# 0.10 Linear programming  (Page 2/4)

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$x\ge y$

means that only values of $x$ that are greater than or equal to the $y$ values are allowed.

$x\le 20$

means that only $x$ values which are less than or equal to 20 are allowed.

The constraints are used to create bounds of the solution.

## The solution

Points that satisfy the constraints are called feasible solutions.

Once we have determined the feasible region the solution of our problem will be the feasible point where the objective function is a maximum / minimum. Sometimes there will be more than one feasible point where the objective function is a maximum/minimum — in this case we have more than one solution.

## Example of a problem

A simple problem that can be solved with linear programming involves Mrs Nkosi and her farm.

Mrs Nkosi grows mielies and potatoes on a farm of 100 m ${}^{2}$ . She has accepted orders that will need her to grow at least 40 m ${}^{2}$ of mielies and at least 30 m ${}^{2}$ of potatoes. Market research shows that the demand this year will be at least twice as much for mielies as for potatoes and so she wants to use at least twice as much area for mielies as for potatoes. She expects to make a profit of R650 per m ${}^{2}$ for her mielies and R1 500 per m ${}^{2}$ on her potatoes. How should she divide her land so that she can earn the most profit?

Let $m$ represent the area of mielies grown and let $p$ be the area of potatoes grown.

We shall see how we can solve this problem.

## Method: linear programming

1. Identify the decision variables in the problem.
2. Write constraint equations
3. Write objective function as an equation
4. Solve the problem

## Writing constraint equations

You will need to be comfortable with converting a word description to a mathematical description for linear programming. Some of the words that are used is summarised in [link] .

 Words Mathematical description $x$ equals $a$ $x=a$ $x$ is greater than $a$ $x>a$ $x$ is greater than or equal to $a$ $x\ge a$ $x$ is less than $a$ $x $x$ is less than or equal to $a$ $x\le a$ $x$ must be at least $a$ $x\ge a$ $x$ must be at most $a$ $x\le a$

Mrs Nkosi grows mielies and potatoes on a farm of 100 m ${}^{2}$ . She has accepted orders that will need her to grow at least 40 m ${}^{2}$ of mielies and at least 30 m ${}^{2}$ of potatoes. Market research shows that the demand this year will be at least twice as much for mielies as for potatoes and so she wants to use at least twice as much area for mielies as for potatoes.

1. There are two decision variables: the area used to plant mielies ( $m$ ) and the area used to plant potatoes ( $p$ ).

• grow at least 40 m ${}^{2}$ of mielies
• grow at least 30 m ${}^{2}$ of potatoes
• area of farm is 100 m ${}^{2}$
• demand is at least twice as much for mielies as for potatoes
• $m\ge 40$
• $p\ge 30$
• $m+p\le 100$
• $m\ge 2p$

## Constraints as equation

Write the following constraints as equations:

1. Michael is registering for courses at university. Michael needs to register for at least 4 courses.
2. Joyce is also registering for courses at university. She cannot register for more than 7 courses.
3. In a geography test, Simon is allowed to choose 4 questions from each section.
4. A baker can bake at most 50 chocolate cakes in 1 day.
5. Megan and Katja can carry at most 400 koeksisters.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
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Abigail
for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
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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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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