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The dual simplex method is frequently the most efficient technique for solving linear programming (LP) problems. This thesis describes an efficient implementation of the sequential dual simplex method and the design and development of two parallel dual simplex solvers. In serial, many advanced techniques for the (dual) simplex method are implemented, including sparse LU factorization, hyper-sparse linear system solution technique, efficient approaches to updating LU factors and sophisticated dual simplex pivoting rules. These techniques, some of which are novel, lead to serial performance which is comparable with the best public domain dual simplex solver, providing a solid foundation for the simplex parallelization. During the implementation of the sequential dual simplex solver, the study of classic LU factor update techniques leads to the development of three novel update variants. One of them is comparable to the most efficient established approach but is much simpler in terms of implementation, and the other two are specially useful for one of the parallel simplex solvers. In addition, the study of the dual simplex pivoting rules identifies and motivates further investigation of how hyper-sparsity maybe promoted. In parallel, two high performance simplex solvers are designed and developed. One approach, based on a less-known dual pivoting rule called suboptimization, exploits parallelism across multiple iterations (PAMI). The other, based on the regular dual pivoting rule, exploits purely single iteration parallelism (SIP). The performance of PAMI is comparable to a world-leading commercial simplex solver. SIP is frequently complementary to PAMI in achieving speedup when PAMI results in slowdown.

This thesis examines linear programming problems, the theoretical foundations of the simplex method, and how a liner programming problem can be solved with the simplex method.

This thesis details a method of enumerating bases of hyperplane arrangements up to symmetries. I consider here automorphisms, geometric symmetries which leave the set of all points contained in the arrangement setwise invariant. The algorithm for basis enumeration described in this thesis is a backtracking search over the adjacency graph implied on the bases by minimum-ratio simplex pivots, pruning at bases symmetric to those already seen. This work extends Bremner, Sikiri c, and Sch urmann's method for basis enumeration of polyhedra up to symmetries, including a new pivoting rule for nding adjacent bases in arrangements, a method of computing automorphisms of arrangements which extends the method of Bremner et al. for computing automorphisms of polyhedra, and some associated changes to optimizations used in the previous work. I include results of tests on ACEnet clusters showing an order of magnitude speedup from the use of C++ in my implementation, an up to 3x speedup with a 6-core parallel variant of the algorithm, and positive results from other optimizations.

The problem of this study was to analyze the effect of using electronic calculators in the teaching of the simplex method upon students' attitudes and achievement in mathematics.

Basic idea of fractal simplex algorithm is based in the theory of matrix counting and knowledge of matrix representation of simplex tabuleao from revised simplex method. My desire is to explain theoretical basics on which this algorithm works and provide solution in language Visual Basic for Applications in application MS Excel 2007. Main benefit I see in the fact, that algorithm can solved specific class of mathematical problems in a way of exactness counting whithout necessity of using decimal numbers.

The development of microwave Gallium Arsenide Metal Semiconductor Field Effect Transistor (MESFET) devices has enabled the miniaturization of pagers, cellular phones, and other electronic devices. With these MESFET devices comes the need to model them. This thesis extracts a small signal equivalent circuit model from a Gallium Arsenide MESFET device. The approach taken in this thesis is to use measured S- parameters to extract a small signal equivalent circuit model by optimization. Small signal models and S-parameters are explained. The Simplex Method is used to optimize the small signal equivalent circuit model. A thorough analysis of the strengths and weaknesses of the Simplex method is performed. / Master of Science

The purpose of this thesis is to efficiently solve real life problems. We study LPs. We study an NLP and an MINLP based on what is known as the generalised pooling problem (GPP), and we study an MIP that we call the cattle mating problem. These problems are often very large or otherwise difficult to solve by direct methods, and are best solved by decomposition methods. During the thesis we introduce algorithms that exploit the structure of the problems to decompose them. We are able to solve row-linked, column-linked and general LPs efficiently by modifying the tableau simplex method, and suggest how this work could be applied to the revised simplex method. We modify an existing sequential linear programming solver that is currently used by Format International to solve GPPs, and show the modified solver takes less time and is at least as likely to find the global minimum as the old solver. We solve multifactory versions of the GPP by augmented Lagrangian decomposition, and show this is more efficient than solving the problems directly. We introduce a decomposition algorithm to solve a MINLP version of the GPP by decomposing it into NLP and ILP subproblems. This is able to solve large problems that could not be solved directly. We introduce an efficient decomposition algorithm to solve the MIP cattle mating problem, which has been adopted for use by the Irish Cattle Breeding Federation. Most of the solve methods we introduce are designed only to find local minima. However, for the multifactory version of the GPP we introduce two methods that give a good chance of finding the global minimum, both of which succeed in finding the global minimum on test problems.

