The finite element method has proven itself to be an effective method of performing both linear and non-linear structural analyses. This numerical method, however, has several limitations, of one which is that the discretization of complex geometry requires human effort. The rise of the application of machine learning models has opened new possibilities of approaching challenges within scientific fields. The neural network, and more specifically, the physics informed neural network, is a promising method and allows for performing analyses without any discretization that is necessary. This type of neural network, which is less of a black box than the standard neural network, utilizes the concept of partial differential equations to accurately predict solutions.

The objective of this thesis is to perform a structural analysis of a geometrically non-linear Timoshenko beam using a physics informed neural network. The network is built using a variational principle (the principle of virtual work) and a force residual. Furthermore, two optimization algorithms, the adaptive weight loss algorithm and the adaptive activation function are separately used in conjunction with the model to examine the potential improvements on the convergence rate.

It is found that the geometrically non-linear Timoshenko beam can be accurately modeled (relative error of below 2% with respect to the finite element output) with a physics informed neural network. This accuracy can be achieved with a model possessing a relatively shallow size of four hidden layers containing eight neurons each. The adaptive weight loss algorithm and the adaptive activation algorithm both improve the convergence rate of the model, though they are not necessary to maintain the practicality of the model, as the convergence rate is adequate without these. It is recommended that the hyperbolic tangent function is utilized in conjunction with the Adam optimizer. The adaptive activation function can be incorporated into the model to improve the convergence rate significantly without substantially increasing the computational cost of the model.

Beams are the fundamental structural engineering element, supporting and stabilizing various structures ranging from suspension bridges to buildings and railways. Modeling and analyzing these structures necessitates a comprehensive understanding of the underlying beam dynamics constituting the structures. Simulating and predicting the beam dynamics is pivotal in ensuring structural integrity, optimizing structure design, and selecting appropriate materials. For instance, in railways, tracks and catenary contact wires are conceptualized as beams, allowing for the application of renowned beam theories like Euler-Bernoulli and Timoshenko. These theories provide a foundation for formulating partial differential equations (PDEs) that govern the dynamic behaviors of these beam systems.

These PDEs could be leveraged to simulate the underlying scenarios. The dissertation introduces physics-informed machine learning (PIML) based approaches tailored to simulate the dynamics of beam structures. The aim is to incorporate the physical laws in the neural networks training for more accurate and realistic simulations, handle noisy data effectively, and improve prediction accuracy while mitigating challenges such as multiscale problems and generalization. Chapter 1 outlines the primary challenges tackled in the dissertation. Chapters 2 through 5 detail the methodologies developed to address each challenge.

Chapter 2 presents a physics-informed neural network (PINN) based methodology to simulate complex beam systems with real-world mate- rial properties. In addition, inverse problems are solved in the presence of noisy data to predict unknown parameters, including force acting on the beam systems. It is essential to consider the real-world material parameters to simulate the dynamics of the modeled system and ensure the digital model represents the ground truth. However, incorporating material characteristics leads to multiscale PDE coefficients in the physical model, posing difficulty in training for PINNs. Subsequently, a frame- work is proposed to incorporate nondimensional PDEs into the PINN loss function. This approach facilitates efficient forward and inverse simulations while robust to noise and uncertainty in measurement data. The efficacy of this approach is demonstrated through simulations of Euler- Bernoulli and Timoshenko beam systems, contributing to the challenge of simulating large-scale systems with multiple interconnected components.

Chapter 3 investigates beam dynamic simulations on Winkler foundations for large spatiotemporal domains using PIML. Predictions on expansive spatiotemporal domains are vital for structural integrity, design optimization, and control mechanisms. A causality-respecting PINN frame- work is introduced, enhancing prediction accuracy. Furthermore, integrating transfer learning addresses the need to re-train the network for different initial conditions and computational domains. Numerical experiments based on Euler-Bernoulli and Timoshenko theories validate the methodology for respecting the causality and generalizing the beam dynamics across similar problems. The approach efficiently predicts beam dynamics under diverse engineering scenarios, reducing computational costs and improving convergence.

