We introduce EETTlib, an instance library for the Energy‐Efficient Train Timetabling problem. The task in this problem is to adjust a given timetable draft such that the energy consumption of the resulting railway traffic is minimized. To this end, the departure times of the trains can be slightly, and their velocity profiles on each trip can be modified. We provide real‐world data originating from two research projects in this field, one with Deutsche Bahn AG, the most important railway company in Germany, the other with VAG Verkehrs‐Aktiengesellschaft, the operator of public transport in the city of Nürnberg, Germany. In both cases, our library contains representative data on the relevant operational constraints and supports various possible choices for the objective function with respect to energy‐efficiency. The resulting benchmark instances can be used by the scheduling and timetabling community to improve their models and algorithms. They are available under https://www.eettlib.fau.de.
Complex subsurface operations are characterized by a life‐threatening environment, a skilled and initiative opponent, and the absence of predictability of the events due to a high level of interdependencies. The reduction of complexity by provision of essential information is crucial for decision‐making and rapid integration and visualization of heterogeneous data is essential for successful mission accomplishment. Currently, only standalone applications are available for the underground operational environment, and collaborative planning and working spaces in command and control are missing. The RApid Data Integration and Visualization (RADIV) process addresses exactly this challenge and ensures the lateral continuity of visualization systems across the entire reality‐virtuality continuum (2D ⇔ 3D ⇔ mixed reality). It provides a comprehensive command and control system for subterranean operations by processing and visualizing data in different views for different purposes. Integration of these data within the Subsurface Operations Mission Tool (SOMT) will increase the decision quality by improved perception and collaboration. Close cooperation and information exchange between operators and action forces is a prerequisite for success by displaying the relevant information within the truly comprehensive common operational picture, thereby enabling more accurate and precise action reducing own losses and collateral damage.
Bilinear terms naturally appear in many optimization problems. Their inherent non-convexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of univariate functions. Each univariate function can again be approximated by a piecewise linear function and modelled via an MIP formulation. In the literature, heterogeneous results are reported concerning which approach works better in practice, but little theoretical analysis is provided. We fill this gap by structurally comparing bivariate and univariate approximations with respect to two criteria. First, we compare the number of simplices sufficient for an $$ \varepsilon $$ ε -approximation. We derive upper bounds for univariate approximations and compare them to a lower bound for bivariate approximations. We prove that for a small prescribed approximation error $$ \varepsilon $$ ε , univariate $$ \varepsilon $$ ε -approximations require fewer simplices than bivariate $$ \varepsilon $$ ε -approximations. The second criterion is the tightness of the continuous relaxations (CR) of corresponding sharp MIP formulations. Here, we prove that the CR of a bivariate MIP formulation describes the convex hull of a variable product, the so-called McCormick relaxation. In contrast, we show by a volume argument that the CRs corresponding to univariate approximations are strictly looser. This allows us to explain many of the computational effects observed in the literature and to give theoretical evidence on when to use which kind of approximation.
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