The Ant Meta-Heuristic Colony Optimization

c law enforcement. Therefore, c congestion was ciency of the improved algorithm with the Dijkstra algorithm. Thus, it could simulate the optimal driving path with better performance, which was targeted and innovative.关键词：蚁群算法;实际路况;最优路径Key words ：ant colony optimization; actual road conditions; optimal path文／张俊豪蚁群算法在最优路径选择中的改进及应用0 引言在国务院发布的《国家中长期科学和技术发展规划纲要（2006-2020年）》中，将交通拥堵问题列为发展现代综合交通体系亟待解决的“三大热点问题”之一。

智能交通系统作为“互联网+交通”的产物，利用先进的科学技术对车、路、人、物进行统一的管控、调配，成为了当下各国缓解交通拥堵的一个重要途径。

路径寻优是智能交通系统的一个核心研究内容，可以有效的提升交通运输效率，减少事故发生频率，降低对城市空气的污染以及提升交通警察的执法效率等。

最著名的路径规划算法是Dijkstra算法和Floyd算法，Dijkstra算法能够在有向加权网络中计算得到某一节点到其他任何节点的最短路径；Floyd算法也称查点法，该算法和Dijkstra算法相似，主要利用的是动态规划思想，寻找加权图中多源节点的最短路径。

近些年，最优路径的研究主要集中以下几个方面：（1）基于A*算法的路径寻优。

A*算法作为一种重要的路径寻优算法，其在诸多领域内都得到了应用。

随着科技的发展，A*算法主要运用于人工智能领域，特别是游戏行业，在游戏中，A*算法旨在找到一条代价（燃料、时间、距离、装备、金钱等）最小化的路径，A*算法通过启发式函数引导自己，具体的搜索过程由函数值来决定。

蚁群算法蚂蚁算法中英文对照外文翻译文献(文档含英文原文和中文翻译)翻译:蚁群算法起源蚁群算法(ant colony optimization, ACO)，又称蚂蚁算法，是一种用来在图中寻找优化路径的机率型算法。

它由Marco Dorigo于1992年在他的博士论文中提出，其灵感来源于蚂蚁在寻找食物过程中发现路径的行为。

蚁群算法是一种模拟进化算法,初步的研究表明该算法具有许多优良的性质.针对PID 控制器参数优化设计问题,将蚁群算法设计的结果与遗传算法设计的结果进行了比较,数值仿真结果表明,蚁群算法具有一种新的模拟进化优化方法的有效性和应用价值。

原理各个蚂蚁在没有事先告诉他们食物在什么地方的前提下开始寻找食物。

当一只找到食物以后，它会向环境释放一种信息素，吸引其他的蚂蚁过来，这样越来越多的蚂蚁会找到食物！有些蚂蚁并没有象其它蚂蚁一样总重复同样的路，他们会另辟蹊径，如果令开辟的道路比原来的其他道路更短，那么，渐渐地更多的蚂蚁被吸引到这条较短的路上来。

最后，经过一段时间运行，可能会出现一条最短的路径被大多数蚂蚁重复着。

为什么小小的蚂蚁能够找到食物？他们具有智能么？设想，如果我们要为蚂蚁设计一个人工智能的程序，那么这个程序要多么复杂呢？首先，你要让蚂蚁能够避开障碍物，就必须根据适当的地形给它编进指令让他们能够巧妙的避开障碍物，其次，要让蚂蚁找到食物，就需要让他们遍历空间上的所有点；再次，如果要让蚂蚁找到最短的路径，那么需要计算所有可能的路径并且比较它们的大小，而且更重要的是，你要小心翼翼的编程，因为程序的错误也许会让你前功尽弃。

这是多么不可思议的程序！太复杂了，恐怕没人能够完成这样繁琐冗余的程序。

然而，事实并没有你想得那么复杂，上面这个程序每个蚂蚁的核心程序编码不过100多行！为什么这么简单的程序会让蚂蚁干这样复杂的事情？答案是：简单规则的涌现。

事实上，每只蚂蚁并不是像我们想象的需要知道整个世界的信息，他们其实只关心很小范围内的眼前信息，而且根据这些局部信息利用几条简单的规则进行决策，这样在蚁群这个集体里，复杂性的行为就会凸现出来。

