With the formation of urban rail transit network, the traffic keeps increasing. For rail transit system, it has become the operational philosophy of rail transit system to provide safe, fast, comfortable and high-quality services for passengers and to promote the sustainable development of urban rail transit through standardized operation. In this section, the sustainable operation of rail transit system is taken as the evaluation object to evaluate the sustainable operation of rail transit system and effectively combine the customer demand with the operational strategy. At first, this study constructs the evaluation model and indices based on PSR; and then, the three-phase evaluation method is proposed to obtain the operational strategy, which is negative entropy flow.
4.1. The Evaluation Model and Indexes of the Sustainable Operation of Rail Transit System Based on PSR
In the rail transit system, due to the problems in service level, operation capacity and social environment, customer requirements cannot be satisfied, and positive entropy flow will be generated. The increasing entropy brings chaos to the existing system, which will weaken the sustainability of the system’s operation. The negative entropy flow can offset the increasing entropy of the system, which makes it have the characteristics of dissipative structure. These characteristics of dissipative structure can maintain the sustainable operation of rail transit system. Therefore, the negative entropy flow needs to be obtained. This study will describe the operation of rail transit system by the model of PSR and find the negative entropy flow, so as to achieve sustainable operation of rail transit system.
The PSR model is divided into three kinds of indices, namely pressure indices, state indices and response indices. In the rail transit system, pressure indices are viewed as customer requirements, which answer the reasons for such changes in this system; state indices refer to the state and environment of system activities, which can be described as the challenges of coping with pressure; “response” describes the system’s actions to address the challenges posed by customer needs, namely operational strategies. In this model, the pressure of customer requirements is the main cause of positive flow. If reasonable and effective strategies are adopted to act on this system, customer requirements will be satisfied. At that moment, the negative entropy flow will be obtained, and system chaos is reduced. Therefore, the reasonable and effective strategies in consideration of customer needs should be proposed to offset the increasing entropy, which can realize the sustainability operation of rail transit system. The PSR model is designed as shown in
Figure 1 and the specific indices are set as follows:
(1). Pressure indices
Due to the limited energy of environmental resources and managers in the operation process, it is customer requirements that become the pressure on the sustainable operation of the system. In order to obtain the information of customer requirements, this study collects them by means of operational data, the sources of network, the hotline of service supervisor, interview, etc. In the process of gathering customer requirements, we find that passengers on the subway can be subdivided into different customer groups by ages, regions, population density, travel reasons, travel time, etc. By ages, it can be divided into: child, youth, adult, old age, etc.; By region, it can be divided into: local, non-province, foreign country, non-local, etc.; By population density, it can be divided into: downtown, suburbs, exurbs, etc.; By travel reasons, it can be divided into: work, school, business, personal shopping, sightseeing, visiting relations, etc.; By travel time, it can be divided into: weekday day, weekend day, weekday night, weekend night, etc. However, not all customer groups are attractive to rail transit system, for the reason that the main customer requirement groups are determined by Mckinsey Matrix, which are: child, youth, adult, old age, local, downtown, work, school, business, personal shopping, sightseeing, weekday day, weekend day, weekday night and weekend night. Through these different groups, pressure indices are summarized, which are shown in
Table 1:
(2). State indices
In the investigation of customer requirements, we find that the different groups have different emphasis, such as: the customer groups of “child” and “old age” have emphasis on security needs and focus on “stable operation”, “security of closing and opening doors”, etc.; the customer groups of “youth” and “work” pay attention to the convenience and timeliness of travel, etc. Based on that, facing to the pressure of customer requirements, five state indices reflect the challenges of the sustainable operation of rail transit system, which are shown in
Table 2.
(3). Response indices
Response indices, that is, operational strategies. To copy with the pressure of customer requirements and the challenges faced by the sustainable operation of rail transit system, it need to strengthen the management of technology, operations, services, resources and cost. For technical management, this system can consider from design standards and informatization; For operational management, the system can discuss from operational rules, passenger flow control, emergency linkage, etc.; For service management, this system can analyze from service process and training standards; For resource and cost management, the system can consider from subway facilities, the pricing mechanism and ticket card management. Due to that, response indices can be summarized in
Table 3.
4.2. The Three-Phase Evaluation Method of the Sustainable Operation of Rail Transit System Based on QFD
Quality function deployment (QFD) [
35,
36] is used to deeply analyze customer requirements (CRs) to meet the market and customers, and then to transform them into technical attributes (TAs).
By the evaluation model and indices of rail transit system in
Section 4.1, this study first, takes customer requirements (pressure indices, Ps) as input variables and obtains state indices (Ss) from Ps. Moreover, then, based on QFD, this study takes state indices (Ss) as CRs and response indices (Rs) as TAs, the rating of Rs can be obtained and the reasoning and effective operational strategies are provided, that is negative entropy flow. Moreover, then, the three-phase evaluation method is proposed. The evaluation methods include fuzzy clustering analysis, evidential reasoning, fuzzy weighted average and expected value. Moreover, the detailed steps are shown in
Figure 2.
In the evaluation method based on QFD, we need to collect the data to obtain the weights of Ss and to determine the relationship matrix between Ss and Rs. In the process of data collection, we conduct a questionnaire survey to get pressure indices, according to the customer groups as following: child, youth, adult, old age, local, downtown, work, school, business, personal shopping, sightseeing, weekday day, weekend day, weekday night and weekend night. Meanwhile, interviews and questionnaires were conducted with experts to analyze “the characteristics and features of Ps” and “the relationship between Ss and Rs”. Based on these data, this study will obtain Ss from Ps by fuzzy clustering analysis, determine the weights of Ss by evidential reasoning and calculate the importance of Rs by fuzzy weighted average and expected value operator. Therefore, the three-phase sustainable operational evaluation method of rail transit system is developed:
Phase 1. Obtaining Ss from Ps by fuzzy clustering analysis
In this phase, fuzzy clustering method can be used to classify customer requirements to reduce the complexity of calculation.
