Linear Programming

Q35 In a LPP, the linear inequalities or restrictions on the variables are called........... FillBlank Q36 In a LPP, the objective function is always........... FillBlank Q37 In the feasible region for a LPP is…………., then the optimal value of the objective function $Z = ax + by$ may or may not exist. FillBlank Q38 In a LPP, if the objective function $Z = ax +$ by has the same maximum value on two corner points of the feasible region, then every point on the line segment joining these two points give the same…………value. FillBlank Q39 A feasible region of a system of linear inequalities is said to be……., if it can be enclosed within a circle. FillBlank Q40 A corner point of a feasible region is a point in the region which is the ............. of two boundary lines. FillBlank Q41 The feasible region for an LPP is always a …………….polygon. FillBlank Q26 . The corner points of the feasible region determined by the system of linear constraints are (0,0),(0,40),(20,40),(60,20),(60,0). The objective function is $Z = 4x + 3y$. Compare the quantity in column $A$ and column $B$.

MCQ Q27 The feasible Solution for a LPP is shown in following figure.

Let $Z = 3x - 4y$ be the objective function. Minimum of $Z$ occurs at MCQ Q28 Refer to question 27. Maximum of $Z$ occurs at MCQ Q29 Refer to question 27, maximum value of $Z +$ minimum value of $Z$ is equal to MCQ Q30 The feasible region for an LPP is shown in the following figure.

Let $F = 3x - 4y$ be the objective function. Maximum value of $F$ is MCQ Q31 Refer to question 30. Minimum value of $F$ is MCQ Q32 Corner points of the feasible region for an LPP are (0,2),(3,0),(6,0), (6,8) and (0,5) . Let $F = 4x + 6y$ be the objective function. The minimum value of $F$ occurs at

Solution

Hence, minimum value of $F$ occurs at any points on the line segment joining the points (0,2) and (3,0) . MCQ Q33 Refer to question 32, maximum of $F -$ minimum of $F$ is equal to MCQ Q34 Corner points of the feasible region determined by the system of linear constraints are (0,3),(1,1) and (3,0) . Let $Z = px + qy$, where $p,$ $q > 0$. Condition on $p$ and $q$, so that the minimum of $Z$ occurs at (3,0) and (1,1) is

