8/7/2019 2001 - Fuzzy Logic Based Decision Support Systems

1/3

Fuzzy Logic Based Decision Support SystemsHeinrich J. Rommelfanger

J. W. Goethe-University Frankfurt am MainInstitute of Statistics and MathematicsMertonstr. 17-23, D-60054 Frankfurt am MainRornmelfanger@wiwi.uni-fiankfurt.de

Keywords: credit rating, expert systems, fuzzy inference, fuzzy logic, linguistic variables, membership functions, rulebased aggregation, soft facts,

1. IntroductionDecisions and evaluations in economic and businessproblems are usually complex. A lot of variables and re-lations have to be specified. The expectations of the fif-ties and sixties of this century - that it would be possibleto model all problems in form of adequate mathematicalsystems - could not be satisfied. Today we know this canoriy be achieved in case of relatively simple problems.However, it has always been necessary to make decisionsin complex situations. A circumstance which is stillvalid, no matter whether it concerns economic, businessrespectively jurisprudential problems or everyday situa-tions like, for example, driving a car during rush hour.Based on qualifications and practical knowledge, indi-viduals attempt to come to a satisfying solution. A personwho obtains extraordinarily good results is called an ,,ex-pert" in this field.The motivation within the last 20 years was to model thedecision process of experts instead of the decision situa-tion. Expert knowledge does normally not consist of con-f m e d theories, it is rather composed of heuristic rulesthe expert obeys during h s own decision making. Expertrules are usually formulated by linguistic terms andtherefore it is difficult to transform them into classicalmathematical terms or to apply them to computer-aidedprocessing.In my contribution 1 want to demonstrate that the fuzzyset theory offers adequate instruments for modeling anddealing with expert rules. By modeling linguistic vari-ables in form of fuzzy sets, it is possible to transform ex-pert rules into mathematical terms. Moreover the fuzzyset theory offers a great variety of operators which areable to aggregate and combine these rules.The advantages of fuzzy logic for controlling technicalprocesses have become well-known, not only in Japanbut during the last decade as well in America andEurope. Now the question arises, whether the procedures,used very successfully in fuzzy control, can also be ap-plied to non-technical expert systems.I am convinced that a transfer of the basic ideas of fuzzycontrol to non-technical decision support systems is pos-

sible. At the lnstitute of Statistics and Mathematics of theUniversity FrankfUrt am Main we are working at the fol-lowing research projects using expert rules and fuzzylogic processing:- Checking the creditability of small business firms- Checking the credit solvency of persons buying a car

on installment plan- Evaluating the capital structure, financial assets and

revenue of firms in order to support the business ofauditors- Analyhcal procedures in the course of audits- Portfolio management- Evaluating suppliersIn this paper 1 malnly want to explain the new ideas bymeans of a rating system for checking the credit solvencyof small business f m . By means of a pilot software Iam going to demonstrate that fuzzy decision support sys-tems offer more information to the user than classical ag-gregation processes. Moreover, they provide a simpleway for explaining the rating result.With the fuzzy logic based decision support system thefollowing goals should be achieved:

construction of a transparent support system, whichcould be accepted by the usersJ reproduction of the behavior of experts concerning

- subjective selection of attributes- classification of the attributes by evaluationclasses- stepwise and rule-based aggregation of the par-tial evaluations to an overall rating judgement

J use of external data (branch of industry, country)J use of linguistic evaluations (poor, medium, good)J use of quantitative and qualitative informationJ processing of data measured on different scales(nominal, ordinal, cardinal)J softening of the crisp borders of evaluation classesJ use of fuzzy inferenceJ use of expert knowledge and of data banksJ creating a rating judgement with accompanyingcommentary.

