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Elements of kanban production control for dynamic job shops

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Elements of kanban production control for dynamic job shops

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Content ELEMENTS OF KANBAN PRODUCTION CONTROL FOR DYNAMIC JOB SHOPS by Blair J. Berkley A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Business Administration) August 1988 Copyright 1988 Blair J. Berkley UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089 This dissertation, written by Blair J. Berkley under the direction of his Dissertation Committee, and approved by all its members, has been presented to and accepted b y The Graduate School, in partial fulfillm ent of re quirem ents for the degree of Ph.D. Corn fa &5I3 D O C T O R OF PH ILOSOPH Y Dean of Graduate Studies Date August 2, 1988 DISSERTATION COMMITTEE Chairperson ii TABLE OF CONTENTS Page Chapter 1 Introduction 1 1.1 Dynamic Kanban Production Control 3 1.2 Kanban-Controlled Lines 8 1.3 Definition of Key Kanban-Line Terms 10 1.4 Outline of Thesis 14 Chapter 2 Stochastic Kanban-Controlled Lines and 22 Tandem Queues 2.1 Model Descriptions 22 2.2 Markov-Numerical Procedure 25 2.3 Simulation Model 32 2.4 The Single-Product Case 32 2.5 The Multi-Product Case 37 2.6 Conclusions 39 Chapter 3 Decomposition of Kanban Lines and Tandem 48 Queues 3.1 Decomposition of Queuing Networks 48 3.2 Phase-Type Distributions 52 3.3 Literature Review: Tandem Queue 54 Decompositions 3.4 Decomposition of Tandem Queues With 56 Exponential Operations Times 3.5 Decompositions of Tandem Queues With 64 Phase-Type Operation Times 3.6 Summary, Conclusions 71 Chapter 4 Kanban-Controlled Finished-Goods Demand 72 4.1 Purpose of Research 72 iii 4.2 Pseudo Stations Page 75 4.3 Decomposition With Pseudo Stations 79 4.4 Experimental Design 80 4.5 Experimental Results 81 4.6 Summary, Conclusions 83 Chapter 5 A Comparison of Tandem Queues And Kanban- Controlled Lines With Nonzero Conveyance Time 98 5.1 Purpose of Research 98 5.2 Experimental Design 100 5.3 Experimental Results 112 5.4 Summary, Conclusions 118 Chapter 6 Setting Kanban Numbers 128 6.1 Problem Description 128 6.2 Literature Review 134 6.3 Purpose of Research 139 6.4 Experimental Design 143 6.5 Experimental Results 145 6.6 Summary, Conclusions 14 7 Chapter 7 Sequencing On Kanban-Controlled Lines 161 7.1 Problem Description 161 7.2 Sequencing Objectives, Conventional Rules, Information 164 7.3 Literature Review 167 7.4 Purpose of Research, Sequencing Rules Considered 172 7.5 Experimental Design 174 iv Page 7.6 Experimental Results 179 7.7 Summary, Conclusions 182 Chapter 8 Summary and Future Research 191 8.1 Summary 191 8.2 Future Research 195 References 198 V LIST OF FIGURES Page(s) 1.1 Four Station Kanban-Controlled Line 17 1-2 Three Station Kanban-Controlled Line 18 1.3 Description of SLAM Network Symbols 19-20 1.4 Two-Card Kanban System 21 2.1 Tandem Queue 41 4.1 Markov State Diagram For Pseudo Station 85 4.2 Station Four With Kanban-Controlled Finished- 86 Goods Demand 6.1 On-Hand And On-Order Levels in Pull Control 149 Systems 6.2 Closed Queuing Network of Production Kanbans 150 At Station Two 6.3 Closed Queuing Network of Conveyance Kanbans 151 Between Stations Two And Three 7.1 A Multi-Product Kanban-Controlled Station 183 LIST OF TABLES vi Page(s) 2.1 Markov Chain States For (P,C) = (1,1) Line 42 2.2 Numerical Results For Balanced Kanban-Controlled Lines With Exponential Operation Times 43 2.3 Numerical Results For Balanced Kanban- Controlled Lines With Erlang (k=2) Operation Times 44 2.4 Numerical Results For Balanced Kanban-Controlled Lines With Erlang (k=3) Operation Times 44 2.5 Numerical Results For Balanced Tandem-Queues With Erlang (k) Operation Times 45 2.6 Simulation Results For Multi-Product Kanban- Controlled Lines With Exponential Operation Times 46-47 4.1 Description of Markov Chain States For (P,C) = (5,2) Line 87 4.2 Description of Markov Chain States For (P,C) = (3,3) Line 88 4.3 Numerical Results For Three-Station Lines With Exponential Operation Times And Kanban- Controlled Finished-Goods Demand 89 4.4 Decomposition Approximation Results For Three- Station Lines 90 4.5 A Comparison of Numerical And Approximation Results For Average Interstage Inventories of Three-Station Lines 91 4.6 Numerical Results For Four-Station Lines With Exponential Operation Times And Kanban- Controlled Finished-Goods Demand 92-93 4.7 Decomposition Approximation Results For Four- Station Lines 94 4.8 A Comparison of Numerical And Approximation Results For Average Interstage Inventories of Four-Station Lines 95 vii 4.9 Numerical Results For Three-Station Lines With Erlang (k=2) Operation Times And Kanban- Controlled Finished-Goods Demand Page (s) 96 4.10 Decomposition Approximation Results For Three Station Lines With Erlang (k=2) Operation Times 96 4.11 A Comparison of Numerical And Approximation Results For Average Interstage Inventories of Three-Station Lines With Erlang (k=2) Operation Times 97 5.1 A Comparison of Shortage Probabilities And Interstage Inventories For Four-Station Lines With CV2 =1.0 120 5.2 A Comparison of Shortage Probabilities And Interstage Inventories For Four-Station Lines With CV2 =0.50 121 5.3 A Comparison of Shortage Probabilities And Interstage Inventories For Six-Station Lines With CV2 - 1.0 122-123 5.4 A Comparison of Shortage Probabilities And Interstage Inventories For Six-Station Lines With CV2 =0.50 .124-125 5.5 Regression Results: Dependent Variable = Approximate Shortage Probability of Last Station -Simulated Shortage Probability of Last Station 126 5.6 Regression Results: Dependent Variable = Approximate Interstage Inventory Preceding Last Station -Simulated Interstage Inventory Preceding Last Station 126 5.7 MAN0VA Results: Dependent Variable = Approximate Shortage Probability of Last Station -Simulated Shortage Probability of Last Station 127 5.8 MANOVA Results: Dependent Variable = Approximate Interstage Inventory Preceding Last Station -Simulated Interstage Inventory Preceding Last Station 127 viii 6.1 Setting Numbers of Production Kanbans on 3- Product 4-Station Lines With Required Production Ratio 1:1:1 Page(s) 152 6.2 Setting Numbers of Production Kanbans on 3- Product 4-Station Lines With Required Production Ratio 2:1:1 153 6.3 Setting Numbers of Production Kanbans on 3- Product 4-Station Lines With Required Production Ratio 3:1:1 154 6.4 Setting Numbers of Production Kanbans on 3- Product 6-Station Lines With Required Production Ratio 1:1:1 155-156 6.5 Setting Numbers of Production Kanbans on 3- Product 6-Station Lines With Required Production Ratio 2:1:1 157.-158 6.6 Setting Numbers of Production Kanbans on 3- Product 6-Station Lines With Required Production Ratio 3:1:1 159-160 7.1 Performance Effectiveness of Scheduling Rules 184 7.2 Results of The Process Times Study 185 7.3 MANOVA Results For Flow Time 186 7.4 MANOVA Results For Work-In-Process Inventory 186 7.5 MANOVA Results For Finished-Goods Inventory 187 7.6 MANOVA Results For Finished-Goods Conveyance- Kanban Waiting Times 187 7.7 Regression Results For Flow Time 188 7.8 Regression Results For Work-In-Process Inventory 188 7.9 Regression Results For Finished-Goods Inventory 189 7.10 Regression Results For Finished-Goods Conveyance-Kanban Waiting Times 189 7.11 Averages and Standard Deviations of Performance Measures 190 ix ABSTRACT This thesis addresses a two-card kanban-controlled line with stochastic operation times and constant withdrawal or conveyance period. The objective is to construct a queuing network model to be used in a hybrid MRP-kanban production-control system to dynamically set kanban numbers in response to changing shop conditions. Because the kanban-controlled line does not have a product-form solution, the approximation strategy of network decomposition is used. Approximate analysis of kanban-controlled lines begins with a relation between kanban lines and tandem queues. It is shown that, under certain circumstances, kanban lines and tandem queues are equivalent. That is, they have equal average throughputs, work-in- process inventories, and material-shortage probabilities. In particular, single-product kanban-controlled lines are equivalent to a tandem queue when finished-goods demand is infinite and the conveyance period of each station is zero. In other cases, the kanban line may be thought of as a generalization of the tandem queue. Experiments conducted exploit this relation along with a tandem- queue decomposition approximation. The basic idea, to estimate the performance measures of multi-product kanban lines with nonzero conveyance period, is to first find the approximate equivalent tandem queue. Pseudo stations are constructed to block the last station, on the tandem queue, as final stations on kanban lines are blocked when no finished-goods conveyance kanbans arrive from the distribution warehouse. The approximation procedure is then applied to the X equivalent tandem queue with attached pseudo station. Results show this approximation strategy to be accurate when the squared coefficient of variation of operation times is 0.50 or larger and the conveyance period is short relative to the number of conveyance kanbans on the line. It is also shown how the approximation method may be used to set the number of production kanbans. In general, simulation and the approximation procedures often arrive at the same number of required production kanbans when the squared coefficients of variation of operation times is as low as 0.50. Additional simulation results show that the FCFS production- ordering kanban sequencing rule minimizes work-in-process inventories. 1 CHAPTER 1 INTRODUCTION Production control is performed continuously in any productive system. The objectives of production control are to route jobs through the system to maximize total output (revenue) while minimizing job-lateness and work-in-process inventory (investment). It includes the activities of establishing order release dates, job sequences or priorities, and lot or batch sizes. In other words, production control determines what is produced, when, and in what quantities. In a dynamic environment, production control must do more than just establish order release dates, job sequences, and batch sizes. All good production-control systems must have the capacity to react to demand and shop conditions that change on a day-to-day, even hour- to-hour, basis. In fact, control means the ability to compare some existing state with an objective and take action that achieves that objective. For example, if a machine fails, the production-control system must be able to schedule-around the broken machine so that other machines remain as fully utilized as possible. There are several alternative production-control systems available. The three most commonly used are MRP (Material Requirements Planning)- driven shop floor control (SFC), finite- loading systems, and kanban. All three systems require that a master production schedule exist, and that production planning and resource planning have been done. It is assumed below that, regardless of the 2 production control system used at the work-station level, MRP is used at the plant level for detailed demand planning and parts ordering. MRP-driven SFC and finite-loading methods are centralized computer information systems. The centralized systems are driven by the planned order releases and due dates of each part generated by MRP. MRP-driven SFC is often an infinite-loading procedure. Each work station is assumed to have infinite capacity. Jobs are scheduled at stations according to, most commonly, due-date based priority rules. Lot-sizes are calculated within MRP-driven SFC systems using various heuristics [54]. On the other hand, finite- loading systems start with specified capacity levels for each work station or resource grouping; this capacity is then allocated to work orders. OPT (Optimized Production Technology) is a proprietary finite-loading procedure. The fundamental principle of OPT is that only bottleneck operations (resources) are of critical scheduling concern. The argument is that production output is limited by the bottleneck operations, and that increased throughput can only come via better capacity utilization of the bottleneck facilities. To maximize output from bottleneck operations, larger lot sizes are run. The result is to reduce the percentage of nonproductive time devoted to setups in these work stations [54], Kanban is a manual, decentralized, method of production control. Each station maintains a material stock point for containers of its output. Withdrawal of a container of material from the stock point of station i, by a subsequent station, triggers the release of a production-ordering kanban to station i. The released kanbans, or 3 cards, serve as work orders and are posted on station i's production- ordering kanban post. The problems of establishing job sequences and batch sizes are left to the work-station supervisors. Production control systems may be classified into push and pull systems. Push and pull systems are distinguished on the basis of the way in which the order-release function is achieved in the control scheme. A pull system is one in which order release occurs due to physical removal of finished inventory. That is, production is authorized (though not necessarily immediately initiated) by material withdrawal from the output inventory of the work station. In contrast, in a push system, production is authorized in advance of the actual realization of demand. The implication is that there is some advance knowledge of the demand that will occur. This advance notice may be very certain as with a firm customer order for delivery to be made in the future. Alternatively, it may be quite uncertain, being based upon a forecast which is associated with some variance or forecast error. By this criterion, the conventional MRP-driven SFC system is a push system and the kanban approach is a pull system [30]. SECTION 1.1 DYNAMIC KANBAN PRODUCTION CONTROL The kanban production-control systems described in the Japanese manufacturing literature appear to be static [40] [47]. While it undoubtedly happens in practice, there is no discussion of how kanban system parameters might be set for different conditions. Companies such as Toyota do not routinely adjust kanban system parameters for 4 at least three reasons. First, Japanese firms prefer to use excess capacity rather than inventory to meet fluctuations in demand. Second, they have cross-trained workers whom they are able to switch from work station to work station to mitigate temporary bottlenecks. Finally, Japanese workers ordinarily remain on the job until each day's work is completed. This allows firms such as Toyota to vary the length of the work-day to meet changing demands. However, the majority of firms either using, or thinking of using, kanban production control do not exhibit Toyota's characteristics. In such firms it is essential to dynamically adjust kanban system parameters. The hybrid MRP-kanban production system addressed below is suitable for dynamic batch-manufacturing shops. This type of system uses MRP at the plant level for detailed demand planning and parts ordering. Each planning period, finished-good demands are forecasted and used to establish a master production schedule. At the work station level, the method of kanban production control is used. The basic approach is to control kanban system parameters on the basis of requirements generated by the MRP system. Finished-goods demands are forecasted and production is to the master production schedule. Product mix and shop load may change substantially from month to month, but production quantities of each item are fairly high [20]. Operations are performed manually and are unpaced. Advantages of Kanban The use of MRP at the plant level permits use of detailed knowledge of demand variations. At the work-station level, kanban production control has several important advantages over MRP-driven 5 SFC. Kanban's advantages include the incentives at the work-station level to control and reduce production lead times and work-in- process inventory. Reducing production lead times increases inventory turnover and, consequently, increases cash flow. When MRP is used, production lead times are planned by the system. As a result, there is no benefit or reward at the work station level for reducing production lead times, or equivalently, work-in-process inventory. On the other hand, if the kanban system is used and responsibility for inventory levels is assigned to the stations, the incentive to reduce production lead times is strong. This is due to the relation (to be discussed in Chapter 6) between production lead times and the required number of kanbans. In a dynamic production environment with uncertainties such as machine breakdowns, worker absenteeism, parts delivery delay, and changes of customer demands, it necessary for a production control system to generate a new schedule quickly in response to new conditions. The conventional push system makes it difficult to quickly adapt to changes caused at some stages or by changes in demand. This is because, to adapt to these changes, the centralized scheduling system must recalculate the production schedule for each process simultaneously. Due to the large number of calculations required, it is virtually impossible to reschedule frequently. As a result, buffer inventories must be held along the production line to absorb the problems created by inaccurate schedules. The kanban method of production control is more flexible than the conventional approach to scheduling. Since detailed schedules 6 are prepared only for final assembly, the task of rescheduling is simplified. Only the final assembly line need be informed about the timing and quantities required by customers. The final assembly line goes to the preceding stages to withdraw the necessary quantities of parts at the necessary time. The adaptability of the kanban system is a competitive advantage. Kanban production control enables a company to revise the final assembly schedule daily to meet customer demand. In contrast, it is almost impossible for a company using the conventional approach to revise the production schedule daily due to the complexity of the scheduling and loading techniques. The added adaptability of the kanban system gives it a greater capacity, than MRP-driven SFC, to function like a true control system. That is, by definition, a (production) control system must have the ability to compare the existing state and possibly take action that achieves the objectives. A third advantage of the decentralized kanban system over centralized MRP systems is the quality of information used for production control. In a kanban system the queue of work is particularly visible. The kanbans posted on each station's production-ordering kanban post represent the entire queue of waiting work. The queue of work is immediately visible to all workers and supervisory staff. A very important consequence of this visibility is that the size of the work queue, the mix of the items in the queue, and eventually, the average residence time for a work order (kanban) in the queue, become apparent to the station crew. 7 A fourth advantage of the kanban system is the relatively small required planning and control staff compared to conventional MRP- driven scheduling systems. This is because the kanban system is a decentralized control system which distributes some of the planning and control functions to the supervisors of work stations. Also, the task of rescheduling is reduced dramatically since there is no need to renew every schedule for every process. In the conventional push production control system, the number of planning and control staff such as schedulers, dispatchers, and expeditors are quite large, since it requires tremendous time and effort to perform scheduling, loading, dispatching, and expediting. Further, rescheduling requires that every schedule for every station be redone. Kanban Systems There are many alternative kanban production systems being used. Almost every implementation of kanban is unique. Some implementations use one type of kanban (card) while other implementations simultaneously use two types of cards [47]. Other systems do not use cards at all. Rather, production orders take the form of color-coded golf balls or flags. As in general inventory theory, kanban systems may be either constant order-quantity or constant order-period. Under the constant order-quantity kanban system, a predetermined fixed quantity will be ordered when the inventory level recedes to the reorder point. Although the order quantity is fixed, the reorder period is irregular [36] [37] [40]. Under the constant order-period system, however, the reorder period 8 is fixed and the quantity ordered depends on the usage over the review period and production lead time [40]. SECTION 1.2 KANBAN-CONTROLLED LINES The two-card, constant replenishment-period kanban system being considered has been described by Monden [40] and Schonberger (Figure 1.4) [47]. Two card systems have both conveyance (C) and production (P) kanbans. Conveyance kanbans may be thought of as production kanbans for the material-handling operation. Let Pi be the number of production kanbans at station i and Ci be the number of conveyance kanbans between stations i and i+1. The method of kanban production control is most conveniently described by a SLAM diagram (Figure 1.3) [44] adapted from Huang et al. [25], In Figure 1.1, queue nodes 1,2,...,13 enable the accumulation of part containers and/or kanbans. Unlabeled go-on nodes denote branching of entities. Assemble (ASM) nodes require that one entity be at each immediately preceding queue node before the paired entities may be moved. Await nodes Al, A2 A6 hold kanbans (and empty containers in the case of nodes A2,A4, and A6) until they are periodically moved by parts carriers. Starting from station 3, the various steps utilizing the kanbans are: 1. Every half hour or so, a parts carrier takes the conveyance kanbans from node A4 to the stock point of station 2. The time between conveyance kanban movements is called the conveyance period. The conveyance period corresponds to the constant withdrawal period. 9 Because the times between periodic parts-carrier visits is assumed to be constant, this is a fixed-period, rather than fixed-quantity, replenishment system. A group of conveyance kanbans being moved simultaneously from a conveyance kanban post to the previous station's stock point is called a conveyance wave. Thus, the time between conveyance waves is the conveyance period. 2. Assuming parts corresponding to the conveyance kanbans are available in queue node 6, the carrier detaches the production kanbans, which were attached to the full containers of parts, and places these kanbans in the kanban receiving post node A3. 3. For each production kanban that he detached, he attaches in its place one of his conveyance kanbans. The assemble node following queue nodes 6 and 7 models the obvious fact that a full container of parts must be available in node 6 for the carrier to remove. If parts corresponding to the conveyance kanbans are not available in node 6, the conveyance kanbans wait in node 7. 4. The carrier returns the full containers and their conveyance kanbans to the input material queue of station 3 given by node 8. 5. Assuming station 3 is idle, a production kanban is available on the production-ordering kanban post (node 9), and a full container of parts (with an attached conveyance kanban) is available in the input material queue, work begins. The assemble node following nodes 8 and 9 requires that 1) station 3 is idle and 2) a full container of parts and a production kanban are available in, respectively, nodes 8 and 9. When work begins, the conveyance kanban is detached from the full container of parts and placed in the conveyance kanban post node A4. 10 6, Full containers of completed parts are moved to the output material queue of station 3 at node 10 so the parts carrier from station four may withdraw them at any time. 7. Periodically, perhaps every hour, the production kanbans are collected from the kanban-receiving post node A5 and placed on the production-ordering kanban post node 9. Physically, nodes A4, 8, 10, 11, and A5 are located at the stock point of station 3. Only the production-ordering kanban post node 9 is physically located in the work station. It should be noted that the network orientation of SLAM [44] requires that service (production) activities immediately follow queue or assemble nodes. Yet in Figure 1.1, a go-on node is required for branching immediately prior to the service (production) activity. This problem, however, is easily overcome. The portions of the network corresponding to step 5 may be coded using the discrete- event orientation of SLAM. The remaining portions of the model may be written using the network orientation. SLAM then provides the framework to combine model segments written in the two orientations. Alternatively, the entire model may be written using the discrete- event orientation. SECTION 1.3 DEFINITION OF KEY KANBAN-LINE TERMS In the chapters below, the term "kanban-control" refers to the particular two-card, constant withdrawal-period method of production control just described. Lines controlled in this fashion are called kanban-controlled lines or just kanban lines. 11 Input and Output Material Queues Stations on kanban-controlled lines have both input and output material container queues. For example, in Figure 1.1, station three's input and output material queues are, respectively, queue nodes 8 and 10. P, the number of production kanbans at a station, gives the maximum number of containers possible in an output queue. C, the number of conveyance kanbans between two stations, gives the maximum number of containers possible in an input queue. Aggregate Interstage Inventory The sum of in-process inventory at a station's output queue and the input queue of the immediately succeeding station is called aggregate interstage inventory. Aggregate interstage inventory may also be referred to as the total work-in-process inventory between two stations. Production Lead Time Production lead time is defined as the average time taken from the time that a production kanban is removed from a container in a station's output material queue to the time that finished material corresponding to that kanban arrives at the output material queue and is available for use by the succeeding station [35]. Lead time includes container operation time, setup time, queuing time, conveyance time, and kanban-collecting time [40], Conveyance Lead Time Conveyance lead time is defined as the time taken from the time that a conveyance kanban is removed from a container in a station's 12 input material queue to the time that finished material corresponding to that kanban arrives at the input material queue. Blocked and Starved Stations Stations on single-product kanban-controlled lines may be either blocked or starved/short. The idle time of each station is partitioned into blocked and starved time. A station is starved/short if it is idle and has a nonempty production-ordering kanban post. A station is blocked if it is idle and is not starved. Note, in the single-product case, it is not possible for a station to be idle and have both a nonempty production-ordering kanban post and a nonempty input material queue. On multi-product lines, however, a station may be idle and have both a nonempty input material queue and a nonempty production- ordering kanban post. This occurs when the raw materials in the input queue do not correspond to any product-type having a work-order on the production-ordering kanban post. Following the definitions given above for the single-product case, an idle station with nonempty production-ordering kanban post (and nonempty input material queue) is said to be starved. Conveyance Period Conveyance period has been defined as the constant period between subsequent-station parts-carrier visits or parts withdrawal. A very small conveyance period generally indicates a just-in-time system while large periods may indicate a job shop environment [55]. The conveyance period may be reduced by shortening the distance 13 between work stations or by improving the material-handling operation. There is another aspect of frequency of moves that is important. This aspect is the relationship of frequency of moves to quality. Both Hall [23] and Shingo [50] have hypothesized that frequent moves lower defect rates because succeeding work stations find defects more quickly, before a large number of defects have been processed [55]. Finite and Kanban-Controlled Finished Goods Demand Finished-goods demand is modeled in two different ways. The first model, as shown in Figure 1.1, assumes that finished-goods demand is "infinite." That is, upon completion of a container of parts at the last station, its production kanban is immediately placed in the production-ordering kanban post. In contrast, when finished-goods demand is "kanban-controlled" by a finished-goods warehouse, distribution system, or customer, production kanbans remain on the container of finished goods in the last station's output queue (node 10 in Figure 1.2). Production kanbans are detached from containers, placed in the kanban receiving post, and subsequently moved to the production-ordering kanban post, only when a finished-goods conveyance kanban arrives from the warehouse. When finished-goods demand is kanban-controlled, it is assumed that a finite number of finished-goods conveyance kanbans are in circulation between the last station and the warehouse (or customer). The number of finished-goods conveyance kanbans, like the number of kanbans at all other stations, is a decision variable. 14 Multi-Product Kanban-Controlled Lines When there is more than one product, let Pi(j) and Ci(j) be, respectively, the number of production and conveyance kanbans at station i of product j. At the assemble nodes of Figures 1.1 and 1.2 preceding production activities, the following first-come-first- served priority rule is used. If a station is idle, the first production kanban is removed from the production-ordering kanban post which also has a conveyance kanban, of the same product type, and a full container of parts in the corresponding input queue. Also, of course, containers are removed from station's output queues only if "pulled" by a conveyance kanban of the same product type. SECTION 1.4 OUTLINE OF THESIS The objectives of the chapters below are to study some of the properties of kanban-controlled lines with stochastic operation times. These properties, such as how line performance changes with the number of kanbans or sequencing rule, must be understood before kanban production control can be applied to dynamic shop environments. The kanban-controlled line is a very challenging queuing system. It is a collection of linked, closed, queuing networks. Production and conveyance kanbans, as well as material containers, cycle in closed queuing networks of respectively, production and conveyance kanbans at each station. The networks are linked at the assemble nodes in Figures 1.1 and 1.2. The finite buffer capacities at each station, due to the finite number of kanbans on the line, cause 15 blocking of entities. Operation and effective operation times, which include operation and blocking times, frequently have low coefficients of variation. Low coefficients of variation, in turn, make it difficult to formulate the lines or stations as continuous- time Markov chains. In Chapter 2 a relation between kanban-controlled lines and tandem queues is studied. It is found that the kanban line is a generalization of the tandem queue. In certain circumstances (one requirement is zero conveyance period), the two types of lines are equivalent. That is, they have equal average throughput, aggregate interstage inventories, and station shortage probabilities. In Chapter 3 the process of tandem-queue decomposition is reviewed. In particular, Perros & Altiok's [2] [43] and Altiok's [1] [4] methods are given. It is important to note the underlying assumptions of these approximation methods. In Chapter 4 pseudo stations are used to model the arrival of finished-goods conveyance kanbans from the distribution warehouse to the last station. With the addition of a pseudo station, kanban- controlled lines, with kanban-controlled finished-goods demand, become equivalent to tandem queues in some instances. Pseudo stations allow results for lines subject to infinite finished-goods demand to be easily extended to lines with kanban-controlled finished-good demand. In Chapter 5 tandem queues and kanban-controlled lines with nonzero conveyance period are compared. It is found that, if the conveyance period is small relative to the number of conveyance !6 kanbans on the line, the kanban-controlled line is approximately equivalent to some tandem queue. The immediate consequence of this observation is that tandem-queue decompositions may be used to quickly obtain performance measures for kanban-controlled lines. The occasions when the tandem-queue decomposition does not accurately predict kanban performance measures are noted and used, in Chapter 8, to suggest a kanban-line decomposition. In Chapter 6 the equivalence of tandem queues and Altiok1s tandem-queue decomposition are used to determine the required number of production kanbans necessary to minimize expected inventory holding and shortage costs. In Chapter 7 four alternative sequencing rules for kanban- controlled lines are evaluated. The objectives are to meet kanban- controlled finished-goods demand while minimizing inventory holding cost. Chapter 8 is a summary of results. Conclusions from Chapter 4 are used to outline the construction of a kanban-line decomposition. The decomposition procedure uses the framework of Altiok1s [1] [4] method but replaces M/Ph/l/N stations with models having either deterministic interarrival or service times. Figure 1.1 Four Station Kanban-Controlled Line Production Kanban Production-Ordering Kanban Post ASM FINISH Production Activity fT[ Conveyance Kanban Post (This Stages's Input) AS Physical units, container, and conveyance kanban Empty container and conveyance kanban Production Kanban A5 Kanban Receiving Post 11 Production-Ordering Kanban Post ASM ASM Production .Activity flf (This Stage's Output) (This Stage's Input) Conveyance Kanban P08t A4 Physical units, container, and conveyance kanban Empty container and conveyance kanban Production Kanban A3 Kanban Receiving Post Production-Ordering Kanban Post ASM ASM Production Activity ( ] (This Stage's Output) (This Stage's Input) Conveyance Kanban Post A2 Physical units, container, and conveyance kanban Empty container and conveyance kanban Production Kanban A1 Kanban Receiving Post Production-Ordering Kanl Post START ASM Production Activity [T] (This Stage's Output) Figure 1.2 18 Three Station Kanban—Controlled Line Demand For Finished Product [Container and Conveyance Finished Product Kanban) Production Kanban AS Kanban Receiving Post Pro due tion-O r de rin g Kanban Post ASM ASM Production Activity fT| (This Stage's Output) .Conveyance | Kanban Post (This Stage's Input) Physical units, container, and conveyance kanban Empty container and conveyance kanban Production Kanban A3 Kanban Receiving Post Production-Ordering Kanban Post ASM ASM v Production \Activity [2] (This Stage's Output) ^Conveyance ) Kanban Post (This Stage's Input) A2 Physical units, container, and conveyance kanban Empty container and conveyance kanban Production Kanban A1 Kanban Receiving Post Production-Ordering Kanban Post ASM Production Activity j T ] (This Stage's Output) QUEUE AWAIT ACTIVITY ASSEMBLE GO-ON Figure 1.3 Description of SLAM Network Symbols A QUEUE node is a location where entities wait for service. When an entity waits at a QUEUE node it is stored in file IFL. The initial number of entities in the queue at the start of the simulation is IQ. The maximum number of entities allowed in the queue is QC. Entities routed to an AWAIT node must wait until a specified GATE is periodically opened. When the GATE is opened, all entities waiting at the AWAIT node for the gate are permitted to pass through the AWAIT node and are routed to each of the M branches emanating from the AWAIT node. Branches are used to model activities. Only at branches are explicit time delays prescribed for entities flowing through the network. The ASSEMBLE node requires that all incoming queues contribute one entity before the paired entities are simultaneously moved through the ASSEMBLE node. GO-ON nodes are used for branching. Branching causes an entity to be duplicated and routed over each of the M branches emanating from the node. Figure 1.3 continued 20 QUEUE IFL AWAIT ACTIVITY ASSEMBLE * ASM GO-ON Code Definition A Activity number IFL File number IQ Initial number of entities in QUEUE QC Queue capacity M Maximum branches Chat an entity can be routed from a node Figure 1.4 Two-Card Kanban System[47] 21 Dual-Card Kanban Flows K a n b a n /c o n ta in e r s to K a n b a n /c o n ta in e r s o th e r W C 's; k a n b a n fro m o th e r W C ’s f r o m / to O ther u sin g W C ’s S to c