Chapter 4 explores the generalization abilities of PIML, essential for practical applications requiring accurate predictions in unexplored regions. The proposed framework exploits the inherent causality in the PDE solutions by merging PIML models with recurrent neural architectures, namely neural oscillators. The neural ordinary differential equations in the form of neural oscillators effectively handle long-time dependencies and address gradient-related issues, fostering improved generalization in PIML tasks. Benchmark equations like viscous Burgers, Allen-Cahn, Schrödinger, and biharmonic Euler-Bernoulli beam equations are used to demonstrate the effectiveness of the proposed approach. Through ex- tensive experimentation with time-dependent nonlinear PDEs, the study showcases superior performance compared to existing state-of-the-art methods. The proposed method provides accurate solutions for extrapolation and prediction beyond the training data by enhancing the generalization capabilities of PIML, promising advancements in complex system simulations.

Chapter 5 follows up on generalization of beam dynamics beyond PIML- based approaches. Computer-aided simulations are crucial for advancing engineering industries, but existing simulators often struggle to generalize beyond their training domain. The chapter proposes a two-stage methodology to tackle this challenge. Firstly, it utilizes specialized simulators tailored to the application, such as causal PINNs and black-box finite element simulations. Secondly, it integrates predictions from the first stage into a recurrent neural architecture, incorporating ordinary differential equations to capture intrinsic dynamics and enhance generalization. The approach efficiently captures causality and generalizes dynamics across various data sources. Numerical experiments cover fundamental structural engineering scenarios, including real-world catenary contact wire uplift predictions, and demonstrate superior performance compared to conventional methods, and promise for diverse industrial applications. This dissertation concludes with Chapter 6.

In particular, this dissertation introduces PIML methodologies for simulating complex beam structures, addressing key challenges such as incorporating real material properties, handling noisy data, and improving prediction accuracy. Chapter 2 introduces a PINN-based methodology that efficiently simulates beam systems and predicts unknown parameters, mitigating the difficulties posed by multiscale PDE coefficients. Chapter 3 tackles the challenge of large-domain beam dynamics predictions on the Winkler foundations by using causality-respecting PINNs and integrating transfer learning to reduce computational costs. Chapter 4 addresses the challenge of out-of-domain predictions in PIML by introducing neural oscillators. Chapter 5 proposes a two-stage methodology to generalize beam dynamics simulations, integrating beam dynamics solvers and recurrent neural-based architectures, showcasing its efficacy in real-world applications such as catenary contact wire uplift predictions.
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The railway track is the most critical element that infrastructure managers have to ensure satisfactory health conditions to provide safe, sustainable, and adequate service quality. The condition of railway tracks is managed through maintenance strategies that benefit enormously from the availability of accurate and updated information from measurement devices. This thesis focuses on estimating the track support condition using Axle Box Acceleration (ABA) measurements. The ABA characteristics under different track support conditions are evaluated, effects of influential parameters in the measurements are analyzed, key performance indicators (KPIs) for condition monitoring of ballast and substructure layers are proposed, and results with traditional techniques such as impact tests are compared. Finally, an enhanced track quality index is proposed to improve the effectiveness of maintenance decision-making. Case studies in The Netherlands, Sweden, and Norway are considered.@en

The railway transportation system is a critical infrastructure that supports the functionality of a country through ways like the transportation of people from point to point to carry out their daily activities i.e., work, study, and leisure; the timely transportation of freight within a country or cross-borders, contributing to economic development, etc., thus it has to be effective in its operation and service provision. Normal operation can get affected when railway systems such as trains, signalling systems, etc. fail to operate, railway infrastructures are damaged, occurrences of collisions or derailment. These events result in inconveniences caused to railway users, loss in service confidence and decline in revenue. Hence, railway organisations continuously put in effort to reduce the occurrence of railway disruptions that are within their means to prevent. However, there have been reports on railway disruptions caused by events that are uncontrollable within the means of the organisations. The railway transportation system faces continuous ‘threats’ despite the continuous effort put in by railway organisations to provide mitigation measures. This thesis has classified these ‘threats’ as expected and unexpected. Expected ‘threats’ refer to events whereby their potential of disrupting the operation of the railway transportation system is anticipated, thus mitigation measures can be implemented in advance so as to eliminate or prevent the event from happening. On the other hand, unexpected ‘threats’ refer to events whereby the occurrences cannot be controlled and sometimes it can be unpredictable. The objective of this thesis is to study how the resilience of a railway organisation, in view of the occurrences of 2 unexpected external events (i.e., climate change and cyber-attack), can be assessed by incorporating the elements of the 4R Resilience Framework to its operation and management. The assessment work is carried out through a semi-qualitative tool created by this thesis which adopts a systematic approach by making use of a 5-stages cycle framework.