Step 1a. Data standardization
Suppose that
state indices are used to describe
pressure indices, which denoted by
and each
has
kinds of indicators. Based on questionnaires to experts, taking
pressure indices as the row vector to get the requirements of original fuzzy matrix
,
,
, where
is a measure value of the
th pressure index described by
th state index. To meet the requirement of original fuzzy matrix, we need data conversion to narrow the data into the interval [0,1], which denoted by
. The conversion formula used standard deviation [
45] as follows:
where,
After that, if
is not in the interval of [0,1], the other formula used range should be carried out as follows:
At this time, it is obvious that each is in the interval of [0,1], so the fuzzy matrix is obtained denoted by .
Step 1b. Establish fuzzy similar matrix.
To research the similarity of customer requirements between
and
, the method of correlation coefficient is used as follows [
45]:
where,
Therefore, the fuzzy similar matrix is obtained, which denoted by , where is the correlation coefficient between th index and th index If the correlation coefficient of two requirements is larger, their characteristic are closer.
Step 1c. Fuzzy equivalence matrix and cluster analysis
The reason seems to be obvious that the fuzzy matrix
T has reflexivity and symmetry but not necessarily has transitivity, the fuzzy equivalent matrix should be calculated by taking the Quadratic method [
45]. we start from fuzzy similar matrix
, as follows,
is the synthesis matrix of
. In this way,
, and there exist a value
, which makes
Then the fuzzy equivalence matrix is the transitive closure , that is .
In addition, we can classify the pressure indices through the cut relation of
. For random
; we get different
-cut relation
, the formula as follows:
The different can be acquired when we vary the value . Moreover, according to that, the right classification can be gained when the proper value is taken.
Phase 2. Calculate the weights of Ss by evidential reasoning
In this phase, evidential reasoning method will be used to evaluate the sustainable operation of rail transit system. The five state indices are obtained by the classified result from Phase 1, which are safety index, reliability index, convenience index, comfort index and economy index. The specific progress is as follows [
65]:
Step 2a. Collecting the initial information of evaluation
Suppose that each state index ( = 1,2,…,) is composed of pressure index denoted by , which is seen as a simple secondary index layer. The relative weight of is shown as , and , . Let pressure index can be evaluated at five different grades described by , which are very important, important, moderate, less important and unimportant. The original assessment set of is SS() = {()}, the belief degree represents the likelihood that the index is assessed to .
By investigation and collection, the single evaluation of each pressure index given by customers, the belief degree
is obtained by:
where,
represents the number of customers who assess the pressure index
to the degree
and
represents the total number of customers who assess
in the evaluation.
Step 2b. Calculate the assessment set of Ss.
By the theory of D–S, the belief degree
which represents the state index
(
= 1,2,…,
) is assessed to the degree
is synthesized. See Formulate (4)–(12):
where,
be a basic probability mass,
be a remaining probability mass,
represents the degree to which other indices can play a role in the assessment,
is caused due to the incompleteness in the assessment
SS(
), S
S(
) will be zero if S
S(
) is complete.
Next, the combined probability masses
SS(
) are generated by aggregating the assessments S
S(
) and S
S(
)as follows:
where,
is the combined probability mass for the grade
by aggregating
assessments for the state index
. Finally, the formula of assessments
SS(
) as follows:
Step 2c. Calculate the weights of Ss.
Let the five different grades denoted by
, (
n = 1,2,…,5), which belongs to a predefined triangular fuzzy set
, that is,
. Therefore, the weights of state indices are as follows:
It is obvious that is a triangular fuzzy number.
Phase 3. Rating the importance of Rs by fuzzy weighted average and expected value
Vangeas and Labib [
66] first suggested the use of fuzzy weighted average in QFD and proposed a model for deriving optimum targets of TAs through the implementation of fuzzy weighted average, whose membership functions are nonlinear in essence and is not explicitly known in most cases. This method can hardly be applied while the derived membership function of fuzzy weighted average is not explicitly known. In order to overcome the above problem, Chen [
35] et al. proposed the method by integrating the fuzzy weighted average method and the fuzzy expected value operator, which is so effective that it is widely used. The specific progress is as follows:
Step 3a. Collecting the data
Based on the weight of each state index obtained by the last phase, the fuzzy relationship
between
and
is just determined. The calculating formulate as follows:
where,
represents the fuzzy relationship between
th state index
and
th operation index
, which belongs to a predefined triangular fuzzy set
,
, that is
;
v is the number of experts participated in the survey.
Step 3b. Calculating the fuzzy importance of s
Let
Where
and
are the membership functions of
and
respectively,
and
are crisp number sets. Based on the fuzzy weighted average method, the fuzzy importance of
s can be obtained as follows:
It is evident that
is a triangular fuzzy number.
Then, the fuzzy weighted average
is defined as the following membership function
, with respected to the fuzzy extension principle [
35]:
Clearly, the equation above can be converted into the equivalent NLP model as follows:
Due to the functions
and
are nondifferentiable, so it is difficult to get the optimal solution in this model. In order to solve this problem, by handling the
cuts of
to replace the above model. Let
,
are the
cuts of
,
, respectively:
Similarly, let
is the
cuts of
. Moreover, then,
can be obtained by the following NLP models:
and
Let
, the models of (3–5) and (3–6) can be transformed into the following LP models:
and
Taking the different value
, the following approximate membership function
of operation index
can be calculated by the models of (20) and (21):
Step 3c. Rating the importance of Rs
Based on the expected value operator, the rating of Rs can be gained by calculate the expected value
:
where,
represents the set of
.