k P o in t L S to c k P o in t M I N IN S ta r t h e r e ^ M illin g W o rk C e n te r D rillin g W o rk C e n te r Kty: S ta n d ard co n tain er R o w path K anban collection box E: em pty C onveyance (C ) kanban R o w path W ork center “d isp atch list” or box F: full P roduction (P ) k an b a n o R o w p ath — — — — — - 22 CHAPTER 2 STOCHASTIC KANBAN-CONTROLLED LINES AND TANDEM QUEUES Introduction This chapter presents an introduction to some properties of stochastic kanban-controlled lines. Markov-numerical analysis is used to study how single-product kanban-controlled lines are related to tandem queues. It is found that, when a kanban line has zero conveyance period and infinite finished-goods demand, so that the last station is never blocked, it is equivalent to some tandem queue. That is, the kanban line and equivalent tandem queue have equal average throughput, aggregate interstage inventories, and station shortage probabilities. Therefore, the kanban-controlled line may be thought of as a generalization of the tandem queue. This relation, along with tandem-queue approximation methods, may be used to estimate the performance of alternative kanban-line configurations. Simulation is also used to determine how multi-product kanban- controlled lines, with zero conveyance period and infinite finished- goods demand, relate to tandem queues. To facilitate Markov analysis, results in this chapter are restricted to balanced four- station lines with exponential or Erlang operation times. SECTION 2.1 MODEL DESCRIPTIONS Both kanban and tandem-queue production lines consist of M single-server stations in series. Parts are conveyed in standard containers and the times to fill a container at station A have an Erlang distribution with shape parameter k^ and a mean of l/li£ > 0 23 (£ =1,2,...,M). Stations have no downtime or, alternatively, downtime is included in operation times. Containers at different stations need not be of the same size but it is assumed exactly one full container from station £ -1 is needed to assemble one container of parts at station £? . It is convenient (without being restrictive) to select the unit of time so that the total average work time required for each container is H time units. Thus, when the production line is perfectly balanced, the average operation time at each station is one time unit. The time unit is also chosen so that the mean output rate of the line exactly matches the mean demand rate for finished goods. Each model assumes there is always a supply of raw material ready to be processed at the first station. The Kanban-Controlled Line The two-card system considered has both (C) conveyance and (P) production kanbans. When each station has the same number of production and conveyance kanbans, such systems are referred to as (P,C) kanban systems. The following convention is used: P£ denotes the number of production kanbans at station £. C£ denotes the number of conveyance kanbans between stations £ and £+1. To allow for different numbers of production and conveyance kanbans, four- station line configurations are given by the vector (P1,C1,P2,C2,P3,C3,P4). The two-card kanban system being considered in this chapter follows the description given in Chapter 1 with the following two exceptions. To simplify the Markov analysis below, it is assumed that entities do not stop at await nodes, or alternatively, the time 24 between carrier visits is zero. In particular, this means that conveyance kanbans do not wait at conveyance kanban posts (for example, node A4 in Figure 1.1); in other words, the conveyance period is zero. The assumption also means that production kanbans do not stop at kanban receiving posts (for example, node A3 in Figure 1.1). Both assumptions have no effect on line performance if parts/kanban carriers do not allow empty queues when immediately preceding await nodes in the network are not empty. For the lines described in Chapter 1, however, kanbans wait at both types of await nodes. Conveyance kanbans wait at conveyance kanban posts for the periodic material-handling operation. As the distance between stations increases, the conveyance period increases and the average stay at conveyance kanban posts increases. Also, production kanbans ordinarily must wait to be periodically moved from the kanban receiving post, located at the station's stock point, to the station's production-ordering kanban post at the work station. But assuming production kanbans do not wait at kanban receiving posts is not likely to impair the generality of the results below. This is because, in practice, a station would never be blocked (i.e., have an empty production-ordering kanban post) if production kanbans were available in its kanban receiving post. This is because, to avoid station idle time, the station supervisor would simply go to his stock point to obtain the work orders or production kanbans available in the kanban receiving post. 25 The Tandem Queue The tandem queue considered is a set of finite queues in series and an infinite queue before the first station (Figure 2.1). Each station £ has a capacity of N£ (£ = 2,3,...,M). Capacity includes both the number of containers in queue and the container being served. Station one has an infinite, never empty, queue of raw material. Four-station line configurations are given by the vector (N2,N3,N4). When a container is filled at the £th station, it proceeds to the (£+l)st queue if there is space available. However, if the (£+l)st queue is full at that time, the container is forced to wait at the £th station until a departure occurs from station £+1. During this time, the £th station remains idle and cannot begin work on any other containers which may be in its queue. In this case, the £th station is said to be blocked, and the container waiting to enter the (£+l)st station that caused the blocking is referred to as the blocking unit. By convention, the blocking unit is held in a fictitious "block" queue, of capacity one, at station £. Containers appear in block queues only when stations are blocked, and stations cannot begin filling the next container as long as their block queue is not empty. Blocking of station M is not possible in this model. Queues 2, 3, and 4 precede, respectively, stations 2, 3, and 4. SECTION 2.2 MARKOV NUMERICAL PROCEDURE With Erlang operation times, both kanban-controlled lines and tandem queues may be formulated as continuous - time Markov chains with 26 discrete state space. The Markov chain states and corresponding steady-state flow-balance equations for both models are generated following the procedure described by Hillier and Boling [24]. Below, details are given only for the four-station kanban-controlled line. The tandem queue specifics may be found in [24], Consider first the four-station kanban-controlled line in Figure 1.1 with exponential operation times (k^ — 1, £ = 1,2,3,4). The state vector may be written as number of containers in queue j is given by Qj, j=l,2,...,13. Each entry PA £ is associated with the production activity at station £ (£ =1,2,3,4). PA£ is set to zero or one if, respectively, station £ is idle or busy. The distinct states of the Markov chain correspond to the feasible values of the state vector satisfying: PAl = 1 if Q1 > 0 PA2 = 1 if Q4 > 0 and Q5 > 0 PA3 = 1 if Q8 > 0 and Q9 > 0 PA4 = 1 if Q12 > 0 and Q13 > 0 Q1 + PAl + Q2 - PI Q3 + Q4 = Cl Q5 + PA2 + Q6 = P2 Q7 + Q8 = C2 Q9 + PA3 + Q10 = P3 Qll + Q12 = C3 Q13 + PA4 = P4 To generate the states and flow balance equations, maintain both a list of states already identified but not analyzed [beginning with state (P1-1,1,0,C1,0,P2,0,0,C2,0,P3,0,0,C3,0,P4,0)] and a list of states that have been identified and analyzed. Begin each iteration by Qj > 0 PA£ e{0,1} j = 1,2,...,13 £ = 1,2,3,4 27 selecting a state to be analyzed from the former list. For this state, identify each of the stations that are in the process of filling a container. For each such station, identify the new state if a service completion were to occur. This is done by moving containers to appropriate queues, processing assemble nodes, and moving containers into production activities if possible; For example, consider the first state (Pl-1,1,0,Cl,0,P2,0,0,C2,0,P3,0,0,C3,0,P4,0). Upon completion of production activity one, we move the full container to queue Q2, (PI-1,0,1,Cl,0,P2,0,0,C2,0,P3,0,0,C3,0,P4,0), process the assemble node at station one (PI,0,0,Cl-1,1,P2,0,0,C2,0,P3,0,0,C3,0,P4,0) and station two, and move into production activities one and two (P1-1,1,0,C1,0,P2-1,1,0,C2,0,P3,0,0,C3,0,P4,0). If this new state has not been identified previously, add it to the list of those to be analyzed. To record transitions, four columns are attached to the end of each state vector. To show that state i moves to state j upon completion of production activity Z , the number i is put in attached column Z of state vector j. Continue these iterations until no states remain to be analyzed. Results of this process when P1=C1=P2=C2=P3=C3=P4=1 are given in Table 2.1. Each row in Table 2.1 corresponds to a different Markov chain state. Fifty six states have been identified. The first 17 columns give the value of the state vector (Q1,PAl,Q2,Q3,Q4,Q5,PA2,Q6,Q7,Q8,Q9,PA3,Q10,Qll,Q12,Q13,PA4) for each 28 state/row. The last four columns record the incoming transitions for each state/row. The flow-balance equations, equating the rates into and out of each state, are immediately obtained from Table 2.1. The rate out of state j is given by ( f i ^PAl + l*2pA2 + 1 * 3PA3 + 1 * ^PA 4)Pj where Pj is the steady-state probability of being in state j. The rates into each state are found from the last four columns of Table 2.1. The flow balance equations are then: State 1 ( P1 )P1 s = + P4P6 2 ( + 1*2 )P2 = l*lPl + P4P7 3 ( yl + 1*2 >p3 = 1*1P2 + P4P8 4 ( + y3 )P4 = P2P2 + P4P9 54 ( y2 + P3 )P54 = U1P30 + P4 P35 55 ( yl + U3 + V P55 = U2P27 + P4P 56 56 ( y1 + P3 + V P56 = 1 * 2P25 The procedure described above is readily adaptable to the case of Erlang service times as well. The only differences arise out of the fact that the states are also distinguished on the basis of individual exponential service phases. PAl, is set to zero or r = 1,2,...,k if, respectively, station £ is idle or in the rth x » exponential service phase. When k^=k2=k3=k4=3, the distinct states 29 of the Markov chain correspond to the feasible values of the state vector satisfying: Qj > 0 j - 1,2,...,13 PA£ e {0,1,2,3 ) £ = 1,2,3,4 PAl > 1 if Q1 > 0 PA2 > 1 if Q4 > 0 and Q5 > 0 PA3 > 1 if Q8 > 0 and Q9 > 0 PA4 > 1 if Q12 > 0 and Q13 > 0 Q1 + 6(PAl) + Q2 = PI Q3 + Q4 = Cl Q5 + 6(PA2) + Q6 = P2 Q7 + Q8 = C2 Q9 + 5(PA3) + QIC1 -P3 Qll + Q12 = C3 Q13 + 6(PA4) = P4 where 6(PA£ ) = 1 if PA£ > 0 and 6(PA£ ) = The process of generating the flow-balance equations is modified as follows. Upon identifying a station £ in the process of filling a container, PA£ is incremented by one and no containers are moved if PA £ < k^ . Containers are moved only upon completion of the last exponential service phase--that is, when PA£ — k^ . An example for state transitions in a four-station line with P1=C1=P2=C2=P3=C3=P4=2 and kj^=k2=k3=k4=3 is given below. U 1) 1-^(0,3,1,0,2,0,0,2,0,2,1,3, 0, 1,1, 1,1) (0,2,1,0,2,0,0,2,0,2,1,3,0,1,1,1,1) 0,0,2,1,1) 0,1,1,1, 2) The balance equations may be written in matrix form as PS = 0 (1) where P = [Pj] (j = 1,2 n) is the steady-state probability vector 30 and S = [S^j] (i,j = 1,2,...,n) is the transition-rate matrix. From equation (1), APS + P = P, where A is an arbitrary constant, and P( AS + I ) = P, i.e. , PW = P (2) where W = AS + I and I is an n-dimensional identity matrix. If A is chosen such that A < R \ where R = max Is.. 1 , then W = [w..] is a stochastic matrix. The vector P i I 11 I t] may then be found to an arbitrary degree of accuracy using the basic Markov chain recursive equation p(m> = p(m'i;)W (3) / \ for suitably large m. Here P is the steady-state probability vector after m iterations. In the results displayed below, the iteration process stopped when every element of the vector P changed by less than 10 ^ . Probabilities were normalized after each iteration to satisfy the requirement £ P. = 1 (4) j=l 3 For reasonable convergence rates, the parameter A should be chosen as large as possible subject to the constraint that W be a stochastic matrix. Stewart [51] has found the value of A = .99R ^ to be suitable in most cases. To accelerate convergence, the method of successive overrelaxation with weighting factor oj = 1.3 was used [14]. 31 Overrelaxation takes advantage of (3) but uses each new pfm^as it is /^\ found for calculating P ^ . P<m> = u>(Vw..P0n) + + (1 ~ co)p5:m“1) (5) J i<j 1 i>j 1J 1 J Overrelaxation was chosen over other numerical methods because the process leaves the matrix W unchanged. Hence, only the nonzero elements of W and the position of these elements need be stored. This is important due to the often high dimension of W. Unfortunately, successive relaxation is not guaranteed to converge except under fairly strong conditions. The sufficient condition for convergence does not hold in general for the problem under consideration primarily because of the normalization requirement (4). Nevertheless, the method did converge in every case attempted when the technique was used. To verify convergence, the steady-state probabilities obtained were substituted back into the balance equations and equalities checked. Once convergence to the steady-state distribution is complete, the mean line output, or throughput, rate R may be found from 5 = ^ M ' J Pj (6) jeE J where E is the set of states corresponding to the last station being in its last exponential service phase. Average inventories at each queue are also easily obtained. 32 SECTION 2.3 SIMULATION MODEL To obtain results for kanban-controlled lines with more than one product type, the model in Figure 1.1 was coded using SLAM and the discrete-event orientation [44]. Initially, a series of simulation runs was made to determine when approximate steady-state was achieved. These runs employed a four-station line with up to ten production and ten conveyance kanbans. In these runs the average throughput for each 1000 time unit block (e.g., 1-1000, 1000-2000) was computed so that the approach to steady-state could be studied. It was assumed that steady-state was reached when the mean throughput of subsequent 1000 time unit blocks varied by less than 1%. This goal was achieved by time 7000 for all runs. The simulation results to follow in this chapter are the result of 10 independent runs of length 12000 time units. Statistical arrays were cleared at time 7000 and steady-state data were collected from time 7000 to 12000. SECTION 2.4 THE SINGLE-PRODUCT CASE Tables 2.2, 2.3, and 2.4 give the number of Markov chain states, average throughput and queues for balanced four-station kanban- controlled lines with, respectively, exponential or Erlang k=2 and k=3 operation times. (Results for lines with three and five stations were also obtained but are similar and so are not shown.) Average inventories 1,2,...,13 in Tables 2.2, 2.3, and 2.4 correspond to average queues at nodes 1, 2.... 13 in Figure 1.1. Table 2.5 gives 33 similar results for tandem queues with exponential (k = 1) or Erlang (k = 2,3) operation times. For tandem queues it is assumed that N2=N3=N4=N and for both tandem queues and kanban-controlled lines ki=k2=k3=k4=k. Several observations are immediately apparent: 1. Function of Production (P) and Conveyance (C) Kanbans Ignoring fictitious block queues, each station £ of a tandem production system has only one container queue--an input queue of maximum size N-l. Stations of kanban-controlled lines, on the other hand, have both input and output container queues. For example, in Figure 1.1, station 3's input and output container queues are, respectively, queue nodes 8 and 10. The value of C, the number of conveyance kanbans between two stations, specifies maximum input queue size. The value P, the number of production kanbans, specifies maximum output material queue size. A station on a (P,C) kanban line is blocked when its output queue has P containers and the input queue at the subsequent station has C containers. Hence, when P=1 at all stations, the kanban-controlled line operates exactly like a tandem queue. The output queue of maximum size P = 1 on the kanban line takes the place of the fictitious block queue on the tandem queue. The kanban-controlled line is then a generalization of the tandem queue: The kanban line is more general because the maximum output queue may be any integer P > 1. By comparing Tables 2.2 & 2.5, 2.3 & 2.5, and 2.4 & 2.5 one can verify the equivalence of tandem queues and kanban-controlled lines 34 when P = 1 and operation times are exponential or Erlang. Kanban- controlled lines and tandem queues are said to be equivalent if they have equal average throughput, blocking and shortage probabilities, and aggregate interstage inventories. For example, the (P,C) = (1,1) line in Table 2.2 is equivalent to a tandem queue with N = 2 in Table 2.5. Average container inventories at the input queues 4, 8, and 12 on the kanban line are equal to the averages of queues 2, 3, and 4 on the tandem queue. Average container inventories at the output material queues 2, 6, and 10 on the kanban line are equal to the average blocking probabilities at stations 1, 2, and 3 of the tandem queue. Note, a station on a kanban line is blocked only when all of its production kanbans are in its output material queue. If there is only one production kanban at each station, the average output queue will be one times the blocking probability. Therefore, the kanban lines and tandem queues have equal blocking probabilities. More generally, any tandem queue (N2,N3,N4) is equivalent to a (F1,C1,P2,C2,P3,C3,P4) line if P = 1 at all stations and N2=C1-1, N3=C2-1, and N4=C3-1. Kanban-controlled lines are usually thought of as storing the majority of work-in-process or interstage inventory in output material queues. Finished material waits in the output queue or stock point of the producing station. Parts carriers of subsequent stations periodically remove, or pull, material as it is needed. In contrast, tandem queues are generally thought of as push systems-- storing the majority of interstage inventory in input queues. 35 Stations continue to push material downstream as long as space is available in the subsequent station's input queue. While identical in the P = 1 case, the two methods of production control represent alternative designs for in-process inventory placement. 2. Equality of Average Throughput, Aggregate Interstage Inventories Conversely, any (PI,Cl,P2,C2,P3,C3,P4) line is equivalent, with respect to average throughput, blocking and shortage probabilities, and aggregate interstage inventories, to a tandem queue. The equivalent tandem queue is found by setting Tandem Kanban N2 = Pl+Cl N3 = P2+C2 N4 = P3+C3 This relation can be checked, for both exponential and Erlang operation times, by comparing Tables 2.2 & 2.5, 2.3 & 2.5, and 2.4 & 2.5. Throughputs of these systems are equal and the relation between average interstage inventories is: Tandem Kanban blocking probability Output material queue at station i of station i + *= + input material queue input material queue at station i+1 of station i+1 The right-hand side of the equality is just the aggregate interstage inventory between stations i and i+1 as defined in Chapter 1. On the left-hand side, to obtain the corresponding aggregate interstage inventory of the tandem queue, the blocking probability of station i must be added to the input material queue of station i+1. This 36 corresponds to adding the average fictitious block queue at station i to the average input queue of station i+1. To check equivalence for a particular case, compare the results in Table 2.2 for (PI,Cl,P2,C2,P3,C3,P4) - (3,2,3,2,3,2,3) and results in Table 2.5 for a tandem queue with N2=N3=N4=N=5. Throughput of both lines are .7818. Average interstage inventories of the tandem queue between stations 1 & 2, 2 & 3, and 3 & 4 are, respectively, .2182+2.5508=2.7690, .1419+2.0317=2.1736, and .0829+1.4976=1.5805. Corresponding interstage inventories for the kanban line are 1.2185+1.5505=2.7690, .8670+1.3066=2.1736, and .5566+1.0239=1.5805. Because of the convergence criteria used in the numerical analysis, results in the tables are only very close approximations of corresponding steady-state performance measures. When comparing equivalent tandem queues and kanban-controlled lines, particularly for larger numbers of Markov chain states, the above equalities may not strictly hold due to the limitations of the numerical procedure. It is not necessary that P1=P2=P3=P4 and C1=C2=C3 or that lines be balanced. For example, the following systems are equivalent: PI Cl P2 C2 P3 C3 P4 = N2 N3 N4 3 1 2 1 3 2 1 = 435 2 3 2 2 1 2 2 = 543 1 2 2 3 1 4 1 = 355 The equivalence of kanban-controlled lines and tandem queues has an important consequence to be discussed in Chapter 5. Finally, note that for conveyance kanbans, the averages at queues 3 & 4, 7 & 8, and 11 & 12 sum to respectively Cl, C2, and C3. 37 Average throughput plus the averages at queues 1&2, 5 & 6, 9 & 10, and 13 sum to, respectively, PI, P2, P3, and P4. SECTION 2.5 MULTI-PRODUCT CASE To this point, the discussion has been limited to the single product case. The purpose of this section is to investigate the function of P and C kanbans in the multi-product environment. With the FCFS priority rule described above, the multi-product kanban-controlled line is not exactly equivalent to either a single product kanban-controlled line or any tandem queue. This is because, in the multi-product case, it is possible to have, for example, station 3 in Figure 1.1 idle while both a production kanban is in queue node 9 and a conveyance kanban is in queue 8. This could happen if the production and conveyance kanbans represent different product types. Similarly, at the output queues a full container of parts may be in queue 10 while a conveyance kanban, corresponding to a different product type, may be in queue 11. Results of the simulation study are shown in Table 2.6. All operation times have exponential distributions. Mean(j) gives the mean average operation time for product j at all stations. However, for configurations 11-15 operation times for product types one and two are not equal. For each product j, it is assumed that Pl(j)-P2(j)-P3(j)-P4(j)-P(j) and Cl(j)=C2(j)-C3(j)=C(j). Given are average throughputs for each product-type and average inventories. The associated standard errors are given in parentheses below each 38 simulation result. Average inventories 1, 2....13 in Table 2.6 correspond to average queues at nodes 1, 2....13 in Figure 1.1. The following observations are apparent: 1. Relative production rates are more influenced by numbers of production kanbans than by numbers of conveyance kanbans. For example, compare configurations 2 and 5 in Table 2.6. When product 1 has twice as many production kanbans as product 2, throughput of product 1 is almost twice the throughput of product 2. In contrast, when product 1 has twice as many conveyance kanbans as product 2, its output is only slightly greater than product 2's. When all products have the same number of conveyance kanbans, it is clear that relative production rates will be approximately the item's proportionate share of the total number of production kanbans. For example, in configuration 19 relative production rates are ,5492/(.5492 + .1538 + .1538) = .6410, .1795, and .1795 while proportionate production kanban shares are, respectively, 4/6 = .6667, 1/6 = .1667, and .1667. These results hold even when average operation times for different product types are not equal. For example, see configurations 11-15. Increasing the number of conveyance kanbans of a product-type somewhat increases its relative production rate. This is because of the matching process at the assemble nodes preceding production activities. Suppose a production kanban of product-type two was followed in production-ordering kanban post node 9 by a production kanban of product-type one. If station three were idle and only a conveyance kanban and a full container of parts, corresponding to 39 product one, were in the input queue node 8, then production would begin on product one (skipping type two). The relative production rate of product-type one increases as its production kanbans pass production kanbans of product-type two. Configurations 8-10 show that relative production rates can be managed precisely by using both production and conveyance kanbans. For example, in configuration 8 when product one has twice as many production and conveyance kanbans as product two, its throughput rate is exactly twice as large. 2. In many cases, average aggregate throughputs and queues are approximately equal to those of the single-product kanban-controlled line. For example, aggregate throughput (.2690 + .2690 + .2690 = .8070) and average queues of configuration 16 are not significantly different from those of the (P,C) = (3,3) line in Table 2.2. Regarding equivalence to tandem queues, one can only suggest the following: Average aggregate throughputs and average queues of multi-product kanban-controlled lines are approximately equal to those of single-product kanban-controlled lines. Therefore, multi product kanban-controlled lines are approximately equivalent to some tandem queue. For example, configuration 13 in Table 2.6 is approximately equivalent to a (N2,N3,N4) = (6,6,6) four-station tandem queue. SECTION 2.6 CONCLUSIONS Markov-numerical results of this chapter have shown a relation between single-product kanban-controlled lines and tandem queues. 40 That is, when conveyance periods are zero and finished-goods demand is infinite, kanban lines are equivalent to some tandem queue. Simulation results for multi-product kanban lines, under similar conditions, are also approximately equivalent to some tandem queue. Figure 2.1 Tandem Queue Infinite Buffer Supply Workstation Storage ... 00-‘-‘-*-*-‘0-**0-‘->-*->0-*0'00-*0*00000000000 00 0000000000000000000 --000000000000000000000 00-0-‘--* ---0 -0 --0 --‘J.-0-.‘^ * 0 ---^ 0 -- 00-‘-‘-*-*-*-‘-‘'‘-,'‘^^^'*-‘-* --‘-‘-‘-‘-> -t-*0-‘000000-'0-‘00-iClOOO-‘0000-‘0000'*00 000!>000000000000000000000000-.0-0-‘0000-‘00-‘0-‘00*0-*00-‘0-‘00-‘ ■ fc t*^ ^ Q O ^ ■* • * -* v^ kvaa ka. i aka . ^k a* O aJ a aak Q O - ak ■* Q Q — * Q Q Q -* ( *0 • * O a k a - ■* lA (0 — k — A 0 Q a -J a - A 0 ^^00-‘--‘-'00 000000000 0*00'*--‘0 00 0000-.-00000000 0 000000 0 000 00 OOOOOI500 0-‘0-‘-‘-*--‘^.*-'‘Oi)flOOOOOOOOOOOOOOO-‘ — - - - - - k - » ii***---Jo-,ooooooooo-‘*-‘- ‘- ‘- ‘-*-*-->-‘- ‘--*-**-‘-*octooooooosoooQ(jooo o o o o o o o o o d o --“ *o o q o o o o o o o o o o o o o o o o o o o o o o c i o o o -‘-‘- o o -‘-‘^ o o -‘--‘ OOO-iO - O O O j- OOO OOOOOO OOOOOOOOOO&OOO-.- — --*— >0 0 0 0 0 0 0 0 0 0(300 00 ^ 0 0 * 0 - - - - ‘ -*” * - * 0 0 0 0 0 & ^ - ‘ 00000 00 0 00 0 -‘-‘-‘ -*-‘-*--‘-*'*-‘-*-‘-*00 00 0000 00 ^xiroSviv:;! £x-x& :-?& x :xx;:;x:xxxx:| xxxxxxrxxx xxx;:;xx;v:x> xjxjxjx-xjxvx oo-ooooooooo-o--*ooooooo>*-‘-‘--*-*oooooaoooooooooooooooo-*.-* .->■-*■- — k a* O a* a. a J aa . - k A' . kk . O -ak Q Q ak — k _k ak a a k O O O O O 0 oo o o o I i l U U M ^ O I & k t K i f A U ^ U - k O I U U U U M M N> M M M — -k ^ ^ -k O O O - > t ) « ( l ) l > I D I « < l l « < < l l » - k 0 I I S O « I U t U l k k | U a k O I 0 0 ) O 9 O I > O N O O U O ' ) ( ! ) O « O M - > O a O ' l ( n O ^ O M - ‘ O M M U K ft ft (*1 ft ft ft <0 U - k U ^ U U J k - i - k ft -k t n - - I O O t n « > 0 3 C ! ) O O ( l 0 O O O O - t k . Q w O U ' - C k C ) O - f k O u 3 M - k t a t 0 O ' 4 m - k O - 4 0 0 ' - j O O O U M O O O ( » - J O O O M W O O O ft Ul U ( f t ft ft OtMCOCkl td C i> (0 U 01 ft (ft ft M M U -* ■ ft MM C O -k JO M ft O O O » O O O O O » B O O w O O J k & i n j k O O ( i l O O O O O Q W k i ! ) i i » i ) ) i j ) u i f t k i i j i t k S ( J i ( ( I O « « l 0 > k 0 U > O O O O O 01 U (ft (ft ft ft ft ft C O (0 6> 01 M ft ft ft Of U 0» (0 M M M M M M J O - * . - k - k ^ - k ^ - k _ _ k - k O J i ( i i O O O O O O O - * s u u i O f f l ( i n u - ‘ ( i i a » M f f i u J J k ' ) ( ) ) O J f t U M O Q O o o o O < ) O a ) 0 ' J ® « ' - i k U « - k O « ) » ' i « ) £ 0 3 li r n a* 0 3 H- a c / 3 rt 03 • rt CD 03 o a ►d o H- a ( 0 H a o' M r o Ko 4^ N> Table 2.2 Numerical Results For Balanced Kanban-Controlled Lines With Exponential Operation Times Average Inventories PI Cl P2 C2 P3 C3 P4 S ta te s Throughput I 2 3 4 5 6 7 8 9 10 11 12 13 1 1 1 1 1 1 1 36 .6312 .0000 .3688 .3233 .6767 .1228 .2461 .4816 .5184 .2241 .1448 .6503 .3497 .3688 1 2 1 2 1 2 1 115 .7007 .0000 .2993 .6829 1.3171 .1020 .1973 .9730 1.0270 .1833 .1160 1.2774 .7226 .2993 1 3 1 3 I 3 1 204 .7477 .0000 .2523 1.0609 1.9391 .0873 .1650 1.4694 1.5306 .1556 .0967 1.8933 1.1067 .2523 1 4 1 4 1 4 1 329 .7818 .0000 .2182 1.4492 2.5508 .0763 .1419 1.9683 2.0317 . 1353 .0829 2.5024 1.4976 .2182 2 1 2 1 2 1 2 115 .7007 .4387 .8606 .2442 .7558 .6928 .6065 .3822 .6178 .9183 .3810 .5425 .4575 1.2993 2 2 2 2 2 2 2 204 .7477 .5209 .7314 .5400 1.4600 .7486 .5036 .8081 1.1919 .9418 .3105 1.1071 .8929 1.2523 2 3 2 3 2 3 2 329 .7818 .5822 .6360 .8670 2.1330 .7870 .4312 1.2575 1.7425 .9558 .2624 1.6819 1.3181 1.2182 2 4 2 4 2 4 2 496 .8077 .6296 .5627 1.2143 2.7857 .8150 .3773 1.7211 2.2789 .9649 .2274 2.2619 1.7381 1.1923 3 1 3 1 3 1 3 204 .7477 .8621 1.3902 .1988 .8012 1.2375 1.0148 .3192 .6808 1.5837 .6686 .4652 .5348 2.2523 3 2 3 2 3 2 3 329 .7818 .9997 1.2185 .4495 1.5505 1.3512 .8670 .6934 1.3066 1.6616 .5566 .9761 1.0239 2.2182 3 3 3 3 3 3 3 496 .8077 1.1081 1.0842 .7358 2.2642 1.4349 .7574 1.1011 1.8989 1.7150 .4773 1.5119 1.4881 2.1923 3 4 3 4 3 4 3 711 .8280 1.1956 .9764 1.0473 2.9527 1.4992 .6728 1.5307 2.4693 1.7538 .4182 2.0624 1.9376 2.1720 4 1 4 1 4 1 4 329 .7818 1.2807 1.9376 .1686 .8314 1,7696 1.4486 .2750 .7250 2.2305 .9877 .4072 .5928 3.2182 4 2 4 2 4 2 4 496 .8077 1.4576 1.7347 .3864 1.6136 1.9277 1.2646 .6083 1.3917 2.3543 .8381 .8726 1.1274 3.1923 4 3 4 3 4 3 4 711 .8280 1.6023 1.5696 .6405 2.3595 2.0495 1.1225 .9805 2.0195 2.4436 .7283 1.3726 1.6274 3.1720 4 4 4 4 4 4 4 980 .8445 1.7227 1.4329 .9219 3.0781 2.1460 1.0095 1.3793 2.6207 2.5111 .6444 1.8945 2.1055 3.1555 5 1 5 1 5 1 5 496 .8077 1.6973 2.4951 .1467 .8533 2.2942 1.8982 .2419 .7581 2.8648 1.3275 .3621 .6379 4.1923 5 2 5 2 5 2 5 711 .8280 1.9034 2.2686 .3395 1.6605 2.4874 1.6846 .5425 1.4575 3.0274 1.1446 .7888 1.2112 4.1720 5 3 5 3 5 3 5 980 .8445 2.0767 2.0788 .5679 2.4321 2.6411 1.5145 .8843 2.1157 3.1491 1.0064 1.2565 1.7435 4.1555 5 4 5 4 5 4 5 1309 .8580 2.2241 1.9178 .8242 3.1758 2.7661 1.3759 1.2558 2.7442 3.2436 .8984 1.7515 2.2485 4.1420 5 5 5 5 5 5 5 1704 .8694 2.3509 1.7796 1.1026 3.8974 2.8697 1.2608 1.6495 3.3505 3.3190 .8116 2.2656 2.7344 4.1306 6 1 6 1 6 1 6 711 .8280 2.1128 3.0591 .1300 .8700 2.8138 2.3582 .2161 .7839 3.4902 1.6817 .3260 .6740 5.1720 6 2 6 2 6 2 6 980 .8445 2.3415 2.8140 .3031 1.6969 3.0354 2.1201 .4899 1.5101 3.6860 1.4696 .7196 1.2804 5.1555 6 3 6 3 6 3 6 1309 .8580 2.5377 2.6042 .5105 2.4895 3.2161 1.9258 .8058 2.1942 3.8368 1.3052 1.1583 1.8417 5.1420 6 4 6 4 6 4 6 1704 .8694 2.7077 2.4228 .7457 3.2543 3.3661 1.7644 1.1531 2.8469 3.9564 1.1742 1.6282 2.3718 5.1306 6 5 6 5 6 5 6 2171 .8791 2.8563 2.2646 1.0036 3.9964 3.4927 1.6282 1.5246 3.4754 4.0535 1.0674 2.1208 2.8792 5.1209 6 6 6 6 6 6 6 2716 .8874 2.9871 2.1254 1.2800 4.7200 3.6008 1.5117 1.9150 4.0850 4:1339 .9787 2.6300 3.3700 5.1126 •p> to Table 2.3 Numerical Results For Balanced Kanban-Controlled Lines With Erlang (k=2) Operation Times Average Inventories PI Cl P2 C2 P3 C3 P4 S ta t e s Throughput 1 2 3 4 5 6 7 8 9 10 11 12 13 1 1 1 1 1 1 1 408 .7307 .0000 .2693 .3357 .6643 .0887 .1807 .4793 .5207 .1614 .1079 .6336 .3664 .2693 1 2 1 2 1 2 1 992 .7978 .0000 .2022 .7129 1.2871 .0688 .1334 .9736 1.0264 .1233 .0789 1.2468 .7532 .2022 1 3 1 3 1 3 1 1960 .8380 .0000 .1619 1.1060 1.8940 .0561 .1058 1.4731 1.5269 ,0999 .0621 1.8520 1.1480 .1619 1 4 1 4 1 4 1 3408 .8649 .0000 .1350 1.5063 2.4937 .0474 .0877 1.9743 2.0257 .0839 .0512 2.4529 1.5471 .1351 2 1 2 1 2 1 2 992 .7978 .4794 .7228 .2334 .7666 .6857 .5165 .3567 .6433 .8695 .3328 .5007 .4993 1.2022 2 2 2 2 2 2 2 1960 .8380 .5742 .5877 .5317 1.4683 .7543 .4076 .7750 1.2250 .9071 .2549 1.0448 .9552 1.1619 2 3 2 3 2 3 2 3408 .8649 .6402 .4948 .8660 2.1340 .7983 .3368 1.2234 1.7766 .9286 .2065 1.6082 1.3918 1.1351 2 4 2 4 2 4 2 5432 .8841 .6887 .4272 1.2217 2.7783 .8288 .2870 1.6882 2.3118 .9422 .1737 2.1812 1.8188 1.1159 3 1 3 1 3 1 3 1960 .8380 .9259 1.2360 .1800 .8200 1.2443 .9177 .2850 .7150 1.5395 .6225 .4124 .5876 2.1619 3 2 3 2 3 2 3 3408 .8649 1.0814 1.0536 .4249 1.5751 1.3774 .7577 .6443 1.3557 1.6396 .4954 .8971 1.1029 2.1351 3 3 3 3 3 3 3 5432 .8841 1.1984 .9174 .7119 2.2881 1.4706 .6452 1.0464 1.9536 1.7044 .4115 1.4190 1.5810 2.1159 4 1 4 1 4 1 4 3408 .8649 1.3592 1.7758 .1470 .8530 1.7839 1.3512 .2378 .7622 2.1865 .9486 .3503 .6497 3.1351 4 2 4 2 4 2 4 5432 .8841 1.5559 1.5599 .3545 1.6455 1.9650 1.1508 .5520 1.4480 2.3379 .7779 .7855 1.2145 3.1159 5 1 5 1 5 1 5 5432 .8841 1.7859 2.3299 .1245 .8755 2.3128 1.8030 .2042 .7958 2.8190 1.2968 .3044 .6956 4.1159 Table 2.4 Numerical Results For Balanced Kanban-Controlled Lines With Erlang (k=3) Operation Times Average Inventories PI Cl P2 C2 P3 C3 P4 S ta te s Throughput 1 2 3 4 5 6 7 8 9 10 11 12 13 1 1 1 1 1 1 1 1488 .7840 .0000 .2160 .3434 .6566 •.0709 .1451 .4793 .5207 .1289 .0871 .6254 .3746 .2160 1 2 1 2 1 2 1 3927 .8454 .0000 .1545 .7298 1.2702 .0527 .1018 .9761 1.0239 .0944 .0602 1.2330 .7670 .1546 2 1 2 1 2 1 2 3927 .8454 .5033 .6513 .2265 .7735 .6861 .4684 .3427 .6573 .8494 .3052 .4780 .5220 1.1546 - p ' ■ o - CJN -P- U> Ni t O t - O v O C O ' J O N m 45 Table 2.5 Numerical Results For Balanced Tandem-Queues With Erlang (k) Operation Times Blocking Probabilities Average Queues S tation Station k S tates Throughput 1 2 3 2 3 4 1 56 0.6312 0.3688 0.2461 0.1448 0.6767 0.5184 0.3497 1 115 0.7007 0.2993 0.1973 0.1160 1.3171 1.0270 0.7226 204 0.7477 0.2523 0.1650 0.0967 1.9391 1.5306 1.1067 1 329 0.7818 0.2182 0.1419 0.0829 2.5508 2.0317 1.4976 1 496 0.8076 0.1923 0.1246 0.0726 3.1560 2.5316 1.8928 1 711 0.8280 0.1719 0.1111 0.0646 3.7570 3.0308 2.2911 1 980 0.8444 0.1555 0.1002 0.0581 4.3552 3.5296 2.6916 1 1309 0.8580 0.1419 0.0913 0.0529 4.9514 4.0283 3.0937 1 1704 0.8694 0.1305 0.0839 0.0485 5.5460 4.5267 3.4969 1 2171 0.8790 0.1208 0.0775 0.0448 6.1393 5.0249 3.9010 1 2716 0.8874 0.1124 0.0721 0.0416 6.7317 5.5230 4.3058 2 408 0.7307 0.2693 0.1807 0.1079 0.6643 0.5207 0.3664 2 992 0.7978 0.2022 0.1334 0.0789 1.2871 1.0264 0.7532 2 1960 0.8380 0.1619 0.1058 0.0621 1.8940 1.5269 1.1480 2 3408 0.8649 0.1350 0.0877 0.0512 2.4937 2.0257 1.5471 2 5432 0.8841 0.1158 0.0748 0.0435 3.0891 2.5232 1.9484 3 1488 0.7840 0.2160 0.1451 0.0871 0.6566 0.5207 0.3746 3 3927 0.8454 0.1545 0.1018 0.0602 1.2702 1.0238 0.7669 Table 2.6 Simulation Results For Multi-Product Kanban-Controlled Lines With Exponential Operation Times Average Throughput Example Operation Times Jroduct Number P (l) C(J) P(2) C(2) P(3) C(3) Mean(1) Mean(2) Mean (3) 1 2 3 1 1 1 I 1 0 0 1.0000 1.0000 0.3741 0.3741 (.0013) (.0012) 2 2 1 1 1 0 0 1.0000 1.0000 0.5051 0.2755 (.0010) (.0004) 3 3 1 1 1 0 0 1.0000 1.0000 0.5879 0.2199 (.0014) (.0007) 4 4 1 1 1 0 0 1.0000 1.0000 0.6453 0.1832 (.0018) (.0004) 5 1 2 1 1 0 0 1.0000 1.0000 0.4113 0.3700 (.0011) (.0010) 6 1 3 1 1 0 0 1.0000 1.0000 0.4397 0.3655 (.0014) (.0011) 7 1 4 1 1 0 0 1.0000 1.0000 0.4623 0.3638 (.0012) (.0010) 8 2 2 1 1 0 0 1.0000 1.0000 0.5378 0.2689 (.0015) (.0007) 9 3 3 1 1 0 0 1.0000 1.0000 0.6334 0.2112 (.0012) (.0004) 10 4 4 1 1 0 0 1.0000 1.0000 0.6957 0.1740 (.0022) (.0006) 11 1 1 1 1 0 0 2.0000 1.0000 0.2430 0.2430 (.0008) (.0008) 12 2 1 1 1 0 0 2.0000 1.0000 0.2997 0.1634 (.0016) (.0010) 13 3 1 1 1 0 0 2.0000 1.0000 0.3342 0.1257 (.0016) (.0006) 14 1 2 1 1 0 0 2.0000 1.oooo 0.2670 0.2401 (.0007) (.0006) 15 1 3 1 1 0 0 2.0000 1.0000 0.2827 0.2347 (.0016) (.0015) 16 1 1 1 I 1 1 1.0000 1.0000 1. 0000 0.2690 0.2690 0.2690 (.0008) (.0008) (.0008) 17 2 1 1 1 1 1 1.0000 1.0000 1.0000 0.3984 0.2140 0.2140 (.0016) (.0008) (.0008) 18 3 1 1 1 1 1 1.0000 1.0000 1.0000 0.4873 0.1800 0.1801 (.0013) (.0006) (.0006) 19 4 1 1 1 1 1 1.0000 1. 0000 1.0000 0.5492 0.1538 0.1538 (.0013) (.0005) (.0005) Ox Table 2.6 continued Example Number 1 2 Average Inventories 10 12 13 14 15 16 17 19 1 0.5187 (.0059) 0.9727 (.0114) 1.4016 (.0268) 1.8147 (.0372) 0.5629 (.0049) 0.5725 (.0075) 0.5769 (.0052) 1.1169 (.0071) 1.7198 (.0364) 2.3140 (.0556) 0.4838 (.0040) 0.9588 (.0187) 1.3724 (.0150) 0.5096 (.0054) 0.5292 (.0093) 1.1204 (.0075) 1.5255 (.0319) 2.0357 (.0338) 2.4810 (.0441) 2 0.7340 (.0105) 1.2479 (.0144) 1.7924 (.0296) 2.3617 (.0417) 0.6494 (.0072) 0.6176 (.0126) 0.6015 (.0079) 1.0715 (.0086) 1.4348 (.0408) 1.8219 (.0609) 0.7885 (.0076) 1.2752 (.0237) 1.8392 (.0175) 0.7234 (.0096) 0.6706 (.0148) 1.0673 (.0100) 1.6559 (.0363) 2.1159 (.0382) 2.6622 (.0479) 3 0.5340 (.0123) 0.4709 (.0117) 0.4112 (.0206) 0.3622 (.0170) 0.9235 (.0144) 1.2693 (.0314) 1.6065 (.0315) 0.7362 (.0116) 0.8956 (.0477) 1.0707 (.0551) 0.5048 (.0056) 0.4609 (.0154) 0.3768 (.0109) 0.8627 (.0193) 1.2098 (.0454) 0.7349 (.0144) 0.6316 (.0273) 0.6287 (.0170) 0.5801 (.0269) 4 1.4660 (.0123) 1.5291 (.0117) 1.5888 (.0206) 1.6378 (.0170) 2.0765 (.0144) 2.7307 (.0314) 3.3935 (.0315) 2.2638 (.0116) 3.1045 (.0477) 3.9293 (.0551) 1.4952 (.0056) 1.5391 (.0154) 1.6232 (.0109) 2.1373 (.0193) 2.7902 (.0454) 2.2652 (.0144) 2.3684 (.0273) 2.3713 (.0170) 2.4199 (.0269) 5 0.7500 (.0068) 1.3450 (.0198) 1.8906 (.0285) 2.4429 (.0505) 0.7816 (.0083) 0.7801 (.0104) 0.7765 (.0091) 1.4394 (.0154) 2.1304 (.0365) 2.9029 (.0514) 0.7337 (.0066) 1.3631 (.0282) 1.8133 (.0262) 0.7298 (.0075) 0.7380 (.0113) 1.4377 (.0166) 1.9844 (.0175) 2.5781 (.0334) 3.1821 (.0434) 6 0.4993 (,0064) 0.8675 (.0218) 1.2996 (.0277) 1.7256 (.0536) 0.4380 (.0105) 0.4089 (.0116) 0.3909 (.0124) 0.7509 (.0176) 1.0229 (.0387) 1.2242 (.0535) 0.5305 (.0088) 0.8706 (.0315) 1.4033 (.0302) 0.4956 (.0099) 0.4569 (.0132) 0.7518 (.0216) 1.1907 (.0214) 1.5788 (.0351) 1.9619 (.0439) 7 0.7961 (. 