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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.
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.bc.ufg.br:tede/3126 |
| Date | 18 March 2013 |
| Creators | Araújo, Pedro Felippe da Silva |
| Contributors | Cruz, José Yunier Bello, Cruz, José Yunier Bello, Sandoval, Wilfredo Sosa, Melo, Jefferson Divino Gonçalves de |
| Publisher | Universidade Federal de Goiás, Programa de Pós-graduação em PROFMAT (RG), UFG, Brasil, Instituto de Matemática e Estatística - IME (RG) |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/masterThesis |
| Format | application/pdf |
| Source | reponame:Biblioteca Digital de Teses e Dissertações da UFG, instname:Universidade Federal de Goiás, instacron:UFG |
| Rights | http://creativecommons.org/licenses/by-nc-nd/4.0/, info:eu-repo/semantics/openAccess |
| Relation | 5637905143957969341, 600, 600, 600, 600, -4268777512335152015, 8398970785179857790, 2075167498588264571, [1] Anton, Howard; Chris Rorres; Algebra Linear com Aplica c~oes: Porto Alegre, Bookman, 2001. [2] Boldrini, Jos e Lu s; Algebra Linear: S~ao Paulo, Harper & Row do Brasil, 1980. [3] Hefez, Abramo; Fernandez, Cec lia S.; Introdu c~ao a Agebra Linear: Rio de Janeiro, SBM, 2012. [4] Elseit, H.A.; Sandblom, C.{L.; Linear Programming and its applications: Springer { Verlag Berlin Heidelberg, 2007. [5] Webster, Roger; Convexity: Oxford University Press, 1994. [6] B arsov, A. S.; Qu e es la programaci on Lineal: Editorial MIR, Traducci on al espa~nol: 1977. [7] Solod ovnikov, A. S.; Sistemas de Desigualdades Lineales: Editorial MIR, Traducci on al espa~nol: 1980. [8] Barbosa, Ruy Madsen; Programa c~ao Linear: S~ao Paulo, Nobel, 1973. [9] Puccini, Abelardo de Lima; Introdu c~ao a Programa c~ao Linear: Rio de Janeiro, Livros T ecnicos e Cient cos Editora S. A., 1978. [10] Beckman, F. S., The Solution of Linear Equations by the Conjugate Gradient Method, in Mathmatical Methods for Digital Computers 1, A. Ralston and H. S. Wilf (editors), John Wiley, New York, 1960. [11] Charnes, A., Optimality and Degeneracy in Linear Programming, Econometrica 20, 1952. [12] Dantzig, G. B., Activity Analysis of Production and Allocation, T. C. Koopmans, John Wiley, New York, 1951. [13] Ford, L. K. Jr., and Fulkerson, D. K., Flows in Networks, Princeton University Press, Princeton, New Jersey, 1962. [14] Hitchcock, F. L., The Distribution of a product from Several Sources to Numerous Localities, J. Math. Phys. 20, 1941. [15] Karmarkar, N. K., A New Polinomial-time Algorithm for Linear Programming, Combinatorica 4, 1984. 112 [16] Koopmans, T. C., Optimum Utilization of the Transportation System, Proceedings of the International Statistical Conference, Washington, D. C., 1947. [17] Lemke, C. E., The Dual Method of Solving the Linear Programming Problem, Naval Research Logistics Quarterly 1, 1, 1954. [18] Kantorovich, L.V. The best use of economic resources, Moscow, 1959. [19] Leontief, Wassily W., Input-Output Economics, 2nd ed., New York, Oxford University Press, 1986. [20] Christodoulos A. Floudas and Panos M. Pardalos, Encyclopedia of Optimization, second edition, Springer, 2009. [21] Manne, A. S., Notes on Parametric Linear Programming, Rand Paper p. 468, The Rand Corporation, Santa Monica, CA, 1953. |
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