Evolving Urbanisation Policies

Using a Statistical Model to Accelerate Optimisation over Agent-based Simulations

Marta Vallejo, David W. Corne and Verena Rieser

School of Mathematical & Computer Sciences, Heriot-Watt University, Edinburgh, U.K.

Keywords:

Agent-based Model, Genetic Algorithm, Statistical Model, Optimisation, Uncertaincy.

Abstract:

Agent-based systems are commonly used in the geographical land use sciences to model processes such as ur-

ban growth. In some cases, agents represent civic decision-makers, iteratively making decisions about the sale,

purchase and development of patches of land. Based on simple assumptions, such systems are able broadly to

model growth scenarios with plausible properties and patterns that can support decision-makers. However, the

computational time complexity of simulations limits the use of such systems. Attractive possibilities, such as

the optimisation of urban growth policies, tend to be unexplored since the time required to run many thousands

of simulations is unacceptable. In this paper we address this situation by exploring an approach that makes

use of a statistical model of the agent-based system’s behaviour to inform a rapid approximation of the ﬁtness

function. This requires a limited number of prior simulations, and then allows the use of an evolutionary algo-

rithm to optimise urban growth policies, where the quality of a policy is evaluated within a highly uncertain

environment. The approach is tested on a typical urban growth simulation, in which the overall goal is to

ﬁnd policies that maximise the ’satisfaction’ of the residents. We ﬁnd that the model-driven approximation

of the simulation is effective at leading the evolutionary algorithm towards policies that yield vastly better

satisfaction levels than unoptimised policies.

1 INTRODUCTION

The main purpose of planning, in the context of ur-

ban and other land-use, is to improve the commu-

nity’s quality of life by creating a better social, eco-

nomical and physical environment. One of the most

urgent research issues within this broad ﬁeld is the

study of mechanisms that can mitigate the ecologi-

cal degradation that is invariably linked with urban

expansion. What makes this particularly difﬁcult is

that the process of urban expansion needs to achieve

effective and acceptable results at many time-scales.

For example, if a growing city builds quickly on the

majority of the green spaces available to it, it will

severely limit its further growth opportunities. One

possible option for maintaining a healthy urban envi-

ronment is by reserving a collection of arbitrary areas

to transform them into recreational parks. However,

the time planning and geographic distribution of these

spaces needs careful consideration to ensure the qual-

ity and quantity of environmental services provided to

the surrounding community (Forsyth and Mussachio,

2005).

Among other important functions, public open

space planning allows local authorities to protect cer-

tain areas from the urbanisation process, and thus

foster the formation of healthier urban environments.

From this point of view, local and central govern-

ments play perhaps the most crucial regulatory role

(moreso than national governments or international

organisations) in the control of land-use change in the

longer term.

Designing feasible long-term plans is not straight-

forward, mainly because of the many and varied un-

certainties that the future entails. However, there is

much active research in this area, whereby researchers

interested in urban planning and sustainability have

investigated a range of agent-based systems and sim-

ilar mechanisms to explore the consequences of dif-

ferent strategies (Parker et al., 2003; Sasaki and Box,

2003; Sanders et al., 1997). One of the most com-

mon interests in such work is the dynamics of urban

growth, which is linked with the relative distribution

of urbanised, industrial and green spaces and their im-

pact on quality-of-life issues, and how these depend

on the broad strategies in place for land-use (Robin-

son et al., 2012). In many agent based systems, a

typical agent represents a local government decision-

maker, or a recent immigrant deciding where to settle

within the growing city. Based on simple assump-

171

Vallejo M., Corne D. and Rieser V..

Evolving Urbanisation Policies - Using a Statistical Model to Accelerate Optimisation over Agent-based Simulations.

DOI: 10.5220/0004261001710181

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 171-181

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

tions, such systems are able broadly to model land-

type distribution scenarios over time and inﬂuence

decision-makers and policy makers.

However, the computational time complexity of

simulations limits the use of these systems merely to

testing one hypothesis at a time. For example, such a

system may be set up and run to examine the potential

effects of a proposed new tariff for land purchase tax.

A more attractive prospect, however, would be to use

the system to help ﬁnd an optimal tariff or, in general,

an optimal policy or strategy for the task in question.

This area of work however tends to be unexplored,

since the time required to run many thousands of sim-

ulations (as is typically required for optimisation) is

not a feasible strategy in practical situations.