0088) 0.7259 (.0165) 0.6455 (.0166) 0.5956 (.0246) 1.3226 (.0180) 1.8095 (.0359) 2.3242 (.0479) 1.1289 (.0214) 1.3707 (.0382) 1.6933 (.0535) 0.8077 (.0127) 0.7342 (.0255) 0.5809 (.0151) 1.2600 (.0217) 1.7888 (.0420) 1.1311 (.0254) 0.9755 (.0198) 0.9198 (.0358) 0.9040 (.0247) 8 1.2039 (.0088) 1.2741 (.0165) 1.3545 (.0166) 1.4044 (.0246) 1.6774 (.0180) 2.1906 (.0359) 2.6758 (.0479) 1.8711 (.0214) 2.6293 (.0382) 3.3067 (.0535) 1.1923 (.0127) 1.2658 (.0255) 1.4191 (.0151) 1.7401 (.0217) 2.2112 (.0420) 1.8689 (.0254) 2.0245 (.0198) 2.0802 (.0358) 2.0961 (.0247) 9 0.9401 (.0051) 1.6472 (.0144) 2.2889 (.0199) 3.0068 (.0329) 0.9477 (.0067) 0.9461 (.0080) 0.9327 (.0088) 1.7184 (.0090) 2.5026 (.0255) 3.3481 (.0448) 0.9395 (.0089) 1.6944 (.0229) 2.3368 (.0306) 0.9174 (.0094) 0.9365 (.0094) 1.7192 (.0097) 2.4025 (.0170) 3.1021 (.0392) 3.7969 (.0485) 10 0.3060 (.0072) 0.5734 (.0140) 0.9058 (.0190) 1.1627 (.0316) 0.2747 (.0079) 0.2495 (.0083) 0.2445 (.0090) 0.4800 (.0087) 0.6601 (.0257) 0.7841 .0473) 0.3283 (.0086) 0.5474 (.0202) 0.8598 (.0321) 0.3093 (.0104) 0.2555 (.0097) 0.4790 (.0107) 0.7706 (.0193) 1.0511 (.0415) 1.3493 (.0491) 11 1.1042 ( . 0 1 1 0) 1.0139 (.0136) 0.8882 (.0129) 0.8631 (.0186) 1.7219 (.0149) 2.3670 (.0274) 2.9489 (.0420) 1.5166 (.0119) 1.9106 (.0308) 2.2898 (.0636) 1.1000 (.0139) 1.0100 (.0180) 0.9024 (.0255) 1.6883 (.0228) 2.4337 (.0327) 1.5179 (.0146) 1.3939 (.0184) 1.3282 (.0380) 1.2714 (.0311) 12 0.8958 (. 0110) 0.9861 (.0136) 1.1118 (.0129) 1.1369 (.0186) 1.2781 (.0149) 1.6330 (.0274) 2.0511 (.0420) 1.4834 (.0119) 2.0894 (.0308) 2.7102 (.0636) 0.9000 (.0139) 0.9900 (.0180) 1.0976 (.0255) 1.3117 (.0228) 1.5663 (.0327) 1.4821 (.0146) 1.6061 (.0184) 1.6718 (.0380) 1.7286 (.0311) 13 1.2521 (.0043) 2.2217 (.0044) 3.1845 (.0031) 4.1759 (.0048) 1.2181 (.0027) 1.1979 (.0042) 1.1754 (.0041) 2.1908 (.0016) 3.1575 (.0027) 4.1292 (.0059) 1.2704 (.0054) 2.2457 (.0074) 3.2105 (.0082) 1.2265 (.0045) 1.2166 (.0050) 2.1908 (.0018) 3.1691 (.0033) 4.1567 (.0060) 5.1421 (.0035) 48 CHAPTER 3 DECOMPOSITION OF KANBAN LINES AND TANDEM QUEUES Introduction This chapter describes phase-type distributions and reviews the process of tandem-queue decomposition. Two decomposition schemes are presented in detail. Perros & Altiok's [43] approximation method for tandem queues with exponential operation times provides an introduction to the general process of decomposition. The decomposition framework in [43] appears to be applicable to a wide range of open queuing networks [2] [28], The description of this method also provides an introduction to modeling effective operation times with phase-type distributions. The second decomposition scheme [1] models general service-time distributions using phase-type distributions and is a generalization of [43], SECTION 3.1 DECOMPOSITION OF QUEUING NETWORKS Two major categories of queuing networks are open networks [26] and closed networks [19]. Entities may enter and/or leave an open network at any node. On the other hand, a fixed and finite number of entities circulate through a closed network and no external arrivals or departures are permitted. The classical results [5], [19], [26] on exponential closed queuing networks assumed an independence between servers. That is, the service time of a job can be a function of only what is occuring at that server (including its queue). The usefulness of these models lies in the fact that the 49 performance measures for such networks have a product-form solution and can be computed very efficiently. Nodes, or work stations, in both open and closed networks may have either finite or infinite queue capacity. Finite queue capacities may cause what is called Type I or production-system blocking [42]: If the queue of the (i+l)st station is full at the moment of the service completion of the unit at the ith station, then the unit waits, keeping the server at the ith station idle until the server at the (i+l)st station is completed. In Type II, or communication-systems blocking, a server is not allowed to start service until space is available in the subsequent buffer. This definition of blocking is better suited for the modeling of communications systems when the "service" includes transmission of data to the next station. Except in special cases, Type I and Type II blocking are not equivalent [42]. In queuing networks with blocking, the total time a unit spends in the service process is called the effective service time. Effective service time includes the operation time and possibly blocking time. In finite buffer systems with blocking, the server- independence requirements of the classical analyses are no longer met. It is this nonclassical feature that considerably complicates the analysis of finite buffer networks--the product form no longer holds, and efficient analysis is not possible. Approximation strategies are often used in the analysis of queuing networks having either very large state spaces, blocking, or general service-time distributions. When approximation works well it 50 provides nearly exact results at a computational cost significantly less than that of simulation studies. Such an approach will be used in the following chapters. For example, to determine optimal numbers of kanbans one must quickly obtain shortage probabilities and average queues (inventories) for each configuration of kanbans. Perhaps the most important approximation strategy in queuing networks is that of aggregation: One solves portions of the model in isolation and gathers the results together to produce a solution of the whole model. This approach has also been referred to as decomposition. One can view the strategy alternatively as decomposition of the whole model or as aggregation of portions of the model. Similar methods have been used for a long time in combinatorics, systems theory, and artificial intelligence. Indeed the motivation for such methods in queuing networks came from similar approaches in general systems theory and in electrical networks [11]. Decomposition methods can be shown to give exact solutions for certain queuing networks having product-form solutions. For example, consider an open, exponential queuing network of the Jackson [26] type--Specifically, there are M nodes with node i having s^ servers, a FCFS queue discipline, and an unlimited queue capacity. Jobs arrive to each node according to a Poisson process, are randomly routed, and have exponential service times. In this case, Jackson [26] showed that, in equilibrium, each node in the network behaves as if it were an independent M/M/s queue. When a queuing network does not have product-form solution (e.g., has finite queues with blocking, nonexponential service times, 51 or queue discipline other than FCFS), the decomposition strategy will cause some error because the aggregation process does not totally capture the interaction between the individual network nodes. The primary justification used for decomposition approximation is the exact decomposition of product-form networks. This justification becomes less credible as the networks to be solved become less similar to a product-form network. Basic Approach The basic steps of node-by-node decomposition are given by Shanthikumar & Buzacott [49]: 1. Analysis of the interactions (arrival and departure processes) between service centers. 2. Decomposition of the queuing network into subsystems of service centers; e.g., single queuing stations. 3. Analysis of the subsystems in isolation. The subsystems are related to their network surroundings by input (arrival) and output (departure) processes. 4. Aggregation of the results obtained for the subsystems. In the first step, if some of the interactions between network nodes are ignored (for tractability reasons), the decomposition procedure will only be approximate. Pseudo-arrival rates and effective service rates are usually introduced to approximate node interdependencies. In the third step, phase-type distributions are often used to approximate general service-time distributions. 52 SECTION 3.2 PHASE-TYPE DISTRIBUTIONS Phase-type distributions are important in queuing theory because their structure can give rise to a Markovian state description. The concept is based on the representation of general distributions by mixtures of convolutions of exponential distributions. If the phase-type distribution given below 1-a 1-a represents a service-time distribution in the context of queuing theory, then the service mechanism can be interpreted as follows: A customer (who may have a zero service time with probability 1 - aQ> goes to the first phase of service with probability aQ . The same customer goes to the second phase after the first phase with probability a^ and goes out with probability 1 - a^ . A customer cannot go to the first phase when there is another customer in any one of the phases. These phases are fictitious, and physically it may not be possible to identify them, but the structure helps in analyzing the system. A k-stage Erlang distribution is a special case of the distribution shown above in which a^ = 1, y ^ — y , for i-1, 2, ...,k. 53 Cox [15] showed that any distribution having a rational Laplace- Stieltjes transform (LST) can be represented by a sequence of exponentially distributed phases with probabilistic branching as shown above. The LST of any distribution function can be approximated arbitrarily closely by a rational function. Therefore, in principle, Coxian distributions may be used to approximate any general distribution. o Let m be the mean and CV^ be the squared coefficient of variation of a random variable of interest. Phase-type distributions of a specific structure are used depending on the value of CV^. For the case where CV^ < 0.50, the formulas of Sauer and Chandy [46] and Marie [39] are used. It is the generalized Erlang distribution 1-a with the representation of ( a ,S) where a = (1,0,...,0) and S is a kxk matrix: ■v ay 0 0 0 0 0 0 p. c d i 0 0 • • • 0 0 -P ay 0 • • • 0 0 S = 0 -y ay 0 0 -y 54 The number of phases, k, is found by 1/k < CV2 < l/(k-l), and 1 - a = 2k»CV2 + (k-2) - Fk2 + 4 - 4kCV2 2(k-l)(CV2+l) y = [l+(k-l)a]/m For the case of CV2 > 0.50, the formulas developed by Marie [39] are used to form a two-phase distribution with the following ( a,S) representation. a - (1,0) and -Ul ayx S = 0 -y2 where for 0.5 < CV2 < 1, y1 - 1/(m*CV2) y2 = 2/m and a = 2(1 - CV2) For CV2 > 1, Vj = 2/m y2 = l/(m*CV2) a = 0.50/CV2 The above formulas are used to obtain the parameters for the phase-type approximations below. For example, these formulas will be used to obtain a two-moment approximate representation of the blocked time ( 3 ,B) at a station. SECTION 3.3 LITERATURE REVIEW: TANDEM-QUEUE DECOMPOSITIONS The process of tandem queue decomposition is an active area of research. All tandem queues considered have finite queues and blocking. The types of tandem queues considered fall into two types. 55 The first type of tandem queue is used to model a transfer line. Transfer lines have deterministic operation times; the time between work-station or machine failures is stochastic. Machine down times may also be random variables. Gershwin [18] reported on an approximation algorithm for obtaining the throughput and mean queue length at each node of a tandem queue assuming deterministic service times and geometrically unreliable servers. Jafari & Shanthikumar [27] give an approximation method for transfer lines with unreliable stages and possible scrapping of workpieces. A second type of tandem queue is used to model unpaced assembly lines with manual operations or communication systems. Operation times are assumed to be stochastic. This type of model is described by Hillier & Boling [24]. To approximate the interdependencies among nodes of an open network with finite buffer capacity and blocking, Takahashi et al. [53] use pseudo-arrival rates and effective service rates. The pseudo-arrival rate is defined as the rate of arrivals for nonblocked time intervals only. The effective service rate is introduced to reflect the phenomenon that a blocked customer holds a server until the next server is released. The service time is expanded to include the actual service time plus the holding time. Using the pseudo-arrival rates and effective service rates, each node is treated as an M/M/l/N queue. Boxma and Konheim [9] studied three basic tandem configurations with a view to developing an approximation algorithm. The three configurations are 1) two single-server queues in tandem 2) a split configuration consisting of a single-server queue in tandem with two 56 parallel single-server queues and 3) a merge configuration consisting of two parallel single-server queues linked to a single queue. The authors also sketched an approximation algorithm for analyzing arbitrary open exponential queuing networks with blocking. Unfortunately, the computational cost of this method increases rapidly with the number of stations in the network. Brandwagn and Jow [10] have given an approximation method for tandem communication networks with Type II blocking. Finally, Altiok [1] [4] and Perros & Altiok [43] have described decomposition procedures for manual assembly lines. The basic procedure is to analyze each station individually, beginning with the last, and work upstream to the first station. When the first queue is finite, and the effective arrival rate to the first station is unknown, the throughput rate is found by treating the entire line decomposition procedure as a fixed-point problem [4]. These methods are described in detail in the next two sections. SECTION 3.4 DECOMPOSITION OF TANDEM QUEUES WITH EXPONENTIAL OPERATION TIMES Perros and Altiok's [43] approximation procedure decomposes an exponential tandem queuing system into individual queues with revised queue capacities and revised arrival and service processes. Individual queues are then analyzed in isolation. In particular, station i (i= 1,2,...,M) is treated as an M/C^ ^+^/l/N.j+l single server queue. The term C^ denotes a phase-type distribution with M-i+1 exponential phases. For convenience, each phase j of the effective service time distribution is numbered sequentially starting 57 from the number corresponding to the queue currently being analyzed. Station capacities N^, maximum buffer plus the unit in service, are increased by one to accomodate blocking units. The arrival process to each station is (heuristically) assumed to be Poisson. The service process is revised by adding exponential phases to the existing exponential service time to reflect possible delays a unit might undergo due to blocking. For example, a unit starting its service at station M-l first receives an exponentially distributed service with mean 1/ Upon completion of service, if queue M is full and the unit is forced to wait until a departure occurs from station M, it receives a second exponentially distributed service with mean 1/ y ^ (due to the memoryless property of the exponential). Thus, the effective service time at station M-l has a two-phase distribution which is shown as M-l M-1,0 The branching probabilities a^ ^ ^ and a^ ^ © reflect the probability that upon completion of service at station M-l, queue M is, respectively, full and not full. Obviously aj y £ X M + aM 1 0 = ‘ Now consider the (M-2)nd station. This station is modeled as an M/C3/1/Nm 2+1 queue. The effective service time distribution is shown as 58 aM-2,M M-2 M-l M-l.M A unit starting its service at station M-2 first receives an exponential service with mean 1/y ^ Upon completion of this service, the unit may depart with probability aj y [ 2 0 * T^ s equivalent to the unit completing its service time at station M-2 when the (M-l)st queue is not full. However, if the (M-l)st queue happens to be full at that instance, the unit will be blocked. The unit will remain blocked, thus blocking station M-2 for a period of time which depends on the state of the (M-l)st station at the time the blocking occurred. The (M-l)st station, at that instance may be either busy serving or blocked by the Mth station. If it is blocked, then the unit will remain blocked for a period of time equal to the residual service time of the Mth server. Obviously, this period of time is exponentially distributed with a mean of 1/y This is reflected in the figure by allowing the blocking unit to join the Mth phase with probability 2 M' ^OW ^ t*ie (M-l)st station is busy serving, then the unit will remain blocked until a departure occurs from the (M-l)st station. This is reflected in the figure by allowing the unit to join the (M-l)st phase with probability a.., _ . In this case, the period of time during which the unit M-2 .M-l_______________ ____________________ 59 will remain blocked is equal to the residual service time at the (M-l)st station (which has mean 1/p plus a possible blocking delay the (M-l)st station might suffer due to the Mth station being full at the moment of service completion at the (M-l)st station. In general, blocking may be backlogged over a number of successive queues. That is, station i may be blocked by station i+1, while station i+1 is blocked by station i+2, which in turn is blocked by station i+3 and so on. Hence, the effective service time of the ith station is represented by a phase-type distribution involving M-i+1 phases. Let A : Arrival rate to the first station P^(n) : Steady state probability n entities are at station i, i= 1,2 M : Service rate of station i : Capacity of station i (including the unit in service) A^ : Effective arrival rate to station i : Total input rate to station i Let be the steady-state conditional probability that upon service completion at the (i-l)st station the queue at the ith station is full. Let w^(j) (i = 1,2,...,M; j — i,i+l,...,M) be the steady-state probability that the ith station is in the jth phase of its effective service time, given that its queue is full. Then the branching probability a„ is approximated by a. . = t t . w. . (j) (1) ij r+1 l+l J 60 The quantities tt ^ and w^(j) are obtained numerically using the queue-length probability distribution of the ith station. This queue-length distribution is obtained by analyzing the ith station as an ^+^/l/N^+l queue. The quantities P-^(n) can be obtained using Neuts' [41] matrix-geometric procedure. In particular, the queue- length distribution of the ith station (i>l) is obtained using P. (0) ot.R^e. n<N.+l x i x i x Pi(n) = ( (2) P. (0)a.R1 ?i (-X.TT1)e. n = N. + 1 X X X X X X X where P^(0) is obtained using normalization. Ri = X . (X. I. - X.e.a. - T.)_1 i > 1 (3) 1 X XI XXX x — Matrix 1^ is an (M-i+1) dimensional identity matrix, is a horizontal vector with M-i+1 elements all equal to zero except the first element being set to one, and e^ is an M-i+1 dimensional column vector with all elements equal to 1. Finally the matrix T^ is given by 61 -y± ^iai5 i+2 yiai,M-l yiai,M 0 "yi+2 i+1 Mi+l“i+l,i+2 yi+lai+l,M-l ^i+l^i+ljM P,..a. * yi+2ai+2,M~l yi+2Si+2,M ‘yM-l yM-laM-l,M M (4) For queue 1, the queue-length distribution is found using Pl(n) - I 1 - p. (1 - p1)a1R^e1 n = 0 n > 0 (5) where p^ The quantities w^(j) and it £ (i = 2,3,. are obtained using ,M; j - i,i+1,...,M) 62 N Wf(j) = Pi(0)a1R1i*vi(j)/P_L(N1) M-1 (6) (7) where v^(j) is a vector with M-i+1 elements, all equal to zero, except the jth element being equal to 1. Perros and Altiok discuss two tandem-queue models. The first model, model 1, has an infinite first queue. Assuming that the first queue is stable, the effective arrival rate to the ith station is A. = A(i = 1,2,...,M). The arrival process to each station is x approximated by a Poisson process with parameter A. such that the effective arrival rate A^ at this station is equal to A , the line's throughput. It should be mentioned that in order to use (2) for P^(n) one must know in advance the total input rate A^ , rather than the effective arrival rate A_^ , where A_^=A. This quantity may be computed numerically using A± = A/[l - Pj .(N1 + 1)] (8) where P^(N^ + 1) is a function of A. . 63 The approximation algorithm for model 1 can be summarized as Step 0 : Analyze the Mth station as an M/M/1/NM+1 queue. * Calculate the probabilities Pjy[(n) * Calculate n using (7) M * Set i = 1 Step 1 : Analyze the M - i station as an queue * Calculate the probabilities ^(n) using (2)-(5) and (8) * Stop if i = M-l, otherwise * Calculate wM i (j), j = M-i... M using (6) * Calculate t t w . using (7) M-i ° Step 2 : Use the results for . WM ^(j)> anc* (1) to characterize the phase-type effective service time distribution of the M-(i+l) station. * Repeat step 1 setting i = i + 1. In general, then, the procedure calculates steady-state probabilities one station at a time, beginning with the last station. In order to characterize the phase-type effective service time distribution at station i, the branching probabilities a „ , obtained from the steady-state distributions of all downstream stations, must be known. Model 2 has a finite first queue so that, in general, the effective arrival rate to the first queue (i.e., line throughput) A^ is unknown and does not equal A. In this case the entire approximation procedure is treated as a fixed-point problem in A^ • 64 That is, the approximation procedure for model 1 is repeated over and over until the value of A^ converges to a single number. The tandem queue described in Chapter 2, and analyzed by Hillier & Boling [24], has an infinite, never empty first queue. Queues of stations two through M are finite. If station one has an exponential service time with mean 1/w^ , then when station one is not blocked, the output process from station one to station two is Poisson with rate y^. Therefore, an M-station Hillier & Boling type tandem queue may be approximated by an M-l station model 2 [43] tandem queue. Only stations two through M are included in the approximation. The arrival process to the second station is assumed to be Poisson with mean rate y^. Since station two has a finite queue the approximation procedure for model 2 [43] is used. Finally, research has shown that Perros and Altiok's procedure is easily generalized to open queuing networks with split and merge configurations [2] [28], SECTION 3.5 DECOMPOSITION OF TANDEM QUEUES WITH PHASE-TYPE OPERATION TIMES Altiok [1] extends the work of Altiok [4] and Perros and Altiok [43] to tandem queues with phase-type service times. The procedure is very similar to that described above. In Altiok [1], each queue is approximated by an M/Ph/l/N queue with an appropriately chosen phase-type service time distribution and arrival rate A . Consider a particular queue and let i r be the conditional probability that a departure occurs given that the next 65 queue is full, it is in fact the probability that the server under consideration gets blocked at a departure instant. Also, let V represent the period of time the server remains blocked once blocking occurs. Clearly, V is a phase-type random variable since the service times are all phase-type random variables. Let the pair ( g ,B) represent the blocked time V with L being the order of B and B° be the vector of deblocking rates with Be + B°= 0. The total time a unit spends in the service place is called the effective service time and it consists of the original service time and possibly the blocked time. Denote the effective service time by U and define it by ! S w.p. 1 - i r S + V w.p. 7 T where S and V are assumed to be independent. Clearly U is a phase- type random variable represented by the pair <\j r \j 'X/ (a ,S) where S is an (L+K)x(L+K) transition rate matrix given below SKxK i T KxL i ~\ ■ " i i °LxL ' BLxL where a = ( a,0), S° (1-1T ) S B , r 0 ^ A with Se + S 0 and T = uS°g' T is the matrix of transition rates from busy states to blocked states. The particular forms of g , B, and T will be clear when the decomposition method is discussed below. Also note that is equivalent to S at queue M where blocking is not experienced. 66 Thus, the individual queues become M/Ph/l/N queues and can be analyzed by the matrix-geometric method of Neuts [41]. Let P(n) be the steady-state probability of having n units in the system. Then {P(n)} can be found from the expressions P(n) = P(0)aRne n < N (10) P(N) = P(0)aRN_1(-XS_1)e n = N < \ , where R = A(AI~-Aea-S) , and e is a vector of ones. Finally P(0) can be found through normalization. The following expression is used in case a queue has an infinite capacity (e.g., the first queue). P(n) = (l-e)aRne n > 0 (11) ' X , 'V-l . and P(0) = 1 - p where p = A^[-a^S^e^] is the traffic intensity. Finally, note that the throughput X is related to the arrival rate X by X(1-P(N)) = X (12) Contrary to standard applications, the model is used with X predetermined while an appropriate arrival rate X is to be found which results in the desired throughput. The throughput is clearly 67 nondecreasing in X. In view of (10) and Cramer's rule, the throughput rate is a rational and hence a continuous function of X . Thus, in view of the mean value theorem, there exists a unique arrival rate for each Xe [0, X ] where X is the maximum max max throughput. This unique arrival rate can be found through the fixed- point iteration represented by (12). Hence, each node is fully characterized in isolation and can be analyzed as shown above. Next, it is shown how the isolated nodes can be aggregated in an interative manner in order to obtain the performance measures of the original system. The Decomposition Method In the decomposition method, one starts by guessing a throughput rate X which is the same for all the queues in the system. Obviously, A = X]_ for N^ = ° ° . The value of X being fixed, each queue is recursively approximated as if it operated as an M/Ph/l/N+1 queue in isolation with a phase-type effective service time distribution. Notice that a job upon completing its actual service at the (i-l)st queue, encounters an extended delay at this queue if the ith queue is full. The distribution of this blocking delay represented by the random variable V described earlier in this section, depends upon the state of the ith server at the moment of blocking. At this moment, the ith server may be either serving or blocked in phase j, 1 < j < Kj+L. of its effective service process. 68 Let W(i,j) = Pr[the ith server is in phase j| there are N - j _ customers in queue i] Once (P(n)}, n = 0,...,N^+1 are computed using (10) at queue i, w(i,j) can be computed using w(i,j) = Pi(0)aiR^i.gj/P±(Ni) j = ls...,Ki+Li (13) where gj is a column vector of all zeros except the jth element being equal to 1. Now, the blocked time V_^ ^ at queue i-1 can be characterized by the pair (S^_j , B^ ^) where Bi-1 = and Bi-1 = W 1-1) W(i,Ki+Li)]. (14) However, the drawback of the above characterization of V. , is that l-l the number of phases in and consequently increase as i approaches the upstream end of the system. This is not desirable especially when dealing with large K^’s. For this reason, after and B_^ ^ are identified pair (g^. ^ ,B^ ^) is replaced by its approximation using the minimum number of phases possible. That is, if CV^ is the squared coefficient of variation of blocked time, then, from Section 3.2, the number of phases required, k, satisfies 1/k < CV^ < l/(k-l). The resultant phase-type distribution of ^ will be a two-stage Coxian (CV^ >0.5) or generalized Erlang (CV^ < 0.5) with or fewer phases and g^ ^ = (1,0,0,...,0). This way, the number of of phases in the effective service 69 distribution will not explode and it will be possible to analyze any number of queues in tandem. The reason why the queue capacity is increased by one in each intermediate queue, excluding the first queue, is that the (i-l)st server acts an an additional space for the ith queue during the period it is blocked by this queue. The blocking unit, although it waits in front of the (i-l)st server, becomes a part of the ith queue. This is not only an accurate way of modeling blocking but it also provides a mechanism to compute the blocking probability ir^ for server i-1. Since X is the effective arrival rate, one can write the following relationship which is essentially Little's law applied at the (N£+l)st position at queue i: ^i-i^i-AVi-P = v ni+i> <i5> where (“8^ ^ -^) is the expected value of the blocked time at node i-1, and e^ ^ is a vector of ones with an appropriate size. Hence tk ^ can be easily computed from (15) for i — 1,...,M-1. The decomposition process starts with the last queue and works upstream until it reaches the first queue. The procedure stops if the first queue has an infinite capacity. Otherwise, the two successive values of the throughput are compared. The method stops if they are sufficiently close. Otherwise the process is repeated with the most recent value of the throughput. After the procedure stops, the actual P^CN^) can be found from P]C(N^C) = P^C^) + ^k^K+l) for k = 2,...,M. On the other hand, P^N^-t-l) gives the approximate 70 arbitrary time probability that server k-1 is blocked. It is clear that Step 2 of the above procedure consists of a fixed-point problem. The method is summarized as follows: Step 0: Initialization * k = M, £ = 0 * Np - Nj+1, i - 2,... ,M * Select (T(£) = if = o o ) Step 1: * Characterize the effective service process at queue k using (9) and analyze it as an M/Ph/l/N queue. If k = 1 * If server 1 is exhausted, go to Step 2. * If Np = < » , compute P^(n) using (11) and stop. * If Np < ° ° , compute P^Cn) , n = 0, . . . , N^ using (10) and go to Step 2. Otherwise * Compute P^(n), n = 0,..., N^+l using (10) * Compute W(k,j) using (13). * Characterize (3 ,B, 1) using (14) and moment k— 1 k - l approximations. * Compute tt k p using (15). * k = k-1 and repeat Step 2. Step 2: * £ = £+1 * Compute * If | A ^ - A^£ ^ J j< e stop; otherwise A^£^=A^£ ^ and go to Step 1. 71 SECTION 3.6 SUMMARY, CONCLUSIONS This chapter has described the queuing-network approximation strategy of decomposition. When finite buffer space generates a blocking phenomenon, phase-type distributions are often used to model effective service-time distributions. The decomposition methods of Altiok and Perros and Altiok have been described in detail. These decomposition procedures may be used to approximate the Hillier & Boling type tandem queues described in Chapter 2. Therefore, they may also be used to approximate equivalent single-product kanban-controlled lines with infinite finished-goods demand and zero conveyance period. The general framework of Altiok1s [1] decomposition, i.e., phase-type operation times and the use of a fixed-point iteration to solve for the unknown throughput rate, also appears to be the most suitable for the development of a decomposition for kanban- controlled lines with nonzero conveyance period. In Chapter 5 Altiok's approximation procedure is combined with the results of Chapter 2. That is, to estimate the average inventories and shortage probabilities of kanban-controlled lines, with nonzero conveyance period, the kanban line's tandem-queue (approximate) equivalent is found. Then Altiok's method is applied to the equivalent tandem queue. Results in Chapter 5 suggest modifications required, within Altiok's framework, to more accurately decompose kanban-controlled lines. 72 CHAPTER 4 KANBAN-CONTROLLED FINISHED-GOODS DEMAND Introduction In this chapter pseudo stations are used to model the arrival of finished-goods conveyance kanbans from a distribution warehouse and the potential blocking of the last station. The service process of the pseudo station is constructed so that it is identical to the arrival process of finished-goods conveyance kanbans. In this way, the pseudo station blocks the last station on a tandem queue in the same way the last station on a kanban line may be blocked due to lack of finished-goods conveyance-kanban arrivals. SECTION 4.1 PURPOSE OF RESEARCH In Chapter 2 it was shown that when kanban-controlled lines have a single product, zero conveyance period, and the last station is never blocked, they are equivalent to tandem queues. Last stations on tandem-queue lines cannot be blocked because there are no following stations. However, in practice, the last station on kanban-controlled lines may be blocked. This difference is due to the arrival process of work to the line. In the case of a tandem queue, only the first station has external arrivals. In contrast, only the last station on a kanban line, with kanban-controlled finished-goods demand, has external arrivals. ? 