An Improved Ant Colony Algorithm for Continuous Optimization Based on Levy Flight

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1 487 A publicatio of CHEMICAL ENGINEERING TRANSACTIONS VOL. 51, 2016 Guest Editors: Tichu Wag, Hogyag Zhag, Lei Tia Copyright 2016, AIDIC Servizi S.r.l., ISBN ; ISSN The Italia Associatio of Chemical Egieerig Olie at DOI: /CET A Improved At Coloy Algorithm for Cotiuous Optimizatio Based o Levy Flight Hogzhag Wag*, Coggag Liag School of Mathematics ad Iformatio Sciece, Pigdigsha Uiversity, Pigdigsha, Hea , Chia @126.com At coloy optimizatio (ACO) is proposed o the study of the foragig behavior of ats o the basis of the proposed ad widely used i the optimizatio. However, it has some shortcomig such as loger time, hardly implemet ad local optimal etc. For overcomig the above shortcomig, combied with the characteristic of Levy flight, based o Levy flight at coloy optimizatio is proposed which used Levy flight istead of local search for improvig the searchig efficiecy. I order to test the performace of the ew algorithm, we apply it to 20 bechmark fuctio test ad compare it with GA, PSO, ACO ad LFACO algorithms. The compariso result shows that LFACO is far better tha the other three algorithms i quality. 1. Itroductio With the developmet of computig techology, it is possible to make radom search i the solutio space. Radom search accordig to the differet search methods ca be divided ito radom search ad blid radom search. Radom search method is a feasible solutio to complete radom search i the solutio space, ad the method is very efficiet whe the scale of the problem is large. A guided radom search rule is accordig to certai strategy i solutio space to focus o the implemetatio of radom search, ad accumulated experiece i the process of radom search, makes the search is itelliget, commoly used itelliget methods such as geetic algorithm (Goldberg, 2002), particle swarm optimizatio (Keedy ad Eberhard, 1995). Biologists foud that by a lot of research, at coloy foragig i the process is the reaso why to fid the shortest path betwee a food source ad their est is due to at idividuals ca o the way after the leave a special substace pheromoes ad become other ats searchig for food or est of clues. Subsequet ats ecouter pheromoe, ot oly ca detect the cotet of pheromoe, but also accordig to the cocetratio of pheromoe to determie the directio of its progress. Iformatio over time will gradually evaporate, so the legth of the path ad the umber of residues i the umber of ats o the choice of a greater impact. I a certai period of time the path by more ats, ats select the path of the greater the probability; at chooses the path, the correspodig path iformatio hormoe cocetratio stregtheed, to promote more ats select the path. Therefore, a large umber of ats through this simple iformatio exchage, to achieve a positive feedback of the iformatio learig mechaism, ad ca quickly fid the shortest path from the food source to the est. A typical represetative of the biological coloy itelliget algorithm is the at coloy algorithm. At Coloy Optimizatio (ACO) is proposed o the study of the foragig behaviour of ats o the basis of the proposed (Dorigo ad Caro, 1999). After years of developmet, the model costructio algorithm relatively mature ad perfect, with the characteristics of strog robustess positive feedback, distributed computig etal, has bee successfully applied to discrete combiatorial optimizatio field, become oe of the most effective meas of optimizatio. However, ACO also has the followig shortcomigs: (1) The algorithm oly through the guidace of pheromoe search optimizatio, search for a loger time; (2) Whe solvig the practical problem by at coloy algorithm, it is ecessary to describe the problem firstly, ad the algorithm is ot powerful eough to describe the complex problem; (3) I the process of searchig, the algorithm is proe to search stagatio pheomeo; Please cite this article as: Wag H.Z., Liag C.G., 2016, A improved at coloy algorithm for cotiuous optimizatio based o levy flight, Chemical Egieerig Trasactios, 51, DOI: /CET

488 I fact, it is ot etirely radom, but i the ature, that the idividual's flight or foragig methods are ot etirely radom, but it obeys the Levy flight (Yag ad Deb, 2009).

