Decision Theory
Introduction
 What makes the difference between good and bad decisions?
 Good decisions may be defined as:
Based on logic,
Considered all possible decision alternatives,
Examined all available information about future, and
Applied decision modeling approach.
 Bad decisions may be defined as:
Not based on logic,
Did not use all available information,
Did not consider all alternatives, and
Did not employ appropriate decision modeling techniques.
Decision Analysis
A set of quantitative decision-making
techniques for decision situations where
uncertainty exists
Decision Theory Elements
 A set of possible future conditions exists that
will have a bearing on the results of the decision
 A list of alternatives (courses of action) for the
manager to choose from
 A known payoff for each alternative under each
possible future condition
Decision Making
States of nature
 Events that may occur in the future
 Decision maker is uncertain which state of
nature will occur
 Decision maker has no control over the states
of nature
Decision Theory
Decision Theory represents a general approach to
decision making which is suitable for a wide range
of operations management decisions, including:
capacity
planning
product and
service design
location
planning
equipment
selection
Five Steps of Decision Making
1. Clearly define the problem at hand.
2. List all possible decision alternatives.
3. Identify possible future events (states of nature)
4. Identify payoff (usually, profit or cost) for each
combination of alternatives and events.
5. Select one of the decision theory modeling
techniques, apply decision model, and make decision.
Thompson Lumber Company (1 of 2)
Step 1. Identifies problem as:
whether to expand product line by manufacturing and
marketing new product which is “backyard storage
sheds.”
Step 2. Generate decision alternatives available.
Decision alternative is defined as course of action or
strategy that may be chosen by the decision maker.
Alternatives are to construct:
(1) a large plant to manufacture storage sheds,
(2) a small plant to manufacture storage sheds, or
(3) build no plant at all.
Step 3. Identify possible future events
Thompson Lumber Company (2 of 2)
Step 4. Express payoff resulting from each possible
combination of alternatives and events.
Objective is to maximize profits.
Step 5. Select decision theory model and apply it to
data to help make decision.
Type of decision model available depends on the
operating environment and the amount of uncertainty
and risk involved.
Payoff Table
 A method of organizing & illustrating the payoffs from
different decisions given various states of nature
 A payoff is the outcome of the decision
Payoff Tables
Payoff Tables can be constructed when there is a finite set
of discrete decision alternatives.
In a Payoff Table The rows correspond to the possible decision
alternatives.
The columns correspond to the possible future events.
Events (States of Nature) are mutually exclusive
The body of the table contains the payoffs.
Payoff Table
States of Nature
Decision
1
2
a
Payoff 1a
Payoff 2a
b
Payoff 1b
Payoff 2b
Payoff Table:Thompson Lumber Company
Types Of Decision Making
Environments
Type 1: Decision Making under Certainty
Type 2: Decision Making under Uncertainty
Type 3: Decision Making under Risk.
Types Of Decision Making Environments
Type 1: Decision Making under Certainty. Decision
maker knows with certainty the consequence of every
decision alternative. (The future state of nature is
assumed to be known.)
Type 2: Decision Making under Uncertainty. Decision
maker has no information about various outcomes.
(There is no knowledge about the probability of the states
of nature occurring)
Types Of Decision Making Environments
Type 3: Decision Making under Risk. Decision maker
has some knowledge regarding the probability of
occurrence of each event or state of nature. (There is
some knowledge about the probability of the states of
nature occurring)
Decision Making Under Uncertainty
Decision Making Under UncertaintySteps of :
• Construct a Payoff Table
• Select a Decision Making Criterion
• Apply the Criterion to the Payoff Table
• Identify the Optimal Solution
Decision Making Under Uncertainty
The decison criteria are based on the decision
maker’s attitude toward life
These include an individual being pessimistic or
optimistic, conservative or aggressive
Decision Making Criteria Under Uncertainty
 Criteria for
uncertainty.
making
decisions
under
 Maximax.
 Maximin
 Equally likely.
