Armin Fügenschuh

Validation of nominations in gas network optimization: models, methods, and solutions

In this article, we investigate methods to solve a fundamental task in gas transportation, namely the validation of nomination problem: given a gas transmission network consisting of passive pipelines and active, controllable elements and given an amount of gas at every entry and exit point of the network, find operational settings for all active elements such that there exists a network state meeting all physical, technical, and legal constraints. We describe a two-stage approach to solve the resulting complex and numerically difficult nonconvex mixedinteger nonlinear feasibility problem. The first phase consists of four distinct algorithms applying mixedinteger linear, mixedinteger nonlinear, nonlinear, and methods for complementarity constraints to compute possible settings for the discrete decisions. The second phase employs a precise continuous nonlinear programming model of the gas network. Using this setup, we are able to compute high-quality solutions to real-world industrial instances that are significantly larger than networks that have appeared in the mathematical programming literature before.

Combinatorial and Continuous Models for the Optimization of Traffic Flows on Networks

A hierachy of simplified models for traffic flow on networks is derived from continuous traffic flow models based on partial differential equations. The hierachy contains nonlinear and linear combinatorial models with and without dynamics. Optimization problems are treated for all models and numerical results and algorithms are compared.

Computational Integer Programming and Cutting Planes

Abstract The study and solution of mixed-integer programming problems is of great interest, because they arise in a variety of mathematical and practical applications. Todays state-of-art software packages for solving mixed-integer programs based on linear programming include preprocessing, branch-and-bound, and cutting planes techniques. The main purpose of this article is to describe these components and recent developments that can be found in many solvers. Besides linear programming based relaxation methods we also discuss Langrangean, Dantzig-Wolfe and Benders’ decomposition and their interrelations.

Single-Car Routing in Rail Freight Transport

Single cars in rail freight service are bundled into trains at classification yards. On the way from their respective origins via intermediate yards to their destinations, they are reclassified several times, which is a time-consuming and personally consuming procedure. The single-car routing problem asks for the design of such routes for a given set of orders origin-destination pairs with associated data on an infrastructure network, such that the number of trains and their travel distances are minimal. A number of hard restrictions must be obeyed, such as restrictions for the train length and weight, and capacity restrictions for the yards, as well as further operational rules. We present a mixed-integer linear programming MILP formulation for this car-routing problem arising at Deutsche Bahn, one of the largest European railway companies. In a further step, we refine the handling of the turnover waiting time for the cars in the yards, which leads to the inclusion of nonlinear constraints in the model. Using adequate linearization techniques, this model can be reduced to a MILP again. Instances of this model turn out to be hard to solve. Further techniques are thus presented to speed up the numerical solution process, among them a tree-based reformulation and heuristic cuts. The different model formulations are computationally compared on a test set of randomly generated instances whose sizes are comparable to real-world instances. Using state-of-the-art MILP solvers, optimal or near-optimal solutions can be computed within a reasonable time frame.

Gas network topology optimization for upcoming market requirements

Gas distribution networks are complex structures that consist of passive pipes, and active, controllable elements such as valves and compressors. Controlling such network means to find a suitable setting for all active components such that a nominated amount of gas can be transmitted from entries to exits through the network, without violating physical or operational constraints. The control of a large-scale gas network is a challenging task from a practical point of view. In most companies the actual controlling process is supported by means of computer software that is able to simulate the flow of the gas. However, the active settings have to be set manually within such simulation software. The solution quality thus depends on the experience of a human planner. When the gas network is insufficient for the transport then topology extensions come into play. Here a set of new pipes or active elements is determined such that the extended network admits a feasible control again. The question again is how to select these extensions and where to place them such that the total extension costs are minimal. Industrial practice is again to use the same simulation software, determine extensions by experience, add them to the virtual network, and then try to find a feasible control of the active elements. The validity of this approach now depends even more on the human planner. Another weakness of this manual simulation-based approach is that it cannot establish infeasibility of a certain gas nomination, unless all settings of the active elements are tried. Moreover, it is impossible to find a cost-optimal network extension in this way. In order to overcome these shortcomings of the manual planning approach we present a new approach, rigorously based on mathematical optimization. Hereto we describe a model for finding feasible controls and then extend this model such that topology extensions can additionally and simultaneously be covered. Numerical results for real-world instances are presented and discussed.

