Skip to contentUnited States Department of Transportation - Federal Highway Administration FHWA Home
Research Home
Public Roads
Featuring developments in Federal highway policies, programs, and research and technology.
This magazine is an archived publication and may contain dated technical, contact, and link information.
Federal Highway Administration > Publications > Public Roads > Vol. 72 · No. 1 > Applying "Fuzzy Concept" To Bridge Management

Jul/Aug 2008
Vol. 72 · No. 1

Publication Number: FHWA-HRT-08-005

Applying "Fuzzy Concept" To Bridge Management

by Adel Al-Wazeer, Bobby Harris, and Waseem Dekelbab

This approach could help account for uncertainties in condition assessments and improve maintenance decisions.

A bridge inspector is examining a steel girder where the paint is peeling off due to surface rust. Using fuzzy numbers, inspectors could account for uncertainties in their condition assessments, ultimately leading to better maintenance decisions.
A bridge inspector is examining a steel girder where the paint is peeling off due to surface rust. Using fuzzy numbers, inspectors could account for uncertainties in their condition assessments, ultimately leading to better maintenance decisions.

Bridges are critical components of the transportation infrastructure and require special attention in terms of maintenance strategies to preserve them and maximize their public benefit. But the aging bridge inventory and increasing traffic pose significant challenges for State highway agencies. Limited resources relative to the demand for overcoming existing deficiencies in highway bridges make it difficult for highway and bridge agencies to carry out maintenance activities on their structures as frequently as necessary to maintain all of them in good condition. Therefore, departments of transportation (DOTs) rely on bridge management systems (BMSs) to help identify bridges most in need of maintenance and apply efficient maintenance strategies to best utilize available resources.

A bridge’s performance depends on a number of factors, including the type, quality, and materials used in construction; and the timing, type, and effectiveness of maintenance, repair, rehabilitation, and replacement (MR&R) actions. To develop preservation policies, bridge owners use a variety of data and models. Necessary data include condition ratings of bridge components, costs of maintenance actions, and economic factors such as discount and inflation rates. Models for optimization and deterioration rates of bridge components assist bridge owners in selecting MR&R actions.

Precise data on the types, timing, costs, and effectiveness of these MR&R actions are challenging to assemble for a number of reasons. One reason is the differences in and complexity of bridge designs. Another is the broad range of materials and methodologies used for bridge work. Further, formats for establishing costs are inconsistent from one agency to another, uniform and easily accessible descriptions and records of actions and costs are lacking, and researchers have performed few followup studies to determine the effectiveness of maintenance actions.

Although bridge engineers continue to make significant strides in simulating reality within the available BMSs, elements of subjectivity and uncertainty still remain in the estimation and predictive ability of BMSs. For example, visual inspections — when an inspector assesses bridge components in person in the field — introduce subjectivity into BMSs. These assessments, based on an inspector’s expert opinion, result in numerical condition ratings used as inputs in a BMS, but are still subjective and can be imprecise.

“The condition of a bridge component, such as a superstructure with five steel bridge girders, is seldom uniform over the total area of the component,” says John Hooks, bridge technology consultant. “Rating of the condition of a bridge is normally done through visual observation and is therefore subject to each inspector’s interpretation of the extent plus severity of different conditions. The variability of ratings from one qualified inspector to another has been documented by FHWA and continues to persist despite extensive efforts to train inspectors using common training curricula and examples.”

To help quantify the uncertainties in inspectors’ assessments, a logic theory known as fuzzy set theory offers a tool to model the condition ratings of bridge components while accounting for the uncertainties. Fuzzy set theory could prove to be a valuable tool for handling uncertainties due to subjective estimates in data using a range of values reflecting the uncertainty in a more realistic way than current practices.

Current Practices

Inspectors in the United States assess bridge conditions in two ways, resulting in two types of ratings: National Bridge Inventory (NBI) condition ratings and element-level condition assessments. Current practices allow inspectors to use visual inspections supported by detailed information such as written descriptions of deterioration, photographs, and measurements to report condition ratings of bridge components using “crisp” (exact) values representing their field condition. For example, in an NBI rating, the conditions of the bridge deck, superstructure, and substructure are scored 0 to 9, where 0 refers to the worst condition and 9 refers to the best.

Likewise, an element-level inspection focuses on the condition of individual elements, such as the deck, girders, and columns. Inspectors rate the element conditions on scales of 1 to 3, 4, or 5, depending on the type of element. Concrete decks, for example, are rated using a scale of 1 to 5, while reinforced concrete columns are rated using a scale of 1 to 4. The inspector distributes the total quantity of the inspected element (such as the linear footage of a steel girder) between the different condition states of the element. (That is, for example, 70 percent of the girder length might be in condition state 2, while 30 percent is in condition state 3.) Unlike NBI ratings, in all cases for element-level inspections, condition state 1 corresponds to best-case conditions, while the condition states with higher numbers correspond to poorer conditions.

Expert opinion, reflected in these visual inspections, plays a key role in estimating the condition ratings. But determining the true values is difficult due to uncertainties in assessing, quantifying, and reporting inspection data based on one person’s expert opinion and engineering judgment. Because human judgment is involved, the possibility of imprecision enters the estimation process. In some cases, uncertainty in the estimates could lead to less-than-optimal decisions about maintenance.

NBI Condition Ratings

9 = Excellent condition
8 = Very good condition
7 = Good condition
6 = Satisfactory condition
5 = Fair condition
4 = Poor condition
3 = Serious condition
2 = Critical condition
1 = Imminent failure condition
0 = Failed condition

Because visual inspections remain integral to assessing the condition of bridge elements, the possibility persists that one inspector might report a different condition state distribution than another inspector for the same element. In particular, opinions could vary in matching field conditions to the parameters outlined in the definition of an element’s condition state. This fact raises a number of questions:

  • Which assessment represents the element’s true condition state distribution?
  • How accurate and confident is each inspector in his or her assessment?
  • Does the assessment provided by one inspector represent the true field conditions while that of another does not? Or is there a degree of truth inherent in each?
  • How can bridge owners factor in the uncertainty?

