question_answer

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 he run to minimize costs while satisfying requirements are

A) 30, 60

B) 60, 30

C) 40, 60

D) 60, 40

Correct Answer: B

Solution :

[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 of 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.