Decision analysis provides a framework for analyzing a wide variety of management models'

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Introduction
Decision analysis provides a framework for
analyzing a wide variety of management models.
The framework establishes
1. A system of classifying decision models
based on the amount of information about the
model that is available
2. A decision criterion (a measure of the
goodness of fit).
Decision analysis treats decisions against nature
(outcomes over which you have no control) and the
returns accrue only to the decision maker.
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In decision analysis models, the fundamental
piece of data is a payoff table.
In this table, the alternative decisions are
listed along the side.
The states of nature are listed across the top.
The center values are the payoffs for all
possible combinations of decisions and states of
nature.
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The decision process is as follows
1. Select one of the alternative decisions (di).
2. After your decision is made, a state of
nature occurs that is beyond your control.
3. The associated return can then be
determined from the payoff table (rij).
The decision that we select depends on our belief
concerning what nature will do (i.e., which state
of nature will occur).
To help us make the decision, several assumptions
about natures behavior will be made. Each
assumption leads to a different criterion for
selecting the best decision.
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Three Classes of Decision Models
The three classes are
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DECISIONS UNDER CERTAINTY
A decision under certainty is one in which you
know (with certainty) which state of nature will
occur.
For example, in the morning you are deciding
whether to take your umbrella to work and you
know for sure that it will be raining when you
leave work in the afternoon.
The payoff table for this model is
It costs 7.00 to have your suit cleaned if you
get caught in the rain.
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All LP, ILP and NLP models as well as other
deterministic models such as the EOQ model can be
thought of as decisions against nature in which
there is only one state of nature.
For example, consider the following LP model
Max 5000E 4000F
s.t. 10E 15F lt 150
20E 10F lt 160
30E 10F gt 135
E 3F lt 0
E F gt 5
E, F gt 5
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For this model, we know (with certainty) exactly
what return we get for each decision.
These returns can thus be listed in the payoff
table
in one column, representing one state
of nature which is certain to occur.
It is easy to solve a model with one state of
nature. Simply select the decision that yields
the highest return.
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DECISIONS UNDER RISK
In most models, there is a lack of certainty
about future events. In quantitative modeling,
the lack of certainty can be dealt with in
various ways.
Definition of Risk Risk refers to a class of
decision models for which there is more than one
state of nature.
In addition, we assume that there is a
probability estimate for the occurrence of each
of the various states of nature.
The probability of state of nature j occurring is
generally estimated using historical frequencies.
Otherwise, subjective estimates are made.
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The expected value of any random variable is the
weighted average of all possible values of the
random variable, where the weights are the
probabilities of the values occurring.
E(X) Spixi
The expected return (ERi) associated with
decision i is
The decision is based on the maximum expected
return. In other words, i is the optimal
decision where
ERi maximum overall i of ERi
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The Newsvendor Model A newsvendor can buy the
Wall Street Journal newspapers for 40 cents each
and sell them for 75 cents.
However, he must buy the papers before he knows
how many he can actually sell. If he buys more
papers than he can sell, he disposes of the
excess at no additional cost. If he does not buy
enough papers, he loses potential sales now and
possibly in the future.
Suppose that the loss of future sales is captured
by a loss of goodwill cost of 50 cents per
unsatisfied customer.
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The demand distribution is as follows
P0 Probdemand 0 0.1
P1 Probdemand 1 0.3
P2 Probdemand 2 0.4
P3 Probdemand 3 0.2
Each of these four values represent the states of
nature. The number of papers ordered is the
decision. The returns or payoffs are as follows
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Where 75 selling price 40 cost of
buying a paper 50 cost of loss of goodwill
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Now, the ER is calculated for each decision i
Of these four ERs, choose the maximum,
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Another way to compare the decisions is to look
at a graph of their risk profiles
The risk profile shows all the possible outcomes
with their associated probabilities for a given
decision and graphically aids in decision making.
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DECISIONS UNDER UNCERTAINTY
In decisions under uncertainty, there is more
than one possible state of nature. However, now
the decision maker is unwilling or unable to
specify the probabilities that the various states
of nature will occur. In this case, there are
several approaches.
Laplace Criterion The Laplace criterion
approach interprets the condition of
uncertainty as equivalent to assuming that all
states of nature are equally likely to occur.
For example, in the newsvendor model, assuming
all states are equally likely means that since
there are four states, each state occurs with
probability 0.25.
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Using the Laplace (equally likely) criterion,
here are the resulting returns
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DECISIONS UNDER UNCERTAINTY
Maximin Criterion The Maximin criterion is an
extremely conservative, or pessimistic, approach
to making decisions.
Maximin evaluates each decision by the minimum
possible return associated with the decision.
Then, the decision that yields the maximum value
of the minimum returns (maximin) is selected.
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Maximin is often used in situations where the
planner feels he or she cannot afford to be wrong.
Consider the following example decision table
Based on the Maximin criterion, you would choose
decision 2. However, is this the best decision?
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DECISIONS UNDER UNCERTAINTY
Maximax Criterion The Maximax criterion is an
optimistic decision making criterion.
This method evaluates each decision by the
maximum possible return associated with that
decision.
