Consider the following set of constraints x1 x2 x3 7 2x1

Consider the following set of constraints: x_1 + x_2 + x_3 = 7 2x_1 - 5x_2 + x_3 lessthanorequalto 10 x_1, x_2, x_3 lessthanorequalto 0 Solve the problem for each of the following objective functions: Maximize z = 2x_1 + 3x_2 - 5x_3.

Solution

maximie z= 2x1+3x2-5x3

max z = 2x1 + x2 + 3x3 + 4x4 s.t 4x1 + 2x2 + 5x3 + 5x4 10 4x1 + 2x2 + 5x3 + 5x4 5 3x1 + 5x2 + 4x3 + x4 8 3x1 + 5x2 + 4x3 + x4 15 x1 + x2 + x3 + x4 = 20 x,x2, x3, x4 0 Rewrite the primal problem: max z = 2x1 + x2 + 3x3 + 4x4 s.t 4x1 + 2x2 + 5x3 + 5x4 10 4x1 2x2 5x3 5x4 5 3x1 5x2 4x3 x4 8 3x1 + 5x2 + 4x3 + x4 15 x1 + x2 + x3 + x4 = 20 x,x2, x3, x4 0 Dual Problem: min z0 = 10w1 5w2 8w3 + 15w4 + 20w5 s.t. 4w1 4w2 3w3 + 3w4 + w5 2 2w1 2w2 5w3 + 5w4 + w5 1 5w1 5w2 4w3 + 4w4 + w5 3 5w1 5w2 w3 + w4 + w5 4 w1, w2, w3, w4 0 w5 unrestricted