A Branch-First, Cut-Second Approach for Locomotive Assignment

Abstract

The problem of assigning locomotives to trains consists of selecting the types and number of engines that minimize the fixed and operational locomotive costs resulting from providing sufficient power to pull trains on fixed schedules. The force required to pull a train is often expressed in terms of horsepower and tonnage requirements rather than in terms of number of engines. This complicates the solution process of the integer programming formulation and usually creates a large integrality gap. Furthermore, the solution of the linearly relaxed problem is strongly fractional. To obtain integer solutions, we propose a novel branch-and-cut approach. The core of the method consists of branching decisions that define on one branch the projection of the problem on a low-dimensional subspace. There, the facets of the polyhedron describing a restricted constraint set can be easily derived. We call this approach branch-first, cut-second. We first derive facets when at most two types of engines are used. We then extend the branching rule to cases involving additional locomotive types. We have conducted computational experiments using actual data from the Canadian National railway company. Simulated test-problems involving two or more locomotive types were solved over 1-, 2-, and 3-day rolling horizons. The cuts were successful in reducing the average integrality gap by 52% for the two-type case and by 34% when more than 25 types were used. Furthermore, the branch-first, cut-second approach was instrumental in improving the best known solution for an almost 2,000-leg weekly problem involving 26 locomotive types. It reduced the number of locomotives by 11, or 1.1%, at an equivalent savings of $3,000,000 per unit. Additional tests on different weekly data produced almost identical results.integer linear programming, branch and cut, decomposition, scheduling, railway

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

“…There are also the train unit circulation problems [2,12,21,23] and the train unit assignment problems [5][6][7][8] belonging to this category. The locomotive assignment problem [9,11,16,22,27,29] is another kind of rolling stock scheduling problem. A common feature of some of the above problems arising in rolling stock scheduling is the use of computable local convex hulls with respect to each train trip.…”

“…A unit type from the diesel family cannot be coupled with a unit from the electric family. To deal with the weak LP relaxation problem above, a class of similar methods has been proposed in several papers [2,6,23,29], which happen to be all in railway rolling stock assignment/scheduling problems. Also in dealing with all the three points listed above, a method for directly computing "train convex hulls" has been given in [14].…”

“…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.…”

Solving Real-Life Locomotive Scheduling Problems

“…Locomotive scheduling problems are among the most important problems in railroad scheduling (Florian et al [1976], Mao and Martland [1981], Smith and Sheffi [1988], Chih et al [1990], Forbes et al [1991], Fischetti and Toth [1997], Nou et al [1997], and Ziarati et al [1997Ziarati et al [ , 1999). Often, locomotive availability is the constraining factor in whether a train departs on time.…”

“…Multicommodity flow based models for planning decisions have been studied by Florian et al [1976], Smith and Sheffi [1988], Nou et al [1997]. Multicommodity flow based models for operational decisions have been developed by Chih et al [1990], and Ziarati et al [1997Ziarati et al [ , 1999. Our multicommodity flow based model for planning decision has more features than any of the existing planning models.…”