Selection Of Inventory Control Points In Multi-Stage Pull Production Systems

Transcription

1 Selection Of Inventory Control Points In Multi-Stage Pull Production Systems Item Type text; Electronic Dissertation Authors Krishnan, Shravan K Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 25/07/ :16:29 Link to Item

2 SELECTION OF INVENTORY CONTROL POINTS IN MULTI-STAGE PULL PRODUCTION SYSTEMS By SHRAVAN K. KRISHNAN A Dissertation Submitted to the Faculty of the DEPARTMENT OF SYSTEMS AND INDUSTRIAL ENGINEERING In Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA

3 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Shravan K. Krishnan entitled Selection of Inventory Control Points in Multi-Stage Pull Production Systems and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date: 4/22/07 Ronald G. Askin Date: 4/22/07 Young Jun Son Date: 4/22/07 Jeffrey Goldberg Date: 4/22/07 Srini Raghavan Final approval and acceptance of this dissertation is contingent upon the candidate s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Date: 4/22/07 Dissertation Director: Ronald G. Askin Date: 4/22/07 Dissertation Director: Young Jun Son

4 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: Shravan K. Krishnan

5 4 ACKNOWLEDGEMENTS I would like to especially thank my advisor Dr. Ronald Askin, whose guidance and valuable suggestions helped me to complete my thesis. I would also like to thank my coadvisor, Dr. Young-Jun Son and my committee members, Dr. Jeffrey Goldberg and Dr. Srini Raghavan for expressing their willingness to serve on my committee. I am forever indebted to my parents and my sister for their continuous support and encouragement. Finally, I extend my thanks to all my friends in the Systems and Industrial Engineering Department at the University of Arizona for all their help, especially Karthik Krishna Vasudevan, who helped me with the Arena models.

6 5 TABLE OF CONTENTS ABSTRACT... 9 CHAPTER 1: INTRODUCTION BACKGROUND A Brief Introduction to Supply Chains Production Control Strategies Reorder Point (ROP) Models Materials Requirements Planning (MRP) Systems Kanban Systems Single-card kanban system Dual-card kanban system CONWIP Systems Advantages of Pull Systems over MRP Requirements of Pull Systems A Brief Introduction to Queuing Theory Queuing system Queuing models MOTIVATION BEHIND OUR RESEARCH CHAPTER 2: LITERATURE REVIEW REORDER POINT RESEARCH MRP RESEARCH KANBAN RESEARCH CONWIP RESEARCH PROBLEM STATEMENT CHAPTER 3: SINGLE PRODUCT, DETERMINISTIC MODEL MODEL 1: CONSTANT CONTAINER SIZE Notation Model Formulation Selecting Control Points Choosing Container Size MODEL 2: SECTION DEPENDENT CONTAINER SIZE... 65

7 6 TABLE OF CONTENTS -- Continued Solution Procedure COMPUTATIONAL RESULTS Experiment Design Discussion of Results Effect of Value Added Structure on Number of Control Sections CHAPTER 4: MULTI-PRODUCT STOCHASTIC MODEL INTRODUCTION ASSUMPTIONS MODEL FORMULATION DETERMINING THE CONTROL POINTS DETERMINING CONTAINER SIZES COMPUTATIONAL RESULTS Experiment Design Discussion of Results Effect of factors on cost and control structure Effect of holding cost distribution CHAPTER 5: MODEL VERIFICATION AND VALIDATION SIMULATION MODEL LOGIC Determining the Number of Kanbans Creation of Orders: Simulation Logic Creation of Entities Corresponding to Containers of Parts and Workstation Logic MODEL VALIDATION Methods to Ensure That Both Models Are Equivalent Single Product, Deterministic model Single Product Stochastic Model Multi Product Stochastic Model CHAPTER 6: SUMMARY AND CONCLUSIONS APPENDIX A: CODE APPENDIX B: DESIGN TABLE FOR EXPERIMENTS IN SECTION REFERENCES

8 7 LIST OF ILLUSTRATIONS Figure 1.1: The supply chain process Figure 1.2: Pure pull, CONWIP, and coordinated push systems Figure 2.1: System under consideration Figure 3.1: Shortest path representation of Model Figure 3.2: Shortest path representation for a two-stage control problem Figure 4.1: Hypothetical structure where stages 4 and 6 are control points Figure 4.2: Graph showing holding cost at each stage for a 20 stage system with linear growth. The dotted line indicates a holding cost ratio of 20, while the solid line indicates a holding cost ratio of Figure 4.3: Graph showing holding cost at each stage for a 20 stage system with delayed capitalization. The dotted line indicates a holding cost ratio of 20, while the solid line indicates a holding cost ratio of Figure 4.4: Graph showing the effect of each factor on system cost Figure 4.5: Graph showing the effect of each factor on system control structure Figure 4.6: Graph showing the combined effect of transportation time and number of products on the average cost Figure 4.7: Graph showing the combined effect of transportation time and number of products on the ratio of total cost to the number of control points Figure 4.8: Graph showing the combined effect of number of stages and number of products on the system control structure Figure 5.1: Arrival of orders Figure 5.2: Signal code 231 logic Figure 5.3: Workstations Figure 5.4: Logic for signal code:

9 8 LIST OF TABLES Table 2.1: Summary of previous relevant research Table 3.1: Comparison of models 1 and Table 3.2: Comparison of results for model 1 and model Table 3.3: Combination of Factors for which model 1 case 1 gives a two-control point 75 Solution Table 3.4: Combination of Factors for which model 1 case 2 gives a two-control point solution Table 3.5: Combination of Factors for which model 2 case 1 gives a two-control point solution Table 3.6: Combination of Factors for which model 2 case 2 gives a two-control point solution Table 3.7: Summary of Experiments in Section Table 4.1: Summary of experiments described in Section Table 5.1: Results of experiments described in Section Table 5.2: Results for a single product with Exponential demand Table 5.3: Results for a single product with Normal demand Table 5.4: Results of experiments described in Section Table A2.1 Effect of factors on number of control points Table A2.2 Effect of factors on system cost

10 9 ABSTRACT We consider multistage, stochastic production systems using pull control for production authorization in discrete parts manufacturing. These systems have been widely implemented in recent years and constitute a significant aspect of lean manufacturing. Extensive research has appeared on the optimal sizing of buffer inventory levels in such systems. However the issue of control points, i.e. where in the multistage sequence to locate the output buffers, has not been addressed for pull systems. Allowable container/batch sizes, optimal inventory levels, and ability of systems to automatically adjust to stochastic demand depend on the location of these control points. We begin by examining a serial production system producing a single part type. Two models are examined in this regard. In the first, container size is independent of the control section, while in the second, container sizes are section dependent. Additionally, a nesting policy is introduced which introduces the additional constraint that the container size in a section is related to the container size in any other section by a power of two. Necessary and sufficient conditions are derived for ensuring that a single, end-of-line accumulation point is optimal. When this is not the case, an algorithm is provided to determine the optimal control points. Effects of factors such as value added structure, fixed location cost, setup and material handling cost, kanban collection time, and material transportation time on the control structure are investigated. Results are extended to

11 10 determine the optimal container size when lead time at a stage is a concave function of container size. The study is then extended to a multi-product case. Queuing aspects are introduced to account for the interaction between the different part types. The queuing model used is a modification of the Decomposition/Recomposition model described in Shantikumar and Buzacott (1981). The models in the chapter do not assume a serial structure any longer. Additionally, general interarrival and service time distributions are considered. The effect of number of products, demand arrival distribution, value added structure, and number of stages on the control structure and system cost is investigated. Finally, a simulation model is developed in Chapter 5 to verify and validate the mathematical models described in Chapters 3 and 4.

12 11 CHAPTER 1: INTRODUCTION 1.1 Background A manufacturing firm can be thought of as a set of resources and procedures involved in converting raw material into products and delivering them to the customers. Thus, from the above definition, among the most important functions of a firm are production and logistics. Production includes decisions relating to the timing and quantity of production. The firm also has to decide when to order raw materials from its suppliers. All these issues come under the realm of inventory control. Single and multi item inventory control has been widely studied in the past, dating back to 1913 when Harris introduced the EOQ model. Several distribution strategies too have been examined but they are beyond the scope of this research. Some of them are discussed in Krishnan (2003) and Krishnan and Askin (2006), which concentrated on the entire supply chain, encompassing production and distribution. A production and distribution system consists of several stages or echelons where each stage aids in the smooth flow of goods from production to consumption. Beginning with raw materials, value is added at each stage as the materials are converted into finished products. The intermediate stages may or may not hold inventory depending on their function. This type of a system is referred to as a Multi-Echelon System or a Supply Chain, as it has come to be known. Over the years, the study of multi-echelon systems

13 12 has generated considerable interest among researchers. The research in this area can find its roots in the classical work of Clark and Scarf (1960) and Scarf (1960). However much of the research carried out in this area focuses mainly on simple two-stage systems. In general a production/distribution system consists of many stages. Viewed from a high level, a typical system for a large manufacturer may, for instance, consist of six stages, three for production, and three for distribution. Such would be the case for raw materials, parts fabrication, and assembly, followed by distribution centers, warehousing, and retailers. However, within fabrication and assembly, there may be many individual valueadded and transport operations. This research focuses on the control of inventory within a facility, but the results also provide insights for multi-echelon systems with physically disbursed stages. In this chapter, we define the relevant terms pertaining to production control strategies. A brief introduction to supply chain management is also provided. In Chapter 2, we review some of the relevant research in this area. In Chapter 3, we answer the important question of where to locate inventory buffers in deterministic, singleproduct pull-type serial production systems. Chapter 4 extends these results to a more general multi-product setting with stochastic demand and interarrival and service times, and a general structure as opposed to a serial structure. Chapter 5 uses simulation models to validate the mathematical models described in Chapters 3 and 4.

14 A Brief Introduction to Supply Chains Various analogous definitions have been proposed for the supply chain. Min and Zhou (2002) define the supply chain as: an integrated system which synchronizes a series of inter-related business processes in order to: (1) acquire raw materials and parts; (2) transform these raw materials and parts into finished products; (3) add value to these products; (4) distribute and promote these products to either retailers or customers; (5) facilitate information exchange among various business entities (e.g. suppliers, manufacturers, distributors, third-party logistics providers, and retailers). Lee and Billington (1995) have a similar definition: A supply chain is a network of facilities that procure raw materials, transform them into intermediate goods and then final products, and deliver the products to customers through a distribution system. Swaminathan et al. (1996) define the supply chain to be: a network of autonomous or semi-autonomous business entities collectively responsible for procurement, manufacturing, and distribution activities associated with one or more families of related products.

15 14 In short, a supply chain can be said to consist of all the stages involved, directly or indirectly, in fulfilling a customer request. (Chopra and Meindl (2001)). In general, the supply chain can be viewed as a network of facilities that can be characterized by a flow of goods and information. Materials flow forward as they are transformed into goods. In most cases the information flows are backward, from retail to suppliers. However, there has been a recent surge in interest in forward flows of information, from supplier to retailer. Refer to Chen (2003) for details. A general supply chain configuration with forward flows of material and information sharing is shown in Fig Flow of information Flow of goods Suppliers Manufacturers Product Assembly Distributors Retailers Customers Inbound logistics Outbound logistics Materials management Physical Distribution Figure 1.1: The supply chain process While our research could be applied to the high level stages shown in Figure 1.1, we are primarily concerned with the operations within a manufacturing facility for fabrication or

16 15 assembly. In the following sections we provide an overview of the systems and control elements within these environments Production Control Strategies Single stage production control models frequently use Reorder Point (ROP) inventory models. These models are based on the EOQ model and will be discussed briefly in the following section. Production and inventory control strategies used in multi-echelon systems can be divided into three basic categories depending on the protocol for authorizing production or distribution and the timing of process execution. A push system is one in which processes are executed based on a centrally coordinated plan, often external to the local process and based on demand forecast. Production authorization is based on upstream conditions or centralized control and the approach is to actually plan production levels. Push systems can be of two types: coordinated push systems and open systems. A coordinated push system is one in which an external coordinator uses forecasts to coordinate production and distribution decisions for all stages. An open system reacts to the arrival of orders. Once released, orders are executed as soon as resources are available. A Materials Requirement Planning (MRP) system is a commonly used push system. A pull system is one in which processes are executed in response to status changes in the downstream process. In a pull-based system, production is demand driven so that it is coordinated with actual customer demand rather than a forecast. In pull systems, the level of inventory in the system is controlled. Production authorization

17 16 is determined by downstream conditions. Commonly used Kanban pull systems control the level of each part in between inventory storage locations. CONWIP (Constant Work- In-Process) is a hybrid system. Orders are released as in a pull system but thereafter, follow a push protocol generally. The three systems are shown Fig The ROP model and Kanban, MRP and CONWIP systems will be discussed in the following sections.

