Assignment Problem Solver (Hungarian Algorithm)

Assignment Problem Solver (Hungarian Algorithm) | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: textarea, select, number
Output type: text
Sample coverage: 4
API ready: Yes

Overview

Solve linear assignment problems using the textbook Hungarian algorithm (Kuhn-Munkres method). Input a 2×2 to 8×8 cost or benefit matrix to calculate the optimal one-to-one assignment that minimizes total expenses or maximizes total yield, complete with row and column reduction steps, König line covers, and substitution-verified totals.

When to use

Assigning staff, contractors, or machines to distinct tasks to minimize overall operational expenses or completion times.
Allocating leads, territories, or projects among team members to maximize total projected revenue or score.
Verifying Operations Research coursework and homework problems requiring step-by-step Hungarian algorithm reductions and line covers.

How it works

1Enter your 2–8 × 2–8 matrix into the text area with one row per line, separating cell values with commas or spaces.
2Select your optimization goal: minimize total cost or maximize total benefit, and configure decimal places if using fractions.
3The solver executes row and column reductions, applies augmenting paths to match zeros, performs König minimum line covers if needed, and outputs the optimal matching pairs and total cost.

Use cases

Resource-to-job allocation: Pairing field engineers with service tickets based on travel time and skill-specific labor costs.
Factory line scheduling: Matching industrial machines to specialized manufacturing runs based on changeover and tooling expenses.
Fleet dispatching: Assigning delivery vehicles to regional routes to achieve minimum aggregate fuel consumption and transit time.

Examples

1. Minimizing Maintenance Task Labor Cost

Operations Coordinator

Background

A facility supervisor needs to assign three technicians to three urgent equipment repair jobs based on varying quoted costs.

Problem

Determine the one-to-one assignment that achieves the lowest aggregate maintenance expense.

How to use

Enter the 3×3 cost matrix, choose 'Minimize total cost' under Goal, and generate the solution.

Matrix: 9, 2, 7
6, 4, 3
5, 8, 1
Goal: min
Decimal Places: 4

Outcome

Worker 1 is assigned to Task 2 (cost 2), Worker 2 to Task 1 (cost 6), and Worker 3 to Task 3 (cost 1), yielding an optimal minimum total cost of 9.

2. Maximizing Territory Sales Revenue

Regional Sales Manager

Background

A sales manager wants to assign three senior sales reps to three regional expansion zones to optimize projected deal values.

Problem

Identify the territory distribution that produces the highest total expected sales revenue.

How to use

Paste the 3×3 projected revenue matrix into the input field, set the Goal option to 'Maximize total benefit', and review the assignment.

FAQ

Can this tool solve rectangular (unbalanced) assignment problems?

Yes, rectangular matrices with unequal rows and columns are automatically padded with zero-cost dummy entries to create a balanced square matrix.

How does the solver handle benefit maximization?

For maximization, the solver internally negates the matrix values, applies the standard Hungarian minimization steps, and converts the resulting assignment back to original total benefit.

What is the matrix size limit?

The solver accommodates matrix dimensions between 2×2 and 8×8.

Can I input decimal numbers in the matrix?

Yes, standard integer and decimal values are supported, with precision configurable up to 8 decimal places.

What does the step-by-step breakdown include?

The output details row and column reduction minima, zero-matching rounds, minimum line cover adjustments when matching is incomplete, and the final paired assignment.

Matrix: 7, 3, 5
2, 9, 4
6, 8, 1
Goal: max
Decimal Places: 4

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Minimizing Maintenance Task Labor Cost

2. Maximizing Territory Sales Revenue

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