 Criterion of realism.
 Minimax regret.
 First four criteria calculated directly from
decision payoff table.
 Fifth minimax regret criterion requires use
of opportunity loss table.
Maximax Criterion (1 of 2)
 Maximax criterion selects the alternative that
maximizes maximum payoff over all alternatives.
 Is based on the best possible scenario.
 First locate maximum payoff for each alternative.
 Select alternative with maximum value.
 Decision criterion locates alternative with highest
possible gain.
 Called optimistic criterion.
 Table shows maximax choice is first alternative:
"construct large plant."
Maximax Criterion (2 of 2)
Example 1: Thompson Lumber Company
 Maximax criterion selects alternative that maximizes
maximum payoff over all alternatives.
 First alternative, "construct a large plant”, $200,000 payoff is
maximum of maximum payoffs for each decision alternative.
Maximin Criterion (1 of 2)
 Maximin criterion finds the alternative that maximizes
minimum payoff over all alternatives.
 Is based on the worst-case scenario.
 First locate minimum payoff for each decision alternative
across all states of nature.
 Select the alternative with the maximum value.
 Decision criterion locates the alternative that has the least
possible loss.
 Called pessimistic criterion.
 Maximin choice, "do nothing," is shown in table.
 $0 payoff is maximum of minimum payoffs for each
alternative.
Maximin Criterion (2 of 2)
Example 1. Thompson Lumber Company
First locate minimum payoff for each alternative, and select the
alternative with maximum number.
Equally Likely (Laplace) Criterion (1 of 2)
 Equally likely, also called Laplace, criterion finds decision
alternative with highest average payoff (here the
probabilities of each state of nature is assumed to be equal)
 Calculate average payoff for every alternative.
 Pick the alternative with maximum average payoff.
 Assumes all probabilities of occurrence for states of nature
are equal.
 Equally likely choice is the second alternative, "construct a
small plant."
 Strategy shown in table has maximum average payoff
($40,000) over all alternatives.
Equally Likely (Laplace) Criterion (2 of 2)
Example 1. Thompson Lumber Company
 Equally likely criterion finds decision alternative with highest
average payoff.
 Calculate average payoff for every alternative.
 Pick alternative with maximum average payoff.
Criterion of Realism (Hurwicz)(1 of 3)
 Often called weighted average, the criterion of
realism (or Hurwicz) decision criterion is a
compromise between optimistic and pessimistic
decision.
 Select coefficient of realism, a, with value between 0
and 1.
– When a is close to 1, decision maker is optimistic
about future.
– When a is close to 0, decision maker is pessimistic
about future.
Criterion of Realism (2 of 3)
Formula for criterion of realism =
a (maximum payoff for alternative) +
(1-a) (minimum payoff for alternative)
 Assume coefficient of realism a = 0.80.
 Best decision would be to construct a large plant.
 This alternative has highest weighted average payoff:
$124,000
Criterion of Realism (3 of 3)
Example 1. Thompson Lumber Company
Coefficient of realism a = 0.80.
$124,000 = (0.80)($200,000) + (0.20)(- $180,000).
Minimax Regret Criterion (1 of 5)
 Final decision criterion is based on opportunity loss.
 Develop opportunity loss (regret) table.
 Determine opportunity loss of not choosing the best
alternative for each state of nature (or the regret by failing
to choose the “best” decision)
Minimax Regret Criterion (2 of 5)
 Opportunity loss, also called regret for any state of
nature, or any column is calculated by subtracting
each outcome in column from best outcome in the
same column.
 The alternative with the minimum of the maximum
regrets for each alternative is selected.
Minimax Regret Criterion (3 of 5)
Example 1. Thompson Lumber Company
• Best outcome for favorable market is $200,000 as
result of first alternative, "construct a large plant."
• Subtract all payoffs in column from $200,000.
• Best outcome for unfavorable market is $0 that is the
result of third alternative, "do nothing."
• Subtract all payoffs in column from $0.
