Optimization for Urban Rail Transit Network Planning Evaluation
Based on Virtual Unit and AHP-DEA Model
Zhaoping Cai
1
and Baoming Han
2
School of traffic and transportation ,Beijing Jiaotong University ,Beijing ,100044 ,P. R. China
1 2
Keywords: Evacuation, data envelopment analysis, the efficiency index, virtual unit.
Abstract: Taking the optimization method for urban rail transit network planning as the background, this paper analyses
the characteristics of existing evaluation methods, and establishes a comprehensive AHP-DEA evaluation
model based on the traditional DEA method. The first stage of the model calculates the weight of criterion
level as to the target level by AHP, and the second stage using DEA to calculate the efficiency index of the
plan level for the criterion level. Finally, the overall weight of each plan is obtained from the two stages. In
addition, the problem of DEA aberration is solved by introducing the virtual unit. Finally, the proposed
method is applied and shows good effectiveness.
1 INTRODUCTION
Urban rail transit is a large-investment and long-cycle
project, and it is difficult to change once built.
Therefore, the decision of network plan is particularly
important. At present, the main evaluation methods in
related literature can be summarized as two major
categories:qualitative and quantitative.
The representative of qualitative method is
analytic hierarchy process (AHP)（Saaty TL,1990）
, in which the subjective decision of decision-makers
playing a decisive role. In the field of urban rail
planning, it’s crucial for experts to make decisions
based on the planning experience of other cities.
However, it is difficult for decision-makers to
achieve satisfactory decisions when faced with too
many decisions. Therefore, the quantitative method is
the mainstream choice in present studies, which
mainly includes: entropy method(Qian B.Y.and Zhao
L.,2017), grey relational analysis(Ren L.,2010),
fuzzy hierarchical evaluation(Li J.F.and Wu
X.P.,2007), conventional data envelopment analysis
(DEA) (Zhang Y.Z., Yan Y.S.，Jiang N.,and Zhang
H.W.,2010)etc.
While analyzing the drawback of traditional
DEA, this paper puts forward a new method
combining the advantage of AHP and DEA，which
not only reserves the qualitative evaluation of the
indicators, but also conveys the preference of
decision-maker. In addition, a virtual DMU is
introduced to optimize the solution dilemma where
DMUs are effective at the same time, which shows
good robustness in solving the evaluation of urban
railways network planning problem.
2 IMPROVED DEA MODEL
2.1 The Conventional DEA model
DEA is a well-known method for efficiency
measurement based on multiple inputs and multiple
outputs which is originated by Charnes, Cooper, and
Rhodes (1978). Assuming there are n DMUs with m
dimensional input vector and s dimensional output
vector, we can define the ith input and output of
DMU
j
as X
ij
and Y
ij
，taking the ratio of output and
input as the efficiency index
j

to seek the best
combination of weight values for the decision makers
, the initial CCR DEA model for evaluating the
efficiency of DMU
0
can be presented as follows(Wei
Q.L., 2004)：
0
0
..
0, 1, 2 ,
1
0, 0
T
j
TT
jj
T
max Y
st
XYj n
X

















(1)
Cai, Z. and Han, B.
Optimization for Urban Rail Transit Network Planning Evaluation Based on Virtual Unit and AHP-DEA Model.
In 3rd International Conference on Electromechanical Control Technology and Transportation (ICECTT 2018), pages 343-346
ISBN: 978-989-758-312-4
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
343

The conventional DEA model can identify
efficient and inefficient DMUs. Those with
efficiency index equal to 1 are called efficient while
DMUs with efficiency index are less than 1 are called
inefficient. Obviously, inefficient DMUs can be
ranked by efficiency index directly, whereas efficient
DMU can’t achieve that because there are generally
more than one efficient DMU，which is called “DEA
aberration”（Wu Y.H., Zeng X.Y., Song J.W.,1999）
. The more inputs and outputs there are, the more
serious the aberration will be.
2.2 The introduction of virtual unit
When it comes to using the CCR DEA model to
evacuate urban rail transit network planning, the
aberration is prominent as a result of few DMUs
along with a large amount of inputs and outputs.
Aiming at this problem, this paper introduces an
optimal virtual DMU with the minimum input and the
maximum output based on the literature (Wang L.,
Liu C.R.,2010), so that DMUs can be further
distinguished. The virtual DMU can be defined as
follows:
Assuming that



