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Selected Advanced Stochastic Programming Models
Brzobohatý, Jan ; Hrabec, Dušan (referee) ; Popela, Pavel (advisor)
This diploma thesis deals with stochastic dominance. The goal is to lay the foundations for defining stochastic dominance, to describe its properties and to explain this concept on simple examples. Another goal is to apply this concept to network problems with random price. Examples in this thesis also contain solutions and python code how to find them.
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Advanced Stochastic Programming Models in Power Engineering
Pavelka, Ondřej ; Štětina, Josef (referee) ; Popela, Pavel (advisor)
This diploma thesis applies stochastic optimization in the field of the power engineering. In the thesis first part, the needed mathematical theory is described, specifically mathematical, linear, nonlinear, integer and stochastic programming. The second part deals with the heating plant, in which heat is generated by gas and biomass boilers. The aim of this thesis is to design a model for the schedule planning of these boilers. The model is based on two stage stochastic programming with scenario approach. Then the model is solved by GAMS software. In the final part of the text, the focus is on the model sensitivity analysis and suggestions for future improvement.
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New Trends in Stochastic Programming
Szabados, Viktor ; Kaňková, Vlasta (advisor) ; Lachout, Petr (referee)
Stochastic methods are present in our daily lives, especially when we need to make a decision based on uncertain events. In this thesis, we present basic approaches used in stochastic tasks. In the first chapter, we define the stochastic problem and introduce basic methods and tasks which are present in the literature. In the second chapter, we present various problems which are non-linearly dependent on the probability measure. Moreover, we introduce deterministic and non-deterministic multicriteria tasks. In the third chapter, we give an insight on the concept of stochastic dominance and we describe the methods that are used in tasks with multidimensional stochastic dominance. In the fourth chapter, we capitalize on the knowledge from chapters two and three and we try to solve the role of portfolio optimization on real data using different approaches. 1
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Empiciral Estimates in Stochastic Programming; Dependent Data
Kolafa, Ondřej ; Kaňková, Vlasta (advisor) ; Dupačová, Jitka (referee)
This thesis concentrates on stochastic programming problems based on empirical and theoretical distributions and their relationship. Firstly, it focuses on the case where the empirical distribution is an independent random sample. The basic properties are shown followed by the convergence between the problem based on the empirical distribution and the same problem applied to the theoretical distribution. The thesis continues with an overview of some types of dependence - m-dependence, mixing, and also more general weak dependence. For sequences with some of these types of dependence, properties are shown to be similar to those holding for independent sequences. In the last section, the theory is demonstrated using numerical examples, and dependent and independent sequences, including sequences with different types of dependence, are compared.
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Nonconvex stochastic programming problems-formulations, sample approximations and stability
Branda, Martin ; Lachout, Petr (advisor) ; Kaňková, Vlasta (referee) ; H.van der Vlerk, Maarten (referee)
Title: Nonconvex stochastic programming problems - formulations, sample approximations and stability Author: RNDr. Martin Branda Author's e-mail address: branda@karlin.mff.cuni.cz Supervisor: Doc. RNDr. Petr Lachout, CSc. Supervisor's e-mail address: lachout@karlin.mff.cuni.cz Abstract: We deal with problems where integer variables may appear, hence no assumptions on convexity are made throughout this thesis. The goal of Chapter 2 is to introduce stochastic programming problems and to outline the most important tasks connected with solving the problems. In Chapter 3, we compare basic formulations of static stochastic programming problems with chance constraints, with integrated chance constraints and with penalties in the objective function. We show that the problems are asymptotically equivalent under mild conditions. We discuss solving the problems using sample approximation techniques and extend some results on rates of convergence. All the formulations and corresponding sample approximations are compared on an investment problem with real features with Value at Risk constraint, integer allocations and transaction costs. Then, stability of financial decision models where two-stage mixed-integer value function appears as a loss variable is studied. In Chapter 4, we study qualitative properties of the...
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Optimization Risk Modelling in GAMS
Kutílek, Vladislav ; Bednář, Josef (referee) ; Popela, Pavel (advisor)
The diploma thesis deals with the possibilities of using the optimization modelling software system GAMS in risk management. According to the assignment, emphasis is placed on a detailed approach to the program for those, who are interested in its use in the field of risk engineering applications. The first part of the thesis contains the knowledge to understand what the GAMS program is and what it is used for. The next part of the work provides instructions on how to download, install, activate the program and what the user interface of the program looks like. Thanks to mathematical programming, it will be explained on a project on the distribution of lung ventilators, what basic approaches may be used in risk modelling in the GAMS program on a deterministic model. The following are more complex wait-and-see models, which contains the probability parameters and here-and-now models, where we work with demand scenarios and verify whether if they meets the requirements of other scenarios or calculate costs for the highest demands. The two-stage model is also one of the here-and-now models, but it is significantly more complex in its size and range of input data, it includes additional price parameters for added or removed pieces of lung ventilators from the order.
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Modelling of Selected Risks in Healthcare
Nováková, Pavlína ; Bednář, Josef (referee) ; Popela, Pavel (advisor)
The diploma thesis deals with the modeling of selected risks in healthcare. Motivated by the current pandemic situation, it focuses on analysis of risks associated with the vaccination center in Brno. The theoretical part is mainly devoted to the issue of risk management with a focus on risks in healthcare, where the methods that are used in the practical part are defined. Furthermore, the thesis presents selected topics of mathematical programming. Especially, the newsvendor problem is introduced as inspiring case for further modelling. The brief description of the covid-19 pandemic situation later serves as one of the data sources. The practical part deals with the description and risk analysis of the vaccination process using the methods "What If?" and the FMEA method. Appropriate decisions are then proposed for selected risk situations using the GAMS optimization system. Based on the results of the calculations, specific recommendations are proposed.
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Optimization Risk Modelling in Strategic Applications
Kovalčík, Marek ; Štětina, Josef (referee) ; Popela, Pavel (advisor)
The aim of this diploma thesis is to design and efficiently implement a framework to support optimization modelling. The emphasis is placed on two-stage stochastic optimization problems and performing calculations on large data. The computing core uses the GAMS system and with using its application interface and Python programming language, the user will be able to efficiently acquire and process input and output data. The separation of the data logic and the application logic then offers a wide range of options for testing and experimenting with a general model on dynamically changing input data. The thesis is also focused on an evaluation of the framework complexity. The framework performance was evaluated by measuring the time required to complete the required task for various use cases, on the increasing sample size of input data.
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Chess preparation optimization
Walica, Roman ; Roupec, Jan (referee) ; Popela, Pavel (advisor)
This work is focused on the preparation for the chess game, the issue of Elo rating system and optimization of the problem associated with the preparation for the chess tournament. You can find here how to obtain and modify appropriate information about potential opponents behind the chessboard. Result of this work is an optimization model which, if you enter relevant data, calculates how much time to spend on preparations for chess variants.
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Optimization in Finance
Sowunmi, Ololade ; Hrabec, Dušan (referee) ; Popela, Pavel (advisor)
This thesis presents two Models of portfolio optimization, namely the Markowitz Mean Variance Optimization Model and the Rockefeller and Uryasev CVaR Optimization Model. It then presents an application of these models to a portfolio of clean energy assets for optimal allocation of financial resources in terms of maximum returns and low risk. This is done by writing GAMS programs for these optimization problems. An in-depth analysis of the results is conducted, and we see that the difference between both models is not very significant even though these results are data-specific.
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