Expert Systems with Applications 36 (2009) 7002–7010

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Expert Systems with Applications

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An integrated scheduling problem of PCB components on sequentialpick-and-place machines: Mathematical models and heuristic solutions

William Ho a,*, Ping Ji b,1

a Operations and Information Management Group, Aston Business School, Aston University, Birmingham B4 7ET, United Kingdomb Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e i n f o

Keywords:Printed circuit board manufacturingSurface mount technologyComponent sequencingFeeder arrangementMathematical modelingGenetic algorithm

0957-4174/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.eswa.2008.08.025

* Corresponding author. Tel.: +44 (0)121 2043342.E-mail addresses: w.ho@aston.ac.uk (W. Ho), mfpj

1 Tel.: +852 27666631.

a b s t r a c t

This paper formulates several mathematical models for determining the optimal sequence of componentplacements and assignment of component types to feeders simultaneously or the integrated schedulingproblem for a type of surface mount technology placement machines, called the sequential pick-and-place (PAP) machine. A PAP machine has multiple stationary feeders storing components, a stationaryworking table holding a printed circuit board (PCB), and a movable placement head to pick up compo-nents from feeders and place them to a board. The objective of integrated problem is to minimize thetotal distance traveled by the placement head. Two integer nonlinear programming models are formu-lated first. Then, each of them is equivalently converted into an integer linear type. The models for theintegrated problem are verified by two commercial packages. In addition, a hybrid genetic algorithm pre-viously developed by the authors is adopted to solve the models. The algorithm not only generates theoptimal solutions quickly for small-sized problems, but also outperforms the genetic algorithms devel-oped by other researchers in terms of total traveling distance.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The wide applicability of PCB has driven researchers and man-ufacturers to concentrate on the PCB assembly process planningin order to improve efficiency and remain competitive. Processplanning consists of two closely related issues: setup managementand process optimization (Ellis, Vittes, & Kobza, 2001). Setup man-agement involves four sub-problems:

� Line assignment: assigning PCBs to different assembly lines;� Machine grouping: grouping placement machines;� PCB grouping: grouping PCBs into families;� PCB sequencing: sequencing the production of PCBs.

On the other hand, process optimization includes three sub-problems:

� Component allocation: allocating component types to differentplacement machines;

� Feeder arrangement: arranging component types to differentfeeders at each machine;

ll rights reserved.

i@polyu.edu.hk (P. Ji).

� Component sequencing: determining the component placementsequence.

The focus of this paper is confined to integrating the second andthird sub-problems of process optimization for the sequential pick-and-place (PAP) machines. The performance of PAP machine isdependent on both component sequencing (i.e., which componentis placed first, second, and so on) and feeder arrangement (i.e.,which feeder stores which type of components). If the arrangementof components to feeders is not made carefully, even the pick andplacement sequencing is solved for optimality, it can result in anextremely poor performance (Altinkemer, Kazaz, Köksalan, &Moskowitz, 2000). So, certainly, the component sequencing andfeeder arrangement problems should be solved simultaneously.

2. Literature review

It is, however, observed that many researchers solved the com-ponent sequencing and feeder arrangement problems for the PAPmachine separately. Ball and Magazine (1988) modeled the com-ponent sequencing problem for the sequential PAP machine asthe rural postman problem. The assumption was that the feederarrangement was given. A heuristic approach was then used tosolve the problem, which assured the solution to be optimal ifthe movement of assembly head was rectilinear. Ji, Leu, andWong (1992) formulated the component sequencing and feeder

Head Movement Sequence Number

Component Number

Component Type

Feeder Number 4 5 6

(6) (2) (3)

Starting Point

(1)

1

2

3

4

5

(5)

(4)

(1)

(6)

1

2

3

(4)

(5)

(1)

(3)

6

7

8

9

10

(5)

(4)

(3)

(2)

1

2

3

4

5

6

7

8

910

11

12

13

14

15

16

17

18

19

20

21

Fig. 1. The assembly sequence of placement head.

Table 1Notation

Indicesi, j: components (i, j = 0, 1, . . . ,n)t: component types (t = 1, 2 . . . ,l)l: feeders (l = 1, 2 . . . ,l)p: placement order or placement position (p = 1, 2 . . . ,n)Distancesd0l: distance traveled from starting point to feeder ldlj: distance traveled from feeder l to the position of component j on the PCBdil: distance traveled from the position of component i to feeder ldi0: distance traveled from the position of component i to starting pointSub-tour elimination constraintui: placement order of component iDecision variablesxij = 1 if component i is placed immediately before component j; 0 otherwisexip = 1 if component i is placed in the pth position; 0 otherwiseytj l ¼ 1 if component j with component type t is stored in feeder l; 0 otherwise

