1.1. American school

1.1.1. ARAS

ARAS is designed to evaluate decision alternatives according to the following steps:

Step 1. Definition of a decision matrix of dimension \(n \times m\), where \(n\) is the number of alternatives, and \(m\) is the number of criteria (1.16).

(1.16)\[\begin{split}\begin{equation} x_{i j}=\left[\begin{array}{llll} x_{11} & x_{12} & \ldots & x_{1 m} \\ x_{21} & x_{22} & \ldots & x_{2 m} \\ \ldots & \ldots & \ldots & \ldots \\ x_{n 1} & x_{n 2} & \ldots & x_{n m} \end{array}\right] \end{equation}\end{split}\]

Step 2. Normalization the decision matrix, where for profit criteria use the equation (1.2), and for cost, criteria use the equation (1.3). In this study, The Sum normalization method was used.

(1.2)\[\begin{equation} r_{ij} = \frac{x_{ij}}{\sum_m^{i=0}x_{ij}} \end{equation}\]

(1.3)\[\begin{equation} r_{ij} = \frac{\frac{1}{x_{ij}}}{\sum_m^{i=0}\frac{1}{x_{ij}}} \end{equation}\]

Step 3. Building a decision matrix \(v_{ij}\) subjected to a weighting and normalization process using the Equation (1.4).

(1.4)\[\begin{equation} v_{ij} = w_{j}r_{ij} \label{weighted} \end{equation}\]

Step 4. Determining values of optimality function using the Equation (1.5).

(1.5)\[\begin{equation} S_i = \sum_{j=1}^{n} v_{ij} \end{equation}\]

Step 5. Calculate the utility degree \(K_i\) based on Equation (1.6).

(1.6)\[\begin{equation} K_i = \frac{S_i}{S_0} \end{equation}\]

where \(S_i\) and \(S_0\) are the optimality criterion values.

1.1.2. COCOSO

COCOSO is designed to evaluate decision alternatives according to the following steps:

Step 1. Definition of a decision matrix of dimension \(n \times m\), where \(n\) is the number of alternatives, and \(m\) is the number of criteria (1.16).

(1.16)\[\begin{split}\begin{equation} x_{i j}=\left[\begin{array}{llll} x_{11} & x_{12} & \ldots & x_{1 m} \\ x_{21} & x_{22} & \ldots & x_{2 m} \\ \ldots & \ldots & \ldots & \ldots \\ x_{n 1} & x_{n 2} & \ldots & x_{n m} \end{array}\right] \end{equation}\end{split}\]

Step 2. Normalization the decision matrix, where for profit criteria use the equation (1.8), and for cost, criteria use the equation (1.9). In this study, The Minimum-Maximum normalization method was used.

(1.8)\[\begin{equation} r_{ij} = \frac{x_{ij} - \min_{i}{x_{ij}}}{\max_{i}{x_{ij}} - \min_{i}{x_{ij}}} \end{equation}\]

(1.9)\[\begin{equation} r_{ij} = \frac{\max_{i}{x_{ij}} - x_{ij}}{\max_{i}{x_{ij}} - \min_{i}{x_{ij}}} \end{equation}\]

Step 3. Calculation of the weighted sum of the comparison sequence and the total power weight of the comparison sequences for each alternative. The values of \(S_i\) are based on the grey relationship generation method (1.10), and for \(P_i\) the values are achieved according to the multiplicative WASPAS setting (1.11).

(1.10)\[\begin{equation} S_i = \sum_{j=1}^{n} (w_j r_{ij}) \end{equation}\]

(1.11)\[\begin{equation} P_i = \sum_{j=1}^{n} (r_{ij})^{w_j} \end{equation}\]

Step 4. Computation of the relative weights of alternatives using aggregation strategies. The formulas determine the strategies (1.12)-(1.14), where the first strategy expresses the average of the sums of WSM and WPM s cores (1.12), the second strategy expresses the sum of WSM and WPM scores over the best (1.13), and the third strategy expresses the compromise strategy of WSM and WPM by using the \(\lambda\) value (1.14). In this study, a \(\lambda\) value of 0.5 was used.

(1.12)\[\begin{equation} k_{i a}=\frac{P_{i}+S_{i}}{\sum_{i=1}^{m}\left(P_{i}+S_{i}\right)} \end{equation}\]

(1.13)\[\begin{equation} k_{i b}=\frac{S_{i}}{\min _{i} S_{i}}+\frac{P_{i}}{\min _{i} P_{i}} \end{equation}\]

(1.14)\[\begin{equation} k_{i c}=\frac{\lambda\left(S_{i}\right)+(1-\lambda)\left(P_{i}\right)}{\left(\lambda \max _{i} S_{i}+(1-\lambda) \max _{i} P_{i}\right)} ; \quad 0 \leqslant \lambda \leqslant 1 \end{equation}\]

Step 5. Establish the final ranking of alternatives based on \(k_i\) values defined using the formula (1.15). The higher the \(k_i\) value, the higher the ranking.

(1.15)\[\begin{equation} k_{i}=\left(k_{i a} k_{i b} k_{i c}\right)^{\frac{1}{3}}+\frac{1}{3}\left(k_{i a}+k_{i b}+k_{i c}\right) \end{equation}\]

1.1.3. CODAS

CODAS is designed to evaluate decision alternatives according to the following steps:

Step 1. Definition of a decision matrix of dimension \(n \times m\), where \(n\) is the number of alternatives, and \(m\) is the number of criteria (1.16).

(1.16)\[\begin{split}\begin{equation} x_{i j}=\left[\begin{array}{llll} x_{11} & x_{12} & \ldots & x_{1 m} \\ x_{21} & x_{22} & \ldots & x_{2 m} \\ \ldots & \ldots & \ldots & \ldots \\ x_{n 1} & x_{n 2} & \ldots & x_{n m} \end{array}\right] \end{equation}\end{split}\]

Step 2. Normalization the decision matrix, where for profit criteria use the equation (1.17), and for cost, criteria use the equation (1.18). In this study, The Linear normalization method was used.