A heuristic method for the operation of shunting yards is created that considers the sorting process of wagons with dangerous goods. This concerns shunting yards where the wagons are distributed manually, and communication takes place by telephone. The first problem is that an entire train is considered dangerous if there is a wagon with dangerous goods on the accompanying wagon list. The wagon list shows where each wagon is positioned within a train composition. Due to safety regulations, a certain distance must be kept between wagons for certain dangerous goods classes. Because entire trains are considered dangerous, adjacent tracks must be left empty. This is at the expense of the capacity of the yard.

The second problem is that the wagon list is not 100% correct due to human errors. Random checks show that the wagon list does not always turn out to be correct and that mistakes are made in creating the list. Therefore, smart camera techniques have been developed to detect which wagon contains which dangerous goods on arrival to confirm the wagon list. This opens up new possibilities concerning the sorting strategy of wagons from the arrival track onto the classification tracks in accordance with their destination.

Research is done to find out what this new sorting strategy should look like. Therefore, the following research question is formulated: ‘What kind of method can distribute wagons, including wagons carrying dangerous goods, on a rail yard, considering safety?’. In order to answer the research question, qualitative research was carried out into the various existing strategies and methods that could be applied. A model is created in which wagons are sorted over classification tracks. The model chooses the track on which the wagons should be placed based on the destination. Specific attention is given to wagons with dangerous goods, for which the model ensures that a specific distance is kept from other wagons on the (neighboring) classification tracks. The model then determines the most appropriate departure time, and the train departs. Different scenarios are tested that the model has to deal with. A distinction is made here between whether the number of classification tracks is equal or not equal to the number of destinations, and whether the rate of dangerous goods over the destinations is equal or not. This makes a total of four scenarios that have been tested.

The effect of using a buffer on the track is examined so that a wagon carrying dangerous goods never stands next to another wagon carrying dangerous goods of a different class if they are not allowed to stand together. This is tested in the model, and the output showed the placement of the wagons and the processing times.

Although monitoring and maintenance of railways is important to ensure safety and avoid delays and financial losses, it is still mainly based on human inspection. The complexity of a railway along with the large area it extends makes manual monitoring difficult and time-consuming. The increasing availability of 3D acquisition technologies has made point clouds a widely used 3D data form. Thus, identifying the key components of a railway and its environment using 3D point cloud data can be the first step for automating this procedure. The past years, machine learning has become the most popular subfield of artificial intelligence used for various different applications, with its subfield of deep learning evolving dramatically. Although deep learning has been vastly researched on 2D data, applying deep learning on 3D point clouds can be challenging due to the irregularity, unstructuredness and unorderedness of such data. To overcome those challenges, recent approaches use projection or voxelization, while the latest methods focus on working directly on raw point cloud data.

In the present research, 2 machine learning methods are implemented to classify 3D point cloud data of railway environments into 7 categories of rails, sleepers, track bed, masts, overhead wires, trees and other. To this end, the ensemble method of Random Forest, and the deep learning method of DGCNN (Dynamic Graph Convolutional Neural Network) are implemented. While Random Forest is a simple and handy ensemble algorithm used for a wide range of applications, DGCNN is a deep learning method based on PointNet, the pioneer method on raw point clouds, and graph CNNs. The methods are validated under 3 case studies, produced by structure from motion photogrammetry, airborne laser scanning and terrestrial laser scanning, and locating in 2 different areas within the Netherlands. For each method, 2 scenarios are developed for classifying colored and uncolored point clouds, respectively. Finally, the 2 methods are combined for the first scenario, as a first attempt to further improve the final results.

