1.Introduction

Artificial bee colony(ABC)is a swarm intelligence optimization algorithm,which was developed by Karaboga[1]in 2005 based on the foraging behavior of honeybees.The ABC algorithm has become a research hot spot for the characteristics of simple structure,few control parameters and easy to implement.Numerical comparisons have confirmed that the performance of ABC is superior to the classical evolutionary algorithms[2,3].Accordingly,ABC has been applied to many applications,such as the numerical optimization problem[4],the scheduling problem[5],and the multi-objective optimization problem[6].

国家市场监督管理总局执法稽查局副局长路政闽在致辞中指出，中国酒业市场巨大，前景广阔，中国的烈酒消费世界第一，啤酒产销量世界第一，葡萄酒消费量接近世界第一。2018年上半年，全国规模以上酒类生产企业完成饮料酒和酒精总产量3111.2万千升，同比增长1.9%，其中完成酒类销售收入4467.1亿元，同比增长12.2%；饮料酒利润总额增速明显，全国规模以上饮料酒生产企业实现利润总额756.1亿元，同比增长30.9%，增速较上年同期加快12.6个百分点。

In recent years,the improvement of ABC mostly focused on the population structure strategy,the selection strategy and the search strategy.Banharnsakun et al.[7]proposed a modified search equation based on the best-sofar solution.Gao and Liu[8]were inspired by differential evolution(DE)to put forward a new solution search equation and the chaotic initialization strategy.Although the above two ABC variants indeed enhance the exploitationability of ABC, they cannot avoid the algorithm falling into local optimum.Then Yan et al.[9]introduced the crossover operator to ABC,which improved the information exchange ability among artificial bees.Liu et al.[10]presented the improved ABC based on local search.Zhou et al.[11]put forward the neighborhood search mechanism to improve the solution search equation of ABC.They argued that the local search ability of ABC can be enhanced.However,the global search efficiency of the ABC algorithm has not been fundamentally improved.

Therefore,improving the local search ability and the global search efficiency have become the two most significant objectives in ABC study.In order to realize these two objectives,we propose the hybrid ABC(HABC)algorithm which includes two strategies.First,the variable neighborhood search factor is added to the solution search equation,which can enhance the local search ability and increase the population diversity.Then,a memory mechanism is put forward,which imitates the natural behavior of honeybees to fundamentally improve the global search efficiency.

The organization of this paper is as follows.Section 2 briefly introduces the standard ABC algorithm.Section 3 presents the HABC algorithm in details.Section 4 validates the superiority of the HABC algorithm in the numerical experiments. Finally, Section 5 makes the conclusion of this paper.

2.Standard ABC algorithm

The standard ABC algorithm divides the artificial bees into three groups,including employed bees,onlooker bees and scout bees.These three groups can play different roles and spontaneously perform role conversion.The employed bees search for the food source and share the information of the food source to onlooker bees through performing dances.Then onlooker bees select a better food source to exploit in a probability,which accords to the qualities of the food sources.Sometimes,employed bees may convert to scout bees,when the food source is abandoned and they need to search for new ones.

（6）高职院校学生对足球运动的热爱程度很高，参与足球活动的积极性也很高，大多数高职院校学生热衷于小型足球活动。

Like the other swarm intelligence optimization algorithms,the standard ABC is an iterative algorithm.The description of standard ABC is described in the following subsections.

2.1 Initialization phase

By introducing the variable neighborhood search factor and the memory mechanism,the HABC algorithm includes the following three advantages.

where i=1,2,...,SN and j=1,2,...,D.SN is the number of food sources,D is the dimension of the food source,are the lower and upper bounds of the dimension j.

2.2 Employed bees phase

In this phase,each employed bee Xi finds a better food sourcein the neighborhood by using

where k=1,2,...,SN is a randomly chosen index and different from i,is a random number in[-1,1].

