Ant Colony Optimization 209

Transcription

1 Ant Colony Optmzaton 9 14 x Ant Colony Optmzaton Benlan Xu, Jhong Zhu and Qnlan Chen Changshu Insttute of Technology Chna 1. Introducton Swarm ntellgence s a relatvely novel approach to problem solvng that taes nspraton from the socal behavors of nsects and of other anmals. In partcular, ants have nspred a number of methods and technques among whch the most studed and the most successful one s the ant colony optmzaton. Ant colony optmzaton (ACO) algorthm, a novel populaton-based and meta-heurstc approach, was recently proposed by Dorgo et al. to solve several dscrete optmzaton problems (Dorgo, 1996, 1997). The general ACO algorthm mmcs the way real ants fnd the shortest route between a food source and ther nest. The ants communcate wth one another by means of pheromone trals and exchange nformaton ndrectly about whch path should be followed. Paths wth hgher pheromone levels wll more lely be chosen and thus renforced later, whle the pheromone ntensty of paths that are not chosen s decreased by evaporaton. Ths form of ndrect communcaton s nown as stgmergy, and provdes the ant colony shortest-path fndng capabltes. The frst algorthm followng the prncples of the ACO meta-heurstc s the Ant System (AS) (Dorgo,1996), where ants teratvely construct solutons and add pheromone to the paths correspondng to these solutons. Path selecton s a stochastc procedure based on two parameters, the pheromone and heurstc values, whch wll be detaled n the followng secton n ths chapter. The pheromone value gves an ndcaton of the number of ants that chose the tral recently, whle the heurstc value s problem-dependent and t has dfferent forms for dfferent cases. Due to the fact that the general ACO can be easly extended to deal wth other optmzaton problems, ts several varants has been proposed as well, such as Ant Colony System (Dorgo,1997), ran-based Ant System (Bullnhemer,1999), and Eltst Ant System (Dorgo,1996). And the above varants of ACO have been appled to a varety of dfferent problems, such as vehcle routng (Montemann,5), schedulng (Blum,5), and travellng salesman problem (Stützle,). Recently, ants have also entered the data mnng doman, addressng both the clusterng (Kanade,7), and classfcaton tas (Martens et al.,7). Ths chapter wll focus on another applcaton of ACO to trac ntaton n the target tracng feld. To the best of our nowledge, there are few reports on the trac ntaton usng the ACO. But n the real world, t s oerved that there s a case n whch almost all ants are nclned to gather around the food sources n the form of lne or curve. Fg. 1 shows the evoluton process of ants searchng for foods. Intally, all ants are dstrbuted randomly

2 1 New Advances n Machne Learnng n the plane as n Fg.1 (a), and a few hours later we fnd that most of ants gather together around the food sources as shown n Fg.1 (b). Tang nspraton from such phenomenon, we may regard these lnear or curvy food sources as tentatve tracs to be ntalzed, and the correspondng ant model s establshed from the optmal aspect to solve the problem of multple trac ntaton. food source 1 food source 1 food source food source (a) Intal dstrbuton of ants (b) The dstrbuton of ants a few hours later Fg. 1. The evoluton process of ant search for foods The remander of ths chapter s structured as follows. Frst, n secton, the wdely used ant system and ts successors are ntroduced. Secton 3 gves the new applcaton of ACO to the trac ntaton problem, and the system of ants of dfferent tass s modeled to concde wth the problem. The performance comparson of ACO-based technques for trac ntaton s carred out and analyszed n Secton 4. Fnally, some conclusons are drawn.. Ant System and Its Drect Successors.1 Ant System Intally, three dfferent versons of AS were developed (Dorgo et al., 1991), namely antdensty, ant-quantty, and ant-cycle. In the ant-densty and ant-quantty versons the ants updated the pheromone drectly after a move from one cty to an adacent cty, whle n the ant-cycle verson the pheromone update was only done after all the ants had constructed the tours and the amount of pheromone deposted by each ant was set to be a functon of the tour qualty. The two man phases of the AS algorthm consttute the ants soluton constructon and the pheromone update. In AS, a good way to ntalze the pheromone trals s to set them to a value slghtly hgher than the expected amount of pheromone deposted by the ants n one teraton. The reason for ths choce s that f the ntal pheromone values are too low, then the search s qucly based by the frst tours generated by the ants, whch n general leads toward the exploraton of nferor zones of the search space. On the other sde, f the ntal pheromone values are too hgh, then many teratons are lost watng untl pheromone evaporaton reduces enough pheromone values, so that pheromone added by ants can start to bas the search. Tour Constructon In AS, m (artfcal) ants ncrementally buld a tour of the TSP. Intally, ants are put on randomly chosen ctes. At each constructon step, ant apples a probablstc acton choce

