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upper bounding subproblem of part t.
x:-' is a nt x (k-1) rnatrix and y:-' is a rt x (k- 1) mauix, i.e. x:" = ( ~f. .T:. . . , .$') and
Y:-' = (y:, y'. . . . , y:.'). coming From the fint (k-1) prima1 solutions of SP, .
n:-' is a (k-1) x m, rnatrix and&' is a (k-l)x qt matrix, i.e. &'= (xfr .$r. ... . &lr)'
and a'-' = (dl. &r , . . , f. coming from the first (k-1) dual solutions of SP, .
The lower bound subproblerns have the sme structures as the part one subpmblem (lower
bound subproblem) in the parallel two pan decomposition method in Chapter 3. except for more
parts. thus having (N-2) more proposals from other subproblerns at each iteration. The upper
bound subproblems also have the same structures as the part two subproblem (upper bound
subproblem) in the pdlel two part decomposition except for more parts, thus having (N-2) more
cuts from other subproblems at each ireration.
The heuristic parallel decomposition structure exhibits an equivalent position of
subproblems. and al1 subproblems have access to information on dl other subproblems and work
together to optimize the whole problem.
The coordination is made through broadcasting proposds and cuts during the iteration.
The proposais. which have the prima1 information of the previous subproblems, are broadcasted
to ai1 other lower bound subproblems and the cuts, which have dual information, are broadcasted
to upper bound subproblems.
At the fint itention of the heuristic method. there is no information flow among the
subproblems since no informarion is available. while Lm's serial method begins with the fint
subproblem having no information from other subproblems. but all other subproblerns are solved