• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Control systems
 Simulation
 Statistics
 Batch size
 Setup time
 Machine failure
 Confidence
 Conclusions
 Glossary
 References
 Biographical sketch














Group Title: comparative simulation study of Kanban, CONWIP, and MRP manufacturing control systems in a flowshop
Title: A comparative simulation study of Kanban, CONWIP, and MRP manufacturing control systems in a flowshop
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 Material Information
Title: A comparative simulation study of Kanban, CONWIP, and MRP manufacturing control systems in a flowshop
Physical Description: Book
Language: English
Creator: Hochreiter, Thomas Alfons, 1973-
Publisher: State University System of Florida
Place of Publication: Florida
Florida
Publication Date: 1999
Copyright Date: 1999
 Subjects
Subject: Manufacturing processes -- Computer simulation   ( lcsh )
Production management -- Computer simulation   ( lcsh )
simulation study -- manufacturing control systems -- Kanban -- CONWIP -- MRP
Industrial and Systems Engineering thesis, M.S   ( lcsh )
Dissertations, Academic -- Industrial and Systems Engineering -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )
 Notes
Summary: ABSTRACT: The globalization of markets due to the improvement of communication and transportation media has had a significant impact on manufacturing technology in recent years. The strong international competition forced companies to establish efficient production facilities ensuring profitability on the long run. The performance of the most prevalent American manufacturing control mechanism, MRP, was questioned after the success of the Japanese Kanban control system during the Just-In-Time era.
Summary: ABSTRACT (cont.): CONWIP, a generalization of the Kanban control system, was introduced as a result of extensive research done to understand manufacturing systems with the aim of improving their efficiency. During an extensive simulation study, the performances of Kanban, CONWIP, and MRP were evaluated for a ten identical machine tandem line with respect to batch size, setup time, and machine failure. The utilization (throughput) was kept constant for all control systems. The parameters were introduced to the models one at a time, thereby increasing the realism and the variability of the manufacturing line. Thus, the performances of the three control mechanisms were explored on three levels of complexity.
Summary: ABSTRACT (cont.): Initially, only the influence of batch size on the performances of the control systems was investigated. Then, the setup time was taken into consideration in addition to the batch size. Last, machine failure was introduced to augment the models' realism resulting in a higher practical applicability. On each level, the performances were evaluated for steady-state, assuming the manufacturing line would run indefinitely. In addition, the response of the performance to machine failure was observed dynamically while keeping batch size and setup time constant.
Summary: ABSTRACT (cont.): Although the performance differences were found to be minute, Kanban and CONWIP were outperformed by the traditional control system, MRP, for experiments with varying batch size and for experiments including both batch size and setup times. On the highest level of variability, with machine failure introduced, Kanban was ranked first, closely followed by CONWIP. The two pull systems easily outranked the push system when evaluated according to average cycle time, maximum cycle time and the standard deviation of cycle time. Kanban performed best for the dynamic response to failure as well, where the system performance was measured by the time taken to recover from failure.
Thesis: Thesis (M.S.)--University of Florida, 1999.
Bibliography: Includes bibliographical references (p. 215-221).
System Details: System requirements: World Wide Web browser and PDF reader.
System Details: Mode of access: World Wide Web.
Statement of Responsibility: by Thomas Alfons Hochreiter.
General Note: Title from first page of PDF file.
General Note: Document formatted into pages; contains xx, 222 p.; also includes graphics.
General Note: Vita.
 Record Information
Bibliographic ID: UF00100669
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 45296482
alephbibnum - 002456831
notis - AMG2162

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Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
    Table of Contents
        Page v
        Page vi
        Page vii
        Page viii
    List of Tables
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
    List of Figures
        Page xiv
        Page xv
        Page xvi
        Page xvii
        Page xviii
    Abstract
        Page xix
        Page xx
    Introduction
        Page 1
        Page 2
        Page 3
    Control systems
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
    Simulation
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
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        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
    Statistics
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
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        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
    Batch size
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
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        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
    Setup time
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
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        Page 125
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        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
    Machine failure
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
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        Page 176
        Page 177
        Page 178
        Page 179
        Page 180
        Page 181
        Page 182
    Confidence
        Page 183
        Page 184
        Page 185
        Page 186
        Page 187
        Page 188
        Page 189
        Page 190
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        Page 200
        Page 201
        Page 202
        Page 203
        Page 204
        Page 205
    Conclusions
        Page 206
        Page 207
        Page 208
        Page 209
        Page 210
        Page 211
        Page 212
    Glossary
        Page 213
        Page 214
    References
        Page 215
        Page 216
        Page 217
        Page 218
        Page 219
        Page 220
        Page 221
    Biographical sketch
        Page 222
        Page 223
Full Text










A COMPARATIVE SIMULATION STUDY OF
KANBAN, CONWIP, AND MRP MANUFACTURING
CONTROL SYSTEMS IN A FLOWSHOP












By

THOMAS ALFONS HOCHREITER


A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY
OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF SCIENCE




UNIVERSITY OF FLORIDA


1999



























Copyright 1999

by

Thomas A. Hochreiter

























To Mum, Dad, and Sis














ACKNOWLEDGMENTS


I would like to express my sincere gratitude to Dr. Suleyman Tufekci for his

assistance and guidance. As my advisor and the chairman of my supervisory

committee he has provided the constructive critique to refine the content of this thesis.

Further, I would like to acknowledge the contribution of the other members of my

committee, Dr. Diane Schaub and Dr. Sherman Bai. During her course in Applied

Probability, Dr. Schaub improved my understanding of statistics while Dr. Bai

provided me with the required background on digital simulation.

In addition, I would like to thank Tim Elftman for the professional administration

of the computer network and Tobias and Henrik Andersson for their initial support on

the material.















TABLE OF CONTENTS



page

A CK N O W LED G M EN T ......................................................... ............ iv
LIST OF TABLES .............................................. ................. .......... ix
LIST OF FIGURES........................................................................ xiv
A B S T R A C T .................................................................................x ix

CH A PTER 1 IN TR OD U CTION ..................................... ........ .............................. 1
1 .1 M o tiv atio n ...................................................................................................... 1
1.2 T h esis O u tlin e .............................................................................................. 3
CHAPTER 2 CONTROL SYSTEMS........................................................................ 4
2.1 Push A nd Pull System s ............................................................. .............. 4
2 .2 K an b an ......................................................................................................... 6
2.2.1 The M echanism ..... .. ............................... ........................ .............. 7
2 .2 .2 C h aracteristics ..... .. ...................................... ........................ .............. 9
2 .3 C O N W IP ........................................................................................................... 10
2 .3 .1 T he M ech anism ... .............................................................. .............. 10
2 .3 .2 C h aracteristics ..... .. ...................................... ...................... .............. 12
2 .4 M R P ............................................................................................................. . .. 1 3
2 .4 .1 T he M ech anism ... .............................................................. .............. 13
2.4.2 Characteristics ................................. ........ 16
2.5 Comparison of CONWIP with MRP............................................................. 17
2.6 Comparison of CONWIP with Kanban......................................................... 19
CH A PTER 3 SIM U LA TION ....................................... ........................ .............. 23
3.1 The Software ............................. .......... .................................... 24
3 .1 .1 E F M L ......................................................................................................... 2 4
3 .1 .2 A ren a .......................................................................................................... 2 6
3.2 T he Sim ulation Study ......................................... ......................... .............. 28
3.2 .1 State O objective ... .. ...................................... ....................... . . .......... 30
3.2.2 C ollect/Prepare D ata ................................. ....................... .............. 30
3 .2 .3 F orm ulate M odels ........................................ ........................ .............. 3 1
3.2.4 V erification of the M odels ................. ................................................ 32
3.2 .5 V alidation ................................. .............................................. ........... . 37
3.2.6 Simulation Experiment Design ................................................ 37
3.2.7 Sim ulation E execution ..................................... ...................... .............. 4 1

v










3.2.8 Output Analysis and Interpretation of the Results ............................... 42
3.2.9 Conclusions and Im plem entation........................................... .............. 42
CH A PTER 4 STA TISTIC S ....................................... ......................... .............. 44
4.1 Transient and Steady-State B ehavior............................................ .............. 45
4.2 Confidence ...... .............. . ..... ............................................. 46
4.2.1 Analysis for Terminating Simulations ................................................. 46
4.2.2 Analysis for Non-Terminating Simulations......................................... 48
4.2.3 Paired-t C confidence Interval .................................................. .............. 49
4.3 M multiple R egression ......................................... .. ......................... ............. 51
4.3.1 Estimating and Testing Hypotheses about the P3 Parameters .................. 54
4.3.2 Usefulness of a Model: R2 and the Analysis of Variance F-Test............ 55
4.3.3 Multiple Coefficient of Determination, R2........................................... 56
4.3.4 V ariance F-T est ..................................... ........................ .......... .. 56
4.3.5 Comparison of two or more Regression Functions.............................. 58
4.3.6 Transform ation .. ................................................ .............. 59
4.3.7 R esidual A analysis ..................................... .. ....................... .............. 60
4.3.8 Influential O observations ......................................................... .............. 62
CH A PTER 5 B A TCH SIZE ....................................... ......................... .............. 63
5 .1 P aram eters ......................................................................................................... 6 4
5.1.1 Process T im e .. ................................................................ ............. 64
5 .1 .2 B atch S iz e ................................................................................................... 6 5
5.1.3 N um ber of C ards ....................................... .. ...................... ............... 65
5.1.3.1 Card Allocation for K anban ................................................ 67
5.1.3.2 C ard A location R ules ......................................................................... 68
5.1.3.3 Deviation of Rules from Optimum................................................ 72
5.1.4 Interarrival Tim e ......................................................... ......... ..... 77
5.2 A average C ycle Tim e........................................................... ................... 78
5.2.1 The Average Machine Utilization ............... .................................. 79
5.2.2 K anban and C O N W IP ............................................................ .............. 83
5.2.3 Findings and Conclusions for Kanban and CONWIP................................ 93
5 .2 .4 M R P ........................................................................................................... 9 5
5.2.5 F findings for M R P ................................................................. .............. 99
5.3 K anban, CON W IP, and M RP ..................................................... .............. 100
5.3.1 Average Cycle Time Dependent on Work in Process........................... 101
5 .3 .2 C o n clu sio n s ..... .. ...................................... ........................ .............. 10 9
CH A PTER 6 SETU P TIM E ................................... ........................ .............. 110
6 .1 P aram eters ....................................................................................................... 1 12
6.1.1 Setup R atio ...................................................................................... . 113
6.1.2 Utilization.................... ........ ... .................... 113
6.2 Average Cycle Time (High Utilization)...... .... .................................. 120
6.2 .1 C om prison ...... .. ...................................... ....................... . . .......... 122
6.2.2 C conclusions .............. ........ ................................ ...................... .. 124
6.3 Average Cycle Time (Low Utilization) .......... .................................. 124
6.3.1 C om prison ...... .. ...................................... ....................... . . .......... 127










6 .3 .2 C o n clu sio n s ..... .. ...................................... ........................ .............. 12 9
6.4 R egression M odels ............................................................... .............. 129
CHAPTER 7 MACHINE FAILURE ..................... .................... 132
7 .1 P a ra m e te rs ....................................................................................................... 1 3 5
7.2 D ynam ics of F failure ..................................... ......................... .............. 136
7.2 .1 Indicators .................................................................................................. 136
7.2.1.1 Tim e Spent in System ....... ...... ...... .................... 137
7.2.1.2 R recovery Tim e ................................. ....................... .............. 138
7.2.2 C onfi guration ... .. ...................................... ....................... .............. 139
7.2.3 Tim e Spent in the System ...... ........ ...... .................... 140
7.2.4 System R recovery .................................... ........................ .............. 146
7 .2 .5 C onclu sion s ..... .. ...................................... ........................ . . ........ .. 150
7.3 Failure in Steady-State .................................. ........................ .............. 151
7.3.1 Param eters........................................................................ ....................... 152
7.3.2 Influence of Interfailure Time and Repair Duration ............................ 153
7.3.3 Conclusions for Interfailure Time and Repair Duration ....................... 156
7.3.4 U tilization ........................................................................................... 157
7.3.5 A average C ycle Tim e.................... ....................................................... 159
7.3.6 Conclusions for Average Cycle Time...... ................... ................. 165
7.3.7 The Maximum Cycle Time ........................................... 166
7.3.8 Conclusions for the Maximum Cycle Time ................... ................. 170
7.3.9 Standard D aviation of Cycle Tim es ..................................... .............. 170
7.3.10 Conclusions for the Standard Deviation of Cycle Times.................... 173
7.3.11 R egression .. ................................................................ ............. 174
7.3.11.1 M models .................................................................. . .......... 175
7.3.11.2 Effects of the Regressors...... ......................... 177
7.3.11.3 M odel V alidation...... .......... ......... .................... 181
7.3.12 Conclusions .................................................... ............ .. 182
CH APTER 8 CON FIDEN CE ................................. ....................... .............. 183
8.1 T ransient B behavior ................................................................ .............. 184
8 .2 B atch siz e ......................................................................................................... 18 8
8.2.1 Prior to Sim ulations.................................. ...................... .............. 188
8.2.2 Succeeding Sim ulations ...... ......... ......... .................... 191
8 .3 S e tu p ........................................................................................................... . .. 1 9 4
8.3.1 Prior to Sim ulations.................................. ...................... .............. 194
8.3.2 Succeeding Sim ulations ...... ......... ......... .................... 195
8 .4 F a ilu re .............................................................................................................. 1 9 7
8.4.1 D ynam ics of Failure ................................. ...................... .............. 198
8.4.1.1 Time Spent in the System....................................... 198
8.4.1.2 Recovery Time ..................................................... 199
8.4.2 M machine Failure in Steady-State .......................................... .............. 199
8.4.2.1 Influence of Interfailure Time and Repair Duration ...................... 200
8.4.2.2 Prior to Sim ulations...... ......... ........ .................... 202
8.4.2.3 Succeeding Simulations ...... .... ...................... 204










