1. The problem is to maximize or minimize a linear objective function subject to a set of linear inequalities or equations called constraints. 2. A typical linear programming problem can be stated as: maximize or minimize $$Z = c_1x_1 + c_2x_2 + \cdots + c_nx_n$$ subject to constraints $$a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \leq b_1,$$ $$a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \leq b_2,$$ and so on, where $x_i \geq 0$. 3. Important rules include that the objective function and constraints must be linear, and the variables are usually non-negative. 4. For example, a company wants to maximize profit $$Z = 40x + 30y$$ where $x$ and $y$ are quantities of two products. 5. Constraints might be $$2x + y \leq 100$$ (resource 1), $$x + 2y \leq 80$$ (resource 2), and $$x, y \geq 0$$. 6. The solution involves finding values of $x$ and $y$ that satisfy constraints and maximize $Z$. 7. This can be solved graphically or using methods like the simplex algorithm. 8. The feasible region is the set of all points satisfying constraints, and the optimal solution lies at a vertex of this region. 9. By evaluating $Z$ at vertices, the maximum profit can be found. 10. This example illustrates how linear programming helps in decision making to optimize objectives under constraints.