In this paper we address this situation by wrapping

optimisation over the agent-based simulation process,

but use a rapidly accelerated model of the agent-based

simulation in place of the real knowledge. This re-

quires a limited number of prior simulations of the

agent-based urban growth system, and then allows the

use of an evolutionary algorithm to optimise urban

growth policies. The approach is tested on a typical

urban growth simulation, in which the overall goal is

to ﬁnd policies that maximise the ’satisfaction’ of the

residents. A ’policy’, in this work, amounts to the

city authorities’ planned schedule for protection of a

speciﬁc set of green spaces.

Note that similar simulation-based approxima-

tions for optimisation are also used in other ﬁelds,

such as user simulations for spoken dialogue sys-

tems (Rieser and Lemon, 2011) or emulators for man-

aging uncertainty in complex models, such as simu-

lated climate models (MUCM).

In our case, the general objective of the model is

to design a planning process which lead us to ﬁnd

the optimal subset of green spaces within the physi-

cal boundaries of the city. A green space can be de-

ﬁned as a natural landscape located close to urban or

peri-urban areas. This deﬁnition can cover concepts

like county parks, areas of outstanding natural beauty,

natural reserves, forest parks or crown lands (Allison,

1975).

The selected agent-based model simulates the

growth of a city over a 50-year time-span. In each

year, different ﬂows of incoming population lead to

pressure for development of new areas. Meanwhile,

individual areas have varying costs, based on a sim-

ple model that values proximity to green areas over

heavily-urbanised areas, and naturally evolve as the

city grows. The model is therefore dynamic in time

and space, and each simulation run will yield a dif-

ferent result. This in turn exacerbates the difﬁculty

of optimising directly over full simulation runs, since

several runs would be needed to evaluate the statisti-

cal properties of even a single policy. However, the

approach we use to obviate the need for full simula-

tion runs also addresses this issue, since it amounts

to importing and exploiting pre-calculated statistical

averages for each time-step in the simulation.

The remainder of the paper is organised as fol-

lows. Section II provides various introductory and

preliminary details, covering the urban planning prob-

lem, the role of agent based simulation, and evolution-

ary algorithms. Section III then provides a detailed

account of the models, assumptions and processes we

employ in our experiments. Section IV focusses on

the sources of uncertainty that are handled by our new

statistical genetic algorithm approach. Computational

experiments are speciﬁed in Section V, and the results

are presented and discussed in Section VI. Section VII

then draws some conclusions and we discuss further

research in Section VIII.

2 PROBLEM DEFINITION

Open green urban areas play an important role in

maintaining a healthy urban environment. Among

all their favourable effects, their crucial impact in the

economy, quality of life and in the local climate of the

cities (Costanza et al., 1998; Nowak and McPherson,

1993) can be highlighted. However their distribution

and location should be carefully studied by develop-

ing an adequate, long-term planning strategy.

2.1 Urban Open Space Planning

There is a lack of agreement about how to implement

and implant a given planning process and which mea-

sures should be selected. The most remarkable points

to discuss are:

• How to select adequate planning criteria.

• Deciding the most suitable size for the open space

according to the current and expected necessities.

• Where the open spaces should be located and how

they should be accessed.

• The design of the potential activities for these ar-

eas according to different age groups and cultures.

A problem that arises when these issues are tack-

led is that there exist a variety of approaches with

clear contradictory main goals. Among all of them,

the present work follows a demand approach where

the planning process should be based on attributes of

the speciﬁc target population. The necessity of provi-

sion of a set of services deﬁnes the pressure over the

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172

available open space. This pressure can be measured

by means of:

• Size of the urban population.

• Subjective personal preferences.

• Residential distribution.

• Density of the population.

2.2 Allocation of Resources

The problem domain of the present paper can be in-

cluded within the ﬁeld of stochastic control theory.

The developed model represents a paradigm of alloca-

tion of resources within a sequential decision-making

simulator where a set of actions can be taken in each

sequence that is followed by the system.

Generally speaking, a sequential planning prob-

lem can be deﬁned as follows: an environment which

can be described as a state-space set S and an action

set A where S and A are both ﬁnite. Each state s ∈ S is

dependent on the previous state of the system and the

action a ∈ A taken. The transition function δ controls

how actions modify the state of its environment.

t+1

= δ(s

,a) (1)

We deﬁne a policy Π such that the mechanism in

charge of selecting the next action is based on the cur-

rent perception of the environment. This perception

can be total or partial:

Π : S → A

Π(s

) = a

(2)

In turn, the action a inﬂuences as well its environ-

ment provoking the change of the current state. The

process starts in the state s

and by means of the se-

quential application of the policy Π, further actions

are chosen.

2.3 Cellular Automata & Agent-based

Modelling

The present study is based on the results collected in

a basic urban growth model created with the use of

Cellular Automata and Agent-Based Modelling tools.

