Hoewel het probleemoplossen in het wiskundecurriculum is opgenomen, hebben docenten vaak behoefte aan meer praktische ondersteuning bij het onderwijzen ervan. De recent in het Nederlands gepubliceerde didactiek ”Building Thinking Classrooms” van Peter Liljedahl met daarin veertien praktisch beschreven principes kan daarbij van betekenis zijn.

Het doel van dit onderzoek is om meer inzicht te verkrijgen in de effectiviteit van twee van de veertien principes ten aanzien van de beoogde leeropbrengst binnen de specifieke lespraktijk. Hierbij gaat het om het maken van aantekeningen met behulp van een ontworpen werkblad en het consolideren van een les door ontworpen lesactiviteiten waarin leerlingen zelfstandig reflecteren. De specifieke lespraktijk waarin dit onderzoek wordt uitgevoerd bestaat uit zes onafhankelijke lessen in ”Thinking Classrooms”. Deze lessen staan in het teken van het onderwijs van het oplossen van algemene wiskundige problemen. De bijbehorende leerdoelen zijn gerelateerd aan het ontwikkelen van metacognitieve vaardigheden en een geschikte wiskundige houding, wat volgens Alan Schoenfeld behoren tot twee van de vier essentiële bekwaamheden voor het wiskundig probleemoplossen. Om het doel van dit onderzoek te bereiken is de volgende hoofdvraag opgesteld.

Welke leerlingactiviteiten binnen de principes “note-taking” en het “consolideren” in een “Thinking Classroom”, ontwikkeld door Peter Liljedahl, dragen bij aan een verhoogde leeropbrengst met betrekking tot het oplossen van algemene wiskundige problemen door leerlingen een gemengde havo/vwo bovenbouwklas?

Dit onderzoek is uitgevoerd door middel van het cyclische ”Design-Based research”. Deze onderwijsbenadering richt zich richt op het ontwerpen, implementeren en evalueren van onderwijsinterventies in echte leeromgevingen. Naar aanleiding van de verschillende iteraties kunnen er patronen worden ontdekt in het leergedrag van de leerlingen. Zo blijkt er dat een individueel werkblad, na het gezamenlijk oplossen van het probleem, een passende niveauverhogende activiteit kan zijn. Dit is het geval wanneer er op het werkblad gevraagd wordt om de stappen voor het vinden van de oplossing van een probleem individueel volledig uit te werken en daarop te reflecteren. Om meer uitgebreide betekenisvolle reflecties van leerlingen te stimuleren blijkt dat het afhankelijk is van de vordering van een groep leerlingen binnen het proces van het probleemoplossen of een docentgestuurd klassengesprek of een leerlinggestuurd groepsgesprek relevanter is. Een bijzonder resultaat hierbij is dat wanneer leerlingen het centrale probleem uit een les al snel hadden opgelost, zij in sommige gevallen uit zichzelf met elkaar gaan discussiëren over hoe de gevonden oplossing toegepast of geoptimaliseerd kan worden en dit vervolgens gaan uitvoeren.

Vervolgonderzoek zou zich kunnen richten op de invloed van deze lessen op de leerprestaties van de deelnemende leerlingen binnen het curriculum. Hierbij krijgen namelijk wiskundige kennis en onderwerpspecifieke heuristieken, de overige twee essentiële bekwaamheden die Alan Schoenfeld beschrijft voor het wiskundig probleemoplossen, meer de nadruk.

In the last decade the graph burning model was developed to model the spread of information between people. Graph burning is a process that is done in rounds with the aim of spreading information to every connected person in a network. Every round one new source of information may be appointed and information spreads from people who have received it, to all of their connections, just like fire spreads. The burning number of a graph, denoted by 𝑏(𝐺), is the parameter that quantifies the speed of this spread of information. It has been conjectured that the burning number for a connected graph on 𝑛 vertices is at most ⌈√𝑛⌉. We prove the burning number conjecture for cat-constructs. Cat-constructs are trees obtained from a path graph 𝑃𝑛 by adding at most two vertices to subtrees of 𝑃𝑛. We show the burning sequence of a cat-construct may contain one fewer source than its burning number if the number of vertices for the cat-construct is more than the first square bigger than 𝑛. With this result

This research is conducted in collaboration with the Sophia Children's Hospital (SCH). The hospital wants to provide their patients with more detailed information about when a patient is approximately scheduled to have a surgery. The first step is to create a model which optimises the operation room (OR) schedule and indicates when different kinds of surgeries are planned. This information, combined with the waiting list, provides insight in when a surgery of a specific patient is scheduled.

