Uncertainties in Mathematical Analysis and Their Use in Optimization Congresso Brasileiro de Sistemas Fuzzy Sorocaba, Brasil 9 de novembro, 2010 Weldon A. Lodwick and Oscar Jenkins University of Colorado Denver Department of Mathematical and Statistics Sciences Weldon.Lodwick@ucdenver.edu Abstract: Fuzzy set theory and possibility theory are potent mathematical languages for expressing transitional (nonBoolean) set belonging and information deficiency (nonspecificity), respectively. We argue that fuzzy and possibility optimization have a most important role to play in optimization. Three key ideas are considered: 1) Normative 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 possibility 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. Congresso Brasileiro de Sistemas Fuzzy 2 Outline Introduction – Why Fuzzy Set Theory, Possibility Theory (in optimization)? Elements of Fuzzy Set Theory and Possibility Theory: A Mathematician’s Point of View I. II. A. B. C. III. Intervals Fuzzy Intervals Possibility Fuzzy and Possibility Optimization A. B. IV. Taxonomy Solution Methods Theoretical Considerations <If we have time> A. B. C. Interval Arithmetic as Function Arithmetic Geometry of convex cones associated with optimization over intervals Orderings in convex cones associated with optimization over intervals Congresso Brasileiro de Sistemas Fuzzy 3 Objectives • To show the clear distinction between fuzzy set theory and possibility theory • To develop two types of possibility analyses 1. Single distribution possibility analysis 2. Dual distribution possibility/necessity analysis – risk • To demonstrate how the differences between fuzzy set theory and possibility theory impact optimization 1. Flexible optimization 2. Possibility or evaluation optimization 3. Possibility/necessity or dual evaluation optimization 4. Mixed optimization Congresso Brasileiro de Sistemas Fuzzy 4 This presentation will state the obvious • Some 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 clarity • My point of view is mathematical and as an area editor for the journal Fuzzy Sets and Systems • There will be some new(er) things – Upper/lower optimization, what we call dual evaluation optimization , dual distribution optimization – Interval analysis of linear functions with non-linear slopes over compact domains in order to compute and do mathematical analysis with intervals and fuzzy intervals <if we have time> – All orders associated with intervals and associated geometry <if there is time> Congresso Brasileiro de Sistemas Fuzzy 5 Point of View Fuzzy sets do not model uncertainty Congresso Brasileiro de Sistemas Fuzzy 6 Uncertainty Models - Possibility Information deficiency - the lack of determinism Congresso Brasileiro de Sistemas Fuzzy 7 Fuzzy – Transition set belonging It has nothing to do with this cartoon from my newspaper except I hope this talk will indeed GET FUZZY Congresso Brasileiro de Sistemas Fuzzy 8 I. Introduction – Why fuzzy 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 than the system itself. For example, an automatic pilot of an airplane models the system’s 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-suited for and most flexible in representing satisficing and epistemic normative criteria. Congresso Brasileiro de Sistemas Fuzzy 9 General Mathematical Approach • Possibility 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, have 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 mathematical analysis, particularly in optimization. This, hopefully, will order approaches to optimization problems. Congresso Brasileiro de Sistemas Fuzzy 10 Prof. H. J. Rommelfanger (2004)"The advantages of fuzzy optimization models in practical use," Fuzzy Optimization and Decision Making, 3, pp. 295), 1. Linear programming is the most widely used OR method. 2. 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. 3. Thus, even with this most highly used and applied operations research method, there is a discrepancy between classical deterministic linear programming and what was/is actually used in practice! Congresso Brasileiro de Sistemas Fuzzy 11 Some Thoughts • Deterministic and stochastic optimization models require 1. Single-value unique distributions for the random variable coefficients, right-hand side values, 2. Deterministic relationships (inequalities, equalities) 3. Real-valued functions or distribution functions to maximize, minimize. 4. Thus, any large scale model requires significant data gathering efforts. If the model has projections of future values, we have no determinism nor certainty. Congresso Brasileiro de Sistemas Fuzzy 12 My Colleagues’ Critique of Fuzzy/Possibilistic Optimization • What is done 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 nonlinear programming problems, we do not need fuzzy nor possibility optimization. • Show me an example for which fuzzy/possibilistic optimization solves the problem and deterministic or stochastic do not. Congresso Brasileiro de Sistemas Fuzzy 13 A (Fuzzy) Mathematician’s 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 branch distinct from real analysis. • Just because stochastic optimization models stated as recourse or chance constraints are translated into realvalued mathematical programming problems does not negate the value of stochastic optimization as a separate branch of optimization. Corollary: Just because fuzzy and possibility optimization translate to a standard linear or nonlinear programming problem does not negate its value as a separate study in optimization theory Congresso Brasileiro de Sistemas Fuzzy 14 The Value of Fuzzy and Possibility Optimization • Flexible programming (what is called fuzzy programming) has no equivalent in classical 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. This 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 as fuzzy set theory and possibility theory. Congresso Brasileiro de Sistemas 15 Fuzzy A (Fuzzy) Mathematician’s 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 problem 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 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. Congresso Brasileiro de Sistemas Fuzzy 16 Some Further Thoughts of Prof. Rommelfanger From an email discussion, Rommelfanger relates the following. “In fact Herbert Simon develops a decision 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 that truly rational research can never be completed. Congresso Brasileiro de Sistemas Fuzzy 17 Some 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 did, we would, perhaps, not have a problem. Congresso Brasileiro de Sistemas Fuzzy 18 PRINCIPLE OF LEAST COMMITMENT A 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 extent of uncertainty and gradualness until one must choose. Corollary 2: (For men) Only make the commitment in marriage when you have to. Congresso Brasileiro de Sistemas Fuzzy 19 II. Elements of Fuzzy Set Theory and Possibility Theory: A Mathematician’s Point of View • Fuzzy set theory is the mathematics of transitional (nonBoolean) set belonging 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), uncertainty Example (Possibility): My evaluation of the age of the outgoing president of Brasil. {45 or 46 or … 59 or 60 or } Note: Lula’s 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. Congresso Brasileiro de Sistemas Fuzzy 20 Fuzzy and Possibility Theorem 1:There is nothing uncertain about a fuzzy set. Theorem 2:There is 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 optimization is not optimization under uncertainty!!!! Fuzzy optimization is flexible (transitional) optimization. We will return to this subsequently. Congresso Brasileiro de Sistemas Fuzzy 21 Fuzzy and Possibility – Dubois/Prade • Fuzzy is conjunctive (and) – a fuzzy entity is more than one thing at once (an element is and isn’t in the set to a certain degree). 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 Lula’s age is a distribution over distinct set of elements {… or 45 or 46 or … 59 or 60 or …}. I would 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. Congresso Brasileiro de Sistemas Fuzzy 22 Probability Alone is Insufficient to Describe All Uncertainty Example: Suppose all that is known is that x∈[1,4]. Clearly, x∈[1,4] implies that the real value that x represents is not certain (albeit bounded). If the uncertainty that x∈[1,4] represents were probabilistic (x is a random variable that lies in this interval), then every distribution having support contained in [1,4] would be equally valid given. Thus, if one chooses the uniform probability density distribution on [1,4], p(x) = 1/3, 1≤x≤4, p(x) = 0 otherwise, we clearly lose information. Congresso Brasileiro de Sistemas Fuzzy 23 View of Uncertainty Congresso Brasileiro de Sistemas Fuzzy 24 Probability Alone is Insufficient to Describe all Uncertainty • The approach that keeps the entire uncertainty considers 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. Congresso Brasileiro de Sistemas Fuzzy 25 Probability Alone 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, one 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. Congresso Brasileiro de Sistemas Fuzzy 26 Structure of mathematical analysis in flexible and possibility optimization Optimization requires: 1. Objective function(s) – way to determine (compute) what an optima is 2. Relationships – a description of how variables and parameters are associated (equality and inequality). 