## 1 Introduction

*et al.*, 2012; Abdullah and Adawiyah, 2014; Stefano

*et al.*, 2015; Gul

*et al.*, 2016; Mardani

*et al.*, 2017; Keshavarz Ghorabaee

*et al.*, 2017b). In the following, we present brief explanations of the SAW (Simple Additive Weighting), WASPAS (Weighted Aggregated Sum Product ASsessment), COPRAS (COmplex PRoportional ASsessment), TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution), VIKOR (from Serbian: VIseKriterijumska Optimizacija I Kompromisno Resenje) and EDAS (Evaluation based on Distance from Average Solution) methods which are used for the evaluation of alternatives. The range of the MCDM methods is wider than the mentioned methods. However, these methods are reviewed because we use them in the current study.

*et al.*(2016a), Goodridge (2016), Kalibatas and Kovaitis (2017) and Muddineni

*et al.*(2017), are some of the recent studies on the SAW method. Abdullah and Adawiyah (2014) presented a review of the applications of simple additive weighting and its fuzzy variants from 2003 to 2013.

*et al.*(2012) proposed the WASPAS method based on a combination of the SAW or weighted sum model (WSM) and weighted product model (WPM). This method has the advantages of both of them. Many researchers have utilized this efficient method in their studies. In some recent studies, Bauðys and Juodagalvienë (2017), Karabasevic

*et al.*(2017), Nie

*et al.*(2017), Peng and Dai (2017) applied the WASPAS method to the MCDM problems. Mardani

*et al.*(2017) presented a systematic review of applications and fuzzy developments of the WASPAS method.

*et al.*, 1994). We should define a set of beneficial and a set of non-beneficial criteria to use this method and determine the performance values of alternatives. This method has also been applied to many real-world MCDM problems. Wang

*et al.*(2016b), Mulliner

*et al.*(2016), Serrai

*et al.*(2017), Rathi and Balamohan (2017), Turanoglu Bekar

*et al.*(2016) and Nakhaei

*et al.*(2016) have done some studies on this method recently. A review of this method and its applications was presented by Stefano

*et al.*(2015).

*et al.*(2018), Cayir Ervural

*et al.*(2018), Chen

*et al.*(2018), Shen

*et al.*(2018), Polat

*et al.*(2017) and Walczak and Rutkowska (2017). A review of developments and applications of this method was made by Behzadian

*et al.*(2012).

*et al.*, 2018), maritime transportation (Soner

*et al.*, 2017), energy management (Sakthivel

*et al.*, 2017) and financial performance evaluation (Chang and Tsai, 2016). Gul

*et al.*(2016) conducted a state-of-the-art literature review on VIKOR applications.

*et al.*(2015). The EDAS method measures the desirability of alternatives based on the distance from an average solution. Like some other MCDM methods, EDAS can be used to handle multi-criteria decision-making problems with non-commensurable and conflicting criteria. Despite the newness of the EDAS method, it has been applied to many practical MCDM problems like supplier selection (Keshavarz Ghorabaee

*et al.*, 2016b; Stević

*et al.*, 2017), life cycle sustainability assessment (Ren and Toniolo, 2018) and evaluation of architectural plans (Juodagalvienë

*et al.*, 2017).

*et al.*, 2016a; Zhang

*et al.*, 2011; Liu

*et al.*, 2010). CRITIC (CRiteria Importance Through Inter-criteria Correlation) is another method which considers within- and between-criterion variation information (Diakoulaki

*et al.*, 1995). The correlation between criteria is used to measure variations between criteria. In this method, lower values of correlation have a positive effect on the weight of each criterion. Some researchers applied CRITIC to practical MCDM problems (Yalçı and Ünlü, 2017; Keshavarz Ghorabaee

*et al.*, 2017a; Adalı and Işık, 2017; Rostamzadeh

*et al.*, 2018).

## 2 Simultaneous Evaluation of Criteria and Alternatives (SECA)

*n*alternatives and

*m*criteria, and the weight of each criterion (${w_{j}}$, $j\in \{1,2,\dots ,m\}$) is unknown. We can define the decision-matrix of this problem as follows:

##### (1)

\[ X=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{x_{11}}\hspace{1em}& {x_{12}}\hspace{1em}& \cdots \hspace{1em}& {x_{1j}}\hspace{1em}& \cdots \hspace{1em}& {x_{1m}}\\ {} {x_{21}}\hspace{1em}& {x_{22}}\hspace{1em}& \cdots \hspace{1em}& {x_{2j}}\hspace{1em}& \cdots \hspace{1em}& {x_{2m}}\\ {} \vdots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \\ {} {x_{i1}}\hspace{1em}& {x_{i2}}\hspace{1em}& \cdots \hspace{1em}& {x_{ij}}\hspace{1em}& \cdots \hspace{1em}& {x_{im}}\\ {} \vdots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \\ {} {x_{n1}}\hspace{1em}& {x_{n2}}\hspace{1em}& \cdots \hspace{1em}& {x_{nj}}\hspace{1em}& \cdots \hspace{1em}& {x_{nm}}\end{array}\right],\]*i*th ($i\in \{1,2,\dots ,n\}$) alternative on

*j*th ($j\in \{1,2,\dots ,m\}$) criterion and ${x_{ij}}>0$.

##### (2)

\[ {X^{N}}=\left[\begin{array}{c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c@{\hskip4.0pt}c}{x_{11}^{N}}\hspace{1em}& {x_{12}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{1j}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{1m}^{N}}\\ {} {x_{21}^{N}}\hspace{1em}& {x_{22}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{2j}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{2m}^{N}}\\ {} \vdots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \\ {} {x_{i1}^{N}}\hspace{1em}& {x_{i2}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{ij}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{im}^{N}}\\ {} \vdots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \hspace{1em}& \ddots \hspace{1em}& \vdots \\ {} {x_{n1}^{N}}\hspace{1em}& {x_{n2}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{nj}^{N}}\hspace{1em}& \cdots \hspace{1em}& {x_{nm}^{N}}\end{array}\right],\]##### (3)

\[ {x_{ij}^{N}}=\left\{\begin{array}{l@{\hskip4.0pt}l}\frac{{x_{ij}}}{{\max _{k}}{x_{kj}}}\hspace{1em}& \text{if}\hspace{2.5pt}j\in \mathit{BC},\\ {} \frac{{\min _{k}}{x_{kj}}}{{x_{ij}}}\hspace{1em}& \text{if}\hspace{2.5pt}j\in \mathit{NC},\end{array}\right.\]*j*th ($j\in \{1,2,\dots ,m\}$) criterion. The standard deviation of the elements of each vector (${\sigma _{j}}$) can get the within-criterion variation information. To capture the between-criterion variation information from the decision-matrix, we need to calculate the correlation between each pair of vectors of criteria. Let us denote by ${r_{jl}}$ the correlation between

*j*th and

*l*th vectors (

*j*and $l\in \{1,2,\dots ,m\}$). Then the following summation (${\pi _{j}}$) can reflect the degree of conflict between

*j*th criterion and the other criteria (Diakoulaki

*et al.*, 1995):

##### (7.1)

\[\begin{aligned}{}\max & {S_{i}}={\sum \limits_{j=1}^{m}}{w_{j}}{x_{ij}^{N}},\hspace{1em}\forall i\in \{1,2,\dots ,n\},\end{aligned}\]##### (7.2)

\[\begin{aligned}{}\min & {\lambda _{b}}={\sum \limits_{j=1}^{m}}{\big({w_{j}}-{\sigma _{j}^{N}}\big)^{2}},\end{aligned}\]##### (7.3)

\[\begin{aligned}{}\min & {\lambda _{c}}={\sum \limits_{j=1}^{m}}{\big({w_{j}}-{\pi _{j}^{N}}\big)^{2}},\end{aligned}\]*ε*is a small positive parameter considered as a lower bound for criteria weights. In this study, this parameter is set to 10${^{-3}}$.

