Modeling, Analysis, and Design of Responsive Manufacturing … · 2020-01-16 · Modeling,...

Modeling, Analysis, and Design of Responsive Manufacturing

Systems Using Classical Control Theory

by

Nga Hin Benjamin Fong

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

In partial fulfillment of the requirements leading to the degree of

Doctor of Philosophy (Ph.D.)

in

Industrial and Systems Engineering (Manufacturing Systems Option)

_____________________________ ____________________________ Co- Chair: Dr. John P. Shewchuk Co-Chair: Dr. Robert H. Sturges

_____________________ _____________________ Dr. F. Frank Chen Dr. Ting-Chung Poon

______________________ Dr. Harry H. Robertshaw

April 15, 2005

Blacksburg, Virginia

Keywords: Classical Control Theory, System Responsiveness, Responsive Manufacturing Systems

Modeling, Analysis, and Design of Responsive Manufacturing Systems

Using Classical Control Theory

by

Nga Hin Benjamin Fong

(ABSTRACT)

The manufacturing systems operating within today’s global enterprises are invariably dynamic and complicated. Lean manufacturing works well where demand is relatively stable and predictable where product diversity is low. However, we need a much higher agility where customer demand is volatile with high product variety. Frequent changes of product designs need quicker response times in ramp-up to volume. To stay competitive in this 21st century global industrialization, companies must posses a new operation design strategy for responsive manufacturing systems that react to unpredictable market changes as well as to launch new products in a cost-effective and efficient way. The objective of this research is to develop an alternative method to model, analyze, and design responsive manufacturing systems using classical control theory. This new approach permits industrial engineers to study and better predict the transient behavior of responsive manufacturing systems in terms of production lead time, WIP overshoot, system responsiveness, and lean finished inventory. We provide a one-to-one correspondence to translate manufacturing terminologies from the System Dynamics (SD) models into the block diagram representation and transfer functions. We can analytically determine the transient characteristics of responsive manufacturing systems. This analytical formulation is not offered in discrete event simulation or system dynamics approach. We further introduce the Root Locus design technique that investigates the sensitivity of the closed-loop poles location as they relate to the manufacturing world on a complex s-plane. This subsequent complex plane analysis offers new management strategies to better predict and control the dynamic responses of responsive manufacturing systems in terms of inventory build-up (i.e., leanness) and lead time. We define classical control theory terms and interpret their meanings according to the closed-loop poles locations to assist production management in utilizing the Root Locus design tool. Again, by applying this completely graphic view approach, we give a new design approach that determine the responsive manufacturing parametric set of values without iterative trial-and-error simulation replications as found in discrete event simulation or system dynamics approach.

Acknowledgements

It has been a challenging and fruitful journey to return to Virginia Tech to study my Ph.D. program in Industrial and Systems Engineering. I am very thankful to have such a unique and multi-disciplinary research committee which they have taught me how to think, learn, and communicate. I would like to take this opportunity to thank them for their guidance and support which greatly enhanced the value of my eight year journey. Dr. John P. Shewchuk, my Co-Chairman of the Committee, has spent tremendous amount of time and effort to guide me through this research journey. From initial research ideas to models implementation, from model validation to dissertation writing, he has provided a lot of good advice and help to make my Ph.D. commencement happening. In particular, his input and concern on validating CCT models to discrete manufacturing world is a crucial part to implement our new methodology from the industrial engineering point of view. Lastly, his high standard writing style has made my dissertation so completed. My heartfelt thanks to him! Dr. Robert H. Sturges, my Co-Chairman of the Committee, is truly a role model in both teaching and research excellence. His innovative ideas, kindness, courage and motivation have made me to realize how fortunate I am to have him to be my co-advisor. From academic research to industrial projects, from departmental politics to my personal issues, we can spend numerous hours to chat and discuss without feeling the time has gone so fast. He has transformed me from a below average graduate student to become awards winning research scholar. In Chinese term, he is my “Inspirational Master”!! Truly, I may never be able to complete translating CCT terminologies to manufacturing world without him. That “special Friday talk” outside Durham in September 2003 has significantly changed my career life. Thanks Dr. Bob! ☺ Dr. Harry H. Robertshaw, my former advisor for my MS in Mechanical Engineering, has guided me through many challenges throughout the past 13 years. From my MS research work to Labor Certification supporting letter, from preliminary research to latest Intel Case Study, it is unquestionable that his willingness to support is vital. I especially appreciate his extra time and effort to assist me to formulate the block diagrams and algebraic expressions for my dissertation work. Definitely, he has spent at least three times amount of time to assist my Ph.D. work than my MS program. Big thanks to him! Dr. F. Frank Chen, my most respectful industrial-managerial, research professor, has taught me one should have long term visions and plans to be success in both academia and industrial world. As the founder of the Center for High Performance Manufacturing (CHPM), Dr. Chen has provided me the complete financial support through my three years of full-time studies. Besides, he gave tremendous advice and help to make my dissertation completed. I look forward to follow his successful foot steps to work for Caterpillar at Peoria, IL. Many thanks to him!! Dr. T.C. Poon, my most admirable professor and long-time friend, has been advising me since I came to Virginia Tech in Fall 1992. Throughout these 13 years in Blacksburg, there are uncountable incidents that required his help and advice to get over the challenges. I really

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appreciated to have him to serve in my dissertation committee. Although the work of modeling manufacturing systems via electrical circuit network has not completed, it is noteworthy to continue these ideas for future work. Hopefully, my upcoming design engineering position will enhance my skills to work with network modeling and frequency response analysis. Thanks very much Dr. Poon! Special thanks to my former boss, Graham Swinfen, an engineering manager at BBA Friction, Inc. in Dublin, VA. Without his persistent help, I would never work at BBA Friction to get my green card and have the opportunity to begin my PhD study at Virginia Tech. From Fall 1997 to Spring 2000, I was able to drive back-and-forth for two hours to go to work-school-work-home regularly. I still remembered how much heat Graham had to take to support of the justification of my continuous education at Virginia Tech giving the time and the financial support from BBA Friction, Inc. Big thanks to Graham! Throughout my eight years of study (3.5 years part-time, 1 year off, 3.5 years full-time), I met so many helpful and interesting colleagues among the student group. Special thanks go to Nathan Ivey and Hitesh Attri in assisting me for the ARENA programming. I wish the best luck to Nate, Hitesh and Radu Babiceanu for their job hunting. It is very thankful to have known Dr. Y.A. Liu and Dr. Hing-Har Lo (Mrs. Liu) through the VT Chinese Bible Study since Fall 1992. They have been acting like my guardian throughout the years – give me the spiritual support and advice while keeping “little Ben” behaves well. ☺ Again, many thanks to Dr. T.C. Poon and Eliza Poon (Mrs. Poon), they are like my older brother and sister via Hong Kong Club and VT Chinese Bible Study. They gave me so many advices and ideas for my daily life, such as school work, buying house, raising kids, career development, retirement plan, etc. Give thanks to the Lord, I have learned so much from these two lovely families! It has been a blessing to get the role to lead and care many young Hong Kong students through the VT Chinese Bible Study and HK Club. Best wishes to my spiritual brothers, Carlos Siu, Henry Yuen, and Winston Ma for their first career challenge as engineers and/or statistical analyst. I enjoyed those uncountable hours to share our joy and sadness while we were in Blacksburg. I especially missed our weekly soccer games at Tech! No doubt in my mind, my life will ever reach to this stage without the support from my family. Thanks be to my Lord to provide me such a lovely and heart-bonded family. I would like to express my heartfelt appreciation to my parents, Hoo-Shin Fong and Shau-Shan Lai, for their unyielding love and support since I was born. My bond with my parents grew even stronger when I left home to England since 1985. They provided me with tremendous mental and financial support through these years. Particularly in the past three years, after their retirement from Hong Kong, they even came to stay with my family to help baby-sit their lovely grandchildren. For sure, their physical support has allowed me to concentrate on my research while my daughters are often crying for milk. I also give thanks to the Lord to giving me such a wonderful and supportive elder brothers, Ricky and Joe. We have grown up together back in Hong Kong, then we all went to Rishworth for high

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school in England, and we all came to States for gaining our higher education. Throughout these years, we have shared and supported each other via visit, phone calls, and prayers. It is so important to maintain this high degree of brother-hood to be able to face all those up-and-down in life. As Ricky said, I shall be thankful to have completed my Ph.D. as a by-product due to my long-waiting green card application process. While Joe always reminds me that getting a Ph.D. is just a beginning for the next chapter of life. Thank you my dearest brothers!! I also give thanks to my two lovely sisters-in-law, Susanna and Florence, for their prayers, supports, and sharing in all these years. Lastly, I would dedicate this dissertation to my beloved wife, Iris (Ching) for her unyielding love, support and care to make me become Dr. Fong! I give thanks to my Lord to give me such an understandable and dedicated wife and mother. Iris and I had gone through so many challenges since we were together in Spring 1995. Our life faced a lot of challenge in the early stage, such as, I worked over 70+ hours weekly in Hazard, KY, then she worked over 70+ hours in Hotel Roanoke, VA. Sure, we were just a cheap-labor whom decided to live in US. By summer 1998, we got married and began our next challenge for the school work at Tech. I began my part-time PhD study while I was working full-time at BBA Friction and she returned to Virginia Tech to study her MS in Accounting and Information Systems in Spring 1998. The most difficult challenge was to study together almost every night at Durham Hall until 4 am while I still had to return to work by 8 am. It is so thankful to have her support and patience to host numerous Friday night gathering with those HK students right after the bible study. We both learned so much and became more mature to take care this young students group. Without a doubt, Iris has sacrificed her career twice to choose a less competitive and lower-paid job to stay in Blacksburg to give me time to finish my PhD program. Her unique encouraging style by keep reminding me “not to waste time and move on” has made me even stronger and more self-confidence to continue to pursue my PhD program. ☺ I give thanks to her to be such a caring mother and daughter-in-law to help taking care our two lovely daughters, Vera and Audrey, and my parents while I was busy preparing my dissertation work. Iris will probably give up her career for the third time when we move to Peoria, IL. But I am sure that the kids must love to see Mommy spend more time at home. ☺ Thank you Lord for giving me such a loving family, I would never complete my PhD study without their support. My final sharing for those whom love to get a PhD degree, you should equip the following elements to be succeeded, such as willingness, hard-work, discipline, persistence, research topic, supportive professor committee and most importantly, communication skills. God Bless America!! ☺

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Table of Contents

Chapter 1 Introduction............................................................................................................1

1.1 Background....................................................................................................................1

1.2 Problem Statement .........................................................................................................5

1.3 Research Objectives.......................................................................................................6

1.4 Contents of Dissertation.................................................................................................7

Chapter 2 Literature Review ..................................................................................................9

2.1 Agile and Responsive Manufacturing............................................................................9

2.2 Early Development in System Dynamics ..........................................................................13

2.3 Recent Applications of System Dynamics...................................................................14

2.4 Input-Output Analysis in modeling production-inventory systems.............................16

2.5 Other approaches to model dynamic manufacturing systems ..................................................20

2.6 Missing link of the existing modeling approaches ......................................................20

Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems......................23

3.1 Modeling and Analysis via System Dynamics……………………………………. ................24

3.1.1 System Dynamics Approach ………………………………………………......24

3.1.2 A Single-Stage Production Control System ……………….. ………………....26

3.1.3 A Basic Kanban System Model ……………….. …………………………… ..28

3.1.4 A Two-Stage Production Control System ……………………………………..31

3.1.5 A Two-Stage Production Control System with Time Delay ………………… .33

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3.2 Modeling and Analysis via Classical Control Theory ………………………….…................36

3.2.1 Fundamentals of Classical Control Theory …………………….. ………….................36

3.2.2 Transfer Function of a Single-Stage Production Control System ………….. ................37

3.2.3 Transfer Function of a Basic Kanban System Model ……………………… ....40

3.2.4 Transfer Function of a Two-Stage Production Control System …………….....48

3.2.5 Transfer Function of a Two-Stage Production Control System w/ Time Delay 54

3.3 Guidelines to Translate Responsive Manufacturing Systems via CCT ………….…..............58

Chapter 4 Model Validation ………………………………………………. .. 61

4.1 Discrete Event Modeling of a Single-Stage Production Control System ....................63

4.2 Comparison between Discrete Event Model and Classical Control Model.................67

Chapter 5 Design of Responsive Manufacturing Systems ………………. .. 72

5.1 Root Locus Analysis and Design of a Two-Stage Production System ….………......73

5.2 Interpretation of CCT Terms to the Manufacturing World ………………………. ...81

5.3 Root Locus Design of a Two-Stage Production System with Time Delay ….……....87

5.4 Guidelines to perform Root Locus Design in Responsive Manufacturing System … 96

Chapter 6 Industrial Case Study ………………………….………………. . 98

Chapter 7 Conclusions and Research Contributions ...................................116

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Chapter 8 Future Research.............................................................................119

References .........................................................................................................122

Vita ....................................................................................................................128

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Lists of Figures Figure 1.1: Existing methods to model and analyze responsive manufacturing systems..........5

Figure 3.1: A single-stage production control system …………………………………… ....27

Figure 3.2: Step response of a single-stage production control system ……………………..28

Figure 3.3: A basic kanban system model ………………………………………………. .....29

Figure 3.4 a & b: A step input response of a kanban system model ………………………...31

Figure 3.5: A two-stage production control system ………………………………………....31

Figure 3.6: Step response of a two-stage production control system ………………….….. ..33

Figure 3.7: A two-stage production control system with a 3rd-order time delay ………….. ..34

Figure 3.8: Step response of a two-stage production system with a 3rd-order time delay …...35

Figure 3.9: Subcomponents of BD representation of a single-stage production system ….. .37

Figure 3.10: Complete block diagram presentation of a single-stage production system …. .38

Figure 3.11: Step-by-step block diagram reduction into a single TF block .. ……………….39

Figure 3.12: Subcomponents of block diagram representation of a basic kanban system …. 40

Figure 3.13: Complete block diagram representation of the basic kanban system ………….41

Figure 3.14: Block diagram reduction into a single transfer function block ………………. .41

Figure 3.15: Step Response of a basic kanban system model with different LT values …….47

Figure 3.16: Subcomponents of BD representation of a two-stage production system …….. 48

Figure 3.17: Complete BD representation of a two-stage production control system ……… 48

Figure 3.18: Family curves of step response for a two-stage production control system …...53

Figure 3.19: BD representation of a two-stage production system with 1st-order Delay ….. .55

Figure 3.20: BD representation of a two-stage production system with 3rd -order Delay …. 55

Figure 3.21: Dynamic Responses between 1st-order time delay and 3rd-order time delay …. 57

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Figure 4.1: A Single-Stage Production Control System …………………………………… .64

Figure 4.2: Step Response of a Single-Stage Production Control System ………………… .64

Figure 4.3: ARENA discrete-event model of a single-stage production control system …....65

Figure 4.4: A single-stage production system (Ship every 2 hr; Production update 4 hr) …..67

Figure 4.5: A single-stage production system (Ship every 4 hr; Production update 4 hr).......68

Figure 4.6: A single-stage production system (Ship every 8 hr; Production update 4 hr)…...68

Figure 4.7: A single-stage production system (Ship every 24 hr; Production update 4 hr) ....69

Figure 4.8: Discrete vs. Continuous in modeling single-stage production system …………. 70

Figure 5.1: A system for Root Locus ……………………………………………………….. 74 Figure 5.2: Root-Locus Plot of a Two-Stage Production Control System …………………. 75

Figure 5.3: Contour Plot of ζ values as a function of LT and WAT ………………………. .78

Figure 5.4: Contour Plot of Breakaway Points as a function of LT and WAT ……………...79

Figure 5.5: Step Response Comparison of a Two-Stage Production Control System……… ..............80

Figure 5.6: Complex s-plane interpretation of varying location of closed-loop poles ……. ..83

Figure 5.7: Transient mode shapes associated with locations of roots in complex s-plane.....86

Figure 5.8: Root-Locus Plot of a two-stage production system with 3rd-order time delay ….89

Figure 5.9: Step Responses of a two-stage system with 3rd-order time delay under

various K values ……………………………………………………………………………. 91

Figure 5.10: Root-Locus Plot of a two-stage production system w/ 3rd-order time delay…... 93

Figure 5.11: Various Step Response of a two-stage production control system

with 3rd-order time delay …………………………………………………………………….94

Figure 6.1: Hybrid Push-Pull Intel Semiconductor Production System …………………… 99

Figure 6.2: System Equations for 1st Stage Process – Fabrication WIP…… ....................... 102

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Figure 6.3: System Equations for 2nd Stage Process – Assembly WIP ................................. 102

Figure 6.4: System Equations for 3rd Stage Process – Finished Inventory............................ 103

Figure 6.5: Block Diagram Representation of a Three-Stage Semiconductor Production

System ................................................................................................................................... 103

Figure 6.6: Simplified BD Representation of a Three-Stage Semiconductor Production

System… ............................................................................................................................... 104

Figure 6.7: Simplified BD Representation with block K of the Three-Stage Semiconductor

System ................................................................................................................................... 107

Figure 6.8: Step Response of a three-stage production system with different parametric

set values................................................................................................................................ 109

Figure 6.9: Root Locus of a three-stage semiconductor production system (Set B) ............. 110

Figure 6.10: Root Locus of a three-stage semiconductor production system (Set D) ........... 110

Figure 6.11: Step Response of a three-stage production system that yields instability

behavior.................................................................................................................................. 113

Figure 6.12: Root Locus of a three-stage semiconductor production system (Set E)............ 113

List of Tables Table 3.1: Box-Behnken Design of Experiment for a Basic Kanban System Model ………. 45

Table 3.2: Box-Behnken Design Factors Effect Responses Summary ……………………...46

Table 3.3: 25-1(Rev V) fractional factorial design factors effect responses summary ……… 51

Table 6.1: Parametric values for a three-stage semiconductor production system ……….. 108

Table 8.1: Common variables used in system modeling ...................................................... 120

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Chapter 1 Introduction N.H. Ben Fong

Chapter 1 Introduction

1.1 Background

Many enterprises have practiced the lean thinking paradigm to enhance the efficiency of their

business processes. In recent years, an agile manufacturing paradigm has been underlined as an

alternative to, and possibly an improvement on, leanness. Christopher and Towill [1] have

described that lean concepts work well where demand is relatively stable and predictable where

product diversity is low. In contrast, when customer requirement for variety is high and volatile,

a much higher level of agility is required. Helo [2] defined agile manufacturing as the capability

of reacting to unpredictable market changes in a cost-effective way, simultaneously prospering

from the uncertainty. In many manufacturing companies, dynamically changing markets are

demanding more differentiated products in lower volumes and within less production lead time.

Any uncertain conditions challenge the dynamic response of manufacturing systems. Enterprises

have to deal with high seasonal rise and fall in demand. Frequent changes of product designs and

complex products need quicker response times in ramp-up to volume. To stay competitive in this

21st century global industrialization, companies require responsive manufacturing systems that

can react to unpredictable market changes as well as launch new products economically and

efficiently. Responsive manufacturing systems yield shorter production lead time, minimal

inventory build-up and related cost, better overall dynamic system behavior and thus lead to

excellent customer satisfaction.

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In order to have responsive manufacturing systems, we must develop a methodology or an

approach which permits engineers to mathematically model, analyze, and design to such systems.

In most manufacturing systems, engineers often categorize models by their computational form,

either analytical or experimental. Analytical models represent a mathematical abstraction of the

real manufacturing systems. A set of equations is formulated that summarizes the aggregate

performance of the system models. Simulation models are experimental and mimic the events

that occur in the real system.

Queuing network analysis is the most common analytical method to find rough-cut or quickly

evaluate average steady-state performance of manufacturing systems [3]. A network of queues is

a system in which materials arrive at a queue, wait until they are processed, and then move to the

next processing stage of a system. Queuing network models are built upon steady-state

probability distributions, often having Poisson arrivals and exponentially distributed processing

time. Gershwin [4] stated that there are limitations in applying queuing network. Blocking would

not occur due to the assumption of infinite buffer size. In addition, queuing models only work for

a limited set of queue disciplines. Furthermore, they do not generally allow a system controller to

observe the queue length or service duration at one station, and change the control policy of the

whole system or of another station. Hence, when a particular processing stage takes an unusual

long operation, the overall system is not permitted to take any kind of action in response.

Moreover, most real manufacturing systems are complex with multiple system feedback loops

under transient conditions, such systems are too complicated to analytically derive mathematical

formulation by queuing network analysis.

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Discrete event simulation (DES) is the most popular tool for modeling and analyzing dynamic

manufacturing systems at a very detailed level. DES is typically characterized by queues, servers,

and probabilistic distributions of parameters such as arrival and service times. Unfortunately, the

DES method often requires too much time to construct models, perform simulation experiments,

and analyze results. In addition, DES is an event-based simulation and modeling tool, it does not

depend on the causality relationships among system variables. There is no analytical solution or

feedback nature for design and analysis purposes. It often requires the use of design of

experiments or trial-and-error iterative simulation replications to generate output performance for

decision-making. Moreover, the transient responses generated by DES generally consider as the

warm-up period behavior and its output performance measures are statistically evaluated at the

steady-state condition. DES does not predict apriori the dynamic characteristics of system

models under transient conditions, such as production settling time, WIP overshoot and lean

finished inventory level. Engineers have to rely on numerous simulation replications and large

numbers of data points to generate solutions for decision-making.

System Dynamics (SD) is an alternative technique developed by Jay W. Forrester [5] to build

dynamic system models. The term SD refers to a business system modeling technique that

employs causal loop diagrams (CLD) and stock-and-flow diagrams (SFD) to describe

information and materials flow. SD is based upon the concept of feedback thinking and control

engineering to the study of economic, business, and organizational systems. It builds on how

information flow, feedback-loops, and time delays within the structure of a system create

dynamic behavior. SD is a simulation-based modeling technique that requires numerous trial-

and-error iterations to study the dynamic behavior of responsive manufacturing systems. It does

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not provide any analytical solution to determine key transient system parameters and their

corresponding output measures. For most control engineers, a mathematical model of a dynamic

system is usually described in terms of differential equations based upon physical laws or

idealized constitutive relationships among system variables. For the latest literature review, there

is no one-to-one correspondence to model dynamic manufacturing systems between the

differential equation formulation and the CLD and SFD structures applied in the SD approach.