• Table illustrates computations and shows complete
opportunity loss table.
Minimax Regret Criterion (4 of 5)
Example 1. Thompson Lumber Company
• Table illustrates computations and shows complete opportunity
loss table.
Minimax Regret Criterion (5 of 5)
Example 1. Thompson Lumber Company
• Once the opportunity loss table has been constructed, locate
the maximum opportunity loss within each alternative.
• Pick the alternative with minimum value
• Minimax regret choice is second alternative, "construct a
small plant." Regret of $100,000 is minimum of maximum
regrets over all alternatives.
Tom Brown Investment Example for
Decision Making Under Uncertainty
Example 2. Tom Brown Investment Decision
Tom Brown has inherited $1000.
He has decided to invest the money for one year.
A broker has suggested five potential investments.
1. Gold.
2. Junk Bond.
3. Growth Stock.
4. Certificate of Deposit.
5. Stock Option Hedge.
Tom has to decide how much to invest in each investment.
The Payoff Table
Example 2. Tom Brown Investment Decision
States
Statesof
ofNature
Nature
Decision
Large
DecisionAlternatives
Alternatives
Large Rise
RiseSmall
SmallRise
RiseNo
NoChange
Change Small
SmallFall
Fall Large
LargeFall
Fall
Gold
-100
100
200
300
00
Gold
-100
100
200
300
Bond
250
200
150
-100
-150
Bond
250
200
150
-100
-150
Stock
500
250
100
-200
-600
Stock
500
250
100
-200
-600
C/D
60
60
60
60
60
C/DAccount
Account
60
60
60
60
60
Stock
2200
150
150
-200
-150
StockOption
OptionHedge
Hedge
00
150
150
-200
-150
The Stock Option Alternative is dominated by the
Bond Alternative
Maximax Criterion
Example 2. Tom Brown Investment Decision
The Maximax criterion
Maximum
Decision Large rise Small rise No changeSmall fall Large fall Payoff
-100
100
200
300
0
300
Gold
250
200
150
-100
-150
200
Bond
500
250
100
-200
-600
500
Stock
60
60
60
60
60
60
C/D
The Maximin Criterion
Example 2. Tom Brown Investment Decision
Decisions
Decisions
Gold
Gold
Bond
Bond
Stock
Stock
C/D
C/Daccount
account
The
TheMaximin
MaximinCriterion
Criterion
LargeRrise
LargeRrise Small
SmallRise
Rise No
Nochange
change Small
SmallFall
Fall Large
LargeFall
Fall
-100
-100
250
250
500
500
60
60
100
100
200
200
250
250
60
60
200
200
150
150
100
100
60
60
300
300
-100
-100
-200
-200
60
60
00
-150
-150
-600
-600
60
60
Minimum
Minimum
Payoff
Payoff
-100
-100
-150
-150
-600
-600
60
60
Minimax Regret Criterion
Example 2. Tom Brown Investment Decision
Regret Table
Decision Large rise Small rise No changeSmall fall Large fall
Gold
600
150
0
0
60
Bond
250
50
50
400
210
Stock
0
0
100
500
660
C/D
440
190
140
240
0
regret
600
400
660
440
Decision Making Under Risk
Decision Making Under Risk
 Common for decision maker to have some idea about the
probabilities of occurrence of different outcomes or
states of nature.
 Probabilities may be based on decision maker’s personal
opinions about future events, or on data obtained from
market surveys, expert opinions, etc.
 When probability of occurrence of each state of nature
can be assessed, problem environment is called decision
making under risk.
Expected Monetary Value (1 of 3)
• Given decision table with conditional values (payoffs)
and probability assessments, determine the expected
monetary value (EMV) for each alternative.
• Computed as weighted average of all possible payoffs
for each alternative, where weights are probabilities
of different states of nature:
EMV (alternative i) =
(payoff of first state of nature) x (probability of first
state of nature) +
(payoff of second state of nature) x (probability of second
state of nature) + . . . +
(payoff of last state of nature) x (probability of last
state of nature)
Expected Monetary Value (2 of 3)
Example 1. Thompson Lumber Company
• Probability of favorable market is the same as probability of
unfavorable market.