73 Tandem Queue Station External Arrival of Work A Station O © • • • ©* External Arrival of Work A Warehouse If finished-goods demand is kanban-controlled, then there are a finite number of finished-goods conveyance kanbans in circulation for each end item. Whenever the warehouse or distribution system requires more finished-goods, it must carry a finished-goods conveyance kanban to the last station. This scheme is shown in Figure 1.2. Finished-goods conveyance kanbans arrive to the stock point of the last station at queue node 11. Assuming products corresponding to the finished-goods conveyance kanbans are available in queue node 11, the carrier detaches the production kanbans, which were attached to the full containers of products, and places these kanbans in the kanban receiving post node A5. For each production kanban that he detached, he attaches in its place one of his finished-goods conveyance kanbans. The carrier returns the full containers and their finished-goods conveyance kanbans to the finished-goods warehouse. This type of production control is called a pull system because finished-goods demands pull parts through the line. 74 The last station on a kanban line may be blocked by the warehouse if, for some reason, the warehouse sends no finished-goods conveyance kanbans to the last station. In this case, no production kanbans would be released from full containers of goods in the output queue, node 10, Figure 1.2, and eventually the last station would have no production kanbans on the production-ordering kanban post. Hence, the last station would be blocked. The objectives of this chapter are to develop a procedure for modeling the process of kanban-controlled finished-goods demands. The idea is to modify Altiok1s tandem-queue decomposition so that the last station is blocked in the same way as if it were subject to kanban-controlled finished-goods demand. If this were possible, Altiok1s tandem queue decomposition could be used to approximate kanban lines with kanban-controlled finished- goods demands. Before describing the procedure to be used, some assumptions must be made. First, there are a finite number of finished-goods conveyance kanbans in circulation between the last station and the distribution warehouse. There is only one product type. It is assumed customer demands, for containers of material, arrive to the warehouse or distribution system according to a Poisson process. In turn, finished-goods conveyance kanbans arrive to the stock point of the last station according to a Poisson process. In particular, it is assumed each finished-goods conveyance kanban arrives, from the warehouse, to the last station according to a Poisson process with rate A. 75 SECTION 4.2 PSEUDO STATIONS The basic idea behind pseudo stations is to construct a pseudo station to be placed after the last station on the tandem-queue line. The pseudo station must be constructed so that it blocks the last station on the kanban line in the same way it would be blocked if subject to kanban-controlled finished-goods demands. Thus, the service rate of the pseudo station must be equal to the arrival rate of finished-goods kanbans. Next, two examples are considered to motivate the construction of the pseudo station. Example 1 Suppose the last station (station three) on the kanban line has five production kanbans. There are two finished-goods conveyance kanbans in circulation between the last station and the distribution warehouse. Each Markov chain state of the pseudo station is given in Table 4.1. The queue nodes in the table, except for node 12, refer to queue nodes in Figure 1.2. States 3 through 7 all have two finished-goods conveyance kanbans residing at the distribution warehouse node 12. States 3 through 7 differ in how many full containers of finished goods are located in the output material queue of station three. Since there are five production kanbans, there may be zero through five containers in the output material queue. States 3 through 7 have, respectively, one through five containers in the output queue node 10. If the five production kanbans are not in the output queue then they must be on station three's production-ordering kanban post 76 node 9 or in the production activity. Since it is assumed there is only a single product-type, if there are any production kanbans, attached to full containers of finished-goods, in the output queue node 10, there cannot be any finished-goods conveyance kanbans in queue node 11. If there were, the conveyance kanban would be matched to a full container of material and returned to the warehouse. Therefore, for states 3 through 7, all two finished-goods conveyance kanbans must be at the warehouse. States 0,1, and 2 correspond to the case when there are no full containers of finished goods in the output material queue node 10. States 0, 1, and 2 have, respectively, 2, 1, and 0 finished-goods conveyance kanbans in queue node 11. Therefore, states 0, 1, and 2 must have, respectively, 0, 1, and 2 finished-goods conveyance kanbans at the warehouse. Each finished-goods conveyance kanban arrives to the warehouse attached to a full container of finished goods. It is assumed that each finished-goods conveyance kanban is sent back to the line at the instant a customer removes its corresponding full container of finished goods from the warehouse. Then each conveyance kanban at the warehouse corresponds to a full container of finished-goods. Since total finished-goods inventory resides at both the output material queue of the last station, node 10, and at the warehouse, states 0 through 7 have, respectively, a total finished-goods inventory of 0 through 7 containers. Let the state of the pseudo station correspond to the total finished-goods inventory. By assumption, each empty container with attached finished-goods conveyance kanban at the warehouse arrives to the last station 77 according to a Poisson process with rate A . . In other words, finished-goods inventory is depleted at a rate A times the number of full containers of material at the warehouse. Let the pseudo station have service rate equal to, for each Markov chain state, the rate at which finished-goods inventory is depleted. These service rates are shown in the last row of Table 4.1. Example 2 Suppose the last station (station 3) on the kanban line has three production kanbans. Also, there are three finished-goods conveyance kanbans in circulation between the last station and the distribution warehouse. Let the state of the pseudo station correspond to the total finished-goods inventory. Then the pseudo station has the states shown in Table 4.2. If A is the throughput rate of the last station, it should be clear that the pseudo station has rate diagram given in Figure 4.1. Each state corresponds to the number of containers in finished-goods inventory. The state of the system is increased by one at rate A , the throughput rate of the last station. The state of the system is reduced by one at a rate equal to the service rate of the pseudo station. The service rate of the pseudo station has been constructed so that it equals the arrival rate of finished-goods conveyance kanbans or the rate at which containers of finished-goods are removed from the warehouse or distribution system. The capacity of the pseudo station is equal to maximum finished- goods inventory which, in turn, is equal to the number of production kanbans at the last station (P) plus the number of finished-goods 78 conveyance kanbans (C). Finally, the pseudo station will block its preceding station, the last station on the line, when it is in state P + C. Refering back to Table 4.1, indeed, the only state in which the last station could be blocked corresponds to state P + C = 7. All other states have less than five production kanbans in the output queue. The other production kanbans must either be on station three's production-ordering kanban post or in the production activity. In either of these cases, the last station, by definition, could not be blocked. To summarize, on kanban lines with kanban-controlled finished- goods demands, the last station may be blocked. For example, if (P,C) — (5,2) and station three were the last station, it is blocked when there are seven containers of finished-goods inventory (state 7, Table 4.1). A pseudo station is constructed to block the last station of a tandem queue in the same way the last station of a kanban line may be blocked. The service rate of this pseudo station is equal to the rate at which finished-goods inventory is depleted. The arrival rate to the pseudo station is equal to the throughput rate of the last station. In the next section, to approximate kanban lines with kanban- controlled finished-goods demand, pseudo stations are attached to the end of equivalent tandem queues. Altiok's decomposition procedure is then applied to the tandem queue. 79 SECTION 4.3 DECOMPOSITION WITH PSEUDO STATIONS Solving the balance equations generated by the rate diagram in Figure 4.1 gives the steady-state distribution P„ 1 < n < C 4 ,(5 )' nl\A/ 0 pn - | CD — L_ A /A\n P Cn-C C! yx J o C < n < P+C where Pn is the long run probability the system is in state n. The normalization requirement may then be used to obtain Pq . The tandem queue decompositions of Perros and Altiok [43] and Altiok [1] are modified by attaching the pseudo station to the end of the line. For example, consider the decompositions of a four-station tandem queue and a four-station kanban-controlled line. For this example it is assumed all stations have exponential operation times and the first station has an infinite supply of raw material. Then the tandem queue is decomposed into Station 2 Station 3 Station 4 m/c3/i/n2+i m/c2/i/n3+i m/m/i/n4+i Recall, when the first station has an infinite supply of raw material and service rate y^ , station one is not included in the decomposition. In this case, the arrival process to station two is Poisson with rate y^ . Note that station four on the tandem queue cannot be blocked. So its effective service time distribution is still exponential. 80 The four-station kanban line, with kanban-controlled finished- goods demand, is decomposed into Station 2 Station 3 Station 4 M/C4/I/N2+I M/C3/I/N3+I M/C2/l/N4+l Pseudo station The pseudo station is analyzed using equation (1). In this case, station four is blocked if, upon service completion, the pseudo station is in state P+C (or finished-goods inventory is full). Because of the addition of blocking time, station four now has a two- phase Coxian effective service-time distribution. The four-station kanban line is decomposed in exactly the same way as a five-station tandem queue. This procedure has been described in Chapter 3. The only change is that the last station is analyzed using equation (1) rather than the M/M/l/N+1 model. SECTION 4.4 EXPERIMENTAL DESIGN In this section, single-product kanban lines with kanban- controlled finished-goods demand are analyzed using Markov-numerical analysis. The results obtained from numerical analysis are then compared to approximate results obtained by decomposition. In this section it is assumed all stations have exponential or Erlang operation times and zero conveyance periods. Each finished-goods conveyance kanban arrives to the last station according to a Poisson process with rate X . The discussion of how kanban lines with 81 nonzero conveyance periods and kanban-controlled finished-goods demand might be approximated is left to the next chapter. The factors employed in the experiments of this chapter are the number of stations, numbers of production and conveyance kanbans, number of finished-goods conveyance kanbans, operation-time distribution, station service rates, and the arrival rate of finished-goods conveyance kanbans from the distribution warehouse. Table 4.3 gives 26 three - station line configurations. Each station in Table 4.3 has an exponential operation time with mean service rate y^(i=l,2,3). There are C3 finished-goods conveyance kanbans. The arrival rate of finished-goods kanbans is A . Similarly, Table 4.6 gives 25 four-station line configurations. C4 is the number of finished-goods conveyance kanbans. Table 4.9 gives six configurations of three-station kanban lines with two-phase Erlang operation times. Each exponential phase has parameter (i-1,2,3). SECTION 4.5 EXPERIMENTAL RESULTS Markov-numerical results are obtained by slightly modifying the procedure given in Chapter 2. An additional queue, number 12, is added to the three-station line in Figure 1.2 to represent the distribution warehouse. The fourth station and the distribution warehouse have SLAM representation given in Figure 4.2. Table 4.3 gives the number of Markov chain states, average line throughput, average inventories, and blocking probabilities at each station for the three-station exponential line. The average 82 inventories are those of the queue nodes given in Figure 1.2 (with the exception of node 12 which represents the distribution warehouse). Note, because finished-goods demand is kanban- controlled, the last station, number 3, may be blocked and so the blocking probabilities for station 3 are nonzero. Tables 4,6 and 4.9 give, respectively, the corresponding results for the four-station exponential and three-station Erlang lines. The average inventories in Table 4.6 are those of the queue nodes given in Figures 1.2 and 4.2. The exception is node 16 which represents the distribution warehouse. Table 4.4 gives the decomposition approximation results for the three-station exponential lines. N1, N2, and N3 are, respectively, the station capacities for the equivalent tandem queues and C3 is the number of finished-goods conveyance kanbans used in the analysis of the pseudo station. X is the arrival rate to the first station of the tandem queue. X is equal to the service rate of the first station on the kanban line. y^, y^, and y^ gi-ve the processing rates of stations 1,2, and 3. Station 3 is a pseudo station. Average interstage inventories between stations 1 & 2, following the procedure given in Chapter 2, are obtained by adding the blocking probability of station 1 to the average input queue of station 2. Similarly, average interstage inventories between stations 2 & 3 are obtained by adding the blocking probability of station 2 to the average input queue of station 3. Tables 4.7 and 4.10 give the corresponding decomposition approximation results for the four-station exponential and three- 83 station Erlang lines. In Table 4.7, C4 is the number of finished- goods conveyance kanbans used in the analysis of the pseudo station. X is the arrival rate to the first station of the tandem queue. X is equal to the service rate of the first station on the kanban line, y^ , > Vig . and y^ give the processing rates of stations 1, 2, 3, and 4. Station 4 is a pseudo station. In Table 4.10, X again corresponds to the arrival rate to the first station of the tandem queue and is equal to the service rate of the first station on the kanban line. The service rate and squared coefficient of variation (CV^) for stations 1 and 2 are given in Table 4.10. Station 3 is a pseudo station with service rate y^ . Approximate results given in Tables 4.4 and 4.7 were obtained using Perros and Altiok's [43] procedure with a pseudo station. Approximate results in Table 4.10 were obtained using Altiok's [1] procedure with two-phase operation times and a final pseudo station. Finally, Tables 4.5, 4.8, and 4.11 summarize the results by comparing the aggregate interstage inventories, obtained by Markov- numerical analysis and decomposition approximation, for the three- station exponential, three-station Erlang, and four-station exponential lines. In general, the approximation and numerical results are very similar. SECTION 4.6 SUMMARY, CONCLUSIONS Results in this chapter have shown that when the arrival process of each finished-goods conveyance kanban is Poisson, kanban- controlled finished-goods demand can be accurately modeled using a 84 pseudo station. The pseudo station is attached to the end of the equivalent tandem queue. The tandem queue, with pseudo station attached, is then decomposed following the procedure given in Chapter 3. The only modification to the decomposition procedure is that the last station is analyzed using formula (1) rather than the M/M/l/N+1 model. Construction of the pseudo station may be easily modified to represent many alternative finished-goods conveyance kanban arrival processes. The only requirement is that the service process of the pseudo station be identical to the arrival process of finished-goods conveyance kanbans. For example, finished-good conveyance-kanban interarrival times that have a general distribution may be approximated by a pseudo station with phase-type service process. As long as the service process can be analyzed as a continuous - time Markov chain, the numerical analysis of the pseudo station can be easily incorporated into Altiok's [1] tandem-queue decomposition procedure. Figure 4.1 Markov State Diagram For Pseudo Station C+1 P+C (C-l)A CA 00 Ui Figure 4.2 Station Four With Kanban-Controlled Finished-Goods Demand Finished Product Demand For Finished Product (C ontainer and Conveyance Kanban) Production Kanban P roduction-O rdering Kanban Post > X ASM (This S tag e's Input) Kanban Receiving Post Production Activity IT l ^ “VThis S tag e's O utput) u Physical u n its, co n tain er, and conveyance kanban / ' V - Conveyance Kanban Post Empty container and conveyance kanban T 87 Table 4.1 Description of Markov Chain States For (P,C) = (5,2) Line Markov chain state 2 3 4 5 6 7 1 0 Warehouse Finished-goods conveyance kanban input queue 12 1 2 2 2 2 2 2 U ] 0 0 0 0 0. 0 1. Output material queue 10 J 0 1 2 3 4 5 Finished-goods inventory 2 3 4 5 6 7 0 Service rate 2X 2X 2X 2X 2X 2X 88 Table 4.2 Description of Markov Chain States For (P,C) = (3,3) Line Markov chain state Warehouse Finished-goods conveyance kanban input queue Output material queue Finished-goods inventory Service rate 3 4 5 6 3 3 3 3 11 ) 0 0 0 0 10 ] 0 1 2 3 3 4 5 6 3A 3A 3A 3A 2 1 2 1 1 2 0 0 2 1 2 A A 0 Table 4.3 Numerical Results For Three-Station Lines With Exponential Operation Times And Kanban-Controlled Finished-Goods Demand Average Inventories Blocking Probabilities Gfo# 4nn Example Number PI Cl P2 C2 P3 C3 v\ u2 U3 X States Throughput 1 2 3 4 5 6 7 8 9 10 11 12 Station 1 2 3 1 2 2 2 2 2 2 1.000 1.000 1.000 0.500 169 0.7204 0.4771 0.8025 0.4647 1.5353 0.6537 0.6259 0.6425 1.3575 0.7467 0.5329 0.5592 1.4408 0.2796 0. 2091 0.1691 2 2 2 2 2 2 2 1.000 1.000 1.000 0.600 169 0.7491 0.5217 0.7291 0.5389 1.4611 0.7449 0.5060 0.7897 1.2103 0.8967 0.3542 0.7515 1.2485 0.2509 0.1642 0.1026 3 2 2 2 2 2 2 1.000 1 .000 1.000 0.750 169 0.7661 0.5497 0.6843 0.5894 1.4106 0.8097 0.4243 0.9083 1.0917 1.0331 0.2008 0.9786 1.0214 0.2339 0.1353 0.0512 4 2 2 2 2 2 2 1 .000 1.000 1.000 1.000 169 0.7737 0.5629 0.6634 0.6150 1.3850 0.8444 0.3818 0.9815 1.0185 1.1353 0.0910 1.2263 0.7737 0.2263 0.1212 0.0193 5 2 2 2 2 2 2 0.995 1.010 0.995 0.500 169 0.7204 0.4820 0.7940 0.4730 1.5270 0.6480 0.6387 0.6284 1.3716 0.7434 0.5326 0.5592 1.4408 0.2760 0.2143 0.1689 6 2 2 2 2 2 2 0.950 1.100 0.950 0.500 169 0.7152 0.5225 0.7247 0.5464 1.4536 0.6043 0.7455 0.5227 1.4773 0.7265 0.5206 0.5696 1.4304 0.2471 0.2584 0.1641 7 2 2 2 2 2 2 0.990 1.000 1.010 0.500 169 0.7216 0.4894 0.7818 0.4850 1.5150 0.6673 0.6111 0.6604 1.3396 0.7496 0.5359 0.5569 1.4431 0.2711 0.2034 0.1704 6 2 2 2 2 2 2 0.950 1.000 1.050 0.500 169 0.7242 0.5392 0.6985 0.5716 1.4284 0.7249 0.5509 0.7370 1.2630 0.7660 0.5443 0.5517 1.4483 0.2377 0.1806 0.1741 9 2 2 2 2 2 2 0.900 1.000 1.100 0.500 169 0.7223 0.6028 0.5947 0.6915 1.3085 0.8047 0.4730 0.8452 1.1548 0.7984 0.5450 0.5554 1.4446 0.1975 0.1519 0.1751 10 3 3 3 3 3 3 1.000 1.000 1.000 0.400 433 0.8136 1,1282 1.0581 0.7579 2.2421 1.4656 0.7208 1.1299 1.8701 1.6952 0.4911 0.9659 2.0341 0.1863 0.1160 0.0638 11 3 3 3 3 3 3 1.000 1.000 i.OOO 0.500 433 0.8261 1.1762 0.9977 0.8218 2.1782 1.5689 0.6050 1.2969 1.7031 1.9404 0.2335 1.3477 1.6523 0.1739 0.0950 0.0234 12 3 3 3 3 3 3 1.000 1.000 1.000 0.750 433 0.8304 1.1931 0.9765 0.8462 2.1538 1.6108 0.5588 1.3789 1.6211 2.1159 0.0537 1.8928 1.1072 0.1697 0.0873 0.0031 13 3 3 3 3 3 3 1 .000 1.000 1.000 1.000 433 0.8307 1.1945 0.9748 0.8482 2.1518 1.6146 0.5547 1.3878 1.6122 2.1511 0.0182 2.1693 0.8307 0.1693 0.0867 0.0007 14 3 3 3 3 3 3 0.995 1.010 0.995 0.500 433 0.8266 1.1927 0.9764 0.8453 2.1547 1.5556 0.6259 1.2689 1.7311 1.9352 0.2340 1.3467 1.6533 0.1692 0.0992 0.0234 15 3 3 3 3 3 3 0.950 1.100 0.950 0.500 433 0.8215 1.3194 0.8159 1.0442 1.9558 1.4435 0.8098 1.0520 1.9480 1.9058 0.2295 •1.3571 1.6429 0.1353 0.1384 0.0228 16 3 3 3 3 3 3 0.990 1.000 1.010 0.500 433 0.8262 1.2070 0.9584 0.8636 2.1364 1.5998 0.5740 1.3423 1.6577 1.9482 0.2338 1.3476 1.6524 0.1654 0.0891 0.0234 17 3 3 3 3 3 3 0.950 1 .000 1.050 0.500 433 0.8224 1.3251 0.8092 1.0350 1.9650 1.7207 0.4569 1.5270 1.4730 1.9852 0.2315 1.3551 1.6449 0.1343 0.0676 0.0233 18 3 3 3 3 3 3 0.900 1 .000 1.100 0.500 433 0.8087 1.4590 0.6426 1.2536 1.7464 1.8625 0.3288 1.7598 1.2402 2.0446 0.2203 1.3826 1.6174 0.1015 0.0455 0.0220 19 2 2 2 2 2 3 1 .000 1.000 1.000 0.400 204 0.7564 0.5339 0.7096 0.5610 1.4390 0.7735 0.4700 0.8437 1.1563 0.9659 0.2777 1.1090 1.8910 0.2436 0.1517 0.0796 20 2 2 2 2 2 3 1 .000 i.ooo 1.000 0.500 204 0.7701 0.5566 0.6733 0.6028 1.3972 0.8279 0.4020 0.9472 1.0528 1.0923 0.1376 1.4598 1.5402 0.2299 0.1280 0.0345 21 2 2 2 2 2 4 1.000 1.000 i .ooo 0.300 239 0.7611 0,5418 0.6971 0.5756 1.4244 0.7927 0.4462 0.8811 1.1189 1.0156 0.2233 1.4630 2.5370 0.2389 0.1435 0.0635 22 2 2 2 2 2 4 1.000 1.000 i.OOO 0.400 239 0.7738 0.5630 0.6632 0.6153 1.3B47 0.8452 0.3811 0.9851 1.0149 1.1514 0.0748 2.0655 1.9345 0.2262 0.121! 0.0177 23 2 2 2 2 2 4 1.000 1.000 1.000 0.500 239 0.7760 0.5669 0.6571 0.6231 1.3769 0.8562 0.3678 1.0108 0.9892 1.1948 0.0292 2.4480 1.5520 0.2240 0.1168 0.0059 24 2 2 2 2 2 5 1.000 1.000 1.000 0.200 274 0.7425 0.5125 0.7451 0.5252 1.4748 0.7299 0.5277 0.7746 1.2254 0.9015 0.3561 1.2878 3.7122 0,2575 0.1736 0.1113 25 2 2 2 2 2 5 1 .000 1.000 1.000 0.300 274 0.7738 0.5630 0.6632 0.6154 1.3846 0-8454 0.3808 0.9865 1.0135 1.1569 0.0694 2.4207 2.5793 0.2262 0.1211 0.0170 26 2 2 2 2 2 5 1.000 1.000 1.000 0.400 274 0.7763 0.5675 0.6562 0.6244 1.3756 0.8581 0.3656 1.0159 0.9841 1.2068 0.0169 3.0591 1.9409 0.2237 0.1161 0.0033 00 VO Table 4.4 Decomposition Approximation Results For Three-Station Lines Example Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 N1 N2 4 4 N3 C3 u 1.000 1.000 1.000 1.000 0.995 0.950 0.990 0.9 5 0 0.9 0 0 1.000 1.000 1.000 1.000 0.995 0.9 5 0 0.990 0.9 5 0 0.900 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1 1.000 1 1.000 1 1.000 1 1.010 0 1.100 0 1.000 1 1.000 1 1.000 1 1.000 1 1.000 1 1.000 1 1.000 1 1.010 0 1.100 0 1.000 1 1.000 1 1.000 1 1.000 1 1.000 1 1.000 1 1.000 1 1,000 1 1.000 1 1.000 1 1.000 1 U2 ^3 .000 0.500 .000 0.600 .000 0.7 5 0 .000 1.000 .995 0 .5 0 0 .950 0.500 .010 0.5 0 0 .050 0.500 .100 0.500 ,000 0.4 0 0 .000 0.500 .000 0 .7 5 0 .000 1.000 .995 0.500 .950 0.500 .010 0.500 .050 0 .5 0 0 .100 0.500 .000 0.400 .000 0.5 0 0 .000 0.300 .000 0.400 .000 0.500 .000 0.200 .000 0.3 0 0 .000 0.4 0 0 Throughput 0.7046 0.7342 0.7543 0.7660 0.7048 0.7026 0.7057 0.7088 0.7082 0.8024 0.8191 0.8270 0.8279 0.8200 0.8194 0.8194 0.8165 0.8042 0.7393 0.7580 0.7434 0.7636 0.7689 0.7195 0.7629 0.7698 In v e n to r ie s 1-2 2 -3 FG 2.0241 1.6764 1.9114 1.9292 1.5084 1.5638 1.8587 1.3719 1.2051 1.8158 1.2808 0.8600 2.0100 1.6975 1.9122 1.8866 1.8675 1.9036 1.9941 1.6508 1.9161 1.8682 1.5435 1.9283 1.6978 1.3977 1.9262 3.1106 2.4234 2.4809 2.9998 2.2168 1.8750 2.9424 2.0975 1.1609 2.9351 2.0808 0.8482 2.9591 2.2615 1.8776 2.6127 2.6311 1.8758 2.9308 2.1497 1.8757 2.6519 1.8807 1.8670 2.3069 1.5524 1.8305 1.9008 1.4350 2.1136 1.8404 1.3232 1.6559 1.8800 1.3869 2.6923 1.8188 1.2763 1.9900 1.8007 1.2414 1.5727 1.9356 1.4723 3.9167 1.8186 1.2730 2.6196 1.7971 1.2331 1.9468 Blocking P r o b a b i l i t i e s S ta t io n 1 2 3 0.2955 0.2167 0.1571 0 .2 6 5 8 0.1741 0.0979 0.2457 0.1 4 3 6 0.0513 0.2344 0.1257 0.0207 0.2917 0.2214 0.1573 0.2604 0.2615 0.1557 0.2871 0.2112 0.1581 0 .2539 0.1891 0.1604 0.2131 0 .1610 0.1600 0 .1976 0 .1254 0.0611 0.1809 0.1007 0.0242 0.1730 0 .0884 0.0037 0.1721 0 .0869 0.0009 0.1759 0 .1046 0.0244 0.1375 0.1407 0.0243 0.1724 0 .0949 0.0243 0 .1406 0 .0734 0.0239 0 .1064 0 .0508 0.0223 0.2607 0.1642 0.0758 0.2420 0 .1368 0.0358 0 .2566 0.1567 0 .0610 0.2364 0 .1279 0.0200 0.2311 0 .1 1 9 8 0.0076 0.2805 0 .1 8 7 3 0.0982 0.2371 0 .1 2 8 4 0.0195 0.2302 0.1184 0.0049 vo O Table 4.5 A Comparison of Numerical And Approximation Results For Average Interstage Inventories of Three-Station Lines Example Markov-Numerical Approximation Number 1-2 2-3 FG 1-2 2-3 FG 1 2.3378 1.9834 1.9737 2.3196 1.8931 1.9114 2 2.1902 1.7163 1.6027 2.1950 1.6825 1.5638 3 2.0949 1.5160 1.2222 2.1044 1.5155 1.2051 4 2.0484 1.4003 0.8647 2.0502 1.4065 0.8600 5 2.3210 2.0103 1.9734 2.3017 1.9189 1.9122 6 2.1783 2.2228 1.9510 2.1470 2.1290 1.9036 7 2.2968 1.9507 1.9790 2.2812 1.8620 1.9161 8 2.1269 1.8139 1.9926 2.1221 1.7326 1.9283 9 1.9032 1.6278 1.9896 1.9109 1.5587 1.9262 10 3.3002 2.5909 2.5252 3.3082 2.5488 2.4809 11 3.1759 2.3081 1.8858 3.1807 2.3175 1.8750 12 3.1303 2.1799 1.1609 3.1154 2.1859 1.1609 13 3.1266 2.1669 0.8489 3.1072 2.1677 0.8482 14 3.1311 2.3570 1.8873 3.1350 2.3661 1.8776 15 2.7717 2.7578 1.8724 2.7502 2.7718 1.8758 16 3.0948 2.2317 1.8862 3.1032 2.2446 1.8757 17 2.7742 1.9299 1.8764 2.7925 1.9541 1.8670 18 2.3890 1.5690 1.8377 2.4133 1.6032 1.8305 19 2.1486 1.6263 2.1687 2.1615 1.5992 2.1136 20 2.0705 1.4548 1.6778 2.0824 1.4600 1.6559 21 2.1215 1.5651 2.7603 2.1366 1.5436 2.6923 22 2.0479 1.3960 2.0093 2.0552 1.4042 1.9900 23 2.0340 1.3570 1.5812 2.0318 1.3612 1.5727 24 2.2199 1.7531 4.0683 2.2161 1.6596 3.9167 25 2.0478 1.3943 2.6487 2.0557 1.4014 2.6196 26 2.0318 1.3497 1.9578 2.0273 1.3515 1.9468 Table 4.6 Numerical Results For Four-Station Lines With Exponential Operation Times And Kanban-Controlled Finished-Goods Demand Blocking P ro b a b ilitie s Example Number PI Cl P2 C2 P3 C3 P4 C4 “ 1 U2 p3 u4 1 2 2 2 2 2 2 2 2 1.000 1.000 1.000 1.000 2 2 2 2 2 2 2 2 2 1.000 1.000 1.000 1.000 3 2 2 2 2 2 2 2. 2 1.000 1.000 1.000 1.000 4 2 2 2 2 2 2 2 2 0.995 1.005.1.005 0.995 5 2 2 2 2 ' 2 2 2 2 0.900 1.100 1.100 0.900 6 2 2 2 2 2 2 2 2 0.985 0.995 1.005 1.015 7 2 2 2 2 2 2 2 2 0.925 0.975 1.025 1.075 8 2 2 2 2 2 2 2 2 0.850 0.950 1.050 1.150 9 2 2 2 2 2 2 2 2 0.700 0.900 1.100 1.300 10 3 1 3 1 3 1 3 1 1.000 1.000 1.000 1.000 11 3 1 3 1 3 1 3 1 1.000 1.000 1.000 1.000 12 3 3 3 3 3 3 3 3 1.000 1.000 1.000 1.000 13 3 3 3 3 3 3 3 3 1.000 1.000 1.000 1.000 14 3 3 3 3 3 3 3 3 0.995 1.005 1.005 0.995 15 3 3 3 3 3 3 3 3 0.900 1.100 1.100 0.900 16 3 3 3 3 3 3 3 3 0.985 0.995 1.005 1.015 17 3 3 3 3 3 3 3 3 0.925 0.975 1.025 1.075 18 3 3 3 3 3 3 3 3 0.850 0.950 1.050 1.150 19 3 3 3 3 3 3 3 3 0.700 0.900 1.100 1.300 20 4 2 4 2 4 2 4 2 1.000 1.000 1.000 1.000 21 4 2 4 2 4 2 4 2 1.000 1.000 1.000 1.000 22 4 2 4 2 4 2 4 2 1.000 1.000 1.000 1.000 23 4 3 4 3 4 3 4 3 1.000 1.000 1.000 1.000 24 4 3 4 3 4 3 4 3 1.000 1.000 1.000 1.000 25 4 3 4 3 4 3 4 3 1.000 1.000 1.000 1.000 \ Station States Throughput 1 2 3 4 0.600 985 0.7307 0.2692 0.1924 0.1413 0.0947 0.750 985 0.7418 0.2583 0.1749 0.1140 0.0460 1.000 985 0.7462 0.2539 0.1676 0.1016 0.0170 0.600 985 0.7310 0.2653 0.1935 0.1451 0.0948 0.600 985 0.7199 0.2001 0.2164 0.2217 0.0893 0.600 985 0.7316 0.2573 0.1811 0.1340 0.0955 0.600 985 0.7290 0.2120 0.1381 0.1051 0.0959 0.600 985 0.7115 0.1631 0.0926 0.0719 0.0906 0.600 985 0.6360 0.0915 0.0351 0.0254 0.0657 1.200 985 0.7357 0.2643 0.1845 0.1290 0.0722 11500 985 0.7439 0.2562 0.1715 0.1082 0.0324 0.400 3409 0.7974 0.2027 0.1409 0.0989 0.0584 0.500 3409 0.8052 0.1949 0.1287 0.0797 0.0208 0.500 3409 0.8057 0.1902 0.1292 0.0830 0.0208 0.500 3409 0.7914 0.1206 0.1460 0.1582 0.0192 0.500 3409 0.8050 0.1827 0.1176 0.0725 0.0208 0.500 3409 0.7956 0.1401 0.0793 0.0474 0.0202 0.500 3409 0.7657 0.0994 0.0454 0.0246 0.0175 0.500 3409 0.6672 0.0470 0.0126 0.0047 0.0096 0.600 3409 0.8003 0.1996 0.1361 0.0916 0.0442 0.750 3409 0.8061 0.1940 0.1273 0.0772 0.0142 1.000 3409 0.8075 0.1926 0.1251 0.0734 0.0030 0.400 5608 0.8211 0.1791 0.1221 0.0825 0.0413 0.500 5608 0.8268 0.1734 0.1134 0.0685 0.0119 0.750 5608 0.8281 0.1721 0.1113 0.0649 0.0010 Table 4.6 continued Average Inventories Example Number l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 0.4938 0.7755 0.4934 1.5066 0.6898 0.5794 0.7074 1.2926 0.8263 0.4429 0.8792 1.1208 0.9380 0.3313 0.7821 1.2179 2 0.5112 0.7470 0.5229 1.4771 0.7267 0.5315 0.7687 1.2313 0.8939 0.3643 1.0011 0.9989 1.0744 0.i838' 1.0110 0.9890 3 0.5183 0.7356 0.5353 1.4647 0.7427 0.5112 0.7969 1.2031 0.9274 0.3264 1.0707 0.9293 1.1716 0.0822 1.2538 0.7462 4 0.4993 0.7659 0.5031 1.4969 0.6905 0.5821 0.7048 1.2952 0.8195 0.4531 0.8654 1.1346 0.9338 0.3315 0.7816 1.2184 5 0.5971 0.6030 0.6906 1.3094 0.7093 0.6362 0.6664 1.3336 0.6979 0.6477 0.6368 1.3632 0.8836 0.3166 0.8002 1.1998 6 0.5116 0.7456 0.5236 1.4764 0.7152 0.5495 0.7447 1.2553 0.8492 0.4228 0.9089 1.0911 0.9461 0.3331 0.7806 1.2194 7 0.5819 0.6301 0.6501 1.3499 0.8192 0.4332 0..9025 1.0975 0.9474 0.3414 1.0375 0.9625 0.9890 0.3329 0.7850 1.2150 8 0.6628 0.5002 0.8131 1.1869 0.9468 0.3044 1.1052 0.8948 1.0786 0.2438 1.2128 0.7872 1.0656 0.3158 0.8142 1.1858 9 0.7919 0.2996 1.1202 0.8798 1.1644 0.1290 1.4586 0.5414 1.3253 0.0966 1.5479 0.4521 1.2723 0.2385 0.9401 1.0599 10 0.8239 1.4404 0.1845 0.8155 1.1563 1.1080 0.2852 0.7148 1.4142 0.8501 0.3789 0.6211 1.6557 0.6086 0.3869 0.6131 11 0.8494 1.4068 0.1939 0.8061 1.2093 1.0469 0.3068 0.6932 1.5167 0.7395 0.4269 0.5731 1.8925 0.3636 0.5041 0.4959 12 1.0700 1.1328 0.6887 2.3113 1.3614 0.8413 0.9963 2.0037 1.5727 0.6300 1.2583 1.7417 1.7438 0.4588 1.0065 1.9935 13 1.0985 1.0965 0.7235 2.2765 1.4157 0.7792 1.0722 1.9278 1.6734 0.5215 1.4261 1.5739 1.9813 0.2135 1.3896 1.6104 14 1.1148 1.0754 0.7449 2.2551 1.4172 0.7810 1.0717 1.9283 1.6596 0.5387 1.4008 1.5992 1.9764 0.2139 1.3886 1.6114 15 1.3775 0.7430 1.1505 1.8495 1.4380 0.8425 1.0625 1.9375 1.3852 0.8953 0.9687 2.0313 1.9190 0.2017 1.4172 1.5828 16 1.1418 1.0409 0.7785 2.2215 1.4679 0.7231 1.1437 1.8563 1.7169 0.4821 1.4893 1.5107 1.9933 0.2137 1.3901 1.6099 17 1.3012 0.8389 0.9999 2.0001 1.6625 0.5217 1.4308 1.5692 1.8860 0.3379 1.7461 1.2539 2.0528 0.2071 1.4088 1.5912 18 1.4678 0.6316 1.2695 1.7305 1.8667 0.3274 1.7695 1.2305 2.0750 0.1958 2.0573 0.9427 2.1507 0.1835 1.4686 1.5314 19 1.7090 0.3380 1.7571 1.2429 2.1462 0.1125 2.2926 0.7074 2.3435 0.0500 2.5278 0.4722 2.3721 0.1147 1.6657 1.3343 20 1.4170 1.7826 0.3667 1.6333 1.8468 1.3528 0.5613 1.4387 2.1863 1.0134 0.7474 1.2526 2.5128 0.6869 0.6662 1.3338 21 1.4481 1.7459 0.3817 1.6183 1.9083 1.2857 0.5965 1.4035 2.3085 0.8854 0.8334 1.1666 2.8547 0.3392 0.9252 1.0748 22 1.4558 1.7368 0.3855 1.6145 1.9243 1.2683 0.6062 1.3938 2.3461 0.8465 0.8646 1.1354 3.0567 0.1358 1.1925 0.8075 23 1.5602 1.6189 0.6082 2.3918 1.9688 1.2102 0.9056 2.0944 2.2825 0.8965 1.1796 1.8204 2.5614 0.6175 0.9472 2.0528 24 1.5934 1.5801 0.6335 2,3665 2.0320 1.1413 0.9635 2.0365 2.4050 0.7683 1.3188 1.6812 2.9015 0.2717 1.3464 1.6536 25 1.6012 1.5710 0.6396 2.3604 2.0477 1.1244 0.9787 2.0213 2.4406 0.7314 1.3678 1.6322 3.1140 0.0580 1.8959 1.1041 VO LO Table 4.7 Decomposition Approximation Results For Four-Station Lines Example Number N1 N2 N3 N4 C4 X U1 U2 y3 U4 Throughput 1 4 4 4 4 2 1.000 1.000 1.000 1.000 0.600 0.7160 2 4 4 4 4 2 1.000 1.000 1.000 1.000 0.750 0.7299 3 4 4 4 4 2 1.000 1.000 1.000 1.000 1.000 0.7368 4 4 4 4 .4 2 0.995 1.005 1.005 0.995 0.600 0.7165 5 4 4 4 4 2 0.900 1.100 1.100 0.900 0.600 0.7128 6 4 4 4 4 2 0.985 0.995 1.005 1.015 0.600 0.7172 7 4 4 4 4 2 0.925 0.975 1.025 1.075 0.600 0.7166 8 4 4 4 4 2 0.850 0.950 1.050 1.150 0.600 0.7025 9 4 4 4 4 2 0.700 0.900 1.100 1.300 0.600 0.6328 10 4 4 4 4 1 1.000 1.000 1.000 1.000 1.200 0.7260 11 4 4 4 4 1 1.000 1.000 1.000 1.000 1.500 0.7345 12 6 6 6 6 3 1.000 1.000 1.000 1.000 0.400 0.7874 13 6 6 6 6 3 1.000 1.000 1.000 1.000 0.500 0.7989 14 6 6 6 6 3 0.995 1.005 1.005 0.995 0.500 0.7998 15 6 6 6 6 3 0.900 1.100 1.100 0.900 0.500 0.7961 16 6 6 6 6 3 0.985 0.995 1.005 1.015 0.500 0.7990 17 6 6 6 6 3 0.925 0.975 1.025 1.075 0.500 0.7908 18 6 6 6 6 3 0.850 0.950 1.050 1.150 0.500 0.7627 19 6 6 6 6 3 0.700 0.900 1.100 1.300 0.500 0.6664 20 6 6 6 6 2 1.000 1.000 1.000 1.000 0.600 0.7935 21 6 6 6 6 2 1.000 1.000 1.000 1.000 0.750 0.8011 22 6 6 6 6 2 1.000 1.000 1.000 1.000 1.000 0.8036 23 7 7 7 7 3 1.000 1.000 1.000 1.000 0.400 0.8142 24 7 7 7 7 3 1.000 1.000 1.000 1.000 0.500 0.8227 25 7 7 7 7 3 1.000 1.000 1.000 1.000 0.750 0.8255 Blocking P robabilities Average Inventories Station 1-2 2-3 3-4 FG 1 2 3 4 2.0020 1.6702 1.3841 1.5110 0.2840 0.2032 0.1495 0.0901 1.9548 1.5834 1.2409 1.1554 0.2701 0.1828 0.1203 0.0460 1.9296 1.5344 1.1508 0.8218 0.2632 0.1722 0.1041 0.0182 1.9867 1.6737 1.4038 1.5124 0.2798 0.2041 0.1531 0.0903 1.6880 1.7382 1.7657 1.5016 0.2080 0.2203 0.2288 0.0887 1.9577 1.6186 1.3446 1.5144 0.2718 0.1920 0.1424 0.0906 1.7739 1.4043 1.1773 1.5126 0.2253 0.1491 0.1141 0.0903 1.5392 1.1321 0.9536 1.4722 0.1735 0.1023 0.0806 0.0846 1.1021 0.6628 0.5349 1.2816 0.0959 0.0402 0.0310 0.0600 1-9701 1.6164 1.3206 1.2041 0.2740 0.1889 0.1307 0.0711 1.9387 1.5539 1.1978 0.8544 0.2655 0.1759 0.1104 0.0336 3.2349 2.7165 2.2411 2.4138 0.2126 0.1498 0.1076 0.0560 3.1597 2.5809 2.0247 1.8148 0.2011 0.1332 0.0842 0.0216 3.1224 2.5835 2.0653 1.8175 0.1962 0.1335 0.0875 0.0217 2.3932 2.6324 2.8398 1.8067 0.1154 0.1398 0.1636 0.0213 3.0666 2.4708 1.9342 1.8152 0.1888 0.1221 0.0772 0.0216 2.6983 2.0346 1.5692 1.7913 0.1450 0.0832 0.0517 0.0206 2.2704 1.5435 1.1451 1.7102 0.1027 0.0484 0.0280 0.0174 1.5402 0.8290 0.5391 1.4479 0.0479 0.0137 0.0060 0.0093 3.1983 2.6564 2.1676 2.0034 0.2065 0.1413 0.0967 0.0438 3.1449 2.5541 1.9847 1.4157 0.1989 0.1299 0.0796 0.0154 3.1262 2.5167 1.9104 0.9466 0.1964 0.1260 0.0735 0.0037 3.8197 3.1990 2.6233 2.6442 0.1858 0.1282 0.0891 0.0404 3.7454 3.0623 2.3977 1.9254 0.1773 0.1158 0.0711 0.0129 3.7192 3.0110 2.3024 1.1633 0.1745 0.1115 0.0645 0.0013 v £ > Table 4.8 A Comparison of Numerical And Approximation Results For Average Interstage Inventories of Four-Station Lines Example Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Markov-Numerical 1-2 2-3 3-4 2.2821 1.8720 1.5637 1. 2.2241 i .7 628 1.3632 1. 2.2003 1.7143 1.2557 0 1 1 2.2628 1.8773 1.5877 1.9124 1.9698 2.0109 2.2220 1.8048 1.5139 1 1.9800 1.5307 1.3039 1 1.6871 1.1992 1.0310 1 1.1794 0.6704 0.5487 1 2.2559 1.8228 1.4712 1 2.2129 1.7401 1.3126 0 3.4441 2.8450 2.3717 2 3.3730 2.7070 2.0954 1 3.3305 2.7093 2.1379 1 2.5925 2.7800 2.9266 1 3.26242.5794 1.9928 1. 2.8390 2.0909 1.5918 1 2.3621 1.5579 1.1385 1 1.5809 0.8199 0.5222 1 3.4159 2.7915 2.2660 2 3.3642 2.6892 2.0520 1 3.3513 2.6621 1.9819 0 4.0107 3.3046 2.7169 2 3.9466 3.1778 2.4495 1 3.9314 3.1457 2.3636 1 Approximation FG 1-2 2-3 3-4 FG .5482 2.2860 1.8734 1.5336 1.5110 .1728 2.2249 1.7662 1.3612 1.1554 .8284 2.1928 1.7066 1.2549 0.8218 .5499 2.2665 1.8778 1.5569 1.5124 .5164 1.8960 1.9585 1.9945 1.5016 .5525 2.2295 1.8106 1.4870 1.5144 .5479 1.9992 1.5534 1.2914 1.5126 .5016 1.7127 1.2344 1.0342 1.4722 .2984 1.1980 0.7030 0.5659 1.2816 .2217 2.2441 1.8053 1.4513 1.2041 .8595 2.2042 1.7298 1.3082 0.8544 .4523 3.4475 2.8663 2.3487 2.4138 .8239 3.3608 2.7141 2.1089 1.8148 .8253 3.3186 2.7170 2.1528 1.8175 .7845 2.5086 2.7722 3.0034 1.8067 8236 3.2554 2.5929 2.0114 1.8152 .7983 2.8433 2.1178 1.6209 1.7913 .7149 2.3731 1.5919 1.1731 1.7102 .4490 1.5881 0.8427 0.5451 1.4479 .0207 3.4048 2.7977 2.2643 2.0034 .4140 3.3438 2.6840 2.0643 1.4157 .9433 3.3226 2.6427 1.9839 0.9466 .6703 4.0055 3.3272 2.7124 2.6442 .9253 3.9227 3.1781 2.4688 1.9254 .1621 3.8937 3.1225 2.3669 1.1633 to Ul Table 4.9 Numerical Results For Three-Station Lines With Erlang (k=2) Operation Times And Kanban-Controlled Finished-Goods Demand Average Inventories Blocking Probabilities Example _ Station Number PI Cl P2 C2 P3 C3 P1 V2 "3 States Throughput 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 1 2 2 2 2 2 2 2,000 2.000 2.000 0.500 881 0.7787 0.4670 0.7543 0.3704 1.6296 0.5751 0.6462 0.4885 1.5115. 0.6552 0.5661 0.4427 1.5573 0.2213 0.1875 0.1698 2 2 2 2 2 2 2 2.000 2.000 2.000 0.750 881 0.8452 0.5894 0.5653 0.5578 1.4422 0.7849 0.3698 0.8307 1.1693 0.9612 0.1935 0.8730 1.1270 0.1547 0.0945 0.0439 3 2 2 2 2 2 2 2.000 2.000 2.000 1.000 881 0.8551 0.6107 0.53410.5964 1.4036 0.8310 0.3139 0.9267 1.0733 1.0694 0.0755 1.1449 0.8551 0.1448 0.0784 0.0132 4 3 3 3 3 3 3 2.000 2.000 2.000 0.400 2521 0.8774 1.16610.9564 0.6738 2.3262 1.4164 0.70610.9667 2.0333 1.6129 0.5098 0.8065 2.1935 0.1225 0.0846 0.0579 5 3 3 3 3 3 3 2.000 2.000 2.000 0.500 2521 0.8941 1.2513 0.8544 0.7786 2.2214 1.5706 0.5352 1.2007 1.7993 1.8898 0.2161 1.2117 1.7883 0.1057 0.0593 0.0174 6 3 3 3 3 3 3 2.000 2.000 2.000 0.750 2521 0.8986 1.2758 0.8255 0.8120 2.1880 1.6233 0.4781 1.2986 1.7014 2.0653 0.0361 1.8019 1.1981 0.1013 0.0520 0.0013 Table 4.10 Decomposition Approximation Results For Three-Station Lines With Erlang (k=2) Operation Times Blocking Probabilities Example Number N! N2 N3 C3 X M1 CV2 m2 CV 2 *3 Throughput Inventories 1-2 2-3 FG Station 1 2 3 1 A A A 2 1.000 1.000 0.500 1.000 0.500 0.500 0.7235 2.0781 1.7805 1.9883 0.2765 0.2159 0.1722 2 4 A A 2 1.000 1.000 0.500 1.000 0.500 0.750 0.785A 1.8815 1.4398 1.2696 0.2146 0.1314 0.0586 3 A A A 2 1.000 1.000 0.500 1.000 0.500 1.000 0.8003 1.8254 1.3282 0.9070 0.1997 0.1094 0.0240 4 6 6 6 3 1.000 1.000 0.500 1.000 0.500 0.400 0.8261 3.1655 2.5534 2.5906 0.1739 0.1176 0.0698 5 6 6 6 3 1.000 1.000 o.soo : 1.000 0.500 0.500 0.8A78 3.0238 2.3020 1.9623 0.1522 0.0879 0.0284 6 6 6 6 3 1.000 1.000 0.500 1.000 0.500 0.750 0.858A 2.9446 2.1468 1.2111 0.1416 0.0726 0.0044 VO C* Table 4.11 A Comparison of Numerical And Approximation Results For Average Interstage Inventories of Three-Station Lines With Erlang (k=2) Operation Times Example Marko-vr -Numerical Approximation Number 1-2 2-3 FG 1-2 2-3 FG 1 2.3839 2.1577 2.1234 2.3546 1.9964 1.9883 2 2.0075 1.5391 1.3205 2.0961 1.5712 1.2696 3 1.9377 1.3872 0.9306 2.0251 1.4376 0.9070 4 3.2826 2.7394 2.7033 3.3394 2.6710 2.5906 5 3.0758 2.3345 2.0044 3.1760 2.3899 1.9623 6 3.0135 2.1795 1.2342 3.0862 2.2194 1.2111 vo ■ v j 98 CHAPTER 5 A COMPARISON OF TANDEM QUEUES AND KANBAN-CONTROLLED LINES WITH NONZERO CONVEYANCE PERIOD Introduction In this chapter, tandem queues are compared to multi-product kanban-controlled lines with nonzero conveyance period. The objective is to determine the conditions under which the tandem- queue approximation can be expected to give accurate results. It is shown the approximation generally overestimates shortage probabilities and average work-in-process inventories on kanban lines. With respect to shortage probabilities, regression results show the amount of overestimation to be decreasing function of both conveyance period and the squared coefficient of variation of operation times. The amount of work-in-process inventory overestimation appears to be an increasing function of conveyance period. SECTION 5.1 PURPOSE OF RESEARCH Chapter 2 presents a relation between stochastic kanban- controlled lines and tandem queues. It was shown that when the kanban line has a single product, infinite finished-goods demand (so that the last station is never blocked), and zero conveyance period, the kanban line and tandem queue are equivalent. That is, they have equal aggregate interstage inventories, throughputs, and station blocking and shortage probabilities. Chapter 4 then presented a 99 method for using pseudo stations to block the last station on kanban lines and simulate kanban-controlled finished-goods demand. However, when the kanban line has multiple products, results in Chapter 2 show that multi-product kanban lines are only approximately equivalent to some tandem queue. Also, the existence of a nonzero conveyance period distinguishes the kanban-controlled line from the tandem queue. That is, in the single-product case with infinite finished-goods demand, when the conveyance period is zero, the kanban line and tandem queue are equivalent. But for nonzero conveyance periods the kanban line is unique: Fully Idle Tandem Kanban Line_____________________Line * Conveyance Period * 0 ° ° Therefore, the kanban line may be thought of as a generalization of the tandem queue. Note that when the conveyance period is infinitely large, the kanban line is completely idle due to lack of parts conveyance. The purpose of this chapter is to determine how closely the tandem-queue decomposition of Altiok approximates the inventories and station shortage probabilities of multi-product kanban lines with nonzero conveyance periods. It is also assumed the kanban lines have infinite finished-goods demand. The objectives are to determine the conditions under which Altiok's tandem- queue decomposition approximation can be expected to give accurate results. Also, identifying the conditions under which the approximation does not 100 perform well will likely suggest modifications necessary to improve the decomposition for kanban-controlled lines. In this section a full factorial experiment is given to compare multi-product kanban-controlled lines, with nonzero conveyance period, to Altiok*s single-product tandem-queue decomposition approximation. It is assumed both types of lines are subject to infinite finished-goods demand so that blocking of the last station is not possible. As shown in Chapter 4, the results of this chapter may be extended to the case of kanban-controlled finished-goods demand by adding a pseudo station to the end of the line. A full factorial experiment was run to compare the kanban line to the tandem-queue decomposition. The four factors employed in the experiments were the product-kanban configuration, number of stations, squared coefficient of variation (CV^) of operation times, and conveyance period. It was assumed that the numbers of kanbans, conveyance periods, and CV^s were equal at all stations. Each product required the same average operation time at each station. Six product-kanban configurations were used: Mean Operation Times SECTION 5.2 EXPERIMENTAL DESIGN Numbers of Kanbans Product Configuration P(l) C(l) P(2) C(2) P(3) C(3) 1 4 4 2 8 8 3 2 2 1 1 4 4 4 2 2 5 2 2 2 2 2 2 6 4 4 4 4 4 4 1 2 3 1.0 1.0 1.2 0.6 1.2 0.6 1.0 1.0 1.0 1.0 1.0 1.0 . 