# Using Feedback Strategies in Simulated Annealing with Crystallization Heuristic and Applications
## 5 SA Applied to Aircraft Design
### Results and Discussion  
SA with crystallization heuristics was used to optimize three aircraft designs with three different positive feedback strategies: I, II, and III (see Table 1). Strategy I performs the positive feedback with a small decrement, focusing on refining the solution. However, if SA stays in a local minima at lower temperature, it will be very difficult to escape from it. Strategy III focuses on exploring the solution space, allowing an easy escape from local minima. Strategy II has a balance between refinement and exploration. The crystallization factor evolution, the acceptance and rejection ratios, as well as the best aircraft for each feedback strategy are shown with respect to the five parameters. Through the optimization process, five parameters showed the largest variations: AA (projected superior area of the wing), DHBA (horizontal distance between leading edges), CF (fuselage chord), A1 (percentage of wingspan designated to Section 1), and TS1 (taper ratio of Section 1). Figure 9 shows a representation of an aircraft and these five parameters. Notice that the yellow region of the wing is defined as Section 1, and the blue region is defined as Section 2. Figure 10 shows the crystallization factor evolution for each positive feedback strategy. Figure 10a shows the expected behavior for the crystallization factors. At higher temperatures, the crystallization factor is small and SA performs exploration. The crystallization factor grows gradually; at lower temperatures, SA performs refinement. However, each parameter might have different sensibilities and, consequently, different crystallization factors. Figure 10b shows the positive feedback strategy II, and the crystallization factors of parameters TS1 and CF do not increase as much as the other three parameters. When  
Figure 8: Flowchart of constraint verification. Starting from a candidate (the new configuration), different constraints are verified in sequence. If any constraint is not satisfied, a new candidate is selected. Figure 9: Schematic representation of the airplane. A set of parameters is shown: CF (fuselage chord), DHBA (horizontal distance between leading edges), and Section 1 (percentage of wingspan represented as A1, and tapper ration represented as TS1). compared to strategy I, these two parameters stay in an exploratory phase during the complete algorithm. Figure 10c shows the positive feedback strategy III and shows a considerable variation at the end of the refinement phase. Notice the difference between the crystallization factors: while the positive feedback strategies I and III reached higher values at the end of convergence (more than 600 in the first method), the positive feedback strategy II barely reached 150. Figure 11 shows the acceptance ratio for each positive feedback strategy--the acceptance ratio is the ratio between the number of accepted candidates and the total number of candidates. If the ratio is equal to 1.0, this means that all candidates were accepted. The ratio is calculated separately for each parameter. As previously shown, it is necessary to increase the number of accepted candidates as much as possible. Usually, the acceptance ratio decreases with the temperature. SA reaches a frozen state when no new candidate is accepted. There is a difference between the positive feedback strategies at lower temperatures. Figure 11a shows a small variation of the acceptance ratios at lower temperatures. Figure 11b shows the smallest acceptance ratio among the three positive feedback strategies. Figure 11c shows a large variation in the acceptance ratios. Each parameter might have its own appropriate positive feedback strategy. Figure 12 shows the total number of accepted solutions. As expected, the number of accepted solutions decreases as the temperature decreases. It is not possible to observe differences between the positive feedback strategies. Figure 13 shows the population dispersion for each positive feedback strategy considering only two intermediary parameters: (1) dimensional sum and (2) payload.