Lecture 2&3 - Decision Makingntdung_ise/Material/Operations...Operations Management Jay Heizer and...

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DECISION ANALYSIS

Part 4 - ADecision Making Tools

TEXT

Operations ManagementJay Heizer and Barry Render

Prentice Hall 1996 6th Edition

REFERENCE

Operations ManagementRoberta Russell and Bernard Taylor

Prentice Hall, 3th Edition 2000 ISBN 0-13-030346-1

S2. Operational Decision-Making Tools: Decision

Analysis.

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REFERENCE

Making Hard DecisionsRobert T Clemen

Duxbury Press 1996 2nd Edition

ISBN 0-534-26034-9

QUANTITATIVE ANALYSIS

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19 82

673

Overview of Quantitative Analysis

u Scientific Approach to Managerial Decision Making

u Consider both Quantitative and Qualitative Factors

Raw DataQuantitative

AnalysisMeaningfulInformation

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The Decision-Making Process

Problem

Quantitative AnalysisLogicHistoric DataMarketing ResearchScientific AnalysisModeling

Qualitative AnalysisWeatherState and federallegislation

New technological breakthroughs

Election outcome

Decision

?

The Quantitative Analysis Approach

Defining the Problem

Developing a Model

Acquiring Input Data

Developing a Solution

Testing the Solution

Analyzing the Results

Implementing the Results


Developing a Model






The Quantitative Approach

The first step in the quantitative approach is to develop a clear, concise statement of

the problem

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Developing a Model







Once the problem to be analyzed in

selected, the next step is to develop a

model. Simply stated, a model is a

representation (usually

mathematical) of a situation.


Developing a Model







Once the model is developed, the data must be obtained

(input data). Obtaining accurate

data for the model is essential, since even

if the model is a perfect representation of reality, improper data will result in misleading results


Developing a Model







Developing a solution involves manipulating the

model to arrive at the best (optimal) solution to the

problem. In some cases, this requires that an equation be solved for the best

decision

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Developing a Model







Before a solution can be analysed and

implemented it needs to be tested

completely. Because the solution depends on the input data and the model, both must

be tested


Developing a Model







Analysing the results starts with

determining the implication of the

solution. Sensitivity analysis determines,

how the solutions will change, with a different model or

input data


Developing a Model







The process of implementing the

results in a company can be very difficult.

Obstacles such as management commitment,

resistance to change etc should be considered.

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Developing a Model






Tile Replacement for the Space Shuttle

To determine when to replace tiles on the

existing space shuttle, NASA needs

a decision making model to analyse

probability values, tile maintenance

policies, and possible outcomes of the

policies


Developing a Model







A decision-making model was developed

for multiple partitions of the

orbiter’s surface. The model determines a maintenance policy for each partition or

zone of tiles

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Developing a Model







The input dtata for each partition

consisted of various probability values,

including the probability of debris striking the surface of the orbiter, the

chance of losing an adjacent tile once the

first one was lost, burn out etc


Developing a Model







The solution included a risk

criticality scale that was developed as a result of the model

for each partition. It was found that 15%

of the tiles contributed 85% of

the risk.


Developing a Model







NASA tested the model and the

solution to make sure that probability

values and possible consequences were

accurate

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Developing a Model







The model revealed that improvements in

the maintenance of the tiles could reduce

the chance of a shuttle disaster or accident caused by defective tiles by

70%


Developing a Model







The tile maintenanceprogramme was implemented to

reduce the potential of disaster. The success of the

implemented solution has caused NASA to

investigate risk management for

other critical areas

Models, and the Techniques of Scientific Management Can Help Managers To:

u Gain deeper insight into the nature of

business relationships

u Find better ways to assess values in such

relationships; and

u See a way of reducing, or at least

understanding, uncertainty that

surrounds business plans and actions

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The Decision-Making Process

Problem Decision

Quantitative Analysis

LogicHistorical DataMarketing ResearchScientific AnalysisModelingQualitative Analysis

EmotionsIntuitionPersonal Experienceand Motivation

Rumors

Roadblocks to Be Faced When Defining a Problem

u Conflicting viewpoints among managers of different departments

u Impact of a problem in one department on other departments in the firm

u The validity of beginning assumptions; and the tendency to state problems in terms of solutions

u The proposed solution becomes outdated by the rapidly changing business environment, where problems appear and disappear virtually overnight

