## 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING

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- ZIB-Report (2)
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- Master's Thesis (1)

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- Mixed-Integer Programming (4) (remove)

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A Polyhedral Study of Event-Based Models for the Resource-Constrained Project Scheduling Problem
(2018)

We consider event-based Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP) that represent an alternative to the common time-indexed model (DDT) of Pritsker et al. (1969) for the case where the underlying time horizon is large or job processing times are subject to huge variations. In contrast to the time-indexed model, the size of event-based models does not depend on the time horizon. For two event-based formulations OOE and SEE of Koné et al. (2011) we present new valid inequalities that dominate the original formulation. Additionally, we introduce a new event-based model: the Interval Event-Based Model (IEE). We deduce linear transformations between all three models that yield the strict domination order IEE > SEE > OOE for their linear programming (LP) relaxations, meaning that IEE has the strongest linear relaxation among the event-based models. We further show that the popular DDT formulation can be retrieved from IEE by certain polyhedral operations, thus giving a unifying view on a complete branch of MIP formulations for the RCPSP. In addition, we analyze the computational performance of all presented models on test instances of the PSPLIB (Kolisch and Sprecher 1997).

The topic of this thesis is the examination of an optimization model
which stems from the clustering process of non-reversible markov processes.
We introduce the cycle clustering problem und formulate it as a mixed
integer program (MIP).
We prove that this problem is N P-hard and discuss polytopal aspects
such as facets and dimension. The focus of this thesis is the development of
solving methods for this clustering problem. We develop problem specific
primal heuristics, as well as separation methods and an approximation
algorithm. These techniques are implemented in practice as an application
for the MIP solver SCIP.
Our computational experiments show that these solving methods result
in an average speedup of ×4 compared to generic solvers and that our
application is able to solve more instances to optimality within the given
time limit of one hour.

In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.