Journal of Transportation Technologies, 2012, 2, 220-229 Published Online July 2012 (
Optimisation of a Bus Network Configuration and
Frequency Considering the Common Lines Problem
Hiroshi Shimamoto1, Jan-Dirk Schmöcker1, Fumitaka Kurauchi2
1Department of Urban Management, Kyoto University, Kyoto, Japan
2Department of Civil Engineering, Gifu University, Gifu, Japan
Email: {shimamoto, schmoecker},
Received May 6, 2012; revised June 10, 2012; accepted June 25, 2012
Public transportation network reorganisation can be a key measure in designing more efficient networks and increasing
the number of passengers. To date, several authors have proposed models for the “transit route network design prob-
lem” (TRNDP), and many of them use a transit assignment model as one component. However, not all models have
considered the “common lines problem,” which is an essential feature in transit network assignment and is based on the
concept that the fastest way to get to a destination is to take the first vehicle arriving among an “attractive” set of lines.
Thus, we sought to reveal the features of considering the common lines problem by comparing results with and without
considering the problem in a transit assignment model. For comparison, a model similar to a previous one was used,
formulated as a bi-level optimisation problem, the upper problem of which is described as a multi-objective problem.
As a result, although the solutions with and without considering the common lines showed almost the same Pareto front,
we confirmed that a more direct service is provided if the common lines problem is considered whereas a less direct
service is provided if it is not. With a small network case study, we found that considering the common lines problem in
the TRNDP is important as it allows operators to provide more direct services.
Keywords: Transit Network Configuration and Frequency Design; Bi-Level Optimisation Formulation; Transit
Assignment Model; Common Lines Problem
1. Introduction
To entice travellers to shift from private cars to public
transportation, many public transport operators have ta-
ken measures such as reducing off-peak fares. Several
researchers have proposed models for determining opti-
mal fares [1,2] or optimal transit frequencies [3]. Another
measure could be to make the network configuration
more efficient because there are many inefficient bus
networks worldwide. Therefore, an optimal public trans-
portation route configuration is necessary as a benchmark
to determine the design of a new public transportation
route configuration. The problem of designing such a
network is referred to as the “transit route network design
problem” (TRNDP); it focuses on the optimisation of bus
routes or frequencies in order to optimise a number of
objectives representing the efficiency of public transpor-
tation networks (such as minimising passengers’ cost or
maximising profit) under operational and resource con-
straints such as the number and length of public trans-
portation routes, allowable service frequencies, and the
number of available buses [4].
Several researchers have proposed models for the
TRNDP. For example, [5] also proposed a bus network
optimisation model, the objective of which was to maxi-
mise the proportion of passengers travelling without
transfers, and solved the model by a parallel ant colony
algorithm. However, passenger behaviour principles
seem not to be described clearly in their model. [6] pro-
posed a model for optimising feeder bus routes, where
the transfer point from the railway to a feeder bus was
fixed and transferring between feeder buses was not al-
lowed. [7] formulated a simultaneous optimisation prob-
lem for railway line configurations and passenger as-
signments as a linear binary integer problem. Because
line frequencies were not determined in their model, they
charged a given transfer penalty as an additional waiting
time, but any additional waiting time due to a transfer
should be defined related to the service frequency (pas-
senger waiting time is short if the service frequency is
high). Another feature of their model is that a branch-
and-bound method was used as a solution algorithm to
obtain an exact solution whereas all previous models had
been solved with heuristic algorithms. However, they
simplified the network to solve the model within a rea-
sonable time. It would be difficult to apply a strict solu-
tion algorithm to a bus network optimisation problem,
opyright © 2012 SciRes. JTTs
which is, generally, more complex than a railway net-
work optimisation problem.