The topological layout of the city is represented

by a Cellular Automata (CA). CA (Newmann, 1966)

was proposed for discrete space-time representation

of problems which obey their local physics. It is based

on the assumption that by means of local interactions,

the model is capable of representing complex phe-

nomena. The dynamics of the CA are generated by

a set of transition functions which deﬁne how cells

can evolve from one state to another.

The inhabitants who populate the city and their

dynamics are modelled with the use of an Agent-

Based Model (ABM) approach. ABM has been used

to understand the interconnections, interdependences

and feedbacks created among a set of heterogeneous

individual entities in order to fulﬁl their goals. Each

agent has an individual decision-making capacity ac-

cording to its personal role. In combination with the

CA approach, agents can be explicitly located and, in

this way, they can inﬂuence its environment and affect

the patterns formed in urban infrastructure.

ABM along with CA taking the role of repre-

senting land-use change dynamics have been applied

broadly in the ﬁeld of urban development. Mention-

able is their use to simulate allocation decisions (Ot-

ter et al., 2001; Brown and Robinson, 2006) or in resi-

dential selection within a non-stationary housing mar-

ket (Devisch et al., 2009; Parker and Filatova, 2008).

On the other hand (Filatova et al., 2009) analyses how

these tools are applied to analyse how prices affect

urban agent behaviour. Finally, (Miller et al., 2004)

studied the role of transportation in the evolution of

an urban region.

2.4 Genetic Algorithm

Genetic Algorithm (GA) (Holland, 1975) it can be

deﬁned as an heuristic that mimics the behaviour of

natural selection postulated by the English naturalist

Charles Darwin in the 19th Century (Darwin, 1861).

This search strategy is based on the assumption that

nature evolves by the course of new generations pre-

serving the species more suited to their environment.

The tools deﬁned by a GA to improve the population

over time are the use of mechanisms like reproduc-

tion, mutation, crossover and selection.

Here we use GA to optimise the green space al-

location problem. GA has been successfully used to

solve complex spatial problems (Pukkala and Kurt-

tila, 2005). However, its performance in uncertain

environments has been questioned (Wu et al., 2006;

Rieser et al., 2011) due to the fact that a simple GA

has insufﬁcient data to deal directly with uncertainty.

This weakness is the main reason why a GA, under

this kind of scenarios, should be deﬁned carefully and

provided with the support of external tools in order to

overcome these difﬁculties and to be totally suitable

for this type of problems.

There exist different attempts and techniques that

can be applied to a GA to give it this extra functional-

ity. In (Qin et al., 2010) a Genetic-Algorithm-Aided

Stochastic Optimisation Model is applied to cope with

the uncertainty related to the study of air quality in

urban areas. Qin makes use of Monte Carlo simu-

EvolvingUrbanisationPolicies-UsingaStatisticalModeltoAccelerateOptimisationoverAgent-basedSimulations

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lation techniques to measure the effectiveness of a

solution. Instead (Wang and Yang, 2009) resorts to

anti-optimisation techniques (local search) to over-

come the uncertainty generated by ageing presented

in many engineering problems and affects remarkably

the ﬁnal performance of the system.

3 MODEL DESCRIPTION

The selected ABM-CA framework is used to repre-

sent a basic urban growth model with a monocentric

spatial structure based on the traditional Alonso’s ur-

ban economic model (Alonso, 1964). The strategy

of this model to explain the modern urbanisation pro-

cess is based on the maximisation of a utility func-

tion. Urban pattern formation is the consequence on

individual urban residence preferences which achieve

an economic competitive equilibrium between hous-

ing and commuting costs.

The physical layout of the city is conﬁgured by a

2-dimensional lattice of 50x50 cells. Each cell cor-

responds to a physical portion of the city and it can

be populated by more than one agent. Each of them

represents a family unit.

Figure 1 represents the development of a city in a

determined sequence of the simulation.

Figure 1: An image which represents the urbanisation dy-

namics where the monocentric city (green cells) spreads

their boundaries. This process convert rural cells (in grey)

into peri-urban areas (white cells).

The evolution of the city is ruled by an inter-

nal schedule or discrete-event box in which the num-

ber of deﬁned events has always associated a deter-

mined time-horizon of ﬁnite duration. The dynamics

of agents and cells allow the model to evolve between

a set of predeﬁned one-directional states. The rules

of transition deﬁne how cells and agents change their

state at each time step.

On the other hand, the types of the cell presented

in the grid can be broadly divided into two main

groups: urbanised and non-urbanised cells.

