RISK BASED LIFE CYCLE COST ANALYSIS FOR PROJECT LEVEL PAVEMENT MANAGEMENT. Eric Perrone, Dick Clark, Quinn Ness, Xin Chen, Ph.D, Stuart Hudson, P.E.

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1 RISK BASED LIFE CYCLE COST ANALYSIS FOR PROJECT LEVEL PAVEMENT MANAGEMENT Eric Perrone, Dick Clark, Quinn Ness, Xin Chen, Ph.D, Stuart Hudson, P.E. Texas Research and Development Inc Dellana Lane, Austin, Texas 78746, U.S.A. Montana Department of Transportation 2701 Prospect Avenue, Helena, Montana 59620, U.S.A Abstract Investment in the provision of highways is not a one time process. An agency charged with providing quality riding surfaces must continually monitor and repair the highway network for it to remain in the best condition possible. Because the funds available to maintain the highway network are limited, it is essential that the decisions made for the maintenance and rehabilitation of the highway network be spent wisely. Network and project level pavement management systems assist in this objective. Pavements require a long term investment of funds over their life to maintain them in the most cost effective manner. Therefore the selection of the most cost effective alternative does not depend solely upon the initial construction investment. The total cost of the pavement over the expected life cycle must be considered to truly evaluate cost effectiveness. This paper describes an advanced method of risk-based life-cycle cost analysis for project level pavement management. 1. INTRODUCTION The life cycle cost of providing a pavement is the discounted costs of required investments that are expected to occur over the life of the pavement. Pavement improvements are meant to correct structural or functional deficiency. Current procedures allow the improvement designer to correct inadequacies in many ways. Unfortunately, these procedures often do not include an estimate of required investment over the life of the pavement to allow the engineer to choose the most costeffective alternative. Life cycle cost analysis is used to address this problem. In classical life cycle cost analysis, a finite number of alternatives are formulated, such that each corrects the deficiency in the pavement structure. Then estimates of the investments required to maintain the pavement over a common analysis period are made. The total investment of each alternative over the analysis period is computed and discounted appropriately to arrive at the expected life cycle cost. Using these estimates, the decision maker is able to choose the most cost effective life cycle alternative that meets the objectives and goals of the project. These estimates of life cycle cost require that the analyst estimate many factors that, by their nature, are inherently uncertain. To estimate a life cycle cost for a pavement investment, the analyst must predict the future performance of a pavement, the costs of

2 materials, appropriate discount rates and other factors which greatly affect the output of any life cycle cost model. In many cases, however, decisions are made using life cycle costs that have neglected the fact that these estimates are not known values but have an inherent variability that can be represented by a distribution of values. This paper discusses the development of a life cycle cost analysis procedure that accounts for this variability in the inputs and provides a clearer picture of its effect on the life cycle costs of pavement treatment alternatives. Since it is important to estimate variability involved in the choice of an alternative, the question remains as to the method of measuring or quantifying the variability across the alternatives. Many factors used for life cycle cost analysis are variable, and it is possible to use historical data and judgment to estimate the distributions of these factors over the life cycle of a pavement. It may not be known exactly how long a particular maintenance or rehabilitation treatment will last, but it is possible to estimate the expected distribution of the life through historical data or local experience. Therefore, the analyst should estimate not only the expected values for variables, but also the distributions of these variables for input into the analysis. Then, using these distributions of the input variables, the analyst can estimate the distribution of the output variable, which is the present value of the investments needed over the life of the pavement (life cycle cost). The distribution of this cost is used to estimate the risk of each alternative. The risk of the design alternative is directly proportional to the standard deviation of the distribution of total life cycle cost. In short, the higher the standard deviation of this distribution, the more uncertain you are about its actual cost, hence more risk. 1.1 Life Cycle Principles Decisions are not made in a vacuum but, rather, with some objective or goal in mind. A transportation agency's primary goal is to provide a network of serviceable and safe pavements by maximizing the benefits obtained from expenditures on the network. This is the goal or objective on which the economic analyses discussed in this paper are based. The application of economic principles occurs at two levels. First there are the management decisions required to determine project feasibility and timing and considering the many possible projects that may be undertaken at any one time. Second is the requirement to choose an alternative that provides the maximum economic benefits for the project once it has been selected. This is the typical distinction between network and project levels of pavement management. Project feasibility is determined at the network level while comparing within-project alternatives is part of the project level pavement management process. Economic analyses take place at both levels of pavement management. The major difference in economic evaluations between these two levels of pavement management relate to the amount and types of information required. However the basic principles

