Linear Programming Test Review Day 6
Arrival Instructions Take out: Your homework, calculator, and the unit outline Pick up: 1. Warm-Up Test Review Day 2. Test Review HW Sheet A sponge Yes, it is different than yesterday s CW sheet 3. Arrival Graph Half Sheet What has holes all around but can still hold water? Warm-Up: Handout Warm-Up Test Review - COLLECTED! THEN do Arrival Problem Half Sheet BE PREPARED TO SHARE Hint: For help writing a constraint, use 1 color to underline everything in the problem about 1 thing that is limiting you. Then put that together in 1 equation.
CHECK YOUR CONSTRAINTS Test Review Warm-Up Juan makes two types of wood clocks to sell at local stores. It takes him 2 hours to assemble a pine clock, which requires 1 oz. of varnish. It takes 2 hours to assemble an oak clock, which requires 4 oz. of varnish. Juan has 16 oz. of varnish in stock and he can work 20 hours. If he makes $3 profit on each pine clock and $4 profit on each oak clock, how many of each type should he make to maximize his profits? Decision Variables: x 1 = # of pine clocks x 2 = # of oak clocks Constraints: (sketch graph) Varnish x 1 + 4x 2 16 Hours 2 x 1 + 2x 2 20 Non- x 1 0 Negativity x 2 0 Objective Function: $3x 1 + $4x 2
Sharing your Warm-up answers We will use the document camera to share what you wrote for your decision variables, constraints and objective functions. Then we will graph the constraints.
COLLECTED HANDOUT Test Review Warm-Up Juan makes two types of wood clocks to sell at local stores. It takes him 2 hours to assemble a pine clock, which requires 1 oz. of varnish. It takes 2 hours to assemble an oak clock, which requires 4 oz. of varnish. Juan has 16 oz. of varnish in stock and he can work 20 hours. If he makes $3 profit on each pine clock and $4 profit on each oak clock, how many of each type should he make to maximize his profits? Decision Variables: x 1 = # of pine clocks x 2 = # of oak clocks Constraints: Varnish x 1 + 4x 2 16 Hours 2 x 1 + 2x 2 20 Non- x 1 0 Negativity x 2 0 Objective Function: Remember to label your axes with units AND decision variables! $3x 1 + $4x 2
X 2 : number of oak clocks Your Graph should look like... Constraints: Varnish x 1 + 4x 2 16 Hours 2 x 1 + 2x 2 20 Non- x 1 0 Negativity x 2 0 You will have to use Matrices or Elimination Method to find this point. X 1 : number of pine clocks
X 2 : number of oak clocks Your Completed Table & Solution Corner Objective Value Points (x 1, x 2 ) Function 3x 1 + 4x 2 (0, 0) 3(0) + 4(0) 0 (10, 0) 3(10) + 4(0) 30 (0, 4) 3(0) + 4(4) 16 (8, 2) 3(8) + 4(2) 32 X 1 : number of pine clocks Juan should make 8 pine clocks and 2 oak clocks for a maximum profit of $32.
1 2 3 4 A B C D E F Today s Date Here Juan s Clocks Profit Maximization 5 Decision Variable Decision 6 Value Pine Clocks (x 1 ) Oak Clocks (x 2 ) 7 8 Objective Function ($) 9 3 4 Total Profit = B8*B6 + C8*C6 10 Constraints Used Available 11 Maximum Varnish 12 Maximum Time 1 4 = B11*B6 + C11*C6 2 2 = B12*B6 + C12*C6 16 20 OR =sumproduct(b12:c12, B6:C6)
Homework Questions
Today s Agenda Formulating the SK8Man problem on paper You need to do this in your notes. Practice with EXCEL Review TEST FRIDAY!!
Title: SK8MAN, Inc. You will need to reference the following table, which is homework packet page 8. Please take that out. Product Type of Maple Shaping Time (min) Screen Printing Time (min) Sporty x 1 Chinese Fancy x 2 North American Pool- Runner x 3 Chinese Pool- Beauty x 4 North American Recluse x 5 Chinese Ringer x 6 Chinese 5 15 4 10 7 4 7 10 8 10 5 6 You also need notebook paper to write down the L.P. formulation.
Product Type of Maple Shaping Time (min) Screen Printing Time (min) Sporty x 1 Chinese Title: SK8MAN, Inc. Fancy x 2 North American Pool- Runner x 3 Chinese Pool- Beauty x 4 North American Recluse x 5 Chinese Ringer x 6 Chinese 5 15 4 10 7 4 7 10 8 10 5 6 Writing the constraints: (Use your answers to the reading guide packet p. 8.) A. Amount of available North American veneers is 840 and each skateboard requires 7 veneers. 7x 7x 840 Therefore the constraint would be. 2 4 B. Amount of available Chinese veneers is 1470 and each skateboard requires 7 veneers. 7( x x x x ) 1470 Therefore the constraint would be. 1 3 5 6
Product Type of Maple Shaping Time (min) Screen Printing Time (min) Sporty x 1 Chinese Title: SK8MAN, Inc. Fancy x 2 North American Pool- Runner x 3 Chinese Pool- Beauty x 4 North American Recluse x 5 Chinese Writing the constraints: (Use your guided reading notes.) C. Amount of available time for shaping is. 2250 Using the information in the table the constraint would be. 3 4 5 6 D. Amount of available time for screen printing is. 2250 Using the information in the table the constraint would be. 3 4 5 6 Ringer x 6 Chinese 5 15 4 10 7 4 7 10 8 10 5 6 5x 15x 4x 10x 7x 4x 2250 7x 10x 8x 10x 5x 6x 2250
Title: SK8MAN, Inc. Nonnegativity constraints: dummy constraints x 1 0 x2 0 x3 0 x 4 0 x5 0 x6 0 Number of trucks available per week is 700 and each skateboard requires two trucks. Therefore, the constraint would be. 2( x1 x2 x3 x4 x5 x6 ) 700
Solving Using Microsoft EXCEL There is no way to find the optimal solution by hand by visualizing a graphical solution so we need Excel SOLVER!
