ON USING FUZZY ARITHMETIC TO SOLVE PROBLEMS WITH UNCERTAIN MODEL PARAMETERS

Transcription

1 ON USNG FUZZY ARTHMETC TO SOLVE PROLEMS WTH UNCERTAN MOEL PARAMETERS Michel Hnss, Ki Willner nstitute A of Mechnics University of Stuttgrt Pfffenwldring Stuttgrt, Germny ASTRACT Fuzzy rithmetic, bsed on Zdeh s extension principle, is presented s tool to solve engineering problems with uncertin model prmeters. Fuzzy numbers re introduced, nd different concepts to prcticlly implement the uncertinty of the prmeters re discussed. As n exmple, rther simple but typicl problem of engineering mechnics is considered. t consists of determining the displcements in two-component mssless rod under tensile lod with uncertin elsticity prmeters. The ppliction of fuzzy rithmetic directly to the trditionl techniques for the numericl solution of the engineering problem, however, turns out to be imprcticble in ll circumstnces. n contrst to the use of exclusively crisp numbers, the results for the clcultions including fuzzy numbers usully differ to lrge extent depending on the solution technique pplied. The uncertinties expressed in the different clcultion results re then bsiclly twofold. On the one hnd, uncertinty is cused by the presence of prmeters with fuzzy vlue, on the other hnd, n dditionl, undesirble uncertinty is rtificilly creted by the solution technique itself. This fuzzy-specific effect of rtificil uncertinties is discussed nd some concepts for its reduction re presented. KEYWORS Uncertinty, fuzzy rithmetic, finite element method. NTROUCTON To chieve relible results for the numericl solution of engineering problems, exct vlues for the prmeters of the problem equtions should be vilble. n prctice, however, exct vlues cn often not be provided. The model prmeters usully exhibit vribility, e.g. due to irregulrities in fbriction when considering the geometricl dimensions or the physicl properties of mteril. Thus, the results obtined for solutions tht just use some specific crisp vlue for the uncertin prmeters cnnot be considered to be representtive for the whole spectrum of possible results. To solve this limittion, the ppliction of fuzzy set theory [5 proves to be prcticl pproch. More specificlly, the uncertinties in the model prmeters cn be tken into ccount by representing the effects of sctter by fuzzy numbers with their shpe derived from experimentl dt. y this technique, one cn demonstrte how initilly ssumed uncertinties re processed through the clcultion procedure leding finlly to fuzzy results tht reflect the relibility of the problem solution. Additionlly, the fuzzy results llow the computtion of crisp vlue s the most likely result for the problem which in generl differs from the result chieved by n initilly non-fuzzy pproch using only crisp prmeters. The implementtion of fuzzy numbers nd fuzzy rithmeticl opertions, however, turns out to be non-trivil problem. Though the generlized mthemticl opertions for fuzzy numbers cn theoreticlly be defined mking use of Zdeh s extension principle [6, rel-world ppliction of fuzzy rithmetic rises two fundmentl problems: first, the problem of prcticlly implementing fuzzy numbers with rbitrrily shped membership functions to be successfully hndled by fuzzy rithmeticl opertions, nd second, the necessity of considering the degree of dependency between fuzzy numbers, to reduce rtificil uncertinties which prove to be chrcteristic phenomenon of binry fuzzy rithmeticl opertions.

