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Public Transport (2023) 15:377–409

https://doi.org/10.1007/s12469-022-00312-5

1 3

ORIGINAL RESEARCH

A simulation‑based optimization approach fordesigning

transit networks

ObioraA.Nnene1 · JohanW.Joubert2· MarkH.P.Zuidgeest1

Accepted: 13 November 2022 / Published online: 18 January 2023

© The Author(s) 2023

Abstract

Public transport network design deals with ﬁnding eﬃcient network solution(s) from

a set of alternatives that best satisﬁes the often-conﬂicting objectives of stakehold-

ers like passengers and operators. This work presents a simulation-based optimiza-

tion (SBO) model for designing public transport networks. The work’s novelty is

in developing such a network design model that fully accounts for the stochastic

behavior of commuters on the transit network. The SBO discipline solves decision-

based problems like the transit network design problem (TNDP) by combining sim-

ulation and optimization models. The proposed model integrates a disaggregated

activity-based travel demand simulation with a multi-objective network optimiza-

tion algorithm. Trip-based travel demand models are commonly used to represent

traveler behavior in the literature. The approach limits its ability to accommodate the

stochastic realities of traveler behavior in a transit network design solution. Using

activity-based simulation instead makes it possible to account for a more realistic

traveler behavior, especially real-time decisions made in response to changing net-

work dynamics which ultimately aﬀect the distribution of demand over time on the

network. The proposed model is applied to the improved design of the integrated

public transport network in the City of Cape Town, South Africa. The results show

SBO can design eﬃcient network solutions that reﬂect the objectives of network

stakeholders.

Keywords Simulation-based optimization· Transit network design· Activity-based

modeling· Multi-objective optimization· Metaheuristics

* Obiora A. Nnene

obiora.nnene@uct.ac.za

Extended author information available on the last page of the article

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1 Introduction

Public transport network design deals with ﬁnding eﬃcient network solution(s)

from a set of alternatives that best satisﬁes the often-conﬂicting objectives of stake-

holders like passengers and operators. The commuter aims to minimize their total

travel time and other associated costs. At the same time, the operator sees costs in

terms of the total resources needed to operate a proﬁtable service. Hence, their goal

is to minimize this cost even at the risk of serving routes the commuter may con-

sider lengthy and unattractive. Therefore, solving a transit network design problem

(TNDP) requires ﬁnding a compromise between these conﬂicting goals. The known

methods for solving the TNDP in the literature are broadly classiﬁed as analytical

and heuristic. The former comprises exact search algorithms which attempt to ﬁnd

the closed form of an objective function in the search for a best possible solution

to the problem. Recent research with analytical solutions in the literature areCon-

stantin and Florian (1995), Lee and Vuchic (2005), Chen etal. (2017), Daganzo and

Ouyang (2019), and Ranjbari etal. (2020). However, these analytical solutions are

limited in solving the TNDP due to the nearly inﬁnite amount of resources needed

to ﬁnd a solution to even relatively small transit network design problems(Chak-

roborty 2003). On the other hand, the TNDP lends itself to heuristic solution tech-

niques, especially metaheuristics, owing to their relatively simple adaptation to

extensive TNDP case studies. Furthermore, metaheuristics are approximate algo-

rithms that can ﬁnd good solution(s) in a reasonable amount of time. Some works

utilizing heuristic approaches include Pattnaik et al. (1998), Fan and Machemehl

(2004), Alrabghi and Tiwari (2015), Huang etal. (2018), Nnene etal. (2019), Yang

and Jiang (2020).

In this paper, the authors present a simulation-based optimization (SBO)

approach for solving the TNDP. This method combines simulation with optimization

models to solve decision-based problems like the TNDP(Gosavi 2015a). The goal

of the paper is to develop a so-called simulation-based transit network design model

(SBTNDM) that integrates simulation with optimization and in the design of transit

networks. The research question revolves around how SBO can be applied to the

optimized design of transit networks, leading to the realization of eﬃcient network

solutions. In the SBTNDM, an activity-based simulation (ABS) evaluates alterna-

tive network solutions by simulating travel demand on them whilst a multi-objective

optimization algorithm searches for eﬃcient network solutions. The paper’s main

contribution is solving the TNDP by integrating a disaggregated activity-based

travel demand simulation with a multi-objective metaheuristic network optimiza-

tion solution framework. Using activity-based simulation makes it possible to fully

account for the microscopic behavior of travelers and other agents on the network.

Also, the network design process can incorporate temporal ﬂuctuations in demand.

The static trip-based travel demand model is commonly used to represent trave-

ler behavior in the literature, as seen in the reviews ofJohar etal. (2016), Durán-

Micco and Vansteenwegen (2022), Ibarra-Rojas etal. (2015). However, they cannot

account for an event such as a route change made in response to unexpected road

closures or other stochastic decisions made by agents in real-time while responding

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Simulation-based optimization for designing transit networks

to changes in network dynamics. Hence, it is advantageous to use ABS as it oﬀers a

more detailed and accurate reﬂection of demand distribution on the network, which

is a critical consideration when operators choose the area to provide a service. Fur-

thermore, a detailed service timetable rather than headway is a crucial input in the

ABS. Therefore, the simulation oﬀers a way to solve the TNDP, which allows for

a feedback loop between travel demand and service supply. However, using the

detailed timetable introduces the need to customize the network decision variable’s

encoding to facilitate the operations of the optimization algorithm. As a result, the

authors deﬁne a custom encoding scheme to address this. This custom encoding is

considered the other contribution of the paper, since representations like vectors and

strings are used more commonly in the literature(Szeto and Wu 2011; Buba and Lee

2018). The proposed solution model is ultimately applied to the design of a transit

network in Cape Town, South Africa, which needs improvement in terms of opera-

tional cost reduction and ridership increase.

In the remainder of this paper, Sect.2 presents a theoretical background for the

proposed model, and Sect.3 presents the mathematical model for the problem and

Sect. 4 outlines the component algorithms of the proposed SBO network design

solution framework. Section5 presents the results of testing the proposed solution

and discusses its application to a large-scale transit network in Cape Town, South

Africa, mainly as it aﬀects passengers and service operators. In the ﬁnal section,

possible areas of future research are highlighted.

2 Literature review

This section starts with an overview of the major developments in the TNDP lit-

erature, speciﬁcally in terms of how previous researchers have tackled the network

design problem. The discourse is then narrowed to the applications of SBO in the

TNDP literature, which is more relevant to our work. Thereafter, we discuss the key

components of the SBTNDM and their operations. With the guiding question of

this research focussing on how the SBO is applied to the TNDP, it is important to

understand solutions trends both in the literature and how they are evolving. Before

the year 2000, many TNDP solution attempts used analytical methods. However,

since then, metaheuristic solution models have gained prominence among research-

ers. Advances in operations research and computational science literature may have

aided this development, making implementing and using metaheuristic procedures

relatively straightforward. TNDP solution algorithms are classiﬁed by the number

of objectives in the problem, namely single and multiple-objective optimization.

The key diﬀerence between single and multi-objective solution algorithms is that in

the former, a linear summation of all objectives is used to reduce many objectives

to a single one. Furthermore, weights must be deﬁned beforehand for each objec-

tive, and the obtained single results reﬂect the weighted objectives(Mauttone and

Urquhart 2009). In contrast, the outcome of multi-objective algorithms is a Pareto

frontier(Knowles etal. 2008), which represents the possible trade-oﬀs between a

problem’s objectives, making it possible to obtain valuable information about these

trade-oﬀs and sensitivity for weighting the various objectives in terms of an optimal

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design solution(Possel etal. 2018). Among other reasons, however, this makes the

multi-objective solution approach more complex than the single-objective version.

Examples of single-objective solution approaches areCipriani et al. (2012); Chen

etal. (2017); Nnene etal. (2017) andCipriani et al. (2020), while multi-objective

solution models include Brands and van Berkum (2014); Heyken Soares et al.

(2019) andMomenitabar and Mattson (2021). Recent review articles on the TNDP

include those byDurán-Micco and Vansteenwegen (2022), Iliopoulou etal. (2019),

Ibarra-Rojas etal. (2015) andJohar etal. (2016).

SBO applications in transportation planning, especially transit network design,

are of interest in this paper. The technique has been applied to diﬀerent aspects of

transportation research like traﬃc signal design control(Osorio and Selvam 2015;

Osorio 2016), in which the authors combine a mathematical model with traﬃc

simulators to identify points on a network with high-level performance in terms

of a stated indicator. Song et al. (2013) performed the minimization of general-

ised cost on a multimodal transport system using a proprietary transport simulation

software VISUM that was combined with a genetic algorithm. Furthermore, Yan

etal. (2013), in their attempt to solve the robust network design problem, used the

Monte Carlo simulation to model travel demand ﬂow with an embedded discrete

choice model to represent passenger choices. While Hassannayebi et al. (2021)

andGao etal. (2022) receptively apply a discrete event simulation to the resched-

uling and passenger capacity analysis on rail services. Lastly,Bal and Badurdeen

(2022) apply SBO to the optimization of circular networks to make location and

allocation decisions when implementing a lease and sell strategy. More relevant to

this paper are the works ofDandl etal. (2021) and Ma and Chow (2022). This is

because the authors use activity-based simulations in their solutions. The former

presents an SBO solution framework, which combines Bayesian optimization with

an agent-based transport system simulation within a tri-level optimization solution

framework. Their research objective was to capture inter-decision dynamics between

mobility service operators and commuters which could then be used to optimize and

analyse policies that relate to service providers. Within their solution framework,

the policymaker represented the highest level, the operator represented the middle

level, and the traveler was at the lowest level. The Bayesian optimization algorithm

was used to maximize social beneﬁts for the authorities and proﬁt for the operator,

while agent-based simulation was used to simulate user behavior on the network.

The model was applied to the case of toll and parking costs for automated mobility-

on-demand systems in Munich, Germany. Also,Ma and Chow (2022) proposed a

bi-level modeling framework for solving the transit frequency setting problem. The

authors used an analytical route cost function representing the upper level and a

lower level represented by an agent-based market equilibrium function which takes

the frequencies of the routes and outputs demand for the transit network represent-

ing the lower level. The problem is applied to the case of the Brooklyn bus network

in New York, USA, which is done with the idea of understanding how the service

performs in competition with dial-a-ride services.

Due to the increased understanding of the power of simulations in evaluating

complex stochastic systems and the advancement of computation science, more

researchers are using them in transit network design. The two works that use ABS

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Simulation-based optimization for designing transit networks

are most relevant to the model proposed in this paper though their problem objective

and application context diﬀer. As such, the authors make a valuable contribution in

applying and advancing the use of SBO technique to solve the TNDP particularly in

the context of public transportation. It is also important to highlight that the research

in this paper is a metaheuristic SBO, combining a multi-objective metaheuristic

algorithm known as the non-dominated sorting genetic algorithm NSGA-II with an

agent-based travel demand simulation known as multi-agent transport simulation

(MATSim). The following section discusses these two modeling components for the

proposed model.

2.1 Modeling components

2.1.1 Non‑dominated sorting genetic algorithm‑II

Deb etal. (2000) is credited for the development of the NSGA-II, a typical multi-

objective evolutionary algorithm (MOEA). Their operations mimic biological phe-

nomena like genetics and bee or ant colonies, and they are thus called bio-inspired

algorithms(Branke etal. 2008; Rangaiah and Bonilla-Petriciolet 2013; Elarbi etal.

2017). This class of algorithms works by enabling the realization of newer and pre-

sumably better generations of solutions from existing ones. To apply the NSGA-II

to the TNDP, an initial population that constitutes the problem’s search space must

be generated. This population is made up of feasible network alternatives or chro-

mosomes, and each chromosome possesses genes or routes. The best-performing

chromosome in the population often represents a near-optimum solution, given that

for very diﬃcult problems like the TNDP it is not feasible to know if a solution is

optimum. The chromosomes or networks must also be encoded in a way that is ame-

nable to the algorithm’s operators. In the literature, string and binary encodings have

been the most common representations used when solving the TNDP. InBuba and

Lee (2018), a string is used to represent the network route, while a tuple is used to

represent the route’s operational frequency as the number of vehicles operated per

hour and the unique identiﬁer for that route. However, in this paper an innovative

encoding based on the JavaScript object notation (JSON) data structure(Crockford

2011) is used to facilitate the simultaneous handling of the route network design

and frequency setting problems. Details of this encoding and how it is used in the

proposed SBTNDM are discussed in Sect.4.3, where the model’s implementation

is described. After encoding the network solutions, they are scored or evaluated

against the objective function(s) of the problem. After this, the initialized solutions

are evaluated against the objective functions and sorted into diﬀerent Pareto fron-

tiers using the non-dominated sorting procedure. A solution is considered to be non-

dominated if it performs better than other solutions in at least one objective and is

not worse than the other objectives. Hence, all non-dominated solutions are ranked 1

and temporarily removed, then the next set of non-dominated solutions are identiﬁed

and ranked till all the solutions are ranked. The rank of each frontier is assigned as

their ﬁtness score and used to indicate the dominance of solutions. After ranking the

solution, a binary tournament selection operator is used to select parents that will

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be used to reproduce the oﬀspring. The operator randomly chooses two solutions,

determines the ﬁtter of both and then adds that one to the mating pool. To achieve

this, the binary tournament operator uses a crowded comparison procedure, which

measures the total distance of both solutions relative to their neighbours. The solu-

tion with a larger crowded distance is considered ﬁtter and selected as the larger

distance indicates a better spread in the Pareto frontier. First, the selected solutions

are compared based on their dominance or rank value. However, if they both belong

to the same front, i.e. they do not dominate each other, the crowding distance is then

used to obtain the better solution. In the next step, the genetic operators, namely

crossover and mutation, are used to create a population of children/oﬀspring of a

size equivalent to that of the ﬁrst parent. These operators are used to generate oﬀ-

spring and introduce diversity in the population, respectively. Thereafter, the proce-

dure is slightly diﬀerent from the ﬁrst generation: the generated oﬀspring and parent

are merged to form a population that is twice the size of the original population in

every subsequent generation. The merged population is evaluated and again ranked

according to the non-dominance and crowding distance criteria, and the better-per-

forming half of the merged populations is selected as the new parent population.

This process goes on iteratively until a speciﬁed termination condition is satisﬁed.

Elitism is introduced in the algorithm by archiving a small percentage of the best-

performing, elite solutions from both the parent and oﬀspring populations during

successive generations, which are reused as part of the parent population in the next

generation.

