1. **Problem statement:** Assign four technicians (T1–T4) to four machines (M1–M4) to minimize total repair time using the Hungarian Method. 2. **Given cost matrix (repair times):** $$\begin{bmatrix}12 & 8 & 10 & 7\\6 & 9 & 7 & 8\\8 & 6 & 9 & 5\\7 & 11 & 8 & 9\end{bmatrix}$$ 3. **Step 1: Row reduction** (subtract the minimum value in each row from all elements of that row): - Row 1 min = 7, subtract 7 from row 1 - Row 2 min = 6, subtract 6 from row 2 - Row 3 min = 5, subtract 5 from row 3 - Row 4 min = 7, subtract 7 from row 4 Result: $$\begin{bmatrix}5 & 1 & 3 & 0\\0 & 3 & 1 & 2\\3 & 1 & 4 & 0\\0 & 4 & 1 & 2\end{bmatrix}$$ 4. **Step 2: Column reduction** (subtract minimum value in each column from all elements of that column): - Column 1 min = 0, subtract 0 - Column 2 min = 1, subtract 1 - Column 3 min = 1, subtract 1 - Column 4 min = 0, subtract 0 Result: $$\begin{bmatrix}5 & 0 & 2 & 0\\0 & 2 & 0 & 2\\3 & 0 & 3 & 0\\0 & 3 & 0 & 2\end{bmatrix}$$ 5. **Step 3: Zero assignments and cover zeros** - Find independent zeros to assign technicians to machines without overlaps. - Assign T1 to M2 (zero at position (1,2)) - Assign T2 to M3 (zero at (2,3)) - Assign T3 to M4 (zero at (3,4)) - Assign T4 to M1 (zero at (4,1)) 6. **Step 4: Final matching and total repair time** - T1->M2: 8 hours - T2->M3: 7 hours - T3->M4: 5 hours - T4->M1: 7 hours Total time = $8 + 7 + 5 + 7 = 27$ hours 7. **Real-life application example:** The Hungarian Method can be used in workforce scheduling to assign workers to tasks minimizing total completion time or costs, such as assigning delivery drivers to routes to minimize total driving time, optimizing resource allocation efficiently.