replacement of (1.10-14) by (1.10-15). Using 2000 generations with a population size of 1000 yields the results shown in Figure 1.25. Figure 1.25(a) illustrates the objective space at the end of the optimization run. Each point in Figure 1.25(a) represents the objectives of a complete design. Nonviable designs have fitness values close to the origin (and are slightly negative in each axis). Viable, but dominated, designs are also apparent. The remaining designs are nondominated. In Figure 1.25(b) only the nondominated designs are shown, and the fitness elements are reciprocated so that mass and loss can be plotted directly. Again, each point represents the mass and loss of an individual design. Note the tradeoff between mass and loss. This will be a recurrent theme in this text. A sample design on the front is also indicated; this design is illustrated in more detail in Figure 1.26. The sample design has a mass of 0.75 kg, and a loss at rated current of 0.67 W. The inductance is just over 1 mH, and flux density at rated current is 0.617 T. The current density at rated current is 1.3 A/mm2.
Figure 1.27 illustrates the gene distribution of the final population of designs. In Figure 1.27(a), the genes are sorted by objective 1. This means that the genes of designs with higher mass are toward the left of the parameter window, and genes of designs with lower mass are toward the right. In Figure 1.27(b), the genes are sorted by objective 2, so that designs with the most loss are toward the left, and designs with the least loss are toward the right.
Figure 1.27 Sample design from Pareto‐optimal front.
Unlike the case of single‐objective optimization, the clustering of all values of a gene to approximately the same value is not expected in multi‐objective optimization because the parameters will vary along the front. In order to illustrate this, consider the slot depth ds. Observe that in Figure 1.27(a) it has a slightly downward slope while in Figure 1.27(b) it has a slightly upward slope. This is because as we move from a low‐mass high‐loss design to a high‐mass low‐loss design the slot depth decreases. The core depth wc can be seen to be approximately constant.
The remaining parameters undergo more interesting behaviors. Consider N*, for example. Observe that in Figure 1.27(a), the nondominated solutions fall into two groups, which are indicated with a darker shaded and lighter shaded ellipses for lower and higher mass, respectively. The direction of decreasing mass is indicated with an arrow. Observe that the designs undergo a bifurcation indicated by a black vertical arrow. This can also be seen in Figure 1.27(b), wherein the sets are again circled. Note that the direction of decreasing mass is now to the left. The designs that are not in the two groups are dominated solutions. The bifurcation of the design space is also readily apparent in the slot width ws, core length lc, and air gap g. Such bifurcations in the design space can be the result of the change of a discrete variable, or the result of the design space moving into or out of a constraint. In this example, if we replace (10.1‐1) with N = N*, the bifurcation disappears. Of course, in doing this, our problem becomes strictly mathematical in nature since N must be an integer in practice.
References
1 1 E. K. Chong and S. H. Zak, An Introduction to Optimization, third edition. Hoboken: John Wiley & Sons, Inc., 2008.
2 2 J. F. Crow, Genetics Notes. New York: Macmillan Publishing Company, 1983.
3 3 J. H. Holland, Adaptation in Natural and Artificial Systems, Boston: MIT Press, 1992.
4 4 D. E. Goldberg, Genetic Algorithms in Search, Optimization, & Machine Learning. Boston: Addison Wesley Longman, Inc., 1989.
5 5 D. E. Goldberg, The Design of Innovation. Boston: Kluwer Academic Publishers, 2002.
6 6 S. D. Sudhoff, MATLAB codes for Power Magnetic Devices: A Multi‐Objective Design Approach, second edition. Available: http://booksupport.wiley.com.
7 7 B. N. Cassimere and S. D. Sudhoff, Population based design of a permanent magnet synchronous machine, IEEE Transactions on Energy Conversion, vol. 24, no. 2, pp. 347–357, 2009.
8 8 Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, third, revised and extended edition. Berlin: Springer‐Verlag, 1999.
9 9 A. Osyczka, Evolutionary Algorithms for Single and Multicriteria Design Optimization. Heidelberg: Physica‐Verlag, 2002.
10 10 G. P. Liu, J. B. Yang and J. F. Whidborne, Multiobjective Optimization and Control. Exeter: Research Studies Press Ltd., 2003.
11 11 K. Deb, Multi‐Objective Optimization using Evolutionary Algorithms. Chichester: John Wiley & Sons, Ltd., 2001.
Problems
1 It is desired to minimize the functionWhat is a possible fitness function (the answer is not unique) if using a canonical GA?
2 The fitness values of the members of a population are 23, 96, 42, 12, 8, 7, and 47. What is the expected number of times the individual with a fitness of 42 will appear in the mating pool? Use roulette wheel selection.
3 Consider two parents, with x1 = 0.2 and x2 = 0.5. Consider simulated binary crossover with ηc = 1. Form 105 children, and plot a histogram of the children arranged in 20 equally spaced bins on the interval 0 to 1. Implement gene repair using ring mapping.
4 Repeat Problem 3 with x2 = 0.25.
5 Suppose a design problem was coded with five genes on a single chromosome, and single‐point crossover was used. During the crossover between two parents, how many different children (in terms of genotype) could be produced?
6 A logarithmically mapped gene has a range from 10−3 to 106. If the value for this gene for a particular individual is 37.6, what is its’ normalized value? What would its’ normalized value be if it were linearly mapped.
7 During an evolution, the minimum, maximum, and average fitness of a population is 1.2, 270, and 40, respectively. If the most fit individual is to be three times more likely than the average fit individual to be selected, and the least fit individual is 1/3 as likely as the average fit individual to be selected, what is the scaled fitness of an individual whose raw fitness is 58.
8 Below are the objective function values for an inductor design. It is desired to minimize both objectives.Individual123456Mass (kg)531424Loss (kW)236541Use Kung’s
