Simplex Linear Programming Calculator

Introduction to simplex linear programming with c, A, and b

This simplex linear programming calculator accepts an objective row c, a constraint matrix A, and a right-hand-side vector b. It constructs a tableau with slack columns, then returns the routine’s reported decision-variable values and objective value. If it selects an entering column but cannot find an eligible leaving row, it reports the model as unbounded under the implemented pivot logic.

A useful simplex entry begins with a clear mapping from your real decisions to the positions in every row. The first coefficient in c and the first coefficient of every A row refer to the same variable. The second positions refer to the next variable, and so on. That ordering is more important than the labels you may use in your own planning model.

The page is designed for small educational models in maximum form. It helps you organize comma-separated coefficients and inspect the resulting values, but it is not a replacement for a reviewed optimization package. In particular, the current JavaScript uses a pivot-sign convention that requires careful interpretation, as explained with the formula and worked example below.

Which simplex linear programming model does this calculator accept?

The simplex form on this page represents a maximization problem with non-negative decision variables and constraints of the form Ax ≤ b. Each row of A supplies the coefficients for one constraint, and the matching entry of b supplies that row’s limit. The form rejects a negative entry in b, because its initial slack-variable tableau assumes non-negative bounds.

For a resource-allocation interpretation, an entry of c is the objective contribution per unit of a decision variable, an entry of A is that variable’s use of a particular resource, and an entry of b is the available amount of that resource. The calculator does not ask what the units are, so you must make the units compatible within each row. For example, coefficients measured in machine-hours per unit should be paired with a machine-hour capacity.

How to use the simplex calculator fields

To use this simplex calculator, first decide what x₁, x₂, and any later variables mean. Keep that order unchanged while entering the objective and every constraint. Then enter the data as follows:

1. Enter c (comma separated) as one row of objective coefficients in decision-variable order.
2. Enter A rows separated by newline with one comma-separated constraint row on each line. Every row must contain the same number of values as c.
3. Enter b values with one bound per line, in the same order as the rows of A.
4. Select Solve to populate the result panel with the routine’s variable values and objective value.
5. Substitute the displayed values back into the constraints and review the sign-convention caution before relying on the result.

Spaces and commas can separate values within an A row, but line breaks define separate constraints. A missing coefficient is not treated as an implied zero. It shortens the row and triggers a validation message. When revising a model, change one coefficient group at a time and retain a copy of the exact inputs, because the result panel does not save a pivot history.

Choosing coefficients for the simplex objective and constraints

Simplex coefficients must describe one consistent variable order across the entire model. A number in the first position of the objective row is not interchangeable with a number in the first position of a constraint row: the objective coefficient describes the value contributed by x₁, while the constraint coefficient describes how much of that row’s resource x₁ consumes.

• Variable order: use the same x₁, x₂, and later-variable order in c and every row of A.
• Constraint direction: enter only rows that genuinely express a less-than-or-equal-to limit. This form does not convert equality or greater-than-or-equal-to constraints.
• Bounds: supply exactly one non-negative b value for each constraint row.
• Units: make every term in a constraint compatible with its bound.
• Model scope: the inputs do not impose integer, binary, minimum-use, or fixed-charge requirements, so returned x values may be fractional.

The objective row c, matrix A, and bounds b therefore have distinct jobs. Their raw values should not be added together. Objective contributions, resource-use rates, and capacities may have different meanings and units even though they all appear as numbers in the tableau.

Formulas for the simplex model and tableau convention

The core simplex formula on this page maximizes the objective row multiplied by the decision vector, subject to the entered matrix limits and non-negativity. The script adds one slack variable for each row of A, placing the corresponding b value in the tableau’s final column.

Here, c is the objective coefficient vector, x is the decision vector, A is the matrix whose rows define resource limits, and b is the vector of those limits. A slack variable turns each entered ≤ constraint into an equality for the initial tableau. The displayed objective value is read from the final entry of the tableau’s objective row, while each displayed x value is read from a column that the routine identifies as a unit column.

