Optimisation of a Bus Network Configuration and Frequency Considering the Common Lines Problem

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 problem” (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.


Introduction
To entice travellers to shift from private cars to public transportation, many public transport operators have taken measures such as reducing off-peak fares.Several researchers have proposed models for determining optimal 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 transportation 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 transportation networks (such as minimising passengers' cost or maximising profit) under operational and resource constraints such as the number and length of public transportation 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 maximise 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] proposed 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 allowed.[7] formulated a simultaneous optimisation problem for railway line configurations and passenger assignments 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 (passenger waiting time is short if the service frequency is high).Another feature of their model is that a branchand-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 reasonable time.It would be difficult to apply a strict solution algorithm to a bus network optimisation problem, which is, generally, more complex than a railway network optimisation problem.
The literature review has so far revealed that many researchers do not describe passenger route choice behaviour accurately.Thus, several papers include a transit assignment model within the TRNDP to consider the passenger route and transfer choice behaviour.[8] combined 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 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 (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 allocation 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 public transportation, and they applied their model to a real network.
As described so far, some TRNDPs have considered the common lines problem, while others have not.Considering the common lines problem implies that passengers can consider the complex route set perfectly, which so far was almost impossible in dense networks.However, 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 aggregate separate bus stops.Therefore, it is important for transit agencies to know how the optimal network configuration differs if passengers come to know the complex route set.Thus, in this study, we explore the importance 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 procedure and a frequency setting procedure in the solution algorithm.
 Capacity constraint conditions were not considered in this paper to save computational costs ([10] confirmed that capacity constraint conditions did not affect 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 optimisation 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 conclusions and identifies future research.

Minimum Cost Hyperpath Searching Problem
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.

Network Representation
To consider the capacities of transit lines together with the common lines problem, the transit network shown in  Generally, the network representation used in public transit assignment models requires more computer memory compared to that used in road traffic assignment 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 components 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).

Notation
We use the following notation regarding the transit as-signment model.Other notation will be shown as appropriate.
A    Y: Vector of hyperpath flows; X: Vector of arc flow; /min).f l : Frequency of line l (1 ptions We adopted the fol common lines problem, similar to previous studies (See, [11,12]):  All bus lines operate with given exponentially distributed 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.

Arc Split Probabiliti
Where there are several arcs le hyperpath, traffic is split according to τ ap .As shown in Figure 1, passengers may be split at stop, failure, or alighting nodes.At stop nodes, because passengers cannot 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, (2)

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 monetary value of the travel the expected waiting time, and the implicit cost associated with the risk of failing to board.We admit that passengers may walk to another bus stop by creating a walking arc between all stop nodes.Thus, the cost for each arc, c a , is defined as: else Using the cost of arc a, c a , the generalised cost of hyperpath p, g p , can be written as follows: where , , The first term of Equation ( 4) represents the "moving cost," which consists of the mon in-vehicle time and the walking co represen

Outline of the Model
olders, the op ed to wish to and movement cost.Addihe operator knows the pas-etary value of the st.The second term t the monetary value of the expected waiting time.Note that α ap and β ip in Equation ( 4) represent the probability that passengers traverse arc a and node k, respectively, both of which are derived from the probability that the elementary path l within hyperpath p is chosen.As Equation ( 4) can be separated by the subsequent node, Bellman's principle can be applied to find the minimum cost hyperpath.For simplicity, we treat t a and f l as constants.

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 sengers' route choice norm (i.e., minimising the movement cost) and that the operator can influence, but not control, passenger route-choice behaviour, then the proposed 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 conflict.Thus, the upper problem is formulated as a multiple objective optimisation problem.In addition to the assumptions 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 xed.

ement:
rpath for a ey can walk origin or intermediate ion of hypereach line, denoted as r = (r 1 , , 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 network configuration.

Model Formulation
The decision variables in the proposed model are the route and frequency of  (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 frequency, implicitly neglecting turning time or waiting time at the depot.The objective function of the operator is to minimise total operational costs (ψ 1 ), and the objective function of the passengers is to minimise total movement cost (ψ 2 ), formulated as: where W: Set of OD pairs; Set of hyperpaths between OD pair rs.Equation ( 13

Solution Algorithm
As shown in Section 3, the proposed model is formulated as 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 generation is described, which consists of "Vehicle Assignment", "Route Design", and "Frequency Setting".A frequency for each line is determined by combining a "Vehicle Assignment" procedure and a "Frequency Setting" procedure.

Vehicle Assignment
Figure 2 illustrates the chromosome for vehicle assignment.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.

Route Design
Although m g me shortcomings for solving the erating indirect routes.In this secfrom the ge an that of creating route 0o 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 m th 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 predetemined origin and destination nodes as 0 and 7, respectively.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 crossover 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 example 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.

Frequency Setting
As a result of vehicle assignment and route design procedures, the number of vehicles assigned t the line length are known.line is defined as: where T l : Travel time of line l; V l : Number of vehicles assigned to Different from [10], who chose the frequency of each four options, the proposed procedures, combi ent and frequency setting, can co re accurately.at and was generated at each n bus stops was 4 min by line l.line from ning vehicle assignm nsider vehicle number constraints mo

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 parameters were 13 yen/min (about 0.1 €/min), 26 yen/min, 0 and 50 yen/min, respectively, for boarding time, waiting time, and walking time based on the SP-mode choice survey of [15] ss and the print of each line was proportional to 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 defined 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 although the proposed model can be applied to a real network [10], we assume above demand distribution on a small grid network in order to better understand the effect 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- 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 considering 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 expensive.
Because it is impractical to show the results of all the Pareto solutions, we compared the results when the operator's cost = 40.Figure 6 illustrates the line configuretion 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 services from node 22 to node 2 are provided if considering the common lines problem whereas only one direct service from node 22 to node 2 is provided and many passengers are forced to transfer if it is not considered.A transfer penalty was not considered in this study; a different result might have been obtained if we had considered one.s ones, formulated s a bi-level optimisation problem, the upper problem of lem is considered.This has implications for transit planners 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 proba 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 considering the common lines problem in passengers' route choice was compared.The output of the TRNDP in a simple network was compared with and without considering 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 structures, possibly reducing operator as well as passenger costs.
In future work, it would be valuable to confirm whether this finding is generalisable with various demand patterns and other networks.Furthermore, more computational efficient algorithms are future research topics, especially in connection with further applications to larger networks.illustrates this crossover.In this example, a route (0→1→4→5→6→7) s generated as a result of the crossover i mined origin and destination nodes as 0 and 7. 2

) Mutation
The following procedure shows the mutation of the modified GA. Step

Figure 1 (
a) was transformed into the graph model shown in Figure 1(b
b kl : Boarding arc of line l on platform k; nodes on a τ ap : Arc split probability on hyperpath p; l(a): A transit line ) represents the total travel time for the r because the left-hand side of Equation (12) re f vehicles required to operate line l.the upper problem of a multi-objective optimisation problem, we use the elitist ic algorithm (NSGA-II) pron the figure, two types of genes are dehicles (A genes) and genes een neighbourhood lines (

Figure 3 .
Figure 3. Alignment of genes in a chromosome.(a) Example network; (b) Alignment of chromosomes.

Figure 5 .
Figure 5. Pareto solutions and the corresponding Pareto front (With/without considering the common lines problem).

Figure 6 .
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.

Figure 7 .Figure 7
Figure 7. Illustration of mutation and crossover.(a) Example of crossover; (b) Exampl of a valid node, this crossover procedure alwa s generates a connected route which either is cyclic or follow-Figure 7(a) with predeter-

1
(Initialise) Set m ← r and h[n] ←  for n ∈ N; tion) 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; g[ Else ← 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 ou ion of the parent chrom →7) is generated as a re nd destin t from node m and g[m] ← n.

Figure 7 (
b) illustrates mutat →6 osome.A route (0→2→4 sult of the mutation with predetermined origin a s a nd ation node s 0 a 7.
that is included in arc a g p : Cost of hyperpath p; ; c a : Arc cost on arc a  A t a : Travel time on arc a  A il : Equal to 1 if elementary path l traverses node i, otherwise 0;