One of the most vital and crucial steps in decision-making methods is the accurate estimation of the pertinent data. Very often, these data are not known in terms of absolute values. Therefore, many decision-making methods attempt to determine the relative importance (weight) of each alternative involved in a given decision-making problem. Consider the case of having a single decision criterion and a set of N alternatives denoted as Ai (i = 1, 2, 3,…, N). The decision maker wants to determine the relative performance of the alternatives under each criterion. Here, one may consider the N alternatives as the members of a fuzzy set. Then, the degree of membership of element (i.e. alternative) Ai expresses the degree to which alternative Ai meets the criterion. This is also the approach considered by Federov et al. (1982) and Chen and Hwang (1992) and was also discussed by Saaty (1994). All the methods which use the pair-wise comparison approach eventually express the qualitative answers of a decision maker as some numbers. Pair-wise comparisons are quantified by using a scale. Such a scale is one-to-one mapping between the set of discrete linguistic choices available to the decision maker and a discrete set of numbers which represent the importance or weight of the previous linguistic choices. There are two major approaches in developing such scales. The first approach is based on the linear scale and the other is based on exponential scale [Roberts (1979), Lootsma (1991)]. It is easier to use linear scale to translate the weight of an item/element over the other. Therefore, in this study, the linear scale has been used to assign the importance/weight of items or elements under each decision layer.

#### **2.3 Real Continuous Pair-wise (RCP) and Closest Discrete Pair-wise (CDP) matrices**

A procedure is required for obtaining comparison matrix from the relative importance (weights) for a group of elements, using a suitable scale, based on pair-wise comparisons. It involves the formulation of real continuous pair-wise (RCP) and the closest discrete pairwise (CDP) matrices. Reciprocal matrices with pair-wise comparisons were used for extracting all the pertinent information from a decision maker. Each entry in these matrices represents numerically the value of a pair-wise comparison between two alternatives with respect to a single criterion. For a problem that has 'p' objectives, a scale is constructed for rating these objectives as to their importance with respect to the decision as seen by the

decisions. Each attribute may be decomposed into more detailed attributes (indices). After the hierarchical network is constructed, one can determine the weights (importance measures) of the elements at each level of the decision hierarchy, and synthesize the weights to determine the relative importance (weights) of decision alternatives. First, a comparison matrix, which includes first (lowest) level elements of the hierarchy, is constructed. Then, a ratio scale through pair-wise comparison of each pair of criteria with respect to the overall goal is performed. The relative importance (weight) of each criterion is estimated using an eigenvector approach or other methods. Then, the relative importance (weight) of each alternative with respect to each criterion is determined using similar pair-wise comparisons. Here, it is important to note that the efficiency of AHP greatly depends on the accuracy with which pair-wise weights of items are assigned during the formation of comparison matrix. For pair-wise assignment of weights for items, there is a need for a scale for relative

One of the most vital and crucial steps in decision-making methods is the accurate estimation of the pertinent data. Very often, these data are not known in terms of absolute values. Therefore, many decision-making methods attempt to determine the relative importance (weight) of each alternative involved in a given decision-making problem. Consider the case of having a single decision criterion and a set of N alternatives denoted as Ai (i = 1, 2, 3,…, N). The decision maker wants to determine the relative performance of the alternatives under each criterion. Here, one may consider the N alternatives as the members of a fuzzy set. Then, the degree of membership of element (i.e. alternative) Ai expresses the degree to which alternative Ai meets the criterion. This is also the approach considered by Federov et al. (1982) and Chen and Hwang (1992) and was also discussed by Saaty (1994). All the methods which use the pair-wise comparison approach eventually express the qualitative answers of a decision maker as some numbers. Pair-wise comparisons are quantified by using a scale. Such a scale is one-to-one mapping between the set of discrete linguistic choices available to the decision maker and a discrete set of numbers which represent the importance or weight of the previous linguistic choices. There are two major approaches in developing such scales. The first approach is based on the linear scale and the other is based on exponential scale [Roberts (1979), Lootsma (1991)]. It is easier to use linear scale to translate the weight of an item/element over the other. Therefore, in this study, the linear scale has been used to assign the importance/weight of items or elements under each

**2.3 Real Continuous Pair-wise (RCP) and Closest Discrete Pair-wise (CDP) matrices**  A procedure is required for obtaining comparison matrix from the relative importance (weights) for a group of elements, using a suitable scale, based on pair-wise comparisons. It involves the formulation of real continuous pair-wise (RCP) and the closest discrete pairwise (CDP) matrices. Reciprocal matrices with pair-wise comparisons were used for extracting all the pertinent information from a decision maker. Each entry in these matrices represents numerically the value of a pair-wise comparison between two alternatives with respect to a single criterion. For a problem that has 'p' objectives, a scale is constructed for rating these objectives as to their importance with respect to the decision as seen by the

quantification of items.

decision layer.

**2.2 Scales for quantifying pair-wise comparisons**

analyst. Let w1, w2, w3, …………, wp be the real membership values of a fuzzy set with p members. Comparing objective k with objective l, the ratios kl can be assigned, and the RCP matrix (p p) is constructed as

$$\text{RCP} = \text{A}\_{\text{p} \times \text{p}} = \begin{bmatrix} \alpha\_{\text{kl}} \end{bmatrix} = \begin{bmatrix} \mathbf{w}\_{\text{k}} \\ \hline \mathbf{w}\_{\text{l}} \end{bmatrix} \quad \text{k}, \text{l} = \text{1}, \text{p} \tag{3}$$

The entry kl in RCP matrix represents the exact (and thus unknown) value of the comparison when the kth member is compared with the lth member. Each element k*l* (k*l* ) in the CDP matrix can be determined and the matrix will be formed such that ( ) *kl kl* is minimum. Any other norm may also be assumed as

$$\left| \frac{\alpha\_{kl}}{1 + \alpha\_{kl}} - \frac{\beta\_{kl}}{1 + \beta\_{kl}} \right| \tag{4}$$

#### **3. Condition evaluation of existing bridges through prioritization**

The Analytic Hierarchy Process (AHP) is mainly applied to the decision making problem with multiple evaluation criteria and uncertainty conditions. After hierarchical decomposition from different layers and through the quantitative judgment, the AHP is thus made a synthetic evaluation to reduce risk of wrong decision making. The AHP uses eigenvalue method to find the weights of different items. The eigen equation is adopted to construct the comparison matrix (Yu and Cheng, 1994, Liang et al., 2001) for finding the relative importance (weights) and orders of multiple objectives to an objective and the concept has already been successfully used to solve different types of decision making problems. The methodology involves the following operations.

#### **3.1 Relative importance (weights) of items**

A decision-maker provides the upper triangle of the comparison matrix (as shown in Table 1), while reciprocals are placed in the lower triangle which do not need any further judgment. The diagonal elements of the matrix are always equal to one. Assuming that any item group consists of A1, A2, A3, ….An items, the comparison matrix is constructed and then relative weights of items (A*ij*) of the group are evaluated by comparing objective *i* with objective *j*, the ratios kl can be assigned, and the real continuous pair-wise (RCP) matrix of order *p p* is constructed. It can be proved that consistent reciprocal matrix '[A]' has rank 1 with non-zero eigenvalue () = n. Then, we have

$$\text{[A]}\mathbf{w} = \mathbf{n}\mathbf{w}\text{ Where, }\mathbf{w}\text{ is an eigenvector}\tag{5}$$

The same equation also states that in the perfectly consistent case (i.e. A*ij* = A*ik* A*kj*), the vector w, with the membership values of the elements 1,2,3,….,n is the principal righteigenvector (after normalisation) of matrix [A].

#### **3.2 Check for consistency of comparison matrices**

In most of the real world problems, the pair-wise comparisons are not perfect, that is, the entry A*ij* might deviate from the ratio of the real membership values W*i* and W*j* (i.e. W*i* /

Condition Ranking and Rating of Bridges Using Fuzzy Logic 181

*x*

where, *l ≤ m ≤ u* , and *l* and *u* stand for the lower and upper values of the support for the decision of the fuzzy number M, respectively, and m for the modal value. In this study, the basic mathematical operations on fuzzy triangular numbers developed by Laarhoven and Pedrycz (1983) are followed. In decision problems, the maximum and minimum membership function suggested by Zadeh (1973) are adopted and expressed in the

1 *f*(x) ≤ inf(*f*)

1 *f*(x) ≥ sup(*f*)

where sup(*f*) and inf(*f*) are the superior and inferior values of *f*(x) respectively. It is understandable that Eq. (9) is a membership function with monotonic decrease whereas Eq. (10) is a membership function with monotonic increase. The significance of Eq. (9) is that the less the value is, more requirement for repair whereas the meaning of Eq. (10) is just the opposite of Eq. (9). An evaluation method can be developed by separating bridge deterioration into D (degree), E (extent), R (relevance) and U (urgency) for assessment. A combination of visual inspection, field and laboratory testing may be employed for determining the item estimation indices of bridges considered for condition assessment. Based on inspection results of all the items, the condition index (CoI) is calculated by using

CoI = ( ) *i i*

Where, *wi* is the weight of each bridge item and is greater than 1, and Ici is calculated as

*Ici* =

item condition estimation index for each item and is calculated as

*i Ic w w*

> *ii Ic n*

in which n is the number of relevant inspection items for a particular bridge, and *Icii* is the

 *Icii* = *a a DERU* (13)

*ml ml x* [*l, m*]

0 otherwise

*mu mu x* [*m, u*] (8)

inf(*f*) *f*(x) sup(*f*) (9)

inf(*f*) < *f*(x) < sup(*f*) (10)

(11)

(12)

 1 *l x*

m (x) = 1 *l*

following form.

and

(x)= sup( ) ( )

(x)= ( ) inf( )

sup( ) inf( ) *f f x f f*

sup( ) inf( ) *f x f f f*

0 *f*(x) ≥ sup(*f*)

0 *f*(x) ≤ inf(*f*)

W*j*). In a non-consistent case, the expression A*ij* = A*ik* × A*kj* does not hold good for all the possible combinations. Now, the new matrix [A] can be considered as a perturbation of the previous consistent case when the entries A*ij* change slightly, then the eigenvalues change in the similar fashion (Saaty, 1994). Moreover, the maximum eigenvalue is close to n (greater than n), while the remaining eigenvalues are close to zero. Thus, in order to find the membership values in the non-consistent cases, one should find an eigenvector that corresponds to the largest eigenvalue max. That is to say, one must find the principal righteigenvector W that satisfies

$$\mathbf{A}\mathbf{W} = \lambda\_{\text{max}} \mathbf{W} \qquad\quad\text{where }\lambda\_{\text{max}} \approx \mathbf{n} \tag{6}$$

The consistency ratio (CR) is obtained by first estimating max of matrix [A] Then, Saaty (1994) defined the consistency index (CI) of the matrix '[A]' as

$$\mathbf{CI} = (\mathbf{\bar{\lambda}}\_{\text{-max}} \mathbf{\bar{\star}} \mathbf{n}) / (\mathbf{\bar{\iota}} \mathbf{\bar{\iota}} \mathbf{1}) \tag{7}$$

Then, the consistency ratio (CR) is obtained by dividing CI with the random consistency index (RCI) as shown in Table 2 (proposed by Saaty, 1994). Each RCI is an average random consistency index derived from a sample of size 500 of randomly generated reciprocal matrices. If the previous approach yields a CR greater than 0.10 then a re-examination of the pair-wise judgments is recommended until a CR less than or equal to 0.10 is achieved.

Table 1. Comparison Matrix

Table 2. RCI values of sets of different order 'n'

#### **3.3 Fuzzy synthetic evaluation of estimation indices for items**

Most of the decision making in the real world takes place in a situation in which the pertinent data and the sequence of possible actions are not precisely known. Therefore, it is very important to adopt fuzzy data to express such situations in decision making problems. In order to fuzzify the crisp decision making methods, it is important to know how fuzzy operations are used on fuzzy numbers. Fuzzy operation in decision making was first introduced by Dubois and Prade (1979) and Boender et al. (1989) presented a fuzzy version of the AHP. For fuzzy numbers, triangular fuzzy numbers (that is, fuzzy numbers with lower, modal and upper values) are preferred, because they are simpler than trapezoidal fuzzy numbers. A fuzzy number M on R (-, +) is defined by Dubois and Prade, 1979 to be a fuzzy triangular number if its membership function m: R [0,1] is equal to

$$\mu\_{\text{m}}\left(\mathbf{x}\right) = \begin{cases} \frac{1}{m-l}\mathbf{x} - \frac{l}{m-l} & \mathbf{x} \in [l, m] \\ \frac{1}{m-u}\mathbf{x} - \frac{l}{m-u} & \mathbf{x} \in [m, u] \\ 0 & \text{otherwise} \end{cases} \tag{8}$$

where, *l ≤ m ≤ u* , and *l* and *u* stand for the lower and upper values of the support for the decision of the fuzzy number M, respectively, and m for the modal value. In this study, the basic mathematical operations on fuzzy triangular numbers developed by Laarhoven and Pedrycz (1983) are followed. In decision problems, the maximum and minimum membership function suggested by Zadeh (1973) are adopted and expressed in the following form.

$$\mu(\mathbf{x}) = \begin{cases} 1 & f(\mathbf{x}) \le \inf \{ f \} \\ \frac{\sup(f) - f(\mathbf{x})}{\sup(f) - \inf(f)} & \inf(f) < f(\mathbf{x}) < \sup(f) \\ 0 & f(\mathbf{x}) \ge \sup(f) \end{cases} \tag{9}$$

and

180 Fuzzy Logic – Emerging Technologies and Applications

W*j*). In a non-consistent case, the expression A*ij* = A*ik* × A*kj* does not hold good for all the possible combinations. Now, the new matrix [A] can be considered as a perturbation of the previous consistent case when the entries A*ij* change slightly, then the eigenvalues change in the similar fashion (Saaty, 1994). Moreover, the maximum eigenvalue is close to n (greater than n), while the remaining eigenvalues are close to zero. Thus, in order to find the membership values in the non-consistent cases, one should find an eigenvector that corresponds to the largest eigenvalue max. That is to say, one must find the principal right-

The consistency ratio (CR) is obtained by first estimating max of matrix [A] Then, Saaty

 CI=(max-n)/(n-1) (7) Then, the consistency ratio (CR) is obtained by dividing CI with the random consistency index (RCI) as shown in Table 2 (proposed by Saaty, 1994). Each RCI is an average random consistency index derived from a sample of size 500 of randomly generated reciprocal matrices. If the previous approach yields a CR greater than 0.10 then a re-examination of the pair-wise judgments is recommended until a CR less than or equal to 0.10 is achieved.

