Linear programming is a process for finding a maximum or minimum value of a linear function when there are restrictions involved. The function that is maximized or minimized is called the objective function. A constraint is an inequality that represents a restriction of the objective function. The set of constraints are modeled by a system of linear inequalities.
The intersection of the constraints is called the feasible region. The maximum and minimum values of the objective function must occur at a corner point, or vertex, of the feasible region.
To solve a linear programming problem:
1. Use the information in the problem to write the constraints as a system of linear inequalities. Note that the constraints and may not be stated but are often implied in the problem.
2. Graph the system of linear inequalities, and find the coordinates of the corner points of the feasible region.
3. Use the information in the problem to write the objective function.
4. Evaluate the objective function at each of the corner points to identify the maximum or minimum value.
Write a system of inequalities made up of the constraints. Let represent the number of necklaces and represent the number of bracelets the jeweler makes.Write one inequality for her time: Each necklace takes 1.5 hours to make, and each bracelet takes 0.75 hours to make. The total time she spends making jewelry can be up to 36 hours.
Graph the system of linear inequalities, and find the coordinates of the corner points of the feasible region.
The constraints and mean that the graph is restricted to the first quadrant.Examine the inequality symbols in the remaining constraints:
The four corner points are , , , and .
Use the information in the problem to write the objective function.The jeweler's profit for selling each necklace is $22, and her profit for selling each bracelet is $14. The function that represents her profit for selling necklaces and bracelets is:
Maximizing the objective function will maximize the jeweler's profit. To maximize the objective function, choose the corner point that results in the greatest value.
The point results in the greatest value for the profit function, so the jeweler should make 6 necklaces and 36 bracelets. Her profit will be $636.