Uncertainty andSensitivity Analysis

H. Scott Matthews12-706 / 19-702

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HW 2 Due Today Please don’t send emails/come by an hour

before class and expect answers - from me or TA’s

Friday recitation: microeconomics review

** How to distribute @RISK CDs? All CEE teaching cluster machines being pre-

installed.

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

Definitions

Uncertainty: The lack of certainty, A state of having limited knowledge where it is impossible to exactly describe existing state or future outcome(s) (Hubbard 07)

Measuring uncertainty: Stated by giving a range Repeated experiments (u = stdev,

stderr)

The steps or path

Sensitivity analysis - computing the effect of changes in inputs on model predictions

Uncertainty propagation - calculating the uncertainty in the model outputs induced by input uncertainty

Uncertainty analysis - comparing importance of input uncertainties in terms of their relative contributions to uncertainty in the outputs

(Henrion and Morgan 1990)

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.

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’

Recap: Example 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 are told that Benefits/Costs of each project

are uncertain by, for example 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 Benefits

B +10% B -10% Base Case Costs

C -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

-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 ($) Best Case Net Benefits ($)

Question 2.4 w/ uncertainty

Best Case:R w/Road(same)

Worst Case:Road Only

General Case:Could beseveral

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, otherExcel 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

• Easier to see results - imagine moving ruler up and down the axis• There is a range from about $3 to $4 million with 3+ options possible

Net

Ben

efit

s ($

M)

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://aitt.acadiau.ca/tutorials/Excel97/Sliderbars.htm

Let’s look at our old TV estimation problemTool gives easy, visual aid for how sensitive

parameters are (but not very quantitative)

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 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 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 - mac with VPC, Parallels, etc] to check sensitivity. One-way (one variable at a time) Tornado and spider diagrams (all at a

time) Two-way (two at a time)

One-Way Sensitivity Analysis

Use 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 Shows simple trend related to one variable

Tornado Diagrams

Shows results of many one-way plots on single chart (one changes while all others held constant)

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

Plug-in notes

See Clemen pp.193- for TopRank tutorial

Two-way SA

Shows a 2-dimensional plot of what happens when we change 2 variables at once Graph generated is a “frontier” of

feasible options between two variables

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 TopRank type analysis typically easier,

however EXCEL TABLE requires no plug in (or expensive software)