The obtained results show that, in this work, DGCNN performs better than Random Forest, and both methods approximate state-of-the-art performance. The contribution of colors is important to improve both the overall accuracy of the models, as well as the classification results of the individual classes. More specifically, Random Forest scenario 1, Random Forest scenario 2, DGCNN scenario 1, DGCNN scenario 2, and the combination of Random Forest and DGCNN scenario 1 result in an overall accuracy of 88.65%, 83.39%, 89.19%, 88.20% and 90.57%, respectively. The corresponding per class F1-scores for all methods and scenarios range between 43% and 94%. Both methods meet difficulties in generalizing on data from different sensor systems and different areas with very different point density and missing data. Nonetheless, the individual methods are already very promising, while they are able to achieve the required accuracy of more than 90% when combined.

Aircraft inspections after unexpected incidents, like lightning strikes, currently require a timeconsuming and costly inspection process, due to the small size of the lightning strike damages. Mainblades Inspections is working on an automated, drone-based solution, that scans the aircraft hull with a high-resolution camera. The objective of this project is to assess the
feasibility of using a deep Convolutional Neural Network (CNN) for (semi-)automated damage detection, with the goal of achieving a high recall (low False Negative Rate (FNR)) on the small damages. The problem is framed as a classification problem on limited and imbalanced data. However, it is not pre-defined if single-label or multi-label classification should be used,
and both approaches are investigated.
The main contribution of this work is to show experimentally how common deep CNN architectures and Deep Learning practices can be used to train classifiers that recognize damages in a specialized domain, with application-specific metrics. We present methods for synthesizing, pre-processing and re-sampling of the necessary dataset. It is shown that pre-trained, parameter-efficient CNN architectures that implement skip-connections, complemented by
global max-pooling before the final layer, are well suited for that dataset. The Xception architecture has been chosen as backbone for the classifier due to its high recall and fast convergence. To mitigate the detrimental influence of imbalanced training data, training data re-sampling that equalizes the class distribution is implemented. It has a positive effect recall, especially
when applied to multi-label classification. When using re-sampling and data augmentation, the performance of multi-label and single-label classification can be brought to the same level. However, the best achieved FNR is 5.4%, with a softmax classifier, combining all regularization methods.
Finally, we investigated how regularization can be used to increase generalization capability with limited training data. Data augmentation is the most effective regularization method, even though its full potential has not been explored yet. Dropout benefits single-label classification
but not multi-label classification. L2-regularization has a moderate positive effect on both. Naively combining the regularization techniques without an exhaustive grid search or automated search on average does not yield any additional gains and shows the limit of manual hyper-parameter tuning.

Railway catenary is the main infrastructure that delivers electric power for train operation. It is a structure commonly constructed along the railway line with contact wires suspended above the track. One or multiple pantographs mounted on the roof of a moving train collects electric current from the catenary through the sliding contact with a contact wire. With the increase of train speed and traffic density in recent years, the catenary is subject to higher impacts from pantographs, leading to critical failures such as the breakage of contact wire. This results in not only an increasing cost for reactive maintenance, but also disruptions of train service that affect many passengers.

To reduce the life cycle cost and failure rate of catenary in practice, planned and predictive maintenance is desired based on the condition monitoring of catenary. However, the monitoring data are underutilized to effectively assess the catenary condition and facilitate maintenance decision-making. This dissertation contributes in improving the dynamic condition assessment of catenary using the data from condition monitoring. New performance indicators (PIs) of catenary are defined in a way that is adaptive to the variations of monitoring data measured under different circumstances, such as the changes of catenary structure, pantograph type and train speed. The relationship between the monitoring data and the contact wire irregularities is studied using historical data and simulations. Data-based approaches are developed for the quantitative assessment of dynamic catenary condition.

