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Linear Programming

Linear Programming is actually one of the most useful topics in math because it is a simple model for problems with limited resources. Many large companies, even the military, use linear programming to solve problems such as finding maximum profits when given certain restrictions on different variables.

There are 5 main steps to solving a Linear Programming problem:

1) Place all of the necessary information into a neat and organized table .

2) Find out what has to be maximized or minimized, and in most cases it is the profit or cost. Then, write the profit/cost equation.

3) Find the variables of the problem and what parts can be controlled. These equations will be known as the constraints. Depending on the problem there can be many or very few constraints.

4) Once the constraints have been established, plot them on a graph to find the vertices of the linear function.

5) After the graph is drawn, the vertices of the graph will tell you the maximum or minimum of the system by plugging them into the profit/cost equation.

Now lets try this problem together.

A magazine company sells two main types of magazines: Healthwise sells for $12 and Superteen sells for $10. It costs the company $9 to produce Healthwise and $8 to produce Superteen. In one week the publishing company can print 200 – 300 copies of Healthwise and from 100 – 250 copies of Superteen, but no more than 500 copies in total. How many of each type should be printed in order for the company to make maximum profit.

The first step is to make a chart to make everything neat and organized. This step is used to make things easier for you to understand.

Magazine Sales Amount Production Costs

Number of Printable Copies

Healthwise $12 $9 200 – 300Superteen $10 $8 100 – 250

Now that the information is organized we can find the profit equation:

X= Healthwise Y = Superteen

The Profit equation will be P = the Profit X makes + the Profit Y makes

Since production costs for Healthwise is $9, we have to subtract it from the Sales amount, which is $12.

So $12 - $9 = $3

The production costs for Superteen are $8 and the price is $10 so the profit will be the two subtracted.

So $10 - $8 = $2

The Profit Equation will be P = 3x + 2y

Now we have to find the constraints. These constraints will explain how to draw the graph.The constraints will be:

x + y ≤ 500 This constraint means that the number of Healthwise magazine (x) plus the number of Superteen magazines (y) is less than or equal to 500, because all together it said in the problem that no more than 500 copies can be made in one week. Therefore x + y has to be less than or equal to 500

200 ≤ x ≤ 300 This constraint explains that only 200 – 300 copies of Healthwise can be printed during one week

100 ≤ y ≤ 250 The last constraint shows the amount of copies that Superteen can make within a week.

To find out how to graph the line x + y ≤ 500, you have to give x a number to solve for y.

X + y = 500 Lets make x = 100 100 + y = 500 y = 500 – 100 y= 400 So the co-ordinates are (100, 400)

Lets get another set of values to draw the line correctly, but this time, lets plug in for y. x + y = 500 Lets make y = 200 x + 200 = 500 x = 500 – 200 x = 300 So the co-ordinates are (300, 200)

As you know x will have two straight lines one on 200 and one on 300. Y will also have two straight lines, one on 100 and the other on 250.

Now we plot the graph, which will look like this:

Notice that the area shaded is shared by all regions of the constraints.

As you can see there is a boxed in area that is shaded which has five labelled intercepts:A (200,100), B (300, 100) and E (200, 250). The intercepts D and E are a little different since you can only see one precise intercept (y = 250). To find the other value plug in the x or y value that you do know and solve for x or y (substitution).

D = y = 250 so x + y = 500 x + 250 = 500

x = 500 – 250 x = 250 The vertices D are (250, 250)

C = x = 300 so x + y = 500 300 + y = 500 y = 500 – 300 y = 200

The Vertices for C are (300, 200)

Now that we have all of the vertices we can plug them into our profit equation and solve for the maximum profit.

