The Use of Fuzzy Optimization Methods for Radiation

1、Uncertainties in Mathematical Analysis and Their Use in OptimizationCongresso Brasileiro de Sistemas FuzzySorocaba, Brasil9 de novembro, 2010Weldon A. Lodwick and Oscar JenkinsUniversity of Colorado DenverDepartment of Mathematical and Statistics Sciences Abstract: Fuzzy set theory and possibility t

2、heory are potent mathematical languages for expressing transitional (non-Boolean) set belonging and information deficiency (non-specificity), respectively. We argue that fuzzy and possibility optimization have a most important role to play in optimization. Three key ideas are considered:1) Normative

3、 decision making/optimization is often satisficing and epistemic. 2) Fuzzy set theory and possibility theory are the mathematical languages well-suited for encapsulating satisficing and epistemic entities, perhaps, the only mathematical languages we have at present. As a consequence, fuzzy and possi

4、bility optimization are powerful approaches to satisficing and epistemic decision making, 3) Semantic and structural distinctions between fuzzy sets and possibility are crucial, especially in fuzzy and possibility optimization.2OutlineIntroduction Why Fuzzy Set Theory, Possibility Theory (in optimiz

5、ation)?Elements of Fuzzy Set Theory and Possibility Theory: A Mathematicians Point of ViewIntervalsFuzzy IntervalsPossibilityIII. Fuzzy and Possibility OptimizationTaxonomySolution MethodsIV.Theoretical Considerations Interval Arithmetic as Function ArithmeticGeometry of convex cones associated with

6、 optimization over intervalsOrderings in convex cones associated with optimization over intervals3ObjectivesTo show the clear distinction between fuzzy set theory and possibility theoryTo develop two types of possibility analyses1. Single distribution possibility analysis2. Dual distribution possibi

7、lity/necessity analysis risk To demonstrate how the differences between fuzzy set theory and possibility theory impact optimization1. Flexible optimization2. Possibility or evaluation optimization3. Possibility/necessity or dual evaluation optimization4. Mixed optimization4This presentation will sta

8、te the obviousSome of what is presented has been known for some time, but perhaps not quite from the same point of view, so we hope to bring greater clarityMy point of view is mathematical and as an area editor for the journal Fuzzy Sets and Systems There will be some new(er) thingsUpper/lower optim

9、ization, what we call dual evaluation optimization , dual distribution optimizationInterval analysis of linear functions with non-linear slopes over compact domains in order to compute and do mathematical analysis with intervals and fuzzy intervals All orders associated with intervals and associated

10、 geometry 5Point of ViewFuzzy sets do not model uncertainty6Uncertainty Models - Possibility Information deficiency - the lack of determinism 7Fuzzy Transition set belongingIt has nothing to do with this cartoon from my newspaper except I hope this talk will indeed GET FUZZY 8I. Introduction Why fuz

11、zy or possibility theory (in optimization)?Many optimization models are satisficing decision makers do not know what is deterministically “the best.” A model is useful if the solutions are good enough.Many optimization models are epistemic they model what we, as humans, know about a system rather th

12、an the system itself. For example, an automatic pilot of an airplane models the systems physics. A fuzzy logic chip that controls a rice cooker is epistemic in that it encapsulates what we know about cooking rice, not the physics of cooking rice.Fuzzy and possibility optimization models are well-sui

13、ted for and most flexible in representing satisficing and epistemic normative criteria.9General Mathematical ApproachPossibility theory as has been articulated by Didier Dubois and Henri Prade, and others, over the last three decades has put it on a solid mathematical foundation. They, and others, h

14、ave also helped put fuzzy set theory that was created by Lotfi Zadeh onto a solid foundation. A solid mathematical foundation is able to order our fuzzy and possibilistic thoughts and applications.We, in this presentation, seek to clarify how fuzzy set theory and possibility theory are used in mathe

15、matical analysis, particularly in optimization. This, hopefully, will order approaches to optimization problems.10Prof. H. J. Rommelfanger (2004)The advantages of fuzzy optimization models in practical use, Fuzzy Optimization and Decision Making, 3, pp. 295), Linear programming is the most widely us