Solving and doing Sensitivity Analysis Using Microsoft EXCEL Use the Excel Sheet and Answer Report to answer questions 1-6. 1) What is the optimal solution? 2) What is the profit for this product mix? 3) Identify the constraints that are binding. 4) Identify the constraints that are non-binding. 5) Interpret the meaning of these binding and non-binding constraints in terms of the problem context. 6) Find the slack for each constraint. Interpret the meaning of slack in terms of the problem context.
Test Review Practice A machine can produce either nuts or bolts, but not both at the same time. The machine can be used at most 8 hours a day. Furthermore, at most 6 hours a day can be used for making nuts and at most 5 hours a day can be used for making bolts. There is a $2 profit for each hour the machine makes nuts and a $3 profit for each hour the machine makes bolts. How many hours per day should the machine make each item in order to maximize profit? What is the maximum profit? Decision Variables: x 1 = x 2 = Constraints: (sketch graph) Objective Function: Optimal Solution:
Quiz Corrections & Practice 1) On a new sheet of notebook paper, work with the person beside you to make corrections to your Unit 3 LP quiz. 2) When you are both finished, staple your corrections on top of your quiz and bring it to me with your stamp sheet to check. 3) Done with Corrections and the Warm-Up worksheet? You may start working on Test Review homework.
Arrival Problem Half Sheet The graph below is created using the following equations: A 3x 2x 48 9x 8x 24 C D E B Find the coordinates of point A, B, C, D and E Discuss with your partner the best approach to figuring this out.
Arrival Problem--Answers The graph below is created using the two equations in blue. Find the coordinates of point A, B, C, D and E C A D E B 3x 2x 48 9x 8x 24 A (0,24) B (16, 0) C (-24/9, 0) D (0,3) E (8,12) Next slide has detailed solutions
Arrival Problem--Solutions The graph below is created using the two equations in blue. Find the coordinates of point A, B, C, D and E C A D A (0,24) B (16, 0) C (-24/9, 0) D (0,3) E (8,12) E 3x 2x 48 B (0,24) and (16,0) are on this line 9x 8x 24 24 (0,3) and,0 9 Next slide has detailed solutions for point E
Arrival Problem--Solutions The graph below is created using the two equations in blue. Find the coordinates of point A, B, C, D and E A Use Matrices or Elimination (see below) to Find the Coordinates of Point E C D E 3x 2x 48 4( 3x 2x 48) B 9x 8x 24 A (0,24) B (16, 0) C (-24/9, 0) D (0,3) E (8,12) 21x 168 x1 8 9(8) 8x 24 x 2 12 2 1
For some review, More about Computer Flips.
Continuing Computer Flips (packet p. 5) Adding an Additional Constraint TAKE NOTES The students at Computer Flips notice that they are getting a lot of returns. Every computer that was returned had a problem with one of the add-ons. They realize that they need to test their finished products before shipping them. The student who will do the testing will work 10 hours per week. It takes her 20 minutes to test a Simplex and 24 minutes to test an Omniplex. Write the additional constraint for testing time. 20x1 24x2 600 minutes OR x1 x2 10 hours 3 5 Which one would you consider to be the best to work with? Adapted with permission from NCSU MINDSET materials
Now, we are ready to continue At the top of a NEW piece of graph paper, write down the following constraints. We will graph using the intercepts. Intercepts on the Next Slide Simplex models Omniplex Models 60x 1 120x 2 2400 Install Minutes + 20x 4x 2 600 Testing Minutes + x 0 1 X 0 2
Now... Determine the x 1 -intercept and x 2 intercept for each of the following constraints. 60x 120x 2400 x intercept: (, 40 ) 0 x intercept:(, ) 20x 24x 600 x intercept: (, 30 0 ) x intercept:(, 0 25 ) 0 20
X 2 : weekly production rate of Omniplex computers The graph would look like... Corner Points (x 1, x 2 ) Objective Function 200x 1 + 300x 2 Value X 1 : weekly production rate of Simplex computers You will have to use Matrices or Elimination Method to find this point. Let s try together! 60x 120x 2400 20x 24x 600 For reference, the Elimination Method work is on the next slide
Elimination to Find Corner Point 60x 120x 2400 60x 120x 2400 20x 24x 600 3( 20x 24x 600) 60x 120x 2400 60x 72x 1800 1 1 1 48x 600 x 2 2 12.5 60x 120(12.5) 2400 60x 900 x 15
X 2 : weekly production rate of Omniplex computers Your Completed Table Corner Points (x 1, x 2 ) Objective Function 200x 1 + 300x 2 Value (0,0) 200(0) + 300(0) 0 (0, 20) 200(0) + 300(20) 6000 (30, 0) 200(30) + 300(0) 6000 (15, 12.5) 200(15) + 300(12.5) 6750 X 1 : weekly production rate of Simplex computers
X 2 : weekly production rate of Omniplex computers Now, Interpret the Solution Corner Points (x 1, x 2 ) Objective Function 200x 1 + 300x 2 Value (0,0) 200(0) + 300(0) 0 (0, 20) 200(0) + 300(20) 6000 (30, 0) 200(30) + 300(0) 6000 (15, 12.5) 200(15) + 300(12.5) 6750 X 1 : weekly production rate of Simplex computers Computer Flips can make 15 Simplex and 12.5 Omniplex Computers per week (on average) for a profit of $6750 (on average).
Practice Write the constraints and the objective function for the following: