USING UNIFORM CROSSOVER TO REFINE SIMULATED

ANNEALING SOLUTIONS FOR AUTOMATIC DESIGN

OF SPATIAL LAYOUTS

Fadratul Hafinaz Hassan

1, 2

and Allan Tucker

Department of Information System & Computing, Brunel University West London,U.K.

School of Computer Science, University Science of Malaysia ,Penang, Malaysia

Keywords: Genetic algorithm, Simulated annealing, Hill-climbing, Cellular automata, Pedestrian simulation, Spatial

layout.

Abstract: The ease of movement of people inside a public space is highly impacted by the public space layout itself.

For example, the flow of a large number of people should be smooth in a well designed public space such as

a stadium or hospital. In extreme cases, people might crush to death during emergency evacuations in

poorly designed spaces. It is vital that design takes into account the smooth flow of pedestrians. In this

paper we describe an initial exploration in using models of pedestrian flow combined with heuristic search

to assist in the automatic design for spatial layout. A two-way pedestrian flow system is simulated and

heuristic search techniques (genetic algorithm, simulated annealing and hill climbing) are used to find

feasible spatial layouts based upon the generated statistics with promising results.

1 INTRODUCTION

Pedestrian flow is an integral part of spatial layout

design. The spatial layout of public space has an

enormous impact upon the ease with which people

can move. There are a number of projects that have

used pedestrian simulation to test pedestrian routes

for airport terminal floor plans, stadium plans,

shopping malls, and galleries (Batty et al., 2003,

Dijkstra and Timmermans, 2002, Pan et al., 2006,

Smedresman, 2006, Zhu et al., 2008). Based on the

previous works, a computer simulation of pedestrian

movement was identified as a useful method to help

designers to understand the relation between spatial

layout and human behaviour. It is vital that spatial

layout design takes into account the smooth flow of

pedestrians which will demonstrate how appropriate

traffic control can effectively address congestion and

safety issue. Pedestrian flow is going to be simulated

by using the concept of cellular automata (CA). This

approach is taken because CA presents a simple

local rule describing the behaviour of each

automaton that can create an approximation of actual

individual behaviour (Dijkstra and Timmermans,

2002). Evolutionary algorithm search method is

preferred for this study because of its principle of

artificial selection: the process involves randomly

mutating a large number of individuals, ranking

them, selecting the best, and iterating over and over

again. According to Smedresman, 2006, they are

particularly adept at sifting through large search

space as in floor plan design domain to find

solutions that may not intuitively present

themselves.

Spatial layout is one of the most challenging

phases of architectural design, and this type of

problem is an NP-hard class which belongs to a

complex combination optimization (Charman et al.,

1994, Honda and Mizoguchi, 1995). In other words,

algorithm for solving this problem needs a very long

running time and a great storage space when the

scale of the problem is large. For solving

combinatorial optimization problem, effective

methods include genetic algorithm, simulated

annealing and hill climbing.

In this paper, we compare simulated annealing

(SA) and hill climbing (HC) with updates that

involve simple genetic algorithm (GA) operators in

order to find solutions to a room layout. This

involves walls/obstacles randomly distributed in a

10-by-10 grid with two types of pedestrian moving

inside the space: one moving from left-to-right and

373

Hassan F. and Tucker A..

USING UNIFORM CROSSOVER TO REFINE SIMULATED ANNEALING SOLUTIONS FOR AUTOMATIC DESIGN OF SPATIAL LAYOUTS .

DOI: 10.5220/0003113503730379

In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 373-379

ISBN: 978-989-8425-31-7

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

another one moving right-to-left based upon the

cellular automata model of Yue et al., 2007. The

aim of the experiments is to compare and understand

how fast and effective the algorithms can generate

(not simply refine as in Smedresman, 2006)

automatic solutions to the spatial layout problem by

using statistics generated from cellular automata

pedestrian simulations.

2 METHODOLOGY

The experiments involve applying simulated

annealing (SA), hill climbing (HC) and a form of

repeated SA that is updated with genetic algorithm

(GA) operator (SA-GAO) to solve the spatial layout

problem. It is not feasible to have a full GA

implementation due to the very complex fitness

function involving several pedestrian simulations.

The methods analysed in this paper have been run

using a pedestrian movement model developed by

Yue et al., 2007. They proposed two pedestrian

movement models on a square lattice for small

system based on cellular automata (CA), i.e. two-

way pedestrian flow and four-way pedestrian flow.

They introduced a technique to simplify tactically

the process into the interaction of four dynamic

parameters, which can reflect the pedestrian

judgment on the surrounding conditions and decide

the pedestrian's choice of action such as moving

ahead, stopping to wait, position exchange, lane

switching, back stepping, etc. In this paper, the two-

way pedestrian flow system is simulated and studied

using the Dynamic Parameters Model from Yue et

al., 2007 to consider direction split and pedestrians’

walking preference.

In this model, pedestrian choose to wait or move

according to the corresponding transition payoff

based on four parameters:

 Direction-parameter indicates the cell’s degree

of approximation to the pedestrian destination;

 Empty-parameter indicates whether the cell is

occupied or empty;

 Forward-parameter describes the proportion of

empty cells in the field ahead of his or her target

position;

 Category-parameter describes the proportion of

the number of empty cells and pedestrians

homogenous with the subject in his or her direction

of destination in the field around his or her target

position.

2.1 Hill Climb and Simulated

Annealing

The pseudocode for our implementation of this

approach is listed in Figure 1.

Figure 1: Pseudo-code for the Hill Climb and Simulated

Annealing.

In order to set this as a HC, the starting

temperature is set to zero. Note that the fitness here

Input: Number of iterations, iteration,

and a random starting layout, startrep,

starting temperature, temperature

oldrep = startrep;

Apply 10 pedestrian simulations to

generate statistics, stats

fit = fitness(stats)

bestfit = fit

for loop = 1:iteration

rep=oldrep;

Apply move operator to rep

Apply 10 pedestrian simulations to

stats

newfit = fitness(stats);

dscore = newfit-fit

if ((bestfit < newfit) OR

(rand(0,1) <e

(dscore/temperature)

)

bestfit = newfit

oldrep = rep;

else

rep = oldrep;

end if

temperature = temperature*0.9

end for

Output: rep

ICEC 2010 - International Conference on Evolutionary Computation

374

uses the statistics generated from 10 repeated runs of

the CA pedestrian simulation. This is done to ensure

that one simulation does not result in a ‘lucky’

fitness score for one layout based upon the starting

positions of the pedestrians.

2.2 Move Operator

The move operator, shown in Figure 2, takes into

account the size of the simulation grid and randomly

moves one object a fraction of this distance

(determined by the parameter changedegree). The

result of the move is then checked to see if the new

coordinates are within the bounds of the grid and do

not result in the object overlapping with others. The

operator is defined fully in Figure 2, where

unidrnd(min,max) is a uniform discrete random

number generator with limits of min and max.

Figure 2: Pseudo-code for Move Operator.

2.3 Fitness Function

The fitness function is calculated based on the

statistics that are generated using the pedestrian

simulation. The statistics take the form of a 3x3

matrix, leftstats, representing the sum of decisions

for left-moving pedestrians and a similar decision

matrix for right-moving pedestrians, rightstats.

Therefore, the middle cell in each grid represents

how many times the pedestrians decided to stay in

the same cell as last time. As we wish to encourage

free flow we wish to increase the fitness for layouts

that results in many cases of left moving pedestrians

moving left and right moving pedestrians moving

right, whilst penalising the fitness of any decisions

where the left-moving pedestrians move right and

vice-versa. For example, consider the two stats

matrices for left and right pedestrians. It is clear that

the leftstats reflect a ‘good’ result as the pedestrians

have generally moved in the desired direction more

often whereas for rightstats this is not the case.

3 1 1 3 3 2

5 3 0 2 2 2

3 2 2 2 3 1

leftstats rightstats

Figure 3: Left and Right statistics – describe as 3x3

matrix.

