### Decision Analysis - The University of Chicago GSB Information Server

```36106 Managerial Decision Modeling
Decision Analysis in Excel
Kipp Martin
University of Chicago
January 31, 2015
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Powell and Baker: Sections 13.1, 13.2, and 13.3
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Course Pack: “Decision Analysis”’ by George Wu
Files used in this lecture:
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wertzTree.xlsx
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wertzTree key.xlsx
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wertzUtility.xlsx
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sarahChangData.xlsx
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sarahChangKey.xlsx
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sarahChangOptimal.xlsx
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sarahChangSensitivity.xlsx
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Lecture Outline
Software Install
Motivation
Example 1: Wertz Game and Toy
The Cost of Uncertainty
Decision Theory
Example 2: Sarah Chang
Sensitivity Analysis
Goal Seek
Utility
Learning Objectives
1. Begin to incorporate uncertainty into an Excel model
2. Learn to model sequential decision problems
3. Learn how to use Precision Tree in Excel
Decision
Outcome
Decision
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Outcome
Software Install – DecisionTools Suite
CRITICAL AND IMPORTANT:
1. Close all programs except Excel.
2. Minimize number of open files in Excel.
Remember my Corollary to Murphy’s law – When it comes to computers,
Murphy was an optimist.
Another Corollary: Excel Add-ins increase the probability that problems
arise.
Software Install – DecisionTools Suite
We will use:
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Precision Tree
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@Risk
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Risk Optimizer
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Software Install – DecisionTools Suite
If you do the default install, you should see desktop icons like:
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Software Install – DecisionTools Suite
Make sure Excel is closed.
Open Excel by “clicking” on the PrecisionTree 6 desk icon.
This should open Excel for you.
Now click on the @Risk icon.
Software Install – DecisionTools Suite
Under File select Options
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Software Install – DecisionTools Suite
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Software Install – DecisionTools Suite
If you have opened Excel by clicking on both @Risk 6 and PrecisionTree 6
you should see
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Software Install – DecisionTools Suite
From now on, you can open Excel and do not need to use the desktop
icons. You should see Tab items for both PrecisionTree and @Risk.
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Software Install – DecisionTools Suite
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Motivation
See http:
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Disaster Planning
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Geothermal Power Plant Equipment Procurement
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Portfolio Management
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Exchange Rate Analysis
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Endangered Species Protection
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Pollution Cleanup
Motivation
Key Concept: most people do not understand
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the concept of an optimal solution
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variables
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parameters
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constraints
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an objective function
Thinking about your problem in the context of these ideas may be
very beneficial.
Motivation
Key Concept: the biggest the benefit of an Excel model is that it forces
a user to:
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think about potential outcomes that result from making a decision
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quantify an outcome (you need numbers and formulas in Excel)
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quantify uncertainty (you need numbers and formulas in Excel)
Motivation
We are going to study Decision Analysis.
Differences from Solver models:
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There are a limited number of possible decisions to make (as
opposed to potentially infinite)
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The outcomes of a decision are uncertain
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Decisions are made sequentially over time
Decision
Outcome
Decision
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Outcome
Motivation
We are going to use PrecisionTree to model these kinds of problems.
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We are going to look at the expected value of our decisions
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This requires a structured approach to modeling which has great
benefits
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Warning: no guarantee that this approach does not lead to a bad
outcome
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Motivation
It is critical to realize that when making decision under uncertainty:
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a poor decision may lead to a good result
See the discussion by former Booth student Zeger Degraeve at
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Example 1: Wertz Game and Toy
Wertz Game and Toy Data (Section 13.2 of Powell and Baker)
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Example 1: Wertz Game and Toy
We look at four criteria:
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maximax payoff
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maximin payoff
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minimax regret
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maximize expected payoff
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Example 1: Wertz Game and Toy
Decision: full line
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Example 1: Wertz Game and Toy
Decision: single line
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Example 1: Wertz Game and Toy
Decision: full line
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Example 1: Wertz Game and Toy
Maximize expected payoff:
Calculate the expected payoff of decision di :
Pj = probability of state of nature j
Vij = value of outcome given decision i and state of nature j
EV (di ) =
N
X
Pj Vij
j=1
Make the decision with the maximum expected payoff.
