annrovp - tdl
TRANSCRIPT
ANALYSIS OF THE PARTS/MACHINES GROUPING
PROBLEM IN GROUP TECHNOLOGY
MANUFACTURING SYSTEMS
by
OLIVER EKEPRE CHARLES, B.S., M.S. in I.E.
A DISSERTATION
IN
INDUSTRIAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
IN
INDUSTRIAL ENGINEERING
Annrovp.d
May, 1981
m '\:S
ACKNOWLEDGMENTS
I wish to express my appreciation to my advisor.
Dr. Brian K. Lambert, for his guidance and help through
out the preparation of this dissertation; I am also
grateful to the other members of my committee, Drs. Lee
Alley, James R. Burns, Richard A. Dudek and Milton L.
Smith for their valuable suggestions. My thanks also
go to the other members of the faculty of the Industrial
Engineering Department whose technical assistance has
made this research possible.
I am indebted to my parents. Chief Charles Owaba,
Mrs. Animi Owaba, my uncle, Mr. Claudius Abere, who have
provided me with the financial support throughout this
research. I am also grateful to Mrs. Sue Haynes who
performed an excellent work in typing this material.
Finally, this work is dedicated to the memory of
my late brother, Olaud A. Charles, who passed away in
the year I was engaged in this research.
11
TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
NOTATION xi
I. INTRODUCTION 1
1.1 The Batchshop Problem and Group Technology 2
1.2 The Grouping Problem 3
1.2.1 Cluster Analysis and GT Problem 4
1.2.2 Group Technology Types 5 1.2.3 Summary of Research
Problem 8
1.3 Review of Past Studies 9 1.4 Critique of Previous Work and
Study Objectives 19 1.5 Outline of Succeeding Chapters 22
II. GROUPING CRITERIA 23
2.1 Introduction 23 2.2 Production Cost of GT 24 2.3 Objective Function in the PCA 25
2.3.1 Set-up Time-Similarity
of Parts Model 26
2.4 Grouping Objective in the PFA 49
III. CHARACTERIZATION OF GT PROBLEMS AND OPTIMALITY CONDITIONS 57 3.1 Introduction 57
111
3.2 Relation Between Similarity of Parts in the Same Groups, H, and Similarity of Parts in Different Groups, L 58
3.3 Characterization of GT Problem Data 67
3.4 Optimality Conditions 73
IV. GROUPING ALGORITHM 76
4.1 Introduction 76 4.2 Heuristic for Initial Solution 77 4.3 Theoretical Background of
Algorithm 88
4.3.1 Grouping Procedure 95 4.3.2 Evaluation of H, L 103 4.3.3 Details of Grouping
Algorithm: The Gradient Technique 106
4.3.4 Computational Experience .... 115
V. NUMBER OF GROUPS AND PRODUCTION COST 122
5.1 Introduction 122 5.2 Study Procedure 122 5.3 Test Problems 123 5.4 Discussion and Presentation
of Results 127 5.5 Additional Machine Cost and
Number of Groups 142 5.6 Set-Up Cost and Number of
Groups 14j 5.7 Production Cost and Number of
Groups 145 5.8 General Observations 147 5.9 Possible Applications 148
VI. CONCLUSIONS 150
6 .1 Summary of Research 150 6.2 Conclusions 150 6.3 Recommendations for Further
Study 154 LIST OF REFERENCES 157
IV
APPENDIX A: LISTING OF FORTRAN IV PROGRAM OF THE GRADIENT ALGORITHM 161
APPENDIX B: OPITZ PART CLASSIFICATION SYSTEM ... 175
APPENDIX C: TABLES OF GROUP HOMOGENEITY (H), NUMBER OF ADDITIONAL MACHINES (A) AGAINST NUMBER OF GROUPS 179
LIST OF TABLES
Table Page
4.1 FLOWCHART DETAIL 106
4.2 SOLUTION VALUES AND ALGORITHM TIME OF THE GRADIENT TECHNIQUE 117
4.3 ALG0RITHr4 TIME VERSUS NUMBER OF PARTS AND GROUPS 119
5.1 SUMMARY OF TEST PROBLEMS 126
VI
LIST OF FIGURES
F^g^re Page
1 Disk-Like Part 12
1.1 Parts Characteristics Matrix (PCM) showing only design and processing features. Opitz coding system is used 14
1.2 Parts Characteristics Matrix (PCM) showing only machines indicated in process route 14
2.1 Set-up task-characteristic Binary Interaction Matrix 32
2.2 Grouping solution resulting from minimizing A 54
2.3 Grouping solution resulting from minimizing L 55
3.1(a) Parts for Example 4.1 60
3.1(b) The parts in Figure 4.1(a) shown in Opitz' code number system 61
3.1(c) Similarity coefficients of the
parts shown in Figure 4.1(a) 61
3. 2 Multiple Population Problem 68
3.3 Universal Population Problem 70
3.4 Natural Population Problem: 0<H<1, 0<L<1 70
3.5 Null-Relation Population Problem: H = 0 , L = 0 70
4.1 Flowchart of the Preference Index Heuristic 81
4.2 Illustration of step operation 91
vi:.
F^g^re Page
4.3 Flowchart of the Gradient Technique 107
4.4 Solution time versus number of parts 120
4.5 Algorithm time versus number of parts and groups 121
5.1 Group homogeneity versus niomber of groups ;n=60,J^ = .6 128
5.2 Group homogeneity versus niomber of groups ; n = 6 0 , f i = .3 128
5.3 Group homogeneity versus number of groups ; n = 8 0 , Q = .6 129
5.4 Group homogeneity versus number of groups; n = 80, = .3 129
5.5 Group homogeneity versus number of groups; n = 100, Q = .6 130
5.6 Group homogeneity versus number of groups; n = 100, ^ = .3 130
5.7 Group homogeneity versus number of groups; n = 120 ,^ = .6 131
5.8 Group homogeneity versus number of groups; n = 120,^ = .3 131
5.9 Group homogeneity versus number of groups; n = 86,Q = .53 132
5.10 Number of additional machines versus number of groups; n = 60, number of each machine tye = 1, PFA 134
5.11 Nianbe- of additional machines versus number of groups; n = 80, number of each machine type = 1, PFA 134
VXIL
liSHEl Page
5.12 Number of additional machines versus number of groups; n = 100, number of each machine type = 1, PFA 135
5.13 Number of additional machines versus number of groups; n = 120, number of each machine type = 1, PFA 135
5.14 Number of additional machines versus number of groups (Bur-bidge's); n = 43, Q = 1.0, number of each machine type = 1, PFA 136
5.15 Number of additional machines versus number of groups (Pur-check's); n = 82, a ^ .45, number of each machine type = 1 136
5.16 Nijmber of additional machines number of groups; n = 60, Q = .3, number of each machine type; uniformly distributed (1, 3) 137
5.17 Nimiber of additional machines versus number of groups; n = 60, Q = .6; number of each machine type: uniformly distributed (1, 3) 137
5.18 Number of additional machines versus niomber of groups n = 80, Q = .3, uniformly distributed (1, 3) number of each machine type 138
5.19 Nimiber of additional machines versus number of groups; n = 80, ^ = .6; uniformly distributed (1, 3) ntimber of each machine type 138
5.20 Number of additional machines versus number of groups; n = 100, Q - .3; uniformly distributed (1, 3) number of each machine type 139
5 . 21 Nijmber of additional machines versus number of groups; n = 100, ^ =.6; uniformly distributed (1, 3) number of each machine type 139
ix
Figure Page
5.22 Number of additional machines versus number of groups; n = 120, U - .3; uniformly distributed (1, 3) number of each machine type 140
5.23 Number of additional machines versus number of groups; n = 120, Q - .3, uniformly distributed (1, 3) number of each machine type 140
5.24 An example of possible relation between set-up cost, additional machine cost, production cost and number of groups; n = 100, Q = .3, number of each machine type = 1, GT approach = PCA 144
X
NOTATION
GT
G
n
N
M
PCA
PFA
V(G)
B. 1
"15 r. 1
A
H
L
T
E
Group Technology
Grouping, Combination of Parts or Partition
Total Number of Farts
Ntjmber of Groups
Nisnber of Machine Type
Nixmber of Parts in Group k
Parts Classification Approach
Production Flow Analysis Approach
Objective Functions Defined in Terms of G
Similarity Coefficient Between Parts i and j
Set of Attributes Possessed by Part i
Ntanber of Objects in a Set
Dissimilarity Coefficient
Process Route for Part i
Production Cost
Set-Up Cost
Cost for Additional Machines
Number of Additional Machines
Group Homogeneity
Link Between Groups
Set-Up Time for Task k
Set of Set-Up Time Elements
Set of Set-Up Task States
XI
^k
^ka
^ijk
ijk
D
V
u
W
d m
mk
T(n,N):
X
P
State of Set-Up Task k
Binary State Expressing the Possible Effect of Characteristic a on Set-Up Task k
Set-Up Task - Characteristic Interaction Matrix
State of Similarity
State of Dissimilarity
Dissimilarity State Vector
Nimiber of Set-Up Tasks
Number of Characteristics of Part in the PCA
Set of Weight of Characteristics of Parts
Relative importance (weight) of the k Characteristic
Difference Between Ntimber of Additional Machines of Type m and Those in the Conventional System
State Specifying Whether Machine Type m Occurs in Group k
Step Size in the Classical Gradient Method
Number of Possible Partitions in a Grouping Problem of n parts N groups
Gradient Vector of the Function V(G)
Set of All Possible V(G)
Grouping Procedure
Set of Partitions in the Neighborhood of a Partition Using the Transfer Step to Create Partitions
Same as Above Except That Interchange Step is Used
Preference Index
XI1
CHAPTER I
INTRODUCTION
Group Technology (GT) is a problem-solving philosophy
based on the premise "that many problems are similar and
that by grouping similar problems, a single solution can be
found to a set of the problems, thus saving time and effort"
[46]. In an attempt to find an efficient solution to the
numerous machine set-up problems that characterize batch
production, researchers have applied Group Technology
principles to organized batch production systems. These
are known as Group Technology Systems. One of the basic
GT design tasks is the formation of production groups such
that parts with similar machine set-up requirements are
processed in one group.
The first formal proposal of Group Technology Sys
tems was presented by a Russian, Mitrofanov (S^] , in the
1940s. Mitrofanov demonstrated that Group Technology can
provide a suitable framework to reduce machine set-up
times. Several researchers since Mitrofanov have consi
dered other problems associated with GT. Opitz of West
Germany [35, 36] and others in Europe and America [2, 14,
19], for instance, dealt with methods of coding data con
cerning parts to be grouped. Other researchers have at
tempted to develop computer based grouping procedures [8,
38, 39].
2
In this research one of the basic problems of Group
Technology, viz; the optimal grouping of parts will be
examined. In particular, criteria of optimization, group
ing procedures, and the effect of number of groups on pro
duction cost will be analyzed.
1.1 The Batchshop Problem and Group Technology
Consider a conventional static batchshop problem
characterized by a set of machines and a known number of
parts. The machines are grouped into subsets. Those that
perform identical functions (drills, for example) laid out
in departments constitute a subset. Each part is described
by an m-dimensional row vector of attributes (attributes
will be interchangeably used with characteristics). These
attributes may be parts design data: shape, size, material
type, accuracy requirement, features to be processed, etc.;
and/or planning data: machines in process route, proces
sing and set-up times, number of units to be produced, etc.
For n parts, the row vectors of characteristics combine to
form Parts-Characteristics Matrix (PCM).
The difference in the characteristics of parts intro
duce a high degree of complexity to the batchshop problem.
There is a complex flow pattern created by the differences
in process route; the functional layout can cause parts to
cover long distances between departments. At each machine
center parts in queues may have different machine set-up
problems. The net effect of this complexity is the dif
ficulty of attaining the level of productivity often
achieved in mass production situations. Previous studies
indicate that the average part in a conventional batchshop
spends only a relatively small percentage of its time in
actual metal removal; a larger percentage of time is spent
in waiting for machines to be set-up, other parts to be
processed or traveling between departments and machine
centers [32]. Thus, the machine set-up and transportation
problems appear to be potential areas where productivity
can be improved. These problems are the primary focus of
the Group Technology approach. The grouping of similar
parts and the machines that process them may allow parts
in a group to use the same machine set-up. Thus, to design
a GT system, a parts/machines grouping problem has to be
solved. This problem, which is the subject of this study,
will now be formally stated.
1.2 The Grouping Problem
Let G be a Group Technology system and g^ be a pro
duction group comprising a set of similar parts. Thus,
V(G) is a performance measure which may be production cost
L L L L a
or other cost-related functions. Now, consider a conven
tional batchshop with n parts presented in the form of a
Parts Characteristics Matrix (PCM) . The set of character
istics of each part are known and remain unchanged (inclu
ding a fixed process route). The grouping problem is that
of defining similarity of parts and assigning the n parts
to production groups {g^} on the basis of similarity, such
that the performance measure V(G) may be optimized.
The problem of grouping objects on the basis of simi
larity has been identified in several disciplines; plant
taxonomy, information storage and retrieval, patients in
hospitals, counseling,to mention just a few. Problems of
this type are known as Cluster Analysis problems [42].
To properly categorize the GT grouping problem, therefore,
a brief outline of the classes of the clustering problem
will be useful.
1.2.1 Cluster Analysis and the GT Problems
Four main classes of the clustering problem
have been reported [42]: Hierarchical, Non-Hierarchical,
Statistical and Non-Statistical. The Hierarchical problem
concerns the grouping of objects at different levels of
similarity so that the resulting cluster is a tree-like
structure. The Non-Hierarchical problem considers all
levels of similarity simultaneously such that the resulting
partition, G, does not have hierarchical structure.
Classification of the clustering problem may also be
in terms of whether the value of attributes are random vari
ables or not. Thus, there is a Statistical grouping problem
where some or all of the attributes (or their values) are
not known with certainty. The grouping of the objects is,
therefore, based on a random sample and the values of attri
butes approximated by a known distribution. In the Non-
Statistical problem all of the attributes and their respec
tive values are known. No statistical distributions are
involved in the grouping process since actual values are
available for use.
In the GT grouping problem all the attributes of parts
and their respective values are known with certainty. A so
lution to the Hierarchical grouping usually results from
solving the Non-Hierarchical problem at several levels of
similarity. Thus, the Hierarchical problem is a much larger
problem to solve. Besides, the reports on production group
formation tend to suggest no tree-like structure [6, 35, 37].
In order to facilitate a (general) solution approach, the GT
problem in this study will be formulated as a Non-Statistical
and Non-Hierarchical grouping problem. Some GT grouping al
ternatives will now be described.
1.2.2 Group Technology Types
GT problems may be classified in three ways:
(1) type of group membership, (2) type of attributes used
in grouping parts, and (3) nature of the problem. Within
the group membership classification there are three cate
gories. The Exclusive Membership type concerns the grouping
of parts such that an individual part is processed in one
and only one group. The Non-Exclusive Membership type differs
from the Exclusive Membership in the sense that parts can be
processed in more than one group. The Hybrid type is a com
bination of the conventional arrangement and either the Ex
clusive Membership or the Non-Exclusive Membership types.
In the attribute-based classification three types
can be identified. There is a Parus Classification Analysis
(PCA) which concerns the use of shapes, sizes, materials,
processing features and accuracy requirements as the basic
attributes for finding production groups. It is based on
the rationale that parts which are similar in the listed
characteristics may have similar set-up problems. The
grouping criterion in the PCA approach is the minimization
of machine set-up times.
Another attribute-based classification is known as
Production Flow Analysis (PFA). It uses information in
process routes to find groups. Parts that use identical
machines form a production family; the grouping objective
is the minimization of additional machines. There is also
a Combined Approach which combines all the characteristics
used in both the PCA and PFA to find production groups.
There are two subclasses of GT problems whose classi
fication is based on the nature of the problem. They may
occur with any of the classes described above. The Uncon
strained problem occurs when parts are to be grouped with
no prior restrictions imposed on the ntimber of groups. The
GT system designer has the freedom of solving for the ntimber
of groups. The Constrained problem occurs when the number
of groups is restricted to a known value before grouping
takes place.
Whether the Constrained or Unconstrained problem occurs
in a particular case is a management choice. In the Uncon
strained problem the system designer has a higher degree of
freedom in terms of an optimal solution. This may be the
case where product design, process planning, plant location,
etc., are variables. In changing over to GT from a conven
tional system, the mentioned factors remain relatively fixed.
Since a large ntimber of groups may mean a substantial increase
in labor, equipment, workspace requirements and energy con-
stimption, management is likely to exercise its influence by
imposing a restriction on the number of groups; these require
ments may involve considerable investment. Consequently, the
more practical problem to solve in industry is likely to be
the Constrained problem. Besides a solution to the Constrains
problem may provide an insight to the Unconstrained.
8
Besides the Constrained problem, the Hybrid, Non-
Exclusive Membership and the Combined types appear equally
interesting. However, there is very little background in
formation available in the literature on these problems.
The Hybrid and Non-Exclusive Membership were suggested by
Purcheck [37] and the Combined by Gallagher and Knight [16],
but were not pursued. Most of the reports of application on
GT make mention of the Exclusive Membership, PCA and PFA.
Because of resource limitations, this research will
be restricted to the Constrained, Exclus5.ve Membership, PCA
and PFA GT types. The research problem is given a formal
definition in the section that follows.
1.2,3 Summary of Research Problem
In compact form, the grouping problem is as
follows: Optimize V(G)
Subject to
n Z n, = n
k=l ^•
(1.1)
, ^ f, for all k (1.2) °k
gPg- = f» fo^ ^^ ^ ^^ k, i 7 k (1.3)
G = {g^} (1-^)
k = 1, 2, 3,...N (1.5)
where
f = an empty set
n ^ = number of parts in group k
n = total number of parts in the grouping problem
^k ^ ^^^ °^ parts in production group k
G = the set of production groups or physical representation of the GT system
and N = fixed ntimber of groups.
Observe that Equations (1.1), (1.2) and (1.3) specify
the Exclusive Membership problem, while Equation (1.5) states
the Constrained problem. In the PCA, the objective function
"optimize V(G)" may imply minimizing total machine set-up cost
or minimizing the ntimber of additional machines in the PFA.
1. 3 Review of Past Studies
The selection of attributes for the purpose of defining
similarity of parts, method of coding grouping data and group
ing procedures for the Unconstrained problem have received
major emphasis in the GT literature. These and other related
subjects will be reviewed in the following paragraphs.
(i) Choice of Attributes (of Parts for Grouping Pur
poses) . The use of design features as the basis for grouping
similar parts was introduced by Mitrofanov [33]. His ori-
ginal idea of Group Technology was to reduce set-up times by
setting up one machine to process parts with similar features.
It did not include the subdivision of parts and machines into
10 mutually exclusive production cells. It was Opitz of West
Germany [34, 34"] who first introduced the concept of pro
duction cells--a set of similar parts and the facilities
that process them. To facilitate the process of sorting
similar parts into groups on the basis of design features,
Opitz developed a comprehensive system for coding and
classifying parts [36]. It was the use of Opitz' system
for coding and classifying parts into groups that was
earlier referred to as Parts Classification Analysis (PCA)
approach.
In contrast to the Opitz', Burbidge's definition of
similarity was based on information from process routes [6,
7]. In this approach, parts which use identical machines or
have similar flow patterns are sorted into one production
group. This is the Production Flow Analysis (PFA) mentioned
earlier. A similar approach was suggested by El-Essawy [12].
Associated with the approaches of Opitz and Burbidge,
respectively, are two formats for coding and presenting
grouping data: Parts Coding and Classification System and
Binary Matrix.
(ii) Parts Coding and Classification Systems. A
coding and classification system is a scheme that specifies
important design characteristics of parts by means of sym
bols. With such a scheme a part can be described by a code
ntimber whose individual digits represent particular charac-
11
teristics. A level of variation of each characteristic is
indicated by the value that appears at the corresponding
digit position. The first coding and classification system
was developed by Opitz [34, 35, 36]. In Opitz' system,
high values in a digit position indicate complex features
while low values represent simple ones. Thus, the simple
part in Figure 1 may be represented by the code ntunber
00100, where, starting from the left-hand side,
0 indicates rotational part with length/diameter <_. 5
0 indicates machined constant diameter
1 indicates smooth bore
0 indicates plane surface machining
0 indicates no other holes.
Since Opitz' system, other researchers have devel
oped several coding and classification systems [12, 4, 14,
18, 23, 28]. Two types of coding systems can be identified:
a monocode and a polycode. In a monocode, the meaning of
a digit in the code number depends on that of the previous
digit (much like the nodes of a decision tree). For the
polycode, the meaning of a digit is independent of other
digits. Polycode systems are suitable for Group Technology
manufacturing purposes, while monocodes are best for design
purposes [4, 11]. The Opitz' system is a polycode utilizing
•" t V
^im
12
Figure 1. Disk-Like Part
13
nine digits. Other polycodes are MICLASS [23], CODE [ll]
and TEKLA [ll]. An example of a monocode is the Brisch
system [l8].
A comparative study of the polycodes by Eckert [ll]
showed numerous commonalities in the basic characteristics
chosen to describe a part. Differences are evident only in
minute details. However, Eckert reported that the MICLASS
and Opitz' systems code the most relevant and greatest con
tent of information. Another polycode system not mentioned
by Eckert, but which appears suitable for production plan
ning, is the SAGT coding system [14]. Researchers have
used coding and classification systems in various GT rela
ted studies. Coding systems have been used in product de
sign [18], grouping of parts [22, 35], determination of
machining parameters [14], and analysis of component statis
tics [17, 28].
In this study, the Opitz system will be used to code
parts in the PCA approach because it has been used by others
[22, 34], it is well-doctimented, and the author is familiar
with the system. The Opitz system is described in greater
detail in Appendix B. An example of Parts Characteristics
Matrix (PCM) coded with Opitz' system is presented in Figure
1.1.
Parts 1
1
2
3
0
3
2
Design & Processing Features
2 3 4 5 6
0
2
1
1
0
2
0
3
1
1
1
2
0
14
n 0
Figure 1.1. Parts Characteristics Matrix (PCM) showing only design and processing features. Opitx coding system is used.
Parts 1
Machines in Process Routes
2 3 4 5 M
1 1 0 0 1
1 0 1 1 0
0 0 1 0 . . 1
1 0 0 0 0
1
2
3
4
0
1
1
1
n 0 0 0
Figure 1.2. Parts Characteristics Matrix (PCM) showing only machines indicated in process route
15
(iii) Binary Matrix for the PFA. The system for
coding the PFA is relatively simple. Most of the data re
quired to form production groups have been presented in the
form of a Binary Matrix [6, 7, 37]. An example is shown
in Figure 1.2. This is another form of the PCM matrix
where the coltimns indicate the machines and the rows repre
sent parts and the elements indicate whether a part uses a
particular machine or not.
Thus,
(PCM)^j -
1, if part i is processed by machine.
0, otherwise.
(iv) Grouping Procedures. The choice of attributes
and a system for coding them provide a basis for the identi
fication of similar parts. The assignment of parts into
groups is accomplished with a grouping procedure. One ap
proach by Mitrofanov [33] makes use of a Composite Part
Concept. A composite part is one whose attributes are care
fully chosen to represent those required in a production
group. All parts with the set or subset of the representa
tive attributes are assigned to a group.
A similar technique called Code Ntimber Field Approach
was utilized by Opitz [35]. This approach is similar to the
Composite Part Concept except that the chosen attributes are
16
represented in form of an Opitz' code number. A complete
grouping of parts may involve the formation of several Code
Ntimber Fields and assignment of parts in an iterative manner
until all parts belong to groups. It is a manual procedure
requiring production know-how and intuition. It only ap
plies to the PCA approach and the technique implicitly as
sumes the Unconstrained problem.
In the PFA, some computer grouping procedures were
reported by Purcheck [37], Carrie [8], Rajagopalan and
Batra [38], respectively. Purcheck's is an optimization
approach using process route as variables and linear pro
gramming techniques to find production groups. The objec
tive is the minimization of additional machines. Purcheck
reported that his approach can solve only small problems
[37].
The procedure reported by Carrie was developed by
Ross in plant taxonomy [39, 40]. This approach requires
the computation of similarity indices or coefficients.
