Study on Evaluation of Setup Economies in Cellular Manufacturing

Description
Cellular Manufacturing is a model for workplace design, and has become an integral part of lean manufacturing systems. Cellular Manufacturing is based upon the principals of Group Technology, which seeks to take full advantage of the similarity between parts, through standardization and common processing.

ABSTRACT
Title of dissertation: EVALUATION OF SETUP ECONOMIES IN
CELLULAR MANUFACTURING
Steven Boyd Kramer, Doctor of Philosophy, 2004
Dissertation directed by: Professor Arjang A. Assad
Decision and Information Technologies
The Robert H. Smith School of Business
This dissertation addresses two research questions relating to the role of setups in
discrete parts manufacturing. The first research topic uses a carefully designed
simulation study to investigate the role of setup economies in the factory-wide
conversion of functional layouts (job shops) to cellular manufacturing. The model-
based literature shows a wide dispersion in the relative performance of cellular
manufacturing systems as compared to the original job-shop configurations, even
when the key performance measure is flow time and the assessment tool used is
simulation. Using a standardized framework for comparison, we show how this
dispersion can be reduced and consistent results can be obtained as to when the
conversion of the job shop is advantageous.
The proposed framework standardizes the parameters and operational rules to
permit meaningful comparison across different manufacturing environments, while
retaining differences in part mix and demand characteristics. We apply this
framework to a test bed of six problems extracted from the literature and use the
results to assess the effect of two key factors: setup reduction and the overall shop
load (demand placed on the available capacity). We also show that the use of transfer
batches constitutes an independent improvement lever for reducing flow time across
all data sets. Finally, we utilize the same simulation study framework to investigate
the benefits of partial transformation, where only a portion of the job shop is
converted to cells to work alongside a remainder shop.
The second research question examines the role of dispatching rules in the
reduction of setups. We use queueing models to investigate the extent of setup
reduction analytically. We single out the Alternating Priority (AP) rule since it is
designed to minimize the incidence of setups for a two-class system. We investigate
the extent of setup reductions by comparing AP with the First-Come-First-Served
(FCFS) rule. New results are obtained analytically for the case of zero setup times
and extended to the case of non-zero setup time through computational studies.
EVALUATION OF SETUP ECONOMIES IN CELLULAR
MANUFACTURING
by
Steven Boyd Kramer
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland at College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2004
Advisory Committee:
Professor Arjang A. Assad, Chair/Advisor
Professor Shapour Azarm
Professor Michael C. Fu
Professor Bharat K. Kaku
Professor Gilvan C. Souza
© Copyright by
Steven Boyd Kramer
2004
ii
ACKNOWLEDGEMENTS
I am deeply indebted to Dr. Arjang Assad for his patience, assistance, and concern
for scholarly research. I also appreciate the guidance of Dr. Bharat Kaku especially in
the formative stage of my research on cellular manufacturing systems. I would also
like to acknowledge the balance of my committee, Professor Shapour Azarm,
Professor Michael Fu and Professor Gilvan Souza who constructively guided my
direction at my proposal toward the addition of analytic contributions and provided
valuable feedback at my defense.
I could not have completed this dissertation without the loving support of my
family and friends. I owe everything to my wife, Susan, daughters, Shannon and
Larkin and son, Keenan. I dedicate this work to them.
iii
TABLE OF CONTENTS
List of Tables ................................................................................................................... v
List of Figures................................................................................................................vii
Chapter 1 INTRODUCTION..........................................................................................1
1.1 The Manufacturing Environment .......................................................................1
1.2 Factory Conversion.............................................................................................8
1.3 Key Trade-offs ..................................................................................................13
1.4 Research Objectives..........................................................................................16
1.5 Plan of the Dissertation.....................................................................................19
Chapter 2 LITERATURE REVIEW............................................................................21
2.1 Conversion Analysis Using Simulation ...........................................................21
2.2 Two-Class, Single-Stage M/G/1.......................................................................48
Chapter 3 FACTORY CONVERSION TO CELLULAR
MANUFACTURING SYSTEMS .............................................................56
3.1 Factory Environment and Notation ..................................................................57
3.2 Job Shop Operation...........................................................................................59
3.3 Standardization Scheme....................................................................................59
3.4 Choice of Data Sets...........................................................................................64
3.5 Metrics and the Simulation Model ...................................................................70
3.6 Simulation Results Comparing Functional and Cellular Layout.....................72
3.7 Sensitivity to Key Operational Factors.............................................................78
3.8 Move times........................................................................................................87
3.9 Discussion on Dispersion of Simulation Results in the Literature..................88
3.10 Summary .........................................................................................................92
Chapter 4 PARTIAL CELLULAR MANUFACTURING SYSTEMS ......................96
4.1 Simulation Analysis of PCMS..........................................................................96
4.2 Summary .........................................................................................................104
Chapter 5 ANALYTIC MODELING OF A SIMPLE SYSTEM WITH
SETUP ......................................................................................................106
5.1 Zero Setup .......................................................................................................107
5.2 Non-Zero Setup...............................................................................................131
Chapter 6 SUMMARY AND DIRECTIONS OF FUTURE RESEARCH..............163
iv
APPENDIX A: SENSITIVITY TO THE SHAPE OF PROCESSING TIME
DISTRIBUTIONS.................................................................................... 170
APPENDIX B: OUTPUT MEASURES FOR SIMULATION RUNS..................... 172
GLOSSARY................................................................................................................ 174
BIBLIOGRAPHY....................................................................................................... 177
v
List of Tables
Table 1-1. Part routing matrix: operation sequence linking part number with
machine type. ....................................................................................................10
Table 1-2. Summarized family and cell requirements. ................................................11
Table 1-3. Machine distribution....................................................................................11
Table 1-4. Partitioned part routing matrix indicating part operation sequences,
part families, cells and machine types per cell.................................................12
Table 1-5. Assumptions for factory conversion research............................................18
Table 1-6. Assumptions for analytic modeling research.............................................19
Table 2-1. Study definitions. .........................................................................................24
Table 2-2. Comparison of factor levels within simulation studies ..............................25
Table 2-3. Difference in operating scenarios may confuse comparative results. ........28
Table 2-4. JS to CMS Conversion Literature Summary ..............................................31
Table 2-5. Deep setup discounts may not be sufficient to guarantee PCMS
success. ..............................................................................................................42
Table 2-6. PCMS Studies..............................................................................................46
Table 2-7. Single-server modeling contributions. ........................................................49
Table 3-1. Choices and parameters values for operational standardization. ...............63
Table 3-2. Data sets used in analysis as reported by source.........................................66
Table 3-3. Data sets characteristics after operational standardization. ........................69
Table 3-4. Comparison of cell designs in source and standardized
configurations....................................................................................................70
Table 3-5. Setup reductions and associated flow ratios for Operational
Standardization..................................................................................................73
Table 3-6. Setup reductions and associated flow ratios for Formation and
Operational Standardization .............................................................................73
vi
Table 3-7. Flow times in cells with smaller batch size or transfer batches .................79
Table 3-8. RL dispatching avoids more major setups in the job shop than FCFS. .....86
Table 3-9. JS to CMS flow ratios in the modeling literature. ......................................91
Table 4-1. Simulation results for best and worst picks at each level of cellular
implementation................................................................................................100
Table 5-1. Four cases defining the parameter space for 1 0 s < Q . ..........................121
Table 5-2. Four cases defining the parameter space for 1 > Q .................................121
Table A-1. Sensitivity of Job Shop and CMS flow times to changes in
distributions of setup and runtime. .................................................................171
vii
List of Figures
Figure 1-1. Single- versus Multi-Stage Processing. ......................................................2
Figure 1-2. Illustrative part routings for parts 8, 9, and 10. .........................................10
Figure 1-3. Pooling loss.................................................................................................15
Figure 2-1. Disparity of results reported in Johnson and Wemmerlöv (1996)...........38
Figure 2-2. Optimal flow time improvements require controlled cell loading............41
Figure 3-1. Comparison of machine utilization for JS and CM...................................76
Figure 3-2. Simulation results for machine types with utilization above 65% in
the JS layout. In the job shop, JS- j denotes machine type j . Within
cells, Cc - j denotes machine type j in cell c . ...............................................77
Figure 3-3. Flow time improvement using unity transfer batches as a function of
operations per part (data set 6)..........................................................................81
Figure 3-4. Job shop loading sensitivity (data set 2). ...................................................83
Figure 3-5. Response of the flow ratio to the two setup parameters............................84
Figure 3-6. Results from standardized approach reduce variability and favor
CM.....................................................................................................................89
Figure 3-7. Results of sensitivity analysis for data set 2. .............................................92
Figure 4-1. Machine utilization ranges during early stages of CMS
implementation................................................................................................102
Figure 5-1. Roots and minimum for ( ) Q f when 2 1 ,
2 1
< µ µ . ................................114
Figure 5-2. Single root of ( ) Q f when 2 1
1
> µ ........................................................115
Figure 5-3. Graph of
1
ì versus Q when 25 . 0
2
= ì . AP and FC indicates
superiority in that region. ................................................................................124
Figure 5-4. Graph of
1
ì versus Q when 10 . 0
2
= ì . .................................................125
Figure 5-5. Graph of
1
ì versus Q when 40 . 0
2
= ì . .................................................126
viii
Figure 5-6. Graph of
1
ì versus Q when 60 . 0
2
= ì . .................................................127
Figure 5-7. Wait time differences (AP-FCFS)*100 when setup is zero and
25 . 0
2
= ì . .......................................................................................................130
Figure 5-8. Wait time differences (AP-FCFS)*100 when setup is zero and
60 . 0
2
= ì . .......................................................................................................131
Figure 5-9. Probability of setup (FCFS% above AP%) when setup is zero and
25 . 0
2
= ì . .......................................................................................................146
Figure 5-10. Probability of setup (FCFS% above AP%) when setup is zero and
60 . 0
2
= ì . .......................................................................................................147
Figure 5-11. Wait time differences (AP-FCFS)*100 for symmetric cases when
2 . 0 =
i
ì ...........................................................................................................148
Figure 5-12. Wait time differences (AP-FCFS)*100 when E(U)=0.001*E(S)
and 25 . 0
2
= ì . ................................................................................................150
Figure 5-13. Wait time when 50 . 0
1
= ì , 25 . 0
2
= ì using AP..................................151
Figure 5-14. Wait time when 50 . 0
1
= ì , 25 . 0
2
= ì using FCFS. ............................152
Figure 5-15. Three zeroes of intersection between wait curves of AP and FCFS
when 50 . 0
1
= ì , 25 . 0
2
= ì and ( ) ( ) S E U E * 01 . 0 = . ..................................153
Figure 5-16. Wait time differences (AP-FCFS)*100 when E(U)=0.01*E(S) and
25 . 0
2
= ì . .......................................................................................................154
Figure 5-17. Wait time differences (AP-FCFS)*100 when E(U)=0.05*E(S) and
25 . 0
2
= ì . .......................................................................................................155
Figure 5-18. Wait time differences (AP-FCFS)*100 when E(U)=1.0*E(S) and
25 . 0
2
= ì . .......................................................................................................156
Figure 5-19. Wait time differences (AP-FCFS)*100 when E(U)=0.01*E(S) and
10 . 0
2
= ì . .......................................................................................................158
Figure 5-20. Wait time differences (AP-FCFS)*100 when E(U)=0.1*E(S) and
10 . 0
2
= ì . .......................................................................................................159
ix
Figure 5-21. Wait time differences (AP-FCFS)*100 when E(U)=1.0*E(S) and
10 . 0
2
= ì . .......................................................................................................160
1
Chapter 1
INTRODUCTION
1.1 The Manufacturing Environment
Buzacott and Shantikumar (1993, p.1) describe a manufacturing system as a
system consisting of “machines and work stations where operations such as
machining, forming, assembly, inspection, and testing are carried out on parts, items,
subassemblies, and assemblies, to create products that can be delivered to customers.”
In discrete part manufacturing systems, each item processed is distinct, although the
processing may take place in batches or distinct packets. The batches are then used as
transfer units between manufacturing areas. This is in contrast to chemical industries
where the processed material may be in the form of continuous fluid. Discrete parts
manufacturing systems arise commonly in “mechanical, electrical, and electronics
industries making products such as cars, refrigerators, electric generators, or
computers .” (p.1).
As an example, we examine a process designed to create a hole through a block of
metal as illustrated in Figure 1-1. The process may require a single operation (single-
stage) as in a drill press drilling a hole, or may require multiple operations linked
together (multi-stage) if the completed hole requires further finishing such as the
addition of a champher and de-burring.
2
Drill
Press
Single-
Stage
Drill
Press
Drill
Press
Multi-
Stage
Drill
Champher De-Burr
Operator:
Sand
Paper
Figure 1-1. Single- versus Multi-Stage Processing.
Multi-stage processes may include internal buffer storage in order to account for
variations in the time between successive outputs of product at a process of each
process step allowing each to work more independently of the other. Single- or multi-
stage processes may be linked together to provide a variety of processing capabilities.
The time between successive outputs of a multi-stage process is usually regulated by
the dynamics of the flow of parts under congestion and may depend crucially on
bottleneck stages that limit the capacity of the overall process. The flow time of a job
is the amount of time a job spends in the system. Specifically, it is the time from when
a job consisting of demand for a certain batch size of a given product is introduced
into the manufacturing facility at the location of its first operation to when the last
operation required on the batch of product has been completed. It includes the time
waiting for processing and material transfer between operations, setup times if
required, as well as the time the batch is being serviced by machines. A bottleneck
process adversely affects the flow time of all parts using that process.
3
One can characterize manufacturing processes based on the way the process flow
is coordinated. A process can be synchronous or asynchronous. Synchronous
processes have a fixed process rate where all work moves at the same rate through the
processing steps in sequence. This is either done continuously, as in automobile
painting operations using continuous conveyors moving at a fixed rate, or discretely,
as in spot welding operations of chassis where automobiles move in and out of robot
welding stations at regular intervals. Synchronous flows eliminate most of the need
for storage between machines, but require tight coordination of customer orders,
material supply and extremely high process quality. Asynchronous processes are
much more common, where work is moved to its next process step when the current
step is completed. Work, since not synchronized, is then staged in an “input” queue
and waits as required for its turn at the next operation.
A key concern of the work presented in this dissertation is the few factory layout
structures used to organize the material flow. The most common type and the one that
naturally aligns with high part variety is the job shop. The term “job shop”
(abbreviated as JS throughout this work) refers to a manufacturing facility comprised
of general-purpose machines organized into a collection of machine centers or
departments grouped on the basis of the operation performed (turning, drilling,
milling, etc.). By providing the appropriate machine types, a small number of
machine departments is sufficient in the factory to accomplish a high variety of part
processing. These machine types can be applied in various sequences to produce a
wide variety of parts. The job shop structure supports a high variety of jobs.
4
Typically, job shops are designed to handle small production batches of custom
products requiring a variety of processing requirements. Accordingly, the equipment
is organized by function as the same general type of operation may be performed by a
number of machines in a wide variety of different ways. For example, when a hole is
needed in a piece of metal, it is sent to the drilling department where a variety of
machines from drill presses to mills to boring machines may reside. We will consider
more details of operations of the job shop below.
Assembly lines (or flow lines) are structures where process equipment is
organized in the order specified by their operations. This organizational principle is
also known as a product focus. Assembly lines minimize material handling since the
next machine needed is in immediate physical proximity. Material handling
automation is commonly employed between process steps to retain part registration,
minimizing setups and reducing labor. This type of structure is biased to the direction
of part flow, so backtracking, where processes must travel opposite the direction of
the standard flow in order to get access to a particular type of machine, is difficult and
very disruptive.
Current industry trends encourage managers to focus their factories to provide
products and services at high quality and low cost. A challenge in discrete parts
manufacturing is to provide customized products to meet individual tastes while
depending on the stability of common processes and equipment (Pine, 1993, p.7).
Factories using general-purpose machines are capable of producing a large variety of
parts by the nature of their process equipment. However, frequent tooling
5
changeovers are required on general-purpose machines to account for part variety that
can be time-consuming and expensive. Below, we outline some of the benefits of an
alternative approach, which we call a cell shop.
1.1.1 Job Shop. In a job shop, a large fraction of the flow time of a given part is
due to wait times. Parts often have to queue up to await their turn at a given machine
or machine center due to limited capacity, wait for material handling devices for
transport to or from a process or wait to join parts being processed in other parts of the
factory. The machines typically require setups due to changeovers between
operations in order to accommodate different part and processing requirements. The
machines in each department share a common queue of incoming work and the length
of this queue accounts for most of the delay at each machine center. If jobs are
assigned at random, the larger the variety of parts types, the more likely it is for setups
to be incurred. Increasing the frequency of setups increases the amount of time
required to complete each job (expected setup plus run time). This increases the time
spent at the machine for each job, and leads to longer queues and wait times. This
relationship is apparent in the familiar M/M/1 queue, where the wait in the queue,
q
W , is a function of the arrival rate and mean service time ( ì and
1 ÷
µ , respectively)
and machine utilization, which is represented in this case by µ ì µ = :
( ) µ µ µ ÷ = 1
q
W . In this dissertation we will consistently associate the wait in the
queue with the time from when a customer arrives in the system until service
commences on that customer. We, therefore, imbed any required setup time in this
6
queue wait. The batch flow time is measured from part batch introduction into the
factory (from receiving) to part batch leaving the factory (sent to shipping).
Material handling also contributes to the flow time of parts in the job shop and
wait times for material handling resources. Parts travel from department to
department to complete their operation sequences traversing the factory. Factory and
department size, part sensitivity, and sequence lengths all exacerbate move times.
1.1.2 Manufacturing Cells. A manufacturing cell is a collection of dissimilar
machines positioned in proximity to work on products of similar shapes and
processing like a production line (Chase, Jacobs and Aquilano, 2004, p.200). We
assume that the nature of manufacturing demands and processing required is similar
to what is found in a job shop. In cell-based production, otherwise know as a cellular
manufacturing system (CMS), parts with similar features use common sequences of
operations and similar tools or fixtures. A group of such related parts defines a part
family. A CMS is therefore closely allied to the concept of group technology: the
concept of grouping similar parts into part families to benefit design and
manufacturing (see Askin and Vakharia, 1990).
In their recent comprehensive monograph on cellular manufacturing, Hyer and
Wemmerlöv (2002, p.18) define a cell using the concept of families:
A cell is a group of closely located workstations where multiple,
sequential operations are performed on one or more families of similar
raw materials, parts, components, products or information carriers.
Typically, a number of different part families occur in the product mix. One of
the challenges in CMS is developing rules for cell formation to associate the part
7
family data with the required machines (see for example Singh and Rajamani, 1996
for a review of the cell formation literature).
The two most basic benefits of cellular manufacturing according to Hyer and
Wemmerlöv (2002, p. 48) are reductions in flow time (due to use of smaller batch
sizes and use of shared tools and fixtures) and inventory (due to the proximity of
equipment). Other benefits of cellular manufacturing according to Chase et al. (2004,
p. 200) are better human relations due to small work clusters, and improved operator
expertise due to learning through repetitions. Other advantages according to the
literature are improved quality and easier control of operations. Physically moving
both machines and associated product family to a cell enables the factory to focus on
that product family. The part family in the cell enjoys unfettered access to a limited
set of resources that are now in proximity to each other aiding quality control.
Moreover, cell-based production makes it easier to incorporate other practices that
improve efficiency such as job sequencing and the use of transfer batches.
The word “cell” is used quite liberally in practice to describe any association or
grouping of machines. In this research, we define a cell as a grouping of machines
used to process a family of one or more parts. We assume that the part families are
pre-specified. In our factory representation, there are NC cells, indexed by
NC n , , 1 … = . Each cell may include more than one of any machine type. Each cell
has a certain number of machine types, with multiple machines of the same type
organized into machine centers. We reserve this term for the cell shop and call the
analogous machine cluster in a job shop a department. Of course, since cells do not
8
contain duplicate machines very frequently, most machine centers just have a single
machine of a given type.
The flow discipline for batches through the machine centers of each cell is
identical to the rules governing the job shop as the batch visits several departments.
Once the batch completes its processing within a given job shop department or cell
machine center, it moves as an entire batch to its next operation or exits the factory if
no further processing is required.
The preceding statement requires modification if a cell uses transfer batches. In
this case, each batch is split up into the transfer batches that then queue up before the
appropriate machine center. Note that because transfer batches constitute the only
aggregation of units recognized within the cell, the identity of the original batch is not
recovered until all of its constituent transfer batches have completed their processing
within the cell. In fact, prior to leaving the cell and prior to being shipped, the work
must be re-batched into its original batch size as required.
1.2 Factory Conversion
The conversion from process layout (job shop) to cellular configuration is a key
question of both theoretical and practical importance in the field. As Cohen and Apte
(1997) describe,
In implementing cellular manufacturing an important task is to create a
plan for smooth transition from process layout to manufacturing cells
layout. Rearranging machines into cells based on part families is also
a major undertaking requiring both considerable time and expense.
9
Once a machine is moved to a cell, it is removed from the general resource pool of
the job shop and confined to processing within the cell. To avoid inter-cell moves as
much as possible, cells are discouraged from accepting work required for parts that
are not assigned to the cell, even if idle machine capacity exists. In this research, we
assume that the cells are independent, so that each part family can be processed
entirely within one cell. Inter-cell moves add to the complexity of flow and work
control and can re-introduce setups. To avoid these drawbacks, we simply disallow
them and assume that the cells formed are independent.
If the entire factory is partitioned as far as possible into cells we call this a cellular
manufacturing system (CMS). This may include a remainder cell or residual job shop
containing exceptional elements.
Example Factory
To illustrate the concept of cells, we present data from Morris and Tersine (1990)
in Table 1-1. This table shows a part routing matrix for a factory with 30 machines
falling into eight machine types. The factory produces 40 distinct parts that fall into
five part families. For each part, the numbers listed along the row specify the order of
the operations required, and the columns specify what machine type is needed for
each such operation. For example, part 10 requires 3 operations (or processing steps),
with the first performed by machine type 8, followed by type 1 for the second
operation, and finally type 2 as the last operation. The path of the part through the
departments is shown in Figure 1-2.
10
Machine Type
P/N 1 2 3 4 5 6 7 8
1 1 2
2 1 2 3
3 1 2
4 1 2 3
5 1 2
6 1 2 3
7 1 2 3 4
8 1 2 3
9 2 3 1
10 2 3 1
11 2 1
12 2 1
13 2 3 1
14 2 1
15 3 4 1 2
16 2 1
17 4 1 2 3
18 2 1
19 1 2 3 4
20 1 2 3 4 5 6
Machine Type
P/N 1 2 3 4 5 6 7 8
21 1 2 3 4 5
22 1 2 3 4 5
23 1 2 3
24 1 2 3 4 5 6
25 1 2 3 4
26 1 2 3 4 5
27 2 3 4 1
28 1 2 3 4 5
29 1 2 3
30 2 3 4 5 1
31 2 3 1
32 2 3 4 5 1
33 1 2 3
34 1 2 3 4 5 6
35 1 2 3
36 1 2 3 4 5
37 1 2 3 4
38 1 2 3 4
39 1 2 3 4
40 1 2 3 4 5
Table 1-1. Part routing matrix: operation sequence linking part number with
machine type.
Job Shop
Dept. 1 Dept. 2 Dept. 3
Dept. 4 Dept. 5
Dept. 6 Dept. 7 Dept. 8
Part Type
8
9
10
8
9
10
Figure 1-2. Illustrative part routings for parts 8, 9, and 10.
Morris and Tersine (1990) grouped the 40 parts listed above into the five families
shown in Table 1-2. They formed the cells so that each family is assigned to a unique
11
cell that is equipped with all the machine types required for the complete processing
of the part family assigned to it.
Family
Part
Types Cell
Machine Types
Required
1 33-40 1 1-7
2 19-26 2 1-8
3 27-32 3 1-5, 8
4 9-18 4 1-2, 6-8
5 1-8 5 3-6
Table 1-2. Summarized family and cell requirements.
The resulting cells are shown in Table 1-3. Five families and cells are identified
in Table 1-4 where the block-diagonal form indicates the complete independence of
cells. The numbers of machines of each type available in the original job shop were
sufficient to equip all cells appropriately. If six cells had been formed then the
addition of new machines would have been necessary (assuming the first five cells
required the machine types shown in Table 1-3). In general, cell formation may
augment or maintain the number of machines in the original job shop.
1 2 3 4 5
1 4 1 1 1 1
2 4 1 1 1 1
3 4 1 1 1 1
4 4 1 1 1 1
5 4 1 1 1 1
6 4 1 1 1 1
7 3 1 1 1
8 3 1 1 1
Machine
Type
Number of
Machines per
Type in the
Job Shop
Cells
Table 1-3. Machine distribution.
12
Machine Type
1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 1 2 3 4 5 8 1 2 6 7 8 3 4 5 6
Family P/N
33 1 2 3
34 1 2 3 4 5 6
35 1 2 3
36 1 2 3 4 5
37 1 2 3 4
38 1 2 3 4
39 1 2 3 4
40 1 2 3 4 5
19 1 2 3 4
20 1 2 3 4 5 6
21 1 2 3 4 5
22 1 2 3 4 5
23 1 2 3
24 1 2 3 4 5 6
25 1 2 3 4
26 1 2 3 4 5
27 2 3 4 1
28 1 2 3 4 5
29 1 2 3
30 2 3 4 5 1
31 2 3 1
32 2 3 4 5 1
9 2 3 1
10 2 3 1
11 2 1
12 2 1
13 2 3 1
14 2 1
15 3 4 1 2
16 2 1
17 4 1 2 3
18 2 1
1 1 2
2 1 2 3
3 1 2
4 1 2 3
5 1 2
6 1 2 3
7 1 2 3 4
8 1 2 3
Cell 5 Cell 1 Cell 2 Cell 3 Cell 4
5
1
2
3
4
Table 1-4. Partitioned part routing matrix indicating part operation sequences, part
families, cells and machine types per cell.
An alternative to a completely converted CMS is what we call a partial cellular
manufacturing system (PCMS). This is a hybrid layout where a number of cells are
formed to work alongside a remainder job shop. In other words, the formation stops
short of full conversion. The parts are therefore manufactured in the cells or in the
13
residual shop; however each cell is dedicated to the manufacture of a unique part
family. Naturally, machines not used in the cells implemented remain in their
residual job shop departments.
The information gathered from industry practice shows that partial
implementation is often the preferred path for implementation. Surveys show that
firms create cells one by one (Wemmerlöv and Hyer 1989, Wemmerlöv and Johnson
1997). In fact, a study by Ahmed, Nandkeolyar and Mahmood (1997) indicates that
practitioners do not exercise full conversions and that successful implementation is
linked to long-term, step-by-step installations.
1.3 Key Trade-offs
A consistent feature of all conversions to a CMS environment is the segregation of
machines of each type from the pooled arrangement of a department to smaller
subsets assigned to the cells. Wolff (1989, p.260) uses the term pooling to refer to the
aggregation of the arrival streams of c separate queues into a single queue where the
server is equipped with the pooled resources of the original queues. He notes that the
pooled queue performs better and goes on to state that “the superiority of pooling can
be shown to be a very general result independent of the nature of the arrival process
and the distribution of service.” Accordingly, we refer to the diseconomies of
segregating a given machine type by assigning them to independent cells as the
pooling loss. This pooling loss always causes an increase in flow time. Therefore, for
the cellular system to outperform the functional layout with respect to flow time, this
pooling loss must be compensated by improvements in such other factors as setup
14
times or move times. In summary, when flow time is the performance measure of
primary interest, the superiority of cellular layout over functional layout is tantamount
to finding the means for overcoming pooling loss.
A simple queueing model based on the well-known M/M/c formulas has been
used to illustrate the nature of the pooling loss as in studies by Suresh (1991, 1992),
Shafer and Charnes (1993, 1995,1997), and Suresh and Meredith (1994). A simple
example will illustrate this modeling approach.
In Figure 1-3, we compare the flow times of two systems -- a pool of four
machines corresponding to a job shop department (solid line), and a system of four
cells, each consisting of a single machine performing a single operation (dashed line).
We model the job shop as an M/M/4 system with µ = 1 for the JS and equate the
arrival rates to both systems. For point A, the flow time for the M/M/4 system equals
1.25 when µ =.65 ( µ ì µ 4 = ) corresponding to an arrival rate of 6 . 2 = ì . When
we segregate the shop into four equal demand streams of ì 4 , the flow time for each
cell equals 2.86 (point B), which is 2.28 times the M/M/4 flow time. In order for the
flow time in the M/M/1 system to be the same as the pooled system, so that
( ) ( ) 25 . 1 1
1 / /
= ÷ = µ ì µ
M M
W , the processing rate must be increased such that the
resultant utilization is 448 . = µ or roughly one and a half times as efficient,
JS CM
45 . 1 µ µ = , as the same process in the JS.
15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1
2
3
4
5
6
Figure 1-3. Pooling loss.
As Figure 1-3 suggests, flow time increases without bounds with the linear
increase in machine utilization. If the M/M/4 is run at µ =.80 , the flow time is 1.75
(point C). After conversion, the M/M/1 flow time is 5.00 (point D) per cell or 2.86
times the M/M/4 flow. Comparing the pairs A-B and C-D, when µ increases from
.65 to .80, the ratio of the flow time increases from 2.28 to 2.86.
The last point is of particular importance since it shows how increased utilization
magnifies the pooling loss. This effect occurs where bottlenecks arise as a result of
conversion to cells, limiting the capacity of the process. In general, in Chapters 3 and
4 of this work, we will see how conversions from JS to CMS are especially sensitive
to the loading of machines in both the cells and the remainder shop. Suresh (1991,
1992) has also alerted readers to “adverse effects in the remainder cell” that are
typically due to loading imbalances.
M/M/1
M/M/4
D
C
A
B
F
l
o
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T
i
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e
Machine Utilization
16
We already mentioned that reductions in setup constitute one of the key factors for
overcoming pooling loss. Major setups are typically incurred when the same machine
switches from one family of parts to another. The frequency with which setup occurs
depend on the demand and service rates as well as the dispatching rule. A dispatching
rule is a priority rule or set of rules used in determining the order of service for
customers waiting in line. In this dissertation, we focus on a dispatching rule that is
designed to minimize the incidence of setups. We do not consider preemptive
dispatching rules because job interruptions will markedly increase the complexity of
workflow control.
1.4 Research Objectives
The research we present has two objectives. First, we investigate the role of setup
economies in the factory-wide conversion of functional layouts (job shops) to cellular
manufacturing. While the literature has chiefly focused on full job shop to cell shop
conversions, we include both complete and partial factory conversion options (where
a sizable residual shop is left in conjunction to the cells). Our second research
objective is to examine the role of dispatching rules in the reduction of setups.
1.4.1 Research Issues and Methodology for Factory Conversions to CMS. Our
research seeks to answer the following questions regarding the results of setup
economies in the cases of factory conversions:
• Can consistent results be obtained as to when the conversion of the job shop to
a cell shop is advantageous?
17
• What are the measured setup economies? When are setup economies large
enough to overcome pooling losses?
• How do other cell factors, including reduced batch sizes and use of transfer
batches, affect flow times achieved in cells?
• Can a partial implementation of CMS provide all or most of the benefits of
full conversion to CMS?
The approach taken to answer these questions is to use a single simulation model
