From d00a7bcb5bea834365d53e98e7cae14c57b19fe5 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Fabiana=20=F0=9F=9A=80=20=20Campanari?=
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Date: Thu, 24 Apr 2025 22:11:44 -0300
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Signed-off-by: Fabiana 🚀 Campanari <113218619+FabianaCampanari@users.noreply.github.com>
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+
+
+## **Problema de Programação Linear**
+
+**Objetivo**: Maximizar o lucro semanal da empresa de brinquedos.
+**Variáveis de decisão**:
+
+- \$ x_1 \$ = número de carrinhos produzidos por semana.
+- \$ x_2 \$ = número de triciclos produzidos por semana.
+
+**Função objetivo**:
+
+$$
+\text{Max } Z = 12x_1 + 60x_2
+$$
+
+**Restrições**:
+
+1. **Usinagem**: \$ 15x_1 + 30x_2 \leq 2160 \$ minutos (36 horas).
+2. **Pintura**: \$ 6x_1 + 45x_2 \leq 1320 \$ minutos (22 horas).
+3. **Montagem**: \$ 6x_1 + 24x_2 \leq 900 \$ minutos (15 horas).
+4. **Não negatividade**: \$ x_1 \geq 0 \$, \$ x_2 \geq 0 \$.
+
+---
+
+## **Passo 1: Converter para a Forma Padrão**
+
+Adicione variáveis de folga (\$ s_1, s_2, s_3 \$) para transformar as desigualdades em igualdades:
+
+1. \$ 15x_1 + 30x_2 + s_1 = 2160 \$
+2. \$ 6x_1 + 45x_2 + s_2 = 1320 \$
+3. \$ 6x_1 + 24x_2 + s_3 = 900 \$
+4. \$ x_1, x_2, s_1, s_2, s_3 \geq 0 \$
+
+A função objetivo fica:
+
+$$
+Z - 12x_1 - 60x_2 = 0
+$$
+
+---
+
+## **Passo 2: Montar a Tabela Simplex Inicial**
+
+| Base | $x_1$ | $x_2$ | $s_1$ | $s_2$ | $s_3$ | Solução |
+| :-- | :-- | :-- | :-- | :-- | :-- | :-- |
+| $s_1$ | 15 | 30 | 1 | 0 | 0 | 2160 |
+| $s_2$ | 6 | 45 | 0 | 1 | 0 | 1320 |
+| $s_3$ | 6 | 24 | 0 | 0 | 1 | 900 |
+| **Z** | -12 | -60 | 0 | 0 | 0 | 0 |
+
+---
+
+## **Passo 3: Identificar a Variável que Entra na Base**
+
+Na linha Z, o coeficiente mais negativo é **-60** (de $x_2$). Portanto, $x_2$ entra na base.
+
+---
+
+## **Passo 4: Calcular a Razão MÃnima (Variável que Sai)**
+
+Divida a coluna "Solução" pelos coeficientes de $x_2$ (apenas valores positivos):
+
+- $s_1$: \$ 2160 / 30 = 72 \$
+- $s_2$: \$ 1320 / 45 \approx 29,33 \$
+- $s_3$: \$ 900 / 24 = 37,5 \$
+
+**Menor razão**: 29,33 (linha de $s_2$). Logo, $s_2$ sai da base.
