## Incentive Games for Neuro-Fuzzy Control

A. Mete Cakmakci and Can Isik
Electrical and Computer Engineering
Syracuse University
Syracuse, NY 13244

In this paper we introduce a two-level modular neuro fuzzy network based on incentive games where the modules are organized as autonomous local optimizers in a leader-follower game hierarchy. Incentive-reaction pairs are used as a measure for the capacity and responsiveness assessment of each follower module. Learning within the follower modules is performed in a traditional error-based manner (e.g., backpropagation) The allocation of targets and incentives to each follower module , on the other hand is independent of connection weights; incentive games are used for that purpose. Two important advantages of the new architecture are its physically significant follower module outputs and the context-based enhancement it makes to backpropagation.

Keywords: neural fuzzy systems, incentive games, system identification

## An Interval Fuzzy Truth-Value Approach to the Generalized Modus Ponens

John Yen and Bogju Lee
Department of Computer Science
Texas A&M University
FAX:(409)-847-8578
yen@cs.tamu.edu

One of the desirable properties of generalized modus ponens in fuzzy logic is
X is A----> Y is B
X is not A
---------------------
Y is unknown

From a truth-value viewpoint, unknown can be represented as the interval [0,1]. However, most existing fuzzy reasoning mechanisms use the truth value 1 to represent unknown''. In this paper, we propose a new approach to fuzzy reasoning based on a possible world approach. We show that our approach obtains the desired interval truth value [0, 1] when the known proposition is the negation of the antecedent. We also compare our approach to Baldwin's and Godo's approaches and discuss the impact of the proposed approach to other desired properties of generalized modus ponens and generalized modus tollens.

## Fuzzy Phase Compensation Algorithm and Its Application

Fangjun Wang
Shanghai Instrumentation Professional Institute
62 Xu Wen Road
Shanghai, China

Charlie Dou
Department of Mathematics,Physical Science and Engineering Technology
West Texes A&U University
WT Box 787
Canyon,TX 79016

Traditional fuzzy logic inference and control algorithm can not be used effectively in phaserelated mechanical velocity control since fuzzy logic is essentially a trail strategy which is based upon observation from the next system response.Experimental results and process applications show that this algorithm is not suitable for fast-reaction system.This report will discuss a modified fuzzy inference/control algorithm - lead and lag fuzzy logic compensation algorithm,which uses lead and lag algorithm to compensate phase error.This algorithm was used on multiple-function food-packing lines successfully.It can trace the phase correctly and reduce phase accumulate-error within an accpetable degree.

## Complex System Inference-Control and Fuzzy Logic Modeling

Charlie Dou
Department of Mathematics,Physical Science and Engineering Technology
West Texas A&M University

J.A Macedo
Department of Industrial Engineering
Texas Tech University

This report systematically reviewed the historical development,theoretical research and practical application of the fuzzy logic inference-control modeling (FLM) for the complex systems- multiple-input-multiple-output (MIMO) system and nonlinear system.The most common and important issues related with fuzzy models: self- organizing FLM,adaptive flm,general purpose FLM,Mamdani-FLM and Tagagi-Sugeno-Kang FLM,fuzzy vector-spaces,hierarchical structured FLM,and fuzzy phase-plane,were exhibited,analyzed,or compared.The survey reveal that the design of inference and control system,especial for MIMO systems and nonlinear systems with uncertainty,can be considered through multiple scenarios,not just thrir rigid dynamic mathematical models but also fuzzy logic models.

## Quality Assurance and Control (QAC) of Uncertainty Models and Simulations of Life Cycle Assessment (LCA)

Odd Andreas Asbjornsen
SMART International Corporation
P.O. Box 8057 Moholt, N-7002 Trondheim, Norway

The Life Cycle Analysis in Systems Engineering and Environmental Engineering is associated with large uncertainties, being of a forecasting, or predictive nature. Various approaches are used to estimate trends in the Life Cycle, either by probabilistic nature or by fuzzy variables. Regardless approach, the models are used for decision support and need to be assured and controlled for their quality in describing the real situation. The paper draws attention to this problem, and suggests methods for cross-checking simulation models, by various methods based on underlying fundamental principles, asymptotic conditions, average properties, submodel interrelations, etc. A generic Systems Engineering Principles approach is used in developing the model of the Life Cycle. This approach is based on analyses and justifications of needs and a definition of requirements to the model and its simulation result.

## Developing Robust System Designs under Uncertainty: A Systematic Approach to Design Evaluation

Chunming Duan
Department of Industrial and Systems Engineering
National University of Singapore
Kent Ridge, Singapore 0511
Internet: isedcm@nus.sg

To develop quality systems, both engineering and management require fundamental principles and methodologies to guide design decision making and advanced planning. This paper presents a systematic approach to support the need for developing better system designs in the face of uncertainties. This approach is developed by integrating statistical decision theory, elements of the systems engineering process, and Taguchi's philosophy of robust design. The result is a structured, systematic methodology for evaluating system design alternatives.

Keywords: Design Evaluation, Robust Design, Systems Analysis, Decision Analysis, Uncertainty Analysis, Design Optimization

## Meaning Acquisition by SCIPS

Burghard B. Rieger
FbII: Dept. of Computational Linguistics
University of Trier
D-54286 TRIER, Germany

The emergence of semantic structure as a self-organizing process is studied in Semiotic Cognitive Information Processing Systems on the basis of word usage regularities in natural language discourse whose linearly agglomerative (syntagmatic) and whose selectively interchangeable (paradigmatic) constraints are exploited by text analysing algorithms. They accept natural language discourse as input and produce a vector space structure as output which may be interpreted as an internal (endo) representation of the SCIP system's states of adaptation to the external (exo) structures of its environment as mediated by the discourse processed. In order to evaluate the sytem's endo-representation against the exo-view of its environment as described by the natural language discourse processed, a corpus of texts -- composed of correct and true sentences with well-defined referential meanings -- was generated according to a (very simple) phrase structure grammar and a fuzzy referential semantics which interpret simple composite predicates of cores (like: on the left, in front, etc.) and hedges (like: extremely nearby, very faraway, etc.). Processed during the system's training phase, the corpus reveals structural constraints which the system's hidden structures or internal meaning representations apparently reflect. The system's architecture is a two-level consecutive mapping of distributed representations of systems of (fuzzy) linguistic entities whose states acquire symbolic functions that can be equaled to (basal) referencial predicates. Test results from an experimental setting with varying fuzzy interpretations of hedges are produced to illustrate the SCIP system's miniature (cognitive) language understanding and meaning acquisition capacity without any initial explicit syntactic and semantic knowledge.

## Weighted Fuzzy Mean Filters for Heavy-Tailed Noise Removal

Chang-Shing Lee and Yau-Hwang Kuo
Institute of Information Engineering
National Cheng-Kung University, Tainan, Taiwan, R.O.C.

Pau-Ta Yu
Institute of Computer Science and Information Engineering
National Chung-Cheng University, Chiayi, Taiwan, R.O.C.
(leecs@cad1.iie.ncku.edu.tw) and (kuoyh@cad1.iie.ncku.edu.tw)

A new fuzzy filter, called Weighted Fuzzy Mean (WFM) filter is proposed and analyzed in this paper. The WFM filter is powerful for removing heavy additive impulse noises from images. By the filtering of each WFM filter, the filtered output signal is the mean value of the corrupted signals in a sample matrix, and these signals are weighted respectively by a membership grade of an associated fuzzy number stored in a knowledge base. The knowledge base contains a set of fuzzy numbers decided by experts or derived from the histogram of referred image. When the probability of occurrence of mixed impulse noises is over 0.3, the WFM filter can recover the noise- corrupted image quite well in contrast with the conventional filters, for examples, the median filters, RCRS, WOS, CWM, an stack filters, based on the Mean Absolute Error (MAE) and Mean Square Error (MSE) criteria. Besides, on the subjective evaluation of filtered images, the WFM filter also results in a higher quality of global restoration.

Keywords: Weighted Fuzzy Mean Filter, Impulse Noise, Fuzzy Number, Knowledge Base, Histogram, Fuzzy Estimator

## Fuzzy Scheduling of Robots in CIM Environment

E. V. Levner
School of Business Administration
Hebrew University of Jerusalem
Jerusalem, ISRAEL 91905

V. B. Kats
Department of Industrial Engineering
Ben Gurion University of the Negev
Beer Sheva, ISRAEL 84105

L. K. Meyzin
Department of Computer Systems
Holon Center for Technological Education
Holon, ISRAEL 58102

A problem of scheduling a transportation robot in a production line with uncertain input data is considered. The data are presented by fuzzy numbers. A heuristic algorithm based on operations over the fuzzy numbers is developed. In the case of exact input data the algorithm produces an exact solution, its complexity being qubic in the number of workstations in the line. The fuzzy algorithm has been tested on real-life problems, and turned out to work better than the random search and FIFO-type heuristics.

Keywords: Scheduling, Heuristics, Fuzzy Algorithms

## Reliability Assessment of Pile-Founded Navigation Structures

Robert C. Patev
Information Technology Laboratory
US Army Engineer Waterways Experiment Station
3909 Halls Ferry Road, Vicksburg, Mississippi, 39180-6199

The US Army Corps of Engineers (USACE) has designed pile-founded navigation structures to provide a consistently high level of safety and serviceability. This has been achieved by utilizing allowable values for pile stresses, pile load capacity, and deflections in their design. The reliability assessment of older in-service pile-founded navigation structures can be developed using the same limit states used in the design of new pile-founded structures. This paper will be to examine the reliability assessment of timber pile-founded navigation structures without the loss of support. The techniques developed to perform the reliability assessment employ a capacity/demand relationship for limit states of axial capacity, lateral deflections, axial deflections, and combined bending. Monte Carlo Simulations and First Order Second Moment techniques were utilized to calibrate the limit states to an instrumented pile load test data from field tests. An example of a river dam monolith subjected to impact loading will also be discussed.

Keywords: Reliability assessment, pile foundations, navigation structures, timber piles, dams

## Modeling Temporal Uncertainty in Microprocessor Systems

S. M. Yuen and K. P. Lam
The Chinese University of Hong Kong
Fax: (852) 2603-5505
smyuen@se.cuhk.hk
kplam@se.cuhk.hk

In microprocessor system diagnosis, temporal reasoning of event changes occurring at imprecisely known time instants is an important issue. The time range approach was proposed to capture the notion of time imprecision in event occurrence. According to this concept, efficient time range constraint reasoning techniques were developed for embedding domain knowledge in a deep-level constraint model. The imprecision in these events contributes to a certain degree of uncertainty in the correctness of a microprocessor system operation. A knowledge-based diagnostic system for microprocessor systems design was designed and developed. The system performs worst-case timing analysis. In particular, for the asynchronous bus operation of the MC68000 microprocessor, the sequence of events during a read cycle was traced through an inference process to determine if any constraint in the model was violated. Although satisfactory results were obtained, the possibility measures implicitly embedded within time ranges were not properly quantified for effective temporal reasoning. To overcome this shortcoming, the fuzzy time point model is proposed. The original time-range representation, specified by two crisp interval end-points, is replaced by the fuzzy time point representation that is specified by a single fuzzy value. The degree of fuzziness of a fuzzy time point has dependency on the functional specification of the corresponding timing parameter. The use of simplistic assumptions on the fuzzy time point model has been shown to enhance the deductive capability of the existing time-range models. The implementation of this system extension has shown promising results in fuzzy time point reasoning.

Keywords: Temporal uncertainty modeling, fuzzy time point reasoning, microprocessor systems diagnosis, knowledge-based systems

## On Fuzzy Decision Processes with Discounted Fuzzy Rewards

Yuji Yoshida
Faculty of Economics and Business Administration
Kitakyushu University
4-2-1, Kitagata, Kokuraminam
Kitakyushu 802, Japan

We deal with a multi-stage decision process with fuzzy transitions, which is termed fuzzy decision process. We consider the fuzzy decision process, where both state and action are assumed to be fuzzy, from a point of view of a dynamic fuzzy system which has been developed by the authors. The discounted total reward is described by a fuzzy number on a closed bounded interval, and a partial order of convex fuzzy numbers, which is called a fuzzy max order, is used to discuss the optimization problem. We characterize the discounted total reward associated with an admissible stationary policy by a unique fixed point of the contractive mapping. Further, we estimate the fuzzy rewards, by introducing a fuzzy expectation generated by a fuzzy goal.

## Fuzziness in Classical Two-Valued Logic

Mark Changizi
University of Maryland
College Park, MD 20742
changizi@wam.umd.edu

Classical two-valued logic has been thought to be inadequate as a logic of fuzzy predicates, because the interpretation of a predicate in classical two-valued logic is just a set of objects, and a set is a precise mathematical entity with nothing fuzzy about it. The aim of this paper is to show that fuzziness may, after all, be successfully modelled in classical two-valued logic via (i) the introduction of a notion of weak negation, where natural language occurrences of not' are not necessarily translated into logic as \neg', and (ii) giving fuzzy predicates recursively enumerable but not recursive interpretations. The model developed here is computer-implementable and well-motivated.

Keywords: Fuzzy, vague, classical logic, two-valued logic

## Distributed Fuzzy Control of Communications

Aivars Celmins
U.S. Army Research Laboratory
Aberdeen Proving Ground, MD 21005-5067
e-mail: celms@arl.mil

This paper deals with the control of battlefield radio communications. The particular properties of battlefield communications in modern warfare make a distributed access control highly desirable. Inputs for such a control are approximate descriptions of the status of the network.The control rules are heuristic because the controlled process does not have a set point.On the other hand, the network nodes have ample computing power that can be used to analyze the input and exercise access control algorithms. These circumstances suggest the use of fuzzy-logic control procedures.The author has developed such control procedures and tested their behavior on a computer model of battlefield communications.

Keywords: radio communications; battlefield communications; distributed control; fuzzy-logic control

## Fuzzy Logic Control of Batch-Feeding Refuse Incineration

Desong Chen
Department of Electrical Engineering
National University of Singapore
10 Kent Ridge Crescent, Singapore 0511

Municipal Solid Waste (MSW) or refuse incineration plant is designated to reduce the volume of the refuse and recover the energy from it. The steam generated from a boiler heated by burning refuse is sent to a turbine to generate the electricity. Batch-feeding style and the uncertainty of refuse composites result in periodical and uncertain fluctuation of refuse combustion and then the steam flow rate. The fluctuation could lead to the unsteady generation of the electricity. Moreover, the incomplete combustion of the refuse is often observed. This leads to the poor efficiency of the refuse volume reduction and the energy recovery. Therefore, the proper controlled combustion of the refuse will be of both environmental and commercial significance. In this paper, we explore the problems of refuse combustion control and discuss how rule-base fuzzy logic control algorithms can be used to damp the fluctuation of the steam flow rate and achieve more complete combustion of the refuse by properly adjusting the grate rotating rates. The extensive simulations based on the data and information from the Ulu Pandan refuse incineration plant, Singapore, show the effectiveness of the rule-base fuzzy logic controller and indicate that the proposed control algorithm has potential of about 10% of increase on the capacity of refuse processing and electricity generation.

Keywords:Waste Treatment, Fuzzy Logic Control, Combustion

## Guard Heuristic by Dynamic Fuzzy Reasoning Model for Chinese Chess

Mei-En Chen and Yo-Ping Huang
Dept. of Computer Science & Engineering
Tatung Institute of Technology
Taipei, Taiwan 104, R.O.C.

This paper provides a novel Chinese Chess model which is based on fuzzy cognitive map (FCM) and rule-based system for reasoning a best legal move. A 10*9 guard matrix is generated by FCM to show the status of each intersection. Each entry in the matrix is called a guard value which indicates how an intersection is guarded by the player's side or attacked by the opponent's. Based on the proposed model, not only the legitimacy of the next move can be improved but also the decision time is much shorter than the game tree searching method of alpha-beta model. An example is given to verify the effectiveness of the proposed methodology.

Keywords: Guard heuristic, fuzzy reasoning, fuzzy cognitive map, guard matrix, Chinese chess

## Rules Discovery in Fuzzy Relational Databases

J.C. Cubero, J.M. Medina, O. Pons, and M.A. Vila
Dept. of Computer Science and Artificial Intelligence
University of Granada. 18071 Granada, Spain.
e-mail: carlos@robinson.ugr.es

In this paper we are concerned with the problem of defining a fuzzy dependency in the framework of a fuzzy relational database. Of primary interest is the development of a fuzzy algebra operator called RULE which extracts the information included in the data of an original relation r. This is a kind of projection which allows us to store that information in a separated relation RULE(r), with fewer tuples.

This process will be done whenever it is possible to detect a 'rules based fuzzy functional dependency'. In order to define it we will need to extend a resemblance relation defined for crisp data to the fuzzy case. It will lead us to define a weak resemblance used to compare antecedent values and a strong resemblance for consequent values. We show how can we detect such dependency through two equivalent approaches: once constructing first relation RULE(r), and another one, working directly with the tuples appearing in r.

Keywords: Fuzzy Databases, Fuzzy Dependencies, Fuzzy Rules.

## Fuzzy DSS Enhance Congitive Support for Strategic Planning in an Uncertain Environment

Nancy Green Hall
Dept. of Decision Sciences, Coles School of Business
Marietta, GA 30061
Kennesaw State College
nhall@kscmail.kennesaw.edu

Recent studies of DSS have raised serious questions regarding the conflict between the ideal of DSS as aids to ill-structured problem-solving and the reality of DSS implementations which tend to focus on moderately- to well-structured problems. Lack of tools to aid certain types of cognitive effort relates directly to this concern. An additional worry is that incorporating inappropriate cognitive tools may do more harm than good. The potential for distortion or miscarriage of the decision process is great in the area of strategic planning. Supporting symbolic representation and reasoning for this area with tools involving fuzzy sets technology has worked very well for Carlssen and Walden with the Finish forest products industry. Augmenting the decision support toolkit with fuzzy-set-theoretic tools as needed can open the door to more successful design and implementation of DSS.

Keywords: fuzzy decision aids, fuzzy DSS, cognitive processing, strategic planning

## Exact Solutions for Interacting Rules Using Conjunctive Logic

Thomas Whalen
Department of Decision Sciences
Georgia State University Atlanta, Georgia 30303-3083
dscthw@gsusgi2.gsu.edu

In this paper, I will derive an exact solution for the membership function of the output of a fuzzy system consisting of n Mamdani-type rules. The consequents of the rules are triangular fuzzy sets that are evenly spaced on a univariate universe of discourse. All the consequents have the same support width, d, which determines the degree to which the consequents overlap. The derivation makes no assumptions about the rule antecedents, since the fuzzy output is expressed as a function of the degree to which each rule is satisfied. Using this function, I derive an exact solution for the defuzzified output of the system using the centroid defuzzification procedure. Finally, a numerical example involving four rules with two input variables illustrates a preliminary investigation into the effect of varying the support width of the consequent fuzzy sets.

## Qualitative and Quantitative Indices for Simulation Systems in Distributed Interactive Simulation

Pamela McCauley Bell
Department of Industrial Engineering & Management Systems
University of Central Florida

Rhonda Freeman
Institute for Simulation and Training
University of Central Florida

The Fidelity Measurement question in Distributed Interactive Simulation (DIS) needs to be answered. It's resolution is crucial to defining and resolving the interoperability issue of "How to compare and contrast simulations of varying fidelities on a quantifiable basis"?" Once such a measure of fidelity is modeled, it would allow high-fidelity simulators to play fairly with low fidelity simulations. The Fidelity Subgroup of the DIS Standards Development Project has developed a Draft Fidelity Description Requirement (FDR) Document, in an hierarchical taxonomy format. This framework provides an object-oriented, modular referent to model. The input and output descriptors are defined verbally, linguistically. Recent research would suggest this linguistic base can easily be transformed into quantifiable measures, indices. This paper provides a methodology for providing quantifiable Fidelity Indices; through the application of Fuzzy Logic.

## Idempotent closing and opening operations in fuzzy mathematical morphology

Bernard De Baets
Department of Applied Mathematics and Computer Science
University of Gent, Krijgslaan 281 (S9), B-9000 Gent, Belgium

In this paper, a logical approach to the fuzzification of binary mathematical morphology is presented. Fuzzy dilation and fuzzy erosion are introduced independently, using the logical operators conjunctor' and implicator'. In this way, duality relationships are not forced from the very beginning. It is shown that by choosing suitable logical operators, all classical duality and other relationships can be preserved. Following a similar line of reasoning, it is possible to obtain the idempotence of the fuzzy closing and fuzzy opening. This important result leads to the introduction of the concepts of B-open and B-closed fuzzy objects. Fundamental classical theorems are generalized for the minimum operator and its residual implicator, and for the Lukasiewicz t-norm and its residual implicator.

## Tail Effects of Uncertainty Modeling in QRA

Marc A. Maes and Luc Huyse
Dept. of Civil Engineering
University of Calgary
Calgary, Alberta, Canada, T2N 1N4

The paper discusses the general feeling of uneasiness engineers and scientists experience when they assume probability models for basic uncertainties governing a risk or reliability analysis. Specifically, the aspect of tail behavior is addressed. It is well known that quantitative risk estimation is critically dependent on the tail behavior. The present paper focuses on tail heaviness and the effect of tail heaviness of basic uncertainties on the overall risk or failure probability. This is of interest to a model-based risk analysis as well as a structural reliability analysis. Two examples are included to illustrate some of the concepts introduced in this paper.

## An Adaptive Rule Extraction with the Fuzzy Self-Organizing Map and a Comparison with Other Methods

Tatsuya Nomura
Human Information Information Processing Laboratories, ATR
2-2, Hikaridai, Seika-cho, Soraku-gun, Kyoro 619-02 Japan

Tsutomu Miyoshi
Information Technology Research Laboratory, Sharp Corporation
2613-1, Ichinomoto-cho, Tenri, Nara 632 Japan

For automatic rule extraction from a set of input-output data examples, decision tree generating methods such as ID3 and Fuzzy ID3 play a mejor role. These methods, however, are difficult to apply when there is a tendency for the examples to change dynamically. This paper presents a new method for adaptive rule extraction with the Fuzzy Self-Organizing Map and the results of simulations to present the effectiveness by a comparison with other methods such as RBF and GA. We got the result that our methodd is superior to other methods for automatic and adaptive rule extraction.

## Fuzzy Logic, Grasp Preshaping for Robot Hands

Kubilay Kaan Aydin
Department of Industrial Engineering
Middle East Technical University, Ankara, 06531, Turkiye
e-mail: aydin@rorqual.cc.metu.edu.tr

Selection of an appropriate hand shape and orientation for grasp preshaping depends on the geometry and physical properties of the target object, along with information on the goal of the task. Features of the target object and the task are mapped onto various grasp attributes. Preshaping of the robot hand is done according to the results of this mapping. A fuzzy logic grasp preshaping methodology is presented in this paper.

## Combinability Functions in Defuzzification

David Swiatlowski
Syracuse Research Corporation
Merill Lane
Syracuse, NY 13210

Can Isik
Electrical and Computer Eng.
Syracuse University
Syracuse, NY 13244-1240
Fax(315)443 2583

In this paper we overview various knowledge-based defuzzification functions from a system modeling and controls perspective, and introduce a new variant which is useful in applications where fuzzy variables are related to deviations from a norm, such as feedback control. We also summarize results of simulation experiments.

Keywords: fuzzy sets, defuzzification, fuzzy control

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

519 Ordering Fuzzy Sets Generated by a Neural Network Algorithm

Les M. Sztandera
Philadelphia College of Textiles and Science
fax: 215-951-2615
sztandera@hardy.texsci.edu
http://larry.texsci.edu/les2.html

Decision making is a process of selecting an optimal course of action from the available alternatives. Ordering fuzzy subsets is an important event in dealing with fuzzy decision problems in many areas. This issue has been of concern for many researchers over the years. This is obvious due to the extensive number of papers published in the last fifteen years dealing with ordering. Some twenty eight methods have been proposed in the publications for ranking fuzzy subsets.

Also, in the last several years, there has been a large and energetic upswing in neuroengineering research efforts aimed at synthesizing fuzzy logic with computational neural networks. The two technologies often complement each other: neural networks supply the brute force necessary to accommodate and interpret large amounts of sensor data and fuzzy logic provides a structural framework that utilizes and exploits these low level results. We have the ways to use either technology as a "tool" within the framework of a model based on the other. As a neural network is well known for its ability to represent functions, and the basis of every fuzzy model is the membership function, so the natural application of neural networks in fuzzy models has emerged to provide good approximations to the membership functions that are essential to the success of the fuzzyapproach. It is the purpose of this paper to evaluate and analyze the performance of available methods of ranking fuzzy subsets on a set of selected examples that cover possible situations we might encounter as defining fuzzy subsets at each node of a neural network. Through this analysis, suggestions as to which methods have better performance for utilization in neural network architectures, as well as criteria for choosing an appropriate method for ranking will be made.

## Determination of Economic Systems Behaviour under Uncertainty

A. I. Serguieva
Technical University-Sofia, 1756 Sofia, Bulgaria
E-mail: ais@bgtus4.vmei.acad.bg

This paper discuses systems of difference equations with fuzzy parameters and presents some solution procedures with the purpose to study the dynamic behaviour of economic systems in case of uncertainty. The trajectories of the endogenous variables are evaluated firstly at contiguous moments of time, and then, simultaneously. The relations between different solutions are shown. We also provide a more informative crisp solution (than the ordinary one) for each level of uncertainty, in case the economic system authorities have no enough time to consider all possibilities. Finally, the author consider essential to provide an algorithm for computing the exact (-cuts of the obtained fuzzy solutions.

Keywords: Econometrics; dynamics; systems of difference equations; nonlinear programming

## Fuzzy Systems Approximation by Wavelet Packets - SISO Case

Armin Shmilovici and Oded Maimon

A Common problem in the design of fuzzy systems, is the selection of the shape and the number of fuzzy sets. Unlike the iterative and computationaly intensive Neuro-Fuzzy optimization methods, we propose the use of a one pass, suboptimal method. The idea is based on the wavelet packet decomposition method, in which a function is expanded into basis functions in a recursive way. An information cost tree is built which tracks the various expansion possibilities. A cost norm is used to prune the tree as a tradeoff between complexity and accuracy. The result is a suboptimal fuzzy system. the algorithm is O(MlogM).

Keywords: Fuzzy systems, Spline wavelets, wavelet transform

## Genetic Algorithm Based Redundancy Resolution of Robot Manipulators

Kubilay Kaan Aydin and Erol Kocaoglan
Department of Industrial Engineering
Middle East Technical University, Ankara, 06531, TURKIYE,
e-mail: aydin@rorqual.cc.metu.edu.tr

This paper presents a genetic algorithm based approach to redundancy resolution of robot manipulators using self-motion topology knowledge. The genetic algorithm presented can work under joint limits and produces end-effector positions with negligible error. Any solution determined by the genetic algorithm is physically realizable, as demonstrated on a PUMA 700 robot manipulator which is configured as a redundant positional manipulator.

## Fuzzy System Representation of Car-Following Phenomena

Mark Luszcz
McCormick, Taylor and Associates, Inc.

Shinya Kikuchi
Univeristy of Delaware
kikuchi@ce.udel.edu

Understanding the way vehicles interact with one another within a traffic stream is an important topic in transportation engineering. By developing accurate representations of these interactions we can improve highway safety, control measures and efficiency of operations. Further understanding of vehicle interaction is important for the development of advanced vehicle control mechanism. The specific situation studied is when a driver is following another vehicle because his/her speed is constrained by the vehicle ahead, (traveling at a speed less than what the driver wish), and roadway or traffic conditions that do not allow passing. This is termed a car-following situation.

Car-following phenomena have traditionally been modeled as a stability problem using deterministic methods. Such approach does not take the inherent vagueness and uncertainty of the driver's decision process into account. When a human is controlling a vehicle in a congested traffic stream, he/she does not perceive and react to the situation in the exact manner assumed by traditional methods. Instead, the conditions at any point in time are perceived in terms of linguistic variables (the headway is a bit long, or the relative speed is about zero). Then, the driver processes that information through an approximate set of rules (e.g., if the vehicle ahead is closing too quickly, slow down) and finally performs some vague action (decelerate somewhat, accelerate hard). In all parts of the perception, decision and reaction process, the relevant variables contain a degree of uncertainty and a model of this process should include. Fuzzy inference based control systems can explicitly model this uncertainty.

