JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL. 18(1): 9-14, 2008©
Discrete Particle Swarm Optimization
Algorithm for Solving Optimal Sensor
Deployment Problem
Milan R. Rapaić, Željko Kanović, Zoran D. Jeličić
Abstract — This paper addresses the Optimal Sensor
Deployment Problem (OSDP). The goal is to maximize the
probability of target detection, with simultaneous cost
minimization. The problem is solved by the Discrete PSO
(DPSO) algorithm, a novel modification of the PSO algorithm,
originally presented in the current paper. DPSO is generalpurpose optimizer well suited for conducting search within a
discrete search space. Its applicability is not limited to OSDP,
it can be used to solve any combinatorial and integer
programming problem. The effectiveness of the DPSO in
solving OSDP was demonstrated on several examples.
Index Terms — Optimal sensor deployment, Particle Swarm
Optimization, Discrete optimization
A
I. INTRODUCTION
wide variety of pr oblems i n modern e ngineering c an
be seen as problems of optimal deployment of a group
of agents w ithin a p redefined s earch s pace. The na ture of
the ag ents, off c ourse, as well as the n ature o f t he s earch
space and the deployment goal may vary significantly from
case t o cas e. T his p aper ad dresses o ne of the pr oblems of
this type, Optimal Sensor Deployment Problem (OSDP).
OSDP is widely studied in literature. Distributive wireless
sensor ne tworks a re be coming i ncreasingly pe rvasive i n
many p ractical applications for either m ilitary o r c ivil
purposes [1]. In g eneral, t he t ask i s t o maximize t he t arget
detection p robability w here maximal deployment c ost has
been specified in advance, or, as it was done in this paper, to
simultaneously m aximize the t arget d etection probability
and m inimize t he de ployment c ost. A r eview o f recent
developments in the field of distributed sensor networks can
be f ound i n [ 2]. Recently, O SDP of m oving s ensors was
addressed in [ 3] and [ 4]. Deployment pr oblem involving
underwater acoustic sensors was analyzed in [5]. Variations
of OSDP were also discussed in [6] and [7].
It is important to r ealize that, although the deterministic
detection m odel i s c onsidered f requently i n l iterature, t he
detection p rocess i s i n f act stochastic i n n ature. Target i s
more likely to be present at certain points of the surveillance
region t han i n ot hers. M oreover, t his pr obability m ay even
Manuscript received August 15, 2008.
Milan R. Rapaić, +381631038044, ([email protected])
Željko Kanović, +381631028539, ([email protected])
Zoran D. Jeličić, +38163559450, ([email protected])
All au thors ar e with C omputing an d C ontrol Dep artment, Facu lty o f
Technical Sciences, University of Novi Sad.
DOI:10.2298/JAC0801009R
change i n t ime. S ensors a re also non-deterministic. The
probability that a sensor a ctually d etects a t arget i s, i n
general, a f unction of di stance be tween t hem, but i t a lso
depends on other factors, such are environmental conditions
and m easurement n oise. To t he be st knowl edge of the
authors of the current paper, the stochastic nature of sensors
themselves was addressed fo r t he fi rst t ime i n [1 ], were
various stochastic sensor models were also discussed.
In most practical a pplications, t he s urveillance r egion i s
continuous. S ometime, howe ver, s ensors c an be pl aced a t
only certain, discrete points within this region. Even if it is
not s o, i t i s c ommonly more c onvenient t o c onsider only a
discrete sets of poi nts wi thin t he s urveillance r egion a s
possible locations for sensor deployment [1], [4], [7]. In the
present paper, this approach was also adopted, with primary
purpose to r educe the computational load w hen calculating
different joined a nd conditional probabilities. B y
discretization of the surveillance region, the OSDP becomes
a c ombinatorial opt imization pr oblem. Moreover, i t ha s
been proved in [ 1] a nd [ 7] t hat combinatorial O SDP is i n
general N P-complete, o r in a nother w ords, that t here i s n o
polynomial-time a lgorithm f or t heir e xact s olution. It i s
therefore of g reat i mportance t o i nvestigate h euristic
approaches t hat would pr ovide ne ar-optimal s olutions in
reasonable a mount of t ime. Genetic a lgorithm ( GA) with
custom-made genetic ope rators a nd e ncoding i s ut ilized i n
[1], while t abu-search ( TS) ba sed m ethod w as pr oposed i n
[7].
