**8. Optimization of the two objective functions using SMOSA**

An ideal situation would be to attain a consistent optimal material gradient, where the maximum one turnover can be made and the displacement is kept to minimal. The problem described in previous sections was solved by multi-objective SA code written in MATLAB software programming and run on Pentium IV, 2500 GHz and 4 GB RAM. The SMOSA was run 5 times and the obtained averaged results were compared to results obtained from RSM.

The parameters which must be specified before running the algorithm are initial temperature, frozen state represented by the final temperature, cooling ratio (annealing), i.e.

the rate at which the temperature is lowered between two cooling cycles, the randomly generated initial solution, and the lower and upper bounds for design variable (*m*). Table 1 shows the user parameters used for the SMOSA.

Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 231

**User Parameters Quantities** 

Lower bound 0.1 Upper bound 10

*<sup>i</sup>*) 0.98

Initial temperature for each objective (*Ti0*) 100

Cooling ratio (

**Table 1.** Optimization parameters associated with the SMOSA.

more data range for cortical density than RSM.

improvements may seem to be negligible.

Pareto front depicted in Fig. 6b.

a wider choice for design.

α

The SMOSA was conducted on two objective functions, *f3* as displacement function and *f1* as cortical density function as in Equation (14). Fig. 6 represents the comparison of Pareto front for two objective functions using SMOSA and RSM (Lin et al., 2009). Comparing Figs. 6a and 6b, the same trend is seen in both figures. It can be seen that the increase in *f1*, results in *f3* to decrease and vice versa. Hence, SMOSA confirms the results obtained by RSM. However, the data ranges obtained by SMOSA do not match exactly with the data range obtained by RSM.

The data range given by SMOSA for *f1* is from 0.51688 to 0.51778 as shown in Fig. 6b while, RSM gives values from 0.5169 to 0.5175 for *f1* (see Fig. 6a). Similarly, the data range given by SMOSA for *f3* is from -3.9054e-5 to -3.5907e-5 and for RSM for *f3* is from 3.5205e-5 to 3.603e-5. The date range for cortical density function given by SMOSA is almost 33% more than the range obtained by RSM in micro scale. This shows the efficiency of the SMOSA in providing

The safety factor (acceptance capability) obtained by SMOSA is 33% higher than the one given by RSM for magnitude of cortical density function. This increasing of data range is interesting for FGM dental implant design and shows that SMOSA has outperformed RSM in cortical density function. SMOSA gives the data range for displacement almost 73% more than the values given by RSM in micro scale. That means SMOSA enables the designer with

In general, SMOSA method gives more selection of material gradient (*m*) for designing of the FGM dental implant but with higher displacement compared to the RSM. The order of quantities (micro-scale) shows the importance of accuracy in the optimization of the FGM implant. Hence, any improvement in quantities may be considerable although such

Fig. 7 illustrates the trend for cortical density and displacement functions with respect to *m*. By inspecting Fig. 7, the acceptable range for *m* is from 0.1 to 0.65. The magnitude of *f1* decreases as *m* increases which means the density of cortical (*Dcortical*) increases. In contrast, with the increase in *m* the quantity of vertical displacement (*f3*) increases. This trend shows the non-dominated solution from the Pareto front plot so that the increase in one function results in the decrease of the other function and vice versa. Such data was extracted from the

Final temperature for *f2* and *f<sup>3</sup>* 1e-6 Final temperature for *f1* and *f<sup>3</sup>* 1e-8

**Figure 5.** Flowchart of the SMOSA optimizer.


Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 231

**Table 1.** Optimization parameters associated with the SMOSA.

230 Simulated Annealing – Single and Multiple Objective Problems

shows the user parameters used for the SMOSA.

**Figure 5.** Flowchart of the SMOSA optimizer.

the rate at which the temperature is lowered between two cooling cycles, the randomly generated initial solution, and the lower and upper bounds for design variable (*m*). Table 1

> The SMOSA was conducted on two objective functions, *f3* as displacement function and *f1* as cortical density function as in Equation (14). Fig. 6 represents the comparison of Pareto front for two objective functions using SMOSA and RSM (Lin et al., 2009). Comparing Figs. 6a and 6b, the same trend is seen in both figures. It can be seen that the increase in *f1*, results in *f3* to decrease and vice versa. Hence, SMOSA confirms the results obtained by RSM. However, the data ranges obtained by SMOSA do not match exactly with the data range obtained by RSM.

> The data range given by SMOSA for *f1* is from 0.51688 to 0.51778 as shown in Fig. 6b while, RSM gives values from 0.5169 to 0.5175 for *f1* (see Fig. 6a). Similarly, the data range given by SMOSA for *f3* is from -3.9054e-5 to -3.5907e-5 and for RSM for *f3* is from 3.5205e-5 to 3.603e-5. The date range for cortical density function given by SMOSA is almost 33% more than the range obtained by RSM in micro scale. This shows the efficiency of the SMOSA in providing more data range for cortical density than RSM.

> The safety factor (acceptance capability) obtained by SMOSA is 33% higher than the one given by RSM for magnitude of cortical density function. This increasing of data range is interesting for FGM dental implant design and shows that SMOSA has outperformed RSM in cortical density function. SMOSA gives the data range for displacement almost 73% more than the values given by RSM in micro scale. That means SMOSA enables the designer with a wider choice for design.

> In general, SMOSA method gives more selection of material gradient (*m*) for designing of the FGM dental implant but with higher displacement compared to the RSM. The order of quantities (micro-scale) shows the importance of accuracy in the optimization of the FGM implant. Hence, any improvement in quantities may be considerable although such improvements may seem to be negligible.

> Fig. 7 illustrates the trend for cortical density and displacement functions with respect to *m*. By inspecting Fig. 7, the acceptable range for *m* is from 0.1 to 0.65. The magnitude of *f1* decreases as *m* increases which means the density of cortical (*Dcortical*) increases. In contrast, with the increase in *m* the quantity of vertical displacement (*f3*) increases. This trend shows the non-dominated solution from the Pareto front plot so that the increase in one function results in the decrease of the other function and vice versa. Such data was extracted from the Pareto front depicted in Fig. 6b.

Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 233

**Figure 7.** Cortical density and displacement functions versus *m* using SMOSA for: (a) *f1*, (b) *f3*.

The results obtained from the optimization of two objective functions may not adequately address the full design requirements and therefore, the three objective functions were developed as defined in Equation (10). Fig. 9a shows the Pareto front surface for three objective functions using RSM. Fig. 9b illustrates the 3D plot of the Pareto front for three objective functions *(f1, f2*, and *f3)* using SMOSA. From Fig. 9, the good harmony and

Fig. 10 represents various aspects of Fig. 9b. This figure shows the trend for *f1*, *f2*, and *f3* in terms of the material gradient. The acceptable range for *m* varies from 0.1 to 0.65 (see Fig. 10). As shown in Fig. 10a, the maximum value for cortical density is obtained for *m=0.63*. In Fig. 10b, the increase in *f2* leads to increase in *m*, and hence, decrease in the cancellous density. This indicates that for material gradient 0.1 (*m=0.1*) cancellous density has the maximum value. In addition, by observing Fig. 10c, when *m=0.1,* the displacement has the minimum value. Finally, based upon the obtained results, the optimal range for material

**9. Optimization of the three objective functions using SMOSA** 

similarity between the results obtained by SMOSA and RSM are depicted.

gradient is almost varying from 0.1 to 0.65 (*0.1 ≤ m ≤ 0.65*).

**Figure 6.** Pareto front for optimization of two objective functions (*f3* and *f1*) using: (a) RSM, (b) SMOSA.

Fig. 8 demonstrates *f1*, *f2* and *f3* with respect to the *m*. When the temperature is high at the beginning of the optimization process, all invalid moves were accepted by the acceptance probability (*P*), as shown in Fig. 8. By decreasing the temperature at each iteration, the probability of accepting invalid moves is reduced and therefore, only qualified points and valid moves (so called non-dominated solutions) were accepted.

As a next step, the multi-objective simulated annealing method was applied on other two objective functions as shown in Equation (15). The result indicates the Pareto front is a cluster of points that are gathered in one point representing an optimal solution. Going back to Figs. 4b and 4c for minimizing both objective functions (*f2* and *f3*), we expect to have one optimal solution as the two functions have a similar trend to reach the optimal point.

