It has increasingly been recognized that our society is undergoing a significant transformation, usually described as a transition from an “Industrial to an Information Society”.
In this span of transformation, one way of simplifying a very complex system perhaps the most significant one – is to allow some degree of uncertainty in its description. It is entails an appropriate aggregation or summary of the various entities within the system. Statement obtained from this simplified system are less precise, but their relevance to the original system is fully maintained. That information loss that is necessary for reducing the complexity of the system to a manageable level is expressed in UNCERTAINTY. The concept of Uncertainty is thus connected with both complexity of information 1
Thus, complexity and information are closely interrelated. During this course of time, the realization of Uncertainty, vagueness and ambiguity in the world has led to the concept of fuzziness.
· Uncertainty – it’s the state of being uncertain, something that you cannot be sure That. Where as, our statistics & probabilistic approaches deal with uncertainty.
· Vagueness – not clear in person’s mind.
· Ambiguity – it’s the state of having more than one possible meaning.
Hence, due to these constraints; the idea of fuzzy sets was born in 1964, and in 1965 Lofti A. Jadeh a well respected professor in the department of Electrical Engineering & computer science at university of Californiapresented a paper on FUZZY SETS 2.
This made to the development of new mathematical formulation called the FUZZY CONCEPTS by himself (Jadeh) & and his followers.
In the last 3 decades, significant progress has been made in the development of Fuzzy sets and Fuzzy logic theory in Engineering, Natural and socio-economic science.
The successful applications of fuzzy sets and fuzzy logic can be attributed to the fact that the fuzzy theory reflects the true situation of the real world, where human thinking is dominated by approximate reasoning logic.
* For example, during the design stage, decision making is mainly based on conceptual understanding. Unfortunately, such information is usually vaguely defined by the experts. Therefore, consideration of imprecise and vague information becomes an important issue in an automated design environment.
2.0. Application to Civil Engineering tiled
Engineering field acts as platform for the application of fuzzy concepts. We know that Civil Engineering field is fundamentally different from other disciplines. It means that the theories never fully satisfy the problem considered. This is because, the civil engineering project is complex and usually large in nature: hence, there will be almost no chance to test the prototype compared to other disciplines. As result, there is uncertainty in application of theoretical solutions. More over, our engineering problems are constraint satisfaction problems.
Example: when we trying to determine the best frame design for a building, one may be constrained by design codes as well as by the design specifications such as functional, architectural and structural behavior requirements. Once these constraints are satisfied the remaining problem may be to find and design that requires the least construction material. These, constraints mush be strictly satisfied without disregarding them.
There by, these complexities, uncertainties and vagueness in decision making in real structure provide the main motivation for the use of fuzzy concepts. Its proven by the experts that the fuzzy concept is a useful tool and can be effectively applied for the problems in civil engineering in order to arrive at the optimal solution.
During the last decade there has been a growing interest in the application of these concepts to engineering problems. Fuzzy concepts provide an easy way of dealing with complex problems, because it can be built with fuzzy models containing vagueness and impreciseness in knowledge representation. Hence, it is suited for applications where the ability to model real world design problem in precise mathematical form is difficult.
Also by integrating Fuzzy concepts with Genetic Algorithms (GA) or Genetic Programming (GP) and Neural Networks (NN), the complex problems can be more efficiently and effectively solved in order to arrive at optimal soln.
So fat, these concepts are effectively used in;
· Structural analysis and Design
- for structural optimization and optimum Design of structures [ Y.Yang & C.K. soh (2000)].
- Computation Morphogenesis of Deserete structures [ H. Kawamura et al. (2004)].
Construction field
- for management problems like construction scheduling of the project [ S.S. Leu et al. (1999)]
- for planning of life cycle of project problems like selection of best construction equipment.*
[V.S.S. Kumar et al. (2004)]
· The field of Hydrology & Water Resource engineering.
- for forecasting rainfall, rainfall runoff, river stage, etc.
[Chandramouli et al. (2001)]
- Hydrologic flow routing [Paresh Deka et. al. (2005)]
· Traffic engineering
- for automatic control of traffic signals based on fuzzy stochastic model [George J. Klir et al.]
· Reliability of structures
- for damage assessment in structures [W. Chiang et al. (2000)]
· Metal structures
- for predicting fatigue & Creep characteristics [J. Harris (2001)]
Fuzzy concepts thrown its wide variety of application for the field of civil engineering. Hence, we the engineers should explore it for its potential application for the problems we face in this engineering world.
3.0. FUZZY CONCEPTS
Fuzzy concepts is mainly of Fuzzy set and Fuzzy logic theories. Before, we get into the discussions of these theories let us discuss classical or crisp sets.
3.1 Classical or crisp sets
A classical set is a collection of objects in a given domain. In which object either belongs to the set or does not belong to the set. Therefore, there is a sharp boundary between members of the set and those not in the set.
