A time‐indexed LP‐based approach for min‐sum job‐shop problems

In this paper we propose two time‐indexed IP formulations for job‐shop scheduling problems with a min‐sum objective. The first model has variables associated to job scheduling patterns. The exponential number of variables calls for a column generation scheme which is carried out by a dynamic programming procedure. The second model is of network flow type with side constraints. This model can be strengthened by adding cutting inequalities of clique type. It turns out that the two models are equivalent, since the dual of the second formulation is equivalent to the compact dual of the first model. However, they require significantly different solution approaches and may behave differently in terms of computing time and memory usage. Good upper bounds are found by a heuristic procedure that randomly generates schedules from fractional solutions. These features allow for an effective pruning of the branch‐and‐bound tree and narrowing the gap between lower and upper bounds. However, the size of both models is critically affected by the time‐indexed formulation which may heavily slow down the computation.