Transcript Slides

Risk Management for
Construction
Dr. Robert A. Perkins, PE
Civil and Environmental Engineering
University of Alaska Fairbanks
Class 3
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Quantitative Risk Analysis
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Probability Basics
Follow Newnan Chapter 10.
Excel
Simulation with Crystal Ball
Certainty in estimating, from RAP site
“Probability of harm….”
• Two minutes of probability and statistics
• Statistics
• Want to know something about a
population
– Often its average, distribution of individuals
within population, range, extremes
• Can’t (usually) measure the population, so
we take a sample
Samples and Populations
• Try to learn something about the
population from the sample
• The larger the sample, the more confident
we will be.
• Also, the less variability we see in the
sample, the more confident we can be.
• We want to know how confidently we can predict the
population values from the sample.
• Often we want to see if two samples come from the
“same population,” or in common terms, “are the same.”
• Are the asphalt lab test results “different” in May than in
July?
• When we are done that analysis, again we want to know
• “How sure are we they are different?” Confidence.
• For those questions we need some probability ideas
Probability
• The characteristic we are looking at is
called the “random variable.”
• It is a variable, perhaps called “y,” but
unlike the “y” in y=mx+b, the value of our y
is not determined by other variables or
constants. It’s value is random, it is a
• “Random variable”
Toss the dice (or die)
• “Random Variable” = value of die
• See Excel RAND
Estimation
• Future event
– Number of rain days
• Often a parameter
– Cost of asphalt in 2010
• Want a “number”
• But “number” is a random variable
Reality
• Either convert to definite number
– Difficult to get, but
– Simple result
• Handle as a probability
– Easier to get, but
– Results need some explanation
• Often a definite number has an imbedded
factor of safety
– Which is based on probability
Forecasting Methods
• Subjective methods
– from within firm
• Statistical Methods
– extrapolation
• Modeling methods
“Precise Estimate”
• Use the best number(s) you have
• Sometimes called a “deterministic”
estimate.
Other Precise
• Breakeven
• Sensitivity
• Examine the impact that variability will
have
Range of Estimates
• Optimistic Estimate
• Most likely
• Pessimistic Estimate
Optimistic  4 * ( Most Likely )  Pess.
Mean 
6
• Has a scientific basis in Beta distribution
• Could do 3 entire estimates using all
optimistic, most likely, or pessimistic value
– then take mean
• Or do one estimate using mean for each
parameter.
• Answers will be slightly different.
Risk and Expected Value
• “Risk” (in this course) implies there are two
or more possible outcomes and we know
the probability associated with each
outcome.
• “Expected Value” is the weighted mean of
the outcomes times probabilities
• The town of Pittsfield needs to budget for snow
removal this winter. Historically it costs Pittsfield
$700,000 in a heavy snow year, $500,000 in an
avenge year, and $200,000 if it is a light snow
year. How much should the town budget? The
town calls the National Weather Service, who
says, "There is a 5% chance it will be a heavy
snow year, a 50% chance it will be an average
snow year, and a 45% it will be a light snow
year."
Expected Value
Projected Chance
Cost
Snow
(probability)
Light
45%
$200,000
Cost*Prob
ability
Medium
50%
$500,000
$250,000
Heavy
5%
$700,000
$35,000
$90,000
Total $375,000
Risk vs. Uncertainty
• Risk implies you know the probability of
the various outcomes
• Uncertainty implies you do not
• Might handle by increasing factors of
safety
• Better to use probability tools
Probability
Distribution of Outcomes
• Sales of 100, 200, 300…600 are equally
probable, each with one-sixth probability
• The total of the probabilities is 1.0
• Uniform Distribution
Relative Frequency
Newman, Lavelle, Eschenbach, 8th
• Experiment, Sample Space, and Event
• Consider the experiment of rolling a (six-faced) die,
the set {1, 2, 3, 4, 5, 6} defines the sample space of
the experiment. Any subset of the sample space is an
event.
– The event of getting odd number {1, 3, 5}
– The event of getting {6}
– The event of getting a number less than 4 {1, 2, 3}
• If, in an n-trial experiment, an event E
occurs m times, then the probability,
P{E} of realizing the event E is defined
mathematically as
m
P{E}  lim ( )
n   n
In our (six-faced) die example,
consider the event of getting the
number 5. If the die is rolled
100,000 times, there is a big chance
we will get the number 5 in about
(100,000/6).
i.e. n = 100,000 and m =
(100,000/6)
P{5} = m/n = 1/6
By Definition
0 ≤ P{E} ≤ 1
P{E} = 0 if the event E is impossible
P{E} = 1 if the event E is certain
In our (six-faced) die example, the
probability that the outcome of the
rolling is 7 is impossible
The probability that the outcome of
the rolling is an integer number from
1 to 6 is certain
Addition Law of Probability
• Consider two events E and F, these two events could
be mutually exclusive if they do not intersect.
– Assume event E = {1, 3, 5} and event F = {2, 4, 6},
these two events do not intersect and they are
mutually exclusive
– On the other hand, the two events E = {1, 2, 3, 4}
and F = {3, 4, 5, 6} intersect in {3, 4} and they are
not mutually exclusive
Addition Law of Probability
• E + F represents the union of E and F
– The union of E = {1, 2, 3, or 4} and F {3, 4, or 5} is
{1, 2, 3, 4, 5}
• EF represents the intersection of E and F
– The Intersection of E = {1, 2, 3, or 4} and F
{3, 4, or 5} is {3, 4}
• How to Calculate P(E+F)
Addition Law of Probability
• E + F represents the union of E and F
– The union of E = {1, 2, 3, or 4} and F {3, 4, or 5} is
{1, 2, 3, 4, 5}
• EF represents the intersection of E and F
– The Intersection of E = {1, 2, 3, or 4} and F {3, 4,
or 5} is {3, 4}
• How to Calculate P(E+F)
Addition Law of Probability
Consider the experiment of rolling a die. The sample
space of the experiment is {1, 2, 3, 4, 5, 6}. For a fair
die, we have
P{1} = P{2} = P{3} = P{4} = P{5} = P{6}
Define E = {1, 2, 3, or 4} and F = {3, 4, or 5}
the outcomes 3 and 4 are common between E and F,
hence EF = {3 or 4}. Thus,
P{E} = P{1} + P{2} + P{3} + P{4} = 1/6 + 1/6 + 1/6 +
1/6 = 4/6
F
P{F} = P{3} + P{4} + P{5} = 3/6
E
1 3 5
P{EF} = P{4} + P{5} = 2/6
4
P{E+F} = P{1} + P{2} + P{3} + P{4} + P{5} = 5/6
2
We can also say:
P{E+F} = P{E} + P{F} – P{EF} = 4/6 + 3/6 – 2/6 = 5/6
Addition Law of Probability
• Note in Example 1 that E and F are NOT mutually exclusive.
They intersect in {3 or 4}. Now, let us modify E and F to make
them mutually exclusive.
Define E = {1, 2, 3, or 4} and F = {5, or 6}
the outcomes 3 and 4 are common between E and F, hence EF
= {}. Thus,
P{E} = P{1} + P{2} + P{3} + P{4} = 1/6 + 1/6 + 1/6 + 1/6 = 4/6
P{F} = P{5} + P{6} = 1/6 + 1/6 = 2/6
P{EF} = 0
P{E+F} = P{1} + P{2} + P{3} + P{4} + P{5} + P{6} = 6/6
We can also say:
P{E+F} = P{E} + P{F} – P{EF} = 4/6 + 2/6 – 0 = 6/6
E
1 3
2 4
5
F
6
Addition Law of Probability
• Conclusion
P{E+F} = P{E} + P{F} – P{EF}
IF E and F are mutually exclusive,
P{E+F} = P{E} + P{F}
Revisit Expected Value
• If events are independent, you can
multiply their probabilities
• Flip a head and role a six
• 1/2 * 1/6 = 1/12
• Probability that the crane and the backhoe
will go down at the same time.
Buy Collision Insurance?
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Cost $800, with $500 deductable
Assume small accident cost $300
Total wreck is $13,000
P of no accident = 0.90, Small = 0.07,
Total = 0.03
Decision Tree
Expected Value
• Buy Insurance
(0.9*0 +(0.07*300) +(0.03*500)
=$36
• Don’t buy
(0.9*0 +(0.07*300) +(0.03*13,000)
=$411
• Of course the EV is less than the cost of
the insurance, $800.
• Roll of die
• “Random Variable” = value of die
• Excel, RAND function
• Sum of probabilities must = 1.0
• Take 15 coins and toss
• Count number of heads in each trial
Number of Heads
Number of Heads
14
12
10
8
6
4
2
15
10
5
0
0
Frequency in 49
trials
Number of Heads, 15 coins
Normal Distribution
• AKA Bell Curve, Gaussian Distribution
• When Random Variable is product of
many independent parameters, normal is
common result.
– Height of men
Distributions
• The coin example was “quantal” ,
“discrete” data.
– heads or tails
• Data is often continuous
– Weight of rat, 234.0 grams
• Look at triangular distribution
Relative
Frequency
$150
Value
$250
$450
Concrete cost next year, $/CY
What is probability
• It will cost $100/CY?
• It will cost $600/CY?
• It will cost $200/CY?
Relative
Frequency
$150
Value
$200
$250
$450
Concrete cost next year, $/CY
Probability Price less than
Probability
1.2
1
0.8
0.6
Probability
0.4
0.2
0
0
200
400
$/CY of Concrete
600
Combining Probabilities
• Most cannot be combined
• Monte Carlo Method
• Inserts random numbers in probability
statements
• Computes outcome
• Repeats 1000 or 10,000 times or more
Example
• Circular Stair
My Estimate
.
Item
Unit Cost
Buy stairs
$5000
Carpenter
time
$35
40
1400
Welder time
$42
20
840
Painter time
$32
20
640
Rent crane
$120
8
960
Total
Units (hr)
Extended
$5000
$8840
• Wrought iron fabricator
– “it depends how busy we are, and material
costs at the time you give us the PO. It may
cost anywhere between $3500 and $7000.”
• Carpenter foreman
– “it varies quite a bit, my guess is 40 hours, but
it could take anywhere from 30 to 70 hours,
but 40 is my best guess.”
• Welder
– “pretty sure” he can complete between 15 and
25 hours.
• Painter
– same.
• The crane shop
– Will charge me $120 if they have a crane, but
if they have to rent one for me it will cost
double that. They do say there is only a 20%
change they will have to rent, this time of
year.
Beta
• Next, here are my guesses inputted into a
beta distribution analysis.
• Total = (Low + 4*Most likely +High)/6
• Total = ( $6,620 + 4*$8,840 + $13,220 )/6
= $9,200
Risk Analysis
• Each parameter is a random variable, and
• we have some idea of the probability
• For example, the amount of carpenter hours is a
random variable. We put 40 hours into the
estimate as if it was a number, but in fact it is not
a number, but may have many values,
depending on what happens in the future.
• What we can put into the estimate is a
“probability distribution” that states the likelihood
of each value of the random variable
Buy Staircase
• The number can be anything between the
two limits and the probability is equal for
all numbers within those limits. This is
called a uniform distribution.
Carpenter time
• She gave us the least, maximum, and
most likely times. The random variable of
the carpenter’s time might be described by
a triangular distribution.
Welder and Painter
• They have given a range that they have some
confidence in, but are by no means sure.
• Let’s translate the “pretty sure” into meaning that they
are about 68% sure they will finish within those limits.
– Of course there is some chance that it could be a lot
longer, and for the moment let’s assume it could be
shorter as well.
• The “normal distribution” or “bell curve” has the property
that 68% is the probably within one “standard deviation”
of the average.
• So let’s approximate the welder and painters times as a
normal distribution with an average (or “mean”) of 20
hours and a “standard deviation” of 5 hours.
• About 65% of the area, that is the probability, lay
between 15 and 25 hours, just like the mechanics told
us.
Crane Cost
• This is figure is not a probably distribution,
the chart just shows it will be one number
80% of the time and 20% the other.
Crystal Ball
• Call Crystal Ball
What is the chance the job will cost less
than my original number, $8840?
Cost between 8 and 10 thousand?
There is a 50% change the job will
cost more than $9548
10% chance job will cost more than
$11,000
Method
Number
% Difference from
point estimate
Point Estimate
$8,840
-
Range Low
Estimate
$6,620
- 25%
Range High
Estimate
Beta
$13,220
49%
$9,200
4%
50% Confidence
$9,548
8%
90% Confidence
(less than)
$11,000
24%
• We are tempted to look at the point estimate and
consider it the “right number,”
• Then judge that the beta and 50% confidence
level are closest to being correct.
• But of course the point estimate itself is unlikely
to be exactly correct.
• My point here is that the difference between the
50% confidence number and the 90%
confidence number is $1500;
• the 90% confidence number is 15% greater than
the 50% confidence number.
Which to use?
• Owners and A/E’s might feel 50%
confidence is “fair”
• Contractors could not stay in business if
they only made a profit 50% of the time
Schedule Risk
•
•
•
•
Similar Process
Easy to input Beta = PERT
Computational issues
Can lay critical path into Excel and use
Crystal Ball or other
• Schedule ties to duration of tasks and thus
to item estimates and job estimates.