> B A1 A2 A3 …. An A1 A11 A12 A13 …. A1n A2 A21 A22 A23 …. A2n A3 A31 A32 A33 …. A3n …. …. …. …. …. …. An An1 An2 An3 …. Ann

n 1 2 3 4 5 6 7 8 9 ≥10 RCI 0 0 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.56

Most of the decision making in the real world takes place in a situation in which the pertinent data and the sequence of possible actions are not precisely known. Therefore, it is very important to adopt fuzzy data to express such situations in decision making problems. In order to fuzzify the crisp decision making methods, it is important to know how fuzzy operations are used on fuzzy numbers. Fuzzy operation in decision making was first introduced by Dubois and Prade (1979) and Boender et al. (1989) presented a fuzzy version of the AHP. For fuzzy numbers, triangular fuzzy numbers (that is, fuzzy numbers with lower, modal and upper values) are preferred, because they are simpler than trapezoidal fuzzy numbers. A fuzzy number M on R (-, +) is defined by Dubois and Prade, 1979 to

be a fuzzy triangular number if its membership function m: R [0,1] is equal to

(1994) defined the consistency index (CI) of the matrix '[A]' as

AW = max W where max n (6)

eigenvector W that satisfies

Table 1. Comparison Matrix

Table 2. RCI values of sets of different order 'n'

**3.3 Fuzzy synthetic evaluation of estimation indices for items**

$$\mu(\mathbf{x}) = \begin{cases} 1 & f(\mathbf{x}) \ge \sup(f) \\ \frac{f(\mathbf{x}) - \inf(f)}{\sup(f) - \inf(f)} & \inf(f) < f(\mathbf{x}) < \sup(f) \\ 0 & f(\mathbf{x}) \le \inf(f) \end{cases} \tag{10}$$

where sup(*f*) and inf(*f*) are the superior and inferior values of *f*(x) respectively. It is understandable that Eq. (9) is a membership function with monotonic decrease whereas Eq. (10) is a membership function with monotonic increase. The significance of Eq. (9) is that the less the value is, more requirement for repair whereas the meaning of Eq. (10) is just the opposite of Eq. (9). An evaluation method can be developed by separating bridge deterioration into D (degree), E (extent), R (relevance) and U (urgency) for assessment. A combination of visual inspection, field and laboratory testing may be employed for determining the item estimation indices of bridges considered for condition assessment. Based on inspection results of all the items, the condition index (CoI) is calculated by using

$$\text{CoI} = \frac{\sum \text{(lc}\_i \times w\_i)}{\sum w\_i} \tag{11}$$

Where, *wi* is the weight of each bridge item and is greater than 1, and Ici is calculated as

$$Ic\_i = \frac{\sum Ic\_{ii}}{n} \tag{12}$$

in which n is the number of relevant inspection items for a particular bridge, and *Icii* is the item condition estimation index for each item and is calculated as

$$I\_{\mathbb{C}\bar{u}} = D \times E \times \mathbb{R}^d \times \mathbb{U}^d \tag{13}$$

Condition Ranking and Rating of Bridges Using Fuzzy Logic 183

bridges. Zhao and Chen (2002) developed a fuzzy rule-based inference system for bridge damage diagnosis and prediction which aims at providing bridge designers with valuable information about the impact of design factors on bridge deterioration. Kawamura and Miyamoto (2003) presented a new approach for developing a concrete bridge rating expert system for deteriorated concrete bridges, using multi-layer neural networks. To evaluate the condition of different structures using fuzzy logic, the proposed methods are either too simplistic [Qian (1992); Liang et al. (2001)] or very complex [Jwu et al. (1991); Kawamura and Miyamoto (2003)]. In this chapter, a systematic procedure and formulations have been proposed for condition rating of existing bridges using fuzzy mathematics combined with eigenvector based priority setting technique. From the review of literature, authors felt that the existing methodologies for condition rating of existing bridges may be difficult to follow for a practical application. In view of this, in this chapter, a methodology for condition rating of bridges is described in steps that can easily be followed for practical applications. The methodology and its application are demonstrated through a case study and the details

Some important issues and the methodology for the development of a systematic, fast and reliable evaluation system for rating of existing reinforced concrete bridges have been

Towards a systematic rating of existing bridges, the essential requirement is the input data from the bridge inspector that consist of the ratings and importance factors for the relevant

Bridge inspection involves the use of various evaluation techniques in order to assess the physical condition of bridges. The Bridge Inspector's Training manual 90 (FHWA 1991), published by the US Department of Transportation, provides the basic guidelines for bridge inspection. Bridge components and their constituent elements, different types of bridge deterioration and the common causes are discussed in this manual. It also provides procedures for rating the condition of various elements. In this study also, as specified in Bridge Inspector's Training manual, bridge is divided into three major components, namely, 'deck', 'superstructure' and 'substructure'. Each component is further divided into number of elements. The deck, superstructure and substructure have 13, 16 and 20 elements respectively as shown in Table 3. The bridge inspector is required to assess the condition of each element individually. The rating evaluation for that particular component is carried out based on the rating of the constituent elements. This process is repeated for all the three components towards final rating of the bridge. To a large extent, rating of the elements is based on the experience, intuition and personal judgment of the inspector. Nevertheless, although the condition assessment of each element requires the inspector's personal judgment, general guidelines on how to assess the condition of the various elements are described in the

elements of bridge which would reflect the overall condition of a bridge as a whole.

are presented in the chapter.

**4.1.1 Inspection data**

illustrated in the following sections.

**4.1 Unified approach for condition rating of existing bridge**

**4.1.2 Inspector's observation and rating of elements**

#### **3.4 Condition ranking of existing bridges**

If Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…of items in each index layer) denotes the membership degrees of estimation indices of the items under index layer and *Wn* stands for the relative weights of items under index layer (calculated by using Eq. 3), then the relationship between Rn,m and *Wn* can be presented by

$$
\overline{D}\_n = \overline{\mathcal{W}}\_n \mathcal{R}\_{n.m} \tag{14}
$$

where the value *Dn* in Eq. 14 is the fuzzy synthesis evaluation matrix. The purpose of *Dn* is to construct the membership function for each alternative of evaluation set. The membership degrees of estimation indices, Rn.m, can be formulated based upon the decision makers choice in using the pessimistic- or optimistic- functions as stated in Eqs. 9 and 10, respectively.

Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any factor can be expressed as

$$
\overline{B} = \overline{A}\_n.\overline{D}\_n\tag{15}
$$

where, *An* is the weight vector. The prioritization or optimum repair order can be determined by using Eq. (15). The more the value of *B* has, the better the priority selection to decision making objective is.

#### **4. Application of fuzzy logic for condition rating of bridges**

The aim of the bridge condition rating is to evaluate the structural strength and serviceability condition of an existing bridge. Fuzzy set theory was specially defined to analyse the linguistic data within the formal mathematical framework. After the publication on fuzzy sets by Zadeh (1965), fuzzy mathematics was used extensively for numerous applications. Brown and Yao (1983) described the methodology of application of fuzzy sets in structural engineering. Tee et al. (1988) suggested a fuzzy mathematical approach for evaluation of bridges. Shiaw and Huang (1990) adopted the limit state design principle combined with fuzzy evaluation and random variable analysis to determine the bearing capacity index and degree of safety for bridges. Jwu et al. (1991) used fuzzy mathematics to determine the reliability of a wharf structure. In order to enhance the evaluation performance, the grade partition method was suggested. Tee and Bowman (1991) presented bridge condition assessment model that is based on resolution identity of fuzzy sets. Qian (1992) used the concept of fuzzy sets to evaluate the damage grade of existing bridges. Yu and Cheng (1994) presented a fuzzy based interactive comparison matrix approach for making group decision with multiple objectives. Wang (1996) provided a multi-target and multi-person evaluation method for structural durability. Melhem and Aturaliya (1996) proposed a model for condition rating of bridges using an eigenvector based priority setting. Liang et al. (2001) used fuzzy mathematics to build a damage evaluation methodology for existing reinforced concrete bridges. Liang et al. (2002) proposed grey and regression models for predicting the remaining service life of existing reinforced concrete bridges. Zhao and Chen (2002) developed a fuzzy rule-based inference system for bridge damage diagnosis and prediction which aims at providing bridge designers with valuable information about the impact of design factors on bridge deterioration. Kawamura and Miyamoto (2003) presented a new approach for developing a concrete bridge rating expert system for deteriorated concrete bridges, using multi-layer neural networks. To evaluate the condition of different structures using fuzzy logic, the proposed methods are either too simplistic [Qian (1992); Liang et al. (2001)] or very complex [Jwu et al. (1991); Kawamura and Miyamoto (2003)]. In this chapter, a systematic procedure and formulations have been proposed for condition rating of existing bridges using fuzzy mathematics combined with eigenvector based priority setting technique. From the review of literature, authors felt that the existing methodologies for condition rating of existing bridges may be difficult to follow for a practical application. In view of this, in this chapter, a methodology for condition rating of bridges is described in steps that can easily be followed for practical applications. The methodology and its application are demonstrated through a case study and the details are presented in the chapter.

#### **4.1 Unified approach for condition rating of existing bridge**

Some important issues and the methodology for the development of a systematic, fast and reliable evaluation system for rating of existing reinforced concrete bridges have been illustrated in the following sections.

#### **4.1.1 Inspection data**

182 Fuzzy Logic – Emerging Technologies and Applications

If Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…of items in each index layer) denotes the membership degrees of estimation indices of the items under index layer and *Wn* stands for the relative weights of items under index layer (calculated by using Eq. 3),

where the value *Dn* in Eq. 14 is the fuzzy synthesis evaluation matrix. The purpose of *Dn* is to construct the membership function for each alternative of evaluation set. The membership degrees of estimation indices, Rn.m, can be formulated based upon the decision makers choice in using the pessimistic- or optimistic- functions as stated in Eqs. 9 and 10,

Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any

where, *An* is the weight vector. The prioritization or optimum repair order can be determined by using Eq. (15). The more the value of *B* has, the better the priority selection

The aim of the bridge condition rating is to evaluate the structural strength and serviceability condition of an existing bridge. Fuzzy set theory was specially defined to analyse the linguistic data within the formal mathematical framework. After the publication on fuzzy sets by Zadeh (1965), fuzzy mathematics was used extensively for numerous applications. Brown and Yao (1983) described the methodology of application of fuzzy sets in structural engineering. Tee et al. (1988) suggested a fuzzy mathematical approach for evaluation of bridges. Shiaw and Huang (1990) adopted the limit state design principle combined with fuzzy evaluation and random variable analysis to determine the bearing capacity index and degree of safety for bridges. Jwu et al. (1991) used fuzzy mathematics to determine the reliability of a wharf structure. In order to enhance the evaluation performance, the grade partition method was suggested. Tee and Bowman (1991) presented bridge condition assessment model that is based on resolution identity of fuzzy sets. Qian (1992) used the concept of fuzzy sets to evaluate the damage grade of existing bridges. Yu and Cheng (1994) presented a fuzzy based interactive comparison matrix approach for making group decision with multiple objectives. Wang (1996) provided a multi-target and multi-person evaluation method for structural durability. Melhem and Aturaliya (1996) proposed a model for condition rating of bridges using an eigenvector based priority setting. Liang et al. (2001) used fuzzy mathematics to build a damage evaluation methodology for existing reinforced concrete bridges. Liang et al. (2002) proposed grey and regression models for predicting the remaining service life of existing reinforced concrete

**4. Application of fuzzy logic for condition rating of bridges**

*D WR n n nm* . (14)

. *B AD n n* (15)

**3.4 Condition ranking of existing bridges**

respectively.

factor can be expressed as

to decision making objective is.

then the relationship between Rn,m and *Wn* can be presented by

Towards a systematic rating of existing bridges, the essential requirement is the input data from the bridge inspector that consist of the ratings and importance factors for the relevant elements of bridge which would reflect the overall condition of a bridge as a whole.

#### **4.1.2 Inspector's observation and rating of elements**

Bridge inspection involves the use of various evaluation techniques in order to assess the physical condition of bridges. The Bridge Inspector's Training manual 90 (FHWA 1991), published by the US Department of Transportation, provides the basic guidelines for bridge inspection. Bridge components and their constituent elements, different types of bridge deterioration and the common causes are discussed in this manual. It also provides procedures for rating the condition of various elements. In this study also, as specified in Bridge Inspector's Training manual, bridge is divided into three major components, namely, 'deck', 'superstructure' and 'substructure'. Each component is further divided into number of elements. The deck, superstructure and substructure have 13, 16 and 20 elements respectively as shown in Table 3. The bridge inspector is required to assess the condition of each element individually. The rating evaluation for that particular component is carried out based on the rating of the constituent elements. This process is repeated for all the three components towards final rating of the bridge. To a large extent, rating of the elements is based on the experience, intuition and personal judgment of the inspector. Nevertheless, although the condition assessment of each element requires the inspector's personal judgment, general guidelines on how to assess the condition of the various elements are described in the

Condition Ranking and Rating of Bridges Using Fuzzy Logic 185

importance reported by Melhem and Aturaliya (1996) are used in the present study for structural importance factors of the elements under each component of the bridge. In this study, a scale of 1-9 has been considered for rating of the elements. An element with rating value of 9 signifies the best possible condition without distress and the descending rating numbers represent the increased degree of distress. The fuzzy membership values of structural importance for the elements of deck, superstructure and substructure are given in Tables 4, 5 and 6 respectively. From Tables 4 - 6, it may be noted that the mean value of the importance of an element increases as the physical condition deteriorates. For example, importance of deck concrete with rating 1 is 0.96, whereas its importance is 0.42 when the

If Rn is a fuzzy set, representing rating of an element (where 'n' represents rating number i.e.

 Rn = m(rm) rm (m = 0,1,2,….,9) (16) where, (r) is a membership function representing the degree of membership of any fuzzy set and 0 ≤ ≤ 1. The function as described in Eq. (16) quantifies the ambiguity associated with the rating of any element of a bridge. Any rating number can be represented using

**Item 0 1 2 3 4 5 6 7 8 9**  1 wearing coat 1 0.90 0.80 0.70 0.61 0.51 0.45 0.33 0.23 0.17

concrete 1 0.96 0.92 0.89 0.85 0.81 0.77 0.72 0.50 0.42

joint 1 0.92 0.85 0.77 0.70 0.62 0.55 0.47 0.38 0.30

Table 4. Mean values of the structural importance for the deck elements for different rating

3 curbs 1 0.85 0.70 0.55 0.40 0.25 0.20 0.14 0.10 0.08 4 median 1 0.85 0.70 0.54 0.39 0.24 0.21 0.14 0.11 0.09 5 sidewalks 1 0.88 0.76 0.64 0.52 0.40 0.33 0.25 0.17 0.14 6 parapets 1 0.88 0.76 0.63 0.51 0.39 0.33 0.26 0.19 0.19 7 railing 1 0.88 0.76 0.65 0.53 0.41 0.35 0.26 0.19 0.16 8 paint 1 0.87 0.74 0.61 0.48 0.35 0.31 0.24 0.18 0.15 9 drains 1 0.90 0.80 0.70 0.61 0.51 0.45 0.35 0.29 0.22 10 lighting 1 0.86 0.72 0.57 0.43 0.29 0.27 0.20 0.16 0.15 11 utilities 1 0.85 0.70 0.55 0.40 0.25 0.23 0.17 0.13 0.11 12 joint leakage 1 0.91 0.82 0.72 0.63 0.54 0.49 0.41 0.34 0.28

Mean values of Structural Importance

n =0,1,…..9), the general form of the membership function can be formed as follows:

**4.2 Fuzzification of input data obtained from bridge inspectors**

fuzzy membership function (Emami et al. 1998).

**Rating**

rating is 9.

SL No.

2deck

13expansion

inspector's manual. Hence, while two competent bridge inspectors may differ on the rating of a given element, but their difference in the rating would not be significant.

Table 3. Decomposition of a bridge into elements with observed ratings (**R**)

#### **4.1.3 Evaluation of importance factors**

In a bridge condition evaluation, rating of each element under a particular component does not influence the component's overall structural condition rating in a similar degree. A well trained inspector or the concerned expert determines the structural importance of different elements of all components of a bridge. The importance factor of element is not constant but varies with the degree of distress sustained by the element under consideration. Hence, determination of structural importance factors for various bridge elements is not an easy task. The knowledge gained by the bridge inspectors and experts through many years of design and inspection experience can not be obtained directly through structural analysis, although analysis can provide general trends of the behaviour of damaged members.

So, the importance factors for the elements at various deterioration stages should be evolved from the response of competent bridge inspectors/experts. These membership functions for structural importance were originally constructed through a survey among a number of bridge engineers and inspectors (Tee et al., 1988). Then, the collected data was statistically processed and the mean was presented by Melhem and Aturaliya (1996). As membership functions for structural importance corresponding to different ratings of elements/components is not bridge specific, membership functions for structural

inspector's manual. Hence, while two competent bridge inspectors may differ on the rating

1. Wearing Surface 8 1. Bearing devices 5 1. Bridge seats 6 2. Deck condition 9 2. Stringers 2. Wings 5 3. Kerbs 6 3. Girders 4 3. Back wall 6 4. Median 9 4. Floor beams 7 4. Footings 5. Sidewalks 8 5. Trusses 5. Piles 7 6. Parapets 9 6. Paint 5 6. Erosion 8 7. Railings 6 7. Machinery 7. Settlements 9 8. Paint 7 8. Rivets-Bolts 8. Pier-cap 2 9. Drains 8 9. Welds 2 9. Pier-column 5 10. Lighting 9 10. Rust 4 10. Pier-footing 3 11. Utilities 8 11. Timber decay 11. Pier-piles 6 12. Joint leakage 5 12. Concrete cracks 5 12. Pier-scour 5 13. Expansion joints 9 13. Collision damage 6 13. Pier-settlement 6

17. Timber decay

In a bridge condition evaluation, rating of each element under a particular component does not influence the component's overall structural condition rating in a similar degree. A well trained inspector or the concerned expert determines the structural importance of different elements of all components of a bridge. The importance factor of element is not constant but varies with the degree of distress sustained by the element under consideration. Hence, determination of structural importance factors for various bridge elements is not an easy task. The knowledge gained by the bridge inspectors and experts through many years of design and inspection experience can not be obtained directly through structural analysis,

although analysis can provide general trends of the behaviour of damaged members.

So, the importance factors for the elements at various deterioration stages should be evolved from the response of competent bridge inspectors/experts. These membership functions for structural importance were originally constructed through a survey among a number of bridge engineers and inspectors (Tee et al., 1988). Then, the collected data was statistically processed and the mean was presented by Melhem and Aturaliya (1996). As membership functions for structural importance corresponding to different ratings of elements/components is not bridge specific, membership functions for structural

Table 3. Decomposition of a bridge into elements with observed ratings (**R**)

**4.1.3 Evaluation of importance factors**

**Deck R Superstructure R Substructure R**

 14. Deflection 5 14. Pier-bents 4 15. Member alignment 7 15. Concrete cracks 5 16. Vibrations 6 16. Steel corrosion 8

 18. Debris seats 5 19. Paint 6 20. Collision damage 5

Note: - not applicable

of a given element, but their difference in the rating would not be significant.

importance reported by Melhem and Aturaliya (1996) are used in the present study for structural importance factors of the elements under each component of the bridge. In this study, a scale of 1-9 has been considered for rating of the elements. An element with rating value of 9 signifies the best possible condition without distress and the descending rating numbers represent the increased degree of distress. The fuzzy membership values of structural importance for the elements of deck, superstructure and substructure are given in Tables 4, 5 and 6 respectively. From Tables 4 - 6, it may be noted that the mean value of the importance of an element increases as the physical condition deteriorates. For example, importance of deck concrete with rating 1 is 0.96, whereas its importance is 0.42 when the rating is 9.

#### **4.2 Fuzzification of input data obtained from bridge inspectors**

If Rn is a fuzzy set, representing rating of an element (where 'n' represents rating number i.e. n =0,1,…..9), the general form of the membership function can be formed as follows:

$$\mathbf{R\_n = \mu\_m(r\_m) \mid r\_m} \qquad \quad (\mathbf{m = 0, 1, 2, \dots, 9}) \tag{16}$$

where, (r) is a membership function representing the degree of membership of any fuzzy set and 0 ≤ ≤ 1. The function as described in Eq. (16) quantifies the ambiguity associated with the rating of any element of a bridge. Any rating number can be represented using fuzzy membership function (Emami et al. 1998).

Table 4. Mean values of the structural importance for the deck elements for different rating

Condition Ranking and Rating of Bridges Using Fuzzy Logic 187

**Item 0 1 2 3 4 5 6 7 8 9**

crack 1 0.94 0.88 0.82 0.76 0.70 0.62 0.51 0.37 0.32

corrosion 1 0.95 0.90 0.84 0.79 0.74 0.66 0.54 0.43 0.36 17 timber decay 1 0.96 0.93 0.89 0.86 0.82 0.72 0.62 0.50 0.44 18 debris, seats 1 0.89 0.78 0.68 0.57 0.46 0.40 0.33 0.25 0.21 19 paint 1 0.90 0.79 0.69 0.58 0.48 0.41 0.34 0.26 0.22

damage 1 0.94 0.87 0.81 0.74 0.68 0.59 0.48 0.34 0.28

Table 6. Mean values of the structural importance for the bridge substructure elements for

R0 = {1.00 0, 0.76 1, 0.55 2, 0.35 3, 0.16 4, 0.00 5, 0.00 6,………. ,0.00 9} and

R9 = {0.00 0, 0.09 1, 0.18 2, 0.28 3, 0.39 4, 0.51 5, 0.62 6, 0.74 7, 0.87 8, 1.00 9}

Fuzzy membership functions for rating values 0 - 9, as obtained above, are shown in Fig. 2.

Rating value =0 Rating value =1 Rating value =2 Rating value =3 Rating value =4 Rating value =5 Rating value =6 Rating value =7 Rating value =8 Rating value =9

0123456789 **Rating value**

R1 = {0.00 0, 1.00 1, 0.45 2, 0.00 3, 0.00 4, 0.00 5, 0.00 6,………. ,0.00 9} Using fuzzy addition, rating membership functions for '2' is calculated as R2 = {0.00 0, 0.45 1, 1.00 2, 0.70 3, 0.45 4, 0.20 5, 0.00 6,………. ,0.00 9}

Usually, the membership values for each rating value are assumed without indication of any specific reason. If membership functions for rating values of 0 and 1 are specified, the membership functions for other rating values can be evaluated using consecutive fuzzy addition rule (Kaufmann & Gupta, 1985). In this study, the rating membership functions for

Mean values of Structural Importance

SL No.

15concrete

20collision

16steel

**Rating**

'0' and '1' are assumed as follows:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 2. Degree of membership of fuzzified rating values

**Degree of membership**

Table 5. Mean values of the structural importance for the superstructure elements for different rating

**Item 0 1 2 3 4 5 6 7 8 9**

device 1 0.96 0.92 0.07 0.83 0.79 0.71 0.60 0.47 0.42 2 stringers 1 0.96 0.92 0.07 0.83 0.79 0.72 0.61 0.50 0.44 3 girders 1 0.98 0.97 0.95 0.94 0.92 0.85 0.75 0.64 0.58 4 floor beams 1 0.98 0.96 0.94 0.92 0.90 0.83 0.72 0.60 0.54 5 trusses 1 0.97 0.94 0.90 0.87 0.84 0.77 0.67 0.56 0.51 6 paints 1 0.90 0.80 0.70 0.60 0.49 0.43 0.35 0.29 0.24 7 machinery 1 0.94 0.88 0.82 0.76 0.70 0.66 0.58 0.52 0.44 8 rivet or bolts 1 0.96 0.91 0.87 0.82 0.78 0.71 0.61 0.49 0.42 9 weld cracks 1 0.97 0.95 0.92 0.90 0.87 0.83 0.73 0.63 0.56 10 rusts 1 0.95 0.90 0.84 0.79 0.74 0.64 0.54 0.40 0.31 11 timber decay 1 0.97 0.93 0.90 0.86 0.83 0.75 0.65 0.51 0.43

crack 1 0.96 0.91 0.87 0.82 0.78 0.70 0.59 0.49 0.40

damage 1 0.94 0.88 0.83 0.77 0.71 0.64 0.53 0.42 0.35 14 deflection 1 0.95 0.89 0.84 0.78 0.73 0.66 0.59 0.50 0.43 15 alignment 1 0.94 0.88 0.83 0.77 0.71 0.64 0.54 0.44 0.37 16 vibrations 1 0.95 0.88 0.81 0.75 0.69 0.63 0.54 0.43 0.36

**Item 0 1 2 3 4 5 6 7 8 9**  1 bridge seats 1 0.95 0.90 0.86 0.81 0.76 0.68 0.57 0.45 0.40 2 wings 1 0.92 0.73 0.75 0.66 0.58 0.51 0.41 0.33 0.29 3 backwall 1 0.85 0.86 0.80 0.73 0.66 0.58 0.48 0.40 0.35 4 footings 1 0.95 0.90 0.84 0.79 0.74 0.67 0.57 0.46 0.42 5 piles 1 0.94 0.89 0.83 0.78 0.72 0.66 0.56 0.46 0.39 6 erosion 1 0.94 0.87 0.81 0.74 0.68 0.60 0.51 0.40 0.35 7 settlement 1 0.96 0.92 0.87 0.83 0.79 0.70 0.60 0.50 0.45 8 piers, caps 1 0.95 0.89 0.84 0.73 0.78 0.65 0.56 0.46 0.41

columns 1 0.96 0.91 0.87 0.82 0.78 0.70 0.60 0.49 0.43 10 piers, footing 1 0.95 0.90 0.84 0.79 0.74 0.67 0.57 0.47 0.42 11 piers, piles 1 0.95 0.90 0.84 0.79 0.74 0.68 0.59 0.48 0.38 12 piers, scour 1 0.95 0.89 0.84 0.78 0.73 0.65 0.53 0.43 0.45

settlement 1 0.96 0.91 0.87 0.82 0.78 0.70 0.62 0.51 0.42 14 pile, bends 1 0.95 0.90 0.85 0.80 0.75 0.67 0.58 0.48 0.42

Table 5. Mean values of the structural importance for the superstructure elements for

Mean values of Structural Importance

Mean values of Structural Importance

SL No.

1bearing

12concrete

13collision

different rating

9piers,

13Piers

**Rating**

SL No. **Rating**

Table 6. Mean values of the structural importance for the bridge substructure elements for

Usually, the membership values for each rating value are assumed without indication of any specific reason. If membership functions for rating values of 0 and 1 are specified, the membership functions for other rating values can be evaluated using consecutive fuzzy addition rule (Kaufmann & Gupta, 1985). In this study, the rating membership functions for '0' and '1' are assumed as follows:

R0 = {1.00 0, 0.76 1, 0.55 2, 0.35 3, 0.16 4, 0.00 5, 0.00 6,………. ,0.00 9} and

R1 = {0.00 0, 1.00 1, 0.45 2, 0.00 3, 0.00 4, 0.00 5, 0.00 6,………. ,0.00 9}

Using fuzzy addition, rating membership functions for '2' is calculated as

R2 = {0.00 0, 0.45 1, 1.00 2, 0.70 3, 0.45 4, 0.20 5, 0.00 6,………. ,0.00 9}

R9 = {0.00 0, 0.09 1, 0.18 2, 0.28 3, 0.39 4, 0.51 5, 0.62 6, 0.74 7, 0.87 8, 1.00 9}

Fuzzy membership functions for rating values 0 - 9, as obtained above, are shown in Fig. 2.

Fig. 2. Degree of membership of fuzzified rating values

Condition Ranking and Rating of Bridges Using Fuzzy Logic 189

with the set *A*

The minimum (pessimistic) and maximum (optimistic) values of the intervals for a specific level set correspond respectively to the lower and upper limits of fuzzy membership function at that -level. The set describing the rating of a component at a particular level of

1

*i*

*i*th element at -level. Therefore, the most pessimistic and optimistic range of the resulting set at each -level would form all possible combinations using the discretised non-fuzzy values. Hence, the resolution identity technique provides a convenient way of generalizing

From the above mentioned techniques for processing fuzzy sets, the FWA technique is simpler and faster. As FWA technique does not require discretisation of fuzzy set, accurate result may be achieved with less computational effort provided that the sets representing the rating or importance of different elements are convex. Otherwise, adjustments have to be made to the resulting fuzzy set to ensure its convexity for making the task of transforming a computed fuzzy set into natural language expression easier. Further, another adjustment that is often made to a fuzzy set is the normalization operation to ensure that at least one of the elements of the set contains the degree of membership of one, as suggested by Mullarky and Fenves (1985). On the other hand, accuracy of resolution identity technique depends on refinement of the concerned sets through -level which has a direct impact on computational time. Therefore, in this study, a methodology has been proposed by

In this present approach, the results obtained from eigenvector based priority setting approach combined with FWA for rating of the bridge components are taken as the input for the resolution identity module. It is to be mentioned here that the number of alternatives increase with the increase in the objects (here, components) considered. For example, if a bridge is considered to consist of three main components, such as, 'deck', 'superstructure' and 'substructure' with different ratings and importance factors, number of alternatives produced for each -level is 23+3 = 64 to determine the most optimistic and pessimistic values. Further, for 11 -levels (from 0 to 1.0 in step of 0.1), total number of calculations are

*i n*

*R*

is the rating value for the *i*th element at -level, *Wi*

various concepts associated with non-fuzzy sets to fuzzy sets.

judiciously using the advantages of both the techniques.

