Balancing Exploration and Exploitation by an Elitist Ant System with Exponential Pheromone Deposition Rule

2008 IEE E Region 10 Colloq uium and the Third I CIIS, Kharagpur, INDIA Dece mber 8-10. Paper ID: 2 50 978-1-424 4-2806-9/08/$ 25.00© 2008 IEEE 1 Ayan Acharya 1 , Deepyam an Maiti 2 , Aritra Baner jee 3 , Amit Kona r 4 1,2,3,4 Department of Electronic s and Te lecommunication Engineerin g, Jadavpur Univer sity, Kolkata: 7 00032 1 masterayan@gmail.co m, 2 deep yamanmaiti@g mail.com, 3 aritraete ce@gmail.co m, 4 konaramit@yahoo .co.in Abstract — The paper presents an ex ponential pheromone deposition rule to modify the basic ant syste m algorithm w hich employs constant deposition rule. A stability analysis using differential equation is carried out to find out the values of parameters that make the ant syste m dynamics stable fo r b oth kinds of deposition r ule. A roadmap of connected cities is chosen as the problem environment w here the shortest route between two given cities is required to be discovered. Si mula tions performed with both forms of d eposition approach using Elitist Ant System model reveal that the ex ponential deposition approach outperforms the classical one by a large ex tent. Exhaustive ex periments are also ca rried out to find ou t the optimum setting of different controlling parameters for exponential deposition ap proach and an empirical relationship between the major controlling param eters of the algorithm and some features of problem environment. Keywords- Ant Colony Optimization, Ant System, Elitist Ant System, Stability Analysis, Exponential Pheromone Deposition. I. I NTRODUCTI ON Ant Colony O ptimization (ACO ) is a parad igm for designing metaheuristic algorithms for combinator ial op timization problems. While roaming from food sources to t he nest and vice versa, ants dep osit on the ground a substance called pherom one . Ants can smell pheromone a nd choose, in probabilit y, paths marke d b y stronger pheromone concentrations. Hence, the pheromone tr ail allows the ants to find their way bac k to the f ood source or to the nest. ACO algorithm simulates this behavior of ant colony to solve difficult NP hard opti mization pro blems . . . . Ant Sy stem (AS) is the ear liest form of a nt colon y optimization algorithm that has been modified b y numero us researchers till date. Elitist Ant Syste m (E AS) model is one such improved model of the p rimary versio n o f a nt s ystem. Our paper extends the AS m odel by introducing an exponential phero mone d eposition app roach, co ntrary to the uniform deposition appro ach used i n cla ssical AS algorit hms. We atte mpt to solve t he dete rministic AS d ynamics using differential equation. T his novel analysis helps in d etermining the range of para meters i n the exponential phero mone deposition rule to confir m stability in phero mone trails. The deterministic solution does not violate the stochastic nature of the AS b ecause a seg ment o f traj ectory here is always select ed probabilisticall y. Our previous work [8] was based on stability anal ysis using difference equation. In this paper, w e have employed differential eq uations which not onl y chara cterize the system more precisely b ut also are m ore po pular than difference equations. The previous p aper, with experi ments perfor med over TSP instances, could not at all highlight the ph ilosophy of t he non uniform deposition rule. T his p aper presents sufficient simulatio n backup to establish the pr oposed algorithm’s superiority o ver the traditional o ne. P roblem environment is also chosen very cleverly to emphasize the efficacy of the pro posed algorithm. Exhausti ve experimentatio ns also help find out t he s uitable values o f parameter for which the proposed algorith m works best and from