In general, we wish to maximise the first column

in leftstats and the third column in rightstats whilst

minimising the other statistics (shaded in the

example, Figure 3). Therefore, we use the following

fitness function:

rightfitness = sum (rightstats(3,1:3))-sum(rightstats(1,1:3)

leftfitness = sum (leftstats(1,1:3))-sum(leftstats(3,1:3)

fitness = rightfitness + leftfitness

Higher fitnesses should reflect simulations

whereby pedestrians have moved in the direction

that they wish more often.

2.4 SA Genetic Algorithm Operators

We extend our work by using GA-style operators. A

full GA implementation is not feasible due to the

very complex fitness function involving several

pedestrian simulations. Therefore, we extend our

work by using GA-style operators on the results of

multiple starts of SA. The initial ‘parents’ are

selected from the best (selection are made based on

more consistent fitnesses value with a good final

layout which is 9.000 or above) solutions generated

Input: Size of simulation grid, W, size

of object, sizobj, current x-coordinate,

oldx, current y-coordinate, oldy

Set the degree of change to make based

upon a fraction of the grid size:

changedegree=W/2;

Choose a random object in the grid, i

[oldx,oldy] = current x and y

coordinates of object i

xchange = unidrnd(-changedegree/2,

changedegree/2)

ychange = unidrnd(-changedegree/2,

changedegree/2)

if ( (oldy+ychange) and (oldx + xchange)

is within grid boundary AND new object

position does not overlap another object

taking into account sizobj)

newx=oldx+xchange;

newy=oldy+ychange;

end if

Move object i to position [newx, newy]

Output: newx, newy

USING UNIFORM CROSSOVER TO REFINE SIMULATED ANNEALING SOLUTIONS FOR AUTOMATIC DESIGN

OF SPATIAL LAYOUTS

375

from a number of SA experiments. We

experimented with different style of combination for

two ‘parents’ that more or less act like uniform

crossover. In the crossover operation, two new

children are formed by exchanging the genetic

information between two ‘parent’ chromosomes.

Multipoint crossover defines crossover points as

places between loci where an individual can be split.

Uniform crossover generalizes this scheme to make

every locus a potential crossover point. A crossover

mask, the same length as the individual structure is

created at random and the parity of the bits in the

mask indicate which parent will supply the offspring

with which bits.

Consider the following two parents with 10

binary variables each:

Parent1 1 1 1 1 1 1 1 1 1 1

Parent2 2 2 2 2 2 2 2 2 2 2

For each variable, the parent who contributes its

variable to the children is chosen randomly with

equal probability. Here, children 1 is produced by

taking the bit from parent 1 if the corresponding

mask bit is 1 or the bit from parent 2 if the

corresponding mask bit is 2. Children 2 is created

using the inverse of the mask, usually.

Mapping 1 2 1 1 2 2 2 1 1 2

After crossover the new individuals are created:

Children1 1 2 1 1 2 2 2 1 1 2

Children2 2 1 2 2 1 1 1 2 2 1

3 RESULTS

The experiments involved running HC, SA and SA-

GAO on the problem of trying to arrange 10 pre-

defined objects in a 10x10 grid with 5 ‘left’

pedestrians and 5 ‘right’ pedestrians. Each algorithm

was run 10 times and the learning curves were

inspected. The final fitnesses and quality of the

layouts were then investigated. Finally, some

inspection of sample simulations on the final layouts

was carried out to look for interesting

characteristics.

3.1 Summary Statistics

Table 1: Summary statistics of final fitness.

Method Min. Max. Mean Std. Dev.

SA1 3.846 9.790 7.615 1.854

SA2 4.786 9.938 8.085 1.477

HC 6.992 9.276 8.586 0.724

Table 1 shows the minimum, maximum and mean

values for the final fitness of each algorithm over ten

experiments, where SA1 represent SA with initial

temperature of 1.0 and SA2 represents a temperature

of 0.2. The statistical values of HC show the

robustness of the solutions with the standard

deviation of 0.724. The standard deviation is

relatively low, which indicates that HC is among the

most consistent approach in finding a good solution

(the mean is also higher). However, the maximum

value is less than SA2 indicating that the SA can

sometimes escape some of the local optima.