Advantages of Using Models

u They are less expensive and disruptive than experimenting with the real world system

u They allow operations managers to ask “What if” types of questions

u They are built for management problems and encourage management input

u They force a consistent and systematic approach to the analysis of problems

u They require managers to be specific about constraints and goals relating to a problem

u They can help reduce the time needed in decision making

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Limitations of Models

u They may be expensive and time-consuming to

develop and test

u They are often misused and misunderstood (and

feared) because of their mathematical complexity

u They tend to downplay the role and value of non-

quantifiable information

u They often have assumptions that oversimplify the

variables of the real world

MAKING DECISIONS UNDER

UNCERTAINTY

Elements of a Decision Analysis

1. Actions2. Chance Occurrences3. Probabilities4. Final Outcomes5. Additional Information6. Decision

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ActionsuAnything that the decision maker can douAn action is something that the decision maker

can control.uThere are often several choices of actionuMany decision problems are sequential in

natureuActions can be made until a final outcome is

reacheduOne the final outcome has been reached a goal

has been achieved or lost (to varying degrees)

Chance Occurrences

uAfter the decision maker takes an action - chance takes an action

uThe action of chance is the chance occurrence

Probabilities

uAll actions of chance are governed by probabilities.

uThe probabilities should be obtained by some method.

uThey can be obtained subjectively (egsomeone's opinion, or objectively through for example a sample survey or analysis

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Final OutcomeuThe decision problem is assumed to be of finite

durationuAfter a sequence of actions have been taken ( by the

decision maker and chance) there is a final outcomeuThe outcome may be viewed as a payoff or reward or

it may be viewed as a loss.uUsually all outcomes are considered as a payoff

(positive or negative)uA payoff is an amount of money (or other measure of

benefit called utility) received at the end of the decision making process

Additional InformationuEach time chance takes over, a random

occurrence takes place.uPrior information may be obtained which allows

us to assess the probability of any chance occurrence.

uOften this is purchased at a price.uThe cost of obtaining this additional information

needs to be subtracted from the final payoffuDeciding whether or not to obtain such

information is part of the decision making process.

DecisionuThe action or sequence of actions we decide to

take is called a decisionuThe decision obtained through a useful

analysis is that set of actions that maximizes the expected final outcome payoff.

uThe decision will often give a set of alternative actions in addition to the optimal solution.

uThe solution to the decision problem - the decision - gives all the information on how to proceed at any given stage or circumstance.

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General FrameworkuA decision maker is faced with a finite number K of

possible actions which will be labelled a1, a2......aK. uAt the time a particular action must be selected, the

decision maker is uncertain about the future of some factor that will determine the consequences of a chosen action.

u It is assumed that the possibilities for this factor can be characterized by a finite number H of states of nature. These will be denoted s1, s2......sH.

uFinally it is assumed that the decision maker is able to specify the monetary rewards or payoffs for each action state of nature combination

Framework for a Decision Problem

1. Decision maker has available K course of action:a1, a2, ...................aK

2. There are H possible uncertain states of nature:s1,s2, ....................sK

3. For each possible action-state of nature combinations there is an associated payoff, Mij, corresponding to action ai and state of nature sj

General Form of Payoff Table for a Decision Problem with K Possible Actions and H States of

Nature Actions States of Nature

s1 s2 . . . sH

a1 M11 M12 . . . M1Ha2 M21 M22 . . . M2H

. . . .. . . . . . . .

aK MK1 MK2 . . . MKH

Mij is the payoff corresponding action ai and state of nature sj

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Definitions

u If the payoff for action aj is at least as high as that for ai, whatever the state of nature, and if the payoff for aj is higher than that for ai for at least one state of nature then action aj is said to dominate ai.

uAny action that is dominated in this way is said to be inadmissible . Inadmissible actions are removed from the list of possibilities prior to further analysis of a decision making problem.

uAny action that is not dominated by some other action and is therefore not inadmissable is said to be admissible.

SOLUTIONS NOT INVOLVING

SPECIFICATION OF PROBABILITIES

Solutions not Involving Specification of Probabilities

uUsually solutions to decision problems require the specification of the outcome probabilities of the various states of nature.

uHowever some decisions problems have no probabilistic content and depend only on the structure of the payoff table

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MAXIMIN CRITERION

uHere the worst possible outcome is considered or each action , whatever state of nature materializes.

uThe worst outcome is the smallest payoff that could conceivably result.

Decision Rule Based of MAXIMIN Criterion

uSuppose that the decision maker has to choose from K admissible actions a1,a12,....aK given H possible states of nature. Let M denote the payoff corresponding to the ith action and jthstate.

uFor each action we seek the smallest possible payoff. For action ai the smallest possible payoff action is given by

Mi* = min(Mi1, Mi2......MiH)

uThe maximin criterion then selects the actionai for which the corresponding Mi

* is the largest

Example

uA manufacturer is planning to introduce a new product. uHe has 4 alternative production processes available to

him ranging from a relatively minor modification of existing facilities to a major CIM implementation.

uThe decision as to which course of action to follow must be made before the eventual demand of the product is known.

uFor convenience the eventual demand of the product is classified as low, medium and high.

uFor each production process the manufacturer calculates the profit over the lifetime of the investment for each of the 3 levels of demand.

