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