The tackles matching tasks to agents efficiently. It's a special case of transportation problems, with one-to-one matching and equal-sized sets. The represents assignment costs, and the goal is to minimize total cost.
The solves assignment problems step-by-step. It creates a reduced cost matrix, finds zeros, and iteratively improves the solution. This method guarantees an , making it a powerful tool for solving real-world matching problems.
Assignment Problem and Hungarian Algorithm
Assignment problem as transportation subset
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A New Approach of Solving Single Objective Unbalanced Assignment Problem View original
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A Comparative Study of Initial Basic Feasible Solution by a Least Cost Mean Method (LCMM) of ... View original
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Proposed Heuristic Method for Solving Assignment Problems View original
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A New Approach of Solving Single Objective Unbalanced Assignment Problem View original
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A Comparative Study of Initial Basic Feasible Solution by a Least Cost Mean Method (LCMM) of ... View original
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Top images from around the web for Assignment problem as transportation subset
A New Approach of Solving Single Objective Unbalanced Assignment Problem View original
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A Comparative Study of Initial Basic Feasible Solution by a Least Cost Mean Method (LCMM) of ... View original
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Proposed Heuristic Method for Solving Assignment Problems View original
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A New Approach of Solving Single Objective Unbalanced Assignment Problem View original
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A Comparative Study of Initial Basic Feasible Solution by a Least Cost Mean Method (LCMM) of ... View original
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One-to-one matching between equal-sized sets assigns tasks to agents efficiently
Supply nodes (agents) and demand nodes (tasks) all have values of 1
Cost matrix represents assignment costs for each agent-task pair
Balanced problem guarantees integer solution unlike general transportation problems
Linear programming for assignments
Decision variables xij (1 if agent i assigned to task j, 0 otherwise) determine optimal assignments
Objective function minimizes total assignment cost ∑i=1n∑j=1ncijxij
Constraints ensure each agent and task assigned once ∑j=1nxij=1, ∑i=1nxij=1
Non-negativity and binary constraints maintain xij≥0, xij∈{0,1}
Hungarian algorithm application
Subtract row minima from each row in cost matrix
Subtract column minima from each column
Draw minimum lines covering all zeros
Create additional zeros by subtracting smallest uncovered element
Find complete assignment using covered zeros
Iterative process improves solution until optimal assignment found
proof uses dual variables and complementary slackness conditions
Reduced cost matrices in assignments
Subtracting row and column minima creates reduced cost matrix preserving optimal solution
Non-negative elements with at least one zero per row and column
Reduced costs show potential objective value improvements
Zero reduced costs indicate potentially optimal assignments
Simplifies optimal solution search in Hungarian algorithm
Allows visual identification of potential assignments (matching zeros)