1. Making choices Dr. Yan Liu
Department of Biomedical, Industrial & Human Factors Engineering
Wright State University

2. Expected Monetary Value (EMV)
One way to choose among risky alternatives is to pick the alternative with the highest expected value (EV). When the objective is measured in monetary values, the expected money value (EMV) is used
EV is the mean of a random variable that has a probability distribution function
(Discrete Variable)
(Continuous Variable)

3. A2 A1 O1 C1 O2 O3 O4 C2 C3 C4
(p1)
Payoff
(1-p1)
(p2)
(1-p2)
EMV(A1)=C1•p1 +C2•(1-p1)
EMV(A2)=C3•p2 +C4• (1-p2)

4. Solving Decision Trees
Decision Trees are Solved by “Rolling Back” the Trees
Start at the endpoints of the branches on the far right-hand side and move to left
When encountering a chance node, calculate its EV and replace the node with the EV
When encountering a decision node, choose the branch with the highest EV
Continue with the same procedures until a preferred alternative is selected for each decision node

5. Lottery Ticket Example
You have a ticket which will let you participate in a lottery that will pay off $10 with a 45% chance and nothing with a 55% chance. Your friend has a ticket to a different lottery that has a 20% chance of paying $25 and an 80% chance of paying nothing. Your friend has offered to let you have his ticket if you will give him your ticket plus one dollar. Should you agree to trade?
Win
$24
EMV=$4
$25
(0.2)
Trade Ticket
EMV(Trade Ticket)=24•0.2+ (-1)•0.8=$4
Lose
Ticket Result
-$1
-$1 $0
(0.8)
Win
$10
EMV=$4.5
$10
(0.45)
Keep Ticket
EMV(Keep Ticket)=100•0.45+ (0)•0.55=$4.5
Lose
Ticket Result $0 $0
(0.55)
Conclusion: You should keep your ticket !

6. Product-Switching Example
A company needs to decide whether to switch to a new product or not. The product that the company is currently making provides a fixed payoff of $150,000. If the company switches to the new product, its payoff depends on the level of sales. It is estimated that there are about 30% chance of high-level sales ($300,000 payoff), 50% chance of medium-level sales ($100,000 payoff), and 20% chance of low-level sales (losing $100,000). A survey which costs $20,000 can be performed to provide information regarding the sales to be expected. If the survey shows high-level sales, then there are about 60% chance of high-level sales and 40% chance of medium-level sales when the company sells the product. On the other hand, if the survey shows low-level sales, then there are about 60%chance of medium-level sales and 40% chance of low-level sales when the company sells the product.

7. Old
$150,000
$130,000
Survey High
High
$300,000 (0.6)
$280,000
(0.5)
New
Medium
$100,000 (0.4)
$80,000
Old
$150,000
Perform Survey
$130,000
Survey Low
Medium
$100,000 (0.6)
(0.5)
$80,000
New
-$20,000
Low
-$100,000 (0.4)
-$120,000
Don’t Perform
Old
$150,000
$150,000
New
High
$300,000 (0.3)
$300,000
Medium
$100,000 (0.5)
$100,000
Low
-$100,000 (0.2)
-$100,000

8. Old $150,000 $130,000 Survey High High $300,000 (0.6) $280,000 (0.5)
EMV= $200,000
Survey High
High
$300,000 (0.6)
$280,000
EMV=
$165,000
(0.5)
New
EMV(U3) =0.6•280, •80,000=$200,000 U3
Medium
$100,000 (0.4)
$80,000
Old
$150,000
Perform Survey U1 D4
$130,000
Survey Low
EMV= $0
Medium
$100,000 (0.6)
(0.5)
$80,000
New
EMV(U4) =0.6•80, •(-120,000)=$0
-$20,000
Low
-$100,000 (0.4) U4
-$120,000 D1
EMV(U1) =0.5•200, •130,000=$165,000 D2
Don’t Perform
Old
$150,000
$150,000
New
High
$300,000 (0.3)
$300,000
Medium
$100,000 (0.5)
$100,000
-$100,000 (0.2)
EMV=
$120,000 U2
Low
-$100,000
EMV(U2) =0.3•300, •(100,000)+0.2•(-100,000)=$120,000
Conclusion: Perform survey. If survey shows high-level sales, then switch the new product ; otherwise, stay with the old product