There is an important implementation-specific caution. The script stores −c in the objective row, but its pivot loop selects a column only when that row contains a positive value. Consequently, an ordinary positive objective row such as the prefilled c = 3, 5 does not trigger a pivot in the current implementation. The routine then stops at its initial slack solution. This behavior is a property of this page’s JavaScript, not a general simplex-method rule, so independently verify any result before describing it as an optimum.

Worked example: the prefilled two-variable simplex model

This worked example uses the prefilled simplex data. The objective row c is 3, 5. The three rows of A are 1, 0; 0, 2; and 3, 2. Their matching b values are 4, 12, and 18. In conventional notation, these inputs seek to maximize 3x₁ + 5x₂ while enforcing x₁ ≤ 4, 2x₂ ≤ 12, and 3x₁ + 2x₂ ≤ 18, with both variables non-negative.

With those exact positive objective coefficients, the current JavaScript initializes its objective row as −3, −5 and searches for a positive entry. It finds none, so this page reports x₁ = 0.0000, x₂ = 0.0000, and an objective value of 0.0000. That is the routine’s actual reported output for the defaults. It demonstrates the implementation’s sign issue and should not be mistaken for the conventional hand-calculated optimum.

To audit a run, substitute every reported x value into each constraint. For the reported zero point, all three left-hand sides are zero, so the point is feasible, but feasibility alone does not establish optimality. Next, calculate the intended objective directly from c and the reported variables. This two-part check helps distinguish a valid feasible point from a correctly optimized solution.

Comparing simplex coefficient changes meaningfully

For this simplex calculator, sensitivity means observing changes in the reported decision variables, objective value, and resource use. It does not mean adding all coefficients and bounds into one scenario total. Those quantities serve different roles and may use different units, so such a sum would have no useful optimization interpretation.

When changing an objective coefficient in c, keep A and b fixed and compare separate submissions. When changing a resource bound in b, recompute the affected row’s left-hand side and compare it with the revised capacity. Changing an A coefficient alters the shape of the feasible region, so it deserves a complete row-by-row feasibility check.

Positive entries in c require special care on this page because of the pivot-sign behavior described above. Different positive objective weights may leave the reported initial zero solution unchanged. That does not prove the real planning problem is insensitive to those weights.

Interpreting the reported simplex result

The result panel lists one value for each variable implied by the number of entries in c, followed by an objective value. If the routine chooses an entering column but cannot choose a leaving row, it displays an unbounded-status message. Input errors, including mismatched row lengths, non-numeric entries, missing bounds, and negative bounds, appear in the same live result area.

A reported zero means the routine left that decision variable at zero in its final tableau reading; it does not prove the variable is irrelevant to the real model. Likewise, an unbounded message is specific to the entered coefficients and the implemented pivot logic. Review the relevant constraint column to determine whether the intended model truly lacks a limiting row.

Keep all three input groups with any result you record. Copying only an objective value is insufficient to reproduce a linear-programming run, because a different objective vector, constraint row, variable order, or bound changes the model.

Limitations of this simplex linear programming page

The limitations of this simplex linear programming calculator matter whenever the output will inform a real decision. This is a compact educational tableau routine rather than a full modeling environment:

• Constraint form: it handles entered ≤ rows with non-negative right-hand-side values and does not transform other constraint types.
• Non-negativity: the displayed model requires x ≥ 0. Variables that may be negative need a different formulation.
• Continuous values: the routine has no integer or binary restrictions, even when a real decision represents a count or yes/no choice.
• Linearity: objective contributions and resource use are assumed to change linearly. Thresholds, nonlinear discounts, and saturation effects are outside the model.
• Pivot convention: because the code stores −c while searching for a positive objective-row entry, conventional positive maximization coefficients can terminate at the initial solution.
• Display precision: the result table rounds values to four decimal places.

Use this page to organize and inspect a small c, A, b setup, then validate consequential operational, financial, legal, medical, or safety decisions with an appropriate reviewed model and trusted solver. Transparent variable definitions, compatible units, and explicit constraint meanings are essential before any linear-programming output can support a decision.