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Recreational mathematics problem solving as an integer programing problems
Künzelová, Barbora ; Jablonský, Josef (advisor) ; Rada, Miroslav (referee)
The bachelor thesis deals with the solution of three selected recreational mathematics problems using discrete optimization. These problems are called Gridspeed Puzzle, Shifty Witnesses and Alien Tiles. The reader first learns the rules of these games of which are then formulated mathematical models that are integer in all cases. The first two games have one model and are being solved one way, but in game Alien Tiles are solved several optimization problems. All models are rewritten MPL modeling language in which they are solved by solver CPLEX. For all selected problems are found optimal integer solution, which is then described and explained to the reader. The solution results of the task Alien Tiles are finally compared and we are able to tell that one form of the desired outcome of the game is more difficult to calculate.
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Pallet loading problem and using one of its heuristics for box placement on pallets in a warehouse
Rybka, Ondřej ; Pelikán, Jan (advisor) ; Rada, Miroslav (referee)
This work concerns new borders, heuristics, algoritms and mathematic models of pallet loading problem (PLP). We try to describe these computational methods and find out if we can use them in real. We maximalize number of boxes placed on rectangular pallets in a particular warehouse by using chosen heuristics. Every box has a rectangular form with the same lenght and width and is fully placed on the pallet. We can rotate with the box by 90% degree until it is fixed as we want and its side lies parallelly with side of the pallet. All instances are setted in model (X, Y, a, b), where X is lenght, Y width of the pallet, a lenght and b width of the box.
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Alternative Method of Solution for LP Problem
Hanzlík, Tomáš ; Kalčevová, Jana (advisor) ; Rada, Miroslav (referee)
Linear programming (LP) stands for an optimization of a linear objective function, subject to linear and non-negativity constraints. For this purpose many methods for LP emerged. The best known is Simplex Method. Another group of methods for LP is represented by Interior Point Methods (IPM). These methods are based on interior points of feasible region of a problem, while Simplex Method uses basic feasible solution of a problem. This thesis focuses on theoretical background of IPM and brings it into relation with algorithms based on IPM. KKT system and its significance are included and the algorithm solving Linear Complementarity Problem is discussed as well. In this thesis, two algorithms based on IPM are introduced and used for solving a sample LP problem.
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The use game theory and real options valuation in investment decisions
Kuchina, Elena ; Dlouhý, Martin (advisor) ; Rada, Miroslav (referee)
This work demonstrates a new approach how to evaluate investment projects under uncertainty. This approach consists of the unification of real option valuation and game theory principles. The application of the theoretical foundation of this "option game" is shown on the illustrative example, that represents two-stage game. Analysis of this example compares the investment project's value in case when the R&D investment is not carried out with the case when this investment is made. In latter case there are two different possibilities whether the investment outcomes are proprietary or shared. The present work shows that in the evaluation of investment projects besides standard NPV there is flexibility value and the value of strategic effect; whereas these two components usually go against each other. Within the framework of the given illustrative example a breakdown of the investment project's value is demonstrated in different cases.
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Application of Game Theory to Oligopolistic Market
Nekola, Jan ; Dlouhý, Martin (advisor) ; Rada, Miroslav (referee)
This work demonstrates the use of models of game theory to oligopolistic market. It is based on the models of Bertrand, Cournot, Stackelberger, which solves, how optimally the companies decide to maximize their benefit. There is illustrative example here, which is an extension of these models. Decision-making takes place in the several stages, where strategy is suggested according to the certain rules of game theory. Second example deals with the signaling game. My goal is to introduce the theory to the illustrative example and show, how games are sensitive to the changing of information or how specific factors affect the game.