9. Decision Path and Strategy
Represents a possible future scenario, starting from the left-most node to the consequence at the end of a branch by selecting one alternative from a decision node and by following one outcome from a chance node. D1 U1 D2 A1 A2 O1 O2 A3 A4
Path 1 ( A1 ) A1 O1
Path 2 ( A2O1 )
Decision Paths: A3 D1 U1
Path 3 ( A2O2A3 ) A2 D2 O2
Path 4 ( A2O2A4 ) A4

10. Decision Path and Strategy (Cont.)
Decision Strategy
The collection of decision paths connected to one branch of the immediate decision by selecting one alternative from each decision node along that path D1 U1 D2 A1 A2 O1 O2 A3 A4
Strategy 1 (A1): Decision path A1 A1 O1
Decision Strategies:
Strategy 2 (A2A3):
Decision paths A2O2A3, A2O1 D1 A3 U1 A2 D2 O2 A4
Strategy 3 (A2A4):
Decision paths A2O2A4, A2O1

11. Risk Profiles Problems with Expected Value (EV) What is Risk Profile
EV does not indicate all the possible consequences
The statistical interpretation of EV as the average amount obtained by “playing the game” a large number of times is not appropriate in rare cases (e.g. hazards in nuclear power plants)
What is Risk Profile
A graph that shows the probabilities associated with possible consequences given a particular decision strategy
Indicates the relative risk levels of strategies
Steps of Deriving Risk Profiles from Decision Trees
Identify the decision strategies
For each strategy, collapse the decision tree by multiplying out the probabilities on sequential chance branches (Don’t confuse it with solving decision trees!)
Keep track of all possible consequences
Summarize the probability of occurrence for each consequence

12. Decision Tree of the Lottery Ticket Example
Decision strategies:
Trade ticket:
2) Keep ticket:
$24(0.2), -$1(0.8)
$10(0.45), $0(0.55)
Decision Tree of the Lottery Ticket Example
Keep Ticket
Trade Ticket
Win
$24
(0.2)
Lose
-$1
$10 $0
(0.8)
(0.45)
(0.55)
Pr(Payoff)
Trade Ticket
Keep Ticket
Payoff($)
Risk Profiles of the Lottery Ticket Example

13. Decision Tree of the Product-Switching Example
Old
$130,000
Survey High
High
(0.6)
$280,000
New
(0.5)
Medium
(0.4)
Perform Survey
$80,000
Old
$130,000
Survey Low
Medium
(0.6)
(0.5)
$80,000
New
Low
(0.4)
-$120,000
Don’t Perform
Old
$150,000
New
High
(0.3)
$300,000
Medium
(0.5)
$100,000
Low
(0.2)
-$100,000
Decision Tree of the Product-Switching Example
1) Don’t perform survey and keep the old product
Decision Strategies:
2) Don’t perform survey and switch to the new product
3) Perform survey, and if survey is high then keep the old product
4) Perform survey, and if survey is high then switch to the new product

14. Don’t Perform New Medium High Low (0.3) (0.5) (0.2) -$100,000 $100,000
Strategy 1): Don’t perform survey and keep the old product
$150,000 (100%)
Strategy 2): Don’t perform survey and switch to the new product
Payoffs
$300,000
$100,000
-$100,000
Probabilities
0.3
0.5
0.2
Don’t Perform
New
Medium
High
Low
(0.3)
(0.5)
(0.2)
-$100,000
$100,000
$300,000
Strategy 3): Perform survey and if survey high then keep the old product
Perform Survey
Survey High
Survey Low
(0.5)
Old
$130,000
$130,000 (100%)
Strategy 4): Perform survey and if survey high then switch to the new product
Perform Survey
Survey High
Survey Low
(0.5)
New
$130,000
Medium
(0.4)
$280,000
(0.6)
High
$80,000
Payoffs
$280,000
$130,000
$80,000
Probabilities
0.3
0.5
0.2

15. Risk Profiles of the Product-Switch Example
Payoff($)
Pr(Payoff)
Risk Profiles of the Product-Switch Example
Strategy 1
Strategy 2
Strategy 3
Strategy 4

