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Contents lists available at ScienceDirect

Int. J. Production Economics

Int. J. Production Economics 123 (2010) 31–41

0925-52

doi:10.1

� Cor

E-m

(P. Ji).

journal homepage: www.elsevier.com/locate/ijpe

Integrated component scheduling models for chip shooter machines

William Ho a,�, Ping Ji b

a Operations and Information Management Group, Aston Business School, Aston University, Birmingham B4 7ET, UKb 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

Article history:

Received 25 July 2005

Accepted 20 May 2009Available online 21 July 2009

Keywords:

PCB assembly

Optimization

Component sequencing

Feeder arrangement

Hybrid genetic algorithm

73/$ - see front matter & 2009 Elsevier B.V. A

016/j.ijpe.2009.05.029

responding author. Tel.: +44121 2043342.

ail addresses: w.ho@aston.ac.uk (W. Ho), mfp

a b s t r a c t

This paper focuses on minimizing printed circuit board (PCB) assembly time for a chip

shooter machine, which has a movable feeder carrier holding components, a movable

X–Y table carrying a PCB, and a rotary turret with multiple assembly heads. The

assembly time of the machine depends on two inter-related optimization problems: the

component sequencing problem and the feeder arrangement problem. Nevertheless,

they were often regarded as two individual problems and solved separately. This paper

proposes two complete mathematical models for the integrated problem of the

machine. The models are verified by two commercial packages. Finally, a hybrid genetic

algorithm previously developed by the authors is presented to solve the model. The

algorithm not only generates the optimal solutions quickly for small-sized problems, but

also outperforms the genetic algorithms developed by other researchers in terms of total

assembly time.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The wide applicability of printed circuit boards (PCBs)has driven researchers or manufacturers to concentrate onthe PCB assembly process planning in order to improvethe efficiency and remain competitive. Besides, customers’contemporary requirements such as smaller in productsize and greater in function and reliability force thesurface mount technology (SMT) to replace the plated-through-hole (PTH) technology. Among several assemblyoperations in an SMT assembly line, the componentplacement is generally the most time-consuming. Inaddition, it is frequently a bottleneck of the line, anddetermines the line’s cycle time (Wilhelm and Tarmy,2003). Evidently, the throughput rate of a PCB manufac-turing company or the company’s competitiveness can beincreased if the component placement process is opti-mized. In order to accomplish this goal, two inter-related

ll rights reserved.

ji@polyu.edu.hk

optimization problems are considered for the chip shootermachine, which is one of the most commonly used SMTplacement machines.

The component sequencing problem (i.e., which com-ponent is placed first, second, and so on) and the feederarrangement problem (i.e., which component type isstored in which feeder) are the two inter-related optimi-zation problems. For the PTH technology, the componentsequencing problem of an auto-insertion machine can besimply formulated as the traveling salesman problem(TSP), and it is not necessary to consider the feederarrangement problem (Chan and Mercier, 1989). However,for the SMT, the assembly time of a placement machine isalso dependent on a feeder to hold which types ofcomponents besides the pick and placement sequence. Ifthe arrangement of component types to feeders is notmade carefully, even if the pick and placement sequencingis optimally solved, it can result in an extremely poorperformance (Altinkemer et al., 2000). Since the chipshooter machine being studied in this paper is an SMTplacement machine, the component sequencing and thefeeder arrangement problems should be studied andsolved simultaneously.

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2. Literature review

For the chip shooter machine, many researchers solvedthe component sequencing and the feeder arrangementproblems separately, and some efforts to solve the twoproblems together were also made.

2.1. The component sequencing problem

Souza and Wu (1995) studied the component sequen-cing problem only for the chip shooter machine. Aknowledge-based component placement system incorpo-rated with the TSP algorithms was developed. Moyer andGupta (1997) also formulated the component sequencingproblem for the machine as the TSP, and developed theboard sequencing heuristic to tackle the problem.

2.2. The feeder arrangement problem

Moyer and Gupta (1996a) formulated the feederarrangement problem for the chip shooter machine asthe quadratic assignment problem (QAP) based on theassumption that the sequence of component placementswas predetermined. Two heuristic methods were pro-posed to solve the problem. Similarly, Dikos et al. (1997)formulated the feeder arrangement problem for themachine as the QAP, and made an assumption that anoptimal component placement sequence was first speci-fied. They employed genetic algorithms (GAs) to find anear optimal feeder arrangement. Klomp et al. (2000)treated the problem of determining an optimal feederarrangement for a line of chip shooter machines as findingthe shortest Hamiltonian path. Heuristic approaches wereemployed to deal with the problem.

