1. Decision Making Under Uncertainty

2. Elements of a Decision Analysis
Although there is a wide variety of contexts in decision making, all decision making problems have three elements:
the set of decisions (or strategies) available to the decision maker
the set of possible outcomes and the probabilities of these outcome
a value model that prescribes results, usually monetary values, for the various combinations of decisions and outcomes.
Once these elements are known, the decision maker can find an “optimal” decision.

3. Expected Monetary Values
The decision must be made before this uncertainty is resolved.
A common way used to make the choice is to calculate the expected monetary value (EMV) of each alternative and then choose the alternative with the largest EMV.
EMV is a weighted average of the possible monetary values, weighted by their probabilities.

4. Decision Trees
All of this can be done efficiently using a graphical tool called a decision tree.

5. Decision Tree Conventions
Decision trees are composed of nodes (circles, squares and triangles) and branches (lines).
The nodes represent points in time. A decision node(a square) is a time when the decision maker makes a decision. A probability node (a circle) is a time when the result of an uncertain event becomes known. An end node (a triangle) indicates that the problem is completed - all decisions have been made, all uncertainty have been resolved and all payoffs have been incurred.

6. Decision Tree Conventions
Time proceeds from left to right. This means that branches leading into a node (from the left) have already occurred. Any branches leading out of a node (to the right) have not yet occurred.
TIME

7. Decision Tree Conventions
Branches leading out of a decision node represent the possible decisions; the decision maker can choose the preferred branch. Branches leading out of probability nodes represent the possible outcomes of uncertain events; the decision maker has no control over which of these will occur. a A B b C

8. Decision Tree Conventions
Probabilities are listed on probability branches. These probabilities are conditional on the events that have already been observed (those to the left). Also, the probabilities on branches leading out of any particular probability node must sum to 1. a b
given history c

9. Decision Tree Conventions
This decision tree illustrates these conventions for a single-stage decision problem, the simplest type of decision problem.
In a single-stage decision problem all decisions are made first, and then all uncertainty is resolved.
Later we will see multistage decision problems, where decisions and outcomes alternate.
Once a decision tree has been drawn and labeled with the probabilities and monetary values, it can be solved easily.

10. Folding Back a Decision Tree
The solution procedure used to develop this result is called folding back on the tree. Starting at the right on the tree and working back to the left, the procedure consists of two types of calculations.
At each probability node we calculate EMV and write it below the name of the node.
At each decision node we find the maximum of the EMVs and write it below the node name.
After folding back is completed we have calculated EMVs for all nodes.

11. An Example Investing in the stock market
An investor has some funds available to invest in one of three choices: a high-risk stock, a low-risk stock, or a savings account that pays a sure $500.
If he invests in the stocks, he must pay a brokerage fee of $200.
His payoff for the two stocks depends on what happens to the market: If the market goes up, he will earn $1700 from the high-risk stock and $1200 from the low-risk stock.
If the market stays at the same level, his payoffs for the high- and low-risk stocks will be $300 and $400, respectively.
Finally, if the stock market goes down, he will lose $800 with the high-risk stock but still gain $100 with the low risk stock.

12. Another Example
The national coffee store Farbucks needs to decide in August how many holiday-edition insulated coffee mugs to order.
These premium mugs sell for $10 and cost $6 each. Farbucks is uncertain of the demand.
They believe that there is a 25% chance that they will sell 10,000 mugs, a 50% chance that they will sell 15,000 and a 25% chance that they will sell 20,000.
Because the mugs are dated, those that are unsold by January 15 are discounted and sold at 20% of the original price.
Build a decision tree to determine if they should order 16,000 or 18,000 mugs.

13. Automobile Industry
In June 2006, the operations manager of a large supplier of parts to the automobile industry has been asked if he would take a contract to supply some subassemblies for a demonstration model.
The quantity to be ordered was not certain; however it would be either 20 or 40 and the number would be known in January 2007, 7 months from now.
The price was $10000 per unit.
He had to respond by next week and commit his company to delivery of the required number of subassemblies by March 2007.

14. Automobile Industry
The operations manager have identified three methods of producing the subassemblies; Process 1 would be the cheapest, if it worked properly. Process 2 was more expensive but sure to work.
If Process 1 were used, they would know by September 2006 if it worked. If not there would still be time to use Process 2; however the investment in Process 1 would be lost.
With either process, the lead time and other work in the company would require that they commit themselves to a fixed production quantity prior to January.
A third possibility was to subcontract the work. A reliable subcontractor was available and if the order were placed now the subcontractor would give them a good price and also be able to wait until the production quantity decision was resolved.