First, an intrinsic wavelength contained in the pantograph-catenary contact force is identified and defined as the catenary structure wavelength (CSW). It is caused by the periodic variations of contact wire stiffness attributed to the cyclic structure of catenary that must regulate the height of contact wire in every spans and interdropper distances. An approach that adaptively extracts the CSWs of pantograph-catenary contact force is proposed based on the empirical mode decomposition algorithm. It extracts the CSW signals corresponding to the span lengths and interdropper distances, respectively, summing to form a characteristic signal of CSWs. The residual signal of the contact force excluding the CSWs is regarded as the non-CSW signal. The mean and standard deviation of the CSWs signal are used as PIs to indicate the condition of the main catenary geometric parameters. A PI based on the quadratic time-frequency representation of the non-CSW signal is proposed for detecting and localizing the local irregularities of contact wire. The proposed PIs are tested by simulation and measurement data and proven effective and adaptive owning to the use of CSWs and non-CSW signal.

Second, the concept of CSW is expanded to the pantograph head acceleration from which the CSWs and non-CSW signal can also be extracted using the same approach developed for the contact force. Considering the characteristics of pantograph head acceleration, the wavelet packet entropy of the CSWs and non-CSW signal is proposed as PIs for detecting contact wire irregularities with different lengths. The entropy of CSWs is used for detecting irregularities with a length longer than 5 m, while the entropy of non-CSW signal is for the short-length local irregularities. An approach to detect and verify contact wire irregularities using the measurement data of pantograph head vertical acceleration from frequent inspections is proposed. The approach is tested using historical inspection data from which irregularities at all lengths are detected and verified. Maintenance resources can thus be specifically allocated to verified detection results to save cost and time.

Third, through analyzing historical inspection data and data-based simulation results, it is found that while the contact wire irregularity deteriorates the pantograph-catenary interaction, the formation of irregularity is also associated with the effects of the interaction like variations of contact and friction forces. Concretely, the contact wire height irregularity with an amplitude of 8 mm can cause considerable increase in the standard deviation of pantograph-catenary contact force. In addition, the irregularity with a certain wavelength can induce the dynamic response with the same wavelength in the contact force. This in turn makes the irregularity part deteriorating faster than the other parts of catenary. At a smaller scale, when the wear irregularity of contact wire has an average wire thickness loss of about 1.5 mm, it can also increase the standard deviation of contact force by more than 5%. Due to the fixing effect at the registration arms and droppers, the wear irregularity commonly contains structural wavelengths of catenary including span lengths and interdropper distances. It is also found that the wear irregularity tends to grow and spread toward in the common or dominant running direction of trains in the specific line. Nevertheless, an existing defect may not affect every pantograph passage and every type of data measured. It is thus advised to measure multiple types of data and perform more frequent inspections to avoid undetected defects.

Last, a data-driven approach using the Bayesian network (BN) to fuse the available inspection data of catenary into an integrated PI is proposed. The BN topology is first structured based on the physical relations between five data types including the train speed, dynamic stagger and height of contact wire, pantograph head acceleration, and pantograph-catenary contact force. Then, tailored PIs are individually defined and extracted from the five types of data as the BN input. As the output of BN, an integrated PI is defined as the overall condition level of catenary considering all defects that can be reflected by the five types of data. Finally, using historical inspections data and maintenance records from a section of high-speed line, the BN parameters are estimated to establish a probabilistic relationship between the input and the output PI. By testing the BN-based approach using new inspection data from the same railway line, it is shown that the integrated PI can adequately represent the catenary condition, leading to considerable reduction in the false alarm rate of catenary defect detection compared with the current practice. The approach can also work acceptably with noisy or partly missing data.

In summary, this dissertation answers how to adequately transform the condition monitoring data of catenary into quantitative assessments of the dynamic catenary condition. The proposed approaches are intended for generic implementations in railway catenaries worldwide.@en

Manufacturing processes often run on a schedule, because each production step takes a certain amount of time, after which a component is transported to the next machine. Mobile robots can be used to automate transportation of components between production steps. To do so, a task scheduler assigns transportation tasks to robots and a path planner calculates trajectories for robots to follow to complete their tasks.

