1. QUANTITATIVE METHODS 1
SAMIR K. SRIVASTAVA

2. 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 maker’s 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.
2/25/2019
Quantitative Methods 1

3. 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
2/25/2019
Quantitative Methods 1

4. 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!
2/25/2019
Quantitative Methods 1

5. 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 < X
2/25/2019
Quantitative Methods 1

6. 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 < X) = Probability (Demand  X-1)
Then,
Expected Marginal Profit = K1 P(A)
Expected Marginal Loss = K2 P(B)
2/25/2019
Quantitative Methods 1

7. 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 (X+1)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 33
2/25/2019
Quantitative Methods 1

8. 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.
2/25/2019
Quantitative Methods 1

9. 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.
2/25/2019
Quantitative Methods 1

10. 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 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?
2/25/2019
Quantitative Methods 1

11. Decision Tree Approach
CASH495 35
145
495
500 40
150
0.66
0.34
0.818
0.182
0.176
0.824
0.6
0.4
The Decision Tree
Find oil
Drill
Find no oil
Test says oil
Don’t Drill
Find oil
Test says no oil
Drill
Take Seismic Test
Find no oil
Don’t Drill
Don’t take Test
Find oil
Drill
Find no oil
Don’t Drill
2/25/2019
Quantitative Methods 1

12. 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
2/25/2019
Quantitative Methods 1

13. Decision Tree Approach
Calculation of EMVs
We start from North-east corner and fold-back.
EMV of the decision “to drill” = * *0.182
= (> )
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?
(495000* *35000 = < )
2/25/2019
Quantitative Methods 1

14. Decision Tree Approach
The Reduced Decision Tree
Take Seismic Test
Don’t take Test
Test says oil
Test says no oil
0.66
0.34
411280
14500
EMV of the decision “to drill”, when no test is taken = * *0.4 = (> 15000)
It is better to go for drilling if the test is not taken.
EMV of taking the test = * *0.34 =
Hence, the decision should be to “Take the Seismic Test”.
Result says oil ---- Drill
Result says no oil Do not drill
2/25/2019
Quantitative Methods 1

15. Decision Tree Approach
Another example
The DTC in New Delhi operates the bus system at Rs 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.
2/25/2019
Quantitative Methods 1

16. Decision Tree Approach
0.4
0.5
0.1
0.2
0.8
0.3
Deficit in ‘000 Rs
600
1800
3000
1500
2400
800
4000
The Decision Tree
2 Years
1440
1890
Expand
Decreased Use
450
2831
1950
0.3
Don’t Expand Now
Don’t Expand
500
Operating Deficit
800
0.7
2040
Sustained Use
Expand
Capital Outlay 300
450
2220
2220
Expand now
2560
Don’t Expand
Increased Use
Sustained Use
Decreased Use
2/25/2019
Quantitative Methods 1

17. 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 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
2/25/2019
Quantitative Methods 1

18. Thank You !
2/25/2019
Quantitative Methods 1