W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010 7003

arrangement problems for the PAP machine as two individual lin-ear assignment problems, each of which was solved with the re-duced matrix method. Foulds and Hamacher (1993) determinedthe sequence of component placements and feeder arrangementfor the PAP machine sequentially. The feeder arrangement prob-lem, which was formulated as a number of one-facility locationmodels, was solved first. Then, the component sequencing prob-lem, which was formulated as the traveling salesman problem(TSP), was solved after. Francis, Hamacher, Lee, and Yeralan(1994) formulated the component sequencing and feeder arrange-ment problems for the PAP machine as the TSP with a special struc-ture in order to minimize the total assembly time. A heuristicmethod called the ‘clock sequence’ was developed to solve theproblem. Once the sequence of component placements was ob-tained, the feeder arrangement problem was also known with ref-erence to the placement sequence. Kumar and Li (1995)determined the sequence of component placements and feederarrangement for the PAP machine. Although they formulated theproblems as an integer quadratic programming model, they didnot solve it to optimality because they found that the model wascomputationally intractable. The authors solved the problems sep-arately instead. First, the component sequencing problem was re-ferred as the TSP. The nearest neighbor, the nearest insertion, thefurthest insertion, and the random generation were used to gener-ate an initial sequence. Then, the 2-optimality, the 3-optimality,and the or-optimality heuristics were used to improve the initialsequence. Second, the feeder arrangement problem was referredas the minimum weight matching problem (MWMP). They usedcommercial package to obtain an optimal solution for the MWMP.Broad, Mason, Rönnqvist, and Frater (1996) formulated the compo-nent sequencing and feeder arrangement problems for the PAP ma-chine as an integer linear programming model. They adopted thesolution procedure, which was developed by other researchers,to solve the feeder arrangement problem first and then the compo-nent sequencing problem. Magyar, Johnsson, and Nevalainen(1999) applied several search heuristics to solve the componentsequencing and feeder arrangement problems separately for thePAP machine. They first determined the feeder arrangement, andthen the order of component placements.

Only few researchers studied both component sequencing andfeeder arrangement problems simultaneously (i.e., the integratedproblem) for the PAP machine. Surprisingly, all of them applied ge-netic algorithms (GAs) to find near-optimal solution to the inte-grated problem (Leu, Wong, & Ji, 1993; Loh, Bukkapatnam,Medeiros, & Kwon, 2001; Ong & Khoo, 1999). This is mainly dueto the success of GAs in solving a wide variety of complex optimi-zation problems such as the TSP and quadratic assignment prob-lem (QAP) (Goldberg, 1989; Gen & Cheng, 1997), and theadvantages of GAs such as simplicity, easy operation, and greatflexibility.

Mathematical modeling is a powerful tool. Without applying it,the optimal solution to a particular problem cannot be obtained.Although heuristic methods like GAs are alternative tools for solv-ing the problem, no one can guarantee that the solution generatedis optimal or even no one knows how good the solution is beforethe optimal solution has been found. According to the literature re-view, only Kumar and Li (1995) and Broad et al. (1996) formulatedmathematical models for the integrated problem. Kumar and Li(1995) presented a nonlinear programming model to the problem.They, however, did not consider the starting point or home posi-tion of placement head. Besides, they did not verify and solve themodel. Broad et al. (1996) did not incorporate sub-tour eliminationconstraint in their model. The solution to their model may be infea-sible. To overcome the inadequacies, this paper not only formu-lates mathematical models for the integrated problem but alsoverifies them by two commercial packages.

3. The sequential pick-and-place machine

This paper focuses on the sequential PAP machine. In this typeof machines, the components are stored in multiple stationaryfeeders. The placement head travels to pick up a component froma feeder at a time, and then place it on the stationary board. ThePAP machine is able to achieve high accuracy, and suitable to oper-ate with large components such as integrated circuits (IC). Theoperation sequence of PAP machine is that the placement headstarts from its original location (starting point or home position),moves to a feeder that carries components, picks up a componentfrom the feeder, then moves to the desired placement location onthe stationary board, and places it there. After that, the head movesback to the previous feeder if the next component is the same typewith the previous one or moves to another feeder to pick up thenext component if it is different from the previous one and placeit on the board. The head repeats this operation procedure untilall components are placed on the board and returns to its startingpoint, waits for the next board, as illustrated in Fig. 1 for 10 com-ponents in a board.

Consider a PCB to be assembled by a PAP machine. The PCB hasn components with l different types. Each component type can bestored in any feeder, and a feeder can only store a unique type of

7004 W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010

components, so l feeders are required. The objective of integratedproblem is to minimize the total traveling distance of placementhead, which includes the distance from the starting point to a fee-der at the beginning (i.e., d0l), the distances from a feeder to a com-ponent’s position on the PCB (i.e., dlj), the distances from acomponent’s position to a feeder (i.e., dil), and the distance fromthe last component’s position to the starting point (i.e., di0). Thenotation used in both individual and integrated mathematicalmodels is summarized in Table 1.