2.1.2 MATSim

MATSim is an activity-based multi-agent simulation framework which models the

microscopic demand of travelers by simulating their daily activity schedule and

decision-making on a transport network. The modeled travelers are called agents

and the simulation is designed to model their travel demand and stochastic deci-

sion making in 24-hour periods. In terms modeling public transit systems, MAT-

Sim organises data in a format that is commonly used by public transit services

worldwide(Horni etal. 2016). A public transport network line modeled in MAT-

Sim will therefore comprise two or more transit routes. Each route serves one direc-

tion of travel and enables transit vehicles to move to and from the depot at the end

and beginning of a day, respectively. The routes also have as an attribute the list of

departures, which gives information about the time a vehicle starts at the ﬁrst

stop on that route. A route also includes a sequential list of transit stops that are

served, alongside operating timetables that indicate when vehicles arrive or leave a

stop. The times are speciﬁed as oﬀsets in time units from the departure at the ﬁrst

stop so that at each subsequent stop, the oﬀset is added to the initial departure time

from the ﬁrst stop. Each departure contains a vehicle’s start time on the route and a

reference to the vehicle. As the timing information is part of the route, it becomes

possible to have routes with identical stop sequences but diﬀerent time oﬀsets. Stop

locations are described by their coordinates and an optional name or id. They must

be assigned to unique lines of the network for the simulation. The hierarchical tree

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Simulation-based optimization for designing transit networks

structure for the schedule ﬁle can be seen in Figure1 while the transit schedule ﬁle

may be seen inFigure2.

To model congestion on a public transit network, MATSim adopts a queue-based

traﬃc ﬂow model. This means that vehicles enter a link from an intersection, join

the end of a waiting queue and remain there until the time required to travel the link

with free ﬂow has lapsed and they are at the head of the waiting queue. In terms of

routing transit demand, an events-based public transport router is used in the MAT-

Sim simulation environment. Its main input are the commuter’s start time, origin

and destination pair (OD). The router mimics reality in its ability to compute alter-

native routes for agents. Their originally scheduled departure can either be due to

the late arrival of a vehicle or to it arriving full and being unable to take more pas-

sengers. This is achieved by taking the transit service’s given schedule as a base in

Fig. 1 A hierarchical tree structure for the transit schedule ﬁle

Fig. 2 The MATSim transit schedule ﬁle with routes and their schedules

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O.A.Nnene et al.

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the ﬁrst iteration, then generating updated information on travel times, vehicle occu-

pancy, and waiting times between subsequent iterations.

Next, the operational steps required to model transport in MATSim are described.

1. Initial demand generation: The initial demand is generated by creating daily activ-

ity plans from socioeconomic and demographic data of agents within a given

transportation area. The demand is usually generated through sampling or discrete

choice modeling and is subsequently converted to activity chains or plans for the

agents.

2. Execution: This involves simulating the generated demand. The plans are executed

sequentially by time of occurrence in a way that respects certain boundary condi-

tions, like the closing hours of a shop or the maximum link and ﬂow capacity of

a road. The constraint represents the physical infrastructure where the activities

and trips will be undertaken(Meister etal. 2010). Another name for this step is

mobility simulation, or mobsim for short.

3. Scoring: After executing the agents’ plans, the plans are evaluated. A score is

obtained by evaluating the plan using a utility function known as a scoring func-

tion. MATSim uses the scores to measure and compare the quality of a passen-

ger’s plan to determine whether it should be dropped or not.

4. Replanning or innovation strategy: Agents adapt their plans in response to changes

in the transit network, allowing the agent to modify their plans as they learn about

prevailing network conditions, making it possible for the agent to maximize their

experience on the public transport network. Details of these steps can be found

inHorni etal. (2016).

5. Termination and post-analysis MATSim: This speciﬁes a termination criterion

that signals the simulation to stop when the condition has been met. Meister

etal. (2010) describe this termination point as an agent-based stochastic user

equilibrium (SUE). The system runs until the score of the agent’s plan does not

meaningfully improve, marking the end of the simulation. Post-analysis involves

collecting and aggregating network performance indicators, passenger mileage

and average trip duration to gain insight into the travel demand and simulated

behavior of agents within the study area.

The input data for a MATSim simulation are the network which contains informa-

tion about nodes and links, plans which is the daily activity chains of all travelers,

transit schedules that consist of routes and their departure times, transit vehicles

which details the operational ﬂeet and their characteristics and lastly, the conﬁgura-

tion ﬁle that is a collection of parameter settings needed to run the simulation. These

are formatted as Extensible Markup Language (XML)(Bray etal. 2006) data struc-

tures. In this work, the transit supply side data such as the network and operational

schedules can be extracted from General Transit Feed Speciﬁcation (GTFS) data of

a transit service. On the other hand, the demand data in MATSim is created from

sources like travel diary surveys, census data and other passenger usage information

sources. In this work the passenger activity chains were derived from automated fare

collection data for the network being designed.

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Simulation-based optimization for designing transit networks

3 Models

The overarching optimization goal of the work is to minimize costs for transit users

and operators of the network. This is depicted in the objective functions in Equa-

tion (1). For the user objective shown in Equation(2), the expression is summa-

tion of total travel time, transfers and a penalty for unsatisﬁed demand. Total travel

time is obtained by multiplying a monetary factor for time by the generalised cost of

travel of the commuters. Generalised cost is the sum of the access time (

trk

a

), waiting

time (

trk

w

), in-vehicle travel time (

tr

k

trv

) and transfer time (

ntr

) where applicable. Due to

the negative perception commuters have of transfers on the transit network(Owais

2015), a time penalty (

𝜙time

) is applied to trips involving transfers in the model.

Lastly, the unsatisﬁed penalty is applied to each network solution for the amount

of travel demand not satisﬁed by the network. It is obtained by multiplying the total

unsatisﬁed demand (

qrk

u

) by a time (

tu

) and monetary factor (

𝛽unsat

) for unsatisﬁed

demand. In Equation(3) the operators are concerned with the total operational cost,

which is the sum of distance (

drk

r

) operated by the vehicle ﬂeet (

nrk

b

) multiplied by

a monetary factor for vehicle mileage and the total vehicle time

trk

r

multiplied by

the ﬂeet and its corresponding monetary factor. Operational distance is the cost that

accrues from wear and tear on the operator’s vehicles as they traverse the desig-

nated routes to satisfy passenger demand and is typically measured in kilometres.

However, operational time consists of personnel cost elements, such as salaries, that

accrue throughout operations. By minimizing these objective functions, the total

cost incurred on the network will be optimized for the stakeholders. Thus, the model

is formulated as follows:

subject to an agent-based route selection model which is based on the conditional

probability of the average route cost for both the user and operator

and some feasibility conditions on route length, frequency and vehicle ﬂeet:

(1)

Min ∶Z1,Z2

(2)

Z

1=𝛽time ⋅

⎛

⎜

⎜

⎝�

r∈R

�

rk∈Rmr

trk

trvqrk

trv +�

r∈R

�

rk∈Rmr

trk

aqrk

a+�

r∈R

�

rk∈Rmr

trk

wqrk

w

⎞

⎟

⎟

⎠

+𝜙time ⋅�

n∈N

ntr +⎛

⎜

⎜

⎝

𝛽unsat ⋅tu⋅�

r∈R

�

rk∈Rmr

qrk

u⎞

⎟

⎟

⎠

(3)

Z

2=

⎛

⎜

⎜

⎝

𝛽dist ⋅�

r∈R

�

rk∈Rmr

drk

rnrk

b+𝛽op ⋅�

r∈R

�

rk∈Rmr

trk

rnrk

b

⎞

⎟

⎟

⎠

(4)

