2.1.1. The Principles of CM-TOPSIS
The procedure of CM-TOPSIS consists of the following seven steps [
16]:
Step 1: Construct the decision matrix , where is the value of the jth attribute of the ith alternative; i = 1, 2, …, m; j = 1, 2, …, n.
Step 2: Normalize the decision matrix
. Two normalization methods are used in CM-TOPSIS [
16], not vector normalization (VN) suggested by Hwang and Yoon [
1].
For the-bigger-the-better attribute, maximum normalization (MN) is used, which is to divide the values of an attribute by the maximum value of the attribute in all the alternatives. The calculation equation of MN is written as follows:
where
is the normalized value of
,
is the maximum value of the
jth attribute in all the alternatives.
For the-smaller-the-better attribute, min-max normalization (MMN) is used to transform it to be the-bigger-the-better attribute. The calculation equation of MMN is written as follows:
where
is the minimum value of the
jth attribute in all the alternatives. It should be noted that the calculation equation of MMN was written incorrectly in [
16].
Step 3: Determine the IS
and the NIS
. As MN and MMN are used in Step 2, all the attributes will change to the-bigger-the-better attributes after normalization and the IS
is
To establish a unique coordination reference standard, an absolute NIS is adopted in CM-TOPSIS [
16]. The NIS
is
Step 4: Calculate the weighted Euclidean distance of each alternative from the IS and the NIS by the following equations:
where
is the weight of the
jth attribute.
Step 5: Calculate the relative closeness (RC) to the IS for each alternative by the following equation:
Step 6: Calculate the coordination degree (CD) of attributes by the following equations:
For unweighted modified TOPSIS:
where
is the angle between the line from the origin to a point (represents an alternative) and the line from the origin to the IS;
is the CD of attributes.
For weighted modified TOPSIS:
Step 7: Calculate the comprehensive evaluation value by the following equation:
where
v is the weight of the CD. If the decision or assessment is to encourage the coordinated development of the attributes,
v must be greater than or equal to 0.5 [
16].
Then, the alternatives can be ranked with respect to their comprehensive evaluation values. A larger comprehensive evaluation value of an alternative indicates that the alternative is relatively better.
2.1.2. The Limitations and Problems of CM-TOPSIS
Although CM-TOPSIS takes into account the coordination level of attributes, which is a significant improvement of TOPSIS, it still has some limitations and problems.
(1) CM-TOPSIS is not based on the original TOPSIS.
The TOPSIS adopted in [
16] is not the original TOPSIS proposed by Hwang and Yoon in 1981 [
1], but modified TOPSIS [
19,
20]. In the original TOPSIS, the attribute weights are used to weight the normalization value
. In this case, Equations (5) and (6) should be written as follows.
Deng et al. [
19] used the weighted Euclidean distances instead of the Euclidean distances that are calculated based on the weighted decision matrix. Equations (5) and (6) are the weighted Euclidean distances. The TOPSIS using weighted Euclidean distances is called modified TOPSIS [
19,
20]. Compared with modified TOPSIS, the original TOPSIS is more frequently used for decision-making and assessment. Therefore, it is significant to establish a coordinated TOPSIS based on the original TOPSIS.
(2) CM-TOPSIS is inapplicable when using VN.
When TOPSIS was proposed, VN was suggested as the normalization method [
1], which is frequently used for TOPSIS [
21,
22,
23]. However, in some studies, VN was replaced by other normalization methods when using TOPSIS, such as MMN [
24,
25,
26].
An important prerequisite for the application of CM-TOPSIS is that the IS is {1, 1, …, 1} and the NIS is {0, 0, …, 0}. If MMN is used for CM-TOPSIS, this prerequisite can be satisfied. However, if VN is used, the IS is not always , and the NIS is not always . In this case, the prerequisite of CM-TOPSIS cannot be satisfied and the calculate formulas of the CD are inapplicable. Since VN is the most frequently used normalization method for TOPSIS, it should be taken into account in establishing coordinated TOPSIS.
(3) The calculation formulas of the CD are incorrect.