We can say, classical sets divide the world into yes & no, true & false and white & black.
i.e, membership in a classical set is a yes & no or true & false or white & black concept.
The classical set can be represented as:
Membership function is;
Where:
C- an arbitrary crisp set
X – an element in the set
U – membership function
Limitation:
It has sharp boundary between members of the set and those not in the set.
3.11 Classical set operations:
There are three basic set operations
Where:
U is a universe discourse
A & B are sets inU.
There three basic operations have several fundamental properties interms of
Laws. They are:
1. Commulative law
2. Associative law
3. distributive law
4. Law of Double complementation
5. Demorgan’s law
6. Law of Excluded middle
7. Law of Contradiction
8. Law of Tautology
9. Law of Absorption
3.2 Fuzzy set theory
Fuzzy set theory generalizes classical set theory.
The best way to introduce fuzzy sets sis to start with a limitation of classical sets. Fuzzy set theory directly addressess this limitation by allowing membership in a set to be a “matter of degree”. The degree of membership in a set is expressed by a number between 0 & 1 ‘0’ means complete Exclusion from the set and ‘1’ means absolute INCKUSION in the set. Such a function which maps object in a domain of concern to their membership value in the set is called as “membership function” (u).
The membership represents a certain degree of belonging of the object in the fuzzy set. The transition from belong to not belonging is gradual, which gives one some means of handling vagueness. Fuzzy sets thus overcome a major weakness of crisp sets.
i.e, Fuzzy sets do not have an arbitrarily established boundary to separate the members from nonmembers. Thus in a fuzzy set membership function can be represented as:
The discussion on classical set and fuzzy set can be represented graphically:
3.2.1. Membership function
What is important about membership function is that it provides gradual transition from regions completely outside a set to regions completely in the set.
There are, numerous types of membership functions, but the most commonly used in practice are triangles, trapezoioh, bell curves, Gaussian and sigmoidal functions.
In which triangles and trapezoid membership functions are more frequently used.
3.2.2. Representation of fuzzy sets
Fuzzy sets can be represented in two ways:
1. by enumerating membership values of those elements in the set completely or partially.
2. by defining the membership function mathematically.
Generally speaking, a fuzzy set ‘A’ be defined through enumeration using expression.
Similarly. When U (universe of discourse) is a set of interval of real members, then:
3.2.3. Fuzzy set operations.
The fuzzy set theory is formulated in terms of the following specific operators.
1. A common fuzzy conjunction (AND) operation is the minimum operator.
Example: Figure shows the result of using min. to form the intersection of two fuzzy sets is:
2. A common fuzzy disjunction (OR) operation is the max. Operator.
Example: Figure shows the result of using max. to form the union of two fuzzy sets as;
3. A complement of fuzzy set A defined by the difference between one and the membership degree in A.
Example: Figure shows Fuzzy set medium.
It should be noted that the range of membership degree is restricted to the set {0, 1}.
3.2.4. A paradox in fuzzy set theory
It doesnot mean that fuzzy set theory should hold all the laws of classical sets.
i.e, failing of the law of excluded middle and the law f contradiction.
We know that fuzzy concept generalizes the black & white concept, by varying the membership to allow “gray areas”, it has to violate two fundamental laws of set theory.
i.e, Law of excluded middle
& the law of contradiction
In other words, it is possible for an element to partially belong to both fuzzy set and sets complement.
Example: Mr. X is a person belongs to the set of bald people to degree 0.2.
i.e, μ bald (X) = 0.2
Based on the definition of complement operator, Mr. X belongs to the complement of bald people to degree 0.8.
i.e, μbald (X) = 0.8
The person partially belongs to the set of bald people as well as the set of people who are not bald.
In fuzzy theory any element partially belonging to fuzzy set, is also partially member of its complement.
Hence, we can say that law of excluded middle & law of contradiction are not axioms of fuzzy theory.
3.2.5. Hedges:
A hedge is a modifier to a fuzzy set. It modifies the meaning of the original set to create a compound fuzzy set.
Very & more or less are two commonly used hedges. Their definitions are listed below.
Figure illustrates their application to the concept of high temperature.
“Very” has the effect of narrowing the membership function.
“More or less” widens the membership function.
This is intuitively appealing because the criteria for “Very high” should be more stringent that those for “thigh”, while the criteria for “more or less high” should be relaxed.
Hence, we can express
In principle, a hedge can be applied to any fuzzy set. In practice, however it is used only when the compound terms is meaningful. For instance, very medium temperature, very excellent.
3.2.6. General aggregation operations:
Aggregation operations on fuzzy sets are operations by which several fuzzy sets are combined to produce a singe set.
