<< Chapter < Page Chapter >> Page >
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"} {}

Got questions? Get instant answers now!

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} {}
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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!!!!

Got questions? Get instant answers now!

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} {}

Got questions? Get instant answers now!

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"} {}

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
Logan 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

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied finite mathematics' conversation and receive update notifications?

Ask