Simplex Method Calculator (Maximization)
Solve linear programming models of the form: maximize Z = c₁x₁ + c₂x₂ + ... + cₙxₙ, subject to Ax ≤ b, b ≥ 0, and x ≥ 0.
What this simplex online calculator does
This tool solves a classic linear programming problem using the primal simplex algorithm. It is designed for quick optimization tasks where you want to maximize an objective function subject to linear constraints. Common examples include profit maximization, production planning, budget allocation, staffing, transportation, and resource optimization.
You choose how many decision variables and constraints you need, enter coefficients, and click calculate. The solver returns the optimal objective value, best variable values, slack variable values, and a readable sequence of simplex pivot steps.
Input format and assumptions
Standard form used by this calculator
- Objective: maximize Z = cᵀx
- Constraints: Ax ≤ b
- Right-hand side values: b ≥ 0
- Decision variables: x ≥ 0
If your model contains equalities, ≥ constraints, or negative right-hand sides, convert it to standard form before solving. This keeps the calculator lightweight and fast while still supporting many practical optimization models.
How to use the calculator step by step
- Set the number of variables and constraints.
- Click Generate Model to create the input tables.
- Enter objective coefficients (the c-values).
- Enter each constraint row (A matrix) and RHS values (b).
- Press Calculate Optimum to run simplex iterations.
- Read the optimal plan and objective value in the result box.
Worked example
Click Load Example and the calculator fills:
- Maximize: Z = 3x₁ + 5x₂
- Subject to:
- 2x₁ + 3x₂ ≤ 8
- 2x₁ + x₂ ≤ 4
- x₁ ≤ 2
The simplex routine finds the best feasible corner point and reports the optimal values. You can then compare alternative models by changing coefficients and rerunning.
Understanding the output
Objective value
This is the maximum achievable value of your objective function under all listed constraints.
Decision variables
These are the recommended values for x₁ through xₙ. If a variable is zero, it means the optimal plan does not need that activity.
Slack variables
Slack indicates unused capacity in each ≤ constraint. A zero slack means a binding constraint, which often identifies the key bottlenecks in your system.
Tips for reliable simplex models
- Keep units consistent across coefficients (hours, dollars, kilograms, etc.).
- Double-check the sign and direction of each inequality.
- Ensure RHS values are nonnegative for this standard implementation.
- Start with small models to validate logic before scaling up.
- Use slack/binding information to identify operational constraints.
Final note
The simplex method remains one of the most important algorithms in operations research, management science, and analytics. This online calculator gives you a practical interface for experimentation, teaching, and real-world optimization tasks without needing specialized software.