Chapter 21 Decision Analysis

1
Chapter 21Decision Analysis

• Problem Formulation

• Decision Making with Probabilities

• Decision Analysis
• with Sample Information

• Computing Branch Probabilities
• Using Bayes Theorem

2
Problem Formulation

• The first step in the decision analysis process
  is problem formulation.

• We begin with a verbal statement of the problem.

• Then we identify
• the decision alternatives
• the uncertain future events
• the consequences associated with
• each decision alternative
• each chance event outcome

3
Problem Formulation

• A decision problem is characterized by decision
  alternatives, states of nature, and resulting
  payoffs.

• The decision alternatives are the different
  possible strategies the decision maker can employ.

• The states of nature refer to future events, not
  under the control of the decision maker, which
  may occur.

• States of nature should be defined so that they
  are mutually exclusive.

4
UCF

• UCF has to decide how many seats the new football
  stadium should have. UCF plans to sell seats
  between 30 to 140 per season.
• UCF has three different projects. One with 30,000
  seats, one with 60,000 seats and one with 90,000
  seats. The financial success of the project
  depends upon the size of the of the stadium and
  the chance event concerning the demand for
  stadium seats.
• The statement of the UCF decision problem is to
  select the size of the new stadium that will lead
  to the largest profit given the uncertainty
  concerning the demand for seats.

5
UCF

• Given the statement of the problem, it is clear
  that the decision is to select the best size for
  the stadium complex.
• UCF has the following 3 alternatives
• d1 a small stadium 30,000
• d2 a medium stadium 60,000
• d3 a large stadium 90,000
• A factor in selecting the best decision
  alternative is the uncertainty associated with
  the event concerning the stadium seats. However
  UCF president decide it would be adequate to
  consider two possible chance event outcomes
• 1. Strong Demand and 2. Weak Demand

6
UCF

• In decision analysis the possible outcomes for a
  chance event are referred to as the STATES OF
  NATURE. The states of nature are defined so that
  one and only one of the possible states of nature
  will occur.
• S1 strong demand
• S2 weak demand

7
Payoff Tables
Given the tree decision alternatives and the two
states of nature, which stadium should UCF choose
? To answer UCF will need to know the
consequences associated with each decision
alternative and each state of nature (PAYOFF)

• The consequence resulting from a specific
  combination of a decision alternative and a state
  of nature is a payoff.

• A table showing payoffs for all combinations of
  decision alternatives and states of nature is a
  payoff table.

• Payoffs can be expressed in terms of profit,
  cost, time, distance or any other appropriate
  measure.

8
PAYOFF TABLE

• The payoff table with profits expressed in
  Hundred thousand of dollars per game is shown in
  Table 1
• We will use the notation Vij to denote the payoff
  associated with decision alternative i and state
  of nature j.

9
Decision Trees

• A decision tree provides a graphical
  representation showing the sequential nature of
  the decision-making process.

• Each decision tree has two types of nodes
• round nodes correspond to the states of nature (3
  in our example) (Chance nodes)
• square nodes correspond to the decision
  alternatives (decision nodes)

10
UCF STADIUM
UCF
UCF
11
Decision Trees

• The branches leaving each round node represent
  the different states of nature while the branches
  leaving each square node represent the different
  decision alternatives.

• At the end of each limb of a tree are the payoffs
  attained from the series of branches making up
  that limb.

12
Decision Making with Probabilities

• Once we have defined the decision alternatives
  and states of nature for the chance events, we
  focus on determining probabilities for the states
  of nature.

• The classical method, relative frequency method,
  or subjective method of assigning probabilities
  may be used.

• Because one and only one of the N states of
  nature can occur, the probabilities must satisfy
  two conditions

P(sj) gt 0 for all states of nature
13
Decision Making with Probabilities

• Then we use the expected value approach to
  identify the best or recommended decision
  alternative.

• The expected value of each decision alternative
  is calculated (explained on the next slide).

• The decision alternative yielding the best
  expected value is chosen.

14
Expected Value Approach

• The expected value of a decision alternative is
  the sum of weighted payoffs for the decision
  alternative.

• The expected value (EV) of decision alternative
  di is defined as

where N the number of states of
nature P(sj ) the probability of state
of nature sj Vij the payoff
corresponding to decision alternative di
and state of nature sj
15
Expected Value UCF

• UCF is optimistic and assign the subjective
  probability of .8 for strong demand and .2 for
  weak demand.
• EV(d1) .8 (8) .2 (7) 7.8
• EV(d2) .8 (14) .2 (5) 12.2
• EV(d3) .8 (20) .2 (-9) 14.2
• Thus, using the expected value approach, the
  large stadium complex with an expected 1,400,000
  in revenue per game is the recommended decision.

