Design of a Long Term Pavement Monitoring System for the Canadian Strategic Highway Research Program

Section 3

CAPITALIZING ON THE BENEFITS OF LONG TERM PAVEMENT PERFORMANCE MONITORING: INJECTION OF INTERIM AND FINAL RESULTS INTO PAVEMENT MANAGEMENT DECISION SYSTEMS

The ultimate objective of the C­LTPP program is to allow people to make better pavement design and management decisions. Pavement management decisions today are made in one of three ways:

· Judgment unaided by formal pavement management techniques.

· Techniques that do not systematically recognize the uncertainties in the mechanism of pavement deterioration or in the environmental or traffic variables. We term these techniques deterministic.

· Techniques that intrinsically recognize uncertainties in the mechanism of pavement deterioration and/or in the environmental or traffic variables. We term these techniques Markovian or semi­Markovian probabilistic techniques.

Rahut (1988) recognizes the need for predictive modeling and recognizes the two basic quantitative models described here. We have attempted to design an approach here so that the two quantitative types of models as well as the subjective procedures can accept input from the C­LTPP program from the day the program is initiated. The key is to use the continuously updated Bayesian statistical approach to process the data obtained from the C­LTPP program as it is produced. The classical statistical approach discussed at length in the SHRP memoranda and embedded in the three statistical software systems reviewed by Rahut (1988) for the SHRP are incapable of supporting interim decision making in the way that the Bayesian approach we propose.

This section will give only the most summary attention to the first type of pavement management decisions: judgmental pavement management decisions. In brief, the results of the study as embodied in the current estimate of the pavement deterioration function f(x,b) and its overall implications will assuredly find their way into the heads of myriad pavement engineers, managers, and students. What such people actually do with that information is both unpredictable and uncontrollable. More significantly, it is unmeasurable. As such, the benefits of the C­LTPP program as embodied in different and better pavement management decisions will be rather invisible to the funding agencies and to users of the pavements if the only applications of the C­LTPP program are subjective and judgmental. In our view, the measurable successes of the C­LTPP program will only be evident when

· they are embedded in explicit, quantitative pavement management systems.

· those explicit, quantitative pavement management systems are capable of showing what practices have changed as a result of the C­LTPP program and how much such changes save highway users.

In brief, if the C­LTPP program does not cause pavement management decisions to be changed from what they would otherwise be, then its value is zero. The Strategic Highway Research Program (2)

"...data of sufficient quality and completeness must be established in the database before the calibration of existing design equations can begin..."

The very essence of this statement is that the long term pavement performance data will allow pavements to be designed and maintained differently than they otherwise would, i.e., pavement design and maintenance decisions will be altered from what they would otherwise be in the absence of the program.

We also believe that pavement design and management decisions are sufficiently difficult and counterintuitive that they can no longer be made subjectively. What is the opportunity cost of a pavement budget dollar spent incorrectly? How much damage will be done to the pavement system as a whole if one pavement dollar is misapplied? In the past when pavement management dollars were abundant relative to need, this question was not so important. Rules of thumb, heuristics, and judgmental operating procedures developed in the past tacitly internalized the low opportunity cost of mistakes and the relative "newness" of much of the highway system. There was enough money when the highway system was relatively new to simply "fix it when it's broke." Nowadays and in the future, there is not enough money to "fix it when it's broke." On the contrary, the highway system is old and aging, and there is not enough money to rehabilitate it as quickly and completely as users might like. Tradeoffs have to be made and priorities set. Opportunity costs have to be clearly understood.

Today and into the future as overall pavement maintenance requirements escalate relative to available pavement management budget, the question of opportunity cost of Pavement dollars misapplied will increasingly lie at the heart of the pavement management issue. One dollar misapplied might induce five, ten, or twenty dollars' worth of cost elsewhere in the pavement system, and the cost might escalate over time as needed maintenance or rehabilitation is foregone for lack of budget. Today's pavement designers and managers are truly trustees of scarce public dollars. As trustees, explicit quantitative pavement management tools are needed to quantify the opportunity cost of budget dollars misapplied and minimize the aggregate opportunity cost over the entire system. This can only be done using a quantitative technique. The remainder of this section outlines the two generic types of quantitative techniques in use today for pavement management and shows how the emerging results of the C­LTPP program can using our Bayesian technique be embedded in those pavement management approaches.

Readers might wonder why we have devoted an entire section of this report to articulating pavement management systems. The answer lies in the realization that pavement monitoring must be explicitly tied to pavement performance prediction and that pavement performance prediction must be explicitly tied to pavement design and maintenance decisions. As Lewis Carroll wrote in Alice in Wonderland: "How will we know when we get there if we don't know where we're going?" By analogy, how will we know what to monitor and how to monitor it if we don't know how it will affect pavement decisions? This section is devoted to designing pavement decision making systems that can specifically exploit the regression and/or Bayesian results derived in the previous section.

3.1 USE OF INTERIM RESULTS FROM PAVEMENT MONITORING BY DETERMINISTIC PAVEMENT MANAGEMENT SYSTEMS

The deterministic pavement management approach takes as elemental the structural portion of the pavement deterioration function y = f(x,b) derived using statistical methods.(3) We reiterate that the independent variables x include variables not under the control of the highway department (or anyone else) such as environmental variables and traffic variables as well as decision variables that are under the control of the highway department (i.e., maintenance procedures used). We will term the variables not under the direct control of the highway department "external variables" and denote them z. We will term the variables that are under the direct control of the highway department "control variables" and denote them u. The independent variables x whose effect was inferred during the statistical analysis are comprised collectively of z and u. Using this notation, we can write the structural model determined from our recommended Bayesian approach y = f(u,z,b).

It is convenient to categorize the highway system into a number of distinct increments or "sections," the sections being indexed 1,2,...,n. Typically the segments correspond to mutually exclusive, collectively exhaustive collections of external variables. That is, the collections of external variables z₁,z₂, ...,z_n collectively span the range of external variables that exist in the highway system under consideration.

The predicted performance of the pavement system when the maintenance strategy u₁,u₂,...,u_n is applied is given by the performance indexes y1,y₂,...,y_n predicted by the system of equations

Equations 1-3

The benefits realized by the users of the pavement system depend on the conditions of the pavements in the system. Therefore, the benefits of the pavement system might be written

Equation 4

where B_i(y_i) represents the contribution to overall benefits realized from pavements in the ith category. The benefits function B_i(.) represents the gross benefits realized by users of the ith pavement net of user costs and taxes those users must pay to the highway agency. The benefits function B_i(.) specifically excludes highway agency costs; they will be explicitly accounted for separately.

The agency costs borne by the citizens who support the highway system depend on the maintenance strategy implemented on each category of pavement. We can postulate a cost function for the ith category of pavement of the form C_i(u_i) and note that the total agency cost is

Equation 5

The net benefits realized by the highway users in the jurisdiction under management can be written

Equation 6

Assuming the highway department has an overall budget limitation T. maximization of net benefits as expressed in (6) occurs subject to a budget constraint

Equation 7

Thus, the highway department's problem in managing the highway system over which they have trusteeship is to select that maintenance strategy u₁,u₂,...,u_n that maximizes the net benefit that accrues to the beneficiaries of the highway system, i.e., to solve the problem:

Equation 8

Deterministic pavement management systems are built around the solution to this problem. We are aware of one such system that "linearizes" the Bi(.) and Ci(.) functions and purports to solve the resulting problem using linear programming. We can envision the use of nonlinear optimization techniques as well. However, the most fruitful avenue of approach is not to actually solve the problem in equation (8) directly but rather to examine the properties of the solution and develop an alternative approach based on the intrinsic properties of the solution.(5)

The optimum solution u₁*,...,u_n* to (8) can be (6) to satisfy the following mathematical relationship

Equation 9

where u is a nonnegative constant and u is strictly positive if the budget constraint in (8) is binding and zero if it is not binding.

The interpretation of equation (8) is profoundly important for the pavement management problem and for the results of the C­LTPP program. In particular, if the budget constraint is not binding (i.e., if there is more than enough money to accomplish all proposed design and maintenance projects), equation (8) tells pavement managers that they should spend money on every category of pavement until the benefit of the marginal of expenditure is exactly equal to the cost of marginal dollar of expenditure, i.e., until the benefit/cost ratio of the marginal dollar of expenditure is unity. This is a rather trivial result; highway departments with enough money should do everything that has net positive value to their constituents and do nothing whose costs exceed the benefits.

