Yaşarcan, H., 2003, Feedback, Delays and Non
Transkript
Yaşarcan, H., 2003, Feedback, Delays and Non
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. v 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. vii 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 viii 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 7.1. Parameter Values for All Runs in this Chapter .................................................... 51 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 8.1. Parameter Values for All Runs in this Chapter .................................................... 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 ix 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 9.1. Parameter Values for All Runs in this Chapter .................................................... 76 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 x 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 xi 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 xiii 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 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 ....... 45 Figure 6.9. 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 2 ......... 46 Figure 6.10. 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 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 respectively for the 1st, 2nd, 3rd and 4th runs) for discrete supply line 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 xvi 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. Causal loop diagram of model in Figure 8.1................................................. 62 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. Causal loop diagram of model in Figure 9.1................................................. 78 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 Figure 12.15. Improved behaviors of Big S, s, Inventory position and Inventory obtained with Equation (12.58) and TSm = 40............................................ 145 Figure 12.16. Improved behaviors of Big S, s, Inventory position and Inventory obtained with Equation (12.60)................................................................... 145 Figure 12.17. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.57)................................................................... 146 Figure 12.18. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.58) and TSm = 40............................................ 146 Figure 12.19. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.60)................................................................... 147 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 Figure 12.21. Improved behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost obtained with Equation (12.58) and TSm = 40 .................................................................. 148 Figure 12.22. Improved behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost obtained with 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 Figure 12.25. Improved behaviors of Big S, s, Inventory position and Inventory obtained with Equation (12.60) and with autocorrelated Customer demand ........................................................................................................ 151 Figure 12.26. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.60) and with autocorrelated Customer demand ........................................................................................................ 151 Figure 12.27. Improved behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost obtained with 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 unreliable supply line model ....................................................................... 157 Figure 12.32. Behaviors of Total revenues, Net total profit and Total cost for the unreliable supply line model ....................................................................... 157 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.................................................................................................... 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 Figure 12.40. Behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost for improvement Equation (12.83), (12.84) and (12.27) ........................................................................ 163 xxiii Figure 12.41. Behaviors of Total purchasing cost, Total ordering cost, Total lost 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 Figure 13.26. Behaviors of Time horizon, Likelihood of accomplishment ratio, Effect 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 Figure 13.28. Behaviors of Time horizon, Likelihood of accomplishment ratio, Effect 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. The final values of the Total revenues, the Net total Profit and the costs at the end of simulation (8 years – 416 weeks), with pure random and with autocorrelated Customer demand................................................. 152 Table 12.3. The final values of the Total revenues, the Net total Profit and the 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 • S = Adjustment flow = 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 Individual delay time 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 averaging time Expected loss Expectation adjustment flow Figure 3.11. Stock-flow diagram of stock management atomic structure 15 Acquisition delay time ± Weight of supply line Supply line Acquisition flow - + - ± Stoc k - Supply line adjustment + + + ± - Control flow + + Stoc k adjustment + Loss flow + ± + Desired supply line Stoc k adjustment time Desired stock + Expectation adjustment flow - + Expected loss - + ± Expected loss averaging time 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 Acquisition delay time 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 Acquisition delay time Stock adjustment time 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 Acquisition delay time Stock adjustment time 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 Acquisition delay time Desired supply line Supply line adjustment Stock adjustment Weight of supply line Desired stock Stock adjustment time 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) The roots of the characteristic equation are: 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: Order of supply line Supply line 1 Supply line 2 Control flow Acquisition flow 1 Supply line Stock Loss flow Acquisition flow 2 Acquisition delay time Supply line adjustment Desired supply line Desired stock Weight of supply line Stock adjustment time 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) The roots of the characteristic equation are: 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 Acquisition delay time Life time Desired supply line Supply line adjustment Equilibrium value of loss Stock adjustment Weight of supply line Desired stock Stock adjustment time 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 Graph 1 (comparison) 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 Graph 1 (comparison) 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 Graph 1 (comparison) 120.00 Time 18:03 160.00 06 Jan 2003 Mon Figure 6.9. 