Linear programming

Linear Programming

What is a linear programming?

Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the "best" production levels for maximal profits under those conditions.

 • Optimization problem consisting in

 • maximizing (or minimizing) a linear objective function 

 • of n decision variables • subject to a set of constraints expressed by linear equations or inequalities.  • Originally, military context: "programme"="resource planning". Now "programme"="problem" 

Examples:

The three inequalities in the curly braces are the constraints. The area of the plane that they mark off will be the feasibility region. The formula "z = 3x + 4y" is the optimization equation. I need to find the(x, y) corner points of the feasibility region that return the largest and smallest values of z.
My first step is to solve each inequality for the more-easily graphed equivalent forms:

                    click this

It's easy to graph the system: 

So the corner points are (2, 6), (6, 4), and (–1, –3).
Somebody really smart proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region. So, to find the solution to this exercise, I only need to plug these three points into "z = 3x + 4y".

(2, 6):      z = 3(2)   + 4(6)   =   6 + 24 =   30 
(6, 4):      z = 3(6)   + 4(4)   = 18 + 16 =   34 
(–1, –3):  z = 3(–1) + 4(–3) = –3 – 12 = –15


Then the maximum of z = 34 occurs at (6, 4),
and the minimum of z = –15 occurs at (–1, –3).

https://www.youtube.com/watch?v=-32jcGMpD2Q

References:

http://www-sop.inria.fr/members/Frederic.Giroire/teaching/ubinet/pdfs/slides-linear-programming-introduction.pdf

http://www.purplemath.com/modules/linprog.htm

Inequality

Inequalities

Definition:

An inequality says that two values are not equal.

a ≠ b says that a is not equal to b

There are other special symbols that show in what way things are not equal.

a < b says that a is less than b
a > b says that a is greater than b
(those two are known as strict inequality)

a ≤ b means that a is less than or equal to b
a ≥ b means that a is greater than or equal to b.

Examples:

    https://youtu.be/y7QLay8wrW8

   

  

https://www.youtube.com/watch?v=0X-bMeIN53I

References:

http://www.mathsisfun.com/definitions/inequality.html

http://www.math.com/school/subject2/lessons/S2U3L5GL.html

Image

Logarithms

Logarithms

Definition:

1. log_a x = N means that a^N = x.
2. log x means log₁₀ x. All log_a rules apply for log. When a logarithm is written without a base it means common logarithm.
3. ln x means log_e x, where e is about 2.718. All log_a rules apply for ln. When a logarithm is written "ln" it means natural logarithm.
    Note: ln x is sometimes written Ln x or LN x.

Common Logarithm

The logarithm base 10 of a number. That is, the power of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 10² = 100.

Natural Logarithm

The logarithms base e of a number. That is, the power of e necessary to equal a given number. The natural logarithm of x is written ln x. For example, ln 8 is 2.0794415... since e^2.0794415... = 8.


Rules
1. Inverse properties:   log_a a^x = x   and   a^(loga ^x) = x
2. Product:  log_a (xy) = log_a x + log_a y

Examples:

Write each of the following in logarithmic form:

Evaluate the following:

 What number is n?

                               

https://www.youtube.com/watch?v=mQTWzLpCcW0

click the link below for more examples!

References:

http://www.themathpage.com/aprecalc/logarithms.htm#ex4

Indices

Indices & the Law of Indices

Introduction

Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.

what's are indices?

Law of Indices

To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3⁴ and 3² can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 3⁵ and 5⁷ as their base differs (their bases are 3 and 5, respectively).

Six rules of the Law of Indices

Rule 1: 


 

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

References  

http://mathematics.laerd.com/maths/indices-intro.php 

Permutation and Combination

Permutation and Combination 

What's is the differences? 

Combination 

A set of objects in which position (or order) is NOT important. 
To a combination, the trio of Brittany, Alan and Greg is THE SAME AS Greg, Brittany and Alan.                        

Permutation 

There are basically two types of permutation:

1. Repetition is Allowed: such as the lock above. It could be "333".
2. No Repetition: for example the first three people in a running race. You can't be first and second.

Formula

Permutation and Combination:

Example 

1. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
A. 24400	B. 21300
C. 210	D. 25200

 

2. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
A. 47200	B. 48000
C. 42000	D. 50400

 

3. How many 3 digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9 which are divisible by 5 and none of the digits is repeated?
A. 20	B. 16
C. 8	D. 24

Answer and Explanation

Question 1
Answer: Option D
Explanation:
Number of ways of selecting 3 consonants from 7
= ⁷C₃
Number of ways of selecting 2 vowels from 4
= ⁴C₂

Number of ways of selecting 3 consonants from 7 and 2 vowels from 4
= ⁷C₃ × ⁴C₂
$=\left(\frac{7\times 6\times 5}{3\times 2\times 1}\right)\times \left(\frac{4\times 3}{2\times 1}\right)=210$


It means we can have 210 groups where each group contains total 5 letters (3 consonants and 2 vowels).

Number of ways of arranging 5 letters among themselves
$=5!=5\times 4\times 3\times 2\times 1=120$

Hence, required number of ways
$=210\times 120=25200$

$\mathrm{Question\; 2}$  
Answer: Option D
Explanation:
The word 'CORPORATION' has 11 letters. It has the vowels 'O','O','A','I','O' in it and these 5 vowels should always come together. Hence these 5 vowels can be grouped and considered as a single letter. that is, CRPRTN(OOAIO).

Hence we can assume total letters as 7. But in these 7 letters, 'R' occurs 2 times and rest of the letters are different.

Number of ways to arrange these letters
$=\frac{7!}{2!}=\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1}=2520$


In the 5 vowels (OOAIO), 'O' occurs 3 and rest of the vowels are different.

Number of ways to arrange these vowels among themselves $=\frac{5!}{3!}=\frac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1}=20$

Hence, required number of ways
$=2520\times 20=50400$




Question 3
Answer: Option A
Explanation:

A number is divisible by 5 if the its last digit is 0 or 5

We need to find out how many 3 digit numbers can be formed from the 6 digits $(2,3,5,6,7,9)$
which are divisible by 5.

Since the 3 digit number should be divisible by 5, we should take the digit 5 from the 6 digits(2,3,5,6,7,9) and fix it at the unit place. There is only 1 way of doing this.

1

Since the number 5 is placed at unit place, we have now five digits(2,3,6,7,9) remaining. Any of these 5 digits can be placed at tens place

5

1

Since the digit 5 is placed at unit place and another one digit is placed at tens place, we have now four digits remaining. Any of these 4 digits can be placed at hundreds place.

4

5

1

Required Number of three digit numbers
$=4\times 5\times 1=20$


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