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Decision TheoryDecision Theory
IntroductionIntroduction 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, andApplied 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 AnalysisDecision Analysis
AA set of quantitative decision-making set of quantitative decision-making techniques for decision situations where techniques for decision situations where uncertainty existsuncertainty 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 MakingDecision Making
States of natureStates of nature Events that may occur in the futureEvents that may occur in the future Decision maker is uncertain which state of Decision maker is uncertain which state of
nature will occurnature will occur Decision maker has no control over the states Decision maker has no control over the states
of natureof nature
Decision Theory represents a general approach to decision making which is suitable for a wide range of operations management decisions, including:
product andservice design
product andservice design
equipment selection
location planning
Decision TheoryDecision Theory
capacityplanning
Five Steps of Decision MakingFive 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 CThompson Lumber Company (1 of 2)ompany (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 CompanyThompson Lumber Company (2 of 2)(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 TablePayoff 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 TablesPayoff 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 TablePayoff Table
States States oof Naturef Nature
DecisionDecision aa bb
11 Payoff 1aPayoff 1a Payoff 1bPayoff 1b
22 Payoff 2aPayoff 2a Payoff 2bPayoff 2b
Payoff Table:Payoff Table:Thompson Lumber CompanyThompson Lumber Company
Types Of Decision Making Types Of Decision Making EnvironmentsEnvironments
Type 1: Decision Making under Certainty
Type 2: Decision Making under Uncertainty
Type 3: Decision Making under Risk.
Types Of Decision Making EnvironmentsTypes 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 EnvironmentsTypes 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 UncertaintyDecision Making Under Uncertainty
Decision Making Under Uncertainty- Decision Making Under Uncertainty- Steps ofSteps of : :
• Construct a Payoff Table• Select a Decision Making Criterion• Apply the Criterion to the Payoff Table
• Identify the Optimal Solution
Decision Making Under UncertaintyDecision 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 Decision Making Criteria Criteria Under UncertaintyUnder Uncertainty
Criteria for making decisions under uncertainty.
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 CriterionMaximax Criterion (1 of 2) (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 CriterionMaximax Criterion (2 of 2) (2 of 2)
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.
Example 1: Thompson Lumber Company
Maximin CriterionMaximin Criterion (1 of 2) (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 CriterionMaximin Criterion (2 of 2) (2 of 2)
First locate minimum payoff for each alternative, and select the alternative with maximum number.
Example 1. Thompson Lumber Company
Equally Likely (Laplace) CriterionEqually Likely (Laplace) Criterion (1 of 2) (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) CriterionEqually Likely (Laplace) Criterion (2 of 2) (2 of 2)
Equally likely criterion finds decision alternative with highest average payoff.
Calculate average payoff for every alternative. Pick alternative with maximum average payoff.
Example 1. Thompson Lumber Company
Criterion of Realism (Hurwicz)Criterion of Realism (Hurwicz)(1 of 3) (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, , with value between 0 and 1.
– When is close to 1, decision maker is optimistic about future.
– When is close to 0, decision maker is pessimistic about future.
Criterion of RealismCriterion of Realism (2 of 3) (2 of 3)
Formula for criterion of realism =
(maximum payoff for alternative) +
(1-) (minimum payoff for alternative)
Assume coefficient of realism 0.80.
Best decision would be to construct a large plant.
This alternative has highest weighted average payoff:
$124,000
Criterion of RealismCriterion of Realism (3 of 3) (3 of 3)
Coefficient of realism 0.80.
$124,000 = (0.80)($200,000) + (0.20)(- $180,000).
Example 1. Thompson Lumber Company
Minimax Regret CriterionMinimax Regret Criterion (1 of 5) (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 CriterionMinimax Regret Criterion (2 of 5) (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 CriterionMinimax Regret Criterion (3 of 5)(3 of 5)
• 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.
Example 1. Thompson Lumber Company
Minimax Regret CriterionMinimax Regret Criterion (4 of 5)(4 of 5)
• Table illustrates computations and shows complete opportunity loss table.
Example 1. Thompson Lumber Company
Minimax Regret CriterionMinimax Regret Criterion (5 of 5)(5 of 5)
• 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.
Example 1. Thompson Lumber Company
Tom Brown Investment Example for Tom Brown Investment Example for Decision Making Under UncertaintyDecision 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.
