Consider the linear program min 2x1 + 3x2 s.t. -2x1 + 3x2 6 3x1 + 2x2

Consider the linear program min 2x1 + 3x2 s.t. -2x1 + 3x2 Ú 6 3x1 + 2x2 Ú 12 x1, x2 Ú 0

(a) Establish that subtracting nonnegative surplus variables x3 and x4 leads to the equivalent standard-form:

min 2x1 + 3x2 s.t. -2x1 + 3x2 - x3 = 6 3x1 + 2x2 - x4 = 12 x1, x2, x3, x4 Ú 0

(b) Solve the original LP graphically in an

(x1, x2) plot, and identify an optimal solution.

Also tag each main constraint with the corresponding surplus variable.

(c) Establish that the dual of the standard form in part

(a) is max 6v1 + 12v2 s.t. - 2v1 + 3v2 … 2 3v1 + 2v2 … 3

-v1 … 0

-v2 … 0

(d) State all applicable complementary slackness conditions between the standard-form of part

(a) and the dual of part (c).