3. Constraint set – a way to determine (compute) how the set of relationships are linked (equations/inequalities are linked or aggregated 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. Congresso Brasileiro de Sistemas Fuzzy 27 Entities of Fuzzy and Possibility Optimization The entities are: 1. Intervals 2. Fuzzy intervals 3. Single possibility distributions 4. Dual possibility/necessity distributions Congresso Brasileiro de Sistemas Fuzzy 28 1. Entities of Analysis - Intervals An 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 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 nonnegative slopes over compact domain [0,1]. • A single-valued linear function with non-negative slope over a compact domain [0,1] (Lodwick 1999): [ x ] f ( x ) (1 x ) x x x wxx x, 0 x 1, (w idth) w x x x Congresso Brasileiro de Sistemas Fuzzy 29 Entities of Analysis - Intervals • When 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 2, an algebra of two points (upper half plane determined by y=x), a subset of in fact. • If an interval is considered as a set or a singlevalued 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. 2 Congresso Brasileiro de Sistemas Fuzzy 30 Entities of Analysis - Intervals b) Two interesting anomalies associated with intervals: i. [a] ≤ [b] and [a] ≥ [b] does not imply [a] = [b]. Consider [2,3]x ≤ [3,6] → x ≤ 1 and [2,3]x ≥ [3,6] → x ≥ 3 These two result in the empty set. But [2,3]x = [3,6] → x = [3/2,2] since [2,3][3/2,2]=[3,6] ii. [2,3][½, ⅓]x = [3,6][½, ⅓] [1,1]x = [3/2,2] [½, ⅓] is not an interval, it belongs to the lower half plane where inverses of intervals live (in a non-interval space). Congresso Brasileiro de Sistemas Fuzzy 31 2. Entities of Analysis - Fuzzy Intervals Triangular Fuzzy Interval – We will call this a fuzzy interval since a fuzzy interval (see next slide) is more general. Congresso Brasileiro de Sistemas Fuzzy 32 2. Entities of Analysis - Fuzzy Intervals Trapazoid Fuzzy Interval: What is an element of this set? Congresso Brasileiro de Sistemas Fuzzy 33 Entities of Analysis - Fuzzy Intervals 1. 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. 2. Thus, fuzzy intervals have or take on a dual nature that of capturing or modeling gradualness of belonging and capturing or modeling non-specificity. Possibility is tied to uncertainty. 3. This dual nature of fuzzy intervals is the source of much confusion. Congresso Brasileiro de Sistemas Fuzzy 34 3. Entities of Analysis – Single Possibility Distribution Possibility models non-specificity, information 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. In 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, A C,in contradistinction to probability. The dual to possibility, necessity is required. Given the C possibility of a set A, the necessity of A is known. Congresso Brasileiro de Sistemas Fuzzy 35 Possibility Distributions - Construction There are at least four ways to construct possibility and necessity distributions: 1. Given a set of probabilities (interval-value probabilities): { p ( x ), x , I index set} P os ( x ) sup p ( x ), N ec ( x ) inf p ( x ) I I 2. Given an unknown probability p(x) such that p ( x ) f ( x ), f ( x ) construct consistent necessity/p ossibility pairs p ( x ) [ N ec ( x ), Pos ( x )]. Jamison/Lodwick 2002, Fuzzy Sets and Systems 3. Given a probability assignment function m whose focal 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) Congresso Brasileiro de Sistemas Fuzzy 36 4. Entities of Analysis – Dual Possibility Distributions A fuzzy interval, generates a possibility and necessity pair. Congresso Brasileiro de Sistemas Fuzzy 37 Possibility Dubois, Kerre, Mesair, and Prade 2000 Handbook, 3 probabilistic views of a fuzzy interval 1. The 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 (necessity). This is our first construction. 