Maximization Simplex Method Calculator
Simplex maximization calculator: maximize profit or any linear objective subject to <= constraints. See each pivot and the optimal solution instantly.
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 |
Maximizing the Objective
This maximization simplex method calculator finds the largest value of a profit or objective function subject to your ≤ constraints. The algorithm selects the most positive Cj − Zj as the entering variable and uses the minimum-ratio test for the leaving variable, repeating until no positive Cj − Zj remains.
Worked Example
For Maximize Z = 3x₁ + 5x₂ with x₁ + 2x₂ ≤ 14 and x₁ + x₂ ≤ 8, the optimum is x₁ = 2, x₂ = 6 and Z = 36. Enter your own coefficients to solve any maximization linear program with full tableau steps.
Worked Example: Maximization Problem
Maximize Z = 3x1 + 2x2 + 4x3 subject to x1 + x2 + 2x3 <= 4, 2x1 + 3x3 <= 5, 2x1 + x2 + 3x3 <= 7, and x1, x2, x3 >= 0. With slack variables added, the starting solution is all decision variables at zero. The calculator finds the maximum Z = 10.5 at x1 = 2.5, x2 = 1.5, x3 = 0.
Optimality Rule for Maximization
In a maximization tableau the entering variable is the one with the largest positive Cj - Zj value, and the tableau is optimal when every Cj - Zj value is zero or negative. If a column has a positive Cj - Zj but no positive entries for the ratio test, the problem is unbounded and Z can increase without limit.
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Frequently Asked Questions
How to maximize using simplex method?
To maximize an objective function, input your coefficients and constraints. The calculator identifies the most negative Cj-Zj value as the entering variable and iterates until all Cj-Zj values are non-positive, indicating optimality.
How to find the maximum value in a simplex tableau?
The maximum objective value is found in the Zj row (or the objective function row) under the Right-Hand Side (RHS) column of the final optimal simplex tableau.
How does it choose the entering variable?
For maximization it selects the column with the most positive Cj-Zj value as the entering variable.
What constraints can I use?
You can use less-than, greater-than, or equal constraints; the calculator adds slack, surplus, or artificial variables as required.
What is the optimal value for the sample problem?
For Maximize Z equals 5x1 plus 4x2 with the sample constraints, the optimal value is Z equals 21.