From cc10e3d4f8d489a9448e6fcbbf42c43fc3109a04 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fabiana=20=F0=9F=9A=80=20=20Campanari?= <113218619+FabianaCampanari@users.noreply.github.com> Date: Thu, 17 Apr 2025 21:23:45 -0300 Subject: [PATCH] Add files via upload MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Fabiana 🚀 Campanari <113218619+FabianaCampanari@users.noreply.github.com> --- ...g Problem Solving by the Simplex Method.md | 139 ++++++++++++++++++ .../answer_3.pages | Bin 0 -> 735801 bytes .../question_3.png | Bin 0 -> 70833 bytes 3 files changed, 139 insertions(+) create mode 100644 class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3-Linear Programming Problem Solving by the Simplex Method.md create mode 100644 class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3.pages create mode 100644 class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/question_3.png diff --git a/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3-Linear Programming Problem Solving by the Simplex Method.md b/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3-Linear Programming Problem Solving by the Simplex Method.md new file mode 100644 index 0000000..eaf31fd --- /dev/null +++ b/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3-Linear Programming Problem Solving by the Simplex Method.md @@ -0,0 +1,139 @@ + +# Resolução de Problema de Programação Linear pelo Método Simplex + +Este relatório apresenta a resolução completa do problema de programação linear usando o método Simplex, simulando uma resolução feita à mão, como seria realizada em um caderno. + +## Problema de Programação Linear + +O problema a ser resolvido é: + +Max. Z = 5x₁ + 4x₂ + +Sujeito a: + +- x₁ + 2x₂ ≤ 6 +- -2x₁ + x₂ ≤ 4 +- 5x₁ + 3x₂ ≤ 15 +- x₁, x₂ ≥ 0 + + +## Preparação do Problema para o Método Simplex + +### Introdução de Variáveis de Folga + +O primeiro passo é converter as restrições de desigualdade em equações, introduzindo variáveis de folga: + +- x₁ + 2x₂ + s₁ = 6 +- -2x₁ + x₂ + s₂ = 4 +- 5x₁ + 3x₂ + s₃ = 15 + +A função objetivo na forma padrão: +Z - 5x₁ - 4x₂ = 0 + +### Construção do Tableau Inicial + +O tableau inicial é formado pelos coeficientes das variáveis nas restrições, as variáveis de folga, e a função objetivo (com coeficientes negativos): + + +| Base | x₁ | x₂ | s₁ | s₂ | s₃ | b | +| :-- | :-- | :-- | :-- | :-- | :-- | :-- | +| s₁ | 1 | 2 | 1 | 0 | 0 | 6 | +| s₂ | -2 | 1 | 0 | 1 | 0 | 4 | +| s₃ | 5 | 3 | 0 | 0 | 1 | 15 | +| Z | -5 | -4 | 0 | 0 | 0 | 0 | + +## Resolução Passo a Passo pelo Método Simplex + +### Primeira Iteração + +**Passo 1: Identificar a variável que entra na base** + +Analisamos a linha Z e escolhemos a variável com coeficiente mais negativo. Neste caso, x₁ tem coeficiente -5, portanto x₁ entrará na base. + +**Passo 2: Identificar a variável que sai da base** + +Calculamos as razões para determinar qual variável sairá da base: + +- Linha 1 (s₁): 6/1 = 6 +- Linha 2 (s₂): não calculamos porque o coeficiente -2 não é positivo +- Linha 3 (s₃): 15/5 = 3 + +A menor razão positiva é 3, correspondente à linha 3, então s₃ sairá da base. + +**Passo 3: Operações de pivoteamento** + +O elemento pivô é 5 (linha 3, coluna x₁). + +1. Dividimos a linha do pivô (linha 3) pelo valor do pivô (5): +-[^2][^1][^12] ÷ 5 = [1, 0.6, 0, 0, 0.2, 3] +2. Eliminamos os outros elementos da coluna pivô: + - Linha 1:[^1][^1][^3] - 1×[1, 