Optimization methods for UMTS radio network planning

The UMTS radio network planning problem poses the challenge of designing a cost-effective network that provides users with sufficient coverage and capacity. We describe an optimisation model for this problem that is based on comprehensive planning data of the EU project Momentum. We present heuristic mathematical methods for this realistic model, including computational results.

A primal heuristic for optimizing the topology of gas networks based on dual information

We present a novel heuristic to identify feasible solutions of a mixed-integer nonlinear programming problem arising in natural gas transportation: the selection of new pipelines to enhance the network’s capacity to a desired level in a cost-efficient way. We solve this problem in a linear programming based branch-and-cut approach, where we deal with the nonlinearities by linear outer approximation and spatial branching. At certain nodes of the branching tree, we compute a KKT point of a nonlinear relaxation. Based on the information from the KKT point we alter some of the binary variables in a locally promising way exploiting our problem-specific structure. On a test set of real-world instances, we are able to increase the chance of identifying feasible solutions by some order of magnitude compared to standard MINLP heuristics that are already built in the general-purpose MINLP solver SCIP.

A fuzzy goal programming approach to analyse sustainable development goals of India

ABSTRACT Energy policy, environmental planning and economic development play a key role in sustainable development. Sustainable development requires suitable and strategic policies satisfying multiple and conflicting objectives. Fuzzy goal programming (FGP) is a well-known approach in multi-criteria decision-making for its practical application. In this article, a FGP approach is proposed to analyse environmental, energy and sustainability goals of India by the year 2030 with reference to the key economic sectors of India. The presented model analyses the improvement opportunities, requirement of efforts and implementation of the sustainable development plans. Numerical illustration is also provided for validation and application of the proposed model.

Variable Speed in Vertical Flight Planning

Vertical flight planning concerns assigning cruise speed and altitude to segments that compose a trajectory, such that the fuel consumption is minimized and the time constraints are satisfied. The fuel consumption over each segment is usually given as a black-box function depending on aircraft speed, weight, and altitude. Without time consideration, it is known that it is fuel-optimal to fly at a constant speed. If an aircraft is under time pressure to speed up, the industrial standard of cost index cannot handle it explicitly, while research literature suggest using a constant speed. In this work, we formulate the vertical flight planning with variable cruise speed into a mixed integer linear programming (MILP) model, and experimentally investigate the fuel saving potential over a constant speed.

Use of Optimization Tools for Routing in Rail Freight Transport

Deutsche Bahn, one of the largest European railway companies, offers mainly two products to commercial and industrial customers for freight transportation. Customers with high demand order unit trains , that are pulled by one or two locomotives from their respective origins to their destinations. In contrast, customers with less demand order a limited amount of single cars , that are first pulled to a classification yard. There they are grouped together with single cars from other customers into a train unit. On the way from their respective origins via intermediate yards to their destinations, the cars are reclassified several times, which is a time-consuming and personnel-intensive procedure. To support the strategic long-term planning process of the single car freight routing, a mathematical optimization tool based on mixed-integer nonlinear programming was developed and is in practice use since 2011. However, real-world constraints have changed over the last years. For example, unit trains and single cars are no longer strictly separated products, but they are more and more integrated: In some unit trains there are still residual capacities that can be used for single cars. For some of these additional new requirements, the existing optimization tool has to be extended slightly by formulating new additional mathematical constraints. For some other requirements, a substantial redevelopment will be necessary in the future. The purpose of this chapter is to review the existing single car routing model, to discuss how it is used in real-life, and to demonstrate how it can be extended to meet the new requirements in the present and future.