Similarity indices indicate the strength of similarity be
tween any pair of parts. Based on such indices, "a nearest
neighbor" method in Cluster Analysis was used to assign
parts to groups. Rajagopalan's approach, which also utili
zes similarity coefficients, differs from Carrie's in that
a graph-theoretic technique replaces the nearest neighbor
grouping method. Like the Code Number Field by Opitz,
17
these PFA approaches are designed for the Unconstrained
problem. Because the use of similarity coefficients is
important in grouping problems, it will be further dis
cussed.
(v) Similarity Coefficients. Most techniques for
solving Clustering problems make use of similarity coeffi
cients. A similarity coefficient is a quantitative repre
sentation of the similarity between a pair of parts. There
are two types: association or distance coefficients. Dis
tance coefficients measure how far apart two objects may be
in terms of the value of corresponding attributes. They
are applicable to attributes whose values can be measured.
On the other hand, association coefficients measure the
degree of similarity between entities with binary or ranked
attributes [3, 44]. Thus, in addition to the PCM matrix,
a grouping problem may be presented in the form of a simi
larity coefficient matrix. It was in the form of a simi
larity coefficient matrix that Carrie [8] and Rajagopalan
[38] represented their respective GT grouping problems.
Both researchers used association coefficients which may be
regarded as a transformation of the PCM matrix. In trans
forming the PCM matrix to a similarity coefficient matrix,
a function is often used. Thus, let R^. be the similarity
coefficient between parts i and j. In Cluster Analysis,
R . has been defined in several ways. One most commonly
used for multi-attribute problems is defined as follows:
18
A(B.nB.) R., - . - \ . (1.6) ^j X(B^uBj)
, • » •
where
B. = set of attributes possessed by object (part) i
B. * set of attributes possessed by j
and, X implies number of objects in the set.
Equation (1.6) expresses R.. as the ratio of the ntimber
of attributes common to parts i and j to the total nuniber
of distinct attributes of both parts. This definition was
the form used by Carrie in GT and has been reported as
being used in several disciplines of Cluster Analysis [5,
25, 41].
The properties of R.. are as follows:
(a) 0 < R^j < 1
(b) R. . =• 1 means maximum similarity
(c) R. . = 0 means minimum similarity
(d) R.. = R..; symmetric property
(e) R. = 0 , for convenience.
The complement of similarity is the dissimilarity. Thus,
if d.. is the dissimilarity between parts i and j, then, ij
19
^ij = 1 - Rij (1-7)
In some Clustering problems, the dissimilarity coefficients,
instead of similarity coefficients, are used [25].
1.4 Critique of Previous Work and Study Objectives
It appears that the two sets of characteristics for
defining similarity in the PCA and PFA, respectively, are
well received by researchers [17, 22, 27]. This seems to
be the case with the methods of coding the attributes.
However, the literature is sparse on experimental or analy
tical studies concerning the behavior of GT systems. Know
ledge of the behavior of different GT alternatives in vari
ous conditions may be valuable for GT system designers.
Though extensive information concerning the benefits of
Group Technology is reported in the literature, it appears
to be accoijnts of empirical examples [2, 32]. The problems
created by this lack of experimental approach was stimmarized
by one GT expert as follows: "Attention has been drawn to
the possible dangers of developing general solutions from
basis of particular examples. In some GT cases, assump
tions have been made empirically or on narrowly-based data,
and have led to fallacious conclusions. Much which has
been written about set-up costs falls into that category...
Each of the hoped for benefits should be tested carefully
20
for validity in light of particular circtimstances of the
manufacturing concerned before a decision to implement
cell production is taken" (Craven [9]).
Several factors may be responsible for this lack
of interest in experimental or analytical studies. One
possible reason is the absence of efficient grouping
techniques suitable for most GT alternatives and range
of problem parameters. For instance, there have been no
reports of industrial applications of the computer grou
ping procedures previously discussed. One possible reason
may be that they are intractable for practical problems.
Besides, they cannot solve uhe Constrained problem. Re
call that a problem is considered constrained when the
ntimber of groups are specified. The importance of the
ntimber of groups as a parameter in GT was referred to in
Section 1.2. Thus, an algorithm for the Constrained
problem may be capable of solving problems over a wide
range of system parameters. The development of a grouping
algorithm for the Constrained problem both in the PFA and
PCA will be pursued in this study.
The grouping problem stated in Section 1.2.4 is a
combinatorial optimization problem; a systematic solution
may require an explicitly-defined objective function. A
function which introduces evaluation difficulties may be
inefficient. On the other hand, one with large storage
21
requirements may be infeasible for computer procedures.
The use of an explicitly-defined grouping objective was
not reported in the GT literature. The discussed proce
dures of Carrie [8] and Rajagopalan [38] did not state
any objective function. That of Purcheck [37] implied
the minimization of additional machines but was not ex
plicitly defined.
In this study, explicitly-defined objective func-
,tions for the PCA and PFA, respectively, will be formu
lated. ^
A solution to the grouping problem only sets the
stage for GT implementation. The success or failure of
such systems may depend on control of operations and the
structure of the problem. The structure of some problems
may be such that similar parts exist in numbers enough to
form profitable production groups. In other cases, parts
may be dissimilar in all respects, therefore unsuitable
for GT. If this is true, then some characterization of
GT problems may be useful in a preliminary attempt to
decide if a conventional batchshop should be converted to
GT. Such a characterization will be one of the study ob
jectives of this research. No such study was reported in
the literature.
A knowledge of any relation between ntimber of groups
and production costs may be valuable in studying GT system
behavior. An increase in ntimber of groups, for instance.
22
may necessitate extra machines, labor and space. Extra
machines may, in turn, affect machine loads, energy con-
stimption, in-process inventory, investment cost, etc.
The possible impact ntimber of groups may have on produc
tion costs was ignored by past studies. One of the ob
jectives of this research is to investigate if a useful
relationship exists between ntimber of groups and produc
tion costs.
In stimmary, then, the objectives of this study are:
(1) Formulation of grouping criteria
(2) Characterization of GT problem data
(3) Development of grouping algorithm for the Constrained problem
(4) Investigate the relationship between ntimber of groups and production costs.
1,5 Outline of Succeeding Chapters
A grouping criterion is important to this study.
The characterization of GT problems and the optimality
conditions for grouping parts may depend on grouping cri
terion. Hence, Chapter II will be devoted to the formula
tion of grouping criteria. In Chapters III and IV, the
characterization of GT problems and the grouping algorithm
will be presented, respectively. In Chapter V, the rela
tion between number of groups and production costs will be
discussed. Finally, the summary of research, conclusions
and recommendations for further study will be outlined in
Chapter VI.
I III I I i i i i m l I I I
M#f
CHAPTER II
GROUPING CRITERIA
2.1 Introduction
The efficiency and effectiveness of a solution
technique for practical grouping problems may depend on
the choice of a grouping criterion. A criterion which
requires a large amount of data may introduce storage
problems, thereby limiting the use of computer grouping
procedures. On the other hand, one which involves eval
uation of many terms may be inefficient. One that appears
obvious is production cost, but the difficulties associ
ated with the evaluation of components like in-process
inventory, penalty cost for late delivery, machine idle
ness cost, etcT, may render its use inefficient. The
purpose of this chapter is to discuss grouping criteria
in terms of production cost, storage requirements, ease
of evaluation and optimality conditions.
In the first section of this chapter, production
cost in GT will be discussed; in the sections that follow,
the formulation of other criteria which may be suitable
for the PCA and PFA, respectively, will be presented.
23
24
2.2 Production Cost in GT
In studying the behavior of production cost in
machine shops, researchers have considered some of the
following cost components:
1. Machining cost
2. Set-up cost
3. In-process inventory cost
4. Penalty cost due to tardiness
5. Machine idleness cost
6.. Transportation cost
7. Investment cost for additional equipment.
Gupta and Dudek [20], for example, considered the
first five in the scheduling aspect while Iwata.and Takano
[24] used the first six in studying the process planning
problem. In the GT grouping problem researchers have con
sidered machine set-up and cost for additional machines
as the most critical [13, 34, 37, 38]. In the PCA approach,
for instance, the minimization of machine set-up is the
grouping criterion; in the PFA approach, it is the minimi
zation of the cost of additional machines [7, 8]. In an
integer programming model to determine which parts to pro
duce in a GT system, Dedich, Soyster and Ham [10] formu
lated production cost as the sum of set-up cost and cost
for additional machines. Thus,
4K its-:
V V.
.•^-«3- 25
^c- S^+A^ (2.1)
where P^ is production cost, S , set-up cost and A is cost
for additional machines. These researchers argued that, for
the exclusive purpose of forming production groups, set-up
and additional machine costs are the most important compo
nents of production cost.
The formulation of alternative grouping criteria
which may relate to machine set-up cost and cost of addi
tional machines will now be discussed.
2.3 Objective Function in the PCA
The ideal information for grouping parts in the PCA
is machine set-up time data. However, time data often
require large amounts of storag e which may preclude compu
ter grouping procedure for large problems. Consider a
moderate-size problem of 400 parts and 70 machines, for
instance. Seventy matrices of 400 x 400 set-up time data
may be needed in order to group parts. In practice, much
larger problems have been reported [22, 35]. Besides, it
is doubtful if such set-up time data for large batchshops
are available in industry. One GT researcher observed that
''...sequential set-up times would be a useful criterion for
part family classification, but the data are seldom avail
able in practice" [47].
26
To overcome these drawbacks, researchers have used
"similarity of parts" as an equivalent criterion. Using
this criterion, parts are grouped so that those processed
in one group are most similar with respect to shape, size,
material, finish requirement and surface to be machined.
Implicit in this approach is the assumption that maximi
zation of group similarity is equivalent to the minimization
of machine set-up time. That this appears to receive uni
versal acceptance may be judged from ntimerous statements
of the following type in the literature: "The similarity
of components within a family allows resetting times be
tween batches to be minimized bv the use of rationalized
tooling arrangements" [27]. A real-life experimental
verification was reported by White and Wilson [47]. These
researchers demonstrated the minimization of machine set
up time by processing parts in subset such that the ma
chine is set-up for each subset of similar parts. The
small size of the problem (6 parts) allowed similar parts
to be identified by inspection. However, a mathematical
model relating set-up time to similarity of parts has not
been reported. Formulation of such a model is now dis
cussed.
2.3.1 Set-Up Time -Similarity of Parts Model
Let f(t, H, Q) be a function relating set-up time
to similarity of parts. Thus,
27
S^ - f(t, H, Q) (2.2)
where
and.
t
H
Q
s.
= time parameter
= a measure of similarity of parts in a group
= parameter relating t to H
= total set-up time.
To explain this model it is asstmied that the total time
it takes to set-up a machine for a set of parts can be
separated into sequence dependent and sequence indepen
dent components. The sequence independent component which •I -w
may include time to adjust workpiece, tool, etc., is a
constant. The sequence dependent may be a variable for"a
given problem; examples may include time to change cutting
tool, jig, fixture, machining parameters, etc. We restrict
the discussion to the sequence dependent set-up time.
Now, consider the problem of setting up a machine
for a given operation. Depending on the particular machine
and the characteristics of the part concerned, some or all
of the following set-up tasks may be performed.
1. Change jig
2. Change fixture
3. Change attachment
4. Change cutting speed
28
5. Change feed rate
6. Change depth of cut
7. Change cutting tool
8. Others.
The total amount of set-up time may depend on the type and
number of set-up tasks performed (or not performed). Thus,
for a given problem, a set-up task may asstime one of two
states: "performed" or "not performed". Let e, represent
the state of task k and E, the set of elements, {e,^}.
e« is defined as
|l, task k is performed
.0, task k is not performed
and
E =
re-
le V
29
where v is the total number of tasks. Let t, represent the
time iinit it takes to perform task k. Also, let T be a row
vector representing the set of time units corresponding to
the set of tasks. Thus,
T - (t^, t2,..., t, ,... t ) (2.3)
th
The set-up time, S.. , for the m operation of part i
after processing part j on the same machine can be expressed
as the vector produc t of T and E.
Expressed in terms of the elements of T and E
^ijm " ^1^1 •*" ^2^2 " • • - k k- * * " vS
V - Z t,e (2.5) k-1 ^ k /
The above equation shows that the set-up time of any opera
tion is the sum of the product of the time taken to perform
individual tasks and the corresponding status of the task.
The value of t, will be independent of the particu
lar part being processed. It depends on the machine used
and can be determined by motion and time study. For any
given machine t, remains approximately constant. Hence,
sequence dependent set-up times for a fixed process route
30
problem depends on the vector E; it will be called Task
Elimination Vector since the number of zero elements pre
sent indicate the number of set-up tasks which do not have
to be performed. E can be expressed as a function of the
similarity of parts as discussed in the following section.
(i) Derivation of Task Elimination Vector, E.
Whether or not a set-up task must be performed, e, , depends
on the similarity in one or more characteristics of two
parts processed in sequence. For example, the task "change
jig" may be eliminated if two parts processed in sequence
are similar in overall size and shape. But similarity in
material type may not cause "change jig" to be eliminated.
Thus, considering "yes" or/"no" answers to questions of the
following type, "can similarity of parts i and j with re-
spect to the a characteristics cause the elimination of
set-up task k when both parts are processed in sequence?",
a binary relation between set-up tasks and characteristics
of parts can be defined. Let q^^ represent the binary
relation.
'l, if task k can be eliminated by similarity in characteristics a
^ka 0, if task k cannot be eliminated by similarity in characteristics a
31
Let Q represent the set of elements {q ^ }. Hence, Q is a
V x u binary matrix where v is the number of tasks and u
the number of characteristics. Matrices that define bi-
nary relations between two sets of quantities are used
extensively in System Analysis [42], An example of Q
involving seven set-up tasks and four characteristics of
parts is presented in Figure 2.1. As shown in the figure,
Q implies a potential state indicating which set-up task
can or cannot be eliminated due to similarity of parts
with respect to characteristics.
The phrases, "potential state" and "similarity of
parts" are important. They imply that the elmination of a
set-up task cannot take place without a state of similarity
Let this state of similarity between parts i and j, to be
processed in sequence, be designated as D... D.. is a Dis-
similarity State Vector with u elements; each element ^^i.
represents the state of dissimilarity of i and j with re-
spect to the k characteristics. Vector representation of
similarity (dissimilarity) is common in Cluster Analysis
[1, 3]. Thus,
1, if part i is dissimilar to part j with respect to the k h charac-
_ teristic 6. ., = \ J* 0, if part i is similar to part j
with respect to the k^^ characteristic
.
-
Machine Set-Up Tasks
Change jig
Change fixture
Change attachments
Change cutting tool
Change cutting speed
Change feed rate
Change dept of cut
Characteristics of
Parts
OV
ER
AL
L SH
AP
E
& S
IZE
I-l
1
0
0
0
0
0
)IM
EN
SIO
NS
&T
YPE
3F
FEA
TUR
ES
TO B
E
0
0
1
1
0
0
0
MA
TE
RIA
L T
YP
E
0
0
0
1
1
1
1 F
INIS
H A
ND
T
OL
ER
AN
CE
0
0
0
0
1
1
1
0 =
1
h d ^ 0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
1
1
1
0 1
0
0
0
1
1
1
Figure 2.1. Set-up task-characteristic Binary Interaction Matrix.
and
33
^ij2
D. . « 13
^ijk
6. . I 3.JUJ
If i and j are dissimilar in all characteristics, then all
the elements of D. . will be 1; they will be zero if i and
j are identical.
E, the Task Elimination Vector, depends on the two
states: Q and D..; Q specifies the potential tasks that can
be eliminated while D.. ensures (or fails to ensure) the
e 1 iminat ion. Thus ,
E = Q X D^. (2.6)
where the operator "x" is a boolean multiplication. With
this understood, E can now be expressed as
E = QDj, j (2.7)
Observe that the definition of Q does not depend on
any particular machine or part. It is a natural state and
34
may remain constant with time. But the vector D.. is a
variable that depends on the similarity of parts processed
in sequence. Thus, Equation (2.7) shows that the number
and type of sequence dependent set-up task to be performed
in a given problem may be controlled by the choice of parts
to be processed in sequence. The choice can be done on the
basis of similarity of characteristics.
To show how similarity of parts relate to set-up
time, replace E with (QD^.) in Equation (2.4). Then,
Sijm = ^(Q°ij) ^2.8)
Since Q and T are constants, the above expression shows that
sequence dependent set-up time, S^. , is directly propor
tional to the state of similarity, D ^ , where i and j are
processed in sequence. It also follows that
S.. = S,. (2.8a) ijm jim
since D.. = D. . and asstiming that the operation ntimber (m)
is the same for both parts. Notice that for the above
equality to hold, S^.^, S.^^must be strictly sequence-
dependent set-up times, all the sequence-independent time
elements (adjustment of cutting tool, machine tool and work-
piece; coolant related set-up tasks, set-up of machines at
the beginning of a schedule period, etc.) cannot be included
in the observations. The computation of S.^^ using Equation
(2.8) is illustrated in Example 2.1.
'*, •/
Example 2.1
T = (4 , 8, 6, 6, 4, 3 , 5)
35
Q -
f 1 0 0 . 0 1
1 0 0 0
0 1 0 0
O l i o
0 0 1 1
0 0 1 1
0 0 1 1
from Figure 2 .1
12
' 1 ^
0
1,
L 0 J
'13
f 0 1
1
0,
0
D 23
' 1 ^
0
0
121 = T(QD^2)=' ^^* ^ ' ^ ' ^ ' ^ ' ^ ' ^^
1 0 0 0'
1 0 0 0
0 1 0 0
O l i o
0 0 1 1
0 0 1 1
0 0 1 1 ^
1] 0
1
.oJ
36
121 = ( . 8, 6, 6, 4, 3, 5)
'V
1
0
1
1
1
1
« 30
/
Similarly, S^^^ = 12 and ^^^ = 24.
These values show that for the first operation of ^
part 1 sequence-dependent set-up time is 30 units when
processed after part 2 and 12 units when processed after
£art 3; both parts are processed on the same machine.
For part 2, set-up time is 24 when processed after part 3.
(ii) Set-up Times and Dissimilarity Coefficients.
The Dissimilarity State Vector, D, may be adequate to re
present the degree of dissimilarity of parts in a group
provided the ntimber of parts in the group is two. With
the ntimber of parts greater than two, some other represen
tation may be more appropriate. One way to do this is to
replace the vector representation with its magnitude. This
magnitude, as often defined in Cluster Analysis, is similar
to the similarity coefficient defined in Section 1.2 [25].
^^n
37
Thus, dissimilarity between any pair of parts (i, j) is
the ratio of the nusoiber of characteristics in which parts
are dissimilar to the total number of characteristics.
Hence,
^^ WD ^ ^io
(2.9)
where
where
• i:
w =
dissimilarity coefficient (magnitude of dissimilarity)
set of weights of the characteris tics
w, eW is the relative importance of * the k* ^ characteristic.
By definition,
"3". , eD. - 1 for all i and k. lok 10
In terms of the elements of the vectors D^. , W, D^^,
u , _ k^l ^k\lk ij " ^
k^l ''k' iok
(2.10)
where u is the number of characteristics. Now, similarity
and dissimilarity are complements [39]. Hence, both quan
tities share common properties. If 5^^^ is the binary
variable defining the similarity of parts i and j with
38
respect to characteristic k, then
«ijk - 1 - \ j k (2.11)
and dij - 1 - \ i (2.12)
where R.. is the magnitude or coefficient of similarity de
fined in terms of the same characteristics of parts as d...
Like the properties of R.. (see Section 1.2), those of d..
are as follows:
(1) 0 < d^j < 1
(2) d. . - 1 means maximum dissimilarity
(3) d. . = 0 means minimum dissimilarity
(4) d^. - d...
Considering the proportional relation of set-up time
to dissimilarity, for two parts in a group, the sequence-
dependent time, S^. , may be proportional to d ^ provided
the weighting parameters, w^, are properly chosen. The de
termination of the elements of the vector W will be dis
cussed subsequently. Thus, S^j^ A d ^ (A means proportional);
or S.. A(l - R..). That is, ijm — ij
_il53L = a constant (2.13) 1-R,j
39
for
0 £ R^. < 1.
Equation (2.13) shows that as similarity of parts increase,
set-up time decreases. Maximum similarity corresponds to
minimum set-up time; the converse also holds true. Hence,
Equation (2.13) provides a basis for defining similarity
based grouping performance measure.
To define such a performance measure, we rely on
the judgment of one expert in Group Technology. He stated:
"In the group or cell concept, setting up time may be re
duced but this clearly depends on 'family homogeneity' and
its relationship to component ordering" [13, page 343].
In Cluster Analysis, homogeneity of family k is interchange
ably used with similarity of parts in group k. It is often
denoted as H, . H, has been defined in several ways. One
most commonly used is the arithmetic average of all the
similarity coefficients in group k [3, 5, 44]. Thus,
^k ^-1 Z Z R^.
« = izUzi i (2.14)
^ n^(n^-l)
ieg^, jeg^, if j, and n^ (the number of parts
in group k) greater than 1
40
For
i ^ - l . H ^ - O
Observe that
0 £ Hj £ 1 since 0 £ R^. <_ 1.
If S, is the total sequence-dependent set-up time for all
parts in group k, then
^K — = a constant. (2.15) 1-
\
where
0 £ Hj < 1.
For a group with two parts. Equations (2.13) and (2.15) are
identical. In a GT system of N groups, H^ may be approxi
mated with H where
^ H.
H = — . (2.16)
Substituting H for H^ and S^ for S^ in Equation (2.15),
^ = a constant (2.17) 1-H
41
where S^ is the total machine set-up time in the GT system
and 0 £ H < 1. Let S be the constant of Equation (2.17), o
then
S^ = S^ (1-H) (2.18)
Since H is dimensionless, S^ must have the same dimension o
as S . Hence, S is defined as the maximum possible set-up
time that can be reduced by the grouping of parts; it cor
responds to H = 0, the worst possible grouping of parts.
Operationally, S may be determined by stimming up all the
sequence dependent set-up times of each operation of all
parts. Thus,
m. n 1 S = 2 2: S. „ ° i-l m=l °°
where S. is the maximtim sequence-dependent set-up time lom
for the m^^ operation of part i. Equation (2.18) can be
written in the following form:
S = S - S H (2.18a) *t o o
In this form, similarity of parts is shown as a necessary
condition for the reduction of set-up times. Thus, (2.18)
may be regarded as the fundamental equation for the design
of GT systems. It provides the theoretical basis for the
42
formation of production groups.- From (2.18a), H may be
regarded as the proportion of set-up times that can be re
duced by grouping similar parts (0 <_ H < 1) . Hence, to
form optimal production groups in the PCA, it may be ade
quate to maximize H, The maximization of H has been used
as grouping criterion in Cluster Analysis [3, 5],
Given that parts must be processed in groups, the
maximization of H is necessary but not sufficient condi
tion for the minimization of set-up times. Sufficient and
necessary condition requires optimal sequencing of parts
within the production groups. Thus, the maximization of
H ensures the best environment where minimum set-UD time
can occur while optimal sequence of production, in a heu
ristic sense, guarantees the corresponding minimtim set-up
time. While the maximization of H is the subject of
Chapter IV, the solution to the sequencing problem is be
yond the scope of this study.
(iii) Relative Importance of Characteristics.
The weight parameters, W, introduced in the definition
of dissimilarity are intended to reflect the relative
importance of the characteristics of parts. Equation
(2.9) is repeated here to facilitate the discussion.
W D^.
ij ' r ^ 10
The weighting of characteristics, though not reported in
Group Technology, has been used in most Cluster Analysis
problems [44]. The values of the elements of W may be
chosen subjectively, as done in Cluster Analysis [44], or
objectively. In GT it may be logical to relate the weight
or importance of a characteristic to the impact it will
independently have on set-up time. One way to do this is
to define the weight of the k characteristic, w, , in
terms of the set-up time that may result if a set of
parts are dissimilar in k, but similar in other charac
teristics. This definition will now be expressed formally.
(iv) Definition of Weight Parameter. Let D..(w,) ij K
denote the Dissimilarity State Vector such that parts i
and j are dissimilar in the k characteristics but simi
lar in others. Expressed in terms of the four characteris
tics shown in Figure 2.1, overall shape and size, surface
to be machined, material type and accuracy requirement.
