The article presents a hierarchical fuzzy rule base for intelligent support of decision making about cause of structural crack of stone building. According to civil engineering practice the causes of structural cracks are classified by the followings diagnoses: static overload; dynamic overload; especial overload; defects of basis and foundation; temperature influence; breach of technological process during the building. Source information needed for decision making is the data of visual investigation of building, icluding simple measuremnts. For decision making we take into account 42 input attributes. The hierarchical system ties 9 fuzzy knowledge bases, which contain 151 rules in total. Cause detection of the crack is carrying out by max-min fuzzy inference with hierarchical knowledge base. Learning of fuzzy rules by genetic algorithms provided a good matching between real causes of cracks and modeling results.
Instant and correct diagnosis of the stone construction cracks makes further investigations, design and reconstruction of buildings successful. The task of diagnosis may be solved correctly by high qualification engineers with huge experience. The number of such experts is lacking, hence the creation of decision making model for diagnosis of structural cracks of buildings is necessity.
One of the most promising ways to processing uncertain expert information is fuzzy
sets theory [
This paper presents a hierarchical fuzzy rule base and corresponding technology
for decision making support about the cause of stone construction crack of building.
The used approach to fuzzy diagnosis model design is based on a conception of
creation and learning the hierarchical fuzzy rule base. The general conception of
identification of multifactor dependences with hierarchical fuzzy rule base is described in
article [
Hierarchical interconnection between input attributes (X) and cause of crack (D) is represented in the form of a fuzzy inference tree (Figure 1). Graph vertices are interpreted in the following way: the squares – possible causes of the crack; the circles – input attributes; the double circles – fuzzy rule bases. Enlarged attributes, to which edges correspond, as going out of nonterminal vertices are interpreted as followings: y1 – state of construction; y2 – destruction of brickwork; y3 – extra support for some cause; y4 – support for basis and foundation defects; y5 – possibility of static overload; y6 – demand to temperature juncture; y7 – support for of crack connected with breach of technological processes; y8 – demand to sedimentary juncture.
The hierarchical structure of decision process makes the diagnostic model more interpretable and more compact. The hierarchical structure reflects expert knowledge and information from a lot of special books and articles about crack dynamics. x15 – {across whole wall (AW), between walls (B), borders of wall (BW), from monolithic inclusion (MI), at supports (S), top of construction (TC), free field (FF), bottom of construction (BC)}; x16 – {vertical (V), oblique (O), horizontal (H)}; x17 – {up, slanting (S), down (D)}; x18 – {hair (H), small (S), average (A), large (L), very large (VL)}; x19 – {short (S), average (A), long (L), very long (VL)}; x24 – {one-sided (OS), through (T)}; x27 – {low (L), excellent (E)}; x28, x30, x41, x42 – {absence (A), uncertainly (U), present (P)}; x33, x39 – {unnecessary (UN), necessary (N)}; x34, x40 – {absence (A), low quality (LQ), quality (Q)}; x37 – {low (L), high (H)}.
The following 24 terms are used for linguistic assessment of enlarged attributes: y1 – {normal (N), weak (W), very weak (VW)}; y2 – {absence (A), medium (M), heavy (H)}; y3 – {absence (A), static overload (SO), dynamic overload (DO), especial overload (EO), defects of basis and foundation (BF), temperature influence (T), breach of technological process of building (TP)}; y4 – {absence (A), low (L), high (H)}; y5, y7 – {absence (A), present (P)}; y6, y8 – {observed (O), ignored (I)}.
Formalisation of linguistic terms of input attributes is carried with bell-shaped membership function with 2 parameters: b – core of the fuzzy set and c – concentration of membership curve.
Natural language expert expressions, which tie up the attributes and output
variable, are formalised in fuzzy rule base form. Tables 1 – 9 show some fragments of the
rule bases. In the tables the symbol "–" is equal to membership function “Do not
care” [
Decision making about diagnosis is carried out according to the following algorithm: 1. Fix the input attributes of the diagnosis object.
2. Make up a fuzzification i.e. find input attributes membership degrees to
linguistic terms and present results in form of bifuzzy sets. Adjective “bifuzzy” [
3. Make up a fuzzy inference for all fuzzy rule bases.
4. Choose the decision from set {d1, d2, d3, d4, d5, d6} with the maximum membership degree.
During the fuzzification the membership degrees of input attributes to terms from
rule base are calculated taking into account crisp and fuzzy values. For crisp source
data, membership degree is calculated by the substitution of the current value of the
input attribute into membership function. It is possible to use the linguistic values for
input attributes. In this case the linguistic values is taken from the term-set of relevant
variable. Hence, the linguistic value became equals to some fuzzy set. For fuzzy
source data, the membership degree of one fuzzy set (the value of an input attribute)
to another fuzzy set (a term from a rule base) must be calculated. According to [
1 1 min b1 b2 c2 c1 2 .
Fuzzy inference is carried according to tree from Figure 1. Operations
fuzzification – defuzzification are not employed for enlarged attributes (Figure 3). The result
of fuzzy inference on the lower level in form of fuzzy set is passed directly into
inference machine at higher level. Fuzzy output value at lower hierarchical level is
considered as input fuzzy value at higher hierarchical level. In this case, membership
functions for terms of the conjuncted variables (enlarged attributes) are unnecessary. We
have selected the following inference options: minimum as t-norm and single winner
rule [
After learning, the misclassification rate is about 4.5%. There are 4 wrong inferred decisions out 89 testing cases. Note, that for these 4 cases the inferred decision with the second rank is correct.
We described the hierarchical fuzzy rule base for decision making support about cause of structural crack of stone building. Different causes of structural cracks are classified by the followings diagnoses: static overload; dynamic overload; especial overload; defects of basis and foundation; temperature influence; breach of technological process during the building. For decision making we use 42 input attributes. The hierarchical system ties 9 fuzzy knowledge bases, which contain 151 rules. The hierarchical structure of decision making process makes the diagnostic model more interpretable and more compact. Learning of fuzzy rules by genetic algorithms provided a good concordance between real causes of cracks and modeling results with misclassification rate at level of 4.5%. The design of our inferring model for stone construction crack diagnosis suggests a general approach to expert systems design in other diagnostic fields.