18 17 Orders Pull system ` Demand forecasts (possibly) CONWIP system Buffer Manufacturing stage Demand information Flow of material Flow of information Coordinated Push system Production/ distribution decisions Coordinator Demand information Material Figure 1.2: Pure pull, CONWIP, and coordinated push systems Base stock system Information

19 Reorder Point (ROP) Models Reorder point models are inventory control models that determine how many parts to order or produce and the timing of orders. The simplest model of this type is the Economic Order Quantity (EOQ) model. This model specifies a (Q, r) inventory policy, which indicates that an order for Q units is placed when the inventory level falls below r units. The assumptions of the model are as follows: 1. Known, static, continuous, deterministic demand 2. The fixed cost to place an order is constant 3. Lead times are deterministic and replenishment is instantaneous 4. Shortages are not allowed 5. All replenishment orders are for the same quantity 6. Continuous inventory tracking The model determines the optimal order quantity and timing by minimizing the total cost. Total cost is the sum of the inventory holding cost, fixed cost of placing and order, and 2AD total purchase cost. Thus, Q* and r D, where A is the fixed cost to place an h order, D is the demand rate in units per time, h is the inventory holding cost per unit-time, τ is the lead time for order delivery. This type of policy is known as a Continuous Review Policy, since inventory level is constantly observed and as soon as it drops to r units, an order for Q units is placed. A periodic review model, on the other hand, is a (R, r, t) model where inventory level is observed every t time periods, and, if it has dropped to r

20 19 units, an order is placed to bring the inventory position to R. EOQ-type models are used for bulk materials and make-to-stock environments. More sophisticated continuous and periodic review models are discussed in Askin and Goldberg [2002] and other production planning textbooks. Section 2.1 contains a review of these models applied to multi-stage production systems Materials Requirements Planning (MRP) Systems An MRP system, as mentioned earlier, is a type of push system. In an MRP system, orders for all the component parts that go into a product are timed so that they coincide with the production schedule of that product. This is the basis for a dependent demand relationship. In the best case, there is very little safety stock carried, since production is carried out only when parts are required. This system is designed for low to intermediate volume production with moderate to high product variety. The various data requirements of an MRP system include: 1. Item master data for each part: this contains data pertaining to the annual demand, order quantity, cost, and scrap rate for each product. In addition, for internally created products, there is a link to engineering drawings and process plans. 2. Inventory status records for each item: contains data relating to on-hand inventory for each item.

21 20 3. Lead times: this helps in determining when to place an order for each item. Lead times are assumed to be deterministic. 4. Bill of materials: bill of materials is a listing of all the materials and components required to make each item produced in the factory. This includes all manufactured parts, subassemblies and end items. 5. Master Production Schedule (MPS): this shows planned production quantities for each end item in each period of the planning horizon. MRP systems suffer from several inherent problems, some of which are discussed in Section Kanban Systems A kanban system, as mentioned earlier, is a type of pull system. In this system, production authorization to produce a part at a workstation comes from the downstream stage. This production authorization comes in the form of a kanban card (virtual or physical). Each kanban card authorizes the production of one container of parts. Each workstation is in charge of keeping a full container of parts in its output buffer for each kanban allocated to it. Thus, the maximum number of parts (units) at any given time at a workstation is given by the product of the number of kanbans and the number of parts authorized by each kanban (usually, container size). There are two important types of kanban systems, single-card kanban systems and dual-card kanban systems. These systems are briefly discussed in the next two sections.

22 Single-card kanban system: This system is preferred when workstations are close to each other. In this system, there is a single set of kanbans, known as production ordering kanbans (POK). When an operator at work station i produces a container of parts (after receiving the authorization to produce it), he puts the kanban associated with it into the container and sends the container to the output buffer. When the operator at workstation i+1 requires the parts, he withdraws the container from the output buffer of workstation i and places the kanban into a collection box. Kanbans are regularly collected from the collection box and arranged in a schedule board. When the operator at workstation i becomes free, he checks the schedule board for kanbans Dual-card kanban system: A dual kanban system is used when distances between workstations are large. In addition to production ordering kanbans (POKs), this system uses withdrawal ordering kanbans (WOKs). The system has two loops, the first loop is the same as in Section the POKs move from the input buffer to the output buffer to the collection box to the schedule board. The WOKs loop from the input buffer of stage i+1 to their own collection box at i+1 to the output buffer of stage i, back to the input buffer of stage i+1. The material handler at stage i+1 checks the collection box frequently. If there are any WOKs present, the material handler moves to the output buffer at stage i to collect containers of parts authorized by the kanban. It removes the POKs from the container and places them in the collection bin and places the WOKs into the container instead. Thus, production and transportation are both controlled by this

23 22 method. Workstations have an input buffer for their raw material and an output buffer for their finished product. A review of literature related to kanban systems is carried out in Section CONWIP Systems CONWIP (Constant Work-In-Process) could be thought of as a hybrid system. Orders are released as in a pull system but thereafter, follow a push protocol generally. Like the kanban system, a CONWIP system maintains a constant WIP in the system. However, unlike a kanban system that has a WIP cap at each stage and for each part, a CONWIP system maintains a WIP cap at the system level, that is, there is a total of N parts in the system at all times (irrespective of the part type). A backlog list is maintained, which contains a list of parts to be manufactured. As soon as one batch completes processing, the next batch on the backlog list for which all raw materials are available is sent for processing. Whereas in a kanban system the internal distribution of jobs among part types and workstations is controlled, only total inventory is controlled in a CONWIP system. This makes it easier to manage the system. Especially when there are a large number of parts to produce, a CONWIP system is easier to manage than a kanban system. On the other hand, a CONWIP system requires more storage space, since, all N jobs need to be potentially accommodated at a workstation. Some of the methods for determining job

24 23 ordering in the backlog list are discussed in Section 2.4. Variants of CONWIP exist such as only controlling the inventory upstream of a bottleneck workstation Advantages of Pull Systems over MRP A pull system has several advantages over an MRP system. Some of these advantages are briefly discussed in this Section. 1. Efficiency - pull systems attain the same throughput as push systems with less average WIP. 2. Ease of control it is easier to set WIP levels than to set release rates (throughput) 3. Robustness - pull systems are less affected by data errors and random events than push systems. They are also self adjusting to minor variations. 4. Low Information System Requirement due to the manner in which they are designed, pull systems have lower information requirements than push systems. 5. Support for improving quality since WIP levels are low in pull systems, they not only require quality in order to prevent disruptions, but also promote it by shortening queues and quickening defect detection. (Source: Askin & Goldberg (2002) and Hopp & Spearman (1996))

25 Requirements of Pull Systems Pull systems, although they have several advantages over MRP systems, operate under some limitations. Some of the requirements of pull systems (specifically kanban) are as follows: 1. Low to Moderate Product Variety a kanban system is suited for high volume production for a few products. As the number of products increases, the number of kanbans increases too, making it more difficult to manage the system. 2. Moderate to High Volume Production a kanban system is suited for high volume production. Excessive changes in demand may lead to inefficiencies. 3. Low Demand Variability while a kanban system can correct itself for small changes in demand, larger variability will require changing the number of kanbans. A CONWIP system takes care of this during the backlog calculations. 4. Reliable Processes (Predictable Lead Time) because pull systems maintain low WIP levels, they require that lead times be predictable. Excessive variations in the lead time may result in inefficiencies in the system 5. Raw material availability (Co-location) pull systems need frequent, small orders of raw material, rather than infrequent bulk orders. This is facilitated by having supplier parks located close to the manufacturing plant.

26 25 6. High Quality Requirements (lower setup times, higher machine availability, lower defect percentage) as a result of low WIP levels maintained, pull systems demand high quality. 7. Small setup times since pull systems react instead of planning, a pull system requires low setup times to rapid response and low inventory levels A Brief Introduction to Queuing Theory Queuing theory is extensively used to analyze manufacturing systems and supply chains. In this Section, we provide a brief introduction to some of the concepts Queuing system: A queuing system protocol provides the basis for how items enter and leave the queue. Some examples of queuing system protocols are: FIFO (First In First Out) also known as FCFS (First Come First Serve) parts are processed in the order in which they arrive. LIFO (Last In First Out) also known as LCFS (Last Come First Serve) similar to a stack where the last part to arrive is served first. Priority Queue where certain parts have higher processing priority than others. Random order Queuing models: Queuing models are used to model systems. They describe the following characteristics of the queuing systems:

27 26 Part arrival process: this describes the arrival distribution, whether parts arrive individually or in batches. Service times Server capacity: the number of available servers. Queue capacity: the number of available spaces in the queue. Queue discipline: as explained in section Kendall introduced a shorthand notation to characterize a range of these queuing models. It is a three-part code a/b/c. The first letter specifies the interarrival time distribution, the second the service time distribution, while the third letter indicates the number of servers. For example, a (b) can be M, which indicates an exponential interarrival (service time) distribution, D, which indicates a deterministic times or G, which indicates a general distribution. The parameter c can be any number equal to or greater than 1. In case the queue has a finite capacity, a fourth letter is added to the notation to indicate queue capacity, for example, M/M/1/5, which means that interarrival and service times are exponential, there is a single server, and the queue has room only for five parts. Some of the performance measures include 1. The distribution of number of customers in the system 2. The distribution of service times of customers in the system and in queue (amount of work in the system). 3. The distribution of the busy period of the server

28 27 An extensive theory survey of queuing theory in manufacturing systems has been carried out by Govil and Fu (1999). Little s Law (1961) relates the queue length to service rate and waiting time for systems in steady state. 1.2 Motivation Behind Our Research Extensive research has been carried out on pull control based production systems as one aspect of lean manufacturing. In such systems, each inventory storage buffer is designated with a desired level of inventory for each part type. This level is typically defined as a desired number of containers with each container holding a designated quantity of the product. This quantity is referred to as the container size. Following the procedure popularized by the successful Toyota Production System, kanbans or cards are used to authorize production. Each kanban corresponds to one container. Kanbans are kept in the output buffer attached to a full container of parts. When the first part from that container is required by the succeeding work area, the container is removed from storage and the kanban is recirculated to the production station to authorize replenishment of those items. The system automatically paces and prioritizes orders within workstations based on downstream consumption. This makes the system easy to implement and selfadjusting. If the proper level of kanbans is selected, and the replenishment and demand processes are highly predictable and deterministic, then the system can be operated such that a completed container of parts reenters the output buffer just in time for its need at

29 28 the downstream workstation. Hence, such systems are often considered part of the JIT (Just-in-Time) production control philosophy. Most of the previous pull-system research focused on determining the number of kanbans, with lesser emphasis placed on container sizes and product sequence in a justin-time (JIT) shop. However, in a multistage system, it is not necessary to include an output buffer at each stage. Within control sections (the area or sequence of workstations between buffers), a push philosophy can be incorporated. Once production is authorized by the removal of a container from the control sections output buffer, a replenishment order is released to the first workstation in the control section. These orders then have authorization to flow through each stage of the control section until again reaching the output buffer, i.e. they are pushed through without waiting for a customer request. The issue of where to locate control points, although important, has not been effectively addressed in the past. In our study, we attempt to find a set of production stages in a JIT system that will act as inventory control points. These control points are the only stages that store inventory. In addition, we determine the number of kanbans and container sizes that should be used in specific conditions. Chapter 3 deals with a single product deterministic case. In Chapter 4, this is extended to a multi-product stochastic case.

