一种装箱问题的高效定位启发式算法

An efficient placement heuristic for three-dimensional rectangular packingKun He,Wenqi HuangÃSchool of Computer Science and Technology,Huazhong University of Science and Technology,Wuhan430074,Chinaa r t i c l e i n f oAvailable online28April2010Keywords:Cutting and packingThree-dimensionRectangular packingContainer loadingHeuristica b s t r a c tBy embodying the spirit of‘‘gold corner,silver side and strawy void’’directly on the candidate packingplace such that the searching space is reduced considerably,and by utilizing the characteristic ofweakly heterogeneous problems that many items are in the same size,afit degree algorithm(FDA)isproposed for solving a classical3D rectangular packing problem,container loading problem.Experiments show that FDA works well on the complete set of1500instances proposed by Bischoff,Ratcliff and Davies.Especially for the800difficult strongly heterogeneous instances among them,FDAoutperforms other algorithms with an average volume utilization of91.91%,which to our knowledge is0.45%higher than current best result just reported in2010.&2010Elsevier Ltd.All rights reserved.1.IntroductionCutting and packing problems are representative combinationaloptimization problems with many important applications in theindustry.This paper addresses the problem of loading a subset ofthree-dimensional(3D)rectangular items into a3D rectangularcontainer,such that the total volume of the packed items is maxi-mized,i.e.the container’s wasted volume is minimized.A layout iscalled feasible,if each packed item is located orthogonally andcompletely in the container and there is no overlapping betweenany two items.This problem is an NP-hard problem,whose1Ddegradation,the0–1knapsack problem,is still NP-hard.This3D rectangular packing problem is also called thecontainer loading problem,because the most common andimportant application of this problem is to load rectangularcargoes into containers,vehicles or ships in the transportationindustry.There are some additional considerations that would betaken into account in the real world[1,2],among which theorientation constraint and the stability constraint are the mostimportant ones.In our opinion,if there is an efficient approach tosolve the basic problem that has no additional constraints,then itis not difficult to make the approach adapted to problemsconsidering some additional ones.Since the orientation constraint has been widely considered bythe researchers,and it has been accepted by the famous BRbenchmarks[1],we take the orientation constraint into accountthat one or two sides of the items may not be used as the height.This situation usually happens when cargoes are boxes full of oilor wine.We do not concern the stability constraint for thefollowing reasons:(1)Stability constraint is not considered aswidely as the orientation one and the stability criteria isinconsistent in the literature.In some cases it requires that eachitem is fully supported,or partially supported with at least a givenpercentage;in other cases it requires that the gravity center ofeach item falls over an underlying item or over the bottom of thecontainer.(2)Stability could become a consequence of the highcargo compactness when the container’s volume utilization ishigh enough.(3)Foam rubber or other stuffing could be used tofill the small empty spaces left,as what has been done in somefreight companies.Many efficient algorithms have been proposed for solving thisclassical3D packing problem.The most prevalent approach iswall building or layering[1,3–9],first proposed by George andRobinson in1980[3].In the past thousand years,people usuallystart packing goods from the bottom and build up the packingin layers,inserting each goods such that it is contiguous withwhat is already packed.Inspired by these human’s experience,thewall building or layering method usually opens a new layeror wall with a width equals to some item dimension,then eachlayer isfilled up by a number of horizontal strips,and each stripis packed in a greedy way.Another efficient approach widelyadopted by the researchers is block arrangement[10–12],firstproposed by Bortfeldt and Gehring in1998[10],which bindsitems of the same or similar size into a larger rectangular