CHAPTER 9 CONCLUSIONS...... ............ ............ .................... 206
9 .1 S u m m a ry ......................................................................................................... 2 0 6
9.2 Future W ork ...... .. .............................. .......... ......... ............ .. 212

GLOSSARY ................................................................................. .. 213

R E FE R E N C E S ................................................................. ............2 15
BIOGRAPHICAL SKETCH............... ...... ...... .......... 222














































viii














LIST OF TABLES


Table page

3-1: Configuration for Kanban, CONWIP, and MRP to verify
correctness of the m odels .......................................................... .............. 33

3-2: Statistics on t-test to verify concurrence of output between EFML
and Arena for Kanban, CONWIP, and MRP.......................................... 33

4-1: For the paired-t test, comparing two systems is reduced to
estimating a single parameter, the difference. .......................... .............. 50

5-1: Additional cards allocated to the system with ten machines ............................ 76

5-2: Multiple regression output for CONWIP with the average cycle
time (Avgct) dependent on the number of cards (Ccards)......................... 87

5-3: The residual standard error of different transformations for the
m multiple regression on K anban ................................................. .............. 90

5-4: Function coefficients describing the dependency of the average
cycle time and the batch size derived by multiple linear
regression for Kanban and CONW IP ....................................... .............. 91

5-5: The derived functions for CONWIP and Kanban to estimate the
average cycle time for given batch size and number of cards
assigned to the system ..................................... ....................... ............. 92

5-6: Combinations of batch size and number of cards for maximum
WIP level 60 and the resulting average cycle times for
K anban and CONW IP. ...... ............ .............. .................... 102

5-7: The increase in cycle time for increasing batch size and constant
W IP .................................................................................................... . . . .. 1 0 4

5-8: The results for the regression analysis modeling the response of
the average cycle time to the WIP for a comparison between
Kanban, CONWIP, and MRP.............. .................... 107










6-1: The setup times and the corresponding setup ratios included in this
stu d y ..................................................................................................... . . . 1 1 3

6-2: Configuration chosen to establish high and low utilization levels ................. 114

6-3: Output for the paired t-tests on difference of high utilization
including setup for Kanban, CONWIP, and MRP................................ 117

6-4: The output of the paired t-tests on the difference between the
average cycle times for Kanban, CONWIP, and MRP............................ 122

6-5: The mean of the average cycle time relative to the mean of the
utilization for Kanban, CONWIP, and MRP. ............. ................. 123

6-6: Output for the paired t-tests on difference of low utilization
including setup for Kanban, CONWIP, and MRP................................ 125

6-7: The regression models for the average cycle time as functions of
the batch size, the number of cards assigned to the system or
the interarrival time for MRP, and the setup ratio and their
corresponding multiple coefficients of determination, R2. ....................... 130

7-1: The configurations chosen for the investigation on the dynamics
o f fa ilu re ...................................................................................................... 1 3 9

7-2: The coefficients of variation, the minimal and the maximal times
after the failure of machine 5 for Kanban, CONWIP, and
M R P ............................................................................................................ 14 3

7-3: The results of the paired t-test for the time after failure at passing
point 11 or departure of the system for Kanban, CONWIP,
and M R P .............................................................................. 145

7-4: Output for the multiple linear regression models fitting the moving
average cycle time dependent on the time after failure for
Kanban, CONWIP, and MRP.............. .................... 148

7-5: Combinations of interfailure time and repair duration resulting in a
constant availability. ............. .... ........... ......................................... 154

7-6: The interfailure times and repair durations in different time units
representing the scenarios for the range of availability
sim u lated ..................................................................................................... 15 7










7-7: The minimum, mean, and maximum values for the low and high
utilization levels as a summary for the simulations completed,
including machine failure for Kanban, CONWIP, and MRP .................. 158

7-8: The output for the paired t-test to establish the difference between
the utilization including machine failure for Kanban,
CON W IP, and M RP. ................. ..................................................... 158

7-9: A summary of statistics on the average cycle time for Kanban,
CONWIP, and MRP including machine failure. .............. ................. 163

7-10: The output for the paired t-test to establish the difference
between the average cycle time including machine failure for
Kanban, CONWIP, and MRP.............. .................... 164

7-11: Statistics on the maximum cycle time including machine failure
for Kanban, CONW IP, and M RP. ...... ........... ..................................... 167

7-12: The output for the paired t-test on the maximum cycle time
including machine failure for Kanban, CONWIP, and MRP .................... 169

7-13: Statistics on the standard deviation of cycle time including
machine failure for Kanban, CONWIP, and MRP. .............................. 171

7-14: The output for the paired t-test for the standard deviation of cycle
time including machine failure for Kanban, CONWIP, and
M . .RP .......................................................................................................... 1 7 2

7-15: The relative increase of the standard deviation in cycle time
including machine failure for Kanban, CONWIP, and MRP .................... 173

7-16: The regression models for the average cycle time including
machine failure for Kanban, CONWIP, and MRP. .............................. 175

7-17: The domain and the corresponding effects for the regressor terms
including machine failure for Kanban's regression model ...................... 178

7-18: The domain and the corresponding effects for the regressor terms
including machine failure for CONWIP's regression model................... 179

7-19: The domain and the corresponding effects for the regressor terms
including machine failure for MRP's regression model.......................... 180

8-1: The configurations for the analysis of the transient behavior for
Kanban, CONWIP, and MRP.............. .................... 184










8-2: Configuration for Kanban and CONWIP to determine confidence
interval prior to simulation and the corresponding utilization
and coefficient of variation as the output...... ................... ................. 189

8-3: Configuration for MRP to determine confidence interval prior to
simulation and the corresponding utilization and coefficient of
variation as the output ....... ............. ............. .................... 189

8-4: Output for confidence interval calculations for CONWIP, Kanban,
an d M R P .................................................................................................... 19 1

8-5: Configuration for CONWIP, Kanban, and MRP resulting in the
highest coefficient of variation of all the simulations run ....................... 191

8-6: Configuration for CONWIP, Kanban, and MRP resulting in the
lowest coefficient of variation of all the simulations run. ........................ 192

8-7: Configuration for Kanban, CONWIP, and MRP including setup
time to determine confidence interval prior to simulation and
the corresponding throughput and coefficient of variation as
th e o u tp u t. ................................................................................................... 19 4

8-8: Output for confidence interval calculations including the setup
time for CONWIP, Kanban, and MRP. ................... ................. 195

8-9: Configuration for Kanban, CONWIP, and MRP including setup
time to determine confidence interval succeeding the
simulations and the corresponding throughput and coefficient
of variation as the output. ...... ......... .......... .................... 195

8-10: The coefficients of variation prior to the simulations and
succeeding the simulations and their difference including
setup for Kanban, CONW IP, and M RP....... .................... ................. 197

8-11: The output for t-tests done for the time after failure at passing
point 11 for Kanban, CONW IP, and MRP. ...................... ...... ........... 198

8-12: The results for the calculation of the confidence intervals for the
time after failure and the moving average of the cycle times
for Kanban, CONW IP, and M RP. ...... ........... ..................................... 199

8-13: The response of the average utilization to different combinations
of interfailure time and repair duration and varying batch size
for a small number of cards [see Table 6-2]assigned to a line
controlled by CONW IP. ...... ........... ............ .................... 200










8-14: The response of the average utilization to different combinations
of interfailure time and repair duration and varying batch size
for a large number of cards [see Table 6-2] assigned to a line
controlled by CONW IP. ...... ........... ............ .................... 201

8-15: The configurations for Kanban, CONWIP, and MRP including
m machine failure prior to sim ulations. ...................................... .............. 203

8-16: The amount of entities processed to ensure good estimation of
indicators including m machine failure...... .... .................. ................. 204

8-17: Output for confidence interval calculations including machine
failure for Kanban, CONW IP, and MRP.......................... ................. 204

8-18: The configurations for Kanban, CONWIP, and MRP including
machine failure succeeding the simulations. .................... ...... ........... 204

8-19: Output for confidence interval calculations including machine
failure for Kanban, CONWIP, and MRP succeeding the
sim ulations ......................................................................................... . 205

9-1: The optimal configurations for the minimal average cycle time for
Kanban, CONW IP, and MRP....... ... .... .................... 210

9-2: The optimal configurations for the minimal average cycle time for
Kanban, CONW IP, and MRP....... ... .... .................... 210

9-3: The optimal configurations for the minimal average cycle time for
Kanban, CONW IP, and MRP....... ... .... .................... 211














LIST OF FIGURES


Figure page

2-1: A push m manufacturing system ............................................................ .............. 5

2-2: A pull m manufacturing system ................................... ...................... .............. 5

2-3: The one-card K anban system .................................... ...................... .............. 7

2-4: A CONW IP production line. ................ ................................................... 11

2-5: Simplified schematic of MRP...................................... 14

2-6 : A M R P produ action line ........................................................................................ 15

2-7 Relative robustness of CONWIP and MRP.................................................... 18

3-1: The object architecture for the EFM L ............................................. .............. 25

3-2: A rena's hierarchical Structure ......................................................... .............. 27

3-3: Flow chart of a sim ulation study....................................................... .............. 29

3-4: The cycle time per entity and the cumulative average cycle time
dependent on the number of processed entities for MRP with
A re n a ........................................................................................................ . . 3 4

3-5: The deviation of the average cycle time between EFML and Arena
for different configurations for CONWIP. .............................................. 35

3-6: The deviation of the average cycle time between EFML and Arena
for different configurations for Kanban.................................................. 36

3-7: The deviation of the average cycle time between EFML and Arena
for different configurations for MRP ............... ................................... 36

4-1: R ejection region for a test of /2 ....................................................... .............. 55

5-1: The ten m machine tandem line .......................................................... .............. 68










5-2: Free body diagram of the ten machine tandem line modeled as a
b e a m ........................................................................................................ . . . 6 8

5-3: Number of cards per machine for 11 cards assigned to a ten
m machine line .............................................................................. . . ......... 70

5-4: Number of cards per machine for 12 cards assigned to a ten
m machine line .............................................................................. . . ......... 70

5-5: Increase in throughput by allocating cards optimally instead of
sim ply applying the rules ......................................................... .............. 73

5-6: The average utilization dependent on the number of cards for
K an b a n ........................................................................................................ 7 4

5-7: The average cycle time dependent on the batch size and number of
cards allocated to the line for the three control systems:
Kanban (1), CONWIP (2), and MRP (3) ............................................... 78

5-8: Utilization dependent on batch size and number of cards for
Kanban (1), CONWIP (2), and MRP (3) ............................................... 82

5-9: Average WIP dependent on the number of cards assigned to a
K anban system .. .................................................................. ............ .. 84

5-10: The average cycle time dependent on the number of cards
assigned to the system for CONW IP ........................................ .............. 85

5-11: The average cycle time dependent on the number of cards
assigned to the system for K anban ........................................... .............. 86

5-12: Unequal residual error variance for initial model fitted to
K an b an ......................................................................................................... 8 9

5-13: The distribution pattern for the residual error of the transformed
multiple regression m odel for K anban............... ................................... 90

5-14: Three dimensional illustration of the In-transformed data points
of the average cycle time, dependent on the batch size and
number of cards, and the data points computed with the
regression m odel for K anban ........................................................................ 93

5-15: Work in process dependent on the interarrival time for different
batch sizes for M R P ... .......................................................... ............. 95


xv










5-16: The average utilization of the line dependent on the interarrival
tim e of the batches for M R P .................................................... .............. 97

5-17: The average cycle time per entity dependent on the interarrival
tim e for M R P ......................................................................................... 98

5-18: The average cycle time dependent on the batch size with a
constant throughput for M RP................................................... .............. 99

5-19: The minimal average cycle time dependent on the average work
in process for the three control system s....... .................... ................. 101

5-20: The average cycle time dependent on different combinations of
batch size and number of cards assigned, simulation and
regression m odel .. ............. ................ ........................................... 103

5-21: A closer look at the minimal average cycle time dependent on
lower average work in process for the three control systems .................. 106

5-22: A closer look at the minimal average cycle time dependent on
higher average work in process for the three control systems ................. 108

6-1: The higher utilization level dependent on the setup ratio and the
batch size for Kanban, CONWIP, and MRP. ................... ...... ........... 116

6-2: Throughput dependent on the setup ratio and the batch size for
Kanban, CONWIP, and MRP.............. .................... 118

6-3: The average cycle time dependent on the setup ratio and the batch
size for Kanban, CONWIP, and MRP. .................................... 120

6-4: The average cycle time dependent on the batch size and setup ratio
for K anban ....................................................................................... . . 12 1

6-5: The mean differences of the average cycle times between Kanban,
CONWIP, and MRP for the high utilization level................................ 123

6-6: The average cycle time dependent on the setup ratio and the batch
size for the high (0.85) and the low utilization level (0.67)..................... 126

6-7: The average cycle time for the low utilization level dependent on
the setup ratio and the batch size for Kanban, CONWIP, and
M R P ............................................................................................................ 12 8

6-8: The mean differences of the average cycle times between Kanban,
CONWIP, and MRP for the low utilization level................................. 129











7-1: Resource states and their occurence times...... .... .................................. 133

7-2: The effect of failure on the entity. ...... ... .... .................... 133

7-3: The points of data collection for the investigation on the dynamics
o f fa ilu re ...................................................................................................... 13 8

7-4: 20 replications showing the first entity passing through the
downstream half of the line after the reactivation of machine 5
for K anban ....................................................................................... . . 14 1

7-5: 20 replications showing the first entity passing through the
downstream half of the line after reactivation of machine 5 for
C O N W IP ..................................................................................................... 14 2