to compare functional and cellular layouts across a test bed of factory environments
extracted from the literature. In our attempt to perform such a comparison, we follow
the established practice of most analytical or simulation conversion studies in using
flow time as the primary performance measure for comparing JS and CMS layouts.
Little’s law then can be used to relate the flow time to inventory measures such as
length and wait time in queue and number of customers in the system. We realize that
the average batch flow time may not directly relate to the total product cost. We
actually capture the flow time of each and then calculate the weighted average, using
the part type demands as the weights. This will be a reasonable surrogate for cost if
there is a linear relationship of cost to piece part flow time. For example, inventory-
related costs are often modeled to be linear in the amount of time each part spends in
the system. In this case our measure would be a surrogate for part costs if all part
types have the same monetary value. Alternatively, we can use a weighted average in
which we weight part types by their contributions to the total cost of goods sold
(GOGS). Our contribution is to control the parameter choices in the data sets in such
18
a way as to make them comparable. We call this approach standardization. Table 1-
5 lists our assumptions in the factory conversion part of this research.
Primary performance measure Average batch flow time
Process flow coordination Asynchronous
Machine selection One machine type specified per operation per part
(no alternates)
Machine input queues Infinite capacity, shared by machine type within job
shop department or cell machine center
Machine operation Sequential processing on the same machine type is
combined within one operation sequence
Machine output queues None: sufficient material handling capacity exists to
move output immediately to next operation
Use of transfer batches Only allowed in cells
Cell resources No inter-cells moves allowed or job shop to cell
moves allowed: all cells assumed to be independent
and capable of processing part family in entirety
Table 1-5. Assumptions for factory conversion research.
To our knowledge, this is the first study where conversion benefits are studied
across data sets selected from different sources in the literature. Our results show that
for a given region of the parameter space, the conversion to cellular layouts
consistently produces an advantage even in the absence of the gains resulting from lot
size redefinition and lower movement or transport times. In addition, we are able to
generate caveats for the implementation process from our PCMS results.
1.4.2 Research Issue and Methodology for Analytic Modeling of a Simple
System with Setup. Our research seeks to answer the following intuitive question
regarding setup economies using models of a simple system:
• What is the impact of the dispatching rule used in the reduction of setups?
The approach taken to answer this question is to apply analytic queueing models
to a system that is simple enough to make exact analysis tractable. The single-stage,
19
single-server system involving two customer classes is the simplest case where setups
occur due to part changeovers. Our choice of this simple system is driven by the
existence of exact results on flow times and the fact that modeling of setups best
matches the manufacturing environment studied in this dissertation. We start by
establishing a baseline using zero setup, evaluating flow times under FCFS versus a
dispatching rule that minimizes the incidence of changeovers. We then extend the
results to the case of non-zero setup. Table 1-6 lists our assumptions for the analytic
section of this research.
Primary performance measure Average batch flow time (batch size = 1)
Setup incidence Incurred when switching from one class of part to the
other (setup magnitude 0)
Table 1-6. Assumptions for analytic modeling research.
New flow time results are provided using different dispatching rules. These
results are obtained analytically for the case of zero setup times and extended to the
case of non-zero setup time through computational studies.
1.5 Plan of the Dissertation
In Chapter 2, we review the literature relevant to the two distinct parts of this
dissertation. We first review the literature on the conversion to cellular manufacturing
using simulation modeling (including both complete transformations and partial
transformations). Next, we review the key sources in the queueing literature that
consider single-machine processing in the presence of setups. In Chapter 3, we
present our study of the full conversion of job shops to cells shops. The first part of
Chapter 3 outlines the factory production environment. Here we describe the choice
20
of data sets included in the test bed, identify the manufacturing characteristics of each
data set, introduce the standardization scheme for the simulation study, and describe
the simulation model. Section 3.6 describes the results of the simulation runs
comparing functional and cellular layouts. Of special importance are sensitivity runs
included to study the effect of batch sizes, transfer batches, factory loads, setup
parameters, and dispatching rules. Chapter 4 provides a brief account of our
investigation of partial cellular implementation.
Chapter 5 is devoted to the analysis of a single-server system with two classes and
switching (setup) costs. Section 5.1 is dedicated to the zero-setup baseline and 5.2 to
the non-zero setup extension. Chapter 6 contains summaries of the key findings of
our research and outlines several directions along which future research can be
conducted. We have also included a short glossary of key terms used for the reader’s
convenience.
21
Chapter 2
LITERATURE REVIEW
This chapter reviews the literature relevant to the two segments of this
dissertation. First, in Section 2.1, we review simulation studies that have dealt with
the conversion of job shops to cellular layouts for both full and partial conversion (in
a partial conversion, a sizable residual shop processes parts along with the cells). In
this chapter, we reserve the term factory conversion for a change in the layout.
The second section, 2.2, reviews the modeling literature for the multi-class,
single-stage processing facilities modeled as queueing systems. Our focus is on
analytic models that can handle setup times.
2.1 Conversion Analysis Using Simulation
The comparison of functional and cellular layouts in the manufacturing of discrete
parts is a topic that has received much research attention over the last decade. This
comparison is often performed when a job shop (JS) is converted to a cellular
manufacturing system (CMS) experiencing the same demand. On the one hand,
reports from industry continue to claim superior performance for cellular layouts,
although the measured improvement seems to vary substantially. For example,
Wemmerlöv and Hyer (1989) reported average flow time reductions of 24% for
cellular layouts, whereas Wemmerlöv and Johnson (1997) reported an average
reduction of 61% in throughput times for 27 respondents. On the other hand,
22
simulation modeling studies in the research literature have produced divergent and at
times contradictory results in evaluating the effect of conversion on flow times. Nor
is the literature of one voice in providing a clear basis or a consistent list of
quantifiable factors that would ensure the benefits of conversion.
The empirical data also shows that partial conversion is also used in practice. A
study by Ahmed, Nandkeolyar and Mahmood (1997) indicates that practitioners do
not opt for full conversions and that successful implementation is linked to long-term,
step-by-step conversion to cellular manufacturing.
To facilitate our review of the literature, we introduce our performance measure
now. Since CM is used to improve the efficiency of a job shop, a job shop will be the
basis for our performance comparisons. For comparative purposes, the flow ratio
(FR) is defined as the ratio of the average batch flow time after cellular conversion to
the average batch flow time of the job shop with the same factory operational
parameters of load, machines and batch size. This definition is slightly different than
that used by Suresh (1992) where the flow ratio related the cellular transformed flow
time to the best job shop flow time which may be measured at a different batch size.
2.1.1 Complete Factory Conversions
In their paper on this subject, Johnson and Wemmerlöv (1996) performed a meta-
analysis of the results of 24 simulation studies designed to investigate the
performance characteristics of conversions from JS to CMS. These authors conclude,
“universal evidence regarding the superiority of cellular versus functional systems can
never be provided due to the data dependency involved.” However, they also remark
23
that whether cellular layouts outperform their functional counterparts depend on a
complex interaction among several key factors including the utilization level, the
degree of resource pooling, setup and move time reductions, and batch sizes used.
To aid in our review of the simulation-based literature on factory conversion, it is
useful to compile a list of factors that can be expected to influence the performance of
job shops as compared to cell shops. We then look at the comparisons provided in
Johnson and Wemmerlöv’s 1996 meta-analysis and examine the different factory
conditions tested. In this chapter, our focus is on the setup reduction as the key
advantage of cells, rather than material handling gains.
We define our terms used in this review in Table 2-1. We then compare the range
of factors and factor settings in five simulation studies in Table 2-2. We follow with
reviews of key studies in the literature (the five in Table 2-2 with others) that use
simulation to investigate factory conversion.
Following the review of the studies, Table 2-4 lists the studies in chronological
order and the overall conclusions drawn for each paper.
24
Operations/part Range in the number of operations per part across all parts
Machine Types Number of distinct machine types
Machines Total machines
Machines/type Range in the number of machines per distinct machine type
Cells Number of cells the JS is converted into (one cell may be a
“remainder” and process unrelated parts)
Batch Size Batch size used in the JS layout and CMS unless stated
otherwise. A list of batch sizes means denotes experimental
factor settings
Major Setup A major setup is incurred if two parts belonging to distinct
families are processed consecutively on the same machine.
Minor Setup Switching between two different part types in the same family.
Typically less than a major setup.
Setup Ratio: s/br Ratio of major setup to mean batch run time per part.
Setup Fraction Ratio of minor to major setup per part.
Dispatching Rule FCFS: First come, first served;
RL: Repetitive Lot (from Jacobs and Bragg, 1988):
(1) A single (pooled) queue is formed for all batches
arriving to be processed at a machine center.
(2) Any arriving batch encountering an available machine
upon entry is immediately routed to the available
machine where it would require the least setup time. If
no machines are available, the batch joins (or forms) a
queue to wait for a machine.
(3) When a machine becomes available, the next job
assigned to it is selected based on the minimum setup
among all jobs in queue. If multiple jobs tie at this
minimum setup value, the FCFS discipline is used to
break the tie.
JS Utilization Source JS average machine utilization as measured by
simulation
Cell Transfer
Batch Size
Transfer batches used only in cells. No transfer batches is
designated by b, the JS batch size, otherwise a value is listed
Arrival Rate
Distribution
Distribution of arrivals with its coefficient of variation in
parentheses.
Setup Time
Distribution
Like arrival distribution above, but for setup
Material
Handling Times
Material handling as a fraction of part run time
Table 2-1. Study definitions.
25
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Parts 40 40 75 50 4
Operations/Part 2-6 1 2-7 3-7 4
Machine Types 8 1 12 10 4
Machines 30 4 63 31 16
Machines/Type 3-4 1, 1, 2 3-4 3-4 4
Cells 5 2 or 3 1-5 5 1-4
Batch Size 50 5-100 2-80
JS: 32-100
CM: 5-100
Setup Ratio:
s /br
0.06 or .1, .5,
1.0
6-0.3 0.3-6.0 1.3-0.2
Setup Fraction:
Min/Maj Setup
0.5 0.1-0.5 0.1-1.0 0, 0.5, 1.0
Dispatching
Rule
RL
FCFS,
SPT
FCFS,
FSP, RL
FCFS or RL-F
in JS, FCFS in
cells
RL
JS Utilization 60-70%
70% with
b=50
62%, 75%
Cell Transfer
Batch Size
b b b b b , b /2, 1
Arrival Rate
Distribution
(CV)
Poisson
(1.0)
deterministic
(0)
Poisson
(1.0)
Poisson
(1.0)
3-Erlang
(.58)
Setup Time
Distribution
(CV)
Normal
(0.08)
Poisson
(1.0)
3-Erlang
(.58)
Part Run Time
Distribution
(CV)
Normal
(0.36)
Gamma
(0.7-1.2)
Poisson
(1.0)
3-Erlang
(.58)
Material
Handling Times
~.15r between
depts.,
0 within cells
3r - 120r
between
depts.,
.75r within
cells
JS only: 0
or 0.6br
Table 2-2. Comparison of factor levels within simulation studies
26
Morris and Tersine (1990) studied the full conversion of a five-cell CMS. They
examined the impact of changes of setup ratio, move time, demand stability and flow
of work within the cells on the conversion. The “demand stability” factor regulated
the sequence of part batch arrivals such that there was a maximum interval between
like part types. The work within cells was random and allowed backtracks or part
sequences were altered to provide unidirectional flow. Morris and Tersine (1990)
considered their shop configuration “supportive” of CM due to the independence of
their cells, use of identical lot sizes in both layouts, and use of RL dispatching.
Their results showed that the setup ratio factor could bring the flow time within
5% of the job shop value. In contrast, their base case resulted in an all-cell shop with
flow time 50% greater than the job shop value. When high setup level was
compounded with other factors such as slow JS move times, and unidirectional flow),
the all-cell flow time was 10% better than that of the job shop. Overall, the authors
concluded limited promise for CM. Looking closely at their experimental setup it is
evident that simply increasing the setup time magnitude for each operation created the
high setup level. Using the same run times, this increase in setup burden added to the
machine utilization of both the job shop and all-cell shop and raised all flow times as
reflected in their mean throughput times (see their Table 4). Operating the cell shop
in this high machine utilization region, as noted in the conclusions of Morris and
Tersine (1989), can distort the apparent impact of setup due to the sensitivity of the
flow time to machine utilization.
27
Suresh (1991) used a single-operation simulation model with parts from three
families. One of the three families represented 50% of the total parts in the factory
and roughly 50% of the total demand. Although deemed a “family” by the author,
there was essentially no similarity between parts. Setup discounting was handled
differently than in Morris and Tersine (1990) – setup was not discounted in the job
shop or in the family of unrelated parts and was discounted by a flat rate of 70% or
90% in the cells (independent of processing sequence). The dispatching rules
included a truncated shortest-setup-plus-run-time (SPTT). The SPTT rule calculated
a due date and gave priority to late jobs followed by shortest discounted setup plus run
time. As each family was moved to a cell, a new batch size for that family was
determined from a pre-selected range (approximately 10% of that originally in the
JS).
Even with a 90% setup discount in the cells and at a setup ratio of 0.6, the all-cell
flow time was 25% greater than the job shop value using FCFS in both job shop and
cells and 9% greater using SPTT in the job shop and cell shop. The study also
showed that SPTT performed better (14%) than FCFS in the job shop using the same
batch size. We observe, therefore, that if SPTT was combined with cell conversion
then it would have resulted in a 6% improvement over the job shop using FCFS. The
authors noted that the flow time of parts in the cells improved even though the overall
factory flow time was inferior to the job shop. The authors attributed this to adverse
effects in the remainder. We understand these “adverse effects” to be pooling losses:
machines were removed from the job shop pool, but the relative load per machine did
28
not change. In Suresh (1991), only when the setup discount was coupled with a
reduction in cell batch size (made feasible for the cells from setup reductions) was the
transformed shop capable of improved factory flow times over the JS.
The results of Suresh (1991) appear to corroborate the conclusion of Morris and
Tersine (1990) that large amounts of setup reduction alone are not sufficient for the
cells shop to overcome the pooling losses and outperform the flow time of the job
shop. Although Suresh included similar factors and levels as in Morris and Tersine,
we note in Table 2-3 that they were handled differently.
Factor Morris and Tersine (1990) Suresh (1991)
Setup
discount
family-based throughout
the shop
flat-rate setup discount
applied to two of three cells
only
Dispatching
Rule(s)
Repetitive Lot FCFS or SPT
Remainder
cell
none
50% of parts in remainder
cell and did not receive
setup discounts
Table 2-3. Difference in operating scenarios may confuse comparative results.
Shambu and Suresh (2000) confirmed Jacobs and Bragg (1988) in showing that
RL is superior to FCFS and SPT dispatching rules. Shambu and Suresh (2000) report
similar results as those in Shambu’s 1993 dissertation. They found that in the cells
RL/SPT (part batch with shortest expected processing time picked from queue) is
only marginally better than SPT (without using RL), but both outperform FCFS. The
authors note that the likelihood of identical parts being processed in succession in a
cell is small so RL rarely impacted the queue. In addition, if the setup fraction is
small then the savings potential due to eliminating the minor setup is minimal.
29
As in Suresh (1991), the flow time of parts in the remainder shop of Shambu and
Suresh (2000) was found to deteriorate with increasing number of cells, even when
the flow time of the cell parts improved over their flow times when in the job shop.
The authors used family-based setup in the residual shop like Morris and Tersine
(1990) and still found increasing flow time. They attribute this decline in
performance of the residual to pooling losses that were not overcome by any residual
shop setup improvements.
The choice of the batch size as a factor in conversion to cells is central to Suresh
and Meredith (1994), who set batch sizes (one size used for all parts) across a range
for the job shop and then reset them the cell shop configurations. Setups were family-
based with the setup fraction ranging from 10% to 100% (no discount).
Their results with both the job shop and all-cell shop using family-based setup
showed up to a 54% batch flow time reduction from a job shop to a cell shop (both
with 10% setup fraction). This was assuming cells used use the same batch size as the
job shop. They report improvements of 58% with batch sizes half that of the job
shop. This was their most extreme result using equal batch sizes, but it was based on
using a job shop with average machine utilization over 95%. At another setting, the
job shop was loaded at approximately 75% machine utilization. The resulting
reduction in batch flow time for the same setup fraction in the cell shop was 16% at
the same batch size used in the job shop and 67% at a batch size 1/6
th
that of the job
shop. As expected, the job shop flow times were best with the lowest move time
setting.
30
Suresh and Meredith (1994) concluded that of the factors they studied influencing
the shop performance, setup and run time reduction had the greatest impact as
opposed to batch size and variability reduction. We note that batch size of the cells
did not have to be reduced from that of the job shop for the factory to realize savings
in flow time (as long as setup fraction was less than 0.5).
Shafer and Charnes (1997) results show that the overall flow time increased with
increasing setup ratio, but decreased with decreasing setup fraction. The flow time
also decreased with transfer batch size. The job shop flow time increased with move
delay. The authors concluded that each of the factors they tested, if set at the
appropriate level, may be sufficient to overcome pooling loss resulting in improved
flow time performance over the job shop. The authors concluded that an all-cell shop
(using transfer batches of size one) can generally reduce job flow time by 45%-65%
over a comparable job shop and showed that without transfer batches less than the
original batch size the flow time could be reduced 11% (assuming 50% setup
fraction).
Table 2-4 summarizes each of studies above in chronological order. The column
labeled “factor” specifies the key factory investigated in the paper. For example, the
first paper listed investigated the effect of move times and the demand distribution on
the conversion. The last column, entitled “limitations,” summarizes our observations
on the study from the perspective of the research questions addressed in this
dissertation.
31
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Table 2-4. JS to CMS Conversion Literature Summary
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Table 2.4 (cont.). JS to CMS Conversion Literature Summary
33
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Table 2.4 (cont.). JS to CMS Conversion Literature Summary
34
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4
Table 2.4 (cont.). JS to CMS Conversion Literature Summary
35
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Table 2.4 (cont.). JS to CMS Conversion Literature Summary
36
Other Relevant Full-Conversion Studies
The following studies provide insights on other factors of a more subtle nature to
the factory conversion literature.
Shafer and Meredith (1993) were mainly interested in transfer batches in a study
of data from industries. Transfer batches were used exclusively in the cells. They
reported improvement in performance largely due to transfer batches. Recognizing
this, looking across their plant-specific results they determined a number of factors
that limit the advantages of cellular manufacturing by limiting the effect of transfer
batches:
1. Short process routes
2. Small batch sizes
3. Short processing times per part
4. Absence of natural part families (reduces ability to form cells, and therefore
the use of transfer batches or cell-based setup reductions)
5. Existence of bottleneck machines (in general cause large queues, also reduces
benefit of transfer batches)
Finally, Seifoddini and Djassemi (1997) compared the effect of part mix changes
to a shop configured as a job shop or all-cell shop. For a fraction of parts, part
operations were changed and then the resulting changed parts were re-assigned to
different cells. For example, one part type was eliminated from the first cell family at
the same time one part type was added to the third cell family. Each part added to a
part family contained machine requirements consistent with its cell machine types (no
inter-cell moves required). Following this example, the first cell experienced a
37
reduction in demand and the third cell experienced an increase in demand. As we
would expect, the smaller cell machine pools were more sensitive to part changes than
the job shop experiencing the same part changes. We conclude from Seifoddini and
Djassemi that CMS sensitivity to changes is not reflected in the factory flow time
measure.
Full Conversion Summary
The literature provides sufficient evidence that given enough potential, move
time, setup or transfer batches are capable of overcoming pooling losses
independently of the other in cellular conversion. We also see the indication that the
use of transfer batches and machine loading may be key factors in cellular conversion.
Review of Meta-Analysis
We now look at the results and then the factor settings compiled in Johnson and
Wemmerlöv (1996) more closely to capture their variety. Figure 2-1 plots the range
of observed flow ratios for 24 studies in the literature summarized by Johnson and
Wemmerlöv sorted by the lowest reported flow ratio. We simply converted the
measure called RAT (reduction in average flow time) reported by Johnson and
Wemmerlöv into flow ratios and used the lowest and highest flow ratios observed by
the authors of each study in the course of their experiments. Consider the vertical line
indicating a flow ratio of 1.0. Any study for which the bar intersects this line includes
results where the CMS and job shop have the same flow times. Similarly, if we draw
two additional lines to mark the boundaries of a 20% band about the 1.0 line, we can
highlight the regions where a given study shows a clear advantage for either the job
38
shop or CMS. We see the results are mixed. Only one study, namely Shafer and
Charnes (1993), reports flow ratios that lie consistently below 1.0, a majority shows
their range of results entering this region, but with the range extending into region that
show a clear advantage for the job shop. While we wouldn’t expect the ranges to be
the same, we find that some studies have no common cell conversion performance.
0 1 2 3 4
Flynn and Jacobs (1986)
Flynn (1987)
Crookall and Lee (1977), Lee (1985)
Morris and Tersine (1994)
Flynn and Jacobs (1987) small shop
Suresh (1991)
Morris and Tersine (1990)
Jensen et al. (1996)
Burgess et al. (1993)
Ang and Willey (1984)
Morris and Tersine (1989)
Shafer and Meredith (1990, 1993, 1990 company C)
Yang and Jacobs (1992)
Leu et al. (1995)
Suresh (1992)
Garza and Smunt (1991)
Shafer and Charnes (1995)
Shafer and Charnes (1993)
Suresh and Meredith (1994)
Suresh (1993)
Moily et al. (1987)
Flow Ratio Ranges
÷ Clear preference for JS Clear preference for CM ÷
Figure 2-1. Disparity of results reported in Johnson and Wemmerlöv (1996).
39
There may be a number of reasons underlying the mixed results on the
comparative performance of functional and cellular manufacturing layouts. The
simulation modeling literature uses flow time to determine the success of the CMS
implementation. Within industry, however, the implementation of cellular
manufacturing may be driven by benefits that are not easily measured by traditional
metrics in computational studies. For example, several key products may be
segregated into cells to provide better control of operations or quality.
Interestingly, comparative results reported in the literature vary widely even when
flow time is taken as the primary performance measure as measured by a simulation
model. Closer examination shows that the studies reflect different values of key input
parameters and use disparate operational rules as seen in Table 2-3 using the
definition of terms in Table 2-2. Given the wide range of manufacturing settings
investigated, it is not surprising that the results of conversion studies are not
consistent.
2.1.2 Partial Implementation of Cells. We now review the literature on partial
conversion where only part of the original JS factory is organized into cells. As
mentioned before, this means that a significant part of the factory continues to operate
as a JS, we call this the remainder shop. The overall hybrid system is also denoted by
PCMS (for Partial CMS). We review the studies that specially focused on partial
conversion and follow with a summary in Table 2-6.
Shunk (1976) was one of the first authors to use simulation for comparing CMS
to JS. He identified experimental results where the flow time for PCMS was superior
40
to both the JS and all-cell settings. However, the study did not offer any insights as to
what lead to this phenomenon. When comparing flow time across the JS to eight- or
nine-cell shop, the minimum flow time generally occurred with three to five cells,
although it ranged from the two-cell to the nine-cell. In some cases the PCMS was
better than the job shop configuration with respect to flow time, while the all-cell
configuration was worse. Curiously, the job shop never exhibited the best flow time.
Burgess, Morgan and Vollmann (1993) compared the configuration of a single
cell with a remainder shop to a job shop, without evaluating the all-cell alternative.
These results are similar to those found in Burgess (1989). The research of Burgess et
al. (1993) focused on the inclusion of labor constraints and we will not be considering
labor constraints in our research. However, the converted shop in their research was
not labor constrained so their insights on cell loading effects are relative to a machine-
constrained shop.
Burgess et al. (1993) varied the fraction of parts sent to the cell. Since the work in
the cell was discounted, the resources (machines and labor pool) appeared to become
more efficient as compared to the job shop. In fact, the machine and labor pool
capacity did not change in the cell, rather the setup requirement for each part entering
the cell was reduced. The resultant factory-wide flow time was reduced even though
the un-discounted part loading sent to the cell increased from 80% to 120% of the
machine capacity. Of course the 120% loading is misleading because it assumes that
the cell parts are paying a full setup which they are not.
41
As shown in Figure 2-2, it took only a 25% setup reduction in the cell to
overcome the pooling losses as long as at least 40% of the parts were routed to the
cell. We would expect the flow curves of Figure 2-2 to rise again when too many
parts were sent to the cell suggesting an optimal loading exists.
0.70
1.00
1.30
1.00 0.75 0.50 0.25 0.10
Cell setup fraction
F
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Cell 45% of parts
Figure 2-2. Optimal flow time improvements require controlled cell loading.
Burgess et al. (1993) concluded that prorating loads to cells in a manner simply
commensurate with the resource fraction found in the job shop results in flow times
that are inferior to the job shop configuration. In other words, prorating
underestimates the load that should go to the cell. They suggest machine loading in
general may be more critical to cellular success than advantages gained through
shorter setup times. For our research, if we were to pick a single cell, favorable
machine loading is something we would look for.
Suresh (1991) included an analysis of a hybrid shop transformation along with the
complete conversion reviewed above. Suresh transformed a job shop into a hybrid
configuration using either a single cell or two cells (operating alongside a residual job
shop). Parts in the job shop and residual were not discounted; parts sent to cells were
42
discounted at a fixed rate of 0.3 or 0.1. We see by the flow ratio results of Suresh
listed in Table 2-5 that using a similar setup discounting scheme as Burgess et al.
(1993), but sending loads commensurate with the machine fraction in the cell, that
pooling loss is not overcome.
30 40 50 100
(FCFS in factory)
Cell parts setup
fraction
Job Shop with four machines
for 100% of parts
n/a 36.9 12.4 12.1 15.1
0.3 1.21 1.19 1.11 1.10
0.1 1.20 1.16 1.08 1.08
0.3 1.69 1.50 1.22 1.16
0.1 1.68 1.46 1.17 1.12
setup/batch run time 1.0 0.8 0.6 0.3
Batch size
One cell with two machines
for 50% of parts + residual
JS Flow
Flow Ratios
One cell with one machine for
20% of parts + residual
Table 2-5. Deep setup discounts may not be sufficient to guarantee PCMS success.
Suresh (1991) concluded that partial (hybrid) situations are clearly unfavorable
when compared to the JS even with high degree of setup reduction. He noted that the
flow time of the cell parts improved over the job shop, yet the overall factory flow
time did not. This indicates that the residual job shop is adversely affected. As we
discussed earlier in this literature review, we expect the effects in the residual from its
own pooling loss.
Shambu and Suresh (2000) compared a job shop to a PCMS with a remainder
shop. Shambu (1993) presents similar results. They showed flow time results
throughout the transition from JS to single cell all the way to five cells (with a
remainder). Unlike the PCMS studies of Burgess et al. (1993) or Suresh (1991), the
43
study by Shambu and Suresh used family-based setups throughout the factory. Setup
discounts were not, therefore, strictly found in cells. This translated into a more even-
handed comparison of factory environments. They showed that job shops using
family-based setups could use smaller batch sizes than those that did not allow
discounts in the job shop, confirming Suresh and Meredith’s results (1994) for total
conversion.
In their environment, it was shown that the a single cell shop (with residual) could
be better than the job shop using the same batch size which was counter to the results
of Suresh (1991). Looking carefully at the flow times, however, the residual flow was
4% worse than the job shop but the single cell flow was low enough to compensate
(45% improvement) weighted by its demand. At five cells, the cells logged an
improvement of 38% over the JS flow time and were paired with a residual that was
84% worse than the JS flow time. The net result was still a 12% improvement for
flow times over the JS. This supports the previous research of Burgess et al. (2000),
and Suresh (1991) suggesting that managing both cell loading and residual loading are
important to optimize factory flow time of the PCMS. Finally, the authors
sequentially picked cells for implementation based on an arbitrary cell numbering
scheme even though they noted that each cell was not equally loaded and therefore
not equal performers with respect to flow time. They concluded from their results
that there were decreasing marginal cell gains as the number of cells formed
increases. We do see differences in the marginal gains in their results, but (and by
their own admission), it is due to loading differences and thus coincidence in cell
44
implementation sequence. This helps motivate our research into the impact of picking
cells to optimize factory flow time.
More recently, Kher and Jensen (2002) presented a study of PCMS based on a
single data set they modified from Vakharia and Wemmerlöv (1990). The authors
measured flow time while serially moving machines (in order of machine number)
from the original job shop to complete pre-defined cells. The significance of the order
of their implementation was not tested. Each machine level of implementation was
run assuming that the cell the machine created or joined enjoyed a level of both setup
and run time reduction. This reduction level was controlled from 5% to 17.5% in
equal 2.5% intervals. These “processing time reductions” were apparently applied as
flat rates to all work within the cells and never to work completed within either the
original job shop or any machines within the residual job shop. The processing
improvements from Morris and Tersine (1989 and 1990) they cite do not include
setup reductions due to family-based processing. The authors used a dispatching rule
that minimized setup incidence (RL), yet did not disclose whether they followed a
family-based setup structure. Another important detail left unspecified was the
amount of setup relative to the run time of work within the factory. In Chapter 3, we
relate these two by introducing the notion of a “setup potential” and show it to be a
key factor in the total factory transformations. Kher and Jensen’s (2002) results
support those in Suresh (1991) that the cell flow improved, but non-cell residual
worsened and the conclusion in Shambu and Suresh (2000) that the remainder shop
flow time deteriorates as cells are added. By sending a machine at a time they also
45
recognized the conclusion of Burgess et al. (1993) that the fraction of the factory load
sent to the cells can be more than the load when in the JS to improve the performance
of the residual job shop.
Table 2-6 mimics Table 2-4 in its structure and summarizes the key studies that
considered partial cellular conversions.
46
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Table 2.6 (cont.). PCMS Studies
48
In summary, the PCMS literature suggests that cell and residual loading are both
important to obtaining good overall factory flow times. An optimum load ratio
between the cell(s) and residual has not been established with the objective of
optimizing the factory flow time. Similarly, even though it has been acknowledged
that cells are not always loaded consistently, the selection of cells to obtain the best
flow time performance has not been pursued systematically in any of these studies.
We address this issue in Chapter 4.
2.2 Two-Class, Single-Stage M/G/1
We present a summary in Table 2-7 of the contributions to flow time statistics of
two-class models followed by details of the models. We list the arrival, setup and
service distributions using “0” for zero, “M” for Markov, and “G” for general. The
models use either first-come-first-served (FCFS) or an alternative dispatching rule,
called alternating priority (AP) defined by Maxwell (1961) and others. The FCFS
rule suffers from the drawback that setups are incurred based entirely on the random
pattern of arrivals. In other words, no attempt is made to avoid setups. Maxwell
(1961) and others have defined an alternative dispatching rule, called alternating
priority (AP). Under this rule, all jobs in queue of a given class are served before
switching to the other class. The server thus alternates between strings of jobs of
either class 1 or class 2 and the idle state, but never switches from class i to class j
( ) i j = if there are jobs of class i still in queue. Clearly, the AP rule is designed to
minimize the incidence of setups. Finally, we list the system performance results
from each model.
49
Input Distributions
Source
Arrival Setup Service
Dispatch
rule
Available Results
Maxwell,
1961
M 0 M AP
Solution for mean
number in system
Gaver,
1963
M G G FCFS
Moment generating
function for flow time
Avi-Itzhak,
Maxwell
and Miller.,
1965
M 0 G AP
Exact solution for
flow time
Miller,
1964
M G
(1)
G AP
Moment generating
function for flow time
Eisenberg,
1967
M G G AP
Moment generating
function for flow time
(1)
Setup forced at the conclusion of each machine idle period
Table 2-7. Single-server modeling contributions.
2.2.1 Single Queue. To analyze the impact of setups, we begin with one of the
simplest exact models: a single-server queueing system with two classes of customers.
Gaver (1963) provides results for this system under the FCFS rule. For the symmetric
cases with equal arrival rates, setup times and service times Gaver provides a closed-
form solution. We assume arrivals follow a Poisson process with rate
i
ì for class i
arrivals ( 2 , 1 = i ),
2 1
ì ì ì + = and with symmetry
2 1
ì ì = . The expected setup paid
on switchover to the other class is ( ) U E . The expected service time paid per part is
( ) S E .
To obtain ( ) U E consider a pair of successive arrivals and note that the occurrence
of setup depends solely on whether these are of the same class or not. Let ( ) j i,
describe the event that the first arrival is of type i and the second arrival of type j
50
( ) 2 , 1 , = j i . Then ( ) ( ) [ ] ( )
¿¿
= =
=
2
1
2
1
, ,
i j
j i P j i U E U E where ( )
ì
ì
ì
ì
j
i
j i P = , . Clearly, no
setup is required if j i = so ( ) [ ] ( ) [ ] 0 2 , 2 1 , 1 = = U E U E . We then obtain
( ) ( ) ( ) [ ]
2 1
2
2 1
U E U E U E + =
ì
ì ì
. The utilization to include expected setup is therefore
( ) ( ) { }
(
¸
(

