#### Operations Research Model Question Paper

11th Standard

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Time : 02:00:00 Hrs
Total Marks : 50
6 x 1 = 6
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. A solution which maximizes or minimizes the given LPP is called

(a)

a solution

(b)

a feasible solution

(c)

an optimal solution

(d)

none of these

3. 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

4. Which of the following is not correct?

(a)

Objective that we aim to maximize or minimize

(b)

Constraints that we need to specify

(c)

Decision variables that we need to determine

(d)

Decision variables are to be unrestricted

5. Network problems have advantage in terms of project

(a)

Scheduling

(b)

Planning

(c)

Controlling

(d)

All the above

6. Given an L.P.P maximize Z=2x1+3x2 subject to the constrains x1+x2≤1, 5x1+5x2≥0 and x1≥0, x2≥0 using graphical method, we observe

(a)

No feasible solution

(b)

unique optimum solution

(c)

multiple optimum solution

(d)

none of these

7. 10 x 2 = 20
8. Draw the logic network for the following:
Activities C and D both follow A, activity E follows C, activity F follows D, activity E and F precedes B.

9. 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
10. Construct a network diagram for the following situation:
A

11. Draw the network for the project whose activities with their relationships are given below:
Activities A,D,E can start simultaneously; B,C>A; G,F>D,C; H>E,F.

12. Construct the network for the projects consisting of various activities and their precedence relationships are as given below:
A, B, C can start simultaneously A<F, E; B<D, C; E, D<G

13. An aeroplane can carry a maximum of 200 passengers. A profit of Rs.1000 is made on each executive class ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. Formulate the mathematical LPP for the above.

14. A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods X and Y. To produce one unit of X, 2 unit of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Yare priced at HOO and H20 per unit respectively, how should the producer use his resources to maximize the total revenue? Formulate the LPP for the above.

15. A dealer whises to purchase a number of fans and sewing machines. He has only Rs.5760 to invest and has a space for atmost 20 items. A fan costs him Rs.360 and a sewing machine Rs. 240. His expectation is he can sell a fan at a profit of Rs.22  and a sewing machine at a profit of ns. Formulate this as an LPP to maximize his profit?

16. Draw a network diagram for the project whose activities and their predecessor relationships are given below

 Activity Predecessor activity A B C D E F - - D A B C
17. Construct the network for the projects consisting of various activities and their precedence relationships are as given below: A, B can start simultaneously
A < D, E; B < F; E < G, D < C, F < H.

18. 3 x 3 = 9
19. Maximize Z = 3x1 + 4x2 subject to x1 – x2 < –1; –x1+x2 < 0 and x1, x2 ≥ 0

20. Solve the following LPP graphically.  $\\ \therefore \quad Maximize\quad Z=3{ x }_{ 1 }+4{ x }_{ 2 }$
subject to the constraints ${ x }_{ 1 }+{ x }_{ 2 }\le 4\quad \quad and\quad { x }_{ 1 },{ x }_{ 2 }\ge 0$

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

22. 3 x 5 = 15
23. Solve the following LPP. Maximize Z = 2 x1 +5x2 subject to the conditions x1+ 4x2 ≤ 24. 3x1+x2 ≤ 21, x1+x2 ≤ 9 and x1, x2 ≥ 0.

24. Solve the following linear programming problems by graphical method.
(i) Maximize Z = 6x1 + 8x2 subject to constraints 30x1+20x2≤300; 5x1+10x2≤110; and x1, x2 > 0.
(ii) Maximize Z =22x1+ 18x2  subject to constraints 960x1+ 640x2≤15360; x1≥ x≤20 and x1, x2 ≥0.
(iii) Minimize Z= 3x1+ 2x1 subject to the constraints 5x1+ x2≥10; x1+x2≥6; x1+ 4, x2 ≥12 and x1, x2≥0.
(iv) Maximize Z= 40x1+ 50x2 = subject to constraints 30x1+x2≤9; x1+2x2≤8 and x1, x2≥0
(v) Maximize Z= 20x1+30x2 subject to constraints 3x1+3x2≤36; 5x1+2x2≤50; 2x1+6x2≤60 and x1,x2
(vi) Minimize Z=20x1+40x2 subject to the constraints 36x1+ 6x2≥108, 3x1+12x2≥36, 20x1+10x2≥100 and x1,x2≥0.

25. Draw the network and calculate the earliest start time, earliest finish time, latest start time and latest finish time of each activity and determine the Critical path of the project and duration to complete the project.

 Jobs 1-2 1-3 2-4 3-4 3-5 4-5 4-6 5-6 Duration 6 5 10 3 4 6 2 9