8/7/2019 2001 - Fuzzy Logic Based Decision Support Systems

2/3

Subsequent to the introduction, the paper is organized in6 chapters which are described with few comments:2. Hierarchical systems of attributes for theevaluation of material business creditabilityComplex and not directly measurable criteria, as for ex-ample the credit solvency of a business firm, may oftenbe explained more transparently and more intelligibly bya hierarchcal system of subaspects. For evaluating thecreditworthiness of a specific firm, at first the aspects onthe bottom level have to be evaluated. Then those evalua-tions of the subattributes need to be aggregated step bystep until the top level of the hierarchcal concept isreached.Usually this aggregation process is accomplished bymeans of aggregation operators depending on parametersor weights whch ask for further specification by the de-cision maker.A disadvantage of these operators is that they portray thecomplex conjunction mechanism of the human mind onlyincompletely. In literature and practice there exist a lot ofexamples, that the weights within an objective systemdepend not only on the objectives but often change withthe obtained values. Therefore it is necessary to look forother ways of modeling the decision process of creditmanagers.3. Aggregation of Attributes by Expert RulesIn the artificial intelligence literature we can find manymodels in which the human decision process is describedby means of rules formulated by experts.1nthese rule-based models the evaluation of attributes isoften described by the linguistic terms ,,poor", ,,mediumL'and ,,good". For every possible situation on the lower h-erarchy level an aggregation rule is defined. Sometimesadditional ratings (- and/or +) are allowed for the aggre-gation results.Objections against those rules can be raised due to thefact that the rules are very inaccurate; the terms ,,good6',,,mediumw, ,,poor" allow a comparably large interpre-tation spectrum. To improve this situation we could try toenlarge the number of valuations for each criterion. This,however would result in an explosive increase of thenumber of rules as for m aspects with r possible valua-tions there exist rm rules. Therefore it is necessary torestrain the number of valuations for each aspect. More-over if the rule map gets too large the expert team willnot be able to guarantee a conscious distinction of eachsituation.4. Description of Expert Rules by Fuzzy SetsSince intervals only allow YesiNo-statements and there-fore different values of the same interval do not portray alinguistic term correspondingly, we propose to model thelinguistic terms by fuzzy sets. The fuzzy set theorymakes it possible to describe the different membershipdegrees according to the categories of ,,poor", ,,medium6'or ,,good" credit solvency as precise as the credit expertcan express it. These membership functions must be

specified as carefully as possible, because they will deci-sively influence the valuation process. Nevertheless wewill never obtain membership functions which are accu-rate in every detail, because a lot of data about similarf m and knowledge of the trade must be collected bythe expert team. Therefore the form of membership func-tions in expert systems will be very simple and the samedesign will be used repeatedly. In practice it is suficientto work with fuzzy numbers or fuzzy intervals of the LR -type-Presenting fuzzy sets as categories of valuation providesthe opportunity to precisely inform users of expert sys- ,tems about their valuation basis. Moreover by comparingdata of different branches of business th~sproceeding al-lows to resort to knowledge already stored in data bases.A way for a practical transformation has been analyzedin a pilot study at the Institute of Statistics and Mathe-matics of the University of Frankfurt am Main. The in-tention was to design membershp functions in the courseof an expert system for supporting the evaluation of thefinancial and operating position of business firms as partof the annual audit.5. Fuzzy Controller and Rule-Based AggregationIn case of modeling the linguistic evaluation terms byfuzzy sets, the aggregation rules are only applied to thosecases, in which the evaluations produce a membershipdegree 1 for all subaspects. Then the corresponding ruleof the rule map is applied and leads to a distinct evalua-tion of the upper-aspect with the membership degree 1,too. Therefore the use of fuzzy sets helps users to under-stand the basic principles of the expert knowledge moreeasily. This ,,understandingwis an essential factor for theacceptance of an expert system and related to that for itssuccessful realization.For all the other cases, where at least one evaluation hasa membership degree smaller than 1, no special ruleshave been stated by the experts. We assume that thegiven rules can be extended to situations in the vicinity.The rules are softened with the consequence that nowmany rules can be used simultaneously in a weakenedmanner.For a real situation we denote the degree of fulfillment(DOF) with the descriptions of state in the rule maps byDOF. According to the proceeding in fuzzy control, DOFis defined as the minimum of the membership degreesattached to the ,,inputsw of this rule. Having examinedvarious operators regarding their ability to describe thehuman conjunction behavior in specific cases, we cameto the conclusion that the minimum operator should beused. Besides others the minimum operator has the ad-vantage that only few rules with positive DOFs exist,whereas by using compensatory operators almost allrules will show positive DOFs and we therefore have toexpect intermediate evaluation. Now all rules with posi-tive DOF contribute to the valuation of sales potential inproportion to their DOFs.