In a hospital, different departments work together to treat the patient as good and efficient as possible. If a patient needs a surgery, not only an OR is needed, but also a bed at a ward which matches the patient's needs. The goal of this thesis is to use the different resources of the hospital as efficiently as possible. This is done by not only optimising the utilisation of the OR, but at the same time levelling the bed occupancy of the different wards. The levelling of the bed occupancy is done by minimising the maximum number of used beds at each ward. Because, if we minimise the maximum, we force that the patients are spread out more evenly over the day.

For each specialty, the patients are divided into patient groups based on historical data using a constrained $k$-means clustering algorithm. For each patient group, information is gathered about the length of stay (LoS) and the surgery duration of patients in this patient group. Next to that, the number of patients in a patient group indicates how often a patient group needs to be scheduled at least.

The probability distribution of the surgery duration is taken into account when deciding at which day, at what time, and in which OR a surgery is planned. A patient group can only be scheduled during OR shifts assigned to the corresponding specialty. At the same time, the levelling of the bed occupancy is taken into account.

After some constraints are linearised, this model can be formulated as a mixed integer linear program (MILP). However, the model has a large number of variables. Therefore, column generation is used to split the model into smaller subproblems per specialty. Some of the pricing subproblems take a lot of time to optimise. For that reason, we set some time limits both on the runtime of the pricing subproblems and the runtime of the entire algorithm. Column generation does not guarantee an optimal solution of our MILP. However, the objective value of our MILP improves over time, when new columns are added to the set of available columns. This indicates that column generation can be used to optimise our model.

In this thesis, several versions of the model are presented. For example, the schedule is different if the bed occupancy is calculated every hour or of every fifteen minutes. Next to that, the model can either be more focussed on maximising the OR utilisation or on levelling the bed occupancy.

"Who do you want to sit next to?'' In this thesis we develop an algorithm that optimizes a classroom lay-out based on this question. Besides the couples that sit next to each other, it also matters who sits in front (or behind). Even the relations across the aisles matter in the lay-out optimization. Each student chooses three other students and divides ten point points between them, thus creating a weighted graph with potential matches. To fill the classroom we have to address three problems: ``Who are the couples?'', ``Where do the couples sit?'' and ``How do the couples sit?''. First, three algorithms for creating couples are explained. Two of these will, ultimately, not be used. The third is the Blossom algorithm. This algorithm is the one we will be using to create the couples. To determine the positions of the couples, we first use the Repeated Nearest Neighbor algorithm to create a string of couples. Secondly, we use a brute-force approach to determine in which direction this string fills the classroom. For the last problem another brute-force approach is used to determine the orientation of each of the couples: ``Who sits at the left / right table?'' Finally, the created algorithm is tested on an actual class.

Erasmus MC is conducting a study on the increased risk of cardiovascular disease in women who have migraine. For this purpose, 50 healthy women and 50 women with migraine in age category of 40-60 years were compared using measurements of their dermal blood flow.
These measurements are done using two different imaging devices: Laser Speckle Imaging and Laser Doppler Imaging. The result is a recording of the fluctuations in the dermal blood flow over a time period influenced by circumstances and various mechanisms and consists of signal like curves. Erasmus MC have processed these measurements and obtained results. However, they are interested in whether applying Fourier Transform could offer additional insights or new (perhaps faster) methods to process the measurements.
The signal like curves can possibly be decomposed into sine and cosine waves using Fourier Transform. The result of the Fourier Transform consists of a visualisation of the intensity of the various components of such a waveform/curve. Based on that, it is possible to estimate the contribution of the mechanisms to blood perfusion. Existing literature and previous studies provide frequency intervals related to several mechanisms that influence the dermal blood flow.
This thesis will research to what extent the mechanisms contribute to the dermal blood flow across the different phases and how their behaviour differs when the circumstances for the measurement are adjusted. Perhaps there are behaviours that are instantly recognizable across all patients. An interesting question to answer would be whether it is possible to directly visualise the risk groups.