t hese r esults we tr y to ascer tain a n a lgebraic relationship between the para meter set of the algorith m and feature set of the problem en vironment. The paper is structured in 6 sections. In sectio n I I, a br ief introduction o f AS and E AS are provided. W e for mu late a scheme for the general solution of the Ant System in secti on III. Stability analysi s with co mplete solut ion for differe nt pheromone deposition rules is undertaken i n section IV. Performance analyses of the proposed and classical A S are compared in sectio n V on elitist mode l. Finally, the conclusions are listed in section VI. II. A NT S YSTEM AND E LITIST A NT S YSTEM The theory of ant s ystem can b est be explained in the context of T ravelling Salesp erson Pro blem ( TSP) ([6]). The basic ACO algorithm f or TSP can be d escribed as follows: procedure ACO algorithm fo r TSPs Ø Set parameters, initialize ph eromone and an ts’ memory while (termination co ndition not met) Ø Construct Solu tion Ø Apply Local Search ( op tional) Ø B est To ur check Ø Update Trails end end ACO algorithm for TSPs Ant System (AS) ([1] ,[2],[3]) basically consists of two levels: Balancing Exploration and Exploitation by an Elitist Ant System w ith Exponential Pheromone Deposition Rule 2008 IEE E Region 10 Colloq uium and the Third I CIIS, Kharagpur, INDIA Dece mber 8-10. Paper ID: 2 50 978-1-424 4-2806-9/08/$ 25.00© 2008 IEEE 2 1. Initialization: 1. Any initial parameters are load ed. 2. Edges are set with an initial phero mone val ue. 3. Each ant is individuall y placed on a r andom city. 2. Main Loop: • Con struct Solution Each ant co nstructs a tour b y successively appl ying the probab ilistic choice functio n: β 0 k k 0 i i k 0 i β α α ij ij ik ik k k: k N i β β α α ij ij i k β β α α ij ij ik ik ik (τ ) . (η ) / (τ ) .( η ) if q < q P ( j) 1 if ( τ ). ( η )= m a x { (τ ). ( η ): k N } wi th q > q ( 1 ) 0 i f (τ ) .(η ) m a x {( τ ) .(η ) : k N } wi th q >q ∈         = ∈       ≠ ∈     ∑ where P i k (j) is the prob ability of selecting nod e j after nod e i for ant k . A node j ∈ N i k ( N i k be ing the nei ghborhood of ant k when it is at node i ) if j is not already visited. η ik is the visibility i nformation ge nerall y taken as the in verse of the length of link (i,k), is the phe romone concentratio n associated with t he link (i,k ). q 0 is a pseudo random factor deliber ately introduced for p ath explorati on and α, β are t he weights for pheromone conce ntration and visibil ity. • Best Tour check: Calculate t he length s of the ants’ tours and co mpare with best tour length so far. If t here is a n improvement, upd ate it. • Update Trails: 1. E vaporate a fixed proportion of the pheromone o n each edge. 2 . For each ant per form the ‘Ant Cycle’ ([3]) pheromone update. First improvement over AS was proposed as the Elitist Ant System (EAS) s trateg y ([2] ,[3],[9]) in which additional reinforcement is pro vided to the b est s olution found f rom the start of the algorithm. Now, let i and j be two succe ssive nodes, on the tour of an ant a nd τ ij (t) be the phero mone concentratio n created by the ant at time t and associated with t he ed ge of the graph joining the nodes i and j. τ ij (t) Fig. 1: Defining τ ij (t) Let ρ >0 b e the phero mone evaporatio n rate, and ∆ ∆ ∆ ∆ τ τ τ τ ij k (t) be the phero mone dep osited by ant k at time t. T he basic pheromone updati ng rule in AS is then given by, τ ij (t)=(1-  )τ ij (t-1)+ 1 ( ) m k ij k t τ = ∆ ∑ (2) In Elitist model, a special preference is gi ven to t he best path found so far. T hus the p heromone upd ate rule for the best so far tour is: τ ij (t)=(1-  )τ ij (t-1)+ 1 ( ) m k ij k t τ = ∆ ∑ + e ∆ τ τ τ τ ij bs (3) where ∆ ∆ ∆ ∆ τ τ τ τ ij k is the amount of pheromone de posited by ant k o n the arcs it has visited