- Email: [email protected]
Adaptive Genetic Algorithm for Optimal Printed Circuit Board Assembly Planning
Adaptive Genetic Algorithm for Optimal Printed Circuit Board Assembly Planning H. Wong, M. C. Leu (2). New Jersey Institute of Technology, NewarWUSA Received on January 15,1993
ABSTRACT We present a novel application of the genetic algorithm approach for solving the problem of planning optimal placcmenllnrtnion sr,quencc and machine setup in primed circuit board (PCB) assembly. The algorithm starts with feasible roluuons and utilizes genetic operators to iteratively generate potentially better solutions in the optimization process, similar to the biological evolution process. We first dunibe the basic algorithm and its application to optimal planning for some popular PCB assembly machines. We then descnie an adaptive genetic algorithm. which has its rates of genetic operators changed automatically during the iterative optimization process. We use a Wikoxon signed rank test to show its p f o m n c e improvement over the fixed-rate genetic algorithm Keywords: Assembly Machines, Algorithms, Optimization
1. INTRODUCTION
Printed circuit boards are used cxtensively in industry for the manufacturing of clearonic and electromechanical products such as computers, process controllers, and industrial robots. Much research has been done in areas such as soldering and inspection (Conway, et. al.. 1991; Mengel. Roth and Sch1992). Although many assembly machines are available commercially for automatic placementlinsenion of board components. the process of planning the placementlinsertion sequence and machine setup b often dom manually. Vlriws ruearchers have started to develop algorithms to automate and optimize the planning process. Ball and Magazine (1988) modeled the PCB component sequencing problem for a moving-head. stationary-board, stationary-feeder machine as a rural postman problem. They solved it with a heuristic which guarantees the solution to be optimum if the assembly head movement is rectilinear. Francis. et al. (1989) formulated the problem of finding bin (feeder) locations and placement sequences in the placement of n parts from tr stationary bins by a Cartesian robot as a traveling salesperson problem with special structure This structure allowed them to develop a solution algorithm which gives asymptotically optimal solutions in n.logn effort. Leipilll and Nevalainen (1989) studied the PCB assembly optimization problem for the Panasert RH machim. They modeled the components sequencing problem as a traveling salesperson problem and the problem of assigning components to feeders as a quadratic assignment problem, and obtained a suboptimal solution with a heuristic approach. Bard, Clayton and Feo (1990) modeled the combined feeder setup and component placement problem for the Fuji CP I1 machine. They divided the problem into three subproblems and solved them with a traveling salesperson heuristic and a Lagrangian rela..ation scheme. Ji. Leu and Wong (1991) decomposed the planning problem for the PCB assembly machine having a moving head. a stationary board, and stationary feeders into two serial assignment problems. each solved with the reduced matrix method. All the above described algorithms tend to be machine dependent, i.e. one algorithm is applicable to one panicular type of machines. Motivated by the success of the genetic algorithm approach in solving difficult, complex optimization problems (Davis, 1985; Grefenstette, 1985). we have explored the feasibility of developing a genetic algorithm for solving PCB assembly planning problems and have obtained promising results. Some of the initial results were described in (Leu. Wong and Ji. 1992).
In this paper we describe our new results in the developmwnt of genetic algorithms for solving the problem of planning optimal placementlinsertion sequence and machine setup in PCB assembly. Thh approach for PCB assembly planning has a major advantage over other methods: a basic algorithm can be easily modified to solve the planning problem for assembly machined having different mechanisms and features. We will first present the basic algorithm that we have developed for assembly time minimization and its application to optimal planning for three popular types of PCB assembly machines. We will then present an adaptive genetic algorithm. which has its rates of genetic operators adapted during the iterative process. and use a Wilcoxon signed rank test to show its improved performance over the fixed-rate, basic genetic algorithm. 2. GENETIC ALGORITHM
The genetic algorithm (Holland, 1975) is a gencral-purpose stochastic rithm which uses a p r o w similar to biological evolution 10 set of feasible solutions (called the initial population) through an iterative process. It has been used uuccesrfully to solve problems in job scheduling, machine learning. and pattern recognition. For a rigorous deoeription of the genetic algorithm we give the following definitions. Definition I : Let X be a set of n continuous positive integers starting from 1. i.e. W = (1, 2, . _ ., n ) and C be a set of n grne5, 1.e.
Annals of the CIRP Vol. 42/7/7993
C * (c,. c2 , . .. , cJ. Each element c; is a gene and c;
f
c, if i # j .
[so),
Definition 2: A link is defined as 3 = s(2). . _ _ ,s(n)], where s(k) is the gene at the &-th position of the link, and 5 is a mapping s : S -B C. s(i)+s(/)ifi#j. Definition 3: The set S is formed by all possible links, i.e. S- p,, 3>, 3,. .._, 3 $ ) . where 4 is the number of all possible permutations of genes forming the links. Definition 4: The set Sw is a subset of S and contains w links, i.e. S = (3,. 3, 33,.. . , X W ) , where w is the number of links in set . , S Definition 5: The cost function is a mapping defined as F: S-13,where 'R is the set of all real numbers. 4 3 ; ) = F;is the cost of solution 3;. Definition 6 : An operator G, is a mapping G, . S,,, -+ where m is the number of parents, ) I is the number of offspring, and i is an index. Definition 7: A mating pool MP, is defined as MP, = ((3,,p,),(S,pJ. ... , (JW,pw))where 3; E S, and p; is the probability for 3; being chosen From MP,. Definition 8: The operation rate of operator G, based on the mating pool Up, is defined as r, = 1 + njw, where n, is the number of offspring generated by G,. Definition9:
R,
The
of the elementl of S, is defined w ... ,raw) where ra; corresponds to the order of F;. S ,
ranking
= (ra,. ra,.
ia calledfu/ly mi&edif the corresponding Rw has been determined. Definition 10: A mating poolMP, is called fully ranked if.$,, is fully ranked. Definition 11 : The cost function of a two-link genetic algorithm b a mapping defined as E SxS + 'R. where 'R is the set of dl red numbers. F(3,. 3;3 =Fl is the cost ofsolution Q,, 3;>. Definition 12 : A mating pool MP, of a two-link genetic algorithm u defined as where 3; E S, MPw = CP,, 3, P,),(3, S, pd. ... (3,. xw 3;' E S , , md'p; is the pr:bability for (3;. ;T:ing chosen From I
.
.
.
MP,. A typical genetic algorithm contains the following steps: 1. Randomly generate a set of initial parents. SmD. that forms a mating pool MPmo. Find F; M,. where i = 1. 2. ... ,m, The probability function p, is chosen to be inverse proportional to ra, and Zp,= 1. Set initial rate for u c h operator 9.j I. 2. .. . , I.