0.6, 0, 0, 0.2, 3] = [0, 1.4, 1, 0, -0.2, 3] + - Linha 2: [-2, 1, 0, 1, 0, 4] - (-2)×[1, 0.6, 0, 0, 0.2, 3] = [0, 2.2, 0, 1, 0.4, 10] + - Linha Z: [-5, -4, 0, 0, 0, 0] - (-5)×[1, 0.6, 0, 0, 0.2, 3] = [0, -1, 0, 0, 1, 15] + +Obtemos o seguinte tableau: + + +| Base | x₁ | x₂ | s₁ | s₂ | s₃ | b | +| :-- | :-- | :-- | :-- | :-- | :-- | :-- | +| s₁ | 0 | 1.4 | 1 | 0 | -0.2 | 3 | +| s₂ | 0 | 2.2 | 0 | 1 | 0.4 | 10 | +| x₁ | 1 | 0.6 | 0 | 0 | 0.2 | 3 | +| Z | 0 | -1 | 0 | 0 | 1 | 15 | + +### Segunda Iteração + +**Passo 1: Identificar a variável que entra na base** + +Na linha Z, o coeficiente de x₂ é -1, então x₂ entrará na base. + +**Passo 2: Identificar a variável que sai da base** + +Calculamos as razões: + +- Linha 1 (s₁): 3/1.4 = 2.143 +- Linha 2 (s₂): 10/2.2 = 4.545 +- Linha 3 (x₁): 3/0.6 = 5 + +A menor razão positiva é 2.143, correspondente à linha 1, então s₁ sairá da base. + +**Passo 3: Operações de pivoteamento** + +O elemento pivô é 1.4 (linha 1, coluna x₂). + +1. Dividimos a linha do pivô (linha 1) pelo valor do pivô (1.4): + - [0, 1.4, 1, 0, -0.2, 3] ÷ 1.4 = [0, 1, 0.714, 0, -0.143, 2.143] +2. Eliminamos os outros elementos da coluna pivô: + - Linha 2: [0, 2.2, 0, 1, 0.4, 10] - 2.2×[0, 1, 0.714, 0, -0.143, 2.143] = [0, 0, -1.571, 1, 0.714, 5.286] + - Linha 3: [1, 0.6, 0, 0, 0.2, 3] - 0.6×[0, 1, 0.714, 0, -0.143, 2.143] = [1, 0, -0.429, 0, 0.286, 1.714] + - Linha Z: [0, -1, 0, 0, 1, 15] - (-1)×[0, 1, 0.714, 0, -0.143, 2.143] = [0, 0, 0.714, 0, 0.857, 17.143] + +Obtemos o seguinte tableau: + + +| Base | x₁ | x₂ | s₁ | s₂ | s₃ | b | +| :-- | :-- | :-- | :-- | :-- | :-- | :-- | +| x₂ | 0 | 1 | 0.714 | 0 | -0.143 | 2.143 | +| s₂ | 0 | 0 | -1.571 | 1 | 0.714 | 5.286 | +| x₁ | 1 | 0 | -0.429 | 0 | 0.286 | 1.714 | +| Z | 0 | 0 | 0.714 | 0 | 0.857 | 17.143 | + +### Verificação de Otimalidade + +Como todos os coeficientes na linha Z são não-negativos, atingimos a solução ótima. + +## Solução Ótima + +Expressando a solução em frações para maior precisão: + +- x₁ = 12/7 ≈ 1.714 +- x₂ = 15/7 ≈ 2.143 +- Valor máximo de Z = 120/7 ≈ 17.143 + + +## Conclusão + +Aplicando o método Simplex, encontramos que a solução ótima para o problema de programação linear é produzir aproximadamente 1.714 unidades do produto 1 e 2.143 unidades do produto 2, resultando em um valor máximo da função objetivo Z = 17.143. + +O método Simplex mostrou-se eficiente, convergindo para a solução ótima em apenas duas iterações. As variáveis de folga s₁ e s₃ são zero na solução ótima, indicando que as restrições correspondentes são ativas (estão no limite), enquanto s₂ = 5.286, indicando que a segunda restrição não é ativa. + diff --git a/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3.pages b/class_8- 1st HandMade EXAM -LP -MathModels - Simplex/QaA_3-Linear Programming Problem Solving by the Simplex Method/answer_3.pages new file mode 100644 index 0000000000000000000000000000000000000000..6e5ddbfe740564915e937d35ca43413b94a320cb GIT binary patch literal 735801 zcmbTdcUTk4A1}NiflwkHOe8=cQ4kT6ppZyFiiBXHHx)f}!~>WFq=^_n5HW-#6k7mkfYF06r2M?zoPCIy*igI+o0t8Rr-H`v(wu1l0*Zw!E z|ID2EvK;7u%rLahe*DKX@3#Oj&;)%2eFcN80W|{<*Z}nVAz-<3Pj%3L@!!i<4j?r! zL|p@_sim#6Qs6IrKn(;2t3kl(>i@Y7=-5g=05MQEw8D947zIW{t!c*id``V4+Ou6~ z67+Zu;}}CP(9$+t4M(h5yUxaTz1`+5PR<14R9`6A}+5 z9ZE|-l98E}ol|(CsQ6^bsZwr5WmWZ=v**rVywq^{N@G)VOUKPye|2_s_uPKa_wbQO 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