Last updated:  |  View sources  |  Methodology
Jump to Calculator

Transportation Problem Calculator with Steps (up to 21×21)

Transportation Problem Calculator

Solve with NW corner, least cost, row/column minima, VAM, Russell, and optional optimization (MODI or Stepping Stone). For large matrices, Results and Step tables are rendered only when you expand/click.

Inputs

Beginner mode adds formulas, VAM penalty tables, MODI/Stepping Stone loop breakdown, and final verification checks.

Bulk paste / import (recommended for large matrices)

Costs: rows separated by new lines; values separated by commas/spaces/tabs. Supply and demand: comma/space separated.

Total supply: — Total demand: — Difference: — Status: —

Results

Minimum total cost
—
Problem status
—
Dimensions (after balancing)
—
Optimization iterations
—
Planning / educational estimate only. This tool solves a classic transportation-cost model under simplified assumptions (divisible shipments; nonnegative costs; supply/demand met exactly after auto-balancing). Real-world logistics may require additional constraints (route capacities, minimum loads, fixed charges, lead times, and feasibility rules). Do not use this output as a guarantee, contract term, or compliance decision. Confirm inputs and decisions with a qualified professional. See our full Disclaimer .

Enter your costs + supply + demand, then click Solve. (Or click Load example.)

Export full report

Export a complete worked solution (copy/paste friendly).

Choose Word-friendly text to paste into Word (no technical skills required). Use HTML for rich formatting.

Embed this calculator

Add this free standards-based calculator to your site — no signup required.

Clicking the button copies an iframe + attribution link (required).

Calculation steps (detailed; tables lazy-load for large matrices)

Steps will appear after you solve. For large matrices, tables are generated only when expanded/clicked.

Understanding the results

How to Read the Solution (Allocation Table)

Transportation problem solution showing allocation quantities and costs in a matrix format
Example final allocation table. Allocated cells show shipped quantities (and optional cost contributions), while unallocated routes are shown as dashes. The total cost is the sum of all (quantity × unit cost).

Understanding the Transportation Matrix

Transportation problem matrix showing 3 sources, 3 destinations, supply, demand, and cost values
Example transportation cost matrix (3×3). Each cell shows the unit shipping cost from a source (S1–S3) to a destination (D1–D3), with supply and demand totals on the margins.
  • Allocation matrix shows how much to ship from each source to each destination (— means 0 shipment).
  • Minimum total cost is computed as ΣΣ(cᵢⱼ × xᵢⱼ) using the balanced matrix used by the solver.
  • If the problem is unbalanced, a dummy source/destination is added to balance totals. Dummy shipments are a modeling device and may not represent real routes.
Common mistakes
  • Mixing cost units (per-mile vs per-trip vs per-unit) inside the same matrix.
  • Treating dummy allocations as real shipments rather than a balancing artifact.
  • Turning off optimization and assuming the initial method is always optimal.
Limitations

This solver addresses the classic transportation model. It does not include route capacity constraints, fixed costs, time windows, integer/lot-size restrictions, multi-commodity constraints, or real-world feasibility checks beyond supply/demand balancing. Always validate against your actual constraints.

Questions people ask

Balanced vs unbalanced, initial methods, MODI vs Stepping Stone, degeneracy, and interpreting results.

What does this Transportation Problem Solver calculate?

This calculator generates a feasible allocation matrix and computes the minimum total transportation cost. It can create an initial solution (VAM, Northwest Corner, Least Cost, Row/Column Minima, Russell) and then improve/prove optimality using MODI (u–v method) or Stepping Stone. It also handles unbalanced transportation problems by automatically adding a dummy row or column.

What is the difference between a balanced and unbalanced transportation problem?

A transportation problem is balanced when total supply equals total demand. It’s unbalanced when totals differ. To solve an unbalanced problem, a dummy destination (if supply > demand) or a dummy source (if demand > supply) is added with zero costs to balance totals. Dummy allocations help the math work but typically represent “unused supply” or “unmet demand,” not real routes.

What is Vogel’s Approximation Method (VAM) and why is it recommended?

VAM is a strong method for finding a near-optimal initial feasible solution. It computes row and column penalties (difference between the two smallest costs), then allocates to the cheapest cell in the row/column with the highest penalty. VAM often produces a starting plan closer to optimal than Northwest Corner or Least Cost, reducing the number of optimization iterations needed.

What is the MODI method in the transportation problem?

