Yaşarcan, H., 2003, Feedback, Delays and Non

Transkript

FEEDBACK, DELAYS AND NON-LINEARITIES IN DECISION STRUCTURES
by
Hakan Yaşarcan
B.S., Industrial Engineering, Dokuz Eylül University, 1993
M.S., Industrial Engineering, Dokuz Eylül University, 1995
Submitted to the Institute for Graduate Studies in
Science and Engineering in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
Graduate Program in Industrial Engineering
Boğaziçi University
2003
ii
FEEDBACK, DELAYS AND NON-LINEARITIES IN DECISION STRUCTURES
APPROVED BY:
Prof. Yaman Barlas
………………………
(Thesis Supervisor)
Prof. Kuban Altınel
………………………
Assoc. Prof. Taner Bilgiç
………………………
Assoc. Prof. Yağmur Denizhan
………………………
Assoc. Prof. Seçkin Polat
………………………
DATE OF APPROVAL: 27.7.2003
iii
Dedicated to H. H. Shri Mataji Nirmala Devi
iv
ACKNOWLEDGEMENTS
I would like to thank Prof. Yaman Barlas, my thesis supervisor, first of all for
teaching me System Dynamics. During the long years of this study, he has not only guided
me, but also taught me the ethics of science. I am thankful to him for his patience and
valuable advice throughout this study.
I wish to thank Prof. Kuban Altınel, a member of my thesis supervising committee,
for his sensible advice and for his positive mood which motivated me to continue this
study. I am thankful to Assoc. Prof. Yağmur Denizhan, another member of my thesis
supervising committee, for teaching me Chaotic Dynamics and for her valuable comments.
I would like to thank Assoc. Prof. Taner Bilgiç for his advice and help especially in the
12th chapter of this thesis. I am thankful to Assoc. Prof. Seçkin Polat for his advice and
suggestions.
I am grateful to Prof. Hasan Akgündüz for helping me understand the depth of a
scientific research. His advice gave me strength and courage to continue my studies. I wish
to thank Hakan Güneri, because he put his desire and attention so that I could finish my
thesis. He is the one who advised and helped me to meet Prof. Hasan Akgündüz.
I would like to thank my parents Müzeyyen Yaşarcan and Hüseyin Yılmaz Yaşarcan
for their continuous support in my whole life. I also wish to thank my brother Tugrul
Yaşarcan, and my sisters Yonca Kayıkçı and Oya Yaşarcan.
I wish to thank Canan Yaşarcan, my wife, who entered in my life, making it more
beautiful.
I am thankful to Neşe Algan, for her help and advice. I also would like to thank all
the Sahaja Yogis and Yoginis for their support.
Finally, I would like to express my deepest gratitude to H. H. Shri Mataji Nirmala
Devi.
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ABSTRACT
FEEDBACK, DELAYS AND NON-LINEARITIES IN DECISION
STRUCTURES
We first define a very general framework for the generic stock management problem.
Models in this framework include feedback loops, delays and non-linearities making their
analysis a challenging task. We use System Dynamics modeling methodology. The
dynamics of complex decision structures are obtained by simulation and supported with
mathematical analysis when it is necessary and possible.
There are papers in the literature, which analyze the importance of proper inclusion
of supply line delay in stock control decisions, but no reported work on other kinds of
delays (such as delays caused by controlling of a primary stock like inventory, via
“secondary” stocks like production capacity or delays in information processing). We
consider all typical delays in the decision control formulations. We develop a Virtual
supply line concept to incorporate information delays and secondary stock control
structures in the decisions. We offer a formulation involving the Virtual supply line as
general way to obtain highly stable, robust and fast response in the primary stock, in
situations involving very complex mixtures of the different types of delays in the stock
management system. We also discuss the implications for the standard inventory
management rules. The second major issue analyzed is the potential instability in the
system when the stock has a decay rate, and the supply line delay is long and high order.
We show that the proper formulation of the anchor term of the decision equations is very
important for stability. We develop an anchor formula that eliminates undesirable
oscillations entirely. Finally, we discuss some goal formation structures involving goal
erosion dynamics. For such systems, we first develop a very general goal formation model
that explains complex ways in which frustration and/or system resistance can develop in
goal seeking, and then offer formulations to avoid such undesirable dynamics in different
unfavorable goal setting environments.
vi
ÖZET
KARAR YAPILARINDA GERİBİLDİRİM, GECİKMELER VE
DOĞRUSAL OLMAYAN İLİŞKİLER
İlk önce, genel stok yönetimi problemi için çok genel bir çerçeve model
tanımlıyoruz. Bu çerçevedeki modeller, analizi zorlaştırıcı, geribildirim döngüleri,
gecikmeler ve doğrusal olmayan ilişkiler içerirler. Sistem Dinamiği yöntemi kullanılmıştır.
Karmaşık yapıların dinamikleri benzetim yolu ile elde edilip, gerekli ve mümkün
olduğunda matematiksel analizle de desteklenmiştir.
Literatürde, tedarik hattı gecikmesinin, stok denetim kararlarında dahil edilmesinin
önemiyle ilgili makaleler vardır, fakat diğer gecikmeler (envanter gibi bir esas stoğun,
üretim kapasitesi gibi ikincil bir stok vasıtasıyla kontrol edilmesinden kaynaklanan
gecikmeler, veya bilgi işlemede oluşan gecikmeler) ile ilgili bilinen herhangi bir çalışma
yoktur. Bu tez, tüm tipik gecikmeleri, kontrol karar formülasyonlarında göz önüne
almaktadır. Bilgi işleme ve ikincil stok denetim gecikmelerini de kararlara dahil etmek için
Sanal tedarik hattı kavramı geliştirilmiştir. Sanal tedarik hattını, farklı türdeki
gecikmelerin karmaşık karışımlarını içeren stok yönetimin sistemlerinde, temel stokta
kararlı, güvenilir ve hızlı yanıt alabilmek için genel bir karar formülasyonu olarak
öneriyoruz, ve standart envanter yönetimi kurallarındaki yerini/karşılığını tartışıyoruz.
İkinci önemli katkı olarak, stoktan bozulma gibi herhangibir çıkış olup, aynı zamanda da
tedarik gecikme süresi uzun ve çok katmanlı olduğu taktirde, ortaya çıkan olası kararsız
dalgalanmalar araştırılmıştır. Gösterilmiştir ki, kararlı dinamik için, karar denklemlerindeki
çapa formülasyonu kritiktir. Bu gibi durumlar için geliştirdiğimiz özel bir çapa
formülasyonu (EVL), istenmeyen kararsız dalgalanmaları yok etmektedir. Son olarak,
karmaşık amaç aşınmalarına yol açan bazı sorunlu amaç oluşturma yapıları tartışılmıştır.
Böyle durumlar için, moral çöküntüsü ve sistem direnci gibi oluşumları açıklayan, ve
çeşitli elverişsiz amaç belirleme ortamlarında istenmeyen dinamikleri engelleyen, genel bir
amaç oluşturma modeli geliştirilmiştir.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS.................................................................................................. iv
ABSTRACT........................................................................................................................... v
ÖZET .................................................................................................................................... vi
LIST OF FIGURES .............................................................................................................xii
LIST OF TABLES............................................................................................................ xxix
LIST OF SYMBOLS/ABBREVIATIONS........................................................................ xxx
1. INTRODUCTION ........................................................................................................... 1
1.1. System Dynamics Methodology ............................................................................ 1
1.2. Human Decisions in System Dynamics Models .................................................... 1
1.3. Feedback, Delays and Non-linearity in Human Decisions .................................... 1
1.4. Systematic Mistakes Made by Decision Makers.................................................... 2
1.5. Purpose and Main Focus of this Research ............................................................. 3
2. REPRESENTATION TOOLS OF SYSTEM DYNAMICS METHODOLOGY ........... 4
2.1. Stock-flow Representation ..................................................................................... 4
2.2. Integral, Differential and Difference Equations..................................................... 5
2.3. Causal-loop Diagrams ............................................................................................ 7
2.4. Dynamic Behavior (Output Behavior) ................................................................... 7
3. USEFUL ATOMIC STRUCTURES............................................................................... 9
3.1. Goal Seeking Atomic Structure.............................................................................. 9
3.2. Delays................................................................................................................... 10
3.2.1. Material Delay Atomic Structure ............................................................. 10
3.2.2. Information Delay Atomic Structure........................................................ 12
3.3. Stock Management Atomic Structure .................................................................. 14
4. PROBLEM DEFINITION............................................................................................. 18
5. LINEAR CONTROL OF A SINGLE STOCK WITH SUPPLY LINE DELAY ......... 21
5.1. Dynamics of Stock Control without Considering Supply Line............................ 21
5.2. The Effects of the Stock Adjustment Time and Acquisition Delay Time on
the Amplitude and the Period of the Oscillations ................................................ 26
5.3. The Role of Supply Line in Stock Control Decisions.......................................... 28
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5.4. Suggestions for Control of a Single Stock with Supply Line and Constant
Outflow ................................................................................................................ 32
6. LINEAR CONTROL OF A DECAYING STOCK WITH SUPPLY LINE
DELAY.......................................................................................................................... 35
6.1. Parameter Values for All Runs in this Chapter .................................................... 37
6.2. Effect of Life Time (Decay Time) ....................................................................... 38
6.3. Trade-off in Stock Adjustment Time Values in a Discrete Supply Line
Delay .................................................................................................................... 39
6.4. Causes of Instability in the Discrete Supply Line Case Combined with
Decaying Outflow ................................................................................................ 41
6.5. Using the Equilibrium Value of Loss to Stabilize the Model .............................. 42
6.6. Suggestions on Control of a Decaying Stock with Supply Line Delay ............... 44
6.7. Some Observations on Controlling a Decaying Stock ......................................... 45
6.7.1. Stability with EVL used as Anchor is Robust........................................... 45
6.7.2. Stability with LF Used as Anchor is Problematic .................................... 46
6.7.3. Comparison of EVL and LF Used as Anchors ......................................... 47
6.7.4. Effects of Using Expected Loss Formulation in Controlling a
Decaying Stock......................................................................................... 48
7. CONTROL OF A DECAYING STOCK WITH UNKNOWN VARIABLE LIFE
TIME.............................................................................................................................. 50
7.2. Case: Life Time and Loss Flow are Observed Directly and Immediately ........... 51
7.3. Case: Life Time cannot be Observed but Loss Flow is Observed
Immediately.......................................................................................................... 52
7.4. Case: Life Time cannot be Observed and Loss Flow is Observed with a
Delay .................................................................................................................... 54
8. LINEAR CONTROL OF A SINGLE STOCK WITH INFORMATION DELAY:
VIRTUAL SUPPLY LINE............................................................................................ 60
8.2. Comparison of Supply Line and Information delay in Stock Control ................. 61
8.2.1. Causal Loop Comparison ......................................................................... 62
8.2.2. Mathematical Equivalency ....................................................................... 63
8.3. Introducing the Notion of Virtual Supply Line in Stock Control ........................ 66
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8.4. Mathematical Equivalency of the Supply Line and Virtual Supply Line
Adjustments in the Decisions............................................................................... 69
8.5. Suggestions on Linear Control of a Single Stock with Information Delay.......... 72
9. LINEAR CONTROL OF A SINGLE STOCK WITH SECONDARY STOCK
CONTROL STRUCTURE ............................................................................................ 73
9.2. Comparison of Secondary Stock Control Structure with Supply Line and
Information Delay Structures ............................................................................... 76
9.2.1. Causal Loop Diagram of the Model with Secondary Stock Control
Structure ................................................................................................... 78
9.2.2. Mathematical Analysis of the Model with Secondary Stock Control
Structure ................................................................................................... 79
9.3. Using Virtual Supply Line Concept in Secondary Stock Control........................ 81
9.3.1. Mathematical Derivation of Virtual Supply Line Formulation for
Secondary Stock Structure ....................................................................... 83
9.3.2. Model and Behavior for Secondary Stock Structure with Virtual
Supply Line .............................................................................................. 87
9.4. Suggestions on Linear Control of a Single Stock with Secondary Stock
Control.................................................................................................................. 89
10. APPLICATION OF “VIRTUAL SUPPLY LINE” CONCEPT IN EXAMPLE
MODELS....................................................................................................................... 90
10.1. A General Inventory-Workforce Model with Three Type of Delays................... 90
10.1.1. Equations of the Example Model with Three Delay Structures ............... 90
10.1.2. Runs of the Example Model with Three Delay Structures....................... 95
10.2. The Inventory-Workforce Model with Non-Linearities in Decisions.................. 97
10.2.1. Production-Inventory Sub-Model and its Equations ................................ 99
10.2.2. Workforce Sub-Model and its Equations ............................................... 103
10.2.3. Problematic Desired Supply Line Equations.......................................... 106
10.2.4. Non-Linear Inventory-Workforce Model with Virtual Supply Line...... 107
11. VIRTUAL SUPPLY LINE AS A STOCK.................................................................. 113
11.1. The Usage of Stock-Type Virtual Supply Line for Information Delay
Structure ............................................................................................................. 113
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11.2. Dependency of Equilibrium Level on Initial Value, and Setting a Proper
Value .................................................................................................................. 114
11.3. Using Virtual Supply Line when Information Delay Stocks cannot be
Observed............................................................................................................. 118
11.4. Some Observations About Virtual Adjustment Time ........................................ 119
11.5. The Stock-Type Virtual Supply Line as a Powerful Control Formulation
when Delay Structure is Complex and Unknown to the Decision Maker ......... 121
12. APPLYING THE RESULTS TO THE INVENTORY MANAGEMENT RULES ... 129
12.1. Management of a Perishable Goods Inventory with Discrete Supply Line
Delay .................................................................................................................. 129
12.1.1. Base Runs of the Perishable Goods Inventory Model............................ 138
12.1.2. Runs with Improved Formulations of s.................................................. 143
12.2. Inventory Management with Unreliable Supply Line........................................ 153
12.2.1. Base Runs of the Unreliable Supply Line Model................................... 156
12.2.2. Runs with Improved Formulations of In-Transit Inventory ................... 158
13. DYNAMICS OF GOAL SETTING ............................................................................ 165
13.1. Simple Goal Structures....................................................................................... 165
13.1.1. Goal as an External Variable.................................................................. 165
13.1.2. Goal as an Internal System Variable ...................................................... 165
13.2. Problematic Goal Structures............................................................................... 166
13.2.1. Capacity Limit on Improvement Rate .................................................... 166
13.2.2. Simple Goal Erosion and Traditional Performance ............................... 170
13.2.3. Goal Erosion and Recovery.................................................................... 175
13.2.4. Goal Erosion, Possible Recovery and Time Limits ............................... 177
13.2.5. Implicit Goal Setting: Short Term Motivation Effect on Weight of
Stated Goal ............................................................................................. 183
13.2.6. Stated Goal Adjustment by Management to Increase Performance....... 187
14. CONCLUSIONS ......................................................................................................... 195
APPENDIX A: MODELING OBJECTS AND SYMBOLS USED IN SYSTEM
DYNAMICS ................................................................................................................ 199
APPENDIX B: ABBREVIATION RULES ADOPTED FOR VARIABLE NAMES..... 200
APPENDIX C: ATOMIC STRUCTURES IN HUMAN SYSTEMS .............................. 201
C.1. First Order Linear Atomic Structure .................................................................. 201
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C.2. Production Process ............................................................................................. 203
C.3. Goal Seeking Atomic Structure.......................................................................... 204
C.4. S-shaped Growth Atomic Structure ................................................................... 205
C.4.1. S-shaped Growth Caused by Transfer from One Stock to Another ....... 205
C.4.2. S-shaped Growth Caused by a Capacity Limit ...................................... 207
C.5. Boom-Then-Bust Atomic Structure ................................................................... 208
C.5.1. Boom-Then-Bust Caused by S-shaped Growth and Decay ................... 208
C.5.2. Boom-Then-Bust Caused by a Delayed Effect of Capacity Limit ......... 210
C.6. Delays................................................................................................................. 212
C.6.1. Material Delay Atomic Structure ........................................................... 212
C.6.2. Information Delay Atomic Structure...................................................... 214
C.7. Oscillating Atomic Structure.............................................................................. 216
C.8. Stock Management Atomic Structure ................................................................ 218
C.9. Goal Setting Atomic Structure ........................................................................... 221
APPENDIX D: NOISE GENERATION .......................................................................... 223
APPENDIX E: A NON-LINEAR LIFE TIME ESTIMATION ADJUSTMENT
RULE FOR SHOCK REDUCTION ........................................................................... 225
APPENDIX F: MATHEMATICAL EQUIVALENCY OF SUPPLY LINE DELAY,
INFORMATION DELAY AND SECONDARY STOCK STRUCTURES FOR
THE GENERAL CASE............................................................................................... 227
F.1. Second Order Supply Line Structure as an Input-Output System...................... 227
F.2. Second Order Information Delay as an Input-Output System ........................... 228
F.3. Secondary Stock Structure with a First Order Supply Line Delay as an
Input-Output System .......................................................................................... 230
APPENDIX G: GENERALIZED VIRTUAL SUPPLY LINE FORMULAS FOR
DELAY STRUCTURES INVOLVING DIFFERENT INDIVIDUAL DELAY
TIMES ......................................................................................................................... 232
APPENDIX H: THE PROBLEMATIC AND NON-PROBLEMATIC VERSIONS
OF THE DESIRED SUPPLY LINE FORMULATION ............................................. 233
REFERENCES .................................................................................................................. 237
xii
LIST OF FIGURES
Figure 2.1.
Stock-flow diagram......................................................................................... 4
Figure 2.2.
Causal-loop diagram and an example ............................................................. 7
Figure 2.3.
Behavior of a variable in time......................................................................... 8
Figure 2.4.
Behavior of two variables with respect to each other (phase plot) ................. 8
Figure 3.1.
Stock-flow diagram of goal seeking atomic structure..................................... 9
Figure 3.2.
Causal-loop diagram of goal seeking atomic structure ................................... 9
Figure 3.3.
Possible behaviors of goal seeking atomic structure..................................... 10
Figure 3.4.
Stock-flow diagram of first order material delay atomic structure ............... 10
Figure 3.5.
Stock-flow diagram of third order material delay atomic structure.............. 11
Figure 3.6.
Stock-flow diagram of infinite order (discrete) material delay atomic
structure......................................................................................................... 11
Figure 3.7.
Possible behaviors of different order material delay structures .................... 12
Figure 3.8.
Stock-flow diagram of first order information delay atomic structure ......... 12
Figure 3.9.
Stock-flow diagram of third order information delay atomic structure ....... 13
Figure 3.10. Possible behaviors of different order information delay structures .............. 14
Figure 3.11. Stock-flow diagram of stock management atomic structure......................... 14
Figure 3.12. Causal-loop diagram of stock management atomic structure ....................... 15
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Figure 3.13. Possible behaviors of stock management atomic structure, under
different parameter settings........................................................................... 16
Figure 4.1.
Stock management and general human decision framework ........................ 18
Figure 5.1.
Stock management model with first order supply line delay ........................ 21
Figure 5.2.
Non-oscillatory and oscillatory behavior runs for TAD equal to 1 and
TSA equal to 8, 4, 2 and 0.5 respectively for the 1st, 2nd, 3rd and 4th
runs................................................................................................................ 24
Figure 5.3.
Stock management model with second order supply line ............................ 25
Figure 5.4.
Runs for TAD equal to 1, and TSA equal to 3.37 (goal seeking), 0.5
(stable oscillation), 0.25 (neutral oscillation) and 0.21 (unstable
oscillation) respectively for the 1st, 2nd, 3rd and 4th runs................................ 25
Figure 5.5.
The effect of the changes in Stock adjustment time (TSA equal to 0.5,
0.4 and 0.3 respectively for the 1st, 2nd and 3rd runs) for TAD equal to 1 ..... 27
Figure 5.6.
The effect of the changes in Acquisition delay time (TAD equal to 3.5,
4 and 4.5 respectively for the 1st, 2nd and 3rd runs) for TSA equal to 1 ......... 27
Figure 5.7.
Stock management model with supply line considered in Control flow....... 28
Figure 5.8.
Stock management model with a second order supply line considered
in CF.............................................................................................................. 31
Figure 5.9.
The effect of the changes in Weight of the supply line (WSL equal to 0,
0.2, 1 and 5 respectively for the 1st, 2nd, 3rd and 4th runs) for TSA equal
to 2 and TAD equal to 11............................................................................... 32
xiv
Figure 5.10. Effect of the Acquisition delay time (TAD equal to 20, 5 and 1
respectively for the 1st, 2nd and 3rd runs) for WSL equal to 1 and TSA
equal to 2 ....................................................................................................... 33
Figure 5.11. Effect of the Stock adjustment time (TSA equal to 10, 1, 0.1 and 0.01
respectively for the 1st, 2nd and 3rd runs) for WSL equal to 1 and TAD
equal to 10 ..................................................................................................... 34
Figure 6.1.
Effect of the Life time (TLf equal to 40, 10 and 3 respectively for the
1st, 2nd and 3rd runs) for WSL equal to zero and TSA equal to 2 .................... 38
Figure 6.2.
Effect of the Life time (TLf equal to 40, 10 and 3 respectively for the
1st, 2nd and 3rd runs) for WSL equal to 1 and TSA equal to 2 ......................... 39
Figure 6.3.
Effect of the Stock adjustment time (TSA equal to 2, 7, 20 and 70
respectively for the 1st, 2nd, 3rd and 4th runs) for discrete supply line
system with WSL equal to 1 and TLf equal to 10 .......................................... 40
Figure 6.4.
Causal loop diagram without a decaying outflow structure.......................... 41
Figure 6.5.
Causal loop diagram with a decaying outflow structure............................... 41
Figure 6.6.
Stock management model with discrete supply line, decaying stock
and EVL used as anchor in Control flow (CF) and in Desired supply
line (SLS*) computation................................................................................ 43
Figure 6.7.
Effect of the Stock adjustment time (TSA equal to 2, 7, 20, 70 and
infinite respectively for the 1st, 2nd, 3rd, 4th and 5th runs) for discrete
supply line model with EVL as the anchor in CF and SLS*, with WSL
equal to 1, and with TLf equal to 10.............................................................. 43
xv
Figure 6.8.
Effect of the Weight of Supply Line (WSL equal to 0, 0.5, 1 and 2
system with EVL (as the anchor in CF and SLS*) and TSA equal to 70 ....... 45
Figure 6.9.
Effect of the Weight of Supply Line (WSL equal to 0, 0.5, 1 and 2
system with EVL (as the anchor in CF and SLS*) and TSA equal to 2 ......... 46
Figure 6.10. Effect of the Weight of Supply Line (WSL equal to 0, 0.5, 1 and 2
system with LF (as the anchor in CF and SLS*) and TSA equal to 70.......... 46
Figure 6.11. Effect of the Weight of Supply Line (WSL equal to 0, 0.5, 1 and 2
system with LF (as the anchor in CF and SLS*) and TSA equal to 2............ 47
Figure 6.12. Comparison of using EVL (1st run with WSL equal to 1) and LF (2nd,
3rd, 4th and 5th runs with WSL equal to 0, 0.15, 0.4 and 1) as anchors
for second order supply line system with TSA equal to 7 ............................. 48
Figure 6.13. Comparison of using EVL (1st run), LF (2nd run) and ELS (3rd and 4th
runs with TEL equal to 2 and 10) as anchors for discrete supply line
system with WSL equal to 1 and TSA equal to 3 ........................................... 49
Figure 6.14. Comparison of using EVL (1st run), LF (2nd run) and ELS (3rd and 4th
runs with TEL equal to 2 and 10) as anchors for discrete supply line
system with WSL equal to 1 and TSA equal to 21 ......................................... 49
Figure 7.1.
Auto-correlated Life time .............................................................................. 50
Figure 7.2.
Behavior of the Stock when Life time is perceived directly .......................... 52
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Figure 7.3.
Shock in Calculated life time as Perceived loss approaches zero,
caused by a phase difference between Stock and Perceived Loss................. 54
Figure 7.4.
Behavior of the Stock when there is a shock in Calculated life time, at
about time 22.70............................................................................................ 55
Figure 7.5.
Reduced shock in TCLf when there is no phase difference between
Stock and Perceived Loss (TSm,S equal to TPD).......................................... 56
Figure 7.6.
LF and TCLf together, shock can be seen at about time 23.46 ..................... 56
Figure 7.7.
Behavior of the Stock for reduced shock (TSm,S equal to TPD) .................. 57
Figure 7.8.
Smoothed life time when there is no phase difference (TSm,S equal to
TPD).............................................................................................................. 57
Figure 7.9.
Life time and Smoothed life time together ..................................................... 58
Figure 7.10. Behavior of the Stock for Smoothed life time................................................ 58
Figure 7.11. Model with decaying Stock and Life time estimation involving
smoothing...................................................................................................... 59
Figure 8.1.
Stock management model with second-order information delay.................. 60
Figure 8.2.
Behaviors of the models with equivalent supply line and information
delay structures are exactly the same ............................................................ 61
Figure 8.3.
Causal loop diagram of model in Figure 5.3................................................. 62
Figure 8.4.
Figure 8.5.
Stock control with Virtual supply line adjustment........................................ 68
xvii
Figure 8.6.
Output equivalency of the supply line model with WSL equal to 1 and
information delay model with WVSL equal to 1 ........................................... 68
Figure 9.1.
Secondary stock control structure ................................................................. 74
Figure 9.2.
Secondary stock control structure example................................................... 75
Figure 9.3.
Behaviors of the models with supply line (first run), information delay
(second run) and secondary stock control (third run with WSSL=0, and
fourth run with WSSL=1) structures.............................................................. 77
Figure 9.4.
Figure 9.5.
Secondary stock control structure with virtual supply line adjustment ........ 88
Figure 9.6.
Optimum behaviors of the models with supply line (first run with
WSL=1), information delay (second run with WVSL=1) and secondary
stock control (third run with WVSL=1 and with WSSL=1) structures,
all with supply line or virtual supply line adjustments ................................. 88
Figure 10.1. Example model using all three types of delay structures.............................. 91
Figure 10.2. Output behaviors (Inventory) of the example model with different
supply line and virtual supply line weight values ......................................... 96
Figure 10.3. Output behaviors (Labor) of the example model with different supply
line and virtual supply line weight values..................................................... 96
Figure 10.4. Graphical function of Effect of schedule pressure........................................ 98
Figure 10.5. Production-Inventory sub-model .................................................................. 99
Figure 10.6. Work in process inventory box as third order supply line (material)
delay .............................................................................................................. 99
xviii
Figure 10.7. Graphical function for Order fulfillment ratio ............................................ 102
Figure 10.8. Workforce sub-model.................................................................................. 104
Figure 10.9. Virtual supply line structure........................................................................ 107
Figure 10.10. Dynamics of Inventory with or without schedule pressure and VSL .......... 109
Figure 10.11. Dynamics of Labor with or without schedule pressure and VSL ................ 110
Figure 10.12. Runs for Inventory in less stable conditions ............................................... 111
Figure 10.13. Runs for Labor in less stable conditions ..................................................... 112
Figure 11.1. Stock control with Virtual supply line stock ............................................... 113
Figure 11.2. Dynamic behavior without any steady-state error (TSA = 2, TID = 12,
WVSL = 1, and -1 unit shock in S at time 5) ............................................... 116
Figure 11.3. Steady-state error resulting from initial value error in Virtual supply
line (TSA = 2, TID = 12, WVSL = 1) ........................................................... 116
Figure 11.4. Result of a shock applied to first stock of Information delay, creating a
steady-state error (TSA = 2, TID = 12, WVSL = 1)...................................... 117
Figure 11.5. Eliminating equilibrium error resulting from initial value error in
Virtual supply line (TSA = 2, TID = 12, WVSL = 1, TVA = 40) .................. 118
Figure 11.6. Eliminating equilibrium error resulting from a shock applied to first
stock of Information delay (TSA = 2, TID = 12, WVSL = 1, TVA = 40) ..... 119
Figure 11.7. Oscillations after a shock applied to first stock of Information delay
(TSA = 2, TID = 12, WVSL = 1, TVA = 10)................................................. 119
xix
Figure 11.8. Long time to restore equilibrium after a shock applied to first stock of
Information delay (TSA = 2, TID = 12, WVSL = 1, TVA = 100) ................. 120
Figure 11.9. Unstable oscillatory behavior when virtual supply line term is ignored
in decisions (TSA = 2, TID = 12, WVSL = 0) .............................................. 120
Figure 11.10. Framework of stock control problem with unknown delay structure ......... 121
Figure 11.11. Stock control with unknown complex delay structures .............................. 123
Figure 11.12. Unstable behavior for Weight of VSL = 0 ................................................... 127
Figure 11.13. Steady-state error in the mean level of Stock for Weight of VSL = 1
and for Virtual adjustment time = infinite................................................... 128
Figure 11.14. A quite stable behavior obtained by the proposed Virtual supply line
formulation ( Weight of VSL = 1 and for Virtual adjustment time = 40)...... 128
Figure 12.1. Perishable goods inventory structure .......................................................... 130
Figure 12.2. Customer demand structure......................................................................... 131
Figure 12.3. Expectations formation structure ................................................................ 132
Figure 12.4. Order decisions structure............................................................................. 134
Figure 12.5. Costs-revenues-profits structure.................................................................. 136
Figure 12.6. Behaviors of Customer demand and Expected weekly demand .................. 138
Figure 12.7. Behavior of Expected weekly deviation ...................................................... 139
Figure 12.8. Behaviors of Big S, s, Inventory position and Inventory (with small
scale) ........................................................................................................... 139
xx
Figure 12.9. Behaviors of Big S, s, Inventory position and Inventory (with large
scale) ........................................................................................................... 140
Figure 12.10. Behaviors of Total revenues, Net total profit and Total cost:
Bankruptcy .................................................................................................. 140
Figure 12.11. Behaviors of Total purchasing cost, Total ordering cost, Total lost
sales cost, Total inventory storage cost and Total perishing cost............... 141
Figure 12.12. Stable behaviors of Big S, s, Inventory position and Inventory (with a
short TAD = 4)............................................................................................. 142
Figure 12.13. Satisfactory behaviors of Total purchasing cost, Total ordering cost,
Total lost sales cost, Total inventory storage cost and Total perishing
cost (with a short TAD = 4) ......................................................................... 142
Figure 12.14. Improved behaviors of Big S, s, Inventory position and Inventory
obtained with Equation (12.57)................................................................... 144
obtained with Equation (12.58) and TSm = 40............................................ 145
Figure 12.17. Improved behaviors of Total revenues, Net total profit and Total cost
obtained with Equation (12.58) and TSm = 40............................................ 146
xxi
Figure 12.20. Improved behaviors of Total purchasing cost, Total ordering cost,
Total lost sales cost and Total inventory storage cost obtained with
Equation (12.57).......................................................................................... 147
Equation (12.58) and TSm = 40 .................................................................. 148
Equation (12.60).......................................................................................... 148
Figure 12.23. Autocorrelated customer demand structure ................................................ 149
Figure 12.24. Behaviors of autocorrelated Customer demand and Expected weekly
demand ........................................................................................................ 150
obtained with Equation (12.60) and with autocorrelated Customer
demand ........................................................................................................ 151
obtained with Equation (12.60) and with autocorrelated Customer
demand ........................................................................................................ 151
Equation (12.60) and with autocorrelated Customer demand..................... 152
Figure 12.28. Supply line and inventory structure for the unreliable supply line
model........................................................................................................... 153
Figure 12.29. Autocorrelated noise structure for adjustment of orders............................. 155
xxii
Figure 12.30. Behaviors of Big S, s, Inventory position and Inventory for the
unreliable supply line model ....................................................................... 156
Figure 12.31. Behaviors of the actual Supply line and the perceived In transit for the
Figure 12.32. Behaviors of Total revenues, Net total profit and Total cost for the
sales cost and Total inventory storage cost for the unreliable supply
line model.................................................................................................... 158
Figure 12.34. Behaviors of Big S, s, Inventory position and Inventory for
improvement Equation (12.83), (12.84) and (12.27) .................................. 160
Figure 12.35. Behaviors of Big S, s, Inventory position and Inventory for
improvement Equation (12.25) and (12.87)................................................ 160
Figure 12.36. Behaviors of Supply line and In transit for improvement Equation
(12.83), (12.84) and (12.27) ........................................................................ 161
Figure 12.37. Behaviors of Supply line and In transit for improvement Equation
(12.25) and (12.87)...................................................................................... 161
Figure 12.38. Behaviors of Total revenues, Net total profit and Total cost for
improvement Equation (12.83), (12.84) and (12.27) .................................. 162
Figure 12.39. Behaviors of Total revenues, Net total profit and Total cost for
improvement Equation (12.25) and (12.87)................................................ 162
sales cost and Total inventory storage cost for improvement Equation
(12.83), (12.84) and (12.27) ........................................................................ 163
xxiii
sales cost and Total inventory storage cost for improvement Equation
(12.25) and (12.87)...................................................................................... 163
Figure 13.1. Graphical function for Effect of desired control flow (ECF*) .................... 167
Figure 13.2. Model with capacity limitation, but without goal erosion .......................... 168
Figure 13.3. Output of the model with stock adjustment (improvement rate)
limitation ..................................................................................................... 170
Figure 13.4. Simple eroding goal structure ..................................................................... 171
Figure 13.5. Behavior of simple eroding goal structure .................................................. 172
Figure 13.6. Simple eroding goal structure with Traditional Performance .................... 173
Figure 13.7. Behavior of eroding goal structure with Traditional Performance ............ 174
Figure 13.8. A general model of goal erosion and recovery ........................................... 176
Figure 13.9. Behavior of goal erosion and recovery model ............................................ 177
Figure 13.10. Model with goal erosion, possible recovery and time limits ...................... 178
Figure 13.11. Graphical function of Effect of motivation (EM) ........................................ 179
Figure 13.12. Behavior for goal erosion and possible recovery with time limits.............. 181
Figure 13.13. Behaviors of Time horizon, Likelihood of accomplishment ratio, and
Effect of motivation ..................................................................................... 182
Figure 13.14. Behavior of system when the Stated goal is sufficiently low (equal to
850) ............................................................................................................. 182
xxiv
Figure 13.15. Behavior of EM when the Stated goal is sufficiently low (equal to
850) ............................................................................................................. 183
Figure 13.16. Graphical function for Effect of short term motivation (ESTM) ................. 184
Figure 13.17. Dynamics of goal erosion with short and long term effects........................ 185
Figure 13.18. Dynamics of goal erosion with short and long term effects when
IGS(0)=ING................................................................................................. 185
Figure 13.19. Behaviors of THS, RLA, EM and ESTM ..................................................... 186
Figure 13.20. Behavior of the system for Stated goal equal to 650 .................................. 186
Figure 13.21. Behaviors of Time horizon, Likelihood of accomplishment ratio, Effect
of motivation and Effect of short term motivation for Stated goal equal
to 650........................................................................................................... 187
Figure 13.22. Behavior of Stated goal (SG) adjustment model when SG(0)=G ............... 189
Figure 13.23. Model with Stated goal (SG) adjustment by management.......................... 190
Figure 13.24. Behaviors of THS, RLA, EM and ESTM for SG(0)=G ................................ 191
Figure 13.25. Behavior for Stated goal (SG) adjustment model when SG(0)=SG* .......... 192
of motivation and Effect of short term motivation for SG(0)= SG* ............ 192
Figure 13.27. Behavior for Stated goal (SG) adjustment model when Time horizon is
insufficient (THS(0)=120)........................................................................... 193
of motivation and Effect of short term motivation for THS(0)=120 ............ 193
xxv
Figure C.1.
Stock-flow diagram of first order linear atomic structure........................... 201
Figure C.2.
Causal-loop diagram of first order linear atomic structure ......................... 201
Figure C.3.
Exponential growth, constant and exponential decay ................................. 202
Figure C.4.
Stock-flow diagram of production process ................................................. 203
Figure C.5.
Linear growth, constant and linear decay ................................................... 203
Figure C.6.
Stock-flow diagram of goal seeking atomic structure................................. 204
Figure C.7.
Causal-loop diagram of goal seeking atomic structure ............................... 204
Figure C.8.
Goal seeking behavior................................................................................. 204
Figure C.9.
Stock-flow diagram of S-shaped growth structure with transfer ................ 205
Figure C.10. Causal-loop diagram of S-shaped growth structure with transfer............... 205
Figure C.11. Simplified causal-loop diagram of S-shaped growth structure with
transfer......................................................................................................... 206
Figure C.12. Two possible dynamics of S-shaped growth model.................................... 206
Figure C.13. Mirror image dynamics (Stock1) of S-shaped growth model ..................... 207
Figure C.14. Stock-flow diagram of S-shaped growth structure with limit..................... 207
Figure C.15. Causal-loop diagram of S-shaped growth structure with limit ................... 207
Figure C.16. Possible behaviors of S-shaped growth structure with limit....................... 208
Figure C.17. Stock-flow diagram of boom-then-bust structure caused by S-shaped
growth and decay ........................................................................................ 208
xxvi
Figure C.18. Causal-loop diagram of boom-then-bust structure caused by S-shaped
growth and decay ........................................................................................ 209
Figure C.19. Two possible dynamics of boom-then-bust structure (S-shaped growth
and decay) ................................................................................................... 209
Figure C.20. Mirror image dynamics (Stock1) of S-shaped growth behavior for
boom-then-bust structure (s-shaped growth and decay) ............................. 210
Figure C.21. Stock-flow diagram of boom-then-bust structure with delayed effect
of capacity limit........................................................................................... 210
Figure C.22. Causal-loop diagram of boom-then-bust structure with delayed effect
of capacity limit........................................................................................... 211
Figure C.23. Possible dynamics of boom-then-bust structure with delayed effect of
capacity limit............................................................................................... 211
Figure C.24. Stock-flow diagram of first order material delay atomic structure ............. 212
Figure C.25. Stock-flow diagram of third order material delay atomic structure............ 212
Figure C.26. Stock-flow diagram of discrete material delay atomic structure ................ 212
Figure C.27. Behaviors of material delay structure for different orders of delay ............ 213
Figure C.28. Stock-flow diagram of first order information delay atomic structure ....... 214
Figure C.29. Stock-flow diagram of third order information delay atomic structure ..... 214
Figure C.30. Behaviors of information delay structure for different orders of delay ...... 215
Figure C.31. Stock-flow diagram of oscillating atomic structure .................................... 216
Figure C.32. Causal-loop diagram of oscillating atomic structure .................................. 216
xxvii
Figure C.33. Growing oscillations for Fraction greater than zero................................... 217
Figure C.34. Neutral oscillations for Fraction equal zero ............................................... 218
Figure C.35. Damping oscillations for Fraction smaller than zero ................................. 218
Figure C.36. Stock-flow diagram of stock management atomic structure....................... 219
Figure C.37. Causal-loop diagram of stock management atomic structure ..................... 219
Figure C.38. Unstable oscillation, neutral oscillation, stable oscillation and goal
seeking behaviors of the stock management structure................................ 221
Figure C.39. Stock-flow diagram of goal setting atomic structure .................................. 221
Figure C.40. Causal-loop diagram of stock management atomic structure ..................... 221
Figure C.41. Eroding goal and goal seeking behaviors.................................................... 222
Figure D.1.
Noise model ................................................................................................ 223
Figure D.2.
Auto-correlated noise .................................................................................. 224
Figure E.1.
Smoothed life time when there is no phase difference (TSm,S equal to
TPD), and with a non-linear adjustment rule .............................................. 225
Figure H.1.
Runs for Inventory with problematic and non-problematic desired
supply line equations................................................................................... 234
Figure H.2.
Runs for Labor with problematic and non-problematic desired supply
line equations .............................................................................................. 234
Figure H.3.
Runs for Inventory with problematic and non-problematic desired
supply line equations with weights equal to one......................................... 235
xxviii
Figure H.4.
Runs for Labor with problematic and non-problematic desired supply
line equations with weights equal to one .................................................... 236
xxix
LIST OF TABLES
Table 3.1.
Example stock management systems ............................................................ 17
Table 5.1.
Critical values for stock control model that ignores supply line................... 26
Table 12.1.
The final values of the Total revenues, the Net total Profit and the
costs at the end of simulation (8 years – 416 weeks), with four
different s formulations ............................................................................... 149
Table 12.2.
costs at the end of simulation (8 years – 416 weeks), with pure random
and with autocorrelated Customer demand................................................. 152
Table 12.3.
costs at the end of simulation (8 years – 416 weeks), with three
different formulations.................................................................................. 164
Table A.1.
Modeling objects and symbols used in dynamic systems modeling........... 199
Table A.2.
Example model illustrating the objects ....................................................... 199
xxx
LIST OF SYMBOLS/ABBREVIATIONS
A
Coefficient matrix
I
Identity matrix
λ
Eigenvalue
AF
Acquisition flow (acquisition flow from supply line to primary
stock)
AFi
Acquisition flow i (acquisition flow from i th stock of supply line
to i+1 st stock of supply line)
CAP
Capacity
CF
Control flow (control flow for the primary stock)
CFX
Control flow that exclude supply line adjustment term
CF*
Desired control flow (desired control flow for the primary stock)
CP
Productivity coefficient
DPSR
Desired production start rate
DT
Time step
EAF
Expectation adjustment flow (adjustment flow of expected loss)
ECF*
Effect of desired control flow (on the improvement rate; i.e.
utilization of capacity)
ELS
Expected loss (stock of expected loss)
EM
Effect of motivation (on the improvement rate, and also on the
weight of stated goal)
EOQ
Economic Order Quantity
ESLS
Expected secondary loss (stock of expected secondary loss)
ESTM
Effect of short term motivation (on the weight of stated goal)
EVL
Equilibrium value of loss (the equilibrium value of the loss flow)
G
Goal (ideal goal)
GAF
Goal adjustment flow (the rate at which implicit goal approaches
some target)
xxxi
IAFi
Information adjustment flow i (i th flow of information delay
structure)
ID
Information delay
IDFIGSi
Information delay for implicit goal structure (i th information delay
stock for implicit goal)
IDS
Information delay (stock of information delay structure)
IDSi
Information delay i (i th stock of information delay structure)
IGS
Implicit goal (stock of implicit goal)
ING
Indicated goal (an average of stated goal and traditional
performance)
LF
Loss flow (loss flow from primary stock)
MAIR
Minimum acceptable improvement rate
MLR
Maximum loss rate
OID
Order of information delay (number of the stocks in the
information delay structure)
OMD
Order of material delay (number of the stocks in the material delay
structure)
OR
Order rate
OSL
Order of supply line (number of the stocks in the supply line)
OSSL
Order of secondary supply line (number of the stocks in the
secondary supply line)
PLS
Perceived loss (stock of perceived loss)
PP
Perceived performance
RL
Reference level (past level in trend computation)
RLA
Likelihood of accomplishment ratio (ratio of time needed to time
available)
RST
Short term accomplishment ratio
S
Stock (primary stock)
Si
Stock i (i th stock in the model)
S*
Desired stock (desired level of primary stock)
SA
Stock adjustment (stock adjustment term in desired control flow)
xxxii
SAF
Secondary acquisition flow (acquisition flow from secondary
supply line to secondary stock)
SAFi
Secondary acquisition flow i (acquisition flow from i th stock of
secondary supply line to i+1 st stock of secondary supply line)
SCF
Secondary control flow (control flow for the secondary stock)
SEAF
Secondary expectation adjustment flow (adjustment flow of
expected loss in the secondary stock structure)
SEVL
Equilibrium value of secondary loss (the equilibrium value of the
loss flow in the secondary stock structure)
SG
Stated goal (managerially stated goal)
SGS
Stated goal (stock of managerially stated goal)
SGS*
Indicated stated goal
SLA
Supply line adjustment (supply line adjustment term in desired
control flow)
SLF
Secondary loss flow (loss flow from secondary stock)
SLS
Supply line (stock of supply line of primary stock)
SLS*
Desired supply line (desired level of supply line of primary stock)
SLSi
Supply line i (i th stock of supply line of primary stock)
SMAF
Smoothing adjustment flow
SMLTS
Smoothed life time (stock of smoothed life time)
SMS
Smoothed stock (stock of smoothed stock)
SMTH3(input,
Third order information delay; it smoothes the given input with the
smoothing time)
given smoothing time.
SR
Shipment rate
SS
Secondary stock
SS*
Desired secondary stock (desired level of secondary stock)
SSA
Secondary stock adjustment (secondary stock adjustment term in
secondary control flow)
SSLA
Secondary supply line adjustment (secondary supply line
adjustment term in secondary control flow)
SSLS
Secondary supply line (stock of supply line of secondary stock)
xxxiii
SSLS*
Desired secondary supply line (desired level of supply line of
secondary stock)
SSLSi
Secondary supply line i (i th stock of supply line of secondary
stock)
STEP(height, τ)
Its output is zero when time < τ
and the output is height when time ≥ τ
TAD
Acquisition delay time (average time that orders spend in supply
line)
TC
Time constant
TCLf
Calculated (estimated) life time
TDF
Time decrease flow (the rate at which the available time decreases)
TEL
Expected loss averaging time
TESL
Expected secondary loss averaging time
TFP
Formation of perception time
TGA
Goal adjustment time
THS
Time horizon (stock of time horizon; available time to reach the
goal)
TID
Information delay time
TLf
Life time (average decay time of loss flow from primary stock)
TMOH
Managerial operating horizon
TPD
Perception delay time
TPF
Traditional performance formation (flow of traditional performance
stock)
TPS
Traditional performance (stock of traditional performance)
TRL
Reference level formation time
TSm
Smoothing time
TSA
Stock adjustment time (time needed to adjust the discrepancy in the
primary stock)
TSAD
Secondary acquisition delay time (average time that orders spend in
secondary supply line)
TSGA
Stated goal adjustment time
xxxiv
TSH
Short time horizon
TSLA
Supply line adjustment time (time needed to adjust the discrepancy
in the supply line stock)
TSLf
Secondary life time (average decay time from secondary stock)
TSm,Lf
Smoothing time for life time averaging
TSm,S
Smoothing time for stock averaging
TSSA
Secondary stock adjustment time (time needed to adjust the
discrepancy between secondary stock and its desired level)
TSSLA
Secondary supply line adjustment time (time needed to adjust the
discrepancy between secondary supply line and its desired level)
TTPF
Traditional performance formation time
TTS
Total secondary delay time (sum of secondary stock adjustment
time and secondary acquisition delay time)
U
Utilization
VACF*
Virtually adjusted “desired control flow” (the desired control
including the virtual supply line adjustment)
VAF
Virtual adjustment flow
VIF
Virtual inflow (inflow of the virtual supply line)
VOF
Virtual outflow (outflow of the virtual supply line)
VSL
Virtual supply line (a converter computing the quantity in the
“virtual supply line”)
VSLID
Virtual supply line for information delay structure
VSLSS
Virtual supply line for secondary stock control structure
VSL*
Desired virtual supply line (desired level of virtual supply line)
VSL*ID
Desired virtual supply line for information delay structure
VSL*SS
Desired virtual supply line for secondary stock control structure
VSLA
Virtual supply line adjustment (the portion of the control coming
from the virtual supply line)
VSLAID
Virtual supply line adjustment for information delay structure
VSLASS
Virtual supply line adjustment for secondary stock control structure
xxxv
VSLS
Virtual supply line stock (a stock computing the quantity in the
“virtual supply line”)
WIPI
Work in process inventory
WSG
Weight of stated goal
WSL
Weight of supply line (ratio of adjustment of supply line of primary
stock to adjustment of primary stock)
WSSL
Weight of secondary supply line (ratio of adjustment of secondary
supply line to adjustment of secondary stock)
WVSL
Weight of virtual supply line (ratio of adjustment of virtual supply
line to adjustment of primary stock)
WVSL,ID
Weight of virtual supply line for information delay structure
WVSL,SS
Weight of virtual supply line for secondary stock control structure
1
1. INTRODUCTION
1.1. System Dynamics Methodology
“System Dynamics” is a methodology for modeling, analyzing and improving
dynamic socio-economic and managerial systems, using a feedback perspective.
System Dynamics method often uses simulation to generate dynamic behaviors of
models. Simulation is a must, since it is hard or impossible to find analytical solutions to
most non-linear feedback models. The results are supported with mathematical analysis
when it is necessary and possible.
The span of applications of the System Dynamics field in general includes:
“corporate planning and policy design”, “public management and policy”, “micro and
macro economic dynamics”, “educational problems”, “biological and medical modeling”,
“energy and the environment”, “theory development in the natural and social sciences”,
“dynamic decision making research” and “complex non-linear dynamics” (Forrester, 1961;
Roberts, 1981; Sterman, 2000).
1.2. Human Decisions in System Dynamics Models
In human systems, systemic dynamic feedback models describe not only the physical
structure of a system, but also the institutional structure of the system, and mimic the
actions of the decision makers that take part within the system, so that these models are
more complex compared with non-human systems. Modeling the behavior and the
decisions of the human beings is a challenging task.
1.3. Feedback, Delays and Non-linearity in Human Decisions
Main aspects of systemic dynamic feedback models are feedback, delays and nonlinearity that are the cause of the complex dynamics. It is almost impossible to build
realistic models of human systems without taking into account the feedback. In dynamic
2
systems, variables are simultaneously causes and effects of each other (i.e. Population is
affected by births and births are affected by population).
Time delays are also very universal in dynamic systems. They detach cause and
effect in time and space (as information/perception delays or material delays), and
contribute to the complexity of the dynamics. Delays prevent our learning of cause and
effect relationships. Human beings are good in learning when the cause and effect
relationship is close in time and space, but have difficulties in delayed cause-effect settings
(Barlas and Özevin, 2001; Sterman, 1987b; Sterman, 1989a; Sterman, 1989b).
In this context non-linearity means that the relation between cause and effect is not
linear, not proportional. It means there are threshold or saturation phenomena between the
cause and effect. Non-linearity is important, since “much of real-life behavior arises from
non-linearities”, and “much of the information we possess about real life is information
about non-linear control policies” (Forrester, 1985). Non-linearity is especially important
in human systems. “In human decision making, the inputs to a decision are perceived nonlinearly. Many variables when in their normal ranges exert little influence, but those same
variables can dominate all others when they move outside their normal ranges” (Forrester,
1985). Only non-linear models have the ability to show rich and complex behavior patterns
that are observed in human systems (Forrester, 1987). Non-linear structures can help the
theory of constructing realistic models of dynamic human systems.
1.4. Systematic Mistakes Made by Decision Makers
The three characteristics mentioned above; feedbacks, delays and non-linearities
make the prediction of the behaviors of the dynamic systems nearly impossible.
Experiments in System Dynamics area show that we are poor decision makers in systemic
dynamic feedback environments (Aybat et al., 2003; Barlas and Özevin, 2001; Morecroft,
1983; Moxnes, 1998; Sterman, 1987b; Sterman, 1989a; Sterman, 1989b; Sterman, 2000).
Decision makers ignore, distort or misperceive feedbacks, time delays and non-linearities
in their decisions. Their natural consequences are undesirable oscillations, boom-then-bust,
unavoidable collapse or other counterintuitive undesirable behavior.
3
1.5. Purpose and Main Focus of this Research
The purpose of this thesis is:
• to identify the basic time-delayed, non-linear feedback decision structures used in the
literature and the typical behavior patterns generated by such structures,
• to evaluate the validity and effectiveness of the typical formulations and structures
used in such decision systems in coping with dynamic complexities, and
• to design improvements in the decision structures/formulations and to analyze the
dynamics of these alternative structures.
The focus area of this thesis is the generic stock management (control) system. As
human decision systems, stock control problems involve complexities caused by feedback,
delays and non-linearities which are important issues in this thesis.
4
2. REPRESENTATION TOOLS OF SYSTEM DYNAMICS
METHODOLOGY
2.1. Stock-flow Representation
Stock-flow representation is the basic convention used in System Dynamics models.
A stock is an accumulation and a flow is the rate that changes the levels of the stocks
(Forrester, 1973):
Stock 1
Convertor 2
Outflow
Inflow
Stock 2
Converter 1
Inflow 2
Figure 2.1. Stock-flow diagram
In Figure 2.1, an arrowhead shows the direction of the flow when its value is positive
and the opposite direction is the direction of the flow when it is negative. (i.e. an inflow
would flow out, if its value is negative). Stocks can only change by way of their in and out
flows.
Flows and converters are expressed as functions of stocks, converters and flows.
Converters are intermediate variables (or auxiliaries). Thin arrows (arcs) that called
“connectors” show the functional relations between variables that are not in form of stockflows. For example to determine the value of the Inflow at time t, the values of the Stock1
and the Converter1 at time t must be known. The general form of a flow and a converter
equation can be stated as follows:
Flowi = f i (Stocks, Convertors, t )
(2.1)
Convertors j = f j (Stocks, Convertors, Flows, t )
(2.2)
5
Main difference between converters and flows is that flows directly flow in or out of
the stocks, while converters can only influence flows and other converters (see Appendix
A for modeling objects and symbols used in dynamic systems modeling).
2.2. Integral, Differential and Difference Equations
The exact integral equation of the first stock of the model in Figure 2.1 is:
t
S1 (t ) = S1 (0 ) + ∫ (Inflow − Outflow ) dt
(2.3)
0
The approximate, numerical version of this integral equation can be given as:
S1 (t ) = S1 (t − DT ) + (Inflow − Outflow) DT
(2.4)
In compact vector form, the equation for any stock is in general;
S (t ) = S (t − DT ) +
(∑
M
i =1
)
( flowi ) DT
(2.5)
where “__” denotes n-dimensional vectors and there are some flows (# of flows = M)
associated with each stock. Note that approximate integral equation is iterative and DT is
the time step in each iteration.
In most cases, the resulting integral equations are non-linear (after all the flow and
converter equations are plugged in). Analytical results are very hard or often impossible to
obtain, so computer simulation is generally used to obtain the behavior of such complex
systems.
If the modeled system is continuous (DT small enough), then the integral Equation
(2.4) can also be stated as the following differential equation:
•
S1 (t ) − S1 (t − dt ) dS1 (t )
=
= (Inflow − Outflow )
dt →0
dt
dt
S 1 = Lim
(2.6)
6
In vector form, Equation (2.6) can be generalized as;
•
S = f (S , t )
(2.7)
Note that, in simulation it is not possible to take DT as zero, so “small enough” DT is
only an approximation to the continuous systems. In simulation, choosing an appropriate
time step (DT) is an important task. Sterman (2000) suggests:
“A widely used rule of thumb is to set the time step between one-fourth and onetenth the size of the smallest time constant in your model. However, in a large model it is
difficult to estimate all the time constants and select an appropriate time step, so you must
always test the sensitivity of your results to the choice of time step and integration method”
(Sterman, 2000).
If the modeled system is discrete (DT equal to one), then the integral Equation (2.4)
reduces to the following difference equation:
S1 (k + 1) = S1 (k ) + Inflow(k ) − Outflow(k )
(2.8)
In vector form, the equation (7) is in general;
S (k + 1) = S (k ) + ∑i =1i ( flowi (k ))
(2.9)
S (k + 1) = f (S (k ), k )
(2.10)
M
or
The order of a model (or a structure), is the number of the stocks in that model or in
that structure. This corresponds to the order of the equivalent differential or difference
equation.
7
2.3. Causal-loop Diagrams
Causal-Loop diagrams show the causal relations between the variables, the sign of
the relations and the feedback loops.
+
Variable 3
-
+
+
-
Variable 1
+
Variable 2
Variable 5
Variable 6
+
Variable 4
+
+
Birth rate
+
+
+
Population
-
-
Birth fraction
Death rate
+
Death fraction
Figure 2.2. Causal-loop diagram and an example
The sign of a loop shows if it is reinforcing or counteracting. A loop is positive (or
reinforcing), if an initial change in a variable is reinforced around the loop, when it comes
back to the same variable. A loop is negative (or counteracting, or balancing), if an initial
change in a variable is counteracted by the loop, so as to create an opposite change in the
variable once the loop is completed.
2.4. Dynamic Behavior (Output Behavior)
Dynamic behavior is the dynamics that results from the given model structure.
Mathematically, it is the solution of the corresponding differential/integral equation.
Dynamic behaviors can be obtained as graphical outputs from simulation runs. Output
graphs may be of different types. The most common plot is time series. Sometimes scatter
8
plots (phase plots) are also useful. Scatter plots are obtained by plotting a variable against
another.
1: X
0.20
1:
1:
-0.15
1:
-0.50
1
1
1
1
0.00
50.00
100.00
Graph 1 (Dynamic behavior)
Time
150.00
00:23
200.00
27 Jan 2003 Mon
Figure 2.3. Behavior of a variable in time
Y v . X: 1 35
0
-35
Page 1
-15
5
X
25
22:10
Wed, Jul 16, 2003
Dy namic behav ior
Figure 2.4. Behavior of two variables with respect to each other (phase plot)
9
3. USEFUL ATOMIC STRUCTURES
There are many human system structures in the System Dynamics literature. Some
most important elementary, generic decision structures are presented in Appendix C in
unified form. Some of these structures are just combinations of others, and furthermore, for
some conditions (parameter values), some of these structures and their behaviors can be
equivalent to others, but their scope is different and they must be treated separately (Barlas,
2002).
Some of the structures presented in Appendix C are also repeated in this chapter,
because they are highly relevant to the scope of this thesis. In this chapter, stock-flow
diagrams, causal-loop diagrams and possible behavior plots are presented for selected
atomic structures. For other atomic structures, please refer to Appendix C.
3.1. Goal Seeking Atomic Structure
Goal seeking structure is the most important single-stock, negative (counteracting)
loop control structure. There is a goal, and the stock is adjusted so as to attain its goal.
Stock
Adjustment flow
Discrepancy
Adjustment time
Goal
Figure 3.1. Stock-flow diagram of goal seeking atomic structure
+
Stock
Adjustment time
Adjustment flow
-
+
-
Goal
Discrepancy +
Figure 3.2. Causal-loop diagram of goal seeking atomic structure
10
The differential equation of the model in Figure 3.1 can be given as:
•
S = Adjustment flow =
Discrepanc y
Goal − S
=
Adjustment time Adjustment time
(3.1)
Possible behaviors of the model in Figure 3.1 are:
1: Stock
1:
2: Stock
3: Stock
8.00
3
3
3
1:
2
4.00
2
2
2
3
1
1
1
1:
0.00
1
0.00
2.50
5.00
Graph 1 (goal seeking)
7.50
Time
19:53
10.00
Fri, Jan 17, 2003
Figure 3.3. Possible behaviors of goal seeking atomic structure
3.2. Delays
3.2.1. Material Delay Atomic Structure
These structures represent the delays experienced by flows on a material (conserved)
stock-flow chain (such as goods ordered and still in supply line).
Stock
Input
Output
Delay time
Figure 3.4. Stock-flow diagram of first order material delay atomic structure
11
For a continuous material delay, the differential equation of the model in Figure 3.4
can be given as:
•
S = Input − Output = Input −
Stock 2
Stock 1
Input
S
Delay time
Acquisition flow 1
Order of material delay
(3.2)
Stock 3
Acquisition flow 2
Output
Individual delay time
Delay time
Figure 3.5. Stock-flow diagram of third order material delay atomic structure
Note that, Order of material delay (OMD) is three for a third order delay. The
differential equations of the model in Figure 3.5 can be given as:


S1

 •   Input − AF1   Input −
 S1  
 
(Delay time / OMD )

•  
 

S1
S2
 S 2  =  AF − AF
=

−
2
   1
  (Delay time / OMD ) (Delay time / OMD ) 
•  
 

S3
S2
S3  
 

−
AF
Output
−
  (Delay time / O ) (Delay time / O ) 
   2
MD
MD 

(3.3)
Stock
Input
Output
Delay time
Figure 3.6. Stock-flow diagram of infinite order (discrete) material delay atomic structure
Order of a delay can be any positive integer number. As OMD approaches infinity, the
material delay is called discrete material delay. The time-lagged differential equation of the
model in Figure 3.6 can be given as:
12
•
S (t ) = Input (t ) − Output (t ) = Input (t ) − Input (t − Delay time )
(3.4)
Possible behaviors of the models in Figure 3.4 (Output), Figure 3.5 (Output2) and
Figure 3.6 (Output3) are:
1: Input
1:
2:
3:
4:
1:
2:
3:
4:
1:
2:
3:
4:
2: Output
1.00
3: Output 2
1
4
4: Output 3
1
4
1
3
3
4
2
2
2
3
0.50
0.00
1
0.00
3
2
4
10.00
Graph 1 (Material delay)
20.00
30.00
Time
01:41
40.00
18 Jan 2003 Sat
Figure 3.7. Possible behaviors of different order material delay structures
3.2.2. Information Delay Atomic Structure
These structures represent delayed awareness about changing conditions, delayed
perceptions or estimations.
Output
Adjustment flow
Delay time
Input
Discrepancy
Figure 3.8. Stock-flow diagram of first order information delay atomic structure
The differential equation of the model in Figure 3.8 can be given as:
13
•
Discrepanc y Input − Output
=
Delay time
Delay time
Information delay 1
Information delay 2
Discrepancy 2
Adjustment flow 2
Adjustment flow 1
Discrepancy 1
(3.5)
Output
Discrepancy 3
Adjustment flow 3
Input
Delay time
Order of information delay
Figure 3.9. Stock-flow diagram of third order information delay atomic structure
The differential equations of the model in Figure 3.9 can be given as:
 Discrepancy1   Input − IDS1 
 

•  
 S 1   (Delay time / O ID )   (Delay time / O ID ) 
  
 

  
 

 •   Discrepancy   IDS − IDS

2
1
2
=
S2  = 


   (Delay time / O ID )   (Delay time / O ID ) 
  
 

•  
 

 S 3   Discrepancy3   IDS 2 − Output 
   (Delay time / O )   (Delay time / O ) 
ID 
ID  

(3.6)
Note that, Order of information delay (OID) is three for a third order delay and order
of a delay can be any positive integer number.
The graphical outputs of the material delay and the information delay structures are
exactly the same (Figure 3.7, Figure 3.10). Possible behaviors of the models in Figure 3.8
(Output) and Figure 3.9 (Output2) are:
14
1: Input
1:
2:
3:
2: Output
1.00
3: Output 2
1
1
1
3
3
2
2
2
1:
2:
3:
1:
2:
3:
3
0.50
0.00
1
0.00
2
3
10.00
Graph 1 (Information delay)
20.00
30.00
Time
02:34
40.00
18 Jan 2003 Sat
Figure 3.10. Possible behaviors of different order information delay structures
3.3. Stock Management Atomic Structure
The stock management structure is a general and more realistic version of the goal
seeking structure seen above. There are three important additions: There is a material delay
(supply line) before control flow actually reaches the stock; there is an outflow (loss) from
the control stock and there is a delay (information delay) before this outflow can be
estimated by the decision maker. Decision making process may also involve information
delays. The three features; supply line, loss flow and estimated loss flow (Expected loss)
are incorporated in the structure and the control flow formulation in Figure 3.11.
Stock
Supply line
Control flow
Acquisition delay time
Supply line adjustment
Desired supply line
Weight of supply line
Loss flow
Acquisition flow
Desired stock
Stock adjustment
Stock adjustment time
Expected loss
Expectation adjustment flow
Figure 3.11. Stock-flow diagram of stock management atomic structure
15
±
Supply line
Acquisition flow
-
+
-
±
Stoc k
-
+
+
+
±
-
Control flow
+
+
Stoc k adjustment
+
Loss flow
+
±
+
Desired supply line
Stoc k adjustment time
Desired stock
+
-
+
Expected loss
-
+
±
Figure 3.12. Causal-loop diagram of stock management atomic structure
The differential equations of the model in Figure 3.11 (for explanation of variable
conventions see Appendix B) can be given as:
 SL
 •   AF − LF  
− LF
S  
  T AD
 •  
 
 SL  =  CF − AF  =  ELS + SA + SLA − SL

 
 
T AD
 •  
 
 ELS  
  LF − ELS
  T

  EAF
EL











(3.7)
where Acquisition flow is typically formulated as the output of a material delay:
AF =
SLS
T AD
(3.8)
Control flow (control decision) is given by:
CF = ELS + SA + SLA
(3.9)
16
Stock adjustment is given by:
S* − S
TSA
SA =
(3.10)
Supply line adjustment is given by:
SLA = WSL •
T • ELS − SLS
SLS * − SLS
= WSL • AD
TSA
TSA
(3.11)
Expectation adjustment flow is given by:
EAF =
LF − ELS
TEL
(3.12)
Note that these equations will be further discussed in the following chapter. Possible
behaviors of the model in Figure 3.11 are:
1: Stock
2: Stock
1:
2.00
1:
0.00
3: Stock
1
3
3
4
4: Stock
4
3
4
3
4
1
2
2
2
2
1
1
1:
-2.00
0.00
25.00
50.00
75.00
Graph 1 (Stock management)
Time
18:09
100.00
25 Jan 2003 Sat
Figure 3.13. Possible behaviors of stock management atomic structure, under different
parameter settings
17
The atomic stock management structure discussed above is very generic, and can be
observed in diverse situations ranging from inventory management to information
processing, from capital investment to agricultural systems. Examples given in Table 3.1
are adopted from Sterman (2000), and enriched by adding information delay examples.
Table 3.1. Example stock management systems
System
Inventory
Stock
Inventory
management
Supply line
Supply line
Control
Acquisition
Loss flow
structure
flow
flow
Goods on
Material
Orders for
Arrivals
Shipments to
order
delay
goods
from
customers
supplier
Capital
Capital plant
investment
Human
Employees
resources
Plant under
Material
New
Construction
Depreciation
construction
delay
contracts
completion
Vacancies
Material
Vacancy
Hiring rate
and trainees
delay
creation
Planting rate
Harvest rate
Consumption
Layoffs and
quits
Agricultural
Crop
Crops in the
Material
commodities
inventory
field
delay
Information
Downloaded
Information
Information
Ordering
Download
Deleting
download
information
ordered to be
delay
new
rate
information
from internet
downloaded
information
from local
and yet not
to be
hard disk
downloaded
downloaded
(a “virtual”
supply line)
Memory
Memorized
Material to
Information
Deciding
Memorizing
Forgetting
management
facts
be
delay
material to
rate
rate
memorized
be
(a “virtual”
memorized
supply line)
18
4. PROBLEM DEFINITION
In human systems, systemic dynamic feedback models describe not only the physical
structure of a system, but also its institutional structure, and mimic the actions of the
decision makers which take part within the system, so that these models are more complex
compared with non-human systems (Forrester, 1961; Forrester, 1994; Sterman, 2000).
Modeling the behavior and the decisions of the human beings is a challenging task.
System
Supply Line
Decision flow
Control variable\Stock
Acquisition flow
Flow
Exogenous variables
Other endogenous variables
Decision Formulations
Expectation Formation
Expectations
Indicated decision flow
Expectation adjustment flows
Expectation delay times
Adjustments
Evaluation and Goal Formation
Evaluations
Desired levels\Goals
Delays\Fractions\Multipliers
Figure 4.1. Stock management and general human decision framework
19
Figure 4.1 shows the most general framework of stock management and human
decision-making, and the basic components involved. Each component box (Evaluation
and Goal Formation, Expectation Formation and Decision Formulations) must be filled
with a proper stock-flow sub-structure and formulation, for the decision model to be
complete.
In the Evaluation and Goal Formation box; the results of the decision (flow) are
evaluated against the goal and further the effectiveness of the goal itself may be evaluated.
In Expectation Formation box; the external inputs (such as demand) and other non-decision
variables are estimated (discussed extensively in Sterman, 1987a). Finally, decision
formulation is completed as a function of the Evaluation and Goal Formation, and
Expectation Formation sub-structures. The resulting decision equation is called “indicated
decision flow” to state the fact that there can be delays and other interventions/effects
before this can become the actual decision flow:
Indicated decision flow = f(Expectations, Evaluations, Goals)
(4.1)
Decision flow = g(Indicated decision flow, other factors)
(4.2)
The framework in Figure 4.1 includes feedbacks, delays and non-linearities. The
management of such a system that can produce complex dynamic behaviors is hard for
human decision makers (Forrester, 1985). In this research, we will evaluate the existing
decision heuristics to see to what extent they can cope with dynamic problems created by
feedbacks, delays and non-linearities. We will analyze the behaviors of the different
decision models, structures, delay types and formulations, and suggest possible
improvements. All three components/structures (Evaluation and Goal Formation,
Expectation Formation and Decision Formulations) will be analyzed in the thesis.
There are some articles in the literature, which analyze the effects of ignoring supply
line (pipeline delay) in decisions in stock control, but there is no reported work on other
kinds of delays (such as delays caused by controlling of a primary stock via “secondary”
stocks –like controlling the production rate by changing the production capacity–, or
delays in information processing). In this research, we will consider all typical delays in
20
the decision control formulations. We pay special attention to the relationship interactions
between delays and non-linearity. We conjecture that appropriate consideration of delays
in decision formulations will increase the efficiency of the linear and non-linear controls.
Since “inventory management” is a typical example of stock management problem,
we compare system dynamics control rules and standard inventory control rules. We
discuss if there can be possible improvements in these control rules by synthesizing their
unique features, including non-linear formulation possibilities.
21
5. LINEAR CONTROL OF A SINGLE STOCK WITH SUPPLY LINE
DELAY
In general, there is a delay between the control action and its result. In most stock
control systems, the orders that are given arrive at the stock after some delay. The orders
that are given, but have not yet reached the stock are said to be in supply line. The supply
line has a material delay structure (Section 3.2.1 and Appendix C.6.1).
5.1. Dynamics of Stock Control without Considering Supply Line
In stock control decisions, it is wrong to ignore the supply line term, if stable
behavior is desirable (Aybat et al., 2003; Barlas and Özevin, 2001; Forrester, 1961;
Özevin, 1999; Sterman, 2000). In the absence of a “supply line adjustment” term, the stock
may oscillate. Furthermore, it can oscillate unstably. In system dynamics literature there is
a famous game called “Beer Game”, which is widely used to illustrate the mismanagement
of the supply line caused by ignoring the supply line (Sterman, 1987b; Sterman 1989a;
Sterman 1989b; Sterman, 2000).
In this section, stock control system with first order supply line delay is going to be
analyzed in the absence of Supply line adjustment term in the Control flow. In model in
Figure 5.1, it is assumed that the delay in observing and reacting to the Loss flow is very
short when compared with other time constants, so instead of including the Expected loss
(which is the expected value of Loss flow), Loss flow itself is directly used in the Control
flow.
Supply line
Control flow
Stock
Loss flow
Acquisition flow
Stock adjustment
Desired stock
Figure 5.1. Stock management model with first order supply line delay
22
The dynamic equations (differential equations) of the model in Figure 5.1 are given
below. For explanation of variable conventions see Appendix B.
•
S = AF − LF
(5.1)
•
SLS = CF − AF
(5.2)
The Acquisition flow (AF), assumed to be the output of a material delay, and a very
simple Control flow (CFX) equation, which ignores the supply line, can be given as
follows:
AF =
SLS
T AD
CFX = LF + SA = LF +
(5.3)
S* − S
TSA
(5.4)
For the purpose of this initial discussion Loss flow (LF) and the Desired stock (S*)
are assumed to be zero without loss of generality. If the equations for the Acquisition flow
and the Control flow are inserted, then the Equation (5.1) and Equation (5.2) become:
•
S=
•
SLS
T AD
(5.5)
S
SLS
−
TSA T AD
(5.6)


 • 
 S  = A• S 


 • 
 SLS 
 SLS 
(5.7)
SLS = −
or in matrix notation:
23
where the coefficient matrix A is:
1 

T AD 


1 
−
T AD 

 0

A=

 1
− T
 SA
(5.8)
At equilibrium points, all the derivatives must vanish:
•
S=
•
SLS = −
SLS
=0
T AD
(5.9)
S
SLS
−
=0
TSA T AD
(5.10)
From Equation (5.9) and Equation (5.10), we find that there is only one equilibrium
point that is:
(S , SLS )equilibriu m = (0,0 )
(5.11)
The characteristic determinant of Equation (5.5) and Equation (5.6) is defined as:
1
−λ
T AD
A −λ •I =
(5.12)
−
1
TSA
−
1
−λ
T AD
where λ represents the eigenvalues of coefficient matrix A. From Equation (5.12), we can
obtain the following characteristic equation:
λ2 +
λ
T AD
+
1
=0
TSA • T AD
(5.13)
24
The roots of the characteristic equation are:
−
λ1,2 =
2
1
T AD
 1 
4
4 • T AD
 −
± 
−1± 1−
TSA • T AD
TSA
 T AD 
=
2
2 • T AD
(5.14)
Keeping in mind that the Acquisition delay time (TAD) and the Stock adjustment time
(TSA) are positive numbers, it can easily be observed from Equation (5.14) that the
eigenvalues can not have real part bigger than zero, meaning that the equilibrium point
(0,0) is always stable (Barlas, 2003). The eigenvalues have imaginary parts when:
TSA
<4
T AD
(5.15)
which means that only in this condition can there be oscillatory behavior. For illustration,
sample runs are shown in Figure 5.2 for different values of TSA. A pulse of -1 unit is
applied on Stock at time 5 to perturb the system from the equilibrium.
1: Stock
1:
2: Stock
3: Stock
4: Stock
1.00
4
1:
0.00
1
2
3
3
2
4
2
1
3
4
1
2
3
4
1
1:
-1.00
0.00
10.00
Graph 1 (oscillation)
20.00
Time
30.00
22:48
40.00
07 Oct 2002 Mon
Figure 5.2. Non-oscillatory and oscillatory behavior runs for TAD equal to 1 and TSA equal
to 8, 4, 2 and 0.5 respectively for the 1st, 2nd, 3rd and 4th runs
25
For a model similar to the one in Figure 5.1 unstable oscillation is only possible for
supply line having more than one stock. The following model is a sample of such a model:
Order of supply line
Supply line 1
Supply line 2
Control flow
Acquisition flow 1
Stock
Acquisition flow 2
Loss flow
Desired stock
Stock adjustment
Figure 5.3. Stock management model with second order supply line
Sample runs of model in Figure 5.3 are shown in the following figure:
1: Stock
1:
2: Stock
3: Stock
4: Stock
2.00
4
4
1:
0.00
1
3
1
3
1
2
1:
-2.00
0.00
2
2
4
3
1
2
3
4
4.00
8.00
12.00
Graph 1 (unstable oscillation)
Time
16:25
16.00
11 Oct 2002 Fri
Figure 5.4. Runs for TAD equal to 1, and TSA equal to 3.37 (goal seeking), 0.5 (stable
oscillation), 0.25 (neutral oscillation) and 0.21 (unstable oscillation) respectively for the
1st, 2nd, 3rd and 4th runs
26
The runs in Figure 5.4 show that unstable oscillation is possible for stock
management model with second order supply line.
The following table summarizes the critical values for oscillations and unstable
oscillations that are found by either mathematical analysis or simulation runs:
Table 5.1. Critical values for stock control model that ignores supply line
Order of supply line delay
(Number of stocks in the supply line)
Oscillation starts for
TSA
<
T AD
Unstable oscillation starts for
TSA
<
T AD
0
1
2
∞ (discrete)
for no value
4
~ 3.37
~ 2.71
for no value
for no value
0.25
~ 0.64
5.2. The Effects of the Stock Adjustment Time and Acquisition Delay Time on the
Amplitude and the Period of the Oscillations
The effects of the time delays on the oscillatory behavior are very clear:
• If Stock adjustment time (TSA) is decreased, the amplitude grows and period shortens.
• If Acquisition delay time (TAD) is increased, the amplitude grows and period gets
longer.
The graphs in Figure 5.5 and Figure 5.6 are the behaviors of the model in Figure 5.3.
The behavior may change if the order of the supply line changes, but the directions of the
effects of the two time constants remain unchanged.
In the following two graphs (Figure 5.5 and Figure 5.6), runs for TAD and TSA are
obtained. A pulse of -1 unit is applied on stock at time 1 to perturb the system from the
equilibrium.
27
1: Stock
1:
2: Stock
3: Stock
1.00
3
1
1:
0.00
3
1
1
1
2
3
2
3
2
1:
-1.00
2
0.00
4.00
8.00
Graph 1 (effect of Tsa)
12.00
Time
23:47
16.00
07 Oct 2002 Mon
Figure 5.5. The effect of the changes in Stock adjustment time (TSA equal to 0.5, 0.4 and
0.3 respectively for the 1st, 2nd and 3rd runs) for TAD equal to 1
1: Stock
1:
2: Stock
3: Stock
3
1.20
2
1
1
1:
0.00
1
2
3
1
3
2
2
3
1:
-1.20
0.00
10.00
Graph 1 (effect of Tad)
20.00
30.00
Time
16:31
40.00
11 Oct 2002 Fri
Figure 5.6. The effect of the changes in Acquisition delay time (TAD equal to 3.5, 4 and 4.5
respectively for the 1st, 2nd and 3rd runs) for TSA equal to 1
28
5.3. The Role of Supply Line in Stock Control Decisions
Supply line
Stock
Control flow
Loss flow
Acquisition flow
Desired supply line
Stock adjustment
Desired stock
Figure 5.7. Stock management model with supply line considered in Control flow
To consider the Supply line in the decision flow formulation, the model in Figure 5.1
is modified. The Equation (5.1) and Equation (5.2) of the model in Figure 5.1 do not
change. Furthermore, the Acquisition flow (AF) Equation (5.3) is not affected either, but
the Control flow (CF) equation changes:
CF = LF + SA + SLA = LF +
(S
= LF +
*
)
(
S * − S SLS * − SLS
+
TSA
TSLA
− S + W SL • SLS * − SLS
TSA
)
(5.16)
where, the Weight of supply line (WSL) is defined as a simple ratio of Stock adjustment time
(TSA) and Supply line adjustment time (TSLA):
WSL =
TSA
TSLA
(5.17)
so, if supply line is going to be considered, one way is to define how much importance is
given to supply line with respect to the stock adjustment (Sterman, 2000).
Again for the purpose of this discussion Loss flow (LF) and the Desired stock (S*) are
assumed to be zero. Furthermore the Desired supply line (SLS*) is also zero because it is
dependent on the Loss flow:
29
SLS * = T AD • LF
(5.18)
Note that the Desired stock is not defined internally in the model like Desired supply
line (SLS*) and can be chosen freely or, we can say that it can be chosen such that it serves
the stock control policy that we apply.
If the Acquisition flow (AF) Equation (5.3), and the Control flow (CF) Equation
(5.16) are inserted to Equation (5.1) and Equation (5.2) the following equations are
obtained:
•
S=
•
SLS = −
SLS
T AD
(5.19)
S + WSL • SLS SLS
−
TSA
T AD
(5.20)
Note that Equation (5.5) and Equation (5.19) are exactly the same. From Equation
(5.19) and Equation (5.20) we find that:
( S , SLS ) equilibrium = (0,0)
(5.21)
This equilibrium point is also exactly the same as the one found in the earlier model
with no supply line consideration.
The coefficient matrix A in Equation (5.7) becomes:

 0

A=

 1
− T
 SA





1 
−
T AD 
1
T AD
−
WSL
TSA
(5.22)
30
The characteristic determinant of Equation (5.19) and Equation (5.20) is:
1
−λ
T AD
A−λ •I =
(5.23)
−
1
TSA
−
WSL
1
−
−λ
TSA T AD
From Equation (5.23) we can obtain the following characteristic equation:
W
1 
1
 λ +
λ2 +  SL +
=0
TSA • T AD
 TSA T AD 
(5.24)
λ1,2 =
W
1
−  SL +
 TSA T AD

 ±

 WSL
1

+
 TSA T AD
2
2

4
 −
TSA • T AD

(5.25)
The eigenvalues have imaginary parts when:
 WSL
1

+
 TSA T AD
2

4
 <
TSA • T AD

(5.26)
Inequality (5.26) can be re-written as:
TSA
4
<
T AD 
T
1 + WSL • AD
TSA




2
(5.27)
If WSL is set to zero, Inequality (5.27) reduces to Inequality (5.15) and if WSL is set to
one Inequality (5.27) becomes:
31
(T AD − TSA )2 < 0
(5.28)
and this condition is impossible so, for WSL equal to one, there cannot be any oscillation
for first order supply line system.
For WSL equal to one, the stock and the supply is considered as a single stock
(“Effective Inventory”) and as it is mentioned in Appendix C.8, a single stock model (first
order) cannot oscillate. If sum of the stock and the supply line stocks cannot oscillate, this
also stabilizes the individual stocks (Sterman, 2000).
Furthermore, it can be shown that linear models like the one in Figure 5.7 cannot
show oscillatory behavior independent of the order of the supply line, in following
condition, whatever the values of the TAD and TSA are:
WSL ≥ 1
(5.29)
Exception for the stability condition given in Inequality (5.29) is when Loss flow is
not constant but dependent on stock that is explored in Chapter 6. To see the effect of the
WSL more clearly, the following model with second order supply line is introduced:
Supply line 1
Supply line 2
Control flow
Acquisition flow 1
Supply line
Stock
Loss flow
Acquisition flow 2
Desired supply line
Desired stock
Stock adjustment
Figure 5.8. Stock management model with a second order supply line considered in CF
32
The following graph is output of model in Figure 5.8 (a pulse of -1 unit is applied on
stock at time one, to perturb the system from the equilibrium):
1: Stock
1:
2: Stock
3: Stock
4: Stock
2.00
1
1:
2
3
0.00
1
2
2
3
4
4
4
3
2
3
4
1
1:
-2.00
0.00
25.00
50.00
Graph 1 (Wsl)
Time
75.00
20:13
100.00
09 Oct 2002 Wed
Figure 5.9. The effect of the changes in Weight of the supply line (WSL equal to 0, 0.2, 1
and 5 respectively for the 1st, 2nd, 3rd and 4th runs) for TSA equal to 2 and TAD equal to 11
As it can be seen from Figure 5.9, WSL brings stability to the system as it increases
till it is equal to one. Above one, it creates “over stability” increasing the time for the stock
to reach to its desired level, which is similar to the effect of increasing TSA.
5.4. Suggestions for Control of a Single Stock with Supply Line and Constant
Outflow
Two different criteria, both stability and settling time (time after which some “small
enough” discrepancy remains between the stock and its goal) the desired level are
important. To reach the desired level quickly and stably the following suggestions must be
considered:
• Set WSL to one.
• Try to decrease TAD if it is possible.
33
• Decrease TSA till the response of the system is quick enough. Note that if the
response of the system is not obviously improved by decreasing the value of the TSA,
it is better to use the larger value since very small TSA may bring instability in real
system (generally, our models do not include imperfections of the real systems that
are not essential for our modeling aim). See Section 6.3 and Section 6.6.
The effects of the TAD and TSA can be seen on the two graphs in Figure 5.10 and
Figure 5.11, when WSL is 1. These outputs are from model in Figure 5.8.
As you can see from Figure 5.11, decreasing TSA does not have linear effect. After a
point, decreasing TSA is useless and furthermore it may create stability problems as it is
mentioned above. For the trade-off in increasing or decreasing TSA, see Section 6.3.
1: Stock
1:
2: Stock
0.00
1
3: Stock
3
2
3
2
3
2
1
3
1:
1
-0.50
1
2
1:
-1.00
0.00
10.00
20.00
Graph 1 (Wsl=1)
Time
30.00
21:52
40.00
09 Oct 2002 Wed
Figure 5.10. Effect of the Acquisition delay time (TAD equal to 20, 5 and 1 respectively for
the 1st, 2nd and 3rd runs) for WSL equal to 1 and TSA equal to 2
34
1: Stock
1:
2: Stock
0.00
3: Stock
4: Stock
1
3
2
4
3
4
2
4
1
3
1:
-0.50
2
1
4
3
1:
-1.00
1
2
0.00
7.50
15.00
22.50
Graph 1 (Wsl=1)
Time
16:39
30.00
11 Oct 2002 Fri
Figure 5.11. Effect of the Stock adjustment time (TSA equal to 10, 1, 0.1 and 0.01
respectively for the 1st, 2nd and 3rd runs) for WSL equal to 1 and TAD equal to 10
35
6. LINEAR CONTROL OF A DECAYING STOCK WITH SUPPLY
LINE DELAY
In this chapter, the basic difference from Chapter 5 is that the Loss flow (LF) is not
constant, but it is linearly dependent on the Stock (S). Loss flow formulation is given as:
LF =
S
TLf
(6.1)
which creates a decaying stock structure. This kind of Loss flow is seen in models of
commercial real estate markets as the demolition of buildings, in models of labor supply
chain as quit rate of labor, in commodity models as discard rate of capital, in population
models as the death rate of the population, and can be seen in many other models and
structures (Sterman, 2000).
Assume Loss flow in model in Figure 5.7 is changed to Equation (6.1). In this case,
Equation (5.19) and Equation (5.20) change to the following equations:
•
S=
•
SLS =
(
SLS
S
−
T AD TLf
)
(
(6.2)
)
S * − S + WSL • SLS * − SLS SLS
S
+
−
TLf
TSA
T AD
(6.3)
where
SLS * = TAD •
S
TLf
(6.4)
36
From Equation (6.2) and Equation (6.3) we find that:
( S , SLS ) equilibrium = (0,0)
(6.5)
This equilibrium point is exactly the same one found in Equation (5.21). The
coefficient matrix A becomes:
1

−

TLf

A=

 1
1 WSL • T AD
−
+

T
T
TSA • TLf
Lf
SA



T AD



WSL
1 
−
−

TSA T AD 
1
(6.6)
The characteristic determinant of Equation (6.2) and Equation (6.3) is:
−
1
−λ
TLf
1
T AD
A −λ •I =
(6.7)
1
1 WSL • T AD
−
+
TLf TSA TSA • TLf
−
WSL
1
−
−λ
TSA T AD
From Equation (6.7) we obtain the following characteristic equation:
 WSL
1
1 
1
+
+
•λ +
=0
 TSA TLf T AD 
•
T
T
SA
AD


λ2 + 
(6.8)
λ1,2 =
W
1
1
−  SL +
+
 TSA TLf T AD


±


 WSL
1
1

+
+
 TSA TLf T AD

2
2

4
 −
 TSA • T AD

(6.9)
37
The eigenvalues have imaginary parts when:
 WSL
1
1

+
+
 TSA TLf T AD

2

4
 <

TSA • T AD

(6.10)
Inequality (6.10) can be re-written as
TSA
4
<
T AD 
1 + WSL • T AD + T AD

TSA TLf





2
(6.11)
It can be seen from Inequality (6.11) that decreasing Life time (TLf; decay time) has
an effect similar to increasing WSL, and it can be further shown that increasing TSA (for
sufficiently large values of TSA), also has a similar effect, which all stabilize the system.
6.1. Parameter Values for All Runs in this Chapter
The values of the model parameters are as follows for all runs in this chapter, unless
it is stated otherwise:
• T AD = 16 [time units ]
• TLf = 10 [time units ]
• S * (0 ) = 0 [items ],
S * (5) = 1 [items ]
In all runs, stocks are initialized at their equilibrium levels and Desired stock (S*) is
increased from zero to one, at time five, to perturb the system from equilibrium.
The values of the other parameters (TSA, WSL, TEL) that are used in this chapter will
be given whenever necessary.
38
6.2. Effect of Life Time (Decay Time)
Following runs (Figure 6.1) are from model in Figure 5.8. This model is modified by
changing its Loss flow formulation from constant to the decaying outflow formulation of
Equation (6.1).
1: Stock
1:
2: Stock
3: Stock
3.00
1
1
1:
2
1.00
2
3
3
2
3
1
3
1
1:
-1.00
0.00
2
30.00
Graph 1 (Tlf)
60.00
Time
90.00
22:57
120.00
05 Dec 2002 Thu
Figure 6.1. Effect of the Life time (TLf equal to 40, 10 and 3 respectively for the 1st, 2nd and
3rd runs) for WSL equal to zero and TSA equal to 2
We prefer to set WSL to 1, since it is easier to manage an effectively first order stock
system, but for the run in Figure 6.1, WSL is set to zero to demonstrate the stabilizing effect
of TLf more clearly. (TSA is set to 2 to have a rather unstable system). As it was shown by
Inequality (6.11) decreasing TLf stabilizes the behavior of the stock.
For the run in Figure 6.2, all parameters except from WSL are the same with
parameters of the run in Figure 6.1. For WSL equal to 1 TLf over stabilizes the response of
the Stock. As TLf decreases, Stock shows slower response as in Figure 6.2.
39
1: Stock
1:
2: Stock
3: Stock
1.00
1
1
2
2
3
1
3
2
3
1:
0.50
3
1:
0.00
1
0.00
2
30.00
Graph 1 (Tlf)
60.00
Time
90.00
23:19
120.00
05 Dec 2002 Thu
Figure 6.2. Effect of the Life time (TLf equal to 40, 10 and 3 respectively for the 1st, 2nd and
3rd runs) for WSL equal to 1 and TSA equal to 2
6.3. Trade-off in Stock Adjustment Time Values in a Discrete Supply Line Delay
Mostly it is not possible, or it is hard to manipulate Acquisition delay time (TAD) and
Life time (TLf; decay time). Sometimes it may be possible to control these parameters
within certain limits. We accept them as externally determined parameters.
On the other hand, Weight of supply line (WSL) and Stock adjustment time (TSA) are
the parameters that can be set and controlled by stock manager. These parameters are not
external. As we discussed before we prefer to set WSL to 1 since it is more easy to manage
an effectively first order stock system.
In Section 5.4, possible values of TSA were mentioned, but no demonstration was
made. Here, the trade-off of selecting appropriate value for TSA will be discussed.
40
In addition to changes made by Equation (6.1), assume also that the supply line in the
model in Figure 5.7 is modified so that it is discrete. The modified delay formulation is:
AF (t ) = CF (t − T AD )
(6.12)
instead of the continuous delay given by SLS / T AD . The following output is from such a
model:
1: Stock
2: Stock
1:
2.00
1:
1.00
3: Stock
4: Stock
1
2
2
1
2
3
3
4
3
4
4
1:
0.00
1
0.00
2
3
4
40.00
Graph 1 (comparison)
80.00
Time
120.00
22:25
160.00
05 Dec 2002 Thu
Figure 6.3. Effect of the Stock adjustment time (TSA equal to 2, 7, 20 and 70 respectively
for the 1st, 2nd, 3rd and 4th runs) for discrete supply line system with WSL equal to 1 and TLf
equal to 10
As it can be seen from the runs in Figure 6.3, if TSA is low, there may be instability in
the system, and when it is high, response is very slow. TSA value must be selected
appropriately to have stable and fast response. For further improvement in the behavior,
see Section 6.5, but first it is better to discuss the reasons behind the unstable behavior that
seems to be contradictory with the findings of Chapter 5 (recall that, in the previous
chapter we mentioned that for WSL equal to 1 there can not be oscillations).
41
6.4. Causes of Instability in the Discrete Supply Line Case Combined with Decaying
Outflow
To see the causes of instability in discrete supply line with decaying stock (and low
TSA value) case, the causal loop diagrams of the model in Figure 5.7 is sketched with and
without decay structures. Note that parameters that are not on any loop are omitted in these
causal loop diagrams.
-
+
Stoc k
+
Supply line
-
Supply line
adjustment +
-
+
Acquisition flow
Desired
supply line
+
Control flow +
-
Loss flow
+
Stoc k
adjustment
-
+
Figure 6.4. Causal loop diagram without a decaying outflow structure
-
-
Stoc k
+
Supply line
+
-
+
Supply line
adjustment
-
+
Acquisition flow
+
Control flow +
Desired
supply line
+
+
+
-
Loss flow
Stoc k
adjustment
+
-
+
Figure 6.5. Causal loop diagram with a decaying outflow structure
42
As it can be seen from the causal loop diagrams, the addition of Loss flow formula
given in Equation (6.1) adds one negative (counteracting) loop and two positive
(reinforcing) loops in the system. The unstable behavior is caused by one of these positive
loops that is shown by thick lines in Figure 6.5. Note that discrete supply line strengthens
the effect of this positive loop. Because discrete delay means infinite-order delay and as it
was shown, the higher the order of the delay, the less stable the system tend to be. Unstable
behavior disappears with lower order supply line systems.
Increasing the value of TSA decreases the gain of the loop that was responsible for
instability. For high enough value of TSA, behavior can be stable.
6.5. Using the Equilibrium Value of Loss to Stabilize the Model
For more stable and faster results, Equilibrium value of loss (EVL) is introduced and
used as the anchor in the formulas of SLS* and CF, instead of Loss flow:
EVL =
S*
TLf
(6.13)
The Control flow (CF) formula in Equation (5.16) becomes:
CF = EVL +
(S
*
)
(
− S + WSL • SLS * − SLS
TSA
)
(6.14)
The Desired supply line formula in Equation (6.4) becomes:
SLS * = T AD • EVL
(6.15)
Using EVL in SLS* and CF instead of LF, removes the two positive loops completely
from Figure 6.5, so instability is eliminated. After the changes made by Equation (6.1),
discrete supply line delay, Equation (6.13), Equation (6.14) and Equation (6.15), the model
obtained is shown in Figure 6.6.
43
Supply line
Stock
Control flow
Acquisition flow
Loss flow
Life time
Desired supply line
Equilibrium value of loss
Stock adjustment
Desired stock
Figure 6.6. Stock management model with discrete supply line, decaying stock and EVL
used as anchor in Control flow (CF) and in Desired supply line (SLS*) computation
The following output is from this improved model:
1: Stock
2: Stock
1:
2.00
1:
1.00
3: Stock
1
2
3
4: Stock
4
5
1
2
3
5: Stock
4
5
1
2
3
4
5
5
4
1:
0.00
1
0.00
2
3
40.00
80.00
Time
120.00
14:27
160.00
08 Dec 2002 Sun
Figure 6.7. Effect of the Stock adjustment time (TSA equal to 2, 7, 20, 70 and infinite
respectively for the 1st, 2nd, 3rd, 4th and 5th runs) for discrete supply line model with EVL as
the anchor in CF and SLS*, with WSL equal to 1, and with TLf equal to 10
44
If runs in Figure 6.3 and runs in Figure 6.7 are compared, it can be seen that, using
EVL as anchor in Control flow (CF) and in Desired supply line (SLS*), (instead of LF),
makes the response of the stock faster and more stable.
Stock adjustment time (TSA) is taken to be infinite in the fifth run of the Figure 6.7,
which means that neither Stock nor Supply line adjustments are made in this run. This run
suggests that, if Life time is low enough to create a fast enough approach, there is no need
to monitor Stock and Supply line. Just adjusting the Control flow to be equal to Equilibrium
value of loss is enough to create fast and stable response, and stock implicitly seeks the
goal (Desired stock). Not that this policy is extremely stable in such a condition.
Note that, if Life time (TLf; decay time) value is big enough, the system becomes
almost equivalent to constant Loss flow case (i.e. zero Loss flow), so Stock and Supply line
adjustments become necessary.
Equilibrium value of loss (EVL) is calculated by dividing the Desired stock (S*) with
the Life time (TLf). See Equation (6.13). In some cases, it may not be possible to know Life
time directly. In such a case, Life time (decay time) must be estimated. A good way is to
divide Stock by its Loss flow and take moving average of this ratio.
6.6. Suggestions on Control of a Decaying Stock with Supply Line Delay
For stability and quick response, consider the following suggestions, in addition to
the suggestions in Section 5.4:
• Try to choose stable anchors and stable sub-anchors. If possible, use the equilibrium
levels as anchors (i.e. anchor of CF formula) or sub-anchors (i.e. anchor of SLS*
formula). If it is not possible to use the equilibrium levels, estimate the equilibrium
levels and then use them as anchors.
• Pay attention to not to form (+) loops involving your anchor. If it is not possible to
avoid, use moving average to weaken such loops containing your anchor (i.e. Use
Expected loss instead of Loss flow, see Section 6.7.4).
45
• To prevent the initial oscillations, or the instability in discrete supply line systems,
try to use relatively high values of TSA.
6.7. Some Observations on Controlling a Decaying Stock
6.7.1. Stability with EVL used as Anchor is Robust
• If WSL is chosen as 1, even for very low TSA values unstable oscillations can not
occur.
• For large enough TSA, results of EVL is insensitive to the values of WSL:
1: Stock
2: Stock
1:
2.00
1:
1.00
3: Stock
1
2
3
4
4: Stock
1
2
3
4
1
2
3
4
4
1:
0.00
1
0.00
2
3
40.00
80.00
120.00
Time
18:13
160.00
05 Jan 2003 Sun
Figure 6.8. Effect of the Weight of Supply Line (WSL equal to 0, 0.5, 1 and 2 respectively
for the 1st, 2nd, 3rd and 4th runs) for discrete supply line system with EVL (as the anchor in
CF and SLS*) and TSA equal to 70
• If TSA is low, decreasing WSL can bring instability to the system, as in the constant
outflow case (models in Chapter 5). Even with low TSA, there must be a significant
change in WSL to have instability, and stable oscillations can be observed before
instability, so EVL usage is robust in the sense that unexpected behaviors do not
arise:
46
1: Stock
2: Stock
1:
5.00
1:
1.00
3: Stock
-3.00
1:
2
1
4
2
1
2
4: Stock
3
3
4
4
2
3
4
3
0.00
40.00
80.00
120.00
Time
18:03
160.00
06 Jan 2003 Mon
for the 1st, 2nd, 3rd and 4th runs) for discrete supply line system with EVL (as the anchor in
6.7.2. Stability with LF Used as Anchor is Problematic
• If TSA is too large, it takes very long time for the stock to reach to its desired level
1: Stock
1:
2: Stock
3: Stock
4: Stock
1.00
1
1
1:
1
0.50
2
3
2
3
2
3
4
4
4
4
3
2
1:
1
0.00
0.00
125.00
250.00
Time
375.00
19:40
500.00
05 Jan 2003 Sun
for the 1st, 2nd, 3rd and 4th runs) for discrete supply line system with LF (as the anchor in
47
• If TSA is low, for stable result WSL must be adjusted accordingly. For low and high
values of WSL, unstable behavior can be observed, so LF is not robust as anchor in
the sense that it requires WSL adjustment. For instance, in Figure 6.11, WSL equal to
0.5 yields stable oscillations; whereas WSL equal to 0 and WSL equal to 1 yield
instability.
1: Stock
1:
2: Stock
3: Stock
4: Stock
5.00
1
1:
2
1.00
1
2
3
4
3
2
3
2
4
3
1:
-3.00
0.00
40.00
80.00
Time
120.00
18:03
160.00
06 Jan 2003 Mon
for the 1st, 2nd, 3rd and 4th runs) for discrete supply line system with LF (as the anchor in
• LF as anchor may give stable results for some WSL values, but those values can not
guarantee stability for other set of parameters. In contrast, note that, for EVL as
anchor, WSL equal 1 guarantees stability for all parameter values.
6.7.3. Comparison of EVL and LF Used as Anchors
For any given parameter settings, EVL can produce better results than LF used as
anchor.
48
1: Stock
2: Stock
1:
2.00
1:
1.00
3: Stock
4: Stock
1
1
3
2
2
3
5: Stock
1
4
2
5
4
3
4
5
5
4
1:
0.00
1
0.00
2
5
3
25.00
50.00
Time
75.00
20:32
100.00
06 Jan 2003 Mon
Figure 6.12. Comparison of using EVL (1st run with WSL equal to 1) and LF (2nd, 3rd, 4th
and 5th runs with WSL equal to 0, 0.15, 0.4 and 1) as anchors for second order supply line
system with TSA equal to 7
6.7.4. Effects of Using Expected Loss Formulation in Controlling a Decaying Stock
To deal with instability, one may want to use the expected value of Loss flow (LF),
which we call as Expected loss (ELS), instead of LF as the anchor in CF and SLS*.
Although expectation can deal with instability to some extent, it creates an additional delay
in the response of the Stock. Its effect on the behavior of the Stock is very similar to the
effect of TSA. Increasing Expected loss averaging time (TEL) and increasing Stock
adjustment time (TSA) creates the same effect on the behavior of the Stock.
•
ELS =
(LF − ELS )
(6.16)
TEL
Modified Control flow (CF) and Desired supply line (SLS*)equation can be given as:
CF = ELS +
(S
*
)
(
− S + WSL • SLS * − SLS
TSA
)
(6.17)
49
SLS * = T AD • ELS
1: Stock
2: Stock
(6.18)
3: Stock
4: Stock
2.00
1:
2
1
1.00
1:
3
1
4
0.00
1:
1
0.00
2
4
2
3
3
1
4
4
2
3
40.00
80.00
120.00
Time
00:02
160.00
07 Jan 2003 Tue
Figure 6.13. Comparison of using EVL (1st run), LF (2nd run) and ELS (3rd and 4th runs
with TEL equal to 2 and 10) as anchors for discrete supply line system with WSL equal to 1
and TSA equal to 3
1: Stock
2: Stock
1:
2.00
1:
1.00
3: Stock
1
4: Stock
1
1
2
2
2
3
3
3
4
4
4
4
1:
1
0.00
0.00
2
3
40.00
80.00
120.00
Time
00:04
160.00
07 Jan 2003 Tue
Figure 6.14. Comparison of using EVL (1st run), LF (2nd run) and ELS (3rd and 4th runs
with TEL equal to 2 and 10) as anchors for discrete supply line system with WSL equal to 1
and TSA equal to 21
50
7. CONTROL OF A DECAYING STOCK WITH UNKNOWN
VARIABLE LIFE TIME
In this part we assume a variable Life time (decay time). One of the hardest case is
“pink noise”, where the variation is not completely random, but it also has autocorrelation
structure. “Pink noise” means “autocorrelated” normal random variates. We obtain pink
noise by firstly generating a normally distributed random variable, and secondly smoothing
it with a desirable autocorrelation coefficient. Lastly, to make it insensitive to simulation
time step (DT), generated pink noise values are hold for one time unit, and their
magnitudes are normalized with respect to DT (see Appendix D for the details about Pink
noise). “Pink noise” is used to generate Life time (TLf) values. The relation is given by:
TLf = 20 + Pink noise
(7.1)
Below, the behavior of the Life time can be seen:
1: Lif e time
1:
30.00
1
1:
20.00
1
1
1
1:
Page 1
10.00
0.00
62.50
125.00
Time
187.50
18:10
250.00
Mon, Apr 21, 2003
Auto-correlated Lif e time
Figure 7.1. Auto-correlated Life time
In this chapter, runs are based on the stock management model in Figure 5.8, which
is modified by changing its Loss flow formulation from constant to the decay formula of
51
Equation (6.1), and by changing its Life time from constant to variable. In this chapter all
the runs use the above graph (Figure 7.1) for the Life time values. Also the decision
formulations of model in Figure 5.8 are modified by Equation (6.13), Equation (6.14) and
Equation (6.15), so Equilibrium value of loss flow is used as anchor in Control flow and
Desired supply line equations.
The values of the model parameters are as follows for all runs in this chapter:
• TSA = 10 [time units ]
• TLf = 10 [time units ]
• WSL = 1
• OSL = 2
• S * (0 ) = −1 [items ],
S * (5) = 1 [items ]
Initial value of Desired stock is set to -1 to make the stock cross the origin (zero) and
it is increased to one, at time five, to perturb the system from equilibrium. Crossing origin
necessitates special care (due to division by zero) in estimation of life time, which is
considered in this chapter. Some parameters are used only in additional equations, as
necessary. These parameters are as follows:
• TPD = 2 [time units ]
• TSm, Lf = 3 [time units ]
• TSm,S = TSm, Lf = 3 [time units ]
7.2. Case: Life Time and Loss Flow are Observed Directly and Immediately
This case is not realistic (normally Life time -TLf - can not be perceived directly, but
estimated by dividing Stock with Loss flow -LF-). We use this case to demonstrate the best
52
possible behavior of the Stock (S), to evaluate later the effectiveness of the estimation
procedures by comparing their results with this benchmark. Note that, direct non-delayed
perception of Loss flow is also an extreme case (if it can be argued that perception delay of
Loss Flow is very small compared with the other time delays -i.e. Acquisition delay timein the system).
1: Stock
2.00
1:
1
1
1
0.00
1:
1
-2.00
1:
0.00
62.50
125.00
Time
Page 1
187.50
17:59
250.00
Mon, Apr 21, 2003
v ariable lif e time
Figure 7.2. Behavior of the Stock when Life time is perceived directly
However, also note that it may not be appropriate to use auto-correlated Life time
(decay time) directly in higher order supply line systems. For example using autocorrelated Life time in discrete supply line system may create sudden shocks in Stock,
which can be eliminated by smoothing the Life time.
7.3. Case: Life Time cannot be Observed but Loss Flow is Observed Immediately
In this case, Life time (TLf; decay time) can be calculated by dividing the Stock by
Loss flow that gives exact values of Life time. We may call this new variable Calculated
life time (TCLf):
TCLf =
S
LF
(7.2)
53
New Equilibrium value of loss becomes:
EVL =
S*
TCLf
(7.3)
Equation (7.2) has problem when Stock crosses the origin. At Stock equals to zero
point Loss flow also becomes zero, and their ratio becomes undefined. What can be done is
that only at this point previous value of Life time can be used. Instead of using the previous
value, smoothed value of the Calculated life time can also be used, but no adjustments
must be made at zero point:
TCLf
 IF (LF = 0 ) THEN TCLf (t − DT )


(t ) = 

S



 ELSE 

 LF 


(7.4)
Alternatively:
 IF (LF = 0 ) THEN 0



 TCLf − SMLTS  
SMLTS = 

 ELSE 


T
Sm, Lf



•
(7.5)
where SMLTS is Smoothed life time and TSm,Lf is Smoothing time for life time. New
Equilibrium value of loss becomes:
EVL =
S*
SMLTS
(7.6)
Using TCLf instead of TLf in EVL formula gives almost the same result with the run in
Figure 7.2. Also using SMLTS in EVL gives very close result when smoothing time is
short. Because these runs do not differ much, they are not shown here.
54
7.4. Case: Life Time cannot be Observed and Loss Flow is Observed with a Delay
Again Calculated life time (TCLf; calculated decay time) is used but this time it is a
ratio of the Stock and the Perceived loss (PLS):
TCLf


 IF (LF = 0 ) THEN T (t − DT )
CLf



(t ) = 



 ELSE S
PLS


(7.7)
where
•
PLS =
LF − PLS
TPD
(7.8)
and TPD is Perception delay time. Equation (7.7) is weak in a sense that it may produce
very small negative or positive values, or very big negative or positive values, especially
when Stock is crossing zero. This may bring sudden shocks to the system. The graph in
Figure 7.3 demonstrates the behavior of Calculated life time when Stock is crossing zero:
1: Calculated life time
1:
80000
1:
35000
1
1:
Page 1
-10000
22.00
1
1
22.25
22.50
Time
1
22.75
18:25
23.00
Wed, Apr 23, 2003
Phase difference
Figure 7.3. Shock in Calculated life time as Perceived loss approaches zero, caused by a
phase difference between Stock and Perceived Loss
55
For convenience, we are also giving the behavior of the Stock in the following figure:
1: Stock
1:
2.00
1
1
1:
1
0.00
1
1:
-2.00
0.00
62.50
125.00
Time
Page 1
187.50
18:18
250.00
Mon, Apr 21, 2003
Figure 7.4. Behavior of the Stock when there is a shock in Calculated life time, at about
time 22.70
One of the reasons for the shock in Figure 7.3 is the phase difference between Stock
and Loss flow (when PLS ≈ 0, S may not necessarily be ≈ 0). We propose to use Smoothed
stock (SMS) in the formula of Calculated life time, instead of Stock:
TCLf


 IF (LF = 0 ) THEN T (t − DT )
CLf



(t ) = 


 ELSE SMS

PLS


(7.9)
where
•
SMS =
S − SMS
TSm,S
(7.10)
Best performance is obtained for Smoothing time for stock (TSm,S) equal to
Perception delay time (TPD). If Perception delay time is not known for sure, it must be
estimated. Several runs that we have made showed that overestimating Smoothing time for
56
stock may create problematic behavior in the estimation of Smoothed life time while
underestimating the parameter is more robust, so it can be in general said that Smoothing
time for stock must be equal or smaller than the Perception delay time. As it can be
observed from Figure 7.5, the shock is significantly reduced. The actual Life time (LF) and
Calculated life time (TCLf) can be seen together in Figure 7.6.
1:
50
1
1:
5
1:
-40
22.00
1
1
22.75
23.50
Time
Page 1
1
24.25
09:11
25.00
Tue, Apr 22, 2003
Without phase difference
Figure 7.5. Reduced shock in TCLf when there is no phase difference between Stock and
Perceived Loss (TSm,S equal to TPD)
1: Life time
1:
2:
30.00
1
1:
2:
20.00
1
2
2
2
1
1:
2:
2
10.00
0.00
Page 1
1
62.50
125.00
Time
187.50
15:15
250.00
Thu, Jul 17, 2003
Auto-correlated Life time
Figure 7.6. LF and TCLf together, shock can be seen at about time 23.46
57
1: Stock
1:
2.00
1
1:
1
1
0.00
1
1:
-2.00
0.00
62.50
Page 1
125.00
Time
187.50
09:11
250.00
Tue, Apr 22, 2003
Figure 7.7. Behavior of the Stock for reduced shock (TSm,S equal to TPD)
The reduced shock can be even further reduced with smoothing Calculated life time.
This is possible if we make a small change in Equation (7.5):
 IF (PLS = 0 ) THEN 0

 TCLf − SMLTS
SMLTS = 
 ELSE 

TSm , Lf


•






(7.11)
1: Smoothed lif e time
1:
30
1
1:
20
1
1
1
1:
Page 1
10
0.00
62.50
125.00
Time
187.50
10:24
250.00
Tue, Apr 22, 2003
Without phase dif f erence
Figure 7.8. Smoothed life time when there is no phase difference (TSm,S equal to TPD)
58
In the following graph Life time and Smoothed life time are plotted together (as it can
be seen our estimation is quite well):
1: Lif e time
1:
2:
30
1
1:
2:
20
2
1
2
2
1
1:
2:
1
2
10
0.00
62.50
Page 1
125.00
Time
187.50
10:24
250.00
Tue, Apr 22, 2003
Auto-correlated Lif e time
Figure 7.9. Life time and Smoothed life time together
1: Stock
1:
2.00
1
1:
1
1
0.00
1
1:
Page 1
-2.00
0.00
62.50
125.00
Time
187.50
10:24
250.00
Tue, Apr 22, 2003
Figure 7.10. Behavior of the Stock for Smoothed life time
When Figure 7.10 is compared with Figure 7.2 the difference is negligible. They are
almost the same, so we can conclude that the estimation is quite good.
59
The shock can be seen as a discontinuity in Smoothed life time around Time equals
23. This can further be removed by using a non-linear estimation adjustment formula (see
Appendix E). One may want to use the formula in Appendix E especially when Perception
delay time is unknown or when it can not be estimated.
One may also observe that the intermediate runs of Stock in Figure 7.4 and Figure 7.7
are also very similar to the benchmark in Figure 7.2 This is mainly because they have
problem only at single point where Stock crosses zero level. One may argue that estimation
techniques given here may fail if Stock is fluctuating often around zero point, but luckily if
Desired stock is equal to zero there is no need to estimate the Life time. In this condition,
Equilibrium value of loss flow is zero independent of the value of Life time, and behavior
of the Stock is very stable.
The final form of the model can be seen in the following figure:
Supply line 1
Control flow
Supply line 2
Acquisition flow 1
Stock
Desired supply line
Loss flow
Acquisition flow 2
Desired stock
Life time
Equilibrium value of loss
Stock adjustment
Smoothed life time
Smoothing adjustment flow for Lf
Calculated life time
Smoothing time
Smoothed stock
Smoothing adjustment flow for S
Smoothing time for life time
Perceived loss
Perception adjustment flow
Perception delay time
Figure 7.11. Model with decaying Stock and Life time estimation involving smoothing
60
8. LINEAR CONTROL OF A SINGLE STOCK WITH
INFORMATION DELAY: VIRTUAL SUPPLY LINE
As it was mentioned in Chapter 5, in general, there is a delay between the control
action and its result. There may also be a delay between control decision and control
action, which we call “information delay”.
Stock
Control flow
Loss flow
Desired stock
Stock adjustment
Information delay 2
Information adjustment flow 2
Desired control flow
Information delay 1
Figure 8.1. Stock management model with second-order information delay
In Figure 8.1, it is assumed that there is a second order information delay, but no
supply line, without loss of generality.
The values of the model parameters are as follows, for all runs in this chapter, unless
• S * = 5 [items ]
61
• TID = T AD = 4 [time units ]
• LF = 2 [items/time unit ]
• OSL = 2
• OID = OSL = 2
Also, in all runs, stocks are initialized at their equilibrium levels and at time four, a
shock of plus one unit is applied to the primary stock, to perturb the system from
equilibrium.
8.2. Comparison of Supply Line and Information delay in Stock Control
Dynamics of the two models with supply line delay and with information delay
(Figure 5.3 and Figure 8.1) are compared in Figure 8.2 (see also Fey, 1974b; Forrester
1973). As it can be seen from the graph in the following figure, both models in Figure 5.3
and in Figure 8.1 have exactly the same behaviors with same parameter values (Stock
represents the Stock of the model in Figure 5.3 and Stock 2 represents the Stock of the
model in Figure 8.1):
1: Stock
2: Stock 2
1:
2:
6.00
1:
2:
5.00
1
1
1
2
2
2
2
1
1:
2:
4.00
0.00
12.50
25.00
Time
37.50
23:36
50.00
17 Nov 2002 Sun
Figure 8.2. Behaviors of the models with equivalent supply line and information delay
structures are exactly the same
62
8.2.1. Causal Loop Comparison
The causal loop diagrams of the two models in Figure 5.3 and in Figure 8.1 can be
sketched as:
+
Acquisition flow 1
+
-
Supply line 2
-
-
Supply line 1
+
Acquisition flow 2
+
+
Control flow
Stoc k
+
Stoc k adjustment
-
Figure 8.3. Causal loop diagram of model in Figure 5.3
+
Information delay 1
+
-
-
-
-
+
Information delay 2
-
+
+
Control flow
+
Stoc k adjustment
-
Stoc k
+
The two causal loop diagrams are very similar in loop structure. If the names of the
variables are ignored, the two causal loop diagrams in Figure 8.3 and in Figure 8.4 are
exactly the same. Note that, parameters that are not on a loop are omitted in these loop
diagrams.
63
8.2.2. Mathematical Equivalency
For simplification, we are treating the supply line and information delay structures as
input-output structures, reduce these structures and re-write the equations in terms of input
and output variables.
First we are giving the full dynamic equations (differential equations) of the two
models. The dynamic equations of the model in Figure 5.3 were as follows:
•
S = AF2 − LF =
•
SLS 2
SLS 2
− LF =
− LF
(T AD / OSL )
(T AD / 2)
SLS 1 = CF X − AF1 = CF X −

