ISE 562; Dr. Problems -1: Problem Structuring and Uncertainty
Decision Theory
ISE 562; Dr. :
– Probability review and Bayes theorem
Copyright By PowCoder代写 加微信 powcoder
– Decision criteria under certainty and uncertainty
– Introduction to utility theory • Today
– The decision problem (inputs and definitions)
– Uncertainty in states of nature vs. payoffs vs. utility lotteries
– Examples (using expected utility vs. expected monetary value; Mars pathfinder
ISE 562; Dr. Between the System Model and the Value Model
Uncertainties:
• States of nature • Payoffs
DIFFERENT!
Uncertainty
Engineering System
Alternatives
Alternative Ranking
Decision Maker(s)
Outcome Descriptions
Outcome Utilities
System Model
Value Model
•Effect of uncertainty on risk attitude
Sure thing vs. p
ISE 562; Dr. for decision problem inputs
• Set of possible actions or decisions, A
• Let aA be an alternative from A
• The set of states of nature called S
• Uncertainty associated with state of nature,
• Represent uncertainty with P() if discrete and f() if continuous
– The distribution may be a prior or posterior
– i.e., P’(), f’(); P”(), f”();
• Each (a,) maps to a consequence
• There is a utility function that maps each consequence to a utility value, U(a,)
9/17/2022 4
ISE 562; Dr. decision problem requires
• A, the set of actions
• S, the set of states of nature
• The probability distribution of the states of nature, (P() if discrete; f() if continuous)
• U(a,), the utility function that associates a utility with each action and state of the nature
• The decision maker should select the action, a, with highest expected utility of all actions in A, EU(a). Let the best choice be denoted, a*.
9/17/2022 5
ISE 562; Dr. calculate EU:
• If discrete, EU(a)=U(a,)P()
• If continuous, EU(a)=U(a,)f()d
• The optimal decision is a* where EU(a*) EU(a) for all aA
Note that if U() linear (or approx. linear) with respect to money, then payoff $ (return, R(a,)) or loss function, L(a,), can be used instead of U(a,).
9/17/2022 6
ISE 562; Dr. expected payoff:
• If discrete, ER(a)=R(a,)P()
• If continuous, ER(a)=R(a,)f()d
• Find a* such that ER(a*)ER(a) for all a For expected loss
• If discrete, EL(a)=L(a,)P()
• If continuous, EL(a)=L(a,)f()d
• Find a* such that EL(a*)EL(a) for all a
9/17/2022 7
ISE 562; Dr. OTE: there is a difference between uncertainty in the states of nature and…
9/17/2022 8
Interest Rates
Office park
Office Bldg
ISE 562; Dr. in payoffs and…
Interest Rates
Office park
Office Bldg
ISE 562; Dr. in utility assessments:
ATTRIBUTE: _______________ ____
____SURE THING
ISE 562; Dr. OTE: there is a difference between uncertainty in the states of nature and…
These are uncertain
Office park 0.5 1.7
alternatives.
Warehouse 1.7
taken by the
9/17/2022 11
variable is the value
Mall 0.7 2.4
Condos state of3n.2atu
Interest Rates
The random
r e ( 1 i . n5 t e r e s t 0 r . a6 t e )
outcomes affecting the
Office Bldg 1.5 1.9 2.5
ISE 562; Dr. in payoffs and…
.15 .25 .60
payoffs (precision of knowledge). The
random variable is the quantity of the
Project Decline Stable Increase
decision variable for
rnative under
Office park
Interest Rates
These are uncertainties on the estimated
the1.7alte
state of nature (in this case Profit).
Office Bldg
1.5 1.9 2.5
ISE 562; Dr. in utility assessments:
ATTRIBUTE: _______________
____ ____SURE THING
These are fixed probabilities to force a choice by the decision maker between a risky___o_utcome and a certain outcome.
These probabilities are a tool for eliciting
utility functions (P(getting best outcome))
End note. 13
ISE 562; Dr. Smith
• States of nature can be continuous random variables, f()
Instead of this:
Rate f() Use this: x
ISE 562; Dr. Smith
• Payoffs as continuous random variables, f(x)
Instead of this:
Stable 2.4 .25
Decline .15
Decline .15
Stable .25
Increase .60
ISE 562; Dr. to calculate the expected value:
Stable 2.4 .25
Decline .15
If the utility function for money is linear use EMV If not linear, could convert into utility like this…
9/17/2022 16
ISE 562; Dr. the mean and look itf(X)
up on u(x):
Decline .15
Stable .25
Increase .60
*for example
ISE 562; Dr. better, take expected value of the utility
function: EU(a)=U(a,)f()d=.92* U(X) 1
Decline .15
Stable .25
*for example
Increase .60
Convolve the pdf of X against the utility function to calculate the EU weighted by the pdf.
ISE 562; Dr. Smith
• Note that there may be many initial alternatives that give the appearance of an overwhelming decision problem.
• First step in defining the set {A} includes screening out “inadmissible” alternatives
• Inadmissible: those alternatives that are impractical or infeasible for technical, budgetary, or other reasons.
9/17/2022 19
ISE 562; Dr. Smith
• Decision making under uncertainty: a simple example using utility theory
9/17/2022 20
ISE 562; Dr. Smith
• Example: Americo Oil is considering making a bid of $110M for shale oil development. Company estimates it has a 60% chance of winning the contract. If it wins, it can choose one of 3 methods of oil extraction: 1) new method; 2) use exising ineff. process; 3) subcontract to smaller companies. Data are summarized in the following table. Cost of the contract proposal is $2M; If company doesn’t bid, will invest with a return guaranteed to be $30M. Construct the decision tree and identify the correct decision.
