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RESEARCH PAPERS

# Design Process Sequencing With Competent Genetic Algorithms

[+] Author and Article Information
Christoph Meier

Institute of Astronautics, Technische Universität München, 85,748 Garching, Germanyc.meier@tum.de

Ali A. Yassine1

Department of Industrial & Enterprise, Systems Engineering (IESE), University of Illinois at Urbana-Champaign, Urbana, IL 61801yassine@uiuc.edu

Tyson R. Browning

M.J. Neeley School of Business, Texas Christian University, TCU Box 298530, Fort Worth, TX 76129t.browning@tcu.edu

The term dependency structure matrix is also commonly used.

All of the situations we model can also be represented using directed graphs (digraphs). We choose the DSM merely as a representation scheme because of its (1) ability to visually highlight iterative loops among coupled components and (2) amenability to matrix-based analysis techniques.

Some DSM literature uses the opposite convention (the transpose of the matrix), with inputs in columns and outputs in rows, and thus feedback below the diagonal; the two conventions convey equivalent information.

In the rest of the paper, the terms “design process sequencing” and “DSM sequencing” are used interchangeably.

In a fully coupled DSM, partitioning will not change the original sequence.

In graph theory, sets of vertices in which every vertex can reach every other vertex are referred to as SCCs. By definition, every vertex can reach itself and forms a SCC of size 1.

Precisely, the complexity is $O(n+a)$, where $n$ is the number of activities (vertices) and $a$ is the number of relationships (arcs) in the DSM.

Crossovers in this context can be understood as the crossing of feedback of one activity with another without exchanging information through the intersection.

As a tool for visualization we used $Microsoft®$ Excel, which has the capability to plot only $215$ values on the $x$-axis. Other tools may be used to plot larger landscapes. We used the algorithm for the generation of permutations, described in (33), because subsequent permutations differ only at one position and can thus be regarded as “neighbors” in the search space. Every permutation $π$ can be regarded as a point in the design space $Φd$ ($x$-axis) with a corresponding point in its objective space $Φo$ ($y$-axis), depending on the defined objective function. Essentially, Fig. 2 plots the relationship $f(π):Φd→Φo$, where $Φo$ corresponds to all real numbers.

However, it is worth noting that the representation is not limited by a binary alphabet and schemata representation is thus different.

Hence, the schema H(∗011∗) has a defining length $δ(H)=4−2=2$.

At premature convergence, the takeover time—the time at which an individual dominates the population—is less than the innovation time—the time for any operator to achieve a solution better than any other solution at this point (62).

Here, good schemata have to compete against each other due to the high SP, causing incomplete schema coverage and greatly reducing the probability of finding a good solution.

In TWR, all individuals who participated in a tournament are also candidates for the following tournament without any constraints. Consequently, the best string is expected to obtain $s$ copies on average. In contrast, TWOR requires $s$ “rounds” of selection, where individuals may participate in a tournament only if all other chromosomes have been part of a tournament in the same round. Thus, the best individual gets exactly $s$ copies for further processing.

Murata and Ishibuchi (37) distinguish between two versions of position-based crossover.

The problem is the so-called One-Max Problem, described in (70).

Mixing time is the number of generations until a population contains $n−1$ copies of the best individual. Selection time is the expected number of generations to obtain one mixing event, where a mixing event occurs if the crossover operator generates an offspring with more BBs of equal size than each parent (71).

Depending on the underlying interests of the tests, the importance of investigating any particular efficiency enhancement technique varies. For instance, we do not want to primarily study CPU run times for a GA but to investigate its scale-up behavior in terms of objective function evaluations. Hence, we do not focus on parallelization strategies in this section; the reader may refer to the references quoted.

Our definition of the term neighborhood is based on the 2-opt heuristic. Using another definition for this term, e.g., a 3-opt neighborhood, would yield a different neighborhood size and computational complexity.

This complexity results from the linear complexity of Eq. 3, which has to be executed $(n2−n)∕2$ times.

This complexity results from the constant complexity of Eq. 4, which has to be executed $(n2−n)∕2$ times.

A BB “level $k$” denotes the processing of BBs of maximum size $k$ within era $k$.

We did not use different population sizes for different runs of a specific test scenario.

The mathematical equations of both objectives can be found in Table 1 (Gebala and Eppinger (40) and Kusiak (41)).

Obviously, these relationships belong to the constructed SCC as well.

These DSM blocks can be regarded as “easy” since the SGA was able to identify the optimum with low population sizes.

1

Author to whom correspondence should be addressed.

J. Mech. Des 129(6), 566-585 (Oct 12, 2006) (20 pages) doi:10.1115/1.2717224 History: Received March 06, 2006; Revised October 12, 2006

## Abstract

In product design, it is critical to perform project activities in an appropriate sequence. Otherwise, essential information will not be available when it is needed, and activities that depend on it will proceed using assumptions instead. Later, when the real information is finally available, comparing it with the assumptions made often precipitates a cascade of rework, and thus cost and schedule overruns for the project. Information flow models have been used to sequence the engineering design process to minimize feedback and iteration, i.e., to maximize the availability of real information where assumptions might otherwise be made instead. In this paper, we apply Genetic Algorithms (GAs) to an information flow model to find an optimized sequence for a set of design activities. The optimality of a solution depends on the objective of rearrangement. In an activity sequencing context, objectives vary: reducing iteration/feedback, increasing concurrency, reducing development lead-time and cost, or some combination of these. We adopt a matrix-based representation scheme, the design structure matrix (DSM), for the information flow models. Our tests indicate that certain DSM characteristics (e.g., size, sparseness, and sequencing objective) cause serious problems for simple Genetic Algorithm (SGA) designs. To cope with the SGA deficiency, we investigate the use of a competent GA: the ordering messy GA (OmeGA). Tests confirm the superiority of the OmeGA over a SGA for hard DSM problems. Extensions enhancing the efficiency of both a SGA and the OmeGA, in particular, niching and hybridization with a local search method, are also investigated.

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## Figures

Figure 1

A small process model shown with digraph and matrix representations

Figure 2

Visualization of a subset of the search space for an 8×8 DSM block; three different landscapes with varied densities and and sequencing objectives are shown. (a) Kusiak objective – 50% density. (b) Kusiak objective – 15% density. (c) Gebala objective – 15% density.

Figure 3

SGA flowchart

Figure 4

Encoding a DSM as a chromosome via integer representation

Figure 5

Illustration of position-based crossover, version 2

Figure 6

Theoretical and empirical predictions of the GA boundaries (67) for an easy problem. (a) Theoretical prediction, according to (67). (b) Empirical results, according to (67).

Figure 7

Illustration of shift mutation

Figure 8

Illustration of a messy chromosome

Figure 9

Demonstration of random keys

Figure 10

Messy chromosomes may have variable length

Figure 11

Use of a competitive template on underspecified chromosomes

Figure 12

OmeGA flowchart

Figure 13

Illustration of the BB filtering phase

Figure 14

Demonstration of the cut and splice operations

Figure 15

Test results for a tightly and loosely coded deceptive function of length 32. (a) SGA. (b) OmeGA.

Figure 16

Comparative performance of the OmeGA and the SGA on deceptive functions

Figure 17

Comparative SGA performance on DSM blocks with varied densities and objective functions

Figure 18

Demonstration of niching to improve SGA performance

Figure 19

Comparative performance of the OmeGA and the SGA with niching

Figure 20

Comparative performance of the OmeGA and SGA with niching as DSM block density varies

Figure 21

Comparative performance of the regular SGA and the global-local SGA (block density=0.5)

Figure 22

Comparative performance of the regular SGA and the global-local SGA (block density=0.15)

Figure 23

Comparative performance of the regular SGA, the global-local SGA, the OmeGA, and the hybrid OmeGA on DSM blocks with varied densities

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