Fig. 6. Pareto front in example 1
curves in both figures indicate the same Pareto set under ($\Delta x_1$, $\Delta x_2$) = (1.0, 5.0). It is found that smooth Pareto fronts are obtained in both cases, because each Pareto design is obtained by using a sequential quadratic programming method.
As $\Delta x_1$ becomes larger, the Pareto front moves in the upper right direction, as shown in Fig. 6 (a). On the other hand, the Pareto front moves in the upper left direction as $\Delta x_2$ becomes larger, as shown in Fig. 6 (b).
A comparison of these figures shows that the variation of $x_1$ has a larger effect on the Pareto set, especially on $f_2$, which is the variation of the normal stresses.
4.2 Example 2: ten-bar truss structure
Consider the ten-bar truss problem shown in Fig. 7, where the representative length L is set at 360, and the applied load P is 100 [10]. The original deterministic design problem is formulated to minimize the structural volume in terms of the member's cross-sectional area subject to the member's stress constraints.
This problem is modified to give the following robust multiobjective optimization problem. Assume that the applied load P and member strength have variations and are treated as random variables. The robust design problem is formulated as three objective function problems, which are defined as the structural volume and the mean and standard deviation of the
Fig. 7. Ten-bar truss design problem
Table 1. Ideal points for each condition in example 1.
| Δx1 | Δx2 | f1' | f2' |
|---|---|---|---|
| 1.0 | 5.0 | 324.02 | 2.46 |
| 1.5 | 5.0 | 325.31 | 3.65 |
| 2.0 | 5.0 | 326.61 | 4.87 |
| 1.0 | 7.0 | 324.05 | 2.53 |
| 1.0 | 10.0 | 324.09 | 2.62 |
tip displacement $d_2$ in terms of the member's cross-sectional area $x_i$, where $x_i$ is assumed to be deterministic. The robust multiobjective optimization problem is formulated as follows:
where $l_i$ is the member length, $\sigma_i$ is the member stress, and $s$ and $\Delta s$ are the mean value and standard deviation of the member strength, respectively. The mean value and standard deviation of the applied load are set at $E[P] = 100.0$ and $\Delta P = 10.0$, respectively. The mean value and standard deviation of the member strength are set at $E[s] = 25.0$ and $\Delta s = 1.0$, respectively.
The obtained Pareto set in the objective function space is shown in Fig. 8 with different viewing angles, where each point is obtained by changing the aspiration level parametrically under the ideal point $f^1 = (1.659 \times 10^4, 1.956, 1.345 \times 10^{-2})^\mathrm{T}$, as determined from each single-objective optimization problem. It is found that a smooth Pareto surface is obtained, because each Pareto design is obtained by using a sequential quadratic programming method. The Pareto set indicates that a trade-off exists between the structural volume, mean of the tip displacement, and variance of the tip displacement.
Next, the correspondences of the selected Pareto designs are investigated in detail. First, designs A, B, C, and D on the Pareto set shown in Fig. 8 are selected, because the Pareto solutions have almost the same volume and mean of the tip displacement, but have different standard deviations of the tip displacement, as shown in Fig. 9. Design A is the most robust