##### (8.1)

\[\begin{aligned}{}\max & Z={\lambda _{a}}-\beta ({\lambda _{b}}+{\lambda _{c}}),\end{aligned}\]##### (8.2)

\[\begin{aligned}{}\text{s.t.}& {\lambda _{a}}\leqslant {S_{i}},\hspace{1em}\forall i\in \{1,2,\dots ,n\},\end{aligned}\]##### (8.3)

\[\begin{aligned}{}& {S_{i}}={\sum \limits_{j=1}^{m}}{w_{j}}{x_{ij}^{N}},\hspace{1em}\forall i\in \{1,2,\dots ,n\},\end{aligned}\]##### (8.4)

\[\begin{aligned}{}& {\lambda _{b}}={\sum \limits_{j=1}^{m}}{\big({w_{j}}-{\sigma _{j}^{N}}\big)^{2}},\end{aligned}\]##### (8.5)

\[\begin{aligned}{}& {\lambda _{c}}={\sum \limits_{j=1}^{m}}{\big({w_{j}}-{\sigma _{j}^{N}}\big)^{2}},\end{aligned}\]*β*($\beta \geqslant 0$). This coefficient affects the importance of reaching the reference points of criteria weights.

## 3 Computational Analysis

##### Table 1

${C_{1}}\in \mathit{BC}$ | ${C_{2}}\in \mathit{BC}$ | ${C_{3}}\in \mathit{BC}$ | ${C_{4}}\in \mathit{NC}$ | ${C_{5}}\in \mathit{NC}$ | ${C_{6}}\in \mathit{NC}$ | ${C_{7}}\in \mathit{NC}$ | |

${A_{1}}$ | 23 | 264 | 2.37 | 0.05 | 167 | 8900 | 8.71 |

${A_{2}}$ | 20 | 220 | 2.2 | 0.04 | 171 | 9100 | 8.23 |

${A_{3}}$ | 17 | 231 | 1.98 | 0.15 | 192 | 10800 | 9.91 |

${A_{4}}$ | 12 | 210 | 1.73 | 0.2 | 195 | 12300 | 10.21 |

${A_{5}}$ | 15 | 243 | 2 | 0.14 | 187 | 12600 | 9.34 |

${A_{6}}$ | 14 | 222 | 1.89 | 0.13 | 180 | 13200 | 9.22 |

${A_{7}}$ | 21 | 262 | 2.43 | 0.06 | 160 | 10300 | 8.93 |

${A_{8}}$ | 20 | 256 | 2.6 | 0.07 | 163 | 11400 | 8.44 |

${A_{9}}$ | 19 | 266 | 2.1 | 0.06 | 157 | 11200 | 9.04 |

${A_{10}}$ | 8 | 218 | 1.94 | 0.11 | 190 | 13400 | 10.11 |

*et al.*(2015). The decision-matrix of the example along with the type of each criterion is shown in Table 1, and the normalized decision-matrix, which is constructed using Eqs. (2) and (3), is represented in Table 2. As can be seen in these tables, there are ten alternatives (${A_{1}}$ to ${A_{10}}$) which need to be evaluated with respect to seven criteria (${C_{1}}$ to ${C_{7}}$).

##### Table 2

${C_{1}}$ | ${C_{2}}$ | ${C_{3}}$ | ${C_{4}}$ | ${C_{5}}$ | ${C_{6}}$ | ${C_{7}}$ | |

${A_{1}}$ | 1 | 0.9925 | 0.9115 | 0.8 | 0.9401 | 1 | 0.9449 |

${A_{2}}$ | 0.8696 | 0.8271 | 0.8462 | 1 | 0.9181 | 0.9780 | 1 |

${A_{3}}$ | 0.7391 | 0.8684 | 0.7615 | 0.2667 | 0.8177 | 0.8241 | 0.8305 |

${A_{4}}$ | 0.5217 | 0.7895 | 0.6654 | 0.2000 | 0.8051 | 0.7236 | 0.8061 |

${A_{5}}$ | 0.6522 | 0.9135 | 0.7692 | 0.2857 | 0.8396 | 0.7063 | 0.8812 |

${A_{6}}$ | 0.6087 | 0.8346 | 0.7269 | 0.3077 | 0.8722 | 0.6742 | 0.8926 |

${A_{7}}$ | 0.9130 | 0.9850 | 0.9346 | 0.6667 | 0.9813 | 0.8641 | 0.9216 |

${A_{8}}$ | 0.8696 | 0.9624 | 1 | 0.5714 | 0.9632 | 0.7807 | 0.9751 |

${A_{9}}$ | 0.8261 | 1 | 0.8077 | 0.6667 | 1 | 0.7946 | 0.9104 |

${A_{10}}$ | 0.3478 | 0.8195 | 0.7462 | 0.3636 | 0.8263 | 0.6642 | 0.8140 |

### 3.1 Analysis of Criteria Weights

*β*($\beta =0.1$, 0.2, 0.3, 0.4, 0.5, 1, 2, 3, 4 and 5). By solving the model, 10 sets of criteria weights are determined. The weights of criteria related to changing the values of

*β*are presented in Table 3, and the variation of the weights is shown in Fig. 1.

*β*parameter are greater than 3 ($\beta \geqslant 3$). Now, to validate the results, the weights of criteria are determined using some other methods. Here three methods including the SD, CRITIC and Entropy methods are chosen for the comparison analysis (Zardari

*et al.*, 2014). It should be noted that the normalization step in the CRITIC method is skipped because we use the normalized decision-matrix (Table 2) for computations. The weights of criteria determined by these methods are presented in Table 4.

##### Table 3

*β*.

β | ||||||||||

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 1 | 2 | 3 | 4 | 5 | |

${w_{1}}$ | 0.001 | 0.001 | 0.035 | 0.068 | 0.081 | 0.133 | 0.148 | 0.153 | 0.156 | 0.158 |

${w_{2}}$ | 0.284 | 0.253 | 0.233 | 0.221 | 0.213 | 0.175 | 0.157 | 0.151 | 0.148 | 0.146 |

${w_{3}}$ | 0.001 | 0.083 | 0.115 | 0.131 | 0.142 | 0.129 | 0.126 | 0.126 | 0.125 | 0.125 |

${w_{4}}$ | 0.001 | 0.001 | 0.001 | 0.001 | 0.010 | 0.104 | 0.159 | 0.178 | 0.187 | 0.193 |

${w_{5}}$ | 0.289 | 0.239 | 0.212 | 0.196 | 0.186 | 0.145 | 0.125 | 0.118 | 0.115 | 0.113 |

${w_{6}}$ | 0.134 | 0.185 | 0.192 | 0.189 | 0.184 | 0.172 | 0.162 | 0.159 | 0.157 | 0.156 |

${w_{7}}$ | 0.289 | 0.238 | 0.211 | 0.194 | 0.183 | 0.143 | 0.122 | 0.116 | 0.112 | 0.110 |

*β*and the results of the other methods (SD, CRITIC and Entropy) are computed. If the value of correlation is greater than 0.6, we can say that there is a strong relationship between the results (Walters, 2009). In Table 5, we present the values of correlations between the results.