In general, any aggregation operation is defined by a function
h = {0, 1}n – [0,1]
when applied to ‘n’ fuzzy sets {Al, A2, ------An} defined on ‘U’, ‘h’ produces an aggregate fuzzy set ‘A’ by operating on the membership grades of each χ € U in the aggregated sets.
Thus,
In order to quality as an aggregation function “h” must satisfy at least the following two axiomatic requirements, which express the essence of the notion of the aggregation.
Axiom 1: If h = {0, 0, -------0} = 0
and h = { 1, 1, ------1} = 1 as boundary conditions.
Axiom 2. For any pair,
i.e, ‘h’ is monotonic nondecreasing in all its arguments the other two additional axioms which we commonly use are:
Axiom 3: ‘h’ is a continuous function.
Axiom 4: ‘h’ is a symmetric function in all its arguments.
i.e,
Fuzzy unions and intersections can be easily qualify as aggregation operations on fuzzy sets.
Note: In application to problems various types of aggregation procedures can be used to in corporate exports judgments.
3.2.7 Properties of fuzzy sets
Convex fuzzy set
The height of a fuzzy set is the largest membership grade affined by any element in that set. A fuzzy set is called normalized when at least one of its elements attains the maximum possible membership grade. If membership grades range in the closed interval between 0 and ‘1’, then at least one element must have a membership grade of ‘1’ for the fuzzy set to be considered normalized such a normalized fuzzy set to be considered as a convex fuzzy set if and only if:
If the above statement is violated, then such a fuzzy set is a nonconvex fuzzy set.
- Cut of a Fuzzy set:
An - cut of a fuzzy set A is a crisp set A that contains all the elements of the universal set U that have a membership grade in A greater that or equal to the specified value of . this definition can be written as:
The value ‘ ‘ can be chosen arbittrarity but is often designated at the values of the membership grades appearing in the fuzzy set under consideration.
Example: consider the concept “moderately approved” regarding the public opinion of a presidential candidate. The universe of discourse is the % of those people supporting the candidate in a poll
Let U = [ 0%, 10%, 20%......... 100%]
The membership function of a fuzzy set
Cardinality of fuzzy set:
The cardinality of a set is the total number of elements in the set. Since, an element can partiality belong to a fuzzy set, a natural generalization of the classical notion of cardinality is to weigh each element by its membership degree, which gives us the following formula for calculating the cardinality of a fuzzy set:
The cardinality of fuzzy sets is useful for answering questions. Therefore, it plays on important role in fuzzy databases and information systems.
3.3 Linguistic variables:
A linguistic variable is defined as a variable whose values are in words, phrases or sentences in a given language.
Having introduced the fundamental concept of a fuzzy set, it is natural to see how it can be used. Like a conventional set, conventional set, a fuzzy set can be used to describe the value of a variable.
For example, if the variable is linguistic one say “site conditions” which can take the values of excellent, fair, poor, etc. The fuformation expressed in words or phrases in this example has a value but it is not clearly defined. These values scan be made susceptible to meaningful classifications.
Jadeh considered these values as fuzzy sets.
i.e, U = {site conditions}
This universal set consists the sub sets as
U = {excellent, fair, poor}
In which membership values may be given for the subsets considering their ranges.
In the manner, fuzzy sets can well be considered for quantifying qualitative factors.
If the Degree of belief in a fuzzy set is say getting narrowed, then the linguistic variables can be described as say very excellent, very fair, very poor, etc.
This can be represented graphically as;
Instead of enumerating all these different linguistic descriptions, they can be generated from a core set of linguistic terms using “modifiers”. In fuzzy theory we call these modifiers as HEDGES as discussed earlier.
NOTE: The interpretation of fuzzy linguistic terms has been clearly given by Jadh [Fuzzy sets and applications (1984)]
3.4. Fuzzy Logic
The foundation of fuzzy logic is mathematically the fuzzy set theory. This logical technique is based on four basic concepts.
1). Fuzzy sets
2). Linguistic variables
3) Possibility distribution
4) Fuzzy if – then rules.
· Fuzzy sets – sets with smooth boundaries
· Linguistic variables – variables whose value are both qualitatively and quantitatively described by a fuzzy set.
· Possibility distribution – constraints on the value of a linguistic variable imposed by assigning it a fuzzy set.
· Fuzzy if – then – a knowledge representation scheme for describing a functional mapping or a logic formula that generalizes an implication in two valued logic.
Frame work of these concepts results in a “Fuzzy logic controller” (FLC).
3.4.1 Fuzzy logic controller [FLC].
A fuzzy logic controller (FLC) is a rule based system that incorporates the flexibility of human decision making by means of the use of fuzzy set theory. The fuzzy rules of FLC incorporate fuzzy linguistic terms described by membership functions. These functions are intended to represent human experts conception of the linguistic terms, thus giving an approximation of the confidence with which precise numeric value is described by a linguistic label.
