Stefan Pickl

Electric load forecasting methods: Tools for decision making

For decision makers in the electricity sector, the decision process is complex with several different levels that have to be taken into consideration. These comprise for instance the planning of facilities and an optimal day-to-day operation of the power plant. These decisions address widely different time-horizons and aspects of the system. For accomplishing these tasks load forecasts are very important. Therefore, finding an appropriate approach and model is at core of the decision process. Due to the deregulation of energy markets, load forecasting has gained even more importance. In this article, we give an overview over the various models and methods used to predict future load demands.

Transshipment and time windows in vehicle routing

Transshipment problems and vehicle routing problems with time windows (VRPTW) are common network flow problems and well studied. Combinations of both are known as intermodal transportation problems. This concept describes some real world transportation problems more precisely and can lead to better solutions. But they are examined rarely as mathematical optimization problems. In this paper two approaches are developed for this kind of merged problems. The source for our considerations is the pickup and delivery problem with time windows as a generalization of the VRPTW. This is extended by transshipment. Thereby loads can be transported by different vehicles on their path from origin to destination. A column generation technique is proposed for solving this problem.

Genetic Networks and Anticipation of Gene Expression Patterns

An interesting problem for computational biology is the analysis of time‐series expression data. Here, the application of modern methods from dynamical systems, optimization theory, numerical algorithms and the utilization of implicit discrete information lead to a deeper understanding. In [1], we suggested to represent the behavior of time‐series gene expression patterns by a system of ordinary differential equations, which we analytically and algorithmically investigated under the parametrical aspect of stability or instability. Our algorithm strongly exploited combinatorial information. In this paper, we deepen, extend and exemplify this study from the viewpoint of underlying mathematical modelling. This modelling consists in evaluating DNA‐microarray measurements as the basis of anticipatory prediction, in the choice of a smooth model given by differential equations, in an approach of the right‐hand side with parametric matrices, and in a discrete approximation which is a least squares optimization pr...

An algorithm to analyze stability of gene-expression patterns

Many problems in the field of computational biology consist of the analysis of so-called gene-expression data. The successful application of approximation and optimization techniques, dynamical systems, algorithms and the utilization of the underlying combinatorial structures lead to a better understanding in that field. For the concrete example of gene-expression data we extend an algorithm, which exploits discrete information. This is lying in extremal points of polyhedra, which grow step by step, up to a possible stopping. We study gene-expression data in time, mathematically model it by a time-continuous system, and time-discretize this system. By our algorithm we compute the regions of stability and instability. We give a motivating introduction from genetics, present biological and mathematical interpretations of (in)stability, point out structural frontiers and give an outlook to future research.

An Anticipatory Extension of Malthusian Model

In this paper, on the base of a new variable — deviation of population from an average value, we propose a new extension of the Malthusian model (see equations (10), (15) and (20)) using differential equations with piecewise constant argument which can be retarded as well as advanced. We study existence of periodic solutions and stability of the equations by method of reduction to discrete equations. Equations (15) and (20) with advanced argument are systems with strong anticipation. Moreover, we obtain a new interpretation of known results for differential equations with piecewise constant argument (6) and (8).

An algorithmic approach to analyse genetic networks and biological energy production: an introduction and contribution where OR meets biology†

An emerging research area in computational biology and biotechnology is devoted to modelling and prediction of gene-expression patterns. In this article, after a short review of recent achievements we deepen and extend them, especially, by emphasizing and analysing the elegant means of matrix algebra. Based on experimental data, ordinary differential equations with nonlinearities on the right-hand side and a generalized treatment of the absolute shift term, representing the environmental effects, are investigated. Then, the genetic process is studied by a time-discretization, in particular, Runge–Kutta type discretization. By a utilization of the combinatorial algorithm of Brayton and Tong, which is based on the orbits of polyhedra, the possibility of detecting stability and instability regions has been shown. The time-continuous and -discrete systems can be represented by means of matrices allowing biological implications, such as thresholds, and interpretations; which are motivated by our gene-environment networks. A specific contribution of this article consists of a careful but rigorous integration of the environment into modelling and dynamics, and in further new sights. Relations to the parameter estimation within modelling, especially, by using optimization, are indicated, and future research is addressed. †With gratitude dedicated to our dear teacher and friend Prof. Dr Alexander Rubinov who passed away in 2006.

Analysis, controllability and optimization of time-discrete systems and dynamical games

Uncontrolled Systems.- Controlled Systems.- Controllability and Optimization.- A.1 The Core of a Cooperative n-Person Game.- A.2 The Core of a Linear Production Game.- A.3 Weak Pareto Optima: Necessary and Sufficient Conditions.- A.4 Duality.- B Bibliographical Remarks.- References.- About the Authors.

Algorithms and the calculation of Nash equilibria for multi-objective control of time-discrete systems and polynomial-time algorithms for dynamic c-games on networks

Abstract We consider a multi-objective control problem of time-discrete systems with given starting and final states. The dynamics of the system are controlled by p actors (players). Each of the players intends to minimize his own integral-time cost of the system’s transitions using a certain admissible trajectory. Nash Equilibria conditions are derived and algorithms for solving dynamic games in positional form are proposed in this paper. The existence theorem for Nash equilibria is related to the introduction of an auxiliary dynamic c -game. Stationary and non-stationary cases are described. The paper concludes with a complexity analysis for that decision process.

Optimization and Multiobjective Control of Time-Discrete Systems

Multi-Objective Control of Time-Discrete Systems and Dynamic Games on Networks.- Max-Min Control Problems and Solving Zero-Sum Games on Networks.- Extension and Generalization of Discrete Control Problems and Algorithmic Approaches for its Solving.- Discrete Control and Optimal Dynamic Flow Problems on Networks.- Applications and Related Topics.

An optimal control problem in cancer chemotherapy

Abstract We consider a mathematical model for the control of the growth of tumor cells which is formulated as a problem of optimal control theory. It is concerned with chemotherapeutic treatment of cancer and aims at the minimization of the size of the tumor at the end of a certain time interval of treatment with a limited amount of drugs. The treatment is controlled by the dosis of drugs that is administered per time unit for which also a limit is prescribed. It is shown that optimal controls are of bang–bang type and can be chosen at the upper limit, if the total amount of drugs is large enough.