This report investigates a scheduling problem where task duration is uncertain. The duration per task has a lower and upper bound, and is dependent on observed duration of other tasks. This tries to closer model real life. We reduce all possible different outcomes to a few extreme scenarios. The report compares two types of heuristcs: one which always chooses the longest duration task first, and one which tries to minimize the uncertainty by choosing tasks that reveal the most information. In the end, we find that the heuristic choosing the longest duration task first competes fairly well with the other type of heuristics.

Packages to encode Machine Learned models into optimization problems is an underdeveloped area, despite the advantages is could provide. The main draw of implementing Machine Learned models into optimization models, is that it allows the optimizer to better account for the human experience.
Maragno D., Wiberg H. et al. constructed an implementation of the encoding with their package OptiCL. In order to verify their implementation and provide principles for (re)designing packages with similar functions, an amount of components of OptiCL were replicated within this paper. The requirements for
the program were first constructed before detailing the implementation process. After the program was implemented, both OptiCL and the found program were tested in order to compare performances. Using the results and an investigation of the two implementations, a framework for encoding similar packages
was provided using the insights gained. Using mathematical formulations supplied by Maragno D., Wiberg H. et al., design principles outlined in this report and research into the encoding of other Machine Learned models, other developers could construct robust packages that allow for easy integration of
valuable information gained from Machine Learning into optimization problems. This in turn allows for frequently used optimization models to account for more human understanding.

In this research the effectiveness of analytical neural networks compared to the maximum likelihood method on the prediction of spatial and DOI positioning of a Gamma detector with a NaI(Tl) scintillator of size 590mm x 470mm x 40mm (x,y,z), with a glass lightguide of size 620mm x 500mm x 4mm and a PMT area of 620mm x 500mm x 40mm with 2-inch round PMTs with a Bialkali photocathode is presented. This is done by training neural networks with different cost function, different amounts of hidden layers and different amounts of neurons per hidden layer, trained on different amounts of training data. The resolution of the predictions of the testing data are compared with those of the maximum likelihood method. It was concluded that the neural network with best spatial resolution, had the Huber loss function as cost function, 4 hidden layers and 512 neurons per hidden layer and was trained on 29,970 datapoints. The FWHM and the FWTM were 3.83 ± 0.54 mm and 12.49 ± 1.19 mm respectively, while the FWHM and the FWTM of the maximum likelihood method were 3.31 mm and 12.13 mm respectively. The resolution of the neural network was lower than that of the maximum likelihood method. The same was done for the DOI resolution, here a neural network with mean squared error as cost function, 4 hidden layers and 64 neurons per hidden layer trained on 9,990 datapoints, gave the best the resolution with FWHM and FWTM equal to 6.00 ±0.50 mm and 11.94 ± 0.94 mm respectively. The FWHM and FWTM of the maximum likelihood method were 6.16 mm and 11.22 mm respectively. This made the DOI resolution of the neural network higher then that of the maximum likelihood method. Finally different ideas were presented to increase the resolution of the neural network. These were: training the neural network on independent data, split the neural network in a spatial part and a DOI part, create a more complex architecture and making use of a convolutional neural network.

The application of machine learning in daily life requires interpretability and robustness. In this paper we try to make the process of building robust and interpretable decision trees more accessible. We do this by making the fitting of these models cheaper and simpler. We build on previous research and see if changing input data or the fitting formulation can create more robust trees that can be computed faster. To investigate this, we test whether data perturbations make heuristic algorithms more robust and whether enforcing constraints on adversarial examples in normal optimal classifica- tion tree MILP formulations can improve robustness. We also provide an altered formulation for the robust OCT model in Vos and Verwer (2021b) that yields better results with shorter runtimes. Finally, we extend the ROCT formulation to be applicable to multi-class classification and regression tasks.

Supply planning is an NP-Hard problem that is often tackled when dealing with supply chain management. It is a problem with many variations but the core idea is to create a plan that resolves as much demand as possible.
There are different approaches in use to solve a planning problem, one of which is by using heuristics. Heuristics are used to get to a relatively good solution within some feasible amount of time compared to trying to get to an optimal solution. Expertise of the problem is often required to come up with good heuristics or to improve existing ones. When such an expert is not readily available alternatives need to be considered, one of which is machine learning.
The goal of this thesis is to create a machine learning model that can improve the heuristics used in an existing planning algorithm at Outperform. This heuristic decides upon a sequence in which products are included in the planning algorithm. Using a different sequence can lead to vastly different results for the algorithm both in quality and time required. Our focus will be on reducing the amount of time required for the algorithm by learning a heuristic that can provide better sequences. 
To achieve this goal, two models are created and evaluated. These models use imitation learning to replicate the sequences that were the fastest. Due to the lack of an expert to provide the fastest sequences, an oracle is created that attempts to search for a sequence as fast as possible within some feasible amount of time. The trained models are evaluated on several supply chains provided by Outperform.

A fundamental tool in radiotherapy treatment planning is the dose calculation algorithm, which models the dose that will be distributed for given beam parameters and patient geometry. Various available algorithms include Monte Carlo simulations (MC) and pencil beam algorithms (PBA), with the former being computationally expensive but offering high precision and the latter sacrificing precision for speed. A recent study presents the deep-learning based Dose Transformer Algorithm (DoTA) which provides MC accuracy at speeds 33 times faster than PBA. However, as currently implemented, DoTA dose computations assume that each ray enters the patient geometry perpendicularly, while clinical treatment plans consist of many diverging rays with angles of entry up to 5°.

In this project, we extend the current model to include angular dependency. The resulting models DoTA-A and DoTA-S improve on DoTA by including angle of entry as an additional input on top of the beam energy and patient geometry. DoTA-A includes the actual angle values as input, while for DoTA-S an expected beam shape is precalculated with a trajectory based on the angle of entry. A training dataset of more than 30.000 samples with MC baseline dose is generated from a public patient dataset, using a 2 mm resolution. The architecture of the models is similar to that of DoTA, with convolutional layers extracting important spatial features from the input geometry and a transformer layer using a self-attention mechanism to weigh token inter-dependence.

The models DoTA-A and DoTA-S are evaluated and compared on different test sets with MC baseline doses. Both models are shown to be more accurate than PBA, with DoTA-S having the best performance by most metrics. We demonstrate the relevance of ray angles in dose calculations by comparing DoTA-A and DoTA-S to perpendicular MC predictions, which were considered ground-truth for DoTA. The models DoTA-A and DoTA-S compute dose distributions at an average speed of 10 ms to 15 ms per dose, with the predictions achieving an average relative error of 1% across various test sets. The average relative error of the perpendicular MC predictions lies around 3%, demonstrating the importance of angle of entry as an input variable in dose calculation algorithms. The gamma pass rates (for δ=1%, Δ=3mm) of a full treatment plan with dose distributions predicted by our models are 97.60% for DoTA-A and 95.74% for DoTA-S, indicating that there is no strictly better model between the two.

In this thesis a novel model is proposed to solve the Robust Tail Assignment problem. The Robust Tail Assignment problem aims to assign aircraft to flights, while minimize expected costs of operating a flight schedule, including expected delay costs. This problem is difficult, because delays can propagate between successive flights in the schedule, creating dependencies between flights assigned to the same aircraft.

Using probability distributions of delay for every individual flight, as well as expected costs associated with delaying flights, the expected delay costs of a full flight schedule can be estimated. The workings of a simulator are described, which can be used to evaluate the total expected costs of solution schedules for the Robust Tail Assignment problem.