The solution pr esented i n t he c urrent pa per i s ba sed on
the Particle Swarm Optimization (PSO) algorithm. Contrary
to t he s olution pr esented i n [ 1], t he opt imization i s not
conducted with respect to t he num ber of s ensors (we
consider the number of sensors t o be f ixed, know n i n
advance). Nevertheless, the presented solution is flexible in
its other aspects; it is not restrictive with respect to the size
and the dimensionality of the detection region, the type and
the num ber of s ensors t o be de ployed, a s w ell a s with
respect to the previously known target probability, which is,
in ge neral, considered to be va riable w ithin t he de tection
area – target is more likely to be active within certain areas
of the detection region than within others. However, due to
fixed num ber of s ensors, t he pr oposed s olution is
computationally cheaper that the one presented in [1]. Since
the or iginal P SO pr esented i n [ 8] assumes that t he s earch
space i s c ontinuous, a n or iginal modification of P SO, t he
Discrete PSO (DSPO), i s p roposed. DP SO e xplores a
discrete search space using the i deas an d p rinciples o f t he
10
RAPAIĆ M., KANOVIĆ Ž., JELIČIĆ Z.: DISCRETE PARTICLE SWARM OPTIMIZATION ALGORITHM FOR SOLVING…
original P SO a lgorithm. T he a ctual i mplementation,
however, takes into a ccount t he s pecific na ture of
combinatorial search space. DPSO is inspired by the binary
PSO de scribed i n [ 9], a nd i s not s pecifically de signed for
the O SDP. I n f act i t can b e u sed, just t he s ame, w ith a ny
combinatorial or integer programming problem.
The r est of t he pa per i s or ganized a s f ollows. The
problem is strictly formulated in section II. Short description
of the classical PSO algorithm is given in section III, while
the DPSO algorithm is presented in section IV. Results are
presented in s ection V. S ection V I contains the concluding
remarks.
II. PROBLEM FORMULATION
A. Formal Statement
Let the surveillance region ( R ) be specified in advance.
Commonly, it is two-dimensional, rectangular area of fixed
size. T he o bject t o b e d etected, t he target, ha s known
probability of accidence in each point r of the surveillance
region. This probability will henceforth be referred to as the
target probability and denoted by tp (r ) . It is assumed that
target probability i s va riable t hroughout t he s earch s pace,
but is constant i n t ime. The s et o f av ailable s ensors S is
also given in advance. Each sensor s in S is described by
its cost sc (s ) , a s w ell a s i ts p robability to detect target in
where α is the design parameter c ontrolling t he r elative
impact of efficiency and cost of the deployment.
Formal statement of the OSDP is of crucial importance. It
establishes a connection between the OSDP and a variety of
other, both theoretical and pr actical pr oblems of seemingly
quite different n ature. An interesting ove rview of s uch
problems, t hat a mong ot her i ncludes signal compression,
numerical integration and clustering, i s gi ven i n [ 10].
Different formulations are also considered in [1].
It w ill b e a ssumed t hat t hat t he d etection p robability of
each s ensor de pends s olely on i ts d istance f rom t he t arget.
Denote the sensor location by rs , the target location by r .
Let || ⋅ || be suitably chosen distance measure (norm). In that
case, t he d etection p robability m odel f or a s ensor w ill be
assumed as

|| r − rs ||2
), || r − rs ||≤ d max
exp(−
,
(4)
sdp (s; r ) = 
2σ 2

d
0
||
||
−
>
r
r
s
max

1
0.9
0.8
0.7
0.6
0.5
each point of the surveillance region, sdp (s; r ) . The number
0.4
of sensors, k , is specified in advance and does not change
during t he opt imization pr ocess. Denote by Dk the s et o f
0.3
all pos sible de ployments of k sensors f rom S . For each
deployment d ∈ Dk , t he d etection p robability i n a c ertain
point r ∈ R , ddp (d; r ) , is defined as the probability that the
target actually present at that point will be detected by any
of the sensors within the deployment. Also, the overall cost
of a deployment, dc(d; r ) , is defined as the sum of costs of
each sensor within the deployment.
