lec6 sensanal 07

Uncertainty andSensitivity Analysis: 

Uncertainty and Sensitivity Analysis H. Scott Matthews 12-706 / 19-702

Admin: 

Admin HW 2 Due Wednesday Please don’t send emails/come by an hour before class and expect answers - from me or TA’s Please also think about sensitivity analysis for pipeline case Friday recitation: microeconomics review How to distribute @RISK CDs? All CEE teaching cluster machines being pre-installed.

Problem of Unknown Numbers: 

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: 

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: 

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: 

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”: 

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?: 

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’

Best and Worst-Case Analysis: 

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: 

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

Question 2.4: 

Question 2.4 Project “R with Road” has highest NB

Question 2.4 w/ uncertainty: 

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’?

Question 2.4 w/ uncertainty: 

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?

Using error/uncertainty bars: 

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: 

Error bar result Easier to see results - imagine moving ruler up and down the axis There is a range from about $3 to $4 million with 3+ options possible Net Benefits ($M)

An Easy Visual Clue - Slider Bars: 

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 problem Tool gives easy, visual aid for how sensitive parameters are (but not very quantitative)

Case: Photo-sensors for lighting: 

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: 

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: 

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

Lighting Case Study - PS sensors: 

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 kWh/yr @ 1/2 9,333 kWh if off 1/3 of the time Cost 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 analysis: 

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?

Sens. Analysis for Photo Sensors: 

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: 

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: 

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: 

Plug-in notes See Clemen pp.193- for TopRank tutorial

Two-way SA: 

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: 

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 TopRank type analysis typically easier, however EXCEL TABLE requires no plug in (or expensive software)