Ant Colony Optimizationwith Immigrants Schemesfor the Dynamic Vehicle Routing ProblemMichalis Mavrovouniotis1and Shengxiang Yang21Department of Computer Science,University of LeicesterUniversity Road,Leicester LE17RH,United Kingdommm251@2Department of Information Systems and Computing,Brunel UniversityUxbridge,Middlesex UB83PH,United Kingdomshengxiang.yang@Abstract.Ant colony optimization(ACO)algorithms have proved tobe able to adapt to dynamic optimization problems(DOPs)when theyare enhanced to maintain diversity and transfer knowledge.Several ap-proaches have been integrated with ACO to improve its performancefor DOPs.Among these integrations,the ACO algorithm with immi-grants schemes has shown good results on the dynamic travelling sales-man problem.In this paper,we investigate ACO algorithms to solve amore realistic DOP,the dynamic vehicle routing problem(DVRP)withtraffic factors.Random immigrants and elitism-based immigrants are ap-plied to ACO algorithms,which are then investigated on different DVRPtest cases.The results show that the proposed ACO algorithms achievepromising results,especially when elitism-based immigrants are used.1IntroductionIn the vehicle routing problem(VRP),a number of vehicles with limited capacity are routed in order to satisfy the demand of all customers at a minimum cost (usually the total travel time).Ant colony optimization(ACO)algorithms have shown good performance for the VRP,where a population of ants cooperate and construct vehicle routes[5].The cooperation mechanism of ants is achieved via their pheromone trails,where each ant deposits pheromone to its trails and the remaining ants can exploit it[2].The dynamic VRP(DVRP)is closer to a real-world application since the traffic jams in the road system are considered.As a result,the travel time be-tween customers may change depending on the time of the day.In dynamic optimization problems(DOPs)the moving optimum needs to be tracked over time.ACO algorithms can adapt to dynamic changes since they are inspired from nature,which is a continuous adaptation process[9].In practice,they can adapt by transferring knowledge from past environments[1].The challenge of such algorithms is how quickly they can react to dynamic changes in order to maintain the high quality of output instead of premature convergence.C.Di Chio et al.(Eds.):EvoApplications2012,LNCS7248,pp.519–528,2012.c Springer-Verlag Berlin Heidelberg2012520M.Mavrovouniotis and S.YangDeveloping strategies for ACO algorithms to deal with premature conver-gence and address DOPs has attracted a lot of attention,which includes local and global restart strategies[7],memory-based approaches[6],pheromone ma-nipulation schemes to maintain diversity[4],and immigrants schemes to increase diversity[11,12].These approaches have been applied to the dynamic travelling salesman problem(DTSP),which is the simplest case of a DVRP,i.e.,only one vehicle is used.The ACO algorithms that are integrated with immigrants schemes have shown promising results on the DTSP where immigrant ants re-place the worst ants in the population every iteration[11].In this paper,we integrate two immigrants schemes,i.e.,random immigrants and elitism-based immigrants,to ACO algorithms and apply them to the DVRP with traffic factor.The aim of random immigrants ACO(RIACO)is to increase the diversity in order to adapt well in DOPs,and the aim of elitism-based im-migrants ACO(EIACO)is to generate guided diversity to avoid randomization.The rest of the paper is organized as follows.Section2describes the problem we try to solve,i.e.,the DVRP with traffic factors.Section3describes the ant colony system(ACS),which is one of the best performing algorithms for the VRP.Section4describes our proposed approaches where we incorporate immigrants schemes with ACO.Section5describes the experiments carried out by comparing RIACO and EIACO with ACS.Finally,Section6concludes this paper with directions for future work.2The DVRP with Traffic JamsThe VRP has become one of the most popular combinatorial optimization prob-lems,due to its similarities with many real-world applications.The VRP is classified as NP-hard[10].The basic VRP can be described as follows:a number of vehicles with afixed capacity need to satisfy the demand of all the customers, starting from and returning to the depot.Usually,the VRP is represented by a complete weighted graph G=(V,E), with n+1nodes,where V={u0,...,u n}is a set of vertices corresponding to the customers(or delivery points)u i(i=1,···,n)and the depot u0and E={(u i,u j):i=j}is a set of edges.Each edge(u i,u j)is associated with a non-negative d ij which represents the distance(or travel time)between u i and u j.For each customer u i,a non-negative demand D i is given.For the depot u0, a zero demand is associated,i.e.,D0=0.The aim of the VRP is tofind the route(or a set of routes)with the lowest cost without violating the following constraints:(1)every customer is visited exactly once by only one vehicle;(2)every vehicle starts andfinishes at the depot;and (3)the total demand of every vehicle route must not exceed the vehicle capacity Q.The number of routes identifies the corresponding number of vehicles used to generate one VRP solution,which is notfixed but chosen by the algorithm.The VRP becomes more challenging if it is subject to a dynamic environment. There are many variations of the DVRP,such as the DVRP with dynamic de-mand[14].In this paper,we generate a DVRP with traffic factors,where eachAnt Colony Optimization with Immigrants Schemes for the DVRP521 edge(u i,u j)is associated with a traffic factor t ij.Therefore,the cost to travel from u i to u j is c ij=d ij×t ij.Furthermore,the cost to travel from u j to u i may differ due to different traffic factor.For example,one road may have more traffic in one direction and less traffic in the opposite direction.Every f iterations a random number R∈[F L,F U]is generated to represent potential traffic jams,where F L and F U are the lower and upper bounds of the traffic factor,respectively.Each edge has a probability m to have a traffic factor, by generating a different R to represent high and low traffic jams on different roads,i.e.,t ij=1+R,where the traffic factor of the remaining edges is set to1 (indicates no traffic).Note that f and m represent the frequency and magnitude of changes in the DVRP,respectively.3ACO for the DVRPThe ACO metaheuristic consists of a population ofμants where they construct solutions and share their information with the others via their pheromone trails. Thefirst ACO algorithm developed is the Ant System(AS)[2].Many variations of the AS have been developed over the years and applied to difficult optimization problems[3].The best performing ACO algorithm for the DVRP is the ACS[13].There is a multi-colony variation of this algorithm applied to the VRP with time win-dows[5].However,in this paper we consider the single colony which has been applied to the DVRP[13].Initially,all the ants are placed on the depot and all pheromone trails are initialized with an equal amount.With a probability1−q0, where0≤q0≤1is a parameter of the pseudo-random proportional decision rule(usually0.9for ACS),an ant k chooses the next customer j from customeri,as follows:p k ij=⎧⎨⎩[τij]α[ηij]βl∈N k i[τil]α[ηil]β,if j∈N k i,0,otherwise,(1)whereτij is the existing pheromone trail between customers i and j,ηij is the heuristic information available a priori,which is defined as1/c ij,where c ij isthe distance travelled(as calculated in Section2)between customers i and j, N k i denotes the neighbourhood of unvisited customers of ant k when its current customer is i,andαandβare the two parameters that determine the relativeinfluence of pheromone trail and heuristic information,respectively.With the probability q0,the ant k chooses the next customer with the maximum proba-bility,i.e.,[τ]α[η]β,and not probabilistically as in Eq.(1).However,if the choice of the next customer leads to an infeasible solution,i.e.,exceed the maximum capacity Q of the vehicle,the depot is chosen and a new vehicle route starts.When all ants construct their solutions,the best ant retraces the solution and deposits pheromone globally according to its solution quality on the correspond-ing trails,as follows:τij←(1−ρ)τij+ρΔτbestij,∀(i,j)∈Tbest,(2)522M.Mavrovouniotis and S.Yangwhere0<ρ≤1is the pheromone evaporation rate andΔτbestij =1/C best,whereC best is the total cost of the T best tour.Moreover,a local pheromone update is performed every time an ant chooses another customer j from customer i as follows:τij←(1−ρ)τij+ρτ0,(3) whereρis defined as in Eq.(2)andτ0is the initial pheromone value.The pheromone evaporation is the mechanism that eliminates the areas with high intensity of pheromones that are generate by ants,due to stagnation be-haviour1,in order to adapt well to the new environment.The recovery time depends on the size of the problem and magnitude of change.4ACO with Immigrants Schemes for the DVRP4.1FrameworkThe framework of the proposed algorithms is based on the ACO algorithms that were used for the DTSP[11,12].It will be interesting to observe if the framework based on immigrants schemes is beneficial for more realistic problems,such as the DVRP with traffic factors,as described in Section2.The initial phase of the algorithm and the solution construction of the ants are the same with the ACS;see Eq.(1).The difference of the proposed framework is that it uses a short-term memory every iteration t,denoted as k short(t),of limited size,i.e.,K s,which is associated with the pheromone matrix.Initially, k short(0)is empty where at the end of the iteration the K s best ants will be added to k short(t).Each ant k that enters k short(t)deposits a constant amount of pheromone to the corresponding trails,as follows:τij←τij+Δτk ij,∀(i,j)∈T k,(4)whereΔτk ij=(τmax−τ0)/K s and T k is the tour of ant k.Here,τmax andτ0are the maximum and initial pheromone value,respectively.Every iteration the ants from k short(t−1)are replaced with the K s best ants from iteration t,a negative update is performed to their pheromone trails,as follows:τij←τij−Δτk ij,∀(i,j)∈T k,(5) whereΔτij and T k are defined as in Eq.(4).This is because no ants can survive in more than one iteration because of the dynamic environment.In addition,immigrant ants replace the worst ants in k short(t)every iteration and further adjustments are performed to the pheromone trails since k short(t) changes.The main concern when dealing with immigrants schemes is how to generate immigrant ants,that represent feasible solutions.1A term used when all ants follow the same path and construct the same solution.Ant Colony Optimization with Immigrants Schemes for the DVRP523 4.2Random Immigrants ACO(RIACO)Traditionally,the immigrants are randomly generated and replace other ants in the population to increase the diversity.A random immigrant ant for the DVRP is generated as follows.First,the depot is added as the starting point; then,an unvisited customer is randomly selected as the next point.This process is repeated until thefirst segment(starting from the most recent visit to the depot)of customers do not violate the capacity constraint.When the capacity constraint is violated the depot is added and another segment of customers starts.When all customers are visited the solution will represent one feasible VRP solution.Considering the proposed framework described above,before the pheromone trails are updated,a set S ri of r×K s immigrants are generated to replace the worst ants in k short(t),where r is the replacement rate.RIACO has been found to perform better in fast and significantly changing environments for the DTSP[11].This is because when the changing environ-ments are not similar it is better to randomly increase the diversity instead of knowledge transfer.Moreover,when the environmental changes are fast the time is not enough to gain useful knowledge in order to transfer it.However,there is a high risk of randomization with RIACO that may disturb the optimization process.A similar behaviour is expected for the DVRP.4.3Elitism-Based Immigrants ACO(EIACO)Differently from RIACO,which generates diversity randomly with the immi-grants,EIACO generates guided diversity by the knowledge transferred from the best ant of the previous environment.An elitism-based immigrant ant for the DVRP is generated as follows.The best ant of the previous environment is selected in order to use it as the base to generate elitism-based immigrants.The depots of the best ant are removed and adaptive inversion is performed based on the inver-over operator[8].When the inversion operatorfinishes,the depots are added so that the capacity constraint is satisfied in order to represent one feasible VRP solution.Considering the proposed framework above,on iteration t,the elite ant from k short(t−1)is used as the base to generate a set S ei of r×K s immigrants,where r is the replacement rate.The elitism-based immigrants replace the worst ants in k short(t)before the pheromone trails are updated.The EIACO has been found to perform better in slowly and slightly changing environments for the DTSP[11].This is because the knowledge transferred when the changing environments are similar will be more useful.However,there is a risk to transfer too much knowledge and start the optimization process from a local optimum and get stuck there.A similar behaviour is expected for the DVRP.524M.Mavrovouniotis and S.Yang5Simulation Experiments5.1Experimental SetupIn the experiments,we compare the proposed RIACO and EIACO with the existing ACS,described in Section3.All the algorithms have been applied to the vrp45,vrp72,and vrp135problem instances2.To achieve a good balance between exploration and exploitation,most of the parameters have been obtained from our preliminary experiments where others have been inspired from literature[11].For all algorithms,μ=50ants are used,α=1,β=5,andτ0=1/n.For ACS,q0=0.9,andρ=0.7.Note that a lower evaporation rate has been used for ACS,i.e.ρ=0.1,with similar or worseresults.For the proposed algorithms,q0=0.0,K s=10,τmax=1.0and r=0.4.For each algorithm on a DVRP instance,N=30independent runs were executed on the same environmental changes.The algorithms were executed for G=1000iterations and the overall offline performance is calculated as follows:P offline=1GGi=1⎛⎝1NNj=1P∗ij⎞⎠(6)where P∗ij defines the tour cost of the best ant since the last dynamic change of iteration i of run j[9].The value of f was set to10and100,which indicate fast and slowly changing environments,respectively.The value of m was set to0.1,0.25,0.5,and0.75, which indicate the degree of environmental changes from small,to medium,to large,respectively.The bounds of the traffic factor are set as F L=0and F U=5. As a result,eight dynamic environments,i.e.,2values of f×4values of m, were generated from each stationary VRP instance,as described in Section2,to systematically analyze the adaptation and searching capability of each algorithm on the DVRP.5.2Experimental Results and AnalysisThe experimental results regarding the offline performance of the algorithms are presented in Table1and the corresponding statistical results of Wilcoxon rank-sum test,at the0.05level of significance are presented in Table2.Moreover,to better understand the dynamic behaviour of the algorithms,the results of the largest problem instance,i.e.,vrp135,are plotted in Fig.1with f=10,m=0.1 and m=0.75,and f=100,m=0.1and m=0.75,for thefirst500iterations. From the experimental results,several observations can be made by comparing the behaviour of the algorithms.First,RIACO outperforms ACS in all the dynamic test cases;see the results of RIACO⇔ACS in Table2.This validates our expectation that ACS need 2Taken from the Fisher benchmark instances available athttp://neo.lcc.uma.es/radi-aeb/WebVRP/Ant Colony Optimization with Immigrants Schemes for the DVRP525 parison of algorithms regarding the results of the offline performancef=10f=100m⇒0.10.250.50.750.10.250.50.75Alg.&Inst.vrp45ACS897.5972.51205.61648.0883.4929.11120.21536.9RIACO841.2902.41089.51482.9834.9867.51016.11375.1EIACO840.1899.81083.81473.5839.8860.61009.11355.5Alg.&Inst.vrp72ACS305.3338.6426.2596.2297.3324.6412.7547.9RIACO294.4322.8401.7562.5280.6303.5375.2489.6EIACO289.9319.4397.8557.0276.2298.5366.7476.5Alg.&Inst.vrp135ACS1427.71567.31967.42745.71383.71519.41820.52536.2RIACO1417.81554.21922.12676.01353.11457.21698.62358.4EIACO1401.31542.11907.62663.11329.11444.31668.52293.8Table2.Statistical tests of comparing algorithms regarding the offline performance, where“+”or“−”means that thefirst algorithm is significantly better or the second algorithm is significantly betterAlg.&Inst.vrp45vrp72vrp135f=10,m⇒0.10.250.50.750.10.250.50.750.10.250.50.75RIACO⇔ACS++++++++++++EIACO⇔ACS++++++++++++EIACO⇔RIACO++++++++++++f=100,m⇒0.10.250.50.750.10.250.50.750.10.250.50.75RIACO⇔ACS++++++++++++EIACO⇔ACS++++++++++++EIACO⇔RIACO−+++++++++++sufficient time to recover when a dynamic change occurs,which can be also observed from Fig.1in the environmental case with f=100.This is because the pheromone evaporation is the only mechanism used to eliminate pheromone trails that are not useful to the new environment,and may bias the population to areas that are not near the new optimum.On the other hand,RIACO uses the proposed framework where the pheromone trails exist only in one iteration.Second,EIACO outperforms ACS in all the dynamic test cases as the RI-ACO;see the results EIACO⇔ACS in Table2.This is due to the same reasons RIACO outperforms the traditional ACS.However,EIACO outperforms RI-ACO in almost all dynamic test cases;see the results of EIACO⇔RIACO in Table2.In slowly and slightly changing environments EIACO has sufficient time to gain knowledge from the previous environment,and the knowledge transferred has more chances to help when the changing environments are similar.However, on the smallest problem instance,i.e.,vrp45,with f=100and m=0.1RIACO performs better than EIACO.This validates our expectation where too much526M.Mavrovouniotis and S.Yang1300 1350 1400 1450 1500 1550 16000100200300400500O f f l i n e P e r f o r m a n c eIterationvrp135 - f = 10, m = 0.1ACS RIACO EIACO 2200 2400 2600 2800 3000 3200 34000100200300400500O f f l i n e P e r f o r m a n c eIteration vrp135 - f = 10, m = 0.75ACS RIACO EIACO 1200 1250 130013501400 1450 1500 1550 16000100200300400500O f f l i n e P e r f o r m a n c e Iteration vrp135 - f = 100, m = 0.1ACS RIACO EIACO 2200 2400 2600 2800 3000 3200 34000100200300400500O f f l i n e P e r f o r m a n c eIterationvrp135 - f = 100, m = 0.75ACS RIACO EIACOFig.1.Offline performance of algorithms for different dynamic test problems 1300 1350 1400 1450 15000.00.20.40.60.8 1.0O f f l i n e P e r f o r m a n c e r vrp135, f = 100, m = 0.1RIACO EIACO ACS 2200 2300 2400 2500 2600 27000.00.20.40.60.8 1.0O f f l i n e P e r f o r m a n c ervrp135, f = 100, m = 0.75RIACO EIACO ACS Fig.2.Offline performance of RIACO and EIACO with different replacement rates against the performance of ACS in slowly changing environmentsknowledge transferred does not always mean better results in dynamic environ-ments.On the other hand RIACO,was expected to perform better than EIACO in fast and significantly changing environments,since the random immigrants only increase the diversity,but that it is not the case.This may be possibly because of too much randomization that may disturb the optimization process and requires further investigation regarding the effect of the immigrant ants.Ant Colony Optimization with Immigrants Schemes for the DVRP527 Third,in order to investigate the effectiveness of the immigrants schemes,fur-ther experiments have been performed on the same problem instances with the same parameters used before but with different immigrant replacement rates, i.e.,r∈{0.0,0.2,0.4,0.6,0.8,1.0}.In Fig.2the offline performance of RIACO and EIACO with the varying replacement rates are presented3,against the ACS performance,where r=0.0means that no immigrants are generated to re-place ants in the k short(t).The results confirm our expectation above,where the random immigrants in RIACO sometimes may disturb the optimization and de-grade the performance.On the other hand,elitism-based immigrants in EIACO improve the performance,especially in slightly changing environments.Finally,the proposed framework performs better than ACS,even if no immi-grants are generated;see Fig.2.The RIACO with r=1.0performs worse than the ACS,whereas the EIACO with r=1.0better than ACS.This is because RIACO destroys all the knowledge transferred to the k short(t)from the ants of the previous iteration with random immigrants,whereas EIACO destroys that knowledge but transfers new knowledge using the best ant from the previous iteration.6ConclusionsDifferent immigrants schemes have been successfully applied to evolutionary al-gorithms and ACO algorithms to address different DOPs[11,16].ACO-based algorithms with immigrants,i.e.,RIACO and EIACO,have shown good perfor-mance on different variations of the DTSP[11,12].In this paper,we modify and apply such algorithms to address the DVRP with traffic factors,which is closer to a real-world application.The immigrant ants are generated either randomly or using the previous best ant as the base and replace the worst ones in the pop-ulation.The aim is to maintain the diversity of solutions and transfer knowledge from previous environments in order to adapt well in DOPs.Comparing RIACO and EIACO with ACS,one of the best performing ACO al-gorithms for VRP,on different test cases of DVRPs,the following concluding re-marks can be drawn.First,the proposed framework used to integrate ACO with immigrants schemes,performs better than the traditional framework,even when immigrant ants are not generated.Second,EIACO is significantly better than RI-ACO and ACS in almost all dynamic test cases.Third,RIACO is significantly bet-ter than ACS in all dynamic test cases.Finally,the random immigrants may disturb the optimization process with a result to degrade the performance,whereas elitism-based immigrants transfers knowledge with a result to improves the performance for the DVRP with traffic factor.An obvious direction for future work is to hybridize the two immigrants schemes.However,from our preliminary results the performance of the hybrid scheme is better than RIACO but worse than EIACO in all dynamic test cases. Therefore,tofind another way to achieve a good balance between the knowledge 3The experimental results of the remaining problem instances and dynamic test cases are similar for EIACO,whereas for RIACO there is an improvement when r>0.0 on the smallest problem instance.528M.Mavrovouniotis and S.Yangtransferred and the diversity generated would be interesting for future work.An-other future work is to integrate memory-based immigrants with ACO,which have also performed well on the DTSP[12],to the DVRP with traffic factors. 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蚂蚁算法（Ant Colony Algorithm）和蚁群算法（Ant Colony Optimization）是启发式优化算法，灵感来源于蚂蚁在觅食和建立路径时的行为。