All organisms are built out of cellular tissue. Being able to recognise abnormalities in these tissues could be useful in recognizing cancerous cells. In this thesis we construct a mathematical model for cellular tissue based on its spatial structure. We consider cells as elements of the network. Touching cells are considered connected. Cells grow at different growth speeds. We determine the point in time when this network is fully connected, meaning there is a path between every pair of cells through touching cells. This point indicates the start of the last phase of the cellular growth, where friction restricts cell movement. We first use a Poisson point process to generate the locations of the cells. To make the model more similar to cellular tissue, we introduce determinantal point processes which have short-ranged repulsion, meaning points repel each other and thus spread. We compare the repulsion of Poisson and Determinantal point processes with real cellular tissue. We conclude that determinantal point processes have significantly higher repulsion than Poisson point processes. We also conclude that the repulsion in cellular tissue is higher than both point processes. Using simulations, we show that the model with determinantal point processes reaches connectivity significantly earlier than the Poisson model. We conclude that the analytically derived connectivity time point from the Poisson model can be used as an upper bound for the determinantal model.

The problem considered in this thesis is the box set optimization problem. In
this problem the goal is to find the optimal dimensions of a number of shipping
boxes which have to provide an optimal fit on a set of items. The application
considered is e-commerce, and hence the number of items is relatively large
compared to other applications found in literature.

The literature describes multiple problem formulations where a small num-
ber of optimal shipping boxes are selected from a large set containing candidate
boxes. However, since the instances considered in e-commerce application are
often too large to be solved in this way, we introduce a simulated annealing
heuristic solution method. In this way we are able to generate box sets con-
taining 10 to 20 boxes with centimeter accuracy for data sets containing up to
XX single item orders.

We pay special attention to an algorithm to very efficiently calculate the
objective value associated to the fit of a selection of candidate boxes on a set of
items. This is done by assigning items to shipping boxes using a pre-calculation
table. Because the heuristic can process a larger set of candidate boxes than
the exact solution method, ultimately results of the heuristic are better than
those found by the exact solution method.

Finally we generate multiple box sets based on the input data provided by
the industry partner bol.com, and compare the results. We conclude that on
almost all tested instances, the total amount of carton used can be reduced by
more than XX in combination with a total send volume reduction of over XX
in comparison with the box set that is currently used. Furthermore, we answer
additional questions. We find that minimizing the total amount of carton used,
instead of minimizing the total volume send, can lead to a result that is most
desirable in practice. We discover that pack-line specific, warehouse specific or
seasonal specific box sets provide none to only minor additional improvements.

Thyroid hormones are fundamental in the development and function of the human body. They are produced by the thyroid and their concentration is regulated by a negative feedback loop involving also the hypothalamus and the pituitary. Many studies have shown that each individual has a unique HPT axis set-point, which means that everyone has his or her own personal level of thyroid hormones. However, when the thyroid is affected by a disease, such as hypothyroidism, a change in the thyroid function modifies the levels of thyroid hormones, therefore they do not match the patient’s set-point anymore. Hence, the aim of this thesis is to investigate the relationship between the thyroid hormones TSH and FT4 in order to predict each individual’s set-point. Moreover, the variation in time of the two hormones in patients under medication is studied as well, which allows to determine around which values the concentrations of TSH and FT4 will stabilize and, if these are not matching the set-point, to adjust the patient’s medication dosage. This research was conducted using patients’ datasets from different hospitals. In particular, datasets of hypothyroid and thyroidectomized patients were included in this study. The analysis of hypothyroid patients’ measurements shows that TSH can be expressed as a negative exponential function of FT4. Furthermore, the individual set-point can be predicted as the point of maximum curvature of that function and this has been validated using data from thyroidectomized patients. Finally, a model of coupled differential equations, taking into account the effects of medication, has been developed to describe how the concentrations of the hormones are changing in time after a patient starts the treatment.

