1. ADDITIONAL NOTES BM014-3-3-DMKG
DECISION MAKING MODELS
Introduction
-Organizations and individuals are faced almost daily with the problems of having to
make decisions. The decision-making process is made difficult by the presence of
uncertainty concerning the surrounding environment.
-For example, a company in the process of formulating an advertising strategy is
uncertain not only of its competitors’ responses but also of the market demand for its
product. Yet, a decision must be made on product pricing, the choice of markets, the
advertising media, and the size of the advertising budget.
-Prior to these decisions, the same company must decide on the scale of its productive
facilities. What machines should be purchased to manufacture the product? Should the
company start out on a small production scale and later expand, or should it remain
small? Should it start out with a large production capacity on the premise that the market
demand will be great?
-In this chapter we shall present the basic concepts of decision theory and illustrate
various methods that have been employed to solve management decision problems.
-Note: Problem formulation or model development need to be completed before solving
and making decisions.
Problem Formulation
-The first step involved identifying:
-Decision alternatives / choices
-Uncertain/chance events /state of nature
-consequences/objectives- e.g max profit/ min cost
-Use technique:
-Payoff tables
-Shows consequences of various combination of decision alternatives &
state of nature/uncertainties
-consequences are known as payoffs (profit, costs, time)
-Example: making investment decision: deciding to purchase a real estate…..
Payoff tables
E.g .1 Mr Azlan deciding to purchase 1 of 3 types of real estate:
d1 = apartment building
d2 = office building
d3 = warehouse
To choose the best 1, depends on the future economic condition(state of nature):
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2. ADDITIONAL NOTES BM014-3-3-DMKG
s1 = Good Economic Condition (GEC)
s2 = Poor Economic Condition (PEC)
Alternatives/Choices GEC(s1) PEC(s2)
Apartment (d1) 15 7
Office (d2) 22 -4
Warehouse (d3) 12 9
His objective is to maximize profit.
Types of Decision Problems
1. Decision making under certainty
In this class of problems, the decision maker (by some means) knows for certain
which event will occur. In the context of the types of problems presented in this
chapter, decision making under certainty is reduced to the trivial task of selecting
the action yielding the highest payoff once we know what event to expect. For
example, referring to example 1 above, if Mr.Azlan has the reliable information
that the economic condition this year will be in a good condition, he would
definitely purchase office building, because it gives the highest payoff. As you
might guess, such decision problems rarely occur.
2. Decision making under uncertainty
Decision making under uncertainty refers to problems in which the decision
maker does not know for certain which event will occur. There are two types of
such problems- probabilistic and non-probabilistic decision problems.
2.a.Nonprobabilistic decision problems
It occurs when management does not have reasonable estimates of the likelihoods
of the occurrence of various events. Thus, certain management may not have
adequate information to assign probabilities to the four possible events.
2.b.Probabilistic decision problem
The decision maker is able to assign probabilities to the various events that may
occur.
2.a.Non-probabilistic decision rules
-When decision maker has less ability in assessing the probabilities ( no information
on the future events),
-or desire a simple best-case and worse-case analysis
-There are 4 approaches to use:
1) Maximax (optimistic) approach- for the risk taker person
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3. ADDITIONAL NOTES BM014-3-3-DMKG
-Choose the best possible pay off (profit).
Step 1 : Identify and List the maximum payoff of each alternative (row-wise)
Step 2: find & select the best possible payoff (e.g. largest profit)- (column-wise)
Note: “minimin” if the payoff are costs
alternatives GEC PEC Maximum p/off
d1 15 7 15
d2 22 -4 22
d3 12 9 12
maximax
So, choose d2 (office) which gives the highest payoff
2) Maximin(conservative) approach-for the risk averse person
-Choose the best from the worst possible payoff (profit)
Step 1: List the minimum payoffs of each alternative (row-wise)
Step 2:Choose alternative that provide overall maximum payoff (column-wise)
Note: “minimax” if the payoff are costs.
alternatives GEC PEC Minimum p/off
d1 15 7 7
d2 22 -4 -4
d3 12 9 9
Maximin
So, choose d3(warehouse) which gives the highest payoff.
3) Minimax Regret Approach/rule
-Minimizing the regrets for not making the best decision
Step 1: Subtract each entry in a column from the largest entry in that column.
opportunity loss (regret) = difference between payoff of the best decision
alternative and the payoff of alternative chose.
Regret / Opportunity loss table
alternative GEC PEC
d1 15 22 - 15 = 7 7 9–7=2
d2 22 22 - 22 = 0 -4 9 – (-4) = 13
d3 12 22 – 12 = 10 9 9–9=0
Choose the highest value
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4. ADDITIONAL NOTES BM014-3-3-DMKG
Regret/opportunity loss table
alternatives GEC PEC Minimax
d1 7 2 7
d2 0 13 13
d3 10 0 10
Minimum regret
So, the best decision is to choose d1 ( apartment)
4) Criterion of Realism ( Hurwicz Criterion)
-Balance, neither purely optimistic, nor pessimistic
-Pay off are weighted by a coefficient of optimism :
( α ) ( max in row) + ( 1 – α) (min in row)
alternatives GEC PEC Criterion of realism (α =0.75)
d1 7 2 (0.75) ( 7) + ( 0.25) (2) = 5.75
d2 0 13 (0.75) (13) + (0.25) ( 0) = 9.75
d3 10 0 (0.75) (10) + (0.25) (0) = 7.5
Best alternative
So, the best alternative will be purchasing office building (d2).
2.b.Probabilistic decision problems
-When the decision maker is able to assign probabilities to the various events, it
is then possible to employ a probabilistic decision rule called the Bayes criterion.
-The Bayes criterion selects the decision alternative having the maximum
expected payoff.
-Some of the textbook referring this Bayes decision to Expected value (EV)
approach which indicates the same interpretation.
-However, EV approach might have slight different in terms of calculation
technique.
-If the decision maker is working with a loss table, the Bayes criterion selects the
decision alternatives having the minimum expected loss.
-The Bayes decision rule (maximizing expected payoffs ) is implemented as
follows:
Step1: For each decision alternative, compute the expected payoff. This is done
by weighting each payoff in the row corresponding to the decision alternative by
the probability of the corresponding event and then summing these terms.
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5. ADDITIONAL NOTES BM014-3-3-DMKG
Step2: Select the decision alternatives having the maximum expected payoff. This
decision is called a Bayes Decision.
Notationally, we shall let R denote payoff (reward) and L denote loss. Also, the
expected payoff if we choose action a will be written ER (a).
Example:
Suppose you are given the payoff table shown in the table below. You are also
told that the probabilities of occurance for the three events, s1, s2, s2 are 0.2, 0.7
and 0.1, respectively. So, P(s1) = 0.2, P (s2) = 0.7, P(s3) = 0.1, where the P
denotes “probability.”
s1 s2 s3
a1 10 15 13
a2 7 20 15
a3 8 20 10
Determine the Bayes decision rules using the maximum expected payoff rule.
Solution:
The expected payoff if we select a1 is computed as follows:
ER(a1) = (0.20) (10) + (0.70) (15) + (0.10) (13) = 13.8
ER(a2) = (0.20) (7) + (0.70) (20) + (0.10) (15) = 16.9
ER(a3) = (0.20) ( 8) + (0.70) (20) + (0.10) (10) = 16.6
The maximum payoff is a2. Thus the Bayes decision is a2.
Expected Value (EV) Approach
By means of EV principle, we find out the expected value of an alternative. This is
repeated for all the alternatives. The formula of this principle is the following:
EV (alternatives d1)
= (payoff of first state of nature) × (Probability of first state of nature)
+(payoff of second state of nature) ×(probability of second state of nature)
+…………………………………………………………………………...
+(payoff of last state of nature) × (probability of last state of nature)
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6. ADDITIONAL NOTES BM014-3-3-DMKG
Mathematically:
n
EV (d1) = ∑ V ij P (Sj)
J=1
Where n = total number of states of nature
Sj = jth state of nature
Vij = Payoff of d1 with respect to Sj
P (Sj) = probability of Sj
The best alternative is that one which will entail highest expected value. The working is
shown in the following table:
GEC PEC(1-p =0.4) EV
(p=0.6)
Apartment(d1) 15 7 15 × 0.6 + 7 × 0.4 = 11.8
Office Building(d2) 22 -4 22 × 0.6 + (-4) × 0.4 = 11.6
Warehouse(d3) 12 9 12 × 0.6 + 9 × 0.4 =10.8
Best alternative
Remark: The expected value of 11.8 (highest in the present case) does not mean that the
chosen alternative, i.e, apartment building will result the profit $11.8miilion; rather it is
one of 15 million and 7 million will result. The expected value means that if the same
decision situation arises a large number of times, then on the average payoff of $11.8
million will result.
Expected Opportunity Loss (EOL) Approach
Firstly we need to from opportunity loss table (the procedure is the same with the
minimax regret approach above), Subtract each entry in a column from the largest entry
in that column.
That alternative is the best which gives the least EOL.
GEC(p=0.6) PEC(1-p=0.4) EOL
Apartment(d1) 22-15 =7 9-7=2 7 × 0.6 + 2 × 0.4 = 5
Office 22-22 =0 9-(-4) =13 0 × 0.6 + 13 × 0.4 = 5.2
Building(d2)
Warehouse(d3) 22-12 =10 9-9 = 0 10 × 0.6 + 0 × 0.4 = 6
Best alternative
Note: the best alternative is d1; Apartment, same as given by EV principle. This is not
coincidence. The best alternative given by both the methods will always be the same.
Expected value of perfect Information (EVPI)
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7. ADDITIONAL NOTES BM014-3-3-DMKG
-Given a probabilistic decision problem, what would it be worth to the decision maker to
have access to an information source that would indicate for certain which of the events
will occur?
-Such an information source would offer perfect information to the decision maker.
-The expected value of such information is referred to as the expected value of perfect
information (EVPI).
-In general, the formula to calculate for EVPI is:
EVPI = (EVwPI – EVwoPI)
Example:
-Suppose Azlan purchase additional information regarding the occurrence of future states
of nature. Azlan hires an economic forecaster to do the analysis.
-Assume that any findings given by the forecaster is completely perfect/correct.
-Assume : study provide “perfect” information, thus company is certain which state of
nature is going to happen.
alternatives GEC PEC
Apartment (d1) 15 7
Office (d2) 22 -4
Warehouse (d3) 12 9
Choose the best
-In GEC, select d2 & gain pay off of $22m
-In PEC, select d3 & gain payoff of 9m What is the EV?
If P (s1) = 0.6 There is a 60% probability that the perfect information will indicate
good economic condition & d2 will provide $22m profit.
If P(s2) = 0.4 There is a 40% probability that the perfect information will indicate
poor economic condition & d3 will provide $9m profit.
EV with perfect info (EVwPI0 = 22 × 0.6 + 9 × 0.4 = $16.8
EV without perfect info (EVwoPI) = $ 11.8 because we choose the highest value.
alternatives GEC(p=0.6) PEC(p=0.4) Expected Value (EV)
Apartment (d1) 15 7 (15)(0.6) + ( 7 ) (0.4) = 11.8
Office (d2) 22 -4 (22) (0.6) + (-4) ( 0.4) = 11.6
Warehouse (d3) 12 9 (12) (0.6) + (9) (0.4) =10.8
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8. ADDITIONAL NOTES BM014-3-3-DMKG
EVPI = EVwPI – EVwoPI
EVPI = $16.8 - $ 11.8
= $5m
Additional EV that can be obtained if perfect info available
Maximum amount that the company should be willing to pay to
purchase the info.
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