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Optimization is the task of finding an optimum solution to a given problem. When the decision variables are discrete we speak of a combinatorial optimization problem. Such a problem is online when decisions have to be made before all data of the problem are known. And we speak of a real-time online problem when online decisions have to be computed within very tight time bounds. This paper surveys the are of combinatorial online and real-time optimization, it discusses, in particular, the concepts with which online and real-time algorithms can be analyzed.
Many optimization problems have several equivalent mathematical models. It is often not apparent which of these models is most suitable for practical computation, in particular, when a certain application with a specific range of instance sizes is in focus. Our paper addresses the Asymmetric Travelling Salesman Problem with time windows (ATSP-TW) from such a point of view. The real--world application we aim at is the control of a stacker crane in a warehouse. We have implemented codes based on three alternative integer programming formulations of the ATSP-TW and more than ten heuristics. Computational results for real-world instances with up to 233 nodes are reported, showing that a new model presented in a companion paper outperforms the other two models we considered --- at least for our special application --- and that the heuristics provide acceptable solutions.
We investigate the problem of designing survivable broadband virtual private networks that employ the Open Shortest Path First (OSPF) routing protocol to route the packages. The capacities available for the links of the network are a minimal capacity plus multiples of a unit capacity. Given the directed communication demands between all pairs of nodes, we wish to select the capacities in a such way, that even in case of a single node or a single link failure a specified percentage of each demand can be satisfied and the costs for these capacities are minimal. We present a mixed--integer linear programming formulation of this problem and several heuristics for its solution. Furthermore, we report on computational results with real-world data.