16、ed OR method.Of the 167 production (linear) programming systems investigated and surveyed by Fandel, ( Fandel, G. (1994), only 13 of these were run as “purely“ linear programming systems. Thus, even with this most highly used and applied operations research method, there is a discrepancy between cla

17、ssical deterministic linear programming and what was/is actually used in practice! 11Some ThoughtsDeterministic and stochastic optimization models requireSingle-value unique distributions for the random variable coefficients, right-hand side values,Deterministic relationships (inequalities, equaliti

18、es)Real-valued functions or distribution functions to maximize, minimize. Thus, any large scale model requires significant data gathering efforts. If the model has projections of future values, we have no determinism nor certainty.12My Colleagues Critique of Fuzzy/Possibilistic OptimizationWhat is d

19、one with fuzzy/possibility optimization can be done by deterministic or probabilistic (stochastic) optimization. Corollary: Since all fuzzy/possibilistic optimization models are transformed into real-valued linear or non-linear programming problems, we do not need fuzzy nor possibility optimization.

20、Show me an example for which fuzzy/possibilistic optimization solves the problem and deterministic or stochastic do not.13A (Fuzzy) Mathematicians Reply Just because the foundation of probability theory was put into the context of real analysis does not negate the value of probability as a separate

21、branch distinct from real analysis.Just because stochastic optimization models stated as recourse or chance constraints are translated into real-valued mathematical programming problems does not negate the value of stochastic optimization as a separate branch of optimization.Corollary: Just because

22、fuzzy and possibility optimization translate to a standard linear or nonlinear programming problem does not negate its value as a separate study in optimization theory14The Value of Fuzzy and Possibility OptimizationFlexible programming (what is called fuzzy programming) has no equivalent in classic

23、al optimization theory with the richness and robustness that fuzzy set theory brings. This is the contribution of fuzzy set theory to optimization.Upper/lower bounds on optimization models arising from necessity (pessimistic) and possibility (optimistic) have no equivalent in optimization theory. Th

24、is is a unique contribution possibility theory to optimization.Possibility theory is a rich and robust mathematical language for representing epistemic and satisficing objective. No classical mathematical language exists for stating epistemic or satisficing objectives and constraints as robust/rich

25、as fuzzy set theory and possibility theory. 15A (Fuzzy) Mathematicians Reply A representation of a mathematical problem always begins in its “native” environment of origin. For example, one always begins a nonlinear problem by stating the problem in its nonlinear environment. After stating the probl

26、em in its full nonlinear setting, one may then turn it into a linear system. Most (all) convergence and error analysis is based on knowing from where the problem came.If the problem is epistemic and/or satisficing, or fuzzy (transitional set belonging) or involves information deficiency, one states

27、the problem in the space from which the problem came. Then, and only then, does one makes the “approximation” or translation into what is possible to solve.16Some Further Thoughts of Prof. Rommelfanger From an email discussion, Rommelfanger relates the following. “In fact Herbert Simon develops a de

28、cision making approach which he calls the Concept of Bounded Rationality. He formulated the following two theses.Thesis 1: In general a human being does not strive for optimal decisions, but s/he tends to choose a course of action that meets minimum standards for satisfaction. The reason for this is

29、 that truly rational research can never be completed. 17Some Further Thoughts of Prof. Rommelfanger Thesis 2: Courses of alternative actions and consequences are in general not known a priori, but they must be found by means of a search procedure.” That is we do not often know ahead of time. If we d

30、id, we would, perhaps, not have a problem.18PRINCIPLE OF LEAST COMMITMENTA useful approach in flexible and possibility (also stochastic) optimization is the Principle of Least Commitment which states:Only commit when you must.Corollary 1: (For fuzzy and possibility optimization) Carry the full exten

31、t of uncertainty and gradualness until one must choose.Corollary 2: (For men) Only make the commitment in marriage when you have to.19II. Elements of Fuzzy Set Theory and Possibility Theory: A Mathematicians Point of ViewFuzzy set theory is the mathematics of transitional (non-Boolean) set belonging

