#### Operations Research Important Questions

11th Standard

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Time : 01:40:00 Hrs
Total Marks : 50

Part A

5 x 1 = 5
1. The critical path of the following network is

(a)

1 – 2 – 4 – 5

(b)

1– 3– 5

(c)

1 – 2 – 3 – 5

(d)

1 – 2 – 3 – 4 – 5

2. In a network while numbering the events which one of the following statement is false?

(a)

Event numbers should be unique

(b)

Event numbering should be carried out on a sequential basis from left to right

(c)

The initial event is numbered 0 or 1

(d)

The head of an arrow should always bear a number lesser than the one assigned at the tail of the arrow

3. In the given graph the coordinates of M1 are

(a)

x1=5, x2=30

(b)

x1=20, x2=16

(c)

x1=10, x2=20

(d)

x1=20, x2=30

4. The maximum value of the objective function Z = 3x + 5y subject to the constraints x > 0 , y > 0 and 2x + 5y ≤10 is

(a)

6

(b)

15

(c)

25

(d)

31

5. The objective of network analysis is to

(a)

Minimize total project cost

(b)

Minimize total project duration

(c)

Minimize production delays, interruption and conflicts

(d)

All the above

6. Part B

5 x 2 = 10
7. Draw a network diagram for the project whose activities and their predecessor relationships are given below:

 Activity: A B C D E F G H I J K Predecessor activity: - - - A B B C D F H,I F,G
8. Draw the event oriented network for the following data:

 Events 1 2 3 4 5 6 7 Immediate Predecessors - 1 1 2,3 3 4,5 5,6
9. Construct the network for each the projects consisting of various activities and their precedence relationships are as given below:

 Activity A B C D E F G H I J K Immediate Predecessors - - - A B B C D E H,I F,G
10. A fruit grower can use two typ~s of fertilizers in his garden, brand P and brand Q. The amounts (in Kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs atIeast 240 kgs of phosphoric acid, at least 270 kg of potash and atmost 310 kg of chlorine. If the grower wants to minimize the amount of nitrogen added to the garden, formulate the above as mathematical LPP.

11. A retired person has Rs. 70,000 to invest and two types of bonds are available in the market for investment. First type of bond yields an annual income of 8% on the amount invested and the second type yields 10% per annum. As per norms, he has to invest a minimum of Rs. 10,000 in the first type and not more than Rs.30,000 in the second type. How should he plan his investment, so as to get maximum returns after one year of investment? Formulate the above as LPP.

12. Part C

5 x 3 = 15
13. A soft drink company has two bottling plants C1 and C2. Each plant produces three different soft drinks S1, S2 and S3. The production of the two plants in number of bottles per day are:

 Product Plant C1 C2 S1 3000 1000 S2 1000 1000 S3 2000 6000

A market survey indicates that during the month of April there will be a demand for 24000 bottles of S1, 16000 bottles of S2 and 48000 bottles of S3. The operating costs, per day, of running plants C1 and C2 are respectively Rs.600 and Rs.400. How many days should the firm run each plant in April so that the production cost is minimized while still meeting the market demand? Formulate the above as a linear programming model.

14. Maximize Z = 3x1 + 4x2 subject to x1 – x2 < –1; –x1+x2 < 0 and x1, x2 ≥ 0

15. Solve the following LPP graphically.$Maximize\quad Z=-{ x }_{ 1 }+2{ x }_{ 2 }$
Subject to the constraints $-{ x }_{ 1 }+3{ x }_{ 2 }\le 10,\quad { x }_{ 1 }+{ x }_{ 2 }\le 6,\quad { x }_{ 1 }{ -x }_{ 2 }\le 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0$

16. Solve the following LPP graphically, Minimize $Z=3{ x }_{ 1 }+5{ x }_{ 2 }$
Subject to the constraints ${ x }_{ 1 }+3{ x }_{ 2 }\ge 3,\quad { x }_{ 1 }+{ x }_{ 2 }\ge 2\quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0.$

17. Construct the network for the projects consisting of various activities and their precedence relationships are as given below:

 Immediate Predecessor A B C D E F G H I Activity B C D,E,F G I H J K L
18. Part D

4 x 5 = 20
19. A company is producing three products P1, P2 and P3, with profit contribution of Rs.20, Rs.25 and Rs.15 per unit respectively. The resource requirements per unit of each of the products and total availability are given below.​​​​​​​

 Product P1 P2 P3 Total availability Man hours/unit 6 3 12 200 Machine hours/unit 2 5 4 350 Material/unit 1kg 2kg 1kg 100kg

Formulate the above as a linear programming model.

20. Solve the following LPP.
Maximize Z= 2 x1 +3x2 subject to constraints x1 + x2 ≤ 30 ; x2 ≤ 12; x1 ≤ 20 and x1, x2≥ 0.

21. One kind of the cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of other ingredients used in making the cakes?

22. The following table use the activities in a building project.

 Activity Duration (days) 1-2 1-3 2-3 2-4 3-4 4-5 21 26 11 13 5 11

Draw the network for the project, calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and find the critical path. Compute the project duration.