Transforming LPP into transportation matrix

Question

I need to transform an LPP into a transportation matrix, however, where I would expect greater than' for constraints on demand, the LPP only has 'less-than' constraints. Any hints on transforming this to a transportation problem.'

Answer 1

Introduce nonnegative slack variables to convert each constraint to an equality. For example, introduce $x_{14}\ge 0$ and replace the first constraint with $x_{11}+x_{12}+x_{13}+x_{14}=13$. Now interpret the slack variables $x_{i4}$ as flow variables into a dummy destination node and the slack variables $x_{3j}$ as flow variables from a dummy source node. Also introduce $x_{34}\ge 0$ to make the bipartite graph complete. The corresponding transportation matrix is $3 \times 4$, and the new entries all have $0$ cost.