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a proposal is passed back to the fint subproblem. The latter scheme has ken implemented by
Lan and Fuller [1995a] for the two stage mode1 structure. a special case of the two part mode1
structure, e.g. two pend models. Lan and Fuller choose part one as the upper bound
subproblem. The senal aigorithm first solves the stage one subproblem and sen& a prima1
proposal to the second stage subproblern. then it solves a new second stage subproblem with a
new column generated. and sen& duai proposais to the stage one subproblem. The stage one
subproblem is then dved with a new cut. The stage one subproblem provides upper bounds
on the optimal value of the whole problem, and the stage two subproblem provides lower
bounds. This sequential process continues until the objective values of both subproblerns have
corne within a predetermined tolerance.
3.3 The Decomposition Algorithm
in this section. the procedure of the panllel decomposition algorithm is fomûlly
described. Various propenies of this algorithm will be discussed in the next section. The
scalar E >O is defined by the user to stop the algorithm when the upper bound $ and the lower
bound :,' are less than E apart. Claims about feasibility and optimality are justified in the next
section. In the statement of the algorithm. the symbols xf2, Y:', nik'' and R~"' appex, and
for the value k=2, they have special meanings: x:, and Y? are nul1 manices having no
columns. while Ill0 ad R,' are nul1 matrices having no rows. The send and receive
statements synchronize the two processors' iteration counten.