This thesis is devoted to option pricing on backward-looking rates. For the last decades, interest rate products were often linked to IBOR rates. IBORs are short-term borrowing rates charged between global banks in the unsecured interbank market. The purpose of this thesis is to compare the Hull-White model to the Black-Karasinski model for the pricing of caps/floors on compounded rates. Both models are so-called short-rate models, which are widely used for interest rate modelling. Due to the IBOR reform, new products are expected to appear in the market. One type of these products is caps/floors linked to the new Risk-Free Rate (RFR). The new RFRs will be in-arrears backward-looking rates and, as a consequence, have an impact on the choice of pricing models.
This thesis considers caps/floors on the new compounded RFR rates. For both models various pricing techniques for caps/floors on compounded rates are investigated. For the Hull-White model, the pricing kernel approach and a Monte Carlo simulation are explored. The pricing kernel approach yields an analytic formula for caps/floors on compounded rates. This formula is also used for the comparison. For the Black-Karasinski model, the pricing kernel approach, the trinomial tree method and the Monte Carlo simulation are considered. The pricing kernel approach yields a semi-analytic formula for caps/floors on compounded rates. However, in practice the computation time of this semi-analytic formula turned out to be substantial. Further, despite the fast computation time of the trinomial tree for LIBOR caps/floors, the trinomial tree method is rather slow for the caps/floors on compounded rates. As a result, the Monte Carlo simulation is the most suitable pricing technique, along the three
explored methods, for caps/floors on compounded rates under the Black-Karasinski model. Therefore, the Monte Carlo simulation is used for pricing caps on compounded rates in the model comparison.
Since there exists no liquid market yet for caps/floors linked to the new RFR, the models are calibrated to a proxy market. First, the Black-Karasinski model is calibrated to the proxy market using a heuristic approach. This heuristic approach is chosen as a compromise between computation time and accuracy. Then, the Hull-White model is calibrated to the Black-Karasinski model using the stripping method. Having both models calibrated, the price of caplets/floorlets on compounded rates are calculated. Thereafter, these prices are inverted to Bachelier implied volatilities for a uniform comparison. From the comparison of the two models, a difference in Bachelier implied volatility is observed in a range of 4 to 4 bps. This is of one order less than the volatility itself.

In intensity modulated proton therapy (IMPT), patients are irradiated with small spots, that deliver a local dose to the tumor. The number of possible spots to choose from is virtually infinite, but practically limited, which requires a spot selection. This spot selection should result in an optimal treatment plan, i.e., to deliver a sufficient dose to the tumor, while sparing the healthy surrounding tissue. These trade-offs make treatment planning in radiotherapy a multi-criteria optimization problem. The current approach for this spot selection by the Erasmus Medical Center (MC) is an iterative resampling method which uses a trial and error principal. A random sampled spot selection is made, bad spots are removed, and new spots are randomly added. The research goal of this project is to improve the current spot selection method, by inducing sparsity on spot selections with the L1-norm, without decreasing the plan quality of the current solution. Sparse solutions are beneficial for optimization problems since they reduce the problem size and have higher probability of producing qualitative solutions. To achieve these goals, the Sparsity-Induced-Spot-Selection (SISS) method was developed. Contrary to the iterative resampling approach, the SISS method uses a top-down approach. Starting with a large spot coverage, it selects as little relevant spots as possible through the use of the L1-norm to induce sparsity, until an acceptable treatment plan using as little spots as possible is made. The developed method was validated on a test set consisting of 10 head and neck patients. Using the SISS method, an average spot selection of 1159 spots was produced, compared to a solution of 1074 spots for the resampling method. For the average patient, 6 out of 10 Organs-at-risk (OAR) received a lower dose with the SISS method than with the resampling method. The remaining OARs all received a marginal dose surplus of 0.6 Gy, with a maximum of 2.6 Gy. The target volumes in the tumor also received a similar dose to the resampling method, with the near-minimum dose of the tumor receiving a dose shortage of 0.2 Gy, and the near-maximum dose of the tumor receiving a dose surplus of 0.5 Gy, both being considered as marginal differences. The SISS method produces a comparable spot selection and shows no decline in plan quality of the dose distributions. Using the L1-norm to induce sparsity on spot selections in treatment planning is feasible. The computation time of the SISS method could be reduced using a voxel reduction, although this reduction in computation time is not guaranteed in practice due to robust optimization being applied.