The literature review has so far revealed that many re-
searchers do not describe passenger route choice behav-
iour accurately. Thus, several papers include a transit
assignment model within the TRNDP to consider the
passenger route and transfer choice behaviour. [8] com-
bined a line planning model and a traffic assignment
model, and demonstrated a solution algorithm based on a
column generation method. However, their assignment
model was very similar to the traditional assignment
model and did not consider the “common lines problem”,
which is an essential feature in a transit network assign-
ment and is based on the concept that the fastest way to
get to a destination is to take the first vehicle arriving
among an “attractive” set of lines (the “attractive” set of
lines is referred to as the “hyperpath”). [6] demonstrated
a model framework for combining the TRNDP and a
transit assignment model considering the common lines
problem. [9] extended their model to determine the allo-
cation of a limited number of environmentally friendly
vehicles and applied the model to a real-size network
(although the details of the transit assignment model are
not described in either paper). [10] expanded the model
of [2] to optimise both the routes and frequencies of pub-
lic transportation, and they applied their model to a real
As described so far, some TRNDPs have considered
the common lines problem, while others have not. Con-
sidering the common lines problem implies that passen-
gers can consider the complex route set perfectly, which
so far was almost impossible in dense networks. How-
ever, due to personalised information technology, such as
smart phones, passengers can nowadays obtain better
knowledge of the complex route set. Also, passengers
will tend to use common lines if transit agencies aggre-
gate separate bus stops. Therefore, it is important for
transit agencies to know how the optimal network con-
figuration differs if passengers come to know the com-
plex route set. Thus, in this study, we explore the impor-
tance of the common lines problem in the TRNDP by
comparing results with and without considering common
lines. The model used in this study is similar to that of
[10], which is formulated as a bi-level optimisation
problem, the decision variables of which are the route
and frequency of each line, but the following aspects
were modified:
The assumption of a fixed origin/destination for each
bus line was relaxed by introducing a dummy origin
and destination node.
Vehicle number constraints were considered more
accurately by combining a vehicle assignment proce-
dure and a frequency setting procedure in the solution
Capacity constraint conditions were not considered in
this paper to save computational costs ([10] con-
firmed that capacity constraint conditions did not af-
fect the output of the TRNDP in a real network).
The remainder of the paper is organised as follows.
Section 2 describes briefly the minimum cost hyperpath
searching problem that is one component of the transit
assignment model proposed by [11]. Section 3 describes
a mathematical formulation of the bus network optimisa-
tion model, and Section 4 shows the solution algorithm.
Section 5 illustrates a case study with a simple network,
comparing the model with and without considering the
common lines problem. Finally, Section 6 provides con-
clusions and identifies future research.
2. Minimum Cost Hyperpath Searching
In this chapter, the minimum cost hyperpath searching
problem, one component of a transit assignment model
[11] that is used in the lower problem of the proposed
model, is presented briefly.
2.1. Network Representation
To consider the capacities of transit lines together with
the common lines problem, the transit network shown in
Figure 1(a) was transformed into the graph model shown
in Figure 1(b). An origin node represents a trip start
node. A destination node represents a trip end node. A
stop node represents a platform at a station. Any transit
lines stopping at the same platform are connected via
boarding arcs and failure nodes. At stop nodes, passen-
gers can either take a bus or walk to neighbourhood bus
stops, and if they take a bus, they are assigned to any of
the attractive lines in proportion to the arc transition
probabilities. A boarding node is a line-specific node at
the platform where passengers board. An alighting node
is a line-specific node at the platform where passengers
alight. Note that boarding and alighting node are se-
paretely defined in order to allow consideration of dwell
time and capacity constraints. line arc represents a transit
line connecting two stations. A boarding arc denotes an
arc connecting a stop node to a boarding node. An
alighting arc denotes an arc from an alighting node to a
stop node. A stopping arc denotes a transit line stopping
on a platform after the passengers alight and before new
passengers board; this arc is created to express the
available capacity on the transit line explicitly. A walking
arc connects an origin to a platform (access), a platform
to a destination (egress), and neighboring platforms
(walk to neighboring platforms).
Generally, the network representation used in public
transit assignment models requires more computer me-
mory compared to that used in road traffic assignment
Copyright © 2012 SciRes. JTTs
Copyright © 2012 SciRes. JTTs
models because of the many arcs and nodes. However,
this may be less of a problem considering recent progress
in computer technology. Moreover, because the compo-
nents of the graph network are simple, it is possible to
convert automatically from Figure 1(a) to Figure 1(b).
