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## Modeling and Analysis of High Volume Manufacturing Systems

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**Discrete vs. Continuous Flow and Repetitive Manufacturing**Systems(Figures borrowed from Heizer and Render)**Operation Process Chart Example for discrete part**manufacturing(borrowed from Francis et. al.)**A typical (logical) Organization of the Production Activity**in Repetitive Manufacturing Assembly Line 1: Product Family 1 Raw Material & Comp. Inventory S1,1 S1,i S1,n Finished Item Inventory S1,2 Fabrication (or Backend Operations) Dept. 1 Dept. 2 Dept. j Dept. k S2,1 S2,2 S2,i S2,m Assembly Line 2: Product Family 2**Advantages and Limitations of the various layout types**(borrowed from Francis et. al.)**Advantages and Limitations of the various layout types**(cont. - borrowed from Francis et. al.)**Production**volume & mix The product-process matrix Low volume, low standardi- zation Multiple products, low volume High volume, high standardization, commodities Few major products, high volume Process type Jumbled flow (job Shop) Commercial printer Void Disconnected line flow (batch) Heavy Equipment Connected line flow (assembly Line) Auto assembly Continuous flow (chemical plants) Sugar refinery Void**A conceptual characterization of the considered environments**• Flow line: A sequence of workstations supporting the production of a single part type. • Each workstation consists of one or more identical servers executing one particular stage of the entire production process. • The part processing time at each workstation follows some general distribution which must be defined in such a way that accounts for the various detractors affecting the station operations; these detractors will include machine downtime, lack of consumables, operator unavailability, experienced set-up times, preventive maintenance, etc. • Finished parts could constitute end items or raw material for some other downstream process. • The operation of the line workstations can be • synchronized through an interconnecting conveyor system, or • decoupled through the installation of some buffering capacity between them.**Line Performance Measures**• Production rate or throughput, i.e., the number of parts produced per unit time • Line capacity, i.e., the maximum sustainable production rate • Line (expected) cycle time, i.e., the average time that is spend by any part into the line (this quantity includes both, processing and waiting time). • Average Work-In-Porcess (WIP) accumulated at different stations • Expected utilization of the station servers. Remark:The above performance measures essentially provide a link between the directly quantifiable and manageable aspects and attributes of the considered operational environments and the primary strategic concerns of the company, especially those of responsiveness and cost efficiency.**Production Authorization Mechanisms**• The issue here is to what extent part production is triggered from actual orders or from forecasted demand. • In a “produce-to-stock” scheme, a certain amount of end-item inventory is maintained in an effort to serve the experienced demand with zero lead time. • “Produce-to-stock” operation is most appropriate for highly commoditized and standardized items. • A key performance measure for “produce-to-stock” production systems is the fill rate, i.e. the percentage of the experienced demand that is actually met from stock. A desired fill rate is attained by maintaining an appropriate safety stock level, that is computed from the statistics of the experienced demand and cycle times. • In a “produce-to-order” scheme, end items are produced in response to particular orders. • “Produce-to-order” operation is more appropriate for (highly) customized items. • A key performance measure for “produce-to-order” production is the attained service level, i.e. the percentage of orders that are served within the quoted lead time. A target service level is attained through informed lead time quotation and some appropriate “capacity buffer”. • Currently, many systems are a hybrid scheme consisting of “produce-to-stock” and “produce-to-order” components. In particular, mass customization is supported by an “assemble-to-order” scheme where end-items are assembled to order from a number of sub-assemblies that are produced to stock.**Shop-Floor / Line Control Mechanisms**• Mechanisms that control the part release and advancement through the line. • They are broadly distinguished into “push” and “pull” mechanisms. • A “push” mechanism releases material into the line according to a target production rate, and material is advanced to downstream stations as early as possible. • Typical instantiations of “push” systems are: the asynchronous transfer line and the synchronous transfer line. • A “pull” system controls the part release and advancement in the line taking into consideration the status of the various workstations in the line. • Typical instantiations of “pull” systems are: the KANBAN and the CONWIP (controlled) production lines. • In general, “pull” systems reduce congestion, since they take into consideration the actual shop-floor status in their decision making, but the same mechanisms will also make them more inert in case of shifts in the production level. • Both mechanisms are effectively implementable in a “produce-to-stock” or “produce-to-order” context.