Transportation Problem Solver (Min-Cost Flow)

Transportation Problem Solver (Min-Cost Flow) | 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, text, number
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Transportation Problem Solver calculates the minimum-cost shipping plan for balanced supply-and-demand networks using successive shortest path augmentations and the MODI (u-v) method. It verifies global optimality by generating reduced costs, dual variable potentials, and an itemized shipment allocation matrix.

When to use

Determining optimal distribution allocations from multiple production sources to regional demand points.
Checking manual operations research homework solutions involving the MODI method, stepping-stone method, or u-v dual variables.
Validating minimum-cost flow calculations and inspecting reduced-cost optimality certificates.

How it works

1Enter the unit cost matrix with one row per supply source (2–8 rows) and comma-separated values for destination columns (2–8 columns).
2Specify the non-negative available capacities in the Supply field and required quantities in the Demand field, ensuring both sums are equal.
3Set the desired decimal precision for flow allocations and dual values, then compute the solution.
4Review step-by-step min-cost flow augmentations, the complete shipping allocation grid, and the MODI dual variables (u, v) certifying optimality.

Use cases

Multi-warehouse logistics route planning to minimize total freight expenses across distribution centers.
Operations research coursework verification for linear programming transport problems and MODI optimality tests.
Supply chain allocation modeling between regional manufacturing plants and retail fulfillment hubs.

Examples

1. 3-Plant to 4-Warehouse Distribution

Logistics Analyst

Background

A manufacturing company operates three production plants with capacities 30, 40, and 30 units, delivering to four regional warehouses requiring 20, 30, 30, and 20 units.

Problem

Determine the lowest-cost shipping schedule across 12 candidate routes with variable unit transit rates.

How to use

Input the 3x4 rate matrix into Costs, set Supply to '30, 40, 30', Demand to '20, 30, 30, 20', and leave Decimal Places at 4.

Costs:
2, 3, 1, 4
4, 2, 5, 3
3, 1, 4, 2
Supply: 30, 40, 30
Demand: 20, 30, 30, 20

Outcome

Finds the global minimum total freight cost of 200 with an exact routing plan and MODI dual potentials confirming optimality.

2. 2x2 Dual Source Allocation

Operations Research Student

Background

A student needs to solve a 2-origin, 2-destination transportation tableau and extract the u and v dual potentials.

Problem

Verify whether a greedy initial allocation achieves the optimal minimum cost of shipment.

How to use

Enter the 2x2 cost rows into Costs, input '10, 20' into Supply, and enter '15, 15' into Demand.

FAQ

What should I do if total supply does not equal total demand?

Add a dummy supply source or dummy demand destination with zero (or penalty) unit shipping costs to balance total supply and total demand before running the solver.

What matrix sizes are supported by the solver?

The tool supports matrices ranging from 2 to 8 supply rows by 2 to 8 destination columns.

What algorithm does this tool use to find the optimal shipping plan?

It models the network as a minimum-cost flow problem, using successive shortest path augmentations with SPFA and computing MODI dual variables (u_i, v_j) for optimality verification.

How do MODI dual variables certify optimality?

An allocation is optimal when the reduced cost (cost minus u_i minus v_j) is zero on all basic routed cells and non-negative on all non-basic cells.

Can I use decimal values for costs, supply, and demand?

Yes, numerical values with decimals are supported, and the decimal precision option adjusts output formatting accordingly.

Costs:
4, 6
8, 3
Supply: 10, 20
Demand: 15, 15

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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. 3-Plant to 4-Warehouse Distribution

2. 2x2 Dual Source Allocation

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