2 > PSfrg replcements FUZZY ARTHMETC efinition of fuzzy numbers siclly, fuzzy numbers cn be considered s specil clss of fuzzy sets showing some specific properties [4. The fuzzy sets themselves result from generliztion of conventionl sets by llowing elements of universl set not only to entirely belong or not to belong to specific set, but lso to belong to the set to certin degree [5, 7. Thus, fuzzy sets cn be expressed by the elements of universl set with certin degree of membership ssigned. The elements belonging to conventionl sets, insted, re chrcterized by degrees of membership tht cn only be equl to zero or unity, i.e. by membership function. On this bsis, closed intervls nd crisp numbers of the form!#"$ %" & ' ()&'* %,+- (1) cn be considered s conventionl subsets of the universl set +- which cn lso be expressed by /. 0' for #"$ %" else for ()& else (2) when the membership function : ;<, %,+-, is used (Figure 1). &? Figure 1: Closed intervl, crisp number & nd fuzzy number of Gussin shpe with men vlue? expressed by their membership functions. Fuzzy numbers, insted, re defined s convex fuzzy sets over the universl set +- with their membership functions where *@ is true only for one single vlue,? %+-. As n exmple, symmetric fuzzy numbers of Gussin shpe re defined by the membership function *)A< #C? E FHG (3) where? G nd denote the men vlue nd the stndrd devition of the Gussin distribution (Figure 1). To define unry or binry rithmeticl opertions with fuzzy numbers, two concepts cn be pplied. On the one hnd, the fuzzy numbers cn be decomposed into sets of intervls for different degrees of membership. The rithmetic for fuzzy numbers cn thus be reduced to intervl rithmetic [1, 4. On the other hnd, Zdeh s extension principle ccording to Eq. (5) cn be pplied which extends the evlution of rithmeticl functions from crisp to fuzzy-vlued opernds. Explicitely, if nd re fuzzy numbers defined by the membership functions /J 0, K +-, nd J2 4L, L M+-, the result of the binry opertion &N)O%P RQ (4) for n rbitrry function O is determined by /J 8 TS UWVYX Z[9\_^ 1 `cbedgfih /J 0 J2 4L jk (5) mplementtion of fuzzy numbers To gurntee the successful inclusion of uncertinties into the solution procedures of engineering problems, the fuzzy numbers tht re used to represent the uncertin model prmeters must be implemented in n pproprite form tht complies t lest with the following requirements: l The form should llow the comprehension of fuzzy numbers with rbitrrily shped membership functions, considering especilly the cse of fuzzy numbers with their shpe derived from mesured dt. l The form should llow prcticl reliztion of the fuzzy rithmeticl opertions between fuzzy numbers by voiding ny loss of informtion in the uncertinty. n the ensuing, two different concepts of implementing fuzzy numbers re presented: the often pplied concept of using tringulr fuzzy numbers nd the more promising pproch of using discretized fuzzy numbers. Considering definite uncertin prmeter, mesured dt for the prmeter re ssumed to be

3 > " PSfrg replcements vilble from which normlized distribution function 0 cn be derived tht expresses the frequency of occurrence of certin mesured vlue for the prmeter within the intervl. n most cses, these dt pproximtely show Gussin distribution. The uncertinty in the prmeter with cn then be modeled by fuzzy number the membership function 0 J of the form #C? 0 E 0 J *)A FHG 0 (6) where? 0 nd G 0 re the men vlue nd the stndrd devition of the Gussin distribution (Figure 2). Figure 2: Normlized distribution function 0 for the uncertin prmeter pproximted by the. membership function 0 J of the fuzzy number Tringulr fuzzy numbers PSfrg replcements Up to now, fuzzy numbers hve primrily been implemented s L-R fuzzy numbers [3. n this form, the membership function of fuzzy number is chrcterized by its scending left nd its descending right brnch which on their prt re expressed by prmeterized functions tht belong to definite clss of bsis functions. For resons of simplicity, liner functions re mostly chosen s the bsis functions. The resulting specil type of L-R fuzzy numbers cn then be referred to s tringulr fuzzy numbers (TFN) [4. The originl fuzzy number with the membership function 0 J in Eq. (6) cn be pproximted by symmetric tringulr fuzzy number with the membership function 0 J tht cn be obtined by postulting 0J? 0 * 0 J? (7) nd 0 J, The membership function J 0 J k (8) 0 of the tringulr fuzzy number is then defined by with /J 0 * b! 7 Nc C " %$ F'& G C? 0 " # (9) 0 (10) which cn lso be expressed in the short form )(? 0 C "? 0? "-,/ * k (11) The membership functions 0 J nd 0 J of the originl fuzzy number nd its pproximtion 4 re illustrted in Figure 3. 0 J 0 J? 0 C "? 0? 05* Figure 3: Originl fuzzy number with Gussin shpe nd its liner pproximtion s tringulr fuzzy number /. Obviously, the use of tringulr fuzzy numbers shows two mjor dvntges. First, the very simple wy of implementtion with liner functions nd only three prmeters tht cn even be reduced to two prmeters in the cse of symmetric fuzzy numbers. And second, the quite uncomplicted reliztion of the elementry fuzzy rithmeticl opertions leding gin to tringulr fuzzy numbers [3. With respect to the ppliction of fuzzy rithmetic to solve engineering problems with uncertinties, the disdvntges of this concept, however, re of higher weight. The tringulr fuzzy numbers re just rough pproximtion of the relly existing uncertinty s cn be seen from Figure 3. Furthermore, to successfully perform even the nonliner elementry rithmeticl opertions, s multipliction nd division, the membership

