Docstoc

Decision Theory

Document Sample
Decision Theory Powered By Docstoc
					                 Chapter 14
               Decision Analysis

Problem Formulation
Decision Making without Probabilities
Decision Making with Probabilities
Risk Analysis and Sensitivity Analysis
Decision Analysis with Sample Information
Computing Branch Probabilities
Utility and Decision Making




                       Dr. C. Lightner           1
                 Fayetteville State University
                  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 will ultimately affect decision
results. States of nature should be defined so that they are
mutually exclusive and contain all possible future events that
could affect the results of all potential decisions.




                          Dr. C. Lightner                             2
                    Fayetteville State University
               Decision Theory Models

Decision theory problems are generally represented as one of the
following:
 – Influence Diagram
 – Payoff Table
 – Decision Tree




                         Dr. C. Lightner                       3
                   Fayetteville State University
                  Influence Diagrams

An influence diagram is a graphical device showing the
relationships among the decisions, the chance events, and the
consequences.
Squares or rectangles depict decision nodes.
Circles or ovals depict chance nodes.
Diamonds depict consequence nodes.
Lines or arcs connecting the nodes show the direction of influence.




                         Dr. C. Lightner                         4
                   Fayetteville State University
                       Payoff Tables

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.




                          Dr. C. Lightner                         5
                    Fayetteville State University
                       Decision Trees

A decision tree is a chronological representation of the decision
problem.
Each decision tree has two types of nodes; round nodes
correspond to the states of nature while square nodes
correspond to the decision alternatives.
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.



                           Dr. C. Lightner                            6
                     Fayetteville State University
     Example: CAL Condominium Complex

A developer must decide how large a luxury condominium
complex to build – small, medium, or large. The
profitability of this complex depends upon the future level
of demand for the complex’s condominiums.




                       Dr. C. Lightner                   7
                 Fayetteville State University
    CAL Condos: Elements of Decision Theory

        States of nature: The states of nature could be defined as
        low demand and high demand.
        Alternatives: CAL could decide to build a small, medium,
        or large condominium complex.
        Payoffs: The profit for each alternative under each potential
        state of nature is going to be determined.

We develop different models for this problem on the following slides.




                             Dr. C. Lightner                        8
                       Fayetteville State University
       CAL Condos: Payoff Table


       THIS IS A PROFIT PAYOFF TABLE

                     States of Nature
Alternatives            Low        High
Small                      8         8
Medium                     5       15
Large                     -11      22
  (payoffs in millions of dollars)


                 Dr. C. Lightner           9
           Fayetteville State University
             CAL Condos: Decision Tree
                                           8



                                           8


                                       5

Medium Complex
                                        15



                                           -
                                           11

                                               22

                       Dr. C. Lightner              10
                 Fayetteville State University
       Decision Making without Probabilities

Three commonly used criteria for decision making when
probability information regarding the likelihood of the states of
nature is unavailable are:
 – the optimistic approach
 – the conservative approach
 – the minimax regret approach.




                          Dr. C. Lightner                           11
                    Fayetteville State University
                    Optimistic Approach
The optimistic approach would be used by an optimistic decision
maker.
The decision with the best possible payoff is chosen.
If the payoff table was in terms of costs, the decision with the
lowest cost would be chosen.
If the payoff table was in terms of profits, the decision with the
highest profit would be chosen.




                            Dr. C. Lightner                          12
                      Fayetteville State University
                   Conservative Approach

The conservative approach would be used by a conservative decision maker.
For each decision the worst payoff is listed and then the decision
corresponding to the best of these worst payoffs is selected. (Hence, the
worst possible payoff is maximized.)
If the payoff was in terms of costs, the maximum costs would be determined
for each decision and then the decision corresponding to the minimum of
these maximum costs is selected. (Hence, the maximum possible cost is
minimized.)
If the payoff was in terms of profits, the minimum profits would be determined
for each decision and then the decision corresponding to the maximum of
these minimum profits is selected. (Hence, the minimum possible profit is
maximized.)



                              Dr. C. Lightner                                    13
                        Fayetteville State University
                 Minimax Regret Approach

1. The minimax regret approach requires the construction of a
   regret table or an opportunity loss table. This is done by
   calculating for each state of nature the difference between each
   payoff and the best payoff for that state of nature.
2. Then, using this regret table, the maximum regret for each
   possible decision is listed.
3. The decision chosen is the one corresponding to the minimum of
   the maximum regrets.




                            Dr. C. Lightner                       14
                      Fayetteville State University
        Solving CAL Condominiums Problem

Suppose that information regarding the probability (or likelihood) that there will be
a high or low demand is unavailable.
 – A conservative or pessimistic decision maker would select the
     decision alternative determined by the conservative approach.
 – An optimistic decision maker would select the decision
     alternative rendered by the optimistic approach.
 – The minimax regret approach is generally selected by a
     decision maker who reflects on decisions “after the fact”, and
     complains about or “regrets” their decisions based upon the
     profits that they could have made (or cheaper costs that they
     could have spent) had a different decision been selected.




                               Dr. C. Lightner                                  15
                         Fayetteville State University
                CAL Condos: Optimistic Decision

       If the optimistic approach is selected:
                              STATES OF NATURE            BEST
       Alternatives           Low              High       PROFIT
       Small                  8                8           8
       Medium                 5                15         15       Maximax
                                                                    payoff
       Large                 -11               22          22
Maximax
decision




                                Dr. C. Lightner                        16
                          Fayetteville State University
                    CAL Condos: Conservative Decision

             If the conservative approach is selected:
                                   STATES OF NATURE             WORST
  Maximi        Alternatives       Low              High        PROFIT
     n                                                                   Maximin
  decision      Small              8                8           8         payoff

                Medium             5                15          5
                Large             -11               22          -11


The decision with the best profit from the column of worst profits is selected.



                                      Dr. C. Lightner                         17
                                Fayetteville State University
      CAL Condos: Minimax Regret Decision

If the minimax regret approach is selected:
Step 1: Determine the best payoff for each state of nature and create a regret
table.
                         STATES OF NATURE
  Alternatives           Low         High
  Small                  8           8
  Medium                 5           15
  Large                 -11          22

                     Best Profit             Best Profit
                     for Low                 for High
                         8                       22


                              Dr. C. Lightner                                    18
                        Fayetteville State University
               CAL Condos: Minimax Regret Decision

        If the minimax regret approach is selected:
        Step 1: Create a regret table (continued).
                                  STATES OF NATURE              For a profit payoff
                                                                table, entries in
          Alternatives            Low         High              the regret table
          Small                   0           14                represent profits
                                                                that could have
          Medium                  3           7                 been earned.
          Large                  19           0

If they knew in advanced that the demand would be low, they would have built a
small complex. Without this “psychic insight”, if they decided to build a medium
facility and the demand turned out to be low, they would regret building a medium
complex because they only made 5 million dollars instead of 8 million had they built
a small facility instead. They regret their decision by 3 million dollars.
                                      Dr. C. Lightner                            19
                                Fayetteville State University
      CAL Condos: Minimax Regret Decision

If the minimax regret approach is selected:
Step 2: Create a regret table (continued).
Step 3: Determine the maximum regret for each decision.
                        STATES OF NATURE                      Max
  Alternatives          Low         High                      Regret
  Small                 0           14                        14
  Medium                3           7                         7
  Large                19           0                         19

                                                       Regret not getting a profit
                                                       of 19 more than not making
                                                       a profit of 0.

                             Dr. C. Lightner                                   20
                       Fayetteville State University
                 CAL Condos: Minimax Regret Decision

           If the minimax regret approach is selected:
           Step 4: Select the decision with the minimum value from the column of max
           regrets.
                                   STATES OF NATURE               Max
             Alternatives          Low         High               Regret
             Small                 0           14                 14 Minima
Minimax      Medium                3           7                  7       x
                                                                        Regret
 Regret
decision
             Large                19           0                  19    payoff




                                        Dr. C. Lightner                                21
                                  Fayetteville State University
                       Generic Example

       Consider the following problem with three decision alternatives
  and three states of nature with the following payoff table
  representing costs:
               States of Nature
                 s1      s2    s3
                                          COST PAYOFF TABLE
          d1     4.5       3        2
Decisions d2     0.5       4        1
          d3      1        5        3




                             Dr. C. Lightner                      22
                       Fayetteville State University
     Generic Example : Optimistic Decision

Optimistic Approach
     An optimistic decision maker would use the optimistic
(maximax) approach. We choose the decision that has the best
single value in the payoff table.

                                          Best
                       Decision           Cost    Maximax
  Maximax               d1                 2       payoff
  decision              d2                 0.5
                        d3                 1



                        Dr. C. Lightner                        23
                  Fayetteville State University
   Generic Example: Conservative Approach

Conservative Approach
    A conservative decision maker would use the conservative
(maximin) approach. List the worst payoff for each decision.
Choose the decision with the best of these worst payoffs.

