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Generalized flower-girl problems
Piskačová, Nikola ; Kopa, Miloš (advisor) ; Lachout, Petr (referee)
This thesis deals with the multi-stage stochastic programming problems. In the first part, there are introduced two-stage and multi-stage stochastic programming problems. Next, two methods how to generate scenarios are described in detail - the moment method and paths-based methods. The second chapter describes the flower-girl problem, its various formulations and other extensions using ro- bustness and endogenous randomness. The practical part follows, where several different formulations of the flower-girl problem are solved. First, the problem when the florist sells roses with an unlimited life for one week is solved. Then this problem is reformulated as a two-stage problem and the results are compared. Then a formulation with a limited rose life of 2 days is presented. The greatest attention is paid to the problem with a limited rose life of 4 days - the basic formulation of the problem and several extended formulations are solved. 1
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Optimization Models of Risk in Energy Systems
Tetour, Daniel ; Cabalka, Matouš (referee) ; Popela, Pavel (advisor)
The diploma thesis deals with mathematical modeling of the resource allocation problem in an energy system with respect to technical parameters of the used resources. The model includes random input variables affecting the amount of demand and constraints related to associated risks. The thesis addresses control of the operation of various types of boilers and also extends the system with a heat storage tank examining its impact on the behavior of the system and achieved results. The optimization model is based on a multi-period two-stage scenario model of stochastic programming and works with simulated data, which combines real data, statistically determined estimates, and the use of logistic regression. The implementation utilizes GAMS software. When comparing the achieved results with the current state, it was found that the heat storage tank has a positive effect on the function of the system as it allows for extended usage of the cheaper unregulated sources by storing surplus heat, and thus helps to reduce the overall costs of the system.
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Advanced Decomposition Methods in Stochastic Convex Optimization
Kůdela, Jakub ; Fabian, Csaba (referee) ; Šmíd,, Martin (referee) ; Popela, Pavel (advisor)
Při práci s úlohami stochastického programování se často setkáváme s optimalizačními problémy, které jsou příliš rozsáhlé na to, aby byly zpracovány pomocí rutinních metod matematického programování. Nicméně, v některých případech mají tyto problémy vhodnou strukturu, umožňující použití specializovaných dekompozičních metod, které lze použít při řešení rozsáhlých optimalizačních problémů. Tato práce se zabývá dvěma třídami úloh stochastického programování, které mají speciální strukturu, a to dvoustupňovými stochastickými úlohami a úlohami s pravděpodobnostním omezením, a pokročilými dekompozičními metodami, které lze použít k řešení problému v těchto dvou třídách. V práci popisujeme novou metodu pro tvorbu “warm-start” řezů pro metodu zvanou “Generalized Benders Decomposition”, která se používá při řešení dvoustupňových stochastických problémů. Pro třídu úloh s pravděpodobnostním omezením zde uvádíme originální dekompoziční metodu, kterou jsme nazvali “Pool & Discard algoritmus”. Užitečnost popsaných dekompozičních metod je ukázána na několika příkladech a inženýrských aplikacích.
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Advanced Optimization of Network Flows
Cabalka, Matouš ; Hrabec, Dušan (referee) ; Popela, Pavel (advisor)
The master’s thesis focuses on the optimization models in logistics with emphasis on the network interdiction problem. The brief introduction is followed by two overview chapters - graph theory and mathematical programming. Important definitions strongly related to network interdiction problems are introduced in the chapter named Basic concepts of graph theory. Necessary theorems used for solving problems are following the definitions. Next chapter named Introduction to mathematical programming firstly contains concepts from linear programming. Definitions and theorems are chosen with respect to the following maximum flow problem and the derived dual problem. Concepts of stochastic optimization follow. In the fifth chapter, we discuss deterministic models of the network interdiction. Stochastic models of the network interdiction follow in the next chapter. All models are implemented in programmes written in the programming language GAMS, the codes are attached.
