Development and Application of a New Modeling Technique for Production Control Schemes in Manufacturing Systems

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Development and Application of a New Modeling Technique for Production Control Schemes in Manufacturing Systems Bashar Hafez Sader Brigham Young University - Provo Follow this and additional works at: Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation Sader, Bashar Hafez, "Development and Application of a New Modeling Technique for Production Control Schemes in Manufacturing Systems" (2005). All Theses and Dissertations This Dissertation is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.

2 DEVELOPMENT AND APPLICATION OF A NEW MODELING TECHNIQUE FOR PRODUCTION CONTROL SCHEMES IN MANUFACTURING SYSTEMS by Bashar H. Sader A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements of the degree of Doctor of Philosophy Mechanical Engineering Department Brigham Young University August 2005

3

4 BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a dissertation submitted by Bashar H. Sader This dissertation has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. Date Carl D. Sorensen, Chair Date William C. Giauque Date Charles R. Harrell Date Spencer P. Magleby Date Robert H. Todd

5

6 BRIGHAM YOUNG UNIVERSITY As chair of the candidate s graduate committee, I have read the dissertation of Bashar H. Sader in its final form and have found that (1) its format, citations, and bibliographical style are consistent and acceptable and fulfill university and department style requirements; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the final manuscript is satisfactory to the graduate committee and is ready for submission to the university library. Date Carl D. Sorensen Chair, Graduate Committee Accepted for the Department Matthew R. Jones Graduate Coordinator Accepted for the College Alan R. Parkinson Dean, Ira A. Fulton College of Engineering and Technology

7

8 ABSTRACT MODELING AND COMPARISON OF PRODUCTION CONTROL SCHEMES IN MANUFACTURING SYSTEMS Bashar H. Sader Mechanical Engineering Department Doctor of Philosophy This dissertation presents dynamic system models of continuous manufacturing systems based on analogies with electrical systems. The developed modeling technique has the capability to explicitly specify production control schemes including control points, material and information flow paths, and logical operations. The model provides standard graphical representations and governing equations to describe both the steady state and transient responses of continuous manufacturing systems. For deterministic systems, these equations can be solved to get closed-form solutions. For stochastic systems, numerical solutions can be obtained for any probabilistic distribution. The electrical analogs provide an excellent tool to model control signals and logical operations. This is especially important for pull control schemes where qualitative descriptions often found in the literature can be ambiguous. The proposed technique is demonstrated by modeling push and a variety of pull systems.

9

10 The developed models are used to study the relative performance of push, CONWIP, and kanban control systems. The results show that the card count distribution significantly affects the performance of a kanban system, and that drawing conclusions on kanban performance requires card count optimization. The results also show that the relative performance of push and CONWIP systems varies with operational factors. The factors studied in this dissertation are the bottleneck utilization, line balance, demand rate variability, and processing rate variability. At some combinations of these factors, the push system is superior to the CONWIP system, with other combinations, the CONWIP system is the superior system. In this work, push systems tend to have better relative performance (compared to CONWIP) at high variability levels in the processing rates and low variability levels in the demand rate, while CONWIP systems tend to have better relative performance (compared to push) at high variability levels in the demand rate and low variability levels in the processing rates. CONWIP systems tend to have higher relative performance at high utilization levels and in lines where a distinct bottleneck exists.

11

12 ACKNOWLEDGEMENTS I am very grateful to Dr. Carl D. Sorensen, my graduate committee chair, for his continuous support for the past five years. This work would not have been possible without his valuable help.

13

14 TABLE OF CONTENTS CHAPTER 1 INTRODUCTION... 1 CHAPTER 2 MODELING CONTROL EXPICITLY IN CONTINUOUS MANUFACTURING SYSTEMS USING ELECTRICAL ANALOGS INTRODUCTION LITERATURE REVIEW Discrete-Event Simulation Queuing Theory System Dynamics Models Limitations THE MODEL THE DETERMINISTIC CASE THE STOCHASTIC CASE VALIDATION AND VERIFICATION OF THE MODEL Conceptual Validity Code Verification Operational Validity CONCLUSION CHAPTER 3 PULL CONTROL SCHEMES IN MANUFACTURING SYSTEMS PART I: MODELING INTRODUCTION LITERATURE REVIEW Push versus Pull Push Systems Pull Systems Kanban CONWIP Pull-Push Hybrid Systems Comments REVIEW OF THE MODELING TECHNIQUE MODELING PULL SYSTEMS CONWIP Shipping-Coupled CONWIP Isolated CONWIP MAXWIP Kanban Numerical Solutions DISCUSSION xiii

15 3.6 CONCLUSION CHAPTER 4 PULL CONTROL SCHEMES IN MANUFACTURING SYSTEMS PART II: COMPARATIVE ANALYSIS INTRODUCTION LITERATURE REVIEW METHODOLOGY The Models The Systems The Factors of Interest The Control Variables The Comparison Framework The Simulation Parameters The Performance Measure RESULTS DISCUSSION CONCLUDING REMARKS CHAPTER 5 CONCLUSION APPENDIX APPENDIX A CLOSED-FORM SOLUTION FOR THE N-STATIONS PUSH SYSTEM DISCUSSED IN SECTION APPENDIX B CLOSED-FORM SOLUTION FOR THE SHIPPING-COUPLED CONWIP SYSTEM SHOWN IN FIGURE APPENDIX C SIMULATION RESULTS FOR PUSH AND SHIPPING- COUPLED CONWIP SYSTEMS xiv

16 LIST OF TABLES Table 2.1: Manufacturing-Electrical Analogies Table 3.1: Manufacturing-Electrical Analogies Table 3.2: System Equations of Push and a Variety of Pull Control Schemes for an N-Work Station line Table 4.1: Simulation Parameters Table 4.2: Kanban Based on CONWIP versus Kanban with Arbitrary Distribution xv

17

18 LIST OF FIGURES Figure 2.1: Ideal Transistor in Parallel with a Capacitor Figure 2.2: Ideal Transistor Voltage-Current Relation Figure 2.3: Single Station Push Manufacturing System Figure 2.4: Three Station Push Manufacturing System Figure 2.5: Deterministic Three Station System without a Bottleneck Figure 2.6: Deterministic Three Station System with a Bottleneck Figure 2.7: Deterministic Three Station System with the Bottleneck Relieved Figure 2.8: Stochastic (cv = 0.05) Three Station System without a Bottleneck Figure 2.9: Stochastic (cv = 0.05) Three Station System with a Bottleneck Figure 2.10: Stochastic (cv = 0.05) Three Station System with the Bottleneck Relieved Figure 2.11: Stochastic (cv = 0.25) Three Station System without a Bottleneck Figure 2.12: Stochastic (cv = 0.25) Three Station System with a Bottleneck Figure 2.13: Stochastic (cv = 0.25) Three Station System with the Bottleneck Relieved Figure 2.14: WIP Increase In Front of the Bottleneck for Three Variability Levels Figure 2.15: Validation and Verification of Simulation Models, After Sargent (1992) Figure 2.16: ProModel WIP versus Our Model WIP Using Uniform Distribution Figure 3. 1: Ideal Transistor in Parallel with a Capacitor Figure 3.2: Ideal Transistor Voltage-Current Relation Figure 3.3: Three Station Push Manufacturing System Figure 3.4: CONWIP Release Mechanism Figure 3.5: Single Station Shipping-Coupled CONWIP System Figure 3.6: Analytical Solution for a Single Station Shipping-Coupled CONWIP System Figure 3.7: Two Station Shipping-Coupled CONWIP System Figure 3.8: Two Station Isolated CONWIP System Figure 3.9: Single Station Release-Shipping-Coupled MAXWIP System Figure 3.10: Single Station Release-Coupled MAXWIP System Figure 3.11: Single Station Kanban System Figure 3.12: Numerical Solution of a Three Station Shipping-Coupled CONWIP System Figure 3.13: Numerical Solution of a Three Station Isolated CONWIP System Figure 3.14: Numerical Solution of a Three Station Release-Shipping-Coupled MAXWIP System xvii

19 Figure 3.15: Numerical Solution of a Three Station Release-Coupled MAXWIP System Figure 3.16: Numerical Solution of a Three Station Kanban System Figure 4.1: Three Station Push System Figure 4.2: Three Station Shipping-Coupled CONWIP Figure 4.3: Three Station Kanban System Figure 4.4: Normalized Inventory Efficiency for Balanced Lines with 90% Utilization Figure 4.5: Normalized Inventory Efficiency for Balanced Lines with 95% Utilization Figure 4.6: Normalized Inventory Efficiency for Unbalanced Lines with 90% Utilization Figure 4.7: Normalized Inventory Efficiency for Unbalanced Lines with 95% Utilization Figure 4.8: Normalized Inventory Efficiency Contours for Balanced Lines with 90% Utilization Figure 4.9: Normalized Inventory Efficiency Contours for Balanced Lines with 95% Utilization Figure 4.10: Normalized Inventory Efficiency Contours for Unbalanced Lines with 90% Utilization Figure 4.11: Normalized Inventory Efficiency Contours for Unbalanced Lines with 95% Utilization xviii

20 CHAPTER 1 INTRODUCTION Many of the improvements in production systems from recent years are in the control aspects. New production control schemes have been introduced to improve the performance of production lines. A production control scheme (also called production control system) specifies when raw materials are released to the production line and in what quantities. It also directs resources when to work and stay idle. Different control schemes can cause significantly different performances in otherwise equivalent production lines. As the number of control schemes is increasing, it becomes more important to develop modeling techniques that not only describe the layout of production systems, but also sufficiently describe the control schemes of them. A manufacturing system has many important variables. However, the most important variables in any manufacturing system are those that describe its overall performance: throughput, work-in-process (WIP), and cycle time. The throughput of a production line is the output rate of the line. The WIP is the inventory between the start and end points of the line. And the cycle time is the time it takes a part (a job) to go through the entire production line from start to end. These three variables are related; knowing two of them provides enough information to calculate the third one. According to Little s law, the WIP in any steady state production line is equal to the throughput of that line times its cycle time. 1

21 Furthermore, for any given production line, it is not possible to simultaneously control both the WIP and throughput. If the throughput of the line is controlled, then the system itself will dictate how much WIP will be in the line. If the amount of WIP is controlled, then the system itself will dictate the throughput. Based on this basic understanding, two categories of production control schemes can be defined. In a push control scheme, the throughput of the line is controlled, while the WIP is monitored. Material requirements planning (MRP) is a typical example of push schemes. In a pull control scheme, the WIP in the line is controlled, while the throughput is monitored. Kanban and CONWIP systems are examples of pull control schemes. Many modeling techniques have been developed to describe manufacturing lines. These techniques provide tools to analyze and predict the performance of production lines. The modeling techniques can be categorized in three main categories: discreteevent models, queuing theory models, and system dynamics models. Discrete-event simulation is a very powerful and versatile tool; but it does not provide governing equations to describe the performance of a manufacturing system. Queuing theory on the other hand does provide governing equations, but it is limited to restrictive cases. System dynamics combines the advantages of the other two techniques: it provides governing equations and it is versatile. However, none of the above modeling techniques provides an explicit description of the control scheme of production lines, nor does any of them include a standard graphical representation that uniquely describes the system. Explicit descriptions of control schemes combined with standard graphical representations would make it easy to identify the characteristics of production lines and thus avoid any confusion in communicating information about manufacturing systems. 2

22 The lack of emphasis on modeling the control aspect of production systems has led to ambiguous definitions of many control schemes, especially pull schemes. Terms like kanban and CONWIP are widely used in the literature without universally accepted definitions. This resulted in comparative studies with contradictory results. For example, it is common in the literature to find studies advocating CONWIP as a superior control system to push, while other studies advocate push as a superior system to CONWIP. This dissertation presents dynamic models for single product continuous manufacturing systems (or discrete systems that can be approximated by continuous behavior). The developed models employ analogies between continuous manufacturing systems and electrical systems. These analogies are intended to provide explicit control points for the manufacturing process and produce a visually understandable graphical representation of the manufacturing system. The graphical model can then be used to write the governing equations of the system. However, the analogy to electrical systems will be carried no further than the graphical representation and basic variables and components. The governing equations are developed for the manufacturing system variables without referring to the equations of the analogous electrical system. The modeling technique is used to model a variety of pull systems and to compare the relative performance of push, CONWIP, and kanban control systems. The comparison is conducted by writing MATLAB codes based on the governing equations. These codes are used to simulate the operation of the systems under study. The modeling technique is introduced in Chapter 2. In Chapter 3, the technique is used to model a variety of pull systems. And in Chapter 4, the models are used to study the relative performance of push, CONWIP, and kanban control systems. Finally, 3

23 conclusions are in Chapter 5. Chapter 2-4 have been written as journal papers, rather than traditional dissertation chapters. The paper in Chapter 2 has been submitted to the Journal of Manufacturing Systems. The papers in Chapters 3 and 4 will be submitted to the International Journal of Production Research. 4

24 CHAPTER 2 MODELING CONTROL EXPICITLY IN CONTINUOUS MANUFACTURING SYSTEMS USING ELECTRICAL ANALOGS 2.1 INTRODUCTION Many of the improvements in production systems from recent years have been made in production control schemes. However, most simulation tools explicitly model processes, not control. In most models, control is an implicit part of the model. Different models have been developed and used to represent manufacturing systems. Most of them are discrete models that deal with the arrival, processing and moving of individual parts and batches. This is in agreement with the fact that the majority of real manufacturing systems are discrete in nature. However, as industry strives to reduce batch sizes and setup times, information on the flow of material becomes more important than information on individual parts. Robinson (1998) notes that with the new policies in industry, we have more islands of continuous processing than islands of automation. This realization gives an increasing value to continuous models of manufacturing systems. In this chapter we present a dynamic model for single product continuous manufacturing systems (or discrete systems that can be approximated by continuous behavior). The basic idea of the model is to employ analogies between continuous manufacturing systems and electrical systems. These analogies are intended to provide 5

25 explicit control points for the manufacturing process and produce a visually understandable graphical representation of the manufacturing system. The graphical model can then be used to write the governing equations of the system. However, the analogy to electrical systems will be carried no further than the graphical representation and basic variables and components. The governing equations are developed for the manufacturing system variables without referring to the equations of the analogous electrical system. After a brief review of the different approaches that are used to model manufacturing systems in the literature, we present the graphical and mathematical models we used to model continuous manufacturing systems. Deterministic and stochastic applications of the model are then presented, followed by a discussion of the validity of the model. 2.2 LITERATURE REVIEW Models of manufacturing systems can be broadly divided into three categories: Discrete-event simulation models, queuing theory models and system dynamic models. While research on queuing theory and discrete-event simulation approaches has been conducted on a wide scale, the system dynamics approach did not receive the same attention. Baines and Harrison (1999) point to the lack of current citations which advocate the system dynamics approach in modeling manufacturing systems. The rest of this section presents a brief literature review on different modeling approaches Discrete-Event Simulation Discrete-event simulation is based on a computer model that imitates the operations carried out in a real system. In a simulation model, different events take place at different points in time, and each event is tracked through the model. An event can be a part 6

26 entering the system, a part being processed at a machine, etc. Random numbers from specified distributions are generated to model the frequency, length or other variables related to an event. A future-event scheduler (FES) keeps track of all current and future events with respect to time. The simulation clock marches up to the first scheduled event and then to the next and so on until all scheduled events have occurred, and statistics are gathered throughout the process. Salama and Hongler (1995) and Law and McComas (1999) note that most simulation studies of manufacturing systems provide information on the long run or steady-state behavior of the systems being modeled, but not on transient responses of the systems. Although most simulation packages are based on discrete-event simulation (Kuo et al. 2001), some efforts have been aimed at utilizing them to model manufacturing processes that are naturally continuous such as chemical production systems or are a hybrid of discrete and continuous systems such as water bottling systems (Klingener 1995 and Kuo et al. 2001) Queuing Theory As the name suggests, queuing theory is simply the science of waiting (Hopp and Spearman 2001). To analyze a queuing theory model, probability distributions need to be associated with the different processes. Most of the research on queuing theory applications to manufacturing systems assumes exponential distributions for processing times (Yeralan and Tan 1997). However, de Souza et al. (1996) note that the queuing theory approach may require complicated mathematical analysis and that it is not applicable in all cases. 7

