As shown in Fig. 3.1, at first glance, the design optimisation framework proposed by Weng may look similar to the one used in the author’s prior work. The key differences however are indicated by two highlighted blocks, which enables an automated decision- making functionality in the optimisation process. This was done by including a fitness function as an overall evaluation criteria (OEC), quantitative termination criteria and a standardised method to adjust each parameters range for the next iteration. Since each iteration use the same procedure, only the first iteration is explained in detail here. Each of the blocks in Fig. 3.1(b) function as follow:
• Problem initialisation: The optimisation procedure starts with the problem initiali- sation, which includes parameter selection, parameter range identification, selecting a suited OA and formulation of a fitness function. The range of a parameter is very important as all the trial machines (as specified by the OA) must be a viable design for performance calculation. The selection of an OA mainly depends on the number of parameters. It is recommended to use an OA with three or more levels per parameter to aid in including possible nonlinear effect. The fitness function is devised according to the optimisation objective and is either maximised or min- imised depending on the objective.
• OA input parameter allocation: For an iteration, the numerical values for each level of a parameter must be determined to conduct the trials. For the first iteration (if a three level parameter OA is used) the maximum (P nmax) and minimum (P nmin)
range value of a parameter are allocated to level-1 and level-3 respectively thus level- 2 will be the mid range value between the two boundaries. The distance between any two levels is known as the level difference (LDni) of the ith iteration. For the
first iteration (LDn1) is determined by the following equation:
LDn1 =
P nmax− P nmin
number of levels + 1 (3.1)
For the subsequent iterations, LDni is reduced after each iteration if the termina-
tion criteria are not met. By reducing the level difference between two levels the parameters range is also reduced. Level placement for the second iteration onwards is discussed later.
• Conduct experiments and result analysis: Once all the OA’s trials have been com- piled and conducted the relative information must be obtained for the fitness func- tion of each trial. The fitness function performance of a given trial is used to build the ANOM’s response table. For this method, the ANOVA is not required. The ANOM’s respond table is formulated using the S/N ratio values of the fitness func- tion.
• Optimal level identification and confirmation experiment: As the S/N ration analy- ses are used, the optimum condition for each parameter is identified by the largest S/N ratio value. Using each of the optimum level conditions a confirmation trial is done under the same conditions as the main OA trials. This is done to determine the fitness value of the current iteration.
• Check the termination criteria: The optimisation is terminated when one or both goals have been achieved. The first and most basic termination criterion is when the fitness function has converged over several iterations. The second termination criterion involves the ratio between the first and current level difference value. As the number of iterations increases, the overall level difference decreases (the tempo of reduction is discussed in the next step). If the LD ratio is larger than the converged value (CV) set by the designer during the problem initialisation another iteration is required. The following equation may be used as a termination criterion for the optimisation procedure:
LDni
LDn1
<CV (3.2)
with CV selected between 0.001 and 0.01. As the parameter level values move closer to each other the current fitness value should be close to the previous value thus converging around the optimum point.
• Reduce the optimisation range: If another iteration is required due to the termina- tion criterion/criteria not being met, the current parameter range for each parameter must be reduced. To reduce a parameter’s range for the next iteration, the current LD is multiplied with a regression rate (RR) factor as follow:
LDni+1=RR LDi (3.3)
The RR is set by the designer between 0.5 and 1 during the initialisation. An RR closer to 1 will results in a slower LD convergence thus a higher number of iterations before termination. For the next iteration, the current optimum value is placed in the level-2 slot. Level-1 and Level-3 are calculated using the new LD
the original range of the parameter as it is possible that it may fall out of bounds. This is especially true when LD is still large and the optimum level is near or is the actual boundary value. Therefore, a process of checking the new level values are necessary to ensure that all level values are within the parameter range.
From the above implementation steps, it is clear that the level difference regression framework can easily be implemented on a wide array of machine design problems. The following two steps replace the designer’s involvement during the optimisation. Firstly by using a simple regression rate formula to reduce the level difference of each parameter for the next iteration and secondly the use of a minimum convergent value requirement before the optimisation is terminated.
As the OEC’s fitness is not influenced by the framework itself, but rather by the objective under investigation, multiple operational states of the design problem can be investigated and optimised simultaneously. Furthermore, as the required trials are set by the selected OA, different simulation/analysis platforms can be used to obtain the required objectives performances before they are combined as set by the OEC.
3.2.1
Limitations of Weng’s Approach
There are some concerns regarding the method. The first concern is the selection of the regression rate for (3.3) and the influence it has on the outcome of the optimisation. By selecting a value closer to 0.5, the level difference will converge at a much faster rate than when using a regression value closer to 1 as the influence of the selected rate affects the tempo at an exponential rate of reduction. This is presented in Fig. 3.4’s example. The graph shows the iteration LD regression curves of the same parameter (with an initial LD of 50) and its iteration-to-iteration reduction using different RR values. Although the use of a lower regression rate is advantageous in terms of computation time, the possibility exists that a poorer machine design could be realised when compared with using a higher regression rate. For a regression rate of 0.5, the level difference is reduced by 50% after each iteration whereas with a rate of 0.95 it is only reduced by 5%. Thus using a larger regression rate ensures a much more thorough analysis of the parameters design range.
A second concern is the confidence in the optimisation results using the same or a fixed regression rate for all the factors. During the initial investigation it was noted that the in- fluence a parameter has on the performance variance is not constant (as seen in Fig. 3.5). Certain factors may only influence the performance variance near the optimum point, D1 for example, but not much when it still has a large level difference. The opposite is also a possibility as in the case of O1 which had a decreasing influence on the performance variance after each iteration. The possibility also exists that a factor hardly influences the performance variance or the outcome of the optimisation (e.g. Rib). For the design case in Fig. 3.5, in the first view, it is advisable for D1 to use a lower regression rate to greatly reduce the level difference and for O1 a higher regression rate to only slightly reduce the level difference for the next view. The use of a varying view-to-view regression rate may lead to a reduction in the number of required iterations while still maintain a high confidence level in the outcome of the optimisation.
Figure 3.5: View-to-view parameter percentage contribution towards performance variance [144]
Lastly, there is also a lack of information on literature about the termination criteria. It is clear that when (3.2) is satisfied and the OEC converges within an acceptable variance range the optimisation can be terminated with ease. There is, however, no mention in literature as to termination if the OEC does not converge once (3.2) is satisfied.