In the discrete form of multi-criteria decision-making (MCDM) problems, we are usually confronted with a decision-matrix formed from the information of some alternatives on some criteria. In this study, a new method is proposed for simultaneous evaluation of criteria and alternatives (SECA) in an MCDM problem. For making this type of evaluation, a multi-objective non-linear programming model is formulated. The model is based on maximization of the overall performance of alternatives with consideration of the variation information of decision-matrix within and between criteria. The standard deviation is used to measure the within-criterion, and the correlation is utilized to consider the between-criterion variation information. By solving the multi-objective model, we can determine the overall performance scores of alternatives and the objective weights of criteria simultaneously. To validate the proposed method, a numerical example is used, and three analyses are made. Firstly, we analyse the objective weights determined by the method, secondly, the stability of the performance scores and ranking results are examined, and finally, the ranking results of the proposed method are compared with those of some existing MCDM methods. The results of the analyses show that the proposed method is efficient to deal with MCDM problems.

Multi-criteria decision-making (MCDM) methods and techniques have been used and developed by many researchers and applied to many real-world problems. The MCDM methods and techniques are usually categorized into two classes: multi-attribute decision-making (MADM) and multi-objective decision-making (MODM). The MODM methods are used to deal with continuous MCDM problems, and the MADM methods are utilized to handle discrete MCDM problems (Rezaei,

Several MCDM methods have been proposed during the past decades. Here, we briefly review some important methods which are used for criteria weight determination and alternative evaluation based on the information of the decision-matrix, and readers are referred to some recent review articles on MCDM methods and their applications for further details (Behzadian

SAW is an old method which is likely the most common and very widely used method for the evaluation of alternatives. MacCrimmon (

Zavadskas

The COPRAS method is an efficient MCDM method in which a ratio based on two measures (the summation of beneficial criteria performance and the summation of non-beneficial criteria performance) is used to evaluate alternatives (Zavadskas

TOPSIS is a value-based compensatory method introduced by Hwang and Yoon (

The VIKOR method, which was proposed by Opricovic (

EDAS is an efficient and relatively new MCDM method introduced by Keshavarz Ghorabaee

Besides the MCDM methods used for evaluation of alternatives, there are some other methods for determination of objective weights of criteria. The Standard Deviation (SD) and Entropy are two common methods which determine the objective weights based on the within-criterion variation information, and assign smaller weights to a criterion if it has similar values across alternatives. These methods have been used in many studies (Keshavarz Ghorabaee

In this paper, we introduce a new method that, unlike many other MCDM methods, can be used for simultaneous evaluation of criteria and alternatives (SECA) in a multi-criteria decision-making problem. A multi-objective non-linear mathematical model is developed to make the evaluation process. The model includes three objective functions. The first objective is related to maximization of the overall performance of alternatives, and the second and third objectives concern within- and between-criterion variation information. The within-criterion variation is measured by the standard deviation, and the between-criterion variation is considered using the correlation. By optimization of the developed mathematical model, the overall performance scores of alternatives and the objective weights of criteria can be determined simultaneously. A numerical example is used in this study for validation of the proposed method. Based on this example, the validity of objective criteria weights and the stability of the overall performance scores and ranking results are analysed. Moreover, a comparison is made between the ranking results of the proposed method with the results of the SAW, WASPAS, COPRAS, TOPSIS, VIKOR and EDAS methods.

The rest of this paper is organized as follows. Section

In this section, a new method is proposed to handle multi-criteria decision-making problems. The aim of this method is to determine the overall performance scores of alternatives and the weights of criteria simultaneously. A multi-objective non-linear mathematical model is formulated in this section to reach this aim. To formulate the mathematical model, two types of reference points are described for criteria weights. The first type is based on within-criterion variation information defined by the standard deviation and the second is related to between-criterion variation information determined based on the correlation measure. The multi-objective model seeks to maximize the overall performance of each alternative and minimize the deviation of criteria weights from the reference points. For maximization of the overall performance of each alternative, a weighted sum model is used as an objective. Also, we take advantage of the sum of squared deviations from the reference points to define the other objectives of the model.

Suppose that we have an MCDM problem with

Based on Eq. (

Let

An increase in the variation within the vector of a criterion (

Based on the above description, a multi-objective non-linear programming model is formulated as follows:

In Model (7), the first equation maximizes the overall performance of each alternative, and the second and third equations minimize the deviation of criteria weights from the reference points for each criterion. Eq. (

To optimize Model (7), we use some techniques of the multi-objective optimization and transform the model to Model (8) as follows:

According to the objective function of Model (8), the minimum of the overall performance score of alternatives (

The overall performance score of each alternative (

The decision-matrix of the numerical example.

23 | 264 | 2.37 | 0.05 | 167 | 8900 | 8.71 | |

20 | 220 | 2.2 | 0.04 | 171 | 9100 | 8.23 | |

17 | 231 | 1.98 | 0.15 | 192 | 10800 | 9.91 | |

12 | 210 | 1.73 | 0.2 | 195 | 12300 | 10.21 | |

15 | 243 | 2 | 0.14 | 187 | 12600 | 9.34 | |

14 | 222 | 1.89 | 0.13 | 180 | 13200 | 9.22 | |

21 | 262 | 2.43 | 0.06 | 160 | 10300 | 8.93 | |

20 | 256 | 2.6 | 0.07 | 163 | 11400 | 8.44 | |

19 | 266 | 2.1 | 0.06 | 157 | 11200 | 9.04 | |

8 | 218 | 1.94 | 0.11 | 190 | 13400 | 10.11 |

In this section, a computational analysis is made to validate the results of the proposed method. This analysis is presented in the three following sub-sections. Firstly, we compare the objective weights resulted from the proposed method with the results of some other methods, secondly the overall performance scores and ranks of alternatives are analysed and finally we make a comparative analysis based on some existing MCDM methods. It should be noted that the model is solved using the LINGO 11 software, and the code and instructions can be found in Keshavarz Ghorabaee (

To make the computational analysis, a numerical example is borrowed from the study of Keshavarz Ghorabaee

The normalized decision-matrix of the numerical example.

1 | 0.9925 | 0.9115 | 0.8 | 0.9401 | 1 | 0.9449 | |

0.8696 | 0.8271 | 0.8462 | 1 | 0.9181 | 0.9780 | 1 | |

0.7391 | 0.8684 | 0.7615 | 0.2667 | 0.8177 | 0.8241 | 0.8305 | |

0.5217 | 0.7895 | 0.6654 | 0.2000 | 0.8051 | 0.7236 | 0.8061 | |

0.6522 | 0.9135 | 0.7692 | 0.2857 | 0.8396 | 0.7063 | 0.8812 | |

0.6087 | 0.8346 | 0.7269 | 0.3077 | 0.8722 | 0.6742 | 0.8926 | |

0.9130 | 0.9850 | 0.9346 | 0.6667 | 0.9813 | 0.8641 | 0.9216 | |

0.8696 | 0.9624 | 1 | 0.5714 | 0.9632 | 0.7807 | 0.9751 | |

0.8261 | 1 | 0.8077 | 0.6667 | 1 | 0.7946 | 0.9104 | |

0.3478 | 0.8195 | 0.7462 | 0.3636 | 0.8263 | 0.6642 | 0.8140 |

In this sub-section, we solve Model (8) with the data provided in Table

According to Fig.

The sets of criteria weights determined by changing

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

0.001 | 0.001 | 0.035 | 0.068 | 0.081 | 0.133 | 0.148 | 0.153 | 0.156 | 0.158 | |

0.284 | 0.253 | 0.233 | 0.221 | 0.213 | 0.175 | 0.157 | 0.151 | 0.148 | 0.146 | |

0.001 | 0.083 | 0.115 | 0.131 | 0.142 | 0.129 | 0.126 | 0.126 | 0.125 | 0.125 | |

0.001 | 0.001 | 0.001 | 0.001 | 0.010 | 0.104 | 0.159 | 0.178 | 0.187 | 0.193 | |

0.289 | 0.239 | 0.212 | 0.196 | 0.186 | 0.145 | 0.125 | 0.118 | 0.115 | 0.113 | |

0.134 | 0.185 | 0.192 | 0.189 | 0.184 | 0.172 | 0.162 | 0.159 | 0.157 | 0.156 | |

0.289 | 0.238 | 0.211 | 0.194 | 0.183 | 0.143 | 0.122 | 0.116 | 0.112 | 0.110 |

The variation of the criteria weights related to

To make a comparison, the correlations between the results of the proposed method in different values of

The objective weights of criteria for comparison.

SD | CRITIC | Entropy | |

0.2206 | 0.1706 | 0.1991 | |

0.0888 | 0.1212 | 0.0199 | |

0.1148 | 0.1104 | 0.0397 | |

0.2934 | 0.2891 | 0.6596 | |

0.0801 | 0.0748 | 0.0162 | |

0.1295 | 0.1647 | 0.0521 | |

0.0727 | 0.0693 | 0.0134 |

The correlation between the results of criteria weight determination.