2. 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. 3. The random set view whereby M encodes the one point coverage function of a random interval, defined by the probability measure on the unit interval (for instance the uniformly distributed one) and a family of nested intervals (the α-levels), via a multi-valued mapping from (0,1] to ℝ, following Dempster. This is our third construction. Congresso Brasileiro de Sistemas Fuzzy 38 III. Fuzzy/Possibilistic Optimization • We turn our attention to optimization • First a classification • Semantics and solutions methods Congresso Brasileiro de Sistemas Fuzzy 39 What We Know About Optimization Optimization 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 problem The normative criterion (or criteria) – the (realvalued) objective function The constraint set – that which defines the limits of resources under which the model operates The relationships (soft/flexible?) Congresso Brasileiro de Sistemas Fuzzy 40 An Example of Fuzzy/Possibility Optimization The radiation therapy problem (RTP) For a given radiation machine, obtain a set of beam angles and beam intensities at these angles so that the delivered dosage kills the tumor while sparing surrounding healthy tissue through which radiation must travel to reach the tumor. Congresso Brasileiro de Sistemas Fuzzy 41 Congresso Brasileiro de Sistemas Fuzzy 42 Linear Accelerator Accelerator Structure Electrons are emitted from the electron gun, reach ~0.99c (c=the speed of light) in the accelerator structure and for photon production (which is all we are concerned with for optimization) the electrons strike a tungsten target which produces the high energy photons (known as bremsstrahlung x-rays) Microwave cavities propagating Electric fields used to accelerate electrons in a linear path. Electron gun Sends electron bunches into the cavities, timed with the E fields. Congresso Brasileiro de Sistemas Fuzzy 43 X-ray Target Converts electron beam to x-rays through bremsstrahlung, (it is retractable). Bending Magnet Achromatic focusing of electron beam before striking target. Flattening Filter Flattens x-ray beam intensity, retractable. Primary Collimator Scattering Foil Flattens electron beam intensity through multiple scattering (it is retractable). Mirror Multi-Leaf Collimator Carrousel Carries flattening filter and scattering foils. Congresso Brasileiro de Sistemas Fuzzy 44 Multi-leaf collimators shape the photon beam to conform to the target (conventional 3D treatment) or to modulate the dose (Intensity Modulated Radiotherapy, IMRT) Shaping the Radiation Beam Multileaf Collimator Congresso Brasileiro de Sistemas Fuzzy 46 y S (, ) b ( , ) r (r , ) x ( , / 2 ) tumor critical organ 1 critical organ 2 body Ss Source S … XsM P T S1 Xsm Xs3 pixel Xs2 Xs1 Congresso Brasileiro de Sistemas Fuzzy 48 Congresso Brasileiro de Sistemas Fuzzy 49 pencil pixel 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2, 19 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 5, 24 0 .172 .828 .5 0 0 .172 .828 0 0 0 .172 0 0 0 0 6, 23 .5 .828 .172 0 0 .5 .828 .172 0 0 .5 .828 0 0 0 .5 10, 27 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 14, 31 0 0 0 .5 0 0 .5 .828 0 .5 .828 .172 .5 .828 .172 0 15, 30 0 0 .828 .5 0 .828 .5 0 .828 .5 0 0 .5 0 0 0 12 17 11 10 B B 9 4 13 16 B C1 C2 B 3 19 B T B 2 20 B B 18 1 1 4 25 26 27 28 Congresso Brasileiro de Sistemas Fuzzy 50 The Flexible/Possibility Optimization Model J m in f ( x ) x (m inim ize total radiation) j j subject to: A x Tideal p t , p t 0, t indices of tum or voxels T A x Tideal q t , q t 0, T A x d ideal rc , rc 0, c k indices of critical tissue T ck k k x0 Congresso Brasileiro de Sistemas Fuzzy 51 Optimization Under Uncertainty - RTP The general uncertainty optimization model for RTP considered here is: ~T x min c subject to ~ ~ ~ : Bx b 0 x x (max beam intensity) where Abody Acriticals B Atumor Atumor ~ b body ~ b criticals and b~ ~ b tumor ~ btumor Congresso Brasileiro de Sistemas Fuzzy 52 Observations About the RTP • Objective function – (1) probability of turning a healthy non-cancerous cell into a cancerous cell. (2) Minimize the maximum radiation to a cancer cell. (3) Minimize total radiation. (4) All three. • Matrix – the classification of a pixel – fuzzy, the location of a pixel (breathing) – fuzzy location is transitional. • Right-hand Side – a target or possibility? • A radiation oncologist would like any solution that satisfies the constraints, though one which gives a minimal damage to health cells is preferable. The best treatment is 100% radiation needed to the tumor and 0% to all other parts – impossible. A radiation oncologist is a satisficer! Congresso Brasileiro de Sistemas Fuzzy 53 Taxonomy – Fuzzy and Possibility Optimization 1. Flexible Optimization – Fuzzy (Zadeh/Bellman 1970, Tanaka/Okuda/Asai 1974, Zimmermann 1976) 2. Single Distribution Optimization – Single Possibility or Single Necessity Optimization (Inuiguchi 1991, Lodwick/Jamison 1997) 3. Dual Distribution Optimization – Paired Possibility Necessity Optimization (Thipwiwatpotjana/Lodwick 2009, Thipwiwatpotjana 2010) – Risk of taking a particular action 4. Mixed Flexible/Evaluation Optimization – Fuzzy and Possibility Random Set Optimization (Lodwick/Jamison 2008, Lodwick/Thipwiwatpotjana 2009, Thipwiwatpotjana 2010) Congresso Brasileiro de Sistemas Fuzzy 54 Harvey Greenberg: A Structural View of Optimization rim opt z c x T B r O Rel b i Ax D m Y This view (rim and body) is useful in deterministic linear programming since it partitions the problem into anatomically useful elements for duality and mathematical programming modeling languages like AMPLE, GAMS since the data (parameters) A, b, and c are separated from the “model.” 55 The General Programming Problem The general deterministic mathematical programming problem we consider here is: z m in f ( x , c ) subject to: g i ( x , a ) bi , i 1, h j ( x , d ) e j , j 1, (1) ,M1 , M 2. c is the objective coefficient rim value b and e are the right-hand side rim values a and d are the body coefficient values 56 1. Flexible Programming a) Soft constraints relationships ≤ and/or = that take on a flexible meaning. For example, we may have a stated constraint to be "Come as close as possible or come as close as possible without violation some hard constraints." The extent of the flexibility will need to be specified. For our RTP example, a constraint on a critical organ in the path of the radiation beam may be, "Come as close as possible to 20 units of radiation for the spinal chord, but never exceed 30 units of radiation." Congresso Brasileiro de Sistemas Fuzzy 57 Flexible Programming b) The objective function expresses an acceptable set of desired targets. For example, for the RTP example, two objective function targets might be, “Try to stay below R units of total radiation delivered to the lungs and minimize the probability of turning a healthy cell into a cancerous one.” The objective function becomes a (flexible) constraint. Congresso Brasileiro de Sistemas Fuzzy 58 Flexible Programming c) The right-hand-side value of a constraint is a fuzzy interval which is semantically a target. For example, for the RTP example, we might have, the constraint, "Do not deliver less than 58 units of radiation to the tumor, preferably between 59 units and 61 units, but never exceed 62 units." This is depicted on the next slide. Congresso Brasileiro de Sistemas Fuzzy 59 When the fuzzy interval right-hand side value is semantically a target, we have a flexible constraint resulting is a flexible programming problem. Congresso Brasileiro de Sistemas Fuzzy 60 Fuzzy Optimization: Decision Making for Flexible (Fuzzy) Programming Given the set of decisions { x domain D n .} Find the optimal decision, that is, sup x h ( F ( x ),..., F ( x )), 1 n w here h :[0,1]n [0,1] (1) is an aggregation operator (such as max). Observe that the decision space is a real-valued set (also called crisp set). Congresso Brasileiro de Sistemas Fuzzy 61 2. Single (Possibility) Distribution Optimization a) Interval, fuzzy interval, possibility cost coefficients of the objective function rim parameter c with real valued coefficient constraint coefficient a, b, d, e ∈ ℝ. In this case we have a family of objective functions and we need to minimize of this family of functions which is typically done using an evaluation function. Here the objective is not a target, but inherently has uncertainty in its cost rim value. Congresso Brasileiro de Sistemas Fuzzy 62 Single (Possibility) Distribution Optimization b) The objective function parameter c ∈ ℝ, a so called “crisp” value where we have interval, probability, possibility, fuzzy interval, and