MA 114 Exam 2 Form A Solutions.pdf - MA 114 Introduction to...

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MA 114 - Introduction to Finite Mathematics Spring 2017 Instructor : Lucas Castle Exam 2 - Form A Solutions 1. Consider the following linear programming problem. Maximize x + 2 y subject to the constraints - 2 x + 4 y 80 6 x + 3 y 30 x 0 , y 0 . (a) [10 points] Construct the simplex tableau associated with the problem above, and identify (by inspection) one feasible solution.
(b) [15 points] Use the simplex method to find a second feasible solution to this linear program- ming problem. Is this solution optimal?
First Row: 80 4 = 20 Second Row: 30 3 = 10 The smallest nonnegative ratio tells us the pivot entry. In this case, we select the (2 , 2)-entry (i.e. the 3 in the second column). We pivot as follows: - 2 4 1 0 0 80 6 3 0 1 0 30 - 1 - 2 0 0 1 0 1 3 R 2 R 2 --------→ - 2 4 1 0 0 80 2 1 0 1 3 0 10 - 1 - 2 0 0 1 0 - 4 R 2 + R 1 R 1 2 R 2 + R 3 R 3 ---------------→ - 10 0 1 - 4 3 0 40 2 1 0 1 3 0 10 3 0 0 2 3 1 20 We now can read off a second solution to the problem: Here, x, v are Group I and y, u, M are Group II. Hence we obtain: x = 0 , y = 10 , u = 40 , v = 0 , M = 20 . This solution is optimal since there are no more negatives on the bottom row of the tableau.
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