According to current inspection practices, a condition rating for a bridge component will have a single rating that the inspector considers to be the true value. But with the fuzzy concept, one can talk about “degrees of truth” represented by “membership grades,” which capture the gray area or uncertainty around a given estimate.

A New Approach

The fuzzy concept takes into account the principle that phenomena in the physical universe do not have sharp boundaries, which results in a degree of vagueness and subjective uncertainty that factors into risk assessments — much like risk assessments due to other types of uncertainty, as in business decisions or in predictions of real-world weather conditions.

Quantifying uncertainty in the inspection data and its effects on bridge management decisions could help DOTs predict future conditions for the elements of a bridge and thereby select optimal maintenance strategies and actions based on better-quantified data.

“Bridge inspection is not an exact science,” says Amjad Waheed, P.E., assistant administrator of the Office of Structural Engineering at the Ohio Department of Transportation (ODOT). “Things are not black and white in the field. At ODOT, if a bridge has a general appraisal of 4 or less [on a scale of 0 through 9, where 9 is the best], it is considered deficient. The decision to appraise a bridge as ‘deficient,’ when that bridge was not deficient previously, is always debatable. The decision depends on subjective judgment based on the individual bridge inspector’s level of training, experience, and critical thinking. During fiscally tight situations, modeling based on the current discrete scale may result in a probabilistically uneven spread of bridge funds. Use of fuzzy logic theory in the bridge management system could result in better modeling of bridge needs that accounts for uncertainty and better allocation of bridge funds at ODOT.”

Fuzzy Set Concepts

In set theory, a “set” is a group of things (members or elements) that share a common characteristic. Each element in the group belongs to the set. For example, in terms of bridge management, a set might consist of the types of materials used in construction of a bridge. Concrete, steel, prestressed concrete, wood, and masonry belong to that set, while straw does not.

Fuzziness in Visual Inspection of Bridge Components

Bridge inspectors try to represent the true field conditions of bridge components using condition assessments based on visual inspections. Assessments that represent the true field conditions score 100 percent (center of the dartboard); assessments that do not represent the true field conditions score 0 percent (outside the area of the dartboard); and assessments that partially represent the true field conditions can be represented by a fuzzy number with a score (called a membership grade) between 0 and 100 percent.

Bridge inspectors try to represent the true field conditions of bridge components using condition assessments based on visual inspections. Assessments that represent the true field conditions score 100 percent (center of the dartboard); assessments that do not represent the true field conditions score 0 percent (outside the area of the dartboard); and assessments that partially represent the true field conditions can be represented by a fuzzy number with a score (called a membership grade) between 0 and 100 percent.

Under conventional logic, every statement is either true or false — there is no gray area. An element or subset (a group of elements from the set) either is or is not a member of the set. Steel is a member of the set of bridge material types; straw is not. With fuzzy set theory, however, values that are known precisely are referred to as “crisp” numbers, while uncertain values that lack the distinction in sharp boundaries can be represented by “fuzzy” sets with partial or subjective memberships of elements in a set. A fuzzy set has no sharp boundary between one element belonging or not belonging. The “fuzziness” in a statement or estimate lies in an interval of confidence (fuzzy range) outside which no estimate is possible.

As an example, if a meteorologist were to estimate that the outdoor temperature in Virginia in August is 30 degrees Celsius, °C (86 degrees Fahrenheit, °F), then his estimate of 30°C would be termed a crisp number. Alternatively, if he were to estimate that the temperature is between 25°C (77°F) and 35°C (95°F), this range would be termed a fuzzy set. He is 100 percent certain that the temperature is not more than 35°C or less than 25°C, so he is making an educated judgment based on his expertise in estimating the temperature. Given that a meteorologist’s expert judgment of a crisp value for the temperature involves subjectivity and limitations in knowledge of the true value for the temperature, the use of a range rather than a crisp value offers a more realistic description of the temperature.

The fuzzy concept accommodates partial truths, where truth values (a value indicating the extent to which a proposition is true) exist between completely true and completely false. In classical logic, the only possible truth values are true and false, but fuzzy logic allows for more truth values across a spectrum that ranges from true to false.

Representing Field Conditions of a Bridge Superstructure Using
Fuzzy Numbers for NBI Condition Ratings 5, 6, and 7
This figure shows a triangular fuzzy number representation for a hypothetical bridge superstructure. The y-axis is labeled “Membership Grade” and shows percentages of the membership ranging from 0 at the origin up to 100 percent. The x-axis is labeled “Condition Variation” and shows an NBI representation of superstructure conditions 5, 6, and 7 modeled using fuzzy numbers. Three overlapping triangles, representing fuzzy numbers corresponding to “Fuzzy Condition 5,” “Fuzzy Condition 6,” and Fuzzy Condition 7,” are plotted on the graph. The apex of each triangle is at 100 percent on the y-axis, with the legs of the triangles intersecting the x-axis. The triangle whose apex is closest to the y-axis is brown and corresponds to “Fuzzy Condition 5.” Further to the right, the second triangle is blue and labeled “Fuzzy Condition 6.” The third triangle, green, is farthest from the origin and labeled “Fuzzy Condition 7.” A pink vertical line drops down between the first and second triangles and is labeled “True Condition of Bridge Superstructure” to represent the field condition of the bridge superstructure.
In this figure, three inspectors describe the field condition of a bridge superstructure by triangular fuzzy numbers representing NBI condition ratings 5, 6, and 7. An NBI condition rating of 6 assigned by one inspector represents 70 percent of the field condition for the superstructure, while the definitions of NBI condition ratings 5 and 7 can represent 10 percent and 20 percent of the field condition for the superstructure.

For example, one inspector might try to match the field condition of a superstructure by giving it an NBI rating of 6 (defined as satisfactory condition with some minor deterioration), while another inspector offers a rating of 7 (defined in NBI as good condition with some minor problems). A third inspector might rate the superstructure 5 (defined as fair condition with minor section loss or cracking). The truth, however, is that the field condition of the superstructure could match 70 percent of NBI rating 6, 20 percent of NBI rating 7, and 10 percent of NBI rating 5. In this case, the first inspector’s NBI rating is actually 70 percent true and 30 percent false. Likewise, the NBI rating by the second inspector is 20 percent true and 80 percent false, and the rating by the third inspector is 10 percent true and 90 percent false. Therefore, the NBI ratings of the three inspectors have truth values between the extremes of true and false, while none of them is completely true or completely false.

According to fuzzy theory, a completely true value has a truth value of 1 (meaning 100 percent true), while a completely false value has a truth value of 0 (meaning 0 percent true). The values in between can have partial membership, representing the degree to which an element belongs to a set, with degrees of truth (membership grade) between 0 and 1. In the example above, NBI condition ratings of 6, 7, and 5 for the superstructure are true, with membership grades of 0.7, 0.2, and 0.1, respectively.

A fuzzy set, therefore, can have elements with membership grade values between 0 and 1. The membership grade represents the truth value, or “strength,” of that element’s membership in the fuzzy set. All possible elements of a fuzzy set have degrees of membership representing the truth value. The membership grade, therefore, represents the degree to which an element belongs to a set.

It is important to clarify the difference between vagueness uncertainty used in the fuzzy approach and statistical uncertainty used in a probabilistic approach. The fuzzy set theory tries to deal with knowledge uncertainty for parameters that are vague in the distinction of their boundaries (not sharply well-defined) as in the case of bridge condition assessments. The probabilistic approach, on the other hand, deals with uncertainty related to randomness of parameters. Statistical uncertainty depends on the sample size of the records (such as the number of inspectors in the case of bridge condition assessment) and uses statistical descriptives such as the mean and standard deviation to represent uncertainty (mean +/- n standard deviations). In the current practice of bridge inspection, regardless of how many trained inspectors make the assessment, each inspector will use crisp (exact) ratings for the conditions of bridge components. Although the probabilistic approach can provide statistical uncertainty that depends on the number and deviation of these crisp observations, it cannot disclose the vagueness in these values. When an experienced inspector makes his judgment, he uses a single (crisp) value judgment without revealing his knowledge (what he thinks) about the possible range of ratings around that crisp value that could represent the real field conditions and what degrees of belief he has for the values in that possible range.

Also worth noting is that other types of uncertainty exist, such as uncertainties related to modeling, ignorance, prediction of future events, and confidence in the decisionmaking process.

Fuzzy Numbers

A fuzzy number is a quantity whose value is uncertain rather than exact (as with crisp numbers). The uncertain quantity has a range of values between the lowest possible limit (below which there are no possible values) and highest possible limit (beyond which there are no possible values). Each value in the range is assigned a specific grade of membership between 0 and 1.

The different membership grades for elements in the range of a fuzzy number combine to form a graphical representation of a membership grade function for the values in the range of a fuzzy number. The shape of the curve for the membership function can be defined by a set of mathematical relationships that maps the membership grade of the values in the range of the fuzzy number. The membership grade function curve might look like a triangle, trapezoid, bell shape, or any general shape — representing a triangular, trapezoidal, bell-shaped, or general-shaped fuzzy number. Each value in the fuzzy number range corresponds to a membership grade value between 0 and 1 in the membership grade function. The membership grades represent the degrees of belief in the truth levels of the values in the range of the fuzzy number.

Expressing a Fuzzy Number Using Partial Membership

Graph. This graph shows a general curve representing a typical fuzzy number. The y-axis is labeled “Membership” and ranges from 0 at the origin up to a value of 1.0. The x-axis represents the values in the fuzzy number corresponding to given membership values between 0 and 1. The possible range of values for the fuzzy number is labeled “Range of Values” and ranges from “Lowest Value” at the left to “Highest Value” toward the right-hand side of the plot area. A general curve (in this case, a convex, bell-shaped curve) is plotted from the point labeled “Lowest Value” on the x-axis, corresponding to a 0.0 membership on the y-axis, extending up and to the right to a peak at a maximum height of 1.0 on the y-axis and then arching back downward and to the right to intersect the x-axis at the point labeled “Highest Value,” corresponding to a 0.0 membership on the y-axis. The curve is labeled “Membership Function.”

The different membership grades for the elements in the range of a fuzzy number combine to form a graphical representation, in this case, a general curve.

Triangular fuzzy numbers can be represented by three values: the most possible value, the lowest possible value, and the highest possible value. The values in the range between the lowest and highest possible values have a membership grade between 0 and 1, with the most possible value having a membership grade of 1. The lowest and highest possible values have membership grades of 0 because they represent the lower and upper limits of the fuzzy range outside which no values belong to the fuzzy number.

The membership grade function for the triangular fuzzy number is represented by a straight line from a membership grade of 0 for the lowest possible value to a membership grade of 1 for the most possible value and continues with another straight line from the membership grade of 1 for the most possible value to a membership grade of 0 for the highest possible value. The two straight lines (representing the membership grade function) and the horizontal line between the lower and upper limits of the fuzzy number form a triangle representing the fuzzy number. The membership grade for a given value in the range between the lowest possible value and the most possible value is evaluated using linear interpolation by finding the membership grade on the straight line corresponding to a given value in the fuzzy range. Similarly, the membership grade for a given value in the fuzzy range between the most possible value and the highest possible value is evaluated using linear interpolation by finding the membership grade on the straight line between the two points corresponding to that value in the fuzzy range.

