Group Sequencing a PCB Assembly System Via

an Expected Sequence Dependent Setup Heuristic

Manuel D. Rossetti 1

Assistant Professor

Department of Industrial Engineering

University of Arkansas

4207 Bell Engineering Center

Fayetteville, AR 72701

and

Keith J. A. Stanford

SAP America, Inc.

Six Concourse Parkway

Suite 1200 Mail Stop 1206

Atlanta, Ga 30328

Manuscript No. 99/129 Version: 3

1email: rossetti@uark.edu, fax: 479-575-8431

Group Sequencing a PCB Assembly System Via

an Expected Sequence Dependent Setup Heuristic

Abstract

In circuit board manufacturing, the production sequencing prob-lem with sequence dependent setups is complicated by the fact thatthe amount of setup required depends on not only the setup directlybefore the current setup, but can depend on all of the preceding se-tups. We present a case study that examines the use of a heuristic forestimating the expected number of setups from the sequence depen-dent setups. Our method is based on estimating the expected numberof setups which may occur given a board-feeder setup configuration.Once estimates for the sequence dependent setups are established,they can be used to measure the similarity between boards in cluster-ing algorithms and in nearest neighbor heuristics for group sequencing.The method is tested on an actual printed circuit board assembly sys-tem. Then, using a simulation of the assembly system, we comparethe sequences generated using the expected number of setups distancemeasure and the Hamming distance measure to optimal sequences.Our results indicate that grouped sequences generated by using theexpected number of setups had significantly better makespan perfor-mance when compared to sequences based on the more traditionalHamming distance for the particular system under study. The signif-icant gains in makespan resulted in only moderate increases in workin process and slight increases in manual station utilization.

Keywords: Sequence dependent setups, group sequencing, heuristics,printed circuit board manufacturing, simulation

1 Introduction

In this paper, we examine the sequencing of printed circuit boards (PCBs)

within a PCB assembly system. The major purpose of this paper is to present

the results of our sequencing methodology as applied to a specific assembly

system. With modifications, our methodology can be applied to other as-

sembly systems of this type; however, the results discussed in this paper are

applicable to this particular case study. The simulation results will indicate

that our methodology shows potential for developing solutions to sequence

dependent setup sequencing problems.

The assembly system under study provides both a case study for the

examination of our methodologies and an opportunity to solve a realistic se-

quencing problem. This paper is derived from an application within a specific

company that required an effective way to sequence printed circuit boards

(PCB’s) through their asynchronous reentrant PCB assembly system in order

to minimize sequence makespan, improve the utilization of the manual por-

tion of the line, and maintain reasonable work in process (WIP) levels. The

assembly system under consideration is typical of printed circuit board lines

both in the types of equipment utilized and in the sequencing issues of the

line. A unique feature of this particular assembly system is its reentrant flow

and the coupling of automated and manual placement lines into one system.

In order to provide context and motivation for our sequencing methodology,

we first present the full details of the assembly system examined in this case

study.

The assembly system consists of an automated surface mounting line

coupled with a manual placement, inspection, and testing line. Figure 1

shows the layout of the line. The boards to be processed by this line may

require both top and bottom side placements of components. The automated

surface mounting line is capable of performing both top-side and bottom-

side placements of components. A set of circuit boards which are processed

together and are of the same type are called a lot. The lot-sizes for the board

1

types are set prior to processing depending upon weekly demand for the

board type. Each lot of circuit boards requiring bottom-side placements are

processed through the surface mounting line twice. All top-side placements

must occur before any bottom-side placements can be made. This assembly

system is reentrant because boards requiring both bottom-side and top-side

placements must be processed twice by the automated surface mount line.

The automated surface mounting line consists of a receiving station, a

glue machine, two surface mounting machines, an inspection station, an oven,

a material handling flipping station, and conveyers which interconnect the

various stations. Circuit boards are loaded on pallets which can hold from 30

unassembled PCB’s to 15 partially assembled (top-side completed) PCB’s.

Up to three pallets can be loaded onto the receiving station. The boards

are indexed from the pallets and then proceed down the line. As soon as a

pallet is empty a full pallet is loaded onto the receiving station ensuring an

unlimited supply of boards. After the receiving station the board proceeds

to the glue station and then to the two surface mount machines. The two

surface mounting machines are label Sp-120 and Ms-102 in Figure 1. The

Sp-120 is the faster of the two machines but is also less flexible than the

Ms-102. Once the PCB has been through the two surface mounting stations,

the board must be manually inspected before entering the oven. After the

oven, the board travels to a material handling flipping station that turns the

board over from top to bottom or vice-versa to orient the board for its next

processing steps. The line operates as an asynchronous automated flow line

with the conveyers acting as finite buffer spaces between the stations. The

boards can not pass each other while on the line.

After processing is completed on the automated surface mounting line, a

lot is processed once through the manual placement, inspection, and testing

line. A large buffer exists between the two lines so that the surface mounting

line can never be blocked, but the manual placement line can be starved.

Each lot of boards to be processed on the manual placement line is loaded

2

onto another receiving station. The boards are indexed down the line to

visit four manual placement stations in series, an oven, and a final inspection

station. There are two inspection stations working in parallel which share a

single buffer. The boards are then processed through a testing station before

exiting the system.

Analysis of the current operation of the assembly system indicated that

the setup time spent to change over the automated surface mounting line

for processing different board types and/or bottom side placements could

require significant amounts of available production time (upwards to half a

day in some cases). Part of the setup time involves the loading of specific

component types into feeders for the surface mount machines. In addition to

the inefficiencies caused by the setups, the system consists of two lines (au-

tomated and manual) that must work together when producing the boards.

For example, the throughput rate of the automated line must ensure a cer-

tain level of utilization within the manual line without causing significant

amounts of WIP. In addition to the current situation, the ramping up of pro-

duction on the line was expected in the near term due to increased demand

for additional board types to be run through the line. The analysis indicated

that an effective solution to the setup/sequencing problem could provide in-

creased capacity and thus more flexibility in handling new demand. In the

following section, we discuss how the assembly system operates with respect

to setups.

2 Specifics of the Setup/Sequencing Problem

Each surface mount machine has a set of feeders that store and feed compo-

nents to the machine. Each feeder has a location that can effect the placement

times for the components onto the boards. The assignment of specific com-

ponents to specific feeders (i. e. locations) in order to minimize placement

times is called the component/feeder assignment problem. The automated

assembly system in this study consists of two surface mount machines (Sp-

3

120 and Ms-102) with the Sp-120 being the faster of the two. Thus, for this

PCB line the basic component/feeder assignment problem is complicated

by the additional goal of establishing a balance between the two machines.

