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 lowgrade 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: (B)

The given data may be put in the following tabular form

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