linear programming - honors icm ghhs...arrival instructions take out: your homework, calculator, and...

Linear Programming

ICM Unit 3

Day 2

Arrival InstructionsTake out: Your homework, calculator, and the unit outline

Pick up:

1. A piece of graph paper

2. Colored pencils basket to share with partner

3. Linear Programming Practice—Classwork Handout

Warm-Up:

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

2. Formulate the Linear Programming bya. 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!!

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 in 1 equation.

Notes: How to Formulate a Linear Programming ProblemUsing the Lego Problem

Step 1: Define the decision variables

x1 :____________________

x2 : ____________________

Step 2: Write the objective function (profit statement)

P(x1 , x2 ):____________________

“production rate”Important to put

# of

# of tables

# of chairs

16x1 + 10x2

Formulating the Lego ProblemStep 3: Write the constraints

_____________

_____________

**Remember don’t be a dummy. Be sure to write the dummy constraints. These are the non-negativity constraints**

____________ ___________

TABLES CHAIRS Resources

6 Large Pieces

8 Small Pieces

2x1 1x2+

+

≤

≤2x12x2

1 22 1 6x x

1 22 2 8x x

1 0x 2 0x

Formulating the Lego Problem

Step 4: Graph the constraints and plot intersection points of the constraints.

Note: More about why the corner points are important during Tomorrow’s lesson.

Document Camera—Volunteer to Guide Us?

Presentation:LEGO Furniture CompanyYour presenters will be:

______________ & ______________

Student PresentersUsing the document camera.

Bring your Work

Arrival InstructionsTake out: Your homework, calculator, and the unit outline

Pick up:

1. A piece of graph paper

2. Colored pencils basket to share with partner

3. Linear Programming Practice—Classwork Handout

Warm-Up:

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

2. Formulate the Linear Programming bya. 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!!

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 in 1 equation.

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.

Using Technology . . . .

•Microsoft EXCEL can help Operations Researchers perform a sensitivity analysis

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

•Next, we will create a template that will help when you actually 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 explore finding a new Optimal Solution when given new profit amounts for tables and/or chairs.

•In addition, a user can 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, there is some blank space.•Write these formulas down in that blank space

•Objective Function

•Large Block Constraint

•Small Block Constraint

P = 16x1 + 10x2

2x1 + 1x2 ≤ 6

2x1 + 2x2 ≤ 8

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

A B C D E F

1 Date: 8/27/20142 Lego Activity3 Profit Maximization4

5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8Objective Function

($)9



Blocks


Blocks13

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


5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8Objective Function

($)9



Blocks


Blocks13

14

16 10

These cells will contain the coefficients

of the LARGE Block Constraint.


A B C D E F


5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8Objective Function

($)9



Blocks


Blocks13

14

1

16 10

2

This cell will contain the number of

available LARGEBlocks.


A B C D E F


5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8Objective Function

($)9



Blocks


Blocks13

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


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

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

•This equation will be typed in as a formula when you are in the computer lab.

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

P = 16x1 + 10x2

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 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 you should turn your paper over and write this on the back of your paper.

Cell F8: = B8*B6 + C8*C6

P = 16x1 + 10x2

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 will then be placed in this cell when we go to the computers.


Cell D11: = B11*B6 + C11*C6

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

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.


You Try! Cell D11: =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

8

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



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


Blocks 2 2 =B12*B6+C12*C6 8

13

14

This is what you should have. Check your template

A B C D E F


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.

A B C D E F


5Decision Variable

Tables

(x1)

Chairs

(x2)

6 Decision Value

7Total Profit

8

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



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


Blocks 2 2 =B12*B6+C12*C6 8

13

14

FINISHED. CHECK YOURS.

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.

Sensitivity Analysis on next slides….

• Save this for another day for Spr ’15 due to moving computer lab to day 3 (due to work keys testing and days lab was available)

Revisiting the Lego Furniture CompanyObtaining Maximum Profit

If a furniture company obtains 6 large and 8 small pieces every day, we determined that the “OPTIMAL SOLUTION” was to produce 2 tables and 2 chairs every day.

Stepping Beyond the Solution.Operations researchers must analyze solutions. One sort of

analysis is called a sensitivity analysis.

Sensitivity analysis: how sensitive a solution is to changes in the parameters of the problem.

LETs Explore . . . . . .

For example, in the LEGO problem one of the parameters is the availability of large pieces.

Altering the number of large pieces might change what the optimal solution will be.

LEGO Follow-up—Beyond the SolutionDiscuss with the people that are sitting around you the

answers to the following questions:

1. Assuming there are still six large pieces available, how would the production rate change if the number of small pieces increased to nine? What would be the optimal profit?

2. Going back to having only eight small pieces but increasing the number of large pieces to seven, how would the production rate change? What would be the optimal profit?

Next slides skipped Fall ‘17….

• Saved for future use

• Skipped slide due to moving computer lab to day 3 (due to half day on day 2)

Turn in your template

• I will hold onto these until tomorrow when we go to the computer lab to work with Excel.

• I will verify that you have correctly written down the required information.

• To do tonight’s homework, you can reference this powerpoint that will be posted on BLACKBOARD

Hint for Tonight’s HW—Read Carefully

• Packet p. 4 Here is what part of the problem 1) states:

“There are 4 hours available to prepare sandwiches. If chicken salad sandwiches take 7 minutes to prepare and roast beef sandwiches take 10 minutes, how many of each type of sandwich should be prepared to maximize the profit?”

Let’s write one of the constraints together:We are limited by the total amount of time to make sandwiches.

Therefore, (time to make roast beef sandwiches) + (time to make chicken salad sandwiches) must be less than the total amount of time available. AND the units of measurement much match.

1 1# of roast beef sandwiches # of chicken salad sandwichesx x

1 210 7 4(60) minutesx x

Practice = Quiz Review Sheet

• Prepare for quiz 1– Complete the back of the warm-up worksheet.

– Done early? Work on Packet p. 3 & 4 (the other part of tonight’s HW)

Instructions For Computer Lab

Before we go . . .

Read the handout/instructions on what to do upon arrival

Watch the demo that I am about to show you

You MUST work with a partner!!!

We will dismiss from the computer lab. So, you will take your belongings.

linear programming - honors icm ghhs...arrival instructions take out: your homework, calculator, and...

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