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Estimated Profits

PRODUCTION LEVEL OF DEMANDPROCESS Low Moderate High

A 70,000 120,000 200,000

B 80,000 120,000 180,000

C 100,000 125,000 160,000

D 100,000 120,000 150,000

Note: Action D as at least as rewarding as C in all cases. It is dominated by C and is termedinadmissable. D is therefore dropped from all further analysis.

Choice of Production Process by Maximin Criterion

PRODUCTION LEVEL OF DEMAND MINIMUMPROCESS Low Moderate High PAYOFF

A 70,000 120,000 200,000 70,000

B 80,000 120,000 180,000 80,000

C 100,000 125,000 160,000 100,000

maximum

SummaryuThe positive feature of the maximin criterion

for decision making is that it produces the largest payoff that can be guaranteed.

u If production process C is used the manufacturer is assured a payoff of at least $100,000 whatever the demand turns out to be.

uThe price here lies in the foregoing of opportunities to receive a very much larger payoff, through the choice of some other action, however unlikely the worst case situation.

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Summaryu In the example the manufacturer may be

virtually certain that a high level of demand will result in which case production process C would be a poor choice, since it yields the lowest payoff at this demand level.

uThe maximin criterion (criterion of pessimism) can be thought of as providing a very cautious strategy for choosing alternative actions.

uSuch actions may, in certain circumstances be appropriate, but only an extreme pessimist would use it invariably

Criterion of Realism (Hurwicz Criterion)

uOften called the weighted average , the criterion of realism (the Hurwicz criterion) is a compromise between an optimistic and a pessimistic decision.

uTo begin with, a coefficient of realism α is selected. The coefficient is between 0 and 1 .

u If α is close to 1, the decision maker is optimistic about the future.

uWhen α is close to 0 the decision maker is pessimistic about the future.

Criterion of Realism (Hurwicz Criterion)

uThe advantage of this approach is that it allows the decision maker to build in personal feelings about relative optimism and pessimism.

uThe formula is as follows:

ucriterion of realism = α (maximum in row + (1- α ) (minimum in row)

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Choice of Production Process by Criterion of Realism (Hurwicz Criterion)

PRODUCTION LEVEL OF DEMAND HurwiczPROCESS Low Moderate High Criterion=0.8

A 70,000 120,000 200,000 174,000

B 80,000 120,000 180,000 160,000

C 100,000 125,000 160,000 138,000

realism

Minimax Regret CriterionuThe decision maker wanting to use the

minimax regret criterion must imagine himself being in the position where choice of action has been made, one of the states of nature has occurred and he can look back on the choice made either with satisfaction or disappointment.

uConsider the example of the manufacturer. Suppose that the level of demand for the new product turns out to be low. In that case, the best choice of action would have been production process C, yielding a payoff of $100,000.

Estimated Profits


A 70,000 120,000 200,000

B 80,000 120,000 180,000

C 100,000 125,000 160,000

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uHad the choice been made the manufacturer would have had 0 regret.

u If process A had been chosen, the resulting profit would have been only $70,000. The extent of the manufacturer’s regret , is the difference between the best payoff that could be obtained ($100,000) and that resulting from the inferior choice of action. Thus the regret would be $30,000.

uA regret table is constructed for each action-state of nature combination by calculating the differences between the each action - state of nature and the best possible choice for that state

Regret Table for Manufacturer

PRODUCTION LEVEL OF DEMAND PROCESS Low Moderate High

A 30,000 5,000 0

B 20,000 5,000 20,000

C 0 0 40,000

uNext we ask for each possible course of action, the largest amount of regret that can result.

uThe minimax regret criterion then selects the action for which the regret is smallest.

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Regret Table for Manufacturer

PRODUCTION LEVEL OF DEMAND PROCESS Low Moderate High

A 30,000 5,000 0

B 20,000 5,000 20,000

C 0 0 40,000

Choice of Production Process byMinimax Regret Criterion

PRODUCTION LEVEL OF DEMAND MAXIMUMPROCESS Low Moderate High REGRET

A 30,000 5,000 0 30,000

B 20,000 5,000 20,000 20,000

C 0 0 40,000 40,000

minimax regret

Decision Rule Based onMinimax Regret Criterion

uSuppose that a payoff is arranged as a rectangular array, with rows corresponding to actions and columns corresponding to states of nature.

u If each payoff in the table is subtracted from the largest payoff in its column, the resulting array is called a regret table.

uGiven the regret table, the action dictated by the minimaxregret criterion is found as follows:

1. For each row (actions), find the maximum regret.2. Choose the action corresponding to the minimum of these

maximum regrets.

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SummaryThe minimax regret criterion for the decision making

produces the smallest possible regret that can be guaranteed . However it has two serious drawbacks:

1. The logic behind the criterion does not provide a compelling framework for analysis for a wide range of decision - making problems.