3 that apply for economic analyses at both the project and network levels of pavement management remain the same. (Haas et al, 1994) These can be summarized as follows: 1. The management level at which the evaluation is to be made must be clearly defined 2. The economic analysis provides the basis (information) for a management decision but does not by itself represent a decision. 3. Criteria, rules, or guides for such decisions need to be separately formulated before the results of the economic decision are applied. 4. The economic analysis itself has no relationship to the method or source of funding for a project. 5. An economic analysis should consider all possible alternatives, within the constraints of time and other planning and design resources. 6. All alternatives should be compared over the same time period. 7. The economic analysis should include all of the benefits and costs of the project alternatives if possible. When these basic economic principles are followed, the effective use of economic analysis methods will help manage the difficulties associated with project selection (network level) and alternative selection (project level). Better management will occur by improving both the timing and cost effectiveness of asset acquisitions, and the quality of the assets purchased. These principles were the basis for the design and development of the life cycle cost analysis software discussed in this paper. 1.2 LCCA Requirements Weston (Weston 1990) lists six necessary steps to follow when performing life cycle cost analysis for a project. These steps identify the minimum information that should be identified for a complete analysis. These are: 1. The benefits and costs of the project must be identified. 2. Estimates of expected cash flows (benefits and costs), including the terminal value of the asset at a specified terminal data must be developed. 3. The appropriate discount rate must be determined. 4. The expected cash flows are discounted to a present value basis to obtain an estimate of the projects current value. 5. The risk of the projected cash flows; and/or information (sensitivity analyses) about the probability distributions of the cash flows must be determined. 6. The present values of the expected cash flow and sensitivity analyses are compared.

4 1.2.1 Identifying Project Benefits and s In evaluating a project, only cash flows (benefits and costs) resulting directly from the project are used. They represent the change in the total cash flow that occurs as a direct result of accepting or rejecting the project. When identifying project costs for inputs into the life cycle cost analysis the following principles are followed (MDT 1994): 1. Only incremental cash flows are used. These are cash flows that result directly from the project being analyzed. Sunk costs are not used because they have already been incurred and cannot be recovered. 2. The costs used for analysis should reflect the opportunity cost of any resource used, measured by the return to those resources in their most productive application elsewhere. This is typically measured by the market price that represents what society is willing to forego in order to obtain a given benefit. (OMB Circular A-1994) 3. Both tangible and intangible benefits and costs should be recognized and accounted for whenever possible. (OMB Circular A-1994) 4. Benefits and costs may be measured in real or nominal terms. A real benefit or cost has been adjusted to eliminate the effect of expected inflation. A nominal benefit or cost is equal to the corresponding real benefit or cost plus a premium for expected inflation. An important note about inflation is that nominal benefits/costs include a factor corresponding to expected or future inflation, not inflation that has been experienced in the past or present. The question of how to take inflation into account during an economic analysis is of great concern to many. Basically, the answer is that inflation is not used in the evaluation, except where substantial evidence exists that real prices will change. Real price change can be defined as the change in the price of an item that does not follow the general trend of inflation of all goods in a society. Some would argue that inflation should be included in the analysis because ignoring inflation leads to underestimating out-of-pocket costs and therefore budgets will be incorrect. This argument indicates a misunderstanding of the objective of an economic analysis, which is to provide management with a tool for the selection of specific options from a set of alternatives. Once the option is selected, a separate budget analysis is required to determine cash flow requirements. The budget analysis generally includes inflation and other factors. (Haas et al, 1994) Therefore it is recommended that only current market prices be used for economic analysis Estimation of Expected Cash Flows The estimation of expected cash flows requires determining two elements: the magnitude of the cash flow and the point in time when that cash flow will occur. Both these elements are not point values and are subject to variation. To estimate these values the analyst should estimate the following parameters for each element of each cash flow in the analysis: The mean (expected value) of the cash flow or when the cash flow will occur.

5 The distribution of the cash flow (for both its location in time and its magnitude). The distribution may be estimated in a number of ways. In this software package the distributions of cash flow timing and magnitude can be input by assuming the distributions are either triangular or normal. Further research is needed to identify other appropriate types of input distribution. These distributions, once identified, will be incorporated into the software in future development. The mean and standard deviation is sufficient to define a normal distribution. For triangular distributions the analyst must specify the expected value, and the range of possible values. The specification of these distributions will be discussed in more detail later in this paper Reduction of Cash Flows to Present Values The life cycle cost model used to prepare the estimates are based upon a cash flow diagram and the time value of money. Each cash flow is appropriately discounted to today s dollars so the present value can be determined Risk of Projected Cash Flows Risk can be described as the amount of uncertainty present in a decision made under unknown or uncertain conditions. For pavement investments, many factors cannot be predicted with certainty; therefore, there is a risk involved with every decision made. Since there are any number of design alternatives available that will provide an adequate pavement structure for the analysis period, the final decision is dependent upon the total life cycle cost of each acceptable design. These costs are based upon uncertain variables and, therefore, contain some amount of risk. This risk must be considered during the decision making process for a truly informed and competent decision to be made. Risk is defined as, the probability that an unfavorable event will occur. Estimates of benefits and costs are typically uncertain because of imprecision in both underlying data and modeling assumptions. Useful information in an analysis includes the key sources of uncertainty; expected value estimates of outcomes; the sensitivity of the results to the important sources of uncertainty; and the probability distributions of the project benefits and costs. (OMB Circular A-1994). As stated above the analyst should determine the distribution of the input variables to the life cycle cost model. These distributions can then be used to perform three types of analyses that, when used together, can present a picture of the risk of a selected alternative. These analyses are simulation, sensitivity, and scenario analyses. A simulation analysis is used to construct the distribution of the total present value of the life cycle cost for an alternative. Basically the life cycle of the project is simulated using a Monte Carlo technique by randomly selecting values from the distributions of all the cash flows expected to occur during the project life cycle and then producing a present value estimate of the life cycle cost. Therefore each simulation run represents a possible outcome of the life cycle for the alternative. These simulations are run many times and the present values tabulated to produce an estimate of the expected distribution of possible life cycle costs for the alternative.

6 The sensitivity analysis basically portrays the sensitivity of the present value of the total life cycle cost to changes in each of the input variables. The life cycle cost for all alternatives is calculated using a range of values for each input variable based upon a user selected range of that variable. The results are plotted to show the relative effects of these variables on the life cycle cost. Scenario analysis uses the input distributions of the cash flows to give worst, best and expected case life cycle costs for each alternative. These estimates show the possible range of life cycle costs for each alternative. Using the output of the three analyses the decision maker can compare each alternative and select the best based upon the agencies goals and other constraints. 2.0 VISUAL/ LCCA SOFTWARE The Visual/LCCA software developed by TRDI for the Montana Department of Transportation incorporates the analysis procedures developed above. A pavement project is a specific length of highway that is to be rehabilitated or reconstructed and each is analyzed for life cycle costs. An alternative is a specific design that meets requirements of the project. Each project can have any number of alternatives. The inputs to the LCCA program can be broken into four categories: 1) Pavement performance inputs - The performance of the pavement section determines when pavement improvement and maintenance costs will occur. This defines the time when cash outlays are required by the agency. To obtain these inputs the software uses the performance models from the network pavement management system and calculates expected life of each available treatment based upon a user defined threshold. The user may also associate maintenance activities with each treatment and specify when those activities will occur with respect to treatment application. 2) inputs - These define the magnitude of the costs that are incurred as each rehabilitation and/or maintenance action is performed on the section over the analysis period. Each treatment and maintenance activity have associated unit costs which are defined by the user. s used in the LCCA Program are: Treatment Initial Administration (Design, Planning etc.) Maintenance User Delay (Calculated based on traffic composition, length of construction etc.) 3) Project Inventory - These are parameters which define the project to be analyzed and do not depend upon the alternative being considered. The project inventory includes the following: Length