3 Ant Colony Optmzaton 11 rule, called random proportonal rule, to decde whch cty to vst next. In partcular, the probablty wth whch ant, located at cty, chooses to go to cty s [ ] [ ] p, f N, [ ] [ ] ln l where 1/ d s a heurstc value that s computed n advance, and are two parameters whch determne the relatve mportance of the pheromone tral and the heurstc nformaton, and N s the set of ctes that ant has not vsted so far. By ths probablstc rule, the probablty of choosng the arc (, ) may ncrease wth the bgger value of the assocated pheromone tral and of the heurstc nformaton value. The role of the parameters and s descrbed as below. If, the closest ctes are more lely to be selected: ths corresponds to a classc stochastc greedy algorthm. If, t means that the pheromone s used alone, wthout any heurstc bas. Ths generally leads to rather poor results and, n partcular, for values of 1 t leads to earler stagnaton stuaton, that s, a stuaton n whch all the ants follow the same path and construct the same tour, whch, n general, s strongly suboptmal. Each ant mantans a memory whch records the ctes already vsted. And moreover, ths memory s used to defne the feasble neghbourhood N n the constructon rule gven by equaton (1). In addton, such a memory allows ant both to compute the length of the tour T t generated and to retrace the path to depost pheromone for upcomng global pheromone update. Concernng soluton constructon, there are two dfferent ways of mplementng t: parallel and sequental soluton constructon. In the parallel mplementaton, at each constructon step all ants move from ther current cty to the next one, whle n the sequental mplementaton an ant bulds a complete tour before the next one starts to buld another one. In the AS case, both choces for the mplementaton of the tour constructon are equvalent n the sense that they do not sgnfcantly nfluence the algorthm s behavour. Update of Pheromone Trals After all the ants have constructed ther tours, the pheromone trals are updated. Ths s done by frst lowerng the pheromone value on all arcs by a factor, and then addng an amount of pheromone on the arcs the ants have crossed n ther tours. Pheromone evaporaton s mplemented by the followng law (1 ), (, ) L () where 1 s the pheromone evaporaton rate. The parameter s used to avod unlmted accumulaton of the pheromone trals and t enables the algorthm to forget bad decsons prevously taen. In fact, f an arc s not chosen by the ants, ts assocated pheromone value decreases exponentally wth the number of teratons. After evaporaton, all ants depost pheromone on the arcs they have crossed n ther tour: l (1)

4 1 New Advances n Machne Learnng where m 1 (1 ), (, ) L (3) s the amount of pheromone ant deposts on the arcs t has vsted. It s defned as follows: 1/ C f arc (, ) belongs to T (4) otherw where C, the length of the tour T travelled by ant, s computed as the sum of the lengths of the arcs belongng to T. By means of equaton (4), the shorter an ant s tour s, the more pheromone the arcs belongng to ths tour receve. In general, arcs that are used by many ants and whch are part of short tours, receve more pheromone and are, therefore, more lely to be chosen by ants n the followng teratons of the algorthm... Eltst Ant System The eltst strategy for Ant System (EAS) (Dorgo,1996) s, n prncple, to provde a strong addtonal renforcement to the arcs belongng to the best tour found snce the start of the algorthm. Note that ths addtonal feedbac to the best-so-far tour s another example of a daemon acton of the ACO meta-heurstcs. Update of Pheromone Trals The addtonal renforcement of tour T s acheved by addng a quantty e / C to ts arcs, where e s a parameter that defnes the weght gven to the best-so-far tour T, and C s ts length. Thus, equaton (3) for the pheromone depost becomes where s defned as n equaton (4) and m e 1 (5) s defned as follows: 1/ C f arc (, ) belongs to T (6) otherw Note that n EAS the pheromone evaporaton s mplemented as n AS..3. Ran-Based Ant System Another mprovement over AS (Bullnhemer,1999) s the ran-based verson of AS ( AS ran ). In AS ran each ant deposts an amount of pheromone that decreases wth ts ran. Addtonally, as n EAS, the best-so-far ant always receves the largest amount of pheromone n each teraton. Update of Pheromone Trals Before updatng the pheromone trals, the ants are sorted by ncreasng tour length and the quantty of pheromone an ant deposts s weghted accordng to the ran of the ant. In each