After producing Vi,it will be compared to Xiwith the greedy selection mechanism.fitidenotes the fitness value of food source i,which is calculated by

2.2.1 气候时间因素。分析以年份、冬夏半年（夏半年3—9月，冬半年9月—次年2月）、月份、节气、工作日对发病例数进行分析，观察天数合计1 461天，结果（见图1～5），结果表明在以年、节气为条件分析病例数，不同的年份、节气差异有统计学意义（年份：P=0.01，节气：P=0.028），发病人数随年份增加，立春、春分、夏至、小寒等节气较其他节气日发病数增多。冬夏半年、月份、工作日差异无统计学意义（月份：P=0.174，冬夏半年：P=0.050，工作日：P=0.204）。

where fiis the objective function value of solution i.

企业若想保证其经济管理的有效性，对于财务会计的工作应给予高度重视，这样才能最大限度的发挥财务会计的作用。而发挥财务会计的作用应该明确企业财务管理的核心实质和目标，然后结合经济信息通过系统的分析后做出系列的计划方案。另外财务会计工作不仅是财务部门的工作，其对于生产流程也有一定的涉及，因此需要将财务会计工作普及化，而这可从以下几个方面进行：一是领导对于企业的资金流动要有全面的了解和认识，以便做好管理工作；二是销售部门在进行销售工作开展时，对于产量和资金流动要很好的掌控；三是市场部门要及时了解市场经济活动的变化，借此给财务部门提供有利的市场经济变动信息以便更好的做好财务会计工作。

2.3 Onlooker bees phase

When employed bees select the food source and return to the hive,they will share the information of the food source to onlooker bees through performing dances.Then onlooker bees select the food source depending on this information in a probability.The probability value piis calculated by

After the food sources have been selected by onlooker bees based on the probability value,each onlooker bee will search a new food source in its neighboring area following(2).Obviously,this searching procedure is similar to the employed bees.

2.4 Scout bees phase

In the scout bees phase,a food source Xiis given a control parameter,which records the number of failed trials and denotes as triali.If a food source cannot be improved within triali,Xiwill be abandoned and the employed bees convert to scout bees.Meanwhile,a new food source will be randomly generated by(1).

The standard ABC algorithm utilizes employed bees,onlooker bees and scout bees to iteratively search the solution space until reach the maximum number of iterations,which is denoted as Gmax.

2.5 Standard ABC algorithm description

Based on the above description,the pseudo-code of the standard ABC is outlined in Algorithm 1.

3.HABC algorithm

In the ABC standard algorithm,the new food source is randomly selected in the neighboring area according to(2)for each employed bee or onlooker bee searches.It is due to such a random strategy of the search equation that the ABC algorithm is poor at local search ability and its global search efficiency is low. In order to solve these problems,the HABC algorithm is proposed which includes two strategies.First,the variable neighborhood search(VNS)factor,which systematicly changes the neighborhood structure to enhance the local search ability and increase the population diversity,is added to(2).Second,a memory mechanism is put forward.In this memory mechanism,the artificial bees are allowed to simulate the memory ability of real honeybees,which can remember their past successful experiences and improve the global search efficiency.The details of the HABC algorithm is described in the following subsections.

3.1 VNS factor

VNS is a heuristic algorithm,which was developed by Mladenovic’in 1997[20].The core idea of VNS is to define the set of neighborhood structures Nl(x)(l=1,2,...,lmax)that can be used in a systematic way to search the solution space and help the search process escape from local optimum.The core idea of VNS is shown in Fig.1.

Fig.1 Core idea of VNS

Ten benchmark functions with different features are considered to test the performance of the HABC algorithm which have been widely discussed in the recent ABC literature[3,4,20].The different features of these benchmark functions include unimodal-seperable(US), unimodal-nonseperable (UN),multimodel-seperable(MS)and multimodel-nonseperable(MN).The definition of benchmark functions are shown in Table 2 which includes its formula,feature,interval,optimal solution and dimensions D.

where l is the number of neighborhood structures,ρlis the radius of Nl(x)monotonically increasing with l.

