Decision Trees
CIS 418
Business problems involve uncertainty: the NEES example
NEES is a company that produces power. It is deciding how much to bid for the salvage rights to a grounded ship, the SS Kuniang.
If successful, the ship could be repaired to haul coal for the company’s power stations. If the bid fails, NEES could purchase a new ship or a tug/barge combination.
The higher the bid the more likely that NEES will win.
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NEES decision is complicated by further uncertainty
U.S. Coast Guard (USCG) judgment about the Marine salvage value of the ship involves an obscure law on shipping in coastal waters. Marine salvage is the process of recovering a ship and its cargo after a shipwreck or other maritime casualty. USCG’s judgment will not be known until after the wining bid.
• If the judgment indicates a low salvage value, then NEES could use the ship for its shipping needs.
• High salvage value means that the ship is considered ineligible of use in domestic shipping unless expensive equipment is installed, i.e. greater expenses for NEES.
How much should NEES bid?
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Decision tree diagrams timing of decisions and revelations of relevant uncertainties
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To complete the tree need to assign $ values (or utilities) to decisions and outcomes
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The value of the ship after realizing the marine salvage value
When all the cash flows are given we can assign terminal values to each branch by adding up cash flows along the branch
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Decision criteria: minimize worst case
Consider bidding $10M
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We assign values to the nodes working from the end branches toward the root of the tree
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The value of an event node is the value corresponding to the worst- case scenario
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The value of a decision node here comes from making the decision that maximizes the net profit
What should be our maximum bid?
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If we just want to minimize the worst case it does not make sense to bid anything over ($11.50-$3.20)M = $8.3M
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If two strategies result in the same “worst case” we might think about another criteria for selecting among the strategies
What should do – bid or purchase the alternative?
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Another possible criteria: Maximizing the expected cash flow
• Expected Utility = EU = Weighted utility
• For example: you can buy a lottery ticket for 10 dollars. If you win you get 100 dollars. If you lose you get 0 dollars. The probability of winning is 8%. Would you buy the ticket?
• We answer by calculating the expected utility:
EU = Pr(winning)100Pr(losing)010
8%10092%082 0
EU 0
• Buy the ticket only if:
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Another possible criteria: Maximizing the expected cash flow
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Event node values are calculated based on expected values
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Decision node values are calculated by selecting the decision that maximizes the value of the node
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Calculating the Expected Utility
60% 4.7 40% 3.2 4.1
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70% 3.5 30% 7.5 4.7
We could also assume an analytical relationship between bid size and the probability of win, and look for the optimal bid size
The problem statement reads:
The higher the bid, the more likely the company will win. They expect that a bid of
$2M would definitely not win $12M would definitely win
$8M has 60% chance of winning
We can write a formula for calculating the probability to win as a function of the bid amount:
Prob_Of_Win (Bid 2) / 10 Simon Business School
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Sensitivity analysis
We use sensitivity analysis to calculate and compare the profit under different bids.
We set the bid price to be our sensitivity parameter (and not as optimization parameter, because we do not use optimization model to find the bid price).
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Let’s take a break to talk about Probability
Probability = chance.
We use probability to measure uncertainty. Probability measurements: 0 to 1 or 0% to 100%.
For example: From a box that contains 2 blue and 3 red balls, we randomly select one ball (with our eyes closed).
What are the chances of getting a blue ball?
23 Blue 5 5 Red
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What are the chances of drawing two blues?
Assumption: no replacements.
Pr(Blue) Blue
Pr(Blue Blue) 1 Blue4
2 5
3 5
Pr(Re d ) Red
3 4Red
2 Blue 4
2
4 Red
Pr(Blue,Blue) 21 1 5 4 10
Pr(Blue,Blue)Pr(Blue)Pr(Blue Blue) Simon Business School CIS-418
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What are the chances of drawing red and then blue?
Assumption: no replacements.
Pr(Blue) Blue
Pr(Blue Blue) 1 Blue4
2 5
3 5
Pr(Re d ) Red
3 4Red
2 Blue 4
2
4 Red
Pr(red,Blue)32 3 5 4 10
Pr(Red,Blue) Pr(Red) Pr(Blue Red ) Simon Business School CIS-418
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Conditional probability
Pr(A B) Pr(Event A would happen if we know that event B happened) Pr(A and B) Pr(B)Pr(A B) Pr(A)Pr(B A)
Bayes’ rule:
Pr(A B) Pr(A and B) Pr(B)
Pr(B A) Pr(A and B) Pr(A)
Interesting video regarding Bayes’ rule:
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Total Probability Theorem
Recall that the probability that the first draw would be of a blue ball is 52 .
What are the chances that the second draw would be of a blue ball?
Answer: Total probability theorem:
Pr(2nd is blue) = Pr(2nd is blue 1st is blue) Pr(1st is blue) + Pr(2nd is blue 1st is red) Pr(1st is red)
Pr(2nd is blue) = 2 1 3 2 2 54545
Meaning:
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A
Mobile Oil Company has recently acquired rights to a new
potential source of natural oil
in
Alaska.
• The current market value of these rights is $90,000. The company could sell these rights now.
• However, if there is natural oil at the site, it is estimated to be worth $800,000; although the company would have to pay $100,000 in drilling costs to extract the oil.
• The company believes there is a 25% probability that the proposed drilling site actually would hit the natural oil reserve.
• Alternatively, the company can pay $30,000 to first carry out a seismic survey at the proposed drilling site.
• The survey is not totally accurate: there is a 20% chance that the survey is favorable when oil is not present (false positive, type I error); and a 40% chance that the survey result is unfavorable when there is oil at the site (false negative, type II error).
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Model formulation
• What is the company objective? – Maximize profit
• What decisions does the Mobile Oil Company face? – Drill
– Sell
– Survey
• How to calculate the expected value of the survey option?
• Go to the excel file “Mobile oil company”
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What is the probability of a favorable survey?
25% 75% 60% 40% 20%
80%
No oil
Favorable
What is the probability of a favorable survey?
Pr(Favorable)=0.25 0.60+0.75 0.20=0.30=30% Pr(unfavorable)=1- Pr(Favorable)=0.70=70%
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Oil
Unfavorable
Favorable
Unfavorable
What is the probability of finding oil given a favorable survey?
25% 75%
60% 40% 20% 80%
Pr(OilFavorable)Pr(OilandFavorable)=0.250.60.5 Pr(NoOilFavorable)10.50.5 Pr(Favorable) 0.3
Pr(OilUnfavorable)Pr(OilandUnavorable)=0.250.40.143 Pr(NoOilUnfavorable)10.1430.857 Pr(Unfavorable) 0.7
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Favorable
Oil
Unfavorable
Favorable
No oil
Unfavorable
What probabilities do we need for the decision tree?
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What is the probability of finding oil if the survey is favorable?
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30%
50% 50%
70%
14.3% 85.7%
Would choose to drill, no matter what the survey results
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Questions
1. Would the decision be different if the survey would cost less?
2. If the accuracy of the survey was different, would that affect the decision?
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