Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 40 / 1 / Solution

Introduce nonnegative slack variables $s_1,\dots ,s_4$ and write $z$ for the objective value. The initial simplex dictionary, recovered from the original PDF rather than the corrupted converted table, isFor this minimization convention, nonnegative objective coefficients give dual feasibility. The dual simplex algorithm repairs a negative basic value while retaining those nonnegative reduced costs. If the leaving row is $x_B=b+{\sum }_j{a}_j{x}_j$, $b<0$, an entering variable must have $a_j>0$; choose the smallest ratio of its objective coefficient to $a_j$.

Choose $s_1$ to leave. Eligible columns are $x_1,x_3$, with ratios $2/1=2$ and $18/6=3$. Thus $x_1$ enters. Solving its row and substituting givesNow only $s_3$ has a negative basic value. Its only eligible entering column is $x_3$, so pivot there, with $s_3$ leaving. The resulting dictionary isSetting the nonbasic variables $s_1,x_2,s_3$ to zero gives a basic feasible solution. The final objective row also proves $z\ge 23$ for every feasible point, because its nonbasic variables are nonnegative. Hence the unique relaxed optimum is

For the integer program, all four original slack variables are integers whenever the decision variables are integers. The fractional row can be writtenThe Gomory fractional cut keeps the fractional parts of these coefficients: $\{-1\}=\{-2\}=0$ and $\{-1/2\}=1/2$. It therefore gives $\frac{1}{2}{s}_3\ge \frac{1}{2}$, or $s_3\ge 1$. Its validity can also be seen directly: the displayed row makes $s_3$ an odd nonnegative integer. In original variables this cut is $2x_1+10x_3\le 18$.

Introduce the new integer-valued slack $u=s_3-1\ge 0$, with initial row $u=-1+s_3$. It is primal infeasible but retains dual feasibility, so one more dual-simplex pivot has $u$ leave and $s_3$ enter. Substitute $s_3=1+u$ to obtainAll basic values at $s_1=x_2=u=0$ are nonnegative integers, including $s_3=1$. The objective row proves a lower bound of $26$ on every point satisfying the valid cut, hence on every integer-feasible point. It is attained atThe strictly positive nonbasic objective coefficients also show uniqueness. One Gomory cut and its reoptimization are sufficient.