3.1 Urban Cells

Urban cells are cells that have been transformed

from native ecosystems into either impermeable sur-

faces or green areas formed normally by non-native

species (Byomkesh et al., 2010).

In the model, when cells receive the permission to

be urbanised, which ﬁguratively means that dwellings

are constructed, they can allocate population that is

represented by agents. Agents decide their residence

location by searching a trade-off between their per-

sonal preferences, quality of life and their economi-

cal restrictions. This search involves the interaction

among different parameters of the model. The de-

cision is represented by the maximisation following

utility function:

maxU = (w, z, x, p : w > 0,z > 0,x ≥ 0, p > 0)

such that:

w − z − kx + p = 0

(3)

In the equation, x represents the distance from the

household to the Central Business District (CBD) that

is located in the centre of the lattice, w is the wage re-

ceived monthly. This quantity is deﬁned by a uniform

random process and does not change through the life

time of the agent, z is the price of the residential good

and k is the constant marginal community cost. Fi-

nally, p represents the agent’s preferences. This pa-

rameter takes into consideration their personal level

of preference for houses located close to green areas.

In the agent deﬁnition a stochastic value which rep-

resents his acceptance to pay more for this kind of

houses, is generated. This parameter is an extension

of the economic competitive equilibrium described

by (Alonso, 1964). Following this utility function

agents populate the grid.

3.1.1 Prices of the Urban Cells

The prices of urban cells represents the amount of

money that agents have to pay regularly as a rental

cost and their values are dependant on the following

factors:

• The Demand. The demand is deﬁned according

to the number of agents living in a given cell.

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174

– The demand for certain preferred locations in-

creases their price.

– The drop in the population in a cell decreases

its price.

– If one cell does not receive any new neighbour

during a determined period of time, the value of

their dwelling is reduced.

• Proximity to Green Areas. The proximity of a

green area is a factor which affects the ﬁnal price

of the houses. This factor increment a 10% the

ﬁnal price of the dwelling.

3.2 Non-urban Cells

Non-urban cells are cells that did not suffer a urbani-

sation transformation. At the beginning of the simu-

lation the model assigns an stochastic value to all the

non-urbanised cells. This parameter called BioValue

represents the ecological value of this parcel of land.

This value is assigned to each cell by a uniformly ran-

dom process U(0,1):

BioNeighbourValue

= 0

for each cell n in neighbourhood(c){

if(BioValue

 0.7)

BioNeighbourValue

+ = 0.01

if(BioValue

 0.3)

BioNeighbourValue

− = 0.01

}

CellValue

= BioValue

+ BioNeighbourValue

In the present model, if the cellValue of a cell is

bigger than 0.7 this cell is classiﬁed as a forest cell,

otherwise it will be assigned an agricultural. Both

types of cell have associated a different price based

on rural land prices in UK (Riley, 2002).

The belonging to each category is dynamic over

the time. Actually the model suffers a continuous

transformation in the state of their cells changing

from forest to agricultural. These uninterrupted trans-

formations represent the ecological degradation pro-

cess of the peri-urban areas provoked by the urban

expansion over the grid.

Local governments can adopt a wide range of

interventionist mechanisms to restrict the ownership

over the land and control its use. Among these mea-

sures the local authority can assume the proper own-

ership of the land and assign them partially or totally

the function of urban green spaces. One sucessful ex-

ample of the use of this mechanishm is the case of

Stockholm city (Passow, 1970). Thus, local govern-

ments act as a response to social requirements over

gardens and parks to provide a set of services based

on the proximity to potential users.

The model delegates the responsibility of select-

ing the best non-urbanised stands to a new special

agent called Municipality. This agent does not inter-

act with the rest of agents. His main goal consists

of managing the purchase and protection of green ar-

eas within the city by means of a monetary income

received periodically called budget.

The location of these areas is a crucial factor for its

future use. The selection is performed sequentially in

each lapse of time and is limited by the budget and

the conﬁguration of the system in this precise mo-

ment. Once the purchase is done, the state of the cell

is changed to protected and the construction of urban

facilities within it is forbidden.

This selection process can be formulated as fol-

lows: if C is deﬁned as the ﬁnite set of cells included

into the lattice, A the subset of cells that can be con-

sidered as a candidate cell to be purchased, P the sub-

set of cells that are protected and U the urbanised cells

such as A ⊂ C, P ⊂ C, U ⊂ C and A ∪ P ∪U =

0, then

the selection of the cells to be acquired in a given se-

quence of time t can be deﬁned as:

∀ cell c ∈ C

price(c)

< budget

∧ c

/∈ {P,U}

=⇒ c

∈ A

(4)

Once the candidate cells are discriminated, the

purchasing process can be formalised as:

∀ cell a ∈ A

max

satis f action

δ(a)

=⇒ a

∈ P ∧ a

/∈ C

budget

t+1

= budget

− price

(a)

(5)

The function δ represents the metric that measures

the level of satisfaction of the population and takes

into account the monitoring of the distance to green

areas. See formula 6.