                                         Worst
                      Decision           Payoff   Maximin
     Maximin           d1                4.5       payoff
     decision
                       d2                4
                       d3                5


                        Dr. C. Lightner                        24
                  Fayetteville State University
      Generic Example: Minimax Regret Decision
  Minimax Regret Approach

               States of Nature
                                                      For a cost payoff
               s1      s2    s3                       table, entries in
                                                      the regret table
          d1   4.5    3        2                      represent
                                                      overpayments
Decisions d2   0.5    4        1                      (i.e. higher costs
                                                      incurred).
          d3    1     5        3

     Best cost for each state of nature.


                            Dr. C. Lightner                                25
                      Fayetteville State University
                                 Example

     Minimax Regret Approach (continued)
          For each decision list the maximum regret. Choose the
     decision with the minimum of these values.

                   States of Nature             Max
                   s1     s2     s3             Regret

             d1     4      0        1             4
   Decisions d2    0       1        0             1
             d3    0.5     2        2              2     Minimax
                                                          regret
Minimax
decision
                               Dr. C. Lightner                     26
                         Fayetteville State University
         Decision Making with Probabilities

Expected Value Approach
 – If probabilistic information regarding the states of nature is
   available, one may use the expected value (EV) approach.
 – Here the expected return for each decision is calculated by
   summing the products of the payoff under each state of
   nature and the probability of the respective state of nature
   occurring.
 – The decision yielding the best expected return is chosen.




                          Dr. C. Lightner                           27
                    Fayetteville State University
         Expected Value of a Decision Alternative

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:
                                       N
                       EV( d i )   P( s j )Vij
                                      j 1



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


                               Dr. C. Lightner                        28
                         Fayetteville State University
                Example: Burger Prince

      Burger Prince Restaurant is contemplating 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 (profits) for the three models is on the next
slide.




                           Dr. C. Lightner                        29
                     Fayetteville State University
               Example: Burger Prince

Payoff Table

           Average Number of Customers Per Hour
                s1 = 80 s2 = 100 s3 = 120

    Model A       $10,000 $15,000                 $14,000
    Model B       $ 8,000 $18,000                $12,000
    Model C       $ 6,000 $16,000                $21,000




                       Dr. C. Lightner                      30
                 Fayetteville State University
                Example: Burger Prince

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, C, and s1, s2, s3
represent the states of nature of 80, 100, and 120.




                          Dr. C. Lightner                         31
                    Fayetteville State University
                Example: Burger Prince
                                                            Payoffs
Decision Tree                                          .4
                                                  s1        10,000
                                                  s2   .2
                                  2               s3        15,000
                                                       .4
         d1                                                 14,000
                                                  s1   .4
         d2                                                  8,000
1                                                 s2   .2
                                  3                         18,000
         d3                                       s3   .4
                                                            12,000
                                                  s1   .4
                                                             6,000
                                                  s2   .2
                                  4                         16,000
                                                  s3
                                                       .4
                                                            21,000
                        Dr. C. Lightner                          32
                  Fayetteville State University
             Example: Burger Prince

Expected Value For Each Decision
                                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)
       Model B d2                   = $11,600
1                              3

                  d3            EMV = .4(6,000) + .2(16,000) + .4(21,000)
     Model C
                                    = $14,000
                               4


    Choose the model with largest EV, Model C.
                             Dr. C. Lightner                          33
                       Fayetteville State University
                CAL Condos Revisited

Suppose market research was conducted in the community where
the complex will be built. This research allowed the company to
estimate that the probability of low demand will be 0.35, and the
probability of high demand will be 0.65. Which decision alternative
should they select.




                         Dr. C. Lightner                        34
                   Fayetteville State University
               CAL Condos Revisited


                  STATES OF NATURE
Alternatives      Low (0.35)  High (0.65)
Small             8            8
Medium            5           15
Large            -11           22




                       Dr. C. Lightner           35
                 Fayetteville State University
                           CAL Condos Revisited


                 STATES OF NATURE
    Alternatives Low       High
                  (0.35)   (0.65)                            Expected value (EV)
    Small         8        8                                8(0.35) + 8(0.65) = 8
    Medium        5       15                                5(0.35) + 15(0.65) = 11.5
    Large        -11       22                              -11(0.35) + 22(0.65) = 10.45


Recall that this is a profit payoff table. Thus since the decision to build a medium
complex has the highest expected profit, this is our best decision.


                                     Dr. C. Lightner                               36
                               Fayetteville State University
      Expected Value of Perfect Information

Frequently information is available which 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.




