Exploring Linear Programming

Transcription

1 Warm Up Exploring Linear Programming Graph the following inequalities. Find and label the solution region. What are the vertices of the solution region? Furniture Company Problem Pretend we are a furniture company that makes only two products: tables and chairs. A table requires two large and two small legos, and a chair requires one large and two small legos. Unfortunately, resources are not unlimited. Suppose the company can only obtain six large and eight small pieces per day. The profit from each table is $16, and the profit from each chair is $10. The production manager (Ms. Hahn) wants to know the rate of production of tables and chairs that maximizes the profit. 1. Build physical models of this problem. Create several combinations and record in the following table. Indicate the profit of each possibility. 2. Which production mix generates the most profit? 3. Why is building four tables an example of an infeasible solution? 4. Would it make a difference if seven large pieces were available instead of six? (Assuming there are still eight small pieces) If so, what is the new optimal solution, and how much profit does it generate?

2 The Linear Programming Process Linear Programming is an application of mathematics to such fields as business, industry, social science, economics, and engineering. The process is used to establish feasible regions and locate maximum and minimum values that take place under certain given conditions. Example 1 Constraints: Objective function: Vertices: A potato chip company makes chips to fill snack-size bags and family-size bags. In one week, production cannot exceed 2,400 units, of which at least 600 units must be for snack-size bags and at least 900 units must be for family size. The profit on a unit of snack-size bags is $12, and the profit on a unit of family-size bags is $18. How much of each type of bag must be processed to maximize profits? Step 2 Write constraints. Step 3 Write objective function. Step 5 Test the vertices of the Step 6 State the solution in a meaningful way.

3 Example 2 Mrs. Smith grows peaches and apples. At least 500 peaches and 700 apples must be picked daily to meet minimum demands from the buyers. The workers can pick no more than 1200 apples and 1400 peaches daily. The combines numbers of peaches and apples that the packaging department can handle is 2300 per day. If Mrs. Smith sells her apples for 25 cents and her peaches for 20 cents each, how many of each should be picked daily for maximum income? What is her maximum income? Step 5 Test the vertices of the Step 6 State the solution in a meaningful way.

4 Example 3 A pizza shop makes $1.50 on each small pizza and $2.15 on each large pizza. On a typical Friday, it sells between 70 and 90 small pizzas and 100 and 140 large pizzas. The total sales have never exceeded 210 pizzas. How many of each size pizza must be sold to maximize profits? Step 5 Test the vertices of the feasible region. Step 6 State the solution in a meaningful way.

5 Example 4 A ski company makes two types of skis and has a fabrication and a finishing department. A pair of downhill skis requires 6 hours to fabricate and 1 hour to finish. A pair of cross-country skis requires 4 hours to fabricate and 1 hour to finish. The fabricating department has 108 hours of labor available per day. The finishing department has 24 hours of labor available per day. The company makes a profit of $40 on each pair of downhill skis and $ 30 on each pair of cross-country skis. How many of each type should the manufacturer produce to maximize profit? Step 5 Test the vertices of the feasible region. Step 6 State the solution in a meaningful way.