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 2 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 Graph 1 (comparison) 250.00 Time 375.00 19:40 500.00 05 Jan 2003 Sun Figure 6.10. 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 LF (as the anchor in CF and SLS*) and TSA equal to 70 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 Graph 1 (comparison) 80.00 Time 120.00 18:03 160.00 06 Jan 2003 Mon Figure 6.11. 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 LF (as the anchor in CF and SLS*) and TSA equal to 2 • 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 Graph 1 (comparison) 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 Graph 1 (comparison) 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 Graph 1 (comparison) 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. 7.1. Parameter Values for All Runs in this Chapter The values of the model parameters are as follows for all runs in this chapter: • T AD = 16 [time units ] • 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 v ariable lif e time 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: Calculated life time 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: 2: Calculated life time 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 v ariable lif e time 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: 2: Smoothed lif e time 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 v ariable lif e time 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 Order of supply line Supply line adjustment Stock Acquisition delay time Desired supply line Loss flow Acquisition flow 2 Desired stock Life time Equilibrium value of loss Stock adjustment Smoothed life time Weight of supply line Stock adjustment 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 time Stock adjustment Information delay 2 Information adjustment flow 2 Information delay time Desired control flow Information delay 1 Information adjustment flow 1 Order of information delay 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. 8.1. 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: • S * = 5 [items ] • T AD = 4 [time units ] 61 • TID = T AD = 4 [time units ] • TSA = 2 [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 Graph 1 (comparison) 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 adjustment flow 2 + - - - - + Information delay 2 Information adjustment flow 1 - + + Desired control flow Control flow + Stoc k adjustment - Stoc k + Figure 8.4. Causal loop diagram of model in Figure 8.1 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) and from above we obtain: • •• • 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) Above equation can be simplified to the following: •• • (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 Stock adjustment time Desired stock Stock adjustment Weight of VSL Desired control flow VSL adjustment Desired VSL Virtual supply line Information delay 2 Virtually adjusted DCF Information delay 1 Information adjustment flow 2 Information adjustment flow 1 Order of information delay Information delay time 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 Graph 1 (comparison) 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) Above equation can be simplified to the following: •• (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 Productivity coefficient Stock adjustment time Stock adjustment Desired control flow 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 Productivity coefficient 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. 9.1. Parameter Values for All Runs in this Chapter The values of the model parameters are as follows for the runs in this chapter, unless it is stated otherwise: • T AD = 6 [time units ] • TID = T AD = 6 [time units ] • TSSA = TSAD = TID / 2 = T AD / 2 = 3 [time units ] • TSA = 4 [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 Graph 1 (comparison) 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 supply line adjustment Secondary supply line - + + Secondary acquisition flow + Secondary control flow - + + Secondary stock Secondary stock adjustment + - + Desired secondary stock + Control flow + Desired control flow Stock + Stock adjustment - Figure 9.4. Causal loop diagram of model in Figure 9.1 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) and from above we obtain: • •• 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) Above equation can be simplified to the following: •• • (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 + TID • LF − C P • TID • SS * 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 + TID • LF − C P • TID • SS * 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 Productivity coefficient Desired stock Weight of VSL Stock adjustment time Stock adjustment Desired control flow Virtually adjusted DCF Desired secondary stock VSL adjustment Virtual supply line Secondary stock adjustment Secondary stock adjustment time Secondary supply line Secondary stock Secondary loss flow Desired VSL Secondary control flow Secondary acquisition flow Secondary acquisition delay time Secondary supply line adjustment Desired secondary supply line Weight of secondary supply line Figure 9.5. 