16. Cumulative Risk Profiles
A graph that shows the cumulative probabilities associated with possible consequences given a particular decision strategy
Payoff($)
Pr(Payoff≤x)
Trade Ticket
Keep Ticket
Cumulative Risk Profiles of the Lottery Ticket Example

17. Dominance Deterministic Dominance
If the worst payoff of strategy B is at least as good as that of the best payoff of strategy A, then strategy B deterministically dominates strategy A
May also be concluded by drawing cumulative risk profiles
Pr(Payoff ≤ x)
Draw a vertical line at the place where strategy B first leaves 0. If the vertical line corresponds to 100% for strategy A, then B deterministically dominates A.
strategy A
strategy B
Payoff

18. Dominance (Cont.) Stochastic Dominance
If for any x, Pr(Payoff ≤ x|strategy B) ≤ Pr(Payoff ≤ x|strategy A), then B stochastically dominates A
strategy A
strategy B
Payoff
Pr(Payoff ≤ x)
There is no crossing between the cumulative risk profiles of A and B, and the cumulative risk profile of B is located at the lower-right to that of A

19. Making Decisions with Multiple Objectives
Summer Job Example
Sam has two job offers in hand. One job is to work as an assistant at a local small business. The job would pay a minimum wage ($5.25 per hour), require 30 to 40 hours per week, and have the weekends free. The job would last for three months, but the exact amount of work and hence the amount Sam could earn were uncertain. On the other hand, he could spend weekends with friends.
The other job is to work for a conservation organization. This job would require 10 weeks of hard work and 40 hours weeks at $6.50 per hour in a national forest in a neighboring state. This job would involve extensive camping and backpacking. Members of the maintenance crew would come from a large geographic area and spend the entire 10 weeks together, including weekends. Sam has no doubts about the earnings of this job, but the nature of the crew and the leaders could make for 10 weeks of a wonderful time, 10 weeks of misery, or anything in between.

20. Decision Elements Objectives (and Measures) Decision to Make
Earning money (measured in $)
Having fun (measured using a constructed 5-point Likert scale; Table 4.5 at page 138)
(5) Best: A large congenial group. Many new friendships made. Work is enjoyable, and time passes quickly.
(4) Good: A small but congenial group of friends. The work is interesting, and time off work is spent with a few friends in enjoyable pursuits.
(3) moderate: No new friends are made. Leisure hours are spent with a few friends doing typical activities. Pay is viewed as fair for the work done.
(2) Bad: Work is difficult. Coworkers complain about the low pay and poor conditions. On some weekends it is possible to spend time with a few friends, but other weekends, are boring.
(1) Worst: Work is extremely difficult, and working conditions are poor. Time off work is generally boring because outside activities are limited or no friends are available.
Decision to Make
Which job to take (In-town job or forest job)
Uncertain Events
Amount of fun
Amount of work (# of hours per week)

21. Influence Diagram Overall Satisfaction Amount Fun Salary of Fun Fun
Job
Decision
Salary
Amount
of Work

22. Decision Tree

23. Analysis of the Salary Objective
EMV(Salary of Forest job) = $2,600
EMV:
EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15( )= $2,422.88

24. Analysis of the Salary Objective
EMV(Salary of Forest job) = $2,600
EMV:
EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15( )= $2,422.88
Risk Profiles:
Strategies:
1) Forest Job 100% $2,600
2) In-Town Job 35% $2,730; 50% $2,320.5; 15% $2,047.5
Conclusion: For the salary objective, the forest job has higher EMV and has no risk
Cumulative Risk Profiles of the Salaries

25. Analysis of the Fun Objective
The ratings in the original 5-point Likert scale only indicate orders of the amount of fun without carrying quantitative meanings.
Therefore, the original ratings are rescaled to points to show quantitative meanings: 5(best) – 100 points, 4(Good) – 90 points, 3(Moderate) – 60 points, 2(bad) – 25 points, 1(worst) – 0 point
EV:
E(Fun of Forest job) =0.10(100)+0.25(90)+0.40(60)+0.20(25)+0.05(0) = 61.5
E(Fun of In-Town job) = 60