7 Width Traffic Level Traffic Composition (%Trucks etc.) Description Discount Rate distribution Traffic Growth rate Of the input parameters, the following have been identified as uncertain variables, therefore distributions instead of point values may be assigned to them in the life cycle cost analysis. The distribution of each variable may be specified by the user. 1) Discount rate. 2) Traffic growth rate 3) Treatment life, the number of years it takes for a pavement to deteriorate to the minimum acceptable level after a treatment is applied. 4) Unit cost of treatments (or material and labor cost) 5) User delay cost (reflected by changes in traffic) For this version of the LCCA program, two distributions for uncertain variables may be used: (1) triangular distribution (2) normal distribution. For triangular distribution, three variables: minimum value (a), most likely value(b), and maximum value (c) need to be specified, while for normal distribution the mean (µ) and the variance (σ 2 ) need to be defined. The user may elect to use either form of the distributions for simulation analysis. 2.1 Use of Input Distributions for Simulation Analysis The simulation analysis randomly calculates a distribution of the life cycle cost as discussed earlier. During each run of the simulation all uncertain variables are randomly selected from their respective input distributions. Then these values are used to calculate a life cycle cost. This process is repeated numerous times and the results tabulated to give the expected distribution of the total life cycle cost of the alternative Simulating a Triangular Distribution To calculate a random variable X from a triangular distribution the following is used. Figure 1 shows the probability distribution function (pdf) of triangular distribution its equation is given by: 2 (x-a) / (b-a) (c-a) (a x b) f(x) = 2 (c-x) / (c-a) (c-b) (b x c) (Eq. 1) 0 otherwise.

8 Figure 1 - A triangular distribution The cumulative distribution function (cdf) for this distribution is given by: 0 (x a) (x-a) 2 / (b-a) (c-a) (a x b) CDF(x) = 1 - (c-x) 2 / (c-a) (c-b) (b x c) (Eq.2) 1 (x c ) The CDF for a triangular distribution is a closed form, meaning that it exists for a finite range of x. If we generate a random number r that is between 0 and 1 and let r be equal to the cumulative probability of a random variable x that has a triangular distribution then the following is true: (x-a) 2 / (b-a) (c-a) (a x b) r = 1 - (c-x) 2 / (c-a) (c-b) (b x c) (Eq. 3) solving for x, we get a random variable generator if r is a random number between 0 and 1: a + [(b-a) (c-a) r ] 1/2 ( 0 r (b-a) / (c-a) ) x = c - [(c-b) (c-a) (1 - r) ] 1/2 ( (b-a) / (c-a) r 1 ) (Eq. 4) Simulating a Normal Distribution For a normal distribution with mean µ and variance σ 2, the commonly used method to generate a normally distributed variable X is the direct method. This method produces exact normal random variates and it is easy to apply and execute. First we generate two uniformly distributed random numbers, r 1 and r 2, that are between zero and one. These are transformed into a normal random variate, Z, with mean 0 and variance 1, using the following: Z = (-2 ln r 1 ) 1/2 sin2πr 2 (Eq. 5)

9 Those standardized normal variates are then transformed into a normal variate X with a distribution having mean µ and variance σ 2 using: X = µ + σz (Eq. 6) Simulation Analysis Procedure The life cycle cost model used to prepare the estimates is based upon a cash flow diagram and the time value of money. To compare life cycle costs for varying design alternatives it is necessary to discount all expected cash flows for each alternative to their present value so they may be compared using a common metric. This discounting uses the expected magnitude, time and discount rate for each cash flow in the life cycle of the design alternative. For situations with constant discount rates over the life cycle the following may be used to calculate the present value of an investment in year n: PV = FV ( 1 + ) d n where: (Eq. 7) PV is the present value FV is the future value, and d is the discount rate used over a period of n years (or discounting periods). This formula is valid only when the discount rate is constant for all years in the analysis. To make the model more general the following formula will be used so that the user may vary the discount rate according to a specified input distribution for each year of the analysis. where: (Eq. 8) PV = FV n ( 1 + d ) i i = 1 all values have been defined above. For each year of the analysis a randomly sampled discount rate d i is used in the formula. The effective discount rate over the analysis period is calculated by multiplying each years discount rate from the beginning of the analysis to the year of the cash flow. Note that this formula is equivalent to the previous formula when the discount rate is held constant over the analysis period. Using this formula it is then possible to reduce each estimated cash flow to its present value. Then the total life cycle cost is the sum of all the present values of expected cash flows over the analysis period minus the discounted salvage value of the pavement at the end of the analysis period. The salvage value is calculated using a straight line depreciation from the last treatment applied to the pavement.

