Network Simplex Method Calculator
Network simplex method calculator for minimum-cost flow problems. Learn how the network simplex uses a spanning-tree basis and node potentials.
Simplex Calculator
How Simplex Method Calculator Works
Enter the LP Problem
Type the objective function coefficients and every constraint row with its right-hand-side value.
Choose Maximize or Minimize
Pick your optimization goal. The tool builds the initial tableau with slack variables automatically.
Run the Pivot Iterations
The calculator identifies pivot column by Cj-Zj, computes ratios, performs elementary row operations until optimal.
Read the Optimal Solution
Final tableau displays optimal variable values, Zj row, and the maximum/minimum objective value.
Sample Simplex Tableau Output
Example tableau iteration for a 2-variable maximization problem
| Basis | x1 | x2 | s1 | s2 | RHS | Cj-Zj |
|---|---|---|---|---|---|---|
| x1 | 14 | 0 | 0 | 1 | 14 | 0 |
| x2 | 7 | 1 | 0 | 0 | 7 | 5 |
| Zj | 35 | 5 | 0 | 0 | 35 |
Network Flow as Linear Programming
Network problems - such as minimum-cost flow, shortest path, and assignment - can be written as linear programs with flow-balance constraints at each node. The network simplex method is a specialized, efficient version of the simplex algorithm for these problems. This calculator solves the linear-programming formulation with the standard simplex method and shows each tableau.
When to Use It
Use it to minimize the total cost of sending flow through a network subject to capacity and balance constraints. Enter the cost objective and the node/arc constraints to get the optimal flow and total cost with full tableau steps.
How the Network Simplex Method Works
The network simplex method is a specialized version of the simplex method for minimum-cost flow problems. Each basic solution corresponds to a spanning tree of the network. Node potentials (the dual values) are computed from the tree arcs, and the reduced cost of each non-tree arc is its cost minus the difference in potentials between its endpoints.
Pivoting on a Network
An arc with a negative reduced cost enters the tree and creates exactly one cycle. Flow is pushed around that cycle until one arc reaches its bound; that arc leaves the tree. Because every pivot uses only additions and subtractions on the graph, the network simplex method is much faster than the general simplex method for transportation, assignment and shortest-path style problems.
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Frequently Asked Questions
What is the network simplex method?
It is a specialized version of the simplex algorithm designed for network flow problems, exploiting the network structure for greater efficiency.
What problems does it solve?
Minimum-cost flow, transportation, assignment, and shortest-path problems can all be formulated and solved as network linear programs.
How is a network problem written as an LP?
Each arc has a flow variable and a cost; each node has a flow-balance constraint requiring inflow to equal outflow plus supply or demand.
Is it faster than the standard simplex method?
For network-structured problems the network simplex method is typically much faster because it uses spanning-tree bases instead of a full tableau.
Does this tool show the steps?
Yes, it solves the linear-programming formulation and displays each simplex tableau to the optimal solution.