Anatomy of a Triangular Fuzzy NumberGraph. This graph shows a triangular fuzzy number, which could offer a practical approach for describing the field conditions of bridge elements during inspections. The y-axis is labeled “Membership Grade” and ranges from 0 at the origin up to a value of 1.0. The x-axis has three points marked on it: “Lower Limit (Membership Grade = 0.0)” on the left, “Most Possible Value (Membership Grade = 1.0),” and “Upper Limit (Membership Grade = 0.0),” toward the right-hand side of the plot area. The range of values on the x-axis between “Lower Limit” and “Upper Limit” are labeled “Range of Values in the Fuzzy Number.” A dotted horizontal line extends to the right from the 1.0 value on the y-axis and meets a vertical dotted line that originates at the “Most Possible Value” point on the x-axis. The point of intersection of the two dotted lines is the apex of the triangle and corresponds to the “Most Possible Value.” The two legs of the triangle extend from that apex point down to intersect the x-axis at the points labeled “Lower Limit” and “Upper Limit.” The hypotenuse of the triangle runs along the x-axis between “Lower Limit” and “Upper Limit.” The triangle legs are labeled “Membership Function.”
A triangular fuzzy number, as illustrated here, could offer a practical approach for describing the field conditions of bridge elements during inspections. The three key values represented in the number are the most possible value and the extreme lower and upper limits.

Can ‘Fuzzy Concept’ Be Applied to Bridge Inspection And Management?

Current bridge management decisions and practices depend on a number of criteria and performance measures, which also depend on factors such as condition ratings, deterioration models, and cost estimates. Most, if not all, of these factors do not have precise or sharp values, but rather uncertain quantities with a range that represents an acceptable margin of error.

A bridge girder element with some sections showing more damage than others (as shown here) might create fuzziness in an inspector’s decision about how to describe the condition states of that element. Fuzzy logic attempts to account for that uncertainty and thereby achieve more accurate condition assessments than is possible simply using one number.
A bridge girder element with some sections showing more damage than others (as shown here) might create fuzziness in an inspector’s decision about how to describe the condition states of that element. Fuzzy logic attempts to account for that uncertainty and thereby achieve more accurate condition assessments than is possible simply using one number.

As noted earlier, visual inspections result in quantification of condition ratings for bridge components using crisp (exact) values. Although these ratings are supported by detailed information such as written descriptions, photographs, and measurements of deterioration, they involve uncertainty in representing the field condition due to many factors, including the inspectors’ judgments.

Because State DOTs use inspection data as the basis for maintenance strategies for bridges, uncertainty in the input data affects uncertainty in the output of the decision criteria. For example, DOTs base maintenance policies for bridge elements in the BMS on uncertain estimates such as condition assessments of elements, costs of actions, deterioration rates, and discount rates (a rate that finds present value from a future value). Calculations based on these input data result in new uncertainty in the output results. Capturing and accounting for the uncertainty of the output decision criteria by using fuzzy numbers could help bridge owners make more informed decisions than they currently can using a single-value result.

Because fuzzy numbers take into account uncertainty in the quantities, they facilitate the depiction of the physical quantities more accurately than is possible using crisp numbers. For example, a bridge inspector rates the condition of a steel girder element on a scale of 1 to 4, where 1 represents the best condition and 4 the worst. The inspector needs to specify the proportion of the girder element in each condition state. To do so, the inspector estimates the portions of the steel girder, in linear meters (or feet), that he deems to be in each condition state. He might define the amount in condition state 3 using a set of possible values estimated as 12, 14, 15, 17, and 18 meters (40, 45, 50, 55, and 60 linear feet). The level of certainty about how confident the inspector is about the possible values can be represented using the fuzzy set concept with a membership grade function that has values in the range of 0 to 1. The inspector might judge that the estimate amounts of 12, 14, 15, 17, and 18 meters (40, 45, 50, 55, and 60 linear feet) have membership grades of 0.0, 0.5, 1.0, 0.5, and 0.0, respectively.

Representing the Quantity (Length) of a Steel Girder in a Condition State (Condition State 3) Using Fuzzy Numbers

 Graph. This graph shows a triangular fuzzy number representation for the estimated length of a steel girder in condition state 3. The y-axis is labeled “Membership” and ranges from 0 at the origin up to a value of 1.0 in increments of 0.5. The x-axis ranges from 40 to 60 feet (12 to 18 meters) in 5-foot (1.5-meter) increments. A dotted horizontal line, labeled “Membership = 1.0,” extends from 1.0 on the y-axis to the point defined at x, y = 50, 1.0. Two lines extend from that point down to intersect the x-axis at the points defined by x, y = 40, 0 and 60, 0, creating a large triangle with the third leg of the triangle running along the x-axis between 40 and 60 feet (12 to 18 meters). Another horizontal line, “Membership = [0.5-1.0]” extends from the y-axis and passes through points at x, y = 45, 0.5 and 55, 0.5, creating a smaller triangle inside the larger one. The top triangle over the 0.5 membership grade represents the range of possible values in condition state 3 with a confidence level (membership) of 0.5 or more.

This graph shows a triangular fuzzy number representation for the estimated length of a steel girder in condition state 3. The inspector determined that 40 feet to 60 feet (12 meters to 18 meters) of the girder could be in condition state 3, with 50 feet (15 meters) as the most possible value in condition state 3. He used the lower limit of 40 feet (12 meters), the upper limit of 60 feet (18 meters), and the most possible value of 50 feet (15 meters) to represent the steel girder length in condition state 3 as a fuzzy number. The top triangle over the 0.5 membership grade represents the range of possible values between 45 and 55 feet (14 meters and 17 meters) in condition state 3 with a confidence level (membership) of 0.5 or more.