Currently, speaker recognition research is mainly based on phonetics and speech signal processing. This research addresses speaker recognition from a new perspective, analysing the transcription of a fragment of speech with text analysis methods. Since text analysis is based on the transcription text only, it can be assumed independent from current automatic speaker recognition software. Hence, it would contribute significantly to the overall evidential value. The analysis is based on the frequencies of non-content, highly frequent words. We study whether information about the identity of the speaker is contained in the transcription of spoken text. The value of evidence is quantified using a score-based likelihood ratio. The score-based approach is chosen because in most forensic cases, there is not enough data from the suspect or of the disputed speech fragment available to model a robust feature-based likelihood ratio. Different methods to model the system from feature vector over score to likelihood ratio have been compared. As a baseline, a distance based method is used, where the score is the distance between the feature vectors. To improve upon this baseline, machine learning algorithms are implemented. The results from SVM and XGBoost are explored. As a third method a feature-based likelihood ratio is calculated and used as a score instead of as a direct likelihood ratio. With this method, both similarity and typicality are taken into account. The model is trained and tested on the FRIDA data set from the Netherlands Forensic Institute, consisting of Dutch conversations from a homogeneous group of 250 individuals. The performance of the likelihood ratio system is evaluated through computing the cost log-likelihood-ratio (Cllr), which is a measure for the accuracy and quality of the likelihood ratios, and the accuracy (A) of the likelihood ratios solely. The performance is also evaluated by inspecting the Tippett, empirical cross-entropy and pool-adjacent-violators plots. Different values for parameters used in the calculation of the likelihood ratios are investigated: the length of the sample, the number of frequent words (number of features) and the number of samples needed to train the model. The distance method showed a strong baseline, with good performance for large sample lengths. The SVM method outperformed the distance method for all parameter settings, with a peak performance of A=0.94 and Cllr=0.24. The XGBoost method showed promising results for smaller samples lengths, but a too large amount of data is needed to obtain good performance for larger sample lengths. The LR score method showed moderate results, but no improvements due to the necessity to estimate high-dimensional distributions. This thesis shows that information about the identity of the speaker is contained in transcriptions of speech. The complete process from data to likelihood ratio is constructed, where the likelihood ratio quantifies the evidential value of a transcribed speech fragment.

Combined chemo-radiation treatments are often used to enhance tumor cell kill, with respect to their separate treatments. Currently, chemo-radiation therapy treatments are prescribed as a separately optimized radiotherapy treatment with the addition of a generic chemotherapy regimen. The dosing of the combined therapies is mostly based on clinical experience. Hence there is a clear need for the optimization of dosing for combined chemo-radiation therapy treatments. The goal of this thesis is to propose novel approaches to optimize cancer treatment efficiency when combining chemotherapy and radiotherapy by using control-theoretical methods as well as radiation therapy models. A positive switched system is a dynamic system consisting of a set of Metzler state-space matrices and a specific switching law, which determines when and how to switch between the subsystems. Positive switched systems seem particularly appropriate to model cancer evolution under different treatments, and therefore to determine optimal treatment scheduling. They are used to model the effect of combined chemo-radiation treatments, thus enabling the systematic design of optimal therapy planning. The proposed model is especially appropriate for heterogeneous tumors, since it describes the impact of therapies on different tumor cell lines, and includes the possibility of mutations. Metastases are also included since the model considers separate tumor compartments, and the possibility of migrations between compartments are modeled as well. Undesired consequences due to toxicities are incorporated, and an upper bound can be set to limit the possible damage. A clinical dataset was obtained from the Erasmus MC, which was used to validate chemotherapy-alone model outcomes. Combined chemo-radiation treatments were also developed for a set of initial tumors. The model shows promising applications for cancer treatment design. The use of positive switched systems for combined chemo-radiation treatments is a new approach to obtain optimal drug usage and radiation fractionation schemes for each body compartment. The achieved findings could have a significant impact on effective treatment planning by choosing optimal drug durations or by using radiotherapy in combination with chemotherapy, for instance, to target certain chemotherapy-resistant areas, to achieve tumor control with minimal side effects.

