CHAPTER - 3 FUZZY SET THEORY AND MULTI CRITERIA DECISION MAKING

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1 CHAPTER - 3 FUZZY SET THEORY AND MULTI CRITERIA DECISION MAKING 3.1 Introduction Construction industry consists of broad range of equipment and these are required at different points of the execution period. Various formulae are available to estimate the equipment required which are based on the standards and site conditions with some factor of safety. But due to the uncertain nature of the project, the available methods fail to provide a realistic estimation. Many non-quantitative factors and approximate numbers such as availability of labor, weather conditions, and approximate number of equipments also influence the construction project (Kumar et al. 2003). Consider a case where it is required to estimate the number of dozers required at the site for a quantity of excavation of about 1000 Cu.m with bucket capacity of around 2 Cum, cycle time of approximately 20 seconds, lead time of somewhat 30 min and swell factor of may be 80%. Apart from these, other vague factors affecting a project include the number of working hours per day, number of working days, weather conditions etc. Without incorporating these factors, estimation would fail to provide a realistic reasoning. Under these conditions, experts qualitatively judge equipments required and there is a possibility that the estimated numbers may increase or decrease at the site. The presence of large number of interacting variables creates a problem for optimization (Kumar et al. 2003). Thus, these seemingly vague but powerful factors have to be incorporated in to the model for achieving optimal solutions. In the past, many analytical and heuristic methods are developed for solving the optimization problems in the construction industry. Analytical methods use mathematical programming techniques such as linear programming and dynamic programming. General mathematical models however are difficult to create, and require a great deal of computational effort. They could not solve the larger and more complex problems encountered in practice and hence, suited for small size projects. Heuristic methods performed over a variety of problems are widely used in practice because of their simplicity and ease of application. However, they proved to be very 21

2 much problem dependent, with varying effectiveness on different cases. Furthermore, there was no other alternate way of finding the best set of heuristic rules to use for a given case. Although, they have provided good solutions, they could not guarantee optimality. Moreover, both kinds of methods generally focused on a single objective optimization. Decision making in construction industry for selection of an optimum crane is the most important scientific and economic effort. The essence of the project manager is to overcome uncertainty and be able to make correct and consistent choice. Crane selection is an important aspect in the construction industry to organize various activities of the project. It is used to determine desirable strategies when a decision maker is faced with several equipment alternatives in an uncertain environment. Equipment selection arises when there are two or more alternatives for equipment or when there is a conflicting action available in the project. Hence, a decision may be defined as a selection of an act, considered to be the best according to some predesignated standards, from the available options. 3.2 Fuzzy Logic Fuzzy logic is an approach for computing uncertain data based on "degrees of truth" rather than the usual "true or false" (1 or 0) (Blockly 1979). The idea of fuzzy logic was first advanced by Zadeh in 1960s. Zadeh was working on the problem of computer understanding of natural language. Natural language (like most other activities in life and indeed the universe) is not easily translated into the absolute terms of 0 and 1. (Whether everything is ultimately describable in binary terms is a philosophical question worth pursuing, but in practice much data we might want to feed a computer is in some state in between and so, frequently, are the results of computing.) Fuzzy logic is a system for dealing specifically with variables that are defined vaguely (Dubois 2000). Fuzzy logic captures quantitative and empirical knowledge of the science; provides simultaneous simulation of multiple processes and non-linear relations; calculates volume and type of sediment accumulation etc. Vagueness is imparted by qualifying adjectives like Approximately, About, Around, etc. Fuzzy logic operates like crisp logic in many ways. Its proponents claim that crisp logic is simply a subset of fuzzy logic (Bortolan and Degani 1985). Unlike classical logic 22

3 which requires a deep understanding of a system, exact equations and precise numeric values, fuzzy logic incorporates an alternative way of thinking, which allows modeling complex systems using a higher level of abstraction originating from our knowledge and experience (Lootsma 1997). It resembles human decision making with its ability to work from approximate data and find precise solutions. One can reason with fuzzy syllogisms. The applicability of fuzzy logic in the field of construction management is quiet notable (Dubois 1980). Fuzzy logic allows expressing this knowledge with subjective concepts such as good weather and a little bit experienced contractor etc., which are mapped into exact numeric ranges. A number of practical and wide-ranging applications are available that fuzzy logic can bring about in the field of construction management (Ross 1997) Fuzzy Logic Definitions The term "fuzzy logic" emerged in the development of the theory of fuzzy sets by Lotfi Zadeh (1965). A fuzzy subset A of a (crisp) set X is characterized by assigning to each element x of X the degree of membership of x in A (e.g. X is a group of people, A the fuzzy set of old people in X). Now if X is a set of propositions then its elements may be assigned their degree of truth, which may be absolutely true, absolutely false or some intermediate truth degree: a proposition may be truer than another proposition. Saying yes (which is the mainstream of fuzzy logic) one accepts the truth-functional approach; this makes fuzzy logic to something distinctly different from probability theory since the latter is not truth-functional (the probability of conjunction of two propositions is not determined by the probabilities of those propositions). 0 a 0 1 b 1 Crisp set 0 c Fuzzy set 0 Fig. 3.1 Demonstration of a crisp set and fuzzy set 23

