New Fuzzy Technologies of Weakly Structured Processes’ Modeling and Simulation Prof. Gia Sirbiladze Iv.Javakhishvili Tbilisi State University, GEORGIA March 21, 2019 

Input Part: On Weakly Structured Processes (WSP) and Fuzzy Technologies • A systems or processes’ model is called weakly structured if in its evolution environment the relationship between its objects cannot be formulated in terms of deterministic or stochastic apparatus • Often, expert knowledge is only source for an identification of WSP models. • Fuzzy modeling of weakly structured processes is a new direction in complex processes analysis March 21 -23, 2019 MACSPro 2019 , Vienna, Austria 

About fuzzy-technologies 1965 L.A. Zadeh’s paper “Fuzzy logic” represents a main basis of “Theory of fuzzy sets”. 1970 First application of fuzzy technologies in control engineering (Europe) 1975 Basics of fuzzy technologies in Japan 1980 Verification of fuzzy technologies in Europe 1985 A wide use of fuzzy technologies in Japan 1990 A wide use of fuzzy technologies in Europe 1995 Fuzzy technologies in USA 2000 Fuzzy technology is a standard technology in the data and sensor signals’ analysis, in financial and business decision making, etc. Fuzzy technology became one of main instrument in complex systems research March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Part I: informational and mathematical elements of fuzzy technologies • Fuzzy logic – a multilevel logic • Fuzzy set and its membership function • Linguistic variables in fuzzy logic • Elements of the possibility theory • Possibilistic Dynamic System for Weakly Structured Processes March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Levels of compatibility (membership) – basis of fuzzy logic • Crisp (Bool) logic: – Proposition can only be true or false. • George is a student (true) • Smoking is healthy (false) – The degree of truth (level) is 0 or 1. • Fuzzy logic: – The degree of membership (level) is from interval [0;1]. • Ann is young (say, with truth level 0.3) • Tom is clever (with truth level 0.9) March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Crisp sets • Classical sets are crisp sets: – An element belongs to the set or not: • Membership function – indicator of a crisp set: 0 ( ) 1 A x A x x A          ( ) 0,1 A x   March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

P Crisp sets P : the set of all people. Y : the set of young people. Y   ( ) 25, age Young y y x x P     1 y ( ) Young y  25 March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Fuzzy sets ( ) [0,1] A x   1 y ( ) Young y  Example March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Fuzzy sets Lotfi A. Zadeh, The founder of fuzzy logic. L. A. Zadeh, “Fuzzy sets,” Information and Control, 8 (1965) 338-353. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Fuzzy sets and membership function If U is a set of certain objects (universe) then fuzzy set A in U is defined as a set of ordered pairs:   ( , ( )) A A x x x U    Membership function : [0,1] A U   March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Example (discrete universe) {1,2,3,4,5,6,7,8} U  A set of training courses in one semester (1,0.1) (2,0.3) (3,0.8) (4,1) (5,0.9) (6,0.5) (7,0.2) (8,0.1) A        A set of important courses of some student 0.5 1 0 2 4 6 8 x : training courses ( ) A x  March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Example (continous universe) Possible age U : the set of positive real numbers   ( , ( )) B B x x x U    4 1 ( ) 50 1 5 B x x          near 50 years 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 x : age ( ) B x  March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

