一种解决矩形布局问题的启发式快速算法

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一种求解矩形块装填问题的启发式快速算法

第２９卷第９期
２０１３年９月
科技通报
ＢＵＬＬＥＴＩＮ０ＦＳＣＩＥＮＣＥＡＮＤＴＥＣＨＮ０ＬＯＧＹ
Ｖｏ１．２９Ｎｏ．９Ｓｅｐ．２０１３
一
种求解矩形块装填问题的启发式快速算法
陈战胜，钮文良，王辉，胡正坤
（１．北京联合大学应用科技学院，北京１０２２００；２．北京交通大学计算机与信息技术学院，北京１０００４４）
摘要：在最低水平线算法基础上，结合工匠砌墙策略，引入矩形块聚类思想，提出了一种矩形块装填问题的改进算法。该算法能使得矩形块装填紧凑，有效提升容器的利用率和提高算法执行速度。实验
仿真结果表明，与ＢＬ、ＳＡ＋ＢＬＦ及ＰＨ相比，本文提出的改进算法对于求解矩形块装填问题十分有效。关键词：矩形装填；最低水平线；砌墙策略；聚类；ＢＬ算法
中图分类号：ＴＰ３９１文献标识码：Ａ文章编号：１００１ — ７１１９（２０１３）０９ — ００６８ — ０４
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ＣｈｅｔｒＺｈａｎｓｈｅｎｆ＇２，ＮｉｕＷｅｎＬｉａｎｇ，ＷａｎｇＨｕｉＩ，ＨｕＺｈｅｎｇｋｕｎ