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Traveling Tournament Problem
Šimpach, Ondřej ; Jablonský, Josef (advisor) ; Rada, Miroslav (referee)
Traveling Tournament Problem is the optimization problem of sport calendar which requires finding the minimum sum of possible travel distances between the teams tournament matches. The tournament is most common type of "double round robin" where distance is crucial for the teams. Concurrently with the issue of finding the optimal solution to a suitable arrangement of sports teams in the table together with respecting the highest possible number of criteria appealing by stakeholders to organizers such as managers of teams and television and radio companies. Thesis provides insight into the complexity of solving large problems, recommendation for their solutions and possible starting-points. In the end there is an alternative option for calculations tournament schedules in the Czech environment event the distances are not so important in this case.
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Optimization over the internet
Šálek, Pavel ; Jablonský, Josef (advisor) ; Rada, Miroslav (referee)
The goal of this thesis is to analyse and evaluate the efficiency of systems used for solving problems of mathematical programming, which are available within the frame of project NEOS (Network Enabled Optimization Server). The most frequent algorithms for solving linear programming and mixed integer programming are described in the beginning of the work -- simplex algorithms, the interior point method, cutting plane method and branch and bound method. These algorithms are used in solvers, which are supported by NEOS server. The efficiency of the algorithms and solvers are tested on collection of chosen problems contained in libraries of NETLIB and MIPLIB.
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Model of the extensive form game with the financial rewards
Erbsová, Markéta ; Dlouhý, Martin (advisor) ; Rada, Miroslav (referee)
The aim of this Bachelor thesis is the analysis and presentation of the original extensive form game with the financial rewards, which is called Výměna (Exchange). In the theoretical part we will describe the basic concepts of Game Theory, especially the game in an explicit form. In the practical part we will deal with the game Výměna (Exchange) in itself. Determine the rules of the game, the graphic form of the game and formulate a mathematical model of the game. Individual parts of the game will be described in more details. No less important part of this Bachelor thesis will be the characteristics of the game, which outline the possible use of the game in practice.
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Výpočetní aspekty metody maximálního polyedru pro rozhodování při neurčitosti
Rada, Miroslav ; Ivánek, Jiří (advisor) ; Dlouhý, Martin (referee)
Metoda maximálního polyedru je metoda výběru nejlepší varianty pro rozhodování při neurčitosti. Její hlavní myšlenkou je vybrat tu variantu, která bude nejčastěji mít nejvyšší střední hodnotu, pokud zvolíme náhodně vektor pravděpodobností, s nimiž nastanou jednotlivé stavy světa. Stručně je tedy v práci popsána rozhodovací situace za neurčitosti, s důrazem na zmíněnou metodu. Cílem práce je popsat a implementovat algoritmus, jakým metoda maximálního polyedru vybírá nejlepší variantu, a posoudit jeho výpočetní aspekty. Každé rozhodovací variantě přísluší mnohostěn n-tic (vektorů) pravděpodobností, ve kterých je daná varianta nejlepší. Součástí práce je proto popis několika známých algoritmů pro exaktní výpočet objemu mnohostěnu. Protože některé algoritmy vyžadují na vstupu mnohostěn zadaný více způsoby, jsou zahrnuty též některé algoritmy pro přepočet reprezentace mnohostěnu. Jako nejvhodnější metoda výpočtu objemu pro mnohostěny vznikající v metodě maximálního polyedru je zvolena metoda HOT, doplněná pro případy jednoduchých mnohostěnů Lawrenceovou metodou. Jako nejvhodnější metoda přepočtu reprezentace je zvolena metoda LRS. V práci je dále sestaven rozhodovací program, který využívá dostupných implementací zvolených metod. Program je testován na náhodně vygenerovaných rozhodovacích úlohách. Na základě testování jsou odhadnuty meze jeho praktické použitelnosti ? velikost úlohy bez potíží výpočetně zvládnutelné (doba výpočtu cca 30 min) danou implementací je cca 15 variant x 14 stavů světa, přičemž výpočetní náročnost se zvyšuje rychleji s počtem stavů světa než s počtem variant. Konkrétní hodnoty mezní velikosti úlohy samozřejmě závisejí na požadované době odezvy a použitém hardware. V práci je dále provedeno srovnání výsledků rozhodování podle různých kritérií a jsou navrženy některé směry zdokonalení implementace metody maximálního polyedru.
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