Question

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 labor required in each shop for the production of each unit of A 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 of the sales of A is Rs. 30 per unit as compared with Rs. 20 per unit 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 LPP.

Answer 1

Hint:
We will assume the weekly production of gadgets as x and y. then, we will write an equation for the total profit. Then we will write the constraints with respect to the foundry and machine shop, this will give us the required answer.

Complete step by step solution:
Given that the firm manufactures two types of gadgets A and B, which first get processed in the foundry and then sent to the machine shop for finishing. This can be shown in the table:

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

Let the weekly production of gadgets A and B be x and y.
Given that the profit of sales of A is Rs. 30 and the profit of sales of B be Rs 20, so profit on x and y number of gadgets A and B are 30x and 20y respectively.
Then the total profit P is,
 \[P = 30x + 20y\]
The production of the gadgets A and B requires 10 and 6 hours respectively in the foundry.
i.e. the weekly production of A and B needs \[10x + 6y\] hours in the foundry and the firm capacity per week is 1000 hours.
 \[10x + 6y \leqslant 1000\]
The production of gadgets A and B requires 5 and 4 hours respectively in the machine shop.
The weekly production of A and B needs \[5x + 4y\] hours in the machine shop.
The capacity of a machine shop per week is 600 hours.
 \[5x + 4y \leqslant 600\]

Hence, \[P = 30x + 20y\]
Gives the constraints
 \[10x + 6y \leqslant 1000\]
And \[5x + 4y \leqslant 600\]

Note:
LPP stands for Linear Programming Problem.
LPP is used on a daily basis. LPP is used for making the processes easy and undercharge. LPP is also used in agriculture. Using LPP, farmers decide what crops to plant and what amount of seeds to plant.