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Simplex Method Applications
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A Business Application: Maximum Profit
ExampleA manufacturer produced three types of plastic fixtures. The timerequired for molding, trimming and packaging is given in theaccompanying table. Note: Times are given in hours per dozenfixtures.
Process Type A Type B Type C Total Time AvailableMolding 1 2 3
2 12,000Trimming 2
323 1 4,600
Packaging 12
13
12 2,400
Profit $11 $16 $15 —
How many dozens of each type of fixture should be produced toobtain a maximum profit?
![Page 3: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/3.jpg)
A Business Application: Maximum Profit
First thing we need to do is assign variables.
Let x represent the number of dozens of type A fixtures.
Let y represent the number of dozens of type B fixtures.
Let z represent the number of dozens of type C fixtures.
What is our object function here?
M = 11x + 16y + 15z⇒ -11x-16y-15z+M=0
![Page 4: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/4.jpg)
A Business Application: Maximum Profit
First thing we need to do is assign variables.
Let x represent the number of dozens of type A fixtures.
Let y represent the number of dozens of type B fixtures.
Let z represent the number of dozens of type C fixtures.
What is our object function here?
M = 11x + 16y + 15z⇒ -11x-16y-15z+M=0
![Page 5: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/5.jpg)
A Business Application: Maximum Profit
First thing we need to do is assign variables.
Let x represent the number of dozens of type A fixtures.
Let y represent the number of dozens of type B fixtures.
Let z represent the number of dozens of type C fixtures.
What is our object function here?
M = 11x + 16y + 15z⇒ -11x-16y-15z+M=0
![Page 6: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/6.jpg)
A Business Application: Maximum Profit
First thing we need to do is assign variables.
Let x represent the number of dozens of type A fixtures.
Let y represent the number of dozens of type B fixtures.
Let z represent the number of dozens of type C fixtures.
What is our object function here?
M = 11x + 16y + 15z⇒
-11x-16y-15z+M=0
![Page 7: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/7.jpg)
A Business Application: Maximum Profit
First thing we need to do is assign variables.
Let x represent the number of dozens of type A fixtures.
Let y represent the number of dozens of type B fixtures.
Let z represent the number of dozens of type C fixtures.
What is our object function here?
M = 11x + 16y + 15z⇒ -11x-16y-15z+M=0
![Page 8: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/8.jpg)
A Business Application: Maximum Profit
What is the system of constraints we get from this?
x + 2y + 3
2 z ≤ 1200023 x + 2
3 y + z ≤ 460012 x + 1
3 y + 12 z ≤ 2400
x ≥ 0, y ≥ 0, z ≥ 0
This becomes ...
x + 2y + 3
2 z + u = 1200023 x + 2
3 y + z + v = 460012 x + 1
3 y + 12 z + w = 2400
x ≥ 0, y ≥ 0, z ≥ 0
![Page 9: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/9.jpg)
A Business Application: Maximum Profit
What is the system of constraints we get from this?
x + 2y + 3
2 z ≤ 1200023 x + 2
3 y + z ≤ 460012 x + 1
3 y + 12 z ≤ 2400
x ≥ 0, y ≥ 0, z ≥ 0
This becomes ...
x + 2y + 3
2 z + u = 1200023 x + 2
3 y + z + v = 460012 x + 1
3 y + 12 z + w = 2400
x ≥ 0, y ≥ 0, z ≥ 0
![Page 10: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/10.jpg)
A Business Application: Maximum Profit
What is the system of constraints we get from this?
x + 2y + 3
2 z ≤ 1200023 x + 2
3 y + z ≤ 460012 x + 1
3 y + 12 z ≤ 2400
x ≥ 0, y ≥ 0, z ≥ 0
This becomes ...
x + 2y + 3
2 z + u = 1200023 x + 2
3 y + z + v = 460012 x + 1
3 y + 12 z + w = 2400
x ≥ 0, y ≥ 0, z ≥ 0
![Page 11: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/11.jpg)
A Business Application: Maximum Profit
What is the system of constraints we get from this?
x + 2y + 3
2 z ≤ 1200023 x + 2
3 y + z ≤ 460012 x + 1
3 y + 12 z ≤ 2400
x ≥ 0, y ≥ 0, z ≥ 0
This becomes ...
x + 2y + 3
2 z + u = 1200023 x + 2
3 y + z + v = 460012 x + 1
3 y + 12 z + w = 2400
x ≥ 0, y ≥ 0, z ≥ 0
![Page 12: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/12.jpg)
A Business Application: Maximum Profit
The initial tableau is
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
Where is our first pivot?