43
44
Di^(wp -
1
0
0 D..(W2) =
0
1
0
0 ^ J
. Dij(w3) =
0
0
1 °ii^V =
0
0
0
With D^. (W| ) defined, w^ may be given by the following ex-3-J
pressIon:
w ^ = T [QD (w^)] (2.19)
Since Q is a constant and D. . (w, ) is specified, w, will de-
pend on the choice of T, the time elements of machine set-up
task. The elements of T may be pooled over all or key ma
chines of the production system. That is, t, eT may be esti
mated as the average of set-up time element corresponding to
the set-up task k of all or key machines. From Figure 2.1,
n 0 0 Ol
1 0 0 0
0 1 0 0
Q = 0 1 1 0
0 0 1 1
0 0 1 1
lo 0 1 1
45
S u b s t i t u t i n g Q and D^j(w^) i n Equation ( 2 . 1 9 ) ,
f l
w, = T 0
0
1
1
0
0
0
0
0
0
1
1
1
1
0'
0
0
0
1
1
1
1'
0
0
.0.
« T
1
0
0
0
0
w^ - (tj^, t2f t^, t^ , t^, t g , ty)
1
0
0
0
0
w = t^ + 4 - (2.20)
iMil:
46
Similarly,
W2 = t3 + t
^3 ' H + ^5 + ^6 + ^7
^4 ' ^5 " *6 + ^7
(2.21)
(2.22)
(2.23)
A close examination of these equations and the matrix Q
shows that the relative importance of a characteristic is
measured by the sum of the times taken to perform the set
up tasks it can affect. This appears reasonable since the
importance of a characteristic should not depend only on
the number of tasks it can affect but also the time taken
to perform them. An illustrative example follows:
Example 2.2: Determination of Relative Importance of Characteristic and Dissimilarity Coefficients
From Example 2.1,
T = (4, 8, 6, 6, 4, 3, 5)
12 , 0^3 = , D 23
'V
Computing the weight of characteristics using
Equations (2.20) to (2.23),
47
Wi
w#
t, + to = 4 + 8 = 12
t3 + t^ = 6 + 6 = 12
Wo = t4 + t^ + tg + t^ « 6 + 4 + 3 + 5 = 18
^A "= tc + t. + t^ « 4 + 3 + 5 - 12 •6
W = ^ (w^, W2, W3, w^) « (12, 12, 18, 12).
In this particular example it is seen that the third char
acteristic is most important; the other three are of equal
importance.
(12, 12, 18, 12)
From Equation (2.19), d,2 =
(12, 12, 18, 12)
0
1
1
1
1
30 52r = .555
Similarly, d^3 = .222 and d23 = .444. These values indicate
that parts 1 and 3 are the least dissimilar (most similar)
pair followed by parts 2 and 3; parts 1 and 2 are the most
dissimilar. These relative values may be used to determine
the grouping of parts or sequence of production at machine
centers.
48
(iii) Asstimptions. The following are the summary
of the asstimptions of the Set-Up Time- Similarity of Parts
model of Equations (2.8) and (2.17).
1. Ntimber of groups, N, remain fixed
2. The process route of parts is known
3. The GT problem is the Exclusive Membership type
4. Similarity of parts is defined in terms of
machine set-up related characteristics (the
PCA approach)
5. Machines are grouped so as to process all the
operations in a group; all the parts use the
same set of machines
6. The time to set up individual set-up tasks
- is independent of the part being processed
7. Initial machine set-up at the beginning o.f. a
schedule period is aimed at the operations of
all parts in the group
8. All n parts are processed in each schedule
period.
The above assumptions describe the ideal conditions
in which the Set-Up Time - Similarity of Parts model apply
With the relaxation of some of these asstimptions, devia
tions from the expected behavior of the model may be ob
served. For instance, if assumptions 6 and 7 are relaxed,
then the equality of Equations (2.8) and (2.8a) may no
49
longer hold in a strict sense. In a group where most of
the parts are dissimilar, the seventh condition may be
difficult to attain. The model expressed in Equation
(2.17) may also need a modification when assumption eight
is relaxed. If some of the n parts which were grouped
initially are not processed in a schedule period, then the
value of the constant, S^. in Equation (2.17) may be modi
fied (see the definition of S ). o
The first to the fourth assumptions merely describe
the problem of this research in the PCA approach. The
significance of the fifth asstimption will be discussed
in a subsequent chapter.
2.4 Grouping Objective for the PFA
The minimization of additional machines is the group
ing criterion in the PFA. Additional machines may be re
quired because two or more parts processed by a machine in
the conventional case belong to different groups in the
Exclusive Membership GT. The ntimber of additional machines
may depend on the ntimber of groups in which individual
types occur and the ntimber of each machine type available
in the conventional system. Thus, if y^ is the number of
additional machines of type m, then
50
y™ *=
d™; d >o m m
m " 10; d <
(2.24)
rJ^
where d is the difference between the number of groups
that require machine type m and those in the conventional
system, b . d is given by
n d = 2: a , - b„ (2.25) m ^_^ mk m
where
^mk
1, if machine type m occurs in group k
0, otherwise.
The total number of additional machines A is defined as
the stim of y^. Hence, •m
M A = S y„ (2.26)
m=l ^
where M = t o t a l ntimber of machines in the conventional
system.
In the case where the ntimber of each machine type
is one (b = 1) in the conventional shop, A is known as m
the number of overlapping attributes or the link between
51
groups in Cluster Analysis. If A equals zero, then there
is no linkor__037erl3^ between the groups. The greater the
value of A, the greater the link. Very often, the measure
of link between, say, group k and the rest of the groups
is defined in terms of similarity coefficients. Thus, if
L, is the link between group k and the rest of the groups,
in terms of similarity coefficients,
E^ E R^. L = ^=^ J=^ i- (2.27)
n^(n.n^)
where ieg , j? gk, n = total number of parts, g^ = group k
and n, = number of parts in group k.
To explain how A and L, may relate, consider parts
i and j where part i belongs to group 1 and part j to
group 2. Let the route of part i be defined by the vector
r. and r. that of part .. For illustration let
r^ = (1, 2, 3, 4)
r. = (2, 3, 4, 5, 6, 7)
b = 1 for all m m N, ntimber of groups = 2
A(i,j) = number of additional machines due to the grouping of i and j.
52
By inspection of the routes, r. and r.,
A(i,j) = 3.
Machines 2, 3, and 4 will be duplicated. Now, R.., the
similarity coefficient of both parts, is defined as follows:
X(rnr.) R. . « =—J— [from Equation (1.6)]
J X ( r U r j )
X(r^ ^j) = 3 ^
' X here means ntimber of machines X(r^ r^) = 7
^ i j - 7-
If parts i and j are the only parts to be grouped, then
L^ - L2 - RjLj 7 7 •
For n parts and N groups, the link between group k and
other groups is defined as the average similarity coef
ficient , R.., for i in group k and j in groups other than
k. For the N groups, the average link, L, is approximated
as follows:
N
T = ^^i^^ (2.28) ^ N
53
Though a mathematical relation between A, the number of
additional machines, and L, the link between groups, is not
immediately clear, the illustrative example tends to sug
gest that the minimization of L may be close to the minimi
zation of A. The minimization of L has been used as an
alternative grouping criterion in Cluster Analysis [3, 44].
To verify the effectiveness of using either L or A
as a grouping criterion, a problem of n = 43, N = 5, and
M = 16, reported by Burbidge [6, page 172] was solved. An
algorithm presented in a subsequent chapter was used to
minimize A and L, respectively. The resulting partitions
are presented in Figures 2.2 and 2.3.
The number of additional machines in the direct mini
mization of A were four and nine for the minimization of L.
However, the minimization of L corresponds to Burbidge's
solution. As displayed in Figure 2.3, the distribution of
parts in the minimization of A may not be suitable for GT
production. A ntimber of 39 parts belong to one group while
the remaining groups have one part each. This is in sharp
contrast with Figure 2.2 where parts are evenly distributed
among the groups. From this example, the use of L may be
better than the direct use of A in problems where the range
of number of operations is very wide. In the Burbidge's
example, the parts in "lone" groups have one machine each
while most of those in the large groups have larger number
54
Parts
4 Ifi 77 ?fi
1 7
, 3 "^
.1 fi 7 P 9
10 n 1? i:^
u T^ 17 i f l 1Q ?n
. ?i 7? ?4 ?«;
9 1
?7 ?P 1 ?Q 1 30 31 37 33 3d 35 3fi 37 3R 39 40 d1 4? 43
5
1
12 91
1
a5
10
q?
1
1
93
1
•
2
q4
1
1
1
1
1 •
•
*
3
1
1
1 1 1
M .
4
T
1
1
1 1
1
A ( • 5
T
1 1
1 1 1
1
1 1
1
1 1
1
1
: H
6
1 1
1
•1
T
1 ]
T
1
1 1
1
• •
1 1
I 7
1
1
1
N
' 8
1 1 1
T
1 1
1 1 1 1 1
1 1
1
1 1
1
1
E i
9
1
1
T
1
1
1 1
*
10
1
•
T
1
•
"
11
T
1
1
1
1
• 12
1
1
1
1
13
•
1
1
14
^ "
1
15
1
1
16
1
1
" 1 1
..,I_
1
•
1
1
1
1 - l .
^Additional machine
Figure 2.2. Grouping solution resulting from minimizing A.
55
Parts
1 s
14 1 0
no -7T
^
'-1 - ^
' f^
I S
/^9
'-]
17
"? )
^ ^q i,q
6. ^e ;
7? 1
•30
26 • ?
" i T O
'^^ ^ T
T -?n
-n . ' > ' '
IS 1 7
•iL
TA 7
U
- L 1 1 1 1
i 1 1
1
2x 1 t
•>
8 1
1
1
1 1 1
1
1 1 1
151 1, 1
] 1
,L. 1 ,1
1
1 1
1 1
1 1
1 t
6|11| 2
1
H 1 1 1 !
1 ' ' - ' 1 '
1 1 1
1
1
_L
i
1
I 1
1 1 1 1
1
1
1
i
1 1
1 1 \ ]
[ 1
1
1
1
1
l | 9 1
1
i
1
^
I
1 _L
I 1 1 1
f 1
f 1 1
\
1
1
M
16
i
1
A (
1
1
T L 1
r ^ L
]
1
1
1 1
i
1
L I
14
J,
N
7
1 1 1
1
E S 10| t
1
1
1 7
1
f
I 1
T
t
i
\
l l r l 2
1
_L 1 T
\
1
1
1
1
1
1
_ | | U | 3
1
1 1
_ L 1
1
,1-
1 1 i 1 1 1
1 ^
\ 1
^ 1 1 1
1 1 1 1
iii
—**—
t 1*6
1 1
1
1
1 1 •"• 1
1 1
i 1 1 1
1 1 1
j 1 \
i 1
1
1 •^
1
—
' i • 1
. . ' . .
-Ul *Addiclonal aachine
Figure 2.3. Grouping solution resulting from minimizing L.
56
of operations. However, it must be pointed out that only
similarity in process routes was considered in the grouping
process. Considering other characteristics such as voltime
(number of units or amount of processing time) of the in
dividual parts it is possible that the distribution of
parts per group of Figure 2.3 may be a better alternative.
The importance of other parameters in the grouping problem
will be discussed in Chapter VI.
Possible relationship between H, the group homogen
eity, and L, the link between groups, will be discussed
in the next chapter.
j^liitBaitete.
CHAPTER III
CHARACTERIZATION OF GROUP TECHNOLOGY PROBLEMS AND OPTIMALITY CONDITIONS
3.1 Introduct ion
The data of a GT problem presented in either the PCA
or PFA format may have structures which may not be obvious
by inspection. A solution to the grouping problem usually
reveals the type of problem structure which may or may not
be desirable for GT systems. For example, if there are no
related parts in a conventional batchshop, there can be no
groups in which similar parts exist. In other problems,
groups may exist such that all parts in a group are similar
with respect to all characteristics, a desirable condition
for GT. A method of characterizing problem data may help
the GT system designer identify the type of problem being
dealt with. A proper characterization may also provide a
basis for developing optimality grouping conditions in some
cases. In this chapter an attempt is made to characterize
GT problems in terms of group homogeneity and the link be
tween groups. Also, grouping optimality conditions for
some special cases will be presented. First, a possible
relation between the similarity of parts within groups, H,
and the link between groups, L, will be explored.
57
58
3.2 Relation Between Similarity of Parts in the Same Groups, H, and Similarity of Parts in Different Groups, L
In Chapter II, H was defined as group homogeneity
for the PCA problem and L as the measure of linkage be
tween groups in the PFA. In general, both H and L can
be defined for the same problem in either the PCA or PFA.
To illustrate this point the definitions of H and L are
repeated for convenience.
1 N H = ^ 2 H, (3.1) ^ k=l ^
where
^k ""k-l
k n^Cn^-1) k
and H, = 0 for n, = 1.
N 1 k=l
L = i I L ^ (3.3)
where
k ^^"V 2 E R-v i^gk» J Stc ,, ,,
= i=l 1=1 ^ (3-^) \ n^ (n-n, )
59
Notice that both H and L are defined in terms of similarity
coefficients, R^A'S. Since any GT problem, either in the
PCA or PFA formats can be transformed into a matrix of
similarity coefficients, both H and L can be computed for
a given problem. For every problem of n parts there are
n(n-l) similarity coefficients (excluding cases with i=j).
Let the set of R. . terms in the definition of H be R(H) and
those in L be R(L) . It will be illustrated, with an example,
that R(H) and R(L) are mutually exclusive sets.
Example 3.1:
Consider the grouping problem of five parts
(n = 5) shown schematically in Figure 3.1a; in
Figure 3.1b, they are presented in Opitz' notation.
Using the relation
R.. = 1 - d.., [see Equation (2.12)] LJ ij
7
A s £EZ± [ s e e Equation ( 2 . 1 0 ) ] ^ij 7 ^ ^
^^^k ^iok
and setting w ^ - 1 for all the seven attributes in Figure
3.1b, the similarity coefficients for all the pairs of
parts are computed as follows:
60
Part 1 Part 2 Part 3
12"
Part 4
J 3%" i
U 12 It
1 ^= c ^ r—
Part 5
Figure 3.1a. Parts for Example 4.1.
P a r t
1
2
3
4
5 .
T-i i^ P H tQ <U OJ
v a. N <U cd - H
>.£ cn o cn
1
(D 1
^
K2)
2
ATTRIBUTES
CO c u
4J
OJ c
Q) cd g XCOrH
w w
1
1
1
5
7
Cd u C Q) C VI OiCU <U Cd e
C cn fH M w
4
4
4
0
0
to (U CO C C (1)
•H cd a Ci-i Cd
•H PH iw ,c u om p Cd o cn
0
3
0
3
3
U CO
J2 I-l iJ o O K
2
2
2
0
0
o cd I - l u 0) 3 >
- O OJ O H J
2
2
2
3
3
cd •H Q) M a cu >. cd S
3
3
3
2
2
61
Figure 3.1(b). The parts in Figure 4.1(a) shown in Opitz' code number system.
1 ^ v ^
1
2
3
4
5
1 -
.86
1.0
0 .0
0 .0
2
.86
-
.86
0.14
0 .14
3
1.0
.86
-
0.0
0.0
4
0.0
.14
0.0
-
0.86
5
0.0
0.14
0.0
.86
-
Figure 3.1(c). Similarity coefficients of the parts shown in Figure 4.1(a).
62
From the PCM matrix of the same figure, the dissimilarity
states for parts 1 and 2 are:
" 121 • 0' " 122 ' 0' " 123 ' °' " 124 "
^125 = 0. ?^26 ' ^' " 127 ^ ^
By definition
^101 •" 102 ' 103 " • • • = " 107 "
Hence,
, ^ (Ix0)-f(lx0)+(lx0)+(lxl)+(lx0)+(lx0)+(lx0) ^ 1 _ , A 12 (lxl)+(lxl)+(lxl)+(lxl)+(lxl)+(lxl)+(lxl) T ""•
R^2 * i - 12 " ^ " -- ^ " - ^
Repeating the same for all pairs of parts, the set of
similarity coefficients, R^., is computed and presented
in Figure 3.1c.
Let the ntimber of groups, N, be 3 and a grouping,
chosen arbitrarily, be
G = {(1, 3); (2); (4, 5)}
R31 " \ . . _ . ^ 1 - ;; = ' - -- « 1.0
H. - 31 " ^ 13 1.0 + 1.0
63
H2 - 0.0 (1x2 = 1)
2 2
H - "1 " "2 " "3 , 1.0 + 0.0 + .86 . g2
3 3
R(H) - { 13' ^ 1 ' 45' 54}
_ 12 '*' ^ 4 '*' ^15 " 32 " 4 •*• 5 L,
.86 + 0.0 + 0.0 4- .86 + 0.0 + 0.0 ^ 233
6
L = 21 " 23 " 24 " 25 ^ 5QQ
^ 4
L « 41 " 42 " 43 '^ 51 ^52 ^53 Q^^
3 "" 6
L « .283 + .500 + .047 £77 3
R(L) = {R3 2' 14' 5 ' 32' 34' 35' 21' 23,
R24, ^25' 41' 42' 43' 51, 52' 53}
R(H) « {R;L3' ^31' ^45' ^54^
By inspection, all the terms in R(H) are different from
those in R(L). Hence,
R(H)rNR(L) - f (3.5)
where f is an empty set.
64
Equation (3.5) is true for any Exclusive Membership problem
defined in terms of similarity coefficients.
A close examination shows that the number of terms
in R(H) and those in R(L) add up to n(n-l) , the total number
of similarity coefficients. Hence,
XR(H) + AR(L) - n(n-l) (3.6)
where XR(H) means number of terms in R(H) and AR(L) means
number of terms in R(L) . Now, the sum of all similarity
coefficients, R^., for any problem is a constant. Thus,
n-1 n 2 I R.. = C (3.7) i=l j=l J
where C is the constant.
Using Equation (3.7) it is shown in the following
that, in terms of R. .'s, H and L may relate. Denote the
stim of all R. . terms in H, as h, and those in the corres-
ponding L, as £. . Then,
h, = Z Z R..; i, jeg, (3.8) ^ i=l j=l ^
E Z i=l j=l
and X-j = Z^ Z R ^ J ; iegj^, j j g| (3.9)
65
Also, if h is the sum of all R terms in R(H) and i the
sum of those in R(L), then
N
^ " yl.^\ (3.10)
N and i ^ ^z^z^ (3 11)
Combining Equations (3 .10) and (3 .11) r e s u l t s in
N h + il = S (h, + £, ) (3 .12)
k=l ^ ^
Equation (3.5) shows that no term in R(H) is in R(L) and
Equation (3.6) shows that the stim of the ntimber of terms
in R(H) and R(L) equals n(n-l) , the total ntimber of elements
in the matrix R. Since the stim of all R..eR is a constant,
then the expression in Equation (3.12) is also a constant.
Hence,
N n n-1 h + il = Z (h, + il, ) = Z Z R. . = C
k=l ^ ^ i=l j=l ^
(3.13)
h + il = C
Equations (3.5) and (3.13) show that h and il are complemen
tary terms. As h increases, il decreases and vice versa.
66
Thus, h and il have a negative correlation. Notice that
H^ is the average of h^ and L^ that of 1, ; similarly, H
is the average of H^ and L that of L, . Although the re
lation established for h and il may not be generally true
for H and L, it appears reasonable to expect that H and L
may also have a negative correlation for some problems.
Such a relation and some of the properties of H and L make
them suitable quantities for the characterization of GT
problems , Recall that H and L have the following lower
bounds:
0 <_ H £ 1 and 0 1 ^ 1 1 •
From these properties, low or high values of H or L can
be easily recognized. The relative values of H and L
may be good preliminary indicators of which GT system
may or may not be economically viable. For instance,
if in the PFA, L is zero, then nonoverlapping groups
exist. Ntimber of additional machines in such a system
will be zero. If, in addition to L = 0, H is high, then
there is an indication that the parts in a group use the
same set of machines. The combination of low L and high
H is a desirable condition in the PFA, A similar case
can be argued for the PCA situation. In general, a mean
ingful characterization of GT problems may be accomplished
using the pair, H and L.
67
3,3 Characterization of 6T Problem Data
In terms of the type of the optimal groupings that
can be attained in a GT problem, four main types of GT
data can be identified. These are (1) Multiple Population
Data, (2) Universal Population Data, (3) Natural Population
Data, and (4) Null-Population Data.
In the Multiple Population Data N groups, {g^, g2,...
gj i---gjj}. exist such that each g^ has a unique set of
attributes. In terms of the link between groups, L, this
means that L = L^ = L2 = ... * I^ * ... » L ^ = 0. Under
the multiple population characterization, two types can be
identified. In one type all the parts in a group are simi
lar with respect to all the attributes. We call this Type
I Multiple Population. In terms of H and L, the type I
Multiple Population is characterized by
H - 1 and L = 0.
In a second type. Type II Multiple Population, parts in a
group are similar in some attributes but dissimilar in
others. In terms of H and L,
0 < H < 1 and L = 0.
Examples of Types I and II are shown in Figures 3.2(a) and
3.2(b), respectively.
68
Parts
1 2
8 5 U 9 3 6 7
10
1
1 1 1 0 n 0 0 0 0 0
2
1 1 1 0 0 0 0 0 n 0
M a c h i n e s
3 4 5
1 1 1 0 0 0 0 0 0 0
0 0 0 1
1 1 0 0 0 0
0 0 0 1 1 1 0 0 0 0
6
0 0 0 0 n 0
1 1 1 1
7
0 0 0 0 n 0
1 1 1 1
8
0 0 0 0 n 0 1 1 1 1
(a) Type I
H = 1
L = 0
Parts
1 2 8 5 4 9 3 6 7
10
1 1 1 0 0 0 0 0 0 0
0
3 1 0 1 0 0 0 0 0 0
0, _
Ms
4 1 0 1 0 0 0 0 0 0 0
Lchine
9
0 0 0 1 1 1 0 0 0 0
IS
5 0 0 0 1 1 1 0 0 0 0
6 0 0
Q 0 0 0 1 1 0 1
7 0 0
Q-J 0 0 0 0 1 1 1
8 0 0
0 ,J 0 0 0
1 1 0 0
(b) Type II
0<H<1
L = 0
Figure 3.2. Multiple Population Problem
69
In the Universal Population problem, all parts are
similar with respect to all characteristics. Hence, the
values of H and L are equal to one (H = 1, L * 1) for all
possible groupings. An example of this type of problem is
presented in Figure 3.3.
A Natural Population problem is one in which, for
all possible groupings, at least one characteristic (attri
butes) of parts will appear in more than one group. Two
types may be identified. In Type I Natural, H « 1 and
0<L< 1. This is a situation where an optimal solution
exists such that all parts in a group are similar in all
characteristics with some characteristics occurring in more
than one group. Type II Natural is different from Type I
in that 0 < H < 1 , 0 < L < 1 . Example of the Natural Population
data is shown in Figure 3.4.
In the Null-Relation problem, no two parts are simi
lar in any respect. In terms of group homogeneity and link
between groups, H = 0 and L = 0 for all possible groupings.
Figure 3.5 is an example of the Null-Relation problem.
In summary, GT problem data may be characterized as
follows:
1. MULTIPLE TYPE I POPULATION
Optimal grouping exist such that H = 1 and L = 0
fe
li K RHI '
^^^mtts-I'SMSfH
Afc
•
1
2
3
4
5
6
«
• *
Maehinea
1
1
1
1
1
1
1
2
1
1
1
1
1
1
3
1
1
1
1
1
1
4
1
1
1
1
1
1
Figure 3 . 3 . Universal Population Problem: H = 1, L - 1
70
P a r t s
1
2
3
7
5
6
4
8
1
1
0
1
1
0
0
0
0
2
1
1
0
0
1
1
0
0
Mac
3
1
0
1
0
0
1
1
1
h i n
4
0
0
1
0
1
1
1
1
es
5
0
0
1
1
0
0
0
1
6
1
1
1
0
0
0
0
0
g
g.
s
Figure 3.4. Natural Population Problem: 0<H<1, 0<L<1
P a r t s
1
2
3
4
1
1
0
0
0
2
1
0
0
0
Ma<
0
1
0
0
ihiE
4
0
0
1
0
les
5
0
1
0
0
6
0
0
0
1
7
0
0
0
1
8
0
0
0
1
Figtire 3.5. Null-Relation Population Problem: H = 0, L = 0
; 4 V , ,
71
2. MULTIPLE TYPE II POPULATION DATA
Optimal grouping exist such that H < 1 and L = 0
3. UNIVERSAL POPULATION
For all possible grouping H « l , L = l
4. TYPE I NATURAL POPULATION DATA
Optimal grouping exist such that H « 1 and 0 < L < 1
5 TYPE II NATURAL POPULATION DATA
Optimal solution exist such that 0 < H < 1 and 0 < L < 1
6. NULL-RELATION POPULATION DATA
For all possible grouping H = 0 and L = 0.