This is a very complex problem which deserves a separate investigation. For

this particular application, we assume that the component/feeder assignment

problem has been solved. In other words, we know in advance which compo-

nents are assigned to which feeders. In general, the sequencing of the boards

on the PCB line will depend upon the solution to the component/feeder and

machine loading problems. Thus, our results are dependent upon the given

assignments of components to feeders. In practice, it is typical to solve the

sequencing problem separately from the component/feeder assignment prob-

lem rather than trying to simultaneously solve both problems. In fact, our

sponsors did not want to change the current setup assignments.

The assignment of components to feeders constitutes a feeder’s setup. In

this application, there are approximately 400 components and 90 feeders. In

our case, there are more components than there are feeders for assignment.

Solutions to the component/feeder assignment and machine loading problems

determine which components are assigned to which feeders. A feeder’s setup

configuration consists of the set of components assigned to the feeder. Each

component is assigned to a distinct feeder. Thus, the components assigned

to a feeder can only be placed at that feeder’s location. Setups that occur be-

tween the production of different board types will occur only at feeders which

have more than one setup configuration and which require a change to a new

configuration, i. e. a component that is needed by the next board type is not

available in the current setup configuration for that feeder and the feeder

has a setup configuration which contains the required component. A feeder

may have more than one setup configuration because of the requirements

of the board types. Given the solution to the component/feeder assignment

problem and the machine loading problem only 32 of the 90 feeders contained

more than one setup configuration for our application. In addition, only 96 of

4

the 400 components were assigned to the 32 feeders that required more than

one setup configuration. The current state of the 32 multiple setup configu-

ration feeders depends on all the previous boards which have been processed.

The number of setup changes required depends not only on the current setup

but also on all the preceding setups. This causes the determination of the

best production sequence to be sequence dependent.

In the next section, we briefly discuss relevant background literature in

order to place our problem in context. In Section 4, we present our solution

methodology for grouping boards and for efficiently sequencing the PCB

assembly system. In Section 5, we evaluate our sequencing methodology

using a simulation representation of the PCB assembly system. Finally, we

summarize our results and suggest areas for future research.

3 Background Literature

French (1982) states that the general problem of determining a schedule

is to find a sequence that is compatible with technological constraints and

allows the jobs to be processed by the machines in a fashion that is optimal

with respect to some performance criteria. The automated surface mounting

line component of our system is a flow line so that all the boards must be

processed by all the stations (machines) on the line in the same order and

the boards can not pass each other on the line. In this case, the scheduling

problem is reduced to determining the optimal ordering of boards to be placed

on the line.

A common simplifying assumption for scheduling is to assume that the

sequence is independent of setups. The setups are assumed to be small in

comparison to the processing times or else they are included in the processing

times of the jobs. The problem of sequencing when the setups are sequence

dependent has been shown to be NP-hard by formulating the problem as

a traveling salesman problem, see for example Baker (1974), Bianco et. al.

(1994), and Pinedo (1995). In those cases, the typical assumption is that

5

the setup depends only on the immediately preceding job. The translation

to a TSP for our PCB assembly system is not direct since the number of

setup changes depends on all preceding setups, see for example (Hashiba and

Chang (1991)) and (Lockett and Muhlemann (1972)). In addition to the

problem of sequence dependent setups, our system requires that the boards

which require bottom-side placement be reprocessed through the automated

surface mounting line before proceeding to the manual placement line. This

characteristic is similar to the reentrant flow lines examined by Gupta (1993)

and has been shown to be NP-hard without the additional sequence depen-

dent setup problem.

Approaches to the solution of sequencing and scheduling problems with

sequence dependent setups include branch and bound methods, integer pro-

gramming formulations, and a variety of both static and dynamic heuristic

procedures. We refer the reader to Baker (1974), French (1982), and Pinedo

(1995) for a general discussion of these approaches. One approach given by

Dilts and Ramsing (1989) is to estimate a sequence independent setup time

for a particular job from its sequence dependent setups. The estimated se-

quence independent setups can then be used in one of the above approaches

to generate production sequences, see also White and Wilson (1977). Doul-

geri and Magaletti (1991) examined a flexible assembly system with similar

characteristics to the one studied here. They determined that successful

heuristics tend to be those that give priority to jobs which can be processed

with the current setup. This type of rule performs better because it relieves

the bottleneck from extra setups.

One way to take advantage of the current setup is to sequence jobs which

are similar in some respects. Group scheduling and sequencing, see for ex-

ample Gongaware and Ham (1984), Ham et al. (1985), and Wemmerlov and

Vakharia (1991) attempts to exploit this fact. Gongaware and Ham (1984)

discuss various methods for placing boards into groups. One common method

is through cluster analysis techniques, see Kusiak (1985), Everitt (1993), and

6

Ignizio and Cavalier (1994). Kusiak (1985) reports on the performance of sev-

eral clustering techniques. The Hamming distance is defined as the number

of different components required to produce two different PCB’s, see for ex-

ample Hashiba and Chang (1991). The Hamming distance is often used to

approximate the distance between two setups in clustering and grouping al-

gorithms. After clustering, scheduling is performed. A variety of methods

have been tried such as first scheduling the group and then scheduling the

members within the group or to first schedule the members within a group

and then scheduling the group. Finally, the performance of schedules is of-

ten evaluated by simulation in order to capture the dynamics of the problem

which were not included in the mathematical modeling of the scheduling and

sequencing problem.

In addition to the assumption that the component/feeder assignment

problem has been solved, we made the following assumptions in order to

make our analysis more tractable. We assume that the components for the

feeders are always in stock, lot sizes for the various board types have been de-

termined, and machine breakdowns do not significantly effect the determina-

tion of production sequences. In addition, we assume that before scheduling,

all board types are equally likely to occupy any position in the production

sequence. We will discuss this latter assumption more fully when we present

our grouping heuristic.

Finally, because the PCB line consists of a series of stations, we attack

the problem of sequencing the line by developing a sequence for the con-

straining station in the line. In this problem, the constraining station is the

surface mount machine because it had the longest processing times compared

to the other machines on the line. This allows us to reduce the problem to a

single machine sequencing problem. Our approach is not to directly solve a

mathematical programming formulation of the problem. Instead, we approxi-

mate the system as a single machine problem, estimate sequence independent

setup costs, group the board types into similar groups, and then sequence the

7

groups according to heuristic procedures. We then examine the performance

of our methodology via a realistic simulation of the line for the generated se-

quences. Even with these assumptions, the simulation results will show that

significant improvements in makespan for this PCB line can be achieved by

our approach. In the next section, we present further details of our solution

methodology.

