In many industries, agile or responsive manufacturing refers to

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 iii 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 iv 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!! ☺ v 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 vi 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 vii Chapter 8 Future Research............................................................................. 119 References ......................................................................................................... 122 Vita .................................................................................................................... 128 viii 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 ix 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 x 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 xi 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. 1 Chapter 1 Introduction N.H. Ben Fong 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. 2 Chapter 1 Introduction N.H. Ben Fong 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 trialand-error iterations to study the dynamic behavior of responsive manufacturing systems. It does 3 Chapter 1 Introduction N.H. Ben Fong 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 inventoryproduction 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 4 Chapter 1 Introduction N.H. Ben Fong Responsive Manufacturing Systems System Dynamics (dynamics, simulation based, with feedback, no analytical solutions, design via iterations) DES (steady-state, simulation based, no analytical, no feedback, design via iterations) Block Diagram Representation Electrical Circuit Model Fluid Model Transfer Function (dynamics, analytical solutions, analytical system design, stability analysis, structure improvement) 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 5 Chapter 1 Introduction N.H. Ben Fong 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; 6 Chapter 1 Introduction N.H. Ben Fong (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 7 Chapter 1 Introduction N.H. Ben Fong manufacturing. Chapter 7 highlights the concluding remarks and research contribution from this dissertation work. Finally, we provide some future research directions in Chapter 8. 8 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 9 Chapter 2 Literature Review N.H. Ben Fong 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. 10 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 2 Literature Review N.H. Ben Fong 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 productioninventory 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 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 2 Literature Review N.H. Ben Fong 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 suboptimal 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 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 2 Literature Review N.H. Ben Fong 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 simulationbased 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 Chapter 2 Literature Review N.H. Ben Fong analysis as described by Towill [6,7] is an alternative method to model dynamic productioninventory 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, goalseeking 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 Chapter 2 Literature Review N.H. Ben Fong 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 stockand-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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong Shipment Time ST Inventory INV B Shipment Rate SR Production Rate PR + Inventory Control + + Desired Production Rate DPR + Adjustment for Inventory AINV + Desired Inventory INV* Inventory Adjustment Time IAT Expected Shipment Rate ESR + 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. 100 90 80 60 50 Inventory INV 70 (Stock of Units) 60 Shipment Rate SR 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 20 10 0 Time (Day) 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 (Units/Day) Production Rate PR 40 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. Lead Time LT Shipment Time ST Production Start Rate PSR + Work-InProcess WIP Production Completion Rate PCR + Finished Inventory FI Shipment Rate SR + Desired Production Rate DPR - - Production Orders PO + Kanban Cycle KC Total Number of Kanban TNK 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 100 90 80 200 180 160 (Stock of Units) (Units/Day) 70 60 50 40 30 20 10 0 0 1 2 3 4 140 120 100 80 60 40 20 0 Production Start Rate PSR Production Completion Rate PCR Work-In-Process WIP Finished Inventory FI Time 5(Day)6 7 8 9 10 0 1 2 3 4 Time (Day) 5 6 7 8 9 10 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 related causal loop variables as shown in Fig.3.5. Lead Time LT Production Completion Rate PCR B Shipment Time ST Shipment Rate SR + Production Start Rate PSR + B Work-InProcess WIP Finished Inventory FI + WIP Control Desired Production Rate DPR + + Adjustment for Work-In-Process AWIP + - Adjustment for FI Control Finished Inventory AFI + + Expected Lead Time ELT Desired Inventory DI* Desired + Work-In-Process + Finished Inventory WIP* Expected Adjustment Time Work-In-Process + Shipment Rate FAT Adjustment Time ESR WAT + Desired Production Completion Rate + DPCR Figure 3.5: A two-stage production control system 31 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong There are two stock elements found in this model that accumulates the work-in-process (WIP) production 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 32 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. 200 70 Work-In-Process WIP 180 160 140 Finished Inventory FI Prod. Completion Rate PCR Shipment Rate SR 60 50 Stock of Units 40 100 80 60 40 10 20 0 1 5 10 Time (Day) 15 20 0 30 20 Figure 3.6: Step response of a two-stage production control system 3.1.5 A Two-Stage Production Control System with Time Delay Delays are inherent in many physical and engineering systems. There are many applications of 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 Units/Day 120 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong [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. Shipment Time ST - Lead Time LT Production Start Rate PSR Work-InProcess WIP - Production Completion Rate PCR B FI Control Finished Inventory FI + + Shipment Rate SR B WIP Control Expected Lead Time ELT - - Desired Production + Rate DPR + Adjustment for + Work-In-Process + Desired AWIP Work-In-Process WIP* + Work-In-Process Adjustment Time WAT Adjustment for Finished Inventory AFI - Desired Inventory DI* + Finished Inventory Adjustment Time FAT + Expected Shipment Rate ESR Desired Production + Completion Rate + DPCR Figure 3.7: A two-stage production control system with a 3rd-order time delay Except for the addition of the third-order time delay to compute the PCR, the rest of the model 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. 300 90 Work-In-Process WIP Finished Inventory FI 250 80 70 60 50 Prod. Completion Rate PCR Shipment Rate SR 200 Stock of Units 150 40 100 30 20 50 10 0 1 5 10 15 20 25 30 35 0 40 Time (Day) Figure 3.8: Step response of a two-stage production system with a 3rd-order time delay The result indicates that the higher the order of the time delay, more dynamics and oscillations 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 Units/Day Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 3.2 Modeling and Analysis via Classical Control Theory In this section, we begin by reviewing some fundamentals of the block diagram (BD) 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. 3.2.1 Fundamentals of Classical Control Theory In the control engineering field, the mathematical model of a system is defined as a set of differential equations. As stated in Ogata [40], the transfer function (TF) of a linear, timeinvariant, 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 Referring to Figure 3.1, we can translate those CLD and SFD terminologies in modeling the single-stage production system using BD representations from CCT as shown in Fig. 3.9 and Fig. 3.10. PR Inventory: SR + _ 1/s INV Expected Shipment Rate: INV* SR ESR Shipment Rate: INV 1 ST SR Adjustment for Inventory: INV + _ 1 IAT AINV Production Rate: Desired Inventory: DPR PR AINV Input INV*0 INV* Desired Production Rate: + DPR + ESR Figure 3.9: Subcomponents of BD representation of a single-stage production system 37 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong + Input INV*0 _ 1 IAT AINV + DPR PR + INV 1/s + ESR _ SR 1 ST 1 1 ST 2 Figure 3.10: Complete BD representation of a single-stage production system As illustrated in Fig. 3.9, each individual subset of block diagrams represents the formulation for each system variable. By linking all these individual subsets together, we will generate the 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 rearrangement called the 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong + INV* _ 1 IAT 1/s ⎛ 1 ⎞ ⎛ 1 ⎞ 1 + (1/s) ⎜ ⎜ ST ⎟ − (1 / s)⎜ ST ⎟ ⎟ ⎜ ⎟ ⎝ 1⎠ ⎝ 2⎠ INV INV* ⎛ 1 ⎞ ⎜ ⎟ ⎝ IAT ⎠ ⎛ 1 1 1 ⎞ ⎜ ⎟ − + s+⎜ ST 1 ST 2 IAT ⎟ ⎝ ⎠ INV Figure 3.11: Step-by-step block diagram reduction into a single TF block ⎛ 1 ⎞ ⎜ ⎟ ⎝ IAT ⎠ ⎛ 1 1 1 ⎞ s+⎜ ⎟ ⎜ ST − ST + IAT ⎟ 2 ⎠ ⎝ 1 −1 INV(s ) = INV * (s ) (3.1) where ⎡ 1 1 1 ⎤ − + τ = ⎢ ⎥ ⎣ ST1 ST2 IAT ⎦ (3.2) Y(s) K = X(s) τ s + 1 (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, K = τ IAT (3.4) The inverse Laplace transform of equation (3.3) with a unit step input is obtained as: -t y(t) = K ⎛ 1 - e τ ⎞ ; ⎜ ⎟ ⎝ ⎠ with lim y(t) = K 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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>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. PSR Work-InProcess (WIP): + _ PCR 1/s WIP Finished Inventory (FI): + _ 1/s FI PCR Shipment Rate (SR): SR 1 ST 1 LT FI SR Production Orders (PO): TNK + _ _ PO WIP Production Completion Rate (PCR): Total Number of Kanban (TNK): WIP PCR FI Desired Production Rate (DPR): Input Container Size TNK PO 1 KC DPR Figure 3.12: Subcomponents of block diagram representation of a basic kanban system 40 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems r + + + _ N.H. Ben Fong TNK _ _ 1 KC _ 1/s 1 LT 1/s FI 1 LT 1 ST Figure 3.13: Complete block diagram representation of the basic kanban system ⎛ 1 ⎞⎛ 1 ⎞ TNK ⎜ ⎟⎜ ⎟ ⎝ KC ⎠ ⎝ LT ⎠ 1 1 ⎤ 1 1 1 1 ⎤ ⎡ 1 ⎡ 1 1 s2 + ⎢ + + s+⎢ + + ⎣ LT KC ST ⎥ ⎦ ⎣ LT ST KC ST KC LT ⎥ ⎦ 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. FI(s) = r(s) ⎛ 1 ⎞⎛ 1 ⎞ TNK ⎜ ⎟⎜ ⎟ ⎝ KC ⎠⎝ LT ⎠ 1 1 ⎤ 1 1 1 1 ⎤ ⎡ 1 ⎡ 1 1 s2 + ⎢ s+⎢ + + + + LT KC ST ⎥ LT ST KC ST KC LT ⎥ ⎣ ⎦ ⎣ ⎦ (3.6) K ω n2 Y(s) = 2 X(s) s + (2 ζω n ) s + ω n2 ( ) (3.7) 41 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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: ωn = 1 1 1 1 1 1 + + LT ST KC ST KC LT (3.8) Damping ratio: ζ = 1 2 1 1 ⎤ ⎡ 1 + + 1 1 1 1 1 1 ⎢ LT KC ST ⎥ ⎣ ⎦ + + LT ST KC ST KC LT 1 (3.9) Damped natural frequency: ω d = ω n 1 − ζ 2 (3.10) Roots of characteristic equation: s 1,2 = - ζω n ± jω n 1 - ζ 2 (3.11) 42 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems Time constant: N.H. Ben Fong (3.12) τ= 1 ζω n = 2 1 1 ⎤ ⎡ 1 ⎢ LT + KC + ST ⎥ ⎦ ⎣ 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 unitstep 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. 4 Settling time (2%criterion): t s = 4τ = ζω n ⎞ π -β ⎟= ⎟ ωd ⎠ (3.13) Rise time (standard): tr = 1 ωd ⎛ ωd tan -1 ⎜ ⎜ - ζω n ⎝ (3.14) where β = tan -1 ωd ζω n (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.): t r, 10, 90% ≅ 0.8 + 2.5 ζ ωn , 0 ≤ζ ≤1 (3.16) The inverse Laplace transform of eq. (3.7) with a unit step input is obtained as: ⎡ ⎛ 1−ζ e -ζω n t sin ⎜ ω d t + tan -1 y(t) = K ⎢1 2 ⎜ ζ ⎢ 1-ζ ⎝ ⎣ 2 ⎞⎤ ⎟⎥ , ⎟⎥ ⎠⎦ for t ≥ 0 (3.17) 43 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong As time goes to infinity, the steady-state value of the kanban system is computed from equations (3.6, 3.7, 3.17): (TNK )⎛ ⎜ lim y(t) = K = t→∞ 1 ⎞⎛ 1 ⎞ ⎟⎜ ⎟ ⎝ KC ⎠ ⎝ LT ⎠ ω n2 (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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 A. Lead Time (LT) B. Kanban Cycle (KC) C. Shipment Time (ST) Low (day) 0.25 0.25 2.5 Medium (day) 0.5 0.5 10 High (day) 1 1 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong LT KC ST LT^2 KC^2 ST^2 LTxKC LTxST KCxST Table 3.2: Box-Behnken Design Factors Effect Responses Summary Undamped Natural Damping Time Settling Steady State Frequency(Wn) Ratio (S) Constant (T) Time (ts) Value(K) -0.7318 0.1632 0.6526 -0.0103 -3.7641 -0.7318 0.1631 0.6526 -0.0103 -3.7641 12.1431 -0.1629 0.0358 0.0209 0.0837 0.1308 0.0552 0.0041 0.0162 -0.9109 0.1308 0.0552 0.0041 0.0162 -0.9109 -8.1435 0.1155 -0.0254 -0.0206 -0.0823 -0.1144 0.1037 0.415 0.2379 0.1984 0.0053 0.0077 0.0121 0.0483 2.6595 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 nonoscillatory 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. 90 80 70 Inventory (Unit) 60 LT=0.25, KC=0.5, ST=5; Inventory=86.96 50 40 30 20 10 0 0 1 2 3 4 5 6 Time (Day) 7 8 9 10 LT=0.5, KC=0.5, ST=5; Inventory=83.33 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. PCR + _ Finished Inventory FI: 1/s FI SR Expected Shipment Rate: DI* SR ESR PSR Work-In-Process WIP: + _ 1/s WIP Adjustment for FI: DI + _ 1 FAT AFI PCR Shipment Rate, SR: FI 1 ST WIP* SR Adjustment for WIP: WIP + _ 1 WAT AWIP Production Completion Rate: WIP 1 LT PCR Desired Production Completion Rate: ESR + DPCR + Production Start Rate, PSR: Desired Inventory: DPR PSR AFI AWIP Input DI*0 DI* Desired Production Rate, DPR: + DPR + DPCR Figure 3.16: Subcomponents of BD representation of a two-stage production control system DPCR ELT DI + _ 1 FAT AFI + + WIP* + _ 1 WAT AWIP + + PSR + 1/s _ WIP DPR 1 LT PCR + 1/s _ FI 1 ST ESR 1 LT SR 1 1 ST 2 Figure 3.17: Complete BD representation of a two-stage production control system 48 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. ⎛ 1 ⎜ ⎝ FAT ⎡ 1 1 1 ⎤ + + s 2 + s⎢ ⎥ LT WAT ST 1 ⎦ ⎣ LT ⎞ ⎞⎛ 1 ⎞⎛ ⎟ ⎟⎜1 + ⎟⎜ WAT ⎠ ⎠ ⎝ LT ⎠ ⎝ ⎡⎛ 1 1 ⎞⎛ 1 1 1 ⎞⎤ + ⎢⎜ + ⎟⎜ ⎜ ST − ST + FAT ⎟ ⎥ ⎟ WAT ⎠ ⎝ ⎠⎦ 1 2 ⎣ ⎝ LT FI(s) = DI(s) (3.19) 2nd order linear DE (non-standard form): 2 K ωn Y(s) = 2 2 X(s) s + (2ζω n ) s + ω n ( ) (3.20) Undamped natural frequency: ω n = ⎢⎜ ⎡⎛ 1 1 ⎞⎛ 1 1 1 ⎞⎤ + ⎟⎜ ⎜ ST − ST + FAT ⎟⎥ ⎟ ⎠⎦ 2 ⎣⎝ LT WAT ⎠⎝ 1 (3.21) Damping ratio: ζ = 1 ⎡ 1 1 1 ⎤ + + ⎢ ⎥ 2ω n ⎣ LT WAT ST1 ⎦ (3.22) Time constant: τ= 1 ζωn = 2 ⎡ 1 1 1 ⎤ + + ⎢ ⎥ ⎣ LT WAT ST1 ⎦ (3.23) Closed-Loop Poles: s1,2 = - ζω n ± jω n 1 - ζ 2 (3.24) Rise time (approx.): t r, 10, 90% ≅ 0.8 + 2.5ζ ωn , 0 ≤ ζ ≤1 (3.25) 49 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong Maximum overshoot % : Mp ≈ e -(ζωn / ωd )π *100 4 ( ) (3.26) Settling time (2% criterion): t s = 4τ = ζω n (3.27) The inverse Laplace transform of equation (3.20) with a unit step input is obtained as: ⎡ ⎛ 1−ζ 2 e -ζω n t sin ⎜ ω d t + tan -1 y(t) = K ⎢1 ⎜ ζ ⎢ 1-ζ 2 ⎝ ⎣ ⎞⎤ ⎟⎥ , ⎟⎥ ⎠⎦ for t ≥ 0 (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): LT ⎞ ⎛ 1 ⎞⎛ 1 ⎞⎛ ⎜ ⎟⎜ ⎟⎜1 + ⎟ FAT ⎠⎝ LT ⎠⎝ WAT ⎠ lim y(t) = K = ⎝ t →∞ 2 ωn (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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 Rise Time Constant (T) (tr) Settling Time (ts) Steady State Value(K) A B C D E AB AC AD AE BC BD BE CD CE DE -0.1324 -0.0528 0.0528 -0.1324 -0.2766 0.0055 -0.0055 -0.0173 0.044 0.0124 0.0055 -0.0196 -0.0055 0.0196 0.044 -0.1081 0.0585 -0.1057 -0.1081 0.3507 -0.0106 -0.0005 -0.0423 -0.0355 -0.0366 -0.0106 0.0693 -0.0005 -0.0885 -0.0355 0.5046 0.1035 0 0.5046 0 0.0601 0 0.2652 0 0 0.0601 0 0 0.0429 0 0.4311 0.9062 -1.1328 0.4311 3.1009 -0.0634 -0.0538 -0.2429 0.2033 -0.3707 -0.0634 0.8623 -0.0538 -1.0264 0.2033 2.0183 0.414 0 2.0183 0 0.2405 0 1.0605 0 0 0.2405 0 0 0.1716 0 0 19.8413 -19.8413 0 7.9365 0 0 -7.9365 0 -8.7302 0 13.4921 0 -13.4921 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 140 120 100 80 ζ =0.35 ζ =0.49 ζ=0.61 Finished Inventory Level (Unit) ζ =1.43 ζ =1.0 60 ζ=0.7 40 20 0 0 5 10 15 Time in days 20 25 30 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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 closedloop 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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). DPCR ELT DI + _ AFI 1 FAT + WIP* + _ + AWIP + 1 + WAT DPR PSR + 1/s _ WIP 1 LT PCR + 1/s _ FI 1 ST ESR 1 LT 1 ST 2 SR 1 Figure 3.17: Complete BD representation of a two-stage production control system 1 1+(LT)s DPCR AWIP + PCR + 1/s _ FI 1 ST 1 SR DI + _ 1 AFI FAT + + ELT WIP* + _ 1 WAT + PSR _ + 1/s ESR DPR WIP PCR 1 ST2 Figure 3.19: BD representation of a two-stage production control system with 1st-order Delay 1 ⎡ (LT)s⎤ ⎢1 + 3 ⎥ ⎣ ⎦ 3 PCR + 1/s _ FI DPCR SR DI + _ 1 AFI FAT + + ELT WIP* + _ 1 WAT AWIP + + PSR _ + 1 ST1 1/s ESR DPR WIP PCR 1 ST2 Figure 3.20: BD representation of a two-stage production control system with 3rd -order Delay 55 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems ⎛ 1 ⎜ FI(s) ⎝ FAT = DI(s) ⎡ 1 1 1 ⎤ + + s 2 + s⎢ ⎥ WAT ST 1 ⎦ ⎣ LT LT ⎞ ⎞⎛ 1 ⎞⎛ ⎟ ⎟⎜ ⎟⎜1 + WAT ⎠ ⎠ ⎝ LT ⎠ ⎝ ⎡⎛ 1 1 ⎞⎛ 1 1 1 ⎞⎤ + ⎢⎜ + ⎟⎜ ⎜ ST − ST + FAT ⎟ ⎥ ⎟ WAT ⎠ ⎝ 1 2 ⎠⎦ ⎣ ⎝ LT 1 ⎞⎛ 1 + ⎟⎜ WAT ⎠ ⎝ LT ⎡⎛ 1 1 + ⎢⎜ WAT ⎣ ⎝ LT ⎞ ⎟ ⎠ ⎞⎛ 1 ⎟⎜ ⎜ ⎠ ⎝ ST 1 N.H. Ben Fong (3.30) FI(s) DI(s) = s 2 ⎡ 1 + s⎢ ⎣ LT 1 + WAT ⎛ 1 ⎜ ⎝ FAT 1 ⎤ + ⎥ + ST 1 ⎦ 1 − ST 2 1 + FAT ⎞⎤ ⎟⎥ ⎟ ⎠⎦ (3.31) ⎡ ⎤⎡ 1 ⎤ LT s = ⎢ ⎥⎢ ⎥ ⎛ 1 ⎞ ⎣ (LT )s + 1 ⎦ ⎣ LT ⎦ 1+ 1 ⎜ ⎟ s ⎝ LT ⎠ 1 ( ) (3.32) For the third-order time delay system, as mentioned in section 3.2.4, we find more dynamics and 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 4thorder differential equation as stated in eq. (3.33) FI(s) DI(s) where, A= B= C= D= ⎞ 1 ⎛ 27 ⎞ ⎛ 1 ⎜ ⎟⎜ ⎟ ⎜ LT 3 + LT 2 (WAT ) ⎟ ⎝ FAT ⎠ ⎝ ⎠ = 4 3 2 s + [A ] s + [B ] s + [C ] s + [D ] ( ) (3.33) 9 1 1 + + LT ST1 WAT 27 9 9 1 + + + 2 (LT )(ST1 ) (LT )(WAT ) (ST1 )(WAT LT ) 27 27 27 9 + + + 3 2 2 LT LT (ST1 ) LT (WAT ) (LT )(ST1 )(WAT ( ) ( ) ) 27 ⎛ 1 1 ⎞⎛ 1 1 1 ⎞ + ⎟⎜ 2 ⎜ ⎜ FAT + ST − ST ⎟ ⎟ LT ⎝ LT WAT ⎠ ⎝ 1 2 ⎠ Given the same initial system parameters, the third-order time delay system yields a higher 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. 180 Finished Inventory Level (Unit) 160 140 120 100 80 1st-Order Delay 60 40 20 0 0 5 10 15 Time in days 20 25 30 3rd-Order Delay Figure 3.21: Dynamic Responses between 1st-order time delay and 3rd-order time delay It is vital to notice here that the representation of a real system by linear feedback models has a 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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 3.3 Guidelines to Translate Responsive Manufacturing Systems via CCT In this chapter, we have demonstrated two alternative modeling and analysis approaches to study 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. To the best of the author’s knowledge, there are no idealized constitutive laws available in 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 stockand-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 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong 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. (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) Convert the cause-and-effect expressions from CLDs and SFDs into different sets of system equations. (The resulting causality expression among variables is algebraically translated into multiple linear system equations.) (3) Construct different sets of functional blocks (i.e., subcomponents of block diagram representation) from those system equations. (Each function block represents each of those original linear system equations.) 59 Chapter 3 Modeling and Analysis of Responsive Manufacturing Systems N.H. Ben Fong (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 Chapter 4 Model Validation N.H. Ben Fong 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 Chapter 4 Model Validation N.H. Ben Fong 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 Chapter 4 Model Validation N.H. Ben Fong 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. Shipment Time ST Production Rate PR + Inventory INV B - Shipment Rate SR + Desired Inventory INV* Inventory Control + Desired Production Rate DPR + Adjustment for Inventory AINV + Inventory Adjustment Time IAT Expected Shipment Rate ESR + Figure 4.1: A Single-Stage Production Control System 100 90 80 60 50 Inventory INV 70 (Stock of Units) 60 Shipment Rate SR 50 40 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 20 10 0 Time (Day) 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 (Units/Day) Production Rate PR 40 Chapter 4 Model Validation N.H. Ben Fong 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 Ship Every X hours (Ship every X hours) Update and Ship 0 Plot Graph2 Parts are Shipped 0 (Team shift every Y hours) Production Create 1 Entity Assign Inital Values Update Inventory Shift Team False True Update Prod_Rate Plot Graph1 Parts in Inventory 0 0 Production Duplicate 0 Figure 4.3: ARENA discrete-event model of a single-stage production control system 65 Chapter 4 Model Validation N.H. Ben Fong 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 Chapter 4 Model Validation N.H. Ben Fong 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., zipsack effects) occurs. Vice versa, the quicker the shipment update, the closer the dynamic characteristics behaves like the continuous differential equation model. 120 100 60 50 Stock (Unit) 80 60 40 20 0 INV-2 PR-2 SR-2 40 30 20 10 0 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 Figure 4.4: A single-stage production system (Ship every 2 hr; Production update 4 hr) 67 Rate (Unit/Day) Chapter 4 Model Validation N.H. Ben Fong 120 100 60 50 80 60 40 20 0 INV-4 PR-4 SR-4 40 30 20 10 0 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 Figure 4.5: A single-stage production system (Ship every 4 hr; Production update 4 hr) 120 100 60 50 Stock (Unit) 80 60 40 20 0 INV-8 PR-8 SR-8 40 30 20 10 0 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 Figure 4.6: A single-stage production system (Ship every 8 hr; Production update 4 hr) 68 Rate (Unit/Day) Rate (Unit/Day) Stock (Unit) Chapter 4 Model Validation N.H. Ben Fong 120 100 60 50 Stock (Unit) 80 60 40 20 0 INV-24 PR-24 SR-24 40 30 20 10 0 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 Figure 4.7: A single-stage production system (Ship every 24 hr; Production update 4 hr) 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 Rate (Unit/Day) Chapter 4 Model Validation N.H. Ben Fong 50 100 40 30 PR(Dis) INV(Cont) SR(Dis) INV(Dis) 60 40 20 0 20 10 0 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 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 Rate (Unit/Day) PR(Cont) SR(Cont) 80 Stock (Unit) Chapter 4 Model Validation N.H. Ben Fong 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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: s1,2 = - ζω n ± jω n 1 - ζ 2 (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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 s1 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 H(s) G(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: ⎛ 1 ⎞⎛ 1 ⎜ ⎟⎜ ⎝ FAT ⎠ ⎝ LT ⎡ 1 1 1 ⎤ + s ⎢ + + ⎥ + LT WAT ST 1 ⎦ ⎣ LT ⎞ ⎞⎛ ⎟⎜1 + ⎟ WAT ⎠ ⎠⎝ ⎡⎛ 1 1 ⎞⎛ 1 1 ⎢⎜ ⎟ ⎜ ST − ST ⎟ ⎜ LT + WAT 1 2 ⎠⎝ ⎣⎝ KG(s) = s2 (5.3) ⎞⎤ ⎟⎥ ⎠⎦ Given that the loop gain, ST1=ST2, Eq.(5.3) reduces to: ⎛ 1 ⎞⎛ 1 ⎜ ⎟⎜ FAT ⎠ ⎝ LT KG(s) = ⎝ ⎡ 1 s2 + s ⎢ + ⎣ LT LT ⎞ ⎞⎛ ⎟⎜1 + ⎟ WAT ⎠ ⎠⎝ 1 1 ⎤ + ⎥ WAT ST 1 ⎦ (5.4) 74 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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. Constant ζ Lines and Constant ωn Circles 2 0.2 0.4 1.5 0.3 0.1 0.6 0.8 ζ=0.9 0.7 0.5 1 Imaginary Axis 0.5 0 ωn=2 ωn=1 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 Real Axis 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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. LT ⎞ ⎛ 1 ⎞⎛ 1 ⎞⎛ ⎜ ⎟⎜ ⎟⎜1 + ⎟ FAT ⎠⎝ LT ⎠⎝ WAT ⎠ ⎝ lim y(t) = K = t →∞ 2 ωn (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 2 ⎡ 1 1 1 ⎤ + + ⎢ ⎥ LT WAT ST1 ⎦ ⎣ τ= ζωn = (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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong C onstant S T= 5 day s and C onstant FA T= 1 day 2.5 0. 0. 58 57 5 WIP Adjustment Time (WAT) in days 2 0 .6 0 .6 1.5 25 0 .6 0 .6 75 5 0 .7 1 07 0 .7 0 .8 5 0 .8 1 0 .9 0 .9 5 0.5 5 0.5 1 1.5 2 2.5 Le ad Tim e (LT) in day s 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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. Constant ST= 5 day s and Constant FAT= 1 day 2.5 -0 . -0 .5 6 53 WIP Adjustment Time (WAT) in days 2 -0 -0 -0 1.5 .6 .6 5 .7 -0 -0 .8 1 .7 5 -0 . 1 9 -1 -1 . 2 -1 . 4 0.5 -1 . 8 -2 . 2 -4 0.5 -3 1 1.5 2 2.5 Le ad Tim e (LT) in day s Figure 5.4: Contour Plot of Breakaway Points as a function of LT and WAT 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 C: LT=1.5, WAT=1.1, FAT=1; Set B: LT=1, WAT=5, FAT=1.224 Set D: LT=1, WAT=1, FAT=1 79 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 120 Set D: ζ=0.7778, Brkaway Pt=-1.1 100 Set B: ζ=0.707, Brkaway Pt=-0.7 Finished Inventory Level in stocks 80 Set C: ζ=0.7073, Brkaway Pt=-0.8879 60 Set A: ζ=1, Brkaway Pt=-0.7 40 20 0 0 5 10 15 Time in days 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 closedloop 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 closedloop 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 splane 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 xaxis 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 2 + 1.5 G + 1 + C E J + Im ag in ary Axis 0.5 0 X X + B A -0.5 L -1 D + F + + K -1.5 + -2 -2 H -1.5 -1 -0.5 0 0.5 Real Axis 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 complexconjugate 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong Negative Real Root (A) 100 Complex Roots w/ Negative Real Parts (C&D) 80 60 Inventory (Unit) Inventory (Unit) 0 5 10 15 20 25 30 80 40 20 0 -20 -40 60 40 20 0 -60 0 5 10 15 20 25 30 Single Root at Origin (B) 101 Complex Roots on Imaginary Axis (G&H) 100 Inventory (Unit) 100.5 Inventory (Unit) 0 5 10 15 20 25 30 50 100 0 99.5 -50 99 -100 0 5 10 15 20 25 30 Time (Day) Time (Day) Positive Real Root (L) 10000 Complex Roots w/ Positive Real Parts (J&K) 1000 Inventory (Unit) Inventory (Unit) 0 5 10 15 20 25 30 8000 500 6000 0 4000 2000 -500 0 -1000 0 5 10 15 20 25 30 Time (Day) Time (Day) Figure 5.7: Transient mode shapes associated with locations of roots in the complex s-plane 86 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 twostage production control system with a 3rd-order time delay from Figure 3.7 on the complex splane 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: FI(s) DI(s) ⎞ 1 ⎛ 27 ⎞ ⎛ 1 ⎜ ⎟⎜ ⎟ ⎜ LT 3 + LT 2 (WAT ) ⎟ ⎝ FAT ⎠ ⎝ ⎠ = 4 s + [A ] s 3 + [B ] s 2 + [C ] s + [D ] ( ) (5.5) 87 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong where, A= B= C= D= 9 1 1 + + LT ST1 WAT 27 9 9 1 + + + 2 (LT )(ST1 ) (LT )(WAT ) (ST1 )(WAT LT ) 27 27 27 9 + + + 3 2 2 LT LT (ST1 ) LT (WAT ) (LT )(ST1 )(WAT ( ) ( ) ) 27 ⎛ 1 1 ⎞⎛ 1 1 1 ⎞ + ⎟⎜ 2 ⎜ ⎜ FAT + ST − ST ⎟ ⎟ LT ⎝ LT WAT ⎠ ⎝ 1 2 ⎠ 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. 180 Finished Inventory Level (Unit) 160 140 120 100 80 1st-Order Delay 60 40 20 0 0 5 10 15 Time in days 20 25 30 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong equation model, there are four closed-loop poles found on the complex s-plane as shown in Figure 5.8 LT=4, ST=8, WAT=1, FAT=1 1.5 ζ =0.7 ζ =0.6 1 ζ =0.9 ζ =0.8 X 0.5 Imag Axes + + 0 X + X -0.5 X + -1 -1.5 -2.5 -2 -1.5 -1 -0.5 Real Axis 0 0.5 1 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 righthand 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong design and improve this 3rd-order time delay, two-stage production control system to give quickresponsive 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 higherorder 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 splane) 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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. Gain K = 0.1 100 Inventory (Unit) Inventory (Unit) 150 100 50 0 0 200 Inventory (Unit) 150 100 50 0 0 Gain K = 0.35 50 0 0 150 Inventory (Unit) 100 50 0 0 10 20 30 10 20 30 Gain K = 0.6 Gain K = 1.0 10 20 Time (Day) 30 10 20 Time (Day) 30 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 nonunity 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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. LT=1, ST=5, WAT=1, FAT=1.4 4 ζ =0.7 3 ζ=0.9 ζ =0.8 X 2 Imag Axes 1 + + 0 X + + X -1 -2 X -3 -4 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Real Axis 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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) 180 Finished Inventory (Unit) 160 140 120 100 Set D 80 60 Set A Set B Set C 40 20 0 0 2 4 6 8 10 12 14 16 18 20 Time (Day) Figure 5.11: Various Step Response of a two-stage production system with 3rd-order time delay 94 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 steadystate 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong 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 Chapter 5 Design of Responsive Manufacturing Systems N.H. Ben Fong (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 SDbased semiconductor manufacturing system into block diagram representation and transfer function. 98 Chapter 6 Industrial Case Study N.H. Ben Fong (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. Wafer Start (WS) Fabrication WIP (FWIP) Net Wafers Outflow (NWO) + - + Die Inflow (DI) Assembly WIP (AWIP) Net Assembly Outflow (NAO) + - Finished Inventory (FI) Shipment Rate (SR) + - Adjustment for Fab WIP (AFWIP) Manufacturing Cycle Time (MCT) Adjustment for AWIP (AAWIP) + Complete Assembly Time (CAT) Adjustment for FGI (AFGI) + - Minimum Order Processing Time (MOPT) - + + + AWIP Adjustment Time (AWAT) Desired Assembly WIP (AWIP*) + FI Adjustment Time (FIAT) + Desired Wafer Start (WS*) + FabWIP Adjustment Time (FWAT) + Safety Stock Percentage (SSP) + Desired Fab WIP (FWIP*) + Desired Finished Inventory (FI*) + + Expected Shimpent Rate (ESR) + Desired Net Wafer Start (NWS*) + + Desired Net Assembly Outflow + (NAO*) - Update Shipments Time (UST) 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 Chapter 6 Industrial Case Study N.H. Ben Fong 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 Chapter 6 Industrial Case Study N.H. Ben Fong 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 firstorder 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 Chapter 6 Industrial Case Study N.H. Ben Fong 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 ⎛ FWIP* - FWIP ⎞ AFWIP = ⎜ ⎟ FWAT ⎝ ⎠ FWIP* = (NWS*) (MCT) NWO = FWIP MCT WS = Max (0, WS*) WS* = AFWIP + NWS* Figure 6.2: System Equations for 1st Stage Process – Fabrication WIP Assembly WIP Stage: AWIP = ∫ DI - NAO NAO = AW IP ⎛ AWIP* - AWIP ⎞ AAWIP = ⎜ ⎟ AWAT ⎝ ⎠ CAT AWIP* = (NAO*) (CAT) NWS* = Max (0, AAWIP +TD) Figure 6.3: System Equations for 2nd Stage Process – Assembly WIP 102 Chapter 6 Industrial Case Study N.H. Ben Fong Finished Inventory Stage: FI = ∫ NAO - SR M O PT ⎛ FI* - FI ⎞ AFI = ⎜ ⎟ ⎝ FIAT ⎠ FI* = (ESR) (MOPT) (SSP) SR = FI NAO* = Max (0, AFI + ESR) 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 TD + NWS* + FWIP* + − MCT AAWIP 1 FWAT + + WS FWIP + − NWO 1 MCT + AWIP 1/s 1 MCT 1/s − NWO FWIP NAO 1 AWAT − 1 CAT AWIP + AWIP* 1 CAT NAO CAT NAO* + + − 1/s 1 MOPT SR 1 MOPT FI ESR AFI + 1 1+(UST) s SR 1 FIAT − FI FI* + SSP 1 1+(UST) s Figure 6.5: Block Diagram Representation of a Three-Stage Semiconductor Production System 103 Chapter 6 Industrial Case Study N.H. Ben Fong 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 + E) NWO = NWS* AEs + ( A + E ) where A = MCT B = CAT E = FWAT F = AWAT C = MOPT D = UST G = FIAT H = SSP V = A+E TD + + AAWIP V AEs + V B Bs + 1 AWIP 1 B C Cs + 1 FI 1 F − AWIP + AWIP* B NAO* + ESR AFI 1 G + 1 Ds + 1 SR 1 C − FI FI* + H 1 Ds + 1 Figure 6.6: Simplified BD Representation of a Three-Stage Semiconductor Production System 104 Chapter 6 Industrial Case Study N.H. Ben Fong 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: ⎛ V ⎞ ⎜ ⎟ ( Ds + 1) FI ABDE ⎠ ⎝ = 4 TD s + [I] s3 + [II] s 2 + [III] s + [IV] where (6.1) [I] = 1 [ ADE(B+C) + BC(AE+DV)] ABCDE [II] = 1 VBCD ⎤ ⎡ AE ( B+C+D ) + DV(B+C) + BCV + ABCDE ⎢ F ⎥ ⎣ ⎦ [III] = ⎡ 1 VBC ⎛ D D ⎞ ⎤ V(B+C+D) + AE + ⎜1 + + ⎟ ABCDE ⎢ F ⎝ C G ⎠⎥ ⎣ ⎦ [IV] = 1 VBC ⎡ V+ (1 − H )⎤ ⎥ ABCDE ⎢ 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 Chapter 6 Industrial Case Study N.H. Ben Fong 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 Chapter 6 Industrial Case Study N.H. Ben Fong AWIP TD + + K AAWIP − V AEs + V B Bs + 1 1 B C Cs + 1 FI 1 F AWIP + AWIP* B NAO* + ESR AFI 1 G + 1 Ds + 1 SR 1 C − FI FI* + H 1 Ds + 1 Figure 6.7: Simplified BD Representation with block K of the Three-Stage Semiconductor System V ⎡ 2 ⎛ 1 1 1 ⎞ (1- H ) ⎤ ⎢s + ⎜ + + ⎟ s + ⎥ AEF ⎣ DG ⎦ ⎝C D G⎠ G(s) = s 4 + [δ ] s3 + [ε ] s 2 + [ λ ] s + [σ ] where (6.3) δ = 1 ⎡ ADE ( B+C ) + BC ( AE+DV ) ⎤ ; ⎦ ABCDE ⎣ ε= λ= σ = 1 ⎡ AE ( B+C+D ) + DV ( B+C ) + BCV ⎤ ; ⎦ ABCDE ⎣ 1 ⎡ V ( B+C+D ) + AE ⎤ ; ⎦ ABCDE ⎣ V ABCDE 107 Chapter 6 Industrial Case Study N.H. Ben Fong 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 1 A = MCT (month) 0.1 B = CAT (month) 7 C = MOPT (month) 0.2 D = UST (month) 1 E = FWAT (month) F = AWAT (month) 0.1 0.3 G = FIAT (month) 1.1 H = SSP TD = 5000 Chips SET B SET C SET D 1 1 1 0.4 0.1 0.1 0.7 1.3 0.94 0.5 0.2 0.1 0.2 1 1 0.2 1 0.4 0.3 0.3 0.4 1.1 1.1 1.1 108 Chapter 6 Industrial Case Study N.H. Ben Fong 7000 6000 5000 Number of Chips 4000 SET A 3000 2000 1000 0 0 SET C SET D SET B 5 10 15 20 25 30 Time (month) 35 40 45 50 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 Chapter 6 Industrial Case Study N.H. Ben Fong 8 6 4 2 0 -2 -4 -6 -8 -8 X ζ =0.8 ζ =0.7 ζ =0.9 Imag Axes X X X -7 -6 -5 -4 -3 -2 Real Axis -1 0 1 2 Figure 6.9: Root Locus of a three-stage semiconductor production system (Set B) 15 ζ =0.7 ζ =0.8 10 ζ =0.9 5 Imag Axes 0 X X X -5 -10 -15 -15 -10 -5 Real Axis 0 5 Figure 6.10: Root Locus of a three-stage semiconductor production system (Set D) 110 Chapter 6 Industrial Case Study N.H. Ben Fong 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 Chapter 6 Industrial Case Study N.H. Ben Fong 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 Chapter 6 Industrial Case Study N.H. Ben Fong 8 6 4 Number of Chips 2 0 -2 -4 -6 x 10 4 -8 0 5 10 15 20 25 30 Time (month) 35 40 45 50 Figure 6.11: Step Response of a three-stage production system that yields instability behavior 20 15 10 Imag Axes 5 0 -5 -10 -15 -20 -8 ζ =0.7 ζ =0.9 ζ =0.8 X X XX -7 -6 -5 -4 -3 Real Axis -2 -1 0 1 2 Figure 6.12: Root Locus of a three-stage semiconductor production system (Set E) 113 Chapter 6 Industrial Case Study N.H. Ben Fong 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: lim f ( t ) = lim sF ( s ) t →∞ s →0 that yields the steady-state value from eq. (6.1) as follows: C ⎛ FI ⎞ lim ⎜ ⎟ = s →0 TD BC ⎝ ⎠ 1+ (1-H ) FG (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 Chapter 6 Industrial Case Study N.H. Ben Fong 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 Chapter 7 Conclusions and Research Contributions N.H. Ben Fong 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 Chapter 7 Conclusions and Research Contributions N.H. Ben Fong (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 Chapter 8 Future Research N.H. Ben Fong 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 Hydraulic Rotational Translational Thermal Current (I) Fluid flow rate (q) Torque (T) Force (F) Heat flow rate (q) Voltage (V) Pressure (P) Angular Velocity (ω) Velocity (v) Temperature (T) Capacitor (C) Volume (Vol) Torsional Spring (kt) Spring (k) 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 Chapter 8 Future Research N.H. Ben Fong 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 References N.H. 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[45] Fong, N.H., Sturges, R.H., and Shewchuk, J.P.(2004), “Classical Control Theory Analysis of a Lean Manufacturing System”, IIE 13th Industrial Engineering Research Conference, Houston, Texas, May 15-19, 2004, IIE Press. [46] Fong, N.H., Sturges, R.H., and Shewchuk, J.P.(2004), “Understanding System Dynamics via Transfer Function in Modeling Production Control Systems”, Proceedings of the 14th International Conference on Flexible Automation and Intelligent Manufacturing, p.1033-1041, Toronto, Ontario, Canada, July 12-14, 2004, NRC Research Press, National Research Council of Canada. [47] Law, A.M. and Kelton W.D. (2000), Simulation Modeling and Analysis, 3rd Edition, McGraw-Hill Companies, Inc. [48] Kelton, W.D., Sadowski, R.P., and Sadowski, D.A. (2001), Simulation with ARENA, 2nd edition, The McGraw-Hill Companies, Inc. [49] MATLAB 5.3 version (1999), The MathWorks, Inc. [50] P. Goncalves, J.Hines, J.Sterman, and C. Lertpattarapong, “The Impact of Endogenous Demand on Push-Pull Production Systems”, Proceedings of the 22nd International Conference of the System Dynamics Society, Oxford, England, July 25-29, 2004, MIT Press. 126 References N.H. Ben Fong [51] Samad, T. (2001), Perspectives in Control Engineering: Technologies, Applications, and New Directions, Institute of Electrical and Electronics Engineers, Inc. [52] Cha, P.D., Rosenberg, J.J., and Dym, C.L. (2000), Fundamentals of Modeling and Analyzing Engineering Systems, Cambridge University Press. [53] Dorny, C.N. (1993), Understanding Dynamic Systems: Approaches to Modeling, Analysis, and Design, Prentice-Hall, Inc. [54] Ogata, K. (1992), System Dynamics, 2nd Edition, Prentice-Hall, Inc. 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 128 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. 129