4. Individual mathematical models

4.1. A component sequencing model

Suppose the assignment of component types to feeders (i.e., thefeeder arrangement problem) is solved beforehand, the componentsequencing model can then be formulated for finding the optimaltotal traveling distance of placement head. In order to achieve thisgoal, a decision variable is defined as

xij¼1 if component i is placed immediately before component j;

0 otherwise:

�

Actually, the component sequencing problem is somewhat similarto the TSP except the objective function. For the PAP machine, theobjective is not to minimize the distance between component iand component j because the placement head is unable to placethe next component on the PCB immediately without picking up acomponent from a feeder first. Therefore, the objective for thePAP machine should be to minimize the summation of different dis-tances including:

� The distance between the position of component i on the PCBand feeder l (if i = 0, it is the distance between the starting pointat the beginning and feeder l).

� The distance between feeder l and the position of the next com-ponent j.

� The distance between the position of the last component i andthe starting point at the end.

An individual mathematical model for the component sequenc-ing problem can be formulated as

Minimize z ¼Xn

i¼0

Xn

j¼1j–i

Xll¼1

ðdil þ dljÞxij þXn

i¼1

di0xi0 ð1Þ

subject to

Xn

i¼0

xij ¼ 1 for j ¼ 0;1; . . . ;n; i–j: ð2Þ

Xn

j¼0

xij ¼ 1 for i ¼ 0;1; . . . ; n; i–j ð3Þ

ui � uj þ nxij 6 n� 1 for i; j ¼ 1;2; . . . ;n; i–j: ð4Þ

All xij ¼ 0 or 1; All ui P 0 and is a set of integers ðM1Þ

In M1, the objective function (1) is to minimize the total travel-ing distance of placement head. If the moving speed of placementhead is incorporated, then the objective can be to minimize the to-tal placement time for assembling all components on a PCB. Con-straint set (2) ensures that exactly one component must beplaced immediately before component j. Constraint set (3) ensuresthat exactly one component must be placed immediately aftercomponent i. These two constraint sets, however, are not sufficient.

Although the solution drawn satisfies both constraint sets (2) and(3), it may still be infeasible due to the occurrence of sub-tours.Constraint set (4) is, therefore, added in order to eliminate sub-tours. Since the starting point must be visited first, it is redundantto include i and/or j = 0 in constraint set (4).

4.2. A feeder arrangement model

If the component placement sequence or xij in M1 is known, weneed to arrange a component type to a feeder, and this is the sec-ond problem to be studied in this paper, called the feeder arrange-ment problem. It is to assign the component types to feeders insuch a way that the total distance traveled by the placement headis minimized. In order to achieve this goal, a decision variable is de-fined as

ytjl¼

1 if component type t of component j is stored in feeder l;

0 otherwise:

�

An individual mathematical model for the feeder arrangementproblem can be formulated as

Minimize z ¼Xn

i¼0

Xn

j¼1j–i

Xlt¼1

Xll¼1

ðdil þ dljÞytjlþ di0 ð5Þ

subject to

Xlt¼1

ytl ¼ 1 for l ¼ 1;2; . . . ;l: ð6Þ

Xll¼1

ytl ¼ 1 for t ¼ 1;2; . . . ;l: ð7Þ

All ytl ¼ 0 or 1 ðM2Þ

In M2, the objective function (5) is to calculate the total distancetraveled by the placement head for assembling all components.Constraint set (6) ensures that exactly one component type is storedin one feeder. Constraint set (7) ensures that exactly one feeder isused to store one component type. The mathematical model forthe feeder arrangement problem is somewhat similar to the QAPexcept the objective function.

5. Integrated mathematical models

5.1. Formulation 1

Before solving M1, it is essential to obtain the solution of feederarrangement problem or M2 first. On the other hand, M2 cannot besolved until the solution of component sequencing problem or M1is known. It is no doubt that the component sequencing and feederarrangement problems are inter-related. Moreover, the objectivefunction in M1 and M2 is to minimize the total distance traveledby the placement head. It can be noticed that the amount of dis-tance traveled is dependent on the position of the next componentto place together with which feeder stores the next component topick up. In order to obtain the optimal solution, both problemsshould, therefore, be considered and solved simultaneously. Other-wise, it can result in poor performance (Altinkemer et al., 2000).

A pure integer nonlinear programming model for the integratedproblem can be formulated as

Minimize z ¼Xn

i¼0

Xn

j¼1j–i

Xlt¼1

Xll¼1

ðdil þ dljÞxijytjlþXn

i¼1

di0xi0 ð8Þ

W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010 7005

subject to

Xn

i¼0

xij ¼ 1 for j ¼ 0;1; . . . ;n; i–j: ð9Þ

Xn

j¼0

xij ¼ 1 for i ¼ 0;1; . . . ;n; i–j ð10Þ

ui � uj þ nxij 6 n� 1 for i; j ¼ 1;2; . . . ;n; i–j: ð11ÞXlt¼1

ytl ¼ 1 for l ¼ 1;2; . . . ;l: ð12Þ

Xll¼1

ytl ¼ 1 for t ¼ 1;2; . . . ;l: ð 13Þ

All xij and yij ¼ 0 or 1;

All ui P 0 and is a set of integers ðM3Þ

Since there is nonlinear term xij ytj lin the objective function and

the model contains both binary variables (i.e., xij and ytjl) and integer

variables (i.e., ui and uj), M3 can be regarded as a pure integer non-linear programming model. The objective function (8) calculates thetotal traveling distance of placement head, whereas the interpreta-tion of constraint sets (9)–(13) was mentioned in M1 and M2.

Since M3 only contains nonlinear function in the form of prod-ucts of binary variables, it can be reformulated as a linear program-ming model by implementing the following steps:

� Introduce a new binary variable w to replace the product termxy.

� Make use of the extra constraints: w 6 x, w 6 y, and w P x +y � 1 to reflect the logical condition: w = 1 if and only if x = 1and y = 1.

The nonlinear term in the objective function is in the form ofproducts of two binary variables. It can, therefore, be rewrittenas a linear one by introducing an extra binary variable wijl as wellas three extra constraint sets. The interpretation of decision vari-able wijl is

wijl ¼1 if component j is placed just after component i and

the type of component j is stored in feeder l;

0 otherwise:

8><>:

M3 can be converted into a pure integer linear programmingmodel as follows:

Minimize z ¼Xn

i¼0

Xn

j¼1j–i

Xll¼1

ðdil þ dljÞwijl þXn

i¼1

di0xi0 ð14Þ

subject to

Xn

i¼0

xij ¼ 1 for j ¼ 0;1; . . . ;n; i–j: ð15Þ

Xn

j¼0

xij ¼ 1 for i ¼ 0;1; . . . ;n; i–j ð16Þ

ui � uj þ nxij 6 n� 1 for i; j ¼ 1;2; . . . ;n; i–j: ð17Þ

Xlt¼1

ytl ¼ 1 for l ¼ 1;2; . . . ;l: ð18Þ

Xll¼1

ytl ¼ 1 for t ¼ 1;2; . . . ;l: ð19Þ

wijl 6 xij for i ¼ 0;1; . . . ;n; for j ¼ 1;2; . . . ;n;

i–j; for l ¼ 1;2; . . . ;l: ð20Þ

wijl 6 ytjlfor i ¼ 0;1; . . . ;n; for j ¼ 1;2; . . . ;n;

i–j; for l; t ¼ 1;2; . . . ;l: ð21Þ

wijl P xij þ ytjl� 1 for i ¼ 0;1; . . . ;n; for j ¼ 1;2; . . . ;n;

i–j; for l; t ¼ 1;2; . . . ;l: ð22Þ

All xij and yij ¼ 0 or 1;

All ui P 0 and is a set of integers ðM4Þ

In M4, the objective function (14), and the constraint sets (20)–(22) are the linear expression of objective function (8) in M3. Theinterpretation of constraint sets (15)–(19) in M4 is the same as thatof constraint sets (9)–(13) in M3.

5.2. Formulation 2

Although the constraint sets (11) and (17) are able to guaranteethat the solution generated is feasible, they increase the complex-ity of models as there are n(n � 1) constraints in this sub-tourelimination constraint. In order to reduce the burden of models,it is essential to find a way to replace the bulky constraint. In thispaper, M3 is re-modeled or the constraint set (11) in M3 is omittedusing another decision variable xip instead of xij. The interpretationof xip is that

xip ¼1 if component i is placed in the pth position;0 otherwise:

�

The idea is to assign n components to n positions or placement or-ders, which means that there are totally n2 decision variables inwhich only n variables are 1 while all others are 0. Since each com-ponent must be placed in exactly one position, no sub-tour will beappeared in this situation. Referring to (Fig. 1), x21 and x32 are both 1because component 2 and component 3 are placed first and second,respectively. A binary integer nonlinear programming model for theintegrated problem can be formulated as follows:

Minimize z ¼Xn

j¼1

Xll¼1

Xll¼1

ðd0l þ dljÞxjlytjlþXn

i¼1

Xn

j¼1j–i

Xn�1

p¼1

Xlt¼1

�Xll¼1

ðdil þ dljÞxipxj;pþ1ytj1þXn

i¼1

di0xin ð23Þ

subject to

Xn

i¼1

xip ¼ 1 for p ¼ 1;2; . . . ; n: ð24Þ

Xn

p¼1

xip ¼ 1 for i ¼ 1;2; . . . ;n: ð25Þ

Xlt¼1

ytl ¼ 1 for l ¼ 1;2; . . . ;l: ð26Þ

Xll¼1

ytl ¼ 1 for t ¼ 1;2; . . . ;l ð27Þ

All xip and ytl ¼ 0 or 1 ðM5Þ

In M5, the objective function (23) calculates the total distancetraveled by the placement head. Constraint set (24) ensures thatexactly one component is placed in one position. Constraint set(25) ensures that one position has exactly one component placed.Constraint set (26) ensures that exactly one component type is

7006 W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010

stored in one feeder. Constraint set (27) ensures that exactly onefeeder is used to store one component type.

Similarly, M5 can be reformulated to a linear programmingmodel. In the objective function (23) of M5, the first nonlinear term(i.e., xj1 ytjl

) is in the form of products of two binary variables. It can,therefore, be rewritten as a linear term by introducing an extra bin-ary variable wj1l as well as three extra constraint sets. The interpre-tation of decision variable wj1l is

wj1l ¼1 if component j is placed first and

the type of component j is stored in feederl;0 otherwise:

8<:

In the objective function (23) of M5, the second nonlinear term(i.e., xip xj;pþ1 ytjl

) is in the form of products of three binary variables.So, the steps for converting it into linear type need to be modified inthis case. The major difference is that four instead of three extra con-straints are introduced. Similarly, a decision variable wij(p+1)l is intro-duced. The decision variable wij(p+1)l is defined as

wijðpþ1Þl ¼

1 if component j is placed just after component iand the type of component j placed in theðpþ 1Þth position is stored in feeder l;

0 otherwise:

8>>><>>>:

Table 2Numbers of variables and constraints in M3 to M6

Number of variables Number of constraints

M3 n2 + 2n + l2 n2 + n + 2l + 2M4 (n2 + 2n + l2) + n2l (n2 + n + 2l + 2) + 3n2lM5 n2 + l2 2n + 2lM6 (n2 + l2) + nl + n(n � 1)2ll (2n + 2l) + 3nl + 4n(n � 1)2l

M5 can be converted into a binary integer linear programmingmodel as follows:

Minimize z ¼Xn

j¼1

Xlt¼1

Xll¼1

ðd0l þ dljÞwj1l

þXn

i¼1

Xn

j¼1j–i

Xn�1

p¼1

Xlt¼1

Xll¼1

ðdil þ dljÞwij;ðpþ1Þl þXn

i¼1

di0xin

ð28Þsubject to

Xn

i¼1

xip ¼ 1 for p ¼ 1;2; . . . ;n: ð29Þ

Xn

p¼1

xip ¼ 1 for i ¼ 1;2; . . . ; n: ð30Þ

Xlt¼1

ytl ¼ 1 for l ¼ 1;2; . . . ;l: ð31Þ

Xll¼1

ytl ¼ 1 for t ¼ 1;2; . . . ;l ð32Þ

wj1l 6 xj1 for j ¼ 1;2; . . . ;n; for l ¼ 1;2; . . . ;l: ð33Þ

wj1l 6 ytj1for j ¼ 1;2; . . . ;n; for l; t ¼ 1;2; . . . ;l: ð34Þ

wj1l P xj1 þ ytjl� 1 for j ¼ 1;2; . . . ;n;

for l; t ¼ 1;2; . . . ;l: ð35Þ

wijðpþ1Þl 6 xip for i; j ¼ 1;2; . . . ;n; i–j;

for p ¼ 1;2; . . . ; n� 1; for l ¼ 1;2; . . . ;l: ð36Þ

wijðpþ1Þl 6 xj;pþ1 for i; j ¼ 1;2; . . . ;n; i–j;

for p ¼ 1;2; . . . ; n� 1; for l ¼ 1;2; . . . ;l: ð37Þ

wijðpþ1Þl 6 ytjlfor i; j ¼ 1;2; . . . ;n; i–j;

for p ¼ 1;2; . . . ; n� 1; for l; t ¼ 1;2; . . . ;l: ð38Þ

wijðpþ1Þl P xip þ xj;pþ1 þ ytjl� 2 for i; j ¼ 1;2; . . . ;n;

i–j; for p ¼ 1;2; . . . ; n� 1; for l ¼ 1;2; . . . ;l: ð39Þ

All xipytlwj1l and w1jðpþ1Þl ¼ 0 or 1 ðM6Þ

In M6, the constraint sets (33)–(35), and the first term in the objec-tive function (28) are the linear expression of the first term in theobjective function (23) of M5. Besides, the constraint sets (36)–(39), and the second term in the objective function (28) are the lin-ear expression of the second term in the objective function (23) ofM5. The interpretation of constraint sets (29)–(32) in M6 is thesame as that of constraint sets (24)–(27) in M5.

6. Computational analysis

In this section, the complexity of four integrated mathematicalmodels (i.e., M3 to M6) is discussed and compared first. Then, themodels with less variables and constraints are solved to optimality.

In order to examine the complexity of models, it is essential tofind out the numbers of variables and constraints in each model.According to M3, it can be seen that the model is very sophisticatedand hard to solve. Not only the objective function is nonlinear, butalso its possible enumeration is huge. M3 has (n2 + n + l2) binaryvariables, n integer variables, and (n2 + n + 2l + 2) constraints. Inthe objective function (8), the possible terms are nl + nl(n � 1) + nor n2l + n. For M4, although the model becomes linear, both num-bers of variables and constraints increase greatly at the same time.Both numbers in M4 are much greater than that in M3. For thenumber of variables, n2l of wijl are introduced in M4 besides(n2 + n + l2) binary variables and n integer variables. For the num-ber of constraints, besides (n2 + n + 2l + 2) constraints, M4 has3n2l constraints more for the constraint sets (20)–(22).

After adopting another decision variable in M5, the bulky sub-tour elimination constraint is omitted. The complexity of M5 islower than that of M3 as both numbers of variables and constraintsreduce significantly. M5 has only (n2 + l2) binary variables, and(2n + 2l) constraints. Since there are two nonlinear terms in theobjective function (23) in M5, two additional decision variablesand seven extra constraints are necessary to be incorporated inthe equivalent linear programming model or M6. As a result, M6becomes enormous and very complex. Both numbers of variablesand constraints are even much greater than that in M4. For thenumber of binary variables, nl of wj1l and n(n � 1)2l of wij(p+1)l

are introduced in M6 besides (n2 + l2) binary variables. For thenumber of constraints, besides (2n + 2l) constraints, M6 has3nl + 4n(n � 1)2l constraints more in which there are 3nl con-straints for the constraint sets (33)–(35) and 4n(n � 1)2l con-straints for the constraint sets (36)–(39). The numbers ofvariables and constraints of four models are listed in Table 2.

For a realistically sized problem of 100 components and 10component types, M3 has 10,300 variables and 10,122 constraints.Comparatively, M5 is a better nonlinear programming formulationbecause it consists of 10,100 variables together with 220 con-straints merely. For the linear programming formulation, M4 ismuch more desirable than M6. M4 has 110,300 variables and310,122 constraints. Both numbers of variables and constraints in

W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010 7007

M6, however, are much greater. It has 9,812,100 variables as wellas 39,207,220 constraints! So, the model may not be solved to opti-mality in a reasonable amount of time.

Table 3Computational time spent for solving M4 and M5

Numbers ofcomponents andtypes

Optimal solution CPU time(hh:mm:ss) by CPLEX for M4

Optimal solution CPU time(hh:mm:ss) by BARON for M5

4 � 4 0.16 s 0.31 s5 � 5 00:00:01 00:00:046 � 6 00:00:41 00:01:527 � 7 00:02:26 01:21:398 � 8 10:30:51 379:02:50

Table 4The data of integrated problem with 10 components and 6 types

Components Types Coordinates (mm) Feeders Coordinates (mm)

x y x y

1 6 30 20 1 10 502 1 30 30 2 10 353 1 30 40 3 10 204 5 30 50 4 30 105 4 30 60 5 50 106 2 50 20 6 70 107 3 50 308 3 50 409 5 50 5010 4 50 60

- Improve Link 1 by Iterated Swap Procedure

- Improve Link 2 by 2-Opt

Local Search Heuristic

Improvement

Input GA Parameters

Evaluation

- Roulette Wheel Method

Selection

- Minimization of Distance Traveled by Placement Head

Two-Link Representation: - Generate Link 1 or Component

Sequencing Using Nearest Neighbor Heuristic

- Generate Link 2 or Feeder

Arrangement Randomly

Initialization

Fig. 2. The flowchart of hyb

Due to the fact that M4 and M5 are the better linear and nonlin-ear formulations in terms of complexity, respectively, these twomodels are solved to global optimality. To solve the models, twocommercial packages are used. First, CPLEX is a well-known pow-erful integer linear programming solver. It is, therefore, applied tosolve M4. Second, BARON is a computational system for solvingnon-convex optimization problems including binary integer non-linear programming models to global optimality. So, it is adoptedto solve M5. By these two commercial packages, the models aretested by several small examples, and both have the same solu-tions to the same examples.

According to (Table 3), it is found that the pure integer linearprogramming model (i.e., M4) is more desirable than the binaryinteger nonlinear programming model (i.e., M5) in terms ofamount of computational time spent. For instance, it spends 10and a half an hours by CPLEX to solve M4 with 8-componentsand 8 types to optimality. But, it takes more than 15 days by BAR-ON to solve M5 with the same problem size to optimality. Since theoptimal solutions of M4 can be obtained more quickly, an inte-grated problem with 10 components and 6 types is formulatedas the pure integer linear programming model or M4 for gettingthe optimal solution. The data of problem is listed in Table 4. Theoptimal assembly sequence of placement head is: starting point?f3 ? c2 ? f3 ? c3 ? f2 ? c4 ? f1 ? c5 ? f1 ? c10 ? f2 ? c9 ? f6 ?c6 ? f5 ? c7 ? f5 ? c8 ? f4 ? c1? starting point, whereas the totaldistance traveled by the placement head is 566.02 mm.

Although the global optimum of both models can be obtained,they are not efficient approaches since the computational timegrows exponentially with the problem size. In addition, the

1. Modified Order Crossover 2. Heuristic Mutation 3. Inversion Mutation

Genetic Operations

Improve New Chromosomes (Offspring):

- Improve Link 1 by Iterated Swap Procedure

- Improve Link 2 by 2-Opt

Local Search Heuristic

- Measure Fitness of Offspring and Compare with That of

Parents

- Retain the Best Population of Chromosomes

Terminate?

