Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / T / 2 / d / Solution

At the current dual prices $(1,1)$, the new product has reduced cost $5-(1+2)=2>0$, so it can increase profit. Its transformed resource column isIncreasing $x_4$ by $t$ changes the old basic variables to $x_1=9+7t$ and $x_3=3-3t$. Thus $x_3$ leaves at $t=1$, givingFor a certificate, take dual prices $(1/3,7/3)$. They price the four products at $3,17/3,26/3,5$, respectively, each at least its profit. Their resource total is $33/3+18(7/3)=53$, equal to the achieved profit. Weak duality proves optimality, and the profit has increased by two.