2010 International Conference on Management Science & Engineering (17th)
November 24-26, 2010
Melbourne, Australia
A Value-function-based Model for Evaluating the Passenger Satisfaction
in the Flexible Community Bus Problem
HU Xiang-pei，WU Li-rong，WANG Wen-juan，LIU Lin-lin
Institute of Systems Engineering, Dalian University of Technology, P.R.China, 116024
Abstract: Measuring the satisfaction is a
complicated task because humans are of bounded
rationality. The relationship between operational costs
and passenger satisfaction has not been established
neither in the definition of satisfaction based on customer
perceptions nor in the studies using the definition based
on specifications in operation research (OR) models. We
will align the passenger requirements and cognition with
the specifications used in OR models by applying the
value function of prospect theory. In our paper, the
reference point hypothesis of prospect theory is applied
to measure the level of satisfaction for the flexible
community bus problem, which can overcome the
one-sidedness of traditional satisfaction evaluation either
defined by specification or by user perceptions. The aim
is to insert as many as unexpected passengers as possible
within certain satisfaction level. The numerical results
proof that the flexible route plan can insert more
unexpected passengers with lower cost and higher
passenger satisfaction compared with the fixed and
timetabled route. It is also revealed the model has strong
adaptability and can be applied in various groups by
adjusting the specified satisfaction level.
Keywords: flexible community bus, passenger
satisfaction, prospect theory, value function
1 Introduction
Flexible Transport Services (FTSs) emphasize on
passengers’ convenience and involves many interactions
with passengers, so how to denote the passengers’
perceptions is extremely important in enhancing the
service quality [1]. Flexibility would produce great
negative effect on passengers because humans have
strong cognition towards variety and uncertainty. So in
order to solve the FTSs problem either in the models or
algorithms, measuring passenger satisfaction is the most
important step to take.
The traditional regular transport services are
generally characterized by a defined itinerary and the
route, bus-stops and timetables are planned in advance [2].
Supported by the National Natural Science Funds for
Distinguished Young Scholar (70725004) and the National
Natural Science Foundation of China (70890080, 70890083 )
The vehicles travel just according to the timetable strictly
which often results in idling or deadhead during some
time bucket. While in reality, many kinds of disruptions
would happen during the route such as travelling time
fluctuation, new requests, vehicle breakdowns,
cancellations, etc. Considering this kind of real-world
problem, the traditional fixed route is not suitable,
especially in such condition as the requests have the
characteristics of higher decentralization and lower
density which may cause vacancy ride and longer
waiting time for the passengers. And this kind of
problem also exists in many such communities as
campus and tourist spot. The problems mentioned above
are the typical issues that Flexible Transport Services are
trying to solve[3].
As far as we are aware, many researchers have done
lots of related work in the models and algorithms for
FTSs, but the researches on passenger satisfaction are,
comparatively speaking, less. Jurgen B. developed a
waiting-strategies-model to solve the real-world vehicle
routing problems, which demonstrated that a proper
waiting strategy could significantly increase the
probability of being able to service the additional
customer, at same time reducing the average detour to
serve the customer [4]. Ichoua et al. have proposed a
diversion method for a dynamic vehicle routing problem
with new requests and have modified a tabu search
algorithm. The authors also proposed that time pressure
is important because vehicles are moving fast and
diversion opportunities may be quickly lost [5]. Denson
used four attributes, pickup time, trip denials, missed
trips, and ride time length, which are the reports on
Californian provider’s service standards, to guide the
development of qualitative indicators for evaluating the
quality of a public Delaware dial-a-ride service from an
external perspective[6]. Knutsson used five dimensions
(i.e. information, dignity, comfort, travel time and fare)
and 40 attributes to explore the index itself. A response
rate highlighted the six qualities attributes which were
the most important for users, but another survey revealed
that the most important dimensions and attributes are not
the same for various groups or uses [7]. These are the
qualities as defined by user perceptions, but different
user groups do not have the same level of satisfaction
with respect to the different service attributes. The OR
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literature on the subject of technical quality in dial-a-ride
problem is quite extensive, even though the same
specifications are almost always used in order to
facilitate a more direct comparison of the algorithms.
