# Risk-Cost Optimized Maintenance Strategy for Steel Bridge Subjected to Deterioration
## 6 Risk-Cost Optimization  
Each item in Equation (1) can be developed based on Equations (2)-(27). A computational procedure of optimization was developed, as shown in Figure 4. The stochastic models of \(R(t)\), including \(M_{R}(t)\), \(V_{R}(t)\), \(\Delta_{S}(t)\), \(D(t)\), and \(C(t)\), were used as inputs. The optimization steps shown in Figure 4 are summarized as follows:1. Set lifetime \(t_{L}\), the failure cost of each failure mode at initial time \(C_{jk0}\), the interest rate \(r\), and the total number of failures mode \(N_{m}\).
2. Formulate stochastic models of \(R(t)\) for different failure modes, as listed in Section 4.
3. Set the total number of maintenance times \(N_{r}\) and let \(i=1\).
4. Set an initial time, \(t_{0}=0\). Then, set the maintenance time for bridge \(t_{r}=t_{0}\)
5. Let \(\Delta t=\frac{t_{L}}{N_{r}}\) and \(t_{i}=\Delta t+t_{i-1}\).
6. For a given time \(t_{r}\), calculate the probability of failure for each failure mode \(p_{f}(t_{r}-t_{i})\) based on [PERSON] and [PERSON] [29], the probability of failure for each component \(p_{fs,c}(t_{r}-t_{i})\) and bridge system \(p_{fs,s}(t_{r}-t_{i})\) using Equations (2)-(5).
7. Check the constraints shown in Equation (1). If the constrain is not satisfied, repeat step (3) with a different \(N_{r}\) until the constrain is satisfied. Otherwise, go to step 8.  
Figure 4: Program for maintenance strategy optimization.  
Rank \(p_{f}(t_{i}-t_{r})\) for all failure modes and \(p_{f_{s,c}}(t_{r}-t_{i})\) for the system. Based on the ranking, the critical component and its failure mode can be determined. This step determines the repair location (based on the component).
* For the critical component identified in step 8, determine the cost of failure \(C_{jk}=C_{JK0}(1+r)^{t_{i}}\).
* Calculate the risk, \(p_{f_{s,s}}\times C_{jk}\), where \(p_{f_{s,s}}\) is determined in step (4) and \(C_{jk}\) is determined in step (7).
* Reset the process, namely, for the critical component, let \(p_{f}(t_{i})=p_{f}(t_{0})\) for all its failure modes. It is assumed that when a critical component has been repaired (e.g., strengthening has been carried out on bridges), the probability of failure for its failure mode is reduced to the value at the initial time.
* Let \(t_{r}=t_{i}\)
* Let \(i=i+1\). Repeat steps (4)-(12), until all the maintenances are completed, i.e., \(i=N_{r}\).
* Repeat steps (2)-(13) for a range of \(N_{r}\) values and find the optimum value corresponding to the minimum risk: output \(N_{r}\) and \(p_{f_{s,c}}\) for each component.  
The search for \(N_{r}\) with given constraints is an iterative procedure. Based on [PERSON] et al. [16], the Genetic Algorithm method [16] is an efficient method in finding the optimum of complex problems with nonlinear constrains. Thus, in this paper, the flowchart in Figure 4 was coded using the Genetic Algorithm through MATLAB. As inputs, \(t_{L}\), \(C_{jk0}\) and \(r\) are case-specific with values summarized in Section 7 for the case study bridge. \(N_{m}=9\) in this study. Since the maintenance of bridge is generally carried out every 5 years, the initial value of \(N_{r}\) can be determined by letting \(N_{r}=\frac{t_{i}}{\Delta t}\), where \(\Delta t=5\)[40].