2. Like the maximin criterion the minimax regret criterion does not allow the decision maker to inject personal views as to likelihood of the occurrence of the states of nature into the decision making process.

EXPECTED MONETARY

VALUE

Expected Monetary Value

u In the majority of decision making problems the decision maker is able to assess the chances of of occurrence of the various states of nature relevant in the determination of the eventual payoff.

uAssuming that a probability of occurrence can be attached to each state of nature we will explore how these probabilities are employed in arriving at the eventual decision.

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uConsidering the manufacturer in the previous example suppose that for all previous new introductions of the product 10% have met low demand , 50% moderate demand and 40% high demand.

u In the absence of any further information it is reasonable to postulate that for this particular market introduction the following probabilities for the states of nature

uProbability of Low Demand = 0.1uProbability of Moderate Demand = 0.5uProbability of High Demand = 0.4

uNote that since one and only one state of nature must occur, these probabilities necessarily sum to 1

- that is the states of nature or mutually exclusive and collectively exhaustive.

It is convenient to add these probabilities to the payoff table as shown:

Payoff Table for Manufacturer


(p=0.1) (p=0.5) (p=0.4)

A 70,000 120,000 200,000

B 80,000 120,000 180,000

C 100,000 125,000 160,000

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u In general if there are H possible states of nature, a probability of must be attached to each.

uFor probabilities p1, p2.......pH such that probability Pj corresponds to state of nature sjthen these probabilities must sum to one ie

uThe general setup for the decision making problem is :

pjj

H

=∑ =

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Actions States of Natures1 s2 . . . sH

(p1) (p2) . . . (PH)

a1 M11 M12 . . . M1Ha2 M21 M22 . . . M2H

. . . .. . . .

. . . .aK MK1 MK2 . . . MKH

Payoffs Mij and State of Nature Probabilities pjfor a Decision Problem with K Possible Actions and

H States of Nature

uWhen choosing an action, the decision maker will see each choice of action as having a specific probability of receiving an expected payoff and will therefore be able to calculate the expected payoff arising from each action.

uThe expected payoff for this action is then the sum of the individual payoffs weighted by their associated probabilities

uThe expected payoff are called that expected monetary values of the actions.

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u If the manufacturer adopts process A he has a probability of 0.1 of receiving a payoff of $70,000; a probability of 0.5 of receiving a payoff of $70,000; and a probability of 0.4 of receiving a payoff of $$200,000

uThe expected monetary value of the three admissible actions are therefore:

A: (0.1)(70,000)+(0.5)(120,000)+(0.4)(200,000) =$147,000.B: (0.1)(80,000)+(0.5)(120,000)+(0.4)(180,000) =

$140,000.A:(0.1)(100,000)+(0.5)(125,000)+(0.4)(160,000) =$136,000.

The action with the highest expected monetary value (Process A) is therefore adopted.



(p=0.1) (p=0.5) (p=0.4)

A 70,000 120,000 200,000 147,000

B 80,000 120,000 180,000 140,000

C 100,000 125,000 160,000 136,000

Expected Monetary Demand (EMV)

Highest EMV

Expected Monetary ValuesuSuppose that a decision maker has K possible

actions a1, a2.......aK and is faced with H states of nature. Let M ij denote the payoff corresponding to the ith action and the jth state and the pj the probability of occurrence of thejth state of nature with

uThe expected monetary value EMV(ai) of the action ai is

uEMV(ai) = p1Mi1 + p2Mi2+...+pHMiH = ΣpjMij

pjj

H

=∑ =

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Summary

uThe expected monetary values associated with the alternative course of action provide the decision maker with a choice criterion that will be extremely attractive for a great many practical problems.

uBy his criterion the action with the highest expected monetary value is adopted.

Decision Tree

uThe analysis of the decision problem by means of a expected monetary value can be conveniently set out diagrammatically though a mechanism called a decision tree.

uFor example consider the manufacturer’s decision making process:

Decision Tree for the ManufacturerActions States of Nature Payoffs

(probabilities)

Low (0.1)

Moderate (0.5)

High (0.4)

Low (0.1)

Low (0.1)

Moderate (0.5)

High (0.4)

Moderate (0.5)

High (0.4)