(1.17)\[\begin{equation} r_{ij} = \frac{x_{ij}}{\max_i x_{ij}} \end{equation}\]

(1.18)\[\begin{equation} r_{ij} = \frac{\min_i x_{ij}}{x_{ij}} \end{equation}\]

Step 3. Building a decision matrix \(v_{ij}\) subjected to a weighting and normalization process using the Equation (1.19).

(1.19)\[\begin{equation} v_{ij} = w_{j}r_{ij} \label{weightedc} \end{equation}\]

Step 4. Determine the negative-ideal solution (point) based on Equation (1.20).

(1.20)\[\begin{equation} ns_j = \min_i {v_ij} \end{equation}\]

Step 5. Calculate the Euclidean and Taxicab distances of alternatives from the negative-ideal solution, shown as follows:

\[\begin{equation} E_i = \sqrt{\sum_{i=1}^m \left ( v_{ij} - ns_j \right)^2} \end{equation}\]

\[\begin{equation} T_i = \sum_{j=1}^m \left | v_{ij} - ns_j \right | \end{equation}\]

Step 6. Construct the relative assessment matrix, shown as follows:

\[\begin{equation} h_{i k}=\left(E_{i}-E_{k}\right)+\left(\psi\left(E_{i}-E_{k}\right) \times\left(T_{i}-T_{k}\right)\right) \end{equation}\]

where \(k \in \left \{ 1,2,\cdots,n \right \}\) and \(psi\) denotes a threshold function to recognize the equality of the Euclidean distances of two alternatives, and is defined as follows:

\[\begin{split}\begin{equation} \psi(x)=\left\{\begin{array}{lll} 1 & \text { if } & |x| \geq \tau \\ 0 & \text { if } & |x|<\tau \end{array}\right. \end{equation}\end{split}\]

In this function, \(\tau\) is the threshold parameter that can be set by decisionmaker. It is suggested to set this parameter at a value between 0.01 and 0.05.

Step 7. Calculate the assessment score of each alternative, shown as follows:

\[\begin{equation} \mathrm{H}_{i}=\sum_{k=1}^{n} h_{i k} \end{equation}\]

Step 8. Rank the alternatives according to the decreasing values of assessment.

1.1.4. COPRAS

COPRAS is designed to evaluate decision alternatives according to the following steps:

Step 1. Calculate normalized decision matrix using equation (1.21).

(1.21)\[\begin{equation} r_{i j}=\frac{x_{i j}}{\sum_{i=1}^{m} x_{i j}} \end{equation}\]

Step 2. Calculate difficult normalized decision matrix, which represents multiplication of the normalized decision matrix elements with the appropriate weight coefficients using equation (1.22).

(1.22)\[\begin{equation} v_{ij} = r_{ij} \cdot w_j \end{equation}\]

Step 3. Determine the sums of difficult normalized values which was calculated previously. Equation (1.23) should be used for profit criteria and equation (1.24) for cost criteria.

(1.23)\[\begin{equation} S_{+i}=\sum_{j=1}^{k} v_{i j} \end{equation}\]

(1.24)\[\begin{equation} S_{-i}=\sum_{j=k+1}^{n} v_{i j} \end{equation}\]

where \(k\) is the number of attributes that must be maximized. The rest of attributes from \(k+1\) to n prefer lower values. The \(S_{+i}\) and \(S_{-i}\) values show level of the goal achievement for alternatives. Higher value of \(S_{+i}\) means that this alternative is better and the lower value of \(S_{-i}\) also points to better alternative.

Step 4. Calculate the relative significance of alternatives using equation (1.25).

(1.25)\[\begin{equation} \label{eq:copras_q} Q_{i}=S_{+i}+\frac{S_{-\min } \cdot \sum_{i=1}^{m} S_{-i}}{S_{-i} \cdot \sum_{i=1}^{m}\left(\frac{S_{-\min }}{S_{-i}}\right)} \end{equation}\]

Step 5. Final ranking is performed according \(U_i\) values (1.26).

(1.26)\[\begin{equation} U_i = \frac{Q_i}{Q^{max}_i} \cdot 100\% \end{equation}\]

Where \(Q^{max}_i\) stands for maximum value of the utility function. Better alternatives has higher \(U_i\) value.

1.1.5. EDAS

EDAS is designed to evaluate decision alternatives according to the following steps:

Step 1. Define a decision matrix of dimension \(n \times m\), where \(n\) is the number of alternatives, and \(m\) is the number of criteria (1.27).

(1.27)\[\begin{split}\begin{equation} X_{i j}=\left[\begin{array}{llll} x_{11} & x_{12} & \ldots & x_{1 m} \\ x_{21} & x_{22} & \ldots & x_{2 m} \\ \ldots & \ldots & \ldots & \ldots \\ x_{n 1} & x_{n 2} & \ldots & x_{n m} \end{array}\right] \end{equation}\end{split}\]

Step 2. Calculate the average solution for each criterion according to the formula (1.28).

(1.28)\[\begin{equation} A V_{j}=\frac{\sum_{i=1}^{n} X_{i j}}{n} \end{equation}\]

Step 3. Calculating the positive distance from the mean solution and the negative distance from the mean solution for the alternatives. When the criterion is of profit type, the negative distance and the positive distance are calculated using equations (1.30) and (1.29), while when the criterion is of cost type, the distances are calculated using formulas (1.32) and (1.31).

