Linear Programming (LP) is a mathematical optimisation technique used to find the best possible outcome—either maximising profit or minimising cost—subject to a set of linear constraints. LP models consist of decision variables, an objective function, and constraints that represent resource limitations. The feasible region contains all solutions that satisfy the constraints, and the optimal solution is the point that gives the best value of the objective function.
LP problems are written in standard form with a linear objective function and non-negative variables. When there are only two decision variables, the graphical method is used: constraints are plotted on a graph, the feasible region is identified, and the objective function line is shifted to find the optimal point.
For problems with more variables, the Simplex Method is applied. It transforms inequalities into equations using slack variables, builds a Simplex tableau, and moves step-by-step between feasible corner points to improve the objective value. The algorithm stops when no further improvement is possible (no negative values in the objective row).
Linear Programming is widely used in engineering, economics, energy planning, transportation, production scheduling, and cost optimisation. It is an essential tool for efficient resource allocation and decision-making.
- المعلم: Abdeldjalil Djouahi