Assumption:
PN
j=1
Pj = 1.0
Example 1: Wertz Game and Toy
Decision: Two Versions
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The Cost of Uncertainty
The basic problem:
Decision
Outcome
Decision
Outcome
Must make a decision before the outcome is known.
It would be nice to know the outcome before making the decision.
For example, if Wertz managers knew the market response would be
good, they would bring out the full line.
How much should Wertz be willing to pay to know the outcome first?
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The Cost of Uncertainty
Expected Value of Perfect Information (EVPI) – the increase in the
expected payoff if one knew with certainty which state of nature would
occur.
The expected payoff with perfect information must be at least as great as
the expected payoff of the optimal solution.
For example, for Wertz the EVPI must be at least 53 which is the
optimal expected payoff.
The Cost of Uncertainty
Expected Value of Perfect Information – make the calculation as
follows:
Step 1: For each possible state of nature, determine the optimal decision
and corresponding outcome
Step 2: Weight each outcome by the probability of the state of nature
associated with that outcome and calculate the expected value
Step 3: Subtract the EV of the optimal decision from the number calculated
in Step 2.
The Cost of Uncertainty
Expected Value of Perfect Information – Wertz:
Step 1: For each state of nature determine the outcome for the optimal
decision.
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Good: best outcome is 300 (full line is the optimal decision)
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Fair: best outcome is 60 (single version is the optimal decision)
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Poor: best outcome is -10 (single version is the optimal decision)
Step 2: Weight each optimal outcome by the probability assigned to that
state of nature
.2 ∗ 300 + .5 ∗ 60 + .3(−10) = 60 + 30 − 3 = 87
Step 3: Subtract the EV of the optimal decision from the number calculated
in Step 2.
EVPI = 87 − 53 = 34
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Example 1 (Follow UP): Wertz Game and Toy
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Example : Wertz Game and Toy
Wertz is a very simple model.
There is only one decision made at time 0
It is trivial to find the optimal decision in an Excel table. Just calculate
the expected payoff for each decision.
However, real life is more complicated, and often involves a sequence of
decisions.
More sophisticated tools are required.
Decision Theory
See Decision Analysis by George Wu.
Four Steps in Decision Making Process
Step 1: Structuring a Decision Problem (alternatives, uncertainties, and
objectives)
Step 2: Assessment and Information Gathering
Step 3: Evaluation of Decision Problem (PrecisionTree used in this step)
Step 4: Sensitivity Analysis (PrecisionTree used in this step)
Decision Theory
Step 1: Structuring a Decision Problem Instead of the Solver (A
(adjustable cells), B (best cell), and C (constraint cells) ) we have:
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What are the alternatives?
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What are the critical uncertainties?
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What are the objectives?
Decision Theory
Step 2: Assessment and Information Gathering: Collect the relevant
information,
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assess the values of the outcomes given decisions that were made
(e.g. in Wertz the outcome of a good market response, given the
decision to produce the full line is 300)
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determine the likelihood of uncertain events (e.g. in Wertz the
probability of a fair market response is 0.5)
Discussion Point: What is meant by the probability of an event?
Decision Theory
What do we mean by probability? See pages 8 and 9 of the Decision
Analysis case.
Probabilities measure the likelihood of uncertain events.
In most cases a probability is a judgement. This does not imply it is
arbitrary.
Sarah Chang says the probability of successfully developing the
microprocessor in six months is .40.
Sarah Chang is saying that she believes this event is just as likely as
drawing a red ball from an urn where 40% of the balls in the urn are red.
Decision Theory
Step 4: Sensitivity Analysis:
Determine how sensitive the optimal solution is to the probabilities and
the outcome values.
We use the sensitivity analysis features of Precision Tree.
For example, how much can probability estimates change before we
change our decisions?
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Example 1: Wertz Game and Toy
We are going to build in Excel something that looks like this:
Good
100
Fair
60
Poor
-10
Single
Version
Good
Two
Versions
200
Fair
50
Poor
-40
Full
Version
300
Good
Fair
40
Poor
-100
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Example 1: Wertz Game and Toy
Icon coding scheme:
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a green square – a point in time where we make a decision
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a red circle – a chance node, an uncertain outcome occurs
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a blue triangle – indicates the end of branch
Time proceeds left to right.
Probabilities and monetary values are placed next to nodes
Expected values are computed by the folding-back process
The optimal path is indicated by TRUE nodes.
Example 1: Wertz Game and Toy
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Example 2: Sarah Chang
Step 1: Structuring a Decision Problem
Step 2: Assessment and Information Gathering
Note the breakdown between Decisions and Outcomes.
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Example 2: Sarah Chang
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Open the file sarahChangData.xlsx Workbook.
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information gathering step.
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open the spreadsheet model – it is currently empty.
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select an arbitrary cell, say C35
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Go to the PrecisionTree tab and select Decision Tree
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Example 2: Sarah Chang
Click OK
Name the model Sarah Chang Decision Model
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Example 2: Sarah Chang
Click on the end blue triangle node.
Select a Decision node and name it Decision to Invest.
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Example 2: Sarah Chang
Next click on the Branches tab and do the following:
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Name the branches Continue and Abandon
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Give the Continue branch a value of =data!E5
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Give the Abandon branch a value of =data!E6
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Example 2: Sarah Chang
At this point, your decision tree should look like:
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Example 2: Sarah Chang
Click on the end blue triangle node for the Continue branch.
Click on Chance button and give the node the name Outcome of R & D.
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Example 2: Sarah Chang
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Name the branches Succeed and Fail
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Give the Succeed branch a probability of =data!F7
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Give the Fail branch a probability of =(1-data!F7)
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Example 2: Sarah Chang
Your tree should now look like this:
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Example 2: Sarah Chang
Now click on the blue triangle at the end of the Succeed outcome.
Create a Decision node named Decision To Propose
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Example 2: Sarah Chang
Click on the Branches tab:
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Name the branches Make a Proposal and No Proposal
Give the Make a Proposal branch a value of -50000 or data!E9
Give the No Proposal branch a value of 0 or data!E10
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Example 2: Sarah Chang
Your tree should look like this
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Example 2: Sarah Chang
Now click on the blue triangle at the end of the Make a Proposal
outcome.
Create a Chance node named Bid Outcome
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Example 2: Sarah Chang
Click on Branches tab:
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Name the branches Win and Lose
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Give the Win branch probability of 90% or data!F11
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Give the Lose branch a probability of 10% or (1 - data!F11)
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Give the Win branch a value of 850000 or data!E11
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Give the Lose branch a value of 0 or data!E13
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Example 2: Sarah Chang
Your Decision Tree should look like this:
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Example 2: Sarah Chang
Wow that was a lot of work!
Sarah may wish to submit a Proposal even if the R&D failed.
Consider the blue triangle termination node at the end of the fail branch
of the Outcome of R&D chance node.
The structure of the decision and outcome nodes at the end of the fail
branch are identical to those at the end of the succeed node.
However, the probabilities and values differ.
Do a copy and paste!
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Example 2: Sarah Chang
Right click on the Decision to Propose node and select Copy Subtree.
Right click on the blue triangle termination node at the end of the fail
branch of the Outcome of R&D chance node.
Select Paste Subtree
Adjust values accordingly: the Win probability is changed from 90% to
5% and the Lose probability is changed from 10% to 95%.
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Example 2: Sarah Chang
Your Decision Tree should look like this. See sarahChangKey.xlsx.
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Example 2: Sarah Chang
The optimal solution: Trace the green TRUE values through the tree.
The optimal strategy for Sarah is to
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First, proceed with the project and continue R&D on the
microprocessor
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If the outcome of R&D is success then make the proposal to the
Olympic committee
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If the outcome of R&D is failure then do not make the proposal to
the Olympic committee
PrecisionTree will generate the above information for you.
PrecisionTree will fold-back the tree for you.
Example 2: Sarah Chang
Click on Decision Analyis button to generate a Policy Suggestion.
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Example 2: Sarah Chang
The optimal policy for Sarah Chang. Notice the coloring scheme.
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Example 2: Sarah Chang
You may wish to place reports in the open workbook instead of creating
a new workbook.
Under Utilities select Application Settings ...
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Optimal Value
How is the 86000 calculated at the outcome node?
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Optimal Value
What is the meaning of the 86000 in the decision tree?