30 29 CHAPTER 2: LITERATURE REVIEW During the last few years there has been considerable interest in studying problems pertaining to inventory management in multistage production/distribution systems. Multistage problems are complex and the same solution procedures used for single-stage systems are not appropriate to be used to solve them. According to Clark and Scarf (1960), in studying single-stage systems, the main assumption that was made was that when the installation requests a shipment of stock, this shipment is delivered in a fixed or random length of time, but the time lag is independent of the order placed. However there are several situations where this assumption may not be valid and hence the need to study multi-echelon systems arose. It has also been shown by Muckstadt and Thomas (1980) that specialized multi-level methods may reduce the total costs substantially over general methods for single stage systems. In this Chapter we first review research in multistage systems using general reorder point policies. Next we will discuss research on MRP, Kanban, and CONWIP systems.

31 Reorder Point Research One of the first studies in this area is the seminal work of Clark and Scarf (1960), who analyze a serial system under a periodic review policy and uncertain end-item demand. They show that for such a system, the problem can be decomposed and the optimal policy at each echelon can be computed separately to get approximate solutions that provide upper and lower bounds on the actual costs. Each echelon in this case is linked by a penalty function, which represents the cost of failing to fulfill a downstream stage s order. The late eighties and the early nineties saw a surge in papers related to these areas. For a comprehensive review of the research in this area, the reader is referred to Axsater (1993). In this Section we summarize some of the recent research in this area and classify the models based on certain characteristics, such as, whether a periodic review or a continuous review policy is used, whether end-item demand is stochastic or deterministic, and whether a finite or infinite horizon is used. In addition, some of the recent models for solving this problem will be discussed. We begin by looking at a serial system with N stages. For this system, it has been shown that the base quantities at different stages follow the integer ratio property, i.e., the base quantity at a stage is an integer multiple of the base quantity at the next stage (assuming that demand occurs at the last stage). Clark and Scarf (1960, 1962) examine the problem of determining optimal purchasing quantities using base-stock policies in a finite-horizon, by decomposing the problem and solving for the optimal quantity at ever stage. They

32 31 show that in the absence of economies of scale, these policies are optimal. A base stock system is similar to a pull system. It can be defined as an (S, S) order up to policy where each stage acts in unison to replace consumed stock. An (s, S) inventory system is one where whenever inventory level falls below s, an order is placed to raise the level up to S. In (S, S), once a retailer sells one unit of product, he places an order with the previous stage to replenish its stock. Thus the stock level is kept constant. The model makes the following assumptions: The cost of purchasing and shipping an item from any stage to the next will be linear, without any setup cost. The only exception to this is at the first stage. At the last stage, a linear holding cost and shortage cost will be operative. Excess demand is backlogged. Federgruen and Zipkin (1984) extend this model to an infinite horizon and prove that the results of Clark and Scarf (1960) are applicable for this case. Roundy (1986) and Maxwell and Muckstadt (1985) devise simple policies that are within 2% of the optimal solution. This results from restricting order intervals to powers-of-two times some base time period. Thus, if T n is the order interval for product n, then: Tn 2 kn where 1 2 and k n is some integer. It is assumed that Zero-Inventory Policy holds, i.e., an order is placed for a product only when its inventory drops to zero. Dong and Lee (2003) prove that the results of Clark and Scarf (1960) can be extended to a time-correlated demand process using Martingale Model of Forecast Evolution (MMFE), a forecasting technique that takes into account past demand and other factors affecting demand.

33 32 A substantial amount of research has gone into determining the size and location of safety stock inventory under various production control strategies. Inderfuth (1991) provides a technique for determining safety stock distribution in serial and divergent production/distribution systems operating under base stock policies. The paper assumes 100% reliable workstations. Inderfuth and Minner (1997) look at divergent systems with normally distributed demands where each stage follows a base stock policy. Inderfuth (1994) contains a review of relevant research in this area. The paper is a survey on concepts and specific approaches for determining safety stocks in divergent inventory systems as a measure of protection against risk. Graves and Willems (2000) develop an approach to optimize the inventory in a supply chain. Each stage in the supply chain is modeled as a network. They make the following assumptions: Each stage in the supply chain operates under a periodic-review, base-stock policy. Production lead times are assumed to be known and deterministic. Demand occurs at stages without successors and is bounded. Each demand node promises a guaranteed service time by which it will satisfy customer demand. For a serial system based with the above assumptions and without any capacity constraints, Simpson (1958) provides a technique to optimize the inventory. van Houtum et al (1996) also review relevant research in the theoretical and numerical analysis of stochastic multistage systems where a periodic review base-stock policy is used.

34 33 Magnanti et al (2005) show that adding a set off redundant constraints and iteratively refining the approximation speeds up the time required on a commercial solver to solve moderately sized safety stock placement problems in acyclic supply chain networks. Minner (2001) analyzes the problem of safety stock placement in reverse supply chains with the integration of internal and external product return and reuse. For assembly systems where two components are assembled into one so that each stage has a unique successor, Crowston, Wagner, and Williams (1973) state that in an optimal solution, an Integer-Ratio property holds, i.e., the batch size at a stage is an integer multiple of the batch sizes at its downstream stages. It was shown by Szendrovits (1981) and Williams (1982) that their results were highly dependent on the assumptions they used. For example, Williams (1982) shows that when the assumption that production occurs at a constant rate at a stage is ignored, the proposition no longer holds. Schmidt and Nahmias (1985) use the decomposition technique of Clark and Scarf to characterize the optimal policy for an assembly system where two components are assembled into one. Rosling (1989) states that under appropriate conditions, a general assembly system is equivalent to a serial system and so the result of Clark and Scarf holds. Chen (2000) extends these results to an assembly system with batch-ordering. Cachon (1995), Graves (1996), Chen and Zheng (1997), Tarim and Miguel (2004) all study distribution systems. DeBodt and Graves (1985) design approximate echelon stock policies for a serial system with setup costs under a continuous review policy with batch ordering. Echelon stock for

35 34 a component is defined as the inventory of that component plus all the inventory of downstream items that use that component. Installation stock, on the other hand, is inventory of the component without considering downstream inventories. Echelon stock policies use echelon stock information to make decisions. The end customer demand is stationary and stochastic and each stage has a deterministic lead time. They assume nested policies, where, whenever one stage reorders, all downstream stages also reorder. Chen and Zheng (1994) provide an exact cost evaluation procedure for such systems. Axsater and Rosling (1993) prove that echelon stock policies are superior to installation stock policies for serial and assembly systems. Axsater (1997) provide an exact cost evaluation technique for a two-stage system with one warehouse and N retailers, where all facilities use a continuous review echelon-stock policy with different reorder points and batch quantities. An echelon stock policy requires centralized demand information. Chen and Zheng (1998) study serial systems with compound Poisson demand and batchordering and provide a near-optimal solution. Chen (1998) determines the relative cost difference between the two policies for a serial system. This cost difference can be thought of as the value of centralized demand information. Mitra and Chatterjee (2003) extend the results of DeBodt and Graves (1985) to fast moving items. Information sharing and coordination are two topics that have been extensively studied in multi stage systems, but more so from a supply chain perspective. Papers by Lee et al. (1997a & b), Gavirneni et al. (1999), Chen et al. (1998, 2000a & b) and more recently,

36 35 Krishnan and Askin (2006) have studied the effect of information sharing and coordination between supply chain partners on the system performance. Table 2.1 below summarizes some of the papers on production planning in multi-echelon systems.

37 36 Table 2.1: Summary of previous relevant research Continuous/ Periodic Review (p/c) stochastic/ deterministic demand (s/d) stationary? nested? batch? lead times Echelon/ Structure Installation Clark and Scarf (1960) P serial Echelon s y n n d Clark and Scarf (1962) P serial Echelon s y n n d DeBodt and Graves (1985) C serial Echelon s y y y d Moinzadeh and Lee (1986) C distribution Installation s y n y d Roundy (1986) C serial/ assembly Echelon d y y n d Lee and Moinzadeh (1987) C distribution installation s y n y d Lee and Moinzadeh (1987) C distribution installation s y n y d Chen and zheng (1994) C/ P serial echelon s y y y d Graves (1996) C serial echelon s y n y d Axsater (1997) C distribution echelon s y n y d Chen (1998) C serial echelon s y y y d Chen and zheng (1998) C serial echelon s y y y d Chen (1999) C serial echelon s y y y d Graves and Willems (2000) P general echelon s y y n d serial/ Chen (2000) C assembly echelon s n y y d Mitra and chatterjee (2004) P distribution echelon s y y n d Dong and lee (2003) C serial echelon s n n y d Mitra and chatterjee serial/ (2003) C assembly echelon s y y y d Liu et al. (2004) C distribution echelon s y y y d other features repairable items repairable items repairable items non stationary demand fast moving items

38 MRP Research A Materials Requirement Planning (MRP) system is a type of push system where production authorization is based on a production plan. In contrast, in pull systems, production authorization depends on realized demand. A significant amount of the early research in this area focused on lot sizing decisions associated with MRP. Several lot sizing models have been developed in the last couple of decades. The simplest lot sizing model is the Economic Order Quantity (EOQ) model, which dates back to Another model used for lot sizing in MRP systems is the Wagner-Whitin (WW) model (Wagner and Whitin, 1958). This model was originally developed for a multi-period single-stage model under a periodic review policy with known time-varying demands. They assume linear holding costs and fixed order costs and that shortages are not permitted. For this cost structure they determine that in an optimal policy production is carried out only in periods with positive starting inventory. Dynamic programming can be used to determine optimal production quantities. However, the WW model has not been frequently used in MRP systems. This is due to the fact that the model has an inherent property that causes nervousness in the system, i.e., alterations or changes in schedules for later periods may cause changes in lot sizes during the early periods. Another model that is used is a heuristic developed by Silver and Meal (1973). It involves computing the cost of holding and setup per period as a function of the number of periods in the current order horizon and using it to calculate the order quantity. Lot-for-lot (LFL) is a strategy where order quantities are equal to the requirements, offset by the lead time. Another popular

39 38 technique known as Periodic Order Quantity (POQ) calculates the optimal order interval, T as a ratio of the EOQ to the demand rate and assigns the first T periods demands to the first period and so on. The Least Period Cost (LPC) method determines the first local minimum cost per unit time and sets the first lot size quantity and proceeds to the next lot size. Other lot sizing techniques have been developed by Veral and LaForge (1985), Blackburn and Millen (1982), Afentakis et al. (1983), Afentakis and Gavish (1983), Afentakis (1987) etc. Texts and articles such as Askin and Goldberg (2002) and Baker (1993) contain a thorough description of these models. Molinder and Olhager (1998) study the effect of different lot sizing techniques on cumulative lead times. Here, cumulative lead time is the total lead time for each bill of material path below the item. Buzacott and Shantikumar (1994) study the influence of accuracy of forecasts, processing time variability, and degree of congestion, along with inventory and shortage costs on system lead time and safety stock, in order to determine whether safety stock or safety time is more useful. They conclude that safety stock is more robust than safety time, except when it is possible to make accurate forecasts over the lead time. Other studies on MRP system performance include, Whybark and Williams (1976), Grasso and Taylor (1984), and Ritzman and King (1991). MRP uses fixed lead times for planning. However in reality, lead times are variable. This lead time variability may result in inefficiency in planning. Melnyk and Piper (1985) suggest the following two options to overcome these problems:

40 39 Track lead time error, i.e., the difference between planned lead times and observed lead times, and update planned lead times appropriately. Minimize the effect of lead time error by constructing planned lead times that provide an appropriate probability that actual lead times will not exceed the planned lead times. Several comparative studies have been carried out comparing the performance of MRP with Kanban, CONWIP, and Reorder Point Policies. Benton and Shin (1998) classify existing MRP/ JIT comparison literature. Recently, Krishnamurthy et al. (2004) have shown using simulation experiments that Kanban may result in significant inefficiencies in a manufacturing setting where a fabrication cell supplies different products to several assembly cells. It is assumed that the assembly cells fix their assembly schedules in advance and share this information with the supplier cells. Axsater and Rosling (1994) have shown that MRP outperforms installation stock (Q, r), Kanban, and order-up-to-r policies for a general acyclic inventory system. More of the literature on these comparative studies will be discussed in the following two sections. The MRP system suffers from several drawbacks, both due to the assumptions that it makes and due to its basic structure. One of the most important assumptions that MRP makes is that there is always sufficient capacity to meet production demands, which is unrealistic. As mentioned earlier, MRP uses fixed lead times for planning. Also, MRP plans for periods, not continuous time, thus requiring inventory to be pre-staged. Our

