-52- hmp654/execmas decision analysis alternatives and states of nature good decisions vs. good...

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Decision Analysis

• Alternatives and States of Nature

• Good Decisions vs. Good Outcomes

• Payoff Matrix• Decision Trees• Utility Functions• Decisions under Uncertainty• Decisions under Risk

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Decision Analysis - Payoff Tables

As director of a home health agency, you are in the process of evaluating the possibleaddition of physical therapy services for your clients. Three options are under consideration:

Option A: The home health agency would contract with an independent practitioner to providePT services on a fee-for-service basis.

Costs: The PT will be paid $60 per home visit.

Option B: The home health agency would hire a staff physical therapist, provide the therapist withan automobile, and cover medical supplies and gasoline.

Costs: Monthly PT salary of $4000Monthly automobile lease payment of $400Medical supplies and gasoline allowance of $7 per visit

Option C: The home health agency would utilize an independent contractor to provide PTservices, provide the contractor with an automobile and fringe benefits, and pay amedical supply and gasoline allowance.

Costs: PT services of $35 per visitMonthly automobile lease payment of $400Fringe benefits (including insurance) of $200 per monthMedical supplies and gasoline allowance of $7 per visit

Under all three alternatives the average payment for a PT home visit is $75 per visit.

The home health agency is trying to decide for which of these three options the maximumprofit would be realized. In searching for the best option, the agency realizes that the demand forservices influences the optimum choice. The agency would like to better understand therelationship between demand and optimum choice. Specific questions that the agency has posedinclude the following:

1. With no knowledge of demand for PT services, which option should be chosen?2. For a given probability distribution of demand for PT services, which option provides maximum

profit?3. How sensitive is the choice of best option to the probability distribution of demand?4. How much value should be placed on a system that can forecast future demand for PT

services?

Assume that, using a combination of marketing analysis, analysis of competitors' volumedata, and some careful assumptions, it has been determined that there are only four possible valuesof monthly demand for PT services: 30, 90, 140, or 150 visits. Assume further that the relativefrequency with which each of these demands occurs is also available from health facilities utilizationdata compiled in a recent statewide study. These relative frequencies provide reasonable estimatesof the probabilities associated with each value of monthly demand, which for the case problem areassumed to equal 0.1, 0.4, 0.2, and 0.3, respectively.

Case Problem - (A) p. 38

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Decisions under Uncertainty

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Decisions under Risk

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Decision Analysis - Payoff TablesDecisions under Risk

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Decisions under Risk

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Decision Analysis - Utility Theory

• Utility theory provides a way to incorporate the decision maker’s attitudes and preferences toward risk and return in the decision analysis process so that the most desirable decision alternative is identified.

• A utility function translates each of the possible payoffs in a decision problem into a non-monetary measure known as a utility.

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Utility

Payoff

1.00

0.75

0.50

0.25

0

risk seeking

risk neutral

risk averse

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• The utility of a payoff represents the total worth, value, or desirability of the outcome of a decision alternative to the decision maker.

• A risk averse decision maker assigns the largest relative utility to any payoff but has a diminishing marginal utility for increased payoffs.

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• A risk seeking decision maker assigns the smallest utility to any payoff but has an increasing marginal utility for increased payoffs.

• A risk neutral decision maker falls in between these two extremes and has a constant marginal utility for increased payoffs.

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Decision Analysis - Utility TheoryConstructing Utility Functions

• Step 1 - Assign a utility value of 0 to the worst outcome (W) in a decision problem and a utility value of 1 to the best outcome (B).

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• Step 2 - For any other outcome x, find the probability p at which the decision maker is indifferent between the following two alternatives:– Receive x with certainty or– Receive B with probability p or W with

probability 1-p

The value of p is the utility that the decision maker assigns to the outcome x.

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– Receive $450 with certainty– Play a game in which the decision

maker can make $5,800 with probability p or lose $2,360 with probability 1-p

Let’s assume that the value of p that makes these two choices equally attractive to the decision maker is 0.7. Then the utility that

the decision maker assigns to the $450 is 0.7.

For example, let’s compute the utility forthe $450 entry that corresponds to alternativeA and state of nature N=30. The problemconsists on finding the value of p that makesthe following two options equally attractive for the decision maker:

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Decision Analysis - Utility TheoryThe Exponential Utility Function

• A sensible value for R is the maximum value of Y for which the decision maker is willing to participate in a game of chance with the following possible outcomes:

• Win $Y with probability 0.5– Lose $Y/2 with probability 0.5

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Decision Analysis - Utility TheoryThe Exponential Utility Function

-52- hmp654/execmas decision analysis alternatives and states of nature good decisions vs. good...

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