Revised Multiphase Simplex Algorithm for Optimization in Engineering Design - Prof. Berdin, Study notes of Mechanical Engineering

Download Revised Multiphase Simplex Algorithm for Optimization in Engineering Design - Prof. Berdin and more Study notes Mechanical Engineering in PDF only on Docsity! 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 r j(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