2.3. The integrated component scheduling problem

Leu et al. (1993) and Ong and Tan (2002) applied GAsto solve the integrated problem, which is an integration ofboth component sequencing and feeder arrangementproblems, simultaneously for the chip shooter machine.Bard et al. (1994) used an iterative approach to determinethe component placement sequence, the feeder arrange-ment, and the retrieval plan for the machine separately.Moyer and Gupta (1996b) developed a heuristic algorithmfor determining the component sequencing and the feederarrangement problems individually. Sohn and Park (1996)formulated the component sequencing and the feederarrangement problems as an integer nonlinear program-ming model for the machine while using the one-headcase as an approximation to the multi-head case. Aheuristic approach was developed to solve the problemssequentially. Yeo et al. (1996) developed a rule-basedframe system to generate the feeder arrangement first forthe machine, followed by the component placementsequence. Crama et al. (1997) also solved the feederarrangement problem for the machine heuristically first,and then solved the component sequencing and thecomponent retrieval problems using heuristic methods.Ellis et al. (2001) used heuristics to generate an initial

placement sequence for the machine first, and then aninitial feeder arrangement. The 2-opt local search heur-istic was used to improve the initial solutions. In Wilhelmand Tarmy’s (2003) approach, each component type wasassigned to a feeder of the machine first. After that, thesequence of component placements was determined bysolving an asymmetric TSP.

One conclusion drawn from the above literatures isthat a complete mathematical model for the integratedproblem of the chip shooter machine has not beenconstructed yet. Although many heuristic approacheswere developed, the effectiveness of these approaches isunknown because the optimal solution has not beenfound. Therefore, in Section 3 of this paper, two mathe-matical models are formulated for the integrated problem.Section 4 describes the hybrid GA for the problem briefly.Section 5 compares the result from the hybrid GA with theoptimal solution, and with the genetic algorithms pro-posed by other researchers. Section 6 concludes the paper.

3. Mathematical models

A chip shooter machine, as illustrated in Fig. 1, has amovable feeder carrier holding components, a rotaryturret with multiple assembly heads, and a movable X–Y

table which locates a PCB. Each assembly head has severalnozzles of different sizes. The operation sequence of themachine is as follows. As the first board of a batch entersthe machine, the first nozzle of the assembly head picksup a component from a feeder. The turret then indexesone step and the next nozzle picks up the secondcomponent. The turret indexes again to pick up the nextcomponent, and so on. At the same time, the PCB is movedto the placement location waiting for the first componentto be placed on the board. When the sixth component isbeing picked up, if the turret has 10 assembly heads, thefirst component is being placed on the board. Theseoperations continue such as the turret indexes one step,the feeder carrier moves to the location containing thenext pick-up component, and the X–Y table moves to thenext placement location. In the assembly of the last fivecomponents, there is no need to pick up components forthe board being assembled. However, the nozzles of theturret can pick up the first five components for the nextboard to be assembled if necessary. For the first fewcomponents assembled in a batch of PCBs, there are onlypick-up movements and no placement movement. For thelast few components of the same batch, there are onlyplacement movements and no pick-up movement. Theseboundary effects can be neglected if the PCB’s quantity ina batch is very large (Leu et al., 1993).

If there are g components between the pick-up and theplacement locations, then there are (2g+2) assemblyheads in the rotary turret. Each PCB has n componentswith m different types. Besides, each of the componenttypes must be stored in a feeder, but a feeder can onlyhold a unique type of components, so m feeders areneeded to hold m types of components.

Three mechanisms of the machine move at differentspeeds, so the traveling time of the X–Y table (i.e., C1ij), the

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Table 1Notation.

Indicesi, j: components (i, j ¼ 1, 2,y, n)

t: component types (t ¼ 1, 2,y, m)

r, s: feeders (r, s ¼ 1, 2,y, m)

p: placement order or placement position (p ¼ 1, 2,y, n)

Traveling timesC1ij: traveling time of the X–Y table

C2rs: traveling time of the feeder carrier

C3: indexing time of the turret

Tp: the longest traveling time among three for assembling the pth

component

Feeder constraintg: number of components between the pick-up and the placement

locations

Decision variablesxip ¼ 1 if component i is placed in the pth position; 0 otherwise

ytir¼ 1 if component type t of component i is stored in feeder r; 0

otherwise

X

Y

Assembly Head

Turret

Feeders

xi,p-1

xi, p+g

Pick-Up Location

Placement Location

Component to be placed;

Component location before placement.

Feeder Carrier

PCB

xj, p

xj, p+g+1

Fig. 1. The schematic diagram of the chip shooter machine.

W. Ho, P. Ji / Int. J. Production Economics 123 (2010) 31–41 33

traveling time of the feeder carrier (i.e., C2rs), and theindexing time of the turret (i.e., C3) are different fromeach other. These three times can be calculated as:

C1ij is the time used by the X–Y table for traveling fromcomponent i (location) to component j (location).

