• # question_answer 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               A)  30, 60                   B)         60, 30   C)         40, 60                   D)         60, 40

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 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.