Decision Analysis

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Chapter 12

• Decision Analysis

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Components of Decision Making (D.M.)

• Decision alternatives - for managers to choose
  from.
• States of nature - that may actually occur in the
  future regardless of the decision.
• Payoffs - payoff of a decision alternative in a
  state of nature.
• The components are given in Payoff Tables.

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A Payoff Table (It shows payoffs of different
decisions at different states of nature)

• Investment States of Nature
• decision Economy Economy
• alternatives good bad
• Apartment 50,000 30,000
• Office 100,000 - 40,000
• Warehouse 30,000 10,000

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Types of Decision Making (D.M.) - 1

• Deterministic D.M. (D.M. under certainty)
• Only one state of nature,
• Payoff of an alternative is known,
• Examples
• Problems for LP, IP, transportation, and network
  flows.

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Types of Decision Making (D.M.) - 2

• D.M. without probabilities (D.M. under
  uncertainty)
• More than one states of nature
• Payoff of an alternative is not known at the time
  of making decision
• Probabilities of states of nature are not known.

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Types of Decision Making (D.M.) - 3

• D.M. with probabilities (D.M. under risk)
• More than one states of nature
• Payoff of an alternative is not known at the time
  of making decision
• Probabilities of states of nature are known or
  given.

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Types of Decision Making (D.M.) - 4

• D.M. in competition (Game theory)
• Making decision against a human competitor.

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Decision Making without Probabilities

• No information about possibilities of states of
  nature.
• Five criteria (approaches) for a decision maker
  to choose from, depending on his/her preference.

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Criterion 1 Maximax

• Pick the maximum of the maximums of payoffs of
  decision alternatives. (Best of the bests)
• Investment States of Nature max
• decision Economy Economy payoffs
• alternatives good bad (bests)
• Apartment 50,000 30,000 50,000
• Office 100,000 - 40,000 100,000
• Warehouse 30,000 10,000 30,000
• Decision

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Whom Is MaxiMax for?

• MaxiMax method is for optimistic decision makers
  who tend to grasp every chance of making money,
  who tend to take risk, who tend to focus on the
  most fortunate outcome of an alternative and
  overlook the possible catastrophic outcomes of an
  alternative.

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Criterion 2 Maximin

• Pick the maximum of the minimums of payoffs of
  decision alternatives. (Best of the worsts)
• Investment States of Nature min
• decision Economy Economy payoff
• alternatives good bad (worsts)
• Apartment 50,000 30,000 30,000
• Office 100,000 - 40,000 - 40,000
• Warehouse 30,000 10,000 10,000
• Decision

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Whom Is MaxiMin for?

• MaxiMin method is for pessimistic decision makers
  who tend to be conservative, who tend to avoid
  risks, who tend to be more concerned about being
  hurt by the most unfortunate outcome than the
  opportunity of being fortunate.

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Criterion 3 Minimax Regret

• Pick the minimum of the maximums of regrets of
  decision alternatives. (Best of the worst
  regrets)
• Need to construct a regret table first.
• Regret of a decision under a state of nature
• (the best payoff under the state of nature)
• (payoff of the decision under the state of
  nature)

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Payoffs
Investment States of Nature decision
Economy Economy alternatives good
bad Apartment 50,000 30,000 Office
100,000 - 40,000 Warehouse 30,000
10,000
Regrets
Investment States of Nature max
decision Economy Economy regret
alternatives good bad Apartment
50,000 0 50,000 Office
0 70,000 70,000 Warehouse 70,000
20,000 70,000
Decision
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Whom Is MiniMax Regret for?

• MiniMax regret method is for a decision maker who
  is afraid of being hurt by the feeling of regret
  and tries to reduce the future regret on his/her
  current decision to minimum. Im concerned more
  about the regret Ill have than how much Ill
  make or lose.