41 40 research focuses on pull systems, such as CONWIP and Kanban, which will be discussed in the following two Sections. 2.3 Kanban Research One of the earliest descriptions of the single kanban system was given by Monden (1983) as being used in the Toyota Production System for serial systems with closely located sequential stages. An overview of various extensions and refinements is given in Askin and Goldberg (2002). A thorough review of the research in Kanban systems is available in Huang and Kusiak (1996). Most research into Kanban systems has emphasized the choice of the number of kanbans to use in such systems. Monden (1983) suggests the following model: k i idi(1 l) ni where k i is the number of kanbans for part type i, i is the total lead time, D i is the demand rate and n i is the container size. l is a safety factor, introduced to counter variability. The container size associated with each part type can be determined using the EOQ expression: 2aD i i ni, h i

42 41 where a i is the setup cost and h i is the holding cost per unit for product i. Schonberger and Schniederjans (1984) suggest that the EOQ model may be inappropriate for the JIT manufacturing environment. They claim that reductions in setup time and a proper value of holding costs will result in an optimal batch size of one. The value of setup time reduction is discussed in papers by Spence and Porteus (1987) and Hahn et al. (1988). Davis and Stubitz (1987) determined the number of kanbans at each station using simulation and response surface techniques. Philipoom et al. (1987) showed experimentally that the lead time demand distribution constitutes a major determinant of the number of kanbans needed. For modeling a kanban system with a fixed parameter specification or to determine the number of kanbans needed, stochastic analytical models with discrete time periods (Deleersnyder et al. 1989) and continuous time (Mitra and Mitrani 1990, Wang and Wang 1990, Askin et al. 1993) have also been presented. Deleersnyder et al. (1989) use a discrete time markov chain to determine the number of kanbans. They take machine reliability, demand variability, and safety stocks into account. Askin et. al (1993) develop a mathematical programming model to determine the number of kanbans under time dependent backorder costs and occurrence based shortage costs by minimizing total system costs. Philipoom et al. (1996) present a solution procedure known as JACKS (JIT Algorithm for Containers, Kanbans and Sequence) that provides an integrated approach to simultaneously determine container sizes, number of kanbans, and product sequence.

43 42 Karmarkar and Kekre (1989) model two-stage, single and dual card kanbans as a Continuous Time Markov Chain in order to study the effect of container size and number of kanbans on the expected inventory holding costs and shortage costs. They conclude that when the number of kanbans was constant, the total inventory cost was a convex function of the container size. In another experiment, the optimal container size was seen to be inversely proportional to the number of kanbans. Berkeley (1996) studies the effect of container size on average inventory and customer service levels in a two-card kanban system processing multiple part types. The study concludes that, as expected, smaller container sizes resulted in smaller average inventories. However, they do not necessarily lead to lower service levels. Some other studies that consider the effect of container sizes on the system performance include, Gupta and Gupta (1989a & b) and Lee (1987). Another area of research in Kanban system compares the performance of kanban systems with other systems, notably MRP systems. Rees et al. (1989) compared an MRP lot-forlot system and a Kanban system in an ill-structured production environment and concluded that the MRP lot-for-lot system is comparatively more cost effective since it carries less inventory. Sarker and Fitzsimmons (1989) concluded that an MRP lot-for-lot handles lumpy demand better than a Kanban system, although difficulties may be caused by stochastic processing times.

44 CONWIP Research The Constant Work In Process (CONWIP) system was proposed by Spearmann et al. (1990) as an alternative to the Kanban production system. Spearmann and Zazanis (1992) present the following conjectures about pull systems: There is less congestion in pull systems 1. Pull systems are inherently easier to control than push systems. 2. The benefits of a pull environment owe more to the fact that WIP is bounded than to the practice of pulling everywhere. The CONWIP system imposes a WIP cap on the entire system, rather than on individual workstations, as in a Kanban system. In fact, CONWIP could be thought of as a hybrid pull-push system, where parts enter the system according to a pull philosophy, but within the system, a push policy is used. A backlog list is used to determine the order in which jobs are processed. Initial studies into the CONWIP system were comparisons of this system with MRP and Kanban systems. Spearmann et al (1990) provide a brief comparison of CONWIP with MRP and kanban systems and conclude that a CONWIP system is more effective than the other two systems. Buzacott and Shantikumar (1991) show that if the value added at each stage in the workstation is negligible, then CONWIP systems exhibit superior performance to Kanban and MRP systems with respect to maximizing customer service, subject to WIP constraint. Spearmann and Zazanis (1992) compare Kanban and

45 44 CONWIP and conclude that CONWIP produces higher mean throughput. Spearmann and Zazanis (1992) and Hopp and Spearmann (1996) suggest that CONWIP systems possess the following advantages over MRP systems: 1. Pull systems are easier to control than push systems. This is because pull systems control WIP and observe throughput, whereas push systems control throughput and observe WIP. It is easier to control the total number of parts currently in the system than to control the production rate. 2. For the same throughput, push systems will have more WIP on an average than CONWIP systems. A corollary of this is that for the same throughput, push systems will have longer average cycle times than an equivalent CONWIP system. 3. A CONWIP system is more robust to errors and breakdowns than a pure push system. Muckstadt and Tayur (1995a, b) show that the CONWIP system has less variable throughput and lower inventory than a Kanban system. Golany et al. (1999) investigate CONWIP based Shop Floor Control (SFC) where products are grouped into families (Group Technology (GT) design). The main reason for GT grouping is to cluster different parts having almost identical routings into families. This creates a flowshop-like environment within each family of parts. The authors propose that within each cell, parts be scheduled as a CONWIP system. They attempt to simultaneously answer two questions: 1. what is the best WIP level? and 2. how to arrange the backlog level for a given system? They formulate a deterministic mathematical programming formulation to

46 45 describe the scenario and solve it using simulated annealing. They compare two systems. In the first, a multi-loop CONWIP system, containers are restricted to stay in given cells, while in the second, a single-loop CONWIP system, containers can circulate anywhere in the system. Based on their results they find the latter system to be superior. A recent study by Yang (2000) uses simulation based techniques to compare Kanban and CONWIP systems. He concludes that CONWIP outperforms Kanban in most cases, consistently producing the smallest mean customer wait and total WIP. Recent textbooks by Hopp and Spearmann (1996) and Askin and Goldberg (2002) contain in depth comparisons of the two systems with each other and with MRP systems. Recent research has focused on determining system parameters, such as, how to order jobs in a backlog list and how many containers to use. CONWIP is a type of closed queueing nework. In these networks, the number of parts in the system never changes. In essence, if a part leaves the system, it is immediately replaced with another part, which keeps the WIP constant. This results in a negative correlation between the parts at each stage. A product form expression for such a system may not be computationally very effective, instead, a technique known as Mean Value Analysis (MVA) (Reiser and Lavenberg 1980, Suri and Hildebrant 1985) is used to determine system paramenters, such as throughput and cycle time. Starting with an initial guess of queue lengths, the system parameters are calculated iteratively, until there is a convergence in the solution. Duenyas and Hopp (1990) and Duenyas et al. (1993) provide approximations to describe the distribution of the output from a CONWIP system. Herer and Masin (1997) develop a

47 46 nonlinear integer model to address the allocation of WIP to different products in a serial system. Optimal job orders and schedules are obtained based on demands and forecasts. The model minimizes total cost. Hopp and Roof (1998) use a process known as Statistical Throughput Control (STC) to set WIP levels in a CONWIP system, where they set a target production rate and cycle time and use it to determine capacity. Zhang and Chen (2001) formulate an integer programming model that minimizes the total setup cost and production smoothness to determine optimal production sequence and lot sizes in a linear system. Ryan et al. (2000) address the issue of determining the number of cards for each product type in a multiproduct job shop type system. Cao and Chen (2005) study an assembly system fed by two CONWIP-based feeder lines and use an MIP model to determine system parameters for this type of a system.

48 Problem Statement Figure 2.1 below shows a single kanban Just-in-time (JIT) system. Circles represent workstations or stages and triangles represent inventory locations (output buffers). Each production-ordering kanban, either physical or as an electronic token, authorizes one container of parts of a given part type. The kanbans flow within a given control section, circulating from the output buffer, where they are attached to a full container, back through all production stages within the control section once they are detached upon withdrawal of the container. A proper design of system control points may improve coordination and reduce total costs. Our study aims to define a characterization of system control points for different environmental conditions and using a set of increasingly complete models. The main problem we wish to address is which stages should serve as the control points for such a system. We assume that production batch and unit load sizes correspond to the container size. We begin by examining a serial production system producing a single part type. Two models are examined in this regard. In the first, container size is independent of the control section, while in the second, container sizes are section dependent. Additionally, a nesting policy is introduced which introduces an additional constraint, the container size in a section is related to the container size in any other section by a factor of two.

49 48 The study is then extended to a multi-product case. Due to the interaction between the different part types, the total time spent by a part in the system increases because of the time spent in queue waiting for the part in the machine to be processed (we assume that this time is zero in the single product models). A variation of the decomposition/ recomposition algorithm of Shantikumar and Buzacott (1981) is proposed. This variation takes into account the hybrid (open/ closed) aspects of our system. Finally, a simulation model is developed to verify our mathematical models. 1a 1b. 1n 2a 2b. 2n. Control Section i Control Section i+1 Material flow Information flow (kanban) Material and information flow Figure 2.1: System under consideration

50 49 CHAPTER 3: SINGLE PRODUCT, DETERMINISTIC MODEL The main problem we wish to address is which stages should serve as the control points for such a system. We assume that production batch and unit load sizes correspond to the container size. In this Chapter we consider a single-product, deterministic model. Two different models are considered. In the first, the container size is constant across all sections. In the second, container sizes are dependent on the control section. Optimality conditions for a single control section for both models are presented. Additionally, a methodology for determining the number and location of control points is provided. 3.1 Model 1: Constant Container Size We initially consider a serial production line, producing a single part type. The batch size at each stage is assumed to be known and fixed across all stages. Demand occurs at stage m and is stochastic and independent in non-overlapping time segments. (For notational simplicity we will assume demand is Normally distributed.) There is a fixed cost associated with setting up a control point. We assume that the lead time at any stage is known. The lead time demand distribution is likewise therefore known for each stage. The number of kanbans is set to provide coverage against a defined upper percentage point (assumed to be close to 1) of the lead time demand distribution. Also, we assume a value added structure so that the holding cost at any stage is greater than the holding cost at its preceding stages. The objective is to select a set of control points and container size

51 50 that minimize the location cost plus setup cost plus inventory holding cost. The system under consideration is shown in Figure Notation The notation that we use is as follows: c i - collection time for kanbans at stage i f( d ) - probability distribution function of demand per time at stage m. E( ) = D and V( ) = 2 f i - Fixed cost per time for locating and maintaining inventory at stage i - Percentage of orders satisfied without delay (service rate) z - Standard Normal Variate such that P{ Z Z } M - Set of all stages {1,., m} h - Inventory holding cost per unit at stage i, i M i L i -Production Lead time at stage i t i - transport time from stage i to stage i+1 X i - Y ij - ij 1 if stage i is a control point 0 otherwise 1 if stage j is a control point that serves stage i 0 otherwise Y defined for j i, i 1,..., m

52 51 a i = Setup cost plus material handling cost per container at stage i n = Container size An important component of system performance is lead time. Assume a control section extends from stage i through j. Using the notation above and the dynamics of pull operation, the lead time, ij, for a container from the time its kanban is removed at stage j until it reenters the buffer at stage j with replenished stock is j j1 (3.1) c L t ij j r r ri ri1 Lead time includes kanban collection at the final stage, production at each stage, and transit between stages within the control section Model Formulation Initially assume that the container size, n, is known. The decision problem then becomes minimization of expected cost per period or a D h ( n 1) Min f X z h X c Y ( L t ) D h ( L t ) i m i i j j j ij i i1 j j j1 im im n 2 jm i j jm (3.2) Subject to: Yij 1 i M (3.3) ji

53 52 Y X i, j M, i j (3.4) ij j Y Y i j M (3.5) ij i1, j, X 1 (3.6) m X [0,1], Y [0,1] (3.7) j ij Here, the objective function, Equation(3.2), minimizes the sum of the total location cost, setup cost, work-in-process (WIP) inventory holding cost and finished goods inventory cost for partial containers, safety stock holding cost. The first term in the objective function represents the fixed cost of locating a control point. The second term represents the total period setup and material cost. The third term represents the WIP inventory holding cost of the partial container available for final demand where we assume units are consumed one at a time. The fourth term represents the safety stock holding cost at each control point. The fifth term represents WIP inventory cost for units flowing through the system. For the system described, expected on-hand inventory in the control buffers is excess of maximum inventory from expected lead time demand. Although in practice the number of kanbans may need to be integer and constant, our model implicitly allows for a fractional number of kanbans to achieve the targeted service level. This relaxation facilitates modeling and solution but could also be used in practice to improve system performance by embedding a planned delay into planning replenishments so as to reduce safety stock to this optimal level. Note that if the system is fully deterministic with 0, the system is completely synchronized and output buffers are always empty.

