Anna DeConti, NCSSM Chapter 11 - wikispaces.netsecondarymath.cmswiki.wikispaces.net/file/view/Linear+Programmin… · 273 Sponsored by NC Math and Science Education Network Module

273 Sponsored by NC Math and Science Education Network

Module 11 Linear Programming

prepared by Anna DeConti, NCSSM

Chapter

11.1. background discussion of mathematics274

11.2. timeline & assignments...............275

11.3. Introductory problem..................276

11.4. vocabulary ..................................................... 277

11.5. examples............................................................278

11.6. worksheet......................................................280

11.7. examples-word problems.........282

11.8. review for quiz........................................284

11.9. group project ........................................... 286

11.10. ties to textbooks............................287

11.11. solution keys ...........................................288

11.12. For all practical purposes worksheet....................................................291

11

Advanced Functions and Modeling Workshop Summer 2004

11.1 Background discussion of mathematics:

Students should be able to do the following before they start working on linear programming problems.

Graph linear inequalities Transform liner inequalities for graphing both by hand and on the calculator. Graph inequalities by hand and on a graphing calculator. Determine points of intersection by hand and using the calculator.

While learning the concepts the students should also become more familiar with using their calculators to determine points of intersection and values of y for a given x. The draw vertical line helps students to visualize the vertical lines.

Before you start the mathematical concepts, discuss the value of this modeling technique: i.e.

Linear programming is used to model optimal allocation of resources for any business or institution. A manufacturing company can use it to determine the optimal number of units it should produce to make the maximum profit. A company, school system, military, government agency can use linear programming to determine the optimal number of personnel it needs in its work force. It can be a very powerful tool in real life situations.

As you work the problems the students will need to use their calculators along with sketches of the feasible region to determine the vertices.

Module 11: Linear Programming 274


11.2. Timeline and Assignments

Linear programming should take 5 days on the 90 minute block.

Day 1 – Introduction problem, vocabulary, Worksheet: Background Examples. HW: Introduction worksheet or Stewart: p731: 1-4

Day 2 – Show video – All Practical Purposes Examples – WS :Word problems and/or Examples from Stewart book: pgs 725-728. HW: 2–3 problems from Stewart pp 731–732: 5–7

or Forrester, Algebra & Trigonometry, pp162 – 165:1–6 or word problems from your Algebra 2 text

Day 3 – Finish word problems

Day 4 – Worksheet Review

Day 5 – Quiz



11.3. Introductory Problem

Introductory Application – adapted from the Mathematics Teacher February 1999, National Council of Teachers of Mathematics, Reston VA.

Animation: Use the animation Product-Mix Problem on Linear Programming in the Algebra section of the TIGER website:

http://www.dlt.ncssm.edu/TIGER

The problem: Suppose a factory manufactures only stools and chairs and that the profit on one stool is $25 and on one chair is $45. Each stool requires one cushion and two arms/legs pieces of stock. Each chair requires two cushions and two arms/legs pieces of stock. Suppose that you have only six cushions and eight arm/leg pieces of stock. How many stools and how many chairs should you build to maximize profit?

Supplies: Lego pieces/plastic cubes (2 different kinds) – 6 for cushions and 8 for arms/legs – for

every 2 students or Colored stock paper cut into 6 pieces of one color and 8 pieces of another.

Strategy: Pair off the students and allow them to see what they can build. Possible solutions:

4 stools 3 stools and 1 chair 2 stools and 2 chairs 3 chairs

Which of these will give the maximum profit? Let them calculate the profits. (Maximum profit will occur when they build 2 stools and 2 chairs.)

Teach the basics: Present the vocabulary and introductory linear programming problems.



11.4. VOCABULARY: Linear Programming

Optimization

Linear Programming

Objective Function

Constraints

Feasible Region

Vertices of the Feasible Region

Determining the Maximum and Minimum Values of the Objective Function



Examples – Background

Name_________________________________________

1. Given the objective function: 3 5M x y= + and the constraints:

63 4

00

x yx y

xy

+ ≤⎧⎪ − ≤⎪⎨ ≥⎪⎪ ≥⎩

a) Graph the constraints. b) Find the vertices of the feasible region. c) Determine the maximum and minimum values of the objective function.



2. Given the following objective function and the vertices of the feasible region, determine the maximum and minimum vales.

