This section begins with the innovation of the DANP-mV model and subsequently explains the problems of the original model and improvements and advantages of the new model. Finally, the overall operation procedure and steps involved for the new model are described.
3.1. Innovation and Improvement of the DANP-mV Model
The DANP-mV model uses a systematic method to propose an improved strategy based on the root cause of a problem [
49]. Additionally, the model has evaluative and selective functions similar to those of traditional MADM methods to solve the selection problem of non-dominant solutions by using influential weights (IWs) [
54,
55]. Unlike the conventional method, DANP-mV puts more emphasis on a systematic way of looking at the problem. To utilize the output of this model, it is necessary to calculate the gap from the aspired level and the influential network relation map (INRM) [
47].
Two novel approaches are involved in the DANP-mV model [
18,
47]. First is the calculation of the gap. This model believes that people naturally pursue an aspired level instead of an ideal point. There are unique results of using this type of gap. Decision makers can clearly recognize the gap for each criterion that needs improvement and the need for continuous improvement to achieve the aspired level. When evaluating a problem in practice, decision makers are sometimes faced with alternatives that are all good or all bad in a situation. In simple terms, if the bucket is full of rotten apples, no matter the choice, picking a better apple is still a rotten apple. To avoid this problem, decision makers can use this gap to identify situations in which all alternatives at each criterion are all good or bad. The second unique concept of this model explains how to systematically develop strategies. After realizing the problem, decision makers must develop improvement strategies to reach the goal. Traditionally, the model assumes that factors are independent of each other. Without interdependence between factors, decision makers can only think of ways to make improvements based on the obvious problem rather than addressing the root causes of the problem. However, in the real world, problems often involve a large and complex system. In this system, many factors have a mutual relationship that affects the goal. Therefore, this model is based on the root causes of the problem and discussed how to improve to achieve goals. This is what we call the systematic way of thinking in this research. Additionally, the INRM plays a central role in this study. The INRM is a map of interrelation between each factor. Unlike other display methods showing relationships based on the correlation coefficient, this map employs an arrow with directionality, and the direction of the arrow is determined by influence levels between the two criteria. Thus, after identifying which factor is problematic, decision makers can identify causal factors and build a comprehensive improvement strategy.
It is essential to determine whether a correlation exists between variables, but it is not very useful for decision makers to know the correlation between independent variables because decision makers want to know where the problems are, why they are problems, and how to solve them. If decision makers want to find the cause of problems, it is essential to identify which factors influence the occurrence of problems. There are ways to find the effects of these relationships, such as through setting thresholds. As previously described, in the DANP-mV model, the intensity of an influence is used to determine the effects of all the criteria. For example, assume again the four independent variables x
1, x
2, x
3, and x
4. As mentioned before, mutual influence relationships are present between the variables. Thus, there is a mutual influence between two criteria; x
1 can affect x
2, and x
2 can influence x
1. However, the effect of this relationship is not the same as its intensity. The thickness of the line represents the effect within the graphics; thicker lines indicate stronger effects. In this case, we assume that the relationship of x
1 affecting x
2, x
3, and x
4 is stronger than the relationship of x
2, x
3, and x
4 affecting x
1, and the relationship of x
3 affecting x
2 and x
4 has a stronger effect than the relationship of x
2 and x
4 affecting x
3, and so forth. Thus, the direction of influence is decided in this way. Finally, we determine that x
4 is the problem point because x4 has the lowest performance and the highest gap. Using this system, decision makers can find causes and formulate improvement strategies from the source in a holistic manner, as shown in
Figure 2.
Although the traditional DANP-mV model has many advantages, it also has a significant drawback; that is, it struggles to handle problems with numerous items. Although the method itself is powerful, if there is a problem with input data, the result of the calculation will be problematic. For example, if the overall structure has seven indicators, an item has 42 questions (7 × 6); if there are ten indicators, the item has 90 questions (10 × 9). When studying a practical issue, the indicators often exceed 15; therefore, the total number of items can exceed 210 questions (15 × 14). In addition, questionnaires rely heavily on the opinions of experts with professional qualities. Because of numerous problems, even experts require considerable time to complete the questionnaire. However, the willingness of time experts to commit to assist in research investigations is limited. Therefore, this study addresses this issue of the DANP-mV model and proposes a new version of the modified DANP-mV model. Then, this model is used to analyze empirical cases.
