As a branch of the decision-making management research field, MCDM involves many methods, such as AHP, ANP, DEMATEL, TOPSIS, and GRAY. The MCDM model has been widely used in online education, e-commerce, supply chain management, energy management, and other industries [
34,
35,
36]. Sadi-Nezhad et al. [
34] proposed the fuzzy analysis network process (FANP) model to evaluate network learning systems and used the model to evaluate the existing network learning platform of some universities. In this study, the best worst method (BWM) and VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) were adopted. The BWM is an improvement based on the AHP, which compares the two-to-two comparisons to the optimal and worst dimensions or indicators compared with the remaining dimensions or indicators. Using such processing reduces the number of steps to be compared, simplifies the calculation process, and greatly reduces the amount of data. Furthermore, the smaller the consistency indicator value is, the higher the reliability of the acquired data [
37]. You et al. [
38] used the BWM method to determine the weights of the evaluation standard of power grid operations and evaluated the operating performance of power grid enterprises. Gupta et al. [
39] and Kaa et al. [
40] used the BWM method to determine the relative importance of green innovation factors, and the research contents were supplier selection based on green innovation and the construction of a green innovation framework. VIKOR is an MCDM method first developed by Opricovic in 1998. It is a method for solving complex decision situations [
41]. This method is used to solve discrete decision-making problems with immeasurable and inconsistent criteria [
42]. Shojaei et al. [
41] used VIKOR technology to propose an evaluation and ranking model to rank the performances of some airports.
3.1. Introduction of BWM
For the best worst method (BWM), the decision maker does not need to compare all of the criteria as in the traditional analytic hierarchy process. After determining the best dimension, the worst dimension, the best criterion, and the worst criterion under each dimension through expert interviews, the method only needs to make pairwise comparisons between the best criterion and the worst criterion, and other criteria. As the BWM provides a more consistent comparison than the AHP, and the metric weights obtained by the BWM are highly reliable [
43]. The BWM could produce a single solution to two or more metric problems when comparing the system to fully meet any number of standards. For comparison systems with three or more inconsistent criteria, weights could be used as intervals where multiple optimal solutions were possible [
44]. The specific steps are as follows:
Step 1: Determine the set of decision indicators. The decision maker identifies n indicators for decision making .
Step 2: Determine the best and worst indicators. The decision maker selects the best (the most desirable, preferred, or important) and the worst (the least desirable or least important) indicators from n indicators.
Step 3: The optimal indicators and other indicators are compared. Decision makers rank the relative importance of the optimal indicators on a scale of 1 to 9. For other indicators, the best-to-other (BO) variable obtained by the allocation is expressed as , where represents the importance of the optimal indicator b compared with indicator j. Obviously, .
Step 4: The other indicators are compared with the worst. Decision makers use a scale from 1 to 9 to show the relative importance of indicators other than the worst. The others-to-worst (OW) variable obtained is expressed as , where represents the degree of importance of indicator j compared with the worst indicator w. Obviously, .
Step 5: The optimal weight
is determined. The optimal weight is determined such that the maximum absolute difference between
and
for all
j is the minimum. The optimal weight can be represented by a maximum and minimum model:
It can be solved by converting this into the following linear programming formula:
For any value of θ, multiply the first constraint in Formula (2) and the second constraint to obtain the solution. The solution space of Formula (2) is the intersection of 2n − 3 (n represents the number of the indicators, and ) linear constraints. Therefore, if there is a sufficiently large , the solution space is nonempty. By solving Formula (2), the corresponding results of the optimal weights and are obtained.
Definition 1: When , it is sufficiently consistent that for all ks, , , and are the preferences of the best indicator for indicator k. Indicator k is the preference of the worst indicator for the worst indicator.
Table 2 shows the maximum value of different values of
(consistency indicator) for
.
Due to the consistency index (
Table 2), the consistency rate (CR) is calculated as follows:
The consistency rate belongs to [0.1]. The closer the value is to 0, the higher the consistency; conversely, the closer the value is to 1, the lower the consistency.