Every subset of selected cells has associated a

level of satisfaction of the population allocated within

the boundaries of the city. The model should select

the conﬁguration of green areas which achieves the

highest possible level of satisfaction according to the

restrictions of the system during the considered period

of time. However, the huge number of possible com-

binatorial selections makes the task of performing an

exhaustive search of all different choices impossible

in a feasible amount of time.

4 SOURCES OF UNCERTAINTY

In the present model uncertainty can emerge from a

wide variety of sources. Apart from the fact that the

EvolvingUrbanisationPolicies-UsingaStatisticalModeltoAccelerateOptimisationoverAgent-basedSimulations

175

implementation of long-term plans always implies to

be able to cope with unpredicted future variations, the

complexity resultant from the multiple interactions

occur between the elements represented in the model

makes their management even more challenging.

Some factors which actively contribute to the in-

crement of the level of uncertainty are mentioned in

the list below:

4.1 Property Prices & Green Areas

In the model, the selection of green spaces exerts

a direct inﬂuence on the prices of the surrounding

urbanised cells. (Tyrv

ainen and Miettinen, 2000;

Thorsnes, 2002) analyse this tendency demonstrating

how prices of home properties increase with the prox-

imity of urban parks. This aspect is included in the

model as the agents’ desire to live close to these areas

and is represented by the agent’s acceptance to pay

more for these speciﬁc locations. The implications of

the inclusion of these personal desires provoke a sig-

niﬁcant growth in the demand and subsequently in the

price. All these factors affect the spatial spread of the

city and the patterns developed over the time.

4.2 Ecological Degradation Process

From the point of view of the non-urbanised cells,

the main parameter which involves a high level of un-

certainty is the relationship created between the land

price dynamics and the cells’ ecological value. Due to

the fact that this ecological value is also inﬂuenced by

the cells that form its neighbourhood as is deﬁned in

the previous code. The consequence of this linkage is

that a signiﬁcant change in a speciﬁc area of the lattice

spreads in all directions and can produce instability in

these eco-values of the surrounding cells.

This is the reason why the growth of the physical

boundaries of the city creates an ecological degrada-

tion process in the surrounding areas. This dynamic

inﬂuences the price of the non-urbanised cells that are

closely located and hence the purchasing process of

protected areas.

4.3 Urbanisation Process

The underlying process of urbanisation is in nature

partially random and mainly determined by two fac-

tors:

• The rules of transition of the cells.

• The demand level.

Firstly, the rules of transition of the cells are based

on preselected probabilities. Regarding the demand

level, the model has deﬁned a constraint in order to

avoid transforming peri-urban into new urban cells if

there is an enough number of non-populated urban

cells in the city. A cell can be unpopulated if any

new adult agent considers this cell proﬁtable enough

to move within it or because all agents allocated in-

side a determined cell have died and any new agent

has move inside.

The model is designed so that it is necessary to

have a minimum population density in all urbanised

cells to achieve the enough level of demand which al-

low the system to build new neighbourhoods. This

characteristic was added to avoid an unrealistic spread

of the city due to the behaviour of the rules of transi-

tion of the cells.

The knowledge of the urbanisation process is cru-

cial: the set of candidate cells to be protected are re-

stricted to the non-urbanised ones and hence, in the

protection procedure, we need to be aware of the com-

plete state of the cells in each time step.

4.4 Flows of Population

Another signiﬁcant characteristic of the model is that

the described city is a non-closed-system. This means

that there are income ﬂows of new population com-

ing from migration as well as new offspring resulted

from the current settled population. The dynamics of

these ﬂows are not ﬁxed and predictable. However

they play a relevant role in the ﬁnal population dis-

tribution within the city because it is not possible to

guess in advance where they are going to be allocated.

Subsequently, the density of each future neigh-

bourhood cannot be totally predicted in advance even

if there exists a general preference to live close to the

city centre in line with the Alonso model. In con-

clusion, this effect means that as we do not know the

population distribution it is not possible to know the

percentage of population directly affected by a deter-

mined location of a new green area.