SLS1
SLS1
S * − S 
=  LF +
−


(T AD / OSL ) 
TSA  (T AD / 2 )
•
SLS 2 = AF1 − AF2 =
SLS1
SLS 2
−
(T AD / 2) (T AD / 2)
(8.1)
(8.2)
(8.3)
And the dynamic equations of the model in Figure 8.1 are as follows:
•
S = CF − LF = IDS 2 − LF
•
IDS 1 = IAF1 =
(
(
(8.4)
)
)
LF + S * − S / TSA − IDS1
CF * − IDS1
=
(TID / OID )
(TID / 2)
•
IDS 2 = IAF2 =
IDS1 − IDS 2
(TID / 2)
(8.5)
(8.6)
Note that Equation (8.1) and Equation (8.4) are identical, once we observe that AF2
in the supply line structure (Figure 5.3) is equivalent to CF in the information delay
structure (Figure 8.1), seen as input-output systems. Thus, the rest is to prove that the
supply line and information delay structures are mathematically equivalent.
64
The input of the supply line structure is Control flow (CFX) and its output is
Acquisition flow2 (AF2) when it is seen as an input-output system. The dynamic equations;
Equation (8.2) and Equation (8.3) are used to obtain a single dynamic equation in terms of
Acquisition flow2 and Control flow. From Equation (8.3) we can obtain the following
equation:
•
SLS1 = (T AD / 2 ) • SLS 2 + SLS 2
(8.7)
and from above we obtain:
•
••
•
SLS1 = (T AD / 2) • SLS 2 + SLS 2
(8.8)
Equation (8.7) and Equation (8.8) can be inserted to Equation (8.2) to obtain the
following equation:
••
•
•
(T AD / 2) • SLS 2 + SLS 2 = CFX
(T / 2) • SLS 2 + SLS 2
− AD
(T AD / 2)
(8.9)
Above equation can be simplified to the following:
••
•
(T AD / 2)2 • SLS 2 + T AD • SLS 2 + SLS 2 = (T AD / 2) • CFX
(8.10)
Equation (8.10) can be re-written for Acquisition flow2 (AF2) by using the
relationship given for Supply line2 (SLS2) and Acquisition flow2 in Equation (8.1):
••
•
(T AD / 2)2 • AF 2 + T AD • AF 2 + AF2 = CFX
(8.11)
The input of the information delay structure is Desired control flow (CF*) and its
output is Control flow (CF) when it is seen as an input-output system. The dynamic
65
equations; Equation (8.5) and Equation (8.6) are used to obtain a single dynamic equation
in terms of Control flow and Desired control flow. From Equation (8.6) we can obtain the
following equation:
•
IDS1 = (TID / 2 ) • IDS 2 + IDS 2
(8.12)
•
••
•
IDS 1 = (TID / 2 ) • IDS 2 + IDS 2
(8.13)
Equation (8.12) and Equation (8.13) can be inserted to Equation (8.5) to obtain the
following equation:
(TID / 2)
2
•
CF * − (TID / 2 ) • IDS 2 − IDS 2
• IDS 2 + IDS 2 =
(TID / 2)
••
•
(8.14)
••
•
(TID / 2)2 • IDS 2 + TID • IDS 2 + IDS 2 = CF *
(8.15)
It is known from Equation (8.4) that Control flow (CF) is equal to Information delay2
(IDS2), so Equation (8.15) can be re-written for Control flow (CF):
••
•
(TID / 2)2 • CF + TID • CF + CF = CF *
(8.16)
Equation (8.11) and Equation (8.16) are exactly same in the form, so they produce
the same output, provided that the given input is the same. Also observe that the inputs
(
(
)
)
CFX and CF* are the same, given by LF + S * − S / TSA in each case. Thus, once the
delays TAD and TID are the same, the outputs are also exactly the same. We can conclude
66
that, mathematically there is no difference between supply line and information delay
structures, but note that their real life meanings are different.
8.3. Introducing the Notion of Virtual Supply Line in Stock Control
In Chapter 5, it shown that, for stable and fast results, supply line must be considered
in decision formulations. This is also true for information delay, but in information delay
structure, Information delay stocks are rates themselves, so they can not be automatically
handled just like supply line stocks. In addition to these facts, it is also shown in Section
8.2 that the supply line and information delay structures have the same effect on the
behavior of the stock, so there must be a similar way to deal with information delay.
Here, we propose to create a conceptual supply line, which we call “Virtual supply
line”, and use this supply line to adjust the control decisions (Desired control flow). It can
be defined as a “virtual” supply line that would create the same output (outflow) as an
equivalent material supply line delay. After adding Virtual supply line and its
corresponding adjustment term in the Desired control flow (CF*), the model in Figure 8.1
turns into the model in Figure 8.5.
For a first-order information delay, the Virtual supply line (VSL) is defined as:
VSL = TID • IDS
(8.17)
For a second-order information delay, the Virtual supply line (VSL) is defined as:
VSL =
TID
• (IDS1 + IDS 2 )
2
(8.18)
For an nth order information delay, the Virtual supply line (VSL) is defined as:
VSL =
n
T
TID OID
• ∑ (IDS i ) = ID • ∑ (IDS i )
n i =1
O ID i =1
(8.19)
67
And in general for an nth order information delay with unequal individual delay
times, the Virtual supply line (VSL) can be defined as given in Appendix G.
The mathematical equivalency of using virtual supply line adjustment in information
delay structures, and using supply line adjustment in supply line structures will be
discussed in Section 8.4 (for cases involving unequal delay times of individual delays, see
Appendix F).
The Desired virtual supply line (VSL*) and Virtual supply line adjustment (VSLA) are
independent of the order of the information delay:
VSL* = TID • LF
(8.20)
(
WVSL • VSL* − VSL
VSLA =
TSA
)
(8.21)
Virtually adjusted desired control flow (VACF*) is added to the model:
VACF * = CF * + VSLA = LF +
(S
*
)
(
− S + WVSL • VSL* − VSL
TSA
)
(8.22)
The differential equations given in Equation (8.4) and Equation (8.6) remain
unchanged, but Equation (8.5) becomes:
(
 * WVSL • VSL* − VSL
 CF +
•
TSA
VACF * − IDS1 
=
IDS 1 =
(TID / 2)
(TID / 2)
(
)
(
*
*

 LF + S − S + WVSL • VSL − VSL

TSA
=
(TID / 2)
) − IDS


1
) − IDS


1
(8.23)
68
Stock
Loss flow
Control flow
Desired stock
Stock adjustment
Weight of VSL
VSL adjustment
Desired VSL
Virtual supply line
Information delay 2
Virtually adjusted DCF
Information delay 1
Figure 8.5. Stock control with Virtual supply line adjustment
1: Stock
2: Stock 2
1:
2:
6.00
1:
2:
5.50
1
1:
2:
2
5.00
1
0.00
2
12.50
1
25.00
Time
2
1
37.50
23:41
2
50.00
17 Nov 2002 Sun
Figure 8.6. Output equivalency of the supply line model with WSL equal to 1 and
information delay model with WVSL equal to 1
Behaviors of the two models in Figure 5.8 and in Figure 8.5 are shown on the same
graph in Figure 8.6. The weights; Weight of supply line (WSL) and Weight of virtual supply
69
line (WVSL) are set to one, which means that both supply line and virtual supply line are
fully considered in the decisions. Stock represents the Stock of the model in Figure 5.8 and
Stock 2 represents the Stock of the model in Figure 8.5. The two stocks show exactly the
same optimum behavior with the same parameter values. Thus, the conclusions that are
mentioned about Supply line control in Chapter 5 are also valid for Virtual supply line
control: for both stable and quick response in the Stock, Virtual supply line must be
considered with a weight 1.
8.4. Mathematical Equivalency of the Supply Line and Virtual Supply Line
Adjustments in the Decisions
The supply line structure with supply line adjustment, and information delay
structure with virtual supply line adjustment are treated as input-output structures. These
structures are going to be reduced and the equations are going to be re-written in terms of
input and output variables.
For supply line structure Equation (8.1) and Equation (8.3) do not change, but
Equation (8.2) becomes:
•
SLS 1 = CF X + SLA −
= CFX +
SLS1
(T AD / 2)
WSL • (T AD • LF − SLS1 − SLS 2 )
SLS1
−
(T AD / 2)
TSA
(8.24)
For information delay structure Equation (8.4) and Equation (8.6) do not change, but
Equation (8.5) becomes:
•
IDS 1 =
VACF * − IDS1 CF * + VSLA − IDS1
=
(TID / 2)
(TID / 2)
CF * +
=
WVSL • (TID • LF − (TID / 2 ) • IDS1 − (TID / 2 ) • IDS 2 )
− IDS1
TSA
(TID / 2)
(8.25)
70
Note that Equation (8.1) and Equation (8.4) are identical, once we observe that AF2
in the supply line structure (Figure 5.3) is equivalent to CF in the information delay
structure (Figure 8.1), seen as input-output systems. Thus, the rest is to prove that the
supply line and information delay structures (with supply line and virtual supply line
adjustments considered in decisions) are mathematically equivalent.
The input of the supply line structure is Control flow (CFX), and its output is
Acquisition flow2 (AF2) as in Section 8.2.2. The dynamic equations; Equation (8.24) and
Equation (8.3) are used to obtain a single equation in terms of Acquisition flow2 and
Control flow. Equation (8.7) and Equation (8.8) that are obtained from Equation (8.3) can
be inserted to Equation (8.24) to obtain the following equation:
••
•
(T AD / 2) • SLS 2 + SLS 2 =
•
(8.26)


•
W SL •  T AD • LF − (T AD / 2) • SLS 2 − 2 • SLS 2 
 − (T AD / 2) • SLS 2 + SLS 2

CF X +
(T AD / 2)
TSA
Note that in the above equation, we use CFX instead of CF, where CF is nothing but
CFX + SLA .
Equation (8.26) can be simplified to the following:
••


 (T AD / 2 )2 • TSA • SLS 2



•
 + T • T + W • (T / 2 )2 • SLS  =  (T AD / 2 ) • TSA • CFX 
2
AD
SA
SL
AD

  + W • T 2 / 2 • LF 
SL
AD

 + (TSA + WSL • T AD ) • SLS 2
 




(
)
(
)
(8.27)
Equation (8.27) can be re-written for Acquisition flow2 (AF2) with using the
relationship given for Supply line2 (SLS2) and Acquisition flow2 in Equation (8.1):
71
••


 (T AD / 2 )2 • TSA • AF 2



•

 + T • T + W • (T / 2)2 • AF  =  TSA • CF X

2
AD
SA
SL
AD

  + WSL • T AD • LF 
 + (TSA + WSL • T AD ) • AF2





(
)
(8.28)
The input of the information delay structure is Desired control flow (CF*), and its
output is Control flow (CF) as in Section 8.2.2. The dynamic equations; Equation (8.25)
and Equation (8.6) are used to obtain a single equation in terms of Control flow and
Desired control flow. Equation (8.12) and Equation (8.13) that are obtained from Equation
(8.6) can be inserted to Equation (8.25) to obtain the following equation:
••
•
(TID / 2) • IDS 2 + IDS 2 =



 CF *


•

 WVSL  T • LF − (T / 2 )2 • IDS

2
ID
• ID
+


T
SA

 − (TID / 2 ) • IDS 2 − (TID / 2 ) • IDS 2  


•


(
)
T
/
2
•
IDS
IDS
−
−
2
2
ID


(TID / 2)
(8.29)
••



 (TID / 2)2 • TSA • IDS 2


•
 T • CF *

2
+ T •T + W

• (TID / 2) • IDS 2  =  SA
ID
SA
VSL

 +W

•
T
•
LF
VSL
ID

 
 + (TSA + WVSL • TID ) • IDS 2




(
)
(8.30)
It is known from Equation (8.4) that Control flow (CF) is equal to Information delay2
(IDS2), so Equation (8.30) can be re-written for Control flow (CF):
••



 (TID / 2 )2 • TSA • CF


•
 T • CF *

2
+ T •T +W

• (TID / 2 ) • CF  =  SA
ID
SA
VSL

 +W

•
T
•
LF
VSL
ID

 
 + (TSA + WVSL • TID ) • CF




(
)
(8.31)
72
Equation (8.28) and Equation (8.31) are exactly same in the form, so they produce
the same output provided that the given input is the same. Also observe that the inputs CFX
(
(
)
)
and CF* are the same, given by LF + S * − S / TSA in each case. Thus, once the delays
TAD and TID are the same, the outputs are also exactly the same. We can conclude that, the
supply line structure with supply adjustment, and information delay structure with virtual
supply line adjustment are mathematically equivalent (as also proven graphically in Figure
8.6). For cases involving unequal delay times of individual delays, see Appendix F.
8.5. Suggestions on Linear Control of a Single Stock with Information Delay
• Information delay must be considered in the control flow to have stable and fast
response in the behavior of the primary stock. For this, Virtual supply line concept
can be used.
• Virtual supply line can be considered if Information delay stocks can be observed. If
direct observation is not possible, try to estimate the Information delay stocks and
then use Virtual supply line in control. Note that estimating Virtual supply line is
better than ignoring it (if estimation is not possible, then consider using the “stocktype” Virtual supply line, proposed in Chapter 11).
73
9. LINEAR CONTROL OF A SINGLE STOCK WITH SECONDARY
STOCK CONTROL STRUCTURE
As it was mentioned in the previous chapters, generally there is a delay between the
control action and its result. This delay may also be in the form of a secondary stock
structure (Fey, 1974a). Delay in the form of a secondary stock can be seen in production
structures (Forrester, 1968), where one has to change the stock of production capacity in
order to change the production rate (see Figure 9.2). Control flow (CF) of the considered
stock (primary stock), is dependent on an other stock (secondary stock), which can be
formulated as:
CF = C p • SS
(9.1)
where Cp is Productivity coefficient and SS is Secondary stock. Within this kind of a
structure, Stock (S) can be controlled by changing the level of the Secondary stock (SS).
This can be seen in production-workforce or production-capital models (Fey, 1974a;
Sterman, 2000).
Secondary stock control structure is in a way similar to both supply line structure and
information delay structure. It is similar to supply line, in the sense that, it is formed as a
material delay structure, so there can be a supply line for this Secondary stock (SS) namely
Secondary supply line (SSLS). It is similar to the information delay structure, in the sense
that, the last stock of the secondary stock structure, Secondary stock (SS), is directly
affecting the Control flow (CF) of the primary Stock (S). As it is important to take supply
line and information delay into consideration in decisions, in the same way it must be
important and possible to consider secondary stock structure in the decisions.
A simple model of secondary stock control structure can be seen in Figure 9.1 and an
example of this model can be seen in Figure 9.2.
74
For the purpose of this discussion, the supply line of the primary stock is omitted
without loss of generality. Note that Loss flow (LF) and Secondary loss flow (SLF) are
constants. Control flow (CF) equation is given in Equation (9.1).
Stock
Control flow
Loss flow
Desired stock
Stock adjustment
Desired secondary stock
Secondary stock adjustment
Secondary stock
Secondary loss flow
Secondary stock adjustment time
Secondary supply line
Secondary control flow
Secondary acquisition flow
Secondary acquisition delay time
Secondary supply line adjustment
Desired secondary supply line
Weight of secondary supply line
Figure 9.1. Secondary stock control structure
The rest of the equations of the model in Figure 9.1 can be given as:
CF * = LF +
SS * =
S* − S
TSA
CF *
CP
(9.2)
(9.3)
75
Inventory
Production
Sales
Desired inventory
Inventory adjustment time
Inventory adjustment
Desired production
Desired capacity stock
Capacity stock adjustment time
Capacity stock adjustment
Capacity on order
Capacity stock
Capacity discard rate
Capacity order rate
Capacity acquisition rate
Capacity acquisition delay time
Capacity on order adjustment
Desired capacity on order
Weight of capacity on order
Figure 9.2. Secondary stock control structure example
SSLS * = TSAD • SLF
SCF = SLF +
(SS
*
)
(
− SS + W SSL • SSLS * − SSLS
TSSA
(9.4)
)
(9.5)
SSLS
TSAD
(9.6)
S = CF − LF
(9.7)
SS = SAF − SLF
(9.8)
SAF =
•
•
•
SSLS = SCF − SAF
(9.9)
76
where SS* is Desired secondary stock, SSLS* is Desired secondary supply line, SCF is
Secondary control flow, SAF is Secondary acquisition flow, TSAD is Secondary acquisition
delay time, TSSA is Secondary stock adjustment time and WSSL is Weight of secondary
supply line.
The values of the model parameters are as follows for the runs in this chapter, unless
• TID = T AD = 6 [time units ]
• TSSA = TSAD = TID / 2 = T AD / 2 = 3 [time units ]
• OID = OSL = 2
• OSSL = OID − 1 = OSL − 1 = 1
• LF = 2 [items/time unit ]
• SLF = 0.4 [ production factors/time unit ]
• C P = 16 [items / ( production factor • time unit )]
• S * (0 ) = 10 [items ],
S * (4 ) = 9 [items ]
Note that, all stocks in all outputs are initialized at their equilibrium levels, and at
time four the Desired stock (desired level of primary stock, S*) is decreased by one unit, to
perturb the system from equilibrium.
9.2. Comparison of Secondary Stock Control Structure with Supply Line and
Information Delay Structures
Behaviors of the three models in Figure 5.3, in Figure 8.1 and in Figure 9.1 are
compared in Figure 9.3. Stock represents the primary stock of the model in Figure 5.3,
Stock 2 represents the primary stock of the model in Figure 8.1, Stock 3 and Stock 4
77
represent the primary stock of the model in Figure 9.1. Stock 3 is from the run with Weight
of secondary supply line (WSSL) equal to zero, and Stock 4 is from the run with Weight of
secondary supply line equal to one.
1: Stock
1:
2:
3:
4:
10.00
2: Stock 2
1
3: Stock 3
3
3
4
1:
2:
3:
4:
1
4
9.00
2
3
4
1
1:
2:
3:
4:
4: Stock 4
2
1
2
4
3
2
8.00
0.00
17.50
35.00
52.50
Time
00:18
70.00
09 Jan 2003 Thu
Figure 9.3. Behaviors of the models with supply line (first run), information delay (second
run) and secondary stock control (third run with WSSL=0, and fourth run with WSSL=1)
structures
As it can be seen from the runs in Figure 9.3, first, second and fourth runs are exactly
the same so if parameters are set accordingly the three models in Figure 5.3, in Figure 8.1
and in Figure 9.1 have exactly the same behavior. Note that, the third run which was also
from model in Figure 9.1 has different behavior. Weight of secondary supply line (WSSL)
was equal to zero in this third run. One can reach the conclusion that secondary stock
control structure must be used with WSSL equal to one, to have structure exactly similar
with supply line and information delay structures. Thus, secondary stock-control structure
shows exactly the same behavior with supply line and information delay structures, only
when WSSL equal to one (fourth run).
78
9.2.1. Causal Loop Diagram of the Model with Secondary Stock Control Structure
In Section 8.2 causal loop diagrams of the two models in Figure 5.3 and in Figure 8.1
were compared and found to be similar (Figure 8.3 and Figure 8.4). The number and the
signs of the loops, the direction and the signs of the causality arrows were exactly the
same. One may wonder if causal loop diagram of the model in Figure 9.1 is the same with
the causal loop diagrams in Figure 8.3 and in Figure 8.4. The answer to this question is
that, although it has some similarities it is not exactly the same (Figure 9.4).
-
-
+
+
+
-
+
+
Secondary stock
+
-
+
+
Control flow
+
Stock
+
Stock adjustment
-
The dashed causality arrows show the effect of the Weight of secondary supply line
(WSSL). For Weight of secondary supply line (WSSL) equal to zero these arrows do not exist.
Note that, parameters, which are not on a loop, are omitted in this causal loop diagram.
79
It is not obvious how the model in Figure 9.1 produces the same behavior with the
models in Figure 5.3 and in Figure 8.1. The causal loop diagram in Figure 9.4 is not same
with the causal loop diagrams in Figure 8.3 and Figure 8.4. Mathematical approach may
reveal how these different structures exhibit the same behavior.
9.2.2. Mathematical Analysis of the Model with Secondary Stock Control Structure
In the previous chapter we already simplified and mathematically proved that supply
line and information delay structures are similar to each other. Now we are going to
perform similar simplifications on the secondary stock control structure.
Resulting dynamic equations of the model in Figure 8.1 (after all flow and converter
equations inserted) are as follows:
•
S = CF − LF = C p • SS − LF
•
SS =
SSLS
− SLF
TSAD
(9.10)
(9.11)
 CF *


− SS  + WSSL • (TSAD • SLF − SSLS )
 C

•
SSLS
P

SSLS = SLF + 
−
TSSA
TSAD
(9.12)
*




  LF + S − S 

 
TSA 
− SS  + WSSL • (TSAD • SLF − SSLS )

CP




SSLS


−
= SLF +
TSSA
TSAD
The input of the secondary stock control structure is Desired control flow (CF*) and
its output is Control flow (CF). Equation (9.11) and Equation (9.12) are used to obtain a
single dynamic equation in terms of Control flow and Desired control flow.
80
Both Secondary stock adjustment time (TSSA) and Secondary acquisition delay time
(TSAD) are set at half of Information delay time (TID). Weight of secondary supply line
(WSSL) is set to one. After replacing the time parameters with Information delay time (TID),
making Weight of secondary supply line one, and doing necessary simplifications,
Equation (9.11) and Equation (9.12) become:
•
SS =
•
SSLS = 2 • SLF +
SSLS
− SLF
(TID / 2)
(9.13)
CF *
SS
SSLS
−
−
C P • (TID / 2) (TID / 2) (TID / 4)
(9.14)
From Equation (9.13) we can obtain the following equation:
•
SSLS = (TID / 2) • SS + (TID / 2) • SLF
(9.15)
•
••
SSLS = (TID / 2) • SS
(9.16)
Equation (9.15) and Equation (9.16) can be inserted to Equation (9.14) to obtain:
••
(TID / 2) • SS = 2 • SLF +
•


CF *
SS
−
−  2 • SS + 2 • SLF 
C P • (TID / 2) (TID / 2) 

(9.17)
••
•
(TID / 2)2 • SS + TID • SS + SS = CF
*
CP
(9.18)
81
It is known from Equation (9.1) that Control flow (CF) is equal to Productivity
coefficient (Cp) times Secondary stock (SS), so Equation (9.18) can be re-written for CF:
••
•
(TID / 2)2 • CF + TID • CF + CF = CF *
(9.19)
Equation (9.19) is exactly equal to Equation (8.16), and it is exactly same in the form
with Equation (8.11). All three equations produce exactly the same output, provided that
the given inputs (CF* or CF) are the same and parameter values are selected accordingly.
We can conclude that, mathematically there is no difference between supply line,
information delay and secondary stock control structures, but note that their real life
meanings are quite different.
9.3. Using Virtual Supply Line Concept in Secondary Stock Control
Information delay and secondary stock control structures give identical simplified
equations in-terms of input (CF*) and output (CF) variables. We conjecture that this must
also be true for virtually adjusted information delay and virtually adjusted secondary stock
structures.
For a secondary stock structure without supply line delay, the Virtual supply line
(VSL) is defined as:
VSL = C P • TSSA • SS
(9.20)
For a secondary stock structure with a first order supply line delay, the Virtual supply
line (VSL) is defined as:
VSL = C P • ((TSSA + TSAD ) • SS + TSSA • (SSLS − TSAD • SLF ))
(9.21)
For a secondary stock structure with an nth order supply line delay, the Virtual supply
line (VSL) is defined as:
82




 (TSSA + TSAD ) • SS

 
n
n
 
TSAD


VSL = C P • +  TSSA • ∑ (SSLS i ) +
• ∑ ((i − 1) • SSLS i )
 
n
i =1
i =2
 

 

(n − 1) 
 −  TSSA + TSAD • 2 • n  • TSAD • SLF


 

(9.22)
And in general for a secondary stock structure with an nth order supply line delay and
with unequal individual delay times, the VSL can be defined as given in Appendix G.
The mathematical derivation of the virtual supply line formulations for secondary
structure will be discussed in Section 9.3.1 (for the mathematical proof of the equivalency
of the three delay structures with unequal individual delay times see Appendix F).
The Desired virtual supply line (VSL*) and Virtual supply line adjustment (VSLA) are
independent of the order of the secondary supply line:
n


VSL* = (TSSA + TSAD ) • LF =  TSSA + ∑ (TSAD i ) • LF
i =1


VSLA =
(
WVSL • VSL* − VSL
TSA
(9.23)
)
(9.24)
Virtually adjusted desired control flow (VACF*) is added to the model:
*
VACF = CF
*
(S
+ VSLA = LF +
*
)
(
− S + WVSL • VSL* − VSL
TSA
)
(9.25)
The differential equations given in Equation (9.10) and Equation (9.11) remain
unchanged, but Equation (9.12) becomes:
83



  VACF *




− SS  + WSSL • (TSAD • SLF − SSLS )

 SSLS 
 CP

•


−

SSLS =  SLF + 
TSSA
TSAD 







*






  CF * + WVSL • VSL − VSL






TSA
− SS  



CP












 SSLS 
 + W • (T

)
•
SLF
SSLS
−
SSL
SAD
−


 SLF +
T
TSAD 
SSA





(
)
(9.26)
9.3.1. Mathematical Derivation of Virtual Supply Line Formulation for Secondary
Stock Structure
Defining virtual supply line for secondary stock control structure is not that obvious,
so we are going to trace back from the resulting input-output equation of information delay
structure. For this derivation we assume TSSA and TSAD equal to each other, without loss of
generality (for the mathematical proof of the equivalency of the three delay structures with
unequal delay times see Appendix F). Equation (8.31) can be re-written for secondary
stock control structure by using Equation (9.1):
••



 (TID / 2)2 • TSA • C P • SS


•
 + T • T + W • (T / 2)2 • C • SS  = T • CF * + W • T • LF
ID
SA
VSL
ID
P
SA
VSL
ID



 + (TSA + WVSL • TID ) • C P • SS




(
)
(
)
(9.27)
We are going to do a series of manipulations, starting by dividing the both sides of
the Equation (9.27) by Production coefficient (CP):
84
••



 (TID / 2 )2 • TSA • SS


*
•
2
 = TSA • CF + WVSL • TID • LF
+ T •T +W
(
)
•
/
2
•
T
SS
ID
SA
VSL
ID


CP

 + (TSA + WVSL • TID ) • SS




(
)
(
)
(9.28)
Divide the both sides of the Equation (9.28) by Stock adjustment time (TSA):
••



 (TID / 2)2 • SS


*
•
 + T + (W / T ) • (T / 2)2 • SS  = CF + (WVSL / TSA ) • TID • LF
ID
VSL
SA
ID


CP

 + (1 + (WVSL / TSA ) • TID ) • SS




(
)
(
)
(9.29)
•
Move the terms that include SS and SS from left hand side to the right side:

 CF * + (WVSL / TSA ) • TID • LF




CP
••


(TID / 2)2 • SS =  − T + (W / T ) • (T / 2)2 • SS• 
ID
VSL
SA
ID







 − (1 + (WVSL / TSA ) • TID ) • SS
(
)
(9.30)
Collect the terms together that are product of (WVSL / TSA ) :
• 


 CF * + (W / T ) •  TID • LF − C P • (TID / 2)2 • SS  
VSL
SA 


••
− C P • TID • SS
2



(TID / 2) • SS = 

CP


•



 − TID • SS − SS
Divide the both sides of the Equation (9.31) by (TID / 2 ) :
(9.31)
85
• 


 CF * + (W / T ) •  TID • LF − C P • (TID / 2)2 • SS  
VSL
SA 


••
 − C P • TID • SS


(TID / 2) • SS = 

(TID / 2) • C P


•


SS
 − 2 • SS − (T / 2)

ID


(9.32)
Add and subtract 2•SLF to the right side of the Equation (9.32), and add and subtract
C P • (TID / 2 )2 • SLF inside the bracket that is multiplied by (WVSL / TSA ) :

 2 • SLF +


• 


2
 TID • LF − C P • (TID / 2) • SS  




− C P • TID • SS
*


(
)
 CF + WVSL / TSA •
 + C • (T / 2)2 • SLF


••
P
ID


(TID / 2) • SS = 
2



 − C P • (TID / 2) • SLF



(
)
T
/
2
•
C
ID
P






•
SS



 − 2 • SS − (T / 2) − 2 • SLF
ID


(9.33)
Reorganize the Equation (9.33):
(TID
 2 • SLF +



 TID • LF − C P • TID • SS





•





 (TID / 2 ) • SS
 

*
 CF + (WVSL / TSA ) •

 − C • (T / 2 ) •  + (T / 2 ) • SLF   

P
ID
ID





 − (TID / 2 ) • SLF   
••


   (9.34)

/ 2 ) • SS = 

 




(TID / 2) • C P






•


 SS + 2 •  (TID / 2 ) • SS + (TID / 2 ) • SLF 





−

(TID / 2)


86
Note that, Equation (9.13) is independent of Virtual supply line adjustment (VSLA)
term, so Equation (9.15) and Equation (9.16) are also valid for virtually adjusted secondary
stock control structure. By using these equations, Equation (9.34) can be re-written as:

 2 • SLF +


T
•
LF
C
•
T
•
SS
−




ID
P
ID
*
 CF + (WVSL / TSA ) •  − C • (T / 2 ) • (SSLS − (T / 2 ) • SLF ) 
P
ID
ID


•

SSLS = 

(TID / 2) • C P





 SS + 2 • SSLS

−
(TID / 2)


(9.35)
Equation (9.35) can be re-arranged as:
 SLF +










*

 CF + (WVSL / TSA ) •  − C • (T / 2 ) •  SSLS

P
ID

 − (T / 2 ) • SLF  


ID



•
− SS 

SSLS = 
CP



(TID / 2)




 (T / 2 ) • SLF − SSLS

SSLS
 + ID

−


(TID / 2)
(TID / 2)


(9.36)
Both Secondary acquisition delay time (TSAD) and Secondary stock adjustment time
(TSSA) are assumed to be equal to (TID / 2 ) , so Desired secondary supply line (SSLS*) can
be stated as:
SSLS * = (TID / 2 ) • SLF
Equation (9.36) can be re-written by using Equation (9.37) as:
(9.37)
87
 SLF +










*

 CF + (WVSL / TSA ) •  − C • (T / 2 ) •  SSLS

P
ID

 − (T / 2 ) • SLF  


ID



•
− SS 

SSLS = 
CP



(TID / 2)






*
 + SSLS − SSLS − SSLS



(
)
(
)
T
/
2
T
/
2
ID
ID


(9.38)
Now, VSL* and VSL can be seen clearly in Equation (9.38):
VSL* = TID • LF
(9.39)
VSL = C P • (TID • SS + (TID / 2 ) • (SSLS − (TID / 2 ) • SLF ))
(9.40)
Recall that both Secondary acquisition delay time (TSAD) and Secondary stock
adjustment time (TSSA) were assumed to be equal to Information delay time (TID / 2 ) . If
(TID / 2) is inserted in Equation (9.21) and (9.23) instead of TSSA and TSAD, it can be seen
that Equation (9.21) becomes equivalent to Equation (9.40) and Equation (9.23) becomes
equivalent to Equation (9.39). Thus, mathematical derivation is complete (for the
mathematical proof of the equivalency of the three delay structures with unequal individual
delay times see Appendix F).
9.3.2. Model and Behavior for Secondary Stock Structure with Virtual Supply Line
Behaviors of the three models in Figure 5.8, Figure 8.5 and Figure 9.5 are shown in
Figure 9.6. Stock represents the Stock of the model in Figure 5.8, Stock 2 represents the
Stock of the model in Figure 8.5 and Stock 3 represents the Stock in Figure 9.5. All stocks
of the three models show exactly the same behavior with the same parameter values, and
they are optimum, as all models involve supply line or virtual supply line adjustments.
88
Stock
Control flow
Loss flow
Desired stock
Weight of VSL
Stock adjustment
VSL adjustment
Virtual supply line
Secondary stock
Secondary loss flow
Desired VSL
Desired secondary supply line
Weight of secondary supply line
Figure 9.5. Secondary stock control structure with virtual supply line adjustment
1: Stock
1:
2:
3:
1:
2:
3:
2: Stock 2
10.00
1
9.50
2
3: Stock 3
3
1
2
3
1:
2:
3:
1
9.00
0.00
12.50
25.00
Time
2
3
1
37.50
00:45
2
3
50.00
15 Jan 2003 Wed
Figure 9.6. Optimum behaviors of the models with supply line (first run with WSL=1),
information delay (second run with WVSL=1) and secondary stock control (third run with
WVSL=1 and with WSSL=1) structures, all with supply line or virtual supply line adjustments
89
9.4. Suggestions on Linear Control of a Single Stock with Secondary Stock Control
The conclusions that are mentioned about Supply line control in Chapter 5 and
Virtual supply line control for Information delay in Chapter 8 are also valid for Virtual
supply line control for secondary stock control structure.
• Secondary stock control structure must be considered in the control flow, to have
optimum (stable and fast) response in the behavior of the primary stock. For this,
Virtual supply line concept can be used.
• In Information delay case, it may be hard to monitor Information delay stocks that
are necessary to calculate Virtual supply line value, but it is easy for secondary stock
control structure, since Secondary stock and Secondary supply line are already
monitored and managed.
• Virtual supply line can also be used successfully for different forms of secondary
stock control structure (i.e. decaying Secondary stock case) without making any
change in Virtual supply line formulations.
90
10. APPLICATION OF “VIRTUAL SUPPLY LINE” CONCEPT IN
EXAMPLE MODELS
In this chapter, different delay types are collected in same models to demonstrate the
effect of virtual supply line on the resulting behavior. We focus on using two models, both
based on the “Inventory-Workforce Model” of Chapter 19 of the book “Business
Dynamics” (Sterman, 2000). The incomplete version of the “Inventory-Workforce Model”
can be seen in Figure 19-5, at page 768 and 769 of the book. For the complete version, one
must use the cd attached to the book.
10.1. A General Inventory-Workforce Model with Three Type of Delays
This model is adopted from Sterman’s model (Sterman, 2000) and modified for
generality. In this version, we remove non-linearities (i.e. schedule pressure, inflow and
outflow asymmetry), and add information delay structure, so in this model all the three
delay structures simultaneously exist. Thus, the purpose is to test our virtually adjusted
decision rules on a generic Inventory-Workforce Model involving all three delay types and
virtual adjustment requirements (Figure 10.1).
Information delay is assumed to be between inventory control department and human
resources department. Production start rate is controlled by changing the labor force. There
is a supply line delay representing the manufacturing process. For simplicity, we assume
without loss of generality that there are no delays in perceiving the quit rates or delay
times, and no immediate layoffs are allowed (Figure 10.1).
10.1.1. Equations of the Example Model with Three Delay Structures
Firstly stock equations are given:
•
Inventory = Production rate − Shipment rate
(10.1)
91
Work in process inv entory
Inv entory
Minimum shipment time
Production rate
Production start rate
Shipment rate
Customer order rate
Manuf acturing cy cle time
Desired inv entory
Weekly productiv ity
Desired WIPI
Inv entory adjustment
Inv entory adjustment time
WIPI adjustment
Desired VSL f or ID
Weight of WIPI
Virtually adjusted DPSR
Inf ormation adjustment time
VSL adjustment f or ID
Perceiv ed DPSR
Weight of VSL f or ID
Virtual supply line f or ID
Inf ormation adjustment f low
Desired VSL f or SS
Desired labor
VSL adjustment f or SS
Weight of VSL f or SS
Labor adjustment
Labor adjustment time
Virtual supply line f or SS
Trainees
Labor
Joining rate
Quit rate
Av erage duration of employ ement
Hiring rate
Av erage time to join to labor
Trainees adjustment
Weight of trainees
Desired trainees
Figure 10.1. Example model using all three types of delay structures
•
Work in process inventory = Production start rate − Production rate
(10.2)
92
•
Perceived DPSR = Information adjustment flow
•
Labor = Joining rate − Quit rate
(10.3)
(10.4)
•
Trainees = Hiring rate − Joining rate
(10.5)
Secondly flow equations are given:


Inventory

Shipment rate = MIN Customer order rate,
Minimum shipment time 

Production rate =
Work in process inventory
Manufacturing cycle time
Production start rate = Weekly productivity • Labor
Information adjustment flow =
Virtually adjusted DPSR − Perceived DPSR
Information adjustment time
(10.6)
(10.7)
(10.8)
(10.9)
Hiring rate = MAX(Quit rate + Labor adjustment + Trainees adjustment ,0 ) (10.10)
Joining rate =
Quit rate =
Trainees
Average time to join to labor
Labor
Average duration of employement
(10.11)
(10.12)
Equations for all other variables and parameters (except from weights that are test
parameters) in alphabetic order are as follows:
Average duration of employement = 40
[weeks]
(10.13)
93
[weeks]
Average time to join to labor = 9
Customer order rate = 4000 + STEP (500,10 )
Desired inventory = 60000
Desired labor =
[widgets / week ]
[widgets ]
Perceived DPSR
Weekly productivity
[ peoples]
Desired production start rate =
Shipment rate+Inventory adjustment+WIPI adjustment
(10.14)
(10.15)
(10.16)
(10.17)
(10.18)
Desired trainees = Average time to join to labor • Quit rate
(10.19)
Desired VSL for ID = Information adjustment time • Shipment rate
(10.20)
Desired VSL for SS =
(Labor adjustment time+Average time to join to labor ) • Shipment rate
Desired WIPI = Manufacturing cycle time • Shipment rate
Information adjustment time = 2
Inventory adjustment =
[weeks]
Desired inventory − Inventory
Inventory adjustment time = 7
Labor adjustment =
[weeks]
Desired labor − Labor
Labor adjustment time = 16
[weeks]
(10.21)
(10.22)
(10.23)
(10.24)
(10.25)
(10.26)
(10.27)
94
Manufacturing cycle time = 9
[weeks]
(10.28)
Minimum shipment time = 1.6
[weeks]
(10.29)
Trainees adjustment = Weight of trainees •
Desired trainees − Trainees
(10.30)
Virtual supply line for ID = Perceived DPSR • Information adjustment time (10.31)
Virtual supply line for SS =
  Labor adjustment time


 

