Primal vs Dual Simplex Method: What's the Difference?

The primal and dual simplex methods solve the same linear program but move toward the optimum from opposite directions.

The Primal Simplex Method

The ordinary (primal) simplex method starts with a feasible solution that is not yet optimal and improves the objective at each pivot until optimality is reached. It keeps every right-hand-side value non-negative (feasible) while driving the Cj minus Zj values toward optimality.

The Dual Simplex Method

The dual simplex method starts with a solution that is optimal but infeasible (the objective row already satisfies the optimality test, but some right-hand-side values are negative). It restores feasibility by choosing the leaving variable first (the most negative RHS) and then the entering variable with a dual ratio test.

Side-by-Side

	Primal Simplex	Dual Simplex
Starts	Feasible, not optimal	Optimal, not feasible
Works toward	Optimality	Feasibility
Picks first	Entering variable	Leaving variable
Best when	Standard problems	Adding constraints to a solved problem

When to Use the Dual Simplex

The dual simplex method is especially useful when you add a new constraint to an already-optimal problem: the old solution stays optimal but may become infeasible, and the dual simplex restores feasibility quickly without restarting. For standard problems, the primal simplex calculator is the usual choice.