So, condition of $p$ and $q$,

so that the minimum of $Z$ occurs at (3,0) and (1,1) is

$p + q = 3p \Rightarrow 2p = q$

$\therefore p = \frac{q}{2}$ MCQ Q1 Determine the maximum value of $Z = 11x + 7y$ subject to the constraints$2x + y \le 6,x \le 2,x \ge 0,y \ge 0$ SA Q2 Maximise $Z = 3x + 4y,$ subject to the constraints $x + y \le 1,x \ge 0,y \ge 0$. SA Q3 Maximise the function $Z = 11x + 7y$, subject to the constraints $x \le 3$, $y \le 2,x \ge 0$ and $y \ge 0$. SA Q4 Minimise $Z = 13x - 15y$ subject to the constraints $x + y \le 7$,$2x - 3y + 6 \ge 0$, $x \ge 0$ and $y \ge 0$. SA Q5 Determine the maximum value of $Z = 3x + 4y$, if the feasible region (shaded) for a LPP is shown in following figure. SA Q6 Feasible region (shaded) for a LPP is shown in following figure. Maximise $Z = 5x + 7y$. SA Q7 The feasible region for a LPP is shown in following figure. Find the minimum value of $Z = 11x + 7y$. SA Q8 Refer to question 7 above. Find the maximum value of Z. SA Q9 The feasible region for a LPP is shown in the following figure. Evaluate $Z = 4x + y$ at each of the corner points of this region. Find the minimum value of Z, if it exists. SA Q10 In following figure, the feasible region (shaded) for a LPP is shown. Determine the maximum and minimum value of $Z = x + 2y$. SA Q11 A manufacturer of electronic circuits has a stock of 200 resistors, 120 transistors and 150 capacitors and is required to produce two types of circuits A and B. Type A requires 20 resistors, 10 transistors and 10 capacitors. Type B requires 10 resistors, 20 transistors and 30 capacitors. If the profit on type A circuit is Rs.50 and that on type B circuit is Rs.60, formulate this problem as a LPP, so that the manufacturer can maximise his profit. SA Q12 A firm has to transport 1200 packages using large vans which can carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs.400 and each small van is Rs.200. Not more than Rs.3000 is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimise cost. SA Q13 A company manufactures two types of screws A and B. All the screws have to pass through a threading machine and a slotting machine. A box of type A screws requires 2 min on the threading machine and 3 min on the slotting machine. A box of type B screws requires 8 min on the threading machine and 2 min on the slotting machine. In a week, each machine is available for 60 h. On selling these screws, the company gets a profit of Rs.100 per box on type A screws and Rs.170 per box on type B screws.
Formulate this problem as a LPP given that the objective is to maximise profit. SA Q14 A company manufactures two types of sweaters type A and type B. It costs Rs.360 to make a type A sweater and Rs.120 to make a type B sweater. The company can make atmost 300 sweaters and spend atmost Rs.72000 a day. The number of sweaters of type B cannot exceed the number of sweaters of type A by more than 100. The company makes a profit of Rs.200 for each sweater of type A and Rs.120 for every sweater of type B. Formulate this problem as a LPP to maximise the profit to the company. SA Q15 A man rides his motorcycle at the speed of 50 km/h. He has to spend Rs.2 per km on petrol. If he rides it at a faster speed of 80 km/h, the petrol cost increases to Rs.3 per km. He has atmost Rs.120 to spend on petrol and one hour's time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem. SA Q16 Refer to question 11. How many of circuits of type A and of type B, should be produced by the manufacturer, so as to maximise his profit? Determine the maximum profit. SA Q17 Refer to question 12. What will be the minimum cost? SA Q18 . Refer to question 13. Solve the linear programming problem and determine the maximum profit to the manufacturer. SA Q19 Refer to question 14. How many sweaters of each type should the company make in a day to get a maximum profit? What is the maximum profit? SA Q20 Refer to question 15. Determine the maximum distance that the man can travel. SA Q21 Maximise $Z = x + y$ subject to $x + 4y \le 8,$ $2x + 3y \le 12,$ $3x + y \le 9$, $x \ge 0$ and $y \ge 0$. SA Q22 A manufacturer produces two models of bikes-model X and model Y. Model X takes a 6 man-hours to make per unit, while model Y takes 10 man hours per unit. There is a total of 450 man-hour available per week. Handling and marketing costs are Rs.2000 and Rs.1000 per unit for models X and Y, respectively. The total funds available for these purposes are Rs.80000 per week. Profits per unit for models X and Y are Rs.1000 and Rs.500, respectively. How many bikes of each model should the manufacturer produce, so as to yield a maximum profit? Find the maximum profit. SA Q23 In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in mg/tablet) are given as below

The person needs atleast 18 mg of iron, 21 mg of calcium and 16 mg of vitamins. The price of each tablet of X and Y is Rs.2 and Rs.1, respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost? SA Q24 A company makes 3 model of calculators; A, B and C at factory I and factory II. The company has orders for atleast 6400 calculators of model A, 4000 calculators of model B and 4800 calculators of model C. At factory I, 50 calculators of model A, 50 of model B and 30 of model C are made everyday; at factory II, 40 calculators of model A, 20 of model B and 40 of model C are made everyday. It costs Rs.12000 and Rs.15000 each day to operate factory I and II, respectively. Find the number of days each factory should operate to minimise the operating costs and still meet the demand. SA Q25 Maximise and minimise $Z = 3x - 4y$ subject to $x - 2y \le 0,$ $- 3x + y \le 4$ $x - y \le 6$ and $x,$ $y \ge 0$. SA

Q1 Maximize $Z = 3x + 4y$
subject to the constraints : $x + y \le 4,x \ge 0,y \ge 0.$ SA Q2 Minimize $Z = - 3x + 4y$ subject to $x + 2y \le 8,3x + 2y \le 12,x \ge 0,y \ge 0.$ SA Q3 Maximize $Z = 5x + 3y$ subject to$3x + 5y \le 15,5x + 2y \le 10,x \ge 0,y0.$ SA Q4 Minimize $Z = 3x + 5y$ subject to$x + 3y \ge 3,x + y \ge 2,x,y \ge 0$ . SA Q5 Maximize $Z = 3x + 2y$ subject to $x + 2y \le 10,3x + y \le 15,x,y \ge 0.$ SA Q6 Minimize $Z = x + 2y$ subject to $2x + y \ge 3,x + 2y \ge 6,x,y \ge 0.$ SA Q7 Minimize and Maximize $Z = 5x + 10y$ subject to $x + 2y \le 120,x + y \ge 60,x - 2y \ge 0,x,y \ge 0.$ SA Q8 Minimize and Maximize $Z = x + 2y$ subject to $x + 2y \ge 100,2x - y \le 0,2x + y \le 200;x,y \ge 0$ . SA Q9 Maximize$Z = - x + 2y$, subject to the constraints:
$x \ge 3,x + y \ge 5,x + 2y \ge 6,y \ge 0.$ SA Q10 Maximize $Z = x + y,$ subject to$x - y \le - 1, - x + y \le 0,x,y \ge 0$ . SA