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
12000
2= 6000
460023
= 6900
240013
= 7200
![Page 13: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/13.jpg)
A Business Application: Maximum Profit
The initial tableau is
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
Where is our first pivot?
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
12000
2= 6000
460023
= 6900
240013
= 7200
![Page 14: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/14.jpg)
A Business Application: Maximum Profit
The initial tableau is
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
Where is our first pivot?
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
12000
2= 6000
460023
= 6900
240013
= 7200
![Page 15: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/15.jpg)
A Business Application: Maximum Profit
The initial tableau is
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
Where is our first pivot?
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
120002
= 6000
460023
= 6900
240013
= 7200
![Page 16: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/16.jpg)
A Business Application: Maximum Profit
The initial tableau is
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
Where is our first pivot?
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
12000
2= 6000
460023
= 6900
240013
= 7200
![Page 17: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/17.jpg)
A Business Application: Maximum Profit
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
23
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
13 0 1
2 −13 1 0 0 600
13 0 1
4 −16 0 1 0 400
-3 0 -3 8 0 0 1 96000
Are we done?
![Page 18: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/18.jpg)
A Business Application: Maximum Profit
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
23
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
13 0 1
2 −13 1 0 0 600
13 0 1
4 −16 0 1 0 400
-3 0 -3 8 0 0 1 96000
Are we done?
![Page 19: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/19.jpg)
A Business Application: Maximum Profit
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
23
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
13 0 1
2 −13 1 0 0 600
13 0 1
4 −16 0 1 0 400
-3 0 -3 8 0 0 1 96000
Are we done?
![Page 20: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/20.jpg)
A Business Application: Maximum Profit
1 2 3
2 1 0 0 0 1200023
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
23
23 1 0 1 0 0 4600
12
13
12 0 0 1 0 2400
-11 -16 -15 0 0 0 1 0
∼
12 1 3
412 0 0 0 6000
13 0 1
2 −13 1 0 0 600
13 0 1
4 −16 0 1 0 400
-3 0 -3 8 0 0 1 96000
Are we done?
![Page 21: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/21.jpg)
A Business Application: Maximum Profit
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
13 0 1
4 − 16 0 1 0 400
-3 0 -3 8 0 0 1 96000
∼
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
1 0 34 − 1
2 0 3 0 1200-3 0 -3 8 0 0 1 96000
∼
0 1 3
834 0 −3
2 0 54000 0 1
4 −16 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
Now are we done?
![Page 22: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/22.jpg)
A Business Application: Maximum Profit
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
13 0 1
4 − 16 0 1 0 400
-3 0 -3 8 0 0 1 96000
∼
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
1 0 34 − 1
2 0 3 0 1200-3 0 -3 8 0 0 1 96000
∼
0 1 3
834 0 −3
2 0 54000 0 1
4 −16 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
Now are we done?
![Page 23: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/23.jpg)
A Business Application: Maximum Profit
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
13 0 1
4 − 16 0 1 0 400
-3 0 -3 8 0 0 1 96000
∼
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
1 0 34 − 1
2 0 3 0 1200-3 0 -3 8 0 0 1 96000
∼
0 1 3
834 0 −3
2 0 54000 0 1
4 −16 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
Now are we done?
![Page 24: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/24.jpg)
A Business Application: Maximum Profit
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
13 0 1
4 − 16 0 1 0 400
-3 0 -3 8 0 0 1 96000
∼
12 1 3
412 0 0 0 6000
13 0 1
2 − 13 1 0 0 600
1 0 34 − 1
2 0 3 0 1200-3 0 -3 8 0 0 1 96000
∼
0 1 3
834 0 −3
2 0 54000 0 1
4 −16 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
Now are we done?
![Page 25: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/25.jpg)
A Business Application: Maximum Profit
0 1 3
834 0 −3
2 0 5400
0 0 14 −1
6 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
∼
0 1 3
834 0 −3
2 0 54000 0 1 −2
3 4 -4 0 8001 0 3
4 −12 0 3 0 1200
0 0 −34
132 0 9 0 99600
∼
0 1 0 1 −3
2 0 0 51000 0 1 −2
3 4 -4 0 8001 0 0 0 -3 6 0 6000 0 0 6 3 6 1 100200
Now are we done?
