Avoiding unnecessary demerging and remerging of multi‐commodity integer flows

Resource flows may merge and demerge at a network node. Sometimes several demerged flows may be immediately merged again, but in different combinations compared to before they were demerged. However, the demerging is unnecessary in the first place if the total resources at each of the network nodes involved remains unchanged. We describe this situation as “unnecessary demerging and remerging (UDR)” of flows, which would incur unnecessary operations and costs in practice. Multi‐commodity integer flows in particular will be considered in this paper. This deficiency could be theoretically overcome by means of fixed‐charge variables, but the practicality of this approach is restricted by the difficulty in solving the corresponding integer linear program (ILP). Moreover, in a problem where the objective function has many cost elements, it would be helpful if such operational costs are optimized implicitly. This paper presents a heuristic branching method within an ILP solver for removing UDR without the use of fixed‐charge variables. We use the concept of “flow potentials” (different from “flow residues” for max‐flows) guided by which underutilized arcs are heuristically banned, thus reducing occurrences of UDR. Flow connection bigraphs and flow connection groups (FCGs) are introduced. We prove that if certain conditions are met, fully utilizing an arc will guarantee an improvement within an FCG. Moreover, a location sub‐model is given when the former cannot guarantee an improvement. More importantly, the heuristic approach can significantly enhance the full fixed‐charge model by warm‐starting. Computational experiments based on real‐world instances have shown the usefulness of the proposed methods

Local convex hulls for a special class of integer multicommodity flow problems

Based on previous work in rolling stock scheduling problems (Alfieri et al. in Transp Sci 40:378–391, 2006; Cacchiani et al. in Math Progr B 124:207–231, 2010; Lin and Kwan in Electron Notes Discret Math 41:165–172, 2013; Schrijver in CWI Q 6:205–217, 1993; Ziarati et al. in Manag Sci 45:1156–1168, 1999), we generalize a local convex hull method for a class of integer multicommodity flow problems, and discuss its feasibility range in high dimensional cases. Suppose a local convex hull can be divided into an up hull, a main hull and a down hull if certain conditions are met, it is shown theoretically that the main hull can only have at most two nonzero facets. The numbers of points in the up and down hull are explored mainly on an empirical basis. The above properties of local convex hulls have led to a slightly modified QuickHull algorithm (the “2-facet QuickHull”) based on the original version proposed by Barber et al. (ACM Trans Math Softw 22:469–483, 1996). As for the feasibility in applying this method to rolling stock scheduling, our empirical experiments show that for the problem instances of ScotRail and Southern Railway, two major train operating companies in the UK, even in the most difficult real-world or artificial conditions (e.g. supposing a train can be served by any of 11 compatible types of self-powered unit), the standard QuickHull (Barber et al. in ACM Trans Math Softw 22:469–483, 1996) can easily compute the relevant convex hulls. For some even more difficult artificial instances that may fall outside the scope of rolling stock scheduling (e.g. a node in a graph can be covered by more than 11 kinds of compatible commodities), there is evidence showing that the “2-facet QuickHull” can be more advantageous over the standard QuickHull for our tested instances. When the number of commodity types is even higher (e.g. >19), or the number of points in a high dimensional space (e.g. 15 dimensions) is not small (e.g. >2000), the local convex hulls cannot be computed either by the standard or the 2-facet QuickHull methods within practical time