Clearly, the target detection probability for a deployment
d ∈ Dk in a poi nt r ∈ R is t he j oined p robability t hat the
target will be de tected by a ny of t he s ensors w ithin t he
deployment a nd t hat t he t arget i s actually present at that
point, or
=
dp (d; r ) ddp (d; r ) ⋅ tp (r ) ,
(1)
where i t wa s a ssumed t hat t arget probability and detection
probability a re s tatistically i ndependent. Efficiency of a
deployment can now be defined as
∫
e(d) = dp (d; r )dr .
(2)
R
The OSDP can be formulated in numerous ways. One can
specify maximal allowable d eployment co st an d m aximize
the ef ficiency. O n t he other ha nd, m inimal a llowable
efficiency can b e s pecified f ollowed b y co st m inimization.
The a pproach a dopted i n t he c urrent work is to combine
efficiency and cost into a single objective function which is
then optimized. The combined optimality criteria is
(3)
I (d=
) e(d) − α ⋅ dc(d) ,
0.2
0.1
0
-3
-2
-1
0
1
2
3
Figure 1 . Gaussian detection p robability as a function of mutual distance
between a s ensor and a target, calculated using equation (4), with σ=1 and
dmax=2.
where σ and d max are parameters s pecific f or each
particular s ensor. T he m odel (4) is know n a s Gaussian
sensor m odel; σ is t he width (standard deviation) of t he
distribution, while the detection range d max determines the
maximal distance at which a sensor is capable of detecting a
target. Other sensor models are considered in [1]. It should
be mentioned that the solution proposed in the current paper
does not depend on the particular sensor model, nor does it,
in general, assume that all sensors obey the same detection
probability model.
It is also assumed that, statistically, sensors do not affect
each ot her. I n ot her w ords, i f d is any de ployment, a nd
d + s is a new deployment, obtained from d by insertion of
an arbitrary new sensor s , than at each point r
ddp (d + s=
; r ) ddp (d; r ) + sdp (s; r ) − ddp (d; r ) ⋅ sdp (s; r ) . (5)
Since the detection pr obability of a n e mpty de ployment i s
zero ( no s ensors – no de tection), t he e quation (5) can b e
used t o c alculate t he t arget d etection p robability f or the
entire deployment recursively.
JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL. 18(1): 9-14, 2008©
B. Discrete Reformulation of the OSDP
In the sequel, it is assumed that the surveillance region is
planar and rectangular. The procedure is directly applicable
to arbitrarily s haped r egions. Evaluation of (2) implies
calculation o f a s urface (t wo-fold) i ntegral. I n or der to
calculate this integral the surveillance region is divided by a
grid of i maginary l ines t o a n et o f e qually s ized c ells. T he
procedure is depicted in Fig. 2. The “width” and “height” of
each cell will be denoted by l x and l y , while the number of
nodes in each d irection will be de noted by N x and N y ,
respectively.
Figure 2 . An ex ample o f s urveillance region with 10x10 node s in the
discretization grid. It is assumed that sensors can b e p laced s olely in th e
nodes of the grid. Target probability is also only calculated on the nodes of
the grid.
Henceforth, i t wi ll be a ssumed t hat t he s ensors c an only
be placed in the nodes of the network. The efficiency (2) of
a de ployment d ∈ Dk can now be approximately evaluated
as
=
ed (d)
Nx N y
∑∑ ddp(d; r(i, j)) ⋅ tp(r(i, j)) ⋅ l
x
⋅ ly ,
(6)
=i 1 =j 1
with r (i, j ) being t he c oordinate of t he node i n t he
intersection of the i-the vertical and j-th horizontal grid-line.
If i t i s assumed that ddp and tp change r elatively s low,
meaning t hat t heir va lue doe s not change significantly
within the single cell of the grid, then the double sum (6) is
a good approximation of the surface integral (2). Finally, the
optimality criteria (3) can be replaced by
=
I d (d)
Nx N y
∑∑ ddp(d; r(i, j)) ⋅ tp(r(i, j)) − β ⋅ dc(d) ,
(7)
=i 1 =j 1
where β =α ⋅ (l x ⋅ l y ) −1 . T he cr iteria (7) is a pproximately
equivalent to the original criteria (3).
III. PSO ALGORITHM
Particle Swarm Optimization (PSO) algorithm is a
modern optimization technique, inspired by social behavior
of a nimals moving i n l arge gr oups – insects and bi rds i n
particular. It w as originally proposed by K ennedy a nd
Eberhart [8] in 1995, and it has developed since trough the
work of m any a uthors [11], [1 2]. It h as b een s uccessfully
11
applied in a variety of engineering problems [13], [14]. An
overview of the algorithm development, improvements and
applications can be found in [9].
The algorithm investigates the search space using a group
of pot ential s olutions – particles. The set o f a ll p articles i s
referred t o as t he swarm. I n e volutionary t erminology, t he
swarm woul d be e quivalent t o the popul ation, w hile t he
particle i s e quivalent t o the individual. Each p article i s
characterized b y i ts position and velocity. The position of
the p article i s a p oint i nside t he s earch s pace, an d i s
effectively t he pot ential s olution r epresented by that
particle. The velocity is d efined as t he d ifference b etween
the current and the previous position. The size of the swarm
(the num ber of pa rticles) i s us ually s pecified in advance,
and is not changed during the search. Initial population and
velocity ar e c hosen r andomly f rom a pr edefine r ange of
values. Let xi [n] and v i [n] respectively d enote the
position a nd ve locity of t he i-th p article a t n-th i teration.
Each particle is also capable of memorizing the best position
it a chieved s o f ar, know n a s t he personal best position,
pi [n] . The swarm as a w hole m emorizes t he b est p osition
ever achieved by any of its particle, the global best position
g[n] .
The basic i dea o f t he P SO a lgorithm i s t o s teer e ach
particle to personal best a nd gl obal be st pos ition. M ore
formally, the velocity is calculated as
v i [n + 1]= w[n] ⋅ v i [n] +
+cp[n] ⋅ rpi [n] ⋅ ( pi [n] − xi [n]) + ,
(8)
+cg[n] ⋅ rgi [n] ⋅ ( g[n] − xi [n])
while the next position is obtained as
(9)
xi [n + 1]
= xi [n] + v i [n + 1] .
Parameters w , cp and cg figuring i n (8) are inertia,
cognitive and social factor (or co efficient), r espectively.
Factors cp and cg are c ommonly know n a s acceleration
factors. The inertia f actor controls t he s tability o f t he
algorithm, a nd was not present in t he or iginal pa per by
Kennedy a nd E berhart [8]. It w as i ntroduced (as q uantity
different t han 1) by S hi a nd E berhart i n [ 11]. It is known
from va rious s tudies, i ncluding [11] and [ 15], that i nertia
value should not be gr eater t han one , a nd t hat i t s hould
decrease as the optimization process develops. It is common
to decrease the inertia linearly from 0.9 to 0.4. Values of the
acceleration co efficients d etermine t he r elative impact of
local t o gl obal knowl edge on the be havior of t he pa rticles.
Larger co gnitive f actor ( cp ) m eans t hat p articles move
autonomously, w ith l ess r egard t o t he r esults obt ained by
other p articles. S uch b ehavior i s b eneficial in early,
exploratory stages of the optimization process. Larger social
factor ( cg ) means that the particles move in strong relation
to each o ther, that they all tend t o e xplore good s olutions
found by a s warm a s a w hole. S uch be havior i s c rucial i n
later, exploitation oriented, stages o f the optimization
process. In t hese l ater s tages, t he al gorithm i s expected to
fine-tune the good solutions it already found. In the original
paper [8] both acceleration factors were set to 2, but it was
later demonstrated by R atnaweera et al i n [ 12] t hat l inear
decrease of c ognitive c omponent f rom 2. 5 t o 0. 5, w ith
12
RAPAIĆ M., KANOVIĆ Ž., JELIČIĆ Z.: DISCRETE PARTICLE SWARM OPTIMIZATION ALGORITHM FOR SOLVING…
simultaneous linear i ncrease of s ocial c omponent f rom 0. 5
to 2. 5 i s m ore a ppropriate. The reasons for t his ar e cl ear
from t he a bove di scussion. T hese conclusions have also
been ve rified by t he a uthors of this paper in t heir r ecent
work [16], [17] and [18].
The optimization stops when pr edefined number of
iterations has been achieved. Other stopping criteria can be
introduced, but they are not used within this paper.
IV. DPSO
The classical PSO algorithm described in previous
section is not applicable if the search space is discrete, that
is if the position of each particle is bound to a discrete set of
values. A modification of the classical algorithm is proposed
in the current paper suitable for optimization within such a
search s pace. The proposed s olution i s i nspired by bi nary
PSO algorithm, explained in [9].
It w ill b e as sumed t hat t he s earch s pace i s m apped into
some r ectangular s ubspace o f  m , w here  is t he s et o f
integers, a nd m is a rbitrary na tural num ber denoting t he
dimension of the s earch-space. The v elocity v ector i s
calculated u sing t he s ame f ormula as in the classical PSO,
formula (8), but is saturated afterword using the hyperbolic
tangent function to obtain a new quantity, that is referred to
as the saturated velocity
1 − exp( − v i [n + 1])
v [n + 1]
(10)
=
υ i [n + 1]
= tanh( i
).
1 + exp(− v i [n + 1])
2
The position of each particle is no longer calculated using
(9), but as
(11)
xi [n + 1]
= xi [n] + round (υ i [n + 1] × ∆x max ) .
Maximal displacement ∆x max is a n ew parameter o f t he
algorithm, a nd s hould be s pecified i n a dvance. The ×
symbol denotes e lement-vise pr oduct of t wo ve ctors,
meaning t hat i f a = [ai ] and b = [bi ] are v ectors, t heir
element-vise pr oduct i s a × b = [ai ⋅ bi ] . The i dea be hind
equations (10) and (11) is simple. If one would calculate the
next pos ition of e ach pa rticle by e quation (9), onl y by
chance would the result be an integer number. One
possibility is to round the resulting position. This would be
satisfactory only if the number of possible positions is very
large, t hat i s i f the problem i s qua si-continuous. By us ing
equations (10) and (11), the saturated velocity v ector i s i n
fact the a mount o f m aximal d isplacement t hat w ill b e
applied i n t he c urrent s tep. By c hanging m aximal
displacement parameter, the al gorithm can b e t uned t o
accommodate va rious pr oblems, r anging f rom bi nary t o
quasi-continuous one s. In bounde d dom ains, i t i s a lso
possible t o r eplace or dinary s um i n (11) by a m odulo-sum
operator. That way, the s earch s paces b ecomes ef fectively
limitless, because its bound points become neighbors.
It i s c onvenient t o l imit t he v elocity t o s ome predefined
maximal value prior to applying the saturation function (10)
. This maximal velocity ( v max ) is another parameter of the
algorithm. In context of the binary PSO it is usually chosen
to be between 4 a nd 6. In the current paper, v max = 6 was
used. The reason for velocity clamping is clear from the Fig.
3 de picting t he s aturation f unction, which m aps a ll very
large positive or ne gative velocities t o a lmost t he same
value, 1 a nd -1, r espectively. If t he v elocity i s allowed to
grow without restrictions, the algorithm would soon become
inert and insensitive to the local properties of the objective
function.
Notice that none of the features of the DPSO is
specifically d esigned t o acco mmodate O SPD. The
modifications made to the cl assical P SO ar e o nly d esigned
to adopt the algorithm to a combinatorial search space. In its
other a spects, t he na ture of t he or iginal P SO i s preserved
within the proposed algorithm.
1.5
1
0.5
0
-0.5
-1
-1.5
-6
-4
-2
0
2
4
6
Figure 3. The saturation function tanh( v / 2) . Notice th at f or v > 6 the
value o f s aturation f unction is almost ex actly 1 . Similar situation is when
v < −6 . In that case, the saturation function is almost -1.
V. RESULTS
In the sequel, a problem of optimal deployment involving
4 s ensors on a pl anar gr id with 100x100 node s i s
considered. All of the cells in the grid are of equal size, with
equal vertices of length l=
l=
l.
x
y
Figure 4. Graphical representation of target probability. Higher probability
is depicted with lighter shades.
The s olution i s i mplemented us ing the programming
language Python 2.5 [19] and its e xtension m odules f or
numerical data processing and data visualization SciPy [20]
and Matplotlib [21]. Regarding t arget pr obability, t wo
separate cas es w ere co nsidered, b oth depicted i n F ig. 4.
Lighter areas are those with higher target probability. In the
first c ase ( Fig. 4a) it was assumed t hat t arget p robability
obeys Gaussian probability distribution, centered at the very
middle of t he s urveillance region, a t node ( 50, 50 ), w ith
standard de viation equal t o 20l . In t he s econd cas e (Fig.
JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL. 18(1): 9-14, 2008©
13
4b), it i s a ssumed that t he t arget distribution is the
superposition of t wo s eparate G aussian di stributions w ith
equal standard deviations of 10l , centered at nodes (25, 25)
and (75, 75).
