Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / T / T2 / a / Solution

With nonnegative production amounts $x_1,x_2,x_3$, the primal linear programme isIntroduce slacks $s_R,s_S\ge 0$. The initial basic variables are the slacks, and the reduced profits of the goods are $5,6,3$. The greatest is six, so $x_2$ enters. The ratio test compares $6/2=3$ and $11/3$, so $s_R$ leaves. The resulting dictionary iswith profit $18-7x_1+3x_3-3s_R$. The greatest positive reduced profit is now three, so $x_3$ enters; its only positive pivot coefficient is two in the second row, and $s_S$ leaves. The final tableau, expressed as equations, isAll nonbasic reduced profits are nonpositive, proving optimality atThe ordered optimal basis is $(x_2,x_3)$, henceThe inverse is read from the two slack columns of the final basic equations.