 • Labor
  + Average time to join to labor 
 (10.32)
Weekly productivity • 


 + Average time to join to labor •  Trainees
 − Desired trainees  




 Desired production start rate 


Virtually adjusted DPSR =  +VSL adjustment for SS

 +VSL adjustment for ID



(10.33)
VSL adjustment for ID =
Weight of VSL for ID •
Desired VSL for ID − Virtual supply line for ID
(10.34)
Desired VSL for SS − Virtual supply line for SS
(10.35)
VSL adjustment for SS =
Weight of VSL for SS •
Weekly productivity = 10
[widgets / (week • people )]
(10.36)
WIPI adjustment =
Weight of WIPI •
Desired WIPI − Work in process inventory
(10.37)
95
All stocks are at their equilibrium levels initially. At time ten, Customer orders is
increased from 4000 to 4500 to disturb the system from its equilibrium point.
10.1.2. Runs of the Example Model with Three Delay Structures
First run:
• Weight of trainees = 0
• Weight of WIPI = 0
• Weight of VSL for ID = 0
• Weight of VSL for SS = 0
Second run:
Third run (the standard system dynamics formulation):
Fourth run:
Fifth run (virtual supply line adjustments optimally considered):
96
Inventory: 1 - 2 - 3 - 4 - 5 80000
1:
3
60000
1:
1
4
2
2
1
3
2
4
5
5
5
4
1
4
5
3
5
1
3
3
4
2
40000
1:
0.00
50.00
100.00
150.00
Page 1
200.00
250.00
22:07 Sat, Jul 05, 2003
Time
Effect of weights
Figure 10.2. Output behaviors (Inventory) of the example model with different supply line
and virtual supply line weight values
Labor: 1 - 2 - 3 - 4 - 5 1:
1:
800
3
450
1
1:
Page 1
100
0.00
4
1
5
2
1
2
5
3
62.50
2
3
4
4
3
4
5
1
125.00
Time
187.50
14:26
5
2
250.00
Thu, Jul 31, 2003
Ef f ect of weights
Figure 10.3. Output behaviors (Labor) of the example model with different supply line
and virtual supply line weight values
97
A standard system dynamics decision formulation run yields the third run, consisting
of oscillations. In this run, supply line of the primary stock and supply line of the
secondary stock are both considered in the decisions, but the information delay and indirect
secondary stock delay effects are ignored. When the information delay and indirect
secondary stock delays are also considered in decisions using our virtual adjustment
formulations, the behavior is improved significantly (fifth run). When all delays are
considered optimally, oscillations are completely eliminated.
10.2. The Inventory-Workforce Model with Non-Linearities in Decisions
We put back the non-linearities (schedule pressure and asymmetry in flows) in the
model that we removed in Section 10.1. The purpose is to test our virtual adjustment
formulation in a complex model involving some non-linearities in an additional loop and in
decision flows. This model is also adopted from Sterman’s model and it is much closer to
the original model, which can be seen in the cd of the book “Business Dynamics”
(Sterman, 2000), with path “\ITHINK\ITHINKMO\CHAP19IT\WIDGETSW.ITM”.
This model has some test variables that we re-remove. The biggest change is that we
add virtual supply line to the decision structures. For our purpose, we are using Workweek
and Weight of virtual supply line as test parameters. From previous chapters, it is known
that when Weight of supply line is zero, the virtual supply line structure has no effect on the
behavior, and when it is one, virtual supply line is fully considered in decisions. For
Workweek we define a new parameter that we call Schedule pressure on\off. When this
new parameter is zero, Workweek is equal to Standard workweek, and when it is one,
Workweek becomes Effect of schedule pressure times Standard workweek. When Schedule
pressure on\off is one (Effect of schedule pressure is active) the Production start rate is
controlled by both Labor and Workweek simultaneously, so the control becomes nonlinear. Effect of schedule pressure is given by:
Effect of schedule pressure = f (Schedule pressure )
(10.38)
98
where
Schedule pressure =
Standard prod start rate
(10.39)
The graphical function of Effect of schedule pressure can be seen in Figure 10.4.
1: Ef f ect of schedule pressure
1:
1
1.25
1
1:
1.00
1:
0.75
1
-1.00
1
0.00
Page 1
1.00
Schedule pressure
2.00
14:19
3.00
Sun, Jul 06, 2003
Figure 10.4. Graphical function of Effect of schedule pressure
The original model has two sub-models; Production-Inventory sub-model and
Workforce sub-model. When sub-models are examined, it can be seen that ProductionInventory sub-model has two input variables; one is the Customer order rate that is an
external variable to the whole model, and the other one is Labor from the Workforce submodel. Expected productivity from Workforce sub-model is also an input for ProductionInventory sub-model but it is taken to be equal to Hourly productivity that is a constant.
The Workforce sub-model has one input that is Desired production start rate from
Production-Inventory sub-model. The two sub-models have two parameters common;
Hourly productivity and Standard workweek.
99
10.2.1. Production-Inventory Sub-Model and its Equations
Work in process inv entory
Inv entory
Inv entory cov erage
Shipment rate
Production rate
Production start rate
Maximum shipment rate
Labor
~
Order f ulf illment ratio
Manuf acturing cy cle time
Hourly productiv ity
Desired shipment rate
Schedule pressure on\of f
Workweek
Desired WIPI
Adjustment f or WIPI
Minimum order processing time
Standard work week
WIPI adjustment time
Production adjustment
f rom inv entory
Saf ety stock cov erage
~
Ef f ect of schedule pressure
Desired inv entory cov erage
Desired production
Standard prod start rate
Desired inv entory
Customer order rate
Expected order rate
Expected productiv ity
Change in expected orders
Schedule pressure
Time to av erage OR
Figure 10.5. Production-Inventory sub-model
Work in process inventory is modeled as a third order material delay (this can be
viewed by double-clicking on Work in process inventory box, when the model is opened
with appropriate software called “Ithink”).
OPP10
Production start rate'
OPP12
OPP11
Noname9
Noname6
Total initial4
Production rate'
rt4
Figure 10.6. Work in process inventory box as third order supply line (material) delay
100
The variable names in Figure 10.6 are irrelevant, but structurally these variables are
meaningful in the sense that they form a third order material delay structure.
Stock equations of the Production-Inventory sub-model are as follows:
•
Inventory = Production rate − Shipment rate
•
Expected order rate = Change in expected orders
•
OPP10 = Production start rate − Noname6
•
OPP11 = Noname6 − Noname9
•
OPP12 = Noname9 − Production rate
(10.40)
(10.41)
(10.42)
(10.43)
(10.44)
and initial values of the stocks are:
• Inventory (0 ) = Desired inventory
• Expected order rate(0 ) = Desired inventory
• OPP10(0 ) = (Total initial4 ) / 3
Flow equations of the Production-Inventory sub-model are as follows:
Production start rate = Labor • Workweek • Hourly productivity
Production rate =
OPP12
rt4
(10.45)
(10.46)
101
Shipment rate = Desired shipment rate • Order fulfillment ratio
Change in expected orders =
Customer order rate − Expected order rate
Time to average OR
(10.47)
(10.48)
Noname6 =
OPP10
rt4
(10.49)
Noname9 =
OPP11
rt4
(10.50)
Equations for all other variables and parameters, except for the variables given
earlier (Effect of schedule pressure and Schedule pressure) and except for the test
parameter (Schedule pressure on\off) are as follows, in alphabetic order:
Adjustment for WIPI =
Desired WIPI − Work in process inventory
WIPI adjustment time
(10.51)
[widgets / week ]
(10.52)
Customer order rate = 10000 + STEP(5000, 10)
Desired inventory = Expected order rate • Desired inventory coverage
(10.53)
Desired inventory coverage =
Minimum order processing time + Safety stock coverage
(10.54)
Desired production =
(10.55)
MAX(0, Expected order rate + Production adjustment from inventory )
Desired production start rate = Adjustment for WIPI + Desired production (10.56)
Desired shipment rate = Customer order rate
(10.57)
Desired WIPI = Desired production • Manufacturing cycle time
(10.58)
102
Hourly productivity = 0.25
[widgets / (hour • people )]
Inventory adjustment time = 12
Inventory coverage =
[weeks ]
(10.60)
Inventory
Shipment rate
Manufacturing cycle time = 8
Maximum shipment rate =
(10.59)
(10.61)
[weeks]
(10.62)
Inventory
Minimum order processing time
Minimum order processing time = 2
[weeks ]
(10.63)
(10.64)
 Maximum shipment rate 

Order fulfillment ratio = f 
 Desired shipment rate 
(10.65)
where the graphical function for Order fulfillment ratio is given in Figure 10.7.
1: Order f ulf illment ratio
1:
1
1
1
1:
1
1:
0
1
Page 1
1
-1.00
0.00
1.00
Maximum_SR/Desired_SR
2.00
18:02
3.00
Sun, Jul 06, 2003
Figure 10.7. Graphical function for Order fulfillment ratio
103
Production adjustment from inventory =
Desired in ventory − Inventory
(10.66)
rt4 = (Manufactur ing cycle time ) / 3
(10.67)
[weeks ]
(10.68)
Safety stock coverage = 2
Standard prod start rate = Labor • Standard work week • Expected productivi ty (10.69)
Standard workweek = 40
[hours / week ]
Time to average OR = 8
(10.70)
[weeks]
(10.71)
Total initial4 = Desired WIPI
WIPI adjustment time = 6
(10.72)
[weeks]
 IF Schedule p ressure on \off = 0

Workweek =  THEN Standard w ork week
 ELSE Effect of schedule p ressure • Standard w ork week

(10.73)





(10.74)
Note that the whole system is initially at equilibrium. The system is disturbed from
equilibrium by increasing Customer Order Rate from 10000 to 15000, at time 10.
10.2.2. Workforce Sub-Model and its Equations
Stock equations of the Workforce sub-model are as follows:
•
Labor = Hiring rate − Quit rate − Layoff rate
(10.75)
•
Vacancies =
Vacancy creation rate − Vacancy closure rate − Vacancy cancellation rate
(10.76)
104
Maximum v acancy cancellation
Vacancy cancellation rate
Av erage lay of f time
Vacancy cancellation time
Vacancies
Maximum lay of f rate
Vacancy creation rate
Vacancy closure rate
Labor
Desired v acancy
cancellation rate
Lay of f rate
Hiring rate
Quit rate
Desired v acancy creation rate
Av erage time to f ill v acancies
Av erage duration of employ ment
Adjustment f or v acancies
Expected time to f ill v acancies
Desired lay of f rate
Desired v acancies
Vacancy adjustment time
Willingness to lay of f
Adjustment f or labor
Desired hiring rate
Expected attrition rate
Expected productiv ity
Desired labor
Standard work week
Figure 10.8. Workforce sub-model
Initial values of the stocks are:
• Labor (0 ) = Desired labor
• Vacancies(0 ) = Desired vacancies
Flow equations of the Workforce sub-model are as follows:
Hiring rate = Vacancies / Average time to fill vacancies
(10.77)
Quit rate = Labor / Average duration of employment
(10.78)
105
Layoff rate = MIN(Desired lay off rate, Maximum lay off rate)
(10.79)
Vacancy creation rate = MAX(0, Desired vacancy creation rate )
(10.80)
Vacancy closure rate = Hiring rat e
(10.81)
Vacancy cancellation rate =
MIN(Desired vacancy cancellation rate, Maximum vacancy cancellation )
(10.82)
Equations for all other variables and parameters are as follows, in alphabetic order
(note that two common parameters; Hourly productivity and Standard workweek are given
before for Production-Inventory sub-model):
Adjustment for labor =
Adjustment for vacancies =
Desired la bor − Labor
Labor adju stment time
Desired va cancies − Vacancies
Vacancy ad justment time
Average duration of employment = 100
Average layoff time = 8
[weeks]
[weeks]
Average time to fill vacancies = 8
[weeks]
Desired hiring rate = Expected a ttrition r ate + Adjustment for labor
Desired labor =
Desired pr oduction s tart rate
Standard w ork week • Expected productivit y
(10.83)
(10.84)
(10.85)
(10.86)
(10.87)
(10.88)
(10.89)
Desired lay off rate = Willingness to lay off • MAX(0,-Desired hiring rate) (10.90)
106
Desired vacancies = Expected t ime to fil l vacancie s • Desired hiring rate (10.91)
Desired vacancy cancellation rate =
MAX(0, - Desired vacancy creation rate )
Desired vacancy creation rate =
Desired hiring rate + Adjustment for vacancies
(10.92)
(10.93)
Expected attrition rate = Quit rate
(10.94)
Expected productivity = Hourly productivity
(10.95)
Expected time to fill vacancies = Average time to fill vacancies
(10.96)
Labor adjustment time = 13
Maximum lay off rate =
[weeks ]
Labor
Average layoff time
Maximum vacancy cancellation =
Vacancies
Vacancy cancellation time
(10.97)
(10.98)
(10.99)
Vacancy adjustment time = 4
[weeks ]
(10.100)
Vacancy cancellation time = 2
[weeks ]
(10.101)
Willingless to lay off = 1
[dimensionless ]
(10.102)
10.2.3. Problematic Desired Supply Line Equations
Before we add virtual supply line structure to the model, we must mention a problem
in the desired levels of the Work in process inventory and the Vacancies. Desired WIPI that
107
is given by Equation (10.58) depends on Desired production, and Desired vacancies that is
given by Equation (10.91) depends on Desired hiring rate. Both Desired production and
Desired hiring rate are varying very fast because they are on control loops. This
contradicts with the suggestions given in Section 6.6. Furthermore, Equation (10.91)
creates “circular connection error” when we try to add virtual supply line structure. We
propose the following two more stable formulas instead of Equation (10.58) and Equation
(10.91):
Desired WIPI = Expected order rate • Manufactur ing cycle time
(10.103)
Desired vacancies =
Expected t ime to fil l vacancie s • Expected attrition rate
(10.104)
In Appendix H we offer extensive comparison of these proposed formulations and
the original equations. We conclude that the modified equations yield more stable results.
For the rest of this chapter we will use the new equations; Equation (10.103) and
Equation (10.104).
10.2.4. Non-Linear Inventory-Workforce Model with Virtual Supply Line
Now we add virtual supply line structure to the model:
Expected order rate
Expected time to f ill v acancies
Weight of VSL
Desired VSL
Expected attrition rate
Virtual Supply Line
Vacancies
Labor
Standard work week
Figure 10.9. Virtual supply line structure
108
Virtual supply line structure can be activated by giving values to Weight of VSL
greater than zero. For Weight of VSL equal to zero, virtual supply line structure is not
active. There are only three new equations, since all the other inputs to this structure is
already defined:
Desired VSL =
 Labor adjustment time


 • Expected order rate
 + Expected time to fill vacancies 
(10.105)
Virtual supply line = Hourly productivity • Standard work week
 (Labor adjustment time + Expected time to fill vacancies) • Labor 


Vacancies −





•

+ Labor adjustment time •  Expected time to fill vacancies 


 • Expected attrition rate







(10.106)
Virtually adjusted DPSR = Desired production start rate
+ Weight of VSL •
(Desired VSL − Virtual Supply Line )
(10.107)
Also the Desired labor formulation in Equation (10.89) changes to include virtual
supply adjustment in decision:
Desired labor =
Standard work week • Expected productivity
(10.108)
Before Desired labor is modified as above, the initial value of Labor must also be
changed, otherwise circular connection occurs:
• Labor (0) =
Standard work week • Expected productivity
Recall the changes proposed in Section 10.2.3; Equation (10.103) and Equation
(10.104) are valid till the end of this chapter. New runs can be defined as:
109
First run (standard system dynamics formulation):
• Schedule pressure on\off = 0
• Weight of VSL = 0
Second run:
Third run (standard system dynamics formulation):
Fourth run:
Inv entory : 1 - 2 - 3 - 4 1:
70000.00
1
2
3
4
1
2
3
4
4
3
2
1:
45000.00
1
1
4
2
1:
Page 1
20000.00
0.00
3
50.00
100.00
Time
150.00
22:47
200.00
Thu, Jul 24, 2003
Figure 10.10. Dynamics of Inventory with or without schedule pressure and VSL
When two outputs in Figure 10.10 and Figure 10.11 are examined it can be obviously
seen that the first run (Schedule pressure on\off = 0, Weight of VSL = 0) is the worst run. In
Figure 10.10 the second run (Schedule pressure on\off = 0, Weight of VSL = 1), the third
110
run (Schedule pressure on\off = 1, Weight of VSL = 0) and the fourth run (Schedule
pressure on\off = 1, Weight of VSL = 1) give fast and stable results, and their results are not
significantly different from each other. In Figure 10.11 the fourth run gives the best result;
no overshoot and fast response.
Labor: 1 - 2 - 3 - 4 1:
1600.00
1
2
3
4
1
2
3
4
1
2
3
4
4
1300.00
1:
3
1:
1000.00
1
0.00
2
50.00
Page 1
100.00
Time
150.00
22:47
200.00
Thu, Jul 24, 2003
Figure 10.11. Dynamics of Labor with or without schedule pressure and VSL
It is not very easy to reach to a general conclusion with these runs. This is because
the system is already very stable with the given parameters, so we are going to assume
different time parameters to force the system towards instability. Furthermore, we are
going to give equal weights to the stocks and their corresponding supply lines to be
consistent with the rest of the thesis (recall that giving equal weight to stock and supply
line produces optimal behavior; fast and stable). The original adjustment time parameters
(weights given to the stocks) in Sterman (2000) are different from each other. The
Inventory adjustment time is 12, while the WIPI adjustment time is 6. The Labor
adjustment time is 13, while Vacancy adjustment time is 4 (see Appendix H). New
parameters that make the system harder to manage are as follows:
[weeks ]
Average time to fill vacancies = 20 [weeks ]
• Manufacturing cycle time = 40
•
• Inventory adjustment time = WIPI adjustment time = 2
[weeks ]
111
• Labor adjustment time = Vacancy adjustment time = 14
[weeks ]
We assume longer Manufacturing cycle time (40 versus 8) and longer Average time
to fill vacancies (20 versus 8), so the conditions for the firm is harder. We assume that
inventory manager wants quick adjustments in his inventory, so Inventory adjustment time
and WIPI adjustment time are 2 (not 12 nor 6). Furthermore, we assume that head of
human resources department is not willing to make quick adjustments in the labor, so
Labor adjustment time and Vacancy adjustment time are 14 (not 13 nor 4). We repeat the
runs with these parameter values:
Inv entory : 1 - 2 - 3 - 4 1:
70000.00
2
4
1
1:
2
3
4
2
3
4
1
3
1
45000.00
1
4
3
1:
20000.00
2
0.00
Page 1
75.00
150.00
Time
225.00
23:14
300.00
Thu, Jul 24, 2003
Figure 10.12. Runs for Inventory in less stable conditions
Now it is easier to conclude. First run (Schedule pressure on\off = 0, Weight of VSL =
0) is complete failure in bringing the system to its desired level. System is quite oscillatory
for the first run, and there is even an error in the average level of the inventory (see Figure
10.12). The fourth run (Schedule pressure on\off = 1, Weight of VSL = 1) is obviously the
best run for both Inventory and Labor (see Figure 10.12 and Figure 10.13). The second run
(Schedule pressure on\off = 0, Weight of VSL = 1) and the third run (Schedule pressure
on\off = 1, Weight of VSL = 0) are both successful in Figure 10.12, but second run is much
better in Figure 10.13.
112
Labor: 1 - 2 - 3 - 4 1:
3500.00
1
3
1
1:
1750.00
2
4
2
3
4
2
3
4
2
3
4
1
1
1:
Page 1
0.00
0.00
75.00
150.00
Time
225.00
23:14
300.00
Thu, Jul 24, 2003
Figure 10.13. Runs for Labor in less stable conditions
The final conclusion is that considering delays are crucial, so virtual supply line must
be taken into consideration in the decision formulations. By taking the delays into account
via virtual supply line, one completely eliminates unwanted oscillatory behavior.
If the system structure and time parameters are clearly known by the decision maker
there may not be need to use additional non-linear controls like flexible workweek; virtual
supply line consideration would be enough. Although additional non-linear control
increases the system performance slightly, it may not be worth the extra cost. But, one
must keep in mind that when system structure and time parameters are not clear, then it
may worth to install an additional non-linear control structure to the system. Note that,
even when used alone, additional non-linear control brings stability to the system.
Finally, it is clear that in all conditions it is worth using virtual supply line, since it
has almost no cost or risk to the system. The main cost encountered by considering virtual
supply line is just to do some more estimations and calculations.
113
11. VIRTUAL SUPPLY LINE AS A STOCK
Virtual supply line can also be defined as a stock. This Virtual supply line can be
used to adjust the control decisions (Desired control flow) like in the previous chapters (i.e.
Section 8.3, Section 9.3 and Chapter 10).
11.1. The Usage of Stock-Type Virtual Supply Line for Information Delay Structure
For example, we change the model in Figure 8.5 for stock-type Virtual supply line.
The Virtually adjusted desired control flow, that is the input of the information delay
structure, is treated to be the inflow of this virtual supply line. The Control flow, that is
output the information delay structure, is treated to be the outflow of this Virtual supply
line.
Stock
Loss flow
Control flow
Desired stock
Stock adjustment
Weight of VSL
VSL adjustment
Desired VSL
Virtual supply line
Virtual outflow
Information delay 2
Virtual inflow
Information delay 1
Figure 11.1. Stock control with Virtual supply line stock
114
Differential equations of the model in Figure 11.1 can be given as:
•
S = CF − LF = IDS 2 − LF
(11.1)
(
)
*
*


 LF + WVSL • VSL − VSLS + S − S  − IDS1

•
TSA
VACF * − IDS1 

(11.2)
IDS 1 = IAF1 =
=
(TID / OID )
(TID / 2)
•
IDS 2 = IAF2 =
IDS1 − IDS 2 IDS1 − IDS 2
=
(TID / OID )
(TID / 2)
(11.3)
•
VSLS = VCF − VAQF = VACF * − CF = VACF * − IDS 2
(
(11.4)
)

WVSL • VSL* − VSLS + S * − S 

= LF +
− IDS 2


TSA


where Desired virtual supply line (VSL*) is:
VSLS * = TID • LF
(11.5)
11.2. Dependency of Equilibrium Level on Initial Value, and Setting a Proper Value
If all the differential equations; Equation (11.1), Equation (11.2), Equation (11.3) and
Equation (11.4) are set to zero, an equilibrium line is found as:


 S 



 IDS1 
=
 IDS 

2

 S* − r
VSLS 

 equilibrium 
 WVSL
where − ∞ < r < ∞ .
r
LF
LF
+ TID






• LF 

(11.6)
115
The equilibrium line can be re-written as:
 S * + WVSL • (TID • LF − r )
 S 






LF
 IDS1 
=

 IDS 
LF
2



VSLS 


r

 equilibrium 

(11.7)
where − ∞ < r < ∞ . Note that desired equilibrium point is:

 S 
S * 



 LF 
 IDS1 
=

 IDS 
2
 LF 

VSLS 



 equilibrium  TID • LF 
(11.8)
which is on the equilibrium line given by Equation (11.6) or Equation (11.7).
As it can be seen from Equation (11.6) and Equation (11.7), if Stock is not equal to
its desired level, in the long-run also Virtual supply line is not equal to its desired level,
and vice versa. To guarantee the intended equilibrium, the initial value of stock-type
Virtual supply line must be set properly. The initial value can be set properly by using the
Virtual supply line formulations given in Section 8.3 and Section 9.3. Specifically for this
example, the initial level of the Virtual supply line stock must be set using Equation (8.18),
which is VSL = (TID / 2 ) • (IDS1 + IDS 2 ) . Given that VSLS(0) is set accordingly, then, the
model in Figure 11.1 can produce exactly the same output behavior like the model in
Figure 8.5 (see Figure 11.2), otherwise it can reach some arbitrary equilibrium value,
yielding a steady-state error.
The dependence on the initial conditions makes the model in Figure 11.1 weak in one
sense. Assume a setting error is done in initial values or somehow a disturbance occurred
in the information delay structure that we do not know, then the result will be biased at
equilibrium. This is illustrated in Figure 11.3 and Figure 11.4. In Figure 11.3, the initial
value of Virtual supply line is one unit above the Desired level of virtual supply line. In
Figure 11.4, a shock of minus one unit is applied to Information Delay1 stock at time five
116
(note that parameter values are same with the values given in Figure 11.2, but no shock is
applied to the primary stock for the runs in Figure 11.3 and Figure 11.4).
1: Desired stock
2: Stock
3: Desired VSL
4: Virtual supply line
8.00
1:
2:
3:
4:
25.00
4
4
3
1:
2:
3:
4:
3
3
4
3
4
6.00
23.00
1
2
1
1
1
2
1:
2:
3:
4:
4.00
2
2
21.00
0.00
10.00
20.00
Time
Page 1
30.00
12:23
40.00
Thu, Jul 31, 2003
Figure 11.2. Dynamic behavior without any steady-state error (TSA = 2, TID = 12, WVSL =
1, and -1 unit shock in S at time 5)
1: Desired stock
1:
2:
3:
4:
2: Stock
3: Desired VSL
8.00
4
4
4
25.00
4
3
1:
2:
3:
4:
3
6.00
2
1
1
1
2
4.00
2
21.00
0.00
Page 1
3
23.00
1
1:
2:
3:
4:
3
10.00
20.00
Time
2
30.00
13:02
40.00
Mon, Jul 28, 2003
Figure 11.3. Steady-state error resulting from initial value error in Virtual supply line (TSA
= 2, TID = 12, WVSL = 1)
117
1: Desired stock
1:
2:
3:
4:
2: Stock
24.00
3
3: Desired VSL
3
4
3
3
4
4
1:
2:
3:
4:
4
14.00
2
2
1:
2:
3:
4:
2
1
2
1
1
10.00
20.00
Time
1
4.00
0.00
Page 1
30.00
13:06
40.00
Mon, Jul 28, 2003
Figure 11.4. Result of a shock applied to first stock of Information delay, creating a
steady-state error (TSA = 2, TID = 12, WVSL = 1)
Note that, for non-stock Virtual supply line like in Section 8.3, the above problem
does not exist. The equilibrium point for the Equation (8.4), Equation (8.23) and Equation
(8.6) can be found as
 S* 
 S 






LF
IDS
=

1


 IDS 
 LF 
2  equilibrium



(11.9)
As it can be seen from Equation (11.9), the equilibrium point is not dependent on
initial conditions, when non-stock Virtual supply line is used. Note that, for stock-type
Virtual supply line, equilibrium is only reached if there is no initialization error in Virtual
supply line, and if there are no disturbances in the stocks of delay structure (see Figure
11.2).
But if delay stocks are not directly observable, neither the non-stock-type nor the
stock-type Virtual supply line would work. This problem is discussed in the following
section.
118
11.3. Using Virtual Supply Line when Information Delay Stocks cannot be Observed
When it is not possible to observe the information delay stocks, the formulations
given for Virtual supply line (VSL) in Section 8.3 cannot be used. The stock-type VSL can
be used, but its initial value can not be determined. For this difficult case, we developed a
rule of thumb to be used, though it is not perfect. Firstly we set the initial value of the VSL
to its desired level VSL*, assuming that it is possible to estimate the total delay duration.
We proved that there might be an error in this equilibrium value resulting in an error in the
equilibrium level of the Stock. In addition to these facts, Equation (11.6) and (11.7) further
say that if VSL goes to its desired level, Stock also goes to its own desired level. So we add
an additional goal seeking adjustment flow to VSL stock, which gradually removes biases
in equilibrium levels. We propose a new dynamic formulation for stock-type VSL:
(
•
)
VSLS = VCF + VAF − VAQF = VACF * + VSLS * − VSLS / TVA − CF
(11.10)
where VAF is Virtual adjustment flow and TVA is Virtual adjustment time. If Equation
(11.10) is applied to the problematic cases presented in Figure 11.3 and Figure 11.4 with
Virtual adjustment time (TVA) equal 14, the following two runs are obtained:
1: Desired stock
1:
2:
3:
4:
2: Stock
3: Desired VSL
4
25.00
3
1:
2:
3:
4:
4
3
4
3
4
6.00
1
1
2
1
2
2
2
4.00
21.00
0.00
Page 1
3
23.00
1
1:
2:
3:
4:
8.00
50.00
100.00
Time
150.00
13:13
200.00
Mon, Jul 28, 2003
Figure 11.5. Eliminating equilibrium error resulting from initial value error in Virtual
supply line (TSA = 2, TID = 12, WVSL = 1, TVA = 40)
119
1: Desired stock
2: Stock
24.00
1:
2:
3:
4:
3: Desired VSL
3
3
3
4
3
4
4
4
1:
2:
3:
4:
14.00
2
1:
2:
3:
4:
2
1
1
1
50.00
100.00
Time
2
1
2
4.00
0.00
Page 1
150.00
13:12
200.00
Mon, Jul 28, 2003
Figure 11.6. Eliminating equilibrium error resulting from a shock applied to first stock of
Information delay (TSA = 2, TID = 12, WVSL = 1, TVA = 40)
11.4. Some Observations About Virtual Adjustment Time
If Virtual adjustment time (TVA) is short, there might be oscillatory behavior:
1: Desired stock
1:
2:
3:
4:
2: Stock
3: Desired VSL
28.00
3
3
3
4
4
3
4
4
1:
2:
3:
4:
14.00
2
1
1:
2:
3:
4:
2
1
2
1
2
0.00
0.00
Page 1
1
50.00
100.00
Time
150.00
13:17
200.00
Mon, Jul 28, 2003
Figure 11.7. Oscillations after a shock applied to first stock of Information delay (TSA = 2,
TID = 12, WVSL = 1, TVA = 10)
120
If TVA is long, the reduction of bias in equilibrium takes longer time:
1: Desired stock
1:
2:
3:
4:
2: Stock
24.00
3: Desired VSL
3
3
3
3
4
4
4
4
1:
2:
3:
4:
14.00
2
2
1:
2:
3:
4:
2
2
1
1
1
50.00
100.00
Time
1
4.00
0.00
Page 1
150.00
13:18
200.00
Mon, Jul 28, 2003
Figure 11.8. Long time to restore equilibrium after a shock applied to first stock of
Information delay (TSA = 2, TID = 12, WVSL = 1, TVA = 100)
1: Desired stock
1:
2:
28.00
1:
2:
14.00
2: Stock
2
2
1:
2:
Page 1
1
1
0.00
50.00
1
1
2
0.00
100.00
Time
150.00
13:20
200.00
Mon, Jul 28, 2003
Figure 11.9. Unstable oscillatory behavior when virtual supply line term is ignored in
decisions (TSA = 2, TID = 12, WVSL = 0)
121
It can be said that Equation (11.10) is robust for a wide range of values of Virtual
adjustment time (TVA). It can be shown that it improves the system performance
significantly by bringing stability to the system when it is applied (compare Figure 11.6
with Figure 11.9, or in the worse cases, even Figure 11.7 and Figure 11.8 are significant
improvements over Figure 11.9).
11.5. The Stock-Type Virtual Supply Line as a Powerful Control Formulation when
Delay Structure is Complex and Unknown to the Decision Maker
In some cases the delay structure might be completely unknown to the decision
maker. It might be complex; including many different types of delay structures (i.e. supply
line, information delay, secondary stock control). Furthermore, it might have some
moderate disturbances such as random delays and random shocks in flows. For this
managerially hard case we propose following framework:
Stock
Loss flow
Acquisition flow
Desired stock
Stock adjustment
Weight of VSL
VSL adjustment
Desired VSL
Virtual supply line
Virtual adjustment time
Virtual outflow
Virtual inflow
Unknown Delay or Secondary Stock Structures
Output of the unknown
delay structure
Estimated delay time of the
unknown delay structure
Input of the unknown
delay structure
Figure 11.10. Framework of stock control problem with unknown delay structure
122
Our assumption are as follows:
• The decision maker knows the input of the unknown delay structure (actually she is
placing the orders that is Virtually adjusted desired control flow -VACF*-).
• The decision maker knows the output of the unknown delay structure (actually the
Acquisition flow of the Stock is the output of the unknown delay structure).
• The decision maker knows or can estimate the total Delay time of the unknown delay
structure.
• The unknown delay structure by and in itself is not a source of unstable oscillations.
We now illustrate the proposed approach on an example. We create a complex delay
structure that has some parameters as normally distributed random variables. In addition
we put some flows that create random shocks in some stocks of this delay structure. This
model can be seen in Figure 11.11. Firstly stock equations are given:
•
Expected productivity = Expectation adjustment flow
•
(11.11)
Secondary stock = Secondary acquisition flow − Secondary loss flow
(11.12)
•
 Secondary control flow + Random shocks1 

Secondary supply line = 
 − Secondary acquisition flow

(11.13)
•
Stock = Acquisition flow2 − Loss flow
(11.14)
Supply line1 = Production + Random shocks2 − Acquisition flow1
(11.15)
•
•
Supply line 2 = Acquisition flow1 − Acquisition flow2
(11.16)
123
Virtual control flow



Virtual supply line =  + Virtual adjustment flow 
 − Virtual acquisition flow 


•
(11.17)
Stock
Loss flow
Desired stock
Stock adjustment
Desired acquisition rate
Weight of VSL
VSL adjustment
Desired VSL
Virtually adjusted DAR
Virtual supply line
Virtual inflow
Acquisition flow 2
Virtual outflow
Unknown Delay Structures
Supply line 2
Acquisition delay time 2
Estimated delay time of the
unknown delay structure
Perceived orders
Acquisition delay time 1
Expected productivity
Acquisition flow 1
Perception delay time 1
Supply line 1
Production
Expectation adjustment time
Productivity
Random shocks 2
Secondary stock
Secondary loss flow
Secondary life time
Perception delay time 2
Secondary acquisition
Random shocks 1
delay time
Perceived secondary loss flow
adjustment
Weight of
Desired
secondary supply line
secondary supply line
Figure 11.11. Stock control with unknown complex delay structures
124
Secondly flow equations are given:
Acquisition flow1 =
Supply line1
Acquisition delay time1
(11.18)
Acquisition flow2 =
Supply line2
Acquisition delay time2
(11.19)
Expectation adjustment flow =
Productivity − Expected productivity
(11.20)
Loss flow = 2
(11.21)
Production = Productivi ty • Secondary stock
(11.22)
Random shocks1 = NORMAL(0,0.02)
(11.23)
Randomshocks2 = NORMAL(0,0.2)
(11.24)
(11.25)
 Perceived secondary loss flow



Secondary control flow =  + Secondary stock adjustment

 + Secondary supply line adjustment 


(11.26)
Secondary acquisition flow =
Secondary loss flow =
Virtual adjustment flow =
Secondary stock
Secondary life time
Desired VSL − Virtual supply line
Virtual acquisition flow = Acquisition flow2
(11.27)
(11.28)
(11.29)
125
Virtual control flow = Virtually adjusted DAR
(11.30)
Equations for all other variables and parameters, in alphabetic order, are as follows:
Acquisition delay time1 = NORMAL(3,0.4)
(11.31)
Acquisition delay time2 = NORMAL(4,0.4)
(11.32)
Desired acquisition rate = Loss flow + Stock adjustment
(11.33)
Desired secondary stock =
Perceived orders
Expected productivity
(11.34)
 Secondary acquisition delay time 

Desired secondary supply line = 
 • Perceived secondary loss flow 
(11.35)
Desired stock = 5
(11.36)
Desired VSL =
Estimated delay time of the unknown dealy structure • Loss flow
(11.37)
Estimated delay time of the unknown dealy structure =
  Acquisition delay time1 + Acquisition delay time2
 

  (11.38)
SMTH3  + Perception delay time1 + Secondary acquisition delay time ,4 
  + Secondary stock adjustment time
 
 

where “SMTH3” represents a third order information delay, which is a three first order
delays cascaded in series. We used SMTH3 to save modeling space in Figure 11.11.
Expectation adjustment time = 5
(11.39)
Perceived orders = SMTH3(Virtually adjusted DAR, Perception delay time1 ) (11.40)
126
Perceived secondary loss flow =
SMTH3(Secondary loss flow, Perception delay time2 )
(11.41)
Perception delay time1 = NORMAL(3,0.2)
(11.42)
Perception delay time 2 = NORMAL(2,0.2)
(11.43)
Productivity = NORMAL(5,0.5)
(11.44)
Secondary acquisition delay time = NORMAL(7,0.5)
(11.45)
Secondary life time = NORMAL(3,0.4)
(11.46)
Secondary stock adjustment =
Desired secondary stock − Secondary stock
(11.47)
Secondary stock adjustment time = 2
(11.48)
Secondary supply line adjustment =
Desired secondary supply line − Secondary supply line
WSSL •
(11.49)
Stock adjustment =
Desired stock − Stock
(11.50)
Stock adjustment time = 2
(11.51)
Virtually adjusted DAR = Desired acquisition rate + VSL adjustment
(11.52)
infinite, for the first and sec ond runs 
Virtual adjustment time = 

for the third run
20,

(11.53)
127
VSL adjustment = WVSL •
Desired VSL − Virtual supply line
Weight of secondary supply line = 0.1
0,
Weight of VSL = 
1,
(11.54)
(11.55)


for the second and third runs 
for the first run
(11.56)
Runs of the model:
• First run: Weight of VSL = 0
• Second run: Weight of VSL = 1 and Virtual adjustment time = infinite
• Third run: Weight of VSL = 1 and Virtual adjustment time = 40
1: Desired stock
1:
2:
12.50
1:
2:
7.50
2: Stock
2
1
1:
2:
1
2
1
1
2
2.50
0.00
Page 1
2
22.50
45.00
Time
67.50
20:28
90.00
Sat, Aug 02, 2003
Figure 11.12. Unstable behavior for Weight of VSL = 0
Third run (Figure 11.14) gives the best result, demonstrating that the usage of stocktype Virtual supply line with given framework in Figure 11.10, greatly increases the
stability of a system that includes unknown delay structures with randomness. The stability
and robustness of the general stock-type Virtual supply line control formulation given in
this chapter can also proven by applying it on the two inventory-workforce models in
128
Section 10.1 and Section 10.2. The dynamic behaviors would be the same given in Figure
10.2, Figure 10.3, Figure 10.12 and Figure 10.13, but they are skipped to conserve space.
1: Desired stock
2: Stock
10.00
1:
2:
2
2
2
1:
2:
5.00
1:
2:
0.00
1
2
0.00
1
1
75.00
150.00
Time
Page 1
1
225.00
14:10
300.00
Thu, Jul 31, 2003
Figure 11.13. Steady-state error in the mean level of Stock for Weight of VSL = 1 and for
Virtual adjustment time = infinite
1: Desired stock
1:
2:
2: Stock
10.00
2
1:
2:
5.00
1:
2:
0.00
1
0.00
Page 1
2
2
1
1
75.00
150.00
Time
2
1
225.00
14:11
300.00
Thu, Jul 31, 2003
Figure 11.14. A quite stable behavior obtained by the proposed Virtual supply line
formulation ( Weight of VSL = 1 and for Virtual adjustment time = 40)
129
12. APPLYING THE RESULTS TO THE INVENTORY
MANAGEMENT RULES
In this chapter, we apply some basic findings of this thesis to the standard inventory
management rules. For this we select the (s, S) rule as a standard inventory control rule.
The (s, S) rule is appropriate since we assume random demand, significant ordering costs
and no review cost. The (s, S) rule is superior to the (s, Q) rule when demand is a random
variable. The costs are lower with the (s, S) rule. Furthermore, as a continuous review rule,
(s, S) is also superior to periodic review rules (n•Q, s, R) and (s, S, R) when costs
associated with reviewing are negligible. The (s, S) rule is also superior to the (S, R) rule
when ordering costs are not negligible (Hax and Candea, 1984).
To apply our findings, we first create problematic inventory cases as was done in
deriving our improved formulations of Chapter 6 – Chapter 11, and then suggest similar
improvements for the (s, S) rule. We are going to assume discrete time models to be
consistent with inventory management literature, so the stock equations will be given as
difference equations (see Section 2.2).
The runs presented in this chapter are also compared with the runs using the (s, Q)
rule, and comparatively similar results are obtained. The only difference is that the (s, Q)
rule incurs higher costs. The runs obtained from the (s, Q) rule are not presented in this
thesis to save space. Although we did not test the periodic inventory review rules, we
foresee that the conclusions would not differ in essence. More rules are compared in a
supply chain context by Gündüz (2003).
12.1. Management of a Perishable Goods Inventory with Discrete Supply Line Delay
In this section, we discuss our results in the context of management of a perishable
goods inventory. For this purpose we build a “perishable goods inventory model”. Discrete
supply line delay with a long acquisition delay time (lead time) is assumed (see Chapter 6).
There are two outflows from the Inventory; one is the rate of decay (Perishing rate), and
the other one is the sales to the customers (see Figure 12.1). Immediately note that the
130
outflow does not necessarily have to be perishing rate, it may be shipments to other
inventories or various branches in other examples. The critical point is that there is an
outflow from the inventory which is proportional to its level. We have shown in Chapter 6
that improper use of such outflows in decision rules may yield severe instabilities. Our
motivation in this chapter is to show that such may be the case with some inventory control
rules, and then discuss improved formulations. Note that the model in Figure 12.1 is not
complete. The other parts of the perishable goods inventory model will be given gradually.
Supply line
Control flow
Inventory
Perishing rate
Acquisition flow
Lost sales
Orders
Perishing fraction
Customer demand
Sales
Figure 12.1. Perishable goods inventory structure
Orders and Customer demand are inputs to the structure given in Figure 12.1 from
the other parts of the perishable goods inventory model, and their equations will be given
when those structures will be presented. The equations of the stock variables are as follows
(note that “k” represents the ‘discrete’ time):
Inventory(k + 1) = Inventory(k ) + Acquisition flow − Sales − Perishing rate
(12.1)
Supply line(k + 1) = Supply line(k ) + Control flow − Acquisition flow
(12.2)
Initial values of the Inventory and the Supply line are assumed to be zero. Flow
equations are given as:
Acquisition flowk +TAD = Control flowk
(12.3)
Control flow = Orders
(12.4)
131
Perishing rate = Perishing fraction • Inventory (k )
(12.5)
Sales = MIN(Inventory(k ) + Acquisition flow − Perishing rate, Customer demand ) (12.6)
Equation (12.6) guarantees that Inventory is always non-negative and we do not sell
more than the Customer demand. The rest of the equations are as follows:
T AD = Acquisition delay time = 20 [weeks ]
(12.7)
Persihing fraction = 0.15 [1 / week ]
(12.8)