Objective function: y 3P x= + ( ) ( ) ( ) ( )3,0 , 4,5 , 1,6 , 7, 5− − − Vertices:

3. Given the graph of the constraints and the objective function, determine the vertices of the feasible region, the values of the objective function and the maximum and minimum values.

P ( , ) 5P x y x y= − + +




Introduction: Linear Programming Worksheet

Name: ________________________________

1. Given the objective function: ya) On the graph below, graph the given constraints and shade the feasible region.

b) What are the vertices (points of intersection) for the feasible region?

c) Using the above points, determine the maximum and minimum values for P. At what points do they occur?

2. A feasible region has vertices at

4 7P x= +

( )4,3 , ( )1,6 and ( )4,0− . Use this to find the maximum and minimum values for each objective function.

y b) ya) 3H x= + 2 3J x= − + c) y2 4K x= − Max ____________ Max ____________ Max ____________ Max Point ________ Max Point ________ Max Point ________ Min _____________ Min _____________ Min _____________ Min Point ________ Min Point ________ Min Point ________

82

00

x yy x

xy

+ ≤⎧⎪ − ≤⎪⎨ ≥⎪⎪ ≥⎩


3. Use your own graph paper to determine the following. Find the maximum and minimum values (if they exist) for the objective functions with the given constraints.

a) 2R x y= − b) y2 3F x= − −

Constraints Constraints

Max ______________ Max ______________Max Point _________ Max Point _________Min ______________ Min ______________Min Point _________ Min Point _________

4. Given the following feasible regions and the objective function, find the maximum and minimum values.

a) b)

⎪⎪⎩

⎪⎪⎨

⎧

≤≥

+≤+−≥

40

321

xy

xyxy

⎪⎪⎩

⎪⎪⎨

⎧

≥≤≥

−≥−

16

12

yy

xyyx

0.60P x y= − + 1.6 3.7 0.5T x y= − −

Max ______________ Max ______________ Max Point _________ Max Point _________ Min ______________ Min ______________ Min Point _________ Min Point _________



Examples – Word Problems

Go back to the introductory problem and rework it in linear programming terms 1. Suppose a factory manufactures only stools and chairs and that the profit on one stool is $25

and on one chair is $45. Each stool requires one cushion and two arms/legs pieces of stock. Each chair requires two cushions and two arms/legs pieces of stock. Suppose that you have only six cushions and eight arm/leg pieces of stock. How many stools and how many chairs should you build to maximize profit?

i) Define the variables

ii) Write the objective function.

iii) If the constraints are: - The number of stools and the number of chairs are greater than 0. - One stool and 2 chairs use a maximum of 6 cushions. - Two stools and two chairs use a maximum of 8 arm/leg pieces.

Write the constraints.

iv) Graph the feasible region.

v) Determine the vertices of the feasible region.

vi) Determine the profit made from each vertex.

vii) What is the maximum profit?



2. (Adapted from Algebra and Trigonometry by Paul Forrester) Danielle and Tim decide to open a music shop in which they will sell guitars and keyboards. They want to find out the maximum amount of money they may have to borrow to purchase new instruments. Each guitar will cost them $150 and each keyboard $350.

a) Define the variables.

b) Write the objective function on the amount of money they have to borrow to pay for the instruments.

c) Danielle and Tim have certain restrictions on the number of instruments they can purchase. i) They can only purchase a maximum of 75 instruments. ii) Because guitars are more popular than keyboards the number of guitars they will

purchase will be at least twice the number of keyboards. iii) To get started they feel they need at least 10 guitars and 7 keyboards. Write the constraints.

d) Graph the feasible region e) Determine the vertices of the feasible region.

f) Determine the amount they need to borrow from each vertex.

g) What is the maximum amount they need to borrow?


Advanced Functions and Modeling Workshop Module 11


11.8. Review for Quiz Linear Programming

Name __________________________ 1. Given the following objective function and constraints graph the constraints and determine

the maximum and minimum values. y Constraints:

What are the vertices of the feasible region? _________________________

Maximum Value ___________ Minimum Value __________

Maximum Point __________ Minimum Point __________

2. Given the objective function and the graphed feasible region, determine the maximum and minimum values.

Objective function

4 2C x= +

⎪⎪⎩

⎪⎪⎨

⎧

−≥−≤+

≥≥

282

00

yxyx

yx

: 2T x y= − + What are the vertices of the feasible region?