¸

+ + =
2 1
2
2 1
U E U E U
ì
ì ì
ì µ where ( ) ( )
2 2 1 1
S E S E ì ì µ + = and for system
stability 1 0 < s U . Gaver’s equation for the expected flow time assuming symmetry
is
( )
( ) ( ) { } [ ] ( )
( )
2 1 4
2 2
U E
S E S U E S E
U
F + + + +
÷
=
ì
. To solve for the general flow
time using the method of Gaver, we must use numerical methods to solve for a
parameter that is a function of the
i
ì ’s, ( )
i
S E ’s, and ( )
i
U E ’s.
2.2.2 Two Queues, One Server: Two Classes with Alternating Priority. This
system can be modeled as a semi-Markov process (SMP) (see Wolff, 1989 p.220) and
analyzed using fundamental results from renewal theory. It is customary to assume
that the SMP is regular which it obtains if the state of the system at any time t is
determined by a finite number of state transitions (jumps).
This type of problem is solved with renewal theory. If we define the states of a
system such that their selection is Markovian, but allow the sojourn time in each state
to be arbitrary then we have a semi-Markov process (SMP) with embedded, discrete-
time Markov chain (EMC) transition probabilities (Wolff, p.221). For an EMC, the
stationary probability of state j ,
j
p , represents the fraction of transitions that are
51
visits to state j . The fraction of time spent in state j ,
j
t , is proportional to the
transition fraction by
j j j
m p t · where
j
m is the sojourn time in state j . The
time-average limit is ( )
j j j ij
t
l m t P t = =
· ÷
lim where
j
l is the mean recurrence time. As
long as · <
¿
= j i
i i
m p (noting that
j
m 1 is the rate into or out of state j ) then state j
is positive recurrent enabling us to use: 0 > =
¿
j
j j
j j
j
m p
m p
t (Wolff, p.223) yielding the
fraction of time the SMP spends in state j .
We start reviewing the two-class, single server model assuming zero setup and an
alternating priority dispatch regime. Maxwell (1961) defines the states using a triple:
the number of items of type-1 in the system, the number of items of type-2 in the
system and an indication of the machine setup: 0 for idle, 1 for setup for type-1 and 2
for setup for type-2. This state definition loses the setup status of the machine upon
entering the idle state, but this information is not required since setups are assumed to
be zero. Maxwell then uses generating functions and relates the expected number of
items of each type in the system to these generating functions. His resulting equation
for mean number in the system is:
( )
( )
( )
( )
( )
( )
( ) ( )( ) [ ]
2 1 2 1
2
2
1
1
1
2
2 1
1 1 1
1 1 1 1
1 µ µ µ µ µ
µ µ µ µ
µ
µ
+ ÷ ÷ ÷
)
`
¹
¹
´
¦
÷
(
¸
(

¸

÷ + ÷
(
¸
(

¸

÷
+
÷
=
S E
S E
S E
S E
L
where ( )
i i i
S E ì µ = and
2 1
µ µ µ + = .
52
Avi-Itzhak, Maxwell and Miller (1965) computes wait times by conditioning on
the job arrival class and the state of the system. A random arrival experiences a wait
time based on the current class of work being processed. If the arrival is of the same
class, then it must wait for the existing job to be completed as well as all jobs of its
class ahead of it in line. If the arrival faces the server working on the other customer
class, then it must wait for processing of all jobs of the other class to be completed as
well as the jobs ahead of it of the same class. Flow times are calculated based on
summing the conditional probabilities that the random jobs arrive within a specific
block of time (a cycle). Fortunately, a closed-form solution is available for this
infinite sum (number of potential cycles to consider). The type-1 mean flow time is:
( ) ( )
( )
( )
( ) ( ) ( )
( )( ) ( )( ) ( )
2 1 2 1 1
2
2
2
1 2
2
1
2
2 1
1
2
1 1
1 1
1 1 1 1 2
1
1 2 µ µ µ µ µ µ
µ ì µ ì
µ
ì
+ ÷ ÷ ÷ ÷
÷ +
+
÷
+ =
S E S E S E
S E F E . We note the
similarity to the P-K formula: the first term is the service time, the second term is the
wait due to FCFS within a cycle and the third term adds the expected wait for the
other class of work to end its processing.
Miller (1964) modified the procedure of Avi-Itzhak et al. (1965) procedure to
handle non-zero setups. Miller’s model assumed setup at the beginning of every busy
period, the unbroken work interval between idle periods, regardless of the type work
ending the previous busy period. The mean flow time is computed by conditioning on
the type of cycle a random arrival sees upon joining the system. The values of mean
flow time are expressed analytically, but numerical methods are required to
approximate the infinite sums encountered. Miller also showed that due to the
53
reduced incidence of changeover in high traffic the system will not saturate if 1 < µ
where ( )
¿
=
i
i i
S E ì µ , independent of the setup magnitude.
Miller (1964) uses a description of the system state that does not keep track of the
last class served prior to an idle period for the machine. Since a setup is incurred at
the start of each busy period, it is unnecessary to record this information in the state.
Naturally if
2 1
µ µ >> , it may be that the job starting the next busy period matches the
class of the last period before going idle. In such a case, Miller would assume that a
setup occurs even though it is not required. In the case of equal Poisson arrival rates,
the probability of two arrivals of the same type in succession (last of previous busy
period and first of next busy period) is 50 . 0
2
1
2
2
=
|
.
|

\
|
× . In the case that the busy
period ends with equal probability of each type then we would expect that 50% of the
subsequent busy periods would not need to start with a setup.
Eisenberg (1967) addressed the case of “setups as needed” by using a richer state
description than Miller (1965). Eisenberg considers the embedded Markov process of
queue lengths at the instant of service completion, and includes the class of service
just completed. Thus, state
i
mn
denotes “server is at line i and m customers are
waiting at line 1 and n customers are waiting at line 2.” This state definition is event
driven: it provides a snapshot of the system whenever a departure occurs. The idle
states are exceptional in this regard: the probabilities of states
1
00
and
2
00
(the idle time)
are the same for the imbedded and general-time probabilities. Solutions to his model
54
also require numerical methods based on known values due to the existence of an
infinite sum.
Eisenberg also provides three limiting cases. First, in the special case of zero
setup times, he provides a wait equation that agrees with Avi-Itzhak et al. (1965).
Next, when service times are assumed to be zero so that only setup remains,
Eisenberg provides both the probability of idle with the machine ready to work on
type-1,
1
00
t , and the mean wait time. The last limiting case is for symmetry where the
following are the same for both classes: ( ) ( ) ( ) ( )
2 2
, , , ,
i i i i i
U E U E S E S E ì . The
symmetric result is consistent with that of Avi-Itzhak et al., and the overall wait time
is the same as for FCFS.
Sykes (1970), Eisenberg (1972) and Takagi (1990) investigate a different
dispatching regime. They all assume that when a queue has been exhausted the server
immediately switches over to the other customer type. Further, the server performs a
setup upon switchover and this is done whether or not any jobs are present at the other
queue. If there are no jobs waiting in that queue after the setup is complete then the
server moves back to the other queue setting up again (again, whether or not there are
jobs waiting). If a customer of class j arrives just as the server initiates a setup for
class i and there are no class i present in the queue to be worked (and none arrive
during the setup time) then he must wait yet another setup delay while the server is
switched back to work on class j .
55
Cooper, Niu and Srinivasan (1999) show that some classes of state-independent
setups (setting up whether or not work is waiting at that queue) yield equal or even
less wait times than their state-dependent (setup only when there is work in the queue)
setup classes. They consider a switchover time, the time required for the server to
travel from queue 1 ÷ k to queue k , in addition to setup time (the time required to
prepare for work at queue k ) and processing time at queue k . If we assume in a
manufacturing setting that the review time immediately after a service completion to
consider if there are jobs immediately available for processing is zero then the
analogous switchover time in Cooper et al. is zero. Left with only setup times and
processing times, they concede that state-independent setup regimes are at best equal
in expected wait time to their state-dependent counterparts and if any variability is
present in the setup distribution then the state-dependent regime encounters less
expected wait than its state-independent counterpart.
In summary, results for general setup and general service time typically require
numerical methods due to the existence of an infinite sum term. Certain
simplifications can be applied (as are done in cyclic models), but restrictions on setup
variability quickly reduce the potential in suitability of such models in the o
manufacturing environment. There still may exist rules between the extremes of
state-dependent and state-independent that allow polling models to be adapted to
manufacturing. For example, one can devise decision rules for setup incidence that
consider the probability of customer arrival type within a given time interval that
corresponds to idle time prior to committing to a setup.
56
Chapter 3
FACTORY CONVERSION TO CELLULAR MANUFACTURING
SYSTEMS
The objective of this part of the research is to use a single simulation model to
compare functional and cellular layouts across a test bed of factory environments
extracted from the literature. In our attempt to perform such a comparison, we follow
the established practice of most analytical or simulation conversion studies in using
flow time as the primary performance measure. We use the flow ratio (FR), which we
define as the ratio of the average batch flow time in the after cellular conversion to the
average batch flow time of the job shop with the same factory operational parameters
of load, machines and batch size. Therefore, measures below 1.0 indicate flow time
superiority for the transformed shop.
It is well known that flow time deteriorates when the size of the machine pool is
reduced, the pooling loss, as described in Chapter 1. Therefore, for the cellular layout
to outperform the functional layout, the pooling loss must be compensated for by
reductions in setup or move times. The key trade-off we consider is between pooling
loss and setup reduction. While a number of well-known studies in the literature have
studied this tradeoff, each has used its own data on demand, manufacturing
capabilities, parts structures, and operating rules. This makes it difficult to compare
the results across the disparate data sets. For this research, we have selected six
57
studies from the literature that provide sufficiently specific information for our
simulation model. We feel that these studies provide us with sufficient diversity in
terms of the parts, machines, and operations, used in the manufacturing simulation.
Having ensured that the same operating rules and measurement procedures apply to
all data sets, we proceeded to choose a common range of key parameter values. We
call this process standardization, although it may also be viewed as a focusing on a
region of the parameter space where the six different data sets we selected can be
compared. Of special importance in this standardization is the use of the same major-
minor setup structure and identical operational rules across all data sets. This
provides a level playing field for our simulation study.
3.1 Factory Environment and Notation
We now describe the main characteristics of the factory environment and
introduce the notation used in our simulation study. Each data set specifies a set of
available machines and a set of demands for parts. The demand is given as a set of
parts, with associated operations sequences, part families, and demand levels. The set
of parts is indexed by i I = 1, , … . Each part i has a unique operations sequence
consisting of ( ) i G operations.
58
For each part i , the following information is available as input:
( )
( )
( )
( )
( )
( )
( )
( ) belongs part which family to part
for size batch
operation its on for time setup major expected ,
operation its on of unit single a for run time expected ,
part of operation for the required type machine ,
, , 1 e index wher operation
part by required operations of number
units/year in part for demand
th
th
th
i i f
i i b
k i k i s
k i k i r
i k k i O
i G k k
i i G
i i V
=
=
=
=
=
= =
=
=
…
We assume that the demand for part i occurs in batches with mean ( ) i ì defined
as part demand divided by batch size, ( ) ( ) ( ) ì i V i b i = . Sequential processing on the
same machine type is combined within one operation sequence so that
( ) ( ) 1 , , + = k i O k i O for all k .
In this research, we do not investigate the effect of move times on the conversion
benefits in much detail. We argue that move times are negligible in cells due to the
proximity of machines. In the job shop, move times may suffer due to congestion
effects or limited transport resources. An investigation of this effect is beyond the
scope of this research. However, we should note that if move times simply reflect
known transport times, then their effect can be studied ex post as described later in
this chapter.
59
3.2 Job Shop Operation
The job shop is configured in a functional layout with J departments, where
department j houses the all the ( ) NM j copies of machine type j . All machines are
available 100% of the time at full capacity. Upon entry, each batch of part i
immediately reports to the department required by the part’s first operation ( ) 1 , i O .
The batch then travels from one department to the next following its operations
sequence, until all of its ( ) i G operations are completed. The batch flow time is
measured from part batch introduction into the factory (from receiving) to part batch
leaving the factory (sent to shipping).
3.3 Standardization Scheme
An important theme of the present study is to pursue a dual objective. On the one
hand, we wish to preserve the main characteristics of the various data sets as studied
in the literature, since these do differ in such key inputs as the number of parts,
number of part families, and the operations required by these parts. On the other
hand, we wish to use uniform operating rules, and a comparable setup structure, batch
size, and job shop load across all data sets. We believe that this is necessary to gain
any general insights. For example, papers in the literature differ in how they account
for setups in the job shop and the cells. We use the same setup structure and measure
setups in the same way in both layouts. In what we call operational standardization,
we ensure consistency in the flow control disciplines and adopt a common range of
parameter as listed in Table 3-1. These values may be compared to Table 3-2, which
60
lists the rules and parameters adopted by each of the sources used in our test bed. We
now discuss and try to justify the choices for each of our baseline parameters.
3.2.1 Batch Size. We use a common batch size in the job shop for all parts. From
the literature, we have noted that batch sizes are generally small for job shops. Batch
sizes used for the job shops studied by Suresh (1991, 1992), Shambu (1993), Suresh
and Meredith (1994), and Shambu and Suresh (2000) were 50 or less. We therefore
used a range of 25-50 for our batch sizes. In this research, we do not use transfer
batches within the job shop: Transfer batches make sense for cells where all machines
are placed in close proximity of one another. This makes manual or automated
machine-to-machine hand-offs reasonable. Job shop departments typically involve
much longer distances and require material handling equipment to transfer goods. In
the cell shop, we use a transfer batch size that is equal to b, 2 b , or 1, where b is the
original batch size used in the JS. Smaller values of batch sizes in the cells were used
in the sensitivity runs.
3.2.2 Setup Structure. We use a major-minor setup structure whereby the setup is
a major setup, a minor setup, or no setup at all. The same setup structure is used in
both the job shop and the cell shop. The incidence of setups is tied to the family
structure of parts types (recall that the I part types are partitioned into F families
numbered f F = 1, , … ). A major setup is incurred if two parts belonging to distinct
families are processed consecutively on the same machine. Switching between two
different part types in the same family incurs a minor setup. Naturally, no setup is
required if a machine processes two batches of the same part type consecutively.
61
3.2.3 Setup magnitudes. Past studies have shown that the relative magnitude of
setups is an important factor in conversion studies [see Morris and Tersine (1990),
Suresh (1991, 1992), Suresh and Meredith (1994), Shafer and Charnes (1997),
Shambu and Suresh (2000).] We therefore control the setup potential, which refers to
the amount of setup reduction that can be realized by cell conversion. Setup potential
involves the choice of two parameters-- the setup ratio and the setup fraction. The
setup ratio is the ratio of major setup, s , to batch run time, r b· . The setup fraction
is the ratio of minor to major setup. We standardize the setup ratio at 1.0. We
selected 1.0 by considering the ranges used in earlier papers: Morris and Tersine
(1989) use values ranging from 0.06 to1.0, while ranges of 0.4-2.3 and 0.3-6.0 are
used in Yang and Jacobs (1992) and Suresh and Meredith (1994), respectively. We
standardize the setup fraction at 0.20. This ratio is consistent with the simulation
studies of Jensen et al. (1996) and within the range of setup fractions of 0.1-0.9 used
in Garza and Smunt (1991) and Suresh and Meredith (1994).
3.2.4 Choice of Dispatching Rule. We use the repetitive lot (RL) dispatching rule
across all departments. This rule is used to minimize the incidence of the setup paid
and Jacobs and Bragg (1988) found this discipline superior to FCFS. Shambu and
Suresh (2000) have confirmed its superiority to both FCFS and SPT in the job shop
and cell environment with setups. It is also an appealing rule to use given our setup
structure. The RL dispatching rule operates as follows:
1. A single (pooled) queue is formed for all batches arriving to be processed at a
machine center.
62
2. Any arriving batch is immediately routed to the available machine where it
encounters the least setup time. If no machines are available, the batch joins
(or forms) a queue to wait for a machine.
3. When a machine becomes available, the next job assigned to it is selected
based on the minimum setup among all jobs in queue. If multiple jobs tie at
this minimum setup value, the FCFS discipline is used to break the tie.
3.2.5 Batch setup and run time
This choice specifies the magnitude of ( br s + ). While this value may depend on
the part, the operation, and the machine type used, we standardize the part processing
time by selecting distributions for the setup and run times. We use the ( ) | Erlang ÷ k
with 2 = k and | = mean of the setup or run time. We chose this distribution
because it has less variability than the exponential (CV = 0.707 versus 1.0). Being
non-symmetric (and skewed to the right), this distribution is more suitable for the time
to complete a task (Law and Kelton, 1991, p.186; Pegden et al., 1995, p. 40). We
provide results of other choices of distributions in Appendix A.
3.2.6 Factory Loading and Measurement. The overall level of utilization in the
job shop has a major impact on the magnitude of pooling losses observed. Based on
the studies used in our test bed, we use a target of 65% for the average machine
utilization in the job shop. For examples, Morris and Tersine (1990) loaded their job
shop at 60%-70%, Garza and Smunt (1991) used 60%, and Suresh and Meredith
(1994) chose 70% for their job shops. Values of other studies appear in Table 3-2.
We reach our target utilization by adjusting the overall factory demand (retaining
63
relative product mix ratios) until the ex-post utilization value reported by the
simulation lands within 2% of this target value. A summary of standardized
parameters is in Table 3-1.
Factor Proposed standard
Batch Size, b 25 to 50, fixed for all parts
Transfer Batch Size b
Part Batch Arrival Rate
Distribution
Poisson, CV=1.0
Setup Time Distribution 2-Erlang, CV=.7
Run Time Distribution 2-Erlang, CV=.7
Setup Ratio = s /br 1.0, fixed for all part operations
Setup Structure
identical = 0 setup
distinct within same family = minor setup
distinct families = major setup
Setup Fraction = Minor/Major
Setup
0.2
Dispatching Rule repetitive lot (RL)
Material Handling unconstrained capacity, 0 move time
Labor unconstrained
Job Shop Average Machine
Utilization
65% ±2%
Machines 100% available at all times
Table 3-1. Choices and parameters values for operational standardization.
3.2.7 Formation Standardization. We expect conversion results to be sensitive to
the particular choice of cells. The configuration of cells formed must therefore be
closely monitored. In formation standardization, we ensure that all data sets use the
same cell formation technique. While there is a vast literature on cell formation
techniques (e.g., Singh and Rajamani, 1996), our interest is to choose a single
algorithm that we can apply to all six data sets. We chose the cell formation
procedure due to Vakharia and Wemmerlöv (1990) because it considers both
sequences and capacities, factors that are left out in earlier cell formation techniques.
64
Vakharia and Wemmerlöv’s method first groups parts by the commonality in their
operations sequences and then proceeds to assign machines to such groups to provide
sufficient capacity to meet demand.
In what follows, the standardized cell configuration refers to the design produced
by the Vakharia and Wemmerlöv algorithm (V-W) when applied to each data set.
This procedure generally results in cells that differ from the CMS configuration in the
original data source. In fact, differences in the number of cells or number of machines
of each type can both arise. In any case, for each data set, we run the simulation
model twice, once for each cell configuration (source and V-W).
3.4 Choice of Data Sets
One of the objectives of this research on factory conversions is to use a single
simulation model to run all the data sets in the test bed we selected. Since sources of
these data sets (as published in the literature) refer to different factory environments
and/or modeling assumptions, the uniformity required for the inputs to our simulation
model is not easily obtained. Of the 24 data sets cited in the Johnson and Wemmerlöv
(1996) overview of modeling studies, we used six in our simulation studies because
they provided information specific enough for our model. We supplemented these
with two data sets from Morris (1988).
We require four eligibility conditions in selecting data sets for our study.
1. The original data source must provide a cell configuration; the number of cells
as well as the assignment of machine types and parts to each cell must be
specified,
65
2. The cell configuration provided must not require inter-cell moves,
3. The number of machines of each type must be specified for both job shop
departments and each cell, and
4. At least one machine type must have more than one copy in the original
functional layout.
Thus, condition (3) excludes a number of data sets in the literature that form cells
based on part-machine incidence, but do not unambiguously define the machine types
used. A number of data sets were eliminated by condition (4).
In constructing our test bed, we sought data sets that provided some details on
operations sequences, setup and run times, arrival and processing distributions, and
available machines as in Table 3-1. Our final test bed therefore uses six data sets
from eight sources in the literature (see details in Table 3-2) - all but two were used in
prior simulation studies by their authors. None of the authors provided an explicit
description of the cell formation technique they employed to configure their cell shop.
The source for data sets 2 and 3 does not provide simulation results for these data sets.
However, this source does supply the required part and machine structure along with a
cell solution; we generated the balance of the operational data.
Table 3-2 lists the operational settings for all data sets as provided in the original
papers. A glance at this table shows considerable differences among these settings,
arguing the case for standardization. Table 3-3 shows the data sets after
standardization.
66
Table 3-2. Data sets used in analysis as reported by source
(blanks denote omissions by source).
67
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Data Set ID 1 2 3 4 5 6a 6b 6c
Parts 60 24 45 50 18 40 40 40
Operations/part 4 2-4 2-6 3-7 1-4 2-6 2-6 2-6
Machine Types 8 6 14 10 4 8 8 8
Machines 24 20 35 31 10 30 30 30
Machines/type 3 3-4 2-3 3-4 2-3 3-4 3-4 3-4
Cells 6 4 4 5 3 5 5 5
Batch size, b 10, 15, 20, 25
JS: 32-100
CM: 5-100
1 50 50 50
Setup ratio:
s /br
.05-.35 0.3-6.0 0.4-2.3 0.06
0.06 or .1, .5,
1.0
0.06
Setup fraction:
min/maj setup
0.1-0.9 0.1-1.0 0.5 0.5 0.5 0.5
Dispatching rule FCFS
FCFS or RL-
F in JS, FCFS
in cells
RL RL RL FCFS
Source JS avg.
mach utilization as
measured by
simulation
60%
70% with
b=50
56-86% 44% 60-70%
Cell transfer batch
size
b b b b, 1 b 1
Arrival rate
distribution
(CV)
deterministic
(0)
Poisson
(1.0)
3-Erlang
(.58)
Poisson
(1.0)
Poisson
(1.0)
Poisson
(1.0)
deterministic
(0)
Poisson
(1.0)
Setup time
distribution
(CV)
deterministic
(0)
Poisson
(1.0)
3-Erlang
(.58)
Poisson
(1.0)
Normal
(0.08)
Normal
(0.08)
Normal
(0.08)
Part run time
distribution
(CV)
0, .33, .66, 1.0
Poisson
(1.0)
3-Erlang
(.58)
Poisson
(1.0)
Normal
(0.36)
Normal
(0.36)
Normal
(.01 per
batch)
Material handling
times
2r or 10r
between depts.,
0 within cells
3r - 120r
between
depts.,
.75r within
cells
.7r
between
depts.,
0 within
cells
5mph + 3 min
load/unload
between depts.,
0 within cells
~.15r between
depts.,
0 within cells
r between
depts.,
0 within
cells
Unique features of
data set
(1) (2) (2) (3) (4) (5)
b, batch size; r, run time per part; CV, coefficient of variation
(1) no minor setup in JS, assumes minor setups in cells due to tooling
(2) not simulated by author
(3) part to same part type required minor setup, included run time productivity improvement factor
(4) designed to test MRP vs. Period Batch Control order-release-and-due-date-assignment systems
(5) cell must be empty before setup changeover
68
In examining Table 3-2, we particularly focus on five factors that are important to
us in this study: batch size, setup ratio, setup fraction, dispatching rule and job shop
loading. There was a wide range of setup ratio. Some studies (1, 5 and 6b) evaluated
the same shop over a range of setup ratios. In the case of study 4, the setup and run
time per part were fixed so when the authors varied the batch size the setup ratio
changed, too. The setup fraction reflects the setup discounting for similar batches
processed in sequence. Studies 1 and 4 tested for this factor explicitly, while the
others used a midpoint value of 0.5.
When using simulation to evaluate their factory performance, each source selected
a certain load for the job shop, and then replicated the same demand for the cell shop.
The average job shop machine utilization varied from 44% to 86% from data sets 6a
and 5, respectively. The authors in study 4 chose the 50 = b case for their job shop
standard for comparison, which resulted in an average machine utilization of 70%.
Our experiments focus on machine-constrained environments; we do not consider
labor constraints. When labor and machine are both limited, then the conversion
study must study the interaction between these two factors as illustrated in Suresh
(1993) and Morris and Tersine (1994). In fact, labor constraints were absent from all
studies in Table 3-2, except for study 6c.
For convenience, we report the material handling time included by some of the
studies for travel between departments. When included, the time varied from 15% of
a single part run time, or 0.15r (data set 6b) to 120r (data set 4), with the average
69
being approximately r. Material handling was always assumed unconstrained so
travel time and not time due to material handling congestion was included.
Upon standardization, data sets 6a-6c collapse into a single data set in our test bed
identified simply as data set 6. Table 3-3 contains the final standardized values of the
parameters in our test bed.
1 2 3 4 5 6
Parts 60 24 45 50 18 40
Operations/Part 4 2-4 2-6 3-7 1-4 2-6
Machine Types 8 6 14 10 4 8
Machines 24 20 35 31 10 30
Cells 6 4 4 5 3 5
Batch Size (b ) 25 25 25 32 25 50
Cell Transfer Batch Size b
Arrival Rate Distribution (CV) Poisson (1.0)
Setup Time Distribution (CV) 2-Erlang (.7)
Part Run Time Distribution (CV) 2-Erlang (.7)
Setup Ratio (s /br ) 1.0
Setup Fraction:min/maj setup 0.2
Dispatching Rule RL
Material Handling Times 0
JS Average Machine Utilization 65%
Data Set ID
Table 3-3. Data sets characteristics after operational standardization.
We did not expect the standardized formation technique to provide the same cell
configurations as specified in the sources. Table 3-4 lists the differences between
configurations in the source and standardized designs.
70
1 2 3 4 5 6
Number of cells from source 6 4 4 5 3 5
Number of cells using
standardized formation
6 4 4 5 2 5
Machines from source 24 20 35 31 10 30
Machines using standardized
formation
24 20 41 32 10 32
Data Set ID
Table 3-4. Comparison of cell designs in source and standardized configurations.
3.5 Metrics and the Simulation Model
The primary metric for the simulation model is average batch flow time. The
simulation also tracks key explanatory output measures including average batch setup
and machine utilization. While the simulation model is capable of measuring move
time, we do not do so here based on our standardized move time of zero. The
expressions used to calculate these measures are listed in Appendix B.
We evaluate all six data sets with the same simulation model. Our model was
designed to possess sufficient generality to apply to both job shop and cellular
configurations. Each data set was first run in its job shop configuration using the
operational standardization. We then evaluate the CMS layout following the cells
designs provided by the data source and ensure that the CMS run uses the same
relative part volumes as the job shop configuration. In keeping with recent industry
survey results (Marsh et al., 1999), we allow for a remainder cell to process non-
related parts.
71
Each experimental condition tested was first warmed-up from an empty factory
for a period long enough for the WIP to stabilize via inspection of time series plots as
developed by Welch (1983). The end state of the warm up period was saved and used
for initial conditions for each of 100 replications starting with different random
number seeds to avoid autocorrelation. Each replication was run long enough for
each part type to have at least 250 completed batches in order for arrival and service
distributions to be adequately represented in the results. For example, data set 2
containing 24 parts and 100-minute flow times was run for approximately 100,000
simulated minutes per replication. The same set of random number seeds used across
replications was used across data sets to reduce variability. Typically, testing a single
data set required 300 simulation runs (each data set run at three levels and replicated
100 times). The comparisons between job shop and CMS flow times under
operationally standardized conditions as listed in Table 3-5 are all based on this run
length and 100 = n replication scheme.
We list both the mean and standard deviation of each statistic in the tables that
follow. The mean for each statistic is calculated as
¿
=
i
i
n x x where each
replication provides a data point and n is the number of replications. The standard
deviation is then calculated as ( )
( )
( ) 1
2
÷
÷
=
¿
n n
x x
x s
i
i
. This data is sufficient to then
calculate confidence intervals. The confidence interval using the t-test as outlined in
Pegden, Shannon and Sadowski (1990, p.177) is calculated as ( ) x s t h
n 2 1 , 1 o ÷ ÷
= .
72
The simulation model is written in GPSS/H (Schriber, 1974; Henriksen and Crain,
1989). The model was run on a 266 MHz AMD K6-based PC running Wolverine
Software’s GPSS/H Professional (Release 3: 1995). The execution time per
replication per level for each data set was roughly two minutes and equal since each
shop was loaded at the same level of congestion.
3.6 Simulation Results Comparing Functional and Cellular Layout
Our goal is to measure the results of conversion and to evaluate their consistency
across data sets. Prior to showing overall flow time results, we examine the measured
setup reduction resulting from the conversion to CMS. We then use this information
as well as congestion effect to explain the overall flow time results.
3.6.1 Setup Reduction Effect. We expect a significant reduction in setups as we
convert to CMS since major setups are eliminated whenever a part family is assigned
to a single cell. Tables 3-5 and 3-6 list the average setup time per batch for both CMS
and JS layouts as reported by the simulation output. In these tables, the setup is
measured as a fraction of the JS flow time per batch (which is normalized to 1.0 for
each data set). Each flow ratio data point is the ratio of the average batch flow time of
the transformed shop to the original job shop for the same replication. The setup
reduction is calculated as (1 – transformed shop setup/job shop setup)*100% for each
replication. We observe in Table 3-5 that the setup reduction is very consistent across
data sets and ranges from 69% to 77% with an average setup reduction of 73% per
batch. The confidence interval using the t-test is calculated as ( ) x s t h
n 2 1 , 1 o ÷ ÷
= so for
73
the setup reduction for data set 1, ( )( ) 00198 . 0 001 . 0 9842 . 1 = = h . We therefore have
95% confidence that the true mean is within 0.00198 of 0.69 or roughly within 0.3%
of our estimate of 69% (0.00198/0.69). Table 3-6 lists the results when formation
standardization is used for each data set. We get similar results indicating that the
standard cell configuration can also reduce setups significantly.
mean stdev mean stdev mean stdev mean stdev
1 0.293 0.002 0.090 0.001 69% 0.001 0.72 0.003
2 0.286 0.003 0.066 0.001 77% 0.002 0.87 0.010
3 0.201 0.004 0.060 0.001 70% 0.002 0.89 0.013
4 0.299 0.002 0.085 0.001 72% 0.002 0.78 0.005
5 0.267 0.001 0.069 0.000 74% 0.001 0.80 0.004
6 0.322 0.002 0.078 0.001 76% 0.001 0.82 0.006
average 73% 0.81
Data
set
Operational Standardization
JS setup CMS setup Setup reduction Flow ratio
Table 3-5. Setup reductions and associated flow ratios for Operational
Standardization
mean stdev mean stdev mean stdev mean stdev
1 0.295 0.002 0.105 0.001 64% 0.001 0.71 0.004
2 0.284 0.003 0.075 0.001 74% 0.002 0.86 0.008
3 0.250 0.004 0.079 0.001 68% 0.002 0.99 0.006
4 0.303 0.002 0.133 0.001 56% 0.002 0.93 0.007
5 0.149 0.004 0.069 0.003 60% 0.003 1.15 0.069
6 0.307 0.003 0.088 0.001 71% 0.003 0.92 0.014
average 66% 0.93
Data
set
Formation and Operational Standardization
JS setup CMS setup Setup reduction Flow ratio
Table 3-6. Setup reductions and associated flow ratios for Formation and
Operational Standardization
3.6.2 Overall Flow Time Improvements. To compare flow times, we ran each
data set with the source and the standardized cell configurations. The results appear
in Table 3-5 and Table 3-6, respectively. The setup reduction realized in the cells
74
tended to overcome pooling losses to outperform job shops by an average of 19%
corresponding to a flow ratio of 0.81. The confidence interval using the t-test is
calculated as ( ) x s t h
n 2 1 , 1 o ÷ ÷
= so for the flow ratio for data set 1,
( )( ) 00595 . 0 003 . 0 9842 . 1 = = h . We therefore have 95% confidence that the true
mean is within 0.00595 of 0.72 or roughly within 1% of our estimate of 0.72
(0.00595/0.72). The formation standardization results show an average improvement
of seven percent from conversion corresponding to a flow ratio of 0.93. This 7%
average improvement increases to 13% if we exclude data sets 5 and 6 containing
bottlenecks (see Figure 3-1 for high utilization levels for these data sets). We remind
the reader that standardized formation results in changes to the number of cells and/or
machines as shown in Table 3-4.
It is useful to compare our results with the findings of Suresh (1991) who
investigated the level of setup required to overcome the pooling loss (Suresh calls
this the breakeven o ). Using an analytical model, Suresh (1991) reduced the
magnitude of the setup ( s ) in the cells by 80% to overcome the pooling loss. A CMS
with this level of setup reduction will then have the same flow time as the job shop.
The results of our tests are more favorable to CMS. We show an average
improvement of 19% in flow time with a corresponding setup reduction of 72%. We
should note that the 80% figure cited from Suresh (1991) corresponds to a simulation
example using FCFS, no setup discounting in the job shop, and a flat-rate discount in
the cells. If we look for operating assumptions closer to ours, we should consider
75
Suresh’s family-based setup configuration for the job shop. The conversion of this
configuration to cells (using the same 80% setup reduction and a lot size of 20)
indicated an improvement of 22%, which is more consistent with our simulation
results.
To gain some insights into the flow times reported in Table 3-5 and Table 3-6, we
can examine the changes in machine utilization in greater detail. Figure 3-1 shows
the average overall utilization levels for JS and CMS for each of the six data sets
(labeled on the horizontal axis). Also shown are the maximum and minimum average
utilization levels realized across all machine types. As expected, the average
utilization for the job shop stays close to the target line of 65%. This is because we
adjust the load on the JS to attain this target utilization within two percent. The
simulation output shows that the average utilization after conversion to CMS is 48%
(this is the lower dashed line in Figure 3-1). Thus, on the average, conversion yields
an overall reduction of 17% in the average machine utilization.
Next, we examine the utilization levels by machine type. Since conversion
involves segregating pools of machines in departments into cells, imbalances may
arise readily unless the cell formation technique takes capacity issues carefully into
account. In fact, the range of machine utilization (computed as the difference between
maximum and minimum levels) increases eight percent when the JS is converted to
CMS using the source formation technique reflecting the machine loading imbalance.
The standardized cell formation technique produces a wider range (25% as compared
to eight percent for the source configuration).
76
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
1
JS
1
CM
1*
JS
1*
CM
2
JS
2
CM
2*
JS
2*
CM
3
JS
3
CM
3*
JS
3*
CM
4
JS
4
CM
4*
JS
4*
CM
5
JS
5
CM
5*
JS
5*
CM
6
JS
6
CM
6*
JS
6*
CM
max
mean
min
JS target
util.
CMS
measured
avg util.
Figure 3-1. Comparison of machine utilization for JS and CM
(the asterisk refers to standardized formation).
In eight of the 12 results tabulated (5 out of 6 from source and 3 out of 6 for
standardized formation), conversion succeeds in reducing both the average and the
maximum utilization. These are the cases that show favorable flow time reductions in
Table 3-5. It is worthwhile to examine the other four cases where the maximum
utilization has not been eased: 2*, 3*, 5*, and 6*. First we note that machine types
utilized less than 65% in the job shop did not have utilization levels exceeding 65% in
any of the cells. We therefore provide additional utilization detail form those machine
types that are utilized more than 65% in the job shop. As seen in Figure 3-2, each of
the four cases where the maximum utilization is not reduced exhibits a bottleneck in
at least one of the cells. Such bottlenecks arise simply because of the way machines
may be distributed among the cells during cell formation.
77
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
J
S
-
1
J
S
-
2
C
4
-
1
C
2
-
1
C
1
-
1
C
1
-
2
C
4
-
2
C
3
-
2
C
2
-
2
(a) Data set #2*
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
J
S
-
6
J
S
-
2
C
5
-
2
C
3
-
6
C
5
-
6
C
5
-
6
C
1
-
2
C
4
-
2
C
1
-
6
C
2
-
2
(d) Data set #6*
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
J
S
-
1
3
J
S
-
2
C
1
-
1
3
C
2
-
1
3
C
3
-
2
C
2
-
2
C
1
-
2
(b) Data set #3*
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
J
S
-
3
J
S
-
2
C
2
-
3
C
2
-
2
C
1
-
3
C
1
-
2
(c) Data set #5*
Figure 3-2. Simulation results for machine types with utilization above 65% in the
JS layout. In the job shop, JS- j denotes machine type j . Within cells, Cc - j
denotes machine type j in cell c .
For example, in data set 2 (using standardized formation), we single out machine
types 1 and 2 in the JS since their utilization exceeds 65%. Additionally, we show the
utilization for these two types wherever they occur in the cells. It is clear from Figure
3-2 (a) that the utilization of machine type 1 is reduced in cells 1, 2 and 4, but
machine type 2’s utilization has increased relative to the job shop to 81% in cell 1 and
78
is reduced in cells 2, 3 and 4. The parts being processed in cell 1 requiring machine
type 2 experience severe congestion resulting in a high flow ratio for the entire factory
as seen in Table 3-5. Example (b) through (d) in Figure 3-2 show similar bottlenecks
in data sets 3, 5 and 6, when standardized formation is used. In summary, these
examples shows that bottleneck effects can dominate the results on flow time in a way
that cannot be captured by system-wide average utilization alone.
3.7 Sensitivity to Key Operational Factors
In this section we investigate the sensitivity of flow time to four key factors. First,
we evaluate the effect of using smaller batch sizes or transfer batches in the cells.
Next we evaluate the effect of job shop loading. Then we study the sensitivity to the
two key parameters of the setup structure. Finally, we compare the effects of the
dispatching rule.
3.7.1 Batch Size Reduction and Transfer Batches. Our results of the last section
matched the batch size in the cells with the original batch size used in the job shop.
However, previous research (e.g., Suresh, 1991) shows that the setup reductions
realized allow us to use smaller batch sizes in the cells than in the job shop and that
this can have a profound effect on the flow time of cells. Moreover, cells can also
make the use of smaller transfer batches possible, since machines are located in close
proximity in cells. We therefore study two changes in the cells: (a) cutting the batch
size to half its original value, and (b) use of transfer batches of size one. The first
choice should provide a good idea of how a 50% reduction of batch sizes affects the
CMS. The latter tests the extreme case of unit transfer batches to assess the maximum
79
potential benefits small transfer batches are capable of producing (from Wagner and
Ragatz, 1994, we know that moving to smaller transfer batch sizes within cells
continues to produce benefits when no additional setup is incurred).
Table 3-7 compares the flow ratios for the job shop with batch size b and the
CMS under four settings: the original batch size b, the reduced batch size 2 b , and
transfer batches of size one used with either b or 2 b as the batch size. In all cases,
the flow time improves when a smaller batch size or a transfer batch of size one is
used.
b mean stdev mean stdev mean stdev mean stdev
1 25 0.72 0.003 0.46 0.003 0.37 0.001 0.28 0.001
2 25 0.87 0.010 0.56 0.016 0.57 0.015 0.41 0.012
3 25 0.89 0.013 0.63 0.011 0.61 0.010 0.49 0.013
4 32 0.78 0.005 0.50 0.004 0.38 0.003 0.30 0.003
5 25 0.80 0.004 0.50 0.003 0.51 0.004 0.36 0.003
6 50 0.82 0.006 0.52 0.007 0.45 0.004 0.34 0.005
average 0.81 0.53 0.48 0.36
JS to CM
reduced b
with TB = 1 Data
set
JS to CM
baseline b
JS to CM
reduced b
JS to CM
baseline b
with TB = 1
Table 3-7. Flow times in cells with smaller batch size or transfer batches
(JS flow time with batch size b provides baseline of 1.00).
For example, in data set 1, direct conversion to CMS reduces the flow time by
28% (flow ratio is 0.72) as compared to the job shop even when the same batch size is
used. The use of batch size of 2 b provides an additional improvement of 26% (0.72-
0.46 = 0.26), the use of unit transfer batches with the original batch size provides a
35% (0.72-0.37 = 0.35) improvement over the advantage of conversion alone.
80
Overall, the batch size reduction to 2 b improves upon the advantage of conversion
alone by 28%. Using transfer batches of size one in the cells provides an average
improvement of 33% over direct conversion (CMS with batch size b). However, if
the batch size is already reduced, this improvement averages 17%. Interestingly,
starting with a batch size of b in the job shop, the two alternatives of reducing the
batch size to 2 b or using transfer batches of size one but retaining b in the cells
produce comparable benefits (0.53 or 0.48). These results are of the same magnitude
as those reported by Smunt et al. (1996) where transfer batches of size one were used
in the first of four stages.
We also expect the improvement from using transfer batches to increase with the
number of operations per part. Figure 3-3 illustrates this relationship for data set 6.
The vertical axis of Figure 3-3 shows the additional improvement in flow ratio due to
transfer batches, as compared to CMS without transfer batches.
81
Slope= 7%, Intercept=15%, R
2
=.79
0%
10%
20%
30%
40%
50%
60%
70%
2 3 4 5 6 7
Operations per part
U
n
i
t
y