54 53 Constraint (3.3) ensures that each stage is served only by one control point. Constraint (3.4) ensures that a stage can be controlled only by a control point. Constraint (3.5) ensures integrity in assigning all workstations within a control section to the same control point. Constraint (3.6) ensures that stage m (the last stage) is always chosen to be a control point. Constraint (3.7) defines the binary restrictions on the variables Selecting Control Points We first reformulate our system design problem defined in Equations (3.2) to (3.7) as a shortest path problem. Consider stage 0 to be the input stage. Also, let M oj be the cost of selecting stage j as the first inventory control point and M jk, the cost of having j and k as two consecutive control points. This is shown in Figure 3.1 below. M 02 M 2m m M 01 M 12 M 1m M 0m Figure 3.1: Shortest path representation of Model 1

55 54 Also define x ij 1 if stages iand j are two consecutive control points 0 otherwise The x ij s can be thought of as arcs connecting successive control points. Then, the above formulation ((3.2) to (3.7)) can be modified as follows: Constraints (3.3) through (3.6) specify that each stage is served by a unique control point. Additionally, if a stage is chosen as a control point, it serves all stages in between it and the previous control point. This is analogous to saying that stage m will have exactly one arc entering into it and none leaving while stages 1 through m-1 will either have no arcs entering into them and leaving them or exactly one entering and one leaving. Finally, stage 0 will have one arc leaving it. The formulation (3.2) through (3.7) can therefore be rewritten in terms of the number of arcs entering and leaving them. Min Mijxij (3.8) im jm Subject to: 1 if i 0 xij xkm 0 if i 0 or m jm km 1 if im where i, j, k M, k i j (3.9) x [0,1] i, j M (3.10) ij

56 55 aid hm( n 1) If L j and t j are fixed, D ( Lj t j1) hj is a constant. If n is fixed, k1 n 2 jm, is a constant for all feasible solutions. Thus, in order to find the optimal solution to the model, we can ignore these terms from the objective function. In general then: M jk fk zhk j 1, k j 0,..., m 1; k j 1,..., m (3.11) im For this model with predetermined lead times, positive echelon holding costs, and constant container size, analysis leads to the following result on locating buffers: Theorem 1: For the cost structure defined in model 1, a single control point is always optimal if for all stages j=1,,m-1, f j z hm 1, m hj ij hm j 1, m. Proof: We will prove the theorem for a two-stage problem and then for a general case using mathematical induction. The proof is simple for a two-stage problem. This problem can be represented as shown in Figure 3.2. M M 01 M 12 Figure 3.2: Shortest path representation for a two-stage control problem

57 56 In the figure it is clear that a single control point is optimal if and only if M02 M01 M12, where the costs can be calculated using Equation (3.11). Therefore, M02 f2 zh2 12, M01 f1 zh1 11, and M12 f2 zh2 22. Substituting in these expressions and rearranging terms, the sufficient condition for a single control point becomes f z h h h (3.12) Thus, the theorem is true for m = 2. For an m+1-stage problem, if a single control point is optimal for all k-stage problems such that k m, we need to consider only the two-arc solutions. This is because any path to a node k m that uses two or more arcs is dominated. Therefore, for an m+1-stage problem, a single control point is optimal if: M0, m1 M0 j M j, m1 j M, j m 1 (3.13) Once again we can use Equation (3.11) to compute the costs. Therefore, M0, m1 fm 1 zhm 1 1, m1, M0 j f j zh j 1, j, and M j, m1 fm 1 zhm 1 j1, m1. Substituting the above terms into Equation (3.13) and rearranging terms gives: f j z hm 1 1, m1 hj 1, j hm 1 j1, m1 (3.14)

58 57 Thus, based on the assumption that the theorem is true for an m-stage problem, it is true for an m+1-stage problem. Hence it is true for all j M. In general, the RHS of Equation (3.14) is negative. Since for a, b > 0, a b a b it can only be positive if the ratio hm/ h j, m > j, is large. If we assume that hm hj h, and noting we are interested in an m state system, Equation (3.14) reduces to: m m1 j j1 m m1 f z h c L t c L t c L t j m r r j r r m r r r1 r0 r1 r0 r j1 r j (3.15) The right hand side in Equation (3.15) is clearly negative. In this case, a single control point is always optimal even if there is no fixed cost to maintain an inventory buffer.

59 Choosing Container Size The formulation in Section assumed that the batch size, n was known. We now consider the selection of n. The formulation remains the same, except for one additional constraint: n n j M (3.16) max j Here, max n j is the maximum container size that stage j can handle. This would typically be dictated by the material handling technology or part form at stage j. We will consider two cases. Case i) Fixed processing time operations: Here, the production lead time, L is independent of the container size. In this case the problems of container size and location of control points are readily seen to be separable in the formulation Equations (3.2) to (3.7) and (3.16) if we assume a single control point. The results of Askin and Goldberg (2002) can be extended to multiple stages to conclude n * 2 i im h ad m. If any constraint (3.16) is violated, then set max n min j n j. Case ii) Variable processing time operations: Here, production lead time is a function of container size n and is given by the equation: L w( s p n) where, i i i

60 59 si - setup time at stage i. p i - processing time per unit at stage i. w - a flow time constant. In this case, the lead time at a stage is dependent on the container size. WIP cost should therefore be included in the objective function. The term D ( Lj t j1) hj is no longer constant in this case. jm For a given n, the model can still be represented as a shortest path problem. Also, since the container size is fixed across all stages, Theorem 1 is valid for this case too. It is likely that a single control point will be optimal, thus, a two stage solution approach seems reasonable. First, determine n. Setting X m = 1, and X i = 0, i < m, converts the objective function into a single variable function that can be readily solved for the optimal batch size. (The upper bounds for each stage must still be checked for feasibility and n adjusted accordingly if necessary.) Given n, the L i can be computed and model 1 applied to ensure a single buffer.

61 60 To determine n, the following heuristics are proposed: Fixed Point Iterative Heuristic: To simplify the notation for a general control section, we define P ij j p and let ri r Z ij 1, if stages i through j form a control section. 0, otherwise The optimal container size can be obtained (as a function of the control structure) by solving the objective function in Equation (3.2). The objective function in Equation (3.2) can first be solved as an unconstrained optimization problem in n and the optimal container size can be determined as max min( n*,min nj ) jm, where n* is the solution from solving Equation (3.2). Noting that L w( s p n) and i i i j j1, differentiating Equation (3.2) c L t ij j r r ri ri1 with respect to n and setting it to zero, gives: m aid h Z ij h m jp ij z 0 2 w w D hi pi im n 2 j1 i j 2 ij im (3.17) Therefore: n* 2D m i1 m hp m j ij hm z wzij 2 w D hi pi j1 i j ij i1 a i (3.18)

62 61 For a single control section incorporating the entire line, n* 2D m i1 m hp m 1m hm z w 2 w D hi pi 1m i1 a i (3.19) In Equations (3.18) and (3.19), 1m is a function of the batch size n*. It is easy to prove that Equation (3.17) is convex in n. Because the control structure depends on the optimal batch size and the optimal batch size itself depends on the control structure (Equation (3.18)), an iterative procedure is considered. Preprocessing (iter = 0): Determine the optimal batch size, assuming a single control point: n 2D 0 i1 m hp m 1m hm z w 2 w D hi pi 1m i1 m a i Iterations: Repeat until n iter = n iter-1 :

63 62 Calculate the optimal control points using n iter for the batch size, by solving the shortest path problem. Set iter = iter + 1 Update n iter for the control structure determined above Remark: The optimal system cost for model 2 is not necessarily a convex function in n. M jk,the cost of having stages j and k as two consecutive control points is the intersection of a monotonically decreasing convex term ( k i j1 ad i ) and a monotonically increasing n concave term ( zh k j 1, k ). Clearly, the function is continuous and unimodal in n and has a unique minimum. The total cost is the sum of costs of all individual control sections from stage 1 to stage m. And, since the sum of unimodal functions is not necessarily unimodal, for any specified set of control points, the total cost does not necessarily have unique global minimum. The optimal cost function could change control sections as n varies. Let M s (n) represent the total cost of having a particular control section structure, where s belongs to the index set S. Let M(n) represent the optimal cost. Therefore, M ( n) Min M ( n). M(n) is not necessarily convex or unimodal. In such a case, we { ss} s cannot be certain if this solution will converge to the global optimum. As such, we can find limits on the feasible range for the optimum n and search over that range. However, we note in Equation (3.18) that the middle term in the denominator is proportional to h 1/ 2 i pij and is therefore bounded in influence by either the first or third terms (depending on p ij ). In our experiments, we have failed to find any cases in which n *

64 63 varied as a function of the control points selected. Nevertheless, we include below a search algorithm that will ensure finding an optimal solution. Bounded Search Algorithm: An alternative solution procedure is to search over n as follows: Let n 2D * i1 max m hm z w 2 w D hi pi i1 m a i (3.20) Min hi P1 m hj P ij im where, max min, 1i jm ij 1 m The search procedure involves increasing n from 1 to i * Min( nmax, nmax ) { ii} and iteratively calculating the total cost and the optimal control structure. This procedure guarantees an optimal n. We will prove that Equation (3.20) gives an upper bound on the optimal batch size, irrespective of the control structure. * Proof: It is sufficient to show that n max from (3.20) is at least as large as n* defined in Equations (3.18) and (3.19). Comparing Equations(3.18) and (3.19), which represent the batch sizes with multiple control sections and one control section respectively, with Equation (3.20), we note that all the difference between the two equations is in the second term of the denominator. We will show that is a lower bound on this term for

65 64 both equations ((3.18) and (3.19)). (Note: We exclude the constant z simplicity since it appears identically in all cases.) wterm for Let 1 h P min j ij 1i jm ij Min hip1 im and 2 m i { } { im} r1 1m Min h m m p m1 c L t m r r r1 r0 r. hm P1 m Consider 1. 1 by definition. Likewise, 1 is the smallest term in any ij hp m j ij Zij j1 ij ij. hm P1 m Now consider 2. 2 since Min hi hm and 1m ij. im ij Assuming that the number of control sections goes up to two and that stages j and m are the two control stages, the relevant term is now: T 2 h p h P h P 1 j j1, m c L t j r m j 1 j m j1, m r1 r j1 j j j1 m m1 j r r r1 r0 h m p c L t m r r r j1 r j r (3.21) A comparison of Equations (3.20) and (3.21) reveals that T2 2 (since a a1 ( a a1) where a a1 0, b b1 0 ). The proof can similarly be extended to b b ( b b ) 1 1 more than two control sections.