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 Graph 1 (comparison) 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 production start rate 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 Inventory adjustment time = 7 Labor adjustment = [weeks] Desired labor − Labor Labor adjustment time 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 Labor adjustment time (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 Inventory adjustment time (10.34) Desired VSL for SS − Virtual supply line for SS Inventory adjustment time (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 Inventory adjustment time (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: • Weight of trainees = 1 • Weight of WIPI = 0 • Weight of VSL for ID = 0 • Weight of VSL for SS = 0 Third run (the standard system dynamics formulation): • Weight of trainees = 1 • Weight of WIPI = 1 • Weight of VSL for ID = 0 • Weight of VSL for SS = 0 Fourth run: • Weight of trainees = 1 • Weight of WIPI = 1 • Weight of VSL for ID = 1 • Weight of VSL for SS = 0 Fifth run (virtual supply line adjustments optimally considered): • Weight of trainees = 1 • Weight of WIPI = 1 96 • Weight of VSL for ID = 1 • Weight of VSL for SS = 1 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 = Desired production start rate 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 Inv entory adjustment time 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 Desired production start 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 • OPP11(0 ) = (Total initial4 ) / 3 • OPP12(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 Inventory adjustment time (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 Labor adjustment time Vacancy adjustment time Willingness to lay of f Adjustment f or labor Hourly productiv ity Desired hiring rate Expected attrition rate Expected productiv ity Desired labor Desired production start rate 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 Desired production start rate Weight of VSL Desired VSL Labor adjustment time Expected attrition rate Virtually adjusted DPSR Virtual Supply Line Vacancies Labor Inv entory adjustment time Hourly productiv ity 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) Inventory adjustment time Also the Desired labor formulation in Equation (10.89) changes to include virtual supply adjustment in decision: Desired labor = Virtually adjusted DPSR 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) = Desired production start rate 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: • Schedule pressure on\off = 0 • Weight of VSL = 1 Third run (standard system dynamics formulation): • Schedule pressure on\off = 1 • Weight of VSL = 0 Fourth run: • Schedule pressure on\off = 1 • Weight of VSL = 1 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 Stock adjustment time Desired stock Stock adjustment Weight of VSL Desired control flow VSL adjustment Desired VSL Virtually adjusted DCF Virtual supply line Virtual outflow Information delay 2 Information adjustment flow 2 Information delay time Virtual inflow Information delay 1 Information adjustment flow 1 Order of information delay 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 4: Virtual supply line 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 4: Virtual supply line 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: 4: Virtual supply line 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 4: Virtual supply line 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 4: Virtual supply line 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 4: Virtual supply line 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 Stock adjustment time Desired stock Stock adjustment Weight of VSL Desired control flow VSL adjustment Desired VSL Virtually adjusted DCF Virtual supply line Virtual adjustment flow 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 time Stock adjustment Desired acquisition rate Weight of VSL VSL adjustment Desired VSL Virtually adjusted DAR Virtual supply line Virtual adjustment flow Virtual adjustment time 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 Expectation adjustment flow Desired secondary stock Production Expectation adjustment time Secondary stock adjustment Productivity Random shocks 2 Secondary stock Secondary loss flow Secondary life time Perception delay time 2 Secondary stock adjustment time Secondary supply line Secondary acquisition flow Secondary control flow Secondary acquisition Random shocks 1 delay time Perceived secondary loss flow Secondary supply line 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 Expectation adjustment time (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) Secondary supply line Secondary acquisition delay time (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 adjustment time 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 Secondary stock adjustment time (11.47) Secondary stock adjustment time = 2 (11.48) Secondary supply line adjustment = Desired secondary supply line − Secondary supply line WSSL • Secondary stock adjustment time (11.49) Stock adjustment = Desired stock − Stock Stock adjustment time (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 Stock adjustment time 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 Acquisition delay time 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 Expectation adjustment flow Number of weeks in a year Expectation adjustment time Customer demand Mean absolute deviation Acquisition delay time 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 ) Expectation adjustment time (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 Acquisition delay time 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: 2: Net total prof it 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 3: Inv entory position 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 3: Inv entory position 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 Figure 12.15. Improved behaviors of Big S, s, Inventory position and Inventory obtained with Equation (12.58) and TSm = 40 1: Big S 1: 2: 3: 4: 2: s 1 1 2 1: 2: 3: 4: 3: Inv entory position 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 Figure 12.16. Improved behaviors of Big S, s, Inventory position and Inventory obtained with Equation (12.60) 146 1: Total rev enues 1: 2: 3: 2: Net total prof it 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 Figure 12.17. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.57) 1: Total rev enues 1: 2: 3: 2: Net total prof it 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 Figure 12.18. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.58) and TSm = 40 147 1: Total rev enues 1: 2: 3: 2: Net total prof it 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 Figure 12.19. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.60) 1: Total purchasin… 2: Total ordering c… 3: Total lost sales … 4: Total inv entory … 5: Total perishing … 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: Total purchasin… 2: Total ordering c… 3: Total lost sales … 4: Total inv entory … 5: Total perishing … 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 Figure 12.21. Improved behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost obtained with Equation (12.58) and TSm = 40 1: Total purchasin… 2: Total ordering c… 3: Total lost sales … 4: Total inv entory … 5: Total perishing … 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 Figure 12.22. Improved behaviors of Total purchasing cost, Total ordering cost, Total lost 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 $ Total purchasing cost 70,500,266,986 $ 976,310 $ 1,333,639 $ 1,252,864 $ Total ordering cost 60,000 $ 297,000 $ 369,000 $ 372,000 $ Total lost sales cost 36,208 $ 114,719 $ 69,864 $ 52,113 $ Total inventory storage cost 22,733,266,986 $ 83,748 $ 164,903 $ 124,501 $ Total perishing cost 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 3: Inv entory position 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 Figure 12.25. Improved behaviors of Big S, s, Inventory position and Inventory obtained with Equation (12.60) and with autocorrelated Customer demand 1: Total rev enues 1: 2: 3: 2: Net total prof it 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 Figure 12.26. Improved behaviors of Total revenues, Net total profit and Total cost obtained with Equation (12.60) and with autocorrelated Customer demand 152 1: Total purchasin… 2: Total ordering c… 3: Total lost sales … 4: Total inv entory … 5: Total perishing … 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 Figure 12.27. Improved behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost obtained with Equation (12.60) and with autocorrelated Customer demand Table 12.2. The final values of the Total revenues, the Net total Profit and the costs at the 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 $ Total purchasing cost 1,252,864 $ 1,284,474 $ Total ordering cost 372,000 $ 369,000 $ Total lost sales cost 52,113 $ 64,574 $ Total inventory storage cost 124,501 $ 143,784 $ Total perishing cost 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 Acquisition delay time 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 3: Inv entory position 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: 2: Net total prof it 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 3: Inv entory position 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 3: Inv entory position 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: 2: Net total prof it 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: 2: Net total prof it 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: Total purchasing cost 1: 2: 3: 4: 2: Total ordering cost 3: Total lost sales cost 4: Total inv entory stora… 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 Figure 12.40. Behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost for improvement Equation (12.83), (12.84) and (12.27) 1: Total purchasing cost 1: 2: 3: 4: 2: Total ordering cost 3: Total lost sales cost 4: Total inv entory stora… 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 Figure 12.41. Behaviors of Total purchasing cost, Total ordering cost, Total lost sales cost and Total inventory storage cost for improvement Equation (12.25) and (12.87) 164 Table 12.3. The final values of the Total revenues, the Net total Profit and the costs at the 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 $ Total purchasing cost 213,277 $ 996,102 $ 996,346 $ Total ordering cost 36,000 $ 177,000 $ 171,000 $ Total lost sales cost 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 Desired control flow Expectation adjustment flow Expected loss averaging time Stock adjustment time 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 Effect of desired CF Effect of motivation Desired control flow Expectation adjustment flow Implicit goal Expected loss averaging time Stock adjustment time Stock adjustment Ideal goal Stated goal Goal adjustment flow Goal adjustment time 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 Effect of desired CF Effect of motivation Desired control flow Expectation adjustment flow Expected loss averaging time Stock adjustment time Implicit goal Stock adjustment Ideal goal Stated goal Inf Delay for IG 2 Inf Delay for IG 1 Goal adjustment flow Information AF 2 Traditional performance Traditional performance formation Information AF 1 Traditional performance formation time Order of information delay Goal adjustment 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 Effect of desired CF Effect of motivation Desired control flow Expectation adjustment flow Expected loss averaging time Stock adjustment time Stock adjustment Implicit goal Inf Delay for IG 2 Traditional performance Inf Delay for IG 1 Traditional performance formation Goal adjustment flow Information AF 2 Information AF 1 Traditional performance formation time Indicated goal Stated goal Ideal goal Order of information delay Goal adjustment time Weight of stated 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: 5: Traditional perf o… 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 Effect of desired CF ~ Effect of motivation Desired control flow Expectation adjustment flow Expected loss averaging time Stock adjustment time Stock adjustment Traditional performance Implicit goal Traditional performance formation Goal adjustment time Traditional performance formation time Formation of perception time Indicated goal Perceived performance Stated goal Weight of 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 5: Traditional perf o… 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 5: Traditional perf o… 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 Short term accomplishment ratio 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 5: Traditional perf o… 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 5: Traditional perf o… 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 5: Traditional perf o… 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 2: Likelihood of accomp… 3: Ef f ect of motiv ation 3 3 3 4: Ef f ect of short term … 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 5: Traditional perf o… 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 ~ Effect of desired CF ~ Expected loss Effect of motivation Desired control flow Expectation adjustment flow Expected loss averaging time Stock adjustment time Stock adjustment Traditional performance Implicit goal Traditional performance formation Goal adjustment time Indicated goal Traditional performance formation time Formation of perception time Ideal goal Perceived performance SG adjustment time Stated goal Weight of stated goal Reference level ~ Effect of short term motivation Likelihood of accomplishment ratio Indicated stated goal Reference level formation time Short term accomplishment ratio 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 2: Likelihood of accomp… 3: Ef f ect of motiv ation 3 1 3 3 4: Ef f ect of short term … 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 5: Traditional perf o… 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 2: Likelihood of accomp… 3: Ef f ect of motiv ation 3 1 3 3 4: Ef f ect of short term … 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 Figure 13.26. Behaviors of Time horizon, Likelihood of accomplishment ratio, Effect of 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 5: Traditional perf o… 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: 2: Likelihood of accomp… 3: Ef f ect of motiv ation 200.00 4.00 1.00 3 4: Ef f ect of short term … 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 Figure 13.28. Behaviors of Time horizon, Likelihood of accomplishment ratio, Effect of 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 The differential equation of the model in Figure C.4 can be given as: • 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 The differential equation of the model in Figure C.6 can be given as: • S = Adjustment flow = Discrepancy Goal − S = Adjustment time Adjustment time (C.5) Possible behaviors of the model in Figure C.6 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 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 The differential equation of the model in Figure C.14 can be given as: • 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 The differential equations of the model in Figure C.17 can be given as: • 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 The differential equation of the model in Figure C.21 can be given as: • 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 Individual delay time 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 The differential equation of the model in Figure C.28 can be given as: • S = Adjustment flow = 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 Individual delay time Delay time Order of information delay Figure C.29. Stock-flow diagram of third order information delay atomic structure 215 The differential equations of the model in Figure C.29 can be given as: 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. The differential equations of the model in Figure C.31 can be given as: • Inflow1 − Outflow1 S1 • = S Inflow − Outflow 2 2 2 (C.18) Productivity • S 2 − Outflow1 = Fraction • S − Consumption multiplier • S 2 1 Possible behaviors of the model in Figure C.31 are: 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 Acquisition delay time Stock adjustment Supply line adjustment Desired supply line Expected loss averaging time Expected loss Stock adjustment time Weight of supply line Expectation adjustment flow Figure C.36. Stock-flow diagram of stock management atomic structure Acquisition delay time ± Weight of supply line Supply line Acquisition flow - + - ± Stoc k - Supply line adjustment + + + ± + - Control flow + Stoc k adjustment + Loss flow + ± + Desired supply line Stoc k adjustment time Desired stock + Expectation adjustment flow + Expected loss - - + ± Expected loss averaging time Figure C.37. Causal-loop diagram of stock management atomic structure 220 The differential equations of the model in Figure C.36 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 (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 Possible behaviors of the model in Figure C.36 are: 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 adjustment time Goal Control flow Goal adjustment flow Stock adjustment time Figure C.39. Stock-flow diagram of goal setting atomic structure - - + ± Control flow + Stock adjustment time Stock + Goal Goal adjustment time + Goal adjustment flow + - ± - 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. The differential equations of the model in Figure C.39 can be given as: 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: 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 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 + and from above we obtain: • 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) From Equation (F.8) we can obtain the following equation: • IDS1 = TID 2 • IDS 2 + IDS 2 (F.9) 229 and from above we obtain: • •• • IDS 1 = TID 2 • IDS 2 + IDS 2 (F.10) Equation (F.9) and Equation (F.10) can be inserted to Equation (F.7), and then simplified to obtain the following equation: •• 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) 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 . 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) From Equation (F.14) we can obtain the following equation: • SSLS = TSAD • SS + TSAD • SLF (F.15) and from above we obtain: • •• SSLS = TSAD • SS (F.16) Equation (F.15) and Equation (F.16) can be inserted to Equation (F.13), and then simplified to obtain the following equation: •• 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) 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 . 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): • Schedule pressure on\off = 0 • Equation (10.58) is used for Desired WIPI • Equation (10.91) is used for Desired vacancies Second run: • Schedule pressure on\off = 0 • Equation (10.103) is used for Desired WIPI • Equation (10.104) is used for Desired vacancies Third run (equations as suggested in Sterman, 2000): • Schedule pressure on\off = 1 • Equation (10.58) is used for Desired WIPI • Equation (10.91) is used for Desired vacancies Fourth run: • Schedule pressure on\off = 1 • Equation (10.103) is used for Desired WIPI 234 • Equation (10.104) is used for Desired vacancies 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. 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