Day 1 Note Linear Programming Answer Key

DAY #1 NOTES Pg. 1 **LINEAR PROGRAMMING** (Warm up: review elimination/substitution) Linear programming is used to help reach the best possible outcome in a situation. So, your objective in these real-life scenarios (goal) will be to do something like maximize your profits or minimize a risk of some type. The equation that describes your goal to do this is called an OBJECTIVE   FUNCTION. Unfortunately, in real life, there are things that constrain you. For example, you can only work so many hours each week, you may have limited funds, perhaps you can only order specific quantities of supplies, etc. These constraints limit your ability to reach your goal. And, there are many, many options for how to use your resources.   Here's an example: An appliance dealer, Andrea Kurtz, wants to purchase a combined total of no more than 100 refrigerators and dishwashers for inventory. Refrigerators weigh   200 pounds each, and dishwashers weigh 100 pounds each. Suppose that the dealer is limited to a total of 12,000 pounds for these two items. If a profit of   $35 for each refrigerator and $20 on each dishwasher is projected, how many of each should be purchased and sold to make the largest profit? (This word problem from:   http://www.cabrillo.edu/~hpayne/Math%2013/Lecture%20Notes/Chapter%204.pdf) HER OBJECTIVE: To make the largest profit! How should she do that??? Notice that she could buy 100 or less refrigerators   and dishwashers in a number of   ways. Let's list some here: | # refrigerators | # dishwashers | total lbs. | Does it meet weight requirements? | How much profit would this combination produce? | |---|---|---|---|---| | 50 | 50 | 15,000 | N | not an option | | 20 | 80 | 18,000 | N | not an option | | 35 | 25 | 8,500 | Y | $1,575 | | 1 | 1 | 300 | Y | $55 | | 30 | 40 | 11,000 | Y | $2,000 | | 62 | 31 | 12,400 | N | not an option | Some of the combinations we listed would be feasible considering her constraints and some would not. Some created higher profits and some less. Linear programming is what helps us decide which combination will return the greatest profit, given these constraints.   **DAY #1 NOTES Pg. 2** First, let’s recap Andrea’s goal. To maximize profits. Let x = # refrigerators and y = # dishwashers she purchases. Can you write an equation that represents the profits she might get once she sells the appliances: **OBJECTIVE FUNCTION** P = 35x + 20y (Answers the question: What am I maximizing/minimizing?) NEXT, our appliance dealer is facing certain limitations. Let’s write equations for these: (Continue to use x for the # of refrigerators and y for the # of dishwashers.) Can you write an equation that would describe the weight limitations she is facing?   200x + 100y ≤ 12,000 Can you write an equation to describe the fact that she can’t buy a negative number of appliances, even though we don’t say it, she must purchase less than 100 appliances total? x + y ≤ 100 x ≥ 0 y ≥ 0 The next steps in a linear programming problem is to put all the constraints onto one graph and identify the points at which the lines cross. (We’ll practice this skill more later). For now, it has been done for   you: [Image of a graph with shaded region and vertices labeled (0,0), (0,100), (60,0), (20,80)] vertices: (0, 0), (0, 100), (60, 0), (20, 80) x + y = 100 200x + 100y = 12,000 Solve: -100x - 100y = -10,000 200x + 100y = 12,000 -------------------------- 100x = 2,000 x = 20 20 + y = 100 y = 80 Any point within the shaded region is considered a viable solution to your problem. Those are all options that are within your constraints.   We call this area the FEASIBLE REGION. (the region that is feasibly possible.) While any point in the feasible region is a possible solution to your problem, the maximum or minimum value will occur at the vertex points. So, to solve the linear programming problem, plug all of the vertex points into your objective function to find out how much profit each situation   will produce. P = 35x + 20y P = 35(0) + 20(0) = 0 P = 35(0) + 20(100) = 2,000 P = 35(60) + 20(0) = 2,100 P = 35(20) + 20(80) = 2,300 Based on your work, how many refrigerators and dishwashers should Andrea purchase in order to maximize her profits? 20 refrigerators & 80 dishwashers