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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

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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

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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

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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©

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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

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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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