66 Model 2: Section Dependent Container Size We no longer assume that the container sizes are constant across stages; each control section selects a container size. All the other assumptions for model 1 apply. Since an entire container must be removed at a time from the output buffer and likewise we assume that an entire container is started into production instantaneously at the beginning of a control section, there should be a nesting relationship. The upstream container size is an integer multiple of that used in its immediate downstream section, i.e. n r n 1 j j j1 for some non-negative integer r j. (Note, following the work or Roundy (1986), r j may be restricted to powers of two with limited loss). Remaining units from the partially emptied upstream container are kept as input staging at the downstream section until needed. The objective function for this model will now be: 1 a (1 1) id hj X j n j rj hm ( nm 1) Min f X Yn 2 2 j j jm im ij j jm ji z h X Y h ( L t ) D j j ij ij i i i1 jm i j im (3.22) where n j is the container size at stage j. In addition to the constraints in model 1, we have: n Y n ; j M; i 1,..., j (3.23) max j ij i

67 66 ni n j Yij ; i M (3.24) ji Objective (3.22) incorporates the residual container quantity at the input to a control section. Constraint (3.23) ensures that the container size in each control section does not exceed the maximum allowable container size for the stages covered. Constraint (3.24) holds the container size constant with a section Solution Procedure Because the container size in this model is not constant, Theorem 1 is not valid for this model. Moreover, we cannot directly estimate the container size as it depends on the control structure and the control structure itself depends on the container size. For this reason, we provide a heuristic technique for this model. We can still formulate the model as a shortest path problem, as in Figure 3.2 if we can estimate container sizes. We first preprocess the data to estimate a container size for each stage using an EOQ model. Working backwards from stage m to 1 n 0 k 1/ 2 2Dak hk m 1/ 2 2D al 0 lk max nk 1, } hk if n n 0 0 k k1 otherwise (3.25) Equation (3.25) ensures that the container size at a stage is greater than that at its downstream stages.

68 67 The arc costs for the two cases of fixed and size dependent lead times are then: Case i) In general: k ad h ( n n ) M f h L t D z h (3.26) l * 0 l j1 k jk k 1 jk k ( * l l l1 ) k j1, k n jk 2 l( j, k ] j 0,..., m 1; k j 1,..., m where, k 2 D al * max max lj1 n jk min n j1,..., nk, hk (3.27) Once we have the optimal container sizes and the control sections, the actual costs are recalculated by replacing the estimated container sizes given by Equation (3.25) with the actual container sizes, in the cost expression. Here, we use two techniques. In the first technique, the nesting property discussed earlier is ignored and the container sizes are calculated according to Equation (3.27) for both models. In the second technique, the container size in the last section is calculated using Equation (3.27). For a range of r j values from the set, {1, 2, 4, 8, 16}, all other section container sizes are simultaneously calculated by minimizing the cost expression given in Equation (3.22) using a onedimensional search technique over n.

69 68 Theorem 1 can be modified for evaluating the shortest path model in this case. Theorem 2: For the cost structure defined in model 2 a single control point is always optimal, i.e., stage m is the only control point, if for all stages j, f j j m m al ald ad l * * 1 1 l j 1 m ( 0, m j, m ) l l h n n D * * * n0, m n0, j n j, m 2. * 0 hj ( n0, j n j1) z h h h 2 m 1, m j 1, j m j1, k Proof: The proof follows the same procedure used in Theorem 1. Case ii) The only difference between Case (i) and Case (ii) is that the lead time depends on the container size in Case (ii). Arguments similar to those used in Theorem 2 can be applied to Case (ii) after including WIP cost. However, the difference is in the calculation of the optimal container size at each control section. In general: k ad h ( n n ) M f h L n D z h l * 0 k l j1 k jk k 1 * jk k ( ) * l l jk k j1, k n jk 2 lj1 j 0,1,..., m 1; k j 1,..., m (3.28) where, * n jk can be determined by minimizing: k ad h ( n n ) f ( n ) h ( L ( n ) t ) D z h l 0 k l j1 k jk k 1 jk l l jk l1 k j1, k n jk 2 lj1 j 0,1,..., m 1; k j 1,..., m (3.29)

70 69 Subject to: n n l ( j, k] (3.30) jk max l n (3.31) 0 m * k k1 kk n max n, n k 1,..., m (3.32) and L ( n ) k ( s p n ). (3.33) l jk l l jk Note that the optimal solution to model 1 is always feasible to model 2. Therefore model 1 costs represent an upper bound on the optimal model 2 costs. However, since the solution to model 2 is only a heuristic, it may not give better solutions than model 1 on all occasions. Thus it is necessary to consider the model 1 solution as well as the model 2 heuristic solution and select the best. This is evident from a comparison of rows 3 and 11 and 4 and 12 respectively in Table 3.1.

71 70 Table 3.1: Comparison of models 1 and 2 Model f h a D Control Points n Cost 1 0 1,1,1,1,1,1 1,1,1,1,1, ,1,1,1,1,1 32,16,8,4,2, ,2,3,4,5,6 1,1,1,1,1, ,2,3,4,5,6 32,16,8,4,2, ,1,1,1,1,1 1,1,1,1,1, ,1,1,1,1,1 32,16,8,4,2, ,2,3,4,5,6 1,1,1,1,1, ,2,3,4,5,6 32,16,8,4,2, ,1,1,1,1,1 1,1,1,1,1, ,1,1,1,1,1 32,16,8,4,2, ,2,3,4,5,6 1,1,1,1,1, ,2,3,4,5,6 32,16,8,4,2, ,1,1,1,1,1 1,1,1,1,1, ,1,1,1,1,1 32,16,8,4,2, ,2,3,4,5,6 1,1,1,1,1, ,2,3,4,5,6 32,16,8,4,2,

72 Computational Results Experiments are conducted to evaluate the conditions under which a single control section is optimal and the comparison of costs and buffer locations when multiple control sections are preferred Experiment Design An experiment is conducted to compare the performance of models 1 and 2 for both cases. For model 2, experiments are run both with and without nesting. The different factors used in the experiment and the levels of each factor are shown below: For cases 1 and 2 f = $0, $100 a = [1, 2, 3, 4, 5, 6], [1, 1, 1, 1, 1, 1], [6, 5, 4, 3, 2, 1] h = [1, 1, 1, 1, 1, 1], [1, 1.2, 1.4, 1.6, 1.8, 2.0],[1, 2, 3, 4, 5, 6] D = 100 units per day n max =, 25 units (maximum allowable container size at each stage) α = 0.99 (service level) σ = 5 units. c = [1, 1, 1, 1, 1, 1], [0.1, 0.1, 0.1, 0.1, 0.1, 0.1] day (collection time) t = [1, 1, 1, 1, 1, 1], [0.01, 0.01, 0.01, 0.01, 0.01, 0.01] day (transfer time)

73 72 For case 1 L = [1, 1, 1, 1, 1, 1], [0.1, 0.1, 0.1, 0.1, 0.1, 0.1] days (case 1 production lead time) For case 2 s = 0.05 p = w = 1 Therefore, a total of 288 runs are carried out for case 1 of each model and 144 runs for case 2. The programs are written in C++. Golden Section Search is used to determine the batch size for case 2 of both models. Each run took less than one second to run Discussion of Results An analysis of the results of the experiments shows that for case 1, for both models 1 and 2, the average cost is around $1495, and for case 2, it is around $1714. The costs for both models are approximately equal because in a majority of the experiments, model 1 and model 2 gave similar solutions. Table 3.2 below summarizes the results of the analysis. The table indicates that model 2 does not increase the number of control points in most cases, especially for Case 1. From row 3 of the table it is evident that this occurs only once for case 1 and only fourteen times for case 2, whereas, from row 4 it is evident that model 2 reduces costs by having different container sizes in different control sections, five times and twenty times for Case 1 and Case 2 respectively.

74 73 For both models, the maximum number of control points found is two for Case 1, while for Case 2, the number of control points is 2 for Model 1 and 3 for Model 2.. Also, from rows 5 and 6 it is clear that on an average, model 1 has approximately the same number of control points as model 2. It is seen that in most cases, model 2 reduces cost without increasing the number of control sections. When the nesting property is enforced, the container size in the one section is either equal to or twice the container size in the following section. When the container sizes in the two sections are equal, model 2 is no different from model 1. When they are different, it is seen that the nested model outperforms model 1 and model 2 without nesting eight times for case 2 Tables 3.3 and 3.4 list the combination of factors for which cases 1 and 2 of model 1 give a two control point solution. It is obvious from the two tables that the holding cost has a significant effect on this, whereas none of the other factors do. It is seen that a two control point solution is optimal only when h goes from 1 to 6 for the stages. Similarly, Tables 3.5 and 3.6 list the combination of factors for which cases 1 and 2 of model 2 give a two control point solution. While, in general, a similar pattern is observed, it can be seen that Model 2 does increase the number of control sections. This increase in the number of control sections may or may not be accompanied by a resulting decrease in cost over Model 1. Further experimentation reveals that there is a relation between the value-added structure and the number of control sections. These experiments are discussed in the following section.

75 74 Table 3.2: Summary of results for experiment in Section # (%) of times model 2 cost < model 1 cost (without nesting) # (%) of times model 2 cost < model 1 cost (with nesting) # (%) of times model 2 CP > model 1 CP # of times models 2 reduces cost by having different batch sizes in different sections (without nesting) Average # of CP for model 1 Average # of CP for model 2 Maximum # of CP for model 1 Maximum # of CP for model 2 Case 1 Case 2 5 (1.74%) 10 (6.9%) 0 (0%) 8 (5.56%) 1 (0.35%) 14 (9.72%)

76 75 Table 3.3: Combination of Factors for which model 1 case 1 gives a two-control point Solution a h nmax L c t Cost N control points cost (assuming single control point) [1,...,6] [1,...,6] inf [1,...,1] [1,...,6] [1,...,6] inf [1,...,1] [1,...,6] [1,...,6] inf [0.1,...,0.1] [1,...,6] [1,...,6] 25 [1,...,1] [1,...,6] [1,...,6] 25 [1,...,1] [1,...,6] [1,...,6] 25 [1,...,1] [1,...,6] [1,...,6] 25 [0.1,...,0.1] [1,...,1] [1,...,6] inf [1,...,1] [1,...,1] [1,...,6] inf [1,...,1] [1,...,1] [1,...,6] inf [1,...,1] [1,...,1] [1,...,6] inf [0.1,...,0.1] [1,...,1] [1,...,6] 25 [1,...,1] [1,...,1] [1,...,6] 25 [1,...,1] [1,...,1] [1,...,6] 25 [1,...,1] [1,...,1] [1,...,6] 25 [0.1,...,0.1] [6,...,1] [1,...,6] inf [1,...,1] [6,...,1] [1,...,6] inf [1,...,1] [6,...,1] [1,...,6] inf [1,...,1] [6,...,1] [1,...,6] inf [0.1,...,0.1] [6,...,1] [1,...,6] 25 [1,...,1] [6,...,1] [1,...,6] 25 [1,...,1] [6,...,1] [1,...,6] 25 [1,...,1] [6,...,1] [1,...,6] 25 [0.1,...,0.1]

77 76 Table 3.4: Combination of Factors for which model 1 case 2 gives a two-control point solution a h c t nmax cost n control points for single control point n cost [1,,6] [1,,6] inf [1,,6] [1,,6] inf [1,,6] [1,,6] [1,,6] [1,,6] [1,,1] [1,,6] inf [1,,1] [1,,6] inf [1,,1] [1,,6] [1,,1] [1,,6] [6, 1] [1,,6] inf [6, 1] [1,,6] inf [6, 1] [1,,6] [6, 1] [1,,6]