16
Expected Value Using Decision Trees
17
Expected Value Approach

• Example Burger Prince

• Burger Prince Restaurant is considering
• opening a new restaurant on Main Street.
• It has three different models, each
• with a different seating capacity.
• Burger Prince estimates that the average
• number of customers per hour will be
• 80, 100, or 120. The payoff table for the
• three models is on the next slide.

18
Expected Value Approach

• Payoff Table

Average Number of Customers Per Hour
s1 80 s2 100 s3 120
Model A Model B Model C
10,000 15,000 14,000
8,000 18,000 12,000
6,000 16,000 21,000
19
Expected Value Approach

• Calculate the expected value for each decision.

• The decision tree on the next slide can assist in
  this calculation.

• Here d1, d2, d3 represent the decision
  alternatives of models A, B, and C.

• And s1, s2, s3 represent the states of nature of
  80, 100, and 120 customers per hour.

20
Expected Value Approach
Payoffs
Decision Tree
.4
s1
10,000
.2
s2
2
15,000
s3
.4
d1
14,000
.4
s1
8,000
d2
.2
3
s2
1
18,000
s3
.4
d3
12,000
.4
s1
6,000
4
s2
.2
16,000
s3
.4
21,000
21
Expected Value Approach
EMV .4(10,000) .2(15,000)
.4(14,000) 12,600
d1
2
Model A
EMV .4(8,000) .2(18,000)
.4(12,000) 11,600
d2
Model B
3
1
d3
EMV .4(6,000) .2(16,000)
.4(21,000) 14,000
Model C
4
Choose the model with largest EV, Model C
22
Expected Value of Perfect Information

• Frequently, information is available that can
  improve the probability estimates for the states
  of nature.

• The expected value of perfect information (EVPI)
  is the increase in the expected profit that would
  result if one knew with certainty which state of
  nature would occur.

• The EVPI provides an upper bound on the expected
  value of any sample or survey information.

23
Expected Value of Perfect Information

• The expected value of perfect information is
  defined as

EVPI EVwPI EVwoPI
where
EVPI expected value of perfect information
EVwPI expected value with perfect
information about the states
of nature EVwoPI expected value without perfect
information about the states
of nature
24
Expected Value of Perfect Information

• EVPI Calculation

• Step 1
• Determine the optimal return corresponding
  to each state of nature.

• Step 2
• Compute the expected value of these optimal
  returns.

• Step 3
• Subtract the EV of the optimal decision
  from the amount determined in step (2).

25
Expected Value of Perfect Information

• Calculate the expected value for the optimum
  payoff for each state of nature and subtract
  the EV of the optimal decision.

• EVPI .4(10,000) .2(18,000) .4(21,000) -
  14,000 2,000

26
Decision Analysis With Sample Information

• Knowledge of sample (survey) information can be
  used to revise the probability estimates for
  the states of nature.

• Prior to obtaining this information, the
  probability estimates for the states of nature
  are called prior probabilities.

• With knowledge of conditional probabilities for
  the outcomes or indicators of the sample or
  survey information, these prior probabilities can
  be revised by employing Bayes' Theorem.

• The outcomes of this analysis are called
  posterior probabilities or branch probabilities
  for decision trees.

27
Decision Analysis With Sample Information

• Decision Strategy

• A decision strategy is a sequence of decisions
  and chance outcomes.

• The decisions chosen depend on the yet to be
  determined outcomes of chance events.

• The approach used to determine the optimal
  decision strategy is based on a backward pass
  through the decision tree.

28
Decision Analysis With Sample Information

• Backward Pass Through the Decision Tree

• At Chance Nodes
• Compute the expected value by multiplying the
  payoff at the end of each branch by the
  corresponding branch probability.

• At Decision Nodes
• Select the decision branch that leads to the
  best expected value. This expected value becomes
  the expected value at the decision node.