101 where P(j) and C(j) are, respectively, the number of production and conveyance kanbans of product j at each station. The mean operation times, Mean(j), refer to the mean operation times of product j at each station. Configurations 1 & 2, 3 & 4, and 5 & 6 have, respectively, one, two, and three product-types. In configurations 3 and 4, the required relative production ratio is 2:1; the required relative production ratio in configurations 5 and 6 is 1:1:1. Configurations 1 and 2 are fairly simple with one product having either four or eight production and conveyance kanbans. If the conveyance period were zero, configurations 1 and 2 would be exactly equivalent to some single-product tandem queue. Thus, it is expected, when the conveyance period is small, the decomposition approximation will provide fairly accurate results for these two configurations. Configurations 3 and 4 were included to see how well the decomposition approximates a two-product kanban line when each product has a different average operation time. Product one has an average operation time of 1.2 time units at each station while product two has an average operation time of only 0.6. Configurations 5 and 6 are included to see how well the decomposition approximates a three-product kanban line. As discusssed in Chapter 2, because of the matching process at the assemble nodes preceding production activities, the multi-product kanban line will not be exactly equivalent to a tandem queue. Configurations 5 and 6 were included to determine how large an effect the matching process of 102 multi-product lines has on the accuracy of the decomposition approximation. The second and third experimental factors included were the number of stations on the line and the CV^ of operation times. Three alternative conveyance periods made up the fourth experimental factor. The factor levels of 1, 2, and 3 periods were chosen for two reasons. First, since kanban lines and tandem queues are equivalent only for zero conveyance period, it is expected the kanban line will be most different for larger conveyance periods. Second, the conveyance periods 1, 2, and 3 are relatively small with respect to the number of conveyance kanbans in configurations 1 through 6. As will be shown in Chapter 6, the number of conveyance kanbans must be sufficient to meet production demand over conveyance lead time. Maximum conveyance lead time is the conveyance period. In each of the configurations 1-6, the inventory represented by the number of conveyance kanbans is sufficient to meet demand even for the longest conveyance period considered (3). The configuration with the least buffer inventory, configuration 3, has three conveyance kanbans representing a total work content of (2 x 1.2) + (1 x 0.6) = 3.0 which is just sufficient, on average, to meet demand over a conveyance period of 3. Conveyance periods larger than 3 would, on average induce shortages on lines of configuration 3. 103 The experimental factors and factor levels are summarized below. Factor Levels 1. Product-kanban configuration 6 2. Number of stations 4,6 3. CV^ of operation times 0.50, 1.00 4. Conveyance period 1,2,3 Two independent simulation runs were obtained for each set of experimental conditions. This gives a total of 6x2x2x3 = 72 x 2 or 144 simulation runs. Simulation Model The method of kanban production control used is the constant withdrawal cycle (i.e., conveyance period), nonconstant quantity system given, for a three - station line, in Figure 1.2 and described in Chapter 1. The model was coded using the discrete-event orientation of SLAM [44]. When the CV^ was 1.0, the operation time for each product at each station was chosen from an exponential distribution. Operation times were chosen from a two-phase distribution when the CV^ of operation times was 0.50. In both the simulation and decomposition models, the formulas of Marie [39] (see Chapter 3) were used to characterize the two-phase distribution. Let m be the mean operation time and CV^ be the squared coefficient of variation. Then the two-phase distribution has the representation 1-a 104 where the nodes 1 and 2 represent exponential phases with respective means 1/y^ and lA^ where y^ = l/(m • CV^) and = 2/m. The branching probability a = 2(1-CV^) denotes the probability the second phase is required [39], The two-phase operation time was generated in the simulation model in the following way: 1. Time = Expon(l/ ) 2. U = Uniform(0,l) 3. If(U > a)Stop 4. Time — Time + Expon(l/ \*2 ) 5. Stop To determine when approximate steady-state was obtained, system output performance measures where graphed over time for each set of experimental conditions. It was found, for the six station line, that all performance measures had reached approximate steady state by time 5000. Below, results for the first 7000 hours, for both the four and six station lines, have been discarded and the steady-state results collected over the next 7000 hours. Approximate Results To approximate a multi-product kanban line, the first step is to calculate a weighted-average operation time and CV^ of operation times. The mean operation times are calculated as the weighted average of the items' mean operation times using the numbers of production kanbans as weights. For example, in configuration number 3 the mean operation time is (P(l)«Mean(l))+(P(2)*Mean(2))/ 105 (P(l)+P(2)) = (2(1.2) + l(0.6))/3 = 1.0. To find the weighted- average CV , the weighted average second moment of operation times is found using the number of production kanbans as weights. The second moments of products 1 and 2 in configuration 3 are, respectively, 2.88 and 0.72 so that the weighted average second moment is (2(2.88)+l(0.72))/3 = 2.16. Since the weighted average mean operation time is 1.0, the weighted average CV2 is (2.16- (1.0)2)/(1.0)2 = 1.16. Because it was assumed the first station on the kanban line has an infinite supply of raw material available, Perros and Altiok’s model 2 tandem-queue approximation, with unknown effective arrival rate, was used [43]. When the weighted-average CV2 of operation times was 1.0, the procedure given in Perros and Altiok [43] was used; for other CV2, the procedure of Altiok [1] with two-phase operation times was used. The capacities (N) of each station on the tandem queue are found using the relation N = P + C given in Chapter 2. Here P and C are the total numbers of production and conveyance kanbans at each station. P and C are obtained by summing the numbers of production and conveyance kanbans for each item. Performance Measures In the applications area of setting the numbers of kanbans, discussed in the next chapter, it is important to be able to estimate both the shortage probabilities and average work-in-process inventories at each station. This is because, in the problem of 106 setting kanban numbers, the costs of additional kanbans (higher inventory carrying cost) are traded-off against the benefits of having more buffer inventory (smaller shortage probabilities and smaller expected shortage costs). Therefore, in the present chapter, the performance measures of interest are the abilities of the decomposition approximation to estimate average interstage inventories and station shortage probabilities. In the analysis of variance calculations, the dependent variables considered were 1. Shortage or stockout probability of the last station given by the approximation minus the shortage probability of the last station obtained from simulation Since the last station is never blocked, one minus the probability the last station is short, or starved for raw material, is the average throughput of the line. 2. Average aggregate interstage inventory between the last and second-to-last stations given by the approximation minus average aggregate interstage inventory between the last and second-to-last stations given by simulation Recall, the average aggregate interstage inventory between the last and second-to -last station on the kanban line includes the inventory in the last station's input material queue and the inventory in the second-to -last station's output material queue. To calculate total line work-in-process, one must estimate the inventory between each pair of stations. However, the experiments below consider only the 107 in-process inventory between the last pair of stations. This is because the results for other pairs of stations are similar and so are not given. Since two simulation runs were obtained for each set of experimental conditions, there are two observations on each of the above two dependent variables for each set of experimental conditions. Hypotheses The first hypotheses relate to the conveyance period. On a kanban-controlled line the number of conveyance kanbans must be sufficiently large to meet demand over conveyance lead time (This is discussed further in Chapter 6.). Average conveyance lead time is one half the conveyance period. If demand over conveyance lead time exceeds the number of conveyance kanbans (which are attached to full containers of parts), a stockout results. Naturally, if the number of conveyance kanbans is held constant and the conveyance period or average conveyance lead time is increased, the stockout or shortage probabilities at each station will increase. On the other hand, material moving through a tandem queue never waits for conveyance to the next station. Upon completion at station i, assuming station i+l's queue is not full, the tandem-queue model assumes material is instantaneously moved to station i+1 (i.e., conveyance period is zero). In contrast, material on the kanban line must wait for periodic conveyance. Therefore, if both the tandem queue and kanban lines have equal amounts of interstage inventories, 108 the kanban line, with nonzero conveyance period, will have higher station shortage probabilities. Intuitively, this is because, as the conveyance period increases there is a greater chance that a container of parts, needed at station i+l, will be waiting for conveyance at station i. The fact that shortage probabilities increase with longer conveyance periods also follows from periodic-review inventory models. In these types of models the safety stock required is a function of the demand distribution and the review period. Longer review periods require larger safety stocks to insure the same stockout probabilities. If the number of conveyance kanbans is held constant and the conveyance period is increased, the number of conveyance kanbans (attached to full containers of material) in each transfer/conveyance batch will increase. But this larger transfer batch covers only the increase in average demand over the conveyance period (as long as the number of conveyance kanbans is sufficiently large). The safety stock is not increased to reflect the longer conveyance period. Therefore, it is expected, regardless of the number of conveyance kanbans on the line, longer conveyance periods will be accompanied by larger station shortage probabilities. Since the tandem-queue decomposition approximation is independent of the conveyance period, it is expected the decomposition will accurately approximate the kanban line only for relatively small conveyance periods. 109 H]_: The conveyance period is a significant factor in determining the ability of the tandem-queue decomposition to approximate kanban- line shortage probabilities and average aggregate interstage inventories. H2: Kanban-line shortage probabilities are an increasing function of the conveyance period. Interstage inventories are contained in both input and output material queues (Chapter 1). The maximum number of full containers in input and output material queues is bounded by, respectively, the number of conveyance and production kanbans. If the conveyance period is lengthened, the time between input material queue replenishments lengthens, and each station has longer to exhaust its input queue. Naturally, then for a given number of conveyance kanbans it is expected that average input queues will be smaller for larger conveyance periods. A corollary of H2 is H3: On kanban lines, average aggregate interstage inventory is a decreasing function of the conveyance period. This follows again from periodic-review inventory theory. If, for a given number of conveyance kanbans, the conveyance period is increased, the average size of the transfer batch will increase. However, because the safety stock is not correspondingly increased simultaneously, the probability that the input material stock will be depleted increases. Hence, average interstage inventories should be smaller for longer conveyance periods. 110 The next hypotheses relate to the CV^ of operation times. Recall that the decompositon approximation analyzes each station as an M/Ph/l/N queue. In particular, it is assumed the arrival process to each station is Poisson. Altiok has found the assumption of Markovian arrivals results in unsatisfactory tandem-queue < y approximations when the CV of operation times are less than 0.25 [1]. This is because, when operation times are less-variable, the output process (rate at which a station completes jobs), and, consequently, the input process at the subsequent station become less-variable. The true variability of interarrival times is then overestimated by the exponential distribution. For small CV^ the approximation tends to overestimate shortage probabilities and average interstage inventories. When applying the tandem-queue decomposition to the kanban line, it is expected the decomposition will again overestimate shortage probabilities and inventories when CV^ of operation times are small. This overestimation may be even more severe for the kanban line than for the tandem queue. This is because, in the absence of a blocking instance, the output process of a station on a tandem queue is equal to the input process at the subsequent station. On the kanban line, however, each station faces an input process with a deterministic interarrival period. This deterministic interarrival period, of course, is the conveyance period. The output process of each station determines only how many containers are in each conveyance batch which may arrive at the end of each conveyance period. It is Ill hypothesized that the assumption of Markovian arrivals will increasingly overestimate the true variability of the arrival process of containers to stations on the kanban line as the CV^ of operation times decreases. H4: Shortage probabilities on the kanban-controlled line are overestimated most by the tandem-queue decomposition when the CV^ of operation times are small. H5: Average interstage inventories on the kanban-controlled line are overestimated most by the tandem-queue decomposition when the CV^ of operation times are small. Finally it is hypothesized that the product-kanban configuration and number of stations are not significant factors in the accuracy of the decomposition. Hg: The product-kanban configuration is not a significant factor in the ability of the decomposition to approximate shortage probabilities and inventories. Configurations 1 through 6 have been constructed so that the number of conveyance kanbans is sufficiently large for each conveyance period considered. The tandem-queue decomposition will accurately approximate the kanban line only when the number of conveyance kanbans is large relative to the conveyance period. If the number of conveyance kanbans is insufficient to meet demand over the conveyance lead time, the shortage probabilities at each station would be very much larger than that predicted by the tandem-queue decomposition approximation. 112 H- j : The number of stations is not a significant factor in the ability of the decomposition to approximate shortage probabilities and inventories. Hypotheses Hg and H7 are important because, if true, say that the decomposition (which is single-product) may be applied to lines having any number of products and stations. Regression Model The following indicator variables are used in the regression approach to obtain MANOVA results. Product-Kanban Configuration 1 2 3 4 5 6 XI 1 0 0 0 0 0 X2 0 1 0 0 0 0 X3 0 0 1 0 0 0 X4 0 0 0 1 0 0 X5 0 0 0 0 1 0 Number of Stations 4 6 X6 0 1 CV2 0.50 1.00 X7 0 1 Conveyance Period 1 2 3 X8 1 0 0 X9 0 1 0 SECTION 5.3 EXPERIMENTAL RESULTS Simulation and approximation results are shown in Tables 5.1, 5.2, 5.3, and 5.4. For four-station lines, the shortage probabilities for stations 2, 3, and 4 are given in Tables 5.1 and 113 5.2 along with average aggregate interstage inventories between stations 1 & 2, 2 & 3, and 3 & 4. Note, to obtain the aggregate interstage inventory results for the approximation, the procedure given in Chapter 2 has been used. That is, the average blocking probability at station i has been added to the average input queue of station i+1. Similarly, for six-station lines, the shortage probabilities for stations 2, 3, 4, 5, and 6 are given in Tables 5.3 and 5.4 along with average aggregate interstage inventories between stations 1 & 2 , 2&3, 3 & 4, 4 & 5, and 5 & 6. The simulation figures given refer to the average of two runs for each configuration. Below the average, given in parentheses, are the corresponding standard errors. MANOVA Results Regression analysis summarizes the results in Tables 5.1, 5.2, 5.3, and 5.4. Analysis of variance tables are given in Tables 5.7 and 5.8 and the corresponding regression coefficients are given in Tables 5.5 and 5.6. The dependent variable in Tables 5.5 and 5.7 is the shortage probability of the last station given by the approximation minus the shortage probability of the last station obtained from simulation. The dependent variable in Tables 5.6 and 5.8 is the average aggregate interstage inventory between the last and second-to -last stations given by the approximation minus the average aggregate interstage inventory between the last and second- to-last station given by simulation. For the purpose of interpreting the regression coefficients in Tables 5.5 and 5.6 it is necessary to 114 observe, from Tables 5.1, 5.2, 5.3, and 5.4 that, in general, the decomposition approximation overestimates the shortage probabilities and interstage inventories of the kanban line. This is particularly true when the GV^ of operation times is 0.50. These observations relate back to hypotheses H4 and H5. : Analysis of variance results in Tables 5.7 and 5.8 permit acceptance of H]_. Regarding shortage probabilities, Table 5.7 shows conveyance period to have a F value of 31.2380. This compares to a critical value, at the .001 significance level, of 7.32. Thus, the conveyance period appears to be a significant factor in determining the difference between the shortage probability for the last station given by the tandem-queue approximation and kanban-line simulation. With respect to average inventories, Table 5.8 shows conveyance period to have a F value of 11.5235. This again compares to a critical value, at the .001 significance level, of 7.32. Table 5.8 also shows the conveyance period is the only significant factor in estimating the difference between the average aggregate interstage inventories given by the tandem-queue approximation and kanban-line simulation. H2: Regression results in Table 5.5 show the indicator variables X8 and X9 to have significant t values. Also, since the coefficient of X9 is a smaller positive number than the coefficient of X8, this implies the dependent variable under consideration is a decreasing function of the conveyance period. Further, because the shortage probability given by the tandem-queue approximation is independent of 115 the conveyance period, it follows that the shortage probability on the kanban line must be increasing. Therefore, H£ is accepted. The obvious implication of this observation is that for very large conveyance periods the tandem-queue approximation significantly underestimates kanban-line shortage probabilities. Note that the significance of the conveyance period depends on the number of conveyance kanbans on the line. For example, compare the shortage probabilities of station four, Short(4), in Table 5.1 for product-kanban configurations three and four. For product-kanban configuration three, conveyance periods of 1, 2, and 3 give Short(4) probabilities of, respectively, .2175, .2474, and .2837. For product-kanban configuration four, the respective shortage probabilities are .1254, .1315, and .1455. Short(4) for configuration three increases more markedly than Short(4) for configuration four when the conveyance period goes from two to three. This is because configuration three has only three conveyance kanbans representing a total work content of (2x1.2) + (1 x 0.6) =3 periods. When the conveyance period reaches three, there is just enough inventory, on average, to meet demand over the conveyance period. On the other hand, configuration four has 6 conveyance kanbans representing a total work content of (4 x 1.2) + (2 x 0.6) = 6 periods. As a result, Short(4) would not increase markedly until the conveyance period reached 6. H3: Regression results in Table 5.6 show the indicator variables X8 and X9 to have significant t values. Also, since the coefficient 116 of X9 is greater than the coefficient of X8, this implies that the dependent variable under consideration is an increasing function of the conveyance period. Further, because the average aggregate interstage inventory given by the tandem-queue approximation is independent of the conveyance period, it follows that the average aggregate interstage inventory on the kanban line must be decreasing. Therefore, H3 is accepted. The obvious implication of this observation is that for very large conveyance periods the tandem- queue approximation significantly overestimates kanban-line aggregate interstage inventories. : Results in Tables 5.2 and 5.4 show the decomposition approximation to generally overestimate shortage probabilities when the CV^ is 0.50. The t statistic associated with the indicator variable X7 in Table 5.5 shows that the amount of overestimation is r \ significantly reduced when the CV of operation times is increased to 1.0. Therefore, is accepted. This is true because, as the CV^ increases, the arrival process to each station becomes more similar to Poisson. H5: Acceptance of should also imply acceptance of H5. However, regression results in Table 5.6 do not permit acceptance of H5. The CV^ does not appear to be a significant factor. However, intuitively, H5 is still appealing. Failure to accept H5 is likely just a consequence of the limited number of simulation runs obtained. In general, the coefficient of variation of inventories over the simulation runs is greater than the variability of shortage 117 probabilities. With more simulation results available it seems likely H5 would also be accepted. Hg : Analysis of variance results in Table 5.7 show the product- kanban configuration to be a significant factor in the ability of the decomposition to approximate shortage probabilities. The F value of 8.3566 compares to a critical value, at the .001 significance level, of 4.10. Therefore, Hg is rejected. Regression results in Table 5.5 provide an explanation. The significant negative t statistic associated with the indicator variable X3 shows that the dependent variable is significantly smaller for configuration three. This is expected. Of all the six configurations, configuration three has the smallest number of conveyance kanbans. For any given conveyance period, one would expect the shortage probabilities on lines of configuration three to be the greatest. If the decomposition approximation is going to consistently overestimate shortage probabilities (particularly for smaller CV^), one would expect this overestimation to be smallest for configuration three. Analysis of variance results in Table 5.8 show the product- kanban configuration not to be a significant factor in the ability of the decomposition to approximate average interstage inventories. The F value of .5145 has a significance level of 0.2351. Hy: Analysis of variance results in Tables 5.7 and 5.8 permit acceptance of Hy. Regarding shortage probabilities, number of stations has a F value of 0.4648 with significance level 0.5034. 118 With respect to average inventories, number of stations has a F value of 0.0 with significance level 1.0. SECTION 5.4 SUMMARY, CONCLUSIONS Results of this chapter show that when the number of conveyance kanbans is sufficiently large, relative to the length of conveyance period, the tandem-queue decomposition closely approximates the average station shortage probabilities and interstage inventories. This is because, when the number of conveyance kanbans is sufficiently large, relative to the conveyance period, periodic conveyance will never induce material shortages. Simulation and approximation results are also closest for larger CV^ of operation times. In general, the decomposition approximation overestimates shortage probabilities on the kanban line. This is due to the decomposition assumption of Markov arrivals which overestimates the true amount of variability in interarrival times. Additionally, results show that the amount of overestimation is decreasing function of the conveyance period. This is because, as the period increases, stockouts become more frequent and the shortage probabilities on the kanban line increase. Since the approximation is independent of the conveyance period, the amount of overestimation decreases with the increasing conveyance period. Eventually, for sufficiently large conveyance periods, the approximation would underestimate true kanban-line shortage probabilities. 119 With respect to average inventories, results show the approximation to consistently overestimate average interstage inventories. Again, this likely due to the assumption, in the decomposition, of Markov arrivals. It has also been shown, in the acceptance of H3, that the amount of overestimation is an increasing function of the conveyance period. Finally, acceptance of H7 allows us to apply the tandem-queue decomposition approximation to kanban lines of any number of stations as long as the number of conveyance kanbans is sufficiently large relative to the kanban line's conveyance period. Further, using the pseudo stations described in Chapter 4, the decomposition may also be used to approximate kanban lines with both nonzero conveyance periods and kanban-controlled finished-goods demand. Table 5.1 2 A Comparison of Shortage Probabilities And Interstage Inventories For Four-Station Lines With CV =1.0 Average Inventories J Conveyance Simulation Approximation Simulation Approximation (1) C(l) P(2) C<2) P(3) C(3) Meun(l) Meun(2) Mean(3) CV Period Short(2) Short(3) Short{4) Short(2) Short(3) Short(4) 1-2 2-3 3-4 1-2 2-3 3-4 4 4 1.0000 1.00 1.0 .0564 (.0005) .1126 (.0122) .1674 (.0080) .0570 .0994 .1570 4.3707 (.1438) 3.3225 (.1018) 2.5315 (.1301) 4.4664 3.6054 2.7558 8 8 1.0000 1.00 1.0 .0368 (.0020) .0731 (.0076) .0936 (.0104) .0320 .0555 .0876 8.5796 (.0895) 6.5707 (.0215) 5.6328 (.0450) 9.0956 7.5587 6.0286 2 2 1 1 1.2000 0.6000 1.00 1.0 .0725 (.0120) .1274 (.0083) .2175 (.0035) .0741 .1301 .2057 3.1709 (.1420) 2.5284 (.0398) I.8030 (.0127) 3.3290 2.6313 1.9464 4 4 2 2 1.2000 0.6000 1.00 1.0 .0494 (.0040) .0764 (.0062) .1254 (.0065) .0435 .0756 .1195 6.3798 (.3625) 5.4608 (.1369) 4.2902 (.0520) 6.7878 5.5698 4.3589 2 2 2 2 2 2 1.0000 1.0000 i.0000 1.00 1.0 .0387 (.0046) .0704 (.0076) .1162 (.0117) .0410 .0712 .1124 6.5112 (.3248) 5.7199 (.1088) 4.0470 (.1314) 6.7772 5.5753 4.3821 4 4 4 4 4 4 1.0000 I.0000 1.0000 1.00 1 .0 .0184 (.0089) .0344 (.0013) .0687 (.0031) .0222 .0384 .0607 13.7325 (1.8087) 10.4856 (.8772) 8.4777 (.5332) 13.7409 11.5406 9.3452 4 4 1.0000 1.00 2.0 .0681 (.0011) .1206 (.0019) .1762 (.0059) .0570 .0994 .1570 3.8413 (.0802) 3.1073 (.0780) 2.2742 (.0737) 4.4664 3.6054 2.7558 8 8 1.0000 1.00 2.0 .0415 (.0019) .0531 (.0036) .0821 (.0006) .0320 .0555 .0876 8.0582 (.0773) 7.3520 (.2199) 6.3224 (.1114) 9.0956 7.5587 6.0286 2 2 1 1 1.2000 0.6000 1 .00 2.0 .0892 (.0027) .1390 (.0050) .2474 (.0031) .0741 .1301 .2057 2.8765 (.0150) 2.3333 (.0172) 1.6758 (.0748) 3.3290 2.6313 1.9464 4 4 2 2 1.2000 0.6000 1 .00 2.0 .0529 (.0040) .0854 (.0052) .1315 (.0064) .0435 .0756 .1195 6.0238 (.2664) 5.0066 (.0452) 3.7231 (.0673) 6.7878 5.5698 4.3589 2 2 2 2 2 2 1.0000 1.0000 i.oooo 1.00 2.0 .0374 (.0059) .0625 (.0047) .1)48 (.0097) .0410 .0712 .1124 6.3495 (.1621) 5.5586 (.1526) 4.1945 (.2800) 6.7772 5.5753 4.3821 4 4 4 4 4 4 1.0000 1.0000 1.0000 1.00 2.0 .0324 (.0007) .0288 (.0087) .0603 (.0105) .0222 .0384 .0607 11.7557 (.4161) 12.3909 (.7239) 8.9444 (.9692) 13.7409 11.5406 9.3452 4 4 1.0000 1 .00 3.0 .0866 (.0014) .1325 (.0063) .1922 (.0065) .0570 .0994 .1570 3.4873 (.0331) 2.8894 (.1064) 2.1960 (.0392) 4.4664 3.6054 2.7558 8 8 1.0000 1.00 3.0 .0284 .0697 .1082 .0320 .0555 .0876 8.4105 6.5311 5.0094 9.0956 7.5587 6.0286 2 (.0002) (.0028) (.0092) (.2329) (.2899) (.1360) 2 I i 1.2000 0.6000 1 .00 3.0 .1020 (.0017) .1810 (.0045) .2837 (.0082) .0741 .1301 .2057 2.7161 (.0481) 2.0295 (.0283) 1.5133 (.0554) 3.3290 2.6313 1.9464 4 4 2 2 1.2000 0.6000 1 .00 3.0 .0498 (.0024) .0930 (.0014) .1455 (.0111) .0435 .0756 .1195 6.1231 (.0830) 4.6698 (.1628) 3.6692 (.3115) 6.7878 5.5698 4.3589 2 2 2 2 2 2 I.0000 1.0000 . 1 .0000 1.00 3.0 .0457 (.0045) .0839 (.0179) .1289 (.0171) .0410 .0712 .1124 6.0383 (.1368) 4.8061 (.5082) 3.6251 (.3110) 6.7772 5.5753 4.3821 4 4 4 4 4 4 1.0000 1.0000 1.0000 1 .00 3.0 .0276 (.0076) .0448 (.0075) .0573 (.0047) .0222 .0384 .0607 12.1907 (1.1871) 10.8209 (1.0945) 8.8132 (.4248) 13.7409 11.5406 9. 3452 N> .Q. Table 5.2 A Comparison 2 of Shortage Probabilities And Interstage Inventories For Four-Station Lines With CV = 0.5 Average Inventories Conveyance Simulation Approximation Simulation Approximation PH) CO) PC2) C(2) P(3) C(3) Mean(l) Mean(2) Mean(3) CV2 Period Short(2) Short(3) Short(4) Short(2) Short(3) Short(4) 1-2 2-3 3-4 i-2 2-3 3-4 4 4 1.0000 0.50 1.0 .0367 .0577 .1059 .0462 .0803 .1262 3.9425 3.2712 2.3968 4.4191 3.6041 2.8016 (.0005) (.0053) (.0030) (.0241) (.0672) (.0317) g 8 I.0000 0.50 1.0 .0161 .0234 .0520 .0251 .0433 .0683 9.0178 7.9300 6.0492 9.0526 7.5705 6.0974 (.0023) (.0065) (.0043) (.4085) (.3947) (.0979) 2 2 ] 1 1.2000 0.6000 0.50 1.0 .0460 .0831 .1250 .0617 .1076 .1693 2.9642 2.3314 1.9179 3.2821 2.6275 1.9874 (.0056) (.0088) (.0012) (.0996) (.1310) (.0709) 4 4 2 2 i.2000 0.6000 0.50 1.0 .0245 .0402 .0619 .0347 ..0601 .0946 6.4793 5.5508 4.4527 6.7456 5.5829 4.4306 (.0023) (.0001) (.0096) (.0164) (.0878) (.2583) 2 2 2 2 2 2 i.0000 1.0000 1.0000 0.50 1.0 .0214 .0356 .0590 .0325 .0563 .0886 6.3198 5.2859 4.3262 6.7325 5.5822 4.4423 (.0010) (.0026) (.0090) (.0005) (.0987) (.5259) 4 4 4 4 4 4 1.0000 1.0000 1.0000 0.50 1.0 .0057 .0246 .0348 .0172 .0297 .0468 13.8822 10.3238 9.0446 13.7003 11.5583 9.4237 (.0034) (.0155) (.0145) (1.7198) (2.5455) (2.3629) 4 4 1.0000 0.50 2.0 .0333 .0497 .1009 .0462 .0803 .1262 3.7125 3.2855 2.4456 4,4191 3.6041 2.8016 (.0046) (.0093) (.0019) (.0628) (. 1559) (.0148) S 8 I.0000 0,50 2.0 .0226 .0316 .0415 .0251 .0433 .0683 7.5549 6.6510 6.0333 9.0526 7.5705 6.0974 (.0002) (.0043) (.0001) (.3183) (.7931) (.0811) 2 2 1 1 1.2000 0.6000 0.50 2.0 .0535 .1051 . .1599 .0617 .1076 .1693 2.6124 2.0112 1.5070 3.2821 2.6275 1.9874 (.0075) (.0085) (.0007) (.0870) (.0895) (.0426) 4 4 2 2 1.2000 0.6000 0.50 2.0 .0199 .0408 .0733 .0347 .0601 .0946 6.3147 5.3637 3.8380 6.7456 5.5829 4.4306 (.0013) (.0020) (.0023) (.0215) (.0418) (.1227) 2 2 2 2 2 2 1.0000 1.0000 1.0000 0.50 2.0 .0283 .0397 .0648 .0325 .0563 .0886 5.8271 4,9983 4.0590 6.7325 5.5822 4.4423 (.0014) (.0029) (.0068) (.1735) (.2620) (.2323) 4 4 4 4 4 4 1.0000 1,0000 1.0000 0.50 2.0 .0109 .0234 .0296 .0172 ,0297 .0468 13.9673) 11.0537 9.8364 13.7003 11.5583 9.4237 (.0033) (.0030) (.0023) (.5174) (.7882) (1.0631) 4 4 1.0000 0.50 3.0 .0531 .0832 .1257 .0462 .0803 .1262 3.1927 2.5439 1.9536 4.4191 3.6041 2.8016 (.0006) (.0009) (.0033) (.0089) (.0570) (.0144) 8 8 I.0000 0.50 3.0 .0168 .0349 .0599 .0251 .0433 .0683 7.7504 6.5772 4.7166 9.0526 7.5705 6.0974 (.0060) (.0032) (.0007) (.2308) (.2263) (.0139) 2 2 1 ! 1.2000 0.6000 0.50 3.0 .0825 .1283 .1893 .0617 ,1076 .1693 2.3103 1.7978 124786 3.2821 2.6275 1.9874 (.0034) (.0048) (.0044) (.0013) (.0467) (.0103) 4 4 2 2 1.2000 0.6000 0.50 3.0 .0311 .0572 .0737 .0347 .0601 .0946 5.3081 4.2837 3.7351 6.7456 5.5829 4.4306 (.0014) (.0048) (.0084) (.0196) (.0141) (.1191) 2 2 2 2 2 2 I.0000 I.0000 I.0000 0.5.0 3.0 .0347 .0483 .0714 .0325 .0563 .0886 5.1296 4.5749 3.6999 6.7325 5.5822 4.4423 (.0032) (.0002) (.0021) (.0723) (.0145) (.1484) 4 4 4 4 4 4 1.0000 1.0000 1.0000 0.50 3.0 .0155 .0248 .0405 .0172 .0297 .0468 11.9075 10.0944 7.6893 13.7003 11.5583 9.4237 (.0013) (.0009) (.0029) (. 3008) (.2565) (.3049) 121 Table 5.3 2 A Comparison of Shortage Probabilities And Interstage Inventories For Six-Station Lines With CV =1.0 1 Conveyance S im u latio n A pproxim ation P (l) C (l) P(2) C(2) P(3) C (3) Mean( I) Mean(2) Mean(3) C V P erio d S h o rt(2 ) S h o rt(3) Short(A ) S h o rt(5 ) S h o rt(6 ) S h o rt(2 ) S h o rt(3 ) Short(A ) S h o rt(5 ) S h o rt(6 ) A A 1.0000 1.00 1.0 . 0A58 (.0057) • 085A (.0057) . 11A0 (.0 0 2 2 ) .1369 (.00A3) .1812 (.0093) ■ 0A77 .0757 .0996 .1275 .1756 8 8 1.0000 1.00 1.0 .0228 (.000A) .03A9 (.0077) .0697 (.0031) .0922 (.0128) .1059 (.0062) .0270 ■ 0A2A .0555 .0709 .0979 2 2 1 1 1.2000 0.6000 1.00 1.0 .0603 (.0110) .0989 (.0055) -125A (.0025) .1660 (.0063) .2352 (.0171) .0616 .098A .1302 .1671 .2300 A A 2 2 1.2000 0.6000 1.00 1.0 • 03A9 (.0 0 A0) ■ 06A6 ( .00A2) .0765 (.0025) .1115 (.0030) .1556 (.0006) .0365 .0576 .0756 .0967 .1336 2 2 2 2 2 2 1.0000 1.0000 1.0000 1.00 1.0 .0389 (.00A3) • 07AA (.0007) .0929 (.0005) .0922 (.0059) .1298 (.0005) • 03A5 • 05AA .0713 .0911 .1258 A A A A A A 1.0000 1.0000 1.0000 1.00 1.0 .0137 (.0010) • 027A (.0126) .0528 (.0031) .0635 (.0016) .0708 (.0018) .0188 • 029A • 038A • 0A90 .0679 A A 1.0000 1.00 2 .0 .0533 (.006A) .0920 ( .00A3) .1215 (.0070) .1552 (.0080) . 20A0 (.0 0 2 9 ) .0A77 .0757 .0996 .1275 .1756 8 8 1.0000 1.00 2 .0 .0198 (.0029) • 0A20 (.0059) .0562 (.0 1 A5) .0701 (.0100) .1169 (.0156) .0270 • 0A2A .0555 .0709 .0979 2 2 1 1 , 1.2000 0.6000 1.00 2 .0 .0770 (. 003 A ) .1189 (.0082) . 1343 (.0073) .1688 (.0 0 5 5 ) . 25A2 (.0 0 1 2 ) .0616 ' -098A .1302 .1671 .2300 A A 2 2 1.2000 0.6000 1.00 2 .0 .0A18 (.00A3) .08A0 (.0002) . 100A (.003A) .1169 (.0OA2) . 1A57 (.0086) .0365 .0576 .0756 .0967 .1336 2 2 2 2 : 2 2 1.0000 1.0000 1.0000 1.00 2 .0 .0303 (.0008) .0519 (.0079) .0751 (.0001) .1095 (.0185) . 1A22 (.0 0 3 9 ) ■ 03A5 .05AA .0713 .0911 .1258 A A A A A A 1.0000 1.0000 1.0000 1 .00 2 .0 .0237 (.0073) .0292 (.0015) .0368 (.0039) .0A12 (.0032) .0622 (.0 0 7 A ) .0188 . 029A .038A .0A90 .0679 A A 1.0000 1.00 3.0 .0720 (.0021) .0951 (.0105) .1225 ( . 00A9) . 159A (.0002) .2236 (.0 0 9 5 ) .0A77 .0757 .0996 .1275 .1756 8 8 1.0000 1.00 3 .0 .0356 (.0037) .0609 (.0120) .0721 (.0196) .0807 (.0122) .1035 (.OOA8) .0270 • 0A2A .0555 .0709 .0979 2 2 1 1 1.2000 0.6000 1.00 3.0 .0891 (.0075) .1388 (.0011) .1752 (.0 0 2 7 ) .2126 (.0091) .3000 (.0020) .0616 • 098A .1302 .1671 .2300 A A 2 2 1.2000 0.6000 1.00 3.0 ■ 0A02 (.0111) .0637 (.0058) .1029 (.0035) .1167 (.0063) .1606 ( .0 0 A1) .0365 .0576 .0756 .0967 .1336 2 2 2 2 2 2 1.0000 1.0000 1.0000 1.00 3.0 .0397 ( -O0A9) .0611 (.0018) .0893 (.0125) .1070 (.0121) .1557 (.0068) . 03A5 • 05AA .0713 .0911 .1258 A A A A A A 1.0000 1.0000 1.0000 1.00 3.0 .0312 (.0018) . 0A33 (.0006) .0529 (.0132) .0596 (.0188) .0912 (.0 0 1 7 ) .0188 .029A .038A . 0A90 .0679 to t o Table 5.3 continued (1) C (l) P (2) C (2) P (3) C (3) Mean(L) 4 4 1.0000 8 8. 1.0000 2 2 1 1 1.2000 4 4 2 2 1.2000 2 2 2 2 2 2 1.0000 4 4 4 4 4 4 1.0000 4 4 1.0000 8 8 1.0000 2 2 1 1 1.2000 4 4 2 2 1.2000 2 2 2 2 2 2 1.0000 4 4 4 4 4 4 1.0000 4 4 1.0000 8 8 1.0000 2 2 1 1 1.2000 4 4 2 2 1.2000 2 2 2 2 2 2 1.0000 4 4 4 4 4 4 1.0000 Conveyance Mean(2) Mean(3) cv^ P erio d 1-2 1.00 1.0 4.4201 (.0657) 1.00 1.0 9.6918 (.0143) 0.6000 1.00 1.0 3.3232 (.1218) 0.6000 1.00 1.0 7:0308 (.0949) 1.0000 1.0000 1.00 1.0 7.0133 (.1 4 8 3 ) 1,0000 1.0000 1.00 1.0 14.8204 (.5341) 1.00 2 .0 4.1167 (.0574) 1.00 2 .0 9.1060 (.1479) 0.6000 1.00 2 .0 2.9663 (.0313) 0.6000 1.00 2 .0 6.6196 (.0705) 1.0000 1.0000 1.00 2 .0 6.7456 (.1117) 1.0000 1.0000 1.00 2 .0 13.1817 (.3877) 1.00 3.0 3.7318 (.0368) 1.00 3.0 7.9650 (.1021) 0.6000 1.00 3 .0 2.8004 (.0 7 5 3 ) 0,6000 1.00 3.0 6.2633 (.2787) 1.0000 1.0000 •1.00 3 .0 6.2254 (.1010) 1.0000 1.0000 1.00 3 .0 12.5801 (.2252) Average In v e n to rie s S im u latio n 2-3 3-4 4-5 5-6 1-2 3.7411 3.3687 2.9727 2.3825 4.7015 (.0 3 4 2 ) (.0298) (.0185) (.0967) 8.6717 6.9487 5.7460 4.9054 9.5058 (.3345) (.1094) (.5666) (.5280) 2.8341 2.5528 2.1820 1.7767 3.5233 (.0587) (.0328) (.0218) (.0252) 5.8575 5.4875 4.6129 3.6852 7.1168 (.1658) (.1541) (.2162) (.1383) 5.3788 4.8711 4.7382 3.7521 7.1006 (.2213) (.1789) (.2 3 2 6 ) (.0686) 12.2574 10.3371 9.5268 8.6554 14.3240 (2. 1659) (.5670) (.4803) (.7557) 3.4688 2.9262 2.5870 2.1050 4.7015 (.0 5 1 4 ) (.1339) (.0728) (.0366) 8.0958 7.1455 6.3362 4.6750 9.5058 (.3 3 9 4 ) (.3397) (.7736) (.5557) 2.5383 2.3700 2.1335 1.5842 3.5233 (.0912) (.0523) (.0511) (.0173) 5.1844 4.9404 4.2469 3.7278 7.1168 (.0 4 7 2 ) (.0515) (.0029) (.1278) 5.7530 5.3280 4.4806 3.5205 7.1006 (.0 3 7 4 (.0424) (.4047) (.0299) 12.5826 10.5765 10.1871 9.1738 14.3240 (.4 0 6 4 ) (1.0387) (.3822) (.0958) 3.2286 2.9237 2.5093 1.8714 4.7015 (.1 4 4 5 ) (.0360) (.0214) (.0822) 6.7719 6.3704 5.7016 5.2225 9.5058 (.1 8 8 9 ) (.4152) (.4046) (.1252) 2.3122 2.0072 1.7741 1.4266 3.5233 (.0132) (.0427) (.0332) (.0252) 5.3211 4.6940 4.0351 3.1614 7.1168 (.2081) (.1 0 2 9 ) (.0769) (.2032) 5.3848 4.5532 4.1533 3.1632 7.1006 (.2 0 8 7 ) (.3 2 4 6 ) (.3383) (.0583) 11.1783 9.7478 8.6680 6.8345 14.3240 (.3 2 2 4 ) (.3562) (.8044) (.1174) A pproxim ation 2-3 3-4 4-5 4.0543 3.6034 3.1572 5-6 2.5287 8.3534 7.5572 6.7639 5.6229 2.9987 2.6305 2.2676 1.7640 6.2038 5.5699 4.9390 4.0381 6.1988 5.5737 4.9523 4.0647 12.674.9 11.5398 10.4064 8.7654 4.0543 3.6034 3.1572 2.5287 8.3534 7.5572 6.7639 5.6229 2.9987 2.6305 2.2676 1.7640 6.2038 5.5699 4.9390 4.0381 6.1988 5.5737 4.9523 4.0647 12.6749 11.5398 10.4064 8.7654 4.0543 3.6034 3.1572 2.5287 8.3534 7.5572 6.7639 5.6229 2.9987 2.6305 2.2676 1.7640 6.2038 5.5699 4.9390 4.0381 6.1988 5.5737 4.9523 4.0647 12.6749 11.5398 10.4064 8.7654 N> w Table 5.4 . . . 2 A Comparison of Shortage Probabilities And Interstage Inventories For Six-Station Lines With CV - 0.5 P (l) C (l) P(2) C(2) 2 Conveyance Sim ulation Approximation P(3) C(3) Mean(l) Mean(2) Mean(3) C V Period S h o rt(2) Short(3) Short(4) Short(5) S hort(6) Short(2) Short(3) Short! 4) Short(5) Short(6) 4 4 1.0000 0.50 1.0 .0393 (.0140) .0540 (.0079) .0628 (.0095) .0762 (.0026) .1071 (.0024) .0389 .0615 .0808 .1031 .1416 8 8 1.0000 0.50 1.0 .0105 (.0032) .0241 (.0037) .0378 (.0078) .0418 (.0026) .0596 (.0040) .0212 .0333 .0435 .0554 .0765 2 2 1 1 1.2000 0.6000 0.50 1.0 .0406 (.0038) .0605 (.0059) .0775 (.0016) .1033 (.0002) .1605 (.0008) .0517 .0822 .1083 .1384 .1899 4 4 2 2 1.2000 0.6000 0.50 1.0 .0181 (.0041) .0288 (.0073) .0394 (.0003) .0526 (.0006) .0779 (.0022) .0293 .0460 .0603 .0769 .1059 2 2 2 2 2 2 1.0000 1.0000 1.0000 0.50 1.0 .0263 (.0046) .0350' (.0007) .0427 (.0005) .0569 (.0028) .0811 (.0023) .0275 .0432 .0566 .0721 .0993 4 4 4 4 4 4 1.0000 1.0000 1.0000 0.50 1.0 .0037 (.0001) .0136 (.0017) .0249 (.0034) .0358 (.0089) .0328 (.0089) .0146 .0228 .0297 .0379 .0523 4 4 1.0000 0.50 2.0 .0273 (.0027) .0453 (.0018) .0658 (.0054) .0897 (.0023) .1217 (.0039) .0389 .0615 .0808 .1031 .1416 8 8 1.0000 0.50 2.0 .0194 (.0085) .0251 (.0001) .0332 (.0098) .0346 (.0094) .0512 (.0022) .0212 .0333 .0435 .0554 .0765 2 2 1 1 1.2000 0.6000 0.50 2.0 .0397 (.0017) .0708 (.0084) .0993 (.0007) .1300 (.0044) .1778 (.0097) .0517 .0822 .1083 .1384 .1899 4 4 2 2 1.2000 0.6000 0.50 2.0 .0270 (.0065) .0478 (.0094) .0519 (.0017) .0612 (.0013) .0767 (.0048) .0293 .0460 .0603 .0769 .1059 2 2 2 2 2 2 1.0000 1.0000 1.0000 0.50 2.0 .0211 (.0029) .0371 (.0054) .0503 (.0015) .0659 (.0067) .0750 (.0116) .0275 .0432 .0566 .0721 .0993 4 4 4 4 4 4 1.0000 I . 0000 1.0000 0.50 2.0 .0102 (.0040) .0136 (.0040) .0176 (.0043) .0212 (.0020) .0370 (.0088) .0146 .0228 .0297 .0379 .0523 4 4 1.0000 0.50 3.0 .0373 (.0034) .0582 (.0071) .0795 (.0015) .1106 (.0018) .1365 (.0060) .0389 .0615 .0808 .1031 .1416 8 8 1.0000 0.50 3.0 . .0156 (.0113) .0203 (.0039) .0333 (.0006) .0374 (.0034) .0611 (.0014) .0212 .0333 .0435 .0554 .0765 2 2 1 1 1.2000 0.6000 0.50 3.0 .0704 (.0009) .1029 (.0022) .1198 (.0031) .1518 (.0006) .2184 (.0017) .0517 .0822 .1083 .1384 .1899 4 4 2 2 1.2000 0.6000 0.50 3.0 .0140 (.0010) .0338 (.0086) .0510 (.0063) .0556 (.0090) .0870 (.0137) .0293 .0460 .0603 .0769 .1059 2 2 2 2 2 2 1.0000 1.0000 1 .0000 0.50 3.0 .0237 (.0002) .0307 (.0023) .0399 (.0011) .0583 (.0022) .0856 (.0027) .0275 .0432 .0566 .0721 .0993 4 4 4 4 4 4 1.0000 1.0000 1.0000 0.50 3.0 .0054 (.0016) .0232 (.0142) .0211 (.0116) .0260 (.0007) .0361 (.0084) .0146 .0228 .0297 .0397 .0523 124 Table 5.4 continued 9 Conveyance P (l) C (l) P(2) C<2) P(3) C(3) H ean(l) Mean(2) Mean(3) C V 4 Period 4 4 1.0000 0.50 1.0 8 8 1.0000 0.50 1.0 2 2 1 1 1.2000 0.6000 0.50 1.0 4 4 2 2 1.2000 0.6000 0.50 1.0 2 2 2 2 2 2 1.0000 1.0000 1.0000 0.50 1.0 4 4 4 4 4 4 1.0000 1.0000 1.0000 0.50 1.0 4 4 1.0000 0.50 2.0 8 8 1.0000 0.50 2.0 2 2 1 1 1.2000 0.6000 0.50 2.0 4 4 2 2 1.2000 0.6000 0.50 2.0 2 2 2 2 2 2 1.0000 1.0000 1.0000 0.50 2.0 4 4 4 4 4 4 1.0000 1.0000 1.0000 0.50 2.0 4 4 1.0000 0.50 3.0 8 8 1.0000 0.50 3.0 2 2 1 1 1.2000 0.6000 0.50 3.0 4 4 2 2 1.2000 0.6000 0.50 3.0 2 2 2 2 2 2 1.0000 1.0000 1.0000 0.50 3.0 4 4 4 4 4 4 1.0000 1.0000 1.0000 0.50 3.0 Average In v e n to rie s Sim ulation Approximation 1-2 2-3 3-4 4-5 5-6 1-2 2-3 3-4 4-5 4.1642 3.4569 3.3442 3.0242 2.4156 4.6396 4.0252 3.5990 3.1780 (.2743) (.3288) (.1144) (.0783) (.0066) 9.9186 7.9943 7.2762 6.7935 5.5708 9.4456 8.3324 7.5655 6.8022 (.4532) (.7684) (.2929) (.3087) (.0812) 3.1145 2.6977 2.5084 2.1465 1.6142 3.4619 2.9674 2.6230 2.2845 (.0998) (.0597) (.0108) (.0119) (.0522) 7.1686 6.1850 5.4541 4.9637 3.9773 7.0560 6.1820 5.5782 4.9785 (.0477) (.3952) (.1259) (.1317) (.0689) 6.5337 5.6597 ' 5.3946 4.7414 3.8001 7.0394 6.1744 5.5771 4.9840 (.1262) (.0375) (.0554) (.0683) (.3426) 14.9031 13.5862 10.6365 8.7137 8.3862 14.2649 12.6577 11.5535 10.4522 (.2257) (.1500) (.3261) (1.4928) (1.8961) 3.8683 3.4345 3.0075 2.6317 2.0323 4.6396 4.0252 3.5990 3.1780 (.1067) (.0308) (.0893) (.0801) (.1003) 8.2507 7.7293 6.9445 7.4548 5.5512 9.4456 8.3324 7.5655 6.8022 (.6188) (.0747) (.6088) (.5166) (.2479) 2.8322 2.4519 2.1100 1.7977 1.4213 3.4619 2.9674 2.6230 2.2845 (.0086) (.0787) (.0078) (.0536) (.0511) 5.9806 4.9738 4.8210 4.4486 3.7743 7.0560 6.1820 5.5782 4.9785 (.2973) (.1933) (.1081) (.0490) (.1416) 6.1413 5.2756 4.6578 4.0027 3.6139 7.0394 6.1744 5.5771 4.9840 (.1111) (.1923) (.0766) (.2261) (.4082) 12.7140 11.8781 12.2224 10.2589 8.4061 14.2649 12.6577 11.5535 10.4522 (.6251) (.0352) (.5022) (.2786) (.3511) 3.5368 3.0725 2.6830 2.2249 1.8986 4.6396 4.0252 3.5990 3.1780 (.1057) (.0105) (.0719) (.0296) (.0272) 8.4529 7.4727 6.8640 6.2292 5.3197 9.4456 8/3324 7.5655 6.8022 (.7368) (.3816) (.4755) (.0771) (.0186) 2.4327 2.0298 1.8578 1.6416 1.2840 3.4619 2.9674 2.6230 2.2845 (.0352) (.0324) (.0560) (.0796) (.0221) 6.1963 5.7262 4.8151 4.4659 3.4575 7.0560 6.1820 5.5782 4.9785 (.1177) (.1474) (.1603) (.2954) (.2787) 5.9387 5.2422 4.7936 4.2731 3.4920 7.0394 6.1744 5.5771 4.9840 (-1595) (.1115) (.2516) (.1095) (.1475) 14.3409 10.8987 10.3246 10.9577 8.3097 14.2649 12.6577 11.5535 10.4522 (.6806) (2.5435) (1.9412) (.9023) (.7350) 5-6 2.5840 5.7036 1.8135 4.1214 4.1359 8.8570 2.5840 5.7036 1.8135 4.1214 4.1359 8.8570 2.5840 5.7036 1.8135 4.1214 4.1359 8.8570 125 126 Table 5.5 Regression Results: Dependent Variable = Approximate Shortage Probability of Last Station - Simulated Shortage Probability of Last Station Predictor Coefficient Standard Deviation t-Ratio Constant 0.012260 0.003977 3.08 XI -0.008600 0.003977 -2.16 X2 -0.000529 0.003977 -0.13 X3 -0.020513 0.003977 -5.16 X4 -0.001321 0.003977 -0.33 X5 -0.000675 0.003977 -0.17 X6 -0.001565 0.002296 -0.68 X7 -0.037571 0.003977 -9.45 X8 0.018967 0.003977 4.77 X9 0.014650 0.003977 3.68 s = 0.01378 R-squared = 71.9% R-squared (adjusted) =69.6% Table 5.6 Regression Results: Dependent Variable = Approximate Interstage Inventory Preceding Last Station - Simulated Interstage Inventory Preceding Last Station Predictor Coefficient Standard Deviation t-Ratio Constant 0.8847 0.1657 5.34 XI -0.0913 0.1657 -0.55 X2 -0.1045 0.1657 -0.63 X3 -0.2559 0.1657 -1.54 X4 -0.1040 0.1657 -0.63 X5 -0.0684 0.1657 -0.41 X6 -0.0008 0.0957 -0.01 X7 -0.0224 0.1657 -0.14 X8 -0.5764 0.1657 -3.48 X9 -0.4570 0.1657 -2.76 s = 0.5741 R-squared = 16.5% R-squared (adjusted) = 9.5% 127 Table 5.7 MANOVA Results: Dependent Variable = Approximate Shortage Probability of Last Station - Simulated Shortage Probability of Last Station Source SS df MS F Between Treatments .0641034 11 .0058276 Product-Kanban Configuration .0079292 5 .0015858 8.3566 Number of Stations .0000882 1 .0000882 0.4648 Squared Coefficient of Variation .0440965 1 .0440965 232.3657 Conveyance Period .0118562 2 .0059281 31.2380 Interactions .0001334 2 .0000667 0.3515 Error .0250499 132 .0001898 Total .0891533 143 Table 5.8 MANOVA Results: Dependent Variable = Approximate Interstage Inventory Preceding Last Station - Simulated Interstage Inventory Preceding Last Station Source SS df MS F Between Treatments 8.5751 11 0.7796 Product-Kanban Configuration 0.8478 5 0.1696 0.5145 Number of Stations 0.0000 1 0.0000 0.0000 Squared Coefficient of Variation 0.0154 1 0.0154 0.0467 Conveyance Period 7.5950 2 3.7975 11.5235 Interactions 0.1168 2 0.0584 0.1772 Error 43.4998 132 0.3295 Total 52.0749 143 128 CHAPTER 6 SETTING KANBAN NUMBERS Introduction In batch-manufacturing shops with time-varying demands, product mix, capacity utilization and, consequently, production lead times may change substantially. It is clear that kanban-system parameters must also change. For example, the number of production kanbans must be sufficient to satisfy requirements over production lead time. This chapter shows how Altiok1s tandem-queue decomposition may be used to set the number of production kanbans. SECTION 6.1 PROBLEM DESCRIPTION The kanban system may be regarded as an evolution from the base- stock system [34]. In the base-stock system, any withdrawal which drops inventories below the base-stock level S causes an order to be placed on the supplying location for the withdrawn amount, to bring the level back up to the base-stock level S. The kanban system takes an analogous view that cards released by withdrawals are released to the shop. The difference between the base-stock and kanban systems is that the units of measurement for inventories in the system are the kanban container size b, and not 1 as in the base-stock case (Figure 6.1). In both kanban and base-stock systems there is a relationship between production (replenishment) lead times, and stock levels. This relationship can be seen as follows: at any time, the quantity 129 on order in a base-stock or kanban system must be equal to the demand over production (or resupply) lead time. This follows since any demand seen before that interval would have been replenished. The quantity of inventory on hand must be the difference between the base stock and the amount on order. Thus, if the amount of inventory on order exceeds the base stock level, a backorder or stockout situation occurs. This trade-off is exactly like the classic "newsboy" problem. The total inventory or, equivalently, the number of kanbans is chosen to minimize the expected holding and shortage costs [31], If the production lead- time distribution is known, this computation is a standard one, and involves setting the inventory level to the appropriate fractile of the distribution of production lead time. In a two-card kanban production-control system there are both production and conveyance kanbans. Conveyance kanbans may be thought of as production kanbans for the material-handling operation. To set the number of production kanbans one must have the distribution of production lead time (L). Production lead time was defined in Chapter 1 as the time taken from the time that a production kanban is removed from a container to the time that finished material corresponding to that kanban is available for use by the succeeding station. In Figure 6.2 it can be seen that production lead time, for example at station 2, is the time from when a production kanban is removed from a full container of material in the output material queue node 6 and placed in the kanban receiving post A3 until the production kanban returns to the output material queue node 6 130 attached to a full container of material. This time includes the time a production kanban spends in the kanban receiving post A3, the production-ordering kanban post node 5, and the production activity. All production kanbans for each item at a station remain in a closed queuing network. In Figure 6.2 this network at station 2 consists of, in order, the output material queue node 6, the following assemble node, the kanban receiving post A3, the production-ordering kanban post node 5, the following assemble node, production activity two, and the closed path between these entities. This closed network is the bold path in Figure 6.2. Conveyance lead time has been defined in Chapter 1 as the time taken from the time that a conveyance kanban is removed from a container in an input material queue to the time that finished material corresponding to that kanban enters the input material queue. In Figure 6.3 it can be seen that conveyance lead time, for example between stations 2 and 3, is the time from when a conveyance kanban is removed from a full container of material in the input material queue node 8 and placed in the conveyance kanban post A4 until the conveyance kanban returns to the input material queue node 8 attached to a full container of material. This time includes the time a conveyance kanban spends in the conveyance kanban post A4 and the stock point of station two (node 7). If transportation time is assumed to be zero, and material corresponding to the conveyance kanban is available in station two's output material queue (node 6), then average conveyance lead time is just the average time a 131 conveyance kanban spends in the conveyance kanban post waiting for periodic conveyance. This average waiting time is just half of the conveyance period. All conveyance kanbans for each item between two stations remain in a closed queuing network. In Figure 6.3 this closed queuing network between stations two and three consists of, in order, the input material queue node 8, the following assemble node, the conveyance kanban post A4, the stock point of station two node 7, the following assemble node, and the closed path between these entities. This closed network is the bold path in Figure 6.3. The distribution of production lead time is necessary to set the number of production kanbans for each item. The distribution of conveyance lead time is necessary to set the number of conveyance kanbans for each item. Establishing a distribution of production lead time is a challenging problem. In a dynamic environment, lead times are a moving target, constantly changing with shop load, production mix, batch-sizing policy, processing-time variability, and random events. In addition, production lead times also depend on the number of kanbans at each manufacturing station. For example, consider a five- station line with two production and two conveyance kanbans at each station. If the number of production kanbans at station three is increased from 2 to 3, to 4, and to 5, average production-ordering kanban post queue time becomes 132 Example 1 Average Production-Ordering Kanban Post Queue Time Number of Kanbans ' Station PI Cl P2 C2 P3 C3 P4 C4 P5 1 2 3 4 5 2 2 2 2 2 2 2 2 2 0.675 0.945 1.114 1.353 1.740 2 2 2 2 3 2 2 2 2 0.688 0.980 2.074 1.291 1.703 2 2 2 2 4 2 2 2 2 0.697 1.006 2.989 1.248 1.677 2 2 2 2 5 2 2 2 2 0.703 1.025 3.904 1.217 1.659 Example 1 shows that as the number of production kanbans is increased at station two, the average production-ordering kanban post queue times at station three increase by almost an integer for each additional production kanban. Since production-ordering kanban post queue time is one component of production lead time, production lead time also increases by almost an integer for each additional production kanban. It is intuitively obvious that as the number of production kanbans is increased, the production lead (incurred by each kanban) time must also increase. This is because as more production kanbans are introduced to the closed queuing network, in Figure 6.2, the average queue sizes at each queue of the network must increase. Example 1 also shows that average production lead time at station i depends not only on the number of production kanbans at station i, but also on the number of production kanbans at all other stations. This can be seen by noting that as the number of production kanbans at station three is increased, average production- ordering queue times at stations one and two increase while queue times at stations four and five decrease. 133 In contrast, conveyance lead time may not be a function of the number of conveyance kanbans. This is because the material-handling operation serves containers (with attached conveyance kanbans) in batches. For example, if material-handling is a fork-lift truck, the truck operator will simultaneously move all the empty containers (with attached conveyance kanbans) from the conveyance kanban post A4 at station three to the stock point node 7 of station two. If the material-handling operation time is independent of the number of containers (conveyance kanbans) in each conveyance batch, then average conveyance lead time is independent of the number of conveyance kanbans between two stations. Further, if average material-handling time is essentially zero, average conveyance lead time reduces to one-half the interval between the periodic batch material-handling operation-- that is, the conveyance period. To summarize, it has been shown that the number of kanbans is set to minimize expected holding and shortage costs. The calculations require knowledge of production and conveyance lead-time distributions. In a dynamic environment, production lead times constantly change with shop conditions. Further, Example 1 shows that production lead times also depend on the number of production kanbans at each station. 134 SECTION 6.2 LITERATURE REVIEW Toyota Motor Company uses the following formula to set the number of kanbans for an item at a work station [52]: n = [DL(l+a)] (1) where n: Number of kanbans D: Average demand (expressed in containers) L: Container lead time a: Safety factor and [x] means the smallest integer greater than or equal to x. If the formula is used to find the number n of production kanbans, then L is average production lead time. If n is the number of conveyance kanbans, then L is average conveyance lead time. Note, there is no reason to require that the number of production and conveyance kanbans be equal at any station. Schonberger contributes to the misunderstanding: "There are precisely one C-kanban and one P-kanban for each container . . . ." [47, p. 224] Similarly, Ebrahimpour and Fathi note that, "... it should be mentioned that in the kanban system the total number of withdrawal cards should be equal to the number of production cards." [16] The number of production kanbans required is a function of the distribution of production lead time while the number of conveyance kanbans required is a function of the distribution of conveyance lead time. An increase in either the number of production or conveyance 135 kanbans both increases average throughput and work-in-process inventory. Kanban production-control systems described in the Japanese manufacturing literature appear to be static [40] [47]. While it undoubtedly happens in practice, there is no discussion of how kanban-system parameters might be set for different conditions. One would expect that as demands and product mix change from period to period, Toyota would use formula (1) to dynamically set the number of production kanbans. Such is not the case at Toyota, as Monden [40, p. 174] explains: "For example, suppose it is expected that the average daily demand of next month will be two times the demand of the current month .... At Toyota . . . the total number of kanbans [is] unchanged." Companies such as Toyota do not have to routinely adjust the number of kanbans from period to period for at least three reasons: They have a large market share and hence demand variations from the forecasted value are a small percentage of the total; they have cross-trained workers whom they are able to switch from work station to work station to mitigate temporary bottlenecks; and their just- in-time shops are so well run that they can handle day-to-day problems as well as variations in demand." [45] The majority of firms either using, or thinking of using, kanban production control do not exhibit Toyota's characteristics. In such firms it is essential to dynamically adjust the number of kanbans. 136 Formula (1) provides little help in the problem of dynamically setting kanban numbers. Ignoring the safety factor a, the formula simply states that the number of kanbans is the smallest number of containers needed to satisfy demand during lead time. Because only average demand and average lead time figures are used, expected inventory holding and shortage costs may not be minimized. Even in a static environment it is not clear how formula (1) might be used. As discussed above, production lead time (L) is a function of the number (n) of production kanbans at a station. Therefore, lead time is a function of n as well as other factors including shop conditions. L(.,n) represents lead time as a function of the number (n) of kanbans as well as other shop conditions. Therefore, solving for n becomes the fixed-point problem n = [DL(.,n)•(1+a)] (2) Solving a fixed-point problem such as (2) iteratively ordinarily requires the rapid evaluation of L(.,n) for each value of n. Unfortunately, no simple function for L(.,n) is available. Karmarkar [30] [31] [32] and Groenevelt and Karmarkar [20] [21] [22] have recently described a forecast-driven dynamic kanban system. The basic approach is to control card counts within the kanban system on the basis of requirements generated by a central MRP system. It is assumed that lead times for each station are fixed and known to the MRP system. Karmarkar notes that a more sophisticated version of this system would also determine the change in production lead times 137 as a result of changes in demand levels and the number of kanbans in each work station [30]. Rees et al. [45] use historical observations at each work station to construct a distribution of lead times. Time series methods are used to estimate the density function of production lead time at each station. The first step is to find the autocorrelation function of lead times and lag k beyond which all autocorrelations are approximately zero. Observations are then collected k periods apart (independent observations) and used to construct a histogram of production lead times. The distribution of lead times is then combined with a forecast of demand to produce a (scaled) distribution for n = DL. The limitations of this time-series method include the requirements of a substantial amount of historical data. Also, the lead-time distributions are assumed to be stable over time and independent of production mix, batch-sizing policy, and numbers of kanbans at each station. Huang et al. [25] simulate a single-product, two-card, kanban- controlled line to investigate the interaction effects between processing-time distributions, bottleneck location, and the number of kanbans. They found, for normal operation times with CV^ = .25, that additional kanbans reduce required worker overtime, but only up to approximately four kanbans; after four, no positive effect is gained. 138 Ebrahimpour and Fathi [16] simulate a single-product, two-card, kanban line to determine the effects of steadily reducing the number of production and conveyance kanbans. They found that when both production and conveyance kanban numbers are reduced simultaneously, finished-goods and work-in-process inventory levels decline. With a constant demand rate, this reduction did not affect sales until a point is reached when finished goods are reduced to a level below demand. When this happens, sales are lost immediately, and in large numbers. The authors also study the effects of reducing only the number of conveyance kanbans in the line. In this case, the level of finished-goods remained constant until the level of work-in-process becomes so low that parts requisitions are severely restricted. Lee [36] has provided the only results to date on a constant order-quantity, variable order-period, kanban system. The fixed order-quantity is called the production kanban size. Lee simulates a multi-product, two-card, kanban line to determine the effects of increasing the production kanban size. The results show that larger production kanban sizes reduce total setup time but also slightly reduce capacity utilization. Bitran and Chang [7] have used a capacitated multi-stage, multi period, integer linear programming model to find the number of kanbans to be detached in each period. This technique assumes production lead time is zero and each stage produces only one type of item. 139 Recent studies have used queuing models to analyze the lead time behavior of production facilities [29] [56], These methods compute average container queues that form in front of work stations depending on production volume, mix, and batching policy. Little's law (L =AW) is then applied to find lead times for widely varying shop conditions. Unfortunately, the queuing models do not immediately extend to kanban-controlled lines. This is because, unlike tandem queues which have only input container queues, kanban- controlled lines have both input and output material queues. Lead times are then defined in terms of waiting times of production kanbans. SECTION 6.3 PURPOSE OF RESEARCH The purpose of the research in this chapter is to investigate an alternative procedure for setting production kanban numbers. An alternative is necessary due to the limitations of the base-stock analysis and time-series methods in a dynamic environment. Recall, to minimize expected inventory holding and shortage costs, in base- stock analysis, requires the distribution of production lead time. If demands, product mix, and capacity utilization and, consequently, the number of kanbans were stationary processes, then the time series methods [45] could be used to construct an empirical distribution of production lead time. The empirical distribution may then be used to minimize expected inventory holding and shortage costs. However, one must assume independence of lead times and the number of kanbans when 140 using the empirical distribution to evaluate alternative kanban configurations. But as shown above, the distribution of lead times, at any station, depends not only on the number of production kanbans at the station, but also on the number of production kanbans at all other stations. In more dynamic environments, with changing demands, product mix, and capacity utilizations, the time series methods due not appear promising. An alternative to minimizing total expected holding and shortage cost is to set the number of kanbans so as to acheive a desired probability of shortage [20]. This procedure is common in the general literature on the subject of safety stocks [12]. In fact, Rees et al. [45] point out that, "... in most JIT shops the cost of shortage significantly exceeds holding costs at most non-final work stations." Therefore, shortage costs are overwhelmingly the largest consideration and the number of kanbans should be set so that the probability of shortage is sufficiently small. The advantage of this procedure is that it reduces the problem of estimating the production lead-time distribution to one of estimating material shortage probabilities at each station. However, a method for estimating shortage probabilities on kanban lines has not yet been proposed. When shortage costs do not overwhelm inventory holding costs, a method for setting kanban numbers must consider both types of costs. Simulation experiments may at times be used to set kanban numbers. This method requires one to simulate, with multiple 141 replications, each alternative kanban configuration, to obtain estimates of inventory holding and shortage costs. The advantages of simulation are the ability to model the kanban-controlled line to any desired degree of accuracy and to use any operation-time distributions required. However, due to the large number of alternative kanban configurations, particularly on longer lines, and the requirement of multiple runs for each configuration, simulation is very (computer) time-consuming. Simulation appears too tediuous to use to set kanban numbers as shop conditions change on a day-to- day basis. The purpose of this chapter is to test the effectiveness of using a queuing model to set production kanban numbers. In Chapter 5 it was shown that in certain circumstances a tandem-queue decomposition approximation may be used to estimate shortage probabilities and average interstage inventories on kanban-controlled lines. In particular, the approximation is accurate when the conveyance period is small relative to the number of conveyance kanbans and the kanban-controlled line is approximately equivalent to a tandem queue. In other words, the number of conveyance kanbans must be sufficiently large to meet demand over conveyance lead time. The problem of setting conveyance kanbans is an easy one. The number of conveyance kanbans for each item must be set sufficiently large to cover production requirements over conveyance lead time. But, this is easy since the material-handling operation serves containers in batches of, it is assumed, unlimited size making 142 conveyance lead time independent of the number of conveyance kanbans on the line. Therefore, below, only the problem of setting the number of production kanbans for each item is addressed. It is assumed the number of conveyance kanbans is sufficiently large so that lack-of-conveyance does not induce material shortages. The proposed method for setting production kanban numbers proceeds as follows. 1. Set production kanban numbers at the smallest level possible while maintaining required relative production rates. For example, if products 1, 2, and 3 have required relative production rates of 1, 2, and 3, then the smallest number of production kanbans at each station is, respectively, 1, 2, and 3 kanbans for items 1, 2, and 3. 2. Estimate the probability of stockout at each station and the average work-in-process holding cost. These estimates are obtained by applying Altiok’s [1] tandem-queue decomposition to the kanban- controlled line. The probability of shortage at station i is estimated by Pj_(0). Shortage cost for station i is assumed to be a constant multiple of P^CO). 3. After comparing holding costs and shortage costs, either stop or add kanbans while maintaining required relative production rates. Repeat step 2. Holding costs are calculated so that one can balance increasing holding costs with smaller shortage costs. The advantage of this process is that the shortage probability of station i reflects shop load, processing-time variability, product 143 mix, and batch-sizing policy of stations 1, 2, . . . , M. Also, no historical data is required. SECTION 6.4 EXPERIMENTAL DESIGN In this section an experiment is given to compare shortage probabilities and average interstage inventories of kanban lines obtained by simulation and the approximate decomposition. Estimates of shortage probabilities and average inventories may be used to find the production kanban configuration which minimizes total inventory holding and shortage costs. The number of conveyance kanbans is not an experimental variable in this chapter. Also, it is assumed the number of conveyance kanbans is sufficiently large for each line's conveyance period. It is assumed the kanban line is subject to infinite finished-goods demand so that blocking of the last station is not possible. Results of this chapter may be extended to the case of kanban-controlled finished-goods demand by adding a pseudo station to the end of the line. The factors considered in the experiments were the product configuration, number of work stations, squared coefficient of variation (CV^) of operation times, and conveyance period. The three-product configurations considered were : Product Configurations Required Mean Average Number of Production Operation Time Configuration Products Ratio Mean(l) Mean(2) Mean(3) 1 3 1:1:1 1.00 1.00 1.00 2 3 2:1:1 0.50 1.00 2.00 3 3 3:1:1 0.66 1.00 2.00 144 Product j is assumed to have a mean operation time of Mean(j) at each station. Tables 6.1, 6.2, and 6.3 give the four-station lines studied, respectively, for product configurations 1, 2, and 3. It is assumed that the numbers of kanbans, mean operation times for each product, CV* 1 of operation times, and conveyance period are the same at all stations. Tables 6.4, 6.5, and 6.6 give the corresponding six-station lines studied, respectively, for product configurations 1, 2, and 3. For product configuration 1, the following kanban configurations are evaluated: Kanban Configurations for Product Configuration 1 Kanban Configuration P(l) C(l) P(2) C(2) P(3) C(3) 1 1 1 1 1 1 1 2 2 12 12 1 3 3 1 3 1 3 1 4 4 1 4 1 4 1 P(j) and C(j) denote, respectively, the number of production and conveyance kanbans for product j at all stations. Notice that for each kanban configuration following the first one, the number of production kanbans has been increased following the procedure outlined in Section 6.3. Since the required relative production ratio is 1:1:1 for product configuration 1, the number of production kanbans in each kanban configuration is the same for all three products. For product configurations 2 and 3, the following kanban configurations are evaluated: 145 Kanban Configurations for Product Configuration 2 Kanban Configuration P(l) C(l) P(2) C(2) P(3) C(3) 1 2 2 1 1 1 1 2 4 2 2 1 2 1 3 6 2 3 1 3 1 Kanban Configurations for Product Configuration 3 Kanban Configuration P(l) C(l) P(2) C(2) P(3) C(3) 1 3 3 1 1 1 1 2 6 3 2 1 2 1 3 9 3 3 1 3 1 Product configuration 2 has a required relative production ratio of 2:1:1. So the number of production kanbans is increased, following the procedure outlined in Section 6.3, maintaining the required relative production ratio. Similarly, product configuration 3 has a required relative production ratio of 3:1:1. So as the number of production kanbans is increased, the number of production kanbans for product one is always three times as large as the number of production kanbans for products 2 and 3. SECTION 6.5 EXPERIMENTAL RESULTS For each product configuration, Tables 6.1 - 6.6 show results for various CV^ of operation times and conveyance periods. For each 9 product configuration and for each combination of CV^- and conveyance period considered, results are given for each kanban configuration. Tables 6.1 - 6.6 give the shortage probabilities, Short(i), for each station i, and the average aggregate interstage inventories between each pair of stations obtained by both simulation and 146 decomposition approximation. Simulation and approximation results were obtained following the procedures given in Chapter 5. Simulation results are the average of two independent simulation runs. The standard error is given in parentheses below each simulation result. Note that the approximation is independent of the conveyance period. Hence, for each product and kanban configuration, simulation results using different conveyance periods, will vary while the approximation is the same for all conveyance periods. In general, the accuracy of the decomposition approximation follows the results in Chapter 5 and so the analysis of variance procedure will not be repeated here. That is, the approximation results are closest to simulation results for small conveyance periods and large CV^ of operation times. The emphasis here is on the usefulness of the decomposition approximation for the practical problem of setting production kanban numbers. For example, suppose the objective were to set the number of production kanbans so that the shortage probability at the last station of the line is less than 0.10. Further suppose there were four stations and the required relative production ratio is 3:1:1. Then Table 6.3 shows that the approximation and simulation procedures arrive at the same number of required production kanbans 3 out of 4 times when the CV^ of operations times is greater than 0.25. The table below shows the number of required production kanbans obtained by both approximation and simulation procedures. 