##### Table 4

SD | CRITIC | Entropy | |

${w_{1}}$ | 0.2206 | 0.1706 | 0.1991 |

${w_{2}}$ | 0.0888 | 0.1212 | 0.0199 |

${w_{3}}$ | 0.1148 | 0.1104 | 0.0397 |

${w_{4}}$ | 0.2934 | 0.2891 | 0.6596 |

${w_{5}}$ | 0.0801 | 0.0748 | 0.0162 |

${w_{6}}$ | 0.1295 | 0.1647 | 0.0521 |

${w_{7}}$ | 0.0727 | 0.0693 | 0.0134 |

##### Table 5

β | SD | CRITIC | Entropy |

0.1 | −0.7745 | −0.6788 | −0.5920 |

0.2 | −0.8896 | −0.7404 | −0.7312 |

0.3 | −0.9345 | −0.7887 | −0.8245 |

0.4 | −0.9534 | −0.8296 | −0.8965 |

0.5 | −0.9577 | −0.8404 | −0.9215 |

1 | −0.6839 | −0.5063 | −0.7397 |

2 | 0.5552 | 0.7391 | 0.4719 |

3 | 0.8011 | 0.9238 | 0.7368 |

4 | 0.8604 | 0.9597 | 0.8040 |

5 | 0.8849 | 0.9725 | 0.8326 |

### 3.2 Analysis of the Performance of Alternatives

*β*as we have used for the analysis of criteria weights. The overall performance scores obtained are presented in Table 6 and the corresponding ranks of alternatives are shown in Table 7. Also, the graphical view of performance scores is represented in Fig. 2.

*β*are greater than 3 ($\beta \geqslant 3$).

*β*, the Spearman’s rank correlation coefficients (${r_{s}}$) between each column of Table 7 are computed. The results are presented in Table 8.

##### Table 6

*β*.

β | ||||||||||

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 1 | 2 | 3 | 4 | 5 | |

${S_{1}}$ | 0.9643 | 0.9631 | 0.9635 | 0.9641 | 0.9625 | 0.9500 | 0.9416 | 0.9388 | 0.9374 | 0.9365 |

${S_{2}}$ | 0.9239 | 0.9196 | 0.9157 | 0.9125 | 0.9113 | 0.9169 | 0.9203 | 0.9215 | 0.9220 | 0.9224 |

${S_{3}}$ | 0.8360 | 0.8295 | 0.8237 | 0.8193 | 0.8120 | 0.7547 | 0.7218 | 0.7108 | 0.7053 | 0.7020 |

${S_{4}}$ | 0.7890 | 0.7738 | 0.7593 | 0.7482 | 0.7378 | 0.6700 | 0.6335 | 0.6213 | 0.6152 | 0.6116 |

${S_{5}}$ | 0.8540 | 0.8370 | 0.8247 | 0.8162 | 0.8075 | 0.7441 | 0.7098 | 0.6983 | 0.6926 | 0.6892 |

${S_{6}}$ | 0.8400 | 0.8180 | 0.8031 | 0.7928 | 0..7836 | 0.7221 | 0.6893 | 0.6784 | 0.6729 | 0.6697 |

${S_{7}}$ | 0.9490 | 0.9421 | 0.9389 | 0.9373 | 0.9342 | 0.9055 | 0.8894 | 0.8840 | 0.8814 | 0.8798 |

${S_{8}}$ | 0.9416 | 0.9347 | 0.9310 | 0.9289 | 0.9252 | 0.8850 | 0.8633 | 0.8560 | 0.8524 | 0.8503 |

${S_{9}}$ | 0.9459 | 0.9242 | 0.9130 | 0.9064 | 0.9009 | 0.8694 | 0.8525 | 0.8469 | 0.8441 | 0.8424 |

${S_{10}}$ | 0.7982 | 0.7841 | 0.7644 | 0.7482 | 0.7378 | 0.6735 | 0.6426 | 0.6324 | 0.6272 | 0.6241 |

##### Table 7

*β*.

β | |||||||||||

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 1 | 2 | 3 | 4 | 5 | ||

Rank | ${A_{1}}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${A_{2}}$ | 5 | 5 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | |

${A_{3}}$ | 8 | 7 | 7 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |

${A_{4}}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |

${A_{5}}$ | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | |

${A_{6}}$ | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |

${A_{7}}$ | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | |

${A_{8}}$ | 4 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | |

${A_{9}}$ | 3 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | |

${A_{10}}$ | 9 | 9 | 9 | 10 | 10 | 9 | 9 | 9 | 9 | 9 |

##### Table 8

*β*.

