Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 2 / 7H (Optimisation)

Introduce slack variables $s_1,s_2,s_3$. In the initial simplex algorithm dictionary, $x_2$ has the largest positive objective coefficient. The ratio test givesso $x_2$ enters and $s_3$ leaves. Solving the third constraint for $x_2$ givesThe objective becomesEvery reduced cost is now nonpositive, so the simplex optimality criterion givesThe other two slacks both equal one.

The dual linear program isThe vectoris dual feasible and has objective value $12$. By weak duality, it and the displayed primal point are optimal; they also satisfy complementary slackness.