Multiple population problems are likely to occur in
large batchshops that manufacture several products. In
many cases, parts that are given different names because
of the ftinctions they perform may, indeed, be identical with
respect to their characteristics. In one case study [22],
for instance, many bushing-like components of the same size
were given such names as pulley spacer, packer, relay tinit,
spacing sleeve, etc. The Type I Multiple may be ideal for
GT production in either the PCA or PFA approaches. Since
it is characterized by H = 1, this is a necessary condition
for the reduction of sequence-dependent set-up time to zero
[see Equation (2.18)]. In general, if a grouping solution
72
is such that H = 1, L » 0 and the average niimber of parts
per group is fairly large, then it may be possible to set
up a mass production system for each group.
Whether the Type JI Multiple is suitable for GT sys
tems or not may depend on the value of H. Since L = 0, in
the PFA, the number of additional machines will be zero.
However, the arrangement of machines and tooling to reduce
set-up time, as claimed by PFA proponents [6, 7], may de
pend on how many parts in a group use the same set of ma
chines and share the same machine set-up related character
istics . A high value of H indicates that parts in a group
use the same set of machines. The converse, for a low
value of H, is also true. Hence, in order that Type II
Multiple be feasible for the PFA approaches, H has to be
of high value. The same argtiment may hold true for the
PGA.
The Universal Population Data may not describe a
batchshop problem. No batchshop problem may have all parts
identical. The condition of all parts being identical de
scribes a mass production system. Hence, if an optimal
grouping is such that H = 1 and L == 1, then the situation
may indicate a mass production system.
Notice, in the above discussion, that the relative
values of H and L are important in a preliminary determin
ation of how good GT systems may be. Thus, H = 1 and L == 0
73
is ideal as in the Type I Multiple problem. But H « 1 and
L = 1 is an indication that GT may not be a good alterna
tive. Inference can be drawn that H = 0 and L = 0, as in
the Null-Relation jproblem, is unsuitable for GT production.
The same argument can be extended to the Natural Population
case.
In the Type I Natural, which is characterized by
H = 1, as L approaches zero, the closer it approaches
the ideal situation (the Type I Multiple). On the other
hand, as L approaches 1, the Type I Natural becomes the
Universal problem which may not be suitable for GT. In
general, the smaller the value of L and the larger the
value of H, the better a batchshop problem may be for GT
production.
3.4 Optimality Conditions for the Grouping Problem
The above discussions suggest that, for some problems,
optimality conditions can be established. For instance, in
the Multiple Population problem, sufficient and necessary
conditions for an optimal solution in the PFA is L - 0 or
A = 0 (A is ntimber of additional machines). This is obvious
since, by definition, the lower bound of L is zero. Also,
if L « 0, then no one attribute occurs in more than one
group. In the PFA, machines are the attributes. Hence,
A = 0 corresponds to a solution with L == 0. Conversely, if
iMMjjSk^^
74
L > 0, then, at least, one machine occurs in more than
one group. In a case where the ntimber of each machine
type is one in the conventional problem, then A will be
greater than zero.
Similarly, H = 1 is the upper bound in the PCA.
Since the grouping objective is the maximization of H and
the Type I Multiple Population is characterized by H = 1,
sufficient and necessary conditions for optimal solution
is H = 1. These optimality conditions will now be for
mally stated without further proofs.
Condition 3.1
In the Multiple Population problem, a suf
ficient and necessary condition for an optimal
solution in the PFA is L = 0,
Condition 3.2
In any grouping problem presented in the
PFA format, if L = 0 then A, the number of addi
tional machines, will also be zero. On the other
hand, if L > 0, then A is also greater than zero
provided the number of each machine type is one.
Condition 3.3
In the Type I Multiple Population problem,
sufficient and necessary conditions for optimal
solution in the PCA is H = 1.
75
The Type I Natural Population problem is also characterized
by H = 1. Hence, Condition 3.3 also applies to the Type I
Natural Population problem.
An algorithm that can maximize H or minimize L is
presented in the next chapter.
CHAPTER IV
GROUPING ALGORITHM
4.1 Introduction
The grouping problem defined in Section 1.2 of
Chapter I may be an explosive combinatorial problem.
This may be judged from the total ntimber of all possible
solutions, T(n,N), for n parts and N groups. T(n,N) is
given by the following expression [3].
N T(n,N) = JP [Z (-1)^-^ (h (i)^] (4.1)
I i=0 ^
For even a small problem of n = 25 and N = 10, T(25, 10) =
1,203,163,392,175,387,500. Thus, traditional enumerative
techniques may be intractible for practical problems. For
instance, a dynamic programming approach to a similar prob
lem in plant taxonomy by Jensen [26] was able to solve only
very small problems. Some authors in Cluster Analysis [21,
44] suggest that no efficient techniques may exist for prob
lems of this type. In this chapter an iterative procedure,
analogous to the classical Gradient Optimization approach,
will be presented.
Iterative techniques have been reported for the Sta
tistical [44] and Non-Hierarchical [3] problem in Cluster
Analysis. Of particular interest to this study is one by
76
I
Rubin [ 41]. Although Rubin's method was intended for the
Hierarchical problem, the grouping objective function was
formulated in terms of similarity coefficients. Most of
the reported techniques have features for specifying ini
tial solutions, a search technique and a stopping rule
[3, 41]. Rubin reported that several local optima exist
in the solution space of the grouping problem and that
the effectiveness of the iterative approach may depend on
the initial solution. Thus, in the algorithm presented
here, a heuristic designed to obtain "good" initial solu
tions is used. The gradient method of solution search is
explored and a feature for checking the quality of local
optima presented. The heuristic will be discussed first.
4.2 A Heuristic for Initial SolutioiT
This heuristic works in two stages. In the first
stage, N indicator groups are chosen. An indicator group
is one with only one representative part. In the second
stage the remaining parts are assigned to the indicator
groups. Both stages make use of a concept called Prefer
ence Index as a decision rule. This concept, which is
unique to this study, will now be discussed.
For now, a Preference Index will be defined for the
ftinction H. A Preference Index, Y .:, may be defined as the
net contribution which part i will make to H when part i is
77
78
assigned to group j. To illustrate, consider a grouping
problem of five parts (n * 5) and two groups (N « 2).
Excluding similarity coefficients of the type R.., each
part has four similarity coefficients. Asstime that, in
the process of group formation, the set of parts in group
one (gj ) is (1, 3} and that in group two (g2) is {4,5}
while part two is a candidate part. The current values
of H^, H2 and H are as follows.
H,=!li:^.u = ! « 1 ^ and
H, + H« RlQ + ROT RAC: + Rr/ H = -i -^ = (-i 31) ( 45 54)
Notice that the similarity coefficients of part two, R., ,
^23' 24 ^^ 25 ^ ® currently not involved in the computa
tion of H. The assignment of part two to g changes H^
jj ^ Rl3 + R31 + R2I " 12 " 23 " 32)
^ 6
while H2 remains the same. That is, R2^ and R2^ do not
contribute to H, the average within group similarity.
This means that the assignment of part two to g contri
butes the average of R2] and R23 (R2] = R] 2 ^^ 23 ~ 32^
to H while the average of R24 and R25 are lost. Similarly,
the assignment of part two to g2 contributes the average
79
of R24 and R25 to H while losing the average of R2, and R^^.
In general, the assignment of a part to a group results in
a contribution and a loss of similarity coefficients to the
average within group similarity, H. It is the net contri
bution resulting from the assignment of a part to a group
that is referred to as Preference Index. Using symbols.
Preference Index of part i to group j , y.., is given by
n. n-n. -, J J-l
k«l il=l 'ij n. ^"^1-1 \^-^^
where keg. and il^g.; n. is the ntimber of parts in group j
excluding part i.
Similarly, Preference Index can be defined between
two parts. Denote Preference Index between parts i and a
as i/;. . = Then,
n-2 ^ _j_ n-2 j _ ^ k^l ^^ kSl ° ^ , i+a+k. (4.3)
' la la 2 (n-2)
^. is the net contribution parts i and a will make if la
assigned to the same group.
For N groups, N Preference Indicies may be computed
for each part. The assignment of parts to groups on the
basis of maximum y^. appear intuitively appealing in a
procedure to maximize H. This is apparent in some of the
ik^^S&ii
80
properties inherent in the definition of y.. or ij;., .
For instance, if Yj *<0, then i may be assigned to a group
other than group j. It is an indication of a negative
contribution. If y. . = 0 for all j , then part i may be
assigned to any group. For Y4 4>0, it is a positive con-
tribution. The same properties may hold true for ^"^,
the Preference Index between two parts.
To group parts using the Preference Index concept,
two initial indicator groups ir and 1^2 ^® formed by as
signing part i to -n-. and k to 1^2 such that ^jy., the simi
larity coefficient, is minimtim. (ir. denotes indicator
group i.) If more indicator groups are desired a compo
site Preference Index, 0., is computed for all candidate
parts:
e. = Z i|;., (4.4) ^ keir ^^
where TT is the current set of parts in indicator groups
and i is a candidate part. The part with mine ^ is assigned
to the next indicator group. The process is repeated until
N indicator groups are formed. The computation of y ^ ,
II;., and 9. are illustrated in Example 4.1. ^ik 1.
The next stage involves the assignment of the remain
ing candidate parts to the formed indicator groups in an
iterative manner. It is done on the basis of maximum y ^ .
• »
Input PCM Matrix, n; Compute R,
For Min R Assign i to IT. & j to ir.
Update _ F, Kount, N
YES
Compute. 9.
I For Min9
Assign i to ''' + i
Compute y .,
I For Maxy^j^,
Assign i to g.
I Update Kount, F
Compute V(G )
81
Figure 4.1. Flowchart of the Preference Index Hueristic.
82
Because the Preference Index concept is central to this
heuristic it will be called the Preference Index Heuristic.
This heuristic is summarized in the flowchart of Figure 4.1.
A detailed outline is presented below. The following nota
tion will be used:
G^ = Completed grouping or partition
F « Current set of candidate parts
TT. « Indicator group j
Kount - Current ntimber of parts in groups
N = Current ntimber of indicator groups
N - Specified number of groups
n = Total number of parts
g. = Group k
V(G ) = Grouping objective function.
(i) Outline of Preference Index Heuristic.
SEGMENT I Formation of indicator groups
STEP 0 Compute similarity indices
STEP 1 : Initialize F, N, Kount
Note: (All ties are broken arbitrarily)
STEP 2 : Select min R., and assign i to TT
and k to 1^2
STEP 3 : Update F, R and Kount
STEP 4 : If N = N, go to Step 6; else, go to 5
g^j^jjg^^-j
STEP 5
SEGMENT II:
STEP 6
STEP 7
STEP 8
83
Compute 6^ for all candidate parts and
indicator groups. Assign i to 7rjj_^ such
that 9jL is minimum. Go to Step 3
Assignment of remaining parts to group
Compute y^. using Equation (4.2) for
all i (candidate parts) and j (indi
cator groups). Select max y.. and
assing L to group j
Update F, Kount
If Kotmt = n, stop; else, go to Step 6.
An illustrative example will now be presented.
Example 4.1.
The problem discussed in Example 3.1 will be
used. The similarity coefficients of Figure 3.1c are
repeated here for convenience.
^j =
1
2
3
4
5
.86
1.0
0.0
0.0
86
86
14
14
1.0
.86
0.0
0.0
0,0
.14
0.0
0.0
.14
0.0
.86
.86
84
The problem is that of assigning the five parts of
Figure 3.1c into three groups in order to maximize H; the
Preference Index Heuristic is to be used.
SOLUTION PROCEDURE:
STEP 0 : Similarity indices were computed as
shown in Chapter III.
STEP 1 : Initialize parameters.
Current set of candidate parts,
F « {1. 2, 3, 4, 5}
Ntimber of indicator groups, N = 0
Current ntimber of parts in groups,
Kount = 0
Ntimber of parts, n = 5
Ntimber of groups, N = 3
STEP 2 : Formation of first two indicator groups
Select min R^. and assign i to first
indicator group TT and j to second in
dicator group 1^2
Min R^j = {R1L4, Ri5> R34' 35^ "
Decision: Choose R^^ arbitrarily and
assign 1 to TT and 4 to 712-
TT = {1}, TT2 = {4}
STEP 3 ' Update parameters.
Candidate parts, F = {2, 3, 5}
N = 2, Kount = 2
85
STEP 4 : Are there enough indicator groups?
(N+N) NO, go to Step 5
STEP 5 : Form next indicator groups using the
preference index, ^. , for parts i
and a [see Equation (4.3)].
For candidate part i = 2
^ _ (R23+R24+^25^ " ^ 13" 14* 15 21 " 21 2
35 _ .86+.14+.14+1.0+0.0+0,0
= .50
Similarly, i|;24 = -.31
®2 " 21 " 24 " '^^ ~ - - " '^^
For c a n d i d a t e p a r t 3
11 3 = . 7 1 , i| 34 = - . 4 8
and 63 = .71 - .48 = .23
For candidate part 5
^51 " ''^^' ^54 " ' "
6. = .33
Minimum {62, 63, 65} = min {.19, .23, .33} = 62 = .19
Decision: Assign part 2 to next indicator group.
.3 = {2}
86
STEP 3
STEP 4
STEP 6
(Repeated): F = {3, 5} , N « 3, Kount = 3
(Repeated): Are there enough indicator
groups? (N = N); YES, go to Step 6
Assign remaining parts to groups using
preference index between candidate part
i and group j, Y^j
For candidate part 3
31 ^31 R32 - R34 + R35
j=n:
1.0 - -86 + 0.0 + 0.0^ -^^29
= .71
In the same manner.
Y32 == --62, y^, = .53 33
For candidate part 5
^51 " -•^^' 52 " - ^ ^^-^ ' --^^ 53
Max (y3^, y32, Y33, Y5i» Y52» Y54) -
Max (.71,
Decision:
STEP 7
STEP 8
-.62, .53, -.33, .81, -.15) = y52 = .81
Assign part 5 to group 2
Tr2 = {4, 5}
Update F and Kount
F = {5}, Kount = 4
Any more candidate parts? YES, there
are because Kount<n. Repeat Step 6
through 8
87
STEP 6 : (Repeated):
^31 = - ^ 32 = --62 y33 = .53
Max (y3^, y32, y33) = Max (.71, -.62, .
=^31 = -71
Decision: Assign part 3 to group 1,
•^-^ = (1, 3}
STEP 7 : (Repeated) :
F = (0), Kount = 5
STEP 8 : (Repeated): Any more candidate parts?
NO .
Compute H and stop,
TT = (1, 3 ) , 772= ^^' ^)>^3 = (2)
GQ = {(1, 3), (4,5), (2)}
53)
H 1
«1
H3
H
V(G^)
=
=
=
=
=
' '13
^45
0.0
^ ^31 2
+ R54 2
H^ + H2 +
3
H = .62
=
=
H3
1+0
.86
+ 2
1
1.0
.86 3
= 1
=
.0
,62
88
4.3 Theoretical Background of Grouping Algorithm
The principle explored here is similar to the one
used in solving classical optimization of nonlinear ftinc
tions in Operation Research. Of particular interest are
the gradient methods which have been extensively discussed
by Gottfried and Weisman [19]. A brief description is pre
sented here to clarify the discussion.
The fundamental equation for generating levels of
state variables in the gradient methods is given by the
following expression:
where
f (Z) = Ftinction to be optimized
Z - State variable
Z .1 = Current level of Z m+1
Z = Previous level of Z m
m = Iteration ntimber
p = Step size
and, Af(Z ) = Gradient of f(Z) which may be defined as
follows:
Af(V =f(2nrfl> - f^V- ^''•^^
89
This is a difference equation which may enable the compu
tation of Af(Z^) for even complex functions, f(Z). Prac
tical methods of estimating ^f(Z ) using Equation (4.6)
exist [19].
To optimize f(Z), one starts with an initial solu
tion, [ZQ,f(ZQ)], compute ^fiZ^) and generate the next
level of state variable, Z^, using Equation (4.5). In
essence, this is a search at a known neighborhood in the
direction of maximum gradient, +Af (Z) , or minimum gradient,
-Af(Z) as desired. The optimum solution corresponds to the
last level of state variable where Af(z) vanishes. The
grouping algorithm will follow the same logic. The con
cepts of state variable, step, gradient^ etc., as applied
to our combinatorial problem will be defined in the course
of the discussion that follows.
A partition, G, was earlier defined as the set of
groups {g^; g2;.-.g^;»»»gN^ °^ combination of parts. Thus,
in the grouping problem G is the State Variable. A parti
cular combination of parts is the level of the State Vari
able. To illustrate, consider the solution to Example 4.1:
Go ' Sl' ^2' S3> ' ^1' 3' 2; 4, 5}
is one level. To move this level, G^, to another level,
say G , a part may be transferred from one group to another
90
G^ = (1; 2; 3, 4, 5},
or two groups interchange respective parts:
G2 = {1, 4; 2; 3, 5}.
This process of systematically changing a state will be
called a Step Operation; a formal definition of Step
Operation follows.
Definition 4.1
With respect to a known level, G , a Step
Operation is the movement of a set of parts from a donor
group, g,, to a receiving group, g . There are two types:
(i) Transfer Step involves the transfer of a set
of parts, all from one group, to another group. The ntimber
of parts in the set may be restricted to one or more. The
transfer step is illustrated in Figure 4.2(a).
(ii) Interchange Step occurs when two groups mutually
interchange an equal ntimber of parts (does not include in
terchanging all the parts from each group). An example is
presented in Figure 4.2(b).
Notice the analogy in creating the level of a state
in the classical gradient methods and the one just described.
Even though the step operations may be different, both can
be regarded as a process of searching in the neighborhood
of a known level. This introduces the concept of neighbor
hood which will now be given a concise definition.
91
G.: Partition before •'" transfer step
Partition after transfer step
(a) Transfer Step
Partition before interchange step
Partition after interchange step
(b) l^asfer Step
4.2. Illustration of step operation
92
Definition 4.2
The Neighborhood of a Known Level or Parti-
tion is the set of all partitions that can be created from
the known level by means of a step operation. Since there
are two types of step operation, a level has two types of
neighborhoods. Thus, if (|) denotes the neighborhood of a
partition, then (^^ is the neighborhood created by the trans
fer step and <()j, that by the interchange step. Observe
from the illustrative levels above, G, and G2, that the set
of partitions in (|>,j, and <^.^ may be different. The signficance
of this will be explored later in the chapter.
The ntimber of levels in the neighborhood of a known
partition depends on the ntimber of parts moved per step
operation. For instance, consider a step operation invol
ving the transfer of one part per step. There are N-1
possible ways of transferring each part from its original
group. For n parts, the total ntimber of transfers, K, is
given by
K = n(N-l).
Thus, if X((J)„) represents the ntimber of levels in the
neighborhood of a partition, then
X ( 0 = K = n(N-l) (4.8)
93
A similar derivation is possible for the number in (|> neigh
borhood. Equation (4.8) shows that the neighborhood of a
partition has a fixed number of other partitions. Hence,
like in the classical gradient methods, the search for a
better level of State Variable can be restricted to any of
the neighborhoods per iteration.
The term "better level" introduces a sense of "dif
ference" or gradient of a ftinction. Let V(G ) be the func
tion to be optimized (in our grouping problem) evaluated at
the level of State Variable, G [V(G) may be H, L or A de
fined in Chapter II] . The gradient of V(G^) will now be
defined.
Definition 4.3
Let A J (j) be the measure of change of the
function V(G ) by transferring part j from group d to group
r of the partition, G^. Thus, if G^^ is the partition or
level of State Variable obtained after the transfer step,
then the A, (j) is the component gradient of V(G^) in the
<t)m neighborhood.
^^^(i) =V(G^i) - V(G^). (4.9)
Equation (4.9) is a difference equation. Rearranging (4.9),
94
Since there are K [see Equation (4.8)] levels in the neigh-
borhood of G , there will be K component gradients. Let
the set of component gradients of G^ be represented by A° .
Hence,
A""- {Adr(J>>
s the Gradient Vector of V(G^). Rewriting Equation (4.10)
in terms of maxA™:
^^Vl> = V(Gm> +=^A'" (4.11)
for the maximization objective, or
m ^(^m+l> = V ^ V - ° A" ( -12)
for the minimization of V(G) . Combining Equations (4.11)
and (4.12),
V(G^^) = V(G^) tmaxA"'. (4.13)
The similarity between Equations (4.9) and (4.6) should be
noted. Recall that Equation (4.6) is the difference equa
tion for computing the gradient of nonlinear functions in
the classical gradient methods. Of greater importance is
Equation (4.13). It provides a basis for a grouping pro
cedure to search in the direction of maximtim gradient,
95
+maxA , or minimum gradient, -maxA°^. Hence, combining
Equation (4.13) with the step operation, a search model
similar to the one described for the classical gradient
techniques may be fabricated.
One characteristic of gradient techniques is that
only local optimum solutions for nonquadratic ftinctions
may be guaranteed. A definition of a local optimum solu
tion for our combinatorial problem will, therefore, be
given. A formal definition of the grouping algorithm and
a proof of optimality will then be presented in the sec
tion that follows.
Definition 4.4
A local optimum solution occurs in the
neighborhood of a level of State Variable, say G^, when
maxA <0.
4.3.1 Grouping Procedure
Like the classical gradient methods, the
grouping procedure starts with a known level of State Vari
able. G , and the corresponding function value, V(GQ). A o
search for better solution in the neighborhood of G^ is
performed. If none better than G^, then stop. Otherwise,
replace G^ with the best level and update V(G), With this
in mind, the grouping procedure will now be given a formal
definition.
.. •BBtBi.n.tik-..;' ,.
96
Definition 4.5; Grouping Procedure
Let X be the set of all solution points
where a solution point is the value of the function V(G ) m
evaluated at the m level of the State Variable, G. The
grouping objective is the maximization of V(G) . Let X
be a subset of X where the points in X form a sequence
XQ, X,, X2,...x , x , ... Let P be the mapping that it-
eratively generates the subset X ; P is defined as follows: IT
(i) X initial point in X is specified
(ii) x^ means V(G^)
(ii) The sequence x^, xm_j,j eX implies that
X . 1 = X + maxA for all m. m+1 m
(iii) maxA >0
(iv) If maxA™£0, stop; else, continue to generate
the sequence in X .
Thus, P:X->X„ is a point to set mapping. P
To show that P will terminate after a finite number of it
erations, it is necessary to demonstrate that X is a finite
set. Earlier in this chapter, the number of all possible
levels of State Variables in the grouping problem was ex
pressed as follows:
T(n.N)=^,[J^(-l)^-^(?)(i)"3
97
where
T(n,N) = number of levels
n « number of parts
N ~ number of groups.
From the above it is clear that T(n,N) is a finite set
when n and N are finite. Since there is a one-to-one
correspondence between a level of State Variable, G, and
function value V(G) , then the ntimber of parts in X (set
of solution values) equals T(n,N). Hence X is a finite
set when n and N are finite. This is stated as a lemma
with no further proof.
LEMMA 4.1
X, the set of all solution values of the
problem of grouping n finite parts into N mutually
exclusive and nonempty subsets, is a finite set.
In order to show that the procedure, P,
will converge at a local optimum in a finite
ntimber of steps it is necessary to define some
possible relation between partitions. (Partition
is interchangeably used with level of State Vari
able. It is a physical representation of the set
of groups.) Two partitions may be identical,
equivalent or distinct. For a given problem, two
partitions G. and G. are identical if (1) both
have the same combination of parts and (2) V(Gj ) =
98
V(G-); Equivalent Partitions if (1) G. and G. J !• J
have different conibination of parts but (2)
V(G^) = V(G.). G^ and G. are Distinct Parti
tions when (1) both are different combinations
of parts and (2) V(G^)+V(G).
THEOREM 4.1
Consider the grouping problem of n parts
and N groups where n and N are finite. An algorithm
using the procedure P (Definition 4.5) to solve this
problem will (1) terminate after a finite ntimber
of steps at (2) a local optimtim.
