ISE 562; Dr. Smith
Decision Problems – 2: Practice
Decision Theory
Copyright By PowCoder代写 加微信 powcoder
ISE 562; Dr.
• Problem structuring practice (inputs and trees)
• Identifying states of nature and their uncertainty
9/20/2022 2
ISE 562; Dr. 1:
The management of First Bank was concerned about the potential loss that might occur in the event of a physical catastrophe such as a power failure or fire. The bank estimated the loss from one of these incidents could be as much as $100M, including losses due to interrupted service and customer relations. One project the bank is considering is the installation of an emergency power generator at its operations headquarters. The cost of the emergency generator is $0.8M, and if it is installed no losses from this type of incident will be incurred. However, if the generator is not installed, there is a 10% chance that a power outage will occur during the next year. If there is an outage, there is a 0.05 probability that the resulting losses will be very large, or approximately $80M in lost earnings. Alternatively, it is estimated there is a 0.95 probability of only slight losses of around $1M. Using decision tree analysis should the bank install the generator?
9/20/2022 3
ISE 562; Dr. are the alternatives?
Install generator
Do not Install generator
9/20/2022 4
ISE 562; Dr. are the states of nature?
Install generator
Prepared for outage
Do not Install generator
P(Slight loss)
P(No Outage)
P(Large loss)
ISE 562; Dr. are the probabilities?
Install generator
Prepared for outage P(Large loss)
Do not Install
P(Slight loss)
P(No Outage)
ISE 562; Dr. are the payoffs?
Install generator
Prepared for outage
$0.8M $80M
Do not Install
P(Outage) =0.10
P(Slight loss) =0.95
P(No Outage) =0.90
P(Large loss) =0.05
ISE 562; Dr. not Install
P(Outage) =0.10
P(Slight loss) =0.95
.10($4.95M) +.90($0M) =$.495M
What are the expected payoffs? (Note: these are costs so actually expected costs)
Install generator
Prepared for outage
P(No Outage) =0.90
1.0($0.8M) =$0.8M
P(Large loss) =0.05
.05($80M) +.95($1M) =$4.95M
ISE 562; Dr. 2: SC is playing UCLA in a major conference game of the season. SC is trailing UC 21 to 14, with 7 seconds left in the game, when SC scores a touchdown. Still trailing 21 to 20, SC can either go for 2 points and win or go for 1 point to send the game into overtime. The conference championship will be determined by the outcome of the game. If SC wins it will go to the with a payoff of $7.2M; if it loses it will go to the Sun Bowl with a payoff of $1.7M. If SC goes for 2 points, there is a 33% chance it will be successful and win (and a 67% chance it will fail and lose). If it goes for 1 point, there is a 0.98 probability of success and a tie and a 0.02 probability of failure. If the teams tie, they will play overtime, during which SC believes it has only a 20% chance of winning because of fatigue.
Should SC go for 1 or 2 points? What would SC’s probability of winning in overtime have to be to make SC indifferent to going for 1 or 2 points?
9/20/2022 9
ISE 562; Dr. are the alternatives?
9/20/2022 10
ISE 562; Dr. are the states of nature?
P(Fail;lose)
P(win in overtime)
P(succeed, overtime)
P(lose in overtime)
P(Succeed; win)
P(Fail; lose)
ISE 562; Dr. are the probabilities?
P(Fail;lose)
P(succeed, overtime)
P(lose in overtime)
P(Succeed; win)
P(Fail; lose)
P(win in overtime)
ISE 562; Dr. are the payoffs?
P(Fail;lose) =0.67
$1.7M $7.2M
P(succeed, overtime)
P(lose in overtime) =0.80
P(Succeed; win) =0.33
P(Fail; lose) =0.02
P(win in overtime) =0.20
ISE 562; Dr. are the expected payoffs?
P(Fail;lose) =0.67
$1.7M $7.2M
P(succeed, overtime)
P(lose in overtime) =0.80
P(Succeed; win) =0.33
P(Fail; lose) =0.02
0.98($2.8M)+0.02($1.7M) = $2.778M
0.33($7.2M)+0.67($1.7M) = $3.515M
P(win in overtime) =0.20
0.20($7.2M)+0.80($1.7M) = $2.8M
ISE 562; Dr. would SC’s probability of winning in overtime have to be to make SC indifferent to going for 1 or 2 points?