汽油的火苗迅速蔓延到脚下，我听见噼啪作响的燃爆声。我的身体也开始燃烧起来，我感受到一种从未有过的畅快淋漓，因为我凭借火，凭借风，终于可以自由地飞回故乡去了。范峥峥也可以和我一起，魂归故里了。

The solution search equation which adds the variable neighborhood search factor is described as follows:

where lbest(i)is the optimal food source in the neighborhood of the Xiwhich obtained by the VNS factor.k is a randomly selected neighboring food source,where k should be different from i.is a random value ranging in[-1,1].

The pseudo-code of the VNS factor is described in Algorithm 2

3.2Memory mechanism

The neuroscience has confirmed that the real honeybees have the ability to remember their foraging behavior.As long as we effectively use the memory ability,we can use the past successful experiences to further guide the foraging behavior of employed bees and onlooker bees.However,this memory ability is not taking into account in the standard ABC algorithm as well as its variants.Each time,the choice of new food source is through random search,which greatly limits the search efficiency.Therefore,a memory mechanism is put forward,which assumes the employed bees and onlooker bees can remember their past successful experiences of foraging behavior.Furthermore,these memorized experiences need to be constantly updated and used to guide the subsequent foraging behavior.Obviously,the memory mechanism can fundamentally improve the global search efficiency through imitating the natural behavior of honeybees.

In the HABC algorithm,the memory of food source is denoted by matrix M.

Fig.2 depicts the convergence curves of these three algorithms.Compared with ABC and HABC,qABC has the faster convergence speed in the early generations in five functions(f1,f4,f6,f9,f10).However,it converges to worse results than HABC as the number of iterations increases.

Initially,M is set to empty.With the search process of HABC,is constantly updated according to(6).For each ,the memorized information includes the following three parts according to:

In order to make a fair comparison,the parameter settings in this paper are the same as that in[4].In detail,the population size SN is set to 25,the maximum number of iterations Gmaxis 10 000 and the maximum evaluation number is 500 000.The parameter limit is calculated with(7).

(ii)Random neighborhood food source

(iii)Random coefficient

还有积薪师父！他的棋室也是星雨喜欢待的地方。星雨他们刚来万花谷的时候，春雨绵绵，乍暖还寒，王积薪听子虚道人乌有先生两个老家伙回来讲，说有四个孩子误打误撞，破了媪妇谱，忙亲自跑到弘道部来，见到袁安、李离、上官星雨三人，又是作揖又是打拱，一定要东方宇轩作证，由他来拜三个少年做老师：“破解媪妇谱是我一生的志业，现在大功告成，你们三个就是我的恩人，小师父在上受我一拜！”

In general,if the values ofandcan improve the quality of solution,we consider them as the successful experience and memorize these values.The structure of memory mechanism is illustrated in Table 1.

Table 1 Structure of memory mechanism

Update index Updating coefficient 1 xjOptimal neighborhood Random neighborhood lbest(i)(1) kj i(1) φj i(1)2 xj lbest(i)(2) kj i(2) φj i(2)............M xj lbest(i)(M) kj i(M) φj i(M)

3.3 HABC algorithm description

The pseudo-code of the HABC is presented in Algorithm 3.

3.4 Advantages analysis of HABC algorithm

In the initialization phase,the standard ABC algorithm randomly generates an initial population of food sources.Let represent the ith food source,and then the position of food sources are produced as

(i)The new solution search equation, that is (6), with thevariable neighborhood search factor has strong local search ability,which can avoid the search process falling into local optimum and make the algorithm fast convergent to the optimal food source.

(ii)The search trajectory of the population is guided by the individual optimal within the neighborhood,and a new food source is generated near the optimal position.This search method based on the elite strategy can effectively improve the exploitation ability and increase the population diversity.

我是一个重感情的人，虽然在这边工作很顺心，但我时常会想念冯阿姨，想起她和蔼的面容，想起她关切的话语，想起我们在一起的点点滴滴。终于我拿起手机拨通了她的电话，原想有好多话跟她说，但当电话那头传来：“小刘，你在那边干得还好吗？”我反倒一句话也说不出来了，眼泪顿时模糊了我的眼睛，有些东西当我们真正拥有的时候也许并不能感觉到它存在的重要性，不懂得珍惜，直到离开了才会去怀念曾经的生活。

(iii)The memory mechanism can fundamentally improve the global search efficiency through imitating the natural behavior of honeybees,which remember their past successful experiences and further guide the subsequent foraging behavior.It is different from the other ABC variants.