_________________________

Maximum Value: ___________________

Minimum Value: ___________________

3. The vertices of a feasible region for a linear programming problem are :

( )1,5 , ( )8,1 ,

( )2, 6− − , ( )6, 3− − .objective function is

Determine the maximum and minimum values and points given that the y .

Maximum Value ___________ Minimum Value __________

Maximum Point __________ Minimum Point __________

2 3T x= +



4. Two manufacturing plants in the local area produce DVD players: RCA and Pioneer. RCA earns a profit of $60 per DVD player sold and Pioneer earns a profit of $50.

RCA uses 10 hours of general labor time to produce one DVD player and Pioneer uses 1 hour, and the combined maximum amount of time for general labor for the two companies is 4000 hours. Machine time for each RCA DVD player is 1 hour and for Pioneer it is 3 hours, and the total machine time for both companies cannot exceed 1500 hours. RCA uses 5 hours for technical labor on each DVD player and Pioneer uses 2 hours, and the maximum total time for technical labor is 2300 hours.

a) Write the objective function for the total profit for the two companies.

b) Write the constraints.

c) Graph the feasible region.

i) What is the maximum value of the objective function? ____________

ii) What does the maximum value mean in our problem?


11.9. Group Project

Your club has decided to volunteer to clean up your school grounds and plant some bushes and trees. You determine that the bushes you want to plant average $15 each and each tree $25. You will definitely buy both bushes and trees. You realize that you cannot plant more than 18 trees. Your school says that you should plant at least 12 plants but no more than 30. The number of trees must be at least ½ the number of bushes.

1. On a poster board make a scale drawing of your school and the current surrounding grounds.

2. On a piece of paper, solve the above problem. a) Define the variables. b) Determine the objective function. c) Write out the constraints. d) Show the graph and shade in the feasible region. e) Determine the vertices of the feasible region. f) Determine the cost for each vertex.

3. Under the given conditions, what is the minimum amount you could spend on the plants?

You will need to go to a garden center or plant catalogue for the following.

4. On another sheet of paper write a 100 word essay on the number and types of shrubs and trees you can purchase.

5. Sketch your new plantings around your school.

6. On another sheet of paper describe how you will obtain the funding for this project.



11.10. Ties to sections in books/Other Resources

Functions Modeling Change: No sections

Algebra and Trigonometry: pp 725-733

Other Algebra 2 books: a) McDougal Littell (2001) – Larson, Boswell, Kanold, Stiff

pp 163–168 b) Holt, Reinhart, Winston (2003) –Schultz, etc

pp 187–194 c) Glencoe/McGraw-Hill (2001) – Collins, etc

pp 153–164

Mathematics Teacher, February 1999, Volume 92 number 2, pp 118–123

NCSSM Algebra 2 website http://www.dlt.ncssm.edu/algebra/

NCSSM Tiger http://www.dlt.ncssm.edu/TIGER

Video For All Practical Purposes #5 Go to www.learner.org

Other Web Sites: Exploremath.com – Interactive (limited use)



Vocabulary: Linear Programming Teacher Worksheet

Optimization: The process of finding the maximum or minimum value of some varying quantity.

Objective Function: The function which is optimized.

Constraints: The linear inequalities which form the feasible region.

Feasible Region: The graph of the system of constraints.

Vertices of the Feasible Region: The points of intersection of the constraint lines.

Determining the Maximum and Minimum values for the Objective Function: The process of using the vertices of the feasible region to determine the values of the objective function. From these, determining which is the largest and which is the smallest.

Linear Programming: The process of using an objective function an constraints to determine the feasible region.



Worked Examples –Background Teacher Worksheet

1. Given the objective function: 3 5M x y= + and the constraints:

63 4

00

x yx y

xy

+ ≤⎧⎪ − ≤⎪⎨ ≥⎪⎪ ≥⎩

a) Graph the constraints. b) Find the vertices of the feasible region. c) Determine the maximum and minimum values of the objective function.

