Dominance-based rough set approach to network bridge management/Dominuojanciu apytiksliu aibiu metodo taikymas tiltu tinklo valdymui/Uz dominejosa stavokla aptuvena novertejuma metodes balstita tiklveida pieeja tiltu kopas parvaldibai/Dominantsuspohine ligikaudne lahenemine teedevorgu sildade hooldusel.
Augeri, Maria Grazia ; Colombrita, Rosario ; Greco, Salvatore 等
1. Introduction
Bridges are subject to decline due to changes in both physical and
mechanical properties of the materials used, but also due to traffic
volume and speed, as well as to environmental aggression (e.g. exposure
to chlorides, freezing, thawing cycles, etc.) and catastrophic events
(floods, earthquakes, landslides, etc.).
In order to optimize the available budget, it is useful for bridge
authorities to implement a management system for identifying the
structures requiring maintenance and the substantial interventions at an
early stage.
It is worth emphasizing the following two aspects:
--timely maintenance intervention leads to a longer bridge
lifetime;
--maintenance costs tend to rise quickly after the deterioration
process has started (Rens et al. 2005).
A Bridge Management System (BMS) is a decision-making process used
to select and prioritize the tasks needed to keep the structure's
functional parameters within acceptable limits given its lifetime cycle.
The priority-setting process requires considerable attention to minimize
costs and disruption of road traffic (Godart, Vassie 2001). An effective
management system needs to carry out an individual analysis of the
structure (Project Level Bridge Management) and to analyze the level of
the network (Network Level Bridge Management). Therefore, decisions need
to be considered at two levels: network and project. The one at network
level defines when it is necessary to intervene while the one at project
level defines the kind of intervention required. Integrating the Project
Level with the Network Level is rather complex. This paper proposes a
Decision Support System (DSS) for managing the Network Level based on
the theory of the Dominance-Based Rough Set Approach (DRSA). This
methodology produces a decision model expressed in terms of easily
understandable "if.... then ..." decision rules which reflects
the decision-maker's choices.
The parameters used in the decisional process take into account not
only the state assessment of the bridge but also the characteristics of
the environment, traffic and road conditions since the network is an
integral part of the territory. The goal is not to define any
maintenance activities for a bridge, but to determine the priorities on
which to base a first approximation of the maintenance program for a
bridge network in a way that the available resources are managed
optimally.
2. Maintenance management of bridges
BMS includes all the activities which help maintain a bridge
network efficiently, assuring safety and usability through the design,
construction and operational phases (Godart et al. 2001). Several
procedures have been developed to optimize bridge network maintenance
where the funds available are limited. Some are based on inspecting and
evaluating bridge condition, whereas more up-to-date procedures use
statistical and mathematical tools to completely design maintenance
plans at the network and project level (Valenzuela et al. 2010).
A complete management system includes:
--database;
--bridge evaluation;
--deterioration prediction and future conditions;
--maintenance alternatives evaluation and their cost;
--optimized maintenance plans.
The database is the sum of information on the bridge network (data,
maintenance activities, state of damage, etc). The database is
periodically updated as a result of inspections.
The bridge evaluation identifies deterioration processes and the
causes of such processes by means of inspections (visual, instrumental,
etc). Subsequently, bridges are classified into several categories.
These are classified by severity which determines the urgency of the
intervention (Bevc et al. 2001).
The condition data are then used to assess load capacity in terms
of structure longevity and towards maximizing the safety and stability
over that period (Bevc et al. 1999).
Future conditions are predicted by means of algorithms based either
on standard statistical methods or on artificial intelligence techniques
(neural networks, genetic algorithms).
Once these bridges are classified, by maintenance alternatives the
optimization procedures are formulated.
Optimization means the best maintenance at the minimum cost, while
maintaining adequate service levels. Optimization is performed over the
period of maintenance interventions; this can vary from a few years to
the entire life-time of the bridge. Several optimization procedures are
available: classical mathematical formulations (scalar, unconstrained or
constrained minimization, linear or quadratic programming, etc.) and
artificial intelligence techniques (neural networks, genetic algorithms)
(Bevc et al. 2001).
BMS is a difficult process with different mathematically complex
stages. There are many studies on this subject in the literature. For
example, Hai (2008) proposes a computer database for maintenance and
management for highway bridges in Vietnam. It includes several
assessment techniques such as lifespan estimation, deterioration
prediction, life cycle cost analysis, cost-benefit analysis and priority
maintenance index (PMI). Miyamoto et al. (2001) have proposed a concrete
bridge management system based on visual inspection and Concrete Bridge
Rating Expert System with machine learning for evaluated bridge
performance, and on genetic algorithms for researching optimized
maintenance.
Other authors have devoted their attention to the assessment of
bridges; for example Valenzuela et al. (2010) proposed an integrated
bridge index (IBI) which depends on four factors: the 'BCI
Index' which reflects bridge damage levels, the "SI
Index" which reflects the importance of the bridge in the road
network, the "HV Index" which reflects hydraulic vulnerability
and the "SR Index" which reflects seismic risk. The index was
calibrated using visual inspection, expert surveys, and regression
analysis.
Other authors have devoted their attention to predicting
deterioration ratio and optional maintenance plans. Frangopol et al.
(2001) showed that bridge management system based on Markovian
deterioration modeling has several limitations that overcome using a
reliability-based approach. Neves et al. (2004) proposed a model for
predicting the performance of deteriorating structures by measuring it
in terms of condition, safety and maintenance cost. This model considers
the interaction between condition and safety by correlating the random
variables of the two associated profiles and their relationship. Liu,
Frangopol (2004) proposed a multi-objective genetic algorithm for
optimal life-cycle maintenance planning of deteriorating bridges where
condition index, safety index, and cumulative life-cycle maintenance
costs were simultaneously considered in the optimization process. Liu,
Frangopol (2005) proposed a multi-objective genetic algorithm in which
structure condition, safety, and cumulative lifecycle maintenance costs
have been considered as separate objective functions subject to
simultaneous optimization.