¼ maxjXj � Xij

Vx;jYj � Yij

Vy

��,

where Xi and Xj are the x-coordinates of the components i

and j, respectively; Yi and Yj are the y-coordinates of thecomponents i and j, respectively; Vx and Vy are speeds ofthe X–Y table in the x and y directions, respectively.

C2rs is the time used by the feeder carrier for travelingfrom feeder r to feeder s.

¼Xs � Xrj j

Vf,

where Xr and Xs are the x-coordinates of the feeders r ands, respectively; Vf is the speed of the feeder carrier. C3 isthe time used by the turret for rotating one step.

The longest one among the three times in one step iscalled the dominating time. So, the objective is tominimize the total assembly time, which is the summa-tion of all dominating times of components such that thehighest productivity of the machine can be achieved.

In the following sub-sections, two individual mathe-matical models for the component sequencing and thefeeder arrangement problems are formulated first. Then,two complete mathematical models are constructed forthe integrated problem. The notation used in the models issummarized in Table 1.

3.1. A component sequencing model

Assume the feeder arrangement problem is solvedbeforehand, the component sequencing problem is fre-quently formulated as the TSP for finding the placement

order of components on a PCB such that the total travelingdistance or time of the X–Y table is minimized. In the TSPmodel, a decision variable, xij, is normally used to indicatethat component i is placed immediately before compo-nent j if xij ¼ 1. However, sub-tours may be formed andthus the bulky sub-tour elimination constraints areessential in the model. Here, xip instead of xij is used asthe decision variable. The interpretation of 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, whichmeans that there are totally n2 decision variables inwhich only n variables are 1 while all others are 0.

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Since each component must be placed in exactly oneposition, no sub-tour will appear in this situation.

For the component sequencing model, only xip isincorporated, whereas the assignment of componenttypes to feeders (i.e., ytir

) is predetermined. After thefeeder arrangement has been generated, the objectivefunction of the model can then be constructed as

Minimize z ¼maxXn

i¼1

Xn

j¼1jai

C1ijxi;p�1xj;p;Xn

i¼1

Xn

j¼1jai

C2rsxi;pþgxj;pþgþ1C3

0B@

1CA

for p ¼ 1;2; . . . ;n.

The above objective function is to minimize the summa-tion of all the dominating times among C1ij, C2rs, and C3 ofthe components. First of all, C1ij calculates the time usedby the X–Y table for traveling from the position ofcomponent i on the PCB to the position of component j,which is placed in the pth position. As described earlier,one nozzle is placing a component on the PCB whileanother nozzle is picking up another component from afeeder at the same time. Here, when the pth component isbeing placed, the type of another component to be placedin the (p+g+1)th position is being picked up from a feeder.It is assumed that the types of the (p+g)th and the(p+g+1)th components are stored in feeders r and s,respectively. So, C2rs calculates the time used by thefeeder carrier for traveling from feeder r to feeder s.Thirdly, C3 is the indexing time of the turret.

After introducing Tp, which is the dominating time forassembling the pth component, and incorporating theconstraint sets for determining the sequence of compo-nent placements, the component sequencing model canbe constructed as

Minimize z ¼Xn

p¼1

Tp (1)

subject to

Tp �Xn

i¼1

Xn

j¼1jai

C1ijxi;p�1xj;p � 0 for p ¼ 1;2; . . . ;n; (2)

Tp �Xn

i¼1

Xn

j¼1jai

C2rsxi;pþgxj;pþgþ1 � 0 for p ¼ 1;2; . . . ;n; (3)

Tp � C3 � 0 for p ¼ 1;2; . . . ;n, (4)

Xn

i¼1

xip ¼ 1 for p ¼ 1;2; . . . ;n; (5)

Xn

p¼1

xip ¼ 1 for i ¼ 1;2; . . . ;n, (6)

All xip ¼ 0 or 1; Tp � 0 and is a set of integers: (M1)

Although the minimax type objective function is trans-formed into the minimization one, three constraint setsare introduced. Constraint set (2) is to calculate thetraveling time of the X–Y table for assembling the pthcomponent. Constraint set (3) is to calculate the travelingtime of the feeder carrier for assembling the pth

component. Constraint set (4) is the indexing time of theturret. For example, when p ¼ 1, if T1Z5, T1Z4, and T1Z3in constraint sets (2), (3), and (4), respectively, then T1 willbecome 5 in order to satisfy all constraints. This is the ideaof obtaining the value of Tp. Besides, constraint set (5) is toguarantee that exactly one component is placed in oneposition, while constraint set (6) is to guarantee that oneposition has exactly one component placed.

