Assignment Problem(3) - Steps to Solve using Branch and Bound

 Example Assignment Problem


Steps in Solving

Solving the assignment problem with the branch and bound algorithm involves a systematic exploration of all possible assignments, aiming to find the least costly solution. Here are the general steps involved in solving the assignment problem using branch and bound:

 1. Formulate the Problem:

   - Define the cost matrix where each element c[i][j] represents the cost of assigning the i-th worker to the j-th job.

   - Determine the objective, typically to minimize the total cost of assignments.

 2. Initial Lower Bound:

   - Calculate an initial lower bound for the entire problem. This can be done by summing the minimum value from each row or each column of the cost matrix, depending on the strategy.

 3. Create the Root Node:

   - Start with a root node representing an empty assignment (no workers are yet assigned to any jobs).

 4. Branching:

   - For the current node, generate child nodes by considering all possible assignments of a worker to different jobs.

   - Each child node represents a partial solution with one more worker assigned to a job.

 5. Bounding:

   - For each child node, calculate a lower bound on the total cost that can be achieved if the partial assignment at that node is expanded to a complete solution. This is often done by reducing the cost matrix and calculating the cost of the cheapest remaining assignment for each worker and job.

   - Prune any branches whose lower bounds exceed the cost of an already known solution.

 6. Selection:

   - Choose the next node to explore. This can be the node with the lowest lower bound (best-first search) or can be based on a depth-first or breadth-first strategy.

 7. Pruning:

   - If a node’s lower bound is greater than the current best solution, prune that node and its descendants.

   - If a node represents a complete assignment (all workers are assigned jobs), and its total cost is lower than the current best solution, update the best solution.

 8. Repetition:

   - Repeat the branching, bounding, selection, and pruning steps until all nodes have been explored or pruned.

 9. Optimal Solution:

   - Once the exploration is complete, the best solution found is the optimal solution to the assignment problem.

 10. Result:

    - The result is a complete assignment of workers to jobs with the minimum total cost.

 Branch and bound is a powerful technique for solving discrete and combinatorial optimization problems like the assignment problem. It systematically explores the search space and is guaranteed to find the optimal solution if allowed to run to completion. However, the efficiency of the algorithm greatly depends on the quality of the bounding mechanism used to estimate lower bounds and the strategy used to select the next node to explore.


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