Dual Simplex Method Calculator
Dual simplex method calculator: solve LP problems with negative right-hand sides. Shows the leaving row, dual ratio test and every tableau until feasible.
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 |
Starting From Infeasibility
The dual simplex method begins with a tableau that is optimal with respect to the objective row but infeasible (some right-hand-side values are negative). It first chooses the leaving variable (most negative RHS), then the entering variable using a dual ratio test, restoring feasibility while keeping optimality. This is ideal when new constraints are added to an already-solved problem.
Primal vs Dual Simplex
The ordinary (primal) simplex keeps the solution feasible and works toward optimality; the dual simplex keeps it optimal and works toward feasibility. Knowing both lets you pick the most efficient path for a given linear program.
Worked Example: Dual Simplex Method
Minimize Z = 2x1 + 3x2 subject to x1 + x2 >= 4 and x1 + 3x2 >= 6. Multiply each constraint by -1 and add slack variables: -x1 - x2 + s1 = -4 and -x1 - 3x2 + s2 = -6. The starting basis s1 = -4, s2 = -6 is infeasible but already optimal for the objective, which is exactly where the dual simplex method starts.
Iteration 1
Choose the leaving row with the most negative right-hand side: s2 = -6. Apply the dual ratio test to the negative entries in that row: |2 / -1| = 2 for x1 and |3 / -3| = 1 for x2. The smallest ratio is 1, so x2 enters. After pivoting, x2 = 2 and the s1 row becomes -2/3 x1 + s1 - 1/3 s2 = -2, which is still infeasible.
Iteration 2 and the Optimal Solution
Now s1 = -2 leaves. The ratios are 1 / (2/3) = 1.5 for x1 and 1 / (1/3) = 3 for s2, so x1 enters. After this pivot all right-hand sides are non-negative: x1 = 3, x2 = 1 and the minimum Z = 9. The solution is both feasible and optimal, so the dual simplex method stops.
Dual Simplex vs Solving the Dual Problem
The dual simplex method is not the same as writing the dual LP and solving it with the ordinary simplex method, although both give the same optimal objective value. The dual simplex works directly on the primal tableau, which makes it the standard tool for sensitivity analysis and for re-solving a problem after a new constraint is added.
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Frequently Asked Questions
What is the dual simplex method?
The dual simplex method is a variant of the simplex algorithm that maintains dual feasibility (optimality condition) while working towards primal feasibility. It is useful when a basic solution is optimal but infeasible.
How to use the dual simplex calculator?
Input your LP problem. The calculator starts with a dual-feasible basis and performs pivot operations to remove primal infeasibilities until an optimal and feasible solution is reached.
How is the dual simplex different from the primal?
The primal simplex keeps the solution feasible and works toward optimality, while the dual simplex keeps it optimal and works toward feasibility.
When should I use the dual simplex method?
It is efficient when you add new constraints to an already-optimal problem, since the tableau stays optimal but may become infeasible.
How does it pick the leaving variable?
It selects the row with the most negative right-hand-side value first, then uses a ratio test on the negative entries to choose the entering variable.