4 3.2.2 Fuzzy sets and membership functions A fuzzy set is a class of objects in which there is no sharp boundary between those objects that belong to the class and those that do not (Klir and Yuan 1995). Some common examples are the sets of all young engineers or the set of small equipments. In a fuzzy set an object may have a grade of membership intermediate between full membership, represented by 1, and non-membership, represented by 0 (Dombi 1990). Mathematically, the transition from regular sets (crisp sets) to fuzzy sets can be explained by assuming a universal set X such that A is a crisp subset of X. Define a function v: X {0,1} such that vx ( ) is 1 if x Aand vx ( ) is 0 when x A. This function is the characteristic function of any crisp subset A of X. Fuzzy sets generalize the characteristic function by allowing all values in the unit interval. This is known as the membership function. Any fuzzy subset F of X is defined by its membership function : X [0,1]. Then ( x) where x F denotes the grade of membership of x in the fuzzy set F. X is called the universe of discourse. A fuzzy subset (or fuzzy set as it is commonly referred to) is defined as a set of ordered pairs F x, ( x) x F, F X. The membership function is denoted by F for the fuzzy set F. Let X be a universe, or a set of elements, x s, and let A be a subset of X. Each element, x, is associated with a membership value to the subset A, A (x). A is an ordinary, non-fuzzy, or crisp set, then the membership function is given by 1 if x belongs to A A x 0 if x does not belong to A Eq. (3.1) In the above Eq. 3.1 there are only two possibilities for an element x, either being a member of A, i.e., A (x) = 1, or not being a member of A, i.e., A (x) = 0. In this case, A has sharp boundaries. On the other hand, if the membership function is allowed to take values in the interval (0,1), depending on its belongingness to the set, such a set is called a fuzzy set. Therefore, such a set does not have sharp boundaries and the membership of x in set A is fuzzy. For example, let x be the level of experience of labor which may range from excellent experience, i.e, x = 1.0, to never 24

5 been to a construction site i.e., x = 0. By dividing the range of labor experience into increments of 0.1, the short experience A, as a linguistic variable, can be defined as Short experience A = [x 1 =1/ A (x 1 )=0, x 2 =0.9/ A (x 2 )=0, x 3 = 0.8/ A (x 3 )=0, x 4 =0.7/ A (x 4 )=0.0, x 5 =0.6/ A (x 5 )=0.0, x 6 =0.5/ A (x 6 ) =0.0, x 7 = 0.4/ A (x 7 )=0.1, x 8 =0.3/ A (x 8 )=0.5, x 9 =0.2/ A (x 9 )=0.7, x 10 =0.1/ A (x 10 )=0.9, x 11 =0/ A (x 11 )=1.0] Eq. (3.2) To define the subset of very experienced contractors, a satisfactory answer is difficult one as the class of Very experienced contractors is not a set in the classical sense, but belongs to a fuzzy, not crisply defined type. The definition of very experienced may involve a spectrum of human perceptions and the class of very experienced contractors and is therefore said to represent a fuzzy set. Or, in short, it can be expressed as Short/small/bad experience, A = (0.4/0.1, 0.3/0.5, 0.2/0.7, 0.1/0.9, 0.0/1.0) Eq. (3.3) Similarly, Quiet small & very small experience can be expressed as Quiet Small (A ) = {x/( A(x) ) 1.25 }; and Very small, A = {x/( A(x) ) 2 }. Eq.(3.4) The fuzziness in the definition of short experience is obvious. It is clear that different values of x; or grades of experience have different membership values, A (x), to the fuzzy set A, short experience. The values of x are 0.4, 0.3, 0.2, 0.1, and 0, and the corresponding membership values are 0.1, 0.5, 0.7, 0.9, and 1.0, respectively. Other values of x have zero membership values to the fuzzy set A. These membership values are generally assigned based on subjective judgment with the help of experts and can be updated with more applications of the method in various projects. If crisp set were used in this example, the value of x would be 0.0 with a membership value of 1.0. And Long/good/high = {1.0/1.0, 0.9/0.8, 0.8/0.7} Eq. (3.5) The medium experience can also be defined as, Medium/Moderate/Middle/Central Experience, A = (0.7/0.2, 0.6/0.7, 0.5/1, 0.4/0.7, 0.3/0.2) Eq. (3.6) 3.3 Basic Set Theoretic Operations for fuzzy sets In order to use fuzzy sets in practical problems, some operational rule similar to those used in classical set theory is defined. Some of the Basic Set Operational rules used are Fuzzy union, fuzzy intersection, fuzzy compliment and fuzzy composition explained in detail as follows (Dombi 1982). 25

