Exercise 2.9 For the following pair of primal-dual problems, determine whether the listed solutions are optimal. min 2a1 + 3x2 2x1 + 3x2 < 30 Tị + 2x2 > 10 max -30y1 + 10y2 -2y1 + -3y1 + + y3 < 2 - y3 < 3 2y2 x2 2 0 Y1, Y3 2 0. x2 2 0 (a) #1 10, x2 = ; Y1 = 0, y2 = 1, y3 = 1. (b) a1 = 20, x2 = 10; y1 = -1, y2 = 4, y3 = 0. 10. %3D (c) ¤1 10 #2 = ; y1 = 0, y2 = }, ys = }. %3D %3D

3.) Solve applying the simplex method to the dual problem. Minimize C= 6y, +8y, +12y, Subject to y, +3y, +3y, 2 6 Y +5y, +5y, 2 4 2y, +2y, +3y, 28 Y1, Y2, y3 20

Question 29 Maximize 2x1 + 4x2 subject to x1 + 3x₂ ��� 12 2x1 + x₂ ≤ 14 120 and ₂ > 0 Using simplex method by introducing slack variables 3 and 4, and M = 2x1 + 4x2, now the inequalities are changed to equations x1 + 3x2 + x3 = 12 (1) 2x1 + x₂ + x4 = 14 (2) 2x14x2 M = 0 (3) To bring 1 and 2 into the solution, which one (1 or 2) is firstly brought into the solution and which equation is used as the pivot equation. O1 and (2) O #2 and (3) O2 and (2) O #1 and (1) O2 and (1)

Use the simplex method to solve the following. Maximize z = 4x1 + 4x2 - 8x3 subject to 5x1 + 5x2 - 10хз 5 95 6х1 + 6x2 + 12х35 128 X120, х2 2 0, Хз 20. ..... Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. Treating x2 as a nonbasic variable, the maximum is when x1 = X2 = X3 , s, = 0 and s2 = %3D B. There is no maximum solution for this problem.