4 Solution Methodology

Our basic approach to the problem is to consider the automated surface

mounting line as the bottleneck of the system and to then simplify the se-

quencing problem by considering the automated surface mounting line as a

single machine. We handle the reentrant nature of the problem by consid-

ering bottom-side placements as a different board type and add precedence

constraints which ensure top-side placements before bottom-side placements.

With top-side and bottom-side placements treated as separate board types,

the board types are grouped to take advantage of similarities in setups and

then sequenced using a nearest neighbor heuristic based on our estimates

of sequence independent setups. In the following section, we formulate a

mathematical program of our sequencing problem in order to justify the use

of heuristics. We then discuss two distance measures for forming groups for

sequencing. This will also include the development of our sequence indepen-

dent setup times from the structure of the feeder arrangements. Finally, we

discuss our basic sequencing procedure.

4.1 Single Machine Formulation

In this section, we present our single machine model of the PCB assembly

system. Because of the large buffer between the automated line and the

manual line, we concentrate on efficiently sequencing the automated line

because it tends to also be the bottleneck for the entire system. Setup occurs

8

only after the last board of a lot is processed through the automated surface

mounting line and a change in board type is necessary. Thus, the processing

time for an individual board is determined by the bottleneck machine (surface

mounting machine) on the line and the entire line can be considered as a

single machine; however, the total processing time associated with a board

type depends upon how many boards are to be produced, i. e. the demand.

In addition to the processing time, we have setups which occur between

different board types. A standard approach to this problem is to model the

situation as a travelling salesman problem (TSP) by estimating sequence

independent setup costs. The analogy to the TSP is to equate the cities

to visit with the boards to sequence and the distance between the cities

(boards) as the estimated sequence independent setups. In order to handle

the reentrant nature of the problem, we consider boards with both top-side

and bottom-side placements to constitute two different board types. We then

add additional constraints so that no bottom-side board type is sequenced

before its corresponding top-side. Our formulation follows Selen and Hott

(1986) and Wilson (1989), and is given as follows:

minN∑

k=1

N−1∑i=0

N∑j=i+1

zikzjk+1 (dijQ + λjcj) (1)

subject to:N∑

i=1

zij = 1, ∀j = 1, . . . , N

N∑j=1

zij = 1, ∀i = 1, . . . , N

N∑j=1

jzij >N∑

j=1

jzitj, ∀i ∈ B, it ∈ T

zik = 0, 1 ∀i, k

where

• N = the number of board types

9

• Q = a value which transforms dij into time units required to go from

board type i to board type j

• λj = demand for a board of type j

• cj = the bottleneck processing time per unit of board type j

• dij = the distance from board type i to board type j (Hamming or

expected setup)

• zij = 1 if board type i is in sequence position j and 0 otherwise

• B = set of all bottom-side board types

• T = set of all corresponding top-side board types

• it = the corresponding top-side board type for i ∈ B

The product zik × zjk+1 equals 1 if board type j follows board type i in the

sequence; otherwise, it equals zero. Thus, the product zikzjk+1 (dijQ + λjcj)

is the total time (setup plus processing) required to change from board type i

to board type j. We have incorporated demand into the objective function to

give a sequence dependent processing time as per Ding (1994). For example,

assume that the distance, dij, between a board type i and a board type j

is 4. Let Q be 4.5 minutes, and the processing time, cj, for a unit of board

type j equal 2 min/unit. The demand, λj, for board type j is 10 units. The

sequence dependent processing time for board type j when it follows board

type i is 38 minutes.

We are minimizing this time across all of the sequence positions. The

first two constraints ensure that the board types are uniquely positioned for

sequencing. The precedence constraint ensures top-side placements before

bottom-side placements in the sequence since∑N

j=1 jzij represents the posi-

tion number in the sequence of a board of type i. In the second summation

in Equation (1), i = 0 represents the current board type setup. This could

10

be the last board type from the previous weeks demand or a common setup.

Finally, if grouping has been done, we would need to ensure that the as-

signment of a sequence position to any item within a group g ∈ G must be

within ng − 1 positions of any other board type within group g, where ng is

the number of boards in the group and G is the set of groups. Constraints

of this type take on the following form:

N∑j=1

jzij −N∑

j=1

jzi′j ≤ nm − 1, ∀i and i′ in group g, and ∀ groups g ∈ G

and would be added to the above mathematical program as appropriate. For

a small number of board types, it is possible to enumerate the solution to

the above formulation. In Section 5, when we refer to the optimal sequence

(OS), we are referring to the enumerated solution to the above problem. In

addition, if we do not include the λj × cj term in the objective function we

call this problem the demand independent (DI) problem; otherwise, we call

it the demand dependent (DD) problem.

4.2 Distance Measures and Grouping Formulation

The problem can be further simplified by grouping board types together

according to some measure of similarity. We are then left with determining

a sequence for the groups. Hashiba and Chang (1991) discuss the Hamming

distance as well as give references for other similarity metrics and clustering

algorithms. The Hamming distance has been used as a measure of difference

between boards for grouping algorithms. The Hamming distance between

two boards is the number of different components placed on the two boards.

The Hamming distance is easily calculated from a component-PCB incidence

matrix. The component-PCB matrix, A = (aij), consists of 0-1 entries, where

aij = 1 indicates that component i is required on board type j. Let �aj denote

the jth column of A. The Hamming distance, hij, between board type i and

board type j is given by

hij = (�ai ⊗ �aj) (�ai ⊗ �aj)T

11

where ⊗ is the exclusive-or operator applied element-wise to the the vectors.

If u and v are elements of the vectors then (u ⊗ v = (u ∩ v) ∪ (u ∩ v)). This

results in a symmetric matrix since the actual feeder assignments are ignored.

Intuitively, the Hamming distance does capture information involving the

similarity between board types; however, it may not adequately explain the

setups required between board types. Instead, we estimate the expected

number of setups which might occur.

Let S be the set of setup configurations, F be the set of feeders, and T be

the set of board-types. Let BS be the relation that represents the setup con-

figurations used by the various board-types. An element of BS is an ordered

pair (t, s) where t ∈ T and s ∈ S. Let FS be the relation that represents the

setup configuration assigned to the feeders. An element of FS is an ordered

pair (f, s) where f ∈ F and s ∈ S. Define FBS as the (feeder, board-type,

setup configuration) relation that represents the setup configuration mapped

to a particular (feeder, board-type) combination. An element of FBS is an

ordered triplet, (f, t, s) where f ∈ F , t ∈ T , and s ∈ S. If a setup config-

uration is not used for a particular (feeder, board-type) combination then

s = null. In terms of relational algebra FBS can be written as

FBS = ((F × T ) LEFT OUTER JOIN (FS JOIN BS))

where × indicates the Cartesian product. An example of the construction

of FBS is presented in the appendix. This relation can be more easily

conceptualized as a matrix. Let B = (bij) be the FBS matrix where an

element bij is the required setup configuration at feeder i ∈ F by board-type,

j ∈ T . Clearly, bij will be a member of S or will be null. If bij = null then

no setup configuration is required. An example FBS matrix (as constructed

from the appendix) is:

12

Boardst1 t2 t3 t4

Feedersf1

f2

f3

null s1 s1 s2

s4 null s3 nulls5 s6 null s5

Let Cik be the number of board types that require a different setup con-

figuration than board-type k’s setup configuration and require a setup con-

figuration on feeder i. Let N ik be the total number of board-types other than

type k that require a setup on feeder i. Both Cik and N i

k can be tabulated

using the steps presented in the appendix.