block todo the tentative packing.By utilizing the block arrangementmethod,Parren˜o proposed an approach that always places acolumn or layer built by same size items into a maximal space,thelargest empty parallelepiped spaces[13,14].Above approaches allutilized the characteristic of the weakly heterogeneous instancesthat many items are in the same size.Therefore,the solutionqualities are in a downtrend when the problems become moreContents lists available at ScienceDirectjournal homepage:/locate/caorComputers&Operations Research0305-0548/$-see front matter&2010Elsevier Ltd.All rights reserved.doi:10.1016/j.cor.2010.04.015ÃCorresponding author.Tel.:+862787543018;fax:+862787545004.E-mail addresses:brooklet60@(K.He),wqhuangwh@(W.Huang).Computers&Operations Research38(2011)227–233and more strongly heterogeneous,i.e.same size items become less and less.In2009,inspired by an old proverb‘‘Gold corner,silver side and strawy void’’,and improved by a new observation‘‘Maximum value in diamond cave’’,Huang and He proposed a caving degree approach(CDA)whose key issue is to pack an outside item into a corner or even a cave in the container such that the item is as compact and close as possible with other items[2,15].Differs from other approaches,CDA is especially good at strongly heterogeneous instances.But unfortunately the searching space of the CDA is quite large because it searches all corners where three orthogonal surfaces of different items meet at each iteration step,and the computation of caving degree is time-consuming for each candidate placement.Above drawback becomes a main barrier if people want to get higher volume utilization within a reasonable time.In this paper,we make improvements on the CDA,largely decrease the searching space by just considering a corner nearest to the edges of the container as the candidate packing place.This strategy embodies the nature of‘‘gold corner,silver side and strawy void’’directly.Besides,the scope of action space in the caving degree approach is modified to be a much smaller and appropriate one in this paper,which simplifies the evaluation of different placements.Furthermore,same size items are bound into different blocks,andfit degree is defined to judge which blockfit the action space best.Experiments show that thefit degree approach(FDA)not only largely speeded up the computa-tion,markedly improved the solution’s quality,but also inherited the advantage of the CDA on the strongly heterogeneous problems.2.Thefit degree algorithmThefit degree algorithm(FDA)contains two phases:the constructive phase and the local search phase.Wefirst introduce FDA’s improvement strategies on the CDA and define several important conceptions used by the FDA.2.1.Improvement strategiesThe caving degree approach(CDA)[2,15]searches all corners, unoccupied space where three surfaces of different items meet,at each iteration step.We can see an example for a2D problem in Fig.1.There are10corners,marked from1to10.For each corner, CDA pseudo places an outside item into the corner and computes the caving degree to evaluate how compact and close the packing item is to other items already in.The more the caving degree is, the more the place looks like a cave for the packing item.This procedure is time-consuming for the reason that each corner should be checked.So in the new algorithm,we just take one corner as the candidate packing place at each iteration step. Then,which corner should we select?Here we comply with the proverb‘‘gold corer,silver side and strawy void’’directly,and try to select a corner nearest to the edges of the container,i.e.farthest to the center of the container,such that items are placedfirst to the corners of the container,then to the sides of the container and last to the center of the container.In Fig.1,we will only take corner6as the candidate packing place at current iteration. Detailed method on how to select a candidate corner will be described later in this section.When placing an item into a corner for the CDA,then aligned with the k pasted surfaces(two surfaces are called pasted if their coinciding area is larger than0)and their normal directions (pointing to the space the packing item is in),an action space is enclosed in the container[2].For example for the2D problem in Fig.2(a),suppose item i is tentative packed in corner6,then the action space of this corner-occupying action(COA)is shown in the dotted rectangle.When computing the caving degree of this action,items intersecting the action space need to be considered (surfaces of the container can be regarded as sixflat items, respectively).The caving degree C i of a COA is as shown in Eq.