Q1 Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs. 60/kg and Food Q costs Rs. 80/kg. Food P contains 3 units/kg of Vitamin A and 5 units/kg of Vitamin B while food Q contains 4 units/kg of Vitamin A and 2 units/kg of Vitamin B. Determine the minimum cost of the mixture. SA Q2 One kind of 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 the other ingredients used in making the cakes. SA Q3 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 craftsman'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, then find the maximum profit of the factory when it works at full capacity. SA Q4 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 each should be produced each day so as to maximise his profit, if he operates his machines for at most 12 hours a day ? SA Q5 A factory manufactures two types of screws A and B. Each type of screw requires 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 5 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 maximise his profit ? Determine the maximum profit. SA Q6 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 manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture 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 ail the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit ? SA Q7 A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type 5 require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is Rs. 5 each for type A and Rs.6 each for type 5 souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit ? SA Q8 A merchant plants to sell two types of personal computers - a desktop model and a portable model that will cost Rs. 25000 and Rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 unite. 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. SA Q9 A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods ${F_1}$ and ${F_2}$ are available. Food ${F_1}$ costs Rs. 4 per unit food and ${F_2}$ costs Rs. 6 per unit. One unit of food ${F_1}$ contains 3 units of vitamin A and 4 units of minerals and one unit of food ${F_2}$ contains 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 the minimal nutritional requirements. SA Q10 There are two types of fertilisers ${F_1}$ and ${F_2} \cdot {F_1}$ consists of 10\% nitrogen and 6\% phosphoric acid and ${F_2}$ consists of 5\% nitrogen and 10\% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If ${F_1}$ costs Rs.6/kg and ${F_2}$ costs Rs. 5/kg, then determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost ? SA Q11 The corner points of the feasible region determined by the following system of linear inequalities:
$2x + y \le 10,x + 3y \le 15,x,y \ge 0\;are\left( {0,0} \right),\left( {5,0} \right),\left( {3,4} \right)$ and (0, 5). Let $Z = px + qy,$ where $p,q > 0$. Condition on p and q so that maximum of Z occurs at both (3, 4) and (0, 5) is

(A) $p = q$

(B) $p = 2q$

(C) $P = 3q$

(D) $q = 3p$ SA

Q1 A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. 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 atleast 240 units of calcium, atleast 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to maximise the amount of vitamin A in the diet ? What is the maximum amount of vitamin A in the diet? SA Q2 A farmer mixes two brands P and Q of cattle feed. Brand P costing Rs.250 per bag contains 3 units of nutritional element A,2.5 units of element B and 2 units of element C. 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 ? SA Q3 A dietician wishes to mix together two. kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg food is given below:

One kg of food X costs Rs.16 and one kg of food Y costs Rs. 20. Find the least cost of the mixture which will produce the required diet ? SA Q4 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 Rs. 7.50 and that on each toy of type B is Rs. 5, then show that 15 toys of type A and 30 toys of type B should be manufactured in a day to get maximum profit. SA Q5 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 maximise the profit for the airline. What is the maximum profit ? SA Q6 Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops D, E, and £ whose requirements are 60, 50 and 40 quintals respectively. The costs of transportation per quintal from the godowns to the shops are given in the following table:

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost? SA Q7 An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distance(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 Rs. 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum ? What is the minimum cost ? SA Q9 Refer to Question 8. If the grower wants to maximise the amount of nitrogen added to the garden, then how many bags of each brand should be added ? What is the maximum amount of nitrogen added? SA Q10 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 dolls A and B, then how many of each should be produced weekly in order to maximise the profit? SA