Highway Project Level Life-Cycle Benefit/Cost Analysis under Certainty, Risk, and Uncertainty: Methodology with Case Study

One of the key steps in the highway investment decision-making process is to realistically estimate project level life-cycle costs and benefits of different types of highway projects. Different highway facilities such as pavements and bridges have different useful service lives. In order to compare the merit of different projects on an equal basis, the life-cycle cost analysis approach needs to be adopted to evaluate the total economic worth of the initial construction cost and discounted future maintenance and rehabilitation costs in the facility life cycle.

Over the last decade, the life-cycle cost analysis approach has been widely used for evaluating highway pavement and bridge projects. As related to pavement project evaluation, the Federal Highway Administration (FHwA) made a concerted effort for the use of life-cycle cost analysis in highway pavement design (FHwA 1998). Hicks and Epps (1999) explored alternative pavement life-cycle design strategies with a logical comparison between conventional mixtures and the mixture containing asphalt rubber pavement materials. Wilde et al. (1999) introduced a life-cycle cost analysis framework for rigid pavement design. Abaza (2002) developed an optimal life-cycle cost analysis model for flexible pavements. Falls and Tighe (2003) enhanced life-cycle cost analysis through the development of cost models using the Alberta roadway maintenance and rehabilitation analysis application. Labi and Sinha (2005) and Peshkin et al. (2005) studied systematic preventive maintenance and the optimum timing strategies to achieve minimum pavement life-cycle costs. Chan et al. (2008) evaluated life-cycle cost analysis practices in Michigan. For bridge project evaluation, Purvis et al. (1994) performed life-cycle cost analysis of bridge deck protection and rehabilitation. Mohammadi et al. (1995) introduced the concept of incorporating life-cycle costs into highway bridge planning and design. Hawk (2003) developed a bridge life-cycle cost analysis software tool for bridge project evaluation.

In recent years, researchers began to utilize the risk-based life-cycle cost analysis approach to establish mathematical expectations of highway project benefits. For instance, Tighe (2001) performed a probabilistic life-cycle cost analysis of pavement projects by incorporating mean, variance, and probability distribution for typical construction variables, such as pavement structural thickness and costs. Reigle et al. (2005) incorporated risk considerations into the pavement life-cycle cost analysis model. Setunge et al. (2005) developed a methodology for risk-based life-cycle cost analysis of alternative rehabilitation treatments for highway bridges using Monte Carlo simulation.

Because of lacking of pertinent information, in many instances it might not be possible to establish a meaningful probability distribution to possible outcomes of a specific input factor such as construction, rehabilitation, and maintenance costs and traffic growth in risk-based life-cycle benefit/cost analysis. That is, the input factors are under uncertainty with no definable probability distributions. Consequently, the mathematical expectation of the input factor cannot be established. Further, risk and uncertainty inherited with input factors for project level life-cycle benefit/cost analysis may vary from project to project. Some projects may only involve risk cases for some input factors, whereas other projects may only experience uncertainty cases for some input factors. In more general situations, a project may face mixed cases of certainty, risk, and uncertainty concerning all input factors for the computation. This necessitates developing a new uncertainty-based methodology for highway project level life-cycle benefit/cost analysis that could rigorously handle such general situations.

In this paper, we will first introduce a methodology for highway project level life-cycle benefit/cost analysis that considers certainty, risk, and uncertainty associated with input factors for the computation. A case study is then conducted to assess impacts of risk and uncertainty considerations in estimating project level life-cycle benefits and on network-level project selection. Discussions of usefulness of the proposed methodology and directions to its refinement are provided in the last section.

In this study, the pavement or bridge life-cycle is defined as the time interval between two consecutive construction events. Maintenance and rehabilitation treatments are performed within the pavement or bridge life-cycle. The pavement and bridge life-cycle agency cost and user cost components are briefly discussed in the following:

The typical life-cycle activity profile for pavements or bridges represents the most cost-effective investment strategy to manage pavement or bridge facilities. If any needed treatment fails to be timely implemented as per the typical life-cycle activity profile, an early termination of the service life is expected. As such, the typical life-cycle activity profile can be used as the base case activity profile and the case with early service-life termination can be considered as an alternative case activity profile. For each type of pavements or bridges, the reduction in life-cycle agency costs of the base case activity profile compared with the alternative case activity profile can be computed as project level life-cycle agency benefits of timing implementing the needed project. Similarly, the decrease in life-cycle user costs according to the base case activity profile against the alternative case activity profile can be estimated as the project level life-cycle user benefits.