and is defi ned as follows: k k 1/C , if arc (i,j ) belo ngs to T 0 , oth er wis e k ij τ   ∆ =     , C k being the lengt h o f the tour T k constructed b y k th ant. e is a pa rameter that d efines the weight given to the best-so-far tour T bs with tour len gth C bs . ∆ τ τ τ τ ij bs in (3) is defined as bs 1/C , if arc (i,j) belongs to T 0 , otherwise bs bs ij τ     ∆ =       . A s uitable choice of parameter e allo ws EAS to find better tour in a smaller number of iteration s compare d to AS . III. D ETERMINISTI C F RAMEWORK FOR S OLUTION OF B A SIC A NT S YSTEM D YNA MICS Now, from (2) , τ ij (t) - τ ij (t-1)= -  τ ij (t-1)+ 1 ( ) m k ij k t τ = ∆ ∑ 1 ( ) m k i j i j k i j d t d t τ ρ τ τ = ⇒ =− + ∆ ∑ 1 ( ) ( ) ( 1 ) (4 ) m k ij k i j D t t ρ τ τ = ∴ + = ∆ + ∑ where we define ( ) ( 1 ) i j i j i j i j d t t D d t τ τ τ τ = − − = Evidently, (4) gives the solution for the ant dynamic s. No w, to solve (4), we have to separ ate the co mplimentary functi on and the particular integral. We now consider t wo different forms of ∆ τ τ τ τ ij k (t) and try to determine the complete solution o f τ ij (t) . Evaluation o f Complimentary Function: The c omplimentary functio n of (4) is obtained by setting 1 ( ) m k ij k t τ = ∆ ∑ to zero . This gives only t he transient b ehavior of the ant system dynamics. Therefore, from (4), ( ) 0 , i j D ρ τ + = ⇒ = − D ρ Thus, the transie nt behavior of the Ant Syste m is given by CF: τ ij (t)=Ae -ρt (5) where A is a co nstant which is to b e deter mined from i nitial condition. Evaluation of Particular Integr al for Both Forms of Deposition Rule: The stead y state solution of the ant system d ynamics is obtained by co mputing particular integral of ( 4). This is given by, 1 1 ( 1 ) ( 6 ) m k i j i j k t D ρ τ τ = = ∆ + + ∑ Case I: When ∆ τ ij k (t)=C k , we obtain from (6) 1 1 m k k P I C D ρ = = + ∑ 1 1 1 . ( 1 ) m k k D C ρ ρ − = = + ∑ 2 2 1 1 ( 1 . .. .. . . ) m k k D D C ρ ρ ρ = = − + − ∑ 1 1 ( 1) m k k C ρ = = ∑ 1 / ( 7 ) m k k C ρ = = ∑ Case II: When ∆ τ ij k (t)=C k (1-e -t/ T ) , we obtain from (6), node i node j 2008 IEE E Region 10 Colloq uium and the Third I CIIS, Kharagpur, INDIA Dece mber 8-10. Paper ID: 2 50 978-1-424 4-2806-9/08/$ 25.00© 2008 IEEE 3 ( 1 )/ 1 1 P I ( 1 ) m t T k k C e D ρ − + = = − + ∑ ( 1 )/ 1 1 1 1 m m t T k k k k C C e D D ρ ρ − + = = = − + + ∑ ∑ ( 1 ) / 1 1 1 / m m t T k k k k C C e D ρ ρ − + = = = − + ∑ ∑ 1 ( 1 )/ 1 1 / / ( ) ( 8 ) m m t T k k k k C C e T ρ ρ = − + = = − − ∑ ∑ IV. S TABILITY A NALYSI S OF A NT S YST EM D YNAMICS WI TH C OMPLET E S OLUTION In this section, we ob tain th e closed form solution of the ant system dynamics for d etermining the co ndition for stabilit y of the dynamics. Case I: For co nstant deposition rule, the co mplete solution can be obtained b y add ing CF a nd PI fro m (5) and ( 7) respectivel y and is given b y, τ ij (t) = Ae -ρt + 1 / m k k C ρ = ∑ . At t=0, τ ij (0) = A+ 1 / , m k k C ρ = ∑ 1 (0 ) / m ij k k C A τ ρ = ⇒ = − ∑ Therefore, the complete solution is, τ ij (t) =[ 1 (0 ) / m ij k k C τ ρ = − ∑ ] e -ρt + 1 / m k k C ρ = ∑ (9) It follo ws from (9 ) that the system is stable for ρ>0 and converges to steady state value 1 / m k k C ρ = ∑ as time increases. The plo t below supports the abo ve observation. Figure 2: τ ij (t) versus t for constant pheromone depositi on Case II: Fo r exponentially incr easing phero mone depo sition, the complete solutio n is, τ ij (t)=Ae -ρt + 1 ( 1 ) / 1 1 / / ( ) m m t T k k k k C C e T ρ ρ = − + = − − ∑ ∑ Now, at t=0, τ ij (0) = A+ 1 1 / 1 1 / / ( ) m m T k k k k C C e T ρ ρ = − = − − ∑ ∑ Therefore, with initial conditio n incorporated , the overall solution is given b y, τ ij (t)= τ ij (0) e -ρt + 1 ( 1 ) t m k k C e ρ ρ − = − ∑ + [ ( 1 / ) ] 1 ( 1 / ) ( 1 ) ( 1 0 ) 1 ( ) T t t T m k k C e e T ρ ρ ρ − − = − + − − ∑ Clearly, the s ystem i s stab le for positive values of ρ and T and converges to 1 / m k k C ρ = ∑ in its stead y state. Figure 3: τ ij (t) versus t for exponential p heromone deposition wit h T=10 A uniform pheromone dep osition by an a nt cannot ensure subsequent ant s to follow t he sa me trajec tory. However, an exponentially increasi ng time function ensures that s ubsequent ants clo se enough to a previously selected trial solution will follow the trajectory, as it can examine gradually thicke r deposition o f phero mones over the trajectory. Natural ly, deception probability ([4]) being less, con vergence ti me should improve. V. S IMULA TION R ESULTS The EAS model is co nsidered here to stud y the perfor mance of the ant s ystem algorith m with expo nential d eposition r ule. As a problem environme nt, we take a network of connected cities where the s hortest route b etween t wo given cities is to be deter mined. Ants be gin their tour at t he starting cit y and terminate their journe y at t he destination city and decid e its next posi tion at eac h i ntermediate step by a prob ability ba sed selection app roach as given in (1). Interpretation of different terms in (1 ) is exactly same as in context o f T SP except the term η ik which i s defined here as η ik =1/(|d ik |+|d kg |) where d ik is defined as the distance between t he cities i and k a nd d kg specifies the distance bet ween cities k and g , g being the destination c ity. α, β ar e the weights for pheromone concentration and visibility as usual. Ants also stop moving if they find a dea d end. In constan t p heromone appro ach, d eposition of excess pheromone in all links of a path i s kep t consta nt. B ut in our approa ch, pheromone depositi on is graduall y increased in t he links near the desti nation city. It i mplies t hat the lin ks lying closer to the destination city rec eive more phero mone compared to those near the starting ci ty. We divide the simulation strateg y in two different levels. In the first level, we run the two competitive algorithms on 10 different city distribu tions and estimate the ran ge of values o f parameters of the prop osed algorithm for which it performs best and outperforms its classical counterpart by largest e xtent. In following section, we tabulate results for only 3 different distributions o wing to space constrain t. 2008 IEE E Region 10 Colloq uium and the Third I CIIS, Kharagpur, INDIA Dece mber 8-10. Paper ID: 2 50 978-1-424 4-2806-9/08/$ 25.00© 2008 IEEE 4 A. Lev el I Results: Results for Environ ment I: Fig 4 : Roadmap for 120 City Distribution The first exper iment is co nducted w ith 120 cities. 20 ants are employed to move t hrough t he graph for 100 iterations. We vary bo th α and β over the range 0 .5 to 5.0 in steps of 0 .5 and best optimum path length was obtained for α=1.5, β=4.0. Length of t he best path found with above parameter setting almost m atches with the theor etical m inima as obtained by applying Dij kstra’s algorith m([16 ]). That path is m arked by bold black line in figure 4 . Table 1 p rovides the variation o f convergence time (number of iteratio ns required for o ptimum path length to co nverge) with the variat ion o f α a nd β. The convergence ti me with α=1.5 and β=4 .0 is near to optimum value (16) which signifies that for the abo ve roa dmap α=1. 5, β=4.0 is the opti mum pa rameter setting which not o nly produces optimum solution b ut also in fairly optimum num ber of iter ations. A 3D plot of op timum pat h length for varying α, β is provided in figure 5. Fig 5 : Variation of Opti mum Pathlength with α a nd β Table 1: Var iation of convergence time with α and β α 0 .5 1.0 1.5 2 .0 2.5 3.0 3 .5 4.0 4.5 5.0 0.5 37 40 4 3 44 46 47 52 60 55 58 1.0 35 38 4 4 47 49 52 53 57 61 64 1.5 38 32 3 5 38 40 44 52 53 55 60 2.0 34 31 3 2 31 38 40 37 44 47 52 2.5 32 28 2 6 32 35 38 32 46 43 55 3.0 29 30 2 3 29 40 35 31 41 44 50 3.5 28 23 2 0 26 28 32 38 47 50 53 4.0 25 16 1 9 22 29 31 40 43 47 55 4.5 29 23 2 5 26 32 37 32 41 53 59 5.0 32 34 3 1 37 40 43 44 46 47 52 In all simulatio ns ab ove, we ass ume T=1 0. T his value of T is guessed fro m the number of links r equired to move from source city to destination cit y which, for most op timal solutions, lies be tween 12 and 1 5. Therefor e, T=10 is a reasonable ap proximation as far a s the philosoph y of exponential dep osition rule is concerned . Still further tuning of T is necessar y and hence e xperiment with var ying value of T in t he neighbor hood of its es timated value is co nducted with the opti mal setting of para meters α, β i.e. α=1.5 and β=4 .0. The result is presented in table 2. Table 2: Var iation of convergence time with T T Convergence Time T Conv ergence Time 7.0 21 10.5 19 7.5 19 11.0 20 8.0 18 11.5 19 8.5 17 12.0 21 9.0 18 12.5 20 9.5 16 13.0 22 10.0 19 A comparative stud y o f t he two competitive algorith ms is carried out next with op timum p arameter settings. The plot depicts the super iority of the p ropo sed method in ter ms of both solution quality and convergence time. In fig ure (6), the red graph sho ws the iteratio n-best paths for expo nential dep osition rule and the blue graph sho ws the sa me for constant dep osition rule. T he line marked green shows t he theoretical mini mum path-length bet ween the source and destination cities. For simulating the constant depo sition algorith m α=1 and β=2 ( as suggested in [3]) are used. e is set to number of nod es p resent in case of T SP ([3]). The closed path fou nd by the ants includes all the cities in TSP. But in our prob lem, the opti mum paths found by the ants d o not consist o f more than 15 citi es. The parameter e , in o ur p roble m, is t herefore set at 15 for simulating both for ms of pheromone d eposition rule. Fig 6 : Comaparative Study o f algorith ms Results for Environ ment II: Environment II is slightly more complicated distributi on with 18 0 cities. Exper iments conducted led to the optimum parameter setti ng α=1.0, β=3.5 , T=11.0 . Fig 7: Roadmap for 180 City Distribution β 2008 IEE E Region 10 Colloq uium and the Third I CIIS, Kharagpur, INDIA Dece mber 8-10. Paper ID: 2 50 978-1-424 4-2806-9/08/$ 25.00© 2008 IEEE 5 Fig 8 : Variation of Opti mum Pathlength with α a nd β Table 3: Var iation of convergence time with α and β α β 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.5 51 48 57 58 58 62 68 76 66 73 1.0 48 52 60 60 63 65 64 67 74 73 1.5 52 43 52 51 48 52 68 62 70 74 2.0 44 43 48 44 51 48 44 55 58 70 2.5 45 40 32 45 48 55 50 58 56 71 3.0 43 42 41 45 56 46 43 54 58 60 3.5 42 28 34 40 38 46 54 57 61 65 4.0 36 29 30 33 40 48 58 52 62 64 4.5 40 35 34 40 45 52 41 52 63 66 5.0 48 48 45 50 52 52 60 64 62 64 Table 4 : Va riation of co nvergence ti m e with T T Convergence Time T Convergence Time 7.0 33 10.5 29 7.5 30 11.0 25 8.0 31 11.5 26 8.5 32 12.0 28 9.0 30 12.5 29 9.5 29 13.0 28 10.0 28 Fig 9 : Comaparative Study o f algorith ms Results for Environ ment III: Figure 10 shows a roadmap of 240 city distribution, an extremely co mplicated grap h. Opti mum per formance is achieved at α=0.5 ,β=3.5 a nd T=12.0 . Fig 10: Ro admap for 24 0 City Distribution Fig 11 : Va riation of O ptimum Pathlength wit h α and β Table 5: Var iation of convergence time with α and β α β 0.5 1.0 1 .5 2.0 2. 