2 488 I fact, it is ot etirely radom, but i the ature, that the idividual's flight or foragig methods are ot etirely radom, but it obeys the Levy flight (Yag ad Deb, 2009). The so-called Levy flight is a step size obeys Levy distributio radom walk. Specifically, idividuals geerally oly i a small area of flight or foragig, but there are a small fractio of a idividual will suddely fly to the distat place. This behavior is very coducive to the search process ad widely used i the CS algorithm (Yag ad Deb, 2010) ad CSB algorithm (Yi ad Liu 2015). I order to improve the search time ad search stagatio, this paper proposes a at coloy algorithm based o Levy flight. I order to Levy flight applicatios i more practical problems, this paper for cotiuous space optimizatio of fuctio optimizatio problems, combied with the ACO algorithm, we proposed a based o Levy flight at coloy optimizatio algorithm (LFACO). This algorithm use Levy flight to perform local search algorithm ad ulike ACO algorithm ad radom search, so local search ability of the algorithm has bee greatly improved. I order to test the performace of the ew algorithm, we apply it to 20 bechmark fuctio test ad compare it with GA, PSO, ACO ad LFACO algorithms. 2. The descriptio of LFACO 2.1 ACO algorithm Assume that solvig the problem of the scale is, the total umber of at coloy is m, the path (I, J) i time t amout of iformatio cocetratio is τ ij (t). A at of K i the process of movig path selectio will accord the iformatio o each path i the pigmet cocetratio ad path of the heuristic iformatio such as factors to calculate the state trasitio probability by which the at selects the path. Here we assume that p ij k (t) represets the probability that i the time t at K is trasferred from the ode I to the ode J, ad computig the probability calculatio is: p ij k (t) = { [τ ij (t)] α [μ ij (t)] β s allowed [τ is (t)] α [μ is (t)] β k 0, otherwise, if j allowed k I the above equatio, allowed k represets a collectio of the at K step allowig the selectio of ode. α is the pheromoe heuristic factor, β is expected heuristic factor, μ ij (t) is the heuristic fuctio. I order to avoid earlier iformatio residues too much ad cause residue iformatio flooded the heuristic iformatio of the pheomeo, i each at fiish or complete traversal of all cities, to update operatio treatmet of residual iformatio. So i the t+1 time the pheromoe cocetratio of the path (I, J) ca be adjusted ad updated as the followig rules: τ ij (t + 1) = (1 ρ) τ ij (t) + τ ij (t) (2) τ ij (t) = m k=1 (t) (3) τ ij k I the above equatios, ρ is the pheromoe evaporatio coefficiet, ad for prevetig ulimited iformatio accumulatio, the rage of P is [0, 1). τ ij (t) is said as the path (I, J) pheromoe icremet, τ ij k (t) is said the iformatio cocetratio the kth at after path (I, J). So the pseudo code of ACO is as Figure 1. (1) Figure 1: The pseudo code of ACO 2.2 Levy Flight Aimals foragig path was cosiderig as a radom or quasi-radom maer i ature. However, various studies have show that the flight behavior of may aimals ad isects obeys the typical characteristics of Levy flights. Broadly speakig, Levy flights are a radom walk whose step legth is draw from the Levy distributio, ofte i terms of a simple power-law formula L(s)~ s ^(-1-β) where 0<β<2. Obviously, the geeratio of step sizes samples is ot trivial usig Levy flights. A simple scheme discussed i detail ca be summarized as followig (Yag, 2010):