32、Example (Fuzzy): Tumorness of a pixel a pixel is both cancerous and non-cancerous at the same time (conjuctive)Possibility theory is the mathematics of information deficiency, non-specificity (non-deterministic), uncertaintyExample (Possibility): My evaluation of the age of the outgoing president of

33、 Brasil. 45 or 46 or 59 or 60 or Note: Lulas age exists, it is a real number (not fuzzy) in counter-distinction with the boundary between cancerous and non-cancerous cells which is inherently transitional.20Fuzzy and PossibilityTheorem 1:There is nothing uncertain about a fuzzy set.Theorem 2:There i

34、s nothing uncertain about a fuzzy set.Proof: Once fuzzy sets are uniquely defined by their membership function, we know the belonging transition precisely. The membership value of 1 means membership with certainty. The membership value of 0 means non-membership with certainty.Corollary: Fuzzy optimi

35、zation is not optimization under uncertainty! Fuzzy optimization is flexible (transitional) optimization. We will return to this subsequently.21Fuzzy and Possibility Dubois/PradeFuzzy is conjunctive (and) a fuzzy entity is more than one thing at once (an element is and isnt in the set to a certain d

36、egree). In image segmentation (fuzzy clustering) a pixel belongs to various classes at once even though a pixel is a distinct non-overlapping unit.Possibility is disjunctive (or) my guess at outgoing President Lulas age is a distribution over distinct set of elements or 45 or 46 or 59 or 60 or . I w

37、ould have a distribution value for these distinct elements (all real numbers in this case). However, the age of outgoing president exists as a real number, but my knowledge (epistemic state) is a possibility distribution.22Probability Alone is Insufficient to Describe All UncertaintyExample: Suppose

38、 all that is known is that x1,4. Clearly, x1,4 implies that the real value that x represents is not certain (albeit bounded). If the uncertainty that x1,4 represents were probabilistic (x is a random variable that lies in this interval), then every distribution having support contained in 1,4 would

39、be equally valid given. Thus, if one chooses the uniform probability density distribution on 1,4, p(x) = 1/3, 1x4, p(x) = 0 otherwise, we clearly lose information. 23View of Uncertainty24Probability Alone is Insufficient to Describe all UncertaintyThe approach that keeps the entire uncertainty consi

40、ders it as all distributions whose support is 1,4 as equally valid. The pair of cumulative distributions that bound all cumulative distributions with this given support is depicted in Figure 1.This pair is a possibility (upper blue distribution)/necessity (lower red distribution)pair.25Probability A

41、lone is Insufficient to Describe all Uncertainty Note that the green line (uniform cumulative distribution) is precisely what in interval analysis would be the most sensible choice when no other information is at hand except the interval itself, “Choose the midpoint when one must choose.” However, o

42、ne does not have to choose a uniform distribution at the beginning of an analysis which is the approach of the Principle of Least Commitment. Analysis with the dual upper/lower bounds does not lose information.26Structure of mathematical analysis in flexible and possibility optimizationOptimization

43、requires:Objective function(s) way to determine (compute) what an optima is Relationships a description of how variables and parameters are associated (equality and inequality).Constraint set a way to determine (compute) how the set of relationships are linked (equations/inequalities are linked or a

44、ggregated by “and” or “or” or t-norms)REMARK: Fuzzy and possibility optimization state each of these components of the structure in a particular mathematical language that is distinct from deterministic and probabilistic statements of the same structure.27Entities of Fuzzy and Possibility Optimizati

45、onThe entities are:IntervalsFuzzy intervalsSingle possibility distributionsDual possibility/necessity distributions281. Entities of Analysis - Intervals29An interval x = a,b = x | a x b. For example, x= 1,4. There are two views of an interval:New type of number (an interval number) described by two