-
2. Sequmtially apply operators Cj with fixed operation rates r j J = I , 2,
... , x.
where r is the total number of operators. and collect the offspnng generated to a set S ,
3. Form MP, from S, after all the operator$ have been applied. For MP,,fi@ F,, where i = 1, 2, ... , u. Let Mpm. = MP, u ,UPmo. Determine ra,'s such that MPmxis fully ranked. 4. Define a new MPmp such that 3;in Up, is the same i = 1, 2. . _ _ , m, Update the probability funitionp,.
as
3;
in MPmxfor
5 Repeatsteps2-4 6. If I specified criterion for ending the genetic search has been met, stop; otherwise go to step 2.
17
3. APPLICATION TO PCB ASSEMBLY PLANNING An important problem in PCB assembly planning is to decide the order in which components should be p l a c c d l m e d . This f l e a s the cycle timc of usembly. In the w the components are directly fed to the assembly head by a tape and the feed time is alwayr shorter than the traveling time of the PCB between two components. the problem can be formulated as a trawling salesperson problem. In this case, the tixed-head, moving-board problem can be reformulated as a moving-head, fixed-board problem. The head visitr each component location once. Most of the commercial assembly machines for through-hole components belong to this category; for example, Amistar AI-6448, Panwnic P l n a r a RT, Universal 62874 and Universal 62418. Figure 1 illustrates this type of machines. The planning problem in this case is to determine the sequence of the assembly head in visiting all component locations such that the total traveling time is minimized. This is the well-known traveling salesperson (TS)problem (Lawier, et al. 1985).
In the use the components are supplied by multiple rtationiuy feeders, the assembly head needs to pick up the components fmm the various feeden and place them on the board. Many assembly machines for surface-mount components belong to this catcgory, such as Amistar FA-2001. Fuji EP-20, and n Prnwnic P u ~ MPA. I n addition to the sequence of placinlJlNcrting
Figure 3. Assembly machine for the MBTD problem The objective of PCB assembly planning is usually to minimize the total assembly time The evaluation function may vary for different types of problems. The total assembly distance UVI be used as the obj&e h c t i o n in the TS and PAP problem, ifthe speed of the X-Ytable or apembly head is constant. The movements of the assembly heads or X-Y tables may have different patterns for different typea of w m b l y machines. For the machines whose X motion and Y motion have the same MNtMt speeds, referred to as the Chcbyshev metric in the literature. the cost function CM be sct as Zmmr(lxi x j r f l , bi -y;+]I). whue 40, b) is the larger of0 and b, and (x;. Yi),( X i + ] , y;+]) are the locations of components in two successive placementsruucrfions. For the machines which always move from one location to another along a straight line at a constant speed. called the Euclidean metric. the evaluation function can be set as
Zdk;-x;+/I~+ lYl -y;+/P. Problem
\\
For a PCB having n components in the TS problem, the assembly sequence is represented by a link with n genes A link is S = [s(f),s(2). ..., Nn)] where HI] E C = ( c,, c l , ._.,c,) V i E N = ( I , 2. ... , n) , s(i) f if i # j , and a c h c; is a board component. The link designates the sequence of assembly of the board components ARer the squence ends, it repeats for another board in the batch.
Figure 1. Assembly machine for the TS problem components on the board, the plu~lingproblem may also include l~Signing components to feeders. The problem will be called the pick-and-place (PAP) problem. figure 2 illustrates a machine for this type of problems.
7
9)be the assembly distance between component c; and component The cost function is: Let D(ci,
which is the total distance of board movement (or. quivalently, the total distance. ofassembly head movement). A PCB of thirty components is used as an example in solving the TS problem. There are thirty links randomly generated and used as the initial population. The Euclidean metric is lusumed in this example: A ncar-aptimal assembly process produced is illustrated in Figure 4. The computation time is 75 seconds for 200 iterations on the Compaq 3 8 6 1 2 0 ~personal computer PCB
45
Feebel5
Wcdcmg Table
Figure 2. Assembly machine for the PAP problem Another type of assembly machine has a moving feeder carria. a moving X-Y table carrying the PCB. and a turret having multiple assembly heads (see Figure 3). There is Ipick-up location and a placement locstion and they are both fixed in space. The P l w u t RH and Universal ORsnter XI 4712B are two commercial machines which bdong to this e~cgory. The problem here ir to simultaneously determine the sequence of assembly of board components and udp the various types of components to feeders such that the total assembly time is minimized. This problem will be called the moving-board-with-himGdehy (MBTD)problem.
,
40
35
-
E E 30
u
F
20
To dovelop a genetic algorithm for planning of PCB assembly. we first consider the link representation. All the above assembly problems involve determining the component assembly sequence, which can be represented by C = (c,), i-1, 2, ... , n, representing the order of rhe placemen?hdon
w-. The initial population in the iterative procwr may be randomly generated. In actual application, howevs. it is desirable and oRen possible to expedite the iterative process by creating some 'good' initial estimates. For aample, an initial estimate may be the scquence generated by UI experienced technician. B d on this md other randomly g~nrmedinitial eslimrtu, the gmnic algorithm CM iteratively improve the estimates and finally arrive at a ncar-optimal solution.
18
o
10
20
ID
40
so
m
70
110
PO
im
X (mm)
Figure 4. Optimal assembly sequence obtained from the genetic algorithm cProbla
For the PAP problem, we w u m e Ih.1the PCB has a total of n components in p different types provided by m feeders, where m 2 p . In applying the genetic algorithm approach to solve this problem. we use two linllr: one is the -bly sequence, with n genes each rcpruatiq a board component number. ud the
o t h a is the feeder aslipnent, with m gena each representing a feeda number.
We assume that the assembly head will rest at a particular l d o n between two successive boards.
In the genetic algorithm, the component assembly sequence and the feeder tunngemcnt are designated by link 1 and link 2. respectively. Each link 1 is defined as 3 [s(l).s(2). .._, s(n)].41)E C = ( cI. c2 , ..., cn) V i E N = (1, 2. .. , n) , s(1) # if i # j , where ck represents a board component. Each link 2 is ddned as 3. = [s*(/). s*(2), ..., s*(m)], s'o] E F = ( f I , j 2 , .... j . ) Vj E M = (1. 2, ... , m ) , ~'(1) # s.0) if i # j , w h m cachjk represents a feeder d each elanent of M is a number designating a component type.