membership value height (m) 1 0 Membership functions (MF’s) • Fuzzy set is completely represented by membership function: it is an expert measure, not probabilistic one. „tall“ in Asia „tall“ in Europe „tall“ in NBA 1.85 March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Linguistic variable • Linguistic variable is a variable which value is a word or sentence from natural or artificial languages. • Linguistic variable can be assigned some linguistic values – term, which are connected, as fuzzy sets, with degrees of membership by corresponding matching functions. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Example on use of linguistic variable If in Vienna temperature is low then it is expected a little snow. Linguistic variable Linguistic variable Linguistic term Linguistic term  low  low March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Definition [Zadeh, 1973] Linguistic variable represents a quintuple :   , ( ), , , x T x U G M name set of terms universe syntax rules semantic rules March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Example : linguistic variable’s representation Linguistic variable -„age“ is represented by quintuple:   , ( ), , , x T x U G M age old, very old, not so old, (age) more or less young, quite young, very young G            [0, 100]     old (old) , ( ) [0,100] M u u u    1 2 old 0 [0,50] ( ) 50 1 [50,100] 5 u u u u                             example of semantic rules: March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Example : linguistic variables qualitative and quantitative presentation Linguistic variable: temperature Linguistic terms (fuzzy sets) : {cold, warm, hot} (x) cold warm hot 20 60 1 x March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Conception of possibility • (D. Dubois and H. Prade (1988) Possibility Theory, New York: Plenum) • Possibility theory as an alternative to probability theory. • When in contrast to the axiomatic of probability theory the source of information (data) is only expert’s knowledge… March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Example of possibility • Suppose we have linguistic variable: x = “age of president”. Suppose also that we do not know the age of the president, however we have some statistical, but incomplete, information about his age. • The possibility distribution of variable x can be represented as follows: • Suppose that linguistic information is represented by its term : “ president is an old man ” . • Possibility distribution is defined by “old man” ‘s membership function: π = µold man March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Possibility distribution – π(x) • Possibility is a representation of expert knowledge state, i.e. a description of expert’s evaluation of a fact or occurrence and which is given in possibility levels. • Possibility is a level from interval [0;1] of expert’s evaluation, which indicates on a potential of event's occurrence. • π(x) is a possibility level (degree), that the president is x year aged. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Possibility of event • Let S be an universe and π be a possibility distribution on S. The possibility of an event A, A  S, is defined as a level of possibility of occurrence x  A : = ∊ = ∊ π(). Pos(A) is a level at which at least one element from A may take a place. In contrast to probability which is an additive sum of the probabilities of the elements of A : = () ∊ . March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Possibility and Necessity of event If A and B are some events from S, then Pos(A  B) = max{Pos(A), Pos(B)}. Necessity is a dual measure of Possibility. It conducts the same information, but codified differently: Nes(A) =1-Pos(S/A); Nes(A  B) = min{Nes(A), Nes(B)}. Nes(A)≤Prob(A) ≤Pos(A). Nes, Pos – imprecise probabilities. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Mathematics, more exactly, mathematical modeling is a loss of certainty!!! This is connected with the complexity of study of uncertain and vague processes in nature and society. With the growth of complexity of information, our ability to make credible decisions from possible alternatives with complex states of nature is being reduced to certain level, below which some dual characteristics such as precision and certainty become mutually conflicting. In such cases, when working on real, complex decision systems using an exact or some stochastic quantitative analysis is often less convenient, concluding that the use of fuzzy methods is necessary ( deep level of detallization of systems parameter; source of input data is only experts’ knowledge,…) March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Construction of possibilistic dynamic system Let be a set of states of linguistic variable’s terms of a weakly structured process, as a set of states of evolutionary system. An expert in the process of evolution evaluates fuzzy terms. In this way on D we have expert knowledge stream as dual trajectory of Weakly Structured Processes (WSP) : March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Possibilistic Dynamic System for Weakly Structured Processes • The possibilistic discrete model for a WSP * ∗ . , , ∗ . , + – possibilistic recurrent system : March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

New Fuzzy Technologies of Weakly Structured Processes’ Modeling and Simulation • On the one hand it is studied new fuzzy- approach for Weakly Structured Processes new models’ identification, filtration, prediction and optimal control problems. • On the other hand it is created software library which represents a new instrument for constructing weakly structured processes analysis’ expert systems. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Part II: Consideration – formation of possibilistic prediction • Formation of prediction in deterministic environment • Formation of prediction in probabilistic environment • Formation of prediction in possibilistic environment. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in incomplete data environment • prediction in incomplete data environment is represented by its two poles: prediction imprecision and prediction uncertainty. • imprecision indicates the degree of prediction variable’s value, while uncertainty indicates prediction’s reliability, probability, possibility, etc. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in incomplete information environment • Informational structure (imprecision+ uncertainty) of a weakly structured process’ prediction modeling defines prediction type forms in incomplete information environment: March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in deterministic environment Syntax of prediction: • Tomorrow the expected temperature in Vienna at 10 o'clock is 12o C. - imprecision here is absolute zero, but it is not evaluated prediction’s uncertainty – i.e. at what degree it is reliable. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in deterministic environment • Tomorrow the expected temperature in Vienna at 10 o'clock will be in the interval [ 8o –15o ] C. - The imprecision here is allowed, which leads to decrease in uncertainty. This means that the recognition is more reliable and probable, but the latter's assessment is not possible in deterministic environment. -Semantics of prediction… . March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

time – exactly 10 o’clock 0 0,5 1 1,5 0 10 20 temperature – in [8o–15o] C interval hr T0 C 8 15 reliability assessment is impossible March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in probabilistic environment • Probably, tomorrow at 10 o'clock the expected temperature in Vienna will be in the interval [ 8o –15o ] C. Probabilistic imprecision is given by random variable – temperature, while uncertainty by probability which necessarily should be numerically assessed. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

time – exactly 10 o’clock 0 0.5 1 1.5 0 10 20 temperature – in [8o–15o] C interval with what probability ? hr T0 C 8 15 March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in probabilistic environment • The probability that tomorrow at 10 o’clock in Vienna the expected temperature will be in the interval [8o– 15o] C is 0.9 Probability imprecision is represented by the interval [8o – 15o] C, while uncertainty – by confidence probability - 0.9 March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