Consider the example FBS matrix given above and suppose a board

of type t3 were to follow a board of type t1 in the sequence. In order to

determine if a setup change is needed on feeder f1, we would need to know

what board type was in sequence before board type t1 because board type

t1 does not require a setup configuration on the feeder since bf1t1 = null. If

a board of type t2 is in front of the board of type t1 then a feeder change is

unnecessary; however, if a board of type t4 is in front of the board of type

t1, then a feeder change must occur.

Unfortunately, in order to determine all the possible feeder changes that

might occur when setting up between two boards all the possible combi-

nations of all feeder locations would have to be evaluated for all possible

sequences. Because of this, we examine the probability, pijk, of a setup occur-

ring at feeder i when a board of type k is sequenced after a board of type j.

The total number of expected setup changes, sjk, that can occur when board

type k is sequenced after board type j is the summation of the pijk over all

feeder locations. Assuming, that all board types are equally likely to occur

in the sequence, the calculation of pijk depends on knowing the number of

board types that require a different setup configuration with respect to the

total that require a setup configuration on the feeder. The Cik and N i

k can be

computed from the FBS matrix as shown in the appendix. For the example

13

matrix, pf1t1t2 = 0.5 since bf1t1 = null and Cf1

t2 = 1, N f1t2 = 2. Let Pi be a

matrix which contains the pijk and let S be a matrix which contains the sjk.

The total expected setup distance matrix is given by S =∑

i∈F Pi.

The method for computing pijk is given by the following algorithm.

Step 1: Set i ∈ F , j ∈ T , k ∈ T

Step 2: if (j = k) OR (bij = bik) OR (bik = null) then pijk = 0 else

goto step 3

Step 3: if (bij �= bik) then pijk = 1 else goto step 4

Step 4: if (bij = null) then pijk =

Cik

N ik

Steps 2-4 are mutually exclusive and cover all possibilities for determining

pijk. In step 2,(j = k) indicates that the same board type follows j (i. e. the

diagonal entries in Pi are zeroes). In step 2, (bij = bik) indicates that the

next setup configuration is the same as the previous setup configuration.

Finally, in step 2 (bik = null) indicates that the board type that follows

the current board type does not require a setup configuration on feeder i.

In each of these cases, a setup change is not required so that pijk = 0. In

step 3, (bij �= bik) and the next setup configuration is different than the

previous configuration. Thus, pijk equals one. In step 4, we are moving from

a board type that did not require a setup configuration on the feeder. Thus,

we can not be certain which board type was before it in the sequence. Our

heuristic takes a probabilistic approach and determines the likelihood of a

setup change. The likelihood is expressed as a ratio of the number of board

types that require a different setup configuration than the next board type

k and the total number of board types that are different than k. This is also

where we use our assumption that any board type is equally likely to appear

in any position in the sequence. This assumption seems reasonable given no

a priori knowledge about the ordering of the board types in the sequence.

14

Applying the algorithm to the example FBS matrix yields the following

expected setup matrix:

Boardst1 t2 t3 t4

Boards

t1t2t3t4

0 1.5 1.5 12 0 1 2

1.5 1 0 1.51 2 2 0

Since the matrix is not symmetric, we take the maximum of sjk and skj

as the final expected setup distance between the two boards. Let D be the

matrix which contains the final expected setup distance between two board

types with elements djk = max(sjk, skj). Even though the problem can be

handled with a non-symmetric matrix, this heuristic simplification allows well

established grouping methods and sequencing methods to be easily applied.

We note that if the board sequences were reversed in the actual production

sequence then only a benefit would occur since we are taking the maximum

distance between the boards.

Before the sequencing can be done the boards must be clustered into

groups based on criteria used to measure the similarity between boards. The

formulation for the clustering problem is:

minK∑

k=1

N−1∑i=1

N∑j=i+1

dijyikyjk

subject to:K∑

k=1

yik = 1, ∀i

yik = 0, 1 ∀i, k

where

• N = the number of boards types

• K = the number of groups

15

• dij = the distance from board i to board j, either Hamming or expected

setup distance

• yik = 1 if board i is in group k and 0 otherwise

The general clustering problem is discussed in Ignizio and Cavalier (1994)

or Everitt (1993). The model attempts to separate the board types into

groups such that board types with a minimal distance between them are

placed within the same groups.

4.3 Application To The Assembly System

The mathematical formulation presented in Section 4.1 assumes a single ma-

chine scheduling problem. There are a variety of heuristic approaches to

developing solutions to this type of problem. In general, the problem is

known to be NP-hard, see for example Appendix B of Ahuja et al. (1993),

since one can show an equivalence to the travelling salesman problem. In our

case study, the number of different board types including bottom-side place-

ments was eleven. Demand projections for the boards indicated that in the

future twice as many new board types are likely to be introduced with the

majority to take advantage of top and bottom side placements. Given that

the problem is NP-hard and that we are primarily concerned with a practical

implementation, we selected to approach the problem using a nearest neigh-

bor heuristic. We felt that the nearest neighbor heuristic used in conjunction

with groupings would lessen the likelihood of the large jumps which can occur

at the end of sequence construction using the nearest neighbor heuristic.

The clustering model was solved for the data available in our system

using both the Hamming distance and the expected setup distance as dij

for the number of groups K = 1, . . . , 6. For both distance measures, the

most effective number of groups was four; however, the actual groupings

were significantly different from each other.

16

The above clustering model was solved using different values for the num-

ber of groups. Figures 2 and 3 present the results for the objective function

for the clustering results for the Hamming distance measure and the expected

setup distance measure. From the figure the separation of the boards into

four groups is very effective. Not much is gained for group sizes greater than

four. In addition, for more than five groups, boards were placed into groups

of size one which defeats the purpose of grouping. Thus, we decided to set the

number of groups to be four. A complete listing of the groups are available

from the authors upon request.

