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.

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Solution

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 $-$ (20x + 24y)
= 5x + 6y

Let Z denote the total profit

Then, Z = $5 x + 6 y$

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.

$∴ 3 x + 2 y \leq 130$

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.

$∴ 4 x + 6 y \leq 260$

Units of products cannot be negative.Therefore,
$x, y \geq 0$

Hence, the required LPP is as follows:

Maximize Z = 5x + 6y

subject to

$3 x + 2 y \leq 130, 4 x + 6 y \leq 260, x \geq 0, y \geq 0$

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