T
B

F
R

i
m
p
r
o
v
e
m
e
n
t
Unity TB ef f ect
Predicted Unity TB ef f ect
Figure 3-3. Flow time improvement using unity transfer batches as a function of
operations per part (data set 6).
3.7.2 Job Shop Loading Sensitivity. Our computational runs have shown that
pooling loss must be linked to the manufacturing load. As mentioned previously,
bottlenecks may occur as the pooled resources of the job shop are segregated into
cells. If such bottlenecks occur, their effect on flow time will be more pronounced as
the overall utilization increases.
We use data set 2 (using standardized formation) to illustrate the case where the
average machine utilization is reduced as a result of conversion, but the maximum
machine utilization deteriorates in the CMS. For this data set, we varied the level of
utilization from 55% to 85% and ran the simulation repeatedly. The results appear in
Figure 3-4. Recall that the JS utilization sets the level of demand since the relative
part demands are adjusted until the average machine utilization gets within 2% of the
82
desired utilization value. Utilization levels above 85% could not be tested for using
this data set since the maximum utilization in the CMS reaches 100%. We see that
the flow time suffers in the CMS when the job shop is loaded at 85%, but for
machines with lower utilization (in the 65% ±10% range), the effect on flow time is
modest. This example shows a point we have observed in other data sets: the flow
time in CMS is more sensitive to machine utilization than in JS. Therefore, cell
layouts may not exhibit superior flow times if bottlenecks appear.
83
0
50
100
150
200
250
300
JS 55% JS 65%
standard
JS 75% JS 85%
Job Shop Average Machine Utilization
A
v
e
r
a
g
e

b
a
t
c
h

f
l
o
w

t
i
m
e
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
M
a
c
h
i
n
e

u
t
i
l
i
z
a
t
i
o
n
JS Flow
CMS Flow
JS Max Util
CMS Max Util
CMS Avg Util
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
JS 55% JS 65%
standard
JS 75% JS 85%
Job Shop Average Machine Utilization
F
l
o
w