27 2.2.3 System Dynamics Models The term system dynamics has been used in the literature of manufacturing systems for at least forty years. Forrester (1958) noted that feedback control systems and explicit quantitative forms are necessary to predict the future course of an existing system. The behavior of dynamic manufacturing systems can be modeled by either discrete or continuous models (Valentin and Ladet 1993). However, the system dynamics approach of modeling manufacturing systems uses mainly continuous models (Baines and Harrison 1999). Continuous models typically deal with flows as opposed to parts in discrete models. It is intuitive that continuous models are better suited for physically continuous manufacturing systems such as the petroleum and chemical production systems, but continuous models can also be used to model some discrete manufacturing systems. Besombes and Marcon (1993) note that the behavior of many flow-shop systems can be approximated by that of a continuous system. Chen and Yao (1992) used a continuous fluid model to study discrete manufacturing systems with random disruptions. They note that although the dynamics of the manufacturing system they study is perhaps best modeled by the queuing model, the fluid model is easier to analyze. Alvarez-Vargas et al. (1994) use a continuous flow model to approximate the behavior of asynchronous manufacturing systems with finite buffers and machines that are subject to random failures. Asynchronous systems refer to lines where there is an unlimited quantity of parts at the input and an unlimited amount of space at the output. In their continuous model, they assume that the machines have constant and deterministic processing rates and that the time to failure (TTF) and time to repair (TTR) 8

28 are exponentially distributed. Their study concludes that continuous flow models can be used as good approximations of asynchronous systems. Salama and Hongler (1995) present a transient analysis for continuous fluid models of manufacturing systems with constant and deterministic processing rates and random failures. They also compare their analysis to a discrete model. They note that continuous models are equivalent to discrete models for large production rates. Brandimarte et al. (1996) approximate discrete models with continuous models. They note that continuous models facilitate the use of optimization techniques and thus help achieving better designs. Besombes and Marcon (1993) and Ferney (2000) apply some system dynamics modeling and analysis techniques to manufacturing systems. They use bond-graphs and state equations to represent continuous manufacturing systems. But they note that the same techniques can be used to model discrete systems at the aggregate level where the behavior can be approximated by that of a continuous system. Besombes and Marcon note that the system dynamics modeling techniques (such as bond-graphs) have the advantage of preserving the structural information of the physical system being modeled by offering a standard symbolic description Limitations While discrete-event simulation provides a powerful tool to understand and improve manufacturing systems, it does not provide closed-form solutions or physically meaningful equations to describe the behavior of the system. Van der Zee (2003) also notes that the lack of attention given to the modeling of control structures is a significant 9

29 shortcoming of traditional simulation. However, discrete-event simulation has the advantage of being able to model a very wide range of scenarios and situations. The other common modeling approach, queuing theory, offers mathematical models and closed form solutions that describe the behavior of some manufacturing systems to a detailed level. Unfortunately, queuing theory models can involve very complicated mathematical analysis. To simplify the analysis (and make it actually possible), restrictive assumptions are needed regarding the probabilistic distributions of the different activities in the system. This limits the use of queuing theory to special cases. Continuous models have been used to describe the behavior of both continuous and discrete systems. However, the models that are found in the literature usually make restrictive assumptions so that closed-form solutions can be found. Typical examples of the assumptions include constant and deterministic arrival and processing rates and exponential TTF and TTR. Tools of system dynamics (like bond-graphs and state equations) have also been used to model manufacturing systems. Although bond-graphs preserve the structural information and facilitate writing state equations for the system being modeled, bondgraphs are not commonly understood or used by manufacturing and industrial engineers. 2.3 THE MODEL The analogy between electrical systems and manufacturing systems begins with the idea that the movement of charge in an electrical circuit is similar to the movement of product through a manufacturing system. Building on this fundamental insight, we can develop analogies between the electrical world and the manufacturing world. There are 10

30 three variables of primary interest in evaluating the performance of manufacturing systems: throughput, work-in-process (WIP), and manufacturing cycle time. If charge is analogous to product, then the flow of product (throughput) is analogous to electrical current. The amount of WIP in a buffer is analogous to the amount of charge on a capacitor, which is proportional to the voltage on the capacitor. The proportionality constant between charge and voltage is the capacitance. For convenience, we make this constant 1, so that the amount of WIP in the buffer is numerically equal to the voltage on the capacitor. Manufacturing cycle time has no analog in the electrical model, as individual charges are not identified in electrical systems. Thus, it is impossible to define a cycle time for any individual charge. However, this is not an insurmountable problem, because the cycle time can be determined from Little s Law (Hopp and Spearman 2001). In order to process materials at a manufacturing station and thus produce material flow, three conditions must be satisfied. First, there must be raw materials to work on. Second, there must be capacity available at the station. Finally, there must be a signal to allow the station to work on that material. The control signal can take a variety of forms. Some common control signals are specific instructions to an operator about what to make when, general instructions to work on any available work as quickly as possible, and signals from other parts of the production line that trigger automatic machinery to begin work. These are not the only possible control signals, but it is important to recognize that every station, manual, semiautomatic, or automatic, has either implicit or explicit rules that control when and how work is done. 11

31 A model of a manufacturing station needs to represent these three conditions. An ideal transistor in parallel with a capacitor provides these requirements as shown in Figure 2.1. Raw materials are represented as voltage on the capacitor. The process equipment is represented by the transistor. The capacity of the station is represented by the current limits for the transistor. The control signal is represented by the control current into the transistor (i cont ). The voltage follower is not part of our manufacturing system, but it is used for electrical convenience to ensure that no current will flow from the capacitor without going through the transistor and to ensure that the voltage across the transistor (V tran ) is equal to the voltage across the capacitor, i.e., V tran =V cap. i cap i cont V tran V cap Transistor - + i pass Voltage Follower Capacitor Figure 2.1: Ideal Transistor in Parallel with a Capacitor 12

32 The electrical circuit works as follows: the transistor passes a current (i pass ) proportional to the control current (i cont ) as long as the voltage across the capacitor is at least equal to the threshold voltage (V th ). If the voltage is less than the threshold voltage, the transistor current is reduced, see Figure 2.2. The maximum value of i pass is obtained when i cont is at its maximum value, i cmax. The control current is usually much smaller than the current passing through the capacitor (i cap ), thus i pass = i cap + i cont i cap. Figure 2.2: Ideal Transistor Voltage-Current Relation 13

33 The electrical analog for the production system is as follows: there should be material in the station (analogous to voltage across the capacitor) to have material flow rate out of it (analogous to current through the transistor). The value of the output rate is determined by a control signal (analogous to a control current) but cannot exceed a maximum value which is a property of the station (analogous to the rated transistor current, i rated ). It should be noted here that a station could have either a single machine or a number of machines working in parallel. It is the total capacity rate of the station that needs to be modeled. If the effective time needed to process a single unit of material is te minutes, and the machine (or parallel machines) can simultaneously work on a maximum of k units, then the maximum possible capacity rate of the station is (k/te) unit per minute. This maximum capacity rate (k/te) is analogous to the rated current of the transistor (i rated ), and the maximum number of product units that can be processed simultaneously (k) is analogous to the threshold voltage of the transistor (V th ). Table 2.1 summarizes the manufacturing-electrical analogies developed in this section. Table 2.1: Manufacturing-Electrical Analogies Manufacturing Systems Electrical Systems Throughput Current WIP Voltage Buffer Capacitor Machine Transistor 14

34 Using the above analogy, we model a single station manufacturing system. The modeling will be based on three assumptions: 1. Single product line. 2. Infinite storing capacity at the station. 3. A push policy is applied. This means that any time material is present, the machine is authorized to work on it at full capacity, which is analogous to the control current being set to its maximum value (i cmax ). Note that for other control schemes, like CONWIP for example, the control current can be used to control the production rate of the station. i in i buf i cont = i cmax WIP - + i out =TH Figure 2.3: Single Station Push Manufacturing System 15

35 Such a system can be modeled as shown in Figure 2.3. Although the remainder of this chapter deals with push systems as outlined in assumption 3, the model technique is not limited to this case. Chapter 3 will show the application to a variety of pull systems, including CONWIP, Kanban and MAXWIP. In fact, the ability to model a variety of control schemes is one of the strengths of this modeling technique. In the above model, i in represents the arrival rate, which is the rate at which materials are released to the system. i in is modeled as a current source, and it cannot be a negative value (i in 0). i buf represents the rate at which materials are flowing in or out of the WIP storage in the station. i out represents the material flow through the machine(s) which is equal to the throughput (TH) of the station, and is always a nonnegative value (i out 0). Note that i in, i buf, i out and WIP are all functions of time. The performance of this system is independent of any amounts of WIP (analogous to voltage) outside it, thus it is grounded on both ends. Assume that the effective time needed to process a single unit of material is te minutes, and that the machine (or parallel machines) can simultaneously work on a maximum of k units. This yields a maximum possible capacity rate of (k/te) unit per minute. However, if there are less than k units in the WIP storage, the station cannot process at maximum capacity. So, we can define i out as: WIP/te : WIP < k i out (t) = TH(t) = (2.1) k/te : WIP k where WIP is the amount of material units available in the station. The WIP includes both the units being processed and the units waiting to be processed. 16

36 Applying Kirchoff s current law (Paul et al. 1992) gives: i = i i. (2.2) in buf + out Since i buf is the rate at which material is flowing in or out of the capacitor, i buf can be defined as: dwip i buf =. (2.3) dt Assume we started with an empty system (i.e., WIP = 0), and consider the system after some time t has elapsed. If WIP is less than k, then Equations (2.1), (2.2) and (2.3) give: dwip WIP + = i in (WIP < k). (2.4) dt te But if WIP is greater than or equal to k, then: dwip k = i in (WIP k). (2.5) dt te Equations (2.4) and (2.5) describe the performance of a single station continuous manufacturing system subject to the assumptions previously mentioned. 17

37 Using Equation (2.1), we can define the utilization of the station. The utilization, u, is defined as the arrival rate to the station divided by the maximum possible processing rate, thus: iin u =. (2.6) (k/te) 2.4 THE DETERMINISTIC CASE In this section, we add two more assumptions to those in section 2.3: 4. The arrival rate (i in ) is a deterministic constant value. 5. The effective processing time (te) is a deterministic constant value. Using these assumptions, and applying the initial condition WIP(0) = 0, Equation (2.4) can be solved to give: WIP(t) = i in te (1-e -t/te ) = uk (1-e -t/te ) (t<ts). (2.7) Equation (2.7) is valid up to the time ts at which saturation is reached (i.e., WIP(ts) = k). This time, ts, can be determined by setting WIP(ts) equal to k, thus getting: ts = -te*ln(1-k/ i in te) = -te*ln(1-1/u). (2.8) Equation (2.5) can also be solved for the condition WIP(ts) = k, which yields: 18

38 WIP(t) = (i in k/te)[t + te*ln(1- k/ i in te)] + k = (k/te)(u - 1)(t - ts) + k (t ts). (2.9) Equation (2.9) is valid after the saturation time, ts. Note from Equation (2.8) that if (k/i in te) is larger than or equal to 1 (utilization is less than or equal to 100%), then no real value exists for ts. This means that WIP will never reach k and Equation (2.7) will be valid for all time. If we evaluate Equation (2.7) at time t, we get WIP(t ) = i in te. Note that the quantity i in te is a constant, so, dwip/dt = 0 as t and Equations (2.2) and (2.3) will give the value of TH as: TH = i in. (2.10) In that case Equation (2.7) can be written as: lim WIP(t) = TH te t (u 1). (2.11) For the steady state case (i.e., t ), Equation (2.7) reduces to Equation (2.11), which is readily recognizable as Little s law. This suggests that Equation (2.7) is a general form of Little s law that covers both the transient and steady state responses, given the previous assumptions. The previous analysis can be carried out for a series line of n stations, where the output rate of machine j is the input rate of machine j+1. MATLAB code was written to numerically solve the resulting linked set of differential equations and plot the results. 19

39 For example, consider a single product continuous manufacturing system that consists of three stations in series with infinite storage capacity, constant effective processing times, and a constant arrival rate (i ar ). Such a system is shown in Figure 2.4. For this example: i in,1 = i ar, i in,2 = i out,1, i in,3 = i out,2, TH = i out,3. The MATLAB code is used to solve the equations for the case of i ar = 10 unit/minute, k 1 = k 2 = k 3 = 10 units and u 1 = u 2 = u 3 =100%. A graphical solution for the first hour starting with an empty system is generated, and a plot showing the WIP levels and i out at each station as a function of time is shown in Figure 2.5. This case represents a balanced capacity line. Note the time delays in both WIP and flow rate between the three stations and note the transient response. If the available capacity rate at station 2 is decreased (by increasing te 2 ) so that its utilization is 110%, then the utilization of machine 3 will automatically decrease to 90.9 %. This unbalanced capacity case is shown in Figure 2.6. Note that WIP 2 increases linearly with time and that machine 2 is the bottleneck (it determines the throughput of the system). Theoretically, WIP 2 will be infinite as t. Nevertheless, at any given finite time, a finite value of WIP 2 can be determined. Thus, this system can be effectively modeled when there are finite time periods where release rate exceeds capacity. This is an advantage of this modeling technique over other models (like queuing theory) in which only steady state behavior can be modeled. Suppose the bottleneck is relieved by increasing k 2 (the maximum number of material units that can simultaneously be processed at station 2) to 11. This case is shown in Figure 2.7. Note that since the line is rebalanced, the station reaches steady state. 20

40 i ar i buf,1 i cont,1 = i cmax Work Station 1 WIP i out,1 i buf,2 i cont,2 = i cmax Work Station 2 WIP i out,2 ibuf,3 i cont,3 = i cmax Work Station 3 WIP 3 i out,3 = TH - + Figure 2.4: Three Station Push Manufacturing System 21

41 Figure 2.5: Deterministic Three Station System without a Bottleneck Figure 2.6: Deterministic Three Station System with a Bottleneck 22

42 Figure 2.7: Deterministic Three Station System with the Bottleneck Relieved Although the above solutions were numerically generated using MATLAB code, analytical solutions can be obtained. For example, in the case where i ar = 10 units/minute, k 1 = k 2 = k 3 = 10 units and u 1 = u 2 = u 3 =100%, it can be shown that WIP 1, WIP 2, WIP 3, i out,1, i out,2 and i out,3 as functions of time are given by: WIP 1 (t) = 10 (1-e -t ), i out,1 (t) = 10 (1-e -t ). (2.12) WIP 2 (t) = 10 (1- e -t - t e -t ), i out,2 (t) = 10 (1- e -t - t e -t ). (2.13) WIP 3 (t) = 10 (1- e -t - t e -t t 2 e -t /2), i out,3 (t) = 10 (1- e -t - t e -t t 2 e -t /2). (2.14) 23

43 The above solutions are subject to empty initial conditions in all three stations. These analytical solutions match the numerical solutions shown in Figure 2.5. In general, for an initially empty, single-product, push, continuous manufacturing system that consists of n stations in series with infinite storage capacity, constant effective processing times, and a constant arrival rate (i ar ) where te 1 = te 2 = = te n = te. (2.15) and u 1 1, u 2 1,, u n 1. (2.16) it can be shown (see Appendix A) that: k 1 j t/te t WIP (t) = i arte (1 e ). k n (2.17) j j!te k j= 0 i k 1 j t/te t (t) = i ar (1 e ). k n (2.18) j j!te out, k j= THE STOCHASTIC CASE To model stochastic processes we allow effective processing times of the stations to be random numbers that come from distributions with known means and standard 24

44 deviations. In other words, assumption 5 from the previous section will be relaxed while assumptions 1, 2, 3 and 4 will still be employed. Equations (2.4) and (2.5) remain the governing equations, however, te varies stochastically with time. Rather than searching for a closed-form solution for a specific distribution, we choose to solve the ordinary differential equations by numerical integration with random values for te. The utilization definition, Equation (2.6), can be rewritten for any station j with stochastic te j and deterministic i in,j as: iin,j (iin,j/k) j uj = =. (2.19) E(k/te j j) E(1/tej) where E represents the expectation operator. It should be noted that E(1/te j ) is not equal to 1/E(te j ). Also note that Var(1/te j ) is not equal to 1/Var(te j ), where Var is the variance operator. For a uniformly distributed random number x on the interval [a,b]: E(x) = (a+b) / 2. (2.20) Var(x) = (b-a) 2 / 12. (2.21) Var(x) Coefficient of variation (x) =. E(x) (2.22) On the other hand, 25

45 ln(b) - ln(a) E (1/x) =. b - a (2.23) 1 ( ln(b) - ln(a)) 2 Var(1/x) =. (2.24) ab (b - a) 2 Var(1/x) Coefficient of variation (1/x) =. E(1/x) (2.25) This means that to achieve a mean available capacity rate, E(k/te), of say 10 parts per minute with a coefficient of variation (cv) of 0.1 at a machine with k = 10 units, we need to choose the interval [a,b] for a uniform distribution in such a way that makes E(1/te) = 1 and the cv of (1/te) = 0.1. It is important to note that the interval [a,b] that meets these requirements will not necessarily give a mean value of te equal to 1 or a cv value of te equal to 0.1. The parameters of any probabilistic distribution for te should be chosen such that (1/te) has the desired expected value and variance. The MATLAB code mentioned in section 3 was modified to use random numbers generated from known distributions instead of deterministic values of te s. The code generates a new value for te (updates it) every deterministic time period tu. In this study, the value for tu is chosen to be E(te). However, if better information is known for a specific system, it should be used. This is also the case with deciding which distribution to use for te. 26

