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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.
The asymmetric travelling salesman problem with time windows (ATSP-TW) is a basic model for scheduling and routing applications. In this paper we present a formulation of the problem involving only 0/1-variables associated with the arcs of the underlying digraph. This has the advantage of avoiding additional variables as well as the associated (typically very ineffective) linking constraints. In the formulation, time window restrictions are modelled by means of ``infeasible path elimination'' constraints. We present the basic form of these constraints along with some possible strengthenings. Several other classes of valid inequalities derived from related asymmetric travelling salesman problems are also described, along with a lifting theorem. We also study the ATSP-TW polytope, $P_{TW}$, defined as the convex hull of the integer solutions of our model. We show that determining the dimension of $P_{TW}$ is strongly {\em NP}--complete problem, even if only one time window is present. In this latter case, we provide a minimal equation system for $P_{TW}$. Computational experiments on the new formulation are reported in a companion paper [1997] where we show that it outperforms alternative formulations on some classes of problem instances.