Linear Programming - Honors ICM GHHS · 2019-05-16 · Answers for Posting Purposes Warm-Up: 1....

Linear Programming

ICM Unit 3

Day 2

Warm-Up and Arrival

Borrow, if needed:

1. A couple Colored pencils to share

2. A ruler

Warm-Up:

1. Read the problem on Side 1 of Handout2. Formulate the Linear Programming by

a. Defining the Decision Variablesb. Writing the Objective Functionc. Writing the Constraints

THEN Repeat these steps for Side 2

STOP – DON’T graph the constraints yet – check them first!!

BE PREPARED TO SHARE

Riddle: What has holes all around but can still hold water? A sponge

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 into 1 equation.

Phones OFF and in the pockets!

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.

Answers for Posting Purposes

Warm-Up:

1. Read the problem statement on side 1 of the CW Handout

2. Formulate the Linear Programming by

a. Defining the Decision Variables

b. Writing the Objective Function

c. Writing the Constraints

1 2: # of brake pads : # of oil filtersx x

1 25.50 8.25x x

Next Slide

Answers for Posting Purposes

Brake Pads Oil FiltersInequality

symbol

x1 + x2 150 Total Items

$0.75 x1 + $1.50x2 $180 Budget

x1 + 0x2 20 Brake Pads

0 x1 + x2 50 Oil Filters

Today’s Agenda and

Learning Outcomes

• HW Questions

• Finding “Corner Points” of Feasible Region

• Exploring what is significant about the corner points

• Formulating Linear Programming problems on paper

• Revisiting the LEGO Furniture Company—sensitivity analysis

• Introducing Microsoft Excel as a tool for finding and analyzing an optimal solution

HW Questions

NOTES: Finding “Corner Points”Take NOTES in your NOTEBOOK.

We will start by using the constraints from last night’s homework, packet p. 2.

Write the equations of the boundary lines for the constraints in your notebook:

In your prior math courses, you learned several ways to solve a system of equations. The one we will review and use in our class is elimination.

1 2

1 2

30 6 600

60

x x

x x

NEXT SLIDE

NOTES: Finding “Corner Points”Take NOTES in your NOTEBOOK.

We will start by using the constraints from last night’s homework, packet p. 2.

Write the equations of the boundary lines for the constraints in your notebook:

In your prior math courses, you learned several ways to solve a system of equations (substitution, elimination, etc). The way we will review now is solving with matrices, and substitution.

1 2

1 2

30 6 600

60

x x

x x

NEXT SLIDE

NOTES: Finding “Corner Points”— Matrices Method

1 2

1 2

30 6 600

60

x x

x x

1

2

30 6 600

1 1 60

x

x

Coefficient Matrix

Variable Matrix

Constant Matrix

A X B

1X A B

To Solve, enter matrices A and B in the calculator, then find X with

10

50

(10, 50)

X

NOTES: Finding “Corner Points”

Take NOTES in your NOTEBOOK.

This confirms what you probably guessed to be the corner point that is the intersection of the two boundary lines.

Now, we are ready to finish out the problem.

Please use the table at the bottom of packet p. 2 to list the corner points of the feasible region.

NEXT SLIDE

Finishing Packet p. 2—Parking Lot Problem

Corner Points(x1 , x2 )

Substitute into Objective Function Value

(10, 50)

(0, 0)

(20, 0)

(0, 60)

7.50(10) + 2.50(50)

7.50(0) + 2.50(0)

7.50(20)

2.50(60)

= $200

= $0

= $150

= $150

Maximum Income is _______ when the attendant accepts ____ cars and ____ buses.

$2005010

BIG CONCLUSION:The optimal solution to a Linear Programming problem

will occur at one of the corner points.

Your Turn to Practice

Find the corner points for the feasible region for the Jerry’s Autoparts problem you started as a warm-up.

THEN Complete the Earthquake Relief Problem on the Back side

This will be collected and counted as a separate CW grade in Powerschools.

Ask for help and work with your partner.

Today: Using Technology with Linear Programming. . . .

•Microsoft EXCEL can help Operations Researchers perform a sensitivity analysis

•Later, we will visit the computer lab to use EXCEL

•Next, we will create a template that will help when you create a Linear Programming worksheet on EXCEL

Fill-in Today’s Date. We will discuss the other boxes as a class.

A B C D E F

1 Date: __________2 Lego Activity3 Profit Maximization4

5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8Objective Function

($)9

10 Constraints Used Available

11Maximum # of Large

Blocks

12Maximum # of Small

Blocks13

14

Using Technology . . . . Microsoft Excel’s “Solver” application allows a user to • find an Optimal Solution• explore finding a new Optimal Solution when given new profit amounts for tables and/or chairs.• find a new Optimal Solution when given new limitations on the resources available.

Later in the computer lab you will conduct such an exploration.

First, a reminder of the Lego formulas. . .

At the bottom of your Lego Template (or on the back of it), there is some blank space.• Write these formulas in that blank space• I recommend leaving space below each formula

to write the corresponding Excel Formulas

P = 16x1 + 10x2 2x1 + 1x2 ≤ 6 2x1 + 2x2 ≤ 8

Objective Function

Large Block Constraint

Small Block Constraint

One more thing to know about Excel...

