Stochastické programování a optimalizace jsou velmi užitečné nástroje pro řešení široké škály inženýrských úloh zahrnujících neurčitost. Diplomová práce se zabývá stochastickým programováním a jeho aplikací při řešení logistických úloh. Teoretická část práce je věnována jak základním pojmům z teorie grafů, tak pojmům souvisejících s matematickým, lineárním, celočíselným a stochastickým programováním. Pozornost je věnována také návaznosti zmíněných pojmů na logistiku. Druhá část se zabývá tvorbou vlastních úloh prezentujících stochastické logistické modely, jejich implementací a výsledky.

Cognitive radio networks (CRN) have been tipped as one of the most promising paradigms for next generation wireless communication, due primarily to its huge promise of mitigating the spectrum scarcity challenge. To help achieve this promise, CRN develop mechanisms that permit spectrum spaces to be allocated to, and used by more than one user, either simultaneously or opportunistically, under certain preconditions. However, because of various limitations associated with CRN, spectrum and other resources available for use in CRN are usually very scarce. Developing appropriate models that can efficiently utilise the scarce resources in a manner that is fair, among its numerous and diverse users, is required in order to achieve the utmost for CRN. 'Resource allocation (RA) in CRN' describes how such models can be developed and analysed. In developing appropriate RA models for CRN, factors that can limit the realisation of optimal solutions have to be identified and addressed; otherwise, the promised improvement in spectrum/resource utilisation would be seriously undermined. In this thesis, by a careful examination of relevant literature, the most critical limitations to RA optimisation in CRN are identified and studied, and appropriate solution models that address such limitations are investigated and proffered. One such problem, identified as a potential limitation to achieving optimality in its RA solutions, is the problem of heterogeneity in CRN. Although it is indeed the more realistic consideration, introducing heterogeneity into RA in CRN exacerbates the complex nature of RA problems. In the study, three broad classifications of heterogeneity, applicable to CRN, are identified; heterogeneous networks, channels and users. RA models that incorporate these heterogeneous considerations are then developed and analysed. By studying their structures, the complex RA problems are smartly reformulated as integer linear programming problems and solved using classical optimisation. This smart move makes it possible to achieve optimality in the RA solutions for heterogeneous CRN. Another serious limitation to achieving optimality in RA for CRN is the strictness in the level of permissible interference to the primary users (PUs) due to the activities of the secondary users (SUs). To mitigate this problem, the concept of cooperative diversity is investigated and employed. In the cooperative model, the SUs, by assisting each other in relaying their data, reduce their level of interference to PUs significantly, thus achieving greater results in the RA solutions. Furthermore, an iterative-based heuristic is developed that solves the RA optimisation problem timeously and efficiently, thereby minimising network complexity. Although results obtained from the heuristic are only suboptimal, the gains in terms of reduction in computations and time make the idea worthwhile, especially when considering large networks. The final problem identified and addressed is the limiting effect of long waiting time (delay) on the RA and overall productivity of CRN. To address this problem, queueing theory is investigated and employed. The queueing model developed and analysed helps to improve both the blocking probability as well as the system throughput, thus achieving significant improvement in the RA solutions for CRN. Since RA is an essential pivot on which the CRN's productivity revolves, this thesis, by providing viable solutions to the most debilitating problems in RA for CRN, stands out as an indispensable contribution to helping CRN realise its much-proclaimed promises. / Thesis (PhD)--University of Pretoria, 2017. / Electrical, Electronic and Computer Engineering / PhD / Unrestricted

Cette thèse a pour but de modéliser et d'optimiser deux composantes du système de gestion de flux de trafic aérien : la prédictibilité et la flexibilité. Cette modélisation est équivalente à établir une relation entre la fenêtre temporelle et les taux d'arrivée des avions. Deux approches sont utilisées : l'analyse des données historiques et la modélisation mathématique. L'analyse des données historique a permis de déterminer la fenêtre temporelle raisonnable mais sans pouvoir apporter les améliorations nécessaires pour y arriver. La modélisation mathématique permet non seulement de définir de façon rigoureuse la prédictibilité et la flexibilité mais également de traiter des vols en différents scénarios de priorités. La combinaison de DC algorithme avec des méthodes de résolutions classiques comme Branch and Bound a nettement amélioré la vitesse de la convergence des solutions et donc elle peut être utilisée pour la phase tactique de gestion de flux du trafic aérien. / This thesis aims to model and optimise two components of the air traffic flow management system : predictibility and flexibility. This modelling is equivalent to establishing a relationship between the time window and the rate of arrival flights. Two approachs are used : the analysis of historical data and mathematical modeling. The analysis of historical data was used to establish the relationship between the time window and arrivla rate of flights. It provided the optimal time window but could not show how to modify the system to lead to that time window. Mathematical modeling can not only define the predictability and flexibility in the rigourous manner but also deal with different scenarios of fligths priorities. The combination of DC algorithm with classical methods like Branch and Bound has significantly improved the speed of convergence of solutions and therefore it can be used for the tactical phase of the air traffic flow management.