26. Cumulative Risk Profiles of the Fun
Analysis of the Fun Objective (Cont.)
Risk Profiles:
Strategies:
1) Forest Job 10% 100; 25% 90; 40% 60; 20% 30; 5% 0
2) In-Town Job 100% 60
Conclusion: For the fun objective, the forest job has higher EV but is more risky
Cumulative Risk Profiles of the Fun

27. Sam’s dilemma: Would he prefer a slightly higher salary for sure and take a risk on how much fun the summer will be? Or otherwise, would the in-town be better, playing it safe with the amount of fun and taking a risk on how much money will be earned? Therefore, Sam needs to make a trade-off between the objectives of maximizing fun and maximizing salary.

28. Trade-off Analysis
Combine multiple objectives into one overall objective
Steps
First, multiple objectives must have comparable scales
Next, assign weights to these objectives (the sum of all the weights should be equal to 1)
Subjective judgment
Paying attention to the range of the attributes (the variables to be measured in the objectives) is crucial; Attributes having a wide range of possible values are usually important (why?)
Then, calculate the weighted average of consequences as an overall score
Finally, compare the alternatives using the overall score

29. Summer Job Example (Cont.)
Convert the salary scale to the same 0 to 100 scale used to measure fun
Set $2730 (the highest salary) = 100, and $ (the lowest salary) =0
Then, Intermediate salary X is converted to: (X )∙100/( ) (Proportion Scoring)
Assign weights to salary and fun (Ks and Kf)
Sam thinks increasing salary from the lowest to the highest is 1.5 times more important than improving fun from the worst to best, hence
Ks=1.5Kf , Because Ks+Kf=1  Ks=0.6, Kf=0.4

30. Overall Score
88.6
84.6
72.6
58.6
48.6
84.0
48.0
24.0

31. EV:
EV(Overall Score of Forest job) =0.10(88.6)+0.25(84.6)+0.40(72.6)+0.20(58.6)+0.05(48.6) = 73.2
EV(Overall Score of In-Town job)
= 0.35(84)+0.50(48)+0.15(24) = 57
Risk Profiles:
The forest job stochastically dominates the in-town job
Conclusion: The forest job is preferred to the in-town job
Cumulative Risk Profiles of the Overall Scores

32. Exercise A B (0.27) A2 A1 $8 $0 $15 (0.5) (0.73) (0.45) (0.55) $4 $10D1 D2 A B
(0.27) A2 A1 $8 $0
$15
(0.5)
(0.73)
(0.45)
(0.55) $4
$10 U1 U3 U2
O21
O11
O22
O12
O31
O32
1. Solve the decision tree in the figure
2. Create risk profiles and cumulative risk profiles for all possible strategies. Is one strategy stochastically dominant? Explain.

33. 1. Solving the decision treeA B
(0.27) A2 A1 $8 $0
$15
(0.5)
(0.73)
(0.45)
(0.55) $4
$10 U1 U3 U2
EV(U2)=$7.5
EV(U2)=0*0.5+15*0.5=$7.5
EV(U1)=$5.08
O21
O11
O22
EV(U1)=8*0.27+4*0.73=$5.08
O12
EV(U3)=10*0.45+0*0.55=$4.5
O31
EV(U3)=$4.5
O32
In conclusion, according to the EV, we should choose A, and if O11 occurs, then choose A1

34. 2. Risk Profiles and Cumulative Risk Profiles
Decision Strategies:
Strategy 1: A - A1 A1 $8
(0.27) D2
$4 (0.73)
$8 (0.27) U1 A
(0.73) $4 D1
(0.5)
Strategy 2: A – A2
(0.27) D2 $0
$0 (0.135)
$4 (0.73)
$15 (0.135) A2 U2
(0.5) U1
$15 A
(0.73) $4 D1
Strategy 3: B D1
(0.45)
$10
$0 (0.55)
$10 (0.45) B U3
(0.55) $0

35. 2. Risk Profiles and Cumulative Risk Profiles (Cont.)
Conclusion: No stochastic dominance exists
Strategy A-A1
Strategy A-A2
Strategy B
Cumulative Risk Profiles