Increasingly so in the future, the budget constraint in equation (8) will be binding. There will not be sufficient design and maintenance funds to undertake every potentially beneficial project. Difficult tradeoffs will have to be made. The highway system is simply too old, and design and maintenance costs have escalated too much. It is not possible to maintain every section of pavement to the point at which the benefit of the marginal dollar of expenditure is equal to the cost of the marginal dollar of expenditure. Rather, budget will be sufficiently scarce that prospective benefits will exceed prospective costs for additional dollars of highway expenditure, but those additional dollars will simply not be available because of competing social or private uses. Benefits that could be realized from additional highway dollars will simply be foregone. Scarce highway dollars will have to be allocated to their highest value use. Highway departments will have to convince their constituents that more budget dollars are needed relative to other potential applications.(7)

Equation (8) guides us precisely as to how to make the necessary difficult tradeoffs. The number u represents the value in excess of agency cost that one additional budget dollar will buy, i.e., it represents the opportunity cost of one additional budget dollar. Optimal deployment of agency dollars dictates that the design and maintenance should be implemented so that the opportunity cost of one additional budget dollar should be identical for every section of pavement in the system. No section should be "gold plated" because other sections will suffer more than they should. No section should be ignored because the benefits foregone will be too large. Every section should be maintained to a precisely equal degree of serviceability, where serviceability should be defined in terms of a marginal benefit.cost ratio. In effect, every section of highway should bear equal "pain" from budget shortfalls (in a benefit/cost sense).

This is a powerful result. Optimal deployment of budget dollars by a highway agency requires that one calculate the marginal benefits and marginal costs of every prospective action on every prospective pavement section. They should then undertake actions in descending order beginning with the largest and proceeding onward until budget is exhausted. Such a strategy optimally deploys highway budget and secures the largest net benefit.

The nature of the optimum design and maintenance strategy­that it should equalize opportunity cost of maintenance foregone on every section of pavement in the system­suggests a clever and efficient way to determine the optimum maintenance strategy and manage highway expenditures to that optimum. To find the optimum, one should not apply brute­force linear or nonlinear programming approaches to (8) but rather should use an iterative approach based on calculating and equalizing the opportunity cost of a budget dollar on every section of pavement in the system. DFI has built methodologies of precisely this type and applied them to a number of disparate problems and would be prepared to do so in support of the Canadian SHRP program. We will outline and recommend such an approach below which considers all the foregoing elements but also explicitly represents inescapable uncertainty. In effect, the probabilistic semi­Markovian procedure we will recommend prioritizes in terms of marginal benefits and marginal costs but allows the highway department to hedge against inevitable uncertainties.

Before proceeding to the probabilistic semi­Markovian approach in the next section, we should reiterate that the primary contribution of the long term pavement performance monitoring program is to deliver increasingly accurate and comprehensive estimates of the pavement deterioration function f(x,b) and the parameters of that function b. In particular, the long term pavement performance monitoring statistical design presented previously can present either the classical regression estimate of the parameters b or alternatively the Bayesian estimate of the expected value of the parameters b.

3.2 SEMI­MARKOViAN PROBABILISTIC PAVEMENT MANAGEMENT APPROACH

We have described the semi­Markovian and the less general Markovian approach to pavement design and management in great detail in Nesbitt and Sparks (1987). In this section, we will simply describe the interface between the semi­Markovian approach and the probabilistic information produced by our Bayesian statistical approach.

Returning to equation (41) in Section 2, the Bayesian statistical approach yields a continuously improving estimate of the probability distribution over pavement performance y as a function of the environmental and traffic variables z that are not under the direct control of the highway department and the maintenance variables u that are under the direct control of the highway department. Using this notation, we can write the structural model determined from our recommended Bayesian approach y = f(u,z,b) + e in which b and e are characterized by an explicit, quantitative, joint probability distribution or more generally {y | z,u}.

Referring to the semi­Markovian approach in Appendix A, the Bayesian probability distribution semi­Markovian is for a given maintenance strategy u precisely equivalent to the discrete probability distribution over pavement deterioration states developed in the paper. That is, the direct output of the Bayesian statistical approach is the direct input to the semi­Markovian probabilistic pavement management model (leaving aside the issue of discretization of pavement states). Because the output from the pavement monitoring process is the input to the semi­Markovian approach, there is strong motivation to

· build an automated semi­Markovian pavement management tool to serve as the centerpiece of pavement management.

· build an automated interface from the Bayesian statistical model that delivers the long term pavement deterioration function results to the semi­Markovian nucleus.

· build a Bayesian statistical module integrated within a statistica/data package such as BMDP, SAS, or SPSS to receive raw pavement monitoring data, produce the posterior probability distribution, and deliver the posterior probability distribution automatically to provincial and state pavement managers.

We will propose just such a procedure in Section 4.

(Continue)

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