1. **Problem Statement:** We need to assign 3 jobs (A, B, C) to 4 machines (W, X, Y, Z) minimizing the total cost, given the cost matrix: $$\begin{array}{c|cccc} & W & X & Y & Z \\ \hline A & 18 & 24 & 28 & 32 \\ B & 8 & 13 & 17 & 19 \\ C & 10 & 15 & 19 & 22 \end{array}$$ 2. **Method:** This is an assignment problem, typically solved by the Hungarian algorithm which finds the minimum cost matching between jobs and machines. 3. **Step 1: Reduce rows by subtracting the minimum value in each row:** - Row A minimum: 18, subtract 18 from each element in row A: $[0, 6, 10, 14]$ - Row B minimum: 8, subtract 8 from each element in row B: $[0, 5, 9, 11]$ - Row C minimum: 10, subtract 10 from each element in row C: $[0, 5, 9, 12]$ Reduced matrix: $$\begin{array}{c|cccc} & W & X & Y & Z \\ \hline A & 0 & 6 & 10 & 14 \\ B & 0 & 5 & 9 & 11 \\ C & 0 & 5 & 9 & 12 \end{array}$$ 4. **Step 2: Reduce columns by subtracting the minimum value in each column:** - Column W minimum: 0, no change - Column X minimum: 5, subtract 5 from each element in column X: $[1, 0, 0]$ - Column Y minimum: 9, subtract 9 from each element in column Y: $[1, 0, 0]$ - Column Z minimum: 11, subtract 11 from each element in column Z: $[3, 0, 1]$ Reduced matrix: $$\begin{array}{c|cccc} & W & X & Y & Z \\ \hline A & 0 & 1 & 1 & 3 \\ B & 0 & 0 & 0 & 0 \\ C & 0 & 0 & 0 & 1 \end{array}$$ 5. **Step 3: Find zero assignments without conflicts:** - Assign Job A to Machine W (cost 0) - Assign Job B to Machine X (cost 0) - Assign Job C to Machine Y (cost 0) 6. **Step 4: Calculate total minimum cost:** - Job A to W: 18 - Job B to X: 13 - Job C to Y: 19 Total cost = $18 + 13 + 19 = 50$ **Final answer:** Assign Job A to Machine W, Job B to Machine X, and Job C to Machine Y for a minimum total cost of 50.