The decision that yields the maximum of these
maximum returns (maximax) is then selected.
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Consider the following example decision table
Based on the Maximax criterion, you would choose
decision 2. However, is this the best decision?
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DECISIONS UNDER UNCERTAINTY
Regret and Minimax Regret Regret measures the
desirability of an outcome. The decision is made
on the least regret for making that choice.
So far, all the decision criteria have been used
on a payoff table of dollar returns as measured
by net cash flows.
The calculated regret indicates how much better
we can do as far as making a choice. Regret is
synonymous with the opportunity cost of not
making the best decision for a given state of
nature.
The following table shows the regret for each
combination of decision and state of nature.
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To build the Regret table, first choose the
maximum value in column 1
Now, subtract every value in that column from
this value
The resulting values are the regrets for the
associated decision and state of nature.
0 - 0 0 ( ) 40 0 (
) 80 0 ( ) 120
0 -40 -80 -120
170 85 0 40
255 170 85 0
85 0 40 80
Repeat these steps for the remaining columns.
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Once the Regret table is built, choose the
maximum value in each row
255 170 85 120
Then, of these maximum values, choose the
smallest i.e., the decision that minimizes the
maximum regret (minimax criterion).
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DECISIONS UNDER UNCERTAINTY
So, using the 3 criteria under uncertainty, we
made the following decisions regarding the
newsvendor data
Maximax Cash Flow Order 3 papers
Minimax Regret Order 2 papers
Note that when making decisions without
probabilities, the three criteria listed above
can result in different optimal solutions.
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The Expected Value of Perfect Information
Newsvendor Model Under Risk
Lets return to the newsvendor model under risk
(with the known probability distribution on
demand) in order to introduce the concept of the
expected value of perfect information.
The newsvendor, without knowing the actual
demand, orders the newspapers based on the
distribution of demand.
At the end of the day, the demand is revealed to
the newsvendor and an actual return can be
determined by his order-size decision and the
demand.
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What if, the newsvendor can purchase perfect
information on the demand for his newspapers
which would enable him to make better decisions?
The question that we need to answer is What is
the largest fee the newsvendor should be willing
to pay for this perfect information?
This fee is called the expected value of perfect
information (EVPI)
EVPI (expected return with new deal)
(expected return with current sequence of events)
The EVPI gives an upper bound on the amount that
you should be willing to pay for the perfect
information.
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With perfect information, the newsvendor will
always order the number of papers that will give
him the maximum return for the state of nature
that will occur.
However, the payment for this information must be
made before the newsvendor learns what the demand
will be.
To calculate the expected return with the new
deal, choose the maximum value for each outcome
(column) and multiply it by its respective
probability. Then, add the resulting products.
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ER(new) 0(0.1) 35(0.3) 70(0.4) 105(0.2)
59.5
ER(current) 22.5
EVPI 59.5 22.5 37.0 cents
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Utilities and Decisions under Risk
Utility is an alternative way of measuring the
attractiveness of the result of a decision. It
is an alternative way of finding the values to
fill in a payoff table.
Previously, we used net dollar return (net cash
flow) and regret as two measures of the
goodness of a particular combination of a
decision and state of nature.
Utility suggests another type of measure.
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THE RATIONALE FOR UTILITY
Consider the following game in which an urn
contains 99 white balls and 1 black ball. A
single ball is drawn from the urn. Each ball is
equally likely to be drawn.
If a white ball is drawn, you must pay 10,000.
If the black ball is drawn, you receive
1,000,000. You must decide whether to play.
The payoff table is
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The probability of a white and a black ball are
0.99 and 0.01, respectively. The expected
returns are
ER(play) -10,000(0.99) 1,000,000(0.01)
-9900 10,000 100
ER(do not play) 0(0.99) 0(0.01) 0
Since ER(play) gt ER(do not play), you should play
if the criterion of maximizing the expected net
cash flow is applied.
So, will you play this game?
This simple example shows that you need to take
care in selecting an appropriate criterion.
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Most people are risk-averse, which means they
would feel that the loss of a certain amount of
money would be more painful than the gain of the
same amount of money.
Utility function in decision analysis measures
the attractiveness of money. Utility can be
thought of as a measure of satisfaction. Two
characteristics are
1. It is nondecreasing, since more money is
always at least as attractive as less money.
2. It is concave (the marginal utility of money
is nonincreasing).
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To illustrate, first suppose you have 100 and
someone gives you an additional 100. Note that
your utility increases by
U(200) U(100) 0.680 0.524 0.156
Now suppose you start with 400 and someone gives
you an additional 100. Now your utility
increases by
U(500) U(400) 0.910 0.850 0.060
This illustrates that an additional 100 is less
attractive if you have 400 on hand than it is if
you start with 100.
The gain of a specified number of dollars
increases utility less than the loss of the same
number of dollars decreases utility.
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Typical risk-averse utility function
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Risk-seeking (convex) function
A gain of a specified amount of dollars increases
the utility more than a loss of the same amount
of dollars decreases the utility.
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Risk-indifferent function
A gain or loss of a specified dollar amount
produces a change of the same magnitude in
utility.