(1.29)\[\begin{equation} PDA_{i j} = \frac{\max \left(0,\left(X_{i j}-A V_{j}\right)\right)}{A V_{j}} \end{equation}\]

(1.30)\[\begin{equation} NDA_{i j}=\frac{\max \left(0,\left(A V_{j}-X_{i j}\right)\right)}{A V_{j}} \end{equation}\]

(1.31)\[\begin{equation} P D A_{i j}=\frac{\max \left(0,\left(A V_{j}-X_{i j}\right)\right)}{A V_{j}} \end{equation}\]

(1.32)\[\begin{equation} N D A_{i j}=\frac{\max \left(0,\left(X_{i j}-A V_{j}\right)\right)}{A V_{j}} \end{equation}\]

Step 4. Calculate the weighted sums of \(PDA\) and \(NDA\) for each decision variant using equations (1.33) and (1.34).

(1.33)\[\begin{equation} \mathrm{A} SP_{i}=\sum_{j=1}^{m} w_{j} P D A_{i j} \end{equation}\]

(1.34)\[\begin{equation} SN_{i}=\sum_{j=1}^{m} w_{j} N D A_{i j} \end{equation}\]

Step 5. Normalize the weighted sums of negative and positive distances using equations (1.35) and (1.36).

(1.35)\[\begin{equation} N S P_{i}=\frac{S P_{i}}{\max _{i}\left(S P_{i}\right)} \end{equation}\]

(1.36)\[\begin{equation} N S N_{i}=1-\frac{S N_{i}}{\max _{i}\left(S N_{i}\right)} \end{equation}\]

Step 6. Calculate the evaluation score (\(AS\)) for each alternative using the formula (1.37). A higher point value determines a higher ranking alternative.

(1.37)\[\begin{equation} A S_{i}=\frac{1}{2}\left(N S P_{i}+N S N_{i}\right) \end{equation}\]

1.1.6. ERVD

ERVD is designed to evaluate decision alternatives according to the following steps:

Step 1. Create a decision matrix.

Step 2. Define reference points \(\mu, j=1,\ldots,n\) for each decision criterion.

Step 3. Normalize the decision matrix using the sum method.

Step 4. Transform the reference points into the normalized scale:

\[\begin{equation} \varphi_j=\frac{\mu_j}{\sum_{i=1}^m d_{i j}} \end{equation}\]

Step 5. Calculate the value of alternative \(A_i\) according to criterion \(C_j\) by increasing value function (for benefit criteria):

\[\begin{split}\begin{equation} v_{i j}=\left\{\begin{array}{l} \left(r_{i j}-\varphi_j\right)^\alpha \quad \text { if } r_{i j}>\varphi_j \\ -\lambda\left(\varphi_j-r_{i j}\right)^\alpha \text { otherwise } \end{array}\right. \end{equation}\end{split}\]

and decreasing value function (for cost criteria):

\[\begin{split}\begin{equation} v_{i j}=\left\{\begin{array}{l} \left(\varphi_j-r_{i j}\right)^\alpha \quad \text { if } r_{i j}<\varphi_j \\ -\lambda\left(r_{i j}-\varphi_j\right)^\alpha \text { otherwise } \end{array}\right. \end{equation}\end{split}\]

Step 6. Determine the ideal and negative ideal solutions \(A^+\) (PIS) and \(A^-\) (NIS), respectively:

\[\begin{equation} A^{+}=\left\{v_1^{+}, \cdots v_n^{+}\right\}, A^{-}=\left\{v_1^{-}, \cdots v_n^{-}\right\} \end{equation}\]

where \(v_j^{+}=\max _i v_{i j}\) and \(v_j^{-}=\min v_{i \vec{j}}\).

Step 7. Calculate the separation measures from PIS and NIS individually with help Minkowski metric:

\[\begin{equation} S_i^{+}=\sum_{j=1}^n w_j \cdot\left|v_{i j}-v_j^{+}\right|, \text {for alternative } i, i=1 \ldots m \end{equation}\]

\[\begin{equation} S_i^{-}=\sum_{j=1}^n w_j \cdot\left|v_{i j}-v_j^{-}\right|, \text {for alternative } i, i=1 \ldots m \end{equation}\]

Step 8. Calculate the relative closeness of each alternative to the ideal solution:

\[\begin{equation} \phi_i=\frac{S_i^{-}}{S_i^{+}+S_i^{-}}, i=1, \ldots, m \end{equation}\]

1.1.7. MABAC

MABAC is designed to evaluate decision alternatives according to the following steps:

Step 1. Define a decision matrix of dimension \(n \times m\), where \(n\) is the number of alternatives, and \(m\) is the number of criteria (1.38).

(1.38)\[\begin{split}\begin{equation} x_{i j}=\left[\begin{array}{llll} x_{11} & x_{12} & \ldots & x_{1 m} \\ x_{21} & x_{22} & \ldots & x_{2 m} \\ \ldots & \ldots & \ldots & \ldots \\ x_{n 1} & x_{n 2} & \ldots & x_{n m} \end{array}\right] \end{equation}\end{split}\]

Step 2. Normalization of the decision matrix, where for criteria of type profit use equation (1.39) and for criteria of type cost use equation (1.40).

(1.39)\[\begin{equation} n_{i j}=\frac{x_{i j}- \min x_{i}}{\max x_{i}- \min x_{i}} \end{equation}\]

(1.40)\[\begin{equation} n_{i j}=\frac{x_{i j}- \max x_{i}}{\min x_{i} - \max x_{i}} \end{equation}\]

Step 3. Create a weighted matrix based on the values from the normalized matrix according to the formula (1.60).

(1.60)\[\begin{equation} v_{i j}=w_{i} \cdot\left(n_{i j}+1\right) \end{equation}\]

Step 4. Boundary approximation area (\(G\)) matrix determination. The Boundary Approximation Area (\(BAA\)) for all criteria can be determined using the formula (1.42).

(1.42)\[\begin{equation} g_{i}=\left(\prod_{j=1}^{m} v_{i j}\right)^{1 / m} \end{equation}\]

Step 5. Distance calculation of alternatives from the boundary approximation area for matrix elements (\(Q\)) by equation (\(equ:qma\)).

