1. Armin Shmilovici Ben-Gurion University, Israel armin@bgu.ac.il

2. Credits for some slides S.Natarajan, Fuzzy Decision Making J.-S.R. Jang, Neuro-Fuzzy and Soft Computing M. Casey, Lectures on Artificial Intelligence – CS364

3. Fuzzy Vs. Probability Walking in the desert, close to being dehydrated, you find two sealed bottles: The first labeled water - 95% probability The second labeled water - 0.95% similarity Which one will you choose to drink from??? 3

4. Uncertainty Information can be incomplete inconsistent, uncertain, or all three. Uncertainty is defined as the lack of the exact knowledge that would enable us to reach a perfectly reliable conclusion. Some Sources of Uncertain Knowledge: Weak implications Imprecise language. Our natural language is ambiguous and imprecise. Unknown data. When the data is incomplete or missing views of different experts.

5. Definition Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise. Fuzzy set theory resembles human reasoning in its use of approximate information and uncertainty to generate decisions. It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems.

6. More Definitions Fuzzy logic is a set of mathematical principles for knowledge representation based on degrees of membership. Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It deals with degrees of membership and degrees of truth. Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true). Instead of just black and white, it employs the spectrum of colours, accepting that things can be partly true and partly false at the same time.

7. Crisp Vs Fuzzy Sets The x-axis represents the universe of discourse – the range of all possible values applicable to a chosen variable. In our case, the variable is the man height. According to this representation, the universe of men’s heights consists of all tall men. The y-axis represents the membership value of the fuzzy set. In our case, the fuzzy set of “tall men” maps height values into corresponding membership values.

8. A Fuzzy Set has Fuzzy Boundaries In the fuzzy theory, fuzzy set A of universe X is defined by function µ A (x) called the membership function of set A µ A (x) : X  {0, 1}, whereµ A (x) = 1 if x is totally in A; µ A (x) = 0 if x is not in A; 0 < µ A (x) < 1 if x is partly in A. This set allows a continuum of possible choices. For any element x of universe X, membership function µ A (x) equals the degree to which x is an element of set A. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element x in set A.

9. Fuzzy Set Representation First, we determine the membership functions. In our “tall men” example, we can obtain fuzzy sets of tall, short and average men. The universe of discourse – the men’s heights – consists of three sets: short, average and tall men. As you will see, a man who is 184 cm tall is a member of the average men set with a degree of membership of 0.1, and at the same time, he is also a member of the tall men set with a degree of 0.4. (see graph on next page)

10. Fuzzy Set Representation

11. Fuzzy Sets Formal definition: A fuzzy set A in X is expressed as a set of ordered pairs: ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 11 Universe or universe of discourse Fuzzy set Membership function (MF) A fuzzy set is totally characterized by a membership function (MF).

12. Fuzzy Sets with Discrete Universes Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A = “sensible number of children” X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0,.1), (1,.3), (2,.7), (3, 1), (4,.6), (5,.2), (6,.1)} ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 12

13. Fuzzy Sets with Cont. Universes Fuzzy set B = “about 50 years old” X = Set of positive real numbers (continuous) B = {(x, mB(x)) | x in X} ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 13

14. Alternative Notation A fuzzy set A can be alternatively denoted as follows: ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 14 X is discrete X is continuous Note that  and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.

15. Fuzzy Partition Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”: ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 15 lingmf.m

16. Linguistic Variables and Hedges The range of possible values of a linguistic variable represents the universe of discourse of that variable. For example, the universe of discourse of the linguistic variable speed might have the range between 0 and 220 km/h and may include such fuzzy subsets as very slow, slow, medium, fast, and very fast. A linguistic variable carries with it the concept of fuzzy set qualifiers, called hedges. Hedges are terms that modify the shape of fuzzy sets. They include adverbs such as very, somewhat, quite, more or less and slightly.

17. Linguistic Variables and Hedges

19. Operations on Fuzzy Sets

20. Note: Membership Functions For the sake of convenience, usually a fuzzy set is denoted as: A =  A (x i )/x i + …………. +  A (x n )/x n where  A (x i )/x i (a singleton) is a pair “grade of membership” element, that belongs to a finite universe of discourse: A = {x 1, x 2,.., x n }

21. Basic definitions & Terminology Support(A) = {x  X |  A(x) > 0} Core(A) = {x  X |  A(x) = 1} Normality: core(A)    A is a normal fuzzy set Crossover(A) = {x  X |  A(x) = 0.5}  - cut: A  = {x  X |  A(x)   } Strong  - cut: A’  = {x  X |  A(x) >  } ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 21

