Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 20C / i / Solution

Use payoff $P=-(2x_1-3x_2-2x_3)$, so maximization of $P$ is equivalent to the required minimization. In the given simplex tableau, $x_1$ can increase the payoff. The simplex ratio test has candidate ratios $3/6=1/2$ in the $z_2$ row and $15/1=15$ in the $z_3$ row; the negative coefficient in the $x_2$ row imposes no upper bound. Thus $x_1$ enters the simplex basis and $z_2$ leaves.

Divide the pivot row by $6$, then eliminate $x_1$ from the other rows. The resulting simplex dictionary isAll slack variables and original variables are nonnegative. The last row therefore proves $P\le 8$ for every feasible point; setting the nonbasic variables $x_3,z_1,z_2$ to zero attains it and leaves all basic variables nonnegative. HenceThe slack vector is $(0,0,29/2)$.