The m-Cost ATSP

Although the m-ATSP (or multi traveling salesman problem) is well known for its importance in scheduling and vehicle routing, it has, to the best of our knowledge, never been studied polyhedraly, i.e., it has always been transformed to the standard ATSP. This transformation is valid only if the cost of an arc from node i to node j is the same for all machines. In many practical applications this is not the case, machines produce with diierent speeds and require diierent (usually sequence dependent) setup times. We present rst results of a polyhedral analysis of the m-ATSP in full generality. For this we exploit the tight relation between the subproblem for one machine and the prize collecting travel-ing salesman problem. We show that, for m 3 machines, all facets of the one machine subproblem also deene facets of the m-ATSP polytope. In particular the inequalities corresponding to the subtour elimination constraints in the one machine subproblems are facet deening for m-ATSP for m 2 and can be separated in polynomial time. Furthermore, they imply the subtour elimination constraints for the ATSP-problem obtained via the standard transformation for identical machines. In addition , we identify a new class of facet deening inequalities of the one machine subproblem, that are also facet deening for m-ATSP for m 2. To illustrate the eecacy of the approach we present numerical results for a scheduling problem with non-identical machines, arising in the production of gift wrap at Herlitz PBS AG.