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69
S. Ashour
Sequencing Theory
Springer-Verlag
Berlin· Heidelberg· New York 1972
Advisory Board
H. Albach· A. V. Balakrishnan' F. Ferschl . R. E. Kalman' W. Krelle . G. Seegmiiller
N. Wirth
Prof. Dr. Said Ashour
Iowa State University
Ames, Iowa 500 10, USA
AMS Subject Classifications (1970): 90B3)
lSBN-13: 978-3-540-05877-9 e-lSBN-13: 978-3-642-80693-3
DOl: 10.1007/978-3-642-80693-3
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© by Springer-Verlag BerlIn' Heidelberg 1972. Library of Congress Catalog Card Number 72-83014.
Softcover reprint of the hardcover 1st edition 1972
PREFACE
The intent of these notes is to provide an appreciation of sequencing theory, and
to develop an awareness of the combinatorial aspects of sequencing problems. An under
standing of the ideas and concepts presented in these notes will provide the necessary
background to enable further study in this rapidly growing subject. These notes have
been used as a part of one-semester C0urses in Operations Research for both senior and
first-year graduate students.
Following the introductory chapter, several classes of scheduling models are
discussed in Chapter 2. Chapter 3 is devoted to discussing the characteristics of
several types of criteria that have been proposed in the literature. Because of the
importance of the combinatorial properties in the analysis of scheduling models,
various combinatorial aspects are treated in Chapter 4. The primary objective of
such a treatment is to provide the reader with an insight into the nature of the
problem. Due to the tremendous number of sequences involved, and the fact that many
sequences have the same schedule time, statistical sampling has elicited a great deal
of theoretical studies. Two such studies, one of which has evolved from my research,
are reported in Chapter 5. Many references to basic sources are cited at the end of
each chapter to provide a guide for further readings.
I am indebted to Professor Santa Arora of the University of Minnesota who intro
duced me to the subject during my graduate career. I have been extremely fortunate
in receiving valuable comments and constructive criticism on an early draft of the
manuscript from Professor Salah E. Elmaghraby of North Carolina State University at
Raleigh. These notes have been prepared at Kansas State University, and I gratefully
acknowledge the support and encouragement received. Special thanks are due to my
graduate student Prakash Bhatt for proofreading the first draft. Further, I wish to
express my deep appreciation to Marie Jirak for her skillful typing of several drafts
of these notes and to Janet Athavichitchanyaraks, Leslie Fletcher, and Jan Gaines for
excellent typing of the final manuscript. Finally, and most importantly, I am grate
ful to my wife for her patience and encouragement during the writing of this manuscript.
March 1972 Said Ashour
TABLE OF CONTENTS
CHAPTER 1: INTRODUCTION. • • • . • • • • • • . • • . • . • • • • • . • . • .. 1
CHAPTER 2: BASIC STRUCTURE • • • • 5
2.1 Problem Definition 5
2.2 Problem Formulation 12
2.3 Problem Assumptions 29
CHAPTER 3: MEASURES OF PERFORMANCE 34
3.1 Characteristics of Criteria 34
3.2 Single Criteria 39
3.3 Multiple Criteria 56
CHAPTER 4: COMBINATORIAL ASPECTS . . . . • • 65
4.1 Characteristics of Sequences 65
4.2 Characteristics of Schedules 88
CHAPTER 5: STATISTICAL ASPECTS 109
5.1 Theoretical Study I 109
5.2 Theoretical Study II 116
CHAPTER 1
INTRODUCTION
A shop production system, in whLch a wide variety of jobs are handled according
to customer specifications, is traditionally found in jobbing-type machine shops.
The complex behavior of the machine shop, which consists of·several work centers
each of which has one or more machines of the same type, can be described as follows.
The work in the machine sh"p is usually initiated by receiving orders from cus
tomers. The order may comprise one or more jobs, each of which should be engineered
before being released to the shop. This includes: (1) establishing the technologi
cal ordering in which the job is to be processed on the machines; (2) specifying the
materials, fixtures and tools which are required to perform the job; (3) determining
the economical batch size, if any, in which the job is to be processed; (4) estimating
the processing time, including job transportatfon, and machine setup times, if any,
for each of the operations comprising the job; and (5) setting the time limit by
which the job should be completed in response to a delivery date dictated by the
customer, or for inventory replenishment. When the job is released to the shop, it
is then routed through the work centers to perform the associated operations. Each
of these operations takes place at one and only one work center. It is natural that
any job may comprise a number of units which may be determined on an economical batch
size basis, or specified in the customer order.