147 Conveyance Period 2 3 CV2 of 1.0 9:3:3 Operation 0.5 6:2:2 6:2:2 Times r \ When the CV^ of operation times is 0.25 the approximation procedure consistently overestimates shortage probabilities and inventories due to the assumption of Markovian arrivals as noted in Chapter 5. Expected inventory holding and shortage costs may be used to set the number of production kanbans. For each kanban configuration, total shortage cost is the sum of shortage costs for each station multiplied by its corresponding shortage probabilities. Total inventory carrying cost is the sum of the carrying cost for inventory between each pair of stations multiplied by the average aggregate interstage inventories. Predictably, the approximation and simulation procedures will arrive at the same number of required production kanbans, to minimize total expected inventory holding and shortage cost, when the conveyance periods are relatively small and CV2 of operation times are large. SECTION 6.6 SUMMARY, CONCLUSIONS Results of this chapter have shown how Altiok's tandem-queue decomposition procedure may be used to set the number of production kanbans on a multi-product kanban line. In general, simulation and approximation procedures arrive at the same number of required production kanbans when conveyance periods are short, relative to the number of conveyance kanbans on the line, and the CV^ of operation times are large. Both simulation and approximation procedures may be applied in a dynamic environment and do not require assumptions regarding stationarity of the production lead time distribution and independence of production lead times and the number of production kanbans. The approximation procedure is, of course, preferable to simulation when it provides accurate results at a small portion of simulation's computing cost. Figure 6.1 On-Hand And On-Order Levels in Pull Control Systems[31] ORDER POINT. ORDER QUANTITY STOCK LEVEL ORDER POINT ON HAND + ON ORDER ON HAND TIME BASE STOCK (S -1 . S) STOCK LEVEL BASE STOCK = ON HAND + ON ORDER ON HAND TIME K A N B A N ( S - b ,S ) ON HAND + ON ORDER STOCK LEVEL ON HAND » TIME 149 150 Figure 6.2 Closed Queuing Network of Production Kanbans At Station Two Demand For Finished Product (Container and Conveyance Kanban) Finished Product Production Kanban A 5 Kanban Receiving Post Production-Ordering Kanban Post ASM ASM Production Activity [ I f ] (This Stage's Output) Conveyance ) Kanban Post (This Stage's Input A4 Physical units, container, and conveyance kanban f \ Empty container and conveyance kanban Production Kanban A3 Kanban Receiving Post >roduction-Ordering Kanban Post Production Activity [2 (This Stage's Output) ^Conveyance j Kanban Post (This Stage's Input) A2 Physical units, container, and conveyance kanban S"\ Empty container and conveyance kanban Production Kanban Al Kanban Receiving Post Production-Or dering Kanban Post A SM Production Activity [T] (This Stage's Output) 151 Figure 6.3 Closed Queuing Network of Conveyance Kanbans Between Stations Two And Three Demand For Finished Product (Container and Conveyance Finished Product Kanban) Production Kanban A5 Kanban Receiving Post Production-Ordering Kanban Post 10 ASM Production .Activity (This Stage's Output) Conveyance ) Kanban Post This Stage's Input) Physical units, container, and conveyance kanbai Empty container and conveyance kanban Production Kanban A3 Kanban Receiving Post Production -Grderin g Kanban Post Production Activity Q] (This Stage's Output) ^Conveyance ) Kanban Post (This Stage's Input) A2 Physical units, container, and conveyance kanban Empty container and conveyance kanban Production Kanban A1 Kanban Receiving Post Production-Ordering Kanban Post _ _\_ ASM Production Activity Q] (This Stage's Output) Table 6.1 Setting Numbers of Production Kanbans on 3-Product 4-Station Lines With Required Production Ratio 1:1:1 Average Inventories CV2 Conveyance Sinulation Approximation Simulation Approximation FC1) C(l) P(2) C(2) P(3) C(3) Hean(l) Hean(2) Mean(3) Period Short(2) Short(3) Short(4) Short(2) Short(3) Short(4) 1-2 2-3 3-4 1-2 2-3 3-4 1 1 1 1 1 1 l.QOOO 1.0000 1.0000 1.0 1.0 .0701 (.0012) .1315 (.0089) . 2088 (.0007) .0707 .1239 .1957 3.1472 (.0132) 2.4082 (.0346) 1.7300 (.0114) 3.3169 2.6314 1.9595 2 1 2 1 2 1 1.0000 1.0000 1.0000 1.0 1.0 .0621 (.0052) .1019 (.0046) .1683 (.0009) .0520 .0905 .1428 4.7809 (.0337) 3.7501 (.0217) 2.7028 (.0209) 5.0430 4.0959 3.1593 3 1 3 1 3 1 1.0000 1.0000. 1.0000 1.0 1.0 .0589 (.0038) .0760 (.0011) .1211 (.0071) .0410 .0712 .1124 6.1210 (.0906) 5.3181. (.1573) 4.0421 (.0350) 6.7772 5.5753 4.3821 4 1 4 1 4 1 1.0000 1.0000 1.0000 1.0 1.0 .0301 (.0007) .0511 (.0074) .0869 (.0125) > .0339 .0587 .0927 8.6323 (.1722) 7.1594 (.5481) 5.7154 (.3396) 8.5156 7.0620 5.6159 1 1 1 1 1 1 1.0000 1.0000 1.0000 1.0 2.0 .0003 (.0070) .1392 (.0089) .2339 (.0044) .0707 .1239 .1957 2.7746 (.0329) 2.2195 (.0958) 1.5773 (.0482) 3.3169 2.6314 1.9595 2 1 2 1 2 1 1.0000 1.0000 1.0000 1.0 2.0 .0768 (.0051) .1221 (.0062) .1705 (.0061) .0520 .0905 .1428 4.6323 (.1392) 3.6141 (.1341) 2.7612 (.0024) 5.0430 4.0959 3.1593 3 1 3 1 3 1 1.0000 1.0000 1.0000 1.0 2.0 .0588 (.0001) .0999 (.0009) .1398 (.0005) .0410 .0712 .1124 6.6343 (.0315) 5.2553 (.0851) 4.1225 (.0400) 6.7772 5.5753 4.3821 4 1 4 1 4 1 1.0000 1.0000 1.0000 1.0 2.0 .0575 (.0001) .0850 (.0013) .1273 (.0059) .0339 .0587 .0927 8.7616 (.0917) 6.8414 (.1174) 5.1027 (.3507) 8.5156 7.0620 5.6159 1 1 1 1 1 1 1.0000 1.0000 1.0000 0.5 1.0 .0351 (.0036) .0732 (.0036) .1229 (.0097) .0584 .1019 . 1601 3.0027 (.0698) 2.3802 (.0160) 1.8089 (.0771) 3.2684 2.6240 1.9939 2 1 2 1 2 1 1.0000 1.0000 1.0000 0.5 1.0 .0357 (.0031) .0511 (.0001) .0848 (.0093) .0418 .0726 .1141 4.4124 (.1788) 3.9174 (.0191) 3.1034 (.1448) 4.9964 4.0970 3.2095 3 i 3 1 3 1 1.0000 1.0000 1.0000 0.5 1.0 .0215 (.0030) .0406 (.0048) .0552 (.0060) .0325 .0563 .0886 6.7337 (.1941) 5.3236 (.1143) 4.1479 (.2504) 6.7325 5.5822 4.4423 4 1 4 1 4 1 1.0000 1.0000 1.0000 0.5 1.0 .0139 (.0062) .0255 (.0004) .0529 (.0037) .0266 .0460 .0724 8.4745 (.5140) 7.3294 (.3845) 5.5558 (.0721) 8.4722 7.0729 5.6829 1‘ 1 1 1 1 1 1.0000 1.0000 1.0000 0.5 2.0 .0484 (.0068) .0895 (.0059) .1450 (.0058) .0584 .1019 .1601 2.5771 (.1069) 2.1026 (.0533) 1.5536 (.0872) 3.2684 2.6240 1.9939 2 1 2 I 2 1 1.0000 1.0000 1.0000 0.5 2.0 .0351 (.0001) .0546 (.0024) .0901 (.0098) .0418 .0726 .1141 4.4033 (.0916) 3.7262 (.0832) 3.0660 (.1489) 4.9964 4.0970 3.2095 3 1 3 1 3 1 1.0000 1.0000 1.0000 0.5 2.0 .0278 (.0018) .0510 (.0060) .0837 (.0048) .0325 .0563 .0886 6.4197 (.0479) 5.1064 (.2506) 3.7703 (.2012) 6.7325 5.5822 4.4423 4 1 4 1 4 1 1.0000 1.0000 1.0000 0.5 2.0 .0258 (.0049) .0381 (.0018) .0656 (.0018) .0266 .0460 .0724 8.3151 (.5218) 6.7379 (.2558) 5.2762 (.2043) 8.4722 7.0729 5.6829 Ul ro Table 6.2 Setting Numbers of Production Kanbans on 3-Product 4-Station Lines With Required Production Ratio 2.1.1 (1) P(2) ? Conveyance Simulation Approximation Average Inventories Simulation Approximation CCD CC2) P(3) C(3) Mean(l) Mean(2) Mean (3) CV Period Short(2) Short(3) Short(4) Short(2) Short(3) Short(4) 1-2 2-3 3-4 1-2 2-3 3-4 2 2 1 1 1 1 0.5000 1.0000 2.0000 1.00 2*0 .0832 (.0054) .1436 (.0089) .2174 (.0005) .0697 .1228 .1954 4.1754 (.1273) 3.2710 (.1330) 2.5108 (.0565) 4.5133 3.5906 2.6727 4 2 2 1 2 1 0.5000 1.0000 2.0000 1.00 2.0 .0532 (.0009) .1090 (.0043) .1645 (.0050) .0516 .0904 .1437 6.8016 (.0579) 5.3096 (.0382) 4.2597 (.2316) 6.8190 5.5443 4.2698 6 2 3 1 3 1 0.5000 1.0000 2.0000 1.00 2.0 .0477 (.0055) .0867 (.0078) .1287 (.0077) .0410 .0716 .1137 9.0Q52 (.0721) 7.1079 (.2753) 5.6119 (.2025) 9.1333 7.5168 5.8978 2 2 1 1 1 1 0.5000 1.0000 2.0000 1.00 3.0 .0914 (.0125) .1559 (.0148) .2431 (.0051) .0697 .1228 .1954 3.8953 (.0939) 3.0601 (.0801) 2.3889 (.0092) 4.5133 3.5906 2.6727 4 2 2 1 2 1 0.5000 1.0000 2.0000 1.00 3.0 .0814 (.0129) .1209 (.0023) .1791 (.0093) .0516 .0904 .1437 6.2661 (.3719) 5.2959 (.1337) 4.0610 (.1162) 6.8190 5.5443 4.2698 6 2 3 1 3 1 0.5000 1.0000 2.0000 1.00 3.0 .0710 (.0137) .1025 (.0010) .1516 (.0040) .0410 .0716 .1137 9.6393 (.4368) 7.2416 (.3550) 5.4815 (.0418) 9.1333 7.5168 5.8978 2 2 1 1 1 i 0.5000 1.0000 2.0000 0.50 2.0 .0434 (.0020) .0843 (.0040) .1488 (.0055) .0582 .1016 .1605 4.0831 (.0626) 3.2994 (.0086) .2.5220 (.0997) 4.4711 3.6045 2.7490 4 2 2 1 2 1 0.5000 1.0000 2.0000 0.50 2.0 .0384 (.0025) .0650 (.0022) .1038 (.0060) .0420 .0729 .1152 6.2244 (.2097) 4.9538 (.2478) 3.9852 (.0387) 6.7815 5.5732 4.3732 6 2 3 1 3 1 0.5000 1.0000 2.0000 0.50 2.0 .0210 (.0050) .0426 (.0016) .0644 (.0048) .0328 .0569 .0899 8.6073 (.3596) 7.2894 (.1657) 5.9415 (.2163) 9.0996 7.5557 6.0184 2 2 1 1 1 1 0.5000 1.0000 2.0000 0.50 3.0 .0597 (.0044) .0957 (.0057) .1551 (.0044) .0582 .1016 .1605 3.5332 (.0320) 2.9661 (.1151) 2.3167 (.0322) 4.4711 3. 6045 2.7490 4 2 2 I 2 1 0.5000 1.0000 2.0000 0.50 3.0 .0447 (.0019) .0677 (.0057) .1115 (.0108) .0420 .0729 .1152 6.1948 (.0885) 5.3750 (.2606) 3.8335 (.0576) 6.7815 5.5732 4.3732 6 2 3 1 3 1 0.5000 1.0000 2.0000 0.50 3.0 .0345 (.0035) .0616 (.0018) .0932 (.0003) .0328 .0569 .0899 9.7032 (.0621) 7.2704 (.1236) 5.5995 (.2012) 9.0996 7.5557 6.0184 2 2 1 1 1 1 0.5000 1.0000 2.0000 0.25 2.0 .0313 (.0030) .0521 (.0026) .0890 (.0014) .0512 .0891 .1403 3.8209 (.0453) 3.2089 (.1140) 2.4857 (.0013) 4.4426 3.6071 2.7842 4 2 2 I 2 1 0.5000 I.0000 2.0000 0.25 2.0 .0224 (.0031) .0281 (.0009) .0507 (.0017) .0364 .0631 .0994 5.8845 (.1016) 5.2910 (.0356) 4.5351 (.2519) 6.7550 5.5821 4.4193 6 2 3 1 3 1 0.5000 I.0000 2.0000 0.25 2.0 .0096 (.0016) .0265 (.0086) .0364 (. 0048) .0282 .0488 .0769 8.9504 (.2824) 7.1454 (.5588) 5.6739 (.1805) 9.0744 7.5685 6.0711 2 2 1 1 1 1 0.5000 I.0000 2.0000 0.25 3.0 .0411 (.0018) .0620 (.0024) .0949 (.0024) .0512 .0891 .1403 3.3206 (.1251) 2.8342 (.0959) 2.3396 (.0717) 4.4426 3.6071 2.7842 4 2 2 1 2 1 0.5000 1.0000 2.0000 0.25 3.0 .0177 (.0044) .0360 (.0047) .0619 (.0070) .0364 .0631 .0994 6.2620 (.1625) 4.9789 (.1425) 3.8723 (.0996) 6.7550 5.5821 4.4193 6 2 3 1 3 1 0.5000 1.0000 2.0000 0.25 3.0 .0247 (.0008) .0367 (.0044) .0473 (.0010) .0282 .0488 .0769 7.6692 (.1186) 5.7072 (.2504) 5.5201 (.1702) 9.0744 7.5685 6.0711 153 Table 6.3 Setting Numbers of Production Kanbans on 3-Product 4-Station Lines With Required Production Ratio 3:1:1 Average inventories 1) CC1) P(2) C(2) P(3) CC3) 2 Conveyance Simulation Approximation Simulation Anoroxiaation Mean(l) Hean(2) Hean(3) CV Period Short(2) Short(3) Short(4) Short(2) Short(3) Short(4) 1-2 2-3 3-4 1-2 2-3 3-4 3 3 1 1 I 1 0.6666 1.0000 2.0000 1.00 2,0 .0680 (.0172) .1119 (.0211) .1799 (.0162) .0563 .0985 .1563 5.1734 (.3500) 4.2777 (.3558) 3.1178 (.3438) 5.6536 4.5716 3.4946 6 3 2 1 2 1 0.6666 1.0000 2.0000 1.00 2.0 .0504 (.0079) .0728 (.0007) .1068 (.0012) .0407 .0709 .1124 8.1106 (.3175) 6.9245 (.0651) 5.6037 (.1695) 8.5447 7.0349 5.5264 9 3 3 1 3 1 0.6666 1.0000 2.0000 1.00 2,0 : .0362 (.0082) .0653 (.0073) .0915 (.0038) .0319 .0554 .0878 11.4820 (.3109) 8.9302 (.1235) 7.4119 (.1858) 11.4438 9.5140 7.5838 3 3 1 1 1 1 0.6666 1.0000 2.0000 1.00 3.0 .0700 (.0032) .1227 (.0005) .1924 (.0011) .0563 .0985 .1563 5.0719 (.0448) 3.9677 (.0355) 3.0629 (.0190) 5.6536 4.5716 3.4946 6 3 2 1 2 1 0.6666 1.0000 2.0000 1.00 3.0 ‘ .0559 (.0060) .0868 (.0019) .1385 (.0136) .0407 .0709 ’ .1124 7.7734 (.1138) 6.3043 (.0488) 5.0631 (.4311) 8.5447 7.0349 5.5264 9 3 3 1 3 1 0.6666 1.0000 2.0000 1.00 3.0 .0*495 (.0011) .0649 (.0002) .1156 (.0022) .0319 .0554 .0878 10.8430 (.3017) 8.9392 (.1749) 7.1065 (.0167) 11.4438 9.5140 7.5838 3 3 1 1 1 1 0.6666 1.0000 2.0000 0.50 2.0 .0446 (.0060) ,0721 (.0188) .1102 (.0070) .0459 .0798 ,1259 4.8740 (.1078) 4.0338 (.4507) 3.1182 (.1065) 5.6129 4.5901 3.5777 6 3 2 1 2 1 0.6666 1.0000 2.0000 0.50 2.0 .0301 (.0037) .0503 (.0019) .0703 (.0032) .0325 .0563 .0888 7.5437 (.3110) 6.2531 (.2718) 5.2184 (. 1573) 8.5086 7.0660 5.6313 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.50 2.0 .0231 (.0043) .0388 (.0094) .0534 (.0015) ,0251 ,0434 .0686 10.6076 (.3601) 8.4884 (.7610) 7.1727 (.4801) 11.4108 9.5529 7.7016 3 3 1 1 L 1 0.6666 1.0000 2.0000 0.50 3.0 .0492 (.0048) .0835 (.0068) .1128 (.0047) ,0459 ,0798 .1259 4.5787 (.0311) 3.6035 (.0881) 3.1089 (.0565) 5.6129 4.5901 3.5777 6 3 2 1 2 1 0.6666 1.0000 2.0000 0.50 3.0 .0254 (.0070) ,0426 (.0100) .0727 (.0135) .0325 .0563 .0888 7.9533 (.3019) 6.6151 (.5726) 5.3549 (.5364) 8.5086 7.0660 5.6313 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.50 3.0 .0244 (.0019) .0217 (.0057) .0520 (.0071) .0251 .0434 .0686 10.7540 (.0995) 9.8586 (.8346) 8.0091 (.4447) 11.4108 9.5529 7.7016 3 3 1 2 i 1 1 0.6666 1.0000 2.0000 0.25 2.0 .0154 (.0016) .0374 (.0052) .0608 (.0042) .0399 .0692 .1089 5.1678 (.1536) 4.1403 (.0965) 3.2146 (.2017) 5.5840 4.5924 3.6122 6 3 1 2 1 0.6666 1.0000 2.0000 0,25 2.0 .0157 (.0017) .0297 (.0049) .0400 (.0088) .0279 .0482 .0759 7.3676 (.0272) 6.0828 (.1344) 5.1054 (.8500) B.4813 7.0730 5.6741 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.25 2.0 .0055 (.0025) .0113 (.0054) .0277 (.0031) .0214 .0370 .0583 12.2594 (.6380) 10.0673 (.8234) 8.4584 (1.3281) 11.3845 9.5628 7.7491 3 3 1 1 1 1 0.6666 1.0000 2.0000 0,25 3.0 .0273 (.0042) .0427 (.0050) .0678 (.0015) .0399 .0692 .1089 4.2334 (.0768) 3.7609 (.1201) 2.9887 (.0079) 5.5840 4.5924 3.6122 6 3 2 1 2 1 0.6666 1.0000 2.0000 0,25 3.0 .0188 (.0002) .0269 (.0027) .0438 (.0051) .0297 .0482 .0759 7.2065 (.4617) 5.9638 (.3701) 4.7629 (.2369) 8.4813 7.0730 5-6741 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.25 3.0 .0119 (.0044) .0158 (.0026) .0366 (.0096) .0214 .0370 .0583 10.1886 (.6933) 8.7996 (.0519) 6.2270 (.8612) 11.3845 9.5628 7.7491 154 Table 6.4 Setting Numbers of Production Kanbans on 3-Product 6-Station Lines With Required Production Ratio 1:1:1 PCD C(l P(2) C(2) P(3) C(3) Mean(l) Mean(2) Mean(3) CV2 Conveyance Period Short(2) Short(3) Simulation Short(4) Short(5) Short(6) Short(2) Approximation Short(3) Short(4) Short(5) Short(6) 1 1 1 1 1 1 1.0000 1.0000 1.0000 1.00 1.0 .0602 .1094 .1385 .1746 .2423 .0590 .0941 .1242 .1592 .2190 2 1 2 1 2 1 1.0000 1.0000 1.0000 1.00 1.0 (.0004) .0468 (.0031) .0768 (.0036) .1039 (.0106) .1276 (.0124) .1827 .0436 .0689 .0906 .1159 .1598 3 1 3 1 3 1 1.0000 1.0000 1.0000 1.00 1.0 (.0030) .0338 (.0090) .0535 (.0009) .0683 (.0080) .0866 (.0048) .1212 .0345 .0544 .0713 .0911 .1258 4 1 4 1 4 1 1.0000 1.0000 1.0000 1.00 1.0 (.0022) .0266 (.0036) .0536 (.0002) .0606 (.0113) .0654 (.0166) .1050 .0286 .0449 .0587 .0750 .1037 1 1 1 1 1 1 1.0000 1.0000 1.0000 1.00 2.0 (.0045) .0652 (.0007). .0998 (.0127) .1429 (.0041) .1769 (.0068) .2371 .0590 .0941 .1242 .1592 .2190 2 1 2 1 2 1 1.0000 1.0000 1.0000 1.00 2.0 (.0039) .0534 (.0085) .0861 (.0066) .1080 (.0123) .1416 (.0127) .1939 .0436 .0689 .0906 .1159 .1598 3 1 3 1 3 1 1.0000 1.0000 1.0000 1.00 2.0 (.0009) .0486 (.0062) .0746 (.0007) .0894 (.0113) .1062 (.0057) .1436 .0345 .0544 .0713 .0911 . 1258 4 1 4 1 4 1 1.0000 1.0000 1.0000 1.00 2.0 (.0004) .0436 (.0024) .0510 (.0110) .0783 (.0062) .0982 (.0085) .1350 .0286 .0449 .0587 .0750 .1037 1 1 1 1 1 1 1.0000 1.0000 1.0000 0.50 1.0 (.0026) .0431 (.0065) .0650 (.0028) .0866 (.0120) .0999 (.0121) .1421 .0491 .0780 .1027 .1312 .1798 2 1 2 1 2 1 1.0000 1.0000 1.0000 0.50 1.0 (.0098) .0279 (.0106) .0395 (.0018) .0548 (.0082) .0688 (.0039) .1035 .0353 .0556 .0730 .0931 .1280 3 1 3 1 3 1 1.0000 1.0000 1.0000 0.50 1.0 (.0059) .0193 (.0009) .0347 (.0040) .0481 (.0020) .0666 (.0089) .0738 .0275 .0432 .0566 .0721 .0993 4 1 4 1 4 1 1.0000 1.0000 1.0000 0.50 1.0 (.0017) .0152 (.0025) .0225 (.0032) .0261 (.0070) .0363 (.0010) .0600 .0225 .0353 .0462 .0588 .0811 1 1 1 1 1 1 1.0000 1.0000 1.0000 0.50 2.0 (.0027) .0455 (.0061) .0667 (.0012) .1010 (.0006) .1291 (.0079) .1638 .0491 .0780 .1027 .1312 .1798 2 1 2 1 2 1 1.0000 1.0000 1.0000 0.50 2.0 (.0069) .0321 (.0010) .0439 (.0083) .0641 (.0031) .0718 (.0031) .1105 .0353 .0556 .0730 .0931 .1280 3 1 3 1 3 1 1.0000 1.0000 1.0000 0.50 2.0 (.0037) .0265 (.0105) .0343 (.0050) .0523 (.0103) .0769 (.0086) .0912 .0275 .0432 .0556 .0721 .0993 4 1 4 1 4 1 1.0000 1.0000 1.0000 0.50 2.0 (.0026) .0236 (.0016) .0369 (.0070) .0476 (.0069) .0526 (.0074) .0770 .0225 .0353 .0462 .0588 .0811 (.0003) (. 0086) (.0021) (.0044) (.0048) I —1 Ln Ui Table 6.4 continued 2 Conveyance PCD CCD P(2) C(2) P<3) C(3) M ean(l) Mean(2) Mean(3) C V P e rio d 1-2 1 1 1 1 1 1 1.0000 1.0000 1.0000 1.00 1 .0 3.2796 (.0493) 2 1 2 1 2 1 1.0000 1.0000 1.0000 1.00 1.0 5.0759 (.0269) 3 1 3 1 3 1 1.0000 1.0000 1.0000 1.00 1 .0 6.8644 (.0860) 4 1 4 1 4 1 1.0000 1.0000 1.0000 1.00 1.0 8.8257 (.0961) 1 1 1 1 1 1 1.0000 1.0000 1.0000 1.00 2. a 3.0040 (.0144) 2 1 2 1 2 1 1.0000 1.0000 1.0000 1.00 2 .0 5.0551 (.0351) 3 1 3 1 3 1 1.0000 1.0000 1.0000 1.00 2 .0 7.0968 (.1035) 4 1 4 1 4 1 1.0000 1.0000 1.0000 1.00 2 .0 9.1591 (.0827) 1 1 1 1 I 1 1.0000 1.0000 1.0000 0 .50 1.0 2.9868 (.1390) 2 1 2 1 2 1 1.0000 1.0000 1.0000 0.5 0 1 .0 4.7950 (.1839) 3 1 3 1 3 1 1.0000 1.0000 1.0000 0.50 1.0 6.8586 (.0 8 7 5 ) 4 1 4 1 4 1 1.0000 1.0000 1.0000 0.50 1 .0 8.4532 (.0176) 1 1 1 1 1 1 1.0000 1.0000 1.0000 0.50 2 .0 2.6463 (.1211) 2 1 2 1 2 1 1.0000 1.0000 1.0000 0.50 2 .0 4.6161 (.0447) 3 1 3 1 3 1 1.0000 1.0000 1.0000 0 .50 2 .0 6.4077 (.1118) 4 1 4 1 4 1 1.0000 1.0000 1.0000 0.50 2 .0 8.6755 (.1936) A verage I n v e n to rie s S im u la tio n A pproxim ation 2-3 3-4 4-5 5 -6 1-2 2-3 3-4 4-5 5-6 2.6853 2.3516 2.0469 L.5143 3.5069 2.9906 2.6293 2.2735 1.7795 (.0207) (.0289) (.0668) (.1 1 1 7 ) 4.2469 3.7300 3.4222 2.5655 5.3003 4.5888 4.0940 3.6035 2.9093 (.1258) (.0309) (.1 6 6 4 ) (.0 2 1 9 ) 6.0497 5.3562 5.0068 4.0300 7.1006 6.1988 5.5737 4.9523 4.0647 (.0990) (.1065) (.1 4 1 6 ) (.4987) 7.6825 7.2080 6.3290 4.8306 8.9042 7.8142 7.0607 6.3101 5.2321 (.3914) (.3060) (.0 5 2 2 ) (.4561) 2.5316 2.2277 1.9181 1.5543 3.5069 2.9906 2.6293 2.2735 1.7795 (.0606) (.0462) (.0389) (.0681) 4.2882 3.7035 3.3114 2.5784 5.3003 4.5888 4.0940 3.6035 2.9093 (.1272) (.0795) ' (.1401) (.0193) 5.9993 5.4550 4.7687 4.0638 7.1006 6.1988 5.5737 4.9523 4.0647 (.0579) (.4641) (.2 2 2 1 ) (.2 5 2 3 ) 8.6538 ■ 7.3575 6.1450 4.9451 8.9042 7.8142 7.0607 6.3101 5.2321 (.0837) (.1240) (.3526) (.2572) 2.5772 2.2364 2.0664 1.6624 3.4453 2.9581 2.6192 2.2860 1.8221 (.1706) (.0431) (.0 5 7 3 ) (.0683) 4.1581 3.8629 3.5050 2.6597 5.2386 4.5611 4.0919 3.6275 2.9695 (.0094) (.2079) (.0 4 6 1 ) (.1 6 6 2 ) 5.8114 5.0840 4.3222 4.1431 7.0394 6.1744 5.5771 4.9840 4.1359 (.0194) (.1907) (.3233) (.0907) 7.8753 7.1399 6.6753 5.2154 8.8437 7.7924 7.0679 6,3470 5.3109 (.7350) (.2910) . (.0418) (.1952) 2.3401 1.9827 1.6996 1.3856 3.4453 2.9581 2.6192 2.2860 ' 1.8221 (.0670) (.0458) (.0 1 8 1 ) (.0348) 3.9952 3.4711 3.3058 2.4311 5.2386 4.5611 4.0919 3.6275 2.9695 (.2732) (.1639) (.0812) (.2218) 5.5847 5.0830 4.0179 3.4173 7.0394 6.1744 5.5771 4.9840 4.1359 (.1629) (.3417) (.0 2 8 3 ) (.2022) 7.0849 6.1327 5.8890 4.1673 8.8437 7.7924 7.0679 6.3470 5.3109 (.4960) (.0409) (.6652) (.0456) 156 Table 6.5 Setting Numbers of Production Kanbans on 3-Product 6-Station Lines With Required Production Ratio 2:1:1 PCD CCD P (2 ) C<2) P(3) C(3) Mean(1) Mean(2) Mean(3) C V 2 Conveyance Period S h o rt(2) Short(3) Sim ulation Short(4) Short(5) S h o rt(6) S h o rt(2) Approximation S h o rt(3) Short(4) S h o rt(5) Short (6) 2 2 1 1 1 1 0.5000 1.0000 2.0000 1.00 2.0 .0679 .1212 .1586 .1901 .2529 .0578 .0924 .1224 .1575 .2182 4 2 2 1 2 1 0.5000 1.0000 2.0000 1.00 2.0 (.0093) .0508 (.0251) .0885 (.0231) .1260 (.0230) .1440 (.0019) .1891 .0431 .0683 .0901 .1157 .1606 6 2 3 1 3 1 0.5000 1.0000 2.0000 1.00 2.0 (.0062) .0351 (.0013) .0524 (.0062) .0762 (.0078) .0916 (.0119) .1294 .0344 .0543 .0713 .0915 .1270 2 2 1 1 1 1 0.5000 1.0000 2.0000 1.00 3.0 (.0082) .0831 (.0043) .1324 (.0002) .1735 (.0099) .1983 (.0025) .2809 .0578 .0924 , 1224 .1575 .2182 4 2 2 1 2 1 0.5000 1.0000 2.0000 1.00 3.0 (.0152) .0626 (.0221) .0833 (.01-88) .1116 (.0203) .1403 (.0278) .2020 .0431 .0683 .0901 .1157 .1606 6 2 3 1 3 1 0.5000 1.0000 2.0000 1.00 3.0 (.0010) .0644 (.0114) .0932 (.0106) .1078 (.0175) .1257 (.0074) .1674 .0344 .0543 .0713 .0915 .1270 2 2 1 1 1 1 0.5000 1.0000 2.0000 0.50 2.0 (.0024) .0437 (.0101) .0801 (.0075) .0995 (.0046) .1307 (.0047) .1676 .0487 .0773 .1017 .1302 .1795 4 2 2 1 2 1 0.5000 1.0000 2.0000 0.50 2.0 (.0132) .0185 (.0210) .0394 (.0195) .0484 (.0156) .0640 (.0152) .0994 .0353 .0556 .0730 .0933 .1289 6 2 3 1 3 1 0.5000 1.0000 2.0000 0.50 2.0 (.0033) .0204 (.0122) .0368 (.0031) .0377 (.0064) .0550 (.0052) .0738 .0277 .0435 .0569 .0727 .1005 2 2 1 1 1 1 0.5000 1.0000 2.0000 0.50 3.0 (.0001) .0714 (.0005) .1035 (.0059) .1209 (.0069) .1439 (.0015) .1867 .0487 .0773 .1017 .1302 .1795 4 2 2 1 2 1 0.5000 1.0000 2.0000 0.50 3.0 (.0308) .0305 (.0393) .0533 (.0325) .0664 (.0199) .0820 (.0208) .1210 .0353 .0556 .0730 .0933 .1289 6 2 3 1 3 1 0.5000 1.0000 2.0000 0.50 3.0 (.0044) .0315 (.0039) .0502 (.0147) .0563 (.0113) .0733 (.0055) .1011 .0277 .0435 .0569 .0727 .1005 2 2 1 1 1 1 0.5000 I . 0000 2.0000 0.25 2.0 (.0010) .0240 (.0036) .0404 (.0011) .0540 (.0046) .0642 (.0086) .1016 .0430 .0681 .0895 .1143 .1571 4 2 2 1 2 1 0.5000 1.0000 2.0000 0.25 2.0 (.0053) .0191 (.0041) .0342 (.0053) .0450 (.0058) .0459 (.0063) .0664 .0307 .0483 .0633 .0807 .1112 6 2 3 1 3 1 0.5000 1.0000 2.0000 0.25 2.0 (.0036) .0110 (.0025) .0187 (.0044) .0194 (.0013) .0353 (.0001) .0441 .0239 .0374 .0489 .0624 .0861 2 2 I 1 1 1 0.5000 1.0000 2.0000 0.25 3.0 (.0039) .0219 (.0047) .0402 (.0036) .0507 (.0039) .0700 (.0061) .1152 .0430 .0681 .0895 .1143 .1571 4 2 2 1 2 1 0.5000 1.0000 2.0000 0.25 3.0 (.0015) .0244 (.0072) .0338 (.0053)' .0475 (.0094) .0487 (.0040) .0681 .0307 .0483 .0633 .0807 .1112 6 2 3 1 3 1 0.5000 1.0000 2.0000 0.25 3.0 (.0033) .0121 (.0021) .0228 (.0009) .0256 (.0073) .0385 (.0026) .0490 .0239 .0374 .0489 .0624 .0861 (.0015) (.0048) (.0032) (.0048) (.0032) P(l) C(l) P(2) C(2) P(3) C(3) Mean(l) Mean(2) Mean(3) CV2 2 2 1 1 1 0.5000 1.0000 2.0000 1.00 4 2 2 2 1 0.5000 1.0000 2.0000 1.00 6 2 3 3 1 0.5000 1.0000 2.0000 1.00 2 2 1 1 1 0.5000 1.0000 2.0000 1.00 4 2 2 2 1 0.5000 1.0000 2.0000 1.00 6 2 3 3 1 0.5000 1.0000 2.0000 1.00 2 2 1 1 1 0.5000 1.0000 2.0000 0.50 4 2 2 2 1 0.5000 1.0000 2.0000 0.50 6 2 3 3 1 0.5000 1.0000 2.0000 0.50 2 2 1 1 1 0.5000 1.0000 2.0000 0.50 4 2 2 2 1 0.5000 1.0000 2.0000 0.50 6 2 3 3 1 0.5000 1.0000 2.0000 0.50 2 2 1 1 1 0.5000 1.0000 2.0000 0.25 4 2 2 2 1 0.5000 1.0000 2.0000 0.25 6 2 3 3 1 0.5000 1.0000 2.0000 0.25 2 2 1 1 1 0.5000 1.0000 2.0000 0.25 4 2 2 2 1 0.5000 1.0000 2.0000 0.25 6 2 3 3 1 0.5000 1.0000 2.0000 0.25 Table 6.5 continued Average Inventories Conveyance Simulation Period 1-2 2-3 3-4 4-5 2.0 4.1897 3.5653 3.0874 2.7908 (.2642) (.4351) (.4114) (.3889) 2.0 ; 6.9252 5.8656 4.7255 4.4118 (.1619) (.1367) (.1288) (.2932) 2.0 10.0041 8.5348 7.9577 7.1349 (.4484) (.1154) (.2020) ,(.2494) 3.0 4.0720 3.3916 2.9617 2.7076 (.1703) (.1832) (.1553) (.2374) 3.0 6.8900 6.1474 5.5043 4.8983 (.0377) (.1462) (.1300) (.3917) 3.0 9.7127 7.8601 7.2115 6.5683 (.0020) (.3680) (.2674) (.1309) 2.0 3.9764 3.3680 3.0183 2.6266 (.3830) (.4791) (.4186) (.2317) 2.0 6.9706 6.0036 5.6529 5.0883 (.0774) (.4373) (.0262) (.1789) 2.0 9.0246 7.6629 7.5144 6.5672 (.1527) (.1310) (.0736) (.2989) 3.0 3.7108 3.0887 2.8353 2.5743 (.3048) (.4035) (.3871) (.2014) 3.0 6.7994 5.7609 5.0615 4.5803 (.2242) (.0629) (.5785) (.4622) 3.0 9.3315 7.7758 7.3213 6.4809 (.1578) (.2976) (.0004) (.4005) 2.0 3.9442 3.5229 3.1112 2.9596 (.0875) (.1002) (.0517) (.1874) 2.0 6.4577 5.0577 4.8,410 4.4971 (.1265) (.3399) (.3318) (.3611) 2.0 8.4969 7.2662 7.4261 6.1194 (.4659) (.3622) (.4844) (.0006) 3.0 3.7502 3.3214 3.0230 2.7364 (.0343) (.1122) (.0878) (.0895) 3.0 5.9136 5.2566 4.6896 4.2941 (. 2188) (.1106) (.0118) (.1166) 3.0 8.6523 7.6344 6.9745 6.1785 (.1913) (.6447) (.0585) (.5119) 5-6 1-2 Approximation 2-3 3-4 4-5 5-6 2.2098 4.7714 4.0823 3.5965 3.1135 2.4336 (.1387) 3.7092 7.1694 6.2176 5.5518 4.8873 3.9377 (.2214) 5.9695 9.5729 8.3655 7.5251 6.6850 5.4759 (.1112) 2.1788 4.7714 4.0823 ' 3.5965 3.1135 2.4336 (.3215) 3.8956 7.1694 6.2176 5.5518 4.8873 3.9377 (.3067) 5.4313 9.5729 8.3655 7.5251 6.6850 5.4759 (.4163) ' 2.1976 4.7081 4.0571 3.6032 3.1538 2.5209 (.2750) 4.1159 7.1071 6.2009 5.5723 4.9472 4.0545 (.1964) 5.7583 9.5122 8.3550 7.5551 6.7578 5.6112 (.0479) 2.1589 4.7081 4.0571 3.6032 3.1538 2.5209 (.1795) 3.7528 7.1071 6.2009 5.5723 4.9472 4.0545 (.0291) 5.0994 9.5122 8.3550 7.5551 6.7578 5.6112 (.4645) 2.4538 4.6690 4.0398 3.6028 3.1711 2.5623 (.1603) 3.6886 7.0686 6.1872 5.5779 4.9727 4.1079 (.0840) 5.3108 9.4745 8.3439 7.5645 6.7885 5.6720 (.0322) 2.1223 4.6690 4.0398 3.6028 3.1711 2.5623 (.0454) 3.7653 7.0686 6.1872 5.5779 4.9727 4.1079 (.0796) 5.2852 9.4745 8.3439 7.5645 6.7885 5.6720 (.8345) rt OQ C T * i-h rt LO c r r t H* r t £ > - rt P (l) 3 C(l) . 3 P(2) 1 C(2) P (3) 1 1 C(3) 1 Mean( 1) 0.6666 Mean(2) 1.0000 Mean(3) 2.0000 .CV2 1.00 6 3 2 1 2 1 0.6666 1.0000 2.0000 1.00 9 3 3 1 3 1 0.6666 1.0000 2.0000 1.00 3 3 1 1 1 1 0.6666 1.0000 2.0000 1.00 6 3 2 1 2 1 0.6666 1.0000 2.0000 1.00 9 3 3 1 3 1 0.6666 1.0000 2.0000 1.00 3 3 1 1 1 1 0.6666 1.0000 2.0000 0.50 6 3 2 1 2 1 0.6666 1.0000 2.0000 0.50 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.50 3 3 1 1 1 1 0.6666 1.0000 2.0000 0.50 6 3 2 1 2 1 0.6666 1.0000 2.0000 0.50 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.50 3 3 1 1 1 1 0.6666 1.0000 2.0000 0.25 6 3 2 1 2 1 0.6666 1.0000 2.0000 0.25 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.25 3 3 1 1 1 1 0.6666 1.0000 2.0000 0.25 6 3 2 1 2 1 0.6666 1.0000 2.0000 0.25 9 3 3 1 3 1 0.6666 1.0000 2.0000 0.25 Table 6.6 continued Average In v e n to rie s Conveyance Sim ulation Period 1-2 2-3 3-4 4-5 2.0 5.4868 4.9398 4.1806 3.8247 (.0857) (.2135) (.2797) (.2809) 2.0 8.4544 7.2102 6.5521 5.3689 (.2419) (-3655) (.0770) (.0539) 2.0 11.7880 10.7775 8.6797 7.7990 (.1127) (.3063) (.3786) (.3839) 3.0 5.2030 4.4177 3.9275 3.5402 (.1120) (.0837) (.0453) (.0508) 3.0 8.3313 7.4599 6.9042 5.7378 (.6022) (.0229) (.0554) (.1040) 3.0 11.8407 10.4868 9.2365 7.7702 (.5647) (1.0690) (.0864) (.0761) 2.0 5.4768 4.9096 4.2321 3.9537 (.1819) (.1453) (.0374) (.2228) 2.0 8.8452 7.6620 7.1136 5.8166 (.0477) (.3345) ■ (.1947) (.3221) 2.0 11.2548, 9.6103 9.2846 8.1629 (.7782) (.6105) (1.1399) (2.3960) 3.0 5.1329 4.3705 3.9682 3.6224 (.1200) (.1603) (.1726) (.0539) 3.0 7.6419 6.7172 5.9331 6.0779 (.2270) (.5409) (.4265) (.0916) 3.0 11.1341 9.9974 9.4219 8.2738 (.0486) (.1506) (.1224) (.8758) 2.0 5.0709 4.5592 4.0655 3.6248 (.2379) (.1748) (.2912) (.1217) 2.0 8.4788 7.7760 6.6253 5.9199 (.0124) (.0354) (.1706) (.2002) 2.0 11.5744 10.1785 7.8561 7.3879 (.0704) (.4081) (1.1009) (.0189) 3.0 4.8456 4.3147 3.6560 3.4492 (.0736) (.0197) (.1224) (.0262) 3.0 7.4513 6.7898 6.0992 5.8473 (..1514) (.0884) (.2128) (.0767) 3.0 10.8835 8.7451 8.0767 8.0661 (.3473) (.3829) (.0056) (.0539) Approximation 5-6 1-2 2-3 3-4 4-5 3.3176 (.0416) 5.9511 5.1419 4.5758 4.0124 4.9884 (.1447) 8.9538 7.8249 7.0402 6.2567 6.4011 (.1279) 11.9624 10.5192 "9.5199 8.5210 2.8857 (.2302) 5.9511 5.1419 4.5758 4.0124 4.5096 (.0133) 8.9538 7.8249 7.0402 6.2567 6.2417 (.0986) 11.9624 10.5192 9.5199 8.5210 3.3968 (.0756) 5.8890 5.1205 4.5874 4.0584 4.9933 (.4741) 8.8936 7.8112 7.0636 6.3191 6.8930 (.4390) 11.9038 10.5106 9.5507 8.5935 2.7715 (.1092) 5.8890 5.1205 4.5874 4.0584 4.6036 (.2277) 8.8936 7.8112 7.0636 6.3191 7.4300 (.1844) U .9038 10.5106 9.5507 8.5935 3.1087 (.0495) 5.8501 5.1038 4.5875 4.0757 4.7926 (.2435) 8.8553 7.7975 7.0682 6.3427 6.6729 (.4079) 11.8660 10.4987 9.5582 8.6208 2.8073 (.0742) 5.8501 5.1038 4.5875 4.0757 5.4955 (.2773) 8.8553 7.7975 7.0682 6.3427 6.7955 (.4978) 11.8660 10.4987 9.5582 8.6208 5-6 3.2120 5.1311 7.0781 3.2120 5.1311 7.0781 3.3070 5.2499 7.2111 3.3070 5.2499 7.2111 3.3480 5.3001 7.2665 3.3480 5.3001 7.2665 160 161 CHAPTER 7 SEQUENCING ON KANBAN-CONTROLLED LINES Introduction In this chapter the performance of four sequencing rules is evaluated on a constant withdrawal period kanban-controlled line. The rules are compared on a line subject to a pull-demand load factor sufficiently small so that each rule is able to produce the same total throughput and product mix. In this environment, results show FCFS most able to meet the just-in-time objectives of satisfying customer demands while minimizing average work-in-process inventories and flow times. Results do not show that SPT-based rules produce significantly better average finished-good conveyance-kanban waiting times. SECTION 7.1 PROBLEM DESCRIPTION The terms scheduling, sequencing, and dispatching are often used interchangeably, but scheduling has a broader meaning. Sequencing/dispatching is concerned with determining which job in queue to do next. Scheduling is involved not only in sequencing decisions, but also in the timing of lot sizes and the allocation of resources to jobs. Scheduling is an important component of production control, and the performance of any productive entity depends largely on the quality of its schedules. 162 Alternative Approaches In general, scheduling systems may be classified into push and pull systems. Push systems require detailed schedules for each production stage or station. Start and stop times for each job and each station are calculated by a centralized MRP-driven shop floor control system. These schedules specify where each product is needed (i.e., which downstream station) and when the parts are needed. The schedule for each station is obtained from the forecast of finished goods. Due dates for every product at every station are obtained by offsetting finished-goods need dates by cumulative planned production lead times. Because parts are sent or pushed, according to schedule, to where they are needed next, this type of production control is called a push system. In contrast, for pull systems a detailed schedule is prepared only for end items. When parts are needed at any station, a parts carrier moves conveyance kanbans upstream to the appropriate stations. In this way, parts are pulled through the line by the last station. Work orders are released to each station by actual part withdrawals. Thus, on a kanban-controlled line, the timing of work orders is established by actual demands. Sequencing on kanban- controlled lines is decentralized: The supervisor at each work station reviews the current work load and decides on the best sequence for processing. The kanbans displayed on the production-ordering kanban post are typically organized into columns by item, and each column is divided 163 into three regions which serve to indicate the urgency of replenishment for the items (Figure 7.1). The regions are characterized as a green safe zone, an amber can-produce zone, and a red must-product zone. Since the number of kanbans in the system for any item is constant, a large number of posted kanbans naturally indicates shortage of finished inventory in that item [31]. Thus, in the decentralized kanban production-control system the supervisor at each work station reviews the current work load and decides on the best sequence for processing. One advantage of this approach is that the information used for production control is current since the status of each job is clearly visible to all workers at the station. Second, there is a reduced need of transferring information to the central scheduling office. Sequencing While the scheduling task is greatly simplified in a kanban system, the sequencing problem remains. Of course, in any batch- processing environment where queues are commonly encountered, some means of resolving priorities must be introduced. As mentioned, the queue of work in a kanban system appears on each station's production-ordering kanban post. The purpose of the research discussed below is to evaluate various sequencing rules that may be used in a kanban production-control environment. 164 SECTION 7.2 SEQUENCING OBJECTIVES, CONVENTIONAL RULES, INFORMATION The primary objectives of the sequencing function are to 1) avoid late job completion and to obtain a high percentage of orders on time, 2) fully utilize the capacity of labor and equipment, 3) minimize flow time or the time that the job spends in the system, from opening a shop order until it is closed and 4) minimize work- in-process inventory, labor overtime, and stockouts of end items. There are conflicts between these objectives, and so one must give a priority to each of these objectives. In a conventional MRP-driven push system the primary concern of the sequencing function is to meet planned due dates. Each work- order released to the floor carries with it an MRP-generated due date. If due dates are not met, rescheduling and perhaps expediting are required. The measures of performance most often used are lateness, tardiness, maximum lateness, number of tardy jobs, as well as mean flow time, capacity utilization, and work-in-process inventory. As a consequence, sequencing is usually performed in a conventional shop by means of due-date driven priority or dispatching rules. Some examples are 1. EDD : Earliest Due Date 2. Dynamic Slack : due date - remaining processing time 3. S/ROP : slack/number of remaining operations 4. Critical Ratio : Time Remaining = Due Date - TNOW Work Remaining Work Remaining 165 On kanban-controlled lines the top objectives are to deliver end products required by customers at the right time and in the right quantities. Japanese consider minimization of work-in-process inventory to be the key to improving manufacturing performance. Inventory is viewed as the source of all problems. Therefore, they would rather leave labor and equipment idle than produce unnecessary inventories [40], To achieve these objectives, one must develop and examine sequencing rules that do not require MRP-generated due dates for each item. This is because, as previously discussed, decentalized kanban production-control systems do not have MRP-generated due dates for items other than end items. To be considered are priority rules that may be implemented with the limited amount of information available. On kanban lines, the following information is known: 1. the time and order of conveyance kanban arrival to the stock point, 2. whether or not conveyance kanbans remain at the stock point, and 3. container processing times. For example, in Figure 1.2, a parts carrier periodically moves conveyance kanbans upstream from station three to station two's stock point at node 7. It is assumed that the time and order of each conveyance kanban arrival to node 7 is known. Assuming parts corresponding to the conveyance kanbans are available in the output material queue node 6, the carrier detaches the production kanbans, which were attached to the full containers of parts, and places these 166 kanbans in the kanban receiving post A3. The time and order at which the production kanbans are released from the output queue is carried on the production kanbans until they are eventually posted on the production-ordering kanban post node 5. In this way the time and order of release may be used to prioritize production kanbans on the production-ordering kanban post. On the other hand, if parts corresponding to the conveyance kanbans are not available in the output material queue node 6 when the parts carrier arrives, the conveyance kanbans must wait in node 7. Thus, conveyance kanbans wait at the stock point of station two (node 7) only if the part demands they represent could not be met from the output material queue. By noting the conveyance kanbans waiting a node 7, it is possible to identify orders that, in some sense, are late. After reviewing the relevant sequencing literature, four alternative sequencing rules, that utilize only the above three types of information, are considered. In all cases, these priority rules are used to choose among production kanbans, on the production- ordering kanban post (e.g. node 5), that also have a corresponding full container of parts in the input material queue (e.g., node 4). This is done because work may begin on a particular production kanban only when its corresponding raw material is available. 167 SECTION 7.3 LITERATURE REVIEW A review of the literature involving sequencing rules in a single-stage conventional (push) job shop can be found in Blackstone [8]. However, research involving sequencing rules in the multi stage job shop is limited. It has been shown that the performance of sequencing rules in a single-stage shop does not generalize to the more complex multi-stage job shop [6]. Research has shown that when a shop is concerned with meeting due dates, sequencing rules which are due-date driven are most appropriate. In the multi-stage job shop, no predominant sequencing rule has been isolated. [17]. The research available on sequencing rules implemented in a pull-system environment is limited to a paper by Lee [36] and Lee & Seah [37]. Lee [36] considers an eight-station flow line with a two- card constant order-quantity, nonconstant cycle withdrawal, system. In this type of production control system, "as soon as the kanban count falls below the reorder level, a withdrawal kanban is dispatched to the upstream processes." [37]. There are 29 product- types each having exponential operation and setup times of mean, respectively, 50 and 10 minutes. Lee evaluates five sequencing rules to choose among jobs available for processing. The rules are: 1. FCFS : First-come-first-served gives highest priority to the first job arriving to the queue. 168 2. SPT : Shortest processing time gives highest priority to the job in queue with the shortest processing time. 3. HPF : Highest pull frequency gives highest priority to the job in queue with the greatest finished-goods demand frequency. 