**Asynchronous Transfer Lines**W1 W2 W3 TH TH TH TH B1 M1 B2 M2 B3 M3 • Some important issues: • What is the maximum throughput that is sustainable through this line? • What is the expected cycle time through the line? • What is the expected WIP at the different stations of the line? • What is the expected utilization of the different machines? • How does the adopted batch size affect the performance of the line? • How do different detractors, like machine breakdowns, setups, and maintenance, affect the performance of the line?**Synchronous Transfer Lines**• The key issue: Assembly Line Balancing (ALB) • Given • a set of tasks to be supported by the line stations • each possessing a nominal processing time, • a number of precedence constraints among these tasks, • and a target throughput, • determine • a partitioning of these tasks to a number of stations that observes the aforementioned specifications, while it minimizes the resulting number of stations (and therefore, the resulting labor cost).**Station 1**Station 2 Station 3 KANBAN-based production lines • Some important issues: • What is the throughput attainable by a certain selection of KANBAN levels? • What is the resulting cycle time? • How do we select the KANBAN levels that will attain a desired production rate? • How do we introduce the various operational detractors into the model?**FGI**Station 1 Station 2 Station 3 CONWIP-based production lines • Some important issues: • Same as those for the KANBAN model, plus • How can we compare the performance of such a system to that of an asynchronous transfer line and/or a KANBAN-based system?**Module Objectives**• Provide an analytical characterization of the operation of HVM systems and their performance, based on queueing-theoretic models • Derive qualitative insights and quantitative results on the attributes and factors that shape the behavior and performance of these systems. • Demonstrate the application of these results to the design and control of the considered class of systems • Focus primarily on flow lines, since they are the main layout used in the context of high-volume, repetitive manufacturing. • Also, flow line dynamics are easier to trace and analyze, and therefore, more enlightening in terms of qualitative and quantitative insights. • However, many of the derived insights and results are extensible to more complex environments either directly or through some appropriate decomposition.**Plan for the remaining part of the module**• Modeling and Performance Analysis of Asynchronous Transfer Lines through a Series of G/G/m queues • Modeling the impact of operational detractors • Employing the above results in line diagnostics • Design of Asynchronous Transfer Lines • Design of Synchronous Transfer Lines (cf. the ALB problem in the module on Sequencing, Dispatching and Scheduling of HVM systems) • Modeling and Performance Analysis of CONWIP-based production lines through Closed Queueing Networks • An integrating framework for bounding and shaping the performance of a production line • Analyzing the impact of batching on the system performance and designing optimized batching policies • Understanding the relative advantages and disadvantages of the various “push” and “pull”-based production systems**M1**TH B TH M2 Mm The G/G/1 model • Modeling Assumptions: • Part release rate = Target throughput rate = TH • Infinite Buffering Capacity • m identical servers • Server mean processing time = te • St. deviation of processing time = e • Coefficient of variation (CV) of processing time: ce = e / te • Coefficient of variation of inter-arrival times = ca**M1**TH B TH M2 Mm Performance measures for the G/G/1 station • Server utilization: • Expected cycle time in the buffer: (Kingman’s approx.) • Expected cycle time in the station: • Average WIP in the buffer: (by Little’s law) • Average WIP in the station: • Squared CV of the inter-departure times:**Some Important Remarks**• The entire analysis of the previous slide is an approximation, since it derives from the provided approximation for CTq. • The provided formula for CTqis exact for the M/M/m and M/G/1 stations. • The station operation will be stable only if u = TH te / m < 1.0, i.e., only if the effective workload released into the station per unit of time is less than the available processing time of the station. Otherwise WIPq, CTqand CT will grow infinitely large. • Notice that the requirement u < 1.0 is also suggested by the provided expression for CTqsince only then CTq takes finite positive values. • The expression for CTq comprises three factors, each depending respectively upon (i) the variability of the processing and inter-arrival times, as expressed by the corresponding SCV’s, (ii) the server utilization, and (iii) the mean processing time. An increase of any of these factors results in an increase of CTq, and also of CT, WIPq and WIP. • Assuming a non-zero mean processing time, CTq=0.0 only if ca2= ce2=0.0; these are essentially the conditions of a paced / synchronous production line. • The SCV of the inter-departure times can be approximated as a linear function of the SCVs of the inter-arrival and processing times. Furthermore, for u1.0, cd2 depends primarily on ce2, while for u0, cd2 depends primarily on ca2.**Analyzing an entire Production Line**TH • Key observations: • A target production rate TH is achievable only if each station satisfies the stability requirement u < 1.0. • For a stable system, the average production rate of every station will be equal to TH. • For every pair of stations, the inter-departure times of the first constitute the inter-arrival times of the second. • Then, the entire line can be evaluated on a station by station basis, working from the first station to the last, and using the equations for the basic G/G/1 model.**Modeling the impact of operational detractors**• Effective processing time = time that the part occupies the server • Effective processing time = Actual processing time + any additional non-processing time • Actual or otherwise natural processing time typically presents fairly low variability (SCV < 1.0). • Non-processing time is due to detractors like machine breakdowns, setups, operator unavailability, lack of consumables, etc. • Detractors are distinguished to preemptive and non-preemptive. Each of these categories requires a different analytical treatment. • Preemptive detractors are outages that take place during the actual processing of the part. Typical examples are machine breakdowns, lack of consumables, operator unavailability, etc. • Non-preemptive detractors are activities that may take place between the processing of two consecutive parts. Typical examples are setups, preventive maintenance, operator breaks, etc. • We want to determine the mean, variance and SCV of the effective processing time from the corresponding attributes of the natural processing time and some additional attributes characterizing the behavior of the various detractors.**Modeling the impact of preemptive detractors**• X = random variable modeling the natural processing time, following a general distribution. • to = E[X]; o2=Var[X]; co=o / to . • T = random variable modeling the effective processing time = where • Ui= random variable modeling the duration of the i-th outage, following a general distribution, and • N = random variable modeling the number of outages during a the processing of a single part. • mr=E[Ui]; r2=Var[Ui]; cr = r / mr • Time between outages is exponentially distributed with mean mf. • AvailabilityA = mf / (mf+mr) = percentage of time the system is up. • Then, te = E[T] = to / A or equivalently re = 1/te = A (1/to) = A ro**Breakdown Example**• Data: Injection molding machine has: • 15 second stroke (to = 15 sec) • 1 second standard deviation (o = 1 sec) • 8 hour mean time to failure (mf = 28800 sec) • 1 hour repair time (mr = 3600 sec) • Natural variabilityco = 1/15 = 0.067 (which is very low)**Example Continued**• Effective variability: Which is very high!**Example Continued**• Suppose through a preventive maintenance program, we can reduce mf to 8 min and mr to 1 min (the same as before) Which is low!**Modeling the impact of non-preemptive detractors**• X = random variable modeling the natural processing time, following a general distribution. • to = E[X]; o2=Var[X]; co=o / to . • NS = average number of parts processed between two consecutive setups • It is also assumed that the number of parts between two consecutive setups follows a geometric distribution, which when combined with the previous bullet, it implies that probability for a setup after any given job = 1/ NS. • Z = random variable modeling the duration of a setup • tS = E[Z]; S2 = Var[Z] • S = random variable modeling the setup time experienced by any given job = • T = random variable modeling the effective processing time = X+S • Then, E[S] = tS / NS ; Var[S] = (S2 / NS) + tS2((NS-1) / NS2); te = E[T] = to+tS / NS ; ;**Setup Example**• Data: • Fast, inflexible machine: (2 hr setup every 10 jobs) • Slower, flexible machine: (no setups) No difference!**Setup Example (cont.)**• Compare mean and variance • Fast, inflexible machine – 2 hr setup every 10 jobs • Slower, flexible machine – no setups • Conclusion: Flexibility can reduce variability.**Setup Example (cont.)**• New Machine: Consider a third machine same as previous machine with setups, but with shorter, more frequent setups • Analysis: • Conclusion: Shorter, more frequent setups induce less variability.**M1**B M2 to1=19 min co12=0.25 mf1=48 hrs mr1=8 hrs MTTR ~ expon. to2=22 min co22=1.0 mf2=3.3 hrs mr2=10 min MTTR ~ expon. 20 parts Ca2=1.0 Example: Employing the presented results for line diagnostics Desired throughput is TH = 2.4 jobs / hr but practical experience has shown that it is not attainable by this line. We need to understand why this is not possible.**Diagnostics example continued:Capacity analysis based on**mean values**Diagnostics example continued:An analysis based on the G/G/1**model i.e., the long outages of M1, combined with the inadequate capacity of the interconnecting buffer, starve the bottleneck!**Example: Designing an asynchronous prod. line**• Design of a new 4-station assembly line for circuit board assembly. • The four consecutive stations and the currently considered “technology options” for them (each option defines the processing rate in pieces per hour, the CV of the processing time, and the cost per unit in thousands of dollars). • The above data correspond to the effective processing times. • Each station can employ only one technology option. • The maximum production rate to be supported by the line is 1000 panels / day. • The desired average cycle time through the line is one day. • One day is equivalent to one 8-hour shift. • Workpieces will go through the line in totes of 50 panels each, which will be released • into the line at a constant rate determined by the target production rate. • Design task: Identify a line configuration that meets the above requirements while • minimizing the equipment cost. • Also, estimate the expected WIP at every station, when the line is operated at • maximum production rate.