From 063d02a1b8cb02012868e2cc1abde24111346e6c 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, 24 Apr 2025 22:12:32 -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> --- .../answer_2.md" | 110 ++++++++++++++++++ .../answer_2.pdf" | Bin 0 -> 133327 bytes .../graphic.png" | Bin 0 -> 188009 bytes .../question_2.png" | Bin 0 -> 70136 bytes 4 files changed, 110 insertions(+) create mode 100644 "class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/answer_2.md" create mode 100644 "class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/answer_2.pdf" create mode 100644 "class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/graphic.png" create mode 100644 "class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/question_2.png" diff --git "a/class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/answer_2.md" "b/class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/answer_2.md" new file mode 100644 index 0000000..93bd29b --- /dev/null +++ "b/class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic Method Problem Linear Programming/answer_2.md" @@ -0,0 +1,110 @@ + + +## **Problema de Programação Linear** + +**Objetivo**: Maximizar a função objetivo. + +**Função objetivo**: + +$$ +\text{Max } Z = x_1 + 5x_2 +$$ + +**Restrições**: + +$$ +\begin{cases} +x_1 + x_2 \leq 5 \\ +x_1 + 2x_2 \leq 6 \\ +2x_1 + x_2 \leq 7 \\ +x_1 \geq 0, \quad x_2 \geq 0 +\end{cases} +$$ + +--- + +## **Passo 1: Converter as Desigualdades em Igualdades para Traçar as Retas** + +1. \$ x_1 + x_2 = 5 \$ + - Se \$ x_1 = 0 \$, então \$ x_2 = 5 \$ + - Se \$ x_2 = 0 \$, então \$ x_1 = 5 \$ +2. \$ x_1 + 2x_2 = 6 \$ + - Se \$ x_1 = 0 \$, então \$ 2x_2 = 6 \Rightarrow x_2 = 3 \$ + - Se \$ x_2 = 0 \$, então \$ x_1 = 6 \$ +3. \$ 2x_1 + x_2 = 7 \$ + - Se \$ x_1 = 0 \$, então \$ x_2 = 7 \$ + - Se \$ x_2 = 0 \$, então \$ 2x_1 = 7 \Rightarrow x_1 = 3,5 \$ + +--- + +## **Passo 2: Desenhar as Retas no Plano Cartesiano** + +1. Trace as retas no plano \$ (x_1, x_2) \$. +2. Identifique a região viável, que é a área delimitada pelas retas e pelos eixos \$ x_1 \geq 0 \$ e \$ x_2 \geq 0 \$. + +--- + +## **Passo 3: Determinar os Vértices da Região Viável** + +Os vértices são os pontos de interseção das retas. Vamos calcular: + +1. Interseção entre \$ x_1 + x_2 = 5 \$ e \$ x_1 + 2x_2 = 6 \$: + - Subtraindo a primeira equação da segunda: +\$ (x_1 + 2x_2) - (x_1 + x_2) = 6 - 5 \$ +\$ x_2 = 1 \$ + - Substituindo \$ x_2 = 1 \$ na primeira equação: +\$ x_1 + 1 = 5 \Rightarrow x_1 = 4 \$ + - Ponto: \$ (4, 1) \$ +2. Interseção entre \$ x_1 + x_2 = 5 \$ e \$ 2x_1 + x_2 = 7 \$: + - Subtraindo a primeira equação da segunda: +\$ (2x_1 + x_2) - (x_1 + x_2) = 7 - 5 \$ +\$ x_1 = 2 \$ + - Substituindo \$ x_1 = 2 \$ na primeira equação: +\$ 2 + x_2 = 5 \Rightarrow x_2 = 3 \$ + - Ponto: \$ (2, 3) \$ +3. Interseção entre \$ x_1 + 2x_2 = 6 \$ e \$ 2x_1 + x_2 = 7 \$: + - Multiplicando a primeira equação por 2: +\$ 2x_1 + 4x_2 = 12 \$ + - Subtraindo a segunda equação da resultante: +\$ (2x_1 + 4x_2) - (2x_1 + x_2) = 12 - 7 \$ +\$ 3x_2 = 5 \Rightarrow x_2 = \frac{5}{3} \$ + - Substituindo \$ x_2 = \frac{5}{3} \$ na primeira equação: +\$ x_1 + 2\left(\frac{5}{3}\right) = 6 \$ +\$ x_1 = 6 - \frac{10}{3} = \frac{18 - 10}{3} = \frac{8}{3} \$ + - Ponto: \$ \left(\frac{8}{3}, \frac{5}{3}\right) \approx (2.67, 1.67) \$ +4. Interseção com os eixos: + - \$ (0, 0) \$ + - \$ (3.5, 0) \$ + - \$ (0, 3) \$ + +--- + +## **Passo 4: Avaliar a Função Objetivo nos Vértices** + +1. \$ (0, 0) \Rightarrow Z = 0 + 5(0) = 0 \$ +2. \$ (3.5, 0) \Rightarrow Z = 3.5 + 5(0) = 3.5 \$ +3. \$ (0, 3) \Rightarrow Z = 0 + 5(3) = 15 \$ +4. \$ (4, 1) \Rightarrow Z = 4 + 5(1) = 9 \$ +5. \$ (2, 3) \Rightarrow Z = 2 + 5(3) = 17 \$ +6. \$ \left(\frac{8}{3}, \frac{5}{3}\right) \Rightarrow Z = \frac{8}{3} + 5\left(\frac{5}{3}\right) = \frac{8}{3} + \frac{25}{3} = \frac{33}{3} = 11 \$ + +--- + +## **Passo 5: Identificar a Solução Ótima** + +O maior valor de \$ Z \$ é 17, que ocorre no ponto \$ (2, 3) \$. + +--- + +## **Resposta Final** + +$$ +\boxed{ +x_1 = 2, \quad x_2 = 3, \quad Z_{\text{máximo}} = 17 +} +$$ + +--- + +Portanto, a solução ótima é \$ x_1 = 2 \$, \$ x_2 = 3 \$, e o valor máximo da função objetivo é 17. + diff --git "a/class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_2-Resolution Graphic 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