A) P = 3x +2y B) P = 3x +2y (200,100) (300,100)

P = 3(200) + 2(100) P = 3(300) + 2(100)

P = 600 + 200 P = 900 +200P = $800 P = $1100

C) P = 3x + 2y D) P = 3x + 2y

(300,200) (250,250)

P = 3(300) + 2(200) P = 3(250) +2(250)

P = 900 + 400 P = 750 +500P = $1300 P = $1250

E) P = 3x + 2yP = 3(200) + 2(250)P = 600 + 500P = $1100

P = 3x + 2y

(200,100) $800(300,100) $1100(300,200) $1300◄ma

x (250,250) $1250(200,250) $1100

The company should make 300 copies of Healthwise and 200 copies of Superteen to make a maximum profit of $1300.

Lets try another question together.

A lumber company converts logs into baseball bats. In a week, the company can turn out 400 bats, of which 100 deluxe bats and 150 regular bats are required on a regular basis. The profit of a deluxe baseball bat is $20 and the profit on a regular baseball bat is $30. How many of each type should the lumber company make to have maximum profit?

Remember to first make a chart to organize the information.

Type of Bat Demand Profit Deluxe 100 $20 Regular 150 $30

X = Deluxe Y = Regular

The profit equation this time will be P = 20x + 30y because there are no production costs.

The constraints are:

x + y ≤ 400 This means that in total the company cannot make anymore than 400 bats in one week.

X ≥ 100 The x is greater than or equal to 100 because that is the minimum amount of deluxe bats that the company can make.

Y ≥ 150 The y is greater than or equal to 150 because it is the minimum amount of regular bats that can be made in one week since there is a demand for at least that amount.

To plot the graph we have to find the co-ordinates for x + y ≤ 400. Remember to pick a number for the x or y value and solve for either x or y.

X + y = 400 x = 100100 + y = 400y = 400 –100y = 300 So the co-ordinates are (100,300)

x + y =400x +100 = 400x = 400 –100x = 300 So the co-ordinates are (300,100)

And of course we know that there will be a line at x = 100 and y = 150

Now you can plot the graph, which will look like this:


To find the coordinates at A, you know it is on the x-axis at 100 and the y-axis is 0, so the A is (100, 0).

To find the coordinates of B, we know that it is 0 on the y-axis and intercepts with the line for x + y = 400. So all we do is sub in 0 for y and solve for x.

y = 0x + y = 400x + 0 = 400x = 400The coordinates at B are (400, 0)

To find the coordinates at C, we know that the two lines intersecting are y = 150, and x + y = 400. Again, sub in 150 for y and solve for x.

y = 150x + y = 400x + 150 = 400x = 400 – 150x = 250The coordinates at C are (250, 150)

The coordinates of D are pretty straight forward sine all we need to do is look at the graph and see that the two lines intersecting at D are x = 100 and y = 150. So the coordinates of D are (100, 150) The vertices are: A (100, 0), B(400, 0), C(250, 150) and D(100, 150). Remember that we plug these intercepts into the profit equation and find the maximum profit.

A) P = 20x + 30y B) P = 20x + 30y(100, 0) (400, 0)

P = 20(100) + 30(0) P = 20(400) + 30(0)P = $2000 P = $8000

C) P = 20x + 30y D) P = 20x + 30y(250, 150) (100, 150)P = 20(250) + 30(150) P = 20(100) + 30(150)P = $9500 P = $6500

Vertices P = 20x + 30y(100, 0) $2000(400, 0) $8000(250, 150) $9500

maximum(100, 150) $6500

To make a maximum profit the lumber company must make 250 deluxe bats and 150 regular bats to make a maximum profit of $9500.

Let’s try one last question together:

A window manufacturing company makes two types of windows, regular and heavy duty. Each regular window takes approximately 3 hours to cut and 1 hour to finish. The heavy-duty windows take 2 hours to cut and 4 hours to finish. Each regular window makes a net profit of $80 and the heavy-duty window makes a net profit of $200. If 4 cutting and 6 finishing workers are used 12 hours per day how many of each window should be made for the company to make a maximum profit?

Draw the organization table

Type of Window Cutting Hours Finishing Hours Profit Regular 3 hours 1 hour $80 Heavy-Duty 2 hours 4 hours $200

The profit equation will be P = 80x + 200y.