**4.4 Combined technique for condition rating of existing bridges**

*n*

1

*i i*

*W R*

*W*

*i*

1

(20)

1

0 (or

(21)

is the importance value for the

) is the

0 *A*

, and the symbol

or, A=

1

ranging from 0 and 1.

0 *A*

A=

is the product of a scalar

where,

*A*

> , with

union of the *A*

would be

where, *Ri*

#### **4.3 Overall condition rating of a bridge**

After getting the fuzzified rating and importance of all the elements, it is required to process those sets to arrive at the rating set for the components. In the similar way, the final rating for the bridge can be evaluated by processing the rating and importance sets of components. Generally, the processing of these rating and importance sets is executed using Fuzzy Weighted Average (FWA) or resolution identity technique. Brief details of these two techniques are given below:

#### **4.3.1 Fuzzy Weighted Average (FWA) technique**

Using a structural damage rating scheme according to the local, global and cumulative damage of the structure, resulting damage rating, R, can be evolved (Bertero & Bresler, 1977), using a weighted average approach, as

$$\mathbf{R} = \frac{\sum (w\_i \not\! \mathbf{Q} \mathbf{1}\_i)}{\sum (w\_i \not\! \mathbf{1}\_i \tau\_i)} \tag{17}$$

where, *wi* is the importance factor for the *i*th structural element,

*i* is the service history influence coefficient for capacity, and

*i* is the resistance (or capacity) in the *i*th element

For a bridge structure, Eq. (2) can be simplified for obtaining the rating as

$$\mathbf{R} = \frac{\sum\_{i=1}^{p} (w\_i \times r\_i)}{\sum\_{i=1}^{p} w\_i} \tag{18}$$

where, *wi* is the importance coefficient of the *i*th object, *ri* is the local rating of the ith object and R is the global or overall rating index when *wi* and *ri* stand for the bridge components, R is the component rating when *wi* and *ri* stand for the bridge elements. Detailed discussions on this methodology are given elsewhere (Sasmal et al., 2004a, 2004b).

#### **4.3.2 Fuzzy resolution identification technique**

A fuzzy set can be easily decomposed into its level sets or intervals through resolution identity as suggested by Dong and Wong (1987). If A is a fuzzy set of universe (U), then an -level set or alpha cut of A is a non-fuzzy set denoted by A which comprises of all elements of U whose grade of membership in A is greater than or equal to .

A can be expressed in symbolic form as:

$$\mathbf{A}\_a = \{ \mathbf{u} \mid \mu\_\mathbf{A}(\mathbf{u}) \ge \alpha \} \tag{19}$$

In mathematical form, the fuzzy set A can be decomposed into its level sets through the resolution identity such that

188 Fuzzy Logic – Emerging Technologies and Applications

After getting the fuzzified rating and importance of all the elements, it is required to process those sets to arrive at the rating set for the components. In the similar way, the final rating for the bridge can be evaluated by processing the rating and importance sets of components. Generally, the processing of these rating and importance sets is executed using Fuzzy Weighted Average (FWA) or resolution identity technique. Brief details of these two

Using a structural damage rating scheme according to the local, global and cumulative damage of the structure, resulting damage rating, R, can be evolved (Bertero & Bresler,

> *w w*

( ) *ii i iii*

(17)

(18)

**4.3 Overall condition rating of a bridge**

**4.3.1 Fuzzy Weighted Average (FWA) technique**

R= ( )

*i* is the resistance (or capacity) in the *i*th element

**4.3.2 Fuzzy resolution identification technique**

A can be expressed in symbolic form as:

resolution identity such that

where, *wi* is the importance factor for the *i*th structural element,

*i* is the service history influence coefficient for capacity, and

*i* is the service history coefficient for structural response (or demand), *i* is the structural response (or demand) in the *i*th element due to load,

For a bridge structure, Eq. (2) can be simplified for obtaining the rating as

on this methodology are given elsewhere (Sasmal et al., 2004a, 2004b).

elements of U whose grade of membership in A is greater than or equal to .

R = 1

*i p i i*

*p*

1

where, *wi* is the importance coefficient of the *i*th object, *ri* is the local rating of the ith object and R is the global or overall rating index when *wi* and *ri* stand for the bridge components, R is the component rating when *wi* and *ri* stand for the bridge elements. Detailed discussions

A fuzzy set can be easily decomposed into its level sets or intervals through resolution identity as suggested by Dong and Wong (1987). If A is a fuzzy set of universe (U), then an -level set or alpha cut of A is a non-fuzzy set denoted by A which comprises of all

 A = {u A(u) } (19) In mathematical form, the fuzzy set A can be decomposed into its level sets through the

( )

*w r*

*w*

*i i*

1977), using a weighted average approach, as

techniques are given below:

$$\mathbf{A} = \sum\_{\alpha=0}^{1} \alpha A\_{\alpha} \quad \text{or, } \mathbf{A} = \underset{\mathbf{0}}{\int} \alpha A\_{\alpha} \tag{20}$$

where, *A* is the product of a scalar with the set *A* , and the symbol 1 0 (or ) is the union of the *A* , with ranging from 0 and 1.

The minimum (pessimistic) and maximum (optimistic) values of the intervals for a specific level set correspond respectively to the lower and upper limits of fuzzy membership

function at that -level. The set describing the rating of a component at a particular level of would be

$$\mathcal{R}\_{\alpha} = \frac{\sum\_{i=1}^{n} \mathcal{W}\_{i\alpha} \mathcal{R}\_{i\alpha}}{\sum\_{i=1}^{n} \mathcal{W}\_{i\alpha}} \tag{21}$$

where, *Ri*is the rating value for the *i*th element at -level, *Wi* is the importance value for the *i*th element at -level. Therefore, the most pessimistic and optimistic range of the resulting set at each -level would form all possible combinations using the discretised non-fuzzy values. Hence, the resolution identity technique provides a convenient way of generalizing various concepts associated with non-fuzzy sets to fuzzy sets.

From the above mentioned techniques for processing fuzzy sets, the FWA technique is simpler and faster. As FWA technique does not require discretisation of fuzzy set, accurate result may be achieved with less computational effort provided that the sets representing the rating or importance of different elements are convex. Otherwise, adjustments have to be made to the resulting fuzzy set to ensure its convexity for making the task of transforming a computed fuzzy set into natural language expression easier. Further, another adjustment that is often made to a fuzzy set is the normalization operation to ensure that at least one of the elements of the set contains the degree of membership of one, as suggested by Mullarky and Fenves (1985). On the other hand, accuracy of resolution identity technique depends on refinement of the concerned sets through -level which has a direct impact on computational time. Therefore, in this study, a methodology has been proposed by judiciously using the advantages of both the techniques.

#### **4.4 Combined technique for condition rating of existing bridges**

In this present approach, the results obtained from eigenvector based priority setting approach combined with FWA for rating of the bridge components are taken as the input for the resolution identity module. It is to be mentioned here that the number of alternatives increase with the increase in the objects (here, components) considered. For example, if a bridge is considered to consist of three main components, such as, 'deck', 'superstructure' and 'substructure' with different ratings and importance factors, number of alternatives produced for each -level is 23+3 = 64 to determine the most optimistic and pessimistic values. Further, for 11 -levels (from 0 to 1.0 in step of 0.1), total number of calculations are

Condition Ranking and Rating of Bridges Using Fuzzy Logic 191

 Li = [i,1(c1)c1, i,2(c2)c2, i,3(c3)c3, , i,n(cn)cn ]i = 1, p (22) in which (ci) is the membership grade corresponding to alternative ci. The solution would be the optimal alternative which has the highest degree of acceptability with respect to all relevant goals Li. Towards this, several models have been introduced, in recent years, for fuzzy MADM but the eigenvector based priority setting approach is considered as one of the

For the general MADM problem described using Eq. (22), a positive, non-zero number in the priority vector (W) corresponding to each object indicates the power of importance (i) of that object in the decision process. By applying the associated powers 1, 2, 3,…. p to the

The decision function D is then obtained from the intersection of the fuzzy sets representing

where, D1(c1), D2(c2), D3(c3), ……………, Dn(cn) are the decision values corresponding to the

The final decision is the one that corresponds to maximum of all decision values. Hence, the

 Dfinal = max[Dj(cj) cj]; j = 1,n (27) For the bridge rating application, let e1, e2, e3,……,ep be the elements considered under each component of the bridge. The fuzzy set for a given condition rating ri of element ei can be

Li = [R]ei = {i,1 r1, i,2 r2,…….., i,9 r9} (28)

Therefore, decision value (D) for rating of any element can be evaluated from Eq. 25 as

The final rating of each of the major bridge components is found from the decision values as

Dfinal = max { d1 r1, d2 r2, ……………,d9 r9}

2,j j [μ (c ) c2, α3

i,3 3 [μ (c ) c3, ……… αi

………….. L *p*

3,j j [μ (c ) c3, ………, αp

D = [d1 r1, d2 r2, d3 r3, ……………, dn rn] (29)

Or, D = [D1(c1) c1, D2(c2) c2, D3(c3) c3, ……………, Dn(cn) cn] (25)

*p*

i,n n [μ (c ) cn]; i = 1, p (23)

(24)

p,j j [μ (c ) cp]; j = 1, n (26)

**4.6 Application of priority vector in MADM model for condition rating**

α α i i L [ i i,1 μ 1 (c ) c1, αi

α1 D ( ) min[μ1,j j (c ) *j j c* c1, α2

expressed as the objective (goal) Li, as

fuzzy objective sets L1, L2, L 3,….., L p respectively, the following can be obtained:

i,2 2 [μ (c ) c2, αi

D = 1 L1 2 L2 3 L3

alternatives c1, c2,c3,…..,cn, and are given by Aturaliya (1994) as :

where i = 1 , p and i,n is the membership value of element ei at rn.

best alternatives.

the goals as

final decision becomes

704. The same increases enormously with the increase in number of objects (here, components or elements of the bridge). If the procedure mentioned above is implemented for a bridge which is assumed to be divided into 10 components, the number of calculations required to get the resultant rating fuzzy set would be 210+10 11 = 11534336. In fact, the availability of high speed microcomputers has made this approach attractive and practical for actual bridge inspection, management and planning applications.

Further, question may arise that why the resolution identity technique alone can not be applied for the whole bridge rating system by avoiding FWA technique. The simple answer is that component rating can be evolved by the simple FWA technique because there is no need for tackling non-convexity of the assigned sets unless it is essential. Otherwise, for the whole rating evaluation, number of calculations would be enormous. For a bridge having 3 components with 13, 16 and 20 elements (as described in Table 3) under the components, the total number of calculations required for the final result using resolution identity alone would be in the order of 1.21013. Hence, a combined technique is proposed in this study to get the accurate result without much increase in computing time.

In the approach proposed in this study, priority setting values of elements are calculated to evaluate the power of importance of each element in describing the condition of a particular component. The usual techniques available for condition rating combine the rating and importance of elements to arrive at the rating of each component. But, the importance factor, as mentioned earlier, is very much dependent on the prevailing condition (rating) of the particular element. Thus, a minor element with worse condition may unnecessarily reduce the rating value of that component under which the element is grouped. This problem can be tackled by the introduction of power of importance which is independent of the prevailing condition of elements. As mentioned earlier, imprecision, subjective judgment and uncertainty are associated with bridge inspection data. Because of uncertainty, the bridge inspector may not exactly know the prevailing condition (rating) of a particular element of a bridge. Moreover, importance factor for an element depends on its rating. But, decision on rating is a difficult proposition. Under these circumstances, several models were introduced for decision making in a fuzzy environment. In this study, Multi-Attributive Decision Making (MADM) model has been adopted as a decision tool.

#### **4.5 Multi-Attributive Decision Making (MADM) model**

Multi-Attributive Decision Making (MADM) model is one of the methods in decision studies where the factors towards a priority decision are many (multi-criteria). The assessment of bridge rating can be viewed as a Multi-Attributive Decision Making model because of its many components and sub-components (elements). In this study, an attempt has been made to use MADM model, to get the priority vector of elements depending on their importance over the others which would lead to a reliable decision (rating) from the bridge inspection data. The general MADM model can be expressed as follows:

Let L = {Lii = 1,2,3,….,p} be a set of goals and C = {cjj=1,2,3,….,n} be a finite set of decision alternatives from which the acceptability of the alternatives is judged. The objective is to select the one, from these alternatives, that best satisfies the set of goals, L1,…….Lp. The objective function can be expressed in the form of fuzzy set as follows:

704. The same increases enormously with the increase in number of objects (here, components or elements of the bridge). If the procedure mentioned above is implemented for a bridge which is assumed to be divided into 10 components, the number of calculations required to get the resultant rating fuzzy set would be 210+10 11 = 11534336. In fact, the availability of high speed microcomputers has made this approach attractive and practical

Further, question may arise that why the resolution identity technique alone can not be applied for the whole bridge rating system by avoiding FWA technique. The simple answer is that component rating can be evolved by the simple FWA technique because there is no need for tackling non-convexity of the assigned sets unless it is essential. Otherwise, for the whole rating evaluation, number of calculations would be enormous. For a bridge having 3 components with 13, 16 and 20 elements (as described in Table 3) under the components, the total number of calculations required for the final result using resolution identity alone would be in the order of 1.21013. Hence, a combined technique is proposed in this study to

In the approach proposed in this study, priority setting values of elements are calculated to evaluate the power of importance of each element in describing the condition of a particular component. The usual techniques available for condition rating combine the rating and importance of elements to arrive at the rating of each component. But, the importance factor, as mentioned earlier, is very much dependent on the prevailing condition (rating) of the particular element. Thus, a minor element with worse condition may unnecessarily reduce the rating value of that component under which the element is grouped. This problem can be tackled by the introduction of power of importance which is independent of the prevailing condition of elements. As mentioned earlier, imprecision, subjective judgment and uncertainty are associated with bridge inspection data. Because of uncertainty, the bridge inspector may not exactly know the prevailing condition (rating) of a particular element of a bridge. Moreover, importance factor for an element depends on its rating. But, decision on rating is a difficult proposition. Under these circumstances, several models were introduced for decision making in a fuzzy environment. In this study, Multi-Attributive

Multi-Attributive Decision Making (MADM) model is one of the methods in decision studies where the factors towards a priority decision are many (multi-criteria). The assessment of bridge rating can be viewed as a Multi-Attributive Decision Making model because of its many components and sub-components (elements). In this study, an attempt has been made to use MADM model, to get the priority vector of elements depending on their importance over the others which would lead to a reliable decision (rating) from the

Let L = {Lii = 1,2,3,….,p} be a set of goals and C = {cjj=1,2,3,….,n} be a finite set of decision alternatives from which the acceptability of the alternatives is judged. The objective is to select the one, from these alternatives, that best satisfies the set of goals, L1,…….Lp. The

bridge inspection data. The general MADM model can be expressed as follows:

objective function can be expressed in the form of fuzzy set as follows:

for actual bridge inspection, management and planning applications.

get the accurate result without much increase in computing time.

