# Multi-Objective, Reliability-Based Design Optimization of a Steering Linkage
## 3 Multi-Objective, Reliability-Based Design Optimization Technique (MORBDO)  
From the literature, an alternative technique to collect the uncertainties into RBDO was proposed. This is anti-optimization used to solve reliability-based design problem. Fuzzy set can be used to describe uncertainties with the help from experts' opinions. In cases where uncertainty exists in the form of fuzzy variables \(\textbf{a}=(\textbf{a}_{1},\,\textbf{a}_{2},\,\ldots,\,\textbf{a}_{\text{l}})\), defining tolerance, backlash, clearance, etc., causes the constraints as a function of \(\textbf{x}\) and \(\textbf{a}\). The constraints consist of inequality and bound constraints. Equality constraints, such as design symmetry, are not included directly in the design problem as they can be treated in the phase of design variable decoding. In reliability-based design with anti-optimization, all constraints should be given for the worst-case scenario or maximum possible (max (\(g_{i}(\textbf{x},\textbf{a})\)). This means all constraints are bucketed to find the worst-case scenario. Then, the upper-level and lower-level optimum design problem is to be formulated as  
\[\text{Min}\,f_{i}(\textbf{x})\,i=1,\,\ldots,\,M \tag{19}\]  
\[\text{Subject to }\underset{\textbf{a}}{\text{max}}g_{i}(\textbf{x}, \textbf{a})\ \leq\ 0,\,\,\,i\,=\,\,1,\ldots,N\]  
\[\textbf{x}_{i}\leq\textbf{x}\ \leq\textbf{x}_{u}\]  
If a simple optimization technique is used to solve the sub-problem (lower-level), only the maximization of the constraint should be solved \(N\) times at each step of modification of \(\textbf{x}\) in the upper level of Equation (18). The anti-optimization and fuzzy set cause the double-loop nested problem due to the need to solve optimization and anti-optimization problems at the same time [12,14]. This technique is well-known in the group of non-probabilistic approaches as the worst-case reliability where the intervals of uncertainties without accurate probability distributions [12,14] are pre-specified. A simple procedure for solving the nested optimization problem can be summarized as  
1. Assign an initial value \(a_{i}^{(k)}\) where k = 0 for \(a\) corresponding to each of inequality constraints \(g_{i}(x,a)\leq 0\), \(i=1,\ldots,N\). 2. Solve problem (18) for fixed \(a=a_{i}^{(k)}\), for the \(i\)th constraint to obtain the optimal solution \(\mathbf{x}^{(k)}\). 3. Fix \(x\) at \(x^{(k)}\) and solve anti-optimization solution (\(\underset{a}{\max}gi(x,a)\)) to find \(a_{i}^{(k)}\) for each inequality constraint, and set \(k=k+1\). 4. Return to Step 2 if the termination condition is not satisfied. The procedure for solving the double-loop nested problem causes computationally expensive evaluation. In the past, several methods have been developed to solve the problem, expecting to increase computational efficiency. The present idea is to reduce the complexity of the double-loop nested problem in such a way to change the single-objective, reliability-based design optimization (18) to multi-objective, reliability-based design optimization problems. The interval of uncertainties using a fuzzy set model and an \(\alpha\)-cut technique as one of the objective functions is also considered as follows  
\[\text{Min }\{f_{i}(x),-\text{WCSV}\}\;i=1,\ldots,M \tag{20}\]  
\[\text{Subject to WCSV}=max(g_{i}(x,a)),\] \[\;i=1,\ldots,N\boldsymbol{x_{i}}\leq x\leq\boldsymbol{x_{u}}\]  
where WCSV is the worst-case scenario value. The complexity reduction herein means that the single-objective design problem (18) that requires a double loop optimization run is altered to the equivalent deterministic multi-objective problem in (19). With the proposed idea, any powerful multi-objective evolutionary algorithm (MOEA) can be used to solve the problem as MOEA can explore a Pareto front within one run. For the processing of fuzzy input values, the fuzzy set model is adopted with \(\alpha\)-cut, which has been proposed in [13].