5 Ant Colony Optmzaton 13 teraton, assume that total W best-raned ants are consdered, and only the ( W 1) bestraned ants and the ant that produced the best-so-far tour are allowed to depost pheromone. The best-so-far tour gves the strongest feedbac wth weght w ; the r th best r ant of the current teraton contrbutes to pheromone updatng wth the value 1/ C multpled by a weght gven by,w r. Thus, the AS ran pheromone update rule s r r where 1/ C and 1/ C. W 1 r ( W r) w r 1 (7).4 Max- Mn Ant System Max-Mn Ant System (MMAS) (St ü tzle & Hoos, ) ntroduces some man modfcatons wth respect to AS. Frst, t strongly explots the best tours found: only ether the teraton-best ant, that s, the ant that produced the best tour n the current teraton, or the best-so-far ant s allowed to depost pheromone. Unfortunately, such a strategy may lead to a stagnaton stuaton n whch all ants follow the same tour, because of the excessve growth of pheromone trals on arcs of a good, although suboptmal, tour. To counteract ths effect, a second modfcaton ntroduced by MMAS s that t lmts the possble range of pheromone tral values to the nterval [ mn, ]. Second, the pheromone trals are ntalzed to the upper pheromone tral lmt, whch, together wth a small pheromone evaporaton rate, ncreases the exploraton of tours at the start of the search. Fnally, n MMAS, pheromone trals are rentalzed each tme the system approaches stagnaton or when no mproved tour has been generated for a certan number of consecutve teratons. Update of Pheromone Trals After all ants have constructed a tour, pheromones are updated by applyng evaporaton as n AS, followed by the depost of new pheromone as follows: best, (8) best best where 1/ C. The ant whch s allowed to add pheromone may be ether the best- best best b so-far, n whch case 1/ C, or the teraton-best, n whch case 1/ C, b where C s the length of the teraton-best tour. In general, n MMAS mplementatons both the teraton-best and the best-so-far update rules are used n an alternate way. Obvously, the choce of the relatve frequency wth whch the two pheromone update rules are appled has an nfluence on how greedy the search s: When pheromone updates are always performed by the best-so-far ant, the search focuses very qucly around T, whereas when t s the teraton-best ant that updates pheromones, then the number of arcs that receve pheromone s larger and the search s less drected. Pheromone Tral Lmts In MMAS, lower and upper lmts mn and on the possble pheromone values on any arc are mposed n order to avod earler searchng stagnaton. In partcular, the mposed

6 14 New Advances n Machne Learnng pheromone tral lmts have the effect of lmtng the probablty when an ant s n cty to the nterval [ mn, ] p of selectng a cty p p p 1. Only p p, wth mn when an ant has ust one sngle possble choce for the next cty, that s N 1, we have p mn p 1. It s easy to show that, n the long run, the upper pheromone tral lmt on any arc s * * bounded by 1/ C, where C s the length of the optmal tour. Based on ths result, MMAS uses an estmate of ths value, 1/ C, to defne : each tme a new best-so-far tour s found, the value of s updated. The lower pheromone tral lmt s set to / mn, where s a parameter (Stützle & Hoos, ). Pheromone Tral Intalzaton and Re-ntalzaton At the start of the algorthm, the ntal pheromone trals are set to an estmate of the upper pheromone tral lmt. Ths way of ntalzng the pheromone trals, n combnaton wth a small pheromone evaporaton parameter, causes a slow ncrease n the relatve dfference n the pheromone tral levels, so that the ntal search phase of MMAS s very exploratve. Note that, n MMAS, pheromone trals are occasonally re-ntalzed. Pheromone tral rentalzaton s typcally trggered when the algorthm approaches the stagnaton behavour or f for a gven number of algorthm teratons no mproved tour s found. 3. ACO for Trac Intaton of Bearngs-only mult-target tracng 3.1 Problem Presentaton Bearngs-only mult-target tracng (BO-MTT) (Nardone, 1984 ; Dogancay, 4, 5) n a tatc system can be descrbed as: gven a tme hstory of nose-corrupted bearng measurements from two oervers, the obectve s to obtan optmum estmaton of the postons, veloctes and acceleratons of all targets. Generally, the whole process of target tracng ncludes trac ntaton, trac mantenance and trac deleton. To the best of our nowledge, however, many reported lterature manly focused on the trac mantenance,.e. target tracng, wthout consderng the trac ntaton process, after the moton of each target s modelled. Actually, trac ntaton plays an mportant role n evaluatng the performance of suequent target tracng, and mproperly ntated tracs may ether lead to target loss or the ncrease of consumpton of lmted resources. In the case of mult-sensor-mult-target BOT, for nstance, two-sensor-two-target BOT at a gven scan, four Lne of Sghts (LOSs) are avalable alone to determne whch LOS belongs to some target of nterest. Usually, such a problem can also be dealt wth the general trac ntaton technques wdely used n the radar tracng feld through ntersectng these LOSs to obtan a group of canddates of true targets poston ponts. However, such an operaton wll result n some ntersectons ncludng both the true target postons and the vrtual target poston called ghost, as shown n Fg.. These ghosts, n fact, do not belong to any target (denoted by poston ponts 3 and 4). Due to ths fact, the orgn uncertanty of obtaned poston canddates should be dscrmnated and ths ssue forms the topc of ths secton. In addton, such a problem becomes harder to handle n the presence of clutter.