MODI (Modified Distribution / u–v method) is an optimization method that tests if a feasible transportation plan is optimal. It assigns potentials uiui and vjvj to basic cells, then computes reduced costs dij=cij−(ui+vj) dij=cij−(ui+vj) for unused routes. If all reduced costs are nonnegative, the solution is optimal. If any are negative, MODI pivots along a loop to reduce total cost.

What is the Stepping Stone method and how is it different from MODI?

The Stepping Stone method checks each unused route by forming a closed loop through allocated cells and computing an opportunity cost ΔijΔij. If ΔijΔij is negative, shifting shipments along that loop reduces total cost. MODI and Stepping Stone usually reach the same optimum; MODI uses potentials and reduced costs, while Stepping Stone explains improvements directly using loop opportunity costs.

How do I know if my transportation solution is optimal?

A solution is optimal when the optimality test finds no improving move. In MODI, this means every unused route has reduced cost dij≥0dij≥0. In Stepping Stone, it means every unused route has opportunity cost Δij≥0Δij≥0. If any value is negative, the plan can be improved by pivoting shipments along a valid loop until no negatives remain.

What is degeneracy in transportation problems (and what is epsilon ε)?

Degeneracy happens when the number of basic allocations is less than m+n−1m+n−1 (sources + destinations − 1). That can prevent MODI/Stepping Stone from computing potentials or loops correctly. Solvers fix it by placing an epsilon (ε) placeholder in a carefully chosen empty cell to complete a valid basis. ε does not change totals or real shipments; it’s a bookkeeping device to keep the algorithm stable.

How is total transportation cost calculated?

Total transportation cost is computed as ∑i∑jcijxij∑i∑jcijxij: multiply each shipped quantity xijxij by its route cost cijcij, then sum across all routes. Only cells with positive shipments contribute. If the model is unbalanced and a dummy row/column is added with zero costs, dummy allocations usually don’t change total cost but do affect interpretation.

How do I paste a large cost matrix from Excel or Google Sheets?

Use the bulk paste/import section. Paste the cost matrix with rows on new lines and values separated by tabs, spaces, or commas (copy/paste from Sheets usually works because it’s tab-separated). Then paste supply (one value per source) and demand (one value per destination). If the paste fails, check for extra blank lines, merged cells, or mismatched counts versus the selected number of sources/destinations.
Sources
  1. Hamdy A. Taha, Operations Research: An Introduction (Pearson). PDF
  2. Hillier & Lieberman, Introduction to Operations Research (McGraw-Hill). PDF
  3. Firas Hamid Thanoon, Proposed Method for Solving Transportation Problems. ResearchGate
  4. Peter Malacký & Radovan Madleňák, Transportation problems and their solutions: literature review. ScienceDirect
  5. GeeksforGeeks overview (educational). Transportation problem intro
How this calculator works (methodology)

Vogel’s Approximation Method (VAM)

VAM computes penalties (difference between the two smallest costs) for each row and column, then allocates where the penalty is largest.

Vogel's Approximation Method penalty calculation showing row and column penalties
VAM penalty = (second-smallest cost − smallest cost) in each row/column. The highest penalty is selected first to reduce the risk of missing a low-cost route.

Method summary

The solver can generate an initial feasible shipping plan using several classic methods (NW corner, least cost, row/column minima, VAM, Russell), and then (optionally) improve/prove optimality using MODI (u–v) or the Stepping Stone method. 

Minimize:  Z = ΣΣ (c_ij * x_ij)

Subject to:
  For each source i:      Σ x_ij = s_i
  For each destination j: Σ x_ij = d_j
  x_ij ≥ 0

MODI reduced cost:
  d_ij = c_ij − (u_i + v_j)

Stepping Stone opportunity cost:
  Δ_ij = sum(+c) − sum(−c) along the closed loop through cell (i,j)

Note: In practice, MODI and Stepping Stone identify the same improving moves; they’re two standard ways to test/improve optimality.

Related and other calculators

These calculators are being prepared. They’ll become clickable once published.

Coming soon Related

Assignment Problem Solver

Optimize one-to-one assignments (often taught alongside transportation problems).

Related

Linear Programming Solver

Solve LP models and understand constraints, objective value, and feasibility.
Coming soon

Minimum Cost Flow Calculator

Network flow optimization for more general routing models.

Coming soon

Simplex Method Calculator

Step-by-step simplex tableaux and optimality interpretation.

Browse all calculators →