decisionth-1

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.


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).


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.



• Table illustrates computations and shows complete opportunity loss table.



• 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 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


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


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 Maximin CriterionThe Maximin Criterion


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


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


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.


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.


Decision Trees for Multi-stageDecision Trees for Multi-stage Decision Making Problems Decision Making Problems

Decision Tree With EMVs ShownDecision Tree With EMVs Shown


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





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





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






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






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






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






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

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