R
a
t
i
o
Flow Ratio
Figure 3-4. Job shop loading sensitivity (data set 2).
3.7.3 Setup Potential. We tested the sensitivity of flow time to the setup potential
by varying both the setup ratio and the setup fraction. We ran all nine combinations
84
of the two factors with three levels per factor. The highest potential occurs when the
setup ratio 2 = br s and setup fraction equals 0.1, while the lowest potential occurs at
the pairing (0.5, 0.4). We chose data set 2 to perform the setup sensitivity runs. We
kept the batch size ( b ) and part processing time ( br s + ) constant when varying the
setup ratio ( br s ) and ran each experiment at the standard 65% target average
machine utilization.
We expected the (2, 0.1) setting to produce results better than the standard (1, 0.2)
setting and expected the CMS flow ratios to increase as the potential for setup
reduction is lowered. The results in Figure 3-5 are consistent with this expectation:
the lowest flow ratio corresponds to the highest setup potential.
min/maj=.4
min/maj=.2
min/maj=.1
s/br=2
s/br=1
s/br=.5
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
Flow Ratio
Setup Fraction
Setup Ratio
1.10-1.20
1.00-1.10
0.90-1.00
0.80-0.90
0.70-0.80
0.60-0.70
0.50-0.60
0.40-0.50
Figure 3-5. Response of the flow ratio to the two setup parameters.
85
3.7.4 Dispatching Rule. Although we chose the repetitive lot (RL) dispatching
rule for our analyses, we recognize that not all shops may use a rule tailored to
minimize the incidence of setup. We, therefore, compare the use of this rule to first-
come-first-served (FCFS) dispatching to understand the dispatching rule’s effect. We
chose one data set from the six (data set 1) and evaluated its flow time at both the JS
and CMS layouts at a common level of demand using the same simulation model.
We set the factory load using the same method as before, but used the FCFS job shop
as the basis: we measured the average machine utilization and then set the demand
relative to the original demand mix such that the average was within two percent of
70%. We chose a slightly different value for the target to keep them clear from the
results of the conversion study above. Since these dispatching rules directly affect the
incidence of setup, we list more detail simulation measurement results in Table 3-8.
As before, we report the average setup as a fraction of the average JS flow time, but
here for a given dispatching rule. We include detailed measures of the incidence of
setup paid: none, minor and major. We do this because it enables us to separate setup
time incurred (which the reader will recall is a function of the ratio of minor to major
setup) from the setup incidence. The flow times are listed along with the calculated
flow ratios. Finally, the average machine utilization measures are listed (the range
data for utilization is similar to that shown in Figure 3-1 above).
86
mean stdev mean stdev
JS 1290 12.34 1169 6.03
CMS 844 2.63 837 2.28
Flow Ratio 0.65 0.005 0.72 0.003
JS 0.295 0.002 0.293 0.002
CMS 0.084 0.001 0.090 0.001
Setup
reduction 72% 0.001 69% 0.001
JS 5% 0.001 6% 0.001
CMS 10% 0.002 12% 0.002
JS 40% 0.003 46% 0.002
CMS 90% 0.002 88% 0.002
JS 55% 0.003 48% 0.002
CMS 0% - 0% -
JS 68% 0.005 66% 0.004
CMS 49% 0.003 50% 0.003
Dispatching Rule
FCFS RL
Average machine
utilization
Setup
Incidence
Major
Flow Time
None
Minor
Setup
Table 3-8. RL dispatching avoids more major setups in the job shop than FCFS.
The RL flow time in the JS is 9% lower than when using FCFS (1169 versus
1290). If we look first at the setup, the impact of either rule seems to be similar. The
fraction of flow time in both the JS and CMS as well as the setup reduction are all
within five percent across dispatching rules. They are, however, fractions of their
respective job shop flow times so the FCFS setup is 0.295*1290=381 and the RL
setup is 0.293*1169=343. This difference is significant with >95% confidence since
the mean difference between the FCFS and RL setup times (381-343=38) is within
1% of its estimate using the paired-t test. The setup incidence reveals that RL
requires fewer major setups (48% as compared to 55%). The reader will recall from
Table 3-1 that this data set contains 60 discrete part types that make up six part
families. The average queue size (not shown in Table 3-8) for the FCFS job shop is
87
0.64 so it is not surprising that the RL dispatching rule rarely has an opportunity to
bring forward a like part from the queue to process in sequence. Although RL isn’t
able to leverage part-to-part sequencing often, it is able to leverage the common
family parts currently in queue generating more minor setups (46% versus 40% using
FCFS). The lower utilization measure is a direct result of the reduced setup paid
using RL. The range of machine utilization across the machine types is roughly
unchanged.
Once the factory is converted to cells, there seems to be little flow time advantage
to RL over FCFS. This may be because the major setup reduction is complete and no
longer a factor. This particular data set has 10 parts per part family and the average
queue size in the cells for FCFS (and RL) was 0.10. The FCFS rule in the cells paid a
minor setup 90% of the time (which corresponds to the number of discrete part types
per cell). Therefore, for RL to improve upon FCFS there must be more than one part
in queue (and of the same type being processed) so the dispatching rule can pull it
forward and avoid the minor setup.
3.8 Move times
While we do not focus on move time effects in this research, it is useful to briefly
explore the magnitude of this effect. We note that when move times are known and
not subject to congestion, these times can be added in ex post. We evaluated this
effect for data set 2 with 31 total machines, 10 machine types, 50 part types, and five
part families forming five cells (Suresh and Meredith, 1994). We set the move time
equal to ( ) br s + o , where o is a multiplier that we can vary, so that the move time is
88
proportional to the standard processing time per batch (major setup plus batch run
time). We used this time every time work was transported between a pair of
departments in the job shop. Since we assume that move times in the CMS are
negligible, the flow ratio should improve as o increases. The value 2 = o
corresponds to the high level of move time used in Suresh and Meredith (1994). We
found that the flow ratio improves 12% each time o is increased by 1. The move is
therefore an independent compensatory factor that can be used to overcome pooling
loss. But the preceding example shows that the magnitude of move times has to be
significant (compared to the batch run time) for it to have an impact.
3.9 Discussion on Dispersion of Simulation Results in the Literature
We now return to the issue that motivated this study: the large dispersion in the
results of simulation studies that compare functional and cellular layouts as shown
previously in Figure 2-1. In Figure 3-6 we add our results. The topmost bar of Figure
3-6 is reserved for the results of our test bed of six data sets. It is immediately clear
that the range of results for our runs is narrower than the results of most of the other
studies and lies consistently in the band that favors CMS. This remains true even
when we compare our results to the first group of bars in Figure 3-6 that represent the
sources of data for our test bed. This shows that standardization can significantly
reduce the dispersion across six different data sets.
The second and third bars in Figure 3-6 show the reduction in flow time for CMS
resulting from the use of reduced batch sizes or the implementation of transfer batches
in cells. For our test bed, the numerical averages reported in Table 3-7 indicate that
89
while retaining the original batch size in CMS produces flow ratios in the range 0.78 -
0.89, using a reduced batch size or transfer batches in the cells can further reduce the
flow ratios to lie in the range 0.37-0.63.
0 1 2 3 4
Flynn and Jacobs (1986)
Flynn (1987)
Crookall and Lee (1977), Lee (1985)
Morris and Tersine (1994)
Flynn and Jacobs (1987) small shop
Suresh (1991)
Jensen et al. (1996)
Burgess et al. (1993)
Ang and Willey (1984)
Shafer and Meredith (1990, 1993, 1990 company C)
Leu et al. (1995)
Suresh (1992)
Shafer and Charnes (1993)
Suresh (1993)
Moily et al. (1987)
Morris and Tersine (1990)
Morris and Tersine (1989)
Yang and Jacobs (1992)
Garza and Smunt (1991)
Shafer and Charnes (1995)
Suresh and Meredith (1994)
Standardized using unit transfer batches in cells
Standardized using reduced batch size in cells
Standardized
Flow Ratio Ranges
÷ Clear preference for JS Clear preference for CM ÷
Figure 3-6. Results from standardized approach reduce variability and favor CM.
90
One may inquire as to the possible sources of the wide dispersion seen in Figure
3-6. Of the 17 data sets where the job shop flow times are superior, eight did not
discount setups at all. On the other hand, ten data sets showed better flow times for
CMS. Seven of these ten data sets used a high ratio of setup to run time (some going
up to 6.0, compared to our baseline values of 1.0). The other three used transfer
batches in the cells. For the specific studies included in our test bed, Table 3-9
compares the flow time results reported in the literature with our results and provides
our choice of the most likely factors that can explain the difference for each study.
91
JS to CM
TB = b
JS to CM
TB = 1
Source mean mean mean stdev mean stdev
1
Garza and
Smunt 1991
1.42 n/a 0.72 0.003 0.37 0.001
low s /br
range
4
Suresh and
Meredith
1994
0.93 n/a 0.78 0.005 0.38 0.003
high JS
utilization
5
Yang and
Jacobs 1992
0.59 n/a 0.80 0.004 0.51 0.004
large material
handling effect
present in JS
6a
Morris and
Tersine 1989
1.19 0.82 0.82 0.006 0.45 0.004
low s /br
high
minor/major
setup
6b
Morris and
Tersine 1990
1.05 n/a 0.82 0.006 0.45 0.004
high
minor/major
setup
6c
Shafer and
Charnes 1995
n/a 0.90 0.82 0.006 0.45 0.004
low s /br
high
minor/major
setup
Source setting
explaining the
difference
Data
Set
ID
Source simulation
results
Standardized
simulation results
JS to CM
TB = b
JS to CM
TB = 1
Table 3-9. JS to CMS flow ratios in the modeling literature.
The results of our runs also allow us to compare the relative impact of utilization
level, setup potential, and batch size reduction. We have shown this in Figure 3-7 for
a single data set (#2). The topmost bar shows the range of flow ratios obtained by
changing the utilization levels, the second bar shows the results for different
combinations of the setup ratios and setup fraction, and the last bar shows the effect of
using a smaller batch size or adopting transfer batches.
92
0.86 0.30
0.56 0.61 1.16
1.16
0.41 0.15 0.31 0.87
0.00
0 1
Batch size
Setup
potential
JS utilization
Flow Ratio
55% to 85%
min/maj setup=0.4 to
s/br=0.5 to 2.0
reduced b with TB=1 effect
baseline b with TB=1 effect
reduced b effect
Figure 3-7. Results of sensitivity analysis for data set 2.
3.10 Summary
In this research, we argue that the wide divergence reported in the literature occurs
because of differences in the choice of demand data, production environments, setup
structures, utilization levels, cell formation, and significant disparities in the operation
of the production system. The present study attempts to study the sources of variation
more systematically by standardizing the operating rules of the factories and adopting
a common set of key parameters ranges, while retaining the differences in demand
and part type characteristics across data sets. By performing a set of baseline runs
with standardized values and a host of sensitivity runs on the level of the standardized
factors individually, we seek to gauge the effect of each factor more reliably.
Of pivotal importance to our computational study is the use of six different data
sets selected from different sources in the open literature, so that the results would not
be tied to a single profile of part types, mix, or demands. To our knowledge, this is
the first study that compares CMS conversion benefits across disparate data sets. In
93
addition, by using consistent operating principles in the simulation runs, we took
utmost care to make the comparison between the job shop and CMS environments
even handed.
Based on over 2000 simulation runs conducted in this study, we can summarize
our main conclusions as follows.
• The conversion of job shops to cells consistently improves flow time by 10%
to 20%, for the test bed used in this study. This result provides a conservative
estimate of the advantages of CMS because it does not take advantage of such
additional factors as reduced batch sizes, transfers batches, or move times.
We conclude that setup reduction can overcome the effects of pooling loss as
long as the magnitude of the setups is not too small and no significant
bottlenecks develop in the cells upon conversion.
• The use of reduced batch sizes, or the implementation of transfer batches, can
each provide cells with an additional improvement in flow time. Typically,
each of these two factors has a significant effect on reducing the flow time for
CMS, and the amount of reduction is usually at least as large as that obtained
by conversion to CMS without any changes in the batch sizes.
• The sensitivity runs show that the overall factory utilization and the potential
for setup reduction can both affect the conversion results obtained. Our tests
indicate that conversion to CMS may not be advantageous if the utilization
level is high or there is not sufficient potential to reduce setups.
94
• The design of cells also has a clear impact on the conversion improvements
obtained. Typically, we observed better performance in cells when the
original source design was used. However, conversion benefits continue to be
present even after we use a uniform cell formation procedure due to Vakharia
and Wemmerlöv (1990). This indicates that careful allocation of machines to
cells to avoid heavy utilization helps to keep the pooling loss within tight
control.
• Our experimental runs support the conclusions of previous authors that RL
dispatching provides less overall setup and supports lower flow times than
FCFS in a job shop with setup. The effect of RL seems to diminish in the
same factory setting once it incorporates cells.
In summary, we believe that this part of the dissertation has shown that the
comparison of job shops and cellular systems with respect to the flow time measure
can produce reasonably consistent results when the same operating rules and key
parameter ranges are used across different data sets. Moreover, our research shows
that setup reduction can overcome pooling losses, even under the conservative
assumptions where batch size remain unchanged and the material transport times in
the job shop are assumed to be negligible. Overall, the conclusions of our research
are consistent with the qualitative insights cited in the literature when comparing
CMS and job shops. However, our research clarifies that the quantitative
comparisons using the flow time metric must be interpreted in the context of the
95
region of the parameter space spanned by the data sets, as well as the particular design
used for the cells.
96
Chapter 4
PARTIAL CELLULAR MANUFACTURING SYSTEMS
Conversion from a job shop environment to cellular manufacturing does not need
to proceed all the way: one can consider a partial implementation of cellular layout.
One can investigate what the benefits of a partial cellular layout may be as compared
to full conversion. For example, we may ask if a few cells can provide most of the
flow time benefits associated with full conversion. To answer this question, we use
the same data sets we analyzed fully in Chapter 3. We consider partial cellular
layouts at all levels ranging between the two extremes of JS (no cells) and CMS (all
cells). For each hybrid layout, we evaluate the flow times in both the cells and the
remainder shop and relate this to congestion effects. We find that cell selection,
sequence of cell application, level of cellular implementation and load balance are all
important considerations in the implementation of partial layouts.
4.1 Simulation Analysis of PCMS
The evaluation of partial layouts follows the schema used in Chapter 3. For each
data set considered, there is a complete cellular layout that is known in advance. This
is the all-cell layout corresponding to full conversion. Suppose that this layout uses
NC cells. We can consider each partial layout as a choice of a subset S of the set
{ } NC T , , 1 … = . Given a subset S of selected cells, let ( ) S FR be the flow ratio of the
configuration represented by the cells in S and the remainder shop handling all parts
97
not assigned to these selected cells. We will use simulation to evaluate ( ) S FR for all
subsets of a fixed cardinality n , where n is successively increased from 1 to NC .
The exhaustive evaluation of all subsets of n cells allows us to rank sort all subsets of
size n with respect to total factory flow time. For each n , we record the best pick as
the subset S of size n that results in the lowest flow ratio and label it ( ) n BP and
denote its flow ratio ( ) n BFR . Similarly, the worst pick subset of cells at level n is
associated with the highest overall flow ratio is denoted by ( ) n WP with flow ratio
( ) n WFR .
Table 4.1 presents the results of this analysis for all six data sets discussed in
Chapter 3. As in Chapter 3, the setup reduction reflects the total setup paid relative to
the total setup paid in the JS layout. At each fixed n , we also compare the best and
worst flow ratios obtained at that level with the best overall pick that gives the lowest
flow ratio across all n . We denote this best overall flow ratio as
( ) { } n BFR BFR min * = with the minimum taken over all n from 1 to NC . This
minimum may be achieved for the all-cell option where NC n = or a partial layout
using a smaller number of cells. We identify the optimum level of cellular
implementation for each data set as the smallest n for which there is no further
marginal reduction in flow ratio. The marginal reduction in flow ratio at any level
NC n < is calculated as
( ) ( )
* 1
1
BFR
n BFR n BFR
÷
÷ ÷
or
( ) ( )
* 1
1
BFR
n WFR n BFR
÷
÷ ÷
and for
NC n = is
( ) ( )
* 1
2
BFR
NC BFR NC BFR
÷
÷ ÷
.
98
In order to assess the impact of the cellular investment at a given implementation
level n , we try to relate the factory flow ratio to the fraction of machines and part
demands sent allocated to the cells. Specifically, these ratios are computed as
follows: We indicate the number of machines sent to cells for the best and worst pick
at level n as ( ) n BM and ( ) n WM , respectively. Therefore, the fraction of machines
sent to the cells is calculated as ( )
¿
j
j
NM n BM and ( )
¿
j
j
NM n WM (we remind
the reader from our notation in Chapter 3 that the number of machines of type j in
the factory is
j
NM ). Similarly, we indicate the total batch demand sent to cells,
( ) ( )
¿
e S F i f
i
ì where ( ) S F is the family of parts assigned to the cells in S , for the best and
worst pick at level n as ( ) n BD and ( ) n WD , respectively. The fraction of batch
demands sent to the cells are calculated as ( )
¿
i
i
n BD ì and ( )
¿
i
i
n WD ì .
To illustrate the contents of Table 4-1, we now review the information presented
for data set 3. We see from the maximum number of cells formed that there are four
cells to choose from. At 2 = n , where we allow two cells to be formed,
( ) { } 4 , 3 2 = BP . The simulation results of that pick list that the overall factory will
enjoy a 70% setup reduction as compared to the original JS. The measured flow ratio
from the simulation is 0.890. This particular pick happens to be equivalent in flow
time to the all-cell pick. In this case only 66% of machines and 47% of batch
demands and have been sent to the (two) cells. If we read the 4 = n data we see that
99
there is no further reduction in flow ratio if we split up the remaining resources and
demands.
The last data set entry, 6
†
, represents a perturbation to data set 6. We created a
bottleneck by shifting the load on a particular machine type: we changed the routing
of the parts requiring machine type 6 common to cells 4 and 5 such that the machine
in cell 4 (when selected) was only 20% utilized. Therefore, whenever cell 4 was
selected the residual was left with type 6 machine utilization in excess of 90%. Data
set 6
†
is a case where the best partial cell option is better than the all-cell option (the
difference in the all-cell and partial option 1,2,3 flow times is significant with >95%
confidence using a paired-t test).
100
mean (%) stdev mean stdev
5 19 0.012 0.937 0.011 17 17
4 18 0.012 0.948 0.010 17 17
1,3 37 0.010 0.862 0.011 33 33
2,3 36 0.010 0.890 0.011 33 33
4,5,6 54 0.008 0.787 0.009 50 50
2,4,6 45 0.009 0.836 0.010 50 50
3,4,5,6 66 0.006 0.734 0.008 67 67
1,3,5,6 58 0.007 0.777 0.009 67 67
6 1,2,3,4,5,6 69 0.004 0.716 0.008 100 100
4 30 0.007 0.949 0.008 25 25
1 31 0.007 0.986 0.009 25 25
2,4 62 0.004 0.897 0.009 50 50
1,3 51 0.005 0.956 0.009 50 50
4 1,2,3,4 77 0.002 0.867 0.010 100 100
3 45 0.004 0.911 0.009 40 26
2 32 0.005 0.984 0.012 23 31
3,4 70 0.002 0.890 0.013 66 47
1,2 45 0.004 0.956 0.012 34 53
4 1,2,3,4 70 0.002 0.890 0.013 100 100
4 25 0.004 0.944 0.004 23 21
2 21 0.005 0.959 0.005 19 20
4,5 45 0.004 0.878 0.005 42 40
2,3 41 0.004 0.904 0.005 39 41
3,4,5 66 0.002 0.807 0.004 61 61
1,2,3 59 0.004 0.841 0.005 58 60
5 1,2,3,4,5 72 0.002 0.781 0.005 100 100
1 50 0.003 0.877 0.004 40 33
3 38 0.004 0.915 0.004 30 33
3 1,2,3 74 0.001 0.798 0.004 100 100
2 28 0.008 0.940 0.004 27 26
4 19 0.008 0.989 0.008 17 18
1,2 53 0.007 0.891 0.009 50 49
3,4 37 0.007 0.967 0.009 37 36
1,2,3 72 0.005 0.835 0.008 70 66
3,4,5 53 0.006 0.920 0.009 50 51
5 1,2,3,4,5 76 0.005 0.824 0.009 100 100
1 26 0.003 0.932 0.006 23 23
4 6 0.095 2.008 0.309 17 17
1,2 53 0.003 0.861 0.008 50 49
3,4 21 0.009 2.123 0.335 37 34
1,2,3 75 0.001 0.749 0.005 70 66
2,3,4 59 0.009 1.171 0.079 63 60
5 1,2,3,4,5 78 0.001 0.766 0.005 100 100
6
†
1
2
3
3
1
2
1
1
2
3
4
5
1
6
1
2
3
Number
of cells
formed
Data
Set
Machines in
Cell(s) (%)
4
1
2
3
2
1
2
Batch
Demands in
Cell(s) (%)
Setup Reduction
Cell Ids:
Best
Worst
Flow Raio
Table 4-1. Simulation results for best and worst picks at each level of cellular
implementation.
101
4.1.1 Cell Selection. To ensure that every potential layout is assessed, we ran the
simulation model exhaustively for all subsets S of the set of cells for each of the six
data sets plus the a perturbed data set 6. The resulting comparison reveals that the
choice of the cells at each level makes a difference. For any n , we observe a
difference in the flow ratios between the best and worst picks. Data set 6 shows this
clearly: at 1 = n the best pick, cell 2, results in flow ratio of 0.940 whereas the worst
pick, cell 4, results in a flow ratio of 0.989.
When we look across results from all the data sets we can compare the last two
columns with the flow ratios. We see that ( ) NC n n BP < , always results in a greater
flow time reduction than the batch demands or machines invested, but this is not the
case with the worst picks. Again, using data set 6 as an example, ( ) 1 BP results in
34% [(1-0.940)/(1-0.824)*100%] of the possible flow ratio reduction for that data set
while requiring only 27% of the machines to be located in cells to work on 20% of the
batch demands. We contrast this with ( ) 1 WP resulting in six percent of flow ratio
reduction [(1-0.989)/(1-0.824)*100%], but requiring 17% of the machines in the cells
working on 25% of the batch demands. So, even though there may be several choices
available that will improve the overall factory flow time, the best pick leverages the
resources of batch arrivals and machines most effectively.
We also observe that ( ) n BP has setup reduction that matches and often exceeds
the setup reduction achieved by ( ) n WP . Although large differences in setup
reduction can account for a portion of the difference between factory flow times, it is
102
not the only source of such differences. A good example is available for data set 1 for
2 = n . The setup reductions achieved by ( ) n BP and ( ) n WP are equal, yet there is a
three percent difference in factory flow times ( ( ) ( ) 2 2 WFR BFR ÷ ). To explain this
disparity we must also review the machine utilization as shown in Figure 4-1.
40.0%
45.0%
50.0%
55.0%
60.0%
65.0%
70.0%
J
S
B
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(
1
)

C
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(
1
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R
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(
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)

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R
e
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(
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)

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a
l
B
P
(
N
C
)
max
mean
min
Data set #1
Figure 4-1. Machine utilization ranges during early stages of CMS implementation.
Both cell utilization levels are well below the original JS. The best choice ( ) 2 BP
shows a lower average and maximum utilization in the residual while the utilization is
comparable in the cells. We get an indication from this example that in comparing
subsets S of the same size, a pair of subsets may show equal performance on the cell
side of the shop but the preferred choice may be the subset that achieves superior
103
performance in the remainder shop. We also note that there can be flow time
differences even when the cells seem to allocate the resources equally. Like the CMS
analysis in Chapter 3, we find that ex post setup reduction information alone is not
sufficient to discern the best cell pick(s).
4.1.2 Effect of Sequence. Full conversions from JS to CMS reported in the
computational studies found in the literature do not address the order in which cells
are implemented. However, the empirical literature clearly shows that firms tend to
implement “one step at a time.” Here we address the sequence question. Using the
same data sets we ask the natural question, “is there always a nested picking order
from a single cell to the all-cell conversion option?” To put it in practical terms, the
manager should be alerted if a cell that appears to be the best choice at a given stage
turns out to be an inferior pick once other cells come into being. In any event, the
manager prefers nested sequences of subsets S with increasing cardinality since
dismantling a cell formed earlier is unattractive.
In our limited number of data sets tested here we found the occurrence of
mutually exclusive sets of cells picked at different levels of CMS implementation
suggesting sequence of cells picked can matter.
We look at data set 1 for an example of this phenomenon: ( ) { } 5 1 = BP , but
( ) { } 3 , 1 2 = BP and then ( ) { } 6 , 5 , 4 3 = BP . While not shown in Table 4-1, it turns out
that in this case there is little difference in the factory flow times of the { } 6 , 5 = S and
the best pick at 2 = n , { } 3 , 1 = S . In fact, the former set was ranked second best in a
104
close contest. Given the best choice for 3 = n level, it is clear that the manger would
prefer the sequence of cells 5, followed by 6, followed by 4 to a blind implementation
of the best subset at each level. Such considerations suggest look-ahead strategies and
the use of a richer set of criteria in selecting the cells for partial implementation.
4.1.3 Stopping Rule. The results of this chapter confirm our statement in Chapter
1 that the best overall flow may be achieved by a hybrid layout, rather than either a
pure JS or all-cell options. In such cases, one should look for rules or strategies to
halt conversion at some intermediate state instead of proceeding to full conversion.
This is apparent in the results of the simulation runs for data sets 3 and 6
†
. In data set
3, ( ) * ) ( 2 BFR NC BFR BFR = = . Any further implementation of cells after 2 = n
will not result in further reduction in flow time. In data set 6
†
further cell picks
(equivalent to all-cell conversion) will actually degrade the factory overall flow time,
( ) ) ( * 3 NC BFR BFR BFR < = .
4.2 Summary
The analysis performed in this chapter provides some insights into implementing
partial cell layouts (hybrids) using the same test bed as in Chapter 3. Below we
summarize some of the lessons learned from the exhaustive computational evaluation
of all partial layouts. We did not pursue this line of investigation any further because
we could not identify general and robust rules that applied across all data sets. Our
observations may be summarized as follows:
105
1. Even when the number of cells to be included in the partial layout is fixed, the
choice of the correct subset of cells can have a significant impact on the flow
time. In short, selection matters.
2. The sequence of best subsets to pick as n increases from 1 to NC is not
necessarily nested, so sequence matters.
3. Factory flow time of a partial cellular implementation may be as good as or
even better than the all-cell option as we have shown in our perturbed data set
6, so it is important to stop short of full conversion where appropriate.
4. The differences in factory flow times are due to the same factors recognized in
the all-cell CMS analysis, setup reduction and machine utilization, but neither
factor alone is sufficient to reliably determine the best subset of cells to select.
The best picks are characterized by large setup reductions along with
reduction of utilization in the residual job shop and the lack of bottlenecks in
the cell(s), so setup reduction and load balance in both the cells and residual
job shop matter.
106
Chapter 5
ANALYTIC MODELING OF A SIMPLE SYSTEM WITH SETUP
The analysis of a job shop under the assumptions of the factory environment
in Chapter 3 presents major challenges in modeling. The simplest model appears
to be a queueing network model with setups. We do not intend to address the
approximations made by queueing models in this work, especially since
adjustments for setups are generally not made in any exact fashion. Instead, in
this chapter, we use analytic models to gain insights into the extent of setup
economies that can be obtained by using dispatching disciplines designed to avoid
unnecessary setups and compare these with first-come-first-serve (FCFS)
protocols. We focus on the simplest queueing model we could find that handles
the effect of setups on flow time exactly. This system involves two customer
classes with general service time distributions and setups are incurred when
switching from one class to the other. The dispatching rule we investigate is
designed to minimize the incidence of setups in a queue with two customer
classes. This will provide a theoretical underpinning for our empirical findings in
Chapter 3, where we found that the dispatching rule selected does make a
difference.
107
5.1 Zero Setup
We start by establishing a baseline in the absence of setups, evaluating flow
times under FCFS versus a dispatching rule that minimizes the incidence of
changeovers. Our comparison involves a system with two customer classes,
where each customer requires a single operation at the service facility. Initially,
we assume that the setup time equals zero, and study the queueing system under
two different dispatching regimes: Alternating Priority (AP) and FCFS. We
already know from Avi-Itzhak et al. (1965) that if the two classes have the same
service distribution, then the mean flow times of both systems are the same
(assuming zero setup). Here, we focus on the asymmetric case where the service
distributions are different. Further, we choose cases where the first and second
moments are easily related and therefore develop our result with the assumption
of exponential service since ( ) ( )
2 2
2
i i
S E S E = . We employ two general results for
our comparison. To measure the AP (two-queue) flow time, we start with the
general result from Eisenberg (1967). We measure the flow time of the FCFS
(single queue) using the familiar Pollaczek-Khintchine (P-K) formula for the
M/G/1. We follow the analytic comparison of AP versus FCFS in the zero setup
case with numerical comparisons at two arrival rate settings.
Because setup times are not involved, there is no difference between service
times paid in either regime, so we focus on the average wait time until service,
versus the flow time, F . We use the notation
q
W for the wait in queue when
108
there is zero setup, consistent with queueing notation. We use the notation
W when the wait includes non-zero setup. The flow time always includes any
setup time paid.
5.1.1 Analytic comparison of AP versus FCFS. From Eisenberg (1967) the
general wait time for AP after removing setup for the class-1 queue is:
( )
( )
( ) ( ) ( )
( )( ) ( )( ) [ ]
2 1 2 1 1
2
2 2
2
1
2
1 1
2
2
1
2
1 1
1 1 1 1 2
1
1 2
1
µ µ µ µ µ µ
ì µ ì µ
µ
ì
+ ÷ ÷ ÷ ÷
÷ +
+
÷
=
S E S E S E
W
AP
q
and for the class-2 queue is:
( )
( )
( ) ( ) ( )
( )( ) ( )( ) [ ]
2 1 2 1 2
2
1 1
2
2
2
2 2
2
1
2
2
2 2
1 1 1 1 2
1
1 2
2
µ µ µ µ µ µ
ì µ ì µ
µ
ì
+ ÷ ÷ ÷ ÷
÷ +
+
÷
=
S E S E S E
W
AP
q
Together, the overall average wait time is:
( )
( )
( )
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
)
`
¹
÷
÷ +
+
¹
´
¦
÷
÷ +
÷
+
)
`
¹
¹
´
¦
÷
+
÷
=
1
2
2 2 1
2
1
2
1
2
1
2
2
2
2
1 2 1
2
2
2
2
2
2
2
1
2
2
2
2
2
1
2
1
2
1
1
1
1
1
1 2
1
1 1 2
1
µ
ì ì µ ì µ
µ
ì ì µ ì µ
µ ì
µ
ì
µ
ì
ì
S E S E
S E S E
D
S E S E
W
AP
q
(1)
where ( )( )
2 1 2 1
1 1 µ µ µ µ + ÷ ÷ = D
but for FCFS,
( )
( ) ( ) [ ]
2
2 2
2
1 1
1 2
1
S E S E W
FCFS
q
ì ì
µ
+
÷
= . (2)
This follows from the Pollaczek-Khintchine formula for the single M/G/1 queue:
( )
( ) µ
ì
÷
=
1 2
2
S E
W
q
. (3)
109
For our case ( ) ( ) ( )
2
2
2 2
1
1 2
S E S E S E
ì
ì
ì
ì
+ =
so ( ) ( ) ( )
2
2 2
2
1 1
2
S E S E S E ì ì ì + = .
Now convert to exponential case using ( ) ( )
2 2
2
i i
S E S E = or ( )
2 2 2
2
i i i
S E µ ì =
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( )
.
1
1
1
1
1
1
1 1
2
1 1
2
2 2
2
2
2
1
1
2 2
2
1 1
2
2
2
1
2
2
2
1
2
1
)
`
¹
¹
´
¦
÷
÷ +
+
÷
÷ +
×
÷
+
÷
+
÷
=
µ
µ µ ì µ µ
µ
µ µ ì µ µ
µ ì µ ì
µ
µ ì
µ
S E S E
D
W
AP
q
(4)
For the exponential case, (2) becomes the following
( ) ( )
µ
µ µ
÷
+
=
1
2 2 1 1
S E S E
W
FCFS
q
. (5)
We can re-write the expression for
AP
q
W in (4) slightly differently:
( )
( ) ( )
( )
( )
( ) ( )
( )
( ) [ ]
( )( )
( ) [ ]
( )( )
( ) ( ) ( ) ( )
( )
.
1
1 1
1 1
1
1 1
1
1
1
1 1 1
1
1
1
1 1 1
1
1 1 2 2 2 2 1 1
2
2
1
2
2
1
2
2
2
1
1 1 2 2
2
2
1
2
2
2
2
2
2 2 1 1
1
2
2
2
1
1
2
1
µ ì
µ µ ì µ µ ì
µ µ ì
µ µ µ
µ µ ì
µ µ µ
µ ì
µ µ ì
µ
µ
µ ì
µ
µ ì
µ
µ ì
µ µ ì
µ
µ
µ ì
µ
µ ì
µ
÷
÷ + ÷
+
÷ ÷
+ ÷
+
÷ ÷
+ ÷
=
÷
÷
+
÷ ÷
+
÷
+
÷
÷
+
÷ ÷
+
÷
=
D
S E S E
D
D
D
D
D
S E
D
D
S E
D
W
AP
q
Consider the bracketed expression within the first term:
( ) ( )( )
( ) ( )
2
2 2 1
2
2
2 2 1
2
2
1 2 1
1 2 1 1
µ µ µ µ µ
µ µ µ µ µ µ µ
+ ÷ + ÷ =
+ ÷ + ÷ = + ÷ D
write ( ) ( )
2 1
1 1 µ µ µ ÷ ÷ = ÷ then
110
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
1
2
2 2
2
1
1
2
2
2
1 2
2
1
2
2 1
2
2
2
1 2
2
1
2
2
2
2 1 1 2 1
2
2 1 2
2
1
2
2
1 2 2 1 1
1 2 1 2 1
2 2 1 2 1
2 1 2 1 2 1 1
µ µ µ µ
µ µ µ µ µ
µ µ µ µ µ µ
µ µ µ µ µ µ µ µ µ µ µ µ
÷ + ÷ ÷ =
÷ + ÷ ÷ ÷ =
÷ + ÷ ÷ ÷ =
+ ÷ ÷ + + ÷ + ÷ = + ÷ D
so
( ) [ ]
( )( ) ( )
( )( ) [ ]
2
2 2 1
2
1
1
2
2
2
1
2 2 1 1
1 1 1
1
µ µ µ
µ ì
µ
µ µ ì
µ µ µ
+ ÷ ÷
÷
=
÷ ÷
+ ÷
D D
D
similarly
( ) [ ]
( )( ) ( )
( )( ) [ ]
2
1 1 2
2
2
2
2
1
2
2
2 2 1 1
1 1 1
1
µ µ µ
µ ì
µ
µ µ ì
µ µ µ
+ ÷ ÷
÷
=
÷ ÷
+ ÷
D D
D
.
Also note that
( )( ) ( )( ) ( )
( )( )
( )( ) ( )
1
2
2
2
2 1
2
2 1
2
2
2
2 1
1
2
2
2
2
2
2 2 1
2
2 2 1
1 1 1
1 1
1 2 2 1 1 2 2 1 1
µ µ µ µ
µ µ µ µ µ
µ µ µ µ µ µ µ µ µ
+ + ÷ ÷ =
+ + ÷ ÷ =
÷ ÷ + + ÷ ÷ = + ÷ ÷
so we can simplify the expression for
AP
q
W
( )
( )( ) [ ] ( )( ) [ ] {
( ) ( ) ( ) ( ) [ ]}
( )
( )( ) ( )( ) [ ] {
( ) ( ) ( ) ( ) [ ]}
1 1 2 2 2 2 1 1
1 2
2
2 2 1
2
1
2
2
2
1
1 1 2 2 2 2 1 1
2
1 1 2
2
2
2
2 2 1
2
1
1 1
2 1 1 2 1 1 4
1
1
1 1
2 2 1 1 2 2 1 1
1
1
S E S E
D
S E S E
D
W
AP
q
µ µ ì µ µ ì
µ µ µ µ µ µ µ µ
µ ì
µ µ ì µ µ ì
µ µ µ µ µ µ µ µ
µ ì
÷ + ÷ +
÷ ÷ + ÷ ÷ +
÷
=
÷ + ÷ +
+ ÷ ÷ + + ÷ ÷
÷
=
also note that
( )( )
( )
( )
2 2
2
2
2 2 1 2 1
2
2 2 1 2 1
2
2 2 1
2 1
2 2 1
2 2 2 1 2 2 1 1
µ µ
µ µ µ µ µ µ
µ µ µ µ µ µ µ µ
÷ ÷ =
÷ + + ÷ ÷ =
+ + ÷ ÷ = + ÷ ÷
D
and similarly ( )( ) ( )
1 1
2
1 1 2
2 1 2 2 1 1 µ µ µ µ µ ÷ ÷ = + ÷ ÷ D
111
so we can re-write the first two terms of
AP
q
W
( )
( ) ( ) ( ) ( ) [ ]
( )
( ) ( ) ( ) ( ) [ ]
( )
( )
( ) ( ) [ ]
( )
( )
( ) ( ) ( ) ( ) [ ]
( )
( )
( )
( )
( ) ( ) ( ) ( ) [ ]
( )
.
1
1 1
1
4
1
1 1
1
1
1
2 1 2 1
1
1
1 1
1
1
2 1 2 1
1
1
1 1 2 2 2 2 1 1
2 1 2 1 2 1
2
2
2
1
1 1 2 2 2 2 1 1
1
2
2 1 2 2
2
1 2
2
2
1
1 1 2 2 2 2 1 1
1 1
2
2 2 2
2
1
µ ì
µ µ ì µ µ ì
µ ì
µ µ µ µ µ µ
µ ì
µ µ
µ µ ì µ µ ì
µ ì
µ ì
µ µ µ µ µ µ
µ µ
µ ì
µ µ ì µ µ ì
µ ì
µ µ µ µ µ µ
µ ì
÷
÷ + ÷
+
÷
÷ +
÷
÷
+
=
÷ + ÷
÷
+
÷
÷ + ÷
÷ +
÷
=
÷ + ÷
÷
+
÷ ÷ + ÷ ÷
÷
=
D
S E S E
D
W
S E S E
D
D
S E S E
D
D D
D
W
AP
q
AP
q
We now try to relate this to
FCFS
q
W by replacing the first expression using the
relation:
( ) ( )
( )
( ) ( ) ( ) [ ]
( )
( ) ( )
( ) µ ì
µ ì µ ì µ µ
µ ì
µ µ ì ì
µ
µ µ
÷
+ + +
=
÷
+ +
=
÷
+
=
1
1
1
2 2 1 1 1 2
2
2
2
1
2 2 1 1 2 1
2 2 1 1
S E S E
S E S E
S E S E
W
FCFS
q
so
( ) ( )
( )
( )
( )
( ) ( ) ( ) ( ) [ ]
( ) µ ì
µ µ ì µ µ ì
µ ì
µ µ µ µ µ µ
µ ì
µ ì µ ì
÷
÷ + ÷
+
÷
÷ +
÷
(
¸
(

¸

÷
+
÷ =
1
1 1
1
4
1
1 1 2 2 2 2 1 1
2 1 2 1 2 1 2 2 1 1 1 2
D
S E S E
D
S E S E
W W
FCFS
q
AP
q
112
therefore
( )
( )( ) ( )( ) { }
( )
( )
.
1
4
1 1
1
1
2 1 2 1 2 1
2 1 1 2 1 2 2 1
µ ì
µ µ µ µ µ µ
µ µ ì µ µ ì
µ ì
÷
÷ +
÷
÷ ÷ + ÷ ÷
÷
= ÷
D
D S E D S E
D
W W
FCFS
q
AP
q
Term within braces is ( ) ( ) ( ) ( )
2 1 1 2 1 2 2 1
1 1 S E D S E D ì µ µ ì µ µ ÷ ÷ + ÷ ÷ .
Use ( )
1 2 1
2 1 1 µ µ µ ÷ = ÷ ÷ D
and ( )
2 1 2
2 1 1 µ µ µ ÷ = ÷ ÷ D
to write above as ( ) ( ) ( ) ( )
2 1 1
2
2 1 2 2
2
1
2 1 2 1 S E S E ì µ µ ì µ µ ÷ + ÷
and substitute
( )
i
i
i
S E
µ
ì =
to get ( )
( )
( )
( )
( )
( )
= ÷ + ÷
1
2
1 1
2
2
2
1
2 2
2
1
2 1 2 1
S E
S E
S E
S E
µ µ µ µ µ µ
( )
( )
( )
( )
( )
( )
(
¸
(

¸

÷ + ÷
1
2
1 2
2
1
2 1 2 1
2 1 2 1
S E
S E
S E
S E
µ µ µ µ µ µ .
The final result is:
( )
( )
( )
( )
( )
( )
( )
( ) . 4 2 1 2 1
1
2 1 2 1
1
2
1 2
2
1
2 1
2 1
(
¸
(

¸

÷ + ÷ ÷ + ÷
×
÷
= ÷
µ µ µ µ µ µ µ µ
µ ì
µ µ
S E
S E
S E
S E
D
W W
FCFS
q
AP
q
(6)
We can now ask when the expression within brackets is negative.
113
If we let
( )
( )
2
1
S E
S E
Q = , then we have an expression ( ) ( ) B A
Q
B
AQ Q f + ÷ + = then
we can re-write (6) as
( ) Q f
C
W W
FCFS
q
AP
q
ì
= ÷ (7)
where ( )
2 1
2 1 µ µ ÷ = A , ( )
1 2
2 1 µ µ ÷ = B and ( ) [ ] µ µ µ ÷ = 1
2 1
D C .
It is well known that the minimum value of
Q
B
AQ + equals AB 2 if
2 1 <
i
µ . So ( ) Q f has minimum value
( ) ( ) [ ] ( )
2
2 2 B A AB B A B A AB ÷ ÷ = ÷ + ÷ = + ÷ .
Observation:
( )
( ) ( ) [ ]
2
1 2 2 1
2 1
2 1 2 1
1
µ µ µ µ
µ ì
µ µ
÷ ÷ ÷
÷
÷ > ÷
D
W W
FCFS
q
AP
q
if both 2 1 <
i
µ .
So, as long as both 2 1 <
i
µ , we have a bound on how much better
AP
q
W can do
as compared to
FCFS
q
W . From this analysis, it is clear that ( ) 0 Min < Q f if
B A = .
Also, if 1 = Q then clearly ( ) 0 = Q f . Since ( ) Q f is U-shaped, we know that
there is another root with 1 < Q and ( ) 0 = Q f as illustrated in Figure 5-1.
If A B < , the roots are
A
B
Q = and 1 with 1 < <
A
B
A
B
if A B < < 0 .
114
A B < if
Q
A B/
( ) Q f
A B/
1
0
2 / 1 ,
2 1
< µ µ
Figure 5-1. Roots and minimum for ( ) Q f when 2 1 ,
2 1
< µ µ .
We now address the case where the condition 2 1 <
i
µ does not hold. The
stability of the queueing system requires that 0 1
2 1
> ÷ ÷ µ µ or 1
2 1
< + µ µ .
Thus, 2 1
1
> µ forces 2 1
2
< µ .
Since ( ) 0 1 = f in all cases, from (7) we see that
FCFS
q
AP
q
W W = for 1 = Q , so
1 = Q is a root for the function f . Since
1 2
2 1 µ µ < < implies that 0 > A and
0 < B , ( ) 0
2
> ÷ = '
Q
B
A Q f for all values of Q. So f is strictly increasing over
[ ) · , 0 and 1 = Q is the only root. As Figure 5-2 shows, this implies that
FCFS
q
AP
q
W W < if 1 < Q
and
FCFS
q
AP
q
W W > if 1 > Q .
115
1 2
2 / 1 , 0 µ µ < < < < A B
Q
( ) Q f
1
0
AQ B A + +
B A+
( ) ( ) B A Q B AQ Q f + ÷ + = /
Figure 5-2. Single root of ( ) Q f when 2 1
1
> µ .
We summarize the preceding discussion in the form of a theorem.
Theorem 1 Consider the two-class single server system with zero setups,
exponential service times, and Poisson arrivals. Let the average wait times for the
AP and FCFS be denoted as
AP
q
W and
FCFS
q
W and set
FCFS
q
AP
q q
W W W ÷ = A .
Then ( ) Q f
C
W
q
ì
= A
where ( ) ( ) B A
Q
B
AQ Q f + ÷ + =
( )
2 1
2 1 µ µ ÷ = A , ( )
1 2
2 1 µ µ ÷ = B , ( ) [ ] µ µ µ ÷ = 1
2 1
D C
( )( )
2 1 2 1
1 1 µ µ µ µ + ÷ ÷ = D ,
( )
( )
2
1
S E
S E
Q = , and assuming
2 1
ì ì > .
If 2 1
1
> µ , then ( ) Q f is strictly increasing and has a single root at 1 = Q .
So 0 < A
q
W if 1 < Q and 0 > A
q
W if 1 > Q .
116
If 2 1
1 2
< < µ µ , so that both
i
µ ’s are less than ½, then ( ) Q f is U-shaped
and has two roots at
A
B
Q = and 1, so that
0 < A
q
W if 1 < < Q
A
B
0 > A
q
W if
A
B
Q s or 1 > Q .
Theorem 1 applies to exponential service. We now extend it for use with non-
exponential service. Previously, we used the relationship between the moments,
( ) ( )
2 2
i i
S kE S E = , with 2 = k for the exponential case. We know that 1 = k for
constant service times. We note how k is related to the coefficient of variation:
( )
( )
( )
( )
2
2
2 2
2
2
1
i
S
i
i
i
i
C
S E
S E
S E
S E
k + =
+
= =
o
.
Then using k ,
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
)
`
¹
¹
´
¦
÷
÷ +
+
÷
÷ +
÷
+
)
`
¹
¹
´
¦
÷
+
÷
=
2
1 1
2
2 2
2
2
2
1
1
2 2
2
1 1
2
2
2
1
2
2
2
1
2
1
1
1
1
1
1
1
2
1 1 2
µ
µ µ ì µ µ
µ
µ µ ì µ µ
µ ì
µ ì
µ
µ ì
µ
S E S E
D
k
k
W
AP
q
and
( ) ( )
µ
µ µ
÷
+
=
1 2
2 2 1 1
S E S E k
W
FCFS
q
.
117
So we have introduced a new factor, 2 k , and therefore know the maximum
benefit received by AP in an exponential service environment will be twice that of
a constant service environment.
The preceding theorem summarizes the two types of behavior exhibited by the
difference
q
W A . We now proceed to map the regions where either AP or FCFS is
superior in the full parameter space of the problem.
Consider any system with parameters
( ) ( ) ( )
2 1 2 1 2 1
, , , , , µ µ ì ì S E S E
when ( )
i i i
S E ì µ = . We define a reference system with parameters
( )
2 1 2 1
, , 1 , , , ì ì ì ì Q Q
where ( ) 1
2
= S E ,
( )
( )
2
1
S E
S E
Q =
and with no loss of generality, assume that
2 1
ì ì > .
It is clear that we can convert any system to the reference system by a simple
re-indexing (if necessary) and re-scaling. Stated otherwise, from the arbitrary
system ( ) ( ) ( )
2 1 2 1 2 1
, ,
~
,
~
,
~
,
~
µ µ ì ì S E S E
we get
( )
( )
|
|
.
|

\
|
2 1
2
1
2 1
, , 1 ,
~
~
, , µ µ ì ì
S E
S E
118
by defining ( )
i i i
S E ì ì
~
= . Note that in such a re-scaling, the
i
µ ’s remain
invariant so the expression for
q
W A changes by the scaling factor alone, that is:
( )
( )
( ) ( )
q q
W S E Q f
S CE
Q f
C
W A = = = A
1
1
~
~
~
ì ì
.
This shows that it is sufficient to map the behavior of the reference system as long
as
q
W A is of interest.
Consider the system with ( )
2 1 2 1
, , 1 , , , µ µ ì ì Q
where Q
1 1
ì µ = and
2 2
ì µ = .
The stability conditions are 1
1 1
< = Q ì µ , 1
2 2
< = ì µ
and 1
2 1 2 1
< + = + ì ì µ µ Q . (8)
We also assume that
2 1
ì ì > . (9)
We consider four cases as listed below. The first three correspond to 2 1
2
< ì
and the last one to 2 1
2
> ì . We discuss each case briefly and then summarize
the results in Table 5-1 and Table 5-2.
Case 1: 2 1
2
< ì
1 2
2 1 µ µ < <
2: 2 1
2
< ì 2 1
1 2
< s µ µ
3: 2 1
2
< ì 2 1
2 1
< < µ µ
4: 2 1
2
> ì
2 1
2 1 µ µ s <
119
Case 1: The stability conditions and the
2 1
ì ì > requirement define the relevant
region as
Q Q
2
1 2
1
2
1
, max
ì
ì ì
÷
< s
)
`
¹
¹
´
¦
with 2 1
2
< ì . (10)
In this case
1 2
2 1 µ µ < < implies that 0 > A and 0 < B , so ( ) Q f as defined in
Theorem 1 is increasing for 0 > Q and has a single root at 1 = Q . So
0 s A
q
W if 1 s Q
and 0 > A
q
W if 1 > Q .
Case 2: 2 1
1 2
< s µ µ . The region is defined by
Q Q 2
1
, max
1
2
2
< s
)
`
¹
¹
´
¦
ì
ì
ì with 2 1
2
< ì . (11)
Since A B s < 0 in this region, ( ) Q f has two roots, at A B Q = and 1 = Q , so
0 < A
q
W if Q lies between these two roots. We need to express the condition
1 < < Q
A
B
as a condition on
1
ì .
( )
( )
Q
A
B
<
÷
÷
=
2 1
1 2
2 1
2 1
µ µ
µ µ
means Q
|
|
.
|

\
| ÷
<
|
|
.
|

\
|
÷
2
2
1
2 1
2
1
ì
ì
µ
or ( )Q
Q
2 2
1
1
2
1
÷ + <
÷
ì
ì
.
So the condition is
( ) [ ] Q Q 2 2
1
1
2
1
÷ +
>
÷
ì
ì with 1 < Q (12)
120
given 2 1
2
< ì .
Note that the right-hand-side is decreasing in Q for 0 > Q , and that its value for
1 = Q equals
2
ì . Since
2 1
ì ì > at all times, the range of validity of this condition
is up to 1 = Q .
Case 3: 2 1
2 1
< < µ µ . The region requires
Q
2
1 2
ì
ì ì < s for 2 1
2
< ì . (13)
This immediately implies that 1 s Q . While ( ) Q f has two roots at 1 = Q and
1 > = A B Q , the latter root does not fall into this region, so we conclude that
0 > A
q
W for 1 0 s < Q .
Case 4:
2 1
2 1 µ µ s < . The region is defined by
Q
2
1 2
1 ì
ì ì
÷
< s for 2 1
2
> ì . (14)
The relation (14) forces
2
2
1
ì
ì ÷
< Q and since
2
ì satisfies 1 2 1
2
s s ì , Q must
satisfy 1 0 < < Q . Since 0 s A and 0 > B in this region, ( ) Q f is strictly
decreasing over ( ) 1 , 0 and ( ) 0 1 = f . So, in this region, we always have:
0 > A
q
W for 1 0 s < Q .
The four cases are summarized in Table 5-1 for the region 1 0 s < Q and in Table
5-2 for Q < 1 .
121
Case Region for 1 0 s < Q
q
W A
(1)
1 2
2 1 µ µ < < Q Q
2
1
1
2
1 ì
ì
÷
< s
2 1
2
< ì
0 s A
q
W
(2)
2 1
1 2
s < µ µ Q Q 2
1
1
2
< s ì
ì
2 1
2
< ì
0 < A
q
W if
( ) [ ] Q Q 2 2
1
1
2
1
÷ +
>
÷
ì
ì
(3)
2 1
2 1
< < µ µ Q
2
1 2
ì
ì ì < s
2 1
2
< ì
0 > A
q
W
(4)
2 1
2 1 µ µ s < Q
2
1 2
1 ì
ì ì
÷
< s
2 1
2
> ì
0 > A
q
W
Table 5-1. Four cases defining the parameter space for 1 0 s < Q .
Case Region for 1 > Q
q
W A
(1)
1 2
2 1 µ µ < < Q Q
2
1 2
1
2
1
, max
ì
ì ì
÷
< s
|
|
.
|

\
|
2 1
2
< ì
0 > A
q
W
(2)
2 1
1 2
s < µ µ
Q 2
1
1 2
< s ì ì
2 1
2
< ì
0 > A
q
W
(3)
2 1
2 1
< < µ µ N/A
(4)
2 1
2 1 µ µ s < N/A
Table 5-2. Four cases defining the parameter space for 1 > Q .
122
Focusing on the sign of
q
W A , we can state the results in the following form.
Theorem 2 For any system with parameters ( ) ( ) ( )
2 1 2 1 2 1
, , , , , µ µ ì ì S E S E with
the conventions
2 1
ì ì > and ( ) 1
2
= S E , the Alternating Priority policy is superior
to FCFS if and only if
1 2
2 1 µ µ < < and 1 0 s < Q
or 2 1
1 2
s < µ µ , 1 0 s < Q and
( ) [ ] Q Q 2 2
1
1
2
1
÷ +
>
÷
ì
ì
where ( ) ( )
2 1
S E S E Q = .
We now illustrate the relevant regions for representative values of the
parameter
2
ì . We start with the choice 4 1
2
= ì . The stability condition is
Q Q 4
3 1
2
1
=
÷
<
ì
ì , so
1
ì must lie below the graph for
Q
y
4
3
= in the
1
ì versus
Q-space. The condition 4 1
2 1
= > ì ì must also be satisfied at all times. The
region of superiority of AP is given by 0 < A
q
W and corresponds to
Q Q 4
3
2
1
1
< < ì for 1 0 s < Q .
For 1 > Q , the region
Q Q 4
3
2
1
1
< < ì is where 0 > A
q
W until Q reaches 3 where
the constraint 4 1
1
> ì becomes binding.
For Case 2, the relevant region is defined by
Q Q 2
1
4
1
1
< < ì for 1 0 s < Q
123
and
Q 2
1
4
1
1
< < ì with 2 1 s < Q .
The condition for 0 < A
q
W is
( ) 1 2
1
1
+
>
Q Q
ì for 1 0 < < Q .
The relevant regions are illustrated in Figure 5-3. Moving on to Figure 5-4,
the regions are shown for 10 . 0
2
= ì . We see that the regions corresponding to
Cases 1 and 2 for 1 < Q have both widened. Conversely, in Figure 5-5, when
2
ì
increases to 0.4, we see that these regions have narrowed compared to the
4 1
2
= ì case. This behavior remains in effect as long as 2 1
2
< ì .
Now consider the scenario when 2 1
2
> ì . When
2
ì exceeds 2 1 , only Case 4
applies and the region is defined by
Q
2
1 2
1 ì
ì ì
÷
< s with
2
2
1
ì
ì ÷
< Q .
For 6 . 0
2
= ì , for example, we have
Q Q 5
2 4 . 0
6 . 0
1
= < s ì with
3
2
6 . 0
4 . 0
= < Q
so the only relevant region lies between the horizontal line at 6 . 0 and the curve
Q 5
2
as shown in Figure 5-6. Within this region 0 > A
q
W and outside this region,
the system is unstable.
124
Figure 5-3. Graph of
1
ì versus Q when 25 . 0
2
= ì . AP and FC indicates
superiority in that region.
ì
2
=1/4
Unstable
µ
1
=1
3/(4Q)
ì
1
Q
1/(2Q)
FC
FC
FC
AP
AP
1/(4Q)
ì1=0.25
1/2Q(Q+1)
FC
125
Figure 5-4. Graph of
1
ì versus Q when 10 . 0
2
= ì .
ì
2
=0.1
Unstable
µ
1
=1
0.9/Q
ì
1
Q
1/(2Q) FC
FC
FC
AP
AP
1/(10Q)
ì1=0.10
1/2Q(4Q+1)
FC
126
Figure 5-5. Graph of
1
ì versus Q when 40 . 0
2
= ì .
ì
2
=.4
Unstable
µ
1
=1
0.6/Q
ì
1
Q
1/(2Q)
FC
FC
FC
AP
AP
2/(5Q)
ì1=0.40
1/2Q(0.25Q+1)
FC
127
Figure 5-6. Graph of
1
ì versus Q when 60 . 0
2
= ì .
5.1.2 Baseline numerical comparisons. We choose two of the preceding
2
ì
settings for our zero-setup baseline, 25 . 0
2
= ì and 60 . 0
2
= ì . Figure 5-7
contains a matrix of discrete values at equal 0.05 intervals of
1
ì and Q where the
numerical value at each location is ( ) 100 * 100 *
q
FCFS
q
AP
q
W W W A = ÷ as defined
in Section 5.1.1. Figure 5-7 therefore resembles Figure 5-3. We label and
italicize the cells that unstable due to
1
µ saturation, “R1,” the cells that are
unstable due to the sum of the
i
µ ’s as “RS,” and cells that violate
2 1
ì ì > , “LV”.
We assist the reader by adding a light shade to the 0 < A
q
W region and darker
shading to the 0 > A
q
W region. We leave the region of 0 = A
q
W un-shaded (for
Q
ì
2
=.6
Unstable
µ
1
=1
2/(5Q)
ì
1
1/(2Q)
0.6/Q
ì1=0.60
1/2Q(-8Q+1)
FC
Q=2/3
128
example at 1 = Q ). The reader will note that although not coincident with our
specific measurement points and therefore not shown without shade, the transition
from 0 < A
q
W to 0 > A
q
W includes the 0 = A
q
W curve. This is not true when
transitioning to a zone of instability or
2 1
ì ì > violation.
We remind the reader that we have assumed ( ) 1
2
= S E , so that ( ) Q S E =
1
and
therefore the expected service time equals ( )
2 1
2 1
ì ì
ì ì
+
+
=
Q
S E . For Figure 5-7,
25 . 0
2
= ì , so ( )
25 . 0
25 . 0
1
1
+
+
=
ì
ì Q
S E . The actual wait difference is useful because
the four largest differences that favor AP in Figure 5-7 are less than 1.5 and all
four occur when 95 . 0 > µ (not shown). AP, therefore, has little positive impact
in the absence of setup when 2 1
2
< ì . If 1 > Q then AP can be significantly
worse than FCFS, but only when
1
ì approaches ( ) Q 4 3 .
For Figure 5-8, 60 . 0
2
= ì , so 60 . 0
2
= µ and ( )
60 . 0
60 . 0
1
1
+
+
=
ì
ì Q
S E . We
simplified Figure 5-8 by trimming off a majority of the unavailable space:
where
2 1
ì ì < and for this case ( ) 2 1
2
> ì where 1 > Q . The load offered by
each class in the absence of setup is
i
µ . AP is biased towards the class that
provides the majority of the load (we will call this the dominant class. Since AP
will not changeover until the current queue is exhausted there is a greater
likelihood that a dominant class arrival will occur continuing the work session
129
than when working on the lesser class. Continuing work on the dominant class is
done at the expense of the other class. The net result for the 2 1
2
> ì case is
higher wait times when using AP where the feasible area for this case starts with
60 . 0 > µ . We will see in Figures 5-9 and 5-10 that AP does require fewer
changeovers as compared to FCFS, but the tradeoff is not always beneficial to the
overall system flow time, especially when there is no setup time at stake.
130
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
5
8
3
1
3
1
L
V
L
V
L
V
L
V
L
V
2
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
3
8
0
1
0
5
L
V
L
V
L
V
L
V
L
V
1.95
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
2
6
4
8
4
.
5
L
V
L
V
L
V
L
V
L
V
1.9
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
2
7
6
9
1
9
1
6
8
.
1
L
V
L
V
L
V
L
V
L
V
1.85
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
7
9
7
1
4
1
5
4
.
8
L
V
L
V
L
V
L
V
L
V
1.8
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
4
1
1
1
0
6
4
4
L
V
L
V
L
V
L
V
L
V
1.75
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
2
5
1
8
0
.
5
3
5
.
2
L
V
L
V
L
V
L
V
L
V
1.7
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
2
6
9
9
1
6
5
6
1
.
4
2
8
.
1
L
V
L
V
L
V
L
V
L
V
1.65
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
5
6
5
1
1
4
4
6
.
8
2
2
.
2
L
V
L
V
L
V
L
V
L
V
1.6
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
2
6
8
8
0
3
5
.
7
1
7
.
4
L
V
L
V
L
V
L
V
L
V
1.55
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
R
S
1
5
4
5
7
2
7
1
3
.
5
L
V
L
V
L
V
L
V
L
V
1.5
R
1
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
4
9
4
9
6
.
1
4
0
.
8
2
0
.
3
1
0
.
4
L
V
L
V
L
V
L
V
L
V
1.45
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
R
S
1
9
7
6
2
.
3
2
9
.
1
1
5
7
.
7
7
L
V
L
V
L
V
L
V
L
V
1.4
R
1
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
1
3
1
8
1
0
2
4
1
2
0
.
5
1
0
.
9
5
.
7
1
L
V
L
V
L
V
L
V
L
V
1.35
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
R
S
2
1
5
5
8
.
5
2
7
1
4
.
3
7
.
7
6
4
.
0
6
L
V
L
V
L
V
L
V
L
V
1.3
R
1
R
1
R
1
R
1
R
1
R
S
R
S
R
S
R
S
8
8
.
6
3
4
.
5
1
7
.
5
9
.
6
2
5
.
3
2
2
.
7
7
L
V
L
V
L
V
L
V
L
V
1.25
R
1
R
1
R
1
R
1
R
S
R
S
R
S
R
S
1
6
4
4
3
.
3
2
0
.
2
1
1
6
.
2
1
3
.
4
7
1
.
7
9
L
V
L
V
L
V
L
V
L
V
1.2
R
1
R
1
R
1
R
S
R
S
R
S
R
S
1
5
9
4
5
3
.
6
2
1
.
8
1
1
.
4
6
.
4
7
3
.
7
4
2
.
1
1
.
0
6
L
V
L
V
L
V
L
V
L
V
1.15
R
1
R
1
R
S
R
S
R
S
R
S
R
S
6
5
.
8
2
0
.
7
1
0
.
2
5
.
7
5
3
.
3
8
1
.
9
9
1
.
1
1
0
.
5
4
L
V
L
V
L
V
L
V
L
V
1.1
R
1
R
S
R
S
R
S
R
S
R
S
8
0
.
8
1
4
.
7
6
.
6
9
3
.
6
9
2
.
1
9
1
.
3
2
0
.
7
8
0
.
4
3
0
.
1
9
L
V
L
V
L
V
L
V
L
V
1.05
R
1
R
S
R
S
R
S
R
S
R
S
0 0 0 0 0 0 0 0 0
L
V
L
V
L
V
L
V
L
V
1
R
S
R
S
R
S
R
S
R
S
-
2
8
-
1
0
-
5
.
4
-
3
.
2
-
2
-
1
.
3
-
0
.
8
-
0
.
4
-
0
.
2
-
0
.
1
L
V
L
V
L
V
L
V
L
V
0.95
R
S
R
S
R
S
R
S
-
7
1
-
2
4
-
1
2
-
7
.
3
-
4
.
6
-
3
-
1
.
9
-
1
.
1
-
0
.
6
-
0
.
3
-
0
L
V
L
V
L
V
L
V
L
V
0.9
R
S
R
S
R
S
-
1
1
3
-
3
8
-
2
0
-
1
2
-
7
.
6
-
4
.
9
-
3
.
2
-
2
.
1
-
1
.
2
-
0
.
6
-
0
.
2
0
.
1
L
V
L
V
L
V
L
V
L
V
0.85
R
S
R
S
-
1
3
3
-
4
9
-
2
6
-
1
6
-
1
0
-
7
-
4
.
7
-
3
.
1
-
1
.
9
-
1
.
1
-
0
.
5
-
0
0
.
2
9
L
V
L
V
L
V
L
V
L
V
0.8
R
S
-
1
2
5
-
5
4
-
3
1
-
1
9
-
1
3
-
8
.
8
-
6
-
4
-
2
.
6
-
1
.
6
-
0
.
8
-
0
.
2
0
.
2
4
0
.
5
4
L
V
L
V
L
V
L
V
L
V
0.75
-
1
0
4
-
5
3
-
3
2
-
2
1
-
1
4
-
1
0
-
7
-
4
.
8
-
3
.
2
-
2
-
1
-
0
.
3
0
.
2
0
.
5
7
0
.
8
4
L
V
L
V
L
V
L
V
L
V
0.7
-
4
7
-
3
1
-
2
1
-
1
5
-
1
1
-
7
.
5
-
5
.
2
-
3
.
5
-
2
.
2
-
1
.
2
-
0
.
4
0
.
1
8
0
.
6
3
0
.
9
6
1
.
1
8
L
V
L
V
L
V
L
V
L
V
0.65
-
2
7
-
1
9
-
1
4
-
1
0
-
7
.
4
-
5
.
2
-
3
.
6
-
2
.
2
-
1
.
2
-
0
.
4
0
.
2
4
0
.
7
4
1
.
1
2
1
.
3
7
1
.
5
3
L
V
L
V
L
V
L
V
L
V
0.6
-
1
7
-
1
2
-
9
.
2
-
6
.
7
-
4
.
7
-
3
.
2
-
2
-
1
-
0
.
2
0
.
4
5
0
.
9
5
1
.
3
4
1
.
6
2
1
.
8
2
1
.
9
2
L
V
L
V
L
V
L
V
L
V
0.55
-
1
0
-
7
.
5
-
5
.
4
-
3
.
8
-
2
.
5
-
1
.
4
-
0
.
5
0
.
2
2
0
.
8
2
1
.
2
9
1
.
6
7
1
.
9
5
2
.
1
5
2
.
2
7
2
.
3
1
L
V
L
V
L
V
L
V
L
V
0.5
-
5
.
3
-
3
.
8
-
2
.
5
-
1
.
4
-
0
.
5
0
.
2
3
0
.
8
5
1
.
3
6
1
.
7
8
2
.
1
2
2
.
3
8
2
.
5
6
2
.
6
9
2
.
7
4
2
.
7
2
L
V
L
V
L
V
L
V
L
V
0.45
-
1
.
9
-
0
.
9
-
0
.
1
0
.
5
8
1
.
1
7
1
.
6
7
2
.
0
8
2
.
4
3
2
.
7
1
2
.
9
2
3
.
0
8
3
.
1
8
3
.
2
2
3
.
2
1
3
.
1
3
L
V
L
V
L
V
L
V
L
V
0.4
0
.
7
7
1
.
3
5
1
.
8
4
2
.
2
7
2
.
6
4
2
.
9
4
3
.
1
9
3
.
4
3
.
5
6
3
.
6
8
3
.
7
5
3
.
7
7
3
.
7
5
3
.
6
7
3
.
5
3
L
V
L
V
L
V
L
V
L
V
0.35
2
.
8
5
3
.
1
8
3
.
4
5
3
.
7
3
.
9
4
.
0
6
4
.
2
4
.
3
4
.
3
7
4
.
3
9
4
.
3
9
4
.
3
4
4
.
2
6
4
.
1
3
3
.
9
3
L
V
L
V
L
V
L
V
L
V
0.3
4
.
5
4
.
6
6
4
.
8
4
.
9
4
.
9
9
5
.
0
6
5
.
1
5
.
1
1
5
.
1
5
.
0
7
5
4
.
8
9
4
.
7
5
4
.
5
6
4
.
3
2
L
V
L
V
L
V
L
V
L
V
0.25
5
.
8
2
5
.
8
7
5
.
9
5
.
9
3
5
.
9
3
5
.
9
2
5
.
9
5
.
8
5
5
.
7
8
5
.
6
9
5
.
5
7
5
.
4
1
5
.
2
3
4
.
9
9
4
.
7
L
V
L
V
L
V
L
V
L
V
0.2
6
.
8
7
6
.
8
6
6
.
8
3
6
.
7
9
6
.
7
4
6
.
6
8
6
.
6
6
.
5
6
.
4
6
.
2
5
6
.
1
5
.
9
1
5
.
6
7
5
.
4
5
.
0
6
L
V
L
V
L
V
L
V
L
V
0.15
7
.
7
1
7
.
6
5
7
.
5
9
7
.
5
1
7
.
4
2
7
.
3
2
7
.
2
2
7
.
0
8
6
.
9
4
6
.
7
7
6
.
5
7
6
.
3
5
6
.
1
5
.
7
9
5
.
4
1
L
V
L
V
L
V
L
V
L
V
0.1
8
.
3
8
8
.
2
9
8
.
2
8
.
1
8
7
.
8
7
7
.
7
4
7
.
5
9
7
.
4
2
7
.
2
3
7
.
0
1
6
.
7
7
6
.
4
8
6
.
1
4
5
.
7
4
L
V
L
V
L
V
L
V
L
V
0.05
1
0
.
9
5
0
.
9
0
.
8
5
0
.
8
0
.
7
5
0
.
7
0
.
6
5
0
.
6
0
.
5
5
0
.
5
0
.
4
5
0
.
4
0
.
3
5
0
.
3
0
.
2
5
0
.
2
0
.
1
5
0
.
1
0
.
0
5
Figure 5-7. Wait time differences (AP-FCFS)*100 when setup is zero and
25 . 0
2
= ì .
Q
ì
1
131
1
134 129 126 125 130 147 212 RS RS RS RS RS RS RS RS RS RS RS RS R1
0.95
132 126 123 121 124 135 170 404 RS RS RS RS RS RS RS RS RS RS RS RS
0.9
129 123 120 117 118 125 145 225 RS RS RS RS RS RS RS RS RS RS RS RS
0.85
126 120 116 114 113 116 128 165 407 RS RS RS RS RS RS RS RS RS RS RS
0.8
123 117 113 110 108 109 116 134 200 RS RS RS RS RS RS RS RS RS RS RS
0.75
119 114 109 106 103 103 106 115 141 263 RS RS RS RS RS RS RS RS RS RS
0.7
115 110 106 102 99 97 98 101 113 148 366 RS RS RS RS RS RS RS RS RS
0.65
111 106 102 97 94 92 90 91 96 108 147 467 RS RS RS RS RS RS RS RS
0.6
LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
3
0
.
3
5
0
.
4
0
.
4
5
0
.
5
0
.
5
5
0
.
6
0
.
6
5
0
.
7
0
.
7
5
0
.
8
0
.
8
5
0
.
9
0
.
9
5
1
Figure 5-8. Wait time differences (AP-FCFS)*100 when setup is zero and
60 . 0
2
= ì .
5.2 Non-Zero Setup
In this section, we introduce a nonzero setup into the comparison of the two
dispatching rules AP and FCFS. We are no longer able to use P-K formula for the
FCFS wait because it assumes independence in the processing times and we know
that the setup times are correlated to the service times by the customer class. The
solution given by Gaver (1963) allows for the processing time correlation by
class. To provide a baseline for comparison, we use the results of the last section
to report measured differences in wait time as well as differences in the incidence
of part changeovers (number of switches). The introduction of setup starts at a
low level. The magnitude of the setup is then increased until it equals the batch
service time, a level that is consistent with our simulation studies in Chapters 3
and 4.
5.2.2 FCFS versus AP in the Non-Zero Setup Environment. We continue
with the comparison started in section 5.1 comparing AP to FCFS now with non-
zero setup.
Q
ì
1
132
The inputs to both FCFS and AP flow time calculations are the same:
• Two streams of Poisson arrivals with mean arrival rates 2 , 1 = i
i
ì ,
ì ì ì = +
2 1
,
ì
ì
i
i
a =
• Distribution function of the service time of a type- i customer: ( ) t F
i
S
,
first moment: ( )
i
S E , second moment: ( )
2
i
S E . Laplace-Stieltjes
transform of distributions: ( ) ( )
}
·
÷
=
0
d t F e z
i
S
zt
i
¸ (15)
Note: If the service time is exponential then ( )
( )
i
i
S zE
z
+
=
1
1
¸ (16)
• Distribution function of the setup time of a type- i customer: ( ) t F
i
U
,
first moment: ( )
i
U E , second moment: ( )
2
i
U E . Laplace-Stieltjes
transform of distributions: ( ) ( )
}
·
÷
=
0
d t F e z
i
U
zt
i
k (17)
Note: If the setup time is exponential then ( )
( )
i
i
U zE
z
+
=
1
1
k (18)
The FCFS wait time (wait in queue prior to setup or service) of Gaver (1963)
is based on a Markov process with a simple integro-differential forward
Kolmogorov equation. The waiting time of a random arrival at t , ( ) t W , depends
on the class of the last service which will determine whether or not a setup is
required. If the arrival is of the same class then there is no setup required,
otherwise a setup must occur prior to service. The joint probabilities result:
( ) ( ) { x t W P t x F s = ,
1
, last demand prior to t in class } 1
133
and
( ) ( ) { x t W P t x F s = ,
2
, last demand prior to t in class } 2
Under suitable conditions, the functions ( ) t x F
i
, will have a limit as · ÷ t . If
we denote the limiting functions by ( ) x F
i
, the Laplace-Stieltjes transforms by
( ) s f
i
, then we have
( )
( ) ( ) { } ( ) ( ) ( ) [ ]
( ) s D
s s F s s F s
s f
1 1 1 2 2 2 1
1
0 0 ¸ k ì ¸ ì ì ÷ + ÷
= (19a)
and
( )
( ) ( ) { } ( ) ( ) ( ) [ ]
( ) s D
s s F s s F s
s f
2 2 2 1 1 1 2
2
0 0 ¸ k ì ¸ ì ì ÷ + ÷
= (19b)
where
( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) s s s s s s s s s D
2 2 1 1 2 1 2 2 1 1
¸ k ¸ k ì ì ¸ ì ì ¸ ì ì ÷ + ÷ + ÷ = (20)
and ( ) ( ) ( ) 1 lim
2 1
0
= +
÷
s f s f
s
. (21)
By taking the limit of (19) we note the probabilities, ( ) x F
1
and ( ) x F
2
, are related
by:
( ) ( ) U F F ÷ = + 1 0 0
2 1
(22)
where ( ) ( ) { }
(
¸
(

