Assignment Problem Solver (Hungarian Algorithm)

Assignment Problem Solver (Hungarian Algorithm) – Step-by-step
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Assignment Problem Solver (Hungarian Algorithm)

Solve worker–task assignments to minimize cost or maximize profit. Includes Hungarian method steps, bipartite matching graph, share + embed, and worked examples.

Calculator type

Switching type updates the URL and preserves your current matrix via the share payload (?wt=…).

Quick intro

The assignment problem is a one-to-one matching: each worker gets exactly one task and each task gets exactly one worker. If your matrix is rectangular (unbalanced), the solver automatically adds dummy rows/columns. Learn unbalanced cases here: unbalanced assignment problem.

Key terms on this page

Short lessons that explain the ideas used by this solver (and keep you inside the CalcTypes learning cluster).

Inputs

Hungarian shows step tables. Brute force verifies small n. Greedy is a baseline and may be suboptimal. See Hungarian algorithm explained.

Cost / profit matrix

Each cell is Worker i → Task j. Mark Forbidden for impossible pairs (see Big M method).
Objective: — Method: — Shape: — Balance: — Status: —

Results

Solution status
—
Total
—
Assignments (real)
—
Notes
—

Enter your matrix and click Solve.

Solution walkthrough (very detailed, current problem)
Solve a problem to generate a detailed explanation here.
Want more practice? Assignment problem examples. For automation: Hungarian algorithm in Python.

Format: ?wt=…&autocalc=1

Embed this calculator

Embedding uses the same wt payload.

Graph (bipartite matching)

Visualizing the assignment as a bipartite graph helps explain why this is a matching problem. Learn more: bipartite matching & assignment.

Solve a matrix to render the graph.
Hungarian method step tables (optional)

Step-by-step tables are available only for Hungarian method. Learn: Hungarian algorithm explained.

Steps will appear after you solve using Hungarian method.

On this page

Try this next

Continue learning inside the assignment-problem calculator cluster:

Maximization Assignment Calculator Profit/output version (maximize) Unbalanced Assignment Calculator Rectangular matrices + dummy interpretation Restricted / Prohibited Assignment Forbidden pairs + feasibility diagnostics

Related reading: assignment vs transportation · unbalanced assignment · Big‑M method

Worked examples (beginner → intermediate → hard → extreme)

Each example includes a “Load this example” button so you can see how the solver behaves. Try different methods (Hungarian vs brute force vs greedy).

Beginner

B1 — Minimize cost (3×3): classic starter 
W1:  9  2  7
W2:  3  6  8
W3:  5  8  4
B2 — Maximize profit (3×3): shows max mode 
D: 12  8 15
E:  7 14  9
F: 10 11 13
B3 — Greedy trap (4×4): why local choices can fail

Intermediate

I1 — Rectangular / unbalanced (4×3): dummy column added

Learn: unbalanced assignment problem.

I2 — Forbidden assignments (4×4): infeasibility risk
I3 — Job scheduling style (5×5): practical scenario matrix

Learn modeling: job scheduling using assignment model.

Hard

H1 — Multiple optimal solutions (ties)
H2 — Negative costs (net savings)

Extreme (still browser-safe)

X1 — 8×8: max size for brute force
X2 — QAP concept demo (facility layout interaction costs)

In Quadratic Assignment Problem (QAP), the total cost depends on pairs of assignments. This calculator solves the linear assignment problem. 

Linear placement costs (Facility → Location)
F1:  9  2  7  8
F2:  6  4  3  7
F3:  5  8  1  8
F4:  7  6  9  4

Learn: quadratic assignment problem (QAP).

More solved problems: assignment problem examples.

Questions people ask about the assignment problem solver

Fundamentals

What is the assignment problem in operations research?

It’s a one-to-one matching problem where you assign n workers to n tasks to minimize total cost (or maximize profit). Each worker and task must be used exactly once (after balancing).

What is an assignment problem solver?

It’s a calculator that returns the optimal worker→task mapping from your matrix, plus the optimized total. This page can also show Hungarian step tables and a bipartite graph visualization.

Is the assignment problem a special case of linear programming?

Yes. It can be formulated as a 0–1 ILP. For broader context, see: linear programming.

Hungarian method

What are the steps of the Hungarian algorithm?

Row reduction → column reduction → cover zeros with minimum lines → if lines < n, adjust by θ and repeat → pick independent zeros to read the assignment. Learn: Hungarian algorithm explained.

When do you stop iterating in the Hungarian algorithm?

Stop when the minimum number of covering lines equals n. Then a perfect matching exists among zeros.

Does this Hungarian algorithm calculator show step-by-step solution?

Yes—solve using Hungarian method and open the “step tables” section to see each reduction and iteration.

Unbalanced / rectangular matrices

Can this solver handle rectangular assignment problems?

Yes. It adds dummy rows/columns to make the matrix square. Learn: unbalanced assignment problem.

Related comparisons

Assignment problem vs transportation problem—what’s the difference?

Transportation moves quantities; assignment is one-to-one. See: assignment vs transportation problem.

How does this relate to bipartite matching?

Assignment is a weighted bipartite matching problem. Learn: bipartite matching & assignment.

Methodology (how this calculator works)

This calculator solves the linear assignment problem (one-to-one matching) to minimize cost or maximize profit. The Hungarian steps are explained in: Hungarian algorithm explained.

If your matrix is rectangular, it becomes an unbalanced assignment problem, and the solver adds dummy rows/columns automatically.

More learning: solved examples · bipartite matching · Python implementation.

IMPORTANT DISCLAIMER (Read before using): This Assignment Problem Solver runs in your browser and is intended for learning, demonstrations, and planning. Real-world allocation often requires constraints beyond a basic matrix (skills, rules, time windows, fairness, priorities). For advanced constraint modeling you may need ILP/MILP techniques such as branch and bound, careful integrality tolerance, and cut methods like Gomory cuts. Do not use this output as the sole basis for high-stakes decisions. See our Disclaimer.
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