• Each state of nature has a 0.50 probability of occurrence.
Expected Monetary Value (3 of 3)
Example 2. Tom Brown Problem
The Expected Value Criterion
Expected
Decision Large rise Small rise No changeSmall fall Large fall Value
-100
100
200
300
0
100
Gold
250
200
150
-100
-150
130
Bond
500
250
100
-200
-600
125
Stock
60
60
60
60
60
60
C/D
0.2
0.3
0.3
0.1
0.1
Prior Probability
(0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130
Expected Value of Perfect Information
Expected value of perfect information: the difference
between the expected payoff under certainty and the
expected payoff under risk
Expected value of
Expected payoff
perfect information = under certainty
-
Expected payoff
under risk
Expected Value of Perfect Information
• Expected value with perfect information is expected or average
return, if one has perfect information before decision has to be
made.
• Choose best alternative for each state of nature and multiply its
payoff times probability of occurrence of that state of nature:
Expected value with perfect information (EV with PI) =
(best payoff for first state of nature)x
(probability of first state of nature)
+ (best payoff for second state of nature)
x (probability of second state of nature)
+ . . . + (best payoff for last state of nature)
x (probability of last state of nature)
EVPI = EV with PI - maximum EMV
It is also the smallest expected regret of any decision
alternative.
EV with PI and EVPI
• Best outcome for state of nature "favorable market" is
"build a large plant" with a payoff of $200,000.
• Best outcome for state of nature "unfavorable market" is "do
nothing," with payoff of $0.
• Expected value with perfect information
(EV with PI) = ($200,000)(0.50) + ($0)(0.50)
= $ 100,000.
• If one had perfect information, an average payoff of
$100,000 if decision could be repeated many times.
• Maximum EMV or expected value without perfect
information, is $40,000.
• EVPI = EV with PI - maximum EMV
= $100,000 - $40,000 = $60,000.
Decision Trees
 Any problem presented in decision table can be
graphically illustrated in decision tree.
 A graphical method for analyzing decision situations
that require a sequence of decisions over time
(decision situations that cannot be handled by
decision tables)
 Decision tree consists of
Square nodes - indicating decision points
Circles nodes - indicating states of nature
Arcs - connecting nodes
These nodes are represented using following symbols:
=
A decision node
Arcs (lines) originating from decision node denote all
decision alternatives available at that node.
О = A state of nature (or chance) node.
Arcs (lines) originating from a chance node denote all
states of nature that could occur at that node.
Decision Tree
State 1
1
State 2
State 1
2
Decision
Node
State 2
Outcome 1
Outcome 2
Outcome 3
Outcome 4
State of Nature Node
Decision Tree
Payoff 1
Decision Point
Chance Event
Payoff 2
2
Payoff 3
1
B
Payoff 4
2
Payoff 5
Payoff 6
Decision Tree
Thompson Lumber Company
• Tree usually begins with decision node.
• Decision is to determine whether to construct large plant,
small plant, or no plant.
• Once the decision is made, one of two possible states of nature
(favorable or unfavorable market) will occur.
Folding Back a Decision Tree
Thompson Lumber Company
• In folding back decision tree, use the following two rules:
– At each state of nature (or chance) node, compute expected
value using probabilities of all possible outcomes at that
node and payoffs associated with outcomes.
– At each decision node, select alternative that yields better
expected value or payoff.
Reduced Decision Tree
Thompson Lumber Company
• Using the rule for decision nodes, select alternative with highest
EMV.
• Corresponds to alternative to build small plant.
• Resulting EMV is $40,000.