one or two of the following - parameters a, d interval, fuzzy interval, possibility and/or right-hand values b, e are possibility. When a, d are possibility, we have uncertainty (lack of specificity) in the model itself. When the values b, e are possibility we do not have a target (semantically or data-wise) but we have uncertainty. While the representation (as a fuzzy interval) is the same, the semantics and methods are radically different. Congresso Brasileiro de Sistemas Fuzzy 63 Single Distribution Optimization To repeat: The constraint depicted in Figure 4, when it occurs as a right-hand side rim value, may also be interpreted as information deficiency, lack of specificity, that is, a possibility distribution. This is uncertainty in the right-hand side rim data. This is radically different than being a target with preferential values even though both target and uncertainty may be (are) represented as fuzzy intervals! Congresso Brasileiro de Sistemas Fuzzy 64 Decision Making for Possibility Given the set of crisp decisions { x domain D n .} and the set of possibility distributions representing the uncertain outcomes from selecting decision x , denoted Ψ x { Fˆ xi , i 1,..., n}, find the decision that produces the “best” (optimal) of possible outcomes with respect to an outcome ordering U. That is, Congresso Brasileiro de Sistemas Fuzzy 65 Single Distribution Optimization su p x U ( Fˆ x1 , ..., Fˆ xn ), (2) w h ere U ( Fˆ x1 , ..., Fˆ xn ) rep resen ts th e "evalu atio n " (o rd erin g ) o f th e set o f p o ssib le o u tco m es Ψ { Ψ F o r ex a m p le, if Fˆ x1 x | x } . 2ˆ x1 3ˆ x 2 , th en each x ( x1 , x 2 )T in d u ces/g en erates a p o ssib ility d istrib u tio n via th e exten tio n p rin cip le. Congresso Brasileiro de Sistemas Fuzzy 66 Flexible and Possibility Optimization • Fuzzy decision making selects from a set of crisp elements ordered by an aggregation operator h whose domain is the set of vectors [0,1]n (see equation (1)). The aggregation operator h may be min or a t-norm. • Possibilistic decision making selects from a set of crisp elements ordered by an evaluation operator U whose domain is the set of distributions Ψ { Fˆ i , i 1, ..., n}, x x (see equation (2)). The evaluation operator U may be the “expected average.” Congresso Brasileiro de Sistemas Fuzzy 67 3. Dual Evaluation Optimization • When we wish to analyze the risk of taking a decision based on lack of information, we use upper/lower bounds on the uncertainty. • Interval data is transformed into upper/lower distributions, probability data is transformed into upper/lower distributions, single possibility is transformed into upper/lower. • There are constructions to transform interval-valued distributions, clouds, and sets of probabilities into upper/lower distributions Congresso Brasileiro de Sistemas Fuzzy 68 4. Mixed (Flexible and Evaluation) Optimization a) Interval-Valued Probability Measure Programming - Any of the coefficients a, b, c, d, e may be interval, fuzzy, possibilistic where there may be a mixture of types within one constraint statement. These mixed values can be and should (must) be transformed into interval-valued probability distributions or clouds or random sets. These are the theories that are supersets of intervals, probabilities, fuzzy sets, possibilities though arguably, random sets are the most general. Congresso Brasileiro de Sistemas Fuzzy 69 Mixed (Flexible and Evaluation) Optimization b) Random Set Programming - Any of the coefficients a, b, c, d, e may be interval, fuzzy, possibility where there may be a mixture of types within one constraint statement may be transformed into random sets. Interval-valued probability optimization, I think, is the most straight forward approach (easiest to implement). Congresso Brasileiro de Sistemas Fuzzy 70 IV. Theoretical Consideration <OSCAR> A. Geometry B. Order Relations Congresso Brasileiro de Sistemas Fuzzy 71 Interval Arithmetic The usual interval arithmetic (called here standard interval arithmetic, SIA) on intervals has no additive or multiplicative identity – SIA was created independently by Warmus (1956) Poland, Sunaga (1958) Japan, Moore (1959) USA. Interval Addition/Subtraction L et X I X X I [2, 3], then I [2, 3] [2, 3] [2, 3] [ 3, 2] [ 1,1], not [0, 0] Congresso Brasileiro de Sistemas Fuzzy 72 Arithmetic N O T E : H ow ever , if let X I [2, 3], and Y = [2,3] then X I Y I [2, 3] [2, 3] I [2, 3] [ 3, 2 ] [ 1,1]. W e seek an arithm etic w hich distinguishe s betw een X I X I and X I Y I even w hen X Congresso Brasileiro de Sistemas Fuzzy I Y . I 73 Arithmetic Multiplicative inverse (lack of in definitional interval arithmetic): X I [1, 2] X I X I [m in{1 / 1,1 / 2}, m ax{1 / 1,1 / 2}] 1 [ ,1] not [1,1] 2 Congresso Brasileiro de Sistemas Fuzzy 74 Arithmetic: Consequences of DIA 1. Sub-distributive: XI(YI + ZI) XIYI + XIZI 2. Multiple instances of a variable, e.g. XIXI + XIYI, means we will, most often, overestimate. 