There is an alternative method called Input-Output Analysis developed by Dennis R. Towill [6,7]

to model dynamic manufacturing systems in block diagram representation. The input-output

analysis mimics the differential equation formulation. As stated in the literature review section,

the block diagram representation of the input-output analysis system models can convert into

transfer functions for control analysis under Laplace Transform domain. Those first-order lag

transfer functions are applied to model and describe supply chain dynamics and inventory-

production control systems. Lately, researchers apply the analogies of electrical circuit and fluid

system to model and study dynamic manufacturing systems behavior. The inter-relationship

among these modeling techniques is illustrated as shown in Figure 1.1

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Responsive Manufacturing Systems

DES (steady-state, simulation based, no analytical, no feedback, design via iterations)

System Dynamics (dynamics, simulation based, with feedback, no analytical solutions, design via iterations)

Block Diagram Representation

Transfer Function (dynamics, analytical solutions, analytical system design, stability analysis, structure improvement)

Electrical Circuit Model

Fluid Model

Figure 1.1: Existing methods to model and analyze responsive manufacturing systems

1.2 Problem Statement

The manufacturing systems operating within today’s global enterprises are invariably dynamic

and complicated. As manufacturing business leans towards globalization, market demand

appears to be highly fluctuated; lean manufacturing philosophies may no longer work well under

these frequently change and unpredictable conditions. It is an awesome challenge for practicing

managers and engineers in attempting to design and improve the overall responsiveness of those

dynamic manufacturing systems. The big picture question here is whether we can develop an

engineering methodology to assist production management to model, analyze, and design

responsive manufacturing systems in this 21st century global industrial market. Can we

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analytically determine key transient manufacturing system parameters, such as production

settling time, WIP overshoot, system responsiveness, and lean finished inventory level? In

addition, without doing any iterative trial-and-error simulation replications, can we assist

management to design, improve, and control the overall dynamic behavior of such

manufacturing systems? The author truly believes that one can find these answers in this

dissertation work. Lastly, this alternative modeling, analysis, and design methodology can be

applied to any manufacturing system in general.

1.3 Research Objectives

The objectives of this dissertation research are as follows:

1) Develop a one-to-one correspondence to model manufacturing systems between the

differential dynamic models (Classical Control Theory) and the CLD and SFD structures

employed in the SD approach;

(2) Derive resulting transfer functions from the particular system block diagram representation to

analytically determine the transient characteristics of the manufacturing systems;

(3) Sensitivity analysis of key manufacturing system parameters that influence the overall

manufacturing system responsiveness and leanness;

(4) Apply the Root Locus technique from Classical Control Theory as a new production

management strategy to better predict and design key manufacturing terms on a complex s-plane

environment;

(5) Define and interpret classical control theory terms as they relate to the manufacturing world;

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(6) Reveal the potential for system instability due to management policies in higher-order system

with time delays dynamics;

(7) Validate the continuous differential equation model as a good approximation of real

discretized manufacturing systems.

1.4 Contents of Dissertation

The rest of the dissertation is outlined as follows. Chapter 2 presents the literature review. We

begin the review by examining agile and responsive manufacturing systems, followed by the

early development of System Dynamics (SD) and its recent applications. We further describe

other modeling approaches, like, input-output analysis, fluid model, and electric circuit modeling.

Chapter 3 describes the modeling and analysis of responsive manufacturing systems. The

fundamental mechanism of SD developed by Jay Forrester is discussed in detail. Four particular

production control system models are presented to translate the SD terminologies into classical

control theory (CCT) approach. The resulting differential equation models permit production

management and industrial engineers to analytically determine the transient characteristics of the

responsive manufacturing systems. The objective of Chapter 4 is to validate the CCT approach

by comparing with discrete event simulation. In Chapter 5, we investigate design issues and

show how we can employ the Root Locus technique and incorporate third-order time delays. To

enhance the validation of this new CCT design and modeling approach, we include an industrial

case study in Chapter 6. In that chapter, we apply the CCT approach developed in this

dissertation to model and design an Intel hybrid push-pull production system for semiconductors

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manufacturing. Chapter 7 highlights the concluding remarks and research contribution from this

dissertation work. Finally, we provide some future research directions in Chapter 8.

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Chapter 2 Literature Review N.H. Ben Fong

Chapter 2 Literature Review

This chapter presents the literature review of the rise of responsive manufacturing systems

modeling and analysis, the development and recent applications of system dynamics (SD), the

input-output analysis in modeling production-inventory systems, and other analogous approaches

to model dynamic manufacturing systems. Finally, we state the missing links of these existing

modeling approaches that lead to our new alternative modeling and design approach.

2.1 Agile and Responsive Manufacturing

In the early 20th century, Henry Ford introduces the well-known mass production system. Ford’s

philosophy is to build a simple, low cost, and fully utilized assembly line system. Such a mass

production system is very inflexible and is not responsive to changing customer demands. It

relies on forecasting future customer demand and scheduling the release of orders. This system

often results in high work-in-process levels and excess finished inventories. In the 1980s, the

Toyota production system or just-in-time (JIT) system is developed to provide better flexibility

through the concept of pull within the factory. JIT production depends on actual customer

demand activating the release of orders into the system to fill the demand. The JIT philosophy

emphasizes making the right products in the right amount at the right time. JIT eliminates excess

inventory, shortens production lead-time, and increases quality in both products and customer

service. In the 1990s, companies begin to implement the concept of lean manufacturing that

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evolved from the Toyota production system. Lean manufacturing is a comprehensive philosophy

where employees continue to strive for improvement to eliminate all non-value added activities.

Although JIT or lean manufacturing has a significant culture impact to improve production

efficiency, its system performance measures are restricted under steady-state conditions.

In the 21st century, due to the highly fluctuating market demand and the frequent change of

product designs, to stay competitive in this global market, manufacturing companies must

possess a new kind of manufacturing system that can be very responsive to volatile global

markets. Helo [2] defines agile manufacturing as the capability of reacting to unpredictable

market changes in a cost-effective way, simultaneously prospering from the uncertainty. In his

paper, three system dynamic simulation models are analyzed to the agility of supply chains. The

analysis recommends smaller order sizes, echelon synchronization and capacity analysis as

methods of improving the responsiveness of the supply chain. Sanchez and Nagi [8] have

reviewed a wide range of recent literature on agile manufacturing. Their paper concludes agile

manufacturing as the solution to a society with an unpredictable and dynamic demand.

Christopher and Towill [1] show the various ways to combine the paradigms of leanness and

agility to enable highly competitive supply chains in a volatile and cost-conscious environment.

The paper emphasizes the important differences between the two paradigms and how one may

benefit from the implementation of the other. In the literature, Naylor et al. [9] define agility as

the use of market knowledge and a virtual corporation to exploit profitable opportunities in a

volatile market place. Whereas leanness is constructing a value stream to eliminate all waste

including time and to enable a level schedule.

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Asi et al. [10,11] define a new manufacturing paradigm called reconfigurable manufacturing

systems (RMS). RMS is designed at the outset for rapid change in the system configuration, their

machines, and controls in order to quickly adjust production capacity and functionality in

response to market changes. This type of system will provide customized flexibility for a

particular part family, and will be open-ended, such that it can be improved and reconfigured,

rather than scrapped and replaced. Mehrabi et al. [12] describe agile manufacturing to focus on

the manufacturing enterprise and the business practices needed to adapt to a changing global

market characterized by uncertainty. It does not provide any operational techniques or any

engineering solutions. In contrast, RMS does not deal with the entire enterprise but only with the

responsiveness of the production system to new market opportunities in an environment of

global competition with suitable market production. The RMS methodologies of rapid system

design and ramp-up, as well as the capability to add incremental capacity and functionality in

response to market demands, is one aspect of agility. Hence, agile manufacturing shares with

reconfigurable manufacturing the ability to improve the overall manufacturing responsiveness.

Consequently, agile manufacturing is complementary to reconfigurable manufacturing.

Asi et al. [10] introduce a control-theory based fluid dynamic model to assist in implementing

the optimum reconfiguration policy and production scheduling of an RMS. They develop and

analyze a simplified dynamic production model whose capacity and/or functionality can change

over time. Asi and Ulsoy [11] further formulate and provide a sub-optimal solution for a general

capacity management using feedback control theory approach under both deterministic and

stochastic market demand.

11


Suri [13,14] defined a new company wide-strategy called Quick Response Manufacturing (QRM)

to pursue the reduction of lead time in all aspects of a company’s operations, both internally and

externally. Internally, QRM focuses on reducing the lead times for all tasks across the whole

enterprise, resulting in improved quality, lower cost, and quick response. From a customer’s

view point, QRM responds to their needs by rapidly designing and manufacturing products

customized to those needs. In addition, Suri has developed a new material control method, called

Paired-Cell Overlapping Loops of Cards with Authorization (POLCA) to provide companies

with significant competitive advantage over the traditional MRP and Kanban systems.

Whether it is agile manufacturing, reconfigurable manufacturing systems, or quick response

manufacturing, companies must be able to react and respond quickly to predict and improve their

overall manufacturing system performance in fast-changing and uncertain global markets. In

other words, the manufacturing system must be responsive in such environments and be able to

operate effectively in transient mode (as well as steady-state). This in turn necessitates news

methods for modeling, analyzing, and designing responsive manufacturing systems. In particular,

it is important to be able to establish the fundamental cause-and-effect relationships among key

manufacturing variables, such as production start rate, production completion rate, WIP level,

Finished Inventory level, desired production rate, production lead time, etc. Unfortunately,

idealized constitutive laws like Newton’s laws in mechanical systems and Kirchhoff’s laws for

electrical systems do not apply in the manufacturing systems world. System Dynamics

developed by Jay W. Forrester [5,15,16] is the most popular modeling technique available to

model and analyze dynamic manufacturing systems, but still leave room for improvement.

12


2.2 Early Development in System Dynamics

The discipline of system dynamics (SD) has been studied over forty years. Dr. Jay W. Forrester

originally developed the framework of SD at the Massachusetts Institute of Technology (MIT) in

the late 1950s [5]. SD builds on how information flow, feedback-loops, and time delays within

the structure of a system create dynamic behavior. SD applies the feedback system thinking and

control engineering concepts to the study of economics, business, and organizational systems

[5,15,16]. Forrester defines SD as the study of the information-feedback characteristic of

industrial activity to show how organization structure, amplification and time delays interact to

influence the success of the enterprises. Forrester argues that mathematical analysis is not

powerful enough to solve the problems of the complex system and we need a simulation

approach. The first major piece of Forrester work published in 1958 gives a succinct explanation

of dynamic behavior in a production-distribution chain. In 1961, this work forms the core of the

book Industrial Dynamics [5]. There are other major publication come in the following years,

include, Urban Dynamics (1969), World Dynamics (1973), and the Collected Papers published

in 1975 [17]. Forrester applies the concepts of feedback loops in the understanding of system

behavior. He discusses the use of mathematical representations coupled with simulation.

Simulation takes the emphasis off mathematics for the sake of analytical solutions. Analytical

solutions are no longer as important as to provide an additional perspective and insight into the

nature that underline system dynamics. To facilitate model simulation, Forrester develops a

dynamic modeling language and simulation tool called DYNAMO [17]. This modeling tool

identifies flows within the system and forms the model about the structure derived from the

interactions of their paths.

13


Although the use of Forrester’s concepts has significant impact to the field of system modeling,

it also produces controversy. Ansoff and Slevin [17] give some criticism on the validity of the

SD technique published by Forrester. Their paper has questioned whether Forrester’s ideas are a

proven theory, plus the span of his potential application that he claimed. Ansoff and Slevin

suggest that SD can give the promise of advantages that may grow from a better understanding

of systems, but it does not adequately convey the essential mathematics in modeling dynamic

systems. In 1980, Anderson and Richardson [17] further state that analytical representation can

be very useful to simulation in system dynamics modeling. The analytical formats will not only

be useful in relating behavior to system structure but their applications can encourage wider

interest in SD from the control theory discipline. Forrester describes that the future SD work

could include the basic structures recurrent in system models to be converted into a generic

library in explicit dynamic form. Edghill and Towill [17] express that explicit dynamic form can

be interpreted as the block diagram representation and Laplace transforms from the control

theory. Computer-based methods are found to be more successful in modeling and simulating

live-system dynamic behavior but they do not provide an analytical approach to analyze the

relationship of cause and effect of the systems. Beginning late 80s, there has been a drift in

emphasis away from the original SD applications that focuses on the design of production-

inventory systems to the latest business consulting process modeling [16,18,19].

2.3 Recent Applications of System Dynamics

O’Callaghan [20] applies system dynamics to model and simulate a kanban-based JIT production

system. The multi-stage manufacturing system model is simulated to show the dynamic behavior

14


subjected to different management policies. Gupta and Gupta [21] further employ SD approach

to model a multi-stage, multi-line, dual-card, JIT-kanban production system. This paper focuses

on the inherent characteristics of the kanban system and investigates the system behavior under

various management policies. Ravishankar [22] at Intel Corporation develops several SD models

to understand the effect of management policies on the performance of a semiconductor

fabrication line. He constructs a resource allocation model within an organization to illustrate

how explicit and implicit policy decisions can have an impact on factory output and equipment

performance. Bianchi and Virdone [23] use a SD approach to re-engineer the manufacturing

processes in a European telecommunication firm. The model is set to estimate potential benefits

of a shift from a push system to a pull system.

Lin et al. [24] give a brief review of the role of SD in manufacturing system modeling. The

limitations of available SD software are identified and the SD generic modeling approach is

stated. Baines and Harrison [18] describe an opportunity for SD in manufacturing system

modeling. They address that it appears to be a lack of applications of continuous simulation

methods for industrial modeling. The reasons may lead to a decline in the general popularity of

SD or whether there is a missed opportunity for SD in manufacturing system modeling. Their

paper reviews problems with SD in the early years. The mathematical equations are too

approximate to be credible to control engineers but too complex to be understood by managers.

A structured classification approach is used to make a survey of the published applications of SD

in the 1990s. Baines and Harrison describe observations and opportunities about the SD

applications for future research. Oyarbide et al. [25] further discusses the difference between SD

principles and discrete event simulation. He develops a SD based computer tool to model an

15


engine production assembly line. The modeling tool has a user interface based on multi

document interface (MDI) approach that is programmed in visual basic (VB). Lai et al. [26]

build an integrated framework of JIT system model in an electronic commerce environment

using SD for modeling and simulation. The model integrates the information flow from the

customer to the supplier and formed a single supply chain. This paper claims that SD approach

can help manager to make policy and decision, and improve the communication in customer,

supplier, and the company. Wikner [27] describes three different approaches to continuous-time

dynamic modeling of variable lead times based on control theory. The three approaches include

first-order delay, third-order delay, and pure delay. He establishes a generic lead-time model

with two parameters, order and average lead-time. The delay model is interpreted as generating

the expected dynamic behavior of a system containing Erlang-k distributed lead times.

2.4 Input-Output Analysis in modeling production-inventory systems

In 1982, Axsäter [28] provides an overview of earlier research using control theory applications

in production and inventory control. In his paper, three areas of control theory applications are

considered: linear deterministic systems, linear stochastic systems, and non-linear deterministic

systems. Axsäter addresses that despite the difficulty to directly apply control theory methods to

production systems, the fundamentals of control theory help on designing and utilizing

production-inventory systems. However, control theory techniques cannot in general contribute

to the problems of lot sizing and machines sequencing. It offers an attractive methodology for

analyzing deterministic dynamic systems at the aggregate level. Towill has been a supporter of

Forrester’s work and most of his work has been involved with developing the Forrester supply

16


chain models [1,7,19,29,30,31]. Towill [29] describes that the big drawback of the SD simulation

is essentially preceded on a trial-and-error basis. He believes that there is a need to examine the

middle ground between the analysis and simulation approaches. He introduces the block diagram

representation to describe an inventory and order based production control system (IOBPCS). He

applies transfer functions from control laws and feedback paths to tune local system parameters

in an industrial dynamic simulation application. Towill defines the demand averaging process

and the production delay as two first-order lag transfer functions. The terms damping ratio and

the undamped natural frequency are briefly introduced in relating to the IOBPCS. Towill [32]

further applies an Input-Output Analysis to identify the man-machine interface prior to computer

simulation for robust system design. The fundamental responses of the SD approach are stated in

the paper. The use of Input-Output Analysis results in a block diagram representation of a

planning department’s decision-making progress.

Edghill and Towill [17] takes a critical review of Forrester’s work that it requires a middle

ground of dynamic system behavior between continuous computer simulated approach and

mathematical approach. Three fundamental flows of dynamic manufacturing characteristics,

include orders, materials and information, are investigated. This paper concludes that the

mathematical models of limited complexity provide the necessary insight to guide the design and

appraisal of live system models built with a continuous computer package. Towill [6,7] has

published two parts detail review on SD in term of its background, methodology, and

applications. Part I shows how servo control theory and cybernetics have influenced SD and

examine the linguistic and numerical information that applies for constructing models. The use

of input-output analysis is an essential SD modeling tool and it mimics the use of differential

17


equations by attempting to balance activities at key points. He comments the role of SD as a

user-friendly software in the business game environment and the relevance of the method as

perceived by an experienced management consultant. In Part II, Towill considers to better

exploit SD in the area of improving business competitiveness by integrating the servo control

theory within the SD framework. The example illustrated requires the smoothing of material

flow within a supply chain through the use of all available marketplace information in contrast to

acting only on distorted orders passed on by the adjacent echelon.

Towill and Del Vecchio [31] further propose the use of filter theory to minimize the total system

stocks in the presence of demand fluctuations as orders proceed along a three-echelons dynamic

supply chain. The simulation results explain the reason behind the selection of a particular sub-

optimal supply chain design as identified via an expert system based on the multi-attribute utility

technique. The overview of the supply chain dynamic model is described in block diagram

representation. Based on the linear control law, each echelon of the supply chain dynamics is

formulated into a single transfer function that consists of first-order time lag or exponential

smoothing of time constant [31]. The complete supply chain can be regarded as the sequence of

amplifiers as shown by the coupling of the each individual transfer functions from different

stages of the production-inventory system. Towill [19] shows various ways to build industrial

dynamics models and exploit in supply chain re-engineering. His paper concludes the improved,

enhanced supply chain dynamics are obtained by adopting a holistic approach in which the basic

disciplines of industrial engineering and business process re-engineering are integrated into a

comprehensive methodology. Disney et al. [33] establish a decision support production system

model coupled with a simulation facility and genetic algorithm based controller to give an

18


enhanced performance with an acceptable trade-off between production smoothing and a high

level of stock turnover. Their paper emphasizes the concept of lean logistics via smart modeling.

By an intelligent design, the chain amplification reduces to a 20-fold improvement. Furthermore,

Disney et al. [34] describe a procedure for optimizing the performance of an industrial designed

inventory control system with three classic control policies. By utilizing sales, inventory, and

pipeline information of the order rate, it gives a desired balance between capacity, demand and

minimum associated stock level. Five selected benchmark performance measures use that

includes inventory recovery to shock demands, in-built filtering capability, robustness to

production lead-time variations, robustness to pipeline level information fidelity, and systems

selectivity. They use these five factors and genetic algorithm to optimize system performance.

Although the focuses on a single supply chain interface, the methodology is applicable to

complete supply chains. Disney et al. [35] further investigate the use of continuous and discrete

time analytical results for studying production and inventory control system design problem

using block diagram representations and transfer functions. A generalized Order-Up-To policy is

chosen to show the equivalence of both continuous and discrete control theory approaches yield

similar qualitative interpretations of the system stability analysis. Finally, Fowler [36] suggests

that concepts such as JIT/Kanban and supply chains are special cases of generic feedback control

principles, while pure MRP is a classic example of feedforward. A hybrid combination of

feedback loop and feedforward control is used to model and analyze a multistage supply chain

model. The resulting system improves the system response rates, eliminates stock fluctuations,

and minimizes total finished inventory.

19


2.5 Other approaches to model dynamic manufacturing systems

Chryssolouris et al [37] describe an analogy between a dynamic manufacturing system and a

mechanical system. The paper attempts to resemble the behavior of a mechanical system under

the excitation of a force that changes over time in the study of an industrial system. The

processing time and the flow times are collected to apply Fourier transform to create a transfer

function to represent the dynamic manufacturing system model. This approach can lead to make

an optimum control policy for the manufacturing system and predict its system performance.

Sader and Sorensen [38] construct a continuous manufacturing dynamic system model using

analogies to electrical systems. They describe the model through the application to a

representative continuous manufacturing line for both deterministic and stochastic cases. The

simulated results are compared to the discrete event simulation approach. As mentioned in the

earlier section, Asi and Ulsoy [10] develop a fluid dynamic analogy to model reconfigurable

manufacturing systems. This analogous dynamic model characterizes the reconfiguration policy

and the production scheduling of an RMS.

2.6 Missing link of the existing modeling approaches

As mentioned in the previous sections, although system dynamics (SD) is built upon the

feedback concepts of control theory, it does not provide any analytical formulation to determine

key transient system parameters and their corresponding output measures due to its simulation-

based modeling nature. Furthermore, SD relies on iterative simulation trials with different

parametric set values to yield specific system dynamic behavior. SD is not capable to predict or

improve system structure for design purposes and system stability analysis. The input-output

20


analysis as described by Towill [6,7] is an alternative method to model dynamic production-

inventory control systems or supply chain systems. By introducing first-order lag functions into

the supply chain system, it mimics the use of differential equations to describe the idealized

constitutive relationships among production system variables. This resulting overall system

transfer function is a powerful tool to analyze and improve supply chain dynamic system

performance. However, for most control engineers, a mathematical model of a dynamic system is

usually described in terms of differential equations based upon physical laws or idealized

constitutive relationships among system variables. It will be a great interest for researchers to

describe the structure of dynamic manufacturing systems via the physical relationships among

system variables instead of using multiple first-order time-delay transfer functions. The fluid

dynamic system analogy for RMS as proposed by Asl et al. [10] is an interest piece of research

work, however, the dynamic structure of the model is based on the first-order lag that is very

similar to the work described by Towill. In addition, backward flow could occur at the fluid

model if control valve is not included to prevent negative production rate. Asl et al. [10] briefly

mentioned the stability boundary issue due to the complex roots of its characteristic equation,

however there is no solid mathematical formulation available to reveal the potential of

manufacturing system instability due to the poor management strategies. Finally, the electrical

dynamic system analogy to a continuous manufacturing systems as described by Sader and

Sorensen [38] is based on the cascaded, three first-order dynamic systems. Hence, the output

response of their three-stations manufacturing system behaves similar to a first-order, goal-

seeking structure, with no oscillation occurs. In a real life manufacturing application, a small

amount of inventory overshoot always occurs during the transient period to give faster

responsive time to meet the specific customer demand.

21


Given these reasons, it is necessary to develop an alternative methodology for modeling,

analyzing, and designing responsive manufacturing systems. In the next section, we tackle

modeling and analysis by translating the terminology from system dynamics to block diagrams

and transfer functions. This allows us to analytically establish key transient system parameters.

Additionally, the resulting differential transfer functions are critical elements for performing

design strategies as discussed in later chapters.

22

Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong

Chapter 3 Modeling and Analysis of Responsive

Manufacturing Systems

In this chapter, we present two alternative approaches for modeling and analyzing responsive

manufacturing systems. Section 3.1 introduces the basic structures and fundamental modes of

System Dynamics (SD) developed by Jay W. Forrester. We apply SD approach to model and

analyze the dynamic behavior of four specific manufacturing models: a Single-Stage Production

Control System, a Basic Kanban System Model, a Two-Stage Production Control System, and a

Two-Stage Production Control System with 3rd Order Time Delay. The models and their

corresponding dynamic analysis have been performed using VENSIM software [39]. Vensim is a

visual modeling tool that allows one to conceptualize, document, simulate, and analyze models

of dynamic systems made from causal loop diagrams and/or stock and flow diagrams. Section

3.2 describes the proposed approach of using Classical Control Theory (CCT) for modeling and

analysis of responsive manufacturing systems. We apply the Block Diagram (BD) and Transfer

Function (TF) techniques to define a one-to-one correspondence in modeling dynamic

manufacturing systems from CLD and SFD structures to differential equations formulation. This

mathematical translation is applied to the previous four production control systems, resulting in a

1st order differential equation model, two 2nd order differential system models, and a 4th order

differential control model, respectively.

23


3.1 Modeling and Analysis via System Dynamics

In this section, we study the basic mechanism and fundamental modes of System Dynamics

originally developed by Dr. Jay W. Forrester at MIT. We have chosen four specific production

control systems to analyze for their dynamic behavior using causal loop diagrams (CLD) and

stock-and-flow diagrams (SFD). We have extracted both a single-stage and a two-stage stock

management structures from Sterman [16]. Their model terminologies have been modified to

become two different production control models. In addition, a SD kanban-based dynamic model

is extracted and modified from O’Callaghan’s paper [20]. Finally, we include a third-order time

delay in the two-stage production control system model. We construct the models using Vensim

software as shown in Figures 3.1, 3.3, 3.5, and 3.7.

3.1.1 System Dynamics Approach

In John D. Sterman’s award winning textbook [16], he introduced several diagramming tools

used in SD to capture the structure of systems, including causal loop diagrams (CLD) and stock-

and-flow diagrams (SFD). CLD represents a closed loop of cause-effect linkages that intends to

capture how system variables interrelate. CLD represents a closed-loop of cause-effect linkages

(causal link) that intend to capture how the manufacturing variables interrelate. SFD provides the

storage element of the manufacturing systems that is accumulating or draining over certain

amount of time. The storage element, like stock or level, is the memory of a system and is only

affected by flows. The stock is an accumulation of any particular manufacturing stage. It

represents the accumulated difference between inflow and outflow rates, illustrating the results

of dynamics within the system over time. Stocks are conserved quantities that can be changed

24


only moving contents in and out. Variables are related by causal links, shown by arrows. Each

causal link is assigned a polarity, either positive (+) or negative (-) to indicate how the dependent

variable changes when the independent variable changes. A positive link indicates that if the

cause increases, the effect increases; whereas a negative link refers to the effect decreases as the

cause increases. However, link polarities only describe the structure of the system but not the

behavior of the variables. They do not describe what happens in terms of the actual changing

value of the variables. For example, the polarity of every link in a diagram, the feedback-loop

identifier uses “+” or “R” to indicate that it is a positive (reinforcing) feedback loop; and use “-

“ or “B” to show it is a negative (balancing) feedback loop.

The most fundamental modes of system dynamic behavior [16] are defined as exponential

growth, goal seeking, and oscillation. Each of these modes is caused by a simple feedback

structure: positive feedback loop yields exponential growth, goal seeking arises from negative

feedback, and negative feedback loops with time delays give system oscillation. More complex

modes such as S-shaped growth and overshoot and collapse arise from the nonlinear interaction

of these fundamental feedback structures. Exponential growth arises from positive feedback. The

larger the quantity, the greater its net increases, further boosting the quantity and guiding even

faster growth. Whereas, negative loops seek balance and equilibrium. Negative feedback loops

act to bring the state of the system in line with a goal or desired state. Like goal-seeking behavior,

oscillations are also caused by negative feedback loops. The state of the system is compared to

its goal, and corrective actions are taken to eliminate any discrepancies. In an oscillatory system,

the state of the system constantly overshoots its goal or equilibrium state, reverses, then

undershoots, and so on. The overshooting proceeds from the presence of significant time delays

25


in the negative loop. The time delays make corrective actions to continue even after the state of

the system reaches its goal, forcing the system to adjust too much, and triggering a new

correction in the opposite direction. In contrast, S-shaped growth begins with an exponential

growth at first, and then it gradually slows until the state of the system reaches an equilibrium

level.

3.1.2 A Single-Stage Production Control System

As shown in Fig. 3.1, our objective is to reach a particular inventory value (i.e., desired inventory,

INV*) of a single-stage production system subjected to a particular customer demand. Given a

new program launch of product, the management policy is to determine a set of system

parameters such that the production will meet the target inventory level within a reasonable

settling time. The production inventory is the accumulation of a difference between the

production rate (PR) and the shipment rate (SR) during a certain shipment time (ST). The

shipment rate is calculated from dividing the total inventory level by the average shipment time.

The production rate (PR) is given by the desired production rate (DPR). Sterman [9] applies a

Max function inside the production rate formulation to prevent any negative production even if

there is a large surplus of inventory presented. The desired production rate (DPR) represents the

rate at which the units of product are to be made to the inventory.

26


InventoryINV Shipment Rate

SR

Adjustment forInventory AINV

ExpectedShipment Rate

ESR

+

-

+

ProductionRate PR

DesiredProduction Rate

DPR+

+

+

InventoryAdjustment Time

IAT

-

ShipmentTime ST

-

DesiredInventory INV*

<Initial DesiredInventory>

<Input>+

B

InventoryControl

Figure 3.1: A single-stage production control system

There are two fundamental decision rules to determine the desired production quantity. First,

production should replace the expected shipment rate (ESR) from the inventory. Second, if there

is any discrepancy between the desired inventory INV* and the actual inventory INV, the

production rate should be controlled by either making more than ESR or making less than ESR

while the inventory level is below or above the target value respectively (i.e., AINV). Hence,

DPR is the sum of ESR and AINV. The adjustment for the inventory AINV generates the

negative (balancing) “inventory control” feedback loop as shown in Fig.3.1. AINV is a linear

adjustment in the discrepancy between INV* and INV over the inventory adjustment time (IAT).

Sterman [16] describes this adjustment time as the time constant for the particular feedback loop.

The IAT represents how fast the production system reacts to correct the discrepancy of inventory

level. In the later section, we show that IAT of this single-stage system model only represents a

portion of the entire system time constant of the actual transfer function. This time delay is

sometimes so short relative to the dynamics of interest, we can assume that there is no delay so

that it is acceptable to let ESR = SR.

27


For simulation purposes, we arbitrarily choose the following parameter values: ST=8 days and

IAT=3 days for the single-stage production control system model in Fig.3.1. A step input of

planned inventory with 100 units is given to the single-stage production control system for a

period of 20 days. The step response of the single-stage production control system is shown in

Figure 3.2.

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Time (Day)

(Sto

ck o

f Uni

ts)

0

10

20

30

40

50

60

(Uni

ts/D

ay)

Inventory INV

Production Rate PR

Shipment Rate SR

Figure 3.2: Step response of a single-stage production control system

Figure 3.2 shows the step response of the single-stage production control system. The inventory

reaches its target value after 13 days. The production rate begins at the highest rate of 50 units

per day and it decays to 20 units per day to match the shipment rate after 12 days.

3.1.3 A Basic Kanban System Model

The Japanese word kanban refers to a “card”. The intent of kanban is to use as a card to signal a

preceding process that the next process requires parts/material. The kanban system can be

considered as an information system that controls lean production. In this section, we use the

28


basic structure of a kanban system model extracted from O’Callaghan’s paper [20]. This kanban

system model has been modified to a two-stage kanban system and constructed using Vensim

software as shown in Fig.3.3. Vensim provides a simple and flexible way of building simulation

models from causal loop diagrams and/or stock and flow diagrams.

Work-In-Process WIP

FinishedInventory FIProduction Start Rate

PSRProduction

Completion RatePCR

Shipment RateSR

Desired ProductionRate DPR

ProductionOrders PO

Lead TimeLT

++ +

-

- -

Kanban CycleKC

-

Shipment TimeST

-

Total Number ofKanban TNK

<ContainerSize>

<Input>

+

Figure 3.3: A basic kanban system model

Referring to Fig. 3.3, the work-in-process (WIP) level is the accumulation of a difference

between the production start rate (PSR) and the production completion rate (PCR) during a

certain production lead- time (LT). The finished inventory (FI) determines the stock level

between the production completion rate less the shipment rate (SR) over an average shipment

time (ST). The total number of kanbans defines the inventory allowed in the entire system. Any

kanban has to be either attached to the stock container (WIP or FI) or the dispatching post (i.e.,

kanban receiving box). Each time a unit is withdrawn from the finished inventory, its kanban is

detached and put in the collection box. Based on a certain time interval, the detached kanbans

found in the collection box will be taken to the dispatching post, where they become production

orders (PO). O’Callaghan states this time interval as the kanban cycle (KC), and it determines

29


how fast the system reacts to changes in production rate. He further asserts that the kanban cycle

is a “time constant”. However, we show in the later section that the kanban cycle of this

particular system model only represents portion of the entire system time constant from the

actual transfer function. Finally, the backlog of PO determines the desired production rate (DPR).

The total number of kanbans (TNK) is defined as the number of kanban multiplied by each

container size. In this kanban system, TNK has to be equal to the sum of the number of WIP

kanbans, the number of FI kanbans and the number of PO kanbans at all times.

We arbitrarily choose the following system parameter values for the kanban system model from

Fig.3.3: number of kanbans=10, container size=10 units, LT=0.5 day, KC=0.5 day, ST=5 days

(assume that production works 20 hours/day). By giving a step input of 100 units’ inventory, the

FI system output response behaves similar to a goal seeking feedback loop structure as shown in

Fig.3.4a [20,21]. The goal is to reach the planned inventory of 100 units. For the given set of

parameters, the steady-state finished inventory reaches 83.33 units instead. The reasons for this

effect are not immediately clear from Fig. 3.3 alone, but will be seen from the corresponding

transfer functions. The WIP level has reached 37 units at its early stage and reduces down to 8.33

units after 3 days. Fig. 3.4b shows the production rates response subjected to the given step input.

The Production Start Rate (PSR) starts producing at a rate of 200 units/day and drops down to

16.67 units/day, whereas the Production Completion Rate (PCR) takes 0.5 day to reach its peak

at 74 units/day and reduces down to 16.67 units/day after 4 days.

30


0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9 10Time (Day)

(Sto

ck o

f Uni

ts)

Work-In-Process WIPFinished Inventory FI

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5 6 7 8 9 10Time (Day)

(Uni

ts/D

ay)

Production Start Rate PSR

Production Completion Rate PCR

Figures 3.4a and 3.4b: A step input response of a kanban system model

3.1.4 A Two-Stage Production Control System

We further modify the single-stage model to become a two-stage production control system by

adding to the preceding stage of production a “work-in-process control” feedback loop and

Figure 3.5: A two-stage pr

related causal loop variables as shown in Fig.3.5.

oduction control system

Work-In-Process WIP

FinishedInventory FIProduction Start

Rate PSRProduction

Completion RatePCR

ShipmentRate SR


DPR

Adjustment forWork-In-Process

AWIP

DesiredWork-In-Process

WIP*

Adjustment forFinished Inventory

AFI


ESR

Lead TimeLT

++

+

+

+

-

-

-

+Finished InventoryAdjustment Time

FATWork-In-ProcessAdjustment Time

WAT

-

-Expected Lead

Time ELT

+

+

Desired ProductionCompletion Rate

DPCR +

+

ShipmentTime ST

-

DesiredInventory DI*


<Input>+

+

+

B

WIP Control B

FI Control

31


There are two stock elem ork-in-process (WIP)

roduction units and its finished inventory (FI) units. WIP accumulates the difference between

production start rate (PSR) and production completion rate (PCR) during a certain production

lead-time (LT). The finished inventory (FI) determines the stock level between production

completion rate (PCR) less shipment rate (SR) over an average shipment time (ST). There are

two balancing feedback loops found to adjust the work-in-process (WIP) and the finished

inventory (FI). Similar to the single-stage model, there is a linear adjustment between desired

work-in-process (WIP*) and WIP over a specified work-in-process adjustment time (WAT) as

the adjustment for work-in-process (AWIP). Likewise, there is an adjustment for finished

inventory (AFI) between desired inventory (DI*) and FI over a specified finished inventory

adjustment time (FAT). In addition, the desired production completion rate (DPCR) is the sum of

AFI and expected shipment rate (ESR). Similarly, the desired production rate (DPR) is

calculated by adding AWIP and desired production completion rate (DPCR). In this model, the

desired work-in-process (WIP*) is the product of DPCR and the expected lead time (ELT). It is

assumed that expected lead time is equal to lead time (ELT=LT), and that there is no time delay

between shipments such that ESR = SR.

For the two-stage production control system from Fig. 3.5, we arbitrarily set the following

system parameter values: LT=3 days, ST=5 days, WAT=1 day, and FAT=1 day. A step input of

desired inventory (DI*) with 100 units is given to this two-stage model for a period of 20 days.

Figure 3.6 shows the step response of the two-stage production control system. The finished

inventory FI rises rapidly to give an inventory of 120.25 units before it gets to the desired

inventory of 100 units after 10 days. The work-in-process WIP also reaches its peak at 195.13

ents found in this model that accumulates the w

p

32


units at the beginning and it decreases to 60 units after 13 days. Both production completion rate

and shipment rate overshoot at a different rate that settles down to 20 units/day after 11 days.

0

20

40

60

80

100

120

140

160

180

200

1 5 10 15 20

Time (Day)

Stoc

k of

Uni

ts

0

10

20

30

40

50

60

70

Uni

ts/D

ay

Work-In-Process WIPFinished Inventory FIProd. Completion Rate PCRShipment Rate SR

Figure 3.6: Step response of a two-stage production control system

.1.5 A Two-Stage Production Control System with Time Delay

y applications of

3

Delays are inherent in many physical and engineering systems. There are man

time delay systems in modeling and analysis reviewed for manufacturing systems and capacity

management [16]. Sterman has described stocks as the source of delays. A delay is a process

whose output lags behind its input. When the input to a delay changes, the output lags behind

and continues at the old rate for some time. Delays in feedback loops create instability and

increase the tendency of systems to oscillate. In the previous two-stage production control model,

we approximated the shipment rate SR as a first-order exponential smoothing. However, as the

order of the time delay goes higher, we can better characterize the production system. Wikner

33


[27] stated that a third-order delay has proved to be an appropriate compromise between model

complexity and model accuracy for most dynamic modeling of production-inventory systems. In

this section, we modify the previous two-stage production control system by formulating the

Production Completion Rate as a third-order delay function of PSR and LT as shown in Fig. 3.7.

Work-In-Process WIP

FinishedInventory FI

Production StartRate PSR

ProductionCompletion Rate

PCR

ShipmentRate SR

Desired ProductionRate DPR

Adjustment forWork-In-Process

AWIP DesiredWork-In-Process

WIP*

Adjustment forFinished Inventory

AFI


ESR

Lead Time LT

+

+

+

-

--

+

B B

WIP Control FI Control

Finished InventoryAdjustment Time

FATWork-In-ProcessAdjustment Time

WAT

-

-

Expected LeadTime ELT

+

+

Desired ProductionCompletion Rate

DPCR +

+

ShipmentTime ST

-

DesiredInventory DI*


<Input>+

+

+

Figure 3.7: A two-stage production control system with a 3rd-order time delay

xcept for the addition of the third-order time delay to compute the PCR, the rest of the model E

structure and system flow remains the same as the two-stage production control system shown in

Figure 3.5. Again, we arbitrarily set the system parameter as follows: LT=3 days, ST=5 days,

WAT=1 day, and FAT=1 day. A step input of desired inventory (DI*) with 100 units is given to

this third-order time delay model for a period of 40 days. Figure 3.8 shows the step response of

the two-stage production system with a third-order time delay. The finished inventory FI yields a

much higher initial overshoot of 174.17 units and it gives three more oscillations of overshoot

before it reaches to the desired inventory of 100 units after 35 days. The work-in-process WIP

34


initially peaks with 279 units and oscillates four more times until it decreases to 60 units after 37

days. Both production completion rate and shipment rate overshoot and oscillate at a different

rate that reduces down to 20 units/day after 35 days.

0

50

100

150

200

250

300

1 5 10 15 20 25 30 35 40

Time (Day)

Stoc

k of

Uni

ts

0

10

20

30

40

50

60

70

80

90

Uni

ts/D

ay

Work-In-Process WIP

Finished Inventory FI

Prod. Completion Rate PCR

Shipment Rate SR

Figure 3.8: Step response of a two-stage production system with a 3rd-order time delay

he result indicates that the higher the order of the time delay, more dynamics and oscillations T

add to the production system structure. Hence, it takes longer lead time to bring the finished

inventory to the desired steady-state final target. For the next section, we will develop a

mathematical equivalence of translating the CLD and SFD terminologies in modeling those four

production control systems using Block Diagram (BD) representation and Transfer Function (TF)

from classical control theory (CCT). This one-to-one correspondence to translate System

Dynamics terminologies into CCT yields a new way to determine an analytical formulation to

design and predict dynamic manufacturing systems in terms of inventory overshoot, lead time,

responsiveness, and production leanness.

35


3.2 Modeling and Analysis via Classical Control Theory

he block diagram (BD)

.2.1 Fundamentals of Classical Control Theory

f a system is defined as a set of

In this section, we begin by reviewing some fundamentals of t

representation and transfer function (TF) from classical control theory (CCT). Our objective is to

develop a one-to-one correspondence to model dynamic manufacturing systems between the

differential equation formulation and the CLD and SFD applied in the System Dynamics (SD)

approach. The resulting mathematical equivalence will offer a new way to determine some key

system parameters and their corresponding performance measures for dynamic manufacturing

systems measures which are not provided by SD and are not possible or difficult to obtain via

other approaches such as discrete event simulation. We apply the CCT approach to model and

formulate the same four production control systems as described from last section. Based on the

result, we can analytically determine some key manufacturing dynamic characteristics without

iterative simulation like inventory overshoot, settling time, responsiveness, and production

leanness.

3

In the control engineering field, the mathematical model o

differential equations. As stated in Ogata [40], the transfer function (TF) of a linear, time-

invariant, differential equation system is defined as the ratio of the Laplace transform of the

output (response function) to the Laplace transform of the input (driving function) under all zero

initial conditions. Like the SD representation of Sterman [16], a block diagram (BD) of a system

is a graphical representation of the cause-and-effect relations operating in a particular system

[40,41]. In a BD, all system variables are linked to each other through functional blocks. The

36


functional block is a symbol for the mathematical operation on the input signal to the block that

produces the output. This links express the information flow from block to block. We typically

enter the transfer functions of the components in the corresponding blocks that are connected by

arrows to indicate the direction of the flow of signals. One of the major advantages of using BD

representation is to form the overall BD for the entire system by connecting the blocks of the

components according to the signal flow in evaluating the overall performance and the

interaction of each system components. In addition, the algebraic representation of the system’s

equations in terms of transfer functions allows easy manipulation for design and analysis

purposes.

3.2.2 Transfer Function of a Single-Stage Production Control System

deling the

Referring to Figure 3.1, we can translate those CLD and SFD terminologies in mo

single-stage production system using BD representations from CCT as shown in Fig. 3.9 and Fig.

3.10.

Figure 3.9: Subcomponents of BD representation of a single-stage production system

PRPRProduction Rate:

Desired Production Rate:

DPRDPR+

+

INVINV1/sInventory:

PR

_

+ESRShipment

Rate:

Expected

AINVAdjustment for Inventory:

IAT1+

_

Desired Inventory: INV*Input

SRShipment Rate: ST

1

SR

INV

DPR

AINV

ESR

SR

INV*

INV

INV*0

37


INVINVInput

PRPRDPRDPR1/s

_

ESR

+ AINV

IAT1 +

_INV*0INV*0

SR

+

+

1ST1

2ST1

Figure 3.10: Complete BD representation of a single-stage production system

As illustrate lation for

variable. By linking all these individual subsets together, we will generate the

nt called the

d in Fig. 3.9, each individual subset of block diagrams represents the formu

each system

complete block diagram as shown in Fig. 3.10. This complete block diagram is mathematically

equivalent to the single-stage model as shown in Fig.3.1. There are a total of two feedback loops

and a feedforward gain found in Fig. 3.10 with the corresponding loop gains, 1/ST1, 1/ST2, and

1/IAT, respectively. The 1/s block represents an integrator. We can simplify this multiple

feedback loops BD into a single transfer function by a step-by-step rearrangeme

block diagram reduction technique. Details of the technique are found in references

[40,41,42,43]. After the BD reduction is applied, Fig. 3.10 representation reduces to a single TF

block as shown in Fig. 3.11. The resulting single TF block can be expressed in the Laplace

transform domain and it yields a first-order closed-loop transfer function of eq. (3.1). Equation

(3.2) gives the time constant of the first-order dynamic system model.

38


+

_ IAT1

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

21 ST1s)/1(

ST1(1/s)1

1/sINV* INVINV

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

⎟⎠⎞

⎜⎝⎛

IAT1

ST1

ST1s

IAT1

21

INV* INVINV

Figure 3.11: Step-by-step block diagram reduction into a single TF block

( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

⎟⎠⎞

⎜⎝⎛

=

IAT1

ST1

ST1 s

IAT1

sINVsINV

21

*

(3.1)

where 1

21 IAT1

ST1

ST1

−

⎥⎦

⎤⎢⎣

⎡+−=τ (3.2)

1s

K X(s)Y(s)

+=

τ (3.3)

The time constant term, τ is a function of ST1, ST2, and IAT, not only IAT. It determines how

quickly the production reacts to changes in the inventory level. In a first-order linear system, it

takes 4τ settling time to reach 98.2% of its steady-state value. Equation (3.3) gives a specific

form of a first-order linear differential equation with nonunity gain, where K is the proportional

gain factor. By comparing eq. (3.1) to eq. (3.3), it yields,

IAT

K τ= (3.4)

The inverse Laplace transform of equation (3.3) with a unit step input is obtained as:

⎟⎠⎞⎜

⎝⎛= τ

-te - 1K y(t) ; with K y(t) lim

t=

∞→ (3.5)

The single-stage production control system subjected to a unit step input yields the following

responses: time constant, τ=3 days, steady-state value, K=1 (i.e., 100% of the desired inventory),

39


and 98% settling time=12 days. Remember, we assume that there is no delay and ESR = SR, thus

the loop gain ST1=ST2. The time constant tells how fast the production system could respond to

any given input. It takes about 12 days for production to reach 98% of its target value as shown

in Fig.3.2. Referring to eq. (3.2), the time constant is a function of IAT, ST1, and ST2. If the Max

function is taken off and we allow ST2<ST1, and if IAT>>ST2, the time constant could go

negative, and the system model could grow exponentially without bound (unstable). In practice,

we would not consider an expected shipment rate higher than an actual shipment rate.

3.2.3 Transfer Function of a Basic Kanban System Model

We make a mathematical equivalence of translating the basic kanban system from Fig. 3.3 into

BD representation as shown in Figs. 3.12 and 3.13. Figure 3.12 shows all the subcomponents of

block diagram representation and each system variable relationship of the kanban system. Figure

3.13 indicates the complete block diagram representation of the kanban system.

PCRPCRProductionCompletion Rate (PCR):

ProductionOrders (PO):

POPO+

WIPWIP1/sWork-In-Process (WIP): _

+

DPRDesiredProductionRate (DPR): KC

1Total Number of Kanban(TNK):

TNKInput

SRShipment Rate (SR): ST

1

PSR

PCR

FI

WIP

TNK

FI

POContainer Size

FIFI1/sFinishedInventory (FI): _

+PCR

SR

LT1

WIP__

Figure 3.12: Subcomponents of block diagram representation of a basic kanban system

40


TNK 1/s FILT1

LT1

1/sKC

1+ + +

__

__

ST1

r

Figure 3.13: Complete block diagram representation of the basic kanban system

⎥⎦⎤

⎢⎣⎡ +++⎥⎦

⎤⎢⎣⎡ +++

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

LT1

KC1

ST1

KC1

ST1

LT1 s

ST1

KC1

LT1 s

LT1

KC1TNK

2

r FI

Figure 3.14: Block diagram reduction into a single transfer function block

As illustrated in Fig.3.13, there are a total of four feedback loops found. The first two feedback

loops contain the loop gain value of unity, whereas the third and the fourth loop have a loop gain

of 1/LT and 1/ST, respectively. The 1/s term is simply an integrator. The complete block

diagram representation as shown in Fig. 3.13 is mathematically equivalent to the kanban system

model built by Vensim as shown in Fig. 3.3. For this multiple feedback loop block diagram as

shown in Fig. 3.13, we can simplify it into a single transfer function by the block diagram

reduction technique. Details of the technique are found in references [40,41,42]. After the block

diagram reduction is applied, Fig. 3.13 representation reduces to a single TF block as shown in

Fig. 3.14.

⎥⎦⎤

⎢⎣⎡ +++⎥⎦

⎤⎢⎣⎡ +++

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

LT1

KC1

ST1

KC1

ST1

LT1 s

ST1

KC1

LT1 s

LT1

KC1TNK

r(s)FI(s)

2

(3.6)

( )

( ) 2nn

2

2n

s 2sK

X(s)Y(s)

ωζωω

++= (3.7)

41


The resulting single TF block can be expressed in the Laplace transform domain, and it yields a

second-order closed-loop transfer function as stated in eq. (3.6). The denominator of eq. (3.6) is

the characteristic equation of the closed-loop kanban system model. Equation (3.7) gives a

specific form of a second-order system differential equation with nonunity gain, where K is the

proportional gain factor, ωn is the undamped natural frequency, and ζ is the damping ratio of the

system. Equation (3.7) can be put into standard form for a second-order system equation by

taking off the K term. The dynamic behavior of a second-order system can be described in terms

of two parameters ωn and ζ. If 0<ζ<1, the closed-loop poles are complex conjugates and its

transient response is oscillatory. This system is called “underdamped”. If ζ=0, the transient

response does not die out, it will oscillate forever. If ζ=1, the system is called “critically

damped”; exponential behavior occurs if ζ>1, the system is called “overdamped”. Overshoot or

oscillation will not occur unless ζ<= 0.707. To better understand the concept of the classical

control theory, please see references [40,41,42,43]. For any linear second-order system, the time

constant is computed as : τ = (1/ζωn). By comparing eq. (3.6) to eq. (3.7), it yields,

Undamped natural frequency: LT1

KC1

ST1

KC1

ST1

LT1 n ++=ω (3.8)

Damping ratio: ⎥⎦⎤

⎢⎣⎡ ++

++=

ST1

KC1

LT1

LT1

KC1

ST1

KC1

ST1

LT1

121 ζ

(3.9)

Damped natural frequency: 2nd 1 ζωω −= (3.10)

Roots of characteristic equation: -1 - s 2

nn1,2 ζωζω j±= (3.11)

42


Time constant:

⎥⎦⎤

⎢⎣⎡ ++

==

ST1

KC1

LT1

2 1 nζω

τ (3.12)

Kanban systems with minimal inventory storage cannot respond instantaneously and will exhibit

transient responses when they are subjected to inputs or disturbances. The performance

characteristics of a kanban system can be specified in terms of the transient response to a unit-

step input. The transient response of a second-order dynamic system often shows damped

oscillation before it gets to the steady state condition. It is common to determine the following

indicators: delay time, peak time, maximum overshoot, rise time, and settling time. Settling time,

ts is the time required for the response curve to reach and stay within a range about the final

value. Rise time, tr is the time needed for the response to rise from 10% to 90% of its goal value.

Settling time (2%criterion):

ns

4 4 tζω

τ == (3.13)

Rise time (standard): dn

d1-

dr

- -

tan1 tω

βπζωω

ω=⎟⎟

⎠

⎞⎜⎜⎝

⎛= (3.14)

where

n

d1- tan ζωω

β = (3.15)

Due to the non-standard form of the second-order system derived in eq. (3.6), we use the

following approximation instead of applying eq. (3.14) to calculate the rise time [41].

Rise time (approx.): 10 , 2.5 0.8 tn

90% 10, r, ≤≤+

≅ ζω

ζ (3.16)

The inverse Laplace transform of eq. (3.7) with a unit step input is obtained as:

0 for t , 1

tant sin -1

e -1K y(t)2

1-d2

t- n

≥⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −+=

ζζ

ωζ

ζω (3.17)

43


As time goes to infinity, the steady-state value of the kanban system is computed from equations

(3.6, 3.7, 3.17):

( )

2n

t

LT1

KC1TNK

K y(t) limω

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

==∞→

(3.18)

Recall from section 3.1.3, we arbitrarily set the system parameter values for the kanban system

model from Fig.3.3 as follows: number of kanbans=10, container size=10 units, LT=0.5 day,

KC=0.5 day, ST=5 days. The kanban system response subjected to a unit step input of 100 units

gives the following characteristics: ωn= 2.19 cycle/day, ζ=0.958, ωd=0.62 cycle/day, τ=0.476

day, tr=1.46 day, ts=1.90 day, and K=83.3 units. Given production runs of 20 hours per day, it

takes 0.456 day to complete a cycle, hence the kanban cycle frequency is 9.13 hours/cycle. ωn is

meaningful only when a system oscillates and therefore it is of little interest alone, whereas ωd is

a characteristic of the system responsiveness, and hence it has great interest to production

planning. The damping ratio, ζ provides a way to determine whether inventory has been made

over or under the target goal. The time constant, τ tells how fast the kanban system respond to

any input or disturbance, and it leads to finding the settling time taken to reach the planned

inventory target within 2%. K gives the final inventory level at the steady state condition. As

shown in Figure 3.4a, there is a steady-state offset of 100-83.33=16.67 units from the planned

inventory. By setting up different system parameter sets, we will find whether the finished

inventory response could reach its final value at 100 units.

Alternatively, one could explicitly write all the relevant partial derivatives but for convenience,

we have decided to apply the Design of Experiment (DOE) [44] technique to study the

44


sensitivity of the key manufacturing system variable(s) that influence(s) the finished inventory

output measures. An advanced factorial design called Box-Behnken Design is used to construct a

balanced incomplete block of three-levels-three-factors design. The three design variables of the

kanban system include lead time (LT), kanban cycle (KC), and shipment time (ST). The design

range of variables is selected as shown in Table 3.1:

Table 3.1: Box-Behnken Design of Experiment for a Basic Kanban System Model

Factor Low (day) Medium (day) High (day)

A. Lead Time (LT) 0.25 0.5 1

B. Kanban Cycle (KC) 0.25 0.5 1

C. Shipment Time (ST) 2.5 10 17.5

In this factorial experiment, there are total of 13 independent runs of parameter sets. The

measured responses of the experiment are: ωn, ζ, τ, ts, and K. We used Excel to generate all

measurement responses into a matrix from the 13 different sets. By applying Matlab, we are able

to compute and determine the most significant factors effect among the five measured response

variables. The effect of the main factors and their interaction factors are all calculated via the

regression coefficients of the matrix as shown in Table 3.2.

45


Table 3.2: Box-Behnken Design Factors Effect Responses Summary

Undamped Natural

Frequency(Wn)Damping Ratio (S)

Time Constant (T)

Settling Time (ts)

Steady State Value(K)

LT -0.7318 -0.0103 0.1632 0.6526 -3.7641KC -0.7318 -0.0103 0.1631 0.6526 -3.7641ST -0.1629 0.0358 0.0209 0.0837 12.1431

LT^2 0.1308 0.0552 0.0041 0.0162 -0.9109KC^2 0.1308 0.0552 0.0041 0.0162 -0.9109ST^2 0.1155 -0.0254 -0.0206 -0.0823 -8.1435

LTxKC 0.2379 -0.1144 0.1037 0.415 0.1984LTxST 0.0053 0.0077 0.0121 0.0483 2.6595KCxST 0.0053 0.0077 0.0121 0.0483 2.6595

The results indicate that both LT and KC are the most significant factors to influence the

undamped natural frequency, ωn. As LT and KC are reduced, the frequency of the kanban system

increases to provide more cycles completed per day. The damping ratio, ζ is most influenced by

the interaction factors of LTxKC. As LT increases and KC decreases, or vice versa, it will affect

the damping ratio significantly. The time constant, 1/ζωn is significantly affected by LT and KC

and their factors’ interaction. As LT and KC increases, the time constant increases. Similarly,

the settling time, ts is also significantly affected by LT and KC and their factors interaction.

Finally, the steady-state value, K is most influenced by ST. As the shipment time takes longer,

the finished inventory is accumulating more units until it reaches the planned inventory.

Final comments on this particular basic kanban system used, as LT and KC are both reduced to

very short periods (i.e., 30 minutes) and as the shipment time is made extremely long (i.e., 100

days), the steady-state value will very closely reach its final value of 99.95 units. However, it is

not economically realistic to bump the production complete rate up to 1700 units per hour with

the WIP level of 45 units at one point and wait for 100 days before shipment is made. For the

46


given kanban system values chosen, the system does not reach the planned inventory under

normal range of system parameters. Although this kanban model behaves like a critically

damped system, as ζ goes to unity, the system should give the fastest time response under non-

oscillatory conditions. Better response can be obtained by setting LT=0.25 day. With KC

unchanged, the system response would change differently as shown in Figure 3.15.

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

Time (Day)

Inve

ntor

y (U

nit)

LT=0.5, KC=0.5, ST=5; Inventory=83.33

LT=0.25, KC=0.5, ST=5; Inventory=86.96

Figure 3.15: Step Response of a basic kanban system model with different LT values

By keeping the same KC = 0.5 day and ST = 5 days, we only reduce LT from 0.5 day to 0.25 day.

The result indicates that the reduced LT setting gives a faster production response time and a

higher steady-state inventory level than the original LT setting with 0.5 day. In addition, the

damping ratio ζ also increases from 0.958 to 1.022. Hence, it yields a faster system response as ζ

gets closer to one with no oscillation found in this basic kanban system.

47


3.2.4 Transfer Function of a Two-Stage Production Control System

Similar to section 3.3.1, the two-stage production control system from Fig 3.5 can translate into a

set of subcomponents of block diagrams as shown in Fig.3.16. It can combine to give a complete

block diagram as shown in Fig. 3.17 that is mathematically equivalent to the two-stage

production control model from Fig. 3.5.

PSRPSRProduction Start Rate, PSR:

Desired Production Rate, DPR:

DPRDPR+

+

FIFI1/sFinished InventoryFI:

_

+ESR

Expected Shipment Rate:

AFIAdjustment for FI:

FAT1+

_

Desired Inventory: DI*Input

SRShipment Rate, SR: ST

1

PCR

SR

FI

DPR

AWIP

DPCR

SR

DI*

DI

DI*0

WIPWIP1/sWork-In-ProcessWIP:

_

+PSR

PCR

AWIPAdjustment for WIP:

WAT1+

_

WIP*

WIP

Desired Production Completion Rate:

DPCRDPCR+

+

ESR

AFI

PCRProduction Completion Rate: LT

1WIP

Figure 3.16: Subcomponents of BD representation of a two-stage production control system

ELT 1/sFI

LT1

WAT1

1/sFAT

1+ + +

__ _+LT1DI + +

+

+

_

1ST1

2ST1

AFI

ESR

DPCRWIP* AWIP

DPR

PSR WIP PCR

SR

Figure 3.17: Complete BD representation of a two-stage production control system

48


There are a total of five feedback loops and one feedforward loop found in Fig. 3.16 with the

corresponding loop gains of unity, 1/ST2, unity, 1/LT, 1/ST1 and unity, respectively. Again, the

1/s block is an integrator. By applying the block diagram reduction technique, Fig. 3.16 reduces

to a single TF block and yields a second-order closed-loop transfer function under the Laplace

domain as stated in equation (3.19). Equation (3.20) is a non-standard form of a second-order DE

with nonunity gain, where K is the proportional gain factor, ωn is the undamped natural

frequency and ζ is the damping ratio of the dynamic system. Equations (3.21)-(3.29) provides

the complete formulation of the dynamic characteristics of the two-stage production system as

shown in Fig. 3.5 [40,41,42,43]. These characteristics will next be evaluated for sample cases.

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡+++

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

FAT1

ST1

ST1

WAT1

LT1

ST1

WAT1

LT1ss

WATLT1

LT1

FAT1

DI(s)FI(s)

211

2

(3.19)

2nd order linear DE (non-standard form): ( )( ) 2

nn2

2n

s 2sK

X(s)Y(s)

ωζωω

++= (3.20)

Undamped natural frequency:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛ +=

FAT1

ST1

ST1

WAT1

LT1

21nω (3.21)

Damping ratio: ⎥⎦

⎤⎢⎣

⎡++=

1n ST1

WAT1

LT1

21 ω

ζ (3.22)

Time constant:

⎥⎦

⎤⎢⎣

⎡++

==

1

n

ST1

WAT1

LT1

2 1 ζω

τ (3.23)

Closed-Loop Poles: -1 - s 2

nn1,2 ζωζω j±= (3.24)

Rise time (approx.): 10 , 2.5 0.8 tn

90% 10, r, ≤≤+

≅ ζω

ζ (3.25)

49


Maximum overshoot % : ( ) 100*e M )/-(

pdn πωζω≈ (3.26)

Settling time (2% criterion):

ns

4 4 tζω

τ == (3.27)

The inverse Laplace transform of equation (3.20) with a unit step input is obtained as:

0 for t , 1

tant sin -1

e -1K y(t)2

1-d2

t- n

≥⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −+=

ζζ

ωζ

ζω (3.28)

As time goes to infinity, the steady-state FI of the two-stage production system is computed from

eqs. (3.19, 3.20, 3.27):

2n

t

WATLT1

LT1

FAT1

K y(t) limω

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

==∞→

(3.29)

According to the section 3.1.4, the two-stage production control system subjected to a unit step

input gives the following responses as shown in Fig.3.6: ωn= 1.155 cycle/day, ζ=0.664, τ=1.30

day, tr=2.13 days, ts=5.22 days, K=100 units and Mp=108.4 units. Assuming production runs 20

hours per day and no time delay between shipment rates (i.e., ST1=ST2), it takes about 0.866 day

to complete a cycle, and hence one production cycle frequency is 17.32 hours/cycle. The

damping ratio, ζ provides a way to determine whether the inventory has been made over or under

the desired production goal during the transient period. In this example, the inventory overshoots

before it reaches the goal value of 100 units. The time constant, τ tells how quickly production

can respond to the given input. We find the settling time of 5.22 days to reach the target value

within 2%. The rise time, tr is the time it takes for the system response to rise from 10% to 90%

of its goal value. It only takes 2.13 days for production to make 90 units of stock.

We again apply a Design of Experiment (DOE) technique [44] to identify some key system

variables that influence some of the output measures of the two-stage production model. For this

50


two-stage system model, we have a total of five system variables, LT, ST1, ST2, WAT, and FAT,

to adjust for six corresponding dynamic output responses, ωn, ζ,τ, tr, ts, and K. A 25-1 (Rev V)

design is used to construct a two-levels-five-factors fractional factorial design matrix. The design

system variables are set as follows: A=Lead Time (1 day or 4 days), B=Shipment Time1 (4 days

or 8 days), C=Shipment Time2 (4 days or 8 days), D=WIP adjustment time (1 day or 4 days), and

E=FI adjustment time (1 day or 4 days). In this experiment, there are total of 16 independent

parameter sets. Excel is used to generate all measurement responses into a matrix from the 16

different sets. By applying Matlab, we can compute and determine the most significant factors’

effect among the six response variables. The main factors’ effect and their interaction factors’

effect are all calculated via the regression coefficients of the matrix as shown in Table 3.3.

Table 3.3: 25-1(Rev V) fractional factorial design factors effect responses summary

Undamped Natural

Frequency(Wn)Damping Ratio (S)

Time Constant (T)

Rise Time (tr)

Settling Time (ts)

Steady State Value(K)

A -0.1324 -0.1081 0.5046 0.4311 2.0183 0B -0.0528 0.0585 0.1035 0.9062 0.414 19.8413C 0.0528 -0.1057 0 -1.1328 0 -19.8413D -0.1324 -0.1081 0.5046 0.4311 2.0183 0E -0.2766 0.3507 0 3.1009 0 7.9365

AB 0.0055 -0.0106 0.0601 -0.0634 0.2405 0AC -0.0055 -0.0005 0 -0.0538 0 0AD -0.0173 -0.0423 0.2652 -0.2429 1.0605 -7.9365AE 0.044 -0.0355 0 0.2033 0 0BC 0.0124 -0.0366 0 -0.3707 0 -8.7302BD 0.0055 -0.0106 0.0601 -0.0634 0.2405 0BE -0.0196 0.0693 0 0.8623 0 13.4921CD -0.0055 -0.0005 0 -0.0538 0 0CE 0.0196 -0.0885 0.0429 -1.0264 0.1716 -13.4921DE 0.044 -0.0355 0 0.2033 0 0

In marked contrast to the previous system, the result indicates that FAT is the most significant

factor to influence the undamped natural frequency, ωn. As FAT decreases, ωn increases, thus

one production cycle gets completed more frequently. Variables LT and WAT have a similar

51


effect on ωn with less magnitude. The damping ratio, ζ is most influenced by FAT. As FAT

increases, ζ increases. The time constant, 1/ζωn is significantly affected by LT and WAT and

their factors’ interaction. The time constant increases as LT and WAT increase. Similarly, the

settling time, ts is also significantly influenced by LT and WAT and their factors’ interaction. It

appears that the regression coefficients are zero for both ST2 and FAT for time constant and

settling time because they are not included in the formulation as stated in eq. (3.23). The FAT

has the most significant impact on rise time, tr. Both ST1 and ST2 have a similar effect on tr and

their own interaction with FAT also influences the rise time. Finally, the steady-state value K is

most affected by ST1 and ST2 and their individual interaction with FAT.

We can study the responsiveness, the inventory overshoot, the rise time, and the steady-state

value by plotting a family of step response curves under different damping ratio values as shown

in Fig. 3.18. It shows that all response curves eventually reach the desired 100 units. As damping

ratio, ζ decreases from 1.43 to 0.35, the gradient of the response curve increases. As ζ drops

below 0.7, overshoot occurs. The magnitude of the excess inventory increases as ζ continues to

drop. At lower damping ratios, the system response curve oscillates at a higher frequency that

leads to multiple excess inventories taken within the same production cycle. The rise time also

gets improved as ζ goes lower, thus it increases the responsiveness of the system. As ζ increases

over 1.0 or more, it takes a much longer time for the system to reach the desired inventory level.

52


0 5 10 15 20 25 300

20

40

60

80

100

120

140

Time in days

Fini

shed

Inve

ntor

y Le

vel (

Uni

t)

ζ=1.43 ζ=1.0

ζ=0.7

ζ=0.61

ζ=0.35

ζ=0.49

Figure 3.18: Family curves of step response for a two-stage production control system

There are some other observations we can make from this two-stage production system response

analysis. The overshoot of the early stage of the transient response can be interpreted as the

excess inventory built up. The reciprocal of rise time, 1/tr can be treated as the responsiveness or

the “agility” of the production system. The shorter tr, the greater the slope of the system curves

response. The greater the slope of curve, the higher excess inventory may occur. The production

management team can use the steady-state value, K to determine whether the finished inventory

level is over or below the desired planned inventory. We can predict this offset inventory and

keep the excess cost low to make our production system more “lean”.

It is not feasible to realize from the SD model the fact that FAT does not play any role in

determining the entire system time constant. However, for the system responsiveness or agility

53


of the system, FAT has the biggest impact on the rising slope of the system response curve. We

can decide the amount of inventory cost and the finished inventory response time by trading off

responsiveness and maximum overshoot. The shipment rate and its interaction with FAT have a

significant affect the final outcome of the finished inventory level. The location of the closed-

loop poles defined in eq.(3.24) is a key element for system stability design and analysis purposes.

3.2.5 Transfer Function of a Two-Stage Production System with Time Delay

We have learned from section 3.1.5 that the higher the order of the time delay added to the

manufacturing system, the higher number of oscillations occurred for its dynamic system

behavior. In this section, we study the difference of the block diagram representation and its

corresponding transfer function formulation among three kinds of time delay included in the

two-stage production control system. The original two-stage production control system is

modeled via a first-order exponential smoothing to formulate the shipment rate. We will show

that this is mathematically equivalent to a two-stage system with first-order time delay as a

second-order differential equation. Lastly, we show to include the third-order time delay to the

two-stage production system will increase the order of the dynamics to a fourth-order differential

equation, better representing the sudden change that actually occur in a real industrial system.

The original two-stage production control system with first-order exponential smoothing is

shown again in Fig. 3.17. Figure 3.19 shows the same production system with a modification of a

first-order time delay inclusion. We again apply the block diagram reduction technique to reduce

the BD representations from Figs. 3.16 and 3.18 into transfer functions as shown in eq. (3.30)

and eq. (3.31). The result indicates that the two different BD representations yield the same

54


mathematical formulation of transfer functions. Thus, for this particular two-stage production

control system, a first-order exponential smoothing gives equivalent system characteristics with a

first-order time delay. The mathematical equivalence check is found in eq. (3.32).

Figure 3.17: Complete BD representation of a two-stage production control system

ELT 1/sFI

LT1

WAT1

1/sFAT

1+ + +

__ _+LT1DI + +

+

+

_

1ST1

2ST1

AFI

ESR

DPCRWIP* AWIP

DPR

PSR WIP PCR

SR

Figure 3.19: BD representation of a two-stage production control system with 1st-order Delay

ELT

1/sFI

WAT1

WAT1

WAT1 1/s

FAT1

FAT1

FAT1+ + +

__ _+

1+(LT)s1

1+(LT)s1

1+(LT)s1

DI + +

+

+

_

1ST1

1ST1

1ST1

2ST1

2ST1

2ST1

AFI

ESR

DPCR

WIP* AWIP

DPR

PSR

WIP

PCR

SR

PCR

ELT

1/sFI

WAT1

WAT1

WAT1 1/s

FAT1

FAT1

FAT1+ + +

__ _+

DI + +

+

+

_

1ST1

1ST1

1ST1

2ST1

2ST1

2ST1

AFI

ESR

DPCR

WIP* AWIP

DPR

PSR

WIP

PCR

SR

PCR

3

3(LT)s1

1

⎥⎦⎤

⎢⎣⎡ +

Figure 3.20: BD representation of a two-stage production control system with 3rd -order Delay

55


(3.30)

(3.31)

(3.32)

or the third-order tim find more dynamics and

(3.33)

where,

iven the same yields a higher

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡+++

⎟⎞

⎜⎛ +⎟

⎞⎜⎛

⎟⎞

⎜⎛ LT111

⎠⎝⎠⎝⎠⎝=

FAT1

ST1

ST1

WAT1

LT1

ST1

WAT1

LT1ss

WATLTFAT DI(s)FI(s)

211

2

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎠⎞

⎜⎝⎛ ++⎥

⎦

⎤⎢⎣

⎡+++

=

FAT1

ST1

ST1

WAT1

LT1

ST1

WAT1

LT1ss

DI(s)

211

2

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛

WAT1

LT1

FAT1

FI(s)

( ) ( ) ⎥⎦⎤

⎢⎣⎡

⎥⎦

⎤⎢⎣

⎡+

=⎞

⎝⎛ LT

11sLT

LT

LT11

s1

⎟

⎠⎜+ s1

F e delay system, as mentioned in section 3.2.4, we

oscillations in the production control system. Its corresponding BD presentation is shown in Fig.

3.19. By reducing the block diagrams step-by-step into a single transfer function, it gives a 4th-

order differential equation as stated in eq. (3.33)

( )( )[ ] [ ] [ ] [ ]D s C s B s A s

WATLTLTFAT

DI(s)FI(s)

234

23

++++⎠⎝⎠⎝=

1127⎟⎟⎞

⎜⎜⎛

+⎟⎞

⎜⎛

( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟

⎠⎞

⎜⎝⎛ +=

+++=

+++=

212

12

123

112

1

ST1

ST1

FAT1

WAT1

LT1

LT27 D

WATSTLT9

WATLT27

STLT27

LT27 C

WATST1

WATLT9

STLT9

LT27 B

WATSTLT++=

119 A G initial system parameters, the third-order time delay system

inventory overshoot peak of 174 units instead of 120 units. In addition, the peak of the WIP

overshoot rises from 195 units to 279 units. The higher order system also takes a longer settling

56


time to reach its desired finished inventory of 100 units (i.e., about 23 days instead of 8 days).

Finally, there are more number of inventory overshoot oscillations found at the higher order

system. Perhaps we change the system parameters as: LT=4; ST=8; WAT=1; FAT=1, we could

observe a significant dynamic oscillation difference between a third-order delay model and a

first-order exponential smoothing model as shown in Fig. 3.20.

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

Time in days

Fini

shed

Inve

ntor

y Le

vel (

Uni

t)

1st-Order Delay

3rd-Order Delay

Figure 3.21: Dynamic Responses between 1st-order time delay and 3rd-order time delay

is vital to notice here that the representation of a real system by linear feedback models has a It

very significant affect on the predicted result. It is therefore necessary to explore the boundaries

of such errors, and make a careful comparison between models and real systems. This will be

described in chapter 5. In addition, we will also introduce the Root Locus design technique from

classical control theory as our new management strategies to design and improve transient

characteristics of agile manufacturing systems.

57


3.3 Guidelines to Translate Responsive Manufacturing Systems via CCT

study

o the best of the author’s knowledge, there are no idealized constitutive laws available in

In this chapter, we have demonstrated two alternative modeling and analysis approaches to

the dynamic behavior of four production control systems. As stated, the proposed Classical

Control Theory (CCT) approach provides a new way to study and analytically formulate

transient manufacturing system variables, such as production settling and lead time, WIP

overshoot, system responsiveness (shape of dynamics in terms of damping ratio) and lean

finished inventory level. In this section, we summarize and provide some basic guidelines using

CCT approach to model and analyze generalized responsive manufacturing systems.

T

describing the causality among dynamic manufacturing system variables in classical control

theory textbooks. In order to mathematically model the responsive manufacturing systems at the

transient condition, we apply System Dynamics (SD) approach to describe the dynamic

relationships among manufacturing variables through the causal-loop diagrams (CLD) and stock-

and-flow diagrams (SFD). The cause-and-effect expressions from the CLDs can convert into

different sets of system equations that represent the physical relationship among different

manufacturing variables, like work-in-process, production start rate, and production completion

rate, etc. The storage characteristic of the SFD in manufacturing systems can describe as an

integrator of products in buffer (i.e., 1/s in Laplace domain) in the differential equation format.

All those different sets of system equations further link to each other through functional blocks.

As mentioned, the functional block is a symbol for the mathematical operation on the input

signal to the block that produces the output. By integrating all those functional blocks of the

corresponding differential system equations, we can graphically represent the dynamic cause-

58


and-effect relationships in block diagram representation. By describing the manufacturing

systems dynamic in this mathematical and graphical expression, we can further apply the block

diagram reduction technique to make the block diagram representation to become a single

function block with specific input and output. The resulting single function block is the transfer

function of the particular responsive manufacturing system model. Once the particular

responsive manufacturing system is described in a differential equation format as a transfer

function, we can analytically determine the key transient characteristics of the particular

manufacturing system. Again, we highlight this classical control theory approach for responsive

manufacturing systems step-by-step as follows:

(1) Apply SD approach to describe the manufacturing system variables through CLD and SFD.

) Convert the cause-and-effect expressions from CLDs and SFDs into different sets of system

) Construct different sets of functional blocks (i.e., subcomponents of block diagram

(Use causality expression to describe relationships among variables. Stocks are used to

define the accumulation elements, like WIP, Finished Inventory, whereas flows are described

as production rates and shipment rate.)

(2

equations. (The resulting causality expression among variables is algebraically translated into

multiple linear system equations.)

(3

representation) from those system equations. (Each function block represents each of those

original linear system equations.)

59


(4) By integrating and linking each of those functional blocks, we graphically represent the

particular responsive manufacturing system dynamics into block diagram representation.

(5) Apply block diagram reduction technique to transform the block diagram into a single

functional block called transfer function in differential equation format. (A step-by-step

algebraic technique to reduce multiple functional blocks diagram into a single block transfer

function.)

(6) Once the transfer function is obtained, we can analytically determine the key transient

characteristics of the particular responsive manufacturing system model.

60

Chapter 4 Model Validation N.H. Ben Fong

Chapter 4 Model Validation

In the 21st century there are more frequent changes of product designs and fluctuating market

demand, so manufacturing companies must seize a new manufacturing system that is responsive

to volatile global business. Discrete event simulation (DES) is still the most popular tool used to

model and analyze performance of any given manufacturing facilities. Through statistical

analysis, DES aids management to make important strategic decisions. However, DES is a very

detailed and precise technique that requires extensive time and effort for development. Hence,

researchers like Forrester, Sterman, Baines, and Harrison [5,15,16,18] promoted another other

kind of simulation and modeling tool, System Dynamics (SD), to take decisions in a more

aggregate way. SD is based upon the concepts of causal loop diagrams and stock-and-flow

diagrams with feedback analysis. The SD approach underlines system structure rather than

collecting statistical data for modeling and analysis purposes. Unfortunately, both DES and SD

rely on trial-and-error or numerical iteration to determine the dynamic characteristics of rapidly

changing manufacturing systems. Fong et al. [45,46] (Appendices I, II) have illustrated the

approach of translating system dynamics manufacturing structure into classical control theory

and differential equation formulation. The resulting transfer function determines key transient

system variables that enable management to more readily predict and analyze important dynamic

characteristics of responsive manufacturing systems.

61


During the process of model building, design engineers and modeling analysts must be

constantly concerned with how closely the model reflects the real system. The process of

determining the degree to which the model corresponds to the real system, or at least accurately

represents the system, is referred to as model validation. Law and Kelton [47] hold that a

simulation model of any complex system can only be an approximation to the actual system

regardless of the amount of effort is spent on the model building. The more time it takes to build

the model, in general the more valid the model should be but there is no such thing as absolute

model validity. Nevertheless, the most valid model is not necessarily the most cost-effective.

Furthermore, a simulation model should always be developed for a particular set of purposes.

Certainly, a model valid for one purpose may not be for another. There is no simple test to

establish the validity of a model. Validation is an inductive process through which the engineers

draw conclusions about the accuracy of the model based on the available information. We can

examine the model structure (i.e., causality relationships among system variables) to see how

closely it corresponds to the actual system definition. Finally, we can analyze the output

performance of the model to see whether the results appear reasonable. Nevertheless, for discrete

manufacturing, like engine blocks, washers, TVs, and computers, simulation is considered the

best approach for obtaining valid models and results.

The main objective of this chapter is to validate the continuous differential modeling approach

developed by classical control theory (CCT) by comparing it with discrete event simulation.

Although differential CCT approach can save time and money to build dynamic models, it is a

continuous and aggregate approximation of any particular real-life discrete manufacturing

systems. To make this validation simple and understandable, we again apply a single-stage

62


production control system to compare the difference in dynamic characteristics between CCT

and DES approaches. We have employed ARENA [48] simulation software to model and

simulate the dynamic behavior of the particular production control system under real-life

discrete-time domain.

4.1 Discrete Event Modeling of a Single-Stage Production Control System

Discrete event simulation (DES) involves the modeling of a system in which the state variables

of the system change instantaneously at only a countable number of points in time. These points

in time are the ones at which an event occurs, where an event is defined as an instantaneous

occurrence that may change the state of the system [47]. In this section, we apply the ARENA

software to model and simulate the single-stage production control system.

From section 3.1.2, the main objective of the single-stage production control system model is to

obtain a particular inventory value (i.e., desired inventory, INV*) subjected to a particular

customer demand. From the aggregate point of view for the new program launch of product, the

management policy is to determine a set of responsive manufacturing system parameters such

that the production will meet the target finished inventory level within a decent amount of

settling time. The single-stage production control system is again showed in Figure 4.1. The

single-stage production control system shown in Figure 4.1 is constructed using System

Dynamics (SD) approach. By applying the block diagram representation and the block diagram

reduction technique, we translate the single-stage production control system into a transfer

function formulation as shown in Eq. (3.1). A step input of finished inventory with 100 units is

63


given to the single-stage production control system for a period of 20 days. The step response of

the single-stage production control system is shown in Figure 4.2.

InventoryINV Shipment Rate

SR

Adjustment forInventory AINV


ESR

+

-

+

ProductionRate PR


DPR+

+

+

InventoryAdjustment Time

IAT

-

ShipmentTime ST

-

DesiredInventory INV*


<Input>+

B

InventoryControl

Figure 4.1: A Single-Stage Production Control System

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Time (Day)

(Sto

ck o

f Uni

ts)

0

10

20

30

40

50

60

(Uni

ts/D

ay)

Inventory INV

Production Rate PR

Shipment Rate SR

Figure 4.2: Step Response of a Single-Stage Production Control System

The result shows that it takes about 12 days to make the Inventory level steady to meet the target

value of 100 production units. At time = 12 days, both production rate and shipment rate obtain

64


equilibrium at a rate of 20 units/day. This continuous single-stage production system model

allows production rate, shipment rate, and inventory level to be updated simultaneously.

Perhaps in the real manufacturing system, the production rate is determined by the amount of

workers scheduled at the particular shift. While the workers are producing the products, the

finished units will probably be kept in a temporary container until the total quantity has met the

preset batch size for shipment. Moreover, in a real manufacturing system, no one could expect

continuous shipment, i.e., shipping units every single minute while production workers are

making the units. Production management has to decide the desired feasible shipment rate, like

ship all finished units at the end of each shift (i.e., 8 hours long), or ship at the end of the day (i.e.,

24 hours). In order to model such a detailed real-life manufacturing scenario, we apply discrete

event simulation. We use ARENA simulation software to describe the same single-stage

production control system as shown in Figure 4.3.

Shipping

Create 1 Entity Values

Assign Inital

Parts in Inventory

Update Prod_Rate

Production

Update Inventory

Duplicate

Production

Plot Graph1

Shift TeamTrue

False

0

0

(Ship every X hours)

(Team shift every Y hours)

Ship Every X hours Update and Ship Parts are ShippedPlot Graph20 0

Ship Every X hours Update and Ship Parts are ShippedPlot Graph20 0

0

Figure 4.3: ARENA discrete-event model of a single-stage production control system

65


Figure 4.3 displays a discrete-event simulation model created in ARENA for the single-stage

production control system. The cause-and-effect interrelationships among the manufacturing

system variables are based upon the original system dynamics model as described in Fig. 4.1.

However, in order to better mimic the real manufacturing shop-floor behavior, we have to

simulate the system model as an event-based system changing at countable points in time instead

of continuous changing time. Firstly, the production lead time is now discretized into increments

of time steps (i.e., minute). Secondly, we define every Y amount of hours to permit updating the

total production work force. For instance, if Y = 4 hours, the management will evaluate the

current level of the finished inventory whether the production workers have overproduced or

under-performed to meet the target level. Hence the management will adjust the production rate

accordingly by adding or taking off workers at each Y hours. Thirdly, the management will

check the current stock level of the finished inventory every X hours to prevent excess inventory

cost occurred at the shipping area. The production manager evaluates every X hours to decide the

shipment rate of the finished product units.

Again, the main objective of this single-stage production control system is to manufacture 100

new products while keeping the appropriate work force according to the particular shipment rate

determined by the production management. In the next section, we demonstrate the differences

among different sets of X shipment hours and Y production hours in comparing the continuous

CCT single-stage production system model.

66


4.2 Comparison between Discrete Event Model and Classical Control Model

To better validate the dynamic behavior of the discrete-event, single-stage production model, as

a production manager, we decide to evaluate our production workers performance at a rate of Y

= 4 hours. At the middle and end of each 8-hr shift, we will allocate workers to either increase or

decrease our production rate according to the finished inventory stock. In addition, we want to

investigate the effect of the shipment update to the finished inventory level as X varies at 2 hours,

4 hours, 8 hours, and 24 hours. We show four plots of the discrete-event simulation result given

the shipment update varying at X= 2 hr, 4 hr, 8 hr, and 24 hr, while the total production work

force evaluated at every 4 hours. For each plot, we display the inventory level, the production

rate, and the shipment rate as shown in Figures 4.4, 4.5, 4.6, 4.7, 4.8. The results indicate that as

the shipment update rate increases (i.e., X hours increases), the higher variation of rate (i.e., zip-

sack effects) occurs. Vice versa, the quicker the shipment update, the closer the dynamic

characteristics behaves like the continuous differential equation model.

0

20

40

60

80

100

120

1 20 39 58 77 96 115 134 153 172 191 210 229 248 267 286 305 324 343 362 381 400 419 438 457

Time Step

Stoc

k (U

nit)

0

10

20

30

40

50

60R

ate

(Uni

t/Day

)

INV-2PR-2SR-2

Figure 4.4: A single-stage production system (Ship every 2 hr; Production update 4 hr)

67


0

20

40

60

80

100

120

1 20 39 58 77 96 115 134 153 172 191 210 229 248 267 286 305 324 343 362 381 400 419 438 457

Time Step

Sto

ck (U

nit)

0

10

20

30

40

50

60

Rat

e (U

nit/D

ay)

INV-4PR-4SR-4


0

20

40

60

80

100

120

1 20 39 58 77 96 115 134 153 172 191 210 229 248 267 286 305 324 343 362 381 400 419 438 457

Time Step

Stoc

k (U

nit)

0

10

20

30

40

50

60

Rat

e (U

nit/D

ay)

INV-8PR-8SR-8


68


0

20

40

60

80

100

120

1 20 39 58 77 96 115 134 153 172 191 210 229 248 267 286 305 324 343 362 381 400 419 438

Time Step

Stoc

k (U

nit)

0

10

20

30

40

50

60

Rat

e (U

nit/D

ay)

INV-24PR-24SR-24


In Figure 4.4, the production rate updates every 4 hours with a shipment rate of every 2 hours.

The steady-state value for production rate and shipment rate is in the range of 20-20.7 units/day

and 20-20.2 units/day, respectively. The finished inventory level is within 100-101 units. As we

increases X hour from 2 to 4 hours, the increase in variation raises very little. As we update the

shipment rate from 4 to 8 hours, we have seen the increase in the range for production rate and

shipment rate as 19.4-20.6 unit/day and 19.6-20.8 unit/day, respectively. The steady-state range

of the final inventory is 98-104 units. If we assume the production take shipment every 24 hours,

it yields the production rate with a range of 17.6-22.7 units/day and 18.2-22 units/day for the

shipment rate. The variation of the final inventory is between 91-110 units. We compare the

discrete event model at a shipment update of X= 4 hr and production update at Y= 4hr with the

continuous classical control model as shown in Figure 4.8.

69


0

10

20

30

40

50

1 17 33 49 65 81 97 113 129 145 161 177 193 209 225 241 257 273 289 305 321 337 353 369 385 401 417 433

Time Step

Stoc

k (U

nit)

0

20

40

60

80

100

Rat

e (U

nit/D

ay)PR(Cont) SR(Cont)

PR(Dis) SR(Dis)INV(Cont) INV(Dis)

Figure 4.8: Discrete vs. Continuous in modeling single-stage production system

Figure 4.8 shows that the state variables of the continuous model like production rate, shipment

rate, and inventory change continuously with respect to time, it yields a greater rate of change of

the state variables. For the discrete model, we have selected to update the production work force

and the shipment every 4 hours, the rate of change of the curves is much slower than the

continuous case. However, as the system reaches to its steady-state, about 12-14 days, the final

steady-state values of inventory, production rate, and shipment rate are insignificantly different

from the CCT continuous system model.

If the shipment rate and the production work force adjust at a shorter interval, say 30 minutes,

the slope of the finished inventory should increase. As the production work force and the

shipment continue to adjust at smaller time intervals, the rising slope of the finished inventory

behaves more like the continuous differential model. In other words, the CCT approach is

optimistic with respect to real discrete manufacturing systems and it always provides a lower

bound on the rise time of the system response. Additionally, the less rapid the management

70


changes the schedule of production work force and shipment, the more of an optimistic CCT

approximation yields (i.e., gives more error to real industrial scenario).

In real life, customer demands often come randomly and unexpected. The CCT approach is

mathematically limited in that it cannot include any stochastic elements. In addition, production

management schedule their amount of work force according to the maximum capacity of the

overall machine throughput rate. The CCT approach cannot model any non-linear elements that

contain saturation due to maximum capacity. For unreliable manufacturing processes and

machine breakdowns, we can still apply CCT approach to model the machine failure as a system

disturbance input. In order to overcome these limitations of the CCT approach, we can apply

Non-Linear Control Theory and Stochastic Linear Control Theory to further enhance model

validity.

In this chapter, we have demonstrated that the classical control theory (CCT) modeling approach

appears to provide valid approximations of real discrete manufacturing systems. To further

validate the CCT approach, we will use it to model, analyze, and design a real three-stage

semiconductor manufacturing system in Chapter 6.

71

Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong

Chapter 5 Design of Responsive Manufacturing

Systems

In this chapter, we introduce a new design methodology to assist management in identifying

operation strategies to better predict the dynamic responses of responsive manufacturing systems

subjected to rapidly changing market demands. Section 5.1 introduces the Root Locus design

technique, originating from classical control theory. In that section, we apply the Root Locus

design tool to improve the overall system responsiveness of a two-stage production control

system. By varying the system loop gain K, the location of the closed-loop poles moves

predictably on the complex s-plane. Hence, management can predict and design the particular

responsive manufacturing systems in terms of production lead time, degree of WIP overshoot

(i.e., damping ratio) and lean finished inventory. In Section 5.2, we further define and interpret

the meanings of those classical control terms, such as damping ratio, closed-loop poles location,

and settling time as they relate to the manufacturing world on a complex s-plane representation.

We give examples of various step response dynamic behaviors according to the different

locations of the closed-loop poles. This complex plane analysis offers a graphical view to

evaluate and understand the overall dynamics and the corresponding parametric sensitivity of

any responsive manufacturing systems. Finally, section 5.3 applies the Root Locus to design a

two-stage production control system with 3rd-order time delay. From section 3.2, we extended

the two-stage production control system with a 3rd-order time delay inclusion and it turned into a

fourth-order differential equation model. The corresponding root locus and dynamic behavior of

this higher-order system behaves very differently. We introduce the Dominant Roots or

72


dominant closed-loop poles concept to design higher-order differential system models. The

resulting root locus reveals the potential for system instability due to poor management policies

through the location of the closed-loop poles.

5.1 Root Locus Analysis and Design of a Two-Stage Production System

In this section, we introduce the Root-Locus technique from classical control theory [40,41,42]

to improve the overall system responsiveness and lead time for the two-stage production control

model. Phillips and Harbor [42] defined: “A Root Locus of a system is a plot of the roots of the

system characteristic equation (the poles of the closed-loop transfer function) as some parameter

of the system is varied”. The location of the system closed-loop poles, as given by Eq. (5.1),

determines the transient characteristics of the two-stage production system.

Closed-Loop Poles: -1 - s 2nn1,2 ζωζω j±= (5.1)

As mentioned, the closed-loop poles of a system are the roots of the characteristic equation. It is

instructive to see how the closed-loop poles move in the complex s-plane as the loop gain of the

system is varied. From a design point of view, an adjustment of the gain value(s) may bring the

closed-loop poles to certain desired locations. We apply the Root-Locus technique to plot the

roots of the characteristic equation for different values of gain K. The root locus is the locus of

roots of the characteristic equation of the closed-loop system as the gain K is varied from zero to

infinity. Such a plot clearly indicates how to modify the open-loop poles such that the response

of the manufacturing system can meet the specific customer requirements. We generally consider

the system of Fig. 5.1 in describing the concept of root locus as stated in Phillips and Harbor [42],

73


with 0 K< ∞. In Figure 5.1, we assume that G(s) comprises both the compensator transfer

function and the plant transfer function. The characteristic equation for this system is given in Eq.

(5.2). A value s

≤

1 is a point on the root locus if and only if s1 satisfies Eq.(5.2) for a real value of

K, with 0 ≤ K< ∞.

1 + KG(s) H(s) = 0 (5.2)

K G(s)

H(s)

+_

Figure 5.1: A system for Root Locus

For more detail on Root-Locus methodology, see Ogata [40] and Phillips and Harbor [42].

According to the closed-loop transfer function of the two-stage production system as derived in

Eq.(3.19), the system equation KG(s) for the root locus analysis as shown in Fig. 5.1 will

become:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +⎟⎟

⎠

⎞⎜⎜⎝

⎛−+⎥

⎦

⎤⎢⎣

⎡+++

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

WAT1

LT1

ST1

ST1

ST1

WAT1

LT1 ss

WATLT1

LT1

FAT1

KG(s)

211

2

(5.3)

Given that the loop gain, ST1=ST2, Eq.(5.3) reduces to:

ST

1WAT

1LT1 ss

WATLT1

LT1

FAT1

KG(s)

1

2⎥⎦

⎤⎢⎣

⎡+++

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

= (5.4)

74


For illustration, we arbitrarily set the system parameters as follows: LT=1 day, ST=5 days,

WAT=5 days, and FAT=5 days. By applying the Matlab [49] rlocus command, we find and plot

the root loci of the two-stage production control system as shown in Fig. 5.2.

-2 -1.5 -1 -0.5 0 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Real Axis

Imag

inar

y A

xis

Constant ζ Lines and Constant ωn Circles

ζ=0.9

0.80.7

0.6 0.5

0.4 0.3

0.20.1

ωn=1ωn=2

Figure 5.2: Root-Locus Plot of a Two-Stage Production Control System

The root-locus branches (i.e., marked as X crosses) starts from the open-loop poles from Eq.(5.4)

at s1= -1.4 and s2= 0 (K=0) at the real axis of the complex s-plane as shown in Fig. 5.2. The gain

K value increases incrementally from 1, at the closed-loop poles derived from Eq.(5.1) at s1= -

1.2 and s2= -0.2, until the increased K value is 18, with a pair of complex-conjugate roots at –0.7

± 1.96i. The arrows indicated in Fig. 5.2 provide the direction of the increasing K values from

the real-axis to the imaginary-axis. As K is varied, the location of the closed-loop poles changes.

The intersection of the horizontal K values and the vertical K values is called the breakaway

point (ζ=1) of the root-locus with s1,2= -0.7 and K value of 2.04. In a second-order dynamic

system with no additional dynamics in the numerator of the transfer function, the breakaway

75


point is simply (s1+s2)/2. The damping ratio ζ of the system is always equal or greater than one

as the closed-loop poles are found along the real-axis on the s-plane. Once the closed-loop poles

have passed the breakaway point, ζ begins to reduce further with pairs of complex-conjugate

roots. Figure 5.2 also plots the grid of constant damping ratio ζ lines and constant undamped

natural frequency ωn lines. The constant ζ lines are radial lines passing through the origin with a

decrement of 0.1 from ζ=1 to 0.1, whereas the constant ωn loci are circles.

In the complex s-plane, we can express the damping ratio ζ of a pair of complex-conjugate poles

in terms of the angle φ which is measured from the negative real-axis with ζ= cosφ, and

determine the distance of the pole from the origin by the undamped natural frequency ωn. As

discussed in the previous section, we quantify the transient dynamic responses of the two-stage

production system by the key parameters ζ and ωn. The importance of introducing Root-Locus

method here is to provide a predictive design technique to improve the overall dynamic behavior

of this manufacturing system. As per classical control theory, we can design a desirable transient

response of a second-order system with a damping ratio around 0.707. It is justified with results

in the next phase. Small values of ζ<0.4 yield excessive overshoot where a system with ζ>0.8

responds sluggishly as shown in Fig. 3.17 previously. The further the closed-loop pole is away

from the origin of the s-plane, the faster the response of the system behaves. However, the

system will become unstable if there is any closed-loop pole found on the right-hand side of the

s-plane.

For our example, in order to bring the system model to ζ=0.707, the K value from the root-locus

analysis is increased to 4.08 with ωn=0.99 at s1,2= -0.7 ± 0.70i as shown in Figure 5.2. Given this

76


particular parametric set of values, the system equation has an initial non-unity loop gain K of

0.24 as computed from Eq.(5.3). As a result, we have to multiply every root-locus gain K from

Matlab by a factor of 0.24. Although K is a function of Finished Inventory Adjustment Time

(FAT), Lead Time (LT), and WIP Adjustment Time (WAT) as given by Eq.(3.29), in order to

keep the same breakaway point while changing the K value, we can only vary K as a function of

FAT.

2n

t

WATLT1

LT1

FAT1

K y(t) limω

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

==∞→

(3.29)

By setting FAT=1.224, we make the system respond with ζ=0.707 at K=4.08*0.24=0.98. In

designing a fast-response manufacturing system, setting ζ to 0.707 is crucial; however, moving

the breakaway point further away from the origin is another vital step. According to Eq.(3.23),

the time response of the system is a function of LT, WAT and ST.

Time constant:

⎥⎦

⎤⎢⎣

⎡++

==

1

n

ST1

WAT1

LT1

2 1 ζω

τ (3.23)

Since we have assumed the shipment time to be 5 days for a transportation schedule, we will

further study the dynamic characteristics of this system model as a function of LT and WAT.

Figures 5.3 and 5.4 show, respectively, contour plots of ζ values and breakaway points against

WAT and LT while keeping FAT=1 day.

77


0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

Le ad Tim e (LT) in day s

WIP

Adj

ustm

ent T

ime

(WA

T) in

day

s

Constant ST=5 day s and Constant FAT=1 day

0.57

0.585

0.6

0.625

0.650.675

0.707

0.750.8

0.850.90.951

Figure 5.3: Contour Plot of ζ values as a function of LT and WAT

Figure 5.3 shows the contour of ζ values as a function of WAT and LT with a constant FAT=1.

A combination choice of LT and WAT along a particular contour line gives the desired damping

ratio. For example, if we pick WAT=1.5, by taking LT=1.1, it results in ζ≈ 0.707 and a

breakaway point of –0.89 (i.e., an improvement from –0.7). Figure 5.3 shows that ζ values

decrease from 1 to 0.57 with increasing values of both LT and WAT. As stated earlier, FAT is

the most significant factor that affects ζ values. With a FAT value higher than 2.0 (not shown

here), changing any value in LT and WAT will not make any impact to reduce ζ to less than

unity (i.e., the system is always overdamped or critically damped). For the contour of breakaway

points as shown in Figure 5.4, the breakaway point moves towards more negative from –0.53

to –4.0 as LT and WAT are reduced, hence it improves the system response as the breakaway

point moves further away from the origin at the complex plane. FAT does not play a role in the

location of the breakaway point. If we arbitrarily pick LT=1 and WAT=1 from Figure 5.4, it

78


gives a breakaway point of –1.1 (i.e., an even further negative value) with a ζ value of 0.78. We

find all of these observations without any simulation and our approach is able to predict directly

the best system response.

Figure 5.4: Contour Plot of Breakaway Points as a function of LT and WAT

0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

Le ad Tim e (LT) in days

WIP

Adj

ustm

ent T

ime

(WA

T) in

day

s

Constant ST=5 days and Constant FAT=1 day

-0.53-0.56

-0.6-0.65

-0.7-0.75

-0.81

-0.9-1

-1.2-1.4

-1.8-2.2

-3-4

In order to verify the findings and the recommended improved parametric set of values for

applying both DOE and Root-Locus, we simulate and compare the step response of the system

model subjected to varied sets of design system parameters as shown in Fig.5.5. We use Matlab

to simulate the following four different sets of system parameters with a constant ST value of 5

days:

Set A: LT=1, WAT=5, FAT=5; Set B: LT=1, WAT=5, FAT=1.224

Set C: LT=1.5, WAT=1.1, FAT=1; Set D: LT=1, WAT=1, FAT=1

79


0 5 10 150

20

40

60

80

100

120

Time in days

Fini

shed

Inve

ntor

y Le

vel i

n st

ocks

Set A: ζ=1, Brkaway Pt=-0.7

Set B: ζ=0.707, Brkaway Pt=-0.7

Set C: ζ=0.7073, Brkaway Pt=-0.8879

Set D: ζ=0.7778, Brkaway Pt=-1.1

Figure 5.5: Step Response Comparison of a Two-Stage Production Control System

The results show that the original parametric set A responds sluggishly due to its ζ value of 1

(i.e., critically damped) with a breakaway point of –0.7. The root-locus analysis suggests to

increase the gain K value to 4.08 and with FAT=1.224. The set B curve has shown a significant

improvement in term of the overall system response. Curve B has a ζ value of 0.707 with the

same breakaway point of –0.7. We further improve the system response by changing the values

of LT, WAT and FAT according to the contour plots. The set C curve gives an even quicker rise

time as compared to curve B due to the improvement of the breakaway point (i.e., increases

from –0.7 to –0.89). Curve D appears to be the best suggested system response. Again, by

studying the dynamic characteristics at the contour plots, we further improve the breakaway

point from –0.89 to –1.1 with even smaller inventory overshoot.

80


5.2 Interpretation of CCT Terms to the Manufacturing World

As stated previously, the root locus is the locus of roots of the characteristic equation of the

closed-loop system as a specific parameter (usually, gain K) varies from zero to infinity. This

technique is very useful since it provides guidelines in which the open-loop poles and/or zeros

(definition of these CCT terms will be described later) should be modified such that the system

responses meet the desired performance specifications. By using the root-locus method, we can

determine the value of the loop gain K that makes the damping ratio, ζ of the dominant closed-

loop poles as prescribed. In the manufacturing world, industrial engineers and managers prefer to

predict and control the critical system variables, like lead time, inventory level, settling time for

production control planning purposes. In this section, we define and interpret some key

manufacturing system variables as they relate to the classical control theory terms described in

the complex s-plane. The aim of this interpretation is to offer a new operation planning strategy

for manufacturing managers to better predict and study the transient behavior of responsive

manufacturing systems. This approach offers a systematic and graphical view to evaluate and

understand the overall dynamics and its corresponding parametric sensitivity of any responsive

manufacturing system model.

Responsiveness is an overall strategy focused on thriving in an unpredictable and dynamic

environment. Lean is a philosophy that seeks to minimize all waste that includes long lead time,

excess WIP inventory, non-value added activities. Responsiveness refers to the dynamics of

manufacturing system behavior in term of damping ratio ζ value. Fowler [31] described an MRP

system as a feedforward system where the production is pulled through the system by a

feedforward scheduling system, referenced to the particular customer demand. He further stated

81


that the Kanban control system may be seen as the production pulled through the system by a

sequence of cascaded feedback control loops. However, there has never been an example to

describe whether these loops are stable or tuned properly. We will now do this with a new

systematic method to study and predict production operation strategies based upon the key

manufacturing variables as they relate to the closed-loop poles location in the complex s-plane.

Again, we pick the Root Locus plot of the two-stage production control system as shown in

Figure 5.2 to perform the investigation. As stated previously, the root-locus branches of this

model (marked as X crosses) starts from the open-loop poles at s1= -1.4 and s2=0 (K=0) at the

real axis of the complex s-plane as shown in Figure 5.6. The gain K value increases

incrementally from 1, at the closed-loop poles at s1= -1.2 and s2= -0.2, until the increased K

value is 18, with a pair of complex-conjugate roots at -0.7 ± 1.96i. We added nine “+ crosses” on

the s-plane to examine different dynamic characteristics according to the location of the closed-

loop poles as shown in Figure 5.6 (next page).

In this section, by varying the location of different closed-loop poles or roots on the complex s-

plane as shown in Figure 5.6, we specifically define and interpret the significance of each

location as it relates to the dynamic behavior found in the manufacturing environment. The x-

axis and the y-axis on Figure 5.6 represent the real axis and the imaginary axis, respectively. The

closed-loop poles can be interpreted as the production buffers (i.e., accumulates products). The

pole located at point A indicates a negative real root of the characteristic equation, Eq. (5.3). This

pole location gives the production inventory level to decay exponentially until it reaches steady

stable condition. Any manufacturing system that contains only real roots, closed-loop poles on

the x-axis will not obtain any inventory overshoot and dynamic oscillation (i.e., ζ ≥1.0).

82


-2 -1.5 -1 -0.5 0 0.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Real Axis

Imag

inar

y A

xis

X X

+

+

+

+

+

+

+

A B

C

D

E

F

G

H

J

+

+

K

L

Figure 5.6: Complex s-plane interpretation of varying location of closed-loop poles The pole located at point B is special case when production maintains at a certain inventory level

steadily (i.e., no rise or fall), thus it is considered as a limited stable condition due to the potential

of inventory growing exponentially. The pair of complex-conjugate roots with negative real parts

located at points C and D give dynamics that makes production inventory oscillate and dies

down to stable condition within its envelope. The points C and D are found at the contour line of

damping ratio, ζ = 0.8. In terms of the speed of the production lead time, points A, C, and D shall

behave similarly except there is no oscillation at point A. As the closed-loop poles get closer to

the right, the speed of the system responsiveness shall decrease. The pair of complex-conjugate

roots with negative real parts located at points E and F is expected to respond slower than the

pair located at points C and D. In addition, the degree of oscillation shall magnify as the damping

83


ratio decreases from ζ =0.8 to 0.5. Their rate of exponential decay will be less than the pair from

C and D but it still leads to a steady stable condition. The pair of complex-conjugate roots on the

imaginary axis located at points G and H yields a limit cycle for the manufacturing system.

Given ζ =0, the production inventory will behave periodically within a particular set range of

inventory level. The production attempts to respond due to the seasonal customer demand,

however given the fixed maximum customer demand, the corresponding production inventory

oscillate within the same envelope. Obviously, operation management does not want to keep a

fixed amount of annual inventory costs to forecast seasonal customer requirements. Rather,

management shall determine a strategy to make shipments as needed. In CCT terms, this type of

manufacturing system is considered to be marginally stable due to the potential to grow

exponentially positive. In practice, of course, saturation of some limited resource will eventually

occur to keep the results finite.

Once the closed-loop poles go beyond the imaginary axis to the right-hand-side of the s-plane,

the manufacturing system will behave unstable without bound or until the plant capacity

saturates. For instance, the pole located at point L has a positive real root. Thus, the production

inventory will grow positively without bound. This is the typical scenario when the management

decided to build as much inventory as production can provide to prevent the unexpected sudden

request by their customers. Truly, this kind of operation policy should not continue to run, given

the concept of Kanban from the Lean manufacturing system. Lastly, the pair of complex-

conjugate roots with positive real parts located at points J and K indicates that production

inventory continues to increase with dynamics and oscillation. This system grows positive and

84


sinusoidal with an increasing size of envelope with no bound. Obviously, it turns into a poor

management policy to let inventory costs add accumulatively in seasons without bound.

We graphically display the dynamic behavior of those six special cases as discussed as shown in

Figure 5.7. Again, the two stable cases, include, a single negative real root (A) and a pair of

complex conjugate roots with negative real parts (C and D). The two marginally stable cases,

contain, a single root at the origin (B), and a single pair of complex conjugate roots on imaginary

axis (G and H). Finally, the two unstable cases comprise, a positive real root (L) and a complex

conjugate roots with positive real parts (J and K).

85


0 5 10 15 20 25 300

20

40

60

80

100Negative Real Root (A)

Inve

ntor

y (U

nit)

0 5 10 15 20 25 30-60

-40

-20

0

20

40

60

80Complex Roots w/ Negative Real Parts (C&D)

Inve

ntor

y (U

nit)

0 5 10 15 20 25 3099

99.5

100

100.5

101Single Root at Origin (B)

Inve

ntor

y (U

nit)

Time (Day)0 5 10 15 20 25 30

-100

-50

0

50

100Complex Roots on Imaginary Axis (G&H)

Inve

ntor

y (U

nit)

Time (Day)

0 5 10 15 20 25 300

2000

4000

6000

8000

10000Positive Real Root (L)

Inve

ntor

y (U

nit)

Time (Day)0 5 10 15 20 25 30

-1000

-500

0

500

1000Complex Roots w/ Positive Real Parts (J&K)

Inve

ntor

y (U

nit)

Time (Day)

Figure 5.7: Transient mode shapes associated with locations of roots in the complex s-plane

86


5.3 Root Locus Design of a Two-Stage Production System with Time Delay

Delays are inherent in many physical and engineering systems. There are different kinds of

delays found as described by Forrester and Sterman [5,16]. They are such as material delays,

pipeline delay or transportation lag, and information delays. For the manufacturing sector,

material delay is a kind that captures the physical flow of material through a delay process. For

example, this delay often happens in a supply chain, distribution business, and construction

management. For multi-stage manufacturing processes, between each station, there is a delay

caused by transportation and order handling. Each station operates individually based on demand

information provided from upstream. As the number of stations increase, the demand signal

amplifies from station to station as orders go through the chain of supply. Forrester described

those oscillations in demand along the chain as the bullwhip effect [2]. In the previous sections,

we stated that the higher the order of the time delay goes, the better the production system can be

characterized. Wikner [27] has described that a third-order delay has proved to be an appropriate

compromise between model complexity and model accuracy for most dynamic modeling of

production-inventory systems. In this section, we investigate the dynamic behavior of the two-

stage production control system with a 3rd-order time delay from Figure 3.7 on the complex s-

plane environment. As recall from Eq.(3.33), the 4th-order differential equation formulated for

the two-stage with a 3rd-order time delay model is as:

( )( )

[ ] [ ] [ ] [ ]D s C s B s A sWATLT

1LT

1FAT

27

DI(s)FI(s)

234

23

++++

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛

= (5.5)

87


where,

( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟

⎠⎞

⎜⎝⎛ +=

+++=

+++=

++=

212

12

123

112

1

ST1

ST1

FAT1

WAT1

LT1

LT27 D

WATSTLT9

WATLT27

STLT27

LT27 C

WATST1

WATLT9

STLT9

LT27 B

WAT1

ST1

LT9 A

Given the set of selected system parameters: LT=4; ST=8; WAT=1; FAT=1, we observe a

significant dynamic oscillation difference between the third-order delay model and the first-order

exponential smoothing model for the two-stage production system from Figure 3.20.