求解矩形Packing问题的基于遗传算法的启发式递归策略

Vol.33,No.9ACTA AUTOMATICA SINICA September,2007 An Improved Heuristic Recursive Strategy Based on Genetic Algorithm for the Strip RectangularPacking ProblemZHANG De-Fu1CHEN Sheng-Da1LIU Yan-Juan1Abstract An improved heuristic recursive strategy combining with genetic algorithm is presented in this paper.Firstly,this method searches some rectangles,which have the same length or width,to form some layers without waste space,then it uses the heuristic recursive strategies to calculate the height of the remaining packing order and uses the evolutionary capability of genetic algorithm to reduce the height.The computational results on several classes of benchmark problems have shown that the presented algorithm can compete with known evolutionary heuristics.It performs better especially for large test problems.Key words Strip packing problems,heuristic,recursive,genetic algorithm1IntroductionMany industrial applications,which belong to cutting and packing problems,have been found.Each application incorporates diﬀerent constraints and objectives.For ex-ample,in wood or glass industries,rectangular components have to be cut from large sheets of material.In warehousing contexts,goods have to be placed on shelves.In newspapers paging,articles and advertisements have to be arranged in pages.In the shipping industry,a batch of objects of var-ious sizes has to be shipped to the maximum extent in a larger container,and a bunch of opticalﬁbers has to be accommodated in a pipe with perimeter as small as possi-ble.In very-large scale integration(VLSI)ﬂoor planning, VLSI has to be laid out.These applications have a similar logical structure,which can be modeled by a set of pieces that must be arranged on a predeﬁned stock sheet so that the pieces do not overlap with one another,so they can be formalized as the packing problem[1].For more extensive and detailed descriptions of packing problems,the reader can refer to[1∼3].A two-dimensional strip rectangular packing problem is considered in this paper.It has the following characteris-tics:a set of rectangular pieces and a larger rectangle with aﬁxed width and inﬁnite length,designated as the con-tainer.The objective is toﬁnd a feasible layout of all the pieces in the container that minimizes the required con-tainer length and,where necessary,takes additional con-straints into account.This problem belongs to a subset of classical cutting and packing problems and has been shown to be non-deterministic polynomial(NP)hard[4,5].Opti-mal algorithms for orthogonal two-dimension cutting were proposed in[6,7].Gilmore and Gomory[8]solved prob-lem instances to optimality by linear programming tech-niques in1961.Christoﬁdes and Whitlock[9]solved the two-dimensional guillotine stock cutting problem to opti-mality by a tree-search method in1977.Cung et al.[10] developed a new version of the algorithm proposed in Hiﬁand Zissimopolous that used a best-ﬁrst branch-and-bound approach to solve exactly some variants of two-dimensional stock-cutting problems in2000.However,these algorithms might not be practical for large problems.In order to solve large problems,some heuristic algorithms were developed. Received June20,2006;in revised form October24,2006 Supported by Academician Start-up Fund(X01109),985Informa-tion Technology Fund(0000-X07204)in Xiamen University1.Department of Computer Science,Xiamen University,Xiamen 361005,P.R.ChinaDOI:10.1360/aas-007-0911The most documented heuristics are the bottom-left(BL), bottom-left-ﬁll(BLF)methods,and other heuristics[11∼13]. Although their computational speed is very fast,the so-lution quality is not desirable.Recently,genetic algo-rithms and its improved algorithms for the orthogonal pack-ing problem were proposed because of their powerful op-timization capability[14∼17].Kroger[14]used genetic algo-rithm for the guillotine variant of bin packing in1995. Jakobs[15]used a genetic algorithm for the packing of poly-gons using rectangular enclosures and a Bottom-left heuris-tic in1996.Liu et al.[16]further improved it.Hop-per and Turton[18]evaluated the use of the BLF heuris-tic with genetic algorithms on the nonguillotine rectangle-nesting problem in1999.In addition,an empirical in-vestigation of meta-heuristic and heuristic algorithms of the strip rectangular packing problems was given by[19]. Recently,some new models and algorithms were devel-oped by[20∼26].For example,quasi-human heuristic[20], constructive approach[21,22],a new placement heuristic[23], heuristic recursion(HR)algorithm[24],and hybrid heuristic algorithms[25,26]were developed.These heuristics are fast and eﬀective,especially,the bestﬁt in[23]not only is very fast,but alsoﬁnds better solutions than some well-known metaheuristics.In this paper,an improved heuristic recursive algorithm that combines with genetic algorithm is presented to solve the orthogonal strip rectangular packing problem.The computational results on a class of benchmark problems show that this algorithm can compete with known evolu-tionary heuristics,especially in large test problems.The rest of this paper is organized as follows.In Section 2,a clear mathematical formulation for the strip rectangu-lar packing problem is given.In Section3,the heuristic recursive algorithm is presented,and an improved heuris-tic recursive algorithm(IHR)is developed in detail.In Section4,the GA+IHR algorithm is puta-tional results are described in Section5.Conclusions are summarized in Section6.2Mathematical formulation of the problemGiven a rectangular board of given width and a set of rectangles with arbitrary sizes,the strip packing problem of rectangles is to pack each rectangle on the board so that no two rectangles overlap and the used board height is min-imized.This problem can also be stated as follows.912ACTA AUTOMATICA SINICA Vol.33Given a rectangular board with given width W,and n rectangles with length l i and width w i,1≤i≤n,let (x li,y li)denote the top-left corner coordinates of rectangle i,and(x ri,y ri)the bottom-right corner coordinates of rect-angle i.Other symbols are similar to[24].For all1≤i≤n, the coordinates of rectangles must satisfy the following con-ditions:1)(x ri−x li=l i and y li−y ri=w i)or(x ri−x li=w i and y li−y ri=l i);2)For all1≤j≤n,j=i,rectangle i and j can not overlap,namely,x ri≤x lj or x li≥x rj or y ri≥y lj or y li≤y rj;3)x L≤x li≤x R,x L≤x ri≤x R and y R≤y li≤h,y R≤y ri≤h.The problem is to pack all the rectangles on the board such that the used board height h is minimized.It is noted that for orthogonal rectangular packing prob-lems the packing process has to ensure the edges of each rectangle are parallel to the x-and y-axes,respectively, namely,all rectangles can not be packed aslant.In addi-tion,all rectangles except the rectangular board are allowed to rotate90degrees.3IHR algorithmThe HR algorithm for the strip rectangular packing prob-lem was presented in[24].It follows a divide-and-conquer approach:break the problem into several subproblems that are similar to the original problem but smaller in size,solve the subproblems recursively,and then combine these solu-tions to create a solution to the original problem.The HR algorithm is very simple and eﬃcient,and can be stated as follows:1)Pack a rectangle into the space to be packed,and divide the unpacked space into two subspaces.2)Pack each subspace by packing it recursively.If the subspace size is small enough to only pack a rectangle,then just pack this rectangle into the subspace in a straightfor-ward manner.3)Combine the solutions to the subproblems for the so-lution of the rectangular packing problem.In order to enhance the performance of the HR algo-rithm,the author in[18]presented some heuristic strate-gies to select a rectangle to be packed,namely,the rect-angle with the maximum area is given priority to pack.In detail,unpacked rectangles should be sorted by nonincreas-ing ordering of area size.The rectangle with maximum area should be selected to packﬁrst if it can be packed into the unpacked subspace.In addition,the longer side of the rect-angle to be packed should be packed along the bottom side of the subspace.It is the disadvantage of the HR algorithm that may have waste space in each layer(See Fig.1).In order to overcome this disadvantage,some layers without waste space areﬁrst considered.Some deﬁnitions are given to clearly describe the idea of the improved algorithm.Deﬁnition1.The reference rectangle is the rectangle that is packedﬁrstly and can form one layer with other rectangles,and its short edge is the height of that layer.Deﬁnition2.The combination layer is the layer that has no waste space,and the rectangles packed into it have the same height as the height of the layer and are spliced together one by one along the direction of W.The sum of the edge length of the rectangles along the W direction is the combination width.From Fig.1to Fig.4,each rectangle at the top of the container is the referee rectangle.Theﬁrst layer in Fig.1 is not a combination layer because it has waste space.The ﬁrst layer in Fig.2is a combination layer because it has no waste space and the spliced rectangles have the same width or length as the height of the layer.Although the second layer in Fig.2has no waste space,it is not a com-bination layer because the two middle rectangles are not spliced along the direction of W.By the deﬁnition of the combination layer,the combination width is W.If some combination layers are found before further computation, then they may decrease the cost of computation because the rectangles already packed into these combination lay-ers will not be considered in thefuture.From the above discussion,it is very important toﬁnd the combination layer.Given a packing ordering,the pro-cedure ofﬁnding the combination layer is given as follows.Find theﬁrst unpacked and unreferenced rectangle as the reference rectangle,and put this rectangle into a two-dimensional array.Then seek downwards from the refer-ence rectangle orderly.If one canﬁnd a rectangle whose length or width is equal to the width of the reference rect-angle,then put this rectangle into the two-dimensional ar-ray,repeat this until a combination layer or no combination layer is found.Repeat the above process until all rectan-gles are packed otherwise no rectangle can be the reference rectangle.In this process,the number of rectangles,which have been packed in the current layers,must be recorded. Finally,the number of all the combination layers must be recorded.The steps of the combination operator can be stated as follows:Combination()RepeatFind theﬁrst unpacked and unreferencedrectangle as the reference rectangle,and put thisrectangle into a two-dimensional array;For i=current position to nIf(the width or the length of rectangle i isequal to the width of the reference rectangle)If(combination width<W)Put the rectangle into the two-dimensionalarray;Else if(combination width=W)Pack all the rectangles of theNo.9ZHANG De-Fu et al.:An Improved Heuristic Recursive Strategy Based on Genetic Algorithm for (913)two-dimensional array on the container;Break;Record the number of all packed rectangles;Until all rectangles are packed or no rectangle is thereference rectangle;Record the number(Num)of all the combination layers;So,the IHR algorithm can be stated as follows:Step1.The combination layers are searched.Step2.Pack the remaining rectangles to the container by HR algorithm.By intuition,the more the number of the combination layers is,the faster the computational speed is.However, it is not always true that more combination layers can ob-tain a better solution.From Fig.1to Fig.4,we know that the best combination layer number is1.Fig.1has no com-bination layer,but layer1,layer3,and layer4waste a little space.Fig.2has one combination layer and is the optimal solution.Fig.3has two combination layers but layer3and layer4waste a little space.Fig.4has three combination layers,but layer4and layer5waste a little space.4GA+IHR4.1Genetic algorithm(GA)GA is a heuristic method used toﬁnd approximate solu-tions to hard optimization problems through application of the principles of evolutionary biology to computer science. It is modeled loosely on the principles of the evolution via natural selection,which use a population of individuals that undergo selection in the presence of variation inducing op-erators such as recombination(crossover)and