3. Since fuzzy arithmetic has traditionally been implemented as definitional interval arithmetic on αcuts of fuzzy intervals, the above problem is compounded in the setting of fuzzy sets over real numbers. Why should we care? When doing fuzzy arithmetic (ala Kaufmann and Gupta), the usual interval arithmetic on alpha levels may, usually do, lead to overestimations and thus not the tightest constraints. This is a characteristic of the arithmetic used NOT the Congresso Brasileiro de Sistemas 75 underlying problem. Fuzzy Constraint Fuzzy Intervals and Gradual Numbers Lodwick 1999 – Constraint Interval Arithmetic (CIA – not to be confused with an institution, the constraint intelligence agency). We start by redefining an interval as the relationship (function, graph) of a single variable X ( X ) { x | w X X x , 0 1, w X x x }. I As a graph, this representation is a set parameterized by a single variable λ where the endpoints are data/inputs and the variable is constrained to lie between 0 and 1, hence the name. As a function it is linear function of a single variable with non-negative slope over a compact set [0,1] Congresso Brasileiro de Sistemas Fuzzy 76 Constrained Arithmetic: Operations Z I [ z, z] X {z | z x w here z I Y I w e w ill drop the superscript I y , x X ( X ), y Y ( Y ), 0 X , Y 1} m in Z , and z 0 X , Y 1 m ax Z 0 X , Y 1 Consequences – operations on functions X X { w x x x w x x x } [0, 0 ] X X { w x x x w x x x , 0 X } [1,1] Congresso Brasileiro de Sistemas Fuzzy 77 Constraint Interval Arithmetic • Remark 1: CIA has an additive identity, namely, [0,0] and a multiplicative identity, namely, [1,1] . And CIA distinguishes between [x]-[x] and [x]-[y], when [x]=[y]. • There is a relationship between CIA and Klir’s earlier 1997 paper “Fuzzy arithmetic with requisite constraints”, FSS 91 (1997). His results are the same for each α-level as those of CIA. However, our parameterization using λx and the use of constrained global optimization on 0≤ λx ≤1 is different and more general since Klir’s arithmetic is case-based. Congresso Brasileiro de Sistemas Fuzzy 78 Gradual Number (2008) Gradual Number (Fortin, Dubois, Fargier, 2008) is an assignment of a unique value for each α-cut (viewed on the y-axis). The next two slides will make this clear. Congresso Brasileiro de Sistemas Fuzzy 79 Gradual Numbers From a fuzzy interval to a gradual number Congresso Brasileiro de Sistemas Fuzzy 80 Gradual Numbers Examples of assignment functions, that is, gradual numbers (r(α ), s(α), and t(α) for 0 ≤ α ≤ 1) Congresso Brasileiro de Sistemas Fuzzy 81 Constraint Fuzzy Arithmetic (CFA) CFA, when applied to fuzzy interval arithmetic α-level by αlevel, is simply CIA applied α –level by α –level. This is illustrated in the following example. A gradual number is an element of a fuzzy set (interval). From the CFA point of view, an element of a fuzzy set (interval) is simply X ( X ( )) { x | w X ( ) X ( ) x , w X ( ) x x , I 0 X ( ) 1, 0 1}. This is simply an assignment of one (and only one) lambda value ( ) for each α –level, a single instantiation (pick one value from each α –level interval) for each α. Thus, I X ( X ( )) is a gradual number and an element of the fuzzy set (interval). Congresso Brasileiro de Sistemas 82 Fuzzy Optimization of Constraint Fuzzy Intervals and Gradual Numbers • Elizabeth (Untiedt) Stock 2010 – "Gradual numbers and fuzzy optimization," PhD Thesis, University of Colorado Denver, Department of Mathematical and Statistical Sciences. • Phantipa Thipwiwatpotjana 2010 - Linear programming problems for generalized uncertainty," Ph.D. Thesis,University of Colorado, Department of Mathematical and Statistical Sciences. Congresso Brasileiro de Sistemas Fuzzy 83

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# The Use of Fuzzy Optimization Methods for Radiation