Linear Programming Simplex Solver

Linear programming (LP) is a mathematical method for finding the best outcome (maximum profit or minimum cost) from a set of linear constraints. It answers questions like: 'What is the optimal way to allocate limited resources?' LP is used in production planning, scheduling, logistics, finance, and many other fields where you need to optimize a linear objective subject to linear limits.

The simplex method is an algorithm for solving linear programming problems. Developed by George Dantzig in 1947, it works by exploring the 'corners' (vertices) of the feasible region—the geometric shape defined by constraints—and moving from one corner to a better one until reaching the optimal solution. It's widely used because it's efficient, reliable, and applicable to problems with many variables and constraints.

Decision variables are the quantities you control (e.g., units to produce, hours to schedule). The objective function is a linear expression of these variables that you want to maximize (like profit) or minimize (like cost). Constraints are linear inequalities or equalities that represent limits on resources, capacities, or requirements (e.g., labor hours ≤ 80, minimum production ≥ 50).

Infeasible means no solution satisfies all constraints simultaneously. For example, if you have constraints x ≤ 10 and x ≥ 15, no value of x can satisfy both. Infeasibility usually indicates contradictory constraints or data entry errors. Review your formulation carefully—check signs, coefficients, and constraint types (≤ vs ≥) for mistakes.

Unbounded means the objective can be improved indefinitely without violating any constraints. For maximization, profit can grow to infinity; for minimization, cost can shrink to negative infinity. This almost always indicates a modeling error—typically a missing constraint that should limit the decision variables. Check if you forgot to include resource limits, capacity bounds, or non-negativity constraints.

Yes! The calculator supports both. Select 'Maximize' for profit/revenue/output problems, or 'Minimize' for cost/time/waste problems. Internally, minimization problems can be converted to maximization (or vice versa) by negating the objective, but the tool handles both natively for your convenience.

Linear programming allows variables to take any non-negative real value, including fractions. If your problem requires whole numbers (like number of employees or machines), you need integer programming (IP), not just LP. LP solutions provide upper bounds (for max) or lower bounds (for min) on the integer problem. Rounding LP solutions doesn't guarantee optimality or feasibility for IP—use specialized IP solvers for exact integer solutions.

Linear programming (LP) allows decision variables to be any non-negative real numbers (including fractions). Integer programming (IP) restricts some or all variables to whole numbers. IP is harder to solve (NP-hard in general), but necessary when decisions must be discrete (e.g., number of projects to fund, machines to buy). LP is often used as a relaxation to get quick bounds and insights before applying more expensive IP algorithms.

This calculator is designed for education, homework, exam prep, and preliminary conceptual planning. It provides accurate LP solutions for the mathematical model you enter. However, real-world optimization involves complexities—uncertain data, nonlinear relationships, strategic considerations—that require domain expertise, validation, and often professional-grade software (like CPLEX or Gurobi). Use this tool to learn LP concepts, test formulations, and build intuition, but combine results with expert judgment for high-stakes decisions.

For homework, check whether your instructor expects exact fractional answers or allows rounding. If rounding is acceptable, round to 2-3 decimal places for clarity. Always include units (dollars, hours, units) and translate variable names into context: instead of 'x₁ = 33.33,' say 'Produce 33.33 units of Product A (or round to 33 if integer).' For reports, present results in a clear table with variable names, optimal values, units, and the optimal objective value, along with interpretation and business context.

Shadow prices (also called dual values) represent the marginal value of relaxing a constraint by one unit. For example, if the shadow price of a labor constraint is $10/hour, increasing available labor from 100 to 101 hours would increase the optimal objective by approximately $10. Shadow prices reveal which resources are bottlenecks and guide investment decisions: prioritize acquiring more of the resource with the highest shadow price. They're valid within a certain range and are key to sensitivity analysis.

A constraint is binding (or active) if it's satisfied exactly at the optimal solution (slack = 0). For example, if the constraint is x₁ + x₂ ≤ 100 and the optimal solution gives x₁ + x₂ = 100, the constraint is binding—it's limiting the objective. Non-binding constraints have slack > 0, meaning they're not currently restrictive. Identifying binding constraints helps you focus improvement efforts on the real bottlenecks.

After formulating, check: (1) Are all objective coefficients and constraint coefficients entered correctly with consistent units? (2) Do inequality signs match the problem ('at most' = ≤, 'at least' = ≥)? (3) Are non-negativity constraints enforced? (4) Does the feasible region make sense (not empty, not unbounded unless expected)? Solve a simple example by hand or graphically (for 2 variables) to verify the calculator's result matches your expectation. If results seem wrong, review your formulation step-by-step.

Yes! If you enable 'Show Tableau,' the calculator can display simplex tableaus at each iteration (depending on UI support). This lets you see which variable enters and leaves the basis, how the objective improves, and when optimality is reached. This feature is invaluable for learning how simplex works internally, verifying manual homework calculations, and preparing for exams that require showing simplex steps.

Unexpected results often indicate: (1) Unit mismatches (mixing hours and minutes, dollars and cents). (2) Typos in coefficients or RHS values. (3) Incorrect objective type (maximize instead of minimize). (4) Missing or contradictory constraints leading to unboundedness or infeasibility. Double-check all inputs against the problem statement. Verify units are consistent. Re-read constraint inequalities carefully. Use the sensitivity analysis and notes/warnings from the solver to diagnose issues.