The fundamental difference between the Taguchi method and algorithm based method may be explained by using the following scenario: a team of engineers is tasked to con- struct a design framework to find an optimum using the least amount of simulations. For the first approach, they can analyse random single-point designs within the domain and move around trying to find an optimum. A second approach would be to use a series of carefully selected designs to identify, with a high degree of certainty, an area within the domain that provides the highest average and thus the highest probability of an optimum design. With the first approach, there is no certainty that the optimum found is the actual optimum. Although the second method may not necessarily pinpoint the precise optimum, the realised design can be very close to true optimum with certainty. The first approach is that of an algorithm based method and the latter the Taguchi method.
2.3.1
Implementation Steps
The Taguchi method is applied in five steps as shown in Fig 2.1(b) with the first step being the most important. These steps are discussed briefly below. To further explain the working mechanism of the method, an example implementation is also provided in Appendix B2 along with the relative formulas required:
• Brainstorming and planning: The idea of upfront thinking to solve complex prob- lems are well known in engineering design. However, it is not always used. In the Taguchi method, brainstorming is seen as the most important step to fully benefit from its advantages. The outcome of the planning is greatly dependent on the na- ture of the project and the associated problems. The focus in this step should be placed on:
– the goal of the design,
– the problems associated with the design,
– the attributes of the parameters (e.g., parameters used as design variables or noise factors),
– how the trial will be conducted
– and how the results will be quantified using one of the three quality character- istics set out by Taguchi.
All these decisions are made not only from understanding the problem at hand but also by experience.
• Construction of experimental trial designs: The experimental design framework re- quired can be constructed according to Fig 2.3. One of the advantages of the Taguchi method is the ability to gain insight of the interactions between two parameters as part of the main design array. Considering the scope of the work and the specific application, this will not be included in this study. Without including the parameter interactions in the design, any standard OA can be selected and parameters can be placed in any column. The range of standard OAs is capable of including 2 to 31 design parameters with two-, three-, four- or five-level for each factor.
2Note: Highly recommended for persons who has no prior knowledge on the implantations of the
• Conduct experiments: It is recommended that the trials formulated by the main OA must be carried out randomly to avoid the influence of experimental set-up. For more self-contained design process such as electrical machine designs, the order in which the trials are conducted has no bearing on the outcome of the results. Fur- thermore, since all the trials are pre-determined and can be executed concurrently, the benefits of parallel computing can be exploited.
• Analyse results to determine optimum conditions: The experimental framework used can affect how the results of each trial is analysed. The type of analysis required can be selected with the aid of Fig 2.4. If outer array design was not included, the standard analysis is then used, whereby the output response is used as that of the Analysis of Mean (ANOM) and Analysis of Variance (ANOVA). In the case that the outer array design was used, the output response is analysed using S/N ratio for the specific quality characteristics. The ANOM is used to identify the optimum condi- tions of each parameter by studying the main effects of each level, which indicates the performance trend over the parameter range. The ANOVA is a statistical tool used to determine the influence each parameter has on the performance outcome. Once the optimum level condition for each parameter is determined the optimum design’s performance can be predicted.
• Run confirmation test using optimum conditions: The optimum level conditions (determined by the ANOM analysis) must be used to confirm the predicted opti- mal design’s performance. The same trial exposure conditions must be used when confirming the predicted optimum.
Figure 2.3: Experimental design framework flow diagram. Adapted from [60]
2.3.2
Orthogonal Array Methodology
Figure 2.4: Data analysis framework. Adapted from [60]
required by the specific OA. A specific OA is also linked to a parameter combination, (lp),
as originally defined for a standard DOE analysis. Table 2.2 can be used to aid in selecting an OA for up to 15 parameters. Some OAs include multi-level parameters, for example, L18 has both two- and three-level parameters. Regardless of the array’s design, they are
analysed using the same approach. When selecting an OA, it is not always necessary to fully populate all the parameter slots. In some designs, an open parameters slot can be used to investigate the interaction between two parameters.
Table 2.1: Standard orthogonal arrays
OA DOE Number of parameters Number of trials
L4 23 3 4 L8 27 7 8 L9 34 4 9 L12 211 11 12 L16 215 15 16 L016 45 5 16 L18 2137 8 18 L25 56 6 25 L27 313 13 27 L32 231 31 32 L0 32 2149 10 32 L36 23313 16 36 L036 211313 24 36 L50 21511 12 50 L54 21325 26 54 L64 263 63 64 L0 64 421 21 64 L81 340 40 81
Table 2.2: Selecting a standard orthogonal arrays Number of parameters Lev els 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 L4 L4 L8 L8 L8 L8 L12 L12 L12 L12 L16 L16 L16 L16 3 L9 L9 L9 L18 L18 L18 L18 L27 L27 L27 L27 L27 L36 L36 4 L0 16 L016 L016 L016 L320 L032 L032 L032 L032 5 L25 L25 L25 L25 L50 L50 L50 L50 L50 L50 L50
Table 2.3 shows the design of the L8 array which has seven columns, eight rows repre-
senting the parameters (A to G) and the number of trials (T1 to T8), respectively. The 1s and 2s beneath each parameter indicate the state or value selected to be investigated. For any given trial, all the parameters are present regardless of its level, for that specific trial. Within a column, the parameter has equal representation over all the trial for each level and is seen as balanced. In the case of an L8 array, level-1 and level-2 are each
represented four times. For an array to be orthogonally balanced, there have to be equal occurrences of parameter combinations between any two columns. For the L8 there are
4 possible combinations (1,1), (1,2), (2,1) and (2,2), each occurring twice between two columns. For an orthogonally balanced array, any parameter can be placed in any column and the analysis of the results will not be affected.
Table 2.3: Orthogonal array L8 Parameters L8 A B C D E F G Output Trials T1 1 1 1 1 1 1 1 Y1 T2 1 1 1 2 2 2 2 Y2 T3 1 2 2 1 1 2 2 Y3 T4 1 2 2 2 2 1 1 Y4 T5 2 1 2 1 2 1 2 Y5 T6 2 1 2 2 1 2 1 Y6 T7 2 2 1 1 2 2 1 Y7 T8 2 2 1 2 1 1 2 Y8
Experimental design using OAs are attractive because it reduces the number of exper- iments and is time efficient. If a full factorial analysis is conducted for the seven two-level parameters of an L8, a total of 128 experiments are required, whereas with the OA only
eight experiments are needed. It should also be noted that an OA based analysis works best when there is minimum parameter interaction or inter-parameter dependency. If there exists interaction between parameters, the OA still possesses the capability of accu- rately identifying the optimum parameter combination. However, depending on both the degree and complexity of the parameter dependency, there might be a difference between the predicted and actual optimum performance. Thus, the use of a confirmation test is highly recommended under such circumstances.
2.3.3
Associated Limitations
The example presented in Appendix B highlights how the method can be used as a parameter screening and a sensitivity analysis tool for both controllable manufacturing
tolerances and uncontrollable noise factors. From the implementation, the two main lim- itations surrounding the Taguchi method is also seen. Firstly, the method can only be used to analyse a single response (which in the example case was either torque ripple or power factor). Secondly, the method only provides a relative optimum with regards to the parameter values used. This is due to the OA using fixed state values for each parameter. For a method to be suited for electrical machine design optimisation, it has to possess the ability to incorporate both multi-objective optimisation criteria and provide the best-suited machine for the given criteria over the wide range of the parameter.
The Taguchi method as presented in the implementation lacks in both requirements, it is, however, possible to overcome these limitations. By implementing an iterative approach as commonly used in electrical machine design the whole range of a parameter can be investigated. The use of multi-objective response criteria is possible by implementing a normalised approach to formulate a new overall evaluation criteria before selecting an analyse approach as set in Fig. 2.4. Although the method may not be ideally suited for machine design, researchers have proposed several ways to overcome the limitations since the attributes of the method are very appealing.