Linear programming is a method for finding the optimal outcome (maximum profit or minimum cost) in a mathematical model whose requirements are represented by linear relationships. When dealing with two variables, we can graphically represent these constraints as lines on a coordinate plane. The feasible region, where all constraints are satisfied, is the intersection of these regions. The optimal solution always lies at one of the vertices (corners) of this feasible region. By evaluating the objective function at each vertex, we can determine the best possible solution.
Match each term to its correct definition:
| Term | Definition |
|---|---|
| 1. Feasible Region | A. A function to be maximized or minimized |
| 2. Objective Function | B. The point at which the objective function achieves its optimal value |
| 3. Constraint | C. A line that bounds the feasible region |
| 4. Optimal Solution | D. A restriction or limitation expressed as an inequality |
| 5. Boundary Line | E. The set of all possible solutions that satisfy all constraints |
Definitions can be rearranged to be interactive
Complete the following paragraph using the words provided (Constraints, Vertices, Feasible Region, Optimal Solution, Linear Programming).
_________ is a method used to optimize a linear objective function subject to linear __________. The ___________ represents all possible solutions that satisfy these constraints. The ____________ always occurs at one of the __________ of the feasible region.
Explain in your own words why the optimal solution in a linear programming problem always occurs at a vertex of the feasible region. Provide an intuitive explanation, not just a mathematical proof.
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