2-Phase Simplex Method Calculator
Two phase method calculator: solve LPP with >= and = constraints using the 2-phase simplex method. Shows Phase 1 artificial removal and Phase 2 tableaus.
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.
When to Use the Two Phase Method
Use the two phase simplex method whenever the problem has >= or = constraints. These constraints do not give an obvious starting basis, so artificial variables are added. Phase 1 removes the artificial variables; Phase 2 then optimizes the real objective function starting from the feasible basis that Phase 1 found.
Worked Example: Two Phase Method
Minimize Z = 4x1 + x2 subject to 3x1 + x2 = 3, 4x1 + 3x2 >= 6, x1 + 2x2 <= 4, x1, x2 >= 0. Subtract a surplus variable from the second constraint, add artificial variables a1 and a2 to the first two rows, and add a slack variable s to the third row. Phase 1 minimizes W = a1 + a2.
Phase 1 and Phase 2 Results
Phase 1 pivots until W = 0, which proves the problem is feasible and drives both artificial variables out of the basis. The artificial columns are then dropped and Phase 2 continues with the original objective. The optimal solution is x1 = 2/5 = 0.4, x2 = 9/5 = 1.8, giving the minimum Z = 17/5 = 3.4. If Phase 1 ends with W > 0, the original problem has no feasible solution.
Two Phase Method vs Big M Method
Both methods solve the same problems and reach the same answer. The two phase method avoids choosing a large penalty value M, so it is numerically more stable and is preferred in software. The Big M method combines everything into one objective and is often quicker by hand. Try both calculators on the same problem to see the difference.
Related calculators
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.