Linear Programming Probability | CUET PG Economics – Notes

Linear Programming: Meaning and Structure

Linear programming (LP) is a mathematical technique used to determine the best possible outcome — either maximisation of profit, output, or welfare, or minimisation of cost, time, or resources — subject to a set of linear constraints. It is one of the most widely used tools of operations research and finds extensive application in economics, particularly in problems of resource allocation, where limited resources such as capital, labour, and raw materials must be distributed among competing uses in the most efficient manner.

Every linear programming problem consists of three essential components. The first is the objective function, which is the linear expression to be maximised or minimised, typically representing profit, cost, or output. The second is the set of constraints, which are linear inequalities or equations representing the limitations on resources. The third is the condition of non-negativity, which requires that all decision variables take values greater than or equal to zero, since negative quantities of output or input have no economic meaning.

A general linear programming problem for maximising profit can be written as:

$$\text{Maximise } Z = c_1 x_1 + c_2 x_2$$

subject to:

$$a_1 x_1 + b_1 x_2 \leq k_1$$ $$a_2 x_1 + b_2 x_2 \leq k_2$$ $$x_1, x_2 \geq 0$$

Here, (Z) is the objective function, \(x_1\) and \(x_2\) are the decision variables, and the inequalities represent the resource constraints.