so)
The distance between feederji and component c ' can be expreued easily in t m s of the (x. y) coordinates of/i and cj as in the Td problem. Let D denote the distance betwcm two points as before. Also. let r denote the rest (Irtarting md end) point, and &j) denote the type of component for component c, The evaluation hmnction, which is the total travd distance, is Lk-/
D(s(n). s*(b(s(n))))+ D(s(0, 4 + D(s(n), r ) A sample PCB with 200 components in 10 differenf component types is urcd u M example in solving the PAP problem. One hundred links are generated randomly as the initial guesses. The total distance is reduced by 1I 84% from the best initial guess in 6,150 iterations. ~ - b o ~ r d - w i t h - t i m c - d c l Prob l v la
Bccau~athe various moving parts of the assembly machine have different speeds in the MBTD problem, assembly time should be used as the objective h a i o n in this problem. The three important times of consideration arc: the traveling time of the PCB, the shifting time of the pick-up head, and the travding time of the f d e r carrier The longest of the three is the time needed in the assembly of each component
We consider M n-component PCB havingp types of components nrppkd by m feeders (m z p ) As in the PAP problem, we create in the genetic algorithm two links: one is the l i for the assembly ~ q u e n c with e n genes, and the other is the link for the feeder assignment with m g e m . For the first few components assembled in a batch of PCB's, there are only pick-up movements md no placement movement. For the lasf few componenlc of the same batch, there are only placement movements and no pick-up movcmrnt. If the quantity of PCB's in I batch b very large, we can neglect thesc boundary dfW. Let the traveling time of the X-Y table, the traveling time of the feeder curier, and the induo'ng time of the turret be denoted by rI, fB md r,, respectively. As in thc PAP problem, b(ci) denotes the component type for component c,. Also, let g denote one plus thc numba of components in the gap between the pickup component and the p l m e n t component in the turret Thc traveling t h e of the X-Y table bcht(een component c,{x,. y,) and
gxj,
Ircrd,
component yj) is for Chebyshev metric. where vx and vu vx ' vy are the speeds of the X-Y table in the x and y directions. rcspcaively. The traveling time of the feeder carrier between feederh(x',. y';) and fceder&x'j. y'j) is
Definition 13 Let m, be the number of initial parents, mj b'e Ihe numba of links in the mating pool Wm, ?a applying G, md the operation rate of G, be 'j= mJmi_,. where J = I. 2. ... ,x, and x k the total number of operators. Definition 14: The ranking of the set &+FS, u (3w+I,Xw+> ... ,3,-k) is ... ,rr+.+&). w h m , S is fully dctincd u Rw+k=Rw u {ru,,+,, r-. ranked with ranking Rw Evcry 3,+i, i = I. 2. ... , k, is generated from operator G, with Fw+, not yet being found, md r+j= max(mI,fa3 . .. , raw) + i. S,+k is d e d ranked if the corresponding Rw+k has bcm determined. Definition IS: A mating poolMfw is called ranked if& L rmked.
The main idea of the adaption method is that we raix the operation ntu of those operators which generate more surviving offspring during the iterations. The number of p a n t s and the number of offspring in each generation are hxcd; namely m, and mx are fixed. To hx the population for each iteration. the operation rates of those operators which perform not as wdl u othcn vc reduced. With a given set of operation rates. the surviving offspring for each operator are counted. If surviving ofipring of an operator are more than those of other operators, the operation rate of this operator is raised with the hope that the algorithm will perform better in the later iterations. The details of this method are d i r e u d bdow. Let T denote a spccified number of iterations. We define Nj as the 'effect number' representing the total number of the offspring genmted by Gi and serving as the links in the initial mating pool A@,,,,in the next itmtion. To adjust the operation rate, we let
where ri' is the operation rate of operator i in the following icentions. Ihe proportional factor in the above relationship should be such that the number of o f f s p a generated in each iteration remains unchanged. Anofher consideration in the proposed method is that some of the operation n t a may keep decreasing to the degree thmt they M no longer valid in the iterative proccsa. Theoretically, UI opmtor generates no offspring if the operation rate is equal to or d e r than 1. So the operation rate of each opentor should be given a lower bound larger than one. The procedure of our genetic algorithm with changing operation rmtes is thus as followr:
I . Randomly generate a set of initial parents. Sma that fomu a mating pool Mpm0.Find Fj and fa,. where i = 1, 2, ... ,m, The probability functionp, is defined to be inversely proportional to ru, md Zp,= 1. Set the initial operation rafe for each opaatwrj.j= 1.2. ... ,I. 2. Let k = I. where k indexes the number of iterations for adapting the operation rates. 3. LetNj-Oforj= 1,2, ... , x
where y i s the speed of the feeder urria
Ac in the PAP problem, we use 3 to denote the placement sequence link and 3.10 denote the feeder assignment link. The time
needed for the placement of
component k is 5,= m W , ( s ( k - l ) .
4k)). rAs*(b(s(k+g-I))), s*(WtW))).
[I).
I n the above expression s(k-I)= s(n) when k = I When s(l)has I > n. where I = hg-1or k g , we replace HI) by s(l-n), which represents M initial component of the next board in the batch. This assumes that there is a large number of the same kind of boards in the batch. The evaluation function is H(3.3.) = Xrb which is
the total assembly time for a board. A board of fiRy components is used to test the algorithm developed for the
MBTD problem. The solution obtained by the algorithm is 12.75 seconds of aunnbly time, which is d y optimal. if not optimal, since the lower bound can be determined to be 12.5 seconds. 4.
THE ADAPTIVE GENETIC ALCORlTEM
An important issue in the genetic algorithm is the operation rate. W "fa of genetic operators should be uxd in generating offspring so that the algorithm an perform efficiently? We describe here a heuristic to ndjust the operation Rtcr during the iterative process. In order to describe the algorithm, we be& with more dc6nition.s:
4. Sequentially apply operators G, with fixed operation rates 9,j =1.2. ... ,x. and enlarge the mating pool to MPm's which are ranked. 7Xe probability functionp, is defined to be inversely pr&artional to ra, in Rm I'
5. ARer all the operators have been applied. find F; for the corresponding dement in , where i = m,+l, m,+2, ... , mr Determine m;s such that ,UPmxis
~y ranfttd. 6. Define a new W msuch that 3;in M f m o = 3 i i n W m z for i
Update the probabiiay functionp,.
-
I, 2,
... .m,
7. Increment Nj.) = 1.2. ... ,x, by I if operator G/ is involved in generating an offspring from Mfmo in this iteration, for all the members in ALPmo.
Increment k by 1. 8. Repeat J l e p ~4-7 while k S i
9. Compute thc new operation rates rj', j = 1.2. described above. 10. If rj' c
... ,x,
using the method
+, where + is the lower bound of operation rate for operator Gp
9 for j -
let
9'
1.2. ... ,I. Then compute the new operation r a t e rj"from 9' such that the number of offspring generated in each itention rCmainr unchanged. Repeat this step until the new operation rates satisfy the lowerbound requirement, i.c. rj' > f o r j = I. 2, ... ,x.
+
11.If a rpccified criterion for mding the genetic vlnh hu bccn met. stop; 0thawiS.Z 80 t0 step 2.
19
m e r i m e n t s and Discussion The easiest way of showing whether one genetic algorithm is better than another is using statistical analysis. Golden and Stewart (1985) used a Wilcoxon signed rank test (Mosteller, 1973) to compare different heuristics that were proposed to solve the traveling salesperson problem. Here we will use this technique to compare the adaptive genetic algorithm with the fixed-rate genetic algorithm. Twenty-four traveling salesperson problems are used to test the proposed adaption method. They are divided into 8 groups, each having three different problems. The numbers of nodes in the eight groups of traveling salesperson problems are 50, 60, 70, 80, 90, 100, 110 and 120. Each problem is designated by an integer between 1 and 24 All the coordinates of the nodes are randomly generated in a square area. The initial population m0 is 40 for all the test problems
1
0
200
400
--
600
Crosrovsr
800
1000
1200
1400
Iteration ----ci
Inve,rmn
Rotalion
1600
--
1800
2000
Mvlarlon
Figure 5 Changes of operation rates for a test problem Four operators are used in sequence: order crossover operator, inversion operator, rotation operator, and mutation operator. The number of offspring generated in each iteration is 80. The initial populations are all randomly generated The initial operation rates for each problem are also randomly chosen, in a reasonable range, and they are the same for the changing-rate and fixed-rate methods. When applying the Wilcoxon signed rank test, the random variable must be first identified. Let q,,,"and y,,," be the minimum traveling distances achieved of problem i after m iterations for the changing-rate method and the fixed-rate method, respectively, and w,be the optimal solution of problem i . Also let a,,, and b,,, be the normalized percentages representing how far the achieved minimum is from the optimal after m iterations for the changing-rate method and the fixed-rate method, respectively Thus, a,,," = lOO(q,,, - y,)/v, and b,,, = IOO(y,,, - q/,)/i~,, for i = I , 2, ... 24. We want to test the hypothesis that the changing operation rate method yields a better solution, using the random variable x!,m= a,,, - b>m. Since the global minima for the test problems are unknown, we use the best solution found from the genetic algorithm in each group as the optimal solution. The first Table 1 illustrates the computed values of the random variable 4,". column contains the numbers that designate the test problems. On the bottom of this Table there are three rows containing the sum of signed ranks, S ,,, the corresponding variable with Gaussian distribution, Z(a), and the error probability, a. Since the total number of problems is 24 in the Wilcoxon signed rank test, the Table 1. Results from the test problems
sum of signed ranks can be approximated by Sufi = Z(a) 70Z(a). An a equal to 50% means these two methods perform about the same An a less than 50% means the changing operation rate method performs better
It is clear that the changing operation rate method converges faster than the fixed operation rate method For all the 24 test problems, the changes of the operation rates are very similar in trend Figure 5 illustrates the typical changes of operation rates as functions of iterations. The crossover rate is the highest in the beginning of the iterative process and it decreases after some iterations. Since the initial estimates are all randomly generated, this observation leads to the conclusion that the crossover operator is most effective when the links are distinct from one another In other words, the crossover operator has a better chance than the other
20
operators to generate good offspring if the parents are quite different in pattern M e r many iterations the better offspring may be similar in pattern, and thus the crossover operator becomes less effective. The rate of mutation operator increases at later stages From the various test problems we also observe that the rate changes of crossover operator and mutation operator depend upon the size of the problem. 5. CONCLUSION
We have developed genetic algorithms for solving the component sequence and feeder assignment problems in printed circuit board assembly The developed algorithms have been applied to studying three types of PCB assembly planning problems: the traveling salesperson problem, the pick-and-place problem, and the moving-board-with-time-delay problem, with the solution objective to minimize the assembly time. The approach is advantageous in that the basic algorithm can be easily modified to solve the planning problems for many types of assembly machines. An adaptive method is proposed to improve the algorithm efficiency. By using a Wilcoxon signed test, the method is shown to be effective in speeding up the convergence of the solution in the iterative process.
REFERENCES [I] Ball, M. 0.. and Magazine, M J , 1988, "Sequencing of Insertion in Printed Circuit Board Assembly,'' Operatiom Research, Vol 36, No. 2, pp. 192201. [Z] Bard, J. F., Clayton, R. W., and Feo, T. A., 1990, "Machine Setup and Component Placement in Printed Circuit Board Assembly," Operations Research Group, College of Engineering, University of Texas, Austin, TX. [3] Conway, P.P., Ogunjimi. A. O.,Sargent, P. M.. Tang, A. C. T.,Walley, D. C., and Williams, D. J.. 1991, "SMD Reflow Soldering: a Thermal Process Model," CIRPAnnuals, Vol. 40/1, pp. 21-24. [4] Davis, L., 1985, "Job Shop Scheduling with Genetic Algorithms," Proceedings of htertmtional Conference on Genetic Algorithms and Their Appkations, pp. 136-140. [5] Francis, R L., Hamacher, H. W., Lee, C-Y,and Yeralan, S., 1989, "On Automating Robotic Assembly Workplace Planning," Tech. Report No. 897, Department of Industrial and Systems Engineering, University of Florida. Gainesville. FL. [6] Golden, B. L , Stewart, W. R., 1985, "Empirical Analysis of Heuristics," The Travelrng Salesman Problem, Chapter 7, John Wiley & Sons Ltd., 207-215 [7] Grefenstette, J. J., Gopal, R., Rosmaita, B., and Van Gucht, D., 1985, "Genetic Algorithm for the Traveling Salesman Problem," Proceeding o/ Ititernatioiial Coifereirce oii Genetic Algorithnis and l'heir Applications, pp 160-168. [8] Holland, J. H., 1975, Adaptattori in Natirral and Arlificial Systems, University of Michigan, Ann Arbor, MI. [9] Ji, Z., Leu, M. C.. and Wong, H , 1991. "Application of Linear Assignment Model for Planning of Robotic PC Board Assembly," ASME Publication AMD-Vol. I~IEEP-VOI.I , pp 35-41. [IOILawer, E. L., Lenstra. J. K., Rinnooy Kan, A H. G., Shmoys, D. B , 1985, The Traveling Salesman Problem, John Wiley & Sons Ltd. [ 1 I ] Leipala, T., and Nevalainen, 0 , 1989, "Optimization of the Movements of a Component Placement Machine," Eitropean Journal of Operatiorial Research, Vol 38, pp. 167-177. [12]Leu, M. C , Wong, H., and Ji, Z., 1992, "Genetic Algorithm for Solving Printed Circuit Board Assembly Planning Problems," Japan-LI.S.A. Svniposium on Flexible Autoniation, pp 1579- 1586. [I3]Mengel, P., Roth, N., and Schwarz, P I 1992, "Applicaiton of Advanced Data Processing Technology for Integrated Inspection in Electronics Assembly," CIRPAnniials, Vol 4111, pp. 29-32. [14] Mosteller, F. and Rourke, R.. 1973, Sturdy Stafistics, Addison-Wesley, Reading, MA.
