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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

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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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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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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