这两种算法都基于模拟蚂蚁的行为，通过模拟蚂蚁的集体智慧来解决组合优化问题。

蚂蚁算法和蚁群算法的基本原理类似，但应用领域和具体实现方式可能有所不同。

下面是对两者的简要介绍：蚂蚁算法：蚂蚁算法主要用于解决图论中的最短路径问题，例如旅行商问题（Traveling Salesman Problem，TSP）。

其基本思想是通过模拟蚂蚁在环境中寻找食物的行为，蚂蚁会通过信息素的释放和感知来寻找最优路径。

蚂蚁算法的核心概念是信息素和启发式规则。

信息素（Pheromone）：蚂蚁在路径上释放的一种化学物质，用于传递信息和标记路径的好坏程度。

路径上的信息素浓度受到蚂蚁数量和路径距离的影响。

启发式规则（Heuristic Rule）：蚂蚁根据局部信息和启发式规则进行决策。

启发式规则可能包括路径距离、路径上的信息素浓度等信息。

蚂蚁算法通过模拟多个蚂蚁的行为，在搜索过程中不断调整路径上的信息素浓度，从而找到较优的解决方案。

蚁群算法：蚁群算法是一种更通用的优化算法，广泛应用于组合优化问题。

除了解决最短路径问题外，蚁群算法还可应用于调度问题、资源分配、网络路由等领域。

蚁群算法的基本原理与蚂蚁算法类似，也是通过模拟蚂蚁的集体行为来求解问题。

在蚁群算法中，蚂蚁在解决问题的过程中通过信息素和启发式规则进行路径选择，但与蚂蚁算法不同的是，蚁群算法将信息素更新机制和启发式规则的权重设置进行了改进。

蚁群算法通常包含以下关键步骤：初始化：初始化蚂蚁的位置和路径。

路径选择：根据信息素和启发式规则进行路径选择。

信息素更新：蚂蚁在路径上释放信息素，信息素浓度受路径质量和全局最优解的影响。

全局更新：周期性地更新全局最优解的信息素浓度。

终止条件：达到预设的终止条件，结束算法并输出结果。

2021576海洋资源已经成为人类开发的重点，但复杂的海洋环境对人类水下作业有着极大的限制，水下机器人正在成为海洋作业的主角，自主式水下机器人（Autono-mous Underwater Vehicle，AUV）依靠自身携带的能源进行水下作业。