This study presents an automated procedure with a large pool of
alternative maintenance solutions establishing optimized tradeoffs
between the competing meritorious objectives. Neves et al. (2006)
proposed a probabilistic multi-objective approach to bridge maintenance
using genetic algorithms which considers single maintenance types. In
this study, the condition index (by visual inspections) and the safety
index (by structural analysis) are used as indicators of the
deteriorating performance of structures. The decision maker choices are
the best possible compromise between available funds, safety and
condition parameters and acceptable levels of deterioration, depending
on the specific situation, the bridge manager preferences and the
on-going maintenance policy. Liu, Frangopol (2006) proposed a
comprehensive mathematical model for probability-based bridge network
performance evaluation using network theories. Elbehairy et al. (2009)
proposed multiple-element bridge management that optimizes repair
decisions. In this study, the proposed system uniquely segments the
problem into smaller sequential optimizations which are solved using the
genetic algorithms technique. Orcesi, Cremona (2011) proposed optimized
maintenance strategies for managing bridges across France based on
Markov chains fitted to condition data. This study evaluated prediction
models for cost analysis and different maintenance strategies.
There is some specific software for managing bridges. Pontis is the
most popular, developed by AASHTO in collaboration with the Federal
Highway Administration (FHWA). Pontis is currently used in more than 40
agencies in the US and is widely adopted in other countries. Pontis
allows both network- and project-level planning where bridges are
represented as an assemblage of structural elements each is being
classified by visual inspections every two years, in condition-state
classes (Estes, Frangopol 2003). Pontis provides optimal maintenance
policies for each state and for each type of element and environmental
condition. Pontis generates simulated scenarios to determine current and
future requirements, predict future performance levels and provide
recommendations. The optimized policies at the network-level are
selected by the software based on minimizing costs over the life-time of
the bridge (Woodward et al. 2001).
3. Proposed methodology
One of the BMS phases is classifying bridges by assigning an
intervention priority level. Accordingly, a first schedule of
maintenance interventions is provided which is often based on linear
equations that combine all the selected parameters, each having a weight
assigned by expertise or literature data.
The aim of this paper is to define maintenance activity priorities
by means of a decision-maker support system, taking into account the
different roles involved in decision-making, each with its own
objective. Thus, a methodology based on decisional rules obtained by the
rough-set theory, the DRSA, has been applied. DRSA highlights both the
methodological and operational point of view.
Using this methodology it is possible to derive a logical behavior
model by observing actions through an inductive learning process (Greco
et al. 2002a, 2002b).
The advantages of this methodology are:
--ability to manage vague or inaccurate data;
--ability to manage qualitative data;
--no need to assign a weight to each criterion;
--it is possible to highlight cause-effect relationships between
the available data, separating the most relevant and strategic
information from the inessential;
--construction of a priority model based on decisional rules such
as "if ... then.
--identification of rules which support each decision;
--facility for the decision-makers to understand how the rules
influence their decisions.
The proposed decision-making support system is a flexible tool. In
fact, it is possible to evaluate and update it periodically as a
consequence of practice, expertise and managing authority's
different policies.
In the first part of the paper, the parameters required to describe
the phenomenon are defined. In the second part, the form of on-field
data gathering is defined. In the third part the proposed
decision-making support system is presented. Finally, the proposed
methodology is applied to a bridge network.
Bridge characteristics are defined by a set of attributes that
describe the state of degradation, the structure, territory, and traffic
and network characteristics. The attributes are divided into condition
attributes, also called criteria ([A.sub.1] to [A.sub.15]) and decision
attributes ([A.sub.16]). The value assigned to each criterion increases
as conditions worsen. Table 1 describes the attributes and the values in
more detail.
3.1. Visual inspection and data acquisition
The data required for implementing the DSS are:
--project;
--thematic maps (hydro-geological risk maps, seismic vulnerability
maps, etc);
--inspections:
a) to identify and classify the various types of structures;
b) to identify the damage and its causes;
c) to prevent collapse.
Each damage depends on several factors: material deterioration,
increase in traffic volume, increase in traffic load, lifespan
reduction, natural disasters, etc.
There are many kinds of inspections: superficial, general,
principal and special. Each country adopts different procedures for such
inspections. Usually, general inspections are carried out every 2 or 3
years.
It is possible to do visual inspections or inspections using
instruments. Visual inspections are performed in the field by compiling
a form that can help identify and classify the damage and the damaged
components. Photographs, sketches and notes are useful. The literature
reports more evolved data gathering, recording and presentation, such as
3D images and virtual reality.
To investigate the causes and magnitude of the damage in detail,
instrument inspections are necessary after visual inspection.
In this study, visual inspections were carried out using 1st level
sheets for the damage state survey. These sheets are classified by
bridge type (masonry arch bridges, reinforced concrete girder bridges,
reinforced concrete arch bridges, steel girder bridges, pre-stressed r.
c. girder bridges). This information facilitates the operator compile
the sheets to obtain an objective description.
Each form contains 6 sections:
--section 1--identifying the bridge, location, road type etc.;
--section 2--geomorphological data, foundation soil;
--section 3--components: slabs, arches, piers, abutments etc.;
--section 4--simplified representation of the bridge,
accessibility, images;
--section 5--survey of structural component damage;
--section 6--survey of non-structural component damage.
The damage to each bridge component has been identified by means of
these sheets corroborated by photographs.
4. Dominance-based rough set approach to prioritize maintenance
To prioritise bridge maintenance for a road system, a
multi-criterion model based on the DRSA theory (Greco et al. 1999, 2001,
2002a, 2005; Slowinski et al. 2005) has been adopted. This is an upgrade
of the Classical Rough Set Approach (CRSA) developed by Pawlak (1991)
which is applicable to multi-criterion issues. The DRSA does not only
allow the representation and analysis of decision-making but, more
generally, of all the phenomena involving monotonicity. DRSA theory grew
out of research in the field of multi-criterion decision-making within
AI techniques.
4.1. Information table and dominance relation
The rough set philosophy assumes that every object in the universe
is described by a set of attributes. This requires inputting a set of
examples representing preferential information by decision makers, while
the analysis output is the model of preferences in terms of decision
rules.
For algorithmic reasons, object information is supplied by a
"data table" whose rows refer to distinct objects and whose
columns refer to different attributes. Each table cell indicates an
evaluation (quantitative or qualitative) of the object located in that
row by an attribute in the corresponding column.
