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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 non-programmable 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:

  Process-I Process- II Process-III
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) Non-feasible solution 5
 

b)

Use Simplex Method to solve the following LPP:

Minimise Z = 30x1 + 20X2

subject to

x1 + x2 <= 8

6x1 + 4x2 >= 12

5x1 + 8x2 = 20

x1,x2 >= 0

12
   

SECTION II

 

Q.6. 

a)

Define slack and floats.
 

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

1-2

1

2-3

5

2-4

3

3-5

4

4-5

2

4-6

5

4-7

9

5-7

4

6-7

2

7-8

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 vice-versa. 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
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