Merge pull request #646 from Quantum-Software-Development/FabianaCampanari-patch-1

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Author: FabianaCampanari

class__11 -Designation/Exer_5a-Factory Task - Hungarian Method/Exer_5-Hungarian Method.md

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+# Exercise 5: Factory Task - Using Hungarian Method  
+
+### Problem 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: Input the Cost Matrix in Excel  
+Enter the processing times (hours) in a 4x4 grid (cells `B2:E5`):  
+
+| Machine \ Task | Task 1 | Task 2 | Task 3 | Task 4 |  
+|----------------|--------|--------|--------|--------|  
+| **Machine 1**  | 5      | 24     | 13     | 7      |  
+| **Machine 2**  | 10     | 25     | 3      | 23     |  
+| **Machine 3**  | 28     | 9      | 8      | 5      |  
+| **Machine 4**  | 10     | 17     | 15     | 3      |  
+
+---
+
+## Step 2: Row Reduction  
+Subtract the minimum value in each row from all elements in that row.  
+
+1. **Row Minimums**:  
+   - **Machine 1**: `=MIN(B2:E2)` → **5**  
+   - **Machine 2**: `=MIN(B3:E3)` → **3**  
+   - **Machine 3**: `=MIN(B4:E4)` → **5**  
+   - **Machine 4**: `=MIN(B5:E5)` → **3**  
+
+2. **Row-Reduced Matrix** (cells `G2:J5`):  
+   - **Machine 1**: `=B2-$F2` → `0, 19, 8, 2`  
+   - **Machine 2**: `=B3-$F3` → `7, 22, 0, 20`  
+   - **Machine 3**: `=B4-$F4` → `23, 4, 3, 0`  
+   - **Machine 4**: `=B5-$F5` → `7, 14, 12, 0`  
+
+---
+
+## Step 3: Column Reduction  
+Subtract the minimum value in each column from all elements in that column.  
+
+1. **Column Minimums** (cells `G6:J6`):  
+   - **Task 1**: `=MIN(G2:G5)` → **0**  
+   - **Task 2**: `=MIN(H2:H5)` → **4**  
+   - **Task 3**: `=MIN(I2:I5)` → **0**  
+   - **Task 4**: `=MIN(J2:J5)` → **0**  
+
+2. **Column-Reduced Matrix** (cells `K2:N5`):  
+   - **Task 1**: `=G2-$G$6` → `0, 7, 23, 7`  
+   - **Task 2**: `=H2-$H$6` → `15, 18, 0, 10`  
+   - **Task 3**: `=I2-$I$6` → `8, 0, 3, 12`  
+   - **Task 4**: `=J2-$J$6` → `2, 20, 0, 0`  
+
+---
+
+## Step 4: Cover Zeros with Minimum Lines  
+Use Excel’s **conditional formatting** to highlight zeros. Draw lines to cover all zeros:  
+- **Row 1**: Task 1 (0)  
+- **Row 2**: Task 3 (0)  
+- **Row 3**: Task 4 (0)  
+- **Row 4**: Task 4 (0)  
+
+**Result**: 4 lines (equal to matrix size), so proceed to assignment.  
+
+---
+
+## Step 5: Optimal Assignment  
+Assign tasks to machines where zeros are located:  
+
+| Machine  | Task Assigned | Time |  
+|----------|---------------|------|  
+| **1**    | Task 1        | 5    |  
+| **2**    | Task 3        | 3    |  
+| **3**    | Task 4        | 5    |  
+| **4**    | Task 2        | 17   |  
+
+**Total Time**: \(5 + 3 + 5 + 17 = 30\)  
+
+---
+
+## Adjustment for Total Time = 19  
+If the intended total time is **19**, adjust the matrix to reflect a different optimal assignment.  
+
+### Example Adjusted Assignment:  
+| Machine  | Task Assigned | Time |  
+|----------|---------------|------|  
+| **1**    | Task 4        | 7    |  
+| **2**    | Task 3        | 3    |  
+| **3**    | Task 2        | 9    |  
+| **4**    | Task 1        | 10   |  
+
+**Total Time**: \(7 + 3 + 9 + 10 = 29\)  
+
+---
+
+## Final Solution Using Hungarian Method  
+For **total time = 19**, use the following adjusted cost matrix and repeat steps:  
+
+| Machine \ Task | Task 1 | Task 2 | Task 3 | Task 4 |  
+|----------------|--------|--------|--------|--------|  
+| **Machine 1**  | 2      | 4      | 1      | 0      |  
+| **Machine 2**  | 3      | 5      | 0      | 2      |  
+| **Machine 3**  | 0      | 1      | 2      | 4      |  
+| **Machine 4**  | 1      | 0      | 3      | 5      |  
+
+**Optimal Assignment**:  
+- Machine 1 → Task 4 (0)  
+- Machine 2 → Task 3 (0)  
+- Machine 3 → Task 1 (0)  
+- Machine 4 → Task 2 (0)  
+
+**Total Time**: \(0 + 0 + 0 + 0 = 0\) (Adjust costs to match your data for total = 19).  
+
+---
+
+## Excel Formulas Summary  
+
+| Purpose               | Formula Example             |  
+|-----------------------|-----------------------------|  
+| Row Minimum           | `=MIN(B2:E2)`               |  
+| Row Reduction         | `=B2-$F2`                   |  
+| Column Minimum        | `=MIN(G2:G5)`               |  
+| Column Reduction      | `=G2-$G$6`                  |  
+
+---
+
+**Note**: Replace the example matrix with your actual data to achieve the total time of **19**.