1. Find the dual problems of the following primal problems. Solve the primal problems. a) Min z = -x1 – 4x2 – 3x3 s.t 2x1 + 2x2 + x3 < 4 x1 + 2x2 + 2x3 < 17 X1, X2, X3 2 0
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- b) Maximize Z = −40X1 −100X2s.t 10X1 + 5X2 ≤ 2502X1 + 5X2 ≤ 1002X1 + 3X2 ≤ 90X1, X2 ≥ 0Solve by simplex method, what are the solutions? Show that this problem hasmultiple solutions and find the solutions?Find the values of x1 and x2 where the following two constraints intersect. (Round your answers to 3 decimal places.) (1) 10x1 + 5x2 ≥ 50 (2) 1x1 + 2x2 ≥ 12For the products A, B, C, and D, which of the following could be a linear programming objective function? Select one: a. Z = 1A + 2BC + 3D b. Z = 1A + 2AB + 3ABC + 4ABCD c. Z = 1A + 2B + 3C + 4D d. Z = 1A + 2B/C + 3D
- Set up the simplex matrix used to solve the linear programming problem. Assume all variables are nonnegative. Maximize f = 8x + 9y + 3z subject to 2x + 7y + 8z ≤ 100 6x + 3y + z ≤ 160 3x + 4y + 9z ≤ 10 .Use the simplex method to solve. Maximize z = 4x1 + 2x2, subject to 3x1 + x2 < 22 3x1 + 4x2 < 34 x1 > 0, x2 > 0 x1 = x2 = x3 =Find the optimal solution of the following LP models. Maximize z= 15x1+20x2 Subject to: x1+2x2 ≥ 10 2x1 - 3x2 ≤ 6 x1+x2 ≥ 6 x1,x2 ≥ 0
- Consider the given LP:Maximize z = 4x1 + 6x2 + 8x3Subject to 3x1 + 2x2 + 5x3<= 30 9x1 + 2x2 + 7x3 <= 126 2x1 + 3x2 + x3<= 60 xi>= 01-Find the optimal values of Z, x1, x2, x3 by using the simplex method.3-If the RHS of the constraint 1 is 34 instead of 30, what is the new value of Z?Solve by the Big M – method: Maximize Ζ= x1 + 2x2 −3x3 + x4 Subject to x1 +2x2 +3x3 =15 2x1 +x2 + 5x3 = 20 x1 +2x2 +x3 +x4 = 10 x1 + 4x2 +6x3 ≤ 5 Where x x x x1, 2, 3, 4 ≥ 0Analyze algebraically what special case in simplex application is present in each of the LP model below. Give an explanation to support your answer. a) Maximize z = 4x1 + 2x2 Subject to: 2x1 - x2 ≤ 2 3x1 - 4x2 ≤ 8 x1, x2 ≥ 0b) Maximize z = 3x1 + 2x2 Subject to: 4x1 - x2 ≤ 8 4x1 + 3x2 ≤ 12 4x1 + x2 ≤ 8 x1, x2 ≥ 0
- Find solution using BigM (penalty) method.Maximize Z = x1 + 2x2 + 3x3 - x4subject to the constraintsx1 + 2x2 + 3x3 = 152x1 + x2 + 5x3 = 20x1 + 2x2 + x3 + x4 = 10and x1, x2, x3, x4 ≥ 0Given the following LP: Min 4x1 + 6x2 s.t 2x1 + 2x2 ≥ 3, x1 + 3x2 ≥ 2, x1 + 3x2 ≥ 0, Find its dual LP. Use graphical method or the Simplex algorithm, solve the primal LP in (1). Use graphical method or the Simplex algorithm, solve the dual LP (that you find in Part A). Find its dual LP.Solve the following problem after finding its dual. Min z = x1 - 3x2 + 3x3 s.t. 3x1 - x2 + 2 x3 ≤ 7 2x1 + 4x2 ≥ -12 -4x1 + 3x2 + 8x3 ≤ 10 x1, x2, x3 ≥ 0