The different ways in which quality is expressed in OR
models include difference between actual and desired
delivery time, time windows, total waiting time [8], total
ride time [9], excess of maximum ride time [10], maximum
ride time [11], etc. These specifications relative to time
service elements are easily quantifiable and statistical
methods from the operations research literature can be
used to measure those attributes. The challenge is how to
measure the qualitative attributes which involves
passengers’ cognition. Coslovich L. et al. used the level
of dissatisfaction as a measure of quality of service for
the customers, which considered the deviation from
desired service time and the excess ride time, the latter
being the difference between the actual ride time and the
shortest time needed to serve the request [12]. Wong, K.I.
presented a modified insertion heuristic to solve the
DARP with multi-dimensional capacity constraints. The
model emphasized on time window to represent quality
of service [13]. The distinctness of the DARP is that
people rather the merchandise are transported which
emphasizes more on the human convenience and thus
service quality must be taking into account. Neither the
quality as defined by customer perception nor quality as
defined by specifications can represent the real quality of
service. There is a gap between the real (waiting)time
and the cognitive (waiting)time, whereas the latter one
can represent the passengers’ satisfaction more precisely.
The paper will combine quality defined by customer
perceptions and that defined by specifications together to
solve the mentioned issue. Here we will apply the value
function of prospect theory to combine the two kinds of
quality, which would link quality to other concepts and
help managers understand the multiple facets of quality
and eventually improve it.
Prospect theory is proposed by Kahneman and
Tversky in 1979 which emphasizes on the
bounded-rationality behavior of the individuals [14]. In
previous study, individual passenger behavior is viewed
as rationality, searching for the optimal outcome, which
deviates from the reality by oversimplifying the model
and ignoring that the passengers are partially rationality.
This is the exact essence of prospect theory. Prospect
theory treats value as a function in two arguments: the
asset position that serves as the reference point and the
magnitude of the change (positive and negative) from the
reference point (Kahneman and Tversky, 1979, 2000) [15].
Prospect theory was initially used in the area of decision
under risk, and our paper only cited the value function of
prospect theory to represent the cognitive degree of
passengers towards loss or gain, and to express the level
of satisfaction. For an earlier application of prospect
theory in the transportation field, see Aviner and Prashker,
2003[16], to auto commuters’ departure time
decision-making and to obtain a better understanding of
how auto commuters use arrival time information in
daily departure time choice, and 2004[17], aspects of
passengers' bounded-rationality behavior are studied and
focus on the decision-making process of the individual
passenger who has to select a bus line. The effect of
information format on individual choice is studied by the
examination of subjects' preferences of bus lines.
In this paper, the focus is on one of the most
interesting problems in FTS service: how to measure the
passenger satisfaction by considering the passengers’
sensibility to the extra waiting time and extra riding
time? We set up a real-time scheduling model
incorporating the reference point hypothesis of value
function which considers the passengers’ cognitive
behavior in extra waiting or riding time. The extra
waiting time relative to the reference point (average
waiting time) and the extra riding time relative to the
reference point (direct riding time) are two key factors
whose effects on the satisfaction of passengers are
investigated in this study through theoretical model
development and statistical analysis with surveyed data.
In this paper, we divide the solution approach into two
phases, before and after the disruption, to solve the
problem. The first phase is called static routing
generation using genetic algorithm and the second one is
to insert the new requests flexibly by C-W algorithm.
The paper is organized as follows. The next section
describes how the scheduling model can be modeled
while incorporating the reference point hypothesis of
value function. Section 3 uses a C-W algorithm to insert
the unexpected passengers and the numerical experiment
is presented in section 4. Conclusions are summarized in
the last section.
2 Problem description and its model
2.1 Problem description
In this paper, we handle one kind of FTSs called
flexible community bus problem, which derives from the
real-world phenomenon that happened in an industrial
community consisting of high-tech companies. The
flexible route is designed to provide service after rush
hours of ending times of companies to pick up
passengers who are working overtime from his nearest
stop to the gate(s) of the community. Thus the density of
customer requests is low and decentralized. The requests
can be sent to the bus scheduling center through internet,
short messages or the receiver at every bus stop. The
real-world events would take into account passengers’
behavior and consider their cognition for extra waiting
time and extra riding time.
