## Chapter 30 – Linear programming Exercise Ex. 30.1

## Chapter 30 – Linear programming Exercise Ex. 30.2

Find graphically, the maximum value of z = 2x + 5y,

subject to constraints given below:

2x + 4y £ 8

3x + y £ 6

X + y £ 4

X ³ 0, y ³ 0

Converting the inequations into

equations, we obtain the lines

2x + 4y = 8, 3x + y = 6, x + y = 4, x =

0, y = 0.

These lines are drawn on a suitable scale

and the feasible region of the LPP is shaded in the graph.

From the graph we can see the corner

points as (0, 2) and (2, 0).

## Chapter 30 – Linear programming Exercise Ex. 30.3

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 unit / kg of Vitamin A and 2

units/kg of vitamin B. Determine the minimum cost of the mixture.

One kind of cake requires 200 g of flour

and 25 g 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.

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 at least 240units of calcium, atleasy 460 units of iron and at most

300 units of cholesterol. How many packets of each food should be used to

minimize the amount of vitamin A in the diet? What is the minimum amount of

vitamin A?

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 unit 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 pre bag?

Note: Answer given in the book is

incorrect.

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:

Food | Vitamin A | Vitamin B | Vitamin C |

X | 1 | 2 | 3 |

Y | 2 | 2 | 1 |

One kg of food X costs Rs. 16 and one kg

of food costs Rs. 20. Find the least cost of the mixture which will

produce the required diet?

A fruit grower can use two types of fertilizer in his

garden, brand P and Q. The amounts (in kg) of nitrogen, acid potash and

chlorine in a bag of each brand are given in the 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.

Kg per bag | ||

Brand P | Brand Q | |

Nitrogen | 3 | 3.5 |

Phosphoric acid | 1 | 2 |

Potash | 3 | 1.5 |

Chlorine | 1.5 | 2 |

If the grower wants to minimize the amount of nitrogen

added to the garden, how many bags of each brand should be used? What is the

minimum amount of nitrogen added in the garden?

## Chapter 30 – Linear programming Exercise Ex. 30.4

If a young man drives his vehicle at 25km/hr, he has to spend Rs 2per km on petrol. if he drives it as a fast of 40 km/hr, the petrol cost increase to Rs 5 per km. He has Rs 100 to speed on petrol and travel a maximum distance in one hour time with less polution . Express this problem as an LPP and solve it graphically. What value do you find hear?

A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and half an hour to make an item B. The maximum time available per day is 16 hours. The profit on an item of A is Rs 300 and on one item of B is Rs 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.

A manufacturer makes two products, A and B. Product A

sells at Rs. 200 each and takes ½ hour to make. Product B sells at Rs.300 each and

takes 1 hour to make. There is a permanent order for 14 units of product A

and 16 units of product B. A working week consist of 40 hours of production

and the weekly turn over must not be less than Rs. 1000. If the

profit on each of product A is Rs. 20 and an

product B is Rs. 30, then how many of each should be produced so that

the profit is maximum? Also find the maximum profit.

If a young man drives his vehicle at 25 km/hr, he has

to spend Rs. 2 per km on petrol. If he drives is at a faster speed

of 40 km/hr, the petrol cost increases to Rs. 5/ per km. He

has Rs. 100 to spend on

petrol and travel within one hour. Express this as an LPP solve the same.

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 hour of machine time and 24 hours of craftman’s time. If the profit on racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it graphically.

A merchant plans 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 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 his profit on the desktop model is Rs 4500 and on the portable model is RS 5000. Make an LPP and solve it graphically.

A cooperative society of formers has 50 hectare of land

to grow two crops X and Y. The profit 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 at to maximise the total profit of the society?

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. 1200 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?

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 craftrnan’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.

- What number of rackets and bats must be made if the factory is to Work at full capacity?
- 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.

A merchant plans 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 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.

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, how many of each should be produced weekly in

order to maximise the profit?

There are two types of fertilisers F_{1} and F_{2}.

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, determine how much of each type of fertiliser should

be used so that nutrient requirement are met at a minimum cost. What is the

minimum cost?

A manufacturer has three machines I, II and III

installed in his factory. Machines I and II are capable of being operated for

at most 12 hours whereas machine III must be operated for atleast 5 hours a

day. She produces only two items M and N each requiring the use of all the

three machines.

The number of hours required for producing 1 unit of

each of M and N on the three machines are given in the following table:

Items | Number of | ||

I | II | III | |

M | 1 | 2 | 1 |

N | 2 | 1 | 1.25 |

She makes a profit of Rs. 600 and Rs. 400 on items M

and N respectively. How many of each item should she produce so as to maximize

her profit assuming that she can sell all the items that she produced? What

will be the maximum profit?

There are two factories located one at place P and the

other at place Q. From these locations, a certain commodity is to be

delivered to each of the three depots situated at A, B and C. The weekly

requirements of the depots are respectively 5, 5 and 4 units of the commodity

while the production capacity of the factories at P and Q are respectively 8

and 6 units. The cost of transportation per unit is given below:

To/from | Cost (in Rs.) | ||

A | B | C | |

P | 160 | 100 | 150 |

Q | 100 | 120 | 100 |

How many units should be transported from each factory

to each depot in order that the transportation cost is minimum. What will be

the minimum transportation cost?

Let x and y units of commodity be

transported from factory P to the depots at A and B respectively.

Then (8 – x – y) units will be

transported to depot at C.

The flow is shown below.

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:

Types of Toys | Machines | ||

I | II | III | |

A | 12 | 18 | 6 |

B | 6 | 0 | 9 |

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, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.

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. What is the maximum profit?

A manufacturer considers that men and women workers are

equally efficient and so he pays them at the same rate. He has 30 and 17

units of workers (male and female) and capital respectively, which he uses to

produce two types of goods A and B. To produce one unit of A, 2 workers and 3

units of capital are required while 3 workers and 1 unit of capital is

required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit

respectively, how should he use his resources to maximize the total revenue?

Form the above as an LPP and solve graphically. Do you agree with this view

of the manufacturer that men and women workers are equally efficient and so

should be paid at the same rate?

## Chapter 30 – Linear programming Exercise Ex. 30.5

## Chapter 30 – Linear programming Exercise Ex. 30RE

A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs 10,5000 and Rs 9,000 respectivley. 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 thatn 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainages from this land. How much land should be allocated to each crop so as to miximise the total profit of the society?

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?

A manufacturer considers that men

and women workers are equally efficient and so he pays them at the same rate.

He has 30 and 17 units of workers (male and female) and capital respectively,

which he uses to produce two types of goods *A*

and *B*. To produce one unit of *A*, 2 workers and 3 units of capital are required while 3

workers and 1 unit of capital is required to produce

one unit of *B*. If *A *and *B* are priced

at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources

to maximize the total revenue? Form the above as an LPP and solve

graphically. Do you agree with this view of the manufacturer that men and

women workers are equally efficient and so should be paid at the same rate?

Corner points

formed for:

Corner Points | Profit | Remarks |

(0, 0) | Z = 100 × 0 + | |

(0, 10) | Z = 100 × 0 + | |

(3, 8) | Z = 100 × 3 + | Maximum |

(17/3, 0) | Z = 100 × 17/3 |

Revenue is

maximum when x = 3, y = 8.

Maximum Profit =

Rs. 1260

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 minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.

Let us plot the constraints and find the feasible region through graph.

Thus the profit is maximum when the dealer buys 8 electronic sewing machines and 12 manual operated sewing machines.

A manufacturing company makes two

types of teaching aids A and B of Mathematics for class XII. Each type of A

requires 9 labour hours of fabricating and 1 labour hour for finishing. Each

type of B requires 12 labour hours for fabricating and 3 labour hours for

finishing. For fabricating and finishing, the maximum labour hours available per

week are 180 and 30 respectively. The company makes a profit of Rs. 80 on

each piece of type A and Rs. 120 on each piece of type B. How many pieces of

type A and type B should be manufactured per week to get a maximum profit?

Make it as an LPP and solve graphically. What is the maximum profit per week?

Thus profit is

maximum and is equal to Rs.1680

The company should

manufacture 12 Type A machines and 6 Type B machines to maximize their

profit.

## Chapter 30 – Linear programming Exercise MCQ

The solution set of the inequation 2x + y > 5 is

- half plane that contains the origin
- open half plane not contains the origin
- whole xy-plane except the points lying on the line

2x + y =5

- none of these

Correct option: (b)

Objective function of a LPP is

- a constraint
- a function to be optimized
- a relation between the variable
- none of these

Correct option: (b)

Objective function of a LPP is always maximized or minimized. Hence, it is optimized.

Which of these following sets are convex?

- {(x, y):x
^{2}+ y^{2}≥1} - {(x, y):y
^{2 }≥ x} - {(x, y):3x
^{2}+ 4y^{2 }≥ 5} - {(x, y): y≥ 2, y ≤ 4 }

Correct option: (d)

Set of points between two parallel lines. Hence, set is connected. Set is convex.

Let X_{1} and X_{2} are optimal solutions of a LPP, then

- X = λ X
_{1}+ (1- λ) X_{2}, λ ∊ R is also an optimal solution - X = λ X
_{1}+ (1- λ) X_{2}, 0 ≤ λ ≤ 1 gives an optimal solution - X = λ X
_{1}+ (1+ λ) X_{2}, 0≤ λ ≤ 1 gives an optimal solution - X = λ X
_{1 }+ (1+ λ) X_{2 }, λ ∊ R given an optional solution

Correct option: (b)

The maximum value of Z = 4x + 2y subjected to the constraints 2x + 3y ≤ 18, x + y ≥ 10; x, y ≥ 0 is

- 36
- 40
- 20
- None of these

Correct option: (d)

From the graph we conclude that no feasible region exist.

The maximum value of the objective function is attained at the points

- given by intersection of inequations with the axes only
- given by intersection of inequations with x-axis only
- given by corner points of the feasible region
- none of these

Correct option: (c)

The maximum value of the objective function is attained at the points given by corner points of the feasible region.

The maximum value of Z = 4x + 3y subjected to the constraints 3x + 2y ≥ 160, 5x + 2y ≥ 200, x + 2y ≥ 80; x, y ≥ 0 is

- 320
- 300
- 230
- none of these

Correct option: (d)

If we put x=0 and y=0 in all the equations then we get contradiction. Hence, region is on open half plane not containing origin. The region is unbounded we can not find the maximum value of the feasible region.

Consider a LPP given by

Minimum Z = 6x + 10y

Subjected to x ≥ 6; y ≥ 2; 2x + y ≥ 10; x, y ≥ 0

Redundant constraints in this LPP are

- x ≥ 0, y ≥ 0
- x ≥ 6, 2x + y ≥ 10
- 2x + y ≥ 10
- none of these

Correct option: (c)

Minimum Z will be at 2x + y ≥ 10.

The objective function Z = 4x + 3y can be maximized subjected to the constraints 3x + 4 y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y ≤ 6; x, y ≥ 0

- At only one point
- At two points only
- At an infinite number of points
- None of these

Correct option: (c)

If the constraints in a linear programming problem are changed

- the problem is to be re-evaluated
- solution is not defined
- the objective function has to be modified
- the change in constraints is ignored

Correct option: (a)

Optimization of objective function is depend on constraints. Hence, if the constraints in a linear programming problem are changed the problem is to be re-evaluated.

Which of the following statement is correct?

- Every LPP admits an optimal solution
- A LPP admits unique optimal solution
- If a LPP admits two optimal solution it has an infinite number of optimal solution
- The set of all feasible solutions of a LPP is not a converse set

Correct option: (c)

Optimal solution of LPP has three types.

- Unique
- Infinite
- Does not exist.

Hence, it has infinite solution if it admits two optimal solution.

Which of the following is not a convex set?

- {(x, y): 2x + 5y <7}
- {(x, y):x
^{2}+ y^{2}≤ 4} - {x:|x|= 5}
- {(x, y): 3x
^{2}+2y^{2}≤ 6}

Correct option: (c)

As |x|=5 will only on x-axis. Hence, set is not connected to any two points between the set.

Hence, it is not convex.

By graphical method, the solution of linear programming problem

Maximize Z= 3x_{1} + 5x_{2}

Subject to 3x_{1} + 2x_{2}≤ 18

x_{1}≤ 4

x_{2}≤ 6

x_{1}≥ 0, x_{2 }≥ 0, is

- x
_{1 }= 2, x_{2}= 0, Z = 6 - x
_{1 }= 4, x_{2}= 6, Z = 36 - x
_{1 }= 4, x_{2}= 3, Z = 27 - x
_{1 }= 4, x_{2}= 6, Z = 42

Correct option: (b)

The region represented by the inequation system x, y ≥ 0, y ≤ 6, x + y ≤ 3 is

- unbounded in first quadrant
- unbounded in first and second quadrants
- bounded in first quadrant
- none of these

Correct option: (c)

As region is on origin side it is always bounded. Also, given that x,y ≥ 0 it is bounded in the first quadrant.

NOTE: Answer not matching with back answer.

The point at which the maximum value of x + y, subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x, y ≥ 0 is obtained is

- (30, 25)
- (20, 35)
- (35, 20)
- (40, 15)

Correct option: (d)

The value of objective function is maximum under linear constraints

- at the centre of feasible region
- at (0, 0)
- at any vertex of feasible region
- the vertex which is maximum distance from (0, 0)

Correct option: (c)

To find maximum or minimum value of the region we use the coordinates of the vertices of feasible region. Hence, the value of objective function is maximum under linear constraints at any vertex of the feasible region.

Note: Answer not matching with back answer.

The corner points of the feasible region determined by the following system of linear inequalities:

2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0),

(3, 4) and (0,5). Let Z = p x + q y, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is

- p = q
- p = 2q
- p = 3q
- q = 3p

Correct option: (d)

Given that Z=px + qy

Maximum value at (3, 4) = maximum value at (0, 5)

3p+4q=5q

q=3p