9/17/2022 21
ISE 562; Dr. Smith
Develop new process
Probability
Profit ($M)
Great success
Use existing (ineff.) process
Great success
Subcontract
Moderate 1.0 250
9/17/2022 22
ISE 562; Dr. ssuming Profit = Profit – Bid Cost
0.3 Great Success
0.6 Moderate
0.1 Failure
0.5 Great Success
0.2 Failure
1 Moderate
0.6 Win Contract
0.4 Don’t win
0.3 Moderate
Use Existing Process
Subcontract
ISE 562; Dr. Smith
• The range of net profits is -212M to $488M
• If the decision maker is risk neutral
– Xo=-212; U(Xo)=U(-212)=0
– X*=488; U(X*)=U(488)=1
– The utility function is the line between the pairs (-212, 0) and (488, 1)
– m=1/(488-(-212))=1/700 (slope)
– U(X)=1/700 (X+212)
• (note at X=-212, U(X)=0; X=488, U(X)=1) 24
ISE 562; Dr. Smith
• Now use the utility function to compute the utilities for all the dollar values in the decision tree or payoff table
0X -212 0 488
ISE 562; Dr. Smith
• U(X)=1/700 (X+212)
25 9/17/2022
ISE 562; Dr. Smith
• Now replace the costs in the decision tree with utility values
ISE 562; Dr. Smith
• Decision is the same • Do you know why?
New process 0
0.3 Great success
0.6 Moderate success
ISE 562; Dr. Smith
• Suppose one (or more) of the payoffs were represented by a probability distribution.
29 9/17/2022
138 0.500 30
ISE 562; Dr.
Great success Moderate
Use existing (ineff.) Great success .50 Moderate .30 Failure .20 Subcontract Moderate 1.0
No bid 0.346
Existing process
Subcontract
1 Moderate success
Great success Moderate
Use existing (ineff.) Great success .50 Moderate .30 Failure .20 Subcontract Moderate 1.0
0.6 Win contract
0.4 Don’t win
0.6426 0.571 0.571
0.1 Failure
0.5 Great success
0.571 0.571
0.3 Moderate success
0.429 0.429
0.2 Failure
0.086 0.086
Develop new process
Probability
Net Profit ($M)
1.000 0.571 0.000
0.571 0.429 0.086
.30 .60 .10
488 188 -212
188 88 -152
• Pick this one (arbitrary) for illustration
Develop new process
Probability
.30 .60 .10
Net Profit Utility ($M)
488 1.000 188 0.571 -212 0.000
88 0.429 -152 0.086
ISE 562; Dr. Smith
• Instead of $488M, there is uncertainty about the true profit. Company analysts believe the true net profit will be somewhere between -100M and 488M.
• A triangular pdf is used to represent the
uncertainty:
f(X) f(X)=(5.786 x 10-6 )(488 – X)
for -100 X 488
-212M -100M
ISE 562; Dr. we calculate EU(a)=U(a,)f()d
EU(new process, great success)
𝐸𝑈𝑎 𝑛𝑒𝑤 𝑝𝑟𝑜𝑐𝑒𝑠𝑠, 𝜃 𝑔𝑟𝑒𝑎𝑡 𝑠𝑢𝑐𝑐𝑒𝑠𝑠
𝑈𝑎,𝜃𝑓𝜃𝑑𝜃 1 𝑥212 5.786𝑥10 488𝑥 𝑑𝑥 0.44
ISE 562; Dr. Smith
1 𝑥212 5.786𝑥10 488𝑥 𝑑𝑥0.44 700
1 𝑥212 488𝑥 𝑑𝑥
700 ⋅ 172872
1 488𝑥𝑥 103456212𝑥 𝑑𝑥
700 ⋅ 172872
276𝑥 𝑥 103456 𝑑𝑥 1 𝑥𝑥
700 ⋅ 172872
700⋅172872 276 2 3 103456𝑥
1 32863872 38738090.67 50486528 1380000 333333.33 700 ⋅ 172872
10345600 0.44
9/17/2022 33
ISE 562; Dr. outcome was 1.0 with result of 0.4999 without uncertainty
Now it is 0.44 with uncertainty and EU drops to 0.41 and the choice of
0.6 Win contract
0.4 Don’t win
0.3 Great success
0.6 Moderate success
0.571 0.571
0.1 Failure
0.5 Great success
0.571 0.571
0.3 Moderate success
0.429 0.429
0.2 Failure
0.086 0.086
1 Moderate success
process changes
0.571 0.571
0.429 0.429
0.086 0.086
0.286 0.286
0.346 0.346
New process 0
No bid 0.346
Subcontract 0
Existing process
ISE 562; Dr. Smith
• If utility function not linear, e.g.
0X -212 0 488
• Procedure is the same except decision can be very different than EMV.
9/17/2022 35
ISE 562; Dr. • Experiment
Modeling cost uncertainty
ISE 562; Dr. Smith
Decision Problems – 2: Practice Problems
Decision Theory
ISE 562; Dr. Smith
• Problem structuring practice (inputs and trees)
• Identifying states of nature and their uncertainty
• Define the decision tree with probabilities and payoffs; find the optimal decision.
• See if you can work the following problems (at least set them up before class.
• (If you are viewing the lecture later, hit “pause” and try the exercises.)
9/17/2022 38
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?
9/17/2022 39
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?
9/17/2022 40
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: 9/17/2022 41
ISE 562; Dr. medical expenses
Probability
Which plan should the employee select?
9/17/2022 42
ISE 562; Dr. Smith
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com