An MIP-based heuristic solution approach for the locomotive assignment problem focussing on (dis-)connecting processes
Introduction
Assigning locomotives to trains is one of the most crucial tasks for a rail freight operator, as it determines the number of locomotives. Since each locomotive is associated with very high capital commitment costs, an efficient solution is necessary. The problem considered in this paper arises from a practical application in European rail freight transport. It can be modeled as a Locomotive Assignment Problem (LAP), which is a large scale combinatorial optimization problem also known as the Engine Scheduling Problem (Florian et al., 1976) or the Locomotive Scheduling Problem (Vaidyanathan and Ahuja, 2015).
For scheduling locomotives, several real-world requirements must be taken into account. One of the most demanding is combining two or more locomotives into a consist (combination of locomotives). In practice, the associated processes of building and busting a consist require time and incur costs. As a result, these connecting and disconnecting processes directly affect the feasibility and costs of a schedule. For example, two locomotives might only be able to pull a train together (i.e., forming a consist), due to the weight of the train. But connecting two locomotives to form a consist needs additional effort (time and costs). That is why time restrictions might be violated or higher costs might be incurred.
Railway planning in Europe has several characteristics that are different from those of other continents, because of the nature of the underlying rail network. The network consists of relatively short connections and is divided into several zones. The zones are not only determined by national borders but are the historically developed results of different technical and legal conditions, which mean that different parts of the network might require different types of locomotives. Due to this segmentation of the network, (dis-)connecting processes between locomotives and cars occur much more frequently than, for example, in North America. In summary, a more detailed consideration of the (dis-)connecting processes between locomotives and locomotives as well as cars and locomotives is necessary for the considered real-life problem. However, refueling strategies are not relevant to the problem under consideration, as only electric locomotives are used.
The proposed solution approach is based on modeling the problem as a multicommodity flow formulation (Ahuja, Liu, Orlin, Sharma, Shughart, 2005, Vaidyanathan, Ahuja, Liu, Shughart, 2008). The main idea is to gradually increase the size of the problem by sequentially solving variants of a generic Mixed-Integer Program (MIP). The solution space can be appropriately restricted for a heuristic solution. For this, suitable possibilities arising from the structure of the problem are considered, such as using predefined consists (which cannot be busted) or ignoring (dis-)connecting processes. Each intermediate solution is used as the initial solution for the next step and, finally, the original problem.
The major contributions of this paper can be summarized as follows:
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To the best of our knowledge, several real-world requirements are integrated into the LAP for the first time, which are necessary for the applicability of the generated solutions in practice. Among these are, in particular, the connecting and disconnecting processes, distinguish between push and pull trains, the limited zones for locomotives, the modeling of tasks as a special case of a train as well as invalid combinations of locomotives.
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Based on an MIP-formulation, a generalized solution framework is presented that allows an analysis and comparison of different solution procedures and provides a guideline for the choice of suitable methods in practice. Furthermore, the existing approaches of Vaidyanathan et al. (2008) and Ahuja et al. (2005) can be modeled as special cases of the generalized MIP by adding or removing certain constraints. Both approaches from the literature are outperformed by the presented framework.
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Finally, the proposed method is able to generate high-quality solutions for a complex real-world problem. Its efficiency is proven for several newly generated instances that possess the relevant characteristics of practical LAP in Europe.
Section 2 gives a brief overview of the relevant literature. This is followed by a detailed problem description and a generalized mathematical formulation in Section 3. Based on this, we present different ways for restricting the size of the problem by transforming the mathematical formulation and a resulting solution framework in Section 4. Section 5 describes the computational tests in detail. The paper is summarized in Section 6, closing with a look at future research questions.
Section snippets
Related work and the general planning process
Although the scheduling of locomotives has a major impact on the overall costs of a rail freight operator, the literature on this is limited. One of the first works considering heterogeneous consists (i.e., consists with different types of locomotives) is Florian et al. (1976). Usually, the problem is modeled as a multicommodity flow problem based on a space-time network (Florian et al. (1976); Ziarati et al. (1997); Ahuja et al. (2005); Vaidyanathan et al. (2008); Piu et al. (2015);
Definition of the problem
In this section, we give a descriptive explanation of the considered problem and the integrated real-life requirements. This is followed by a presentation of the underlying space-time network. Finally, the notation used is summarized and a generalized mathematical problem formulation is presented.
Solution approach
The basic concept of our solution approach is sequentially solving the adapted formulations of Vaidyanathan et al. (2008) (Consist Flow Formulation; CFF) and Ahuja et al. (2005) (Locomotive Flow Formulation; LFF). Both formulations are covered by our generalized formulation presented in Section 3.3. This means we are able to create a heuristic solution by CFF and use this as initial solution for LFF. For this reason, we first describe the necessary transformation processes in Section 4.1. Since
Computational analysis
In this section we present the computational experiments. First, we describe our experimental design, which is followed by a detailed depiction of the real-life instance. After that, we explain some newly generated instances. Finally, we present the results of our tests.
Conclusions and further research
This paper has presented a generalized multicommodity flow formulation for the locomotive assignment problem dealing with various practical requirements in European rail freight transport. A main focus was the correct modeling of (dis-)connecting processes of locomotives with other locomotives as well as cars. Based on an MIP formulation, we introduced a heuristic solution framework for increasing the problem size gradually. We discussed several possibilities for restricting, which can speed up
CRediT authorship contribution statement
Martin Scheffler: Conceptualization, Methodology, Software, Investigation, Writing - original draft, Visualization. Janis S. Neufeld: Conceptualization, Validation, Writing - review & editing, Supervision. Michael Hölscher: Methodology, Validation, Writing - review & editing.
Acknowledgement
The authors would like to thank the anonymous reviewers as well as Udo Buscher for their valuable comments and suggestions for improving the quality of the paper. They are also grateful to Florian Linß for assistance in the preparation of the figures.
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2022, Computers and Operations ResearchCitation Excerpt :More recently, Ortiz-Astorquiza et al. (2021) propose a novel hybrid formulation that combines features from both locomotive-based and consist-based representations of the problem to solve the LAP at the Canadian National Railways (CN). Scheffler et al. (2020) propose a formulation that generalizes the locomotive- and consist-based models proposed by Ahuja et al. (2005b) and Vaidyanathan et al. (2008a), respectively, while satisfying practical requirements commonly found in European freight railroads. We refer to Piu and Speranza (2014) for a survey on the LAP.