In th~scontext we want to remark that there is an es-sential difference between fuzzy control applications andnon-technical evaluation and decision problems. Usually

8/7/2019 2001 - Fuzzy Logic Based Decision Support Systems

3/3

technical control processes are rapidly repeated. There-fore, it is sufficient when an approximately correct actionis carried out, because the correction will follow immedi-ately. On the contrary decision support systems require adefinite decision for every section which evidently has tobe correct. As a consequence the procedure in decisionsupport systems has to be handled more carefully than infuzzy control. Not only the linguistic evaluation termshave to be defined more carefully but also the calculationof the DOFs and the influence of the DOFs on the finalresult require an exact empirical examination.

In fuzzy control the total result is calculated by apply-ing the Maximum-operator for aggregating the evalua-tions of the rules with positive DOF. But in decision sup-port systems corrections should be considered. On theother hand it seems absurd to add the DOFs if two ormore rules turn up with the same ,,outputa, because itwould then be possible to get DOF-values greater then 1.W e propose to ad opt a middle course and suggest the useof the algebraic sum.In order to ,,compressu the data to a unique valuation,the well known defuzzification procedures can be ap-plied. The best-known are the center of gravity methodand the center of area method. But in hierarchical sys-tems defuzzifying is not necessary, because it is better touse directly the fuzzy valuations as inputs for the nextaggregation step.7. F i n a l R e m a r k sThe application of linguistic variables and fuzzy conjunc-tion methods offer an appropriate method to model thehuman reflection process. By doing s o expert systems areconstructed which actually deserve this name. There arestill a lot of questions left to be answered, but I am con-vinced that thls concept aims at the right direction. In themeantime, the first applications are used in practice.

References1. Bagus, T.: Wissensbasierte Bonitatsanalyse fiir da sFirmenkundengeschaft der Kreditinstitute. Peter Lang-Verlag Frankfurt a. M. 19922. Kahlert, J.; F rank, H .: Fuzzy-Logik und Fuzzy-Control. Vieweg Verlag, Braunschweig~Wiesbaden

19933. Miiller, C.: Entwicklung eines wissensbasietrenSystems zur Unterstiitzung der AnalytischenP ~ f u n g s h a n d l u n g e n irn Rahmen derJahresabschluBpr-fung. Peter Lang Verlag, Frankfurtam Main 19964. Rommelfanger, H.: Fuzzy Decision Support-Systeme

- Entscheiden bei Unscharfe. Springer-Verlag BerlinHeidelberg 1988, second edition 19945. Rornmelfanger, H.: Fuzzy Logic-Based Processing ofExpert Rules Used for Checking the Credit Solvencyof Small Business Firms or for Supporting Analytic

Procedures of Auditors. In: Ribeiro R.R. a.o.(Eds.)Soft Computing in Financial Engineering. Physica-Verlag, Heidelbe rg 1999, 37 1-2446. Scheffels, R.: Fuzzy Logik in derJahresabschluBpriifung. - Entwicklung eineswissensbasierten Systems zur Analyse derVermogens-, Finanz- und Ertragslage. DeutscherUniversitatsverlag, Wiesbaden 19967. Sugen o, M. (Ed.): Industrial Application s of FuzzyControl. North Holland Amsterdam 19858. Tilly, T.: Fuzzy-Loglk. Franzis-Verlag, Miinchen19919. Yager, R.R.; Zadeh, L.A.: An Introduction to FuzzyLogic Applications in Intelligent Systems. KluwerDordrecht 199210. Zirnrnermam, H.J.: Fuzzy Sets, Decision Makingand Expert Systems. Kluwer Academic Publishers,Boston 1987