We study the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from conveyor belts onto pallets. 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. 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 strengthen the hardness of the FIFO Stack-Up by considering practical cases and the distribution of the pallets onto the sequences. We introduce a digraph model for this problem, the so called decision graph, which allows us to give a breadth first search solution. Further we apply methods to solve hard problems to the FIFO Stack-Up problem. In order to evaluate our algorithms, we introduce a method to generate random, but realistic instances for the FIFO Stack-Up problem. Our experimental study of running times shows that the breadth first search solution on the decision graph combined with a cutting technique can be used to solve practical instances on several thousands of 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.