ME 6103 – Optimization in Engineering Design (Bras)
REVISED MULTIPHASE SIMPLEX (MULTIPLEX) ALGORITHM
Find v so as to
lexmin uT = {c(1)Tv, ..., c(k)Tv, ..., c(K)Tv}
subject to
Av = b
v ≥ 0 where v =
x
x
d
d
1) Initialization. Let vB = = d-). Thus, B = I, B-1 = I, and = b.
Set k = 1. Initially, all variables are unchecked.
2) Develop the pricing vector. Determine:
(k)T = cB(k)T B-1
3) Price out all UNCHECKED, non-basic columns. Compute:
rj(k) = (k)T aj - cj(k) for all j N’
where N’ is the set of non-basic and unchecked variables.
4) Selection of entering non-basic variable. Examine those rj(k) as computed in step 3. If none are
positive, proceed to step 8. Otherwise, select the non-basic variable with the most positive rj(k) (ties
may be broken arbitrarily) as the entering variable. Designate this variable as vq.
5) Update the entering column. Evaluate:
q = B-1 aq
6) Determine the leaving basic variable. The leaving variable row is designated as i=p. Using the
present representation of = B-1b and the values of q, as derived in step 5, determine
p
j,p
min
i|
i,q
0
i
i,q
Again, ties may be broken arbitrarily. The basic variable associated with row i=p is the leaving
variable, vBp.
If none exists, the entering variable is unbounded (all i,q ≤ 0). This condition will not occur as
long as v ≥ 0 and c(k) ≥ 0 for all k. In such case no elements of uT can become less than 0 and
therefore are always bounded.
7) Pivot. Replace the column ap in B by aq and compute the new basis inverse B-1. Return to step 2
8) Convergence check. If either one (or both) of the following conditions holds, STOP as the optimal
solution has been found.
a) if all rj(k) as computed in step 3 are negative, or
b) if k = K (where K = the number of priority levels, or terms in uT).
Dr. Bert Bras Telephone 404-894-9667 Fax 404-894-9342 E-mail bert.bras@me.gatech.edu