(1). Firstly,it is decided by paste number k i for how many surfaces of other items the packing item pastes;then it is decided by the minimum distance d i between the packing item and other items that intersect the action space but do not paste the item(l i,w i,h i are the lengths of the three dimensions for the packing item),and lastly,it is decided by pasted ratio r i for how proportion the surface area of the packing item be pasted.C i¼100k iþ10ad iþr i¼100k iþ10expÀd i3ffiffiffil ipw i h i!þr ið1ÞSo in Fig.2(a),when computing the minimum distance d i,we need to consider distances between item i and{item a,b,e,f,g,the right side and the bottom side of the container}.But in fact, we could shrink the action space like as shown in Fig.2(b).Fig.1.Corners in the caving degree approach.Fig. 2.Shrinkage on the action space:(a)action space in the caving degree approach and(b)action space in thefit degree approach.yer arrangement strategy:(a)one layer yx-arrangement and(b)two layers yx-arrangement.K.He,W.Huang/Computers&Operations Research38(2011)227–233 228When computing the minimum distance d i ,we only need to consider the distance between item i and {the bottom side and the right side of the action space}.In this way the computation is simplified markedly.Another main modification is that instead of placing one item at each iteration step,we bind same size items into one or several layers to do the tentative packing.For example in Fig.3,suppose there are 12items in the same size,if we want to place them to corner 6in the horizontal orientation,and if we first try to place as many items as possible in the y direction,then there are two possible arrangements:one layer arrangement and two layers arrangement,as shown in Fig.3(a)and (b).This strategy tries to utilize the characteristic of weakly heterogeneous problems to get higher volume utilizations.2.2.DefinitionsSome important conceptions used in the FDA are defined in this subsection.Items in the same size are bound into a larger cuboid block which is built by one layer or several layers,and fit degree is defined to evaluate that to what degree the packing block fit the candidate action space.Definition 1.(A CTION S PACE ).At current iteration,placing a dummy cuboid into the container orthogonally and without overlapping,and the size of the cuboid is large enough such that each of its surfaces pastes at least one item already in (two surfaces are called pasted if the coinciding area is larger than 0),then the empty space it occupies is called an action space.The six surfaces of the container can be regarded as six flat items.Then in Fig.2(b)for the 2D problem,the dotted rectangle space is an action space with its upper,lower,left and right sides pasting at least one item.There are some other action spaces for current iteration,as shown with dotted rectangles in Fig.4.An action space has eight corners for the 3D problem,each having a different corner direction,as shown in Fig.5.Note that this conception of corner is different from what is defined for the caving degree approach.In fact,it contains the one for the CDA.For example in Fig.2(b),the 4corners of current action space contain corner 2,6,and 7in Fig.1,but its lower-right corner is a new one that is not considered in the CDA.Definition 2.(D ISTANCE V ECTOR OF A C ORNER ).Suppose the distance in the x ,y ,z directions between a corner and its corresponding corner of the container,which has the same corner direction,is d x ,d y and d z ,and then a nondecreasing order of d x ,d y and d z is called the distance vector of this corner,marked as (d 1,d 2,d 3).Two distance vectors can be compared in the lexicographical order.For example if d a ¼(1,2,8)and d b ¼(2,2,3),then d a o d b .Definition 3.(C ANDIDATE C ORNER OF AN A CTION S PACE ).A corner with the minimum distance vector is the candidate corner of this action space.In case of a tie,a corner with smaller corner direction is selected.For example in the 2D problem of Fig.6,the distance vectors of corner 1,2,3and 4are (d y 1,d x 1),(d x 2,d y 2),(d y 3,d x 3)and (d y 4,d x 4).Therefore,corner 1is the candidate corner for this action space.Definition 4.(C ORNER O CCUPYING A CTION ,COA).It is an action that places a cuboid block,bound by outside items of the same size,into an action space such that one of the block’s vertices coincides with the space’s candidate corner,and it satisfies the problem’s constraints.The size of the block is decided by the following factors:(1)Which action space is selected (the candidate corner isdetermined then);(2)Which size of item is selected (the number of outside items inthis size is determined then);(3)For an item in size (l ,w ,h ),there are six possible itemorientations whose dimensions on x ,y ,and z axes are (l ,w ,h ),(l ,h ,w ),(w ,l ,h ),(w ,h ,l ),(h ,l ,w ),(h ,w ,l );We need to select one feasible orientation;(4)There are six types of arrangements for binding same sizeitems into larger blocks,marked as xyz ,yxz ,xzy ,zxy ,yzx and zyx .Here xyz means that we first try to bind as many items as possible in the x direction,and then in the y direction;if the maximal number that we could bind in the z direction is k ,then items in the z direction could be 1,2to k ,corresponding to one layer arrangement,two layers arrangement,and k layers arrangement.So,the block is determined by the type of arrangements and how many layers in the third dimension for this type.For example for the 2D action space shown in Fig.3,the candidate corner is corner 6,and there are 12outside items of the same size.If we place them in the horizontal orientation,then the yx arrangement is as shown in Fig.3,and the xy arrangement is as shown in Fig.7.Note that the fifth arrangement shown in Fig.7is as the same as that shown in Fig.3(b),and we just need to consider it for one time.ad efcb gadefcb gFig.4.Some action spaces at current iteration.Fig.5.Corners and corner directions.Fig.6.Distance vector of the corners.K.He,W.Huang /Computers &Operations Research 38(2011)227–233229So,there are many different COAs at current iteration.Then,which COA should we select to do?Fit degree is defined to evaluate different placements.Definition 5.(A VERAGE O CCUPATION ).The average occupation of a corner-occupying action is as shown in Eq.(2).Here L i ,W i ,H i are the three dimensions of the action block,the cuboid block being packed into the action space.L D ,W D ,H D are the three dimensions of the action space,and n i is the total number of items bound in the block.u i ¼1ffiffiffiffin i 3p L i W i H iL D W D H Dð2ÞSo,(L i W i H i )/(L D W D H D )is the volume utilization of the action space,and coefficient (n i )À1/3is used to balance the volume utilization and the bind number n i .For bind number n i ¼n x Ân y Ân z where n x ,n y or n z means the bind number in x ,y or z direction,coefficient (n i )À1/3means the average bind number in one direction.Here we give an example to show the specific impacts of coefficient (n i )À1/3.If action a could place a block bound by 8items into an action space and achieves a volume utilization of 60%for the action space,and action b could place a block bound by 27items into the action space and achieves a volume utilization of 75%,then the average occupations for action a and action b are 0.6/81/3¼30%and 0.75/271/3¼25%,respectively.Thus action a has the larger average occupation,even though its volume utilization is lower.However,if the volume utilization of action a is just 40%,then its average occupation is 0.4/81/3¼20%,and in this case action b has the larger average occupation.Table 1shows the specific impact of the coefficient (n i )À1/3for this example.Definition 6.(F IT D EGREE ).The fit degree of a corner-occupying action is as shown in Eq.