Solution:

a) Transform , to Transform to Graph:

6x y+ ≤ 6y x≤ − + 3 4x y− ≤ 3 4y x≥ −

Window: [0,10], [0,10]

b) Vertices: ( ) ( ) ( ) ( )0,0 , 4 3,0 , 2.5,3.5 , 0,6 c) Values for the objective function:

Vertex 3 5M x y= +

( )0,0 0

( )4 3,0 4

( )2.5,3.5 25

( )0,6 30

Maximum value is 30 at Minimum value is: 0 at( )0,6 ; ( )0,0

2. Given the following objective function and the vertices of the feasible region, determine the maximum and minimum vales.

Objective function: y 3P x= + ( ) ( ) ( ) ( )3,0 , 4,5 , 1,6 , 7, 5− − − Vertices:



Solution:

Vertex y3P x= +

( )3,0 9

( )4,5 17

( )1,6− 3

( )7, 5− − –26

Maximum value is 17 at while the minimum value is: –26 at3. Given the graph of the constraints and the objective function, determine the vertices of the

feasible region, the values of the objective function and the maximum and minimum values.

( )4,5 ( )7, 5− −

( , ) 5P x y x y= − + +

Solution: The vertices of the feasible region are: ( ) ( ) ( ) ( )0,0 , 0,1 , 4, 4 , 10, 10−

Vertex 5P x y= − + +

( )0,0 5

( )0,1 6 Maximum value

( )4, 4 5

( )10, 10− –15 Minimum value



Introduction: Linear Programming Worksheet Solutions

1. Given the objective function: ya) On the graph below, graph the given constraints and shade the feasible region.

4 7P x= +

Window [0,10], [0,10]

b) What are the vertices (points of intersection) for the feasible region?

Solution: )

c) Using the above points, determine the maximum and minimum values for P. At what points do they occur?

Solution: Vertex y

( ) ( ) ( ) (0,0 , 0, 2 , 3,5 , 8,0

4 7P x= +

( )0,0 0 Minimum value

( )0, 2 14

( )3,5 47 Maximum value

( )8,0 32



( )4,3 , ( )1,6 and ( )4,0−2. A feasible region has vertices at . Use this to find the maximum and minimum values for each objective function.

a) y b) y3H x= + 2 3J x= − + c) y2 4K x= − Max: 15 Max: 16 Max: –4 Max Point: Max Point: ( )4,3 ( )1,6 Max Point:

Min –12 Min 1 Min –22 Min Point Min

( )4,3

( )4,0− Point ( )4,3 Min

3. Use your own graph paper to determine the following. Find the maximum and minimum values (if they exist) for the objective functions with the given constraints.

Point ( )1,6

a) 2R x y= − b) y2 3F x= − −

Constraints Constraints

Vertices of feasible region

⎪⎪⎩

⎪⎪⎨

⎧

≤≥

+≤+−≥

40

321

xy

xyxy

⎪⎪⎩

⎪⎪⎨

⎧

≥≤≥

−≥−

16

12

yy

xyyx

( ) ( ) ( ) ( )2 3,5 3 , 4,11 , 1,0 , 4,0−

( 2 3,5 3) 3(4,11) 3(1,0) 2(4,0) 8

RRRR

− == −

==

−

Vertices of feasible region ( ) ( ) ( ) ( )0,1 , 1,1 , 6,6 , 5 2,6

(0,1) 3(1,1) 5(6,6) 30(5 2,6) 23

FFFF

= −= −= −

= −

Max: 8 Max: –3 Max Point: Max Point: ( )4,0 ( )0,1

Min –3 Min –30 Min Point ( )4,11 , ( )2 3,5 3− Min Point ( )6,6



4. Given the following feasible regions and the objective function, find the maximum and minimum values.

a. b. 0.60P x y= − + 1.6 3.7 0.5T x y= − −

(0,0) 0.5(2,4) 12.1(8, 2) 19.7

TTT

= −= −

− =

(0,0) 0.6(0,8) 7.4(8,0) 8.6

PPP

== −=

Max: 8.6 Max: 19.7 Max Point: Max Point:( )8,0 ( )8, 2−

Min –7.4 Min –12.1 Min Point Min Point ( )0,8 ( )2,4



Examples – Word Problems SOLUTIONS

1. Suppose a factory manufactures only stools and chairs and that the profit on one stool is $25 and on one chair is $45. Each stool requires one cushion and two arms/legs pieces of stock. Each chair requires two cushions and two arms/legs pieces of stock. Suppose that you have only six cushions and eight arm/leg pieces of stock. How many stools and how many chairs should you build to maximize profit?

i) Define the variables

x: stools y: chairs

ii) Write the objective function. y

iii) If the constraints are: - The number of stools and the number of chairs are greater than 0. - One stool and 2 chairs use a maximum of 6 cushions. - Two stools and two chairs use a maximum of 8 arm/leg pieces.