0 0.5 1 1.5 0 5 10 15 time – exactly 10 o’clock 0 0.5 1 1.5 0 10 20 temperature – in [8o–15o] C interval confidence probability =0.9 8 15 hr T0 C March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in possibilistic environment • It is possible, that tomorrow at 10 o’clock in Vienna the expected temperature will be approximately in the interval [8o – 15o] C. Imprecision changed. It is represented by fuzzy interval: approximately [8o–15o] C. The possibilistic (subjective) uncertainty arose, which necessarily should be assessed. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

0 0.5 1 1.5 0 5 10 15 time – exactly 10 o’clock 0 0.5 1 1.5 0 10 20 temperature – approximately in [8o–15o] C interval With what possibility? 6 8 15 17 C 0 hr March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in possibilistic environment • Possibility level that, tomorrow at 10 o’clock, in Vienna the expected temperature will be approximately in [8o – 15o] C interval is 0.7 Fuzzy uncertainty is represented by possibilistic level 0.7, which is not probability or statistic frequency. It represents an activity level of expert’s knowledge, degree, bound with given fact… March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

0 0.5 1 1.5 0 5 10 15 time – exactly 10 o’clock 0 0.5 1 1.5 0 10 20 temperature – approximately in [8o–15o] C interval possibilistic level of prediction– 0.7 6 8 15 17 C 0 hr March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in possibilistic environment • It is of high possibility that tomorrow at 12 o’clock in Vienna the expected temperature will comprise a winter average. Imprecision here is increased with winter average temperature, instead we obtain increase in reliability – with high possibility, which necessarily should be assessed. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

time – exactly 12 o’clock 0 0.5 1 1.5 0 10 20 temperature – winter average temperature high possibilistic level of prediction 3 7 16 18 C 0 0 2 0 10 20 hr 12 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 0.9 March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Formation of prediction in possibilistic environment • It is possible that tomorrow approximately at 12 o’clock in Vienna the expected temperature will be equaled to a winter average temperature. Again, we increased imprecision, which has dual nature: time + linguistic variable. We obtain possibilistic recognition of high reliability, which has only verbal form… March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

0 0.5 1 1.5 0 10 20 temperature – winter average temperature possibilistic level – ? 3 7 16 18 C 0 სთ 0 0.5 1 1.5 0 5 10 15 20 12 March 21-23, 2019 MACSPro 2019 , Vienna, Austria time – approximately 12 o’clock hr 

Formation of prediction in possibilistic environment • It is of high possibility that tomorrow at noon in Vienna the expected temperature will comprise a winter average temperature. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

time - noon 0 0,5 1 1,5 0 10 20 temperature – winter average temperature high possibility 3 7 16 18 C 0 0 0,5 1 1,5 0 0,5 1 1,5 0.9 0 0,5 1 1,5 0 5 10 15 20 12 17 15 hr March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Application of New Fuzzy Technologies • Precision problems of expert’s knowledge streams (teaching process and others); • Estimated early prediction of the local origin of tornadoes in USA and Canada; • Decision problems of strategic management (application in business start-up processes); • Facility location-selection problems and products transportation problems under uncertain environment; (Georgian companies with consumer distribution networks); and others. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Future Application of New Fuzzy Technologies (corporation with biggest Georgian business groups) 1. Big Data Streams Analytics in uncertain environment (platform selection, prediction problems and others); 2. Big Data Analytics using Fuzzy Multiple Criteria Decision-Making or Models of Fuzzy Multi-Scale Decision Making; 3. Fuzzy rule-based knowledge representation in Big Data Processing; 4. Fuzzy models for large dimension problems; 5. Studies on scalability by new fuzzy models. March 21-23, 2019 MACSPro 2019 , Vienna, Austria 

Thank you for your attention March 21-23, 2019 MACSPro 2019 , Vienna, Austria