Fig. 1 illustrates an example of base case and alternative case activity profiles for the steel-box beam bridge and the method for estimating project level life-cycle agency benefits and user benefits by keeping the typical life-cycle activity profile for the bridge. For the base case life-cycle activity profile, agency costs in the $T$ -year bridge service life consist of initial bridge construction cost $C_{\mathrm{CON}}$ in year 0, first deck rehabilitation cost $C_{\text{DECK}\mathrm{REH}1}$ in year $t_1$ , deck replacement cost $C_{\text{DECK}\mathrm{REP}}$ in year $t_2$ , second deck rehabilitation cost $C_{\text{DECK}\mathrm{REH}2}$ in year $t_3$ , and annual routine maintenance costs. The annual routine maintenance costs between two major treatments in the bridge life-cycle will gradually increase over time due to the combined effect of higher traffic demand, aging materials, climate conditions, and other nonload related factors. Different geometric gradient growth rates are used for intervals between year 0 and $t_1$ , $t_1$ and $t_2$ , $t_2$ and $t_3$ , and $t_3$ and $T$ , respectively.

For the alternative life-cycle activity profile, it is assumed that the deck replacement project (with the cost of $C_{\text{PROJECT}}$ ) is actually implemented $y_1$ years after year $t_2$ as the base case profile, namely, $C_{\text{DECK}\mathrm{REP}}$ in year $t_2$ is replaced by $C_{\text{PROJECT}}$ in year $t_2+y_1$ . This will defer the second deck rehabilitation by $y_1$ years. Because of postponing deck replacement and the second deck rehabilitation, the bridge service life may experience an early termination of $y_2$ years. As for the annual routine maintenance costs, different geometric gradient growth rates are used for intervals between year 0 and $t_1$ , $t_1$ and $t_2+y_1$ , $t_2+y_1$ and $t_3+y_1$ , and $t_3+y_1$ and $T-y_2$ , correspondingly. In particular, the annual routine maintenance cost profiles for the base case and alternative case profiles are identical from year 0 to year $t_2$ . The project level life-cycle agency benefits are estimated as the reduction in bridge life-cycle agency costs quantified according to the base case activity profile compared with the alternative case activity profile.

The primary user cost items include vehicle operating costs, travel time, vehicle crashes, and vehicle air emissions. For each user cost item, the base case and alternative case annual user cost profiles in bridge life cycle follow a pattern similar to the profile of annual routine maintenance costs in bridge life cycle. In either the base case profile or alternative case profile, the first-year user cost amounts immediately after the major treatments including bridge construction, first deck rehabilitation, deck replacement, and second deck rehabilitation are directly computed on the basis of the unit user cost in constant dollars per vehicle mile of travel (VMT) and the annual VMT. The unit user cost per VMT is estimated according to average travel speed and roadway condition as in Li and Sinha (2004). Geometric growth rate is then applied to the first-year user cost amount for each interval between two major treatments to establish the annual user cost amounts for subsequent years within the interval. Additional work zone related costs are estimated using the procedures in FHwA (1998, 2000) and AASHTO (2003), and added to the annual user 235 cost amounts for the years in which major treatments are implemented. This ultimately establishes the base case and alternative case annual user cost profiles for vehicle operating costs, travel time, vehicle crashes, and vehicle air emissions, respectively.

For each user cost item, the annual user cost profiles for the base case and alternative case are identical from year 0 to $t_2$ and are different for the remaining years in the bridge life cycle. The travel demand in terms of annual VMT for a specific year after year $t_2$ could be different between the base case and alternative case due to the fact that the traffic volume, i.e., annual average daily traffic (AADT) and/or travel distance associated with the bridge might change for the two cases. The consumer surplus concept is employed to separately compute the user benefits by comparing the base case and alternative case annual user cost profiles for intervals from year $t_2$ to $t_2+y_1$ , $t_2+y_1$ to $t_3$ , $t_3$ to $t_3+y_2$ , $t_3+y_2$ , $T-y_2$ , and $T-y_2$ to $T$ . The total project level life-cycle user benefits are the aggregation of individual user benefit items associated with reductions in vehicle operating costs, travel time, vehicle crashes, and vehicle air emissions in the bridge life cycle. With equal weights assigned for agency benefits and user benefits, the total project level life-cycle benefits by keeping the typical life-cycle activity profile for the bridge are established by combining the two sets of benefits.