In this case study the decision support system was collated from a
set of 100 bridges whose features represent most of the bridges found on
Italian roads.
The row objects are bridges and the columns are the criteria which
characterize the bridges, as shown in Table 2.
Formally, a data table in the 4-tuple S = (U, Q, V, f) where U is a
finite set of objects (universe), Q = [[q.sub.1], [q.sub.2],...,
[q.sub.n]} is a finite set of attributes, [V.sub.q] is the domain of
attribute q, V = [U.sub.q][member of]Q[V.sub.q] and f:UxQ [right arrow]
V in a total function such as f(x, q) [member of][V.sub.q] for each q
[member of] Q, x [member of] U, called "information function".
The set Q is, in general, divided into set C of condition attributes and
a decision attribute d. In multi-criteria classification condition
attributes are "criteria". The notion of criterion involves a
preference order in its domain while the domains of attributes are not
preference-ordered.
In this case, all the condition attributes are criteria because it
is possible to order them according to increasing preference of
maintenance activity.
Furthermore, decision attribute d makes a partition of U into a
finite number of classes Cl = [[Cl.sub.t], t [member of] T}, T = [1, n}.
Each x [member of] U belongs to one and only one class [Cl.sub.t]
[member of] Cl. The classes from Cl are preference-ordered according to
the increasing order of class indices, i.e. for all r, s [member of] T,
such as r > s, the objects from [Cl.sub.r] are preferred to the
objects from [Cl.sub.s].
In the presented case, the set of decision D attributes is a
singleton given by the attribute "degree of urgency of the
maintenance activity" which divides the set U of 100 bridges into
four classes:
--[Cl.sub.1]: bridges that keep to the inspection schedule;
--[Cl.sub.2]: bridges requiring prior inspection;
--[Cl.sub.3]: bridges requiring urgent intervention;
--[Cl.sub.4]: bridges requiring partial or total closure.
4.2. Dominance-based approximation
In multi-criteria classification, due to the preference order in Cl
classes the sets requiring approximation are not particular classes but
upward unions (1) and downward unions (2) of classes, respectively:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
Union [Cl.sup.[greater than or equal to].sub.t] is the set of
objects belonging to class [Cl.sub.t] or to a more preferred class,
while [Cl.sup.[less than or equal to].sub.t] is the set of objects
belonging to class [Cl.sub.t] or to a less preferred class.
Note, that for t = 2, n [Cl.sup.[greater than or equal to].sub.t] =
U - [Cl.sup.[less than or equal to].sub.t]_1, i.e. all the objects not
belonging to class [Cl.sub.t] or better, belong to class [Cl.sub.t] - 1
or worse.
It the case study the upward union classes are:
--the union upward [Cl.sub.1.sup.[greater than or equal to]] is
formed by bridges with necessity of inspection visual "at
least" not advance;
--the union upward [Cl.sub.2.sup.[greater than or equal to] is
formed by bridges with necessity of inspection visual "at
least" advance;
--the union upward [Cl.sub.3.sup.[greater than or equal to]] is
formed by bridges with necessity "at least" of programming for
urgent action;
--the union upward [Cl.sub.4.sup.[greater than or equal to]] is
formed by bridges that need "at least" immediate closure or
the traffic reduced.
The downward union classes are:
--the union upward [Cl.sub.1.sup.[less than or equal to]] is formed
by bridges with necessity of inspection visual "at most" not
advance;
--the union upward [Cl.sub.2.sup.[less than or equal to]] is formed
by bridges with necessity of inspection visual "at most"
advance;
--the union upward [Cl.sub.3.sup.[less than or equal to]] is formed
by bridges with necessity "at most" of programming for urgent
action.
--the union upward [Cl.sub.4.sup.[less than or equal to]] is formed
by bridges that need "at most" immediate closure or the
traffic reduced.
In this application, the upward union classes
[Cl.sub.1.sup.[greater than or equal to]] and the downward union classes
[Cl.sub.4.sup.[less than or equal to]] contain all 100 bridges
considered: in fact for all these bridges degree of maintenance urgency
is always at least scheduled and at most immediate partial or total
closure. Usually, classification issues concern data collection for a
given class [Cl.sub.t] dividing the universe U into class [Cl.sub.t]
(set of positive examples) and its complement U--[Cl.sub.t] (set of
negative examples), t = 1,..., n. However, such bipartitions do not take
into account the preference order among classes. Thus, multi-criteria
classification requires another type of bipartition which divides the
universe into upward and downward unions of classes and
[Cl.sub.t.sup.[greater than or equal to]] and [Cl.sup.[less than or
equal to].sub.t-1], t = 1,..., n. Each object from the upward union
[Cl.sub.sup.[greater than or equal to].sub.t] is preferred to each
object from the downward union [Cl.sup.[less than or equal to].sub.t-1].
Data collection for upward union [Cl.sup.[greater than or equal
to].sub.t] considers all objects positive and all objects belonging to
[Cl.sup.[less than or equal to].sub.t-1] as negative. Analogously, data
collection for downward union [Cl.sup.[less than or equal to].sub.t-1]
consider all objects belonging to [Cl.sup.[less than or equal
to].sub.t-1] as positive and all objects belonging to [Cl.sup.[greater
than or equal to].sub.t] as negative.
In this approach to data collection, the dominance principle is
applied as follows.
Let [[greater than or equal to].sub.q] be a weak preference
relation of U (often called outranking) representing a preference
applied to the set of objects associated with criterion q; x [[greater
than or equal to].sub.q] y means 'x is at least as good as y with
respect to criterion q'. If x [[greater than or equal to].sub.q] y
for all q[member of]P, then x dominates y with respect to P[subset or
equal to]C (for short x P-dominates y) denoted by x[D.sub.P]y. Assuming,
without loss of generality, that domains of all criteria are ordered
such that preference increases with the value, x[D.sub.P]y is equivalent
to:f(x, q) [greater than or equal to] f(y, q) for all q [member of] P.
Observe that for each x [member of] U, x[D.sub.P]x, that is, P-dominance
is reflexive.
