Elements of Operational Research 
Time: 3 Hours 
March – 2004 
Marks: 100 


N.B.: 
(1) 
Answer six questions attempting three from each section. 


(2) 
Answers to the two sections must be written in separate answer books. 


(3) 
Figures to the right indicate full marks for the question. 


(4) 
Graph papers, statistical tables, log tables will be supplied on request. 


(5) 
Use of nonprogrammable hand held calculator is allowed. 



SECTION I 

Q.1. 
a) 
What is Operations Research? 
4 

b) 
Explain four models in Operations Research. 
8 

c) 
What are the limitations to Operations Research? 
4 




Q.2. 

A company has four factories situated in different locations in the country. The cost of production (Rs. per unit), the sale price (Rs. per unit), shipping cost (Rs. per unit), monthly capacities and requirements are given below:
Factory 
Sales Agency 
Monthly Capacity 
Cost of Production 
1 
2 
3 
4 
A 
7 
5 
6 
4 
10 
10 
B 
3 
5 
4 
2 
15 
15 
C 
4 
6 
4 
5 
20 
16 
D 
8 
7 
6 
5 
15 
15 
Monthly Requirements:
Units 
8 
12 
18 
22 
Sales Price 
20 
22 
25 
18 
Find the monthly production and distribution schedule which will maximise profits 
17 




Q.3. 
a) 
Explain Hungarian method for solving assignment problem. 
5 

b) 
To stimulate interest and provide an atmosphere for intellectual discussion a finance faculty in management school decides to hold special seminars on four contemporary topics  Leasing, Portfolio, Management, Private Mutual funds, Swaps and Option. Such seminar should be held once per week in the afternoons. However, scheduling these seminars (one for each topic, not more than one seminar per afternoon) has to be done carefully so that the number of students unable to attend a particular seminar is kept minimum. A careful study indicates that number of students who cannot attend a particular seminar.

Leasing 
Portfolio Management 
Private Mutual Fund 
Swaps and Option 
Monday 
50 
40 
60 
20 
Tuesday 
40 
30 
40 
30 
Wednesday 
60 
20 
30 
20 
Thursday 
30 
30 
20 
30 
Friday 
10 
20 
10 
30 
Find an optional Schedule of seminar. Also find out total number of students who will be missing at least one seminar. 
11 




Q.4. 
a) 
Explain the following with reference to Simplex Method of solving an LPP : (i) Degeneracy (ii) Shadow prices 
6 

b) 
A company produces 2 products X and Y each of which requires 3 types of processing. The details are as below:

ProcessI 
Process II 
ProcessIII 
Product X: (hr/unit) 
12 
3 
8 
Product Y : (hr/unit) 
12 
6 
4 
Available capacity per day (hr) 
840 
300 
480 
Profit per unit: Rs. 500, Rs. 700 respectively.
How many of each should be produced to maximise the profits? Solve graphically. 
10 




Q.5. 
a) 
Explain the following with reference to Simplex Method of solving an LPP :(i) Surplus variable (ii) Nonfeasible solution 
5 

b) 
Use Simplex Method to solve the following LPP:
Minimise Z = 30x_{1} + 20X_{2}
subject to
x_{1} + x_{2} <= 8 
6x_{1} + 4x_{2} >= 12 
5x_{1} + 8x_{2} = 20 
x_{1},x_{2} >= 0 

12 


SECTION II 

Q.6. 
a) 
Define slack and floats. 
6 

b) 
For the following work of a repair work, find:
(i) 
Critical path 
(ii) 
Earliest start time, earliest finish time, latest start time and latest finish time. (iii) Total float and free float. 
Activity 
Duration 
12 
1 
23 
5 
24 
3 
35 
4 
45 
2 
46 
5 
47 
9 
57 
4 
67 
2 
78 
2 

10 




Q.7. 

The detailed activities in a building project are given below:
Activity 
Preceding Activity 
Normal 
Crash 
Time (days) 
Cost (Rs.) 
Time (Days) 
Cost (Rs.) 
A 
 
9 
12000 
6 
18000 
B 
A 
14 
14000 
4 
24000 
C 
A 
4 
2000 
3 
2400 
D 
C 
6 
44000 
4 
56000 
E 
 
14 
1600 
13 
1800 
F 
E 
6 
4000 
6 
4000 
G 
B,D 
5 
4000 
3 
4800 
H 
F,G 
2 
12000 
1 
14000 
(i) 
Obtain critical path. 
(ii) 
Indirect cost per day Rs. 5000. 
(iii) 
Find the minimum time schedule and corresponding cost. 

17 




Q.8. 
a) 
Describe any two methods used for decision making with uncertainty. 
4 

b) 
A person wants to invest in two independent investment schemes A and B, but he can undertake only one at a time due to certain constraints. He can choose A first and then stop, or if A is not successful then take B or viceversa. The probability of success of A is 0.9, while for B it is 0.4. The investment in both the schemes requires an initial capital outlay of Rs. 10,000 and both return nothing if the venture is unsuccessful. Successful completion of A will return Rs. 20,000 (over cost) and successful completion of B will return Rs. 24,000 (over cost). Draw decision tree and determine the best strategy. 
12 




Q.9. 
a) 
A firm advertises that it will pay Rs. 15,000 at the end of 8 years who deposit annually Rs. 1000. What interest rate is implicit in the offer? 
6 

b) 
Obtain the expected NPV if the cost equals 12% per annum, cost of project is Rs. 5000, Life = 3 years, probability distribution of cash flows is:
Year 
Cash Flow 
Probability 
1 
1000 
0.1 

1500 
0.2 

2000 
0.4 

2500 
0.2 

3000 
0.1 


1.0 
Year 
Cash Flow 
Probability 
2 
1900 
0.2 

2500 
0.3 

2750 
0.2 

3150 
0.3 


1.0 

10 




Q.10. 
a) 
The cash price of a washing machine is Rs. 20,000. If a person pays Rs. 2,000 at the time of delivery and the remaining in 5 annual installments at the rate of 16 percent per year, what is the annual instalment? 
6 

b) 
A retailer purchases cherries every morning at Rs. 50 a case and sells them for Rs. 80 a case. Any case remaining unsold at the end of the day can be disposed of next day at a Salvage Value of Rs. 20 per case. Past sales have ranged from 15 to 18 cases per day. The following is the record of sales for the past 120 days:
Cases sold : 
15 
16 
17 
18 
Number of Days : 
12 
24 
48 
36 
Find how many cases the retailer should purchase per day to maximise his profit. 
10 