This paper examines the potential of fuzzy set theory for representing the car-following phenomena and its application to platoon operation. The proposed model has simplified the model developed first time by Chakroborty by using 36 rules (as compared to 396 used by Chakroborty). The proposed model directly addresses the fact that the distance headway maintained between vehicles is a function of speed by utilizing empirical speed-headway data. This allows the model to maintain an accurate headway at any speed. An additional advantage of using this speed-headway data is that drivers with different personalities may be simulated (i.e., aggressive, moderate, or conservative).

The major focus of the proposed model is to simulate the propagation of vehicle-to-vehicle interaction along a platoon of vehicles. The model results are compared the actual driver behavior observed on the highway using a test vehicle. The model allows analysis of the asymptotic stability of the platoon under various conditions, such as different types of drivers are in the platoon )aggressive and conservative). The proposed model also performed well in the examination of emergency conditions and traffic signal release and stopping situations.

## The Lindley Paradox, Information and Generalized Functions

K. Breitung
Dept. of Civil Engineering
University of Calgary
Calgary, Alberta, Canada, T2N 1N4

In testing a sharp point hypothesis there is a difference between frequentist and Bayesian results. Even for sample sizes increasing to infinity, Bayesian methods accept the point null hypothesis for values where the frequentist method leads to rejection. This is called the Lindley paradox. Here it is attempted to explain this. The reason appears to be not a specific feature of Bayesian methods, but a misuse of the theorem of Bayes.

## Uncertainties in Loads for Ships Structures

Khaled I. Atua and Bilal M. Ayyub
Department of Civil Engineering
University of Maryland at College Park, MD 20742, USA

Structural reliability analysis requires characterizing uncertainties in strength and loads for selected failure modes. The objective of this paper is to study the uncertainties associated with the vertical wave-induced bending moment on ship hull girders. Also uncertainties in whipping effects are investigated. Spectral analysis was used to calculate the maximum life time wave loads on a ship with and without whipping. This type of analysis requires knowing a set of basic random variables. The uncertainties in wave-induced and whipping bending moments were investigated using Monte Carlo simulation by randomly generating these basic random variables and calculating wave and whipping loads on a ship. A parameteric study was performed to investigate the effect of variability in these basic random variables on the uncertainties of wave loads

Keywords: Reliability, Loads, Waves, Bending, Whipping, Uncertainty, Spectral

## Fuzzy-Based Analysis of Structures

Ru-Jen Chao and Bilal M. Ayyub
Department of Civil Engineering
University of Maryland at College
Park, MD 20742, USA

The analysis of existing structures requires engineers to model two types of uncertainty, cognitive and non-cognitive types. The objective of this paper is to reexamine structural analysis methods by considering the cognitive type of uncertainty. Two approaches based on the displacement method for structural analysis are proposed: (1) fuzzy arithmetic approach, and (2) permutations approach. The first approach only obtains approximate solutions. The second approach produces the exact solution but requires more computing time. The behavior of fundamental structural systems was investigated, and the results based on the second approach showed that if the modulus of elasticity is a triangular fuzzy number, the member forces can be either fuzzy numbers or crisp values depending on the structural system type. Modified fuzzy division and subtraction were also proposed for solving simultaneous equations with fuzzy coefficients using fuzzy arithmetic. Keyword: cognitive, fuzzy arithmetic, permutation, structural analysis, triangular fuzzy number, uncertainty

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

668 Distributions with Fuzziness and Randomness Ru-Jen Chao and Bilal M. Ayyub Department of Civil Engineering, University of Maryland at College Park, MD 20742, USA Both cognitive and noncognitive uncertainties can be present in the same variable. The non-cognitive uncertainty of a variable can be described by its own probability density function (PDF); whereas the cognitive uncertainty of a random variable can be described by the membership function for its fuzziness and its a-cuts. A PDF called fuzzy-random PDF is proposed in this paper based on considering the combined effects of both cognitive and non-cognitive uncertainties for the variable. The variable is assumed to have a fuzzy mean and a non- fuzzy standard deviation. The fuzzy-random PDF is defined as the marginal density function of the multiplication of its normalized membership function and its random distribution. Relationships for the means and variances among the fuzzy-random distribution, normalized membership function, and random distribution were developed. Moments method and discrete method were proposed for dealing with the fuzzy-random PDF.

Keywords: cognitive, fuzzy, random, marginal density function, normalized membership function, probability density function, uncertainty

## An Empirical Study of Attractor Strength as a Function of Decision Structure: A Refinement of Yager's Measures of Decision Confidence

Carol Hartz, Brian Schott and Thomas Whalen
Georgia State University, Atlanta, Georgia USA 30303

Previous empirical studies indicate that differences exist between self-reported measures of post-decision confidence and Yager's predicted measures of decision tranquility. The purpose of this article is to demonstrate that some systematic differences are a result of attractors, round numbers that appear reasonable at or near special cases although they depart from a broader pattern. The results indicate that the strength of the attractors varies as a function of the decision structure. The research is a preliminary attempt to apply fuzzy mathematics to cognitive psychology. Subjects reported their confidence in a choice based on alternatives using random sets. Subjects'frequencies of responses were compared by decision type. Three decision structures that appear to increase attractor strength were identified. Univariate t tests of the differences of mean response frequencies between decision structure types were significant at the .05 level.

Keywords: Fuzzy set, decision making, confidence, tranquility, attractor

## Catalyzing Database Inference with Fuzzy Relations

John Hale and Sujeet Shenoi
Mathematical and Computer Sciences
University of Tulsa
Tulsa, Oklahoma 74104-3189, USA

Inference analysis plays a major role in database security and knowledge discovery. Common sense knowledge, typically expressed in imprecise or fuzzy terms, can be introduced as catalytic relations to existing databases. Analyzing the augmented databases materializes new rules and latent compromising inference channels based on common knowledge and existing database data. This paper shows how fuzzy relations can be used to catalyze new inferences in database systems. A knowledge discovery tool for analyzing catalytic inference in Oracle databases is described.

Keywords: Database inference, knowledge discovery, database security, fuzzy relational databases

## Reliability-Based Design of Unstiffned Panels for Ship Structures

Ibrahim Assakkaf and Bilal M. Ayyub
Department of Civil Engineering
University of Maryland, College Park, MD

The main objective of structural design is to insure safety, function, and performance of an engineering system for target reliability levels and for specified time period. As this must be accomplished under conditions of uncertainty, probabilistic analyses are necessary in the development of such probability-based design of unstiffened panels for ship structures. The load and resistance factor design (LRFD) format was developed in this paper for unstiffened panels. Partial safety factors were determined to account for the uncertainties in strength and load effect. In developing these factors, Monte Carlo simulation was utilized to assess the probabilistic characteristics of strength models by generating the basic random variables that define the strength and substituting in these models; while the First-Order Reliability Method (FORM) was used to determine the partial safety factors based on prescribed probabilistic characteristics of load effects.

Keywords: Reliability, Panels, Plates, Steel, Ship, Probability, Design, Load Factors, Strength Factors, Safety Factors

## Default Reasoning with Qualified Syllogisms

Daniel G. Schwartz
Florida State University
Tallahassee, Florida 32306-4019
schwartz@cs.fsu.edu

Prior works by the author have introduced the system QUAL (herein Q) of qualified syllogisms. An example of such a syllogism is Most birds can fly; Tweety is a bird; therefore, it is likely that Tweety can fly''. Q provides a formal language for expressing such syllogisms, together with a semantics which validates them. Also introduced in the prior works is the notion of a path logic. Reformulating Q as a path logic allows for the expression of modifier combination rules, such as From likely P and unlikely P, infer uncertain P''. The present work builds on this, showing how to incorporate Q into a system for default reasoning. Here is introduced the notion of a dynamic reasoning system (DRS), consisting of a path logic, together with a semantic net, or more exactly, a taxonomic hierarchy that allows for multiple inheritance.

The taxonomic hierarchy enables definition of a specificity relation, which can then be used in default reasoning (more specific information takes priority over less specific). Modifier combination rules prescribe what to do when defaults are applied in the context of multiple inheritance. Propositions derived in this manner all bear qualitative likelihood modifiers, representing the extent to which the proposition is believed.

Keywords: default reasoning, fuzzy quantifiers, fuzzy likelihood, usuality, qualified syllogisms, dynamic reasoning systems

## A Fuzzy Expert System For Qualitative Chemical Analyses of Minerals Using Color Recognition

Anita Maria da Rocha Fernandes and Rogerio Cid Bastos
Universidade Federal de Santa Catarina
Florianopolis - Brazil
Fax : 55 + (048) 231 - 9770
(anita@inf.ufsc.br) and (rogerio@inf.ufsc.br)

The process relating to qualitative chemical analyses is based on sample analyses of minerals. Several considerations are performed by chemists, involving features such as hardness, luster, etc. Besides, solubility tests have a problem - how to determine suitable solvents to specific samples (based on color recognition). The process demands an expert knowledge and involves pon- derations and cognitive inferences, peculiar to human reasoning. This kind of uncertainty is difficult to be modeled with traditional models. The color recog- nition is a complex process, because it is variable from observer to observer. This work proposes an Expert System, based on fuzzy logic, which has a knowledge base structured using a classification tree with a membership function. To solve the color problem, the HSI method is used, because of its similarity with the way human observers colors.