mutation.In order to run GA,we must be able to create an initial pop-ulation of feasible solutions.The initial population is very important for achieving a good solution.There are many ways to do this based on the form of the problems.The evolution starts from a population of completely random in-dividuals and happens in generations.In each generation, theﬁtness of the whole population is evaluated.Multiple individuals are stochastically selected from the current pop-ulation and are modiﬁed to form a new population,which becomes current population in the next iteration of the algorithm.Further detailed theoretical and practical de-scriptions of genetic algorithm,the interested reader can refer to[27].Combining GA with IHR,the GA+IHR algorithm to solve the strip rectangular packing problem can be stated as follows:GA+IHR()Sort all rectangles by non-increasing ordering of area size;Combination();For i=0to NumInitialization();For j=1to NumberFor k=1to N/2Select two individuals in the parentsrandomly,then crossover with probabilityP c or copy with probability(1−P c)tocreate two middle oﬀspring;Mutate the middle oﬀspring withprobability P m;Compare the parent and the middle oﬀspring,if theﬁtness of the middle oﬀspring is less thanthe parent s,we accept it as new oﬀspring,otherwise we accept it with probability P b oraccept the parent with probability(1−P b);Select the best solution from the parents;Select the worse solution from the newgeneration;Replace the worse solution with the bestsolution;Save the best solution acquired from combinationlayer i to array A;Select the best solution from array A;where Num is the number of all the combination layers; Number is the iteration number of genetic algorithm;N is the number of population;P c=0.8,P m=0.2,but P m will increase as the parent chromosomes become more alike,P b=0.33.Theﬁtness value of genetic algorithm is calculated by HR algorithm.The required container length is the sum of combination height and currentﬁtness value.4.2InitializationThe GA is used to optimize the solution for unpacked rectangles.Theﬁtness value of GA is calculated by HR algorithm.In this paper,a string of integers,which forms an index into the set of rectangles,is used,and then the HR strategy is used to create the sequence of theﬁrst indi-viduals,and the sequence is divided into two equal parts, the two parts of theﬁrst individuals are then permuted to obtain N−1individuals(N is the size of population).The method of the permutation is to produce a point in each range randomly and exchange the position of the point for its neighbor.This initialization method can keep the diver-siﬁcation of each individual in the population.At the same time,it can keep the individual,which has betterﬁtness. From the experiment results,we know that the method has better eﬀect.4.3CrossoverThe role of the crossover operator is to allow the advanta-geous traits to spread throughout the population such that the population as a whole may beneﬁt from this chance dis-covery.The steps of the crossover operator are as follows:1)Choose two individuals Parent1and Parent2from the parents randomly.2)Get the items of individual Child1from Parent1and ly,if the sequence number of the Child1is odd,ﬁnd an item orderly in Parent1until the item is diﬀerent from all the items in Child1,otherwise ﬁnd an item orderly in Parent2until the item is diﬀerent from all the items in Child1.When the number of Child1 is n,a new individual is created successfully.3)Similarly,get the items of individual Child2from Parent2and ly,if the sequence number of the Child2is odd,ﬁnd an item orderly in Par-ent2until the item is diﬀerent from all the items in Child2, otherwiseﬁnd an item orderly in Parent1until the item is diﬀerent from all the items in Child2.When the number of Child2is n,a new individual is created successfully.As an example of crossover,suppose two individuals are already selected:Parent1:53267841Parent2:86517324 According to the steps of the crossover operator,the sequences of the children can be obtained:Child1:58362174Child2:856312744.4MutationIn each individual A,two diﬀerent points are chosen ran-domly,and the sequence within two points is inversed,then914ACTA AUTOMATICA SINICA Vol.33in the appointed iteration step,judge theﬁtness of the new individual B,if theﬁtness of B is less than that of A,B is accepted.As an example of mutation,suppose two mutation points (3and6)are already selected:A:24587136After mutation,we can get the new individualB:24178536Mutation is adaptive,that is,the mutation rate increases as the parent chromosomes become more alike.4.5ReplacementAfter the operations of crossover and mutation,a set of solutions are produced.For keeping the bestﬁtness and quickening the speed of convergence,a best solution is se-lected from the set of solutions,and it is saved to the next generation in each iteration step.5Computational resultsIn order to compare the relative performance of the pre-sented GA+IHR with other published heuristic and meta-heuristic algorithms,several test problems taken from the literature are used.Perhaps the most extensive instances given for this problem are found in[19],where21prob-lem instances are presented in seven diﬀerent sized cate-gories ranging from16to197items.The optimal solu-tions of these21instances are all known.Table1(see next page)presents an overview of the test problem Class1from [19].As we wanted to extensively test our algorithm,other test problems were generated at random.Table2shows an overview of the test problem Class2generated randomly with known optimal solution.The problem Class2can be accessed in[23].In order to verify the performance of GA+IHR,two best meta-heuristic GA+BLF and SA+BLF[23],Bestﬁt[13],and HR[19]are selected.The computational results are listed in Tables3and4.20iterative times are chosen for GA and 80iterative times are chosen in the mutation operation.On this test problem Class1,as listed in Table3,Gap of GA+IHR ranges from0.83to4.44with the average Gap 2.06.The average Gaps of GA+BLF,SA+BLF,Bestﬁt, and HR are4.57,4,5.69,and3.97,respectively.The av-erage Gap of GA+IHR is lower than those of GA+BLF, SA+BLF,Bestﬁt,and HR.And as listed in Table4,the average running time of GA+IHR is also lower than those of GA+BLF and SA+BLF,but is larger than that of Best ﬁt and HR.The packing results can be seen in Fig.5and Fig.6,where L denotes the optimal height.The heights of C11,C12,C13,and C72using GA+IHR are20,21,21,and 241,respectively.GA+IHR canﬁnd the optimal heights for C11,C23,and C32.In order to extensively test the performance of our al-gorithm for randomly generated instances,especially for larger instances,12problem instances ranging from10to 500were generated at random.The computational results are listed in Table5.For such problem,100iterative times are chosen for the GA and10iterative times are chosen in the mutation operation.Although our algorithm can ob-tain a better solution,it needs much time,especially for large problems.So for N12,40iterative times are chosen for the GA and10iterative times are chosen in the muta-tion operation.From Table5,we observe that the running time is acceptable.What is more,the Gap is better thanothers.Fig.5Packed results of C1forGA+IHRFig.6Packed results of C72for GA+IHR6ConclusionsIn this paper,the GA+IHR algorithm for the orthogo-nal stock-cutting problem has been presented.IHR is very simple and intuitive,and can solve the orthogonal stock-cutting problem eﬃciently.GA is an adaptive heuristic search algorithm.It has the capability of global search within the solution space.The idea of combination layers to reduce the number of unpacked rectangles has been used. During the process of iteration search,HR is called repeat-edly to calculate the height of an individual.As we know,ﬁnding the optimal solution is more diﬃcult for the packing problem as increasing the size of problem.But it can be overcome by using the characteristic of GA.The computa-tional results have shown that we can obtain the desirable solutions within acceptable computing time by combining GA with IHR.So GA+IHR can compete with other evolu-tion heuristic algorithms,especially for large test problems, it performs better.So GA+IHR may be of considerable practical value to the rational layout of the rectangular objects in the engineeringﬁelds,such as the wood,glass, and paper industry,the ship building industry,and textile and leather industry.Future work is to further improve the performance of GA+IHR and minimize the inﬂuence of the parameters selection,and extend this algorithm for three-dimensional rectangular packing problems.No.9ZHANG De-Fu et al.:An Improved Heuristic Recursive Strategy Based on Genetic Algorithm for (915)Table1Test problem Class1Problem category Number of items:n Optimal height Object dimensionC1(C11,C12,C13)16(C11,C13),17(C12)2020×20C2(C21,C22,C23)25(C21,C22,C23)1515×40C3(C31,C32,C33)28(C31,C33),29(C32)3030×60C4(C41,C42,C43)49(C41,C42,C43)6060×60C5(C51,C52,C53)73(C51,C52,C53)9090×60C6(C61,C62,C63)97(C61,C62,C63)120120×80C7(C71,C72,C73)196(C71,C73),197(C72)240240×160Table2Test problem Class2Problem category Number of items:n Optimal height Object dimension N1104040×40N2205030×50N3305030×50N4408080×80N550100100×100N66010050×100N77010080×100N88080100×80N910015050×150N1020015070×150N1130015070×150N12500300100×300Table3Gaps of GA+BLF,SA+BLF,Bestﬁt,HR,and GA+IHR for the test problem Class1C1C2C3C4C5C6C7Average GA+BLF4753445 4.57 SA+BLF46533344Bestﬁt11.67 6.79.9 3.87 2.93 2.5 2.23 5.69HR8.33 4.45 6.67 2.22 1.85 2.5 1.8 3.97 GA+IHR 3.33 4.44 2.22 1.67 1.110.830.83 2.06Table4Average running time of GA+BLF,SA+BLF,and GA+HRC1C2C3C4C5C6C7Average GA+BLF 4.619.2213.8359.93165.96396.463581.97604.57 SA+BLF 3.22711.06418.4452.13530.151761.0219274.413107.2 Bestﬁt0.00.00.00.000.0030.0050.0070.005 HR000.030.140.69 2.2136.07 5.59 GA+IHR0.88 1.52 2.249.5630.0465.56426.0476.55Table5Gaps of GA+BLF,SA+BLF,Bestﬁt,HR,and GA+IHR for the test problem Class2n Optimal heightGA+BLF SA+BLF BF Heuristic GA+IHRh Time(s)h Time(s)h Time(s)h Time(s)N1104040 1.02400.2445<0.01450.68 N22050519.2528.1453<0.0154 3.32 N3305052 2.65239.552<0.0151 6.18 N440808312.6838483<0.018313.09 N55010010652.31062281050.0110333.01 N6601001032611033101030.0110250.12 N7701001066711065541070.0110457.04 N8808085114285810840.018231.36 N9100150155443115517151520.01152185.94 N102001501542×10415460661520.021511154.18 N113001501558×1041553×1041520.031513763.17 N125003003134×1053126×1043060.063045864.27 Average Gap(%)-- 3.72- 3.85- 4.35- 3.46-916ACTA AUTOMATICA SINICA Vol.33References1Lodi A,Martello S,Monaci M.Two-dimensional packing problems:a 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Prob-lems,2002,2(1):33∼5622HiﬁM,Hallah R.A hybrid algorithm for the two-dimensional layout problem:the cases of regular and irregular shapes.International Transactions in Operational Research,2003, 10(3):195∼21623Burke E,Kendall G,Whitwell G.A new placement heuristic for the orthogonal stock-cutting problem.Operations Re-search,2004,52(4):655∼67124Zhang D F,Kang Y,Deng S.A new heuristic recursive algo-rithm for the strip rectangular packing puters &Operations Research,2006,33(8):2209∼221725Zhang D F,Deng A S,Kang Y.A hybrid heuristic algo-rithm for the rectangular packing problem.Lecture Notes in Computer Science,2005,3514:783∼79126Zhang D F,Liu Y J,Chen S D,Xie X G.A meta-heuristic algorithm for the strip rectangular packing problem.Lecture Notes in Computer Science,2005,3612:1235∼124127Davis L.Handbook of Genetic Algorithms.New York:Van Nostrand Reinhold,1991ZHANG De-Fu Associate professor atSchool of Information Science and Technol-ogy,Xiamen University.He received hisbachelor and master degrees in computa-tional mathematics from Xiangtan Univer-sity in1996and1999,respectively,and hisPh.D.degree in computer software and itstheory from Huazhong University of Sci-ence and Technology in2002.His researchinterest covers computational intelligence andﬁnancial data mining.Corresponding author of this paper. E-mail:dfzhangl@CHEN Sheng-Da Master student atXiamen University.He received his bach-elor degree from Jimei University in2004.His research interest is computational in-telligence.E-mail:cshengda@LIU Yan-Juan Master student in Xia-men University.She received her bache-lor degree from Shijiazhuang University ofEconomics in2004.Her research interest iscomputational intelligence.E-mail:jjj514@。

