Optimizing PCB assembly for family setup strategy

Research articleOptimizing PCB assembly for familysetup strategyY ongzhong Wu and P.JiIndustrial and Systems Engineering,The Hong Kong Polytechnic University,Kowloon,People’s Republic of China,andWilliam HoAston University,Aston,UKAbstractPurpose–The purpose of this paper is to investigate the optimization for a placement machine in printed circuit board(PCB)assembly when family setup strategy is adopted.Design/methodology/approach–A complete mathematical model is developed for the integrated problem to optimize feeder arrangement and component placement sequences so as to minimize the makespan for a set of PCB batches.Owing to the complexity of the problem,a specific genetic algorithm(GA)is proposed.Findings–The established model is able tofind the minimal makespan for a set of PCB batches through determining the feeder arrangement and placement sequences.However,exact solutions to the problem are not practical due to the complexity.Experimental tests show that the proposed GA can solve the problem both effectively and efficiently.Research limitations/implications–When a placement machine is set up for production of a set of PCB batches,the feeder arrangement of the machine together with the component placement sequencing for each PCB type should be solved simultaneously so as to minimize the overall makespan.Practical implications–The paper investigates the optimization for PCB assembly with family setup strategy,which is adopted by many PCB manufacturers for reducing both setup costs and human errors.Originality/value–The paper investigates the feeder arrangement and placement sequencing problems when family setup strategy is adopted,which has not been studied in the literature.Keywords Assembly,Printed circuits,Process planning,Process efficiencyPaper type Research paper1.IntroductionPrinted circuit board(PCB)assembly consists of mounting electronic components onto a bare board by the so-called placement machines(Figure1).A typical placement machine has a feeder carrier,a PCB holder,and a placement device. The feeder carrier consists of many feeder slots where the component feeders are located.The component feeders are used to provide the machine with a continuous supply of components.Generally,the placement device picks components from the feeders and places them onto the board.According to different operation modes,the placement machines can be classified into different types,e.g.sequential pick-and-place machines,turret-type machines,collect-and-place machines,multi-station machines(Figure1shows the first two machine types).Optimization of the feeder setup and component placement sequence are two most important factors for improving the efficiency of placement machines.They correspond to two fundamental subproblems for machine optimization,i.e.the feeder arrangement problem(FAP)and the placement sequencing problem(PSP).The FAP is to decide the assignment of different component feeders to the slots on the feeder carrier,while the PSP is to decide the placement sequence of the components.Nowadays,many PCB manufacturers face with the difficulties caused by the wide variety of products.High-mix environment often requires frequent machine setups.While the batch sizes are small,setup time may constitute as great as 50percent of the total production time(Jain et al.,1996). Therefore,families of similar PCBs are clustered so that setups are only required between switching of families.This setup strategy is referred to as family setup strategy. However,the family setup strategy makes the already complicated machine optimization problems even more difficult.When a family setup is adopted,the machine is required to be set up for a family of boards.In this situation, the feeder arrangement and placement sequence should be optimized with respect to multiple boards.The current issue and full text archive of this journal is available at /0144-5154.htmAssembly Automation29/1(2009)61–67q Emerald Group Publishing Limited[ISSN0144-5154][DOI10.1108/01445150910929875]This research investigates the optimization of feeder arrangement and placement sequencing for a PCB placement machine adopting a family setup strategy,i.e.the placement machine is set up (feeder arrangement)once for a set of PCB batches.The paper is organized as follows.Literature relating to machine optimization problems in PCB assembly is surveyed in Section 2.In Section 3,a complete mathematical model for the problem is established.In Section 4,the model is verified through solving some problem instances.A specific genetic algorithm (GA)for the problem is proposed and described in Section 5.Finally,the paper is concluded in Section 6.2.Literature reviewThere are many planning problems in PCB assembly owing to the great variety of production environments concerning shop layout,production mix,setup policy,etc.These problems can be classified according to different planning levels on the planning hierarchy.This paper focuses on the optimization of a single placement machine.For the optimization of an assembly line with multiple machines,one may refer to Sze et al.(2001)and Wan and Ji (2001).Numerous researchers have investigated machine optimization problems in PCB assembly,i.e.feeder arrangement and PSP .Owing to the complexity of these two problems,some researchers focused on only one of them,while assuming the solution to the other is given in advance (Drezner and Nof,1984;Ball and Magazine,1988;Moyer and Gupta,1996a,b;Ahmadi and Mamer,1999;Klomp et al.,2000;Kim and Park,2004;Li et al.,2007).However,most researchers agree that these two problems are intertwined with each other to influence the efficiency of the machine,and should be considered at the same time.Some researchers focused on an approach that consists of tackling both problems by iterating between (heuristic)both solutions.Among these,Foulds and Hamacher (1993)used an iterative approach for a sequential pick-and-place machine.The PSP was solved by a TSP heuristic and the feeder arrangement problem was solved by a QAP heuristic.Egbelu et al.(1996)also considered a sequential pick-and-place machine and solved the two problems with two heuristics iteratively.Crama et al.(1990)solved the two problems iteratively for a collect-and-place machine,with each problem solved by a heuristic.Leipa ¨la ¨and Nevalainen (1989)stated that the iterative approach was fast but obtained inferior solutions.For this reason,many other researchers proposed approaches for solvingthe two problems simultaneously.Broad et al.(1996)established an integrated integer programming (IP)model for the two problems for a sequential pick-and-place machine.The model was solved by a binary IP package.Leu et al.(1993)proposed a two-link GA to simultaneously solve the feeder arrangement and PSP for both a sequential pick-and-place and a turret-type machine.Ho and Ji (2003,2004)modified the two-link GA proposed by Leu et al.for both a sequential pick-and-place and a turret-type machine and obtained better results.Moyer and Gupta (1996a,b)proposed a heuristic called AAT for solving the two problems simultaneously for a turret-type machine.They also stated that on average,their approach was superior to the GA proposed by Leu et al.(1993).Y eo et al.(1996)proposed a rule-based approach to simultaneously solve the feeder arrangement problem and PSP for a turret-type machine.Ellis et al.(2001)proposed a heuristic for solving the two problems simultaneously for a turret-type machine.They used a construction procedure with a set of rules to generate an initial component placement sequence and feeder arrangement,and an improvement procedure to improve the initial solution.Some researchers studied the optimization problems for some more advanced placement machines.Kulak et al.(2007)proposed GA for collect-and-place machines with revolving heads.Ho et al.(2007)also proposed a GA to solve the feeder arrangement and PSP simultaneously for a dual-gantry collect-and-place machine.Wang et al.(1999)proposed a GA to optimize feeder setup for a multi-station placement machine.Recently,Grunow et al.(2003)established an IP model for the optimization of a multi-station machine.It can be seen from the survey that numerous researchers have investigated the machine optimization problems in PCB assembly.However,all the researchers considered optimizing a machine for a single board type.T o the best of our knowledge,no researchers have considered the situation when a machine is set up for production of multiple boards,i.e.the family setup strategy is adopted.3.A mathematical modelSince the investigated problem has not been investigated by other researchers,a mathematical model for the problem is required.In this paper,the sequential pick-and-place machine (as shown in Figure 1(a))is studied.For a sequential pick-and-place machine,the placement head can move in the X -Y plane.After the placement head travels to a feeder and picks a component,it moves to the placing location on the board and places the component.Then it travels to another feeder to pickFigure 1Different PCB placement machine types(a) Sequential pick-and-place (b) Turret-typeFeedersup another component.The same operations are repeated until all required components are placed on the board.As discussed earlier,optimization of placement machines consists of solving the feeder arrangement problem,which deals the assignment of different component feeders to different feeder slots on the machine,and the PSP,which decides the sequence for the components to be placed.These two problems intertwine with each other to influence the process time for a specific board.When the machine is set up for producing different board types,the feeder arrangement problem should consider the characteristics of all the board types.At the same time,different placement sequences corresponding to different board types should also be determined.The optimization of feeder arrangement and placement sequencing should aim to minimize the makespan for all PCB batches.Here,an integrated mathematical model for the stated problem is established.The parameters and variables used in this model are defined as follows:1Parameters:.t ijs ,time elapsed between placing component i and placing component j when feeder f (j )is stored in slot s ;.n ,total number of components on all board types;.m ,number of board types;.n h ,number of components on a PCB of type h ;.q i ,batch size of the PCB that consists of component i ;.n ,total number of component types;.c ,total number of feeder slots on the machine;.C h ,set of components on a PCB of type h ;and .G h ,arbitrary (n h 21)components on a PCB of type h .2Decision variables:.x ij ,0-1variables.x ij ¼1if component j is placed immediately after component i (i ,j ¼1,...,n );.y f (i ),s ,0-1decision variables.y f (i ),s ¼1if a feeder type for component i is stored in slot s ;and .u i ,auxiliary integer variables,indicating the placement order of component i .With the above parameters and variables,an IP model can be formulated as follows:Model 1:min X n i ¼1X n j ¼1X c s ¼1q i t ijs x ij y f ðj Þ;sð1Þs :t :j [C hXx ij ¼1;i [C h ;i –j ;;h #mð2Þi [C hXx ij ¼1;j [C h ;i –j ;;h #mð3Þu i 2u j þn h x ij #n h 21;i [G h ;;j [G h ;i –j ;;h #m ð4Þs #cXy ks ¼1;k #n ð5Þk # nXy ks #1;s #cð6ÞObjective function (1)is to minimize the makespan,i.e.totalplacement time for all the PCBs.Constraints (2)-(4)guarantee that the placement sequence for each board type constitutes aHamiltonian cycle.Constraint (2)requires that for each boardtype there must be exactly one motion to leave each component location;constraint (3)requires that for each board type there must be exactly one motion entering each component location;constraint (4)is a sub-tour elimination constraint,which guarantees that for each subset of component locations,the sequence of motions must not form a cycle.Constraint (5)requires that each type of components must be assigned to exactly one pickup slot location.This constraint may not hold for some cases where a feeder may occupy more than one feeder slots.However,for the machine investigated in this paper,most of the components to be placed are those small ones which require one-slot feeders.Accordingly,constraint (6)requires that each pickup slot location must contain at most one type of components.There is a nonlinear term x ij y f ðj Þ;s in the objective function,which can be eliminated by introducing an auxiliary integer variable w ijs .Let w ijs be an n £n £c dimensional binary decision variable where w ijs ¼1if and only if after the i th component is placed,the j th component from the s th feeder slot will be placed immediately.Clearly,the operating time t ijs is spent if and only if w ijs ¼1.Thus,the problem can be reformulated as:Model 2:minX n i ¼1X n j ¼1X c s ¼1q i t ijs w ijs ð7Þs :t :j [C hXx ij ¼1;i #C h ;i –j ;;h #mð8Þi [C hXx ij ¼1;j #C h ;i –j ;;h #mð9Þu i 2u j þn h x ij #n h 21;j [G h ;i –j ;;h #mð10Þs #cXy ks ¼1;k #n ð11Þk # nXy ks #1;s #cð12Þw ijs #x ij ;i #n ;;j #n ;;s #c ð13Þw ijs #y f ðj Þ;s;i #n ;;j #n ;;s #cð14Þw ijs $x ij þy f ðj Þ;s 21;i #n ;;j #n ;s #cð15ÞThe time elapsed between placing component i and placingcomponent j when the feeder f (j )is stored in slot s ,i.e.t ijs ,is assumed to be proportionate to the sum of the chebyshev distance between component i and slot s ,d is and the chebyshev distance between component j and slot s ,d js .The Chebyshev distance,which is defined as the greater one of the distances along x and y direction,is adopted because movements along x and y directions are separated for the placement head.Let d ijs ¼d is þd js .Then the objective function in Model 2can be reformulated as:minX n i ¼1X n j ¼1X c s ¼1q i d ijs w ijs ð16ÞAs can be seen,the established mathematical model is an integer linear programming (ILP)model,which can be solvedby many existing software packages.In order to verify the established model,a set of test instances are generated and solved,which is described in the next section.4.Exact solutions to the problemIn this section,some testing instances of the problem are generated.A commercial package,CPLEX 10.2,is utilized to solve these instances which are formulated as the ILP using Model 2.4.1Testing instancesT otally,seven problems of different sizes are created for testing.T able I summarizes the characteristics of these problem instances.For instance,one,which is the smallest,there are two PCB batches with the same batch size.Because each board type has the same batch size,so reasonably,let q i ¼1.There are five components on each PCB.These components belong to six different component types.There are exactly six slots on the machine.So each slot will be assigned to exactly one feeder type.The component location and type data for the two boards are shown in T ables II and