β | |||||||||||

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 1 | 2 | 3 | 4 | 5 | ||

β | 0.1 | 1 | 0.9758 | 0.9515 | 0.9240 | 0.9240 | 0.8788 | 0.8788 | 0.8788 | 0.8788 | 0.8788 |

0.2 | 0.9758 | 1 | 0.9879 | 0.9726 | 0.9726 | 0.9152 | 0.9152 | 0.9152 | 0.9152 | 0.9152 | |

0.3 | 0.9515 | 0.9879 | 1 | 0.9848 | 0.9848 | 0.9515 | 0.9515 | 0.9515 | 0.9515 | 0.9515 | |

0.4 | 0.9240 | 0.9726 | 0.9848 | 1 | 1 | 0.9605 | 0.9605 | 0.9605 | 0.9605 | 0.9605 | |

0.5 | 0.9240 | 0.9726 | 0.9848 | 1 | 1 | 0.9605 | 0.9605 | 0.9605 | 0.9605 | 0.9605 | |

1 | 0.8788 | 0.9152 | 0.9515 | 0.9605 | 0.9605 | 1 | 1 | 1 | 1 | 1 | |

2 | 0.8788 | 0.9152 | 0.9515 | 0.9605 | 0.9605 | 1 | 1 | 1 | 1 | 1 | |

3 | 0.8788 | 0.9152 | 0.9515 | 0.9605 | 0.9605 | 1 | 1 | 1 | 1 | 1 | |

4 | 0.8788 | 0.9152 | 0.9515 | 0.9605 | 0.9605 | 1 | 1 | 1 | 1 | 1 | |

5 | 0.8788 | 0.9152 | 0.9515 | 0.9605 | 0.9605 | 1 | 1 | 1 | 1 | 1 |

*β*are greater than 1. Generally, we can say that $\beta =3$ is also a good threshold for determination of the overall performance scores and ranks of alternatives.

### 3.3 Comparative Analysis

*β*is set to 3 for computations.

##### Table 9

SAW | WASPAS | COPRAS | TOPSIS | VIKOR | EDAS | SECA | |

${A_{1}}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${A_{2}}$ | 2 | 2 | 2 | 2 | 5 | 2 | 2 |

${A_{3}}$ | 6 | 6 | 6 | 9 | 6 | 6 | 6 |

${A_{4}}$ | 10 | 10 | 10 | 10 | 10 | 10 | 10 |

${A_{5}}$ | 7 | 7 | 7 | 8 | 7 | 7 | 7 |

${A_{6}}$ | 8 | 8 | 8 | 7 | 8 | 8 | 8 |

${A_{7}}$ | 3 | 3 | 3 | 3 | 2 | 3 | 3 |

${A_{8}}$ | 4 | 4 | 4 | 5 | 3 | 4 | 4 |

${A_{9}}$ | 5 | 5 | 5 | 4 | 4 | 5 | 5 |

${A_{10}}$ | 9 | 9 | 9 | 6 | 9 | 9 | 9 |

${r_{s}}$ | 1 | 1 | 1 | 0.867 | 0.927 | 1 | – |

*β*is $\beta =3$. However, this parameter can be set according to the preferences of decision-makers and characteristics of the decision-making problem. The appropriate value of

*β*can be estimated by making a sensitivity analysis based on the decision-matrix of the problem.

## 4 Conclusions

*β*) in an appropriate value, we can get stable weights for criteria and performance scores for alternatives. A comparison has also been made between the results of SECA and those of some existing methods. This comparison also shows the validity and efficiency of the proposed method. Future research can extend SECA for MCDM problems in fuzzy and other uncertain environments.