# Assignment Problem Solver (Hungarian Algorithm)

Textbook Hungarian algorithm for 2–8 × 2–8 assignment problems: row/column reductions, augmenting-path zero matching, König line covers, and substitution-verified optimal assignments.

> Canonical page: https://elysiatools.com/en/tools/assignment-problem-solver

- **Category:** Math & Numbers

- **Keywords:** assignment problem, hungarian algorithm, kuhn-munkres, optimal assignment, cost matrix, matching, operations research, job assignment, worker task, combinatorial optimization

## 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.

## Inputs

- **Cost matrix (one row per line)** (textarea): Cost (or benefit) matrix, one row per worker, one column per task. 2–8 rows × 2–8 columns; rectangular matrices are padded with zero-cost dummies.
- **Goal** (select)
- **Decimal Places** (number)

## 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

- Enter your 2–8 × 2–8 matrix into the text area with one row per line, separating cell values with commas or spaces.
- Select your optimization goal: minimize total cost or maximize total benefit, and configure decimal places if using fractions.
- The 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.

## Frequently asked questions

### 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.

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