Conclusions

In this thesis, we have investigated simulation-based design optimization methods. The methods that are described in the literature can be categorized into Objective Driven Design Optimization (ODDO) methods and Meta-model Driven Design Optimization (MDDO) methods. Methods in the first category are primarily driven by obtaining the best objective value (see also Driessen [2006]). This thesis concentrates on MDDO methods, which are primarily focused on obtaining meta-models that approximate the simulation model for the entire design space. The meta-models can be used for many types of analyses, e.g., identification of the most important design parameters, design optimization and sensitivity analysis. The MDDO approach consists of the following six steps:

1. Problem setup 2. Screening

3. Design of Computer Experiments (DoCE) 4. Black box evaluations

5. Meta-modeling 6. Analysis

In this thesis, we have contributed to steps 3, 5, and 6.

In many design optimization problems, the design space is restricted by non-box design parameter constraints, i.e., constraints on combinations of design parameters. These constraints may originate from three sources:

• A-priori knowledge of which combinations of design parameter values might lead to a good design and which will probably not

• A-priori knowledge of physical restrictions

• A-priori knowledge of the combinations of design parameters for which the black box cannot be evaluated.

In Chapter 2, we have introduced a new method for solving the Maximin design problem (step 3 of the MDDO method). This sequential method is based on Non Linear Programming, and is specifically suited to cope with many design parameter constraints. We called our method SFDP**. We have compared this method with Trosset’s [1999] method and a method that is inspired by Drezner and Erkut’s [1995] method for solving the p-dispersion problem, which we called SFDP*.

We conclude from our analysis in Chapter 2 that our methods have proven to be an effective way of constructing a space-filling non-box constrained design. The three methods increase in solution time as the problem size increases. The SFDP* problem size increases dramatically as the number of design sites increases. The problem size of Trosset’s formulation also increases dramatically as the number of design constraints increases. Moreover, it seems that the quality of the solution found by Trosset’s method deteriorates for high-dimensional problems. The problem size of SFDP** is relatively insensitive to the number of design sites and design constraints, but the number of problems that have to be solved increases linearly in the number of design sites. Which method to use, depends on problem size, as well as on the available solver and computational capacity. We propose the following use of the methods presented. When the problem is small enough, first solve SFDP*, then solve SFDP** to increase the average distance between design sites. If the problem size is too large for SFDP* but small enough for solving Trosset’s formulation, then first solve Trosset’s formulation and subsequently solve SFDP**. When the problem size is even too large for Trosset’s formulation, solve SFDP** only.

In Chapter 3, we have introduced a new method for meta-modeling (step 5 of the MDDO method). The method is a variant of Symbolic Regression. A Symbolic Regression meta-model consists of operators, functions, and constants that are combined to form a formula (the meta- model). Symbolic Regression models usually use a genetic algorithm in order to find the best combination and order of building blocks. Our variant, however, uses Simulated Annealing.

Programming algorithms. Further, our variant takes the complexity of the meta-model into account, in order to avoid overfitting. Complexity is defined by a measure that estimates the minimal order of the polynomial that is necessary to approximate the model with acceptable accuracy. For the two test cases that we discussed, the Symbolic Regression meta-models have a quality comparable to Kriging meta-models, but they are much less complex and therefore more interpretable. Compared to other Symbolic Regression methods, we found that the combination of Simulated Annealing and least squares results in a comparable solution using fewer function evaluations than Genetic Programming methods.

In Chapter 4, we have investigated three sources of systematic errors that should be taken into account when applying the MDDO method: the simulation model error, the meta-model error, and the implementation error. The simulation error receives little attention in the literature, while in practice this error may have a significant impact. The robust counterpart methodology can be used to obtain robust solutions, i.e., solutions that are less sensitive with respect to these three errors. For two meta-model types (i.e., linear regression and Kriging models) and for different types of errors, we have developed solvable robust counterpart optimization problems. For an overview, see Table 12, page 88. The practical examples described in Section 6.4 show that small errors could have a large impact on the quality of a design. Therefore, it is always advisable to investigate the robustness of the proposed design.

In Chapter 5, we have investigated the application of MDDO in a large-scale environment. We assumed that the black box could be split up into multiple connected black boxes. This prevents a waste of computer time while searching for interaction factors that cannot exist due to the black box structure. The suggested CMM approach has many advantages compared to existing large-scale techniques, such as sensitivity analysis, global optimization and robust design. Moreover, the resulting meta-models at part-level can be re-used in following studies. Coordination is one of the many issues raised by the CMM approach. Three coordination methods are proposed and compared, namely parallel simulation, sequential simulation and sequential modeling.

In Chapter 6, we have presented a number of practical cases in which the theory from the preceding chapters is used. We have investigated ship design, the design of a metal forming process, and several applications in TV tube design. We conclude that the MDDO approach is very useful in these applications. Further, we see that the extensions that are described in this thesis are applicable in industry.