4.1. The Framework of the Proposed Model
By employing a hybrid multi-criteria group decision-making evaluation method, this paper provides a comprehensive analysis of inland port evaluation, as illustrated in
Figure 2. As can be seen in
Figure 2, we present a theoretical framework for calculating indicator weights by employing the DEMATEL–BBWM approach, and the framework builds upon the innovative evaluation index system for inland port development.
The BBWM method introduces a novel approach to establishing the ranking of index credibility, hence enhancing the dependability of the indicators’ weights for group decision-making. The DEMATEL method exhibits enhanced capability in analyzing the causal relationship between indicators, rendering it more congruent with the DRPP indicator framework system given within this paper.
Consequently, the DEMATEL method is utilized to determine the weights of the first-level indicators in evaluating the development of inland ports, and the BBWM method is adopted for determining the weights of the second-level indicators, which fully reflects the interrelatedness and independence of the indicators.
Finally, relying on the evaluation results, the construction priority of each inland port is clarified. By constructing a performance–importance matrix, the specific challenges encountered during the inland port development process are identified, which help us to propose diversified development strategies for each inland port.
4.2. DEMATEL for Weight Determination of the First-level Evaluation Indicators
The DEMATEL method is widely recognized as a highly successful approach for discerning the constituent elements of the causal chain within a complicated system [
13,
33]. The methodology facilitates the identification of crucial elements inside intricate systems through the evaluation of the interrelationships among components and the creation of a visual structural model. The proposed methodology offers a solution to address the issue of ambiguous key indicators resulting from their interplay.
The coupling effect link among the demand, risk, power, and potential factors, is evident in the evaluation problem of inland port development, which aligns well with the application scenario of the DEMATEL approach.
Step 1. Construct a set of indicators for the evaluation of inland port development and to identify the interactions among the indicators.
According to the opinions of relevant experts in the field of inland ports, we decided on the specific degree of mutual influence between each level of indicators of demand, risk, power, and potential. Moreover, we constructed the direct influence matrix
D of the first-level indicators, in which the rows and columns are distributed according to the order of the mentioned indicators as shown in Equation (1) [
33]:
where
represents the degree of influence of indicator
i to indicator
j. The degree is classified into four levels: 0, 1, 2, and 3, where the scores of 0, 1, 2, and 3 represent ‘‘No influence’’, ‘‘Low influence’’, ‘‘High influence’’, and ‘‘Very high influence’’, respectively [
33,
34].
Step 2. Normalize the direct impact matrix and calculate the integrated impact matrix.
The direct impact matrix of the first-level indicators for the evaluation of inland port development is normalized, and each row of data is standardized separately to the interval [−1, 1]. By this step, we can obtain the matrix
, where λ represents the standardized coefficient. On this basis, the comprehensive impact matrix
T is calculated, as shown in Equation (2) [
33,
34]:
Step 3. Calculate various indicator values.
The degree of influence, being influenced, centrality, and causality are computed for each first-level indicator using matrix T. The values of each indicator are computed in the following manner.
The degree of influence
is the sum of the elements of the rows of matrix
T, indicating the degree of influence of indicator
i to other indicators, as in Equation (3):
The degree of being influenced
is the sum of the elements of the columns of matrix
T, indicating the degree of influence of other indicators to indicator
i, as in Equation (4):
The centrality of indicator i is the sum of and , . The higher centrality of the indicator indicates the higher importance of the indicator in the overall evaluation.
The causality of indicator i is the difference between and , . The higher centrality of the indicator indicates the higher importance of the indicator in the overall evaluation. A positive indicates that the indicator i is a cause indicator, and a negative indicates that the indicator i is a contributing indicator.
Step 4. Determine the weights of first-level evaluation indicators based on centrality and causality.
Based on the centrality and causality degree of each indicator, the distance of each indicator from the origin is calculated, . The longer distance means that the indicator has a stronger influence in the overall system of indicators. Thus, we should give higher weight to this kind of indicators. Moreover, the resulting distance matrix is normalized to obtain the final weights of the first-level indicators for the evaluation of inland port development.
4.3. Bayesian Best–Worst Method for Weight Determination of the Second-level Evaluation Indicators
The basic principle of the best–worst method is to obtain the final indicator weights by first identifying the best and worst indicators in a set of decision indicators, then using a number between 1 and 9 to determine the preference of the best indicator over all other indicators and the preference of all indicators over the worst criterion [
28], and to calculate them through a series of comparisons. While the original best–worst method was limited to solving the case of targeting the decision maker uniquely, Mohammadi and Rezaei in 2020 proposed a new method for multi-criteria group decision-making based on the best–worst method by considering the case of group decision-making, which is known as the Bayesian best–worst method. The basic idea of the Bayesian best–worst method is to provide a priori and a posteriori probabilistic explanation for the inputs and outputs of the best–worst method, which ultimately leads to a comprehensive analysis of group decision-making [
14]. Since its introduction, the Bayesian best–worst method has been widely used for multi-criteria group decision-making problems in many fields [
35].