III,respectively.The locations of the feeder slots on the machine are shown in T able IV .All the coordinates are based on the same origin point (0,0).Based on the coordinate data,Chebyshev distances between each feeder slot and each component can be calculated,as shown in T able V .4.2Solutions and analysisCPLEX 10.2is used to solve the testing instances.The software was run on a desktop computer with Intel Xeon 2.4GHz CPU and 3G RAM.The results are summarized in T able VI.For instance,one,CPLEX spent 2.61s to obtain the optimal solution.Following the solution,feeder types f 1,f 2,f 3,f 4,f 5,f 6are assigned to feeder slots s 2,s 5,s 6,s 1,s 3,s 4,respectively.The component placement sequence for board type 1is c 1!c 4!c 5!c 3!c 2.The component placement sequence for board type 2is c 10!c 9!c 6!c 7!c 8.The total distance traveled by the placement head is 505mm.Table IV Coordinate data of feeder slots on the machineSlots Coordinates (mm)(s )x y 110502103531020430105501067010Table I Characteristics of the problem instancesInstance Number of boardtypes Total number of components (number of component on each board)Total number of component types Number of feeder slots on the machine1210(5,5)662215(7,8)683320(6,6,8)884325(7,8,10)8105430(6,6,8,10)101064120(30,30,30,30)153074260(60,60,60,80)2030Table II Component coordinate and type data for board 1Components Types Coordinates (mm)(i )(k )x y 163020213030313040453050543060Table III Component coordinate and type data for board 2Components Types Coordinates (mm)(i )(k )x y 6250207350308350409550501045060Table V Chebyshev distances between each feeder slot and each componentSlotComponent 12345613020201020402202020202040320202030304042020304040405202540505050640404020102074040402020208404040303030940404040404010404040505050Table VI Computational results for testing instances by CPLEXInstance Optimal solution CPU time (hh:mm:ss)150500:00:02262800:00:3731,55300:02:2643,02101:15:1957,37269:36:566No result for seven days No result for seven days 7No result for seven daysNo result for seven daysIn Model 2,there are totally n 2c þn 2þ nc binary variables plus n integer variables.So the complexity of the model grows exponentially with the problem size.From the results shown in T able VI,it can be seen that CPLEX spent seven days without finding the optimal solutions for instances six and seven,which are more realistic instances.The exact solutions seem to be not practical for solving real-sized problems.5.A genetic algorithmFrom the experimental results discussed in Section 4,it can be seen that the complexity of the investigated problem is great.In order to solve the problem efficiently,a heuristic method is required.In this paper,a GA is proposed for solving the investigated problem.GA are a population-based metaheuristic which aims to find near-optimal solutions in reasonably computational time.GA have been used to solve different types of optimization problems that are difficult for exact algorithms.One may refer to (Man et al.,1999)for more detailed description of GA.Nevertheless,the adoption of GA in this paper does not necessarily mean that GA is the best heuristic method for solving the parison of different heuristic methods is out of the scope of this paper.5.1Description of the proposed GAThe key issues in developing a GA are the representation scheme,genetic operators,fitness evaluation,and reproduction process.In this section,these issues are described in details.5.1.1Representation schemeFor the investigated problem,a solution shall include a common feeder assignment and several placement sequences that correspond to different board types.In order to represent the unique solution structure,a multi-link chromosome is proposed,as shown in Figure 2.As shown in Figure 2,each chromosome consists of several links.The first link represents the feeder arrangement solution,while the following links represent the component placement sequences for different board types. 5.1.2Crossover and mutationSince each link in the chromosome is similar to the permutation type,genetic operators for permutation representation can be used for the proposed GA.In this paper,the order crossover (OX)and the swap mutation is adopted.The OX operator was designed by Davis (1991)and is suitable especially for order-based permutation problems.The process of the operator is described as follows:.choose two crossover points at random,and copy the segment between them from the first parent into the first offspring;.starting from the second crossover point in the second parent,copy the remaining unused numbers into the first child in the order that they appear in the second parent,wrapping around at the end of the list;and .create the second offspring in an analogous manner,with the parent roles reversed.The swap mutation works simply by randomly picking two genes in the string and swapping their values.It should be noted that for the proposed multi-link representation (Figure 2),the genetic operators,i.e.the crossover and the mutation,need to be performed on each link separately to ensure the feasibility of the offspring.5.1.3Fitness evaluationThe objective of the proposed GA is the same as that described in Model 2.The objective function (16)is used to calculate the objective values of each chromosome.Let F h be the objective value if the feeder arrangement and placement sequences are represented by the chromosome X h ,then the fitness function of X h is defined as:eval ðX h Þ¼1F hð17Þ5.1.4General processThe selection mechanisms are the driving force for GA.There are two places in GA where selection occurs:when choosing parents to produced offspring,and when choosing which individuals will survive.If the combined selection pressure is too strong,the GA is likely to converge too quickly to aFigure 2One representation scheme for the proposed GAlink 1(feeder arrangement)link 2(placement sequence for board 1)link 3(placement sequence for board 2)component 9 on board 2 is placed secondsuboptimal region of the space.As a consequence,one of the two selection mechanisms(i.e.either parent or survival selection)is generally chosen to be random selection(de Jong, 2006).In the proposed GA,the parents are randomly selected. The replace-worst replacement strategy is adopted for selecting the chromosomes for survival,that is,a new generation is formed by selecting the best individuals from the parents and the offspring.The general framework of the proposed GA can be shown in Figure3.The general process for the proposed GA is described as follows.At the beginning of the algorithm,an initial population of chromosomes is randomly generated.Then, for each generation,randomly select pairs of chromosomes and apply crossover operator to each pair of parents to produce offspring.The number of parents is determined by the crossover rate C r,which represents the percentage of population to be chosen as parents.The generated children are then mutated according to a certain mutation possibility M r.For the swap mutation,M r represents the possibility for a chromosome to be selected for mutation.After the reproductive operations,the best chromosomes from the original chromosomes and the offspring are selected for survival in the next generation.5.2Experimental results for the GAIn this section,the same testing instances in Section4are used to test the performance of the proposed GA.The population size for the GA is set to100.The crossover rate,C r,is set to0.3and the mutation rate,M r,is set to0.1. The algorithm is run until1,000generations are produced. The program is coded in Microsoft Visual Cþþ6.0,and runs on the same computer described in Section 4.For each instance,the algorithm is run20times.T able VII summarizes the computational results.From the results,it can be seen that the proposed GA obtains the optimal solutions for all the instances that can be solved by CPLEX in all20runs,while the computation times are much shorter than those required by CPLEX.Although it cannot guarantee optimal solutions for large problems(instances six and seven),the standard deviations of objective values for20 runs are not large,indicating that the GA has relatively stable performance on solving the problems.In addition,the computational times are paring with exact solutions by CPLEX,the proposed GA approach can solve the investigated problem both effectively and efficiently.6.ConclusionThe family setup strategy is adopted when PCB manufacturers want to save setup cost.This is significant especially setup cost is relatively large.In this paper,the optimization problem for PCB placement machines adopting family setup strategy is studied.In this sense,the approach established in this paper forms the basic requirement for PCB manufacturers to make strategic decisions on setup policy. When a family setup strategy is adopted,the feeder arrangement of the machine should be optimized for several board types simultaneously while the placement sequences should be optimized with respect to each board type.In this paper,a complete mathematical model for the investigated problem is developed for obtaining overall optimal solutions. The model has been verified through solving some problem instances.However,the great complexity makes the exact solutions impractical for realistic problems.Therefore,a specific GA has been proposed.In the proposed GA,a multi-link representation scheme is used to represent the combined solution.Experimental results show that the GA can solve the problem both effectively and efficiently.The model developed in this paper adopts some simplified assumptions,e.g.the moving speed of the placement head is assumed to be constant(in fact,the speed may vary a bit according to the component size).As the result,the established approach should be further examined through real case studies,which form a logical extension to this research.Figure3General framework for the proposed GA for the CAPTable VII Computational results for the proposed GAInstance Objective a SD b CPU time c1505(optimal)00.012628(optimal)00.0131,553(optimal)00.0343,021(optimal)00.2357,372(optimal)0 1.49616,60134.6914.58759,363178.8158.24Notes:a Average objective value for20runs(in mm);b SD of objectivevalues for20runs;c average CPU time(in min)ReferencesAhmadi,R.H.and Mamer,J.W.(1999),“Routing heuristics for automated pick and place machines”,European Journal of Operational Research,Vol.117,pp.533-52.Ball,M.O.and Magazine,M.J.(1988),“Sequencing of insertions in printed circuit board assembly”,Operations Research,Vol.36,pp.192-201.Broad,K.,Mason,A.,Ro¨nnqvist,M.and Frater,M.(1996),“Optimal robotic component placement”,Journal of the Operational Research Society,Vol.47,pp.1343-54. Crama,Y.,Kolen,A.W.J.,Oerlemans,A.G.and Spieksma, F.C.R.(1990),“Throughput rate optimization in the automated assembly of printed circuit boards”,Annals of Operations Research,Vol.26,pp.455-80.Davis,L.(1991),Handbook of Genetic Algorithms, Van Nostrand Reinhold,New Y ork,NY.de Jong,K.A.(2006),Evolutionary Computation:A Unified Approach,The MIT Press,London.Drezner,Z.and Nof,S.(1984),“On optimizing bin picking and insertion plans for assembly robots”,IIE Transactions, Vol.16,pp.262-70.Egbelu,P.J.,Wu,C.T.and Pilgaonkar,R.(1996),“Robotic assembly of printed circuit boards with component feeder location consideration”,Production Planning&Control, Vol.7,pp.162-75.Ellis,K.P.,Vittes,F.J.and Kobza,J.E.(2001),“Optimizing the performance of a surface mount placement machine”, IEEE Transactions on Electronics Packaging Manufacturing, Vol.24No.3,pp.160-70.Foulds,L.R.and Hamacher,H.W.(1993),“Optimal bin location and sequencing in printed circuit board assembly”, European Journal of Operational Research,Vol.66, pp.279-90.Grunow,M.,Gu¨nther,H.O.and Schleusener,M.(2003),“Component allocation for printed circuit board assembly using modular placement machines”,International Journal of Production Research,Vol.41No.6,pp.1311-31.Ho,W.and Ji,P.(2003),“Component scheduling for chip shooter machines:a hybrid genetic algorithm approach”, Computers&Operations Research,Vol.30,pp.2175-89. Ho,W.and Ji,P.(2004),“A hybrid genetic algorithm for component sequencing and feeder arrangement”,Journal of Intelligent Manufacturing,Vol.15,pp.307-15.Ho,W.,Ji,P.and Wu,Y.(2007),“A heuristic approach for component scheduling on a high-speed PCB assembly machine”,Production Planning&Control,Vol.18No.8, pp.655-65.Jain,S.,Johnson,M.E.and Safai,F.(1996),“Implementing setup optimization on the shopfloor”,Operations Research, Vol.44,pp.843-51.Kim,K.M.and Park,T.H.(2004),“PCB assembly optimization of chip mounters for multiple feederassignment”,SICE Annual Conference,Sapporo, pp.1425-30.Klomp,C.,Klundert,J.V.D.,Spieksma,F.C.R.and Voogt,S. (2000),“The feeder rack assignment problem in PCB assembly:a case study”,International Journal of Production Economics,Vol.64,pp.399-407.Kulak,O.,Yilmaz,I.O.and Gu¨nther,H.O.(2007),“PCB assembly scheduling for collect-and-place machines using genetic algorithms”,International Journal of Production Research,Vol.45No.17,pp.3949-69.Leipa¨la¨,T.and Nevalainen,O.(1989),“Optimization of the movements of a component placement machine”,European Journal of Operational Research,Vol.38,pp.167-77. Leu,M.C.,Wong,H.and Ji,Z.(1993),“Planning of component placement/insertion sequence and feeder setup in PCB assembly using genetic algorithm”,Journal of Electronic Packaging,Vol.115,pp.424-32.Li,S.,Hu,C.and Tian,F.(2007),“Enhancing optima feeder assignment of the multi-head surface mounting machine using genetic algorithms”,Applied Soft Computing,Vol.8, pp.522-9.Man,K.F.,T ang,K.S.and Kwong,S.(1999),Genetic Algorithm:Concepts and Design,Springer,New Y ork,NY. Moyer,L.K.and Gupta,S.M.(1996a),“SMT feeder slot assignment for predetermined component placement paths”,Journal of Electronics Manufacturing,Vol.6, pp.173-92.Moyer,L.K.and Gupta,S.M.(1996b),“Simultaneous component sequencing and feeder assignment for high speed chip shooter machines”,Journal of Electronics Manufacturing,Vol.6,pp.271-305.Sze,M.T.,Ji,P.and Lee,W.B.(2001),“Modeling the component assignment problem in PCB assembly”, Assembly Automation,Vol.21No.1,pp.55-66.Wan,Y.F.and Ji,P.(2001),“A tabu search heuristic for the component assignment problem in PCB assembly”, Assembly Automation,Vol.21No.3,pp.236-40.Wang,W.,Nelson,P.C.and Tirpak,T.M.(1999),“Optimization of high-speed multistation SMT placement machines using evolutionary algorithms”,IEEE Transactions on Electronics Packaging Manufacturing,Vol.22No.2, pp.137-46.Y eo,S.H.,Low,C.W.and Y ong,K.H.(1996),“A rule-based frame system for concurrent assembly machines”, International Journal of Advanced Manufacturing T echnology, Vol.12,pp.370-6.Corresponding authorY ongzhong Wu can be contacted at:w.yz@.hkT o purchase reprints of this article please e-mail:reprints@ Or visit our web site for further details:/reprints。