Linear Programming Calculator
linear programming calculator. Get instant, accurate results. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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Formula: Maximize Z = cx subject to Ax ≤ b, x ≥ 0
Additional inputs: Constraint 3: x coeff, Constraint 3: y coeff, Constraint 3: ≤ RHS.
Worked example — Optimal Z
Maximize Z = cx subject to Ax ≤ b, x ≥ 0
Solves 2-variable LP graphically by evaluating the objective at each vertex of the feasible region.
Problem:Max 3x+5y, s.t. x+2y≤14, 3x+y≤14, x+y≤8
Solution:Evaluate Z at each vertex of feasible region
Result:Optimal Z
Linear programming (LP) is a mathematical optimization technique that finds the maximum or minimum of a linear objective function subject to linear inequality constraints. It is widely used in manufacturing (production planning), logistics (transportation routing), finance (portfolio optimization), and workforce scheduling to allocate limited resources efficiently.
The optimal solution is the point within the feasible region that maximizes or minimizes the objective function Z. By the Fundamental Theorem of Linear Programming, if an optimal solution exists it will always occur at a corner (vertex) of the feasible region. The coordinates give the optimal values of the decision variables, and Z gives the best achievable objective value.
Linear programming is the mathematical framework — defining the objective and constraints. The simplex method is a specific algorithm used to solve LP problems. The simplex method starts at a vertex of the feasible region and iteratively moves to adjacent vertices that improve the objective until no further improvement is possible or unboundedness is detected.
LP models assume proportionality (contribution of each variable is proportional to its value), additivity (total objective and constraint values are sums of individual contributions), divisibility (decision variables can take non-integer values), and certainty (all coefficients are known constants). When divisibility is violated, integer programming must be used instead.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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