Pn(k)=Pn(k∣E{𝜏(x({rn

k}))})

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386

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3.1 Sets

N = set of nodes on the network (-)

R = set of transit routes (-)

Rmr

= set of segments

rk

that serves demand on route r (-)

3.1.1 Decision variables

r = route on the network (-);

rk

= segment

rk

that serves demand on route r (-);

B = Total ﬂeet size (-);

3.1.2 Parameters

Z1

= user cost objective function (-);

𝛽time

= monetary unit value for user travel time (’000);

tr

k

trv

= travel time on route segment

rk

(hr);

qr

k

trv

= travel demand on route segment

rk

(pax);

trk

a

= access time on route segment

rk

(hr);

qrk

a

= passengers boarding on route segment

rk

(pax);

trk

w

= waiting time on route segment

rk

(hr);

qrk

w

= passengers waiting on route segment

rk

(pax);

𝜙time

= time penalty associated with transfers (-);

ntr

= transfers on a route r (-);

𝛽unsat

= monetary unit value for unsatisﬁed travel (’000);

trk

u

= time penalty for unsatisﬁed travel

rk

(hr);

qrk

u

= volume of unsatisﬁed travel demand

rk

(pax);

Z2

= operator cost objective function (’000);

𝛽dist

= monetary unit value for vehicle mileage (’000);

drk

r

= length of route segment

rk

(km);

nrk

b

= bus operating on a route segment (-);

𝛽op

= monetary unit value for vehicle operating time (’000);

n = index of the agent (-);

Pn(k)

= agent-based probabilistic route choice model (-);

E = mean traﬃc conditions on the network (-);

𝜏(x)

= network costs as a result of x (-);

x = network conditions (-);

{rn

k}

= all individual agent route demands on the network (-);

(5)

nr

k

b

<

B

(6)

rtot

≤

Rmax

(7)

dmin

≤d

r

k

r

≤d

max

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387

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Simulation-based optimization for designing transit networks

dmin

= minimum route length (km);

dmax

= maximum route length (km);

rtot

= number of designed routes (-);

Rmax

= maximum number of routes that are allowed on the network (-);

The objectives are subject to an agent-based SUE (Horni et al. 2016) which

describes the individual traveler’s behavior on a public transportation network.

Equation(4) is a probabilistic choice model that is used as a proxy for the agent-

based SUE with the assumption that travelers base their route choice on the average

route costs on the network. This traﬃc assignment method is based on decoupling

the steady ﬂow of passengers on a network in the static and dynamic contexts to that

of the individual traveler. Flötteröd and Rohde (2011) and Zhou and Taylor (2014)

show that it is challenging to model traﬃc ﬂow dynamics in complex networks, but

disaggregating the OD matrix into individual trip makers allows for vehicle assign-

ment to each trip maker. On this premise, the user equilibrium (UE) and SUE can

then be extended to a so-called disaggregate or particle case, where the particle

represents the microscopic or single traveler with their route choices replaced with

random variables. Hence, each traveler can draw routes from this choice distribu-

tion and the resulting distribution of traﬃc conditions regenerates the choice dis-

tribution. This method when combined with stochastic network loading that uses

time-dependent trip departures and an extension of choice dimensions beyond the

traditional ones (route and mode choice) used in UE to accommodate destination

choice and others, leads to the realization of an agent-based model that describes

fully the disaggregate behavior of agents on the network. However, the complex-

ity of the model means that it rather lends itself to simulation rather than an ana-

lytical solution. Hence, simulation ensures that each agent can optimize their plans

on the network by modifying either their departure time, route, mode and destina-

tion choices. These choice dimensions deﬁne the variation that occurs in the agent’s

plans during simulation. The process is repeated till the average score of the popu-

lation is stabilized or attains equilibrium which is also SUE as the optimization is

performed in terms of individual scoring functions and within each traveler’s set

of plans. This is achieved based on a co-evolutionary algorithm (Meneghini et al.

2016) which optimizes each agent’s plan in competition for network resources with

other agents, while respecting deﬁned constraints.

In the description of the SBTNDM it is important to highlight that the route

and ﬂeet size are used as the problem’s decision variables. The latter serves as a

proxy for the operator’s total budget. In the TNDP literature, a decision variable is

a resource that is subject to the transit stakeholders’ choice in terms of its allocation

(Curtin 2004). The limits or bounds of their availability are usually deﬁned by a

feasibility constraint, which is a parameter that deﬁnes the limiting conditions of the

decision variable(s) in a TNDP. They generally deﬁne the feasibility of the optimiza-

tion problem and ensure that solutions are obtained within reasonable resource limi-

tations. The feasibility constraints for the model are those on vehicle ﬂeet size, num-

ber of routes and total route length as seen in Equations(5) through Equation(7).

These constraints are used to set the allowed limiting conditions for the allocation

of resources on the transit network. Equation(5) is the ﬂeet size constraint that rep-

resents the limits of the operator’s resources. This ensures that an optimal network

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does not utilize more vehicles than the available number vehicles. Furthermore, the

resources at an operator’s disposal determines what service frequency they can pro-

vide. Hence, the constraint on ﬂeet size signiﬁcantly aﬀects the level of service that

can be provided with a transit network design solution. Equation(6) deﬁnes a con-

straint on the maximum number of routes in the designed solution. This ensures that

the maximum number of routes determined according to the current vehicle ﬂeet

size is not exceeded. The maximum number of routes has a big impact on ﬂeet size

and driver scheduling.

Lastly, Equation (7) is a feasibility constraint on transit service route length.

Usually, public transit operators will not run a service on routes that users can con-

veniently traverse by walking. Operators also avoid developing excessively long

routes (Cipriani et al. 2012), as they make schedule adherence diﬃcult and may

require too many transfers, which users ﬁnd unappealing(Walker 2011).

3.1.3 Modeling assumptions

The following assumptions are made in the model development:

1. At the level of the network, a ﬁxed total travel demand context is assumed.

2. A complete trip or satisﬁed demand may be in two forms: boarding–alighting

(B–A) or boarding–connection–alighting (B–C–A). The former is a direct trip

without transfer, while the latter is a trip satisﬁed with one transfer required. This

speciﬁcation aligns with how demand coverage is deﬁned in this article: demand

that is satisﬁed with zero or one transfer. It is assumed that commuters generally

ﬁnd a trip less attractive beyond one transfer and that this would lead them to

search for alternative, more direct routes or even in some cases to change their

mode of travel(Owais 2015).

3. In this work automated fare collection data is used to create the daily trip chains

of the commuters which is subsequently converted to the initial demand used in

the MATSim simulation.

4. In agent-based travel demand models, demand is generated from people’s activi-

ties at diﬀerent locations based on various land uses; however, in this work it

was not possible to obtain information concerning activities or activity locations

outside the transit network. Consequently, activities refer strictly to transactions

like passenger boarding, alighting transfers and others that occur on the network.

4 Solution procedure

Three steps are taken in the solution framework to realize the SBTNDM. The ﬁrst is

a heuristic route network generation algorithm (NGA), which is used to generate ini-

tial candidate transit networks. Secondly, an agent-based simulation route network

evaluation procedure (NEP) is used to score the quality of each generated transit net-

work. Finally, an NSGA-II network search algorithm (NSA) is used to search for the

Pareto-optimal set of network solutions. The reader is referred to Figure3, in which

the interaction between the three components of the model are shown.

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4.1 Network generation algorithm

The ﬁrst stage of the SBTNDM involves creating a pool of feasible transit networks

from which the ﬁrst population of solutions will be initialized. An ad-hoc heuristic

algorithm was developed for the network generation exercise. Its inputs include: (1)

an existing transit network and its constituent routes, (2) the network size parameter

(number of routes) and (3) feasibility criteria for route length (

rlen

), route directness

(

rdir

) – minimum deviation from the shortest path between a given origin destination

pair and route overlap

roverlap

considered in this work as the maximum coincidence

between the links of a route and the shortest path. These parameters are used to

deﬁne the feasibility conditions for acceptable routes. The network generation heu-

ristic is developed with the Java programming language(Arnold and Gosling 2000),

JGrapht (Michail et al. 2019)—an open-source graph creation and manipulating

library—and XML(Bray etal. 2006). The existing transit network data is presented

as one of the outputs of a General Transit Feed Speciﬁcation (GTFS) feed(Wong

2013), which involves extracting and reformatting the transit network and routes

from the GTFS data. The network is then converted into a GraphML ﬁle(Brandes

etal. 2002), a unique XML format for graphs. The conversion makes it possible to

read the network as a graph with nodes, links and their attributes, and the graph

can be manipulated with the JGrapht tool and graph theory operations. As part of

the reformatting activity, the OD stops for existing network routes are extracted and

Fig. 3 Flow diagram of SBTNDM

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used in the NGA. The steps taken to generate the feasible candidate networks with

the NGA are:

1. Read in OD pair data: The algorithm starts by reading in the OD pairs extracted

from the existing network routes.

2. Generate multiple paths between each OD pair: Next, the k-shortest paths algo-

rithm byYen (1971) is used to create a user-speciﬁed number of paths for all the

OD or node pairs so that multiple routes can be enumerated between each OD

pair. The k-shortest path algorithm typically generates multiple paths in increasing

order of magnitude relative to a weighted cost factor. In this work, the path length

in kilometres for each route is used as the cost factor. Therefore, if x paths are

generated between an OD pair, the ﬁrst path corresponds to the Dijkstra shortest

path (SP)(Johnson 1973), and its length is equal to the beeline distance between

the node pairs. The created paths, which will hereafter be referred to as alternate

paths, are usually longer than the shortest path in increasing order of magnitude.

3. Check route length feasibility conditions for all routes: At this stage, the route

length

rlen

feasibility is checked for both the shortest path and alternate paths to

verify that a maximum and minimum route length condition is satisﬁed.

4. Check other feasibility conditions on the alternate paths: After satisfying the route

length feasibility, other checks are carried out only on the alternate paths. The

ﬁrst one veriﬁes the directness

rdir

of the route, checking that an alternate path

does not deviate excessively from the geometry of the shortest path.

(a) Check for route directness: This is important because users consider route

deviations unappealing; hence, the deviation should be very small. How-

ever, it is sometimes necessary for a route to deviate to adjoining areas

where a major transit route does not run to help cover demand in those

areas. A factor of 1.2 is used in this work.

(b) Check for route overlap: The second feasibility condition is for route over-

lap

roverlap

, checking whether there is a similarity between the links of the

shortest path and the alternate path. A minimum value of 0.5 has been used

in this work, implying that each satisfactory alternate route must contain at

least half of the shortest path’s constituent links.

(c) Check if the route exists already: Lastly, a ﬁnal check is made to ensure that

the alternate path does not currently exist in the list of stored routes.

5. Save the feasible routes for the current OD pair: If all the above-stated conditions

are met, the alternate path is saved as a candidate route in a list created for the

speciﬁc OD pair. This process is then repeated for all the OD pairs, with each

OD pair having its own unique list wherein the routes generated for that OD are

saved.

6. Perform stratiﬁed sampling of routes in all saved lists of routes: After generat-

ing the feasible routes, candidate network solutions are created by ﬁrst using

a stratiﬁed sampling technique to select routes from each OD and combining

them into a network. In stratiﬁed sampling, a population is divided into various

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sub-populations, and individuals are then selected from each group or strata to

make up a random sample. See further details of stratiﬁed sampling inDorofeev

and Grant (2006). In this work, drawing from this sampling technique, the list

of routes generated for each OD pair is considered a stratum. The sampling is

then achieved by randomly choosing the routes from each stratum and combin-

ing them into networks to ensure that the order of the existing network OD pairs

is retained after sampling. Through this process, a pool of feasible networks is

generated. From this pool of feasible networks, the ﬁrst population is initialized in

the NSGA-II. In cases where it is not paramount to retain the order of the routes,

the feasible networks generated for all the OD pairs can be placed in a single pool.

Other sampling techniques, like random sampling, may then be used to generate

the required number of networks.

7. Convert the sampled routes to a network transit schedule input ﬁle: The ﬁnal

step in the route generation process is to convert the candidate route networks to

MATSim transit schedule ﬁles, the appropriate input format for the optimization

algorithm. However, for the NSGA-II to operate on the solutions, a unique encod-

ing will be deﬁned. Details of this will be revealed when the NSA is discussed.

4.2 Network evaluation procedure

In this step of the model, MATSim is used to evaluate the generated network solu-

tions, and a parallel implementation of a MATSim public transit scenario is set up

for this purpose. This is called by the SBTNDM during the evaluation process, and

MATSim is called each time a new solution is to be evaluated. Inputs for the NEP

include:

1. the initialized population of network alternatives,

2. a synthetic population of agents and their travel demands for a 24-hour activity

plan created from the automated fare collection data,

3. an initial schedule of transit operations on the routes of the network, comprising

a timetable with its detailed ﬂeet schedule and vehicle departures and

4. a ﬂeet of transit vehicles that will operate the schedules.

The network evaluation step outputs an objective function value or score, which is

mapped to each evaluated network. The optimization algorithm then uses the score

to rank the performance of each solution in the next step - NSA. Before evaluating a

new solution, the subsisting transit schedule data ﬁle is overwritten, as it would have

been altered during the NSGA-II reproduction. The MATSim simulation process

then begins by executing and optimizing the users’ initial demand. At the end of

the simulation, the resulting events ﬁles are analyzed to evaluate the objective func-

tions in Equations(2) and(3), respectively, with parameter values obtained from the

events ﬁle. A score or objective function value is obtained from the analysis and that

score is assigned to the current network solution, which is returned to the optimiza-

tion module for further processing. The MATSim scenario used in this paper was

parallelized. The parallel implementation of the simulation is discussed next.

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One way to account for the randomness associated with stochastic processes

is to simulate the process multiple times and use the mean result of the diﬀerent

simulation runs. In this research, the simulation experiment ran multiple instances

of MATSim in each evaluation of the candidate network solutions. To satisfactorily

describe the stochastic behavior of passengers on the transit network, multiple runs

of the simulation are required in each iteration of the optimization process. MAT-

Sim has multi-threading capabilities, which means that it can run in parallel when

extensive simulations or a high number of iterations are required. The paralleliza-

tion is achieved by setting MATSim’s numberOfThreads feature in the global

module within the conﬁguration ﬁle. Internally, each simulation or run is comprised

of a user-speciﬁed number of MATSim iterations. Therefore, the number of itera-

tions required to achieve equilibrium in every run of the simulation, was experimen-

tally determined to be 80 iterations. Figure4 shows the number of iterations for this

model.

