1. Sensitivity Analysis and Uncertainty
H. Scott Matthews
12-706/73-359/19-702
Lecture 6

2. Admin HW 2 Due today Still no word on books
We think they’re due today?
Unfortunately cant keep pushing back lectures
I will send with private link to PDF for Wednesday reading (can’t post)

3. Problem of Unknown Numbers
If we need a piece of data, we can:
Look it up in a reference source
Collect number through survey/investigation
Guess it ourselves
Get experts to help you guess it
Often only ‘ballpark’, ‘back of the envelope’ or ‘order of magnitude needed
Situations when actual number is unavailable or where rough estimates are good enough
E.g. 100s, 1000s, … (102, 103, etc.)
Source: Mosteller handout

4. Definition: “Base Case”
Generally uses single values and our ‘best guesses’
Sensitivity Analysis acknowledges uncertainty exists
Incorporate variables instead of constant assumptions
If our ‘Net Benefits’ remain positive over a wide range of reasonable assumptions, then robust results

5. How many variables?
Choosing ‘variables’ instead of ‘constants’ for all parameters is likely to make model unsolvable
Partial sens. Analysis - change only 1
Equivalent of dy/dx
Do for the most ‘critical’ assumptions
Can use this to find ‘break-evens’

6. Best and Worst-Case Analysis
Analogous to “upper and lower bounds” used in estimation problems
Does any combination of inputs reverse the sign of our answer?
If so, are those inputs reasonable?
E.g. using very conservative ests.
Might want NB > 0, but know when NB < 0
Similar to ‘breakeven analysis’

7. Question 2.4 from Boardman
3 projects being considered R, F, W
Recreational, forest preserve, wilderness
Which should be selected?

8. Question 2.4
Project
“R with Road”
has highest NB

9. Question 2.4 w/ uncertainty
What if we note that Benefits/Costs of each project are uncertain by plus or minus 10%?
e.g. instead of Project R having benefits of $10 million, could be as low as $9 million or as high as $11 million
Repeat for all project combinations
Now which project is ‘best’?

10. Question 2.4 w/ uncertainty
Best Case:
R w/Road
(same)
Worst Case:
Road Only
General Case:
Could be
several
But difficult to determine that from this chart - can we do better?

11. Using error/uncertainty bars
Show ‘original’ point as well as range of uncertainty associated with point
Range could be fixed number, percentage, standard deviation, other
Excel tutorial available at:
See today’s spreadsheet on home page
Graphs original points, and min/max deviations from that as error bars…

12. Error bar result
Easier to see ‘best case/base case/worst case’ results - imagine moving
a straight edge vertically up and down the axis to see result.

13. Uncertainty
Investment planning and benefit/cost analysis is fraught with uncertainties
forecasts of future are highly uncertain
applications often made to preliminary designs
data is often unavailable
Statistics has confidence intervals – economists need them, too.
Talked about this last time with respect to social investments, and having no clue how to approach them

14. Sensitivity Analysis (SA)
Most of our discussions and examples have been simple (e.g. y=mx+b)
Life is not as simple as this
E.g. y=ax+b+e
We need to be able to create models and methods that can incorporate our uncertainties
“SA” is just a fancy phrase for saying “tell me what happens to the decision if the inputs change”

15. An Easy Visual Clue - Slider Bars
Use built-in excel functions to visually see effects of incremental changes in variables
“Scroll bars” (form toolbar)
Tutorial at:
Let’s look at our old TV estimation problem

16. EXCEL’s TABLE function
One- and two-input data tables
Sort of a built-in tool for sensitivity analysis (without fancy graphs/etc).
See PDF posted for examples and instructions

17. Case: Photo-sensors for lighting
Save electricity by installing sensors in areas where natural light exists
Sensors ‘see’ light, only turn light fixtures on when needed
MAIN Posner 2nd floor hallway uses watt fluorescent bulbs for 53 fixtures
How could we make a model to determine whether this makes sense?
Assume only one year time frame

18. Photo-sensors for lighting
Assume we only care about ‘one year project’
Costs = Labor cost, installation cost, electricity costs, etc.
Assume each bulb costs $6
Benefits = ? How should we set up model?
Assume equal, set up as ‘show minimum cost’ option
Case 1 ‘Status quo’: assume lights used as is
On all the time, bulbs last 10,000 hours ~ burn out once per year)
Case 2 ‘PS’: pay to install sensors now, bulbs off between 1/3 and 1/2 of time

19. Lighting Case Study - Status Quo
Costs(sq) - lights on all the time
Labor cost: cost of replacing used bulbs
“How many CMU facilities employees does it take to change a light bulb?” - and how long does it take?
Assume labor cost = $35/hr, 15 mins/bulb
26.5 hours to change all bulbs each year, for a total labor cost of $927.50! Also, bulb cost $636/yr
Electricity: 106*15W ~ 14,000 kWh/yr (on 24-7)
Cost varies from cents/kWh ~ $350-$1045
Cost Replacing bulbs is same ‘order of magnitude’ as the electricity! (Total range [$1,911 - $2,608])

20. Lighting Case Study - PS sensors
Costs(ps) - probably ‘off’ 1/3 - 1/2 of time
Labor cost: cost of installing sensors = ‘unknown’
Labor cost: cost of installing new bulbs
Could assume 1/2 - 2/3 of bulbs changed per year instead of ‘all of them’ [Total $464 - $700]
Bulbs cost [$318-$424]
Electricity: 106*15W ~ 7,000 1/2
9,333 kWh if off 1/3 of the time
Cost varies [ cents/kWh] ~ $175-$700
Total cost (w/o sensors) ~ [$955 - $1,740]
How much should we be WTP for sensors if time horizon is only one year?

21. PS sensors analysis WTP [$170 - $1,700] per year (NB>0)
We basically ‘solved for’ benefit
But our main sensitive value was elec. Cost, so range is probably [$919 - $1,129] per yr.
No overlap in ranges - PS always better
Should consider effects over several years
Could do a better bulb replacement model
Use more ranges - Bulb cost, labor, time
Check sensitivity of model answer to changes
Find partial sensitivity results for each
Look at spreadsheet model
This is fairly complicated - easier way?

22. Sens. Analysis for Photo Sensors
Several built-in options from the plugins [Win only] or treeplan [Win/Mac]) to check sensitivity.
One-way (one variable at a time)
Two-way (two at a time)
Tornado (all at a time)

23. One-Way Sensitivity Analysis
or treeplan plugin.
Makes graph that varies a single variable from a low- to high-end range and shows output (eg NPV) as a function of variable
The resulting graph shows how “sensitive” your answer is to changes in the one variable

24. Tornado Diagrams Shows results of many one-way plots on single chart
Length of bars tells you how sensitive output is to each variable
Bigger bar = more sensitive
Software typically “sorts” most to least
Looks like a tornado
Spider diagrams are similar

25. Two-way SA Treeplan plug-in cant do this
Shows a 2-dimensional plot of what happens when we change 2 variables at once
Graph generated is a “frontier”

26. Plug-in notes See Clemen pp.193- for TopRank tutorial
See course web page for instructions o using treeplan plugins (eg SensIt)
Read the PDFs for install instructions!
Should install ok on most cluster computers