For unweighted modified TOPSIS, taking two-dimensional space as an example, the CD is illustrated in
Figure 1. In
Figure 1, point N represents an alternative that is described by two attributes. The values of the two attributes are X
N and Y
N, respectively. Point A (1, 1) represents the IS and origin O (0, 0) represents the NIS. OA is the
line (coordination line). A point located on the
line means that the attributes represented by the point are completely coordinated. The size of ∠NOA indicates the coordination level of attributes. Assuming that point N is located on the
line, it can be easily obtained that X
N = Y
N. Plugging X
N = Y
N into Equation (8) gives
According to the calculated result of Equation (16), point N should be located on the X or Y axis, which is contradictory to the assumption (N is located on the
line). Thus, Equation (8) is incorrect. According to the calculation formula of the angle between two vectors in Euclidean space, the correct calculation formula of
is
Yu et al. (2018) used
to indicate the coordination level of attributes [
16]. In two-dimensional space, the maximum
is
, however, in multi-dimensional space, the maximum
is larger than
. In this case,
is less than zero, which is irrational. Thus, Equation (9) is incorrect. Equation (11) can be used to replace Equation (9).
For weighted modified TOPSIS, the calculation formula of
is interrelated with the attribute weights, as shown in Equation (10). Assuming that the normalization values of all the attributes are equal for an alternative, Equation (10) can be written as
Equation (18) indicates that even if the normalization values of all the attributes are equal, the attributes are not coordinated. This means that the attributes of the alternative represented by the IS {1, 1, …, 1} are also not coordinated. For example, a student gets full marks in both Chinese and mathematics. Only considering these two courses, no matter what weight is, it can be obtained that , and . This indicates that the student’s learning is very unbalanced in Chinese and mathematics, which is inconsistent with the actual scores. Therefore, the calculation formula of for weighted modified TOPSIS is incorrect.
Equations (5) and (6) can be written as
In this case, an equivalent IS is
, an equivalent NIS is {0, 0, …, 0}, and an equivalent point that represents an alternative is
. Obviously, if an equivalent point is located on the line from the equivalent NIS to equivalent IS, the attributes of the alternative represented by the point are completely coordinated. According to the calculation formula of the angle between two vectors in Euclidean space,
can be calculated by the following equation:
Assuming that the normalization values of all the attributes are equal for an alternative, Equation (21) can be written as
Equation (22) indicates that if the normalization values of all the attributes are equal, the attributes are completely coordinated. Thus, Equation (10) should be replaced by Equation (21) when using weighted modified TOPSIS.
If the original TOPSIS is adopted to establish coordinated TOPSIS, Equation (21) should be written as
(4) The coordination level of attributes should be independent of weight.
The weight of an attribute represents the importance of the attribute in decision-making or assessment, but not the importance in the evaluation of coordination level. The coordination level of attributes should be independent of weight, and be evaluated based on the values of attributes. However, Equations (10), (21), and (23) all indicate that the coordination level of attributes is interrelated with the weights, which may cause an irrational result. For example, a student’s learning performance is always evaluated based on the scores in the exam, experiment and class discussion. Assuming that there are three students for evaluation, the scores of the three students are listed in
Table 1. The weights of the exam, experiment, and class discussion are 0.6, 0.3, and 0.1. An absolute NIS is adopted, which is
. Equations (11) and (23) are adopted to calculate the CD. The coefficients of variation (CV) are also calculated for comparison. In terms of the scores, it can be easily judged that the coordination level of Zhang in the three aspects is higher than that of Wang. The CV for the scores of Zhang is 0.088, which is much smaller than that of Wang. However, from
Table 1, the CD for the scores of Wang is 0.947, which is larger than that of Zhang. This means that the coordination level of Wang in the three aspects is higher than that of Zhang. A student with a high coordination level of learning should not have a bad performance in class discussion. The contradictory results are caused by the attribute weight. The small weight of 0.1 results in that the score of class discussion has little effect on the coordination level. Even if the score of class discussion is small, a high coordination level can be also obtained. Therefore, the coordination level of attributes should be independent of weight. If the decision-maker believes that the importance of attributes in the evaluation of coordination level is the same as that in decision-making or assessment, the weight should be taken into account in the coordination level. In this case, Equations (21) and (23) can be used.