The Complete Enumeration Method in the context of assignment problems refers to a solution approach that involves systematically evaluating all possible assignments and selecting the one that results in the best outcome, often the minimal cost or maximal profit. This method is a brute-force approach, ensuring that every possible assignment combination is checked to find the optimal solution.

Steps in the Complete Enumeration Method for Assignment Problems:

1. List All Possible Assignments:
   The first step is to list out all possible ways to assign tasks or jobs to workers. If there are n tasks and n workers, the total number of possible assignments is n! (n factorial). This can be thought of as creating a set of permutations of the workers to the tasks.

2. Evaluate Each Assignment:
   For each possible assignment, calculate the total cost (or profit, depending on the problem) based on the given cost matrix. Each element of the matrix represents the cost of assigning a particular task to a particular worker.

3. Select the Optimal Assignment:
   After evaluating all possible assignments, choose the one that minimizes the total cost (or maximizes the profit, depending on the nature of the assignment problem).

Example:

Suppose you have 3 workers (A, B, C) and 3 tasks (T1, T2, T3), and the cost matrix looks like this:

The Complete Enumeration Method will evaluate all 3! = 6 possible assignments:

1. Assignment 1: A-T1, B-T2, C-T3 → Total cost = 10 + 12 + 10 = 32
2. Assignment 2: A-T1, B-T3, C-T2 → Total cost = 10 + 15 + 25 = 50
3. Assignment 3: A-T2, B-T1, C-T3 → Total cost = 15 + 10 + 10 = 35
4. Assignment 4: A-T2, B-T3, C-T1 → Total cost = 15 + 15 + 20 = 50
5. Assignment 5: A-T3, B-T1, C-T2 → Total cost = 20 + 10 + 25 = 55
6. Assignment 6: A-T3, B-T2, C-T1 → Total cost = 20 + 12 + 20 = 52

The optimal solution in this case is Assignment 1 (A-T1, B-T2, C-T3) with a total cost of 32.

Advantages:

• Guaranteed Optimal Solution: Since every possible assignment is considered, the method guarantees finding the best possible solution.
• Simplicity: The approach is straightforward and easy to implement, especially for small problems.

Disadvantages:

• Computationally Expensive: The time complexity of the Complete Enumeration Method is O(n!), which grows very quickly as the number of tasks and workers increases. This makes it impractical for large problems.
• Inefficient for Large Problems: For problems with many workers or tasks, the number of possible assignments becomes so large that it’s infeasible to check all of them.

In summary, the Complete Enumeration Method is an exact but computationally expensive way of solving assignment problems, best suited for small instances where finding the optimal solution is critical.