In other words, if one moves from the right side of the Figs. 4b and 4c to the left side, the cancellous density increases while the displacement decreases. This situation may be considered as an optimal state. The optimal material gradient is taken as 0.1 *(m=*0.1*)*. The values for *f3* and *f2* are equal to -3.9068e-5 and 0.8540, respectively. Hence, the best material gradient for these two objective functions is given when *m=0.1*.

**Figure 6.** Pareto front for optimization of two objective functions (*f3* and *f1*) using: (a) RSM, (b) SMOSA.

Fig. 8 demonstrates *f1*, *f2* and *f3* with respect to the *m*. When the temperature is high at the beginning of the optimization process, all invalid moves were accepted by the acceptance probability (*P*), as shown in Fig. 8. By decreasing the temperature at each iteration, the probability of accepting invalid moves is reduced and therefore, only qualified points and

As a next step, the multi-objective simulated annealing method was applied on other two objective functions as shown in Equation (15). The result indicates the Pareto front is a cluster of points that are gathered in one point representing an optimal solution. Going back to Figs. 4b and 4c for minimizing both objective functions (*f2* and *f3*), we expect to have one

In other words, if one moves from the right side of the Figs. 4b and 4c to the left side, the cancellous density increases while the displacement decreases. This situation may be considered as an optimal state. The optimal material gradient is taken as 0.1 *(m=*0.1*)*. The values for *f3* and *f2* are equal to -3.9068e-5 and 0.8540, respectively. Hence, the best material

optimal solution as the two functions have a similar trend to reach the optimal point.

valid moves (so called non-dominated solutions) were accepted.

gradient for these two objective functions is given when *m=0.1*.

**Figure 7.** Cortical density and displacement functions versus *m* using SMOSA for: (a) *f1*, (b) *f3*.

## **9. Optimization of the three objective functions using SMOSA**

The results obtained from the optimization of two objective functions may not adequately address the full design requirements and therefore, the three objective functions were developed as defined in Equation (10). Fig. 9a shows the Pareto front surface for three objective functions using RSM. Fig. 9b illustrates the 3D plot of the Pareto front for three objective functions *(f1, f2*, and *f3)* using SMOSA. From Fig. 9, the good harmony and similarity between the results obtained by SMOSA and RSM are depicted.

Fig. 10 represents various aspects of Fig. 9b. This figure shows the trend for *f1*, *f2*, and *f3* in terms of the material gradient. The acceptable range for *m* varies from 0.1 to 0.65 (see Fig. 10). As shown in Fig. 10a, the maximum value for cortical density is obtained for *m=0.63*. In Fig. 10b, the increase in *f2* leads to increase in *m*, and hence, decrease in the cancellous density. This indicates that for material gradient 0.1 (*m=0.1*) cancellous density has the maximum value. In addition, by observing Fig. 10c, when *m=0.1,* the displacement has the minimum value. Finally, based upon the obtained results, the optimal range for material gradient is almost varying from 0.1 to 0.65 (*0.1 ≤ m ≤ 0.65*).

Optimum Functionally Gradient Materials for Dental Implant Using Simulated Annealing 235

**Figure 10.** Mass densities and displacement functions in optimization of three objectives in terms of

material gradient using SMOSA for: (a) *f1*, (b) *f2*, (c) *f3*.

**Figure 8.** Distribution of mass densities and displacement functions with respect to the *m* using SMOSA: (a) *f1*, (b) *f2*, (c) *f3*.

**Figure 9.** 3D Pareto front for three objective functions *(f1, f2* and *f3)* using: (a) RSM, (b) SMOSA.

SMOSA: (a) *f1*, (b) *f2*, (c) *f3*.

**Figure 8.** Distribution of mass densities and displacement functions with respect to the *m* using

**Figure 9.** 3D Pareto front for three objective functions *(f1, f2* and *f3)* using: (a) RSM, (b) SMOSA.

**Figure 10.** Mass densities and displacement functions in optimization of three objectives in terms of material gradient using SMOSA for: (a) *f1*, (b) *f2*, (c) *f3*.