This thesis is about the problem of searching an interval that enclose all op- timal values of the objective function in interval linear programming, so called the optimal value range. The solution to this problem is sometimes reduced to solving just a few linear programs but in general it is a hard problem. Af- ter we get familiar with interval arithmetics and when we extend it to linear programming, we define important sets and their properties, B-stability and other connected subproblems. We will extend B-stability to generalized interval linear programming and we will examine methods for computing the optimal value range and we will compare them numerically on random systems. The goal is to implement all mentioned methods in MATLAB/INTLAB and based on numerical results provide one function that will solve this problem, possible efficiently. 1

Steelmaking industry, one of the most electricity-intensive industrial processes, is seeking for new approaches to improve its competitiveness in terms of energy savings by taking advantage of the volatile electricity prices. This fluctuation in the price is mainly caused by the increasing share of renewable energy sources, the liberalization of energy markets and the growing demand of the energy. Therefore, making the production scheduling of steelmaking processes with knowledge about the cost of the energy may lead to significant cost savings in the electricity bills. With this aim in mind, different models are developed in this project in order to improve the existing monolithic models (continuous-time based scheduling) to find an efficient formulation of accounting for electricity consumption and also to expand them with more detailed scheduling of Electric Arc Furnace stage in the production process. The optimization of the energy cost with multiple electricity sources and contracts and the production planning are usually done as stand-alone optimizers due to their complexity, therefore as a new approach in addition to the monolithic model an iterative framework is developed in this work. The idea to integrate the two models in an iterative manner has potential to be useful in the industry due to low effort for reformulation of existing models. The implemented framework uses multiparametric programming together with bilevel programming in order to direct the schedule to find a compromise between the production constraints and goals, and the energy cost. To ensure applicability heuristic approaches are also examined whenever full sized models are not meeting computational performance requirements. The results show that the monolithic model implemented has a considerable advantage in terms of computational time compared to the models in the literature and in some cases, the solution can be obtained in a few minutes instead of hours. In the contrary, the iterative framework shows a bad performance in terms of computational time when dealing with real world instances. For that matter a heuristic approach, which is easy to implement, is investigated based on coordination theory and the results show that it has a potential since it provides solutions close to the optimal solutions in a reasonable amount of time. Multiparametric programming is the main core of the iterative framework developed in this internship and it is not able to give the solutions for real world instances due to computational time limitations. This computational problem is related to the nature of the algorithm behind mixed integer multiparametric programming and its ability to handle the binary variables. Therefore, further work to this project is to develop new approaches to approximate multiparametric technique or develop some heuristics to approximate the mp-MILP solutions.

Discrete event systems (DES) are a special class of dynamical systems with discrete-valued state space and event-driven transitions. DES are ubiquitous in today's world and are used in different sectors such as manufacturing systems, transport networks and computer networks. They offer unique capabilities, such as flexibility and adaptability; at the same time, they can be challenging to model and analyze. Moreover, the complexity of DES is scaled up when disturbances are present. Many different kinds of real life DES can be modeled using dioid algebra which is a powerful tool for describing nonlinear behaviors using linear system models. Dioid algebra is an exotic algebra of formal series which can be understood as a set of only positive numbers without negatives. This special algebraic structure is useful in modeling DES because such systems feature variables that cannot be inverted with respect to some variables. Nonlinear behaviors of DES are able to be modeled as linear systems in terms of dioid algebra in order to use classical control techniques in scheduling and control of DES.This dissertation presents the scheduling and control of DES using a special dioid called max-plus algebra, which is a set of real numbers with the operation of maximum and addition replacing the usual classical operations of addition and multiplication, respectively. This dissertation also studies the behavior of DES when disturbances are present. Two different paths to the scheduling of DES are presented: using dioid algebra and using linear programming methods. The control of DES with disturbances and uncertainties is also explored, particularly, the solutions of the disturbance decoupling problem and the modified disturbance decoupling problem using various controller structures are presented. Disturbance decoupling in this dissertation means the scheduling of the DES will not not be affected by the presence of the disturbances. On the other hand, modified disturbance decoupling means the scheduling will not be worse than the delays caused by the disturbances in industrial just-in-time (JIT) standards. JIT means that the operations start with just enough time to be completed by the desired schedule in order to minimize waste and costs in work in progress and material storage.The applicability of the approach presented in this dissertation is demonstrated in real-world processes including a large-scale high throughput screening (HTS) system in drug discovery and an optimal scheduler for an airport's runways. The main contributions of this dissertation are max-plus and mathematical programming solutions for scheduling and control of discrete event systems with disturbances. The results present a theoretical scheduling prior to exhaustive scheduling algorithms in large-scaled industrial systems.