¸

+ + =
2 1
2
2 1
U E U E U
ì
ì ì
ì µ and ( ) ( )
2 2 1 1
S E S E ì ì µ + = .
This is exactly the same utilization measure obtained using conditional
probabilities as outlined in Section 2.2.1.
134
The expected wait is prior to setup or service is
( ) ( ) W E W E
2 1
+ (23)
where ( ) ( )
( )
ds
s df
W E
s
1
0
1
1 lim ÷ =
÷
and ( ) ( )
( )
ds
s df
W E
s
2
0
2
1 lim ÷ =
÷
and
( )
( ) ( ) { } ( ) ( )
( )
( )
( ) ( ) [ ( ) ( ) ( ) ( ) {
( ) ( ) ( ) ( ) ( ) ( )]
( ) ( ) ( ) ( ) } 2 2 2
2 2
4 4 2
1 2
1
0 1 0
2 2 1 1
2
2 2
2
1 1
2
2
2
1 1 2 2 1
2 2 1 1 2 1 2 1
2
1
1 1 1 2 2 2 1
1
÷ + + + +
+ + + +
+ + ÷ ×
÷
+
÷
+ + ÷
=
S E S E S E S E
U E U E S E U E S E U E
S E U E S E U E S E S E
U
U
S U E F S E F
W E
ì ì ìì ìì
ì ì
ì
ì
ì
ì ì
(24)
( )
( ) ( ) { } ( ) ( )
( )
( )
( ) ( ) [ ( ) ( ) ( ) ( ) {
( ) ( ) ( ) ( ) ( ) ( )]
( ) ( ) ( ) ( ) } 2 2 2
2 2
4 4 2
1 2
1
0 1 0
2 2 1 1
2
2 2
2
1 1
2
2
2
1 1 2 2 1
2 2 1 1 2 1 2 1
2
2
2 2 2 1 1 1 2
2
÷ + + + +
+ + + +
+ + ÷ ×
÷
+
÷
+ + ÷
=
S E S E S E S E
U E U E S E U E S E U E
S E U E S E U E S E S E
U
U
S U E F S E F
W E
ì ì ìì ìì
ì ì
ì
ì
ì
ì ì
(25)
using ( )
( ) ( ) ( ) { }
( ) { } ( ) ( ) s s s s
s s U
F
1 1 1 2 2
1 1 1
1
1
0
¸ k ì ¸ ì ì
¸ k ì
+ + ÷
÷
= (26)
and from (20) ( ) ( ) 0 1 0
1 2
F U F ÷ ÷ = (27)
Numeric methods are required to solve for the positive real root of ( ) s D which is
required to eliminate the singularity of (19).
The AP system state definition Eisenberg (1967) uses is based on service
completions. For AP, Eisenberg provides an expression for the probability that a
135
service completion by an arbitrary customer is followed by a changeover. To
compute this probability, he uses numerical methods even in the case of zero
setup times. The changeover probability is a result of the AP flow time
calculations by Eisenberg (1967) which we review after the FCFS flow time
calculations by Gaver (1963).
FCFS and AP Flow Time Calculations. Due to the complexity of the
computations we provide the necessary background for the reader to replicate
results. For both FCFS and AP calculations we provide step-by-step details of the
computations leading to the mean flow time. We also include a description of the
imbedded state probabilities for the AP model. The changeover probability is
pointed out after each wait equation is stated.
FCFS Flow Time Calculations.
1 Determine the positive root of ( ) s D . Using Newton-Raphson method:
1.1 Set 9 . 0 = s as the first guess of the root.
1.2 If ( ) c < s D , stop and retain positive root, s . Otherwise compute a new
estimate for the root using ( ) ( ) s D s D s s ' ÷ = .
If exponential setup and service distributions,
( )
( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]
( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]
2 2
2
2
1
1
2
2
1
1
2 1 2 1
2 1
2 1
2 1
2
1 1 1 1
1 1 1 1 1 1
s s
S sE
s
S sE
s
S sE S sE
U sE U sE S sE S sE S sE S sE
s D
+ ÷ +
+
+
+
+
+
÷
+
÷
+ + + +
÷
+ +
=
ì ì
ì ì ìì ìì
ì ì ì ì
(28)
136
and
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
. 2
1 1
2
1 1 1
1 1 1 1 1 1
1 1 1 1 1
2
2
2 2
1
1
1 1
2 2 1 1
2 1 1
2
2 2 1
1
1 2 1
2
2 2 1
1
2
1
1
2
2 1
s
S sE
S sE
S sE
S sE
S E S E
S sE U sE S sE
U E
S sE U sE S sE
U E
S sE U sE U sE
S E
S sE U sE U sE
S E
S sE
S E
S sE
S E
s D
+
+
+ +
+
+ ÷ ÷
+ +
(
¸
(
+ + +
+
+ + +
+
+ + +
+
+ + +

¸

+
+
÷
+
÷ = '
ì
ì
ì
ì ì
ìì ìì
ì ì
(29)
2 Calculate utilization including expected setup:
( ) ( ) { }
(
¸
(

¸

+ + =
2 1
2
2 1
U E U E U
ì
ì ì
ì µ where ( ) ( )
2 2 1 1
S E S E ì ì µ + = (30)
3 ( )
( ) ( ) ( ) { }
( ) { } ( ) ( ) s s s s
s s U
F
1 1 1 2 2
1 1 1
1
1
0
¸ k ì ¸ ì ì
¸ k ì
+ + ÷
÷
= using s from step 1.2 (31)
If exponential setup and service distributions,
( )
( )
( ) ( )
( ) ( ) ( )
1 1
1
2
2
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
S sE U sE S sE
s
S sE U sE
U
F
+
·
+
+
)
`
¹
¹
´
¦
+
+ ÷
)
`
¹
¹
´
¦
+
·
+
÷
=
ì ì ì
ì
(32)
4 ( ) ( ) 0 1 0
1 2
F U F ÷ ÷ = (33)
5 ( ) ( )
( )
ds
s df
W E
s
1
0
1
1 lim ÷ =
÷
137
( ) ( ) { } ( ) ( )
( )
( )
[ {
]
( ) ( ) ( ) ( )}
2 2 1 1
2
2 2
2
1 1
2
2
2
1 2 2 1 1
2 1 2 1 2 1 2 1
2
1
1 1 1 2 2 2 1
2 2 2
) ( ) ( ) ( ) ( 2 ) ( ) ( 2
) ( ) ( 2 ) ( ) ( 2 ) ( ) ( 2
1 2
1
0 1 0
S E S E S E S E
U U E S E U E S E U E
U E S E S E U E U E U E
U
U
S U E F S E F
ì ì ìì ìì
ì ì
ì
ì
ì
ì ì
+ + ÷ + +
+ + + +
+ + ×
÷
+
÷
+ + ÷
=
(34)
6 ( ) ( )
( )
ds
s df
W E
s
2
0
2
1 lim ÷ =
÷
( ) ( ) { } ( ) ( )
( )
( )
[ {
]
( ) ( ) ( ) ( )}
2 2 1 1
2
2 2
2
1 1
2
2
2
1 2 2 1 1
2 1 2 1 2 1 2 1
2
2
2 2 2 1 1 1 2
2 2 2
) ( ) ( ) ( ) ( 2 ) ( ) ( 2
) ( ) ( 2 ) ( ) ( 2 ) ( ) ( 2
1 2
1
0 1 0
S E S E S E S E
U U E S E U E S E U E
U E S E S E U E U E U E
U
U
S U E F S E F
ì ì ìì ìì
ì ì
ì
ì
ì
ì ì
+ + ÷ + +
+ + + +
+ + ×
÷
+
÷
+ + ÷
=
(35)
7 Wait in queue prior to processing (does not include setup) is
( ) ( ) W E W E
2 1
+ (36)
8 Wait in queue prior to service (comparable to AP wait) is
( ) ( ) ( ) ( ) [ ]
2 1
2
2 1
2 1
U E U E W E W E W
FCFS
+ + + =
ì
ì ì
(37)
9 Flow time is
( ) ( ) ( ) ( ) ( ) ( )
(
¸
(

¸

+ +
(
¸
(

¸

+ + + =
1
1
2
2
2
2
1
1
2 1
U E S E U E S E W E W E F
ì
ì
ì
ì
ì
ì
ì
ì
(38)
The probability that an arbitrary customer is followed by a changeover is
2
2 1
2
ì
ì ì
. (39)
138
AP Imbedded Markov State Probabilities. Recalling from Chapter 2,
Eisenberg considers the imbedded Markov process of queue lengths at the instant
of service completion, and includes the class of service just completed. Thus,
state
i
mn
denotes “server is at line i and m customers are waiting at line 1 and n
customers are waiting at line 2.” The imbedded process is described as follows.
• State is ( ) n m i , ; where i is customer type of service just completed, m
and n are numbers of customers present in queues 1 and 2, respectively.
• Equilibrium probability that an arbitrary service completion leaves the
system in state ( ) n m i , ; is
i
mn
t .
Now we define the transition probabilities of the imbedded Markov chain
( ) ( ) [ ]
' ' '
, ; , ; n m i n m i P ÷ . Using equilibrium equations:
( ) ( ) [ ]
¿¿¿
=
·
=
·
=
÷ =
2
1 0 0
' ' ' '
' '
, ; , ;
i m n
i
mn
i
n m
n m i n m i P t t (40)
and normalization condition, ì ì t
i i
m n
i
mn
a = =
¿¿
·
=
·
= 0 0
, (41)
the fraction of all possible states left by customer type- i completions (noting
¿¿¿
=
·
=
·
=
=
2
1 0 0
1
i m n
i
mn
t ). The generating functions of the imbedded state probabilities
are ( )
¿¿
·
=
·
=
÷
0 0
,
m n
n m i
mn
i
v y v y t t . (42)
139
The transition probabilities for the process are defined as =
ij
p prob( i type-1
customers and j type-2 customers arrive during the service time of a type-1
customer)
( ) [ ]( ) [ ]
( )
( )
}
·
+ ÷
=
0
2 1
1
2 1
d ! ! t F e j t i t
S
t i i ì ì
ì ì (43)
• =
ij
q prob( i type-1 customers and j type-2 customers arrive during the
service time of a type-2 customer
( ) [ ]( ) [ ]
( )
( )
}
·
+ ÷
=
0
2 1
2
2 1
d ! ! t F e j t i t
S
t i i ì ì
ì ì (44)
• =
ij
r prob( i type-1 customers and j type-2 customers arrive during the
changeover from 2 to 1)
( ) [ ]( ) [ ]
( )
( )
}
·
+ ÷
=
0
2 1
1
2 1
d ! ! t F e j t i t
U
t i i ì ì
ì ì (45)
• ( ) ÷ v y R , generating function of transition probabilities (of type-1 and type-2
arrivals) during type-1 setup so
( ) ( ) v y v y R
2 2 1 1 1
, ì ì ì ì k ÷ + ÷ =
(46)
• =
ij
h prob( i 1-customers and j 2-customers during changeover from 1 to 2
( ) [ ]( ) [ ]
( )
( )
}
·
+ ÷
=
0
2 1
2
2 1
d ! ! t F e j t i t
U
t i i ì ì
ì ì (47)
• ( ) ÷ v y H , generating function of transition probabilities (of type-1 and type-2
arrivals) during type-2 setup so
( ) ( ) v y v y H
2 2 1 1 2
, ì ì ì ì k ÷ + ÷ = (48)
140
( ) z
i
| is the Laplace-Stieltjes transform of the customer type- i busy period
distribution function in isolation where
( ) ( ) ( ) z z z
i i i i i
| ì ì ¸ | ÷ + = . (49)
Note: If the service time is exponential then
( )
( ) ( )
4 1 1
2
1
2
(
(
¸
(

¸

÷
|
|
.
|

\
|
+ + ÷ + + =
i
i
i
i
i
i
i
z z
z µ
µ
µ
µ
µ
µ
| (50)
Let ÷ g ratio of number of times the system is emptied by completing service
on type-2 customer to type-1 (a constant). We must solve for g because it relates
the limits of the generating functions used in the mean wait equation. These
generating functions are boundary conditions for the states of the system and are
defined as:
( ) ( ) [ ] ( ) [ ] ( )
( ) [ ] ( ) [ ] 1 ,
,
2 2 1 1 2 2 1
2 2 1 1 2 2 1
2
2 2 1
1
÷ ÷ ÷ +
÷ + ÷ ÷ =
v a v v gR
v a v g v v R v
ì ì | ì ì |
ì ì | ì ì | q ì ì | q
(51)
( ) ( ) [ ] ( ) [ ] ( )
( ) [ ] ( ) [ ] 1 ,
,
1 1 2 2 1 1 2
1 1 2 2 1 2 2
1
1 1 2
2
÷ ÷ ÷ +
÷ + ÷ ÷ =
y a y y H
y ga y y y H y g
ì ì | ì ì |
ì ì | ì ì | q ì ì | q
(52)
In solving for g , we also solve for the limiting value of the generating
function ( ) 1
1
q . The limits of the generating functions are related using
( ) ( ) [ ]
2
2
1
1
1 1 a g a ÷ = ÷ q q . (53)
Only one value of g leads to a consistent solution of the functional equations.
We build the functional equations with many different sizes of their arguments by
141
first initiating them with either 0
0
= v or 0
0
= y . We use the fact that 1 lim =
· ÷
i
i
v
which implies ( ) ( ) 1 lim
1 1
q q =
· ÷
i
i
v and therefore they converge regardless of the
starting point. This is only true when g is chosen correctly. The solution is
calculated as follows.
Select two arbitrary values of 2 , 1 : = k g g
k
. Since g is a ratio of incidences,
restrict 0 > g . For each value of g compute two limiting ( ) v
1
q values, ( ) 1
1
q , by
calculating it with two different initial conditions: 0
0
= y and 0
0
= v per the
procedure below and define the result as follows: ( ) ( ) { }
k k
g g v v = = = = I , 0 1 0
0
1
0
q
and ( ) 0
0
= I y
k
similarly.
Using
k
g g = set 1 = k
1 Set ( ) 0 0 , 0 = = y j , let ( ) ( )
j
v j
1 1
1 q q = and ( ) ( )
j
y j
2 2
1 q q = , ( ) 1 0 1
2
= q
1.1 ( ) ( ) [ ] j y j v
1 1 2
ì ì | ÷ =
1.2 ( ) ( ) [ ] j v j y
2 2 1
1 ì ì | ÷ = +
Starting iterations are therefore:
( ) 0 0 = y , ( ) [ ]
1 2
0 ì | = v , ( ) ( ) [ ] [ ] [ ]
1 2 2 2 1 2 2 1
0 1 ì | ì ì | ì ì | ÷ = ÷ = v y , and
( ) ( ) [ ] [ ] [ ] [ ]
1 2 2 2 1 1 1 2 1 1 2
1 1 ì | ì ì | ì ì | ì ì | ÷ ÷ = ÷ = y v .
1.3
( ) ( )
( ) ( ) [ ]
( ) ( ) ( ) j v j y H
j v a j g
j v a j
,
1
1 1
2
2
2
1
÷
+ ÷ =
q
q
where
( ) ( ) [ ] ( ) ( ) [ ] j v j y j v j y H
2 2 1 1 2
, ì ì ì ì k ÷ + ÷ =
142
since
( ) [ ] ( )
( ) ( ) [ ]
( ) [ ] y y H
y a y g
y a y
i
i
1 1 2
1 1 2 2
2
1 1 2 2 1 2 2
1
,
1
ì ì |
ì ì | q
ì ì | ì ì | q
÷
÷ ÷
+ ÷ ÷ = ÷
and
( ) g = 0
2
q
1.4 ( ) ( )
( ) ( )
( ) ( ) [ ] j v j y gR
j y a j
j y a j
, 1
1 1
1 1 1 1
1
1
1
2
+
+ ÷
+ + ÷ = +
q
q where
( ) ( ) [ ] ( ) ( ) [ ] j v j y j v j y R
2 2 1 1 1
1 , 1 ì ì ì ì k ÷ + + ÷ = +
since ( ) [ ] ( )
( ) ( )
( ) [ ] v v gR
v a v
v a v
,
1
2 2 1
2 2 1 1
1
2 2 1 1 2 2 1
2
ì ì |
ì ì | q
ì ì | ì ì | q
÷
÷ ÷
+ ÷ ÷ = ÷
1.5 Assign ( ) ( ) { }
k k
g g j v = = = I 1 0
1
0
q
1.6 Repeat steps (1.1 – 1.5) until sign of convergence: ( ) ( ) c < ÷ ÷ 1 j v j v
1.7 Retain ( ) ( )
k k
g g v j v = = = = I , 0 1 0
0
1
0
q since ( ) j 1
1
q at the last value of j
represents ( ) 1
1
q
2 Reset ( ) 0 0 , 0 = = v j , let ( ) ( )
j
v j
1 1
1 q q = and ( ) ( )
j
y j
2 2
1 q q = and ( ) 1 0 1
1
= q
2.1 ( ) ( ) [ ] j v j y
2 2 1
ì ì | ÷ =
2.2 ( ) ( ) [ ] j y j v
1 1 2
1 ì ì | ÷ = +
Starting iterations are therefore:
( ) 0 0 = v , ( ) ( ) [ ] [ ]
2 1 2 2 1
0 0 ì | ì ì | = ÷ = v y , ( ) ( ) [ ] [ ] [ ]
2 1 1 1 2 1 1 2
0 1 ì | ì ì | ì ì | ÷ = ÷ = y v ,
and ( ) ( ) [ ] [ ] [ ] [ ]
2 1 1 1 2 2 2 1 2 2 1
1 1 ì | ì ì | ì ì | ì ì | ÷ ÷ = ÷ = v y .
2.3 ( ) ( )
( ) ( )
( ) ( ) [ ] j v j y gR
j y a j
j y a j
,
1
1 1
1
1
1
2
÷
+ ÷ =
q
q where
( ) ( ) [ ] ( ) ( ) [ ] j v j y j v j y R
2 2 1 1 1
, ì ì ì ì k ÷ + ÷ =
143
since ( ) [ ] ( )
( ) ( )
( ) [ ] v v gR
v a v
v a v
,
1
2 2 1
2 2 1 1
1
2 2 1 1 2 2 1
2
ì ì |
ì ì | q
ì ì | ì ì | q
÷
÷ ÷
+ ÷ ÷ = ÷ and
( ) 1 0
1
= q
2.4
( ) ( )
( ) ( ) [ ]
( ) ( ) ( ) 1 ,
1 1
1 1 1 1
2
2
2
1
+
+ ÷
+ + ÷ = +
j v j y H
j v a j g
j v a j
q
q
where
( ) ( ) [ ] ( ) ( ) [ ] 1 1 ,
2 2 1 1 2
+ ÷ + ÷ = + j v j y j v j y H ì ì ì ì k
since
( ) [ ] ( )
( ) ( ) [ ]
( ) [ ] y y H
y a y g
y a y
i
i
1 1 2
1 1 2 2
2
1 1 2 2 1 2 2
1
,
1
ì ì |
ì ì | q
ì ì | ì ì | q
÷
÷ ÷
+ ÷ ÷ = ÷
2.5 Assign ( ) ( ) { }
k k
g g j y = = = I 1 0
1
0
q
2.6 Repeat steps (2.1 – 2.5) until sign of convergence: ( ) ( ) c < ÷ ÷ 1 j y j y
2.7 Retain ( ) ( ) { }
k k
g g y j y = = = = I , 0 1 0
0
1
0
q since ( ) j 1
1
q at the last value of
j represents ( ) 1
1
q
3 Set 2 = k , repeat steps 1 and 2.
4 The convergence is linearly dependent on g so we evaluate the differences
in ( ) 1
1
q starting with 0
0
= v and 0
0
= y at the two arbitrary values of g and
then get
*
g g = by
( ) ( ) [ ] ( ) ( ) [ ]
( ) ( ) [ ] ( ) ( ) [ ] 0 0 0 0
0 0 0 0
0 1 0 1 0 2 0 2
0 1 0 1 2 0 2 0 2 1 *
= I ÷ = I ÷ = I ÷ = I
= I ÷ = I ÷ = I ÷ = I
=
v y v y
v y g v y g
g (54)
5 Set
*
g g = , repeat steps in section 1 of this procedure above to determine
( ) ( ) { }
*
0
1
0
, 0 1 0 g g v j v
k
= = = = I q which represents ( ) 1
1
q and using (32) we
get ( ) 1
2
q .
144
At this point we can calculate the idle state probabilities using
( ) ( ) [ ] ( ) [ ]
1
1
2 1
2 1 1
00
1 1
1
a U E U E g ÷ + + +
÷ ÷
=
q ì
µ µ
t . (55)
The total idle fraction is then ( ) g + 1
1
00
t . (56)
The wait prior to service for a class-1 customer with AP dispatching and non-zero
setup is finally:
( )( )( )
( ) ( )( ) [ ] { ( ) [ ]
( ) ( ) ( ) [ ]
( )
( )( ) [ ] ( ) ( ) ( ) [ ]
( ) ( )( ) [ ] ( ) ( ) ( ) [ ]
)
`
¹
÷ + + + ÷ ÷ ÷ ÷ |
.
|