Warehouse operations have become an important part of the retail industry today. For many companies in the retail industry, the profit margin can be as little as one cent for every dollar sold. Because of these extremely small profit margins, it is important that companies save as much in costs along the way as they can. One such way is to cut down on warehouse costs. In this paper, we shall study the costs of storing and retrieving products in a pick, pack and ship warehouse for an office supply company. In particular, we will examine possible ways to improve the flow of products into and out of reserve racking, by implementing different storage assignment policies. One way of finding the best possible storage assignments would be to formulate a cost equation along with constraints, and to minimize the cost using linear programming techniques. We shall study the Simplex Method, and we will use it to find optimal locations for a given set of storage tasks. However, the set of products to be stored and retrieved in our warehouse changes from day to day, and is subject to customer demand. Because there is no closed form solution or algorithm to find the optimal storage assignment policy under a random situation, we will use simulation techniques to study different storage assignment policies that could be applied in the warehouse. We will study the efficiency along with the maintenance requirements for each of the different policies and compare them with the current policy being used in the warehouse today.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-09-23T11:12:32Z No. of bitstreams: 2 Araújo, Pedro Felippe da Silva.pdf: 1780566 bytes, checksum: d286e3b501489bf05fab04e9ab67bb26 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-09-23T11:34:23Z (GMT) No. of bitstreams: 2 Araújo, Pedro Felippe da Silva.pdf: 1780566 bytes, checksum: d286e3b501489bf05fab04e9ab67bb26 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-09-23T11:34:23Z (GMT). No. of bitstreams: 2 Araújo, Pedro Felippe da Silva.pdf: 1780566 bytes, checksum: d286e3b501489bf05fab04e9ab67bb26 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-03-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Problems involving the idea of optimization are found in various elds of study, such as, in Economy is in search of cost minimization and pro t maximization in a rm or country, from the available budget; in Nutrition is seeking to redress the essential nutrients daily with the lowest possible cost, considering the nancial capacity of the individual; in Chemistry studies the pressure and temperature minimum necessary to accomplish a speci c chemical reaction in the shortest possible time; in Engineering seeks the lowest cost for the construction of an aluminium alloy mixing various raw materials and restrictions obeying minimum and maximum of the respective elements in the alloy. All examples cited, plus a multitude of other situations, seek their Remedy by Linear Programming. They are problems of minimizing or maximizing a linear function subject to linear inequalities or Equalities, in order to nd the best solution to this problem. For this show in this paper methods of problem solving Linear Programming. There is an emphasis on geometric solutions and Simplex Method, to form algebraic solution. Wanted to show various situations which may t some of these problems, some general cases more speci c cases. Before arriving eventually in solving linear programming problems, builds up the eld work of this type of optimization, Convex Sets. There are presentations of de nitions and theorems essential to the understanding and development of these problems, besides discussions on the e ciency of the methods applied. During the work, it is shown that there are cases which do not apply the solutions presented, but mostly t e ciently, even as a good approximation. / Problemas que envolvem a ideia de otimiza c~ao est~ao presentes em v arios campos de estudo como, por exemplo, na Economia se busca a minimiza c~ao de custos e a maximiza c~ao do lucro em uma rma ou pa s, a partir do or camento dispon vel; na Nutri c~ao se procura suprir os nutrientes essenciais di arios com o menor custo poss vel, considerando a capacidade nanceira do indiv duo; na Qu mica se estuda a press~ao e a temperatura m nimas necess arias para realizar uma rea c~ao qu mica espec ca no menor tempo poss vel; na Engenharia se busca o menor custo para a confec c~ao de uma liga de alum nio misturando v arias mat erias-primas e obedencendo as restri c~oes m nimas e m aximas dos respectivos elementos presentes na liga. Todos os exemplos citados, al em de uma in nidade de outras situa c~oes, buscam sua solu c~ao atrav es da Programa c~ao Linear. S~ao problemas de minimizar ou maximizar uma fun c~ao linear sujeito a Desigualdades ou Igualdades Lineares, com o intuito de encontrar a melhor solu c~ao deste problema. Para isso, mostram-se neste trabalho os m etodos de solu c~ao de problemas de Programa c~ao Linear. H a ^enfase nas solu c~oes geom etricas e no M etodo Simplex, a forma alg ebrica de solu c~ao. Procuram-se mostrar v arias situa c~oes as quais podem se encaixar alguns desses problemas, dos casos gerais a alguns casos mais espec cos. Antes de chegar, eventualmente, em como solucionar problemas de Programa c~ao Linear, constr oi-se o campo de trabalho deste tipo de otimiza c~ao, os Conjuntos Convexos. H a apresenta c~oes das de ni c~oes e teoremas essenciais para a compreens~ao e o desenvolvimento destes problemas; al em de discuss~oes sobre a e ci^encia dos m etodos aplicados. Durante o trabalho, mostra-se que h a casos os quais n~ao se aplicam as solu c~oes apresentadas, por em, em sua maioria, se enquadram de maneira e ciente, mesmo como uma boa aproxima c~ao.