In scientific literature, path planning and task scheduling are treated as separate problems. Typically, path planners use initial and goal positions as a given and task schedulers consider tasks of fixed duration. In this thesis, the integration of task scheduling and path planning is treated, resulting in three contributions. First, a novel cost function to minimise the duration of paths is proposed for an existing Mixed-Integer Linear Programming (MILP) formulation of the multi-robot path planning problem. Second, a new MILP formulation is developed for the scheduling problem of tasks with deadlines to be completed by mobile robots. Third, two methods for integrating the path planner and task scheduler are proposed. Through simulations these methods are compared with an integration method that uses rule-based scheduling, which assigns tasks to robots in a predefined order.

Simulation results show that the novel path planning objective function improves upon that from the literature by striking a balance between duration, distance, energy use and computation time. Furthermore, integration methods using the MILP scheduler have trouble outperforming a rule-based approach, caused by a practical limit on the computation time. Moreover, compared with single-robot path planning the use of multi-robot path planning is shown to improve performance at the cost of higher computation time. However, this improvement is negligible for environments with a lot of free space, where robots can easily avoid each other without making detours.

These results lead to the conclusion that the MILP path planner with the new path planning objective function performs well when integrated with a task scheduler. For restrictive physical environments, multi-robot path planning can further improve results. On the other hand, for practical applications a scheduling method other than the MILP scheduler should be employed to integrate task scheduling and path planning. The scheduling method should at least outperform rule-based scheduling with a practical limit on computation time, even if an optimal solution is not guaranteed.

An essential component of railway infrastructures is track ballast. As railway track is used frequently by passing rolling stocks, its performance degrades over time. At certain degradation levels, maintenance interventions must be carried out to improve the track performance so to meet technical and safety regulations. In this way, the risk of accident or derailment can be minimized and the railway interoperability is ensured. Furthermore, the responsibility of designing maintenance plan belongs to infrastructure managers. To help them, predictive strategies based on optimization can suggest the optimal schedule to maintain the track over a certain time period. In this way, track performance and maintenance costs can be explicitly optimized over the whole life cycle of the track.

However, a railway network typically consists of multiple track sections, each of them with different degradation level and parameters. Hence, the optimization of track maintenance can be considered as a large-scale problem which has a large number of decision variables. For such kind of problem, the conventional centralized optimization is very difficult or even not tractable to solve due to limitations on the computational time and resources. One way to overcome this issue is by applying the so-called distributed optimization scheme. In such approach, the original optimization problem is partitioned into multiple smaller, tractable subproblems. Therefore, the optimization is tractable and more preferred for real-life implementations.

This thesis develops distributed optimization approaches for track maintenance operations planning problem. Three different schemes are compared: Parallel Augmented Lagrangian Relaxation (PALR), Alternating Direction Method of Multipliers (ADMM), and Distributed Robust Safe But Knowledgeable (DRSBK). As these distributed approaches basically designed for convex problems, extension techniques to handle non-convex nature of the proposed optimization problem are implemented. Furthermore, some case studies are defined to evaluate the algorithms from both performance and numerical perspectives. In simulations of small, medium, and large-scale instances, it is shown that in most cases, DRSBK can outperform the other distributed approaches, by providing the closest-to-optimum solution to the centralized optimization problem while having the shortest computation time.

In this thesis, the effect of traffic light control on urban networks is investigated, with the main focus on the optimization algorithms used to solve the optimization problems resulting from a model predictive control approach. This will be done by using a macroscopic traffic flow model (the S-Model) to simulate the real-time traffic flows inside a network. A microscopic emission model (the VT-micro emission model) is further added to also include impact of the vehicle behaviour (accelerating, decelerating, driving at a constant speed, etc.) on the total amount of emission gasses that are exhausted by the vehicles inside the traffic network.