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Reinsurance optimization using stochastic programming and risk measures
Došel, Jan ; Branda, Martin (advisor) ; Cipra, Tomáš (referee)
Title: Reinsurance optimization using stochastic programming and risk measures Author: Jan Došel Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathe- matical Statistics Abstract: The diploma thesis deals with an application of a stochastic progra- mming in a reinsurance optimization problem in terms of a present regulatory framework of the insurance companies within the European Union, i.e. Solvency II. In this context, the reinsurance does not only transfer a portion of the risk to the reinsurer but also reduces an amout of required capital. The thesis utilizes certain risk measures and their properties, premium principles and non-linear in- teger programming. In the theoretical part, there are basic terms from Solvency II, reinsurance, risk measures and the comonotonicity of random variables descri- bed and the optimization problem itself is derived. The approach is then applied in the practical part on data of Czech Insurers' Bureau using the GAMS software. Finally, a stability of the solution is tested depending on several parameters. Keywords: reinsurance optimization, stochastic programming, Solvency II, risk measures 1
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Stochastic models in theory of the firm
Vaněk, Petr ; Kopa, Miloš (advisor) ; Hendrych, Radek (referee)
The goal of this bachelor's thesis is the stochastic extension of deterministic models belonging to the theory of the firm. The thesis deals specifically with finding optimal solutions for deterministic and stochastic problems of production maximization, cost minimization and profit maximization. At first, basic concepts of theory of the firm are introduced in this work and also there are listed deter- ministic optimization problems with their solutions. Then these deterministic models are extended by random input prices and random demand. A stochastic programming solution is proposed for each extension. The end of this bachelor's thesis deals with the practical stochastic problem of production maximization, which illustrates the dependence of the optimal solution on the input parameters of the model. 1
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Optimal Value of Loans via Stochastic Programming
Kaňková, Vlasta
A question of mortgage leads to serious and complicated problems of financial mathematics. On one side is a bank with an aim to have a “good” profit, on the other side is the client trying to invest money safely, with possible “small” risk.Let us suppose that a young married couple is in a position of client. Young people know that an expected and also unexpected unpleasant financial situation can happen. Many unpleasant financial situation can be caused by a random factor. Consequently stochastic methods are suitable to secure against them. The aim of the suggested model is not only to state a maximal reasonable value of loans, but also to endure unpleasant financial period. To this end we employ stochastic optimization theory. A few suitable models will be introduced. The choice of the model depends on environment of the young people. Models will be with “deterministic” constraints, probability constraints, but also with stochastic dominance constraints. The suggested models will be analyzed both from the numerical point of view and from possible method solution based on data. Except static one-objective problem we suggest also multi–objective models.
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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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Vertex coloring algorithms in scheduling problems under uncertainty
Hájek, Štěpán ; Branda, Martin (advisor) ; Lavička, Karel (referee)
This thesis concerns solutions to problems that arise in optimizing fixed interval scheduling under situations of uncertainty such as when there are random delays in job process times. These problems can be solved by using a vertex coloring with random edges and problems can be formulated using integer linear, quadratic and stochastic programming. In this thesis is propo- sed a new integer linear formulation. Under certain conditions there is proved its equivalence with stochastic formulation, where is maximized the schedule reliability. Moreover, we modified the proposed formulation to obtain bet- ter corresponding to real life situations. In a numerical study we compared computational time of individual formulations. It turns out that the propo- sed formulation is able to solve scheduling problems considerably faster than other formulations. 1
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Generalized Leontiev models
Hála, Petr ; Kopa, Miloš (advisor) ; Cipra, Tomáš (referee)
"his thesis de-ls with veontiev¡s input -nd output model of the e onomy -nd its potenti-l extensionsF et the eginning of the thesis -si formul-tions -nd h-r- teristi s of the veontiev¡s model -re summ-rized with emph-sis on its solv- ilityF sn the third -nd fourth h-pterD we present the simplest modi( -tions with -ddition-l restri tions or o je tive fun tionF sn the sixth h-pter - dyn-mi model with dis rete time is derivedD -g-in with emph-sis on the formul-tion of the onditions of existen e of solutionF "he l-st h-pter presents - sto h-sti gener-liz-tion of the veontiev¡s model using pro - ilisti onstr-ints -nd the s en-rio -ppro- hF "he thesis is - omp-nied y its own ex-mple of veontiev¡s sto h-sti modelF
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