A matrix-based clustering approach for the decomposition of design problems

Abstract

In matrix-based modeling, a design problem is represented by a rectangular matrix that captures the dependency relationships between design functions and parameters. To decompose such a matrix-based design problem, the two-phase method, which adapts hierarchical cluster analysis, has been proposed in literature. Yet, the clustering strategy in general is still challenging since the decomposition of design problems requires simultaneous clustering of two types of objects (i.e., design functions and parameters). In this paper, the notion of coupling is advanced by characterizing the coupling between two objects of different types. Accordingly, the two-phase method is revised via two methodical components: coupling-based dependency analysis and tree-based partitioning analysis. While the coupling-based dependency analysis concatenates different types of coupling for organizing the dependency structure, the tree-based partitioning analysis utilizes the coupling information from dependency analysis to identify design sub-problems and their interactions. Due to a better understanding of the coupling notion, the revised approach is able to simplify the algorithmic procedure and narrow down the solution search space without compromising the solution quality. Two examples (i.e., a relief valve system and a powertrain system) are used to demonstrate and justify the utility of the revised method.

Appendix: Steps of hierarchical cluster analysis

This appendix explains the basic steps of the single linkage clustering method (one type of hierarchical cluster analysis) that is used in the revised two-phase method of this paper. Particularly, the method takes a coupling matrix as the input and yields the tree (or dendrogram) as the output. The algorithmic procedure is based on Romesburg (2004). The coupling matrix shown in Table 4 is used for illustration.

Table 4 Coupling matrix

Step 1: Pick two objects that yield the highest value in the coupling matrix. In this case, objects c and d have the highest coupling value (i.e., 0.92). Thus, these two objects are picked.

Step 2: Form the branch of the tree by combining the picked objects, as shown in Fig. 16. The vertical axis of Fig. 16 is labeled with the coupling values. The newly formed branch combines two objects at their coupling value. For instance, objects c and d are combined at the coupling value of 0.92 in Fig. 16. The newly formed branch is labeled as ‘cd’.

Fig. 16

Newly formed tree branch

Step 3: Update the coupling matrix to represent the newly formed branch. Such modification is achieved through the average distance formulation as follows.

$$ r_{(ij)k} = {\frac{{r_{ik} + r_{jk} }}{2}}\quad 1 \le k \le n\quad k \ne i\quad k \ne j $$

where r _ik is the coupling value between objects i and k in the coupling matrix, and the subscript ij is referred to the newly combined branch. Table 5 has shown the updated coupling matrix after combining objects c and d.

Table 5 Updated coupling matrix after combining objects c and d

Step 4: Repeat Steps 1 to 3 until the coupling matrix cannot be further reduced. The resulting tree of the coupling matrix in Table 4 is shown in Fig. 17.

Fig. 17

Tree of the coupling matrix