由于在整个过程中无法补充能源，因此利用路径规划与安全避障技术对AUV导航控制，是其能否精确、安全和完整地完成水下作业的关键。

AUV 路径规划问题已经成为了一个研究热点[1]，主要涉及两方面问题：一是对海洋环境进行三维建模；二是选取合适的算法进行全局路径规划。

海洋环境建模主要有两类方法：一类是规则地形模型，主要利用正方形、矩形等规则形状进行组合来表示海底表面；另一类是不规则地形模型，将三角形、多边形等不规则形状作为模型单元的基础[2]。

文献[3]使用Voronoi图法简化三维水下环境，生成全局路线图；文献[4]将Delaunay三角模型应用于被测地标，建立拓扑模型。

文献[5]利用八叉树模型来反映AUV工作环境，但主要应用于较大障碍物之间的路径规划，不适合存在许多小障碍物的环境；文献[6-7]不考虑水深，将三维空间简化为二维栅格模型，节省了空间，但却丢失了环境信息；文献[8-9]将三维空间划分为若干平面，然后利用二维栅格模型将每个平面栅格化，有效实现三维栅格建融合粒子群与改进蚁群算法的AUV路径规划算法朱佳莹，高茂庭上海海事大学信息工程学院，上海201306摘要：针对传统蚁群算法在处理自主式水下机器人AUV（Autonomous Underwater Vehicle）三维路径规划问题时存在初期寻径能力弱、算法收敛速度慢等问题，提出一种融合粒子群与改进蚁群算法的AUV路径规划算法PSO-ACO（Particle Swarm Optimization-improved Ant Colony Optimization）。

基于空间分层思想建立三维栅格模型实现水下环境建模；综合考虑路径长度、崎岖性、危险性等因素建立路径评价模型；先使用粒子群算法预搜索路径来优化蚁群算法的初始信息素；再对蚁群算法改进状态转移规则、信息素更新方式并加入奖惩机制实现全局路径规划。