Output the Best Solution

Yes

No

rid genetic algorithm.

Selected sub-string

7008 W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010

number of components on a PCB in the real-world situations isquite large, normally several hundreds. It is unacceptable tospend several days or even hours to solve the integrated prob-lem. To solve the problem efficiently, a heuristic method shouldbe adopted.

7. A hybrid genetic algorithm

GA, developed by John Holland in the 1960s, is a stochastic opti-mization technique. Similar to simulated annealing (SA) and tabusearch (TS), GA can avoid getting trapped in a local optimum bythe aid of mutation operation. Actually, the basic idea of GA is tomaintain a population of candidate solutions that evolves undera selective pressure. Hence, it can be viewed as a class of localsearch based on a solution-generation mechanism operating onattributes of a set of solutions rather than attributes of a singlesolution by the move-generation mechanism of the local searchmethods, like SA and TS (Osman & Kelly, 1996).

Since the component sequencing and feeder arrangement prob-lems are considered simultaneously, a simple GA may not performwell in this situation. It was proved that the individual problemsare already very hard to solve (Crama, Flippo, Klundert, & Spi-eksma, 1997). The GA adopted here is, therefore, hybridized withseveral heuristics in order to improve the solution further. It is,however, found that none of the previous researchers used the hy-brid GA or HGA to solve the integrated problem for the PAPmachine.

The flowchart of HGA for the integrated problem is shown inFig. 2. Since both problems are considered simultaneously, eachchromosome or solution includes two path representations or links(Fig. 3). The first link denotes the component sequencing, whereas

Assembly Sequence 1 2 3 4 5 6 7 8 9 10

Component Number 2 3 9 4 5 10 8 7 6 1

Link 1

Component Type 4 5 1 6 2 3

Feeder 1 2 3 4 5 6 Link 2

Fig. 3. The two-link representation for a chromosome.

Select 2 genes randomly

Parent: 2 3 9 4 5 10 8 7 6 1

Offspring 1: 2 3 7 4 5 10 8 9 6 1

Swap the neighbors of 2 genes to form 4 more offspring

Offspring 2: 2 7 3 4 5 10 8 9 6 1

Offspring 3: 2 3 4 7 5 10 8 9 6 1

Offspring 4: 2 3 7 4 5 10 9 8 6 1

Offspring 5: 2 3 7 4 5 10 8 6 9 1

Fig. 4. The iterated swap procedure.

the second link represents the feeder arrangement. After theparameters (i.e., the population size, iteration number, crossoverrate, and mutation rate) have been set up, the HGA generates aninitial population in which the first links are generated from thenearest neighbor heuristic while the second links are generatedrandomly. During this initialization step, each chromosome is im-proved as follows: the iterated swap procedure (ISP) (Fig. 4) is per-formed on the first link while the 2-opt local search heuristic isapplied to the second link. Actually, the principle of ISP is very sim-ilar to that of the 2-opt local search heuristic, except that some in-stead of all two swaps are examined to generate offspring. It candefinitely reduce the computational time because the number ofcomponents is quite large, normally several hundreds. Each chro-mosome is then measured by an evaluation function, which wasdescribed thoroughly in Section 5. The roulette wheel selectionoperation (Goldberg, 1989) is performed to select some chromo-somes for the genetic operations including the modified ordercrossover (Fig. 5), the heuristic mutation (Fig. 6), and the inversionmutation (Fig. 7). After an offspring is produced, the first link is im-proved by the ISP while the second link is improved by the 2-optlocal search heuristic. The fitness of offspring will be measuredand may become a member of population if it possesses a rela-tively good quality. These steps form an iteration, and then the rou-lette wheel selection is performed again to start the next iteration.The HGA will not stop unless the predetermined number of itera-tions is conducted.

Parent 1: 2 3 9 4 5 10 8 7 6 1

Parent 2: 1 2 3 4 5 6 7 8 9 10

Proto-child: 4 5 10 8

Find the gene right prior to the first gene of sub-string from

the second parent, and place it in front of the sub-string in

the proto-child.

Proto-child: 3 4 5 10 8

Find the gene right behind the last gene of sub-string from

the second parent, and place it just after the sub-string in

the proto-child.

Proto-child: 3 4 5 10 8 9

The remaining genes, that is, the genes not in the

proto-child yet, form a sequence. Place the genes into the

unfilled positions of proto-child from the left to the right

according to the sequence in the second parent.

Offspring: 1 2 3 4 5 10 8 9 6 7

Repeat the steps above to produce the second offspring by

exchanging the two parents

Fig. 5. The modified order crossover operator.

Select 3 genes at random

Parent: 2 3 9 4 5 10 8 7 6 1

Generate neighbors for all possible permutations of the

selected genes, and all neighbors generated are regarded as

the offspring.

Offspring 1: 2 3 9 4 5 7 8 10 6 1

Offspring 2: 2 3 10 4 5 9 8 7 6 1

Offspring 3: 2 3 10 4 5 7 8 9 6 1

Offspring 4: 2 3 7 4 5 9 8 10 6 1

Offspring 5: 2 3 7 4 5 10 8 9 6 1

Fig. 6. The heuristic mutation operator.

