Question

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	Rs 25	Rs 30
Labour cost per unit	Rs 16	Rs 20
Raw material cost per unit	Rs 4	Rs 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.

Answer 1

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 16x  and Rs 20y respectively.
Total labour cost to manufacture product A and product B is Rs (16x+20y)
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 4x and Rs 4y respectively.
Total raw material cost to manufacture product A and product B is Rs (4x + 4y)
Hence, total cost price to manufacture product A and product B = Total labour cost + Total raw material cost
                                          = 16x + 4x + 20y + 4y 
                                          = 20x + 24y
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 25x and Rs 30y   respectively.
Total selling price = 25x + 30y 
∴  Total profit  = Total selling price − Total cost price = 25x + 30y

-(20 x + 24y) 

                            =5x + 6y

​Let Z denote the total profit

Then, Z = 5x + 5y 

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 3x hours and 2y hours respectively.
The weekly capacity of department 1 is 130.

\[\therefore 3x + 2y \leq 130\]

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 4x hours and 6y hours respectively.
The weekly capacity of department  2 is 260.

\[\therefore 4x + 6y \leq 260\]

Units of products cannot be negative.Therefore,

\[x, y \geq 0\] ​

Hence, the required LPP is as follows:
Maximize Z = 5x + 6y
subject to

\[3x + 2y \leq 130, \]

\[ 4x + 6y \leq 260, \]

\[x \geq 0, y \geq 0\]