The document provides an introduction to the simplex method for solving linear programming problems (LPP). It discusses how to convert an LPP into standard form, which involves writing it as an optimization problem with a linear objective function subject to linear equality and inequality constraints, with all variables nonnegative. It describes how to add slack and surplus variables to convert inequality constraints into equalities. The document also covers elementary row operations, row equivalence of matrices, row echelon form, reduced row echelon form, and how these concepts relate to the standard form of a system of linear equations.

1. Chapter 02 - Simplex method introduction

2. 2
nn xcxcxcZ  ...2211
0,...,0,0
0,...,0,0
...
;
.
...
...
21
21
2211
22222121
11212111





m
n
mnmnmm
nn
nn
bbb
xxx
bxaxaxa
bxaxaxa
bxaxaxa
 The standard form of an LPP with n variables and m unknowns is as
follows:
 Maximize
 Subject to:

3. 3
 Main features of the LPP standard form:
 Maximize or Minimize the objective function.
 All constraints are equations.
 All decision variables are nonnegative.
 The right hand side of each constraint is nonnegative.

4. 4
 The above standard LPP can be written in matrix form as:
 max(min) Z=CX
 Subject to: Ax=b, x>=0, b>=0.
 Where,















mnmm
n
n
aaa
aaa
aaa
A
...
.
...
...
21
22221
11211
 n
n
cc
x
x
x
x ...cc,
b
.
b
b
b,
.
21
m
2
1
2
1

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














5. 5
 max
 Subject to:
 Can be written in matrix form with:
54321 325 xxxxxZ 
5,...,1,0
7x43
822
5321
4321



ix
xxx
xxxx
i
 11325,
7
8
,,
10143
01221
5
4
3
2
1



















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
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






 cb
x
x
x
x
x
xA

6. 6
 The above standard form is required in order to start the simplex
method. The question is how to convert a given problem from
nonstandard form into standard form?, i.e. how to convert an
inequality constraint into an equality?.
 The answer to this question is to add the so called slack variables and
surplus variables.
 Back to the example of inspection, which had the constraints:
 Minimize
 Subject to:
21 3640 xxZ 
0,0
4535
10
8
21
21
2
1




xx
xx
x
x

7. 7
 Minimize Subject to:
 The first constraint is converted to where
represents the number of grade 1 inspectors who are not utilized.
 The second constraint is converted to Where
represents the number of grade 2 inspectors who are not utilized.
 are called slack variables.
 The third constraint is converted to
 is called a surplus variable. It represents the number of extra pieces
inspected over the min. required minimum.
21 3640 xxZ 
0,0
4535
10
8
21
21
2
1




xx
xx
x
x
0,8 331  xxx 3x
0,10 442  xxx 4x
43 and xx
0,4535 5521  xxxx
5x

8. 8
 If the problem contains a variable which is not restricted in sign, then we
do the following: If is unrestricted in sign, then set
 The value of (positive or negative) depends on the values of
 Example: Convert the following problem into a standard LPP:
 Solution: 1. Set
2. Multiply the last constraint by -1, to get:
3. Add a slack variable ( ) to 1st constraint to become:
4. Add a surplus variable ( ) to the 2nd constraint to become:
5. Replace by in the original problem , to get the standard form:
1x 321 xxx 
1x 32 xandx
edunrestrictx0,x,x
-52x-x-3x
2x
7x:to.
32xZ
321
321
321
321
321





xx
xxS
xxMax
.0xand, 54543  xxxx
.523 321  xxx
6x .0,7 66321  xxxxx
7x .27321  xxxx
3x 54 xx 

9. 9
 The standard form:
0,...,,x
5223x-
2x
7x:.
332xZ
721
5421
75421
65421
5421





xx
xxx
xxxx
xxxxTS
xxxMaximize

10. 10
 Two systems of linear equations are said to be equivalent if both systems
have the same solution set.
 The following are called elementary row operations that can be applied to
rows of a matrix:
1. Exchange any two rows.
2. Multiply a row by a nonzero constant.
3. Multiply a row by a constant and add the result to another row.
 Notice that these operations can also be applied to equations in a system
of equations. They are called in this case elementary operations.
 Two matrices are said to be row equivalent if one of them can be obtained
from the other by a sequence of elementary row operations.

11. 11
 A matrix is said to be in reduced row-echelon form if it satisfies all four of
the following conditions.
 If there are any rows of all zeros then they are at the bottom of the matrix.
 If a row does not consist of all zeros then its first non-zero entry (i.e. the left
most non-zero entry) is a 1. This 1 is called a leading 1.
 In any two successive rows, neither of which consists of all zeroes, the leading
1 of the lower row is to the right of the leading 1 of the higher row.
 If a column contains a leading 1 then all the other entries of that column are
zero.
 A matrix is said to be in row-echelon form if it satisfies items 1- 3 of the
reduced row-echelon form definition.

12. 12
 Notice that the only real difference between row-echelon form and
reduced row-echelon form is that a matrix in row-echelon form is only
required to have zeroes below a leading 1 while a matrix in reduced row-
echelon from must have zeroes both below and above a leading 1.
 A standard form of a system of equations is as follows:

13. 13
 The following matrix is called the augmented matrix of the above system:
 If two augmented matrices are row equivalent, then they represent two
equivalent systems of linear equations.

14. 14
 And the following is called is called the coefficient matrix: