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How can optimization maximize a company's profit while respecting limits on materials and machine time?

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optimization problem — optimization Constraint set bounded by the five lines x₁ = 0, x₂ = 0, x₁ = 8, x₂ = 5, and x₁ + x₂ = 10. These enclose an infinite number of points that represent feasible solutions.

To maximize profit while respecting limits on resources like materials and machine time, companies utilize linear programming. This mathematical modeling technique allows businesses to determine the optimal production levels for various commodities by balancing goals against constraints.12

The Optimization Process

Define the Objective Function: A company first identifies a linear function representing total profit, such as $x_{1} + 2 x_{2}$ , where $x_{1}$ and $x_{2}$ are the quantities of different products.13
Establish Constraints: Limitations on resources—such as raw materials, labor, and machine time—are expressed as a system of linear inequalities (e.g., $x_{1} + x_{2} \leq 10$ ). These constraints define a "feasible region" of possible production combinations.123
Find the Optimal Solution: The goal is to find the point within the feasible region that yields the highest value for the objective function. Mathematically, this optimal solution typically occurs at a vertex, or "extreme point," of the region.13

For complex problems involving many variables, the simplex method is the standard algorithm used to efficiently test these vertices and identify the maximum profit.32