DECISION MODELING WITH
MICROSOFT EXCEL
Chapter 8
DECISION
ANALYSIS
Part 1
Copyright 2001
Prentice Hall
DECISION ANALYSIS
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.
In decision analysis models, the fundamental
piece of data is a payoff table.
In this table, the alternative decisions are listed
along the side.
State of Nature
2
…
m
Decision
1
d1
r11
r12
…
r1m
d2
r21
r22
…
r2m
…
…
…
…
dn
rn1
rn2
…
…
rnm
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.
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 nature’s behavior will be
made. Each assumption leads to a different
criterion for selecting the “best” decision.
DECISION ANALYSIS
Three Classes of Decision Models
The three classes are
Decisions Under Certainty
Decisions Under Risk
Decisions Under Uncertainty
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:
Take Umbrella
Do Not
Rain
0
-7.00
It costs $7.00 to have your suit cleaned if you get
caught in the rain.
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 < 150
20E + 10F < 160
30E + 10F > 135
E – 3F < 0
E+F>5
E, F > 5
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.
Decision
State of Nature
E = 0, F = 0
-
E = 5, F = 4
41,000
…
44,000
…
…
E = 6, F = 3.5
…
Value assigned to any infeasible decision.
It is easy to solve a model with one state of
Return Simply
is defined
to be
objective
nature.
select
thethe
decision
thatfunction
yields the
value (5000E
highest
return. + 4000F).
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.
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
m
ERi = Srijpj = ri1p1 + ri2p2 + … + rimpm
j=1
The decision is based on the maximum expected
return. In other words, i* is the optimal decision
where:
ERi* = maximum overall i of ERi
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.
The demand distribution is as follows:
P0 = Prob{demand = 0} = 0.1
P1 = Prob{demand = 1} = 0.3
P2 = Prob{demand = 2} = 0.4
P3 = Prob{demand = 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:
Decision
State of Nature (Demand)
0
1
2
3
0
0
-50
-100
-150
1
-40
35
-15
-65
2
-80
-5
70
20
3
-120
-45
30
105
Payoff = 75(# papers sold) –
40(# papers ordered) – 50(unmet demand)
Where 75¢ = selling price
40¢ = cost of buying a paper
50¢ = cost of loss of goodwill
Now, the ER is calculated for each decision i :
ER0 = 0(0.1) – 50(0.3) – 100(0.4) – 150(0.2) = -85
ER1 = -40(0.1) + 35(0.3) – 15(0.4) – 65(0.2) = -12.5
ER2 = -80(0.1) – 5(0.3) + 70(0.4) + 20(0.2) = 22.5
ER3 = -120(0.1) – 45(0.3) + 30(0.4) – 105(0.2) = 7.5
State of Nature (Demand)
Decision
0
1
0
1
2
0
-50 ER’s,
-100
Of these
four
-40 the maximum,
35
-15
choose
2
-80
3
Prob.
3
ER
-150
-85
-65
-12.5
-5
70
20
22.5
-120
-45
30
105
7.5
0.1
0.3
0.4
0.2
and order 2 papers
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.
The Cost of Lost Goodwill:
A Spreadsheet Sensitivity Analysis
This decision is based on the cost of lost goodwill,
whose value is much less certain than selling
price and purchase cost.
Now, perform a sensitivity analysis to determine
what would happen to the optimal decision if the
cost of lost goodwill were different.
An Excel spreadsheet is highly suitable to
calculate the payoff matrix and expected returns.
Here is the spreadsheet model for the decision
table:
=$B$1*MIN($A7,B$6)-$B$2*$A7$B$3*MAX(B$6-$A7,0)
=SUMPRODUCT(B7:E7,$B$12:$E$12)
Using the Data Table command, a table of
expected returns can easily be generated for a
range of Goodwill Costs.
After copying the
entire “Base
Case” into a new
worksheet, enter
0 in cell A16, then
click on Edit – Fill
– Series – Series
in Columns with a
step value of 5.
=F7=F8=F9=F10
Click on Data – Table
and enter $B$3 as the
column input cell.
Now, graph the result of the Data Table to help
sort out all the numbers.
Highlight the range of data (A16:E46) and click
on the Chart Wizard icon. In the resulting dialog,
choose the Line graph and click Next.
In the next dialog, click on
the Series tab and indicate
that the Category (X) axis
labels are found in
A16:A46.