启发式算法介绍

启发式算法介绍
启发式算法（Heuristic Algorithm）是一种基于直观或经验构造的算法，主要用于解决复杂的优化问题。

其基本思想是模拟人类或自然界中蕴含的智慧和经验来寻找问题的最优解。

相对于传统的数学方法，启发式算法更加注重在近似解空间中进行搜索，从而能够快速找到较好的结果。

启发式算法有许多类型，包括但不限于遗传算法、鱼群算法、蚁群算法、粒子群算法等。

这些算法都提供了不同的机制来解决不同的问题，并且通常具有良好的适应性和可扩展性。

启发式算法常被应用于组合优化、约束优化、排队论、路径规划、生产调度等领域，并被证明在某些情况下能够为问题提供更好的解决方案。

然而，启发式算法也存在一些局限性。

例如，它在某些特殊情况下可能会得到很坏的答案或效率极差，但造成这些特殊情况的数据组合可能永远不会在现实世界出现。

因此，在使用启发式算法时，需要综合考虑其效果和实际问题的需求，选择合适的算法。

总之，启发式算法是一种基于经验和直观的算法，通过模拟自然界或人类的智慧来寻找问题的最优解。

它能够快速地找到较好的结果，但也需要考虑其局限性和适用范围。

矩形布局的启发式优化策略

ｐｓｃｎｅｄｕｔａｇｓｉｇｅｅｓｌｒｇｏｒａｄｘａｄｔｓａｅＴｅｅｕｆｈｕｅｃｌｘｍｐｏｅａｊｓｄｔｃｎｅｔｓｌｆａｉｅｅｉｎｏｍｐｎｓｐｃ．ｈｓｌｏｅｍｒａｅａｌｂａｅｏｈｉｎｂｆｎｅｉｒｔｔｎｉｅｓｏａｔｅｉａｔｆｅｔｎｌｐｓｄｕｔｎａｋｎｒｂｅｉｒｇｌｒＡｐｌｉｇｔｅｆａｉｌｇｏｈｗｓｈｔｈｔｍｐｃｏｃａｇｅｏｅａｊｓｒｍｅｔｏｐｃｉｇｐｏｌｍｅｕａ．ｐｙｎｅｓｅｒｉｎｔｓｈｂｅ
０引言
矩形布局问题作为一个具有Ｎ —ｈｄ的组合最优Ｐａｒ化问题，广泛存在于板材切割、排样、大规模集成电路设计、服装裁剪和印刷排版等领域。由于在有限时间内无法获得全局最优解，许多学者提出各种启发式算法ｌ１于解决不同领域的矩形布局问题，但很难找１用到一种对于各种不同特点的布局问题都有很好适应性的求解方法。通过对不同算例计算结果进行分析和比
Ａｂｓｒｃ：ｎｏｄｒｏｓｌｅｔｅｅｔｎｌａｋｎｒｂｅｏｅｂｓｓｆｅｓｌｇｏｌｏｉｍ，ｈｃａｇｅｔａｔＩｒｅｖｃａｇｅｃｉｇｐｏｌｍｎｔａｉｏａｉｅｒｉｎａｇｒｈｔｅｒｔｎｌｔｏｈｒｐｈｆｂｅｔｅ
间。算例结果表明，矩形调整对布局结果影响有规律，利用可行域算法求解矩形布局问题，简便、快捷、灵活、适

启发式算法详细讲解

启发式算法详细讲解
启发式算法（Heuristic Algorithm）也被称为启发算法或者近似算法，是一种通过启发式搜索的方式来解决问题的算法。

启发式算法与精确算法不同，它不保证最优解，但通常能够在合理的时间内找到较好的解。

启发式算法的基本思想是根据问题的特性和经验，使用一些启发式的规则或策略来指导搜索过程，以此来引导算法在搜索空间中找到可能更接近最优解的解。

具体来说，启发式算法通常包含以下步骤：
1. 初始解生成：通过某种方法生成一个初始解，可以是随机生成、基于经验的启发式规则生成等。

2. 邻域搜索：在当前解的周围搜索邻域解，通过一系列的局部搜索操作，如交换、插入、删除等，来生成新的解。

3. 评估函数：对新生成的解进行评估，评估函数用来衡量解的好坏程度，可以是目标函数值、代价函数值、质量评估值等。

4. 更新解：根据评估函数的结果，更新当前解为评估值更好的解。

5. 终止条件：根据预设的终止条件，判断是否终止搜索过程。

终止条件可以是找到满足要求的解或达到最大迭代次数等。

启发式算法的性能依赖于初始解的生成和邻域搜索操作的设计，以及评估函数的准确性。

在实际应用中，针对不同的问题，可以使用不同的启发式算法。

常见的启发式算法有贪婪算法、模拟退火算法、遗传算法、禁忌搜索等。

需要注意的是，启发式算法不能保证找到全局最优解，但可以在合理的时间内找到接近最优解的解。

启发式算法常常应用于那些NP难问题或解空间很大的问题中，可以在较短的时间内找到近似最优解，是一种非常实用的算法设计思想。

矩形件优化排样的一种启发式算法

矩形件优化排样的一种启发式算法
陈仕军;曹炬
【期刊名称】《计算机工程与应用》
【年(卷),期】2010(046)012
【摘要】对大规模矩形件正交排样问题,提出了一种快速高效的启发式排放算法.对当前的可排放位置(水平线),用贪婪算法从未排矩形件中选择可排放于该水平线的最优矩形件组合块;根据各个排放位置与其对应的矩形件组合块的匹配程度,选择最优的可排放位置(最优水平线)优先排放.在排放时,为了便于后续排放,先将待排放位置对应的矩形件组合决从低到高进行排序,再排放.对E.Hopper提供的规模最大的一类实例进行计算,排样率都在99%以上,平均排样率达到了99.38%,平均计算时间只用了1.12秒.与相关文献最好结果进行了比较,结果表明该文算法解决大规模的矩形件排样具有高效性.
【总页数】4页(P230-232,238)
【作者】陈仕军;曹炬
【作者单位】华中科技大学,数学与统计学院,武汉,430074;华中科技大学,数学与统计学院,武汉,430074
【正文语种】中文
【中图分类】TP301.6
【相关文献】
1.一个实用的矩形件优化排样启发式算法 [J], 罗意平;刘军;李兵;蒋庄德
2.一种矩形件优化排样算法的研究 [J], 张伟;安鲁陵;孙金虎
3.一种快速的有约束矩形件优化排样模型 [J], 彭文
4.矩形件优化排样改进的启发式算法与系统 [J], 罗意平;刘军
5.一种“一刀切”式矩形件优化排样混合算法 [J], 陈仕军;曹炬
因版权原因，仅展示原文概要，查看原文内容请购买。

一种求解矩形件排样问题的启发式算法

一种求解矩形件排样问题的启发式算法
陈学松;曹炬;方仍存
【期刊名称】《锻压技术》
【年(卷),期】2004(29)5
【摘要】研究在一定的矩形板材上排放所需要小矩形的优化排样问题 ,提出一种基于 4块结构排放模式的启发式算法 ,并且建立了动态规划模型。

通过对在计算机上随机产生的试验数据的模拟计算 ,获得了比遗传算法更好的解。

【总页数】3页(P26-28)
【关键词】矩形件;启发式算法;排样问题;计算机;遗传算法;优化排样;动态规划模型;求解;随机;矩形板
【作者】陈学松;曹炬;方仍存
【作者单位】广东工业大学应用数学学院;华中科技大学数学系
【正文语种】中文
【中图分类】TG386;O224
【相关文献】
1.分布估计算法求解矩形件排样优化问题 [J], 马康;高尚
2.矩形件排样最优化问题求解 [J], 张青;刘芳
3.基于蚁群优化算法求解矩形件排样问题 [J], 童科;毛力
4.矩形件优化排样的一种启发式算法 [J], 陈仕军;曹炬
5.求解矩形件排样问题的十进制狼群算法 [J], 罗强;饶运清;刘泉辉;李世红
因版权原因，仅展示原文概要，查看原文内容请购买。

求解带有预放置矩形块的布局问题的启发式算法

求解带有预放置矩形块的布局问题的启发式算法
刘景发;黄文奇
【期刊名称】《计算机应用研究》
【年(卷),期】2007(024)008
【摘要】在超大规模集成电路设计中,一些特别重要的部件,如RAM、ROM、CPU 等经常被优先放置,而其他元件则被两两互不重叠地放置在芯片的剩余区域.这类问题能被形式化为带有预放置矩形块的布局问题.基于占角和最大穴度优先的放置策略,为该问题的快速求解提供了一种高效的启发式算法.算法的高效性通过应用于标准电路MCNC得到了验证.
【总页数】3页(P119-121)
【作者】刘景发;黄文奇
【作者单位】南京信息工程大学,计算机与软件学院,南京,210044;华中科技大学,计算机学院,武汉,430074;华中科技大学,计算机学院,武汉,430074
【正文语种】中文
【中图分类】TP311
【相关文献】
1.求解带动不平衡约束的卫星舱布局问题的启发式算法 [J], 刘景发;高泽旭;龙羽正;姚永雷;刘文杰;刘朝霞
2.求解VLSI布局问题的启发式算法 [J], 陈矛;黄文奇
3.一种求解矩形块布局问题的拟物拟人算法 [J], 黄文奇;陈端兵
4.求解二维正交矩形布局问题的动态填空启发式算法 [J], 孙宝金;贺良华
5.求解风力发电机布局问题的超启发式算法研究 [J], 迟宗正;董绍正;郭童;任志磊;周宽久;郭禾
因版权原因，仅展示原文概要，查看原文内容请购买。

启发式算法

该可行解与最优解的偏离程度一般不能被预计，但在实际情况下，启发式算法通常能够在合理时间内得到非常不错的解。
启发式算法的特点是能够在搜索过程中利用问题自身的特性信息，从而指导搜索朝着更有希望的方向前进。
发展历程及现状
启发式算法的发展历程可以追溯到20世纪50年代，当时人们开始尝试使用启发式方法来求解一些复杂的优化问题。
随着计算机技术的快速发展，启发式算法得到了广泛的应用和研究，出现了许多不同类型的启发式算法，如模拟退火算法、遗传算法、蚁群算法等。
目前，启发式算法已经成为解决复杂优化问题的重要工具之一，在各个领域都得到了广泛的应用。
应用领域与前景
• 启发式算法的应用领域非常广泛，包括生产调度、交通运输、网络通信、数据挖掘、生物信息学等。
01
模拟生物进化过程，通过选择、交叉、变异等操作寻找全局最
优解。
粒子群优化算法
02
模拟鸟群觅食行为，通过个体和群体的历史最优位置来更新粒
子的速度和位置。
蚁群算法
03
模拟蚂蚁觅食过程，通过信息素的积累和更新来寻找最优路径
。
混合启发式算法
遗传模拟退火算法
结合遗传算法和模拟退火算法的特点，既保持种群多样性又避免陷入局部最优。
启发式算法
汇报人： 2024-02-06
目录
• 启发式算法概述 • 启发式算法分类 • 经典启发式算法介绍 • 启发式算法设计原则与技巧 • 实际应用案例分析 • 挑战、发展趋势及未来方向
01
启发式算法概述
定义与特点
启发式算法是一种基于直观或经验构造的算法，它能够在可接受的花费（指计算时间、占用空间等）下给出待解决组合优化问题的一个可行解。
实际应用效果

求解矩形Packing问题的砌墙式启发式算法

ＡｂｓｒｃＡｖｌａｄｅｆｃｉｉｋａｉｇｈｅｒｓｉｌｏｉｈｍｏｗｏｄｍｅｉｎａｅｔｎｌｒｔａｔｎｏｅｎｆｅｔｖｅｂｒｃｌｙｎｕｉｔｃａｇｒｔｆｒｔ — ｉｎｓｏｌｒｃａｇｕａＰａｃｎｇｐｏｂｌｍｓｐｒｓｎｅｋｉｒｅｉｅｅｔｄ．Ｔｈｓａｇｏｉｈｓｍａｎｙｂａｅｎｂｉｋａｉｇｈｕｒｓｉｔａｅｉｓｉｌｒｔｍｉｉｌｓｄｏｒｃｌｙｎｅｉｔｃｓｒｔｇｅ
ｉｓｉｅｙａｌｒｅｎｍｂｒｏｘｅｉｎｅｃｕｕａｅｙｂｉｋａｅｓｄｒｎｈｒｃｓｆｕｌ — ｎｐｒｄｂａｇｕｅｆｅｐｒｅｃｓａｃｍｌｔｄｂｒｃｌｙｒｕｉｇｔｅｐｏｅｓｏｉｂｄｉｇｔｅｗａｌｓｅｉｌｙｈｕｌｉｇｗａｌｔａｅｙｂｓｄｏｈｅｅｅｃｒｃｓｐｅｅｔｄＴｈｎｈｌ，ｅｐｃａｌ，ｔｅｂｉｎｌｓｒｔｇａｅｎｔｅｒｆｒｎｅｂｉｋｉｒｓｎｅ．ｄｅ
Ｋｅｗｏｄｙｒｓ
ｔｅｒｈｇｎｌｅｔｎｕａＰａｋｎｐｏｌｍ；ｈｕｉｔｃｂｉｋａｉｇｕｅｌｃｌｈｏｔｏｏａｒｃａｇｌｒｃｉｇｒｂｅｅｒｓｉ；ｒｃｌｙｎｒｌ；ｏａ
策略，思想主要来源于砖匠在砌墙过程中所积累的经验：于基准砖的砌墙规则．国际上公认的大量的Ｂｎｈ其基对ｅｃ— ｍａｋ问题例的计算结果表明，算法的计算速度不仅比著名的现代启发式算法快，且获得更优的高度．ｒ该而

