The Hungarian algorithm, while typically used for minimizing costs in assignment problems, can also be applied to maximization problems. Here’s how we can handle assignment problems where the objective is to maximize something like profit or total output:
Conversion to Minimization:
Similar to transportation problems, a common approach is to convert the maximization assignment problem into an equivalent minimization problem. This allows you to leverage the well-established Hungarian algorithm, which is designed for minimization. Here’s the conversion strategy:
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Identify the Maximization Objective: Clearly define what you want to maximize, be it profit per assignment (profit_ij) or total output achieved by assigning resources to tasks.
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Find the Highest Unit “Benefit”: Identify the highest value among your profit per assignment or other relevant measure you’re trying to maximize. Let’s call this highest value ‘M’.
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Subtract Unit Values from M: In your cost matrix, subtract each cell’s unit value (profit_ij) from ‘M’. This essentially transforms profit into a cost where a higher value initially represents a lower “cost” (since it’s further from M). Lower costs in the Hungarian algorithm translate to better assignments for maximization.
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Solve as Minimization Problem: With the adjusted cost matrix (where higher values represent lower costs), use the Hungarian algorithm to find the minimum cost assignment. Remember, minimizing cost in this transformed matrix translates to maximizing the original objective in the actual problem.
Example: Maximizing Total Output
Imagine you have a team with different skill sets and a set of tasks with varying difficulty levels. You want to assign team members to tasks such that the total output achieved is maximized.
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The cost matrix shows the expected output when a specific team member is assigned to a particular task (higher values represent higher expected output).
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Solve the problem using the Hungarian algorithm, but instead of minimizing actual costs, you’re minimizing the negative output values (obtained by subtracting the actual output from the maximum possible output ‘M’).
Benefits of Conversion Approach:
- Leverages Existing Algorithm: By converting to minimization, you can utilize the well-developed Hungarian algorithm for efficient solution.
- Intuitive Interpretation: Even after solving the minimization problem, it’s easy to translate the results back to the original maximization problem, as minimizing negative output translates to maximizing actual output achieved.
Alternative Methods:
While conversion is a common approach, some specialized algorithms are designed specifically for maximization assignment problems. These algorithms might be more efficient for certain scenarios, but familiarity with the conversion method offers a broader applicability using the established Hungarian algorithm.