46 Now consider the same example from section 2.4, but now with te 1, te 2 and te 3 as random numbers generated from uniform distributions. The parameters of the uniform distributions for te s are chosen such that the desired level of utilization and variability are achieved. Figures 2.8, 2.9 and 2.10 show graphical results for 3 different scenarios, which are the same scenarios seen in Figures 2.5, 2.6 and 2.7 respectively, but this time with stochastic te s. The system parameters assumed are shown on each graph. At this relatively low level of variation (cv = 0.05), it can be seen that the graphs are similar to the no variability case. While the stations maintained average output flow rates close to the deterministic case, they accumulated higher levels of WIP. Figure 2.8: Stochastic (cv = 0.05) Three Station System without a Bottleneck 27

47 Figure 2.9: Stochastic (cv = 0.05) Three Station System with a Bottleneck Figure 2.10: Stochastic (cv = 0.05) Three Station System with the Bottleneck Relieved 28

48 The effect of a higher variability level in the system can be shown for the same example with coefficients of variation of 0.25 at each station. We are still trying to achieve a 100% utilization level at each station. Figures 2.11, 2.12 and 2.13 represent the same scenarios seen in Figures 2.8, 2.9 and 2.10 respectively but at this higher variability level. The system parameters assumed are shown on each graph. Note that although the stations still maintained average output flow rates close to the previous cases, the WIP levels became considerably higher. Actually, at high levels of variability, the WIP levels may become so high that for practical reasons they can be thought of as infinite. This is in agreement with the literature that gives an infinite WIP level at a utilization level of 100% for exponentially distributed te s (cv = 1.0) (Hopp and Spearman 2001). Figures 2.5 through 2.13 show that it is the combination of high levels of utilization and high variability levels that generates high levels of WIP. The problem of high WIP levels can be dealt with by reducing utilization (adding more capacity), reducing variability, or a combination of both. Although no closed-form solutions are found for the stochastic case, general trends (like the average rate of WIP increase) can be predicted from the corresponding deterministic case closed-form solution. For example, note that the average slope of WIP 2 increase in Figures 2.9 and 2.12 is the same as that found in Figure 2.6. See Figure The same stream of random numbers was used in each of the simulations whose results are shown in this section. This was to make sure that differences in the results were due to the changes in system parameters and not a result of using a different set of random numbers. The goal was to see the effect of different levels of variability and not to accurately estimate the WIP levels and flow rates. To estimate the WIP levels and flow 29

49 rates given any set of system parameters, a number of replications using different streams of random numbers should be used. Figure 2.11: Stochastic (cv = 0.25) Three Station System without a Bottleneck 30

50 Figure 2.12: Stochastic (cv = 0.25) Three Station System with a Bottleneck Figure 2.13: Stochastic (cv = 0.25) Three Station System with the Bottleneck Relieved 31

51 Figure 2.14: WIP Increase In Front of the Bottleneck for Three Variability Levels 2.6 VALIDATION AND VERIFICATION OF THE MODEL The validation and verification process is closely related to the modeling process. In the modeling process, we start with the problem entity, which is the real system to be modeled, then we use mathematical, verbal, or graphical representation (or a combination of them) to come up with a conceptual model. When the conceptual model is implemented on a computer we get the computerized model which is then used to produce results to simulate the real system. When going from the problem entity to the conceptual model, conceptual model validity needs to be established to show that the conceptual model is a reasonable description of the real system. After the computerized 32

52 model is obtained, a verification process is needed to ensure that the computer programming is correct. The last stage is to determine if the output of the model has sufficient accuracy to represent the performance of the real system, this is referred to as operational validity, (see Figure 2.15). For a detailed description of validation and verification processes of manufacturing system models, the reader is referred to Sargent (1992). Problem Entity Operational Validity Conceptual Validity Computerized Model Conceptual Model Code Verification Figure 2.15: Validation and Verification of Simulation Models, After Sargent (1992) When establishing the validity of any model, one should keep in mind the intended purpose of that model. The validation process of a model that is built to represent the detailed performance of a specific manufacturing system is different from the validation 33

53 process required for a model built to predict the overall performance of a general manufacturing system Conceptual Validity For the purpose of our model, the conceptual model is the graphical representation combined with the governing equations. The conceptual validity is established by noticing that many researchers have been using continuous models to represent manufacturing systems (see section 2.2). Also, the results of the deterministic steady state case compare well with the results predicted by Little s law and show that the conceptual model is reasonable. The results of the stochastic case at high variability and high utilization compare well with predictions found in the literature Code Verification Comparing the closed-form solutions of the example presented in section 2.4 with the output of the MATLAB code shows that the two solutions are equivalent. This verifies that the programming is correct. Since no closed-form solutions are obtained for the stochastic case, the verification is established by noticing that the results at low variability level are very similar to the deterministic case Operational Validity Since the purpose of this model is to study the general behavior of manufacturing systems using an approach that explicitly models control schemes, we do not need to compare the output of our model to real data from a specific system. Instead, we decided to compare our model to a commonly used simulation tool, namely, ProModel. A formal comparison of discrete-event simulation with continuous simulation is beyond the scope 34

54 of this dissertation. However, we compared some results from ProModel to results from our model to see how different they would be from each other. A three-station manufacturing line model was built in ProModel. Each station had 10 parallel machines. The arrival rate was set to 1 unit per 0.1 minute. Note that in discreteevent simulation this is not the same as 10 units per minute. Having 10 parallel machines (each with one tenth of the station capacity) instead of a one big machine, and the way arrival rate was defined helps make ProModel behave in a way that is more similar to continuous simulation. As a result of having 10 parallel machines at each station, a variability pooling adjustment was necessary in defining cv at each machine so that the required cv for the whole station can be achieved. We compared the results for 12 scenarios using uniform distributions for te s. The scenarios involved two levels of variability and different utilization levels. Each scenario was replicated 10 times and the averages of replications were used for the comparison. Each replication simulated a period of time of 10 hours. Figure 2.16 shows the results of comparing WIP levels at each of the three stations. All the scenarios produced matching results within one standard deviation, and the highest percentage difference was 8.8 %. This shows strong agreement between our model and ProModel. Although very small, the discrepancies noted in WIP levels between ProModel and our model may result from different sources: 1. The conceptual difference between discrete processing (where each unit has its own processing time) and continuous processing (where there is a flow rate of units). This factor is still important even if there is no variability in the system. 35

55 20 15 Our Model WIP ProModel WIP Figure 2.16: ProModel WIP versus Our Model WIP Using Uniform Distribution 2. In ProModel, a random number is generated to determine the processing time of each individual unit, while in our model a random number determines the processing rate over a fixed period of time, tu. 3. Since ProModel needs to generate more random numbers than our model, it is not possible to use the same set of random numbers in both models. The first source of discrepancy can be minimized by increasing the number of parallel machines at each station in ProModel. However, when we increased the number of parallel machines from 10 to 100, the effect was very small. The second source of discrepancy is inherent to the continuous modeling approach. We found that the choice of tu has a significant effect on the WIP levels. The mean value of the processing times is used as a reasonable choice for tu, however, the best choice for tu depends on the real 36

56 system being modeled and is subject of further research. The third source of discrepancy can be handled by increasing the number of replications and the simulated operation time. We used 10 replications and simulated an operation time of 10 hours. However, more replications and longer operation times can be used at the expense of longer simulation times. 2.7 CONCLUSION This chapter presents a dynamic system model of continuous manufacturing systems. It employs electrical analogies and components to build a visually understandable model of continuous manufacturing systems. The model includes the control aspects of manufacturing system. The control aspects are very important especially when modeling pull systems. Compared to discrete-event simulation and queuing theory approaches, the developed model has the advantage of providing physically meaningful equations (which is not the case with discrete-event simulation) without limiting itself to any specific probabilistic distribution (as in queuing theory). It also has the advantage of producing closed form solutions that cover both the transient and steady states in the deterministic case. Although the deterministic case is theoretical and can hardly exist in a real life manufacturing system, its closed-form solution provides a good insight. In stochastic cases, the model is capable of handling variability in the processing and arrival rates. It should be noted here that the way we used to handle variability in the example mentioned in this chapter is not unique. The best way to handle stochastic effects in one real system might not be the best way to handle it in another real system. 37

57 When compared to a discrete-event simulation tool, our model produced matching results within one standard deviation with a highest percentage difference of 8.8 %. 38

58 CHAPTER 3 PULL CONTROL SCHEMES IN MANUFACTURING SYSTEMS PART I: MODELING 3.1 INTRODUCTION As industry moves in the direction of leaner manufacturing, new production control systems are being introduced and modeled. From the traditional push system (which is still widely used), to kanban and CONWIP, the flow of information within systems is getting more attention. Understanding, modeling, and clearly communicating the control system in any manufacturing line are important steps towards improving that line. As opposed to other modeling techniques, like queuing theory and discrete-event simulation, system dynamics has the advantage of explicitly preserving the structural and control information of the physical systems being modeled (Besombes and Marcon 1993). Using system dynamics, Chapter 2 presented a model for single product continuous manufacturing systems (or discrete systems that can be approximated by continuous behavior). Analogies between continuous manufacturing systems and electrical systems were employed to provide visually understandable graphical representations of push manufacturing systems. The graphical models were then used to write governing equations of the systems. In this chapter, we use the previously developed modeling technique to model a variety of continuous pull systems. Our goal in this chapter is to show that this modeling 39

59 technique can accurately preserve the control structures and facilitate writing system equations for pull systems as well as push systems. In Chapter 4, we use the same technique to quantitatively compare different control systems. 3.2 LITERATURE REVIEW Choosing a control system is a basic part of managing any manufacturing system. A control system specifies when raw materials are released to the manufacturing system and in what quantities. It also directs resources when to work or stay idle (Grosfeld-Nir et al. 2000). This section presents a brief literature review on different production control systems Push versus Pull Production control systems can broadly be divided into two categories: Push and pull systems. Spearman et al. (1989) mention that a basic difference between push and pull systems is the fact that pull systems control work-in-process (WIP) and measure throughput while push systems try to establish throughput (by controlling release rates) and measure WIP. Although these terms (push and pull) are widely used, Grosfeld-Nir et al. (2000) state that there are no generally accepted definitions for these terms. Spearman et al. (1990) notice that a particular manufacturing system can have elements of both push and pull Push Systems In a typical push system, the raw material for a specific job is pushed (released) to the floor on a date that is computed by subtracting an established lead time from the date that job is needed (Spearman and Zazanis 1992). The need for a job might be based on actual demand or anticipated demand. Gstettner and Kuhn (1996) mention that since lead times 40

60 have to be approximated in push systems, infeasible production schedules can be generated and can cause high WIP levels. Hopp and Spearman (2001) notice that strictly speaking, the release times in push systems depend on production schedules that are not affected by the amount of WIP already in the system or the capacity of the system. However, they also mention that push systems can be modified to generate schedules that consider capacity requirements Pull Systems In a pull system, raw materials for jobs are not released to the system based on schedules, but the release of one job is triggered by the completion of another job (Spearman et al. 1990). When finished jobs are pulled from the last station (through demand), new jobs are released to replenish the outgoing stock. Grosfeld-Nir et al. (2000) mention that many authors define a pull system as one in which information is transmitted backwards from subsequent stations to the preceding stations. In a pull system, information about external demand goes to the last station of the system and then travels backwards. However, the path this information takes and the series of reactions that follow can vary depending on the specific control system. Many variations of pull systems have been discussed in the literature. Gstettner and Kuhn (1996) notice that details must be included when describing a pull system in order to avoid confusion. Some of the most commonly discussed pull systems are kanban and CONWIP Kanban Kanban, which is Japanese for card or marker, is the most widely recognized of all pull systems and is sometimes referred to as the Toyota Production System (Marek et al. 41

61 2001). However, Monden (1998) considers referring to kanban as the Toyota Production System is incorrect. He states that while the Toyota Production System actually makes products, the Kanban system is an information system that controls production quantities in all stations. Although kanban has been described in detail by many authors (see for example Monden 1998), Hopp and Spearman (2001) notice that kanban as a production control system has been confused with manufacturing philosophies and concepts like just-in-time (JIT) and total quality control. Gstettner and Kuhn (1996) discuss variations of the kanban system. However, they mention that the basic idea in all kanban variations is to use cards (or some other signal) to control the amount of WIP at each station. Each station is assigned a specific number of cards that are attached to the jobs in that station. The number of cards that is assigned to each station is a decision variable whose value is chosen by those in charge of the system. No job is allowed in the station without a card attached to it. This means that the number of cards assigned to each station represents a cap on the number of jobs in that station. When a job is pulled from a station by a downstream station, the card of the preceding station is removed and the card of the downstream station is attached. A work station is authorized to work only if raw materials and cards are available. The total system WIP is limited to the summation of cards in all stations CONWIP CONWIP (constant WIP) is a production control system that has both push and pull characteristics (Spearman and Zazanis 1992). Instead of controlling the amount of WIP at each station, CONWIP only controls the total WIP in the system to keep it at a desired 42

62 constant value. The total number of cards that is assigned to the system is a decision variable whose value is chosen by those in charge of the system. In a CONWIP system, each job is required to have a card attached to it. This means that the total number of jobs in the system is limited to the number of cards. When a job is pulled from the last station, the card is detached and transferred to the first station in the line. When both a card and raw materials are available at the front of the line, a new job is started. Once in the system, jobs are pushed downstream at all stations (Hopp and Spearman 1991). In order to maintain the WIP level constant in a CONWIP system, Grosfeld-Nir and Magazine (2002) notice that enough raw materials should always be available for the first station. They also notice that the release of raw materials to the CONWIP system should be instantaneous Pull-Push Hybrid Systems Although CONWIP has both push and pull mechanisms, Spearman and Zazanis (1992) consider it a pull system because it controls WIP and not throughput. However, many pull-push hybrid systems have been described in the literature. Gstettner and Kuhn (1996) discuss lines in which some segments are controlled using CONWIP while other segments are controlled using a push policy. Another hybrid is to control a specific station of the system using kanban while work is pushed everywhere else. Grosfeld-Nir and Magazine (2002) define a control system called G-MaxWIP, in which there is a gate to control the release of raw materials to the line. In this system, all the resources, except for the gate, work unconstrained. Although the gate is assumed to have infinite supplies of raw materials, the gate stops releasing materials once the total 43

63 WIP in the system reaches a maximum allowable level. This maximum allowable level is a decision variable whose value is decided by those in charge of the system However, in contrast with CONWIP, the release of materials to G-MaxWIP does not have to be instantaneous. This allows the total WIP to drop below its maximum level if the release rate is lower than the throughput. Grosfeld-Nir and Magazine state that push and CONWIP are two special cases of G-MaxWIP. If the maximum WIP level is allowed to be infinite, the G-MaxWIP becomes a push system. If the release rate is allowed to be infinite (instantaneous release), the G-MaxWIP becomes a CONWIP system Comments Terms like push and pull are widely used in the literature without generally accepted unique definitions. Describing a specific production control system requires describing details about the paths that jobs and information take, the release mechanisms, and when resources should work or stay idle. Many variations of push and pull systems exist, and the differences between them are usually stated in qualitative and often ambiguous terms. 3.3 REVIEW OF THE MODELING TECHNIQUE The modeling technique used in this chapter is based on analogies between electrical systems and manufacturing systems described in Chapter 2. The analogy begins with the idea that the movement of charge in an electrical circuit is similar to the movement of product through a manufacturing system. If charge is analogous to product, then the flow of product (throughput) is analogous to electrical current. The amount of WIP in a buffer is analogous to the amount of charge on a capacitor, which is proportional to the voltage on the capacitor. 44

64 In order to process materials at a manufacturing station and thus produce material flow, three conditions must be satisfied. First, there must be raw materials to work on. Second, there must be capacity available at the station. Finally, there must be a signal to allow the station to work on that material. The control signal can take the form of specific instructions to an operator about what to make when, general instructions to work on any available work as quickly as possible, signals from other parts of the production line that trigger automatic machinery to begin work, or many other forms. A model of a controlled manufacturing station needs to include all three conditions. An ideal transistor in parallel with a capacitor provides these requirements as shown in Figure 3.1. Raw materials are represented as voltage on the capacitor. The process equipment is represented by the transistor. The capacity of the station is represented by the current limits for the transistor. The control signal is represented by the control current into the transistor (i cont ). The voltage follower is not part of our manufacturing system, but it is used for electrical convenience to ensure that no current will flow from the capacitor without going through the transistor and to ensure that the voltage across the transistor (V tran ) is equal to the voltage across the capacitor, i.e., V tran =V cap. The electrical circuit works as follows: the transistor passes a current (i pass ) proportional to the control current (i cont ) as long as the voltage across the capacitor is at least equal to the threshold voltage (V th ). If the voltage is less than the threshold voltage, the transistor current is reduced, see Figure 3.2. The maximum value of i pass is obtained when i cont is at its maximum value, i cmax. The control current is much smaller than the current passing through the capacitor (i cap ), thus i pass = i cap + i cont i cap. 45