在发展乡村旅游时，与其利用稀缺的政府资金进行实验性探索，不如借鉴参考他乡邻村的实践经验，有效规避风险。然而，归根结底还应结合自身特点，不可照搬照抄成功案例，沦为“东施效颦”。因此参考其他乡村旅游的发展方式时，要分析两个主体的相似程度，审视模式的适用性。

4.Computational results and discussion

4.1 Benchmark functions and parameter settings

In detail,the neighborhood Nl(x)denotes the set of solutions in the lth neighborhood ofx.Thus,the construction of Nl(x)is the significant part of the VNS algorithm.Using the lp metrics,we can define the neighborhoodsNl(x)in numerical optimization problem as follows:

Table 2 Benchmark functions for test

(i)Optimal neighborhood food source

Moreover,compared with the standard ABC algorithm,the HABC algorithm introduces two new parameters,such as:lmax(the number of neighborhood structures)and M(memory size). Thuswe conduct the experiments to test the influence of the different values of lmaxand M on the performance of the HABC algorithm,where lmax={1,3,5},M={1,2,3,5,10}.

In each of the following experiments,30 independent runs on 10 benchmark functions are carried out,and the best results are marked as bold.

刚走到大街上，就看见村东边燃放了一个钻天猴，这个钻天猴钻得真叫一个高，摇头摆尾像火箭一样直冲上天去了，到了最高处还叭地响了一下。我心里就踏实了。这是我跟刘铁头约定的暗号，这说明他已经得手了，他和李金枝已经像钻天猴一样钻上天了。

4.2 Experimental results

4.2.1 Effect of neighborhood structures size lmax

In the HABC algorithm,a new control parameter is introduced which refers to the number of neighborhood structures,that is,lmax.In this subsection,we design the experiment with different values of lmax(lmax=1,lmax=3,lmax=5)and the influence of lmaxis analyzed in terms of best,worst,mean and standard deviation(SD)benchmark function values.The results of comparison are shown in Table 3.As shown in Table 3,the value of lmaxhas great influence on the performance of HABC.The larger value of lmax,the better algorithm performance.When lmax=5,HABC produces the optimal results in terms of best,worst,mean and SD in seven functions(f2,f3,f4,f6,f8,f9,f10),and finds the globally optimal solution in six functions(f1,f3,f4,f5,f9,f10).In addition,when lmax=1,HABC gives the best SD value in function f1.

When lmax=3,HABC presents the best mean value in function f5,and the better value in the column “Best”as well as “Worst”of the table on function f7.It is obvious that lmax=5 can find the best results for most functions.