A B C D E F

1 Date: __________

2 Lego Activity

3 Profit Maximization

4

5 Decision VariableTables

(x1)

Chairs

(x2)

6 Decision Value

7 Total Profit

8Objective Function

($)

9



Blocks


Blocks

13

14

We will leave Cell B6 and C6 empty – to act as our variables for Excel.

Eventually…after you input everything and Run Excel Solver, Excel will enter

the solutions into these cells.

Next…We will set up the template with our current values.

A B C D E F

1 Date: __________

2 Lego Activity


4


(x1)

Chairs

(x2)

6 Decision Value

7 Total Profit

8Objective Function

($)

9



Blocks


Blocks

13

14

These cells will contain the coefficients of the Objective

Function.

Today…We will set up the template with our current values.

A B C D E F

1 Date: __________

2 Lego Activity


4


(x1)

Chairs

(x2)

6 Decision Value

7 Total Profit

8Objective Function

($)

9



Blocks


Blocks

13

14

16 10

These cells will contain the coefficients of

the LARGE Block Constraint.


A B C D E F

1 Date: __________

2 Lego Activity


4


(x1)

Chairs

(x2)

6 Decision Value

7 Total Profit

8Objective Function

($)

9



Blocks


Blocks

13

14

1

16 10

2This cell will contain the number of available

LARGE Blocks.


A B C D E F

1 Date: __________

2 Lego Activity


4


(x1)

Chairs

(x2)

6 Decision Value

7 Total Profit

8Objective Function

($)

9



Blocks


Blocks

13

14

1

16 10

2 6

Similarly for the Small Block Constraint You Try!

This is what you should have. Check your template

A B C D E F

1 Date: __________

2 Lego Activity


4


(x1)

Chairs

(x2)

6 Decision Value

7 Total Profit

8Objective Function

($)

9



Blocks


Blocks

13

14

1

16 10

2

2 2

6

8

Using Technology . . . .

•The next step is to define to Excel the equation for calculating the profit, which is our objective function.

•You will type this equation in as a formula when you are in the computer lab.

KEY INFO: We must tell Excel our formulas by using Cell Names, Math Symbols, and Numbers

•Then after you run Excel Solver, the computer will determine the profit made based on the parameters entered.

P = 16x1 + 10x2


(x1)

Chairs

(x2)

6 Decision Value

7 Total Profit

8Objective Function

($)

9



Blocks


Blocks

13

1

16 10

2

2 2

6

8

The objective function will be written using EXCEL language, which will then be placed in this cell when we go to the computers.

Due to space limitations, write this at the bottom of the page OR turn your paper over and write this on the back of your paper.

Cell F8: = B8*B6 + C8*C6P = 16x1 + 10x2

A B C D E F

Using Technology . . . . •Now, we will discuss how to define the LARGE Block Constraint inequality to Excel.

•The left side of the inequality will be typed in as a formula when you are in the computer lab.

•Then after Solver is run, the computer will determine the actual number of LARGE blocks that were used to obtain the Optimal Solution.

2x1 + 1x2 ≤ 6

5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8Objective Function

($)9



Blocks


Blocks13

14

1

16 10

2

2 2

6

8

The left side of the LARGE Block Constraint inequality will be written using EXCEL language, which you will type in this cell when we go to the computers. Due to space limitations, write this at the bottom of the page OR turn your paper over and write this on the back of your paper.

Cell D11: = B11*B6 + C11*C62x1 + 1x2 ≤ 6

5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8Objective Function

($)9



Blocks


Blocks13

14

1

16 10

2

2 2

6

8

In a similar manner, the left side of the Small Block Constraint inequality will be written using EXCEL language, which will then be placed in this cell when we go to the computers. Due to space limitations, write this at the bottom of the page OR turn your paper over and write this on the back of your paper.

You Try!

Cell D12: = B12*B6 + C12*C6

2x1 + 2x2 ≤ 8

A B C D E F

1 Today’s Date2 Lego Activity3 Profit Maximization4

5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total

Profit

8Objective Function

($) 16 10

=B8*B6+

C8*C6

9



Blocks 2 1

=B11*B6+

C11*C6 6


Blocks 2 2

=B12*B6

+C12*C6 8

13

14

Finally, just for cosmetic purposes we put an inequality symbol in these cells. This serves no operational purpose to EXCEL, but is there for the user’s benefit.

Check your Template Next!

A B C D E F

1 Today’s Date2 Lego Activity

3Profit

Maximization4

5Decision

VariableTables

(x1)

Chairs

(x2)

6Decision

Value

7 Total Profit

8Objective

Function ($) 16 10 =B8*B6+C8*C69


11Maximum # of

Large Blocks 2 1 =B11*B6+C11*C6 6

12Maximum # of

Small Blocks 2 2 =B12*B6+C12*C6 813

FINISHED. CHECK YOURS.

Remember, we leave Cell B6 and C6 empty – to act as our variables.

After you input everything and Run Solver, Excel will put the

solutions in these cells.

NEXT . . .(Time permitting)

• Start on Packet p. 3.• Finishing this is your HW

• EXPECTATIONS– You are working with the people

sitting around you.– Ask for help from your neighbors or

from me.– Remember, you are here to learn and

I am here to help you learn. Doing work in class makes completing the HW easier when nobody is around to help.

Linear Programming - Honors ICM GHHS · 2019-05-16 · Answers for Posting Purposes Warm-Up: 1....

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Transcript of Linear Programming - Honors ICM GHHS · 2019-05-16 · Answers for Posting Purposes Warm-Up: 1....