78 77 Table 3.5: Combination of Factors for which model 2 case 1 gives a two-control point solution For single control point a h nmax L c t Cost n control points n cost [1,...,6] [1,...,6] inf [1,...,1] [1,...,6] [1,...,6] 25 [1,...,1] [1,...,6] [1,...,6] 25 [0.1,...,0.1] [1,...,1] [1,...,6] inf [1,...,1] [1,...,1] [1,...,6] inf [1,...,1] [1,...,1] [1,...,6] inf [0.1,...,0.1] [1,...,1] [1,...,6] 25 [1,...,1] [1,...,1] [1,...,6] 25 [1,...,1] [1,...,1] [1,...,6] 25 [0.1,...,0.1] [6,...,1] [1,...,6] inf [1,...,1] [6,...,1] [1,...,6] inf [1,...,1] [6,...,1] [1,...,6] inf [1,...,1] [6,...,1] [1,...,6] inf [0.1,...,0.1] [6,...,1] [1,...,6] inf [0.1,...,0.1] [6,...,1] [1,...,6] 25 [1,...,1] [6,...,1] [1,...,6] 25 [1,...,1] [6,...,1] [1,...,6] 25 [0.1,...,0.1]

79 78 Table 3.6: Combination of Factors for which model 2 case 2 gives a two-control point solution for single control point a h c t n control points n cost [1,,6] [1,,6] [1,,6] [1,,6] [1,,6] [1,,6] [1,,6] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [1,,1] [1,,6] [6, 1] [1,,2] [6, 1] [1,,2] [6, 1] [1,,2] [6, 1] [1,,2] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6] [6, 1] [1,,6]

80 Effect of Value Added Structure on Number of Control Sections From theorems 1 and 2, it is clear that the number of control points is dependent on the value added structure and to a much lesser extent on the lead times. In this section we perform a small experiment on a 20-stage pull system. Fixed parameters for the experiment are as follows: f = $0 a = [1, 2, 3,,20] D = 100 units per day n max = (maximum allowable container size at each stage) α = 0.99 (service level) σ = 5 units. c = [0.1, 0.1,.,0.1] day (collection time) t = [1, 1,,1] day (transfer time) The value added structure is carried for the experiment. We consider only case 1 of model 1 because of the similarity between the control structures obtained using models 1 and 2. The results are displayed in Table 3.7. For the parameters described above, when the increase in holding costs is gradual, there are only 2 control points. For example, whether the holding costs go from 1 to 20 in steps of 1 or from 1 to 1000 in steps of 50, a two control point solution is optimal. However, when the holding costs are [1, 2, 19, 24] stages 1, 4, and 20 are selected as control points. When the holding costs are [1, 2,

81 80 19, 25 (and above)], the optimal solution changes to 2, 19, and 20 as control points. Clearly, the holding cost at the final stage has a significant impact on the control structure. In a postponement type strategy where the maximum value-added strategies are carried out at the end, the model tries to push more control points to the end. When the holding cost increases in batches, with the first five stages having a holding cost of 1, the next five having a holding cost of 50, the next five 100, and the final five 150 ([1,...,1,50,,50,100,,100,150, 150]), the result is similar to the gradually increasing holding cost case. A two control point solution is optimal with control points at stages 5 and 20. When we have a structure where the holding cost for the first ten stages is 50 and for the next ten stages is 100, a single control point is optimal. However, when the holding cost for the next ten stages increases above 130, two control points are setup at stages 10 and 20. The same solution is obtained when only the holding cost at the final stage is changed to 130. and locations will stay the same or move downstream. From Equation (3.14) it is clear that a single control point is optimal if: h m z f z h j j ij 1m 1, m for all j M. In all of our test cases, the algorithm places a control point at the first stage where this constraint is violated.

82 81 Table 3.7: Summary of Experiments in Section 4.3 Holding cost structure n control points cost [1,,20] [1,50,,950,1000] E6 [1,2, 19,24] [1,2, 19,25] [1,,1,50,,50,100,,100,150,,150] [50,,50,100,,100] [50,,50,130,,130] [50,,50,100,,100,130] [1,,1,10,100,1000]

83 82 CHAPTER 4: MULTI-PRODUCT STOCHASTIC MODEL 4.1 Introduction In the previous chapter, we discussed the issue of locating control points for a shop manufacturing a single part type, where the part visits each machine once. Queuing aspects were not considered. Processing times were assumed to be deterministic. In this chapter, we extend the models in the previous chapter to a multi-product serial system. Service times and demand interarrival times are now assumed to be random. We assume that sufficient average capacity is available. Since more than one product type is produced, the products compete with each other for the limited machine resources. This results in an increase in the production lead time mean and variance at each stage as compared to the models in the previous chapter. This increase is due to the additional time spent by each part, waiting for service. We still assume a serial product flowing through all stages, i.e., product merge/split issues are not considered. We model the system as a GI/G/1 queue and determine the approximate waiting times by using a Decomposition/Recomposition Technique modeled after Shantikumar and Buzacott (1981). We then build a simulation model to test the accuracy of our expression. This model is described in Chapter 5. Although an approximation, we believe that an open queueing network model for flow between stages within a control section will be accurate unless the system is consistently operating above capacity. Since we have assumed sufficient average capacity, this situation will not occur too often.

84 Assumptions All products have the same set of control points. This assumption is reasonable because it makes the system easier to operate when the manufacturing system produces a large number of parts. All products have the same serial flow. The modeling assumption is applicable for p distinct products or p customized versions of a single product type. Sufficient average capacity is available. Raw materials are always available. The waiting time for a transporter is assumed to be known. FCFS queuing discipline is assumed.

85 Model Formulation The notation used in this model is as follows: p - product index (p = 1 P) c i - collection time for kanbans at stage i f ( ) p d - probability density function of demand per time for product p at stage m. E ( ) D, V( ) 2 p p p p p f i - Fixed cost per time for locating and maintaining inventory at stage i - Percentage of orders satisfied without delay (service rate) z - Standard Normal Variate such that P{ Z Z } M - Set of all stages {1,., m} h - Inventory holding cost per unit per period of product p at stage i, i M ip L ip - Production Lead time at stage i for product p (total time spent by each unit of product p at machine i) t i - transport time from stage i to stage i+1 (including time to wait for move if applicable) X i - Y ij - ij 1 if stage i is a control point 0 otherwise 1 if stage j is a control point that serves stage i 0 otherwise Y defined for j i, i 1,..., m

86 85 aip - Setup cost plus material handling cost per container of product p at stage i n p - Container size for product p ijp - Lead time from stage i to stage j for product p, defined for i being the first stage in a control section ip - Processing time for each unit of product p at machine i Sip - Setup time for part p at machine i W q i - Expected average waiting time (in queue) at machine i ip - Average arrival rate of product p containers at machine i ip - Average service rate of product p containers at machine i i - Traffic intensity at machine i In order to determine the control structure, we need to know the container size for each product. We use a model, similar to the ones described in Karmarkar et al (1985) and Karmarkar (1987). We first determine the expected waiting time and processing time for each part type at each machine separately. The equations are as follows: L W ( S n ) (4.1) ip qi ip p ip j j1 (4.2) c L t ijp j rp r ri ri1 ip D n p p (4.3)

87 86 i P (4.4) p1 ip ip 1 1 L W ( S n ) ip qi ip p ip (4.5) i E P p1 i ip ip, (4.6) where E i is the expected service time at machine i (4.7) P ip i p1 ip Thus, the lead time equation (4.1) now includes the time spent by a part waiting for service. Equation (4.2) states that ijp, the total lead time for a container from the time its kanban is removed at stage j until it reenters the buffer at stage j with replenished stock, is the sum of kanban collection at the final stage, production at each stage, and transit between stages within the control section. Equations (4.3) through (4.7) determine the queuing parameters (See Karmarkar (1987) for details). The expected waiting time can be approximated by the following GI/G/1 approximation: 2 2 Ca C i i i Wq ( S ) i ip npip 2 1 i (4.8) For an M/M/1 system, Equation(4.8) reduces to: W qi i (1 ) i i (4.9)

88 87 Here, 2 C is the squared coefficient of variance (scv) of service times at stage i and is i given by: C 2 i Var[ ] (4.10) i 2 { E[ i ] } E[ i ] is given in Equation (4.6). Var[ i ] is determined from the distribution of the service process at each machine. To determine 2 C a i, the scv of interarrival times, each workstation is treated as a separate GI/G/1 queue. Arrivals to a workstation are treated as a renewal process. The mean time between arrivals is still the reciprocal of the effective arrival rate. If 2 Cd j is the scv of the time between departures at j, 2 C a i, the scv of interarrival times can be calculated by solving the following system of linear equations: m m e k ki k a k i a i k ki ki k k ki i k1 k1 (4.11) p (1 ) C C p ( p C 1 p ) 1 i m (for derivation details the reader is referred to Shantikumar and Buzacott [1981]) where e i is the external arrival rate to workstation i. Note that in our situation, for all workstations except the last one, i.e., e i is zero 0 for i M, i m e P i D p p1 n p i m (4.12)

89 88 pki is the probability with which a job transfers from station k to station i on completion at k. Since we have assumed a serial production system, p ki is either 1 or 0 and m pki 1 i M (4.13) k1 In our situation, there exists a unique j M p ji 1 for each i. Let j(i) indicate j p ji 1. Equation (4.11) reduces to: (1 ) C C ( C ) i M (4.14) e j( i) j( i) a i a j( i) j( i) i j ( i) i j ( i) Therefore, C 2 a i e i j( i) { j( i) ( C C ) } j ( i ) a C j ( i ) a j ( i ) i (4.15) Additionally, the following rules apply: Rule 1: Arrivals at the first stage in the last control section are generated by customer demand, aggregated into batches of size defined by the container size. Rule 2: At the first stage in all other control sections, arrivals are generated by the first stage in the next control section. Rule 3: At all other stages, arrivals are generated by the previous stage. Note that, pj 1, i 1 where Yij 1 X 1 1. (4.16) i

90 89 We summarize the flow relations as: If 1 1 then: Xi p j1, i 1 for j smallest index > i and X j 1 p ji 0 otherwise (4.17) If Xi 1 0 then: pi 1, i 1 p ji 0 j i -1 (4.18) Consider a hypothetical case in a six-stage serial system where stages 4 and 6 are chosen as control points. This situation is shown in Figure 4.1. Conceptually, in this system, arrivals at stages 2, 3, and 6 are generated by stages 1, 2, and 5 respectively. Arrivals at stage 1 are generated by stage 5 and arrivals at stage 5 are generated by end customer demand i Workstation i buffer Flow of material Flow of information Figure 4.1: Hypothetical structure where stages 4 and 6 are control points Therefore, if stage i is the first stage in the last control section (Case 1), then:

91 90 2 For deterministic arrivals, C 0. Otherwise, for each part type, p, a i 2 Ca i is the scv of an entire container. Therefore, C 2 a i, p, the scv for each product at the first stage in the last CS, can be calculated from the distribution of the sum of n p end customer demands. For exponential end customer demand arrivals, the distribution of the entire container is a gamma random variable with a mean of (n/d p ) and a variance of (n/d p 2 ). Therefore: C 2 1 a i, p n p 2 C a i is the weighted sum of the scv for each product: C 2 a i P D p1 P p1 p D / n p p (4.19) If stage i is the first stage in a control section and stage l is the first stage in the next control section (Case 2), then: C e { ( ) } 2 i l l C C l a C l al a i i (4.20) If stage i is not the first stage in a control section (Case 3) then: C e i j { i 1( C C ) } i 1 a C i1 ai1 a i i (4.21) In order to combine Equations (4.19) through (4.21), we define a new variable, ij as follows:

92 91 ib 1 if stage i is the first stage in control section b 0 otherwise Then, C 2 ai e i i 1{ i 1( C C ) } i 1 a C i1 ai1 (1 ib) i m1 e i l, b1 l { l ( C C ) } l a C l al l i 1 ( ib) 1 i m, i b i (4.22) To determine ib Let b i represent the control section in which stage i is. b i is a function of the total number of control points before stage i. For example, in Figure 1, b is 1 for stages 1, 2, and 3 and 2 for stages 4, 5, and 6. b i ( X j) 1 (4.23) ji The subscript i in Equation(4.23) indicates that b is calculated for each stage. ib is therefore 1 if both, i and b i are 1 and zero otherwise. Therefore: ib bi max{0, bi b i1 if i1 } otherwise (4.24)

