Chapter 6 The Revised Simplex Method

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# Chapter 6 The Revised Simplex Method - PowerPoint PPT Presentation

Chapter 6 The Revised Simplex Method. This method is a modified version of the Primal Simplex Method that we studied in Chapter 5.

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Presentation Transcript
Chapter 6The Revised Simplex Method
• 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 {aij} is very sparse, namely most of its elements are equal to zero.
Bottom line:
• Don’t update all the columns of the simplex tableau: update only those columns that you need!
Standard Form
• opt=max
• ~ 
• bi ≥ 0 ,
• for all i.
Canonical Form

As in the standard format, bi≥0 for all i.

System P

(6.4)

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.
6.2 The Transformation
• How can we compute S’ from S ?
• From Linear Algebra we know that any finite sequence of pivot operations is equivalent to (left) multiplication by a matrix.
• In other words,

S’ = TS

• The question is then:

T = ??????

What is T ???
• 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.

Analysis
• 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.

Since S’ = TS, it follows that

S’ = [I,D’] = TS = [TB,TD]

hence

I = TB

from which we conclude that

T = B-1

Notation:
• IB = Indices of the basic elements (in canonical form)
• ID = indices of the nonbasic variables in increasing order
• cB = Initial cost vector of the basic variables
• cD = Initial cost vector of the nonbasic variables
r = reduced costs vector
• D = columns of the coefficient matrix in the initial simplex tableau corresponding to the current nonbasic variables.
Behind the Formula (NILN)
• Each column of the coefficient matrix in the new tableau is equal to B-1 times the corresponding initial column, i.e

new column = B-1 initial column

• This is also true for the right-hand-side vector, i.e

new RHS = B-1 initial RHS

• Observe that the z-row is not included in this formulation (why?)

rD =

correction

• C’B = (0,0,...,0)
(NILN)But how do we compute B-1 ?