Question

An oil company required 12000, 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 200, 400 and 100 barrels per day of high-grade, medium-grade and low-grade oil, respectively. If refinery A costs ₹ 400 per day and refinery B costs ₹ 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: (A)

The given data may be put in the following tabular form

Refinery	High grade	Medium grade	Low grade	Cos t per day
A	100	300	200	Rs 400
B	200	400	100	Rs 3100
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
$100x+200y\ge 12000$
$300x+400y\ge 20000$
$200x+100y\ge 15000$
and $x,y\ge 0$
The feasible region of the above LPP is represented by the shaded region in the given figure. The corner 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 following table

Point (x , y )	Value of the objective function Z = 400x + 300y
$A_2(120,0)$	$Z=400\times 120+300\times 0=48000$
$P(60,30)$	$Z=400\times 60+300\times 30=33000$
$B_3(0,150)$	$Z=400\times 0+300\times 150=45000$

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