93 Determining the Control Points Assuming that the container sizes for each product are known, the optimal set of control points can be determined by solving the following problem: Min a D h ( n 1) P P ip p mp p fix i im im p1 np p1 2 P z h X Y h ( L t ) D jp j ij ijp ip ip i1 jm p1 i j im p1 P (4.25) Subject to: Yij 1 i M (4.26) ji Y X i, j M, i j (4.27) ij j Y Y i j M (4.28) ij i1, j, X 1 (4.29) m X [0,1] j M (4.30) j Y [0,1] i, j M (4.31) ij Here, the objective function, Equation (4.25), minimizes the sum of the total location cost, setup cost, work-in-process (WIP) inventory holding cost and finished goods inventory cost for partial containers, safety stock holding cost. The first term in the objective function represents the fixed cost of locating a control point. The second term represents the total period setup and material cost. The third term represents the WIP

94 93 inventory holding cost of the partial container available for final demand where we assume units are consumed one at a time. The fourth term represents the safety stock holding cost at each control point. Note that in evaluating the safety stock, we assume that the number of kanbans is continuous. A more accurate technique of calculating the number of kanbans is described in Section The fifth term represents WIP inventory cost for units flowing through the system. Once again we will use the shortest path transformation for locating the control points. Consider stage 0 to be the input stage. Also, let M oj be the cost of selecting stage j as the first inventory control point and M jk, the cost of having j and k as two consecutive control points. Also define x ij 1 if stages iand j are two consecutive control points 0 otherwise The x ij s can be thought of as arcs connecting successive control points. Then, the above formulation ((4.25) to(4.31)) can be modified as a shortest path problem as shown in Chapter 3 as: Min Mijxij (4.32) im jm Subject to: 1 if i 0 xij xkm 0 if i 0 or m jm km 1 if im where i, j, k M, k i j (4.33)

95 94 x [0,1] i, j M (4.34) ij P a P ipdp hmp ( np 1) If n p is fixed, k1 n 2, is a constant for all feasible solutions. im p1 p p1 Thus, in order to find the optimal solution to the model, we can ignore these two terms from the objective function. In general: k P P M f h ( L t ) D z h (4.35) jk k ip ip i1 p kp j1, kp i j1 p1 p1 Theorem 1 can be modified for evaluating the shortest path model in this case. Theorem 3: For the cost structure defined in Equation (4.35), a single control point is always optimal, i.e., stage m is the only control point, if for all stages j, P f j z hmp 1 mp hjp 1 jp hmp j 1, mp p1. m P j P m P hipdp ( Lip ti 1) hipdp ( Lip ti 1) hipdp ( Lip ti 1) i1 p1 i1 p1 i j1 p1 Proof: The proof follows the same procedure used in Theorem 1.

96 Determining Container Sizes The formulation in Section 4.4 assumes that the container sizes are known. The formulation for determining the optimal container sizes remains the same, with additional constraints: n n i M (4.36) p max i n 0 p 1... P (4.37) p i 1 i M (4.38) Equation (4.38) can be rewritten as: P D p ( Sip np ip ) 1 (4.39) p1 n p The decision variables here are n p, X j, and Y ij. We consider two cases here: Case 1 Equal container sizes: Every product has the same container size n. The above formulation is modified to: a D h ( n 1) Min ( ) 2 P P P P ip p mp hip Lip ti 1 Dp z hjp X j Yijijp im p1 n im p1 jm p1 i j p1 (4.40) Subject to: Yij 1 i M (4.41) ji Y X i, j M, i j (4.42) ij j

97 96 Y Y i j M (4.43) ij i1, j, X 1 (4.44) m X [0,1] j M (4.45) j Y [0,1] i, j M (4.46) ij n n i M (4.47) max i n 0 (4.48) i 1 i M (4.49) Constraint (4.49) can be rewritten as: P p1 D n P (1 ) p p1 S ip ip (4.50) We can solve (4.40) as an unconstrained optimization problem. The optimal container size is then (the derivation is similar to the single product one shown in Chapter 3): n opt 2 P im p1 a D j ' ( ) P rp Wq i P ri ' ij ip ip p qi jm i j p1 ijp im p1 z Z h ( D W ) ip ip (4.51)

98 97 Constraint (4.50) gives us a lower limit on the container size, n min ( n min P Dp Sip p1 ). P (1 ip) p1 Therefore, the optimal container size, * n n n min opt max(, ) where W ' q i dwq i dn. Because the waiting time at a stage depends on its location in the ' control section, it is convenient to calculate W numerically. Here, q i P p 2 2 p p Ca ( n) C ( n) i i p 1 n qi q i P ip p 2 D p 1 p p p1 n W ( n) W ( n 1) ( S n) P Dp 2 2 p p Ca ( n 1) C ( 1) 1 1 i n i p n ( Sip p ( n1)) P 2 D p 1 p p p1 n 1 The solution procedures in this case are similar to the ones described in Section Note that when setup costs are zero, the optimal solution is to have the same n for all p products. This is clear from the objective function(4.40). Here, the objective function is minimized when each n p is 1. D Case 2 Product dependent container size: The container size once again does not vary across sections but each product has a different container size. The optimal container sizes can be determined by solving the formulation given in Equations (4.25) to (4.31) and (4.36) (4.38). Due to the interaction between the different products, finding a closed

99 98 form is not as trivial as in Case 1 and the container sizes and the control structure can be determined by solving the Mixed Integer Non-Linear Programming (MINLP) problem. 4.6 Computational Results Experiments are conducted to answer the following questions: Does the control structure differ for a stochastic model and a deterministic model? Is the control structure dependent on the number of products? As the number of products increases, does the number of control points decrease? The experiment design is described in Section Experiment Design The following parameters are used: P = 1, 2, 5, and 20 (number of products) D p = D/p (Daily demand for each product) we assume that every product has the same demand rate. End customer demand distribution = deterministic, exponential with rate D/p S ip = 0 (Setup time in days for a container of parts) t = 0, 0.5 (transportation time in days for a container of parts) c = 0.1 (collection time in days for a kanban) m = 6, 20 (total number of stages)

100 99 All products are assumed to have the same Uniform processing time distribution per unit. To calculate the upper and lower bounds, utilization factors of 60% and 90% are used. Utilization is the ratio of number of arrivals of containers per day to the number containers of parts served during the day. Therefore, P D 1 P p D p i S ip D pip 0.6 or 0.9 p1 np 1/( Sip npip ) p1 n p (4.52) Since the setup time in our experiments is equal to zero at all stages and the demand per product is known, the processing time per product can be easily computed from the above equation. Holding cost ratio: ratio of h m to h 1 = 6 and 20 Value added structure: increase in h from stage 1 to stage m = linear and delayed capitalization. Figure 4.2 plots the holding cost increase for the delayed linear pattern and Figure 4.3 for the capitalization pattern.

101 Stage h(20) h(6) Holding cost Figure 4.2: Graph showing holding cost at each stage for a 20 stage system with linear growth. The dotted line indicates a holding cost ratio of 20, while the solid line indicates a holding cost ratio of 6.

102 Stage h(20) h(6) Holding cost Figure 4.3: Graph showing holding cost at each stage for a 20 stage system with delayed capitalization. The dotted line indicates a holding cost ratio of 20, while the solid line indicates a holding cost ratio of 6. This results in a total of 128 design points. The container size for each product is ' determined assuming a single control point. For simplicity, we assume W 0. q i

103 Discussion of Results The effect of various factors on the system cost and control structure is discussed in this Section Effect of factors on cost and control structure: Table 4.1 summarizes the results of the experiments. Column A and B list the main effect of each factor on system cost and the number of control points located respectively. Column C lists the total number of solutions with multiple control points for each level while Column D lists the maximum number of control points obtained for each factor level. A complete design table is provided in Appendix 3.

104 103 Table 4.1: Summary of experiments described in Section A B C D Avg. cost Avg. # CP # of multiple CP solutions Max CP Prod Stage h-ratio h-pattern Arr. Dist. t linear delayed comp det expo Figure 4.3 is a plot of the effect of the various factors on system cost. As expected, the average cost goes up with the number of products even though the total demand across all products remains the same. This is due to the increase in the number of parts in the system at a given time. Similarly, the average cost increases with an increase in the number of stages, holding cost ratio, and the transportation time. Also, the cost with deterministic demand is lower than the cost with exponential demand. The average cost for a delayed compensation model is lower than that for the models with linear increase in holding costs. The reasons are discussed in Section

105 M e a n 104 Main Effects Plot for cost Data Means prod stage h-ratio h-pattern arr dist t lin del_comp expo det Figure 4.4: Graph showing the effect of each factor on system cost. Figure 4.5 is a plot of the effect of the various factors on the number of control points. From the Figure, it is obvious that the average number of control points increases with the number of products. Section discusses the true difference in cost, taking into account the time average of inventory value. The average number of control points located also increases with the number of stages, holding cost ratio, and the transportation time. Additionally, there is an increase in the average number of control points when the arrival distribution changes from deterministic to exponential.

106 Mean 105 prod Main Effects Plot for num cp Data Means stage h-ratio h-pattern arr dist t lin del_comp expo det Figure 4.5: Graph showing the effect of each factor on system control structure Effect of holding cost distribution: The average number of control points increases as we move from a linear increase in holding costs to a delayed differentiation scenario. This is consistent with the observations for the single product model (see Section 3.3.3). The average cost however goes down for a delayed differentiation model. It is clear that in a delayed differentiation model, cost is reduced by locating more control points at earlier stages where the holding costs are lower Effect of number of products: Since all products have the same set of control points, an increase in the number of products obviously results in a corresponding

107 106 increase in the total holding cost at the control points. Figure 4.6 plots the effect of the number of products on total cost for different values of the transportation time. From the figure, it is clear that the total cost is directly related to the number of products held at that stage and the lead time demand that is to be satisfied. Therefore, having more control points not only decreases the number of parts held at each control point but also reduces the transportation lead time that needs to be accounted for. Figure 4.7, which plots the effect of the number of products on the ratio of total average cost to the number of control points for different values of the transportation time shows that the model tries to keep the ratio constant as the number of products increases. Figure 4.8 plots the combined effect of number of products and number of stages the system control structure. It is clear from the figure that the average number of control points increases consistently with an increase in the number of products, irrespective of the number of stages in the manufacturing system.

108 Total Cost 107 Effect of number of products on cost t=0.0 t= # of products Figure 4.6: Graph showing the combined effect of transportation time and number of products on the average cost.

109 M e a n 108 Interaction Plot for cost/cp Data Means prod t 0.5 Figure 4.7: Graph showing the combined effect of transportation time and number of products on the ratio of total cost to the number of control points.

110 M e a n 109 Interaction Plot for num cp Data Means prod stage 20 Figure 4.8: Graph showing the combined effect of number of stages and number of products on the system control structure.

111 110 CHAPTER 5: MODEL VERIFICATION AND VALIDATION A simulation model is built in Arena to verify the validity of our mathematical models. A separate model is constructed for each control system architecture. We first discuss the single part deterministic model. Additionally, comparative results for the models in Chapters 3 and 4 with the simulation model will be presented. 5.1 Simulation Model logic In this Section, the logic behind the simulation model is presented Determining the Number of Kanbans We use the following equation to determine the number of kanbans in any control secti on: k j D * c j Yij ( Li ti 1) z hj X j c j Yij ( Li ti 1) i j jm i j n (5.1) The first term in the numerator of Equation (5.1) represents the total lead time demand, and the second term represents the safety stock. Since each kanban authorizes the

112 111 production of one container of parts, the expression is normalized by the container size. The number of kanbans in process at any time at the control section controlled by stage j is determined using Little s law, k, where p j j ij p k j is the number of kanbans in process, ij is the total processing time for a container of parts, and j is the arrival rate of orders for containers. Here, i is the first stage in the control section controlled by stage j Creation of Orders The logic for the single-part, single control point model will be discussed in this section. Figure 5.1 shows the arrival of orders in the system. We model the arrival of containers of parts, rather than individual parts in this simulation model. Since the demand for parts is D units every day and each container n parts, this corresponds to the arrival of an order for a container order every n/d days on an average. The orders enter an orders queue and wait there until they receive a signal code of 231. The logic for this signal code is shown in Figure 5.2 and is explained in Section

113 112 Figure 5.1: Arrival of orders Figure 5.2: Signal code 231 logic