Table 3 Performance of HABC algorithm with different lmaxvalues

Function f2 Parameter Best Worse Mean SD Best Worse Mean SD Function f1 lmax=1 4.81E-15 3.53E-14 6.17E-15 4.72E-18 0.099 7 0.114 1 0.103 5 0.582 8 lmax=3 2.96E-15 4.89E-15 3.04E-15 6.51E-18 0.046 2 0.072 9 0.056 6 0.339 4 lmax=5 0 6.24E-16 1.02E-16 3.46E-17 0.010 5 0.026 4 0.015 3 0.125 7 Function f3 Function f4 Parameter Best Worse Mean SD Best Worse Mean SD lmax=1 0 0 0 0 2.27E-15 5.01E-15 3.44E-15 7.67E-18 lmax=3 0 0 0 0 4.08E-16 7.36E-16 5.80E-16 4.11E-18 lmax=5 0 0 0 0 0 5.46E-16 1.25E-16 2.09E-18 Function f5 Function f6 Parameter Best Worse Mean SD Best Worse Mean SD lmax=1 0 6.13E-19 3.78E-19 1.36E-18 5.03E-14 2.67E-13 8.52E-14 1.33E-09 lmax=3 0 3.85E-20 0.92E-20 7.70E-18 6.16E-15 5.29E-14 9.67E-15 5.05E-10 lmax=5 0 2.37E-20 1.03E-20 4.52E-19 8.44E-16 3.07E-14 4.73E-15 6.06E-11 Function f7 Function f8 Parameter Best Worse Mean SD Best Worse Mean SD lmax=1 4.09E-14 7.51E-14 5.19E-14 3.68E-14 1.34E-07 5.06E-07 3.91E-07 3.63E-12 lmax=3 2.27E-14 3.39E-14 3.18E-14 5.54E-15 7.60E-08 9.23E-08 8.04E-08 7.71E-13 lmax=5 2.56E-14 3.52E-14 3.01E-14 2.28E-15 5.71E-08 5.71E-08 5.71E-08 5.64E-13 Function f9 Function f10 Parameter Best Worse Mean SD Best Worse Mean SD lmax=1 –1.011 2 –1.004 1 –1.010 3 3.31E-15 0.399 6 0.436 7 0.411 1 0 lmax=3 –1.025 5 –1.020 8 –1.023 5 1.70E-15 0.398 8 0.418 2 0.405 8 0 lmax=5 –1.031 6 –1.031 6 –1.031 6 6.78E-16 0.397 9 0.397 9 0.397 9 0

It can be concluded that the performance of HABC is improved with the increase of lmax.In the theory,the HABC algorithm will find the globally optimal solution when lmax→+∞,but in fact,too large value of lmaxwill decrease the search efficiency of the algorithm,so we set lmax=5 in the following experiments.

4.2.2 Effect of memory size M

It can be concluded that HABC are superior to ABC and qABC both in final solutions and convergences peed.

Table 4 shows that the HABC algorithm has the similar performances with different values of M.For all M values,we can obtain the optimal results in terms of best,worst and mean values in function f3,f8,f9and f10.In the other six benchmark functions,M=2 reaches the best mean values in five functions(f2,f4,f5,f6,f7),and finds the best SD values in three functions(f2,f4,f5).In addition,M=1 produces the best mean and SD values in function f1.M=3 presents the best SD value in function f7.For most functions,M=1,3 and 5 can produce similar results to M=2.Although M=10 gets the worst results in function f1and f2,it still produces the competitive solutions in function f6and f7.

It can be concluded that the final results of HABC are very similar with different values of M,which means the HABC algorithm is insensitive to parameter M’s setting.Even so,M=2 can achieve the best performance in most functions,so we set M=2 in the following experiment.

4.2.3 Performance test and comparison

In order to further validate the effectiveness of the proposed algorithm,the results of HABC are compared with the standard ABC and the quick ABC(qABC)[4].The relevant results are shown in Table 5.It can be observed that,the same optimal results are obtained from ABC,qABC and HABC algorithms in terms of best,worst,median and mean values in four functions(f3,f8,f9,f10).Except for function f8,the SD values of HABC are not greater than ABC and qABC,which indicates the HABC algorithm has strong robustness.

采用SPSS 18.0软件对数据进行分析处理，计量资料以（均数±标准差）表示，采用t检验；计数资料以（n，%）表示，以P＜0.05表示差异具有统计学意义。

Table 4 Performance of HABC algorithm with different M values

Function f2 Parameter Best Worse Mean SD Best Worse Mean SD Function f1 M=1 0 6.05E-17 3.14E-17 1.70E-17 0.049 2 0.084 1 0.063 7 0.476 3 M=2 0 6.24E-16 1.02E-16 3.46E-17 0.010 5 0.026 4 0.015 3 0.125 7 M=3 0 2.79E-16 9.35E-17 2.25E-17 0.035 4 0.059 3 0.044 1 0.342 5 M=5 0 6.58E-16 4.87E-16 3.88E-17 0.009 6 0.062 6 0.042 9 0.289 8 M=10 0 9.21E-16 6.64E-16 4.39E-17 0.057 0 0.087 2 0.075 2 0.301 5 Function f3 Function f4 Parameter Best Worse Mean SD Best Worse Mean SD M=1 0 0 0 0 0 4.60E-15 2.72E-16 5.63E-18 M=2 0 0 0 0 0 5.46E-16 1.25E-16 2.09E-18 M=3 0 0 0 0 0 3.25E-15 1.37E-15 1.09E-17 M=5 0 0 0 0 0 4.43E-15 2.79E-15 2.72E-17 M=10 0 0 0 0 0 6.94E-15 4.82E-15 4.63E-17 Function f1 Function f2 Parameter Best Worse Mean SD Best Worse Mean SD M=1 0 5.14E-20 3.90E-20 5.03E-19 9.01E-16 3.62E-14 6.59E-15 1.75E-11 M=2 0 2.37E-20 1.03E-21 4.52E-19 8.44E-16 3.07E-14 4.73E-15 6.06E-11 M=3 0 7.06E-21 3.48E-21 5.77E-19 8.57E-16 5.84E-14 5.25E-15 3.94E-11 M=5 0 2.59E-20 1.79E-20 6.36E-19 8.89E-16 6.68E-14 5.46E-15 7.13E-11 M=10 0 3.82E-20 2.26E-20 6.98E-19 9.23E-16 1.55E-14 6.03E-15 7.48E-11 Function f7 Function f8 Parameter Best Worse Mean SD Best Worse Mean SD M=1 2.56E-14 3.52E-14 3.04E-14 2.26E-15 5.71E-08 5.71E-08 5.71E-08 2.08E-13 M=2 2.56E-14 3.52E-14 3.01E-14 2.28E-15 5.71E-08 5.71E-08 5.71E-08 5.64E-13 M=3 2.69E-14 3.77E-14 3.22E-14 2.02E-15 5.71E-08 5.71E-08 5.71E-08 6.72E-13 M=5 2.56E-14 3.52E-14 3.06E-14 2.39E-15 5.71E-08 5.71E-08 5.71E-08 7.09E-13 M=10 2.56E-14 3.52E-14 3.37E-14 2.31E-15 5.71E-08 5.71E-08 5.71E-08 6.81E-13 Function f9 Function f10 Parameter Best Worse Mean SD Best Worse Mean SD M=1 –1.031 6 –1.031 6 –1.031 6 6.78E-16 0.397 9 0.397 9 0.397 9 0 M=2 –1.031 6 –1.031 6 –1.031 6 6.78E-16 0.397 9 0.397 9 0.397 9 0 M=3 –1.031 6 –1.031 6 –1.031 6 6.78E-16 0.397 9 0.397 9 0.397 9 0 M=5 –1.031 6 –1.031 6 –1.031 6 6.78E-16 0.397 9 0.397 9 0.397 9 0 M=10 –1.031 6 –1.031 6 –1.031 6 6.78E-16 0.397 9 0.397 9 0.397 9 0

Table 5 Comparison of the HABC algorithm with ABC and qABC

Function Algorithm Best Worst Median Mean SD ABC 2.81E-16 5.39E-15 3.87E-16 4.64E-16 5.35E-17 f1 qABC 2.04E-16 3.77E-15 3.12E-16 4.64E-16 7.82E-17 HABC 0 6.24E-16 2.54E-16 1.02E-16 3.46E-17 ABC 0.093 2 0.141 1 0.125 9 0.108 8 0.190 6 f2 qABC 0.138 6 0.250 7 0.184 3 0.177 8 0.407 0 HABC 0.010 5 0.026 4 0.018 9 0.015 3 0.125 7 ABC 0 0 0 0 0 f3 qABC 0 0 0 0 0 HABC 0 0 0 0 0 ABC 2.38E-16 8.25E-16 4.04E-16 4.96E-16 7.38E-17 f4 qABC 2.11E-16. 7.73E-16 3.95E-16 4.66E-16 6.35E-17 HABC 0 5.46E-16 2.28E-16 1.25E-16 2.09E-18 ABC 5.69E-19 8.77E-18 5.35E-18 1.85E-18 1.01E-17 f5 qABC 3.16E-09 6.02E-08 4.40E-08 1.21E-08 4.31E-08 HABC 0 2.37E-20 1.92E-20 1.03E-20 4.52E-19 ABC 3.52E-12 1.19E-10 4.76E-11 5.29E-11 7.85E-10 f6 qABC 4.80E-12 2.52E-10 6.31E-11 5.52E-11 1.47E-10 HABC 8.44E-16 3.07E-14 5.26E-15 4.73E-15 6.06E-11