The assembly time of the chip shooter machine isdependent on all the C1ij, C2rs, and C3. Definitely, thesethree traveling times must be incorporated together in theobjective function of the component sequencing model.Focusing on the traveling time of the X–Y table onlycannot represent the actual situation, and therefore theTSP may not be desirable for the component sequencingproblem.

3.2. A feeder arrangement model

Assuming that the sequence of component placementsis predetermined, some researchers formulated the feederarrangement problem as the QAP for assigning thecomponent types to feeders so that the number of feedercarrier’s movements or the total traveling time of thefeeder carrier is minimized. As the case for the componentsequencing model, the three traveling times should beconsidered simultaneously in the feeder arrangementmodel. So, the QAP may not be desirable in this case.

In the model, only ytiris incorporated, whereas the

sequence of component placements (i.e., xip) is known inadvance. It is to determine which component type isstored in which feeder. Since it is assumed that thenumber of feeders available is equivalent to that ofcomponent types required, a single component type canonly be assigned to a feeder. The interpretation of ytir

isthat

ytir¼

1 if component type t of component

i is stored in feeder r;

0 otherwise:

8><>:

After xip has been known, the objective function of thefeeder arrangement model is constructed as

Minimize z ¼ max C1ij;Xn

i¼1

Xn

j¼1jai

Xmt¼1

Xmr¼1

Xms¼1

C2rsytirytjs

;C3

0B@

1CA

for p ¼ 1;2; . . . ;n.

The above objective function calculates the total assemblytime for assembling all components on a PCB. It is thesummation of all the dominating times among C1ij, C2rs,and C3 of the components. Since the placement order ofeach component is known, the times used by the X–Y

table for traveling from the position of component i to thatof component j (i.e., C1ij) can be directly obtained. It isnoted that the index j in C1ij refers to component j to beplaced in the pth position. When the position of the pthcomponent is being moved to the placement location, thefeeder holding the type of component to be placed in the(p+g+1)th position is being moved to the pick-up location.Here, the indices i and j in C2rs refer to components i and j,

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respectively. Also, component j is placed in the (p+g+1)thposition, and component i is placed immediately prior tocomponent j. If the type of component i (i.e., ti) is stored infeeder r and the type of component j (i.e., tj) is assigned tofeeder s, C2rs is equivalent to the time used by the feedercarrier to travel from feeder r to feeder s.

By introducing Tp and incorporating the constraint setsfor determining the assignment of component types tofeeders, the feeder arrangement model can be formulatedas

Minimize z ¼Xn

p¼1

Tp (7)

subject to

Tp � C1ij � 0 for p ¼ 1;2; . . . ;n, (8)

Tp �Xn

i¼1

Xn

j¼1jai

Xmt¼1

Xmr¼1

Xms¼1

C2rsytirytjs� 0 for p ¼ 1;2; . . . ;n;

(9)

Tp � C3 � 0 for p ¼ 1;2; . . . ;n, (10)

Xmt¼1

ytr ¼ 1 for r ¼ 1;2; . . . ;m, (11)

Xmr¼1

ytr ¼ 1 for p ¼ 1;2; . . . ;m. (12)

All ytr ¼ 0 or 1; Tp � 0 and is a set of integers: (M2)

Similar to M1, the minimax type objective function istransformed into a simple objective function (7) whilesubjecting to three constraint sets (8), (9), and (10) in M2.This transformation requires an introduction of variableTp. Besides, two constraint sets should be incorporated todetermine which feeder holds which component type.Constraint set (11) ensures that exactly one componenttype is stored in one feeder. Constraint set (12) ensuresthat exactly one feeder holds one component type.

3.3. The integrated component scheduling models

Before solving the component sequencing model (i.e.,M1), it is essential to obtain the solution of the feederarrangement problem (i.e., M2) first. On the other hand,M2 cannot be solved until the solution of M1 is known.Therefore, it is no doubt that the component sequencingand the feeder arrangement problems are inter-related.And, it is more suitable to consider the two problemssimultaneously rather than separately, as many research-ers did.

For the integrated problem, the objective of thecomplete model is to minimize the summation of all thedominating times among C1ij, C2rs, and C3 of all

components on the PCB. It can be formulated as

Minimize z ¼ maxXn

i¼1

Xn

j¼1jai

C1ijxi;p�1xj;p;

0B@

Xn

i¼1

Xn

j¼1jai

Xmt¼1

Xmr¼1

Xms¼1

C2rsxi;pþgxj;pþgþ1ytirytjs

;C3

1CA

for p ¼ 1;2; . . . ;n.

When the pth component is being moved to the place-ment location, the feeder holding the type of (p+g+1)thcomponent is being moved to the pick-up location, asshown in Fig. 1. The pick-up and the placement operationstake place after the end of the movements.

After adding the decision variable Tp, and the con-straints for determining the component sequencing andthe feeder arrangement, the complete model for theintegrated problem in the form of minimization isformulated as

Minimize z ¼Xn

p¼1

Tp (13)

subject to

Tp �Xn

i¼1

Xn

j¼1jai

C1ijxi;p�1xj;p � 0 for p ¼ 1;2; . . . ;n; (14)

Tp �Xn

i¼1

Xn

j¼1jai

Xmt¼1

Xmr¼1

Xms¼1

C2rsxi;pþgxj;pþgþ1ytirytjs� 0

for p ¼ 1;2; . . . ;n, (15)

Tp � C3 � 0 for p ¼ 1;2; . . . ;n, (16)

Xn

i¼1

xip ¼ 1 for p ¼ 1;2; . . . ;n, (17)

Xn

p¼1

xip ¼ 1 for i ¼ 1;2; . . . ;n, (18)

Xmt¼1

ytr ¼ 1 for r ¼ 1;2; . . . ;m, (19)

Xmr¼1

ytr ¼ 1 for t ¼ 1;2; . . . ;m. (20)

All xip and ytr ¼ 0 or 1; Tp � 0 and is a set of integers:

(M3)

M3 is a pure integer nonlinear programming model. Theinterpretation of the objective function and the constraintsets can be found in M1 and M2. M3 has (n2+m2) binaryvariables, n integer variables, and (5n+2m) constraints. Ineach of the constraint sets (14) and (15), the terms aren(n–1) and nm2(n–1), respectively. For a realistically sizedproblem of 100 components and 10 component types, M3has 10,100 binary variables, 100 integer variables, and 520constraints. In addition, M3 has 9,900 and 990,000 termsin each of the constraint sets (14) and (15), respectively.

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Due to the fact that nonlinear programming model isvery difficult to solve, it is desirable to reformulate M3into linear programming, if possible. Since the nonlinearterms in the constraint sets (14) and (15) are in the form ofproducts of binary variables, M3 can be reformulatedas a linear one by implementing the following twosteps: (a) replace the product term xy by a binary variablew; (b) impose the logical condition: w ¼ 1 if and only ifx ¼ 1 and y ¼ 1 by means of the extra constraints: wrx,wry, and wZx+y–1.

In the constraint set (14) of M3, the nonlinear term isin the form of products of two binary variables. So, it canbe rewritten as a linear constraint by introducing an extrabinary variable wijp and three extra constraint sets. Theinterpretation of the decision variable wijp is

wijp ¼

1 if component j is placed just after component i and

component j is placed in the pth position;

0 otherwise:

8><>:

On the other hand, in the constraint set (15) of M3, thenonlinear term is in the form of products of four binaryvariables. So, the steps for converting it into linear typeneed to be modified in this case. The major difference isthat five instead of three extra constraints are introduced.Similar to that for the constraint set (14), a decisionvariable wij(p+g+1)rs is introduced. The interpretation of thedecision variable wij(p+g+1)rs is

Table 2Numbers of variables and constraints in M3 and M4.

M3 M4

No. of variables n2+m2+n (n2+m2+n)+[n2(n–1)]+[n2m2(n–1)]

No. of constraints 5n+2m (5n+2m)+[3n2(n–1)]+[5n2m2(n–1)]

wijðpþgþ1Þrs ¼

1 if the type of component i is stored in feeder r and the type of

component j placed in the ðpþ g þ 1Þth position is stored in feeder s;

0 otherwise:

8><>:

The mathematical model in the form of linear functioncan be formulated as follows:

Minimize z ¼Xn

p¼1

Tp (21)

subject to

Tp �Xn

i¼1

Xn

j¼1jai

C1ijwijp � 0 for p ¼ 1;2; . . . ;n; (22)

Tp �Xn

i¼1

Xn

j¼1jai

Xmr¼1

Xms¼1

C2rswijðpþgþ1Þrs � 0 for p ¼ 1;2; . . . ;n,

(23)

Tp � C3 � 0 for p ¼ 1;2; . . . ;n, (24)

Xn

i¼1

xip ¼ 1 for p ¼ 1;2; . . . ;n, (25)

Xn

p¼1

xip ¼ 1 for i ¼ 1;2; . . . ;n, (26)

Xmt¼1

ytr ¼ 1 for r ¼ 1;2; . . . ;m, (27)