4 J J J S > 0J functions of the resulting fuzzy numbers hve to be pproximted by liner functions fter ech of those opertions to void chnge of the type of bsis function. Within the solution procedure of n engineering problem with uncertin prmeters, these multiple pproximtions cuse n enormous loss of informtion of the initilly induced uncertinty. iscretized fuzzy numbers the elements of the support of the discrete fuzzy number re finlly given by P P Q 0J nd Q T H) kk k k (15) n cse of Gussin fuzzy numbers, the elements PSfrg replcementsfor / re to be neglected. Motivted by the generl prctice of smpling nlog signls for computer-bsed signl processing, the membership functions of rbitrry shpe cn be discretized, leding to discrete fuzzy sets for which the fuzzy rithmeticl opertions cn then be defined using Zdeh s extension principle. siclly, two different wys of obtining discretized fuzzy numbers seem to be possible: discretizing the membership functions by subdividing either the bsciss or the ordinte into intervls of definite length. Splitting up the bsciss, i.e. the -xis, however, turns out unsuitble due to some undesirble effects tht cn be observed when evluting fuzzy rithmeticl opertions for such fuzzy numbers. As the mjor effect, the resulting fuzzy sets show shpes tht vry depending on how the -xis is subdivided. Thus, if the intervls re chosen rther lrge to reduce computtionl efforts, the fuzzy sets might show no convexity ny more nd cn consequently no longer be considered s fuzzy numbers lthough, strictly speking, they still re. Those problems cn effectively be voided if the -xis is subdivided into number of segments, eqully spced s illustrted in Figure 4. The fuzzy number cn then be pproximted by the discrete fuzzy number which cn be expressed in the form R kk kc (12) k kkc * i H@ kk k nd *k (13) ntroducing the strictly monotonous functions 0 nd 0 to denote the scending nd the descending brnch of the membership function 0 J in the form J # 0 J * 0 J % 7 0 J " # (14) 0 J 0 J? 0 0 J Figure 4: Approximtion of the fuzzy number by the discrete fuzzy number. Using discretized fuzzy numbers of type (12) nd (13) to represent fuzzy numbers like, in Eqs. rithmeticl opertions with fuzzy numbers cn successfully be implemented by defining the opertions to be executed seprtely for the elements of ech degree of membership [3. Out of the number of possible combintions between these elements, the number of vlid combintions tht finlly led to the proper results cn be determined by mking use of Zdeh s extension principle. For exmple, for the elementry rithmeticl opertion of dding two strictly positive fuzzy numbers nd using their discrete forms nd, the discrete form & of the resulting sum & * cn be obtined by combining only the elements of the scending brnches of both fuzzy numbers on the one side nd the elements of the descending brnches on the other. Explicitly, if nd re defined ccording to Eq. (12) over nd L, & is given over S by & S k kkc S R (16) S k kkc S with ) * L nd S Y kkk k * L (17)

5 > PSfrg replcements n generl, however, the specific ppernce of the formul in Eqs. (16) nd (17) depends on the type of rithmeticl opertion to be relized nd on whether the domin covered by the fuzzy numbers is strictly positive, strictly negtive or both positive nd negtive. The presented principle of implementing fuzzy numbers nd fuzzy rithmeticl opertions is very similr to the well-known method of decomposing fuzzy number into number of -cuts,, for which the rithmeticl opertions cn then be defined using intervl rithmetic [4. The mjor dvntge of the present method, however, is its extensibility to the hndling of fuzzy sets with incoherent -cuts. Such fuzzy sets re the results of some rther problemtic rithmeticl opertions, e.g. the division by fuzzy number with its support contining both positive nd negtive elements. With respect to the convexity PSfrg of fuzzy replcements numbers s defined on the bsis of coherent - cuts, the results of such opertions would not be considered s fuzzy numbers ny more. With the implementtion presented bove, however, those results cn still be chrcterized by n scending nd descending brnch (not necessrily left nd right one) nd cn thus be used for further processing. Reserch work on the hndling of those degenerted fuzzy numbers is currently in process, nd it lredy shows some promising results. As n exmple, the fuzzy number with membership function of Gussin shpe (?, ) nd its inverse J re shown in Figure 5. of the rod s components re to be determined (Figure 6). The components of the rod re chrcterized by the length prmeters nd, the cross sections nd nd the Young s moduli nd quntifying the elsticity of the components. The externl loding consists of the tensile force cting t the end of the rod. To determine the displcements nd, the finite element method cn be pplied, leding finlly to the eqution & * & CN& CN& & with the stiffness prmeters & (18) / F k (19) Figure 6: Two-component rod under tensile lod. When mteril uncertinty in elsticity is considered, the Young s moduli nd re no longer crisp, but behve s fuzzy prmeters nd with their vlues given by fuzzy numbers. Thus, the globl stiffness mtrix in Eq. (18) becomes fuzzy-vlued mtrix! with the fuzzy stiffness prmeters & / F k (20) Solving the system eqution (18) for the unknown fuzzy-vlued displcement vector ", one obtins >R >R Figure 5: Fuzzy number (Gussin shpe, [.?, ) [ nd its inverse ENGNEERNG PROLEM As rther simple but typicl engineering problem, two-component mssless rod under tensile lod is considered where the displcements nd "! $# (21) which finlly leds to the following expressions for nd : &% & (22) & * & (' k (23) Since this solution is chieved by consecutive symbolic simplifiction of Eq. (21), this expression constitutes the cnonicl solution of Eq. (18), nd will be refered to s the exct solution.

