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Linear Programming

Long Answer Type

21.

Solve the following linear programming problem graphically:
Minimise Z = 3x + 2y
subject to the constraints x + y ≥ 8, 3x + 5 y ≤ 15, x ≥ 0, y ≥ 0

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22.

Solve the following linear programming problem graphically:
Minimise Z = x + y
subject to the constraints x - y ≤ - 1, - x + y ≤ 0, x, y ≥ 0.

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23. A furniture dealer deals in the sale of only tables and chairs. He has Rs. 5000 to invest and a space to store at most 60 pieces. A table costs him Rs. 250 and a chair Rs. 50. He can sell a table at a profit of Rs. 50 and a chair at a profit of Rs. 15. Assuming that he can sell all the items that he buys, how should he invest his money in order that he may maximise his profit?

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24. A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profits from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society?

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25. A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 8000 on each piece of model A and Rs. 12000 on each piece of Model B. How many pieces of Model A and Model B should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

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26. One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g 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 the other ingredients used in making the cakes.

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27.

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman’s time.
(i) What number of rackets and bats must be made if the factory is to work at full capacity?
(ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

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28. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs.17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of cach should be produced each day as to maximise his profit, if he operates his machines for at the most 12 hours a day?

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29.
If a young man rides his motor - cycle at 25 km per hour, he has to spend Rs. 2 per km on petrol ; if he rides it at a faster speed of 40 km per hour, the petrol cost increases to Rs. 5 per km. He has Rs. 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as a linear programming problem and then solve it.

Let the young man ride x km at the speed of 25 km per hour and y km at the speed of 40 km per hour. Let f be the total distance covered, which is to be maximized.
∴ f = x + y is the objective function.
Cost of travelling per km is Rs. 2 at the speed of 25 km per hour and cost of travelling per km is Rs. 5 at the speed of 40 km per hour.
∴ total cost of travelling = 2x + 5y
Also Rs. 100 are available for petrol
∴ 2x + 5y ≤ 100
Time taken to cover x km at the speed of 25 km per hour = $straight x over 25 hour$
Time taken to cover y km at the speed of 40 km per hour = $straight y over 40 hour$
Total time available = 1 hours
$therefore space space space space space space we space have space space straight x over 25 plus straight y over 40 less or equal than 1 or space space space 8 straight x plus 5 straight y less or equal than 200 Also comma space straight x greater or equal than 0 comma space space straight y greater or equal than 0$

∴ we are to maximize f = x + y
subjecl to the constraints
2x + 5y ≤ 100
8x + 5y ≤ 200
x ≥ 0, y ≥ 0.
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Let us draw the graph of the line 2 x + 5 y = 100
For x = 0, 5 y - 100 or y = 20
For y = 0, 2 x = 100 or x = 50
∴ line meets OX in A (50, 0) and OY in L (0, 20)
Again we draw the graph of the line
8x + 5y = 200
For x = 0, 5 y = 200 or y = 40
For y = 0, 8 x = 200 or x = 25
∴ line meets OX in B (25, 0) and OY in M (0, 40)

Since feasible region is the region which satisfies all the constraints,
∴ feasible region is the quadrilateral OBCL. The comer points are
$straight O left parenthesis 0 comma space 0 right parenthesis comma space space straight B left parenthesis 25 comma space 0 right parenthesis space straight C open parentheses 50 over 3 comma space 40 over 3 close parentheses comma space space space straight L left parenthesis 0 comma space 20 right parenthesis At space straight O left parenthesis 0 comma space 0 right parenthesis comma space space straight f space equals space 0 space plus space 0 space equals space 0 At space straight B left parenthesis 25 comma space 0 right parenthesis comma space space straight f space equals space 25 plus 0 space equals space 25 At space straight C open parentheses 50 over 3 comma space 40 over 3 close parentheses comma space straight f space equals space 50 over 3 plus 40 over 3 space equals space 30 At space straight L space left parenthesis 0 comma space 20 right parenthesis comma space space straight f space equals space 0 plus 20 space equals space 20 therefore space space space space space space space space maximum space value space of space straight f space equals space 30 space space at space open parentheses 50 over 3 comma space 40 over 3 close parentheses$
∴ the young man covers the maximum distance of 30 km when he rides $50 over 3$ km at the speed of 25 km per hour and $40 over 3$ km at the speed of 40 km per hour.

86 Views

30.

A firm is engaged in producing two products A and B. Each unit of product A requires two kg of raw material and four labour hours for processing whereas each unit of product B requires three kg of raw material and three hours of labour, of the same type. Every week, the firm has an availability of 60 kg of raw material and 96 labour hours. One unit of product A sold yields Rs. 40 and one unit of product B sold gives Rs. 35 as profit.
Formulate this problem as linear programming problem to determine as to how many units of each of the products should be produced per week so that the firm can earn the maximum profit. Assume that there is no marketing constraints so that all that is produced can be sold.

134 Views