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Chapter review

Solve the following linear programming problems using the simplex method.

Maximize z = 5x 1 + 3x 2 size 12{z=5x rSub { size 8{1} } +3x rSub { size 8{2} } } {}

subject to x 1 + x 2 12 2x 1 + x 2 16 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} #<= {} {} # "12" {} ## 2x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} #<= {} {} # "16"{} } } {}

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

x 1 = 4 size 12{x rSub { size 8{1} } =4} {} , x 2 = 8 size 12{x rSub { size 8{2} } =8} {} , y 1 = 0 size 12{y rSub { size 8{1} } =0} {} , y 2 = 0 size 12{y rSub { size 8{2} } =0} {} , z = 44 size 12{z="44"} {}

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Maximize z = 5x 1 + 8x 2 size 12{z=5x rSub { size 8{1} } +8x rSub { size 8{2} } } {}

subject to x 1 + 2x 2 30 3x 1 + x 2 30 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} #<= {} {} # "30" {} ## 3x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} #<= {} {} # "30"{} } } {}

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

x 1 = 6 size 12{x rSub { size 8{1} } =6} {} , x 2 = 12 size 12{x rSub { size 8{2} } ="12"} {} , y 1 = 0 size 12{y rSub { size 8{1} } =0} {} , y 2 = 0 size 12{y rSub { size 8{2} } =0} {} , z = 126 size 12{z="126"} {}

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

subject to 4x 1 + 2x 2 + 5x 3 32 2x 1 + 4x 2 + 3x 3 28 size 12{ matrix { 4x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # +{} {} # 5x rSub { size 8{3} } {} #<= {} {} # "32" {} ## 2x rSub { size 8{1} } {} # +{} {} # 4x rSub { size 8{2} } {} # +{} {} # 3x rSub { size 8{3} } {} #<= {} {} # "28"{} } } {}

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 = 6 size 12{x rSub { size 8{1} } =6} {} , x 2 = 4 size 12{x rSub { size 8{2} } =4} {} , x 3 = 0 size 12{x rSub { size 8{3} } =0} {} , y 1 = 0 size 12{y rSub { size 8{1} } =0} {} , y 2 = 0 size 12{y rSub { size 8{2} } =0} {} , z = 24 size 12{z="24"} {}

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Maximize z = x 1 + 6x 2 + 8x 3 size 12{z=x rSub { size 8{1} } +6x rSub { size 8{2} } +8x rSub { size 8{3} } } {}

subject to x 1 + 2x 2 1200 2x 2 + x 3 1800 4x 1 + x 3 3600 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} #<= {} {} # "1200" {} ## 2x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # "1800" {} ## 4x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # "3600"{} } } {}

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 = 450 size 12{x rSub { size 8{1} } ="450"} {} , x 2 = 0 size 12{x rSub { size 8{2} } =0} {} , x 3 = 1800 size 12{x rSub { size 8{3} } ="1800"} {} , y 1 = 750 size 12{y rSub { size 8{1} } ="750"} {} , y 2 = 0 size 12{y rSub { size 8{2} } =0} {} , y 3 = 0 size 12{y rSub { size 8{3} } =0} {} , z = 14 , 850 size 12{z="14","850"} {}

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

subject to 4x 1 + x 2 + x 3 1800 2x 1 + 2x 2 + x 3 2000 4x 1 + 2x 2 + x 3 3200 size 12{ matrix { 4x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # "1800" {} ## 2x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # "2000" {} ## 4x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #<= {} {} # "3200"{} } } {}

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 = 200 size 12{x rSub { size 8{2} } ="200"} {} , x 3 = 1600 size 12{x rSub { size 8{3} } ="1600"} {} , y 1 = 0 size 12{y rSub { size 8{1} } =0} {} , y 2 = 0 size 12{y rSub { size 8{2} } =0} {} , y 3 = 1200 size 12{y rSub { size 8{3} } ="1200"} {} , z = 9600 size 12{z="9600"} {}

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Minimize z = 12 x 1 + 10 x 2 size 12{z="12"x rSub { size 8{1} } +"10"x rSub { size 8{2} } } {}

subject to x 1 + x 2 6 2x 1 + x 2 8 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} #>= {} {} # 6 {} ## 2x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} #>= {} {} # 8{} } } {}

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

x 1 = 2 size 12{x rSub { size 8{1} } =2} {} , x 2 = 4 size 12{x rSub { size 8{2} } =4} {} , z = 64 size 12{z="64"} {}

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Minimize z = 4x 1 + 6x 2 + 7x 3 size 12{z=4x rSub { size 8{1} } +6x rSub { size 8{2} } +7x rSub { size 8{3} } } {}

subject to x 1 + x 2 + 2x 3 20 x 1 + 2x 2 + x 3 30 size 12{ matrix { x rSub { size 8{1} } {} # +{} {} # x rSub { size 8{2} } {} # +{} {} # 2x rSub { size 8{3} } {} #>= {} {} # "20" {} ## x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #>= {} {} # "30"{} } } {}