Once we have decided upon using groups and a nearest neighbor approach,

our solution methodology is straight forward:

Step 1: Establish effective groupings

Step 2: Choose an available board type, t, according to nearest neigh-

bor from any available group.

Step 3: Sequence the rest of the boards types from t’s group

Step 4: Go to step 2 until all groups have been sequenced.

The first board chosen should be a top-side board since one can not sequence

a bottom-side before its corresponding top-side. An available group to se-

quence from is determined by considering that an unavailable group is one

that

• contains bottom-side boards and not the respective top-sides or

• contains bottom-side boards whose corresponding top-side boards have

not been previously sequenced as part of another group.

These two conditions ensure that bottom-side boards are sequenced after

their corresponding top-side boards. The criteria for selecting the nearest

neighbor can be either the Hamming distance or the estimated expected

17

setup distance. Both the Hamming distance and the expected setup distance

can be modified to include λj × cj.

The above methodology allows a solution to our assembly system problem

to be developed. Given the above methodology, the question remains as to

how well the estimated expected setup distance criteria compares to the

Hamming distance criteria in terms of efficiencies gained from the generated

groups and whether or not demand affects the results.

5 Evaluation of Methodology

In this section, we discuss the analysis and evaluation of the performance

of the proposed heuristics in the context of the operation of the PCB flexi-

ble assembly system. Our motivation here is not to perform an exhaustive

comparison of these heuristics. Rather, we are interested in how the sched-

ules generated by our methodology for this case study perform under realistic

operating conditions. In order to perform this analysis, we developed a simu-

lation model of the system and ran the model using the sequences produced

by our methodology. We will first briefly describe the development of the

simulation model, then we will cover our experimental designs, and finally

we discuss the results of the experiments.

5.1 Simulation Model Development

The PCB flexible assembly system shown in Figure 1 is a new line designed to

handle non-standard size boards with demand for boards to be increased in

the near future. The demand for boards should increase in both the number

of different board types and the quantity of boards. Only twenty six weeks

of board demand data was available for analysis. Because of the new nature

of the line and because increasing levels of future demand were expected, we

used weekly demands conditioned on the demand for a board being greater

than zero. In other words, during each week each board type must have de-

18

mand. Demand distributions were fit for each board type. The distributions

are available from the authors upon request. Individual time studies were

taken of the various work stations along the lines and service time distri-

butions were fit. The service time of the ovens and the material handling

flipping station were assumed to be deterministic. The service time of the

glue machine was assumed to have a deterministic linear relationship with

respect to the number of components to be placed on the boards. The spe-

cific distributions are available from the authors upon request. Setup time

collection was more difficult due to the uncertainty of when setups would

take place. Repeated interviews with manufacturing engineering personnel

and shop personnel performing the setup changes indicated that the setup

time per setup change was Normal (4.5,1) minutes/change. A setup change

is defined as changing a feeder setup for the next board in the production

sequence. These setups are completed one at a time and are not done simul-

taneously.

Since we are primarily concerned with the performance of a sequence for

production, the simulation was of the finite horizon (terminating) type. A

weeks demand for circuit boards was simulated through the assembly system

with the simulation terminating when the entire week’s demand is produced.

Upon termination, performance statistics on makespan, utilization, and WIP

were recorded. Each simulated week represents a replication. For each repli-

cation the line starts in the empty and idle condition with all placement

machines in the common setup state. The model was validated by observing

actual makespan times and comparing to simulated performance. The details

of the model and the code are available from the authors.

5.2 Experimental Design

We are primarily interested in exploring the following questions:

• What affect does sequencing with respect to groupings have on the

performance of the sequencing heuristics?

19

• Does the expected setup distance metric perform as well as the Ham-

ming distance metric when used as the criteria for grouping and se-

quencing?

• How does randomly changing demand affect the performance of heuris-

tics that do not take demand into account?

• How do the sequences compare with respect to performance measures

other than makespan such as WIP and utilization?

The factors of interest were thus the grouping method, distance measures,

and the demand dependency. The grouping method defines how the boards

are clustered into families. The levels were no groups (NG), current groups

(CG), and clustered groups (CLG). The boards are first separated into types

such that top-sides and bottom-sides of the same board are considered to be

different board types. No groups means that the board types are not placed

into groups. The current grouping (CG) refers to board type’s with top-side

and bottom-side placements being placed within the same group. In other

words, if we consider board types with top and bottom side placements as

two different board types then the current grouping has them together in

the same group. Clustered groups refers to groups which are formed by

the clustering algorithm when considering boards types with both top and

bottom side placements as separate board types.

The distance measure is defined as the method used to approximate the

sequence independent setup changes required from board type to board type.

The levels were Hamming distance (HD) and expected setups (ES). When

approximating the sequence independent setup changes, we may or may not

include information concerning the demand as described in Section4.1. The

demand dependency indicates whether or not the sequence to be generated

for a given demand takes the demand into account or just the fixed setup

estimates. That is, whether or not demand through the form of λj × cj is

included when the grouping is performed. Each week will have a different

20

amount of demand for each of the board types. Because of this, if the se-

quence is demand independent (DI) then the sequence will be fixed for all

distances and groupings for each simulated week; however, if the sequence

is demand dependent (DD) then the sequence will be constructed based on

a sequence dependent processing time and may vary from week to week in

the simulation experiments. The sequence dependent processing time is the

estimated setup time plus the expected processing time for a lot.

In addition, the heuristic sequences will be compared to sequences which

are optimal (OS) with respect to minimizing the expected sequence makespan

based upon the estimation method for the sequence dependent setup, weekly

demand, and grouping method. In other words, given the estimated se-

quence dependent setups and resulting groups, we can optimally determine

the sequence with the formulation presented in Section4.1. For the case of

no groups (NG), the generated (OS) sequences only depend upon distance

measure used (HD) or (ES) and the demand dependency (DD) or (DI). The

optimal sequences were generated by a complete enumeration of all possi-

ble sequences for each week of demand. Table 1 illustrates the experimental

design and indicates for each factor/level combination whether or not the

sequences are fixed (FS) or variable (VS). If the sequencing is demand de-

pendent, the sequence is allowed to vary since the sequence dependent setup

time will become a sequence dependent processing time which is made up

of the expected processing time of the weekly demand and the estimated

sequence dependent setup time.

Starting from a common feeder setup, each simulation experiment pro-

ceeds as follows. First, circuit board demands are generated for each board

type. A sequence is constructed using a sequencing method which depends

upon the levels of the various experimental factors. The sequence is simu-

lated through the PCB flexible assembly system. The lot’s makespan times

through the stages of the system are collected as well as the average number

of lots in the system and the utilization of work stations along the manual

21

line. Because each simulation starts from a common feeder setup, the ac-

tual number of setup changes can and was also recorded for each generated

sequence.