ABSTRACT We present a novel application of the genetic algorithm approach for solving the problem of planning optimal placcmenllnrtnion sr,quencc and machine setup in primed circuit board (PCB) assembly. The algorithm starts with feasible roluuons and utilizes genetic operators to iteratively generate potentially better solutions in the optimization process, similar to the biological evolution process. We first dunibe the basic algorithm and its application to optimal planning for some popular PCB assembly machines. We then descnie an adaptive genetic algorithm. which has its rates of genetic operators changed automatically during the iterative optimization process. We use a Wikoxon signed rank test to show its p f o m n c e improvement over the fixed-rate genetic algorithm Keywords: Assembly Machines, Algorithms, Optimization
1. INTRODUCTION
Printed circuit boards are used cxtensively in industry for the manufacturing of clearonic and electromechanical products such as computers, process controllers, and industrial robots. Much research has been done in areas such as soldering and inspection (Conway, et. al.. 1991; Mengel. Roth and Sch1992). Although many assembly machines are available commercially for automatic placementlinsenion of board components. the process of planning the placementlinsertion sequence and machine setup b often dom manually. Vlriws ruearchers have started to develop algorithms to automate and optimize the planning process. Ball and Magazine (1988) modeled the PCB component sequencing problem for a moving-head. stationary-board, stationary-feeder machine as a rural postman problem. They solved it with a heuristic which guarantees the solution to be optimum if the assembly head movement is rectilinear. Francis. et al. (1989) formulated the problem of finding bin (feeder) locations and placement sequences in the placement of n parts from tr stationary bins by a Cartesian robot as a traveling salesperson problem with special structure This structure allowed them to develop a solution algorithm which gives asymptotically optimal solutions in n.logn effort. Leipilll and Nevalainen (1989) studied the PCB assembly optimization problem for the Panasert RH machim. They modeled the components sequencing problem as a traveling salesperson problem and the problem of assigning components to feeders as a quadratic assignment problem, and obtained a suboptimal solution with a heuristic approach. Bard, Clayton and Feo (1990) modeled the combined feeder setup and component placement problem for the Fuji CP I1 machine. They divided the problem into three subproblems and solved them with a traveling salesperson heuristic and a Lagrangian rela..ation scheme. Ji. Leu and Wong (1991) decomposed the planning problem for the PCB assembly machine having a moving head. a stationary board, and stationary feeders into two serial assignment problems. each solved with the reduced matrix method. All the above described algorithms tend to be machine dependent, i.e. one algorithm is applicable to one panicular type of machines. Motivated by the success of the genetic algorithm approach in solving difficult, complex optimization problems (Davis, 1985; Grefenstette, 1985). we have explored the feasibility of developing a genetic algorithm for solving PCB assembly planning problems and have obtained promising results. Some of the initial results were described in (Leu. Wong and Ji. 1992).
In this paper we describe our new results in the developmwnt of genetic algorithms for solving the problem of planning optimal placementlinsertion sequence and machine setup in PCB assembly. Thh approach for PCB assembly planning has a major advantage over other methods: a basic algorithm can be easily modified to solve the planning problem for assembly machined having different mechanisms and features. We will first present the basic algorithm that we have developed for assembly time minimization and its application to optimal planning for three popular types of PCB assembly machines. We will then present an adaptive genetic algorithm. which has its rates of genetic operators adapted during the iterative process. and use a Wilcoxon signed rank test to show its improved performance over the fixed-rate, basic genetic algorithm. 2. GENETIC ALGORITHM
The genetic algorithm (Holland, 1975) is a gencral-purpose stochastic rithm which uses a p r o w similar to biological evolution 10 set of feasible solutions (called the initial population) through an iterative process. It has been used uuccesrfully to solve problems in job scheduling, machine learning. and pattern recognition. For a rigorous deoeription of the genetic algorithm we give the following definitions. Definition I : Let X be a set of n continuous positive integers starting from 1. i.e. W = (1, 2, . _ ., n ) and C be a set of n grne5, 1.e.
Annals of the CIRP Vol. 42/7/7993
C * (c,. c2 , . .. , cJ. Each element c; is a gene and c;
f
c, if i # j .
[so),
Definition 2: A link is defined as 3 = s(2). . _ _ ,s(n)], where s(k) is the gene at the &-th position of the link, and 5 is a mapping s : S -B C. s(i)+s(/)ifi#j. Definition 3: The set S is formed by all possible links, i.e. S- p,, 3>, 3,. .._, 3 $ ) . where 4 is the number of all possible permutations of genes forming the links. Definition 4: The set Sw is a subset of S and contains w links, i.e. S = (3,. 3, 33,.. . , X W ) , where w is the number of links in set . , S Definition 5: The cost function is a mapping defined as F: S-13,where 'R is the set of all real numbers. 4 3 ; ) = F;is the cost of solution 3;. Definition 6 : An operator G, is a mapping G, . S,,, -+ where m is the number of parents, ) I is the number of offspring, and i is an index. Definition 7: A mating pool MP, is defined as MP, = ((3,,p,),(S,pJ. ... , (JW,pw))where 3; E S, and p; is the probability for 3; being chosen From MP,. Definition 8: The operation rate of operator G, based on the mating pool Up, is defined as r, = 1 + njw, where n, is the number of offspring generated by G,. Definition9:
R,
The
of the elementl of S, is defined w ... ,raw) where ra; corresponds to the order of F;. S ,
ranking
= (ra,. ra,.
ia calledfu/ly mi&edif the corresponding Rw has been determined. Definition 10: A mating poolMP, is called fully ranked if.$,, is fully ranked. Definition 11 : The cost function of a two-link genetic algorithm b a mapping defined as E SxS + 'R. where 'R is the set of dl red numbers. F(3,. 3;3 =Fl is the cost ofsolution Q,, 3;>. Definition 12 : A mating pool MP, of a two-link genetic algorithm u defined as where 3; E S, MPw = CP,, 3, P,),(3, S, pd. ... (3,. xw 3;' E S , , md'p; is the pr:bability for (3;. ;T:ing chosen From I
.
.
.
MP,. A typical genetic algorithm contains the following steps: 1. Randomly generate a set of initial parents. SmD. that forms a mating pool MPmo. Find F; M,. where i = 1. 2. ... ,m, The probability function p, is chosen to be inverse proportional to ra, and Zp,= 1. Set initial rate for u c h operator 9.j I. 2. .. . , I.