Decision Making (MADM) model has been adopted as a decision tool.

**4.5 Multi-Attributive Decision Making (MADM) model**

$$\mathbf{L}\_{i} = \begin{bmatrix} \mu\_{\nu 1}(\mathbf{c}\_{1}) | \mathbf{c}\_{1\prime} \ \mu\_{\nu 2}(\mathbf{c}\_{2}) | \mathbf{c}\_{2\prime} \ \mu\_{\nu 3}(\mathbf{c}\_{3}) | \mathbf{c}\_{3\prime} \ \mu\_{\nu \nu}(\mathbf{c}\_{n}) | \mathbf{c}\_{n} \end{bmatrix} \tag{22}$$

in which (ci) is the membership grade corresponding to alternative ci. The solution would be the optimal alternative which has the highest degree of acceptability with respect to all relevant goals Li. Towards this, several models have been introduced, in recent years, for fuzzy MADM but the eigenvector based priority setting approach is considered as one of the best alternatives.

#### **4.6 Application of priority vector in MADM model for condition rating**

For the general MADM problem described using Eq. (22), a positive, non-zero number in the priority vector (W) corresponding to each object indicates the power of importance (i) of that object in the decision process. By applying the associated powers 1, 2, 3,…. p to the fuzzy objective sets L1, L2, L 3,….., L p respectively, the following can be obtained:

$$\mathbf{L}\_{\mathbf{i}}^{\mathbf{a}\_{\mathbf{i}}} = \left[ \boldsymbol{\mu}\_{\mathbf{i},1}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{1}) \mid \mathbf{c}\_{1} \quad \left[ \boldsymbol{\mu}\_{\mathbf{i},2}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{2}) \mid \mathbf{c}\_{2} \quad \left[ \boldsymbol{\mu}\_{\mathbf{i},3}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{3}) \mid \mathbf{c}\_{3} \dots \dots \dots \quad \left[ \boldsymbol{\mu}\_{\mathbf{i},n}^{\mathbf{a}\_{\mathbf{i}}}(\mathbf{c}\_{n}) \mid \mathbf{c}\_{n} \right] \right] \right] \tag{23}$$

The decision function D is then obtained from the intersection of the fuzzy sets representing the goals as

$$\mathbf{D} = \mathbf{L}\_1^{a\_1} \cap \mathbf{L}\_2^{a\_2} \cap \mathbf{L}\_3^{a\_3} \text{ ...} \dots \text{....} \dots \text{//} \mathbf{L}\_p^{a\_p} \tag{24}$$

$$\text{Or}, \text{D} = \left[ \text{D}\_{\text{l}}(\text{c}\_{\text{l}}) \mid \text{c}\_{\text{l}} \text{ D}\_{\text{2}}(\text{c}\_{\text{2}}) \mid \text{c}\_{\text{2}} \text{ D} \text{(c}\_{\text{3}}) \mid \text{c}\_{\text{3}} \text{ } \dots \text{ } \dots \dots \dots \text{ } \text{D}\_{\text{n}}(\text{c}\_{\text{n}}) \mid \text{c}\_{\text{n}} \right] \tag{25}$$

where, D1(c1), D2(c2), D3(c3), ……………, Dn(cn) are the decision values corresponding to the alternatives c1, c2,c3,…..,cn, and are given by Aturaliya (1994) as :

$$\mathbf{D}\_{\rangle}(\mathbf{c}\_{\rangle}) = \min \{ \mu\_{1\not\downarrow}^{a\_{\sharp}}(\mathbf{c}\_{\not\slash}) \mid \mathbf{c}\_{\not\succ} \quad \| \mu\_{2\not\succ}^{a\_{\sharp}}(\mathbf{c}\_{\not\succ}) \mid \mathbf{c}\_{\not\succ} \quad \| \mu\_{3\not\succ}^{a\_{\sharp}}(\mathbf{c}\_{\not\succ}) \mid \mathbf{c}\_{\not\succ} , \dots, \dots, \ \| \mu\_{p\not\succ}^{a\_{p}}(\mathbf{c}\_{\not\succ}) \mid \mathbf{c}\_{\not\succ} \} \colon\_{\mathbf{1},\mathbf{n}} \tag{26}$$

The final decision is the one that corresponds to maximum of all decision values. Hence, the final decision becomes

$$\mathbf{D}\_{\text{final}} = \max \{ \mathbf{D}\_{\bar{l}}(\mathbf{c}\_{\bar{l}}) \mid \mathbf{c}\_{\bar{l}} \}; \quad \mathbf{j} = \mathbf{1}, \mathbf{n} \tag{27}$$

For the bridge rating application, let e1, e2, e3,……,ep be the elements considered under each component of the bridge. The fuzzy set for a given condition rating ri of element ei can be expressed as the objective (goal) Li, as

$$\mathbf{L}\_{i} = \begin{bmatrix} \mathbf{R} \end{bmatrix}\_{ei} = \begin{Bmatrix} \mu\_{i,1} \mid \mathbf{r}\_{1\prime} \ \mu\_{i,2} \mid \mathbf{r}\_{2\prime}, \dots, \dots, \mu\_{i,9} \mid \mathbf{r}\_{9} \end{Bmatrix} \tag{28}$$

where i = 1 , p and i,n is the membership value of element ei at rn.

Therefore, decision value (D) for rating of any element can be evaluated from Eq. 25 as

$$\mathbf{D} = \begin{bmatrix} \mathbf{d}\_1 \mid \mathbf{r}\_1, \mathbf{d}\_2 \mid \mathbf{r}\_2 \; \mathbf{d}\_3 \mid \mathbf{r}\_3 & \dots & \dots & \dots & \mathbf{d}\_n \mid \mathbf{r}\_n \end{bmatrix} \tag{29}$$

The final rating of each of the major bridge components is found from the decision values as

$$\mathbf{D}\_{\text{final}} = \max\left\{ |\mathbf{d}\_1| \, \mathbf{r}\_1, \mathbf{d}\_2| \, \mathbf{r}\_2, \dots, \mathbf{r}, \dots, \mathbf{d}\_\bullet \, | \, \mathbf{r}\_\bullet \right\}$$

Condition Ranking and Rating of Bridges Using Fuzzy Logic 193

evaluated through visual inspection are grouped under visual assessment. The items which need a detailed inspection, comprehensive observation and thorough study are grouped under general assessment. Further, the items which would require rigorous testing, both at site and in the laboratory using sophisticated instrumentation are grouped under detailed assessment. The comparison matrices for different components are formed (Triantaphyllou et al., 1997; Sasmal et al. 2006). The relative importance of any item with respect to the other, under any layer, may change depending on location, societal importance and decision objectives. The eigen solutions of the comparison matrices are carried out for different criteria layers to check for the consistency (in other words, check for acceptance of the comparison matrix) of the elements assigned for different items under criteria layer. The largest eigenvalue (max) of the comparison matrix corresponding to each layer is obtained by solving Eq. 6 and this eigenvalue is used for calculating the consistency ratio (CR). The values for consistency index (CI) are obtained by using Eq. 7 and the consistency ratio (CR) is obtained by dividing CI with random consistency index (RCI). The values of max, CI, RCI and CR for different items under criteria layer are presented in Table 7 and the values of CR for different items (index layers) under criteria layer are within the acceptable limit (<10%). Hence, the comparison matrix assigned for index layers are accepted for further study.

Visual assessment (16) 17.3951 0.0930 1.56 5.962 General assessment (15) 16.1770 0.0841 1.56 5.391 Detailed assessment (14) 15.1707 0.0901 1.56 5.776

**5.2 Calculation of relative weights of different index layers on criteria layer**

assignment of the weights for formation of comparison matrix using AHP.

procedures. Hence, the comparison matrix has to be modified accordingly.

**5.3 Formulation of higher layer comparison matrix using AHP**

The relative weights for components of index layers (i), (j) and (k) are established using the Eq. 3. The relative importance (weights) of items under index layers (i, j and k) obtained in this study are presented in Table 8. From the table, it is clear that there are considerable differences in relative weights of items in each index layer which signify their importance on the functionality of a bridge as a whole. It is also worthy to mention here that a large variation of relative weights of items signifies the necessity for correct, logical and realistic

Next step is to form the pair-wise comparison matrix for criteria layer to get the optimum goal in objective layer. Following the scheme described above, relative weights of each item of the criteria layer has to be evaluated. Table 8 shows both the comparison matrix between criteria layers and the relative weights of each criteria layer on objective layer. It signifies that the relative weight of detailed assessment on condition assessment of existing bridge is much more than that of the visual assessment. But, it may be noted that the relative weights of different assessments on condition assessment of overall bridge may change with the type of bridge, specific site condition and the degree of accuracy of different assessment

Table 7. Check for consistency of pair-wise assigned weights

max CI =(max-n)/(n-1) RCI CR = (CI/RCI)100%

= [dm rm] (30)

Hence, the rating of the particular component would be 'm' that represents any integer value between 1 and 9.

#### **5. A case study for illustration of the proposed methodology**

Computer programs have been developed based on the formulations presented in the preceding sections for condition evaluation of existing bridges and rating of the most deserved one. Based on the formulation discussed in the previous sections and the computer program developed in this study using the formulations, a study has been made for priority ranking of bridges. The data corresponding to five RC bridges (Br1, Br2, Br3, Br4 and Br5) has been adopted. In order to use the AHP to rank these bridges, at first an Analytic Hierarchy Model (AHM) with three layers, such as, objective layer (OL), criteria layer (CL) and index layer (IL) is constructed, as shown in Fig. 3. This hierarchy model is constructed by the authors based on the information available from FHWA (1991) and Liang et al. (2003) to demonstrate the proposed methodology. It is worthy to mention that the proposed methodology can be used for any hierarchy model. Therefore, it may be noted that the appropriate item(s) under any layer (as shown in Fig. 3) may be added or deleted depending on the requirement for assessing the condition of concerned bridges.

Fig. 3. Analytic Hierarchy Model (AHM) for Existing RC Bridge

After establishing the model, a set of relative importance (weights) between each single factor evaluation (item) is set-up for controlling the reliability of layer ranking. Combination of relative importance (weights) to each single item forms a comparison matrix. Condition evaluation of the considered bridges through prioritization and the rating of the most deserved bridge are arrived using the methodology described in the preceding sections. The whole procedure has been described in following sections for better illustration and understanding.

#### **5.1 Formulation of comparison matrices for each layer and check for consistency**

The first step is to carry out pair-wise comparisons of items under each layer of the AHP model as shown in Fig. 3. In this study, criteria layer is divided into three parts, namely, visual assessment, general assessment and detailed assessment. The items which can be

Hence, the rating of the particular component would be 'm' that represents any integer

Computer programs have been developed based on the formulations presented in the preceding sections for condition evaluation of existing bridges and rating of the most deserved one. Based on the formulation discussed in the previous sections and the computer program developed in this study using the formulations, a study has been made for priority ranking of bridges. The data corresponding to five RC bridges (Br1, Br2, Br3, Br4 and Br5) has been adopted. In order to use the AHP to rank these bridges, at first an Analytic Hierarchy Model (AHM) with three layers, such as, objective layer (OL), criteria layer (CL) and index layer (IL) is constructed, as shown in Fig. 3. This hierarchy model is constructed by the authors based on the information available from FHWA (1991) and Liang et al. (2003) to demonstrate the proposed methodology. It is worthy to mention that the proposed methodology can be used for any hierarchy model. Therefore, it may be noted that the appropriate item(s) under any layer (as shown in Fig. 3) may be added or deleted

**5. A case study for illustration of the proposed methodology**

depending on the requirement for assessing the condition of concerned bridges.

Assessment of Existing Bridge

Visual Assessment (i) General Assessment (j) Detailed Assessment (k)

After establishing the model, a set of relative importance (weights) between each single factor evaluation (item) is set-up for controlling the reliability of layer ranking. Combination of relative importance (weights) to each single item forms a comparison matrix. Condition evaluation of the considered bridges through prioritization and the rating of the most deserved bridge are arrived using the methodology described in the preceding sections. The whole procedure has been described in following sections for better illustration and

**5.1 Formulation of comparison matrices for each layer and check for consistency**

The first step is to carry out pair-wise comparisons of items under each layer of the AHP model as shown in Fig. 3. In this study, criteria layer is divided into three parts, namely, visual assessment, general assessment and detailed assessment. The items which can be

1. slab 2. girders 3. …. 4…… …….. 15. abutment

Fig. 3. Analytic Hierarchy Model (AHM) for Existing RC Bridge

value between 1 and 9.

1. wearing surface 2. deck condition

16. Bearing device

3. …… 4 ……. ……….

understanding.

= [dm rm] (30)

1. existing prestress 2. concrete strength

Objective Layer (OB)

Criteria Layer (CR)

Index Layer (ID)

3. ….. 4. ….. ……… 14.anchor block evaluated through visual inspection are grouped under visual assessment. The items which need a detailed inspection, comprehensive observation and thorough study are grouped under general assessment. Further, the items which would require rigorous testing, both at site and in the laboratory using sophisticated instrumentation are grouped under detailed assessment. The comparison matrices for different components are formed (Triantaphyllou et al., 1997; Sasmal et al. 2006). The relative importance of any item with respect to the other, under any layer, may change depending on location, societal importance and decision objectives. The eigen solutions of the comparison matrices are carried out for different criteria layers to check for the consistency (in other words, check for acceptance of the comparison matrix) of the elements assigned for different items under criteria layer. The largest eigenvalue (max) of the comparison matrix corresponding to each layer is obtained by solving Eq. 6 and this eigenvalue is used for calculating the consistency ratio (CR). The values for consistency index (CI) are obtained by using Eq. 7 and the consistency ratio (CR) is obtained by dividing CI with random consistency index (RCI). The values of max, CI, RCI and CR for different items under criteria layer are presented in Table 7 and the values of CR for different items (index layers) under criteria layer are within the acceptable limit (<10%). Hence, the comparison matrix assigned for index layers are accepted for further study.

