# A Small Manufacturing Firm Produces Two Types of Gadgets a and B, Which Are First Processed in the Foundry, Then Sent to the Machine Shop for Finishing. - Mathematics

A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of and B, and the number of man-hours the firm has available per week are as follows:

 Gadget Foundry Machine-shop A 10 5 B 6 4 Firm's capacity per week 1000 600

The profit on the sale of A is Rs 30 per unit as compared with Rs 20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.

#### Solution

Let x and y number of gadgets A and B respectively being produced in order to maximize the profit.
Since, each unit of gadget A takes 10 hours to be produced by machine A and 6 hours to be produced by machine B and each unit of gadget B takes 5 hours to be produced by machine A and 4 hours to be produced by machine B.
Therefore, the total time taken by the Foundry to produce x units of gadget A and units of gadget B is

$10x + 6y$ This must be less than or equal to the total hours available.
Hence,  10x + 6y ≤ 1000.
This is our first constraint.
The total time taken by the machine-shop to produce x units of gadget A and units of gadget B is 5x + 4y. This must be less than or equal to the total hours available.
Hence, 5x + 4y ≤ 600
This is our second constraint.
Since and y are non negative integerstherefore
$x, y \geq$ 0
It is given that the profit on the sale of A is Rs 30 per unit as compared with Rs 20 per unit of B. Therefore, profit gained on x and y number of gadgets A and is Rs 30x and Rs 20y respectively.
Let Z denotes the total cost
Therefore, Z= Rs (30x + 20y)
Hence, the above LPP can be stated mathematically as follows:
Maximize Z = 30x + 20y
subject to
10x + 6y ≤ 1000,
5x + 4y ≤ 600
x, y ≥ 0

Concept: Mathematical Formulation of Linear Programming Problem
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 30 Linear programming
Exercise 30.1 | Q 1 | Page 14