A fuzzy set theoretic framework for knowledge-based simulation

Computers and Industrial Engineering Vol . 25, Nos 1-4, pp . 119-122, 1993

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0360-8352/93$6 .08+0 .00 Copyright © 1993 Pergamon Press Ltd

A Fuzzy Set Theoretic Framework for Knowledge-Based Simulation Steven Hill Rogers and Adedeji B . Badiru, Ph .D. Expert Systems Laboratory The School of Industrial Engineering The University of Oklahoma Norman, OK 73019

Abstract - Complex

systems such as an industrial enterprise operating in a physical, biological, socioeconomic environment generally have many ill defined or imprecisely known parameters which are typically approximated by exact numbers or represented with random variables . Fuzzy set theory is a promising alternative for explicitly treating these aspects of a system . This paper describes a fuzzy set

2. Previous Work A simulation model specification details the goals and requirements for that model. Zeigler [2] reviews the system-theoretic representation of simulation models, emphasizing those models expressed in discrete event simulation languages . This set-theoretic specification is a rigorous basis for model expression, but userfriendly languages which map to the set-theoretic formalism are required to expedite the specification of complex models in program independent form .

theoretic formalism for knowledge-based simulations . A prototype system is being implemented in Mathematica.

1.

Zadeh [3] originated the concept of fuzzy sets as a mechanism for handling imprecise information . He extended classical set theory, in which set membership is a binary function (0 or 1), to allow continuous set membership functions over the real interval [0, 1] . He generalized set operations to this new type of set and later extended fuzzy set theory to include inferencing mechanisms for systems analysis [4] .

Introduction

This work explores the integration of simulation and artificial intelligence techniques into a knowledgebased systems modeling environment . We propose the use of fuzzy set theory as the foundation for specification of systems models, as a method to represent intelligent behavior within simulation models, and for assisting in the interpretation of simulation results .

Elms [5] explored the knowledge engineering aspects of modeling and simulation, summarizing the knowledge representation requirements for modeling and simulation to be generality, consistency, and finiteness . He considered frames to be the most appropriate knowledge representation scheme for modeling and simulation because of their greater generality than such simpler schemes as lists and production rules. The usefulness of frames for knowledge representation in knowledge-based simulation was demonstrated by Doukidis [6]. He implemented an object-oriented discrete event simulation system in Lisp, showing that frames provide a common structure for representing both simulation entities and events .

Fuzzy set theory and the fuzzy logic derived from it [1] provide a mechanism for approximate, human-like reasoning . Such reasoning is more flexible and robust than reasoning based on non-fuzzy, or crisp, logic when subjected to imprecise, real-world inputs . This paper describes the application fuzzy set theory in the development of a prototype knowledge-based simulation environment. This approach is particularly relevant to the design and operation of modem industrial facilities when environmental risk is one of the performance criteria . The environmental risks associated with the industrial operations have been receiving increasing attention in recent years . Risk estimation is a complex process in which many uncertain and ambiguous factors must be considered . There are few tools available to assist the decision maker with this task and a knowledge-based simulation environment based upon a fuzzy set theory would significantly advance the state-of-the-art for environmental risk assessment

As an example of how AI techniques might improve the usefulness of simulation output, Hutrion [7] described the integration of an Artificial Neural Network into a simulation model Model input variables such as resource levels and work load and model output in the form of the mean and variance of the time requited to complete a day's work were used to train the neural network . The trained network was

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able to predict simulation output based upon input values with a high level of accuracy for conditions not in its training set . This allowed queries about the system to be answered immediately without making additional simulation urns, which might take hours or even days .

3 . Approach 3 .1. Overview This research explores the use of fuzzy set theory as a vehicle for improving the fidelity and usability of computer simulation models . Potential applications of fuzzy sets and logic in knowledge-based simulation are : 1) Declarative specification of simulation using rules formulated with linguistic variables; 2) Fuzzy logic to emulate the behavior of intelligent entities within simulations; 3) Interpretation of simulation results in linguistic terms; and

imprecise values such as : "LARGE" "SMALL" "NEAR" and "FAR". Fuzzy linguistic variables map linguistic concepts to sets of numeric values with membership functions. They represent imprecise quantitative concepts . Membership functions define a mapping from the domain of interest to the continuous range of 0 to 1 . While arbitrary functional forms may be used, triangular fuzzy numbers (T.F.N.) and trapezoidal fuzzy numbers (TrF.N .) are often used for simplicity. A fuzzy number is a fuzzy subset of the real number line. Standard algebraic operations extend to cover Tr.F.N .s . Addition and subtraction of two Tr .F.N.s yield a Tr .F.N ., but multiplication, division, maximum, and minimum do not necessarily yield a Tr.F.N. However, the result may be closely approximated by a Tr.F.N. Fuzzy logic, or continuous valued logic, allows not just "True" or 'False", but various degrees of "Maybe" . Logical AND for fuzzy sets is the minimum over their respective membership functions, illustrated below by the shaded area Fuzzy AND

4) Learning and fuzzy inferencing about model behavior. Fuzzy set theory is used to generalize Zeiglees classical set theoretic formalism for the specification of simulation models. A significant weakness of Zeigler's formalism is that his set theoretic formulation is not user friendly and must be translated to other forms for most users. A fuzzy set theoretic formulation allows the use of declarative rules and linguistic variables, enhancing the comprehensibility of the model specification . Directed graphs provide an alternate representation of the system for additional insight and model validation /verification . A prototype simulation system is being developed to investigate the feasibility of the proposed fomtalism . This system will be used to address problems in the transformation of a declarative, near natural language specification to a machine executable form and evaluate fuzzy inferencing as a model execution mechanism to supplement conventional stochastic discrete event simulation.