Fuzzy rules take the form. If (conditions or inteceden) and THEN (actions or consequent). Where conditions ad actions are linguistic labels applied to input and output variables, respectively.
A set of such fuzzy rules conciliates the Fuzzy rule as of FLC. The system uses this rule base to produce precise output values according to the input values.
Control process is divided in to three stages.
· FUZZIFICATION: to calculate the Fuzzy input (i.e, to evaluate the input variable w.r. to to corresponding linguistic terms I the condition side).
· FUZZY INFERENCE: to calculate the Fuzzy output (i.e, to evaluate the activation strength of every rule base and combine their action sides.
· DEFUZZIFICATION: to calculate the actual output (i.e, to correct the Fuzzy output into precise numerical value).
· So, this is just an outlook discussion of Fuzzy concepts, if yougo n depth, explore it & understanding; it will help in applying these concepts to a complex problem, so that the problem can be tuned to meet all constraints or in over coming all the uncertainties, ambiguity & vagueness .
3.5. Considering the CASE STUDY : A Fuzzy logic approach to selection of Cranes.
3.5.1 INTRODUCTION
Selection of a crane on construction projects is a central element in the planning phase of the life cycle of the project. An appropriately selected crane is the life blood of many multistory construction projects and contributes largely to the efficiency, timeliness, and profitability of the project. An error in selection can lead to large and unnecessary expenses arising from operational inadequacy or Failure, and can produce an unsafe working environment. Decision to select a particular crane depends on many input parameters such as site conditions, economy, safety and weather conditions and their variability. Many of these parameters are qualitative, and subjective judgments implicit in these terms cannot be directly incorporated into the classical decision making process.
Therefore, it is important to find some way of incorporating an expert’s knowledge on these qualitative Factors into the selection process. One way of achieving this, by the development of fuzzy logic to decision making process. In this, expert’s opinion interns of membership values of fuzzy sets are aggregated by modified pessimistic aggregation procedure and final selection of a crane is achieved by max-min criteria.
The objective of this paper is to provide a methodology to select the most appropriate crane using a fuzzy logic approach by considering the viewpoints of the experts in relevant fields.
3.5.2. Classification of cranes.
A crane is defined as a mechanism for lifting and lowering loads with a hoisting mechanism. On many construction sites a crane is needed to lift small to medium loads such as concrete skips, reinforcement, and form work. As the lifting needs of the construction industry have increased and diversified, a large number of general and special purpose cranes have been designed and manufactured.
Mobile cranes:
A mobile crane is crane capable of movement under its own power without being restricted to predetermined travel. Mobility is provided by mounting or integrating the crane with trucks by providing crawlers. Truck-mounted cranes have the advantage of being able to move under their own power to the construction site. Additionally, mobile cranes can move about the site, and are often been able to do the work of several stationary units.
Tower Cranes:
The tower crane is a crane with a fixed vertical most and equipped with a winch for hoisting and lowering loads. Tower cranes are designed for situations which require operation in congested areas. Congestion may arise from the nature of the site or from the nature of the construction project. There is no limitation to the height of a high rise building that can be constructed with a tower crane. The very high line speeds, up to 304.8m/min, available with some models yield good production rates at any height. They provide a considerable horizontal working radius, yet require a small work space on the ground. Some machines can also operate in winds of upto 72.4km/hr, which is far above mobile crane wind limits.
Derrick Cranes:
A derrick is a device for raising, lowering, and moving loads laterally. The simplest form of the derrick is called a Chicagoboom and is usually installed by being mounted to building during construction. This derrick arrangement becomes a guy derrick when it is mounted to a mast and a stiff leg derrick when it is fixed to a frame.
An appropriately selected crane contributes, efficiency, timeliness, and profitability to the project. This is achieved using fuzzy logic approach.
3.6. Translation of qualitative factors into fuzzy sets:
The qualitative factors that are commonly used in the selection of cranes need to be translated into mathematical measures. For example, high temperature, inadequate external access, and poor design co-ordination can be considered as qualitative factors. The resulting quantitative indices for these variables are known as fuzzy variables.
Let U be a universe of linguistic variable set of element’x’s, and A be a subset of U. that is, variability of a linguistic variable. Each element, ‘x’, is associated with a membership value ma (x) , to the subset ‘A’, in which ma (x) is the grade of membership if x in A, and us a real number.
Therefore, variability of qualitative factor
A= (x, ma(x) ), xEA&AEU.
For example, the linguistic variable “Weather conditions” is interpreted , using the following membership function. The “universal set” is “weather conditions” as a variable. The linguistic quantification for this variable can be expressed by fuzzy sets. Dividing the range into increments of 0.1, the following fuzzy set ‘poor’ can be assigned for the variable ‘poor weather conditions.