$70,000

$120,000

$200,000

$80,000

$120,000

$180,000

$100,000

$125,000

$160,000

Process A

Process B

Process C

EMV = $147,000

EMV = $140,000

EMV = $136,000

EMV = $147,000

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DECISION TREES

Analyzing Problems with Decision Trees: The Five Steps

1 Define the problem

2 Structure or draw the decision tree

3 Assign probabilities to the states-of-nature

4 Estimate the payoffs for each possible

combination of alternative and state-of- nature

5 Solve the problem by computing expected

monetary values (EMV) for each state-of -

nature node

Decision TreesuThe sequential approach to decision making is very well

modelled and visualized by a decision tree.uA decision tree is a set of nodes and branches.uAt a decision node, the decision maker takes an action:

the action is the choice of a branch to be followed.uThe branch leads to a chance node, where chance

determines the outcome.uThen either the final outcome is reached (the branch ends)

or the decision maker takes another action and so on.uA decision node is marked by a square and a chance node

by a circle. These are connected by the branches of the decision tree

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uThe decision tree is completed by assigning probabilities to each possible state of nature (that is the two possible actions of chance)

uNote that since one and only one of the states of nature must occur, these probabilities must sum to one, that is, the states of nature are mutually exclusive and collectively exhaustive

as follows........


(probabilities)

Low (0.1)

Moderate (0.5)

High (0.4)

Low (0.1)

Low (0.1)

Moderate (0.5)

High (0.4)

Moderate (0.5)

High (0.4)

$70,000

$120,000

$200,000

$80,000

$120,000

$180,000

$100,000

$125,000

$160,000

Process A

Process B

Process C

EMV = $147,000

EMV = $140,000

EMV = $136,000

EMV = $147,000


Developing a Model






Using Decision Trees Analysis on R&D Projects

The Canadian subsidiary of ICI discovered a new process to reduce

paper mill pollution. The company had to decide whether or not

to invest funds in research and

development for the new process

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Developing a Model







A traditional decision tree model was used. Instead of expected monetary values, the model used expected net

present value, which converts future

monetary flows into today's dollars


Developing a Model







ICI collected both probability and

monetary values. The probability data

included the probability of

technical success, the probability of a

significant market for the new process, and the probability of a commercial success


Developing a Model







The solution was obtained using

traditional decision tree analysis

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Developing a Model






Using Decision Trees Analysis on R&D Projects ICI tested the solution

by analysing various risks of the the process,

including whether or not the new process

could be developed, the market for the new

process, the accuracy of the conditional

probabilities in the decision tree and

various expenses and monetary flows.


Developing a Model







The estimated net present value form the decision tree analysis was $3.2 million. If the new

project was successful, the net present value could be as high as $25

million


Developing a Model







The decision analysis moved the R&D

project forward. As a result of the analysis,

it was decided to investigate the

process further. After field testing,

however, difficulty with pulp mills

resulted in the project being cancelled.

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EXAMPLE

uA manufacturer needs to decide if it is profitable to market a new Product.

uHis decision will obviously need to take into account the investment he needs to make in the new product.

SOLUTION

uThe first step of the solution is to prepare the payoff table which tabulates the possible payoffs we would receive if we took certain actions and certain chanceoccurances followed.

Payoff Table for New Product Introduction

Product is:Action Successful Not Successful

Market the Product +$100,000 -$20,000Do not Market the Product 0 0

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Decision Tree for New Product Information

Final Outcome

$100,000

-$20,000

$0

Product is Successful

Product is not Successful

Market

Do not Market

We now need to assignprobabilites to the two possible states of nature………..


Final Outcome

$100,000

-$20,000

$0

Success Probability

= 0.75

Failure Probability

= 0.25

Market

Do not Market

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Solving a Decision TreeuThe solution of decision tree problems is

achieved by working backwards from the final outcome.

uThe method is called averaging out and folding back.

uWorking backwards from the final outcomes all chance occurrences are averaged out ie the expected value is found for each node.

Solving a Decision Tree

uAt each chance node the expected monetary value of all branches leading out of the node is calculated (folding back the tree)

uAt each decision node the action that maximizes the (expected) payoff is chosen.

uNon optimal branches are clipped.

Expected ValuesuExpected Value of X, Denoted E(X) is

E X x P xallx

( ) ( )= ∑The outcome as you leave the chance node is a random variable with two possible values 100,00 and -20,000. the probability of the outcome 100,00 is 0.75 and the probability of outcome -20,000 is 0.25. Therefore the expected value is:

E(outcome at chance node) = (100,000) (0.75) + (-20,000) (0.25)= 70,000

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uThe expected value associated with the chance node is thus 70,000.

uAt the decision node the best branch is chosen and the other sub-optimal branches are clipped.

uThus at the decision node the two values 70,000 and 0 are compared.

uSince 70,000 is greater than 0 the expected monetary outcome of the decision to market the product is greater than the monetary outcome of the decision not to market the product

uWe follow the rule of choosing the decision that maximizes the expected payoff so we choose to market the product..