1,2,...,
min
1
x
xi m
ij
i
jn


，



1,2,...,
max
1
yyrs
rj
r
jn


， then the virtual
DMU can be presented by

n+1 n+1
X Y、
,where

T
mn
xxxX ,...,,
211


、

T
sn
yyyY ,...,,
211


,  the
improved DEA model based on virtual DMU can be
presented as followed:
0
0
..
0, 1, 2 , 1
1
0, 0
T
TT
jj
T
max Y
st
XYj n
X

















(2)
In the model, the input and output of the original
decision unit is replaced by the virtual unit, so that the
efficiency index is reduced relative to the virtual
unit., therefore the decision units (network plans) can
be ranked accordingly.
Structure
evaluation
Operation
effect
Project
implementation
Urban
development
Total length
Number of transfer nodes
Density of center line
Average distance between stations
Section non-equilibrium factor of
passenger flow
Average speed of motor vehicle
Load of network
Total daily passenger volume
Rail transit travel ratio
Travel time saved by rail transit
Project facility value
Investment estimation
Rationality of staging construction
Anastomosis with land use
Coordination with urban layout
Meet the needs of urban
development
Plan 1
Plan 2
Plan n
...
Evacuation
Index
system

Figure 1: The evaluation index system
2.3 The AHP-DEA Comprehensive
Evacuation Model
In the process of urban rail network planning
decision, allowing for factors such as land use and
urban layout coordination, decision-makers often
have subjective preferences, which directly affect the
final decision. Allowing for it, the paper introduces
AHP to reflect preference of decision-makers and
constructs a AHP-DEA comprehensive valuation
model based on virtual units. The structure of the
model is shown in Figure 2.
The AHP-DEA Comprehensive Evacuation Model
Establish judgement
matrix
Classify the indexes into
input and output
Calculate The Weight of
Criterion Level
Calculate efficiency
index of the plan level
Comprehensive
model
The optimum plan
AHP DEA
Virtual unit
Figure 2: The structure of the AHP-DEA comprehensive
evaluation model.
ICECTT 2018 - 3rd International Conference on Electromechanical Control Technology and Transportation
344

2.4 Calculate Weight of Criterion Level
The first step is establishing the judgment matrix

according to the “1-9 scale method”, the maximum
eigenvalue 

and eigenvector W are obtained ，
where AW=λ
m
W, then the n component of W is the
weight of the n factors correspondingly.After
calculating, consistency check is necessary to ensure
the reasonablity of 

.
2.5 The efficiency index of the plan level
to the criterion level
As for criterion i (i=1, 2, 3, 4, representing structural
evaluation, project implementation, operation effect
and urban development, respectively)， classifying
the indexes of evaluation system by taking the cost
index and the benefit index as input and output
respectively, we can define the rth input and output
of DMU
j
as x
rj
and y
rj
, regard each plan as a DMU,
and establish DEA model based on virtual unit for
each criterion by the method referenced in Section 2
：
,
max
ij r rj
rj
Euy

,,
,
0, 1, 2,.. 1
1
0, 0; 1, 2,
rrj rrj
rj rj
rrj
rj
rr
vx y j n
vx
uvr







 








(3)
Where 

represents the evacuation index of the
j scheme for criterion i.
2.6 Comprehensive Evacuation
Calculating the overall weight of each plan M
j
based
on W
i
and E
ij
obtained above:
i
i
ijj
weM



