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Can you solve me this equation in operation research major and Make the solution more concise so that it is clear that I am the one who solved the question and not you.

Certainly! You provided an image showing a problem that requires solving using the **Hungarian Algorithm**. The problem involves assigning four jobs (J1, J2, J3, J4) to four workers (W1, W2, W3, W4) in such a way that the total cost is minimized.

### Step-by-Step Solution Using the Hungarian Algorithm:

1. **Matrix Representation**:
   - Write down the cost matrix provided:

     \[
     \begin{bmatrix}
     81 & 72 & 15 & 92 \\
     77 & 69 & 58 & 83 \\
     72 & 72 & 69 & 52 \\
     85 & 11 & 95 & 98
     \end{bmatrix}
     \]

2. **Row Reduction**:
   - Subtract the smallest element of each row from all elements in that row:

     \[
     \begin{bmatrix}
     66 & 57 & 0 & 77 \\
     19 & 11 & 0 & 25 \\
     20 & 20 & 17 & 0 \\
     74 & 0 & 84 & 87
     \end{bmatrix}
     \]

3. **Column Reduction**:
   - Subtract the smallest element of each column from all elements in that column:

     \[
     \begin{bmatrix}
     47 & 46 & 0 & 77 \\
     0 & 0 & 0 & 25 \\
     1 & 9 & 17 & 0 \\
     54 & 0 & 84 & 87
     \end{bmatrix}
     \]

4. **Optimal Assignment**:
   - Cover all zeros using a minimum number of horizontal and vertical lines. If the number of lines is equal to the number of rows (or columns), the assignment is optimal. If not, adjust the matrix and repeat the process.
   - In this case, you can cover all zeros using 4 lines, so we can move to the assignment step.
   - Assign zeros in such a way that no two assignments share the same row or column:

     - **J1** to **W3** (cost = 15)
     - **J2** to **W1** (cost = 77)
     - **J3** to **W4** (cost = 52)
     - **J4** to **W2** (cost = 11)

5. **Calculate the Total Cost**:
   - The total cost of the optimal assignment is:

     \[
     15 (J1, W3) + 77 (J2, W1) + 52 (J3, W4) + 11 (J4, W2) = 155
     \]

### Final Answer:
The minimal total cost for assigning the jobs to the workers is **155**.

---

Would you like any more details or explanations?

### Additional Questions:
1. What other methods could be used to solve this type of assignment problem?
2. How does the Hungarian Algorithm compare with other algorithms in terms of efficiency?
3. Could you explain the concept of "dual variables" in the context of the Hungarian Algorithm?
4. What are some real-world applications of the Hungarian Algorithm?
5. How would the solution change if there were an additional job or worker?

### Tip:
The Hungarian Algorithm is particularly powerful for solving minimization problems in assignments, making it widely used in operational research and combinatorial optimization.

Using the Hungarian Algorithm to solve the following problem: Four jobs (J1, J2, J3, and J4) need to be executed by four workers (W1, W2, W3, and W4), one job per worker. The matrix below shows the cost of assigning a certain worker to a certain job. The objective is to minimize the total cost of the assignment. The cost matrix is: [[81, 72, 15, 92], [77, 69, 58, 83], [72, 72, 69, 52], [85, 11, 95, 98]].

Learn how to apply the Hungarian Algorithm to solve an assignment problem where four jobs need to be assigned to four workers, aiming to minimize the total cost. This solution involves cost matrix manipulation through row and column reductions, leading to an optimal job-worker assignment. Suitable for Operational Research students.

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