                          Dr. C. Lightner                         37
                    Fayetteville State University
       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).



                            Dr. C. Lightner                          38
                      Fayetteville State University
                 Example: Burger Prince

  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




                           Dr. C. Lightner                       39
                     Fayetteville State University
                   Sensitivity Analysis

Some of the quantities in a decision analysis, particularly the
probabilities, are often intelligent guesses at best.
It is important to accompany any decision analysis with a sensitivity
analysis.
Sensitivity analysis can be used to determine how changes to the
following inputs affect the recommended decision alternative:
  – probabilities for the states of nature
  – values of the payoffs
If a small change in the value of one of the inputs causes a change
in the recommended decision alternative, extra effort and care
should be taken in estimating the input value.


                          Dr. C. Lightner                        40
                    Fayetteville State University
                    Sensitivity Analysis

One approach to sensitivity analysis is to arbitrarily assign different
values to the probabilities of the states of nature and/or the payoffs
and resolve the problem. If the recommended decision changes,
then you know that the solution is sensitive to the changes.
For the special case of two states of nature, a graphical technique
can be used to determine how sensitive the solution is to the
probabilities associated with the states of nature.




                           Dr. C. Lightner                          41
                     Fayetteville State University
         CAL Condos: Sensitivity Analysis

This problem has two states of nature. Previously, we stated that
CAL Condominiums estimated that the probability of future low
demand is 0.35 and 0.65 is the probability of high demand. These
probabilities yielded the recommended decision to build the medium
complex.
In order to see how sensitive this recommendation is to changing
probability values, we will let p equal the probability of low demand.
Thus (1-p) is the probability of high demand. Therefore
EV( small) = 8*p + 8*(1-p)= 8
EV( medium) = 5*p + 15*(1-p) = 15 – 10p
EV( large) = -11*p + 22*(1-p) = 22 – 33p

                          Dr. C. Lightner                         42
                    Fayetteville State University
         CAL Condos: Sensitivity Analysis

Next we will plot the expected value lines for each decision by
plotting p on the x axis and EV on the y axis.
EV( small) = 8
EV( medium) = 15 – 10p
EV( large) = 22 – 33p




                          Dr. C. Lightner                         43
                    Fayetteville State University
 CAL Condos: Sensitivity Analysis


25



20


EV( small)
15



10
                   Dr. C. Lightner           44
             Fayetteville State University
         CAL Condos: Sensitivity Analysis

Since CAL condominiums list payoffs are in terms of profits, we
know that the highest profits is desirable.
Look over the entire range of p (p=0 to p=1) and determine the
range over which each decision yields the highest profits.




                         Dr. C. Lightner                          45
                   Fayetteville State University
 CAL Condos: Sensitivity Analysis


25



20


                                             EV( small)
15



10
         B1                             B2
              Dr. C. Lightner                             46
        Fayetteville State University
            CAL Condos: Sensitivity Analysis

 Do not estimate the values of B1 or B2 (the points where the intersection of lines
 occur). Determine the exact intersection points.
 B1 is the point where the EV( large) line intersects with the EV( medium) line:
 To find this point set these two lines equal to each other and solve for p.
22-33p= 15-10p
 7= 23p
 p=7/23= 0.3403 So B1 equals 0.3403

B2 is the point where the EV( medium) line intersects with the EV( small) line:
15-10p = 8
7 = 10p
p = 0.7         So B2 equals 0.7



                               Dr. C. Lightner                                    47
                         Fayetteville State University
 CAL Condos: Sensitivity Analysis


25



20


                                              EV( small)
15



10
         0.3403                         0.7
              Dr. C. Lightner                              48
        Fayetteville State University
         CAL Condos: Sensitivity Analysis

From the graph we see that if the probability of low demand (p) is
between 0 and 0.3403, we recommend building a large complex.
From the graph we see that if the probability of low demand (p) is
between 0.3403 and 0.7, we recommend building a medium
complex.
From the graph we see that if the probability of low demand (p) is
between 0.7 and 1, we recommend building a large complex.

From this sensitivity analysis we see that if CAL Condos estimate of
0.35 for the probability of low demand was slightly lower, the
recommended decision would change.

                          Dr. C. Lightner                        49
                    Fayetteville State University
              End of Chapter 14




       See your textbook for more
 examples and detailed explanations
of all topics discussed in these notes.



                    Dr. C. Lightner           50
              Fayetteville State University

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:4
posted:1/3/2013
language:English
pages:50