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1、Correspondence autuor（通讯作者：）: tel/fax XXX; e-mail: XXXAbstractShuffled frog-leaping algorithm （SFLA） has long been considered as new evolutionary algorithm of group evolution, and has a high computing performance and excellent ability for global search. Knapsack problem is a typical NP-complete prob

2、lem. For the discrete search space, this paper presents the improved SFLA, and solves the knapsack problem by using the algorithm. Experimental results show the feasibility and effectiveness of this method.Keywords: shuffled frog-leaping algorithm; knapsack problem; optimization problem0 Introductio

3、nKnapsack problem（KP） is a very typical NP-hard problem in computer science, which was first proposed and studied by Dantzing in the 1950s. There are many algorithms for solving the knapsack problem. Classical algorithms for KP are the branch and bound method （BABM）, dynamic programming method （分支界定

4、法和动态规划法）, etc. However, most of such algorithms are over-reliance on the features of problem itself, the computational volume of the algorithm increases by exponentially, and the algorithm needs more searching time with the expansion of the problem. Intelligent optimization problem for solving NP ar

5、e the ant colony algorithm, greedy algorithm, etc. Such algorithms do not depend on the characteristics of the problem itself, and have the strong global search ability. Related studies have shown that it can effectively improve the ability to search for the optimal solution by combining the intelli

6、gent optimization algorithm with the local heuristic searching algorithm. Shuffled frog-leaping algorithm is a new intelligent optimization algorithm, it combines the advantages of meme algorithm based on genetic evolution and particle swarm algorithm based on group behavior. It has the following ch

7、aracteristics: simple in concept, few parameters, the calculation speed, global optimization ability, easy to implement, etc. and has been effectively used in practical engineering problems, such as resource allocation, job shop process arrangements, traveling salesman problem, 0/1 knapsack problem,

8、 etc. However, the basic leapfrog algorithm is easy to blend into local optimum, and thus this paper improved the shuffled frog-leaping algorithm to solve combinatorial optimization problems such as knapsack problem. Experimental results show that the algorithm is effective in solving such problems.

9、1 The mathematical model of knapsack problemKnapsack problem is a NP-complete problem about combinatorial optimization, which is usually divided into 0/1 knapsack problem, complete knapsack problem, multiple knapsack problem, mixed knapsack problem, the latter three kinds can be transformed into the

10、 first, therefore, the paper only discussed the 0/1 knapsack problem. The mathematical model of 0/1 knapsack problem can be described as:where: n is the number of objects; wi is the weight of the ith object（I = 1, 2n）; vi is the value of the ith object;xi is the choice status of the ith object; when

11、 the ith object is selected into knapsack, defining variable xi = 1,otherwise xi = 0; C is the maximum capacity of knapsack.2 The basic shuffled frog-leaping algorithmIt generates P frogs randomly, each frog represents a solution of the problem, denoted by Ui, which is seen as the initial population

12、. Calculating the fitness of all the frogs in the population, and arranging the frog according to the descending of fitness. Then dividing the frogs of the entire population into m sub-group of, each sub-group contains n frogs, so P =m*n. Allocation method: in accordance with the principle of equal

13、remainder. That is, by order of the scheduled, the 1, 2, ., n frogs were assigned to the 1,2, ., N sub-groups separately, the n+1 frog was assigned to the first sub-group, and so on, until all the frogs were allocated. For each sub-group, setting UB is the solution having the best fitness, UW is the

14、 solution having the worst fitness, Ug is the solution having the best fitness in the global groups. Then, searching according to the local depth within each sub-group, and updating the local optimal solution, updating strategy is:where, S is the adjustment vector of individual frog, Smax is the lar

15、gest step size that is allowed to change by the frog individual. Rand is a random number between 0 and 1. 3 The improved shuffled frog-leaping algorithm for KPA frog is on behalf of a solution, which is expressed by the choice status vector of object, then frog U = （ x1, x2, , xn ）, where, xi is the

16、 choice status of the i-th object; when the i-th object is selected into knapsack, defining variable xi = 1,otherwise xi = 0; f （i）, the fitness function of individual frog can be defined as:3.1 The local update strategy of frogThe purpose of implementing the local search in the frog sub-group is to

17、 search the local optimal solution in different search directions, after searching and iterating a certain number of iterations, making the local optimum in sub-group gradually tend to the global optimum individual. Definition 1 Giving a frogs status vector U, the switching sequence C（i,j） is define