(1.43)\[\begin{split}\begin{equation} Q=\left[\begin{array}{cccc} v_{11}-g_{1} & v_{12}-g_{2} & \ldots & v_{1 n}-g_{n} \\ v_{21}-g_{1} & v_{22}-g_{2} & \ldots & v_{2 n}-g_{n} \\ \ldots & \ldots & \ldots & \ldots \\ v_{m 1}-g_{1} & v_{m 2}-g_{2} & \ldots & v_{m n}-g_{n} \end{array}\right]=\left[\begin{array}{cccc} q_{11} & q_{12} & \ldots & q_{1 n} \\ q_{21} & q_{22} & & q_{2 n} \\ \ldots & \ldots & \ldots & \ldots \\ q_{m 1} & q_{m 2} & \ldots & q_{m n} \end{array}\right] \end{equation}\end{split}\]

The membership of a given alternative \(A_i\) to the approximation area (\(G\), \(G^{+}\) or \(G^{-}\)) is established by (1.44).

(1.44)\[\begin{split}\begin{equation} A_{i} \in\left\{\begin{array}{lll} G^{+} & \text {if } & q_{i j}>0 \\ G & \text { if } & q_{i j}=0 \\ G^{-} & \text {if } & q_{i j}<0 \end{array}\right. \end{equation}\end{split}\]

Step 6. Ranking the alternatives according to the sum of the distances of the alternatives from the areas of approximation of the borders (1.45).

(1.45)\[\begin{equation} S_{i}=\sum_{j=1}^{n} q_{i j}, \quad j=1,2, \ldots, n, \quad i=1,2, \ldots, m \end{equation}\]

1.1.8. MAIRCA

MAIRCA is designed to evaluate decision alternatives according to the following steps:

Step 1. Define a decision matrix of dimension \(n \times m\), where \(n\) is the number of alternatives, and \(m\) is the number of criteria (1.46).

(1.46)\[\begin{split}\begin{equation} x_{i j}=\left[\begin{array}{llll} x_{11} & x_{12} & \ldots & x_{1 m} \\ x_{21} & x_{22} & \ldots & x_{2 m} \\ \ldots & \ldots & \ldots & \ldots \\ x_{n 1} & x_{n 2} & \ldots & x_{n m} \end{array}\right] \end{equation}\end{split}\]

Step 2. Determining the preference for choosing alternatives using the vector \(P_{Ai}\) using the formula (1.47).

(1.47)\[\begin{equation} P_{A i}=\frac{1}{n} ; \sum_{i=1}^{n} P_{A i}=1, i=1,2, \ldots, n \end{equation}\]

If the decision-maker is neutral in choosing an alternative, the vector \(P_{Ai}\) should have the same values (1.48).

(1.48)\[\begin{equation} P_{A 1}=P_{A 2}=\ldots=P_{A n} \end{equation}\]

Step 3. Creating a theoretical ranking matrix \(T_p\). The elements of this matrix are the multiplied priorities of alternatives by the criteria weights. The form of this matrix can be represented by the formula (1.49).

(1.49)\[\begin{split}\begin{equation} T_{p}=\left[\begin{array}{cccc} t_{p 11} & t_{p 12} & \ldots & t_{p 1 m} \\ t_{p 21} & t_{p 22} & \ldots & t_{p 2 m} \\ \ldots & \cdots & \ldots & \ldots \\ t_{p n 1} & t_{p n 2} & \ldots & t_{p n m} \end{array}\right] = \left[\begin{array}{cccc} P_{A 1} \cdot w_{1} & P_{A 1} \cdot w_{2} & \ldots & P_{A 1} \cdot w_{m} \\ P_{A 2} \cdot w_{1} & P_{A 2} \cdot w_{2} & \ldots & P_{A m} \cdot w_{m} \\ \ldots & \ldots & \ldots & \ldots \\ P_{A n} \cdot w_{1} & P_{A n} \cdot w_{2} & \ldots & P_{A n} \cdot w_{m} \end{array}\right] \end{equation}\end{split}\]

When the preferences determined for the alternatives by the decision-maker are equal, the theoretical ranking matrix is represented by a theoretical ranking vector using the formula (1.50).

(1.50)\[\begin{equation} T_p = \left[\begin{array}{cccc} t_{p 11} & t_{p 12} & \ldots & t_{p 1 n} \end{array}\right]= \left[\begin{array}{llll} p_{A 1} . w_{1} & p_{A 1} \cdot w_{2} & \ldots & p_{A 1} \cdot w_{n} \end{array}\right] \end{equation}\]

Step 4. Create the real rating matrix, which is shown by the formula (1.51).

(1.51)\[\begin{split}\begin{equation} T_r = \left[\begin{array}{cccc} t_{r 11} & t_{r 12} & \ldots & t_{r 1 m} \\ t_{r 21} & t_{r 22} & \ldots & t_{r 2 m} \\ \ldots & \ldots & \ldots & \ldots \\ t_{r n 1} & t_{r n 2} & \ldots & t_{r n m} \end{array}\right] \end{equation}\end{split}\]

The values of the real rating matrix are determined depending on the criterion of profit type or cost type, sequentially according to the formulas (1.52) and (1.53).

(1.52)\[\begin{equation} \label{equ:trpr} t_{r i j}=t_{p i j} \cdot\left(\frac{x_{i j}-\min x_{j}}{\max x_{j}-\min x_{j}}\right) \end{equation}\]

(1.53)\[\begin{equation} \label{equ:trcs} t_{r i j}=t_{p i j} \cdot\left(\frac{x_{i j}-\max x_{j}}{\min x_{j}-\max x_{j}}\right) \end{equation}\]

Step 5. Calculating the total gap matrix (\(G\)) by taking the difference between the theoretical grade matrix (\(Tp\)) and the actual grade matrix (\(Tr\)) using the formula (1.54).