As mentioned above, Formula (2) can produce multiple optimal solutions. If you want to find the minimum and maximum values in the set
and minimize and maximize the set
, the problem can be expressed by the following formula:
Formula (4) can be converted into the following linear equation:
Formula (5) is a linear problem with a unique solution. It can obtain the optimal weights and .
For this model, the closer the value of is to 0, the higher the consistency.
3.2. VIKOR
The VIKOR method is an effective tool for multiple criteria decision-making technology. The method is used for decision makers who cannot or do not know how to clearly express preferences or the situation of inconsistencies and conflict between the evaluation principles. The VIKOR method can address these problems so that decision makers can accept compromise solutions. Therefore, Chitsaz and Banihabib [
45] stated that VIKOR provided a compromise ranking to decision makers based on “proximity” to “ideal” solutions.
First, the ideal solution and the negative ideal solution are defined. The ideal solution is the optimal value of all the evaluation criteria in each evaluation criterion, and the negative ideal solution is the worst value of all of the evaluation criteria in each evaluation criterion. All scenarios are evaluated according to each standard function, and the ordering is performed based on the proximity of the ideal solution. This method uses the
-metric as the aggregate function:
In the above formula,
j is the scheme number;
i is the evaluation criterion number;
represents the performance value of the
jth alternative on the
ith criterion; and
and
represent the best value and the worst value of all standard functions, respectively.
P is the distance parameter of the aggregate function (generally 1, 2, or ∞; this paper takes 1),
n is the number of criteria,
represents the
ith standard weight, and
represents the distance from the solution to the ideal solution.
In Formulas (7) and (8) above, represents a set of revenue type criteria and represents a set of cost type criteria. Thus, the positive ideal solution and the negative ideal solution are calculated.
The second step is to calculate the group benefit
(optimal solution) and individual regret
(worst solution) of the comprehensive evaluation of the scheme.
where
J = 1, 2, 3, …,
j, where
represents the weight of the
ith indicator;
represents the group benefit of the alternative, where the smaller the value of
, the greater the group benefit; and
represents individual regret, where the smaller the value of
is, the smaller the individual regrets.
The third step is to calculate the benefit ratio
Q generated by each scheme.
where
v represents the coefficient of the decision mechanism. If
v > 0.5, it means that the decision is made according to the principle of benefits first; if
v ≈ 0.5, the decision is made according to the principle of balanced compromise; and if
v < 0.5, it means that the decision is made according to the principle of the cost supremacy decision. In this paper,
v = 0.5, that is, the tradeoffs between benefits and costs are balanced.
The fourth step is to sort the alternatives according to .
The fifth step is to sort according to the value of when the following two conditions are met, and is the minimum winning unit.
Condition 1: When Q(a) − Q(b) ≥ 1/(J − 1), Q(a) is the Q of the second scheme based on Q, Q(b) is the Q of the first scheme based on Q, and J is the number of all schemes. The difference between the two Qs in which the order is only one bit must exceed the value of 1/(J − 1) to determine the optimal scheme to sort the first scheme. If there are more than two schemes, the first scheme is sorted. The plan is compared with other scenarios to determine whether it meets Condition 1.
Condition 2: Admitted assurance of decision-making. After sorting according to Q, the S of the first-ranked scheme must be ranked better than the S of the second-ranked scheme or its R must be ranked better than the R of the second-ranked scheme. If there are more than two schemes, the first scheme is compared in order with other schemes to determine whether it meets Condition 2.
If the first-ranked scheme satisfies both Condition 1 and Condition 2, the first-ranked scheme is optimal. If the first-ranked scheme and the sorted second-ranked scheme or another scheme satisfy only Condition 2 but Condition 1 is not satisfied, the schemes that do not satisfy Condition 1 but satisfy Condition 2 are all optimal schemes.