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TECHNICAL PAPERS

Analysis of Decomposability and Complexity for Design Problems in the Context of Decomposition

[+] Author and Article Information
Li Chen1

United Technologies Research Center, 411 Silver Lane, MS-129-85, East Hartford, CT 06108chenl@mie.utoronto.caDesign and Manufacturing Integration Laboratory, Department of Mechanical and Industrial Engineering, The University of Toronto, 5 King’s College Road, Toronto, ON, Canada M5S 3G8chenl@mie.utoronto.ca

Simon Li

United Technologies Research Center, 411 Silver Lane, MS-129-85, East Hartford, CT 06108Design and Manufacturing Integration Laboratory, Department of Mechanical and Industrial Engineering, The University of Toronto, 5 King’s College Road, Toronto, ON, Canada M5S 3G8

From here forward, the term “design problem” will be used interchangeably with the term “design model” in the context of problem decomposition.

The term “redecomposition” refers to the rerunning of decomposition subject to new specifications for the setting of decomposition criteria (see (1,9)).

A decomposable system refers to a system that can be completely decomposed without any induced interactions, while a nearly decomposable system implies the presence of non-negligible interactions. In this work, the term of decomposable is used interchangeably with that of nearly decomposable.

1

Corresponding author.

J. Mech. Des 127(4), 545-557 (Aug 21, 2004) (13 pages) doi:10.1115/1.1897405 History: Received January 19, 2004; Revised August 21, 2004

The current practice in problem decomposition assumes that (i) design problems can be rationally decomposed a priori and (ii) decomposition can usefully result in complexity reduction a priori. However, these assumptions are not always true in reality. In response to this concern, this paper introduces the notions of decomposability and complexity to problem decomposition. In particular, a full scale of decomposability analysis and complexity analysis in the context of decomposition are presented along with approaches and algorithms. These new analyses not only address the viability and validity of decomposition, but also help achieve an optimal number of subproblems during decomposition, which is usually determined by trial and error or a priori. Furthermore, a procedure able to combine these new analyses into our two-phase decomposition framework is described. This effort leads to an enhanced decomposition method able to find the most appropriate decomposition solution to a complex design problem.

Copyright © 2005 by American Society of Mechanical Engineers
Topics: Design , Couplings
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References

Figures

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Figure 1

Sample of incidence matrix and problem decomposition

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Figure 2

Example of highly coupled incidence matrix

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Figure 7

Sample block to illustrate the presence of zero coupling value

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Figure 8

Results for the sample matrix after tree-based dependency analysis

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Figure 9

Tree-cutting step for the sample matrix

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Figure 11

Example for complexity measure with decomposition solutions

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Figure 12

A formal two-phase decomposition method (9)

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Figure 13

Flowchart for achieving the optimal number of blocks in decomposition

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Figure 14

Original matrix and the resulting diagonal matrix for the example

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Figure 15

Results of the decomposability and complexity analysis

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Figure 16

Plots of the coupling and complexity changes

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Figure 10

Matrix patterns versus (a) before decomposition, (b) ideal decomposition, (c) coordination-based decomposition (1)

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Figure 6

External and internal coupling in the coupling graph

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Figure 5

Column-coupling graph for the example problem in Fig. 1

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Figure 4

Coupling measure between Col8 and Col9

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Figure 3

Coupling measure between Col2 and Col10

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