Simplex Method - Maximization Case

Consider the general linear programming problem

Maximize z = c₁x₁ + c₂x₂ + c₃x₃ + .........+ c_nx_n
subject to
a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + .........+ a_1nx_n ≤ b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + .........+ a_2nx_n ≤ b₂
.........................................................................
a_m1x₁ + a_m2x₂ + a_m3x₃ + .........+ a_mnx_n ≤ b_m
x₁, x₂,....., x_n≥ 0
Where:
c_j (j = 1, 2, ...., n) in the objective function are called the cost or profit coefficients.
b_i (i = 1, 2, ...., m) are called resources.
a_ij (i = 1, 2, ...., m; j = 1, 2, ...., n) are called technological coefficients or input-output coefficients.

Converting inequalities to equalities

Introducing slack variables to convert inequalities to equalities
a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + .........+ a_1nx_n + s₁ = b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + .........+ a_2nx_n + s₂ = b₂
..............................................................................
a_m1x₁ + a_m2x₂ + a_m3x₃ + .........+ a_mnx_n + s_m = b_m
x₁, x₂,....., x_n≥ 0
s₁, s₂,....., s_m ≥ 0
An initial basic feasible solution is obtained by setting x₁ = x₂ =........ = x_n = 0
s₁ = b₁
s₂ = b₂
..............
s_m = b_m

The initial simplex table is formed by writing out the coefficients and constraints of a LPP in a systematic tabular form. The following table shows the structure of a simplex table.

Structure of a simplex table

	cj	c1	c2	c3	---	cn	Right Hand Side
cB	Basic variables B	x1	x2	x3	---	Sn	Solution values b (=XB)
cB1	x1	a11	a12	a13	---	a1n	b1
cB2	x2	a21	a22	a23	---	a2n	b2
cB3	x3	a31	a32	a33	---	a3n	b3
---	-----	----	----	-----	---	----	-----
cBm	xn	am1	am2	am3	---	amn	bm
Z		z1	Z2	Z3	---	Zn
cj- zj		c1-z1	C2-z2	C3-z3	---	Cn-zn

Where:
_Cj = coefficients of the variables in the objective function.
c_B = coefficients of the current basic variables in the objective function.
B = basic variables in the basis.
X_B = solution values of the basic variables(Right Hand Side.
c_j- z_j = index row.

Source Link : Simplex Method - Maximization Case

Simplex Method

Simplex Method Introduction

Simplex Method is an algorithm for solving the classical linear programming problem; developed by George B.Dantzig in 1947. The simplex method is an iterative procedure, solving asystem of linear equations in each of its steps, andstopping when either the optimum is reached, or thesolution proves infeasible. The basic method remainedpretty much the same over the years, though therewere many refinements targeted at improvingperformance (eg. using sparse matrix techniques),numerical accuracy and stability, as well as solvingspecial classes of problems, such as mixed-integerprogramming.
Basic Terminology - Simplex Method

Slack variable

It is a variable that is added to the left-hand side of a less than or equal type of constraint to convert the constraint into an equality. In economic terms, slack variables represent left-over or unused capacity.

Specifically:
a_i1x₁ + a_i2x₂ + a_i3x₃ + .........+ a_inx_n ≤ b_i can be written as
a_i1x₁ + a_i2x₂ + a_i3x₃ + .........+ a_inx_n + s_i = b_i
Where i = 1, 2, ..., m

Surplus variable

It is a variable subtracted from the left-hand side of a greater than or equal to type constraint to convert the constraint into an equality. It is also known as negative slack variable. In economic terms, surplus variables represent overfulfillment of the requirement.
Specifically:
a_i1x₁ + a_i2x₂ + a_i3x₃ + .........+ a_inx_n ≥ b_i can be written as
a_i1x₁ + a_i2x₂ + a_i3x₃ + .........+ a_inx_n - s_i = b_i
Where i = 1, 2, ..., m

Artificial variable

It is a non negative variable introduced to facilitate the computation of an initial basic feasible solution. In other words, a variable added to the left-hand side of a greater than or equal to type constraint to convert the constraint into an equality is called an artificial variable. 

From: Operation Research Simplified 

Finding Vertical and Horizontal Asymptotes of a function as well as Oblique Asymptotes

Value of "x" for which the Function is discontinuous

Increment Method of Differentiation (Example 4)

Increment Method of Differentiation (Example No.3)

Increment Method of Differentiation (Example No.2)

Increment Method of Differentiation (Example No.1)

Find the derivative by Increment Method(Differential Calculus)

Limits Solution on Love and Rainville Textbook Page 18 No. 29-38

Limits Solution on Love and Rainville Textbook on Page 18 No. 29-38

Limits Examples

Differential Calculus (Graph of y=x|x|)

Graph \[y = x|x|\]

Table of Values:

Graph

Note: Scaling on x and y axis are different.

Graphics are Generated by Wolfram|Alpha.

Differential Calculus(Example 4 of the Graph of Signum Function)

Graph \[y = {x^2}{\mathop{\rm sgn}} (x - 1)\]

Table of Values:

Graph:

Note: Scaling on x and y axis are different.


Thanks to Wolfram|Alpha for the Graphics.

Differential Calculus(Example 3 of the Graph of Signum Function)

Graph \[y = {x^2} + {\mathop{\rm sgn}} (x - 1)\]

This is it's table of values of  from x -5 to 5.

Graph

Note that scaling on x and y axis are different.

Thanks to Wolfram|Alpha for the Graphics.

Differential Calculus (Example 2 of Signum Function Graph)

\[{\rm{Graph}}\] \[y = x - |x|\]

Table of Values

Graph

Differential Calculus(Example 1 of the Graph of Signum Function)

Graph \[y = \frac{1}{2}(1 - {\mathop{\rm sgn}} (x)){x^2}\]

Here's what I get when I make table of values for this:

You should not be confused about the Signum Function sgn(x) because basically it only give
-1,0 and 1. Because it is a sign function, meaning it decides the sign.
If x<0,     sgn(x)=-1
   x=0       sgn(x)=0
   x>0       sgn(x)=1
Those value of sgn(x) are the range that you have to put on sgn(x) depend on the sign of
the x's.

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Differential Calculus (What is a Signum Function?)

From: Wikipedia

In mathematics, the sign function or signum function (from signum,Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as sgn.

Definition: 
The signum function of a real number x is defined as follows:

Properties:

Any real number can be expressed as the product of its absolute value and its sign function:

It follows that whenever x is not equal to 0 we have

Similarly, for any real number x,

The signum function is the derivative of the absolute value function (up to the indeterminacy at zero): Note, the resultant power of x is 0, similar to the ordinary derivative of x. The numbers cancel and all we are left with is the sign of x.

 

So basically Signum Function sgn(x) has a domain of any Real Numbers 

and a range of {-1,0,1}.