The project level life-cycle benefits in perpetuity can be quantified on the basis of the base case and alternative life-cycle activity profiles. As the base case life-cycle activity profile represents the most cost-effective investment strategy, investment decisions are always made with the intention to keep abreast of the base case life-cycle activity profile. For the base case life-cycle activity profile in perpetuity, the base case typical facility life-cycle is assumed to be repeated an infinite number of times. For the alternative case life-cycle activity profile in perpetuity, early termination of service life may occur in the first life-cycle, in the first and second life cycles or in the first several life cycles. After experiencing early service life terminations, the base case typical facility life cycle is expected to be resumed back for the subsequent life cycles in perpetuity horizon. This is because that the base case life-cycle profile represents the most cost-effective investment strategy that the decision maker always aims to achieve. Without loss of generality, the alternative case life-cycle profile in perpetuity in this study adopts early terminations for the first two life cycles and the base case life-cycle profile is used for subsequent life cycles in perpetuity horizon. The reduction in project level life-cycle agency costs between the base case and the alternative case life-cycle activity profiles in perpetuity is computed to establish project level life-cycle agency benefits in perpetuity.

Similarly, the reduction in project level life-cycle user costs between the base case and the alternative case life-cycle annual user cost profiles in perpetuity for vehicle operating costs, travel time, vehicle crashes, and vehicle air emissions can be separately computed and summed up to establish project level life-cycle user benefits in perpetuity. With equal weights considered for agency benefits and user benefits, they can be directly added to establish overall project level life-cycle benefits in perpetuity.

As a practical matter, the input factors under risk considerations may not be readily characterized using reliable probability distributions. Consequently, a meaningful mathematical expectation for each factor cannot be established and this invalidates risk-based analysis. Shackle’s model introduced herein is well suited to handle each input factor under uncertainty where no probability distribution can be readily established for a number of possible outcomes (Shackle 1949).

In general, Shackle’s model overcomes the limitation of inability to establish the mathematical expectation of possible outcomes of each input factor for project level life-cycle benefit/cost analysis according to the following procedure. First, it uses degree of surprise as a measure of uncertainty associated with the possible outcomes in place of probability distribution. Then, it introduces a priority index by jointly evaluating each known outcome and the associated degree of surprise pair. Next, it identifies two outcomes of the input factor maintaining the maximum priority indices, one on the gain side and the other on the loss side from the expected outcome $X_{\left(E\right)}$ . The expected outcome could be the average value or the mode of all known possible outcomes, but it is not the mathematical expectation as outcome probabilities are unknown. The two outcomes need to be standardized to remove the associated degrees of surprise. The absolute deviations of two outcomes relative to the expected outcome are terms as standardized focus gain $x_{\mathrm{SFG}}$ and standardized focus loss $x_{\mathrm{SFL}}$ from the expected outcome $X_{\left(E\right)}$ . This model yields a triple $⟨x_{\mathrm{SFL}},X_{\left(E\right)},x_{\mathrm{SFG}}⟩$ for each input factor under uncertainty. More details of Shackle’s model are in Ford and Ghose (1998), Young (2001), and Li and Sinha (2004, 2006).

To simplify the application of Shackle’s model for uncertainty-based analysis, the grand average of simulation outputs from multiple iterations of replicated simulation runs can be used as the expected outcome $X_{\left(E\right)}$ for an input factor under uncertaintywhere $X_i=\text{a}$ simulation output representing a possible outcome; $N=\text{number}$ of iterations in each simulation run; and $M=\text{number}$ of replicated simulation runs. If higher valued outcomes are preferred for an input factor, the absolute deviation of the average value of simulation outputs that are lower than the expected outcome can used as standardized focus loss value $x_{\mathrm{SFL}}$ and the absolute deviation of the average value of simulation outputs that are equal or higher than the expected outcome can used as standardized focus gain value $x_{\mathrm{SFG}}$ for the input factor under uncertaintywhere $N_r=\text{number}$ of simulation outputs in the $r\text{th}$ simulation run such that $X_i<X_{\left(E\right)}$ if a higher outcome value is preferred for the input factor.

In some cases, lower outcome values are preferred for an input factor such as the discount rate. The $N_r$ for computing the standardized focus loss value $x_{\mathrm{SFL}}$ and the standardized focus gain value $x_{\mathrm{SFG}}$ thus refers to number of simulation outputs in the $r\text{th}$ simulation run such that $X_i>X_{\left(E\right)}$ .