The constraints are:

3x +2y ≤ 48 It takes three hours to cut a regular window and two hours to cut a heavy duty one. The forty-eight comes from the 4 cutting workers multiplied by how long they worked which was 12 hours at maximum.

x+ 4y ≤ 72 It takes one hour to finish a regular and 4 hours to finish a heavy-duty window. The seventy-two is the 6 workers multiplied by the 12 hours because that is the maximum time that the finishers can work.

To plot the graph we have to draw the two constraints so lets sub in the values for x and y.

3x +2y = 48 3x +2y =483(10) + 2y = 48 3x + 2(3)= 482y = 48 – 30 3x = 48 - 62y = 18 3x = 42y = 9The co-ordinates are (10,9) x = 14

The co-ordinates are (14,3)

x + 4y = 72 x + 4y = 724 + 4y = 72 x + 4(15) = 724y = 72 – 4 x = 72 - 604y = 68 x = 12

y = 17The coordinates are (4, 17) The coordinates are (12, 15)

Here’s what the graph should look like:


A) (0, 0)B) y = 0

3x + 2y = 483x + 0 = 483x = 48x = 16The coordinates of B are (16, 0)

C) 3x + 2y = 48x + 4y = 72For this particular intercept, we use the method of elimination to solve for x and y.(3x + 2y = 48)(-2)-6x – 4y = -96 + x + 4y = 72 -5x = -24 x = 4.8

x + 4y = 724.8 + 4y = 724y = 72 – 4.84y = 67.2 y = 16.8

So, the coordinates for C are (4.8, 16.8)

D) x = 0x + 4y = 720 + 4y = 72y = 18The coordinates at D are (0, 18)

P = 80x + 200y(0,0) $0(16,0) $1280(4.8, 16.8) $3744

maximum(0,18) $3600

To make a maximum profit the company would have to sell 4.8 regular windows and 16.8 heavy-duty windows to make a profit of $3744.

Now you can try some of the practice problems on your own.

Practise Questions

Let’s Refresh Your Memory…( A review of finding max. and mins., and y = mx + b)

1) Determine the maximum and minimum values for each of the following:a) (x,y) = 3x + 5y vertices at (4,8), (2,4), (1,1), (5,2) b) (x,y) = x + 4y vertices at (0,7), (0,0), (6,2), (5,4)

2) The size of shoe a person needs varies linearly with the length of his or her foot. The smallest adult shoe size is Size 5, and it fits a 9 inch long foot. An 11inch foot requires a Size 11 shoe.

a) Write the particular equation, which expresses shoe size in terms of foot length.

b) If your foot is a foot long, what size shoe do you need?c) Bob Lanier of the Detroit Pistons wears a Size 22 shoe.

How long is his foot?d) Plot a graph of adult shoe size versus foot length. Use the

given points and calculated points. Be sure that the domain is consistent with the information in this problem.

3) If you jump out of an airplane at high altitude but do not open your parachute, you will soon fall at a constant velocity called your “terminal velocity.” Suppose that at t=0, you jump. When t=15 seconds, your wrist altimeter shows that your distance from the ground, d, is 3600 meters. When t=35 seconds, you have dropped to d=2400 meters. Assume that you are at your terminal velocity by the time t=15.

a) Explain why d varies linearly with t after you have reached your terminal velocity.

b) Write the particular equation expressing d in terms of t.c) If you neglect to open your parachute, when will you hit the ground?d) According to your mathematical model, how high was the airplane when you jumped? e) The plane was actually at 4200 meters when you jumped.

How do you reconcile this fact with your answer in part d?

Now we’re ready!

4) A lumber company can convert logs into either lumber or plywood. In a given week the mill can turn out 400 units in production, of which 100 units of lumber and 150 units of plywood are required by regular customers. The profit on a unit of lumber is $20 and the profit on plywood for one unit is $30. How many units of each type should the mill produce per week in order for maximum profit?

