2-Phase Simplex Method Calculator
2-Phase simplex method calculator for LP problems with artificial variables. Solve Phase 1 and Phase 2 automatically.
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 |
How the Two-Phase Method Works
The 2-phase simplex method solves linear programs that contain ≥ or = constraints and therefore need artificial variables. Phase 1 minimizes the sum of the artificial variables to find a feasible basic solution. If that minimum is zero, Phase 2 drops the artificials and optimizes the original objective using the standard simplex iterations.
Two-Phase vs Big M
Both the two-phase method and the Big M method handle artificial variables and reach the same optimum. The two-phase approach avoids the large penalty constant M, which keeps the arithmetic clean and avoids numerical issues - a common reason instructors prefer it for hand calculations.
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Frequently Asked Questions
When to use the 2-phase simplex method?
The 2-phase simplex method is used when the linear programming problem contains constraints with 'greater than or equal to' (≥) or 'equal to' (=) signs, requiring artificial variables to find an initial basic feasible solution.
How does the 2-phase simplex calculator work?
In Phase 1, the calculator minimizes the sum of artificial variables to find a feasible basis. If the minimum is zero, Phase 2 begins, dropping artificials and optimizing the original objective function.
When is the two-phase method needed?
It is needed when a linear program has greater-than or equal constraints, which require artificial variables to start the simplex algorithm.
What happens in Phase 1?
Phase 1 minimizes the sum of the artificial variables. If that minimum is zero, a feasible solution exists and Phase 2 begins; if it is positive, the problem is infeasible.
Is the answer the same as the Big M method?
Yes, both methods reach the same optimal solution. The two-phase method just avoids the large penalty constant M.