NIS can be represented by the following relationship:  
\[H_{j}\ =\ max(w_{ij}),D_{j}\ =\ min(w_{ij})\]  
where \(H_{j}\) is the positive ideal solution (PIS), and \(D_{j}\) is the negative ideal solution (NIS). The distance of the obtained PIS and NIS can be calculated according to  
\[d_{i}^{+}\ =\ \left[\sum_{j\ =\ 1}^{k}\left(w_{ij}-H_{j}\right)^{2}\right]^{1 /2},\ d_{i}^{-}\ =\ \left[\sum_{j\ =\ 1}^{k}\left(w_{ij}-D_{j}\right)^{2}\right]^{1/2}\]  
where \(d^{+}\) is the distance to the PIS, and \(d^{-}\) is the distance to the NIS. Figure 1: Multi-criteria decision making methods used in 1990–2014. From the perspective of alternatives, the desired minimisation of distance from the PIS is (\(d^{+}\)), and the maximisation of distance from the NIS is (\(d^{-}\)). The relative distance from the PIS is the basic criterion for setting the rank of an alternative. This criterion, by using the relationship below, considers the two identified distances from the previous step:  
\[c_{i}\ =\ \frac{d_{i}^{-}}{d_{i}^{-}+d_{i}^{+}}\]  
where \(c_{i}\) is the relative distance from the PIS. The final step involves ranking based on the relative distance to the PIS alternative. The best-rated alternative (subject) is the alternative with the highest value achieved. Within the terms of each MCDM method, the last important step is determining the weights via an individual indicator, which directly determines the overall results of the assessment. The authors in [46] classified approaches for determining weights into four groups: subjective, expert, objective, and integrated (which represents a combination of the previous approaches).