7-6: 20 replications showing the first entity passing through the
downstream half of the line after reactivation of machine 5 for
M R P ............................................................................................................ 14 3

7-7: The average time after failure at the passing points for Kanban,
CON W IP, and M RP. ................. ..................................................... 144

7-8: The moving average of the cycle time dependent on the time after
failure for five replications per Kanban, CONWIP, and MRP ................ 146

7-9: The time after failure for which the exponentially smoothed
average of the cycle times exceeds the average cycle time by
less than 10% for Kanban, CONWIP, and MRP .............. ................. 149

7-10: The average of the average utilizations per batch size and
replication versus the configuration for increasing interfailure
tim es and repair durations ...... ......... ....... .................... 155

7-11: The average cycle time versus the batch size and the setup ratio
for the six availability levels for Kanban...... .................... ................. 160

7-12: The average cycle time versus the batch size and the setup ratio
for the six availability levels for CONW IP. ..................... ................. 161

7-13: The average cycle time versus the batch size and the setup ratio
for the six availability levels for M RP...... .... ................................... 162

7-14: Box plots of the maximum cycle time including machine failure
for Kanban (1), CONWIP (2), and MRP (3). ................... ...... ........... 168










7-15: Boxplots of the standard deviation of cycle time including
machine failure for Kanban (1), CONWIP (2), and MRP (3). ................. 171

7-16: The Cook's Distance versus the index of the data points for the
regression model for Kanban, including machine failure........................ 176

8-1: Cycle time and average cycle time dependent on the number of
processed entities for CONWIP............... ........................ 185

8-2: Cycle time and average cycle time dependent on the number of
processed entities for Kanban. ...... ....... .................... 186

8-3: Cycle time and average cycle time dependent on the number of
processed entities for M RP....... ....... ........ .................... 187

8-4: Correlogram for MRP indicating the correlation dependent on the
lag num ber. ....................................................................................... . . 190

8-5: The coefficient of variation dependent on the interarrival time for
M R P ............................................................................................................ 1 9 3

8-6: The coefficient of variation and the utilization dependent on the
interarrival time for MRP with batch size one and setup time
2 0 0 ...................................................................................................... . . . 1 9 6


xviii













Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

A COMPARATIVE SIMULATION STUDY OF
KANBAN, CONWIP, AND MRP MANUFACTURING
CONTROL SYSTEMS IN A FLOWSHOP


By

Thomas Alfons Hochreiter

May, 1999

Chairman: Dr. Suleyman Tufekci
Major Department: Industrial & Systems Engineering

The globalization of markets due to the improvement of communication and

transportation media has had a significant impact on manufacturing technology in

recent years. The strong international competition forced companies to establish

efficient production facilities ensuring profitability on the long run. The performance

of the most prevalent American manufacturing control mechanism, MRP, was

questioned after the success of the Japanese Kanban control system during the Just-In-

Time era. CONWIP, a generalization of the Kanban control system, was introduced as

a result of extensive research done to understand manufacturing systems with the aim

of improving their efficiency.

During an extensive simulation study, the performances of Kanban, CONWIP,

and MRP were evaluated for a ten identical machine tandem line with respect to batch

size, setup time, and machine failure. The utilization (throughput) was kept constant

for all control systems. The parameters were introduced to the models one at a time,










thereby increasing the realism and the variability of the manufacturing line. Thus, the

performances of the three control mechanisms were explored on three levels of

complexity. Initially, only the influence of batch size on the performances of the

control systems was investigated. Then, the setup time was taken into consideration in

addition to the batch size. Last, machine failure was introduced to augment the

models' realism resulting in a higher practical applicability. On each level, the

performances were evaluated for steady-state, assuming the manufacturing line would

run indefinitely. In addition, the response of the performance to machine failure was

observed dynamically while keeping batch size and setup time constant.

Although the performance differences were found to be minute, Kanban and

CONWIP were outperformed by the traditional control system, MRP, for experiments

with varying batch size and for experiments including both batch size and setup times.

On the highest level of variability, with machine failure introduced, Kanban was

ranked first, closely followed by CONWIP. The two pull systems easily outranked the

push system when evaluated according to average cycle time, maximum cycle time

and the standard deviation of cycle time. Kanban performed best for the dynamic

response to failure as well, where the system performance was measured by the time

taken to recover from failure.


xx














CHAPTER 1
INTRODUCTION



1.1 Motivation


Primarily due to rapid development of technology in the past thirty years, the

market structure throughout the world has changed considerably. Local markets have

become accessible to foreign investors, who are not only able to perform well in their

newly established territory, but, who are even able to excel because of superior

technology. Successful companies embedded globalization in their expansion

strategies, consistently seeking for new markets abroad. Consequently, manufacturing

companies are facing global competition, forcing them to keep up with new concepts

and even to proactively incorporate improvement into their daily production routine.

In 1972 the American Production and Inventory Control Society (APICS) strongly

promoted material requirements planning (MRP) in an effort to strengthen the

American manufacturing industry and its standing in the international arena. MRP

was hoisted to the most prevalent production control system on a national level. After

the successes of Just-In-Time (JIT) its dominant appearance in industry was

questioned. The Japanese had introduced their superior products manufactured with

the support of the Kanban control system enhancing their global competitiveness. An

enormous amount of research was directed towards the new system giving rise to a

rich body of literature documenting various concepts.










In 1990 another system, striving to maintain a constant work in process

(CONWIP), was presented, able to prove its usefulness in theory and in industry. The

extensive research produced ample knowledge of system's behavior and good

understanding of the factors involved. The newly evolved science, Factory Physics,

attempts to describe and formalize the characteristics of the extreme probabilistic

systems.

However, the models analyzing and comparing the different control systems

analytically are based on too many simplifying and unrealistic assumptions. The

results can merely serve as approximations of real systems, a very limiting attribute

for their practical applicability. Simulation has established itself as a very powerful

alternative to the analytical modeling process. With the reduction in computer

hardware prices and the increase of processor speed, simulation has become a popular

tool in recent years. It enables modeling with great precision resulting in a very good

representation of real systems and trustworthy output data. The simulations software

available allows the study of manufacturing systems dynamically, giving the analyst a

feeling for the system in addition to generating realistic results.

In this research paper the three control systems Kanban, CONWIP, and MRP are

analyzed by means of a comparative simulation study. Ever since the introduction of

Kanban to the world of production, MRP has been discredited as an inferior control

system. However, despite its significant success, Kanban is not flawless. CONWIP is

investigated as a highly praised alternative. An evaluation of their performance with

respect to batch size, setup time and failure should unveil the superior control system

for the chosen manufacturing line.















1.2 Thesis Outline


Chapter 2 highlights the mechanisms and characteristics of the control systems,

Kanban, CONWIP, and MRP. A comparison regarding specific attributes reveals

basic differences that support the existence of all three control systems. Chapter 3

introduces simulation as the alternative to analytical modeling of manufacturing

systems. It denotes the important aspects of a simulation study. Chapter 4 serves as a

reference to both, statistical analysis methods unique to simulation, and methods

common to general data interpretation. In Chapter 5, the influence of batch size on the

performance of the control systems is demonstrated. In Chapter 6, setup time is

included in the investigations. Chapter 7 deals with the manufacturing system with the

highest degree of realism, including batch size, setup time and failure. The response

of the system to failure dependent on time is analyzed as well. Chapter 8, summarizes

calculations performed to ensure a high accuracy of the output data on a 95%

confidence level while Chapter 9 encompasses the conclusions and suggestions for

future work.














CHAPTER 2
CONTROL SYSTEMS


A brief theoretical background on the three manufacturing control systems is

given in this Chapter. The purpose is to primarily elaborate on the characteristics

unique to the individual control systems and their differences and to secondarily

explain their most important mechanisms.




2.1 Push And Pull Systems



Spearman and Hopp [HOP96, p.316] give a very describing quote of Taiichi

Ohno, the father of Just-in-Time (JIT), to distinguish the meaning of the two terms,

push and pull:

Manufacturers and workplaces can no longer base production on desktop planning

alone and then distribute, or push, them onto the market. It has become a matter of

course for customers, or users, each with a different value system, to stand in the

frontline of the marketplace and, so to speak, pull the goods they need, in the amount

and at the time they need them [OHN88, xiv].

This global perspective can be applied to any individual manufacturing system.

The following definition gives a general and thus abstract explanation of the words:

A push system schedules the release of work based on demand, while a pull system

IuII/v i:- e' the release of work based on system status [HOP96, p.317].











This means that a push system releases an entity to the line according to the

exogenous master production schedule (MPS). The release time is not modified for a

change in the manufacturing system [see Figure 2-1]. Information flows from the

MPS downstream towards the finished goods inventory.




Raw Finished
Material Goods


~---- -- --------- ------------ --- - -


--- Physical Flow
--.-- Information Flow

Figure 2-1: A push manufacturing system.



A pull system, however, only allows an entity to enter the system when a signal

generated by a change in the line status calls for it. This change results in the most

cases from the departure of an entity from the line [see Figure 2-2]. Information flows

from the finished goods inventory, the customer, upstream towards the raw material

inventory.




Raw Finished
Material Goods


-- - --- ------ -- - - --- --- -


--- Physical Flow
----- Information Flow


Figure 2-2: A pull manufacturing system.










The performance of the two systems is dependent on scheduling rules as well.

Here the most prevalent one, fist come first serve (FCFS), will be assumed

throughout. Extensive simulations done by Hum and Lee for JIT systems reveal no

dominant rule. However, the results seem to indicate that FCFS is not necessarily

justified, its weakness becomes most apparent under tight production conditions.

According to them, the user should not arbitrarily adopt a scheduling rule. Instead, the

nature of the scheduling rule and the production environment should be understood

[HUM98].

As the release of material to the line is initiated by the MPS in MRP, the

manufacturing system is controlled by the release rate of material resulting in a

specific throughput. The pull systems on the other hand only allow material into the

system when a card is liberated, a consequence of a reduction in work in process

(WIP). Thus, they control the system by managing the WIP and putting an upper

boundary on the material present in a line.

Kanban and CONWIP are the pull systems discussed here. Their performance will

be compared with the performance of MRP, the most prevalent push system.

Before a comparison of their characteristics can be made, Kanban, CONWIP, and

MRP are discussed as a basis of a practical control system in the following chapters.




2.2 Kanban

Mostly the Toyota-style Kanban system is discussed as a pull system and it is

hardly surprising that the term pull is commonly viewed as synonymous with Kanban










[SCH82]. There is an immense Kanban literature often comparing its performance to

a push system driven by unreliable demand forecasts [BER92].

In a Kanban system, production is triggered by demand. When a part is removed

from the final inventory point, the last workstation in the line is given authorization to

replace the part. This workstation in turn sends an authorization signal to the upstream

workstation to replace the part it just used. This process continues upstream,

replenishing the downstream void by requesting material from the antecedent

workstation. To control information transfer, the operator requires both parts and an

authorization signal, a card, to work.




2.2.1 The Mechanism

The Kanban system simulated here makes use of one inventory storage point and

requires only one card per station. The Kanban system developed at Toyota makes use

of a two-card system requiring a production card and a move card per station [see

HOP96, p.163]. Figure 2-3 illustrates the one-card Kanban system.







.------ P3 1 ----
5p


0 Workstation E Outbound Stockpoint D Standard Container P Kanban Card


Figure 2-3: The one-card Kanban system.










The operator finds a card in the hold box at workstation J (1). He/she gets material

from the outbound stockpoint of the upstream workstation I (2). The card attached to

the material is removed and placed into the hold box of the upstream workstation (3).

The material enters the manufacturing process and the card in the hold box is attached

to the product placed in the outbound stockpoint (4). The operator at the upstream

workstation I finds the card in his/her holdbox and starts processing (5). The same

cycle is followed for the upstream machines until the raw material inventory is

reached [see Figure 2-2]. A Kanban system can be seen as a closed queuing network

with blocking. Jobs circulate around the network indefinitely. However, unlike the

CONWIP system [see 2.3.1], the Kanban system limits the number of entities per

workstation, since the number of production cards at a station establishes a maximum

WIP level for that station. Each production cards acts exactly like a space in a finite

buffer in front of the workstation. The upstream workstation is blocked when the

buffer is full [HOP96, p.325].

Berkley shows that a common model of a Kanban system is equivalent to a

traditional tandem production line with finite buffers. His model assumes that kanbans

travel instantly to their destinations when they are detached from a part, and that the

kanbans and parts travel in quantities of one [BER91]. Gstettner and Kuhn describe

and classify different Kanban systems. They analyze the system with respect to

production rate and average work in process [GST96].










2.2.2 Characteristics

As the amount of material in the system is limited to the number of cards

assigned, there is a natural upper bound of material in process.

Due to the presence of the cards the involvement of the operators in controlling

the flow of material is enhanced. This involvement and active participation paired

with a proactive thinking enables continuous improvement not necessarily given for

the push systems.

A Kanban system suits a stable material flow best. The product mix should be

fairly stable and not too large as the cards are unique to certain products and

expensive in their introduction to a system.

Kanban is not useful in an environment with expensive items that are rarely

ordered, since it would require at least one of each kind of item to be in inventory at

all times.

The performance is very sensitive to the number of cards assigned to the system

and their specific allocation. Gstettner and Kuhn show that the distribution of cards

has a significant effect on the performance of Kanban systems. According to them,

the different types of Kanban control mechanisms show equivalent performance data,

if the distribution pattern is adapted accordingly [GST96].

In most Kanban systems the number of cards assigned to specific workstations is

fixed resulting in blockages or starvation. Blocking occurs when all the cards are

attached to full containers in the outbound stockpoint, while starvation occurs when at

least one production Kanban is in the hold box waiting for a container from the

upstream workstation while the machine at that station is idle. Gupta and Al-Turki










have developed an algorithm to implement a flexible Kanban system adjusting the

number of cards to stochastic processing times and a variable demand environment

[GUP97].

