Vitamins A and B are found in two different foods F 1 and F 2 . One unit of food F 1 contains 2 units of vitamin A and 3 units of vitamin . One unit of food F 2 contains 4 units of vitamin A and 2 units of vitamin B . One unit of food F 1 and F 2 cost Rs 50 and 25 respectively. The minimum daily requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that any thing in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F 1 and F 2 at the minimum cost which meets the daily minimum requirement of vitamin A and B . Formulate this as a LPP .

Let x and y units of food F 1 and food F 2 were mixed. Clearly, x ≥ 0 and y ≥ 0 One unit of food F 1 contains 2 units of vitamin A and one unit of of food F 2 contains 4 units of vitamin A . Therefore, x and y units of food F 1 and food F 2 respectively contains 2 x and 4 y units of vitamin A . It is given that the minimum daily requirements for a person of vitamin A is 40 units. Hence, 2 x + 4 y ≥ 40 One unit of food F 1 contains 3 units of vitamin B and one unit of food F 2 contains 2 units of of vitamin B . Therefore, x and y units of F 1 and F 2 respectively contains 3 x and 2 y units of vitamin B . It is given that the minimum daily requirements for a person of vitamin B is 50 units. Hence, 3 x + 2 y ≥ 50 One unit of food F 1 and food F 2 cost Rs 50 and 25 respectively. Therefore, x and y units of food F 1 and food F 2 costs Rs 50 x and Rs 25 y respectively. Let Z denote the total cost Then, Z = Rs ( 50 x + 25 y ) Hence, the required LPP is Minimise Z = 50 x + 25 y Subject to 2 x + 4 y ≥ 40 3 x + 2 y ≥ 50 x ≥ 0 , y ≥ 0

A firm manufactures two products, each of which must be processed through two departments, 1 and 2 . The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows: Product A Product B Weekly capacity Department 1 3 2 130 Department 2 4 6 260 Selling price per unit ₹ 25 ₹ 30 Labour cost per unit ₹ 16 ₹ 20 Raw material cost per unit ₹ 4 ₹ 4 The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP .

Let x and y units of product A and B were manufactured respectively. Labour cost per unit to manufacture product A and product B is Rs 16 and Rs 20 respectively. Therefore, labour cost for x and y units of product A and product B is Rs 16 x and Rs 20 y respectively. Total labour cost to manufacture product A and product B is Rs ( 16 x + 20 y ) Raw material cost per unit to manufacture product A and product B is Rs 4 and Rs 4 respectively. Therefore,raw material cost for x and y units of product A and product B is Rs 4 x and Rs 4 y respectively. Total raw material cost to manufacture product A and product B is Rs ( 4 x + 4 y ) Hence, total cost price to manufacture product A and product B = Total labour cost + Total raw material cost = 16 x + 4 x + 20 y + 4 y = 20 x + 24 y Selling price per unit for product A and product B is Rs 25 and Rs 30 respectively. Therefore, total selling price for product A and product B is Rs 25 x and Rs 30 y respectively. Total selling price = 25 x + 30 y ∴ Total profit = Total selling price − Total cost price = 25 x + 30 y - ( 20 x + 24 y ) = 5 x + 6 y Let Z denote the total profit Then, Z = 5 x + 5 y One unit of product A and product B requires 3 hours and 2 hours respectively at department 1 . Therefore, x units and y units of product A and product B require 3 x hours and 2 y hours respectively. The weekly capacity of department 1 is 130 . ∴ 3 x + 2 y ≤ 130 One unit of product A and B requires 4 hours and 6 hours respectively at department 2 . Therefore, x units and y units of product A and product B require 4 x hours and 6 y hours respectively. The weekly capacity of department 2 is 260 . ∴ 4 x + 6 y ≤ 260 Units of products cannot be negative. Therefore, x , y ≥ 0 Hence, the required LPP is as follows: Maximise Z = 5 x + 6 y Subject to 3 x + 2 y ≤ 130 , 4 x + 6 y ≤ 260 , x ≥ 0 , y ≥ 0

HARD

12th CBSE

IMPORTANT

Earn 100

A company sells two different products $A$ and $B$ . The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of $45000$ man-hours. It takes $5$ hours to produce a unit of $A$ and $3$ hours to produce a unit of $B$ . The market has been surveyed and company officials feel that the maximum number of units of $A$ that can be sold is $7000$ and that of $B$ is $10, 000$ . If the profit is $Rs 60$ per unit for the product $A$ and $Rs 40$ per unit for the product $B$ , how many units of each product should be sold to maximize profit? Formulate the problem as $LPP$ .