-
2. Sequmtially apply operators Cj with fixed operation rates r j J = I , 2,
... , x.
where r is the total number of operators. and collect the offspnng generated to a set S ,
3. Form MP, from S, after all the operator$ have been applied. For MP,,fi@ F,, where i = 1, 2, ... , u. Let Mpm. = MP, u ,UPmo. Determine ra,'s such that MPmxis fully ranked. 4. Define a new MPmp such that 3;in Up, is the same i = 1, 2. . _ _ , m, Update the probability funitionp,.
as
3;
in MPmxfor
5 Repeatsteps2-4 6. If I specified criterion for ending the genetic search has been met, stop; otherwise go to step 2.
17
3. APPLICATION TO PCB ASSEMBLY PLANNING An important problem in PCB assembly planning is to decide the order in which components should be p l a c c d l m e d . This f l e a s the cycle timc of usembly. In the w the components are directly fed to the assembly head by a tape and the feed time is alwayr shorter than the traveling time of the PCB between two components. the problem can be formulated as a trawling salesperson problem. In this case, the tixed-head, moving-board problem can be reformulated as a moving-head, fixed-board problem. The head visitr each component location once. Most of the commercial assembly machines for through-hole components belong to this category; for example, Amistar AI-6448, Panwnic P l n a r a RT, Universal 62874 and Universal 62418. Figure 1 illustrates this type of machines. The planning problem in this case is to determine the sequence of the assembly head in visiting all component locations such that the total traveling time is minimized. This is the well-known traveling salesperson (TS)problem (Lawier, et al. 1985).
In the use the components are supplied by multiple rtationiuy feeders, the assembly head needs to pick up the components fmm the various feeden and place them on the board. Many assembly machines for surface-mount components belong to this catcgory, such as Amistar FA-2001. Fuji EP-20, and n Prnwnic P u ~ MPA. I n addition to the sequence of placinlJlNcrting
Figure 3. Assembly machine for the MBTD problem The objective of PCB assembly planning is usually to minimize the total assembly time The evaluation function may vary for different types of problems. The total assembly distance UVI be used as the obj&e h c t i o n in the TS and PAP problem, ifthe speed of the X-Ytable or apembly head is constant. The movements of the assembly heads or X-Y tables may have different patterns for different typea of w m b l y machines. For the machines whose X motion and Y motion have the same MNtMt speeds, referred to as the Chcbyshev metric in the literature. the cost function CM be sct as Zmmr(lxi x j r f l , bi -y;+]I). whue 40, b) is the larger of0 and b, and (x;. Yi),( X i + ] , y;+]) are the locations of components in two successive placementsruucrfions. For the machines which always move from one location to another along a straight line at a constant speed. called the Euclidean metric. the evaluation function can be set as
Zdk;-x;+/I~+ lYl -y;+/P. Problem
\\
For a PCB having n components in the TS problem, the assembly sequence is represented by a link with n genes A link is S = [s(f),s(2). ..., Nn)] where HI] E C = ( c,, c l , ._.,c,) V i E N = ( I , 2. ... , n) , s(i) f if i # j , and a c h c; is a board component. The link designates the sequence of assembly of the board components ARer the squence ends, it repeats for another board in the batch.
Figure 1. Assembly machine for the TS problem components on the board, the plu~lingproblem may also include l~Signing components to feeders. The problem will be called the pick-and-place (PAP) problem. figure 2 illustrates a machine for this type of problems.
7
9)be the assembly distance between component c; and component The cost function is: Let D(ci,
which is the total distance of board movement (or. quivalently, the total distance. ofassembly head movement). A PCB of thirty components is used as an example in solving the TS problem. There are thirty links randomly generated and used as the initial population. The Euclidean metric is lusumed in this example: A ncar-aptimal assembly process produced is illustrated in Figure 4. The computation time is 75 seconds for 200 iterations on the Compaq 3 8 6 1 2 0 ~personal computer PCB
45
Feebel5
Wcdcmg Table
Figure 2. Assembly machine for the PAP problem Another type of assembly machine has a moving feeder carria. a moving X-Y table carrying the PCB. and a turret having multiple assembly heads (see Figure 3). There is Ipick-up location and a placement locstion and they are both fixed in space. The P l w u t RH and Universal ORsnter XI 4712B are two commercial machines which bdong to this e~cgory. The problem here ir to simultaneously determine the sequence of assembly of board components and udp the various types of components to feeders such that the total assembly time is minimized. This problem will be called the moving-board-with-himGdehy (MBTD)problem.
,
40
35
-
E E 30
u
F
20
To dovelop a genetic algorithm for planning of PCB assembly. we first consider the link representation. All the above assembly problems involve determining the component assembly sequence, which can be represented by C = (c,), i-1, 2, ... , n, representing the order of rhe placemen?hdon
w-. The initial population in the iterative procwr may be randomly generated. In actual application, howevs. it is desirable and oRen possible to expedite the iterative process by creating some 'good' initial estimates. For aample, an initial estimate may be the scquence generated by UI experienced technician. B d on this md other randomly g~nrmedinitial eslimrtu, the gmnic algorithm CM iteratively improve the estimates and finally arrive at a ncar-optimal solution.
18
o
10
20
ID
40
so
m
70
110
PO
im
X (mm)
Figure 4. Optimal assembly sequence obtained from the genetic algorithm cProbla
For the PAP problem, we w u m e Ih.1the PCB has a total of n components in p different types provided by m feeders, where m 2 p . In applying the genetic algorithm approach to solve this problem. we use two linllr: one is the -bly sequence, with n genes each rcpruatiq a board component number. ud the
o t h a is the feeder aslipnent, with m gena each representing a feeda number.
We assume that the assembly head will rest at a particular l d o n between two successive boards.
In the genetic algorithm, the component assembly sequence and the feeder tunngemcnt are designated by link 1 and link 2. respectively. Each link 1 is defined as 3 [s(l).s(2). .._, s(n)].41)E C = ( cI. c2 , ..., cn) V i E N = (1, 2. .. , n) , s(1) # if i # j , where ck represents a board component. Each link 2 is ddned as 3. = [s*(/). s*(2), ..., s*(m)], s'o] E F = ( f I , j 2 , .... j . ) Vj E M = (1. 2, ... , m ) , ~'(1) # s.0) if i # j , w h m cachjk represents a feeder d each elanent of M is a number designating a component type.
