Question

A company produces two different products. One of them needs $\frac{1}{4}$ of an hour of assembly work per unit, $\frac{1}{8}$ of an hour in quality control work and Rs 1.2 in raw materials. The other product requires $\frac{1}{3}$ of an hour of assembly work per unit, $\frac{1}{3}$ of an hour in quality control work and Rs 0.9 in raw materials. Given the current availability of staff in the company, each day there is at most a total of 90 hours available for assembly and 80 hours for quality control. The first product described has a market value (sale price) of Rs 9 per unit and the second product described has a market value (sale price) of Rs 8 per unit. In addition, the maximum amount of daily sales for the first product is estimated to be 200 units, without there being a maximum limit of daily sales for the second product. Formulate and solve graphically the LPP and find the maximum profit.

Answer 1

Let x and y be the no. of units of Product I and II to be produced daily, respectively.
To maximize : Z = Rs {(9 -1.2)x + (8 - 0.9)y} = Rs (7.8x + 7.1y).
Subject to the constraints : $x,y\ge 0;$
$\frac{x}{4}+\frac{y}{3}\le 90$ i.e., $3x+4y\le 1080,$
$\frac{x}{8}+\frac{y}{3}\le 80$ i.e., $3x+8y\le 1920,$
$x\le 200.$
$\underset{–}{CornerPoints}$ $\underset{–}{ValueofZ}$
O(0, 0) 0
A(200, 120) $2412\leftarrow$ Max. Value
B(0, 240) 1704
C(200, 0) 1560
D(80,210) 2115
So clearly, maximum profit is Rs 2412.