 Inventory(k ) + Acquisition flow  
, 0  (12.9)
Lost sales = MAX Customer demand − 
 − Perishing rate
 

Customer demand is a normal random variable. The structure and equations for
Customer demand are as follows:
Mean
Customer demand
W hite noise
Seed
Stdeviation
Figure 12.2. Customer demand structure
Customer demand = Mean + White noise
(12.10)
Mean = 200 [items ]
(12.11)
White noise = NORMAL (0, Stdeviation, Seed )
(12.12)
Stdeviation = 10 [items ]
(12.13)
Seed = 3
(12.14)
132
Expectations must be formed prior to calculating order decisions (Order quantity, ‘s’,
‘S’, Inventory position, In transit and Orders). Customer demand and Acquisition delay
time are inputs to the expectations formation structure.
Expected weekly demand
Expected annual demand
Number of weeks in a year
Customer demand
Mean absolute deviation
Deviation adjustment flow
Stdeviation for lead time
Expected stdeviation
Deviaiton adjustment time
Figure 12.3. Expectations formation structure
Equations of the expectation formation:
Expected annual demand =
(12.15)
Number of weeks in a year • Expected weekly demand (k )
Number of weeks in a year = 52 [weeks / year ]
Expected weekly demand (k + 1) = Expected weekly demand (k )
+ Expectation adjustment flow
(12.16)
(12.17)
Expectation adjustment flow =
Customer demand − Expected weekly demand (k )
(12.18)
Expectation adjustment time = 12 [weeks ]
(12.19)
Initial value of the Expected weekly demand is 200.
133
Equations of the deviation formation:
Stdeviation for lead time = Expected stdeviation • (T AD )0.5
(12.20)
Equation of Stdeviation for lead time comes from the sum of the variances of the
independently and identically distributed random variables (note that we assumed
independently and identically distributed normal customer demand). Expected stdeviation
can be found from the Mean absolute deviation by using the following formula given in
the literature (Montgomery and Johnson, 1976; Plossl, 1985):
Expected stdeviation = 1.25 • Mean absolute deviation(k )
Mean absolute deviation(k + 1) = Mean absolute deviation(k )
+ Deviation adjustment flow
(12.21)
(12.22)
Deviation adjustment flow =
 ABS(Customer demand − Expected weekly demand (k ))


 − Mean absolute deviation (k )

Deviation adjustment time
Deviation adjustment time = 40 [weeks ]
(12.23)
(12.24)
The Mean absolute deviation formulation is efficient and generally used in practice
instead of the standard deviation (Montgomery and Johnson, 1976). The formulation of
Mean absolute deviation is also suitable for computation during simulation. Initial value of
the Mean absolute deviation is 8.
The order decisions structure is given in Figure 12.4. Stock and flow equations of the
order decisions structure are as follows:
In transit (k + 1) = In transit (k ) + Inflow − Outflow
(12.25)
134
In-transit inventory equation is based on the definition of “outstanding orders”
(Axsäter, 2000) and definition of “on order” (Silver et al., 1998).
Inflow = Orders
(12.26)
Outflow = Acquisition flow
(12.27)
Initial value of the In transit is zero.
In transit
Acquisition flow
Outflow
Inflow
Inventory
Inventory position
Expected weekly demand
Orders
Perishing rate
s
Big S
Order quantity
Ordering cost
Expected annual demand
Safety stock
Inventory carrying cost
Perishing fraction
Inventory perishing cost
Safety factor
Unit cost
Stdeviation for lead time
Inventory storage cost
Weekly inventory
storage charge
Number of weeks in a year
Figure 12.4. Order decisions structure
The rest of the equations:
 Big S − Inventory position,
Orders = 
 zero,
for Inventory position < s 

otherwise

Inventory position = Inventory (k ) + In transit (k )
(12.28)
(12.29)
135
Big S = s + Order quantity
(12.30)
where ‘Big S’ is nothing but ‘S’ of the (s, S) rule (the simulation program does not let two
variables to have the same name). A simple common sense approach to modify the
standard s formula would be to add the Perishing rate (or the Expected perishing rate) to
the expected demand rate:
s = T AD • (Expected weekly demand (k ) + Perishing rate ) + Safety stock
(12.31)
We will discus pros and cons of this formulation, and possible modifications in the
following paragraphs. Expected weekly demand and Perishing rate were already given
respectively in Equation (12.17) and Equation (12.5). Safety stock follows as:
Safety stock = Safety factor • Stdeviation for lead time
(12.32)
where Stdeviation for lead time was given in Equation (12.20) and Safety factor is chosen
arbitrarily as:
Safety factor = 2 [dimensionless ]
(12.33)
The Order quantity given below in Equation (12.34) is not derived to be optimum in
any sense, but is just a reasonable approximate modification of the standard EOQ
(Economic Order Quantity) formula when there is a perishing cost (the same goes for our
definition of carrying cost in Equation (12.36) involving the perishing cost). These
approximations are fine for our purpose, since we are not focusing on exact profit
maximization in this thesis. Note that the Order quantity given below is not constant, but
changes as Expected annual demand changes (Equation (12.15)).
 2 • Ordering cost • Expected annual demand 

Order quantity = 
Inventory carrying cost


(12.34)
Ordering cost = 3000 [$ / order ]
(12.35)
136
Inventory carrying cost = Inventory perishing cost + Inventory storage cost (12.36)
Inventory perishing cost =
Number of weeks in a year • Perishing fraction • Unit cost
Inventory storage cost =
Weekly inventory storage charge • Number of weeks in a year • Unit cost
(12.37)
(12.38)
Unit cost = 12 [$ / item]
(12.39)
Weekly inventory storage charge = 0.05 [1 / week ]
(12.40)
We also add a structure to calculate the costs, the revenues and the profits:
Total purchasing cost
Total lost sales cost
Weekly purchasing cost
Acquisition flow
Unit cost
Weekly lost sales cost
Total ordering cost
Weekly ordering cost
Total inventory storage cost
Total cost
Stockout cost
Lost sales
Weekly inventory storage cost
Inventory
Ordering cost
Control flow
Unit selling price
Net total profit
Weekly inventory
storage charge
Unit cost
Total revenues
Total perishing cost
Weekly revenues
Sales
Perishing rate
Weekly perishing cost
Figure 12.5. Costs-revenues-profits structure
Stock equations of the costs-revenues-profits structure:
Total inventory storage cost (k + 1) = Total inventory storage cost (k )
+ Weekly inventory storage cost
(12.41)
137
Total lost sales cost (k + 1) = Total lost sales cost (k ) + Weekly lost sales cost
(12.42)
Total ordering cost (k + 1) = Total ordering cost (k ) + Weekly ordering cost
(12.43)
Total purchasingcost(k + 1) = Total purchasingcost(k ) + Weekly purchasingcost (12.44)
Total perishingcost(k + 1) = Total perishingcost(k ) + Weekly perishingcost
(12.45)
Total revenues(k + 1) = Total revenues(k ) + Weekly revenues
(12.46)
The initial values of the Total inventory shortage cost, the Total lost sales cost, the
Total ordering cost, the Total purchasing cost, the Total perishing cost and the Total
revenues are all zero. Flow equations are as follows:
Weekly inventory storage cost =
(12.47)
Weekly inventory storage charge • Unit cost • Inventory (k )
Weekly lost sales cost = Stockout cost • Lost sales
Ordering cost ,
Weekly ordering cost = 
0,
for Orders > 0

otherwise

(12.48)
(12.49)
Weekly purchasing cost = Unit cost • Acquisition flow
(12.50)
Weekly perishing cost = Unit cost • Perishing rate
(12.51)
Weekly revenues = Unit selling price • Sales
(12.52)
Rest of the equations:
Net total profit = Total revenues(k ) - Total cost
(12.53)
138
Total cost = Total inventory storage cost (k ) + Total lost sales cost (k )
+ Total ordering cost (k ) + Total purchasing cost (k )
(12.54)
Note that we did not add Total perishing cost to the Total cost formula, since it is
already implicit in the formula as a fraction of the Total purchasing cost, and influences
the Total revenues negatively since perished goods are not sold.
Stockout cost = 5 [$ / item ]
(12.55)
Unit selling price = 45 [$ / item ]
(12.56)
The perishable goods inventory model is now complete. The following section gives
the outputs of the model and its analysis.
12.1.1. Base Runs of the Perishable Goods Inventory Model
We choose a simulation duration of 8 years (416 weeks). The following are the
outputs of the complete perishable goods inventory model:
1: Customer demand
1:
2:
235
1:
2:
200
1:
2:
165
2
1
1
0.00
Page 1
2: Expected weekly demand
104.00
1
1
2
2
2
208.00
Week
312.00
19:20
416.00
Thu, Aug 21, 2003
Figure 12.6. Behaviors of Customer demand and Expected weekly demand
139
1: Stdev iation
2: Mean absolute dev iation
3: Expected stdev iation
13.00
1:
2:
3:
3
1:
2:
3:
1
10.00
3
3
1
1
3
1
2
2
2
2
1:
2:
3:
7.00
0.00
104.00
Page 2
208.00
Week
312.00
16:29
416.00
Thu, Aug 21, 2003
Figure 12.7. Behavior of Expected weekly deviation
1: Big S
1:
2:
3:
4:
1:
2:
3:
4:
2: s
3: Inventory position
4: Inventory
100000
50000
3
1:
2:
3:
4:
1
0
0.00
Page 1
2
4
104.00
208.00
Week
312.00
16:31
416.00
Thu, Aug 21, 200
Figure 12.8. Behaviors of Big S, s, Inventory position and Inventory (with small scale)
140
1: Big S
2: s
3: Inventory position
4: Inventory
100000000
1:
2:
3:
4:
1:
2:
3:
4:
50000000
3
4
1:
2:
3:
4:
0
1
0.00
2
3
4
1
104.00
2
3
1
2
4
208.00
Week
Page 2
312.00
16:31
416.00
Thu, Aug 21, 2003
Figure 12.9. Behaviors of Big S, s, Inventory position and Inventory (with large scale)
The behaviors of Big S, s, Inventory position and Inventory are unstable oscillations
shown in Figure 12.8 and Figure 12.9 (these two graphs are exactly the same; only their
scales are different to focus on the initial and the ending dynamics). This instability in the
system results in very high costs:
1: Total rev enues
1:
2:
3:
2: Net total prof it
3: Total cost
9.323e+010
3
1:
2:
3:
1:
2:
3:
0
-9.323e+010
Page 1
1
2
3
1
2
3
1
2
3
1
2
0.00
104.00
208.00
Week
312.00
16:31
416.00
Thu, Aug 21, 2003
Figure 12.10. Behaviors of Total revenues, Net total profit and Total cost: Bankruptcy
141
At the end of the 8 years the Total cost is too high (93,234,131,201 $) and the Total
revenues is relatively negligible (3,405,736 $) resulting in very high loss (93,230,725,465
$). The biggest portion of the Total cost is the Total purchasing cost (70,500,266,986 $),
and the second biggest portion is the Total inventory storage cost (22,733,768,007 $). The
Total ordering cost (60,000 $) and Total lost sales cost (36,208 $) are negligible. Very
huge portion of Total purchasing cost is the Total perishing cost (68,201,304,020).
1: Total purchasin… 2: Total ordering c… 3: Total lost sales … 4: Total inv entory … 5: Total perishing …
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
Page 2
7.05e+010
5
3.525e+010
4
0
1
0.00
2
3
4
5
1
104.00
2
3
4
5
1
208.00
Week
2
3
4
5
1
312.00
16:31
2
3
416.00
Thu, Aug 21, 2003
Figure 12.11. Behaviors of Total purchasing cost, Total ordering cost, Total lost sales
cost, Total inventory storage cost and Total perishing cost
The unstable fluctuations and the high costs result from using the Perishing rate as
part of the anchor in the formula of s (see Equation (12.31)). Using the Perishing rate in
the formula of s can give acceptable results when the Supply line is continuous and/or
when the Acquisition delay time (lead time) is short, but when Supply line is discrete and
the Acquisition delay time is long, Perishing rate must not be directly used. The simple
common sense formula that we try in Equation (12.31) works fine if Acquisition delay time
is short and/or the delay is continuous (see Figure 12.12 and Figure 12.13). These results
were more generally proven earlier in Chapter 6 and proper improved formulations were
derived. To prevent unstable oscillations under discrete and long delays, the formula of s
above must be re-formulated in light of those improvement suggestions.
142
1: Big S
1:
2:
3:
4:
2: s
3: Inv entory position
4: Inv entory
2000
1
1
1
1
3
3
3
1:
2:
3:
4:
2
1000
2
2
2
3
4
4
1:
2:
3:
4:
4
4
0
0.00
104.00
208.00
Week
Page 2
312.00
18:01
416.00
Thu, Aug 21, 2003
Figure 12.12. Stable behaviors of Big S, s, Inventory position and Inventory (with a short
TAD = 4)
1: Total rev enues
1:
2:
3:
3: Total cost
4000000
1
1:
2:
3:
2000000
1
2
2
1
2
1:
2:
3:
Page 1
0
1
0.00
2
3
3
3
3
104.00
208.00
Week
312.00
18:01
416.00
Thu, Aug 21, 2003
Figure 12.13. Satisfactory behaviors of Total purchasing cost, Total ordering cost, Total
lost sales cost, Total inventory storage cost and Total perishing cost (with a short TAD = 4)
143
12.1.2. Runs with Improved Formulations of s
The unstable behaviors seen in Figure 12.8 and Figure 12.9, and the high costs seen
in Figure 12.10 and Figure 12.11 can be eliminated by changing the formula of s that was
given in Equation (12.31). An improved formula may totally ignore Perishing rate, may
use Perishing rate after sufficiently smoothing it, or may use a formula similar to EVL
(Equilibrium value of loss) proposed in Section 6.5. As a matter of fact, Nahmias and
Wang derive an approximate cost-optimum algorithm very similar to adding the EVL to the
demand rate in obtaining the improved s formula (Nahmias and Wang, 1979). But we are
unable to adopt and test their approach in this thesis because they are unable to obtain a
closed formula for Order quantity and Safety stock. They suggest numerical approximation
procedures that are very difficult to apply in our simulation models. For benchmarking, we
start with a very simple s formula that totally ignores Perishing rate:
s = T AD • Expected weekly demand (k ) + Safety stock
(12.57)
The second alternative formula for s using Smoothed perishing rate is:
s = T AD • (Expected weekly demand(k ) + Smoothed perishing rate) + Safety stock (12.58)
where Smoothed perishing rate is given as:
Smoothed perishing rate(k + 1) = Smoothed perishing rate(k )
+
Perishing rate − Smoothed perishing rate(k ) (12.59)
Smoothing time
And the third improved formula for s using Average long run value of perishing rate
is:
 Expected weekly demand (k )

 + Safety stock
s = T AD • 
 + Average long run value of perishing rate 
where Average long run value of perishing rate is given as:
(12.60)
144
Average long run value of perishing rate =
(12.61)
Order quantity 

Perishing fraction •  Safety stock +

2


Although it is impossible to apply exactly the Nahmias and Wang algorithm, we used
their formula after simplification, for testing purpose. We assume a given constant Safety
stock level, and then adopt this simplified formula for discrete time (Nahmias and Wang
assume continuous time). After solving this simplified and adopted formula numerically,
we get constant values for Order quantity and Average long run value of perishing rate.
These values are very close to the dynamic values that we obtain with our proposed
formula in Equation (12.60) and Equation (12.61). The Inventory dynamics and cost
dynamics are very close to each other. They produce slightly better results with respect to
each other with different seeds. Although our aim was not optimization, our simple
straightforward formulas are producing very satisfactory results. Note that here we are not
presenting the adopted lengthy formulas to save space.
Runs for the three different improvement formulations of s are as follows:
1: Big S
1:
2:
3:
4:
2: s
1
1:
2:
3:
4:
1:
2:
3:
4:
1
1
2
3
2
3750
4: Inv entory
3
1
3
2
2
3
4
4
0
0.00
Page 2
7500
104.00
4
208.00
Week
4
312.00
19:17
416.00
Thu, Aug 21, 2003
Figure 12.14. Improved behaviors of Big S, s, Inventory position and Inventory obtained
with Equation (12.57)
145
1: Big S
1:
2:
3:
4:
2: s
4: Inv entory
7500
1
3
1
3
1
3
2
2
2
3
2
1
1:
2:
3:
4:
3750
4
1:
2:
3:
4:
4
4
0
0.00
104.00
208.00
Week
Page 2
4
312.00
19:19
416.00
Thu, Aug 21, 2003
with Equation (12.58) and TSm = 40
1: Big S
1:
2:
3:
4:
2: s
1
1
2
1:
2:
3:
4:
4: Inv entory
7500
3
1
2
3
1
3
2
2
3
3750
4
1:
2:
3:
4:
0
0.00
Page 2
4
4
4
104.00
208.00
Week
312.00
19:20
416.00
Thu, Aug 21, 2003
with Equation (12.60)
146
1: Total rev enues
1:
2:
3:
3: Total cost
3250000
1
1:
2:
3:
1625000
3
1
2
3
3
1
1:
2:
3:
3
0
1
0.00
2
2
2
104.00
208.00
Week
Page 1
312.00
19:17
416.00
Thu, Aug 21, 2003
obtained with Equation (12.57)
1: Total rev enues
1:
2:
3:
3: Total cost
3250000
1
1:
2:
3:
3
1625000
1
Page 1
3
0
1
0.00
2
3
1
1:
2:
3:
3
2
2
2
104.00
208.00
Week
312.00
19:19
416.00
Thu, Aug 21, 2003
obtained with Equation (12.58) and TSm = 40
147
1: Total rev enues
1:
2:
3:
3: Total cost
3250000
1
1:
2:
3:
1625000
3
1
3
3
1
1:
2:
3:
3
2
2
2
0 1
0.00
104.00
208.00
Week
Page 1
312.00
19:20
416.00
Thu, Aug 21, 2003
obtained with Equation (12.60)
1:
2:
3:
4:
5:
1400000
1:
2:
3:
4:
5:
1
700000
1
1
1:
2:
3:
4:
5:
0
Page 2
1
0.00
2
3
4
5
104.00
2
5
3
4
208.00
Week
5
2
3
2
5
3
4
312.00
19:17
4
416.00
Thu, Aug 21, 2003
Figure 12.20. Improved behaviors of Total purchasing cost, Total ordering cost, Total lost
sales cost and Total inventory storage cost obtained with Equation (12.57)
148
1:
2:
3:
4:
5:
1400000
1
1:
2:
3:
4:
5:
700000
1
5
5
1
1:
2:
3:
4:
5:
5
0
1
0.00
2
3
2
5
2
4
3
2
4
104.00
3
208.00
Week
Page 2
4
4
3
312.00
19:19
416.00
Thu, Aug 21, 2003
sales cost and Total inventory storage cost obtained with Equation (12.58) and TSm = 40
1:
2:
3:
4:
5:
1400000
1
1:
2:
3:
4:
5:
700000
1
1
1:
2:
3:
4:
5:
5
5
5
0 1
0.00
Page 2
2
3
4
104.00
2
3
4
2
3
208.00
Week
5
2
4
3
312.00
19:20
4
416.00
Thu, Aug 21, 2003
sales cost and Total inventory storage cost obtained with Equation (12.60)
149
Table 12.1. The final values of the Total revenues, the Net total Profit and the costs at the
end of simulation (8 years – 416 weeks), with four different s formulations
(12.57)
(12.58)
Total revenues
(12.31)
3,405,736 $
2,699,143 $
3,102,838 $
3,262,598 $
Net total profit
Equation of s
(12.60)
-93,230,725,465 $
1,227,367 $
1,165,433 $
1,461,121 $
Total cost
93,234,131,201 $
1,471,776 $
1,937,406 $
1,801,478 $
70,500,266,986 $
976,310 $
1,333,639 $
1,252,864 $
Total ordering cost
60,000 $
297,000 $
369,000 $
372,000 $
36,208 $
114,719 $
69,864 $
52,113 $
22,733,266,986 $
83,748 $
164,903 $
124,501 $
68,201,304,020 $
251,243 $
494,709 $
373,504 $
The outputs of the perishable goods inventory model using different improvements
for s are all stable (last three columns, Table 13.1). The improved formula for s in Equation
(12.60) that uses Average long run value of perishing rate (formulation similar to EVL) is
better than the others. All three are effective in eliminating unwanted oscillations and
reducing the undesirable costs, and are thus far superior to the simple common sense
formulation (first column, Table 13.1). The Equation (12.58) that uses the Smoothed
perishing rate necessitates special care. If the Smoothing time is not appropriate it may not
completely eliminate oscillations, so if we are not sure, we must consider using Equation
(12.60). Equation (12.57) is not suggested since it increases the lost sales and its Net total
profit is less compared to the others. Equation (12.57) serves as a benchmark for the other
formulations. It is practically the lower bound for the Net total profit.
Equation (12.60) is simple and very robust. We also made runs for this suggestion
with autocorrelated demand. Even with high Autocorrelation (0.98), the resulting behavior
is quite robust (see Figure 12.23, Figure 12.24, Figure 12.25, Figure 12.26, Figure 12.27
and Table 12.2). New Customer demand is defined to be a random variable with
autocorrelation (see Chapter 7 and Appendix D). The structure and equations for the new
Customer demand is as follows:
Pink noise
Stdeviation
Seed
Autocorrelation coefficient
Mean
Dummy adjustment flow
Customer demand
Figure 12.23. Autocorrelated customer demand structure
150
Customer demand = Mean + Pink noise
(12.62)
Mean = 200 [items ]
(12.63)
Pink noise(k + 1) = Pink noise(k ) + Dummy adjustment flow
(12.64)
Dummy adjustment flow = Autocorrelation coefficient • Pink noise (k )
(12.65)
+ NORMAL( 0 ,Stdeviation,Seed) − Pink noise (k )
Autocorrelation coefficient = 0.98
(12.66)
Stdeviation = 10 [items ]
(12.67)
Seed = 3
(12.68)
1: Customer demand
1:
2:
2: Expected weekly demand
310
2
1:
2:
200
2
1
2
1
1
2
1
1:
2:
90
0.00
Page 1
104.00
208.00
Week
312.00
20:09
416.00
Thu, Aug 21, 2003
Figure 12.24. Behaviors of autocorrelated Customer demand and Expected weekly demand
151
1: Big S
1:
2:
3:
4:
2: s
4: Inv entory
8500
1
1
2
3
2
1
2
3
1
1:
2:
3:
4:
3
4250
2
3
4
1:
2:
3:
4:
4
4
4
0
0.00
104.00
208.00
Week
Page 2
312.00
20:09
416.00
Thu, Aug 21, 2003
with Equation (12.60) and with autocorrelated Customer demand
1: Total rev enues
1:
2:
3:
3: Total cost
3250000
1
1:
2:
3:
1625000
3
1
3
2
3
1
1:
2:
3:
3
0
Page 1
1
0.00
2
2
104.00
208.00
Week
312.00
20:09
416.00
Thu, Aug 21, 2003
obtained with Equation (12.60) and with autocorrelated Customer demand
152
1:
2:
3:
4:
5:
1400000
1
1:
2:
3:
4:
5:
700000
1
5
1:
2:
3:
4:
5:
5
1
5
0
1
0.00
2
3
5
2
4
3
104.00
2
4
3
208.00
Week
Page 2
2
4
4
3
312.00
20:09
416.00
Thu, Aug 21, 2003
sales cost and Total inventory storage cost obtained with Equation (12.60) and with
autocorrelated Customer demand
end of simulation (8 years – 416 weeks), with pure random and with autocorrelated
Customer demand
Equation of s and Customer
demand pattern
(12.60) with pure random Customer
(12.60) with autocorrelated
demand
Customer demand
Total revenues
3,262,598 $
3,199,211 $
Net total profit
1,461,121 $
1,337,381 $
Total cost
1,801,478 $
1,861,831 $
1,252,864 $
1,284,474 $
Total ordering cost
372,000 $
369,000 $
52,113 $
64,574 $
124,501 $
143,784 $
373,504 $
431,351 $
Table 12.2 shows that the performance of Equation (12.60) is very robust even under
very highly autocorrelated demand. This is an important property of our decision rule.
153
12.2. Inventory Management with Unreliable Supply Line
In this section, we discuss our results in the context of inventory management with
an unreliable supply line. For this we build a model that we call “unreliable supply line
model”. The suppliers are assumed to adjust the orders that they receive from the inventory
manager, depending on their needs, so sometimes they supply less and sometimes more
than we order. This implies that there is some uncertainty in knowing the exact quantity in
the supply line, hence uncertainty in determining the inventory position. Note that,
unreliable suppliers is just an example that would cause uncertainty in the supply line;
there can be other causes such as accidents, breakdowns and strikes. We have shown in
Chapter 11 that using standard ordering rules in such uncertain and complex supply lines
can yield severe instabilities, and also bias in stock levels. We have also derived a proper
stock type Virtual supply line (VSLS) formulation that takes care of instability as well as
bias (Chapter 11). Our motivation in this chapter is to show that similar instabilities and
bias may exist with some inventory control rules, and then offer improved formulations. A
first order continuous supply line delay is assumed, so supply is continuous process
making it impossible to trace exactly each individual order. Unreliable supply line model
can be seen in Figure 12.28:
Supply line
Control flow
Adjusted orders
Inventory
Acquisition flow
Sales
Lost sales
Customer demand
Figure 12.28. Supply line and inventory structure for the unreliable supply line model
The equations of the stock variables are as follows:
Inventory(k + 1) = Inventory(k ) + Acquisition flow − Sales
(12.69)
Supply line(k + 1) = Supply line(k ) + Control flow − Acquisition flow
(12.70)
154
Initial values of the Inventory and the Supply line are assumed to be zero. Flow
equations are given as:
Supply line
T AD
(12.71)
Control flow = Adjusted orders
(12.72)
Sales = MIN(Inventory (k ), Customer demand )
(12.73)
Acquisition flow =
The rest of the equations:
T AD = Acquisition delay time = 8 [weeks ]
(12.74)
Lost sales = MAX(Customer demand − Inventory , 0 )
(12.75)
Adjusted orders and Customer demand are inputs to the structure given in Figure
12.28 from the other parts of the unreliable supply line model. To save space, we use
exactly the same customer demand and expectation formation structures that are given for
the perishable goods inventory model in Section 12.1 (see Figure 12.2 and Figure 12.3).
Costs-revenues-profits structure is also exactly the same as in Figure 12.5, but the value of
Unit selling price is assumed to be 24 [$/item] instead of 45 [$/item], which was given by
Equation (12.57). Also there is no perishing cost associated with this model. Order
decision structure is also similar with the structure given in Figure 12.4, but there are slight
modifications because Perishing rate does not exist in the unreliable supply line model.
Equation (12.37) is deleted. Equation (12.57) is used for s instead of Equation (12.31).
Equation (12.36) is changed and becomes:
Inventory carrying cost = Inventory storage cost
(12.76)
We assume that the adjustments that suppliers do on orders are random but
autocorrelated, so Adjusted orders is assumed to be a Normal random variable with
155
autocorrelation (see Chapter 7 and Appendix D). The structure and equations for Adjusted
orders is very similar to the autocorrelated customer orders structure and are given as
follows:
Pink noise 2
Autocorrelation coefficient 2
Dummy adjustment flow 2
Adjusted orders
Seed 2
Stdeviation 2
Orders
Figure 12.29. Autocorrelated noise structure for adjustment of orders
Orders + Pink noise 2 (k ),
Adjusted orders = 
0,
for Orders > 0 

otherwise

Pink noise2 (k + 1) = Pink noise2 (k ) + Dummy adjustment flow2
Dummy adjustment flow 2 = Autocorrel ation coefficien t 2 • Pink noise 2 (k )
+ NORMAL (0, Deviation 2 , Seed 2 ) − Pink noise 2 (k )
(12.77)
(12.78)
(12.79)
Autocorrelation coefficient 2 = 0.90 [dimensionless ]
(12.80)
Stdeviatio n 2 = 0.10 • Orders
(12.81)
Seed 2 = 9
(12.82)
Initial value of the Pink noise2 is zero.
The unreliable supply line model is now complete. The following section gives the
outputs of the model and its analysis.
156
12.2.1. Base Runs of the Unreliable Supply Line Model
We choose a simulation duration of 8 years (416 weeks) like in Section 12.1.
Customer demand, Expected weekly demand, Stdeviation, Mean absolute deviation and
Expected stdeviation are exactly the same as given in Figure 12.6 and Figure 12.7, so they
are not repeated here. The rest of the outputs are as follows:
1: Big S
1:
2:
3:
4:
2: s
4: Inv entory
3650
1
1
1
1
3
1:
2:
3:
4:
1:
2:
3:
4:
1825
2
2
4
0
0.00
3
2
Page 1
2
4
4
104.00
3
208.00
Week
3
4
312.00
22:09
416.00
Thu, Aug 21, 2003
Figure 12.30. Behaviors of Big S, s, Inventory position and Inventory for the unreliable
supply line model
Inventory becomes depleted since Inventory position is perceived to be high, but
actually it is not. This is caused by the error in In-transit inventory due to noise in the
supply line (see Figure 12.31). Note that In-transit inventory formula given in Equation
(12.25) is based on the definition of “outstanding orders” in (Axsäter, 2000) and definition
of “on order” in (Silver et al., 1998). For healthy decisions, In-transit inventory and actual
Supply line must be (approximately) equal to each other, but because the supply line is
unreliable, we do not receive exactly what we order. This may result in very high costs (see
Figure 12.32). Note that, for other values of noise seed, Seed2, Inventory position may be
perceived to be lower than the actual.
157
1: Supply line
1:
2:
4000
1:
2:
2000
2: In transit
2
1
1:
2:
2
2
2
1
0
0.00
1
208.00
Week
104.00
Page 2
1
312.00
22:09
416.00
Thu, Aug 21, 2003
Figure 12.31. Behaviors of the actual Supply line and the perceived In transit for the
unreliable supply line model
1: Total rev enues
1:
2:
3:
1:
2:
3:
3: Total cost
2000000
914000
3
1
3
1:
2:
3:
1
-172000
0.00
3
2
2
104.00
Page 1
3
1
1
2
208.00
Week
2
312.00
22:09
416.00
Thu, Aug 21, 2003
Figure 12.32. Behaviors of Total revenues, Net total profit and Total cost for the
unreliable supply line model
At the end of the 8 years, Total cost (598,263 $) is more than Total revenues
(426,554 $) resulting in loss (171,709 $).
158
1: Total purchasing cost
2: Total ordering cost
3: Total lost sales cost
4: Total inv entory stora…
1000000
1:
2:
3:
4:
1:
2:
3:
4:
500000
3
1
1:
2:
3:
4:
0
1
0.00
Page 2
2
3
1
2
4
104.00
3
3
2
4
208.00
Week
1
2
4
312.00
22:09
4
416.00
Thu, Aug 21, 2003
Figure 12.33. Behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost
and Total inventory storage cost for the unreliable supply line model
The bias between perceived In-transit inventory and the actual Supply line can be
minimized as discussed earlier in Chapter 11, and proposed in the following section.
12.2.2. Runs with Improved Formulations of In-Transit Inventory
We propose two different formulations for In transit inventory. There are two flows
of the In transit inventory; Inflow and Outflow. The formulation of the Inflow that is given
in Equation (12.26) remains unchanged. One of the proposed formula is taken directly
from Chapter 11. It adds a new flow to the In-transit inventory stock that continuously
adjusts it towards its desired equilibrium level (see Section 11.3 and note that the concepts
of Virtual supply line as a stock and In-transit inventory play the same role). The second
improvement proposal is new, not discussed before. It does not change In-transit inventory
formulation and only modifies the original Outflow. For the first improvement suggestion,
the updated In-transit inventory and the new flow formulations are as follows:
In transit(k +1) = In transit(k ) + Inflow − Outflow+ In transit adjustment flow
(12.83)
In transit adjustment flow = (Desired in transit − In transit(k )) / In transit adjustmenttime (12.84)
159
where
Desired in transit = Acquisition delay time • Expected weekly demand (k )
(12.85)
In transit adjustment time = 3 [weeks ]
(12.86)
We choose In transit adjustment time to be 3 weeks, after several experimental runs
that show that for inventory management systems with (s, S) rule, short In transit
adjustment time gives better results.
For the second improvement, updated Outflow formulation is given as follows:
Outflow =
In transit (k )
T AD
(12.87)
The formula in the second improvement suggestion is simpler and does not
necessitate a parameter selection. Both of the formulas have the effect of driving the
estimated In-transit stock towards its desired equilibrium level. The first proposal does this
explicitly while the second one does it implicitly (when computed, the equilibrium level of
In-transit also turns out to be Acquisition delay time•Expected weekly demand). These two
alternative formulations may be superior to each other for different values of parameters.
Second case is especially preferred if the Ordering cost is low. The first case creates an
artificial gap between In-transit inventory and Supply line, when Ordering cost is low.
This is because the ordering period becomes shorter, and even though we do not order
enough in these short periods we perceive that the In-transit is full. This further cause us to
order less and again the In-transit is full, since we artificially force it to have a value,
which is its desired value. So, the first formulation given in Equation (12.83) and Equation
(12.84) is not adequate for the inventory management systems when orders are small. The
second formulation is robust with all parameter values. The second formulation given in
Equation (12.87) can also be used in the stock type Virtual supply line (VSLS) discussed in
Chapter 11.
160
Runs for the different new formulations are as follows:
1: Big S
1:
2:
3:
4:
2: s
4: Inv entory
3650
1
1:
2:
3:
4:
1825
1:
2:
3:
4:
1
2
3
1
3
2
1
2
3
3
2
4
4
4
4
0
0.00
104.00
208.00
Week
Page 1
312.00
23:16
416.00
Thu, Aug 21, 2003
Figure 12.34. Behaviors of Big S, s, Inventory position and Inventory for improvement
Equation (12.83), (12.84) and (12.27)
1: Big S
1:
2:
3:
4:
2: s
4: Inv entory
3650
1
1
1
1
3
3
3
3
1:
2:
3:
4:
1825
2
2
2
2
4
4
1:
2:
3:
4:
0
0.00
Page 1
4
4
104.00
208.00
Week
312.00
22:11
416.00
Thu, Aug 21, 2003
Figure 12.35. Behaviors of Big S, s, Inventory position and Inventory for improvement
Equation (12.25) and (12.87)
Inventory and Inventory position are naturally changing over time in two figures
(Figure 12.34 and Figure 12.35). The major improvement is that the big difference between
161
In-transit inventory and actual Supply line is greatly reduced (see Figure 12.36 and Figure
12.37).
1: Supply line
2: In transit
4000
1:
2:
2
1:
2:
1
2000
1
1
2
2
2
1
1:
2:
0
0.00
104.00
208.00
Week
Page 2
312.00
23:16
416.00
Thu, Aug 21, 2003
Figure 12.36. Behaviors of Supply line and In transit for improvement Equation (12.83),
(12.84) and (12.27)
1: Supply line
1:
2:
4000
1:
2:
2000
2: In transit
1
1
1:
2:
1
2
1
2
2
0
0.00
Page 2
2
104.00
208.00
Week
312.00
22:11
416.00
Thu, Aug 21, 2003
Figure 12.37. Behaviors of Supply line and In transit for improvement Equation (12.25)
and (12.87)
162
1: Total rev enues
1:
2:
3:
3: Total cost
2000000
1
3
1
1:
2:
3:
914000
3
1
1:
2:
3:
-172000
2
2
3
1
2
3
2
0.00
104.00
208.00
Week
Page 1
312.00
23:16
416.00
Thu, Aug 21, 2003
Figure 12.38. Behaviors of Total revenues, Net total profit and Total cost for improvement
Equation (12.83), (12.84) and (12.27)
1: Total rev enues
1:
2:
3:
3: Total cost
2000000
1
3
1:
2:
3:
1
914000
3
1
Page 1
1
-172000
0.00
2
2
3
1:
2:
3:
2
3
2
104.00
208.00
Week
312.00
22:11
416.00
Thu, Aug 21, 2003
Figure 12.39. Behaviors of Total revenues, Net total profit and Total cost for improvement
Equation (12.25) and (12.87)
163
1:
2:
3:
4:
1000000
1
1:
2:
3:
4:
1
500000
1
1:
2:
3:
4:
0
1
2
3
2
4
0.00
2
2
3
4
104.00
3
3
208.00
Week
Page 2
4
4
312.00
23:16
416.00
Thu, Aug 21, 2003
and Total inventory storage cost for improvement Equation (12.83), (12.84) and (12.27)
1:
2:
3:
4:
1000000
1
1:
2:
3:
4:
1
500000
1
1:
2:
3:
4:
0
1
0.00
Page 2
2
3
2
4
3
104.00
2
2
4
3
208.00
Week
4
4
3
312.00
22:11
416.00
Thu, Aug 21, 2003
and Total inventory storage cost for improvement Equation (12.25) and (12.87)
164
end of simulation (8 years – 416 weeks), with three different formulations
Equations of In-
(12.25) & (12.27)
(12.83), (12.84) & (12.27)
(12.25) & (12.87)
transit stock
Total revenues
426,554 $
1,978,302 $
1,966,311 $
Net total profit
-171,709 $
681,346 $
654,120 $
Total cost
598,263 $
1,296,956 $
1,312,191 $
213,277 $
996,102 $
996,346 $
Total ordering cost
36,000 $
177,000 $
171,000 $
325,758 $
2,477 $
4,975 $
23,228 $
121,377 $
139,870 $
Total inventory
storage cost
In Table 12.3, the results of the runs for different formulations of In-transit (and
inventory position) are summarized. The equations that are changed from run to run are
given as a reference. The standard formulation of In-transit uses Equation (12.25) and
Equation (12.27) that cannot manage the unreliable supplier case effectively, incurring
huge lost sales (first column, Table 13.2). The two proposed formulations yield quite
satisfactory results. Although their costs are higher than the standard case, their revenues
are even higher, generating much higher positive Net total profit. The formulation using
Equation (12.83), Equation (12.84) and Equation (12.27) comes directly from Chapter 11.
The formulation using Equation (12.25) and Equation (12.87) is derived in this section.
The outputs of these two improvement formulations are very similar to each other (last two
columns, Table 13.2). Depending on the parameter values, one of them may become
superior to the other, but after many runs, we can say that second improvement formulation
(Equation (12.25) and Equation (12.87)) is superior to the first one (Equation (12.83),
Equation (12.84) and Equation (12.27)) in the sense that it is more robust (insensitive to
the parameter values). We can state in general that they are both very successful in
managing the uncertainty caused by unreliable supply lines (comparing the last two
columns of Table 13.2 with the first column).
165
13. DYNAMICS OF GOAL SETTING
Goal setting is crucial for stock management systems. There must be a base for the
decisions and the control actions. In a stock management system, the results (the levels of
the stocks) of the decisions (flows) are evaluated against their goals and further the
effectiveness of the goals themselves can/must be evaluated (see Figure 4.1).
13.1. Simple Goal Structures
In systems that have simple goal structure there is just one single, well defined goal
for each stock that is managed. Goal may be externally determined, or set by decision
maker, but once it is determined, it is not challenged by the other internal structures of the
system. We can say that goals in simple goal structures do not “erode”. These type of goals
can be seen mostly in short-term decision systems. Goals in the previous chapters of the
thesis, fall in this category. One may further divide this category into two sub-categories:
13.1.1. Goal as an External Variable
The simplest form of a goal is when it is externally determined. In this case goal
formation is beyond the scope of the model. The given goal is assumed as “optimum”, and
the modeler only focuses on bringing the Stock to this desired level (goal), with a stable
and quick response. This kind of goal was seen in the previous chapters. For example, the
goal of Stock (S), that is Desired stock (S*) was externally given as zero in Chapter 5, and it
was given as zero initially and is increased to one at time five in Chapter 6.
13.1.2. Goal as an Internal System Variable
When goal is an internal factor it must be set so that it improves the system
performance. Either the goal is set to an optimum, or its formula, that is the way the goal is
formed, is determined. As soon as the goal or its formula is specified, the goal functions
just like in the external goal case. The goal is assumed to be the optimum level, and the
decision maker focuses on bringing the Stock to this desired level (goal). This kind of a
166
goal was also seen in the previous chapters. For example, the goal of Supply line (SLS),
that is Desired supply line (SLS*), is determined internally by multiplying Acquisition
delay time (TAD) with Loss flow (LF) or with Equilibrium value of loss (EVL) in Chapter 6,
and the goal of Secondary stock (SS), that is Desired secondary stock (SS*), is determined
by dividing the Desired control flow (CF*) with Productivity coefficient (CP).
13.2. Problematic Goal Structures
In systems that have problematic goal structures, there may be one or more
endogenously system created, unintentional, implicit goals that system seeks instead of the
explicitly set goal. This type of goal may “erode” if there is a belief in the system that to
reach the set goal is hard or impossible. “People find the tension created by unfulfilled
goals uncomfortable and often erode their goals to reduce cognitive dissonance” (Senge,
1990; Sterman, 2000). In such a control system, not only the size of the applied control, but
also motivation must be considered as a factor. The human element in the system must be
motivated accordingly to prevent goal erosion. When motivation is ignored even control
actions themselves may result in conflicting pressures that force the Stock in the opposite
direction. This type of goal can be seen in long-term decision structures.
13.2.1. Capacity Limit on Improvement Rate
In problematic goal structures there can also be a capacity limitation on the size of
the applied control. This limitation may prevent Stock from being adjusted fast enough
towards its goal. Furthermore, it may create frustration in the human element in the system.
This capacity is utilized fully when desired control is higher than the capacity, and it is
utilized as much as needed when desired control is less than the capacity. Utilization is not
only affected by the desired control but it is also affected by the motivation. If there is a
belief that it is possible to reach the goal, the effect is equal to one, otherwise it is less than
one depending on the level of the de-motivation. Utilization (U) can be given as
multiplication of two effects; Effect of motivation (EM) and Effect of desired control flow
(ECF*):
167
U = E M • ECF *
(13.1)
For the base run and for the simple goal erosion models we set Effect of motivation
(EM) to one. Later EM is going to be defined as a graphical function for more complicated
models. Effect of desired control flow (ECF*) can be given as a function of Desired control
flow (CF*) and Capacity (CAP) ratio:
ECF *
 CF * 