Decision Trees for Multi-stage
Decision Making Problems
Decision Tree With EMVs Shown
Thompson Lumber Company
Decision Making in Southern Textile
Southern Textile Company
STATES OF NATURE
DECISION
Expand
Maintain status quo
Sell now
Good Foreign
Poor Foreign
Competitive Conditions
Competitive Conditions
$ 800,000
1,300,000
320,000
$ 500,000
-150,000
320,000
Southern Textile Company
STATES OF NATURE
DECISION
Good Foreign
Poor Foreign
Competitive Conditions
Competitive Conditions
Expand
$ 800,000
Maintain status quo
1,300,000
Maximax Solution
Sell now
320,000
Expand:
Status quo:
Sell:
$ 500,000
-150,000
320,000
$800,000
1,300,000  Maximum
320,000
Decision: Maintain status quo
Southern Textile Company
STATES OF NATURE
DECISION
Good Foreign
Poor Foreign
Competitive Conditions
Competitive Conditions
Expand
$ 800,000
Maintain status quo
1,300,000
Maximin Solution
Sell now
320,000
Expand:
Status quo:
Sell:
$ 500,000
-150,000
320,000
$500,000  Maximum
-150,000
320,000
Decision: Expand
Southern Textile Company
STATES OF NATURE
Minimax Regret Solution
Good Foreign
Poor Foreign
GOOD CONDITIONS
POOR CONDITIONS
DECISION
Competitive Conditions
Competitive Conditions
$1,300,000 - 800,000 = 500,000
$500,000 - 500,000 = 0
1,300,000 - 1,300,000 = 0 $ 800,000
500,000 - (-150,000)$ =500,000
650,000
Expand
1,300,000
320,000 = 980,0001,300,000
500,000 - 320,000 -150,000
= 180,000
Maintain
status- quo
Sell now
Expand:
Status quo:
Sell:
320,000
320,000
$500,000  Minimum
650,000
980,000
Decision: Expand
Southern Textile Company
STATES OF NATURE
DECISION
Good Foreign
Poor Foreign
Competitive Conditions
Competitive Conditions
Hurwicz Criteria
Expand
Maintain status
quo
a = 0.3
Sell now
$ 800,000
1,300,000
1320,000
a=
$ 500,000
0.7 -150,000
320,000
Expand:
$800,000(0.3) + 500,000(0.7) = $590,000  Maximum
Status quo: 1,300,000(0.3) -150,000(0.7) = 285,000
Sell:
320,000(0.3) + 320,000(0.7) = 320,000
Decision: Expand
Southern Textile Company
STATES OF NATURE
Foreign
Equal Likelihood Good
Criteria
DECISION
Competitive Conditions
Poor Foreign
Competitive Conditions
Two states of nature each weighted 0.50
Expand
$ 800,000
$ 500,000
Maintain
status quo $800,000(0.5)1,300,000
-150,000
Expand:
+ 500,000(0.5) = $650,000
 Maximum
Sell now
320,000
320,000
Status quo:
Sell:
1,300,000(0.5) -150,000(0.5) = 575,000
320,000(0.5) + 320,000(0.5) = 320,000
Decision: Expand
Southern Textile Company
STATES OF NATURE
DECISION
Good Foreign
Poor Foreign
Competitive Conditions
Competitive Conditions
Expected Value
Expand
$ 800,000
$ 500,000
Maintain status
quo = 0.70 1,300,000
p(good)
p(poor) = 0.30 -150,000
Sell now
320,000
320,000
EV(expand)
EV(status quo)
EV(sell)
$800,000(0.7) + 500,000(0.3) = $710,000
1,300,000(0.7) -150,000(0.3) = 865,000
320,000(0.7) + 320,000(0.3) = 320,000
Decision: Status quo
 Maximum
EVPI Example
Good conditions will exist 70% of the time, choose
maintain status quo with payoff of $1,300,000
Poor conditions will exist 30% of the time, choose
expand with payoff of $500,000
Expected value given perfect information
= $1,300,000 (0.70) + 500,000 (0.30)
= $1,060,000
EVPI = $1,060,000 - 865,000 = $195,000