 Gomory fractional cut

For an all-integer dictionary row $x_B+{\sum }_j{a}_j{x}_j=b$ with nonnegative variables, this cut follows because $x_B+{\sum }_j\lfloor a_j\rfloor x_j$ is an integer and ${\sum }_j\{a_j\}{x}_j\ge 0$. Consequently that integer is at most $\lfloor b\rfloor$. Negative coefficients use the usual fractional part $u-\lfloor u\rfloor$, so $\{-1/5\}=4/5$. Normalize a resulting inequality sensibly before introducing a new integer-valued slack variable.