Abstract
Product family optimization involves not only specifying the platform from which the individual product variants will be derived but also optimizing the platform design and the individual variants. Typically these steps are performed separately, but we propose an efficient decomposed multiobjective genetic algorithm to jointly determine optimal platform selection, platform design, and variant design in product family optimization. The approach addresses limitations of prior restrictive component sharing definitions by introducing a generalized two-dimensional commonality chromosome to enable sharing components among subsets of variants. To solve the resulting high-dimensional problem in a single stage efficiently, we exploit the problem structure by decomposing it into a two-level genetic algorithm, where the upper level determines the optimal platform configuration while each lower level optimizes one of the individual variants. The decomposed approach improves scalability of the all-in-one problem dramatically, providing a practical tool for optimizing families with more variants. The proposed approach is demonstrated by optimizing a family of electric motors. Results indicate that decomposition results in improved solutions under comparable computational cost, and generalized commonality produces families with increased component sharing under the same level of performance.
An earlier version of this chapter appeared in A. Khajavirad, J.J. Michalak, and T.W. Simpson (2009) “An Efficient Decomposed Multiobjective Genetic Algorithm for Solving the Joint Product Platform Selection and Product Family Design Problem with Generalized Commonality”, Structural and Multidisciplinary Optimization, 39(2):187–201 (© AIAA 2009), reprinted with permission.
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Notes
- 1.
Here, without loss of generality, we assume each component is represented by a single design variable (i.e., gene); however, the algorithm is applicable to the case which each component includes different number of design variables.
- 2.
In our discussion, we assume that all products include the same number of components as candidates for commonality. However, this representation can be modified to include a general case in which any subset of components may be absent in a variant by setting the corresponding gene in the 2D chromosome to a distinct integer number (e.g., zero) and omitting those genes from the design variable chromosome of that variant. In addition, for the components that are only present in p′ < p products, their commonality genes can take any integer value between 1 and p′.
- 3.
In case of discrete variables, the average value should be further rounded to the closest discrete level. Moreover, this constraint can be imposed using other strategies such as generating a random number in the upper level and sending it to the lower levels.
- 4.
The user-defined tolerance for considering two design variables to be equal should be set using knowledge about the problem, including the physical interpretation of the variable values and knowledge about the sensitivity of performance to the value of these variables. Thus, setting of the user-tolerance is necessarily case-specific.
- 5.
It should be noted that the commonality consistency constraints are only imposed for finding the optimal solution faster, and the method can identify optimal solutions without these constraints as well.
- 6.
To estimate the tooling cost savings more precisely, CI should be reformulated to include coefficients representing the amount of cost saving due to sharing each component; however, this extension has no effect on the optimization approach, and all coefficients are assumed to be equal in this chapter.
- 7.
Data exchange necessary to enforce consistency constraints is handled through Message Passing Interface (MPI) library. Details of the implementation and a copy of the code are available through the authors or at http://www.cmu.edu/me/ddl.
- 8.
It should be noted that the parallelization method applied herein is the direct benefit of the proposed decomposition scheme and should not be confused with general types of parallel GAs (e.g., fine-grain, coarse-grain, and master–slave models), which are derived from the evolutionary nature of the GA and are independent of the specific problem being solved. Generic parallel GAs could additionally be used to solve subproblems in the proposed decomposition if the optimization of an individual product is too complex for a single GA or if further speedup is desired, but we do not pursue this possibility here.
- 9.
Power equality constraint is a second-order equation as a function of current and has two positive roots if any. Therefore, the roots are compared with respect to feasibility and objective value and the better one is picked as the current value.
- 10.
Since GAs are generally inefficient for handling equality constraints directly and need a large population size for finding feasible solutions, the torque equality constraint has been implemented using an adaptive coefficient strategy in the constrained dominated approach.
- 11.
Because of dynamic penalty function parameters for constraint handling that depend on the MaxGen parameter, the algorithm was restarted in each case from the same starting point.
- 12.
We set the maximum allowed number of function evaluations to twice the number of function evaluations required for solving the 10 product case using the decomposed algorithm.
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Acknowledgments
This work is supported in part by the Pennsylvania Infrastructure Technology Alliance, a partnership of Carnegie Mellon, Lehigh University, and the Commonwealth of Pennsylvania’s Department of Community and Economic Development (DCED). Dr. Simpson acknowledges support from the National Science Foundation under CAREER Award No. DMI-0133923. Any opinions, findings, and conclusions or recommendations presented in this chapter are those of the authors and do not necessarily reflect the views of the sponsors.
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Khajavirad, A., Michalek, J.J., Simpson, T.W. (2014). Solving the Joint Product Platform Selection and Product Family Design Problem: An Efficient Decomposed Multiobjective Genetic Algorithm with Generalized Commonality. In: Simpson, T., Jiao, J., Siddique, Z., Hölttä-Otto, K. (eds) Advances in Product Family and Product Platform Design. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7937-6_11
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