#### 4.2. Maintenance Cost Model

The total lifecycle cost of a bridge can be derived from the function of the used materials, including the specifications, environment, and time, and can be generally divided into the initial investment cost, maintenance cost, and the demolition and disposal cost. Here, the maintenance cost is the direct cost that is paid by those who manage the bridge and comprises the periodical inspection cost presented in the detailed guideline, as well as the maintenance cost calculation based on the maintenance scenario analysis. In this study, we considered three expected maintenance cost terms which are the inspection, repair/reinforcement, and rehabilitation (or failure) in this maintenance cost. Furthermore, when the maintenance work is performed, the economic usage of the structure is possible through the improvement of the condition and performance. However, if the optimal maintenance period (or lifetime) is missed, a large-scale construction method or the replacement of the whole structure would be required. To solve these problems, a preventive bridge maintenance scenario strategy needs to be established.

In this study, we considered potential maintenance scenarios for the bridge that is being used and proposed a maintenance cost calculation model that is used for the monitoring techniques in addition to the precise safety diagnosis [

23]. The monitoring-based maintenance cost calculation model used the particle filtering technique for updating the existing deterioration model and determining the time for maintenance to predict the maintenance of the target bridge before the damage occurred.

The total expected maintenance cost that is incurred up to a specific lifetime (

t) in the general operation stage is formulated in Equation (16);

$E[{C}_{MAI,EXI}^{TOT}\left(t\right)]$ denotes the total expected maintenance cost by the existing method which is a function of the specific lifetime (

t);

${C}_{INS,EXI}^{TOT}\left(t\right)$ is the inspection cost;

${C}_{R\&R,EXI}^{TOT}\left(t\right)$ is the expected repair/reinforcement cost, and

${C}_{F,EXI}^{TOT}\left(t\right)$ is rehabilitation (or failure) cost. A detailed definition of the three terms is presented in Equations (17) to (19);

In Equation (17), T is the target lifetime of bridge; the total expected inspection cost is derived as the sum of the regular inspection cost (${C}_{INS,EXI}^{rins}\left(t\right)$), precise inspection cost (${C}_{INS,EXI}^{pins}\left(t\right)$), and precise safety diagnosis cost (${C}_{INS,EXI}^{psdia}\left(t\right)$). In Equation (18), ${t}_{i}$ is the time at which the repair and reinforcement methods are applied to the target bridge, and q is the discount rate; ${C}_{R\&R,EXI}\left({t}_{i}\right)$ and ${P}_{R\&R,EXI}\left({t}_{i}\right)$ are the repair and reinforcement cost and the repair and reinforcement application probability. In Equation (19), ${C}_{F,EXI}\left({t}_{k}\right)$ and ${P}_{F,EXI}\left({t}_{k}\right)$ are the expected rehabilitation (or failure) cost and application probability, respectively; ${t}_{k}$ is limit state approaching time which is E grade and after in this paper.

Meanwhile, the total expected maintenance cost for a specific time (t) is when the particle filtering method using monitoring techniques (

$E[{C}_{MAI,PF}^{TOT}\left(t\right)]$) is defined in Equation (20). Moreover, a detailed definition of three terms is presented in Equations (21) to (23);

In Equation (20), $E[{C}_{MAI,PF}^{TOT}\left(t\right)]$ denotes total expected maintenance cost by particle filtering method which is the function of the specific lifetime (t); ${C}_{INS,PF}^{TOT}\left(t\right)$ is the inspection cost; ${C}_{R\&R,PF}^{TOT}\left(t\right)$ is the expected repair/reinforcement cost, and ${C}_{F,PF}^{TOT}\left(t\right)$ is rehabilitation (or failure) cost. In Equation (21), ${C}_{SEN}\left({t}_{0}\right)$ denotes the initial (${t}_{0}$) sensor installation cost, and ${C}_{INS,PF}^{psdia}\left(t\right)$ is derived from the sum of the precise safety diagnosis cost. In Equation (22), ${C}_{R\&R,PF}\left({t}_{i}\right)$ is the repair and reinforcement method cost linked to the updated deterioration model at the repair and reinforcement application time (${t}_{i}$) of the target bridge, ${P}_{R\&R,PF}\left({t}_{i}\right)$ is the repair and reinforcement implementation probability linked to the updated deterioration model at the time (${t}_{i}$); In Equation (23), ${C}_{F,PF}\left({t}_{k}\right)$ and ${P}_{F,PF}\left({t}_{k}\right)$ are the expected rehabilitation (or failure) cost and application probability.

In calculating the maintenance cost incurred in the actual bridge, the proposed preventive maintenance cost analysis method with the monitoring technique-based particle filtering is advantageous. This is because the particle filtering technique is used to reduce the uncertainty (${\sigma}_{t}^{exi}$) of the existing deterioration model in the life cycle management analysis. In addition, the comparison of the maintenance cost before and after the monitoring-based particle filtering update denotes that the monitoring-based preventive maintenance is more advantageous than not only using the existing maintenance method, but also using Bayesian method in terms of cost efficiency.