The correct answer is option 'B', Vogel's approximation method. This method is known for providing very good solutions to the assignment problem. Let's understand why Vogel's approximation method is preferred over other methods.

1. Introduction to the Assignment Problem:
The assignment problem is a mathematical optimization problem that aims to find the best assignment of a set of tasks to a set of resources. It is widely used in various fields, including operations research, logistics, and scheduling.

2. Northwest Corner Rule:
The Northwest corner rule is a basic and simple method to solve the assignment problem. It starts by allocating the maximum possible value to the cell in the top-left corner of the cost matrix and then proceeds by allocating the remaining values in a sequential manner. While this method is easy to apply, it often does not provide optimal solutions and may result in higher costs.

3. MODI Method:
MODI (Modified Distribution) method is an iterative technique used to improve the initial feasible solution obtained from other methods, such as the Northwest corner rule. It involves calculating the opportunity costs for unoccupied cells and then selecting the cell with the highest opportunity cost for improvement. Although the MODI method can enhance the initial solution, it may still not guarantee an optimal solution.

4. Stepping-stone Method:
The stepping-stone method is another iterative procedure that determines the optimal solution by examining all possible routes in the transportation table. It involves identifying closed paths and evaluating the potential of each route in terms of reducing the total cost. While the stepping-stone method is effective, it requires a significant amount of computation and can be time-consuming for large-scale problems.

5. Vogel's Approximation Method:
Vogel's approximation method, also known as the Vogel's approximation algorithm (VAM), is a well-known and widely used method for solving the assignment problem. It focuses on selecting the most advantageous allocation in each row and column based on the difference between the two smallest costs. By considering the penalties associated with each allocation, Vogel's method tends to provide more accurate and efficient solutions compared to other methods.

In summary, while the Northwest corner rule, MODI method, and stepping-stone method are commonly used in solving the assignment problem, Vogel's approximation method is preferred due to its ability to generate very good solutions. It considers the differences in costs and penalties, allowing for better allocation decisions and ultimately leading to optimal or near-optimal solutions.