The minimum cost hyperpath searching problem is now
described. If the common lines problem is not considered,
it is easy to obtain a minimum cost path by applying a
shortest path searching algorithm, such as the Dijkstra
method, with the graph network shown in Figure 1(b).
2.2. Notation
We use the following notation regarding the transit as-
signment model. Other notation will be shown as appro-
Ap: Set of arcs on hyperpath p;
L: Set of line arcs;
Ll: Set of line arcs on line l;
Ul: Set of platforms on transit line l;
WA: Set of walking arcs;
BA: Set of boarding arcs;
Sp: Set of stop nodes on hyperpath p;
Vp: Set of elementary paths on hyperpath p;
OUTp(i): Set of arcs that lead out of node i on hyper-
path p;
wkl: Stopping arc of line l on platform k;
O1 2
3 45
Line I
Frequency = 1/5 minute
Capacity = 500 passengers / vehicle
Line II
Frequency = 1/10 minute
Capacity = 1,000 passengers/ vehicle
12 12 12
15 8
Line II
Line I
Figure 1. Network representation. (a) Example transit network; (b) Graph network.
bkl: Boarding arc of line l on platform k;
lows satisfying flow con-
cluded in l,
2.3. Assum
lowing assumptions regarding the
ading out of nodes on a
τap: Arc split probability on hyperpath p;
l(a): A transit line that is included in arc a
gp: Cost of hyperpath p;
; ca: Arc cost on arc a A
ta: Travel time on arc a A
ξ: On-board value of time;
;ζ: Value of time for walking
η: Value of time for waiting;
: Set of feasible hyperpath f
λlp: Probability of choosing any particular elementary
path l of hyperpath p;
αap: Probability that traffic traverses arc a;
βip: Probability that traffic traverses node i;
δal: Dummy variable, equal to 1 if arc a is in
herwise 0;
εil: Equal to 1 if elementary path l traverses node i,
otherwise 0;
Y: Vector of hyperpath flows;
X: Vector of arc flow;
/min). fl: Frequency of line l (1
We adopted the fol
common lines problem, similar to previous studies (See,
All bus lines operate with given exponentially dis-
tributed headways, and a mean equal to the inverse of
the line frequency. The distributions of the lines are
assumed to be independent of each other.
Passengers arrive randomly at every stop node and
decide whether to take a bus or walk. If they take a
bus, they always board the first arriving vehicle of
their choice set.
2.4. Arc Split Probabiliti
Where there are several arcs le
hyperpath, traffic is split according to τap. As shown in
Figure 1, passengers may be split at stop, failure, or
alighting nodes. At stop nodes, because passengers can-
not simultaneously choose between taking a bus and
walking to other platforms, the arc split probability is
defined with boarding arcs and walking arcs separately,
as shown in Equation (1). If boarding arcs are included in
hyperpaths, the arc spilt probability is proportional to the
line frequency,
() ,
la ipp
ap p
f F
 
2.5. Cost of Hyperpaths
In this paper, the cost of a hyperpa
generalised cost that consists of three elements: the
time, the monetary value of
th is represented as a
ip la
a OUTi
monetary value of the travel
the expected waiting time, and the implicit cost associ-
ated with the risk of failing to board. We admit that pas-
sengers may walk to another bus stop by creating a
walking arc between all stop nodes. Thus, the cost for
each arc, ca, is defined as:
Using the cost of arc a, ca, the generalised cost of hy-
perpath p, gp, can be written as follows:
aAkS kp
ip la
aOUT i
lp app
apal lpp
ipil lpp
The first term of Equation (4) represents the “moving
cost,” which consists of the mon
in-vehicle time and the walking co
3.1. Outline of the Model
olders, the
oped to wish to
and movement cost. Addi-
he operator knows the pas-
etary value of the
st. The second term
t the monetary value of the expected waiting
time. Note that αap and βip in Equation (4) represent the
probability that passengers traverse arc a and node k,
respectively, both of which are derived from the prob-
ability that the elementary path l within hyperpath p is
chosen. As Equation (4) can be separated by the subse-
quent node, Bellman’s principle can be applied to find
the minimum cost hyperpath. For simplicity, we treat ta
and fl as constants.