65 i cap i cont V tran V cap Transistor - + Capacitor i pass Voltage Follower Figure 3.1: Ideal Transistor in Parallel with a Capacitor The electrical analog for the production system is as follows: there should be material in the station (analogous to voltage across the capacitor) to have material flow rate out of it (analogous to current through the transistor). The value of the output rate is determined by a control signal (analogous to a control current) but cannot exceed a maximum value which is a property of the station (analogous to the rated transistor current, i rated ). It should be noted here that a station could have either a single machine or a number of machines working in parallel. It is the total capacity rate of the station that needs to be modeled. 46

66 Figure 3.2: Ideal Transistor Voltage-Current Relation If the effective time needed to process a single unit of material is te minutes, and the machine (or parallel machines) can simultaneously work on a maximum of k units, then the maximum possible capacity rate of the station is (k/te) unit per minute. This maximum capacity rate (k/te) is analogous to the rated current of the transistor (i rated ), and the maximum number of product units that can be processed simultaneously (k) is analogous to the threshold voltage of the transistor (V th ). 47

67 i ar i buf,1 i cont,1 = i cmax Work Station 1 WIP i out,1 i buf,2 i cont,2 = i cmax Work Station 2 WIP i out,2 ibuf,3 i cont,3 = i cmax Work Station 3 WIP 3 i out,3 = TH - + Figure 3.3: Three Station Push Manufacturing System 48

68 This modeling technique was previously used to model single product lines with infinite storage capacities when a push policy is applied (Sader and Sorensen 2003). A push policy means that any time material is present, the machine is authorized to work on it at full capacity. Note that this is analogous to the control current being set to its maximum value. Their model of a 3-station, single product, push manufacturing system is shown in Figure 3.3. Table 3.1 summarizes the manufacturing-electrical analogies they developed. Table 3.1: Manufacturing-Electrical Analogies Manufacturing Systems Electrical Systems Throughput Current WIP Voltage Buffer Capacitor Machine Transistor Raw Material Input Rate Current Source 3.4 MODELING PULL SYSTEMS A basic requirement of pull systems is the existence of real time control signals that provide the stations with information about when to work or stay idle. This information varies over time to reflect the current state of WIP in the system and the current level of external demand. In the electrical analog model reviewed in section 3.3, the control current into the transistor provides real time control signal. Using additional electrical components, such as voltage comparators and current dividers, the value of the control current can be adjusted to reflect the current state of WIP in the system as well as the 49

69 current level of external demand. The details of these extra components are not included in the models, as the models need only the value of the control current. The electrical analog of any specific manufacturing system can be constructed to accurately model the control scheme that is used in the system. In this section we construct electrical analogs of different pull systems and use these analogs to write the equations that describe these systems. As an example, we carry out a detailed analytical solution for one of the CONWIP systems CONWIP Since the amount of WIP in a CONWIP system needs to remain constant at its target value (known as the total card count), there must be a release mechanism capable of providing large amounts of raw material over very short time to instantaneously replenish any outgoing finished goods. This release mechanism can be modeled by defining a release station that is responsible for releasing raw materials to the manufacturing line. The release station can be represented by the same electrical analog used to represent a manufacturing station in a push system, an ideal transistor in parallel with a capacitor. However, to reflect the existence of large amounts of raw material (charge), the capacitor will be replaced by a battery. Since a very high release rate is required, it will be assumed that the transistor rated current is very high such that raw material (charge) can pass through it in zero time. In a manufacturing system, this means that the raw material inventory (RMI) contains enough raw materials that it will never run out of stock, and that the time needed to transfer materials from the RMI to the first station is negligible. These assumptions are reasonable if raw materials can be supplied to the RMI at a much higher rate than the production rate and if the RMI is sufficiently close to the line. 50

70 The control signal that goes to the release station should be based on real time information about the total WIP (WIP t ) level in the system so that any outgoing finished goods can immediately be replenished. To model this control signal, we feed the total voltage across the system downstream of the release station to a voltage comparator. This total voltage is proportional to the total charge (WIP t ) in the system. The voltage comparator compares the total voltage to a predetermined target value. The target voltage depends on the target WIP level (WIP ) for the CONWIP system. The voltage comparator will then produce a control current (i cont,rmi ) that depends on the difference between the target WIP level and the total WIP currently in the system. This control current is fed to the transistor to determine how much raw material (charge) to release so that the total WIP is kept at its desired constant target level. The electrical analog of a CONWIP release station is shown in Figure 3.4. i cont,rmi α (WIP -WIP t ) Comparator Release Station RMI WIP t WIP - + i in,1 Figure 3.4: CONWIP Release Mechanism 51

71 Except for the release station and the last station, all stations in a CONWIP system are essentially push stations. Thus, all intermediate stations in a CONWIP system will be modeled in the same way they were modeled in push systems, ideal transistors in parallel with capacitors with the control currents set to their maximum values. Regarding the throughput (TH) of CONWIP systems, models based on two different assumptions need to be considered. The first model is based on the assumption that demand on the final product might be less than the maximum capacity of the line. In this case, it is necessary to model a shipping station that includes a finished good inventory (FGI). Since the throughput of this system is related to the shipping rate, we refer to this system as a shipping-coupled CONWIP. The second model is based on the assumption that the demand on the final product always exceeds the maximum capacity rate of the line. In this case, it is not necessary to include a shipping station in the model since the throughput of the line is simply the output of the last station. The last station in this case is another push station that is authorized to work as fast as it can. Since the throughput of this system is completely determined by the system itself without any external influence, we refer to this system as an isolated CONWIP. The two different models are discussed below Shipping-Coupled CONWIP Since demand in this case can be less than the maximum capacity of the line, a shipping station needs to exist where finished goods can be stored and then shipped to satisfy demand. Notice that the throughput of such a CONWIP system depends not only on the capacity of the line and the amount of WIP in it, but also on the level of external demand. The shipping station can be represented by the same electrical analog used to 52

72 model a push station, an ideal transistor in parallel with a capacitor. However, instead of setting the control current to its maximum value, the control current should be a function of the external demand. The external demand is represented by a current source that goes into a current divider. The divider scales down the demand current as control currents are assumed to be much smaller than other currents as described in section 3. The resulting control current, which is proportional to the demand, is then fed to the transistor of the shipping station to determine the rate at which finished goods should be shipped. Notice that it is necessary to have voltage (finished goods) on the capacitor as well as a control signal (demand) to produce any throughout. The shipping station is assumed to have an effective processing time te s minutes and can simultaneously work on a maximum of k s units, thus the maximum possible throughput of the of the station (and the whole line) is (k s / te s ) unit per minute. The maximum capacity rate of the shipping station (k s / te s ) is assumed to be much larger than the maximum capacity rate of all work stations and the external demand. A single station shipping-coupled CONWIP system is shown in Figure 3.5. The total WIP (WIP t ) in this system is composed of the WIP in the work station (WIP w ) plus the finished goods in the shipping station (FGI). Thus: WIP = WIP FG I. (3.1) t w + 53

73 i cont,rmi α (WIP -WIP t ) Comparator Release Station RMI WIP - + i in,1 i buf,1 i cont,1 = i cmax Work Station WIP w - + i out,1 ifgi D Current Divider i cont,fgi α D Shipping Station FGI - + TH= i out,fgi Figure 3.5: Single Station Shipping-Coupled CONWIP System 54

74 If the total WIP in the system (WIP t ) is less than the target WIP (WIP ), the release station will instantly release enough raw materials to the first station to raise the total WIP to its target value. This means that even if the total WIP in the system is below the target immediately before start ( t = 0 - ), the total WIP will be equal to the target immediately after start ( t = 0 + ). If the total WIP is more than the target value, the release station will not pass any raw materials. Finally, if the total WIP is equal to the target value, the release station will pass a raw materials rate that is equal to throughput (TH) of the system so that the total WIP will remain at its target value. So, we can now define the release rate (i in,1 ) as: TH : WIP t = WIP i in,1 (t) = (3.2) 0 : WIP t > WIP Let the effective processing time of the work station be te and assume that the machine (or parallel machines) can simultaneously handle a maximum of k units. Noticing that the work station is essentially a push station, i.e., always authorized to work as fast as possible, the output rate (i out,1 ) of the work station can be defined as: i WIP (t) = k/te /te : WIP k w w out,1 (3.3) : WIP w > k The shipping station will ship finished goods at a rate that depends on the external demand rate (D) and the amount of finished goods available in the FGI. Since the output 55

75 rate of the shipping station (i out,fgi ) is the throughput of the line (TH), it cannot exceed the demand rate (D). Thus we can write: min[d, FGI/tes ] : FGI k s TH(t) = i out,fgi (t) = (3.4) min[d, k s/tes ] : FGI > k s where min is the minimum of the quantities in the brackets. Since i buf,1 and i fgi are the rates of change of WIP w and FGI respectively, we can write: dwip i = w buf,1. (3.5) dt dfgi i fgi =. (3.6) dt Applying Kirchoff s current law (Paul et al. 1992) gives: i = i i. (3.7) in,1 buf,1 + out,1 i = i i. (3.8) out,1 fgi + out, fgi As an example, we solve Equations (3.1) through (3.8) to get a closed-form solution subject to the following assumptions: 1) Demand and effective processing times are deterministic constants. 56

76 2) The system starts with all WIP t at the work station, which is consistent with starting up an empty system by releasing WIP t units of raw materials just before startup. 3) The demand rate (D) is less than the maximum capacity rates of both the work station and the shipping station (i.e., D < (k/te) and D < (k s / te s )). 4) The WIP target level (WIP ) is greater than the total number of units that can simultaneously be handled in the work station and the shipping station (i.e., WIP > (k + k s )). Assumptions 2, 3, and 4 are not necessary to get a closed-from solution. The solution can be carried out for any other set of assumptions. However, a set of assumptions regarding the initial conditions and the relation between the different system parameters will always be required to get any specific solution. The detailed solution for the mentioned single station shipping-coupled CONWIP system subject to the above assumptions is shown in Appendix B. From that appendix we find that: For 0 t < t cd : t tes s FGI(t) = k (1 e te ). (3.9) te t tes WIP (t) WIP k (1 e te s w = ). (3.10) te 57

77 t te k s TH(t) = (1 e ). (3.11) te for t cd t < t ck : k Dte FGI(t) = ( D)(t + tes ln (1 )) + Dtes. (3.12) te k k Dte WIPw (t) = WIP ( D)(t + tesln (1 )) Dtes. (3.13) te k TH(t) = D. (3.14) and for t ck t: (t t cr4 ) te FGI(t) = WIP Dte (k Dte)e. (3.15) WIP w (t) = (k Dte)e (t t te cr4 ) + Dte. (3.16) TH(t) = D. (3.17) where t cd and t ck are critical times defined in Appendix B. Figure 3.6 shows a plot of the above solution. 58

78 Figure 3.6: Analytical Solution for a Single Station Shipping-Coupled CONWIP System The three segments of the found solution, 0 t < t cd, t cd t < t ck, and t t ck can be easily seen. Notice that the first and third segments of the WIP w and FGI solutions are exponential while the second segment is linear. For throughput, the first segment is exponential, while the second and third segments are linear. The form of the solution (linear and/or exponential segments) depends on the initial conditions and the assumptions that were made. Different assumptions and initial conditions will lead to different forms. 59

79 The critical times for the analyzed system are functions of important system parameters, like the ratio of demand rate to the maximum possible capacity rate of the work station, and the ratio of the maximum possible capacity rate of the shipping station to the maximum possible capacity rate of the work station, see Appendix B. For deterministic single station CONWIP systems, closed-form solutions can be obtained for any set of initial conditions. If the effective processing times or the demand rate are allowed to be random variables, a numerical solution is necessary. However, the closed-form solution for the deterministic single station system gives good insight about the important parameters in any CONWIP system. The same modeling technique can be used to model CONWIP systems with multiple work stations. The additional work stations will have the same graphical representations and equations as the single work station modeled above has. For example, Figure 3.7 shows a two station shipping-coupled CONWIP system. The equations that describe the performance of the system are: = WIP + WIP FG I. (3.18) WIP t w,1 w,2 + TH : WIP t = WIP iin,1 (t) = (3.19) 0 : WIP t > WIP WIPw,1/te1 : WIPw,1 k1 i out,1(t) = (3.20) k1/te1 : WIPw,1> k1 60

80 i cont,rmi α (WIP -WIP t ) Comparator Release Station RMI WIP - + i in,1 i buf,1 i cont,1 = i cmax Work Station 1 WIP w,1 i out,1 - + i buf,2 i cont,2 = i cmax Work Station 2 WIP w,2 i out,2 - + D Current Divider i cont,fgi α D i fgi Shipping Station FGI TH= i out,fgi - + Figure 3.7: Two Station Shipping-Coupled CONWIP System 61

81 WIPw,2 /te 2 : WIPw,2 k 2 i out,2 (t) = (3.21) k 2/te 2 : WIPw,2 > k 2 min[d, FGI/tes ] : FGI k s TH(t) = i out,fgi (t) = (3.22) min[d, k s/tes ] : FGI > k s dwipw,1 i buf,1 =. (3.23) dt dwipw,2 i buf,2 =. (3.24) dt dfgi i fgi =. (3.25) dt i = i i. (3.26) in,1 buf,1 + out,1 i = i i. (3.27) out,1 buf,2 + out,2 i = i i. (3.28) out,2 fgi + out, fgi In the above equations, k 1, k 2, and k s are the maximum number of units that can simultaneously be handled at work station 1, work station 2, and the release station 62

82 respectively. te 1, te 2, and te s are the effective processing rates of work station 1, work station 2, and the release station respectively. Equations (3.18) through (3.28) can be used to get closed-form solution for the system. Unfortunately, the mathematical analysis becomes more tedious and more critical times need to be considered. Some of these critical times need to be evaluated using numerical methods as they cannot be evaluated analytically due to the complexity of the equations. It is therefore recommended to use numerical solutions for the system equations. Numerical solutions also have the advantage of allowing for randomness in the effective processing times and demand rates Isolated CONWIP The basic assumption for this model is that demand will always be larger than the maximum capacity of the line. Thus, the output rate of the last station will be the throughput (TH) of the system and no shipping station or finished goods inventory (FGI) need to be considered. A release station with the same characteristics mentioned in the previous subsection is still needed to maintain the WIP at its predetermined target value (WIP ). Except for the release station, all stations in this system are authorized to work at full capacity whenever they have material to work on. This system can be modeled in the same way as the previous system (the shippingcoupled CONWIP). However, no shipping station will be included in the model. The graphical representations and system equations used in the previous subsection can still be used here. 63

83 i cont,rmi α (WIP -WIP t ) Comparator Release Station RMI WIP - + i in,1 ibuf,1 i cont,1 = i cmax Work Station 1 WIP w,1 i out,1 - + i buf,2 i cont,2 = i cmax Work Station 2 WIP w,2 - + TH= i out,2 Figure 3.8: Two Station Isolated CONWIP System 64

84 Figure 3.8 shows a two-work station isolated CONWIP system. If we assume that the effective processing times of the work stations are te 1, and te 2 respectively, and the maximum number of units that they can simultaneously handle are k 1 and k 2 respectively, then the basic equations of the shown system are: = WIP WIP. (3.29) WIP t w,1 + w,2 TH : WIP t = WIP iin,1 (t) = (3.30) 0 : WIP t > WIP WIPw,1/te1 : WIPw,1 k1 i out,1(t) = (3.31) k1/te1 : WIPw,1> k1 WIPw,2 /te 2 : WIPw,2 k 2 TH(t) = i out,2 (t) = (3.32) k 2/te 2 : WIPw,2 > k 2 Notice that no external information enters this CONWIP system. Raw materials are always available in any quantities needed and the throughput is completely determined by the system and is independent of demand. Such a CONWIP system was modeled by Hopp and Spearman (2001) using the mean-value analysis (MVA) technique (a closednet queuing model) MAXWIP The MAXWIP system we describe here is not the G-MaxWIP system described by Grosfeld-Nir and Magazine (2002) and mentioned in the literature review (section 3.2). 65