Keywords: qualitative chemical analyses, fuzzy expert system, fuzzy classification tree, HSI method

## An Autonomous Approach to the Mountain-Clustering Method

P.J. Costa Branco, N. Lori, and J.A. Dente
Laboratorio de Mecatronica - CAUTL
Instituto Superior Tecnico
Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
Fax: 351-1-8482987

The paper presents an autonomous approach for the clustering algorithm based on a mountain function proposed by Yager and Filev. It intends to answer the parameter selection problem and attenuate the effects of the granularity of the griding in algorithm's performance using a cluster realocation procedure. The solving of those problems has greatly enhanced the possibility of achieving an autonomous clustering process. The proposed clustering approach is explained in detail and examples of its performance are analyzed.

Keywords: Clustering, mountain method, data structure

## Medical Image Compression with Neural Nets

Alexander Steudel, Stefan Ortmann, and Manfred Glesner
Darmstadt University of Technology
Institute of Microelectronic Systems
Karlstrasse 15, D-64283 Darmstadt, Germany
Fax.: (+49) 6151 16-4936
{alex|stefano|glesner}@microelectronic.e-technik.th-darmstadt.de

A nonlinear 5-layer artificial neural autoencoder network for image data compression is constructed and trained using the back propagation algorithm and medical CT-images. The influence of linear and nonlinear pre/postprocessing operations is studied as well as an alternative compression scheme. Important implementational issues of neural networks are addressed as well as autoencoder issues. One of the results of this work is a compression/decompression tool that provides maximum flexibility and can be used independent from the training environment.

Keywords: Autoencoder, Back Propagation, Compression, Neural Networks

## Linear Statistical Inference for Fuzzy Data

Wolfgang Naether
Freiberg University of Mining and Technology
naether@mathe.tu-freiberg.d400.de

The paper presents some steps towards linear statistical inference for fuzzy data. In establishing best linear unbiased estimators (BLUE) it is necessary to consider a suitable notion of expectation and variance for random fuzzy sets. As methodological guide, we use the Frechét-approach which leads for a given metric to an associated expectation and variance. Especially the well known Aumann expectation appears as Frechét-expectation and the associated variance has the advantage, that at least special fuzzy number data can formally be handled like Euclidean vectors. Application to linear regression shows that only in special cases the fuzzified version of classical BLUE keeps their optimality.

Keywords: Best linear unbiased estimation, Expectation and variance of random fuzzy sets

## Generation of Polyline Reference Sets on Quantized Supports

Tibor Lapohos, George K. Knopf, and Ralph O. Buchal
University of Western Ontario,
London, Ontario Canada N6A 5B9

A key step in fuzzy relational modeling is the generation of appropriate reference sets. The membership functions used to define these reference sets are often based on heuristics. However, more attention must be paid to how these reference sets are created if the actual statistical characteristics of the system signals are to be reflected in the shape of the membership functions. A new algorithm which automatically generates reference sets without neglecting important information hidden in the data statistics is described in this paper. The method employs a Scalar Quantization (SQ) algorithm to partition the universe of discourse, and a polyline technique to generate the membership functions on the partitioned universe. The performance of the reference set generator algorithm is analyzed by modeling and simulating a stochastic process.

Keywords: fuzzy relational modeling, fuzzy reference set design, Fuzzy c-Means, scalar quantization, polyline algorithm

## Hybrid Arithmetic

Scott Ferson and Lev Ginzburg

Kaufmann's formulation of hybrid numbers, which simultaneously express fuzzy and probabilistic uncertainty, allows addition and subtraction, but offers no obvious way to do multiplication, division or other operations.We describe another, more comprehensive formulation for hybrid numbers that allows the full suite of arithmetic operations, permitting them to be incorporated into complex mathematical calculations.There are two complementary approaches to computing with these hybrid numbers.The first is extremely efficient and yields theoretically optimal results in many circumstances.The second more general approach is based on Monte Carlo simulation using intervals or fuzzy numbers rather than scalar numbers.

Keywords: fuzzy numbers, probability distributions, Kaufmann, multiplication

## Quality Assurance for Monte Carlo Risk Assessment

Scott Ferson
Applied Biomathematics
100 North Country Road, Setauket, NY 11733

Three major problems inhibit the routine use of Monte Carlo methods in risk and uncertainty analyses:

1. correlations and dependencies are often ignored,
2. input distributions are usually not available, and
3. mathematical structure of the model is questionable.
Most practitioners acknowledge the limitations induced by these problems, yet rarely employ sensitivity studies or other methods to assess their consequences.This paper reviews several computational methods that can be used to check a risk assessment for the presence of certain kinds of fundamental modeling mistakes, and to assess the possible error that could arise when variables are incorrectly assumed to be independent or when input distributions are incompletely specified.

Keywords: correlation, dependency, input distributions, mathematical structure

## Modeling Geometric Uncertainties of Continuum Structures for Reliability Assessment Purposes

Rafi L. Muhanna and Bilal A. Ayyub
Department of Civil Engineering
University of Maryland at College Park, MD

A prototype computational methodology for reliability assessment of continuum structures using finite element analysis with instability failure modes is described in this paper. Examples were used to illustrate and test the methodology. Geometric and material uncertainties were considered in the finite element model. A Computer program was developed to implement this methodology by integrating uncertainty formulations to create a finite element input file, and to conduct the reliability assessment on a machine level. A commercial finite element package was used as a basis for the strength assessment in the presented procedure. A parametric study for a stiffened panel strength was also carried out. The developed method is expected to have significant impact on the reliability assessment of structural components and systems. This impact can extend beyond structural reliability into the generalized field of engineering mechanics.

Keywords: reliability, reliability assessment, uncertainty, structures, finite element method, geometric, simulation, probability, continuum

## Scaling Genetically Guided Fuzzy Clustering

Lawrence O. Hall and Burak Ozyurt
University of South Florida
hall@csee.usf.edu

This paper describes improvements to previous work on the use of genetic algorithms and evolutionary strategies to generate fuzzy partitions of unlabeled data. It was found that genetically guided clustering could be used in some domains to produce fuzzy partitions for which the objective function does not get trapped in local extrema. Gray code representation, two point crossover, tournament selection, and variable crossover and mutation rates combine for improved performance in terms of the final partition and the required population size. Also, methods to allow this approach to scale to problems such as magnetic resonance imaging (22000 pixels in 3 dimensions) are detailed. Initially, using properly chosen subsamples of the full data allows the genetically guided approach to move to good regions in the search space quickly. Examples are given to show how local extrema of an objective function (and thereby poor data partitions) can be avoided in domains with a large number of patterns.

## The Fuzzy Median and the Fuzzy MAD

Paul R. Kersten
Naval Air Warfare Center, Weapons Division
Target Recognition Section, Code 455540D
China Lake, CA 93555

An extension of two key robust statistics to fuzzy sets is presented and applied to the fuzzy c-means clustering algorithm. Examples of a robust statistic are the median and the median absolute deviation from the median or MAD, a robust estimate of dispersion. These extensions are derived, and the fuzzy median is applied to the fuzzy c-means clustering algorithm. The modified clustering algorithm shows improved performance in clustering data sets generated by heavy-tailed distributions like the Cauchy distribution. Approved for public release; distribution unlimited.

## Learning from Data by Coherent Probabilistic Reasoning

Giulianella Coletti
Dipartimento di Matematica - Universita di Perugia
Via Vanvitelli 1 - 06100 PERUGIA (Italy)

Romano Scozzafava
Dip. Metodi e Modelli Matematici - Universita' "La Sapienza"
Via Scarpa 16 - 00161 ROMA (Italy)

We apply the theory of (de Finetti) coherent inference to the handling of uncertainty in the process of automatical medical diagnosis. Given some possible diseases (that could explain an initial piece of information) and a tentative probability assessment on them, the data base consists of con- ditional probabilities P(E|K), where each K is a disease and each evidence E comes from a suitable test. The coherence of the whole assessment is checked. The doctor can now update the probability of each disease and check again coherence of the whole assessment, since the diseases do not constitute, in general, a partition (so that the usual Bayes theorem can- not be applied). These steps can be iterated until a degree of belief suf- ficient to make a diagnosis is reached: the coherence condition acts as a control tool on every stage.

Keywords: Coherence, Subjective probability, expert systems, medical diagnosis

## The Use of Fuzzy Rules in Classification of Normal Human Brain Tissues

Anandhkumar Namasivayam and Lawrence O. Hall
Department of Computer Science and Engineering
University of South Florida
Tampa, FL. 33620
hall@csee.usf.edu

Automatic classification and tissue labeling of 2-D magnetic resonance images of the human brain may involve a preliminary clustering stage. Segmenting large multi-dimensional data sets like those from magnetic resonance images is very time consuming. Better performance at the clustering stage is achieved if partial classification of the image can be done before applying clustering. We show the use of fuzzy rules to do this partial classification to be very effective. Fuzzy rules can pre-classify a major portion of the image giving a clustering algorithm a lesser number of pixels to operate upon. Furthermore, as the pre-classification stage is itself fuzzy, it can be directly used to initialize a fuzzy clustering algorithm, giving it a much needed headstart. In this paper we present an approach to using fuzzy rules to pre-classify magnetic resonance images of the normal human brain. Good segmentation of normal brain into tissues of interest is obtained much faster than with clustering alone.

