Question

An oil company required $13000$, $20000$ and $15000$ barrels of high grade, medium grade and low grade oil respectively. Refinery $A$ produces $100,300$ and $200$ barrels per day of high grade, medium grade and low grade oil respectively. While, refinery $B$ produces $200400$ and $100$ barrels per day of high grade, medium grade and low grade oil respectively. If refinery $A$ costs Rs $400$ per day and refinery $B$ costs Rs $300$ per day to operate, then the days should each be run to minimize costs, while satisfying requirements are

• A. 30, 60
• B. 60, 30
• C. 40, 60
• D. 60, 40

Answer 1

Answer: (B)

The given data may be put in the following tobular form

Refinery	High grade	Medium grade	Low grade	Cost per day
A	100	300	200	Rs 400
B	200	400	100	Rs 300
Minimum requirement	12000	20000	15000

Suppose, refineries A and B should run for $x$ and $y$ days respectively to minimize the total cost. The mathematical form of the above is
Minimize $Z=400x+300y$
Subject to $100+200y\ge 12000$
$300x+400y\ge 20000$
$200x+100y\ge 15000$
and $x,y\ge 0$
The feasible region of the above LPP is represented by the shaded region in the given figure. The comer points of the feasible region are $A_2(120,0),P(60,30)$ and $B_3(0,150).$ The value of the objective function at these points are given in the table.

Point $(x,y)$	$z=400x+300y$
$A_2(120,0)$	48000
$p(60,30)$	33000
$B_3(0,150)$	45000

Clearly, Z is minimum when $x=60,y=30$ . Hence, the machine A should run for 60 days and machine B should run for 30 days to minimize the cost, while satisfying the constraints.