29
Decision Analysis With Sample Information

• Example Burger Prince

• Burger Prince must decide whether to
  purchase a
• marketing survey from Stanton Marketing for
  1,000.
• The results of the survey are "favorable" or
• "unfavorable". The conditional
• probabilities are

P(favorable 80 customers per hour) .2
P(favorable 100 customers per hour) .5
P(favorable 120 customers per hour) .9
30
Computing Branch ProbabilitiesUsing Bayes
Theorem

• Bayes Theorem can be used to compute branch
  probabilities for decision trees.

• For the computations we need to know

• the initial (prior) probabilities for the states
  of nature,

• the conditional probabilities for the outcomes or
  indicators of the sample information given each
  state of nature.

• A tabular approach is a convenient method for
  carrying out the computations.

31
Computing Branch ProbabilitiesUsing Bayes
Theorem

• Step 1

For each state of nature, multiply the prior
probability by its conditional probability for
the indicator. This gives the joint
probabilities for the states and indicator.

• Step 2

Sum these joint probabilities over all states.
This gives the marginal probability for the
indicator.

• Step 3

For each state, divide its joint probability by
the marginal probability for the indicator. This
gives the posterior probability distribution.
32
Posterior Probabilities
Favorable
State Prior Conditional Joint
Posterior
.08 .10 .36
.148 .185 .667
80 100 120
.4 .2 .4
.2 .5 .9
.08/.54
1.000
Total .54
P(favorable) .54
33
Posterior Probabilities
Unfavorable
State Prior Conditional Joint
Posterior
.32 .10 .04
.696 .217 .087
80 100 120
.4 .2 .4
.8 .5 .1
.32/.46
1.000
Total .46
P(unfavorable) .46
34
Decision Analysis With Sample Information

• Decision Tree (top half)

s1 (.148)
10,000
s2 (.185)
15,000
4
d1
s3 (.667)
14,000
s1 (.148)
8,000
d2
s2 (.185)
5
2
18,000
s3 (.667)
I1 (.54)
12,000
d3
s1 (.148)
6,000
s2 (.185)
6
16,000
s3 (.667)
1
21,000
35
Decision Analysis With Sample Information

• Decision Tree (bottom half)

s1 (.696)
10,000
1
I2 (.46)
s2 (.217)
7
15,000
d1
s3 (.087)
14,000
s1 (.696)
8,000
d2
s2 (.217)
8
3
18,000
s3 (.087)
12,000
d3
s1 (.696)
6,000
s2 (.217)
9
16,000
s3 (.087)
21,000
36
Decision Analysis With Sample Information
EMV .148(10,000) .185(15,000)
.667(14,000) 13,593
d1
4
17,855
d2
EMV .148 (8,000) .185(18,000)
.667(12,000) 12,518
5
2
I1 (.54)
d3
EMV .148(6,000) .185(16,000) .667(21,000)
17,855
6
1
EMV .696(10,000) .217(15,000)
.087(14,000) 11,433
7
d1
I2 (.46)
d2
EMV .696(8,000) .217(18,000)
.087(12,000) 10,554
8
3
d3
11,433
EMV .696(6,000) .217(16,000) .087(21,000)
9,475
9
37
Expected Value of Sample Information

• The expected value of sample information (EVSI)
  is the additional expected profit possible
  through knowledge of the sample or survey
  information.

EVSI EVwSI EVwoSI
where
EVSI expected value of sample information
EVwSI expected value with sample
information about the states
of nature EVwoSI expected value without sample
information about the states
of nature
38
Expected Value of Sample Information

• EVwSI Calculation

• Step 1
• Determine the optimal decision and its
  expected return for the possible outcomes of the
  sample using the posterior probabilities for the
  states of nature.

• Step 2
• Compute the expected value of these
  optimal returns.

39
Decision Analysis With Sample Information
d1
13,593
4
17,855
d2
12,518
5
2
I1 (.54)
d3
17,855
6
EVwSI .54(17,855) .46(11,433)
14,900.88
1
11,433
7
d1
I2 (.46)
d2
10,554
8
3
d3
11,433
9,475
9
40
Expected Value of Sample Information

• If the outcome of the survey is "favorable,
• choose Model C.

• If the outcome of the survey is unfavorable,
  choose Model A.

EVwSI .54(17,855) .46(11,433) 14,900.88
41
Expected Value of Sample Information

• EVSI Calculation

Subtract the EVwoSI (the value of the optimal
decision obtained without using the sample
information) from the EVwSI.
EVSI .54(17,855) .46(11,433) - 14,000
900.88

• Conclusion

Because the EVSI is less than the cost of the
survey, the survey should not be purchased.
42
End of Chapter 21