It should be noted that the iterations operate sequentially and not in parallel,

because in the simulation each new iteration uses the results of the previous one as

input. In essence, succeeding iterations learn from preceding ones until an equilib-

rium is achieved in the simulation. However, as multiple runs are required in this

case, they can be set up in parallel. Each parallel MATSim simulation is executed

in its own Java virtual machine (JVM)(Arnold and Gosling 2000), because each

simulation needs to use a unique pseudo-random number generator (PRNG)(Rahi-

mov etal. 2011; Matsumoto and Nishimura 1998). In the end, the various results are

averaged and used. The collection of multiple runs is referred to as an ensemble of

runs, and counts as one evaluation of the candidate networks. The MATSim ensem-

ble can be seen in Figure5.

Fig. 4 Number of MATSim iterations for the SBTNDM; convergence occurs after 80 iterations

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Simulation-based optimization for designing transit networks

In this work, 30 runs of the simulation are used in each ensemble. This value is

obtained by experimentation. This means that to evaluate each network, 30 parallel

runs of MATSim are executed.

4.3 Network search algorithm

The ﬁnal stage of the model describes how the network optimization progresses and

how a Pareto set of transit network solutions is realized. The main inputs used here

are the feasible candidate solutions from the NGA and the objective function scores

from the NEP, implying that at diﬀerent stages of its operation, the NSA will call the

NGA and NEP sub-routines. The generated network routes are converted to MAT-

Sim transit schedule ﬁles, which contain both the transit routes and their schedules.

The format of these ﬁles is Extensible Markup Language (XML)(Bray etal. 2006).

An important step in evolutionary algorithms is to encode the phenotype of each

solution. To this end, the transit schedule ﬁle which is initially in XML format, is

converted to a JSON data structure, facilitating the eﬃcient manipulation of the

transit schedules with the genetic operators (selection, crossover and mutation) dur-

ing the reproduction process. However, this encoding scheme makes it necessary to

customize the NSGA-II operators to enable them to manipulate the JSON format.

As stated previously, the major advantage of this approach is that it accommodates

the encoding of each network with a detailed operational schedule, thereby facilitat-

ing the simultaneous handling of the route network design and frequency setting

sub-problems of the TNDP. The optimization process then starts with initializing

the pool of feasible solutions in the NSGA-II. The initial population is evaluated

with the NEP, thereafter, the NSGA-II’s crowding comparison operator is used to

rank the solutions, based on the objective function scores obtained from the evalu-

ation step. Subsequently, pairs of the best performing solutions are selected from

the population and encoded as JSON ﬁles to serve as parents in the reproduction

Fig. 5 The parallel implementation of MATSim used in the SBTNDM

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process. The single point crossover and polynomial mutation operators are then

used to perform the actual reproduction of oﬀspring. The crossover and mutation

operators adapt the genetic programming (GP) strategy which solves the problem

of ﬁxed length solution encoding commonly used in genetic algorithms by deﬁning

non-linear structures with diﬀerent sizes and shapes. This is applicable to the XML

and JSON decision variables, since the latter are tree-like structures. The method

allows the direct manipulation of the encoded network variable with a crossover or

mutation point corresponding to a node on the network while the genetic material or

routes are swapped between nodes as demonstrated in Figures6a, 7,8a.

The ﬂexibility of this approach entails that further customization, such as mul-

tiple point operations is possible. The crossover and mutation operators are con-

trolled by probabilities set to 0.75 and 0.25, respectively. For each oﬀspring cre-

ated, a check is done of its topology to ensure it is logical. MATSim allows for the

check to be done using its network cleaner function. After creating a new popula-

tion of oﬀspring solutions, the oﬀspring are merged with the parent population. The

process continues iteratively with the better-performing solutions selected in each

generation, thereby guaranteeing continuous improvement of the solutions until the

termination criterion (number of generations) is reached. The latter is set to ensure

that the algorithm stops once the criterion is satisﬁed. Lastly, the set of solutions

obtained in the ﬁnal generation are decoded by converting them from the JSON for-

mat back to the MATSim network and schedule ﬁles for further analysis.

Fig. 6 Parent networks chosen

from the pool of feasible net-

work solutions

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Simulation-based optimization for designing transit networks

Fig. 7 Oﬀspring networks after

crossover

Fig. 8 Oﬀspring networks after

mutation

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5 Results anddiscussion

In this section two tests are performed on the SBTNDM. The ﬁrst tests are that of

computational time and the performance of the algorithm in terms of its result qual-

ity. Two indicators – hypervolume and generational distance are used for the latter

test. The numerical tests, on the other hand, involve the application of the model to

a real network case study in the city of Cape Town, South Africa. These tests will

demonstrate the robustness of the model and its practical application in the improve-

ment of large-scale networks. The tests are conducted with the national Centre for

High Performance Computing’s Lengau cluster, which is a Dell Linux HPC cluster

with a total of 1368 nodes and 32832 cores and allowing access to 240 nodes at a

time. On this resource, one experiment took approximately 15 minutes to run. To

show how the SBTNDM scales in terms of the network size, the number of routes

in the network is varied from 10 to 50. The plot for computational time of the model

can be seen in Figure9. Computation time is observed to increase as the number of

routes or network size increases.

5.1 Algorithm performance

In the computational tests, attributes of the model’s solutions such as their spread

and convergence are measured. The tests also give parameter values that can be used

when the model is applied to design scenarios. The result of an MOEA is a near-

optimal solution set, which is also called an approximation set and considered as an

approximation of the often unknown Pareto front which is also called the reference

set. The quality of MOEA solutions is therefore measured based on the proximity of

the approximation set to the reference set in the search space(Coello etal. 2007), if

the latter is known or available. When the reference set is not known as is the case

with many TNDPs, it is possible to measure the quality of an MOEA’s solutions by

checking factors like the convergence or the spread of solutions across the obtained

Fig. 9 Plot of the computational time vs. network size

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Simulation-based optimization for designing transit networks

Pareto front. Two performance indicators, namely hypervolume and generational

distance, are used in this paper to measure the quality of the obtained solutions.

5.1.1 Hypervolume

The Hypervolume indicator measures the volume of a problem’s search space that is

dominated by the approximation set(Bringmann and Friedrich 2013). The indicator

is calculated relative to a reference point known as the nadir point which is usu-

ally the worst-case objective value for each objective function(Hadka 2017). Some

advantages of the hypervolume indicator are that:

1. it is easily adapted to problems with many objectives,

2. it is a measure of both convergence and diversity in an MOEA and

3. it does not require prior knowledge of the Pareto front to guide the search for a

solution that approximates the former.

The main limitation of this indicator is that it is computationally expensive. In terms

of its behavior, a higher hypervolume value indicates a better solution or approxima-

tion set, because it dominates a greater portion of the search space. Figure10 shows

a plot of the indicator after 50 generations of the SBTNDM. The ﬁgure shows that

the value of the indicator steadily increases as the algorithm’s generations increase,

implying that the SBTNDM’s solutions improve in successive generations, which

matches the known behavior of the hypervolume indicator. The results also show

that the indicator converges close to 50 generations; hence, the number of genera-

tions required to get near-optimal network solutions with the SBTNDM is 50.

5.1.2 Generational distance

The generational distance (GD) indicator is used to measure the convergence of the

solution set obtained from an MOEA(Liu etal. 2019). It is obtained by measuring

the average distance between each solution in the approximation set and the nearest

one in an MOP’s reference set. Smaller values of the indicator are considered better.

Fig. 10 Plot of the hypervolume

indicator

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When the approximated set is a subset of the reference set, the GD is equal to zero.