(3),where paste number k i means how many surfaces of the block are pasted by the surfaces of the action space,other-paste number p i means the number of other blocks pasting with the action block (the six surfaces of the container are regarded as six special blocks),u i means the average occupation,and paste ratio r i equals the block’s pasted area with the action space divided by the block’s total surface area.F i ¼/k i ,p i ,u i ,r i Sð3ÞFit degree is defined to describe that to what extend the packing block fit the action space.Note that paste number and paste ratio is different from what is defined in the caving degree approach.Firstly,the observation unit is blocks instead of single items.Secondly,the relation is between the packing block and the action space.Thirdly,we use the volume utilization of the action space to replace the function of minimum distance d i .Above strategies not only simplify the computation but also embody how compact and close the packing block is to other blocks already in.Two fit degrees can be compared in the lexicographical order.The larger the fit degree is,the better the placement is.2.3.The constructive phaseBased on above definitions,the constructive FDA can be described as follows.It is a greedy algorithm that attempts to produce good placements by always examining an action space nearest to the edge of the container and then placing a rectangular block that best fit the space.So,the whole packing procedure is from the edge to the center.The constructive FDA:1.At current iteration step,select an action space based on following parameters of the action space in a lexicographic order:(a)distance vector of the candidate corner:smaller is better (b)volume:larger is better(c)coordinate x ,y ,z of the lower-left-near corner:smaller isbetter(d)coordinate x ,y of the upper-right-far corner:smaller isbetter2.If there is no feasible COA for this action space,then go to step 1to select the next action space;3.A best COA is selected to do based on some parameters of the action block in a lexicographic order:(a)fit degree:larger is better(b)volume of one item in the block:larger is better (c)length of the long side for one item:larger isbetterFig.7.The possible xy arrangements.Table 1–1/3K.He,W.Huang /Computers &Operations Research 38(2011)227–233230(d)length of the short side for one item:larger is better(e)coordinate x,y,z of the lower-left-near corner:smaller isbetter(f)coordinate x,y,z of the upper-right-far corner:smaller isbetter(g)item orientation number:smaller is better4.Repeat1–3,until all outside items have been packed into thecontainer without overlapping,or none of the remainders could be packed in.The constructive algorithm always selects a remote action space and does a corner-occupying action with the largestfit degree.At step1,rules(2)–(4)break tie to select a unique action space,so the candidate corner to befilled is determined too.At step2,rules (2)–(4)determine the size of the outside item,rule(7)determines the item orientation,rules(5)and(6)determine the three dimensions of the block,and so the type of arrangements and how many layers in the third dimension are determined too.Above rules break tie to make the algorithm deterministic.2.4.The local search phaseThe local search FDA observes the top N COAs that have better developing foreground than the remainders,and selects one action with the best pseudo utilization to do.Definition7.(P SEUDO U TILIZATION).At current iteration step,do a COA and pseudo execute the constructive FDA,and then thefinal volume utilization of the container is called the pseudo utilization of this COA.So,each COA has corresponding pseudo utilization.Starting from the beginning when there is no item in the container,the local search FDA executes as follows:The local search FDA:(1)At current iteration step,select an action space according tothe rules in step1of the constructive FDA;(2)If there is no feasible COA for this action space,then go to step1to select the next action space;(3)Sort all COAs according to the rules in step3of theconstructive FDA,and compute the pseudo utilization for each of the top N actions;(4)A COA with the largest pseudo utilization is selected to do,incase of a tie,select one sorted on the front;(5)Repeat1–4,until all outside items have been packed into thecontainer without overlapping,or none of the remainders could be packed in;(6)Output the best placement result that has the highest volumeutilization.The larger the value of N is,the longer the running time is,and the higher