QUANTITATIVE METHODS 1

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• QUANTITATIVE METHODS 1

SAMIR K. SRIVASTAVA
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Decision Theory

• Why we need Decision Theory
• Ubiquity of decision problems in different
  fields in day to day life
• Options of alternative ways of action
• Decision makers attitude towards risk

• Classification of Decision Problems
• The information on outcomes are deterministic,
• i.e. known with certainty.
• The information on outcomes are probabilistic
• The probabilities may be known or unknown
• This is commonly referred to as Decision Theory.
• Here, our background in probability theory comes
  handy.

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Some Basic Concepts
Single stage decision problem or sequential In
real life, almost all problems are sequential.
Decision An action taken by a decision maker that
impacts future outcomes. State of Nature A future
event NOT under the control of decision
maker Certainty The decision in which only one
state of nature exists Consequence An interaction
between a decision and a state of nature
Payoff Benefit that accrues from a consequence
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Marginal Analysis

• Why we need Marginal Analysis
• Recall Practice Problem at the end of Session 4
• There are several alternative courses of action.
• Computations become tedious as the no. of values
  the random variable can take increases
• Suppose the newspaper man finds from past data
  that the demand may vary from 31, 32,50.
• Marginal Analysis provides a simple method to aid
  decision-making!

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Marginal Analysis

• How?
• Let us assume that all the demands have equal
  chances of occurrence
• So, each demand has a probability of 1/20
• The problem is to decide the no. of copies in
  order to maximize profit.
• Marginal Analysis proceeds by examining whether
  ordering an additional copy is worthwhile or not!
• Ordering Xth copy may have two consequences
• The copy can be sold
• The copy cannot be sold.

Marginal Profit 0.30 Marginal Loss 0.50
Demand ? X Demand lt X
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Marginal Analysis
Let us use the following notations K1 Marginal
Profit K2 Marginal Loss P(A) Probability
(Demand ? X) 1 - Probability (Demand ? X-1)
P(B) Probability (Demand lt X) Probability
(Demand ? X-1) Then, Expected Marginal Profit
K1 P(A) Expected Marginal Loss K2 P(B)
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Marginal Analysis

• Ordering the Xth copy is worthwhile only when the
  expected profit is more than the expected loss
• i.e. when K1 P(A) ? K2 P(B)
• Hence, K1 1-F(X-1) ? K2 F(X-1)
• F (X-1) ? K1 / (K1 K2) .(1)
• It is worthwhile to order Xth Copy.
• Similarly, it is not worthwhile to order (X1)th
  Copy, if
• F (X) ? K1 / (K1 K2) .(2)
• Thus, X lies in the K1 / (K1 K2) th fractile
  of the demand function.

• 0.625
• 43 here

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Decision Tree Approach

• A Decision Tree is a graphic model of a decision
  process
• It is an analysing process (ROLLBACK PROCESS)
  that starts from RIGHT and works back towards
  LEFT, i.e., Future Decision is made first and
  then rolled back to become part of earlier
  decisions

Represents a Decision Node.
Represents a Chance Node.
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Decision Tree Approach

• Procedure
• Calculate the expected monetary value (EMV) at
  each node
• EMV Probability on each branch emanating from
  the node multiplied by the payoff at the end of
  the branch
• Select the decision that maximizes the EMV
• DECISION Max (all EMV for all the branches
  emanating from the node). Simultaneously, prune
  all the branches with lower EMV.

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Decision Tree Approach
Example Consider that a decision-maker is
confronted with the decision of drilling for oil
for a particular region.The chances of getting
the oil as per geologist's report is known to be
0.6. To start with, he has Rs 1.5 lacs with him.
The consequences of drilling and getting oil and
that of drilling and not getting oil in terms of
cash left after decision are known to be Rs 5
lacs and Rs 40000 respectively. The decision
maker has got an option to undertake a seismic
test that will increase his knowledge about the
oil content in the region. The test costs Rs
5000 however, the benefit is that it predicts
correctly for 90 of the time if oil is actually
there and 70 correctly if oil is not there. What
should he do and why?
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Decision Tree Approach
The Decision Tree
Find oil
Drill
Find no oil
Test says oil
Dont Drill
Find oil
Test says no oil
Drill
Take Seismic Test
Find no oil
Dont Drill
Dont take Test
Find oil
Drill
Find no oil
Dont Drill
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Decision Tree Approach
Calculation of probabilities
P(A) Probability of finding oil 0.6 P(B)
Probability of not finding oil 0.4 P(C/A)
Probability test predicts correctly when oil is
actually there 0.9 P(D/A) Probability test
predicts incorrectly when oil is not there
0.1 P(C/B) Probability test predicts
incorrectly when no oil is there 0.3 P(D/B)
Probability test predicts correctly when oil is
not there 0.7
P(C) Probability that test says oil is
there P(D) Probability that test says no oil is
there P(A/C) Probability of finding oil, given
positive test results P(B/C) Probability of not
finding oil, given positive test results P(A/D)
Probability of finding oil, given negative test
results P(B/D) Probability of not finding oil,
given negative test results
0.66 0.34 0.818 0.182 0.176 0.824
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Decision Tree Approach

• Calculation of EMVs
• We start from North-east corner and fold-back.
• EMV of the decision to drill 4950000.818
  350000.182
• 411280 (gt 145000)
• Once the test says oil, it is better to go for
  drilling.
• Similarly, when test says no oil, not drilling
  is a better option. Why?
• (4950000.0176 0.82435000 115960 lt 145000)

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Decision Tree Approach

• EMV of the decision to drill, when no test is
  taken 5000000.6 400000.4 316000 (gt
  15000)
• It is better to go for drilling if the test is
  not taken.
• EMV of taking the test 4112800.34
  1450000.34 320745
• Hence, the decision should be to Take the
  Seismic Test.
• Result says oil ---- Drill
• Result says no oil ----- Do not drill

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Decision Tree Approach
Another example
The DTC in New Delhi operates the bus system at
Rs 400000 deficit annually. The municipal
corporation has decided to raise bus fares to
offset the deficit. The director believes that
this will decrease ridership unless system
capacity is increased. She suggests that expanded
services be offered simultaneously with the fare
increase to offset negative community reaction
and perhaps increase ridership. An influential
member suggests an alternative plan. He would
increase the fare now, but delay the capacity
expansion decision for two years. If expansion is
delayed, the director is sure that ridership will
either decrease or be sustained at current
levels. If service is expanded two years after
fare increase, ridership may increase, be
sustained or decrease. If service is not expanded
in two years, then ridership will either be
sustained or decrease, not increase. The director
decides to use a decision tree analysis to
evaluate this problem over an eight year planning
horizon.
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Decision Tree Approach
Deficit in 000 Rs 600 1800 3000 1500 2400 600 180
0 3000 1500 2400 800 2400 4000
0.4 0.5 0.1 0.5 0.5 0.2 0.4 0.4 0.2 0.8 0.2 0.5 0
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The Decision Tree
2 Years
1440
1890
Expand
Decreased Use
450
2831
1950
0.3
Dont Expand
Dont Expand Now
500 Operating Deficit 800
0.7
2040
Expand
Sustained Use
Capital Outlay 300
450
2220
2220
Dont Expand
2560
Expand now
Increased Use
Sustained Use
Decreased Use
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Practice Problem

• A newspaper seller gets his copies from the
  newspaper office every morning. He buys each copy
  for Rs 1.50 and sells it for Rs 2.00. However, he
  has to tell the office in advance how many copies
  he will buy. The office takes back the copies he
  is not able to sell and pays him only Rs 1.20 per
  copy. How many copies should he order every day?
• He has estimated the p.d.f. of the daily demand
  as
• f(D) 0.1 for D 30
• 0.2 for D 31
• 0.2 for D 32
• 0.3 for D 33
• 0.1 for D 34
• 0.1 for D 35

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Thank You !