As an extension of Shackle’s model dealing with the input factor under uncertainty, a decision rule is introduced to help compute a single value $X$ for the input factor based on the triple $⟨x_{\mathrm{SFL}},X_{\left(E\right)},x_{\mathrm{SFG}}⟩$ that can be used for estimating project benefits. Assuming that the decision-maker only tolerates loss from the expected outcome for the input factor under uncertainty by $\Delta X$ and if higher outcome values are preferred, the decision rule is set asWhen lower outcome values are preferred for an input factor, the decision rule is revised toIf the standardized focus loss $x_{\mathrm{SFL}}$ from the expected outcome $X_{\left(E\right)}$ does not exceed $\Delta X$ , the expected outcome value will be utilized for the input factor for the computation. This will produce an identical input factor value for both uncertainty-based and risk-based analyses. If the standardized focus loss $x_{\mathrm{SFL}}$ from the expected outcome $X_{\left(E\right)}$ exceeds $\Delta X$ , a penalty is applied to derive a unique value for the input factor. Different tolerance levels $\Delta X$ ’s may be used for different input factors under uncertainty.

Project level impact assessments compare project level life-cycle benefits separately estimated using the deterministic, risk-based, and the uncertainty-based project level life-cycle cost analysis approaches. For the application of deterministic project level life-cycle benefit/cost analysis, project benefits are calculated by assuming that all input factors are under certainty and each input factor has a single value. These values are directly used for the computation.

For the application of risk-based project level life-cycle benefit/cost analysis, project benefits are calculated by assuming that input factors regarding unit rates of construction, rehabilitation, and maintenance treatments, traffic growth rates, and discount rates are all under risk. The remaining input factors such as pavement or bridge service life and timing of treatments are still treated as being under certainty with single values. Monte Carlo simulations are executed to establish the grand average values of simulation outputs as mathematical expectations of input factors under risk. The single values of input factors under certainty and the grand average values of input factors under risk are used for the computation.

For the application of the uncertainty-based methodology, project benefits are calculated by assuming that the input factors regarding unit rates of construction, rehabilitation, and maintenance treatments, traffic growth rates, and discount rates are all under uncertainty or under mixed cases of risk and uncertainty. The remaining input factors are still considered under certainty with single values. For the input factor under risk, the grand average value as the mathematical expectation is established using Monte Carlo simulation outputs. For the input factor under uncertainty, the grand average value of simulation outputs is adjusted according to the preset decision rule. The single values of input factors under certainty, the grand average values of input factors under risk, and the adjusted grand average values of input factors under uncertainty are used for the computation.

In order to assess the network level impacts of adopting different approaches for project benefit estimation, the three sets of project benefits computed using the deterministic, risk-based, and uncertainty-based approaches are separately applied to a stochastic optimization model for network-level project selection. The network level impacts are assessed by cross comparison of the overall benefits of selected projects and consistency matching rates of project selection using the three different sets of project benefits with the actual project selection and programming practice. This section briefly discusses the stochastic optimization formulation for finding the optimal subset of highway projects from all candidate projects to achieve maximized total project level life-cycle benefits where there is stochasticity in the available budget.

Consider a state transportation agency that carries out highway network-level project selection over a future project implementation period of $t_{\Omega }$ years. The agency makes first round of investment decisions many years prior to project implementation using an estimated budget for all years. With time elapsing, updated budget information on the first few years of the multiyear project selection and programming period becomes available that motivates the agency to refine the investment decisions. In each refined decision-making process, the annual budget for the first few years that can be accurately determined is treated as a deterministic value, while the budget for the remaining years without accurate information is still handled as a stochastic budget.

Table 3 lists project level life-cycle benefits of some pavement and bridge projects. On average, the present worth amounts of project level life-cycle benefits estimated using deterministic, risk-based, and uncertainty-based analysis approaches for the 7,380 projects are 4.18, 7.14, and 6.64 million dollars per project (in 1990 constant dollars), respectively. The average benefit-to-cost ratios are 3.24, 5.54, and 5.16, correspondingly. The significant difference between the project benefits estimated using the deterministic analysis approach and risk-based analysis approach is mainly attributable to large standard deviations of input factors considered for probabilistic risk assessments. The comparable results of project benefits computed using the risk-based analysis approach and uncertainty-based analysis approach are intuitive. This is because the grand average of simulation outputs for each input factor under uncertainty is adjusted only if the deviation between the grand average as the expected outcome and standardized focus loss value exceeds the preset threshold level. The input factor values for risk-based and uncertainty-based analyses will be identical if no adjustment is made.