5 3.0 3.5 4.0 4 .5 5.0 0.5 60 67 64 68 71 76 77 87 73 86 1.0 64 57 69 70 68 73 73 80 86 83 1.5 66 52 57 64 62 68 80 78 80 85 2.0 55 57 57 50 65 69 59 70 71 71 2.5 55 48 51 59 60 65 56 71 69 86 3.0 46 55 53 54 62 59 58 64 71 77 3.5 35 47 49 55 47 60 60 77 79 76 4.0 44 43 55 53 49 50 61 61 70 79 4.5 50 40 54 47 58 56 58 65 78 84 5.0 50 55 58 65 66 68 70 69 78 74 Table 6: Var iation of convergence time with T T Convergence Time T Convergence Time 7.0 37 10.5 35 7.5 35 11.0 34 8.0 36 11.5 34 8.5 38 12.0 32 9.0 37 12.5 35 9.5 35 13.0 34 10.0 35 Fig 12: Co maparative Study of algorit hms B. Level II Re sults: Experiments per formed abo ve reveal that the propo sed algorithm perfor ms b est for α l ying b etween 0.5 and 1.5 and β lying b etween 3 .5 and 4 .0, no m atter ho w co mplex the environment is. In secondar y level o f our simulation strategy, we vary α and β over the a bove mentioned range i n steps o f 0.1 and try to estimate their re lation with t wo features of problem environ ment: i) the node density and ii) standa rd deviation of le ngths o f smalle st arc associated with each node. We performed experiments on r oadmaps with 120,140 ,160,180,200,220 and 240 number of cities. Fo r each of above roadmaps, we c hose seven different distributions and recorded the values of α and β for best performance. Table Curve 3D V4.0 , a curve fitting tool, was then employed to fit 2008 IEE E Region 10 Colloq uium and the Third I CIIS, Kharagpur, INDIA Dece mber 8-10. Paper ID: 2 50 978-1-424 4-2806-9/08/$ 25.00© 2008 IEEE 6 a curve thro ugh 49 d ata points for e ach of α and β and obtain an algebraic relation between α or β and the features of problem environment. The results are displa yed in the following two figures (fig 13 and 14) along with th e approximated equation s estab lishing the relation b etween two sets of para meters . This exhaustive experime ntation allo ws determination of opti mum values of α and β when t he features of problem environ ment are kno wn in advance. Fig 13: Plot of α Function: Cosine Series Bivariate Order 4 [ x": x scaled 0 to π, y": y scaled 0 to π; x ≡ number of nodes in 200 sq unit area,y ≡ standard deviation] α=a+bcos(x")+ccos(y")+dcos(2x")+ecos(x")cos(y")+ fcos(2y")+ gcos(3x")+hcos(2x")cos(y")+icos(x")cos(2y") +jcos(3y")+ kcos(4x")+lcos(3x")cos(y")+mcos(2x")cos(2y") + ncos(x")cos(3y")+ocos(4y") Co-efficient values: a=0.935,b=-0.237,c=-0.020,d=-0.011,e=-0.028, f=0.028, g=-0.002, h=0.027, i=0.0006, j=-0.039 k=-0.006, l=0.056, m=-0.022, n=0.047, o=-0.020 Fig 14: Plot of β Function: Sigmoid Series Bivariate Order 4 [ x': x scaled -1 to +1, y': y scaled -1 to +1 S i=2..n (x')=-1+2/(1+exp(-(x'+1-(i-1)*(2/n))/0.12)),S 1 (x')= x'] β=a+bS 1 (x')+cS 1 (y')+dS 2 (x')+eS 1 (x')S 1 (y')+fS 2 (y')+ gS 3 (x')+ hS 2 (x')S 1 (y')+iS 1 (x')S 2 (y')+jS 3 (y')+kS 4 (x')+ lS 3 (x')S 1 (y') + mS 2 (x')S 2 (y')+nS 1 (x')S 3 (y')+oS 4 (y') Co-efficient values: a=3.742, b=0.323, c=0.422, d=-0.090, e=0.414 f=-0.124, g=-0.105, h=-0.12, i=-0.131, j=-0.111 k=0.019, l=-0.196, m=0.100, n=0.007, o=-0.139 VI. C ONCLUSIONS A ND S COPE OF F UTU RE W ORK The paper presents a novel appr oach o f stability analysi s as well as a new kind of pheromone deposition rule which outperfor ms t he traditional appr oach of pheromone deposition used so far i n all variants of a nt syste m algor ithms. Our future effort is focused in comparing the two kinds of depositi on approa ch with other models of ant system li ke M ax- M in Ant System ( MM AS ) and Ra nk-Based Ant S ystem a nd esti mate the opti mum para meter setting of p roposed deposition approa ch for these models. R EFERENCES [1] C. Blum and M. Dorigo, “Search bias in ant colony: On the role of competition balanced sy stems,” IEEE Transactions on Evolutionary Computation, vol. 9, no.2, pp. 159-174, 2 005. [2] Marco Dorigo, Vittorio Maniezzo and A lberto Colorni “The Ant Syste m: Optimization by a colony of coo perating agents” I EEE Transactions on Sy stems, Man, and Cybernetics–Part B, Vo l.26, No.1, 1996, pp.1-13 [3] M. Dorigo and C. 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