The beauty of linear programming theory arises from linear programming duality that provides a short certificate that a given pair of primal and dual feasible solutions are optimal. One of the central fundaments of this duality result is the so-called Farkas Lemma that establishes that a given system of linear inequalities has a solution if and only if another system of linear inequalities is i...

There is a long history of modeling projects to meet time and cost objectives. Most of these models look at adjusting the level of resources available to the project in order to crash the time required to complete certain activities. These models usually take the activities and the graph structure of the project as given and fixed, but in practice there is often significant discretion in how ac...

1 The Integer Programming formulation, and its Linear Relaxation In this section we show, that the minimisation of the extended energy function E(x) = c∈C ψ c (x c) + C(L(x)) (1) can be formulated as an Integer Program (IP). First the objective function is linearised using a vector z of binary indicator variables to represent the assignment of labels.