so)
The distance between feederji and component c ' can be expreued easily in t m s of the (x. y) coordinates of/i and cj as in the Td problem. Let D denote the distance betwcm two points as before. Also. let r denote the rest (Irtarting md end) point, and &j) denote the type of component for component c, The evaluation hmnction, which is the total travd distance, is Lk-/
D(s(n). s*(b(s(n))))+ D(s(0, 4 + D(s(n), r ) A sample PCB with 200 components in 10 differenf component types is urcd u M example in solving the PAP problem. One hundred links are generated randomly as the initial guesses. The total distance is reduced by 1I 84% from the best initial guess in 6,150 iterations. ~ - b o ~ r d - w i t h - t i m c - d c l Prob l v la
Bccau~athe various moving parts of the assembly machine have different speeds in the MBTD problem, assembly time should be used as the objective h a i o n in this problem. The three important times of consideration arc: the traveling time of the PCB, the shifting time of the pick-up head, and the travding time of the f d e r carrier The longest of the three is the time needed in the assembly of each component
We consider M n-component PCB havingp types of components nrppkd by m feeders (m z p ) As in the PAP problem, we create in the genetic algorithm two links: one is the l i for the assembly ~ q u e n c with e n genes, and the other is the link for the feeder assignment with m g e m . For the first few components assembled in a batch of PCB's, there are only pick-up movements md no placement movement. For the lasf few componenlc of the same batch, there are only placement movements and no pick-up movcmrnt. If the quantity of PCB's in I batch b very large, we can neglect thesc boundary dfW. Let the traveling time of the X-Y table, the traveling time of the feeder curier, and the induo'ng time of the turret be denoted by rI, fB md r,, respectively. As in thc PAP problem, b(ci) denotes the component type for component c,. Also, let g denote one plus thc numba of components in the gap between the pickup component and the p l m e n t component in the turret Thc traveling t h e of the X-Y table bcht(een component c,{x,. y,) and
gxj,
Ircrd,
component yj) is for Chebyshev metric. where vx and vu vx ' vy are the speeds of the X-Y table in the x and y directions. rcspcaively. The traveling time of the feeder carrier between feederh(x',. y';) and fceder&x'j. y'j) is
Definition 13 Let m, be the number of initial parents, mj b'e Ihe numba of links in the mating pool Wm, ?a applying G, md the operation rate of G, be 'j= mJmi_,. where J = I. 2. ... ,x, and x k the total number of operators. Definition 14: The ranking of the set &+FS, u (3w+I,Xw+> ... ,3,-k) is ... ,rr+.+&). w h m , S is fully dctincd u Rw+k=Rw u {ru,,+,, r-. ranked with ranking Rw Evcry 3,+i, i = I. 2. ... , k, is generated from operator G, with Fw+, not yet being found, md r+j= max(mI,fa3 . .. , raw) + i. S,+k is d e d ranked if the corresponding Rw+k has bcm determined. Definition IS: A mating poolMfw is called ranked if& L rmked.
The main idea of the adaption method is that we raix the operation ntu of those operators which generate more surviving offspring during the iterations. The number of p a n t s and the number of offspring in each generation are hxcd; namely m, and mx are fixed. To hx the population for each iteration. the operation rates of those operators which perform not as wdl u othcn vc reduced. With a given set of operation rates. the surviving offspring for each operator are counted. If surviving ofipring of an operator are more than those of other operators, the operation rate of this operator is raised with the hope that the algorithm will perform better in the later iterations. The details of this method are d i r e u d bdow. Let T denote a spccified number of iterations. We define Nj as the 'effect number' representing the total number of the offspring genmted by Gi and serving as the links in the initial mating pool A@,,,,in the next itmtion. To adjust the operation rate, we let
where ri' is the operation rate of operator i in the following icentions. Ihe proportional factor in the above relationship should be such that the number of o f f s p a generated in each iteration remains unchanged. Anofher consideration in the proposed method is that some of the operation n t a may keep decreasing to the degree thmt they M no longer valid in the iterative proccsa. Theoretically, UI opmtor generates no offspring if the operation rate is equal to or d e r than 1. So the operation rate of each opentor should be given a lower bound larger than one. The procedure of our genetic algorithm with changing operation rmtes is thus as followr:
I . Randomly generate a set of initial parents. Sma that fomu a mating pool Mpm0.Find Fj and fa,. where i = 1, 2, ... ,m, The probability functionp, is defined to be inversely proportional to ru, md Zp,= 1. Set the initial operation rafe for each opaatwrj.j= 1.2. ... ,I. 2. Let k = I. where k indexes the number of iterations for adapting the operation rates. 3. LetNj-Oforj= 1,2, ... , x
where y i s the speed of the feeder urria
Ac in the PAP problem, we use 3 to denote the placement sequence link and 3.10 denote the feeder assignment link. The time
needed for the placement of
component k is 5,= m W , ( s ( k - l ) .
4k)). rAs*(b(s(k+g-I))), s*(WtW))).
[I).
I n the above expression s(k-I)= s(n) when k = I When s(l)has I > n. where I = hg-1or k g , we replace HI) by s(l-n), which represents M initial component of the next board in the batch. This assumes that there is a large number of the same kind of boards in the batch. The evaluation function is H(3.3.) = Xrb which is
the total assembly time for a board. A board of fiRy components is used to test the algorithm developed for the
MBTD problem. The solution obtained by the algorithm is 12.75 seconds of aunnbly time, which is d y optimal. if not optimal, since the lower bound can be determined to be 12.5 seconds. 4.
THE ADAPTIVE GENETIC ALCORlTEM
An important issue in the genetic algorithm is the operation rate. W "fa of genetic operators should be uxd in generating offspring so that the algorithm an perform efficiently? We describe here a heuristic to ndjust the operation Rtcr during the iterative process. In order to describe the algorithm, we be& with more dc6nition.s:
4. Sequentially apply operators G, with fixed operation rates 9,j =1.2. ... ,x. and enlarge the mating pool to MPm's which are ranked. 7Xe probability functionp, is defined to be inversely pr&artional to ra, in Rm I'
5. ARer all the operators have been applied. find F; for the corresponding dement in , where i = m,+l, m,+2, ... , mr Determine m;s such that ,UPmxis
~y ranfttd. 6. Define a new W msuch that 3;in M f m o = 3 i i n W m z for i
Update the probabiiay functionp,.
-
I, 2,
... .m,
7. Increment Nj.) = 1.2. ... ,x, by I if operator G/ is involved in generating an offspring from Mfmo in this iteration, for all the members in ALPmo.
Increment k by 1. 8. Repeat J l e p ~4-7 while k S i
9. Compute thc new operation rates rj', j = 1.2. described above. 10. If rj' c
... ,x,
using the method
+, where + is the lower bound of operation rate for operator Gp
9 for j -
let
9'
1.2. ... ,I. Then compute the new operation r a t e rj"from 9' such that the number of offspring generated in each itention rCmainr unchanged. Repeat this step until the new operation rates satisfy the lowerbound requirement, i.c. rj' > f o r j = I. 2, ... ,x.
+
11.If a rpccified criterion for mding the genetic vlnh hu bccn met. stop; 0thawiS.Z 80 t0 step 2.