The modified DANP-mV model is intended to maintain the effect of the original model but address the problem of numerous items. The output of the original model has three results: the INRM, the IWs, and the identified criterion of poor performance. The modified DANP-mV model maintains these three results, but the differences lie in the illustration of the INRM and the calculation of IWs.
We first explain differences in the INRM. When drawing the INRM for the original model, mutual influence between two criteria is represented by a line. However, in practice, numerous criteria lead to the complexity of the line graph and ultimately cause decision makers to be unable to clearly identify relationships from the INRM. Thus, past scholars have shown that the INRM of all criteria can be divided into two levels: dimensions and criteria. The dimension level shows the influences between each dimension, and the criteria level shows the influences of the criteria within the dimension. Therefore, interpreting the management implications requires explaining the influence relationships of the dimensions first and those of the criteria within the dimension second. When using the modified model in the data survey, we do not first obtain the effect relationship values of all criteria and then separate them into two levels. Instead, we directly divide the influences of dimensions and criteria at the beginning of the survey. In this way, if the indicator structure is composed of three dimensions and each dimension contains five criteria (total 15 criteria) and each dimension has six question items (3 × 2), the criteria comprise 60 items (5 × 4 + 5 × 4 + 5 × 4). The total number of questions is reduced from the original 210 items to 66 items. Next, the differences between IWs are described. When the weights are obtained, the original model first obtains the global weight of each criterion and then calculates the local weight of each dimension and criterion. The new model does this in the reverse order. We first obtain the local weights of dimensions and criteria within the dimension and subsequently calculate the global weight of each criterion.
In addition to the advantage of the dramatic reduction of the number of items, the new model has characteristics that more closely approximate a real environment. Organizational operations in practice mostly use a hierarchical management structure, especially large enterprises. Each department has different businesses and responsibilities, and cross-sector business typically involves only middle and high-level management, whereas employees in lower positions are often only responsible for internal affairs of the department. Therefore, the new model has characteristics closer to those of practical problems.
3.2. Procedure of Modified DANP-mV Model
The DANP-mV model is a hybrid research tool that consists of two technologies: DANP and modified VIKOR [
54,
55,
56,
57]. This method mainly uses questionnaires to collect data and establish a database. Questionnaires contain two parts: degree of influence between factors and degree of satisfaction of each indicator. Determining the degree of influence between factors relies on experts with relevant professional experience on the subject. Therefore, the respondents are limited. Depending on the topic, determining the satisfaction degree can employ experts or the public.
Because the new model is based on this version and modified, it is called the modified DANP-mV model. Its operation is divided into 12 steps. In terms of the direct-influence relation matrix, the questionnaire design of the new model is different from that of the previous version. The new version is divided into dimensions and criteria within dimensions to collect data. Therefore, the operations of the first to eighth steps are respectively divided into these two parts for calculation. Thus, the local weights of dimensions and criteria within dimensions based on the DANP technique can be gained. “Factor” collectively describes both the dimensions and criteria in the following steps because the calculation procedure is the same. The global weight of all criteria can be obtained through the ninth step. Researchers can use this value to understand cross-dimensional information. The INRM can be drawn according to the sixth step, and the gaps of each alternative can be obtained through the tenth to twelfth steps. The entire process is shown in
Figure 3.
In the following description of the method steps, in brief, the reader can regard formula 1 to formula 11 as a module (the module is shown in
Figure 4 as a dotted box), After learning the influence of the dimension through the questionnaire, dimension weight can be calculated by using this module. Similarly, after obtaining the impact relationship of the criteria through the questionnaire, the weight of the criteria can be calculated by using this module.