4. SPT/LATE 5. HPF/LATE SPT/LATE and HPF/LATE are two-tier hierarchical rules. SPT/LATE, for example, gives highest priority to the job with the shortest processing time as long as all jobs in the queue are not late. Otherwise, priority is based on the amount of job lateness. Lee terms a job "late" if, "it is not possible to proceed with the processing of a job in the subsequent process because of the inability to pull from the present station." [36] Job "tardiness" is taken to be, "the interval between the time when an attempt is made to pull a specific job from the final process station and the time when the pull is realized." [36] The five sequencing rules are first compared when the line has a load factor (ratio of demand to process capacity) of 0.94. The data obtained for the first 150 days = 3600 hours of each simulation run were discarded and results were collected over the next 300 days — 7200 hours. Lee's results are reproduced in Table 7.1. Lee concludes from Table 7.1 that the poor maximum queue time and maximum tardiness render SPT unacceptable for practical applications. In a second series of experiments Lee compares FCFS, SPT/LATE, HPF/LATE, at load 169 factors of up to 187.5% of capacity. His results show SPT/LATE to be much more effective than FCFS and HPF/LATE with respect to jobs drawn (i.e., throughput), process utilization, mean job tardiness, and mean job queue time. Because the late component in SPT/LATE causes the rule to have a maximum job tardiness between that of SPT and FCFS, Lee recommends SPT/LATE to be the best overall rule. Lee and Seah [37] use Lee's [36] model to compare FCFS and SPT/LATE when, again, the line has a load factor of 0.94. Operation times may be exponential, constant, or normal. Their results are reproduced in Table 7.2. Because SPT/LATE has better performance than FCFS with respect to jobs drawn, process utilization, mean job tardiness, and mean queue time, Lee and Seah recommend SPT/LATE. When the pull-demand load factor is close to one, one might expect the use of SPT alone will cause significant relative production-rate bias. Consider the following example of a six- station line producing ten different items. Each item has one production and one conveyance kanban at each station and an exponential operation time. Since all items have the same number of kanbans, all items should have equal throughput. Finished-goods conveyance kanbans arrive at a rate equal to 95% of line capacity. Average throughputs, obtained from three simulation experiments, are shown below for each item and for three different conveyance periods. It is assumed all stations use the same conveyance period. 170 Example Mean Operation Conveyance Period Product Time 2 25 . 50 1 1 .19500 .04000 .02000 2 2 .13460 .04000 .02000 3 3 .06580 .04000 .02000 4 4 .01500 .04000 .02000 5 5 .00300 .04000 .02000 6 6 .01000 .03960 .02000 7 7 .00000 .02300 .02000 8 8 .00000 .00000 .02000 9 9 .00000 .00000 .01580 10 10 .00000 .00000 .00120 Results show, particularly for smaller conveyance periods, that items with smaller average operation times are consistently produced at greater rates than items with longer average operation times. In conclusion, if the pull-demand load factor is sufficiently close to 1.0, so that the queue of finished-goods conveyance kanbans at the last station becomes large, pull lines operating with SPT-based priority rules will produce only those items with the shortest processing times. However, if the load factor is sufficiently small so that the queue of finished-goods conveyance kanbans at the last station may be periodically exhausted, relative production-rate bias will disappear. All priority rules will produce the same product mix and total number of items. In both Lee [36] and Lee & Seah [37] some relative production- rate bias is evident. Since in Tables 7.1 and 7.2 the outputs of the sequencing rules are different, it is obvious the pull-demand load factor applied was not sufficiently small to eliminate relative production-rate bias. It is also clear, particularly since SPT and SPT/LATE have the greatest throughputs (jobs drawn), that the SPT- 171 based rules are selectively choosing the shortest jobs and passing on the long jobs. Because SPT always takes the smallest job, one would expect its throughput to be greater than the throughput of an unbiased rule such as FCFS if pull demand were sufficiently large. Note that the maximum tardiness for SPT in Table 7.1 (3,265.6) is almost half the length of the computer run to obtain steady-state results. Lee [36] uses the "late" component of the two-tiered rule SPT/LATE to substantially improve its maximium tardiness performance over SPT. However, Lee's definition of "late" does not appear to be operational in a strictly decentralized manual kanban system. Recall, Lee terms a job late if, "it is not possible to proceed with the processing of a job in the subsequent process because of the inability to pull from the present station." [36] Presumably, this definition would mean, for example, that a container of parts is late if there were a production kanban posted on station three's production-ordering kanban post (node 9, Figure 1.2) while a corresponding conveyance kanban was still waiting at the stock point of station two (node 7). In practice, while station two would be aware of conveyance kanbans waiting at its stock point, it would not know the status of production kanbans at subsequent stations-- whether or not a corresponding production kanban were on the production-ordering kanban post of station three (node 9) or not. The corresponding production kanban may still be waiting in the 172 output material queue (node 10) until it is released to the production-ordering kanban post (node 9) by an external demand. SECTION 7.4 PURPOSE OF RESEARCH, SEQUENCING RULES CONSIDERED The purpose of this research is to investigate the performance of four sequencing rules in a decentralized, manual, kanban production-control environment. The sequencing rules considered require only the three types of available information given in Section 7.2. The definition of "late" is modified for a decentralized environment. Also, the type of production control system, in contrast to Lee [36] and Lee & Seah [37], is the constant withdrawal cycle (i.e., conveyance period), nonconstant quantity, system. The four rules considered are FCFS, SPT, SPT/LATE, and SPT/WAVE. Finally, the four sequencing rules are compared over three different conveyance periods. First-come-first-served (FCFS) is used as follows. Suppose there are several production kanbans on station two's production- ordering kanban post (node 5) that also have a corresponding conveyance kanban and a full container of raw material in the input material queue (node 4). Then the first kanban arriving at the production-ordering kanban post is removed for processing. Shortest processing time (SPT) is used in the same way as FCFS except that the production kanban with the shortest associated container processing time is removed first for processing. 173 SPT/LATE is used as follows: SPT is used to choose among production kanbans on the production-ordering kanban post (node 5, Figure 1.2) which, in addition to having a corresponding conveyance kanban and full container of raw material in the input material queue (node 4), also have a corresponding conveyance kanban waiting at the stock point (node 7). The conveyance kanban at the stock point represents demand from the following station that could not be met, when the parts carrier from the following station arrived, from the output material queue (node 6). Thus, the conveyance kanban at node 7 represents a late order. If no jobs have a corresponding conveyance kanban waiting at the stock point, then SPT alone is used. Note, SPT/LATE does not require any information regarding the status of subsequent stations. This SPT/LATE rule is similar to Lee's in spirit, but is modified for a strictly decentralized manual kanban system. The aim of SPT/WAVE is to use SPT while eliminating the possibility of relative production-rate bias. SPT is used to choose among demands (conveyance kanbans) arriving in the same conveyance wave. The earliest conveyance wave is given highest priority. For example, conveyance kanbans arriving to the stock point of station two (node 7) release production kanbans from the output material queue (node 6) to the production-ordering kanban post (node 5). The time of conveyance kanban arrival is used to place production kanbans on the production-ordering kanban post. Production kanbans released by the most recent conveyance wave are posted in the green safe zone 174 (Figure 7.1). Production kanbans released by earlier conveyance waves are posted in the amber can-produce and red must-produce zones. SPT is then used to choose among production kanbans posted in the same zone. Since all production kanbans from each wave must be removed before kanbans released by later conveyance waves, the possiblility for relative production-rate bias is eliminated. Note, however, as the period between conveyance kanban movements lengthens, the time of conveyance kanban movement becomes less valuable in distinguishing among kanbans because more and more arrive in the same conveyance wave. Note, when the conveyance period becomes short, SPT/WAVE will approach the performance of FCFS. Alternatively, as the conveyance period becomes longer and longer, SPT/WAVE will approach the performance of SPT (as long as there is no relative production-rate bias). SECTION 7.5 EXPERIMENTAL DESIGN A full factorial experiment was run to determine the performance of FCFS, SPT, SPT/LATE, and SPT/WAVE. The three factors employed in the experiments were the sequencing rule, conveyance period, and the coefficient of variation (CV) of operation times. It was assumed that the conveyance periods and CV at all stations were equal. 175 Factor Levels Sequencing rule FCFS, SPT, SPT/LATE, SPT/WAVE Conveyance period 2 , 4 , 6 Coefficient of variation 0.10, 0.25 Ten independent simulation runs were obtained for each set of experimental conditions resulting in 4x3x2x10 or 240 simulation runs. Three alternative conveyance periods are included for several reasons. First, in practice, the conveyance period is a dynamic parameter that is gradually reduced over time as the kanban system is refined. In general, "more frequent moves (smaller conveyance periods) indicate a just-in-time system while infrequent moves (longer conveyance periods) indicate a job shop environment." [55] Secondly, it was of interest to observe the general performance of a kanban-controlled line under different conveyance periods. For example, how would average work-in-process inventory respond to longer conveyance periods? A final objective was to see if the preferred sequencing rule in some way depends on the conveyance period. Simulation Model The kanban line considered is a six-station flow line with kanban-controlled finished-goods demand. The method of production control is the constant withdrawal cycle (i.e., conveyance period), nonconstant quantity, system given, for a three-station line, in Figure 1.2 and described in Chapter 1. The model was coded using the 176 discrete-event orientation of SLAM [44]. There are ten different products and each one has one production kanban and one conveyance kanban at every station. It was assumed that each product required the same average operation time at every station. The expected operation times required for each product was chosen from an exponential distribution of mean 1.0 when the product's finished-good conveyance kanban arrived to the stock point of the last station. The operation time for each item at every station was chosen from a normal distribution. Setup times are assumed to be included in the operation time. A series of initial simulation runs was made to determine 1) the maximum load factor such that the throughput for each sequencing rule was identical and 2) when approximate steady-state was achieved. That is, the load factor should be sufficiently small so that the queue of finished-goods conveyance kanbans would be periodically exhausted and each sequencing rule would produce the same mix and total number of products. On the other hand, it was desirable to have the load factor as large as possible: It was found, for smaller load factors, that SPT and SPT/LATE produced identical sequences (i.e., no product would ever be "late"). Results showed that when the coefficients of variation were 0.10 and 0.25 all sequencing rules could produce a throughput of, respectively, 0.98 and 0.95 units per period (having identical product mix) for all three conveyance periods considered. Therefore, the interarrival periods of each 177 group of 10 finished-goods conveyance kanbans (one for each product) were set at 10 x 1/.98 - 10.2041 and 10 x 1/.95 = 10.5263. To determine when approximate steady-state was reached, system parameters were graphed over time for each set of experimental conditions. It was found that all performance measures had reached approximate steady-state by time 5000. Below, results for the first 7000 hours have been discarded and steady-state results collected over the next 7000 hours. Performance Measures One of the most important measures of kanban-line performance is its average work-in-process inventory. Larger in-process inventories imply greater conveyance batch sizes and longer delays in detecting defective processes. Average work-in-process is defined as the sum of the aggregate interstage inventories between stations 1 & 2, 2 & 3, 3 & 4, 4 6 c 5, and 5 6 c 6. Mean flow time gives a measure of the aggregate amount of time jobs spend in queue waiting to be worked on. Flow time is defined as the time from when a production kanban is posted on the production- ordering kanban post of station one until finished goods corresponding to that kanban arrive in the output material queue of the last station. Average waiting time of finished goods conveyance kanbans gives a measure of service level. It shows how quickly the line responds to pull demands, or what portion of demand may be met from stock. 178 This performance measure is used as an alternative to Lee's "tardiness" [36]. Average finished-goods inventory is related to the demand service level. In general, the larger the average finished-goods inventory, the shorter the average waiting time of finished-goods conveyance kanbans. Due to the choice of demand load factors used, all four sequencing rules have exactly the same average throughput and station utilizations for a given conveyance period and coefficient of variation combination. Therefore, throughput and station utilizations are not included as performance measures. Hypotheses The first two hypotheses relate to average work-in-process inventory and flow time. Naturally, the objective is to minimize both of these averages. H]_: FCFS minimizes average work-in-process inventory. H2: FCFS minimizes average flow time. Lee [36] and Lee & Seah [37] found, for their constant withdrawal quantity system, that SPT and SPT/LATE had substantially smaller average tardiness figures than FCFS. In other words, on average SPT and SPT/LATE satisfy finished-goods demands quicker than FCFS. Below, a service-level hypothesis is formulated in terms of waiting times of finished-goods conveyance kanbans. H3: SPT, SPT/LATE, and SPT/WAVE have shorter average finished-goods conveyance kanban waiting times than FCFS. 179 H3 is essentially the same as H4: FCFS has the smallest average finished-goods inventories. A final objective is to study the performance of the alternative rules over a set of withdrawal cycle times (conveyance periods). H5: The best sequencing rule is independent of conveyance period. Regression Model The following indicator variables are used in the regression approach to obtain MANOVA results. Sequencing Rule FCFS SPT SPT/LATE SPT/WAVE Xx 1 0 0 0 X2 0 1 0 0 X3 0 0 1 0 Conveyance Period 2 4 6 X4 1 0 0 X5 0 1 0 Coefficient of Variation 0.10 0.25 X6 1 0 SECTION 7.6 EXPERIMENTAL RESULTS Tables 7.3, 7.4, 7.5, and 7.6 show the MANOVA results for, respectively, flow time, work-in-process inventory, finished-goods inventory, and finished-goods conveyance kanban waiting times. Tables 7.7, 7.8, 7.9, and 7.10 give the coefficients for the main- effects variables, defined in Section 7.5, for the corresponding regression models. Table 7.11 gives the averages and standard deviations for the performance measures under study. 180 Analysis of variance results in Tables 7.3 and 7.4 show that the sequencing rule and conveyance period significantly affected average flow time and work-in-process inventory levels. Longer conveyance periods make the importation of raw material to each station less frequent and so reduce the size of average input material queues. It follows that, because containers spend less time in input material queues, flow times are shorter for longer conveyance periods. As the conveyance period gets very long, each station on the line eventually becomes starved for raw material due to lack of conveyance. Flow times would also become very large. Note this implies, for any given line configuration and required throughput rate, there is some optimal conveyance period. Longer conveyance periods reduce work-in- process inventory and hence average flow times. However, excessively long conveyance periods will cause work-station starvation and reductions in the throughput rate. The consequence of this phenomenon is observed on kanban lines. Station supervisors, who are evaluated on the basis of their service levels and average in-process inventory, frequently delay material- handling trips to upstream stations to obtain raw material, until the last possible moment [30]. The significant negative t statistics associated with the indicator variable for FCFS (XI) in Tables 7.7 and 7.8 permit acceptance of and H2. These results are consistent with Lee & Seah [37]. Similarly, Tables 7.5 and 7.9 show that FCFS has significantly smaller average finished-goods-inventories than any of 181 the three SPT-based rules. Therefore, may also be accepted. These results are also consistent with Lee [36]. Tables 7.6 and 7.10 show the sequencing rule not to be a particularly significant factor in determining finished-goods conveyance kanban waiting times. The sequencing-rule F value in Table 7.6 of 5.0643 has a significance level of 0.0021. However, using Table 7.10, H3 must be rejected: SPT is shown to have significantly longer average finished-goods queue times than the other rules. The indicator variable for SPT, X2, has a significant t value of 6.22. Undoubtedly, this average has been inflated by the longer jobs continually being passed over. This can be seen in Table 7.11. Note for CV = 0.25 and a conveyance period of 6, SPT has a much longer (2.0761) average finished-goods conveyance-kanban waiting time than the other three rules. From Tables 7.7 and 7.8, it is clear that FCFS has significantly better average flow times and work-in-process inventories than the SPT-based rules. However, Tables 7.3 and 7.4 show that there are significant interaction effects. The critical F value of 2.85, for a significance level of .001 and df = 11/222, compares to F values of, respectively, 6.29 and 11.47. Therefore, H5 cannot be accepted. However, Table 7.9 shows that FCFS has the smallest average finished- goods inventories and Table 7.5 shows the interaction effects to be insignificant. Therefore, H5 may be accepted with respect to finished-goods inventory. Regression results in Tables 7.6 and 7.10 show that the sequencing rule is a marginally significant factor in 182 predicting finished-goods conveyance-kanban waiting times. From the significant t value of 6.22 in Table 7.10, for the indicator variable X2 associated with SPT, one can say SPT has significantly longer average finished-goods conveyance kanban waiting times than the other three rules. However, it is not possible to conclude that FCFS has shorter average finished-goods conveyance-kanban waiting times than SPT/LATE or SPT/WAVE. SECTION 7.7 SUMMARY, CONCLUSIONS The experiments conducted in this chapter have compared the performance of four sequencing rules on a constant withdrawal cycle (conveyance period) kanban-controlled line. The rules are compared at a pull-demand load factor sufficiently small so that no relative production-rate bias occurs. That is, each of the rules have produced the same total throughput and same output product mix. In this regard, the results differ from Lee [36] and Lee & Seah [37]. When pull-demand load is sufficiently less than line capacity, results have not shown SPT, SPT/LATE, or SPT/WAVE to be superior to FCFS. In general, the only result of producing the shortest- processing- time job first is to increase average work-in-process inventories and hence average flow times. Use of the SPT-based rules then contradicts the just-in-time philosophy of trying to minimize work-in-process. Further, the evidence does not show that SPT or SPT/LATE produced significantly better average finished-goods conveyance kanban waiting times than FCFS. Therefore, in summary, FCFS appears to be the best overall rule. Figure 7.1 A Multi-Product Kanban-Controlled Station[31] o o o o o o CARD TRIGGERS PRODUCTION LABOR MACHINES OUTPUT QUEUES A A A A A B B B C C D D D D D D E E E SAFE ZONE CAN BUILD MUST BUILD PART WITHDRAWAL RELEASES CARD B A C D CARDS POSTED ON BOARD : INPUT (PRODUCTION) QUEUE Table 7.1 Performance Effectiveness of Scheduling Rules[36] Process Job tardiness (hr) Job Queue Time (hr) Output Kanban Scheduling rule Jobs drawn utilization (%) Mean Maximum Standard Deviation Mean Maximum inventory (hr) FCFS 2185 64.1 497.1 1722.8 459.4 7.9 23.1 99.6 SPT 2656 75.7 79.7 3265.6 274.4 3.7 958.4 200.2 HPF 2110 58.2 466.1 2351.6 508.4 8.4 539.4 95.7 SPT/LATE 2641 75.5 92.9 2662.5 321.2 5.6 388.5 120.5 HPF/LATE 2082 58.6 489.7 1848.2 572.3 8.7 529.4 96.2 oo 4> Table 7.2 Results of The Process Times Study[37] Scheduling Distribution Jobs Process Job tardiness (hr) Queue time (hr) Work-in- rule drawn utilization progress (%) Mean Standard Deviation Mean Standard Deviation (hr) FCFS Exponential 72.8 64.1 497.1 459.4 7.9 8.1 99.6 FCFS Constant 87.8 82.6 231.2 170.1 7.9 5.3 64.0 FCFS Normal (0.2)* 87.6 80.3 222.0 183.9 6.8 5.6 96.4 FCFS Normal (0.4)* 86.8 78.5 259.3 203.1 6.9 5.9 88.1 SPT/LATE Exponential 88.0 75.5 92.9 321.2 5.6 14.1 120.5 SPT/LATE Constant 88.7 83.2 210.1 187.4 8.1 6.5 69.0 SPT/LATE Normal (0.2)* 89.8 81.5 81.6 274.3 5.5 17.2 180.7 SPT/LATE Normal (0.4)* 89.4 81.8 71.0 239.2 5.7 14.3 209.7 * Figure in brackets is the coefficient of variation 00 l/l Table 7.3 186 MANOVA Results For Flow Time Source SS df MS F Between Treatments 11763.49 17 691.97 2918.82 Sequencing Rule 34.86 Conveyance Period 11593.85 Coefficient of Variation 118.37 Interactions 16.41 3 2 1 11 11.62 5796.93 118.37 1.49 49.01 24452.16 499.30 6.29 Error 52.63 222 .24 Total 11816.11 Table 7.4 239 MANOVA Results For Work-In-Process Inventory Source SS df MS F Between Treatments 12446.71 17 732.16 1993.16 Sequencing Rule 92.12 Conveyance Period 12208.72 Coefficient of Variation 99.51 Interactions 46.35 3 2 1 11 30.71 6104.36 99.51 4.21 83.59 16617.63 207.89 11.47 Error 81.55 222 .37 Total 12528.26 239 Table 7.5 187 MANOVA Results. For Finished-Goods Inventory Source SS df MS F Between Treatments 160.38 17 9.4339 37.2143 Sequencing Rule 158.53 3 52.8440 208.4560 Conveyance Period 0.03 2 0.0168 0.0663 Coefficient of Variation 1.50 1 1.4950 5.8974 Interactions 0.32 11 0.0287 0.1131 Error 56.28 222 0.2535 Total 216.65 239 Table 7.6 MANOVA Results For Finished--Goods Conveyance-Kanban Waiting Times Sour ce SS df MS F Between Treatments 30.07 17 1.7689 4.8545 Sequencing Rule 5.54 3 1.8454 5.0643 Conveyance Period 3.20 2 1.6002 4.3915 Coefficient of Variation 3.01 1 3.0133 8.2696 Interactions 18.32 11 1.6656 4.5710 Error 80.89 222 0.3644 Total 110.96 239 Predictor Table 7.7 Regression Results For Flow Time Coefficient Standard Deviation t-Ratio 188 Constant 77.1937 0.1333 578.92 XI -1.0143 0.1778 -5.71 X2 0.4568 0.1778 2.57 X3 0.3618 0.1778 2.04 X4 16.2724 0.1721 94.53 X5 8.2316 0.1721 47.82 X6 -1.4153 0.1540 -9.19 s =0.4869 R-squared = 99.6% R-squared (adjusted) = 99.5% Table 7.8 Regression Results For Work-In-Process Inventory Predictor Coefficient Standard Deviation t-Ratio Constant 65.0913 0.1660 392.16 XI -1.7099 0.2213 -7.73 X2 0.9684 0.2213 4.38 X3 0.6361 0.2213 3.10 X4 16.2297 0.2143 75.74 X5 8.1251 0.2143 37.92 X6 1.0268 0.1917 5.36 s = 0.6061 R-squared = 99.3% R-squared (adjusted) = 99.3% 189 Table 7.9 Regression Results For Finished-Goods Inventory Predictor Coefficient Standard Deviation t-Ratio Constant 6.4133 0.1379 46.51 XI -1.8329 0.1838 -9.97 X2 -0.1010 0.1838 -0.55 X3 -0.0430 0.1838 -0.23 X4 0.0145 0.1780 0.08 X5 0.0027 0.1780 0.02 X6 0.2073 0.1592 1.30 s = 0.5035 R-squared = 74.02 R-squared (adjusted) = 72.0% Table 7.10 Regression Results For Finished-Goods Conveyance-Kanban Waiting Times Predictor Coefficient Standard Deviation t-Ratio Constant 0.2186 0.1653 1.32 XI 0.0477 0.2204 0.22 X2 1.3717 0.2204 6.22 X3 0.0052 0.2204 0.02 X4 -0.2440 0.2134 -1.14 X5 -0.2405 0.2134 -1.13 X6 -0.3598 0.1909 -1.88 s = 0.6036 R-squared = 27.1% R-squared (adjusted) = 21.5% Table 7.11 Averages and Standard Deviations of Performance Measures FCFS SPT SPT/LATE SPT/WAVE CV Period Mean SD Mean SD Mean SD Mean SD FLOW TIME 0.10 2 92.655 0.1277 92.858 0.1236 92.889 0.1034 92.710 0.1056 0.10 4 83.875 0.0895 84.523 0.0433 84.536 0.0621 84.269 0.1144 0.10 6 75.012 0.0857 75.628 0.0432 75.615 0.0508 75.651 0.0332 0.25 2 93.314 0.2400 94.382 0.2128 94.501 0.1560 93.363 0.2666 0.25 4 84.817 0.2379 86.379 0.1366 86.340 0.1939 85.401 0.2855 0.25 6 76.165 0.1971 77.619 0.0414 77.474 0.1032 77.321 0.1963 WORK-IN-PROCESS 0.10 2 83.6893 0.1718 84.0023 0.1422 83.9883 0.1313 83.7384 0.1375 INVENTORY 0.10 4 74.2708 0.1185 75.2225 0.0744 75.1749 0.1058 74.8438 0.1744 0.10 6 64.7197 0.1020 66.1545 0.0535 66.1027 0.0636 65.9944 0.0830 0.25 2 81.0541 0.2901 82.6334 0.2415 82.7221 0.1928 81.2140 0.3087 0.25 4 72.2315 0.3528 74.6621 0.1657 74.4210 0.2211 73.2000 0.3271 0.25 6 63.4142 0.2270 66.0277 0.0413 65.6531 0.1331 65.2151 0.2601 FINISHED-GOODS 0.10 2 4.6601 0.2809 6.5923 0.1040 6.5888 0.1015 6.5768 0.1034 INVENTORY 0.10 4 4.6526 0.2813 6.5898 0.0983 6.6110 0.1020 6.5908 0.1061 0.10 6 4.6820 0.2855 6.5749 0.1106 6.5617 0.0973 6.6162 0.1035 0.25 2 4.6165 0.2779 6.4635 0.0819 6.4240 0.1018 6.4200 0.0822 0.25 4 4.5201 0.2544 6.4679 0.0848 6.3950 0.0880 6.4194 0.0879 0.25 6 4.5975 0.2866 6.2675 0.0693 6.3936 0.0820 6.4177 0.0835 FINISHED-GOODS 0.10 2 0.0326 0.0038 0.0222 0.0013 0.0223 0.0016 0.0237 0.0018 CONVEYANCE KANBAN 0.10 4 0.0317 0.0033 0.0220 0.0024 0.0199 0.0018 0.0221 0.0015 WAITING TIME 0.10 6 0.0281 0.0033 0.0319 0.0048 0.0249 0.0023 0.0196 0.0015 0.25 2 0.0953 0.0096 0.1215 0.0214 0.0570 0.0058 0.0568 0.0054 0.25 4 0.1197 0.0126 0.1229 0.0148 0.0614 0.0081 0.0566 0.0053 0.25 6 0.1044 0.0159 2.0761 0.8902 0.0608 0.0068 0.0578 0.0072 VO o 191 CHAPTER 8 SUMMARY AND FUTURE RESEARCH SECTION 8.1 SUMMARY This thesis has addressed a two-card kanban-controlled line with constant withdrawal or conveyance period. The objective is to construct a queuing model capable of estimating average work-in- process inventories, throughputs, and material-shortage probabilities. Some method of estimating these performance measures is needed to implement kanban production control in a dynamic environment. For example, as product mix changes, the number of production and conveyance kanbans must be set to maintain required relative-production ratios while minimizing inventory holding and shortage cost. The kanban-controlled line described in Figure 1.1 is a complicated queuing system. It is a collection of linked, closed, queuing networks. Production and conveyance kanbans, as well as material containers, cycle in closed queuing networks of, respectively, production and conveyance kanbans at each station. The finite buffer capacities of each station, due to the finite number of kanbans on the line, cause blocking of entities. Because the kanban line does not have a product-form solution, efficient exact analysis is not possible. Approximate analysis of kanban-controlled lines begins with a relation between kanban lines and tandem queues. It has been found 192 that, under certain circumstances, kanban lines and tandem queues are equivalent. That is, they have equal average throughputs, work-in- process inventories, and material-shortage probabilities. In particular, single-product kanban-controlled lines are equivalent to a tandem queue when finished-goods demand is infinite and the conveyance period of each station is zero. The conveyance period is the constant period between movements of material containers between stations. In other cases, the kanban line may be thought of as a generalization of the tandem queue. Experiments in the previous chapters have exploited this relation along with a tandem-queue decomposition approximation. The basic idea, to estimate the performance measures of multi-product kanban lines with nonzero conveyance periods, is to first find the (approximate) equivalent tandem queue. The decomposition approximation is then applied to the equivalent tandem queue. There are two requirements for this approximation strategy to be successful. First, the kanban line must be sufficiently similar to its (approximate) equivalent tandem queue. Second, the decomposition must accurately approximate the equivalent tandem queue. Numerical results have shown that multi-product kanban lines remain approximately equivalent to a tandem queue, regardless of the number of products on the line, as long as the conveyance period is zero. It has also been shown how kanban lines, with kanban- controlled finished-goods demand, may be approximated by tandem queues with specially constructed pseudo stations. 193 On the other hand, results show that kanban lines distinguish themselves from tandem queues when the conveyance periods become large. On a tandem queue, of course, material never waits for conveyance. Upon completion at a station on a tandem queue, material is assumed to be instantaneously conveyed to the next downstream station. If the conveyance period is so long, for the number of conveyance kanbans on the line, that lack-of-conveyance induces material shortages, one must expect the tandem-queue approximation to underestimate shortage probabilities and overestimate average work- in-process inventories. Prior research has shown, when the squared coefficient of variation (CV^) of operation times is small (CV^ < 0.25), decomposition produces large errors when approximating tandem queues. This is because M/Ph/l/N models are used to analyze each station. When the CV^ of operation times is large, the assumption of Markov arrivals provides a good approximation to the arrival process. However, when the CV^ of operation times is small, the assumption of Markov arrivals overestimates the true variability of interarrival times. This causes the approximation to overestimate shortage probabilities and average inventories. In summary, when either of the two requirements above is not satisfied, the decomposition approximation does not accurately estimate kanban-line performance measures. Results have shown that, when the CV^ of operation times is small, the decomposition approximation overestimates station shortage probabilities and 194 average inventories. This is because the approximation is overestimating the true variability of interarrival times. Of course, on a kanban line, true interarrival times are deterministic and correspond to the conveyance period. Further, as conveyance periods increase, kanban lines become less similar to their (approximate) equivalent tandem queues. Therefore, as conveyance periods increase, the approximation, which is independent of the conveyance period, tends to underestimate true kanban-line shortage probabilities and overestimate average inventories. Combining these factors, for small CV^ of operation times, the approximation overestimates station shortage probabilities and average inventories. But as the conveyance period increases, the amount of overestimation decreases. For sufficiently large conveyance periods, the approximation eventually underestimates true shortage probabilities and overestimates average inventories. The approximation method was applied to the problem of setting production kanban numbers. To set kanban numbers, average work-in- process inventories and shortage probabilities must be estimated so that total inventory holding and shortage costs may be minimized. In general, simulation and approximation procedures arrive at the same number of required production kanbans when conveyance periods are short, relative to the number of conveyance kanbans on the line, and when the CV^s of operation times are large. Finally, the unique scheduling environment of kanban lines was described. On kanban lines there are no MRP-generated due dates 195 to drive conventional dispatching rules such as slack or critical ratio. Therefore, new sequencing rules have been constructed using only the information available in a manual, kanban, production- control environment. Four sequencing rules, FCFS and three SPT-based methods, were then evaluated in a simulation experiment. Results have not shown the SPT-based rules to be superior to FCFS. In general, the only result of producing the shortest-processing-time job first is to increase work-in-process inventories and hence average flow times. SECTION 8.2 FUTURE RESEARCH One objective of future research is to revise the arrival process used in the decomposition. One reason for doing this is to eliminate the errors caused by the assumption of Markov arrivals. A second objective is to make the decomposition dependent on the conveyance period. Conveyance period appears to be an important decision variable that strongly affects station shortage probabilities and average work-in-process inventories. Choice of conveyance period is somewhat analogous to the choice of review period in a periodic review inventory model. However, because of the dependence of stations on kanban-controlled lines, the choice of conveyance period at each station is tied to the choice of conveyance period at all other stations. Further research is necessary to determine how conveyance periods of all stations interact and how conveyance periods might be set. 196 This thesis has used a single-product tandem-queue decomposition approximation to estimate performance measures of multi-product kanban-controlled flow lines. Further research is necessary to extend the approximation method to the case of more general lines with split and merge configurations. Also, in the multi-product case, different products may not have the same routings. Some method is needed to obtain an aggregate measure of each station's load that may be used in a queuing model. In a dynamic job shop with time-varying product-demands, station production lead times change on a day-to-day, even hour-to-hour, basis. Therefore, in theory, the number of kanbans required also changes on a day-to-day or hour-to-hour basis. Future research is needed to determine the required frequency of kanban-system parameter changes. More than likely, the optimal frequency is some function of the level of excess capacity and safety stock on the line. Much remains to be done in the area of sequencing on multi- product kanban-controlled lines. As in any batch-manufacturing environment where queues are commonly encountered, some method of establishing job or production-kanban priorities is needed. However, the vast majority of sequencing literature does not immediately extend to kanban lines. This is because most conventional sequencing or priority rules require MRP-generated due dates that are not available on kanban lines. Further, while MRP-driven shop-floor control is a "push" method of production control, kanban is a "pull" method of production control. It is unknown if the large number of 197 results available for push lines might be extended to kanban or pull lines. 198 REFERENCES [1] Altiok, T. Approximate analysis of queues in series with phase- tvpe service times and blocking. 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International Journal of Production Research. 1977, 15(6), 553-564. [53] Takahashi, Y., Miyahara, H., & Hasegawa, T. An approximation method for open restricted queueing networks. Operations Research. 1980, 28(3), 594-602. 202 [54] Vollmann, T.E., Berry, W.L., & Whybark, D.C. Manufacturing Planning and Control Systems. Homewood, IL: Dow-Jones Irwin, 1984. [55] Wacker, J.G. The complementary nature of manufacturing goals by their relationship to throughput time: A theory of internal variability of production systems. Journal of Operations Management. 1987, 7(2), 91-106. [56] Zipkin, P. Models for design and control of stochastic multi item batch production systems. Operations Research. 1986, 34(1), 91-104. UMI Number; DP22641 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, th e se will be noted. Also, if material had to be removed, a note will indicate the deletion. Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Dissertation Publishing UMI DP22641 Microform Edition © ProQ uest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQ uest LLC. 789 E ast Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6

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Creator Berkley, Blair J. (author)

Core Title Elements of kanban production control for dynamic job shops

School Graduate School

Degree Doctor of Philosophy

Degree Program Business Administration

Degree Conferral Date 1988-08

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