19
m e r i m e n t s and Discussion The easiest way of showing whether one genetic algorithm is better than another is using statistical analysis. Golden and Stewart (1985) used a Wilcoxon signed rank test (Mosteller, 1973) to compare different heuristics that were proposed to solve the traveling salesperson problem. Here we will use this technique to compare the adaptive genetic algorithm with the fixed-rate genetic algorithm. Twenty-four traveling salesperson problems are used to test the proposed adaption method. They are divided into 8 groups, each having three different problems. The numbers of nodes in the eight groups of traveling salesperson problems are 50, 60, 70, 80, 90, 100, 110 and 120. Each problem is designated by an integer between 1 and 24 All the coordinates of the nodes are randomly generated in a square area. The initial population m0 is 40 for all the test problems
1
0
200
400
--
600
Crosrovsr
800
1000
1200
1400
Iteration ----ci
Inve,rmn
Rotalion
1600
--
1800
2000
Mvlarlon
Figure 5 Changes of operation rates for a test problem Four operators are used in sequence: order crossover operator, inversion operator, rotation operator, and mutation operator. The number of offspring generated in each iteration is 80. The initial populations are all randomly generated The initial operation rates for each problem are also randomly chosen, in a reasonable range, and they are the same for the changing-rate and fixed-rate methods. When applying the Wilcoxon signed rank test, the random variable must be first identified. Let q,,,"and y,,," be the minimum traveling distances achieved of problem i after m iterations for the changing-rate method and the fixed-rate method, respectively, and w,be the optimal solution of problem i . Also let a,,, and b,,, be the normalized percentages representing how far the achieved minimum is from the optimal after m iterations for the changing-rate method and the fixed-rate method, respectively Thus, a,,," = lOO(q,,, - y,)/v, and b,,, = IOO(y,,, - q/,)/i~,, for i = I , 2, ... 24. We want to test the hypothesis that the changing operation rate method yields a better solution, using the random variable x!,m= a,,, - b>m. Since the global minima for the test problems are unknown, we use the best solution found from the genetic algorithm in each group as the optimal solution. The first Table 1 illustrates the computed values of the random variable 4,". column contains the numbers that designate the test problems. On the bottom of this Table there are three rows containing the sum of signed ranks, S ,,, the corresponding variable with Gaussian distribution, Z(a), and the error probability, a. Since the total number of problems is 24 in the Wilcoxon signed rank test, the Table 1. Results from the test problems
sum of signed ranks can be approximated by Sufi = Z(a) 70Z(a). An a equal to 50% means these two methods perform about the same An a less than 50% means the changing operation rate method performs better
It is clear that the changing operation rate method converges faster than the fixed operation rate method For all the 24 test problems, the changes of the operation rates are very similar in trend Figure 5 illustrates the typical changes of operation rates as functions of iterations. The crossover rate is the highest in the beginning of the iterative process and it decreases after some iterations. Since the initial estimates are all randomly generated, this observation leads to the conclusion that the crossover operator is most effective when the links are distinct from one another In other words, the crossover operator has a better chance than the other
20
operators to generate good offspring if the parents are quite different in pattern M e r many iterations the better offspring may be similar in pattern, and thus the crossover operator becomes less effective. The rate of mutation operator increases at later stages From the various test problems we also observe that the rate changes of crossover operator and mutation operator depend upon the size of the problem. 5. CONCLUSION
We have developed genetic algorithms for solving the component sequence and feeder assignment problems in printed circuit board assembly The developed algorithms have been applied to studying three types of PCB assembly planning problems: the traveling salesperson problem, the pick-and-place problem, and the moving-board-with-time-delay problem, with the solution objective to minimize the assembly time. The approach is advantageous in that the basic algorithm can be easily modified to solve the planning problems for many types of assembly machines. An adaptive method is proposed to improve the algorithm efficiency. By using a Wilcoxon signed test, the method is shown to be effective in speeding up the convergence of the solution in the iterative process.
REFERENCES [I] Ball, M. 0.. and Magazine, M J , 1988, "Sequencing of Insertion in Printed Circuit Board Assembly,'' Operatiom Research, Vol 36, No. 2, pp. 192201. [Z] Bard, J. F., Clayton, R. W., and Feo, T. A., 1990, "Machine Setup and Component Placement in Printed Circuit Board Assembly," Operations Research Group, College of Engineering, University of Texas, Austin, TX. [3] Conway, P.P., Ogunjimi. A. O.,Sargent, P. M.. Tang, A. C. T.,Walley, D. C., and Williams, D. J.. 1991, "SMD Reflow Soldering: a Thermal Process Model," CIRPAnnuals, Vol. 40/1, pp. 21-24. [4] Davis, L., 1985, "Job Shop Scheduling with Genetic Algorithms," Proceedings of htertmtional Conference on Genetic Algorithms and Their Appkations, pp. 136-140. [5] Francis, R L., Hamacher, H. W., Lee, C-Y,and Yeralan, S., 1989, "On Automating Robotic Assembly Workplace Planning," Tech. Report No. 897, Department of Industrial and Systems Engineering, University of Florida. Gainesville. FL. [6] Golden, B. L , Stewart, W. R., 1985, "Empirical Analysis of Heuristics," The Travelrng Salesman Problem, Chapter 7, John Wiley & Sons Ltd., 207-215 [7] Grefenstette, J. J., Gopal, R., Rosmaita, B., and Van Gucht, D., 1985, "Genetic Algorithm for the Traveling Salesman Problem," Proceeding o/ Ititernatioiial Coifereirce oii Genetic Algorithnis and l'heir Applications, pp 160-168. [8] Holland, J. H., 1975, Adaptattori in Natirral and Arlificial Systems, University of Michigan, Ann Arbor, MI. [9] Ji, Z., Leu, M. C.. and Wong, H , 1991. "Application of Linear Assignment Model for Planning of Robotic PC Board Assembly," ASME Publication AMD-Vol. I~IEEP-VOI.I , pp 35-41. [IOILawer, E. L., Lenstra. J. K., Rinnooy Kan, A H. G., Shmoys, D. B , 1985, The Traveling Salesman Problem, John Wiley & Sons Ltd. [ 1 I ] Leipala, T., and Nevalainen, 0 , 1989, "Optimization of the Movements of a Component Placement Machine," Eitropean Journal of Operatiorial Research, Vol 38, pp. 167-177. [12]Leu, M. C , Wong, H., and Ji, Z., 1992, "Genetic Algorithm for Solving Printed Circuit Board Assembly Planning Problems," Japan-LI.S.A. Svniposium on Flexible Autoniation, pp 1579- 1586. [I3]Mengel, P., Roth, N., and Schwarz, P I 1992, "Applicaiton of Advanced Data Processing Technology for Integrated Inspection in Electronics Assembly," CIRPAnniials, Vol 4111, pp. 29-32. [14] Mosteller, F. and Rourke, R.. 1973, Sturdy Stafistics, Addison-Wesley, Reading, MA.