Step 1: Build the individual direct relation matrix E
The direct relation matrix is established through a pairwise comparison. Because the degree of influence between two factors is not necessarily the same, there is no reciprocal relationship between the pairwise comparisons of the matrix. The scale of the questionnaire uses an integer score of 0, 1, 2, 3, or 4, expressing the range from absolutely no influence (0) to very high influence (4) through natural language in linguistics [
58]. Thus, this matrix is an
n × n nonnegative matrix. Then, the direct-influence relation matrix of the
H experts can be obtained through the questionnaire as shown in Equation (1). It can be expressed as
Eh = [
eijh]
n×n for
h = 1, 2, …,
H:
Step 2: Calculate the average direct-influence relation matrix A
The average scores of the
H experts are
. The average matrix is called the average direct-influence relation matrix
A, and the degree of influence it receives from other factors is given by Equation (2):
Step 3: Examination of consistency
The consistency value can be calculated according to Equation (3), which represents the consensus of the experts. It verifies whether the overall system has reached stability. The threshold for the value is 5%. If the value is less than 5%, confidence reaches more than 95%, meaning that the system has stabilized. However, if the value is greater than 5%, the system is not yet stable, and we must return to the first phase to determine whether the data collection is correct and whether the number of experts is sufficient:
Step 4: Obtain the normalized average direct-influence relation matrix N
The normalized average direct-influence relation matrix
N is acquired by normalizing matrix
A. Matrix
N is easily derived from Equations (4) and (5), whereby all principal diagonal factors are equal to zero [
59,
60]:
Step 5: Obtain the total influence relationship matrix T
A continuous decrease of the indirect effects of problems moves with the powers of the matrix
Nq = [0]
n×n for
(
I −
N)
−1, where
I is an
n × n matrix [
61]. The total influence relation matrix
T is an
n × n matrix, and is defined by
T = [
tij]
n×n for
i,
j = 1, 2, …,
K, …,
n, as shown in Equation (6):
Step 6: Illustrate the total INRM from the INRM of dimensions and criteria
The total influence relation matrix
T of the INRM can be acquired using Equations (7) and (8) to generate each row sum and column sum, respectively, in the matrix
T:
where
oi is the sum of a row in the total influence relation matrix
T and represents the total effects (both direct and indirect) of factor
i on the all other factors
. Similarly,
rj is the column sum in the total influence relation matrix
T and represents the total effects of factor
j received from all other factors
. The term (
oi +
ri) offers an index of the strength of the total influences given and received; that is, (
oi +
ri) indicates the degree of importance that factor
i plays in the system. In addition, (
oi −
ri) provides an index of the degree of the cause of total influence. If (
oi −
ri) is positive, then criterion/perspective
i is a net causer, and if (
oi −
ri) is negative, then factor
i is a net receiver. Finally, drawing the INRM uses (
oi −
ri) as a y axis and (
oi +
ri) as an x axis to illustrate the scatter plot [
62]. The party with a stronger influence relationship indicates the direction of the arrows between all influence factors. For instance, the influence of the first factor on the second factor is greater than the effect of the second factor on the first factor; therefore, the direction of the arrow is from the first factor to the second factor.
Because the new method investigates the dimensions and criteria separately during the questionnaire survey phase, we can directly obtain the total influence relation matrix of dimensions and criteria through the calculations in the first through sixth steps. Here, TD is the total influence relation matrix of dimensions, and TcD is the total influence relation matrix of the criteria belonging to this dimension. The complete INRM integrates the INRMs of dimensions and criteria.
Step 7: Transpose and normalize the total influence relation matrix
Use Equation (9) to normalize the total influence relation matrix
T; then, the normalized total influence relation matrix
Tα can be calculated. Subsequently, the transpose matrix
Tα and the transposed total influence relation matrix
wα can be obtained as shown in Equation (10):
Step 8: Obtain local weights for dimensions wld and criteria wld_c
Limit the transposed total influence relation matrix
wα by raising it to the
z power until the
wα has converged and become a stable matrix, in which
z represents any number of power as shown in Equation (11). The local weights of dimension
wld and criteria
wld_c within the dimension are obtained; these are called the IWs in the DANP model:
Step 9: Find global weights of all criteria
Global weights of all criteria
are obtained by integrating the local weights of dimensions with criteria, as shown in Equation (12).
Step 10: Derive the aspiration levels and worst value
Define the best value of n criteria, called the aspiration level, shown as faspired and the worst value shown as fworst. The aspiration level refers to the upper bound of the semantic scale in the questionnaire. By contrast, the worst case refers to the lower limit of the semantic scale in the questionnaire. In this study, the performance ranges from 0 to 10; 0 indicates very bad, and 10 indicates very good. These are used with natural language in the linguistic questionnaire. Thus, faspired is 10, and fworst is 0.
Step 11: Normalize performance of k alternatives, and calculate the gap
The performance values of the
j factors of the
k alternatives are normalized, and the distance between these performance values and aspiration level is calculated at the same time. These normalized distances are gaps represented by
rkj, as shown in Equation (13):
Step 12: Determine the mean group utility Sk for the gap
Traditional VIKOR considers two types of gap: mean group utility and maximum regret degree. Because the modified DANP-mV model focuses on improving the performance of alternatives to achieve the aspiration level, the gap of this model adopts mean group utility
Sk. This value can be calculated using Equation (14):