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8/10/2019 4th Session PPT
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Risks and uncertainity
Insurable risks
Non-insurable risks
Investment decisions under risk:
Strategies and state of nature. Outcome and pay-off matrix.
Risk-Return evaluation.
Risk preference.
risk seekerrisk averter
risk indifferent
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Adjustments for risk and criteria for decisions:
1. Finite-horizon method
2. Risk discounting model.
3. The shackle approach.
4. The probability theory approach.
5. Sensitivity analysis.
6. Simulation.7. Hedging.
8. Decision tree.
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Investment decisions under
uncertainity
The maximin criteria.
The minimax regret criteria.
The hurwicz alpha index. The maximax criteria.
The laplace criterion.
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Treatments of Risks and Uncertainty in
Projects
The availability ofpartialor imperfect
information about a problem leads to two
new category of decision-making techniques
Decisions under risk (In terms of a probability function)
Decisions under Uncertainty (No probability function is
secure)
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Decisions under risk
Decisions under risk are usually based on one
of the following criteria
Expected Value
Combined Expected value and variance
Known Aspiration level
Most likely occurrence of a future state
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Expected Value Criterion
Expressed in terms of either actual money or its utility
Decision Makers attitude towards the worth or utility of
money is important
The final decision should ultimately be made by consideringall pertinent factors that affect the decision makersattitude
towards the utility of money
The drawback of this is that use of expected value criterion
may be misleading fro the decisions that are applied only a
few number of times i.e small sample sizes
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Example 1
A preventive maintenance policy requires making decisions about when a
machine (or a piece of equipment) should be serviced on a regular basis in
order to minimize the cost of sudden breakdown
The decision situation is summarized as follows. A machine in a group of nmachines is serviced when it breaks down. At the end of T periods,
preventive maintenance is performed by servicing all n machines. The
decision problem is to determine the optimum T that minimizes the total
cost per period of servicing broken machines and applying preventive
maintenance
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Let ptbe the probability that machine would break down in period t
Let ntbe the random variable representing the number of broken machines in
the same period.
C1is the cost of repairing a broken machine C2 the preventive maintenance of the machine
The expected cost per period can be written as
Where E{nt}is the expected number of broken machines in period t.
nt is a binominal random variable with parameter (n,pt), E{nt}=npt
The necessary condition for T*to minimize EC(T) are
EC(T*-1)>= EC(T*) and EC(T*+1)>= EC(T*)
T
ncnEcTEC
T
tt
1
121
)(
8/10/2019 4th Session PPT
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To illustrate the above formulation, suppose
c1=Rs.100, c2=Rs.10 and n=50
The values of pt and EC(T) are tabulated below
T*
T pt Cumulative pt EC(T)
1 0.05 0 500
2 0.07 0.05 375
3 0.10 0.12 366.7
4 0.13 0.22 400
5 0.18 0.35 450
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Expected Value-Variance Criterion
We indicated that the expected value criterion is suitable formaking long-run decisions
To make it work for the short-run decision problems Expected
Value-Variancecriterion is used
A possible criterion reflecting this objective is Max E[Z]-k*var[z]
Where z is a random variable for profit and k is a constant
referred to as risk aversion factor
Risk aversion factor k is an indicator of the decision makersattitude towards excessive deviation from the expected
values.
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Applying this criteria to example 1 we get
Ct is the variance of EC(T) This criteria has resulted in a more conservative decision that applies
preventive maintenance every period compared with every third period
previously
T pT pT2 Cum. pT Cum. pT2 EC(T)+varcT
1 0.05 0.0025 0 0 500
2 0.07 0.0049 0.05 0.0025 6312
3 0.10 0.0100 0.12 0.0074 6622
4 0.13 0.0169 0.22 0.0174 6731
5 0.18 0.0324 0.35 0.0343 6764
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Aspiration Level Criterion
This method does not yield an optimal decision in the sense
of maximizing profit or minimizing cost
It is a means of determining acceptable courses of action
Most Likely Future Criterion
Converting the probabilistic situation into deterministic
situation by replacing the random variable with the singlevalue that has the highest probability of occurrence
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Decisions under uncertainty
They assume that there is no probability distributionsavailable to the random variable.
The methods under this are
The Laplace Criterion The Minimax criterion
The Savage criterion
The Hurwicz criterion
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Laplace Criterion
This Criterion is based on what is known as theprinciple of
insufficiency
ai is the selection yielding the largest expected gain
Selection of the action ai
*corresponding
where 1/n is the probability that
ia
n
j
jiavn
1
,1
max
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Example 2
A recreational facility must decide on the level of supply it must stock to meet the needsof its customers during one of the holiday. The exact number of customers is not known,but it is expected to be of four categories:200,250,300 or 350 customers. Four levels ofsupplies are thus suggested with level i being ideal (from the view point of the costs) if thenumber of customer falls in category i. Deviation from these levels results in additionalcosts either because extra supplies are stocked needlessly or because demand cannot besatisfied. The table below provides the costs in thousands of dollars
a1, a2, a3 and a4 are the supplies level
Customer Category
3 4
a1 5 10 18 25
a2 8 7 8 23
a3 21 18 12 21
a4 30 22 19 15
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Solution by Laplace Criterion
E{a1} = (1/4)(5+10+18+25) = 14.5
E{a2} = (1/4)(8+7+8+23) = 11.5
E{a3} = (1/4)(21+18+12+21) = 18.0
E{a4} = (1/4)(30+22+19+15) = 21.5
Thus the best level of inventory according to
Laplace criterion is specified by a2.
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Minimax (Maxmini) Criterion
This is the most conservative criterion since it is based on
making the best out of the worst possible conditions
If the outcome v(ai,j)represents loss for the decision maker,
then, for, aithe worst loss regardless of what jmay be is max
j [v(ai,j)]
The minimax criterion then selects the action ai associated
with min ai max j [v(ai,j)]
Similarly if v(ai,j)] represents gain, the criterion selects the
action aiassociated with max ai min j [v(ai,j)]
This is called themaxmini criterion
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Applying this criterion to the Example 2
Thus the best level of inventory according to this criterion isspecified by a3
Customer Category
MaxSupply 3 4
a1 5 10 18 25 25
a2 8 7 8 23 23
a3 21 18 12 21 21
a4 30 22 19 15 30
Minimax value
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Savage Minimax Regret criterion
This is an extremely conservative method
The Savage Criterion introduces what is called as regret matrix
which is defined as
r(ai,j)={jijk
aavav
k
,,max
jkaji avav k ,min,
if v is profit
if vis loss
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Applying this criteria to Example 2
The regret matrix is shown below
Thus the best level of inventory according to this criterion is specified by a2
Customer Category
MaxSupply 1 2 3 4
a1 0 3 10 10 10
a2 3 0 0 8 8
a3 16 11 4 6 16
a4 25 15 11 0 25
minimax
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Hurwicz Criterion
This Criterion represents a range of attitudes from the most optimistic to the most
pessimistic
The Hurwicz criterion strikes a balance between extreme pessimism and extreme
optimism by weighing the above two conditions by the respective weights and
1- , where 0
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Applying this criterion to Example 2 Set =0.5
Resolving with =0.75 for selecting between a1 and
a2
min max min+(1-)max
5 25 15
7 23 15
12 21 16.5
15 30 22.5
minimum