![Page 26: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/26.jpg)
A Business Application: Maximum Profit
0 1 3
834 0 −3
2 0 5400
0 0 14 −1
6 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
∼
0 1 3
834 0 −3
2 0 54000 0 1 −2
3 4 -4 0 8001 0 3
4 −12 0 3 0 1200
0 0 −34
132 0 9 0 99600
∼
0 1 0 1 −3
2 0 0 51000 0 1 −2
3 4 -4 0 8001 0 0 0 -3 6 0 6000 0 0 6 3 6 1 100200
Now are we done?
![Page 27: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/27.jpg)
A Business Application: Maximum Profit
0 1 3
834 0 −3
2 0 5400
0 0 14 −1
6 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
∼
0 1 3
834 0 −3
2 0 54000 0 1 −2
3 4 -4 0 8001 0 3
4 −12 0 3 0 1200
0 0 −34
132 0 9 0 99600
∼
0 1 0 1 −3
2 0 0 51000 0 1 −2
3 4 -4 0 8001 0 0 0 -3 6 0 6000 0 0 6 3 6 1 100200
Now are we done?
![Page 28: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/28.jpg)
A Business Application: Maximum Profit
0 1 3
834 0 −3
2 0 5400
0 0 14 −1
6 1 -1 0 200
1 0 34 −1
2 0 3 0 12000 0 −3
4132 0 9 0 99600
∼
0 1 3
834 0 −3
2 0 54000 0 1 −2
3 4 -4 0 8001 0 3
4 −12 0 3 0 1200
0 0 −34
132 0 9 0 99600
∼
0 1 0 1 −3
2 0 0 51000 0 1 −2
3 4 -4 0 8001 0 0 0 -3 6 0 6000 0 0 6 3 6 1 100200
Now are we done?
![Page 29: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/29.jpg)
A Business Application: Maximum Profit
0 1 0 1 − 3
2 0 0 51000 0 1 − 2
3 4 -4 0 8001 0 0 0 -3 6 0 6000 0 0 6 3 6 1 100200
x = 600
y = 5100
z = 800
u = 0
v = 0
w = 0
M = 100200
We have a maximum profit of$100,200 when we produce 600dozen of type A, 5100 dozen oftype B and 800 dozen of type C.
![Page 30: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/30.jpg)
A Business Application: Maximum Profit
0 1 0 1 − 3
2 0 0 51000 0 1 − 2
3 4 -4 0 8001 0 0 0 -3 6 0 6000 0 0 6 3 6 1 100200
x = 600
y = 5100
z = 800
u = 0
v = 0
w = 0
M = 100200
We have a maximum profit of$100,200 when we produce 600dozen of type A, 5100 dozen oftype B and 800 dozen of type C.
![Page 31: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/31.jpg)
A Business Application: Maximum Profit
0 1 0 1 − 3
2 0 0 51000 0 1 − 2
3 4 -4 0 8001 0 0 0 -3 6 0 6000 0 0 6 3 6 1 100200
x = 600
y = 5100
z = 800
u = 0
v = 0
w = 0
M = 100200
We have a maximum profit of$100,200 when we produce 600dozen of type A, 5100 dozen oftype B and 800 dozen of type C.
![Page 32: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/32.jpg)
A Nutrition Problem
ExampleSuppose you wanted to make rice and soybeans a staple of your diet.The object is to design the lowest-cost diet that provides certainminimum levels of protein, calories and vitamin B12. One cup ofuncooked rice costs 21 cents and contains 15 grams of protein, 810calories and 1
9 mg of vitamin B12. One cup of uncooked soybeanscosts 14 cents and contains 22.5 grams of protein, 270 calories and 1
3mg of vitamin B12. The minimum daily requirements are 90 grams ofprotein, 1620 calories and 1 mg of vitamin B12. Design this lowestcost diet that meets these requirements.
![Page 33: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/33.jpg)
The Table
Category Rice Soybeans RequirementProtein 15 22.5 90Calories 810 270 1620
Vitamin B1219
13 1
Cost 21 14
![Page 34: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/34.jpg)
The System
Let x be the number of cups of rice and let y be the number of cups ofsoybeans.