Click Next and enter titles
or labels for the graph.
Click Finish when done to
display the graph.
Here is the resulting graph.
50
15
0
14
0
13
0
12
0
11
0
10
0
90
80
70
60
50
40
30
20
10
Expected Return (cents)
0
0
-50
Order 0
-100
Order 1
Order 2
-150
Order 3
-200
-250
-300
Goodw ill Cost (cents)
For a Goodwill Cost less than 125 cents, the
optimal decision is to order 2 papers.
For a Goodwill Cost of 125 cents, alternative
optima exists: order 2 or 3 papers.
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.
Using the Laplace (equally likely) criterion, here
are the resulting returns:
Each state of nature has equal
probability of occurring = 0.25
Choose
the max.
return
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.
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 1. However, is this the best
decision?
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.
Consider the following example decision table:
Based on the Maximax criterion, you would
choose decision 2. However, is this the best
decision?
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.
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.
State of Nature
Decision
0
1
2
3
85
170
255
0
0-
1
0 – (-40 ) = 40
0
85
170
2
0 – (-80 ) = 80
40
0
85
3
0 – (-120 ) = 120
80
40
0
0 =0
Repeat these steps for the remaining columns.
Once the Regret table is built, choose the
maximum value in each row:
Decision
State of Nature (Demand)
0
1
2
3
Max. Regret
0
0
85
170
255
255
1
40
0
85
170
170
2
80
40
0
85
85
3
120
80
40
0
120
Then, of these maximum values, choose the
smallest [i.e., the decision that minimizes the
maximum regret (minimax criterion)].
DECISIONS UNDER UNCERTAINTY
So, using the 3 criteria under uncertainty, we
made the following decisions regarding the
newsvendor data:
Criteria
Decision
Maximin Cash Flow
Order 1 paper
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.
DECISION ANALYSIS
The Expected Value of Perfect Information:
Newsvendor Model Under Risk
Let’s 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.
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.”
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.
State of Nature
Decision
0
1
2
0
0
-50
-100
-150
1
-40
35
-15
-65
2
-80
-5
70
20
3
-120
-45
30
105
0.1
0.3
0.4
0.2
Prob.
3
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
DECISION ANALYSIS
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.
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:
DECISION
Play
Do Not Play
STATE OF NATURE
White Ball
Black Ball
-10,000
0
1,000,000
0
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) > 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.
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).
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.
Typical risk-averse utility function:
A gain
A loss
in
in
utility
utility
of
of
0.06
0.075
Utility
1.0
0.910
0.850
0.775
0.680
0.524
100
200
300
400
500
600 Dollars
Go $400
from $400
to
Go from
to
results
in
$300 $500
results
in
Risk-seeking (convex) function:
Utility
1.0
A gain
in
A loss
utility
in
of
utility
0.330
of
0.185
0.590
0.260
0.075
100
200
300
400 Dollars
Go from
Go from
$200$200
to
to
A gain of a specified
amount
$100$300
results
results
in ofindollars increases
the utility more than a loss of the same amount of
dollars decreases the utility.
Risk-indifferent function:
Utility
1.0
0.90
0.60
0.30
100
200
300
400 Dollars
A gain or loss of a specified dollar amount
produces a change of the same magnitude in
utility.
CREATING AND USING A UTILITY FUNCTION
To begin, arbitrarily select the endpoints of the
utility function.
For convenience, set the utility of the smallest net
dollar return equal to 0 and the utility of the
largest net return equal to 1.
In the newsvendor example, the smallest return
is –150 and the largest is +105. Therefore
U(-150) = 0
U(+105) = 1
Now, assume that the decision maker starts with
these values and wants to find the utility of 10
[U(10)].
Select an unbiased probability p for the following
two alternatives:
1. Receive a payment of 10 for sure.
2. Participate in a lottery in which a payment
of 105 will be received with probability p or
a payment of –150 with probability 1-p.
If p = 1, then alternative 2 is preferred (a
payment of 105 is better than a payment of 10).
If p = 0, then alternative 1 is preferred (a
payment of 10 is better than a loss of 150).
Somewhere between 0 and 1, there is a value for
p such that the decision maker is indifferent
between the two alternatives.
This value for p will vary from person to person
depending on how attractive the various
alternatives are to them. This value of p is called
the utility for 10.
For example, choose p = 0.6, the expected value
of the lottery is:
0.6(105) + 0.4(-150) = 3
In this case, the manager is seeking risk since a
sure payment of 10 (larger than the expected
return of 3) is required to compensate her for the
loss of the possibility of making more than the
expected return.
Now, choose p = 0.8, the expected value of the
lottery is:
0.8(105) + 0.2(-150) = 54.0
In this case, the manager is adverse to risk since
an expected value larger than the sure payment
of 10 is required to compensate for the riskiness
of the lottery.
The larger the value of p, the more risk-adverse
the manager is, because she requires a larger
expected value of the lottery to compensate her
for its riskiness.
By solving the equation
p(105) + (1-p)(-150) = 10
255p – 150 = 10
p = 160/255 = 0.6275
We find the value of p for which the expected
value of the lottery is equal to the sure payment
of 10. So,
p > 0.6275
adverse to risk
p = 0.6275
p < 0.6275
indifferent to risk
seeking risk
To completely assess the entire utility function,
repeat this procedure for all other possible dollar
returns.
Another (and more popular) way to assess utility
functions is to use an exponential utility function.
This function has a predetermined shape (i.e., it’s
concave, risk averse) and requires the
assessment of only one parameter.
The function has the following form:
U(x) = 1 - e-x/r
Where
x is the dollar amount that will be converted
to utility.
r is a constant that measures the degree of risk
aversion (the larger the value of r, the less
risk-averse, the smaller the value of r, the
more risk-averse the company or person is).
One way to determine r, is to first determine the
dollar amount r such that the manager is
indifferent between the following two choices:
1. A 50-50 gamble where the payoffs are a
gain of r dollars or a loss of r/2 dollars
2. A payoff of zero
Now, suppose the newsvendor is indifferent
between a bet where he wins $100 or loses $50
with equal probability and not betting at all. In
this case, his r is $100.
The second way to determine r is based on
empirical evidence from many corporations net
income, equity, and net sales to the degree of risk
aversion, r.
This evidence shows that r is approximately equal
to 124% of net income, 15.7% of equity, and
6.4% of net sales.
For example, a large company with net income of
$1 billion would have an r of 1.24 billion, whereas
a smaller company with net sales of $5 million
would have an r of 320,000.
Return to the newsvendor example:
Because the newsvendor owns a very small
business, he opts for using the predetermined
exponential utility function approach.
In addition, he uses the 50-50 gamble approach
to determine his r value ($100).
Using Excel, develop a spreadsheet model of the
utility of all the dollar payoffs.
In this payoff table, the entries are the utility of
the net cash flow associated with each
combination of a decision and a state of nature
(i.e., the utility of the net cash flow is substituted
for the respective cash flow.
=1-EXP(-’Base Case’!B7/$B$14)
=1-EXP(-A17/$B$14)
=SUMPRODUCT(B7:E7,$B$12:$E$12)
Enter –150 in the first cell, then
click on Edit – Fill – Series and
enter the following parameters:
Here is a graph of the utility function:
Utility(x)
1.00
-2.00
-3.00
-4.00
X
90
70
50
30
10
-10
-30
-50
-70
-90
-110
-130
-1.00
-150
Utility
0.00
On the basis of the criterion of maximizing the
expected utility, the newsvendor would order 2
papers.
Consider the following auto insurance example:
Ten years after graduating from Standford’s
Graduate School of Business, Carol Lane
purchased a Lexus. The annual premium from
her insurance company for collision insurance
would be $1000 with $250 deductible.
There is only a 0.5% chance she will cause a
collision in the next year. In this case, she can
expect about $50,000 worth of damage.
Since she owns the Lexus outright, she is not
required to purchase collision insurance.
Should she buy the insurance or not?
Here is the spreadsheet model:
=B2 + B1
=B6
=B1
=SUMPRODUCT(E3:F3,$E$6:$F$6)
The Expected Returns indicate that it would be
less expensive for her, on average, to not buy the
insurance.
However, to be sure, she performs an utility
analysis. She decides that she is risk-averse,
and estimates her r value to be $10,000.
=1-EXP(E3/$F$8)
Based on the expected utility analysis, Carol feels
it is worth buying the insurance to have peace of
mind that she won’t have to pay $50,000 if she
causes a collision.
End of Part 1
Please continue to Part 2