求解二维正交矩形布局问题的动态填空启发式算法

ｇｏｒｉｔｈｍｃａｎａｃｈｉｅｖｅ１００％ｐｌａｎｅｕｔｉｌｉｚａｔｉｏｎ，ｇｒｅａｔｌｙｉｍｐｒｏｖｅｔｈｅｅｉｃｆｉｅｎｃｙｏｆｔｈｅＢＬａｌｇｏｒｉｔｈｍ，ａｎｄｃａｎｂｅａｐｐｌｉｅｄｔｏｌａｒｇｅ－ｓｃａｌｅｐａｃｋｉｎｇｐｒｏｂｌｅｍ．
ｍａｔｉｃａｌｍｏｄｅｌｏｆｔｈｅｐｒｏｂｌｅｍｔｏｉｍｐｒｏｖｅｔｈｅｒｕｌｅｏｆＢＬａｌｇｏｒｉｔｈｍ．Ｆｏｒｉｎｄｆｉｎｇｔｈｅｓｐａｒｅｐｌａｎｅｄｕｒｉｎｇｔｈｅｐａｃｋｉｎｇｐｒｏｇｒｅｓｓ，ｉｔ
第３４卷第６期
２０１７年６月
计算机应用研究
ＡｐｐｌｉｃａｔｉｏｎＲｅｓｅａｒｃｈｏｆＣｏｍｐｕｔｅｒｓ
Ｖｏ１．３４Ｎｏ．６
Ｊｕｎ．２０１７
求解二维正交矩形布局问题的动态填空启发式算法
ｄｏｉ：１０．３９６９／ｊ．ｉｓｓｎ．１００１．３６９５．２０１７．０６．０１９
Ｄｙｎａｍｉｃｆｉｌｌｂｌａｎｋｈｅｕｒｉｓｔｉｃｓａｌｇｏｒｉｔｈｍｆｏｒｓｏｌｖｉｎｇ２Ｄｏｒｔｈｏｇｏｎａｌｒｅｃｔａｎｇｕｌａｒｐａｃｋｉｎｇｐｒｏｂｌｅｍ

求解二维矩形装箱问题的启发式算法

第２４卷第３期２０１８年３月
计算机集成制造系统
ＣｏｍｐｕｔｅｒＩｎｔｅｇｒａｔｅｄＭａｎｕｆａｃｔｕｒｉｎｇＳｙｓｔｅｍｓ

Ｖｏ１．２４Ｎｏ．３Ｍａｒ．２０１８
求解二维矩形装箱问题的启发式算法
尚正阳，顾寄南，丁卫，ＥｎｏｃｋＡ．Ｄｕｏｄｕ
收稿日期：２０１６—１１—１０；修订日期：２０１７～０３—２１。Ｒｅｃｅｉｖｅｄ１０Ｎｏｖ．２０１６；ａｃｃｅｐｔｅｄ２１Ｍａｒ．２０１７．基金项目：国防基础科研资助项目（ＪＣＫＹ２０１３４１４Ｃ００１）。Ｆｏｕｎｄａｔｉｏｎｉｔｅｍ：ｐｒｏｊｅｃｔｓｕｐｐｏｒｔｅｄｂｙｔｈｅＮａｔｉｏｎａｌＤｅｆｅｎｓｅＢａｓｉｃＲｅｓｅａｒｃｈＦｏｕｎｄａ
关键词：矩形装箱；布局优化；空间利用率最大；启发式算法中图分类号：ＴＰ３９１：Ｕ６７３文献标识码：Ａ
Ｈｅｕｒｉｓｔｉｃａｌｇｏｒｉｔｈｍｆｏｒ２ＤｒｅｃｔａｎｇｌｅｐａｃｋｉｎｇｐｒｏｂｌｅｍＳＨＡＮＧＺｈｅｎｇｙａｎｇ，ＧＵＪｉｎａｎ，ＤＩＮＧＷｅｉ，ＤＵＯＤＵＥｎｏｃｋＡ（ＭｅｃｈａｎｉｃａｌＩｎｆｏｒｍａｔｉｏｎＲｅｓｅａｒｃｈＣｅｎｔｅｒ，ＪｉａｎｇｓｕＵｎｉｖｅｒｓｉｔｙ，Ｚｈｅｎｊｉａｎｇ２１２０１３，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｔｏｓｏｌｖｅｔｈｅｔｗｏ—ｄｉｍｅｎｓｉｏｎａｌＲｅｃｔａｎｇｕｌａｒＰａｃｋｉｎｇＰｒｏｂｌｅｍ（ＲＰＰ）ｅｆｆｉｃｉｅｎｔｌｙ，ａｒｅｃｕｒｓｉｖｅｈｅｕｒｉｓｔｉｃａｌｇｏ— ｒｉｔｈｍｎａｍｅｄＢｅｓｔＲｅｓｉｄｕａｌＳｐａｃｅＡｌｇｏｒｉｔｈｍ（ＢＲＳＡ）ｗａｓｐｒｏｐｏｓｅｄ．Ｂａｓｅｄｏｎｔｈｅｉｄｅａｔｏｍａｋｅｔｈｅｉｔｅｍｓｐｌａｃｅｍｅｎｔｍｏｒｅｃｏｍｐａｃｔａｎｄｔｈｅｒｅｍａｉｎｉｎｇｓｐａｃｅｓｍｏｏｔｈｅｒ，ｔｈｒｅｅｓｔａｇｅｓｔｈａｔｗｅｒｅｓｐａｃｅｄｉｖｉｓｉｏｎ，ｐｌａｃｅｍｅｎｔｐｏｓｉｔｉｏｎｓｅｌｅｃｔｉｏｎａｎｄｏｐｔｉｍａｌｓｏｌｕｔｉｏｎｓｅａｒｃｈｉｎｇｗａｓａｄｏｐｔｅｄ。ＢａｓｅｄｏｎｔｗｏｓｅｔｓｏｆｃｌａｓｓｉｃａｌｂｅｎｃｈｍａｒｋｉｎｓｔａｎｃｅｓＣ２１ａｎｄＮ１３，ＢＲ— ＳＡｗａｓｃｏｍｐａｒｅｄｗｉｔｈｍａｎｙａｌｇｏｒｉｔｈｍｓ．ＴｈｅｅｘｐｅｒｉｍｅｎｔａｌｒｅｓｕｌｔｓｓｈｏｗｅｄｔｈａｔＢＲＳＡｗａｓａｂｌｅｔｏｏｂｔａｉｎａｌｌ１００ｆｉｌｌｉｎｇｒａｔｅｓｆｏｒＣ２１ａｎｄｎｉｎｅｏｐｔｉｍａｌｓｏｌｕｔｉｏｎｓｆｏｒＮ１３ｉｎｓｈｏｒｔｅｓｔｔｉｍｅ，ｗｈｉｃｈｗａｓｓｕｐｅｒｉｏｒｔｏｔｈｅｃｕｒｒｅｎｔａｌｇｏ— ｒｉｔｈｍｓｉｎｂｏｔｈｃｏｍｐｕｔａｔｉｏｎａｌｅｆｆｉｃｉｅｎｃｙａｎｄｅｆｆｅｃｔｉｖｅｎｅｓｓ．Ｋｅｙｗｏｒｄｓ：ｒｅｃｔａｎｇｕｌａｒｐａｃｋｉｎｇ；ｌａｙｏｕｔｏｐｔｉｍｉｚａｔｉｏｎ；ｓｐａｃｅｕｔｉｌｉｚａｔｉｏｎｍａｘｉｍｉｚａｔｉｏｎ；ｈｅｕｒｉｓｔｉｃａｌｇｏｒｉｔｈｍｓ