Minimize C = 21x + 14y subject to the constraints
15x + 22.5y ≥ 90810x + 270y ≥ 162019 x + 1
3 y ≥ 1x ≥ 0, y ≥ 0
We can simplify some. Note that:We can multiply the first inequality by 2 to get rid of thefractions.All of the coefficients in the first one are now divisible by 15.All of the coefficients in the second one are divisible by 270.We can multiply the third one by 9 to get rid of the fractions.
![Page 35: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/35.jpg)
The System
Let x be the number of cups of rice and let y be the number of cups ofsoybeans.
Minimize C = 21x + 14y subject to the constraints
15x + 22.5y ≥ 90810x + 270y ≥ 162019 x + 1
3 y ≥ 1x ≥ 0, y ≥ 0
We can simplify some. Note that:We can multiply the first inequality by 2 to get rid of thefractions.All of the coefficients in the first one are now divisible by 15.All of the coefficients in the second one are divisible by 270.We can multiply the third one by 9 to get rid of the fractions.
![Page 36: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/36.jpg)
The System
Let x be the number of cups of rice and let y be the number of cups ofsoybeans.
Minimize C = 21x + 14y subject to the constraints
15x + 22.5y ≥ 90810x + 270y ≥ 162019 x + 1
3 y ≥ 1x ≥ 0, y ≥ 0
We can simplify some. Note that:We can multiply the first inequality by 2 to get rid of thefractions.
All of the coefficients in the first one are now divisible by 15.All of the coefficients in the second one are divisible by 270.We can multiply the third one by 9 to get rid of the fractions.
![Page 37: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/37.jpg)
The System
Let x be the number of cups of rice and let y be the number of cups ofsoybeans.
Minimize C = 21x + 14y subject to the constraints
15x + 22.5y ≥ 90810x + 270y ≥ 162019 x + 1
3 y ≥ 1x ≥ 0, y ≥ 0
We can simplify some. Note that:We can multiply the first inequality by 2 to get rid of thefractions.All of the coefficients in the first one are now divisible by 15.
All of the coefficients in the second one are divisible by 270.We can multiply the third one by 9 to get rid of the fractions.
![Page 38: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/38.jpg)
The System
Let x be the number of cups of rice and let y be the number of cups ofsoybeans.
Minimize C = 21x + 14y subject to the constraints
15x + 22.5y ≥ 90810x + 270y ≥ 162019 x + 1
3 y ≥ 1x ≥ 0, y ≥ 0
We can simplify some. Note that:We can multiply the first inequality by 2 to get rid of thefractions.All of the coefficients in the first one are now divisible by 15.All of the coefficients in the second one are divisible by 270.
We can multiply the third one by 9 to get rid of the fractions.
![Page 39: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/39.jpg)
The System
Let x be the number of cups of rice and let y be the number of cups ofsoybeans.
Minimize C = 21x + 14y subject to the constraints
15x + 22.5y ≥ 90810x + 270y ≥ 162019 x + 1
3 y ≥ 1x ≥ 0, y ≥ 0
We can simplify some. Note that:We can multiply the first inequality by 2 to get rid of thefractions.All of the coefficients in the first one are now divisible by 15.All of the coefficients in the second one are divisible by 270.We can multiply the third one by 9 to get rid of the fractions.
![Page 40: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/40.jpg)
The System
Minimize C = 21x + 14y subject to the constraints
2x + 3y ≥ 123x + y ≥ 6x + 3y ≥ 9x ≥ 0, y ≥ 0
Since this is not in standard form ...
−2x− 3y ≤ −12−3x− y ≤ −6−x− 3y ≤ −9x ≥ 0, y ≥ 021x + 14y + M = 0
![Page 41: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/41.jpg)
The System
Minimize C = 21x + 14y subject to the constraints
2x + 3y ≥ 123x + y ≥ 6x + 3y ≥ 9x ≥ 0, y ≥ 0
Since this is not in standard form ...
−2x− 3y ≤ −12−3x− y ≤ −6−x− 3y ≤ −9x ≥ 0, y ≥ 021x + 14y + M = 0
![Page 42: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/42.jpg)
The System
Minimize C = 21x + 14y subject to the constraints
2x + 3y ≥ 123x + y ≥ 6x + 3y ≥ 9x ≥ 0, y ≥ 0
Since this is not in standard form ...