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Optimal Value
You should understand:
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How the numbers at each decision node are calculated (and what
they mean)
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How the numbers at the outcome nodes are calculated
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How the numbers at the terminal nodes are calculated
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Optimal Value
Example Calculations:
The Outcome of R&D node has a value of \$86,000. This comes from:
86000 = .4 ∗ 515000 + .6 ∗ −200000
The terminal node at the top of the tree has a probability value of
36.0%. We reach this point in the tree given successful R&D and
winning the bid. These are independent events so the probability is
.36 = .4 ∗ .9
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Optimal Value
Example Calculations:
This terminal node also has a value of \$600,000. At this terminal node,
we won the bid with a return of \$850,000 but paid \$50,000 to prepare
the bid and \$200,000 to continue the R&D. This gives
\$600, 000 = \$850, 000 − \$50, 000 − \$200, 000.
Now, how do you interpret the \$86,000 at the root decision node?
Some Tips
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Separate the model from the data
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Separate the model from the data – you get the idea
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Make use of copy and paste – but be careful of cell references
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Make sure probabilities sum to 1.0
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Do not play with the formulas generated by PrecsionTree (blue, red,
green)
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You should only edit through the user interface
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Some Tips
If you get a Value error at nodes instead of True or False make sure:
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The probabilities sum to one at each chance node. This is the
number one cause of getting a Value error. You should see an
error message about probabilities not summing to 1.0 in small print
at the bottom of the Excel spreadsheet.
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Do a File save. The Value error may go away.
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Recalculate the spreadsheet. Even better, make sure under
Workbook Calculations, that Automatic is checked.
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Always enter probabilities and values through Precision Tree. Do
not enter them directly into the cells.
Have these tips at your side for the exam.
Sensitivity Analysis
Sensitivity Analysis: See pages 12-13 of the Decision Analysis case.
How sensitive is the model to the probability estimate of winning the bid?
Select the Sensitivity Analysis menu item from the Analysis group.
We are going to do sensitivity analysis on cell data!F11
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Sensitivity Analysis
Add the cell data!F11 we use for sensitivity analysis.
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Sensitivity Analysis
Vary the probability of win the bid (assuming successful R&D) from .4 to
.9 in increments of .05 (11 steps).
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Sensitivity Analysis
At approximately what probability should we make the Continue decision?
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Sensitivity Analysis
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Goal Seek
Using Excel GoalSeek we can calculate exactly the probability of the bid
being accepted at which we are indifferent to the decision to abandon the
R&D of the chip.
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Goal Seek
The tradeoff probability is approximately 64.71%
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Goal Seek
Oil Drilling Example (practice midterm) – the optimal decision is to not
hire the geologist. How much would the report have to cost in order to
be indifferent between a hire and no hire decision?
Redford Example (midterm part c) – how much would the value of selling
the partnership have to be in order to be indifferent between building and
selling the rights?
Utility
Motivation: Assume the probability of making one million dollars is 10%
and the probability of losing one million dollars is 10%.
The contribution to the expected value of winning one million is
100, 000 = .1 ∗ 1, 000, 000
The contribution to the expected value of losing one million is
−100, 000 = .1 ∗ (−1, 000, 000)
These two events are of equal magnitude in the expected value
calculation.
Problem: in real life people care about their tail!
Clarification: By tail, I mean tail of their expected payoff distribution.
Utility
Key Concept: If you maximize expected value (payoff) there may be a
nontrivial probability of losing money.
Key Concept: If you maximize expected value (payoff) bad things may
happen.
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In Wertz, if you make the decision to produce two versions, there is
a 30 percent chance you will lose 40 thousand dollars
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If Sarah Chang follows the optimal policy recommended by Precision
Tree there is a 60 percent chance she will lose 200,000 dollars.
If you are risk averse you may wish to use a utility function.
Utility
In PrecisionTree the default behavior is to use expected value or payoff.
You can change that by going to Settings and then Model Settings.
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Utility
If you are risk averse (concave function), select an exponential utility
function. In this case we have the utility function f (x) = 1 − exp(−x/10).
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Utility
The utility function U(x) = 1 − e(−x/10).
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Utility
Wertz (tt wertzUtility.xlsx) with maximizing expected utility.
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Utility
Now the optimal decision is to go with the single version.
This is a safer strategy. We lose at most 10,000 dollars.
The expected utility of this decision is:
.2∗(1−e(−100/10))+.5∗(1−e(−60/10))+.3∗(1−e(10/10)) = 0.183266995
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Utility
When applying this material in the public sector it may be difficult to
measure the value of outcomes even when probabilities are reasonably
estimated.
Consider “black swan” events.
Blizzards are good examples.
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