Q. 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 $200 \,\, 400$ 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
Bihar CECEBihar CECE 2010
Solution:
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=400\,x+300\,y $
Subject to $ 100\,+200\,y \ge 12000 $
$300\,x+400\,y \ge 20000 $
$ 200\, x+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 $(x, y)$
$z=400 x+300 y$
$A_{2}(120,0)$
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.
| 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 |
| Point $(x, y)$ | $z=400 x+300 y$ |
|---|---|
| $A_{2}(120,0)$ | 48000 |
| $p(60,30)$ | 33000 |
| $B_{3}(0,150)$ | 45000 |