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**Local convex hulls for a special class of integer multicommodity flow problems.**
*(English)*
Zbl 1352.90102

Summary: Based on previous work in rolling stock scheduling problems, 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 C. B. Barber et al. [ACM Trans. Math. Softw. 22, No. 4, 469–483 (1996; Zbl 0884.65145)]. 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., loc. cit.] 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.

### MSC:

90C35 | Programming involving graphs or networks |

90B35 | Deterministic scheduling theory in operations research |

### Citations:

Zbl 0884.65145### Software:

Qhull
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\textit{Z. Lin} and \textit{R. S. K. Kwan}, Comput. Optim. Appl. 64, No. 3, 881--919 (2016; Zbl 1352.90102)

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### References:

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