PROOF
In the mapping P
X. .T = X. + maxA L+1 1
Rearrangement of this equation results in
'i+1
Thus, Equation (4.14) shows that
x..n - X. « maxA^>0 (4.14)
X. ,1 >X. , V i . 'i+l"'!
The set of points in X is, therefore, strictly
an ascending sequence
^o^V^2---^Vi+l^ (4.15)
99
To prove the first part of the theorem, we assert
that the partitions corresponding to x.ex , G , G-, , G^ 1 p o 1 Z
...G^, G _|_ ... are all distinct partitions. Sup
pose by contradiction, that they are not. Then there
exists, at least, two identical or equivalent parti
tions G^, G. generated by the procedure P, By defi
nition of identical or equivalent partitions.
^i "^j-
But this is not in conformity with the strictly as
cending condition established in Equation (4,15).
Thus, P can only generate distinct partitions. With
only distinct partitions, no cycling can take place.
Hence, the maximtim ntimber of points in X cannot be P
greater than these in X (X is a subset of X). By ?
Lemma 1, X is a finite set, X must, therefore, be
a finite set as required by the theorem. Conse
quently, ? must terminate after a finite ntimber of iterations.
For the second part of the- theorem, let X^sX^
be the terminal point generated by P. The maxA <0,
is the required stopping rule. But maxA^<0 is, by
definition, a local optimum. Hence, P must ter
minate at a local optimum, completing the proof of
Theorem 4,1.
100
From Definition 4.5 and Theorem 4.1 it appears that
P is a versatile procedure for solving the grouping problem.
For Instance, it is clear that replacing the equation
V(G»ri.i) - V(G^) + maxA'»
with
m V<G^l> = V(G^) - maxA
P can be used to minimize V(G) . Since the only restriction
in the definition of a criterion of optimization is that
V(G) be defined in terms of characteristics of parts, P
may therefore be used to optimize H, L or A. Also, in the
definition of P, no restriction was placed on the type of
step operation for generating levels of State Variable.
Recall that the transfer step generates partitions in <j),j,
neighborhood and the interchange in (|)j. Thus, P can be used
to search in either neighborhoods. The iterative procedures
reported in other disciplines of Cluster Analysis use only
the transfer step [3, 5, 41]. The interchange step is
unique to this study. The interchange step was introduced
because of the following preliminary observation. The
"local optimtim" in Theorem 4.1 turned out to be the global
solution in some cases while in others it is not. On
critical examination of those that are not, one common
characteristic was observed: two or more groups require
101
the simultaneous interchange of some of the respective
members (parts). Using only the transfer step in such
a situation, P converges to a local optimum different
from the global.
To illustrate, consider a solution to a hypothe
tical problem of eight parts presented in form of simi
larity coefficients below; the objective is to maximize
H. The ntimber of groups are three. Asstime that while
using the transfer step in
Parts
1
2
3
4
5
6
7
8
1
-
.67
.20
0
.67
0
.25
.25
2
.67
-
.17
.2
.5
0
.5
.5
3
.2
.17
-
.4
.4
.5
.4
.17
4
0
.2
.4
-
0
.25
.5
.5
Parts 5
.67
.5
.4
0
-
.25
.17
.2
6
0
0
.5
.25
.25
-
0
.25
7
.25
.5
.4
.5
.17
0
-
.2
8
.25
.5
.17
.5
.2
.25
.2
-
.th . the procedure P the solution obtained at the m itera
tion is G^ = (g^; g2; 83)-
gi = (1, 5); g2 = (2, 7); g3 =(3, 4, 6, 8)
and H = V(G ) = .504. m
102
A further search in the neighborhood of G results in m
maxA™ - -..044<0. Thus, P terminates and (G , V(G )) is
the local optimum. Now, let the global solution be
(G*, V(G*)) where
G* = (g*i; g*2; g*3)
It turned out that g*j = (1, 5); g*2 - (3, 6) « (2, 4, 7,
8) and V(G*) = .523. The difference between G and G* m
should be noticed. Groups 2 and 3 interchange the respec
tive sets of parts: (2, 7) and (3, 6) in order to arrive
at G*. The transfer step aloiie is inadequate here because,
in the context of the problem, 2 is highly similar to 7 and
3 to 6. Also, the set of parts (2, 7) are more similar to
other members of go without the set (3, 6). Hence, in order
for the "local optimtim" of P to coincide with the global
solution most of the time, the search may be done alter
nately in <t>m and <t>-r neighborhoods.
From the ongoing discussion, it is clear that the
efficiency of P may depend partly on a good initial solu
tion. Fewer iterations will result from using a good
starting solution. It was for this reason that the Pre
ference Index Heuristic was developed. Another factor
that may affect efficiency is the number of terms in the
evaluation of H, L and A. For N groups, H and L have N
c^Jb^^^Ll^Li •
103
terms, respectively [see Equations (3.1) and (3.3) in
Chapter III]. In the following section it is shown that
only two terms need computation per function evaluation
using P to optimize H or L.
4.3.2 Evaluation of H and L
Let A^^r^. V be the component gradient of
partition G resulting from transferring the set of parts
(ij...) from group d to r. From Equation (4.9), if G _-,
is the resulting partition, then
V(ij...) =V(«nri-l) -^(«) m
In terms of group homogeneity, H
A /.. N = H - H T , (4.16) dr(ij...) new old
^^^^^ \ew = (^mfl) ^^ «old = ^ V • ^°^'
H = l [H +H2... +H^ + H ... + % ] ( -17)
where H, and H are the similarity of the donor and re-d r
ceiving groups in the transfer step. Substituting H of
Equation (4.17) in Equation (4.16),
104
new ^drCij...) ' h [% + H 2 . . . +Hjj+ (H^ + H^]
- I [H^ + H2. . . + Hj, + (H^ + Hp] ^^ . (4.18)
In the creat ion of G^ ^ from G , using the transfer s tep ,
the terms H , H2,. . . H^ were not af fected. Only H, and H
change. Hence,
[H, + H + . . . H„] = [H, + Ho + H i 1 2 N-* new ^ 1 "2 * * * n-'old*
(4.19)
By substituting (H^ + H2 ... + H^^)^^^ of Equation (4.19) in
Equation (4.18),
H, + H« . . . + H,j (H ,+H ) A a rC—±——± -il nlH 4. d r^newi ^dr(ij...) LC ^ )old + ^ ]
^(H^+H2 ... H^)^^^ + a V » ^ ^ i . ^l^d^Vnew '• N N ^ N
(«d Void N
_ ^Vnew " ^Vold + ^\^new " ^\^old A _ yji n e w Vi. Vd>JLVJ. I i. new i. yjA.\j. fi on\
dr(ij...) N N ''• "
Put
and
Y . ^Vnew - ^Vold (4.20a) d N
^ _ ^Vnew ' ^Vold (4.20b) h: N
105
then
^dr(ij...) = Y ^ + \ (4.21)
Equation (4.21) defines the component gradient of transfer
ring a set of parts between two groups, d and r, in terms
of the change in the group similarity of the respective
groups. A similar argtiment holds for the interchange
step. Thus,
^dr(ij...; kl...) = d + \ (*-22)
where ^i;i-(ij . . . • kl ) ^^ ^^^ gradient due to the inter
change of the set of parts (ij . . . } and (kl..,} by groups
d and r, respectively. Equations (4.21) and (4.22) also
hold for the minimization of link between groups, L,
since it is defined in terms of individual groups. For
large ntimbers , these equations can be used to reduce com
putational requirement per function evaluation in the
optimization of H and L. This is not the case with the
minimization of A since it is not defined as a stim of
terms that depend on individual groups (see Section 2.4).
In the next section the details of an algorithm based
on the procedure P are presented.
106
'•'•' ?echniquf '"°"^'"^ Algorithm: The Gradient
Because of the close analogy between the pro
cedure P, defined in Definition 4.5, and the classical gra
dient methods, the grouping algorithm will be called the
Gradient Technique. The basis of the Gradient Technique
is P, A complete solution can be accomplished in three
stages. In the first stage, initial solution is specified
using the Preference Index Heuristic. In the second, the
transfer step is used to obtain a local optimum while, in
the third stage, the interchange step is used to check the
quality of the solution in the second stage. Improvement
is made if the solution of the second stage is of poor
quality. If desired, stages two and three are carried
through alternately until both converge to the same solu
tion. Notice that, if preferred, stage one or stages one
and two can be used independently.
The outline of the Gradient Technique has been pre
sented in the flowchart of Figure 4.3. The details of each
labeled block of the flowchart are given in Table 4.1.
TABLE 4.1: FLOWCHART DETAIL
BLOCK 0: Specify initial solution (see detail of
Preference Index Heuristic in 4.2)
BLOCK A: Computation of Gradient Vector
1. Compute A^^(j...) for all (j...), d, r
using Equation (3.22)
107
Specify i n i t i a l partit ion, GjL and V(G.)
Loop 1 NO
Compute Gradient Vector A^; create partitions
J^ith Transfer Step
B YES,
Loop 2
Replace G. with
i iJpdate solution value
V(G^^^) = V(G^)+maxA^
Compute A^ using the Interchange Step to create partitions
Loop 1: Search in P neighborhood
Loop 2: Search in _ neighborhood
YES
Replace G. with
. t Update solution value
•
V(G^_^^) = V(G^)+maxA^
YES
Figure 4 . 3 . Flowchart of the Gradient Technique.
"P " 1 1 : J
108
2. Set A^ = {A^^(j...)}
- S®^ d*r*(j*...) = '^^^^
BLOCK B: Optimality Test
Check if A^^^^(.^ ^ < 0
BLOCK C: Update Current Best Solution
1. Remove the set of parts (j*...) from
group d*
2. Assign (j*...) to group r*
(the resulting partition is G..^)
BLOCK D: Update Current Solution Value
V(G,^P =V(G,) +id*r*(j*...)
BLOCK E: Computation of Gradient Vector: Inter
change step is used to create partition.
1. Compute A^^^j^^^ . ^^ ^ for all d, r
and set of parts {j, m. ..}eg^, {k, l...}eg^
2. A^ = { jr(jm. ..; kl...)}
3- Vr*(jlml...; k*l*...) = " ^ "
BLOCK F: Optimality Test
Check if ^dr*(3*m*...; k*l*...) -
BLOCK H: Update of current Best Partition
1. Remove the set of parts {j*m*...} from
group d* and assign to group r*
109
2. Remove the set of parts {k*l*...} from
group r* and assign to group d*
BLOCK I: Update Solution Value
V(Gi^P - V(G ) + A^^^^^^^ . ^^^ ^
BLOCK T: Check if the search in both neighborhoods,
<l>m and < j, converge to the same solution.
The gradient algorithm will now be illtistrated with
an example. The same problem described in Figures 3.1(a),
3.1(b) and 3.1(c) used to illustrate the Preference Index
Heuristic will be solved. From Figure 3.1(c):
Parts 3 4
R-.eR =
Parts
1
2
3
4
5
1
-
.86
1.0
0
0
.86 1.0 0.0 0.0
.86 .14 .14
.86 - 0.0 0.0
.14 0.0 - .86
.14 0.0 .86
As in example 4.1, the objective is to maximize H,
MiUitliSfeii-j
110
SOLUTION PROCEDURE:
Let the initial solution be G and the
value of H corresponding to G^ be designated as
V(GQ); the gradient of V(G ) is A°
STEP 0 : Specify initial solution, G . From o
the solution of Example 4.1, G = o
Ul, 3); (4, 5); 2)
V(GQ) = .62 (the Preference Index
solution)
STEP A : Search for a better solution in the
neighborhood of <J)„ (the transfer step
is used).
From Equation 4.21, the component gradient
of transferring part 1 from group 1 to
group 2
^12(1) " 1 " 2
.here Y, ^^^^^^" f^-^^^^ from (4.20a)
,,, Y^ . ^"2)new -<Vold f,,, (4.20b)
CH.,) =0.0 because only part 3 remains ^ I' new
in group 1 after the transfer
of part 1
(H,),,,, !ii;i3i = hi^i^ - 1.0
t.Vi;
I l l
Y B 0.0 - 1.0 -J, ^1 ^J »-.33
rw - 14 " 41 " 15 " 51 "*• ^45 " 54 **2 new 5
_ 0.0 + 0.0 + 0.0 + 0.0 + .86 + .86
.29
(H2)old - ! ^ l l ^ = 81+-86 , S6
Y2 « '^^ I - ^ - -.19
A 2(l-) = --33 + (-.19) « -.52
Similarly, other component gradients are as follows:
A^2(3) = -.52, A 3 3_ = -.05, A^3^3^ = -.05
A2i(4) = -.50, A2i(5) = -.50, A23(4) - -.24,
^23(5) = --^^
^31(2) ' infeasible, A32(2) - infeasible
A° ={-.52, -.05, -.05, -.50, -.50, -.24,
-.24}
maxA° = -.05
112
STEP B : Test for optimality of the solution
obtained in the <i>^ neighborhood.
Is maximum A° < 0?
YES, because maxA° = -.05 < 0
Decision: Solution is locally opti
mal in the (|),p neighborhood, current
best solution is
GQ = {(1, 3); (4, 5); 2}, V(G^) = ,62
STEP E : Search for a better solution in the
neighborhood of ({) (the interchange
step is used) with G as the current
best solution.
From Equation (4,22), the component gradient re
sulting from groups 1 and 2 interchanging parts
1 and 4, respectively,
A^2(l. 4) = Y^ + Y2
where Y and Y2 are as defined in Step A,
^ I' new 2 Z
^13 " 13 _ 1.0 + 1.0 _ ^Vold " ^ = 2 = -Q
113
(Ho) - ^^5 ^ ^51 , 0.0 + 0.0 . 0 0 ^2''new Z 2 "•"
fH ^ - ^^5 "*" 54 _ .86 + .86 «. ^^2^old 2 2 ' -^^
V _ 0.0 - .86 _ oo
A3 2( » ^) * --^^ --29 » - . 62
S imi lar ly , other gradient components are:
A^2<1» 5) = - . 6 2 , A^2(3' ^> * - - ^ 2 '
Ai2(3. 5) = - . 6 2 , A^3(l, 2) = - .05
A^3(3, 2) = - . 0 5 , A23(4, 2) = - . 24
A23(5, 2) = - . 2 4
A'' = {-.62, -.62, -.62, -.62, -.05, -.24,
-.24}
maxA° = -.05
STEP F Test for optimality in the neighborhood
o f <j)-jp
maxA*'^ 0? YES. maxA° = -.05 < 0.0
Decision: Solution has converged to a
local optimum.
Current best solution remains G^
^o = {(1, 3); (4, 5); 2} , V(G^) = .62
114 STEP T : Check if best solutions obtained in
(.p and 0j are the same.
Notice that the solutions obtained in Steps
B and F are the same. Thus, the algorithm
will terminate as indicated in the flow
chart. .
FINAL SOLUTION:
GQ = {(1, 3); (4, 5); 2}
H = V(G^) = .62.
By totaling entimeration G has been verified as the
global partition. It should be noted that in this partti-
cular example, the Preference Index Heuristic also resulted
in the global optimal solution.
In the above example H is maximized. As earlier
stated, L and A can also be minimized by the Gradient Tech
nique. The possible relation between H and L was demonstra
ted in Chapter III and that between L and A (additional
machines) discussed in Chapter II. Since H and L may have
negative correlation, a starting solution for the maximiza
tion of H may also be used for the minimization of L. Thus,
the Preference Index Heuristic may be used to specify star
ting solution in the minimization of L or A.
115
4.3.4 Computational Experience
The dependence of the efficiency of the Gra
dient Technique on the starting solution and number of terms
involved in function evaluation was previously mentioned.
Another factor may be that of problem size. A large ntimber
of parts may adversely affect the efficiency of the algori
thm because the number of component gradients computer per
iteration depends on the number of parts.
There is also a problem of effectiveness since P guar
antees only a local optimtim solution. Although the combina
tion of the transfer and interchange steps may result in
global optimal solutions, it may depend on the number of
parts moved in a step operation. To verify the effective
ness and efficiency ntimerically, therefore, 400 hypotheti
cal problems were solved by the Gradient Technique and then
by total enumeration. The grouping objective was the maxi
mization of H. Both the Gradient Algorithm and entimeration
procedure were coded in FORTRAN IV and run on the Itel com
puter at the Texas Tech Computer Center. The problems were
generated randomly in the PFA format. There were six machines
for each problem and the number of operations which a part can
have was uniformly distributed between two and four; each ma
chine was equally likely to perform any one operation.
116
Since only small problems can be solved by enumeration,
the number of parts vary from 7 to 15 and the number of
groups from 2 to 4. The Gradient Algorithm, the number of
parts in a set per step operation, were limited to one in
the transfer but two in the interchange step. The solu
tions to the first twenty problems have been tabulated in
Table 4.2. For these twenty problems, the Gradient Tech
nique converged at the global solution when the transfer
and interchange steps were combined. However, only eight
out of the twenty were globally optimal when the transfer
step was used alone. For the 400 problems, 95 percent
were solved globally when both steps were combined, but
about 60 percent when only the transfer step was used.
These results tend to suggest the Gradient Algorithm as
an effective approach. It also shows that the interchange
step is a significant feature.
Also shown in Table 4.2 are the computation times
for the entimeration as compared to the algorithm. These
have been plotted in Figure 4.4. From the data and the
plot it appears that the Gradient Technique is relatively
efficient for the range of problems solved.
In addition to the 400 small problems, larger prob
lems ranging from 40 to 120 parts (the generation of these
problems is explained in Chapter V) were solved with the
Gradient Algorithm. Only the interchange step was used
Iii S ' l l f f '^''^'^ T° ° ^ .'•ISJ'A'
117 -ini!
14-1 . 0
km
7
7
7
8
8
8
9
9
9
10
10
10
11
11
12
12
13
14
15
o u o. 9 9 m u
2
3
4
2
3
4
2
3
4
2
3
4
2
3
TOTAL ENUMERATION
Time (Sec.)
.020
.110
.140
.050
.360
.710
.100
1.21
3.48
.22
4.06
16.42
.47
13.48
1.02
44.49
144.30
4.84
10.49
Global Solution
(H*)
.634
.702
.554
.703
.745
.733
.701
.723
.764
.705
.724
.736
.688
.704
.673
.693
.699
.675
.668
GRADIENT ALGORITHM
Transfer and Interchange Steps
Time Solution (Sec.) (H)
.010
.010
.010
.010
.010
.020
.020
.020
.020
.020
.030
.020
.030
.030
.030
.050
.060
.040
.060
Transfer Step Only Time Solution (Sec.) (H)
.634
.702
.554 j
.703
.745
.733
.701
.723
.764
.705
.724
.736
.688
.704
.673
.693
.699
.675
.668
.010
.010
.010
.010
.010
.010
.010
.010
.010
.010
.020
.020
.020
.020
.020
.030
.030
.030
.030
.634
.702
.554
.654
.745
.729
.628
.688
.764
.616
.668
.736
.592
.704
.673
.682
.688
.675
.490
118
in order to reduce cost of experimentation since many
problems at this range were solved as further discussed
in Chapter V. A preliminary observation indicates that
it takes over three minutes to solve problems with n > 80
when the interchange step and the transfer step are com
bined.
The algorithm time appears to increase polynomially
with respect to the ntimber of parts and groups as indicated
in Table 4.3 and Figure 4.5. The dependence of computation
time on number of transfer steps per iteration is one pos
sible reason; each transfer step requires some amount of
computation time. As discussed in Section 4.3, the ntimber
of transfer steps per iteration depends on ntimber of parts
and ntimber of groups (see Equation 4.8). Thus, algorithm
time increases with ntimber of parts (n) as shown in Figure
4.5a. As n increases above — , some groups will have only ,
one member; the transfer of a part from a group with only
one part results in an infeasible solution. Hence, at
N > , the ntimber of transfer steps per iteration (or com
putation time) reduces as N increases. Thus, algorithm
time exhibits a concave curve with respect to ntimber of
groups.
The use of the Gradient Algorithm as a tool for
studying the behavior of GT systems will be illustrated
in the following chapter.
119
TABLE 4.3. ALGORITHM TIME VERSUS NUMBER OF PARTS AND GROUPS
• •
Number of Groups, N
2
6
10
14
18
22
26
30
34
38
42
46
50
54
58
62
66
70
i
n = 40
.51
.64
.86
.910
.840
.620
.56
.34
.31
Algorithm
n = 60
1.69
2.32
2.50
2.77
3.29
3.84
4.55
2.68
2.21
1.73
1.53
1.32
1.06
0.98
Time (Seconds) for
n = 80
3.81
4.76
5.66
6.06
6.42
7.83
8.14
7.80
7.68
7.10
6.3
5.16
4.31
3.68
3.46
3.00
2.69
2.42
n = 100
7.98
10.43
11.29
12.09
12.92
13.29
15.51
14.82
17.25
16.21
18.05
17.78
12.84
12.62
10.25
10.25
9.29
7.79
n = 120
13.88
18.03
18.41
19.94
20.96
20.11
22.16
21.66
22.85
28.49
28.86
31.07
30.45
53.31
22.68
22.64
22.64
20.69
150.0
140.0-
130.0.
120.0.
140.0.
100.0-
120
'J 80.OJ c o S 20.0J
03
Enumeration
--•--. Algorithm
14 15
Ntimber of Parts
Figure 4.4. Solution time versus number of parts
121
(a) Algorithm Time Versus Number of Parts
N = 2 N = 10
+ - V + 4 4 N = 30
N = 50
" N = 70
20 40 60 80 100 120
Number of Parts (n)
10 20 30
(b) Algorithm Time Versus Number of Groups
40 50 60 70
Ntimber of Groups (N)
Figure 4 . 5 . Algorithm time versus ntimber of p a r t s and groups.
^ ^ CHAPTER V
NUMBER OF GROUPS AND PRODUCTION COST
5.1 Introduction
As stated in Chapter I, the theme of this research
is development of tools to enable efficient and systematic
study of the behavior of GT systems. Noting that batchshop
problems are extremely complex, it is apparent that a com
prehensive study of most of the factors that may impact GT
systems will require a combination of several analytical
tools. Such an endeavor is beyond the scope of this study.
However, in this chapter, we wish to demonstrate that the
Gradient Algorithm may be suitable as one of the tools to
study the behavior of GT systems. In particular, the use
of the Gradient Algorithm to study how set-up time and
ntimber of additional machines may respond to changes in .
number of groups will be presented.
5.2 Study Procedure
In Chapter II an inverse relation between set-up time,
S , and group homogeneity H was established. Thus,
S = S^(l-H) t o
where 0 < H < 1 , S is a constant and S the sequence depen-— — ' o ^
dent component of set-up time. Thus, to study any relation
between S. and number of groups, N, it may be adequate to Km
observe how H varies with N,
122
•fafM,
I 123 i
To observe how H or A. number of additional machines
change with N, the following procedure may be desirable;
several batchshop problems collected from different manu
facturing environments may be coded in the combined formats
of the VZk and PFA. Another alternative is to generate hy
pothetical problems in the combined formats. For each
batchshop problem select several levels of ntimber of groups,
N = 2 , 3, 4,.... Notice that each level of N corresponds
to one constrained GT grouping problem. Thus, in each
batchshop problem, the Gradient Algorithm can be used to
solve several constrained problems first to minimize H (the
PCA approach) and then minimize A or the alternative L (in
the PFA approach). In each case (PCA, PFA) , both H (must
be defined in terms of design features) and A are computed
and recorded against the corresponding ntimber of groups.
The sets of data (H, N) , (A, N) in either the PCA or PFA
can then be analyzed to verify how H or A responds to changes
in N. Some examples illustrating this procedure will now be
presented.
5.3 Test Problems
In order to illustrate the described procedure eight
hypothetical problems were generated. There were four
levels of number of parts (60, 80, 100, 120) and two levels
of Variety of Parts: low and high variety. Operationally.
124
variety of parts is defined as the proportion of parts with
distinct process routes in the PFA; in the PCA it is the
proportion of parts with distinct Opitz code ntimbers. Let
n be the symbol of variety of parts. Thus, n- .3 was con
sidered low variety and f2 = .6, a high variety.
