# 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. - Mathematics

Sum

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 of A and B to produce during the given time period. Formulate this as a LPP.

#### Solution

Let x and y units of products A and B were manufactured respectively.
The contribution to profit is Rs 2 for each unit of A and Rs 3 for each unit of B.
Therefore for x units of A and y units of B,the contribution to profit would be Rs 2x and Rs 3y respectively.

​Let Z denote the total profit

Then, Z = Rs (2x + 3y)

Total hours required for grinding, turning, assembling and testing are $x + 2y, 3x + y, 6x + 3y, 5x + 4y$  respectively.

The available capacities of these operations in hours for the given period are grinding 30, turning 60, assembling 200 and testing 200.

$x + 2y \leq 30, 3x + y \leq 60, 6x + 3y \leq 200, 5x + 4y \leq 200$

Units of products cannot be negative.Therefore,

$x, y \geq 0$

Hence, the required LPP is as follows:
Maximize Z = 2x + 3y
subject to

$x + 2y \leq 30,$
$3x + y \leq 60,$
$6x + 3y \leq 200,$
$5x + 4y \leq 200$

Concept: Introduction of Linear Programming
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 30 Linear programming
Exercise 30.1 | Q 8 | Page 16