46、real numbers a, and b, a b, the lower and upper bound x = a,b Warmus, Sunaga, and Moore approach.A set x = x | a x b which we will represent as a function (the set of single-valued linear functions with non-negative slopes over compact domain 0,1.A single-valued linear function with non-negative slo

47、pe over a compact domain 0,1 (Lodwick 1999):Entities of Analysis - IntervalsWhen an interval is represented and operated on by its lower/upper bounds a,b as a new type of number, then the ensuing algebraic structure is more limited than necessary. It is an algebra of vectors in , an algebra of two p

48、oints (upper half plane determined by y=x), a subset of in fact.If an interval is considered as a set or a single-valued function with non-negative slope over a compact 0,1 domain, then the ensuing algebraic structure is that of sets or functions which is richer. 30Entities of Analysis - Intervalsb)

49、 Two interesting anomalies associated with intervals: i. a b and a b does not imply a = b. Consider 2,3x 3,6 x 1 and 2,3x 3,6 x 3These two result in the empty set.But 2,3x = 3,6 x = 3/2,2 since 2,33/2,2=3,6 ii. 2,3, x = 3,6, 1,1x = 3/2,2, is not an interval, it belongs to the lower half plane where

50、inverses of intervals live (in a non-interval space).312. Entities of Analysis - Fuzzy IntervalsTriangular Fuzzy Interval We will call this a fuzzy interval since a fuzzy interval (see next slide) is more general.322. Entities of Analysis - Fuzzy Intervals33Trapazoid Fuzzy Interval: What is an eleme

51、nt of this set?Entities of Analysis - Fuzzy Intervals A fuzzy interval can be automatically translated into a possibility distribution and thus may be a model for both the lack of specificity as well as transition depending on the semantics.Thus, fuzzy intervals have or take on a dual nature - that

52、of capturing or modeling gradualness of belonging and capturing or modeling non-specificity. Possibility is tied to uncertainty. This dual nature of fuzzy intervals is the source of much confusion.343. Entities of Analysis Single Possibility DistributionPossibility models non-specificity, informatio

53、n deficiency and is a mathematical structure developed by L. Zadeh in 1978 (first volume of Fuzzy Sets and Systems). Since possibility is not additive, a dual to possibility, necessity, is required to have a more complete mathematical structure. Necessity was developed by Dubois and Prade in 1988. I

54、n particular, if we know the possibility of a set A, it is not known what the possibility of the complement of a fuzzy set A, ,in contradistinction to probability. The dual to possibility, necessity is required. Given the possibility of a set A, the necessity of is known.35Possibility Distributions

55、- ConstructionThere are at least four ways to construct possibility and necessity distributions: Given a set of probabilities (interval-value probabilities):2.Given an unknown probability p(x) such thatJamison/Lodwick 2002, Fuzzy Sets and Systems3.Given a probability assignment function m whose foca

56、l elements are nested, construct necessity/possibility distributions which are the plausibility/belief functions of Demster and Shafer theory. Here probabilities are know on sets (not elements of sets)364. Entities of Analysis Dual Possibility DistributionsA fuzzy interval, generates a possibility a

57、nd necessity pair. 37Possibility Dubois, Kerre, Mesair, and Prade 2000 Handbook, 3 probabilistic views of a fuzzy intervalThe imprecise probability view whereby M encodes a set of (cumulative) probability measures shown in Figure 3 between the dashed blue line (possibility) and the dotted green line

58、 (necessity). This is our first construction.The pair of PDFs view whereby M is defined by two random variables x and x with cumulative distributions in blue and green of Figure 3. This is our fourth construction.The random set view whereby M encodes the one point coverage function of a random inter

59、val, defined by the probability measure on the unit interval (for instance the uniformly distributed one) and a family of nested intervals (the -levels), via a multi-valued mapping from (0,1 to , following Dempster. This is our third construction.38III. Fuzzy/Possibilistic OptimizationWe turn our at

60、tention to optimizationFirst a classificationSemantics and solutions methods39What We Know About OptimizationOptimization models are normative models that are most often constrained. Thus, from this perspective, there are three key parts to a fuzzy, possibility, and interval optimization problemThe