Université Libre de BruxellesInstitut de Recherches Interdisciplinaireset de Développements en Intelligence Artificielle Ant Colony Optimization:Overview andRecent AdvancesMarco Dorigo and Thomas St¨u tzleIRIDIA–Technical Report SeriesTechnical Report No.TR/IRIDIA/2009-013May2009IRIDIA–Technical Report SeriesISSN1781-3794Published by:IRIDIA,Institut de Recherches Interdisciplinaireset de D´e veloppements en Intelligence ArtificielleUniversit´e Libre de BruxellesAv F.D.Roosevelt50,CP194/61050Bruxelles,BelgiumTechnical report number TR/IRIDIA/2009-013The information provided is the sole responsibility of the authors and does not necessarily reflect the opinion of the members of IRIDIA.The authors take full responsability for any copyright breaches that may result from publication of this paper in the IRIDIA–Technical Report Series.IRIDIA is not responsible for any use that might be made of data appearing in this publication.Ant Colony Optimization:Overview and Recent AdvancesMarco Dorigo and Thomas St¨u tzleMay20091IntroductionAnt Colony Optimization(ACO)[57,59,66]is a metaheuristic for solving hard combinatorial optimization problems.The inspiring source of ACO is the pheromone trail laying and following behavior of real ants,which use pheromones as a communication medium.In analogy to the biological example,ACO is based on indirect communication within a colony of simple agents, called(artificial)ants,mediated by(artificial)pheromone trails.The pheromone trails in ACO serve as a distributed,numerical information,which the ants use to probabilistically construct solutions to the problem being solved and which the ants adapt during the algorithm’s execution to reflect their search experience.Thefirst example of such an algorithm is Ant System(AS)[55,63,64,65],which was pro-posed using as example application the well known traveling salesman problem(TSP)[6,99,128]. Despite encouraging initial results,AS could not compete with state-of-the-art algorithms for the TSP.Nevertheless,it had the important role of stimulating further research both on algorithmic variants,which obtain much better computational performance,and on applications to a large va-riety of different problems.In fact,there exist now a considerable number of applications of such algorithms where world class performance is obtained.Examples are applications of ACO algo-rithms to problems such as sequential ordering[76],scheduling[18],assembly line balancing[19], probabilistic TSP[7],2D-HP protein folding[132],DNA sequencing[25],protein–ligand docking [98],packet-switched routing in Internet-like networks[47],and so on.The ACO metaheuris-tic provides a common framework for the existing applications and algorithmic variants[57,59]. Algorithms which follow the ACO metaheuristic are called ACO algorithms.The(artificial)ants in ACO implement a randomized construction heuristic which makes prob-abilistic decisions as a function of artificial pheromone trails and possibly available heuristic in-formation based on the input data of the problem to be solved.As such,ACO can be interpreted as an extension of traditional construction heuristics,which are readily available for many com-binatorial optimization problems.Yet,an important difference with construction heuristics is the adaptation of the pheromone trails during algorithm execution to take into account the cumulated search experience.The rest of this chapter is organized as follows.In Section2,we briefly overview construction heuristics and local search algorithms.In Section3,we present a specific version of the ACO metaheuristic that focuses on applications to N P-hard problems.Section4outlines the inspiring biological analogy and describes the historical developments leading to ACO.In Section5,we illustrate how the ACO metaheuristic can be applied to different types of problems and we give an overview of its successful applications.Section6gives an overview of recent developments in ACO and Section7concludes the chapter.2Approximate approachesMany important combinatorial optimization problems are hard to solve.The notion of problem hardness is captured by the theory of computational complexity[79,123]and for many important problems it is well known that they are N P-hard,that is,the time needed to solve an instance1procedure Greedy Construction Heuristics p=empty solutionwhile s p not complete1Other approximate methods are also conceivable.For example,when stopping exact methods,like Branch &Bound,before completion[10,95](for example,after some given time bound,or when some guarantee on the solution quality is obtained through the use of lower and upper bounds),we can convert exact algorithms into approximate ones.procedure IterativeImprovement(s∈S)s′=Improve(s)while s′=s dos=s′s′=Improve(s)endreturn send IterativeImprovementFigure2:Algorithmic skeleton of iterative improvement.Figure3:Schematic illustration of a2-exchange move.The proposed move reduces the total tour length if we consider the Euclidean distance between the points.2.2Local search algorithmsLocal search algorithms start from a complete initial solution and try tofind a better solution in an appropriately defined neighborhood of the current solution.In its most basic version,known as iterative improvement,the algorithm searches the neighborhood for an improving solution.If such a solution is found,it replaces the current solution and the local search continues.These steps are repeated until no improving neighbor solution can be found and the algorithm ends in a local optimum.An outline of an iterative improvement algorithm is given in Figure2.The procedure Improve returns a better neighbor solution if one exists,otherwise it returns the current solution, in which case the algorithm stops.The choice of an appropriate neighborhood structure is crucial for the performance of local search algorithms and has to be done in a problem specific way.The neighborhood structure defines the set of solutions that can be reached from s in one single step of the algorithm.An example neighborhood for the TSP is the k-exchange neighborhood in which neighbor solutions differ by at most k edges.Figure3shows an example of a2-exchange neighborhood.The2-exchange algorithm systematically tests whether the current tour can be improved by replacing two edges.To fully specify a local search algorithm,it is necessary to designate a particular neighborhood examination scheme that defines how the neighborhood is searched and which neighbor solution replaces the current one.In the case of iterative improvement algorithms,this rule is called the pivoting rule [157]and examples are the best-improvement rule,which chooses the neighbor solution giving the largest improvement of the objective function,and thefirst-improvement rule,which uses thefirst improved solution found when scanning the neighborhood to replace the current one.A common problem with local search algorithms is that they easily get trapped in local minima and that the result strongly depends on the initial solution.3The ACO metaheuristicArtificial ants used in ACO are stochastic solution construction procedures that probabilistically build a solution by iteratively adding solution components to partial solutions by taking into account(i)heuristic information about the problem instance being solved,if available,and(ii)(artificial)pheromone trails which change dynamically at run-time to reflect the agents’acquired search experience.A stochastic component in ACO allows the ants to build a wide variety of different solutions and hence to explore a much larger number of solutions than greedy heuristics.At the same time, the use of heuristic information,which is readily available for many problems,can guide the ants towards the most promising solutions.More important,the ants’search experience can be used to influence,in a way reminiscent of reinforcement learning[149],the solution construction in future iterations of the algorithm.Additionally,the use of a colony of ants can give the algorithm increased robustness and in many ACO applications the collective interaction of a population of agents is needed to efficiently solve a problem.The domain of application of ACO algorithms is vast.In principle,ACO can be applied to any discrete optimization problem for which some solution construction mechanism can be conceived. In the following of this section,wefirst define a generic problem representation that the ants in ACO may exploit to construct solutions,and then we define the ACO metaheuristic.3.1Problem representationLet us consider minimization problems2and define a general model of a combinatorial optimization problem.Definition3.1A model P=(S,Ω,f)of a combinatorial optimization problem consists of •a search space S that is defined by afinite set of decision variables,each with afinite domain, and a setΩof constraints among the variables;•an objective function f:S→I R+0that is to be minimized.The search space is defined by afinite set of variables X i,i=1,...,n,each having an associated domain D i of values that can be assigned to it.An instantiation of a variable consists in an assignment of a value v j i∈D i to variable X i and it is denoted by X i=v j i.A feasible solution s∈S is an assignment to each variable of a value in its domain such that all the problem constraints inΩare satisfied.IfΩis empty,then the problem is unconstrained and each decision variable can take any value from its domain,independent of the other variables.In this case,P is an unconstrained problem model;otherwise it is called constrained.A feasible solution s∗∈S is called a global minimum of P if and only if we have that f(s∗)≤f(s)∀s∈S.We denote by S∗⊆S the set of all global minima.This model of a combinatorial optimization problem can be directly used to derive a generic pheromone model that is exploited by ACO algorithms.To see how,let us call the instantiation of a variable X i with a particular value v j i of its domain a solution component,which is denoted by c j i.Ants then need to appropriately combine solution components to form high-quality,feasible solutions.To do so,each solution component c j i will have an associated pheromone variable T ij. We denote the set of all solution components by C and the set of all pheromone variables by T. Each pheromone variable T ij has a pheromone valueτij;this value indicates the desirability of choosing solution component c j i.Note that,as said before,the pheromone values are time-varying and therefore they are a function of the algorithm iteration t.In what follows we will,however, omit the reference to the iteration counter and write simplyτij instead ofτij(t).As an example of this formalization,consider the TSP.In this case,the solution components are the moves from one city to another one.This can be formalized by associating one variable to each city.Each variable X i has then associated n−1values,j=1,...,n,j=i.As a result,with each edge between a pair of cities is associated one pheromone valueτij.An instantiation of the decision variables corresponds to a feasible solution,if and only if the set of edges corresponding to the values of the decision variables forms a Hamiltonian cycle.(Note that for the TSP it is easily possible to guarantee that ants generate feasible solutions.)The objective function f(·)computes for each feasible solution the sum of the edge lengths,that is,the length of the Hamiltonian cycle.procedure ACO algorithm for combinatorial optimization problemsInitializationwhile(termination condition not met)doC onstructAntSolutionsA pplyLocalSearch%optionalG lobalUpdatePheromonesendend ACO algorithm for combinatorial optimization problemsFigure4:Algorithmic skeleton for ACO algorithms applied to combinatorial optimiza-tion problems.The application of a local search algorithm is a typical example of apossible daemon action in ACO algorithms.3.2The metaheuristicA general outline of the ACO metaheuristic for applications to static combinatorial optimization problems,3is given in Figure4.After initializing parameters and pheromone trails,the main loop consists of three main steps.First,m ants construct solutions to the problem instance under consideration,biased by the pheromone information and possibly by the available heuristic information.Once the ants have completed their solutions,these may be improved in an optional local search phase.Finally,before the start of the next iteration,the pheromone trails are adapted to reflect the search experience of the ants.The main steps of the ACO metaheuristic are explained in more detail in the following.Initialization.At the start of the algorithm,parameters are set and all pheromone variables are initialized to a valueτ0,which is a parameter of the algorithm.ConstructAntSolutions.A set of m ants constructs solutions to the problem instance being tackled.To do so,each ant starts with an initially empty solution s p=∅.At each construction step,an ant extends its current partial solution s p by choosing one feasible solution component c j i∈N(s p)⊆C and adding it to its current partial solution.N(s p)is the set of solution components that may be added while maintaining feasibility and it is defined implicitly by a solution construction process that the ants implement.If a partial solution cannot be extended maintaining feasibility,it depends on the particular construction mechanism whether the solution construction is abandoned or an infeasible,complete solution is constructed.In the latter case, infeasible solutions may be penalized in dependence of the degree to which they violate problem constraints.The choice of the solution component to add is done probabilistically at each construction step. Various ways for defining the probability distributions have been considered.The most widely used rule is that of Ant System(AS)[65]:ταij·[η(c j i)]βp(c j i|s p)=3Static problems are those whose topology and costs do not change while they are being solved.This is the case,for example,for the classic TSP,in which city locations and intercity distances do not change during the algorithm’s run-time.In contrast,in dynamic problems the topology and costs can change while solutions are built. An example of such a problem is routing in telecommunications networks[47],in which traffic patterns change all the time.ApplyLocalSearch.Once complete candidate solutions are obtained,these may further be improved by applying local search algorithms.In fact,for a wide range of combinatorial optimiza-tion problems,ACO algorithms reach best performance when coupled with local search algorithms [66].More generally,local search is one example of what have been called daemon actions[57,59]. These are used to implement problem specific or centralized actions that cannot be performed by individual ants.GlobalUpdatePheromones.The pheromone update is intended to make solution components belonging to good solutions more desirable for the following iterations.There are essentially two mechanisms that are used to achieve this goal.Thefirst is pheromone deposit,which increases the level of the pheromone of solution components that are associated with a chosen set S upd of good solutions.The goal is to make these solution components more attractive for ants in the following iterations.The second is pheromone trail evaporation,which is the mechanism that decreases over time the pheromone deposited by previous ants.From a practical point of view,pheromone evaporation is needed to avoid a too rapid convergence of the algorithm towards a sub-optimal region.It implements a useful form of forgetting,favoring the exploration of new areas of the search space.The pheromone update is commonly implemented as:τij=(1−ρ)τij+s∈S upd|c ji ∈sg(s)(2)where S upd is the set of solutions that are used to deposit pheromone,ρ∈(0,1]is a parameter called evaporation rate,g(·):S→I R+is a function such that f(s)<f(s′)⇒g(s)≥g(s′).It determines the quality of a solution and it is commonly called evaluation function.ACO algorithms typically differ in the way pheromone update is implemented:different spec-ifications of how to determine S upd result in different instantiations of update rule2.Typically, S upd is a subset of S iter∪{s gb},where S iter is the set of all solutions constructed in the current iteration of the main loop and s gb is the best solution found since the start of the algorithm(gb stands for global-best).4HistoryThefirst ACO algorithm to be proposed was Ant System(AS).AS was applied to some rather small TSP instances with up to75cities.It was able to reach the performance of other general-purpose heuristics like evolutionary computation[55,65].Despite these initial encouraging results, AS did not prove to be competitive with state-of-the-art algorithms specifically designed for the TSP.Therefore,a substantial amount of research in ACO has focused on ACO algorithms which show better performance than AS when applied,for example,to the TSP.In the following of this section wefirst briefly introduce the biological metaphor on which AS and ACO are inspired,and then we present a brief history of the early developments that have led from the original AS to more performing ACO algorithms.4.1Biological analogyIn many ant species,individual ants may deposit a pheromone(a chemical that ants can smell) on the ground while walking[43,80].By depositing pheromone,ants create a trail that is used, for example,to mark the path from the nest to food sources and back.Foragers can sense the pheromone trails and follow the path to food discovered by other ants.Several ant species are capable of exploiting pheromone trails to determine the shortest among the available paths leading to the food.Deneubourg and colleagues[43,80]used a double bridge connecting a nest of ants and a food source to study pheromone trail laying and following behavior in controlled experimentalconditions.4They ran a number of experiments in which they varied the ratio between the length of the two branches of the bridge.The most interesting,for our purposes,of these experiments is the one in which one branch was longer than the other.In this experiment,at the start the ants were left free to move between the nest and the food source and the percentage of ants that chose one or the other of the two branches was observed over time.The outcome was that,although in the initial phase random oscillations could occur,in most experiments all the ants ended up using the shorter branch.This result can be explained as follows.When a trial starts there is no pheromone on the two branches.Hence,the ants do not have a preference and they select with the same probability either of the two branches.It can be expected that,on average,half of the ants choose the short branch and the other half the long branch,although stochastic oscillations may occasionally favor one branch over the other.However,because one branch is shorter than the other,the ants choosing the short branch are thefirst to reach the food and to start their travel back to the nest.5But then,when they must make a decision between the short and the long branch,the higher level of pheromone on the short branch biases their decision in its favor.6Therefore,pheromone starts to accumulate faster on the short branch,which will eventually be used by the great majority of the ants.It should be clear by now how real ants have inspired AS and later algorithms:the double bridge was substituted by a graph,and pheromone trails by artificial pheromone trails.Also, because we wanted artificial ants to solve problems more complicated than those solved by real ants,we gave artificial ants some extra capacities,like a memory(used to implement constraints and to allow the ants to retrace their solutions without errors)and the capacity for depositing a quantity of pheromone proportional to the quality of the solution produced(a similar behavior is observed also in some real ants species in which the quantity of pheromone deposited while returning to the nest from a food source is proportional to the quality of the food source[9]).In the next section we will see how,starting from AS,new algorithms have been proposed that, although retaining some of the original biological inspiration,are less and less biologically inspired and more and more motivated by the need of making ACO algorithms better or at least competitive with other state-of-the-art algorithms.Nevertheless,many aspects of the original Ant System remain:the need for a colony,the role of autocatalysis,the cooperative behavior mediated by artificial pheromone trails,the probabilistic construction of solutions biased by artificial pheromone trails and local heuristic information,the pheromone updating guided by solution quality,and the evaporation of pheromone trail,are present in all ACO algorithms.4.2Historical developmentAs said,AS was thefirst ACO algorithm to be proposed in the literature.In fact,AS was originally a set of three algorithms called ant-cycle,ant-density,and ant-quantity.These three algorithms were proposed in Dorigo’s doctoral dissertation[55]andfirst appeared in a technical report[63,64] that was published a few years later in the IEEE Transactions on Systems,Man,and Cybernetics [65].Other early publications are[34,35].While in ant-density and ant-quantity the ants updated the pheromone directly after a move from a city to an adjacent one,in ant-cycle the pheromone update was only done after all the ants had constructed the tours and the amount of pheromone deposited by each ant was set to be a function of the tour quality.Because ant-cycle performed better than the other two variants, it was later called simply Ant System(and in fact,it is the algorithm that we will present in the following subsection),while the other two algorithms were no longer studied.The major merit of AS,whose computational results were promising but not competitive with other more established approaches,was to stimulate a number of researchers,mostly in Europe,to develop extensions and improvements of its basic ideas so as to produce better performing,and often state-of-the-art,algorithms.4.2.1Thefirst ACO algorithm:Ant System and the TSPThe TSP is a paradigmatic N P-hard combinatorial optimization problem,which has attracted an enormous amount of research effort[6,94,99].The TSP is a very important problem also in the context of Ant Colony Optimization because it is the problem to which the original AS wasfirst applied[55,63,64,65],and it has later often been used as a benchmark to test new ideas and algorithmic variants.In AS each ant is initially put on a randomly chosen city and has a memory,which stores the partial solution it has constructed so far(initially the memory contains only the start city). Starting from its start city,an ant iteratively moves from city to city,which corresponds to adding iteratively solution components as explained in Section3.2.When being at a city i,an ant k chooses to go to an as yet unvisited city j with a probability given by Equation1.The heuristic information is given byηij=1/d ij and N(s p)is the set of cities that ant k has not yet visited.The solution construction ends after each ant has completed a tour,that is,after each ant has constructed a sequence of length n,corresponding to a permutation of the city indices.Next,the pheromone trails are updated.In AS this is done by using Equation2,where we haveS upd=S iter(3) andg(s)=1/f(s),(4) where f(s)is the length of the tour s.Hence,the shorter the ant’s tour is,the more pheromoneis received by edges(solution components)belonging to the tour.7In general,edges which are used by many ants and which are contained in shorter tours will receive more pheromone and therefore are also more likely to be chosen in future iterations of the algorithm.4.2.2Ant System and its extensionsAs previously stated,AS was not competitive with state-of-the-art algorithms for the TSP.Re-searchers then started to extend it to try to improve its performance.Afirst improvement,called the elitist strategy,was introduced in[55,65].It consists in giving the best tour since the start of the algorithm(called s gb)a strong additional weight.In practice, each time the pheromone trails are updated by Equation2,we have that S upd=S iter∪{s gb}and that g(s),s=s gb,is given by Equation4.For s gb we have that g(s gb)=e/f(s gb),where e is a positive integer.Note that this type of pheromone update is afirst example of daemon action as described in Section3.2.Other improvements were rank-based Ant System(AS rank),MAX–MIN Ant System(MM AS), and Ant Colony System(ACS).AS rank[30]is in a sense an extension of the elitist strategy:it sorts the ants according to the lengths of the tours they generated and,after each tour construc-tion phase,only the(w−1)best ants and the global-best ant are allowed to deposit pheromone. The r th best ant of the colony contributes to the pheromone update with a weight given by max{0,w−r}while the global-best tour reinforces the pheromone trails with weight w.This can easily be implemented by an appropriate choice of S upd and g(s)in Equation2.MM AS[144,147,148]introduces upper and lower bounds to the values of the pheromone trails,as well as a different initialization of their values.In practice,in MM AS the allowed rangeof the pheromone trail strength is limited to the interval[τmin,τmax],that is,τmin≤τij≤τmax∀τij, and the pheromone trails are initialized to the upper trail limit,which causes a higher explorationat the start of the algorithm.In[144,148]it is discussed how to set the upper and lower pheromone trail limits in a principled way.Pheromone updates are performed using a strong elitist strategy: only the best solution generated is allowed to update pheromone trails.This can be the iteration-best solution,that is,the best in the current iteration,or the global-best solution.The amount of pheromone deposited is then given by g(s b)=1/f(s b),where s b is either s ib,the iteration-best solution,or s gb.In fact,in MM AS the iteration-best ant and the global-best ant can be used alternately in the pheromone putational results have shown that best results are obtained when pheromone updates are performed using the global-best solution with increasing frequency during the algorithm execution[144,148].As an additional means for increasing the explorative behavior of MM AS(and of ACO algorithms,in general),occasional pheromone trail reinitialization is used.MM AS has been improved also by the addition of local search routines that take the solution generated by ants to their local optimum just before the pheromone update.ACS[60,61,75]improves over AS by increasing the importance of exploitation of information collected by previous ants with respect to exploration of the search space.8This is achieved via two mechanisms.First,a strong elitist strategy is used to update pheromone trails.Second, ants choose a solution component(that is,the next city in the TSP case)using the so-called pseudo-random proportional rule[61]:with probability q0,0≤q0<1,they move to the city j for which the product between pheromone trail and heuristic information is maximum,that is,j=arg maxc ji ∈N(s p){τij·ηβij},while with probability1−q0they operate a biased exploration inwhich the probability p ij(t)is the same as in AS(see Equation1).The value q0is a parameter: when it is set to a value close to1,as it is the case in most ACS applications,exploitation is favored over exploration.Obviously,when q0=0the probabilistic decision rule becomes the same as in AS.Also,as in MM AS,in ACS only the best ant(the global-best or the iteration-best ant)is allowed to add pheromone after each algorithm iteration;the former is the most common choice in applications of ACS.The amount of pheromone deposited is then given by g(s b)=ρ/f(s gb), whereρis the pheromone evaporation.Finally,ACS also differs from most ACO algorithms because ants update the pheromone trails while building solutions(as in ant-quantity and in ant-density).In practice,ACS ants“eat”some of the pheromone trail on the edges they visit.This has the effect of decreasing the probability that the same path is used by all ants(that is,it favors exploration,counterbalancing this way the other two above-mentioned modifications that strongly favor exploitation of the collected knowledge about the problem).Similarly to MM AS,ACS also usually exploits local search to improve its performance.One could continue by enumerating the modifications that have been proposed in various other ACO algorithms that have been reported in the literature.Instead,we give an overview of the various developments on ACO algorithms for N P-hard problems in Table1.There we give for each of the main ACO variants that have been proposed,the main references to these algorithms, the year in which they have been proposed and whether they have been tested on the TSP.In fact,(published)tests of most ACO variants have been done on the TSP,which again confirms the central role of this problem in ACO research.4.2.3Applications to dynamic network routing problemsThe application of ACO algorithms to dynamic problems,that is,problems whose characteristics change while being solved,is among the main success stories in ACO.Thefirst such application [131]concerned routing in circuit-switched networks(e.g.,classical telephone networks).The proposed algorithm,called ABC,was demonstrated on a simulated version of the British Telecom network.The main merit of ABC was to stimulate the interest of ACO researchers in dynamic。