5.3 Statistical Analysis of Results

In order to analyze the performance of the heuristics, we utilized a two-staged

sequential ranking and selection procedure described in Dudewicz and Dalal

(1975) and Law and Kelton (1993). The ranking and selection procedure

requires a probability of correct selection parameter, P ∗, and a specification

of an indifference amount, d∗. The procedure ensures that with a probability

of at least P ∗, the expected performance of the selected heuristic will be less

than the true expected performance plus the indifference amount. For our

experiments, P ∗ = 0.95 and d∗ = 120 minutes. In other words, we say that

we are indifferent if the differences in makespan for the competing heuristics

are less than 120 minutes apart. An initial 40 replications were chosen in

order to obtain the first stage sample variances for the makespans of the

sequencing heuristics.

The ranking and selection procedure assumes that the individual re-

sponses for each replication are normally distributed. The procedure is known

to be robust to departures in normality assumptions. Examination of the

data indicated only minor departures from this assumption. Based on the

sample variances, the number of second stage experiments were determined.

The procedure computes weighted means based on the first and second stage

results. We let X(1) and X(2) be the mean from the first and second stages

respectively. The weighted mean is given as X = W (1)X(1) + (1−W (1))X(2)

where the W (1) is a complicated function of the number of samples in each

stage, the indifference amount, and a constant that depends on the number of

comparisons being made as well as the probability of correct selection. Law

and Kelton (1993) presents the computational formulas and shows why the

weights take their form to ensure the correct selection probability. Table 2

22

gives the final weighted means and the rankings of the heuristics in ()’s in

terms of makespan. In the table, lower values are preferred and n represents

the total number of replications. The significant conclusion to be drawn is

that sequences obtained from the expected setup heuristic out perform the

sequences obtained from the Hamming setup heuristic. In addition, the ex-

pected setup heuristic with clustered groups was the best across all demand

characteristics.

We found that generating sequences that are demand dependent, i. e.

incorporating demand by forming a demand dependent processing time did

not improve the performance of the heuristics with respect to the optimal

sequence. Table 3 displays the performance for the first 40 replications with a

95% confidence interval indicated. Only the first 40 replications were used to

allow the use of the paired-t confidence interval. The values in the table are

estimates for the expected difference between the heuristic’s makespan and

the optimal sequence (OS) case’s makespan. Thus, the results are relative

to the (OS) case. Lower values are preferred to larger ones. Sequencing

independent of demand dominated the performance of sequencing dependent

on demand in all cases except when the sequencing was done with respect to

clustered groups. Clustered groups performed approximately the same across

the demand factor primarily because the groupings constrained the variety of

sequences which could be generated and thus the actual sequences generated

were similar across this factor. For our sequence dependent environment,

minimizing makespan depends much more critically on minimizing setup time

than it does on incorporating demand into the processing times. Results for

other systems can be different if processing times are large in comparison to

setup times.

Based upon the makespan ranking results and the performance of the

heuristics with respect to the optimal makespans, it is clear that fixed se-

quences (demand independent) had the best performance. For our system,

makespan is the more important performance measure; however, we were also

23

interested in how the sequences performed in terms of work in process (WIP)

and utilization of the manual line. Table 4 presents the mean WIP results.

The results indicate that the current grouping method achieved the lowest

WIP levels. WIP is defined as the number of lots in the system, where each

lot consists of a batch of a given board type. An interesting result is that se-

quencing using the expected setups and no grouping in a demand dependent

environment achieved very low levels of WIP. This was due to the fact that

the sequences generated by this method are very similar to the sequences

produced by the current grouping method.

In the following tables and figures, we show how the WIP and utilization

performed when compared to sequencing with the current groupings (CG)

using the Hamming distance (HD) in the nearest neighbor heuristic. This

represents a baseline for comparison since this is essentially the status quo

alternative. Figure 4 presents box plots for the WIP difference results. For

example, D1 = CGES − CGHD is the difference between the WIP under

current groupings when sequencing using the expected setup distance criteria

and the WIP under current groupings when sequencing using the Hamming

distance criteria. In the figure, lower values are consider better. We see

that the WIP level is essentially the same if the current groupings are used

with expected setups as the distance measure. The case of no groupings

performed somewhat in the middle and the clustered groupings with both

Hamming and expected setups as distance criteria performed the worst.

Table 5 indicates a 95% confidence interval for the differences in WIP.

Using the clustered groups based on the expected setup distance measure

increased the WIP level in the system on average by 1.456 lots. Some increase

in WIP should be expected since the procedures are minimizing makespan.

It is unknown why clustered grouping increased the WIP more than the other

methods. Figure 5 presents the box plots for the utilization difference results.

Table 5 presents a 95% confidence interval for the differences in utilization.

In this case, higher values indicate improved utilization. From the table and

24

figure, we see a slight trend in improved utilization, but the differences are

not statistically significant at the current sample size.

Next, we present results that illustrate the performance of the sequencing

methods with respect to the total number of actual feeder changes that were

experienced within each replication of the simulation experiments. As indi-

cated in Table 1, the sequences may or may not vary for each experiment.

This is because the sequences can be dependent upon the demand due to

the calculation of the sequence dependent processing times. If the sequences

vary, then the total number of actual feeder changes experiences will also

vary for each replication of the experiments. The sequences were generated

based on the estimated sequence dependent setups. Once the sequence is

generated, the actual number of setups, or feeder changes, that are required

to produce the sequence can then be calculated. Table 6 tabulates the statis-

tics for the required number of setups and where appropriate, including the

average (x), the standard deviation (s), the minimum, the median (x), and

the maximum.

Sequencing dependent on demand (redefining the sequence dependent

setup time as the sequence dependent processing time) resulted in an overall

higher number of setups to be required. This is expected, since the nearest

neighbor search is allowed to take a different path for each week’s demand.

The processing time which is added to the setup time may mask the nearest

neighbor’s setup time, which results in a neighbor being chosen that may

have a higher setup than another neighbor.

The optimal sequence case, minimizing the expected makespan, using the

expected distances as the estimates for the sequence dependent setups, gener-

ated sequences that required fewer setups than did those using the Hamming

distance. Using the common feeder setup as a starting point, we also enu-

merated all possible sequences to find that the sequence with the smallest

number of setups had 124. As indicated, the sequences using the expected

distance are within 2 percent of the optimal, 128 versus 124, for either no

25

groupings or clustered groupings. The optimal sequences, using the hamming

distance and no groupings required a average of 154 setup changes. When

the clustered groupings are used with the hamming distance, the number of

required setup changes was reduced. The data also indicates that for both

the demand independent sequences and the optimal sequences the expected

distance heuristic produced sequences that required fewer setups.

6 Summary

In this paper, we examined a realistic PCB assembly system in order to

develop a solution to efficiently sequencing the line in terms of reducing

makespan while maintaining adequate WIP and utilization levels. The prob-

lem is complicated by the fact that the setup of the line depends not only

on the current board type, but on all previously sequenced board types.

Our solution consisted of modeling the assembly system as a single machine

sequencing problem and estimating sequence independent setup costs. In ad-

dition, we grouped similar board types together to reduce the problem size

and simplify the sequencing procedure.

Our method of estimating sequence independent setup costs is based on

estimating the expected number of setups given component feeder and board

types. Our expected setup distance measure can easily be incorporated into

board clustering algorithms to capture the similarities between boards in

terms of required setups. In addition, we showed that the expected setup

distance measure can be incorporated into group sequencing heuristics(e. g.

nearest neighbor) to produce sequences. For our case study, the sequences

generated based on the expected setup distance measure performed signifi-

cantly better than sequences which utilize the Hamming distance in terms of

makespan. We showed that the expected setup distance produced sequences

that required fewer setups and that are closer to the optimal when compared

to the Hamming distance for our problem. This would imply that it predicted

the actual number of setups better.

26

In terms of WIP, no groupings (NG) performed slightly worse than the

current groupings, and clustered groups performs the worst in terms of WIP.

In terms of utilization, no groups and clustered groups are perhaps slightly

better than the current groups. Because of the significant improvement in

makespan, we would recommend the use of clustered groups and group se-

quencing based on the expected setup distance measure. For this system, if

WIP and utilization become a concern then we would recommend no groups,

demand independent, using expected setups as a viable alternative since it

was ranked third in Table 2 but also performed well in terms of WIP and

utilization.

The development of the expected setup distance measure opens up many

possible avenues for future research. First and foremost would be the detailed

examination of this heuristic as compared with other heuristics on randomly

generated problem instances. Other approaches such as genetic algorithms

or tabu search would be natural methods for addressing this problem and

do not rely on grouping. We would suggest examining the effect of the total

number of board types on both the clustering algorithms and the resulting

groups. In addition, the sensitivity of the sequencing heuristics to increases in

the number of board types should be performed. In developing the expected

setup matrix, we ensured symmetry by taking djk = max(sjk, skj). The effect

of a non-symmetric matrix should be explored as well as other operators to

ensure symmetry such as taking djk = avg(sjk, skj). Another method would

be to predict the expected setups, sequence to find the actual setups, readjust

the estimated setups, and then resequence as suggested in Prabhakar (1974).

Finally, it would be interesting to examine the effectiveness of this method

with no groupings at all as compared to the optimal non-grouping sequences.

We incorporated the expected setup distance measure into the nearest

neighbor heuristic. Many other heuristics have been developed, see for ex-

ample Bianco et. al. (1994) and Wemmerlov and Vakharia (1991). The per-

formance of these heuristics using the expected setup distance measure should

27

be examined. For example, the results of Gupta (1993) on reentrant flow-

shops and Das et. al. (1995) on sequence dependent setup times should be

explored in combination with our expected setup cost heuristic. Finally, the

effect of using expected setups could be tested on other performance criteria

such as due date tightness.

Acknowledgments

We would like to thank Jay Endahl and COMDIAL Communications of Char-

lottesville Virginia for access to their manufacturing facilities and for valued

assistance during this research.

Appendix

In this appendix, we present an example calculation of the expected setup

matrix. Let S, F , T , BS, FS, and FBS be defined as in Section 4.2. In

the following example suppose that there are 3 feeders, 4 board-types, and 6

setup configurations as given in Table 7.

From this example, we see that board type t1 uses setup configurations

s4 and s5. Feeder f1 has both setup configurations s1 and s2 assigned to it.

In forming FBS, F × T forms a relation with all possible combinations of

elements of F with elements in T . This results in a relation with 12 rows. For

this example, FS JOIN BS results in a relation with 8 rows, those board type

and feeder combinations with matching setup configurations. The resulting

FBS relation is given in Table 8. FBS can be rearranged into a matrix

form.

Boardst1 t2 t3 t4

Feedersf1

f2

f3

null s1 s1 s2

s4 null s3 nulls5 s6 null s5

28

Let Cik be the number of board types that require a different setup configura-

tion than board-type k’s setup configuration and require a setup configura-

tion on feeder i. Let N ik be the total number of board-types other than type

k that require a setup on feeder i. To calculate N ik, we need to count the

board types that are different than board type k and which have a non-null

setup configuration.

Step 1: Set i ∈ F

Step 2: Set k ∈ T

Step 3: for each j ∈ T where j �= k and bij �= null, add 1 to N ik

To calculate Cik, we need to count the board types that have setup configu-

rations that are different than board type k’s setup configuration and that

are non-null.

Step 1: Set i ∈ F

Step 2: Set k ∈ T

Step 3: for each j ∈ T where bij �= null and bij �= bik, add 1 to Cik

N ik can also be determined with a parameterized query on the FSB relation:

SELECT COUNT(t)

FROM FSB

WHERE ((f = i) AND (t �= k) AND (s �= null));

Cik can also be determined with a parameterized query on the FSB relation:

SELECT COUNT(t)

FROM FSB

WHERE ((f = i) AND (s �= null) AND

29

( s NOT IN (SELECT s FROM BS WHERE t = k))));

Using these algorithms, we can determine that N f1t2 = 2 since board types

t3 and t4 require setup configurations s1 and s2, and board type t1 does not

require a setup configuration on f1. Also, Cf1t2 = 1 since of the other board

types that require a setup configuration on feeder f1 only board type t4 has

a different setup configuration than board type t2. Tabulating the remaining

Cik and N i

k we have:

BoardsCi

k t1 t2 t3 t4

Feedersf1

f2

f3

3 1 1 21 2 1 21 2 3 1

BoardsN i

k t1 t2 t3 t4

Feedersf1

f2

f3

3 2 2 21 2 1 22 2 3 2

To calculate the pijk, we apply the algorithm given in Section 4.2. The re-

sulting matrices are:

P f1 =

0 0.5 0.5 10 0 0 10 0 0 10 1 1 0

P f2 =

0 0 1 01 0 1 01 0 0 01 0 1 0

P f3 =

0 1 0 01 0 0 1

0.5 1 0 0.50 1 0 0

30

Consider the first row of P f1 . Clearly, pf1t1t1 = 0 since we are transitioning to

the same board type. For the other elements of the row, we are transitioning

from t1 which does not have a required setup configuration. Thus, we must

invoke step 4 of the algorithm:

pf1t1t2 =

Cf1t2

Nf1t2

= 12

pf1t1t3 =

Cf1t3

Nf1t3

= 12

pf1t1t4 =

Cf1t4

Nf1t4

= 22

= 1

The summation of the P i matrices will yield the expected setup matrix given

in Section 4.2.