Mascolo et al. show that the performance of a multi-stage Kanban system can be

derived from evaluating a set of subsystems. The subsystems result from a

decomposition of the original line, where each set is being associated with a particular

stage. Numerical results show that the method is fairly accurate [MAS96].




2.3 CONWIP

The CONWIP (CONstant Work In Process) control system strives to maintain a

constant work in process. It was first introduced by Spearman et al. in 1990 and can

thus be classified as a very new control concept [SPE90].




2.3.1 The Mechanism

CONWIP can be considered a special case of Kanban, where the entire line

constitutes one workstation. Departing jobs send production cards back to the

beginning of the line to authorize release of new jobs.










Raw Finished
Material Goods

A B .. L
E 3 1T 0

2




0 Workstation I Parts Buffer D] Standard Container P Card

Figure 2-4: A CONWIP production line.


The finished product is taken out of the inventory that is fed by workstation L (1).

The production card is sent back to workstation A to authorize the release of a new

job (2). The operator at the upstream workstation A finds the card, gets the raw

material from the inventory and starts processing the unit (3). In a Kanban system,

each card is used to signal production of a specific part. CONWIP production cards

are assigned to the production line and are not part number specific. Part numbers are

assigned to the cards at the beginning of the production line. The numbers are

matched with the cards by referencing a backlog list. When work is needed for the

first process center in the production line [see Figure 2-4, (3)], the card is removed

from the queue and marked with the first part number in the backlog number for

which raw materials are present [SPE90].

Here, the following simplifying assumptions are made for CONWIP:

1. The production line consists of a single routing, along which all parts flow, and

2. WIP can be measured in units (i.e., number of jobs or parts in the line).










Spearman and Hopp [HOP96, p.324] remark that a CONWIP system resembles a

closed queuing network, in which entities never leave the system, but instead circulate

around the network indefinitely. In reality, the entering jobs are different from the

departing jobs. Assuming that all jobs are identical, this difference does not matter for

modeling purposes. Gstettner and Kuhn mention that the model developed by

Spearman et al. [SPE90] can be refined and adapted to different production

environments as done by Duenyas and Hopp [DUE92] and Duenyas et al. [DUE93]

[GST96]. Huang and Wang show by means of simulation that the CONWIP

production control system is very efficient for the production and inventory control of

semi-continuous manufacturing, such as that found in a steel rolling plant [HUA97].




2.3.2 Characteristics

As does Kanban, CONWIP controls the total amount of work in process in the

system. The WIP is limited to the number of cards assigned to the entire line instead

of to the individual machines.

If a machine fails in a CONWIP line, the amount of material downstream of it will

eventually be flushed out of the system by the demand process. These demand events

will cause the release of new entities to the system. If the machine fails for a long

period of time, these entities and the entities already in the system upstream of the

failed machine will accumulate in the buffer immediately upstream of the failed

machine. The release of the new jobs to the system stops once no more cards are

released from entities departing the system [BON97].










There is no blocking in CONWIP lines since buffers are assumed big enough to

hold all parts that circulate in the line [GST96].

In CONWIP systems information about demand is sent directly from the last to

the first station. The entity goes through all the workstations in the line carrying the

information about necessary production.




2.4 MRP

The promotion of material requirements planning (MRP) by the American

Production and Inventory Control Society (APICS) in 1972 boosted this production

control paradigm to the most prevalent system today. Only after the successes of JIT

and Kanban its dominant appearance in industry was questioned.




2.4.1 The Mechanism

As can be derived from its name, MRP plans material requirements. It deals with

the two dimensions of production control: quantities and timing. The system must

determine appropriate production quantities of all types of items, from final products

that are sold, to components used to build final products, to inputs purchased as raw

materials. It must also determine production timing that facilitates meeting order due

dates.



























Figure 2-5: Simplified schematic of MRP.


The data from the bill of material (BOM) and the master production schedule

(MPS), as the source of demand for MRP, is processed in several steps to produce the

planned order releases and notices such as change notices and exception notices [see

Figure 2-5]. The BOM describes the relationship between end items and lower level

items while the MPS gives the quantity and due dates for all parts to obtain the gross

requirements. The schematic is presented to illustrate that all the information needed

for the entire manufacturing system originates from the MPS.










Raw Finished
Material Goods



F- I I I I






O Workstation ] Unlimited Parts Buffer D Entity

Figure 2-6: A MRP production line.


The order is released at the raw material post (1) as planned with the help of the

MPS [see Figure 2-6]. As the entity is released independent of the amount of the

material in the buffer preceding Workstation A, the buffer size may not be limited to a

specific amount of entities. Mostly constraints are given by physical space on the

manufacturing floor. When workstation A is finished with processing the entity, it

pushes it on to the next workstation, B (2). This process continues downstream until

the entity departs the system at the finished goods post.

To be able to address the huge problem of coordinating thousands of orders with

hundreds of tools for thousands of end items made up of additional thousands of

components manufacturing resources planning (MRP II) was developed [HOP96,

p. 143]. It provides a general control structure that breaks the production control

problem into a hierarchy based on time scale and product aggregation, thus, primarily

taking the capacity of the manufacturing system into account. MRP II brings together

many functions to generate a truly integrated manufacturing management system










including demand management, forecasting, capacity planning, rough-cut capacity

planning, dispatching and input/output control.




2.4.2 Characteristics

MRP provides a simple method for ordering materials based on needs, as

established by a master production schedule and bills of material. As such, it is well

suited for use in controlling the purchasing of components. However, in the control of

production MRP shows deficiencies [HOP96, p. 143]. This is especially true for

manufacturing systems that require proper exploitation of capacity resources by

taking bottlenecks into consideration.

According to Spearman and Hopp the real reason for MRP's inability to perform

well is the faulty underlying model. The key calculation is performed by using fixed

lead times to derive releases from due dates. These lead times are functions of the part

number only. They are not affected by the status of the plant. More importantly, the

lead times do not consider the loading of the manufacturing system. An MRP system

assumes that the time for a part to travel through the plant is the same whether the

plant is empty or overflowing with work, which is only true for infinite capacity.

Furthermore, to ensure the coordination of parts at assembly, there is a strong

incentive to increase the lead times to provide a buffer against unforeseen

obstructions. However, as inflating lead times introduces more material into the

system, it increases congestion and consequently the cycle times. Instead of delivering

on time, the products are delayed even more [HOP96, p. 175].










As quoted by the APICS literature, MRP's bad performance in industry was

blamed on inaccurate data, including bills of material and inventory records. MRP

requires a high standard of data integrity to function properly [LAT81].




2.5 Comparison of CONWIP with MRP

As mentioned previously [see 2.1], a push system controls throughput and

observes WIP, while a pull system controls WIP and observes throughput. WIP is

directly observable, while throughput can only be determined indirectly. The jobs on a

shopfloor can be physically counted and maintained according to the WIP cap. In

contrast, the release rate for MRP must be set with respect to capacity. If the rate is

chosen too high, the system will be congested with material resulting in high cost due

to insufficient throughput and high WIP. As estimating capacity is very difficult,

optimizing a push system is much more intricate [HOP96, p.325].

Concerning the efficiency, Spearman and Hopp state the following law:

For a given level of throughput, a push system will have more WIP on average than

an equivalent CONWIP system [HOP96, p.327].

The law is supported by a calculation for a simple example of a five machine

tandem line and exponentially distributed process times with mean one hour.

According to Spearman and Hopp MRP systems have more variable cycle times

than equivalent CONWIP systems [HOP96, p.327]. As the total amount of WIP in a

line is fixed, the WIP level at the individual stations are negatively correlated. As the

WIP level increases at one station, it decreases at all the other stations, which tends to

dampen the fluctuations in cycle time. In contrast, WIP levels at the individual











stations are independent of one another for MRP. The WIP level at one station reveals

no information about the WIP levels at the other stations. The overall WIP level may

become extremely high or even low, resulting in great variability of the cycle times

that are directly dependent on the WIP level.

Spearman and Hopp state another law to express the robustness of the two

systems:

A CONWIP system is more robust to errors in WIP level than MRP is to errors in

release rate.

The law is verified with the help of a simple profit function dependent on the

throughput and the WIP level expressed in terms of percent error. The coefficients are

calculated from empirical data, revealing the functions given in Figure 2-7 [HOP96,

p.329].





--- Conwip
Mrp


2 --
0- 60 -- --------------------.




100

Control [% of optimal]


Figure 2-7 Relative robustness of CONWIP and MRP.



The profit function for CONWIP is very flat between WIP levels as low as 40%

and as high as 160% of the optimal level. The MRP function declines steadily when










the release rate is chosen at a level below the optimum and falls off sharply when the

release rate is set even slightly above the optimum level.




2.6 Comparison of CONWIP with Kanban

Both CONWIP and Kanban are pull systems since new order releases are

triggered by external demand. As both systems control the WIP and limit the level by

an upper bound, they show similar performance relative to the push system, MRP.

Gstettner and Kuhn reveal in their comparisons between Kanban and CONWIP

that Kanban is more flexible with respect to a certain objective than CONWIP. Not

only does the absolute number of cards matter, but, the card distribution is another

parameter that influences performance. Selecting a favorable card distribution showed

that in a Kanban system a given production rate is reached with less WIP than in a

CONWIP system [GST96]. However, Spearman et al. point out that by allowing WIP

to collect in front of the bottleneck, CONWIP can function with lower WIP than

Kanban [SPE90].

As there is no blocking in CONWIP lines it can easily be understood that a

CONWIP system with n cards will have a higher production rate than a Kanban

system with n cards [SPE92].

According to Spearman and Hopp the most obvious difference is that Kanban

requires setting more parameters than does CONWIP [HOP96, p.330]. In a one-card

system a card count must be established for every workstation, in a two-card system

twice as many. In a CONWIP system the amount of cards is set for the entire line,

which needs to be established only once. Coming up with the optimal card count










requires a combination of analysis and continual adjustment, making it a great deal

easier to find the right configuration for the CONWIP system.

Cards are part number specific in a Kanban system and only line specific in a

CONWIP system. Instead of being matched to a specific part at the upstream

workstation, the cards are matched against a backlog [see 2.3.1], which gives the

sequence of parts to be introduced into the line. Thus, in its pure form, a Kanban

system must include standard containers of WIP for every active part number in the

line to which the cards can be matched. For a high number of parts, although only

occasionally produced, this implies a very high overall WIP level swamping the

manufacturing system [HOP96, p.330]. Gstettner and Kuhn elaborate on this

difference as well, neglecting special release mechanisms in the CONWIP system

which are based on a MPS [GST96]. In a paper Spearman et al. mention that although

the backlog affords the opportunity for control, it also provides a tremendous

challenge. The backlog sequence is the key to assuring adequate capacity when there

are significant setups and to optimizing synchronization of production of part

components [SPE90].

Hall points out, that Kanban is applicable only in repetitive manufacturing

environments [HAL83]. Spearman and Hopp explain repetitive manufacturing by

systems where material flows in fixed paths at steady rates [HOP96, p.331]. They

mention that large variations in either volume or product mix destroy this flow, at

least when parts are viewed individually, and hence seriously undermine Kanban. In

another publication Spearman et al. mention that the JIT environment provided by

CONWIP can accommodate a changing product mix as it is suitable for short runs of










small lots. Furthermore, they find this environment to be more predictable than its

pendant provided by Kanban [SPE89]. A CONWIP system is more robust due to the

planning capability introduced by the process of generating a work backlog.

Spearman and Hopp mention prevalent employee issues differentiating CONWIP

and Kanban. The pull mechanism at every workstation results in great operator stress

as described by Klein [KLN89]. When the operator receives a card having to wait for

the material to start processing, the void has to be replenished as quickly as possible

upon arrival of this material. This is only true for the first workstation in a CONWIP

system. The other station function according to a push system where the operators are

subjected to less pacing stress [HOP96, pp.332-333].

The previous comparisons illustrate the advantages of CONWIP over MRP and

Kanban. Most fundamentally, the differences between the pull and the push systems

can be utilized as an advantage to building a manufacturing system that encompasses

the positive attributes of the different mechanisms. The result is an integration of the

systems to compensate for the weaknesses on both sides. According to Titone

integration of various functions into a total comprehensive manufacturing strategy

leads to world-class manufacturing and profits. Using MRP II for planning and JIT for

the execution combines two powerful tools into an efficient manufacturing system

[TIT94]. Wang et al. introduce an experimental push/pull production planning and

control software system which is designed as an alternative to a MRP II system for

mass manufacturing enterprises in China [WAN96].

Bonvik et al. compare a two-boundary hybrid system to conventional systems.

The system is a hybrid of basestock and Kanban control. Basestock control limits the










amount of inventory between each production stage and the demand process. Each

machine tries to maintain a certain amount of material in its output buffer, subtracting

backlogged finished goods demand, if any [KIB88]. For the hybrid system demand

information is propagated directly as in basestock control and inventory at the

individual workstations is limited as in Kanban control. The hybrid control policy

demonstrated superior performance in achieving a high service level target with

minimal inventories [BON97].

The three control mechanisms were evaluated by means of simulation as the

analytical methods available serve as approximations limited to special cases not

applicable to more complex systems.














CHAPTER 3
SIMULATION


Simulation refers to a broad collection of methods and applications to mimic the

behavior of real systems, usually on a computer with appropriate software. Since

computers and software have evolved tremendously in recent time, simulation has

become very powerful and popular [KEL98, p.3]. Simulation, like most analysis

methods, involves systems and their models. A system is a facility or process, either

actual or planned. It is a collection of elements that cooperate to accomplish some

stated objectives. A model is a collection of symbols and ideas that approximately

represent the functional relationship of the elements in a system [BA198, p.2]. The

system is studied to measure its performance, improve its operation or to determine an

optimal design. As sometimes the primary goal is to focus attention on understanding

how a system works, the results after the modeling process may become irrelevant.

