Simplex Method Tableau Calculator
Simplex tableau calculator: builds the initial tableau, computes Zj and Cj-Zj, runs the minimum ratio test and pivots to the optimal simplex table.
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 |
Building the Simplex Tableau
The simplex tableau organizes the objective coefficients, constraint rows, slack and artificial variables, and the right-hand-side column. This tableau calculator computes the Zj row (sum of Cb × column) and the Cj − Zj row used to select the entering variable, then updates the table after each pivot so you can follow standard operations research notation.
Initial and Final Simplex Tableau
The initial simplex tableau is built from the standard form of the problem, with slack (and, if needed, surplus and artificial) variables forming the starting basis. Each iteration produces a new tableau, and the final simplex tableau is the one where the optimality condition holds. The calculator shows the initial tableau, every intermediate tableau and the final optimal tableau, so you can copy any of them into your solution.
Cj, Zj and Cj − Zj
Cj are the objective coefficients, Zj measures the contribution lost by bringing a variable into the basis, and Cj − Zj is the net improvement. The optimal tableau is reached when no Cj − Zj is positive (for maximization). Every value is shown for each iteration.
Parts of a Simplex Tableau
A simplex tableau (simplex table) lists the basic variables (CB column), the coefficients of every variable in each constraint, the right-hand side (RHS) column, the Zj row and the Cj - Zj row. Zj for a column is the sum of each CB value multiplied by that column's entry, and Cj - Zj tells you how much the objective would change if that variable entered the basis.
Reading the Tableau Step by Step
Pick the pivot column from the Cj - Zj row (most positive for maximization), then run the minimum ratio test RHS / pivot-column entry using only positive entries to pick the pivot row. The pivot element is where they cross. Divide the pivot row by the pivot element and use row operations to make every other entry in that column zero. The calculator highlights each of these steps.
When Is a Tableau Optimal?
For maximization, the simplex tableau is optimal when all Cj - Zj values are less than or equal to zero; for minimization, when all are greater than or equal to zero. A zero Cj - Zj for a non-basic variable in the optimal tableau means there are alternative optimal solutions.
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Frequently Asked Questions
How to calculate Zj in simplex method?
Zj = sum of (Cbi × aij) for each column j, where Cbi is the objective coefficient of the current basic variable in row i, and aij is the corresponding tableau element in that column.
What is a simplex tableau?
A simplex tableau is a tabular representation of the linear programming problem used to perform the simplex algorithm iterations. It contains the objective function row, constraint coefficients, and the right-hand side values.
What does the Zj row mean?
Zj measures the amount of objective value lost by bringing a variable into the basis; it is the sum of the basic-variable coefficients times the column entries.
How do I read Cj-Zj?
Cj-Zj is the net change in the objective if a variable enters the basis. A positive value (for maximization) means the solution can still be improved.
When is the tableau optimal?
For maximization, the tableau is optimal when no Cj-Zj value is positive.