Given P [subset or equal to] C and x [member of] U, the
"granules of knowledge" used in DRSA for approximation of the
unions [Cl.sub.t.sup.[greater than or equal to].sub.t] and
[Cl.sub.t.sup.[less than or equal to].sub.t] are:
--a set of objects dominating x, called P-dominating set (3):
[D.sup.+.sub.p](x) = {y [member of] U : y[D.sub.p]x}, (3)
--a set of objects dominated by x, called P-dominated set (4):
[D.sup.-.sub.p](x) = {y [member of] U : x[D.sub.p]y}. (4)
In the case study, for example [D.sup.+.sub.p](x) is composed of
all bridges with a degree of inspection urgency "at least"
equal to x, while [D.sup.-.sub.p] (x) is composed of all bridges that
have a degree of inspection urgency "more than" equal to x.
For example, if the criteria were "type of damage" and
"seismic zone", both evaluated on three scales of high,
moderate and low, and bridge x is evaluated as moderate regarding
"type of damage" as well as "seismic zone", then:
[D.sup.+.sub.p](x) is composed of all moderate or low bridges
regarding type of damage and seismic zone, and [D.sup.-.sub.p](x) is
composed of all moderate or high bridges regarding type of damage and
seismic zone.
Given the set of criteria P [subset or equal to] C, the inclusion
of object x [member of] U in the upward union of classes
[Cl.sup..[greater than or equal to].sub.t], t = 2, ..., n, creates an
inconsistency in the dominance principle if one of the following
conditions holds:
--x belongs to class [Cl.sub.t] or better but it is P-dominated by
object y belonging to a class worse than [Cl.sub.t],
--x belongs to a worse class than [Cl.sub.t] but it P-dominates
object y belonging to class [Cl.sub.t] or better.
If, given the set of criteria P [subset or equal to] C, the
inclusion of x [member of] U in [Cl.sub.t.sup.[greater than or equal
to]], t = 2, ..., n, creates an inconsistency in the dominance
principle, we say that x belongs to [Cl.sub.t.sup.[greater than or equal
to]], with some ambiguity. Thus, x belongs to [Clt.sup.3] without any
ambiguity with respect to P [subset or equal to] C, if x [member of]
[Cl.sub.t.sup.[greater than or equal to]], and there is no inconsistency
in the dominance principle. This means that all objects P-dominating x
belong to [Cl.sub.t.sup.[greater than or equal to]].
It is possible that y [member of] U belongs to class
[Cl.sub.t.sup.[greater than or equal to]], with eventually some
ambiguity, if one object x [member of] [Cl.sub.t.sup.[greater than or
equal to]] exists such as y dominates x with respect to the set P
[subset or equal to] C, or y [member of] [D.sup.+.sub.p](x). For
example, if "bridge y dominates bridge x", with the latter
belonging to the ascending union of classes [Cl.sub.3.sup.[greater than
or equal to]] of bridges with not less than urgent intervention, it is
possible that y belongs to the ascending union of classes
[Cl.sub.3.sup.[greater than or equal to]] too, if with some ambiguity.
In simpler words, if bridge y is no worse than bridge x for all criteria
(i.e. y dominates x) then the maintenance urgency of y should be no less
than that of x. Some ambiguity is possible if y or some other bridge
that dominates x has lower maintenance urgency for specific reasons not
taken into account (for example criteria not considered in the general
case).
Saying that y [member of] U belongs to [Cl.sub.t.sup.[greater than
or equal to]] does not necessarily mean that it actually belongs to this
class. In the previous example, it is possible that y belongs to
[Cl.sub.3.sup.[greater than or equal to]] but, if its maintenance
urgency is 2 (prior inspections), y belongs to class
[Cl.sub.2.sup.[greater than or equal to]]. This is due to the ambiguity
between x and y with respect to criteria set P.
For P [subset or equal to] C, the set of all objects belonging to
[Cl.sub.t.sup.[greater than or equal to]] without any ambiguity
constitutes the P-lower approximation (5) of [Cl.sub.t.sup.[greater than
or equal to]], denoted by [P.bar]([Cl.sup.[greater than or equal
to].sub.t]), and the set of all objects that possibly belong to
[Cl.sub.t.sup.[greater than or equal to]] constitutes the P-upper
approximation (6) of [Cl.sub.t.sup.[greater than or equal to]], denoted
by [bar.P]([Cl.sup.[greater than or equal to] .sub.t]):
[P.bar]([Cl.sup.[greater than or equal to].sub.t]) = {x [member of]
U: [D.sup.+.sub.p](x) [subset or equal to] [Cl.sup.[greater than or
equal to].sub.t]}, for t = 1,...,n, (5)
[bar.P]([Cl.sup.[greater than or equal to].sub.t]) = {x [member of]
U: [D.sup.-.sub.p](x) [intersection] [Cl.sup.[greater than or equal
to].sub.t] [not equal to] [empty set]}, for t = 1,...,n, (6)
Analogously, it is possible to define P-lower approximation (7) and
P-upper approximation (8) of [Cl.sup.[greater than or equal to].sub.t]
as follows:
[P.bar]([Cl.sup.[less than or equal to].sub.t]) = {x [member of] U:
[D.sup.-.sub.p](x) [subset or equal to] [Cl.sup.[less than or equal
to].sub.t]}, for t = 1,...,n, (7)
[bar.P]([Cl.sup.[less than or equal to].sub.t]) = {x [member of] U:
[D.sup.+.sub.p](x) [intersection] [Cl.sup.[less than or equal to].sub.t]
[not equal to] [empty set]}, for t = 1,...,n, (8)
All the objects belonging to [Cl.sub.t.sup.[greater than or equal
to]] and [Cl.sub.t.sup.[greater than or equal to]] with some ambiguity
constitute the P-boundary (9, 10) of [Cl.sub.t.sup.[greater than or
equal to]] and [Cl.sub.t.sup.[less than or equal to]], denoted by
[Bn.sub.P]([Cl.sub.t.sup.[greater than or equal to]]) and
[Bn.sub.P]([Cl.sub.t.sup.[greater than or equal to]]), respectively. It
is possible to represent them in terms of upper and lower approximations
as follows:
[Bn.sub.p]([Cl.sup.[greater than or equal to].sub.t]) =
[bar.P]([Cl.sup.[greater than or equal to].sub.t]) -
[P.bar]([Cl.sup.[greater than or equal to].sub.t]), for t = 1,..., n,
(9)
[Bn.sub.p]([Cl.sup.[less than or equal to].sub.t]) =
[bar.P]([Cl.sup.[less than or equal to].sub.t]) - [P.bar]([Cl.sup.[less
than or equal to].sub.t]), for t = 1,..., n, (10)
From a data collection point of view, P-lower approximations of
unions of classes represent certain knowledge provided by criteria from
P [subset or equal to] C, while P-upper approximations represent
possible knowledge and the P-boundaries contain doubtful knowledge
(Greco et al. 2002b).