3.515=[Pw(7.2)+(1-Pw)1.7].98 + 1.7(.02) =7.056Pw + 1.67 – 1.67Pw + 0.034 5.384Pw = 3.515 – 1.67 – .034
Pw = 0.337
If probability of winning in overtime = .337 then indifferent between 2 or 1 point. (if >.337 go for 1 point)
9/20/2022 15
ISE 562; Dr. 3: The company has 3 health care plans for staff to choose from:
Plan 1: monthly cost of $32 with a $500 deductible; participants pay the first $500 of medical costs for the year; the insurer pays 90% of all remaining expenses.
Plan 2: monthly cost of $5 but a deductible of $1200 with the insurer paying 90% of medical expenses after the insurer pays the first $1200 in a year.
Plan 3: monthly cost of $24 with no deductible; the participants pay 30% of all expenses with the remainder paid by the insurer.
estimates her annual medical expenses
are defined by the following probability distribution: 9/20/2022 16
ISE 562; Dr. medical expenses
Probability
$100 .15 500 .30 1500 .35 3000 .10 5000 .05 10000 .05
Which plan should the employee select?
9/20/2022 17
ISE 562; Dr. are the alternatives?
Plan 1 Plan 2 Plan 3
ISE 562; Dr. are the states of nature?
Annual medical expenses =$100
Annual medical expenses =$500 Annual medical expenses =$1500 Annual medical expenses =$3000 Annual medical expenses =$5000 Annual medical expenses =$10000
9/20/2022 19
ISE 562; Dr. are the probabilities?
P(medical expenses =$100)=.15 P(medical expenses =$500)=.30 P(medical expenses =$1500)=.35 P(medical expenses =$3000)=.10 P(medical expenses =$5000)=.05 P(medical expenses =$10000)=.05
9/20/2022 20
ISE 562; Dr. are the payoffs (costs)?
If she chooses Plan 1 and the state of nature is $100, what is her 1st year cost?
Monthly cost = $32 x 12 months = $384
How much will she pay due to deductible?
$100 < $500 deductible so she will pay $100.
What does the insurer pay?
$100 < $500 deductible, so insurer pays zero
So payoff (cost) = $384+$100= $484
9/20/2022 21
ISE 562; Dr. Smith
Payoff Table
Cost P(cost) Plan 1 Plan 2 Plan 3
$100 .15 500 .30 1500 .35 3000 .10 5000 .05 10000 .05
ISE 562; Dr. she chooses Plan 1 and the state of nature is
$10000, what is her 1st year cost?
Monthly cost = $32 x 12 months = $384
How much will she pay due to deductible?
$10000 > $500 deductible so she will pay $500.
What does the insurer pay?
$10000 > $500 deductible, so insurer pays 90% of difference or .90(9500)=8550 and mary pays 10% or .10(9500) = $950
So Mary’s cost = $384+$500+$950= $1834
9/20/2022 23
ISE 562; Dr. Smith
500 .30 1500 .35 3000 .10 5000 .05
Payoff Table
Cost P(cost) Plan 1 Plan 2 Plan 3 $100 .15 $484
Class exercise: fill in the table
9/20/2022 24
ISE 562; Dr. Smith
Payoff Table
Class exercise: fill in the table…
ISE 562; Dr. Smith
288+.3(expense)
Payoff Table
Cost P(cost) Plan 1 Plan 2 Plan 3
$100 .15 500 .30 1500 .35 3000 .10 5000 .05 10000 .05
$160 $318 $884 $560 $438 $984 $1290 $738
$1134 $1440 $1334$1640 $1788
$1834 $2140 $3288 384+500+.1
60+1200+.1
ISE 562; Dr. are the expected payoffs?
Cost P(cost) Plan 1 Plan 2 Plan 3
$100 .15 500 .30 1500 .35 3000 .10 5000 .05 10000 .05
$160 $318 $884 $560 $438 $984 $1290 $738
$1134 $1440 $1334$1640 $1788
$1834 $2140 $3288
EMV[Plan 1] = .15(484)+.30(884)+…+.05(1834) = $954
EMV[Plan 2] = .15(160)+.30(560)+…+.05(2140) = $976
EMV[Plan 3] = .15(318)+.30(438)+…+.05(3288) = $810
9/20/2022 27
ISE 562; Dr. are the expected payoffs?
Cost P(cost) Plan 1 Plan 2 Plan 3
$100 .15 500 .30 1500 .35 3000 .10 5000 .05 10000 .05
She should select Plan 3 to
minimize expected medical costs.
$984 $1290
$1134 $1440
$1334$1640 $1834 $2140
$1788 $3288
ISE 562; Dr. 1 $954
Plan 2 $976
The decision tree
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com