22. Set-Theoretic Operations Subset: Complement: Union: Intersection: ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 22

23. Set-Theoretic Operations ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 23 subset.m fuzsetop.m

24. MF Formulation Triangular MF: ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 24 Trapezoidal MF: Generalized bell MF: Gaussian MF:

25. MF Formulation ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 25 disp_mf.m

26. Equality Fuzzy set A is considered equal to a fuzzy set B, IF AND ONLY IF (iff):  A (x) =  B (x),  x  X A = 0.3/1 + 0.5/2 + 1/3 B = 0.3/1 + 0.5/2 + 1/3 therefore A = B

27. Inclusion Inclusion of one fuzzy set into another fuzzy set. Fuzzy set A  X is included in (is a subset of) another fuzzy set, B  X:  A (x)   B (x),  x  X Consider X = {1, 2, 3} and sets A and B A = 0.3/1 + 0.5/2 + 1/3; B = 0.5/1 + 0.55/2 + 1/3 then A is a subset of B, or A  B

28. Cardinality Cardinality of a non-fuzzy set, Z, is the number of elements in Z. BUT the cardinality of a fuzzy set A, the so-called SIGMA COUNT, is expressed as a SUM of the values of the membership function of A,  A (x): card A =  A (x 1 ) +  A (x 2 ) + …  A (x n ) = Σ  A (x i ),for i=1..n Consider X = {1, 2, 3} and sets A and B A = 0.3/1 + 0.5/2 + 1/3; B = 0.5/1 + 0.55/2 + 1/3 card A = 1.8 card B = 2.05

29. Empty Fuzzy Set A fuzzy set A is empty, IF AND ONLY IF:  A (x) = 0,  x  X Consider X = {1, 2, 3} and set A A = 0/1 + 0/2 + 0/3 then A is empty

30. Alpha-cut An  -cut or  -level set of a fuzzy set A  X is an ORDINARY SET A   X, such that: A  ={  A (x) ,  x  X}. Consider X = {1, 2, 3} and set A A = 0.3/1 + 0.5/2 + 1/3 thenA 0.5 = {2, 3}, A 0.1 = {1, 2, 3}, A 1 = {3}

31. Normalized Fuzzy Sets A fuzzy subset of X is called normal if there exists at least one element x  X such that  A (x) = 1. A fuzzy subset that is not normal is called subnormal. All crisp subsets except for the null set are normal. In fuzzy set theory, the concept of nullness essentially generalises to subnormality. The height of a fuzzy subset A is the large membership grade of an element in A height(A) = max x (  A (x))

32. Fuzzy Sets Core and Support Assume A is a fuzzy subset of X: the support of A is the crisp subset of X consisting of all elements with membership grade: supp(A) = {x   A (x)  0 and x  X} the core of A is the crisp subset of X consisting of all elements with membership grade: core(A) = {x   A (x) = 1 and x  X}

33. Fuzzy Set Math Operations aA = {a  A (x),  x  X} Let a =0.5, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then A a = {0.25/a, 0.15/b, 0.1/c, 0.5/d} A a = {  A (x) a,  x  X} Let a =2, and A = {0.5/a, 0.3/b, 0.2/c, 1/d} then A a = {0.25/a, 0.09/b, 0.04/c, 1/d} …

34. Classical Decision Making Alternative actions {drill/not drill} Alternative states of nature {oil in ground} Relations between action/state/outcome Utility or objective function(s) Weakness: Assume utility is known and accurate Assume relations are known and accurate Assumptions not correct ->solution not accurate!

35. 35 Fuzzy Logic  Simulating the process of human reasoning  framework to computing with words and perception, using linguistics variables.  Deals with uncertainties  Dreative decision-making process

36. 36 Fuzzy Decision Making A model for decision making in a fuzzy environment. Object function and constraints are characterized as their membership functions. The intersection of fuzzy constraints and fuzzy objection function.