Final Outcome(payoff)

$100,000

-$20,000

$0

Success Probability

= 0.75

Failure Probability

= 0.25

Market

Do not Market

Expected Payoff = $70,000

Non-optimal decision is clipped

Arrow points where to go

u In reality possible outcomes may be finally divided into degrees of success

as follows

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Payoff Table for New Product Introduction

Product is:Action A B C D E F G H

Market $150,000 $120,000 $100,000 $80,000 $40,000 $0 -$20,000 -$50,000 Not Market 0 0 0 0 0 0 0 0

A = Extremely SuccessfulB = Very Successful

C = SuccessfulD = Somewhat Successful

E = Barely SuccessfulF = Break-evenG = UnsuccessfulH = Disastrous


Final Outcome(payoff)

$150,000

-$20,000

$0

Market

Do not Market

Expected Payoff = $77,500

0.10.20.30.10.10.1

0.050.05

$120,000$100,000$80,000$40,000$0

-$50,000

Optimal Decision

Sensitivity AnalysisuOften a decision maker will be uncertain about

estimates of the payoffs for each action -state of nature combination and on estimated probabilities of occurrences for the states of nature.

u It is therefore useful to ask under what range of specification of a decision problem a particular action will be optimal under the expected monetary value criterion.

uSensitivity Analysis seeks to answer such questions, the most straightforward case being where a single problem specification is allowed to vary while all other specification are held fixed

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THE VALUE OF SAMPLE

INFORMATION

The Value of Information

u In decision making the question often arises as to the value of information.

uHow much would we be willing to pay for additional information.

uThe first step to answering this question is to find out how much we should be willing to pay for perfect information.


u If we could determine the value of perfect information this would give us an upper bound on the value of the imperfect information.

uSince all sample information is probabilisticin nature the value of sample information is less than the value of perfect information

uSince we do not know what the perfect information is we can only compute an expected value of the perfect information in a given decision making situation.

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uThe expected value is a mean computed over prior probabilities of the various states of nature.

u It assumes however that at any given point when we actually take an action we know its exact outcome.

uBefore we (hypothetically) buy the perfect information we do not know what the state of nature will be and therefore we must average payoffs using our prior probabilities

The Expected Value of Perfect Information(EPVI)

EPVI = The expected monetary value of the decision situation when perfect information is available minus the expected value of the decision situation when no additional information is available.

uThis definition of the expected value of perfect information is logical:

it says that the (expected) maximum amount we should be willing to pay for perfect information is equal to the difference between our expected payoff from the decision situation when we have the information and our expected payoff from the decision situation without the information.

uThe expected value of the information is equal to what we stand to gain from this information.

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ExampleuSuppose that a manufacturer is involved in a

price war with one of his competitors for a similar product.

uProfits depend upon the competing selling price set by the competitor for the product.

uThe table shows the payoffs (in millions of dollars) to the manufacturer over a given period of time, for a given price set by the manufacturer and its competitor.

Manufacturer’s Payoffs (Millions of Dollars)

Competitor’s Price (state of nature)$200 $300

8 9 4 10

Manufacturer’s Price (Action)

$200$300

uAssuming that there is a certain probability that the competitor will choose the low ($200) price and a certain probability that the competitor will choose the high price.

uSuppose that the probability of the low price is 0.6 and that the probability of the high price is 0.4.

uA decision tree can be constructed for this situation.

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Decision Tree of Competitive Pricing

Payoff

Price $200

$300

0.6

0.4

0.6

0.4

Competitor’s Price$200

$200

$300

$300

$8 million

$9 million

$4 million

$10 million

expected payof = 8.4 million

expected payoff =6.4 million

Solving the tree:

If we set our price at $200 the expected payoff is $8.4 million.

If we set our price at $300 the expected payoff is $6.4 million.

The optimum action is therefore to set the price at $200.

Is it worthwhile to obtain more information ?

uObtaining new information may mean hiring a consultant who is knowledgeable about the operating philosophy of our competitor or analyzing the competitors past pricing strategy.

uThe important question is : What do we stand to gain from the new information ?

uWe know that without any additional information the optimum strategy is to set the price at $200 obtaining an expected payoff of $8.4 million.

What is the expected payoff with perfect information ?

39

uWe do not know what the perfect information may be but assuming that the prior probabilities are a true reflection of the long term proportion of the time our competitor sets either price we know that 60% of the time our competitor sets his price low and 40% of the time he sets it high.

uWe therefore average the maximum payoff for each case to give the payoff that would be obtained from under perfect information using our probabilities

E(Payoff under perfect information) = (maximum payoff if the competitor chooses $200)

x (Probability that the competitor will choose $200)

+

(maximum payoff if the competitor chooses $300) x

(Probability that the competitor will choose $300)=(8)(0.6) + (10)(0.4) = $8.8 million

Manufacturer’s Payoffs (Millions of Dollars)

Competitor’s Price (state of nature)$200 $300

8 9 4 10

Manufacturer’s Price (Action)

$200$300

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u If we could get perfect information we could expect (on average) to make a profit of $8.8 million

uWithout perfect information we expect to make $8.4 million

uEVPI = E(payoff under perfect information -E(payoff without perfect information)

uTherefore EPVI = 8.8 - 8.4 = $0.4 millionuTherefore $400,000 would be the maximum

amount we would be willing to pay for additional information.