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33

Figure 1: PCB Assembly System Layout

Clustering ResultsHamming Distance

0

200

400

600

800

1000

1200

1400

1600

1 2 3 4 5 6

Groups

Se

t U

p C

ha

ng

es

Figure 2: Clustering Results For Hamming Distance

34

Clustering ResultsExpected Setups

0

100

200

300

400

500

600

700

1 2 3 4 5 6

Groups

To

tal

E[S

etu

ps

]

Figure 3: Clustering Results For Expected Setups

-0 .8

-0 .4

0.0

0.4

0.8

1.2

1.6

2.0

D1 D2 D3 D4 D5

D1 = CGES - CGHD

D2 = CLGES - CGHD

D3 = CLGHD - CGHD

D4 = NGES - CGHD

D5 = NGHD - CGHD

Figure 4: WIP Differences Box Plots

35

-0 .035

-0 .025

-0 .015

-0 .005

0.005

0.015

0.025

0.035

0.045

0.055

0.065

0.075

D1 D2 D3 D4 D5

D1 = CGES - CGHD

D2 = CLGES - CGHD

D3 = CLGHD - CGHD

D4 = NGES - CGHD

D5 = NGHD - CGHD

Figure 5: Utilization Differences Box Plots

Table 1: Experimental Design Table

Distance Grouping Demand Demand OptimalMeasure Method Independent Dependent Sequence

Hamming None FS VS VSCurrent FS VS VS

Setups Clustered FS VS VS

Expected None FS VS VSCurrent FS VS VS

Setups Clustered FS VS VS

36

Table 2: Final Weighted Makespans and Rankings

Distance Grouping Demand Demand OptimalMeasure Method Independent Dependent Sequence

Hamming None 2015.8 (2) 2100.4 (3) 1976.4 (4)n = 41 n = 41 n = 41

Current 2315.1 (6) 2376.0 (5) 2272.9 (6)n = 41 n = 41 n = 41

Setups Clustered 2066.8 (4) 2066.1 (2) 1944.9 (3)n = 41 n = 41 n = 41

Expected None 2029.3 (3) 2478.0 (6) 1929.2 (2)n = 74 n = 74 n = 101

Current 2227.2 (5) 2369.9 (4) 2160.3 (5)n = 41 n = 41 n = 41

Setups Clustered 1955.9 (1) 1980.1 (1) 1836.4 (1)n = 42 n = 46 n = 45

Table 3: Makespan Differences Relative To (OS) Case

Distance Grouping Demand DemandMeasure Method Independent Dependent

Hamming None 39.48 ± 8.44 125.44 ± 13.46Current 55.29 ± 11.0 114.56 ± 9.56

Setups Clustered 121.27 ± 20.62 120.86 ± 16.6

Expected None 68.72 ± 22.64 514.89 ± 23.4Current 56.16 ± 15.1 211.16 ± 9.6

Setups Clustered 113.14 ± 13.4 114.67 ± 15.54

37

Table 4: Mean Work In Process

Distance Grouping Demand Demand OptimalMeasure Method Independent Dependent Sequence

Hamming None 1.044 ± 0.04 0.669 ± 0.06 2.042 ± 0.06n = 100 n = 100 n = 100

Current 0.349 ± 0.04 0.254 ± 0.04 0.304 ± 0.04n = 100 n = 100 n = 100

Setups Clustered 1.347 ± 0.05 1.44 ± 0.05 2.239 ± 0.14n = 100 n = 100 n = 26

Expected None 1.077 ± 0.04 0.282 ± 0.04 2.24 ± 0.05n = 100 n = 100 n = 100

Current 0.337 ± 0.04 0.2503 ± 0.04 0.305 ± 0.04n = 100 n = 100 n = 100

Setups Clustered 1.814 ± 0.04 1.392 ± 0.04 2.329 ± 0.04n = 100 n = 100 n = 90

Table 5: WIP and Utilization Comparisons

Sample size n = 100

WIP UtilizationCGES - CGHD −0.012 ± 0.036 0.00655 ± 3.656 × 10−3

CLGES - CGHD 1.465 ± 0.036 0.01992 ± 3.736 × 10−3

CLGHD - CGHD 0.998 ± 0.052 0.02224 ± 3.943 × 10−3

NGES - CGHD 0.728 ± 0.043 0.02946 ± 4.121 × 10−3

NGES - CGHD 0.695 ± 0.045 0.01779 ± 4.102 × 10−3

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Table 6: Number of Required Setups

Distance Grouping Demand Demand OptimalMeasure Method Independent Dependent Sequence

Hamming None x = 169 x = 197.5 x = 155.9s = 0 s = 6.1 s = 1.4

min = 169 min = 183 min = 153x = 169 x = 200 x = 156

max = 169 max = 202 max = 157Current x = 247 x = 252.7 x = 230.7

s = 0 s = 1.2 s = 10.1min = 247 min = 250 min = 222x = 247 x = 253 x = 226

max = 247 max = 255 max = 257Setups Clustered x = 168 x = 167.5 x = 139.9

s = 0 s = 1.8 s = 14.5min = 168 min = 163 min = 124x = 168 x = 168 x = 153

max = 168 max = 170 max = 154

Expected None x = 143 x = 247.3 x = 128s = 0 s = 16.6 s = 0

min = 143 min = 196 min = 128x = 143 x = 253 x = 128

max = 143 max = 255 max = 128Current x = 227 x = 253.2 x = 226

s = 0 s = 1.2 s = 0min = 227 min = 250 min = 226x = 227 x = 253 x = 226

max = 227 max = 255 max = 226Setups Clustered x = 151 x = 168.8 x = 128

s = 0 s = 5.7 s = 0min = 151 min = 140 min = 128x = 151 x = 167 x = 128

max = 151 max = 180 max = 128

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Table 7: Example Relations

S F T BS FSs1 f1 t1 t1 s4 f1 s1

s2 f2 t2 t1 s5 f1 s2

s3 f3 t3 t2 s1 f2 s3

s4 t4 t2 s6 f2 s4

s5 t3 s1 f3 s5

s6 t3 s3 f3 s6

t4 s2

t4 s5

Table 8: Example FBS

f t sf1 t1 nullf1 t2 s1

f1 t3 s1

f1 t4 s2

f2 t1 s4

f2 t2 nullf2 t3 s3

f2 t4 nullf3 t1 s5

f3 t2 s6

f3 t3 nullf3 t4 s5

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