Write the constraints.

8

iv) Graph the feasible region.

25 45P x= +

00

2 62 2

xy

x yx y

⎧ ≥⎪⎪⎪⎪ ≥⎪⎨⎪ + ≤⎪⎪⎪ + ≤⎪⎩

Window: [0,5,1], [0,5,1]

v) Determine the vertices of the feasible region.

Vertices: ( ) ( ) ( )0,3 , 2, 2 , 4,0

vi) Determine the profit made from each vertex.

(0,3) $135(2,2) $140(4,0) $100

PPP

===



vii) What is the maximum profit?

Maximum profit is $140.

2. (Adapted from Algebra and Trigonometry by Paul Forrester) Danielle and Tim decide to open a music shop in which they will sell guitars and keyboards. They want to find out the maximum amount of money they may have to borrow to purchase new instruments. Each guitar will cost them $150 and each keyboard $350.

a) Define the variables.

x = number of guitars y = number of keyboards

b) Write the objective function on the amount of money they have to borrow to pay for the instruments.

150 350B x y= +

c) Danielle and Tim have certain restrictions on the number of instruments they can purchase. i) They can only purchase a maximum of 75 instruments. ii) Because guitars are more popular than keyboards the number of guitars they will

purchase will be at least twice the number of keyboards. iii) To get started they feel they need at least 10 guitars and 7 keyboards. Write the constraints.

107

752

xy

x yx y

⎧ ≥⎪⎪⎪⎪ ≥⎪⎨⎪ + ≤⎪⎪⎪ ≥⎪⎩

d) Graph the feasible region

Window: [0,75,10] [0,75,10]

e) Determine the vertices of the feasible region.

Vertices: )( ) ( ) (14,7 , 68,7 , 50, 25



f) Determine the amount they need to borrow from each vertex.

g) What is the maximum amount they need to borrow?

$16,250

(20,10) $6,500(65,10) $13,250(50,25) $16,250

BBB

===




Review for Quiz Linear Programming Solutions

1. Given the following objective function and constraints graph the constraints and determine the maximum and minimum values.

y Constraints:

4 2C x= +

⎪⎪⎩

⎪⎪⎨

⎧

−≥−≤+

≥≥

282

00

yxyx

yx

What are the vertices of the feasible region? )

Maximum Value 16 Minimum Value 0

Maximum Point ) Minimu Point 2. Given the objective function and the graphed feasible region, determine the maximum and

minimum values. Objective function

( ) ( ) (2, 4 , 0,0 , 4,0

( ) (2, 4 , 4,0 m ( )0,0

: 2T x y= − + What are the vertices of the feasible region?

( ) ( ) ( )0,0 , 6,14 , 0,10

Maximum Value: 10

Minimum Value: 0



3. The vertices of a feasible region for a linear programming problem are : ( )1,5 , ( )8,1 ,

( )2, 6− − , ( )6, 3− − .objective function is

Determine the maximum and minimum values and points given that the y .

Maximum Value 19 Minimum Value –22

Maximum Point Minimum Point4. Two manufacturing plants in the local area produce DVD players: RCA and Pioneer. RCA

earns a profit of $60 per DVD player sold and Pioneer earns a profit of $50.

RCA uses 10 hours of general labor time to produce one DVD player and Pioneer uses 1 hour, and the combined maximum amount of time for general labor for the two companies is 4000 hours. Machine time for each RCA DVD player is 1 hour and for Pioneer it is 3 hours, and the total machine time for both companies cannot exceed 1500 hours. RCA uses 5 hours for technical labor on each DVD player and Pioneer uses 2 hours, and the maximum total time for technical labor is 2300 hours.

a) Write the objective function for the total profit for the two companies.

x = RCA y = Pioneer y

b) Write the constraints.

c) Graph the feasible region.

2 3T x= +

( )8,1 ( )2, 6− −

60 50P x= +

10 40003 1500

5 2 2300

x yx yx y

⎧ + ≤⎪⎪⎪⎪ + ≤⎨⎪⎪ + ≤⎪⎪⎩

i) What is the maximum value of the objective function? 38,000

ii) What does the maximum value mean in our problem?

The maximum profit of $3800 is made when RCA makes 300 DVD’s and Pioneer makes 400.


Group Project — Teacher Guidelines

The students should be placed in groups of 3 or four.

Problem: Your club has decided to volunteer to clean up your school grounds and plant some bushes and trees. You determine that the bushes you want to plant average $15 each and each tree $25. You will definitely buy both bushes and trees. You realize that you cannot plant more than 18 trees. Your school says that you should plant at least 12 plants but no more than 30. The number of trees must be at least ½ the number of bushes.

1. On a poster board make a scale drawing of your school and the current surrounding grounds.

This will vary by school and by teacher. Nonetheless, have the students do this. It gives them another opportunity to look at scaling and ratio/proportion.

2. On a piece of paper, solve the above problem. a) Define the variables.

x = bushes y = trees

b) Determine the objective function.

y

c) Write out the constraints.

15 25C x= +

181230

12

yx yx y

y x

⎧ ≤⎪⎪⎪⎪ + ≥⎪⎪⎪⎨ + ≤⎪⎪⎪⎪ ≥⎪⎪⎪⎩

d) Show the graph and shade in the feasible region.

Window: [0,50,10] [0,50,10]



e) Determine the vertices of the feasible region.

Vertices: ) (0,18) $450 (0,12) $300 (8,4) $220 (20,10) $550 (12,18) $630

f) Determine the cost for each vertex.

Costs:

3. Under the given conditions, what is the minimum amount you could spend on the plants?

Plantings: 8 shrubs, 4 trees Cost: $220

4. You will need to go to a garden center or plant catalogue for the following.

5. On another sheet of paper write a 100 word essay on the number and types of shrubs and trees you can purchase.

6. Sketch your new plantings around your school.

7. On another sheet of paper describe how you will obtain the funding for this project.

Rubric for Grading:

Activity Number of points

( ) ( ) ( ) ( ) (0,18 , 0,12 , 8, 4 , 20,10 , 12,18

( )( )( )( )( )

0,18 $450

0,12 $300

8,4 $220

20,10 $550

12,18 $630

C

C

C

C

C

=

=

=

=

=

1. Scale drawing 15 2. Define variables 5 Objective function 5 Constraints 15 Graph 10 Vertices 15 Cost for each vertex 5 3. Minimum amount 5 5. 100 Word essay 10 6. Sketch of plantings 10 7. Funding 5

Total 100



Linear Programming Name:

For All Practical Purposes – Tape 5 1. Limits or fixed quantities are called __________________________.

2. The Process of making the best choice is called __________________________.

3. The recipe for cranapple juice is: __________________________ quarts of cranberry juice

__________________________ quarts of apple juice.

4. The recipe for appleberry juice is: __________________________ quarts of cranberry juice

__________________________ quarts of apple juice.

5. How many quarts of cranberry juice are available for use? __________________________

6. How many quarts of apple juice are available for use? __________________________

7. How much profit does the company make on cranapple juice? __________________________

8. How much profit does the company make on appleberry juice? __________________________

9. What was x used to represent? __________________________

10. What was y used to represent? __________________________

11. What is the expression for the total juice profit: __________________________

12. For the total production of cranapple juice ______ quarts of cranberry and ______ quarts of apple juice are used.

13. For the total production of appleberry juice ______ quarts of cranberry and ______ quarts of apple juice are used

14. What is the inequality for the amount of cranberry juice used? _________________________

15. What is the inequality for the amount of apple juice used? _________________________

16. Why should and also be used on the graph?

__________________________________________________________________

17. Any value of x and y that satisfy the constraints is called a _________________.

0x≥ 0y ≥



18. What is the equation for the apple juice line? __________________________

19. What are the four corner points: __________ __________ __________ ___________

20. Why wouldn’t the point (0, 0) be a good choice? __________________________

21. What principle helps narrow the search in linear programming problems and what does it mean?

22. Which corner gave the highest profit? ___________ What is the profit? ______________

23. In a full sentence answer the question: What production yields what maximum profit.


Anna DeConti, NCSSM Chapter 11 - wikispaces.netsecondarymath.cmswiki.wikispaces.net/file/view/Linear+Programmin… · 273 Sponsored by NC Math and Science Education Network Module

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