(1.54)\[\begin{split}\begin{equation} G=T_{p}-T_{r}= \left[\begin{array}{cccc} t_{p 11}-t_{r 11} & t_{p 12}-t_{r 12} & \ldots & t_{p 1 m}-t_{r 1 m} \\ t_{p 21}-t_{r 21} & t_{p 21}-t_{r 21} & \ldots & t_{p 2 m}-t_{r 2 m} \\ \ldots & \ldots & \ldots & \ldots \\ t_{p n 1}-t_{r n 1} & t_{p n 2}-t_{r n 2} & \ldots & t_{p n m}-t_{r n m} \end{array}\right] \end{equation}\end{split}\]

Step 6. Calculating the final values of the criterion functions (\(Q_i\)) for the alternatives using the sum of the rows of the gap matrix (\(G\)) using the formula (1.55). The alternative with the lowest value of \(Q_i\) has the highest ranking.

(1.55)\[\begin{equation} \label{equ:qima} Q_{i}=\sum_{j=1}^{m} g_{i j} \quad i=1,2, \ldots, n \end{equation}\]

1.1.9. MARCOS

MARCOS is designed to evaluate decision alternatives according to the following steps:

Step 1. Based on the decision matrix, create an augmented decision matrix with the ideal solution (AI) defined in the last row and the anti-ideal (AAI) solution defined in the first row. This can be represented by the Equation (1.56).

(1.56)\[\begin{equation} M = \left[\begin{array}{cccc} x_{aa1} & x_{aa2} & \dots & x_{aan} \cr x_{11} & x_{12} & \dots & x_{1n} \cr x_{21} & x_{22} & \dots & x_{2n} \cr \dots & \dots & \dots & \dots \cr x_{m1} & x_{m2} & \dots & x_{mn} \cr x_{ai1} & x_{ai2} & \dots & x_{ain} \end{array}\right] \end{equation}\]

The ideal and anti-ideal solution values for the cost (C) and benefit (B) criteria are defined as follows:

(1.57)\[\begin{split}\begin{equation} AAI = \left \{ \begin{array}{cc} \min_i x_{ij} & if \quad j \in B\\ \max_i x_{ij} & if \quad j \in C \end{array} \right . \end{equation}\end{split}\]

(1.58)\[\begin{split}\begin{equation} AI = \left \{ \begin{array}{cc} \max_i x_{ij} & if \quad j \in B\\ \min_i x_{ij} & if \quad j \in C \end{array} \right . \end{equation}\end{split}\]

Step 2. Normalization of the extended decision matrix using Equation (1.59).

(1.59)\[\begin{split}\begin{equation} n_{ij} = \left \{ \begin{array}{cc} \frac{x_{ai}}{x_{ij}} & if \quad j \in C \\ \frac{x_{x_{ij}}}{x_{ai}} & if \quad j \in B \end{array}\right . \end{equation}\end{split}\]

Step 3. Create a weighted matrix based on the values from the normalized extended matrix according to the formula (1.60).

(1.60)\[\begin{equation} v_{i j}=w_{i} \cdot\left(n_{i j}+1\right) \end{equation}\]

Step 4. Calculating the degrees of utility of alternatives Ki relative to the ideal and anti-ideal solution using Equations (1.61), (1.62).

(1.61)\[\begin{equation} K_{i}^{+} = \frac{S_i}{S_{ai}} \end{equation}\]

(1.62)\[\begin{equation} K_{i}^{-} = \frac{S_i}{S_{aai}} \end{equation}\]

where \(S_i\) \((i=1,2,\dots,m)\) represents the sum of the elements of weighted matrix \(V\), Equation (1.63).

(1.63)\[\begin{equation} S_i = \sum_{i=1}^n v_{ij} \end{equation}\]

Step 5. Determination of the utility function for the decision options considered according to (1.64).

(1.64)\[\begin{equation} f\left(K_{i}\right)=\frac{K_{i}^{+}+K_{i}^{-}}{1+\frac{1-f\left(K_{i}^{+}\right)}{f\left(K_{i}^{+}\right)}+\frac{1-f\left(K_{i}^{-}\right)}{f\left(K_{i}^{-}\right)}} \end{equation}\]

where \(f(K_{i}^{-})\) denotes the utility function relative to the anti-ideal solution, while \(f(K_{i}^{+})\) denotes the utility function relative to the ideal solution, which can be determined using the Equations respectively (1.65) and (1.66).

(1.65)\[\begin{equation} f\left(K_{i}^{-}\right)=\frac{K_{i}^{+}}{K_{i}^{+}+K_{i}^{-}} \end{equation}\]

(1.66)\[\begin{equation} f\left(K_{i}^{+}\right)=\frac{K_{i}^{-}}{K_{i}^{+}+K_{i}^{-}} \end{equation}\]

1.1.10. MOORA

MOORA is designed to evaluate decision alternatives according to the following steps:

Step 1. Normalize the decision matrix based on the Equation (1.67).

(1.67)\[\begin{equation} r_{ij} = \frac{x_{ij}}{\sqrt{\sum_{i=1}^{m}{x_{ij}^2}}} \end{equation}\]

where \(x_{ij}\) can be called the value of the \(i-th\) alternative for the \(j-th\) criterion.

Step 2. Determine weighted normalized decision matrix based on Equation (1.68).

(1.68)\[\begin{equation} v_{ij} = r_{ij} w_{j} \end{equation}\]

where \(w_j\) can be called the weight for \(j-th\) criterion.

Step 3. Calculate the value of \(y_i\) based on the values from the normalized weighted decision matrix \(v_{ij}\) by using Equation (1.69).