Continued

Function Algorithm Best Worst Median Mean SD ABC 2.74E-14 3.89E-14 3.05E-14 3.26E-14 3.51E-15 f7 qABC 2.95E-14 4.13E-14 3.27E-14 3.60E-14 3.72E-15 HABC 2.56E-14 3.52E-14 2.99E-14 3.01E-14 2.28E-15 ABC 5.71E-08 5.71E-08 5.71E-08 5.71E-08 3.32E-13 f8 qABC 5.71E-08 5.71E-08 5.71E-08 5.71E-08 7.83E-13 HABC 5.71E-08 5.71E-08 5.71E-08 5.71E-08 5.64E-13 ABC –1.031 6 –1.031 6 –1.031 6 –1.031 6 6.78E-16 f9 qABC –1.031 6 –1.031 6 –1.031 6 –1.031 6 6.78E-16 HABC –1.031 6 –1.031 6 –1.031 6 –1.031 6 6.78E-16 ABC 0.397 9 0.397 9 0.397 9 0.397 9 0 f10 qABC 0.397 9 0.397 9 0.397 9 0.397 9 0 HABC 0.397 9 0.397 9 0.397 9 0.397 9 0

In the other six benchmark functions,the best solutions are obtained by HABC.In addition,ABC finds the better mean values than qABC in function f2,f5,f6and f7.It is worth mentioning that,only HABC finds the globally optimal solutions in function f1,f4and f5,which means HABC can effectively solve the complex numerical optimization problems.

在实际运行过程中，也暴露出如下缺点：（1）相较于人工湿地与水平潜流人工湿地，垂直流人工湿地具有较高的建设投资费用；（2）管理相对难于表面流人工湿地；（3）应用区域比较局限，仅适合温暖地区使用，不适合寒冷地区应用；（4）有滋生蚊蝇可能性；（5）有机物的去除能力不如水平潜流人工湿地。

wheredenotes the memory of the jth dimension parameter of the food source i(i=1,2,...,SN and j=1,2,...,D),M is a new control parameter which defines the memory size(m=1,2,...,M).

Fig.2 Convergence curves of three algorithms

Meanwhile,HABC converges to best final solutions with the fast convergence speed in function f2,f3,f5,f7 and f8.

In this experiment,the effect of another new parameter M in HABC is tested.As can been seen from the definition of matrix M,the smaller value of M,the less memory of the artificial bees about successful trails.In contrast,the larger value of M,the artificial bees can memorize the more successful trails.Therefore, five HABC variants with M={1,2,3,5,10}are checkout.Table 4 presents the experimental results.

The main reason for this is that the HABC algorithm redefines the solution search equation by adding the variable neighborhood search factor which increases the population diversity and enhances the local search ability.Moreover,the memory mechanism indeed improves the global search efficiency.

5.Conclusions and future studies

This paper proposes a novel optimization algorithm called the HABC algorithm.Compared with other ABC variants,HABC contains two innovations.First,the local search ability and the population diversity are improved by adding the variable neighborhood search factor to solution search equation.Second,the global search efficiency is enhanced by memory mechanism which imitates the natural behavior of honeybees,and it is not taken into account in the standard ABC as well as its variants.

In the numerical experiments,we test the two new parameters,and give the best parameter values.Then,the comparison results on 10 benchmark functions demonstrate that the HABC algorithm can obtain better final solutions and faster convergence speed.In the next step,we will focus on applying the HABC algorithm to combinatorial optimization problems, such as the allocation problems and scheduling problems.

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