An ideal situation in a conventional shop is for two
parts which are identical with respect to design features
to have the same process route. If this were so, then a
solution to the grouping problem in either the PCA or PFA
approaches would have resulted in the same GT system for a
given problem. However, both Eckert [11] and Purcheck [37]
independently observed that this may not be the case; parts
with similar design features may be assigned different pro
cess routes in large conventional shops. Thus, in the
generated problems, even though two parts may be identical
in Opitz code number, they may or may not be assigned the
same process route.
The design features and process routes of parts were
generated to reflect the complexity of parts in a general
purpose shop. Past research in component statistics [16,
28] shows high frequency for simple features as compared
to complex ones. In the Opitz notation, this means more
lower valued numbers may appear on the digits of the code
number of parts. Ajreanencx count of the numbers on the di-
gits of well over a- thousand code numbers of parts in two case
studies show the following empirical distributions [22, 34].
125
Ntimbers Appearing on the Digits of Code Number of
Parts
0
1
2.3
4,5
6,7,8,9
Relative Frequency
.45
.20
.20
.11
.04
Similarly, the following distribution was observed for
ntimber of operations per part from case studies [37].
Ntimber of Operations Relative Frequency
2,3,4 .50
5,6,7 .35
8,9,10,11,12,13,14,15
There were 40 machine types for each problem. The ntimber
df each machine type was maintained at 1 in one case and
uniformly distributed between 1 and 3 in another case.
Based on the above distributions, the hypothetical problems
were generated in the combined formats of the PFA and PCA.
The parts studied were restricted to rotational parts
because they contribute the majority in batchshops (70 per
cent to 80 percent of parts) [16, 22, 28]. In addition to
the hypothetical problems, three case study problems were
taken from the literature [22, 34]. A summary of all
problems is presented in Table 5.1. The experiment was
run on the Itel computer at the Texas Tech Computer Center;
the Gradient Algorithm was coded in FORTRAN IV.
126
TABLE 5.1. SUMMARY OF TEST PROBLEMS
Problem
1
2
3
4
5
6
7
8
9
10
11
Number of Parts (n)
60
60
80
80
100.
100
120
120
43
82
86
Variety of Parts (Jl)
.3
.6
.3
.6
.3
.6
.3
.6
1.0
.62
.53
Approach
PCA, PFA
If fl
tt ft
ft ft
ft ft
If If
If tl
M II
PFA
PFA
PCA
Comments
Hypothetical
Burbidge [ 6]
Purcheck [37]
Harworth [22]
127
The sets of data (H,N), (A,N) for each problem have
been plotted in Figures 5.1 to 5.19 and the values have
been tabulated and presented in Appendix C.
5.4 Discussion and Presentation of Results
The observed results will be discussed in two parts.
The first part concerns the relation between H, group
homogeneity, and N, ntmiber of groups; the second will deal
with the relation between A and N.
(i) Relation Between H and N:
The graphs of Figures 5.1 to 5.9 show a concave
relation between H and N in the PCA approach. This obser
vation, which is true for all the problems solved, may be
explained as follows.
The concave structure tends to suggest the existence
of a specific ntmiber of families of similar parts. We re
present this number by N*. An optimal solution to the
grouping problem entails the assignment of these families
to respective groups. However, if a constrained problem
is such that N, ntimber of groups, is less than N*, then
the members of a group may be a mixture of parts from more
than one family. Thus, group similarity, H, may become
relatively small. As N approaches N*, more families get
assigned to individual groups. In effect, H is improved.
At N = N* each group contains just one family of similar
parts and H reaches a peak value.
; i , .
128
H
1.0
.8
.6-
.4
.2
0.0
« . « « -«*" PCA
• «
PFA . • • ~ •
• •
• • • •
0 10 20 30 40 50 60 70 N
Figtire 5.1. Group homogeneity versus ntimber of groups; n = 60, n = .6
H
1.0
.8
.6.
.4-
.2
0.0
' PCA
. - • • « • • PFA
% • • •
• • •^UU
0 10 20 30 40 50 60 70 N
Figure 5.2. Group homogeneity versus ntimber of groups; n = 60, ^ - .3
129
1.0
.8
.6
H -4
.2
0,0
* « * **,««« PCA
y Ktf
a •
PFA • . • • - • • • ' ' # • » . »
0 10 20 30 40 50 60 70
n ^ SO ^ = 6 '°'' homogeneity versus number of groups;
rt
1.0
.8
.6-
.2-
0.0
» * • ' - • • * , * »
PCA
PFA ' * • • • • .
• •
•
0 10 20 30 40 50 60 70
Figure 5,4. Group homogeneity versus ntimber of groups; n = 80, fi = .3
130
H
1.0
.8 .
.6
.4
.2
0.0
»-• •* .. PCA
• »
PFA
0 10 20 30 40 50 60 70
N
Figure 5.5. Group homogeneity versus number of arouDs-n = 100, Q = .6 to . »
1.0
.8
H
•"it PCA ' x»
««
.2
0.0 I
• * • • • •
PFA
• • •
0 10 20 30 40 50 60 70
N
Figure 5.6. Group homogeneity versus ntimber of groups; n » 100, fi - .3
H
1.0
.8
.6
.4j
.2
0.0
« # « » ^'
ar « '
-. PCA « w"" •I'm,
m m
M9\t
. • • • * • • •
PFA • • « • •
• • • • •
131
0 10 20 30 40 50 60 70
N
Figure 5.7. Group homogeneity versus ntimber of groups; n = 120, « = .6
H
1.0
.8.
.6.
.4.
.2j
0.0
IC^'JI- PCA
PFA
10 20 30 40 50 60 70 N
Figure 5.8. n = 120, fi
Group homogeneity versus number of groups;
= .3
132
H
1.0
.8
.6
.4
.2
0.0
" • " ^ " "
, - ,»««Jf«ir«*?^^ * *
ar » ar
ar M
ar
X K • i f
<
0 10 20 30 40 50 60 70
N
Figure 5.9, Group homogeneity versus ntimber of groups; n = 86, fi = .53 (Harworth's problem).
133
At N>N* additional groups may be created only by
taking members from the families. In doing so, only the
members with less relation to their respective families
may be removed. The group homogeneity of the established
families improves as a result, but that of the newly created
groups may be poor. The net effect may be a slow reduction
of H as N slightly increases beyond N*. This may be respon
sible for the flattened appearances of the peak portions of
the concave curve (Figures 5.1 to 5.9). The average number
of parts per group ( /N) may also be responsible for the
decrease of H for N>N*. As N increases at this range, some
groups may have only one member. By definition, H, = 0
when Uj = 1. Hence, as N approaches n (the ntimber of parts)
H approaches zero. In Section 5.7, it will be illustrated
that, from cost point of view, the smallest N which yields
a high value of H may be better than N*.
In the PFA approach the concave relation between H
and N is not so pronounced as depicted by the graphs of
Figures 5.1 to 5.9. This may be because A (ntimber of addi
tional machines) instead of H (which is defined in terms of
the attributes used in the PCA) was optimized.
(ii) Ntimber of Groups and Additional Machines. The
graphs of Figures 5.10 to 5.20 show that the ntimber of ma
chines increase with ntimber of groups. This is because as
N approaches 1, most of the parts belong to the same group
and few machines are duplicated.
134
200 <
150 •
100 -
50 .
0
• *
^ * • • ' f t
• • • - f t
« • • ar
• - •
' ' • 1 • • , . I
0 10 20 30 40 50 60
N
70
Figure 5.10. Ntimber of additional machines versus ntimber of groups; n = 60, ntimber of each machine type = 1, PFA.
300
250 -
200 -
150
100
50
• - •
<b ^ •
9.. .* • •
• ft IT
ft
. ^ -^ ft ft g ,
•
ft «
^ •
K m
. ^M
• ar
^ I I t • • .
0 10 20 30 40 50 60 70
N
Figure 5.11. Number of additional machines versus ntimber of groups; n = .80, number of each machine type = 1, PFA.
135
Figure 5.12. Number of additional machines versus number of groups; n = 100, number of each machine type = 1, PFA.
A
300
250-
200-
150-
100-
50-
jr X
• • ft*
ft -<o •
• ft
ft
« «
• «
« f t . *
«
0 10 20 30 40 50 60 70 N
Figure 5.13. Number of additional machines versus number of groups; n = 110, ntimber of each machine type fi = 1, PFA.
136
F i g u r e 5 .14 , Ntimber of a d d i t i o n a l machines v e r s u s ntimber of groups ( B u r b i d g e ' s ) ; n = 43 , fi = 1 .0 , ntimber of each machine type = 1, PFA,
300
0 10 20 30 40 50 60 70
N F i g u r e 5 . 1 5 . Ntimber of a d d i t i o n a l machines v e r s u s ntimber of groups (Purcheck ' s) ; n = 82, fi = . 4 5 , number of each machine type = 1,
137
500
400 ^
300 •
200
100 -
PCA « «
« « K i t ai « « ftjiV ft « a a
< • ! * > l l NT
I k » »
• * • • >
• • « « PFA
0 10 20 30 40 50 60 N
70
Figtire 5 .16. Ntimber of addit ional machines versus ntimber of groups; n - 60, fi = . 3 , ntimber of each machine type: uniformly distr ibuted (1 , 3 ) .
100 -
Fi- ure 5 17 Number of additional machines versus number of groups; n = 60 fi - .6; number of each machine type; tiniformly distributed (1, i) .
138
500
400 .
A 300 ,
200 .
100 .
,*•» "ft
^pg4^„„'*«*««—*
• • • •
• • • PFA
. « . . . » * » « *
0 10 20 30 40 50 60 70
N
Figure 5.18. Ntimber of additional machines versus ntimber of groups n = 80, fi = .3; uniformly distributed (1, 3) ntimber of each machine type.
500
400 .
300 -
200
100.
JC*
JIfft
PCA ^ ,
PFA ..
• •
0 10 20 30 40 50 60 70 N
Figure 5 19 Number of additional machines versus l ^ l r of groups; n ^ 80, fi. -6; uniformly distri-(1, 3) number of each machine type.
^ i i ^ j H ^ ' l
DUU
4 0 0 .
3 0 0 .
200-
1 0 0 -
0
PCA
, a - * -
PFA . # • • • <* •
ft • * *
. -
. f - . ^ r , " ' : " ' , , ,
0 10 20 30 40 50 60 70 N
Figure 5 .20. Ntimber of addit ional machines versus ntimber of groups; n .= 100, fi - . 3 ; uniformly d i s tr ibuted ( 1 , 3) ntimber of each machine type.
500
400.
300
200.
100
Figure 5 .21 . Ntimber of addit ional machines versus number of groups; n = 100, Q = .6; tiniformly d i s tr ibuted ( 1 , 3) number of each machine type.
140
500
400
300
200
100-
0 10 20 30 40 50 60 70
N
Figure 5.22. Ntimber of additional machines versus number of groups; n - 120, fi = .3; uniformly distributed (1, 3) ntimber of each machine type.
500
400.
300.
200
100"
PCA,-' m* «• »^
,XJ*« • • •
PFA
** - • • • • •
0 10 20 30 40 50 60 70 N
Figure 5.23. Number of additional machines versus number of groups; n = 120, fi " -3. uniformly distributed (1, 3) number of each machine type.
L\i.V
141
On the other hand, as N approaches the number of
parts, n. many groups may have only one member. This means
that, in the Exclusive Membership GT, theoretically each
part is processed by an exclusive set of machines at N « n.
The number of additional machines will be maximal in this
situation. It follows from these extreme cases that, for
l<N<n, every additional group may result in more sets of
parts which, being processed by the same machine in the
conventional system, now belong to different groups in
the Group Technology situation.
One Important observation is the large ntimber of
additional machines resulting from relatively small changes
in ntimber of groups in the PCA approach. For instance, in
Figure 5.22, there is an increase of ten additional machines
for every additional group created (4£N£20) in the PCA.
This tends to suggest that, in the GT approach, large ntimber
of groups may not be economically viable; the cost of ma
chines will likely offset any gains that may result from the I
reduction of machine set-up times. Even if the ntimber of
groups is small (say 8) , it may be necessary to re-plan the
process route of some members of the groups so as to reduce
machine duplication. This fact was mentioned by some GT
researchers [6, 7].
From the plotted curves of Figures 5.10 to 5.23, a
linear relation tends to hold between A and N. This
i&A^
142
linearity appears pronounced in the cases where ntimber of
each machine type is one (Figures 5.10 to 5.15); the curves
deviate from linearity at the higher values of N in those
cases where each machine type is uniformly distributed
(see Figures 5.16 to 5.23). The cause of the linear re
sponse of A to N is not obvious to the author.
As expected, there are fewer additional machines in
the PFA as compared to the PCA approach for the problem.
This observation which is true for all the test problems
supports the effectiveness of the Gradient Technique. Re
call that it is only in the PFA that A is minimized. The
curves of Figures 5.10 to 5.23 also indicate higher number
of additional machines for the high variety as opposed to
low variety problems since high variety of process routes
increases the tendency for the duplication of machines. In
the following sections these results will be used to explain
possible behavior of additional machine cost, set-up cost
and production cost with respect to ntmiber of groups.
5.5 Additional Machine Cost and Number of Groups
Let A(N) be the ntimber of additional machines in
terms of number of groups; A^(N) , the corresponding cost
per schedule period. Thus
A (N) = r^C^A(N) (5.1)
143
where C^ is the unit cost per machine and r, is depre
ciation rate of machines per schedule period. If it is
assumed that C^ and r^ are constants, then A (N) and A(N)
may exhibit the same behavior. The curves of A (N) may,
therefore, be similar to those in Figures 5.10 to 5.20.
To illustrate, the resulting additional machines in the
solution to Problem 5, Table 5.1 is used to compute A (N)
where C - $20,000 and r v - .01 per schedule period (see
Figure 5.24). It is clear, from Equation (5.1), that if
C changes with the number of groups, then the behavior a
of A (N) will deviate from linearity.
5.6 Set-Up Cost and Ntimber of Groups'
Similarly, let S (N) be the total set-up time with Km
respect to ntimber of groups and S (N) the corresponding
cos t .
S^(N) = CgS^(N) (5 .2 )
From the set-up time model of Equation (2.18) and denoting
the sequence independent component as S^, then total set
up time S (N) may be expressed as follows: Vm
S^(N) = S^[1-H(N)] + S^
Substituting S^(N) in Equation (5.2),
S (N) - C rS^(l-H(N))) + S^] (5.3) c s «J
144
50
45
40
35 -o o
^ 30
% 25 I o CJ
20
15 ,
10 .
5 •
0
A^(N)
^..^.^^^-^l-^*'"''
10 15 20 25 30 35 40 45 50
Ntimber of Groups (n)
Figure 5.24. An example of possible relation between set-up cost, additional machine cost, production cost and number of groups; n - 100, fi = .3, ntimber of each machine type - 1, GT approach =» PCA.
145 ^ ^
S^(N) may be a convex curve since 0 < H(N) < 1. Using the
values of H(N) of the solutions to Problem 5 in Table 5,1,
and setting
Cg = $15 per hour per schedule period
SQ = 2000 hours
and S^ = 500 hours.
The function S (N) is illustrated in Figure 5.24.
As N approaches 1, even if families of similar parts
exist, both similar and dissimilar parts may belong to one
group; changeover time or set-up cost may be relatively
high at this range. An increase of N from a very low value
may result in the assignment of highly similar parts to
production groups. Set-up cost may, therefore, decrease
until the number of production groups equal the ntimber of
families that exist. As N increases further from this
point, fewer similar parts may remain in a group to share
the same machine set-up. Consequently, set-up cost may
increase.
5.7 Production Cost and Ntimber of Groups
In Chapter II, it was discussed that additional ma
chine cost and set-up cost may be the critical components
of production cost in a GT system. Hence, the relation of
production cost to number of groups may be a combination
146
of those of S^(N) and A (N) L&t v rK\ A^ ^ . Q\ / V* A^\iij . Let F^(N) denote production
cost in terms of N. Thus,
P e W = S^W + A^(N) (5 4j
where l<N<n. In the cases where S^(N) is convex,
P^(N) may also be convex since A^(N) may be an increasing
function. Depending on the relative values of S (N) and
A^(N), optimal number of groups, N*, may exist. This is
shown in Figure 5.24 where P^(N) corresponding to Problem
5 of Table 5.1 is plotted.
However, if costly machines form the bulk of addi
tional machines the unit cost of machines [C in Equation
(5.1)] will be large. The curve A^(N) in Figure 5.24 will,
therefore, shift to the left causing the cost for additional
machines to be the dominant factor of production cost, P (N) .
In this situation, the ctirve, P (N), may increase monotoni-
cally making the GT approach unprofitable for any ntimber
of groups.
Also, the profitability of the GT approach may depend
on the value of S^ [see Equation (2.18) or (5.3)], the
maximtim possible set-up time that can be reduced by the
grouping of parts. If S is large, and family of similar
parts exist, set-up cost reduction will be large enough
to offset the cost of additional machines. A large batch-
shop where families of highly similar parts exist may be
147
ideal for GT; this situation favors large values of
So-
From the observed results (Figures 5.1 to 5.24), it
appears that the formation of small number of production
groups may be better than large ones. Small values of N
tend to yield lower values of set-up cost, S (N), [or
higher values of H(N)]. Large values of N will increase
cost of additional machines without any additional reduc
tion in set-up cost.
5.8 General Observations
From the results of this study some general obser
vations concerning the feasibility of changing over to GT
systems from the traditional arrangement have become ob
vious. In the discussion of the set-up time-similarity .
of parts model one of the asstimptions was that all parts '
in a group use the same set of machines. This means that
parts in a group be similar not only in design character
istics but also in process routes. Now, consider a situa
tion in the PCA where this assumption is relaxed; that is,
parts which are similar in design characteristics are pro
cessed by different machines. Set-up time for every opera
tion in the group will be sequence independent. No set-up
time reductions due to grouping can take place.
NiAi^i.
148
The same argument holds for the PFA. The grouping
of parts which are similar in process routes but dissimi
lar in set-up related characteristics may not result in
set-up time reduction. This is obvious from the set-up
time-similarity of parts model presented in Chapter II.
This suggests that both the PCA and PFA GT approaches may
be viable alternatives only if a set of parts similar in
the combined set of characteristics exist in the conven
tional shop. Thus, it appears reasonable to infer that,
if a conventional system is suitable for GT production,
then the PCA, PFA or even the combined approaches may not
differ significantly as previous researchers tend to claim
[6, 12, 16]. An optimal partition in the PCA approach may
be the same solution in the PFA as well as in the combined
approach.
5.9 Possible Application of Research Results
The results of this research may be of value both
for further theoretical studies as well as industrial ap
plications. As demonstrated in this chapter, the Gradient
Algorithm may be used as a tool to investigate the behavior
of GT systems under various manufacturing environments. As
stated in Chapter I, there is a need for experimental stu
dies of the behavior of Group Technology productions
systems.
149
The same procedure outlined in Section 5.2 may be
used to perform a feasibility study of a particular indus
trial batchshop problem in order to determine the possibil
ity of implementing GT. Grouping parts in the PCA or PFA
approach possible set-up cost savings and additional machine
cost may be determined. This will be based entirely on the
prevailing conditions of the problem which is in contrast
from the currently practiced "rule of thtimb" philosophy
that what works for company A will also work for company B
[45]. Considering a planning horizon, the determined set
up savings, additional machine cost and other expenses may
then be used in a cost-benefit analysis. The decision to
adopt or reject GT system can, therefore, be made in a more
objective manner.
Using a graphical approach, as shown in Figure 5.24,
the procedure of Section 5.2 may also be used to determine
optimal ntimber of groups for a particular problem.
The system of characterizing GT solutions discussed
in Chapter III may enable a GT system designer to decide
preliminarily which problem may be good for GT production.
For instance, if Multiple Type I (H=l, L=0) or Natural
Type I (H=l, 0<L<1) were identified then the designer may
infer that GT may be an attractive alternative. An expla
nation was presented in Section 3.3 of Chapter III.
CHAPTER VI
CONCLUSIONS
6.1 Summary of Research
This research dealt with the problem of grouping
parts for GT production systems. The grouping problem
was formulated as a combinatorial optimization problem;
the different classes were identified, its relation to
Cluster Analysis discussed, and the current state of the
art in the GT approach reviewed.
In an attempt to solve the grouping problem effi
ciently, a criterion of optimization was formulated.
Also, possible forms of solution to the grouping problem
were characterized and a grouping algorithm as a solution
technique presented. Using this algorithm as the tool,
a procedure for studying the behavior of production cost
in a GT system was outlined. To illustrate this procedure
the possible effect of ntimber of groups on production cost
was investigated.
Finally, study results were discussed and possible
applications of research information suggested; the con
clusions and recommendations for further research follow.
6.2 Conclusions
In the analysis of the grouping problem some conclu
sions can be drawn concerning grouping criterion, optimality
150
151
conditions, a grouping algorithm and ntimber of groups
These are discussed in order.
1. Grouping Criterion:
(1) The relation between set-up time and the simi
larity of parts discussed in Chapter II suggests that the
maximization of group homogeneity, H, and the minimization
of set-up time, S^, are equivalent grouping criteria. The
models presented in Equations (2.8) and (2.18) support
this conclusion.
(ii) A solution to the grouping piroblem may be char
acterized by the relative values of group homogeneity, H,
and the link between groups, L. A solution with high value
of H and low value of L is a preliminary indication that
the resulting groups may be suitable for GT production.
This observation was explained in Sections 3.2 and 3.3 of
Chapter III. On the other hand, simultaneous occurrence of
high values of H and L, or low values of H and L indicates
that the solution may not be suitable for GT production.
The universal population problem with H = 1 and L « 1 and
the null-relation population problem (H = 0, L * 0) are
good examples of unsuitability to GT.
152
2. Optimality Conditions:
The existence of optimality criteria may
influence the type of solution technique that can be
developed. For instance, in cases where optimality con
ditions are not known the only way to ensure an optimal
solution is by entimerative technique which can be inef
ficient. In the analysis of the GT grouping problem,
it has been observed that optimality conditions exist
for some classes of problems. In the PFA approach, a
global optimal solution in the Multiple Population prob
lem requires that L be equal to zero. In the PCA approach
a global optimal solution for the Multiple Type I is char
acterized by H = 1. These observations have been explained
in Section 3.4.
3. Grouping Algorithm:
The Gradient Algorithm may be used to solve
the grouping problem in either the PCA or PFA as shown
in Theorem 4.1 and verified by the numerous examples of
Chapters IV and V. It may also be a suitable technique for
studying the behavior of Group Technology systems. This »
has been demonstrated in Chapter V.
153
4. Ntimber of Groups:
(1) The number of additional machines required
for the Exclusive Membership GT problem depends on the
ntimber of groups; additional machines increase with ntimber
of groups. This observed relation is supported by the
results of all the twenty-four test problems discussed in
Section 5.3. It is also consistent with observations made
from empirical examples by past researchers [6, 45]. In
the case where ntimber of each machine type is one, the
relation between additional machines and number of groups
may be approximated by a linear model as shown in the
graphs of Figure 5.10 to 5.15. A straight line fit appears
obvious by inspection.
(ii) In a situation where families of parts exist
and each constrained problem is solved optimally, the re
lation between set-up time and number of groups may be ap
proximated by a convex curve in some cases. It follows
that if unit set-up cost is constant, then total set-up
cost may also have a convex relation with number of groups.
Explanation to these observations have been given in Sec
tions 5.4 and 5.6.
From the above stated observations, it appears rea
sonable to add that number of groups is an important para
meter in the GT system. The relation between number of
groups and additional machines, number of groups and set
up time appear significant in all the observed cases.
154
6.3 Recommendations for Further Study
1. Because of resource and time constraints, at
tention was focused primarily on the Exclusive Membership
GT problem using the Gradient Technique. It is apparent
that the method of this research may be used to investigate
^^^ Non-Exclusive Membership and the Hybrid QT jr-rnnp-fp
problems. It is recommended here that these alternatives •rwi^wfWMW*^
be studied using either the Gradient approach or other ap
proaches like Integer programming, Dynamic programming
and Branch and Bound.
2. The Gradient Technique established in this
study may be combined with discrete simulation languages
in a general methodology for studying several factors of
GT production. For instance, such a methodology may be
suitable for investigating the possible impact which pro
duction control functions may have on GT systems. The
development of such a methodology was not pursued because
of resource constraints. Its development may be worthwhile
not only for further investigations but also for a compre
hensive design of GT systems. The combination of the Gra
dient Technique (or other methods) and discrete simulation
languages may enable the comparison of several GT alterna
tives (including no GT approach) for any given problem.