\
|
+
÷ + + + ÷ ÷ |
.
|

\
| +
+
+ ÷ ÷ +
÷ + + ÷ ÷
×
+ + ÷ ÷ ÷ ÷
=
2
2 1
2
1
2
2
2
2 1
2
2 1 2 1
2
2 2 1
2
1 1
2
2
2
2 1
2
2 1
2 1
2 1 1 2 1 2 2
2 1 1 2
2
2 1
2
2 1 1
2 1 2 1 2 1 2 1
1
1 1 1 1
2
1 1 1
2
1 1
1 1 1
2 1 1
1
U E U E
C
S E S E
C C
C C U E
C C U E
C C
W
µ µ µ µ µ µ µ µ
ì µ ì µ µ µ µ µ
µ µ µ µ
µ µ µ µ µ µ
µ µ µ µ µ µ
(57)
using ( ) [ ]
1
1
1 a C ÷ ÷ q ì , ( )
1 1
1 U CE C + ÷ , ( )
2 2
U CE g C + ÷
and
2
W is the same equation with the subscripts switched.
The overall expected wait time is the convex combination of the expected wait
times of the two classes:
2 2 1 1
W a W a W
AP
+ = . The probability that an arbitrary
customer is followed by a changeover is ( ) ( )
1
1 1
00
1 2 a ÷ q t .
Changeover Comparisons. Each cell in Figure 5-9 and 5-10 contains the
FCFS probability of setup above the AP probability of setup. We see in both
figures that AP always requires fewer changeovers than FCFS in the zero setup
case. The FCFS probability is invariant to Q since from equation (39) the
145
probability of a random arrival requiring a changeover is
2
2 1
2
ì
ì ì
. The AP
probabilities monotonically decrease with increasing Q at any
1
ì (increasing
utilization) and approach zero at saturation. Queue sizes grow with load;
therefore, AP has a greater probability of a non-empty queue of the class currently
being serviced from which to draw at higher utilization levels. This fact will lead
to an increase in system capacity when compared with FCFS when setup is non-
zero.
146
4
8
.
6
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.
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1
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1
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1
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1
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1
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1
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1
0
.
3
R
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1
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1
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1
0
.
5
0
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4
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4
0
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3
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7
0
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6
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0
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6
0
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5
5
0
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0
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7
5 1
0
.
9
5
R
1
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1
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1
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1
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1
R
1
Figure 5-9. Probability of setup (FCFS% above AP%) when setup is zero and
25 . 0
2
= ì .
Q
ì
1
147
47 47 47 47 47 47 47
25 24 22 20 17 13 8
47 47 47 47 47 47 47 47
25 25 23 21 18 15 11 4
48 48 48 48 48 48 48 48
26 25 24 22 20 16 13 7
49 49 49 49 49 49 49 49 49
27 26 25 23 21 18 14 10 4
49 49 49 49 49 49 49 49 49
27 26 25 24 22 19 16 12 8
49 49 49 49 49 49 49 49 49 49
28 27 26 25 23 21 18 15 11 5
50 50 50 50 50 50 50 50 50 50 50
28 28 27 25 24 22 20 17 13 9 4
50 50 50 50 50 50 50 50 50 50 50 50
29 28 27 26 25 23 21 19 16 12 8 3
0.6
LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV LV
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
3
0
.
3
5
0
.
4
0
.
4
5
0
.
5
0
.
5
5
0
.
6
0
.
6
5
0
.
7
0
.
7
5
0
.
8
0
.
8
5
0
.
9
0
.
9
5
1
RS
RS RS RS RS RS RS RS RS
RS RS
RS RS RS RS RS RS RS RS
RS RS RS RS RS RS RS RS
RS RS RS RS
RS RS RS
RS RS RS RS RS RS RS
RS RS RS RS RS RS RS RS
RS RS RS RS RS RS RS RS RS RS RS RS
RS RS RS RS RS RS RS RS RS RS
R1
RS RS
RS RS RS RS RS RS RS RS RS RS RS RS
0.8
0.75
0.7
0.65
1
0.95
0.9
0.85
Figure 5-10. Probability of setup (FCFS% above AP%) when setup is zero and
60 . 0
2
= ì .
We add setup time in a way consistent with our analyses of Chapters 3 and 4,
using the setup fraction, as a ratio of the expected batch service time (with a batch
size of one). We evaluate a range of setup magnitudes starting with a very low
setup fraction of 0.001. Our highest level is 1.0, the level we use in our
operational standardization in Chapter 3 and Chapter 4. We compare the baseline
AP-FCFS wait differences of Figure 5-7 to non-zero setup using numerical
methods. We identify regions of interest that we explain as follows. The FCFS
system stability is limited as stated in Chapter 2 by
( ) ( ) ( ) ( ) [ ] 1 0
2 1
2 1
2 2 1 1
< + + + = s U E U E S E S E U
ì
ì ì
ì ì ,
but the AP system is only limited by ( ) ( ) ( ) 1 0
2 2 1 1
< + = s S E S E S E ì ì ì . We
identify this disparity in system capacity for the non-zero setup cases in the
figures by AP.
Q
ì
1
148
The first comparison is made for the symmetric cases where
2 1
ì ì = ,
( ) ( )
2 1
S E S E = , and ( ) ( )
2 1
U E U E = . Figure 5-11 with 2 . 0 =
i
ì , is characteristic
of the symmetric comparisons. We immediately see that in the presence of setup
AP always requires less wait than FCFS and without setup ( 0 . 0 =
i
U ), there is no
difference between AP and FCFS wait. We also note that the AP wait is
monotonically better than FCFS with both increasing setup and service. As setup
is introduced AP will minimize the changeovers and in the symmetric case
provide lower wait times. Given any fixed
i
ì , as setup and service times
increase so does the utilization and, thus, expected lengths of the queues. AP by
avoiding changeovers is able to provide a stable system in areas where FCFS is
saturated.
2.0 0 -0.8 -8.1 -91.71 -938.3 -1.E+3 -2.E+3 -4.E+3 -1.E+4 AP AP AP AP AP AP AP AP AP AP AP
1.9 0 -0.5 -5.08 -56.58 -511 -741.1 -1.E+3 -2.E+3 -2.E+3 -4.E+3 -1.E+4 AP AP AP AP AP AP AP AP AP
1.8 0 -0.33 -3.34 -36.92 -310.3 -435.3 -603.2 -835.8 -1.E+3 -2.E+3 -3.E+3 -4.E+3 -1.E+4 AP AP AP AP AP AP AP
1.7 0 -0.22 -2.28 -25.04 -201.2 -277 -374.6 -502.3 -673.3 -909.7 -1.E+3 -2.E+3 -3.E+3 -4.E+3 -1.E+4 AP AP AP AP AP
1.6 0 -0.16 -1.59 -17.45 -136.3 -185.6 -247.3 -325.5 -425.9 -557 -732.3 -974 -1.E+3 -2.E+3 -3.E+3 -5.E+3 -1.E+4 AP AP AP
1.5 0 -0.11 -1.14 -12.38 -95.21 -128.7 -170.1 -221.4 -285.6 -366.7 -470.7 -606.2 -786.9 -1.E+3 -1.E+3 -2.E+3 -3.E+3 -5.E+3 -1.E+4 AP
1.4 0 -0.08 -0.82 -8.91 -67.9 -91.46 -120.3 -155.6 -199 -252.9 -320.1 -404.8 -513.1 -653.9 -841.3 -1.E+3 -1.E+3 -2.E+3 -3.E+3 -5.E+3
1.3 0 -0.06 -0.59 -6.45 -49.14 -66.1 -86.73 -111.9 -142.5 -180 -226 -282.8 -353.6 -442.5 -555.9 -703.1 -898.6 -1.E+3 -2.E+3 -2.E+3
1.2 0 -0.04 -0.43 -4.69 -35.9 -48.33 -63.42 -81.74 -104 -131 -163.8 -203.8 -252.7 -313 -387.9 -481.8 -601.1 -755.6 -960.6 -1.E+3
1.1 0 -0.03 -0.32 -3.4 -26.37 -35.59 -46.81 -60.42 -76.91 -96.84 -121 -150.1 -185.4 -228.3 -280.7 -345 -424.5 -523.9 -650.1 -813.1
1.0 0 -0.02 -0.23 -2.46 -19.41 -26.31 -34.75 -45 -57.44 -72.46 -90.58 -112.4 -138.7 -170.3 -208.5 -254.7 -310.9 -379.6 -464.4 -570.1
0.9 0 -0.02 -0.16 -1.76 -14.26 -19.46 -25.86 -33.68 -43.18 -54.68 -68.55 -85.23 -105.2 -129.2 -157.9 -192.4 -233.7 -283.6 -344 -417.7
0.8 0 -0.01 -0.11 -1.24 -10.43 -14.36 -19.25 -25.25 -32.59 -41.51 -52.29 -65.25 -80.78 -99.34 -121.5 -147.8 -179.3 -216.8 -261.7 -315.6
0.7 0 -0.01 -0.08 -0.86 -7.57 -10.55 -14.29 -18.94 -24.67 -31.66 -40.13 -50.34 -62.59 -77.21 -94.61 -115.3 -139.8 -168.8 -203.2 -244.2
0.6 0 -0.01 -0.05 -0.57 -5.42 -7.69 -10.57 -14.19 -18.69 -24.22 -30.96 -39.11 -48.89 -60.59 -74.48 -90.96 -110.4 -133.4 -160.5 -192.4
0.5 0 -0.01 -0.03 -0.37 -3.84 -5.56 -7.78 -10.62 -14.17 -18.59 -24 -30.57 -38.49 -47.96 -59.22 -72.55 -88.3 -106.8 -128.6 -154.1
0.4 0 0 -0.02 -0.21 -2.67 -3.98 -5.69 -7.93 -10.77 -14.32 -18.71 -24.07 -30.54 -38.3 -47.55 -58.49 -71.4 -86.55 -104.3 -125.1
0.3 0 0 -0.01 -0.12 -1.82 -2.82 -4.16 -5.93 -8.22 -11.11 -14.69 -19.11 -24.45 -30.88 -38.56 -47.66 -58.38 -70.96 -85.67 -102.8
0.2 0 0 -0.01 -0.05 -1.23 -1.99 -3.06 -4.48 -6.33 -8.7 -11.66 -15.33 -19.8 -25.19 -31.63 -39.28 -48.3 -58.87 -71.21 -85.57
0.1 0 0 0 -0.02 -0.84 -1.43 -2.28 -3.43 -4.96 -6.92 -9.41 -12.49 -16.26 -20.82 -26.29 -32.8 -40.45 -49.44 -59.92 -72.1
0
.
0
0
.
0
0
1
0
.
0
1
0
.
1
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
.
0
1
.
1
1
.
2
1
.
3
1
.
4
1
.
5
1
.
6
1
.
7
1
.
8
1
.
9
2
.
0
Figure 5-11. Wait time differences (AP-FCFS)*100 for symmetric cases when
2 . 0 =
i
ì .
Figures 5-12 through 5-15 show a progression of the effects of setup when
25 . 0
2
= ì . With minimum setup added ( ) ( ) ( ) S E U E * 001 . 0 = we see the equality
E(U
i
)
E(S
i
)
149
at 1 = Q has been replaced entirely by AP. The dominance of AP, where only AP
yields the lesser wait as compared to FCFS, is quickly realized. We note that only
5% setup is needed for AP to dominate the 1 > Q region as shown in Figure 5-13.
As we expect from the stability limits, the AP win area increases with setup
magnitude, especially approaching the region of instability.
150
R
1
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1
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1
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1
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5
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1.95
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1
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1.6
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8
1
.
3
2
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.
7
4
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0
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2
.
3
4
2
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5
4
2
.
6
6
2
.
7
2
2
.
7
L
V
L
V
L
V
L
V
L
V
0.45
-
2
-
1
-
0
.
2
0
.
5
2
1
.
1
1
1
.
6
1
2
.
0
4
2
.
3
9
2
.
6
6
2
.
8
9
3
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0
5
3
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1
5
3
.
2
1
3
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1
9
3
.
1
2
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L
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L
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L
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0.4
0
.
7
1
1
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2
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1
.
7
9
2
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2
2
2
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1
3
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1
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3
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3
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3
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5
4
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3
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3
.
7
3
3
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3
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1
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0.35
2
.
8
1
3
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1
3
3
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4
2
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6
6
3
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8
7
4
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2
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3
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3
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3
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2
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1
1
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9
2
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.
4
7
4
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6
3
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7
7
4
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8
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9
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9
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8
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7
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0.25
5
.
7
9
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8
5
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8
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7
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2
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9
9
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6
9
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0.2
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.
8
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1
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7
8
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7
3
6
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6
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5
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6
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4
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3
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6
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2
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5
.
9
5
.
6
7
5
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3
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5
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0
6
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0.15
7
.
7
7
.
6
4
7
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5
8
7
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5
7
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4
1
7
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3
1
7
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2
7
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0
8
6
.
9
3
6
.
7
6
6
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5
7
6
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3
5
6
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0
8
5
.
7
7
5
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4
1
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V
L
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0.1
8
.
3
7
8
.
2
8
8
.
2
8
.
1
7
.
9
8
7
.
8
7
7
.
7
3
7
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5
8
7
.
4
2
7
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2
3
7
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0
1
6
.
7
6
6
.
4
8
6
.
1
4
5
.
7
5
L
V
L
V
L
V
L
V
L
V
0.05
1
0
.
9
5
0
.
9
0
.
8
5
0
.
8
0
.
7
5
0
.
7
0
.
6
5
0
.
6
0
.
5
5
0
.
5
0
.
4
5
0
.
4
0
.
3
5
0
.
3
0
.
2
5
0
.
2
0
.
1
5
0
.
1
0
.
0
5
Figure 5-12. Wait time differences (AP-FCFS)*100 when E(U)=0.001*E(S) and
25 . 0
2
= ì .
Q
ì
1
151
We note an interesting pattern in the figures illustrating the wait differences.
Figure 5-12 at the 65 . 0
1
= ì setting shows the AP-FCFS wait to be negative just
prior to the region of saturation. This pattern is also seen in Figure 5-16 for three
settings: 40 . 0
1
= ì , 45 . 0
1
= ì , and 50 . 0
1
= ì , but not in Figure-17. To explain
this pattern we show the actual wait times for AP and then FCFS for 50 . 0
1
= ì at
four levels of setup magnitude in Figures 5-13 and 5-14. We then follow with a
plot of the difference in flow time for the 50 . 0
1
= ì .
0.1
1
10
100
0
.
0
5
0
.
1
5
0
.
2
5
0
.
3
5
0
.
4
5
0
.
5
5
0
.
6
5
0
.
7
5
0
.
8
5
0
.
9
5
1
.
0
5
1
.
1
5
1
.
2
5
1
.
3
5
1
.
4
5
AP: U=1.0
AP: U=0.05
AP: U=0.01
AP: U=0.0
`
Q
W
AP
Figure 5-13. Wait time when 50 . 0
1
= ì , 25 . 0
2
= ì using AP.
152
0.1
1
10
100
0
.
0
5
0
.
1
5
0
.
2
5
0
.
3
5
0
.
4
5
0
.
5
5
0
.
6
5
0
.
7
5
0
.
8
5
0
.
9
5
1
.
0
5
1
.
1
5
1
.
2
5
1
.
3
5
1
.
4
5
FC: U=1.0
FC: U=0.05
FC: U=0.01
FC: U=0.0
Q
W
FCFS
Figure 5-14. Wait time when 50 . 0
1
= ì , 25 . 0
2
= ì using FCFS.
If we superimpose the wait time curves we find that the FCFS curves are
steeper at the same ( )
1
S E Q = . This is because the wait time, driven by
congestion, is a function of both service and setup times and AP pays less setup
than FCFS. This steeper slope near saturation causes the wait curves to intersect.
We show the case of ( ) ( ) S E U E * 01 . 0 = and identify three points of intersection
of the two curves. This does not happen with greater setup magnitude because the
FCFS wait curve is shifted up, intersecting the AP wait curve in only one place.
Figure 5-15 shows the three points of intersection for the 50 . 0
1
= ì , 25 . 0
2
= ì
and ( ) ( ) S E U E * 01 . 0 = case.
153
0.1
1
10
0
.
0
5
0
.
1
5
0
.
2
5
0
.
3
5
0
.
4
5
0
.
5
5
0
.
6
5
0
.
7
5
0
.
8
5
0
.
9
5
1
.
0
5
1
.
1
5
1
.
2
5
1
.
3
5
1
.
4
5
Wait
(FCFS-AP)+1
Q
Figure 5-15. Three zeroes of intersection between wait curves of AP and FCFS
when 50 . 0
1
= ì , 25 . 0
2
= ì and ( ) ( ) S E U E * 01 . 0 = .
154
R
1
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9
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1.9
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3
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2
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1.75
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1
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3
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+
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6
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4
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1.65
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1.6
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1.55
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5
8
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1
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4
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1.5
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2
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3
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1.45
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3
1
0
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1
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2
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1.4
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1
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3
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9
3
3
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4
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1.35
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1
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1
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1
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1
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1
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9
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5
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P
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3
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2
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1
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1.15
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5
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1
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7
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1
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7
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0.95
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6
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0.9
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0.85
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1
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0.8
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0.75
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4
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1
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2
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1
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0.65
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3
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7
4
1
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1
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3
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0.6
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7
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3
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5
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9
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1
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4
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9
5
1
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3
1
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5
6
1
.
7
1
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Figure 5-16. Wait time differences (AP-FCFS)*100 when E(U)=0.01*E(S) and
25 . 0
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Figure 5-17. Wait time differences (AP-FCFS)*100 when E(U)=0.05*E(S) and
25 . 0
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5
Figure 5-18. Wait time differences (AP-FCFS)*100 when E(U)=1.0*E(S) and
25 . 0
2
= ì .
Q
ì
1
157
Under certain circumstances FCFS will provide less wait than AP even when
setup time is non-zero. There are two regions, one characterized by
( ) ( )
1 2
S E S E > with
2 1
ì ì > and the other ( ) ( ) 5 . 0
2 1
< = Q S E S E with
2 1
ì ì > .
Both of these regions decrease in size with increasing setup as shown in Figures
5-12 and 5-16 through 5-18 and 5-19 through 5-21 such that when ( ) ( ) S E U E = ,
AP dominates the entire feasible space.
158
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Figure 5-19. Wait time differences (AP-FCFS)*100 when E(U)=0.01*E(S) and
10 . 0
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Figure 5-20. Wait time differences (AP-FCFS)*100 when E(U)=0.1*E(S) and
10 . 0
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Figure 5-21. Wait time differences (AP-FCFS)*100 when E(U)=1.0*E(S) and
10 . 0
2
= ì .
Q
ì
1
161
With the addition of setup, we make a number of observations:
1. There is an area where AP provides enough savings in the setups realized to
increase the capacity of the system, relative to what the FCFS can achieve. In
fact the region is stable when using AP, but unstable for FCFS. Recalling
from Chapter 2, the FCFS system utilization is ( ) P E ì where the processing
time, S U P + = , includes the setup time and therefore is greater than
( ) S E ì µ = when the setup, U , is non-zero. The AP rule self-regulates the
incidence of setup: in high traffic the queue is longer so there is less likelihood
of switchover at the end of a service and in the limit when 1 = µ there is zero
probability of switchover at the end of a service. Thus, the AP system
saturation is a function only of ( ) S E ì µ = , regardless of the setup magnitude.
2. AP always requires fewer changeovers than FCFS. The FCFS probability is
invariant to Q since the probability of a random arrival requiring a
changeover is a function of only
1
ì and
2
ì . The AP probabilities
monotonically decrease with increasing Q at any
1
ì (increasing utilization)
and approach zero at saturation. This fact will lead to an increase in system
capacity for AP when compared with FCFS when setup is non-zero.
3. For the symmetric case where
2 1
ì ì = , ( ) ( )
2 1
S E S E = , and ( ) ( )
2 1
U E U E =
AP always requires less wait than FCFS ( 0 . 0 >
i
U ). We also note that at any
2 1
ì ì = the AP wait is monotonically better than FCFS with both increasing
setup and service.
162
4. There is always an area where AP provides less wait than FCFS. The region
where AP wait is less than FCFS is much larger when setup is present. When
the setup equals the service magnitude, AP dominates the entire feasible
space. This may also suggest that 1 = Q has much less significance with non-
zero setup.
163
Chapter 6
SUMMARY AND DIRECTIONS OF FUTURE RESEARCH
In this dissertation, we addressed two questions concerning the role of setup
economies in discrete parts manufacturing. First, using simulation as the tool of
choice, we design and conduct a computational study to evaluate the impact of
setup reduction on the factory flow time in the setting of factory conversion from
a job shop to full or partial cellular layout. A key component of the design is the
construction of a framework for experimentation and a standardized test bed of
scenarios with sufficient uniformity as to make meaningful comparisons possible.
In the second segment of the dissertation, we focus on a queueing system that is
simple enough so that the exact analysis of the extent of setup incidence and
economies can be computed exactly. We use the results of analytic models of this
system to gain insights into the role of the dispatching rule in the queueing
system.
We now re-state the research questions in Chapter 1 and summarize the
findings of Chapters 3-5 in the form of responses to these questions.
Factory Conversions to Cellular Manufacturing Systems
• Can consistent results be obtained as to when the conversion of the job
shop can be expected to prove advantageous?
164
• What are the measured setup economies? When are setup economies
large enough to overcome pooling losses?
The conversion of job shops to cells consistently improves flow time by 10%
to 20%, for the test bed used in our research. This result provides a conservative
estimate of the advantages of CMS because it does not take advantage of such
additional factors as reduced batch sizes, transfers batches, or move times. We
find that conversion to cells consistently reduces setup on the order of 65% to
75% for the test bed we used. We conclude that setup reduction can overcome the
effects of pooling loss as long as the magnitude of the setups is not too small and
no significant bottlenecks develop in the cells upon conversion.
• How do other cell factors including reduced batch sizes and the use of
transfer batches affect flow times achieved in cells?
The use of reduced batch sizes, or the implementation of transfer batches, can
each provide cells with an additional improvement in flow time. Typically, each
of these two factors has a significant effect on reducing the flow time for CMS,
and the amount of reduction is usually at least as large as that obtained by
conversion to CMS without any changes in the batch sizes.
Our sensitivity runs show that the overall factory utilization and the potential
for setup reduction can both affect the conversion results obtained. Our tests
indicate that conversion to CMS may not be advantageous if the utilization level is
high or there is not sufficient potential to reduce setups.
165
The design of cells also has a clear impact on the conversion improvements
obtained. Typically, we observed better performance in cells when the original
source design was used. However, conversion benefits continue to be present
even after we use a uniform cell formation procedure due to Vakharia and
Wemmerlöv (1990). This indicates that careful allocation of machines to cells to
avoid heavy utilization helps to keep the pooling loss within tight control.
Regarding dispatching rules, our experimental runs support the conclusions of
previous authors that Repetitive Lot dispatching provides less overall setup and
supports lower flow times than FCFS in a job shop with setup. The effect of RL
seems to diminish in the same factory setting once it incorporates cells.
• Is there value in considering a partial implementation of CMS?
Although we could not identify general and robust rules that applied across all
data sets, we observed that the factory flow time of a partial cellular
implementation may be as good as or even better than the all-cell option, so it is
important to stop short of full conversion where appropriate. In addition, other
considerations include the following. Even when the number of cells to be
included in the partial layout is fixed, the choice of the correct subset of cells can
have a significant impact on the flow time. The sequence of best subsets to pick
as n increases from 1 to NC is not necessarily nested, so sequence matters. The
differences in factory flow times are due to the same factors recognized in the all-
cell CMS analysis, setup reduction and machine utilization, but neither factor
alone is sufficient to reliably determine the best subset of cells to select. The best
166
picks are characterized by large setup reductions along with reduction of
utilization in the residual job shop and the lack of bottlenecks in the cell(s), so
setup reduction and load balance in both the cells and residual job shop matter.
To our knowledge, this is the first simulation study that compares cell shop
conversion benefits across disparate data sets. We believe that this dissertation
has shown that the comparison of job shops and cellular systems with respect to
the flow time measure can produce reasonably consistent results when the same
operating rules and key parameter ranges are used across different data sets.
Moreover, our research shows that setup reduction can overcome pooling losses,
even under the conservative assumptions where batch size remain unchanged and
the material transport times in the job shop are assumed to be negligible. Overall,
the conclusions of our research are consistent with the qualitative insights cited in
the literature when comparing cell shops and job shops. However, our research
clarifies that the quantitative comparisons using the flow time metric must be
interpreted in the context of the region of the parameter space spanned by the data
sets, as well as the particular design used for the cells.
By investigating the efficacy of implementing partial cell layouts (hybrids)
using the same test bed, we are able to define considerations for the cell
implementation process. We find the selection of the subset of cells picked at any
level of cellular implementation has an impact on factory flow time and that a
partial cellular implementation may be as good as or even better than the all-cell
option.
167
Analytic Modeling of a Simple System with Setup
• What is the role of dispatching rules in the reduction of setups?
We find that the Alternating Priority (AP) dispatching rule that minimizes
setup incidence, and therefore, changeover incidence, can outperform the FCFS
rule over significant regions of the two-class parameter space even when the setup
time is taken to be zero (the metric for this comparison is average wait time in
queue). We characterize the region of superiority of AP over FCFS analytically
and provide bounds on the relative performance of the two rules.
When setup enters the comparison between these rules, we determine the
extent of the difference in setup paid as well as the difference in setup incidence
between AP and FCFS. We are able to identify regions where AP is always the
better choice as well as regions where AP increases the service capacity due to
reduction in the setups incurred. For the symmetric case of non-zero setup where
2 1
ì ì = , ( ) ( )
2 1
S E S E = , and ( ) ( )
2 1
U E U E = AP always requires less wait than
FCFS. We also note that at any
2 1
ì ì = the AP wait is monotonically better than
FCFS with both increasing setup and service. For the non-symmetric case we also
note that by the time the setup is equal to the service in magnitude, AP dominates
the entire feasible region. This may also suggest that 1 = Q has much less
significance with non-zero setup.
168
Directions for Future Research
The following topics are offered as potentially fruitful areas of research that
would extend the findings of this dissertation.
1. Analytic comparison of rules in the presence of non-zero setups: In the
case of non-zero setups, further research should pursue the derivation of
analytic results that constitute a counterpart to the analysis of Section 5.1.
We think there is opportunity to examine regions of dominance for the AP
rule using formulas for non-zero setup. This would also help explain the
behavior of FCFS and how it can dominate AP even in the presence of
setup.
2. Extension from two classes of customers to multiple classes. This
research would extend the results of Sections 5.1 and 5.2 to the multi-class
case. Analytically, this requires extending the results of Eisenberg (1967)
to the multi-class case. While the mathematics of following Eisenberg’s
specific approach becomes extremely cumbersome, simpler schemes of
analysis or approximate results may still reveal useful insights. Naturally,
simulation remains open as a tool for performance evaluation for all such
extensions.
3. Alternative rules for multiple customer classes: A quick search of the
literature reveals that the analysis of queues with multiple classes in the
presence of setups has let to a stream of research involving cyclic polling
rules (where customers are serviced in a pre-determined order). Such
169
rules may be viewed as alternatives to extensions of the AP rule to the
multi-class case (greater than two classes) such as the Repetitive Lot rule
(Jacobs and Bragg, 1988) or its variants discussed in this dissertation.
Further study is needed to evaluate such extensions. In particular, cyclic
policies can be compared to dynamic policies that incorporate dynamic
information into the switching decision. Of special interest is how setup
impacts the comparative advantages of these policies.
4. Discount factors to reflect setup economies. Some studies use flat-rate
discounts coupled with FCFS in analytic models to represent the effects of
setup economies in job shops and cell shops. Further research is required
to explore where this approximation can introduce severe distortion,
especially as magnified by bottlenecks or increased congestion in the
system.
170
APPENDIX A: SENSITIVITY TO THE SHAPE OF PROCESSING
TIME DISTRIBUTIONS
The runs presented in the body of this research use a 2-Erlang distribution for
both setup time and run time. The CV for this distribution is 0.707. To test the
sensitivity of the flow time results to the shape of these distributions, we varied
the CV while staying in the k-Erlang family and retaining the same mean. Of
course, CV=1.0 corresponds to an exponential distribution (k=1) and CV=0.25
(k=16) captures the shape the normal curve. We also tested the effect of skewness
by comparing the 2-Erlang with distributions from the beta(
1
o ,
2
o ) family, each
skewed in a different direction.
Below in Table A-1 we tabulated the results of these runs for two data sets.
Each cell with a dual entry shows the flow time for the job shop on top and CMS
directly below it. Although the shape of the distribution affects both the job shop
and CMS flow times, these values move together so that the flow ratio remains
insensitive to the changes.
171
Data Set #2 Data Set #6
JS flow
CMS flow
0.250 0.707 1.000
JS flow
CMS flow
0.250 0.707 1.000
0.250
148
127
148
127
148
130
0.250
7582
6163
7586
6208
7595
6253
0.707
149
128
149
130
149
130
0.707
7613
6225
7612
6261
7634
6303
1.000
150
129
150
130
151
131
1.000
7657
6267
7644
6309
7659
6343
149
130
7612
6261
148
127
7581
6148
149
129
7607
6263
CV
Setup
CV Run CV Run
CV
Setup
2-Erlang
CV=0.707
Beta(5.5,1.4)
CV=0.180
Beta(1.4,5.5)
CV=0.705
Table A-1. Sensitivity of Job Shop and CMS flow times to changes in
distributions of setup and runtime.
172
APPENDIX B: OUTPUT MEASURES FOR SIMULATION RUNS
Our additional input parameters for Chapters 3 and 4 are as follows.
= T duration of simulation window for releasing batches
= P number of batch orders released during simulation release window, T
The output statistics gathered by the simulation are as follows.
= TQ time at which last of P released batches is completed (simulation horizon)
( ) = p FT flow time of the
th
p batch released within release window, T
( ) P p , , 1 … = [flow time measured from order release to shipping]
( ) = p ST total setup incurred for the production of the
th
p batch ( ) P p , , 1 … =
( ) = p RT total run time incurred for the production of the
th
p batch ( ) P p , , 1 … =
( ) = j SQ total setup time accrued on machine type j during TQ
( ) = j RQ total run time accrued on machine type j during TQ
The output measures are then calculated as follows.
The average batch flow time is
( ) P p FT
P
p
¿
=1
(B-1)
173
Average time a batch spent in setup
( ) P p ST
P
p
¿
=1
(B-2)
Average time a batch spent being run
( ) P p RT
P
p
¿
=1
(B-3)
Average machine utilization for type j
( ) ( ) ( ) ( ) j NM TQ j RQ j SQ · + (B-4)
Overall average machine utilization for the factory (JS or CMS)
( ) ( ) ( ) ( )
¿ ¿
= =
+
J
j
J
j
j NM TQ j RQ j SQ
1 1
(B-5)
Maximum machine utilization for the JS configuration
j
max ( ) ( ) ( ) ( ) [ ] j NM TQ j RQ j SQ · + (B-6)
The minimum calculations are analogous. For the CMS, the maximum and
minimum utilization values consider machine types over all cells, so that equation
(B-6) is computed once for each cell.
174
GLOSSARY
Alternating Priority a dispatching rule from Maxwell (1961) designed
to minimize setup incidence in a single-server
queue with two customer classes: all jobs in queue
of a given class are served before switching to the
other class. The server thus alternates between
strings of jobs of either class 1 or class 2 and the
idle state, but never switches from class i to class
j ( ) i j = if there are jobs of class i still in queue
Cell a collection of different machines positioned in
proximity to work on a family of parts with similar
shapes and processing requirements
Cellular Manufacturing manufacturing part families using cells
Flow Ratio ratio of the average batch flow time after cellular
conversion to the average batch flow time of the
job shop with the same factory operational
parameters of load, machines and batch size
175
Job Shop a manufacturing facility comprised of general-
purpose machines organized into a collection of
machine centers (departments) grouped on the basis
of the operation performed
Major-Minor Model a setup structure whereby the setup is a major
for Setup setup, a minor setup, or no setup at all. A major
setup is incurred if two parts belonging to distinct
families are processed consecutively on the same
machine. Switching between two different part
types in the same family incurs a minor setup. No
setup is required if a machine processes two
batches of the same part type consecutively
Part Family parts with similar features and common sequences
of operations requiring similar tools or fixtures
Pooling Loss the diseconomies of segregating a given machine
type by assigning them to independent cells
Remainder Shop that part of the factory that is not converted to cells
and continues to operate as a job shop
Repetitive Lot Dispatching a dispatching rule from Jacobs and Bragg (1988)
designed to minimize setup: (1)a single (pooled)
176
queue is formed for all batches arriving to be
processed at a machine center, (2) Any arriving
batch encountering an available machine upon
entry is immediately routed to the available
machine where it would encounter the least setup
time. If no machines are available, the batch joins
(or forms) a queue to wait for a machine, (3)When
a machine becomes available, the next job assigned
to it is selected based on the minimum setup among
all jobs in queue. If multiple jobs tie at this
minimum setup value, the FCFS discipline is used
to break the tie.
Setup Fraction the ratio of minor to major setup
Setup Ratio the ratio of major setup to batch run time
Transfer Batch lot quantities moved between workstations or
production areas – typically equal to or smaller than
the production lot size
177
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