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The discovery of transcription factor binding site (TFBS) motifs remains an important and challenging problem in computational biology. This thesis presents MITSU, a novel algorithm for TFBS motif discovery which exploits stochastic methods as a means of both overcoming optimality limitations in current algorithms and as a framework for incorporating relevant prior knowledge in order to improve results. The current state of the TFBS motif discovery field is surveyed, with a focus on probabilistic algorithms that typically take the promoter regions of coregulated genes as input. A case is made for an approach based on the stochastic Expectation-Maximisation (sEM) algorithm; its position amongst existing probabilistic algorithms for motif discovery is shown. The algorithm developed in this thesis is unique amongst existing motif discovery algorithms in that it combines the sEM algorithm with a derived data set which leads to an improved approximation to the likelihood function. This likelihood function is unconstrained with regard to the distribution of motif occurrences within the input dataset. MITSU also incorporates a novel heuristic to automatically determine TFBS motif width. This heuristic, known as MCOIN, is shown to outperform current methods for determining motif width. MITSU is implemented in Java and an executable is available for download. MITSU is evaluated quantitatively using realistic synthetic data and several collections of previously characterised prokaryotic TFBS motifs. The evaluation demonstrates that MITSU improves on a deterministic EM-based motif discovery algorithm and an alternative sEM-based algorithm, in terms of previously established metrics. The ability of the sEM algorithm to escape stable fixed points of the EM algorithm, which trap deterministic motif discovery algorithms and the ability of MITSU to discover multiple motif occurrences within a single input sequence are also demonstrated. MITSU is validated using previously characterised Alphaproteobacterial motifs, before being applied to motif discovery in uncharacterised Alphaproteobacterial data. A number of novel results from this analysis are presented and motivate two extensions of MITSU: a strategy for the discovery of multiple different motifs within a single dataset and a higher order Markov background model. The effects of incorporating these extensions within MITSU are evaluated quantitatively using previously characterised prokaryotic TFBS motifs and demonstrated using Alphaproteobacterial motifs. Finally, an information-theoretic measure of motif palindromicity is presented and its advantages over existing approaches for discovering palindromic motifs discussed.

In this paper we investigate a manufacturer's sustainable sourcing strategy that includes recycled materials. To produce a short life-cycle electronic good, strategic raw materials can be bought from virgin material suppliers in advance of the season and via emergency shipments, as well as from a recycler. Hence, we take into account virgin and recycled materials from different sources simultaneously. Recycling makes it possible to integrate raw materials out of steadily increasing waste streams back into production processes. Considering stochastic prices for recycled materials, stochastic supply quantities from the recycler and stochastic demand as well as their potential dependencies, we develop a single-period inventory model to derive the order quantities for virgin and recycled raw materials to determine the related costs and to evaluate the effectiveness of the sourcing strategy. We provide managerial insights into the benefits of such a green sourcing approach with recycling and compare this strategy to standard sourcing without recycling. We conduct a full factorial design and a detailed numerical sensitivity analysis on the key input parameters to evaluate the cost savings potential. Furthermore, we consider the effects of correlations between the stochastic parameters. Green sourcing is especially beneficial in terms of cost savings for high demand variability, high prices of virgin raw material and low expected recycling prices as well as for increasing standard deviation of the recycling price. Besides these advantages it also contributes to environmental sustainability as, compared to sourcing without recycling, it reduces the total quantity ordered and, hence, emissions are reduced.

Piecewise-smooth stochastic systems are widely used in engineering science. However, the theory of these systems is only in its infancy. In this thesis, we take as an example the Brownian motion with dry friction to illustrate dynamical properties of these systems with respect to three interesting topics: (i) weak-noise approximations, (ii) first-passage time (FPT) problems and (iii) functionals of stochastic processes. Firstly, we investigate the validity and accuracy of weak-noise approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example the Brownian motion with pure dry friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided that the singularity of the path integral is treated with some heuristics. We also consider a smooth regularisation of this piecewise-constant SDE and study to what extent this regularisation can rectify some of the problems encountered in the non-smooth case. Secondly, we provide analytic solutions to the FPT problem of the Brownian motion with dry friction. For the pure dry friction case, we find a phase transition phenomenon in the spectrum which relates to the position of the exit point and affects the tail of the FPT distribution. For the model with dry and viscous friction, we evaluate quantitatively the impact of the corresponding stick-slip transition and of the transition to ballistic exit. We also derive analytically the distributions of the maximum velocity till the FPT for the dry friction model. Thirdly, we generalise the so-called backward Fokker-Planck technique and obtain a recursive ordinary differential equation for the moments of functionals in the Laplace space. We then apply the developed results to analyse the local time, the occupation time and the displacement of the dry friction model. Finally, we conclude this thesis and state some related unsolved problems.