In such a methodology, the Gradient Technique may be used
to evaluate several aspects of the production control
functions.
155
3. The use of weighting parameters in order to
reflect the relative importance of attributes of parts was
demonstrated in the PCA approach (see Section 2.3). An
investigation of how weights may be assigned to machines ""r^'^Tn—\w\^ -,m •!»
in order to properly reflect ^h. oi^n'Yr immrrnnrr _nf
machines in the PFA grouping problem m^^ fc^ IIP- JJ!
Weights may be defined in terms of cost or machine loads.
4. In Chapter V it was illustrated that an opti
mal number of groups exist for some GT problems. The re-
lations between the number of groups and the cost of addi-
tional machines, as well as b^tween^^j^e^ntmb^
^^^ SQt-up costs, may be used to formulate a model for
analytic dp prrg-irtaMnn of optimal ntmber, of groups. Devel
opment of such a model was not pursued because a more ex
tensive experimentation will be required to determine the
parameters of the relation between number of groups and
production cost. A study of such a model is reconmiended.
5. The central focus of this study has been the
batchshop machine set-up problem. It is obvious that the
processing times-based scheduling problem is also important.
For sequence-dependent set-up times, a solution to the set
up problem may not imply a solution to the processing time-
based scheduling problem and vice versa. It appears that
tbjB-xelatiY^ values of processing and set-up times per
operation mav be critical in the decision asto which
wrn^i
156
problem to solve: the set-up problem as pursued in the
GT approach or the scheduling problem as sometimes pursued
in the conventional arrangement. In a batchshop where
processing times, relative to set-up times, are the domi
nant components of schedule time, it may be better to solve
the processing time-based scheduling problem. On the other
hand, if set-up time per operation is much higher, then
the GT approach may be a better alternative. This suggests
that some critical ratio of processing time/set-up time
per operation that makes the GT approach the better alter
native may exist. The study of such a critical ratio is
recommended.
6. Throughout this study only similarity of parts
(defined in terms of process routes or material type, size
and shape, accuracy requirement and features to be processed)
has been considered in the grouping process. While this
approach is ideal for the machine set-up problem, considera
tion of additional parameters may be necessary for the
batchshop problem. The grouping of parts with consideration . t I ini[-ii mil " — " ^~~-' •" "" ' ' ' •— ' ' ' " " *"" I •ai imiM.i ^
to^^tha^vnlnmp of ln4toy[uaLa§rt ^ ^ ^ ^ o^^ example. The modi
fication of the grouping procedure developed in this study
such that parameters like voltime, due dates, value of
parts, etc., are taken into consideration may be worthwhile.
REFERENCES
1. Alley, Lee, "Ranking Group Characteristics by Relative "Typicalities and Subject Assignment According to Largest Group Similarity," Unpublished Paper, University of Nebraska, Dept. of Computer Science, November 1970.
2. Abou-Zied, Mohammed Raafat, "Group Technology and the Manufacturing Systems of the Jobshons," Industrial Engineering, Voin32, May Vil^
j 3. Anderberg, R., Cluster Analysis for Applications, Academic Press, New York, 1963.
X 4. Bergen, Jay H. , "Parts Classification as a Basis for Programmed Process Planning," Hertec Corp, 1975.
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11. Eckert, Roger, L. , "Codes and Classification Systems," American Machinist, December 197 ),
157
158
''• "'""inal^ks'-VEffe^^^^^^ ' " "^-Po-nt Flow System Design " Irodultiorp"^'^ '° Production ^ 1972. ir-roduction Engineer. 51, 5, May •"
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>19. Gottfried, B. S. and Weisman, J., Introduction to Optimization Theory. Princeton-Hall, Inc., Engle- ' wood Cliffs, 1973,
V 20, Gupta, J. N. and Dudek, R. A., "Optimality Criteria for Flowshop Schedules," AIIE Transactions, Vol. 3, 3, September 1971.
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— 22\ Harworth, E. A. , "Group Technology Using the Opitz System," The Production Engineer, January 1968.
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159
25. Jardine, N. and Sibson, R., Mathematical Taxonomy. John Wiley & Sons Limited, tiew York, 1963.—
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ted Production Control System," Production Engineer, July/August 1975.
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160
'36.
»/42;
pxeces. Fart I, Pergamon Press, 1970.
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??oi rf?T '^S* °*^i8? of Group Technology Produc- /, i P tion Cells," .Tmimnl of rymbBi.maLlL,ii. d ^ \uni -PntLJ^
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APPENDIX A
LISTING OF FORTRAN IV PROGRAM OF THE GRADIENT ALGORITHM
161
iifiilrf'riir
\
APPENDIX A
PROGRAM LISTING
The following program
1) generates hypothetical batchshop problems in
the combined PCA and PFA format
2) computes similarity coefficients for either
the PCA or PFA
3) optimizes either H or L
4) and computes number of additional machines.
162
VARIABLES
NOP
C
N
NUMG
W
NOM
ntimber of operations
cost of machines
ntimber of parts
number of groups
relative importance of characteristics
ntimber of machines
SUBROUTINES
PINDEX: this subroutine accomplishes the Preference
Index Heuristic
TSTEP: this accomplishes the Gradient Algorithm
using only the transfer step
PCASIM: computes similarity coefficients with set-up
related characteristics
MMkk.s.
163
fn^TX \*\ IV 1 L^VTL ?1 '*\V "J'^Tz a M32^
0002 2301
0?»T»
CC«^?N/U<I /S I' FR (I 30 f 130 ) , .rt I "'OJ , « l 70», 2VL»l30) ,MPI5(70) , IB' iF*f l? 'J ) , •>oipTS(TO»l23).N,NU.^-.fICTJ.ITC?,Mnvc^,jjj,rT
0005 0005 0007 n'^^^ 0 0 0 1
oon 00 11 0012 0013 0014. 0015 OOIS 0^17 001? 00 1? 0020 ' ' 021 0022
0023 0024' 00 ' 5
0027 OP'S 002"? 00 ?0 •"»0 •> I no 32 0013 10 3'* rir)":^ 00 35 nn-i? 00 3^ 00 3P nc.^'i 0041 01^2 0043 004.4
c r ^ >., r ^ ^
C
r ^ ^ J s l MCiMJ nOTT*»T7' J TC«?J=2 '•E^^^S CP'^f'^ir.c L TFl.A';»l .•'!gA"S JSE FJNCTIPV X IPLAG*. ' '*e».*'^ 'JS5 PMNrtTO*! ^
<FL4G=0 «€*• !$ 7M«= 3PA A03P04CH KCLA'^al ^?A*'S THP 9C4 4PO'^4r-<
M'J^*?«2 M«70 ^IU^l»? MijM 2a ? Nr*«a40 C P ^ V * . * IFLAfJ ' l ITCOsO
10=0 I X * 7 5 9 3 rn '3 j»2
?90 F r R M A T f S t l )
?0 2
5 22
.*<0
or» 302 I « l » 4 5 W ( I ) = l . O on 5?2 t»lf^' '^'« C-ILL «»AN0U( I X , V C D
«f n s V = L * 3 . o > i . o PFAo 3 0 3 , ( : ( n . i « i . ^ ' ' 3 ' i )
CALL ^ANOUf r x , Y ! : L ) PaVPL CiLL 9A^•nn( TX.YSL) I F ( Y P L . ' ; T . . 2 0 ) r ; p T-" S'15 < s P * 3 0 0 .3 * ' 'O0 . 0
c 'n=< i r T"? .*« 04
K !<aP*«,0.* ' l .T cr ! )= '<
f.nu "o iMT *'^*'»I fCf I ) t,of c r o M ^ T ( ! l O , r i 5 . o )
X='*! »in'.sC " ' •" / ** x ^ l \ ' » ? . o >|rMt a»'n'4+"y xMf7M«Mn'* P l » . 5 F 2 » . 3 5 P 4 a . 4 5 P 9 » . ^ * FC=...^5 F O * . ^ ' '
164
F^on.^M IV z i.rvFL 21 Map,
^^^^ IF ( 10 )<?00, 3 0 0 , 5 0 1 0046 qno on 302 T « 1 , * J - » M I W 4 7 30 TO 304 004^ ooi on 303 r»i,MO't 5049 w f n « c i n 2*'53 303 IF(C(rj ,L5.S0.)W{rj»l .0 00^1 "^nu Cn»'TTMUP 00^2 3 0 7 FnoMAT(F lO.O)
C GENc^ATF OtST lMPr o^-jTrc C
^053 or 10 t= l ,MOP '^'?*'* CALL PANDiJ( l y . v P L ) C0^5 I F f Y P L - F l ) ' 5 F , « « ? , S 6 0056 55 Xt»a3.0 0057 Y P « 2 . 0 00 5" GH f n 5 " O'J^^ 5<S I F r Y £ L . G T . F 2 } ' ; 0 TO 5* OOftO XP-«3.0 OOftl Yt»a5,o 00*!? vO TP '•9 00*'3 5a XPssi.o 00 64 YP=<?.0 •'^065 SQ CALL "Ai^lOUMX.VPL) 00^5 NCt>=YFL *X '»*Y ' 00S7 CALL "ANDLK r x , Y P L ) 00*.9 KaYPL*X!Mr'«+l ,0 00/S9 ^o/ \pT(r , K ) a l 0070 . MPP=N0P-1 OO-'l • OP Ll J a l . M P P 0077 30 CALL "»1N0',j( I X , Y £ L ) 0073 '<lsYEL*XV':".»* i . o no7v i F f K i . E o . K ) :;o r p ?o 00 •'5 KsKl 007S 11 M049T(r , . i < l )a l 0 0 7 7 10 COl^JTIM!Je
'^^r•• » ' l l ? * .
c JENE^ATE PCUTES PpR PE '^ IMIMG OA.?" "
00' 'S XTaMOP Ony^ MMs.MPP+1 00-30 00 21 IaMM,M 0031 C iLL aAMpiJf r x , Y E L ) 003? K a v p ^ ^ Y T + l , T 00«3 OP 22 ; < l » l , M T * 00^4 22 '^OAPTf r , X l ) =«PaffTC«:,.< n 00!»<5 2L CP''"''I^!i/^
- " OF^'E? ^TIG*! OP 3 C i N'PP'JTE",
C GPVEOATP " ^ f S T OIGIT ATT^ I'"jT = 'f / •
00?.b or 500 I ^ l f M O " O0.?7 CALL PAMOU( TX ,Y=L) 0 0 « " 500 >*01PT( T , 4 1 ) aYPL** i .3
C GE"E'ATE AT- ' IT 'PS P C ' F ^ A P l t N O "^IJI"'"'^
OO?,? OP 50 1 I= l , ' l '^w
165
pnoT^AM TV -; LPVPL 21 'J»MV . n^pr , , 1 , 7 4
00<»3 OP 5 0 1 J « ? , 7 0001 CALL P4Nn'l( I X , Y F L ) 0092 i r r v E L . L E . F A l G n TT «;o? OOOl I F ( Y E L . L E . P 9 ) G n TO 503 0094 I F f Y E L . L S . F r i GH TP 504 0OO5 I F f Y E L . L E . F O I G P TO 505 0095 CALL 9ANnU( TX,YEL) 0C97 *^P4RT(T ,40«'J)avSL*4.0+«.0 noaq (jn T-> 501
0103 'ir rn - o! 0101 503 -'OAi Tri ,J>40)al 0132 GO IP 501 0 1 0 3 «?04 CALL PA IOIH IX,VPL) 0 1 0 4 ^OAOTd , j * 4 0 ) a Y P L * ; ? . * 2 . 3 0 1 0 5 GO 7P 501 0105 505 CALL RANOU( IX,YEL) 0107 ««P^PTrI ,J-|.40) =vFL*2.3+4.3 0109 * 0 l CPMTIMijP 0109 OP 311 I*MV,.N 0113 CALL PANn«j{ r x . v E L ) -^ l l l . K3YFL*XT+l.O 0112 OP s i l J = l , 7 0 1 1 3 5 1 1 MPAOTf r ,J><.01 a'«OAPTr<,J*40» 0 1 1 4 'OT^T 267,M,NO'* 0 1 1 5 PPt.VT 5 2 0115 PR PIT 259 0117 OP 250 Ia l ,M 0113 2 5 0 PPT^J^ . a 2 , r , ( M P A P T ( t , J ) ,J=L, . \ 'OMl ) 0119 PPTMT ?66 0120 PPPIT 305
C C CALL TWE SU«PPJTTMP THAT CP'""JT?S « T v i L \ S [ T Y 1*155* {- TMTFUM^ 3P SET i;o ?. ELAT=0 ZH^PHCJ^." I^TICS C
0121 CALL PC4SI" ( '^P* .PT, . i ) O l?2 00 265 rLaM(JMl,.MU><2t.MiJ»«3 0123 NUMG* IL 0124 I J a l L - l 0 1 2 5 521 IK'O 01 25 0127 0125 012^ 0130 C l ? l '^l?2 0133 0 1 3 4 0135 01 ?5 0137 0133 0139 0140 0141 0142 0143
30 31
CALL r^UTMf 0 , X T I ) CALL "IMOEX CALL T5TEO CALL CP'JT'M l , « T l ) XlaO X?aO VaO.O on Z7 IaL,^"»*«G MM-.Mpy, f l )
rp(vo iG(n .GT. i ) ~r X a l . 3 GC ' "' 3 1 XaMM«(MN-U / ? YaMV«(^'-*'N) XHTsMJ r ) /X 5fL i«*L( ! ) / Y XlaXMl+.Xt X 2 a X 2 * X L l
rr"
166
FOPT^AM IV ' LEVEL 21 MA!' 0^r= s no2'* 014'. Oli5 014*. 0147 014a 0149 0150 0151 0152 0153 0154 015«? 0155 0157 0155 015? 0160 0161 0162 0163 0154 0165 ' 166 0167 0168
0169 01 "'O 0171 0172 0173
O f 4 0175
» 7
265 13 " 3
39 43 52 53 54
52 154 15'' 1.00
20«! 251
266 267
269 273 305 269 305
VaV+'XHl XaNlJMr, V ! * V / X x i»x i /x X2-X2/X
V l « X I OTTMCalc PT IvE«e OP I SIJ K cn»'ri*Mj pcoMi^T( FPP'-'A 7{
FCR*«AT( PPOMAT{ Frc«ATf FPR^AT( PPPMATf FnpMAT( FPO'UTf pnPMATf FCP'^ATf
PHQMAT FPP"MT( FCP'^ATf
i r 6 , « 3AP FCR»«AT( FPPMATf FPR»«AT( FOR^ATf cnpM^Tf
1 6 X , » A 0 0 ST"P P*!0
- X L l
Tl TIMC
4 , T L e ^x,r
2 X , ' 20Y, 2 X / / lOX, l i l t IX,I 30X, OCX, 30X, iM,r {2X , I H U 2 6 * , TS A 6 X , 4 * A 40F2 I X , ' t *
I r • f F l 2 . ^ )
/lO-^.O • v i , A , c : \ ' « , p . ^ r ' « E , j j j
5 . ? X , 2 0 ( 1 2 , I X ) ) 1 2 X , 1 5 ) V » « , F 1 2 . ^ , ' ' - « ' • , = 1 2 . - , • • AFTEP • , ! ' » » • rTEPAriOM«:») / / / / / / / ) • A0nITI0^51L « OF - v c s a ' . T i : ) P l l . 5 , 2 f = l 5 . 0 , 5 X , F l « ? . 3 , f l O ) 5 , 4 7 1 2 ) •T!4P SQUJriClN CnV'/EPGEn TO \H ^:>Tl'*U'* • ) I 7MF S^LUTTH*} WAS ''PT C0»'\":?3F'» » ) • ^F= Ct' l^H IN vexT Si- .T ' ) 1 0 , 1 ? 0 A I ) 4 3 F 3 . 0 )
•T-HIS IS A • , r 6 , « ''*'?T« \.vn • , NO • , r 6 , « ^AC-»I»JFS "POBLE^ M 0 1 3 ) LG^P ITH'^ 5IJW Tf-tPat , p | 3 , r , i S r : 2 \ ' 0 S » ) . 3 ) PAflTS • , 4 0 X , ' A T P T 3 U T P 5 M nc rpn i jas • , 5X, •-» • , U X, » A"T 4 / C ' f '•0<?T • , t » X , M L G TIMC (SEC) • , • TTFISTIO' IS •)
167
FPPT^AM IV Z LEV^L 21 PI^IDEx OATr a »1024
0001 0002 0 0 0 3
0 0 0 4 0 0 0 5 0006
0007 0009 00 OO
oon 0011 0 0 1 2 0 0 1 3 0 0 1 4 3 0 1 5 0015 0017 0013
r r r
SUPOOIITIME opUOEx OI"«FNSinM I P O S ( 1 2 0 ) C 0 M M C N / 5 L < l / S ! M F P ( l 3 0 , 1 3 0 ) , X L ( 7 0 ) , ^ f 7 0 ) ,
2 V L ( 1 3 0 ) , M P I 5 I 7 0 ) , I 9 U F 4 ( 1 3 0 ) . ' P & P T S { 7 0 , l 2 3 > f .MfWIMGf I C B J , ITE^ , « 0 V E 5 , J J J , ZZ
I'^TE3B» PARTS rm 50 I « l , * l
60 I P ' J F A ( n « 0
Cj^vo'jrp I»"l Tt \L POPPPPP'JCE IM*^**
3 E ^ T a i O 0 . o X « ( M - 2 ) * 2 OP 1 L a l , M SU*< 1*0 .0 OG 3 ! = l , M I F ( L - I ) 4 , 3 , 4
4 SU»*«0.0 00 2 J « l f N
7 SU«aS'JM*SIMFP(Lf J ) * S I « P R ( I» J) su^asi^M-sr'p" n . , I ) *2 .o 5 i r « S I ^ F P { L , I )-S'JM/X
su'^i'SU'^i+s r^p= f L . r ) SB^P^H FPP PA5TS WITH M I M r ' J v poce«=Mr.S TMOE*
C r
0019 00 20 0021 0022 0023 00 24 0025
0025 00 27 00 2S 0029 00 30 0031 00-^2 ' 0 33' ' 0 34 00 •*5
0037 0035
003<» 0O40 0 0 4 1
0042
c c c C
I F ( 3 E S T - S U " * ) 3 , 3 , 7 t 71 LRESTaL
l E E S T a l BESTaSlJ^
^ CONTINUE VLlL)«SIJ»»l
I CONTIMUE
FOR** F IRST TWO IVOICATOR GROUPS
N P I G ( l ) « l . N P I G t 2 ) * l 0 4 R T S ( ! , 1 ) « L 3 5 S T P A R T S ( 2 f l ) « I 5 6 S T t e ' j F A ( L 5 E 5 T ) a l i a ! J F A { I 5 E S T ) a ? <CUNTa? ».i5RU3 = 2 • - • ( l )aO.O W ( 2 ) » 0 . 0 XL( l ) a V L ( L 5 ' = S T ) X L ( 2 ) = V L ( I « F S T ) IFf MGPUO.FQ.M'T'GJG? TO
C C C C
aO
n 7
rpMPUTP THE PPEFEREMPE OF Pi'^T Tp'c'JPP?NT PIOICATPR GRCM'
•^FSTalOO.O OP 70 I = l » ^ I F ( I ? M F A < I ) . G T . 3 ) G 0 t o ' 0
SL"* l '0 .0
168
PfwnAM IV Z LEVEL 21 DIMPFX OATC a » 1 0 2 4
0043 0044 0045 0 0 4 6 0047 004? 0049 0050 o r S I
0052 0 0 5 3 00 54 0055 0 0 5 6 •00^7 00 5 j 00 59 " 0 6 3
0 0 6 1 0062 0063 0 0 6 4 0065 0 0 5 6 0057 0065 ao6«> 0070 0 0 7 1 00 72 007-5 0074 0075 0076 00 7T 0079 0079 0050 0 0 « l 00«»2 0P<^3 nc\o(t
00*^5 00«5 /^naij OOP? 00 50
00 90 00-51
72
25
70
r
r r c 7
•
a
l l
<3 10
r
r
OP 72 x«i,Mf?R!jo *P«»4PTSCK,l) su*»»vLr n • V I . ( v » ! } - s i ^ ^ ^ ( i , > r ' ) * 2 . a SL'*««SI'-1FRI I ,NNJ-SU«/X SU»l«SUMl*Stlw IF C "IE ST-SUM I ) 7 0 , 7 0 , 25 IBESTrl
9EST»SUM1 rprTTNtiE
FPRM PE>*AIMI»«G IVPICATpo GRP'J«*5
»IGPUPaMGRU«»*l PAP TS (MGP'J" , 1 ) a 19 FS T XCUflT«<OU*!T-»-i IRUFA(IREST)a^f r ,RUP N P I G I N G R U " ) ' ! HrNGRUP ) « 0 . 0 X L ( M G P U P ) a V L ( I 3 F 5 T ) IF(NGPU<».LT.MU^G)Gn TO ^0 IP(XPU»!T .EO.M)Gn T? ^ l
ASSIGM PPVATMIVI PARTS T^ 0'O'Jf»S
CPMTlMtJF a F S T a - l O O . O OP 10 1 = 1,M rF ( I 3 U F A C n .GT.0)G.3 TO 10 OP q <al,MJ^«G SU** 4 - 0 . 0 S»J'»laHf K) SU'«2aXL(K) NN*»iPTG(K) XlaWN X7a*J-* !M- l 00 S J a U M M M l a P A P T S { X , J ) SUM 2a SU « 2 - 5 I « P9 ( I , •! I ) SU^4»SU^4*S t ' *PR( I , M l ) SU^ lsS 'J * * ! *? I ^ ^ " I I f >.' I ) X 4 a V L ( I ) -s ; j '<4 SU'*2aSU'*2+X4 X a ( M * | * l )«NV./-> Y a ( M N * l )*(»t-(MM-».l n SUM3*SU ' *4 /X l - x< . /X2 I F { 9 E S T - ^ U ' * 3 » l l » 9 , ^ 0CSTaSU«3 I ' E S T a ! IGP-*^ X H l a S U « l XH?aSU'^2 CONTINUE CCNTIVUS
APO OAPT v t T H ulGHEST IMPPX m
N'^JaMOl'^ (IG"» ) * l
NPtG( IGR)-=NM
• ) 0
i M-'
169
cnoTRAM IV G LEV«"L 21 "IMOEX '^iT" s 91024
00*»2 OOO? 0094 0095 0096 0097 0095 0099 01 10 0101 0132 0103 0104 0135 01«5 010? 0109 0109 0113 0111 0112 0113 0 1 1 4 0115
51
703
703
P f tR ' 'S ( IGR, * lM)a ioEST I R U F A ( I R F S T ) a I 3 P KCUMTaKCUNT+l H f l G P J a X H l XL( IGR)«XH2 I F { K 0 U N T . L T . M ) G 0 T*" " COMTpiuE X l a O . O X 2 a 0 . 0 XaMiJMQ
OP 700 Lal,N'.r«G NMaMPIG(L) I F f M V . F O . D G O TO 703 X X a N N « ( N N - l ) X X a X X / 2 . 0 X l a H { L ) / X X * ' X l YYa-MM*(N-NM) X 2 » X L ( L ) / y v * x 2 CPNTT*!'fE X l a X l / X X?aX2/X ZZaXl RETURN pMr)
170
FPOTRAM IV G LSVFl 21 TSTEP OATE a a 1024
0001
0002
0O03 0C04 0005 0 0 0 6 0007 O' Ofl 0 0 0 «
0013 0 0 1 1 0012 0 0 1 3 0 0 1 4 00 15 0015 0 0 1 7
0019 001? 00 20 00 21 0022 0 0 2 3 0 0 2 4 0025 00 25 0027 0025 0029 0030 0031 0032 0 0 3 3 00 34 0035 0036 00 37 0035 00 39 nnt^-) 0 0 4 1 0042
C C r c c
43
c C c
36
3 1
32 33
C r r
SUBROUTINE TSTEP
THIS S•JRPC?JTI^'E TESTS THF O 'TTMALfTy pc SIL-JTI^NS ANO •'AKFS NECESSARY
COMM«M/ 9LK1 / S T'PR ( 1 3 0 , 1 30 ) , XL ( ' 0 ) , H( 7 0 ) , 2VL f 1 3 0 ) , M P I G C 7 0 ) , I 5 U F a ( l 3 0 ) , ' « » A P T S ( " ' 0 , 1 2 3 ) , N , N U * * G , r P 5 J , I ' ' : 9 , - 0 V « < > , J J J , ! ?