**A baseline design:Meeting the desired prod. rate with a low**cost**Reducing the line cycle time by adding capacity to Station 2****An alternative option:Employ less variable machines at**Station 1 This option is dominated by the previous one since it presents a higher CT and also a higher deployment cost. However, final selection(s) must be assessed and validated through simulation.**Analyzing CONWIP-based flowlines with single-machine**stations as Closed Queueing Networks (CQNs) Mean Value Analysis - the key underlying ideas: A CONWIP-based flow line with single-machine stations and its WIP level set to W, can be modeled by a closed queueing network (CQN) with general processing distributions, W jobs in it, and the following structure: II. In a CQN with W jobs and exponential processing times, the expected number of jobs observed at the various stations by a job arriving at some station Sj, is equal to the expected number of jobs observed at any random time at the same stations when the system is operated with W-1 jobs in it. III. Assuming that this effect applies in an approximate sense for more general distributions of the processing times, we proceed to develop an algorithm that will compute the performance measures of interest iteratively, for various W levels, starting with W=0. M1 M2 Mn**Notation**n = number of stations te(j) = mean effective processing time at station j ce2(j) = SCV for effective processing time at station j TH(W) = the line throughput when operated with WIP level W CT(W) = expected job cycle time through the line CTj(W) = expected job cycle time at station j when the WIP level is W WIPj(W) = expected WIP level at station j when the WIP level is W uj(W) = utilization of the server at station j when the WIP level is W**Deriving the algorithm iteration**CTj(W) = E[remaining processing time for the job at the server of Sj] + (E[number of jobs at station Sj]-E[number of jobs in service])te(j) + te(j) But E[remaining processing time for the job at the server of Sj] = Prob(Server of Sj busy)E[remaining process time | busy] = uj(W-1)E[remaining process time | busy] (b) E[number of jobs at station Sj] WIPj(W-1) E[number of jobs in service] uj(W-1) (d) uj(W-1) = TH(W-1) te(j)**Deriving the algorithm iteration (cont.)**Combining the results of the previous slide: But then, Obviously, for W=0, CT(0) =TH(0) = WIPj(0) = 0 Furthermore, application of the above formulae for W=1 gives: (from Little’s law)**Bounding the line throughput:Upper bounds**• For W, TH(W)rbminj{1/te(j)}, the bottleneck rate of the line. • rbcan also be achieved with finite WIP in a deterministic setting, i.e., in a line with ce(j) = 0, j, and synchronized with pace tb=1/rb. • However, by Little’s law, a line with raw process time To, in order to produce at rate rb, will need a WIP level of Wo=rbTo; this WIP level is known as critical WIP. • An interpretation of Wois given by the following formula: • i.e., Wois the level of WIP that we must maintain in the system in order to maintain the bottleneck utilization at 100%. Otherwise, the bottleneck will starve. • If W<Wo, then in a deterministic setting we can pace the jobs through the system in such a way that CT=To. Hence, the maximal line throughput will be**t1**t2 = = t3 = t4 = t5 = 1.0 TH=rb=1 W=Wo=rbTo=5 CT=To=5 TH=rb=1 W=6 CT=To+tb=6 TH=W/CT =4/5 W=4 CT=To=5 Example: Attaining the throughput upper bound with balanced, deterministically paced line**Bounding the line throughput:Lower bounds (under global**non-idleness) • Clearly, 1/Tois a lower bound to TH(W) under global non-idleness, since this is the rate of a line with only one job in it, and therefore, no parallelism. • This bound is also achievable under any other finite WIP level W, by a non-idling policy that moves all W jobs as a single batch from station to station. Indeed, for that policy**Example: Attaining the throughput lower bound through**batching t1 t2 t3 t4 W=3 = = = = 2.0 T = 0 T = 6 T = 12 T = 18 T = 24 TH = W / (W To) = 3 / 24= 1 / 8**The W-TH(W) space**TH(W) Ideal Operational Point rb 1/To 1/To 1 Wo=rbTo W**CT(W)**To 1/rb To Ideal Operational Point 1 Wo W The W-CT(W) Space The depicted curves are induced from those depicted in the W-TH(W) space, through Little’s law. In particular, upper bounds in the W-TH(W) space provide lower bounds in the W-CT(W) space, and vice versa, as follows:**Practical System Performance**• The “ideal” performance is attained in an optimized, deterministic setting. this performance is typically compromised by • the variability that is inherent in the system operations; • the impact of the applied control policies (e.g., the batching policy that defines the lower bounds for the system throughput - Notice, however, that these policies might be justified from some broader considerations). • A “benchmark” case: Maximizing the variability inherent in the system operation • Single-machine stations • Exponential processing times • Balanced line, i.e., te(j) = t, j • Key feature: All feasible states for this line are equiprobable. • Mean Value Analysis (exact, since processing times are exponential): Remarks: As expected, TH(1) = rb/Wo =1/To and TH() = rb. A performance that is worse than that of the “benchmark” case is a strong indication of mismanagement / bad practice.