Value of Bayesian Pre-Posterior Information for Enhancing Life-Cycle Cost Analysis of Highway Investments

Abstract

Assessing risk in life-cycle cost and benefit estimates of highway investments is recommended by major organizations such as the World Bank and the U.S. Federal Highway Administration. This challenging task needs methodological support. Mutually exclusive investment alternatives can differ in terms of the costs of construction, maintenance, rehabilitation, and end-of-life value. Due to many causal factors and the long life of highway infrastructure, these items cannot be estimated with certainty. To go beyond the study of the sources of uncertainty, a method is needed to check the economic feasibility of acquiring additional information for deeper insight. This paper reports on research on the value of Bayesian pre-posterior information for refining the life-cycle cost analysis of uncertain costs and benefits for evaluating highway investment alternatives. Example applications demonstrate how the Bayesian pre-posterior analysis can be applied to check the feasibility of obtaining new information for enhancing the life-cycle cost analysis of highway investments. The value of Bayesian pre-posterior information is illustrated for reducing risk. Also, depending upon the specifics of uncertain states, a change in the choice of the investment alternative for implementation can be investigated. The product of this research can potentially upgrade highway infrastructure planning and management practices.

1. Introduction

Many well-known funding, research, and infrastructure sustainability agencies are promoting life-cycle analysis as a basis for highway infrastructure investment decision-making. Also, they are recommending the study of risk in estimates of life-cycle costs and benefits. Examples are the World Bank Institute [1], the Federal Highway Administration (FHWA) [2,3], the Transportation Research Board (TRB) [4], and the Infrastructure Sustainability Institute (ISI) [5]. The importance of life-cycle analysis and treating risk in investment decisions can be inferred from observations of the sources of uncertainty in life-cycle analysis and the economic impacts of using uncertain estimates (

Table 1. Sources of uncertainty in life-cycle cost estimates and economic impacts.

Life-Cycle Cost Component	Sources of Uncertainty	Socio-Economic Impacts
Initial construction cost.	Cost overruns potentially caused by many factors, including incompletion and/or design errors, and quality of data issues.	As the dominant component of the life-cycle cost, the choice of the best alternative cannot be ensured. Adverse effect on financial resource allocation decisions.
Cost of maintenance and rehabilitation cycles.	Limitations of predictive models to forecast conditions under which service quality is delivered and the associated cost of corrections.	Unexpected magnitude of rehabilitation and corrective actions. Adverse effect on budgets in future periods.
User cost of congestion, safety, and vehicle operating cost.	Uncertain traffic forecasts due to data and methodology issues.	Economic impacts of high user costs to motorists, businesses, and society.
End-of-life value (a negative cost)	Predictive model limitations in producing end-of-life economic estimates.	Effect on life-cycle cost estimates. The choice of the best alternative cannot be ensured.

). Further information on sources of uncertainty is provided following Table 1.

Given that mainly public funds are invested in highway infrastructure and uncertain socio-economic and socio-technical factors can compromise well-intended decisions, the importance of treating risk in life-cycle estimates of costs and benefits is acknowledged by public agencies. For public–private partnership (PPS) infrastructure, this policy is also followed for the reasons noted above [2,3,4,5,6].

The initial construction cost of a highway accounts for the highest component of life-cycle costs due to many expensive items (e.g., land, materials, and labor). But the importance of other items cannot be overlooked. Given the long life of the infrastructure (30 or more years), following the initial construction, cycles of rehabilitation keep the quality of service acceptable to road users. The need for the rehabilitation of alternatives depends upon design factors, including trade-offs between the initial construction cost and the cost of rehabilitation cycles. To offer an acceptable level of service to road users, meet the safety and sustainability objectives, and ensure the fiscally responsible use of monetary resources during the life of the facility, the study of life-cycle costs and benefits is becoming almost mandatory.

Developing highway life-cycle cost estimates is a complex task due to many sources of uncertainty in every future cost component. Transportation researchers and those in professional practice are aware of uncertain cost overruns. They also recognize that the predictions of future values of variables noted in Table 1 are subject to uncertainties. Among other causes, the long time horizon of 30 or more years and uncertain future traffic contribute to uncertainty in predicting impacts.

Although advances have been made in cost-estimation methods, knowledge deficiencies remain in treating risk [7,8,9,10,11,12]. Because of the many potential causes of construction cost overrun, estimates may be uncertain. Cost overrun is defined as the difference between the actual cost and the planned/estimated budget. The actual cost is the total funds that the spending agency has paid for the construction, and the estimated budget is the money assigned to the project before the commencement of construction.

Avoiding cost overrun in initial construction continues to be a challenging research subject [13,14,15,16,17,18]. Although a contingency item is almost always included in construction cost estimates, cost dispute reports suggest that overrun frequently becomes necessary to complete the project [19]. The tracking of U.S. and Canadian disputed transportation infrastructure cost cases shows several causes of claims and disputes, including design-related issues (e.g., design information issued late, incomplete design, incorrect design) [20].

Other costs that are incurred during the long service life of the infrastructure also cannot be estimated with certainty. For the maintenance and rehabilitation parts of the life-cycle tasks, cost estimates are developed using predictive models that require refinement [21,22,23,24,25]. For the last part, the end-of-life value (a negative cost) estimate based on the serviceability condition of the infrastructure is difficult to predict [26]. Estimates of road user costs (i.e., costs of vehicle operation, congestion, and safety) are obtained using predictive models, which require refinement regarding the effect of uncertain future operating conditions (e.g., traffic volume) [2,27].

For a highway project, the benefits of an investment alternative are calculated as the reduction in road users plus the transportation agency costs of the “do-nothing” option attributable to the investment alternative under study. Given that all cost items cannot be predicted with certainty, it is necessary to treat the net benefits of highway infrastructure investments as stochastic.

This paper advances methods to treat risk in economic factors for highway projects. Specifically, the paper describes research on the role of Bayesian pre-posterior analysis in refining life-cycle cost and benefit estimates for use in the evaluation of highway investment alternatives. If social and economic factors are converted to their monetary equivalents, they can be added to the economic factors.

Following the problem definition, the limitations of the available probability-based methods are described. Next, the capability of the Bayesian method to overcome a methodological gap is noted. The theoretical foundation of this method is described, and examples demonstrate how the Bayesian pre-posterior analysis can be applied to check the feasibility of acquiring additional information for improving life-cycle analysis of highway investments. The ultimate result will be enhanced highway infrastructure planning and management.

2. Problem Definition

To promote the practice of life-cycle cost analysis and the illustration of the concept, the U.S. Federal Highway Administration (FHWA) provided detailed life-cycle cost estimates for design alternatives that are used as building blocks for a highway project [2]. A report by the Ontario Hot Mix Producers Association (OHMPA) funded by the Ontario Ministry of Transportation (MTO) contains additional data used in this research [28]. The OHMP report refers to the 1995 Ontario Provincial Auditor recommendation that the MTO should incorporate improved life-cycle costing procedures into design and construction decisions.

The life-cycle costs noted in the FHWA report are in the 1996 U.S. constant dollar. Although costs for various items occur in different years over a 35-year life, they were converted to present values (termed present worth in this paper) with the use of the 4% real (inflation-free) interest rate (also called the discount rate). The choice of a 4% real interest rate made by the FHWA researchers reflects the prevailing desired inflation-free rate of return. The formula used to convert future cost items into present worth is presented as Equation (1).

P = F/(1 + i)ⁿ

(1)

where

P is the present worth of the future cost item;

F is the future cost item that will occur in the future year n;

i is the interest rate;

n is the number of years.

For an appreciation of the cost estimates in the 2023 U.S. dollar, a multiplier of 1.942 is applied to convert 1996 estimates to the 2023 dollar. This multiplier is obtained from the consumer price index (CPI) for 1996 and 2023 [29]. The formula applied is shown in Equation (2).

Cost in 2023 dollar = Cost in 1996 dollar [(CPI for 2023)/(CPI for 1996)]

(2)

Based on design features and the corresponding costs noted above, three alternatives of a highway project (i.e., A1, A2, and A3) are defined, which differ in terms of initial cost, rehabilitation cycle cost, road user cost, and end-of-life value estimate. The alternatives have unique design/route location attributes.

To account for risk, cost overruns (CORs) are applied to transportation agency costs (i.e., initial construction, rehabilitation and maintenance, and end-of-life value (a negative cost)). The CORs are based on research reported by Alfasi [30] and Berechman and Chen [31]. The application of cost overrun results in three uncertain states of cost (S1, S2, and S3) for each investment alternative (called uncertain states of nature in decision theory terms). Since these states are uncertain, probabilities are applied to their occurrence. Considering that the predicted road user costs are uncertain, the same probabilities are applied to these estimates.

To explain the risk states further, the cost overruns are defined as follows:

• S1 Low risk of cost overrun (COR ≤ 1) (i.e., no cost overrun): 1.0 is used.
• S2 Medium risk of cost overrun (1 < COR ≤ 1.2): 1.2 is used.
• S3 High risk of cost overrun (COR > 1.2): 1.5 is used.