(1.69)\[\begin{equation} y_i = \sum_{j=1}^g v_{ij} - \sum_{j=g+1}^n v_{ij} \end{equation}\]

where type of beneficial and cost criteria are represented as follows \(j = 1, 2, \dots, g\) and \(j = g + 1, g + 2,\dots,n\).

1.1.11. OCRA

OCRA is designed to evaluate decision alternatives according to the following steps:

Step 1. Create a decision matrix.

Step 2. Normalize the decision matrix using the min-max method.

Step 3. Create normalized weighted decision matrix.

Step 4. Determination of preferences for cost-type and profit-type criteria sequentially according to the Equations (1.70),(1.71).

(1.70)\[\begin{equation} \overline{\bar{I}}_{i}=\bar{I}_{i}-\min \left(\bar{I}_{i}\right) \end{equation}\]

(1.71)\[\begin{equation} \overline{\overline{O_{i}}}={\overline{O_{i}}}^{-} \min \left(\bar{O}_{i}\right) \end{equation}\]

where \(\bar{I}_{i}\) is a measure of relative performance for the \(i-th\) alternative and cost-type criteria, and \(\bar{O}_{i}\) is a measure of of relative performance for the \(i-th\) alternative and profit-type criteria.

Step 5. Determine the overall preference of the considered alternatives using the Equation (1.72).

(1.72)\[\begin{equation} P_i = \overline{\bar{I}}_{i} + \overline{\overline{O_{i}}} - \min \left ( \overline{\bar{I}}_{i} + \overline{\overline{O_{i}}}\right ) \end{equation}\]

1.1.12. PROBID

PROBID is designed to evaluate decision alternatives according to the following steps:

Step 1. Create a decision matrix.

Step 2. Normalize the decision matrix using the vector method.

Step 3. Create normalized weighted decision matrix.

Step 4. Sort the normalized weighted decision matrix by criteria taking into account their type. This will a matrix of successively Positive Ideal Solutions (1st, 2cd, …, mth PIS) will be formed. It can be presented by using the following formula:

\[\begin{split}\begin{equation} \begin{aligned} A_{(k)} & =\left\{\left(\operatorname{Large}\left(v_j, k\right) \mid j \in J\right),\left(\operatorname{Small}\left(v_j, k\right) \mid j \in J^{\prime}\right)\right\} \\ & =\left\{v_{(k) 1}, v_{(k) 2}, v_{(k) 3}, \ldots, v_{(k) j}, \ldots, v_{(k) n}\right\} \end{aligned} \end{equation}\end{split}\]

where \(k \in \{1,2, \ldots, m\}\), \(J\) is the set of benefit criteria and \(J^{\prime}\) is the set of cost criteria.

Then, find the average value of each objective column as follow:

\[\begin{equation} \bar{v}_j=\frac{\sum_{k=1}^m v_{(k) j}}{m} \quad \text { for } j \in\{1,2, \ldots, n\} \end{equation}\]

The average solution is then given by

\[\begin{equation} \bar{A}=\left\{\bar{v}_1, \bar{v}_2, \bar{v}_3, \ldots, \bar{v}_j, \ldots, \bar{v}_n\right\} \end{equation}\]

Step 5. Iteratively calculate the Euclidean distance of each solution to each of the m ideal solutions as well as to the average solution. The distance to ideal solutions is found as:

\[\begin{equation} S_{i(k)}=\sqrt{\sum_{j=1}^n\left(v_{i j}-v_{(k) j}\right)^2} \end{equation}\]

Step 6. Determine the overall positive-ideal distance and negative-ideal distance as follow:

\[\begin{split}\begin{equation} S_{i(\text { pos-ideal })}=\left\{\begin{array}{l} \sum_{k=1}^{(m+1) / 2} \frac{1}{k} S_{i(k)} \quad i \in\{1,2, \ldots, m\} \text { when } m \\ \text { is an odd number } \\ \sum_{k=1}^{m / 2} \frac{1}{k} S_{i(k)} \quad i \in\{1,2, \ldots, m\} \text { when } m \\ \text { is an even number } \end{array}\right. \end{equation}\end{split}\]

\[\begin{split}\begin{equation} S_{i(\text { neg-ideal })} \quad=\left\{\begin{array}{l} \sum_{k=(m+1) / 2}^m \frac{1}{m-k+1} S_{i(k)} \\ i \in\{1,2, \ldots, m\} \text { when } m \text { is an odd number } \\ \sum_{k=m / 2+1}^m \frac{1}{m-k+1} S_{i(k)} \\ i \in\{1,2, \ldots, m\} \text { when } m \text { is an even number } \end{array}\right. \end{equation}\end{split}\]

Positive-ideal distance and negative-ideal distance for sPROBID:

\[\begin{split}\begin{equation} S_{i(\text { pos-ideal })}=\left\{\begin{array}{l} \sum_{k=1}^{m \backslash 4} \frac{1}{k} S_{i(k)} \quad i \in\{1,2, \ldots, m\} \text { when } m \geq 4 \\ S_{i(1)} \quad i \in\{1,2, \ldots, m\} \text { when } 0<m<4 \end{array}\right. \end{equation}\end{split}\]

\[\begin{split}\begin{equation} S_{i(\text { neg-ideal })}=\left\{\begin{array}{l} \sum_{k=m+1-(m \ 4)}^m \frac{1}{m-k+1} S_{i(k)} \\ \quad i \in\{1,2, \ldots, m\} \text { when } m \geq 4 \\ S_{i(m)} \quad i \in\{1,2, \ldots, m\} \text { when } 0<m<4 \end{array}\right. \end{equation}\end{split}\]

Step 7. Calculate the pos-ideal/neg-ideal ratio (\(R_i\)) and then the performance score (\(P_i\)) of each solution as follows:

\[\begin{equation} R_i=\frac{S_{i(\text { pos-ideal })}}{S_{i(\text { neg-ideal })}} \end{equation}\]

\[\begin{equation} P_i=\frac{1}{1+R_i^2}+S_{i(\mathrm{avg})} \end{equation}\]

1.1.13. RIM

RIM is designed to evaluate decision alternatives according to the following steps:

Step 1. Define the following values, which determine the problem’s context and the problem itself.

