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LINGUISTIC VARIABLES A linguistic variable is a variable whose values are not numbers but words or sentences in a natural or artificial language (Zadeh, 1975a, p. 201). A fuzzy restriction on the base variable, e.g. long, is characterized by a compatibility function that associates with each value of the base variable a number in the interval [0, 1]. This value represents the variable's compatibility with the fuzzy restriction. The notion of compatibility is distinct from that of probability: The subject should be interpreted as a subjective indication of the extent to which a particular value or expression fits one's conception of the base variable. Zadeh (1981) has extended this idea in the form of a possibility theory. Fundamentals: -- Semantic and syntactic rules -- The use of connectives such as: AND OR NEITHER -- and hedges such as: VERY QUITE MORE_OR_LESS More formally, a linguistic variable is characterized by a quintuple, say [, T(), U, G, M], in which stands for the name of the variable, T() denotes the the term set of , with each value being a fuzzy variable denoted generically by X, and ranging over a universe of discourse U that is associated with the variable u. G is a syntactic rule (usually a grammar) for generating names, X, of values of , and M is a semantic rule for associating with each X its meaning, M(X), which is a fuzzy subset of U (Zadeh, 1975b, 314) A fuzzy subset of A of a universe of discourse U is characterized by a membership function A: U [0, 1], with A(u) representing the grade of membership of u in A. For a more formal definition see Bellman and Zadeh (1970, 5). The support of A is the set of points in U at which A is positive, the height of A is the supremum of A over U, and a crossover point of A is a point in U whose membership in A is 0.5 (Zadeh 1975a, 219) Example: "Arrive in Pittsburgh at approximately 6 pm," can be represented as c(x) = (1 + a | x - 6 | m)-1; a = 2-1, m = 1, = 1 / (1 + a | x - R | m) REFERENCES: AbdelRahman M. (1991) Artificial Intelligence Expands Sensor Applications, EC&M, December, 20-23 Bandler W., Kohout L.J. (1985) Probabilistic Versus Fuzzy Production Rules in Expert Systems, International Journal of Man- Machine Studies, 22:3, 347-353 Bellman R.E., Zadeh L.A. (1970) Decision-Making in a Fuzzy Environment, Management Science, 17:4, B-141 - B-164 Brubaker D.I. (1993) Everything You Always Wanted To Know About Fuzzy Logic, EDN, March 31, 103-106 Conner D. (1993) Designing a Fuzzy-Logic Control System, EDN, March 31, 76-82, 84, 86, 88 Dubois D., Prade H. (1987) Upper and Lower Images of a Fuzzy Set Induced by a Fuzzy Relation -A Fresh Look at Fuzzy Inference and Diagnosis, Technical Report, No. 265, pp. 36-66, Université P. Sabatier, Toulouse, France Fusaro D. (1993) Why All the Fuss About Fuzzy Logic, Control, February, 28-30, 32 Lim M-H., Takefuji Y. (1990) Implementing Fuzzy Rule-Based Systems on Silicon Chips, IEEE Computer, 5:1, 31-45 Lundberg C.G. (1988) On the Structuration of Multi-Activity Task Environments, Environment and Planning A, 20:12, 1603-1621 Roth E.M., Mervis C.B. (1983) Fuzzy Set Theory and Class Inclusion Relations in Semantic Categories, Journal of Verbal Learning and Verbal Behavior, 22:5, 509-525 Togai M., Wang P.P. (1985) Analysis of a Fuzzy Dynamic System and Synthesis of Its Controller, International Journal of Man- Machine Studies, 22:3, 355-363 Zadeh L.A. (1975a) The Concept of a Linguistic Variable and its Application to Approximate Reasoning - I, Information Sciences, 8:3, 199-249 Zadeh L.A. (1975a) The Concept of a Linguistic Variable and its Application to Approximate Reasoning - II, Information Sciences, 8:4, 301-357 Zadeh L.A. (1975a) The Concept of a Linguistic Variable and its Application to Approximate Reasoning - III, Information Sciences, 9:1, 43-80 Zadeh L.A. (1981) Possibility Theory and Soft Data Analysis, in Cobb L., Thrall R.M. (Eds.) Mathematical Frontiers of Social and Policy Sciences, pp. 69-129, Westview Press, Boulder, CO Zadeh L.A. (1982) Fuzzy Probabilities and Their Role in Decision Analysis, Proceedingsof the Fourth MIT/ONR Workshop on Distributed Information and Decision Systems, place and date unknown Zadeh L.A. (1983) A Computational Approach to Fuzzy Quantifiers in Natural Languages, Computation and Mathematics with Applications, 9:1, 149-184 Zadeh L.A. (1983) Linguistic Variables, Approximate Reasoning and Dispositions, Medical Information, 8:3, 173-186 Zadeh L.A. (1983) A Theory of Commonsense Knowledge, Memo- randum No. UCB/ERL M83/26, Electronics Research Labora- tory, College of Engineering, University of California, Berkeley Zadeh L.A. (1983) The Role of Fuzzy Logic in the Management of Uncertainty in Expert Systems, Fuzzy Sets and Systems, 11, pp. 199-227 DUQUESNE UNIVERSITY GRADUATE SCHOOL OF BUSINESS ADMINISTRATION Exercise 642: 1 One of the problems in the battle between human and model (machine) is the problem of LOGIC. Models tend to embrace a form of logic called Boolean; where a variable takes on a precise value only: say 73 for age, 432 miles for distance, and 6 foot 2 for height. A person reasoning about these variables most likely would say "quite old," "about 400 miles," and "tall" or "very tall." Construct a model in LOTUS, SYMPHONY, QUATTRO, EXCEL (or another spreadsheet) that can convert numerical information about a concept given by a person into a linguistic variable. Also, include an option to apply a hedge on the variable.
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