In the life-cycle risk analysis problem described in this paper, three mutually exclusive alternatives (i.e., A1, A2, and A3) for the highway transportation project are evaluated.

Table 2. Life-cycle costs (2023 USD).

	S1	S2	S3
Alternative A1
Construction cost	94.67	113.61	142.01
User cost	67.68	67.68	67.68
Rehab cost	27.96	33.56	41.95
End-of-life value	−2.43	−2.43	−2.43
Total (M USD)	187.88	212.42	249.21
Alternative A2
Construction cost	106.81	128.17	160.22
User cost	54.38	54.38	54.38
Rehab cost	27.19	32.63	40.78
End-of-life value	−5.34	−5.34	−5.34
Total (M USD)	183.04	209.84	250.04
Alternative A3
Construction cost	107.78	129.34	161.67
User cost	51.95	51.94	51.94
Rehab cost	30.72	36.87	46.08
End-of-life value	−5.34	−5.34	−5.34
Total (M USD)	185.11	212.81	254.35

and Figure 1 present the life-cycle costs of alternatives in the 2023 U.S. constant dollar. For each alternative, three cost estimates are shown that correspond to uncertain states S1, S2, S3.

As expected, the initial construction cost accounts for a high proportion of life-cycle cost. As noted in the FHWA report [2], the approximately equal maintenance costs are low and will not affect the relative position of alternatives; these are not shown. Road user costs differ for the three alternatives, but for each alternative, their incidence is not affected by the magnitude of transportation agency cost overruns. However, these cannot be predicted with certainty and are therefore considered stochastic. The end-of-life values (negative costs) are very small.

The present worth of life-cycle costs of alternatives that correspond to cost states are presented in

Table 3. Life-cycle present worth (PW) of costs and net present worth (M USD) (2023 USD).

Alternative	S1 PW (Costs)	S2 PW (Costs)	S3 PW (Costs)	S1 NPW	S2 NPW	S3 NPW
A1	187.88	212.42	249.21	66.52	41.98	5.19
A2	183.04	209.84	250.04	71.36	44.56	4.36
A3	185.11	212.81	254.35	69.29	41.59	0.05

. Also shown are the net present worth (NPW) of alternatives under cost states. As noted above, in computing the present worth of costs, the FHWA researchers applied a 4% (real) interest rate and a 35-year analysis period. Alternative A1 has three rehabilitations and Alternatives A2 and A3 have two rehabilitations.

The net present worth (NPW) for each alternative-stochastic state combination shown in Table 3 is computed as follows: (PW of cost of “do-nothing” − PW of cost of the alternative). The life-cycle cost for the “do-nothing” option is USD 254.402M (in present worth). This consists of agency and user costs of the existing route, which are high. The NPWs are shown in Figure 2 to illustrate the effect of unfavorable conditions, especially under S3.

Given the availability of the above information on the example highway project, the life-cycle risk analysis problem can now be defined as follows. The NPW for each alternative shown in Table 3 depends on the unknown cost states (i.e., S1, S2, and S3). An examination of these NPWs suggests that under S1, A2 is the choice; under S2, A2 is the choice; and under S3, A1 is the choice. Since the cost states (i.e., S1, S2, S3) are stochastic, the probability of the occurrence of each is needed for the calculation of the expected NPW for each alternative. The alternative with the highest expected NPW will be the choice. The question is how to assign probabilities. In the following section, available methods are reviewed and illustrated regarding their capability to answer this question.

Another question related to life-cycle risk analysis is as follows. The cost states and the probability of their occurrence are defined by the analyst based on the available information. If it is possible to search for additional information through such means as a market survey, simulations, retaining a specialist/consultant, etc., the analyst can potentially replace previous probabilities (called prior probabilities in decision theory terminology) with revised probabilities (called posterior probabilities). There is a need for a theory and an associated method to enable the analyst to compute posterior probabilities by considering the reliability of the additional information. In this paper, the application of the Bayesian statistical theory is described for this purpose.

The decision to acquire additional information on uncertain factors requires a method to assess the economic feasibility of this action. As noted earlier, the focus of this research is to illustrate how to quantify the value of Bayesian pre-posterior information so that the economic feasibility of additional information can be assessed before such an action is undertaken.

3. Existing Risk Analysis Methods and Their Limitations

The variables of the decision model as applied in this research are

• Alternatives (A1, A2, A3);
• The cost states (S1, S2, S3);
• The gain (i.e., NPW) for each A and S combination.

