Overview of Congestion as It relates to Capacity and Variability Operations Management Dr. Mark P. Van Oyen Filename: congestion-lec.ppt
Capacity Planning • Capacity is the upper limit or ceiling on the load that an operating unit can handle. • Our basic questions in this capacity discussion is How much is needed? • Objective: introduce complexity of cycle times and service levels when there are statistical variations.
Capacity • Design capacity (maximum) • maximum obtainable output • e.g. truck service facility designed with 10 bays for a theoretical capacity of 5 trucks/day. Total Cap = 50 • Actual output • rate of output actually achieved--cannot exceed effective capacity due to • multiple jobs competing for common resources, • lack of training or effective organization, • equipment breakdown, • worker absenteeism • product mix, • scheduling difficulties,
Efficiency/Utilization Example • Truck repair/maintenance facility. • 1 “unit” = 1 truck • Design capacity = 50 trucks/day • Actual output = 36 units/day • Utilization= Actual output / Design capacity = • 36 units/day / 50 units/ day = 72% • (Very similar to Efficiency)
Efficiency is a measure • Efficiency (and utilization) measures are common in firms using cost-accounting. They give an incentive for • Keeping workers and resources busy = activated (difference between properly utilized and activated workers and resources) • Producing more than is requested in anticipation of future demand • Keeping larger than absolutely necessary inventories of raw materials, WIP, and finished goods inventory (FGI) • Large production batch sizes • Is That the RIGHT THING TO DO?
Silly Western Thinking? THE GOAL, page 87: • [Jonah] ``..you assumed that if you trim capacity to balance with market demand you won't affect throughput or inventory,'' he [Jonah] says. ``But, in fact, that assumption--which is practically universal in the western business world--is totally wrong.'‘ • ``How do you know it's wrong?'' [Rogo] • [Jonah] ``For one thing, there is a mathematical proof which could clearly show that when capacity is trimmed exactly to marketing demands, no more and no less, throughput goes down, while inventory goes through the roof,'' he says. ``And because inventory goes up, the carrying cost of inventory--which is operational expense--goes up. So it's questionable whether you can even fulfill the intended reduction in your total operational expense, the one measurement you expected to improve.''
When the Wait Gets Too Long … • The G/G/1 queueing model can explain some basic factory dynamics: • Why do Westerners want to balance capacity with demand? • Capital expense. Labor expense. • Example: Semiconductor industry, multi-million dollar machines. $2 Billion to build a new wafer fab. B = balking probability of not joining the line – taking your business elsewhere (use Erlang loss model when Balking/rejection significant)
Queuing Systems • Queue is a line of waiting customers who require service from one or more servers (service providers represent service capacity - machines, tools, workers, etc.). • Queueing system = waiting room + customers + server + workstations ArrivalsQueue Server(s) Departures
Queueing Happens … But Why? • Queues form whenever current demand (temporarily) exceeds existing capacity to serve. • Variability of Demand: patterns are irregular or random (measure: Std. Dev of job interarrival times) • Service times vary among “customers” or “jobs” (measure: Std. Dev of job service times) • = Statistical Fluctuations • Dependent Events. • Machine breaks down, causes backup • Wait (as distinct from queueing) happens when a part must wait for its counterparts before being assembled • Managers try to strike a balance between efficiently utilizing resources (which comes at the price of high WIP and long cycle-times) and keeping customer satisfaction high (which usually requires lower utilization levels). • Waiting increases when the variability of arrival times and/or service times increases.
How many examples of queues can you think of? • “Simple” … waiting in line for • Lunch/dinner at cafeteria • Video rental • Banking • Buying groceries • Complex • Service at sit-down restaurant • Amusement parks (network of queues) • Waiting for train/bus or elevator ride • Traffic congestion & waiting for traffic signals • ordering a Harley Davidson, then waiting for it to come.
Definitions of Queueing System Variables • = average arrival rate (AKA system Throughput) • 1/ = average time between arrivals of customers • µ = average service rate for each server (capacity) • 1/µ = average service time • r = /µ = server utilization • = long run fraction of time server is busy) • P0 = probability of empty system (arrival does not wait) = 1 - r • WIP = L = average number of customers in the system • CT = W = average cust. waiting time (queue + service)
Efficiency • Throughput = = production rate measured in completed jobs per unit time. Maximizing this is the first goal of many managers when they can sell all they can make – capacity constrained production (requires high r) • CT = Cycle Time = average value in hours of [time job completed – time job/raw materials enter system]. When demand is limited, this is increasingly important (need low r) • L = WIP (work in process inventory) = amount of product in system. Visually seen as the length of the waiting line. Managers want to keep performance good, but WIP low (need low r)
Utilization versus Cycle-time Tradeoff The M/M/1 queueing model: both WIP (L) and Cycle Time (Waiting time, W) follow the increasing curve shown below (which goes to infinity as utilization, r, approaches 1). Here, throughput is equal to the arrival rate – linear in r. Inventory, WIP Throughput, TH
Capacity versus Cycle-time Tradeoff The Total-Cost Curve is U-Shaped Total Cost Cost per Unit Time Capacity Cost Cycle Time Costs Excess Capacity = the amount that service capacity (rate) exceeds demand (rate) = ( - )
When Efficiency is a Red Herring • The rigidity and simplicity of Efficiency Measuresfrequently conflict with overall profit maximization. Profit is a necessary condition for most operations to exist. • THE GOAL (Goldratt) - makes strong arguments about the meaninglessness of efficiency metrics that are not directly linked to the goals of the firm (and a primary goal is to make money). • Caution: THE GOAL has a fallacy in their profit argument – it establishes profitability as necessary for a corporation and then jumps (in a logical fallacy) to profit being both necessary and sufficient as THE goal. Can’t there be more than one goal?
