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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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?
Chine Reply
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?
Chalton Reply

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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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