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class_7-✍🏻 HandMade Excercise prep EXAM -LP -MathModels - Simplex/✍️1-Buildin Mathematical Model/✍️1-Buildin Mathematical Model.md

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+# 📊 Linear Programming Model — Production Optimization
+
+## ✅ Problem Statement
+
+A company, after going through a production streamlining process, ended up with the availability of 3 productive resources: **R1**, **R2**, and **R3**.
+
+A study on resource usage showed the possibility of producing two products: **P1** and **P2**. After evaluating costs and consulting the sales department, it was found that:
+
+- **P1 yields a profit of 120 monetary units per unit**
+- **P2 yields a profit of 150 monetary units per unit**
+
+The production department provided the following **resource usage** table:
+
+| Product | R1/unit | R2/unit | R3/unit |
+|---------|---------|---------|---------|
+| **P1**  |    2    |    3    |    5    |
+| **P2**  |    4    |    0    |    3    |
+
+And the **monthly resource availability**:
+
+| Resource | Monthly Availability |
+|----------|----------------------|
+| **R1**   | 100                  |
+| **R2**   | 90                   |
+| **R3**   | 120                  |
+
+---
+
+## 🎯 Objective
+
+Mathematically model the **Linear Programming (LP)** problem to **maximize profit** under resource constraints.
+
+---
+
+## 🧠 Step-by-Step Modeling
+
+### 1. 🧮 Decision Variables
+
+Let:
+
+x₁ = quantity produced of product P1
+x₂ = quantity produced of product P2
+
+Or in LaTeX (for use in documents):
+
+```latex
+x_1 = \text{quantity produced of product P1} \\
+x_2 = \text{quantity produced of product P2}
+
+
+
+⸻
+
+2. 📈 Objective Function
+
+Maximize total profit:
+
+\text{Maximize } Z = 120x_1 + 150x_2
+
+
+
+⸻
+
+3. 📏 Resource Constraints
+
+Each resource has limited availability:
+	•	R1 constraint:
+
+2x_1 + 4x_2 \leq 100
+
+	•	R2 constraint:
+
+3x_1 \leq 90
+
+	•	R3 constraint:
+
+5x_1 + 3x_2 \leq 120
+
+
+
+⸻
+
+4. 🚫 Non-Negativity Constraints
+
+We cannot produce a negative quantity of products:
+
+x_1 \geq 0, \quad x_2 \geq 0
+
+
+
+⸻
+
+🧾 Complete Mathematical Model
+
+\boxed{
+\begin{cases}
+\text{Maximize } Z = 120x_1 + 150x_2 \\
+2x_1 + 4x_2 \leq 100 \\
+3x_1 \leq 90 \\
+5x_1 + 3x_2 \leq 120 \\
+x_1 \geq 0, \quad x_2 \geq 0
+\end{cases}
+}
+
+
+
+⸻
+
+## 📌 Summary Tables
+
+### 🔢 Profit per Product
+
+| Product | Profit per Unit (u.m.) |
+|:--------|:----------------------:|
+| **P1**  | 120                    |
+| **P2**  | 150                    |
+
+---
+
+### 🧰 Resource Usage per Unit
+
+| Product | R1/unit | R2/unit | R3/unit |
+|:--------|:-------:|:-------:|:-------:|
+| **P1**  |   2     |   3     |   5     |
+| **P2**  |   4     |   0     |   3     |
+
+---
+
+### 📦 Monthly Resource Availability
+
+| Resource | Available Units |
+|:---------|:----------------:|
+| **R1**   |      100         |
+| **R2**   |       90         |
+| **R3**   |      120         |
+
+
+⸻
+
+🧠 Notes
+	•	This LP model can be solved using methods such as the Simplex Algorithm.
+	•	Can also be implemented in software such as Python (PuLP), MATLAB, or Excel Solver.
+