Transcript Treatments of Risks and Uncertainty in Projects

Treatments of Risks and
Uncertainty in Projects
• The availability of partial or 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)
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
Expected Value Criterion
• Expressed in terms of either actual money or its utility
• Decision Maker’s attitude towards the worth or utility of
money is important
• The final decision should ultimately be made
by
considering all pertinent factors that affect the decision
maker’s attitude 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
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 n machines 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
• Let pt be the probability that machine would break down in period t
• Let nt be the random variable representing the number of broken
machines in the same period.
• C1 is the cost of repairing a broken machine
• C2 the preventive maintenance of the machine
• The expected cost per period can be written as
 T 1

 c1  Ent   c2 n 

EC(T )   t 1
T
•
•
•
•
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*)
• 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
1
2
3
4
5
pt
0.05
0.07
0.10
0.13
0.18
Cumulative pt
0
0.05
0.12
0.22
0.35
EC(T)
500
375
366.7
400
450
Expected Value-Variance Criterion
• We indicated that the expected value criterion is suitable
for making “long-run” decisions
• To make it work for the short-run decision problems
Expected Value-Variance criterion 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
maker’s attitude towards excessive deviation from the
expected values.
• Applying this criteria to example 1 we get
T
pT
pT 2
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
Cum. pT
• 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
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 single
value that has the highest probability of occurrence
Decisions under uncertainty
• They assume that there is no probability
distributions available to the random variable.
• The methods under this are
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•
•
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The Laplace Criterion
The Minimax criterion
The Savage criterion
The Hurwicz criterion
Laplace Criterion
• This Criterion is based on what is known as the principle
of insufficiency
• ai is the selection yielding the largest expected gain
• Selection of the action ai* corresponding
1 n

max   v ai ,  j 
 n j 1

ai
• where 1/n is the probability that
Example 2
•
A recreational facility must decide on the level of supply it must stock to meet the
needs of 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 of supplies are thus suggested with level i being ideal (from
the view point of the costs) if the number of customer falls in category i. Deviation
from these levels results in additional costs either because extra supplies are
stocked needlessly or because demand cannot be satisfied. The table below
provides the costs in thousands of dollars
Customer Category
•
1
2
3
4
a1
5
10
18
25
a2
8
7
8
23
a3
21
18
12
21
a4
30
22
19
15
a1, a2, a3 and a4 are the supplies level
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•
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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.
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,θj) represents loss for the decision
maker, then, for, ai the worst loss regardless of what θj
may be is max θj [v(ai,θj)]
• The minimax criterion then selects the action ai
associated with min ai max θj [v(ai,θj)]
• Similarly if v(ai,θj)] represents gain, the criterion selects
the action ai associated with max ai min θj [v(ai,θj)]
• This is called the maxmini criterion
• Applying this criterion to the Example 2
Customer Category
Supply
1
2
3
4
Max
a1
5
10
18
25
25
a2
8
7
8
23
23
a3
21
18
12
21
21
a4
30
22
19
15
30
• Thus the best level of inventory according to this
criterion is specified by a3
Minimax value
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,θj) =
{
max vak , j  vai , j 
if v is profit
vai , j   minvak , j 
if v is loss
ak
ak
• Applying this criteria to Example 2
• The regret matrix is shown below
Customer Category
Supply
1
2
3
4
Max
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
• Thus the best level of inventory according to this criterion is specified
by a2
Hurwicz Criterion
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•
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 α and 1- α, where 0<= α<=1
If v(ai,θj) represents profit, select the action that yields
•




max max vai , j   1    min vai , j 
j
ai 
j





If v(ai,θj) represents cost, select the action that yields
•




min min vai ,  j   1    max vai ,  j 
ai
j
j






• Applying this criterion to Example 2 Set α =0.5
min
max
αmin+(1-α)max
5
25
15
7
23
15
12
21
16.5
15
30
22.5
minimum
• Resolving with α =0.75 for selecting between a1
and a2