The cost states and probabilities are as follows:

• S1 Low risk of cost overrun (Multiplier 1.0), used with probability P1;
• S2 Medium risk of cost overrun (Multiplier = 1.2), used with probability P2;
• S3 High risk of cost overrun (Multiplier = 1.5), used with probability P3.

The basic method for the risk analysis of estimated NPWs corresponding to A and S combinations is to compute probability-weighted expected values:

$$EXP\left({NPW}_i\right)={\sum }_{j=1}^n [{NPW}_{ij}\times P_j]$$

(3)

where

EXP (${NPW}_i$) = the expected net present worth of alternative $A_i$; $i$ = 1, 2,…, m;

${NPW}_{ij}$ = the NPW of alternative $A_i$, under uncertain cost state $S_j$; $j$ = 1, 2,…, n;

$P_j$ = probability of state of nature (i.e., cost state) $S_j$.

Several existing methods can be applied to evaluate mutually exclusive alternatives under risk. The expected value results obtained from the four methods are shown in

Table 4. Expected NPW results of risk analysis methods (2023 USD).

Alternatives	Expected Discrete Equal Probability	Monte Carlo Uniform Probability	Monte Carlo Triangular Probability	Monte Carlo Normal Probability
A1	19.49	19.98	20.32	19.51
A2	20.63	20.95	20.35	20.65
A3	19.03	19.95	20.1	19.05

and Figure 3. The first method is based on the use of discrete probabilities to states of nature S. In this method, the analyst applies subjective probabilities. Depending upon the ata, sensitivity analysis using different values of probabilities may lead to changes in the best alternative answer. The results of equal discrete probabilities assigned to states of cost overrun are presented.

Next, the results of three versions of the Monte Carlo simulation method are shown in Table 4 and Figure 3. Commonly used continuous probability distribution functions used in Monte Carlo simulation are uniform, triangular, and normal probability distribution functions. The formulas and required inputs for these functions are noted.

• Uniform probability distribution function:

f(x) = 1/(b − a) for x ϵ |a,b| and 0 otherwise

(4)

Mean: ½(a + b)

(5)

Median: ½(a + b)

(6)

Mode: any value in (a,b)

(7)

where x is the random variable, a is the minimum value, and b is the maximum value. The inputs are a and b.

• The triangular probability distribution function and the inputs are as follows.

$$f\left(x\right)={\displaystyle \frac{2\left(x-a\right)}{\left(b -a\right)\left(c -a\right)}}\mathrm{for} a \le x \le c$$

(8)

$$f\left(x\right)={\displaystyle \frac{2\left(b-x\right)}{\left(b-a\right)(b-c)}} \mathrm{for} c \le x \le b$$

(9)

$$f\left(x\right)=0 \mathrm{for} x<a \mathrm{and} x>b$$

(10)

where

$$c\in \left|a,b\right| \mathrm{i}\mathrm{s}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}$$

(11)

The mean is

$${\displaystyle \frac{1}{3\left(a+b+c\right)}}$$

(12)

The lower limit a, the upper limit b, and the mode c (i.e., the highest frequency value) are the inputs.

• Normal probability distribution function and the inputs are shown below.

$$f(x)={\displaystyle \frac{1}{\sigma \sqrt{2\pi } }} e^{-{\displaystyle \frac{1}{2}} {( {\displaystyle \frac{x-\mu }{\sigma }} )}^2}$$

(13)

where x is the random variable, μ is the mean, and σ is the standard deviation. The mean value and the standard deviation are the inputs.

A description of the Monte Carlo simulation method and the mathematical formulas for the above probability distributions are reported in references [32,33,34]. The cumulative distribution function corresponding to each probability function is used by the Monte Carlo simulation method to randomly sample the probability distribution.

In the condition of minimal data availability for risk analysis, the uniform and triangular distributions are used. The uniform distribution represents an equal likelihood for all possible outcomes of a random variable within the analyst-specified range to occur. In scientific terms, this probability distribution is considered the maximum entropy probability function for a stochastic variable [32]. Although the uniform probability distribution is used for simulating the values of variables, random-number-based sampling produces high probabilities for values in the central part of the range.

As noted above, the continuous triangular probability distribution function is defined by three values: the minimum value, the maximum value, and the peak (i.e., the mode or most likely) value. Due to the following characteristics, this probability distribution function has been widely applied. In real-life analyses under uncertainty, the analyst is likely to estimate the range (i.e., maximum and minimum values), and the most frequent outcome. The analyst may be able to set these without knowing the mean and the standard deviation of the values of the variables of interest, which are difficult to obtain. Also, this function enables the analyst to avoid assumed extreme values due to definite upper and lower limits. Another favorable feature of this function is the treatment of skewed probability distributions [32,33].

In risk analysis, the continuous normal distribution is commonly used in situations when the mean and St. deviation can be estimated.

The results presented in Table 4 and Figure 3 based on the example highway data show that Alternative A2 has the highest expected NPW. In the discrete probability case, equal probabilities are assigned to all states. As can be observed, the results of the equal discrete probabilities are not identical to the Monte Carlo simulation results for the uniform probability case. As noted above, in the Monte Carlo method, the probability distributions are sampled randomly, resulting in higher NPWs for values in the central part of the range. But in the discrete probability application case, probabilities are applied directly.

If the analyst is curious about a change in the alternative with the highest NPW or the magnitude of the probability-weighted expected NPW, the discrete probabilities can be altered. In the example Monte Carlo simulation results shown in Table 4 and Figure 3, the mode of the triangular probability distribution is set at the middle of the range of NPW values. The mode value can be altered to observe changes in the relative position of alternatives and the expected NPW values. In the Monte Carlo simulation with the normal probability distribution function, the mean value is the midpoint of the range of NPW values, and an assumed St. deviation is applied. A sensitivity analysis can be carried out by changing the St. deviation value.

Although the methods illustrated above can be used to identify the relative position of competing alternatives in terms of expected NPW, there is little indication about how to investigate the economic feasibility of acquiring additional information to lower the risk under highly unfavorable states. Even if additional information acquisition is under consideration, existing methods cannot be of assistance regarding how much can be spent on such a study from an economic feasibility perspective.

4. Addressing Methodological Gaps with Bayesian Statistical Decision Method

The methodological gap for the life-cycle risk analysis of highway investments can be addressed by the Bayesian pre-posterior analysis method, which is a part of applied statistical decision theory [35]. The principles of statistical decision theory are “prescriptive” and do not require calibration and validation. These enable a decision-maker (or a decision system) to identify the optimal course of action in situations when the outcomes are not known with certainty.

Decision-making under uncertainty is a specialized subject that requires the study of probabilistic states of nature and gains/payoffs that are predicted to occur for the combinations of applicable actions and states of nature. Bayesian analysis is a statistical approach that allows one to use prior information and offers the ability to update probabilities using new information. These probabilities have important roles in risk analysis.

The field of Bayesian decision theory as a part of the broader statistical decision theory has enabled various disciplines to model their problems and obtain logical answers. Applications have been reported in the business, engineering, and medical fields [36,37,38,39,40,41,42,43,44,45,46]. If a decision is to be made under uncertainty and there is the opportunity to learn from new observations to modify probabilities of the uncertain phenomenon, Bayesian theory can assist in modeling the problem. The applications of this theory to transportation planning and related subjects (i.e., telecommuting decisions and risk analysis of intelligent transportation system investments) are described in references [36,37,38,39,40,41].

4.1. Role of Posterior Analysis

Equation (3) defined earlier enables the analyst to compute the expected NPWij of an alternative Ai using the probabilities Pj of unknown states Sj. This basic step in risk analysis is intended to compute probability-weighted expected values of NPW for each alternative. The resulting expected NPW in essence is the prior analysis part of the Bayesian method, which does not include a role for new information for revising probabilities.

Raiffa and Schlaifer [35] describe the basic variables of the Bayesian pre-posterior analysis method as follows. An alternative is to be selected under uncertainty, defined by a set of probabilistic states of nature. That is, probabilities must be assigned to each state in the set of states of nature. A set of alternative information acquisition means (termed experiments in the decision theory terminology) e₀, e₁, …, are available, and the decision-maker may elect to use one experiment from the set for the purpose of obtaining more information about the actual state of nature prior to the selection of an action (i.e., an investment alternative in this research). These include e₀, which represents no new information acquisition. The set of outcomes or results of the information acquisition experiment are r₀, r₁ … The result r₀ applies to e_o.

For use in the posterior analysis, the NPWs are mainly called gains (Gs) from here on. The G (e, r, A, S) represents the decision maker’s preferences for all e, r, A, S combinations. Actions are taken in the following sequence. An information acquisition means e is selected, a result r is observed, a particular A is selected, and finally, a particular state of nature, S, occurs. The space of all possible combinations of actions and events is (e, r, A, S). A single-valued gain function G(e, r, A, S) is defined, which, in accordance with utility theory, consists of the gains G(e, r) and G(A, S). In this research, these gains are in dollars.

To solve the Bayesian decision problem, the following probability distribution functions are required. The prior probability P′(S) for each state of nature is required before observing the outcome r of the additional information acquisition activity (experiment) e. The conditional measure P(r|S, e) is to be assigned, which represents the probability that the outcome r will be observed if the experiment e is performed, and S is the true state of nature. This implies that the decision-maker should define the reliability of each possible information outcome r for each information acquisition activity e in predicting the true state of nature S.

The marginal measure P(r|e) is computed using the following equation:

P(r|e) = ΣP′(S) P(r|S, e)

(14)

The posterior probability P″(S|r, e) can now be calculated using the Bayes Theorem:

P″(S|r, e) = [P(r|S, e)P′(S)]/(P(r|e)

(15)

This equation reflects the Bayesian philosophy that each e can be characterized by a conditional probability distribution P(r|S, e) (a reliability indicator). The relationship between the prior and posterior distribution is defined by the Bayes Theorem (Equation (15)).

4.2. Solving the Decision Problem

The decision tree for comparing the posterior and prior analyses is shown in Figure 4. The sequences of (e, r, A, S) represent the decision problem. The decision variables are e and A, and the random variables are r and S. It is solved by moving from right to left. The value of a sequence of actions is represented by the gain G (e, r, A, S). The prior branch is solved using Equation (3). The equations for solving the posterior branch and the comparison of results of both branches are presented in the following section. For each r, the best alternative can be found, and next, for an e, the best alternative can be identified. To find the optimal e, both branches of the decision tree are analyzed. A comparison of the results of posterior and prior branches leads to the quantification of the value of the additional information (i.e., the value of pre-posterior information).

5. Pre-Posterior Analysis

The mode of analysis concerned with the evaluation of alternative courses of action to determine the most appropriate information acquisition alternative e is known as pre-posterior analysis. In this research, pre-posterior analysis is used to establish if it is desirable to obtain additional information on the uncertain cost states S and the amount of money that can be spent for this purpose. Also, the optimal investment alternative can be identified by using the expected gain (i.e., expected NPW) result.

The pre-posterior analysis steps are as follows [35]. The likelihood of different states of nature, S, is expressed in the form of prior probability distribution P′(S). For each experiment e, the conditional probability characteristics P(r|S, e) are defined. The marginal measures P(r|e) for each experiment are computed using Equation (14). For the null experiment e₀, the marginal probability is equal to one (i.e., P(r₀|e₀) = 1.0). The posterior probability P″(S|r, e) is computed for each combination of S and r (Equation (15)). For each combination of e, r, A, and S, its gain is found: G (e, r, A, S).

The expected gain for each alternative A in the posterior branch, for each (e, r) combination, is as follows:

G*(A, r, e) = ∑_S P″(S|r, e) G(e, r, A, S)

(16)

However, for the prior branch, where no new information is acquired,

G*(A, r₀, e₀) = ∑_sP′ (S) G(e₀, r₀, A, S)

(17)

For each (e, r) combination, the optimal alternative is determined, and its associated gain is noted:

G*(r, e) = Max_AG*(A, r, e)

(18)

For each information acquisition activity (experiment) e, the expected gain can be computed:

G*(e) = ∑_rP(r|e) G*(r, e)

(19)

The optimal experiment e* is that e for which G*(e) is a maximum. That is,

G*(e*) = Max_eG*(e)

(20)

6. Value of Information

The value of information part of the Bayesian decision theory enables the assessment of how beneficial new information could be before actually obtaining it. The application of this information helps decision-makers to determine if acquiring that information is worth the cost. It is a unique tool used to evaluate the potential benefits of acquiring new information before deciding. In a decision analysis context, when variables are quantified in monetary terms, as is the case in this research, this tool helps determine if the cost of obtaining that information is justified by the expected monetary improvement in the decision outcome.

To clarify further, using the Bayesian method, the decision-maker can assess the economic feasibility of obtaining additional information on sources of uncertainty. That is, the increase in gain can be found so that the decision-maker can decide how much can be invested in additional information acquisition. If the cost of obtaining additional information is known, the feasibility can be established.

The computational steps are noted next.

(1) From the posterior branch, for e and each r, find Max_AG*(A,r,e) (Equation (18)). Call it Ar.

(2) For e₀ in the prior branch, find MaxG*(A). Call it A′.

(3) For each r, find (Ar-A′). This is Vt(e,r), the terminal value for the (e,r) combination.

Vt(e,r) = Max_AG*(A,r,e) − MaxG*(A)

(21)

The subscript t represents terminal values.

(4) The expected value of additional information is computed as follows.

Vt*(e) = ∑_r[P(r|e)(Ar-A′)] = ∑_rP(r|e)V_t(e,r)

(22)

The superscript * represents the expected (i.e., probability-weighted) value.

The expected value of additional information can be interpreted as the amount of money that can be spent on acquiring the additional information for the purpose of reducing risk (i.e., to reduce the risk of not obtaining the Max. gain). In this research, this is the only answer required, and there is no need to compute the expected net gain of information acquisition v*(e) by considering the expected cost of acquiring additional information. However, if the analyst is interested in this item, the equation is shown below.

v*(e) = V_t*(e) − c_s*(e)

(23)

where

c_s*(e) is the expected cost of acquiring the additional information.

Examples of Vt*(e) applications are presented in the following sections.

7. Application of the Bayesian Pre-Posterior Decision Model

7.1. Methodological Framework

The process for applying the pre-posterior model shown in Figure 5 consists of the inputs and the computations required for obtaining the results on the value of information and the alternative with the highest expected gain. The inputs are the prior probabilities, the conditional probabilities, and the gain matrix. The equations required for computational steps are noted in Figure 5. Six example applications of the pre-posterior method presented next illustrate how to improve life-cycle risk analysis.

7.2. Example 1

The inputs are the prior probabilities P′(S), the conditional probabilities P(r|S, e), and the gain matrix G(A,S). The NPWs shown in Table 3 form the gain matrix. The prior probabilities for the occurrence of S1, S2, and S3 are based on Cauchey’s probability distribution function calibrated with cost overrun data from British Columbia (Canada) [31]. The Cauchy distribution is characterized by three parameters (location, scale, and shape). The location parameter defines the mean value, and the scale parameter results in a shorter or taller graph. A smaller scale parameter results in a taller and thinner curve [34]. The prior probabilities are P′(S1) = 0.42, P′(S2) = 0.55, P′(S3) = 0.03. The conditional probabilities are set on the basis that if additional information is to be considered, it should be reasonably reliable (i.e., P(r|s, e) = 0.7).

Based on methodological steps (Figure 5), Figure 6 shows inputs, intermediate steps, and outputs. Computations are carried out for the following factors:

• Marginal probabilities P(r|e) and posterior probabilities P″(S|r, e).
• Maximum gain obtainable from each branch of analysis.
• Value of information by using (Ar-A′) and marginal probabilities.

An examination of the gain (i.e., NPW) matrix shows that alternative A2 is the choice under S1 and S2, but A1 is the choice under S3 (the high-cost state). Completion of computations shows that Alternative A2 should be the choice, and risk cannot be reduced by acquiring additional information (i.e., Vt* = 0). This is the same answer as obtained with Equation (1). The expected gain (i.e., NPW(A2)) = USD 54.61M(2023 USD).

7.3. Example 2

In this case, as shown in Figure 7, the stochastic states have equal prior probabilities (i.e., P(S1) = P(S2) = P(S3) = 0.333) and the conditional probabilities P(r|S, e) are the same as for Example 1. The results show that the value of information is USD 0.040M and A2 is the choice. The expected gain (A2) = USD 40.053 (2023 USD).

7.4. Example 3

In this example, prior probabilities are the same as in Example 2 (i.e., P(S1) = P(S2) = P(S3) = 0.333). But the conditional probabilities are increased from P(r|S) = 0.7 to P(r|S = 0.8) with the understanding that the information acquisition is of higher reliability than in Example 2. Figure 8 shows the inputs and results. In this case the Vt*(e) = USD 0.095M, and Alternative A2 with an exp. gain of USD 40.108M is the choice. Compared to Example 2, both the value of information and the expected gain have improved. Alternative 2 remains the choice.

7.5. Examples 4, 5, and 6

The analyst is interested in the analysis of a very high probability assigned to S3 (i.e., 0.9), which represents a high risk. Also, it is of interest to study this high-risk situation using additional information acquisition methods that vary in reliability (i.e., P(r|s, e) from 0.5 to 0.8. In Example 4 (Figure 9), the conditional probability is P(r|S,e) = 0.5. The results show a modest Vt*(e) = USD 0.004M (i.e., USD 4000). Under r2 and r3, alternative A1 is the choice, but in the case of r1 (that corresponds to low-risk state S1), Alternative A2 is the choice. Based on the application of marginal probabilities, A1 is the choice (with an expected gain of USD 10.1M).

In Example 5 (Figure 10), a higher reliability method with P(r|S,e) = 0.7 is used to obtain additional information. The Vt*(e) rises to USD 0.121M, and Alternative A1 is the choice (with an expected gain of USD 10.217M). Finally, in Example 6 (Figure 11), the use of a much higher reliability method, with P(r|s,e) of 0.8, results in a value of information equal to USD0.197M, and Alternative A1 is the choice (with an expected gain of USD 10.293M). On balance, based on the expected gain, A1 is the best alternative in high-risk cases.

An explanation of the change in the preferred alternative from A2 (in example cases 1 to 3) to A1 in the high-risk cases 4 to 6 is as follows. The NPWs (i.e., gains) presented in Table 3 indicate that A2 should be selected under states S1 and S2. But if S3 (the highest cost state) becomes true, A1 is the preferred alternative. The assignment of a very high probability to S3 (i.e., P(S3) = 0.9) causes A1 to be the choice.

8. Value of Information Comparisons

Figure 12 illustrates the value of information comparisons for example cases 1 to 6. In Example 1, the probability of high risk is almost zero, and therefore, there is little need for additional information. This is the only case with Vt*(e) equal to zero. On the opposite end of the scale, Example 6 represents a condition with a very high probability of the occurrence of the high-risk state S3, and therefore, the need to probe further is elevated. Coupled with the high need, a relatively high-reliability method can be used to obtain additional information. It is logical that Example 6 has the highest value of information.

In Examples 2 and 3, equal probabilities of uncertain states suggest a situation where the analyst has no basis to believe which future state is likely to affect decision-making. Therefore, it is logical that positive Vt*(e) answers are obtained from the pre-posterior analysis. Also, in relative terms, it is reasonable to obtain a higher value of information for Example 3 due to the use of a more reliable method compared to Example 2.

Example 4 represents a situation where the probability of the high-risk state S3 is very high, and therefore, there is a need for further information acquisition. But the use of a low-reliability method results in a low value of information. In Examples 5 and 6, the value of information rises due to the application of higher-reliability methods.

In Examples 1 to 3, investment alternative A2 is the choice due to low and moderate probabilities assigned to S3 (the high-risk state). On the other hand, in Examples 4 to 6, the very high probability of high-risk state S3 results in expected values that make alternative A1 a more reasonable choice.

Discussion

According to the scope of this study, methods are advanced to treat risk in economic factors for highway investment projects. Specifically, the paper describes research on the role of Bayesian pre-posterior analysis in enhancing life-cycle cost and benefit estimates for use in the evaluation of highway investment alternatives. The developed methods can be adapted to include social and economic factors if they are converted to their monetary equivalents and are added to economic factors (e.g., monetary benefit of emission reduction). The developed methods can potentially be applied to investments in other transportation facilities and systems. As for the limitations of the Bayesian methodology, the analyst requires a well-researched gain (i.e., NPW) matrix and reasonable estimates of conditional probabilities. These may not be readily available to a potential user.

In the problem definition part, methodological gaps are identified for addressing risk in life-cycle estimates of costs and benefits. The application of existing risk analysis methods highlights their limitations regarding the necessary information needed for highway investment decision-making under uncertainty. The Bayesian decision-theoretic method is adapted to fill the methodological gap for assessing the feasibility of additional information for use in risk reduction and checking the relative position of investment alternatives.

Following the explanation of variables and formulas, the example cases illustrate the value of Bayesian pre-posterior information for enhancing the life-cycle analysis of highway investments to avoid the risk of not obtain the maximum NPW and to check on the choice of the best alternative for implementation. As the results of the example cases show, the decision-maker is guided on how much can be spent on additional information for refining the life-cycle estimates.

9. Conclusions

In adapting the Bayesian approach to highway investment decision-making under uncertainty, three requirements are met. The first two are the life-cycle analysis of investments and accounting for risk in estimates of economic factors. The third requirement is to use the Bayesian pre-posterior information as an aid to decision-making following a check on its economic feasibility.

In professional practice, the first two requirements are now considered almost mandatory. The need for the third requirement arises due to a gap in knowledge regarding the beneficial role of additional information to reduce risk.

The research reported in this paper provides the theoretical foundation of the method for assessing the value of Bayesian pre-posterior information and illustrates how to enhance the life-cycle analysis of highway investments under uncertainty. The proposed method for decision analysis illustrated with six example cases evaluates the anticipated reduction in risk using the potential role of additional information on uncertain variables.

The Bayesian approach is an advanced form of the decision-theoretic method due to its ability to treat stochastic states of nature and offers the decision-maker the opportunity to find out how much can be spent on additional information in support of decision-making. Also, depending on the specifics of uncertain states, the availability of additional information may change the choice of investment alternative for implementation.

The product of this research can potentially enhance highway infrastructure planning and management. The developed methods can be adapted for application to other transportation infrastructure.