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

Time in days

Fini

shed

Inve

ntor

y Le

vel (

Uni

t)

1st-Order Delay

3rd-Order Delay

Figure 3.21: Dynamic Responses between 1st-order time delay and 3rd-order time delay

Again, we apply the Root Locus technique to investigate the effect of the closed-loop poles

location for the dynamic production control system. Since it is a fourth-order differential

88


equation model, there are four closed-loop poles found on the complex s-plane as shown in

Figure 5.8

-2.5 -2 -1.5 -1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5Im

ag A

xes

Real Axis

LT=4, ST=8, WAT=1, FAT=1

ζ=0.9 ζ=0.8

ζ=0.7ζ=0.6

X X

X

X

+

+

+

+

Figure 5.8: Root-Locus Plot of a two-stage production control system with 3rd-order time delay

The root-locus branches starts from the open-loop poles location where s1= 0, s2= -1.74, and s3,s4

= -0.819±0.909i, (K=0) on the complex s-plane as shown in Fig. 5.8. The gain K value increases

in the direction of the arrows from the starting poles location (K=0) to the stopping poles

location (K=8). As observed in Figure 5.8, the complex conjugate pair of poles enters the right-

hand side of the s-plane as K increases over 1.305 (not explicitly shown here). As the gain K

continues to increase to infinity (K→∞), the production system grows oscillatory with no bound

(i.e., unstable). Whereas, the two real roots (poles) continue to go towards to the further left of

the s-plane (i.e., stable) as the gain K value increases to infinity. The major challenge here is to

89


design and improve this 3rd-order time delay, two-stage production control system to give quick-

responsive production lead time, less oscillation, and lean final inventory level. Although we

have derived the fourth-order differential equation for this particular production system, there is

no standard formulation that describes damping ratio, time constant, rise time and settling time

for higher-order control systems. In MATLAB programming software [49], we can apply a

function command, rltool, from the Control System Toolbox, to determine the corresponding

gain K value of the corresponding higher-order manufacturing system that yields fast production

lead time and less inventory overshoot on the rlocus plot as shown in Figure 5.8. For higher-

order dynamic systems, it is a good rule of thumb to set the damping ratio between 0.7 and 0.9 to

obtain a fast responsive system behavior [40,41,42,43]. As the gain K value is varied, the

location of the closed-loop poles changes. However, there are altogether four closed-loop poles

or roots to be adjusted in this fourth-order system model. The challenge is to select the particular

K gain value such that the overall production control system yields the best system performance.

Fortunately, there is a concept called Dominant Roots or dominant closed-loop poles from

classical control theory [40,41,42] to help in determining the most influential pole(s) for the

system. We have seen that a time constant τ is a measure of the decay rate of an exponential e-at,

where τ=1/a. The time constant corresponds to the characteristic roots, s= -a, or a complex pair,

s= -a ± ib. If a stable dynamic system model has several roots with different real parts, the root

having the largest time constant (i.e., the one lying the farthest to the right on the complex s-

plane) is the root whose exponential term dominates the overall system response. This particular

root is called the dominant root. There can be two dominant roots (complex pair) existing in a

system because they both have the same real part and same time constant. We illustrate different

90


step input responses of the two-stage production system with a 3rd-order time delay under various

gain K values as shown in Figure 5.9.

0 10 20 300

50

100Gain K = 0.1

Inve

ntor

y (U

nit)

0 10 20 300

50

100

150Gain K = 0.35

Inve

ntor

y (U

nit)

0 10 20 300

50

100

150Gain K = 0.6

Time (Day)

Inve

ntor

y (U

nit)

0 10 20 300

50

100

150

200Gain K = 1.0

Time (Day)

Inve

ntor

y (U

nit)

Figure 5.9: Step Responses of a two-stage system w/ 3rd-order time delay under various K values

Given the set of selected system parameters: LT=4; ST=8; WAT=1; FAT=1, we show four

different step response curves by varying K values as shown in Figure 5.9, As the gain K values

varies from the root locus, the location of the closed-loop poles (roots) changes on the complex

s-plane. As described earlier, the location of the closed-loop poles determines the overall system

responsiveness for the two-stage production control system. The multiple oscillated and dynamic

step response as shown in Figure 3.20 is set with K=1.0. By varying K values via rltool from

MATLAB, we are able to improve the system response by minimizing the inventory overshoot

91


and reducing the production lead time as shown in Figure 5.9. In this particular case, by reducing

the K value, we decrease the amount of oscillation and improve the system settling time.

However, if we reduce K value too low, like K=0.1, the higher-order system behaves sluggishly

like a first-order dynamic system as shown on the top left hand corner of Figure 5.9. Obviously,

we would like the system respond similar to the case when K=0.35. As recall from the previous

section, the gain K is a function of Finished Inventory Adjustment Time (FAT), Lead Time (LT),

and WIP Adjustment Time (WAT). Plus, this fourth-order system equation has an initial non-

unity loop gain K of 2.109. Hence, in order to set K=0.35 from the rltool in MATLAB, we have

to multiply 0.35 by a factor of 2.109 to give 0.7383. The dynamic structure of this fourth-order

differential model is very similar to its previous 2nd-order system except for the third-order delay

term of 1/LT3. Again, in order to keep the same breakaway point while changing the K value, we

only vary K as a function of FAT. By setting FAT=2.8571, we make the system respond with a

settling time of 12 days at ζ = 0.67 and Kactual = 0.7383. Its resulting closed-loop poles are s1,2 =

-0.4521±0.4884i, s3,4= -1.2354±0.3747i. Given the same breakaway point, we can improve the

system response by changing the K value from 0.35 to 0.300676, thus FAT becomes 3.3258. The

production lead time reduces to 10 days with no oscillation with ζ = 0.83 and Kactual=0.6342. Its

corresponding closed-loop poles are s1,2= -0.6203±0.4148i, s3,4= -1.067.

The lead time has been reduced from 12 days to 10 days because the dominant pair of roots go

further left from the imaginary axis (i.e., from -0.45 to -0.62). In addition, the damping ratio

increases from 0.67 to 0.83, especially the improved roots, s3 and s4, are located on the real axis

with real parts to give fast response time. Finally, we can further reduce the production lead time

and system settling time by changing the parametric set of system variables to LT=1, ST=5,

92


WAT=1, and FAT=1.4. For this particular parametric set of values, we find a different initial

non-unity loop gain K of 54. The root-locus branches start from the open-loop poles with arrows

at s1=0, s2= -5.1042, s3,4= -2.5479±2.345i (K=0) and go towards the poles ending at K=13 as

shown in Figure 5.10.

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2-4

-3

-2

-1

0

1

2

3

4

Real Axis

Imag

Axe

s

LT=1, ST=5, WAT=1, FAT=1.4

ζ=0.9

+

+

+

+

X

X X

X ζ=0.8

ζ=0.7

Figure 5.10: Root-Locus Plot of a two-stage production control system with 3rd-order time delay

It is shown that the root-locus plot in Figure 5.10 is very different from the Figure 5.8. All the

closed-loop poles can be found on the real axis depending on their selected K values. Whereas,

the pair of dominant poles in the previous case could never reach ζ>0.9 nor the real axis as gain

K varies. As mentioned, the starting open-loop poles location and its breakaway points are

determined by the given set of design system variables in terms of LT, ST, WAT, and FAT. By

changing the K value to 0.7143 with FAT=1.4, we make the system respond with ζ=0.89 and

93


Kactual =0.7143*54=38.57. Its corresponding closed-loop poles are located at s1,2= -1.5087

±0.7675i and s3,4= -3.5913±0.7513i. The resulting production lead time has significantly

improved and reduced from 10 days to 4.3 days. In order to verify the findings and the

recommended improved parametric set of system variables applying Root Locus technique, we

simulate and compare those four particular step responses of the two-stage production control

model with a 3rd-order time delay subjected to varied sets of design system parameters as shown

in Figure 5.11.

Set A: LT=4, ST=8, WAT=1, FAT=1; (Multiple oscillations with very long settling time)

Set B: LT=4, ST=8, WAT=1, FAT=2.8571; (Single Overshoot with reduced lead time, 12 days)

Set C: LT=4, ST=8, WAT=1, FAT=3.326; (No Overshoot with improved lead time, 10 days)

Set D: LT=1, ST=5, WAT=1, FAT=1.4; (No Overshoot, quick-response lead time, 4.3 days)

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120

140

160

180

Time (Day)

Fini

shed

Inve

ntor

y (U

nit)

Set A

Set B

Set C

Set D

Figure 5.11: Various Step Response of a two-stage production system with 3rd-order time delay

94


We have demonstrated a new operation design method that permits production management to

better predict and design manufacturing system variables that yield responsive production lead

time and minimal inventory build-up (i.e., leanness) under transient manufacturing conditions.

This Root Locus design approach gives a completely graphic view to understand the sensitivity

on the location of the closed-loop poles by varying the loop gain K values to the dynamic system

models in the complex s-plane. We are able to determine responsive production parametric

values without iterative trial-and-error simulation as found in discrete event simulation or the

system dynamics approach.

The Root Locus design method we demonstrated here is based on varying the loop gain K values,

plus changing the manufacturing system parameters to reduce the time constant of the dominant

poles. In many industrial cases, however, the adjustment of the gain K values alone may not

provide sufficient alternation of the manufacturing system structure to meet the specific customer

requirement. As we learned from this section, as gain K values increases, it improves the steady-

state behavior but it could make the system in a poor stability region (even unstable) as K goes to

infinity. If it is the case, it may be necessary to redesign the dynamic manufacturing system

structure to alter the overall transient behavior to meet the management specification. Such a

redesign approach is called System Compensation technique found in classical control theory

[40,41,42]. We can easily perform this redesign approach using rltool from MATLAB by adding

selective poles and zeros into the original differential equation system model. In general, by

adding poles (more feedback loops) to the system, we destabilize the dynamic behavior with the

addition of production buffers. Whereas, adding zeros is similar to anticipating the customer

demand (i.e., forecasting with feedforward loops) to the system, in result, it has a stabilizing

95


influence. In root locus plotting, the gain K always begins at its open-loop location as K

increases until it reaches either to a zero location or infinity. System compensation is definitely

something we shall consider for responsive manufacturing systems applications in the near future.

5.4 Guidelines to perform Root Locus Design in Responsive Manufacturing Systems In this chapter, we have demonstrated how to apply Root Locus design technique from classical

control theory to improve the overall dynamic responsiveness of responsive manufacturing

systems according to the closed-loop poles locations on the complex s-plane. We here

summarize and give some basic guidelines for those whom may interest to apply Root Locus to

design and improve their particular manufacturing system dynamic behavior.

(1) Model the responsive manufacturing system into transfer function (i.e., differential equation

format).

(2) Apply Root Locus design technique to identify the root locus of the particular dynamic

manufacturing system behavior by varying the loop gain K values on a complex s-plane (you

can use MATLAB Control Toolbox to perform this function task).

(3) Select your desired closed-loop poles location (i.e., damping ratio in the range of 0.7-0.9)

and record your corresponding system loop gain K value.

(4) Adjust your key manufacturing system variables to match the same loop gain K value.

(5) Plot your resulting dynamic step response according to your selected parametric set values.

96


(6) Shorten your production lead time by moving your dominant closed-loop poles location

further to the left-hand side of the complex s-plane and adjust your key manufacturing

system variables to match the new loop gain K value.

(7) Verify your dynamic step response behavior according to your selected parametric set values.

97

Chapter 6 Industrial Case Study N.H. Ben Fong

Chapter 6 Industrial Case Study

The purpose of this chapter is to enhance the validation of applying classical control theory

(CCT) methodology to responsive manufacturing systems. By using a real industrial case study,

we aim to further validate the potential of using CCT approach to model, design, and improve

the overall responsiveness of manufacturing systems. We have chosen a hybrid push-pull

production system for semiconductor manufacturing to represent a particular Intel Corporation

plant facility as extracted from the proceedings of the 22nd International Conference of the

System Dynamics Society (July 2004) written by Goncalves et.al. [50]. In their industrial case

study, Goncalves, Hines, Sterman, and Lertpattarapong undertook a year-long, in-depth research

project to develop and analyze a manufacturing model of producing semiconductors via the

System Dynamics (SD) approach. Their case study addressed the causes of oscillatory behavior

in capacity utilization at a semiconductor manufacturer and the role of endogenous customer

demand in influencing the company’s production and service level. For more a detailed

understanding of this 41-pages case study, please read Goncalves et al. [50].

We intend to make use of this particular Intel hybrid push-pull semiconductor production system

in the following way:

(1) Demonstrate the use of classical control theory approach to convert this real industrial SD-

based semiconductor manufacturing system into block diagram representation and transfer

function.

98


(2) Apply the Root Locus design technique to study the sensitivity of the closed-loop pole

locations and the dynamic behavior of this higher-order differential equation system model.

(3) Investigate any new findings and difficulties to design this industrial higher-order

responsive manufacturing system.

We will find that the CCT approach suggests new ways to control the studied production system

and to represent its improvement potential. This hybrid push-pull semiconductor production

consists of three stages (i.e., Fabrication WIP for Wafers, Assembly WIP for dies, and Finished

Inventory for chips) as shown in Figure 6.1. The push system is found at the upstream stages and

a pull system is at the downstream stages.

FabricationWIP (FWIP)

FinishedInventory

(FI)Wafer Start(WS) Net Wafers

Outflow (NWO)

Shipment Rate(SR)

Desired FinishedInventory (FI*)

Desired FabWIP (FWIP*)

Adjustment forFGI (AFGI)

Adjustment forFab WIP (AFWIP)

Desired WaferStart (WS*)

ManufacturingCycle Time (MCT)

FabWIPAdjustment Time

(FWAT)

+

-

FI AdjustmentTime (FIAT)

+

-

+

- +

-

+

AssemblyWIP (AWIP)

Net AssemblyOutflow (NAO)

Desired NetAssembly Outflow

(NAO*)

DesiredAssembly WIP

(AWIP*)

Adjustment forAWIP (AAWIP)

CompleteAssembly Time

(CAT)

AWIP AdjustmentTime (AWAT)

+

+

-

-

Die Inflow (DI)+

Desired Net WaferStart (NWS*)

Minimum OrderProcessing Time

(MOPT)

ExpectedShimpent

Rate (ESR)

UpdateShipments Time

(UST)

-

+

+

+

<Total Demand(TD)>

+-

++

+

-+

+

+

+-

+

Safety StockPercentage (SSP)

+

Figure 6.1: Hybrid Push-Pull Intel Semiconductor Production System

The push system characterizes the front-end: weekly updates from the total demand and the

adjustment from Fabrication and Assembly WIP serve as the basis for the desired wafer

production rate (i.e., Wafer Start). In contrast, the back-end operates as a pull system, with

assembly/testing, packing, and distribution based on current customer demand. Production

99


decisions often rely on customer demand. The current customer demand drives the particular

shipment and its assembly completion, whereas the demand forecasts influence production start

rate. All the incoming orders are logged by the Intel’s information system and tracked until they

are shipped to customers or cancelled. If the finished products (i.e., microprocessors) are

available in Finished Inventory (FI), orders can be filled immediately. Hence, incoming customer

orders “pull” the available microprocessors from FI. Consequently, the replenishment of FI

shipped to customers “pull” microprocessors from the Assembly WIP (AWIP).

The current customer demand drives the pull characteristic of assembly WIP and finished

inventory. The actual shipment operates in a pull mode, with shipment being determined by the

desired rate. However, if there is not enough finished inventory, the system will ship out only

what it is available. The Finished Inventory (FI, units) is the accumulation of difference between

Net Assembly Outflow and Shipment Rate. The shipment rate (SR, units/month) depends on the

stock of FI and the minimum order processing time (MOPT, month) via a simple first-order

delay process. The expected shipment rate (ESR, units/month) is computed under the feedback

of the current shipment rate with a first-order delay of Update Shipment Time (UST, month).

The desired net chips (Adjustment finished inventory, AFI (units/month)) is adjusted above or

below recent shipment to close any gap between the desired finished inventory FI* (units) and

the actual FI proportional to the Finished Inventory Adjustment Time (FIAT, month). The

desired finished inventory is calculated by the product of ESR and MOPT with a safety stock

percentage (SSP) factor. The Desired Net Assembly Outflow (NAO*, units/month) is the

summation of AFI and ESR.

100


At the Assembly WIP process stage, the Adjustment Assembly WIP (AAWIP, units/month) is

adjusted between the desired Assembly WIP (units) and the current Assembly WIP (units)

proportional to an Assembly WIP Adjustment Time (AWAT, month). The desired Assembly

WIP (AWIP*, units) is the product of desired Net Assembly Outflow (NAO*, units/month) and

Complete Assembly Time (CAT, month). The desired Net Wafer Start (NWS*, units/month) is

the summation of AAWIP (units/month) and the total demand by the customer (TD, units/month).

The stock level of Assembly WIP (AWIP, units) is the accumulation of difference between Die

Inflow (DI, unit/month) and Net Assembly Outflow (NAO, units/month). NAO is computed as a

first-order delay between Assembly WIP proportional to Complete Assembly Time.

The wafers produced in the fabrication process stage are pushed into the Assembly WIP where

they are stored until orders for specific products pull them from AWIP into Finished Inventory

for shipment. While the Net Wafer Outflow (NOW, units/month) depletes fabrication WIP

(FWIP, units), wafer start (WS, unit/month) replenishes it. The Net Wafer Outflow is a first-

order time delay of Manufacturing Completion Time (MCT, month) from the Fab WIP (FWIP,

units). The decision on actual production rate, WS, is based directly on the desired Wafer Start

(WS*, units/month). The Fab planners determine the desired wafer start considering the desired

Net Wafer Start (NWS*, units/month) requested by the Assembly Stage and an adjustment for

fabrication WIP (AFWIP, units/month). The AFWIP is calculated as the difference between the

current Fab WIP and the desired Fab WIP (FWIP*) proportional to a FWIP Adjustment Time

(FWAT, month). The FWIP* is the product of the desired Net Wafer Start (NWS*, units/month)

and the Manufacturing Completion Time (MCT).

101


To simplify the non-linear mathematical expression for the non-negativity constraints to prevent

negative production at Wafer Start, Desired Net Wafer Start, and Desired Net Assembly Outflow,

we assume that there is no backlog in this industrial case study. The system variables, WS,

NWS*, and NAO* always have positive productive rates. For the detail descriptions of this case

study, please refer to Goncalves et al. [50].

Following the guidelines stated in Section 3.3, we now convert the cause-and-effect expressions

from CLDs and SFDs into different sets of system equations as shown in Figures 6.2, 6.3, 6.4.

Fabrication WIP Stage:

FWIP = WS - NWO∫

FWIPNWO = MCT

WS = Max (0, WS*) WS* = AFWIP + NWS*

FWIP* - FWIPAFWIP = FWAT

⎛ ⎞⎜ ⎟⎝ ⎠

FWIP* = (NWS*) (MCT)

Figure 6.2: System Equations for 1st Stage Process – Fabrication WIP

Assembly WIP Stage:

AWIP = DI - NAO∫

AWIPNAO = CAT

NWS* = Max (0, AAWIP +TD)

AWIP* - AWIPAAWIP = AWAT

⎛ ⎞⎜ ⎟⎝ ⎠

AWIP* = (NAO*) (CAT)

Figure 6.3: System Equations for 2nd Stage Process – Assembly WIP

102


:

Finished Inventory Stage

FI = NAO - SR∫

FISR = M OPT

NAO* = Max (0, AFI + ESR)

FI* - FIAFI = FIAT

⎛ ⎞⎜ ⎟⎝ ⎠

FI* = (ESR) (MOPT) (SSP)

ESR = Delay1(SR, UST)

Figure 6.4: System Equations for 3rd Stage Process – Finished Inventory

By constructing sets of functional blocks according to the above system equations, we integrate

and link each of those functional blocks to generate our complete block diagram representation

of a three-stage semiconductor production system as shown in Figure 6.5

1MCT

1FWAT

1MCT

1/s 1/s

1CAT

1MOPT

1AWAT

1MOPT

1FIAT

11+(UST) s

11+(UST) s

MCT

SSP

CAT

TD NWS*FWIP* FWIP NWO

WS

FWIP

AWIP

NAOAWIP*

AAWIP NAO

FISR

ESR

FI

FI*

NAO*

+

+ +

++ +

− − −

−

NWO

+

+

+

+

+

−

− AWIP

1CAT

1/s

SR+

AFI

Figure 6.5: Block Diagram Representation of a Three-Stage Semiconductor Production System

103


To simplify the algebraic expression from Figure 6.5, we use basic capital letters (A,B,C, etc.) to

represent individual key production system variables. In addition, we apply the block diagram

reduction technique to make the mathematical relationship between Net Wafer Outflow (NOW)

and Desired Net Wafer Start (NWS*) to the following:

( )

( )A + ENWO

NWS* AEs + A + E=

where A = MCT

B = CAT

C = MOPT

D = UST

E = FWAT

F = AWAT

G = FIAT

V = A E+

H = SSP

1C

1F

1G

1Ds + 1

H

B

TD AWIP

AWIP*

AAWIP

FI

ESR

FI

FI*

NAO*

+

−

+

+

+

+

− AWIP

SR+

AFI

VAEs + V

BBs + 1

CCs + 1

1B

1Ds + 1

Figure 6.6: Simplified BD Representation of a Three-Stage Semiconductor Production System

104


After algebraic simplification, the block diagram representation of a three-stage semiconductor

production system is shown in Fig. 6.6. We again apply the block diagram reduction technique to

reduce the algebraic expression into a single, 4th-order differential equation, transfer function as

stated as below:

( )4 3 2

V Ds + 1FI ABDETD s [I] s [II] s [III] s + [IV]

⎛ ⎞⎜ ⎟⎝ ⎠=

+ + + (6.1)

where

[ ]1[I] = ADE(B+C) + BC(AE+DV)ABCDE

( )1 V[II] = AE B+C+D DV(B+C) BCV + ABCDE F

⎡ ⎤+ +⎢ ⎥⎣ ⎦

BCD

1 VBC[III] = V(B+C+D) + AE + 1ABCDE F C G

⎡ ⎤⎛ ⎞+ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

D D

( )1 VBC[IV] = V + 1 HABCDE FG

⎡ ⎤−⎢ ⎥⎣ ⎦

We observe a new finding from equation (6.1) that there is a Laplace s term found in the

numerator of the transfer function. From the classical control theory [40,41,42], we know that a

zero exists whenever the differential equation contains numerator dynamics. This particular

numerator dynamic is caused by the first-order delay from the expected shipment rate while the

finished inventory (FI) is sending back as a velocity feedback (or tachometer feedback) through

its derivative action. In manufacturing terms, a zero is acting like a forecast for product demand.

105


It usually improves the stability of the manufacturing system as the varying loop gain K

increases until it reaches the location of a zero.

Next, we apply the Root Locus design technique per Section 5.4 guidelines, to study the

sensitivity of the closed-loop poles location of this three-stage production control system. Given

the complexity of this, multiple feedback and forward loops, system transfer function as stated in

eq. (6.1), it is not an easy task to derive its corresponding open-loop system transfer function for

Root Locus analysis. Instead, we can add a gain block K in the simplified block diagram from

Fig. 6.6, such that we can adjust the gain K value to bring the closed-loop poles to certain desired

locations. The simplified system block diagram with an additional block K is displayed in Figure

6.7. Referring to eq. (5.2) from Section 5.1, we assume that G(s) comprises both the compensator

transfer function and the plant transfer function. The characteristic equation for the system is

computed as follows, where H(S) =1:

1+KG(s) H(s) = 0 (6.2)

We can algebraically derive the new system transfer function from the block diagram in Figure

6.7. Its corresponding characteristic equation, G(s) for this new system is computed as in eq.

(6.3).

106


1C

1F

1G

1Ds + 1

H

B

TD AWIP

AWIP*

AAWIP

FI

ESR

FI

FI*

NAO*

+

−

+

+

+

+

− AWIP

SR+

AFI

VAEs + V

BBs + 1

CCs + 1

1B

1Ds + 1

K

Figure 6.7: Simplified BD Representation with block K of the Three-Stage Semiconductor System

( )

[ ] [ ] [ ] [ ]

2

4 3 2

1- HV 1 1 1s + s AEF C D G DG

G(s) = s + s + s + s +δ ε λ σ

⎡ ⎤⎛ ⎞+ + +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(6.3)

where

( ) (1 = ADE B+C + BC AE+DVABCDE

δ ⎡ ⎤⎣ ⎦) ;

( ) ( )1 = AE B+C+D DV B+C BCVABCDE

ε ⎡ ⎤+ +⎣ ⎦ ;

( )1 = V B+C+D AEABCDE

λ ⎡ ⎤+⎣ ⎦ ;

V = ABCDE

σ

107


There are total of four closed-loop poles (roots) found in this particular system. In addition, there

are two numerator dynamics (i.e., zeros) found according to where we added the gain block K.

Assume that production management wants to investigate the overall production lead time and

its dynamic behavior to manufacture a target of 5000 chips. Given the four different sets of

system parameters shown in Table 6.1, we use MATLAB [49] to investigate each individual step

response dynamic behavior and its corresponding root loci. These system parameters are

disguised to maintain confidentiality for Intel Corporation. Figure 6.8 shows the step response of

the three-stage semiconductor production system with four different sets of system parameters,

labeled as Set A, Set B, Set C, and Set D.

Table 6.1: Parametric Values for a three-stage semiconductor production system

SET A SET B SET C SET DA = MCT (month) 1 1 1 1B = CAT (month) 0.1 0.4 0.1 0.1C = MOPT (month) 7 0.7 1.3 0.94D = UST (month) 0.2 0.5 0.2 0.1E = FWAT (month) 1 0.2 1 1F = AWAT (month) 0.1 0.2 1 0.4G = FIAT (month) 0.3 0.3 0.3 0.4H = SSP 1.1 1.1 1.1 1.1TD = 5000 Chips

108


0 5 10 15 20 25 30 35 40 45 500

1000

2000

3000

4000

5000

6000

7000N

umbe

r of C

hips

Time (month)

SET A

SET B

SET C SET D

Figure 6.8: Step Response of a three-stage production system with different parametric set values

As shown in Figure 6.8, Set A curve yields a sluggish dynamic response (i.e., overdamped with

ζ>1) with an offset of 400 chips less than the target 5000 units as it reaches the steady-state at

time 20 months. Set B curve responds with a less rising slope but gives a surplus of 1400 chips at

time 50 months. Set C curve gives a much steeper rising slope and it reaches its steady-state at

time 10 months. Unfortunately, the system passes the target value with a surplus of 1800 chips.

Set D curve yields the fastest rising slope with no surplus made. It reaches the target value of

5000 chips at time 8 months. Among all four different parametric sets values, Set D gives the

best overall dynamic response that leads to shorter production lead time, minimal WIP overshoot,

fastest rising slope at the transient period. Figures 6.9-6.10 show the corresponding root loci

from the parametric set values of Set B and Set D.

109


-8 -7 -6 -5 -4 -3 -2 -1 0 1 2-8

-6

-4

-2

0

2

4

6

8

Real Axis

Imag

Axe

s

ζ=0.9

X X X X

ζ=0.8 ζ=0.7

Figure 6.9: Root Locus of a three-stage semiconductor production system (Set B)

-15 -10 -5 0 5-15

-10

-5

0

5

10

15

Real Axis

Imag

Axe

s

ζ=0.9

X X X

ζ=0.8

ζ=0.7

Figure 6.10: Root Locus of a three-stage semiconductor production system (Set D)

110


Figure 6.9 shows that there are two zeros found on the root locus plot, located at -6.7 and +0.15

respectively. The location of the four closed-loop poles with K=0 begins at s1 = -6.0, s2 = -2.5, s3

= -2.0, s4 = -1.4286 (i.e., all real roots on the real-axis). The arrows indicate the direction of the

four closed-loop poles movement as the gain K values varies from 0 to infinity. The breakaway

points of the pair of complex conjugate roots are located at -2.273. The major difference between

this three-stage semiconductor production system from the previous two-stage production system

with a 3rd-order delay is that the two dominant closed-loops (i.e., closest to the imaginary axis)

will terminate their movement once they reach the zeros at -6.86 and +0.097 respectively. By

applying rltool from Matlab, we determine that as K>2.142, the dominant closed-loop pole will

reach a positive value on the real axis. Hence, the manufacturing system could lead unstable or

marginally stable response.

We can improve the overall dynamic response by moving the breakaway point and the dominant

poles further away from the imaginary axis as shown in Figure 6.10. The location of the four

closed-loop poles are found at s1 = -11, s2,s3 = -10, and s4 = -1.0638 when K=0. As gain K

increases, the repeated roots, s2 and s3 breakaway from each other and stay with the same real

value of -9.1345 with increasing complex conjugate values as K continues to increase. The two

zeros are located at -13.7 and +0.182. By comparing Figure 6.9 to Figure 6.10, the breakaway

point of the complex conjugate pair has moved from -2.273 to -10. Hence, Set D responds much

faster than Set B with a shorter production lead time. Again, the arrows indicate the direction of

the four closed-loop poles movement as the gain K values varies from 0 to infinity Although all

the closed-loop poles are located at the right-half plane, if gain K goes beyond 17, the

manufacturing system will behave unstable or marginally stable. Obviously, it takes much higher

111


gain K value to bring the manufacturing system to be unstable, hence Set D is the best parametric

set values among the four choices.

We address that the root locus of any dynamic system could behave differently according to the

system parametric set values chosen. For instance, if we chose the following parametric set

values (i.e., Set E): MCT=1 month, CAT = 5 months, MOPT= 0.6 month, UST=8 months,

FWAT=1 month, AWAT=0.2 month, FIAT=0.2 month, and SSP=1.1. The three-stage

semiconductor manufacturing system behaves very different as shown in Figure 6.11 and its

corresponding root locus plot is displayed in Figure 6.12.

Figure 6.11 indicates that the three-stage production system gives unstable and oscillatory

behavior after 20 months. The envelope of the oscillations gets larger as time continues to

increase. In Figure 6.12, it shows that the four closed-loop poles are located at s1 = -2, s2 = -1,667,

s3 = -0.2, and s4 = -0.125 with K=0. As gain K increases, the dominant closed-loop poles will

move towards the right-half plane. As a result, it makes the system become unstable and

oscillatory.

112


0 5 10 15 20 25 30 35 40 45 50-8

-6

-4

-2

0

2

4

6

8 x 104

Num

ber o

f Chi

ps

Time (month)

Figure 6.11: Step Response of a three-stage production system that yields instability behavior

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2-20

-15

-10

-5

0

5

10

15

20

Real Axis

Imag

Axe

s

ζ=0.9

X X X X

ζ=0.8

ζ=0.7

Figure 6.12: Root Locus of a three-stage semiconductor production system (Set E)

113


In this particular industrial case study, due to the complexity of the differential transfer function

as shown in Eq. (6.1), it may not be so clear to identify which particular production system

variables influence the most as the loop gain K is varying on the Root Locus plot. By applying

the Final Value Theorem from classical control theory [40,41], we can analytically determine the

steady-state function as:

( ) ( )t s 0lim t = lim sF sf→∞ →

that yields the steady-state value from eq. (6.1) as follows:

( )s 0

FI Clim BCTD 1 1-FG

→

⎛ ⎞ =⎜ ⎟⎝ ⎠ + H

)

(6.4)

Equation (6.4) shows that the steady-state final value is not influenced by variables A, D, and E.

This result allows us to determine the final finished inventory without adjusting the

manufacturing system values of (A)MCT, (D)UST, and (E)FWAT.

To fully utilize the results of Chapter 6, consider the dominant closed-loop poles of Fig. 6.9 with

location of a ± bi on the complex plane. We can solve for b in terms of the characteristic

equation of G(s), so that, (b = , , ,f δ ε λ σ . By adjusting the original manufacturing variables, we

can position b to give our desired dynamic behavior. We can either apply the Design of

Experiments or analytically determine the partial derivative of b with respect to ( ), , ,δ ε λ σ to

find the most significant system variables that most influence b. This approach is in fact the Pole

Placement technique from Control Theory.

114


Moreover, since we have inserted a gain block K into the original System Dynamics model from

Fig. 6.1, we have merely asked the production management to compute K(TD+AAWIP) instead

of the original case when K=1. This single step has created a new state variable (i.e., control

handle) that could not be seen otherwise. Given K≠1, we can now adjust the overall dynamic

characteristics without changing any original manufacturing variables and can make the system

model to behave to our desired specifications. These tasks are left for future work.

In conclusion, poles tend to destabilize a system and that the closer they are to the imaginary axis,

the more destabilizing they are. Adding zeros makes the production system more stable, however,

a too stabilized system could give a very sluggish and slow response. A zero in the right-half

plane on the other hand leads to unstable operation which could not be predicted by typical

System Dynamics (SD) approach by Goncalves et. al.[50].

In this chapter, we have applied the guidelines from Section 3.3 and 5.4 to model, analyze, and

design a real manufacturing system. This provides evidence that our classical control theory

(CCT) approach can apply to higher-order responsive manufacturing systems. However, as the

order of the manufacturing system goes higher with more processing variables, it becomes a

greater challenge for engineers to model and translate the system model into differential

equations. Nevertheless, the ideas and the concepts of applying CCT approach hold as the real

life manufacturing system gets more complicated, and even more control opportunities are

developed.

115

Chapter 7 Conclusions and Research Contributions N.H. Ben Fong

Chapter 7 Conclusions and Research

Contributions In this research, we have demonstrated a new alternative method to model, analyze, and design

responsive manufacturing systems using classical control theory (CCT). This new approach

permits manufacturing engineers and production managers to translate System Dynamics (SD)

models into differential equation system models. The resulting transfer functions provide

engineers a new way to analytically determine production lead time, settling time, WIP

overshoot, system responsiveness, and lean finished inventory. Moreover, we have introduced

the Root Locus design technique to offer a new production operation strategy that engineers and

management can predict and improve the overall system responsiveness without running

numerous simulation replications as found in discrete event simulation and system dynamics

approach. The Root Locus technique provides a complete graphical overview to identify the

critical desired dynamic responses according to the location of the closed-loop poles. In addition,

we have revealed the potential for manufacturing system instability due to poor management

policies through the movement of the closed-loop poles.

It is known that continuous differential system modeling is only an approximation of discrete

manufacturing systems. We have shown a comparison in modeling and analyzing a single-stage

production control system between discrete event simulation (i.e., ARENA) and CCT approach.

Depending on the discrete time step to update particular production work force and shipment rate,

CCT appears to give valid approximations of real manufacturing systems. To further enhance the

116


validation of using CCT approach, we include a real industrial case study in the semiconductor

manufacturing industry. We have applied CCT to model a three-stage push-pull semiconductor

production system with a first-order time delay expected shipment rate. As the order of the

differential system model increases, the analytical formulation becomes less observable to

compute those key manufacturing system variables, such as WIP overshoot, production settling

time, and lean finished inventory. Nevertheless, we are still able to model and formulate the

multi-stage production system into a higher-order differential equation model. By deriving the

resulting transfer function, we can apply the Root Locus technique to adjust and design the

dynamic characteristics of the system model to the desired specifications.

The major contributions of this dissertation research are as follows:

(1) An alternative way to model dynamic manufacturing systems in terms of block diagram

representations and transfer functions using classical control theory approach;

(2) Analytical formulation of key responsive manufacturing system variables, such as production

lead time, inventory overshoot, rise time, system responsiveness in terms of damping ratio, and

lean finished inventory level;

(3) A new operation design method that permits production management to better predict and

design responsive manufacturing system variables that yield shorter settling time and minimal

inventory build-up (i.e., leanness) under transient conditions;

(4) An interpretation of classical control theory terms as responsive manufacturing system

variables in the complex s-plane environment;

117


(5) Mathematical formulation and graphical evaluation of the difference in dynamic system

behavior between a 1st-order delay and a 3rd-order delay for the same two-stage production

control system;

(6) A method for establishing the likelihood of manufacturing system instability due to poor

management policies via location of the closed-loop poles;

We have addressed the limitation of applying CCT to approximate the real discrete

manufacturing systems. The CCT approach can only apply for time-invariant, linear systems. To

include maximum capacity (i.e., system saturation) and account for randomness, we have to

apply Non-Linear Control Theory and Stochastic Control Theory or other methods such as

simulation.

We conclude that we can apply Classical Control Theory as a good approximation to model,

analyze, and design responsive manufacturing systems to obtain shorter production lead time,

minimal inventory build-up and related cost, and better overall dynamic system behavior. Again,

this alternative modeling, analysis, and design methodology can be applied to any manufacturing

system in general.

118

Chapter 8 Future Research N.H. Ben Fong

Chapter 8 Future Research

In studying classical control theory, a mathematical model of a dynamic system is defined as a

set of equations that adequately predicts the behavior of a system to a set of known inputs

[40,51,52,53,54]. A system may be represented in many different ways of mathematical models,

depending on one’s perspective. Whether the systems are mechanical, electrical, fluid, thermal,

economic, or even biological, may be described in terms of differential equations. These

differential equations are derived by using physical laws or idealized constitutive relationship

among particular system variables. It is important to realize that deriving the reasonable

mathematical models plays a vital role for analyzing control systems.

Idealized constitutive laws are essential to describe the causality among dynamic system

variables. They are such as Newton’s laws found in the mechanical systems and Kirchhoff’s

laws for the electrical systems. A complicated physical dynamic system can be approximately

modeled by a network of simply physical elements (or lumps) in term of differential equations.

These equations are obtained by formulating a set of mathematical equations by summation of

through-variables at any junction, summation of across-variables within any closed loop, and a

mathematical representation of each element of the system. The dynamical order (i.e., 1st or 2nd

order, etc) of the system is governed by the number of independent energy storing elements. The

major components of lumped-element models are energy sources, passive energy storage

119


elements, and passive dissipative elements. Variables required to formulate various lumped

element models and symbols commonly used to denote them are listed in Table 8.1.

Table 8.1: Common variables used in system modeling

System Through-Variable Across-Variable Accumulated-Variable

______________________________________________________________________________

Electrical Current (I) Voltage (V) Capacitor (C)

Hydraulic Fluid flow rate (q) Pressure (P) Volume (Vol)

Rotational Torque (T) Angular Velocity (ω) Torsional Spring (kt)

Translational Force (F) Velocity (v) Spring (k)

Thermal Heat flow rate (q) Temperature (T) Thermal Capacity (C)

Little’s Law (named for John D. C. Little at MIT, 1961) is perhaps the most widely recognized

principle of manufacturing systems [44,45]. At every work-in-process (WIP) level, WIP is equal

to the product of throughput (TH) and cycle time (CT). However, the WIP levels and cycle time

referred to are average values that assume the system is under steady-state conditions. Due to the

dynamic nature of fast-changing market demand, there is a strong desire to draw an analogy

between responsive manufacturing systems and electrical circuit systems to analyze transient

system behavior. In the near future, we should consider modeling responsive manufacturing

systems directly from electrical circuit theory. The advantage of modeling manufacturing

systems in electrical circuit theory is due to well-established techniques and tools that permit

production management to improve their operation strategies in terms of supply chain filtering,

120


frequency responsive analysis for seasonal demand, and integrated hybrid circuits with the use of

operational amplifiers.

Furthermore, we can consider an in-depth study of applying System Compensation techniques to

further improve the overall dynamic behavior of responsive manufacturing systems. Tools like

Feedback compensation, Cascade control, Velocity feedback control, State-variable feedback,

and Disturbance Rejection (i.e., machine breakdowns) could be implicitly included in those

production systems that are constructed by the System Dynamics approach.

As stated previously, to better enhance model validity, we can consider the advanced control

theory such as Non-Linear Control Theory and Stochastic Control Theory to include maximum

machine capacity (i.e., system saturation) and stochastic random processes found in typical

manufacturing plants.

Lastly, real industrial management often works with multiple customer demands and different

product varieties while trying to improve their own manufacturing system performance, such as

reduce scrap rate, shorten lead time, and minimize inventory costs. In that scenario, we can

consider implementing the multiple-input-multiple-output (MIMO) principles from the State

Space Method of Modern Control Theory.

121

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127

Vita N.H. Ben Fong

Vita Nga Hin Benjamin Fong (Square Head) was born in Hong Kong on July 25, 1970. He grew up

in Hong Kong where he completed his 9th grade in St. Joseph’s College. He left home in

September 1985 to follow his two elder brothers’ foot steps to study high school at Rishworth

School, West Yorkshire, England. In Fall 1988, Ben came to US to study his Bachelor of Science

in Mechanical Engineering at the University of Texas at Arlington and completed his program in

May 1992. In August 1992, he began his new journey at Blacksburg, Virginia. He completed his

Master of Science in Mechanical Engineering (Vibration and Control Theory) at Virginia Tech

(VPI&SU) in December 1994.

Ben began his career as a Process Engineer at Trus Joist MacMillan in Hazard, Kentucky. After

working regularly for 70+ hours at that (painful) engineered-lumber start-up plant, he left and

began his second career as a Manufacturing Engineer at BBA Friction, Inc. in Dublin, Virginia.

While working at BBA Friction, he was given the opportunity to start his part-time PhD program

in Industrial and Systems Engineering at Virginia Tech in Fall 1997. He held various positions as

Manufacturing Engineer, Operations Engineer, and Process Engineering Specialist at BBA

Friction and later became the expert in applying Six Sigma Tools to improve overall process

variation and product development. Ben got married with his lovely wife, Iris in August 1998. In

Fall 2000, due to the company reconstructed, Ben decided to leave and began his third career as

a Process Development Engineer at Haleos, Inc. in Blacksburg, VA (manufacturer of wafer

microphotonic components and optical interconnect).

After the September 11 attack, the entire US economy hit bad especially in the field of fiber

optical and telecommunication industries. Ben got laid off and returned to Virginia Tech as a

full-time student in Spring 2002. After working with Dr. Bob as a GTA in Spring 2002, Ben

began to work as a GRA and was financially supported by the Center for High Performance

Manufacturing (CHPM) until January 2005. Ben and Iris have two lovely daughters, Vera and

Audrey, born in October 2002 and December 2004, respectively. Thankfully, Ben has

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Vita N.H. Ben Fong

successfully defended his dissertation in April 15, 2005 and has already accepted an offer from

Caterpillar, Inc in January 2005. Ben will begin his new career as a Senior Engineer at the

Advanced Technology Analysis & Support Division of the CAT Electronics Division in

Mossville, Illinois.

Ben’s next five year career goal is to pursue an engineering management position, possibly

explore to the International business in China or Europe. Nevertheless, Ben will love to teach as

an Adjunct Professor to share his experiences with the students while working full-time at

Caterpillar, Inc. Lastly, Ben has great desire to continue doing research and submitting journal

publications regularly with his respectful committee professors.

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