In scheduling a set of jobs through the shop, the objective is to ensure that
the prescribed technological ordering of each job is satisfied, and there are no
conflicts between jobs while attempting to meet job due-dates or other criteria.
The ideal situation in such a shop is that machines are kept continuously busy,
manpower is just sufficient to handle the work load, materials and in-waiting
inventories, and end-product inventory, if any, are kept at reasonable levels,
and job due-dates are set that they can be met. In practice, however, the shop
usually accommodates itself to the current situations regardless of scheduling
procedures or rules. Indeed, as Pounds [6] has pointed out, it is possible for
shop management not to be aware of the existence of any scheduling problem because
2
the other components of the total shop production system have acted to shield
management from strongly interdependent scheduling problems.
The majority of academic research on shop scheduling has centered on the
sequencing problem. The sequencing problem is concerned with determining the
sequence in which a set of jobs is to be performed on each of a number of machines.
This problem is, of course, only a part of the overall production control problem in
a shop production system. The sequencing problem is preceded by the planning area
which involves the functions c~ material procurement, technological processes, pro-
ductive capacity, routing requirements, processing time estimates, and shop loading.
On the other hand, the sequencing problem is followed by the control area which
involves the regulations and adjustment of production activities to conform to plans
Because of the strong interaction between the sequencing problem and its preceding
and succeeding areas, it is not immediately obvious how one can either isolate the
sequencing problem or assess precisely the impact of sequencing decisions on the
other decisions, and vice versa. In fact, the areas of capacity planning, shop
loading, and job sequencing are strongly interrelated to the extent that decisions
in one area do strongly influence decisions in the other two.
Although the most interesting production control problems arise in the totality
of the capacity planning, shop loading, and job sequencing areas; the sequencing
theory itself is worth studying for several reasons stated by Elmaghraby [3].
First, a meaningful study of a multi-stage system such as the
more common production system necessitates intimate knowledge of each
component. The sequencing phase is one such component, and its detailed
study is therefore an essential element of the study of the complete
system.
Second, the advent of the computer as an operational tool for the
management of productive systems tends to eliminate a great deal of the
"fat" previously found in these systems. This, in turn, emphasizes the
effect of hitherto secondary decisions, such as those concerned with
sequencing.
Third, continued and expanded education in operations research and
management sciences is beginning to bear fruit in the form of improved
management information systems. This has led to great strides in the
other aspects of the production function, such as inventory, loading,
financing, and so on. Further improvement in some of these areas is,
in fact, constrained by the possible improvement in the sequencing area.
From a pedagogical point of view, ••• , sequencing theory is an
excellent demonstration of the applicability of several theoretical
models (such as linear programming and mathematical abstractions (such
3
as combinatorial analysis) to everyday problems. It has proved
invaluable in rendering these conceptual formulations understandable
to the students.
Traditional scheduling procedures are typically designed to cope with the
complexities of shop scheduling, where each job is unique and no prescribed techno-
logical ordering exists. These procedures depend on graphical devices such as Gantt
chart or its concepts to help maintain control of the shop operations. The Gantt
chart, an innovation of the scientific management era, has been developed by Henry
Gantt. A complete discussion of the basic Gantt charts has been presented by Clark
[1]. Essentially, a Gantt chart is a device for portraying the information regarding
the progress of each job in the shop and the current capacity of machines. It also
pinpoints potential and actual problems in time for corrective action to be taken.
Although Gantt charts are useful in developing feasible schedules, they are not, of
course, a substitute for decision making. Scheduling decisions include, for example,
the time at which a job should be released to the shop, and the sequence in which the
jobs should be performed on each machine to meet one or more measures of performance.
