5. Capacity and
Waiting
Dr. Ron Lembke
Operations Management
How much do we have?

Design capacity: max output designed for
 Everything

goes right, enough support staff
Effective Capacity
 Routine
maintenance
 Affected by resources allocated
 We can only sustain so much effort.
 Output level process designed for
 Lowest 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 = 0.857
 Efficiency: 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 consistently
 First
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

R*Q
FC+VC*Q
Break-Even
Point
Volume, Q
Cost Volume Analysis
Solve for Break-Even Point
 For profit of P,
 QBE =
FC
R – VC
FC = $50,000 VC=$2, R=$10
QBE = 50,000 / (10-2) = 6,250 units

747-400 vs 777
Monthly Debt Operating $/ton mile
747
$1,367,000
$50,000
$1.45
777
$1,517,000
$50,000
$1.38
Break-even:
747 ($1,367,000+$50,000)/(2-1.45)=
2,576,364 ton/miles per month
777 ($1,517,000+$50,000)/(2-1.38)=
2,527,419 ton/miles per month
Capacity Tradeoffs
120,000
4-door
cars
150,000
Two-door cars

Can we make combinations in between?
Adjust for aircraft size
777 – 124 tons per flight
2,576,364/124 = 20,777 full miles/month
747 – 104 tons per flight
2,527,419/104 = 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 processes

 Carry
unused overhead for a long time
 May 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 shifts
 Outsourcing non-core activities
 Training 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 Capacity
1997: Cars and
vans?
That’s crazy talk
First time in North
America
292,000 Camrys
89,000 Siennas
89,000 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 rides
 Buy souvenirs
 Do more rides per day
Benefits of Interactivity
Slow me down before going again
 Create buzz, harvest email addresses

Dumbo
False Hope
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 =
f (t )  e
 t
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

Avg service time = 1/ = 1/12
 Utilization

service rate =  = 12
= = /
10/12 = 5/6 = 0.833
Avg. # of customes

Lq 
    
2


Lq = avg # in queue =
Ls = avg # in system =

Ls  Lq 

 Lq  
Probability of # in System

Probability of no
customers in system

Probability of n
customers in system

P0  1 


Pn  P0  

n
Average Time


Wq = avg time in the
queue
Ws = avg time in
system
Wq 
Lq

Ws  Wq 
1

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 =
2
202
400
Lq 


 0.762
     3535  20 525


Ls = avg # in system = Lq + /
=
0.762 + 0.571 = 1.332
How Long is the Wait?
 Time
Wq 
Lq = 0.762, λ=20
Wq = 0.762 / 20 = 0.0381 hours
Wq = 0.0381 * 60 = 2.29 min
Lq

 Total
Ws  Wq 
waiting for service =
1

time in system =
Wq = 0.0381 * 60 = 2.29 min
μ=35, service time = 1/35 hrs = 1.714 min
Ws = 2.29 + 1.71 = 4.0 min
What did we learn?
Memoryless property means exponential
distribution, Poisson arrivals
 Results hold for simple systems: one line,
one server

 Average
length of time in line, and system
 Average number of people in line and in
system
 Probability of n people in the system