4.3. Quality of sorting and reducts
For every P [subset or equal to] C and t [member of] T, the quality
of approximation of partition Cl by set of attributes P, or in short,
quality of sorting was defined (11):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)
This quality expresses the ratio of all P-correctly sorted objects
in the information table. Note, that enlarging the set of criteria P,
the quality of sorting not increase but decrease. In fact, any new
criteria that were ambiguous before become non-ambiguous.
In the case study, for example, sorting quality is 0.98; meaning
the information table contains "ambiguous objects". In Table 3
the bridge 25 has the same characteristics as bridge 46 but it is made
of better material which requires higher maintenance than bridge 46.
This means that the conditional criteria alone do not explain the
decision maker's choices.
Each minimal subset P [subset or equal to] C such as
[[gamma].sub.P](Cl) = [[gamma].sub.C](Cl) is called a reduct of Cl and
is denoted by [RED.sub.Cl]. Again, it is possible that a data table has
more than one reduct. The intersection of all the reducts is known as
the core, denoted by [CORE.sub.Cl] (12):
[Core.sub.[gamma]](P) = [intersection][Red.sub.[gamma]](P). (12)
Is not possible to remove from the data sample set criteria from
[CORE.sub.Cl], without impairing the knowledge to be discovered. This
means that in set C there are indispensable criteria included in the
core, exchangeable criteria included in some reducts but not in the
core, and redundant criteria being neither indispensable nor
exchangeable, thus not included in any reduct.
In the case study there are 13 reducts. Table 4 shows the criteria
included in the reducts and the core which shows that there are 5
indispensable criteria (material, type of damage, damaged surface,
seismic zone, hydrogeological instability), 8 exchangeable criteria and
0 redundant criteria.
4.4. Extraction of decision rules
The dominance-based rough approximations of upward and downward
unions of classes allow to induce a generalized description of objects
contained in the data matrix in terms of "if..., then ..."
decision rules.
For a given upward or downward union of classes,
[Cl.sub.t.sup.[greater than or equal to]] or [Cl.sub.S.sup.[less than or
equal to]], the decision rules induced under a hypothesis that objects
belonging to [P.bar]([Cl.sup.[greater than or equal to].sub.t]) or
[P.bar]([Cl.sup.[less than or equal to].sub.S])are positive and all the
others negative, suggest an assignment to "class [Cl.sub.t] or
better", or to "class [Cl.sub.S] or worse", respectively.
On the other hand, the decision rules induced under a hypothesis that
objects belonging to the intersection [bar.P]([Cl.sup.[less than or
equal to].sub.S]) [intersection] [bar.P]([Cl.sup.[greater than or equal
to].sub.t]) are positive and all the others negative are suggesting an
assignment to some classes between [Cl.sub.S] and [Cl.sub.t] (s < t).
In multi-criteria classification, it is meaningful to consider the
following three types of decision rules:
--certain D [greater than or equal to] -decision rules, providing
lower profile descriptions for objects without ambiguity: if f(x,
[q.sub.1]) [greater than or equal to] [rq.sub.1] and f(x, [q.sub.2])
[greater than or equal to] [rq.sub.2] and ... f(x, [q.sub.p]) [greater
than or equal to] [rq.sub.p], then x [member of] [Cl.sub.t.sup.[greater
than or equal to]], for example:
if "Bridge Age" is [less than or equal to] 2, "rate
of heavy traffic" is [less than or equal to] 1 and "strategic
viability" is [less than or equal to] 1, then the recommended
degree maintenance urgency is at least 2 (prior inspection), i.e. bridge
x [member of] [Cl.sub.2.sup.[greater than or equal to]];
--certain D [less than or equal to] -decision rules, providing
lower profile descriptions for objects without ambiguity: if f(x,
[q.sub.1]) [less than or equal to] [rq.sub.1] and f(x, [q.sub.2]) [less
than or equal to] [rq.sub.2] and ... f(x, [q.sub.p]) [less than or equal
to] [rq.sub.p], then x [member of] [Cl.sub.t.sup.[less than or equal
to]], for example:
if "damaged surface" is [greater than or equal to] 2,
"hydrogeological instability" is [greater than or equal to] 2
and "average daily traffic" is [greater than or equal to] 2,
then the recommended degree of maintenance urgency is at most 2 (prior
inspection), i.e. bridge x [member of] [Cl.sub.2.sup.[less than or equal
to]];
--approximate D [greater than or equal to][less than or equal to]
-decision rules providing simultaneously lower and upper profile
descriptions for objects belonging to
[Cl.sub.s][union][Cl.sub.s]+1[union] ... [union][Cl.sub.t] without being
able to discern class: if f(x, [q.sub.1]) [greater than or equal to]
[rq.sub.1] and f(x, [q.sub.2]) [greater than or equal to] [rq.sub.p],
and ... f(x, [q.sub.k]) [greater than or equal to] [rq.sub.k] and f(x
[q.sub.k+1]) [less than or equal to] [rq.sub.k+1] and ... f(x,
[q.sub.2]) [less than or equal to] [rq.sub.p], then x [member of]
[Cl.sub.s][union][Cl.sub.s+1][union] ... [union] [Cl.sub.t], for
example:
if "material" is [less than or equal to] 3,
"environmental condition" is [greater than or equal to] 2,
"seismic zone" is [less than or equal to] 2, "static
scheme" is [greater than or equal to] 3, "hydrogeological
instability" is [greater than or equal to] 2, then the recommended
degree of maintenance urgency must be between 2 (prior inspection) and 1
(scheduled inspection) i.e. bridge x [member of]
[Cl.sub.1][union][Cl.sub.2].