ITCPaO JJaO J J J « 0 MPVES«0 IlaO J J J « J J J ^ l
rPMOgTF MPJ ^Al (IF5 OF -i AMO L 'JiTMPUT PA^T I. Jf
00 30 I a l , M M l a i g i j F A d ) HlaH(Ml ) »WaM«»rGlNl) I F f M N . E O . D G O ' 0 30 XlaNN*( M-M-l ) / 2 X2aM^J*( »i-NM) H C L O l a M { N l ) / X l x n L 0 1 » X L ( M l ) / X 2 S U ^ l a H f M l ) SUM2'XL (MlJ SUM4aO,0 DP 31 KaU. - l v M2aPAPTSf'^' l »K) I F ( M 2 ..ME. I ) GO TP 36 <KaK GO TO 3 1 S U ' * l s S U M t - S r M P P ( T , " ! 2 ) SU'*2«SU'*2*5IMFP ( I , V 2 ) SU«4aSUM4*S I'^FP ( I ,M2) CPMTINUF S U * « 4 » V L ( I ) - 5 U ^ 4 SU*'2«SUM?-SUM4 X H l ' S U ^ l XLl«SUM2 H2aSUf ' t K a f ' " N - l ) * ( V * ' - 2 ) / 2 Ya(* !M-1 ) * ( M - W » * - l ) I K r M M . G T . 2 ) G t : TP 32 uMPWl«0 . 0 GP TO *«*» HNPWlaSU'n /X XNEWl»SU.»'2/V
CO.MPUTF MFW 7ALIJF5 PP H ^^0 TO ENTER »AOT I IN GPCU" J
, AFT=5 ATTE'^'TIM',
0043 0044
!^FSTa0.3 on 34 Jal ,MU'^G
171
ppO TR AM
0045 0046 0 0 4 7 0045 0 0 4 9 0050 0 0 5 1 0052 00 53 0054 0055 0055 0057 0055 0059 C060 0 0 6 1 0062 0063 00 64 0063 0066 0OA7 00 6-? 00 69 <so70 00 71
00 • ' I 0073 00 74 0075 0075 0077 0075 0079 O C O 0051 0032 0033 00 «4 00fl5 CO 56 •^0P7 C0*»5 00'»o 0000 0 0 9 1
0092 0093
IV G L^VEL
5
1}
7
a
c
1
2 49 • ' 5
37
3'* 30
C
c
c
c r
21 TSTE"*
H3«H(JJ I F ( I R U F A d l .EO.JJGC Tn 34 SU»«l»H{.II SU^2«XL(J ) SUW3«0,0 NN«NPIG(J ) IF (>!M-1 1 5 , 5 , 6 H " L 0 2 « 0 . 0 GO TP 7 Xla»'M<i( V»J-1 ) / ? H n L 0 2 » H < J ) / X 1 X2«MN*r N-MM) X P L 0 2 » X L ( J ) / X 2 OP 5 < a l , M \ ' N4«PARTS( J *K) SU»«laSUMl*S I*«PR( I , M 4 ) SUM3aSUM3-«-S IMPP J I ,.M4) SUM2aSU^2-S IMF" { I , ' : 4 ) SUM3aVL(n-S'JM-» SU** 2a SU "2<• S!J*'' Xs(MN*^l )**n«/? Y a ( N N * l ) * r : - N N - l ) XH2aSUMl Xl2aSU'»2 H4aSU*n H*»EW2aSUMl/X XNEW2aS'JM2/Y
COMPUTE THE "^ATE CF CHAMGE P« P A ' T I T I
GC TO ( 1 , 2 ) , r n « j 0FLTA=H»lFWl-HPL0l**'MFW2-Hnt.02 GO TO 4 9 0ELTAaX0LDl-XMEWl+tpLO2-*Mcv/7 IFf 0 F L T A ) 3 4 , 3 4 , 3 5 I F ( ? E S T - 0 E L T A ) 3 " ' , 3 4 , 3 ' » IGPaJ SESTaOELTA XH3aXH2 X L 3 * X l ? I I « l
XLaKK V laXWl
V2aXLl l?ESTa l CP*'TP'nE CPMTIMUE IFf I I .EQ.O)PETijPM MlatRUPA( I 3 E 3 T ) <Ka<L
REASSIGN I T^ G^'^'P - I ^ H MAXTMJ-4
I F n « . E g . * l " I G ( M l ) ) G C T- 47
M 7 a V P I G { M l ) - l
PE-PVF PART I PPP** ^Ln ^ ' O ' ^
O^TE a a iO?4
RATE TP : - (ANG'
0094 OP *6 TLarx ,M7
172
FPPTRAM IV G LFV^L 21 T ^ T - P ' ^ ' ' ' OATE a 3 13^4
°^ll , P - ^ f t T S f M l . I D a P A P T S f M l . l L * ! ) 0095 45 CPMTIN?UF ''0*»^ * ' N P I G ( v i l * M P I G f N l ) - l
C APD PAPT T TO MFW GP'^U'*
0005 I « U F A ( I 5 E S T ) a r G P '^''''9 MJaMPIGdGR ) * ' ' ' l ' ^ ' ' MOIG(IG'»)aMJ - 1 5 1 PARTS( IGP,MJ) =r3FST '^102 MPVESaM'^VPS*!
C r r C
0103 HJIGR)aXH3 0104 X L ( I G R ) a X L 3 0 1 0 5 H ( N l ) a V l 0106 X L ( M l ) s V 2 0 1 0 7 ITFRal Ol''*^ I F C ' P V E S . G T . O ) ' ; n TP 43 0109 40 OFTURM 0110 EMO
U*»OATE VALJPS ""P AFFPCT = 0 l^O-.J^S
173
pppTPAV IV z LFVEL 21 PC AS I " OAT= a ?1024
0001 0 0 0 2 0003
0004 0005 0005 0307
00 09 0010 00 l l 0012 0 0 1 3 0 0 1 4 0015 0016 0 0 1 7 0015 nn i^
0 0 2 0 00 21 0022 0023 0024 00 25
SURP?UTIME PnASlM(MP4PT,w) OI>»ENSnM '«P4RT{13O,47),Wf«50) CPMMOM/ 5LK1 /S I"FP ( I 30 , 1 30 J , XL d O ) , H( "'O),
2 V L ( 1 3 0 ) , M P I G ( 7 0 ) , I P U « : M 1 3 0 ) ,
*«? AOTS( 7 0 , 1 2 0 ) , .M, MUMG , 103 J , I ^^R , ' ' 'nvP5,J J J , r ? V « 0 , 0 7 « M « ( M . I ) T-^TALaO.O OP ^07 K a l , 5
^ « 7 YaV*.!-, f |<+4P) YaY*2 . 0 N l a M - l OP 505 I a l , M l I l -«I*1 OP 50P J » I l , N S«l«a0.0 OP 50n Kal,"?
50O I F C I P A ' T d , < * 4 0 ) . = 0 . ' " A P T f J.Ki-^O) ) <;!J' aS J*«l-V( X*40 ) S r ' F R ( I , J ) a S ' j « / f Y - S l J M ) TPTALaTCTAL*Sr 'FR(r , J ) * ? . 0 S I ^ P 9 » J , I ) a S r ' P ' ( ! , J )
50'« COMTIV'JE TPTAL«TPTAL/Z pppiT 9 0 2 , r n T A L
•^02 F 0 t ? M A T ( 2 0 X , F l 5 . 6 ) QCTIJP M
EMP
174
«"?TRAW
0001 0002 0003 0004 0005 0005 0007 0005 OO' o
IV 1 ; LEVPL
1 2
21 PA MP'J
SU9P"'JTIME 'ANOUf IX,VFL) I Y s I X « l 2 2 0 7 0 3 l 2 5 I F ( r Y ) l , 2 , 2 IY« IY*2 14745364 7+I
YEL«IY Y « : L « Y E L * 0 . 46566 130-9 IXalY RpTJJRM END
'^^: a '1024.
APPENDIX B
OPITZ PART CLASSIFICATION SYSTEM
175
ijiik
176 APPEmiX B
OPITZ PART CLASSIFICATION SYSTEM
The shape, size, processing features, material, tol
erance, etc., of a part are described by a nine-digit clas
sification number system. The first five digits are the
main form while the last four are supplementary (optional).
Figure 7 shows the general layout of the Opitz system
while Figure 8 represents the form as used in this study.
The first digit designates whether a part is rota
tional or nonrotational and the overall size. If a part
classification code has zero in digit one it is rotational
and its length to diameter ratio (L/D) is less than or
eqtial to .5. A value of 6 on the first digit indicates
nonrotational with the longest to shorter dimensions less
than or equal 3 (A/B £ 3) . The meaning of other digits
are shown in Figure 7. Each digit can assume values
ranging between 0 and 9. A number within a digit has a
unique meaning as shown in Figure 8. Higher values denote
complexity on all the digits except digit one.
For this study only two of the supplementary codes
are appended to the main form code (see Figure 8). The
values from 0 to 9 in digit 6 indicate increasing order of
accuracy requirement. Those in digit 7 show variation in
material type. The remaining digits are as Opitz's origi
nal system.
177
<
O ''•J
1 A : Y s . r : : T
i 5 " ^ i « 2 . ' ' » ' • » ' ! iO ! « » - ? •^>'.2tvC
j * " ' S S A ' ' ' »
1
1, S N C ' I . * . J ' ^ ' C
s
c '1-i 'd o o N 4J
a. o <+4 o
o CO
0)
•ri
fl4
178
*<i^i rsr ia iy^
TSAa-^ xS'szT^zoY
'Ji s
s ^
, "* •--! ^ s ^ '^ < <
I ^ ^ — Z I X -w Vt
< • ^
= ' - ^ 1 .- 1 <• i .- 1 ^
JI
^
i" ^ -•
> > ^ >
• s i i -
— a >
1— - i - • 1
i^iiSI'l^ 1 i
1 -.-••'1 -->C inoct-." ! •--
1 — ^ 1 i * ^ p? — t " ^ ^ ~
s ^ I -j 1 r ^ "
• 2 ; ^ ' « — I ^ C " ^
..: 2 {Z<J. 1 •!-? ~ ^ 1 ^ — ' • • * ^
a 1 ' 1 - 1
C ^ I ^ L— '^ 1 — C
— — ", j " i •t' ~ * r : ' s > « ' x — ' ^ - " - — ' - t 1 :; >" • i - .?tj « ' 1 * w 1 •«. C I •'•*»
: . ; i ; < : ' — > | 4 C i < N ^ { x : 1 . . Z Z ^ ' 1 ^ ^ Z I li^ '. '^ [lim — 1 S S
' — £ — i : ' « 1 —'.r C ' — 1 :»i i .-^ ) < • 1 . - , ^
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^ 2 ^
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! • < t^ 7* } S ^^ ?!? . ^ —^
i r r < •= ^ **" "T
> - :.:
^ ^ -2 "^
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s TJ 5 ;2
— -^ i -
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# ^ 1 2 a 1 3 '3 c ^ ' ^ ^ — > 1 -5 S
.• c ; ^ u ^ C I '•^ ^ ? ^ 1 tj 3 CJ ' C ^
^ * 1 ^ ^
— — j 5 i 1
:7
"^ 5
^ 5
: « c ^ U 1 — 5
^ 1 V —
j ? -^ : • —
— s 1 v: l-j-ccsc 13! 3s^3^»iz : -" i'3-cuT 5Gr3 act'i ss^r t i-iz-3..3<
— 1 — 1 .->j 1 -^ 1 » r ! . - 1 , ;
- 1 = 1 :
i
H
M
^
T. T . TT
- 1 -1
' " I «- '*C --"•.'*.
• • t
3
^
•j, 1
r j
l^l\Vr% - ^ — §'
- £ 1 -:'"" 1 r ^ '• l ^ • ^! ^ • » ' - W - ^ " ' M
2 1 '.4 C ^ 1 C
— !:;:>— 1 - - 1 r ' r. '
^ C
^ ^ ^ • ^ -* c * * * ^ ' ^m
' — 1 ^ C ! ^ * 3 ^ 1 ^ ^ ^ «'-^
« • M 1 «• w ' • • ^ W « *
'"• .^ 1 ^ '^ ' r.-- i _r "* .c ^ ^—^
' " ;£ = *» i i r
• CQ
'a o CJ
Cd
r-l
a a 3 CO
0 15 i j
4= 4J •"^ ^ ^
' e CL) 4-J cn > N
CO
&0 •H
O • u
I
1 . 5... i
' • 1
^1^ ^ 1 * '
^Vi^Z S« • - - . — — 3 S j -T f»i ;ii. 1 - i - i 1 ^
1
• * * r f « ' ^ ^
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"2
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s •••» 1 i -
3 1 j J1 j .
1 1 ^ ^> 1
.^ ^ ^ — >
1 '
••ZZZa:ZX 10, 2 —
• S d C T 3 0 " : 3 9Cr.\ 3 3 C "
t
' ^ I ^
! H 3
1 " 1 ^ ^ 1 ^ • •
i — i^ — "z"^
— :C r
i i^ •w •H
1 ^ i 1
' CO 1 : CJ J J-l
3
1 i ! 1 i 1 ! 1 I - L 1
! - ! \C :- I
1 •" 1
j « • • • 1 Wl
* i ' - —
^ ! i 5 " 1 Wl 1 "./ i 1 S 1 . < 1 > ^
•w ^ i I
.' f:^ "Z ^ \ r 1M 'N •" " ~ •" ' ^ •" 1 — I s ' —
> 1=:: :H:: i '' << i c i - i ^ i - l ^ l - i -1 — - n r r - •*-••" ^ ' ^ 1
i ' ' 1 ! i ! 1 i
1 1 •; i — 1 .••*• - " ^ ^
• = 1 -c c ; r
j ^ ; = 1 "
i bO I -H ! fi^
t
1 ! 1
1
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:;>'• 'T^'.fj'^t^
APPENDIX C
TABLES OF GROUP HOMOGENEITY (H) , NUMBER OF ADDITIONAL MACHINES (A) AGAINST NUMBER OF GROUPS
179
180
( i ) Group Homogeneity (H) and Nuniber of Additional Machines (A): n * 60; Ntmber of Each Machine Type i s Uniformly Distributed Between 1 and 3.
N
2
4
6
8
10
12
14
16
18
20
22
24
26
78
30
32
34
36
38
40
42
44
46
48
50
52
54
"~r^: 1 PFA
H
.1972
.1764
.2372
.2128
.2617
.2660
.3275
.3388
.3350
.3261
.3404
.3595
.3608
.3565
.3042
.2717
.2604
.2081
.1401
.1020
.0726
.0383
.0333
.0263
.0301
.0237
.0164
A
0
0
1
3
3
3
3
6
7
9
16
22
28
40
44
48
50
57
63
67
71
77
87
95
103
111
117
n « . T -*" PFA
H
.1590
.1679
.1262
.1467
.2018
.1827
.1843
.1915
.1825
.1615
.1585
.1569
.1498
.1663
.2287
.1366
.1240
.1104
.0855
.0702
.0681
.0618
.0547
.0425
.0320
.0293
.0217
A
0
0
1
2
7
11
15
20
27
30
39
41
47
56
63
64
66
70
73
78
83
87
91
99
106
115
124
fi = , t^^ PCA
H .6005
.8059
.5827
.7561
.8535
.9200
.9420
.9693
.9859
.9873
.9884
.9798
.9813
.9801
.9692
.8750
.7647
.6666
.5789
,5000
.4285
.3636
.3043
.2500
.2000
.1538
.1111
A 0
3
29
47
56
65
80
83
88
91
101
111
115
123
129
130
130
132
135
135
135
135
135
135
139
138
138
fi = . ,6 PCA
H .4133
.7939
.7251
.8558
.8837
.9327
.9140
.9249
.9312
.8794
.8811
.8612
.8146
.7259
.7058
.6635
.5266
.4451
.4023
.3554
.3623
.3104
.2665
.2137
.1822
.1367
.1111
A 0
3
10
17
27
. 33
51
67
73
92
102
121
132
143
141
144
152
152
152
152
153
155
157
159
162 163
164
181
( i i ) Group Homogeneity (H) and Number of Additional Machines (A): n = 80; Nuniber of Each Machine Type i s Uniformly Distributed Between 1 and 3.
N
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
""^r^T: PFA
H
.1997
.1884
.1729
.1777
.1823
.1805
.1684
.1894
.2191
.2139
.2084
.1995
.1751
.1976
.2010
.1865
.1952
.1685
.1754
.1757
.1741
.1820
.1625
.1545
.1377
.1412
.1298
.1154
.0947
A
0
0
0
0
2
4
6
7
8
10
15
18
27
34
41
50
59
66
72
84
87
89
94
97
104
107
111
119
126
ii « . r-" PFA
H
.1568
.2168
.1331
.1764
.1627
.1867
.1788
.1840
.1913
.1887
.2028
.1872
.1740
.1793
.1732
.1695
.1707
.2011
.1985
.2058
.1758
.1623
.1519
.1450
. 1359
.1149
.1058
.0828
.0800
A
0
1
1
2
5
7
10
15
14
18
30
45
55
65
87
94
104
107
112
123
130
134
137
136
139
143
146
154
159
46 *"* J rT"-^ PCA
H
.2860
.6649
.8033
.8531
.8897
.8769
.9175
.9561
.9621
.9682
.9712
.9736
.9798
.9812
.9890
.9596
.9764
.9628
.8952
.8351
.7264
.7175
.6676
.6416
.5760
.5153
.4703
.4285
.3793
A
0
4
24
44
61
77
93
115
128
138
151
158
162
167
169
177
176
189
194
191
199
200
192
203
206
216
213
213
213
"ir^ • 0 PCA
H
.2301
.6589
.7718
.7618
.7621
.7190
.8316
.8253
.8453
.8888
.8987
.8522
.8782
,8777
.8709
.8353
.8144
.8036
.7614
.6611
.6916
.5315
.5151
.4809
.4816
.4246
.3837
.3343
.3243
A
0
2
10
23
42
60
72
88
95
101
123
144
150
171
185
196
207
221
224
236
240
240
240
234
236
236
237
237
242
182
( i i i ) Group Homogeneity (H) and Ntmiber of Addit ional Machines (A): n = 100; Ntimber of Each Machine Type i s Uniformly Dis t r i buted Between 1 and 3 .
N
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
u u ^" • ^
PFA
H
.2461
.1899
.1946
.1888
.1901
.1839
.1858
.1644
.1689
.1641
.1691
.1781
.2059
.2167
.2157
.2107
.2253
.2150
.2036
.2069
.2126
.2011
.1925
.1911
.1958
.1905
.1730
.1515
.1441
\
A
0
0
1
2
4
4
5
8
10
14
17
20
22
30
34
44
54
63
75
84
92
98
102
118
131
131
135
139
145
U = .1 PFA
H
.1798
.1833
.1849 '
.1981
.2026
.1874
.1922
.2111
.1856
.2325
.2248
.2001
.2075
.2047
.1998
.1984
.1958
.1984
.1900
.1996
.1932
.1780
.1700
.1860
.1045
.1510
.1356
.1229
.1197
b
A
0
0
1
1
3
3
7
9
9
31
17
30
47
50
60
66
73
84
100
115
123
147
157
161
182
185
190
192
193
li - . J PCA
H
.2466
.5155
.8656
.9020
.9230
.8953
.9454
.9533
.9586
.9621
.9658
.9691
.9714
.9754
.9770
.9788
.9791
.9802
.9812
.9711
.9433
.9379
.9228
.9055
.8774
.7965
.6971
.6908
.6508
A
0
7
12
39
69
103
118
141
153
171
183
197
202
212
216
232
247
256
257
261
275
278
288
284
283
285
284
289
286
u = . ,b PCA
H
.5884
.6632
.8625
.8974
.9186
.8914
.9424
.9226
.8982
.8785
.8895
.8989
.8669
.8462
.8420
.8509
.8729
.8507
.8518
.8292
.8074
.8067
.7401
.6868
.5784
.5352
.5591
.5272
.4748
A
0
7
20
42
53
79
98
113
135
140
143
159
175
187
206
225
235
254
272
270
296
310
318
325
336
343
337
339
342
183
( iv ) Group Homogeneity (H) and Ntmiber of Addit ional Machines (A): n = 120; Number of Each Machine Type i s Uniformly Dist r i b u t e d Between 1 and 3.
•
N
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56 1? o
58
!.Z = . J PFA
H
.2758
.2215
.2098
.1830
.2100
.2022
.1925
.1868
.1784
.1706
.1786
.1941
.1904
.1949
.1917
.1887
.1848
.1851
.1840
.1840
.1816
.1857
.1877
.1820
.1900
.1808
• .1881
.1890
.1896
1
A
0
0
1
4
4
9
14
17
18
23
25
24
29
32
38
38
44
52
55
62
74
87
88
101
123
133
155
177
187
il = A PFA
H
.1258
.1625
.1715
.1421
.1580
.1626
.1499
.1517
.1446
.1380
.1526
.1587
.1595
.1338
.1304
.1587
.1642
.1701
.1764
.1804
.1802
.1777
.1720
.1713
.1627
.1614
.1665
.1637
.1591
3
A
0
0
0
4
4
6
9
14
19
24
27
34
39
47
54
64
80
84
99
107
120
133
141
151
177
187
206
214
237
il — . 3 PCA
H
.2600
.5818
.8673
.9011
.9214
.9347
.9113
.9265
.9586
.9630
.9373
.9718
.9626
.9765
.9605
.9627
.9649
.9682
.9707
.9600
.9561
.9576
.9679
.9735
.9532
.9642
.9362
.9142
.8689
A
0
14
17
42
75
96
133
161
186
209
234
267
294
281
304
304
316
335
348
347
367
381 364
371
390
387
384
392
402
ii = . b PCA
H
.5732
.7900
.7818
.7802
.7349
.8106
.8752
.8037
.8187
.8392
.8317
.8493
.8818
.8705
.9005
.8956
.8930
.8710
.8884
.8790
.8647
.8524
.8457
.8468
.8046
.7891
.7409
.7309
.7714
A
0
2
17
23
47
57
73
93
107
121
133
151
179
183
221
203
246
253
257
262 ^H /^ /^
282
311
322
332
359
370
373
392 401
184
(V) Number of Additional Machines: Type i s 1.
Ntmiber of Each Machine
N
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38 W W
40
42
44
46
n =
a- .3 8
11
16
19
23
27
39
40
45
46
67
74
91
101
107
110
119
120
126
131
135
142
154
60 a-.6
2
9
21
36
47
56
59
73
75
74
86
102
112
121
127
132
133
137
138
143
152
157
162
n »
«=.J
11
8
20
27
32
37
40
49
53
62
64
71
77
86
93
103
117
123
134
144
146
160
165
80
K «.b
7
19
29
37
49
49
66
78
94
103
121
129
131
141
160
167
186
200
203
211
207
210
214
n =
M =.J
5
12
21
24
29
38
48
55
60
68
78
85
91
97
102
122
123
136
146
157
170
176
187
100
u^,b 3
15
23
48
38
45
55
76
86
92
102
124
140
133
150
167
189
191
208
215
229
246
267
n =
u-.:i 2
14
22
39
48
51
48
55
60
68
83
91
101
104
112
119
129
144
151
171
175
187
204
120
li=.b 3
13
22
30
43
60
66
79
96
106
115
124
140
148
161
172
173
188
200
224
239
252
270
185
(v i ) So lu t ion to Case Study Problems.
No. of Burb idge ' s Prob. Purcheck 's Prob. Harworth's Prob n = 43 , M = 16 n = 82, M = 19 n = 85
PFA PFA PCA
A A H
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
2
9
11
13
15
19
22
26
32
36
42
47
50
56
64
70
77
83
92
1
3
7
10
15
19
25
35
52
57
63
69
75
81
89
99
108
119
126
134
148
156
164
.3094
.6565
.8162
.8194
.8558
.8526
.8496
.8472
.8456
.8447
.8438
.8427
.8328
.8209
.7912
.7555
.7395
.7075
.7672
.7656
.7060
.6926
.6583
.d i^.