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These are the important key points of lecture slides of Introduction to Operations Research are:Primal Simplex Method, Revised Simplex Method, Practical Applications, Standard Form, Convenient Form, Canonical Form, Number of Pivot Operations, Identity Matrix, Multiplication By A Matrix, Basic Variables
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This method is a modified version of the Primal Simplex Method that we studied in Chapter 5. It is designed to exploit the fact that in many practical applications the coefficient matrix {a (^) ij } is very sparse , namely most of its elements are equal to zero.
Bottom line:
max
..
x
s t Z cx Ax b x
More Convenient Form
Canonical Form
a 11 x 1 + a 12 x 2 + ... + a 1 n xn + x (^) n + 1 = b 1 a 21 x 1 + a 22 x 2 + ... + a 2 n x (^) n + + xn + 2 = b 2 ..... ..... ... .... .... ..... ..... ... .... .... a (^) m 1 x 1 + am 2 x 2 + ... + amn xn + + x (^) n + m = b (^) m
x (^) j ≥ 0 , j = 1,..., n + m
max z = (^) j = 1 c (^) j
n
As in the standard format, b (^) i≥0 for all i.
1 2 3 1 2 4 1 5 1 2 3 4 5
x x x x x x x x x x x x x
max x
Z = 4 x 1 (^) + 3 x 2
System P
Observation
After any iteration of the simplex method the columns of the m basic variables comprise the columns of the mxm identity matrix. The order in which these columns are arranged for this purpose is important. This order is specified in the BV column of the simplex tableau.
BV Eq. # (^) x 1 x 2 x 3 x 4 x 5 RHS x 3 1 0 1 1 0 - 2 10 x 4 2 0 1 0 1 - 1 15 x 1 3 1 0 0 0 1 15 Z Z (^0) - 3 0 0 4 60
Example 6.1.
S ' =
0 1 1 0 1 0 1 0 0
0 1 0
− 2 − 1 1
b ' = (10 1515 , , ) c ' = ( , , , ,0 3 0 0 − 4 ) Z'=60. Docsity.com
Observation 1: After any number of iterations of the simplex method, the columns of the coefficient matrix corresponding to the basic variables at that iteration, comprise the identity matrix. Observation 2: Initially, the last m columns of the coefficient matrix comprise the identity matrix.
If we group the columns of the basic variables into I and the nonbasic variables into D’, then S’ = [I,D’] If we do the same for the initial matrix S, we have S = [B,D] where B is the matrix constructed from the columns of the initial matrix corresponding to the current basic variables.
BV Eq. # (^) x 1 x 2 x 3 x 4 x 5 RHS x 3 1 2 1 1 0 0 40 x 4 2 1 1 0 1 0 30 x 5 3 1 0 0 0 1 15 Z Z (^) - 4 - 3 0 0 0 0
Example 6.2.
BV Eq. # (^) x 1 x 2 x 3 x 4 x 5 RHS x 2 1 0 1 1 0 - 2 10 x 4 2 0 0 - 1 1 1 5 x Z (^1 3) Z^10 00 03 00 1
1 0 2 1 1 1 0 0 1
B −^1 =
1 0 − 2 − 1 1 1 0 0 1
S ( )^2 = B −^1 S
B −^1 S =
1 0 − 2 − 1 1 1 0 0 1
2 1 1
1 1 0
1 0 0 0 1 0 0 0 1
=
0 0 1
1 0 0
1 0 − 2 − 1 1 1 0 0 1
= S (2)