• Criteria weights: \(w_j, j \in \{1, 2, \ldots N\}\) and the sum of the criteria weights should be equal to one: \(\sum^N_{j = 1} w_j = 1\).

• Decision matrix: \(X = [ x_{ij} ]_{M \times N}\) which contains information about \(M\) alternatives evaluated under \(N\) criteria.

• The Criteria Range: \(t_j = [t_{j}^{(min)}, t_{j}^{(max)}]\), \(j \in \{1, 2, \ldots N\}\) which defines the arbitrary chosen bounds of the criteria.

• The Reference Ideal: \(s_j = [s_{j}^{(min)}, s_{j}^{(max)}]\), \(j \in \{1, 2, \ldots N\}$ and $[s_{j}^{(min)}, s_{j}^{(max)}] \subset [t_{j}^{(min)}, t_{j}^{(max)}]\). Reference Ideal define most preferred interval of values for each criterion. It can be either derived from criteria range, or define expected outcome of decision process.

Step 2. After defining the problem we should normalize the decision matrix \(X\) using the RIM normalization function \(f(x,[A, B],[C, D])\) defined as (1.73). This normalization requires a definition of the criteria range \([A. B]\) and the reference ideal \([C, D]\).

(1.73)\[\begin{split}\begin{equation} f(\ldots) = \left\{\begin{array}{lll} 1 &\textit{IF}& x \in[C, D] \\ 1-\frac{d_{\min }(x,[C, D])}{|A-C|} &\textit{IF}& x \in[A, C] \wedge A \neq C \\ 1-\frac{d_{\min }(x,[C, D])}{|D-B|} &\textit{IF}& x \in[D, B] \wedge D \neq B \end{array}\right., \end{equation}\end{split}\]

where \([A, B]\) is range of criteria, \([C, D]\) is the reference ideal, and \(x \in [A, B]\), \([C, D] \subset [A, B]\). Function \(d_{min}(x, [C, D])\) is defined as (1.74).

(1.74)\[\begin{equation} d_{min}(x, [C, D]) = min(|x - C|, |x - D|) \end{equation}\]

This normalization allows to map value \(x\) to range \([0, 1]\) in the criteria domain with regard to the ideal reference interval. The normalization process is defined as follows (1.75).

(1.75)\[\begin{equation} Y = [ y_{ij} ]_{M \times N} = [ f(x_{ij}, t_j, s_j) ]_{M \times N} \end{equation}\]

Step 3. Calculate the weighted normalized matrix \(Y^\prime\) using (1.76).

(1.76)\[\begin{equation} Y^\prime = Y \otimes W = [ y_{ij} \cdot w_{j} ]_{M \times N} \end{equation}\]

Step 4. Compute the variation to the normalized reference ideal for each alternative \(A_i\) using Equations (1.77) and (1.78).

(1.77)\[\begin{equation} I_i^{+}=\sqrt{\sum_{j=1}^n\left(y^{\prime}{ }_{i j}-w_j\right)^2}% \quad i \in \{1, 2, \ldots M\}, \quad j \in \{1, 2, \ldots N\} \end{equation}\]

(1.78)\[\begin{equation} I_i^{-}=\sqrt{\sum_{j=1}^n\left(y^{\prime}\right)^2}% \quad i \in \{1, 2, \ldots M\}, \quad j \in \{1, 2, \ldots N\} \end{equation}\]

Step 5. Calculate the relative index of each alternative \(A_i\), using the Equation (1.79).

(1.79)\[\begin{equation} R_i = \frac{I_i^-}{I_i^+ + I_i^-} %\quad i \in \{1, 2, \ldots M\} \end{equation}\]

Order the alternative \(A_i\) in descending order with regard to \(R_i\). The alternatives with the bigger value of \(R_i\) are more preferred ones.

1.1.14. SPOTIS

SPOTIS is designed to evaluate decision alternatives according to the following steps:

Step 1. Define the bounds of the problem - min and max bounds of classical MCDM problem must be defined to transform MCDM problem form ill-defined to well-defined.

\[\begin{equation} \left[S_{n}^{\min }, S_{n}^{\max }\right]=\left[x_{1}, x_{2}\right] \end{equation}\]

where, \(n\) - criterion number, \(x_1\) - min bound, \(x_2\) - max bound.

Step 2. Define the ideal solution point - define vector which includes maximum or minimum from bounds for specific criterion depending on criterion type. For profit type, the max value should be taken, for cost type, min value.

\[\begin{equation} S^{\star}=\left(S_{1}^{\star}, S_{2}^{\star}, S_{3}^{\star}\right) \end{equation}\]

Step 3. Compute normalized distance matrix - for each alternative \(A_{i}\) (i= 1, \(\ldots\) , M),compute its normalized distance with respect to ideal solution for each criteria \(C_{j}\) (j= 1, \(\ldots\) , N ).

\[\begin{equation} d_{i j}=\frac{\left|A_{i j}-S_{j}^{*}\right|}{\left|S_{j}^{\max }-S_{j}^{\min }\right|} \end{equation}\]

Step 4. Compute normalized averaged distance - for each criteria \(C_{j}\) (j= 1, \(\ldots\) , N ) take into account its weight and calculate final preference by executing following Equation.

\[\begin{equation} \bar{p}_{j}=\sum_{j=1}^{N} w_{j} d_{i j} \end{equation}\]

1.1.15. TOPSIS

TOPSIS is designed to evaluate decision alternatives according to the following steps:

Step 1. Normalize the decision matrix by using min-max normalization. The values of benefit type criteria are normalized using the (1.80) formula, while the values of cost type criteria are normalized using the (1.81) formula.