States of Nature
Decision AlternativesLarge Rise Small Rise No Change Small Fall Large Fall
Gold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D Account 60 60 60 60 60Stock Option Hedge200 150 150 -200 -150
States of Nature
Decision AlternativesLarge Rise Small Rise No Change Small Fall Large Fall
Gold -100 100 200 300 0Bond 250 200 150 -100 -150Stock 500 250 100 -200 -600C/D Account 60 60 60 60 60Stock Option Hedge200 150 150 -200 -150
The Payoff TableThe Payoff Table
The Stock Option Alternative is dominated by the Bond Alternative
Example 2. Tom Brown Investment Decision
The Maximax criterion MaximumDecision Large rise Small rise No changeSmall fall Large fall PayoffGold -100 100 200 300 0 300Bond 250 200 150 -100 -150 200Stock 500 250 100 -200 -600 500C/D 60 60 60 60 60 60
The Optimal decision
Maximax CriterionMaximax Criterion
Example 2. Tom Brown Investment Decision
The Maximin Criterion Minimum
Decisions LargeRrise Small Rise No change Small Fall Large Fall Payoff
Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60
The Maximin Criterion Minimum
Decisions LargeRrise Small Rise No change Small Fall Large Fall Payoff
Gold -100 100 200 300 0 -100Bond 250 200 150 -100 -150 -150Stock 500 250 100 -200 -600 -600C/D account 60 60 60 60 60 60
The Optimal decision
The Maximin CriterionThe Maximin Criterion
Example 2. Tom Brown Investment Decision
Decision Large rise Small rise No changeSmall fall Large fall regretGold 600 150 0 0 60 600Bond 250 50 50 400 210 400Stock 0 0 100 500 660 660C/D 440 190 140 240 0 440
The Optimal decision
Minimax Regret CriterionMinimax Regret CriterionExample 2. Tom Brown Investment Decision
Regret Table
Decision Making Under RiskDecision Making Under Risk
Decision Making Under RiskDecision 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 ValueExpected Monetary Value (1 of 3) (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)
• 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 ValueExpected Monetary Value (2 of 3) (2 of 3)Example 1. Thompson Lumber Company
The Expected Value Criterion ExpectedDecision Large rise Small rise No changeSmall fall Large fall ValueGold -100 100 200 300 0 100Bond 250 200 150 -100 -150 130Stock 500 250 100 -200 -600 125C/D 60 60 60 60 60 60Prior Probability0.2 0.3 0.3 0.1 0.1
(0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130
The Optimal decision
Expected Monetary Value (3 of 3)Expected Monetary Value (3 of 3)
Example 2. Tom Brown Problem
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 ofperfect information
Expected payoffunder certainty
Expected payoffunder risk= -
Expected Value of Perfect InformationExpected 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 EVPIEV 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 TreesDecision 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 ofSquare nodes - indicating decision pointsCircles nodes - indicating states of natureArcs - connecting nodes
These nodes are represented using following symbols: = A decision nodeArcs (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
1
2
State 1
State 2
State 1
State 2
Alternative 1
Alternative 2
Decision Node
Outcome 1Outcome 1
Outcome 2Outcome 2
Outcome 3Outcome 3
Outcome 4Outcome 4
State of Nature Node
Decision Tree
State of nature 1
B
Payoff 1
State of nature 2
Payoff 2
Payoff 3
2
Choose A’1
Choose A’2
Payoff 6State of nature 2
2
Payoff 4
Payoff 5
Choose A’3
Choose A’4
State of nature 1
Choose A
’
Choose A’2
1
Decision PointChance Event
Decision TreeDecision Tree
• 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.
Thompson Lumber Company
Folding Back a Decision TreeFolding Back a Decision Tree
• 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.
Thompson Lumber Company
Reduced Decision TreeReduced Decision Tree
• Using the rule for decision nodes, select alternative with highest EMV.
• Corresponds to alternative to build small plant. • Resulting EMV is $40,000.
