Integral simplex using decomposition with primal cutting planes

This paper concentrates on the addition of cutting planes to the integral simplex using decomposition (ISUD) of Zaghrouti et al. (Oper Res 62(2):435–449, 2014 ). This method solves the set partitioning problem by iteratively improving an existing feasible solution. We present the algorithm in a primal language and relate it to existing augmenting methods. The resulting theoretical properties, stronger than the ones already known, simplify termination proofs and deepen the geometrical insights on ISUD in particular. We show that primal cuts, that is, cutting planes that are tight at the current feasible integer solution, can be used to improve the performance of the algorithm, and further that such cutting planes are enough to solve each augmentation problem. We propose efficient separation procedures for well-known polyhedral inequalities, namely primal clique and odd-cycle cuts. Numerical results demonstrate the effectiveness of primal cutting planes; tests are performed on small and large-scale set partitioning problems from aircrew and bus-driver scheduling instances up to 1600 constraints and 570,000 variables.