6 3.3.1 Fuzzy Union ( ) The union, of fuzzy sets A and B of a universe, X corresponds to the connective or and its membership function is A B (x) = max [ A (x), B (x)] Eq. (3.7) Fuzzy Intersection ( ) The intersection, of fuzzy subsets A and B correspond to the connective and and its membership function is A B (x) = min [ A (x), B (x)] Eq. (3.8) For example, consider superintendent experience as a linguistic variable, to be expressed by the fuzzy subset C = (1.0/1.0, 0.9/0.8, 0.8/0.6, 0.7/0.4, 0.6/0.2} And long labor experience is represented by Eq.3.3. then, the labor or superintendent experience can be expressed by the union of the fuzzy subsets B and C, and is given by B C = [1/1, 0.9/0.9, 0.8/0.7, 0.7/0.4, 0.6/0.2] On the other hand, the labor and superintendent experience can be expressed by the intersection of the fuzzy subsets B and C, and is given by B C = [1/1, 0.9/0.8, 0.8/0.6, 0.7/0.2, 0.6/0.1] Fuzzy Complement ( A (x)) The complement of a fuzzy subset A is denoted by A, and its membership function is A (x) = 1- A (x) Eq. (3.9) Fuzzy Relation A fuzzy relation, R, or Cartesian product, AB, between two fuzzy subsets A (subset of a universe X) and B (subset of a universe Y) has the following membership function : R (x i, y j ) = AxB (x i,y j ) = min ( A (x i ), B (y j )) Eq. (3.10) The relation is usually expressed in matrix form as R0S (x i, y j ) = min ( R (x i, y j ), S (s i, y j )) Eq. (3.11) 26

7 3.3.5 Fuzzy Composition If R is a fuzzy relation from X to Y, and S is a fuzzy relation from Y to Z, the composition of R and S is a fuzzy relation that is described by the following membership function. RoS (x i, z k ) = max (min ( R (x i,y j ), S (y j z k ) ) ) Eq. (3.12) Eq basically evaluates a fuzzy relation between the fuzzy subsets X and Z using the fuzzy relations of X and Z to the common fuzzy subset Y. The triangular function represented by x(a, b, c) and trapezoidal by x(a,b,c,d) have three, four parameters a (minimum), b (middle) and c (maximum) and a (minimum), b, c (core) and d (maximum) that determine the shape of the triangle or trapezoidal. Figure 4.2 shows the triangular function of x(a, b, c) and trapezoidal function x(a,b,c,d) x 1 C 0 Trapezoidal Triangular membership a b c 0 a b c bound bound d Support of Fuzzy Fig 3.2 Typical membership functions and their properties x A trapezoidal membership function is specified by four parameters given by (x, a, b, c, d) The function for triangular and trapezoidal are described as shown in Eq and Eq , x a ( x a)/( b a), x ( a, b) A ( c x)/ c b), x ( b, c) 0, x c Eq. (3.13) 0, x a ( x a) /( b a), x ( a, b) A Eq. (3.14) 1, x ( b, c) ( d x) / d c), x ( c, d) 27

8 Similar definitions for Gaussian and generalized Bell can be obtained. However triangular and trapezoidal functions are simple and most frequently used. The membership functions are not restricted to these four. One can have their own tailor- made functions. The functions above were mere one dimensional in nature. In principle one can even have multi- dimensional membership functions. Therefore, for sets A and X, we can define the fuzzy set A in X as a set of ordered pairs given by x ( x) Where A, a x X. Eq A ~ x is the membership function or degree of truth of x which maps X to the membership space [0,1] denoted by ~ x: X [0,1 ] with x ~ x Therefore, conventional set becomes a special case called crisp set. A. A 3.4 Linguistic variables and Linguistic values A linguistic variable is a variable whose value is expressed in terms of spoken language. These terms are imprecise and are represented by fuzzy sets. Let t be a variable that denotes Experience of Engineer over an interval [0, T]. Let X be the domain of real numbers. Let there be three fuzzy sets L, M and H that have the member ship functions L, M, H respectively. Each of these fuzzy sets can be referred to as Less, Medium and High. Experience can be treated as a linguistic variable. Less, Medium and High are linguistic values of the linguistic variable Experience. Therefore t can be Less, Medium and High (Dombi 1990). Broadly speaking, fuzzy logic works on the following steps: Fig 3.3. A Fuzzy Logic Working Steps 28

9 x Ax Fuzzy B x y Fuzzy Defuzzyfy crisp fuzzy fuzzy crisp inference Fig 3.3.B Fuzzy Logic Working Steps The fuzzyfication maps the crisp value into the fuzzy set. A common approach is the singleton fuzzyfication where a binary membership is used. Eq (3.16) represents the singleton method. 1 if x = x (continuous) or nearest x d (discrete) A x 0 otherwise Eq. (3.16) Fuzzification To convert numeric data in real-world domain to fuzzy numbers in fuzzy domain, It is the process of converting a crisp quantity into fuzzy quantity incorporating the needed flexibilities wherever possible Fuzzy inference systems (FIS) (Mamdani) Mamdani inference system is shown in Fig 3.4. To compute the output of this FIS given the inputs, one must go through six steps: 1. determining a set of fuzzy rules 2. fuzzifying the inputs using the input membership functions, 3. combining the fuzzified inputs according to the fuzzy rules to establish a rule strength, 4. finding the consequence of the rule by combining the rule strength and the output membership function, 5. combining the consequences to get an output distribution, and 6. defuzzifying the output distribution (this step is only if a crisp output (class) is needed). 29