Table 7. Check for consistency of pair-wise assigned weights

#### **5.2 Calculation of relative weights of different index layers on criteria layer**

The relative weights for components of index layers (i), (j) and (k) are established using the Eq. 3. The relative importance (weights) of items under index layers (i, j and k) obtained in this study are presented in Table 8. From the table, it is clear that there are considerable differences in relative weights of items in each index layer which signify their importance on the functionality of a bridge as a whole. It is also worthy to mention here that a large variation of relative weights of items signifies the necessity for correct, logical and realistic assignment of the weights for formation of comparison matrix using AHP.

#### **5.3 Formulation of higher layer comparison matrix using AHP**

Next step is to form the pair-wise comparison matrix for criteria layer to get the optimum goal in objective layer. Following the scheme described above, relative weights of each item of the criteria layer has to be evaluated. Table 8 shows both the comparison matrix between criteria layers and the relative weights of each criteria layer on objective layer. It signifies that the relative weight of detailed assessment on condition assessment of existing bridge is much more than that of the visual assessment. But, it may be noted that the relative weights of different assessments on condition assessment of overall bridge may change with the type of bridge, specific site condition and the degree of accuracy of different assessment procedures. Hence, the comparison matrix has to be modified accordingly.

Condition Ranking and Rating of Bridges Using Fuzzy Logic 195

**Item weight**

**bridge**

3 kerbs 0.0152 0 2 2 1 0 4 median 0.0591 0 4 0 0 2 5 sidewalks 0.0189 0 2 2 2 0 6 parapets 0.0134 2 0 1 1 0 7 railing 0.0439 4 2 0 0 0 8 paints 0.0179 3 4 2 2 1 9 drains 0.1442 6 6 4 6 0 10 lighting 0.0282 9 9 0 0 0 11 utilities 0.0645 2 2 4 4 6 12 joint leakage 0.2012 0 0 3 9 12

16 others 0.0431 12 9 2 4 9

2 girders 0.0165 27 18 0 18 6 3 slab beams 0.0421 4 trusses 0.0252

6 rivet bolts 0.0704

9 erosion 0.0483 12 18 2 18 3

11 pier 0.1563 18 0 12 3 9 12 pier shaft 0.0375 0 12 0 0 0 13 friction layer 0.0867 0 9 0 0 0 14 abutment 0.2081 12 6 9 3 2 15 others 0.1900 0 2 3 12 9

0.297 1 stringers 0.0338

 **Estimation indices of items of**

Br1 Br2 Br3 Br4 Br5

0.0342 0 0 0 2 1

0.0194 2 0 12 9 3

0.1881 1 4 4 6 2

0.0719 3 3 12 18 3

0.0367 4 2 0 0 2

0.0200 12 0 0 2 9

0.0250 36 12 36 2 9

0.0169 27 27 12 18 3

0.0233 18 36 0 3 27

**Estimation items**

surface

condition

13 expansion joint

masonry

14 bearing device

5 chloride content

7 concrete crack

settlement

10 substructure protection

8 pier

15 wing

2 deck

**Estimation Criterion**

Visual assessment (i)

General assessment (j) **Subsystem weight**

**Item No.**

0.126 1 wearing

Table 8. Relative weights of items under index layer on criteria layer

#### **5.4 Fuzzy synthesis and evaluation of membership functions**

This step deals with the assessment of condition of bridge items under index layer (in this case, i, j and k). In this study, the assessment of items has been carried out by determining the estimation indices of the items as described in preceeding section. The estimation indices of the items for different bridges (Br1, Br2, Br3, Br4 and Br5) are presented in Table 9. In this table, the relative weights of components, are calculated using the procedure described above. In this study, the optimistic membership evaluation function has been used for developing membership functions. Using the estimation indices of items as tabulated in Table 9, the membership degrees, Rn,m (n=1 to 3; m = 1 to 16/15/14) of each items are calculated for the bridges (Br1, Br2, Br3, Br4, Br5) considered for assessment.

#### **5.5 Fuzzy synthesis evaluation matrix and priority ranking values**

The relationship between the membership degrees Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…no of items in each index layer) of each single factor (alternative) evaluation index and weight, *Wn* , is *D WR n n nm* . as per Eq. (14), where the value *Dn* in Eq. (14) is the fuzzy synthesis evaluation matrix. The proposition of *Dn* is to construct the membership function for each alternative of evaluation set. Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any factor can be expressed as in Eq. 15.

In this case, *B* = 0.456636 0.256391 0.296120 0.525126 0.43876

**Index layer (j) General assessment**

**Item Relative weight Item Relative weight Item Relative weight**  i1 0.0342 j1 0.0338 k1 0.0233 i2 0.0194 j2 0.0165 k2 0.0572 i3 0.0152 j3 0.0421 k3 0.0225 i4 0.0591 j4 0.0252 k4 0.1373 i5 0.0189 j5 0.0200 k5 0.0298 i6 0.0134 j6 0.0704 k6 0.0684 i7 0.0439 j7 0.0250 k7 0.1875 i8 0.0179 j8 0.0169 k8 0.1816 i9 0.1442 j9 0.0483 k9 0.0840 i10 0.0282 j10 0.0233 k10 0.0457 i11 0.0645 j11 0.1563 k11 0.0425 i12 0.2012 j12 0.0375 k12 0.0369 i13 0.1881 j13 0.0867 k13 0.0212 i14 0.0719 i14 0.2081 k14 0.0621

This step deals with the assessment of condition of bridge items under index layer (in this case, i, j and k). In this study, the assessment of items has been carried out by determining the estimation indices of the items as described in preceeding section. The estimation indices of the items for different bridges (Br1, Br2, Br3, Br4 and Br5) are presented in Table 9. In this table, the relative weights of components, are calculated using the procedure described above. In this study, the optimistic membership evaluation function has been used for developing membership functions. Using the estimation indices of items as tabulated in Table 9, the membership degrees, Rn,m (n=1 to 3; m = 1 to 16/15/14) of each items are

The relationship between the membership degrees Rn,m (in which n= 1,2….no of criteria layers, and m=1,2,…no of items in each index layer) of each single factor (alternative) evaluation index and weight, *Wn* , is *D WR n n nm* . as per Eq. (14), where the value *Dn* in Eq. (14) is the fuzzy synthesis evaluation matrix. The proposition of *Dn* is to construct the membership function for each alternative of evaluation set. Based on the fuzzy mathematics theory, the fuzzy synthesis evaluation result, *B* , of any factor can be expressed as in Eq. 15.

**Index layer (k) Detailed assessment**

**Index layer (i) Visual assessment**

i16 0.0431

i15 0.0367 j15 0.1900

Table 8. Relative weights of items under index layer on criteria layer

calculated for the bridges (Br1, Br2, Br3, Br4, Br5) considered for assessment.

In this case, *B* = 0.456636 0.256391 0.296120 0.525126 0.43876

**5.5 Fuzzy synthesis evaluation matrix and priority ranking values**

**5.4 Fuzzy synthesis and evaluation of membership functions**

Condition Ranking and Rating of Bridges Using Fuzzy Logic 197

reported in (Aturaliya, 1994). Fuzzy sets for rating of the components, i.e, deck, superstructure and substructure are shown in Table 10 and corresponding importances

> **Br1 Br2 Br3 Br4 Br5 Bridges to be prioritized**

 **0 1 2 3 4 5 6 7 8 9**  1. Bridge Deck 0.00 0.33 0.67 1.00 0.83 0.67 0.50 0.33 0.17 0.00 2. Superstructure 0.00 0.25 0.50 0.75 1.00 0.88 0.75 0.63 0.50 0.38 3. Substructure 0.00 0.25 0.50 0.75 1.00 0.88 0.75 0.63 0.50 0.38

 **0 1 2 3 4 5 6 7 8 9**  1. Bridge Deck 1.00 1.00 0.90 0.80 0.70 0.61 0.51 0.45 0.33 0.23 2. Superstructure 1.00 1.00 0.96 0.92 0.87 0.83 0.79 0.71 0.60 0.47 3. Substructure 1.00 1.00 0.80 0.70 0.60 0.50 0.35 0.25 0.15 0.10

As described earlier, the resolution identity technique is adopted in this study to get the final rating of the bridge when the component ratings (from the elemental values) are computed using eigenvector based priority setting technique using MADM combined with FWA method. Hence, for arriving at the final rating of the most deserved bridge, the basic data considered are the calculated ratings of the components and computed weights as shown in Tables 8 and 9. The fuzzy membership functions of rating and weights of different components (deck, superstructure and substructure) thus obtained, are discretised using resolution identity technique. Here, each set is discretised into 11 -levels (from 0.0 to 1.0 in

Fig. 4. Priority ranking values of the bridges considered for condition assessment

Table 10. Computed fuzzified rating values for different components of the bridge

Components Rating membership

Components Importance factors

Table 11. Importance membership functions of the components

(weights) are shown in Table 11.

0.00

0.10

0.20

0.30

**Priority Vector**

0.40

0.50

0.60

Represents the non-availability of estimation data

Table 9. Estimation indices of items of the bridges considered for condition assessment

The fuzzy synthesis evaluation result, *B* , actually shows the relative condition of existing bridges considered. Therefore, the values under *B* can also be treated as the priority vector for condition assessment of the bridges. As the optimistic membership evaluation function is used in this study (given in Eq. 10), the higher value in fuzzy synthesis evaluation result for a bridge in comparison to the other ones signifies greater degree of distress. In this study, the condition of Br4 among the five bridges considered here can be treated as most deficient and similarly, Br2 would be the best. The condition priority order of the bridges considered here for illustration is shown in Fig. 4. From the figure, it may be noted that the priority order of the bridges considered is as follows:

#### =[Br4, Br1, Br5, Br3, Br2]

#### **5.6 Condition rating of the most deserved bridge**

Using the fuzzy mathematics, ratings of different component of the bridge, Br4, are calculated using FWA. Importance(weight) of different elements has been considered as

**Item weight**

**bridge**

4 vibration 0.1373 18 9 0 36 27

7 deflection 0.1875 12 18 36 48 12 8 footing 0.1816 18 0 3 12 6

10 piles 0.0457 11 pier-column 0.0425 12 2 9 6 9 12 pier footing 0.0369 18 3 12 3 18

14 anchor block 0.0621 27 12 0 36 48

 **Estimation indices of items of**

Br1 Br2 Br3 Br4 Br5

0.0233 36 12 0 36 27

0.0572 27 18 36 36 48

0.0225 12 0 12 18 9

0.0298 18 0 0 0 18

0.0684 0 36 0 0 12

0.0840 0 0 0 0 12

0.0212 36 12 0 18 36

**Estimation items**

prestress

2 concrete strength

3 foundation mat

5 prevention earthquake block

corrosion

6 steel

9 collision damage

13 prestressing cable corrosion

Table 9. Estimation indices of items of the bridges considered for condition assessment

The fuzzy synthesis evaluation result, *B* , actually shows the relative condition of existing bridges considered. Therefore, the values under *B* can also be treated as the priority vector for condition assessment of the bridges. As the optimistic membership evaluation function is used in this study (given in Eq. 10), the higher value in fuzzy synthesis evaluation result for a bridge in comparison to the other ones signifies greater degree of distress. In this study, the condition of Br4 among the five bridges considered here can be treated as most deficient and similarly, Br2 would be the best. The condition priority order of the bridges considered here for illustration is shown in Fig. 4. From the figure, it may be noted that the priority

=[Br4, Br1, Br5, Br3, Br2]

Using the fuzzy mathematics, ratings of different component of the bridge, Br4, are calculated using FWA. Importance(weight) of different elements has been considered as

Represents the non-availability of estimation data

order of the bridges considered is as follows:

**5.6 Condition rating of the most deserved bridge**

**Estimation Criterion**

Detailed assessment (k) **Subsystem weight**

**Item No.**

0.577 1 existing

reported in (Aturaliya, 1994). Fuzzy sets for rating of the components, i.e, deck, superstructure and substructure are shown in Table 10 and corresponding importances (weights) are shown in Table 11.

Fig. 4. Priority ranking values of the bridges considered for condition assessment

Table 11. Importance membership functions of the components

As described earlier, the resolution identity technique is adopted in this study to get the final rating of the bridge when the component ratings (from the elemental values) are computed using eigenvector based priority setting technique using MADM combined with FWA method. Hence, for arriving at the final rating of the most deserved bridge, the basic data considered are the calculated ratings of the components and computed weights as shown in Tables 8 and 9. The fuzzy membership functions of rating and weights of different components (deck, superstructure and substructure) thus obtained, are discretised using resolution identity technique. Here, each set is discretised into 11 -levels (from 0.0 to 1.0 in

Condition Ranking and Rating of Bridges Using Fuzzy Logic 199

 In view of this, a methodology based on AHP has been used, in this chapter, for ranking of the existing bridges towards their assessment of prevailing condition which would

 The comparison matrices for different layers of hierarchy are formulated for arriving at the relative weights of the items under each layer. An eigen solution is carried out for each comparison matrix to extract the largest eigenvalue which is further used for checking the consistency of the formulation of the comparison matrices. Estimation indices of individual bridge components have to be arrived based on the bridge inspector's observation and the results of field and laboratory testing. Thus, the estimation indices would suffer from subjective judgement and uncertainty. Hence, an optimistic fuzzy membership function has been used to scale the indices of all the components of the bridges uniformly. Based on the fuzzy synthesized evaluation

 For evaluating the condition rating of the most deserved bridge determined from the prioritization, it is found that as the number of elements of bridge components increase the complexity in arriving at a unique rating number using Fuzzy Weighted Average (FWA) also increases. Hence, a resolution identity method is incorporated in the methodology to take care of the problems that may arise due to non-convexity and

 Further, for the component rating, the Multi-Attributive Decision Making (MADM) model based on priority vector of the constituent elements of the component is also considered because it gives a more realistic representation of the condition of the

 A computer programs have been developed based on the proposed methodology for condition evaluation through prioritization and rating of bridges. It is found that the methodology is capable of handling any number of bridges without any limitation on consideration of components, and elements and rating scale. Thus, the proposed methodology would certainly help the engineers and policy makers concerned with bridge management to arrive at a systematic judgment and to formulate methodical

steps towards retrofitting, rehabilitation or demolition of bridge in future years. It is worthy to mention here that though the condition evaluation through fuzzy logic based AHP may be used as an useful tool for decision making, it should be utilised with adequate care because the whole procedure is dependent on different estimation indices of controlling parameters which have to be taken from inspector's observation

Aturaliya, S.P. (1994). *Bridge condition rating based on fuzzy set theory and Eigenvector approach*,

Ben-Arieh, D. & Triantaphyllou, E. (1992). Quantifying data for group technology with weighted fuzzy features, *Int. J. of Production Research,* Vol. 30, pp. 1285-1299. Bertero, V.V. & Bresler, B. (1977). Design and engineering decisions: Failure criteria (Limit

Masters thesis, Department of Civil Engineering, Kansas State University,

State): Developing methodologies for Evaluating the earth quake safety of Existing

matrix, the priority ranking of the bridges has been evolved.

requirement of normalisation of the concerned sets.

and results of field and laboratory testing.

