An Improved Shuffled Frog-Leaping Algorithm for Knapsack Problem

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1 A Improved Shuffled Frog-Leapig Algorithm for Kapsack Problem Zhoufag Li, Ya Zhou, ad Peg Cheg School of Iformatio Sciece ad Egieerig Hea Uiversity of Techology ZhegZhou, Chia Abstract. Shuffled frog-leapig algorithm (SFLA) has log bee cosidered as ew evolutioary algorithm of group evolutio, ad has a high computig performace ad excellet ability for global search. Kapsack problem is a typical NP-complete problem. For the discrete search space, this paper presets the improved SFLA (ISFLA), ad solves the kapsack problem by usig the algorithm. Experimetal results show the feasibility ad effectiveess of this method. Keywords: shuffled frog-leapig algorithm, kapsack problem, optimizatio problem. 1 Itroductio Kapsack problem (KP) is a very typical NP-hard problem i computer sciece, which was first proposed ad studied by Datzig i the 1950s. There are may algorithms for solvig the kapsack problem. Classical algorithms for KP are the brach ad boud method (AM), dyamic programmig method, etc. However, most of such algorithms are over-reliace o the features of problem itself, the computatioal volume of the algorithm icreasig by expoetially, ad the algorithm eeds more searchig time with the expasio of the problem. Itelliget optimizatio problem for solvig NP are the at coloy algorithm, greedy algorithm, etc. Such algorithms do ot deped o the characteristics of the problem itself, ad have strog global search ability. Related studies have show that it ca effectively improve the ability to search for the optimal solutio by combiig the itelliget optimizatio algorithm with the local heuristic searchig algorithm. Shuffled frog-leapig algorithm is a ew itelliget optimizatio algorithm, it combies the advatages of meme algorithm based o geetic evolutio ad particle swarm algorithm based o group behavior. It has the followig characteristics: simple i cocept, few parameters, the calculatio speed, global optimizatio ability, easy to implemet, etc. ad has bee effectively used i practical egieerig problems, such as resource allocatio, job shop process arragemets, travelig salesma problem, 0/1 kapsack problem, etc. However, the basic leapfrog algorithm is easy to bled ito local optimum, ad thus, this paper improved the shuffled frog-leapig algorithm to D. Ji ad S. Li (Eds.): Advaces i CSIE, Vol. 2, AISC 169, pp sprigerlik.com Spriger-Verlag erli Heidelberg 2012

2 300 Z. Li, Y. Zhou, ad P. Cheg solve combiatorial optimizatio problems such as kapsack problem. Experimetal results show that the algorithm is effective i solvig such problems. 2 The Mathematical Model of Kapsack Problem Kapsack problem is a NP-complete problem about combiatorial optimizatio, which is usually divided ito 0/1 kapsack problem, complete kapsack problem, multiple kapsack problem, mixed kapsack problem, the latter three kids ca be trasformed ito the first, therefore, the paper oly discussed the 0/1 kapsack problem. The mathematical model of 0/1 kapsack problem ca be described as: max xv i i i= 0 xw i i C (xi = 1 or 0, i= 1, 2,..., ) i= 0 (1) where: is the umber of objects; w i is the weight of the i-th object(i=1,2 ); v i is the value of the i-th object; x i is the choice status of the i-th object; whe the i-th object is selected ito kapsack, defiig variable x i =1,otherwise x i =0; C is the maximum capacity of kapsack. 3 The asic Shuffled Frog-Leapig Algorithm It geerates P frogs radomly, each frog represets a solutio of the problem, deoted by U i, which is see as the iitial populatio. The it calculates the fitess of all the frogs i the populatio, ad arrages the frog accordig to the descedig of fitess. The dividig the frogs of the etire populatio ito m sub-group of, each sub-group cotais frogs, so P =m*. Allocatio method: i accordace with the priciple of equal remaider. That is, by order of the scheduled, the 1, 2,..., frogs were assiged to the 1,2,..., N sub-groups separately, the +1 frog was assiged to the first sub-group, ad so o, util all the frogs were allocated. For each sub-group, settig U is the solutio havig the best fitess, U is the solutio havig the worst fitess, U g is the solutio havig the best fitess i the global groups. The, searchig accordig to the local depth withi each sub-group, ad updatig the local optimal solutio, updatig strategy is: S = { ) max} { } mi it( rad ( U U ), S, U U 0 max it( rad ( U U )), Smax, U U 0 (2)