85 The MAXWIP we choose to model is similar to the CONWIP systems described in the previous subsection. However, in this system, the raw materials are not necessarily available at all times. The MAXWIP manufacturing system has a raw material arrival rate (i ar ) that comes from outside the system. The arrival rate might depend on an external supplier or a scheduling department that the manufacturing system has no control over. However, instead of allowing the arrival rate to enter the system directly (as in push systems), the arrival rate is directed to a release station with a raw material inventory (RMI). This is done to control the amount of WIP in the manufacturing system. If the system is started with a total WIP (WIP t ) that is less than the target level (WIP ), the release station will instantly pass enough raw materials to raise the total WIP to its target value provided that enough raw materials exist in the RMI. If the RMI does not have enough raw materials to raise the total WIP to its target value, it will immediately release whatever quantity available. If after start the total WIP is still below its target (which happens only if the RMI is empty), then the release station will pass whatever raw material arrival rate it gets. If the total WIP is equal to the target value, the release station will pass a raw materials input rate (i in,1 ) that is equal to throughput (TH) of the system so that the total WIP will remain at its target value. However, if the release station cannot sustain this input rate because it becomes empty, then it will pass whatever raw material arrival rate it gets. If the total WIP in the system is more than its target value, there will be no release rate (i.e., i in,1 = 0). This system has the same goal as a CONWIP system, to maintain the total WIP at a predetermined level. But this is a more realistic system since it allows for the possibility 66

86 that raw materials might not be available all times. Under the MAXWIP control scheme, we can model a manufacturing system while it is being drained. If the raw material arrival rate goes to zero, the system will simply consume all the raw materials in the RMI and then shut down. Notice that in MAXWIP, the target WIP level is a cap on the WIP, and that WIP might go below this cap, but this is not allowed in CONWIP models. Similar to CONWIP, models based on two different assumptions need to be considered. The first model is based on the assumption that demand on the final product might be less than the maximum capacity of the line. In this case, it is necessary to model a shipping station that includes a finished goods inventory (FGI). Since the throughput of this system is related to both the shipping rate and the raw materials release rate, we refer to this system as a release-shipping-coupled MAXWIP. The second model is based on the assumption that the demand on the final product always exceeds the maximum capacity rate of the line. In this case, it is not necessary to include a shipping station in the model since the throughput of the line is simply the output of the last station. The last station in this case is another push station that is authorized to work as fast as it can. Since the throughput of this system is related to the release rate (and not to the shipping rate), we refer to this system as a release-coupled MAXWIP. Except for the release station, release-shipping-coupled MAXWIP systems and release-coupled MAXWIP systems have the same graphical representations and equations as those of shipping-coupled CONWIP systems and isolated CONWIP systems respectively. The release station of a MAXWIP system can be represented by an ideal transistor in parallel with a capacitor. To allow for immediate release of raw materials, it will be 67

87 assumed that the transistor rated current is very high such that raw material (charge) can pass through it in zero time. In a manufacturing system, this means that the time needed to transfer materials from the RMI to the first station is negligible. This assumption is reasonable if the RMI is sufficiently close to the line. The arrival rate (i ar ) of raw materials to the release station is modeled as a current source coming from outside the system. The control signal that goes to the release station is similar to the one used in CONWIP systems. It comes from a voltage comparator and provides information about the difference between the current total WIP in the system and the target total WIP level. Figure 3.9 shows a single station release-shipping-coupled MAXWIP system. The equations of the shown release station can be written as: i ar : WIPt < WIP TH : WIPt = WIP, RMI 0 iin,1 (t) = (3.33) i ar : WIPt = WIP, RMI = 0 0 : WIPt > WIP i = i i. (3.34) ar in,1 + rmi drmi i rmi =. (3.35) dt The rest of the system equations are identical to those of a shipping-coupled CONWIP system. 68

88 i ar i cont,rmi α (WIP -WIP t ) Comparator i rmi Release Station RMI WIP - + i in,1 i buf,1 i cont,1 = i cmax Work Station WIP w - + i out,1 i fgi D Current Divider i cont,fgi α D Shipping Station FGI - + TH= i out,fgi Figure 3.9: Single Station Release-Shipping-Coupled MAXWIP System 69

89 For a release-coupled MAXWIP system, the release station will be the same as that described above. The rest of the stations will be identical to those of an isolated CONWIP system. Figure 3.10 shows a single work station release-coupled MAXWIP system i ar i rmi i cont,rmi α (WIP -WIP t ) Comparator WIP Release Station - + RMI i in,1 i buf,1 i cont,1 = i cmax Work Station WIP w - + TH= i out,1 Figure 3.10: Single Station Release-Coupled MAXWIP System 70

90 Notice that push and CONWIP systems are special (extreme) cases of MAXWIP systems. If the desired WIP level (WIP ) in a release-coupled MAXWIP system is allowed to be infinite, it becomes a push system. If the raw materials arrival rate in a release-coupled MAXWIP system is assumed to be infinite, it becomes an isolated CONWIP system. Finally, if the raw materials arrival rate in a release-shipping-coupled MAXWIP system is assumed to be infinite, it becomes a shipping-coupled CONWIP system. Another view of the MAXWIP system is to think of the arrival rate (i ar ) as the arrival of forecasted orders instead of raw materials, and to think of the RMI as a backlog where orders can accumulate. In this view of the MAXWIP, the release station is assumed to always have enough raw materials, but the backlog can go empty if there are no forecasted orders. Such a MAXWIP system can be modeled using the same graphical representation and equations used above Kanban For kanban systems, the goal is to limit the WIP in each station to a predetermined target level specified for each station (known as the station card count). For the first work station, the WIP is always maintained at its target level by a release mechanism capable of providing large amounts of raw materials over a very short time to instantaneously replenish any outgoing goods from the station. For kanban systems, it is usually assumed that there are enough raw materials in an RMI close enough to the line that any needed quantity of raw materials can be transferred to the line in virtually zero time. But since we cannot assume that a work station can process materials in zero time, maintaining the WIP in subsequent work stations (other than the first one) at the target levels all the time 71

91 cannot be guaranteed. Thus, for all stations other than the first, the WIP targets are caps on the WIP, but the actual WIP at any given time can go below these targets. Each work station in the kanban system we model receives information from the downstream station about the current WIP level in that downstream station. When the WIP in the downstream station is below its target level, the upstream station will work as fast as it can to raise the WIP in the downstream station to its target level. The speed at which the upstream station can work depends on the amount of WIP it has, its effective processing time, and the maximum number of units it can simultaneously handle. The last station of our kanban system is a shipping station that has a finished goods inventory (FGI) and receives information about the rate of external demand. The throughput of the line is the output rate of the shipping station, and it depends on the demand rate, the amount of finished goods in the FGI, the effective processing time of shipping, and the maximum number of units that can simultaneously be handled. The release station of this kanban system is modeled in the same way as the release station in a CONWIP system. However, the value of the control current of the release station depends on the amount of WIP and target level for the first work station only and not the whole line. Kanban work stations are modeled the same as work stations in push systems, but the control current will not always be set to its maximum value. Instead, the control current will come from a voltage comparator that compares the WIP (voltage) on the downstream buffer (capacitor) to its target value and issues a current that depends on the difference between them. The shipping station is modeled in the same way as the shipping station of a shipping-coupled CONWIP system. 72

92 i cont,rmi α (WIP,w -WIP w ) Comparator Release Station RMI WIP w WIP,w - + i in,1 i buf,1 i cont,1 α (FGI -FGI) Comparator Work Station WIP w FGI FGI - + i out,1 i fgi D Current Divider i cont,fgi α D Shipping Station FGI - + TH= i out,fgi Figure 3.11: Single Station Kanban System 73

93 Figure 3.11 shows a single work station kanban system. If we assume that the effective processing times of the work station and the shipping station are te, and te s respectively, the maximum number of units that they can simultaneously handle is k and k s respectively, and the target WIP levels for them are WIP,w and FGI respectively, then the basic equations of the shown system are: i out,1 : WIPw = WIP,w i in,1(t) = (3.36) 0 : WIPw > WIP,w min[wipw /te, k/te] : FGI < FGI i out,1 (t) = min[wipw /te, k/te, TH] : FGI = FGI (3.37) 0 : FGI > FGI min[d, FGI/tes ] : FGI k s TH(t) = i out,fgi (t) = (3.38) min[d, k s/tes ] : FGI > k s The system described in this subsection is often referred to as a one-card kanban system. This system assumes that the transfer time between any two stations is negligible. Once a station finishes working on a part, this part becomes immediately available for the next station to work on. In a two-card kanban system, the transfer process is dealt with as another station and is assigned a number of cards. Notice that a two-card kanban system can easily be modeled using the electrical analogs discussed here. We just need to model the transfer process as an additional process that has its own effective processing time and target WIP level. 74

94 Table 3.2 shows the system equations of push and a variety of pull control schemes for an N-work stations line Numerical Solutions Based on the electrical model, it is relatively easy to derive the system equations for any control scheme. The system equations can then be solved to get a quantitative description of the model. For example, the system equations for the five models discussed in this chapter have been programmed and numerically solved in MATLAB. The programs allow for any number of work stations. They also allow for stochastic processing times and demand rates. In the following examples, we consider balanced deterministic systems that consist of three work stations. All the work stations have the same following parameters: k 1 = k 2 = k 3 =1 unit and te 1 = te 2 =te 3 =1 minute. Whenever a shipping station exists, it is assumed to have k s =1 unit and te s =0.2 minute. All the systems are assumed to start empty and the solution shows the first three hours of operation. Figure 3.12 shows a numerical solution of a shipping-coupled CONWIP system. The demand rate is set at 0.9 unit/minute and the target total WIP is set at 8 units. The solution shows that after a transient period of about 1 hour, each of the three work stations has only 0.9 unit of WIP, and the rest of the WIP, 5.3 units, is accumulated at the shipping station as finished goods. Notice that the throughput of the system is equal to the demand rate (0.9 unit/minute), and not to the capacity of the system (1 unit/minute). Figure 3.13 shows the solution to an isolated CONWIP system with a target total WIP level of 8 units. In contrast with the shipping-coupled CONWIP, the throughput of this system depends only on the system parameters since the demand is assumed to be higher 75

95 Table 3.2: System Equations of Push and a Variety of Pull Control Schemes for an N-Work Station line Release Rate to Work Station 1 Output Rate of Work Station j Throughput Push i in,1(t) = i ar WIP (t) = k j/te /te : WIP k w, j j w, j j i TH(t) = i out, j out, N j : WIPw, j> k j Shipping-Coupled CONWIP i in,1 TH (t) = 0 : WIP = WIP t : WIP > WIP t i out, j WIP (t) = k j/te w, j j /te j : WIP : WIP w, j w, j k j > k j min[d, FGI/tes] TH(t) = min[d, ks/tes ] : FGI k : FGI > k s s Isolated CONWIP i in,1 TH (t) = 0 : WIP = WIP t : WIP > WIP t WIP (t) = k j/te /te : WIP k w, j j w, j j i TH(t) = i out, j out, N j : WIPw, j> k j 76 Release-Shipping-Coupled MAXWIP i in,1 iar TH (t) = iar 0 : WIP < WIP t : WIP = WIP, RMI 0 t : WIP = WIP, RMI = 0 t : WIP > WIP t i out, j WIP (t) = k j/te w, j j /te j : WIP : WIP w, j w, j k j > k j min[d, FGI/tes TH(t) = min[d, ks/tes ] : FGI k : FGI > k s s Release-Coupled MAXWIP i in,1 iar TH (t) = iar 0 : WIP < WIP t : WIP = WIP, RMI 0 t : WIP = WIP, RMI = 0 t : WIP > WIP t WIP (t) = k j/te /te : WIP k w, j j w, j j i TH(t) = i out, j out, N j : WIPw, j> k j Kanban i in,1 i (t) = 0 out,1 : WIP : WIP w,1 w,1 = WIP > WIP,1,1 i out, j min[wip (t) = min[wip 0 w, j w, j /te, k /te ] j /te, k /te, i j j j j j out,(j+ 1) : WIP ] : WIP : WIP w,(j+ 1) w,(j+ 1) w,(j+ 1) < WIP = WIP > WIP,(j+ 1),(j+ 1),(j+ 1) min[d, FGI/tes] TH(t) = min[d, ks/tes ] : FGI k : FGI > k s s

96 than the maximum system capacity. The solution shows that after a very short transient period, the throughput reaches its maximum value of 1 unit/minute. Notice that at steady state, stations two and three have 1 unit of WIP each, the rest of the WIP is kept at the first station. Figure 3.14 shows a numerical solution of a release-shipping-coupled MAXWIP system. The demand rate and raw materials arrival rate are both set at 0.9 unit/minute, the target total WIP is set at 8 units. Notice that since the system starts empty (including the RMI), the system cannot reach its target WIP level. Each station, including the shipping station, has only enough WIP to sustain a throughput of 0.9 unit/minute, and the RMI is empty. The total WIP in the system is 2.88 units. Figure 3.15 shows a numerical solution to a release-coupled MAXWIP system with a target total WIP level of 8 units. Similar to the previous system, each station has only enough WIP to sustain a throughput of 0.9 unit/minute, and the RMI is empty. However, notice that this system has slightly less WIP (2.7 units) since no finished goods are kept. Both MAXWIP systems behave like push systems in this case. However, both MAXWIP systems would behave like CONWIP systems under different condition, for example, higher raw materials arrival rate or different initial conditions. Figure 3.16 shows a numerical solution of a kanban system. The demand rate is set at 0.9 unit/minute and the target WIP level for each station, including the shipping station, is set at 2 units. The solution shows that after a transient period of about 45 minutes, each station reaches its target WIP level and the throughput is equal to the demand rate. Notice that the first station is full from the start time, while the other stations fill up in reverse order. However, different parameters and demand rates would produce different patterns. 77

97 Although all the above solutions are for deterministic cases, stochastic numerical solutions are easy to get using the same MATLAB programs. Instead of setting the effective processing times and demand and arrival rates to deterministic values, they can be random numbers drawn from a specific stochastic distribution. Figure 3.12: Numerical Solution of a Three Station Shipping-Coupled CONWIP System 78

98 Figure 3.13: Numerical Solution of a Three Station Isolated CONWIP System 79

99 Figure 3.14: Numerical Solution of a Three Station Release-Shipping-Coupled MAXWIP System 80

100 Figure 3.15: Numerical Solution of a Three Station Release-Coupled MAXWIP System 81

101 Figure 3.16: Numerical Solution of a Three Station Kanban System 3.5 DISCUSSION In a push control scheme, there is theoretically no need for real time information to be transferred through the system. All stations are already authorized to work at their full capacity whenever they have enough materials regardless of the state of other stations in the system. This simplifies the modeling process of push systems since only the flow of materials and not information need to be modeled. However, in pull systems the case is different. The rate at which a station is authorized to work might depend on the state of other station(s). This means that when describing a pull system, we not only need to specify the flow of materials, but also the flow of information. Furthermore, we also need 82

102 to describe the logical operations that are performed on this real time information. This of course, complicates the modeling of pull systems. We believe that the lack of a modeling technique that is capable of precisely describing the material flow, information flow, and logical operations is the main source of ambiguity often found in the literature of pull manufacturing systems. The modeling technique used in this chapter has the capability to describe pull systems by employing electrical circuits as a modeling tool. Electrical circuits provide simple but accurate tools to model both flows and logical operations. Electrical circuits also have the advantage of providing standard and explicit graphical representation for the control system being modeled. We demonstrated this technique by modeling five different pull systems. The same technique can be used to model any pull system. With the increasing number of control systems that are being introduced, it is important to identify the conditions under which one of these control systems may become superior to the other systems. Although many studies have been conducted to compare the performance of different control systems, many of the reported results are contradictory. We believe that the lack of a suitable modeling technique is a main reason for such contradictory results. In Chapter 4, we use the modeling technique discussed in this chapter to compare the performance of push, shipping-coupled CONWIP, and kanban systems. 3.6 CONCLUSION In this chapter, we have shown that the modeling technique proposed in Chapter 2 can accurately and sufficiently describe a variety of pull systems. The electrical analogs provide an excellent tool to model the control signals and paths of information flow in 83

103 pull systems. It should be noted here that although we did not cover all pull systems, the technique itself is capable of describing any control scheme. A closed-form solution for a single station deterministic CONWIP system is derived based on the basic equations of the system. The solution reveals some important parameters of the system that carry physical meaning. However, when attempting to get closed-form solutions for pull systems with multiple stations, the mathematical analysis becomes tedious and it becomes hard to see the physical meaning of these parameters. For multiple station pull systems we recommend solving the system equations numerically. The numerical solutions have the advantage of allowing for randomness in the system. The goal in this chapter was not to compare the performance of the different control schemes, but to model them graphically and derive their system equations. However, in Chapter 4, we use numerical solutions that are based on the system equations discussed in this chapter to compare push, shipping-coupled, and kanban control systems. 84

104 CHAPTER 4 PULL CONTROL SCHEMES IN MANUFACTURING SYSTEMS PART II: COMPARATIVE ANALYSIS 4.1 INTRODUCTION With the increasing number of production control systems that are being introduced, it is important to identify the conditions under which one of these control systems becomes superior to the others. Although many studies have been conducted to compare the performance of different control systems, many of the results reported are contradictory. It is also common to find studies that advocate one control system and consider it generally superior to all other systems. However, many of these studies do not provide enough information about the models and systems they compare to reproduce their results. In Chapter 3, we used a modeling technique previously proposed by the authors (Sader and Sorensen 2003) to model a variety of pull systems. It was shown that the proposed technique can sufficiently and accurately describe manufacturing systems including their control aspects. In this chapter, we use that technique to compare the relative performance of push, CONWIP, and kanban control systems. The comparison is conducted by writing MATLAB codes based on the system equations previously derived (see Chapter 3). These codes are used to simulate the operation of the systems under study. 85

105 After a brief literature review of some comparative studies, we describe the methodology of our study. The results of our simulations are then reported followed by a discussion and concluding remarks. 4.2 LITERATURE REVIEW Many authors have studied the relative performance of different manufacturing control schemes, often with contradictory results. In this section, we briefly focus on the three schemes that are commonly compared: Push, CONWIP, and kanban. Spearman and Zazanis (1992) study a CONWIP manufacturing line and compare it to push and kanban lines. The three lines are assumed to have the same number of stations with independent and exponentially distributed processing times. For the push line, the control variable is the arrival rate of raw materials, and it is assumed to be Poisson distributed (exponentially distributed inter-arrival times). For the CONWIP system, the control variable is the desired total WIP level. The control variables for the kanban system are the card counts at each of the processing stations. The CONWIP and kanban lines are assumed to have unlimited raw materials availability and enough demand. This means that the throughputs of the CONWIP and kanban lines are never limited by the lack of demand or the shortage of raw materials. The CONWIP and kanban lines are also assumed to have the same total maximum WIP. Using queuing theory analysis, Spearman and Zazanis (1992) conclude that under the above mentioned assumptions, a CONWIP system will have less WIP than an equivalent push system at the same throughput level. They also state that the throughput of an exponential kanban system will not exceed that of an equivalent CONWIP system, almost surely. 86

106 Based on simulation studies and the same assumptions used by Spearman and Zazanis (1992), Spearman et al. (1990, written after, but published before the Spearman and Zazanis 1992 paper) notice that the superiority of CONWIP systems over equivalent push systems is particularly true at high utilization levels and in lines where a distinct bottleneck station exists. Gstettner and Kuhn (1996) compare balanced CONWIP and kanban systems based on numerical solutions to queuing theory models. All the processing stations in both lines are assumed to have exponentially distributed processing times. It is also assumed that there is unlimited demand at the end of the lines and unlimited raw materials before the first station in each line. They notice that the card distribution in a kanban system has a significant effect on the performance of the system, and they suggest an algorithm to optimize the card distribution. They mention that by using a favorable card distribution in the kanban system, a given throughput can be reached with less WIP than that required by a CONWIP system. However, they notice that the total card count for the kanban system will be higher than that of the CONWIP system. They mention that this seems to contradict the findings of Spearman and Zazanis (1992), but they notice that Spearman and Zazanis assume that kanban has the same total card count as CONWIP and they did not examine the effect of different card distributions. Grosfeld-Nir and Magazine (2002) study manufacturing systems with exponentially distributed processing times and different production control schemes. For their push system, they assume that the release rate is a deterministic constant. For their CONWIP system, they assume that the demand rate is always higher than the maximum production capacity rate. They study the effect of the number of stations in the manufacturing line on 87

107 its relative performance. The performance measure they use is the amount of WIP necessary to reach a given throughput. Based on their simulations, they conclude that push systems outperform CONWIP systems as long as the line has more than three stations. They recognize that this contradicts the results obtained by Spearman and Zazanis (1992), but they point out to the difference in defining the arrival rate to the push system (deterministic versus Poisson) as a possible reason for the discrepancy. Huang et al. (1998) conducted a simulation study to compare the performance of push, CONWIP, and kanban systems in a cold rolling plant where the processing times are normally distributed with coefficients of variation of 0.2. Their results show that CONWIP is the most efficient of these systems based on different performance measures including the amount of WIP necessary for a given throughput. In a review paper, Framinan et al. (2003) review different contributions that have been made to the area of CONWIP comparison with other production control schemes. They point out to the lack of a common framework for comparison as a partial explanation of the often contradictory results found in the literature. They also state that the issues of CONWIP operation, application, and comparison are relevant subjects of further research. In addition to the lack of a common framework for comparison, many comparative studies define the systems that are being compared in ambiguous terms. Names like CONWIP and kanban are often used without precise descriptions of the specific systems considered. It is also common to find generalizations about the relative performance of different control schemes based on analysis of models with restrictive assumptions. 88

108 4.3 METHODOLOGY In this section, we describe the models and systems in this study. We also define the factors of interest, the control variables, the comparison framework, the simulation parameters, and the performance measure The Models We believe that one of the main reasons for the contradictory results found in the literature about the relative performance of production control schemes is the lack of a good way to accurately and sufficiently describe the systems that are being compared. This can cause different studies to appear contradictory to each other when they are actually studying different schemes. Since many definitions exist for terms like push and pull, and many variations have emerged for kanban and CONWIP, a comparative study should unambiguously describe the specific systems subject to study. In addition to describing the physical characteristics of a production line (i.e., number of stations, processing times, arrival rates, buffers, etc.), a good description of a manufacturing system should also include the control scheme of the line. The control scheme determines aspects like the raw materials release mechanism, the conditions under which processing stations are authorized to work and at what rate, the shipping mechanism of finished goods, and the flow of information in the system. In a previous paper (Sader and Sorensen 2003), we proposed a modeling technique for continuous manufacturing systems and discrete systems that can be approximated by continuous behavior. That modeling technique is based on analogies between electrical and manufacturing systems. We believe that the proposed technique provides an accurate and 89

109 sufficient description of manufacturing systems, and hence is used in this chapter to model the three systems that are being compared The Systems The three control schemes compared in this chapter are: Push, shipping-coupled CONWIP, and kanban. In the push system, the raw materials release mechanism releases raw materials in a rate that is based on real or estimated demand. This release rate is the arrival rate to the system. The raw materials become part of the total WIP in the system as soon as they are released. All the stations in a push system are authorized to work as fast as possible on any available materials. The push system studied in this chapter has three work stations and a shipping station and is shown in Figure 4.1. The shipping station in the push system works exactly like the other stations (as fast as possible). The output of the shipping station is the throughput of the line. The shipping-coupled CONWIP system is assumed to have infinite amount of raw materials available at the release station which has infinite capacity rate. Obviously, this infinite amount of raw materials is not considered part of the total WIP in the system since it has not been released yet. The amount of WIP in a CONWIP system is always constant at a desired value (WIP ). This desired value is known as the total card count. This can be achieved because the release mechanism is capable of releasing any raw materials required to keep the WIP at its desired level at all times. Once released to the first station, raw materials become part of the total WIP. Similar to the push system, all the work stations are authorized to work as fast as they can on any available materials. However, the shipping station can ship materials only if there is enough demand for them. 90

110 i ar i cont,1 = i cmax Work Station 1 WIP w,1 - + i cont,2 = i cmax Work Station 2 WIP w,2 - + i cont,3 = i cmax Work Station 3 WIP w,3 - + i cont,fgi = i cmax Shipping Station FGI TH - + Figure 4.1: Three Station Push System 91

111 i cont,rmi α (WIP -WIP t ) Release Station RMI Comparator WIP - + i in,1 i cont,1 = i cmax Work Station 1 WIP w,1 - + i cont,2 = i cmax Work Station 2 WIP w,2 - + i cont,3 = i cmax Work Station 3 WIP w,3 - + D Current Divider i cont,fgi α D Shipping Station FGI TH - + Figure 4.2: Three Station Shipping-Coupled CONWIP 92

112 i cont,rmi α (WIP,1 - WIP w,1 ) Release Station RMI Comparator WIP w,1 WIP,1 - + i in,1 i cont,1 α (WIP,2 - WIP w,2 ) Work Station 1 WIP w,1 Comparator - + WIP w,2 WIP,2 i cont,2 α (WIP,3 - WIP w,3 ) Comparator Work Station WIP w,2 WIP w,3 WIP,3 i cont,3 α (FGI - FGI) Work Station 3 WIP w,3 FGI Comparator FGI - + D Current Divider i cont,fgi α D Shipping Station FGI TH - + Figure 4.3: Three Station Kanban System 93

113 The shipping-coupled CONWIP we choose to study allows back orders. If the demand rate at time period t i exceeds the rate that the shipping station can provide, the excess demand is added to the demand for period t i+1. The output of the shipping station is the throughput of the line. The CONWIP system studied in this chapter has three work stations plus a shipping station and is shown in Figure 4.2. Each station in the kanban system has a maximum WIP level, known as the card count. The card count of station j is denoted as WIP,j. The release mechanism of the kanban system is similar to that of the shipping-coupled CONWIP system. However, instead of releasing enough raw materials to keep the total WIP level at a desired value, the kanban release mechanism keeps the WIP level in the first work station at a desired value (WIP,1 ). A work station in a kanban system is authorized to work on available materials only if the station downstream of it has a WIP level below its card count (WIP w,(j+1) < WIP,(j+1) ). The shipping station in the kanban system works exactly like that of the shipping-coupled CONWIP system with back orders. The kanban system studied in this chapter has three work stations plus a shipping station and is shown in Figure 4.3. Detailed descriptions of the Push, shipping-coupled CONWIP, and kanban systems can be found in Chapter The Factors of Interest Different factors can affect the performance of manufacturing systems, these include, but are not limited to, utilization level, number of stations, balanced or unbalanced capacities, bottleneck position (if unbalanced), variability levels in processing and demand rates, and other factors. If a factor is not of interest for a particular study, its 94

114 effect is isolated by keeping it at one level. For example, the number of stations in this study is not considered a factor of interest, and hence all the systems in this chapter have three work stations and a shipping station. The factors of interest and their levels in this study are as follows: 1) Balance of capacity rates. Two levels are considered. The first level is when the three work stations have the same capacity rate. The second level is when the middle work station is a bottleneck (has the lowest capacity rate) while the other two work stations have equal capacity rates. In both cases, the shipping station is assumed to have a higher capacity rate than the rest of the stations. 2) Utilization level. Two bottleneck utilization levels are considered: 90% and 95%. 3) Processing rates variability level. Four levels are considered for the coefficient of variation of the processing rates (C e ): 0.0, 0.17, 0.34, and ) Demand rate variability level. Four levels are considered for the coefficient of variation of the demand rate (C a ): 0.0, 0.17, 0.34, and The Control Variables For the push system (Figure 4.1), the control variable is the arrival rate (i ar ) measured in unit/minute. Different arrival rates produce different WIP and throughput (TH) levels. However, in a steady state push system, the average arrival rate is equal to the average throughput rate. For the shipping-coupled CONWIP system (Figure 4.2), the control variables are the total card count (WIP ) and the demand rate (D). For the kanban system (Figure 4.3), the control variables are the card counts at each station (WIP,1, WIP,2, WIP,3, and FGI ) and the demand rate (D). Different card counts and demand rates produce different WIP 95

115 and throughput levels. Notice that in a CONWIP system, the total card count is always equal to the total WIP in the system. In a kanban system, the card count of the first station is always equal to the WIP in that station, but this is not generally true for the other stations The Comparison Framework Using the system equations defined in Chapter 3, A MATLAB code was written to simulate each of the three mentioned control systems (Push, shipping-coupled CONWIP, and kanban). Each simulation run consists of 30 hours of simulated production time. Although the longest startup transient period noticed is less than 2 hours, the first 6 hours are dropped from the analysis to eliminate the effect of any startup transient response. For each simulation run, the three systems are simulated using the same set of random numbers, and the simulation results are based on 10 replications. Since there are two levels of balance, two bottleneck utilization levels, four processing rates variability levels, and four demand rate variability levels, a total of 2*2*4*4 = 64 combinations is needed. For each one of the 64 combinations, the three systems are simulated with the same demand and processing rates using the following procedure: 1) Set the parameters of the MATLAB code of the push system to reflect the desired combination of the factors of interest. 2) Run the push system and record the WIP levels at each station, the total WIP level, and the throughput. 3) For the CONWIP system, set the demand rate equal to the arrival rate of the push system in step 1. Use the same processing times used for the push system. 96

116 4) Run the CONWIP system with a small total card count (WIP ) and notice that the resulting throughput is less than the demand rate. 5) Slightly increase the card count of the CONWIP system and rerun the simulation. Notice that the throughput has increased. 6) Keep repeating step 5 until the throughput rate is equal to the demand rate (within %). Record the WIP levels at each station, the total WIP level, and the throughput. 7) For the kanban system, set the demand rate equal to the arrival rate of the push system in step 1. Use the same processing times used for the push system. 8) Using the recorded WIP values of the CONWIP system in step 6, calculate the WIP share of each station as a percentage of the total WIP. 9) Define the card count for each station in the kanban system in terms of its percentage of the total card count using the percentages calculated in step 8. WIP WIPw, j. WIP, j. WIP = (4.1), j CONWIP 10) Repeat steps 4, 5, and 6 for the kanban system instead of the CONWIP system. For each of the 64 combinations, the three systems have the same throughput, but they can have different total WIP levels. However, although the CONWIP and kanban systems may have different total WIP levels and different total card counts, the above procedure allocates card counts to each kanban station in the same proportion as that of the WIP in the CONWIP system. 97

117 4.3.6 The Simulation Parameters For the balanced lines, it is assumed that each one of the three work stations has an average capacity rate of 1 unit/minute. It is also assumed that the effective processing times are uniformly distributed. The four levels of variability (C e = 0.0, 0.17, 0.34, and 0.50) are for the processing rates, not the processing times. The capacity rate of the shipping station is assumed to be 1.5 unit/minute with a processing time that is also uniformly distributed. The three work stations and the shipping station are assumed to have the same variability level in their processing rates. The demand rate (arrival rate for the push system) is assumed to have an average of 0.90 part/minute or 0.95 part/minute depending on the required utilization level. In both cases, the demand rate is uniformly distributed with the required level of variability (C a = 0.0, 0.17, 0.34, or 0.50). For the unbalanced lines, the middle work station (the bottleneck) is assumed to have an average capacity rate of 1 unit/minute. The first and third work stations have an average capacity rate of 1.25 unit/minute. The shipping station has an average capacity rate of 1.5 unit/minute. The three work stations and the shipping station are assumed to have the same variability level in their processing rates. The demand rate (arrival rate for the push system) has an average of 0.90 unit/minute or 0.95 unit/minute depending on the required bottleneck utilization level. All the processing times and the demand rates are uniformly distributed. Table 4.1 shows the simulation parameters. Notice that the factors of interest (the first four rows) have multiple levels. The factors that are not of interest have only one level. 98

118 Table 4.1: Simulation Parameters Level 1 Level 2 Level 3 Level 4 C e C a Bottleneck Utilization (Demand Rate) Line Balance Bottleneck Capacity Rate Non-bottleneck Capacity Rate Number of Work Stations Balanced 1 unit/minute 1.25 unit/minute 3 Bottleneck at Work Station The Performance Measure Manufacturing systems can be compared based on different performance measures. For real-life manufacturing systems, the profitability of the system is the most meaningful measure of relative performance. However, using the profitability as a performance measure requires detailed information about the financial aspects of the specific production line being studied (e.g., inventory costs, unit cost, marginal profits, etc.). And even then, the results are valid only for that specific line. In this chapter, we define a modified inventory efficiency as our performance measure. Hopp and Spearman (2001) define the inventory efficiency of a production line as the ratio of the total inventory of an equivalent perfect system to the total inventory of the system being measured. They define the equivalent perfect system as a deterministic system with the same average throughput, average processing rates and average arrival 99

