LP solution space in Equation Form

For the sake of standardization . the algebraic representation of the LP
solution Space is made under two condition:

1. All the constraints ( with the exception of the nonnegativity restrictions ) are equations with
a nonnegative right-hand side .
2- All the variables are Nonnegative .

Converting Inequalities into Equations
in(≤) constraints the right-hand side  can be thought of as representing the limit on the availability of a resource in
which case the left hand side would represent  the usage of this limited resource by the activities ( variables) of
the model. the difference between the right hand side and the left hand side of the (اق من) constraints thus the unused
or slake amount of resource.

To convert a ( ≤  ) inequality to an equation a nonnegative slake variable is added to the left-hand side of the
constraints . 'For example in the reddy mikks model ,

the constraints associated with the use of raw material M1 is given as

   6X₁  +     4X₂   ≤ 24

defining s_{1  as the slak or unused amount of M1 ,the constraint can be converted to the following equation}                                                      6X₁  +     4X₂   + S_{1 =  24 ,} S₁ ≥  0

next a( ≥) constraint normally sets a lower limit on 

the on the activities of LP model. AS such the amount by which the left hand side exceeds the

minimum limit represents a surplus.

The conversion from ( ≥) to (=) is achieved by subtracting a nonnegative surplus variable from

the left hand side of the inequality .

For example in the diet model ( Example 202-2 ) , the constraint representing the 

minimum feed requirements is given as  X₁  +     X₂   ≥  800

Defining  S_{1   as the surplus variable, the constraint can be converted to the following eqution} 

X₁  +     X₂   - S_{1 =  800 ,} S₁  ≥   0

Note importantly that the slack and surplus variable , _{s1 and S1}

_{are always  non-negative.}

The only remaining requirement is for the right-hand side of the resulting equation to be non-negative.

The condition can always be satisfied  by multiplying both sides of the resulting equation by -1 where necessary.

For example  the constraint   -X₁  +     X₂   ≤   -3

is equivalent to the equation 

  -X₁  +     X₂  + S_{1  = -3,}S₁ ≥   0

Now multiplying both sides by -1 will render a non-negative right-hand side as desired that is

X₁  -     X₂   - _s1  =3

Example 2.2-2 This example solves the reddy mikks Model

the simplex method

the graphical method  shows that the optimum LP solution is always associated 

with a corner point of solution space.

this result is key to the development of the general algebraic simplex method for solving any LP

model.

The transition from the geometric corner-point solution to the simplex method entails a computational
procedure that determines the corner points algebraically.

This is accomplished by first converting all the inequality constraints into equations and then manipluating 
the resulting equations in systematic manner.

A main feature of the simplex method is that it solves the LP in iterations.

Each iteration moves the solution to a new corner point that has the potential
to improve the value of the objective fuction .

The process ends when no further improvements can be realized .

The simplex method involves tedious and voluminous computations .

Which makes the computer  an essential tool for solving LP Problems.

The computational rules of the simplex method are thus designed to facilitate automatic computations.