Figure 4: Distribution of Final Fitness for each Search

Method.

Figure 4 shows the final fitness distribution for

each search method. From Table 1 and this figure, it

is clear that the distribution of the fitnesess for the

SA algorithms for both temperatures is spread more

than for the HC. SA is sometimes empirically better

at avoiding local minima than hill-climbing though

HC is the most consistent approach based on Table

Figure 5: Max, Min and Mean learning curves for SA

temperature = 1.0.

Figure 6: Max, Min and Mean learning curves for SA

temperature = 0.2.

"< 0" "0‐2" "2‐4" "4‐6" "6‐8" "8‐10" "10‐12"

Fitness

Frequency

SA1

SA2

1 168335

1

127

190

253

316

379

442

2

ICEC 2010 - International Conference on Evolutionary Computation

376

The graphs in Figures 5 and 6 show the best,

worst and mean learning curves for the SA

algorithm over 10 reruns with two initial starting

temperatures. The total iteration for every learning

curve is 500 iterations. These curves characterise the

typical SA with noisy search at high temperatures in

the early phases followed by smoother learning in

the later stages. It seems the higher starting

temperatures result in more diverse final fitnesses.

This could be due to the fact that the fitness

landscape is very noisy and difficult to negotiate,

resulting in many local optima.

Figure 7: Max, Min and Mean learning curves for HC.

However, the HC curves in Figure 7 show far

less variations in final fitness that is surprising.

Despite the mean final fitness of HC being better

than SA, it is clear that the SA does indeed escape

sub-optimal local peaks in the fitness on some

occasions with the maximum final fitness being

higher than HC. Clearly this is likely to be due to

HC techniques exploiting the best available solution

for possible improvement but neglecting exploring a

large portion of the search space (Michalewicz &

Fogel, 2000). Additionally, the higher starting

temperature for SA seems to result in better

solutions but at the expense of greater variance in

the final fitness.

We then expanded our work to SA-GAO. Four

different mappings were run and each was run 10

times. Seven ‘parents’ were selected from the

consistent resulting solutions of previous SA

experiments where each of their fitness values is

9.000 or above. The highest ‘parent’ fitness is

10.048 as shows in Table 2.

Table 2: Selected ‘parents’ and their fitness value for SA-

GAO experiments.

Parents Fitness

Parent1 9.382

Parent2 10.048

Parent3 9.010

Parent4 9.124

Parent5 9.780

Parent6 9.438

Parent7 9.414

Table 3: ‘Children’ with highest fitness values generated

from SA-GAO of four different combinations.

Mapping Max. Fitness

1122211222 10.434

1112221112 10.460

1112111211 10.398

1212121212 9.672

Table 3 shows different mappings of uniform

crossover and the highest fitness value of the

children for each combination. The fitnesses of the

‘children’ scored better compared to the ‘parents’

highest fitness value. This implies that recombining

the best of the SA solutions can indeed improve the

overall layout without having to implement a full

GA which would not be feasible for such an

expensive fitness function.

Table 4: ‘Children’ with highest fitness values generated

from SA-GAO of one hundred random combinations.

Mapping Max. Fitness

2122112111 9.642

2212221212 Child1: 9.714,

Child2: 9.476

1211122122 9.178

2121221121 9.846

2121122212 9.322

2221121212 9.092

2212211112 9.656

2122112111 9.466

1121111221 9.046

2122211111 9.200

2122222111 9.378

1222211122 9.112

We next tried to experiment with one hundred

random combination of mappings. The highest

fitness of the children is 9.846 as shown in Table 4

generated from ‘2121221121’ mapping. Note that

mapping ‘2212221212’ generates both ‘child1’ and

‘child2’ with good fitness (higher than 9.000). This

result shows that it may be necessary to run quite a

large number of recombination of parents to ensure

improved fitness. It may also imply that further

mutations are required to fine-tune these new

solutions.