If the bridge inspector wants to consider an interval of confidence based on his degree of belief, he can represent the interval using a given level of membership grade. This will include all elements in the fuzzy set that have an equal or higher level for that membership grade. The fuzzy subset that represents an interval of confidence associated with a 0.5 (50 percent) membership grade in the above example will be 14, 15, and 17 meters (45, 50, and 55 feet), which have membership grades of 0.5 or higher.

Modeling Failure Costs for Bridge Elements Using Fuzzy Numbers

Bridge Element

Unit

Total Element Quantity

Failure Unit Cost

Most Possible Cost Estimate

Least Failure Cost Estimate

Highest Failure Cost Estimate

Concrete Deck

Each
(Square Meters)

300

$600

$590

$610

Steel Girder

Meters

100

$4,000

$3,950

$4,050

Reinforced Concrete Abutment

Meters

24

$8,000

$7,950

$8,050

Reinforced Concrete Column

Each

4

$10,000

$9,950

$10,050

Joint Seal

Meters

24

$500

$490

$510

The table shows triangular fuzzy number representations for the failure costs of five bridge elements (concrete deck, steel girder, and so on) on a hypothetical bridge.

Applying the Principles Of Fuzzy Numbers to Describing Field Conditions

In the field, inspectors use the definitions of each condition state for bridge elements to match field conditions. For each element, an inspector will divide the total quantity (length, area, and so on) into portions that show similar conditions. Inspectors who are not 100 percent certain about an exact value for a condition assessment of a bridge element but believe the assessment will fall within a range of values might opt in the future to use fuzzy numbers to represent the condition. A triangular fuzzy number offers a range of values that includes a most possible value between lower and upper limits for matching field conditions to a better quantified condition assessment of bridge elements.

In trying to reflect the reality of the field conditions using fuzzy numbers, inspectors could face the challenge of matching field conditions to the condition state definitions using a range of possible values with a most possible value in the range instead of using one crisp value. The challenge will be in determining lowest, most possible, and highest values of element quantities in each condition state for each element. For each quantity of an element that looks similar in condition, an inspector can ask three questions: What is the most possible condition state that can represent the field condition for that quantity? What is the condition state above which no condition can represent that quantity of the element? What is the condition state below which no condition can represent that quantity of the element? The answers to these questions will result in the most possible, highest, and lowest condition states representing the field condition for that quantity of the element.

To illustrate application of fuzzy concepts in field inspections, consider the following example for calculating the bridge health index (BHI), a measure used by some State DOTs to assess bridge condition performance and make maintenance decisions. The health index is a performance measure for a bridge based on visual inspection results for the condition states of all the bridge’s elements and the economic consequences of failure of those elements. The health index uses a scoring scale of 0–100 to compare the current economic value of the bridge elements to the total economic value of the elements, where a value of 0 represents the worst bridge condition, with all bridge elements in their worst condition states, and 100 represents the best bridge condition, with all elements in their best condition states (as good as new) and have their total economic values. The current economic values of bridge elements are usually less than or equal to the total economic values, where the bridge elements are as good as new. The economic consequences, or costs, of failure are used as weights to the different elements of the bridge in combining the element-level condition assessment into a bridge-level condition measure.

 Graph. This graph plots the triangular fuzzy number representation for the failure costs of a concrete deck. The y-axis is labeled “Membership” and ranges from 0 at the origin up to a value of 1.0. The x-axis is labeled “Concrete Deck Failure Cost” and has three points marked on it: “$590” (the lower limit nearest the origin), “$600” (the most possible value), and “$610” (the upper limit, toward the right-hand side of the plotted area). A dotted horizontal line extends to the right from the 1.0 value on the y-axis and meets a vertical dotted line that originates at the $600 point on the x-axis. The point of intersection of the two dotted lines is the apex of a triangle. The two legs of the triangle extend from that apex point down to intersect the x-axis at the points labeled $590 and $610. The hypotenuse of the triangle runs along the x-axis between $590 and $610.
This graph plots the triangular fuzzy number representation for the failure costs of a concrete deck. The failure cost ranges from $590 per square meter to $610 per square meter of the concrete deck, with $600 per square meter as the most possible value for the failure cost.

BHI is calculated as the summation of the current values of bridge elements divided by the summation of the total values of bridge elements multiplied by 100. The total value for a bridge element is calculated by multiplying the total element quantity by the failure cost of the element. Weight factors representing the different condition states of the element are used for calculating elements’ values in their current conditions. The current value of a bridge element is the weighted sum of the element quantity distribution in the condition states of the element multiplied by the element failure cost.

Calculating Total Element Value From Element Quantity and Failure Cost

Bridge Element

Unit

Total Element Quantity

Failure Unit Cost

Total Element Value

Most Possible Cost Estimate

Least Failure Cost Estimate

Highest Failure Cost Estimate

Most Possible Estimate

Least Estimate

Highest Estimate

Concrete Deck

Each (Square Meters)

300

$600

$590

$610

$180,000

$177,000

$183,000

Steel Girder

Meters

100

$4,000

$3,950

$4,050

$400,000

$395,000

$405,000

Reinforced Concrete Abutment

Meters

24

$8,000

$7,950

$8,050

$192,000

$190,800

$193,200

Reinforced Concrete Column

Each

4

$10,000

$9,950

$10,050

$40,000

$39,800

$40,200

Joint Seal

Meters

24

$500

$490

$510

$12,000

$11,760

$12,240

By multiplying the total element quantity by the three fuzzy number values for unit failure cost (most possible, least, and highest failure cost), an engineer would arrive at three corresponding estimates for the (most possible, least, and highest) total element value.