Often, simulation analysts find that the process of defining how a system works,

which must be done before developing a model, provides great insight into the

mechanisms of the system.

From a practical viewpoint, simulation is the process of designing and creating a

computerized model of a real or proposed system for the purpose of conducting

numerical experiments to improve the understanding of the behavior of that system

for a given set of conditions [KEL98, p.7].










Here, the purpose of the simulation was to evaluate the behavior of the system

under different sets of conditions by using the models to carry out groups of

experiments. The simulations primarily provided estimates of the statistics of system

performance. The systems, Kanban, CONWIP, and MRP, were modeled by a ten

identical machine tandem line and exponential distributed process time with mean 20

seconds. Indeed, the modeling process gave great insight into the mechanisms of the

systems creating a feeling for their behavior.

Yavuz and Satir reviewed selected published research on Kanban-based

operational planning and control in assembly and flow lines. Their article focuses on

simulation models and distinguishes between explorative and comparative type

research. Operational and experimental design features are summarized in tabular

format giving a good overview of work done in this area [YAV95].




3.1 The Software

Two simulation tools were used to conduct the experiments: EFML and Arena.




3.1.1 EFML

The Emulated Flexible Manufacturing Laboratory (EFML) was developed in the

Department of Industrial & Systems Engineering at the University of Florida. The

originating concept was to develop a hands-on environment where students and

companies could test and study manufacturing operations in a factory setting, giving










students and managers the ability to test the performance of a manufacturing facility,

which could be distributed over several computers.

The EFML is composed of a network of personal computers linked together

through the Virtual Manufacturing Software, which enables the communication of the

computers via the TCP/IP protocol and the internet. The software is written with

Borland's Delphi Developers Toolkit based on an object oriented architecture. The

objects machine, dispatch/raw material inventory storage, repair and maintenance

facility, transportation, assembly line, and finished goods inventory storage can be

assigned to different computers to construct a complete factory. The object

architecture is illustrated in

Figure 3-1.


Figure 3-1: The object architecture for the EFML.










As the dispatch object releases material to the shop floor, based on predetermined

release times, the behavior of each factory component can be observed in real time.

According to Mijon the advantage of the EFML over traditional simulation software

is the visual interface providing meaningful output. This output lets the viewer see

where the problem is arising and potentially the reason for its occurrence [MIJ97, p.

3].

The EFML is an evolving system which is continuously improved, adding more

features to increase the realism of the system and to enhance user friendliness even at

the time of writing this thesis.




3.1.2 Arena

Arena combines the ease of use found in high-level simulators with the flexibility

of simulation languages down to general-purpose procedural languages like the

Microsoft Visual Basic programming system, FORTRAN, or C. It does this by

providing alternative and interchangeable templates of graphical simulation modeling-

and-analysis models that one can combine to build a fairly wide variety of simulation

models. For ease of display and organization, modules are typically grouped into

panels to compose a template. By switching templates one can gain access to a whole

different set of simulation modeling constructs and capabilities. In many cases,

modules from different panels and templates can be mixed together in the same

model. The modules in Arena templates are composed of SIMAN components. Arena

maintains its modeling flexibility by being fully hierarchical, as depicted in Figure

3-2.















User-Created Templates
.-- Commonly used constructs.
Company-specific processes.
Company-specific templates.
Etc.


Application Solution Templates .
Call$im -5
BP$im
Etc.



Common Panel
2 Many common modeling constructs.
SVery accessible, easy to use.
o Reasonable flexibility. E
0 E)
-s i-
-- Support, Transfer Panels
Access to more detailed modeling for greater flexibility. <



Blocks, Elements Panels 0-
All the flexibility of the SIMAN simulation language. I
I--


User-Written Visual Basic, CIC++, FORTRAN code
SThe ultimate in flexibility.
0o C/C++/FORTRAN requires compiler.





Figure 3-2: Arena's hierarchical Structure.




Arena includes dynamic animation in the same work environment. It also provides


integrated support, including graphics, for some of the statistical design and analysis


issues that are part of a good simulation study [KEL98, p. 13].


The models for Kanban, CONWIP, and MRP were created with the Blocks and


Elements Panels to utilize all the flexibility of the SIMAN simulation language.


EFML and Arena served as the framework for the simulation study, which is


introduced next.






28






3.2 The Simulation Study

Issues related to design and analysis and representing the model in the software

certainly are essential to a successful simulation study. However, there are more

aspects that should be taken into consideration. Following the flowchart in Figure 3-3

should improve the chances of conducting a successful study.















































Figure 3-3: Flowchart of a simulation study.


The simulation study does not necessarily have to exactly follow the given

flowchart, there is no general formula to guarantee success. It rather gives a rough

path to follow. Here, the identification of a problem can be omitted directly

proceeding to the second step, stating the objective.













3.2.1 State Objective

The objective is to compare the performance of the three manufacturing control

systems: Kanban, CONWIP, and MRP. The comparison should involve three main

parameters influencing the performance of a manufacturing system:

* Batch size,

* Setup time, and

* Machine failure.

To observe the influence of the individual parameters without any blurring

interaction between one another, the central parameter of this study, batch size, is

introduced first. The complexity of the models is increased steadily by adding setup

time and failure in two further steps. This process allows to build new investigations

on the knowledge gained during prior steps improving the realism with the increasing

number of parameters.

After determining the objective of this study, the focus had to be directed on the

input data.




3.2.2 Collect/Prepare Data

The data is produced by the random number generator provided by the software

packages. The Arena random number generator was tested by applying the chi-square

test of uniformity to the numbers generated. The null hypothesis of uniformity was

not rejected at level a = 0.10 revealing that the numbers generated didn't behave in a










way significantly different from the expectations for truly independent and identically

distributed random variables [BA198, p.60]. Similar behavior was expected from

EFML. As previously mentioned, the exponential distribution function was chosen as

the input distribution function. This distribution function is commonly used for

simulations on manufacturing systems as it has the remarkable memoryless property,

where the past history of a random variable, that is distributed exponentially, plays no

role in predicting its future [KLE75, p. 66]. Unlike most other probability

distributions, the shape of the exponential distribution is governed by a single

quantity. Further, it is a distribution with the property that its mean equals its standard

deviation [MCC94, p.250].




3.2.3 Formulate Models

The models of the systems were built according to the descriptions previously

given. Figures 1-3, 1-4, and 1-6 depict the graphical models of Kanban, CONWIP,

and MRP respectively. For each control system 4 models were created to enable

simulations on the 4 levels including the following parameters:

* Batch size,

* Batch size and setup time,

* Batch size, setup time, and failure (dynamic response), and

* Batch size, setup time, and failure (in steady state).

A few assumptions were made to simplify the simulation process, unfortunately

resulting in a less realistic system. The most important assumptions were the

following:










* The 10 stages are in series, i.e., each stage has only one supplier and one

consumer,

* There is an infinite supply of raw parts at the input of the production system,

* The systems are saturated, there are always demands for finished parts,

* Information is transmitted instantly,

* Transportation within and between workstations is instantaneous,

* The system produces a single part type,

* Kanbans are associated with batches and not with individual entities, and

* Any kanban detached at the output of a stage is immediately available for the

upstream stage, there is no return delay.

More assumptions may result implicitly from those given above.




3.2.4 Verification of the Models

The three basic models were verified with the EFML output. EFML was verified

formally. However, the output data has not been verified before with another

simulation software, therefore making this verification process an especially

interesting task.

For both Kanban and CONWIP 25 replications were run on Arena and EFML. For

MRP 30 replications were carried out. The configurations are given in Table 3-1. The

interarrival time corresponds to batch interarrival times.













Table 3-1: Configuration for Kanban, CONWIP, and MRP to verify correctness of the
models.

Control System Process Time Batch Size Number of Cards Interarrival Time
Kanban 20 4 20
CONWIP 20 4 20
MRP 20 5 105


A paired t-test [see 4.2.3] was performed on the output data to test the following

hypothesis:

Ho: True mean of average cycle time differences is equal to 0, and

Ha: True mean of average cycle time differences is not equal to 0,

to calculate the 95% confidence interval. The statistics are given in Table 3-2.


Table 3-2: Statistics on t-test to verify concurrence of output between EFML and
Arena for Kanban, CONWIP, and MRP.

System t-value df p-value Interval Estimate of Average
mean of diff. Cycle Time
Kanban 0.4048 24 0.6892 (-3.6486; 5.4292) 0.8903 1608.517
CONWIP 0.2335 24 0.8174 (-2.2046; 2.7671) 0.2812 1848.7355
MRP -0.164 29 0.8709 (-115.2965; 98.1806) -8.5580 4166.046


All of the intervals include the value 0 resulting in the failure of rejecting the null

hypothesis. The 95% confidence intervals indicate a small deviation of the average

cycle times for Kanban and CONWIP.

For MRP the interval calculated is considerably bigger, even evaluated relative to

the average cycle time. Here CONWIP presents a very small deviation. The reason for

the strong deviation of MRP is the varying average cycle time, even after a big

amount of entities have passed through the system. The half-width for the confidence

interval indicated [see Table 8-4], that 10,000 entities would result in an accurate











estimation of the cycle time. Although Figure 3-4 reveals that the average cycle time

for 10,000 entities produced has approached a fairly stable value, it is still varying for

bigger numbers.







6000




7 4000
I-



2000-

Cycle Time
-- Average Cycle Time

0-
0 5000 10000 15000 20000 25000 30000
Number of Processed Entities

Figure 3-4: The cycle time per entity and the cumulative average cycle time
dependent on the number of processed entities for MRP with Arena.



Even after 20,000 entities processed the average is moving, indicating that the

random generator has an influence on the output for Arena. The same behavior is

expected for EFML, as both simulation tools don't generate true random numbers.

This fact could explain great deviations even for a high number of replications

completed [see Figure 3-6].

To get an impression of how the systems would behave for different

configurations, more simulations were run for varying batch size (1, 2, 4, 8, and 10)

and number of cards (10, 15, 18, 10, and 22) or length of interarrival time (22 645).











The difference was measured as the percentage deviation in average cycle time,

Atcycle


t EFML Arena
Atc = 100 cycle cycle
cycle 1t cEFML
cycle


where tyEm is the average cycle time for EFML and t .e" is the average cycle time


for Arena.

As not enough replications were run to evaluate the output data statistically, scatter

diagrams were constructed to visualize the results.


0.8
0.6
0.4 *
E 0.2 -
oU 0 .

S-0.4

-0.6
.- -0.8 -
-1
-1.2
Batch Size


Figure 3-5: The deviation of the average cycle time between EFML and Arena for
different configurations for CONWIP.


















10

4.


5 6 7 8 9 10


Batch Size


Figure 3-6: The deviation of the average cycle time between EFML and Arena for
different configurations for Kanban.


20

15

10

5

0

-5

-10

-15


A


2 3 4 5 6 1


9 10


Batch Size


Figure 3-7: The deviation of the average cycle time between EFML and Arena for
different configurations for MRP.




Figures 2-5, 2-6, and 2-7 indicate a fairly random output. The shift of data points


for Kanban [see Figure 3-6] can probably not be associated to a software error, as the


points lie above (batch size eight) and below (batch size four) the x-axis. The


3 1


-

-

-










calculations done earlier reveal no significant difference between the outputs for batch

size four and 20 cards assigned. Faulty input data would probably result in a bigger

difference than 2%. The shift may again be attributed to the random generators.




3.2.5 Validation

The models under consideration were representing systems existing in theory

only. Too many parameters were omitted to enable the simulation of a real system,

making validation impossible. Yavuz and Satir mention, that the modeling of real-life

manufacturing environment and usage of empirical data would provide a practical

means of validation for the simulation models developed. The validation was missing

in most of the articles reviewed. Validation would unravel intricacies of

manufacturing that are demystified through mostly gross assumptions [YAV95].




3.2.6 Simulation Experiment Design

Experiments are performed by investigators in virtually all fields of inquiry,

usually to discover something about a particular process or system. Literally, an

experiment is a test. A designed experiment is a test or series of tests in which

purposeful changes are made to the input variables of a process or system to observe

and identify the reasons for changes in the output response [MON91, p.1].

The progression of choice of factors and levels included in the experiments is

discussed at the beginning of the chapters covering the different stages of simulation:

* batch size,










* batch size and setup time,

* and batch size, setup time and failure.

The discussions comprise the following factors, henceforth called system parameters:

* Total number of cards assigned to the entire line, c [see 5.1.3],

* Batch size, b [see 5.1.2],

* The ratio of setup time to process time, r, [see 6.1.1],

* Time between failures (interfailure time), t nfl [see 7.1], and

* Repair duration, trepr [see 7.1].

The levels were determined according to practical applicability, primarily

concerning average machine utilization. First, a high and a low level per factor was

established. Then, the interval [low, high] was divided into segments with a certain

amount of intermediate levels. As the average run time of one replication was

approximately two minutes, the amount of levels was held high, mostly equal to ten.

The total cost could be selected as the primary response variable as the cost affects

the basic goal of a company: making profit. The optimization of manufacturing

resources can reduce costs considerably resulting in a higher profit margin or even a

higher revenue as other market segments are conquered, in turn increasing the overall

market share. This elevates cost to one of the most important indicators if not the most

important indicator for the efficiency of a manufacturing system.

One characteristic makes cost even more useful. It can serve as an overall

indicator, that takes different aspects into consideration, consolidating all the

indicators. However, when several indicators are accumulated to be represented by

one quantifier, the question, how to weigh the individual components, arises. The










weights are most diverging for different industries. Even within one industry, they

may differ considerably, representing the company's unique environment.