SamplinguThe expected value of sample information

(EVSI) is equal to the expected value of perfect information minus the expected cost of sampling errors.

uThe expected cost of sampling errors is known from sampling theory - and the resulting loss of payoff due to making less than optimal decisions.

uThe expected net gain of sampling information is equal to the expected value of sampling information minus the cost of sampling

uAs the sample size increases, the expected net gain from the sample first increases, as our new information is valuable and improves our decision making ability.

uThen the expected gain decreases because we are paying for the information at a constant rate, while the information content in each additional data point becomes less and less important as we get more and more data.

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Expected Net Gain from Sampling as a Function of Sample Size

max

Sample Size

n max

Expe

cted

Net

Gai

n

ALLOWING FOR RISK

Utility Analysis

UTILITYuOf ten the rewards obtained from a decision

are not (easily) quantifiable. ( companies reputation, hidden costs etc).

uCritics have argues that using money to measure outcomes is a mistake and that people will invariably take those actions that maximize their welfare or utility and that actions that maximize monetary benefit or minimize monetary cost may well not coincide with those that maximize utility

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Doubts about the Expected Monetary Value Criterion

Events oil price rises oil price rises Expected Monetary Valuemoderately sharply EMVActions

Investing $100 million in vineyards

Investing $100 million in the auto industry

15 15 15(0.5) + 15 (0.5) = 15

Prob .= 0.5 Prob. = 0.5

25 5 25(0.5) + 5 (0.5) = 15

optimum ?

uThe concept of utilities derives from the seemingly non-quantifiable way we deal with rewards.

uFor most, the value of $ 1000 for example is not constant.

uThe value you attach to money - the utility of money is not a straight line, but a curve.

Question

u If you can get $5,000 for certain or you could get a lottery ticket with 0.5 chance of winning $10,000 or 0.5 chance of losing $2,000 Which would you choose?

uThe expected payoff from the lottery ticket is $9,000, almost twice as much as the certain choice of $5,000.

uBut few people would choose the lottery ticket.uThis shows that the possible risk of losing $2,000

dollars is not worth the possible gain of $20,000.

uSuch behavior is called risk aversion

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uRisk aversion produces a concave risk function where the utility of one dollar earned (one dollar to the right of zero) is less than the value of a dollar lost ( one dollar to the right of zero.

uRisk seeking would produce a convex function ( a function with an increasing slope) where an added dollar is worth more than the pain of a lost dollar.

uRisk neutrality produces a straight line. A dollar is worth a dollar no matter what ! The pain of losing a dollar is the same as the reward of gaining a dollar

Uti

lity

Amount of Money

UTILITY FUNCTIONRisk Averse

Uti

lity

Amount of Money


utility of additional

$1000

utility of additional

$1000

$100 $1 million

additional $1000

additional $1000

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Uti

lity

Amount of Money


utility of -$1

utility of +$1

-$1 +$10

Uti

lity

Amount of Money

UTILITY FUNCTIONRisk Prone

Uti

lity

Amount of Money

UTILITY FUNCTIONRisk Neutral

45

uAlso the utility function can be mixed. For example an individual (or company) may avoid risk when their wealth is small but take more risks when their wealth is great.

Uti

lity

Amount of Money

UTILITY FUNCTIONMixed Risk

A Method for Assessing UtilityuOne way of assessing a utility curve is:

u1. Identify the maximum payoff in a decision problem and assign it the utility 1 U(Max Value ) = 1

u2. Identify the minimum payoff in a decision problem and assign it the utility 0 U(Min Value ) = 0

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A Method for Assessing Utility3. Conduct the following game to determine the

utility of any intermediate value. Ask the person whose utility you are trying to assess to determine the probability p such that he expresses indifference between 2 choices to receive the payoff R with certainty or have a probability p of receiving the maximum value and probability 1-p of receiving the minimum value. The determined p is the utility of the value R. This is done for all values of R for which we want to assess the utility.

Uti

lity

Amount of Money

ASSESSMENT OF A UTILITY FUNCTION

min R1 R2 R3max

0

1

p1

p2

p3

ExampleuSuppose that a company is

investing in CIM. The company has several different options (each investment has a different level of risk). The possible payoffs are $1,500, $4,300, $22,000, $31,000 and $56,000.

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Solution

uThe minimum payoff of $1,500 is assigned the utility 0.

uThe maximum payoff of $56,000 is assigned the utility 1.

uSuppose that the investor says he is indifferent between receiving $4,300 for certain and receiving $56,000 with probability of 0.2 and $1,500 with probability of 0.8. This means that the utility of the payoff $4,300 is 0.2.

