3-3 Linear Programming

1. 3-3 Linear Programming

2. Linear Programming- is the process of optimizing a linear objective function. Objective Function- a system of linear inequalities that is optimized Constraints- the boundaries created by the linear inequalities Feasible Region- is the graph that is created by graphing the linear inequalities. Optimization- finding the maximum or minimum value of some quantity.

3. Step 1 Determine variables, objective function, and constraints Step 2 Graph the constraints. Step 3 Find the vertices Step 4 Test the vertices in objective function • bounded region – find both max and min • unbounded region – find either max or min, but not both pick a point in the solution set an eliminate the “min” or “max” Step 5 Answer the question

4. Objective Function: C = –x + 3y Constraints: x > 2 x < 5 y > 0 y < –2x + 12 Objective Function: C = x + 5y Constraints: x > 0 y < 2x + 2 5 > x +y

5. 1. Behrens Breakfast Bar Company makes fruit-filled breakfast bars and granola bars. For each case of breakfast bars, you make $40 profit. For each case of granola bars, you make $55 profit. To make one case of breakfast bars you need 2 machine hours and 5 labor hours. To make one case of granola bars you need 6 machine hours and 4 labor hours. If there are a maximum of 150 machine hours and a maximum of 155 labor hours, how many of each case should be produced to maximize profits?

6. 2. Shieve Shirt City produces short-sleeved and long sleeved shirts. A short-sleeved shirt requires 30 minutes of labor, a long-sleeved shirt requires 45 minutes of labor, and 240 hours of labor are available per day. The maximum number of shirts that can be packaged in a day is 400. If the profits on a short-sleeved shirt and a long-sleeved shirt are $11 and $16, respectively, find the maximum possible daily profit. 3. Cars and trucks are made in Cruz Car Company that is divided into two shops. Shop 1 performs the basic assembly operation working 6 person-days on each truck and only 3 person-days on each car. Shop 2 performs finishing operations, working 4 person-days on each car or truck that it produces. Shop 1 has 150 person-days per week available, while Shop 2 has 120 person-days per week. The manufacturer makes a profit of $500 on each truck and $350 on each car. How many of each should be produced each week to maximize the profit?

7. 4. Funk's Furniture makes chairs and sofas from prepackaged parts. The table gives the number of packages of wood parts, stuffing, and material required for each chair or sofa. The packages are delivered weekly and the manufacturer has room to store 1300 packages of wood parts, 2000 packages of stuffing, and 800 packages of fabric. The manufacturer earns $200 per chair and $350 per sofa. How many chairs and sofas should they make each week to maximize profit?

8. 5. Mcghee & Manis TV R Us Company makes color television sets. The company produces a bargain set which sells for $100 profit and a deluxe set which garners $150 profit. On the assembly line, the bargain set requires 3 hours of time to produce while the deluxe set requires 5 hours. The cabinet shop spends one hour on the cabinet for the bargain set and three hours on the cabinet of the deluxe set. Both set require two hours of time for testing and packaging. On particular production run, the Mark & Marcus TV R Us Company has available 3900 work hours on the assembly line, 2100 work hours in the cabinet shop, and 2200 works hours in the testing and packaging department. How many sets of each type should it produce to maximize its profit? What is the maximum profit?

9. 6. At Mr. Murphy's Moo-Moo Cattle Ranch, they use two brands of cattle feed. Brand X cost $24 per bag and contains two units of nutritional supplement A, 2 units of supplement B, and 2 units of supplement C. Brand Y costs $20 per bag and contains 1 unit of supplement A, 9 units of supplement B, and 3 units of supplement C. Find the number of bags of each brand that should be mixed to minimize your costs, but meet the minimum requirements of nutrients A, B, and C which are 12 units, 36 units, and 24 units, respectively.

10. 7. At Diego's Delicate Flowers and Fruit Farm there are 150 acres of land available to grow grapes and berries. It takes one day to trim an acre of grapes, two days to trim an acre of berries, and there are only 240 days available throughout the year to do this trimming. To pick an acre of grapes requires .3 days while .1 days are needed to harvest an acre of berries. There are only 30 days per year available for harvesting. Find the number of acres for each crop that she should plan in order to maximize her profit assuming that for grapes her profit margin per acre is $140 while for berries she realizes a profit of $235 per acre.

11. 8. A Nicole's Nutrition Center sells health food to mountain-climbing teams. The Trailblazer mix package contains one pound of corn cereal mixed with four pounds of what cereal and sells for $9.75. The Frontier mix contains two pounds of corn cereal mixed with mixed with three pounds of wheat cereal and sells for $9.50. The center has available 60 pounds of corn cereal and 120 pounds of wheat cereal. How many packages of each mix should the center sell to maximize its income?

12. 9. Wood's Wood Furntiture Company can sell all the tables and chairs that it produces. Each table requires 1 hour in the assembly center and 1 1/3 hours in the finishing center. Each chair requires 1 1/2hours in the assembly center and 1 1/2hours in the finishing center. The company’s assembly center is available 12 hours per day, and its finishing center is available 15 hours per day. If the profit on each table is $100 and $50 on each chair, how many of each should be produced to maximize profit?