# Optimization for Bridge Network Maintenance Planning

Bridge network maintenance planning has to deal with multiple bridges in a highway network under limited annual maintenance budgets. Thus, MCDM approaches can be used to help bridge owners, authorizers and/or maintenance managers to make rational decisions on maintenance actions applied to each of individual bridges in the highway network. In this study, the multiple attribute utility theory (MAUT) is adopted. As a matter of fact, MAUT focuses on the development of the multiple attribute utility functions to model and represent the decision maker’s preferential structures. The multiple attribute utility functions combine all of the marginal utility functions associated with individual attribute of each alternative. The marginal utility functions for each attribute can be built up by either direct interrogation with decision makers or by indirect methods, as well as by using the analytic
hierarchy process (Saaty, 1980) that has been mainly used in USA. The decomposition forms of the multiple attribute utility functions may be (1) additive, (2) multiplicative and (3) multi-linear forms (Keeney and Raiffa, 1993). The additive form requires mutual preferential independence, that is, every subset of criteria is preferentially independent from the remaining criteria. A subset of criteria is considered to be preferentially independent form the remaining criteria if and only if the decision maker’s preferences on the alternatives differ only with respect to the criteria, and are independent on the remaining other criteria. It must be noted that very complex decomposition forms are of not interest from a practical point of view (Vincke, 1992). MAUT also employs an interactive and iterative procedure involving policy analyst and decision makers to specify the weight and marginal utility function corresponding to each criterion. Finally, the total utility of each alternative can be used as an objective function in traditional mathematical programming in order to make final decisions (Doumpos and Zopounidis, 2002).

In this study, each of individual bridges in a highway network may be treated as a subset of criteria with a marginal utility function associated with the probabilities that each of the above five maintenance alternatives (including “do nothing”) may be conducted at certain time (year). Since the mutual preferential independence requirement can be easily satisfied in this case, the additive form of the multiple attribute utility function is used to form a single-objective function for optimization. The objective function of multiple attribute utility may be also weighted by using RIF of individual bridges in the network, where RIF is defined as the sensitivity of the bridge network reliability to the change in the individual bridge system reliability (Liu and Frangopol, 2005). RIF in this paper reflects the sensitivity of the bridge network reliability in terms of the network connectivity to the change in the individual bridge system reliability due to maintenance actions, and must be developed as a function of the bridge system reliability profiles, network reliability, and network topology. RIF may also be expanded to include traffic capacity and impacts of bridge maintenance activities on economy, environment and society, when considering user’s satisfaction and critical bridge performance of a bridge network (Liu and Frangopol, 2005). Consequently, the optimization problem in bridge network maintenance planning can be formulated as follows:

Maximize ^ Dij x RIFt x Pij (1)

І

Subject to: £C. < Cbudget (2)

is a binary design variable, i. e. the value of Dj can be either 0 or 1; is the reliability importance factor for bridge i;

is the probability of the maintenance alternative j applied to bridge ; is the cost associated with the selected maintenance alternative for bridge i; is the annual maintenance budget at certain year;

The binary design variable, Dj represents the decision on selecting the maintenance alternative j applied to bridge i, that is, Dj = 0 means the maintenance alternative j will not be applied to bridge i, and Dj = 1 means the maintenance alternative j is selected to be applied to bridge i. In addition, it should be noted that the values of RIFi and Pj usually vary during the entire service lifetime of bridge i. This is because RIFi is normalized by considering all bridges in a highway network that experience the aging and deterioration with time (Liu and Frangopol, 2005). On the other hand, Pj is normalized by considering all of the five maintenance alternatives applied to bridge i in a certain year, and is dependent on the results from the DP procedure that is combined with Monte Carlo simulations (Liu and Frangopol, 2006). Moreover, Ci is related to the cost in Table 3, but the actual values of Ci should
be assigned in order to obtain an optimal bridge network maintenance planning. Finally, this combinatorial optimization problem can be easily solved by either traditional mathematical programming or the advanced heuristic search methods such as Genetic Algorithms (GAs).

Case Study

A numerical example involving five highway bridges is provided to demonstrate the application of the proposed DSS in bridge network maintenance planning. Table 4 presents the values of RIFj, Py and Ci for combinatorial optimization that is subject to a budget constraint of C budget = 10,000 The optimization results from a traditional mathematical programming are also summarized in Table 4.

Table 4: Example Values of RPFi, Py and Ci

 Maintenance Bridge Bridge Bridge Bridge Bridge Cost Alternatives E-17-HE E-17-HR E-17-LE E-16-MU E-16-NM (Py) P) (Py) P) (Py) (C) Minor Concrete Repair 0.40 0.30 0.15 0.20 0.15 4,500 Silane Treatment 0.20 0.20 0.40 0.10 0.35 48 Cathodic Protection 0.25 0.30 0.20 0.30 0.20 2,600 Rebuild 0.10 0.15 0.05 0.30 0.05 12,000 Do Nothing 0.05 0.05 0.20 0.10 0.25 0 Sum of Py 1.00 1.00 1.00 1.00 1.00 – Reliability Importance Sum of RIFj Factor (RIF,) 0.38 0.27 0.19 0.11 0.05 1.0 Selected minor cathodic silane cathodic silane C budget Maintenance concrete protection treatment protection treatment Alternative repair 10,000 Sum of Cj Cost (C) 4,500 2,600 48 2,600 48 9,796

Conclusions

This paper presented a decision support system (DSS) for bridge network maintenance planning using the multiple attribute utility theory (MAUT). The combinatorial optimization problem was developed with a single-objective function of the probabilities that the maintenance alternatives may be applied to each of the individual bridges in a highway network. The probabilities in the single-objective function had to be obtained from a Dynamic Programming (DP) procedure, considering individual bridge condition index, safety index and life-cycle maintenance cost. The single-objective function was also weighted by the Reliability Importance Factors (RIF), which had to be the functions of individual bridge system reliability profiles, bridge network reliability, and network topology. The constraint of the optimization problem was the limited annual maintenance budget. A numerical example was provided to demonstrate the application of the proposed DSS in bridge network maintenance planning.