Solution

uSuppose that the investor says he is indifferent between receiving $22,000 for certain and receiving $56,000 with probability of 0.7 and $1,500 with probability of 0.3. This means that the utility of the payoff $22,000 is 0.7.

uFinally the investor indicates an indifference between receiving $31,000 for certain and receiving $56,000 with probability of 0.8 and $1,500 with probability of 0.2. The utility of the payoff $31,000 is 0.8.

Solution

uThe utility curve can now be plotted.

uWhatever decision problem facing the investor, the utilities rather than the actual payoffs are used as the values for the analysis.

uThe analysis is therefore based on maximizing the investors expected utility rather than the expected monetary outcome

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Uti

lity

Dollars ($000)

INVESTOR’S UTILITY

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1.0

MANUFACTURING EXAMPLE



(p=0.1) (p=0.5) (p=0.4)

A 70,000 120,000 200,000 147,000

B 80,000 120,000 180,000 140,000

C 100,000 125,000 160,000 136,000


Highest EMV

49


(probabilities)

Low (0.1)

Moderate (0.5)

High (0.4)

Low (0.1)

Low (0.1)

Moderate (0.5)

High (0.4)

Moderate (0.5)

High (0.4)

$70,000

$120,000

$200,000

$80,000

$120,000

$180,000

$100,000

$125,000

$160,000

Process A

Process B

Process C

EMV = $147,000

EMV = $140,000

EMV = $136,000

EMV = $147,000

Incorporating RiskuThe manufacturer’s risk can be identified by

the procedure outline.uFor each payoff the utility is determined and a

utility function is developed.uThe payoffs are then replaced in the decision

making process by the utilities.uThe decision in then based on the expected

utility criterion.

EXPECTED UTILITY CRITERION

uSuppose that a decision maker has K possible actions a1, a2.......aK and is faced with H states of nature. Let U ij denote the utility corresponding to the ith action and the jth state and the pj the probability of occurrence of thejth state of nature

uThe expected utility EU(ai) of the action ai isuEU(ai) = p1Ui1 + p2Ui2+...+pHUiH = ΣpjUij

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Expected Utility CriterionuGiven a choice between alternative actions, the

expected utility criterion dictates the choice of the action for which expected utility if highest.

uUnder generally reasonable assumptions, it can be shown that this criterion should be adopted by the rational decision maker.

u If the decision maker is indifferent to risk, the expected utility criterion and the expected monetary value criterion are equivalent.

UTILITY TABLE

Payoff Utility

70000 080000 0.18100000 0.42120000 0.58125000 0.62160000 0.84180000 0.98200000 1

Uti

lity

Dollars ($000)

MANUFACTURER’S UTILITY

0 25 50 75 100 125 150 175 2000

0.2

0.4

0.6

0.8

1.0

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(p=0.1) (p=0.5) (p=0.4)

A 70,000 120,000 200,000 147,000

B 80,000 120,000 180,000 140,000

C 100,000 125,000 160,000 136,000


Highest EMV



(p=0.1) (p=0.5) (p=0.4)

A 0 0.58 1 0.690

B 0.18 0.58 0.98 0.700

C 0.42 0.62 0.84 0.688

Expected Utility (EU)

Highest EU


(probabilities)

Low (0.1)

Moderate (0.5)

High (0.4)

Low (0.1)

Low (0.1)

Moderate (0.5)

High (0.4)

Moderate (0.5)

High (0.4)

$70,000

$120,000

$200,000

$80,000

$120,000

$180,000

$100,000

$125,000

$160,000

Process A

Process B

Process C

EMV = $147,000

EMV = $140,000

EMV = $136,000

EMV = $147,000

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Decision Tree for the ManufacturerActions States of Nature Utility

(probabilities)

Low (0.1)

Moderate (0.5)

High (0.4)

Low (0.1)

Low (0.1)

Moderate (0.5)

High (0.4)

Moderate (0.5)

High (0.4)

0

0.58

1

0.18

0.58

0.98

0.42

0.62

0.84

Process A

Process B

Process C

EU = 0.690

EU = 0.700

EU = 0.688

EU = 0.7

Uti

lity

Dollars ($000)

MANUFACTURER’S UTILITY

0 25 50 75 100 125 150 175 2000

0.2

0.4

0.6

0.8

1.0

SUMMARYuThe expected utility criterion is the most generally

applicable and intellectually defensible of the criteria introduced here for attacking decision making problems

uThe main drawback arises from the difficulty of eliciting information about which gambles are regarded equally attractive as particular assured payoffs.

uThis type of information is essential when determining the utility function.

u In situation where indifference to risk can be assumed the expected monetary value criteria remains applicable

Lecture 2&3 - Decision Makingntdung_ise/Material/Operations...Operations Management Jay Heizer and...

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