Merge pull request #462 from Quantum-Software-Development/FabianaCampanari-patch-1

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Author: FabianaCampanari

class_7-✍🏻 HandMade Excercise prep EXAM -LP -MathModels - Simplex/✍️3-LP solved Method Simplex/LP_Simplex_pythonCode.md

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+# Linear Programming Problem – Simplex Method
+
+This repository presents a complete, step-by-step solution to a **Linear Programming (LP)** problem using the **Simplex Method**, along with a basic **Python implementation**.
+
+## 🧮 Problem Statement
+
+**Maximise:**
+
+Z = 4x₁ + 3x₂
+
+**Subject to:**
+
+x₁ + 3x₂ ≤ 7  
+2x₁ + 2x₂ ≤ 8  
+x₁ + x₂ ≤ 3  
+x₂ ≤ 2  
+x₁ ≥ 0, x₂ ≥ 0
+
+## ✅ Standard Form Conversion
+
+Introduce slack variables: s₁, s₂, s₃, s₄
+
+The system becomes:
+
+x₁ + 3x₂ + s₁ = 7  
+2x₁ + 2x₂ + s₂ = 8  
+x₁ + x₂ + s₃ = 3  
+x₂ + s₄ = 2  
+x₁, x₂, s₁, s₂, s₃, s₄ ≥ 0
+
+## 📊 Initial Simplex Tableau
+
+| Base | x₁ | x₂ | s₁ | s₂ | s₃ | s₄ | RHS |
+|------|----|----|----|----|----|----|-----|
+| s₁   | 1  | 3  | 1  | 0  | 0  | 0  | 7   |
+| s₂   | 2  | 2  | 0  | 1  | 0  | 0  | 8   |
+| s₃   | 1  | 1  | 0  | 0  | 1  | 0  | 3   |
+| s₄   | 0  | 1  | 0  | 0  | 0  | 1  | 2   |
+| **Z**| -4 | -3 | 0  | 0  | 0  | 0  | 0   |
+
+## 🔄 Iterations Overview
+
+### 🔹 Iteration 1
+
+- **Entering variable**: x₁ (most negative in Z row)  
+- **Leaving variable**: s₃ (minimum ratio = 3)  
+- Pivot to bring x₁ into the basis.  
+
+Updated tableau shows next candidate as:  
+- **Entering variable**: x₂  
+- **Leaving variable**: s₁ or s₄ (tie – choose s₁)
+
+### 🔹 Iteration 2
+
+After pivoting x₂ into the basis, tableau is updated again.  
+Now the most negative coefficient in the Z row is for s₃, but:  
+- No valid pivot is possible (no positive coefficients in that column).  
+- Hence, no further improvement is feasible.
+
+## 🏁 Final Optimal Solution
+
+The optimal solution was reached at the end of Iteration 1:
+
+x₁ = 3  
+x₂ = 0  
+Z(max) = 12
+
+**All constraints are satisfied.**
+
+---
+
+## 🐍 Python Code – Simplex Solver (Basic Version)
+
+```python
+from scipy.optimize import linprog
+
+# Coefficients of the objective function (to maximise Z = 4x₁ + 3x₂)
+# Convert to minimisation: -Z
+c = [-4, -3]
+
+# Coefficients of the inequality constraints (Ax ≤ b)
+A = [
+    [1, 3],
+    [2, 2],
+    [1, 1],
+    [0, 1]
+]
+
+b = [7, 8, 3, 2]
+
+# Bounds for x₁ and x₂: both ≥ 0
+x_bounds = (0, None)
+bounds = [x_bounds, x_bounds]
+
+# Solve the problem
+res = linprog(c, A_ub=A, b_ub=b, bounds=bounds, method="simplex")
+
+# Output results
+if res.success:
+    print("Optimal solution found:")
+    print(f"x₁ = {res.x[0]:.2f}")
+    print(f"x₂ = {res.x[1]:.2f}")
+    print(f"Maximum Z = {(-res.fun):.2f}")
+else:
+    print("No solution found:", res.message)