Thompson Lumber Company
Decision Trees for Multi-stageDecision Trees for Multi-stage Decision Making Problems Decision Making Problems
Decision Tree With EMVs ShownDecision Tree With EMVs Shown
Thompson Lumber Company
Decision Making in Southern TextileDecision Making in Southern Textile
Southern Textile CompanySouthern Textile Company
STATES OF NATURESTATES OF NATURE
Good ForeignGood Foreign Poor ForeignPoor Foreign
DECISIONDECISION Competitive ConditionsCompetitive Conditions Competitive ConditionsCompetitive Conditions
ExpandExpand $ 800,000$ 800,000 $ 500,000$ 500,000Maintain status quoMaintain status quo 1,300,0001,300,000 -150,000-150,000Sell nowSell now 320,000320,000 320,000320,000
Southern Textile CompanySouthern Textile Company
STATES OF NATURESTATES OF NATURE
Good ForeignGood Foreign Poor ForeignPoor Foreign
DECISIONDECISION Competitive ConditionsCompetitive Conditions Competitive ConditionsCompetitive Conditions
ExpandExpand $ 800,000$ 800,000 $ 500,000$ 500,000Maintain status quoMaintain status quo 1,300,0001,300,000 -150,000-150,000Sell nowSell now 320,000320,000 320,000320,000Maximax SolutionMaximax Solution
Expand: $800,000Status quo: 1,300,000 MaximumSell: 320,000
Decision: Maintain status quo
Southern Textile CompanySouthern Textile Company
STATES OF NATURESTATES OF NATURE
Good ForeignGood Foreign Poor ForeignPoor Foreign
DECISIONDECISION Competitive ConditionsCompetitive Conditions Competitive ConditionsCompetitive Conditions
ExpandExpand $ 800,000$ 800,000 $ 500,000$ 500,000Maintain status quoMaintain status quo 1,300,0001,300,000 -150,000-150,000Sell nowSell now 320,000320,000 320,000320,000Maximin SolutionMaximin Solution
Expand: $500,000 MaximumStatus quo: -150,000Sell: 320,000
Decision: Expand
Southern Textile CompanySouthern Textile Company
STATES OF NATURESTATES OF NATURE
Good ForeignGood Foreign Poor ForeignPoor Foreign
DECISIONDECISION Competitive ConditionsCompetitive Conditions Competitive ConditionsCompetitive Conditions
ExpandExpand $ 800,000$ 800,000 $ 500,000$ 500,000Maintain status quoMaintain status quo 1,300,0001,300,000 -150,000-150,000Sell nowSell now 320,000320,000 320,000320,000
Minimax Regret SolutionMinimax Regret Solution
$1,300,000 - 800,000 = 500,000 $500,000 - 500,000 = 01,300,000 - 1,300,000 = 0 500,000 - (-150,000) = 650,000
1,300,000 - 320,000 = 980,000 500,000 - 320,000 = 180,000
GOOD CONDITIONS POOR CONDITIONS
Expand: $500,000 MinimumStatus quo: 650,000Sell: 980,000
Decision: Expand
Southern Textile CompanySouthern Textile Company
STATES OF NATURESTATES OF NATURE
Good ForeignGood Foreign Poor ForeignPoor Foreign
DECISIONDECISION Competitive ConditionsCompetitive Conditions Competitive ConditionsCompetitive Conditions
ExpandExpand $ 800,000$ 800,000 $ 500,000$ 500,000Maintain status quoMaintain status quo 1,300,0001,300,000 -150,000-150,000Sell nowSell now 320,000320,000 320,000320,000
Hurwicz CriteriaHurwicz Criteria
= 0.3 1 - = 0.7
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 CompanySouthern Textile Company
STATES OF NATURESTATES OF NATURE
Good ForeignGood Foreign Poor ForeignPoor Foreign
DECISIONDECISION Competitive ConditionsCompetitive Conditions Competitive ConditionsCompetitive Conditions
ExpandExpand $ 800,000$ 800,000 $ 500,000$ 500,000Maintain status quoMaintain status quo 1,300,0001,300,000 -150,000-150,000Sell nowSell now 320,000320,000 320,000320,000
Equal Likelihood CriteriaEqual Likelihood Criteria
Two states of nature each weighted 0.50
Expand: $800,000(0.5) + 500,000(0.5) = $650,000 Maximum
Status quo: 1,300,000(0.5) -150,000(0.5) = 575,000
Sell: 320,000(0.5) + 320,000(0.5) = 320,000
Decision: Expand
Southern Textile CompanySouthern Textile Company
STATES OF NATURESTATES OF NATURE
Good ForeignGood Foreign Poor ForeignPoor Foreign
DECISIONDECISION Competitive ConditionsCompetitive Conditions Competitive ConditionsCompetitive Conditions
ExpandExpand $ 800,000$ 800,000 $ 500,000$ 500,000Maintain status quoMaintain status quo 1,300,0001,300,000 -150,000-150,000Sell nowSell now 320,000320,000 320,000320,000
Expected ValueExpected Value
p(good) = 0.70 p(poor) = 0.30
EV(expand) $800,000(0.7) + 500,000(0.3) = $710,000
EV(status quo) 1,300,000(0.7) -150,000(0.3) = 865,000 Maximum
EV(sell) 320,000(0.7) + 320,000(0.3) = 320,000
Decision: Status quo
EVPI ExampleEVPI Example
Good conditions will exist Good conditions will exist 70%70% of the time, choose of the time, choose maintain status quo with payoff of maintain status quo with payoff of $1,300,000$1,300,000
Poor conditions will exist Poor conditions will exist 30%30% of the time, choose expand of the time, choose expand with payoff of with payoff of $500,000$500,000
Expected value given perfect informationExpected value given perfect information
= $1,300,000 (0.70) + 500,000 (0.30)= $1,300,000 (0.70) + 500,000 (0.30)= $1,060,000= $1,060,000
EVPIEVPI == $1,060,000 - 865,000 = $195,000$1,060,000 - 865,000 = $195,000