When Efficiency is a Red Herring • Queueing theory teaches us that in the presence of uncertainty, there is a clear tradeoff between performance (waiting times, queue length) and server utilization • Utilization must be set according to system capacity and the desired quality of service • Generally, higher quality of service requires lower utilization • Typically, we want to balance and set the production flow rate according to capacity, but not necessarily match flow to demand, which will vary. If 100% of capacity is needed to meet the demand that we intend to supply, then there is a problem! (Goldratt)
Fundamental Law of Production:Little’s Law • Having used a basic queueing model to describe the dynamics of congestion, • There is another jewel that comes from queueing theory: Little’s Law. • This is the most generally useful universal law that governs production systems. • It applies to systems in the long run over time … provided that job/customers that enter the system are counted in the number that exit it.
Little’s Law • LAW: For any “stable” queueing system (even a network of queues) • L = * W(queueing notation), that is, • WIP = TH * CT(production systems notation), • Insights: • Can be applied to financial flows • We often measure TH and Inventory (WIP), so now we can get Cycle Time easily!
Little’s Law Example: Gas Tanks • Given: • An automotive assembly plant produces 300,000 cars per year • The length of the assembly line from the first task to the last is 1.5 miles • Gas tanks are regularly sent in batches to the plant, stored in inventory, and then used in the assembly operation. On average, the time from gas tank arrival at the plant to the completion of the finished car is 3.5 days. • How much gas tank WIP (raw material or in production) does the plant carry?
Little’s Law Example: Gas Tanks • WIP = TH * CT • (1) TH = 300,000 tanks/year • (3) CT = 3.5 days * (1 year/ 365 days) = .0096 years • WIP = 300,000 tanks/year * .0096 years • = 2,877 tanks • This is a lot, but we are producing 822 cars/day on average.
Little’s Law Example: Cash Flow • Given: • A firm carries an average annual finished goods inventory (FGI) worth $1 Billion. • Average annual sales is currently $4 Billion. • We focus on cash flow rather than product flow. • What is the average cycle time of a dollar that is put into FGI? Note: this is an aggregate model for a typical product.
Little’s Law Example: Cash Flow • Given: • WIP = $1 Billion. • TH = $4 Billion. • What is the average cycle time of a dollar that is put into FGI? • WIP = TH * CT • 1 B. = 4 B. * CT • so the CT = WIP / TH = 1/4 year • CT = 3 mo's on average that our dollar spends sitting in inventory.
Little’s Law & Buying a House • Problem:Dean and Sue want to buy a home in a subdivision. There are about 100 homes in the subdivision and the average “dwell” time (including avg. time home is on market – about 3 months) is 5 years before they move out. Dean wants the house, but Sue is worried that, with 4 houses currently up for sale, too many people are selling their homes – a bad sign. • --->WIP = TH * CT • Dean says that homes on the market are those in “queue,” where service means that the queue is sold (each home gets its own server!). Then, • CT = avg. time house in queueing system = on the market = 1/4 year • TH = rate at which home go up for market = sales rate in subdivision TH = 100 Homes * (1/5 sales/year). • Notice that WIP tells us the average number of homes on market • WIP = TH * CT = 20 homes/year * ¼ year = 5 homes on avg. • So, this should give them both a little confidence that all is normal (or better!).
Influence of Variability • Variability Law: Increasing variability always degrades the performance of a production system. • Examples: • process time variability decreases throughput and increases Cycle Time & WIP. • higher demand variability requires more safety stock for same level of customer service • higher cycle time variability requires longer lead time quotes to attain same level of on-time delivery
Variability Buffering • Buffering Law: Systems with variability must be buffered by some combination of: 1. inventory 2. capacity 3. time. • Interpretation: If you cannot pay to reduce variability, you will pay in terms of high WIP, under-utilized capacity, or reduced customer service (i.e., lost sales, long lead times, and/or late deliveries).
Variability Buffering Examples • Ballpoint Pens: • can’t buffer with time (who will backorder a cheap pen?) • can’t buffer with capacity (too expensive, and slow) • must buffer with inventory • Ambulance Service: • can’t buffer with inventory (stock of emergency services?) • can’t buffer with time (violates strategic objectives) • must buffer with capacity • Organ Transplants: • can’t buffer with WIP (perishable) • can’t buffer with capacity (ethically anyway) • must buffer with time
Buffer Flexibility • Buffer Flexibility Corollary: Flexibility reduces the amount of variability buffering required in a production system. • Examples: • Flexible Capacity: cross-trained workers • Flexible Inventory: generic stock (e.g., assemble to order, delayed product differentiation) • Flexible Time: variable lead time quotes