Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

# An Automobile Manufacturer Makes Automobiles and Trucks in a Factory that is Divided into Two Shops. Formulate this as a Lpp. - Mathematics

Sum

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

#### Solution

Let x number of trucks and number of automobiles were produced to maximize the profit.
Since, the manufacturer makes profit of Rs 30000 on each truck and Rs 2000 on each automobile.
Therefore, on x number of trucks and y number of automobiles profit would be Rs 30000x and Rs 2000y respectively.
Total profit  = Rs (30000x + 2000y)
​Let Z denote the total profit
Then, Z = 30000x + 2000y
Since, 5 man-days and 2 man-days were required to produce each truck and automobile at shop A.
Therefore, 5x man-days and 2y man-days are required to produce trucks  and automobiles at shop A.
Also,
Since 3 man-days were required to produce each truck and automobile at shop B.
Therefore, 3x man-days and 3y man-days are required to produce trucks and y automobiles​.
As, shop A has 180 man-days per week available while shop B has 135 man-days per week.

∴ $5x + 2y \leq 180, 3x + 3y \leq 135$

Number of trucks and automobiles cannot be negative.

∴ $x, y \geq 0$
Hence, the required LPP is as follows :
Maximize Z = 30000x + 2000y
subject to

$5x + 2y \leq 180,$

$3x + 3y \leq 135,$

$x \geq 0, y \geq 0$

Concept: Introduction of Linear Programming
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 30 Linear programming
Exercise 30.1 | Q 10 | Page 16