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 | |||

4 | |||

5 |

As it can be seen it Table

In this sub-section, we obtain the overall performance score of each alternative (

As we can see in Fig.

To analyse the stability of the ranks of criteria in different values of

The overall performance scores of alternatives in different values of

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

0.9643 | 0.9631 | 0.9635 | 0.9641 | 0.9625 | 0.9500 | 0.9416 | 0.9388 | 0.9374 | 0.9365 | |

0.9239 | 0.9196 | 0.9157 | 0.9125 | 0.9113 | 0.9169 | 0.9203 | 0.9215 | 0.9220 | 0.9224 | |

0.8360 | 0.8295 | 0.8237 | 0.8193 | 0.8120 | 0.7547 | 0.7218 | 0.7108 | 0.7053 | 0.7020 | |

0.7890 | 0.7738 | 0.7593 | 0.7482 | 0.7378 | 0.6700 | 0.6335 | 0.6213 | 0.6152 | 0.6116 | |

0.8540 | 0.8370 | 0.8247 | 0.8162 | 0.8075 | 0.7441 | 0.7098 | 0.6983 | 0.6926 | 0.6892 | |

0.8400 | 0.8180 | 0.8031 | 0.7928 | 0..7836 | 0.7221 | 0.6893 | 0.6784 | 0.6729 | 0.6697 | |

0.9490 | 0.9421 | 0.9389 | 0.9373 | 0.9342 | 0.9055 | 0.8894 | 0.8840 | 0.8814 | 0.8798 | |

0.9416 | 0.9347 | 0.9310 | 0.9289 | 0.9252 | 0.8850 | 0.8633 | 0.8560 | 0.8524 | 0.8503 | |

0.9459 | 0.9242 | 0.9130 | 0.9064 | 0.9009 | 0.8694 | 0.8525 | 0.8469 | 0.8441 | 0.8424 | |

0.7982 | 0.7841 | 0.7644 | 0.7482 | 0.7378 | 0.6735 | 0.6426 | 0.6324 | 0.6272 | 0.6241 |

The ranks of alternatives in different values of

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

Rank | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

5 | 5 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ||

8 | 7 | 7 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ||

10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ||

6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | ||

7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ||

2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | ||

4 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ||

3 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | ||

9 | 9 | 9 | 10 | 10 | 9 | 9 | 9 | 9 | 9 |

Variations in the overall performance scores related to

The correlation (

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

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 |

As can be seen in Table

In this sub-section, the results of the proposed method are compared with those of some existing MCDM methods. We use six MCDM methods including SAW, WASPAS, COPRAS, TOPSIS, VIKOR and EDAS to make this comparison. Based on the results of the previous sections, the value of

The methods which are used for comparative analysis need a set of weights to rank the alternatives. In the comparative analysis, we use the set of weights which has been determined by the proposed method with

According to Table

The ranking results of different MCDM methods and the correlation values.

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

1 | 1 | 1 | 1 | 1 | 1 | 1 | |

2 | 2 | 2 | 2 | 5 | 2 | 2 | |

6 | 6 | 6 | 9 | 6 | 6 | 6 | |

10 | 10 | 10 | 10 | 10 | 10 | 10 | |

7 | 7 | 7 | 8 | 7 | 7 | 7 | |

8 | 8 | 8 | 7 | 8 | 8 | 8 | |

3 | 3 | 3 | 3 | 2 | 3 | 3 | |

4 | 4 | 4 | 5 | 3 | 4 | 4 | |

5 | 5 | 5 | 4 | 4 | 5 | 5 | |

9 | 9 | 9 | 6 | 9 | 9 | 9 | |

– |

For using the proposed method, the suggested value of

It should be noted that the elements of the decision-matrix should be greater than zero (

Many MCDM methods have been developed in the past decades. Most of these methods were designed to evaluate some alternatives with respect to a given set of criteria weights. There have also been some methods for determination of the subjective and objective weights of criteria. In this study, we have proposed a new method for simultaneous evaluation of criteria and alternatives (SECA). A multi-objective non-linear mathematical model has been formulated for the proposed method. The objectives of the model have been defined to maximize the overall performance of alternatives considering the within- and between-criterion variation information of decision-matrix. By using the proposed method, we can determine the overall performance scores of alternatives and objective weights of criteria simultaneously. For validation of SECA, the objective weights of criteria and the overall performance of alternatives determined by the method have been analysed. The results show that by setting the parameter of the method (