In dit verslag beschrijven we met behulp van een wiskundig model hoe een malaria infectie zich ontwikkelt in een persoon. We gebruiken hiervoor een stelsel van gewone differentiaalvergelijkingen, wat beschreven wordt in een artikel van Mary Bushman. Hierbij volgen we veranderingen in verschillende aantallen die een rol spelen bij een infectie, zoals het aantal geïnfecteerde en ongeínfecteerde rode bloedcellen, het aantal parasieten en de immuniteit, die wordt opgedeeld in een aangeboren deel en een adaptief deel. Hierbij kijken we naar de invloed van medicijngebruik en de infectiehistorie op een infectie. We gebruiken een numerieke rekenmethode om het stelsel op te lossen. We gebruiken de functie ’ode45’ in het programma MATLAB om tot een oplossing te komen. Bij een infectie door zowel vatbare parasieten sterven de vatbare parasieten eerder uit. Dit heeft te maken met de ontwikkeling van de adaptieve immuniteit tegen een bepaald soort parasiet. Het toedienen van medicijn vroeg in de infectie heeft meer effect dan later. Als laatste hebben we gekeken naar de invloed van eerdere infecties op de ontwikkeling van een nieuwe infectie. Doordat het immuunsysteem een soort parasiet al eerder is tegengekomen, geeft dit een nadeel voor de parasiet. Dit zorgt ervoor dat de parasieten sneller uit zullen sterven.

Cholera is an infectious disease caused by the bacterium Vibrio cholerae. Infections can occur directly via contact with infected people, but also indirectly via infected water. Because of these direct, but especially indirect infections, it is useful to investigate the spread of the disease using mathematical epidemic models. In this report a compartmental epidemic model for cholera is discussed: the SIRB model. This model consists of four compartments: susceptibles, infected, recovered and the bacterial concentration in water. The interaction between the groups is given by a system of differential equations. The equilibrium solutions and their stabilityare mathematically analysed. The main focus of this report, however, is on two different spatial models. The patches model consists of multiple patches connected by migration of people. Different measures against cholera are studied using this model. The second spatial model that has been numerically implemented is a diffusion SIRB model. The influence of the diffusion coefficients on the solution patterns will be researched for different initial conditions. The results presented in this report are building blocks for further research into compartmental models that simulate the spatial dynamics of cholera.

This thesis provides you with basic information on graph theory as well as phylogenetic networks, it studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks, based on the manuscript 'Rooting for phylogenetic networks' . Undirected phylogenetic networks can be oriented to become a directed network. In this manuscript the authors come up with multiple algorithms for orienting phylogenetic networks meeting different characteristics. A network is built up of reticulation vertices (where lineages merge) and tree vertices (where lineages separate). A network can be binary, meaning every node in the body of the network has a degree of three or a network can be non-binary, meaning there are no restrictions to the amount of edges a vertex can have.
When a network is binary, an algorithm is described that given the location of the root as well as a set of reticulation points, is used to find an orientation (Algorithm 1). If a network is non-binary, an algorithm is described that given a location of the root as well as the indegree of each vertex, is used to find an orientation (Algorithm 2 ). Once an orientation is found for a certain undirected network this orientation can be checked to see whether it meets the characteristics of a certain network class. Three network classes are considered. First of all an orientation can be tree-child, meaning that every non-leaf vertex has a child that is not a reticulation. Secondly an orientation can be stack-free, meaning that no reticulation has a child which is a reticulation. And last, an orientation can be valid, meaning that it is stack-free and deleting a single reticulation edge and suppressing its endpoints does not give parallel arcs. When we either did not find an orientation in the class we wanted or want to know all the orientations for a certain network in a certain class, we use Algorithm 3 or 4. The location of the root and the set of reticulation points were necessary input for Algorithm 1; whereas Algorithms 3 and 4 do not require this information. Algorithm 3 returns you the first found orientation in the class you wanted and Algorithm 4 returns you a collection of all orientations in the class. All orientations found by the program are returned in an output format that can be read by a network visualisation website.

In this thesis we are going to study outlier detection methods and propose a new method. Classical outlier detection is typically based on the assumption that the data is from a Gaussian/normal distribution. When the underlying distribution of a random sample is heavy tailed, so not normal , it is likely to have some extreme observations which would be identified as outlier using the classical procedure. This paper aims to address this issue by proposing a procedure
to identify real ‘outliers’ for heavy tailed data set. We first dive in the some existing methods and see how they work, try to understand them, simulate them and see their shortcomings in the case of a heavy tailed distribution. Then we study Extreme Value Theory (EVT) which we shall use to set up our proposed method of detecting outliers. Once we have constructed the proposed method, we are going to simulate and compare it with the existing methods. The goal in the case of normality is that the new method is not worse than the existing ones, at least not extremely, and in the case of a heavy tailed function to work better.