3.2 Exploring the Final Layouts

We now explore some of the characteristics of the

final layouts discovered by the algorithms. The

blue/down arrows in figure 8 (a)-(e) represent the

final positions of wall layout; 500 iterations using

SA, HC and SA-GAO algorithm. Meanwhile, the

black and pink arrows represented left and right

pedestrians moving in each ways. From Figure 8 (a)

168

335

HC

USING UNIFORM CROSSOVER TO REFINE SIMULATED ANNEALING SOLUTIONS FOR AUTOMATIC DESIGN

OF SPATIAL LAYOUTS

377

and (b), we can see clearly that the bad layouts are

generated from the series of lowest fitness. The

higher the fitness, the better the layout as in Figure 8

(c), (d) and (e). Notice in 8 (a) (the lowest fitness for

SA with starting temperature of 0.2) the large wall

created by the objects on the right were blocking any

left-right movement. It may be that more complex

operators are required to escape from this point. The

lowest scoring HC in Figure 8 (b) features a long

wide corridor in the centre which results in an

“almost dead end” at the left. Notice the completed

“corridor” in Figure 8 (c) from left to right with a

width of 1 cell allowing free flow of both left and

right pedestrians. Also note the alternative routes.

Figure 8 (d) shows the highest scoring layout of HC

with the existence of single cell corridors allowing

some free flowing movements. Finally, notice in the

Figure 8 (e) for the highest fitness of SA-GAO that

there are two “corridors” at the top and bottom of

the final layout.

(a) SA (temperature = 0.2) Final layout with lowest fitness of 4.786. (b) HC final layout with lowest fitness of 6.992.

(c) SA (temperature = 0.2) Final layout with highest fitness of 9.938. (d) HC final layout with highest fitness of 9.276.

(e) SA-GAO final layout with highest fitness of 10.460.

Figure 8: (a)-(e). SA, HC and SA-GAO final layouts.

ICEC 2010 - International Conference on Evolutionary Computation

378

4 CONCLUSIONS

In this paper we explored using pedestrian flow

simulations combined with heuristic search to assist

in the automatic design for spatial layout planning.

Using pedestrian simulations, the activity of crowds

can be used to study the consequences of different

spatial layouts.

Based on the results that have been observed in

this paper, we have demonstrated that simple

heuristic searches appear to deal with the NP-hard

spatial layout design problem to some degree, at

least on the very much simplified problem addressed

here. Both SA and HC are able to automatically find

adequate solutions to this problem when

incorporated with the pedestrian simulator.

Moreover, the solution is further improved when we

paired ‘parents’ and apply a GA style operator using

our method SA-GAO. Whilst it is not guaranteed

that the optimal solution will be found, this does not

mean that useful and unexpected designs cannot be

learnt using these types of approaches. Indeed, the

real positive outcome of the experiments here is that

we found certain characteristics that may not have

been immediately expected. We have found several

key results:

 The highest fitnesses produced useful layouts,

passageways (diagonal or horizontal) and clustered

objects. These demonstrably show smoother flow

when running the simulations and exploring the

statistics of movement;

 SA has more variations in final fitness. Whilst

HC cannot ‘escape’ local optima, SA does

sometimes manage to do this with better final

solutions. In general, the distribution of final

fitnesses is higher for SA though more adventurous

solutions are explored;

 SA-GAO generated better solutions compared

to SA solutions: the SA-GAO children show higher

fitnesses than their parents. This implies that

solutions with lower fitnesses may still offer useful

information and when these are recombined in a

constructive way, they generate better overall

layouts than if no recombination is used.

We feel that approaches that combine heuristic

search with simulation should offer the ability to

find novel design solutions in more complex design

layouts with larger spaces, more objects, different

constraints and different pedestrian goals. In general,

we found that SA-GAO treats combinations of two

existing solutions as being ‘near’, making the

‘children’ share the properties of their parents, so

that a child of two good solutions is more probably

good than a random solution as in HC and SA.

Future work will involve extending our work by

make use of real world data to validate the

discovered layouts. We have access to large amounts

of pedestrian flow data in existing public buildings

and private offices. We will use the data to further

test our algorithms on layouts discovered from more

complex real-world spaces.

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OF SPATIAL LAYOUTS

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