## On Estimation of Temporal Fuzzy Sets for Signal Analysis: FCM vs. FMLE Approaches

Bogdan R. Kosanovic
Laboratory for Computational Neuroscience
Department of Electrical Engineering
University of Pittsburgh

Luis F. Chaparro
Department of Electrical Engineering
University of Pittsburgh

Robert J. Sclabassi
Department of Neurological Surgery
University of Pittsburgh

Estimation of temporal fuzzy sets that model dynamic processes is discussed. It has been found that although poles of attraction can be estimated fairly well with different fuzzy partitioning algorithms, membership function estimates may fail in accurately describing dynamic changes within the observed signals. Two types of fuzzy partitioning algorithms are compared: fuzzy c-means (FCM) and fuzzy maximum likelihood (FMLE). The simulations performed on quasi-stationary Gaussian signals suggest that the membership functions estimated by FMLE fail to follow continuous changes of dynamics, while those estimated by FCM provide a good compromise between precision and physical relevance.

Keywords: temporal fuzzy sets, signal processing, fuzzy clustering, dynamic systems

## Seismic Analysis of Coupled Shear Wall Systems Using Probability and Random Vibration Concepts

J.S.Kuang and Jawahar Puvvala
Dept of Civil and Structural Engineering
The Hongkong University of Science and Technology
Clear Water Bay, Kowloon, Hongkong

Reinforced concrete coupled shear wall systems are widely accepted as a rational and economical part of tall buildings. The purpose of the system is to provide lateral resistance against external horizontal loads arising from wind and earthquakes. Much progress has been made in the last decade on the static and dynamic analysis of coupled shear walls.One method of analysis of plane coupled sherar wall is to replace the discrete connecting beam between the piers by an equivalent continuous system of laminae. Dynamic analysis has also been done to determine the fundamental frequency on order to apply the response spectrum. In modern earthquake engineering the response spectrum plays a vital role in the most commonly used method of analysis of coupled shear wall systems. Based on the free vibration analysis of coupled shears and mode superposition method of dynamic analysis of coupled shear walls, a method is established in this paper with the contribution of response spectrum using probability and random cibration concepts.

The modified probabilistic and random vibration concepts uses a new method to determine mean and standard deviation of peak response for a coupled shear wall subjected to earthquake excitation. The main practical application fo this method to determine the response mean frequency of coupled shear wall systems. This frequency determines the average number of cycles over a unit duration of time. The effectivenes and flexibility of this method is that it provides the designer a simple basis for specifying the earthquake loading. The technique results in a considerable reduction in computaionsl effort. The procedure is demonstrated using hypothetical structure subjected to earthquke induced base excitations. Computed results based on this method are close agreement with simulation results obtained from time-hsitory dynamic analysis. None of the existing methods provide any menas for computing the mean and standard deviation of peak repsonse of coupled shear wall systems. This aspect of formulatin is unique and furtherance of state of art.

## Robust Phase Reversal Tone Detection Using Soft Computing

Francesco Beritelli, Salvatore Casale, and Marco Russo
Universita' di Catania (ITALY)
Viale Andrea Doria 6 - 95125 Catania - Italy
Fax: +39-95-338280
E-mail: mrusso@iit.unict.it

The paper presents a simple robust algorithm for the recognition of a 2100 Hz tone with periodic phase reversal and the disabling of an echo canceller based on Soft Computing. The authors have used a novel tool that is able to extract fuzzy knowledge using a hybrid technique based on Genetic Algorithms and Neural Networks. The approach proposed, compared with signal detection solutions existing in literature, is certainly more efficient in terms of robustness to channel noise and can therefore be usefully applied in all cases in which signals are to be detected with very low SNRs.

## Fuzzy numerical functions as special fuzzy multivalued mappings

Elena Tsiporkova-Hristoskova, Bernard De Baets, and Etienne Kerre
Department of Applied Mathematics and Computer Science
University of Gent, Krijgslaan 281 (S9)
9000 Gent, BELGIUM

The concept of a fuzzy numerical function is introduced as a special fuzzy multivalued mapping associating to each element of the universe of discourse a fuzzy (real) number. The notion of a truncated fuzzy number is introduced and its properties are studied. It is proven that the direct image of a fuzzy singleton under a fuzzy numerical function always is a truncated fuzzy number. A strict ordering in the set of truncated fuzzy numbers is constructed and it is shown that the class of fuzzy numbers, endowed with this ordering, forms a lattice. Addition, subtraction, multiplication, division, maximum and minimum of fuzzy numerical functions are defined and the distributivity of the direct image of a fuzzy singleton with respect to these operations is obtained. Carefully chosen definitions of lower and upper semi-continuity of fuzzy numerical functions are presented. The behaviour of addition, subtraction, maximum and minimum of lower (resp. upper) semi-continuous fuzzy numerical functions is investigated.

Keywords: fuzzy numerical function, fuzzy multivalued mapping, fuzzy number, truncated fuzzy number, lower and upper semi-continuity

## Constructing Possibility Measures

Bernard De Baets, Gert De Cooman
Department of Applied Mathematics and Computer Science
Department of Electrical Engineering
University of Gent, Krijgslaan 281 (S9), B-9000 Gent, Belgium

In this paper, we address some aspects of the extension problem for possibility measures: given the values that a (fuzzy) set mapping takes on a family of (fuzzy) sets, is it possible to extend this mapping to a possibility measure? This problem is shown to be equivalent to a special system of relational equations. When the family of sets considered is a (semi)partition, two important solutions are identified. It is shown that these solutions, and their fuzzifications, play a central part in the treatment of the more general extension problem. This role is shown to be even more conspicuous when the family of fuzzy sets considered is a T-(semi)partition, a notion introduced and studied for the first time in this paper.

## Effects of the Shape of Fuzzy Membership Functions on Fuzzy Inference

J. R. Boston
Department of Electrical Engineering and the Bioengineering Program
University of Pittsburgh, Pittsburgh, PA 15261USA

This paper investigates the effect of the shapes of membership functions on a fuzzy inference system to detect a signal in a noisy waveform. The detector, which uses values of features derived from the waveform, can classify the waveform as signal or noise, or it can be uncertain, that is, it can decide that no conclusion regarding presence or absence of a signal can be drawn. Piece-wise linear membership functions were used, and analytical expressions for the dependence of classification on the membership function parameters were obtained. These results were verified in a simulation, using sensory evoked potential signals and simulated noise. The performance of the system was compared to a Bayesian maximum likelihood detector. By varying membership function parameters, the fuzzy detector can be made comparable to the Bayesian detector or it can almost completely eliminate errors, at the cost of a large number of uncertain classifications.

## Signal Representation Using Fuzzy Morphology

Chin-Pan Huang and L. F. Chaparro
Department of Electrical Engineering
University of Pittsburgh

In this paper, we propose a signal representation based on fuzzy morphology and adaptive structuring functions which is analogous to polynomial transforms. For completeness, we use a set of normalized and ordered structuring functions derived from orthogonal polynomials defined on a window. After windowing the given signal, the geometric decomposition of a windowed signal is achieved by applying fuzzy morphological opening sequentially using each of the structuring functions to fit the signal. The resulting representation is made to resemble an orthogonal expansion by constraining the result of opening to equal the adapted structuring function. Properties of the geometric decomposition permit us to develop an iterative procedure to calculate the adaptation parameters. The analysis and synthesis procedures are illustrated for images.

Keywords: Fuzzy sets, mathematical morphology and data compression

## Safety Analysis of Tension Elements Using Various Reliability Methods

Jesa H. Kreiner
California State University
Fullerton, CA 92634, USA
jkreiner@fullerton.edu

Chandra S. Putcha
California State University
Fullerton, CA 92634, USA
cputcha@fullerton.edu

This work is an extension of the earlier research done by the authors in this area (1) wherein these principles were applied to a steel beam with an external moment. The main methods used for estimation of reliability in the previous as well as the present research are: the First Order Second Moment (FOSM) method, the Advanced First Order Second Moment (AFOSM) and the Point Estimate Method (PEM). For each method, the reliability of the tension element is calculated.The safety indices (b) are calculated for the various types of probabilistic combinations of resistance and load. Results obtained by these three methods are compared to one another. Subsequently, relevant conclusions are drawn.

Keywords: safety, reliability, tension, safety index, probability of failure, normal, log-normal

## Robust Handwritten Word Recognition with Fuzzy Sets

Paul Gader and Jung-Hsien Chiang
Dept. of Electrical and Computer Engineering
University of Missouri - Columbia
gader@sun1.ece.missouri.edu

A hybrid fuzzy neural system is used to improve a handwritten word recognition algorithm. The word recognition algorithm matches digital images of handwritten words to strings in a lexicon. This algorithm requires a module to assign character class membership values to images of segments of handwritten words. Many of these images are not characters. It is shown that a hybrid neural system consisting of a cascade of a Kohonen Self Organizing Feature Map (SOFM) followed by Choquet fuzzy integrals can yield improved performance over a multi-layer feedforward network (MLFN). The hybrid method scored a word recognition rate of 85% compared to 77% for the MLFN method.

Keywords: Handwriting Recognition, Self-Organizing Feature Maps, Fuzzy Integrals, Character Recognition, Backpropagation

## Adaptivity in a Hierarchical Fuzzy Model

Robert J. Hammell II and Thomas Sudkamp
Department of Computer Science & Engineering
Wright State University
Dayton, Ohio 45435

A hierarchical architecture for fuzzy modeling and inference has been developed to allow adaptation based on system performance feedback. A general adaptive algorithm is presented and its performance examined for three types of adaptive behaviour: continued learning, gradual change, and drastic change. In continued learning, the underlying system does not change and the adaptive algorithm utilizes the real-time data and associated feedback to improve the accuracy of the existing model. Gradual and drastic change represent fundamental alterations to the system being modeled. In each of the three types of behaviour, the adaptive algorithm has been shown to be able to reconfigure the rule bases to either improve the original approximation or adapt to the new system.

