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Optimalizační modely pro podporu strategického rozhodování
Ulverová, Michaela ; Škapa, Stanislav (oponent) ; Popela, Pavel (vedoucí práce)
Diplomová práce se zabývá možnostmi matematického modelování rozpočtů veřejných vysokých škol. V práci jsou nejprve diskutovány vnější podmínky financování veřejných vysokých škol, je zde uvedena základní legislativa a tyto podmínky jsou ilustrovány na konkrétních datech. V další části práce jsou analyzovány a pomocí obecného schématu popsány finanční toky uvnitř vysoké školy. S využitím analýzy existujících dat, předpisů a vzorců byl krok za krokem sestaven matematický model rozpočtu vysoké školy. Sestavený model představuje úlohu nelineárního vícestupňového stochastického programování založenou na scénářích, zahrnující dále lineární a síťová omezení a zohledňující možnosti více účelových funkcí a s nimi související parametrické analýzy. Model byl implementován v systému GAMS s rozhraním v prostředí MS Excel. Cílem sestavení matematického modelu nebylo nabídnout nástroj, který bude používán automaticky pro rozdělování finančních prostředků na VŠ, ale poskytnout jeho uživatelům širší možnosti výpočtových experimentů a testů a získat tak lepší vhled do problému.
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Heuristic algorithms in optimization
Šandera, Čeněk ; Popela, Pavel (oponent) ; Roupec, Jan (vedoucí práce)
The thesis deals with stochastic programming and determining probability distributions which cause extreme optimal values (maximal or minimal) of an objective function. The probability distribution is determined by heuristic method, especially by genetic algorithms, where whole population approximates desired distribution. The first parts of the thesis describe mathematical and stochastic programming in general and also there are described various heuristic methods with emphasis on genetic algorithms. The goal of the diploma thesis is to create a program which tests the algorithm on linear and quadratic stochastic models.
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Stochastic programming models with applications
Novotný, Jan ; Michálek, Jaroslav (oponent) ; Popela, Pavel (vedoucí práce)
The thesis deals with stochastic programming and its application to aggregate blending, an optimization problem within the area of civil engineering. The theoretical part is devoted to the derivation of basic principles of stochastic programming (optimization under uncertainty). The applied part presents a development of suitable mathematical models for aggregate blending, their implementation and results. The thesis contains original results achieved in solution of the project GA CR reg. n. 103/08/1658 Advanced optimum design of composed concrete structures and it contains theoretical results of the project from MSMT of the Czech Republic no. 1M06047 Centre for Quality and Reliability of Production.
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Convexity in stochastic programming model with indicators of ecological stability
Houda, Michal
We develop an optimization model dealing with construction expenses that are prescribed as a result of the EIA (Environmental Impact Assessment) process. The process is an obligatory part of every large construction project and evaluates possible influences of the project to the environment, including population health, natural and other socio-economic aspects; the result of the process is a set of recommendation and arrangements the construction must meet. Our optimization model incorporates uncertainties in model parameters; we represent them through their probabilistic distribution. Furthermore, to overcome a problem with quantifying subjective utility function of ecological impacts, we measure them by so-called indicators of ecological stability. The resulting problem is stochastic programming problem formulated as (C)VaR model used traditionally in finance area. In our contribution we deal with convexity properties of this problem – these are especially important from the theoretical as well as from the computational point of view.
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Using indicators of ecological stability in stochastic programming
Houda, Michal
When building bigger construction the EU law impose the so-called EIA process - evaluation of possible influences of the construction on the environment and population health, grouped into several categories. Outputs of the EIA process are recommendations to the investors compensating the negative impacts of the constructions by additional arrangements. In our contribution we develop an innovative approach to model the expenses devoted to obey the EIA rules by stochastic programming tools: especially, we represent uncertainty in parameters by their probabilistic distributions, and subjective utility function representing the ecological demands is modelled via so-called indicators of ecological stability. The model takes into account budget limitations, several legislative obligations, and other ecological aspects; the goal is to help choose the optimal compensating constructions and arrangements. The resulting stochastic programming model is seen as parallel to V@R problem.
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Dependent Data in Economic and Financial Problems
Kaňková, Vlasta
Optimization problems depending on a probability measure correspond to many economic and financial applications. The paper deals with the case when an empirical measure substitutes the theoretical one. Especially the paper deals with a convergence rate of the corresponding estimates. ``Classical" results for independent samples are recalled, situations in which the case of dependent sample can be (from the mathematical point of view) reduced to independent case are mentioned. A great attention is paid to weak dependent samples fulfilling the Phi-mixing condition.
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Empirické odhady a stabilita ve stochastickém programování
Kaňková, Vlasta
It is known that optimization problems depending on a probability measure correspond to many applications. It is also known that these problems belong mostly to a class of nonlinear optimization problems and, moreover, that very often an ``underlying" probability measure is not completely known. The aim of the research report is to deal with the case when an empirical measure substitutes the theoretical one. In particular, the aim is to generalize reults dealing with convergence rate in the case of empirical esrimates. The introduced results are based on the stability results corresponding to the Wasserstein metric. A relationship berween tails of one-dimensional marginal distribution functions and exponentional rate of convergence are introduced. The corresponding results are focus mainly on ``classical" type of problems corresponding to the cases with penalty and recourse. However, an integer simple recourse case and some special risk funkcionals are discussed also.
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Hranice výrobních možností a stochastické programování
Chovanec, Petr
Již ze své podstaty se metoda obálky dat (DEA) nemůže vypořádat s nepřesnostmi v datech jako jsou chyby měření. Abychom zlepšili tuto metodu, uvažujeme $alpha$-stochastickou efficienci a ukážeme příbuznost s problémem stochastického programování. Dva typy pravděpodobnostních nerovností jsou použity pro vytvoření nových kritérií pro tento typ efficience.
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Notes on approximation of stochastic programming problem
Šmíd, Martin
In stochastic optimization problems, expectation of random function is often being minimized. Since the expectation can rarely be evaluated exactly an approximation has to be done. In the present paper, three types of approximation are dealt with: discretization, Monte Carlo and Quasi Monte Carlo. Convergence rate of the approximation error is evaluated and some upper bounds of the error are given.
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