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

The primal simplex method and the dual simplex method solve the same linear programs and stop at the same optimal tableau. The difference is the direction they approach it from. The primal method keeps the solution feasible and works towards optimality; the dual method keeps the solution optimal and works towards feasibility. Knowing which one to use saves iterations and, in many problems, removes the need for artificial variables altogether.

Two Conditions Every Final Tableau Must Meet

A simplex tableau is solved when two conditions hold at the same time:

• Primal feasibility: every right-hand-side (RHS) value is ≥ 0, so the basic solution satisfies all constraints.
• Optimality (dual feasibility): no Cj − Zj value can improve the objective. For maximization, all Cj − Zj ≤ 0; for minimization, all ≥ 0.

The two methods differ only in which condition they protect while they work on the other.

The Primal Simplex Method

The ordinary simplex method starts from a feasible corner, usually the origin with slack variables as the basis, and moves to better neighbouring corners.

1. Choose the entering variable from the Cj − Zj row.
2. Choose the leaving variable with the minimum ratio test, which keeps every RHS ≥ 0.
3. Pivot and repeat until the optimality condition holds.

The RHS column never becomes negative; the Cj − Zj row improves until it passes the optimality test. A complete example is in how to solve an LPP by the simplex method.

The Dual Simplex Method

The dual simplex method starts from a tableau that already passes the optimality test but has one or more negative RHS values.

1. Choose the leaving variable: the row with the most negative RHS.
2. Choose the entering variable with the dual ratio test: among the negative entries of that row, the smallest |(Cj − Zj) ÷ entry|.
3. Pivot and repeat until every RHS is ≥ 0.

The Cj − Zj row keeps passing the optimality test throughout; the RHS column improves until it is feasible. A complete example with every tableau is in the dual simplex method guide.

Side-by-Side Comparison

	Primal simplex	Dual simplex
Starting tableau	Feasible, not yet optimal	Optimal, not yet feasible
Maintains	Feasibility (RHS ≥ 0)	Optimality (Cj − Zj test)
Works towards	Optimality	Feasibility
Chooses first	Entering variable	Leaving variable
Ratio test	RHS ÷ positive pivot-column entries	Cj − Zj ÷ negative pivot-row entries
Stops when	Cj − Zj passes the test	All RHS ≥ 0
Typical use	Standard maximization problems	Minimization with ≥ constraints; adding a constraint to a solved problem

The Same Problem Both Ways

Consider: minimize Z = 2x1 + x2 subject to 3x1 + x2 ≥ 3, 4x1 + 3x2 ≥ 6, x1 + 2x2 ≤ 3.

Primal approach (Big M). The two ≥ constraints each need a surplus and an artificial variable, giving a tableau with seven columns (x1, x2, two surplus, two artificial and one slack). It takes two pivots, with an M term in many Cj − Zj entries, to reach the optimum.

Dual approach. Multiply the two ≥ constraints by −1 so that every constraint is ≤ with its own slack variable. The tableau has only five columns, starts optimal because the costs 2 and 1 are non-negative, and reaches feasibility in two pivots with no M at all.

Both reach the same answer: x1 = 3/5, x2 = 6/5, minimum Z = 12/5. On this problem the number of pivots is the same, but the dual approach uses a smaller tableau and simpler numbers.

When to Use Each Method

Use the primal simplex method when:

• The problem is a maximization with ≤ constraints and non-negative RHS values, so slack variables give a feasible start.
• You are solving a problem from scratch and the textbook expects standard tableaus.

Use the dual simplex method when:

• The problem is a minimization with ≥ constraints and non-negative costs. Flipping the constraints gives an optimal but infeasible start with no artificial variables.
• You add a new constraint to a problem you have already solved and the old optimum violates it. The tableau stays optimal but becomes infeasible, and a few dual pivots repair it.
• A right-hand-side value changes during sensitivity analysis and a basic variable becomes negative.

Use neither directly when the starting tableau is both infeasible and non-optimal. Then use the Big M method or the two phase method, or first fix one of the two conditions.

Primal and Dual Problems vs Primal and Dual Methods

The words primal and dual have two related meanings. The dual problem is a separate linear program built from the primal problem’s data, explained in duality in linear programming. The dual simplex method is an algorithm that works on the primal tableau. The connection is that the dual simplex method keeps the dual problem feasible at every step, which is where its name comes from.

Common Mistakes

• Using the dual simplex method on a non-optimal tableau. Its rules only work when the Cj − Zj row already passes the optimality test.
• Choosing the entering variable first in the dual method. The leaving row always comes first.
• Using positive entries in the dual ratio test. Only negative entries in the leaving row qualify.
• Concluding that a problem is infeasible too early. In the dual method, infeasibility is proved only when the leaving row has no negative entries.

Check Your Answer

Both methods end at the same optimum, so any simplex solver can confirm your result. Enter the problem in the dual simplex method calculator, which solves it with the standard simplex method and Big M, to check the final values. For standard maximization problems, the simplex calculator with steps shows every primal tableau.