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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
1 768
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
The Quantitative Approach
The first step in the quantitative approach is to develop a clear, concise statement of
the problem
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Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
The Quantitative Approach
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.
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
The Quantitative Approach
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
The Quantitative Approach
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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Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
The Quantitative Approach
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
The Quantitative Approach
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
The Quantitative Approach
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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Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Tile Replacement for the Space Shuttle
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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Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Tile Replacement for the Space Shuttle
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Tile Replacement for the Space Shuttle
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.
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Tile Replacement for the Space Shuttle
NASA tested the model and the
solution to make sure that probability
values and possible consequences were
accurate
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Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Tile Replacement for the Space Shuttle
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%
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Tile Replacement for the Space Shuttle
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
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
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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
PRODUCTION LEVEL OF DEMANDPROCESS Low Moderate High
(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.
Payoff Table for Manufacturer
PRODUCTION LEVEL OF DEMANDPROCESS Low Moderate High
(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........
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
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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Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Using Decision Trees Analysis on R&D Projects
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Using Decision Trees Analysis on R&D Projects
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Using Decision Trees Analysis on R&D Projects
The solution was obtained using
traditional decision tree analysis
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Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
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.
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Using Decision Trees Analysis on R&D Projects
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
Defining the Problem
Developing a Model
Acquiring Input Data
Developing a Solution
Testing the Solution
Analyzing the Results
Implementing the Results
Using Decision Trees Analysis on R&D Projects
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………..
Decision Tree for New Product Information
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..
Decision Tree for New Product Information
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
Decision Tree for New Product Information
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.
The Value of 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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The Value of Information
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 ?
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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 FUNCTIONRisk Averse
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 FUNCTIONRisk Averse
utility of -$1
utility of +$1
-$1 +$10
Uti
lity
Amount of Money
UTILITY FUNCTIONRisk Prone
Uti
lity
Amount of Money
UTILITY FUNCTIONRisk Neutral
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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
Payoff Table for Manufacturer
PRODUCTION LEVEL OF DEMANDPROCESS Low Moderate High
(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
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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
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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Payoff Table for Manufacturer
PRODUCTION LEVEL OF DEMANDPROCESS Low Moderate High
(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
Payoff Table for Manufacturer
PRODUCTION LEVEL OF DEMANDPROCESS Low Moderate High
(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
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 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