启发式优化算法综述

启发式优化算法综述启发式优化算法是一类基于启发式思想的算法，用于解决优化问题。

与传统优化算法不同，启发式优化算法通过启发性的探索和策略，能够在大规模优化问题中找到接近最优解的解决方案。

本文将对启发式优化算法进行综述，并介绍其中几种常见的算法。

1.启发式优化算法概述启发式优化算法是一类基于启发式思想的算法，通过对问题的空间进行启发性地探索和，找到问题的最优解或近似最优解。

与传统的优化算法（如数学规划算法）相比，启发式优化算法更适用于大规模优化问题，尤其是在空间非常庞大或者问题非常复杂的情况下。

2.启发式思想启发式是一种通过规定一定的策略，在解空间上进行有针对性地的方法。

它通过选择最有希望的方向进行，以期望达到更好的结果。

启发式的关键在于找到有效的启发信息，用于指导过程。

启发性信息可以通过问题的特点、领域知识、经验等方式得到。

3.常见的启发式优化算法以下是几种常见的启发式优化算法的简要介绍：（1）遗传算法（Genetic Algorithm，GA）遗传算法是一种模拟生物遗传和进化过程的优化算法。

它通过选择、交叉、变异等操作对解空间中的个体进行和优化。

遗传算法的核心思想是模拟自然界的生物进化过程，通过遗传交叉和变异操作产生新的解，并利用适应度评价函数对解进行评估，进而选择适应度较高的个体进行下一代的繁殖。

（2）粒子群优化算法（Particle Swarm Optimization，PSO）粒子群优化算法是基于鸟群觅食行为的一种优化算法。

它模拟了粒子在解空间中最优解的过程。

每个粒子根据自身的位置和速度，在解空间中进行，并通过与邻域粒子的信息交流，不断更新自己的位置和速度。

粒子群优化算法具有全局能力和较强的收敛性。

（3）模拟退火算法（Simulated Annealing，SA）模拟退火算法是一种模拟金属退火冷却过程的优化算法。

它通过随机性的接受劣解来避免陷入局部最优解，并逐渐降低温度，从而收敛到全局最优解。

矩形件优化排样的混合启发式方法

ｔｎ，０２４（３：３ —３．ｉｓ２１，８１）２４２９ｏ
Ａｂｔａｔｓｒｃ：Ｂａｅｎｈｕｉｔｅｕｓｖｌｏｔｍｎｅｅｉｌｏｉｍ，ｈｓｐｐｒｒｓｎｓａｌｏｔｍｆｏｖｎｓｄｏｅｒｓｉｒｃｒｉｅａｇｒｈａｄｇｎｔａｇｒｈｔｉａｅｅｅｔｎａｇｒｈｏｌｉｇｃｉｃｔｐｉｓｒｃａｇｌｒｐｃｉｇｐｏｌｍ．ｒｃｒｉｅａｇｒｈｏｅｒｔｓｒｐｓｄａｄａｌｅｔｎｌａｔａｅｃｎｅｄｔｅｔｎｕａａｋｎｒｂｅＡｕｓｖｌｏｔｍｆｕｓｉｉｐｏｏｅｎｌｒｃａｇｅｐｒｓｒｏｖｎｅｅｉｈｉｃｏ
一
刀切” 的矩形件排样。Ｈｐｅ等ｏｐｒ随机生成了不同
并对多种元启发式求种类和数量的小矩形件排放到板材上，使得所需的规模的矩形件排样测试实例，ｕ等总板材数最少，即使材料的利用率达到最大。此外，解这些实例的性能做了比较和分析。Ｃｉ提出一在排放过程中，还要满足下料的工艺要求，例如，对种启发式与动态规划相结合的方法来解决钢桥建设能用于 “ 一刀切” 问题。刘景玻璃或木材的切割，通常要满足“ 一刀切 ”即从板材中的矩形件下料问题，，的一端，沿直线方向切割到另一端。矩形件排样问题是公认的Ｎ．Ｐ难问题，内外很多学者进行了研国究，并提出了一些有效的启发式或近似算法。曹炬等ｎ出了背包算法，提能够快速地找到近似最优解。
Ｋｅｒｓｅｔｎｌａｋｎ；ｅｕｓｅａｇｒｈｏｅｒｓｉ；ｅｅｃａｇｒｈｙｗｏｄ：ｒｃａｇｅｐｃｉｇｒｃｒｉｌｏｔｍｆｕｔｇｎｔｌｏｔｍｖｉｈｉｃｉｉ

基于蚁群算法的带平衡约束矩形布局问题的启发式求解

龙源期刊网
基于蚁群算法的带平衡约束矩形布局问题的启发式求解
作者：季美,肖人彬
来源：《计算机应用》2010年第11期
摘要:
以卫星舱布局问题作为研究背景,求解了带平衡约束的矩形布局问题。

采用启发式策略设
计了分区域分步布局法,该策略将圆形卫星舱承重板分成4个区域,分区域同步进行布局。

当所布矩形和区域都确定时,采用最左最底填充策略进行布局。

该方法通过不干涉约束,使布局紧凑,通过控制系统质心的位置,使系统保持平衡。

在启发式策略的基础上,设计了蚁群算法搜索优化定位次序,从而得到优化的布局。

数值仿真结果表明,该布局方法具有优良的计算性能。

关键词:
矩形布局;平衡约束;启发式策略;蚁群算法。

rectpack 算法 -回复

rectpack 算法-回复什么是rectpack算法？Rectpack算法是一种基于矩形排列优化问题的算法，其目标是找到最优的方式来将一组矩形布局在一个矩形容器中，最大程度地减少空白面积，以实现最优的空间利用。