Keywords: fuzzy model, adaptivity, rule base completion

## Reliability Assessment of Systems Operating in Variable Conditions

Patrick Huamin Liu, Viliam Makis, and Andrew K. S. Jardine
Department of Industrial Engineering
University of Toronto
Canada, M5S 1A4

The performance and reliability of a system are strongly influenced by its operating conditions. However, existing reliability models are unable to calculate the reliability of a system operating in variable conditions. This paper presents a new approach to this problem and discusses two typical cases: (1) one item operating sequentially under different conditions, where the transition points are deterministic; (2) a system consisting of two parallel sub-systems sharing a load, where the failure rate of each sub-system is dependent on the load it shares. We show that if the underlying failure time distribution of the item (or sub-system) can be represented by the accelerated failure time model (AFTM), the system reliability at any time can be obtained easily by using the formulas derived in this paper.

## On Robustifying the C-Means Algorithms

Jongwoo Kim and Raghu Krishnapuram
Department of Electrical and Computer Engineering
University of Missouri, Columbia, MO 65203
kimjw@sun1.ece.missouri.edu; raghu@ece.missouri.edu

Rajesh Dave
Department of Mechanical and Industrial Engineering
New Jersey Institute of Technology, Newark, NJ 07102
dave@shiva.njit.edu

The Hard and Fuzzy C-Means algorithms are commonly used in many applications. However, they are highly sensitive to noise and outliers. In this paper, we reformulate the Hard and Fuzzy C-Means algorithms and combine them with a robust estimator called the Least Trimmed Squares to produce robust versions of these algorithms. To find the optimum trimming ratio of the data set and to eliminate the noise from the data set, we develop an unsupervised algorithm based on a cluster validity measure. We illustrate the robustness of these algorithm with examples.

Keywords: Fuzzy C-Means, Least Trimmed Squares, Robust Methods, Noisy Data, Clustering

## Fuzzy Modelling for Automatic Telephone Answering Systems

K. Brownsey, M. Zajicek and P. Palau
Oxford Brookes University
Fax: +44 1865 483666
Email: kwb@brookes.ac.uk

This paper is concerned with the use of fuzzy sets for automatic telephone answering systems for large organisations or organisations providing large amounts of vaguely structured information. Currently available commercial systems are generally based on hierarchical dialogue systems, reflecting a hierarchical information structure. The authors have been investigating alternative approaches where the information structures are less well defined. In these cases, callers often have a degree of vagueness about what information they require. The paper looks at the idea of using fuzzy sets for modelling both the domain of interest and the users emerging goals, as perceived by the system. It then goes on to describe two algorithms for traversing the information structures. Conclusions and further work are then discussed.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

139 Using Reliability-Based Models to Assess Rehabilitation Needs Mary Ann Leggett, PhD USAE Waterways Experiment Station 3909 Halls Ferry Road, Vicksburg, MS 39180 leggetm@ex1.wes.army.mil Due to increasing budget constraints, funding available to maintain, rehabilitate, improve, or replace aging US Army Corps of Engineers (USACE) structures is declining with respect to funding requirements. Therefore, the available funds must be selectively invested to achieve maximum benefits. To compete for scarce appropriation resources, USACE districts are now required to justify civil works rehabilitation project funding by demonstrating a need for improvement in reliability or efficiency. A risk-based benefit-cost model is utilized to establish funding justification. This analysis model requires input to assess the current condition of a structure and its degradation rate. This methodology incorporating both reliability and economic aspects would aid in forming a nationwide planning system. Keywords: reliability assessment, risk analysis, rehabilitation, probabilistic methods, structural analysis

## Linguistic Context Spaces and Modal Logic for Approximate Reasoning and Fuzzy-Probability Comparison

Boris Kovalerchuk George Klir
Dept. of Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803-6409, borisk@unix1.sncc.lsu.edu
Dept. of Systems Science and Industrial Engineering,  SUNY, Binghamton, NY 13902-6000

Effective inference under uncertainty in Artificial Intelligence depends on context.InferencesbasedonBayesianconditional probabilities use context effectively.However,the concept of context is not sufficiently developed for reasoning based upon the variousothertheoriesofuncertainty,suchasfuzzy set theory, Dempster-Shafer theory ,or rough set theory.Inthis paper,wedevelopaconceptof "context space" forfuzzy systems theory. Such a description of context spaceallowsone to usefully construct fuzzy setsforspecificapplications,and thus improves the foundation for fuzzysystemstheory.In addition,theproblemofestablishingmembershipfunctions (MFs) for context spaces is considered. It is shown that Hisdal's operational procedures and modal logic are preferable when used jointly with a complete and exactly defined context space as introduced in the paper. Finally, the theory of fuzzy sets is compared with probability theory in connection with the problem of MF acquisition.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

643 Elimination of Semantic Ambiguity in Fuzzy Relational Models Michinori Nakata A generalized possibility-distribution-fuzzy-relational-model is proposed considering semantic ambiguity for values of membership attribute and ambiguity contained in values of membership attribute. And then the extended relational algebra is shown. In order to eliminate the semantic ambiguity, the concept of membership is introduced into each attribute. This clarifies the origin of membership attribute values. What the value of membership means depends on the property of attributes. In order to eliminate ambiguity contained in values of membership attribute those values are expressed by fuzzy values. This clarifies what relationships fuzzy data values have with their membership attribute values. Therefore there is no semantic ambiguity for the values of membership attributes and no ambiguity in the values of membership attributes in our extended relational model.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

543 Fuzzy Scheduling in Compilers Optimizations O. Hammami Computer Architecture Lab. The University of Aizu Fukushima, 965-80 JAPAN Cache memories are essential components in all existing commercial microprocessors. In order to attain best performance, cache memories have to be managed either with hardware support or compiler support. The compiler approach makes use of specialized cache memories management instructions to generate an optimal management. This is done by generating an optimal scheduling of these specialized instructions for the program being compiled. Up to now, conservative approaches have been used to tackle this issue despite the occurence of impredictable real time events and the fact that many variables are imprecise. This explains the unstable performances of these algorithms varying from excellent to very poor.\\ We propose to make use of a fuzzy scheduling approach to deal with the problem. Keywords: cache,compiler,fuzzy,instructions,scheduling

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

430 Multiresponse Quality Design and Possibilistic Regression Young-Jou Lai Department of Industrial and Manufacturing Systems Engineering Kansas State University Manhattan, KS 66506 Bitmail: Lai509u@ksuvm.ksu.edu Multiresponse quality design techniques are used to identify settings of proce performance close to target values in the presence of multiple quality character quality characteristics and thus their functional relations are imprecise to som measuring errors, incomplete knowledge, vagueness of definitions and so on. Here possibilistic regression models are used to model these imprecise natures and in relationships. We first integrate and extend existing possibilistic regression predictive quality characteristics or responses. We then propose a mulitple obj an appropriate combination of process parameter settings based on the obtained p predictive responses. We not only optimize the most possible responses values, or deviations from the most possible values.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

705 Development of Interval Based Methods for Fuzziness in Continuum Mechanics Rafi L. Muhanna Department of Civil Engineering University of Maryland at College Park, MD 20742, USA muhannar@eng.umd.edu Robert L. Mullen Department of Civil Engineering Case Western Reserve University Cleveland, OH 44106, USA Accounting for uncertainties in mechanics problems has previously been accomplished by probabilistic methods. Such methods can require highly repetitive computations to analyze the behavior of mathematical models. In addition, knowledge of the probability distribution of state variables is often incomplete. In this paper, a new treatment of uncertainties in continuum mechanics based on fuzzy set theory is introduced. Uncertainties or fuzzy numbers here-in are viewed through the concept of presumption l evel of the uncertainty ,[0, 1], which gives an interval of confidence A = [a1 (), a2()]. The interval approach of treating uncertainties in continuum mechanics is applied to both geometric and material uncertainties in number of examples. Results de monstrate sharp inclusion of the interval solution in comparison with the exact solutions. Keywords: interval calculus, fuzzy sets, interval finite element, fuzzy finite element, possibility

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

95 Feature-Based Target Recognition with Bayesian Inference Jun Liu and Kuo-Chu Chang George Mason University Fairfax, VA 22030 jliu@c3i.gmu.edu, kchang@c3i.gmu.edu The problem of target classification with high-resolution, fully polarimetric, synthetic aperture radar (SAR) imagery is considered. This paper summarizes our recent work in SAR target recognition using a feature-based Bayesian inference approach. The approach works on the selected features. Features are chosen such that the separabilities of the original data are well maintained for later classification. Once the original data is mapped into feature space, the conditional probability distributions of features given the target are estimated statistically, which are then used to calculate the probabilities that a target belongs to one of the given classes based on the observed features. The target is assigned to the class with the highest probability. A comparison between the above technique and the traditional statistical approaches such as nearest mean and Fisher pairwise is illustrated based upon performance on a fully polarimetric ISAR image data set Keywords: Pattern recognition, Bayesian network, Automatic target recognition, Uncertainty

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

655 New Semantics for the Membership Degree in Fuzzy Databases Noureddine MOUADDIB, Nathalie BONANNO Centre de Recherche en Informatique de Nancy Equipe EXPRIM - CNRS (URA 262) B_t. Loria, Campus Scientifique BP 239, 54506 Vandoeuvre-les-Nancy, FRANCE Fax : (33) 83 41 30 79 E-Mails : {mouaddib, bonanno}@loria.fr This paper proposes a definition of a fuzzy relational schema that include Fuzzy Integrity Constraints (FIC). Unlike others approaches, we assign a degree to each tuple of the relation which measures of the compatibility between tuples and constraints defined on the relation. Among FIC, we are interested in Fuzzy Explicit Integrity Constraints (FEIC) that are not supported by the data model. To specify them, we propose an extension of an assertion language used to describe integrity constraints in relational databases. Using the Possibility Theory, we finally explain the method to calculate the compatibility degree within the limit of individual constraints. K Fuzzy relational databases, Fuzzy Integrity Constraints, Membership degree