10 The following is a more detailed description of this process. Fig 3.4 A two input, two rule Mamdani FIS with crisp inputs Creating fuzzy rules Fuzzy rules are a collection of linguistic statements that describe how the FIS should make a decision regarding classifying an input or controlling an output. Fuzzy rules are always written in the following form: if (input1 is membership function1) and/or (input2 is membership function2) and/or then output n is output membership function. For example, one could make up a rule that says: if temperature is high and humidity is high then room is hot. There would have to be membership functions that define what we mean by high temperature (input1), high humidity (input2) and a hot room (output1). This process of taking an input such as temperature and processing it through a membership function to determine what we mean by "high" temperature is called fuzzification Defuzzyfication Defuzzyfication is a mathematical process used to convert a fuzzy set or sets to a real number. It is necessary as fuzzy sets generated by fuzzy inference in fuzzy arithmetic must be mathematically combined to come up with one number as the 30

11 output of a model. It is the process of calculating a scalar value from the fuzzy output. From composition we obtain a single fuzzy set. Defuzzification aggregates the set into a single value. The common techniques used are the centroid and maximum methods. In the centroid method, the scalar value of the output variable is computed by finding the variable value of the center of gravity of the membership function for the fuzzy value. The general formula is X ( centroid) a x( x) dx b a ( x) dx b Eq. (3.17) where [a, b] is the interval of the aggregated membership function. In the maximum method, the value at which the fuzzy subset has its maximum truth value is chosen as the value for the output variable. Centroid of the Area, the most prevalent and physically appealing of all the defuzzification methods [Sugeno,1985; Lee, 1990]. Fig. 3.5 shows the centroid method of defuzzification. Fig. 3.5 Defuzzification using the Centre-of gravity method The results obtained in section and are defuzzified to obtain the centroidal value. This is achieved by the macro function defuzzy(cell ref. of the solution, required -cut (0,1]). Here, the facility of incorporating the alpha cut is given so as to give the user the freedom to specify his/her own level of satisfaction. Defuzzyfication can be achieved by the following methods: 1) max-membership principle (height method) 2) Centroid method 3) Weighted average method 4) Mean-max membership method 5) Center of sums method 6) Center of largest area method 7) First (or last) or maxima method 31

12 3.4.5 Fuzzy Maximum and Fuzzy Minimum Consider two discrete fuzzy numbers, I ~ and J ~ defined on universe X and Y respectively. The fuzzy maximum between these two numbers is defined as ~ ~ ( z) max ~ max( I, J ) zmax( x, y) I J and the minimum is given as min( ~ ( x), ( y) ~ ~ ( z) min ~ min( I, J ) zmax( x, y) I J max( ~ ( x), ( y) Eq. (3.18) Eq. (3.19) if I ~ or J ~ is continuous then max or min is replaced by supremum and infimum respectively. Fig. 3.6 depicts pictorially the fuzzy numbers and the fuzzy minimum and maximum of them. Fig. 3.6 Fuzzy Numbers and their minimum and maximum 3.5 Fuzzy Numbers To qualify as a fuzzy number, a fuzzy set A ~ on R must possess at least the following three properties: A must be a normal fuzzy set; A (alpha-cut of A; {x A(x) >= }) must be a closed interval for every in (0,1]; The support of A, 0+A (strong 0-cut of A; {x A(x) > 0}), must be bounded. Therefore, every fuzzy number is a convex fuzzy set. Although triangular and trapezoidal shapes of membership functions are most often used for representing fuzzy 32

13 numbers, other shapes may be preferable in some applications, and they need not be symmetrical. This can include symmetric or asymmetric "bell-shaped" membership functions, or strictly increasing or decreasing functions (e.g. sigmoids) that capture the concept of a "large number" or a "small number". Figure 3.7 shows the example of a fuzzy number approximately 10. µ(x) x Fig 3.7 Demonstration of Fuzzy Number Approximately Cuts for Fuzzy Sets Alpha cuts are used to decompose a fuzzy set into a weighted combination of classical sets using the resolution identity principle. The principle is important in fuzzy set theory because it establishes a bridge between fuzzy sets and crisp sets. Hence, it has been used as the foundation for generalizing concepts and methods based on crisp sets into those based on fuzzy sets. Alpha cuts can be used to describe intermediate fuzzy conclusions generated by fuzzy inference. -cuts for fuzzy sets are given by a fuzzy set (A) defined on X and any number α[0,1], the α-cut A is the crisp sets. Such that A ={ x } that contains all A(x) the elements of the universal set X whose membership grades in (A) is greater than or equal to the specified value of α Strong α -Cut Strong α-cut is the crisp set that contains all the elements of the universal set X whose membership grades in (A) are greater than α. It is represented as A ={ x A(x) }. Eq. (3.20) 33

14 3.5.3 Level Set The set of all levels α[0,1] that represent distinct α-cuts of a given fuzzy set A is called a level set of A. ( A) { A( x) for somex X} Eq. (3.21) Where denotes the level set of fuzzy set A defined on X Support of a Fuzzy Set The support of Fuzzy Set A within a universal set X is the crisp set that contains all the elements of X that have non-zero membership grades in A. (or) the support of A is exactly the same as the strong α-cut of A for α=0. This can be represented as S(A) or Supp(A) or 0+ A Core of a Fuzzy Set The 1 cut is called the core of the Fuzzy set (A), represented as 1 A Height of a Fuzzy Set 1-Cut = 1 A. Eq. (3.22) Height of a Fuzzy Set h(a) is the largest membership grade obtained by any element in that set. h(a) = SupA(x) x X Eq. (3.23) Normal and Subnormal Fuzzy Sets A Fuzzy Set (A) is called Normal when h(a) =1; and Subnormal when h(a)<1. The height of A may also be viewed as the supremum of α for which α A 0. Fig. 3.8 Types of Fuzzy Sets, depiction of Core, Support, and boundary of the fuzzy sets 34

15 It is the crisp domain in which we perform all computations with today s computers. The conversion from fuzzy to crisp sets can be done by two means, one of which is alpha-cut sets. Given a fuzzy set A ~, the alpha-cut (or lambda cut) set of A ~ is defined by A x A( x ~ ). By virtue of the condition on ~ A x ( ) in above equation, i.e., a common property, the set A is now a crisp set. In fact, any fuzzy set can be converted to an infinite number of cut sets. If the fuzzy number A ~ is having a trapezoidal membership function, specified by a foursome a, a, a, a the membership can be calculated as: ~ A a a a / a a a a/ a a a a a a 1 3 a a a a 1 a a Then the cut can be expressed by the following interval. U a a a, a a a ~ ~ L ~ A A, A Eq. (3.24). Eq. (3.25) Here, if 2 a a 3, then A ~ is reduced to triangular fuzzy number, specified by a 1 2,3 4, a, a ; and if a a a a then, A ~ reduces to an real number. 3.6 Fuzzy Arithmetics, Fuzzy sets and membership functions Fuzzy arithmetic is the process of extending the arithmetics of normal mathematics to fuzzy mathematics (Kandel 1986). In this, there are mainly two type of operations i.e. arithmetic operations on fuzzy sets and arithmetic operations on fuzzy intervals. The first one considers the fuzzy sets where as the later considers intervals. Even though both operations aim at a single objective of obtaining the arithmetic value of the two fuzzy variables, there are some differences which make both the methods unique in their own domain. Fuzzy arithmetic is based on two properties of fuzzy numbers: 1. Each fuzzy set, and thus each fuzzy number, can fully and uniquely be represented by its alpha-cuts; 2. Alpha-cuts of each fuzzy number are closed intervals of real numbers for all alpha in (0,1]. 35

16 3.6.1 Arithmetic operations on fuzzy sets If fuzzy numbers are represented by continuous membership functions and # is any of the four basic arithmetic operations on the Fuzzy sets A and B, we define a fuzzy set A#B by defining its alpha-cut, (A#B), as (A#B) = A # B for any in (0,1]. Therefore, A#B = Union of all in [0,1]) (A#B). Since (A#B) is a closed interval for each in (0,1], and A and B are fuzzy numbers, A#B is also a fuzzy number. The other method for developing fuzzy arithmetic, which is based on the extension principle, where standard arithmetic operations on real numbers are extended to fuzzy numbers, is as follows. (A#B)(z) = supremum(z=x#y) {min[a(x),b(y)]} Eq. (3.26) for all z in the set of real numbers. More specifically, we define for all z in R: (A+B)(z) = supremum(z=x+y) {min[a(x),b(y)]} Eq. (3.27) (A-B)(z) = supremum(z=x-y) {min[a(x),b(y)]} Eq. (3.28) (A*B)(z) = supremum(z=x*y) {min[a(x),b(y)]} Eq. (3.29) (A/B)(z) = supremum(z=x/y) {min[a(x),b(y)]} Eq. (3.30) To demonstrate the applicability of the method let us consider performing arithmetic calculations on two fuzzy numbers approximately 2.1 & somewhat Firstly, define the membership function for the linguistic parameters approximately & somewhat. Approximately = {0.5/0,0.64/0.6,0.78/0.85,0.9/0.95,1/1,1.05/0.95,1.20/0.65,1.5/0} Somewhat = {0.15/0,0.4/0.3,0.56/0.5,1/1,1.25/0.75,1.4/0.55,2/0} Then, let A=approximately 2.1, B= somewhat therefore using the software, the fuzzy sets A, B can be transformed to: A= {1.05/0,1.344/0.6,1.638/0.85,1.89/0.95,2.1/1,2.205/0.95,2.52/0.65,3.15/0} B ={1.56/0,4.16/0.3,5.824/0.5,10.4/1,13/0.75,14.56/0.55,20.8/0} This is achieved by the macro function fuzzify (cell ref. of approximately, cell ref. of 2.1) for A and fuzzify(cell ref. of somewhat, cell ref. of 10.4) for B in an Excel worksheet. 36

17 From the principles of extension principle using Eq.3.29 above, therefore, A*B = {1.64/0,2.1/0,2.56/0,2.95/0,3.28/0,3.44/0,3.93/0, 4.37/0,4.91/0,5.59/0.3,6.12/0, 6.81/0.3,7.83/0.5,7.86/0.3,8.74/0.3,9.17/0.3,9.54/0.5,10.48/0.3,10.92/0,11.01/0.5,12.23/0.5,12.84/0.5,13.1/0,13.65/0,13.98/0.6,14.68/0.5,15.29/0,17.04/0.85,17.47/0.6,18.35/0,19. 57/0.55,19.66/0.95,21.29/0.75,21.84/1,22.93/0.95,23.85/0.55,24.57/0.75,26.21/0.65,27.3 /0.75,27.52/0.55,27.96/0,28.66/0.75,30.58/0.55,32.1/0.55,32.76/0.65,34.07/0,36.69/0.55, 39.31/0,40.95/0,43.68/0,45.86/0,45.86/0,52.42/0,65.52/0} This is achieved by the developed macro function fmul (cell ref. of A, cell ref. of B) in Excel worksheet. The graphical representation of the obtained results from the extension principle is obtained by pressing the macro key ctrl+shift+p. Fig. 3.9 shows the same results plotted by the developed macro using the shortcut key A*B Fig Results of A*B as per extension principle If the desired value is 24, hence for comparing the results the alpha cut is applied at various levels using the centroid defuzzification technique and the obtained value for A*B can be as represented pictorially as shown in Fig Desired Value A * B Fig Comparison of defuzzified results obtained from extension principle 37

18 3.7 Extension Principle The principle of fuzzifying crisp functions is called Extension Principle. It is a basic identity that allows extending the domain of a function from crisp points to fuzzy sets in a universe. Let a simple relation y = f(x) between one independent variable x and one dependent variable y, where f is of analytic form, x, y are deterministic. This relation is a single-input single-output process where the transfer function represents the mapping provided by the general function f as x f ( x) y. Eq. (3.31) But in a typical case if x is a fuzzy variable or a fuzzy set, and function f may or may not be fuzzy then the mapping has to be extended. Here, X, Y be two universes, A ~, B ~ are two fuzzy sets in X and Y respectively; and f be a function such that f: X Y be a function from crisp set X to crisp set Y. When f is an one-to-one mapping, then ~ ( y) ~ [ f B A 1 ( y)], but if f is not one-to-one then take the maximum i.e. ~ ( y) B 1 max ~ [ f ( y)], A 1 x where 1 ( y ) f ( y) yy Eq. (3.32) y Y Eq. (3.33) x X suchthat f f denotes the set of all points y To identity 3.33 is called the extension principle. For example if denote general multiplication then the multiplication between the two fuzzy numbers A ~, B ~ denoted by A ~ B ~, be defined on universe Z using the extension principle as ~ ( ~ y A B xyz A ~ z) ( ~ ( x) ( )) B Eq. (3.34) where denotes the supremum of the set. If more than one of the combinations of the input variables, X1, X2, are mapped to the same variable in the outer space, Y; i.e. if the mapping is not one-to-one, then take the maximum membership grades of the combinations mappings to the same output variable, which can be shown as x ~ ( X1, X 2) max [min{ 1( X1), 2( X 2)}] Eq. (3.35) A Y f ( X1, X 2 ) Eq and 3.35 help us to develop a procedure for extending crisp domains to fuzzy domains. 38

19 Consider a simple relation y=f(x) between one independent variable x and one dependent variable y, where f is of analytic form, x,y are deterministic. This relation is a single-input single-output process where the transfer function represents the mapping provided by the general function f as x f ( x) y. But in a typical case if x is a fuzzy variable or a fuzzy set, and function f may or may not be fuzzy then the mapping has to be extended. Using the extension principle developed by Zadeh (1975), elaborated by Yager (1986) enables us to extend the domain of a function on fuzzy sets. The extension principle can also be useful in propagating fuzziness through generalized relations that are discrete mappings of ordered pairs of elements from input universes to ordered pairs of elements in an output universe. 3.8 The Analytic Hierarchy Process Part of the analytic hierarchy process (AHP) (Saaty, 1980, 1994) deals with the structure of an M N matrix. This matrix, say matrix A, is constructed using the relative importance of the alternatives in terms of each criterion. The vector (a i,1, a i,2, a i,3,...,a i,n ), for each i=1,2,...,m, in this matrix is the principal eigenvector of an N N reciprocal matrix which is determined by pair wise comparisons of the impact of the M alternatives on the i-th criterion. Some evidence is presented in Saaty (1980) which supports the technique for eliciting numerical evaluations of qualitative phenomena from experts and decision makers. According to AHP the final preference, P i, of alternative A i is also given by Eq. (3.39) However, now the a ij value expresses the relative performance value of alternative A i when it is examined with the rest of the other alternatives in terms of criterion C j. In the maximization case, the best alternative is the one which corresponds to the highest P i value. The similarity between the WSM and the AHP is clear. The AHP uses relative values instead of absolute measures of performance which may or may not be readily available. In the original version of the AHP the performance values a ij are normalized so they sum up to one. That is, the following relation is always true in the AHP case: m aij 1 for any j= 1, 2, 3,..., N. Eq. (3.36) i1 Thus, it can be used in single or multi-dimensional decision making problems. 39

20 Belton and Gear (1983) proposed a revised version of the AHP model. They demonstrated that an unacceptable rank reversal may occur when the AHP is used. Instead of having the relative values of the alternatives A1, A2, A3,..., AM sum up to one (e.g. equation (5) to hold), they propose to divide each relative value by the maximum quantity of the relative values in each column of the M N matrix A. 3.9 Decision Making under Certainty Whenever there is only one choice for crane, it is called selection under certainty. When selection is made under certainty, the decision maker chooses the crane with the highest productivity from the available resources. In such a situation, the decision maker knows which state to expect and chooses the alternative with the highest utility, in the given state of nature i.e., the outcome for each action can be determined and ordered precisely. The alternative that leads to the outcome yielding the highest utility is chosen. The decision-making problem becomes an optimization problem of maximizing the utility function Decision Making under Uncertainty Crane selection in uncertain environment deals with a situation of selection under uncertainty, in which the objective functions as well as the constraints are fuzzy (Dubois and Prade 1985). The fuzzy objective function and constraints are characterized by its membership function. The optimization of the objective function and constraints are analogues to the non-fuzzy environment and the constraints are non interactive. Therefore the decision in a fuzzy environment is an intersection of fuzzy constraints and fuzzy objective function and is represented by the and operation. The relationship between constraints and objective functions in a fuzzy environment is completely symmetric. It is the confluence of goals and constraints (Hersh et al.1979 and Saaty 1974). A fuzzy decision is represented as shown in the Fig. 5.4 where the objective function and the constraints are characterized by the membership function. The membership function (x D ) of the decision is then ( x) ( x) ( x). where (x O ) is membership function of the objective function and (x C ) is membership function of the constraints. D o c 40

21 Fig Solution space of a fuzzy decision Therefore, the fuzzy set is characterized by its membership function and is the minimum of µ O (x) and µ C (x). If the decision maker wants to have a crisp decision proposal, then the highest degree of membership in the fuzzy set is suggested (Dubois et al. 1996). This is called the maximizing decision and is defined by x max arg max min O ( x), ( x). Eq. (3.37) This is as shown in the Fig.3.12 C Fig Exhibit of maximizing decision 41

22 When a fuzzy goal G and a fuzzy constraint C in a space of alternative X is known, then G and C combined to form a decision D which is a fuzzy set resulting from the intersection of G and C. This is represented as min, D where: G C µ D is a membership function of fuzzy set decision µ G is a membership function of fuzzy goal µ C is a membership function of fuzzy constraint. In general form, G 1,G 2,..G n are n goals and C 1, C 2,. C m are m constraints. Then the resultant decision is the intersection of the given goals and constraints. That is D G1 G2... G C C2... and correspondingly D N 1 C m,... Let min G, G,... G, C, 1 2 C 2 n 1 C m C i 1... m where x εx be the membership function of the constraint and i G j 1... n where x ε X be the membership function of the objective function. A i decision is then defined by its membership function. D ( x) ( x) ( x) i 1... m, j 1,... n C i G where denotes an contextdependent aggregator. j 3.11 Decision Making under Risk This situation arises, when the decision maker chooses to consider several cranes with varying probabilities of selection (Kumar et al 2002). The probability of selection may be determined objectively from the past records. However, past records may not be available in many cases. In such cases, the decision maker on the basis of experience and judgment is able to assign subjective probabilities to the various outcomes. Maximum likelihood principle and expectation principle are used to make decisions under this category (Zadeh 1975). Here, the decision maker does not know exactly which state will occur and only knows a probability function of the states. This type of selection is very difficult. When the only knowledge concerning the outcomes consists of their conditional probability distributions, one for each action, the problem of selection, becomes an optimization problem of maximizing the expected utility. 42

23 3.12 Multi Criteria Decision Making for Crane Selection The multi criteria decision making process involves the following steps: Identification of sufficient alternatives Identification of all the criteria that are available to the decision maker Determination of the consequences resulting from the different combination of alternatives and Choosing the best possible alternative on the basis of some criteria. There are several procedures, on the basis of which selection is made. The selection of appropriate alternative depends on factors like the size of project, nature of work, cost of crane, and quantity of work as shown in fig 3.13 Fig 3.13 Selection of an alternative. The selection of an alternative depends on many factors. The cost for unproductive time may impact severely on the ultimate cost of construction. The cost of keeping it idle is greater for large capacity one than for a smaller one. The contrary is true for the costs of production. When selecting new alternative, it is advisable to consider the lifecycle cost of the crane, not just the initial purchase price. What's really important is to minimize all of the direct and indirect expenses throughout the life of the crane. Choosing the economical equipment in the market proves unfavorable in the future. Direct and indirect costs associated with the equipment while purchasing are as shown in Fig

24 Fig Factors for selecting an optimum crane There are five broad areas that should be considered when selecting crane as shown in Fig below. Operating environment and Equipment performance Health and safety Maintenance Appearance Fig Concerted area when buying/leasing equipments In multi-criteria decision problems, relevant alternatives are evaluated according to a number of criteria. Each criterion induces a particular ordering to the alternatives and a procedure is to be evolved for one overall preference ordering (Mac Crimmon 1973). There is similarity between these decisions problems and problems of 44