5 CASE STUDY

5.1 Conﬁguration of the GA

5.1.1 Encoding

A GA evolves through time a constant size popula-

tion of individuals as well called chromosomes cho-

sen randomly from the set of candidate solutions. In

the present paradigm an individual is encoded as a se-

quential selections of cells grouped in a predeﬁned

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176

number of time steps. Each of these selections rep-

resents a gene of the individual and can contain from

0 to n protected cells chosen by the Municipality in

one of the sequences of time. The superior limit n

is bounded by the maximum budget available in this

time step.

Linked with each subset of cells selected in each

interval, its budget is stored. The budget can be de-

ﬁned as the amount of money that the Municipality

can assign to the purchase of urban parks. It consists

of the remaining funds resulted from the last transac-

tion made by the Municipality plus possible new in-

comes. The individual value of the budget is always

insufﬁcient to buy any non-urbanised cell.

5.1.2 Selection Scheme & Mutation Process

There exist many selection schemes for GA, among

them the present model uses tournament selec-

tion (Goldberg, 1990) that is a robust and simple to

code selection mechanism for GA based on the idea

of holding a tournament between a group of competi-

tors randomly selected among the population.

Mutation is a tool used to maintain the diversity.

The mutation process alters one or more values of the

genes inherited from the parent. In our case a muta-

tion consists of changing which cell/s will be trans-

formed into a green area in a determined sequence of

time. Additionally if the price of the swapped cells is

different, the associated budget value is updated.

5.1.3 Fitness Function

As we have already mentioned, optimising the selec-

tion of the best recreational areas in a period of time

with the only use of a classical GA approach is not

possible under a high uncertainty environment.

In the previous section we listed the different

sources of uncertainty that the optimisation procedure

should cope with. In spite of all of them, we should

be able to assess how feasible a given solution can be

in the future. In a GA paradigm this is done by the use

of a ﬁtness function. This function can be nonlinear,

discontinuous or even nondifferentiable.

To measure the value of a determined green area,

the model can use different kind of metrics according

to which aspect or aspects want to be emphasized. In

the current model the quality of a solution or satisfac-

tion represents the accumulative satisfaction achieved

by each person settled on the city with respect to

the topological distribution of green areas in the grid.

Concretely it is associated with the distance between

him and a green area.

(Giles-Corti et al., 2005) states that the distance

to a green area inﬂuences the frequency of use and

the activities undertaken. According to this crite-

rion, green areas can be classiﬁed into the following

groups:

• Access within a short walk. If the green area can

be reached within less than 300 meters.

• Access within a long walk. The distance objective

is ranged from 300 to 600 meters.

• Access with help of any means of transport. If the

distance is larger than 600 meters.

The same study concludes that people do not gen-

erally use a green area if it is located beyond a thresh-

old of 300-400m. Following this approach the pro-

cess of calculation the ﬁtness function can be deﬁned

as follows:

If A is the set of agents living in the city, P is the

set of protected cells and C is the set of cells deﬁned

in the grid such as P ⊂ C, then for a given time t:

∀ agent a ∈ A

∀ protected cell c ∈ P

δ(a) = min

distance

(a,c)

s(a) =











3 if δ(a) = 1

2 if δ(a) = 2

1 if δ(a) = 3

0 otherwise

Θ =

∑

i=0

s(a

)

(6)

We deﬁne δ as the function that calculates the dis-

tance from the location of a given agent a to the clos-

est green area in the grid using Manhattan distance.

Besides we deﬁne s as the function which calculates

the individual satisfaction achieved by a given agent

a. Finally, Θ represents the total satisfaction achieved

by the population of size n in the lattice in time step t.

This ﬁtness function is, in turn, linked directly

with the spatial spread of the city and the population

density of each stand. However, to be able to use a ﬁt-

ness function, it is necessary to know the location of

the entire population in each time step and make this

information available to the algorithm as input data.

5.2 Data Collected from Simulation

Runs

The optimisation method overcomes this aforemen-

tioned lack of knowledge by the use of data gath-

ered from the non-optimised version of the simula-

tion. This data should describe what is the most likely

topological development of the city and the popula-

tion evolution in terms of number and location that

can occur in the future.

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The collected data retrieved during the multiple

runs of the non-optimised simulation includes the fol-

lowing elements:

• Due to the fact that only non-urbanised cells can

be candidate to be protected it is necessary to

know when each cell is more likely to be ur-

banised. This data is used to know if in a given

instant of time a cell can receive the status of pro-

tection or not. The non-optimised simulation an-

notates in each run when the different cells are ur-

banised.

• Density distribution. The simulation collects sta-

tistical data about the amount of agents living in

the city and their precise location in the grid in

every time step. This information is necessary to

calculate the ﬁtness function.