5) An accountant’s analysis shows that you have $40 000 to invest in stocks and bonds. The least that you are allowed to invest in stocks is $6 000 and you cannot invest more than $22 000 in stocks. You may also invest no more than $30 000 in bonds. The

interest in the stocks is 8% tax-free and the interest on the bonds is percent tax-free. How much should you invest in each type to maximize your profit? What is the income form the $40 000 invested.

6) A stationary company makes two types of notebooks: a deluxe notebook, with subject dividers that sell for $1.25 and a regular notebook that sells for $.90. The production cost is $1.00 for each deluxe notebook and $0.75 for each regular notebook. The company has the facilities to manufacture between 2000 and 3000 deluxe and 3000 and 6000 regular, but not more than 7000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production costs?

7) A coffee company purchases mixed lots of coffee beans and grades them into premium, regular and unusable beans. The company needs at least 280 tons of premium grade and 200 tons of regular grade coffee beans. The company can purchase ungraded coffee beans from two suppliers in any amount desired. Samples from the two suppliers contain the following percentages of premium, regular and unusable beans;

Sample Premium Regular Unusable A 20% 50% 30% B 40% 20% 40%

If supplier A charges $125 per ton and B charges $200 per ton how much should the company purchase from each supplier to fulfill its needs at minimum cost?

8) Gary Smelting Company receives a monthly order for at least 40 tons of iron, 60 tons of copper and 40 tons of lead. It can fill the order by smelting either alloy A or alloy B. Each railroad carload of A will produce 1 ton of iron, 3 tons of copper and 4 tons of lead after smelting. Each railroad carload of B will produce 2 tons of iron, 2 tons of copper and 1 ton of lead after smelting. If the cost of smelting one carload of the alloy A is $350 and the cost of smelting one carload of the alloy B is $200, how many carloads of each should be used to fill the order at the minimum cost to Gary Smelting? What is the minimum cost?

9) A furniture company makes two types of desks, one plain and one fancy. Each plain desk takes 3 hours of work to assemble

and 1 hour to finish. Each fancy desk takes 2 hours of work to assemble and 4 hours to finish. The 4 assembly workers and 6 sanding workers are each used 12 hours per day. Each plain desk has a net profit of $80 and each fancy desk has a net profit of $200. If the company can sell all the desks it makes, how many of each kind should be produced each day in order to make maximum profit.

10) Almosttexas makes two types of calculators. Deluxe sells for $12 and Top of the Line sells for $10. It costs Almosttexas $9 to produce a deluxe and $8 to produce a Top of the Line calculator. In one week, Almottexas can produce 200 to 300 deluxe calculators and 100 to 250 Top of the Line calculators, but no more than 500 in total. How many of each type should the company make to have the maximum profit?

11) A carpenter makes tables and chairs. Each table can be sold for a profit of $30 and each chair for a profit of $10. The carpenter can afford to spend up to 40 hours working and it takes six hours to make a table and three hours to make a chair. Customer demand requires her to make at least three times as many chairs as tables. Tables take up four times as much storage space as chairs and there is no room for more than 4 tables per week. How many tables and chairs will she have to make, to make maximum profit?

12) A nutrition centre sells health food to mountain climbing teams. The Trailblazer mix package contains one pound of corn cereal mixed with four pounds of wheat cereal and sells for $9.75. The Frontier mix package contains two pounds of corn cereal with three pounds of wheat cereal and sells for $9.50. The centre has 60 pounds of corn cereal and 120 pounds of wheat cereal available. How many packages of each mix should the centre sell to maximize its income?

13) Angie and Nicole run very successful lemonade stand for most of the summer. They made both lemonade and fruit punch. Although the ingredients for the fruit punch cost more, they sold both drinks for the same price. Their profit for a cup of lemonade was $0.20, but only $0.15 a cup for fruit punch. Every morning they made drinks for the day. One morning they discovered that they were low on supplies. The lemonade recipe requires ½ a cup of sugar per quart, whereas the fruit punch only required ¼ a cup per quart. They had about 4.5 cups of sugar on hand and in addition they only had enough oranges to make 8 quarts of fruit punch. The friends want to make as much profit as they can,

how much profit will they make if they only made lemonade? Only fruit punch? What would the profit be if they made both drinks?

14) A manufacturer of CB radios makes a profit of $20 on a deluxe model and $15 on a standard model. The company wishes to produce at least 70 deluxe models and at least 100 standard models per day. To maintain high quality, the daily production should not exceed 200 radios. How many of each type should be produced daily in order to maximize the profit?

15) A manufacturer of tennis rackets makes a profit of $15 on each oversized racket and $8 on each standard racket. To meet dealer demand, daily production of standard rackets should be between 30 and 80, and production of oversized rackets should be between 10 and 30. To maintain high quality, the total number of rackets produced should no exceed 80 per day. How many of each type of racket should be manufactured daily to maximize the profit?

16) Put your knowledge to the test! Create a scenario involving the production of Discmans and Walkmans. Using the constraints below, create your own linear programming question!

Constraints:5x + 3y ≤ 39003x + y ≤ 21002x + 2y ≤ 2200

Profit = $75x + $50y

LINEAR PROGRAMMING QUIZLINEAR PROGRAMMING QUIZ3) An Agricultural farm consists of 240 acres of cropland. The

farmer wants to plant part or all of the acreage in corn or oats. The profit per acre in corn production is $40 and the profit in oats is $30. An additional restriction is that the total hours of labor during the production is no more than 320. Each of the acres of land in corn production uses 2 hours of labor during the production period whereas the production of oats requires only 1 hour per acre. How many acres of land should be planted in corn and how many in oats to make the absolute profit?

4) Manufacturing cabinet company makes two types of cabinets

drawers, one plain and one fancy. Each plain drawer takes 2 hours of work to assemble and 1 hour to sand. Each fancy drawer

takes 1 hour of work to assemble and 4 hours to sand. The 4 assembly workers and the 6 sanding workers are used 12 hours per day. Each plain drawer makes a net profit of $3 and the fancy drawers make a net profit of $5. If the company can sell all the drawers it makes, how many of each kind should be produced each day in order to maximize the profit.

5) Kellogg’s of Toronto, Ontario is going to produce a new cereal from a mixture of bran and rice that contains at least 88 grams of protein and at least 36 mg of iron. Knowing that bran contains 80 grams of protein and 40 mg of iron per kilogram and that rice contains 100g of protein and 30 mg of iron per kilogram, find the minimum cost of producing this new cereal if bran cost 50 cents per kilogram and rice cost 40 cents per kilogram.

6) Sanyo makes stereo receivers. It produces a 30-watt receiver that it sells for $100 profit and a 50-watt receiver that it sells for a $150 profit. The 30-watt receiver requires 3 hours to manufacture and the 50-watt receiver takes 5 hours to manufacture. The cabinet shop spends 1 hour on a 30-watt receiver and 3 hours on a 50-watt receiver. Packing takes 2 hours for both receivers. Per week, Sanyo has available 3 900 work hours for manufacturing, 2 100 hours for cabinet making and 2 200 hours for packing. How many receivers of each type should Sanyo produce per week to maximize its profit and what is the maximum profit per week?

7) A cement manufacturer produces two types of cement, namely granules and powder. She cannot make more than 1600 bags a day due to a shortage of vehicles to transport the cement out of the plant. A sales contract means she must produce at least 500 bags of powered cement per day. She is further restricted by a shortage of time; the granulated cement requires twice as much time to make as the powered cement. A bag of powered cement is $3 per bag and a bag of granulated cement is $4. What is the maximum profit she can make?

8) The Buick Manufacturing plant in Flint, Michigan must fill orders for Park Avenues from two dealers. The first dealer, Farmington Buick, has ordered 20 Park Avenues and the second dealer Brighton Buick has ordered 30. The manufacturer has the cars stored in two different areas, southeast Flint and southwest Flint. Both dealers can spend no more than $800 each on shipping costs. The shipping costs, per car are $15 from southeast Flint to Brighton, $13 from southeast Flint to Farmington, $14 from southwest Flint to Brighton and $16 from southwest Flint to

Farmington. With these conditions find the number of cars shipped from each area to each dealer if the total shipping cost is to be a minimum. What is the cost?

9) Seibering Tire Company of Akron, Ohio has 1 000 units of raw rubber to use in production radical tires for cars and tractors. Each car tire requires 5 units of rubber and each tractor tire requires 20 units of rubber. Labor costs are $8 for a car tire and $12 for a tractor tire. If Seiberling does not want to pay more than $1500 in labor costs and wishes to make a profit of $10 per car tire and $25 per tractor tire, how many of each type of tire should be made in order to make the most the most profit.

10) You invested a certain amount of money in at 12.5 percent and another sum at 14 percent. If your total investment was $60 000 and your total yearly interest for one year on both investments was $8 190, how much did you invest at each rate?

11) RCA Manufacturing Company in Ft. Wayne, Indiana, makes a $60 profit on each 19-inch TV it produces and a $40 profit on each 13-inch TV. A 19-inch TV requires 1 hour on machine X, 1 hour on machine Y, and 4 hours on machine Z. The 13-inch TV requires 2 hours on X, 1 hour on Y, and 1 hour on Z. In a given day, machines X, Y, and Z can work a maximum of 16, 9, and 24 hours, respectively. How many 19-inch TV’s and how many 13-inch TV’s should be produced per day to maximize the profit?

Helpful Links1) www.iln.net

This is a great online resource for not only linear programming, but for many other topics in mathematics as well.

2) www.cut-the-knot.com

3) http://www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/13/ wa/HWCDA/file?tg=MATH&fileid=32258&flt=CABThis website offers a basic, straight forward approach to linear programming. The homepage also has great resources for students, teachers, and parents on a wide variety of subjects.

4) www.cs.sunysb.edu/~algorith/linear-programming.shtml

For Cool Links Go to:

http://www.cs.sunysb.edu/~algorith/linear-programming.shtml

http://www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/13/wa/HWCDA/file?tg=MATH&fileid=32258&flt=CAB

http://www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/13/wa/HWCDA/file?tg=MATH&fileid=32258&flt=CAB

http://www.cut-the-knot.com/

http://www.iln.net/

1) www.mathforum.org

2) www.cloudnet.com/%7Eedrbsass/edmath.htm#algebra

Appendix AAnswers to Practice Questions

1a)

A) (4,8) 12 + 40 = 52 ←maximumB) (2,4) 6 + 20 = 26C) (1,1) 3 + 5 = 8 ←minimumD) (5,2) 15 + 10 = 25

b)

A) (0,7) 0 + 28 = 28B) (0,0) 0 + 0 = 0C) (6,2) 6 + 8 = 14D) (5,4) 5 + 16 = 21


2) a) y = 1/3x + 9b) Size 13c) 39 inches

d)


3) a) You will be falling at a constant velocity

Vertices f(x) = 3x + 5yVertices f(x) = x + 4y

http://www.cloudnet.com/~edrbsass/edmath.htm#algebra

http://www.mathforum.org/

b) d = -60t + b c) at t = 75 seconds d) 4500m e) Your velocity was accelerating until you reached your terminal velocity.


4)x = lumbery = plywoodP = $20x + $30y

A) (100, 150) 2000 + 4500 = $6500B) (250, 150) 5000 + 4500 = $9500C) (100, 300) 2000 + 9000 = $11000

←maximum

Graph

Appendix AAnswers to Practice Problems

5) s = Stocks b = Bonds Constraints: 60000 ≤ s ≤ 22000 P = .08s + b b ≤ 30000

s + b ≤ 40000

A) (6000, 0) $480B) (22000, 0) $1760C) (22000, 18000) $19760D) (10000, 30000) $30800 ← maximumE) (6000, 30000) $30480

Vertices Profit: P = 20x + 30yVertices Profit: P = .08s + b

GRAPH


6) d = Deluxe r = Regular Constraints: 2000 ≤ d ≤ 3000 P = $.25d + $.15r 3000 ≤ r ≤ 6000

d + r ≤ 7000

A) (2000, 3000) $950B) (3000, 3000) $1200C) (3000, 4000) $1325 ← maximumD) (2000, 5000) $1250

GRAPH

Appendix A

Answers to Practice Problems

7) a = Supplier A b = Supplier B Constraints: .2a + .4b ≥280 Cost = $125a + $200b .5a + .2b ≥ 200

A) (150, 625) $143,750 ← minimum

Vertices Profit: P = $.25d + $.15rVertices Cost: C = $125a + $200b

B) (1400, 0) $280,000

GRAPH


8) a = Alloy A b = Alloy B Constraints: a + 2b ≥ 40 Cost = $350a + $200b 3a + 2b ≥ 60 4a + b ≥ 40

A) (10, 15) $6500B) (4, 24) $6200 ← minimumC) (40, 0) $14000

GRAPH


9) x = Plain Desksy = Fancy Desks Constraints: 3x + 2y ≤ 48Profit = $80x + $200y x + 4y ≤ 72

x, y ≥ 0

Vertices Cost: C = $350a + $200b

A) (0, 0) $0B) (16, 0) $1280C) (4.8, 16.8) $3744 ← maximumD) (0, 18) $3600

GRAPH


10) x = Deluxe y = Top of the Line Constraints: x + y ≤500 Profit = $3x + $2y 200 ≤ x ≤ 300

100 ≤ y ≤250

A) (200, 100) $800B) (300, 100) $1100C) (300, 200) $1300 ← maximumD) (250, 250) $1250E) (200, 250) $1100

GRAPH

Vertices Profit: P = $80x + $200yVertices Profit: P = $3x + $2y


11) x = Table y = Chair Constraints: x ≤ 4 Profit = $30x + $10y 6x + 3y ≤ 40

y = 3x

A) (0, 0) $0B) (2.6, 8) $158 ← maximumC) (0, 13.3) $133

GRAPH


12) t = Trailblazer Products f = Frontier Products Constraints: t ≥ 0 Cost = $9.75t + $9.50f f ≥ 0

t + 2f ≤ 60 4t + 3f ≤ 120

A) (0, 0) $0B) (30, 0) $292.50C) (12, 24) $345 ← maximumD) (0, 30) $285

GRAPH

Vertices Profit: P = $30x + $10yVertices Income: I = $9.75t + $9.50f


13) x = Lemonade y = Fruit Punch Constraints: 1/2x + 1/4y ≤ 4.5 Profit = $.80x + $.6y y ≤ 8 All Lemonade: 9 quarts, Profit = $7.20All Fruit Punch: 8 quarts, Profit = $4.80Maximum Profit: $8.80; 5 quarts of lemonade, 8 quarts of fruit punch

GRAPH


14) x = Deluxe y = Standard Constraints: x + y ≤ 200 Profit = $20x + $15y x ≤ 70

y ≤ 100

A) (70, 100) $2900B) (100, 100) $3500 ← maximum

Vertices Profit: P = $20x + $15y

C) (70, 130) $3350

GRAPH


15) x = Oversized y = Standard Constraints: x + y ≤ 80 Profit = $15x + $8y 10 ≤ x ≤ 30

30 ≤ y ≤ 80

A) (10, 30) $390B) (30, 30) $690C) (30, 50) $850 ← maximumD) (10, 70) $710

GRAPH


16) Sanyo makes Discmans and Walkmans. A Walkman makes a profit of $50, and a Discman makes a profit of $75. The Walkman requires 3 hours to manufacture and the Discman takes 5 hours to manufacture.

The finishing shop spends 1 hour on a Walkman and 3 hours on a Discman. Packing takes 2 hours for both. Per week, Sanyo has

available 3900 work hours for manufacturing, 2100 hours for finishing, and 2200 hours for packing. How many Discmans and how many

Walkmans should Sanyo produce per week to maximize its profit, and what is the maximum profit per week?

Vertices Profit: P = $15x + $8y

Appendix BAnswers to Linear Programming Quiz

1) x = Corny = Oats Constraints: x + y ≤ 240Profit = $40x + $30y 2x + y ≤ 320

x, y ≥ 0

A) (0, 0) $0B) (160, 0) $6400C) (80, 160) $8000 ← maximumD) (0, 240) $7200

GRAPH


2) x = Plain y = Fancy Constraints: 2x + y ≤ 48 Profit = $3x + $5y x + 4y ≤ 72

x, y ≥ 0

A) (0, 0) $0B) (24, 0) $72C) (120/7, 96/7) $120 ← maximumD) (0, 18) $90

17 plain drawers, and 13 fancy drawers per day.

GRAPH

Vertices Profit: P = $40x + $30yVertices Profit: P = $3x + $5y


3) x = Bran y = rice Constraints: 80x + 100y ≥ 88 Cost = $.50x + $.40y 40x + 30y ≥ 36

x, y ≥ 0

A) (.6, .4) $0.46 ← minimumB) (1.1, 0) $0.55C) (0, 1.2) $0.48

GRAPH


4) x = 30 watt y = 50 watt Constraints: 3x + 5y ≤ 3900 Profit = $100x + $150y x + 3y ≤ 2100

2x + 2y ≤ 2200

A) (0, 0) $0B) (1300, 0) $130, 000C) (800, 600) $170, 000 ← maximumD) (300, 600) $120, 000E) (0, 700) $105, 000

Vertices Cost: C = $.50x + $.40yVertices Profit: P = $100x + $150y

GRAPH


5) x = Granules y = Powder Constraints: x + y ≤ 1600 Profit = $4x + $3y x = 2y

y ≥ 500

A) (250, 500) $2500B) (1100, 500) $5900 ← maximumC) (3200/6, 3200/3) $5333.33

GRAPH


6) Farmington Buick x = Southeast Flint y = Southwest Flint Constraints: x + y ≤ 20

13x + 16y ≤ 800

A) (0, 0) $0B) (20, 0) $260 ←minimumC) (0, 20) $320

GRAPH

Vertices Profit: P = $4x + $3yVertices Cost = $13x + $16y

Brighton Buickx = Southeast Flinty = Southwest Flint Constraints: x + y ≤ 30

15x + 14y ≤ 800

A) (0, 0) $0B) (30, 0) $450C) (0, 30) $420 ← minimum

GRAPH


7) x = Car tires y = Tractor tires Constraints: 5x + 2y ≤ 1000 Profit = $10x + $25y 8x + 12y ≤ 1500

A) (0, 0) $0B) (187.5, 0) $1875C) (0, 125) $3125 ← maximum

GRAPH

Vertices Cost = $15x + $14yVertices Profit: P = $10x + $25y


8) x = 12.5% y = 14% Constraints: x +y = $60, 000 Interest = .125x + .14y = $8190 .125x + .14y = $8190

A) (14k, 46k) $1750 + $6440 = $8190

GRAPH


9) X = Number of 19” TV’s Y = Number of 13” TV’s P = 60x + 40y

Constraints X ³ 0 Y ³ 0x + 2y £ 16

x + y £ 9

Vertices Interest: I = .125x + .14y

4x + y £ 24

Vertices Profit: P = 60x + 40yA) (0,0) $0B) (6,0) $360C) (5,4) $460 ← MaximumD) (0,8) $320

GRAPH

Bibliography

Fworkowski, Earl W. Algebra and Trigonometry with Analytic Geometry. Kent Publishing Company:Boston.1989.

“Linear Programming” [Online] www.cs.sunysb.edu/~alorith/linear-programming.html.

“Linear Programming”[Online] www.cut-the-knot.com.

“Linear Programming” [Online] www.iln.net.



“Linear Programming” [Online]www.pento.dynds.org/~olivierd/compgeo/project_front.html.

linear programming - hrsbstaff home pagehrsbstaff.ednet.ns.ca/pbetuik/linear...

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