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Maximize
Z = 3x + 4y, subject to the constraints,
.
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Minimum
Z = ?3x + 4y subject to the constraints, x + 2y
3x + 2y
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Maximum Z = 5x + 3y, subject to 3x
+ 5y 5x + 2y
,
.
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Minimize P = 3x + 5y, such that
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Miximize
Z = 3x + 2y subject to
x + 2y
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Minimize
Z = x + 2y, subject to
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Minimize
and Maximize Z = 5x + 10y, subject to x + 2y
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Minimize
and Maximize Z = x + 2y, subject to x + 2y
-
Maximize
Z = ?x + 2y, subject to the constraints
-
Maximum
Z = x + y subject to
x ? y
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Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixure contain at least 8 units of vitamin A and 11 units of vitamin B. Food costs Rs.6/kg. and food Q costs Rs.80/kg. Food P contains 3 units/kg of vitamin A and 5 unit/kg. Of vitamin B while food Q contains 4 units/kg of vitamin A and 2 units/kg of B.
Determine the minimum cost of the mixture.
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One kind of cake requires 200 g of flour and 25g of fat, and another kind of cake requires 100g 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 the other ingredients, used in making the cakes.
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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 hour 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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A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 horus on machine B to produce a package of nuts while it takes 3 hours on machine A and 1 hours 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 may packages of each should be produced each day so as to maximize his profit, it he operates his machines for at the most 12 hours a day ?
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A factory manufactures two types of services. A and B; each type requiring the use of two machines an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs.7 and screws B at a profit of Rs.10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit Determine the maximum profit.
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A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/ cutting machine and 3 hours on the sprayer to manufacturer a pedestal lamp, while it takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacturer a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs.5 and that from a shade is Rs.3. Assuming that the manufacturer can sell all the lamps and shades that manufacturer can sell all the lamps and shades that he produces, how should be schedule his daily production in order to maximize his profit ?
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A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A required 5 minutes each for cutting and 10 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each fror assembling. The profit is Rs.5 each for type A and Rs.6 each for type B souvenirs How many souvenirs of each type should the company manufacture in order to maximize the profit ?
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A merchant plant to sell two types of personal computers – a desktop model and a portable model that will cost Rs.25,000 and Rs.40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs.70 lakhs and if his profit on the desktop model is Rs.4500 and on portable model is Rs.5000.
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A diet is to contain at lest 80 units of vitamins A and 100 units of minerals. Two food F1 and F2 are available. Food F1 costs Rs.4 per unit food and F2 costs Rs.6 per unit. A unit of food F1 contains at least 3 units of vitamin A and 4 units of minerals. A unit of food F2 contains at least 6 units of vitamin A and 3 units of minerals Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets and minimal nutritional requirements.
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There are two types of fertilizers F1 and F2¬, F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a fearmer finds that the needs at least 14kg. f nitrogen and 14kg. Of phosphoric acid for her crop. If F1¬ costs Rs. 6 kg and F2costs Rs.5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost ?
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The corner points of the feasible region determined by the following system of linear inequalities :
2x + y 10, x + 3y x, y are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy where p, q > 0. Conditions on P and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is
(A) p =q (B) p = 2q
(C) p = 3q (D) q = 3p
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(Diet problem) A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A, while each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol, and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A ?
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A farmer mixes two brands P, and Q of cattle feed. Brand P, costing Rs.250 per bag, contains 3 unitso f nutritional element A, 2.5unitso f element B and 2 units of element C whereas brand Q costing Rs.200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units,45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag ? What is the minimum cost of the mixture per bag ?
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A
dietician wishes to mix together two kinds of foods X and Y in such a way that
the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8
unis of vitamin C. The vitamin contents of one kg. food is given below :
|
Food
|
Vitramin
A
|
Vitamin
B
|
Vitamin
C
|
|
X
Y
|
1
2
|
2
2
|
3
1
|
One
kg. of food X costs Rs.16 and one kg. of ood Y costs Rs.20. Find the least cost
of the mixture which will produce a required diet ?
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A
manufacturer makes two types of toys A and B. Three machines are needed for
this purpose and the time (in minutes) required for each toy on the machines is
given below :
Each
machine is available for a maximum of 6 hours per day. If the profit on each
toy of type A is Rs7.50 and that on each toy of type B is Rs.5, show that 15
toys of type A and 30 of type B should be manufactured in a day to get maximum
profit.
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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 th executive class. Determine how many tickets, of each type must be sold in order to maximize the profit for the airline. What is the maximum profit ?
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Two
godowns A and B have grain capacity of 100 quintals and 50 quintals
respectively. They supply to 3 ration shops, D, E and F, whose requirements are
60, 50 and 40 quintals respectively.
The
cost of transportation per quintal from the godowns to the shops are given in
the following table :
How
should the supplies the transported in order that the transportation cost is
minimum cost ?
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Oil
company has two depots A and B with capacities of 7000/ and 4000/respectively.
The company is to supply oil to three petrol pumps, D, E and F, whose
requirements are 4500/, 3000/ and 3500/ respectively. The distances (in km)
between the depots and the petrol pumps is given in the following table :
Assuming
that the transportation cost of 10 litres of oil is Re. 1 per k, how should the
delivery be scheduled in order that the transportation cost is minimum ? What
is the minimum cost ?
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A
fruit grower can use two types 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 mix are given in the table. Tests indicate that the garden needs at
least 240 kg. of phosphoric acid, at least 270 kg. of potash and at most 310
kg. of chlorine.
If
the grower wants to minimize the amount of nitrogen added to the garden, how
many bags of each mix should be used. What is the minimum amount of nitrogen
added?
|
|
Kg. per bag
|
|
Brand P
|
Brand Q
|
|
Nitrogen
|
3
|
3.5
|
|
Phosphoric acid
|
1
|
2
|
|
Potash
|
3
|
1.5
|
|
Chlorine
|
1.5
|
2
|
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A toy company manufactures two types of dolls. A and B, Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further the production level of Dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of Rs.12 and Rs.16 per doll respectively on dols A and B, how many of each should be produced weekly in order to maximize the profit ?
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