7 Ant Colony Optmzaton 15 Y trac of target trac of target sensor 1 sensor O Fg.. The generated ghosts n case of two-sensor-two-target BOT X 3. Motve In the mage detecton feld, the Hough transform (H-T) has been recognzed as a robust technque for lne or curve detecton and also have been largely appled by scentfc communty (Bhattacharya, ; Shapro, 5). The basc dea of H-T s to transform a pont ( x, y) n the Cartesan coordnate system onto a curve n the (, ) parameter space, whch s formulated as x cos y sn (9) where s the dstance from the lne through ( x, y ) to the orgn, and s the angle to the normal wth the x axs. The angle vares from to 18, whle the may be ether postve or negatve. So, t s oerved that, f a set of ponts n the Cartesan coordnate le on the same lne, all curves each correspondng to a pont must ntersect at a same pont denoted by (, ) n the parameter space. Inspred by ths phenomenon, the H-T technque can be utlzed to ntalze the trac of target whch maes a unform rectlnear moton. 3.3 Soluton to Mult-Target Trac Intaton by ACO In ths secton, we wll nvestgate the problem of mult-target trac ntaton. Frst, a obectve functon s presented to descrbe the property of the mult-target trac ntaton. Second, a novel ACO algorthm, called dfferent tass of ants, s modelled to ntate the tracs of nterest. As noted before, f there are n curves n the parameter space, at most C n ntersectons are obtaned n general. However, n a real tracng scenaro, these curves wll not strctly ntersect the pont but several ponts dstrbuted n the parameter space due to the exstence

8 16 New Advances n Machne Learnng of measurement error. Even so, these ponts are stll dstrbuted n a small regon, and thus such a small area could be deemed as an obectve functon to be optmzed. For the case of two gven tracs, the correspondng ntersectons n the parameter space are plotted n Fg.3, and for the upper left expanded subfgure, whch corresponds to target 1, the mnmum and mum values of could be obtaned and then denoted by mn and, respectvely. Smlarly, the related mnmum and the mum values of are also found and denoted by mn and, respectvely ( m ) ( m ) ( mn, mn ) target ( rad ) target 3 S 1 1 (, ) (, ) 1 ( m ) target 1 target ( mn, mn ) S ( rad ) ( rad ) Fg. 3. A case of determnaton of obectve functon n the parameter space As a result, two rectangular blocs are formed and the area of each s calculated as and the obectve functon J s defned as S ( ) ( ), (1) mn mn J mn M r 1 S r( r1 r r3 r4 ) s. t r m r1 r r3 r4, m1 m m3 m4, (11) 1,..., 4; where r 1 r r 3 r 4 or m1 m m3 m4 s the possble trac n the trac space, M s the number of tracs to be ntalzed. Afterwards, the ants of dfferent tass wll be nvestgated, and t has the followng characterstcs: 1) The number of tass s equal to the one of tracs to be ntated, or equal to the one of targets of nterest. ) The tradtonal ACO algorthm bulds solutons n an ncremental way, but the proposed system of dfferent tass of ants bulds solutons n parallel way. Especally, n the proposed system of ants of dfferent tass, the thought of both collaboraton and competton between ants s consdered and ntroduced. For nstance, ants of the same tas search for foods n a collaboratve way, whle ants of dfferent tass wll compete wth each other durng establshng solutons.

9 Ant Colony Optmzaton 17 3) Ants of the same tas are dedcated to fndng ther best soluton, and a set of all best solutons found by ants of dfferent tass consttute the solutons to Eq. (11) we descrbe. 4) In the system of ants of dfferent tass, the search space depends not only on the measurement returns at the next scan but also on the pror nowledge of target moton. The determnaton of search space In the case of bearngs-only two-sensor- M -target tracng, the samplng data of the frst four scans are utlzed sequentally to ntate tracs, and then total four search spaces,.e., 1,, 3, and 4, are obtaned sequentally. Suppose that the pror nowledge about target moton, such as the mnmum and mum veloctes denoted by v respectvely, s nown and then utlzed to construct an annular regon whose nner and outer raduses are determned by r 1 v mn T and r v T, respectvely, where T denotes the samplng nterval. For nstance, f an ant s now located at poston n 1, then the ant wll vst the next poston located n the shadow secton covered by both the annular regon and, whch s denoted by n Fg.4. vmn and 1 r 1 r 3 4 Fg. 4. The determnaton of search spaces Trac Canddate Constructon Usng the Ants of Dfferent Tass Intally, M ants of dfferent tass are placed randomly on poston canddates n the frst search space 1, then each ant of a gven tas vsts probablstcally the poston canddate n the next search space. Suppose that an ant of a gven tas s s now located at poston n (1 3 ), then the ant wll vst poston n the next search space by applyng the followng probablstc formula:

10 18 New Advances n Machne Learnng s, 1 arg, f q q s 1,,, J otherwse, (1) and J s a random varable selected accordng to the followng probablty dstrbuton s, 1, s,,, f 1 P( ) s, l 1 (13) l, l s 1, l, l, l otherwse s where, denotes the pheromone amount deposted by ants of tas s on tral (, ),, s the total pheromone amount deposted by all ants of dfferent tass on tral (, ), shows the repulson on the foregn pheromones left on the tral (, ), q s a random number unformly dstrbuted between and 1, and q s a parameter whch determnes the relatve mportance of the explotaton of good solutons versus the exploraton of search spaces. Accordng to the search spaces dscussed above, Fg. 5 plots the process of how the heurstc value s calculated from search spaces 1 to, namely, f an ant wll move from postons to, the correspondng heurstc value can be defned as ( d r ), (14) r s equal to ( r r1 ) /.,, exp ( r r1 ) where d, denotes the dstance between postons and, and Note that f poston falls out of, we set,, and the search falure s declared for the current ant. r 1 r d,, r o r 1 r d, r r Fg. 5. The calculaton of heurstc value

11 Ant Colony Optmzaton 19 Update of Pheromone The pheromone update s performed n two phases, namely, local update and global update. Whle buldng a soluton, f an ant of tas s carres out the transton from postons to, then the pheromone level of the correspondng tral s changed n the followng way: (1 ), (15) s s s,, s where s the ntal pheromone level of ants of tas s. Once all ants of dfferent tass at a gven teraton have vsted four canddate postons each from dfferent samplng ndces, the pheromone amount on each establshed trac wll be updated globally. Here, we use the best-so-far-soluton found by ants of the same tas,.e. the best soluton found from the start of the algorthm run, to update the correspondng pheromone tral. We adopt the followng rule p s s s,, (1 ),, 1. (16) s, where s the pheromone amount that ant of tas s deposts on the tral (, ) t has, traveled at the current teraton, and p s the number of ants. In the case of bearngs-only s, mult-sensor-mult-target tracng, s set to a constant., 4. A Comparson of ACO-Based Methods for Trac Intaton 4.1 The Problem Two cases are nvestgated here, namely two and three tracs ntaton problems. For each scenaro, the performance of trac ntaton s nvestgated both n clutter-free envronments and n clutter envronments, respectvely. Two fxed sensors used to measure the targets bearngs are located at (, ) and (18 m, ) respectvely n a survellance regon. The standard devaton of the bearng measurements for each sensor s taen as.1, and the samplng nterval s set to be T 1s. The case n whch each target maes a unform rectlnear moton s consdered, and the ntal state of each target s llustrated n Table 1. Scenaros 1 Targets x y x y (m) (m) (m/s) (m/s) Table 1. The ntal poston and velocty of each target n the two consdered scenaros

12 New Advances n Machne Learnng 7 6 target 1 target ghost target 1 target target 3 ghost 5 45 Y (m) 4 Y (m) X (m) X (m) Fg. 6. The target poston canddates n a clutter-free envronment (left: Scenaro 1, rght: Scenaro ) target 1 target clutter and ghost 8 7 target 1 target target 3 clutter and ghost 45 6 Y (m) Y (m) X (m) X (m) Fg. 7. The target poston canddates n clutter envronments (left: Scenaro 1, rght: Scenaro ) Fgs.6 and 7 depct a part of poston canddates obtaned by ntersectng LOSs at each scan, and our obect s to dscrmnate the true postons of each target of nterest. Here, we use two ACO-based technques, namely the Ant System (called the tradtonal ACO) and the system of ants of dfferent tass (called the proposed ACO). Other parameters related to the two ACO-based methods are llustrated n Table Parameter Value Parameter Value.1.3. M 3M v mn 1 m / s.8 v 4 m / s q.7 a 15 m / s.5 N 5 Table. The Parameter Settngs for ACO-related Methods

13 Ant Colony Optmzaton 1 4. Evaluaton Indces Two performance ndces are ntroduced to evaluate the system of ants of dfferent tass,.e. The probablty of false trac ntaton: assumng N Monte-Carlo runs are performed, we defne the probablty of false trac ntaton as N N F f n, (17) 1 1 where f denotes the number of false ntated tracs at the th Monte-Carlo run, and n s the total number of ntated tracs. The probablty of correct ntaton of at least tracs: f at least (1 M ) tracs are ntated correctly, ts correspondng probablty s where l s a bnary varable and defned as N 1 C l N, (18) 1 l at the th Monte-Carlo run. f at least tracs are ntated correctly otherwse (19) 4.3 Results All results n Tables 3 to 6 are averaged over 1, Monte-Carlo runs. Accordng to the evaluaton ndces we ntroduce, the tradtonal ACO algorthm performs as well as the proposed one, as llustrated n Tables 3 and 4, n clutter-free envronments. However, n the presence of clutter, the proposed ACO algorthm shows a sgnfcant mprovement over the tradtonal one wth respect to the probablty of false trac ntaton, as shown n Tables 5 and 6. Evaluaton ndces The tradtonal ACO The proposed ACO Pro. of false trac ntaton ( F ).1. Pro. of correct ntaton of at least tracs( C ) C C Table 3. Performance comparson for two-trac-ntaton problem n clutter-free envronments

14 New Advances n Machne Learnng Evaluaton ndces The tradtonal ACO The proposed ACO Pro. of false trac ntaton ( F ) C Pro. of correct ntaton of at least tracs( C ) C C Table 4. Performance comparson for three-trac-ntaton problem n clutter-free envronments Evaluaton ndces The tradtonal ACO The proposed ACO Pro. of false trac ntaton ( F ) Pro. of correct ntaton of at least tracs( C ) C C Table 5. Performance comparson for two-trac-ntaton problem n clutter envronments Evaluaton ndces The tradtonal ACO The proposed ACO Pro. of false trac ntaton ( F ) C Pro. of correct ntaton of at least tracs( C ) C C Table 6. Performance comparson for three-trac-ntaton problem n clutter envronments Among 1, Monte-Carlo runs, only the cases of all tracs beng ntated successfully are nvestgated and called effectve runs later. For the obectvty of comparson, we select the worst case, n whch the mum runnng tme for each ACO algorthm s evaluated, from the effectve runs.

15 Ant Colony Optmzaton 3 Fg. 8 depcts the trends of obectve functon evoluton wth the ncreasng number of teratons n scenaro. Compared wth the tradtonal ACO algorthm, the proposed one requres fewer teratons for convergence n clutter-free or clutter envronments. Accordng to Tables 3 and 4, although the performance of the tradtonal ACO algorthm s comparable to that of the proposed one, we fnd that the proposed ACO one seems more practcal due to less runnng tme needed. Fgs. 9 and 1 depct varyng curves of pheromone on the true targets tracs, t s oerved that the amount of pheromone on each true trac ncreases n a moderate way, whch means most ants prefer choosng these tracs and regarded them as optmal solutons The proposed ACO The tradtonal ACO 35 The proposed ACO The tradtonal ACO 9 3 Obectve value (m.rad) Obectve value (m.rad) Iteraton Iteraton Fg. 8. Obectve functon curves (left: In clutter-free envronments; rght: In clutter envronments) Pheromone amount.75 On trac 1 On trac On trac 3 Pheromone amount On trac 1 On trac On trac Iteraton Iteraton Fg. 9. Pheromone curves n clutter-free envronments (left: The proposed ACO; rght: The tradtonal ACO)

