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Locomotive planning has been a popular application of classical optimization models for decades, but with very few success stories. There are a host of complex rules governing how locomotives should be used. In addition, it is necessary to simultaneously manage locomotive inventories by balancing the need for holding power against the need for power at other yards. At the same time, we have to plan the need to return foreign power, and move power to maintenance facilities for scheduled FRA appointments. An additional complication arises as a result of the high level of uncertainty in transit times and delays due to yard processing, and as a result we may have to plan additional inventories in order to move outbound trains on time despite inbound delays. We describe a novel modeling and algorithmic strategy known as approximate dynamic programming, which can also be described as a form of "optimizing simulator" which uses feedback learning to plan locomotive movements in a way that closely mimics how humans plan real-world operations. This strategy can be used for strategic and tactical planning, and can also be adapted to real-time operations. We describe the strategy, and summarize experiences at Norfolk Southern with a strategic planning system.

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We present a general optimization framework for locomotive models that captures different levels of detail, ranging from single and multicommodity flow models that can be solved using commercial integer programming solvers, to a much more detailed multiattribute model that we solve using approximate dynamic programming (ADP). Both models have been successfully implemented at Norfolk Southern for different planning applications. We use these models, presented using a common notational framework, to demonstrate the scope of different modeling and algorithmic strategies, all of which add value to the locomotive planning problem. We demonstrate how ADP can be used for both deterministic and stochastic models that capture locomotives and trains at a very high level of detail.

For decades, locomotive planning has been approached using the classical tools of mathematical programming; the result has been very large-scale integer programming models that are beyond the capabilities of modern solvers but still require a host of simplifying assumptions that limit their use for analyzing important planning problems. The primary interest of Norfolk Southern was in developing a model that could assist it with fleet sizing. However, the cumulative effect of the simplifications required to produce a practical integer programming formulation resulted in models that underestimated the required fleet. We use the modeling and algorithmic framework of approximate dynamic programming, which uses an intuitive balance of simulation and optimization with feedback learning, to produce a highly detailed model that calibrates accurately against historical metrics. The result was a model that can be used to plan fleet size and mix, be sensitive to a wide range of operating parameters, and adapt to many scenarios.