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3.2 . Fuzzy Modeling Classical or "Crisp" sets have a binary membership function, i .e . an object is either a member of a set or it isn't. Fuzzy sets have a graded membership function taking with values on the continuous interval 0 to 1 . Fuzzy linguistic variables provide a medium for computer manipulation of poorly bounded human concepts . They allow the systematic manipulation of

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HILL Rooaas and BenIRU : Fuzzy Set Theoretic Framework

Fuzzy relations are fuzzy subsets showing how different domains relate to each other . They may be represented by a relation matrix . For example, a "Resemblance" relation for two sets of cars might be described by the following matrix :

This research will utilize MathemoAca [8] which, in addition to being a powerful system for doing both symbolic and numeric mathematics by computer, has an embedded programming language supporting the rule based, functional, and object-oriented paradigms . Mathematica has a LISP-like syntax making it more verbose than APL or I, but the resulting code is more readable as a result. An example of the Marlumadca's power is a short program to calculate the mean of a list of numbers.

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IMean[lisLList]:=Apply[Plus, list]/Length[list] For a fuzzy input x and fuzzy relation R . the compositional role of inference may be represented using Mathematica's generalized inner product function with the following expression.

In classical propositional calculus, inferences are expressed by expressions such as IF A THEN B

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where A and B are propositional variables . In fuzzy conditional statements, A and B become fuzzy sets . In the compositional rule of inference, IF-THEN rules may be represented by fuzzy relations . Let R be a fuzzy relation mapping U to V, if x is fuzzy subset of U, then y is the fuzzy subset of V defined by R . This may be expressed by y =x ° R, or

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0 .8 0 .9 0.2 [0 .2 1 .0 0.31 ° 0.6 1 .0 0 .4 - [0.6 1 .0 0.41 0 .5 0 .8 1 .0 3 .3. Methodology Development The proposed fuzzy modeling system will support multiple representations of a system model, including near natural language production rules and directed graphs, to allow the modeler to utilize the most convenient mode for a particular phase of model creation, validation, or use. The system will provide facilities for creating and modifying fuzzy linguistic variables and their associated adjectives, production title bases, and a network representation. The modeler/user should be able to easily switch between the different modalities as required . Queries about system behavior may be made in a natural language formal, with the system prompting the user for additional information as required

Inner[Min, x, R, Maxi The prototype simulation environment will be implemented as a shell on top of Mathematica, allowing the modeler access to Marhematica's extensive analysis and graphical capabilities . This simulation environment will be called PRISM for Package foR Intelligent Simulation in Mathenwrica. PRISM's architecture is illustrated below .

The user interface is based upon the Mathematica notebook paradigm which combines text, graphics (which may be animated), and executable code. The production ode model representation encodes the system description as a knowledge base of facts and rules while the network provides an equivalent visual representation . The experimental frame contains experimental conditions for model execution . The libraries contain model components and complete models for use in constructing new models . The simulation engine executes crisp simulation models and the fuzzy inferencing engine executes fuzzy models . The fuzzy learning module constructs rules from either real world or simulated data . A simulation model could be run under various conditions defined in one or rare experimental frames while the learning module observed the input parameters and the associated output data to "learn" the model behavior. The fuzzy inferencing engine could then make inferences about model behavior based upon new input parameters without the necessity of running additional simulations.

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Proceedings of the 15th Annual Conference on Computers and Industrial Engineering

PRISM Architecture NeMOrk MXlel Represertaion

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4. Conclusion

The proposed methodology will facilitate the specification, development, validation and use of knowledge-based simulation models by a variety of users. This methodology will allow the formal model to be represented in terms of natural language phrased as fuzzy production rules . This is especially important when simulation models are to be used as a decision support tool for complex problem areas such as the environmental risk assessment of industrial systems where models must have credibility with decision makers and the public who are not simulation specialists.

5. References

1 . Zadeh, L. A., 1992, "The Calculus of Fuzzy lumen Rules", Al Expert, March 1992, pp . 23-27 . 2 . Zeigler, Bernard P ., 1984, "System-Theoretic Representation of Simulation Models", HE TRANSACTIONS, Vol . 16, No . 1, pp . 19-34. 3 . Zadeh, L. A ., 1965, "Fuzzy Sets", Information and Control, Vol. 8, pp . 338-353. 4. Zadeh, L . A ., Jan 1973 . "Outline of a New Approach to the Analysis of Complex Systems and Decision Processes," IEEE Trans . Syst., Man, Cybem ., Vol . SMC-3, No. 1, pp. 28-44.

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5. Elms, M. S ., 1986, "1Le Applicability of Artificial Intelligence Techniques to Knowledge Representation in Modeling and Simulation", in Modeling and Simulation Methodology in the Artificial Intelligence Era, Eds M . S . Elm, T. I. Oren, and B . P . Zeigler, New York, Elsevier Science Publishers B . V. (NorthHolland), pp. 19-40 . 6. Doukidis, O . I ., 1988, "Using Lisp for Developing Discrete Event Simulation Models", in Artificial Intelligence, Expert Systems, and Languages in Modeling and Simulation, Eds C. A . Kulikowski, R. M. Huber, and O. A. Ferrate', New York, Elsevier Science Publishers B .V . (North-Holland), pp. 31-42. 7 . Hurrion, R . D., 1992, "Using a Neural Network to Enhance the Decision Making Quality of a Visual Interactive Simulation Model", J. Opl. Res . Soc ., Vol. 43, No . 4, pp . 333-341 . 8 . Wolfram Research, Inc., 1992, Mathematics, Version 2.1, Champaign, Illinois .