Step 1. Construct a collection of the second-level indicators , and determine the best and the worst criteria from C.
Step 2. Conduct pairwise comparisons between the best indicator and other indicators.
In this step, the decision maker shows his/her preferences by a number between 1 and 9; the higher the number, the greater the relative importance between the criteria. The resulting best-to-others vector is , where denotes the preference of the best criterion over other criteria .
Step 3. Conduct pairwise comparisons between the other indicators and the worst indicator.
Similar to step 2, conduct the pairwise comparison between the other criteria and the worst criterion. The resulting others-to-worst vector is , where denotes the preference of other criteria over the worst criterion .
Step 4. Estimating the probability distribution of each individual optimal weight and the overall optimal weight given and , where k represents the decision makers and k = 1, …, K.
The joint probability distribution is sought as in Equation (5):
The probability of each variable then can be computed using the following rule under probability theory:
where
x and
y are two arbitrary random variables.
To build a Bayesian model, a graphical model is plotted as in
Figure 3.
It is clear that the variable
depends on both
and
, while
, in turn, depends on
, while either
or
are independent of
according to the direction of the arrow. This independence feature can be described as in Equation (7):
Applying Bayes’ theorem to Equation (1), we can obtain the following Equation (8):
Step 5. Derive the prior distribution and calculate the posterior distribution to obtain the weights of each indicator.
To compute the above equation, we need to determine the distributions of each related element.
and
are the inputs of BWM. According to the Bayesian structure as in
Figure 3, they can be modeled by a multinomial distribution due to the property of the integer, which is shown in Equations (9) and (10):
For the multinomial distribution of weight
w, the Dirichlet distribution is used as the prior distribution because of its non-negativity and sum-to-one properties, as in Equation (11):
Therefore, when
is given, one can expect each and every
to be in its proximity. The models of
are as in Equation (12):
For the non-negative parameter
, the gamma distribution is adopted to model the distribution of
due to its wide range of applications in prior distributions, as in Equation (13):
where a and b are the shape parameters of the gamma distribution, and in choosing the parameters of the gamma distribution, we can use either great likelihood estimation or Bayesian estimation.
Finally, the prior distribution over
can be expressed as in Equation (14):
where the parameter
is set to be 1.
However, the model we built up until now does not output a closed-form solution. To deal with this, we introduced Markov chain Monte Carlo (MCMC) to compute the posterior distribution in Equation (8). Moreover, we used the “just another Gibbs sampler” (JAGS) to generate randoms samples in Equation (8).
Step 6. Calculate the credal ranking for indicators.
When faced with multi-criteria decision-making problems, the relative significance of indicators is frequently determined by comparing whether one indicator’s (mean) weight is greater than that of another indicator. However, in the group decision-making process, the concept of credal ranking needs to be introduced. By constructing the posterior distribution of indicator weights, credal ranking can calculate the degree to which one indicator is superior to another and evaluate the superiority relationship among indicators in a more objective manner. Compared with interval-based ranking, fuzzy ranking, and ranking based on gray-scale relational analysis, credal ranking is based on the Dirichlet distribution, while other ranking methods often explore the superiority or inferiority relationship among indicators by constructing two numbers/intervals.
Based on Mohammadi and Rezaei’s idea [
14], we introduce the calculation process of credal ranking in this study as follows.
For the indicator set , the credal raking is the set of confidence orders for each pair of indicators , while .
A new Bayesian test is constructed with the goal of determining the certainty of each confidence order. The test is designed with the posterior distribution of
. The mathematical equation of
is more important than
is shown as in Equation (15):
where
represents the posterior probability of
,
I denotes a logic parameter that evaluates to 1 when the subscript condition of
I is true, and 0 otherwise.
This integral can be estimated from samples obtained by means of Markov chain Monte Carlo (MCMC). With Q samples from the posterior distribution, the confidence level can be expressed as Equations (16) and (17):
where
denotes the
qth sample of
in the MCMC sample.
Thus, for each pair of indicators, the confidence level at which one indicator is more important than the other can be calculated as above. Confidence rankings can also be transformed into traditional rankings in this way. Obviously, . Therefore, an indicator is considered to be more important than when and only when .