3 489 L(s)~ u 1 v β (4) I which u N(0, σ u 2 ), v N(0, δ v ) are ormal distributio. 2.3 The descriptio of LFACO For improvig the performace of ACO, a possible method is to chage the pheromoe volatilizatio coefficiet ρ usig Levy distributio while ρ is a costat i ACO, so we propose a ew improved ACO algorithm based o Levy Flight (LFACO) i which each movemet of each at idividual obey the levy distributio. I LFACO, the equatio (4) is itroduced ito the equatio (2) ad gets the followig equatio for updatig the locatio of each at idividual: τ ij (t + 1) = (1 ρ i,j ) τ ij (t) + τ ij (t) (5) I the eq(5), ρ i,j levy(β), levy(β) T(1+β)si (βπ/2) T( 1+β 2 )β2(β 1)/2 u 1 v β ( τ ij (t + 1) τ ij (t)) ad u N(0, σ u 2 ), v N(0,1), σ u = 1/β.Combied the above aalysis, the step of LFACO is followig as figure 2. Begi Iitializig the oopuatio Costructig the matrix of iformatio Performig the movig operator for each idividual Performig the Levy flight as local search for each idividual Selectig the best idividual accordig the fitess N Updatig the matrix of iformatio stop Y Ed Figure 2: The descriptio of LFACO. Fuctio Optimizatio Fuctio Optimizatio is ofte expressed i the followig form: mi f(x) subject to L X U (6) I which X = {x 1, x 2, x 3,, x } is a vector, ad L = {l 1, l 2, l 3,, l } is the lower boud of X, while U = {u 1, u 2, u 3,, u } is the upper boud of X. I this work, we test ad verify the effectiveess of the proposed LFQPSO algorithm i this chapter by usig the followig 15 bechmark fuctios of which its defiitio, domais ad optimal value are respectively defied as: (1)The defiitio of f 1 is followig: f 1 (x) = 2 i=1 x i (7) Its domais is 5.12 x i 5.12, i = 1,2,3,, (2)The defiitio of f 2 is followig: f 2 (x) = i=1 (ix 2 i ) (8)

4 490 Its domais is 5.12 x i 5.12, i = 1,2,3,, (3)The defiitio of f 3 is followig: f 3 (x) = 1 [100(x 2 i x i+1 ) 2 i=1 + (x i 1) 2 ] (9) Its domais is x i 2.048, i = 1,2,3,, Its argumet ad optimal value are x = (1,1,1,,1), f(x ) = 0 (4)The defiitio of f 4 is followig: f 4 (x) = 10 + i=1 (x 2 i 10cos (2πx i )) (10) Its domais is 5.12 x i 5.12, i = 1,2,3,, (5)The defiitio of f 5 is followig: x i 2 f 5 (x) = i=1 cos ( x i i=1 + 1 (11) 4000 i ) Its domais is 1 x i 1, i = 1,2,3,, (6)The defiitio of f 6 is followig: f 6 (x) = i=1 x i i+1 (12) Its domais is x i , i = 1,2,3,, (7)The defiitio of f 7 is followig: f 7 (x) = 20 + e 20e i=1 x i 2 Its domais is x i , i = 1,2,3,, (8)The defiitio of f 8 is followig: e 1 i=1 cos (2πx i) (13) f 8 (x) = max( x i ), i = 1,2,3,, (14) Its domais is 100 x i 100, i = 1,2,3,, (9)The defiitio of f 9 is followig: f 9 (x) = max( x i ), i = 1,2,3,, (15) Its domais is 100 x i 100, i = 1,2,3,, (10)The defiitio of f 10 is followig: f 10 (x) = 2 i=1 x i + ( i=1 0.5ix i ) 2 + ( i=1 0.5ix i ) 4 (16) Its domais is 5 x i 10, i = 1,2,3,, (11)The defiitio of f 11 is followig: f 11 (x) = si 2 (πy 1 ) + 1 i=1 [((y i 1) 2 (1 + 10si 2 (πy i + 1)))] + (y 1) 2 (1 + si 2 (2πy )), y i = 1 + x i 1, i = 4 1,2,3,, (17) Its domais is 10 x i 10, i = 1,2,3,, Its argumet ad optimal value are x = (1,1,1,,1), f(x ) = 0 (12)The defiitio of f 12 is followig: f 12 (x) = 2 ( x j i=1 i j=1 ) (18) Its domais is x i , i = 1,2,3,, (13)The defiitio of f 13 is followig:

5 491 f 13 (x) 2 2 = 1 cos (2π i=1 x i ) i=1 x i (19) Its domais is 100 x i 100, i = 1,2,3,, (14)The defiitio of f 14 is followig: f 14 (x) = i=1 (si(x i )) i=1(exp ( x 2 i )) (20) Its domais is 10 x i 10, i = 1,2,3,, (15)The defiitio of f 15 is followig: f 15 (x) = i=1(x i 1) 2 x i x i 1 i=2 + (+4)( 1) 6 Its domais is 2 x i 2, i = 1,2,3,, Its argumet ad optimal value are x = i( i + 1), i = 12,3,,, f(x ) = 0 (21) 4. Experimet Result I order to compare the results of LFACO algorithm, GA, PSO, ACO ad LFACO algorithm use the R programmig laguage implemetatio, ad all the program results i 3.3GHz Core Duo processor, 4GB of ram ad widows 7 operatig system of PC operatio. For the 15 bechmark fuctios, the four algorithms are ru 100 times respectively, ad the average experimetal results are show i Table 1 ad table 2. As Table 1 show, for all 20 bechmark test fuctios, the covergece speed of PSO is the best, ad of LFACO is better tha that of GA ad ACO, which shows that LFACO ca ideed faster tha ACO i optimizig the optimal searchig process. Next we focus o aalysis the result of LFACO ad the rest of the three algorithms. Table 2 show the result is the best LFACO optimizatio, ad although the executio time of the PSO algorithm best as Table 1 show, but the aalysis table 2, we ca see that PSO result is the worst i four algorithms, which shows the o a algorithm i which some idexes ca reflect the best. Table 1: Rutime of GA, PSO, ACO ad LFACO o the 15 bechmark fuctios 5. Coclusio Fuctio GA PSO ACO LFACO f f f f f f f f f f f f f f f I order to improve performace ACO algorithm i fuctio optimizatio, i this paper the Levy flight itroduced to ACO algorithm ad get the ew update of the idividual equatios. Based o the above discuss, we propose the ew algorithm of based o Levy Flight of the ACO algorithm (LFACO) ad detailed describe its steps ad processes. I order to verify the LFACO algorithm, 15 bechmark fuctios are used to test the

6 492 LFACO which is compared with GA, PSO algorithm ad ACO algorithm. The compariso results show that for the fuctio optimizatio problem LFACO i effectiveess as well as ru time is far better tha other three algorithms. Table 2: the average ad stadard variatio of GA, PSO, ACO ad LFACO o the 15 bechmark fuctios Fuctio GA PSO ACO LFACO f ± ± ± ±0.00 f ± ± ± ±0.00 f ± ± ± ±2.97 f ± ± ± ±1.48 f ± ± ± ±0.00 f ± ± ± ±0.00 f ± ± ± ±0.03 f ± ± ± ±1.11 f ± ± ± ±0.00 f ± ± ± ±0.60 f ± ± ± ±0.00 f ± ± ± ±0.00 f ± ± ± ±0.32 f ± ± ± ±0.00 f ± ± ± ±33.85 Ackowledgmet Fud Project: 2011 Hea provicial sciece ad Techology Departmet of sciece ad techology foudatio project; Exact solutios of higher order oliear partial differetial equatio; Project approval umber: Referece Dorigo M., Caro G.D., 1999, At coloy optimizatio: A ew meta-heuristic, Proceedigs of the 1999 Cogress o Evolutioary Computatio, 2: Goldberg D.E., 2002, the Desig of Iovatio: Lessos from ad for Competet Geetic Algorithms. Norwell, MA: Kluwer Academic Publishers. ISBN Keedy J., Eberhart R., 1995, Particle swarm optimizatio, IEEE Iteratioal Coferece o Neural Networks, 4, , DOI: /ICNN Yag X.S., Deb S., 2009, Cuckoo Search via lévy flights. Proceedig of world cogress o Nature & Biologically Ispired Computig, Idia, Yag X.S., Deb S., 2010, Egieerig optimizatio by Cukcoo Search, It. J. Mathematical Modelig ad Numerical Optimizatio, 1(4), Yi L., Liu Y.G., 2015, Biclusterig of the gee expressio data by coevolutio cuckoo search. Iteratioal Joural Bioautomatio, 19(2):

Study on Coal Consumption Curve Fitting of the Thermal Power Based on Genetic Algorithm

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