|Class 12 Maths Linear Programming||Introduction|
Optimization is the way of life. We all have finite resources and time and we want to make the most of them. Different type of problems which seek to maximize or minimize profit or cost, form a general class of problems called optimization problems. So, an optimization problem may involve finding maximum profit, minimum cost, or minimum use of resources etc.
Linear programming problems are one of the important class of optimization problem. Linear Programming is a mathematical modeling technique in which a linear function is maximized or minimized when subjected to various constraints. A typical example would be taking the limitations of materials and labor, and then determining the best production levels for maximal profits under those conditions.
In general, a linear programming problem is specified as follows:
(i) n variables x1, x2, x3,... ,xn.
(ii) m linear inequalities in these variables.
E.g., 2x1 + 5x2 ≤ 10, 0 ≤ x1 ≤ 2, etc.
(iii) We may also have a linear objective function.
E.g. Maximize/Minimize 3x1 + 5x2 + 7x3
Goal: Find values for the xi’s that satisfy the constraints and maximize/minimize the objective function.