Important Questions on Linear Programming

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 7

To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients-calcium, protein and calories and the person's diet consists of only two food items, $I$ and $II$ , whose price and nutrient contents are shown in the table below:

	Food $I$ (per $l b$ )	Food $II$ (per $l b$ )	Minimum daily requirementfor the nutrient
Calcium	$10$	$5$	$20$
Protein	$5$	$4$	$20$
Calories	$2$	$6$	$13$
Price $(Rs)$	$60$	$100$

What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a $LPP$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 8

A manufacturer can produce two products, $A$ and $B$ , during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products $A$ and $B$ are given below.

	$A$	$B$
Grinding	$1$	$2$
Turning	$3$	$1$
Assembling	$6$	$3$
Testing	$5$	$4$

The available capacities of these operations in hours for the given time period are: grinding

30

; turning

60

, assembling

200

; testing

200

. The contribution to profit is

Rs 20

for each unit of

A

and

Rs 30

for each unit of

B

. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount ofAandBto produce during the given time period. Formulate this as a

LPP

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 9

Vitamins

A

and

B

are found in two different foods

F_{1}

and

F_{2}

. One unit of food

F_{1}

contains

2

units of vitamin

A

and

3

units of vitamin . One unit of food

F_{2}

contains

4

units of vitamin

A

and

2

units of vitamin

B

. One unit of food

F_{1}

and

F_{2}

cost

Rs 50

and

25

respectively. The minimum daily requirements for a person of vitamin

A

and

B

40

and

50

units respectively. Assuming that any thing in excess of daily minimum requirement of vitamin

A

and $B$ is not harmful, find out the optimum mixture of food

F_{1}

and

F_{2}

at the minimum cost which meets the daily minimum requirement of vitamin

A

and

B

. Formulate this as a

LPP

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 10

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop $A$ , which performs the basic assembly operation, must work

5

man-days on each truck but only

2

man-days on each automobile. Shop $B$ , which performs finishing operations, must work

3

man-days for each automobile or truck that it produces. Because of men and machine limitations, shop $A$ has

180

man-days per week available while shop

B

has

135

man-days per week. If the manufacturer makes a profit of

Rs 30000

on each truck and

Rs 2000

on each automobile, how many of each should he produce to maximize his profit? Formulate this as a

LPP

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 11

A firm manufactures two products, each of which must be processed through two departments, $1$ and $2$ . The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

	Product $A$	Product $B$	Weekly capacity
Department $1$	$3$	$2$	$130$
Department $2$	$4$	$6$	$260$
Selling price per unit	$₹ 25$	$₹ 30$
Labour cost per unit	$₹ 16$	$₹ 20$
Raw material cost per unit	$₹ 4$	$₹ 4$

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a $LPP$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 12

An airline agrees to charter planes for a group. The group needs at least

160

first class seats and at least

300

tourist class seats. The airline must use at least two of its model

314

planes which have

20

first class and

30

tourist class seats. The airline will also use some of its model

535

planes which have

20

first class seats and

60

tourist class seats. Each flight of a model

314

plane costs the company

Rs 100, 000

and each flight of a model

535

plane costs

Rs 150, 000

. How many of each type of plane should be used to minimize the flight cost? Formulate this as a

LPP

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 13

Amit's mathematics teacher has given him three very long lists of problems with the instruction to submit not more than

100

of them (correctly solved) for credit. The problem in the first set are worth

5

points each, those in the second set are worth

4

points each, and those in the third set are worth

6

points each. Amit knows from experience that he requires on the average

3

minutes to solve a

5

point problem,

2

minutes to solve a

4

point problem, and

4

minutes to solve a

6

point problem. Because he has other subjects to worry about, he can not afford to devote more than

3 \frac{1}{2}

hours altogether to his mathematics assignment. Moreover, the first two sets of problems involve numerical calculations and he knows that he cannot stand more than

2 \frac{1}{2}

hours work on this type of problem. Under these circumstances, how many problems in each of these categories shall he do in order to get maximum possible credit for his efforts? Formulate this as a

LPP

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 14

A farmer has a

100

acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is

Rs 1

per kilogram for tomatoes,

Rs 0.75

a head for lettuce and

Rs 2

per kilogram for radishes. The average yield per acre is

2000 kgs

for radishes,

3000

heads of lettuce and

1000

kilograms of radishes. Fertilizer is available at

Rs 0.50

per

kg

and the amount required per acre is

100 kgs

each for tomatoes and lettuce and

50

kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is

5

man-days for tomatoes and radishes and

6

man-days for lettuce. A total of

400

man-days of labour are available at

Rs 20

per man-day. Formulate this problem as a

LPP

to maximize the farmer's total profit.

Important Questions on Linear Programming

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