37. 37 Example Objective function: x should be larger than 10 Constraint: x should be in the vicinity of 11

38. 38 Fuzzy decision-making method Representation of the decision problem Fuzzy set evaluation of the decision alternatives Selection of the optimal alternative The method consists of three main steps: Different classical decision-making

39. 39 Fuzzy Decision-Solution

40. Example-Job Selection Choose one of four alternative job offers {a1,a2,a3,a4} Following criteria High Salary (G1) Interesting Job (C1/G2) Close driving time (C2) Salaries in Eu {40k,45k,50k,60k} Time in minutes {42, 12, 18, 4}

41. Job Selection – Cont. Fuzzify the goals and constraints according to your personal utility functions. For example, for a linear salary utility function (membership function) such as 37K is a 0.0 membership in “High Salary”, and 64k is a 1.0 membership in “High Salary”,then G1=0.11/a1+0.3/a2+0.48/a3+0.8/a4 C1 is fuzzy by its nature, for example C1=0.4/a1+0.6/a2+0.2/a3+0.2/a4 For a linear driving time utility function (membership function) such as 8 minutes is a 0.0 membership in “Close driving time ”, and 50 minutes is a 1.0 membership in “Close driving time ”,then C2=0.1/a1+0.9/a2+0.7/a3+1.0/a4 Find D by minimizing across alternatives D=0.1/a1+0.3/a2+0.2/a3+0.2/a4 Choose alternative which maximize D -> a2 Note that a2 is the best, but its not good (only 0.2)

42. 42 MULTIOBJECTIVE DECISION MAKING Most simple decision processes are based on a single objective, such as minimizing cost, maximizing profit, minimizing run time, and so forth. Often, however, decisions must be made in an environment where more than one objective function governs constraints on the problem, and the relative value of each of these objectives is different. Most simple decision processes are based on a single objective, such as minimizing cost, maximizing profit, minimizing run time, and so forth. Often, however, decisions must be made in an environment where more than one objective function governs constraints on the problem, and the relative value of each of these objectives is different. For example, we are designing a new computer, and we want simultaneously to minimize the cost, maximize CPU, maximize Random Access Memory (RAM), and maximize reliability. For example, we are designing a new computer, and we want simultaneously to minimize the cost, maximize CPU, maximize Random Access Memory (RAM), and maximize reliability.

43. 43 Multi-objective Decision Making A = {a 1,a 2,…,a n }: set of alternatives O = {o 1,o 2,…,o r }: set of objectives The degree of membership of alternative a in O j is given below. Decision function: The optimum decision a *

44. 44 b i is a parameter measuring how important is the objective to a decision maker for a given decision b i is a parameter measuring how important is the objective to a decision maker for a given decision D = M(O 1,b 1 ) ∩ M(O 2,b 2 ) ∩.. ∩ M(O r,b r ) D = M(O 1,b 1 ) ∩ M(O 2,b 2 ) ∩.. ∩ M(O r,b r ) This function is represented as the intersection of r tuples of a decision measure, M(O i,b i ), involving the objectives and preferences, This function is represented as the intersection of r tuples of a decision measure, M(O i,b i ), involving the objectives and preferences, The classical implication satisfies the requirements. Of preserving the linear ordering the preference set, and at the same time relates the two quantities in a logical way where negation is also accommodated. The classical implication satisfies the requirements. Of preserving the linear ordering the preference set, and at the same time relates the two quantities in a logical way where negation is also accommodated. The decision measure for a particular alternative, a, can be replaced with a classical implication of the form M(O i (a i ), b i ) = b i --> O i (a) = b¯ i V O i (a) The decision measure for a particular alternative, a, can be replaced with a classical implication of the form M(O i (a i ), b i ) = b i --> O i (a) = b¯ i V O i (a)

45. 45 Multi-objective Decision Making Define a set of preferences {P} Parameter b i is contained on set {P}

46. 46 EXAMPLE – Multi-Objective Decision EXAMPLE – Multi-Objective Decision A construction engineer on a construction project must retain the mass of soil form sliding into a building site. Alternatives : A construction engineer on a construction project must retain the mass of soil form sliding into a building site. Alternatives : MSE - mechanically stabilized embankment Conc - a mass concrete spread wall Gab -a gabion wall Four objectives that impact his decision: Four objectives that impact his decision: Cost C Maintainability M Standard S Environment E A = { MSE, Conc, Gab} = {a 1, a 2, a 3 } O = {C, M, S, E} = {O 1, O 2, O 3, O 4 } P = {b 1, b 2, b 3, b 4 } [0,1]

47. 47 O 1 = { 0.4/MSE + 1/Conc + 0.1/Gab } O 1 = { 0.4/MSE + 1/Conc + 0.1/Gab } O 2 = { 0.7/MSE +.8/Conc + 0.4/Gab} O 2 = { 0.7/MSE +.8/Conc + 0.4/Gab} O 3 = {0.2/MSE +.4/Conc + 1/Gab } O 3 = {0.2/MSE +.4/Conc + 1/Gab } O 4 = {1/MSE + 0.5/Conc + 0.5/Gab } O 4 = {1/MSE + 0.5/Conc + 0.5/Gab } Mse conc Gab 1.5 0 O1 O2 O3 O4