A plot of GD against the number of generations is shown in Figure11.

If the approximation set contains a single solution that is too close to the refer-

ence set relative to other solutions in the set, the GD measurements may be unre-

alistically low, and for this reason GD is often combined with other quality or per-

formance indicators. The results show a convergence of indicator values after 45

generations, and the behavior of the indicator observed in the ﬁgure is in line with

its expected behavior.

5.2 Numerical results

The results discussed here were obtained by applying the SBTNDM to the design

of the integrated Public transit network (IPTN) in Cape Town, South Africa. The

IPTN which is a public transit network planned in anticipation of the future eﬀect

of urban growth on travel demand in Cape Town. It is a long term plan which is

expected to be implemented in phases and intended to be fully functional by 2032.

The plan involves a signiﬁcant expansion of the city’s existing public transporta-

tion network. This is logical as the population of the city is expected to grow by

approximately 37% by the target year. The current network comprises a bus network

known as the Golden Arrow Bus Service (GABS), a bus rapid transit (BRT) net-

work and a rail network. When completed, it is expected that BRT and rail would

form the backbone of the IPTN. This work focuses on the improvement of the BRT

service. The network consists of 472 nodes and about 46 operational routes. The

service is intended as the backbone for a planned larger and fully Integrated Rapid

Transit Network (IRTN) in Cape Town, which comprised of other land-based public

transport modes like a bus and rail service that currently operate with low eﬃciency.

The BRT system oﬀers a restricted tap-in and tap-out access to passengers at the

terminals and with dedicated bus lines in high congestion areas like the central busi-

ness district. However, it shares network links with other road-based public transport

modes like GABS in other areas within the network. Some ineﬃciencies have been

identiﬁed, as the service experiences low ridership on some routes and there is also

Fig. 11 Plot of the generational

distance indicator

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Simulation-based optimization for designing transit networks

a need to reduce the total operational costs of the system. An image of the network

is given inFigure12.

The application of the SBTNDM to this network optimization problem will,

therefore, focus on the respective objectives of reducing user costs to attract more

commuters and reducing the service operator’s total costs. Two main data sources

are utilized, being the automated fare collection data from which the agent popula-

tion and their attributes are obtained and the GTFS data for the service from which

the network, schedules and transit vehicles are obtained. One of the disadvantages

of evolutionary algorithms is the randomness and uncertainty in the ﬁnal solutions,

meaning, that the produced solutions may vary with each run of the algorithm. This

further means that though the solution scores might be the same, the detailed route

structures and timetable schedules within the solutions might be diﬀerent and, in

actual operations, such small details can matter. From the perspective of transit

operators who use the proposed optimization algorithm, it would be diﬃcult to rely

on the sets of solutions, since, they vary with each run. To prevent this, the results

discussed here are from seeded runs of the model. In computational science, random

seeds are used to generate a series of pseudo-random numbers that can replicate

the state of an experiment or simulation(Gosavi 2015b). It implies that if all input

parameters are kept constant, a simulation’s results would be the same if set to run

with a given seed, and diﬀerent if the random seed changes. The resulting Pareto set

Fig. 12 The MyCiTi BRT network

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of solutions obtained by the SBTNDM is further evaluated, and the evaluation of

each network solution is done with MATSim. The Pareto front can be seen in Fig-

ure13. The plot shows the solutions plotted with the base case of the MyCiTi BRT

network.

A visual observation of the plot reveals that among the solutions in the Pareto

front, the network with the highest travel time also has the lowest operator cost and

vice versa, which is indicative of the trade-oﬀ between both the users’ and opera-

tors’ perspectives. Users prefer direct trips that reduce their travel time, while opera-

tors prefer longer and slightly more circuitous routes that increase the volume of

demand they can potentially satisfy while reducing their average service costs. Bal-

anced network solutions occur within the marked cluster in the middle of the plot in

Figure13. In the context of this work, the networks are considered to be balanced,

as they exhibit the least conﬂict between the previously mentioned objectives, i.e,

they are the best compromise solutions between the stated objectives. These solu-

tions are, therefore, regarded as the best trade-oﬀ network solutions to the problem.

The above discussion, however, contrasts with the current situation of the MyCiTi

BRT network, indicated as base network in Figure13, which shows both higher user

and higher operator objective cost values compared to the models’ solutions. This

is indicative of the earlier mentioned operational issues of low ridership and high

operational costs on the network. In Table 1 the objective function scores for the

Pareto solution set and the base case are presented. The base network is clearly infe-

rior to the Pareto set of solutions as it performs worse than all solutions in the user

cost objective.

Among the solutions obtained from the SBTNDM, network 1 has the lowest user

cost or objective 1 score and will be referred to as the user-centric solution. This net-

work also has the highest operator cost or objective 2 score. By contrast, network 14

has the lowest operator cost and will be called the operator-centric solution. As indi-

cated earlier, the best compromise between the objectives occurs between solutions

Fig. 13 Pareto front plotted with the base case network

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6 through 11. However, network solution 8 shows the least diﬀerence in the values

of both objectives. Hence, the balanced network is considered as one that shows the

least conﬂict or the best compromise between the commuter’s and operator’s per-

spectives. To depict this clearly, the above objective scores are normalized by rescal-

ing them to a scale of 1 to 10 using the standard formula below.

where:

xmin

= minimum objective function value

xmax

= maximum objective function value

ymin

= normalized scale minimum value

ymax

= normalized scale maximum value

xi

= objective function value to normalize

z = expected normalized objective function value

Subsequently, the normalized scores are ordered and plotted against one another.

Table2 shows a plot of the normalized objective function scores.

This facilitates the results being plotted on a similar scale, and the normalized

scores are ordered and plotted against one another. Figure 14 shows a plot of the

normalized objective function scores for the Pareto solutions.

Having identiﬁed these three network solutions – 1 (user-centric), 8 (balanced)

and 14 (operator-centric) – as proxies for the perspectives mentioned above, they

are then isolated for further analysis. The analysis is carried out to measure their

performance in terms of diﬀerent network performance indicators. The indicators

used include total satisﬁed travel demand, total operational cost, network utilization

percentages, unsatisﬁed demand, vehicle mileage and vehicle hours. The results of

the analysis are presented in Table3.

(8)

z=ymin +(xi−xmin )

⋅

(ymax −ymin)∕(xmax −xmin )

Table 1 Raw objective function

values Network User cost (hours) Operator cost (’000)

1 519.79 24,767.52

2 520.51 24,536.12

3 525.85 24,304.72

4 532.02 24,073.32

5 535.50 23,976.82

6 544.26 23,745.42

7 552.67 23,648.92

8 555.06 23,514.02

9 560.16 23,417.52

10 572.36 23,321.01

11 587.91 23,186.12

12 602.20 23,089.61

13 609.32 22,993.11

14 633.37 22,979.07

Base network 647.55 24,228.14

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Table 2 Normalized objective

function values Network User cost Operator cost

1 1.00 10.00

2 1.06 8.84

3 1.48 7.67

4 1.97 6.51

5 2.24 6.02

6 2.94 4.86

7 3.61 4.37

8 3.79 3.69

9 4.20 3.21

10 5.17 2.72

11 6.40 2.04

12 7.53 1.56

13 7.59 1.97

14 8.09 1.07

15 10.00 1.00

Fig. 14 Network solutions on the Pareto front showing diﬀerent compromise solutions