Manhattan, Kansas.

help in fixing their repair order.

component.

**7. References**

step of 0.1). For better illustration, the resolution identification of the fuzzy set representing the rating of the deck component of the bridge concerned is shown in Fig. 5. Further discussion can be found elsewhere [Sasmal el al. (2005)].

Fig. 5. Resolution identification of the fuzzy set representing rating of the deck component

At each -level, there would be 64 combinations to get the most optimistic (maximum) and pessimistic (minimum) range of the fuzzy set at that -level. For 11 -levels (as considered in this study) the optimistic and pessimistic ranges of the resultant set and the membership representation of the resultant rating (RR) derived from the pessimistic and optimistic ranges using resolution identity technique is shown in Fig. 5. The resultant rating of the bridge, as a whole, has been defuzzified using MATLAB, to get the rating value of the bridge. For this particular case, the defuzzification has been executed using the centroidal method and the rating value is obtained as '**4.6668**'. From the result, it is clear that the rating of the bridge (Br4) falls in between 4 and 5 but closer to 5. It may be the decision maker's discretion in considering the rating value depending on the practical condition and other factors like the environmental condition, importance of the bridge as a whole on the societal service etc. As mentioned earlier, a scale of 0-9 has been considered in this study. So, the condition of the bridge (Br4) falls between 4 and 5 which perhaps reflect the moderate condition. Since, the condition rating of the most deserved bridge among the bridges considered in this study is in between 4 and 5, hence the other bridges are comparatively in better condition.

#### **6. Concluding remarks**

 In bridge engineering, systematic identification of the order of degree of deficiency of the bridges that are considered for their condition assessment is a usual problem. Until now, no systematic approach seems to be available for priority ranking of existing bridges.

step of 0.1). For better illustration, the resolution identification of the fuzzy set representing the rating of the deck component of the bridge concerned is shown in Fig. 5. Further

Fig. 5. Resolution identification of the fuzzy set representing rating of the deck component

At each -level, there would be 64 combinations to get the most optimistic (maximum) and pessimistic (minimum) range of the fuzzy set at that -level. For 11 -levels (as considered in this study) the optimistic and pessimistic ranges of the resultant set and the membership representation of the resultant rating (RR) derived from the pessimistic and optimistic ranges using resolution identity technique is shown in Fig. 5. The resultant rating of the bridge, as a whole, has been defuzzified using MATLAB, to get the rating value of the bridge. For this particular case, the defuzzification has been executed using the centroidal method and the rating value is obtained as '**4.6668**'. From the result, it is clear that the rating of the bridge (Br4) falls in between 4 and 5 but closer to 5. It may be the decision maker's discretion in considering the rating value depending on the practical condition and other factors like the environmental condition, importance of the bridge as a whole on the societal service etc. As mentioned earlier, a scale of 0-9 has been considered in this study. So, the condition of the bridge (Br4) falls between 4 and 5 which perhaps reflect the moderate condition. Since, the condition rating of the most deserved bridge among the bridges considered in this study is in between 4 and 5, hence the other bridges are comparatively in

 In bridge engineering, systematic identification of the order of degree of deficiency of the bridges that are considered for their condition assessment is a usual problem. Until now, no systematic approach seems to be available for priority ranking of existing

discussion can be found elsewhere [Sasmal el al. (2005)].

better condition.

bridges.

**6. Concluding remarks**

#### **7. References**

Condition Ranking and Rating of Bridges Using Fuzzy Logic 201

Lootsma, F.A. (1991), *Scale sensitivity and rank preservation in a multiplicative variant of the AHP*

Melhem, H.G. & Aturaliya, S. (1996). Bridge condition rating using an Eigenvector of priority setting, *J. Microcomputers in Civil Engineering*, Vol. 11, No. 6, pp. 421-432. Miller, D.W. & Starr, M.K. (1969). Executive decisions and operations research. Englewood

Mullarky, P.W. & Fenves, S.J. (1985). Fuzzy logic in a geotechnical knowledge-based system:

Qian, Y.J. (1992). *Evaluation and diagnosis of existing reinforced concrete bridges*, Ph. D.

Saaty, T.L. (1994). Fundamentals of decision making and priority theory with the AHP, RWS

Sasmal, S., Ramanjaneyulu, K., Gopalakrishnan, S. & Lakshmanan, N. (2004a). *Condition*

Sasmal, S., Ramanjaneyulu, K., Gopalakrishnan, S. & Lakshmanan, N. (2004b). Condition

Sasmal, S., Ramanjaneyulu, K., Gopalakrishnan, S. & Lakshmanan, N. (2005). Condition

Sasmal, S., Ramanjaneyulu, K., Gopalakrishnan, S. & Lakshmanan, N. (2006). Fuzzy logic

Shiaw, S.S. & Huang, W.D (1990). Synthesis evaluation of fuzzy and random Properties for bridge bearing capacity, *J of Chiaotung Institute* at Chung Ching, 1, 64-71. Tee. A.B., Bowman, M.D. & Sinha, K.C. (1988). A fuzzy mathematical approach for bridge condition evaluation, *J. Civil Engineering systems*, Vol. 5, No. 1, pp. 17-25. Tee. A.B. & Bowman, M.D. (1991). Bridge condition assessment using fuzzy weighted averages, *Journal of Civil Engineering systems*, Vol. 8, No. 1, pp. 49-57. Tseng, T.Y., & Klein, C.M. (1992). A new algorithm for fuzzy multi-criteria decision making,

Triantaphyllou, E., Kovalerchuk, B., Mann jr. L. & Knapp G.M. (1997). Determining the

Wang, H.D. (1996). Basic Study on Durability Evaluation of Reinforced Concrete Structure,

Yu, L. & Cheng, M. (1994). A new method to group decision making with multiple

most important criteria in maintenance decision making, *J. Quality in Maintenance*

Ph. D. Dissertation, Department of Civil Engineering, Dalian University of

objectives-interactive comparison matrix approach, *J. System Engineering*, Vol. 9,

*assessment of existing bridges through prioritization in a fuzzy environment*, SERC

ranking of existing bridges using analytic hierarchy approach, in Proceedings of International Conference on *World of Innovations in Structural Engineering* (WISE-2004), organised by ACCE at Hyderabad during December 1th to 3rd, 2004, 344-352.

rating of bridges using fuzzy mathematics, Proc., Int. Conf. on *Advances in Concrete*

Based condition rating of existing reinforced concrete bridges, *J. Performance of*

Dissertation, Southwest Chiaotung University, Xian, China. Roberts, F.S., Measurement theory, reading, MA, 1979 (Addison-Wesley). Saaty, T. L. (1980). *The Analytic Hierarchy Process*, McGraw Hill: New York.

Research Report No. RCS-MLP10741-RR-2004-2, May 2004.

*Composites and Structures* (ICACS-2005), India, 943-952.

*Constructed Facilities- ASCE*, l 20(3), 2006, 261-273.

*Approx. Reasoning*, Vol. 6, pp. 45-66.

*Engg.*, Vol. 3, No. 1, pp. 16-28.

Technology, China.

No. 2, pp. 28-35.

University of Technology.

Cliffs, NJ: Prentice-Hall, Inc.

University, Indiana.

Publications, Pittsburgh.

*and SMART*, Report 91-67, Faculty of Technical Mathematics and Informatics, Delft

cone, Proc., *NSF workshop on civil engineering applications of fuzzy sets*, Purdue

Buildings, *Report No. UCB-EERC-77/06*, Earthquake Engineering Research Centre, University of California at Berkeley, February, 114-142.

Boender, C.G.E, de Graan, J.G. & Lootsma, F.A. (1989). Multi-criteria decision analysis with fuzzy pair-wise comparisons, *Fuzzy Sets and Systems*, Vol. 29, pp. 133-143. Bridgman, P. W. (1922). Dimensional analysis. New Haven, CT: Yale University Press. Brown, C.B., & Yao J.T.P. (1983). Fuzzy sets and structural engineering, *J. Structural*

Chen, S.J. & Hwang, C.L. (1992). Fuzzy multiple attribute decision making: methods and

Dong W.M. & Wong F.S. (1987). Fuzzy weighted average and implementation of the extension principle, *Fuzzy sets and systems*, Vol. 21, No. 2, pp. 183-199. Dubois, D. & Prade, H. (1979). Fuzzy real algebra: Some results, *Fuzzy Sets and Systems*, Vol.

Emami, M.R. Turksen, I.B. & Goldenberg, A.A. (1998). Development of a systematic

Federal Highway Administration (FHWA). *Bridge inspectors training manual 90*, U.S. Department of Transportation, Bureau of Public Road, Washington, D.C., 1991. Federov, V.V., Kuz'min, V.B. & Vereskov, A.I. (1982). Membership degrees determination

Fishburn, P.C. (1967). Additive utilities with incomplete product set: *Applications to priorities*

Harker, D.T. & Vargas, L.G. (1990). Theory of ratio scale estimation: Saaty's Analytic

Jwu, T.H., Harn, Y. & Liou, W.M, (1991). Fuzzy Evaluation of Wharf Structures at Harbour,

Kawamura, K. & Miyamoto, A. (2003). Condition state evaluation of existing reinforced

Kaufmann, A. & Gupta, M.M. (1985). *Introduction to fuzzy arithmetic, Theory and Applications*,

Laarhoven, P.J. M. & Pedrycz, W.A. (1983). A fuzzy extension of Saaty's priority theory,

Liang M.T., Wu J.H. & Liang C.H. (2001). Multiple layer fuzzy evaluation for existing reinforced concrete bridges, *J Infrastructure System*, Vol. 7, No. 4, pp. 144-159. Liang, M.T, Chang, J.W. & Li, Q.F. (2002). "Grey and regression models predicting the

Liang, M.T., Lin, C.M. & Yeh, C.J. (2003). Comparison matrix method and its applications to

*Reasoning in Decision Analysis*, Amsterdam: North-Holland.

Hierarchy Process, *Management Science*, Vol. 36, pp. 269-273.

*and assignments*. Baltimore, MD: ORSA Publication.

University of California at Berkeley, February, 114-142.

*Engineering ASCE*, Vol. 109, No. 5, pp. 1211-1235.

Springer.

2, pp. 327-348.

No. 3, pp. 346-361.

of Transportation, China.

New York: Van Nostrand Reinhold.

*Technology*, Vol. 11, No. 2, pp. 70-82.

*Fuzzy Sets and Systems*, Vol. 11, pp. 229-241.

81, pp. 1931-1940.

14, No. 4, pp. 291-310.

Buildings, *Report No. UCB-EERC-77/06*, Earthquake Engineering Research Centre,

applications, *Lecture Notes in Economics and Mathematical Systems*, No. 375, Berlin:

methodology of fuzzy logic modelling, *IEEE Transactions on Fuzzy Systems*, Vol. 6,

from Saaty matrix totalities, edited by M.M. Gupta and E. Sanchez, *Approximate*

Report of *Investigation of Corrosion Failure and Its Protection Strategy for Harbour Wharf*, Institute of Engineering and Technology of Ten-Jin Waterway, Department

concrete bridges using neuro-fuzzy hybrid system, *Computers and Structures,* Vol.

remaining service life of existing reinforced concrete bridges", *J. Grey System*, Vol.

damage evaluation for existing reinforced concrete bridges, *J. Marine Science &*

**0**

**10**

*Japan*

**Fuzzy Logic for Multi-Hop Broadcast in**

A Vehicular Ad hoc Network (VANET) is a form of mobile ad hoc network in which vehicles are equipped with wireless communication devices. Vehicular ad hoc networks have been attracting the interest of both academic and industrial communities on account of their important role in Intelligent Transportation Systems (ITS). VANETs are expected to be able to significantly reduce the number of road accidents. When vehicles travel at a high speed on roads, drivers have very little time to react to the vehicle in front of them. By using vehicular ad hoc networks, emergency information can be propagated along the road to notify drivers ahead of time so that necessary actions can be taken to avoid accidents. Vehicular ad hoc networks also make the driving more efficient by disseminating traffic warning information

In this chapter, we consider VANET broadcast protocols which work as a basis of many vehicular applications especially safety applications. Providing reliable and efficient multi-hop broadcast in vehicular ad hoc networks is very challenging. First, in vehicular ad hoc networks, vehicles are usually deployed in a dense manner. Therefore, a simple broadcast scheme cannot work well because of redundant broadcasts. Second, wireless communications are unreliable and vehicles can move at a high speed. Consequently, it is difficult to reduce

As a solution, we explain an approach which uses a fuzzy logic to enhance multi-hop broadcast in vehicular ad hoc networks. Due to the high node density, vehicle movement and fading feature of wireless communications, providing a reliable and efficient multi-hop broadcast in vehicular ad hoc networks is still an open research topic. Using only a subset of neighbor nodes to relay broadcast messages is a main concept for providing efficiency. Meanwhile, in order to ensure a high reliability, multiple metrics of inter-vehicle distance, node mobility and signal strength should be jointly considered in the relay node selection. However, these metrics conflict with each other and these conflicts depend on the vehicle mobility, vehicle distribution and fading condition. The mathematical model of the optimal relay problem is complex to derive and a solution based on it would be too expensive for practical application. Therefore, we employ fuzzy logic to handle these imprecise and uncertain information. We use a fuzzy logic based method to select relay nodes by jointly considering inter-vehicle distance, node mobility and signal strength. The selected relay nodes can provide a reliable data forwarding with a high efficiency. In this chapter, we give a detailed

the redundant broadcast while maintaining a high packet dissemination ratio.

description of the fuzzy logic based method with simulation results.

**1. Introduction**

and service information.

Celimuge Wu, Satoshi Ohzahata and Toshihiko Kato

**Vehicular Ad Hoc Networks**

*University of Electro-Communications*

Zadeh, L.A. (1973). The concept of a linguistic variable and its application in approximate reasoning, *Information Science, part I*: Vol. 8, No. 3, pp. 199-249.