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 = 10 size 12{x rSub { size 8{1} } ="10"} {} , x 2 = 10 size 12{x rSub { size 8{2} } ="10"} {} , x 3 = 0 size 12{x rSub { size 8{3} } =0} {} , z = 100 size 12{z="100"} {}

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Minimize z = 40 x 1 + 48 x 2 + 30 x 3 size 12{z="40"x rSub { size 8{1} } +"48"x rSub { size 8{2} } +"30"x rSub { size 8{3} } } {}

subject to 2x 1 + 2x 2 + x 3 25 x 1 + 3x 2 + 2 x 3 30 size 12{ matrix { 2x rSub { size 8{1} } {} # +{} {} # 2x rSub { size 8{2} } {} # +{} {} # x rSub { size 8{3} } {} #>= {} {} # "25" {} ## ital "xl"1 {} # +{} {} # 3x rSub { size 8{2} } {} # +{} {} # 2 ital "xl"3 {} #>= {} {} # "30"{} } } {}

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 = 15 / 4 size 12{x rSub { size 8{1} } ="15"/4} {} , x 2 = 35 / 4 size 12{x rSub { size 8{2} } ="35"/4} {} , x 3 = 0 size 12{x rSub { size 8{3} } =0} {} , z = 570 size 12{z="570"} {}

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A department store sells three different types of televisions: small, medium, and large. The store can sell up to 200 sets a month. The small, medium, and large televisions require, respectively, 3, 6, and 6 cubic feet of storage space, and a maximum of 1,020 cubic feet of storage space is available. The three types, small, medium, and large, take up, respectively, 2, 2, and 4 sales hours of labor, and a maximum of 600 hours of labor is available. If the profit made from each of these types is $40, $80, and $100, respectively, how many of each type of television should be sold to maximize profit, and what is the maximum profit?

x 1 = 0 size 12{x rSub { size 8{1} } =0} {} , x 2 = 40 size 12{x rSub { size 8{2} } ="40"} {} , x 3 = 130 size 12{x rSub { size 8{3} } ="130"} {} , y 1 = 30 size 12{y rSub { size 8{1} } ="30"} {} , y 2 = 0 size 12{y rSub { size 8{2} } =0} {} , y 3 = 0 size 12{y rSub { size 8{3} } =0} {} , z = 16 , 200 size 12{z="16","200"} {}

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A factory manufactures three products, A, B, and C. Each product requires the use of two machines, Machine I and Machine II. The total hours available, respectively, on Machine I and Machine II per month are 180 and 300. The time requirements and profit per unit for each product are listed below.

A B C
Machine I 1 2 2
Machine II 2 2 4
Profit 20 30 40

How many units of each product should be manufactured to maximize profit, and what is the maximum profit?

x 1 = 0 size 12{x rSub { size 8{1} } =0} {} , x 2 = 30 size 12{x rSub { size 8{2} } ="30"} {} , x 3 = 60 size 12{x rSub { size 8{3} } ="60"} {} , y 1 = 0 size 12{y rSub { size 8{1} } =0} {} , y 2 = 0 size 12{y rSub { size 8{2} } =0} {} , z = 3300 size 12{z="3300"} {}

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A company produces three products, A, B, and C, at its two factories, Factory I and Factory II. Daily production of each factory for each product is listed below.

Factory I Factory II
Product A 10 20
Product B 20 20
Product C 20 10

The company must produce at least 1000 units of product A, 1600 units of B, and 700 units of C. If the cost of operating Factory I is $4,000 per day and the cost of operating Factory II is $5000, how many days should each factory operate to complete the order at a minimum cost, and what is the minimum cost?

x 1 = 60 size 12{x rSub { size 8{1} } ="60"} {} , x 2 = 20 size 12{x rSub { size 8{2} } ="20"} {} , z = 340 , 000 size 12{z="340","000"} {}

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For his classes, Professor Wright gives three types of quizzes, objective, recall, and recall-plus. To keep his students on their toes, he has decided to give at least 20 quizzes next quarter. The three types, objective, recall, and recall-plus quizzes, require the students to spend, respectively, 10 minutes, 30 minutes, and 60 minutes for preparation, and Professor Wright would like them to spend at least 12 hours(720 minutes) preparing for these quizzes above and beyond the normal study time. An average score on an objective quiz is 5, on a recall type 6, and on a recall-plus 7, and Dr. Wright would like the students to score at least 130 points on all quizzes. It takes the professor one minute to grade an objective quiz, 2 minutes to grade a recall type quiz, and 3 minutes to grade a recall-plus quiz. How many of each type should he give in order to minimize his grading time?

x 1 = 12 size 12{x rSub { size 8{1} } ="12"} {} , x 2 = 0 size 12{x rSub { size 8{2} } =0} {} , x 3 = 10 size 12{x rSub { size 8{3} } ="10"} {} , z = 42 size 12{z="42"} {}

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Questions & Answers

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
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 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
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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

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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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