the volume utilization tends to be.putational resultsWe evaluate the performance of the proposed heuristic via computational experiments.The above algorithms were coded and compiled in Java2Platform(Standard Edition,(J2SE)V1.5.0_14)and were run sequentially on a PC with Intel(R)Xeon(R)2.33GHz CPU.Test instances are1500problems generated by Bischoff,Ratcliff and Davies[1,5],called BR benchmarks by other researchers.The whole set of instances comprises15classes of instances,classified depending on the number of items in different sizes,each class consisting of100instances.With respect to the item types vary from3for BR1to100for BR15, the instances vary from weakly heterogeneous to strongly heterogeneous,i.e.same size items become less and less. Generally speaking,BR1to BR7are taken as the weakly heterogeneous groups,and BR8to BR15are taken as the strongly heterogeneous groups.parisons with other algorithmsAlgorithms having computed all the1500BR instances include H_BR[1],H_B_al[4],GA_GB[16],TS_BG[10],HGA_BG[6], PGA_GB[7],GRASP[9],GRASP’[13]and VNS[14].In addition, there are some other algorithms having computed at least one group of the instances:PTS_B_al[11],PH_M_al[8]and H_B[17] for the weakly heterogeneous groups BR1to BR7and CDA[2]for the strongly heterogeneous groups BR8to BR15.Table2shows the average volume utilization on each of the15 BR classes for above algorithms and for the local search FDA (N¼100).The average volume utilizations on three groups,BR1to BR7,BR8to BR15and BR1to BR15,are shown at the bottom of the table.As for the average utilization on all the1500instances, PGA_GB proposed by Gehring and Bortfeldt held the highest record of88.97%since2002[7],until Parren˜o et al.reported new results in2008[13]and2010[14]successively.To our knowledge,current best result reported in the literature is 92.89%achieved by Parren˜o et al.in2010[14].As for the weakly heterogeneous group,PH_M_al proposed by Mack et al. held the highest record of93.78%since2004,until Parren˜o et al. achieved an average utilization of94.53%in2010[14].As for the strongly heterogeneous group,Gehring and Bortfeldt reported a result of87.69%in2002[7],we reported a result of87.97%in 2009[2],and current best result reported in the literature was achieved by Parren˜o et al.in2010with an average utilization of 91.46%[14].Taking FDA into account,current highest volume utilizations for each class and each group appear in bold in Table2.As can be seen from Table2,FDA achieved an average volume utilization of 91.91%on the strongly heterogeneous group,which is0.45% higher than current best result.Moreover,each value that FDA obtained on each of the8strongly heterogeneous classes is also higher than current best result.3.2.Detailed computational results on the FDAIn this subsection,we report some detailed computational results on the FDA.We tested the constructive FDA and the local search FDA with parameter N¼30,50and100such that FDA could get high volume utilizations within reasonable times. Through a small-scale test,we found that if parameter N was set to the number of all candidate actions,the utilization was just 0.02%higher than that of N¼100,but the running time increased by three times.Therefore,we did not do any computation using a parameter larger than100.The detailed results are shown in Table3.As a reference, utilizations of the constructive VNS and the local search VNS are also listed.Values for each class and each group for the constructive FDA appear in bold if they are higher than that of the constructive VNS.Also,values for the local search FDA(N¼30, 50or100)appear in bold if they are higher than that of the local search VNS.At the constructive phase,the average utilization of FDA on BR1to BR15is86.77%,which is0.19%higher than that of the compared one.FDA was run on a2.33GHz PC,while VNS was run on a 1.5GHz PC.The average running time of the constructive FDA is0.34s,which is very short.Although Parren˜oK.He,W.Huang/Computers&Operations Research38(2011)227–233231et al.did not report the running time on their constructive VNS,it can be inferred from their algorithm’s description that it is also very short.Another point we could find from Table 3is that almost each value of the FDA is higher