A. OSDP involving sensors of the same type
A pr oblem i nvolving 4 i dentical s ensors i s considered
first. All sensors have equal Gaussian probabilities (4), with
standard deviation σ = 10l and maximal range d max = 30l .
Since all sensors are the same, it is assumed that they are of
equal p rice. T herefore, al l p ossible deployment strategies
have the same cost, an d t he co st t erm o f t he o ptimality
criteria (7) can be neglected ( β = 0 ). PSO parameters were
chosen t o be
w = 0.8 and cp
= cg
= 2 ; m aximal
displacement was chosen t o b e t he s ame al ong each
dimension ∆xmax =
10 ; the number of particles was chosen
to be 30, the number of iterations was set to 100.
Figures 5. a nd 6. de pict de tection pr obability f or t he
deployment obtained by DPSO a lgorithm f or t he t arget
probabilities shown in Fig 4a and 4b, respectively.
B. OSDP involving sensors of different type
Let us c onsider t he pr oblem of optimal deployment of 4
different sensors. The assumption is that types of sensors are
known i n a dvance, a nd a lso t hat t he number of sensors of
each type is not bounded (optimal deployment may involve
all four sensors of the same type, but it also possible that all
selected sensors be of different type). The target probability
will be assumed to be the one presented in Fig 4a. Table 1
shows the characteristics of different sensor types, including
standard de viation of t heir G aussian di stribution, m aximal
detection range, and relative price.
Table 1. Sensor types
std. dev.
Type 1
5l
Type 2
5l
Type 3
10l
Type 4
15l
max. range
10 l
20 l
25 l
40 l
Figure 5. Op timal s urveillance r egion co verage, with 4 eq ual s ensors an d
target probability as depicted in figure 4a.
Figure 6. Op timal s urveillance r egion co verage, with 4 eq ual s ensors an d
target probability as depicted in figure 4b.
rel. price
0.3
0.4
0.5
1
Parameters of the PSO were the same as above. The solution
is depicted in Fig 7. All selected sensors are of type 4. It is
interesting, and of course expected, that i f t he pr ice of a
single sensor type would decrease dramatically, it would be
selected even if it is not the most efficient one. Indeed, if the
price of t he s ensor t ype 3 w ould decrease to 0.01 relative
units (98% decrease) all selected sensors are of type 3. The
detection probability of the optimal deployment in this case
is depicted in Fig. 8.
Figure 7. Optimal surveillance region coverage, with 4 different sensors as
described in Table 1. Target probability is as depicted in figure 4a.
VI. CONCLUSION
This paper addressed t he OSDP problem involving
stationary s ensors w ithin a s urveillance r egion w ith t arget
probability that vary from one point of the region to another,
but t hat i s s tationary i n t ime. A n ovel combinatorial
optimizer, DPSO, i s p resented, a nd t he obtained results
testify t hat it is ef fective and pr omising t echnique. Several
14
RAPAIĆ M., KANOVIĆ Ž., JELIČIĆ Z.: DISCRETE PARTICLE SWARM OPTIMIZATION ALGORITHM FOR SOLVING…
issues r emain o pen, an d d eserve f urther r esearch. F irst, i t
would be i nteresting t o a pply DPSO to several o ther
variations of OSDP, but a lso t o other r elated pr oblems
discussed in [10]. Also, it would be interesting to investigate
its behavior when applied parallel to the conventional PSO
in solving hybrid optimization problems, where some of the
variables a re c hosen f rom a c ontinuous, w hile others ar e
chosen f rom a combinatorial s earch s pace. Finally, a l arge
number of va riations t o the original PSO ha ve been
proposed i n l iterature. M ost of t hese va riations a re we ll
applicable to t he D PSO, an d t heir ef fectiveness i n t he
combinatorial optimization context should be investigated.
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
Figure 8. Optimal surveillance region coverage, with 4 different sensors and
target probability as depicted in figure 4a. The price of the third sensor type
was reduced to only 2% of its original price.
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Python Pr
ogramming L
anguage, o
fficial web
s
ite,
http://www.python.org.
Open source library of scientific tool written in Python, SciPy, official
web site, http://www.scipy.org.
Matplotlib, 2D plotting library for Python, official web site,
http://matplotlib.sourceforge.net.