基于改进蚁群算法的优化方法及其应用IntroductionMetaheuristic algorithms are popular techniques used for solving complex optimization problems such as the traveling salesman problem, portfolio optimization, and many others. One of the famous metaheuristic algorithms is the Ant Colony Optimization (ACO) algorithm, which simulates the behavior of ants in finding the shortest path between their colony and a food source. However, the traditional ACO algorithm has some limitations that affect its performance in solving complex optimization problems. In this article, we will introduce an improved version of the ACO algorithm and its applications in various optimization problems.Chapter 1: Basic Ant Colony Optimization AlgorithmThe ACO algorithm is a population-based search algorithm that imitates the behavior of ants in finding the shortest path between their nest and food source. The algorithm consists of a set of ants that move through a graph and deposit pheromone trail on the edges they traverse. The pheromone trail acts as a form of communication among ants, and those edges with the highest pheromone concentration are more likely to be chosen by other ants.The basic steps of the traditional ACO algorithm are as follows:1. Set the number of ants, the initial pheromone concentration, and the heuristic value for each edge.2. Each ant selects a starting node and iteratively selects the next node based on a probabilistic rule that combines the pheromone trail and the heuristic value of each edge.3. After an ant completes a tour, the pheromone trail on each edge is updated based on the length of the tour. Edges with shorter tour length receive more pheromone.4. Repeat steps 2 and 3 until a stopping criterion is met.Chapter 2: Limitations of Basic ACO AlgorithmAlthough the traditional ACO algorithm is effective in solving many combinatorial optimization problems, it has some limitations that may affect its performance in solving more complex problems. Some of the limitations are:1. Premature Convergence: The ACO algorithm tends to converge prematurely to a local optimum, which means that it fails to explore the search space adequately, leading to suboptimal solutions.2. Stagnation: The algorithm can get stuck in a local optimum due to the lack of exploration.3. Inefficient Parameter Tuning: The performance of the ACO algorithm highly depends on parameter values such as the pheromone evaporation rate, the initial pheromone value, and the visibility of theedges. The selection of appropriate parameter values can be challenging and time-consuming.Chapter 3: Improved Ant Colony Optimization AlgorithmTo address the limitations of the basic ACO algorithm, several improved versions have been proposed. One of the popular improved ACO algorithms is the Max-Min Ant System (MMAS) algorithm that ensures better exploration and avoids premature convergence.The MMAS algorithm introduces several enhancements that improve the performance of the basic ACO algorithm. These enhancements include:1. Pheromone Updating Rule: The MMAS algorithm uses a max-min strategy to update the pheromone trail. Each edge's pheromone concentration is bounded by a maximum and minimum value to ensure proper pheromone evaporation and allow better exploration of the search space.2. Pheromone Initialization: The initial value of the pheromone concentration is set to a higher value than the traditional ACO algorithm to encourage global exploration.3. Dynamic Parameter Tuning: The algorithm uses a dynamic parameter tuning mechanism that adjusts the parameter values based on the current state of the search. This tuning mechanism helps to find a balance between exploration and exploitation.The MMAS algorithm has been successfully applied in many optimization problems such as the Traveling Salesman Problem, Quadratic Assignment Problem, and many others.Chapter 4: Applications of Improved ACO AlgorithmThe improved ACO algorithm has been applied in many real-world optimization problems such as:1. Wireless Sensor Network Optimization: The optimization of Wireless Sensor Networks (WSNs) is a challenging task due to the complex nature of the network topology. The ACO algorithm has been used to optimize the WSN topology for better energy efficiency, coverage, and connectivity.2. Vehicle Routing Problem: The Vehicle Routing Problem (VRP) is a combinatorial optimization problem where a set of vehicles has to visit a set of customers while minimizing the total distance traveled. The ACO algorithm has been used to optimize the route taken by the vehicles to minimize the total distance traveled.3. Image Segmentation: Image segmentation is a critical task in computer vision that involves dividing an image into separate regions. The ACO algorithm has been used to segment medical images for better diagnosis and treatment.ConclusionThe Ant Colony Optimization algorithm has been successfully applied in many optimization problems, but its performance can be further improved by introducing several enhancements. The Max-Min Ant System algorithm is an improved version of the ACO algorithm that ensures better exploration and avoids premature convergence. The improved ACO algorithm has been applied in many real-world optimization problems such as Wireless Sensor Network Optimization, Vehicle Routing Problem, and Image Segmentation.。