Until 1954 there has been virtually no reported attempt to treat the shop
scheduling problem analytically with the explicit purpose of optimizing some measure
of performance. Since that time, however, the study of the shop scheduling problem
and its contexts has attracted researchers from every field. This is due to the
development of operations research techniques and their application to an area
previously dominated by intuition and judgment; and due to the increased awareness
of management to the potential gains realizable when good scheduling practices are
employed. Theoretical models representing simple shop scheduling problems have been
formulated and analyzed utilizing operations research techniques such as comb ina to-
rial analysis, dynamic programming, integer programming, reliable heuristics, and
network analysis approaches. The principal difficulty with these approaches, however,
has been that the computational effort required incrp.as~s rapidly even with a rela-
tively small size problems and simplified assumptions. Shop scheduling problems have
been formulated as queueing models. However, due to the difficulty in solving such
models analytically, various groups have pursued the study of shop scheduling
problems utilizing computer simulation techniques. Several references in which
4
excellent expositions on possible approaches to shop scheduling problems, are cited
below.
REFERENCES
[l] Clark, W., The Gantt Chart: A Working Tool of Management, Sir Isaac Pitman &
Sons Ltd., London, England, 1952.
[2] Conway, R. W., W. L. Maxwell and L. W. Miller, Theory of Scheduling, Addison
Wesley Publishing Company, Reading, Massachusetts, 1967.
[3] Elmaghraby, S., "The Machine Sequencing Problem - Review and Extension," Naval
Research Logistics Quarte:ly, Vol. 15, No.2, 1968, pp. 205-232.
[4] Mellor, P., "A Review of Job Shop Scheduling," Operational Research Quarterly,
Vol. 17, No.2, 1966, pp. 161-171.
[5] Muth, J. F., and G. L. Thompson, eds., Industrial Scheduling, Prentice-Hall,
Inc., Englewood Cliffs, New Jersey, 1963.
[6] Pounds, W. F., "The Scheduling Environment," Chapter 1 in Industrial Scheduling,
(eds. J. F. Muth and G. L. Thompson), Prentic-Hall, Inc., Englewood Cliffs,
New Jersey, 1963.
[7] Sisson, R. L., "Methods of Sequencing in Job Shops - A Review," Operations
Research, Vol. 7, No.1, 1959, pp. 10-29.
[8] Sisson, R. L., "Sequencing Theory," Chapter 7 in Progress in Operations Research,
(ed. R. M. Ackhoff), John Wiley & Sons, New York, N.Y., 1961.
[9] Spinner, A. H., "Sequencing Theory - Development to Date," Naval Research
Logistics Quarterly, Vol. 15, No.2, 1968, pp. 319-330.
[10] Thompson, G. L., "Recent Developments in the Job-Shop Scheduling Problems,"
Naval Research Logistics Quarterly, Vol. 7, No.4, 1960, pp. 585-589.
CHAPTER 2
BASIC STRUCTURE
Different scheduling problems naturally lead to different models. In this
chapter we shall discuss several classes of these models. In order to provide an
insight into the structure of the problem, we shall introduce an explicit mathe
matical formulation which can be represented by Gantt chart, and a linear graph.
Finally, several assumptions restricting the analytical models, which have been
developed to date, will be discussed.
2.1 Problem Definition
The classical shop scheduling process may be characterized in terms of the
following context. A machine shop is equipped with various groups of machines.
Each machine group consists of a number of identical machines with respect to their
ability to process the expected work in the shop. The amount of work constitutes
several jobs to be performed on the machines. Each job may consist of a single unit
or a batch of identical units, usually determined on an economical basis. Any job
requires one or more operations, each of which is to be processed on a single machine.
An operation is determined entirely by the specification of the job and the machine
involved. The operations comprising each job are processed on the machines in a
specified routing, hereafter referred to as a machine ordering. The machine order
ings are usually imposed by technological requirements which always dictate the basic
manufacturing process and types of machines, fixtures and tools to be used.
Associated with each operation is a processing time which is required for the
complete performance of this operation. The processing time of an operation may
consist of: (1) the job transportation time--the time required to transport the job
from one machine to another according to the prespecified machine ordering; (2) the
machine setup time--the time required to prepare the machine for performing that
operation; (3) the operation running time--the time required to perform the job on
a particular machine; and (4) the machine teardown time--the time required to reset
the machine after the operation has been completed. As the jobs flow through the
shop, the associated operations must compete with one another for each of the
Description:The intent of these notes is to provide an appreciation of sequencing theory, and to develop an awareness of the combinatorial aspects of sequencing problems. An under standing of the ideas and concepts presented in these notes will provide the necessary background to enable further study in this