18、d:where, Ui said the state of object i becomes from the selected to the cancel state, or in turn; Ui= Uj, object i and object j exchange places, that object i and object j are selected or deselected at the same time. UiUj, object i is selected or canceled, or in turn. Then the new vector of switchin

19、g operation is:Definition 2 Selecting any two vectors Ui and Uj of frog from the group, D, the distance from Ui to Uj is all exchange sequences that Ui is adjusted to Uj. where, m is the number of adjusting.Based on the above definition, the update strategy of the individual frog is defined as follo

20、ws:where, l is the number of switching sequence D（UB,UW） for updating UW; lmax is the maximum number of switching sequence allowed to be selected; s is the switching sequence required for updating UW.3.2 The global information exchange strategyDuring the execution of the basic shuffled frog-leaping

21、algorithm, the operation of updating the feasible solution was is executed repeatedly, it is usually to meet the situation that updating fail, the basic shuffled frog-leaping algorithm updates the feasible solution randomly, but the random method often falls into local optimum or reduces the rate of

22、 convergence of the algorithm. Obviously, the key that overcoming the shortcomings of basic SFLA in evolution is: it is necessary to keep the impact of local and global best information on the frog jump, but also pay attention to the exchange of information between individual frogs. In this paper, f

23、irst two jumping methods in basic SFLA are improved as follows:Pn= PX + r1*（PgXp1 （t） +r2*（PWXp2 （t） （ 5）Pn= Pb + r3*（PgXp3 （t） （ 6）Where, Xp1（t）,Xp2（t）,Xp3（t） are any three different individuals which are different from X. Meanwhile, removing the sorting operation according to the fitness value of

24、frog individual from basic SFLA, and appropriately limiting the third frog jump. Thus, we get an efficient modified SFLA basing on the improvements of above. In the modified algorithm, the frog individual in the subgroup generates a new individual （ the first jump）by using formula （5）,if the new ind

25、ividual is better than its parent entity then replacing the parent individual. otherwise re-generating a new individual （the frog jump again）by using （6）.If better than the parent ,then replacing it. or when r4 FS （the pre-vector, its components are 0.2 FSi 0.4）,generating a new individual （the thir

26、d frog jump ） randomly and replacing parent entity.The new update strategy will enhance the diversity of population and the search through of the worst individual in the iterative process, which can ensure communities evolving continually, help improving the convergence speed and avoid falling into

27、local optimum, and then expect algorithm both can converge to the nearby of optimal solution quickly and can approximate accuracy, improved the performance of the shuffled frog-leaping algorithm.4 Simulation experimentTwo classical 0/1 knapsack problem instances were used in the paper, example 1 was

28、 taken from the literature 11, example 2 was taken from the literature 12. The comparison algorithm used in the paper was branch and bound method for 0/1 knapsack problem. Under the same experimental conditions, two instances of simulation experiments were conducted 20 times, the average statistical

29、 results were shown in Table 1 and Table 2.5 ConclusionThe shuffled frog-leaping algorithm is a kind of search algorithm with random intelligence and global search capability, this paper improved shuffled frog-leaping algorithm and solved the 0/1 knapsack problem by using the algorithm. Experiments

30、show that the improved algorithm has better feasibility and effectiveness in solving 0/1 knapsack problem.AcknowledgementsThis work was supported by XXX（基金号）. Our special thanks are due to Prof. XXX （name）, XXX （affiliation）, for his helpful discussion with preparing the manuscript.References:7 Eusu

31、ff MM, Lansey KE. Optimization of water distribution network design using the shuffled frog leaping algorithmJ. Water Resource Planning and Management, 2003, 129（3）: 2102258 Ying-hai LI, Jian-zhong ZHOU, Jun-jie YANG. An improved shuffled frog-leaping algorithm based on the selection strategy of thr

32、esholdJ. Computer Engineering and Applications, 2007, 43（35）: 19219 Xue-hui LUO, YANG Ye, LI Xia. Improved shuffled frog-leaping algorithm for TSPJ. Journal of Communication, 2009, 30（7）: 13013510 Zong-yi XUAN, Cui-jun ZHANG. Solving the KP based on shuffled frog-leaping algorithmJ. Science Technology and Engineering,2009,9（15）: 43634