In this thesis the optimization algorithms will be investigated with respect to the reduction of the cost function as well as the computation time needed compute the (sub)optimal solution. The cost function can consist of the Total Time Spent (TTS) by the vehicles inside the network, the Total Emissions (TE) the vehicles exhaust while inside the network, or a combination of both.

We make use of Matlab for implementation of the models and the optimization algorithms and SUMO as a traffic simulator, which will be used to simulate a real-time traffic network.

The traffic flow model and the emission model are both non-smooth and non-convex, which in general would require the use of a global optimization algorithm together with multiple starting points. We also use a smoothening function on the models to work with a local optimization algorithm that works with derivatives of the cost function and both models. By using multiple starting points for this method, we hope to obtain similar results. The optimization algorithms that are implemented and investigated in this thesis are: the Genetic Algorithm (GA), the Simulated Annealing (SA) algorithm, the Pattern Search (PS) algorithm, and the Resilient backPROPagation (RPROP) algorithm.

To compare the results of the four optimization algorithms, four different scenarios are considered. The final results show that all control methods perform better than the FT controller. Overall the GA algorithm performs best without using a multi-start approach, with PS and SA having similar results. The RPROP method is either close to the other methods (0-2%) or quite far off (10-15%), depending on the scenario. When looking at computation time, PS is the fastest. It is twice as fast compared to the GA in most scenarios. SA however takes around fifteen times the amount of computation time compared to the GA. RPROP has varying results again compared to the GA algorithm. A better cost function for the TTS as well as a more optimized algorithm for the RPROP method can resolve both of these issues but future work might need to prove this.

Investment projects in railway infrastructure networks are needed to maintain a high condition of the network, maximize the capacity of the network, and to keep the risk low. Railway organizations are aiming to achieve these goals, but also want to minimize the costs of the investments. The desires of stakeholders in railway networks such as the government, passengers, and freight users are often not aligned. Decision makers in this field need to make trade-offs between the conflicting objectives. In the decision-making process regarding investment projects, it is important that the multiple objectives are considered explicitly. A multi-objective optimization problem formulation is proposed for the investment project portfolio optimization problem. Solving a multi-objective optimization problem means to identify the set of Pareto optimal solutions. These Pareto optimal solutions are used by decision makers to gain insight in the effects on the objectives, and to make well-founded trade-offs.

An approach is proposed to include the uncertain nature of investment projects in the problem formulation, without increasing the complexity of the problem too much. This is done by introducing lower and upper bounds for uncertain variables. By optimizing not only the expected values of the objective functions, but also the lower and upper bound, one is able to identify the solutions that give satisfactory results for all the scenarios. Project portfolios that are Pareto optimal for the lower bound, expected value, and upper bound, are considered as robust choices with respect to the uncertainties in the objective values.

Railway infrastructures typically consist of many assets with a long lifespan. The prediction horizon and the level of detail considered in the optimization problem need to be chosen carefully to avoid extremely large computation times. Even for a modest level of detail, the problem size is too large to check all the possible solutions within acceptable time. The computation time is also dependent on the algorithm that is used to search for the Pareto optimal set. Next to the computation time, several performance indicators are selected to measure the quality of the approximation of the Pareto optimal set that is provided by an algorithm.

Two algorithms with different approaches are proposed to approximate the Pareto optimal set, the repeated ε-constraint algorithm and the widely used genetic algorithm NSGA II.
Some benchmark problems are introduced to gain insight into the working of the proposed algorithms.
A case study is performed for an artificial data set for investment projects in a railway network.
The selected performance indicators are used to analyze the results of the case studies for the proposed algorithms with different settings.
The repeated ε-constraint algorithm is able to find a close representation of the Pareto optimal set, and has a very stable performance due to the established search method of the CPLEX solver. However, obtaining these results requires an excessive amount of computation time. The genetic algorithm NSGA II is much faster, and for carefully tuned settings, the quality of the approximation set is fine. The coverage of the NSGA II is considered sufficient, so due to the reduced computation time, the NSGA II is recommended to use in the decision support process of selecting investment project portfolios in railway infrastructure networks.