Q.

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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a

30, 60

b

60, 30

c

40, 60

d

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 Question Image 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 Question ImageFor high grade barrel:
Production by Refinery A in x days Question ImageProduction by Refinery B in y days Question ImageTotal minimum required productionQuestion ImageFor medium grade barrel:
Production by Refinery A in x days Question ImageProduction by Refinery B in y days Question ImageTotal minimum required productionQuestion ImageFor low grade barrel:
Production by Refinery A in x days Question ImageProduction by Refinery B in y days Question ImageTotal minimum required productionQuestion ImageJust keep in mind that x and y cannot be negative, Question ImageSo, the linear programming problem is expressed as,
Minimize, Question ImageSubject to: Question ImageQuestion ImageNow draw the corresponding lines to the equations that have been created by converting all of the inequalities that represent the constraints.
Question ImageQuestion ImageAs 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 Question Imageat the intersections A, B and C respectively.
Question ImageAs 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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