Dans cette thèse, nous étudions des Hoist Scheduling Problems (HSP) qui se posent fréquemment dans des lignes automatiques de traitement de surface. Dans ces lignes, des ponts roulants sont utilisés pour transporter les pièces entre les bains. Ainsi, les ponts roulants jouent un rôle essentiel dans la performance de ces lignes ; et un ordonnancement optimal de leurs mouvements est un facteur déterminant pour garantir la qualité des produits et maximiser la productivité. Les lignes que nous étudions comportent un seul pont roulant mais peuvent être des lignes de base ou des lignes étendues (où des bains sont à fonctions et/ou capacités multiples). Nous examinons trois Hoist Scheduling Problems : l’optimisation robuste d’un HSP cyclique, l’ordonnancement dynamique d’une ligne étendue de type job shop et l’ordonnancement cyclique d’une telle ligne.Pour l’optimisation robuste d’un HSP cyclique, nous définissons la robustesse comme la marge dans le temps de déplacement du pont roulant. Nous formulons le problème en programmation linéaire en nombres mixtes à deux objectifs pour optimiser simultanément le temps de cycle et la robustesse. Nous démontrons que le temps de cycle minimal augmente avec la robustesse, et que par conséquent la frontière Pareto est constituée d’une infinité de solutions. Les valeurs minimales et maximales des deux objectifs sont établies. Les résultats expérimentaux à partir de benchmarks et d’instances générées aléatoirement montrent l’efficacité de l’approche proposée.Nous étudions ensuite un problème d’ordonnancement dynamique dans une ligne étendue de type job shop. Nous mettons en évidence une erreur de formulation dans une un modèle existant pour un problème similaire mais sans bains multi-fonctions. Cette erreur peut rendre l’ordonnancement obtenu sous-optimal voire irréalisable. Nous construisons un nouveau modèle qui corrige cette erreur. De plus il est plus compact et s’applique au cas avec des bains à la fois à capacités et à fonctions multiples. Les résultats expérimentaux menés sur des instances avec ou sans bains multi-fonctions montrent que le modèle proposé conduit toujours à une solution optimale et plus efficace que le modèle existant.Nous nous focalisons enfin sur l’ordonnancement cyclique d’une ligne étendue de type job shop avec des bains à fonctions et capacités multiples. Nous construisons un modèle mathématique en formulant les contraintes de capacité du pont roulant, les intervalles des durées opératoires, et les contraintes de capacité des bains. Nous établissons également des contraintes valides. Les expériences réalisées sur des instances générées aléatoirement montrent l’efficacité du modèle proposé. / This thesis studies hoist scheduling problems (HSPs) arising in automated electroplating lines. In such lines, hoists are often used for material handing between tanks. These hoists play a crucial role in the performance of the lines and an optimal schedule of the hoist operations is a key factor in guaranteeing product quality and maximizing productivity. We focus on extended lines (i.e. with multi-function and/or multi-capacity tanks) with a single hoist. This research investigates three hoist scheduling problems: robust optimization for cyclic HSP, dynamic jobshop HSP in extended lines and cyclic jobshop HSP in extended lines.We first study the robust optimization for a cyclic HSP. The robustness of a cyclic hoist schedule is defined in terms of the free slacks in hoist traveling times. A bi-objective mixed-integer linear programming (MILP) model is developed to optimize the cycle time and the robustness simultaneously. It is proved that the optimal cycle time strictly increases with the robustness, thus there is an infinite number of Pareto optimal solutions. We established lower and upper bounds of these two objectives. Computational results on several benchmark instances and randomly generated instances indicate that the proposed approach can effectively solve the problem.We then examine a dynamic jobshop HSP with multifunction and multi-capacity tanks. We demonstrate that an existing model for a similar problem can lead to suboptimality. To deal with this issue, a new MILP model is developed to generate an optimal reschedule. It can handle the case where a multi-function tank is also multi-capacity. Computational results on instances with and without multifunction tanks indicate that the proposed model always yields optimal solutions, and is more compact and effective than the existing one.Finally, we investigate a cyclic jobshop HSP with multifunction and multi-capacity tanks. An MILP model is developed for the problem. The key issue is to formulate the time-window constraints and the tank capacity constraints. We adapt the formulation of time-window constraints for a simpler cyclic HSP to the jobshop case. The tank capacity constraints are handled by dealing with the relationships between hoist moves so that there is always an empty processing slot for new parts. Computational experiments on numerical examples and randomly generated instances indicate that the proposed model can effectively solve the problem.

Das Ziel dieses Projektes ist, die bestehende tägliche Produktionsablaufplanung in der Handgalvanik bei dem Lohngalvanikbetrieb C. Jentner GmbH mit Hilfe eines mathematischen Modells zu optimie-ren. Hierfür wurde das Flexible-Job-Shop-Modell von Ziaee ([1], S. 91-95) ausgewählt und auf die Gegebenheiten vor Ort angepasst. Es gehört zu den MILP-Problemen (mixed integer linear programming- gemischt ganzzahlige Programmierung). Bei der Verwendung des Modells für die Praxis stellt die Modellgröße, die benötigt wird, um die Vorgänge in der Produktion ausreichend abbilden zu können, ein Problem dar. Diese führt zu langen Lösungszeiten, die für den täglichen Einsatz in der Produktionsablaufplanung ungeeignet sind. Zur Lösung dieses Problems wurde ein problemspezifisches Verfahren basierend auf Aggregations- und Dekompositionstechniken entwickelt. Durch Anwendung dieses Verfahrens kann die Problemgröße für den Solver klein und so die Lösungszeit in einem für die tägliche Produktionsablaufplanung annehmbaren Rahmen gehalten werden.

Renewable power production is becoming a necessity to improve society and overcome the challenges of climate change. In Sweden, large-scale solar PV power is growing year-on-year and today comprises 1 percent of electricity production. Solar power, however, is an intermittent form of electricity production which, whilst being renewable, contributes to increasing grid instability. For the grid to stay in balance, at grid frequency 50 Hz, electricity must be consumed at the time of production. If there is a surplus of production or a sudden decrease in consumption, the frequency will deviate from the nominal value. When introducing larger quantities of intermittent power production, the power system inertia decreases, and the frequency becomes prone to deviate. To combat this, the Swedish TSO Svenska Kraftnät procures ancillary services which aid the grid when needed. For solar PV power to be able to contribute to these marketplaces, a battery storage solution system (BESS) is utilized. This thesis aimed to investigate the economic feasibility of co-locating a solar photovoltaic (PV) park with a battery energy storage system (BESS) and to determine the optimal size of the BESS. The study utilized a linear optimization model to simulate the operation of a 14 MW solar PV park with different sizes of BESS ranging from 1 MWh to 14 MWh. The analysis considered the revenue generated by providing different services to the electricity grid, such as energy arbitrage and frequency regulation. The results indicate that co-locating a solar PV park with a BESS increases revenue, and the optimal BESS size for a 14 MW solar PV park is between 1 and 8 MWh. Above this range, the revenue recedes due to the limitations of the grid connection, which restricts the BESS from participating in the ancillary service markets. The analysis considers the running costs associated with power discharge to the grid and initial BESS investment. The study did not account for ancillary market bids that are not accepted, which could have a significant impact on the revenue generated. The ongoing trend of lowering battery prices could further boost the economic assessment and increase interest in all battery sizes, resulting in larger battery system installations in general.

In telecommunications, there is currently a lot of work being done to migrate to the cloud, and a lot of specialized hardware is being exchanged for virtualized solutions. One important part of telecommunication networks that is yet to be moved to the cloud is known as the base-band unit, which sits between the antennas and the core network. The base-band unit has very strict latency requirements, making it unsuitable for out-of-the-box cloud solutions. Ericsson is therefore investigating if cloud solutions can be customized in such a way that base-band unit functionality can be virtualized as well. One such customization is to describe the functionality of a base-band unit using a directed acyclic graph (DAG), and deploy it to a cloud environment using Kubernetes. This thesis sets out to take applications represented using a DAG and deploy it using Kubernetes in such a way that the network latency is reduced when compared to the deployment generated by the default Kubernetes scheduler. The problem of placing the applications onto the available hardware resources was formulated as an integer linear programming problem. The problem was then implemented using Pyomo and solved with the open-source solver GLPK to obtain an optimized placement. This placement was then used to generate a configuration file that could be used to deploy the applications using Kubernetes. A mock application was developed in order to evaluate the optimized placement. The evaluation carried out in this thesis shows that the optimized placement obtained from the solution could improve the average round-trip latency of applications represented using a DAG by up to 30% when compared to the default Kubernetes scheduler.