Add files via upload

Signed-off-by: Fabiana 🚀  Campanari <[email protected]>

Author: FabianaCampanari

class_7-✍🏻 HandMade Excercise prep EXAM -LP -MathModels - Simplex/✍️2-Lp solved by the Graphic Method/2a-graphicalSolutionmd.md

@@ -0,0 +1,121 @@
+# 📈 Graphical Solution to the Linear Programming (LP) Problem
+
+**Objective:**
+
+Maximize  
+$$
+Z = 4x_1 + 3x_2
+$$
+
+**Subject to:**
+
+$$
+\begin{cases}
+x_1 + 3x_2 \leq 7 \\
+2x_1 + 2x_2 \leq 8 \\
+x_1 + x_2 \leq 3 \\
+x_2 \leq 2 \\
+x_1 \geq 0, \quad x_2 \geq 0
+\end{cases}
+$$
+
+---
+
+## Step 1: Plot the Constraints
+
+Convert inequalities into equalities to draw the lines:
+
+1. $x_1 + 3x_2 = 7$
+   - If $x_1 = 0 \Rightarrow x_2 = \frac{7}{3} \approx 2.33$
+   - If $x_2 = 0 \Rightarrow x_1 = 7$
+
+2. $2x_1 + 2x_2 = 8$
+   - If $x_1 = 0 \Rightarrow x_2 = 4$
+   - If $x_2 = 0 \Rightarrow x_1 = 4$
+
+3. $x_1 + x_2 = 3$
+   - If $x_1 = 0 \Rightarrow x_2 = 3$
+   - If $x_2 = 0 \Rightarrow x_1 = 3$
+
+4. $x_2 = 2$ → horizontal line
+
+---
+
+## Step 2: Identify the Feasible Region
+
+- The feasible region is the intersection of all shaded regions that satisfy the constraints.
+- Must include $x_1 \geq 0$ and $x_2 \geq 0$.
+
+---
+
+## Step 3: Find Intersection Points (Vertices)
+
+1. Intersection of $x_1 + 3x_2 = 7$ and $2x_1 + 2x_2 = 8$:
+   - Multiply first by 2: $2x_1 + 6x_2 = 14$
+   - Subtract: $4x_2 = 6 \Rightarrow x_2 = 1.5$, $x_1 = 2.5$
+   - Point: **(2.5, 1.5)**
+
+2. Intersection of $x_1 + 3x_2 = 7$ and $x_1 + x_2 = 3$:
+   - Subtract: $2x_2 = 4 \Rightarrow x_2 = 2$, $x_1 = 1$
+   - Point: **(1, 2)**
+
+3. Intersection of $x_1 + x_2 = 3$ and $x_2 = 2$:
+   - $x_1 = 1$
+   - Point: **(1, 2)**
+
+4. Intersection of $2x_1 + 2x_2 = 8$ and $x_2 = 2$:
+   - $x_1 = 2$
+   - Point: **(2, 2)**
+
+5. $x_1 + x_2 = 3$ and $x_1 = 0 \Rightarrow x_2 = 3$ ❌ (Invalid since $x_2 \leq 2$)
+
+6. $x_1 + 3x_2 = 7$ and $x_1 = 0 \Rightarrow x_2 = 7/3 \approx 2.33$ ❌ (Invalid since $x_2 \leq 2$)
+
+---
+
+## Step 4: Evaluate Objective Function at Each Vertex
+
+Feasible Vertices:
+
+- A: (0, 0)
+- B: (0, 2)
+- C: (1, 2)
+- D: (2, 2)
+- E: (2, 0)
+- F: (3, 0)
+
+| Point | $x_1$ | $x_2$ | $Z = 4x_1 + 3x_2$ |
+|:-----:|:-----:|:-----:|:-----------------:|
+| A     | 0     | 0     | 0                 |
+| B     | 0     | 2     | 6                 |
+| C     | 1     | 2     | 10                |
+| D     | 2     | 2     | 14 ❌             |
+| E     | 2     | 0     | 8                 |
+| F     | 3     | 0     | 12                |
+
+---
+
+## Step 5: Check Feasibility
+
+- **(2,2)** violates: $x_1 + 3x_2 = 2 + 6 = 8 > 7$ ❌
+- All others: ✅
+
+---
+
+## ✅ Final Step: Choose the Best Feasible Point
+
+| Point | $Z$ | Feasible |
+|:-----:|:---:|:--------:|
+| A     | 0   | Yes      |
+| B     | 6   | Yes      |
+| C     | 10  | Yes      |
+| E     | 8   | Yes      |
+| F     | 12  | ✅ Best   |
+| D     | 14  | No       |
+
+---
+
+## 🏁 Conclusion
+
+- **Optimal solution:** $x_1 = 3$, $x_2 = 0$
+- **Maximum value:** $Z = 12$