48. 48

49. 49

50. 50 Multiple-Choice Multiple-Criteria

51. 51 Example – Cooling an nuclear Reactor (Step1:representation of decision problem) Decision Alternatives A = ｛ A 1,A 2,A 3 ｝ A 1 ： flooding the reactor cavity) A 2 ： depressurizing the primary system A 3 ： doing nothing Decision Criteria C= ｛ C 1,C 2,C 3,C 4,C 5 ｝ C 1 ： feasibility of the strategy C 2 ： effectiveness of the strategy C 3 ： possibility of adverse effects C 4 ： amount of information needs C 5 ： compatibility with existing procedures

52. 52 Step1:representation of decision problem(Cont.) C1C1 C2C2 C3C3 C4C4 C5C5 A1A1 A2A2 A3A3

53. 53 Step1:representation of decision problem (Cont.) C1C1 C2C2 C3C3 C4C4 C5C5 A1A1 A2A2 A3A3 Hierarchical structure of an example problem

54. 54 Fuzzy set evaluation of decision alternatives The step consist of three activities (1)Choosing sets of the preference ratings for the importance weights of the decision Preference ratings include Linguistic variable and triangular fuzzy number Example ： The preference ratings for the importance weights of the decision criteria can be defined as follows:T importance= ｛ very low, low, medium, high,very high ｝

55. 55 Fuzzy set evaluation of decision alternatives(Cont.) Where a,b and c are real numbers and a ≦ b ≦ c The triangular fuzzy numbers are used as membership functions corresponding to the elements in term set

56. 56 00.250.50.751 Linguistic variable and triangular fuzzy number 1 VLLM HVH

57. 57 Fuzzy set evaluation of decision alternatives Choosing sets of the preference ratings for the importance weights of the decision

58. 58 Fuzzy set evaluation of decision alternatives(Cont.) (2) Evaluating the importance weights of the criteria and the degrees of appropriateness of the decision alternatives (3) Aggregating the weights of the decision criteria

59. 59 Fuzzy set evaluation of decision alternatives(Cont.) Substituting S it and W t with triangular fuzzy numbers, that is, S it = (o it, p it, q it ) and W t =(a t, b t, c t ), F i is approximated as where For i =1,2,…,n and t = 1,2,..,k

60. 60 Evaluating and Aggregating 0.23750.53750.8625 0.21250.50.7625 0.2750.56250.8

61. 61 This step includes two activities Prioritization of the decision alternatives using the aggregated assessments Choice of the decision alternative with highest priority as the optimal Selection of the optimal alternative

62. 62 Selection of the optimal alternative(Cont.) α = index, the degree of optimism of the decision-maker 0 ≦ α ≦ 1 Larger Fi means the higher appropriateness of the decision alternative α The total integral value method Let the total integral value for triangular fuzzy number F=(a, b, c)

63. 63 Selection of the optimal alternative If α =0 then A3 is Best! If α =0.5 then A3 is Best! If α =1.0 then A1 is Best! 0.54375 (2)0.3875 (2)0.7 (1) 0.49375 (3)0.35625 (3)0.63125 (3) 0.55 (1)0.41875 (1)0.68125 (2)

64. 64

65. 65 Fuzzy Ordering- contd y 1 = 5, y 2 = 2, y 1 >= y 2, usually no ambiguity in the ranking, so we have CRISP ordering Situations where issues or actions are associated with uncertainty, random or fuzzy, ranking is necessary Uncertainty in rank is random – we can use pdf ( probability density function), use Gaussian normal function involving standard deviation and solve Plot of pdf function is there in the previous slide Example of uncertain ranking – x 1 height of Italians and x 2 is the height of Swedes Are Swedes taller than Italians ? Assign membership function µ 1 for Swedes and membership function µ 2 for Italians.

66. 66 Fuzzy Ordering

67. 67

68. Fuzzy Expert Systems Used to capture human procedural knowledge in the form of linguistic variable Fuzzy rules: If oven’s temperature=HIGH then cooking time=SHORT A fuzzy expert system can chain rules together via the process of inferencing. Many successful application, especially for control of nonlinear systems.

69. Fuzzy Reasoning Single rule with multiple antecedent Rule: if x is A and y is B then z is C Fact: x is A’ and y is B’ Conclusion: z is C’ Graphic Representation: ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה ל'/אב/תשע"ה 69 AB T-norm XY w A’B’C2C2 Z C’ Z XY A’B’ x is A’y is B’z is C’