Decomposition Analysis of an Automotive Powertrain Design ...

epresentation facilitates decomposition of NLP problems and MDO optimization problems
because of a subtle but important distinction - no precedence relationships among variables are
specified a priori. In MDO problems, a directed graph is often used to partition the sequence of
analysis tasks based on precedence relationships specified prior to partitioning (for example,
Rogers 1989, Rogers and Bloebaum 1994, Altus et al. 1995). Directed graphs are not readily
amenable to representing the NLP problem because the precedence relationships they require do
not exist in an NLP problem until after the problem is solved. Only then are dependent and
independent variables distinguished. Hence, we contend that model-based decomposition
facilitates identification of broader set of coordination strategies for a given problem - those found
in the NLP literature and those found in the MDO literature.
Model-based decomposition proceeds with the four phases shown in Figure 1: model
development, partitioning, coordination strategy development, and, solution. If the objective
function is identified prior to the partitioning process, we use decomposition analysis (Wagner op
cit.) because an optimal design problem (ODP) is constructed and analyzed for decomposability.
If the objective is selected after partitioning, such that a decomposable optimization problem is
synthesized, we use decomposition synthesis (Krishnamachari 1996).
Model development consists of formulating the system model by defining variables,
parameters, functional relationships, and constraints, and by translating desirable defining
properties into partition metrics which define what linking properties are sought (variables vs.
functions), the maximum and minimum number of linking properties, the minimum number of
subproblems desired, the relative balance of subproblems, and any a priori preferences such as
including a specific variable as a linking variable. Wagner (op cit.) classifies in detail defining
properties for a broad class of NLP and MDO decomposition methods.
The partitioning process consists first of constructing the functional dependence table
(FDT) of the NLP problem which is an incidence matrix of Booleans denoting whether a function
is dependent on a variable. The undirected graph representation is then constructed and the optimal