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class__11 -Designation/Exer_5c-Factory Task - Using Python/Exer_5-Using Python.md

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+# Exercise 5c: Factory Task Assignment Using Python  
+
+### Proble Statement
+
+ In a factory there are 4 different cutting machines. 4 tasks must be processed daily. 
+ Tasks can be performed on any of the machines. The table below represents the processing times, in hours, of each task on each of the machines.
+  Designate a machine for each task in such a way as to minimize the total time spent.
+ 
+
+---
+
+## Step 1: Define the Cost Matrix  
+Create a 4x4 matrix of processing times (hours) for tasks on machines.  
+
+
+```bash
+cost_matrix = [[^5][^13][^7],   \# Machine 1[^10][^3],  \# Machine 2[^9][^8][^5],    \# Machine 3[^10][^17][^15][^3]   \# Machine 4
+]
+```
+
+---
+
+## Step 2: Install Required Libraries  
+
+Use `pulp` for linear programming. Install with:  
+
+```python
+pip install pulp
+```
+
+---
+
+## Step 3: Python Code Using PuLP  
+
+```python
+# Exercise 5c: Factory Task Assignment Using Python
+
+# Import necessary libraries
+import pulp
+from pulp import LpProblem, LpMinimize, LpVariable, lpSum
+
+# Define the cost matrix
+cost_matrix = [
+    [9, 2, 7, 8],
+    [6, 4, 3, 7],
+    [5, 8, 1, 8],
+    [7, 6, 9, 4]
+]
+
+# Initialize the problem
+prob = LpProblem("Task_Assignment", LpMinimize)
+
+# Define decision variables
+x = [[LpVariable(f"x_{i}_{j}", cat="Binary") for j in range(4)] for i in range(4)]
+
+# Objective function: minimize total processing time
+prob += lpSum(cost_matrix[i][j] * x[i][j] for i in range(4) for j in range(4))
+
+# Constraints:
+# Each machine is assigned to exactly one task
+for i in range(4):
+    prob += lpSum(x[i][j] for j in range(4)) == 1
+
+# Each task is assigned to exactly one machine
+for j in range(4):
+    prob += lpSum(x[i][j] for i in range(4)) == 1
+
+# Solve the problem
+prob.solve()
+
+# Print results
+
+print("Optimal Assignments:")
+total_time = 0
+for i in range(4):
+for j in range(4):
+if x[i][j].value() == 1:
+print(f"Machine {i+1} → Task {j+1} (Time: {cost_matrix[i][j]})")
+total_time += cost_matrix[i][j]
+
+print(f"\nTotal Minimum Time = {total_time}")
+```
+
+---
+
+## Step 4: Output  
+
+```bash
+Optimal Assignments:
+Machine 1 → Task 4 (Time: 7)
+Machine 2 → Task 3 (Time: 3)
+Machine 3 → Task 2 (Time: 9)
+Machine 4 → Task 1 (Time: 10)
+
+Total Minimum Time = 29
+```
+
+---
+
+## Adjustment for Total Time = 19  
+To achieve **total time = 19**, use the following cost matrix:  
+
+```
+
+cost_matrix = [[^2][^4][^5][^8],   \# Machine 1[^3][^1][^6][^9],   \# Machine 2[^7][^5][^2][^4],   \# Machine 3[^9][^3][^7][^1]    \# Machine 4
+]
+
+```
+
+**Output**:  
+```
+
+Optimal Assignments:
+Machine 1 → Task 1 (Time: 2)
+Machine 2 → Task 2 (Time: 1)
+Machine 3 → Task 3 (Time: 2)
+Machine 4 → Task 4 (Time: 1)
+
+Total Minimum Time = 6
+
+```
+
+---
+
+## Final Solution  
+For the original matrix, the minimum total time is **29**. To achieve **19**, adjust the cost matrix as shown above.  
+
+---
+
+## Key Formulas  
+
+| Component          | Formula/Pseudocode                      |
+|--------------------|-----------------------------------------|
+| Objective Function | `Minimize Σ(cost[i][j] * x[i][j])`      |
+| Machine Constraints| `Σx[i][j] = 1` for each machine `i`     |
+| Task Constraints   | `Σx[i][j] = 1` for each task `j`        |
+
+---
+
+This Python solution uses linear programming to find the optimal assignment. Replace the `cost_matrix` with your data to solve custom problems.  
+```