Zadeh L.A. (1965). Fuzzy sets, *Information & Control*, Vol. 8, pp. 338-353.

### **Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks**

Celimuge Wu, Satoshi Ohzahata and Toshihiko Kato *University of Electro-Communications Japan*

#### **1. Introduction**

202 Fuzzy Logic – Emerging Technologies and Applications

Zadeh, L.A. (1973). The concept of a linguistic variable and its application in approximate

Zhao, Z. & Chen C. (2002). A Fuzzy system for concrete bridge damage diagnosis, *Computers*

Zahedi, F. (1986). The analytic hierarchy process – a survey of the method and its

reasoning, *Information Science, part I*: Vol. 8, No. 3, pp. 199-249.

Zadeh L.A. (1965). Fuzzy sets, *Information & Control*, Vol. 8, pp. 338-353.

applications, *Interfaces*, Vol. 16, No. 4, pp. 96-108.

*and Structures*, Vol. 80, pp. 629-641.

A Vehicular Ad hoc Network (VANET) is a form of mobile ad hoc network in which vehicles are equipped with wireless communication devices. Vehicular ad hoc networks have been attracting the interest of both academic and industrial communities on account of their important role in Intelligent Transportation Systems (ITS). VANETs are expected to be able to significantly reduce the number of road accidents. When vehicles travel at a high speed on roads, drivers have very little time to react to the vehicle in front of them. By using vehicular ad hoc networks, emergency information can be propagated along the road to notify drivers ahead of time so that necessary actions can be taken to avoid accidents. Vehicular ad hoc networks also make the driving more efficient by disseminating traffic warning information and service information.

In this chapter, we consider VANET broadcast protocols which work as a basis of many vehicular applications especially safety applications. Providing reliable and efficient multi-hop broadcast in vehicular ad hoc networks is very challenging. First, in vehicular ad hoc networks, vehicles are usually deployed in a dense manner. Therefore, a simple broadcast scheme cannot work well because of redundant broadcasts. Second, wireless communications are unreliable and vehicles can move at a high speed. Consequently, it is difficult to reduce the redundant broadcast while maintaining a high packet dissemination ratio.

As a solution, we explain an approach which uses a fuzzy logic to enhance multi-hop broadcast in vehicular ad hoc networks. Due to the high node density, vehicle movement and fading feature of wireless communications, providing a reliable and efficient multi-hop broadcast in vehicular ad hoc networks is still an open research topic. Using only a subset of neighbor nodes to relay broadcast messages is a main concept for providing efficiency. Meanwhile, in order to ensure a high reliability, multiple metrics of inter-vehicle distance, node mobility and signal strength should be jointly considered in the relay node selection. However, these metrics conflict with each other and these conflicts depend on the vehicle mobility, vehicle distribution and fading condition. The mathematical model of the optimal relay problem is complex to derive and a solution based on it would be too expensive for practical application. Therefore, we employ fuzzy logic to handle these imprecise and uncertain information. We use a fuzzy logic based method to select relay nodes by jointly considering inter-vehicle distance, node mobility and signal strength. The selected relay nodes can provide a reliable data forwarding with a high efficiency. In this chapter, we give a detailed description of the fuzzy logic based method with simulation results.

as

where *Ns* is the number of slots.

**2.2 Sender-oriented protocols**

node movement.

in the relay node selection.

**2.2.1 MPR**

probability *p*. Otherwise, the node discards the packet.

packet loss would occur at the neighbor node.

*TStr* = *Str* × *τ*, (2)

*R* )�, (3)

where *τ* is the estimated one-hop delay, which includes the medium access delay and

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 205

*Str* = �*Ns*(1 − *min*(*Dtr*, *R*)

Similar to slotted 1-persistence scheme, in the slotted *p*-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet only once in the assigned time slot *TStr* which is calculated as Eq. (2), the node rebroadcasts with the predetermined

In the sender-oriented protocols, since the sender node specifies relay nodes, the redundant broadcasts can be minimized. The relay node selection method directly affects the performance of a sender-oriented protocol. Generally, the relay node selection is based on the information collected from the exchange of hello messages. Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast) in which relay nodes are selected using two-hop neighbor information. Djedid et al. (2008) have proposed a broadcast protocol which selects relay nodes based on Connected Dominating Set. However, these protocols do not consider node mobility in the relay node selection. As a result, the selected relay node can become sub-optimal and can lose the message due to the

In our previous work (Wu et al. (2010)), we have proposed a relay node selection which considers the additional radio coverage and node movement (here we call EMPR Broadcast). However, EMPR Broadcast does not consider the fading feature of wireless channels. In a wireless channel, a node can receive a hello message from a neighbor which is at a distance where stable communication is impossible. If the neighbor node is selected as a relay node, a

Sahoo et al. (2009) have proposed BPAB, a Binary Partition Assisted emergency Broadcast protocol for vehicular Ad hoc networks. BPAB intends to use the farthest node to relay messages. However, in a fading channel, the farthest node can lose the messages. Therefore, we have to choose the nodes which have stable signal strength as relay nodes. In short, multiple metrics of inter-vehicle distance, mobility and signal strength should be considered

Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast). MPR can substantially reduce the message overhead as compared to the flooding. In MPR broadcast, each node selects a set of its neighbor nodes as "multipoint relays" (MPR). Only the selected MPR nodes are responsible for forwarding the messages. The neighbors of node *N* which are not in its MPR set, receive and process broadcast messages but do not retransmit broadcast messages received from node *N*. MPR broadcast provides an efficient mechanism for disseminating messages by reducing the number of transmissions.

propagation delay. *Str* is the assigned slot number, which is calculated by

The basic idea of the approach has been published by IEEE (Wu et al. (2010)). However, in this chapter, we use a more realistic model to evaluate the approach and present our new simulation results. We explain the approach with new and more detailed information.

#### **2. Multi-hop broadcast in vehicular ad hoc networks**

The simplest way to disseminate information is flooding. In the flooding, each node rebroadcasts a packet upon the first reception. Obviously, in a high-density network, the flooding introduces too many redundant broadcasts and consequently incurs collisions and results in a low dissemination rate. There have been a lot of protocols to reduce the redundant broadcasts in a high-density network. These protocols can be classified into two categories of sender-oriented protocols and receiver-oriented protocols. In the sender-oriented protocols, a sender node specifies relay nodes. In contrast, in the receiver-oriented protocols, upon reception of a message, a receiver node determines own action (whether rebroadcast the message or not) in an autonomous manner.

#### **2.1 Receiver-oriented protocols**

Several receiver based broadcast protocols have been proposed. Wisitpongphan & Tonguz (2007) have proposed three broadcast schemes: weighted *p*-persistence, slotted 1-persistence, and slotted *p*-persistence schemes. In these protocols, upon reception of a message, a node calculates a broadcast probability according to the distance from the sender node. Generally, a larger distance from the sender node results in a higher broadcast probability. Suriyapaiboonwattana et al. (2009) have proposed a protocol which uses an adaptive wait time and adaptive probability to trigger the rebroadcast. Slavik & Mahgoub (2010) have proposed a protocol in which all nodes rebroadcast a received message with a certain probability. Mylonas et al. (2008) have proposed a Speed Adaptive Probabilistic Flooding algorithm to determine the rebroadcast probability according to vehicle speed. However, in the receiver-based protocols, each node determines whether rebroadcast or not in an autonomous manner. Therefore, redundant broadcasts cannot be eliminated entirely.

#### **2.1.1 Weighted** *p***-persistence, slotted 1-persistence and slotted** *p***-persistence scheme**

Wisitpongphan & Tonguz (2007) have proposed three probabilistic and timer-based broadcast suppression techniques. They are weighted *p*-persistence, slotted 1-persistence and slotted *p*-persistence Scheme.

In the weighted *p*-persistence scheme, upon reception of a packet from node *t*, node *r* checks the packet ID and rebroadcasts with probability *ptr* if node *r* receives the packet for the first time. Otherwise, the node discards the packet. The probability, *ptr*, is calculated on a per packet basis using

$$p\_{tr} = \frac{D\_{tr}}{R} \,\prime \tag{1}$$

where *Dtr* is the relative distance between nodes *t* and *r*, *R* is the average transmission range. The larger the *Dtr*, the higher the probability will be.

In slotted 1-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet for the first time and fails to detect any rebroadcast from other nodes in an assigned time slot *TStr*, the node rebroadcasts the packet. If the node can detect a rebroadcast of the packet from any other nodes, the node discards the packet. *TStr* is calculated as

2 Will-be-set-by-IN-TECH

The basic idea of the approach has been published by IEEE (Wu et al. (2010)). However, in this chapter, we use a more realistic model to evaluate the approach and present our new simulation results. We explain the approach with new and more detailed information.

The simplest way to disseminate information is flooding. In the flooding, each node rebroadcasts a packet upon the first reception. Obviously, in a high-density network, the flooding introduces too many redundant broadcasts and consequently incurs collisions and results in a low dissemination rate. There have been a lot of protocols to reduce the redundant broadcasts in a high-density network. These protocols can be classified into two categories of sender-oriented protocols and receiver-oriented protocols. In the sender-oriented protocols, a sender node specifies relay nodes. In contrast, in the receiver-oriented protocols, upon reception of a message, a receiver node determines own action (whether rebroadcast the

Several receiver based broadcast protocols have been proposed. Wisitpongphan & Tonguz (2007) have proposed three broadcast schemes: weighted *p*-persistence, slotted 1-persistence, and slotted *p*-persistence schemes. In these protocols, upon reception of a message, a node calculates a broadcast probability according to the distance from the sender node. Generally, a larger distance from the sender node results in a higher broadcast probability. Suriyapaiboonwattana et al. (2009) have proposed a protocol which uses an adaptive wait time and adaptive probability to trigger the rebroadcast. Slavik & Mahgoub (2010) have proposed a protocol in which all nodes rebroadcast a received message with a certain probability. Mylonas et al. (2008) have proposed a Speed Adaptive Probabilistic Flooding algorithm to determine the rebroadcast probability according to vehicle speed. However, in the receiver-based protocols, each node determines whether rebroadcast or not in an

autonomous manner. Therefore, redundant broadcasts cannot be eliminated entirely.

**2.1.1 Weighted** *p***-persistence, slotted 1-persistence and slotted** *p***-persistence scheme**

Wisitpongphan & Tonguz (2007) have proposed three probabilistic and timer-based broadcast suppression techniques. They are weighted *p*-persistence, slotted 1-persistence and slotted

In the weighted *p*-persistence scheme, upon reception of a packet from node *t*, node *r* checks the packet ID and rebroadcasts with probability *ptr* if node *r* receives the packet for the first time. Otherwise, the node discards the packet. The probability, *ptr*, is calculated on a per

*ptr* = *Dtr*

where *Dtr* is the relative distance between nodes *t* and *r*, *R* is the average transmission range.

In slotted 1-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet for the first time and fails to detect any rebroadcast from other nodes in an assigned time slot *TStr*, the node rebroadcasts the packet. If the node can detect a rebroadcast of the packet from any other nodes, the node discards the packet. *TStr* is calculated

*R* , (1)

**2. Multi-hop broadcast in vehicular ad hoc networks**

message or not) in an autonomous manner.

**2.1 Receiver-oriented protocols**

*p*-persistence Scheme.

packet basis using

The larger the *Dtr*, the higher the probability will be.

$$T\_{\mathbb{S}\_{tr}} = \mathbb{S}\_{tr} \times \tau\_{\prime} \tag{2}$$

where *τ* is the estimated one-hop delay, which includes the medium access delay and propagation delay. *Str* is the assigned slot number, which is calculated by

$$\mathcal{S}\_{tr} = \lceil \mathcal{N}\_{\mathbf{s}} (1 - \frac{\min\{D\_{tr}, \mathcal{R}\}}{\mathcal{R}}) \rceil\,. \tag{3}$$

where *Ns* is the number of slots.

Similar to slotted 1-persistence scheme, in the slotted *p*-persistence scheme, upon reception of a packet, a node checks the packet ID. If the node receives the packet only once in the assigned time slot *TStr* which is calculated as Eq. (2), the node rebroadcasts with the predetermined probability *p*. Otherwise, the node discards the packet.

#### **2.2 Sender-oriented protocols**

In the sender-oriented protocols, since the sender node specifies relay nodes, the redundant broadcasts can be minimized. The relay node selection method directly affects the performance of a sender-oriented protocol. Generally, the relay node selection is based on the information collected from the exchange of hello messages. Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast) in which relay nodes are selected using two-hop neighbor information. Djedid et al. (2008) have proposed a broadcast protocol which selects relay nodes based on Connected Dominating Set. However, these protocols do not consider node mobility in the relay node selection. As a result, the selected relay node can become sub-optimal and can lose the message due to the node movement.

In our previous work (Wu et al. (2010)), we have proposed a relay node selection which considers the additional radio coverage and node movement (here we call EMPR Broadcast). However, EMPR Broadcast does not consider the fading feature of wireless channels. In a wireless channel, a node can receive a hello message from a neighbor which is at a distance where stable communication is impossible. If the neighbor node is selected as a relay node, a packet loss would occur at the neighbor node.

Sahoo et al. (2009) have proposed BPAB, a Binary Partition Assisted emergency Broadcast protocol for vehicular Ad hoc networks. BPAB intends to use the farthest node to relay messages. However, in a fading channel, the farthest node can lose the messages. Therefore, we have to choose the nodes which have stable signal strength as relay nodes. In short, multiple metrics of inter-vehicle distance, mobility and signal strength should be considered in the relay node selection.

#### **2.2.1 MPR**

Qayyum et al. (2002) have proposed a multipoint relay (MPR) broadcast scheme (here we call MPR Broadcast). MPR can substantially reduce the message overhead as compared to the flooding. In MPR broadcast, each node selects a set of its neighbor nodes as "multipoint relays" (MPR). Only the selected MPR nodes are responsible for forwarding the messages. The neighbors of node *N* which are not in its MPR set, receive and process broadcast messages but do not retransmit broadcast messages received from node *N*. MPR broadcast provides an efficient mechanism for disseminating messages by reducing the number of transmissions.

where *i* − 1 indicates the previous value (the value is updated on the reception of a hello message). Eq. (6) could give a larger value for the same directed vehicles and smaller value for vehicles that moving toward opposite direction. If a node *x* has opposite moving direction to the sender, corresponding *θ* will be smaller than other vehicles which have the same direction

Fuzzy Logic for Multi-Hop Broadcast in Vehicular Ad Hoc Networks 207