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 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

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30, 60

60, 30

40, 60

60, 40

answer is B.

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Detailed Solution

Concept-In order to minimize z under the given conditions, we need to determine the values of x and y. Examining a linear equation is needed to solve this problem.
For this, we will assume that

is the objective function and also that R is the feasible region (convex polygon) for the LPP. Then, at least one of the feasible region's corner points (vertexes) has the minimal value of z.
Let us assume that x is the days of refinery A and y is the days of refinery B. Total costs incurred will be equal to

For high grade barrel:
Production by Refinery A in x days

Production by Refinery B in y days

Total minimum required production

For medium grade barrel:
Production by Refinery A in x days

Production by Refinery B in y days

Total minimum required production

For low grade barrel:
Production by Refinery A in x days

Production by Refinery B in y days

Total minimum required production

Just keep in mind that x and y cannot be negative,

So, the linear programming problem is expressed as,
Minimize,

Subject to:

Now draw the corresponding lines to the equations that have been created by converting all of the inequalities that represent the constraints.
Question Image

only lies in the first quadrant. From the graph, we have noticed that the shaded region is the feasible region.
At coordinates (120, 0) and (0, 65), we obtain the intersection of the line (1) with the coordinate axes.
Similarly, at coordinates (66.66, 0) and (0, 50), we obtain the intersection of the line (2) with the coordinate axes and at (120, 0) and (0, 65), we obtain the intersection of the line (3) with the coordinate axes.
The feasible region with corner points A (0, 150), B (60, 30) and C (120, 0) is the common shaded region.
Let us just calculate values for the objective function

at the intersections A, B and C respectively.
Question Image

As a result, we discover that the minimum value of the objective function Z is 33000 at (60, 30).
Hence, the correct answer is option 2) 60, 30

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