Sensitivity Analysis and Uncertainty

H. Scott Matthews12-706/73-359/19-702Lecture 6

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HW 2 Due todayStill no word on books

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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

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

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 y/x Do for the most ‘critical’ assumptions

Can use this to find ‘break-evens’

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’

Question 2.4 from Boardman

3 projects being considered R, F, W Recreational, forest preserve, wilderness Which should be selected?

Alternative Benefits($)

Costs($)

B/CRatio

NetBenefits ($)

R 10 8 1.25 2R w/ Road 18 12 1.5 6F 13 10 1.3 3F w/ Road 18 14 1.29 4W 5 1 5 4W w/ Road 4 5 0.8 -1Road only 2 4 0.5 -2

Question 2.4

Base Case Net Benefits ($)

-4 -2 0 2 4 6 8

R

R w/ Road

F

F w/ Road

W

W w/ Road

Road only

Project“R with Road”has highest NB

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’?Alternative Base Case

BenefitsB +10% B -10% Base Case

CostsC -10% C +10%

R 10 11 9 8 7.2 8.8R w/ Road 18 19.8 16.2 12 10.8 13.2F 13 14.3 11.7 10 9 11F w/ Road 18 19.8 16.2 14 12.6 15.4W 5 5.5 4.5 1 0.9 1.1W w/ Road 4 4.4 3.6 5 4.5 5.5Road only 2 2.2 1.8 4 3.6 4.4

Question 2.4 w/ uncertainty

Best Case:R w/Road(same)

Worst Case:Road Only

General Case:Could beseveral-4 -2 0 2 4 6 8 10

R

R w/ Road

F

F w/ Road

W

W w/ Road

Road only

Worst Case Net Benefits ($) Best Case Net Benefits ($)

But difficult to determine that from this chart - can we do better?

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: http://phoenix.phys.clemson.edu/tutorials/excel/advgraph.html

See today’s spreadsheet on home page Graphs original points, and min/max deviations from that as error bars…

Error bar result

Base Case Net Benefits ($)

R

R w/ Road

FF w/ Road W

W w/ RoadRoad only

-4

-2

0

2

4

6

8

10

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

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 unavailableStatistics has confidence intervals – economists need them, too.

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=x

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”

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: http://plato.acadiau.ca/aittmodules/excel/viewlets/scrollbar/scrollbar.viewlet/scrollbar_viewlet_swf.html

Let’s look at our old TV estimation problem

EXCEL’s TABLE function

One- and two-input data tablesSort of a built-in tool for sensitivity analysis (without fancy graphs/etc). See PDF posted for examples and instructions

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 106 15-watt fluorescent bulbs for 53 fixtures

How could we make a model to determine whether this makes sense? Assume only one year time frame

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

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/bulb26.5 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 2.5 - 7.5 cents/kWh ~ $350-$1045Cost Replacing bulbs is same ‘order of magnitude’ as the electricity! (Total range [$1,911 - $2,608])

Lighting Case Study - PS sensorsCosts(ps) - probably ‘off’ 1/3 - 1/2 of time

Labor cost: cost of installing sensors = ‘unknown’

Labor cost: cost of installing new bulbsCould 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 kWh/yr @ 1/2

9,333 kWh if off 1/3 of the timeCost varies [ 2.5 - 7.5 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?

PS sensors analysisWTP [$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?

Sens. Analysis for Photo Sensors

Several built-in options from the plugins (@RISK-TopRank [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)

One-Way Sensitivity Analysis

Use @RISK 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

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 tornadoSpider diagrams are similar

Two-way SA

Treeplan plug-in cant do thisShows a 2-dimensional plot of what happens when we change 2 variables at once Graph generated is a “frontier”

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