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Linear Programming Linear Programming is a branch of math that uses linear inequalities to solve practical decision-making problems involving maximums and minimums. Frequently used in business and government, these problems try to determine optimum values based on given constraints or inequalities. Once the feasible region is shaded, the coordinates of the corners of this region are found. The maximum or minimum will be found using these corners. Let’s look at an example. Calculator Pro is a business that creates scientific calculators and graphing calculators. Their profit is determined by: P = 6x + 15y, where x is the # of scientific calculators and y is the # of graphing calculators. Constraints are graphed and the feasible region is shown below: 40 y 30 20 10 x 50 100 We can find the maximum Profit using the corner points of the feasible region and the profit equation that is given. Point (x, y) P = 6x + 15y Let’s look at another example. Furniture Recovery is a business that reupholsters sofas and chairs. Their overall operation cost is determined by: C = 35x + 40y, where x is the # of sofas covered and y is the # of chairs covered. Constraints are graphed and the feasible region is shown below: 30 y 25 20 15 10 5 x -5 5 10 15 20 25 30 -5 We can find the minimum Cost using the corner points of the feasible region and the profit equation that is given. Point (x, y) C = 35x + 40y More Linear Programming Problems 1. A tea and coffee distributor sells two types of gift packs: the Early Riser and the Afternoon Delight. The Early Riser pack has 18 servings of tea and 15 servings of coffee. The Afternoon Delight pack has 15 servings of tea and 6 servings of coffee. The distributor can package no more than 330 servings of tea and 210 servings of coffee per day. The profits for the Early Riser and the Afternoon Delight are $7 and $5 respectively. How many of each pack should be assembled per day to maximize profit? 40 y 30 20 10 x 5 10 15 20 2. A manufacturer two models of children’s buggies: a 4-wheel model and a 3-wheel model. The 4-wheel model requires 13 minutes of prep time and 17 minutes of assembly time. The 3-wheel model requires 8 minutes of prep time and 3 minutes of assembly time. In a day, there is a maximum of four hours for prep and 187 minutes for assembly. The profits for the 4-wheel model and a 3-wheel model are $25 and $14 respectively. How many of each type should this manufacturer produce to maximize profit? y 60 50 40 30 20 10 x 5 10 15 20 3. ACE ELECTRONICS PROBLEM The Ace Electronics Company makes CD’s and DVD’s. The process involves pressing and packing. Each CD requires 6 minutes to be pressed and 10 minutes packing. Each DVD requires 5 minutes to be pressed and 5 minutes packing. The Pressing Machine is only available for a maximum of 120 minutes per day. The Packing Machine is only available for a maximum of 160 minutes per day. The profit on a CD is $3 and on a DVD is $4. Assuming that they sell all the CD’s and DVD’s that they make, how many of each should they make to maximize the profits? y x Name all points for the corners. Use all points in the profit equation. 4. “BAD SANTA” Santa is making cookies. He loves Elf and Reindeer cookies. The Elf cookies take 5 minutes to mix the dough, 6 minutes to form the cookies, and 9 minutes to bake. The Reindeer cookies take 3 minutes to mix, 6 minutes to form the cookies, and 4 minutes to bake. Bad Santa hasn’t much time to do his cookie production. He has a maximum of 120 minutes to mix, 150 minutes to form cookies, and 180 minutes to bake them. This is Bad Santa’s only income. If he makes $3 on each Elf cookie and $5 on each Reindeer cookie, how many of each would he have to make to maximize his earnings? y x Name all points and use each point to determine maximum earnings. 5. MRS. CLAUS AND HER PROBLEM ELVES Mrs. Claus is in charge of the North Pole Toy Factory. She has two elf workers. Jerome assembles dolls and Mathilda dresses them. There are Santa dolls and Rudolph dolls. The Santa dolls take 2 minutes to assemble while the Rudolph dolls take 3 minutes to assemble. The Santa dolls take 3 minutes to dress while the Rudolph dolls take 1 minute to dress. Jerome has a maximum time of 15 minutes to assemble today. Mathilda has a maximum time of 19 minutes to dress them today. What is the maximum number of dolls that they can assemble and dress in the time period allotted? y x 6. SUNNY GARDEN GIFT SHOP The shop makes 2 types of gift packs of JAM and MARMALADE. 1. The “Morning Glory” pack has 4 jars of Marmalade and 1 jar of Jam. 2. The “Berry Patch” pack has 1 jar of Marmalade and 4 jars of Jam. 3. They have 50 jars of MARMALADE 4. They have 90 jars of JAM. 5. The Profit on the “Morning Glory” package is $3.00 6. The Profit on the “Berry Patch” package is $4.00 How many of each package should be made to maximize profit? y x
"2010 Linear Programm"