3.1. Evaluation Index Weighting Based on AHP Weight Method
After determining the evaluation index system of public transport comprehensive service quality, to reflect the evaluation results more accurately, the weight of the indicators must be allocated reasonably. In order to make up for the shortcomings of a single weighting method, we first adopted the combination of AHP in the subjective weighting method and the entropy weighting method in the objective weighting method to determine the subjective and objective weights of the evaluation indicators, and then combined the linear weighting method to AHP and the entropy weight method to obtain the comprehensive weight of the index.
Assuming that the weight vector obtained by AHP is
], the weight vector obtained by the entropy weight method is
], using the linear weighting method to obtain the comprehensive weight vector
].
Symbolic notations used in (1) are defined as follows.
: Comprehensive weight of the i-th evaluation index;
: Subjective weight of the i-th evaluation index;
: Objective weight of the i-th evaluation index;
: Weighting factor for subjective weights; and
: Weighting factor for objective weights.
In order to make the distribution of weights more reasonable, we used the difference coefficient method to determine the values of
and
[
11], and the calculation formula is as follows in (2).
Symbolic notations used in (2) are defined as follows.
: The number of evaluation indicators; and
: The corresponding components in the subjective weight vector sorted obtained by the i-th index from minimum to maximum.
3.2. Application Performance Index System of Pure Electric Buses Based on Extensible Cloud Model
After the analytic hierarchy process, the subjective weight of each evaluation index was determined. Then, the entropy weight method was used to determine the objective weight of each assessment index. On this basis, the extension cloud model was used for evaluation. The evaluation method of the extension cloud model is to use mathematical expressions to associate the randomness and fuzziness of things and realize the uncertainty conversion between qualitative concepts and quantitative values. The specific steps are shown in
Figure 3.
- (1)
Extension cloud theory
Matter–element extension theory refers to matter–element as the basic element for describing things, expressed as
, where
is the name of the described thing,
is the characteristic of the thing, and
is the characteristic value of the thing [
27]. It can study problems from both qualitative and quantitative perspectives. Thus, the change process of things can be represented more objectively. In the traditional matter–element evaluation model,
is often regarded as a fixed value, ignoring its own randomness and ambiguity, which easily causes partial information loss. Therefore, in this work, we introduced the normal cloud model into the matter–element extension theory for analysis.
The normal cloud model can be represented by three eigenvalues: expected , entropy , and super entropy .
represents the cloud distribution center value corresponding to cloud droplets at a certain evaluation level, which can best reflect the classification level of the transfer evaluation index.
represents the value range of a certain evaluation level, reflecting the randomness of data collection in the evaluation process.
represents the randomness of the membership degree of a certain evaluation level, which reveals the correlation between the randomness and ambiguity of the evaluation index level in the transfer evaluation process.
The extension cloud model uses the normal cloud model (
,
,
) to replace the eigenvalue
of the matter in the matter–element extension theory, so as to realize the mathematical description of the randomness and fuzziness in the evaluation process. The extension cloud model is shown as follows in (3):
Symbolic notations used in (3) are defined as follows.
: Buses to be evaluated;
: The i-th evaluation index for application performance index of pure electric buses; and
(, , ): The cloud description for evaluation index .
- (2)
Calculation of characteristic parameters
Assuming that the upper and lower critical values of the evaluation level corresponding to the comprehensive bus service quality evaluation index
are
and
, respectively, the calculation formulas of the parameters
,
, and
are shown in (4).
Symbolic notations used in (4) are defined as follows.
: A constant determined according to the degree of fuzziness, and its value is generally set as 0.1 [
28,
29].
- (3)
Determination of cloud membership in extension cloud model
Here, we consider each index value
x as a cloud droplet, and generate a normally distributed random number
with expected value
and standard deviation
. After determining the number of cloud droplets
, the cloud membership degree
between each index value
and the normal cloud model is calculated.
Symbolic notations used in (5) are defined as follows.
: Membership degree between index value x and extension cloud model; and
: Random numbers with a normal distribution.
According to (5), the cloud membership degree between each evaluation index value and the normal cloud model can be obtained, and a comprehensive judgment matrix
is formed:
Symbolic notations used in (6) are defined as follows.
: Cloud membership degree between the valuation index for application performance index of pure electric buses and the j-th level normal cloud model ; and
: Evaluation level.
- (4)
Determination method of the evaluation level for the application performance
According to the comprehensive weight value obtained above, combined with the comprehensive judgment matrix, the comprehensive certainty
and comprehensive evaluation score
for the application performance index of pure electric buses can be calculated:
Symbolic notations used in (8) are defined as follows.
: Value of the j-th component corresponding to the vector ; and
: The score value of the evaluation level j. The score values corresponding to the evaluation levels 1 to 5 in this paper are 1, 2, 3, 4, and 5, respectively.
Due to the randomness in solving the membership degree, it is necessary to solve it multiple times to reduce the influence of random factors. The calculation method of the expected value
, the entropy
, and the reliability factor
θ of the comprehensive evaluation score is shown in (9).
Symbolic notations used in (9) are defined as follows.
: Number of operations;
: The comprehensive evaluation score obtained by the i-th calculation; and
θ: The degree of dispersion of the evaluation results. Its value is inversely proportional to the credibility.