3 A Improved Shuffled Frog-Leapig Algorithm for Kapsack Problem 301 U = U + S (3) q w where, S is the adjustmet vector of idividual frog, S max is the largest step size that is allowed to chage by the frog idividual. rad is a radom umber betwee 0 ad 1. 4 The Improved Shuffled Frog-Leapig Algorithm for KP A frog is o behalf of a solutio, which is expressed by the choice status vector of object, the frog U=( x 1, x 2,, x ), where, x i is the choice status of the i-th object; whe the i-th object is selected ito kapsack, defiig variable x i =1,otherwise x i =0; f (i), the fitess fuctio of idividual frog ca be defied as: f() i = xv (4) i= 1 i i 4.1 The Local Update Strategy of Frog The purpose of implemetig the local search i the frog sub-group is to search the local optimal solutio i differet search directios, after searchig ad iteratig a certai umber of iteratios, makig the local optimum i sub-group gradually ted to the global optimum idividual. Defiitio 1. Givig a frog s status vector U, the switchig sequece C(i,j) is defied: 1, whe i = j ad U i chaged Ci (, j) = 0, whe i j ad Ui = Uj 2, whe i j ad U i U j (5) where, U i said the state of object i becomes from the selected to the cacel state, or i tur; U i = U j, object i ad object j exchage places, that object i ad object j are selected or deselected at the same time. U i U j, object i is selected or caceled, or i tur. The the ew vector of switchig operatio is: U' = U + C( i, j) Defiitio 2. Selectig ay two vectors U i ad U j of frog from the group, D, the distace from U i to U j is all exchage sequeces that U i is adjusted to U j. { } DU (, U ) = Ci (, j) Ci (, j)... Ci (, j ) (6) i j m where, m is the umber of adjustig. m

4 302 Z. Li, Y. Zhou, ad P. Cheg ased o the above defiitio, the update strategy of the idividual frog is defied as follows: {, max} l = mi it[ rad D ], l (7) { (, )} s = D U U (8) q w U = U + S (9) where, l is the umber of switchig sequece D(U,U ) for updatig U ; l max is the maximum umber of switchig sequece allowed to be selected; S is the switchig sequece required for updatig U. 4.2 The Global Iformatio Exchage Strategy Durig the executio of the basic shuffled frog-leapig algorithm, the operatio of updatig the feasible solutio was is executed repeatedly, it is usually to meet the situatio that updatig fail, the basic shuffled frog-leapig algorithm updates the feasible solutio radomly, but the radom method ofte falls ito local optimum or reduces the rate of covergece of the algorithm. Obviously, the key that overcomig the shortcomigs of basic SFLA i evolutio is: it is ecessary to keep the impact of local ad global best iformatio o the frog jump, but also pay attetio to the exchage of iformatio betwee idividual frogs. I this paper, first two jumpig methods i basic SFLA are improved as follows: P= PX + r1*(pg Xp1 (t)) +r2*(p Xp2 (t)) (10) P = P b + r3*(p g Xp3 (t)) (11) where, Xp1(t), Xp2(t), Xp3(t) are ay three differet idividuals which are differet from X. Meawhile, we remove the sortig operatio accordig to the fitess value of frog idividual from basic SFLA, ad appropriately limit the third frog jump. Thus, we get a efficiet modified SFLA based o the improvemets of above. I the modified algorithm, the frog idividual i the subgroup geerates a ew idividual ( the first jump)by usig formula(10), if the ew idividual is better tha its paret etity the replacig the paret idividual. Otherwise re-geeratig a ew idividual (the frog jump agai) by usig (11). If better tha the paret, the replacig it. Or whe r 4 FS (the pre-vector, its compoets are 0.2 FSi 0.4), geeratig a ew idividual (the third frog jump) radomly ad replacig paret etity. The ew update strategy will ehace the diversity of populatio ad the search through of the worst idividual i the iterative process, which ca esure commuities evolvig cotiually, help improvig the covergece speed ad avoid fallig ito local optimum, ad the expect algorithm both ca coverge to the earby of optimal solutio quickly ad ca approximate accuracy, improved the performace of the shuffled frog-leapig algorithm.

5 A Improved Shuffled Frog-Leapig Algorithm for Kapsack Problem Simulatio Experimet Two classical 0/1 kapsack problem istaces were used i the paper, example 1 was take from the literature_[4], example 2 was take from the literature_[5]. The compariso algorithm used i the paper was brach ad boud method for 0/1 kapsack problem. Uder the same experimetal coditios, two istaces of simulatio experimets were coducted 20 times, the average statistical results were show i Table 1. Table 1. Compariso of two algorithms istace The set of solutios Capacity of value ISFLA AM Average ruig time (ISFLA/AM) / / / / / / Coclusio The shuffled frog-leapig algorithm is a kid of search algorithm with radom itelligece ad global search capability, this paper improved shuffled frog-leapig algorithm ad solved the 0/1 kapsack problem by usig the algorithm. Experimets show that the improved algorithm has better feasibility ad effectiveess i solvig 0/1 kapsack problem. Refereces 1. Eusuff, M.M., Lasey, K.E.: Optimizatio of water distributio etwork desig usig the shuffled frog leapig algorithm. ater Resource Plaig ad Maagemet 129(3), (2003) 2. Li, Y.-H., Zhou, J.-Z., Yag, J.-J.: A improved shuffled frog-leapig algorithm based o the selectio strategy of threshold. Computer Egieerig ad Applicatios 43(35), (2007) 3. Luo, X.-H., Yag, Y., Li, X.: Improved shuffled frog-leapig algorithm for TSP. Joural of Commuicatio 30(7), (2009) 4. He, Y.-C.: Greedy geetic algorithm ad its applicatio for KP. Computer Egieerig ad Desig 28(11), (2007) 5. u, Z.-H.: Algorithm Desig ad Aalysis, pp Higher Educatio Press, eijig (1993)