lineaariohjelmoinnissa

Lineaariohjelmointi, also known as linear programming, is a mathematical optimization technique which seeks to maximize or minimize a linear objective function subject to a set of linear equality and inequality constraints. The standard form of a linear program is: maximize or minimize cᵀx subject to Ax ≤ b and x ≥ 0, where c and x are vectors, A is a matrix and b is a vector of constants. The solutions of linear programs lie at extreme points, or vertices, of the feasible region defined by the constraints.

The method was first formalized in the 1940s by Hungarian mathematician George Dantzig, who developed the simplex algorithm. Subsequent research yielded polynomial‑time interior point methods, most notably Karmarkar’s algorithm in 1984, which broadened the practical applicability of linear programming to high‑dimensional problems.

Linear programming is widely used in operations research, economics, engineering, and management science. Typical applications include resource allocation, production planning, transportation and logistics, network design, and portfolio optimization. The flexibility of the linear model allows it to accommodate a broad range of practical constraints such as supply limits, demand requirements, and capacity restrictions.

Several software packages support linear programming, including open‑source solvers like GLPK and COIN‑OR CBC, and commercial solvers such as CPLEX, GUROBI, and Xpress. Most of these solvers implement the simplex algorithm, interior‑point methods, or a hybrid of both. In addition, linear programming problems can be extended to integer linear programming, where some or all variables are constrained to take integer values, enabling the modeling of combinatorial problems.

Overall, lineaariohjelmointi provides a powerful, well‑established framework for decision making under linear constraints, and remains a foundational tool in applied mathematics and computational optimization.