−2x− 3y ≤ −12−3x− y ≤ −6−x− 3y ≤ −9x ≥ 0, y ≥ 021x + 14y + M = 0
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The Matrix and Pivots
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -6-1 -3 0 0 1 0 -921 14 0 0 0 1 0
∼
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -6
-1 -3 0 0 1 0 -921 14 0 0 0 1 0
∼
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -613 1 0 0 − 1
3 0 321 14 0 0 0 1 0
∼
-1 0 1 0 -1 0 -3−8
3 0 0 1 − 13 0 -3
13 1 0 0 − 1
3 0 3493 0 0 0 14
3 1 -42
![Page 44: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/44.jpg)
The Matrix and Pivots
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -6-1 -3 0 0 1 0 -921 14 0 0 0 1 0
∼
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -6
-1 -3 0 0 1 0 -921 14 0 0 0 1 0
∼
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -613 1 0 0 − 1
3 0 321 14 0 0 0 1 0
∼
-1 0 1 0 -1 0 -3−8
3 0 0 1 − 13 0 -3
13 1 0 0 − 1
3 0 3493 0 0 0 14
3 1 -42
![Page 45: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/45.jpg)
The Matrix and Pivots
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -6-1 -3 0 0 1 0 -921 14 0 0 0 1 0
∼
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -6
-1 -3 0 0 1 0 -921 14 0 0 0 1 0
∼
-2 -3 1 0 0 0 -12-3 -1 0 1 0 0 -613 1 0 0 − 1
3 0 321 14 0 0 0 1 0
∼
-1 0 1 0 -1 0 -3−8
3 0 0 1 − 13 0 -3
13 1 0 0 − 1
3 0 3493 0 0 0 14
3 1 -42
![Page 46: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/46.jpg)
The Matrix and Pivots
-1 0 1 0 -1 0 -3−8
3 0 0 1 − 13 0 -3
13 1 0 0 − 1
3 0 3493 0 0 0 14
3 1 -42
∼
1 0 -1 0 1 0 3− 8
3 0 0 1 − 13 0 -3
13 1 0 0 − 1
3 0 3493 0 0 0 14
3 1 -42
∼
1 0 -1 0 1 0 3− 7
3 0 − 13 1 0 0 -2
23 1 − 1
3 0 0 0 4353 0 14
3 0 0 1 -56
![Page 47: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/47.jpg)
The Matrix and Pivots
1 0 -1 0 1 0 3
− 73 0 − 1
3 1 0 0 -223 1 − 1
3 0 0 0 4353 0 14
3 0 0 1 -56
∼
1 0 -1 0 1 0 37 0 1 -3 0 0 623 1 −1
3 0 0 0 4353 0 14
3 0 0 1 -56
∼
8 0 0 -3 1 0 97 0 1 -3 0 0 63 1 0 -1 0 0 6
-21 0 0 14 0 1 -84
![Page 48: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/48.jpg)
The Matrix and Pivots
8 0 0 -3 1 0 97 0 1 -3 0 0 63 1 0 -1 0 0 6
-21 0 0 14 0 1 -84
∼
8 0 0 -3 1 0 91 0 1
7 − 37 0 0 6
73 1 0 -1 0 0 6
-21 0 0 14 0 1 -84
∼
0 0 − 8
737 1 0 15
71 0 1
7 − 37 0 0 6
70 1 − 3
727 0 0 24
70 0 3 5 0 1 -66
![Page 49: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/49.jpg)
The Conclusion
0 0 −8
737 1 0 15
71 0 1
7 −37 0 0 6
70 1 −3
727 0 0 24
70 0 3 5 0 1 -66
We minimize the cost at $.66 when we use 67 cups of rice and 3 3
7 cupsof soybeans.
![Page 50: Simplex Method Applications](https://reader033.vdocuments.us/reader033/viewer/2022060520/62966043cf97002df83ea284/html5/thumbnails/50.jpg)
The Conclusion
0 0 −8
737 1 0 15
71 0 1
7 −37 0 0 6
70 1 −3
727 0 0 24
70 0 3 5 0 1 -66
We minimize the cost at $.66 when we use 67 cups of rice and 3 3
7 cupsof soybeans.