4
1
(4)
3 NUMERICAL EXAMPLES
In this section, we present a numerical example
Urban rapid rail transit network planning in
Changsha taken from related literature(Meng
X.D.,2007) to apply the new proposed AHP-DEA
model for selecting the most efficient DMU, The data
for this example is given in table 1.
The specific steps are as follows:
①Establish the evaluation index system for urban
rail transit network planning (as shown in Figure 1);
②Calculate the weight of the criterion level as to
total level:
Table 1: Evaluation index of urban rail transit network planning in Changsha.
T
yp
e Index name Plan 1 Plan 2 Plan 3
Input index
Total len
g
th 172.1 163.9 172.2
Number of transfer nodes 14 10 12
Section non-equilibrium factor of passenger flow 2.36 2.59 2.65
Project facility value 6.68 7.83 8.14
Investment estimation 8.61 7.9 7.84
Output index
Densit
y
of center line 31 29 31
Avera
g
e station s
p
acin
g
2.1 2.07 2.24
Average speed of motor vehicle 80 75 80
Loa
d
of networ
k
3.51 4.06 3.48
Total dail
y
p
assen
g
er volume 376.6 364.6 372.2
Rail transit travel ratio 34.76 36.39 37.49
Travel time save
d
by
rail transit 11.88 11.81 12.12
Rationality of staging construction 8.22 7.9 7.47
Anastomosis with land use 8.61 7.9 7.84
Coordination with urban layout 6.39 6.42 6.62
Meet the needs of urban develo
p
ment 8.74 7.7 7.55
Optimization for Urban Rail Transit Network Planning Evaluation Based on Virtual Unit and AHP-DEA Model
345

The judgement matrix
11 53
134
A=
1/5 1/3 1 1/2
1/3 1/4
1
21






is constructed and then the maximum eigenvalue


and eigenvector W are obtained: λ
m
=4.085 ,
  
TT
W= = 0.6894, 0.6699, 0.1514, 0.2287
i

.Where 

is the weight of the criterion as to total
level correspondingly.
③Calculate the efficiency index of the plan level
as to criterion level;
Table 2 is the efficiency index of each plan for
each criterion calculated based on the method
referenced in Section 3.4:
④Comprehensive evacuation. Table 3 gives the
calculation result of overall weights of plan 1、plan
2 and plan 3：
Table 2: Efficiency index of plan level to criterion level.
Evaluation criteria Plan 1 Plan 2 Plan 3
structure evaluation 0.952 0.935 1.000
operation effect 1.000 0.911 0.891
Project Implementation 1.000 0.902 0.908
urban development 1.000 0.884 0.920
Table 3: The overall weight.
DMU Plan 1 Plan 2 Plan 3
overall weight 1.709 1.596 1.637
As we see from Table 4：M
1
>M
3
>M
2
, plan 1 is
the most efficient DMU ranked as the top position,
which is consistent with the result of the literature
(Meng X.D.,2007) obtained by improved multi-
objective decision making model, as well as the final
result of Changsha urban rail transit network planning.
It is proved that the AHP-DEA comprehensive
evaluation model based on virtual unit proposed in
this paper is feasible. And for the decision makers,
this model could be further applied in performance
evaluation.
4 CONCLUSION
Based on the existing DEA model(CCR), this paper
introduces the AHP to reflect the preference of
decision-maker in evacuation of urban rail transit
network planning, and a comprehensive evacuation
AHP-DEA model is proposed for finding the
optimum plan. Furthermore, confronted with the
problem that the traditional DEA model may appear
all effective DMUs, when there are multiple inputs
and multiple outputs (especially the number of DMU
is far less than the number of indexes), a virtual unit
is introduced in order to distinguish DMUs, which
provides a good solution to DEA aberration, thus the
proposed model is of strong practicability compared
with tradition model. However, the current model
doesn’t consider the select of input and output data in
detail, which is an issue in the latest literatures.
Further important future research directions would be
selecting the more efficient data for the model by
additional restraints or developing models to deal
with fuzzy data.
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Li J.F., Wu X.P.,2007. Comprehensive Evacuation of
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Zhang Y.Z. Yan Y.S. ， Jiang N., Zhang H.W.,2010.
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