In Intensity Modified Proton Therapy, the number of energy layers and the number of beamlets determine the radiation time and the plan calculation time. The purpose of this research project is to test sparsity inducing terms to reduce both the energy layers and the beamlets.

Cancer is a disease that one of every three people will get in The Netherlands. One of the treatment methods for this disease is radiotherapy. Approximately half of all cancer patients will get radiotherapy at some point of their treatment. During radiotherapy cancer cells are destroyed with ionizing radiation, but healthy cells get destroyed too. When a patient gets treated with radiotherapy, the goal is to find a treatment plan which will destroy all of the cancer cells and as few healthy cells as possible. To reach this goal we want to make a unique treatment plan for every patient, because every patient is anatomical unique. We use a wish-list to generate this unique optimal treatment plan. This wish-list contains all of the demands of the physician. All of the demands can be written into cost-functions. We will use inverse multicriteria optimisation to find the most relevant cost-functions for every organ and the tumour (planning target volume (PTV)). The relevance of a cost-function can be obtained by determining the weight of a cost-function. We start with a non-linear problem and we use the Karush-Kuhn-Tucker conditions. We did not receive the desired solutions.Afterwards, we tried to find the optimal weights for a linear problem by writing it in the form of an absolute duality gap minimization problem. This gave the results we were hoping for.

In a Vehicle Routing Problem with Time Windows (VRPTW), orders have to be picked up and delivered within certain time windows. In practice, planners often allow violations of these time windows, when the solutions with violations have better objective values. This is done by changing the problem into a Vehicle Routing Problem with Soft Time Windows (VRPSTW). In this thesis, the effect of soft time windows on the objective values of a real-world VRP is analysed for multiple single-day cases of a distribution company. The used algorithm is a heuristic, based on local search methods. At first, a sensitivity analysis is performed on the sensitivity of the number of planned tasks to the time window tolerance. The addition of time window tolerance increases the number of planned tasks in most of the cases. Furthermore, when the tolerance increases, the number of planned tasks generally increases as well, until the number of planned tasks is equal to the number of tasks that is planned when the time windows are completely removed. In the second part of the experiments, trade-offs between the cost function and the amount of violation are investigated, by varying the slope of the linear penalty function (for a fixed tolerance). In general, a higher penalty leads to a lower amount of violation and a higher cost (without penalties), but this is not a monotone correlation. For the smaller cases, the costs can be lowered by 4.8% to 9.0%. To reach this minimum cost, an average violation of 1 to 7 minutes per task has to be accepted. For the bigger cases, only the lowest penalty values lead to a cost improvement. Here, the lowest cost value is achieved by choosing the zero function as the penalty function, which means that the corresponding solution contains a large amount of violation.

The coarse grid of numerical weather prediction and climate models requires parametrization models to resolve atmospheric processes that are smaller than the grid size. For parametrization development, these processes are simulated by a high resolution model. At the Royal Netherlands Meteorological
Institute, the Dutch Atmospheric Large-Eddy Simulation (DALES) is used. This three-dimensional high resolution model uses advection schemes that are too diffusive when steep gradients are present. In this thesis, an advection scheme based on the Discontinuous Galerkin (DG) method is implemented
for DALES.

The DG method is known to be dispersive. To remove those non-physical oscillations, the moment limiter of Krivodonova is used. Krivodonova constructed the limiter for one- and two-dimensions. In this thesis the moment limiter and limiting order are derived for three-dimensions.

DALES is a model based on the finite difference method and uses operational splitting. Therefore, the DG advection scheme needs a mapping from each cell average to all nodal values that are needed for one DG cell, and a mapping back, which we called mapping a and b respectively. Mappings a that are discussed are taking the cell average as value for all nodal points of the DG cell (cell average a), and taking the L -projection of the cell average to the continuous finite element space (L -projection). This thesis describes mappings b that calculate cell averages of nodal DG values (cell average b)
and calculate the cell averages of the tendencies of DG values (cell average of tendency). Using cell average a combined with cell average of tendency, made the DG method as diffusive as the first order upwind scheme. Substituting the cell average a method with the L -projection, the DG method became
very dispersive, meaning that there was not enough diffusion. At last, cell average b was tested with the L -projection. Its numerical results showed that the speed of the advection was slower than the theoretical velocity. Therefore, a method is suggested which does not need mappings. An option could
be a supergrid that takes multiple DALES cells as a DG cell.