该算法被广泛应用于计算机图形学、计算几何学和背包问题等领域，用于解决诸如装箱问题、切割问题和排样问题等具有实际应用价值的经典问题。

Rectpack算法的核心思想是通过一系列的启发式策略和搜索技巧，尽可能地提高装箱效率。

其基本步骤包括：1. 定义问题：首先，需要明确问题的定义，包括矩形的尺寸、容器的尺寸以及可能的限制条件等。

例如，如果装箱问题需要考虑矩形之间的匹配关系或者规定某些矩形不能相邻等。

2. 初始化容器：根据问题定义，初始化一个矩形容器，该容器将作为矩形的布局空间。

容器的大小通常与问题有关，可以是一个预先定义的固定大小，也可以是根据矩形尺寸动态调整。

3. 初始排列：选择一种初始排列策略，将所有待布局的矩形按照一定的顺序放置在容器中。

常用的策略有贪心算法、随机算法和启发式搜索等。

初始排列的质量将直接影响后续算法的优化效果。

4. 矩形布局：在初始排列的基础上，通过迭代调整和优化，尝试不断改善装箱效果。

这可以通过多种方式实现，例如移动和旋转矩形，尝试不同的布局策略和调整算法，以及应用优化算法来寻找最佳解决方案。

5. 评估和优化：在每一步布局之后，评估当前装箱效果的质量。

常见的评估指标包括空白面积比例、紧密度、平均面积利用率等。

根据评估结果，可以进行进一步优化调整，迭代更新布局。

6. 终止条件：根据预设的停止条件，决定是否终止算法。

停止条件可以是固定的迭代次数，达到一定的装箱质量阈值，或者运行时间超过设定阈值等。

一般来说，在算法运行时间和最终布局质量之间需要做取舍。

7. 输出结果：在算法终止后，将最终得到的布局结果输出。

结果可以是一个最优布局，也可以是多个近似最优解，根据问题复杂度和应用需求决定。

启发式算法分析

启发式算法分析1. 启发式算法的概念意思是什么启发式算法是一种基于经验和猜测来解决问题的算法，中文意思是“智能搜索算法”。

它通过结合现有知识和预先定义的规则来搜索最优解，从而解决复杂的问题。

2. 启发式算法的应用启发式算法是一种基于经验和直觉而非完全依赖于精确步骤的搜索算法。

它以一种模糊的、非确定的方式进行搜索，常被用来解决复杂的问题，如搜索、排序、路径规划、位置规划、视觉等。

启发式算法的应用非常广泛，可以用于模拟计算机游戏、图像处理、数据挖掘、网络优化、机器学习、语音识别、自然语言处理等领域。

3. 启发式算法的优缺点优点：1、启发式算法可以快速得到比较优质的解决方案，计算效率高；2、在求解复杂问题时，启发式算法可以帮助我们找到最优或接近最优的解决方案；3、启发式算法可以减少计算量，比如贪心算法可以减少计算复杂度，以达到更快的运行速度；4、在处理复杂结构数据时，启发式算法可以更好地处理；缺点：1、启发式算法往往不能保证求解问题的最优结果，也就是不能保证给出的结果是最优的；2、启发式算法往往只能解决特定类型的问题，在其它问题上效果可能不尽如人意；3、启发式算法可能会受到数据的影响，如果数据的质量较差，那么启发式算法的结果也可能不尽如人意；4、启发式算法的可扩展性较差，在扩展时可能会遇到问题。

4. 启发式算法的发展趋势启发式算法是以经验为基础的算法，它们旨在以更高效的方式找到最优解。

近年来，启发式算法的发展趋势主要表现为：（1）深度学习算法的发展；（2）网络架构的改进；（3）采用更多的搜索策略；（4）使用先进的知识表示算法；（5）更强大的计算能力；（6）更精确的模型；（7）对环境变量的更有效的应用。

5. 启发式算法的研究现状启发式算法是一种基于经验的算法，它模仿人类的思维方式，通过搜索、评估和解决问题的过程来解决特定问题。

它具有计算效率高、计算复杂度低、容易理解和实现等优点，在人工智能、模式识别、操作系统、科学计算等领域有着广泛的应用。

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ｓｓｗｅｅｅｕｅｎｄｖｒｅｔａｕｒｐｅｌｏｔｍｗａｅｙｅｆｃｉｅｅｍｘｔｄ．ａｅｉｄｈｔｏｒｐｏｏｓａｇｒｈｃｉｆｄｉｓｖｒｆｔ．ｅｖ
Ｋｅｒｓｐｃｎｙｗｏｄ：ａｋｇ；ｓｅｌｅ；ｈｕｉｔｌｏｔｍ；ｒｃａｇｅｐｃｉｇｐｏｌｍｉｔｐｉｋｅｒｓｉａｇｒｈｃｉｅｔｎｌａｋｎｒｂｅ
大规模集成电路的布局与布图规划等，这类问题在
研究时都可以形式化为矩形Ｐｃｉ问题。矩形Ｐｃ— ａｋｇｎａｋ
ｉ是一个ＮｎｇＰ难问题，寻找这类问题的快速解法，具有广泛的实际意义，因此吸引了国内外许多学者的研
于正交矩形Ｐｃｉｇ题，有一般性和全局性，易ａｋ问ｎ具极扩展为三维Ｐｃｉｇ三维装箱）多个矩形容器的全ａｋｎ（和
第２卷１
第３期
计算机技术与发展
ＣＯＭＰＵＴＥｒＲＥＣＨＮＯＬＯＧＹＡＮＤＤＥＶＥＬＭＥＮＴＯＰ
２１０１年３月来自Ｖｏ．Ｎｏ３１２ｌ．Ｍａ．２１ｒ０１
一
种解决矩形布局问题的启发式快速算法
王石
ＡｂｔａｔＡｔｐｉｅｓａｋｎｌｏｉｍｓｐｏｏｅｏｖｎｓｒｃ：ｓｅｌｔｃｉｇａｇｒｔｋｈｗａｒｐｓｄｔｓｌｉｇ２Ｄｅｔｎｕｌｒｐｃｎｒｂｅｏｒｃａｇａａｋｇｐｏｌｍ．Ｔｈｌｏｉｍｓｂａｅｎａｓｅｌｋｉｅａｇｒｈｗａｓｄｏｔｐｉｅｔｈｕｓｃｒｌｎｒｂｂｆｓｃｂｃｔｃｎｌｏｉｍ．ｏｅｚｒ－ｗａｔｄｐｏｌｍ，ｔｅａｇｒｔｍｐｎｓｌｓｎｆｎｉｇｔｅｏｔｅｒｔｕｅａｄｐｏａｉｔａｋｒｋｇａｇｒｈｉｉｉｉａｉｔＦｒｔｅｏｈｓｅｒｂｅｈｏｉｌｈｓｅｄｅｓｏｄｎｐｉｌＯｉｈｍａ－Ｓｌｔｏｈｎｏｈｒｍｅｏｓｔｓｅｎｔａｅａｒｙｇｏｒｏｍａｃｎｓｌｉｇｎｎ —ｚｒ —ｗａｔｒｂｅ．Ｔｗｏｗｅｌｎｗｎｔｓａｕｎｔａｔｅｔｄ．Ｉａｏｄｍｏｓｒｔｄｆｉｌｏｄｐｆｒｎｅｏｏｖｎｏｅｅｏｉｈｌｅｓｅｐｏｌｍｓｌｋｏｔ－ｅｃ
可以很好地求解非零浪费问题。采用国际公认的两个算例进行测试，明文中算法是非常高效的。证
关键词：填问题；装阶梯式；发式；启矩形排样
中图分类号：Ｐ０．Ｔ３１６文献标识码：Ａ文章编号：６３６９２１）３０３ — ４１７ — ２Ｘ（０１０ — ０６０
究，ＨｆＭ，ｏｐｒＥＭａｅｏＳ，ｕｎＷＱ，如ｉ．Ｈｐｅ．，ｒｌ．Ｈａｇｉｔｌ
率最高。为了使放人的矩形块最大化，常任一放人通
容器的矩形块的边都和容器的边平行，因此该问题也被称为正交矩形Ｐｃｉｇａｋ问题。文中提出的算法适用ｎ
（中科技大学软件学院，华湖北武汉４０７）３０４
摘要：对二维矩形Ｐｃｉ问题，出了一种沿阶梯线轮廓进行布局矩形的启发式算法。该算法基于“ 针ａｋｇｎ提阶梯式堆码 ” 的
启发式规则，能够快速地对矩形块进行紧靠布局。为避免算法陷入局部最优，法采用随机回溯策略在选择矩形和阶位算上扩大搜索范围。结果表明，法对于浪费面积为零的矩形全Ｐｃｉ算ａｋｎｇ问题，能够在极短的时间内找到最优解，同时它也
Ｏ引言
生产实践中，经常遇到一类最优化问题，即如何将若干矩形最大化的装入一个较大的矩形容器中，如货物装载、木材下料、建筑装修布局、工程调度… ，以及超
构造算法，之后引入随机策略进行混合求解。
１问题描述
矩形Ｐｃｉａｋｇ问题描述为：ｎ已知一个宽为，高为Ｈ的矩形容器Ｓ以及ｎ，块矩形块Ｒ，其中矩形。Ｒ …Ｒ，的宽为，高为ｈ。需要把这ｎ块矩形块互不嵌入且尽可能多地放进容器Ｓ中，即要使容器的面积利用