蚁群算法外文文献翻译(含：英文原文及中文译文)英文原文The ant colony optimizationOriginAnt colony algorithm (ant colony optimization, ACO), also known as ant algorithm is a probability -based algorithm is used to find the optimal path in the graph . It in his doctoral thesis proposed by Marco Dorigo in 1992 , inspired by the behavior of the path of the ants found in the process of looking for food . Ant colony algorithm is a simulated evolutionary algorithm , preliminary studies show that the algorithm has many excellent properties. Optimization design for PID controller parameters , the results of the results of the ant colony algorithm design and genetic algorithm are compared , the numerical simulation the results show that ant colony algorithm is a new kind of simulated evolutionary optimization effectiveness and value.principleEach ant began to search for food without first telling them where the food was. When a food is found, it will release a pheromone to the environment, attracting other ants, so that more and more ants will find food! Some ants do not always repeat the same path as other ants. Theywill In a different way, if the road to development is shorter than the other roads, then gradually more ants are attracted to this shorter road. Finally, after a period of time, it may appear that the shortest path is repeated by most ants.Why are small ants able to find food? Are they intelligent? Let's imagine if we want to design an AI program for ants, how complex is this program? First, if you want ants to be able to avoid obstacles, you must According to the appropriate terrain, it is programmed to allow them to avoid obstacles, and secondly, to allow ants to find food, they need to be traversed by all points in space; again, if the ant is to find the shortest path, then Y ou need to calculate all possible paths and compare their sizes, and more importantly, you must be careful programming, because the mistakes of the program may make you work harder. This is an incredible process! Too complicated, I am afraid no one can complete such a cumbersome and redundant program.However, the facts are not as complicated as you think. The above program has only 100 lines of code for each ant's core program! Why is such a simple program to make ants do such complicated things? The answer is: the emergence of simple rules. In fact, each ant does not need to know the information of the whole world as we imagine. They actually only care about the immediate information in a very small range, and use these simple rules to make decisions based on these local information. Inthe collective, complex behaviors will emerge. This is the law of the interpretation of artificial life and complexity science! So what are these simple rules?1 ScopeThe range observed by ants is a checkered world. An ant has a parameter that is a speed radius (typically 3). The range it can observe is a 3*3 checkered world, and the distance that can be moved is also in this range. Inside.2, the environmentThe ant's environment is a virtual world. There are obstacles, other ants, and pheromones. There are two kinds of pheromones. One is the food pheromone that is found by the ants who find food. One is to find the nest. The ants sprinkled the pheromone of the nest. Each ant can only sense the environmental information within its range. The environment makes the pheromone disappear at a certain rate.3, foraging rulesFind out if there is food in the range that each ant can sense, and if so, go straight. Otherwise, if there are pheromones and if there are more pheromones in the perceivable range, then it will go to places with a large number of pheromones, and each ant will make mistakes with a small probability, and not go to the information. Most points move. The rules for the ant to find the nest are the same as above, except that it respondsto the pheromone of the nest and does not respond to the food pheromone.4, move rulesEach ant moves in the direction of the most pheromone, and, when there is no pheromone guidance around it, the ant will continue to move in the direction of its original movement, and there will be a random, small disturbance in the direction of movement. In order to prevent the ant from turning in circles, it will remember which points it has recently passed. If it is found that the next point to be taken has recently passed, it will try to avoid it.5, obstacle avoidance rules:If the ant is moving in the direction of obstacles, it will randomly choose another direction, and if there are pheromone guidelines, it will follow the rules of foraging.6, broadcast pheromone rules:Each ant spreads the most pheromones when he first finds food or nests. As he goes further, the number of pheromones sown is decreasing.According to these rules, there is no direct relationship between ants, but each ant interacts with the environment, and through the pheromone ties, the ants are actually associated with each other. For example, when an ant finds food, it does not directly tell other ants that there is food. Instead, it broadcasts pheromones to the environment. When other antspass near it, they will feel the existence of pheromone, and according to The pheromone guide found the food.problem:Having said so much, how did an ant find food? When no ants find food, there is no useful pheromone in the environment. Why do ants find food relatively efficiently? This is due to the ants' movement rules, especially It is a movement rule when there is no pheromone. First of all, it must be able to maintain a certain inertia as much as possible so that the ant moves forward as far as possible (starting, this front is a randomly fixed direction), rather than spinning or shaking in situ; second, the ants must have some randomness. Although there is a fixed direction, it cannot move in a straight line like a particle, but it has a random disturbance. This makes the ant move with a certain purpose, try to keep the original direction, but there are new temptations, especially when it comes to obstacles, it will immediately change direction, this can be seen as a selection process, also It is the environmental obstacles that make ants correct in one direction, but not in other directions. This explains why a single ant can still find a well-kept food in a complex map such as a maze. Of course, when an ant finds food, most ants will find food quickly along the pheromone. However, it is not ruled out that at the beginning, some ants randomly selected the same path. As more pheromones are released by the ants on this path, more ants also choose this path. However, thispath is not optimal (ie, shortest). Therefore, after the number of iterations is completed, the ant finds that it is not an optimal solution but a suboptimal solution. The result of this case may be of practical significance. Not much.How does the ant find the shortest path? This is due to the pheromone, and to the environment, specifically the computer clock. Where there are many places of information, it is obvious that there will be more ants here, so more ants will come together. If there are two roads leading from the nest to food, the number of ants that follow the two roads is the same (or there are more ants on the longer roads, which does not matter). When the ant arrives at the end point along a road, he will return immediately, so that short ant roads will be shorter and shorter, which means that the frequency of repetition will be faster, and therefore there will be more ants walking through the unit time. The number of pheromones will naturally drop. Naturally there will be more ants being attracted to it, so that more pheromones will be sprayed. The long road is the opposite. Therefore, more and more ants are gathering to The shortest path was found and the shortest path was found. Some people may ask the question of the shortest path locally and the shortest part of the overall situation. In fact, the ants gradually approach the shortest path of the whole world. Why? It is because ants make mistakes, that is, they will not go to places with high pheromone according to a certain probability.Going for another way, this can be understood as an innovation. If this innovation can shorten the road, then according to the principle just described, more ants will be attracted.ExtendedWhat do you find after following the trail of ants? Through the above principle of narrative and practical operation, it is not difficult to find that ants have intelligent behavior, due entirely to its simple rules of conduct, and these rules are integrated into the following two aspects. Features:1, diversity2, positive feedbackThe diversity ensures that the ants do not walk into the dead-end and loop indefinitely while feeding, and the positive feedback mechanism ensures that relatively good information can be preserved. We can think of diversity as a creative ability, and positive feedback as a learning reinforcement. The power of positive feedback can also be compared to the authoritative opinion, and diversity is the creativeness of breaking the authority. It is precisely these two carefully combined cleverness that has led to the emergence of smart behavior.In terms of extension, the evolution of nature, the progress of society, and the innovation of mankind are all inseparable from these two things. The diversity guarantees the system's ability to innovate, and positivefeedback ensures that good features can be strengthened, and both must be just right. Combine. If there is excess diversity, that is, if the system is too active, this is equivalent to excessive random movement of ants, and it will be trapped in chaos. On the contrary, if diversity is not enough and positive feedback mechanism is too strong, then the system will be like a pool of stagnant water. In terms of ant colonies, this is manifested in the fact that the behavior of ants is too rigid. When the environment changes, ant colonies still cannot adjust properly. Since complexity and intelligence behaviors emerge according to the underlying rules, since the underlying rules have diversity and positive feedback characteristics, then perhaps you would ask where these rules come from? Where does diversity and positive feedback come from? My own opinion: Rules come from the evolution of nature. The evolution of nature is based on the clever combination of diversity and positive feedback. And why is this clever combination? Why is the world presented before your eyes so lifelike? The answer is that the environment has created all of this. The reason why you see a vivid world is because those who cannot adapt to the diversity of the environment and The combination of positive feedback has been dead and eliminated by the environment!Ant Colony Algorithm Features1) Ant Colony Algorithm is a self-organizing algorithm. In system theory, self-organization and its organization are two basic categories oforganizations. The difference is whether organizational or organizational instructions come from within or outside the system. From the inside of the system, they are self-organizing. Outside the system is his organization. If the system has no specific external intervention during the process of obtaining space, time, or functional structure, we say that the system is self-organizing. In an abstract sense, self-organization is the process of making the system increase without external action (that is, the system changes from disorder to order). The ant colony algorithm satisfactorily satisfies this process. The ant colony optimization is used as an example. At the beginning of the algorithm, a single artificial ant finds the solution disorderly. After a period of evolution of the algorithm, artificial ants increasingly tend to find a near optimal solution through the role of the information hormone. This is a no Orderly process.2) The ant colony algorithm is an essentially parallel algorithm. Each ant searches independently of each other and only communicates through the information hormone. So ant colony algorithm can be regarded as a distributed multi-agent system. It begins independent search at multiple points in the problem space, which not only increases the reliability of the algorithm, but also makes the algorithm have a strong global search capability. .3) The ant colony algorithm is a positive feedback algorithm. From the process of foraging of real ants, it is not difficult to see that ants canultimately find the shortest path and rely directly on the accumulation of information hormones on the shortest path. However, the accumulation of information hormones is a positive feedback process. For the ant colony algorithm, there is exactly the same information hormone in the environment at the initial moment, giving a slight disturbance to the system, making the trajectory concentration on each side not the same, and the solution constructed by ants has advantages and disadvantages. The feedback method used by the algorithm It is to leave more information hormones in the path of better solutions, and more information hormones attract more ants. The process of positive feedback makes the initial differences continue to expand, while guiding the entire system. Evolution toward the optimal solution. Therefore, positive feedback is an important feature of the ant algorithm, which enables the algorithm evolution process to proceed.(4) Ant colony algorithm has strong robustness. Compared with other algorithms, the ant colony algorithm does not require a high initial route. That is, the result of the ant colony algorithm does not depend on the selection of the sub-original route, and no manual adjustment is required during the search. Secondly, the number of parameters of the ant colony algorithm is small, the setting is simple, and the ant colony algorithm is easy to apply to other combinatorial optimization problems.The ant colony optimization algorithm was originally used to solvethe TSP problem. After years of development, it has penetrated into other fields, such as graph coloring, large-scale integrated circuit design, routing problems in communication networks, load balancing, and vehicle scheduling. Wait. The ant colony algorithm has been successfully applied in several fields, and the most successful one is in the application of combinatorial optimization.In the process of network routing, the traffic distribution of the network constantly changes, and the network links or nodes randomly fail or rejoin. The ant colony's self-catalysis and forward feedback mechanism exactly match the solution characteristics of this type of problem. Therefore, the ant colony algorithm has been applied in the network field. The parallel and distributed nature of ant colony foraging behavior makes the algorithm particularly suitable for parallel processing. Therefore, implementing parallel execution of the algorithm has great potential for solving a large number of complex practical application problems.If there is a large number of individuals without intelligence in a group, the intelligent behavior they exhibit through simple cooperation between them is called Swarm Intelligence. Communication on the Internet is nothing more than the result of the interaction of neurons (human brains) through the Internet. Optical cables and routers are merely extensions of axons and synapses. From the perspective ofself-organization phenomenon, there is no essential difference between the intelligence of the human brain and the ant colony. A single neuron has no intelligence and no single ant, but the system formed by connection is an agent.中文译文蚁群算法起源蚁群算法(ant colony optimization, ACO),又称蚂蚁算法,是一种用来在图中寻找优化路径的机率型算法。

