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- Global existence for a degenerate haptotaxis model of cancer invasion (2015)
- We propose and study a strongly coupled PDE-ODE system with tissue-dependent degenerate diffusion and haptotaxis that can serve as a model prototype for cancer cell invasion through the extracellular matrix. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations for a two-dimensional setting.

- Performance Analysis in Robust Optimization (2015)
- We discuss the problem of evaluating a robust solution. To this end, we first give a short primer on how to apply robustification approaches to uncertain optimization problems using the assignment problem and the knapsack problem as illustrative examples. As it is not immediately clear in practice which such robustness approach is suitable for the problem at hand, we present current approaches for evaluating and comparing robustness from the literature, and introduce the new concept of a scenario curve. Using the methods presented in this paper, an easy guide is given to the decision maker to find, solve and compare the best robust optimization method for his purposes.

- Minimizing the Number of Apertures in Multileaf Collimator Sequencing with Field Splitting (2015)
- In this paper we consider the problem of decomposing a given integer matrix A into a positive integer linear combination of consecutive-ones matrices with a bound on the number of columns per matrix. This problem is of relevance in the realization stage of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf collimators with limited width. Constrained and unconstrained versions of the problem with the objectives of minimizing beam-on time and decomposition cardinality are considered. We introduce a new approach which can be used to find the minimum beam-on time for both constrained and unconstrained versions of the problem. The decomposition cardinality problem is shown to be NP-hard and an approach is proposed to solve the lexicographic decomposition problem of minimizing the decomposition cardinality subject to optimal beam-on time.

- Robust storage loading problems with stacking and payload constraints (2015)
- We consider storage loading problems where items with uncertain weights have to be loaded into a storage area, taking into account stacking and payload constraints. Following the robust optimization paradigm, we propose strict and adjustable optimization models for finite and interval-based uncertainties. To solve these problems, exact decomposition and heuristic solution algorithms are developed. For strict robustness, we also present a compact formulation based on a characterization of worst-case scenarios. Computational results show that computation times and algorithm gaps are reasonable for practical applications. Furthermore, we find that the robustness concepts show different potential depending on the type of data being used.

- Discrete Parallel Machine Makespan ScheLoc Problem (2015)
- Scheduling-Location (ScheLoc) Problems integrate the separate fields of scheduling and location problems. In ScheLoc Problems the objective is to find locations for the machines and a schedule for each machine subject to some production and location constraints such that some scheduling object- ive is minimized. In this paper we consider the Discrete Parallel Machine Makespan (DPMM) ScheLoc Problem where the set of possible machine loc- ations is discrete and a set of n jobs has to be taken to the machines and processed such that the makespan is minimized. Since the separate location and scheduling problem are both NP-hard, so is the corresponding ScheLoc Problem. Therefore, we propose an integer programming formulation and different versions of clustering heuristics, where jobs are split into clusters and each cluster is assigned to one of the possible machine locations. Since the IP formulation can only be solved for small scale instances we propose several lower bounds to measure the quality of the clustering heuristics. Ex- tensive computational tests show the efficiency of the heuristics.

- A new solution approach for solving the 2-facility location problem in the plane with block norms (2015)
- Motivated by the time-dependent location problem over T time-periods introduced in Maier and Hamacher (2015) we consider the special case of two time-steps, which was shown to be equivalent to the static 2-facility location problem in the plane. Geometric optimality conditions are stated for the median objective. When using block norms, these conditions are used to derive a polygon grid inducing a subdivision of the plane based on normal cones, yielding a new approach to solve the 2-facility location problem in polynomial time. Combinatorial algorithms for the 2-facility location problem based on geometric properties are deduced and their complexities are analyzed. These methods differ from others as they are completely working on geometric objects to derive the optimal solution set.

- Competitive Algorithms for Multistage Online Scheduling (2015)
- We study an online flow shop scheduling problem where each job consists of several tasks that have to be completed in t different stages and the goal is to maximize the total weight of accepted jobs. The set of tasks of a job contains one task for each stage and each stage has a dedicated set of identical parallel machines corresponding to it that can only process tasks of this stage. In order to gain the weight (profit) associated with a job j, each of its tasks has to be executed between a task-specific release date and deadline subject to the constraint that all tasks of job j from stages 1, …, i-1 have to be completed before the task of the ith stage can be started. In the online version, jobs arrive over time and all information about the tasks of a job becomes available at the release date of its first task. This model can be used to describe production processes in supply chains when customer orders arrive online. We show that even the basic version of the offline problem with a single machine in each stage, unit weights, unit processing times, and fixed execution times for all tasks (i.e., deadline minus release date equals processing time) is APX-hard. Moreover, we show that the approximation ratio of any polynomial-time approximation algorithm for this basic version of the problem must depend on the number t of stages. For the online version of the basic problem, we provide a (2t-1)-competitive deterministic online algorithm and a matching lower bound. Moreover, we provide several (sometimes tight) upper and lower bounds on the competitive ratio of online algorithms for several generalizations of the basic problem involving different weights, arbitrary release dates and deadlines, different processing times of tasks, and several identical machines per stage.

- On the History of Differential-Algebraic Equations (2015)
- To write about the history of a subject is a challenge that grows with the number of pages as the original goal of completeness is turning more and more into an impossibility. With this in mind, the present article takes a very narrow approach and uses personal side trips and memories on conferences, workshops, and summer schools as the stage for some of the most important protagonists and their contributions to the field of Differential-Algebraic Equations (DAEs).

- A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor (2015)
- A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed. The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion equation for the extracellular proton concentration on the macroscale. In a more general context the existence and uniqueness of solutions for local and nonlocal SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model both, in its local version and the case with nonlocal path dependence. Numerical simulations are performed to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.

- A stochastic model featuring acid induced gaps during tumor progression. (2015)
- In this paper we propose a phenomenological model for the formation of an interstitial gap between the tumor and the stroma. The gap is mainly filled with acid produced by the progressing edge of the tumor front. Our setting extends existing models for acid-induced tumor invasion models to incorporate several features of local invasion like formation of gaps, spikes, buds, islands, and cavities. These behaviors are obtained mainly due to the random dynamics at the intracellular level, the go-or-grow-or-recede dynamics on the population scale, together with the nonlinear coupling between the microscopic (intracellular) and macroscopic (population) levels. The wellposedness of the model is proved using the semigroup technique and 1D and 2D numerical simulations are performed to illustrate model predictions and draw conclusions based on the observed behavior.