10 2.2 Simulation Process There are four steps to the sampling simulation. This section will describe each step briefly to illustrate the simulation process Step 1 Sample Project Variables from Input Distributions The first step in the simulation process is to generate estimates for the discount rate and the traffic growth rate for each year of the analysis period. Note that these two variables are used for all alternatives being studied since the selection of the discount and traffic growth rates are independent of the alternative chosen Step 2 Construct Cash Flow Diagrams for Each Alternative Based upon defined alternatives the software will construct cash flow diagrams for each alternative. The process begins by sampling the design lives of each treatment based upon performance models or user input and determining when each major cash flow will occur. To determine the time at which a treatment n will be applied the following is used: T n 1 n 1 n = t i + ε i i = 1 i = 1 Where: (Eq. 9) T n is the time in years after the analysis period begins until the application of treatment n. t i is the expected life of each treatment preceding treatment n. This value is fixed at the expected life value input from either performance models or the user. ε i is the number of years the actual life of treatment i deviates from the expected life of treatment i. This value has a mean of zero and varies according to the triangular or normal distribution specified by the user during the definition of each treatment. n is the number of treatments from initial construction to treatment n. After the timings of the cash flows are found the costs are determined using the following: Cn = a jcj + a jε j Where: (Eq. 10) C n is the total cost for treatment n at time i (see previous formula). a j is the units for cost component j within treatment n. c j is the unit cost for cost component j for treatment n at time i. ε j is the deviation of each unit cost from its expected value. This value is determined from the input distribution for each unit cost. As with treatment life this has a mean of zero and either a standard deviation or range as specified by the user.

11 2.2.3 Step 3 Calculate Discounted Present Values for Each For each cost calculated in step two the discounted present values are computed and summed to arrive at the life cycle cost of each treatment Step 4 Repeat Simulation Process Until Stable Output Distribution is Found These three steps are repeated until the maximum number of iterations or the output distribution becomes stable within a specified tolerance. 3. EXAMPLE Table 1 shows two possible alternatives for a rehabilitation project in Montana. The project is 5.8 miles long with a pavement width of 32 feet including shoulders. Alternative 1 consists of two thin resurfacing treatments to maintain ride quality until a complete reconstruction can be done in year 8. The second alternative is a complete reconstruction for the initial treatment followed by a thick overlay after an expected life of 20 years. The cost of each treatment along with its associated maintenance activities is shown in Table 2. Table 1: Alternative Definitions Treatments Assigned Year Strategy 1 - Thin Overlay Strategy 2 - Reconstruction 0 Thin Resurfacing (Design life 4 years) Reconstruction (Design life 20 years) 4 Thin Resurfacing (design life years) 8 Reconstruction (design life years) Thick overlay (design life 15 years) 28 Thin Resurfacing (design life 4 years)

12 Table 2. Management System Total (Unit $/SY) Treatment Construction Administrative Maintenance User Delay ** Thin Resurfacing $1,200, ($11.76) $250, ($2.45) ---- $99, ($0.97) Reconstruction $2,400, ($23.52) $350, ($3.43) ---- $199, ($1.95) Crack Seal (Years 3,10,17) $11, ($0.11) $6, ($0.06) Seal and Cover (Years 7, 14) $40, ($0.39) $200, ($1.96) $11, ($0.11) Thick Overlay $1,800, $350, $132,689 ($17.63) ($3.43) Crack Seal (Years 3,10,17) $11, ($0.11) $6, ($0.06) Seal and Cover (Years 7, 14) $40, ($0.39) $200, ($1.96) $11, ($0.11) **Calculated using Pennsylvania DOT method (PennDOT 1994) 3.1 Uncertainty in treatment costs and design lives. Currently there is a lack of adequate cost history and performance history to estimate variability in the cost and design lives of the treatments for both strategies. standard deviations were estimated based on a maximum likely change of approximately 10%. Performance variability was estimated based upon the opinion of MDT staff engineers given the existing condition of the pavement project. Subjective opinion was used to arrive at the standard deviations for each element listed in Table 3 below. It is hoped that as more data is accumulated through the Pavement Management System and through other activities currently underway at the department that both costs and performance variability estimates can be made through the use of historical data. Table 3: Specification of variability for the example project Element Mean (µ) Standard Deviation (σ ) Thin Resurfacing Maximum Likely Value (µ+3σ) Minimum Likely Value (µ-3σ) Design Life 4 Years 0.33 Years 5 Years 3 Years Construction $1,200,000 $120,000 $1,560,000 $840,000 Administrative $250, $25,000 $325,000 $175,000