On the left side of D [greater than or equal to][less than or equal
to] -decision rule it is possible to have f(x, q) [greater than or equal
to] rq and f(x, q) [less than or equal to] r'q, where rq [less than
or equal to] r'q, for the same q [member of] C. Moreover, if rq =
r'q the two conditions boil down to 'f(x, q) = rq'.
An object x [member of] U supports decision rule r if its
description matches both the condition and decision part of the rule.
Decision rule r covers object x if it matches the condition part of the
rule. Each decision rule is characterized by its strength defined as the
number of objects supporting the rule. In the case of approximate rules,
strength is calculated for each possible decision class separately. If a
univocal rule is supported by objects from the lower approximation of
the corresponding decision class only, then the rule is called certain
or deterministic. If, however, a univocal rule is supported by objects
from the upper approximation of the corresponding decision class only,
then the rule is called possible or probabilistic. Approximate rules are
supported, in turn, only by objects from the boundaries of the
corresponding decision classes. Generating decision rules from decision
tables is a complex task and a number of procedures have been proposed
to simplify it. Existing induction algorithms use one of the following
strategies (Stefanowski 1998):
--generate a minimal set of rules covering all objects from a
decision table,
--generate an exhaustive set of rules consisting of all possible
rules for a decision table,
--generate a set of "strong" decision rules, called a
satisfactory set of rules, each of which apply to many objects but not
necessarily to all the objects in the decision table.
For the induction of decision rules free software is also available
called 4eMka2 [http://idss.cs.put.poznan. pl/site/4emka.html]. This
software solves multi-criteria sorting problems using rough set theory
and decision rule induction and is freely available on the internet.
In this case, 1183 "strong" decisional rules were
generated, as follows:
--187 recommend a maintenance urgency degree [less than or equal
to] 1 (scheduled inspection);
--188 recommend a maintenance urgency degree [less than or equal
to] 2 (prior inspection);
--96 recommend a maintenance urgency degree [less than or equal to]
3 (urgent intervention);
--214 recommend a maintenance urgency degree [greater than or equal
to] 4 (reduced traffic or bridge closure);
--323 recommend a maintenance urgency degree [greater than or equal
to] 3 (scheduled urgent intervention);
--175 recommend a maintenance urgency degree [greater than or equal
to] 2 (prior inspection).
For each rule, the number and identity of data table objects
supporting that rule are known. They are given in Table 5.
5. Applying SSDs to the bridges of a secondary suburban road
This study focused on applying current methodology to the bridges
of the Italian national road, owned by the National Road Agency (ANAS),
to estimate maintenance urgency. There are 14 bridges whose main
characteristics are reported in Table 6.
5.1. Data acquisition
For each bridge, the data on construction year, static scheme,
average daily traffic and heavy traffic ratio were provided by the ANAS
database. The foundation soil characteristics and hydrogeological
instability data were obtained from thematic maps. The PGA values were
obtained by the Italian Technical Code. A GIS was used to locate the
bridges on the net and to process alternative routes.
Damage typology, damaged surface, damaged components and the
presence or not of anti-seismic devices were assessed by visual
inspections (section 5.2).
5.2. Bridge assessment by visual inspection
In this study, visual inspections were carried out using the First
Level Sheets for the Degradation Survey described in section 3.2. For
example, for bridge 1 the following phenomena were detected: corrosion
of reinforcement and spalling both in the deck and in beam-abutment
connection, joint deterioration and intrusive vegetation. Therefore the
"type of damage" criterion corresponds to 2, the "damaged
surface" criterion is 1 and the "damaged components"
criterion is 3.
5.3. Determination of the urgency degree of maintenance activities
Once the condition criteria values were known, the following Table
7 was compiled:
The decision support system above was used. At this stage of the
research, the proposed DSS makes use of simple software to receive the
DRSA output. Table data are inputted and the recommended urgency degree
for inspections and decisional rules behind them are outputted. For
example, let us evaluate the maintenance urgency of bridge No. 1 which
has the characteristics described in Table 8.
From the rules above, the DSS suggests a maintenance urgency of 2
(prior inspection) returning 4 rules which recommend a degree [greater
than or equal to] 2 (prior inspection), 37 rules for a degree [less than
or equal to] 2 (prior inspection) and 46 rules for a degree [less than
or equal to] 3 (urgent intervention). The inspection urgency degree
returned by the DSS is that which satisfies all the decisional rules,
being 2 in this case. If it is impossible to satisfy all the rules
returned by the DRSA, the rules supported by a larger number of
"objects" in the decision table are considered, until the rule
set allows for a unique urgency value which satisfies all the decisional
rules.
The decisional rules allow the decision maker to understand the
DRSA's recommended urgency.
Obviously, it is unreasonable to give the decision maker a large
number of rules (88 for bridge 1), thus, for each class only the most
supported rules have been reported. For bridge 1 these are:
--if "type of damage" [less than or equal to] 2 then
urgency at least 3 (support 80);
--if "type of damage" [less than or equal to] 2 and
"damaged surface" [less than or equal to] 1 then urgency at
least 2 (support 32);
--if "seismic zone" [greater than or equal to] 3 then
urgency at most 2 (support 34);
--if "age of the bridge" [greater than or equal to] 2 and
"damaged elements" [greater than or equal to] 3 then urgency
at most 2 (support 34).
From these rules, it is clear that the decision criteria are those
related to structure damage (type of damage, damaged surface, damaged
components), seismic zone (PGA value) and bridge age.
It is possible that the decision maker take into account the
suggestion given by the DSS or that he prefers to carry out the
maintenance to improve the bridge and reduce urgency.
Proceeding on the same for all other bridge is possible to classify
the bridges of the network depending on the degree of urgency of
maintenance activities.
6. Conclusion
1. In this paper a Decision Support System for bridges maintenance
management at the network level, based on Dominance Rough Set Approach,
is proposed. It allows to set the order of bridges according to their
maintenance urgency, on the basis of parameters related to bridge
damage, characteristics of the territory, traffic and the network.
2. Using the Dominance Rough Set Approach, a decision model
expressed in terms of easily understandable "if ... then ..."
rules has been generated. The decision rules allow to control the
decision process and to avoid the "black box" effects of many
alternative methods. The starting point of the methodology is
represented by the "exemplary decisions" with which the
decision maker expresses his preferences.