Xmr¼1

ytr ¼ 1 for t ¼ 1;2; . . . ;m, (28)

wijp � xi;p�1 for i; j; p ¼ 1;2; . . . ;n; iaj, (29)

wijp � xjp for i; j; p ¼ 1;2; . . . ;n; iaj, (30)

wijp � xi;p�1 þ xjp � 1 for i; j; p ¼ 1;2; . . . ;n; iaj, (31)

wijðpþgþ1Þrs � xi;pþg for i; j; p ¼ 1;2; . . . ;n; iaj;

for r; s ¼ 1;2; . . . ;m, (32)

wijðpþgþ1Þrs � xj;pþgþ1 for i; j; p ¼ 1;2; . . . ;n; iaj;

for r; s ¼ 1;2; . . . ;m, (33)

wijðpþgþ1Þrs � ytirfor i; j; p ¼ 1;2; . . . ;n; iaj;

for r; s ¼ 1;2; . . . ;m, (34)

wijðpþgþ1Þrs � ytjsfor i; j; p ¼ 1;2; . . . ;n; iaj;

for r; s ¼ 1;2; . . . ;m, (35)

wijðpþgþ1Þrs � xi;pþg þ xj;pþgþ1 þ ytirþ ytjs

� 3,

for i; j; p ¼ 1;2; . . . ;n; iaj; for r; s ¼ 1;2; . . . ;m. (36)

All xip ytr ; wijp and wijðpþgþ1Þrs ¼ 0 or 1;

Tp � 0 and is a set of integers: (M4)

In M4, constraint sets (22), and (29) to (31) are the linearexpression of the constraint set (14), whereas constraintsets (23), and (32)–(36) are the linear expression of theconstraint set (15). Although the model becomes linear,both numbers of variables and constraints in M4 increasegreatly. For the number of binary variables, n2(n–1) of wijp

and n2m2(n–1) of wij(p+g+1)rs are introduced in M4 besides(n2+m2) binary variables and n integer variables. For thenumber of constraints, besides (5n+2m) constraints, M4has [3n2(n–1)]+[5n2m2(n–1)] constraints more in whichthere are [3n2(n–1)] constraints for constraint sets (29) to(31) and there are [5n2m2(n–1)] constraints for constraintsets (32) to (36). For a realistically sized problem of 100components and 10 component types, M4 has100,000,200 variables, and 497,970,520 constraints. Thenumbers of variables and constraints of both M3 and M4are summarized in Table 2.

To verify the models, two commercial packages areused. BARON is adopted to solve the pure integernonlinear programming model or M3, whereas CPLEX is

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applied to solve the pure integer linear programmingmodel or M4. By these two commercial packages, M3 andM4 are tested by several small examples, and they canboth generate the same optimal solutions to the sameproblems. The computational time spent on solving themodels on a 2.26 GHz computer is represented in Table 3.It is found that M3 is more desirable than M4 in terms ofthe amount of computational time spent. Because of this,an integrated problem with 10 components and 6component types is formulated as M3. The data of theproblem are listed in Table 4. The optimal sequence of

Table 3Computational times spent by BARON and CPLEX for solving M3 and M4.

Numbers ofcomponentsand types

Optimal solution CPUtime (hh:mm:ss) byBARON for M3

Optimal solution CPUtime (hh:mm:ss) byCPLEX for M4

4�4 00:00:01 00:00:01

5�5 00:00:14 00:00:20

6�6 00:02:27 00:07:19

7�7 00:47:20 04:37:39

8�8 14:30:25 328:07:51

Table 4Data of the integrated problem with 10 components and 6 component

types.

Components Types Coordinates (mm) Feeders Coordinates (mm)

x y x y

1 6 100 80 1 100 300

2 5 120 80 2 115 300

3 1 140 80 3 130 300

4 3 160 80 4 145 300

5 4 180 80 5 160 300

6 5 100 40 6 175 300

7 1 120 40

8 3 140 40

9 4 160 40

10 2 180 40

Remarks: The number of components between the pick-up and the

placement locations in the turret, g ¼ 2; the table’s speed in both x and y

directions, Vx ¼ Vy ¼ 60 mm/s; the linear speed of the feeder carrier, Vf

¼ 60 mm/s; the indexing time of the head, C3 ¼ 0.25 second per index.

Table 5Calculation of the three traveling times in the integrated problem.

