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This chapter covers principles of the simplex method to Linear Programming. After completing this chapter students should be able to: solve linear programming maximization problems using the simplex method and solve the minimization problems using the simplex method.

Maximization by the simplex method

Solve the following linear programming problems using the simplex method.

Maximize z = x 1 + 2x 2 + 3x 3 size 12{z=x rSub { size 8{1} } +2x rSub { size 8{2} } +3x rSub { size 8{3} } } {}

subject to x 1 + x 2 + x 3 12 2x 1 + x 2 + 3x 3 18 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # "12" {} ## 2x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # +{} {} # 3x rSub { size 8{3} } {} #<= {} {} # "18"{} } } {}

x 1 , x 2 , x 3 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} } ,x rSub { size 8{3} }>= 0} {}

x 1 = 0 size 12{x rSub { size 8{1} } =0} {} , x 2 = 9 size 12{x rSub { size 8{2} } =9} {} , x 3 = 3 size 12{x rSub { size 8{3} } =3} {} , z = 27 size 12{z="27"} {}

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Maximize z = x 1 + 2x 2 + x 3 size 12{z=x rSub { size 8{1} } +2x rSub { size 8{2} } +x rSub { size 8{3} } } {}

subject to x 1 + x 2 3 x 2 + x 3 4 x 1 + x 3 5 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} #<= {} {} # 3 {} ## x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # 4 {} ## x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # 5{} } } {}

x 1 , x 2 , x 3 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} } ,x rSub { size 8{3} }>= 0} {}
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A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and $20 of capital, and each acre of corn requires 16 hours of labor and $40 of capital. The farmer has at most 800 hours of labor and $2400 of capital available. If the profit from an acre of wheat is $80 and from an acre of corn is $100, how many acres of each crop should she plant to maximize her profit?

Wheat 80 acres, corn 20 acres; Profit $8400

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A factory manufactures chairs, tables and bookcases each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 600 hours; the second at most 500 hours; and the third at most 300 hours. A chair requires 1 hour of cutting, 1 hour of assembly, and 1 hour of finishing; a table needs 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; and a bookcase requires 3 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is $20 per unit for a chair, $30 for a table, and $25 for a bookcase, how many units of each should be manufactured to maximize profit?

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The Acme Apple company sells its Pippin, Macintosh, and Fuji apples in mixes. Box I contains 4 apples of each kind; Box II contains 6 Pippin, 3 Macintosh, and 3 Fuji; and Box III contains no Pippin, 8 Macintosh and 4 Fuji apples. At the end of the season, the company has altogether 2800 Pippin, 2200 Macintosh, and 2300 Fuji apples left. Determine the maximum number of boxes that the company can make.

600 boxes; 400 of Box I, 200 of Box II, and none of Box III

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Minimization by the simplex method

In problems 1-2, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

Minimize z = 6x 1 + 8x 2 size 12{z=6x rSub { size 8{1} } +8x rSub { size 8{2} } } {}

subject to 2x 1 + 3x 2 7 4x 1 + 5x 2 9 size 12{ matrix { 2x rSub { size 8{1} } {} # +{} {} # 3x rSub { size 8{2} } {} #>= {} {} # 7 {} ## 4x rSub { size 8{1} } {} # +{} {} # 5x rSub { size 8{2} } {} #>= {} {} # 9{} } } {}

x 1 , x 2 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} }>= 0} {}

Initial Simplex Tableau

image needed!!!!

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Minimize z = 5x 1 + 6x 2 + 7x 3 size 12{z=5x rSub { size 8{1} } +6x rSub { size 8{2} } +7x rSub { size 8{3} } } {}

subject to 3x 1 + 2x 2 + 3x 3 10 4x 1 + 3x 2 + 5x 3 12 size 12{ matrix { 3x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # +{} {} # 3x rSub { size 8{3} } {} #>= {} {} # "10" {} ## 4x rSub { size 8{1} } {} # +{} {} # 3x rSub { size 8{2} } {} # +{} {} # 5x rSub { size 8{3} } {} #>= {} {} # "12"{} } } {}

x 1 , x 2 , x 3 0 size 12{x rSub { size 8{1} } ,x rSub { size 8{2} } ,x rSub { size 8{3} }>= 0} {}

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In the next two problems, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

Minimize z = 4x 1 + 3x 2 size 12{z=4x rSub { size 8{1} } +3x rSub { size 8{2} } } {}

subject to x 1 + x 2 10 3x 1 + 2x 2 24 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} #>= {} {} # "10" {} ## 3x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} #>= {} {} # "24"{} } } {}

x , x 2 0 size 12{x,x rSub { size 8{2} }>= 0} {}

x 1 = 4 size 12{x rSub { size 8{1} } =4} {} , x 2 = 6 size 12{x rSub { size 8{2} } =6} {} , z = 34 size 12{z="34"} {}

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A diet is to contain at least 8 units of vitamins, 9 units of minerals, and 10 calories. Three foods, Food A, Food B, and Food C are to be purchased. Each unit of Food A provides 1 unit of vitamins, 1 unit of minerals, and 2 calories. Each unit of Food B provides 2 units of vitamins, 1 unit of minerals, and 1 calorie. Each unit of Food C provides 2 units of vitamins, 1 unit of minerals, and 2 calories. If Food A costs $3 per unit, Food B costs $2 per unit and Food C costs $3 per unit, how many units of each food should be purchased to keep costs at a minimum?

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Questions & Answers

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
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At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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