As mentioned above, many kinds of disruptions
would happen during the route such as travelling time
fluctuation, new requests, vehicle breakdowns, and
cancellations. Different strategies would be utilized
according to certain kind of disruption. Because the
disruption of new requests is the most common
phenomena in transportation service and it concerns
human convenience, here we only consider disruption of
new requests [18].
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Before considering mathematical formulations, we will
define the problem. We considered the demand response
transport problem of single vehicle with capacity
constraints (e.g. the number of passengers on a bus has
an upper bound.). The passengers are supposed to have
the same destination (i.e. the gate of the community). As
is shown in fig. 1, the stop 0 stands for the initial and
destination of one route (i.e. the gate of the community).
The traditional timetabled route is 0-1-2---8 which is
established by static optimization method. In our paper,
as shown in fig. 1, we only consider the stops which are
demanded by passengers ahead and generate a flexible
route as shown in the right route in fig. 10-1-4-7(as only
stop 1、4、 7 exist passengers ). And the new unexpected
passenger occurred at stop 5 while the bus located
between stop 1 and stop 4.
4
3
2
average waiting time at stop i
actual pick-up time for passengers at stop i
responded pick-up time at stop i
advance passengers at stop i
unexpected passengers at stop i
the travel time from stop i to 0 under the
static condition
Tdepot the time when the bus arrives at the depot
Tewi
extra waiting time for the passenger at stop i
Teri
extra riding time for the passengers at stop i
Tsi
the times for serving stop i
Finally, we consider the satisfaction level (SL) as a
measure of quality of service for the passengers. The SL
concludes two parts, one is the value for extra waiting
time, and the other is value for extra riding time. Here we
will first apply value function of prospect theory to
measure satisfaction level. Then more formally, we
define the traditional dissatisfaction level (DL)
(represented by time directly) for passengers at stop i as:
Tiw
Tip
Tir
Pis
Pid
Ti
5
Tewi = Tiw − (Tip − Tir )
6
1
Teri = Ti − (Tdepot − Tip )
7
0
DLi = ⎡⎣Tiw − (Tip − Tir ) ⎤⎦ + ⎡⎣Ti − (Tdepot − Tip ) ⎤⎦
8
Timetabled route
(1)
(2)
(3)
Then we convert the above quality specifications to
satisfaction level based on passengers’ cognition by
combining with value function of prospect theory which
is illustrated in Fig.2.
Demand-responsive route
Value
function (V)
Re
fe
r
en
ce
po
in
t
Vgain
0
Waiting time (riding time) deviation
in a loss region
Flexible route
Waiting time (riding time) deviation
in a gain region
Fig.1 Example of a flexible routing
The objective of our problem is to maximize the
quality of service for passengers, which is presented by
satisfaction embodied not directly by extra waiting time
and extra riding time, but the passenger-perceived value
caused by such extra time.
2.2 Conceptual background
We define the transportation network in terms of a
complete undirected weighted graph G = (V, E)
where
the
vertex
set
V = {v0 , v1 , v2 , ……， vn } represents all the
possible bus stops (the depot is ,in the following, referred
to as v0 ). The arc set E = vi , v j : vi , v j ∈ V stands
{(
)
}
for the shortest path between the stops.
Let i be the stop index, the variables used in the
model are summarized as follows for reference:
Vlose
Fig.2 Value function in prospect theory
⎧ Tα , T ≥ 0
V (T ) ⎨
β
⎩ − λ ( −T ) , T ≤ 0
(4)
A value function is postulated for measuring the
satisfaction level of passengers as shown in Fig. 2. It is a
function of the extra waiting time (extra riding time) and
the gain and loss derived from two reference points.
Choosing the proper reference point is of ultimate
importance in order to represent the passengers’
cognition precisely. And different decisions depend on
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different reference points. Here we define the two
reference points as average waiting time and fixed riding
time according to our survey. Take the waiting time for
example, the origin represents the reference point which
in this paper we use average waiting time to represent.
The net value would be positive if the passengers’
waiting time is less than average waiting time. On the
other hand, the net value would be negative if the
passengers wait longer than average waiting time. If the
passengers’ waiting time is the same as Tjm, then the net
value is zero (therefore Tjm is an indifference point).
Another property of prospect theory is that the marginal
increase in subjective value decreases as the amount of
gain increases. The same relation holds for absolute
values of losses. Namely, the value function is concave
with respect to the origin in the gain region and convex
in the loss region. As the figure shows, the proposed
value function represents how the passengers respond
differentially to gains and losses [19].
In accordance with the principle of diminishing
sensitivity, it is concave for gains ( α ≤ 1 ) and convex
for losses ( β ≤ 1 ). Moreover, it is steeper for losses than
cognitive value can be counteracted or complemented by
each other. Formulation (4) is the value function of
prospect theory to convert extra time to cognitive value.
We define two reference points, one is the average
waiting time at every stop, and the other is fixed riding
time, which is denoted by formulation (1) and (2).
Formulation (6) indicates the insertion condition that the
total cognitive value for passengers is no less than the
value of set satisfaction level, or the insertion request
would be rejected. Here the range of the satisfaction
level value represents the adaptability of the model. The
condition that every stop is visited no more than one time
is expressed by formulation (7). The number of
passengers on a bus has an upper bound 30 which is
indicated by formulation (8).
for gains ( λ > 1 ) according to the principle of loss
we
define
aversion.
According
to[15],
α = 0.88 , β = 0.88 , λ = 2.25 .
So we combine the prospect theory with service of
quality, citing the concept loss aversion to replace the
weighted values, which is closer to passengers’ real
cognition by decreasing the influence of human
elements.
2.3 Model generation
According to the problem, we convert the real time
to cognitive time (here cognitive time is the value of real
time perceived by certain passengers) and generate the
value-function based model as follows:
n
max ∑ (V (Tewi ) + V (Teri ))
(5)
⎧ Tα , T ≥ 0
V (T ) = ⎨
β
⎩−λ (−T ) , T ≤ 0
(4)
Tewi = Tiw − (Tip − Tir )
(1)
i
s.t.
3 Solution approach
Teri = Ti − (Tdepot − Tip )
(2)
V (Tewi ) + V (Teri ) ≤ SL
(6)
Tsi ≤ 1
(7)
n
∑ (P
i =1
is
+ Pid ) ≤ 30
Fig.3 Process of solution approach
(8)
In formulation (5), the objective function consists of
two parts: cognitive value for extra waiting time for
passengers at stop i; cognitive value for extra riding time
for passengers at stop i. Here we hold that two kinds of
In this paper, we divide the solution approach into
two phases, before and after the disruption, which is
outlined in Fig. 3. Before disruption occurs, genetic
algorithm is applied to static route generation, which is
only for the static passengers before setting out. A buffer
time zone and a threshold of the request numbers are set.
During this process, the bus is waiting at the depot. Once
the waiting time exceeds the buffer time or the requests’
number exceeds the threshold, the initial route will be
calculated and the bus will start off.
When disruption happens, which is that a new
request occurs, the system should first judge whether the
stop has been finished within the current circle. If YES,
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the request must wait until next circle, and if NO, insert
the new request within current route with C-W algorithm,
and then call the model to determine accepting or
rejecting it, which is called real-time passenger insertion.
In this phase, the new requests would be inserted
dynamically to divert the bus from its current route in
order to satisfy as many passengers as possible. In the
real-time routing process, the most important issues are
quick response and quick optimization to serve more
dynamic passengers. In our research, we propose a
value-function-based model to evaluate the satisfaction,
which ensures that all passengers’ degrees of satisfaction
are above certain level. The model we proposed is
adaptive in application. For example, if the passengers in
one community are more sensitive to time or these
passengers are VIPs, we can adjust the parameter in
function (6) in order to fit different applications.
route, respectively. The degrees of serviced dynamic
requests are represented by service/total in column three.
The next two columns report the average extra riding
time and average extra waiting time within one circle.
Each of the 5 rows represents a test class and
summarizes the results of the comparison. The number of
total passengers varied, as well as the unexpected
passengers.
As it can be expected, the flexible route leads to
better results, that is to say the total cost is greatly
decreased. The cost has been saved more than 50% in
row two. Although extra waiting time is greater, but can
be counteracted by the decreased extra riding time. Here
the saved net time is determined by the parameter of SL
Tab.1 Results for comparing fixed and flexible route
Run
4 Numerical experiment
In this section we present some numerical results of
the solution approach based on the value-function-based
model, in order to test its performance, especially the
adaptability of the proposed model. We have built 5
different instances of the problem, which are based on
the real data collected through survey of the high-tech
zone. In experiment one, we compared the results of
flexible route based on value-function-based model for
evaluating satisfaction with the traditional timetabled
route using four indexes, the cost, percentage of serviced
passengers, extra riding time and extra waiting time,
respectively. In experiment two, different satisfaction
levels are set within one group data to validate the
adaptability of the model. The structure of the instances
is displayed in the following.
Our routes are based on Dalian high-tech zone, as
well as the stops. The distance of one fixed circle is 20
kilometers (average cost is 20 RMB) and there are 22
stops overall. The number of passengers, both static and
unexpected passengers, is base on the survey. We analyze
five groups of data, and the number of total passengers is
7, 12, 10, 15 and 14 respectively, with 5 static passengers
in each group.
We set two kinds of experiments; one is to test and
verify the performance of the flexible community bus
compared with the fixed timetabled route in the aspect of
both cost and satisfaction level. The other is to test the
adaptability of the value-function-based model. Here we
set Tiw =600, SL= -600, and the numerical results are
shown in Tab.1. And we test the data of group four,
setting SL=0, -400, -600 and -800, in order to test the
performance under different satisfaction level. The
results are shown in Tab. 2.
4.1 Performance of the flexible community bus
The first column reports the data sequences.
Column two is the cost (we transformed the distance to
cost denoted with RMB). R1 column represents for
results the fixed timetabled route and R2 for the flexible
Cost
(RMB)
R1
R2
Passengers
(Serviced/Total)
R1
R2
Ter(Seconds)
Tew(Seconds)
R1
R2
R1
R2
1
20
13.58
6/7
6/7
0
-260
192
143
2
20
8.48
7/12
6/12
0
-696
9
-253
3
20
9.82
5/10
9/10
0
-478
375
414
4
20
11.81
12/15
11/15
0
-729
-87
-79
5
20
12.56
11/14
8/14
0
-102
172
28
Tab.2 Results under different satisfaction level
LS
Cost
(RMB)
R1
R2
Passengers
(Serviced/Total)
R1
R2
T er(Seconds)
T ew(Seconds)
R1
R1
R2
R2
0
20
20.44
12/15
15/15
0
-192
-87
196
-400
20
14.71
12/15
12/15
0
-642
-87
49
-600
20
11.81
12/15
11/15
0
-729
-87
-79
-800
20
9.59
12/15
7/15
0
-899
-87
-70
we set in the model. We also need to balance the cost and
the percentage of serviced passengers. From row one and
three, we can see that the cost can be saved greatly
without sacrificing the benefits of more dynamic
passengers.
4.2 Adaptability of the value-function-based model
As we mentioned above, the model has strong
adaptability with different setting of parameter SL. This
experiment aims to validate adaptability of the model.
Here we set four different levels of satisfaction (SL), 0,
-400, -600 and -800 respectively. Take the fourth group
of data as the input, the results are shown in Tab. 2.
There would be less passengers being serviced within
one circle with the increase of the SL, while the cost is
reduced. The average waiting time is decreased no
deeper than the saved average riding time.
5 Conclusions
This paper designs a value-function-based model
for evaluating the satisfaction for flexible community bus.
The proposed model takes passengers’ behavior and their
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cognition for extra waiting time and extra riding time
into account. The model has strong adaptability and can
be applied into different situations according to the
characteristics of different user groups. In this paper, we
applied the model in high-tech zone to test its validity.
The effectiveness of the proposed model and solution
approach have been demonstrated through the numerical
experiment compared with the traditional timetabled
method.
There are also limitations in this paper. First, we
apply the value function of prospect theory directly to
denote the cognition. A new form should be established
according to the certain application area through
investigation, which would make the value conversion
more precise. Second, we have just started to consider
one vehicle routing problem, and the multi-vehicle
problem has to be investigated in further research to
improve its performance on dealing with more
complicated problems [20, 21].
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