C1ij C2rs C3 Tp

p ¼ 1 0.6667 (c5-c10) 0.25 (f3-f4) 0.25 0.6667

p ¼ 2 0.3333 (c10-c9) 0.25 (f4-f5) 0.25 0.3333

p ¼ 3 0.3333 (c9-c8) 0.25 (f5-f6) 0.25 0.3333

p ¼ 4 0.3333 (c8-c7) 0.25 (f6-f5) 0.25 0.3333

p ¼ 5 0.3333 (c7-c6) 0.25 (f5-f4) 0.25 0.3333

p ¼ 6 0.6667 (c6-c1) 0.25 (f4-f3) 0.25 0.6667

p ¼ 7 0.3333 (c1-c2) 0.25 (f3-f2) 0.25 0.3333

p ¼ 8 0.3333 (c2-c3) 0.25 (f2-f1) 0.25 0.3333

p ¼ 9 0.3333 (c3-c4) 0.25 (f1-f2) 0.25 0.3333

p ¼ 10 0.3333 (c4-c5) 0.25 (f2-f3) 0.25 0.3333

Total assembly time 4.0000

component placements is (10, 9, 8, 7, 6, 1, 2, 3, 4, 5).Besides, the optimal feeder arrangement is that compo-nent types 2, 4, 3, 1, 5, and 6 are stored in feeders 1, 2, 3, 4,5, and 6, respectively. The minimum assembly time is 4 s.Table 5 illustrates the calculation of the three traveling,and shows how Tp is obtained. According to the aboveexample, Tp is a positive variable. It can be converted intoan integer variable by multiplying it with a constant value(e.g., 10,000).

Although the models can be solved to global optim-ality, it is not an efficient method due to the fact that thecomputational time grows exponentially with the pro-blem size. As a result, it is necessary to develop a heuristicmethod to solve the integrated problem for the chipshooter machine efficiently. However, a heuristic may notachieve the optimal solution.

4. A hybrid genetic algorithm

Due to the complexity of the integrated problem, aheuristic method is desirable to be adopted. The success ofGAs in solving a wide variety of complex optimizationproblems (Goldberg, 1989; Gen and Cheng, 1997) and theadvantages of GAs such as simplicity, easy operation, andgreat flexibility have prompted us to apply GA. Never-theless, a simple GA may not perform well in this situationbecause the component sequencing and the feederarrangement problems are considered simultaneously.Actually, the individual problems are already very hardto solve (Crama et al., 1997). The GA adopted here istherefore hybridized with several heuristics in order toimprove the solution further. Recently, hybrid geneticalgorithms have been widely applied to tackle hardoptimization problems (Huang, 2007; Farahani andElahipanah, 2008; ElMekkawy and Liu, 2009; Lova et al.,2009).

The flowchart of the hybrid GA or HGA for theintegrated problem is shown in Fig. 2. Since bothproblems are considered simultaneously, each chromo-some or solution includes two path representations orlinks (Fig. 3). The first link denotes the componentsequencing, whereas the second link represents the feederarrangement. After the parameters have been set up, theHGA generates an initial population in which the firstlinks are generated from the nearest neighbor heuristic(NNH) while the second links are generated randomly.During this initialization step, each chromosome isimproved as follows: the 2-opt local search heuristic isapplied to the second link while the iterated swapprocedure (ISP) (Fig. 4) is performed on the first link.The principle of the ISP is very similar to that of the 2-optlocal search heuristic, except that some instead of all twoswaps are examined to generate offspring. It can definitelyreduce the computational time because the number ofcomponents is quite large, normally several hundreds.Each chromosome is then measured by an evaluationfunction, which has been described thoroughly in Section3. The roulette wheel selection operation is performed toselect some chromosomes for the genetic operationsincluding the modified order crossover (Fig. 5), the

ARTICLE IN PRESS

Initialization

- 1st Link: NNH; - 2nd Link: Random Generation.

Termination?

Best Solution Outputting

Yes

No

GA Parameters Inputting

Selection

The Modified Order Crossover

The Heuristic Mutation

The Inversion Mutation

Evaluation (Parents)

Improvement (Parents)- 1st Link: ISP;- 2nd Link: 2-opt Local Search.

Improvement (Offspring) - 1st Link: ISP; - 2nd Link: 2-opt Local Search.

Evaluation (Offspring)

Best Population Retaining

Fig. 2. The flowchart of the HGA.

W. Ho, P. Ji / Int. J. Production Economics 123 (2010) 31–4138

heuristic mutation (Fig. 6), and the inversion mutation(Fig. 7). After an offspring is produced, the second link isimproved by the 2-opt local search heuristic while thefirst link is improved by the ISP. The fitness of theoffspring will be measured and may become a member of

the population if it possesses a relatively good quality.These steps form an iteration, and then the roulette wheelselection is performed again to start the next iteration.The HGA will not stop unless the predetermined numberof iterations is conducted.

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Assembly Sequence

Component Number 10

Link 1

Component Type 2Feeder

Link 2

9 8 7 6 1 2 3 4 5

1 2 3 4 5 6

1 2 3 4 5 6 7 8 9 10

4 3 1 5 6

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

Fig. 4. The iterated swap procedure.

Fig. 5. The modified order crossover operator.

Fig. 6. The heuristic mutation operator.

Fig. 7. The inversion mutation operator.

W. Ho, P. Ji / Int. J. Production Economics 123 (2010) 31–41 39

5. Performance analysis

In this section, the performance of the HGA isevaluated by comparing it to the optimal solutions ofseveral problems mentioned in Section 3, which wereobtained using either BARON or CPLEX. Then, thecomparison between the HGA and the GAs developed byLeu et al. (1993) as well as Ong and Tan (2002) is carriedout. In addition, the effect of the population size on theeffectiveness of the HGA is studied.

5.1. Comparison to optimal solutions

The same problems as mentioned in Section 3 aresolved by the HGA. It is found that the HGA achieves theoptimal solutions to all problems quickly. Furthermore,the longest computational time spent is only 5 s for the8-component problem. Compared with the time spent onfinding the optimal solution for the 8-component pro-blem, the HGA is much more efficient. It saves more than14 h when compared with BARON, and saves about 14days when compared with CPLEX.

5.2. Comparison to other approaches

The performance of the HGA is evaluated using the PCBexample in Leu et al. (1993). The HGA parameters are:population size ¼ 25, iteration number ¼ 1000, crossover

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Table 6A comparison of the experimental results.

Leu et al. (1993) Ong and Tan (2002) HGA

Best one in the initial solution (s) 70 About 60 30

Population size 100 10 25

Iteration number About 1,750 5,000 323

Final best solution (s) About 51.5 26.9 26

0 500 1000 1500 2000 2500 300050

55

60

65

70

75

80

Iteration number

Tota

l ass

embl

y tim

e (s

econ

ds)

Population size = 5

Population size = 25

Population size = 50

Fig. 8. The effect of population size (100-component problem).

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000105

110

115

120

125

130

135

140

145

150

155

Iteration number

Tota

l ass

embl

y tim

e (s

econ

ds)

Population size = 5

Population size = 25

Population size = 50

Fig. 9. The effect of population size (200-component problem).

W. Ho, P. Ji / Int. J. Production Economics 123 (2010) 31–4140

rate ¼ 0.4, and mutation rate ¼ 0.2. According to Table 6, itis found that the performance of the HGA is superior tothat of the GAs used in Leu et al. (1993) and Ong and Tan(2002) in three aspects. Firstly, the best chromosome(30 s) in the initial population obtained by the HGA isbetter than that of the simple GAs (both more than 60 s).Secondly, the HGA can obtain a better solution with feweriterations, 323 vs. 1,750 or 5,000. Finally and the mostimportantly, the HGA obtained a better solution than the

previous methods, 26 s vs. 51.5 or 26.9 s. This improve-ment of just 0.9 s when compared with Ong and Tan(2002) is significant in the PCB assembly industry. Sincethe component placement is the bottleneck of theassembly line as mentioned in Section 1, a minorreduction in the cycle time will save a significantproduction time. For example, to produce 100,000 boards,a reduction of 0.9 s in the cycle time will save 1,500 min or25 working hours. Therefore, the productivity of a PCBmanufacturing company can be enhanced if our HGA isadopted.

5.3. Effect of population size

Population size plays a vital role since it determinesthe effectiveness of the HGA. The larger the populationsize, the higher the chance of finding the optimal solutionor a better solution. In order to identify the effect of thepopulation size, the HGA program is run with a 100-component problem and a 200-component problem eachof which uses three different population sizes: 5, 25, and50. Figs. 8 and 9 show the performance of the HGA for the100-component and 200-component problems, respec-tively. It can be seen that the curves representing thepopulation size of 50 are the lowest. This shows that theHGA with a larger population size can obtain a better finalsolution.

6. Conclusions

This paper considered the component sequencing andthe feeder arrangement problems simultaneously for thechip shooter machine, with the objective of minimizingthe total assembly time. Two individual mathematicalmodels were formulated for the individual componentsequencing and feeder arrangement problems first. It wasnoticed that the problems are inter-related. One cannot besolved unless the solution of the other one is obtainedbeforehand. Due to their inseparable relationship, a pureinteger nonlinear programming model and a pure integerlinear programming model for the integrated problemwere formulated. The nonlinear and the linear program-ming models were verified using the commercialpackages, and both of them generated the same optimalsolutions to the same problems. Different types of modelstook different times for computation. In terms of theamount of computational time spent for solving themodel to global optimality, the nonlinear type ispreferred.

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W. Ho, P. Ji / Int. J. Production Economics 123 (2010) 31–41 41

Although the optimal solution could be found usingthe commercial packages, it was found that the computa-tional time grows exponentially with the problem size. Asa result, a HGA was adopted to solve the integratedproblem. It was found that the performance of the HGA isdesirable not only because it can reach the globaloptimum of several problems with small sizes quickly,but also it can generate better solutions than the otherapproaches in terms of the total assembly time. Further-more, the HGA with a larger population size can obtain abetter final solution.

Acknowledgment

The work described in this paper has been supportedby the Research Grants Council of Hong Kong, China(Project no.: PolyU 5259/04E).

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