than that of the compared one.Furthermore,differs from the VNS,the average utilization of the constructive FDA on the strongly heterogeneous groups,BR8to BR15,is higher than what was obtained on the weakly heterogeneous groups,BR1to BR7.This is a salient feature of FDA,since most of other algorithms usually obtain lower utilizations when the problems become more and more heterogeneous.At the local search phase,the average utilizations of VNS on BR1to BR7,BR8to BR15,and BR1to BR15are 94.53%,91.46%and 92.89%,with the average running times of 28,531and 296s,respectively.Set parameter N to 30,within similar running time,FDA obtained an average utilization of 92.40%on BR1to BR15,which is 0.49%lower than that of the VNS.However,the average utilization that FDA obtained on the strongly heterogeneous group is 91.67%,which is 0.21%higher.It can be observed from the table that the larger the value of parameter N was,the higher the utilization FDA obtained,and the longer the time was.Finally,FDA got higher utilizations on each of the strongly heterogeneous class when N was set to 100.In summary,the average utilizations of FDA on BR1to BR7,BR8to BR15,and BR1to BR15are 93.53%,91.91%and 92.67%,with the average running times of 10,1178and 633s,respectively.Table 3Detailed computational results on the 1500BR instances.Problem set (number of item types)Constructive Local search Constructive FDA FDA (N ¼30)FDA (N ¼50)FDA (N ¼100)VNS (%)VNS (%)Utilization (%)Time (s)Utilization (%)Time (s)Utilization (%)Time (s)Utilization (%)Time (s)Weakly heterogeneous BR1(3)84.3494.9385.320.0192.430.6492.70 1.1292.92 1.16BR2(5)85.6195.1985.960.0193.30 1.3293.72 2.3493.93 2.54BR3(8)85.8194.9987.270.0293.57 2.6293.71 4.2693.71 5.14BR4(10)87.0794.7187.090.0293.50 3.9393.54 6.3293.687.66BR5(12)86.4694.3387.250.0293.41 5.4393.609.1293.7310.38BR6(15)88.2194.0487.000.0493.317.993.5613.5793.6316.66BR7(20)85.9693.5387.190.0493.0213.493.0921.9993.1429.54Strongly heterogeneous BR8(30)85.9692.7886.810.0692.6832.8692.7849.8192.9282.94BR9(40)86.2392.1987.190.1192.1660.0892.2894.2492.49160.77BR10(50)85.7291.9286.890.2592.04121.7292.21188.8992.24298.95BR11(60)85.8591.4686.840.2591.71180.0891.89274.9991.91497.79BR12(70)85.1891.2086.650.8291.60302.5291.65540.1691.83861.37BR13(80)85.4091.1186.830.8791.37475.7191.45749.0891.561775.79BR14(90)84.8790.6486.68 1.0090.99748.9291.171205.6791.302218.17BR15(100)85.4190.3886.59 1.5590.842060.4390.882752.5291.023531.71Average (1–7)85.8794.5386.720.0293.22 5.0493.428.3993.5310.44Average (8–15)86.2091.4686.810.6191.67497.7991.79731.9291.911178.44Average (1–15)85.5892.8986.770.3492.40267.8492.55394.2792.67633.37Table 2Comparisons with other algorithms (%).ProblemH_BR [1](1995)H_B_al [4](1995)GA_GB [16](1997)TS_BG [10](1998)HGA_BG [6](2001)PGA_GB [7](2002)PTS_B_al [11](2003)PH_M_al [8](2004)GRASP [9](2005)H_B [17](2006)CDA [2](2008)a GRASP’[13](2008)VNS [14](2010)FDA (2010)BR1(3)83.3781.7686.7792.6387.8188.1093.5293.7089.0789.39–93.2794.9392.92BR2(5)83.5781.7088.1292.7089.4089.5693.7794.3090.4390.26–93.3895.1993.93BR3(8)83.5982.9888.8792.3190.4890.7793.5894.5490.8691.08–93.3994.9993.71BR4(10)84.1682.6088.6891.6290.6391.0393.0594.2790.4290.90–93.1694.7193.68BR5(12)83.8982.7688.7890.8690.7391.2392.3493.8389.5791.05–92.8994.3393.73BR6(15)82.9281.5088.5390.0490.7291.2891.7293.3489.7190.70–92.6294.0493.63BR7(20)82.1480.5188.3688.6390.6591.0490.5592.5088.0590.44–91.8693.5393.14BR8(30)80.1079.6587.5287.1189.7390.26––86.13–88.4191.0292.7892.92BR9(40)78.0380.1986.4685.7689.0689.50––85.08–88.1490.4692.1992.49BR10(50)76.5379.7485.5384.7388.4088.73––84.21–87.9089.8791.9292.24BR11(60)75.0879.2384.8283.5587.5387.87––83.98–87.8889.3691.4691.91BR12(70)74.3779.1684.2582.7986.9487.18––83.64–87.9289.0391.2091.83BR13(80)73.5678.2383.6782.2986.2586.70––83.54–87.9288.5691.1191.56BR14(90)73.3777.4082.9981.3385.5585.81––83.25–87.8288.4690.6491.30BR15(100)73.3875.1582.4780.8585.2385.48––83.21–87.7388.3690.3891.02AVG(1–7)83.3781.9788.3091.2690.0690.4392.6593.7889.7390.55–92.9494.5393.53AVG(8–15)75.5578.5984.7183.5587.3487.69––84.13–87.9789.3991.4691.91AVG(1–15)79.2080.1786.3987.1588.6188.97––86.74––91.0592.8992.67aCDA was published online in 2008.K.He,W.Huang /Computers &Operations Research 38(2011)227–233232。