13 Table 3: Specification of variability for the example project (cont.) Element Mean (µ) Standard Deviation (σ ) Maximum Likely Value (µ+3σ) Minimum Likely Value (µ-3σ) User Delay $99, $10,000 $129,517 $69,517 Reconstruction Design Life 20 Years 1.66 Years 25 Years 15 Years Construction $2,400,000 $240,000 $3,120,000 $1,680,000 Administrative $350, $35,000 $455,000 $245,000 User Delay $199, $19,903 $179,130 $218,936 Thick Overlay Design Life 15 Years 0.66 Years 17 Years 13 Years Construction $1,800,000 $180,000 $2,340,000 $1,260,000 Administrative $350, $35,000 $455,000 $245,000 User Delay $132,689 $13,269 $172,495 $92, Analysis Results The results for the simulation and scenario analysis are summarized in Table 4. The results from the analysis show that the expected present value of alternative #2 is considerably lower than that of alternative 1. The expected distribution of the life cycle costs is also smaller as shown by the standard deviation. An example of the sensitivity analysis for discount rate is shown in Figure 3 and the sensitivity analysis graph for construction cost is shown in Figure 4. Finally a graph of the possible distributions of total life cycle costs for each alternative is shown in Figure 5. Table 4 - Simulation Analysis Results Alternative Mean Life St. Dev Best Worst Cycle 1 (Thin Resurface) $4,677,955 $302,302 $4,085,443 $5,270,467 2 (Reconstruct) $3,669,242 $168,943 $3,338,114 $4,000,370 Difference(1-2) $1,008,713 $133,359 From the scenario and simulation analysis results shown in table 4 we can see that alternative 2 has a lower expected net present value over the analysis period. More significant however is the fact that alternative 2 also has a smaller standard deviation as estimated during the analysis. This indicates that the alternative has significantly less risk.

14 Figure 2 - Simulation Analysis Results Figure 3 - Sensitivity Analysis of Discount Rate

15 As shown in this graph alternative 2 always has a smaller net present value over the life cycle regardless of the discount rate experienced. Overall the net present value of the project is affected the most by the discount rate. 4.0 Conclusions The risk based life cycle cost methodology and corresponding Visual/LCCA software developed for the Montana DOT is an effective tool for project level pavement management life cycle cost evaluations. The departments recent implementation of a comprehensive pavement management process and software system is collecting the types of data needed to support the required input distribution estimates to utilize the full power of this program. Regardless of the availability of historical data to estimate variations in costs and pavement performance the Visual/LCCA package is an excellent tool for inspecting the sensitivity of analysis results to critical variables such as discount rate and pavement performance. It allows for a variety of "what-if" types of analyses in an easy to use environment and provides clear results that can be used to identify risk in decision making. REFERENCES Haas, R., Hudson, W.R., and Zaniewski, J., Modern Pavement Management Krieger Publishing Company. Montana Department of Transportation, Pavement Management Section, Guidelines for Economic Analysis at the Project Level Office of Management and Budget, Guidelines and Discount Rates for Benefit - Analysis of Federal Programs, Circular No. A-94, Office of Management and Budget, Executive Office of the President, Washington, D.C. Commonwealth of Pennsylvania Department of Transportation, Bureau of Maintenance and Operations, 1996 Edition. Pavement Policy Manual, Pub. 242, Weston, J. F., and Brigham. E., Essentials of Managerial Finance, Los Angeles, Univ. of California, L.A. CA. KEYWORDS LCCA, Risk, Simulation, life cycle costing, sensitivity