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## Development of LP Problems

Development of LP Problems

Example 1A firm produces two kinds of products A and B and sells them at a gain of Rs. 2 on class A and Rs. 3 on class B. All products are processed on two machines G and H. Class A needs 1 minute of processing time on G and 2 minutes on H; Class B needs 1 minute on G and 1 minute on H. The machine G is obtainable for not more than 6 hours 40 minutes while machine H is obtainable for 10 hours for any working day. Prepare the problem as a linear programming problem.

Assume

x

_{1}be the number of products of class Ax

_{2 }be the number of products of class BAfter comprehending the problem, the known information can be methodically arranged in the form of the table given below.

Class of products (minutes)

Machine

Class A (x

_{1 }units)Class B (x

_{2 }units)Available time (mins)

G

1

1

400

H

2

1

600

Profit per unit

Rs. 2

Rs. 3

As the gain on class A is Rs. 2 per product, 2 x

_{1 }will be the gain on selling x_{1 }units of class A. Likewise, 3x_{2}will be the gain on selling x_{2}units of class B. Thus, total gain (profit) on selling x_{1}units of A & x_{2}units of class B is given throughMaximize Z = 2 x

_{1}+3 x_{2 }(objective function)As machine G takes 1 minute time on class A and 1 minute time on class B, the totality of minutes needed on machine G is given by

_{ }x_{1}+ x_{2}.Likewise, the total number of minutes needed on machine H is given by 2x

_{1 }+ 3x_{2}.But, machine G is obtainable for not more than 6 hours 40 minutes (400 minutes). Consequently,

x

_{1}+ x_{2 }≤ 400first constraintIn addition, the machine H is obtainable only for 10 hours that is 600 minutes, thus,

2 x

_{1 }+ 3x_{2 }≤ 600second constraintAs it is not feasible to manufacture negative quantities

x

_{1 }≥ 0 and x_{2 }≥ 0non-negative restrictionsTherefore

Maximize Z = 2 x

_{1 }+ 3 x_{2}Subject to restrictions

x

_{1 }+ x_{2 }≤ 4002x

_{1 }+ 3x_{2 }≤ 600& non-negativity constraints

x

_{1 }≥ 0 , x_{2 }≥ 0A firm manufactures two products A and B which have raw materials 400 quintals and 450 labour hours. It is identified that 1 unit of product A needs 5 quintals of raw materials and 10 man hours and gives a gain of Rs 45. Product B needs 20 quintals of raw materials, 15 man hours and gives a gain of Rs 80. Generate the LPP.

Answer

Assume

x

_{1}- number of units of product Ax

_{2}- number of units of product BProduct A

Product B

Availability

Raw materials

5

20

400

Man hours

10

15

450

Profit

Rs 45

Rs 80

Therefore

Maximize Z = 45x

_{1}+ 80x_{2}Subject to

5x

_{1}+ 20 x_{2 }≤ 40010x

_{1}+ 15x_{2}≤ 450x

_{1 }≥ 0 , x_{2 }≥ 0Example 3A firm produces 3 products A, B and C. The profits are Rs. 3, Rs. 2 and Rs. 4 correspondingly. The firm has 2 machines and below is given the necessary processing time in minutes for each machine on each product.

Products

Machine

A

B

C

X

4

3

5

Y

2

2

4

Machine X and Y have 2000 and 2500 machine minutes. The firm must produce 100 A's, 200 B's and 50 C's type, but not more than 150 A's.

Answer

Assume

x

_{1}- number of units of product Ax

_{2}- number of units of product B_{3}- number of units of product CProducts

Machine

A

B

C

Availability

X

4

3

5

2000

Y

2

2

4

2500

Profit

3

2

4

Max Z = 3x

_{1}+ 2x_{2}+ 4x_{3}Subject to

4x

_{1}+ 3x_{2}+ 5x_{3}≤ 20002x

_{1}+ 2x_{2}+ 4x_{3}≤ 2500100 ≤ x

_{1}≤ 150x

_{2}≥ 200x

_{3}≥ 50Example 4A company possesses 2 oil mills A and B which have varied production capacities for low, high and medium grade oil. The company signs a contract to deliver oil to a firm every week with 12, 8, 24 barrels of each grade correspondingly. It costs the company Rs 1000 and Rs 800 per day to run the mills A and B. On a day A generates 6, 2, 4 barrels of each grade and B generates 2, 2, 12 barrels of each grade. Formulate an LPP to find out number of days per week each mill will be operated so as to meet the contract cost-effectively.

Answer

Assume

x

_{1}- no. of days a week the mill A has to workx

_{2}- no. of days per week the mill B has to workGrade

A

B

Minimum requirement

Low

6

2

12

High

2

2

8

Medium

4

12

24

Cost per day

Rs 1000

Rs 800

Minimize Z = 1000x

_{1}+ 800 x_{2}Subject to

6x

_{1}+ 2x_{2}≥ 122x

_{1}+ 2x_{2}≥ 84x

_{1}+12x_{2 }≥ 24x

_{1 }≥ 0 , x_{2 }≥ 0Example 5

A company has 3 operational divisions processing, weaving and packing with the capacity to manufacture 3 different kinds of clothes that are suiting, shirting and woolen yielding with the gain of Rs. 2, Rs. 4 and Rs. 3 per meters correspondingly. 1m suiting needs 2 mins in processing 3mins in weaving and 1 min in packing. In the same way 1m of shirting needs 1 min in processing 4 mins in weaving and 3 mins in packing whereas 1m of woolen requires 3 mins in each division. In a week total run time of each department is 60, 40 and 80 hours for weaving, processing and packing department correspondingly. Develop a LPP to find the product to maximize the profit.

AnswerAssume

x

_{1}- number of units of suitingx

_{2 }- number of units of shirtingx

_{3 }- number of units of woolenSuiting

Shirting

Woolen

Available time

Weaving

3

4

3

60

Processing

2

1

3

40

Packing

1

3

3

80

Profit

2

4

3

Maximize Z = 2x

_{1}+ 4x_{2}+ 3x_{3}Subject to

3x

_{1}+ 4x_{2}+ 3x_{3}≤ 602x

_{1}+ 1x_{2}+ 3x_{3}≤ 40x

_{1}+ 3x_{2}+ 3x_{3 }≤ 80x

_{1}≥0, x_{2}≥0, x_{3}≥0Need Linear Programming LP Assignment Help - Homework Help?

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