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Optimized sequencing of analysis components in multidisciplinary systems

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Abstract

System analysis of complex engineering systems is synthesized from a collection of analysis components that have data dependencies on each other. Sequencing interdependent analysis components in order to reduce the execution time has been addressed by multidisciplinary design optimization researchers. Representation of interdependency of analysis components is accomplished as a design structure matrix or as a graph made of nodes and edges. Sequencing of interdependent analysis components that form a directed acyclic graph is trivial. Aggregation (i.e., group of components) of some of the components into a single super-component that can render a directed cyclic graph to a directed acyclic graph is important in sequencing. Identification of components that form an aggregation is the first step in sequencing. We argue that the best form of aggregation is the strongly connected component of the graph. Challenge essentially is in sequencing within aggregations. An aggregation having n components presents a search space of n! candidate sequences. The current state of the art is to use evolutionary algorithms for this search. An aggregation requires repeated traversal (cycle/loop) of components within it for convergence. The central aim of sequencing is to reduce/minimize the overall execution time for achieving convergence through iterations. Several objective functions have been proposed for the associated optimization problems like minimize the number of feedback paths, minimize the weighted sum of feedback paths, minimize feedback and crossovers, etc. These are proxy objectives as they are not backed by mathematically established relation between the proxy objective and the aim. An objective method of predicting the number of iterations based on the sensitivity of components is proposed here. It is shown that the best sequence that takes least time to execute has a particular ordering of components, which we call one-hop-sequence. The one-hop-sequencing of components is easily achieved using a small extension to Tarjans depth first search algorithm, a standard tool in graph theory. Extended TDFS does not use sensitivity information and is much faster than evolutionary algorithms that use sensitivity information. System analysis can have simple aggregation, recursive aggregations (i.e., aggregation within aggregation) or overlapping aggregations. One-hop-sequence is shown to be the best sequence for all three cases. After sequencing of the components is done, we investigate whether an inner aggregation must retain its loop or it must be severed for speed up. This step uses sensitivity information and can offer further speed up. The proposed methodology is implemented as a tool named CASeq. Ideas discussed here may be useful to other design structure matrix applicable domains like software design, systems engineering, organizational design, product development, multidisciplinary design, product architecture, project management, building construction, manufacturing and so on.

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Notes

  1. A strongly connected component of a directed graph G = (V, E) is a maximal set of vertices S ⊂ V, such that ∀ u & v ε S, we have both u → v and v → u; that is, vertices u and v are reachable from each other within the set S.

  2. TDFS is a standard search technique (Cormen et al. 2001; http://mathworld.wolfram.com/GraphTheory.html; Smith and Eppinger 1997; http://algowiki.net/wiki/index.php/Tarjan's_algorithm) to efficiently discover the aggregations (strongly connected components).

  3. Example: (A[{B C} D]) is an implementable looping structure whereas ({A[B C} D]) is a non implementable looping structure (overlapping aggregation).

  4. Authors would like to thank one of the reviewers for suggesting this.

  5. It shall be noted that the context of the problem in Hisham et al. (2006, 2007) is slightly different, and refers to a product development project and the components refer to project activities. The problem of interest is to find the optimal sequence of components to minimize project execution duration. This context throws an additional logical constraint to the overall search space-“it is infeasible to switch the direction of a coupling from a feedback to a feed-forward” (i.e., internal order within a family of sequences is important there).

  6. Depth first search timestamps each vertex. Each vertex V v has two timestamps: the first timestamp x records when V v is first discovered and second timestamp y records when the search finishes (represented as x/y).

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Acknowledgments

We thank Centre for Aerospace Systems Design & Engineering, IIT Bombay, India for the research environment and Aeronautics Research and Development Board (ARDB), India for this project grant. We are also thankful to Mr R Mahesh for his support in generating the diagrams used in this paper. We are extremely grateful to one of the reviewers for insightful comments that helped improve the paper.

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Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

    A. S. Shaja & K. Sudhakar

Authors
  1. A. S. Shaja
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  2. K. Sudhakar
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Correspondence to A. S. Shaja.

Appendices

Appendix 1

1.1 ETDFS

ETDFS is an extension to Tarjan depth first search .

Line numbers 1 to 21 is the standard TDFS (Smith and Eppinger 1997; http://algowiki.net/wiki/index.php/Tarjan's_algorithm). A small extension (line number 22-reverse the vertices which are popped from the stack) will lead to one-hop sequence within each aggregation.

Appendix 2

2.1 Topological sorting

Topological-sort(G)

  1. 1

    call DFS(G) to compute finishing times f(v)Footnote 6 for each vertex v

  2. 2

    as each vertex is finished, insert it onto the front of a linked list

  3. 3

    return the linked list of vertices.

Appendix 3

3.1 A comparison of proposed method with proxy objective function-based approaches

Most of the proxy objective functions-based approaches advocate minimizing feedback loops. The example below shows that sequence that minimizes feedback loops need not execute faster. Consider two sequences of five components 2, 3, 4, 5 and 8. Figure 28a uses the sequence (52834) and has only one feedback loop. Figure 28 uses the sequence ((45)283) and has two feedback loops. Strategy based on minimizing feedback loops will identify (52834) as the best sequence. But for sensitivities as in Fig. 20 and decade fall of p = 4 the sequence (52834) takes 35,880 s for executing, while sequence ((45)283) takes only 31,500 s. i.e., Convergence is reached faster by implementing an inner loop (45).

Fig. 28
figure 28

a Minimum feedback case. b One of the candidate for best sequence

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Shaja, A.S., Sudhakar, K. Optimized sequencing of analysis components in multidisciplinary systems. Res Eng Design 21, 173–187 (2010). https://doi.org/10.1007/s00163-009-0082-5

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