This problem involves determining the optimal assignment of agencies to minimize the travel distances, which is essentially a type of assignment problem. The distance matrix is provided, and we need to allocate agencies $A, B, C, D, E$ to locations $a, b, c, d, e$ such that the total travel distance is minimized.

Distance Matrix:

Problem Analysis

This is a classical linear programming problem that can be solved using the Hungarian method or an optimization tool such as linear programming in Python or Excel Solver. The objective is to find the assignment that minimizes the total distance, taking into account that each agency must be assigned exactly once.

Steps for Solving

1. Set up the assignment problem: Each agency (row) needs to be assigned to one city (column) to minimize the cost (distance).
2. Use an algorithm like the Hungarian method: This algorithm finds the minimum-cost assignment by iteratively reducing the matrix and finding zero-cost matches.
3. Calculate the total distance for the optimal assignment.

Would you like me to solve it step-by-step and provide the solution? If so, I can either explain it manually or provide code to find the solution. Let me know how you'd like to proceed!

Related Questions:

1. How does the Hungarian method work to solve assignment problems?
2. Can this problem be solved using a different optimization technique, such as linear programming?
3. What happens if there are more or fewer agencies than locations?
4. Can we solve this problem using software tools like Excel or Python?
5. How do real-world constraints like availability or capacity affect assignment problems?

Tip: In assignment problems, ensure that the number of agents (rows) equals the number of tasks (columns). If not, add dummy rows or columns with zero costs.