(1.80)\[\begin{equation} r_{ij} = \frac{x_{ij} - \min(x_j)}{\max(x_j) - \min(x_j)}\ \end{equation}\]

(1.81)\[\begin{equation} r_{ij} = \frac{\max(x_j) - x_{ij}}{\max(x_j) - \min(x_j)} \end{equation}\]

Step 2. Building a decision matrix \(v_{ij}\) subjected to a weighting and normalization process using the Equation (1.82).

(1.82)\[\begin{equation} v_{ij} = w_{j}r_{ij} \end{equation}\]

Step 3. Derive a positive ideal solution \(PIS\) and a negative ideal solution \(NIS\). The ideal positive solution is calculated as the maximum value for each criterion (1.83), while the ideal negative solution is calculated as the least value for each criterion (1.84).

(1.83)\[\begin{equation} v_{j}^{+} = \{v_{1}^{+}, v_{2}^{+}, \dots, v_{n}^{+} \} = \{\max_{j}(v_{ij}) \} \end{equation}\]

(1.84)\[\begin{equation} v_{j}^{-} = \{v_{1}^{-}, v_{2}^{-}, \dots, v_{n}^{-} \}= \{\min_{j}(v_{ij}) \} \end{equation}\]

Step 4. Determine the Euclidean distance for each normalized weighted alternative from the \(PIS\) (1.85) and \(NIS\) (1.86) solution.

(1.85)\[\begin{equation} D_{i}^{+} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{+})^{2}} \end{equation}\]

(1.86)\[\begin{equation} D_{i}^{-} = \sqrt{\sum_{j=1}^{n}(v_{ij}-v_{j}^{-})^{2}} \end{equation}\]

1.1.16. VIKOR

VIKOR is designed to evaluate decision alternatives according to the following steps:

Step 1. Determinate the best \(f_{j}^{*}\) and the worst \(f_{j}^{-}\) value for the function of a particular criterion. For profit criteria, the Equation is used (1.87).

(1.87)\[\begin{equation} f_{j}^{*} = \max_i f_{ij},\; \; \; f_{j}^{-} = \min_i f_{ij} \end{equation}\]

where in the case of the cost criteria, the following Equation is used (1.88).

(1.88)\[\begin{equation} f_{j}^{*} = \min_i f_{ij},\; \; \; f_{j}^{-} = \max_i f_{ij} \end{equation}\]

Step 2. Calculate \(S_{i}\) and \(R_{i}\) with using Equations (1.89) and (1.90).

(1.89)\[\begin{equation} S_{i} = \sum_{j=1}^{n}w_{j}(f_{j}^{*}-f_{ij})/(f_{j}^{*}-f_{j}^{-}) \end{equation}\]

(1.90)\[\begin{equation} R_{i} = \max_j \left [ w_{j}(f_{j}^{*}-f_{ij})/(f_{j}^{*}-f_{j}^{-}) \right ] \end{equation}\]

Step 3. Calculate \(Q_{i}\) with using Equation (1.91).

(1.91)\[\begin{equation} Q_{i} = v(S_{i}-S^{*}) / (S^{-}-S^{*}) + (1 - v)(R_{i}-R^{*}) / (R^{-}-R^{*}) \label{VikorQi} \end{equation}\]

where:

\(S^{*} = min_{i} S_{i},\; \; \; S^{-} = max_{i} S_{i}\),

\(R^{*} = min_{i} R_{i},\; \; \; R^{-} = max_{i} R_{i}\),

\(v\) means the weight adopted for the strategy of ‘’most criteria’’.

Step 4. Ranked alternatives \(S\), \(R\) and \(Q\) are ordered in ascending order. Three ranked lists are the outcome.

Step 5. A compromise solution is proposed considering the conditions of good advantage and acceptable stability within the three vectors obtained in the previous step. The best alternative is the one with the lowest value and the leading position in the ranking \(Q\).

1.1.17. WASPAS

WASPAS is designed to evaluate decision alternatives according to the following steps:

Step 1. Create a decision matrix.

Step 2. Normalize the decision matrix using the max method.

Step 3. Calculate WSM and WPM as follow:

\[\begin{equation} W S M=\sum_{j=1}^n \bar{x}_{i j} w_j \end{equation}\]

\[\begin{equation} W P M=\prod_{j=1}^n\left(\bar{x}_{i j}\right)^{w_j} \end{equation}\]

where \(w_j\) denote the weights for the criteria, and \(x_ij\) denote the values of the decision options from the normalized decision matrix.

Step 4. Calculation of total relative importance for each alternative as follow:

\[\begin{equation} Q_i=\lambda WSM+(1-\lambda) WPM=\lambda \sum_{j=1}^n \bar{x}_{i j} w_j+(1-\lambda) \prod_{j=1}^n\left(\bar{x}_{i j}\right)^{w_j} \end{equation}\]

1.1.18. WPM

WPM is designed to evaluate decision alternatives according to the following steps:

Step 1. Create a decision matrix.

Step 2. Normalize the decision matrix using the sum method.

Step 3. Calculate WPM as follow:

\[\begin{equation} W P M=\prod_{j=1}^n\left(\bar{x}_{i j}\right)^{w_j} \end{equation}\]

where \(w_j\) denote the weights for the criteria, and \(x_ij\) denote the values of the decision options from the normalized decision matrix.

1.1.19. WSM

WSM is designed to evaluate decision alternatives according to the following steps:

Step 1. Create a decision matrix.

Step 2. Normalize the decision matrix using the sum method.

Step 3. Calculate WSM as follow:

\[\begin{equation} W S M=\sum_{j=1}^n \bar{x}_{i j} w_j \end{equation}\]

where \(w_j\) denote the weights for the criteria, and \(x_ij\) denote the values of the decision options from the normalized decision matrix.