3. This methodology is like a "glass box", since it is
possible to map out from where each rule is derived. The model generated
is flexible and could be updated by varying the exemplary decisions set
required to calibrate the model.
4. A sample application of the proposed model is also reported.
Putting as input the characteristics of some bridges of a rural Italian
road, the built Decision Support System gives back the intervention
urgency for each bridge and only the more important decision rules that
can help decision maker to understand the reasons of the suggestions.
5. This is the first application of the Dominance Rough Set
Approach to this type of issue and other new developments will be
presented in future.
doi:10.3846/bjrbe.2014.05
Received 14 November 2011; accepted 2 February 2012
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Maria Grazia Augeri (1) ([mail]), Rosario Colombrita (2), Salvatore
Greco (3), Paola Sapienza (4)
Dept of Civil and Environmental Engineering, University of Catania,
viale Andrea Doria, 6, Catania, 95125 Italy
E-mails: (1) maugeri@dica.unict.it; (2) saroc@dica.unict.it; (3)
salgreco@unict.it; (4) paola.sapienza@dica.unict.it
Table 1. Attributes description
Attribute Characteristic Refer
[A.sub.1] Bridge age Approval date of the
design project
[A.sub.2] Material Used in
constructing the
bridge
[A.sub.3] Environmental Structural exposure
conditions
[A.sub.4] Foundation soil Physical and
mechanical
characteristics of
soil
[A.sub.5] Damage type Structure
degradation
[A.sub.6] Damaged surface Damage extent
[A.sub.7] Damaged components Importance of the
damaged component in
the overall
structure
[A.sub.8] Seismic zone Peak ground
acceleration (PGA)
which a measure of
seismic risk at the
site
[A.sub.9] Anti-seismic Dampers or isolators
devices which dissipate
earthquake energy
[A.sub.10] Static scheme Structural
components
configuration
[A.sub.11] Hydrological Morphological
instability processes caused by
soil and soil and
sub-soil water flow
which produce
superficial
degradation or even
a landslide
[A.sub.12] Average daily Level of daily
traffic (ADT) traffic over the
bridge
[A.sub.13] Heavy traffic rate Percentage of heavy
vehicles with
respect to total
traffic over the
bridge
[A.sub.14] Strategic The bridge is on a
viability main itinerary
subject to emergency
vehicle transit
[A.sub.15] Alternative Presence of an
viability alternative
availability itinerary to the
bridge without an
excessive increase
in the travel time
[A.sub.15] Decision attribute Level of maintenance
urgency
Attribute Value
[A.sub.1] 1 low The bridge was designed in the 90s
2 moderate The bridge was designed 70s-90s
3 high The bridge was designed
earlier in the 70s
[A.sub.2] 1 low Bridge is made of masonry
2 moderate Bridge is made of steel
3 high Bridge is made of reinforced concrete
4 very high For pre-stressed concrete
[A.sub.3] 1 low Non-aggressive environment
2 moderate Moderately aggressive environment
3 high Very aggressive environment
[A.sub.4] 1 low Rocky soil
2 moderate Granular (uncohesive) soil
3 high Limey or clayey (cohesive) soil
[A.sub.5] 1 low No damage or superficial damage
2 moderate Cracks, corrosion or imperfect bearings
3 high Large deformations, ruptures,
displacements, instability, pre-
stressed cable damage, minimum hydraulic
clearance is not met.
[A.sub.6] 1 low Little or no surface damage
(i.e. < 10% of the total
component surface)
2 moderate Damaged surface is between
10% and 60% of the total
component surface
3 high Damage surface is more than 60% of the
total component surface
[A.sub.7] 1 low No damaged components
2 moderate Damaged components are non-structural or
secondary structural
3 high Damaged components are main structural
components (piers, spandrels, spans)
[A.sub.8] 1 low PGA less than 0.15 g
2 moderate PGA between 0.15 g and 0.25 g
3 high PGA greater than 0.25 g
[A.sub.9] 1 yes Anti-seismic devices are present
2 no Anti-seismic devices are not present
[A.sub.10] 1 low Arch bridge
2 moderate Continuous beam
3 high Series of simply-supported beams
[A.sub.11] 1 low No hydrological instability
2 moderate Moderate hydrological risk
3 high High hydrological risk
[A.sub.12] 1 low ADT < 6000 vpd
2 moderate ADT < 6000 < 20 000 vpd
3 high ADT > 20 000 vpd
[A.sub.13] 1 low Less than 10%
2 moderate Between 10% and 20%
3 high More than 20%
[A.sub.14] 1 yes Bridge is strategic
2 no Bridge is not strategic
[A.sub.15] 1 yes An alternative itinerary is present
2 no An alternative itinerary is not present
[A.sub.15] 1 low No intervention is required. It is
possible to have next inspection within
the scheduled time
2 moderate No intervention is required. It is
necessary to anticipate the next
inspection (prior inspection)
3 high Urgent intervention is required
4 very high It is necessary to close the bridge or
to reduce the traffic on he bridge
Attribute Remarks
[A.sub.1] The more recent the
project, the more
