5. Capacity and Waiting
Dr. Ron LembkeOperations Management
How much do we have? Design capacity: max output designed for
Everything goes right, enough support staff Effective Capacity
Routine maintenanceAffected by resources allocatedWe can only sustain so much effort.Output level process designed forLowest cost per unit
Loss of capacity
Utilization and Efficiency
Capacity utilization = actual output design capacity Efficiency = actual output effective capacity
Efficiency can be > 1.0 but not for long
Scenario 1
Design Capacity 140 tons Effective Capacity 124 tons –
landing gear could fail in bad weather landing With 120 ton load
Utilization: 120/140 = 0.857Efficiency: 120/124 = 0.968
Economies of Scale
Cost per unit cheaper, the more you make Fixed costs spread over more units
Dis-economies of scale
Congestion, confusion, supervision Running at 100 mph means more
maintenance needed Overtime, burnout, mistakes
Marginal Output of Time Value of working n
hrs is Onda As you work more
hours, your productivity per hour goes down
Eventually, it goes negative.
Better to work b instead of e hrs
S.J. Chapman, 1909, “Hours of Labour,” The Economic Journal 19(75) 353-373
Learning Curves
time/unit goes down consistentlyFirst 1 takes 15 min, 2nd takes 5, 3rd takes 3
Down 10% (for example) as output doubles We can use Logarithms to approximate this
cost per unit after 10,000 units? If you ever need this, email me, and we can
talk as much as you want
Break-Even Points
FC = Fixed Cost VC = variable cost
per unit QBE = Break-even
quantity R = revenue per unit
FC+VC*Q
Volume, Q
R*Q
Break-EvenPoint
Cost Volume Analysis
Solve for Break-Even Point For profit of P, QBE = FC R – VCFC = $50,000 VC=$2, R=$10QBE = 50,000 / (10-2) = 6,250 units
747-400 vs 777Monthly Debt Operating$/ton mile
747 $1,367,000 $50,000 $1.45777 $1,517,000 $50,000 $1.38Break-even:747 ($1,367,000+$50,000)/(2-1.45)=
2,576,364 ton/miles per month777 ($1,517,000+$50,000)/(2-1.38)=
2,527,419 ton/miles per month
Capacity Tradeoffs
Can we make combinations in between?
150,000Two-door cars
120,0004-doorcars
Adjust for aircraft size
777 – 124 tons per flight2,576,364/124 = 20,777 full miles/month747 – 104 tons per flight2,527,419/104 = 24,302 full miles/month
# Flights / month
747:20,777 miles/2,869 = 7.24 fully loaded flights= 8 full flights
777:24,302 miles/2,869 = 8.47 fully loaded flights= 9 full flights
Time Horizons
Long-Range: over a year – acquiring, disposing of production resources
Intermediate Range: Monthly or quarterly plans, hiring, firing, layoffs
Short Range – less than a month, daily or weekly scheduling process, overtime, worker scheduling, etc.
Adding Capacity
Expensive to add capacity A few large expansions are cheaper (per
unit) than many small additions Large expansions allow of “clean sheet of
paper” thinking, re-design of processesCarry unused overhead for a long timeMay never be needed
Capacity Planning How much capacity should we add? Conservative Optimistic
Forecast possible demand scenarios (Chapter 11)
Determine capacity needed for likely levels Determine “capacity cushion” desired
Capacity Sources
In addition to expanding facilities:Two or three shiftsOutsourcing non-core activitiesTraining or acquisition of faster equipment
What Would Henry Say? Ford introduced the $5 (per day) wage in 1914 He introduced the 40 hour work week “so people would have more time to buy” It also meant more output: 3*8 > 2*10
“Now we know from our experience in changing from six to five days and back again that we can get at least as great production in five days as we can in six, and we shall probably get a greater, for the pressure will bring better methods.