119 rate but without any variability. They define the total inventory of the system as the summation of the raw materials inventory (RMI), work-in-process (WIP), and the finished goods inventory (FGI). However, for our three control systems, the raw materials are not considered part of the manufacturing system until they are released to the first station, at that point we consider them as WIP and not raw materials. Also, since the systems we are comparing can have slightly different throughputs (within %), we include the throughput (TH) in our performance measure. We redefine the inventory efficiency as: TH IE = (4.2) WIP + FGI Notice that when comparing systems that have the same throughput levels, the difference in inventory efficiencies becomes a function of WIP and FGI only. Since the best possible performance of each of the three manufacturing systems considered in this chapter is achieved when there is no variability in the system (C e =0.0 and C a =0.0), we use the inventory efficiency of the deterministic case (IE d ) as a reference value to normalize the inventory efficiencies of the stochastic cases. This reference value (IE d ) is found to be the same for all three systems (see section 4.4). We define the normalized inventory efficiency (IE * ) of a system that has an inventory efficiency IE and whose equivalent deterministic system has an inventory efficiency IE d as: 100

120 IE IE * IE = (4.3) d Notice that if the system whose performance is being measured has the same throughput as that of its equivalent deterministic system, then IE * becomes the same as the inventory efficiency defined by Hopp and Spearman (2001). Also notice that IE * cannot be more than 1.0, and is equal to 1.0 for the deterministic case. 4.4 RESULTS Using the procedure mentioned in section 4.3.5, the 64 combinations have been simulated for the push, shipping-coupled CONWIP, and kanban systems. However, after conducting few simulation runs, it became obvious that the kanban system is not given the chance to get to its best possible performance level. Because our kanban system is forced to have a card count distribution that is based on the WIP distribution of the CONWIP system, the kanban performance we get from our simulation does not represent the best possible performance that the kanban system can reach. This was verified by trying different card distributions for the same kanban system. It was noticed that some arbitrary card distributions produced better performance (higher IE * ) than that of an equivalent kanban system whose card distribution was generated based on the WIP distribution of the CONWIP system. (See Table 4.2) 101

121 Table 4.2: Kanban Based on CONWIP versus Kanban with Arbitrary Distribution Kanban Kanban CONWIP (Based on CONWIP) (Arbitrary Distribution) WIP Card Count WIP Card Count WIP Work Station Work Station Work Station Shipping station Total TH IE IE* Although we did not expect the procedure mentioned in section to allow the kanban system to run at its optimum performance, we did expect it to produce good performance. But since the results of the simulation showed that this assumption was not valid, it would be meaningless to include the kanban results in our comparative analysis. As performed, the study does not us tell if kanban is better or worse than CONWIP. Therefore, no further results on the kanban system are presented. The results of the 64 combinations for the push and shipping-coupled CONWIP systems are shown in Appendix C. Using these results, we plot the normalized inventory efficiency (IE * ) as a function of the coefficient of variation of the demand rate (C a ) for each combination of utilization levels, balance levels, and processing rates variability levels. The plots are shown in Figures 4.4 through 4.7. Notice in the figures that when the demand rate and processing rates are all deterministic (C e =0.0 and C a =0.0), the CONWIP system and the push system (and the kanban system, although not shown) have the same performance, IE * =1.0. Figure 4.4 shows the normalized inventory efficiency (IE * ) of the push and CONWIP systems as a function of C a for the balanced lines at a utilization level of 90% and four 102

122 levels of C e. The figure shows that with the deterministic processing rate (C e =0.0), the CONWIP system is superior to the push system at high variability demand rates. However, with the stochastic processing rates (C e =0.17, 0.34, and 0.50), the push system is the superior system. The magnitude of this superiority changes with C e and C a in a fashion that suggests the existence of a coupling effect between C a and C e on the relative performance. By coupling effect, we mean that the degree to which one of the two factors will affect the performance depends on the value of the other factor. For example, notice that when C e = 0.17, the effect of changing C a on the performance of both systems is more pronounced than the effect of changing C a when C e is equal to 0.0 or Figure 4.5 shows the normalized inventory efficiency (IE * ) of the push and CONWIP systems as a function of C a for the balanced lines at a utilization level of 95% and four levels of C e. The figure shows that with deterministic or low variability processing rates (C e = 0.0 or 0.17), the CONWIP system is superior to the push system, especially at high C a. But with higher processing rates variability (C e = 0.34 and 0.50), both systems have very close performance levels. This figure also suggests the existence of a coupling effect between C a and C e on the relative performance. In Figures 4.4 and 4.5, the deteriorating effect of variability on the performance of both the push and CONWIP systems is very clear. At the 90% utilization level, when C e = C a =0.5, the performance of both systems drop by about 60% compared to their performance in the deterministic case. At the 95% utilization level, the deteriorating effect of high variability (C e = C a =0.5), is even more severe, with a drop of about 75% in the performance of both systems compared to their performance in the deterministic case. 103

123 C e =0.0, u=0.90, balanced C e =0.17,u=.90, balanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a 104 C e =0.34, u=.90, balanced C e =0.50, u=0.90, balanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a Figure 4.4: Normalized Inventory Efficiency for Balanced Lines with 90% Utilization

124 C e =0.0, u=.95, balanced C e =0.17, u=0.95, balanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a 105 C e =0.34, u=.95, balanced C e =0.50, u=0.95, balanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a Figure 4.5: Normalized Inventory Efficiency for Balanced Lines with 95% Utilization

125 C e =0.0, u=0.90, unbalanced C e =0.17,u=.90, unbalanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a 106 C e =0.34, u=.90, unbalanced C e =0.50, u=0.90, unbalanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a Figure 4.6: Normalized Inventory Efficiency for Unbalanced Lines with 90% Utilization

126 C e =0.0, u=.95, unbalanced C e =0.17, u=0.95, unbalanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a 107 C e =0.34, u=.95, unbalanced C e =0.50, u=0.95, unbalanced IE * Push CONWIP 0.80 IE * Push CONWIP C a C a Figure 4.7: Normalized Inventory Efficiency for Unbalanced Lines with 95% Utilization

127 Figure 4.6 shows the normalized inventory efficiency (IE * ) of the push and CONWIP systems as a function of C a for the unbalanced lines at a utilization level of 90% and four levels of C e. Figure 4.7 shows the normalized inventory efficiency (IE * ) of the push and CONWIP systems as a function of C a for the unbalanced lines at a utilization level of 95% and four levels of C e. Both figures show that the CONWIP system is superior to push system, especially at the higher utilization level. In addition, Figures 4.6 and 4.7 also show some coupling effects between C a and C e. Although the deteriorating effect of variability on the performance of both systems is clear in Figures 4.4 through 4.7, it can be seen that the deteriorating effect of variability is more severe at the higher level of utilization in both the balanced and unbalanced lines. Figures 4.4 through 4.7 also show that the deteriorating effect of increasing C e while keeping C a at zero is more severe than the effect of increasing C a while keeping C e at zero. To better see the effect of C a and C e on the relative performance of the push and CONWIP systems, we generate contour plots that show the ratio of the performance of the push system to that of the CONWIP system as it changes with C a and C e. Figures 4.8 through 4.11 show smoothed contours of (IE * p/ie * c), where IE * p is the normalized inventory efficiency of the pull system and IE * c is the normalized inventory efficiency of the CONWIP system. Notice that IE * p/ie * c is a measure of relative and not absolute performance. An improved relative performance of one of the systems does not mean that its absolute performance has improved, it just means that its performance compared to the performance of the other system has improved. A value of (IE * p/ie * c) that is less than 1.0 indicates that the CONWIP system is superior to the push system, while a value of 108

128 (IE * p/ie * c) that is more than 1.0 indicates that the push system is superior to the CONWIP system. Figure 4.8 shows contours of (IE * p/ie * c) for the balanced lines at a utilization level of 90%. The figure clearly shows that there is a coupling effect between C a and C e on the relative performance of the systems. At the point (C e =0.34,C a =0.5), the push system has the most pronounced superiority over the CONWIP system with IE * p/ie * c value of But as we move away from this point in any direction within the area (C e 0.5, C a 0.5), the relative performance of the CONWIP system improves. For most of the C e, C a combinations, the push system remains superior to the CONWIP system, however, at low levels of C e and high levels of C a, the CONWIP system becomes superior to the push system with IE * p/ie * c values between 0.85 and Figure 4.9 shows contours of (IE * p/ie * c) for the balanced lines at a utilization level of 95%. The figure shows that the push system has its best relative performance at high levels of C e and low levels of C a where its performance is slightly better than that of the CONWIP system. The CONWIP system has its best relative performance at low levels of C e and high levels of C a, and in that area, its performance is significantly better than that of the push system. Figure 4.10 shows contours of (IE * p/ie * c) for the unbalanced lines at a utilization level of 90%. The figure shows that the push and CONWIP systems have close performance levels with some advantage for the CONWIP system. The advantage of the CONWIP system is more pronounced at high levels of C a. 109

129 Figure 4.8: Normalized Inventory Efficiency Contours for Balanced Lines with 90% Utilization 110

130 Figure 4.9: Normalized Inventory Efficiency Contours for Balanced Lines with 95% Utilization 111

131 Figure 4.10: Normalized Inventory Efficiency Contours for Unbalanced Lines with 90% Utilization 112

132 Figure 4.11: Normalized Inventory Efficiency Contours for Unbalanced Lines with 95% Utilization Figure 4.11 shows contours of (IE * p/ie * c) for the unbalanced lines at a utilization level of 95%. The figure shows that the CONWIP system is superior to the push system at all levels of variability (except for the deterministic case, where they have equal performance). The superiority of the CONWIP system is very significant at high levels of C a and low levels of C e. 113

133 Figures 4.8 through 4.11 suggest that there is a coupling effect between all the factors, and not only C a and C e. We can summarize the results of our simulation as follows: 1) In the kanban system, the card count distribution (and not only the total card count) has a very significant effect on the performance of the system. Using a card count distribution that is based on the WIP distribution of a CONWIP system with the same parameters does not produce good performance. 2) The relative performance of the push and CONWIP systems changes with the utilization level, balance level, and variability in processing and demand rates. At some combinations of these factors, the push system is superior to the CONWIP system, while at other combinations, the CONWIP system is the superior system. 3) The factors mentioned in number 2 above affect the relative performance of push and CONWIP systems in a fashion that suggests the existence of a coupling effect between the factors. The following are general trends that are noticed (although some exceptions exist due to the coupling effect): a. The push system tends to have its best relative performance at high variability levels in the processing rates and low variability levels in the demand rate. b. The CONWIP system tends to have its best relative performance at high variability levels in the demand rate and low variability levels in the processing rates. 114

134 c. Higher utilization levels tend to improve the relative performance of the CONWIP system. d. The existence of a distinct bottleneck tends to improve the relative performance of the CONWIP system. 4) For deterministic systems, push, CONWIP, and kanban systems have the same performance, provided that the total card count of the CONWIP system and the card count distribution of the kanban system are properly chosen. 5) The deteriorating effect of variability on the performance of our push and CONWIP systems increases with increasing the utilization level. 6) Adding variability in effective processing rates to an initially deterministic system degrades performance more than adding equivalent variability in the demand rate. 4.5 DISCUSSION We believe that ambiguously defined systems and improper generalizations are main reasons for the contradictory results found in the literature regarding the relative performance of different production control systems. By using the modeling technique proposed by these authors in a previous paper, and by offering a detailed description of our systems, parameters, framework, and measure of performance, we hope to put our simulation results in context with the assumptions behind them and make them reproducible. Although our kanban system results are not valid for comparative analysis and hence are not reported in this dissertation, they are still useful. Our simulation verifies the results of Gstettner and Kuhn (1996) about the importance of the card count distribution 115

135 in kanban systems. A meaningful comparative study of kanban systems must be based on optimized card count distribution. It is common in the literature to find studies advocating CONWIP as a superior control system to push or push as a superior control system to CONWIP. However, our results show that such generalizations are not proper. A change in the utilization level, balance level, demand rate variability level, or processing rate variability level can change the relative performance of the two systems, and in many cases can even cause the system that was superior before the change to become the inferior system. Any attempt to establish that one control system is universally superior to the other system is an improper generalization. Since it is uncommon to find two real production lines with the exact same parameters, the choice of control system for a production line should be based on the parameters of the specific line under consideration. Furthermore, since the parameters of a production line can change over time, a comparative analysis should include the expected change in the relative performance of the control systems if one or more of the parameters change. Although we are not ready at this point to offer a formal explanation for the coupling effects noticed between the different factors on the relative performance of push and CONWIP systems, some of the noticed trends can (at least partially) be explained. For example, imagine a two station production line with high variability in the processing rates and low variability demand rate that is being controlled as a CONWIP system. Because of the high variability in the processing rates, on occasion, the second station has a low processing rate while the first station has a high processing rate. In this case, the 116

136 WIP accumulates in front of the second station. And because the amount of total WIP in the system is limited, the first station is partially blocked (does not have enough materials to work at full capacity). Subsequently, if the second station has a high processing rate, but the first station has a low processing rate, the second station becomes partially starved after a while, and the WIP accumulates in front of the first station. The reason for the starvation of the second station is the fact that the first station was blocked in the first time period and was not allowed to work ahead. The result of this scenario is lost production time that the system cannot make up for. Notice that if the same line is controlled as a push system and the same scenario happens, the first station is not blocked in the first time period because there is no limit on how much WIP can be in the system. And hence, the second station is not starved in the second time period, and the lost production time is minimized. This may explain the tendency of a push system to have its best relative performance (compared to CONWIP) at high variability levels in the processing rates and low variability levels in the demand rate. To understand the tendency of a CONWIP system to have its best relative performance (compared to a push system) at high variability levels in the demand rate and low variability levels in the processing rates, consider the following example. Imagine a production line with high variability in the demand rate and low variability processing rates that is being controlled as a push system. Because of the high variability in the demand (arrival) rate, on occasion, the arrival rate is low. In this case, the whole system is starved and drained (no WIP is left in the system). Subsequently, if the arrival rate is high, the raw materials still have to go through the entire empty line before they 117

137 can be shipped as finished goods. The result of this scenario is lost production time that the system cannot make up for. Notice that if the same line is controlled as a CONWIP system and the same scenario happens, the line is not drained because the line always keeps the same amount of WIP, and thus can work ahead, even if the demand rate is low. And hence, as soon as the demand rate is high again, the system is able to give throughput, and the lost production time is minimized. The noticed tendency of a CONWIP system to have better relative performance (compared to a push system) at high utilization levels, and in lines where a distinct bottleneck exists agrees with the findings of Spearman and Zazanis (1992). They mention the ability of CONWIP systems to work ahead (because they cannot become starved) and to accumulate WIP where it is needed as possible explanations for these tendencies. For the deterministic systems (C e =0.0 and C a =0.0), our results show that push, CONWIP, and kanban systems have the same performance. This is an expected result. Any steady state deterministic push system will only keep the minimum amount of WIP necessary to meet the arrival rate, (notice that for steady state to exist, the demand rate cannot be greater than the capacity rate of any station in the system). The performance of such a push system can be matched by a CONWIP system with a total card count that is equal to the total WIP in the push system. The same performance can also be matched by a kanban system with a card count total and distribution that is equal to the WIP distribution in the push system. Notice that in Figures 4.8 through 4.11 we are able to offer partial explanations for the behavior noticed around the upper left corners (low C e s and high C a ), lower left 118

138 corners (low C e s and low C a ), and lower right corners (high C e s and low C a ). However, around the upper right corners (high C e s and high C a ), we do not offer any explanation. The coupling effects around those corners complicate the behavior and in many cases, make it counterintuitive. Our simulation results show that higher levels of utilization increase the negative effect of variability on the performance of manufacturing systems. This is in agreement with queuing theory solutions. See Hopp and Spearman (2001) for more details. The simulation results also show that adding variability in effective processing rates to an initially deterministic system degrades performance more than adding equivalent variability in the demand rate. We believe that this result cannot be generalized, and that it is best explained based on the concept of variability propagation. In systems with deterministic processing rates, any variability in the demand rate does not have the chance to grow through the system, and all stations see the same level of variability in their arrival rates. However, if there is some variability in the processing rates, then each station sees more variability in its arrival rate than the station upstream of it, even if the first station in the line sees no variability at all. The effect of this propagation of variability depends on the number of stations in the line and the specific levels of variability. 4.6 CONCLUDING REMARKS When comparing the performance of different production control systems, it is important to provide clear descriptions of the models, systems, parameters, assumptions, and performance measures that are used. The lack of such descriptions and the tendency 119

139 to make improper generalizations about production control systems are main reasons for the contradictory results often found in the literature. The card count distribution (and not only the total card count) significantly affects the performance of kanban systems. A meaningful comparative study of kanban systems must be based on optimized card count distribution. The relative performance of push and CONWIP systems changes with the utilization level, balance level, and variability level in demand and processing rates. At some combinations of these factors, the push system is superior to the CONWIP system, at other combinations, the CONWIP system is the superior system. Any attempt to establish that one control system is universally superior to the other system is an improper generalization. For deterministic systems, push and CONWIP systems have the same performance, provided that the total card count of the CONWIP system is properly chosen. For stochastic systems, the existence of coupling effects between different factors complicates the relative behavior of the systems and in many cases, makes it counterintuitive. However, some general trends are noticed. For example, a push system tends to have its best relative performance (compared to CONWIP) at high variability levels in the processing rates and low variability levels in the demand rate, while a CONWIP system tends to have its best relative performance (compared to push) at high variability levels in the demand rate and low variability levels in the processing rates. It is also noticed that a CONWIP system tends to have higher relative performance at high utilization levels and in systems where a distinct bottleneck exists. 120