25 multi person decision making. In both cases, multiple ordering of relevant alternatives are involved and have to be integrated into one global preference ordering. The difference is that the multiple orderings represent either preference of different people or ratings based on different criteria. The number of criteria in multi criteria decision making is virtually always assumed to be finite, and in addition, that the number of considered alternatives is also finite. The basic information involved in multi criteria decision making is expressed in matrix form. Then it is converted into single-criterion decision problems by finding a global criterion. In general the entries of the matrix are fuzzy numbers, and weights are specified in terms of fuzzy numbers on [0,1]. Then using the operations of fuzzy addition and fuzzy multiplication to calculate the weighted overage by suitable formula (Kaufmann and Gupta 1988). Fuzzy multi criteria analysis has become more and more obvious that comparing different ways of action as to desirability, judging and the suitability of products, or determining optimal solutions in decisions problems can in many cases not be done by using a single criterion or a single objective function (Zelene 1982). Hence multi criteria decision making has led to numerous evaluation schemes (e.g., in the areas of cost benefit analysis and marketing) and to the formulation of vector maximum problems of mathematical programming (Kondel 1986). Two major areas have evolved, both of which concentrate on decision making with several criteria: Multi Objective Decision Making (MODM) and Multi Attribute Decision Making (MADM). The main difference between these two directions is: The former concentrates on continuous decision spaces, primarily on mathematical programming with several objective functions; the latter focuses on problems with discrete decision spaces. There are some exceptions to this rule (eg., integer programming with multiple objectives), but for our purposes this distinction seems to be appropriate. Multi objective decision making was first dealt by Hwang and Masud (1979) and multi attribute Decision Making was first dealt by Hwang and Yoon (1981). Fuzzy set has contributed to Multi Objective Decision Making as well as Multi Attribute Decision Making (Chen and Hwang 1992). There are three main steps in 45

26 utilizing a decision making technique involving numerical analysis of a set of discrete alternatives: 1. Determining the relevant criteria and alternatives. 2. Attaching numerical measures to the relative importance (i.e., weights) of the criteria and to the impacts (i.e, the measures of performance) of the alternatives in terms of these criteria. 3. Processing the numerical values to determine a ranking of each alternative In this thesis sensitivity analysis is given utmost importance in preparing the data as said above in step 2. Consider a decision making problem with M alternatives and N criteria. In this paper alternatives will be denoted as Ai (for i = 1,2,3,...,M) and criteria as Cj (for j = 1,2,3,...,N). We assume that for each criterion Cj the decision maker has determined its importance, or weight, Wj. It is also assumed that the following relationship is always true: N Wj 1 Eq. (3.38) i1 Furthermore, it is also assumed that the decision maker has determined aij (for i = 1,2,3,...,M and j = 1,2,3,...,N); the importance (or measure of performance) of alternative Ai in terms of criterion Cj. Then, the core of the typical MCDM problem examined in this paper can be represented by the following decision matrix as seen in Table 3.1 Table 3.1 Decision Matrix Criteria C 1 C 2 C 3... C N Alternatives W 1 W 2 W 3... W N Weights A 1 a 11 a 12 a a 1N A 2 a 21 a 22 a a 2N A 3 a 31 a 32 a a 3N A M a M1 a M2 a M3... a MN 46

27 Some decision methods for instance, the AHP require that the a ij values represent relative importance. Given the above data and a decision making method, the objective of the decision maker is to find the best alternative or to rank the entire set of alternatives (Choobineh and Li 1993). Let P i (for i = 1,2,3,...,M) represent the final preference of alternative A i when all decision criteria are considered. Different decision methods apply different procedures in calculating the values P i. Without loss of generality, it can be assumed by a simple rearrangement of the indexes that the M alternatives are arranged in such a way that the following relation ranking is satisfied that is, the first alternative is always the best alternative and so on: P1 P2 P3... PM The Weighted Sum Model The simplest and still the widely used MCDM method is the weighted sum model (WSM). The preference P i of alternative A i (i = 1,2,3,...,M) is calculated according to the following formula (Fishburn, 1967) p N a w For i = 1,2,3,....,M. Eq. (3.39) 1 ij j, i1 Therefore, in the maximization case, the best alternative is the one which corresponds to the largest preference value. The supposition which governs this model is the additive utility assumption. However, the WSM should be used only when the decision criteria can be expressed in identical units of measure (e.g., only dollars, or only pounds, or only seconds, etc.) The Weighted Product Model The weighted product model (WPM) is very similar to the WSM. The main difference is that instead of addition in the model there is multiplication. Each alternative is compared with the others by multiplying a number of ratios, one for each criterion. Each ratio is raised to the power equivalent to the relative weight of the corresponding criterion. In general, in order to compare alternatives A p and A q (where M p,q 1) the following product has to be calculated: w j A N p a pj R A Eq. (3.40) a q ji qj 47

28 . If the ratio R(A p /A q ) is greater than or equal to one, then the conclusion is that alternative A p is more desirable than alternative A q for the maximization case. The best alternative is the one which is better than or at least equal to all other alternatives. The WPM is sometimes called dimensionless analysis because its structure eliminates any units of measure. Thus, the WPM can be used in single and multi-dimensional decision making problems. 48