5.3 Deﬁned Assumptions

Once the statistical data is gathered, it is necessary

to deﬁne two common assumptions used as a point

of departure in all the components of the optimisa-

tion framework. The involved parameters in these as-

sumptions are the budget and the ecological scenario.

5.3.1 The Budget

The stochastic budget assigned to the municipality in

each sequence of time is decided in advance and is

shared by all individual solutions of the GA. This fact

is critical because of the accumulative nature of the

budget and the possible interdependence between the

sequence of choices undertaken.

Dealing with the budget in the optimisation phase

as currently is deﬁned, can arise a potential problem

in the gene mutation process. We recall that the in-

formation encoded in a gene is composed by a set of

cells to be protected and the remaining budget in a

given time step.

Due to the fact that non-urbanised cells can have

different prices, a single modiﬁcation in the selected

cells of a gene can inﬂuence substantially in the

amount of money used in the future selection of cells.

This change could entail the appearance of inconsis-

tencies encoded of the future choices into the chro-

mosome if the former and the new cell involved in

the mutation process are different enough in price that

the remaining budget for posterior purchases could

lead to a signiﬁcant different scenario. This prob-

lem is caused because the election of the budget in

each sequence of time, is not independent from on-

ward choices and one decision could condition future

purchases.

5.3.2 Ecological Scenario

The second assumption refers to the ecological con-

ﬁguration of the lattice at its point of departure. The

scenario is deﬁned by the initial random generation

of the ecological values in each cell. An example of

ecological values assigned to the grid can be seen in

Figure 2.

The optimisation process shares the same initial

scenario for all individuals. The same values are

the base for the non-optimised simulations in which

statistical data is gathered and also for the ﬁnal test

where the real optimality achieved by the GA out-

come is measured.

Figure 2: Environmental values and the effect of the urban-

isation process. The range of colours from green to black

depicts the possible ecological values of the cell. Notice

that in the centre of the lattice, where the city is located, the

black eco-values represent the biological degradation or the

metropolitan area.

5.4 Methodology Followed

The main components which form the data ﬂow of the

optimisation process are:

• The non-optimised simulation.

• The GA algorithm.

• The test component.

The point of departure of the following workﬂow

consists of the deﬁnition of the initial model conﬁgu-

ration as it was mentioned in the previous section and

the assignation of the values for the budget and the

ecological values of the lattice.

Secondly, the non-optimised version of the simu-

lation should be run the necessary amount of times to

retrieve enough comprehensive knowledge about the

dynamics of the model: the topological spread of the

city and population density.

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Once this data is gathered, the GA algorithm can

be carried out. In this phase the GA population is

generated and evolved for 2500 iterations, assuring

its convergence. For each new generation, the possi-

ble candidate cells should satisfy the following con-

straints to cope with the restrictions derived from the

management of an uncertain future:

• New selected cells must not have been already se-

lected for this individual neither in the past nor in

the future in order to avoid inconsistencies.

• The model tries to verify that in a real situation the

selected cells are unlikely to be already urbanised.

Due to the GA cannot infer a priori what is the

possible state of these cells, it should resort to the

collected knowledge stored previously in the sys-

tem. This data depicts the behaviour of the city

and based on it the GA algorithm should discern

what is the most plausible topological layout of

the city in this concrete sequence of time.

The concept of tolerance is used to decide the

level of conﬁdence in which a cell can be selected

to be protected. The more tolerance is permitted,

the more inconsistencies can occur when the real

simulation performs the testing of the solution.

The value for this value used in the experiments

is zero, that means that it is not allowed to protect

the cell if in any of the simulations performed in

the non-optimised phase this cell was urbanised at

this point in time.

• Every time a new individual solution is generated,

the optimisation component should be able to ap-

ply to it the ﬁtness function to measure the quality

of the new solution. However, this ﬁtness function

requires to know which amount of population will

be affected by the choices included into the solu-

tion from the beginning to the end of the simula-

tion.

For this purpose the optimisation component

makes again use of the collected data, estimating

the density of the population in the surrounding

areas and calculating the ﬁtness using Formula 6.

Once the optimisation phase has been concluded

and the ﬁnal individual solution with the highest ﬁt-

ness is generated, the test phase is carried out. The test

component uses the output data from the GA phase to

check the viability of the protected cells. This output

data deﬁnes the cells that have to be selected in each

lapse of time.

These simulations run in a modiﬁed version of the

model where the selection of green spaces is done

in a deterministic way meanwhile the rest of factors

and interactions maintain its dynamical nature and its

complex and unpredictable behaviour. The main pur-

pose of this test step consists of:

• Measuring the real satisfaction of the population.

• Detecting inconsistencies and incompatibilities of

the selected cells.

6 COMPUTATIONAL RESULTS

The results were calculated as averaged over 20 re-

peated optimisations, all of them in compliance with

the assumptions and restrictions commented in previ-

ous sections.

Figure 4 plots two functions which represent the

efﬁciency achieved during 500 ticks of the simulation

cycle. The function in red depicts the amount of sat-

isfaction achieved by the population when the cell se-

lection is performed randomly meanwhile the results

achieved in a test simulation by the GA approach are

represented in blue.

Numerically, the same achieved results are shown

in Table 3 where the data has been discretised and

averaged with a periodicity of 50 ticks of the clock.

The ﬁrst column shows the non-optimised satisfac-

tion, the second represents the GA-optimised satisfac-

tion and the third column calculates the percentage of

improvement of applying GA.

Figure 3: Efﬁciency grouped per each 50 generations.

From these results we can state that GA outper-

forms clearly the non-optimised version, fact that

only on its own is not really noteworthy. However

our main goal with this development was to ﬁnd a

easy methodology to use the Genetic Algorithm ap-

proach into a high uncertainty environment and over-

coming the computational time complexity inherited

of this kind of optimisation. We develop a new robust

methodology able to deal with uncertainty without the

help of any other extra tool or pre-knowledged distri-

bution.

One limitation of the stochastic random approach

is that the efﬁciency achieved depends strongly on the

EvolvingUrbanisationPolicies-UsingaStatisticalModeltoAccelerateOptimisationoverAgent-basedSimulations

179

Figure 4: Discretised version of the satisfaction achieved by

the two approaches.

extension of the land analysed. On the other hand, a

possible drawback of the present approach is that it

requires to perform a large number of simulations be-

fore the optimisation can be carried out even if indi-

vidually each of them are not costly signiﬁcant.

7 CONCLUSIONS

This paper reports on results from a proof-of-concept

study, which show that statistical model approxima-

tions can be used for policy optimisation. In particu-

lar, we show how we can capture and represent uncer-

tainty in an Agent-based Model using data from sim-

ulated runs and ﬁnd optimal urban planning policies

using genetic algorithms.

The strategy is tested in a monocentric urban

model based on Alonso’s model where the main ob-

jective of the experiment is to distribute a set of green

protected areas throughout the lattice with the goal of

achieving the maximum satisfaction from the inhab-

itants of the city. An individual is considered to be

’satisﬁed’ if a green area is placed close enough to the

location of his residence.

The main observation that we draw from the work

presented here is that the appropriate prior use of un-

optimised simulations was effective in guiding the

GA to achieve successful outcomes. The speciﬁc ap-

proach we took is potentially applicable to a wide

range of applications which concern sequential deci-

sion making and require time-consuming simulations

to evaluate decisions.

8 FURTHER RESEARCH

The results on our case study suggest there is consid-

erable promise in our approach. The ability to suc-

cessfully address a wider range of optimization prob-

lems of this kind could lead to a new generation of

tools for use in urban planning.

However, in the meantime, various aspects of the

approach need further investigation. Among them are

three main directions:

Evaluation of Statistical Simulation-based Ap-

proaches for ABM Optimisation. Related research

ﬁelds, such as optimisation of natural dialogue strate-

gies, make use of similar simulation techniques to ap-

proximate real-world behaviour. In the case of spo-

ken dialogue systems, for example, user simulations

are build from small data set of real user interactions.

Similarly, we simulate the (uncertain) behaviour of an

ABM by estimating transition probabilities, i.e. pos-

sible impact of planning decisions, from a small set of

model runs. In future work, we want to explore how

evaluation techniques for user simulations can be ap-

plied to estimate the quality and policy impacts of our

ABM simulations (Rieser and Lemon, 2011).

Improving GA to Include Uncertainty for Sequen-

tial Decision Making Problems. In the previous ex-

periments we have used a variant of genetic algo-

rithms which does not explicitly encode uncertainty

endured by the model environment. In future work,

we plan to investigate advanced evolutionary algo-

rithms, such as X Classiﬁer Systems (Wilson, 1995)

for sequential decision tasks, which explore similari-

ties between evolutionary approaches and Reinforce-

ment Learning and other a priori more traditional ap-

proaches like Reinforcement Learning.

Improve the Complexity of the Urban Model. In

particular, we plan to increment the complexity of our

current metric including factors like size of the urban

park and quality. We will also develop a new eco-

logical metric based on preserving the ecosystems of

the surrounding areas of the city and conduct exper-

iments to compare the trade-off between our current

metric and the new one.

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