Crowther, World’s Work, 1926
Toyota Capacity1997: Cars and vans? That’s crazy talkFirst time in North America
292,000 Camrys89,000 Siennas89,000 Avalons
Decision Trees
Consider different possible decisions, and different possible outcomes
Compute expected profits of each decision Choose decision with highest expected
profits, work your way back up the tree.
Draw the decision tree
Everyone is Just Waiting
Everyone is just waiting
Retail Lines
Things you don’t need in easy reach Candy Seasonal, promotional items
People hate waiting in line, get bored easily, reach for magazine or book to look at while in line
Deposit slips Postal Forms
In-Line Entertainment
Set up the story Get more buy-in to ride Plus, keep from boredom
Disney FastPass Wait without standing
around Come back to ride at
assigned time Only hold one pass at a time
Ride other ridesBuy souvenirsDo more rides per day
Benefits of Interactivity
Slow me down before going again Create buzz, harvest email addresses
False HopeDumbo
Peter Pan
Queues
In England, they don’t ‘wait in line,’ they ‘wait on queue.’
So the study of lines is called queueing theory.
Cost-Effectiveness
How much money do we lose from people waiting in line for the copy machine?Would that justify a new machine?
How much money do we lose from bailing out (balking)?
Service Differences Arrival Rate very variable Can’t store the products - yesterday’s
flight? Service times variable Serve me “Right Now!” Rates change quickly Schedule capacity in 10 minute intervals,
not months How much capacity do we need?
We are the problem Customers arrive randomly. Time between arrivals is called the “interarrival
time” Interarrival times often have the “memoryless
property”: On average, interarrival time is 60 sec. the last person came in 30 sec. ago, expected time
until next person: 60 sec. 5 minutes since last person: still 60 sec.
Variability in flow means excess capacity is needed
Memoryless Property
Interarrival time = time between arrivals Memoryless property means it doesn’t matter how long
you’ve been waiting. If average wait is 5 min, and you’ve been there 10 min,
expected time until bus comes = 5 min Exponential Distribution Probability time is t =
tetf )(
Poisson Distribution
Assumes interarrival times are exponential Tells the probability of a given number of
arrivals during some time period T.
Simeon Denis Poisson "Researches on the probability
of criminal and civil verdicts" 1837
looked at the form of the binomial distribution when the number of trials was large.
He derived the cumulative Poisson distribution as the limiting case of the binomial when the chance of success tend to zero.
Larger average, more normal
Queueing Theory Equations
Memoryless Assumptions:Exponential arrival rate = = 10
Avg. interarrival time = 1/ = 1/10 hr or 60* 1/10 = 6 min
Exponential service rate = = 12 Avg service time = 1/ = 1/12
Utilization = = / 10/12 = 5/6 = 0.833
Avg. # of customes
Lq = avg # in queue =
Ls = avg # in system =
2
qL
q
qs
L
LL
Probability of # in System
Probability of no customers in system
Probability of n customers in system
10P
n
n PP
0
Average Time
Wq = avg time in the queue
Ws = avg time in system
q
q
LW
1
qs WW
Example
Customers arrive at your service desk at a rate of 20 per hour, you can help 35 per hr. What % of the time are you busy?How many people are in the line, on average?How many people are there in total, on avg?
Queueing Example
λ=20, μ=35 so Utilization ρ=20/35 = 0.571 Lq = avg # in line =
Ls = avg # in system = Lq + /= 0.762 + 0.571 = 1.332
762.0525400
2035352022
qL
How Long is the Wait?Time waiting for service =
Lq = 0.762, λ=20Wq = 0.762 / 20 = 0.0381 hoursWq = 0.0381 * 60 = 2.29 min
Total time in system = Wq = 0.0381 * 60 = 2.29 min μ=35, service time = 1/35 hrs = 1.714
minWs = 2.29 + 1.71 = 4.0 min
q
q
LW
1
qs WW
What did we learn?
Memoryless property means exponential distribution, Poisson arrivals
Results hold for simple systems: one line, one serverAverage length of time in line, and systemAverage number of people in line and in
systemProbability of n people in the system