In the previous chapter, we discussed about the **graphical method for
solving linear programming problems (LPP)**. Although the graphical method is
an invaluable aid to understand the properties of linear programming
models, it provides very little help in handling practical problems.

In this chapter, we concentrate on the **simplex method** for solving linear
programming problems with a larger number of variables.

Many different methods have been proposed to solve linear programming
problems, but simplex method has proved to be the most effective. This
technique will nurture your insight needed for a sound understanding
of several approaches to other programming models, which will be studied
in subsequent chapters.* Simplex Method* is applicable to any problem that
can be formulated in terms of linear objective function, subject to
a set of linear constraints. Often, this method is termed Dantzig's
simplex method, in honour of the mathematician who devised the approach.

In the following section, we introduce you to the standard vocabulary
of the simplex method.

## Basic Terminology - Simplex Method Linear Programming (LP)

### Slack variable

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

Specifically:

a_{i1}x_{1} + a_{i2}x_{2} + a_{i3}x_{3} + .........+ a_{in}x_{n} ≤
b_{i} can be written as

a_{i1}x_{1} + a_{i2}x_{2} + a_{i3}x_{3} + .........+ a_{in}x_{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_{i1}x_{1} + a_{i2}x_{2} + a_{i3}x_{3} + .........+ a_{in}x_{n} ≥ b_{i} can be written as

a_{i1}x_{1} + a_{i2}x_{2} + a_{i3}x_{3} + .........+ a_{in}x_{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.