= f
 CAP 


(13.2)
1: Ef f ect of desired CF
1:
1
1:
1
1:
0
1
1
1
Page 1
1
-0.50
0.00
0.50
Desired control f low/Capacity
1.00
9:54
1.50
Tue, Jun 10, 2003
Figure 13.1. Graphical function for Effect of desired control flow (ECF*)
We assume an outflow that represents the contradictory forces that are trying to
decrease the level of the Stock. This outflow is estimated with a delay in the system.
Firstly, the base model that does not have goal erosion is going to be sketched. The
performance of the structures that have goal erosion can be compared with the
performance of the base run obtained from the following model:
168
Max loss rate
Stock
Loss flow
Control flow
Life time
Utilization
~
Capacity
Effect of motivation
Expected loss
Effect of desired CF
Stock adjustment
Ideal goal
Stated goal
Figure 13.2. Model with capacity limitation, but without goal erosion
Note that Expected loss (ELS) is standard first order smoothing of Loss flow (LF)
which is proportional to the Stock (S). Loss flow and the Expected loss are not critical for
the dynamics generated in Figure 13.3 and they are not seen in the simple models of goal
erosion in literature. We keep Loss flow and its expectation since they will play important
role in the dynamics after Section 13.2.4, as will be explained later.
The model equations can be given as:
•
S = CF − LF
•
ELS = EAF =
LF − ELS
TEL
(13.3)
(13.4)
CF = U • CAP
(13.5)
CF * = ELS + SA
(13.6)
169
SA =
SG − S
TSA

 S
, MLR 
LF = MIN

 TLf


(13.7)
(13.8)
The model parameter values are:
• CAP = 12
• EM = 1
• TLf = 70
• MLR = 10
• TEL = 2
• TSA = 4
• G = 1000
• SG = G = 1000
• S (0 ) = 100
The Stated goal (SG) that is set by the top management, is assumed to be equal to the
Ideal goal (G) that is believed to be the best goal for the system (for this base model we
omit the Implicit goal that is the endogenously system created internal goal of the system).
We choose Maximum loss rate (MLR) to be less than Capacity (CAP), otherwise it is
not possible to fulfill the Ideal goal (G), which is not an interesting case.
Additional equations, additional values of parameters, any changes in the structure
and any changes in the values will be provided whenever necessary.
The model in Figure 13.2 with the above equations and settings produce the
following output:
170
1: Ideal goal
1:
2:
3:
1000
2: Stated goal
1
2
1
2
3: Stock
1
2
3
1
2
3
3
1:
2:
3:
1:
2:
3:
Page 1
500
0
3
0.00
100.00
200.00
Time
300.00
18:20
400.00
Mon, Jun 16, 2003
Figure 13.3. Output of the model with stock adjustment (improvement rate) limitation
The key feature of this model is that the improvement rate of the system doe not (can
not) automatically increase in proportion to the discrepancy (SG-S), because there is
capacity limit on the improvement rate. The critical role of this limit on potential
frustration caused by unrealistically high goals will be seen in the following sections.
13.2.2. Simple Goal Erosion and Traditional Performance
We call, the internally created goal Implicit Goal (IGS). This goal is unintentional
and mostly unknown by the top management. In some cases, decision maker may be aware
of this goal, and treat it as the short term goal. Whether it is known or not known by the
decision maker, Implicit Goal is the reason for the goal erosion. System will create, erode
and follow this intermediate goal, thinking that the Stated goal (SG) is too high to satisfy.
Though there may be hope to catch the Stated goal in the future, as system starts following
the Implicit Goal, it may forget the Stated goal entirely.
In the simple goal erosion model the system (Stock) seeks its goal that is the Implicit
goal, and the goal in turn seeks the Stock (see Appendix C.9 and Sterman, 2000). We
assume that the Implicit goal is equal to the Stated goal initially, since there is no other
thing to challenge the Stated goal. But later system adjusts its goal (Implicit goal) towards
171
the Stock as the actual achievement (Stock) creates a stronger belief, and the Stated goal is
forgotten after some time.
Max loss rate
Stock
Loss flow
Control flow
Life time
Utilization
~
Capacity
Expected loss
Implicit goal
Stock adjustment
Ideal goal
Stated goal
Goal adjustment flow
Figure 13.4. Simple eroding goal structure
The only addition is the Implicit goal (IGS) and only modification is the change in
the formula of Stock adjustment (SA):
•
IGS = GAF =
SA =
S − IGS
TGA
IGS − S
TSA
One parameter and one initial value are also needed:
• TGA = 14
• IGS (0 ) = SG = 1000
(13.9)
(13.10)
172
1: Ideal goal
1:
2:
3:
4:
1:
2:
3:
4:
1000
2: Stated goal
1
2
1
3: Implicit goal
2
1
4: Stock
2
1
2
3
500
3
4
3
4
3
4
4
1:
2:
3:
4:
0
0.00
25.00
Page 1
50.00
Time
75.00
18:37
100.00
Mon, Jun 16, 2003
Figure 13.5. Behavior of simple eroding goal structure
If parameter settings are different, the mid-point that the Implicit goal and the Stock
meet may change.
Note that it is oversimplification to assume that Implicit goal (IGS) seeks the Stock. It
is better to state that Implicit goal tends toward the past performance of the system.
Organizations and individual people are highly affected from their past achievements in
determining their goals. The past performance (or past achievements) can be called
“Traditional Performance” (Forrester, 1975).
A little more complicated model for simple goal erosion may be obtained by
modifying the model in Figure 13.4. Traditional performance is added as a first order
delayed version of stock. We furthermore assume Implicit goal (IGS) to be a third order
information delay instead of first order. This small change does not affect the general
behavior, but it creates a more realistic initial behavior for the Implicit goal. An immediate
fall in Implicit goal (IGS) is unrealistic, and with this change, Implicit goal holds at least
for a short time before falling. The modified model for simple goal erosion can be given
as:
173
Max loss rate
Stock
Loss flow
Control flow
Life time
Utilization
~
Capacity
Expected loss
Implicit goal
Stock adjustment
Ideal goal
Stated goal
Inf Delay for IG 2
Inf Delay for IG 1
Information AF 2
Traditional performance
Traditional
performance formation
Information AF 1
formation time
Figure 13.6. Simple eroding goal structure with Traditional Performance
In this model, the Stock (S) seeks the Implicit goal (IGS), the Implicit goal seeks the
Traditional performance and the Traditional performance (TPS) seeks the Stock.
Instead of Equation (13.9), we have the following set of differential equations:
174
 TPS − IDFIGS1

•
 

 

TGA / 3
 IDFIGS 1   IAF 1  

 

 

 

 

•
  IDFIGS1 − IDFIGS 2 

 
 IDFIGS 2  =  IAF 2  = 

TGA / 3
 

 

 

 

 
 •
 

IGS
−
IDFIGS
IGS
GAF
 
2

 


 
 
TGA / 3


(13.11)
Also the differential equations of the Traditional Performance (TPS) is:
•
TPS = TPF =
S − TPS
TTPF
(13.12)
Additional parameter and initial value are given below:
• TTPF = 40
• TPS (0 ) = S
1: Ideal goal
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
1000
2: Stated goal
1
2
3: Implicit goal
1
4: Stock
2
1
5: Traditional perf o…
2
1
2
3
500
4
1:
2:
3:
4:
5:
Page 1
5
0
0.00
25.00
3
4
5
50.00
Time
3
4
5
75.00
9:49
3
4
5
100.00
Wed, Jul 02, 2003
Figure 13.7. Behavior of eroding goal structure with Traditional Performance
175
The addition of the Traditional Performance to the system changes the behavior in
such a manner that the Stock and its goal (that is the Implicit goal) may cross each other,
and Implicit goal further erodes. This is because the Traditional Performance is a longterm delayed average of the Stock, which itself determines the Implicit goal after a third
order delay.
13.2.3. Goal Erosion and Recovery
Both behaviors in Figure 13.3 and Figure 13.7 are two extremes; in the first one there
is no goal erosion, and in the second one the Implicit goal completely ignores the Stated
goal. In a more complex model, Implicit goal (IGS) would be a weighted average of Stated
goal (SG) and Traditional Performance (TPS) (Forrester, 1975; Sterman, 2000).
The weighting factor can be affected by many factors like the power of the
leadership of the top management. We call this weight as Weight of stated goal (WSG). If
the leadership of the top management is charismatic enough the system responds faster,
and the system may recover from goal erosion. Simplest case is when the Weight of stated
goal is assumed to be constant.
For this model, the first equation of the equation set for Implicit goal (IGS) in
Equation (13.11) is modified to include Indicated goal (ING):
•
IDFIGS 1 = IAF 1 =
ING − IDFIGS1
TGA / 3
ING = WSG • SG + (1 − WSG ) • TPS
The value of the Weight of stated goal (WSG) is assumed as:
• WSG = 0.3
(13.13)
(13.14)
176
Note that the following model would create same behavior as the model in Figure
13.2 when Weight of stated goal is one, and same behavior as the model in Figure 13.6
when Weight of stated goal is zero:
Max loss rate
Stock
Loss flow
Control flow
Life time
Utilization
~
Capacity
Expected loss
Stock adjustment
Implicit goal
Inf Delay for IG 2
Inf Delay for IG 1
Traditional
Information AF 2
Information AF 1
formation time
Indicated goal
Stated goal
Ideal goal
Figure 13.8. A general model of goal erosion and recovery
This model first creates a goal erosion and then slow recovery as the Traditional
Performance (TPS) improves. Amount of the erosion and the speed of the recovery
depends on the value of the Weight of stated goal (WSG). When it is high, the erosion is
less and recovery is faster, and when it is low, the erosion is more and the recovery is
slower. But note that in this model eventually the system would always recover toward its
177
goal, unless the Weight of stated goal (WSG) is zero. This may not be realistic, as will be
discussed in the following section.
1: Ideal goal
1:
2:
3:
4:
5:
1000
2: Stated goal
1
2
3: Implicit goal
1
1
1:
2:
3:
4:
5:
2
1
2
3
3
3
1:
2:
3:
4:
5:
4: Stock
2
4
5
5
5
4
500
0
4
4
3
0.00
Page 1
5
125.00
250.00
Time
375.00
10:36
500.00
Wed, Jul 02, 2003
Figure 13.9. Behavior of goal erosion and recovery model
13.2.4. Goal Erosion, Possible Recovery and Time Limits
Now we consider a more realistic case; where there is a time limit. In most real
problems, not only reaching the goal, but reaching it within a time horizon is important.
We assume that system evaluates possibility to reach to the Stated goal (SG) in the
remaining period of time. If it is possible, the Effect of motivation is one, otherwise it is
less than one. When system feels that it will not be able to fulfill its goals, first it slows
down, and then it gradually stops its efforts, simulating the “giving up” phenomenon. Time
limit and motivation effect play role in making the Utilization (U) to be very low. When
Utilization (U) is low enough, Control flow (CF) may fall below Loss flow (LF) resulting
in decrease in Stock (S) level.
There are two time periods in the system. One is Time horizon (THS) and the other
one is Short time horizon (TSH). Time horizon is the time left to fulfill the Stated goal.
Short time horizon is the time representing the time period that is perceived as a “short”
178
time by the system, so once system feels that it is possible to catch the Stated goal in this
short time period, it does not give up (even if the actual time horizon is passed).
Max loss rate
Stock
Loss flow
Control flow
Life time
Utilization
~
Capacity
Expected loss
~
Stock adjustment
Implicit goal
Traditional
formation time
Indicated goal
Stated goal
Ideal goal
Likelihood of
accomplishment ratio
Short time horizon
Time horizon
Time decrease
Time constant
Figure 13.10. Model with goal erosion, possible recovery and time limits
179
We did not change the structure for Implicit goal (IGS), but to save modeling space a
macro formulation is used:
IGS = SMTH3( ING, TGA , SG )
(13.15)
In the above equation, “SMTH3” represents a third order information delay.
Indicated goal (ING) is the input of the delay structure, Goal adjustment time (TGA) is the
delay duration, and Stated goal (SG) is the initial value of the stocks of the delay structure,
so Equation (13.15) is exactly the same as the Equation set (13.11) combined with
Equation (13.13).
The Effect of motivation (EM) is a function of Likelihood of accomplishment ratio
(RLA):
E M = f (R LA )
(13.16)
1: Ef f ect of motiv ation
1:
1.00
1:
0.50
1
1
1
1:
Page 1
0.00
0.00
0.75
1.50
Ratio_of _likelihood_of _accompli
1
2.25
12:33
3.00
Tue, Jun 17, 2003
Figure 13.11. Graphical function of Effect of motivation (EM)
where Likelihood of accomplishment ratio (RLA) is:
180
R LA
 SG − PP

=  CAP − ELS

THS








(13.17)
and Perceived performance (PP) is a delayed version of Stock (S):
PP = SMTH3(S , TFP )
(13.18)
“SMTH3” represents a third order information delay just like in Equation (13.15).
Stock (S) is the input of the delay structure, and Formation of perception time (TFP) is the
delay duration. Thus, the ratio:
SG − PP
CAP − ELS
(13.19)
is an estimation of how long it would take to reach Stated goal (SG), if we improve at full
Capacity (CAP). And when we normalize this by dividing by Time horizon (THS), we
obtain Likelihood of accomplishment ratio (RLA), which determines the effect of time
horizon on motivation.
Differential equation of Time horizon (THS) is assumed to exponentially decaying
towards the Short time horizon (TSH):
•
THS = −TDF =
TSH − THS
TC
(13.20)
The above equation means that the time horizon to reach the goal gradually decays,
but never goes below some “short” time horizon.
Additional parameters and initial value are:
• TFP = 10
181
• TSH = 20
• TC = 200
• THS (0 ) = TC = 200
We assume that the Effect of motivation (EM) has an effect on Weight of stated goal
(WSG) where the equation can be given as a constant (that is in between zero and one)
times EM:
WSG = 0.3 • E M
(13.21)
In Figure 13.12, there are three phases; first, there is goal erosion until about time 32,
then the system and the implicit goal together improve until about time 175, and after that
point there is the frustration dynamics caused by passage of too much time.
1: Ideal goal
1:
2:
3:
4:
5:
1000
2: Stated goal
1
2
3: Implicit goal
1
4: Stock
2
1
2
1
2
3
1:
2:
3:
4:
5:
4
5
3
4
500
3
5
4
5
3
1:
2:
3:
4:
5:
Page 1
4
0
0.00
125.00
250.00
Time
375.00
11:00
5
500.00
Wed, Jul 02, 2003
Figure 13.12. Behavior for goal erosion and possible recovery with time limits
As the Time to reach to stated goal (THS) decays, the Likelihood of accomplishment
ratio (RLA) increases and de-motivates the system, so the Effect of motivation (EM) falls.
This results in giving up the improvement efforts. If Stated goal (SG) was sufficiently
lower than the Ideal goal (G) then system could recover. For example consider the run
182
below. Counter-intuitively keeping Stated goal low may increase the performance of the
system. This is because of the increased motivation effect (in Section 13.2.6, we will
introduce an advanced model in which the management is first able to lower the Stated
goal (SG), build up motivation and then gradually increase it towards the ideal goal).
1: Time horizon
200
4.00
1.00
1:
2:
3:
2: Likelihood of accomplishment … 3: Ef f ect of motiv ation
3
1
3
1
100
2.00
0.50
1:
2:
3:
2
2
1
2
3
1
2
0
0.00
0.00
1:
2:
3:
3
0.00
125.00
250.00
Time
Page 1
375.00
11:00
500.00
Wed, Jul 02, 2003
Figure 13.13. Behaviors of Time horizon, Likelihood of accomplishment ratio, and Effect
of motivation
1: Ideal goal
1:
2:
3:
4:
5:
1000
2: Stated goal
1
3: Implicit goal
4: Stock
1
2
2
1
2
2
3
3
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
Page 1
500
3
0
0.00
4
1
4
4
3
4
5
5
5
5
125.00
250.00
Time
375.00
12:28
500.00
Wed, Jul 02, 2003
Figure 13.14. Behavior of system when the Stated goal is sufficiently low (equal to 850)
183
1: Time horizon
1:
2:
3:
200
4.00
1.00
1:
2:
3:
100
2.00
0.50
2: Likelihood of accomplishment … 3: Ef f ect of motiv ation
3
1
3
3
3
1
1
2
1
2
1:
2:
3:
Page 1
0
0.00
0.00
2
2
0.00
125.00
250.00
Time
375.00
12:28
500.00
Wed, Jul 02, 2003
Figure 13.15. Behavior of EM when the Stated goal is sufficiently low (equal to 850)
13.2.5. Implicit Goal Setting: Short Term Motivation Effect on Weight of Stated Goal
In the previous section, it is assumed that the Weight of stated goal depends only on
(de)motivation caused by time horizon pressure. Even when there is enough time, Implicit
goal may seek Traditional performance caused by low values of Weight of stated goal, so
we assume that Weight of stated goal depends also on another (de)motivation effect (Effect
of short term motivation). Modified equation of Weight of stated goal is as follows:
WSG = E M • E STM
(13.22)
where ESTM represents Effect of short term motivation.
There may be several factors affecting this second type motivation. Factors like
frustration (created by unfulfilled goals), or exhaustion (created by the efforts that aim to
close the gap between Stock and the goal) make Weight of stated goal to fall, and
motivation (created by being close enough to the goal), or shared vision (constant
projection of goal) make it rise (Senge, 1990). There may be many other factors, but here
we define Effect of short term motivation such that it depends on the possibility of
accomplishing the Stated goal (SG) in the Short time horizon (TSH):
184
1: Ef f ect of short term motiv ation
1:
1.00
1:
0.50
1
1
1
1:
0.00
0.00
1.00
Page 1
2.00
1
3.00
17:19
4.00
Fri, Jul 18, 2003
Figure 13.16. Graphical function for Effect of short term motivation (ESTM)
where Short term accomplishment ratio (RST) is:
RST
 SG − S

=  CAP − ELS
TSH









(13.23)
where the ratio:
SG − S
CAP − ELS
(13.24)
is an estimation of how long it would take to reach Stated goal (SG), if we improve at full
Capacity (CAP). And when we normalize this by dividing by Short time horizon (TSH), we
obtain Short term accomplishment ratio (RST), which determines the effect of short time
horizon on motivation. The final form of Effect of short term motivation (ESTM), Short term
accomplishment ratio (RST) and Weight of stated goal (WSG) can be seen in Figure 13.23.
With these latest additions, the behavior becomes:
185
1: Ideal goal
1000
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
2: Stated goal
1
2
3: Implicit goal
1
4: Stock
2
1
2
1
2
500
3
1:
2:
3:
4:
5:
4
5
3
4
5
3
4
0
0.00
175.00
350.00
Time
Page 1
5
3
5
700.00
Sat, Jul 19, 2003
525.00
21:17
4
Figure 13.17. Dynamics of goal erosion with short and long term effects
Till now, the initial value of Implicit goal (IGS) is assumed to be equal to Stated goal
(SG) for various reasons. At this section we generalize and assume that the internal system
can decide on the initial value of Implicit goal, so the Implicit goal is set to the Indicated
goal (ING) initially (IGS(0) = ING), which in turn is a weighted average of the Stated goal
(SG) and Traditional performance (TPS).
1: Ideal goal
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
1000
2: Stated goal
1
3: Implicit goal
1
4: Stock
2
1
2
1
2
500
3
4
5
3
4
5
3
4
0
0.00
Page 1
2
175.00
350.00
Time
5
525.00
17:27
3
4
5
700.00
Wed, Jun 18, 2003
Figure 13.18. Dynamics of goal erosion with short and long term effects when
IGS(0)=ING
186
1: Time horizon
1:
2:
3:
4:
200
4.00
1.00
2: Likelihood of accomp… 3: Ef f ect of motiv ation
4: Ef f ect of short term …
3
1
3
2
1:
2:
3:
4:
100
2.00
0.50
2
1
2
1:
2:
3:
4:
1
1
2
0
0.00
4
0.00
0.00
4
175.00
3
4
350.00
Time
Page 1
3
525.00
11:41
4
700.00
Thu, Jul 03, 2003
Figure 13.19. Behaviors of THS, RLA, EM and ESTM
When system does not believe that the Stated goal (SG) is reachable, it may show no
improvement at all as in Figure 13.18. It can be seen in Figure 13.19 that Effect of short
term motivation (ESTM) is zero for all time, so first of all the system must have a belief that
Stated goal is reachable in order to give weight to Stated goal while determining its
internal goal (that is the Implicit goal). The following run show that it is necessary to lower
the Stated goal to have a satisfactory performance:
1: Ideal goal
1:
2:
3:
4:
5:
1000
2: Stated goal
1
3: Implicit goal
2
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
Page 1
4: Stock
1
1
2
1
2
3
4
5
2
3
4
5
500
3
3
0
0.00
4
4
5
5
175.00
350.00
Time
525.00
12:08
700.00
Thu, Jul 03, 2003
Figure 13.20. Behavior of the system for Stated goal equal to 650
187
1: Time horizon
1:
2:
3:
4:
200
4.00
1.00
1:
2:
3:
4:
100
2.00
0.50
3
3
3
4
3
4
1
1
4
1
1:
2:
3:
4:
1
2
0
0.00
2
4
2
0.00
0.00
Page 1
175.00
350.00
Time
2
525.00
12:08
700.00
Thu, Jul 03, 2003
Figure 13.21. Behaviors of Time horizon, Likelihood of accomplishment ratio, Effect of
motivation and Effect of short term motivation for Stated goal equal to 650
Above dynamics suggest that there must be a feedback from the system while setting
the Stated goal. If there is no improvement, Stated goal must be lowered to motivate the
system, and later it can be increased gradually to make the system reach to the Ideal goal
eventually. This will be discussed in a more advanced model in the following section.
13.2.6. Stated Goal Adjustment by Management to Increase Performance
If top management knows the exact behavior, or rules of behavior of the system then
it can set the Stated goal accordingly to have the best possible performance. More
interesting and real case is when the management does not have such perfect knowledge.
“Planning too often seems to be a process of arbitrarily setting a goal. The goal
setting is then followed by the design of actions which intuition suggests will reach the
goal. Several traps lie within this procedure. First, there is no way of determining that the
goal is possible. Second, there is no way of determining that the goal has not been set too
low and that the system might be able to perform far better. Third, there is no way to be
sure that the planned actions will move the system toward the goal” (Forrester, 1975).
188
We assume that top management does not know Implicit goal, Weight of stated goal,
motivation effects, Short time horizon, Capacity and Loss flow. It can only perceive the
system performance (Stock) over time, so only thing that it can do is to adjust the Stated
goal accordingly. If Stock is not improving it can lower the Stated goal, and otherwise it
can increase it with respect to the performance. We further assume there is no learning
effect for the top management in the model (with Stated goal adjustment) in Figure 13.23.
In the model in Figure 13.23, top management decides adjusting the Stated goal
(SGS) by considering several factors; Stated goal can not be bigger than Ideal goal (G), it
can not be lower than a minimum goal (level) determined by the top management. If the
level determined by the trend in Stock is in acceptable region, then Stated goal must be
equal to this level. The formula of Indicated stated goal (SGS*) can be given as:
SGS * = MIN(G , MAX(Goal achievable by trend , Minimum acceptable goal )) (13.25)
where
Goal achievable by trend = S + Trend • TMOH
(13.26)
Minimum acceptable goal = S + MAIR • TMOH
(13.27)
Trend =
S − RL
TRL
RL = SMTH3( S , TRL )
(13.28)
(13.29)
TMOH is Managerial operating horizon, MAIR is Minimum acceptable improvement
rate, RL is Reference level, and TRL is the Reference level formation time. Top management
must take decisions and/or make plans for future so that these plans (Stated goal) can guide
the system. Since plans are stated for near future or distance future may effect the
motivation badly, this time span, that is the Managerial operating horizon (TMOH), must
also be well stated. In this thesis, we will not go in details of selecting an appropriate
189
Managerial operating horizon, but select and use reasonable values that serve our aim in
this section.
Stated goal (SGS) is adjusted towards Indicated stated goal (SGS*) with a first order
delay:
•
SGS =
SGS * − SGS
TSGA
(13.30)
where TSGA is Stated goal adjustment time, which is another time parameter. The selected
values of the parameters are:
• TMOH = 40
• MAIR = 2
• TRL = 10
• TSGA = 20
1: Ideal goal
1:
2:
3:
4:
5:
1000
2: Stated goal
1
3: Implicit goal
1
2
2
3
1:
2:
3:
4:
5:
1:
2:
3:
4:
5:
Page 1
4: Stock
1
500
3
2
0
0.00
4
3
4
1
2
3
4
5
5
5
4
5
125.00
250.00
Time
375.00
21:38
500.00
Mon, Jul 14, 2003
Figure 13.22. Behavior of Stated goal (SG) adjustment model when SG(0)=G
190
Stock
Loss flow
Control flow
Capacity
Life time
Utilization
Max loss rate
~
~
Expected loss
Stock adjustment
Implicit goal
Traditional
Indicated goal
formation time
Ideal goal
SG adjustment time
Stated goal
Reference level
~
Effect of short term motivation
Likelihood of
accomplishment ratio
Indicated stated goal
Reference level formation time
Minimum acceptable goal
Short time horizon
Time horizon
Time decrease
Time constant
Goal achievable by trend
Manager's operating horizon
Min acceptable improvement rate
Figure 13.23. Model with Stated goal (SG) adjustment by management
191
1: Time horizon
1:
2:
3:
4:
200.00
4.00
1.00
3
1
3
3
4
3
4
4
1:
2:
3:
4:
1
100.00
2.00
0.50
1
4
1:
2:
3:
4:
1
2
0.00
0.00
2
2
0.00
0.00
2
125.00
Page 1
250.00
Time
375.00
21:38
500.00
Mon, Jul 14, 2003
Figure 13.24. Behaviors of THS, RLA, EM and ESTM for SG(0)=G
We observe that the state gradually reaches the ideal goal in an oscillatory manner
(Figure 13.22). These oscillations are caused by short-term frustrations (Figure 13.24)
immediately followed by the management’s appropriately lowering of stated goal, which in
turn encourages the participants to work toward this more realistic goal. Effect of
motivation is always one, since Likelihood of accomplishment ratio is always smaller than
one (Figure 13.24). So, long term motivation (time horizon) has no effect on dynamics.
Oscillations in Implicit goal and Stock are caused by the oscillations in Effect of short term
motivation, which is result of the oscillations in Stated goal, and in turn Stated goal is
affected from the improvement rate of Stock (Figure 13.22 and Figure 13.24).
In these runs, it can be observed that the responsive top management states its goal
so that the motivation of participants in the system is assured. The performance of the
Stock (Figure 13.22) is quite good, when compared with the base run of the simplest model
in Figure 13.3. Note that, the base run is assuming that there is no performance lost due to
motivation effects, so it is the best possible performance for the improvement of the Stock.
Unrealistically, there is an immediate fall in Stated goal in Figure 13.22, because the
Stated goal is initially se to Ideal goal. If top management is responsive, it is better to
assume that top management will not choose an extremely high Stated goal initially, but
prefer an intermediate level instead:
192
1: Ideal goal
2: Stated goal
1000
1:
2:
3:
4:
5:
1
3: Implicit goal
4: Stock
1
1
2
2
3
1:
2:
3:
4:
5:
500
3
0
4
2
3
4
5
5
5
4
5
2
1:
2:
3:
4:
5:
4
3
1
0.00
125.00
250.00
Time
Page 1
375.00
22:08
500.00
Mon, Jul 14, 2003
Figure 13.25. Behavior for Stated goal (SG) adjustment model when SG(0)=SG*
1: Time horizon
1:
2:
3:
4:
200.00
4.00
1.00
3
1
3
3
4
3
4
4
1:
2:
3:
4:
100.00
2.00
0.50
1
1
4
1:
2:
3:
4:
0.00
0.00
2
2
0.00
0.00
Page 1
1
2
2
125.00
250.00
Time
375.00
22:08
500.00
Mon, Jul 14, 2003
motivation and Effect of short term motivation for SG(0)= SG*
The behaviors are not affected much when the initial level of the Stated goal is
changed, so the conclusion here is that this system dynamics is not sensitive to the initial
value of Stated goal. But what if the Time horizon (THS) is not enough for the system to
reach the Ideal goal? Even though the Stated goal is adjusted to motivate the system, in
193
this case system does not have the capacity to reach to the Ideal goal in the given time
period.
1: Ideal goal
1:
2:
3:
4:
5:
2: Stated goal
1000
1
3: Implicit goal
4: Stock
1
1
1
2
2
3
1:
2:
3:
4:
5:
500
3
0
3
5
4
2
5
3
4
5
4
5
2
1:
2:
3:
4:
5:
4
0.00
125.00
250.00
Time
Page 1
375.00
22:32
500.00
Mon, Jul 14, 2003
Figure 13.27. Behavior for Stated goal (SG) adjustment model when Time horizon is
insufficient (THS(0)=120)
1: Time horizon
1:
2:
3:
4:
200.00
4.00
1.00
3
3
4
4
1:
2:
3:
4:
100.00
2.00
0.50
3
3
1
4
2
4
2
1:
2:
3:
4:
0.00
0.00
1
1
2
0.00
0.00
Page 1
2
1
125.00
250.00
Time
375.00
22:32
500.00
Mon, Jul 14, 2003
motivation and Effect of short term motivation for THS(0)=120
194
For the behavior runs in Figure 13.27 and in Figure 13.28 the Time horizon (THS)
and Time constant (TC) are set to 120 instead of 200. Even though the Stock (S) can not
reach to the Ideal goal (G) in the given time period, it does not totally collapse and sustains
a satisfactory level, so even in very unfavorable conditions, properly adjusting the Stated
goal (SGS) level increases the system performance significantly. This last version of our
goal formation model is the most general one that can represent subtle ways in which
frustration and/or system resistance can develop in goal seeking. The model also describes
that can avoid such undesirable dynamics in different unfavorable goal setting
environments.
195
14. CONCLUSIONS
Dynamic decision making structures may contain feedbacks, delays and nonlinearities. Management of such systems that can produce complex dynamic behaviors are
hard for human decision makers. In this research, we evaluate the existing decision
heuristics to see to what extent they can cope with dynamic problems created by
feedbacks, delays and non-linearities, and suggest formulation improvements.
We use system dynamics modeling methodology and computer simulation to analyze
and improve the model structures considered in the thesis. The results are supported with
mathematical analysis when it is necessary and possible.
We first sketch a most general framework of stock management and human decisionmaking (Figure 4.1). This framework shows the basic components involved in stock
management decision-making, which are “Evaluation and Goal Formation”, “Expectation
Formation” and “Decision Formulation”. The remaining chapters focus on different
components of this framework.
We next summarize the standard approach of System Dynamics literature to the
control of a single stock subject to a supply line delay. We show that proper inclusion of
the supply line in decisions is a must to have stable and fast response in the control stock.
The optimum way to consider supply line is to give it equal weight with the control stock,
a conclusion already implicit in some published literature.
A problematic version of stock management structure is a decaying stock involving a
discrete (high order) supply line delay with a long delay time. We show that the standard
System Dynamics approach that would use Loss flow (or its estimation) as an anchor may
fail to create stable dynamics. We develop Equilibrium value of loss (EVL) as a proper
anchor in Control flow (CF) and Desired supply line (SLS*) formulations. The stability of
this formulation is discussed and demonstrated in Chapter 6. In the following chapter we
discuss the robustness of our Equilibrium value of loss formulation, for a non-constant Life
time (decay time). We develop and discuss extensively, a procedure to estimate the Life
196
time for a very general case. We conclude that Equilibrium value of loss (formulated as
Desired stock over Life time) is extremely robust even with autocorrelated Life time. The
results of Chapter 6 and Chapter 7 constitute one of the main contributions of this thesis.
Another major issue in the thesis is the role of information delays in the dynamics of
stock management structures. The standard System Dynamics decision formulations ignore
information delays in the decisions. Firstly, we prove that control of a stock with supply
line delay and control of a stock with information delay are identical provided that the
delay times are equal. Then we develop a Virtual supply line concept and show that it is
possible to consider information delays in the decisions using this notion. Again the two
structures (including supply line delay and information delay) are proven to be identical
when delays are properly considered in the decisions. Using Virtual supply line (VSL) in
decisions is essential, as it guarantees stable and fast responses in the stock.
An immediately related but more difficult complication is secondary stock control
structures on the dynamics of stock management system. In such a structure, primary stock
is controlled via a secondary stock (i.e. controlling the production rate by changing the
production capacity). A secondary stock structure can also be seen as a delay between
control decision and actual control. We mathematically prove that when secondary stock
structure is seen as an input-output system, it is identical to supply line delay and
information delay structures provided that parameters are selected appropriately. Based on
this equivalency, we mathematically derive Virtual supply line (VSL) formula for the
secondary stock structure. We show again that considering Virtual supply line in the
decisions increases system performance by bringing stability and fast response in the
primary stock. The results of Chapter 8 and Chapter 9 (the notion of Virtual supply line
and its proper inclusion in the stock management formulation) are the second major
contribution of the thesis. Conceptualizing complex delay structures (involving
information delays and secondary stock control) as a simple delay box and modeling it as a
Virtual supply line (VSL) is a significant original contribution.
We apply our Virtual supply line (VSL) formulations on two examples. We first show
that a stock control structure containing all the three type of delays (supply line delay,
information delay and secondary stock control structure) can be managed optimally by
197
considering all the delays in the decisions. We show that our approach (using Virtual
supply line adjustment terms in the decisions) is much better than the standard System
Dynamics approach that ignores information delay and secondary stock in the decisions.
We next consider the System Dynamics approach that adds a non-linear additional control
loop (i.e. schedule pressure) to eliminate the unwanted oscillations caused by a secondary
stock structure. Although this additional non-linear control loop gives satisfactory results
in controlling the primary stock, it is not that successful in controlling the secondary stock.
Furthermore, it will bring additional costs (overtime and under-time costs) when
implemented. We show that our approach (Virtual supply line) is successful both in
controlling the primary stock and secondary stock simultaneously. Furthermore, our
approach has no implementation costs. We further suggest that if extreme stability is
desired in the primary stock, one may consider using our approach (Virtual supply line)
and additional non-linear control loops together. We observe that whenever realistic
examples are considered, all the three components of the general decision framework
appear naturally: “Decision Formulations”, “Evaluation and Goal Formation” and
“Expectation Formation”.
The following chapter further discusses the Virtual supply line concept and
introduces what we call ‘stock type’ Virtual supply line for the cases where it is impossible
to observe the state (stock) variables of the information delay and secondary stock
structures. Stock type Virtual supply line (VSLS) can also be used for the cases where it is
not possible to observe supply line stocks. We create an example that includes nonobservable supply line delay, information delay and secondary stock structures at the same
time. We further assume that these structures contain random delay times and random
shocks in the stock variables. On this example, we demonstrated that the most advanced
stock type Virtual supply line can manage a stock with such a complex supply line quiet
satisfactorily. This result is another major contribution of the thesis.
We also discuss potential applicability of our major results in the standard (s, S) rule
of inventory management. First we demonstrate the implementation of our Equilibrium
value of loss (EVL) formulations by assuming a perishable goods inventory. The simplest
common sense approach would suggest to use Perishing rate (decay rate) directly in the
formula of “s”. This works quite well for small values of lead time (Acquisition delay
198
time), but it creates instability when lead time is long. For this problem we suggest an
approximate formula that calculates the long term average Perishing rate based on the
Equilibrium value of loss (EVL) formulations. Using this formula in the formula of “s”
brings stability immediately. The suggested approximate solution methodology gives quite
satisfactory results even with autocorrelated demand. Next, we demonstrate the
implementation of the stock type Virtual supply line (VSLS). For this, we assume an
unreliable supply line case where individual order tracing is impossible. The standard way
to compute In-transit inventory level does not work, which creates a gap between the
calculated In-transit inventory level and actual Supply line level. The proposed formulas
(based on Virtual supply line notion) increase the system performance by preventing the
accumulation of this gap over time.
The last major topic in the thesis is the “Evaluation and Goal Formation” component
of the decision framework. Firstly we present the simple goal setting structures, and then
we develop problematic structures with capacity limitations on improvement rate. We then
discuss the simple goal erosion dynamics present in System Dynamics literature, based on
our capacity-limited model. We then develop a non-linear model for more general goal
erosion. We show that to increase the system performance in the problematic case, the
Stated goal must be adjusted such that it increases the motivation of the whole system. The
final major contribution of the thesis is this most general goal formation and goal seeking
model that can represent subtle ways in which frustration and/or system resistance can
develop in goal seeking and help avoid such undesirable dynamics in different unfavorable
goal setting environments.
To conclude, the stated objective of the thesis was achieved. Dynamic decision
making structures containing feedback, delays and non-linearities were evaluated, and
alternative improved formulations were specifically developed as summarized above. The
general framework developed in Chapter 4, with its three decision components, provides a
platform to carry out further research on dynamics of decision making in nonlinear
feedback environments. Various other decision formulations (linear and non-linear) can be
tested. The structures and formulations that we propose must also be validated in large
scale real life models of stock management problems.
199
APPENDIX A: MODELING OBJECTS AND SYMBOLS USED IN
SYSTEM DYNAMICS
Table A.1. Modeling objects and symbols used in dynamic systems modeling
Name of the object
Object
Explanation
Stock
Stock
Stocks are accumulations. They can only change by way of their in
and out flows. Also called states.
Conv ey or
Conveyor
A special type of stock representing a pure discrete delay. The inflow,
after spending given delay time in the stock, flows out of it.
Flow
A flow is the rate that changes the levels of the stocks.
Flow
Converter
Conv erter
Converters are intermediate computation variables (or auxiliaries).
Connectors show the functional relations between variables that are
Connector
not in form of stock-flow relation.
Table A.2. Example model illustrating the objects
Example Model
Example Equations
•
Stock
Inflow
Stock = (Inflow − Outflow )
Outflow
Converter 1
Converter 3
Converter 2
Inflow = f 1 (Converter3 )
Outflow = f 2 (Stock ,Converter1 )
Converter1 = constant
Converter2 = constant
Converter3 = f 3 (Stock ,Converter2 )
f1, f2 and f3 must be specified by the modeler.
200
APPENDIX B: ABBREVIATION RULES ADOPTED FOR VARIABLE
NAMES
In this thesis we use the following abbreviation rules:
• All variables are abbreviated in all-capital letters.
• Desired levels and desired values are indicated by appending a “*”.
• All stock variables end with “S”.
• All flow variables end with “F”.
• Parameters are abbreviated by an initial capital letter followed by a subscript index.
Parameter conventions:
O: Order of a structure (number of stocks or states in it)
T: Time constants
W: Weight coefficients
201
APPENDIX C: ATOMIC STRUCTURES IN HUMAN SYSTEMS
There are many human systems models in the System Dynamics literature, and from
literature the following elementary generic linear and non-linear decision structures are
crystallized. Some of these structures are just a combination of others, and furthermore, for
some conditions (parameter values), some of these structures and their behavior can be
equivalent with each other, but their scope is different and they are treated separately
(Barlas, 2002).
We give simpler structures first and then more complicated ones, but there is no
absolute order in the following classification of atomic structures.
C.1. First Order Linear Atomic Structure
Stock
Fraction 1
Inflow
Fraction 2
Outflow
Figure C.1. Stock-flow diagram of first order linear atomic structure
+
Fraction 1
+
+
Inflow
+
Stock
+
-
-
Fraction 2
Outflow
+
Figure C.2. Causal-loop diagram of first order linear atomic structure
The differential equation of the model in Figure C.1 can be given as:
•
S = Inflow − Outflow
(C.1)
202
where S represents the Stock and Inflow and Outflow can be given as:
Inflow = Fraction 1 • S
(C.2)
Outflow = Fraction 2 • S
(C.3)
Possible behaviors of the model in Figure C.1 are:
1: Stock
1:
2: Stock
3: Stock
40.00
1
1:
20.00
1
1
2
2
2
2
3
3
1:
0.00
0.00
2.50
Graph 1 (Exponential)
3
3
5.00
7.50
Time
17:00
10.00
17 Jan 2003 Fri
Figure C.3. Exponential growth, constant and exponential decay
First order linear atomic structure can be divided into two smaller structures called
exponential growth structure and exponential decay structure. If Fraction2, that is the
fraction of the decay part is zero, then a pure exponential growth structure is obtained. If
Fraction1, that is the fraction of the growth part is zero, then a pure exponential decay
structure is obtained. If both are non-zero, then the larger one determines the behavior of
the whole system as exponential growth or exponential decay. If they are equal to each
other, then Inflow and Outflow also become equal, Stock stays constant at its initial level.
203
C.2. Production Process
Stock 1
Production rate
Stock2
Productivity
Figure C.4. Stock-flow diagram of production process
•
S 1 = Production rate = Productivity • S 2
(C.4)
Assume Stock2 (S2) is constant. If Productivity is positive then Stock1 (S1) linearly
increases. If Productivity is negative, it serves as a Consumption multiplier, and this time,
S1 linearly decreases. If it is zero then S1 stays at its initial level. Possible behaviors of the
model in Figure C.4 are:
1: Stock 1
1:
2: Stock 1
3: Stock 1
120.00
3
3
3
3
1:
100.00
1
2
2
2
2
1
1
1
1:
80.00
0.00
2.50
Graph 1 (production)
5.00
Time
7.50
18:43
10.00
Fri, Jan 17, 2003
Figure C.5. Linear growth, constant and linear decay
204
C.3. Goal Seeking Atomic Structure
Stock
Adjustment flow
Goal
Discrepancy
Adjustment time
Figure C.6. Stock-flow diagram of goal seeking atomic structure
+
Stock
Adjustment time
Adjustment flow
-
+
Goal
Discrepancy +
Figure C.7. Causal-loop diagram of goal seeking atomic structure
•
Discrepancy
Goal − S
=
Adjustment time Adjustment time
(C.5)
1: Stock
1:
2: Stock
3: Stock
8.00
3
3
3
1:
2
4.00
2
2
2
3
1
1
1
1:
0.00
1
0.00
2.50
Graph 1 (goal seeking)
5.00
Time
Figure C.8. Goal seeking behavior
7.50
19:53
10.00
Fri, Jan 17, 2003
205
If Stock (S) is initially above the Goal, it decreases in an exponentially decaying way
till it reaches the Goal. If S is initially below the Goal, it increases in a negative
exponential way till it reaches the Goal. If S is initially on the Goal, it stays there forever.
Note that the behavior of the absolute value of Discrepancy is always exponential decay.
C.4. S-shaped Growth Atomic Structure
C.4.1. S-shaped Growth Caused by Transfer from One Stock to Another
Stock 1
Stock 2
Transfer rate
Transfer fraction
Ratio
Total
Figure C.9. Stock-flow diagram of S-shaped growth structure with transfer
+
+
Stock 1
-
Transfer rate
+
-
Stock 2
+
Transfer fraction
+
+
Ratio +
-
-
+
Total +
Figure C.10. Causal-loop diagram of S-shaped growth structure with transfer
In this simple structure the Total of the stocks are assumed to be conservative so the
causal-loop diagram can be re-sketched as:
206
+
+
Stock 1
Stock 2
Transfer rate
+
-
-
+
Transfer fraction
Ratio
+
+
-
Total
Figure C.11. Simplified causal-loop diagram of S-shaped growth structure with transfer
The differential equations of the model in Figure C.9 can be given as:
•
 S 
S 1 = −Transfer rate = Transfer fraction • S1 = − f  2  • S1
 Total 
(C.6)
•
 S 
S 2 = Transfer rate = Transfer fraction • S1 = f  2  • S1
 Total 
(C.7)
Note that these differential equations are non-linear even when f is linear. Behaviors
of the two stocks are mirror image of each other, since the differential equation of the one
stock, is equal to (-1) times the differential equation of the other one. S2 shows either Sshaped growth or goal seeking behaviors. Possible behaviors of the model in Figure C.9
are:
1: Stock 2
1:
2: Stock 2
100.00
2
1:
2
2
1
2
1
50.00
1
1:
0.00
1
0.00
25.00
Graph 2 (S-shaped growth)
50.00
75.00
Time
22:36
100.00
17 Jan 2003 Fri
Figure C.12. Two possible dynamics of S-shaped growth model
207
1: Stock 1
1:
100.00
2: Stock 1
1
1
1:
50.00
1
2
2
1:
0.00
0.00
1
2
25.00
50.00
75.00
Graph 1 (Mirror image of S-shaped) Time
22:36
2
100.00
17 Jan 2003 Fri
Figure C.13. Mirror image dynamics (Stock1) of S-shaped growth model
C.4.2. S-shaped Growth Caused by a Capacity Limit
Stock
Inflow
Fraction 1
Outflow
Ratio
Fraction 2
Capacity
Figure C.14. Stock-flow diagram of S-shaped growth structure with limit
+
+
+
Inflow
+
Stoc k
+
-
Frac tion 1
-
-
Outflow
+
Frac tion 2
+
Ratio
Capacity
Figure C.15. Causal-loop diagram of S-shaped growth structure with limit
208
•
S = Inflow − Outflow = Fraction1 • S − Fraction 2 • S = f (S / Capacity ) • S − Fraction 2 • S
(C.8)
Note that, this differential equation is non-linear, even when f is linear. S either
shows S-shaped growth or goal seeking behaviors. Possible behaviors of the model in
Figure C.14 are:
1: Stock
1:
2: Stock
3: Stock
4: Stock
120.00
4
3
2
3
4
2
3
4
1
2
3
4
1
2
1:
60.00
1
1:
0.00
1
0.00
30.00
60.00
90.00
Time
21:55
Graph 1 (S-shaped growth)
120.00
17 Jan 2003 Fri
Figure C.16. Possible behaviors of S-shaped growth structure with limit
C.5. Boom-Then-Bust Atomic Structure
C.5.1. Boom-Then-Bust Caused by S-shaped Growth and Decay
Stock 1
Stock 2
Transfer rate
Decay rate
Ratio
Transfer fraction
Decay fraction
Total
Figure C.17. Stock-flow diagram of boom-then-bust structure caused by S-shaped growth
and decay
209
+
-
+
+
Transfer rate
Stock 1
+
Transfer fraction
-
-
-
Decay rate
+
Ratio +
-
+
+
+
Stock 2
+
Total
Decay fraction
+
Figure C.18. Causal-loop diagram of boom-then-bust structure caused by S-shaped growth
and decay
•
S 1 = −Transfer rate = Transfer fraction • S1 = − f (S 2 / (S1 + S 2 )) • S1
(C.9)
•
S 2 = Transfer rate − Decay rate = f (S 2 / (S1 + S 2 )) • S1 − Decay fraction • S 2 (C.10)
Behavior of S2 is either boom-then-bust or exponential decay. Behavior of S1 is
either mirror image of S-shaped growth or goal seeking. Runs of the model in Figure C.17:
1: Stock 2
1:
100.00
1:
50.00
2: Stock 2
1
1
1
2
1:
0.00
1
0.00
2
2
50.00
Graph 2 (Boom-then-bust)
2
100.00
150.00
Time
23:00
200.00
17 Jan 2003 Fri
Figure C.19. Two possible dynamics of boom-then-bust structure (S-shaped growth and
decay)
210
1: Stock 1
1:
100.00
1:
50.00
2: Stock 1
1
1
2
2
1:
0.00
2
2
1
100.00
1
150.00
Graph 1 (Mirror image of S-shaped) Time
23:00
0.00
50.00
200.00
17 Jan 2003 Fri
Figure C.20. Mirror image dynamics (Stock1) of S-shaped growth behavior for boom-thenbust structure (s-shaped growth and decay)
C.5.2. Boom-Then-Bust Caused by a Delayed Effect of Capacity Limit
Stock
Inflow
Outflow
Fraction 1
Ratio
Fraction 2
Capacity
Effective ratio
Adjustment flow
Delay time
Figure C.21. Stock-flow diagram of boom-then-bust structure with delayed effect of
capacity limit
211
+
+
+
+
+
+
-
Fraction 1
Ratio
-
+
Effective ratio
Fraction 2
Delay time
-
-
+
Capacity
+
Adjustment flow
-
Outflow
-
Stock
Inflow
Figure C.22. Causal-loop diagram of boom-then-bust structure with delayed effect of
capacity limit
•
S = Inflow − Outflow = Fraction1 • S − Fraction2 • S = f (Effective ratio) • S − Fraction2 • S (C.11)
•
Effective ratio = Adjustment flow = =
S / Capacity − Effective ratio
Delay time
(C.12)
Stock (S) either shows boom-then-bust or decline-then-rise behaviors. Possible
behaviors of the model in Figure C.21 are:
1: Stock
1:
2: Stock
3: Stock
4: Stock
120.00
3
2
4
3
1
4
2
3
4
1
2
3
4
2
1:
60.00
1:
0.00
1
1
0.00
40.00
Graph 1 (Boom-then-bust)
80.00
120.00
Time
23:33
160.00
17 Jan 2003 Fri
Figure C.23. Possible dynamics of boom-then-bust structure with delayed effect of
capacity limit
212
C.6. Delays
C.6.1. Material Delay Atomic Structure
These structures represent the delays experienced by flows on a material (conserved)
stock-flow chain (such as goods ordered and still in supply line).
Stock
Input
Output
Delay time
Figure C.24. Stock-flow diagram of first order material delay atomic structure
For a continuous material delay, the differential equation of the model in Figure C.24
can be given as:
•
S = Input − Output = Input −
Stock 2
Stock 1
Input
S
Delay time
Acquisition flow 1
Order of material delay
(C.13)
Stock 3
Acquisition flow 2
Output
Delay time
Figure C.25. Stock-flow diagram of third order material delay atomic structure
Stock
Input
Output
Delay time
Figure C.26. Stock-flow diagram of discrete material delay atomic structure
213
Note that, Order of material delay (OMD) is three for a third order delay. The
differential equations of the model in Figure C.25 can be given as:


S1
 Input −

• 
 S1 
(
Delay time / OMD )


 •   Input − AF1  



S1
S2
 S 2  = AF − AF


=
−

2
   1
(
)
(
)
Delay
time
O
Delay
time
O
/
/


MD
MD
 •   AF2 − Output  

S3
S2
S3 


 
 (Delay time / O ) − (Delay time / O ) 
MD
MD 

(C.14)
Order of a delay can be any positive integer number. As OMD approaches infinity, the
material delay is called discrete material delay. The time-lagged differential equation of the
model in Figure C.26 can be given as:
•
S (t ) = Input (t ) − Output (t ) = Input (t ) − Input (t − Delay time )
(C.15)
Possible behaviors of the models in Figure C.24 (Output), Figure C.25 (Output2) and
Figure C.26 (Output3) are:
1: Input
1:
2:
3:
4:
1:
2:
3:
4:
1:
2:
3:
4:
2: Output
1.00
3: Output 2
1
4
4: Output 3
1
4
1
3
3
4
2
2
2
3
0.50
0.00
1
0.00
2
3
4
10.00
Graph 1 (Material delay)
20.00
30.00
Time
01:41
40.00
18 Jan 2003 Sat
Figure C.27. Behaviors of material delay structure for different orders of delay
214
If Input to a material delay, which is the inflow of the first stock of the delay, is
changed, then the Output, that is the outflow from the last stock of the delay, follows it
after a period of time. In Figure C.27, we can observe this fact. First run is the Input itself,
second run is Output from first order material delay, third run is Output from third order
material delay and the fourth run is Output from discrete material delay.
C.6.2. Information Delay Atomic Structure
These structures represent delayed awareness about changing conditions, delayed
perceptions or estimations.
Output
Adjustment flow
Input
Delay time
Discrepancy
Figure C.28. Stock-flow diagram of first order information delay atomic structure
•
Information delay 1
Discrepancy 1
Discrepancy Input − Output
=
Delay time
Delay time
Information delay 2
Discrepancy 2
Adjustment flow 2
Adjustment flow 1
(C.16)
Output
Discrepancy 3
Adjustment flow 3
Input
Delay time
Figure C.29. Stock-flow diagram of third order information delay atomic structure
215
 Discrepancy1   Input − IDS1 
 

•  
 S 1   (Delay time / OID )   (Delay time / OID ) 
  
 

  
 

 •   Discrepancy   IDS − IDS

2
1
2
S2  = 
=

   (Delay time / OID )   (Delay time / OID ) 
  
 

•  
 

 S 3   Discrepancy 3   IDS 2 − Output 
   (Delay time / O )   (Delay time / O ) 
ID 
ID  

(C.17)
Note that, Order of information delay (OID) is three for a third order delay and order
of a delay can be any positive integer number.
The graphical outputs of the material delay and information delay structures are
exactly the same (Figure C.27, Figure C.30). Possible behaviors of the models in Figure
C.28 (Output) and Figure C.29 (Output2) are:
1: Input
1:
2:
3:
2: Output
1.00
3: Output 2
1
1
1
3
3
2
2
2
1:
2:
3:
1:
2:
3:
3
0.50
0.00
1
0.00
2
3
10.00
Graph 1 (Information delay)
20.00
30.00
Time
02:34
40.00
18 Jan 2003 Sat
Figure C.30. Behaviors of information delay structure for different orders of delay
216
If Input to an information delay (that is the goal of the first stock of the delay
structure) is changed, then, the Output that is the last stock of the information delay
structure follows it after a period of time. First run of the model in Figure C.30 is the Input
itself, second run is Output from first order information delay and third run is Output from
third.
C.7. Oscillating Atomic Structure
Stock 1
Inflow 1
Outflow 1
Productivity
Stock 2
Outflow 2
Inflow 2
Fraction
Consumption multiplier
Figure C.31. Stock-flow diagram of oscillating atomic structure
-
+
±
Outflow 1
Stoc k 1
+
Inflow 1
Outflow 2
-
±
+
Productivity
Stoc k 2
+
Frac tion
-
Consumption
multiplier
±
±
Inflow 2
±
Figure C.32. Causal-loop diagram of oscillating atomic structure
217
Note that the polarity of the loop between Stock2 and Inflow2 depends on the sign of
the Fraction. If Fraction is positive, then the loop polarity is also positive (reinforcing), if
it is negative, then the loop polarity is also negative (counteracting). Stocks are free to take
negative and positive values, but Productivity and Consumption multiplier are assumed to
be positive.
 •   Inflow1 − Outflow1 

 S1  

• =
 S   Inflow − Outflow 
2
2
 2 
(C.18)
 Productivity • S 2 − Outflow1



=

 Fraction • S − Consumption multiplier • S 
2
1

1: Stock 1
1:
2:
2: Stock 2
10.00
1
2
1:
2:
0.00
1
2
1
2
1
2
1:
2:
-10.00
0.00
10.00
20.00
30.00
Graph 3 (Growing oscillations)
Time
11:48
40.00
18 Jan 2003 Sat
Figure C.33. Growing oscillations for Fraction greater than zero
218
1: Stock 1
1:
2:
2: Stock 2
1.00
1
2
1
1:
2:
0.00
1
2
2
1
1:
2:
2
-1.00
0.00
10.00
Graph 1 (Oscillation)
20.00
30.00
Time
11:45
40.00
18 Jan 2003 Sat
Figure C.34. Neutral oscillations for Fraction equal zero
1: Stock 1
1:
2:
2: Stock 2
1.00
1
2
1:
2:
0.00
1
2
1:
2:
1
2
1
2
-1.00
0.00
10.00
Graph 2 (Damping oscillations)
20.00
30.00
Time
11:47
40.00
18 Jan 2003 Sat
Figure C.35. Damping oscillations for Fraction smaller than zero
C.8. Stock Management Atomic Structure
Basically, stock management atomic structure consists of a primary stock, on which
the control is applied, and a material delay between the control decision (or control action)
and actual control.
219
In simpler form, it may be possible to control a stock directly without a delay. This
very simple form (one-stock system) can not show oscillatory behavior, even if the flow
formulations are non-linear. For oscillations at least two stocks (second order system) is
needed.
The delay in the stock management structure (Figure 4.1) can be in the form of
supply line, information delay and secondary stock control structures or may be a mixture
of these forms.
Stock
Supply line
Control flow
Loss flow
Acquisition flow
Desired stock
Stock adjustment
Desired supply line
Expected loss
Figure C.36. Stock-flow diagram of stock management atomic structure
±
Supply line
Acquisition flow
-
+
-
±
Stoc k
-
+
+
+
±
+
-
Control flow
+
Stoc k adjustment
+
Loss flow
+
±
+
Desired supply line
Stoc k adjustment time
Desired stock
+
+
Expected loss
-
-
+
±
Figure C.37. Causal-loop diagram of stock management atomic structure
220
 SL
 •   AF − LF  
− LF
S  
  T AD
 •  
 
 SL  =  CF − AF  =  ELS + SA + SLA − SL

 
 
T AD
 •  
 
 ELS  
  LF − ELS
  T

  EAF
EL











(C.19)
where Acquisition flow is typically formulated as the output of a material delay:
AF =
SLS
T AD
(C.20)
Control flow (control decision) is given by:
CF = ELS + SA + SLA
(C.21)
Stock adjustment is given by:
SA =
S* − S
TSA
(C.22)
Supply line adjustment is given by:
SLA = WSL •
T • ELS − SLS
SLS * − SLS
= WSL • AD
TSA
TSA
(C.23)
Expectation adjustment flow is given by:
EAF =
LF − ELS
TEL
(C.24)
221
1: Stock
2: Stock
1:
2.00
1:
0.00
3: Stock
1
4
3
4
3
4: Stock
3
4
3
4
1
2
2
2
2
1
1
1:
-2.00
0.00
25.00
Graph 1 (Stock management)
50.00
75.00
Time
18:09
100.00
25 Jan 2003 Sat
Figure C.38. Unstable oscillation, neutral oscillation, stable oscillation and goal seeking
behaviors of the stock management structure
To be able to obtain unstable behavior, Order of supply line must be bigger than one.
C.9. Goal Setting Atomic Structure
Stock
Goal
Control flow
Figure C.39. Stock-flow diagram of goal setting atomic structure
-
-
+
± Control flow
+
Stock
+
Goal
+
+
-
±
-
Figure C.40. Causal-loop diagram of stock management atomic structure
222
Causal loop diagram is sketched under “parameters are positive” assumption. Stocks
are free to take negative and positive values. Note that, positive loop can only be activated
through high delays between Control flow and Goal and/or, Goal adjustment flow and
Stock.
G −S 

 •   CF  
  TSA 
S  
= 
•=

 G   GAF   S − G 
  
  T

 GA 
(C.25)
Possible behavior of the model in Figure C.39 are:
1: Goal
1:
2:
2: Stock
10.00
1
1
1:
2:
6.00
1:
2:
2.00
2
1
2
1
2
2
0.00
10.00
20.00
30.00
Graph 1 (Untitled)
Time
19:58
40.00
25 Jan 2003 Sat
Figure C.41. Eroding goal and goal seeking behaviors
223
APPENDIX D: NOISE GENERATION
We use the following structure and equations to generate noise:
Autocorrelation coefficient
Pink noise
Dummy adjustment flow
Deviation
Seed
Figure D.1. Noise model
Pink noise means “autocorrelated” normal variates. The approximate integral
equations of the above model can be given as:
Pink noise(t ) = Pink noise(t − DT ) + DT • Dummy adjustment flow
(D.1)
Pink noise(0 ) = 0
(D.2)
where
Dummy adjustment flow =
 IF (MOD(t ,1) = 0 )




 Autocorrelation coefficient • Pink noise




+ NORMAL(0, Deviation, Seed ) − Pink noise  

 THEN



DT




 ELSE 0

(D.3)
224
Using “IF THEN ELSE” with “MOD” prevents Pink noise to change its value every
DT, so Pink noise is constant between two integer TIME points. Simplified equation for
Pink noise can be given as:
Pink noise(t ) = Autocorrelation coefficient • Pink noise(t − 1)
+ NORMAL(0, Deviation, Seed )
(D.4)
Autocorrelation coefficient takes values between 0 and 1. If it is equal to zero the
outcome becomes white noise, if it is one the outcome is random walk and if it takes values
between 0 and 1 the outcome is pink noise.
We assume zero mean for the “NORMAL” function and use a seed to be able to
generate comparable runs. The values of the parameters for Chapter 7 are set as follows:
• Autocorrelation coefficient = 0.95
• Deviation = 1
• Seed = 0
Pink noise run for Chapter 7 can be seen below:
1: Pink noise
1:
10
1
1:
0
1
1
1
1:
Page 1
-10
0.00
62.50
125.00
Time
Autocorrelated noise
Figure D.2. Auto-correlated noise
187.50
18:03
250.00
Mon, Apr 21, 2003
225
APPENDIX E: A NON-LINEAR LIFE TIME ESTIMATION
ADJUSTMENT RULE FOR SHOCK REDUCTION
The shock in Figure 7.8 can further be reduced by using a non-linear estimation
adjustment rule. In Equation (7.11) adjustments are linear. In the non-linear formula below,
adjustments are linear when discrepancy is in a certain range, but when discrepancy is
outside of the range, which possibly may correspond to a shock value, then the adjustments
are limited. The new adjustment formula for Smoothed life time can be given as:

 IF (PLS = 0 ) THEN 0


TCLf − SMLTS

− 0.4 • SMLTS 
< −0.4 THEN

 ELSE IF
SMLTS
TSm, Lf


•


TCLf − SMLTS
0.4 • SMLTS
SMLTS = 

ELSE IF
> 0.4 THEN
SMLTS
TSm, Lf




 TCLf − SMLTS 




 ELSE 

TSm, Lf




(E.1)
1:
30
1
1:
20
1
1
1
1:
Page 1
10
0.00
62.50
125.00
Time
187.50
17:35
250.00
Fri, Apr 25, 2003
Without phase dif f erence
Figure E.1. Smoothed life time when there is no phase difference (TSm,S equal to TPD), and
with a non-linear adjustment rule
226
If Figure E.1 is compared with Figure 7.8, it can be observed that the shock that is
seen as a discontinuity in Smoothed life time around Time equals 23, is further reduced, and
discontinuity is very negligible this time.
227
APPENDIX F: MATHEMATICAL EQUIVALENCY OF SUPPLY
LINE DELAY, INFORMATION DELAY AND SECONDARY STOCK
STRUCTURES FOR THE GENERAL CASE
F.1. Second Order Supply Line Structure as an Input-Output System
The Equation (8.24) and Equation (8.3) are modified to include different delay times:
•
SLS 1 = CF X +
W SL • ((T AD1 + T AD 2 ) • LF − SLS1 − SLS 2 ) SLS1
−
TSA
T AD1
•
SLS 2 =
SLS1 SLS 2
−
TAD1 TAD 2
(F.1)
(F.2)
From Equation (F.2) we can obtain the following equation:
•
T AD1
• SLS 2
T AD 2
(F.3)
••
•
T AD1
• SLS 2
T AD 2
(F.4)
SLS1 = T AD1 • SLS 2 +
•
SLS 1 = T AD1 • SLS 2 +
Equation (F.3) and Equation (F.4) can be inserted to Equation (F.1), and then
simplified to obtain the following equation:
••



 T AD1 • TSA • SLS 2


•

  TSA • CFX
 + ((T
 (F.5)
AD1 / T AD 2 + 1) • TSA + W SL • T AD1 ) • SLS 2  = 
 + W • (T

SL
AD1 + T AD 2 ) • LF 

 + (TSA / T AD 2 + WSL • (1 + T AD1 / T AD 2 )) • SLS 2 




228
As was seen in Section 8.4, we again see this system as an input-output system, with
input Control flow (CFX) and output Acquisition flow2 (AF2). Equation (F.5) can be rewritten for AF2, which is equal to (SLS 2 / T AD 2 ) :
••


 T AD1 • T AD 2 • TSA • AF 2




  (T AD1 + T AD 2 ) • TSA  •
  TSA • CFX

+
•
AF
=
2

 +W •T
  + W • (T
SL
AD1 • T AD 2 
SL
AD1 + T AD 2 ) • LF 




 + (TSA + WSL • (T AD1 + T AD 2 )) • AF2 


(F.6)
Equation (F.6), Equation (F.12) and Equation (F.18) are identical provided that
parameter values are chosen such that
(TSSA = TID1 = T AD1 ) , (TSAD
= TID 2 = T AD 2 ) ,
(WVSL,SS = WVSL,ID = WSL ) and (WSSL = 1) . Note that input values are also same. i.e. CFX
of supply line structure, and CF* of information delay and secondary stock structures are
(
(
)
)
all equal to LF + S * − S / TSA .
F.2. Second Order Information Delay as an Input-Output System
The Equation (8.25) and the Equation (8.6) are modified to include the different
delay times:
•
IDS 1 =
CF * +
WVSL • ((TID1 + TID 2 ) • LF − TID1 • IDS1 − TID 2 • IDS 2 )
− IDS1
TSA
TID1
•
IDS 2 =
IDS1 − IDS 2
TID 2
(F.7)
(F.8)
•
IDS1 = TID 2 • IDS 2 + IDS 2
(F.9)
229
•
••
•
IDS 1 = TID 2 • IDS 2 + IDS 2
(F.10)
••



 TID1 • TID 2 • TSA • IDS 2


*

  TSA • CF
  (TID1 + TID 2 ) • TSA +  •

•
IDS
=
+
2

  + W • (T + T ) • LF 
 W • T • T
VSL
ID1
ID 2


VSL
ID1
ID 2

 

 + (TSA + WVSL • (TID1 + TID 2 )) • IDS 2 


(F.11)
As was seen in Section 8.4, we again see this system as an input-output system, with
input Desired control flow (CF*) and output Control flow (CF). Equation (F.11) can be rewritten for CF, which is equal to IDS 2 :
••



 TID1 • TID 2 • TSA • CF


*

  TSA • CF
  (TID1 + TID 2 ) • TSA +  •

•
CF
=
+

  + W • (T + T ) • LF 
 W • T • T
VSL
ID1
ID 2


VSL
ID
1
ID
2




 + (TSA + WVSL • (TID1 + TID 2 )) • CF 


(F.12)
= TID 2 = T AD 2 ) ,
(
(
)
)
230
F.3. Secondary Stock Structure with a First Order Supply Line Delay as an InputOutput System
The Equation (9.26) and the Equation (9.11) are used without making any changes.
We present the same equations again:
(
)
*






   CF * + WVSL • VSL − VSL  / C − SS  
P






TSA


•



SSLS = 
 SSLS 
 + W • (T
)
SLF
SSLS
•
−
SSL
SAD
−

 SLF + 
T
TSAD 

SSA




•
SS =
SSLS
− SLF
TSAD
(F.13)
(F.14)
•
SSLS = TSAD • SS + TSAD • SLF
(F.15)
•
••
SSLS = TSAD • SS
(F.16)
••



 TSAD • TSSA • TSA • C P • SS


*
•

  TSA • CF
  (TSSA + WSSL • TSAD ) • TSA 

C
SS
•
•
=
+
 P
  + W • (T
 +W •T •T
+ TSAD ) • LF 
VSL
SSA
SAD


VSL
SSA



 + (TSA + WVSL • (TSSA + TSAD )) • C P • SS 


(F.17)
231
As was seen in Section 9.3.1, we again see this system as an input-output system,
with input Desired control flow (CF*) and output Control flow (CF). Equation (F.17) can
be re-written for CF, which is equal to (C P * SS ) :
••



 TSAD • TSSA • TSA • CF


*

  (TSSA + WSSL • TSAD ) • TSA  •   TSA • CF

CF
•
=
+

  + W • (T
 +W •T •T

)
T
•
LF
+
VSL
SSA
SAD


VSL
SSA
SAD

 

 + (TSA + WVSL • (TSSA + TSAD )) • CF 


(F.18)
= TID 2 = T AD 2 ) ,
(
(
)
)
Thus, three delay structures are proven to be equivalent.
232
APPENDIX G: GENERALIZED VIRTUAL SUPPLY LINE
FORMULAS FOR DELAY STRUCTURES INVOLVING DIFFERENT
INDIVIDUAL DELAY TIMES
For an nth order information delay with unequal individual delay times, the Virtual
supply line (VSL) can be defined as:
VSL =
OID
n
i =1
i =1
∑ (TIDi • IDSi ) = ∑ (TIDi • IDSi )
(G.1)
The Desired virtual supply line (VSL*) given in Equation (8.20) and Virtual supply
line adjustment (VSLA) given in Equation (8.21) are also same for this formulation given in
Equation (G.1).
For a secondary stock structure with an nth order supply line delay with unequal
individual delay times, the Virtual supply line (VSL) can be defined as:
n



  TSSA + ∑ (TSAD ) • SS

i




i =1







n
n

  T


(
)
(
)
+
•
SSLS
T
•
SSLS
SAD1 ∑
i
i
  SSA ∑


i =2
i =1
+ 


n

VSL = C P •  

  TSAD 2 • ∑ (SSLS i ) + L + TSAD (n −1) • SSLS n 

i =3

 



n
n

  T


(
)
(
)
+
•
T
T
•
T
SAD1 ∑ SAD i
  SSA ∑ SAD i


i =1
i =2
− 
• SLF 

n
 


T
•
  SAD 2 ∑ (TSAD i ) + L + TSAD (n −1) • TSAD n 

i =3

 

(G.2)
The Desired virtual supply line (VSL*) given in Equation (9.23) and Virtual supply
line adjustment (VSLA) given in Equation (9.24) are also same for this formulation given in
Equation (G.2).
233
APPENDIX H: THE PROBLEMATIC AND NON-PROBLEMATIC
VERSIONS OF THE DESIRED SUPPLY LINE FORMULATION
There is a problem in the desired levels of the Work in process inventory and the
Vacancies. Desired WIPI that is given by Equation (10.58) depends on Desired
production, and Desired vacancies that is given by Equation (10.91) depends on Desired
hiring rate (suggested by Sterman, 2000). Both Desired production and Desired hiring
rate are varying very fast because they are on control loops. This contradicts with the
suggestions given in Section 6.6. Equation (10.103) and Equation (10.104) were proposed
instead of Equation (10.58) and Equation (10.91).
We obtain the following simulation results:
First run (equations as suggested in Sterman, 2000):
• Equation (10.58) is used for Desired WIPI
• Equation (10.91) is used for Desired vacancies
Second run:
Third run (equations as suggested in Sterman, 2000):
Fourth run:
234
Inv entory : 1 - 2 - 3 - 4 1:
80000.00
1
1
2
1
3
3
4
2
3
4
4
1:
2
50000.00
1
4
2
1:
20000.00
3
0.00
50.00
100.00
Time
Page 1
150.00
20:47
200.00
Sun, Jul 06, 2003
Figure H.1. Runs for Inventory with problematic and non-problematic desired supply line
equations
Labor: 1 - 2 - 3 - 4 1:
2000.00
1:
1500.00
2
3
3
4
2
1
4
1:
Page 1
1000.00
1
0.00
3
4
1
2
3
4
1
2
50.00
100.00
Time
150.00
20:47
200.00
Sun, Jul 06, 2003
Figure H.2. Runs for Labor with problematic and non-problematic desired supply line
equations
235
When Schedule pressure on\off is zero (first and second runs), second run is
obviously better than the first run, so first two runs suggest using Equation (10.103) and
Equation (10.104), instead of Equation (10.58) and Equation (10.91).
When Schedule pressure on\off is one (third and fourth runs), reaching a conclusion
is not easy. Third run seems to be fast and a bit oscillatory, and the fourth run seems to be
slow and stable. Before reaching a conclusion let us mention about another problem. The
reason for the fourth run to be slow is because in adjustments more weight is given to
supply lines than the stocks themselves. The Inventory adjustment time is 12, while the
WIPI adjustment time is 6. The Labor adjustment time is 13, while Vacancy adjustment
time is 4. We propose to repeat the same runs with weights equal to one, so the following
settings are done:
Inventory adjustment time = WIPI adjustment time = 6
[weeks ]
(H.1)
Labor adjustment time = Vacancy adjustment time = 4
[weeks ]
(H.2)
Inv entory : 1 - 2 - 3 - 4 1:
80000.00
1
2
1:
3
4
1
2
3
4
2
3
4
50000.00
4
1
1
3
2
1:
Page 1
20000.00
0.00
50.00
100.00
Time
150.00
21:09
200.00
Sun, Jul 06, 2003
Figure H.3. Runs for Inventory with problematic and non-problematic desired supply line
equations with weights equal to one
236
Labor: 1 - 2 - 3 - 4 1:
2000.00
3
4
1:
2
1500.00
3
2
4
3
4
2
3
4
1
1
2
1:
1000.00
Page 1
1
0.00
50.00
100.00
Time
150.00
21:09
200.00
Sun, Jul 06, 2003
Figure H.4. Runs for Labor with problematic and non-problematic desired supply line
equations with weights equal to one
Now it is more obvious that second and fourth (Equation (10.103) and Equation
(10.104) are used) runs are better than the first and the third runs (Equation (10.58) and
Equation (10.91) are used). Using new equations stabilizes both Inventory and Labor. One
can also conclude that using flexible workweek (Schedule pressure on\off is one, third and
fourth runs) is better than constant workweek (Schedule pressure on\off is zero, first and
second runs).
237
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Yaşarcan, H., 2003, Feedback, Delays and Non

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