A wide variety of functions is available enabling a controller to construct a model

perfectly fitting the needs. Unfortunately, often weights contain error terms and other

parameters that are determined by subjective estimation, making a cost analysis at this

point questionable.

Constructing functions for different scenarios would certainly give more insight

into the problem [AFY98]. But, the gain in investigating other factors was classified

as more important. Furthermore, the regression models can be transformed into cost

functions without greater effort. The construction of more complex models would

certainly be an interesting topic for another thesis that would probably be most

rewarding when written in cooperation with industry.

Consequently, the performance measures were selected as the response variables.

The control systems were evaluated on several criteria utilizing the following

performance indicators:

* Work in process, WIP,

* Throughput, Th,

* Average utilization, u [see 5.2.1],

* Average cycle time, tcycle,

* Time spent in the system (analysis of dynamic response), t system [see 7.2.1

Indicators, Time Spent in System], and

* Recovery time (analysis of dynamic response), trecove, [see 7.2.1 Indicators,

Recovery Time].










The relationship,

Th = WIP
Th=-_-
cycle

is known as Little's Law and is often referred to in manufacturing literature, being

originally derived for a basic queuing system. It was found to be independent upon

any specific assumptions regarding the arrival distribution, the service time

distribution, the number of servers in the system or upon the particular queuing

discipline within the system [KLE75, p. 17]. The formula existed as a "folk theorem"

for many years before Little established its validity in a formal way [LIT61]. The

formula is a useful tool as it can be used to calculate the third unknown indicator

when two indicators are known, independent of system configurations.

The three basic principles of experimental design,

1. randomization,

2. blocking, and

3. replication

were taken into consideration in the following manner:

1. As the system variables and statistics were reinitiated after every replication and

the random number generators were assumed to produce numbers confidently,

behaving like numbers following a true random distribution, the order of the runs

was not randomized.

2. The simulation software and the computer hardware provided an identical

environment for every experiment performed, making experiment blocking

[MON91, p. 9], unnecessary.










3. As most of the simulations were performed for non-terminating systems [see

3.2.7] a large number of entities was produced rather than completing several

replications of the same configuration [see 4.2.2]. Only the analysis done on the

dynamic behavior to failure involved a terminating system [see 4.2.1]. Here, the

number of replications was established prior to the bulk of experiments [see

8.4.1].




3.2.7 Simulation Execution

Depending on the starting and stopping conditions, terminating or non-terminating

simulations can be executed as a natural reflection of how the target system actually

operates. The terminating simulation ends according to some model-specific rule or

condition. For instance, a manufacturing line operates as long as it takes to produce

500 completed assemblies specified by order. According to Kelton et al. the key

notion is that the time frame of the simulation has a well-defined and natural end, as

well as a clearly defined way to start up. A steady-state of non-terminating simulation,

on the other hand, is one in which the quantities to be estimated are defined in the

long run, i.e., over a theoretically infinite time frame [KEL98, p. 177]. For a

manufacturing line that never stops or restarts, a non-terminating simulation is

appropriate.

After initial reflections on parameter settings and several model modifications,

preliminary calculations of the confidence intervals [see 4.2 and CHAPTER 8] were

conducted. These computations were done to ensure high accuracy on the estimation

of the performance indicators. After the completion of the simulations on each of the










levels, the confidence on the indicators was reevaluated. All the calculations carried

out on the confidence were aggregated and documented in a separate chapter not to

disrupt the analysis of the data.




3.2.8 Output Analysis and Interpretation of the Results

The output analysis and interpretation forms the major part of this documentation.

Since simulation was the modeling tool in question, statistical output analyses were

considered in a comprehensive manner. Yavuz and Satir, and Chu and Shih found

these issues to be treated rather lightly in many studies reviewed [YAV95] [CHU92].




3.2.9 Conclusions and Implementation


At the end of the three chapters encompassing the discussions on the stepwise

introduction of batch size, setup time, and machine failure the conclusions drawn

from prior investigations are presented. Conclusions presented within the chapters, are

clearly marked by a heading.

Unfortunately, a few additional factors have to be taken into consideration to

enable simulations of an authentic manufacturing line. However, some findings may

be translated into implementations able to improve productivity and efficiency of a

real production system.

Before proceeding to the actual discussions of the simulations, a fairly

comprehensive but short theoretical background on the statistical analysis methods

used is given in the next chapter. The summary of the statistical theory in one chapter






43



can serve as a review for some readers, but should primarily serve as the source of

reference making explanations within the chapters redundant. Thus, several

clarifications are reduced to one only, and the obstruction of narration is eliminated.














CHAPTER 4
STATISTICS


A simulation is a computer-based statistical sampling experiment [BA198, p.97].

The results of a simulation have to be analyzed with the appropriate statistical

techniques to reveal their full potential. Statistics cannot prove that a factor has a

particular effect. They only provide guidelines as to the reliability and validity of

results. Properly applied, statistical methods do not allow anything to be proved

experimentally, but, they do allow us to measure the likely error in a conclusion or to

attach a level of confidence to a statement. Thus, the primary advantage of statistical

methods is that they add objectivity to the decision-making process. Unfortunately,

the output processes of virtually all simulations are non-stationary and auto-

correlated. Thus, classical statistical techniques based on identical independent

distributed (lID) observations may not be directly applicable. Sometimes, special

techniques have to be applied to ensure the statistical independence of the output data.

Let x1,, x12 ,..,x m be a realization of the random variables X1,X2,..., Xm resulting

from a simulation run of m replications using the random numbers Ul1,u12,.... If the

simulation is run with different sets of random numbers u21,u22,..., a different

realization x21,x22,..., X2m of the random variables X1,X2,..., Xm will be obtained. For

different runs of a simulation, different random numbers are used for each replication.










The statistical counters are reset at the beginning of each replication, which uses the

same initial conditions. Suppose that we make n independent runs of length m,

resulting in the observations:

,11 ... 11 ... 1C m
21 ... X *... nm

xCn ... x z ... x

The observations from a particular replication (row) are not IID due to the nature

of the random generators. However, the observations in the ith column are IID

observations of the random variable X,, i=1, 2, ..., m. This independence across runs

allows the statistical methods discussed below to be used. The goal it to make use of

the observations to draw inferences about the random variables X1,X2 ,..., Xm, the

parameters influencing the performance of the different control systems [BA198,

p.98].




4.1 Transient and Steady-State Behavior

For the output stochastic process X1,X2,... let

F,(xI) =P(X, < xJI), i=1, 2, ...,


where x is a real number an I represents the initial conditions. 1F (x I) is called the

transient distribution of the output process at time i for initial conditions I.

For fixed x and I, the probabilities F, (x I), F2 (x I),... are just a sequence of numbers.


If 1F (xI) -' >F(x)for all x and all initial conditions I, then F(x) is called the










steady-state distribution of the output process X, X2,... Here, if the distributions are

approximately the same after k steps in time, then steady-state is said to start at time k.

However, steady-state does not mean the random variables Xk+1, Xk+2,...will take on

the same value in a particular simulation run. It means that they will have

approximately the same distribution [BA198, p.98].

As mentioned earlier, statistics can not prove the correctness of a certain

statement. Instead, they allow statements to be made with a certain confidence.




4.2 Confidence

The statistical analysis methods differ according to whether simulations are

terminating or non-terminating [see 3.2.7].




4.2.1 Analysis for Terminating Simulations

The data set is given by n independent replications of a terminating simulation.

Each replication is initiated with the same conditions and a different random generator

seed and terminated by a certain event. Thus, independence of the observations is

achieved by a different string of random numbers.

Let Xbe the observation of the ith replication, i=1, 2, ..., n. It is assumed that the

X,'s are comparable for different replications. Consequently, the X,'s can be defined

as identical independently distributed random variables.

For n data points X,, X2, ..., Xn, the sample mean is an unbiased point estimator for

the mean of X represented by the following formula:










n
X,
X(n) =--
n

The 100(1-(x)% confidence interval for the mean is given by





where s2(n) is the sample variance given by

2
[X, -X (n)]
s2()= -=1
n-1

with n-1 degrees of freedom.

Let h be the half-width of the confidence interval of the point estimate,

h __ 2 *n)

n

To ensure the desired accuracy of the estimation,

h < X(n),

where y is a given parameter, 0 < y< 1, here y= 0.1 by default.

After an initial simulation with n replications this condition may not be satisfied.

Additional n2 replications have to be run to reduce the initial half-width h, to the

desired half-width h2 [BA198, p. 103]

For moderately large n1, the sample statistics will remain relatively unchanged

with respect to n, thus,

tnl-1,1-a/2 t n2-1,1-a /2


s 2 (n ) S2 (n2 :










X(n) -X(n2)

Consequently,








4.2.2 Analysis for Non-Termin ating Simulations

Let YI Y,, ... be an output string from a single replication of a non-terminating

simulation. P(Y, y)= ((y) >P(Y : y)=F(y),

where Y is the steady state random variable with distribution F. Due to the initial

conditions, the observations near the beginning of the simulation usually are not

representative of the steady-state behavior. For given observations Y,, Y2, ..., Ymthe

following formula gives a good point estimate of E(Y):

m

Y (n,l)= 1=+1


where I/ stands for the warm-up period and m for the number of observations. I/ and m

are determined such that

Y(nm,/)> E(Y).

The Method of Batch Means is applied to ensure the accurate calculation of a

point estimate for non-terminating systems.

A replication results in observations Y,, Y2, ..., Ym after removing the warm-up

period 1. The m observations are divided into n batches of length k, thus, mnnk. Let

Y' (k) be the sample mean of the k observations in the jth batch. Let













Y(n, k) =1
n m

be the grand sample mean. Then Y(n, k) can be used as the estimate point for E(Y).

The batch size k can be determined by a correlation analysis. k is set equal to the

lag length resulting in a minimal correlation of the data. Should

m
n = -
k

be non-integer, the excess amount of data, e,


e =m- -[n
k

can be truncated.




4.2.3 Paired-t Confidence Interval

The following assumptions have to be made:

1. Each system provides an equal amount of data (n replications),

2. Observations are independent within the systems.

The following descriptions will refer to the two systems as System A and System












Table 4-1: For the paired-t test, comparing two systems is reduced to estimating a
single parameter, the difference.


Replication System A System B Difference
1 xai xb1 dX
2 Xa2 Xb2 d2

n Xan Xbn dn


The confidence interval on the quantity 8, which is the expected value of d, will

enable a comparison between the two systems. Thus, the problem of comparing two

systems is reduced to estimating a single parameter, namely d, [see Table 4-1]. The

resulting confidence interval is referred to as apaired-t confidence interval.

This method is particularly appealing as the following assumptions can be

omitted:

1. Variance of xa = variance of xb (assumption for the two-sample-t method),

2. xa, and xb, are independent.

The confidence interval requires Xai and xa2 to be independent, but correlations

across rows are permitted. The procedure for computing the confidence interval on 8

is exactly the same as for the single-system case:


Sd






n-
snd) *










The half-width for a (1-c) confidence interval on 8 centered at d is then given by

h = t,,_a /2s(d).

The statistic d is an estimate of the difference in the measured performance of the

two systems: if the two systems perform identically, the expected value of d is 0. If

the computed confidence interval contains 0, a difference between System A and

System B can not be reliably stated. However, if the interval does not contain a 0, a

difference between the two systems can be stated with the appropriate confidence

level. If the confidence interval does not contain 0, the two systems differ and the

appropriate system can be selected based on the sign of d .

The authors elaborate on the fact, that if the interval on the difference between the

systems contains 0, the two systems are not necessarily the same. Additional

replications may be required to discern any difference [PEG95, pp. 177].

Another powerful tool to analyze data is regression. As regression describes

statistical relations between variables, it also enables estimation and prediction of data

points.




4.3 Multiple Regression

A regression model is a formal means of expressing the two essential ingredients

of a statistical relation:

1. A tendency of the dependent variable to vary with the independent variable in a

systematic fashion, and

2. A scattering of points around the curve of statistical relationship [NET90, p. 27].










Probabilistic models that include terms involving x2, X3 (or higher-order terms), or

more than one independent variable are called multiple regression models. The

general form of these models is

y =-O + 31xi + P2x2 + ...+ kxk + .

The dependent variable y is written as a function of k independent

variables xl, x2,..., k. x*,x 2,..., xk can be functions of variables as long as the

functions do not contain unknown parameters. The random error term, e, is added to

make the model probabilistic rather than deterministic. The value of the coefficient Pf,

determines the contribution of the independent variable x, and /o is the y-intercept.

The coefficients / Po, ..., fkA are usually unknown because they represent population

parameters

y = PO + Axi + 2 +... + k +k
-, Randomerror
Deterministic part of model

The Least Squares Approach is used to fit the multiple regression models. The

estimated model

P=O +fAx, +...+Pk Xk

minimizes

SSE = (y-)2,

where SSE stands for the sum of square errors.

The sample estimates ,..., k are obtained as a solution to a set of

simultaneous linear equations.










Model Assumptions:

1. For any given set of values of x,, x2, ..., xk, the random error e has a normal

probability distribution with mean equal to 0 and variance equal to o2.

2. The random errors are independent in a probabilistic sense [MCC94, p.744].

0-2 represents the variance of the random error, e. Thus it is an important measure

of the usefulness of the model for the estimation of the mean and the prediction of

actual values of y. If 72 = 0, all the random errors will equal 0 and the predicted

values, 5y, will be identical to E(y), that is, E(y) will be estimated without error. On

the other hand a large value of o'2 implies large values of e and larger deviations

between the predicted values, y), and the mean value, E(y). Thus, o'2 plays a major

role in making inferences about f0 3, ..., 1k, in estimating E(y), and in predicting for

specific values ofx,, x2, ..., k..

Since the variance of the random error will rarely be known, the results of the

regression analysis are used to estimate its value with the following formula

2 2(
n- (k+) '

(k+1) indicating the number of P parameters. This will be referred to as the mean

square for error (MSE). To enable a meaningful interpretation, the standard deviation

s is introduced as a measure of variability



n-(k +1)










4.3.1 Estimating and Testing Hypotheses about the P3 Parameters

Some of the P3 parameters have practical significance in the models formulated in

the following chapters. Thus, their values will be estimated and hypotheses will be

tested about them. Considering the model

y= 30 + P3x + Px2 +e

the following hypothesis could be performed using a t-test:

null hypothesis Ho: /= 0 (No curvature in the response curve.)

against the

alternative hypothesis H,: f32< 0 (Concavity exists in the response curve.).

The t-test utilizes a test statistic analogous to that used to make inferences about

the slope of the straight-line regression model. The t statistic is formed by dividing the

sample estimate, /2, of the parameter, /3, by the estimated standard deviation of the

sampling distribution of /2, S, s


Test statistic: t =
s


For relevant estimated model coefficients f/ the estimated standard deviation

s A and the calculated t values will be given. To find the rejection region for the test

the upper-tail value for t is retrieved from the t-table. This is a t, such that P(-t < -t)

= a. This value can then be used to construct rejection regions for either one-tailed

[see Figure 4-1] or two-tailed tests.

















Rejection Area

Figure 4-1: Rejection region for a test of /2


The numbers given in the following chapters list the two-tailed significance levels

for each t value. The null hypothesis, that the parameter equals to zero, would be

rejected in favor of the alternative hypothesis, that the parameter does not equal to

zero, at any cx level larger than the given number. A 100(1-a)% confidence interval

for a 13 parameter is given by

, -ta/2S

where t/2is based on n-(k+l) degrees of freedom and n observations and (k+1) /

parameters in the model [MCC94, p.746].




4.3.2 Usefulness of a Model: R2 and the Analysis of Variance F-Test

Conducting t-tests on each /P parameter in a model is not a good way to determine

whether a model is contributing information for the prediction of y. When conducting

a series of t-tests to determine whether the independent variables are contributing to

the predictive relationship, it is most likely that an error would be made in deciding

which terms to retain in the model and which to exclude. This may result in including

a large number of insignificant variables and excluding some useful ones. Thus, a










global test that encompasses all the P parameters is needed. Furthermore, it would be

useful to find a statistical quantity that measures how well the model fits the given

data. As this statistical quantity R2, the multiple coefficient of determination, can be

used to calculate the F value. R2 will be introduced first.




4.3.3 Multiple Coefficient of D determination, R2

As the name multiple coefficient of determination indicates, R2 is the equivalent

of r2, the coefficient of determination for the straight-line model [see MCC94, p. 697].

It is defined as the following

R2 1 y y)2 _- Explained variability
Y, (y y)2 Total variablity

where ) is the predicted value of y for the model. R2 represents the fraction of the

sample variation of the y values that is explained by the least squares prediction

equation. R2 = 0 implies a complete lack of fit of the model to the data and R2 = 1

implies a perfect fit with the model passing through every data point. Thus, the larger

the value of R2, the better the model fits the data [MCC94, p. 759].




4.3.4 Variance F-Test


The following test would formally test the global usefulness of the model:

Ho: =1 = ...= = 0

(All model terms are unimportant for predictingy.,

Ha : At least one of the coefficients /, is nonzero










(At least one model term is useful for predicting y).

The test statistic used to test this hypothesis is an F statistic, which can be

calculated with the following formula:

F- R2/k
(1-R2)/I[n-(k +1)]'

where n is the sample size and k is the number of terms in the model. The formula

indicates that the F statistic is the ratio of the explained variability divided by the

model degrees of freedom to the unexplained variability divided by the error degrees

of freedom. The larger the proportion of the total variability accounted for by the

model, the larger the F statistic.

To determine when the ratio becomes large enough that the null hypothesis can be

rejected and the model is more useful than no model at all for predicting, the

calculated F value is compared to a tabled F value:

Rejection region: F > F,, where F is based on k numerator and n-(k+l)

denominator degrees of freedom.

McClave et al. caution the reader that a rejection of the null hypothesis leads to

the conclusion, with 100(l-a)% confidence, that the model is useful. However, useful

does not necessarily mean best. Another model may prove even more useful in terms

of providing more reliable estimates and predictions. Thus, this global F-test is

usually regarded as a test that the model must pass to merit further consideration

[MCC94, p.762]. It will only be used in this sense in the following chapters.










4.3.5 Comparison of two or more Regression Functions

Instead of fitting separate regressions for separate data sets, only one regression is

fitted. This regression gives rise to the same response functions otherwise obtained.

This has the following advantages:

1. Inferences can be made more precisely by working with one regression model

containing indicator variables since more degrees of freedom will then be

associated with the mean standard error (MSE)[NET90, p.355],

2. One regression run on the computer will yield both fitted regressions, and

3. Tests for comparing the regression functions for the different classes of the

qualitative variable can be clearly seen to be tests of regression coefficients in a

general linear model [NET90, p.358].

Here the data sets of the different control systems are accumulated to produce one

data set. Indicator variables (or binary variables) that take on the values 0 and 1 are

used to quantitatively identify the classes of the qualitative variables distinguishing

the control systems. To prevent computational difficulties a qualitative variable with c

classes will be represented by (c-1) indicator variables [see NET90, p.351].

Assuming that a first order model is to be employed it would give rise to the

following function:

y =3o +31x, +/3211i

where

x, = independent variable, and

{1 controlsysteml
0 controlsystem2










The response function of this regression model is:

E(y)= P0 + 1,x, + 2P1,

which can be interpreted as:

E(y) = (o + 2)+ ,11

for the control system 1, and as:

E(y) = o + 1x,

for the control system 2. Thus, P2 measures the differential effect of the type of

control system. It shows how much higher (lower) the mean response line is for the

class coded 1 than the line for the class coded 0, for any given level of x,

This approach is completely general. If three control systems are to be compared,

additional variables are simply added to the model. Furthermore, the differentiation is

not only limited to the y-intercept, but can be introduced to distinguish gradients or

coefficients of variables with higher order.

However, the following assumption has to be made:

The error term variances in the regression models for the different populations are

equal, otherwise transformations may be used to approximately equalize them.




4.3.6 Transformation

Simple transformations of either the dependent variable y or the independent

variable x, or of both, are often sufficient to make the simple regression model

appropriate for the transformed data. Unequal error variances and non-normality of

the error terms frequently appear together. To reduce the departure from a simple










linear regression model a transformation on y is needed, since the shapes and spreads

of the distributions of y need to be changed. Such a transformation on y may help to

linearize a curvilinear regression relation at the same time. At other times, a

simultaneous transformation on x may also be needed to obtain or maintain a linear

regression relation. However, it is very unlikely that such a transformation will be

needed in the following chapters.

Box and Cox [COX58] have developed a procedure for choosing a transformation

from the family of power transformations on y. This procedure is useful for correcting

unequal error variances. The family of power transformations is of the form:

y, = yY ,

where y is a parameter to be determined from the data. The family encompasses the

following and widely used transformation:

y' = loge y.

The criterion for determining the appropriate parameter yof the transformation of

y in the Box-Cox approach is to find the value of that minimizes the error sum of

squares SSE for a liner regression based on that transformation.




4.3.7 Residual Analysis

When regression analysis is applied deviations from the initial assumptions may

result in incorrect reliabilities stated. The departures have to be detected and taken

into account should they be big enough to alter the results. Fortunately, experience

has shown that least squares regression analysis produces reliable statistical tests,










confidence intervals, and prediction intervals as long as the departures from the

assumptions are not too great [MCC94, p.784].

As the assumptions [see 3.2.3] concern the random error component, e, of the

model, a first step is to estimate the random error. Since the actual random error

associated with a particular value of y is the difference between the actual y value and

its unknown mean, the error is estimated by the difference between the actual y value

and the estimated mean. This estimated error is called the regression residual, denoted

by .

S = actual random error

= (actual y value) (mean of y)

=y E(y)

= y-(0 +P 1x1 + 1 2 +.." k xk)

S = estimated random error (residual)

= (actual y value) (estimated mean of y)

= y-y

= y-( + 1x,+1 2 x2 ...+ kxk).

As the true mean of y (i.e., the true regression model) is not known, the actual

random error can not be calculated. However, because the residual is based on the

estimated mean (the least squares regression model), it can be calculated and used to

estimate the random error and to check the regression assumptions. These checks are

generally referred to as residual analyses [MCC94, p.784].










4.3.8 Influential Observations


When using regression, some subset of the observations may be found to be

unusually influential. Sometimes these influential observations are relatively far away

from the vicinity of the rest of the data. Dennis R. Cook developed an excellent

diagnostic, the Cook's distance. This is a measure of the squared distance between the

usual least squares estimate of P based on all n observations and the estimate obtained

when the ith point is removed, say, f, [NET90, p. 403].

The next chapter comprises a discussion of the influence of the batch size on the

performance of the three manufacturing systems. A comparison between the systems

introduces the chapter to give the reader a brief overview of the material. Then, the

two pull systems are discussed in more detail to explain their behavior. The push

system, MRP, is introduced separately due to its different attributes. After dealing

with the material in more detail on a level where the interdependence of factors is

more evident the discussion continues on a higher level by returning to the

comparison of the systems.














CHAPTER 5
BATCH SIZE


Avoiding setups and facilitating material handling are the two primary reasons for

watching jobs together in a manufacturing system. If large lots of similar products are

run in batches, equipment setups are infrequently needed. If setups are long, large lots

result in substantially more effective capacity. Furthermore, for process batches equal

to move batches the material that is moved between workstations in large batches

requires less handling than if it is moved in small lots [HOP96, p. 288].

The entities arrive at a workstation in a batch. While the first entity of that batch

enters the machine, the remaining entities have to wait to be processed. The batch can

be transported to the next stage in the system, only when the last entity of a batch is

completed. Here, transportation is assumed infinitely fast resulting in zero

transportation time.

A variety of single stage models and analytical techniques have been reviewed by

Chaudhry and Templeton [CHA83]. The literature covers single stage manufacturing

systems only, not applicable to a ten machine tandem line. Gold investigates

sophisticated batch service systems in push and pull manufacturing environments as

single stage systems by using embedded Markov chain techniques [GOL92]. Kim et

al. focus on production scheduling in semiconductor wafer fabrication taking batch

sizes into account. They use simulation to evaluate new scheduling rules [KIM98].










Schoening and Kahnt show how to extend the methodology of Mitra and Mitrani

[MIT90] to model a one-card Kanban system with batch servers [SCG95]. However,

in all three cases the batches could be processed simultaneously by batch servers, such

as plating baths, drying facilities, and heat-treating ovens, not quite transferable to the

tandem line with sequentially processing machines.

The model parameters and their levels are introduced to elaborate on the input

data prior to the discussion of the simulation results.




5.1 Parameters

The process time was established at 20 seconds throughout all the simulations

while the following parameters were varied to evaluate the performance of the

manufacturing systems:

* Batch size,

* Total number of cards assigned to the line, and

* The interarrival time for MRP.

The levels of these parameters or factors are discussed briefly.




5.1.1 Process Time

A workstation which processes a batch size r can be modeled as an r-stage

Erlangian server. In such a system a customer enters the server, proceeds one stage at

a time through the sequence of r stages and departs at the end. Only then, a new

customer enters. The total time that a customer spends in this service facility is the










sum of r independent identically distributed random variables, each chosen from an

exponential distribution. The probability distribution function of the service time is an

Erlangian distribution [KLE75, pp. 123-124]. Consequently, the process time for a

batch of size r is distributed according to an r-stage Erlangian distribution with a

mean of the individual process time, viz. 20 seconds. The batch size and the mean

process time per entity were given as an input.




5.1.2 Batch Size

The following batch sizes were selected:

1, 2, ..., 10, and 20.

Initially, neutral experiments were conducted to establish differences of system

behavior for batch size 20. The results were found to be compliant with the results

obtained for batch size 1 to 10. Thus, batch size 20 was omitted for further

experiments.




5.1.3 Number of Cards

The second design parameter portrays the number of cards assigned to the entire

line. Naturally, this parameter applies to the pull systems only. Its pendant for MRP is

the interarrival time. The parameter merely indicates the total amount of cards in the

system. It does not specify the number of cards assigned to individual machines.

Huang and Wang determine the number of cards in a CONWIP system, 0, by

applying Little's Law:










0 =ut,

where p is the average throughput of the production line and t is the average time for

a card to pass through the production line. The formula is expanded to approximate

the number of cards in a production line in series containing a bottleneck [HUA98].

Optimizing the number of kanbans in a line has been a popular research topic.

According to Bonvik et al. most kanban implementations set the parameters by rules

of thumb or simple formulas [BON97]. Sugimori et al. state Toyota's formula as an

example:

c> DL(1+ a)
P

where c is the number of cards, D is the demand rate, L the replenishment lead time, a

a safety factor, andp the number of parts in a container [SUG77]. During factory

operation, the kanban numbers are steadily decreased by reducing the safety factor.

According to Bonvik et al. the fact that the formula is based on standard lead times is

less than satisfying, as it does not reflect the lead time consequences of shop floor

congestion and limited machine capacities [BON97].

Liberopoulos and Dallery use an iterative heuristic to optimize the number of

cards assigned to a conventional single-stage Kanban control system (KCS). They

show that the computational complexity of optimizing a single-stage generalized

Kanban control system (GKCS) is the same as that of optimizing the KCS, which can

be considered a special case of the GKCS [LIB95]. However, the algorithm was

found to be rather complex, making use of an analytically tractable approximation

method or simulation for initialization. Dallery and Liberopoulos introduce the

extended Kanban control system (EKCS) as a KCS accommodating N stages in










another publication [DAL95], which was recently generalized to assembly structures

by Chaouiya et al. [CHY98]. However, these discussions have a pure comparative

nature, not incorporating the number of cards assigned to the system.

Unlike CONWIP, the Kanban control system does not only vary with the number

of cards assigned to the entire system, but, its performance is dependent on the

number of cards assigned to the individual machines. To ensure a comparison of an

optimal Kanban with CONWIP and MRP, some card allocation studies had to be

carried out prior to the actual simulations.




5.1.3.1 Card Allocation for Kan ban

Card allocations can not be carried out according to a generally applicable

algorithm. Some rules have been documented, applicable to specific manufacturing

systems. Gsettner and Kuhn make use of a heuristic to determine the optimal

allocation for a given production rate in a Kanban line with m stations. The

production rate is calculated analytically underestimating the true production rate

systematically. The procedure starts with assigning one card to every station. The

number of cards is then increased at each station on a trial basis. The distribution

which shows the best ratio between change in production rate and WIP is finally

accepted (greedy procedure) [GST96].

The next sub-chapter constitutes an endeavor to specify general allocation rules

relevant to the ten machine tandem line.










5.1.3.2 Card Allocation Rules

To visualize the material and to avoid ambiguity, the rules are explained with the

assistance of statics, essential to any engineering education. The ten machine tandem

line [see Figure 5-1] can be modeled as a beam supporting ten weights of equal

distance to one another [see Figure 5-2].







Figure 5-1: The ten machine tandem line.



W1 W2 W3 W4 W5 W6 W7 W8 W9 W10




Arm + Moment
.4-Center Point -+

Figure 5-2: Free body diagram of the ten machine tandem line modeled as a beam.


The moment of a force is its tendency to produce rotation of the body on which it

acts, about some axis. The measure of a moment is the product of the force and the

perpendicular distance between the axis of rotation and the line of action of the force.

This distance is called the moment arm [see Figure 5-2]. The intersection of the axis

of rotation with the plane of the force and its moment arm is called the center of

moments [JEN83, p. 15]. As it is a point, it is referred to as center point here. All the

forces of a system may be regarded as the component of their resultant force. Hence,

about any point, the moment of the resultant force (total weight) equals the algebraic










sum of the moments of the separate forces (weights). This principle is known as

Varignon's Theorem [JEN83, p. 17].

For the ten machine production line, the weight refers to the number of cards

assigned to a machine. The weight increases with increasing number of cards

allocated. Thus, the balance of the line can be expressed as the moment of the

resultant force, a consequence of a specific card allocation.

As the rules are not applicable to all manufacturing lines, the following

assumptions were made:

1. Identical machines,

2. All machines comprise the bottleneck,

3. Objective: maximum throughput, and

4. Center point: median of line (between machine 5 and 6).



Applying the statics analogy to the manufacturing line the following rules result:

1. Increase weight of last machine last,

2. Positive moment preferred to negative moment:

Increase weight on positive side of center point first,

Start increasing weight with smaller arm first,

3. Establish balance on line:

Symmetric structure relative to center point,

moment close to the absolute minimum (zero): same weight with certain

arm on either side (positive and negative) of center point,










Small difference (one card) in weight between the machines for the entire line,

and

4. Minimize number of consecutive machines with same weight.

All rules are to be applied simultaneously. However, the importance of the rules

decreases with increasing number. Thus, if the rules contradict one another, the rules

with lower number override the rules with higher number. Initially, all machines get

assigned the same amount of cards. Then, any additional cards are allocated according

to the rules. All the additional cards previously positioned may have to be reallocated

for one more card assigned to the line, thus satisfying an additional rule. For example:

the card remaining from allocating one card to each machine, the 11th card, is

assigned to the 6th machine [see Figure 5-3].


1 1 1 1 1 2 1 1 1 1




Figure 5-3: Number of cards per machine for 11 cards assigned to a ten machine line.


However, when a 12th card is assigned, the 11th card previously assigned has to be

reallocated to machine seven while the 12th card is assigned to machine four [see

Figure 5-4].



1 1 1 2 1 1 2 1 1 1



Figure 5-4: Number of cards per machine for 12 cards assigned to a ten machine line.










To test the correctness of the rules, some simulations were carried out. For these

simulations batch size, setup time, and failure were not taken into consideration. It

was assumed, that the card allocations were optimal independent of the above

mentioned parameters.

The rules were found to result in a good approximation of the optimum. However,

an approximation was not good enough to compare Kanban with the other two

systems, both being able to perform at their optimal settings. Consequently, more

simulations were run to establish optimal card allocations for 10 to 70 cards assigned

to the line. Assuming that the performance of the line could not be improved

otherwise, one rule was kept: small difference (one card) in weight between the

machines for the entire line.

The following number of combinations, m, had to be run for 10 to 19 cards being

assigned:

9 10
m = =1023.


As it was found that even the optimal allocations for the interval 10 to 19 cards

could not be applied to the lines with 20 to 70 cards, simulations had to be run for the

following intervals:

1. [10, 19],

2. [20, 29],

3. [30, 39],

4. [40, 49],

5. [50, 59], and

6. [60, 69],










plus one last replication for 70 cards assigned. This resulted in

n = 6m+l = 6(1023)+1 = 6139

experiments.

Thus, 6139 replications were completed resulting in the data to evaluate the rules

quantitatively.




5.1.3.3 Deviation of Rules from Optimum

The performance of the line for 10 to 70 cards assigned was measured by the

throughput. The percentage increase in throughput, I, for allocating optimally, Tho,

instead of allocating according to the rules, Thr, was calculated according to the

following formula:

I Th Th, 100%.



Figure 5-5 indicates an increase in most of the cases. Only in a few cases the rules

resulted in the optimal allocation. Naturally, there was no increase in throughput for

10, 20, ..., 70 as with these numbers only one allocation was possible under the given

assumptions [see p. 68].














18
16
14
12
c 10 -- Rules
S3- -0- Max
S6
4 0-

0
10 20 30 40 50 60 70
# of Cards

Figure 5-5: Increase in throughput by allocating cards optimally instead of simply
applying the rules.


To put these percentage increases in a relative context, the maximal increases, i.e.

the increases from the worst possible allocation to the optimal allocation, are

indicated in Figure 5-5 (Max) as well. The graph shows all maximal increases for the

first interval, 10 to 20 cards assigned, and only the maximal increase for 25, 35, ...,

65 cards assigned per consecutive allocation interval. These numbers were expected

to show the greatest deviation in throughput as they give rise to the greatest amount of

different possible allocations, a:

10
a = = 252,


where five additional cards had to be assigned after an equal amount of cards was

allocated to all the machines.

The graph illustrates the good approximation of the optimum by the rules. This is

especially true for a bigger number of cards assigned. It can clearly be seen that the











maximal increase decreases with an increasing amount of cards in the system. This

can be ascribed to the following:

* the machines are busy most of the time as enough cards have been allocated to

them,

* the increase of utilization per additional card assigned to the system decreases

with an increasing amount of cards allocated [see Figure 5-6], and

* the ratio,


r = -c,
C2


where c1 is the smallest number of cards assigned to any machine on the line and

c2 the largest number of cards assigned to a machine, decreases as the

difference, d = c2 c1, is kept constant and equal to 1.





0.9
0.85 -

0.75
=' 0.7 0000O0
S0.65 600"
& 0.6
.0.55 00
< 0.5 -<0
0.45
0.4 4..


10 20 30 40
Number of Cards


50 60


Figure 5-6: The average utilization dependent on the number of cards for Kanban.










The optimal card allocations for maximum throughput, minimal work in process

and minimal average cycle time were carefully studied.

The optimal allocations for minimizing WIP and average cycle time were found to

be very close to the rules applied. Note that these rules are different from those given

above, as the primary objective to achieve minimal WIP and minimal average cycle

time is to liberate the system of WIP. This is most efficiently done by placing more

cards towards the end of the line to pull material out of the system. Less cards at the

beginning of the line would result in raw material only being pulled into the system

for processing, not keeping any excess material in the buffers.

However, trying to achieve maximal throughput resulted in a great variability of

where the additional cards should be placed. Table 5-1 shows an extraction of the list

obtained to illustrate this interesting phenomenon. The systems with the same amount

of additional cards were grouped together. These additional cards were indicated as

ones in their respective rows. Looking at the table unveils no obvious pattern. Mmedan


and Mbeg7nn.ng are discussed below.













Table 5-1: Additional cards allocated to the system with ten machines.

# of cards assigned to M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 Mmedian Mbeg nn.ng
the system
11 0 0 0 0 0 1 0 0 0 0 1 6
21 0 0 0 1 0 0 0 0 0 0 -2 4
31 0 0 0 0 0 0 0 1 0 0 3 8
41 1 0 0 0 0 0 0 0 0 0 -5 1
51 0 0 0 0 0 0 0 0 0 1 5 10
61 0 0 0 0 0 0 0 0 0 1 5 10
12 0 0 0 1 0 0 1 0 0 0 0 11
22 0 0 1 0 0 0 0 0 1 0 1 12
32 0 0 0 0 0 1 1 0 0 0 3 13
42 0 1 0 0 0 0 0 0 1 0 0 11
52 0 0 1 0 0 0 1 0 0 0 -1 10
62 0 0 1 0 0 0 0 0 1 0 1 12
13 0 0 1 0 1 0 1 0 0 0 -2 15
23 0 0 1 0 1 0 1 0 0 0 -2 15
33 0 1 0 0 0 1 0 1 0 0 0 16
43 0 0 1 0 1 0 0 0 1 0 0 17
53 0 0 0 0 1 1 1 0 0 0 2 18
63 0 0 1 0 1 0 0 0 1 0 0 17



As the research on card allocation was not the main topic of this research paper, a

very limited amount of time was spent trying to find patterns that could explain this

variation. Some calculations were done to express the findings mathematically. The

interest was focused on the balance of the system. Mm/edan represents the moment of


the line with the center point at the median (between machine 5 and machine 6):

10
Almedan= 2 ,w
1=1

where w, stands for the weight of machine i [see 5.1.3, Card Allocation Rules] and /

stands for the arm of machine i. This was expected to be close to zero at all times,

assuming the correctness of the rules. As can be seen in Table 5-1, this number

greatly varies and sometimes equals to the maximum arm, /5= 5.


Mbegnn.ng quantifies the moment of the line for additional cards with the center


point at the beginning of the line, such that 1,=i:










10
M e.. = id'
Ilfbeginrnng = X dz,
1=1

where diC is the difference between the amount of cards of the different machines in

the system [see 5.1.3, Card Allocation Rules]. This formula indicates the position of

weight on the line. For 33 and 43 [see Table 5-1] Mmedan is the same and indicates a

balanced line. However, Mb eg.nng shows, that the weight is distributed differently,

viz. more towards the end of the line for 43 cards assigned. Comparing MI edan and

M/begimnmg for the different allocations, shows no evident pattern. More research could

be conducted to find explanations for this behavior.




5.1.4 Interarrival Time

This parameter stands for the time interval between two consecutive batch

arrivals. Its inverse is the arrival rate. The interarrival time was favored to the arrival

rate as it is understood more intuitively. Furthermore, it served as a direct input value

for the software applied.

The selected levels resulted from setting the utilization interval [u,,,,un ,,x ] for

MRP equal to the utilizations for the pull systems. The levels selected divided the

intervals into nine partitions.

As the average cycle time represents one of the primary indicators of the

performance of a manufacturing line, its response to a change in batch size is

discussed first.










5.2 Average Cycle Time

The following graph shows the influence of the batch size and the number of cards

allocated to the system on the average cycle time for the three control systems:

Kanban (1), CONWIP (2), and MRP (3) [see Figure 5-7].































Figure 5-7: The average cycle time dependent on the batch size and number of cards
allocated to the line for the three control systems: Kanban (1), CONWIP (2), and
MRP (3).


The average cycle time increases with increasing batch size. For the vertically

aligned data points, the number of cards assigned increases from bottom to top. As the

material is pulled into the system in batches the last member of each batch has to wait










until all the other members are processed. As the batch size increases, this waiting

time increases.

The lowest values of the average cycle time per batch size were obtained for the

least number of cards assigned to the system, viz. ten. Ten cards theoretically enable

all the machines to be busy simultaneously. Furthermore, Kanban requires this

minimal amount to function. For the upper bound at most 200 entities were chosen:

WPmax = bc = (10)(20) = 200,

where b is the batch size and c is the number of cards assigned to the system.

However, another constraint enforces even stronger limitations on the systems: the

average utilization of the machines.




5.2.1 The Average Machine Utilization

Little's Law relates the three parameters: throughput, cycle time, and work in

process. This interdependence has proven practically to be the only stable observation

for the turbulent stochastic manufacturing systems. Thus, it can easily be used to

make conclusions about one of the parameters when one is kept constant and the other

one is known.

The three parameters constitute ideal indicators of performance for a production

system. The production engineers are most definitely interested in reducing work in

process to decrease cycle time and increase the throughput of the line. Thus, these

parameters serve as quantitative indicators enabling state of the art process control.











From these indicators other indicators can be derived. One of these indicators

would be the machine utilization. The utilization, u, can be determined independently

of the throughput, but, they are directly related:

Th
average
Thheory


where Thmerage is the average throughput derived from the systems under study, and


Ththeot is the theoretical throughput, which can be determined by the following

formula:

Th 1
Th theory -
process

where tprocess is the process time of the bottle neck machine in minutes. Here, the

machines are identical and can all be considered bottle neck machines with a process

1
time of 20 seconds or minute resulting in the following:
3


Ther = 1 = 3

3

entities per minute.

The utilization gives a relative performance of a machine and can be calculated

for the entire line. The average utilization, u of the line can be calculated by the

following formula:

10
10

10




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