15. Automobile Industry
If the orders are placed after August 2006, a higher price would be charged but as long as the subcontractor had a firm order on the number of units required by January 2007, they would meet the delivery date.
The probability of success with Process 1 was assessed by the engineers who would be involved in the work as 0.5.
The operations manager assessed the probability of an order quantity of 40 to be 0.4, after talking with the automobile company.

16. Automobile Industry Incremental costs Process 1 Process 2
Testing process $20,000
Per unit production cost if successful 4,000
Process 2
Per unit production cost 6,000
Subcontracting costs (per unit)
Order placed before August 1, ,000
Order placed after August 1, ,000
Assumptions:
If he produces 20 units and 40 units are ordered, the remaining orders are obtained by subcontracting at $9,000 per unit
If he produces 40 units and 20 units are ordered, the excess can be disposed of at $2,000 per unit

18. Value of Information Investing in the stock market
An investor has some funds available to invest in one of three choices: a high-risk stock, a low-risk stock, or a savings account that pays a sure $500.
If he invests in the stocks, he must pay a brokerage fee of $200.
His payoff for the two stocks depends on what happens to the market: If the market goes up, he will earn $1700 from the high-risk stock and $1200 from the low-risk stock.
If the market stays at the same level, his payoffs for the high- and low-risk stocks will be $300 and $400, respectively.
Finally, if the stock market goes down, he will lose $800 with the high-risk stock but still gain $100 with the low risk stock.

19. Value of Information 1500 100 -1000 1000 200 -100 500 up (0.5)
same (0.3)
100
down (0.2)
-1000
high-risk stock
up (0.5)
1000
low-risk stock
same (0.3)
200
down (0.2)
-100
savings account
500

20. Value of Information
Expected Value of Perfect Information
(EVPI)

21. Value of Information True Market State Economist’s Prediction Up Flat
Conditional probabilities characterizing economist’s forecasting ability
True Market State
Economist’s Prediction Up
Flat
Down
“Up”
0.80
0.15
0.20
“Flat”
0.10
0.70
“Down”
0.60
P(Economist says “Flat” | Flat) = 0.70
P(Economist says “Up” | Down) = 0.20

22. Value of Information

23. Value of Information

24. Value of Information

25. Posterior probability for
Value of Information
Posterior probabilities for market trends depending on economist’s information
Posterior probability for
Economist’s Prediction
Market Up
Market Flat
Market Down
“Up”
0.8247
0.0928
0.0825
“Flat”
0.1667
0.7000
0.1333
“Down”
0.2325
0.2093
0.5581
P(Market Up | Economist says “Up”) =
P(Market Flat | Economist says “Up”) =

26. Value of Information
Expected Value of Imperfect Information
(EVII)

27. Value of Information Marketing a new product at Acme
Acme Company is trying to decide whether to market a new product.
As in many new-product situations, there is much uncertainty about whether the product will “catch-on”.
Acme believes that it would be prudent to introduce the product to a test market first.
Thus the first decision is whether to conduct the test market.

28. Value of Information
Acme has determined that the fixed cost of the test market is $3 million.
If they proceed with the test, they must then wait for the results to decide if they will market the product nationally at a fixed cost of $90 million.
If the decision is not to conduct the test market, then the product can be marketed nationally with no delay.
Acme’s unit margin, the difference between its selling price and its unit variable cost, is $18 in both markets.

29. Value of Information
Acme classifies the results in either market as great, fair or awful.
Each of these has a forecasted total units sold as (in 1000s of units) 200, 100 and 30 in the test market and 6000, 3000 and 900 for the national market.
Based on previous test markets for similar products, it estimates that probabilities of the three test market outcomes are 0.3, 0.6 and 0.1.

30. Value of Information
Then based on historical data on products that were tested then marketed nationally, it assesses the probabilities of the national market outcomes given each test market outcome.
If the test market is great, the probabilities for the national market are 0.8, 0.15, and 0.05.
If the test market is fair. then the probabilities are 0.3, 0.5, 0.2.
If the test market is awful, then the probabilities are 0.05, 0.25, and 0.7.
If Acme decides to conduct a test market they will base the decision to market nationally on the test market results.
In this case its final strategy will be a contingency plan, where it conducts the test market, then introduces the product nationally if it receives sufficiently positive test market results and abandons the product if it receives negative test market results.

31. Value of Information
Expected Value of Sample Information
(EVSI)