## 3 Bridge Element Maintenance Decision Optimization Model
### Sequential Decision Making Model for Bridge Project-Level Maintenance Decisions
#### 3.2.3 Maintenance Effectiveness Matrix  
In this study, we established a maintenance effectiveness matrix as the transition probability matrix based on the actual defect conditions of bridge elements to enhance the accuracy of decision making. For bridge maintenance benefit analysis, maintaining and improving the technical condition of bridges is the most significant aspect of enhancing benefits. [PERSON] et al. (2016) compared the bridge deck condition levels before and after the application of maintenance action, and they analyzed the impact of bridge deck maintenance measures on the condition levels. They established a maintenance effectiveness matrix. In this paper, via referring to their research findings, we define the effectiveness matrix as the transition probability matrix after implementing maintenance plans, and this represents the repair effects of maintenance actions on bridge elements. Different maintenance plans applied to elements in different states result in varying repair effects, thus leading to different transition probabilities. The calculation formula for the effectiveness matrix of executing a maintenance plan \(k\) is as follows:  
\[M_{A_{k}}=\begin{bmatrix}\frac{C_{k11}}{C_{1}}&0&0&0&0\\ \frac{C_{k21}}{C_{2}}&\frac{C_{k22}}{C_{2}}&0&0&0\\ \frac{C_{k11}}{C_{3}}&\frac{C_{k22}}{C_{3}}&\frac{C_{k23}}{C_{3}}&0&0\\ \frac{C_{k11}}{C_{4}}&\frac{C_{k22}}{C_{4}}&\frac{C_{k13}}{C_{4}}&\frac{C_{k 14}}{C_{4}}&0\\ \frac{C_{k11}}{C_{5}}&\frac{C_{k22}}{C_{5}}&\frac{C_{k23}}{C_{5}}&\frac{C_{k 13}}{C_{5}}&\frac{C_{k14}}{C_{5}}\\ \end{bmatrix}. \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multirow{2}{*}{\begin{tabular}{c} **Bridge** \\ **Deterioration** \\ **Types** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Maintenance** \\ **Plan** \\ **Description** \\ \end{tabular} } & \multicolumn{4}{c}{**Maintenance Measures**} \\  & & & & & **Measure** \\  & & & & **Number** & & \\ \hline \multirow{2}{*}{\(\text{A}_{1}\)} & \multirow{2}{*}{Defect Types 1} & Repairing & \multirow{2}{*}{\begin{tabular}{c} \(\text{a}_{1}\) \\ Elements \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} Maintenance Measures 1 \\ Maintenance processes 2 \\ Maintenance processes 3 \\ Maintenance processes 1 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ \hline \end{tabular} } \\ \hline \multirow{2}{*}{\(\text{A}_{3}\)} & \multirow{2}{*}{Defect Types 3} & Repairing & \multirow{2}{*}{\
\begin{tabular}{c} \(\text{c}_{2}\) \\ Elements \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} Maintenance processes 1 + Maintenance processes 2 + Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ Maintenance processes 3 \\ \hline \end{tabular} } \\ \hline \end{tabular}
\end{table}
Table 2: Examples of bridge element maintenance strategy set \(A\).  
Assuming that, after executing a maintenance plan, the state of an element can only transition to a lower state, then the elements of the matrix can be denoted as \(\frac{C_{kij}}{C_{i}}\), where \(C_{i}\) represents the quantity of elements in state \(i\); \(C_{kij}\) represents the quantity of elements transitioning from state \(i\) to state \(j\) after executing maintenance plan \(k\); \(M_{A_{k}}\) is the element in the \(i\)th row; the \(j\)th column represents the probability of elements in state i transitioning to state \(j\) after implementing maintenance plan \(A_{k}\); and \(k\) is the number of maintenance plans, and it is also determined by the actual number of defects present in the elements.