3. Bus Network Configuration and
Frequency Optimisation Model
In the proposed model, we consider two stakeh
erator and passengers, and both are assum
minimise the total travel time
tionally, if it is assumed that t
Copyright © 2012 SciRes. JTTs
sengers’ route choice norm (i.e., minimising the move-
ment cost) and that the operator can influence, but not
control, passenger route-choice behaviour, then the pro-
posed model can be formulated as a Stackelberg game or
a bi-level optimisation problem, where the operator is the
leader and the passengers are followers. If the transit
operator tries to minimise the total travel time, the level
of service will decrease, causing an increase in passenger
cost; thus, the objectives of the stakeholders often con-
flict. Thus, the upper problem is formulated as a multiple
objective optimisation problem. In addition to the as-
sumptions shown in Section 2.3, we make the following
assumptions in the proposed model.
First, regarding the bus operation service:
The position of bus stops is given and fixed, but not
all the bus stops have to be used.
Express service is not considered (i.e., all buses stop
rpath for a
ey can walk
origin or intermediate
ion of hyper-
each line, denoted as r = (r1,
, f|L|), respectively. The pro-
at all stops they pass en-route).
Travel time between bus stops is constant.
The maximum number of lines is fi
Dwell time is not considered.
Second, regarding passenger mov
Passengers choose the minimum cost hype
given bus network configuration, and th
to a different bus stop from an
bus stop if it is cheaper (see the definit
path cost in Equation (4)).
The OD demand is fixed regardless of the bus net-
work configuration.
3.2. Model Formulation
The decision variables in the proposed model are the
route and frequency of
r2, ···, r|L|) and f = (f1, f2, ···
posed model is formulated as
min, , , 1,2,
rf yrf mM
such that
0 otherwise
rsp rs
s (11)
ll l
M: Number of objective functions in the upper prob-
|L|: Number of lines (fixed);
Minimum cost from the origin of the hyperpath (r)
Travel demand between OD pair rs;
ime on line l;
esents the objective function, de-
fine case where
all the choose the
sh hort-
the destination (s);
C: Upper value of travel t
NV: Number of available vehicles.
Equation (10) repr
ed later, and Equation (11) describes th
passengers between each OD pair
ortest hyperpath. Passengers are assigned to the s
hyperpath based on Markov Chain assignments (see
[11]). Equation (12) concerns vehicle number constraints.
The number of vehicles required to operate a certain line
is assumed to be proportional to the line length and fre-
quency, implicitly neglecting turning time or waiting
time at the depot. The objective function of the operator
is to minimise total operational costs (ψ1), and the objec-
tive function of the passengers is to minimise total
movement cost (ψ2), formulated as:
ll l
rf fCr
2,, ()
rs Wp H
yr fygy
W: Set of OD pairs;
Set of hyperpaths between OD pair rs.
Equation (13) represents the total travel time for the
r because the left-hand side of Equation (12)
ref vehicles required to operate line
the upper problem of a multi-objective optimisation
problem, we use the elitist
ic algorithm (NSGA-II) pro-
n the figure, two types of genes are de-
hicles (A genes) and genes
een neighbourhood lines (B
presents the number o
4. Solution Algorithm
As shown in Section 3, the proposed model is formulated
problem. To solve the upper
non-dominated sorting genet
posed by [13], which is an expanded genetic algorithm
(GA) that requires fewer parameters than other methods.
In this section, only a solution algorithm within one ge-
neration is described, which consists of “Vehicle As-
signment”, “Route Design”, and “Frequency Setting”. A
frequency for each line is determined by combining a
“Vehicle Assignment” procedure and a “Frequency Set-
ting” procedure.
4.1. Vehicle Assignment
Figure 2 illustrates the chromosome for vehicle assign-
ment. As shown i
fined: genes representing ve
representing boundaries betw
genes). The number of A and B genes are equivalent to
the number of available vehicles and maximum number
of operated lines, respectively. Using these genes, the
number of vehicles for each line is equivalent to the
number of A genes sandwiched between two B genes.
Figure 2 illustrates an example of vehicle assignment. In
this example, two vehicles are assigned to line 1, three
vehicles are assigned to line 2, but no vehicle is assigned
to line 3.
Copyright © 2012 SciRes. JTTs
… …
Number of Vehicles(Number of Lines)-1
… …
Line 1:
2 Vehicles
Line 2:
3 Vehicles
Line 3:
0 Vehicle
Figure 2. Chromosome illustration for vehicle assignment.
4.2. Route Design
Although m
me shortcomings for solving the
erating indirect routes. In this sec-
from the
an that of creating route
o each line and
Then, the frequency of each
any researchers examining the TRNDP use a
enetic algorithm (GA) as a solution algorithm, the
original GA has so
TRNDP, such as gen
tion, a modification of the GA procedure for route search
under fixed origin and destination nodes is described
following Inagaki et al. [14], using the example network
shown in Figure 3(a). In the modified GA procedure, the
number of genes in a chromosome is the same as the
number of nodes in a network N. Each gene m can only
take the values of the nodes to which direct links from
the node m exist; that is, a link connecting nodes m and n
is represented by assigning node ID n to the mth gene.
Thus, the alignment of the genes in a chromosome can
provide the ID of nodes that make up a route, if one
keeps moving (“jumping”) from gene m to gene n (in
Figure 3(b), these are the genes with a square).
Thus, the chromosome defined here consists of two
types of genes, those contributing to the representation of
the route and those not. The Proposition in the Appendix
1 shows that we can always obtain a valid route
nes that contribute to the route description, unless a
cyclic route is obtained, which occurs if the same node
ID appears in at least two of these genes. Figure 3(b)
represents the route (0 1 4 6 7), with predete-
mined origin and destination nodes as 0 and 7, respec-
tively. For the genes that are not needed for the route
description, a random node ID among the available node
IDs is selected. With this chromosome definition, one
can trace a unique route from the origin to destination
nodes. There are also the well-known elements of cross-
over and mutation within GA optimisation, as described
in (14) (See, Appendix 2).
Note that the modified GA procedure might not create
all the possible routes with equal probability. For exam-
ple in Figure 3(a), the probability of creating route
0-1-3-7 would be higher th
1-4-6-7 because with above definition, the probability
of connecting from node 3 to 7 equals to one (there is
only one arc connecting nodes 3 and 7) whereas the
probability of connecting node 4 to 6 is 0.33. However,
the latter route overlaps also with a number of would
have a route with higher overlapping rate (e.g. 0-1-
4-5-6-7) than the former route. Therefore, since the
modified GA procedure implicitly generates routes that
overlap less with higher probability, this procedure is
expected to create more variety of routes.
4.3. Frequency Setting
As a result of vehicle assignment and route design pro-
cedures, the number of vehicles assigned t
the line length are known.
line is defined as:
Tl: Travel time of line l;
Vl: Number of vehicles assigned to
Different from [10], who chose the frequency of each
four options, the proposed procedures, com-
bient and frequency setting, can
core accurately.
and was generated at each
n bus stops was 4 min by
line l.
line from
ning vehicle assignm
nsider vehicle number constraints mo
5. Numerical Example
The proposed model was applied to the simple network
shown in Figure 4. Bus stops were assumed to exist
each node, and passenger dem
node. The travel time betwee
bus and 12.5 min by walking, and the value of time pa-
rameters were 13 yen/min (about 0.1 €/min), 26 yen/min,
4 7661 467
Node ID
Pr e de ter m i ned origin n ode
Predet e r mined dest inat ion node
Figure 3. Alignment of genes in a chromosome. (a) Example
network; (b) Alignment of chromosomes.
Copyright © 2012 SciRes. JTTs
and 50 yen/min, respectively, for boarding time, waiting
time, and walking time based on the SP-mode choice
survey of [15]. Also, to rhe
origin and destination of eaus line is fixed, a dummy
ks with zero cost
nodes in Figure
tions with and 77 solutions without considering the
commn lineroblemere obtained. sser
cost he ur leftlution lowehereahe
operator cost hose tions higheris reln-
ship tween two eholde is revfor-
tionsown ihe lowght of figure.he
ss and the print of each line was proportional
elax the assum
ch b
ption that t
origin node (Node 25) is introduced, which is an origin
of all lines of buses. Further, dummy lin
connect the dummy origin node to all of the bus stops.
Similarly, a dummy destination node (Node 26) and
zero-cost dummy links connecting all the bus stops with
the dummy destination node are added.
With this network, all the demands were assumed to
be generated from node 22, with destinations spread to
all the other nodes. The demand volume was generated
with a uniform random number, taking the value from 0
to 1; 200, 100, and 50 times the random number was de-
fined as the demand to node 12, the grey
and other nodes, respectively. As a result, the demand
pattern shown in Table 1 was obtained. Also, the number
of available vehicles and maximum number of operated
lines were set as 30 and 10, respectively. Note that al-
though the proposed model can be applied to a real net-
work [10], we assume above demand distribution on a
small grid network in order to better understand the ef-
fect of considering the common lines in TRNDP.
Figure 5 shows the Pareto solutions with and without
considering the common lines problem. In total, 96 solu-
12 34
56 78 9
10 1113 14
15 161718 19
20 212223 24
25 26
Figure 4. Test network.
Table 1. Assumed OD volume (Ori gin node is 22).
Destination Demand Destination Demand DestinationDemand
0 66 8 44 16 43
1 37 48
2 30 10 19 18 3
9 5 17
3 27 11 8 19 10
4 97 12 130 20 32
5 41 13 3 21 23
6 42 14 4 23 17
7 93 15 21 24 7
os p wThe paeng
for tppe sos isr, ws t
of tsoluis . Thatio
bethestakrsersed solu
shn ter ri sidethe T
extreme case is a solution with zero operator cost, where
the operator does not provide any bus services and all the
passengers have to walk to their destination. Furthermore,
contrary to expectations, solutions with and without con-
sidering the common lines problems showed almost the
same Pareto front. We suspect that this is due to a fairly
low number in GA iterations (1000 iterations with 100
individuals) exploring only a very small range of the
possible solutions. Nevetheless we find some interesting
differences in the network structures as described in the
following. The reason for the fairly low number in GA
iterations is due to the computational cost of the lower
problem. Especially when considering common lines is
the lower problem becomes computationally relatively
Because it is impractical to show the results of all the
Pareto solutions, we compared the results when the oper-
ator’s cost = 40. Figure 6 illustrates the line configure-
tion and headways (inverse of frequency) from the output
with and without considering the common lines problem.
The thickne
the operational frequency, as shown in the figure.
When considering the common lines problem, many
lines were concentrated in the centre of the network
(22-17-12-7-2), a “trunk with feeders” network, whereas
only one line ran in the centre of the network, a “trunk
and branch” network when not considering the common
lines problem. The reason for such a difference is that a
“trunk with feeders” network brings benefit to those who
consider the common lines problem because they can
take a line coming first among the bundle. On the other
hand, a “trunk with feeders” network does not bring
benefit to those who do not consider the common lines
problem, because they stick to a single line. Instead, a
“trunk with branch” network is more attractive to them
because they have more chance to transfer to a branch
line from the feeder line. As a result, many direct ser-
vices from node 22 to node 2 are provided if considering
the common lines problem whereas only one direct ser-
vice from node 22 to node 2 is provided and many pas-
sengers are forced to transfer if it is not considered. A
transfer penalty was not considered in this study; a dif-
ferent result might have been obtained if we had con-
sidered one.
6. Conclusions
This study revealed the effect of considering the common
Copyright © 2012 SciRes. JTTs
Copyright © 2012 SciRes. JTTs
05000001000000 1500000 2000000
Passengers' Cost
W/O Common LinesWith Common Lines
Figure 5. Pareto solutions and the corresponding Pareto front (With/without considering the common lines problem).
