# Illustrative Example

The approach presented in this paper is applied for the maintenance optimization of 100 bridge deck projects from different bridges within a network of a highway agency. The objective here is to optimize the prioritization of the bridge deck projects for maintenance, considering simultaneously their condition rating, maintenance cost, and user costs, assuming a constraint on the allocated funding for the current year, which is assumed as \$1.5 Million. The physical condition of the bridge decks is assessed using the 1-5 condition rating scale described earlier.

The deterioration of the bridge decks is predicted using the Bogdanoff’s cumulative damage model described earlier, assuming a unit-jump and stationary deterioration model. A constant duty cycle is assumed throughout the service life of bridge decks that consists in one-year exposure to chloride-induced corrosion due to deicing salts, freeze-thaw cycles, and traffic loading. The transition probability matrix has the following elements (Lounis 2000): pn=0.7, p22=0.765, p33=0.85, p^=0.9, and p55=0.98, p66=0.98, and p77=1. The probability mass function of the current network condition is shown in Fig.3(a), and is given by the following initial condition vector:

p0=[0.06 0.34 0.31 0.19 0.08 0.01 0.01] (10a)

Using Eq. 2(b), the predicted deterioration of the bridge deck network after 10, 20 and 30 years is shown in Fig. 3(b). For example, after 30 years, the probabilistic distribution of the condition of the deck network is given by the vector:

p30=[0. 0. 0.008 0.089 0.547 0.273 0.082]

Hence, after 30 years, no deck of the network is in damage state 1 or 2 (i. e. no damage or minor damage), and about 90% of the deck network is in damage state 5, 6 or 7 (i. e. poor to failed states).

For bridge decks with condition ratings (or damage states) 1 or 2, no action or some preventive maintenance may be needed. For decks in damage states 3 and 4, some intermediate maintenance actions may be required (patch repair, asphalt overlay). For the decks with damage states 5 or higher, the possible maintenance actions include deck overlay using latex-modified concrete or low-slump dense concrete, cathodic protection, partial or complete replacement of the deck. To illustrate the application of the approach, the maintenance optimization is carried on a group of bridge decks with damage states 5 or higher, i. e. the threshold condition rating CRth is assumed equal to 5.

In this example, the maintenance alternatives are assumed optimized for the individual deficient bridge decks on the basis of life cycle cost minimization. The present value costs of the maintenance alternatives for the ten most damaged bridge decks are summarized in Table 2. The user costs are assumed to represent the sum of all costs incurred by the users during the maintenance activity, which include the delay costs, accident costs, and vehicle operating costs. These user costs depend primarily on the duration of the maintenance activity, average daily traffic, accident rate increase due to traffic detour or/and lane closure, and are summarized in Table 2.

 10 years 30 years 50 years —– ,n_ L J J
 0.6 0.5 0.4 0.3 0.2 0.1

 1 2 3 4 5 6 7 Damage state

(a) Current network condition (b) Network condition after 10, 30, and 50 years

Fig.3. Current and future network condition predicted using Bogdanoff CD model

The normalized values (normalized with regard to the maximum value) of the three selected objective functions are shown in Fig. 4. Table 2 and Fig. 4 illustrate the conflicting nature of these criteria and the difficulty in prioritizing, as the project with the highest urgency in terms of condition rating (Project #1) is neither the same in terms of maintenance cost (Project #9) nor in terms of user cost (Project #4). If single objective-based optimizations are undertaken, the bridge deck projects will be prioritized for maintenance following these three different schemes, depending on the selected objective:

• Damage condition rating – based prioritization: the projects will be ranked in terms of decreasing condition rating, i. e. the project with the highest condition rating will given first priority, and end up with the project which exhausts the available budget;

• Maintenance cost-based prioritization: the projects will be ranked in terms of increasing cost, i. e. the project with the lowest maintenance cost will be given first priority, ending with the project at which the available budget is exhausted;

• User cost-based prioritization: the projects will be ranked in terms of decreasing user costs, i. e. the project with the highest user cost will be given first priority, ending with the project at which the available budget is exhausted.

The required total fund to address all maintenance needs for the 10 deck projects is \$2.891 million, which is well in excess of the available budget of \$1.5 million. From Table 2, the “ideal” (but non­existing) maintenance solution is associated with the following “ideal” objective vector f*=[f1 f2 .

f3max]T= [7, 75000, 153000]T.

Table 2. Multi-objective-based maintenance optimization of bridge decks

 Bridge Deck Project # Damage Condition Rating Maintenance Costs (\$1,000) User Costs (\$1,000) Euclidean Metric MOI2 Chebyshev Metric MOI„ 1 7 620 120 1.260 1.000 2 6 832 132 1.130 1.000 3 5 350 124 0.424 0.363 4 5 364 153 0.382 0.382 5 5 125 76 0.583 0.579 6 5 150 76 0.587 0.579 7 5 100 56 0.730 0.729 8 5 125 20 1.002 1.000 9 5 75 20 1.000 1.000 10 5 150 84 0.528 0.519 Average= 5.3 £=\$2891,000 £=\$861,000 MIN=0.382 MIN=0.363

Using Eq. (9), the values of the multi-objective optimality indices MOI2 and MOI„, corresponding to the Euclidean and Chebyshev metrics, respectively, are determined for the bridge deck projects and are summarized in Table 2. Using the min. MOI2 criterion, the “satisficing” solution is found to be Project # 4, however using the min. MOI„ criterion, the “satisficing” solution is found to be Project #

3. Fig. 5, however, illustrates the similarity between the rankings of projects for maintenance obtained using both the min. MOI2 and min. MOI„ criteria for the other deck projects.

Considering now the budgetary constraint, the scheduling of projects maintenance will be as follows:

(i) Euclidean metric-based prioritization: Projects #4, #3, #10,#5, #6, #8, and #9, for a total cost of \$1.439 million. The other projects may delayed until the next year; however, a detailed analysis may be required.

(ii) Chebyshev metric-based prioritization: Projects #3, #4, #10, #5, #6, #7, #8, and #9 for a total cost of \$1.439 million.

From the above example, both the Euclidean and Chebyshev (or minimax) criteria for multi-objective optimization yield the same prioritization of bridge deck projects for maintenance. It should be pointed out that for both metrics, project #6 has always a higher priority than project #8, because the latter is not a “true” Pareto optimum, as it is dominated by solution (or project) #6. It is also possible to use the proposed approach by introducing different weighting factors on the different objectives, as discussed in the previous sections.

Conclusions

This paper illustrated that the bridge deck maintenance optimization problem can be formulated as a multi-objective optimization problem. The major merits of the approach are: (i) consideration of all possible (even conflicting) objective functions; (ii) ability to put more emphasis on the more relevant objectives (e. g. condition improvement); and (iii) rational decision-making regarding the selection of bridge projects for maintenance. The prioritization of the bridge decks is based on the satisfaction of several conflicting objectives simultaneously, including improving the physical condition, reducing the maintenance costs and user costs. The proposed multi-objective optimization approach provides a decision support tool for effective bridge management that enables decision-makers to select all relevant objectives in planning the maintenance of their bridge network. The development and integration of the proposed models for maintenance optimization will lead to an effective approach to bridge maintenance management, which optimizes the allocation of maintenance funds, as well as improves the risk management of bridge decks. The solutions obtained achieved a satisfactory trade­off between several competing criteria, including the maximization of the bridge deck condition, minimization of maintenance costs, and minimization of user costs. The proposed multi-objective optimality index can be used as an effective optimality criterion for the prioritization of deteriorated bridge decks for maintenance