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Criterion 4 Hurwicz

• Pick the maximum of Hurwicz values of decision
  alternatives. (Best of the weighted averages of
  the best and the worst)
• Hurwicz value of a decision alternative
• (its max payoff)(?) (its min payoff)(1-?)
• where ? (0???1) is called coefficient of
  optimism.

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Payoffs
Investment States of Nature decision
Economy Economy alternatives good
bad Apartment 50,000 30,000 Office
100,000 - 40,000 Warehouse 30,000
10,000
Hurwicz Values with ?0.4
Investment decision Hurwicz Values
alternatives Apartment 50,000(0.4)30,000(
0.6) 38,000 Office 100,000(0.4)?40,000(0.6)
16,000 Warehouse 30,000(0.4)10,000(0.6)
18,000
Decision
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Whom Is Hurwicz Method for?

• Hurwicz method is for an extreme risk taker
  (?1), an extreme risk averter (?0), and a
  person between the two extremes (? is somewhere
  between 1 and 0) .

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Criterion 5 Equal Likelihood

• Pick the maximum of the average payoffs of
  decision alternatives. (Best of the plain
  averages)
• Average payoff of a decision alternative

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Payoffs
Investment States of Nature decision
Economy Economy alternatives good
bad Apartment 50,000 30,000 Office
100,000 - 40,000 Warehouse 30,000
10,000
Average Payoffs
Investment decision Average Payoffs
alternatives Apartment (50,00030,000) /
2 40,000 Office (100,000?40,000) / 2
30,000 Warehouse (30,00010,000) / 2 20,000
Decision
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Whom Is Equally Likelihood for?

• Equally likelihood method is for a decision maker
  who tends to simply use the average payoff to
  judge an alternative.

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Dominated Alternative

• If alternative As payoffs are lower than
  alternative Bs payoffs under all states of
  nature, then A is called a dominated alternative
  by B.
• A dominated alternative can be removed from the
  payoff table to simplify the problem.
• Investment States of Nature
• decision Economy Economy
• alternatives good bad
• Apartment 50,000 30,000
• Office 100,000 - 40,000
• Warehouse 30,000 10,000

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Decision Making with Probabilities

• The probability that each state of nature will
  actually occur is known.
• States of Nature
• Investment Economy Economy
• decision good bad
• alternatives 0.6 0.4
• Apartment 50,000 30,000
• Office 100,000 - 40,000
• Warehouse 30,000 10,000

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CriterionExpected Payoff

• Select the alternative that has the largest
  expected payoff.
• Expected payoff of an alternative
• nnumber of states of nature
• Piprobability of the i-th state of nature
• Vipayoff of the alternative under the i-th state
  of nature

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Example
Decision Alts Econ Good 0.6 Econ Bad 0.4 Expected payoff
Apartment 50,000 30,000
Office 100,000 -40,000
Warehouse 30,000 10,000
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Expected Opportunity Loss (EOL)

• Each decision alternative has an EOL which is the
  expected value of the opportunity costs
  (regrets).
• The alternative with minimum EOL has the highest
  expected payoff.

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Payoffs 0.6 0.4
Investment States of Nature decision
Economy Economy alternatives good
bad Apartment 50,000 30,000 Office
100,000 - 40,000 Warehouse 30,000
10,000
Opp Loss Table 0.6 0.4
Investment States of Nature
decision Economy Economy alternatives
good bad Apartment 50,000
0 Office 0 70,000 Warehouse
70,000 20,000
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Example (cont.)

• EOL (apartment)
• 50,0000.6 00.4 30,000
• EOL (office)
• 00.670,0000.4 28,000
• EOL (warehouse)
• 70,0000.620,0000.4 50,000
• Minimum EOL 28,000 that is associated with
  Office.

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(Max Exp. Payoff) vs. (Min EOL)

• The alternative with minimum EOL has the highest
  expected payoff.
• The alternative selected by (Max expected payoff)
  and by (Min EOL) are always same.

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Expected Value of Perfect Information (EVPI)

• It is a measure of the value of additional
  information about states of nature (in addition
  to the information in payoff table).
• It tells up to how much you would pay for the
  additional information.
• Information is perfect if what it says will
  occur.

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An Example

• If a consulting firm offers to provide perfect
  information about the future with 5,000, would
  you take the offer?
• States of Nature
• Investment Economy Economy
• decision good bad
• alternatives 0.6 0.4
• Apartment 50,000 30,000
• Office 100,000 - 40,000
• Warehouse 30,000 10,000

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Another Example

• You can play the game for many times.
• What is your rational strategy of guessing?
• Someone offers you perfect information about
  landing at 65 per time. Do you take it? If
  not, how much you would pay?

Land on Head Land on Tail
Guess Head 100 - 60
Guess Tail - 80 150
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Calculating EVPI

• EVPI
• EVwPI EVw/oPI
• (Exp. payoff with perfect information)
• (Exp. payoff without perfect information)

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Expected payoff with Perfect Information

• EVwPI
• where nnumber of states of nature
• hihighest payoff of i-th state of nature
• Piprobability of i-th state of nature

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Example for Expected payoff with Perfect
Information

• States of Nature
• Investment Economy Economy
• decision good bad
• alternatives 0.6 0.4
• Apartment 50,000 30,000
• Office 100,000 - 40,000
• Warehouse 30,000 10,000
• hi 100,000 30,000
• Expected payoff with perfect information
• 100,0000.630,0000.4 72,000

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Expected payoff without Perfect Information

• Expected payoff of the best alternative selected
  without using additional information. i.e.,
• EVw/oPI Max Exp. Payoff

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Example for Expected payoff without Perfect
Information
Decision Alts Econ Good 0.6 Econ Bad 0.4 Expected payoff
Apartment 50,000 30,000 42,000
Office 100,000 -40,000 44,000
Warehouse 30,000 10,000 22,000
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Expected Value of Perfect Information (EVPI) in
above Example

• EVPI
• EVwPI EVw/oPI
• 72,000 - 44,000
• 28,000

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Example Revisit (1)
0.5 0.5
Land on Head Land on Tail
Guess Head 100 -60
Guess Tail -80 150

• Up to how much would you pay for a piece of
  information about result of landing?

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Example Revisit (2)

• Maximum expected payoff
• Your best decision is
• EVw/oPI
• EVwPI
• EVPI
• Minimum EOL

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Maximum average payoff per game, or EVwPI
125
avg. regret
Min EOL, EVPI
avg. regret
EOL
average payoff
Max EMV, EVw/oPI
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average payoff
EMV
Alt. 2, Guess Tail
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Alt. 1, Guess Head
Alternatives
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EVPI is equal to (Min EOL)

• EVPI is the expected opportunity loss (EOL) for
  the selected decision alternative.

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EVPI is a Benchmark in Bargain

• EVPI is the maximum amount the decision maker
  would pay to purchase perfect information.

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Value of Imperfect Information

• Expected value of imperfect information
• (discounted EVwPI) EVw/oPI
• (EVwPI ( of perfection)) EVw/oPI

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Construct a Decision Table

• Determine states of nature (columns) and decision
  alternatives (rows).
• Payoff is usually net profit
• Profit Revenue Cost ( in) ( out)

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Decision Tree

• Decision tree is used to help make a series of
  decisions.
• A decision tree is composed of decision nodes
  (square), chance nodes (circle), and payoff nodes
  (final or tip nodes).
• A decision tree reflects the decision making
  process and the possible payoffs with different
  decisions under different states of nature.

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Making Decision on a Decision Tree

• It is actually a process of marking numbers on
  nodes.
• Mark numbers from right to left.
• For a chance (circle) node, mark it with its
  expected value.
• For a decision (square) node, select a decision
  and mark the node with the number associated with
  the decision.

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Example p.560-561
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Example p.566-567