3.2 m i nutes6 m in utes10 m inutes
4 m inut es6. 67 m inut es12 m inutes
8 m inut es16 minut esOrigin of demand
58 9
11 13
15 18 19
012 3
6 7
10 12
16 17
21 22 23 24
(a) (b)
Figure 6. Bus routes and headways (Solution at operator’s cost = 40). (a) Considering the common lines problem; (b) Without
considering the common lines problem.
s ones, formulated
s a bi-level optimisation problem, the upper problem of
lem is considered. This has implications for transit plan-
ners and the design of bus stops. Our results suggests that,
lines problem in the transit route network design problem
(TRNDP). A similar model to previou
more direct service is optimal if the common lines prob-
which was described as a multi-objective problem, was
used, but several assumptions were relaxed. The lower
problem of the model is a passenger assignment model
and the output of the TRNDP with and without consid-
ering the common lines problem in passengers’ route
choice was compared. The output of the TRNDP in a
simple network was compared with and without consid-
ering the common lines problem. It was confirmed that a
if bus stops for several lines are arranged in such a way
as to make it easy for passengers to interchange between
lines, this could result in very different network struc-
tures, possibly reducing operator as well as passenger
In future work, it would be valuable to confirm
whether this finding is generalisable with various de-
mand patterns and other networks. Furthermore, more
computational efficient algorithms are future research
topics, especially in connection with further applications
to larger networks.
pp. 105-124.
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Appendix 1
of the genes within a chromosome de-
.2 ensures that the destination is reached
s takes value d, which is the ID
of the destination node. Let N1 indicate the number of
describe the route until destination d is
The procedures of crossover and mutation in the
fied GA procedure are described using following notations.
rmined origin node;
in the modified GA is shown
assover is executed only when
a m number, is smaller than a
1.1. Prop
The alignment
fined in Section 4
if the route is not cy
1.2. Proof
Suppose none of the gene
genes that
reached. Because no node takes value d, the search for a
feasible route will continue until all bus stops in the net-
work have been visited so that N1 will be equal to N, the
number of genes or bus stops. Because no cyclic route is
included, this means that each bus stop is visited only
once. However, because N equals the number of bus
stops, this means that one gene must take value d or at
least one bus stop must be visited twice, both of which
contradict our assumptions.
Appendix 2
r: The predete
s: The predetermined destination node;
N: The set of nodes (representing for bus stops) in a
g[*]: Gene of parent;
h[*]: Gene of offspring;
Mrate: Mutation rate;
Kmax: Threshold numb
1) Crossover
e crure Thossover proced
elo bw. Note that the cros
erated randorepeatedly gen
ven crossover rate.
Step 1 (Initialise)
Set m r and h[n]
for n N;
Step 2 (Roulette selection)
rSelect two parentsandomly g1 and g2;
ly and h[m]
Step 3 (Crossover)
2) randomSelect one parent i (i = 1,
[m], m gi[m];
of loop)
Step 4 (Termination
Repeat Step 3 until m = s;
Step 5 (Interpolation of the empty genes)
If g[m] =
(m N), then,
select one node n going
ke the value
t from node m and g[m]
As the genes of the offspring will only ta
4 7661 467
632 371 75
1 731 4576
Paren t 1
Offspr ing
Node ID
Paren t 2
4 766146 7
3 732 46 76
Offsprin g
Nod e ID
0Predete rmined origin node
7Pr edeterm ined destination no de
Figure 7. Illustration of mutation and crossover. (a) Exam-
ple of crossover; (b) Exampl
of a valid node, this crossover procedure alwas generates
a connected route which either is cyclic or follow-
Figure 7(a)
with predeter-
e of mutation.
ing the above proposition. illustrates this
crossover. In this example, a route (014567)
s generated as a result of the crossoveri
mined origin and destination nodes as 0 and 7.
2) Mutation
The following procedure shows the mutation of the
modified GA.
Step 1 (Initialise)
Set m r and h[n]
for n N;
Step 2 (Muta
Generate a random number rnd;
If rnd < Mrate, then, select one node n going out from
node m and h[m] n and m n;
m]; h[m] g[m] and m
Step 3 (Termination of loop)
Repeat Step 2 until m = s;
Step 4 (Interpolation of the empty genes)
] = If g[m
(m N), then, select one node n going
ion of the parent chro-
m7) is generated as a
rend des-
t from node m and g[m] n.
Figure 7(b) illustrates mutat
6osome. A route (024
sult of the mutation with predetermined origin a
s andation nodes 0 a 7.
Copyright © 2012 SciRes. JTTs