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

662 Fuzzy and Probabilistic Interpretation of Spectral Information M. Kudra and H. Bohlig Institute of Physical and Theoretical Chemistry University of Leipzig Linnestr.2, D-04103 Leipzig e-mail: kudra@rz.uni-leipzig.de E. Geide Institute of Physical Chemistry University of Hamburg Bundesstr. 45, D-20146 Hamburg The paper shows on a spectroscopical example, how expert knowledge can be modeled by fuzzy sets or in the probabilistic sense as conditional density functions. Fuzzy sets, so-called fuzzy observations, are used to model the information available about frequency positions of main vibrations. These vibrations appear in infrared or RAMAN spectra as absorption bands. The calculation of such vibrations by means of Normal Coordinate Analyses (NCA) often leads to a fitting of calculated frequency positions to experimental ones. Fuzzy methods can be used to evaluate calculated eigenfrequencies in the light of fuzzy observations. In the probability sense, this evaluation can be managed by Baysian methods. Finally, comparing fuzzy and probability approach, we conclude that both methods lead to comparable formulas and results.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

613 A General Framework for Comparing Numerical Uncertainty Theories Elisabeth Umkehrer and Kerstin Schill Institut fuer Medizinische Psychologie, Goethestr.31, 80336 Muenchen, Germany Ludwig-Maximilians-Universitaet Muenchen Deciding which of the existing uncertainty theories is the appropriate one to use in the formalization of a given problem is still a difficult task. It is possible to compare these theories either using pragmatic considerations (eg. efficiency) or experimentally applying them on a set of problems; however, up to now there is no general frame in which we can compare the uncertainty theories with respect to their meanings. Our work aims to develop a general formalism for representing and reasoning with uncertain knowledge, which provides such a framework and is not restricted in using one specific uncertainty theory as a basis. This formalism is based on the work of Carnap's {\it Logical Foundation of Probability}. But, instead of propositions, we regard distinctions as elementary notions. Up until now, the following theories can be expressed within the framework: Bayes theory, Fuzzy set theory, Dempster/Shafer theory (Belief functions) and Upper/Lower probability theory. Keywords:Uncertain knowledge, Bayes, Belief functions, Fuzzy, Upper/Lower Probability

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

437 Analysing Uncertain Data in Decision Support Systems Kerstin Schill Institut fuer Medizinische Psychologie, Goethestr.31, 80336 Muenchen, Germany Ludwig-Maximilians-Universitaet Muenchen Decision support systems use two basic strategies: the pursuit of a small set of hypotheses and the sequential partitioning of hierarchical hypothesis structures. We present an alternative method based on the maximization of information gain. In each step, we evaluate the difference between the actual and potential future evidence distributions. The data "promising" the maximum information gain are then inquired by the system. In simple situations, the new method behaves like traditional strategies but in divergent and inconsistent evidence situations, it avoids the drawbacks induced by the predetermined standard strategies by adapting itself continuously to the actual data configuration. Our method can be extended to layered hierarchical data structures, where its behavior is reminiscent of the cognitive phenomenon of "restructuring". Keywords: uncertain data, decision support systems, control strategies

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

466 Reliability Analysis of a Reinforced Concrete Drainage Structure Robert C. Patev and Mary Ann Leggett Information Technology Laboratory US Army Engineer Waterways Experiment Station 3909 Halls Ferry Road, Vicksburg, Mississippi, 39180-6199 USA A reliability assessment was performed to examine potential modes of unsatisfactory performance for a six- barrel gravity drainage structure and pumping plant located under a 33-ft-high levee section. The structure is composed of reinforced concrete which has suffered severe structural deterioration and exposure of reinforcement. This structure is a critical element in the levee system that protects a major metropolitan area from river and bayou flooding. Loss of this structure during a project flood event would lead to a high probability of loss of life because of its proximity to a highly populated area. Using soil-structure interaction analyses, the outer culvert wall was modeled as a beamcolumn subjected to lateral and vertical earth pressures, and hydrostatic water pressures. Monte Carlo Simulations and First Order Second Moment techniques were utilized to examine the reliability of the culvert for normal operating conditions and for project floods of fifty and one hundred years. Keywords, Structural reliability, reinforced concrete, drainage structure, culvert, reliability assessment

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

715 Belief Updating A. Slobodova Institute of Control Theory and Robotics Slovak Academy of Sciences Bratislava, SLOVAKIA, 842 37 e-mail:utrraslo@savba.savba.sk Uncertainty is present in most task that require intelligent behaviour. Probability theory has by far the longest tradition in problems connected with uncertainty, but the Dempster-Shafer theory provides more general model. A central problem of this theory is conditioning. Spiess paper presents a new approach to a solution of this problem by establishing a link between conditional events and discrete random sets. Conditional events were introduced as sets of equivalent events under conditioning. These sets are targets of a multivalued mapping and conditional belief functions were introduced. We study properties of these functions in the cases that belief functions were obtained by Bayesian conditioning from an unconditional belief function. Keywords: conditional event, discrete random sets, conditional multivalued mapping, conditional belief functions, properties

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

363 Efficient Fuzzy Logic Architectures Suitable for Silicon Compilation Tony Wicks, Meyer Nigri, and Philip Treleaven University College London Department of Computer Science Gower Street London, WC1E 6BT UK Fax: +44 (0)171 387 1397 Email : twicks@cs.ucl.ac.uk The increasing complexity of fuzzy logic systems and their application to an ever widening set of problems leads to a requirement for automatic system generation methods. Fuzzy logic has proved to be a good alternative to traditional methods, with uses found in control, pattern matching and expert reasoning systems. Its use in embedded system controllers, in particular, brings about a need for efficient hardware solutions. This arises at a time when Application Specific Integrated Circuits (ASICs) are proving to be an increasingly attractive implementation route, in terms of cost, speed and reliability. Compilation methods which map a high level fuzzy logic description directly down to silicon are being developed. Efficient fuzzy logic architectures are therefore sought for the outcome of this mapping. This paper discusses the hardware requirements for fuzzy logic and develops a range of flexible implementation schemes which are suitable for silicon compilation.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

497 Extending the Application of Fuzzy Sets to the Problem of Agricultural Sustainability E.G. Dunn, J.M. Keller, L.A. Marks, J.E. Ikerd, P.D. Gader, and L.D. Godsey University of Missouri-Columbia USA email:ssedunn@muccmail.missouri.edu : While there is no consensus on a definition, it is widely recognized that the concept of sustainability has economic, environmental, and social dimensions. This multidimensionality has impeded the development of empirical applications of the concept of sustainability. In this paper, the specific problem of modeling agricultural sustainability is described. It is argued that fuzzy methods offer important advantages in overcoming at least three of the inherent difficulties in modeling agricultural sustainability: 1) the problem of noncommensurate units can be effectively handled through the use of linguistic variables; 2) information that is vague or imprecise can still be included in a problem when this information is modeled as a fuzzy set; and 3) the interrelationships between the dimensions of sustainability can be incorporated into a model by means of a fuzzy rule base. The application of fuzzy sets to the problem of agricultural sustainability is illustrated with a brief example of a mixed crop-livestock farming system in Missouri.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

503 Multiple Criteria Decision Making (MCDM) Using Fuzzy Logic: An Innovative Approach to Sustainable Agriculture L.A. Marks, E.G. Dunn, J.M. Keller, and L.D. Godsey Dept. of Agricultural Economics and Dept. of Electrical and Computer Engineering, University of Missouri-Columbia USA email: ssedunn@muccmail.missouri.edu For a farming system to be considered sustainable, certain economic, environmental, and social criteria need to be met. An approach which integrates measures of achievement in these three areas has been lacking in the past. It is argued that the combination of multiple criteria decision making (MCDM) and fuzzy logic provides a promising theoretical framework for the evaluation of alternative farming systems. In this paper, three well-known MCDM methods are described to illustrate a range of problems that may arise using conventional MCDM methods. These problems include: 1) the treatment of noncommensurate units; 2) the ranking procedure for a solution; and, 3) the degree of discrimination between attribute values, and, hence alternatives. The combination of MCDM and fuzzy logic provides a superior methodology which overcomes some of these problems.

#### Proceedings of the International Symposium on Uncertainty Modeling and Analysis and NAFIPS '95 (ISUMA '95)

595 Signal Parameter Estimation When the Parameters Are Fuzzy Variables Bulent Baygun Schlumberger-Doll Research Old Quarry Road Ridgefield, CT 06877 fax: (203)-438-3819 e-mail: baygun@ridgefield.sdr.slb.com We present a methodology for estimation of the fuzzy parameters of a stochastic signal. The proposed approach minimizes a fuzzy average decision error probability by a proper choice of decision regions. We use a scalar index, called the total distance criterion (TDC) ranking index, in order to rank the fuzzy average decision error probabilities of different decision rules. We identify the optimal decision rule which minimizes the TDC index of the fuzzy average decision error probability. As an example we apply the general approach proposed here to the classification of the fuzzy mean of a Gaussian random variable. The optimal decision regions are specified explicitly and closed form probability expressions are given for arbitrary symmetric membership functions. keywords: parameter estimation, fuzzy parameters, hypotheses testing, fuzzy probability