16 4 New Advances n Machne Learnng On trac 1 On trac On trac Pheromone amount Iteraton Pheromone amount On trac 1 On trac On trac Iteraton Fg. 1. Pheromone curves n clutter envronments (left: The proposed ACO; rght: The tradtonal ACO) 5. Concluson Ths chapter manly ams to ntroduce some wdely used ACO algorthms and ther orgns, such as the AS, EAS, MMAS, and so on. It s found that all concerns are focused on the pheromone update strategy. Some uses the best-so-far-ant or the teraton-best ant ndependently/nteractvely to update the tral that ants travelled. Meanwhle, the update law may dffer a bt for dfferent ACO algorthms. Among the four ACO algorthms, two versons have receved great populartes n varous applcatons,.e. AS and MMAS. Another contrbuton n ths chapter s the extenson of the general ACO algorthm to the system of ants of dfferent tass, and ts behavour s modelled and mplemented n the trac ntaton problems. Smulaton results are also presented to show the effectveness of the novel ACO algorthm. Accordng to the example presented n ths chapter, we beleve that the general framewor of AS can be modfed to solve varous optmal or non-optmal problems. 6. References B. Bullnhemer; R. F. Hartl & C. Strauss. (1999). A new ran based verson of the ant system: A computatonal study, Central Eur. J. Oper. Res. Econ., Vol. 7, No. 1, 5 38, ISSN X. C. Blum. (5). Beam-ACO hybrdzng ant colony optmzaton wth beam search: An applcaton to open shop schedulng, Comput. Oper. Res., Vol. 3, No. 6, , ISSN Davd Martens; Manu De Bacer & Raf Haesen. (7). Classfcaton Wth Ant Colony Optmzaton, IEEE Trans. on Evolutonal Computaton, Vol. 11, No. 5, October 7, , ISSN X. Kutluyll Dogancay. (4). On the bas of lnear least squares algorthm for passve target localzaton, Sgnal Processng, Vol. 84, No. 3, , ISSN

17 Ant Colony Optmzaton 5 Kutluyll Dogancay. (5). Bearngs-only target localzaton usng total least squares, Sgnal Processng, Vol. 85, No. 9, , ISSN M. Dorgo; V. Manezzo & A. Colorn. (1991). Postve Feedbac as a Search Strategy, Techncal Report 91 16, Poltecnco d Mlano, Mlano, Italy. M. Dorgo; V. Manezzo & A. Colorn. (1996). The ant system: optmzaton by a colony of cooperatng agents, IEEE Trans. on System, Man, and Cybernetcs-part B, Vol.6, No. 1, 9-4, ISSN M. Dorgo & L. M. Gambardella. (1997). Ant colony system: A cooperatve learnng approach to the travelng salesman problem, IEEE Trans. on Evolutonal Computaton, Vol.1, No. 1, 53-66, ISSN X. P. Bhattacharya; A. Rosenfeld & I. Wess. (). Pont-to-lne mappngs as Hough transforms, Pattern Recognton Letters, Vol. 3, No. 4, , ISSN Parag M. Kanade & Lawrence O. Hall. (7). Fuzzy Ants and Clusterng, IEEE Trans. on System, Man, and Cybernetcs-part A, Vol. 37, No. 5, September 7, ,ISSN R. Montemann; L. M. Gambardella; A. E. Rzzol & A. Donat. (5). Ant colony system for a dynamc vehcle routng problem, J. Combnatoral Optm., Vol. 1, No. 4, , ISSN S.C. Nardone; A.G. Lndgren & K.F. Gong. (1984). Fundamental propertes and performance of conventonal bearngs-only target moton analyss, IEEE Transactons on Aerospace and Electronc Systems, Vol. 9, No. 9, , ISSN T. Stützle & H. H. Hoos. (). MAX-MIN ant system, Future generaton computer systems, Vol.16, , ISSN X. V. Shapro. (6). Accuracy of the straght lne Hough transform: the non-votng approach, Computer Vson and Image Understandng, Vol. 13, No. 1, 1-1, ISSN

18 6 New Advances n Machne Learnng