Table 3 Aggregate transit network performance indicators for the identiﬁed scenarios

Indicators Base network Solution 1 (User) Solution 8 (Balanced) Solution 14

(Operator)

Satis. demand (pax) 24,928 34,216 31,694 29,590

utilization (%) 64.63 88.71 82.17 76.72

Veh. dist (km) 52,619.35 48,567.20 45,215.15 42,452.99

Veh. time (hr) 2,057.47 1,618.91 1,507.17 1,348.43

Op. cost (’000 ) 24228.14 24767.52 23514.02 22979.07

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403

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Simulation-based optimization for designing transit networks

In the table, the existing base network satisﬁes the smallest amount of demand,

though its operational cost is less than that of solution 1 but more than the balanced

and operator solutions. It should be noted that the increased eﬀectiveness in terms of

travel demand satisfaction, which is visible in the other networks when compared to

the base network is attributable to latent demand that arises as a result of the network

improvements achieved from optimizing the BRT system. On the other hand, solu-

tion 1 has the highest satisﬁed demand and network utilization, as well as the high-

est operational cost. This is similar to an optimization scenario in which the users’

objectives are prioritized and more passenger demand on direct routes is served.

Therefore, circuitous routes and those running through transfer points will be mini-

mal or excluded where necessary. This also means that, on average, passengers will

travel shorter distances to their destination, which will encourage more people to use

the service. However, the increase in ridership leads to an attendant increment in the

operational frequencies, because operators would like to maintain the attractiveness

of their service and encourage continued patronage from commuters by sustaining

a good level of service in terms of travel time. Typically, increased operational fre-

quency is a major cost driver for operators, as they must deploy more resources (per-

sonnel and ﬂeet) on the network. In contrast, network solution 14 shows an opposite

trend to that of solution 1, as it shows the lowest operator cost and lowest total net-

work utilization and is similar to a case in which the operator’s objective is prior-

itized. The results show that the operator has less vehicle mileage and operational

hours than the user-centric solution, but it also satisﬁes less demand. This may be

because trying to maximize network coverage by using circuitous routes may ulti-

mately discourage passengers who prefer the direct routes. A network that is skewed

in favor of the operator will primarily contain routes that are longer than those pre-

ferred by users. Lastly, an optimal transit network solution would contain a mix of

direct routes and other, more circuitous ones. Hence, the solutions earlier referred to

as the best compromise solutions, respectively, represent a balance between the user

and operator perspectives. As direct routes reduce operators’ ability to cover demand

along more circuitous paths, an optimized solution must compromise between the

needs of commuters and service operators, which is reﬂected in the middle column

of Table3 for solution 8, where the indicators have values between the user and

operator perspectives. The results show that the solution does indeed oﬀer the best

compromise because it minimizes costs for all stakeholders. The outcomes discussed

above are reinforced in Table4, where the balanced network solution has indicator

values that show a compromise between the users’ and operator’s perspectives.

Table 4 Average performance indicators at route level for the identiﬁed scenarios

Indicators Base network Solution 1 (User) Solution 8

(Balanced) Solution 14

(Operator)

Route density (pax/route) 541.92 743.83 689.00 643.26

Avg. op. cost (’000) 526.70 538.42 511.17 499.55

Avg. veh. time (hr/route) 44.73 35.19 32.76 29.31

Avg. veh. dist (km/route) 1,143.90 1,055.81 982.94 922.89

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404

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Fig. 15 Network map showing the local area around the CBD

Fig. 16 Network design results

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405

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Simulation-based optimization for designing transit networks

To see the details of the earlier discussed network results at a sizeable scale, a

local area of the network results, covering the Central Business District in Cape

Town and its surroundings are highlighted in the map (see Figure15). The details

are then shown in Figure16a through Figure16c, respectively, while an overlay of

all three networks is also visible inFigure16d. Lastly, these results show that sig-

niﬁcant design improvements have been achieved with the SBTNDM during the

optimization process. Furthermore, the balanced solution is the most attractive for

all stakeholders, and it also oﬀers better access to public transit services. Ultimately,

depending on what a policymaker wishes to achieve, they can easily apply decision

support tools such as a multi-criteria decision analysis to the obtained results to

arrive at other trade-oﬀ solutions from the set of non-dominated solutions to match

their priority. These results show that SBO can indeed yield reasonable network

solutions when used in the TNDP process and that ABS can play an essential role in

the process. In conclusion, the potential of the SBTNDM to use big data and other

technological advances currently unfolding in the transit sector make it viable for

modeling large-scale and complex transport scenarios.

6 Limitations

One limitation of the model presented in this work is the fact that an ABM is data-

intensive and computationally expensive in terms of the resources required to simu-

late the model, due to the microscopic level of data needed to build the models.

However, future improvements in the computational performances and speeds will

most likely address this limitation. Another limitation of this work is that passen-

ger activities are derived strictly from automated fare collection data on the system.

While being suﬃcient for the work, it is useful to expand the activities beyond net-

work-related activities.

7 Conclusion

Simulation-based optimization oﬀers a new approach to tackle the TNDP. It solves

the problem through the combination of optimization and simulation models.

Though simulations are computationally resource-hungry, given the number of

evaluations required to solve a problem, they have the distinct advantage that the

stochastic behavior of agents and other random and realistic occurrences on the net-

work can now be simulated within the network design solution model. Furthermore,

advances in computational sciences entails that increasingly large scenarios can now

be simulated in shorter time frames. In this work the authors present the so-called

SBTNDM which combines activity-based travel simulation known as MATSim with

the NSGA-II. Results of applying the model to the MyCiTi BRT in Cape Town,

South Africa reveals that the designed network solutions perform better than the net-

work in terms of travel time reduction for users and operational cost reduction for

operators of the service. Practically, the SBTNDM improves public transport net-

works in line with the objectives of the network user and operator. As a decision

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406

O.A.Nnene et al.

1 3

support tool, the model will be useful in guiding policymakers in Cape Town in

making policy decisions that are relevant to the transportation context and realities

of today. Overall, the use of SBO in solving the TNDP makes it possible to broaden

the potential design objectives, variables and performance measures that can be

used in the network design and optimization process. This enables the implemen-

tation of network optimization studies in increasingly complex scenarios, such as

those that are sensitive to time of day, pricing elasticities and/or that require detailed

traveler behavior. In terms of research directions that might extend from this work

in the future, the primary consideration should be to extend the application of the

SBTNDM to a multi-modal network context to study modal integration. There is

also potential to study the transit network frequency setting problem with SBTNDM.

Acknowledgements The authors acknowledge the Centre for High Performance Computing (CHPC),

South Africa, and the University of Cape Town’s ICTS High Performance Computing team for providing

computational resources for this research project.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,

which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as

you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-

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are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the

material. If material is not included in the article’s Creative Commons licence and your intended use is

not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission

directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen

ses/ by/4. 0/.

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Authors and Aliations

ObioraA.Nnene1 · JohanW.Joubert2· MarkH.P.Zuidgeest1

Johan W. Joubert

johan.joubert@up.ac.za

Mark H. P. Zuidgeest

mark.zuidgeest@uct.ac.za

1 Department ofCivil Engineering, Centre forTransport Studies, University ofCape Town,

Rondebosch, CapeTown7700, WesternCape, SouthAfrica

2 Department ofIndustrial andSystems Engineering, Centre forTransport Development,

Lynwood Road, Pretoria0002, Gauteng, SouthAfrica

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1.

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3.

4.

5.

6.

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