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The image contains a cost matrix for an assignment problem. The goal of this problem is to assign tasks (A, B, C, D) to resources (P, Q, R, S) in a way that minimizes the overall cost. The matrix is as follows:

\[
\begin{array}{c|c c c c}
   & P  & Q  & R  & S  \\
\hline
A  & 12 & 15 & 18 & 8  \\
B  & 13 & 10 & 9  & 14 \\
C  & 10 & 12 & 15 & 13 \\
D  & 7  & 8  & 9  & 14 \\
\end{array}
\]

To solve this using the **Hungarian method** for the assignment problem:

1. **Step 1: Row Reduction**  
   Subtract the minimum value from each row.

2. **Step 2: Column Reduction**  
   After the row reduction, subtract the minimum value from each column.

3. **Step 3: Assignment**  
   Attempt to assign zeros in the reduced matrix to minimize the cost, ensuring that each task is assigned exactly once.

Let me proceed with these steps to find the solution.After applying the **row reduction** and **column reduction**, we get the following matrices:

### Row-reduced matrix:
\[
\begin{array}{c|c c c c}
   & P  & Q  & R  & S  \\
\hline
A  & 4  & 7  & 10 & 0  \\
B  & 4  & 1  & 0  & 5  \\
C  & 0  & 2  & 5  & 3  \\
D  & 0  & 1  & 2  & 7  \\
\end{array}
\]

### Column-reduced matrix (after further minimizing the columns):
\[
\begin{array}{c|c c c c}
   & P  & Q  & R  & S  \\
\hline
A  & 4  & 6  & 10 & 0  \\
B  & 4  & 0  & 0  & 5  \\
C  & 0  & 1  & 5  & 3  \\
D  & 0  & 0  & 2  & 7  \\
\end{array}
\]

Now, we attempt to make optimal assignments by assigning tasks to zeros in the matrix, ensuring that each task is assigned only once. I'll proceed to analyze the matrix and find the optimal assignments.The optimal assignment of tasks to resources using the **Hungarian Method** is as follows:

- Task A is assigned to resource S (A → S).
- Task B is assigned to resource R (B → R).
- Task C is assigned to resource P (C → P).
- Task D is assigned to resource Q (D → Q).

The total minimized cost for this assignment is **0** after reduction.

Would you like further details or explanations on any step? 

Here are

Solve the assignment problem based on the cost matrix for tasks A, B, C, and D and resources P, Q, R, and S.

Learn how to solve the assignment problem using the Hungarian method. This step-by-step solution minimizes the cost matrix for tasks A, B, C, and D assigned to resources P, Q, R, and S. The solution involves row and column reduction and is suitable for undergraduate students.

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