Since condition state assessment and failure costs of bridge elements include subjectivity in the visual inspection and cost assumptions, bridge managers can use fuzzy concept to model the subjectivity and quantify the impact on the results. The following steps demonstrate how to apply fuzzy concepts to calculate the BHI. A hypothetical bridge is composed of five elements: a concrete deck, steel girders, reinforced concrete abutments, columns, and a joint seal. The economic consequences of failure for the elements are based on a unit failure cost, which is an uncertain quantity that can be represented by a triangular fuzzy number with a most possible value, least possible value, and highest possible value.

To calculate the BHI, the next step is to compare the economic values of bridge elements based on the current conditions and their total economic values when they are assumed to be as good as new. A total economic value for each element is calculated using its total quantity and unit failure cost value (fuzzy number). Multiplying a crisp value (the element quantity) by the unit failure cost of the element (a fuzzy number) results in fuzzy numbers for the total economic value of each element, represented by most possible, least, and highest economic values.

Next, the bridge engineer could calculate the current economic values of each bridge element based on its condition state distribution and failure cost. In element-level visual inspections, the inspector distributes the element quantity across a number of possible condition states, depending on the element type. In this example, the bridge deck and the steel girder have five possible condition states, the reinforced concrete abutment and column have four condition states, and the joint seal has three condition states. Each quantity of an element attributed to a given condition state is assigned a weighting factor. Although element condition states are distinct, weighting factors are defined based on a continuous line from 100 percent in the best state to 0 percent in the worst state. The condition states between the best and worst are given weighting factors based on linear interpolation on the continuous line from 100 to 0 percent (best to worst condition state). Therefore, the condition state weighting factor depends on the condition state S of the element and the possible number of condition states N for the element [condition weighting factor, WF = (N-S)/(N-1)]. For example, the weighting factor for a quantity of the concrete deck or the steel girder in condition state 3 will be 0.50 [WF = (5-3)/(5-1) = 0.5], while a quantity of the reinforced concrete abutment or column in condition state 3 will be 0.33 [WF = (4-3)/(4-1) = 0.33]. Similarly, the weighting factor for a quantity of a joint seal in condition state 3 will be 0.0 [WF = (3-3)/(3-1) = 0.0].

As noted earlier, the condition states attributed to the quantity distributions of the bridge elements can be uncertain; therefore, the inspector could employ fuzzy numbers to model the condition states, with most possible, worst, and best condition assessments. Each assessment is associated with a weighting factor, as described above. Therefore, each quantity distribution will be associated with most possible, lowest, and highest weighting factors representing the above-mentioned condition assessments.

Next, a bridge engineer can calculate the current economic values of the bridge elements based on their existing condition state distributions. The current economic value for each element is calculated by multiplying the quantity in each condition by the condition weighting factor, and adding the values over all the condition state distributions of the element. Then the engineer would multiply the result by the failure unit cost of the element. Because the condition assessment is modeled using fuzzy numbers corresponding to the most possible, worst, and best condition assessments, the current element values will be modeled similarly using fuzzy numbers that have most possible, least, and highest estimated values.

Weighting Factors for Possible Condition States

Bridge Element

Number of Condition States

Weighting Factors for Possible Condition States

Condition State 1

Condition State 2

Condition State 3

Condition State 4

Condition State 5

Concrete Deck

5

1.00

0.75

0.50

0.25

0.00

Steel Girder

5

1.00

0.75

0.50

0.25

0.00

Reinforced Concrete Abutment

4

1.00

0.67

0.33

0.00

 

Reinforced Concrete Column

4

1.00

0.67

0.33

0.00

 

Joint Seal

3

1.00

0.50

0.00

 

The table shows the weighting factors for the possible condition states for the elements of a hypothetical bridge. The condition state of the element and the number of condition states for each element drives the assigned weighting factor.

Using Condition State Weighting Factors in Calculating Bridge Health Index

Bridge Element

Quantity Distribution

Condition State Estimation

Condition State Weighting Factor

Most Possible

Worst Possible 

Best Possible

Most Possible

Lowest

Highest

Concrete Deck

300

3

4

2

0.50

0.25

0.75

Steel Girder

60

2

3

1

0.75

0.50

1.00

30

3

4

2

0.50

0.25

0.75

10

4

5

3

0.25

0.00

0.50

Reinforced Concrete Abutment

12

1

2

1

1.00

0.67

1.00

12

2

3

1

0.67

0.33

1.00

Reinforced Concrete Column

3

1

2

1

1.00

0.67

1.00

1

2

3

1

0.67

0.33

1.00

Joint Seal

8

2

3

1

0.50

0.00

1.00

16

3

3

2

0.00

0.00

0.50

This table, used for calculating the bridge health index, shows the fuzzy modeling of the condition state distributions and their weighting factors for the bridge elements on a hypothetical bridge.

To calculate the BHI, the engineer compares the summation of the bridge elements’ current economic values to the summation of their total economic values. Again, because the current and total economic values are modeled using fuzzy numbers, the resulting BHI also reflects fuzzy numbers, with most possible value, least possible value, and highest possible value. To calculate the lowest possible value of the health index, the engineer divides the lowest current economic value by the highest total economic value for the bridge, and multiplies the result by 100. The engineer calculates the highest possible value of the health index by dividing the highest current economic value by the lowest total economic value for the bridge, and multiplying the result by 100. The most possible value of the BHI is calculated by dividing the most possible current economic value by the most possible total economic value for the bridge, and multiplying the result by 100.

Using Condition State Weighting Factors in Calculating Bridge Health Index

Bridge Element

Quantity Distribution

Condition State Estimation

Condition State Weighting Factor

Most Possible

Worst Possible 

Best Possible

Most Possible

Lowest

Highest

Concrete Deck

300

3

4

2

0.50

0.25

0.75

Steel Girder

60

2

3

1

0.75

0.50

1.00

30

3

4

2

0.50

0.25

0.75

10

4

5

3

0.25

0.00

0.50

Reinforced Concrete Abutment

12

1

2

1

1.00

0.67

1.00

12

2

3

1

0.67

0.33

1.00

Reinforced Concrete Column

3

1

2

1

1.00

0.67

1.00

1

2

3

1

0.67

0.33

1.00

Joint Seal

8

2

3

1

0.50

0.00

1.00

16

3

3

2

0.00

0.00

0.50

This table, used for calculating the bridge health index, shows the fuzzy modeling of the condition state distributions and their weighting factors for the bridge elements on a hypothetical bridge.

The example shows how fuzzy set theory can be used as a tool to represent the vagueness in assessment boundaries by quantifying the subjectivity in the expert judgment of condition assessment and cost estimations and their effect on bridge management decisions based on BHI. A slight change in the condition assessment of bridge elements and their unit failure costs can result in a wide range of values for the BHI. The 2001 Federal Highway Administration (FHWA) study, Reliability of Visual Inspection for Highway Bridges, Volume I: Final Report (FHWA-RD-01-020), showed that at least 48 percent of the condition ratings for bridge elements are assigned incorrectly. The same study showed that 95 percent of condition ratings vary approximately two rating points from the average value. Ultimately, the FHWA study emphasizes the reality of uncertainty in assigning condition ratings. Furthermore, it demonstrates the potential value in considering ranges rather than crisp values for condition assessments to quantify the subjectivity in the assessment process and reflect a more realistic representation of field conditions. Based on this study, the calculation of the BHI (65 in this example) using crisp values for the condition assessment would not reflect the subjectivity in the condition assessment and cost assumptions, and therefore would not be a realistic measurement representing the true quantification of bridge health.

Calculating the Current Element Value

Bridge Element

Quantity Distribution

Condition State Estimation

Current Element Value

Most Possible

Worst Estimation 

Best Estimation

Most Possible Value

Least Estimation

Highest Estimation

Concrete Deck

300

3

4

2

$90,000

$44,250

$137,250

Steel Girder

60

2

3

1

$250,000

$148,125

$354,375

30

3

4

2

10

4

5

3

Reinforced Concrete Abutment

12

1

2

1

$160,000

$95,400

$193,200

12

2

3

1

Reinforced Concrete Column

3

1

2

1

$36,667

$23,217

$40,200

1

2

3

1

Joint Seal

8

2

3

1

$2,000

$0

$8,160

16

3

3

2

The table shows the fuzzy representation of the current element values with most possible, least, and highest estimated values for the bridge elements. Because the condition state assessments are represented using fuzzy numbers, the current element values also are modeled using fuzzy numbers. The most possible, least, and highest current value for each element is calculated as the sum of the products for the quantity distribution of the element and the weighting factors corresponding to the most possible, worst, and best condition state of the element, respectively.

The fuzzy concept approach offers a range of calculated lower and upper limits for the BHI (37 to 90 in this example) based on fuzzy number modeling of the condition assessment and failure costs. The range reflects an estimation of the BHI that measures a bridge’s condition based on more realistic values for the condition assessment and failure costs estimation of its individual elements. The range also accounts for the subjectivity of visual inspections and failure cost estimations. Furthermore, the fuzzy concept allows for choosing confidence intervals at any degree of membership grade, which means that the range of values can be narrowed for higher confidence intervals. In the abovementioned example, the health index range of 37 to 90 represents all possible values for the BHI between the lower and upper limits, while, for example, a BHI with confidence level of 0.5 (50 percent) or more ranges from 51 to 78, and a BHI with confidence level of 0.9 (90 percent) or more ranges from 63 to 68. This can help decisionmakers choose the BHI range corresponding to the target confidence level set by the agency.

Calculating the Bridge Health Index

Bridge Health Index Calculation

Most Possible Estimate

Least
Possible Estimate

Highest
Possible Estimate

Summation of Current Elements’ Values for All Bridge Elements

$538,667

$310,992

$733,185

Summation of Total Elements’  Values for All Bridge Elements

$824,000

$814,360

$833,640

Bridge Health Index

65

37

90

Calculating the bridge health index involves comparing the sum of the bridge elements’ current economic values to the sum of their total economic values. Shown here are the most possible, least possible, and highest possible estimates. This example shows that a slight change in the condition assessment of bridge elements and their unit failure costs can result in a wide range of values (37 to 90) for the bridge health index.

Fuzzy Numbers in Life-Cycle Cost Analysis

Another potential application for fuzzy numbers involves modeling uncertainty in various input parameters used in BMSs. For example, to determine the life-cycle cost of a bridge project, DOTs need to consider initial construction cost; components and timing of maintenance, rehabilitation, and replacement activities; and the cost of traffic delays for bridge users. These components are not exact values but rather involve uncertainty in their estimates. Furthermore, in order to calculate life-cycle costs, DOTs need to discount these components into their net present values. The uncertainty in the discount rate adds to the uncertainties. Adding up the net present values to calculate the life-cycle cost yields even more uncertainty.

Therefore, a judgment of a decisionmaker to select a preservation scenario for a bridge based on a single value of the life-cycle cost of the project could prove less than optimal. A fuzzy range of life-cycle cost estimates, with various levels of confidence values, could help factor in the uncertainty and enable bridge owners to make more informed decisions.