Selected sub-string

Parent: 2 3 9 4 5 10 8 7 6 1

Flip the selected sub-string to form an offspring.

Offspring: 2 3 9 8 10 5 4 7 6 1

Fig. 7. The inversion mutation operator.

Table 5A comparison of experimental results

Leu et al.(1993)

Ong and Khoo(1999)

HGA HGA

Number of extrafeeders

N/A N/A N/A 3

Population size 100 100 25 25Iteration number 6150 6150 3000 1000Final best solution

(cm)About 6129 5673.7 5660.5 4855.5

W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010 7009

8. Performance analysis

In this section, the performance of HGA is evaluated by compar-ing it to the optimal solutions of several problems mentioned inSection 6 first. Then, the comparison between the HGA and theGAs developed by Leu et al. (1993) and Ong and Khoo (1999) is car-ried out. It is unable to compare with the GA proposed by Loh et al.(2001) because there is no data provided.

8.1. Comparison to optimal solutions

The same problems as mentioned in Section 6 are solved by theHGA. It is found that the HGA achieves the optimal solutions to allproblems quickly. Furthermore, the longest computational timespent is only 9 s for the 8-component problem. Compared withthe time spent on finding the optimal solution for the 8-compo-nent problem, the HGA is much more efficient. It saves more than10 h when compared with CPLEX, and saves about 15 days whencompared with BARON.

8.2. Comparison to other approaches

The performance of HGA is evaluated using the PCB example(refer to Leu et al. (1993)) in which there are 200 componentsand 10 component types. The HGA parameters are set as: popula-tion size = 25, iteration number = 3000, crossover rate = 0.4, andmutation rate = 0.2. According to Table 5, it is found that the per-formance of HGA is superior to that of GAs (Leu et al., 1993; Ong& Khoo, 1999) in three aspects. Firstly, the HGA can obtain a better

solution with a smaller population size, 25 only, while the othertwo used 100. Secondly, the HGA can obtain a better solution notonly with a smaller population size, but also with a fewer itera-tions, 3000 vs. 6150. Finally, and most importantly, the HGA ob-tained a better solution than any previous methods, 5660.5 cmvs. 5673.7 cm or 6129 cm. If the traveling speed of placement headis assumed to be 60 mm/s, the improvement is 2.2 s when com-pared with Ong and Khoo (1999). Since the component placementis the bottleneck of PCB assembly line (Ong & Tan, 2002; Wilhelm& Tarmy, 2003), a minor reduction in the cycle time will save a sig-nificant production time. For example, to produce 100,000 boards,a reduction of 2.2 s in the cycle time will save 3667 min or 61working hours. The productivity of a PCB manufacturing companycan, therefore, be enhanced if our HGA is adopted.

In the above experiment, it is assumed that the number of feed-ers provided is exactly the same as the number of component types(i.e., 10). Each component type can only be stored in exactly onefeeder. Herein, the 200-component problem is solved again usingthe HGA in which three additional feeders are available or thereare totally 13 feeders. In this case, three types of components canbe assigned to two feeders. In the 200-component problem, com-ponent types 2, 6, and 8 are the most frequently used, these threetypes of components can, therefore, be stored in two feeders.According to Table 5, it is noticed that the solution is much better(4855.5 cm) if there are additional feeders. Since the three mostfrequently used component types are assigned to more than onefeeder, they can be retrieved from a closer feeder. So, the total dis-tance traveled by the placement head can be reduced.

9. Conclusions

Mathematical modeling is a powerful tool in today’s life. Thispaper presented several nonlinear and linear programming modelsfor an integrated scheduling problem in a sequential pick-and-place machine. After the mathematical models for the individualcomponent sequencing and feeder arrangement problems hadbeen formulated, it was found that the problems are inter-related.One cannot be solved unless the solution of the other is obtainedbeforehand. Due to their close relationship, we formulated twointeger nonlinear programming models and two integer linear pro-gramming models for solving the integrated problem for thesequential pick-and-place machine. In terms of amount of compu-tational time spent, the model of linear type is more desirable.

Although the optimal solution can be found using the commer-cial packages, it was noticed that the computational time growsexponentially with the problem size. In order to solve the problemefficiently, a HGA was adopted. The algorithm is so called becausethe nearest neighbor heuristic, the iterated swap procedure, andthe 2-opt local search heuristic are incorporated to improve thesolution. It was shown that the performance of HGAs was superiorto that of GAs proposed by previous researchers in terms of totaltraveling distance. Furthermore, the solution was even much bet-ter when component types could be stored in more than onefeeder.

7010 W. Ho, P. Ji / Expert Systems with Applications 36 (2009) 7002–7010

Acknowledgement

The work described in this paper was partially supported by agrant from the Research Grants Council of Hong Kong SpecialAdministrative Region, China (Project No. PolyU 5259/04E).

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