Integer Programming Models and Parameterized Algorithms for Controlling Palletizers

We study the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from conveyor belts onto pallets. This is done by palletizers or palletizing robots. Given k sequences of labeled bins and a positive integer p, the goal is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put it onto a pallet located at one of p stack-up places. Each of these pallets has to contain bins of only one label, bins of different labels have to be placed on different pallets. After all bins of one label have been removed from the given sequences, the corresponding stack-up place becomes available for a pallet of bins of another label. All bins have the same size. The FIFO Stack-Up problem asks whether there is some processing of the sequences of bins such that at most p stack-up places are used. In this paper, we introduce three digraph models, namely the processing graph, the decision graph, and the sequence graph, and two linear programming models for the problem. We show that there is a processing of some list of sequences with at most p stack-up places if and only if the sequence graph of this list has directed pathwidth at most p− 1. This connection allows us to show that the FIFO Stack-Up problem is NP-complete in general. Based on these characterizations we give parameterized algorithms for various parameters which imply efficient solutions for the FIFO Stack-Up problem restricted to small parameter values. We also consider an approximation result for the problem. Our experimental study of running times shows that a breadth first search solution on the decision graph can be used to solve practical instances on several thousand bins of the FIFO Stack-Up problem. Further we analyze two integer programming approaches implemented in CPLEX and GLPK. As expected CPLEX can solve the instances much faster than GLPK and our pallet solution approach is much better than the bin solution approach.