140 In this chapter, we studied the relative performance of push and shipping-coupled CONWIP systems over a limited range of factors. Lines with different control systems, more variability levels, more utilization levels, different stochastic distributions, and more stations are subjects for future research. 121

141

142 CHAPTER 5 CONCLUSION The modeling technique developed in this dissertation has the capability to sufficiently and accurately describe continuous manufacturing systems (or discrete systems that can be approximated by continuous behavior). The technique provides standard graphical representations and governing equations to describe both the steady state and transient responses of manufacturing systems. For deterministic systems, these equations can be solved to get closed-form solutions. For stochastic systems, numerical solutions can be obtained for any probabilistic distribution. The developed technique defines building blocks to describe the three main components of a manufacturing system: the release mechanism, the work stations, and the shipping mechanism. By defining these three components, any manufacturing system can be modeled. In this dissertation, four release mechanisms (for push, CONWIP, MAXWIP, and kanban), two work stations (one for kanban and one for all other systems), and two shipping mechanisms (one for systems where demand might be less than the capacity of the line and one for systems where demand is always larger than the capacity of the line) were discussed. Other variations of these components can also be modeled using the same technique. In order to model a manufacturing system using the proposed technique, the modeler must answer specific questions regarding the above three components before a model can 123

143 be built. This results in a model that is not only capable of explicitly describing the control scheme of a manufacturing system, but actually requires such description. The technique s requirement to explicitly describe the control aspects of manufacturing systems is especially useful for pull production control schemes where qualitative descriptions often found in the literature can be ambiguous. As opposed to push systems, real time information and continuous logical operations are necessary for pull systems. Electrical circuits lend themselves to include these control aspects in the models of manufacturing systems. When comparing the performance of different production control systems, it is important to provide clear descriptions of the models, systems, parameters, assumptions, and performance measures that are used. The lack of such descriptions and the tendency to make improper generalizations about production control systems are main reasons for the contradictory results often found in the literature. The modeling technique developed in this dissertation has overcome the problem of poor descriptions of the systems; and the comparative analysis has avoided generalizations. The results of the comparative study conducted in this dissertation show that card count distribution (and not only the total card count) significantly affects the performance of kanban systems. A meaningful comparative study of kanban systems must be based on optimized card count distribution. The results also show that the relative performance of push and CONWIP systems changes with the utilization level, balance level, and variability level in demand and processing rates. At some combinations of these factors, the push system is superior to the CONWIP system, at other combinations, the CONWIP 124

144 system is the superior system. Any attempt to establish that one control system is universally superior to the other system is an improper generalization. For deterministic systems, push and CONWIP systems are found to have the same performance, provided that the total card count of the CONWIP system is properly chosen. For stochastic systems, the existence of coupling effects between different factors complicates the relative behavior of the systems and in many cases, made it counterintuitive. However, some general trends are noticed. For example, a push system tends to have its best relative performance (compared to CONWIP) at high variability levels in the processing rates and low variability levels in the demand rate; while a CONWIP system tends to have its best relative performance (compared to push) at high variability levels in the demand rate and low variability levels in the processing rates. It is also noticed that a CONWIP system tends to have higher relative performance at high utilization levels and in systems where a distinct bottleneck exists. In this dissertation, models of production control schemes in manufacturing systems were developed. The modeling technique was demonstrated by modeling push, and a variety of pull systems. The developed models were also used to compare push, CONWIP, and kanban production control schemes. Future research can include issues like: modeling other production control schemes, studying kanban systems with optimized card count distributions, comparing other production control schemes over expanded variability ranges, studying different approaches of handling variability in manufacturing systems, and other issues. 125

145

146 REFERENCES Alvarez-Vergas, R., Y. Dallery and R. David, A study of the continuous flow model of production lines with unreliable machines and finite buffers, Journal of manufacturing systems, 1994, v 13, n 3, pp Baines, T. S., and D. K. Harrison, An opportunity for system dynamics in manufacturing system modeling, Production Planning and Control, 1999, 10 (6): Besombes, B., and E. Marcon, Bond-graphs for modelling of manufacturing systems, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, 1993, 3: Brandimarte, P., A. Sharifnia and B. F. Turkovich, Continuous flow models of manufacturing systems: A review, Annals of the CIRP, 1996, v 45, n 1, pp Chen, H., and D. D. Yao, A fluid model for systems with random disruptions, Operations Research, 1992, v 40, supp. n 2, pp S239-S247. de Souza, R., R. Huynh, M. Chandrashekar, and D. Thevenard, A comparison of modelling paradigms for manufacturing line, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, 1996, Part 2 (of 4), pp Ferney, M., Modelling and controlling product manufacturing systems using bondgraphs and state equations: Continuous systems and discrete systems which can be represented by continuous models, Production Planning and Control, 2000, v 11, n 1, pp

147 Forrester, J. W., Industrial dynamics- a major breakthrough for decision makers, Harvard Business Review, 1958, v 36, pp Framinan, J. M., P. L. Gonzalez and R. Ruiz-Usano, The CONWIP production control system: review and research issues, Production Planning and Control, 2003, v 14, n 3, pp Grosfeld-Nir, A., M. Magazine and A. Vanberkel, Push and pull strategies for controlling multistage production systems, International Journal of Production Research, 2000, v 38, n 11, pp Grosfeld-Nir, A., and M. Magazine, Gated MAXwip: a strategy for controlling multistage production systems, International Journal of Production Research, 2002, v 40, n 11, pp Gstettner, S., and H. Kuhn, Analysis of production control systems kanban and CONWIP, International Journal of Production Research, 1996, v 34, n 11, pp Hopp, W. J., and M. L. Spearman, Throughput of a constant work in process manufacturing line subject to failures,, International Journal of Production Research, 1991, v 29, n 29, pp Hopp, W. J., and M. L. Spearman, Factory physics, Irwin/McGraw-Hill, Huang, M., D. Wang and W. H. IP, A simulation and comparative study of the CONWIP, kanban and MRP production control systems in a cold rolling plant, Production Planning and Control, 1998, v 9 n 8 pp

148 Klingener, J. F., Combined discrete-continuous simulation models in promodel for windows, Proceedings of the 1995 Winter Simulation Conference, 1995, pp Kuo, S. S., E. J. Chen, P. L. Selikson and Y. M. Lee, Modeling continuous flow with discrete-vent simulation, Proceedings of the 2001 Winter Simulation Conference, 2001, v 2, pp Law, A. M., and M. G. McComas, Simulation of manufacturing systems, Proceedings of the 1999 Winter Simulation Conference, 1999, v 1, pp Marek, R. P., D. A. Elkins and D. R. Smith, Understanding the fundamentals of kanban and CONWIP pull systems using simulation, Proceedings of the 2001 Winter Simulation Conference, 2001, pp Monden, Y., Toyota production system: an integrated approach to just-in-time, Engineering and Management Press, Paul, C. R., S.A. Nasar, and L.E. Unnewehr, Introduction to electrical engineering, McGraw-Hill, Robinson, J. A., A manufacturing paradigm that combines discrete and continuous methods: Part 2, Production and Inventory Management Journal, 1998, v 39, n 4, pp Sader, B. H., and C. D. Sorensen, Deterministic and stochastic dynamic modeling of continuous manufacturing systems using analogies to electrical systems, Proceedings of the 2003 Winter Simulation Conference, 2003, pp

149 Salama, Y., and M. O. Hongler, Continuous versus discrete flow of parts in a production dipole. Exact transient analysis, IEEE Symposium on Emerging Technologies and Factory Automation, 1995, v 3, pp Sargeant, R. G., Validation and verification of simulation models, Proceedings of the 1992 Winter Simulation Conference, 1992, pp Spearman, M. L., W. J. Hopp and D. L. Woodruff, A hierarchical control architecture for constant work-in-process (CONWIP) production systems, Journal of Manufacturing and Operations Management, 1989, n2, pp Spearman, M. L., D. L. Woodruff and W. J. Hopp, CONWIP: a pull alternative to kanban, International Journal of Production Research, 1990, v 28, n 5, pp Spearman, M. L., and M. A. Zazanis, Push and pull production systems: issues and comparisons, Operations Research, 1992, v40, n3, pp Valentin, C., and P. Ladet, Flow modelling in a class of hybrid (continuous-discrete) systems, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, 1993, v 3, pp van der Zee, Durk-Jouke, Modeling control in manufacturing systems, Proceedings of the 2003 Winter Simulation Conference, 2003, pp Yerlan, S., and B. Tan, A station model for continuous materials flow production systems, International Journal of Production Research, 1997, v 35, n 9, pp

150 APPENDIX 131

151

152 APPENDIX A CLOSED-FORM SOLUTION FOR THE N-STATIONS PUSH SYSTEM DISCUSSED IN SECTION 2.4 In this appendix, an induction proof is presented to show that, for an initially empty, single-product, push, continuous manufacturing system that consists of N stations in series with infinite storage capacity, constant effective processing times, and a constant arrival rate (i ar ) where te 1 = te 2 = = te N = te. (A.1) and u 1 1, u 2 1,, u N 1. (A.2) the behavior of the system is described by: k 1 j t/te t WIP (t) = i arte (1 e ). k N (A.3) j j!te k j= 0 133

153 and i k 1 j t/te t (t) = i ar (1 e ). k N (A.4) j j!te out, k j= 0 For the first station in the system (k=1), i in,1 = i ar. (A5) and the governing equation is: dwip dt 1 WIP1 + = i ar. (A6) te If the station is initially empty (WIP 1 (0)=0), Equation (A6) can be solved to get: t/te WIP (t) = i te(1 e ). (A7) 1 ar and so, i out,1 WIP1 t/te (t) = = i ar (1 e ). (A8) te So, for k=1, Equations (A3) and (A4) are true. 134

154 Now, we show that if Equations (A3) and (A4) are true for k=m, where m<n, they must also be true for k=m+1. For k=m, WIP m m 1 j t/te t (t) i arte (1 e = ). (A9) j j!te j= 0 and, i out, m m 1 j WIPm t/te t (t) i arte (1 e = = ). (A10) j te j!te j= 0 The governing equation for station m+1 is: m 1 j dwipm 1 WIPm 1 t/te t iin,m 1 i out,m i arte (1 e = + = = ). (A11) j dt te j!te j= 0 which can be solved to get: WIP m j t/te t (t) = i arte (1 e ). (A12) j j!te m+ 1 j= 0 135

155 and so, i m j WIPm = 1 t/te t (t) = = i ar (1 e ). (A13) j te j!te out, (m+ 1) j= 0 So, Equations (A3) and (A4) are true for k=1; and if they are true for k=m, then they must also be true for k=m+1. This proves that Equations (A3) and (A4) are true for any k N. 136

156 APPENDIX B CLOSED-FORM SOLUTION FOR THE SHIPPING-COUPLED CONWIP SYSTEM SHOWN IN FIGURE 3.5 A single station shipping-coupled CONWIP system is shown in Figure 3.5. The total WIP (WIP t ) in this system is composed of the WIP in the work station (WIP w ) plus the finished goods in the shipping station (FGI). Thus: WIP = WIP FG I. (B1) t w + If the total WIP in the system (WIP t ) is less than the target WIP (WIP ), the release station will instantly release enough raw materials to the first station to raise the total WIP to its target value. This means that even if the total WIP in the system is below the target immediately before start ( t = 0 - ), the total WIP will be equal to the target immediately after start ( t = 0 + ). If the total WIP is more than the target value, the release station will not pass any raw materials. Finally, if the total WIP is equal to the target value, the release station will pass a raw materials rate that is equal to throughput (TH) of the system so that the total WIP will remain at its target value. 137

157 So, we can now define the release rate (i in,1 ) as: i TH (t) = 0 : WIP = WIP t in,1 (B2) : WIP t > WIP Let the effective processing time of the work station be te and assume that the machine (or parallel machines) can simultaneously handle a maximum of k units. Noticing that the work station is essentially a push station, i.e., always authorized to work as fast as possible, the output rate (i out,1 ) of the work station can be defined as: i WIP (t) = k/te /te : WIP k w w out,1 (B3) : WIP w > k The shipping station will ship finished goods at a rate that depends on the external demand rate (D) and the amount of finished goods available in the FGI. Since the output rate of the shipping station (i out,fgi ) is the throughput of the line (TH), it cannot exceed the demand rate (D). Thus we can write: min[d, FGI/tes ] : FGI k s TH(t) = i out,fgi (t) = (B4) min[d, k s/tes ] : FGI > k s where min is the minimum of the quantities in the brackets. 138

158 Since i buf,1 and i fgi are the rates of change of WIP w and FGI respectively, we can write: dwip w i buf,1 =. (B5) dt dfgi i fgi =. (B6) dt Applying Kirchoff s current law (Paul et al. 1992) gives: i = i i. (B7) in,1 buf,1 + out,1 i = i i. (B8) out,1 fgi + out, fgi We solve Equations (B1) through (B8) to get a closed-form solution subject to the following assumptions: 1) Demand and effective processing times are deterministic constants. 2) The system starts with all WIP t at the work station, which is consistent with starting up an empty system by releasing WIP t units of raw materials just before startup. 3) The demand rate (D) is less than the maximum capacity rates of both the work station and the shipping station (i.e., D < (k/te) and D < (k s / te s )). 139

159 4) The WIP target level (WIP ) is greater than the total number of units that can simultaneously be handled in the work station and the shipping station (i.e., WIP > (k + k s )). At time t = 0, the initial conditions will be as follows: WIP w (0) = WIP. (B9) FGI(0) = 0. (B10) Applying these initial conditions to equations (B1) through (B8), we obtain the following governing equation: dfgi FGI k + =. (B11) dt te te s Equation (B11) can be solved to get: t tes s FGI(t) = k (1 e te ). (B12) te Using Equations (B1) and (B4), we can also get: 140

160 WIP w t tes s (t) = WIP k (1 e te ). (B13) te t te k s TH(t) = (1 e ). (B14) te Equations (B11) through (B14) are valid only as long as: WIP w > k, FGI/te s <D, and FGI<k s. If any condition of these three conditions becomes invalid, a new solution needs to be found. We can define a critical time associated with each one of the three conditions to examine which one of them (if any) will first become invalid. Critical time t cw is the time at which WIP w =k, critical time t cd is the time when D = FGI/te s, and the third critical time t cf is the time when FGI=k s. Using Equations (B12), (B13), and (B14), it can be shown that: t cw = te ln(1 α). (B15) s t cd = te ln (1 β). (B16) s t cf = te ln (1 γ). (B17) s where: α = (WIP k)/te k/te s. (B18) 141

161 D β =. (B19) k/te k s/tes γ =. (B20) k/te Note that for t cw, t cd, and t cf to exist (as real values), α, β, and γ respectively must be less than 1. Using assumptions 3 and 4, we can show that: α > γ > β t > t t. (B21) cw cf > cd β < 1 t exists. (B22) cd So, at time t = t cd Equations (B11) through (B14) become invalid. At that time, a new governing equation needs to be derived and solved. It can be shown that after t = t cd, the governing equation becomes: dfgi k = - D. (B23) dt te which can be solved to obtain: k FGI(t) = ( D)(t + tes ln (1 β)) + Dtes. (B24) te 142

162 WIP w k (t) = WIP ( D)(t + tes ln (1 β)) Dtes. (B25) te TH(t) = D. (B26) Equations (B23) through (B26) are valid as long as WIP w > k. So we define critical time t ck to be the time when WIP w = k. Using Equation (B25), it can be shown that: t ck = WIP k Dte k D te s - te s ln(1- β). (B27) Using assumptions 3 and 4, we can show that a real value exists for t ck and that this value is always larger than t cd. After t = t ck, Equations (B23) through (B26) become invalid and a new governing equation needs to be derived and solved. It can be shown that the new governing equation is: dwip WIP + = D. (B28) dt te which can be solved to obtain: WIP w (t) = (k Dte)e (t t te ck ) + Dte. (B29) 143

163 (t t ck ) te FGI(t) = WIP Dte (k Dte)e. (B30) TH(t) = D. (B31) The above equations are valid for any time t t ck. 144

164 APPENDIX C SIMULATION RESULTS FOR PUSH AND SHIPPING- COUPLED CONWIP SYSTEMS 145

165 146