Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00002327/00001
## Material Information- Title:
- Modeling uncertainty using probabilistic based possibility theory with applications to optimization
- Creator:
- Jamison, Kenneth David
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- vii, 146 leaves : ; 28 cm
## Subjects- Subjects / Keywords:
- Probability measures ( lcsh )
Fuzzy numbers ( lcsh ) Mathematical optimization ( lcsh ) Fuzzy numbers ( fast ) Mathematical optimization ( fast ) Probability measures ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 142-146).
- General Note:
- Department of Mathematical and Statistical Science
- Statement of Responsibility:
- by Kenneth David Jamison.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 41470592 ( OCLC )
ocm41470592 - Classification:
- LD1190.L622 1998d .J36 ( lcc )
## Auraria Membership |

Downloads |

## This item has the following downloads: |

Full Text |

MODELING UNCERTAINTY USING
PROBABILISTIC BASED POSSIBILITY THEORY WITH APPLICATIONS TO OPTIMIZATION by Kenneth David Jamison M.S., University of California at Riverside, 1981 B.S., University of California at Riverside, 1978 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 1998 This thesis for the Doctor of Philosophy degree by Kenneth David Jamison has been approved by Weldon Lodwick Stephen Billups Jan Mandel Burt Simon William Wolf Date Jamison, Kenneth David (Ph.D., Applied Mathematics) Modeling Uncertainty Using Probabilistic Based Possibility Theory with Applications to Optimization Thesis directed by Professor Weldon Lodwick ABSTRACT It is shown that possibility distributions can be formulated within the context of probability theory and that membership values of fuzzy set theory can be interpreted as cumulative probabilities. The basic functions and opera- tions of possibility theory are interpreted within this setting. The probabilistic information that can be derived from possibility distributions is examined. This leads to two functionals that provide estimates for the expected value of a random variable, the expected average of a single possibility distribution and the estimated expectation that requires two special possibility distributions to compute. Secondly, the space of fuzzy numbers is examined. It is shown that this space can be partitioned into a vector space and that the expected average functional motivates a norm on this space. It is shown that for most applications, Cauchy sequences converge in this space. Thirdly, applications of this theory to problems in optimization are examined. The concept of a fuzzy function is formulated. The minimum and minimizer of a fuzzy function are described. Objectives in fuzzy optimization are examined. Lastly, the ideas m of the thesis are applied to the linear programming problem with uncertain coefficients. This abstract accurately represents the content of the candidates thesis. I recommend its publication. Signed Weldon Lodwick IV ACKNOWLEDGMENTS I wish to thank Weldon Lodwick for his direction, support and en- couragement, the fuzzy set theory study group (Weldon, Steve Russell, Chris Mehl and Ram Shanmugan) for their inspiration, Burt Simon for his challeng- ing comments and Steve Billups for his very thoughtful review of this thesis. Most of all I thank my wife Liza ONeill. Without her love and support I would not have been able to persevere. CONTENTS 1. Introduction and Overview............................................... 1 1.1 Overview ............................................................. 7 2. Possibility Theory..................................................... 10 2.1 Possibility Measures and Possibility Distribution Functions.............................................. 13 2.2 Membership Values as Cumulative Probabilities........................ 16 2.3 Possibility Distributions when Membership Values are Cumulative Probabilities................................. 26 2.4 Set Representations.................................................. 36 2.5 Functions of Possibilitity Distributions - Extension Principles................................................ 42 2.6 Probabilistic Based Possibility Distributions for Random Vectors ................................................. 52 2.7 The Information Contained in a Probabilistic Based Possibility Distribution........................ 55 3. The Space of Fuzzy Numbers ............................................ 77 3.1 Fuzzy Numbers........................................................ 77 3.2 A Normed Space of Fuzzy Number Equivalence Classes................................................. 80 3.3 An Isometry between (Â£, ||-||fyt) and BV[0,1] as a Subspace of Li[0,1]................................................. 85 3.4 Convergence in (Â£, || ||fyt) ....................................... 91 4. Optimization of Fuzzy Functions........................................ 94 vi 4.1 Fuzzy Functions............................................. 94 4.2 The Minimum and Minimizer of a Fuzzy Real-valued Function................................. 102 4.3 Method of Minimum Regrets ................................. 107 5. Fuzzy Linear Programming using a Penalty Method.............. 115 5.1 Problem Formulation........................................ 116 5.2 Properties of the Fuzzy Optimization Problem.................................................... 119 5.3 An Algorithm............................................... 122 5.4 Example.................................................... 124 6. Conclusion................................................... 129 Appendix A. Details of Fuzzy Linear Programming Example................................... 132 B. Formulas for Implementation of Fuzzy Linear Programming..................................... 138 References...................................................... 142 vii 1. Introduction and Overview Suppose a decision maker (DM) is interested in optimizing (in some sense) the function f(x) = a2x b (1.1) where a and b are fixed but unknown (uncertain) parameters. How does the DM model the unknown parameters a and b? Given the model, how does the DM evaluate and interpret f(x)=a2x 6? How is a decision reached? Two common techniques for estimating the parameters a and b are interval analysis and probability theory. In this thesis we will consider a third technique, called possibility theory. The advantage of interval analysis is that the interval solution is guar- anteed to contain all solutions (see Moore [32]). Interval analysis also offers the advantage of being computationally tractable. It is often straightforward to evaluate the function using the known methods of interval arithmetic (Moore [32]). For example, if aÂ£ [1,3] and bÂ£ [4,6] then /(l) Â£ [1,3]2 [4,6] = [1,9] [4,6] = [-5,5], (1.2) 1 The evaluation gives an interval which represents the range of possible val- ues of the function, though most often this range is overestimated (see Moore [32]). The DM is then faced with the problem of optimizing an interval val- ued function, i.e. the DM must decide on an ordering of the set of intervals. One approach is to assume a uniform probability distribution over any given interval and optimize the expected value, i.e. optimize the midpoint of the interval. For example, f(2)G[l,3]22 [4,6] = [2,18]-[4,6] = [-4,14] would be con- sidered greater than f(l)G [5, 5] since the midpoint (14-4)/2=5 is greater than the midpoint (5-5)/2=0. A disadvantage of interval analysis is that it gives the worst case and interval solutions can be large (useless) if care and computationally intensive approaches are not used. Another disadvantage is that it gives no consideration to the likelihood of a particular outcome. It only considers the range of all possible outcomes. Probability theory considers not only the range of possible outcomes but the likelihood a particular outcome may occur. For example, suppose a and b are modeled as the independent random variables X and Y with the following probability density functions: fx(x) = < x 1 for x G [1,2] 3 x for x G [2, 3] and fY(y) = < y-4 for yG [4, 5] 6 y for yG [5, 6] (1,3) Then the probability density function for the random variable Z=f(l)=X2 Y 2 is the following: fz(z) = / ------- f^6+z(w 1)(6 w2 + z)dw zG[-5,-4] X/gyy (w ~ 1)(6 w2 + z)dw + f/5+z(w 1 ){w2 z 4)dw zG[-4,-3] Iy^(w ~ Xl6 ~w2 + z)dw + 1 )(w2 z 4)dw zG[-3,-2] f^5+z(w ~ 1)(6 w2 + z)dw + f/6+z(3 w)(6 w2 + z)dw + -l)(w2 z 4)dw ze[-2,-l] X/yfy (3 w)(6 w2 + z)dw + J^^(w l)((rc2 z) 4)dw +(3 w)(w2 z 4 )dw zG[-l,0] X/Sf (3 w){6 w2 + z)dw + /XJ/(3 w)(w2 z 4)dw zG[0,3] X/s+y(3 w)(6 -w2 + z)dw + 3 w)(w2 z 4)dw zG[3,4] Xyi+y(3 w)(w2 z 4 )dw zG[4,5] Using this approach, the DM might seek to optimize the expected value of f(x). For example, the expected value of Z=f(l), E(Z), is -.8333. A disadvantage of this approach is that it may be impossible, difficult and/or impractical to obtain the probability distribution for the random variable f(x) for each x to be considered in the optimization problem. In addition, some problems do not admit to a probabilistic formulation (e.g. locating a tumor for radiation therapy). In this thesis we examine a third approach called possibility theory. 3 It is argued that possibility theory is a compromise between interval analysis and probability theory. In essence, we perform interval arithmetic over families of intervals where the families of intervals are parameterized by a probability distribution. The result is a family of intervals representing the possible values of f(x) that preserves some of the characteristics of the probability distribu- tion. There are two principle advantages to be gained from this approach. One advantage is that problems formulated in possibilitistic terms are more com- putationally tractable. Another advantage is that possibility theory allows for computations with imprecise probabilities. For example, a possibility distribu- tion can be formulated from a family of confidence intervals. These advantages will become clear in the sequel. Consider the example above where a and b were modeled as indepen- dent random variables X and Y with density functions given in (1.3). We will show that two parameterized families of intervals can be constructed for X and Y to arrive at the following: [1, 2 + 1] for a Â£ [.75,1] [1,2 ^1 Â§y(2-2yP + l))) + 1] for a Â£ [0, .75] and YL = [4, 2 + 4] for a Â£ [.75,1] 4,2(l-4Y(2-2^(-a + l))j +4 for a Â£ [0, .75] 4 and XR = [3-2 (J\y/T^) 3] for a G [.75,1] [3 2 ^1 ^{2-2^a + l))^ 3] for a G [0, .75] and YR = 6 2 (^1 2 Y (2 2^(-a + f))j 6j for a G [.75, f] [6-2 (/17^7) 6] for a G [0, .75] Then two parameterized families of intervals can be constructed for Z=f(l) using interval arithmetic as follows: ZR = and [(3-2^17r^) (2^17^7 + 4) ,5] for a G [.75,1] (3-2(1-1^2-27-0 + !)^ - (2 (l 1^2-277^7^) +4^ ,5] for a G [0, .75] Zi -5, (2 + l) (6 277^7)] for a G [.75,1] -5, (2 (l |\/(a 2+(-a + 1+) + I)' - [e 2 [1- iy4- 2^+4+!)H] for a [0, .75] The DM must now rank these families of intervals in order to reach a decision. We will show that in general these two families of intervals provide an 5 upper and a lower bound for the cumulative probability distribution function of Z = /(l). If the goal of the DM is to optimize the expected value, this characterization allows for the calculation of an estimated expected value (in this case an actual value). For this example, the calculation of the estimated expected value of Z, which we denote as EE(Z), is as follows: The focus of this thesis is to explore the application of possibility theory to the problem of incorporating uncertainty into math modeling, par- ticularly problems in optimization. The thesis will examine the use of possibil- ity distributions as a method of modeling unknown parameters in real valued 6 functions. The contributions of this work to the mathematical foundation and application of possibility theory include (i) a formulation of possibility theory within the framework of probability theory (ii) a method for selecting among families of possibility distributions over the real line based on an estimate of the expected value (iii) a norm on the space of fuzzy numbers based on this estimated expected value (iv) an isometry between this normed spaced and the subspace of L1 [0,1] (Lebesgue integrable functions on the interval [0,1]) consisting of functions of bounded variation (v) convergence properties of this normed space (vi) a new definition of a fuzzy function and several implications of this definition and (vii) an approach to linear programming problems where the coefficients are stated in terms of fuzzy numbers. Many of these results are a compilation and continuation of research that will appear in publications Jamison&Lodwick [16], Jamison [17] and Jamison&Lodwick [15]. Additional aspects of this research have appeared in Lodwick&Jamison [28] and Lod- wick&Jamison [29]. 1.1 Overview The second chapter of this paper develops topics in possibility theory as used here. The chapter begins with known definitions and basic implications from possibility theory. Then it is shown that a possibility distribution can be 7 interpreted as a cumulative probability distribution. Several results are pre- sented showing how the image of a possibility distribution under an arbitrary function is defined and how the resulting possibility distribution is interpreted. Chapter three focuses on possibility distributions over the real line. Of particular interest are fuzzy numbers, i.e. unknown numbers characterized by possibility distributions. There are possibilistic distributions over linguistic variables but these are not examined here. The mathematics of fuzzy numbers is presented. It is shown how fuzzy numbers can be added, subtracted, multi- plied, etc. An equivalence relation on the space of fuzzy numbers is presented and it is shown that this results in a vector space. A norm on this space is defined. This norm is motivated from an estimated expected value functional. The convergence properties in this normed space are examined. In chapter four, applications to certain optimization problems are ex- amined. First, the concept of a fuzzy function between finite dimensional vector spaces is discussed. A fuzzy function is defined as a possibility distribution, satisfying certain properties, over the space of bounded functions. The image of a fuzzy function is shown to be a fuzzy vector. In particular, if the range space is the real line, the image is a fuzzy number. Second, the case of minimizing an unconstrained fuzzy real-valued convex function is examined. The minimum of a fuzzy function is defined 8 and it is shown to be a fuzzy number. The minimizer of a fuzzy function is defined. It is shown that the possibility distribution for the minimizer has a connectedness property. A minimization approach is examined based on the estimated expected value functional of chapter two. It is called the method of minimum regrets, and seeks to minimize the estimated expected value of the possible error. In chapter five, the theory and methods of chapters two, three and four are applied to the linear programming problem where all of the coefficients are replaced by fuzzy numbers. The problem is first formulated in possibilistic terms and converted to a problem of maximizing the estimated expected value of an unconstrained fuzzy function. It is shown that the resulting problem is concave. Conditions to insure a bounded solution are developed and an upper bound on the decision variables is determined. An algorithm is provided. Chapter six concludes by summarizing the results of this thesis and suggesting areas of future research. 9 2. Possibility Theory L. Zadeh [48] introduced the concept of a fuzzy set in the 1960s. The basic idea is to extend the notion of set membership to include degrees of partial membership. This is generally characterised by a membership function which is a mapping from the universe of discourse into the unit interval. For example, if A is a fuzzy subset of set X, the membership function for A is ha ' X > [0,1]. For xÂ£ X, fiA(x) is called the membership value of x and is interpreted as the degree to which x is a member of fuzzy set A where 1 represents full membership and 0 represents complete lack of membership. If Ha(x) = 0 or 1 for all x, A is called a crisp set (in this case /jla is just the characteristic function of set A). Zadeh defined a set of rules of combination for fuzzy sets, each of which is a generalization of the same concepts for crisp sets. So, for example, the standard union, intersection and complement operations for fuzzy sets A and B, defined in terms of their membership functions are I^aub(x) = max(iia(x)7iib(x))} nAnB(x) = min/iB(x))} and /iAc(x) = 1 /jLA(x) where Ac is the complement of A. Note that these are the usual operations if A and B are crisp sets. 10 Possibility theory, also introduced by L. Zadeh [49], is a natural ex- tension of his work with fuzzy sets. As a tool in optimization problems that involve uncertainty, the basic idea is to consider the set of all possible alter- natives for an unknown parameter of the problem. But the set of possible alternatives may not be well defined. Thus, a fuzzy set of alternatives is used. To do this, each element in the universal set is assigned a membership value (or possibility level) in the interval [0,1]. In possibility theory, as formulated by Zaden, the membership value of an element is interpreted as the degree of possibility that the parameter is that element. If the membership value is zero, it is impossible that the parameter is the element. If the membership value is one the element is considered a possible value of the parameter without reser- vation. Membership values produce an ordering on the universal set in terms of each elements acceptability as a possible value of the unknown parameter. The mapping of each member of the universal set to its membership value is called a possibility distribution function for the unknown parameter. The typical process for utilizing possibility theory in optimizing a function with ill-defined parameters is as follows: First, each unknown param- eter of the function is represented using a possibility distribution. Second, for a given point in the decision space the function is evaluated over the set of possible parameter values using the rules of combination of possibility theory. 11 This produces a possibility distribution for the function evaluation. Third, the possibility distribution is assigned a value. The process of assigning a value to a possibility distribution is called defuzzification (taken from fuzzy set theory). Fourth, this value is used to order the possibility distributions so that an optimal decision can be made. It is the second of these steps that makes possibility theory an interesting alternative to probability theory in modeling uncertainty. For many problems, it is fairly straight forward to calculate the image of a function of possibility distributions, where it may be very difficult if probability distributions are used. The process just described is controversial. The basic problem is that membership values have not been well defined except as a tool for ordering the universal set. Very different membership values can be used to arrive at the same ordering. But most defuzzification methods are dependent on the membership values. Thus the decision derived from the process may be arbitrary, since it depends on what membership values were assigned. Also, it is difficult to interpret the meaning of the number derived from defuzzification. In [12], Dubois&Prade state ...when we scan the fuzzy set literature, including Zadehs own papers, there is no uniformity in the interpretation of what a membership grade means. This situation has caused many a critique by fuzzy set opponents, and also many a misunderstanding within the held itself. Most 12 negative statements expressed in the literature turn around the question of interpreting and eliciting membership grades. This thesis begins with an interpretation of membership values based on probability theory. The fundamental properties of possibility theory are then examined from this perspective. The advantage of this approach is that it gives a concrete meaning to membership values and the value derived from the defuzzification method proposed. 2.1 Possibility Measures and Possibility Distribution Functions Before introducing the probabilistic interpretation of membership val- ues, the formal definitions and resulting implications from possibility theory are provided. Note that the basis for selecting membership values is not discussed in this section. Definition 1 (Klir&Yuan [21]) Given a universal set X and a nonempty family C of subsets of X, a fuzzy measure on (X, C) is a function <7 = [ 0,1] (2.1) that satisfies the following requirements: {gl) g(0) = 0 and g(X) = l (boundary condition) (g2) V A,Be C, if AC B then g(A)< g(B) (monotonicity) 13 (g3) for any increasing sequence Ai C A2 C ... in C, if (J^r A; G C, then OO Yira g{Ai) = g{{j Ai) (2.2) %Voo . 2=1 (continuity from below) (g4) for any decreasing sequence Ai D A2 D ... in C, if f\Ai Ai G (7, then OO lim flf(At-) = flf(fl Ai) (2.3) %too . ' 2 = 1 (continuity from above). An immediate consequence of this definition is that if A, B, AU B and A fl B G (7, then every fuzzy measure satisfies the inequalities, 9{A n B) < min(g(A),g(B)) and g(A 1J B) > max(g(A),g(B)). (2.4) Fuzzy measures include probability measures as well as belief and plausibility measures of evidence theory (see Klir&Yuan [21]). Possibility the- ory is based on the following two fuzzy measures. Definition 2 (Klir&Yuan [21]) Let nec denote a fuzzy measure of (Y, C). Then nec is called a necessity measure iff nec I P| Ak = inf nec(Ak) (2-5) \kei< ) keK for any family {A& | k G K} in C such that p|keK Ak G (7, where K is an arbitrary index set. 14 Definition 3 (Klir&Yuan [21]) Let pos denote a fuzzy measure of (X}C). Then pos is called a possibility measure iff pos [J Ak = sup pos ( Ak) (2.6) \kei< / keK for any family {A& | k G K} in C such that (JkeK X G C, where K is an arbitrary index set. When C = V(X), the power set of X, possibility and necessity mea- sures occur in pairs. Theorem 1 (Klir&Yuan [21]) If pos is a possibility measure on V(X). Then its dual set function nec, which is defined by nec(A) = 1 pos(Ac) (2-7) is a necessity measure called the necessity measure associated with pos. Definition 4 (Klir&Yuan [21]) Given a possibility measure pos on power set "P(X), the function fi : X > [0,1] such that p(x) = pos ({}) for all xG X is called the possibility distribution function associated with pos and p(x) is called the possibility level or membership value of x. Theorem 2 (Klir&Yuan [21]) Every possibility measure pos on a power set "P(X) is uniquely determined, for each A G V(X), by the associated possibility distribution function via the formula pos(A) = supxeA/i(x). (2.8) 15 Theorem 3 (Wang&Klir [46]) If /x is a possibility distribution function for possibility measure pos then supxeXfJ,(x) = 1. (2.9) Conversely, if a function /x : X > [0,1] satisfies (2.9), then /x can determine a possibility measure pos uniquely, and /x is the possibility distribution function of pos. Definition 5 (Dubois&Prade [10]) A possibility distribution is called normal if 3a G X such that pos({a}) = l. 2.2 Membership Values as Cumulative Probabilities Given a possibility distribution function, there are various ways mem- bership values have been interpreted. For example, membership values have been interpreted as probabilities over nested sets as in Dubois&Prade [11] or as upper bounds on probability distributions as in Dubois&Prade [9]. A good overview of the various interpretations of membership grades in fuzzy set theory as well as possibility theory can be found in Dubois&Prade [12]. In this thesis we adopt an interpretation that is closely related to the random set interpreta- tion and the interpretation as the upper bounds on probability distributions. This interpretation provides a straight forward method for constructing pos- sibility distributions. This interpretation will give context to the operations 16 proposed in this thesis. The need for a concrete interpretation of membership values was discussed in the introduction to this chapter. The prototype for the following discussion is a parameterized real valued function where the parameters are not known with precision. However, other applications of possibility theory will be considered. Definition 6 Let X be a space and x a random vector on X. A possibility nest for if is a relation >p on X X X such that (1) >p is a linear ordering of X with the property that V x,y G X either x >p y, y >p x or x =p y and (2) \/x G X, {y G X \ y >p x} is measurable with respect to the probability measure for x and (3) \/B C X, 3 an at most countable subset C CB with the property that Vx G B 3xn, xm G C such that xn
This third property requires some examination. The assumption is
a point to set mapping / : T > "P (3^) by |

Full Text |

PAGE 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PAGE 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. |