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3.4 Linear Programming Day #1. Do Now (Or Forever Hold Your Peace…). Take a copy of the slides, a copy of the “ LEGO ” problem, and a bag of Legos from the front. Work with the person sitting next to you. (Only one bag of Legos per partnership!)
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Do Now (Or Forever Hold Your Peace…) • Take a copy of the slides, a copy of the “LEGO” problem, and a bag of Legos from the front. Work with the person sitting next to you. (Only one bag of Legos per partnership!) • Complete the FRONT PAGE ONLY of the “LEGO My Programming Problem” worksheet
Homework Answers 1/2. 3. No, it is not possible. (80,50) does not satisfy the system. 4. • 5. 6.
Solving the Problem Mathematically Let’s go through this mathematically! Define our variables. Let t=______number of tables_______________________ and c=________________number of chairs_________________. The objective function is an equation for a quantity that we wish to minimize or maximize. In this problem, that quantity is________profit__________ and the equation is__________p=15c+20t______________.
The constraints are limitations created by scarce resources, and they are expressed as inequalities. A chair requires 1 large Lego and 2 small Legos. A table requires 2 large Legos and 2 small Legos. We have 6 large Legos and 8 small Legos This gives us the following two constraints: _________________________________________ and ________________________________________________ We have two additional constraints. What might they be? _____________________________________and___________ ______________________________________
7) The “corner principle” says that the optimal solution will always lie on one of the vertices (corners!) of the feasible region. Find all of the corner points, and identify the profit for each corner point. (tables, chairs) p=15c+20t (0,0) p=$0 (3,0) p=$60 (0,4) p=$60 (2,2) p=$70 How many tables and chairs maximize profit? 2 tables and 2 chairs
Solving Linear Programming Problems • Identify variables. • Write the objective function (what will you minimize or maximize?) • Identify the constraints (inequalities). • Graph the constraints to get your feasible region • Determine which solution is the optimal one in the feasible region. It will be one of the vertices of the feasible region, so check the corner points in the objective function.
George Dantzig“Father of Linear Programming” Logistical planning during WW 2
Class Example A farmer has 90 acres on which he may raise peanuts and corn. He has accepted orders requiring at least 10 acres of peanuts and 5 acres of corn. He also must follow a regulation that the acreage for corn must be at least twice the acreage for peanuts. If the profit is $100 per acre of corn and $200 per acre of peanuts, how many acres of each will give him the greatest profit?
Solution 1) Identify the variables. Since the exercise is asking for the number of acres of each crop required for the optimal profit, my variables will stand for the number of ounces of each: c=number of acres of corn p=number of acres of peanuts 2) Write the objective function. The optimization equation will be the profit relation P=100c+200p, which we would like to maximize.
Continued 3) Identify the constraints. The farmer only has 90 acres to work with. Orders require that he plant at least 10 acres of peanuts and 5 acres of corn, and that he must plant at least twice as much corn as peanuts. This gives these constraints.
Cheese and Crackers…. 4) Graph the feasible region. 5) Corner points (20,10) (80,10) (60,30) Test all of these by plugging them into P=100c+200p to get the maximum profit of $12,000 for 60 acres of corn and 30 acres of peanuts
Practice! ... There have been many studies of elite performers -- concert violinists, chess grand masters, professional ice-skaters, mathematicians, and so forth -- and the biggest difference researchers find between them and lesser performers is the amount of deliberate practice they've accumulated. Indeed, the most important talent may be the talent for practice itself. K. Anders Ericsson, a cognitive psychologist and an expert on performance, notes that the most important role that innate factors play may be in a person's willingness to engage in sustained training. He has found, for example, that top performers dislike practicing just as much as others do. (That's why, for example, athletes and musicians usually quit practicing when they retire.) But, more than others, they have the will to keep at it anyway.
You Try! A biologist needs at least 40 fish for her equipment. She cannot use more than 25 perch or more than 30 bass. Each perch costs $5, and each bass costs $3. How many of each fish should she use in order to minimize costs?
Extra Practice Allison is a college student who is majoring in mathematics education. She would like to earn some money to help with expenses. She has two part-time jobs. Allison earns $8 an hour tutoring at the Academic Support Center on campus, and $10 an hour working at a local restaurant. Although she would like to work more hours at the tutoring center, she likes earning the higher pay at a local restaurant. She can tutor between 2 and 8 hours per week. What is Allison’s maximum earning potential if she works a total of no more than 20 hours per week?