probable anti-seismic
design rules have been
applied
[A.sub.2] The better the materials'
mechanical
characteristics, the
lower the vulnerability
of the bridge
[A.sub.3] The higher the value, the
worse the bridge
conditions
[A.sub.4] The characteristics of
the soil influence bridge
stability and longevity
[A.sub.5] The higher the value, the
worse the bridge
conditions
[A.sub.6] The higher the value,
the worse the bridge
conditions
[A.sub.7] The higher the value, the
worse the bridge
conditions
[A.sub.8] The higher the value, the
greater the seismic risk
[A.sub.9] The presence of anti-
seismic device increases
the bridge
characteristics
[A.sub.10] The higher the value the
worse the ability of the
structure to respond to
an earthquake
[A.sub.11] The higher the value the
greater the risk
[A.sub.12] The higher the value the
greater the stress
suffered by the structure
[A.sub.13] The higher the value the
greater the stress
suffered by the structure
[A.sub.14]
[A.sub.15]
[A.sub.15]
Table 2. Data table
Bridge Criteria
[A.sub.1] [A.sub.2] [A.sub.3] [A.sub.4] [A.sub.5]
1 3 1 1 1 1
2 3 1 2 2 1
3 3 1 3 1 1
98 2 4 1 1 3
99 2 4 2 3 3
100 1 4 2 1 3
Bridge Criteria
[A.sub.6] [A.sub.7] [A.sub.8] [A.sub.9] [A.sub.10]
1 1 1 1 2 1
2 1 1 2 2 1
3 1 1 1 2 1
98 2 3 3 2 3
99 3 3 2 2 2
100 1 3 3 1 2
Bridge Criteria
[A.sub.11] [A.sub.12] [A.sub.13] [A.sub.14] [A.sub.15]
1 3 1 3 1 1
2 2 2 3 2 1
3 1 1 2 1 2
98 1 2 2 2 2
99 2 1 2 1 2
100 2 2 2 2 1
Bridge
Decision
attribute
1 2
2 1
3 1
98 4
99 4
100 3
Table 3. Ambiguous object
Criteria
Bridge
[A.sub.1] [A.sub.2] [A.sub.3] [A.sub.4] [A.sub.5]
25 3 2 2 2 1
46 3 3 2 2 1
Criteria
Bridge
[A.sub.6] [A.sub.7] [A.sub.8] [A.sub.9] [A.sub.10]
25 1 1 1 2 3
46 1 1 2 2 3
Criteria
Bridge
[A.sub.11] [A.sub.12] [A.sub.13] [A.sub.14] [A.sub.15]
25 2 2 3 2 1
46 2 2 3 2 1
Bridge Decision attribute
25 2
46 1
Table 4. Reducts and core
Criteria Reduct
#1 #2 #3 #4 #5 #6 #7
[A.sub.1] X X X X X X
[A.sub.2] X X X X X X X
[A.sub.3] X X X
[A.sub.4] X
[A.sub.5] X X X X X X X
[A.sub.6] X X X X X X X
[A.sub.7]
[A.sub.8] X X X X X X X
[A.sub.9]
[A.sub.10] X X X
[A.sub.11] X X X X X X X
[A.sub.12] X X X X
[A.sub.13] X X
[A.sub.14] X X X
[A.sub.15] X X X
Criteria Reduct Core
#8 #9 #10 #11 #12 #13
[A.sub.1] X X X X
[A.sub.2] X X X X X X X
[A.sub.3] X X
[A.sub.4] X X X X
[A.sub.5] X X X X X X X
[A.sub.6] X X X X X X X
[A.sub.7] X
[A.sub.8] X X X X X X X
[A.sub.9] X X
[A.sub.10] X
[A.sub.11] X X X X X X X
[A.sub.12] X X X X
[A.sub.13] X X X X X
[A.sub.14] X X
[A.sub.15]
Table 5. Decision rules
Rules If...
1 (Environmental conditions [greater than or equal to] 3) &
(Damage type [greater than or equal to] 3)
538 (Material [greater than or equal to] 4) & (Foundation soil
[greater than or equal to] 3)
713 (Bridge age [less than or equal to] 1) & (Material [less
than or equal to] 2) & (Environmental conditions <1)
900 (Bridge age [less than or equal to] 1) & (Material [less
than or equal to] 3) & (Environmental conditions [less than
or equal to] 1)
Rules Then ... Support Cases supported
1 Urgency at most 4 3 22, 44, 67
538 Urgency at most 2 4 82, 88, 95, 99
713 Urgency at least 1 2 17, 40
900 Urgency at least 2 4 17, 40, 61, 70
Table 6. Main bridge characteristics
Bridge Length, m Material
1 70.00 Reinforced concrete
2 106.00 Masonry + Reinforced concrete
3 12.00 Masonry +Reinforced concrete
4 23.00 Masonry +Reinforced concrete
5 10.00 Reinforced concrete
6 14.00 Reinforced concrete
7 14.00 Reinforced concrete
8 14.00 Reinforced concrete
9 26.00 Reinforced concrete
10 14.00 Reinforced concrete
11 26.00 Reinforced concrete
12 29.00 Reinforced concrete
13 35.00 Reinforced concrete
14 12.00 Masonry
Table 7. Data table
Bridge Criteria
[A.sub.1] [A.sub.2] [A.sub.3] [A.sub.4] [A.sub.5]
1 2 3 1 1 2
2 2 3 1 1 2
3 3 3 1 1 2
4 3 3 1 1 2
5 3 1 1 1 3
6 2 3 1 1 1
7 2 3 1 1 1
8 2 3 1 1 1
9 2 3 1 3 2
10 2 3 1 2 2
11 2 3 1 2 2
12 2 3 1 1 2
13 2 3 1 3 2
Bridge Criteria
[A.sub.6] [A.sub.7] [A.sub.8] [A.sub.9] [A.sub.10]
1 1 3 3 2 3
2 1 3 3 2 3
3 3 3 2 2 1
4 3 3 2 2 1
5 1 1 2 2 1
6 1 1 2 2 3
7 1 1 2 2 3
8 1 1 2 2 3
9 1 2 2 2 3
10 1 2 2 2 3
11 1 2 2 2 3
12 2 2 2 2 3
13 1 3 3 2 3
Bridge Criteria
[A.sub.11] [A.sub.12] [A.sub.13] [A.sub.14] [A.sub.15]
1 1 1 2 1 1
2 1 1 2 1 1
3 1 1 2 1 1
4 1 1 2 1 1
5 1 2 1 1
6 1 1 2 1 1
7 1 1 2 1 1
8 1 1 2 1 1
9 1 1 2 1 1
10 1 1 2 1 1
11 3 1 2 1 1
12 3 1 2 1 1
13 1 1 2 1 1
Table 8. Characteristics of bridge No. 1
Criteria Bridge characteristics
Bridge age designed between the 70s and
90s
Material reinforced concrete
Environmental conditions non-aggressive environment
Foundation soil rocks
Damage type corrosion
Damaged surface <10% total component surface
Damaged components structural component (deck)
Seismic zone 0.252 g
Anti-seismic devices no
Static scheme series of simply-supported
beams
Hydrogeological instability no risk
Average Daily Traffic <60 00 vpd
Heavy traffic volume between 10% and 20%
Strategic viability no
Alternative viability yes
