Variable and constraint reduction techniques for the temporal bin packing problem with fire-ups

The aim of this letter is to design and computationally test several improvements for the compact integer linear programming (ILP) formulations of the temporal bin packing problem with fire-ups (TBPP-FU). This problem is a challenging generalization of the classical bin packing problem in which the items, interpreted as jobs of given weight, are active only during an associated time window. The TBPP-FU objective function asks for the minimization of the weighted sum of the number of bins, viewed as servers of given capacity, to execute all the jobs and the total number of fire-ups. The fire-ups count the number of times the servers are activated due to the presence of assigned active jobs. Our contributions are effective procedures to reduce the number of variables and constraints of the ILP formulations proposed in the literature as well as the introduction of new valid inequalities. By extensive computational tests we show that substantial improvements can be achieved and several instances from the literature can be solved to proven optimality for the first time.