25个经典的元启发式算法元启发式算法（metaheuristic algorithm）是一种用于解决复杂优化问题的算法。

它们通常基于自然界的现象或者数学模型，能够快速、有效地寻找到全局最优解或者接近最优解。

下面我们来看看25个经典的元启发式算法。

1.蚁群算法（Ant Colony Optimization）:模拟蚂蚁寻找食物的行为，用于解决组合优化问题。

2.遗传算法（Genetic Algorithm）:模拟生物进化的过程，用于解决优化问题。

3.粒子群算法（Particle Swarm Optimization）:模拟鸟群觅食的行为，用于解决连续优化问题。

4.模拟退火算法（Simulated Annealing）:模拟固体退火的过程，用于解决组合优化问题。

5.蚁狮算法（Ant Lion Optimizer）:模拟蚁狮捕食的过程，用于解决连续优化问题。

6.蝗虫优化算法（Grasshopper Optimization Algorithm）:模拟蝗虫觅食的行为，用于解决优化问题。

7.人工蜂群算法（Artificial Bee Colony Algorithm）:模拟蜜蜂寻找花蜜的行为，用于解决优化问题。

8.蝇算法（Fly Algorithm）:模拟蝇类觅食的行为，用于解决优化问题。

9.骨骼优化算法（Skeleton-based Optimization Algorithm）:模拟骨骼结构的优化过程，用于解决优化问题。

10.人工鱼群算法（Artificial Fish Swarm Algorithm）:模拟鱼群觅食的行为，用于解决优化问题。

11.基因表达式编程算法（Gene Expression Programming Algorithm）:模拟基因调控的过程，用于解决优化问题。

12.蚁狮算法（Ant Lion Optimizer）:模拟蚁狮捕食的过程，用于解决连续优化问题。

13.蝗虫优化算法（Grasshopper Optimization Algorithm）:模拟蝗虫觅食的行为，用于解决优化问题。

蚁群算法变异算子Ant Colony Optimization (ACO) is a metaheuristic algorithm inspired by the foraging behavior of ants in nature. It simulates the ant's ability to find the shortest path between their nest and a food source through pheromone deposition and trail following. However, to maintain diversity and avoid premature convergence, a crucial component of ACO is the mutation operator.蚁群优化（ACO）是一种元启发式算法，它受到自然界中蚂蚁觅食行为的启发。

它模拟了蚂蚁通过释放信息素和跟随轨迹来找到巢穴和食物源之间最短路径的能力。

然而，为了保持多样性并避免过早收敛，ACO中的一个关键组成部分是变异算子。

The mutation operator in ACO introduces controlled random perturbations to the ants' behavior, allowing them to explore alternative paths and potentially discover better solutions. This randomness is carefully balanced with the exploitation of existing knowledge, ensuring that the search process remains both efficient and effective.蚁群优化中的变异算子通过引入受控的随机扰动来影响蚂蚁的行为，使它们能够探索替代路径并可能发现更好的解决方案。

蚂蚁算法步骤Ant Colony Optimization (ACO) is a metaheuristic algorithm inspired by the foraging behavior of ants in nature. 蚁群优化算法（ACO）是受到自然界中蚂蚁觅食行为启发的一种元启发式算法。

It was initially proposed by Marco Dorigo in the early 1990s and has since been widely applied to solve combinatorial optimization problems. 最初是由Marco Dorigo在上世纪90年代初提出的，并且已经被广泛应用于解决组合优化问题。

The main idea behind ACO is to simulate the way ants communicate with each other through pheromone trails to find optimal paths between their nest and food sources. ACO的主要思想是模拟蚂蚁通过信息素路径相互通信以找到从巢穴到食物源的最佳路径。

In this algorithm, artificial ants iteratively construct solutions by moving from one solution component to another based on a combination of heuristic information and pheromone trails. 在这种算法中，人工蚂蚁通过根据启发信息和信息素路径的组合从一个解决方案组件移动到另一个解决方案组件来迭代构建解决方案。

The pheromone trails are updated by the artificial ants based on the quality of the solutions they find, with stronger solutions resulting in higher pheromone deposition. 信息素路径根据人工蚂蚁找到的解决方案的质量进行更新，更强的解决方案会导致更高的信息素沉积。