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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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chapter 6 the revised simplex method
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.
slide2
Bottom line:
    • Don’t update all the columns of the simplex tableau: update only those columns that you need!
standard form
Standard Form
  • opt=max
  • ~ 
  • bi ≥ 0 ,
  • for all i.
slide5
Canonical Form

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

system p
System P

(6.4)

observation
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
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
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
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.

slide15
Since S’ = TS, it follows that

S’ = [I,D’] = TS = [TB,TD]

hence

I = TB

from which we conclude that

T = B-1

notation
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
slide22
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
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?)
slide25

rD =

correction

  • C’B = (0,0,...,0)
niln but how do we compute b 1
(NILN)But how do we compute B-1 ?
  • Bad news:
  • We have to compute it as we go along
  • Good News:
  • We do not have to compute it from scratch
  • Observation:

S’ = B-1S = B-1 [M,I] = [B-1M,B-1I] = [B-1M,B-1]

  • Hence, B-1 is equal to the matrix comprising the last m columns of the LHS matrix.