Graph. This figure shows a triangular fuzzy number representation for the bridge health index (BHI). The y-axis is labeled “Membership” and ranges from 0 at the origin up to a value of 1.0 in increments of 0.5. The x-axis, labeled “Estimated Bridge Health Index,” shows values for the BHI and ranges from 37 to 90. A dotted horizontal line, labeled “Membership = 1.0,” extends from 1.0 on the y-axis to the point defined at x, y = 66, 1.0, labeled BHI1.0 = 66. Two lines extend from that point down to intersect the x-axis at the points defined by x, y = 37, 0 and 90, 0, creating a large triangle with the third leg running along the x-axis between 37 and 90 feet. Another horizontal line, labeled “BHI0.5 = [51–78], Membership = [0.5–1.0]” extends from the y-axis and passes through points at x, y = 51, 0.5 and 78, 0.5, creating a smaller triangle inside the larger one.
This figure shows a triangular fuzzy number representation for the BHI of a hypothetical bridge. The crisp value for the BHI is 65 (most possible value with membership of 1.0). The range of values for the BHI as calculated based on the fuzziness of the inputs (condition, failure cost) is between 37 and 90. For a confidence level more than 0.5 (50 percent), the BHI values range between 51 and 78. For a confidence level of 0.9 (90 percent), the BHI values range between 63 and 68.

 

Modeling Life-Cycle Cost Alternatives Using Fuzzy NumbersGraph. This figure shows the triangular fuzzy number representation for modeling life-cycle cost alternatives for a bridge. The y-axis is labeled “Membership” and ranges from 0 at the origin up to a value of 1.0. The x-axis is labeled “Net Present Value of LCC Alternatives” and is intersected by the legs of three plotted triangles at the origin. The triangle whose apex is closest to the origin is purple and is labeled “Alternative A.” Further to the right, the second triangle is blue and labeled “Alternative B.” The third triangle, green, is farthest from the origin and labeled “Alternative C.” The figure shows that alternative A has the lowest most possible net present value but the largest upper limit. Alternative B has a medium most possible value but the smallest lower limit. Alternative C has the highest most possible value but a smaller upper limit than alternative A.
This figure shows the triangular fuzzy number representation for modeling three life-cycle cost (LCC) alternatives, A, B, and C, for a bridge. The figure shows that alternative A has the lowest most possible net present value but the largest upper limit. Alternative B has a medium most possible value but the smallest lower limit. Alternative C has the highest most possible value but a smaller upper limit than alternative A.

Next Steps

Inspectors and engineers can use the fuzzy concept as a tool to improve the quality of knowledge about bridges and make more informed decisions. As illustrated in the examples above, applying the fuzzy concept to assorted bridge management activities potentially can produce more realistic estimates than is possible using crisp values. Fuzzy modeling of bridge management practices can show the uncertainties in inputs and parameters, such as visual inspection results, cost data, deterioration rates, disparities between inspectors due to human factors, and can provide a better national understanding of the state of quantifying uncertainty for all subjective aspects of bridge management. Understanding these uncertainties can help DOTs minimize the risk of making less-than-optimal maintenance decisions.

The use of a range to represent the subjectivity in the condition assessment of bridge elements based on fuzzy modeling can help normalize the data across inspectors. At first, inspectors could find it challenging to represent field conditions in terms of fuzzy numbers because current BMSs do not support the fuzzy concept. Therefore, development of new tools and inspection procedures, manuals, and software explaining methods for conducting condition assessments using fuzzy numbers will be necessary before the fuzzy concept can be deployed.

The goal is to emphasize the need for more quantitative definitions in the inspection process that will help build well-defined boundaries between conditions. An FHWA scan team emphasized this objective in the report, Bridge Evaluation Quality Assurance in Europe (FHWA-PL-08-016), recommending detailed and more illustrative manuals for describing bridge conditions to improve accuracy and consistency of inspections. (Specifically, the report called for quantitative classification of cracks in concrete structures for different damage classes.)

More research is needed to move the science of the fuzzy concept into practical systems that support existing bridge management practices, including BMSs and life-cycle cost analysis. “As I read about the concept, I thought that the basic theory and principles of fuzzy logic might have good application to bridge inspection and management,” says FHWA’s Office of Bridge Technology Director M. Myint Lwin. “However, tools, training, and a plan to put the concept into practice would be needed to properly and effectively introduce the fuzzy concept to bridge engineering practice. In addition, current practices would need to be changed to adapt to this new method of collection and evaluation.”

Lwin adds: “In my view, the next steps would be to apply the fuzzy concept to bridge condition evaluation and rating, to apply the concept to bridge management systems, and to document the necessary changes to the current practices on condition evaluation and rating, the coding guide, the sufficiency rating, etc., to make this concept more than just theory. The benefits of the fuzzy concept also must be documented to convince decisionmakers to invest in tools, training, and changes to current practices, procedures, and systems.”

On a related front, the Long-Term Bridge Performance (LTBP) Program at FHWA is on a new mission to better understand bridge performance. “The LTBP program has a great potential in eliminating uncertainties by collecting and documenting quality measured field data, which will lead to the development of improved deterioration models, a better understanding of bridge performance, and the next generation of bridge management systems,” says Dr. Hamid Ghasemi, manager of LTBP Program.


Adel Al-Wazeer, Ph.D., is a senior research engineer with Science Applications International Corporation (SAIC), working at FHWA’s Turner-Fairbank Highway Research Center. Al-Wazeer holds a doctorate in civil engineering from the University of Maryland, College Park and is a recipient of the Eisenhower Grant for Research Fellowship and International Road Federation Fellowship.

Bobby Harris is senior project manager at SAIC. He serves as the contractor’s coprincipal investigator for FHWA’s Bridge Management Information Systems (BMIS) Laboratory. He has 20 years of experience in transportation information technology design and implementation, with 9 years’ involvement in bridge management research and data mining.

Waseem Dekelbab, Ph.D., PE., is a senior bridge research engineer at SAIC and serves as principal investigator for the BMIS Laboratory.

For more information, contact Adel Al-Wazeer at 202–493–3202 or adel.al-wazeer@fhwa.dot.gov.

ResearchFHWA
FHWA
United States Department of Transportation - Federal Highway Administration