To be able to incorporate expected delay costs in a mathematical model, the construction of a multi-commodity flow network is described, which uses departure and arrival states for flight rotations, corresponding to discrete amounts of delay. The amount of flow through edges of this network represents the probability of these states transitioning into other states. By activating and deactivating edges, based on the assignment of aircraft to rotations, this network can be used in a model to approximate the total expected delay costs of a model solution.

The proposed robust flow model uses such a state network in a MIP model, that can be solved using an iterative solver to find good solutions to the Robust Tail Assignment problem. Delay costs are imposed on edges in the network, to quantify the expected delay costs. In the model, the network size is reduced by only considering connections between rotations that have high probabilities of propagating delay. This reduces the accuracy of the model, but shortens the run-time of the optimization process significantly.

Several experiments are done to test the run-time and performance of the robust flow model. The model proved hard to solve to optimality, but is able to find good solutions, if the model parameters are well tuned. Recommendations are given for using the model, as well as future research directions.

The field of robust optimization deals with problems where uncertainty influences the optimal decision. Some of these problems can be formulated in a ‘two-stage’ formulation, such as the location transportation problem. To solve such a problem, a column-and-constraint-generation algorithm has been introduced in which constraints are iteratively added to mixed-integer pro- gram based on different uncertain scenarios. However, if these scenarios are randomly chosen, this problem can grow too large to efficiently solve for. For most problems, there is some mini- mal set of scenarios needed to find the optimal solution, and it is important to find the ‘right’ scenarios early. In this study, we attempt to predict these scenarios for a location transportation using machine learning. Using customer demand data for different instances of the problem, we train a logistic regression classifier, a neural network and a random forest classifier to predict important scenarios for newly generated problems. We find that when applying these machine learning tools, we reach an average reduction of scenarios added to the problem ranging from 8% to 24%. Even though we do not spend much effort on training perfect models, we see that there is a strong indication that machine learning can be used to increase the efficiency of the algorithm.

Robust optimisation is shown to be extremely important in a wide range of applications including real life. Many research projects are dedicated to this relatively young and active research field and show the significant value of robust optimisation. Since many researchers have developed complex methods dedicated to robust optimisation, the aim of this project is to find out whether or not these complex methods were helpful or that the same or even better results could have been obtained with rather simpler methods. This is done by analysing one famous paper about multi-stage robust optimisation and comparing the ”here and now” decisions of this approach with the ”here and now” decisions of a much simpler method.

Climate change challenges the resiliency of our cities, and lack of spaces in densely paved areas makes it impossible to implement adaptation and mitigation strategies on the street level. However, buildings have often unused roofs, which can be converted into sustainable roofing options. Green roofs are systems which allow for the growth of different types of vegetation, mitigating heat and capturing water. Blue roofs are layers for water collection, which provide temporary storage and slow release of rainwater. Yellow roof is another name to indicate roofs with photovoltaic (PV) panels. Some areas of the city may be more subject to flooding and heat. Some buildings may instead receive more solar radiation. Therefore, prioritizing which option to install where may be the key to the success of sustainable roofing placement. In this study, we construct an optimization model that evaluates the benefits of the roofing option in each potential location and chooses the combination of roofs and roofing options which maximizes the total societal benefits. The reduction of flooding and heat risks and the production of clean energy are modeled considering both climate variables and the city’s characteristics. However, the derived data and parameters are uncertain by nature. Climate data is used in ensembles and is integrated into the problem by formulating the model into a stochastic and robust version. We take Schiedam as our case study area and solve the model. We also perform uncertainty analysis, making use of clustering and classification tree regression. This analysis allows to understand which parameters are driving which type of solutions, and to understand how robust the constructed model is. Moreover, we formulate the same problem in the shape of a multi-time step model, where the decision maker can make the optimal placement decision every year. In this case, it is possible to simplify the decision process into a one-branch tree for which we present a methodology. The results show that our model is robust in terms of placement choice when parameters change, and we discover which parameters drive the main variations. The total benefits are instead much more reliant on the uncertainty of the problem.