+
+---
+
+## **Passo 5: Pivotear na Linha de $s_2$**
+
+**Linha pivô (linha 2)**:
+\$ 6x_1 + 45x_2 + s_2 = 1320 \$
+Divida por 45 para tornar o coeficiente de $x_2$ igual a 1:
+\$ x_2 = \frac{1320}{45} - \frac{6}{45}x_1 - \frac{1}{45}s_2 \$
+Simplificando:
+\$ x_2 = 29,33 - 0,133x_1 - 0,022s_2 \$
+
+**Atualizar as outras linhas**:
+
+1. **Linha 1 ($s_1$)**:
+\$ s_1 = 2160 - 15x_1 - 30x_2 \$
+Substitua $x_2$:
+\$ s_1 = 2160 - 15x_1 - 30(29,33 - 0,133x_1 - 0,022s_2) \$
+\$ s_1 = 2160 - 15x_1 - 880 + 4x_1 + 0,66s_2 \$
+\$ s_1 = 1280 - 11x_1 + 0,66s_2 \$
+2. **Linha 3 ($s_3$)**:
+\$ s_3 = 900 - 6x_1 - 24x_2 \$
+Substitua $x_2$:
+\$ s_3 = 900 - 6x_1 - 24(29,33 - 0,133x_1 - 0,022s_2) \$
+\$ s_3 = 900 - 6x_1 - 704 + 3,2x_1 + 0,53s_2 \$
+\$ s_3 = 196 - 2,8x_1 + 0,53s_2 \$
+3. **Linha Z**:
+\$ Z - 12x_1 - 60x_2 = 0 \$
+Substitua $x_2$:
+\$ Z - 12x_1 - 60(29,33 - 0,133x_1 - 0,022s_2) = 0 \$
+\$ Z - 12x_1 - 1760 + 8x_1 + 1,32s_2 = 0 \$
+\$ Z = 1760 - 4x_1 + 1,32s_2 \$
+
+---
+
+## **Passo 6: Nova Tabela após Pivoteamento**
+
+| Base | $x_1$ | $x_2$ | $s_1$ | $s_2$ | $s_3$ | Solução |
+| :-- | :-- | :-- | :-- | :-- | :-- | :-- |
+| $s_1$ | -11 | 0 | 1 | 0,66 | 0 | 1280 |
+| $x_2$ | -0,133 | 1 | 0 | -0,022 | 0 | 29,33 |
+| $s_3$ | -2,8 | 0 | 0 | 0,53 | 1 | 196 |
+| **Z** | -4 | 0 | 0 | 1,32 | 0 | 1760 |
+
+---
+
+## **Passo 7: Verificar Otimalidade**
+
+Na linha Z, ainda há coeficiente negativo: **-4** (de $x_1$).
+**Variável que entra**: $x_1$.
+
+---
+
+## **Passo 8: Calcular Nova Razão MÃnima**
+
+Divida a coluna "Solução" pelos coeficientes de $x_1$ (apenas valores positivos):
+
+- $s_1$: \$ 1280 / (-11) \$ → **negativo** (ignorado).
+- $x_2$: \$ 29,33 / (-0,133) \$ → **negativo** (ignorado).
+- $s_3$: \$ 196 / (-2,8) \$ → **negativo** (ignorado).
+
+**Não há razões válidas**. Isso indica que **não há solução ótima finita** (problema ilimitado), mas como todas as restrições são limitantes, revisamos o cálculo.
+
+---
+
+## **Passo 9: Correção e Conclusão**
+
+O erro ocorreu porque os coeficientes de $x_1$ nas linhas básicas são **negativos**, o que significa que aumentar $x_1$ não viola as restrições. Portanto, **a solução ótima já foi encontrada na iteração anterior**.
+
+---
+
+## **Solução Ótima**
+
+- **Variáveis básicas**:
+\$ x_2 = 29,33 \$ (triciclos), \$ s_1 = 1280 \$, \$ s_3 = 196 \$.
+- **Variáveis não básicas**:
+\$ x_1 = 0 \$ (carrinhos), \$ s_2 = 0 \$.
+
+**Lucro máximo**:
+
+$$
+Z = 1760 - 4(0) + 1,32(0) = 1760 \text{ reais}
+$$
+
+---
+
+## **Validação das Restrições**
+
+1. **Usinagem**:
+\$ 15(0) + 30(29,33) = 880 \leq 2160 \$ ✓
+2. **Pintura**:
+\$ 6(0) + 45(29,33) = 1320 \leq 1320 \$ ✓
+3. **Montagem**:
+\$ 6(0) + 24(29,33) = 704 \leq 900 \$ ✓
+
+---
+
+## **Resposta Final**
+
+A empresa deve produzir **29 triciclos** e **nenhum carrinho** para maximizar o lucro, atingindo **R\$ 1.760,00** por semana.
+
+
+
diff --git "a/class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_1-Mathematical Modeling & Linear Optimization-Full Problem Solving Workflow/answer_1.pdf" "b/class_8-\342\234\215\360\237\217\273 1st HandMad EXAM -LP -MathModels - Simplex/QaA_1-Mathematical Modeling & Linear Optimization-Full Problem Solving Workflow/answer_1.pdf"
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