Final answer:
The objective of this LP problem is to maximize the value of Z = 5x₁ + 12x₂ while satisfying the given constraints.
Step-by-step explanation:
The objective of the given linear programming (LP) problem is to maximize the value of the objective function Z = 5x₁ + 12x₂. This is a typical goal in many optimization problems where one seeks to find the maximum value of profits, outputs, or some other metrics of interest under certain constraints. In this scenario, the constraints given by the inequalities x₁ + x₂ ≥ 25, 3x₁ + x₂ ≥ 45, 5x₁ + 7x₂ ≥ 20, and 7x₁ + 13x₂ ≤ 83, along with the non-negativity restrictions (x₁, x₂ ≥ 0), define the feasible region within which the optimal solution (maximized value of Z) must be found. The problem can be solved using methods such as the simplex algorithm or by graphical representation if it's in two dimensions.