6 k > G G 2 2 When considering the usul cse of globl stiffness mtrix! with lrge dimension, symbolic simplifiction of the system solution is imprcticl. For this reson, the finite element problem is usully solved numericlly using specil computer progrms. n the following, two different wys of numericl solution shll be presented. n the first method, the finite element problem cn be solved ccording to Eq. (21), i.e. by determining t first the inverse globl stiffness mtrix! nd then forming the mtrix product! #. This procedure leds to % P & % P & * * & Q & C & Q & C P & Q ' P & Q ' & (24) (25) P & * & Q Finlly, three different results, *) nd cn be obtined for the fuzzy-vlued displcement n the second method, since the globl stiffness t the end of the rod, depending on the solution mtrix is usully symmetric nd positive definite, technique pplied (Figure 7). This fuzzy-specific the finite element problem in Eq. (18) cn be solved effect of getting different results for different solution procedures is explined nd discussed in the effectively by n T decomposition of! where denotes lower tringulr PSfrg mtrixreplcements with next section. digonl terms being unity nd is mtrix of digonl form. The problem is then solved by forwrd nd bck substitution procedures ccording to # T " k (26) As n exmple, the finite element problem is now solved for definite prmeter configurtion, where the first component of the rod is ssumed to be steel nd the second luminum with the geometry prmeters nd the externl loding specified by H b b Mk (29) The uncertin Young s moduli re considered to hve membership functions of Gussin shpe with + J? F k <!"#?? $Yk %< '&(!"#? k (30) The dvntges of the T decomposition re the following: 1. For constnt mtrix! the cost intensive prt of the clcultion, nmely the decomposition, cn be performed t the outset for ll right-hnd sides #. This proves to be especilly useful for different lod cses or trnsient clcultions. 2. An often encountered bnd width structure of! is preserved by nd cn be stored in-plce with on the min digonl. Formulting this method for the problem considered, one obtins & C & C & & * & & & * & & & * & (27) k (28) *) ', ', >, >, -, - [mm Figure 7: Fuzzy-vlued displcement t the end of the rod. Exct solution:, solution by. / T decomposition:, solution by inversion: *). When defuzzifiction of the fuzzy-vlued result is pplied to determine crisp-vlued equivlence 10 using the center-of-re method [2 with 0 + J + J (31)

7 ( the resulting crisp number 0 cn be considered s the expected vlue for when uncertinty in the model prmeters is ssumed. As for the exct so- lution ccording Eq. (23), defuzzifiction of leds to )Ykg F % k (32) 0 Considering the cse when no uncertinty is implied, the non-fuzzy solution of the problem with the crisp Young s moduli? F k <'? $k %<& results in the crisp displcement )k_ F $ (33) (34) which equls the vlue with the mximum degree of membership of the support of, i.e. where + J. Thus, due to the generlly symmetric shpe of the fuzzy-vlued result, cused by the nonliner rithmeticl opertions in the solution procedure, the expected result 0 for the problem under the ssumption of given symmetric uncertinty in the elsticity prmeters differs from the result of the crisp solution with no uncertinty ssumed by 0.25 %. Although this difference is not very significnt in this simple exmple, it more nd more becomes n importnt fctor when rel-world pplictions re considered nd the dimension of the corresponding finite element problem increses. SCUSSON The reson for the observed fuzzy-specific effect of getting different results depending on the solution procedure will become clerer when considering the following simple exmple where tringulr fuzzy numbers in the form lower bound pek vlue upper bound,.1032 ccording to Eq. (11) re used for resons of simplicity. Let be vrible of fuzzy vlue given by the tringulr fuzzy number e (E F,.1032 (35) nd let be nother fuzzy vrible of different origin given by the tringulr fuzzy number (W F,!.10 2 (36) with the sme men vlue nd the sme uncertinty rnge s the vrible. Evluting the expression L * (37) by using fuzzy rithmetic for tringulr fuzzy numbers [4 then leds to L (EC F,.1032 (38) However, by pplying some preceding symbolic clcultion to the expression in Eq. (37) one obtins L * which cn be rewritten s * (39) (40) when the priori knowledge is included tht the quotient of two identicl vribles must be equl to crisp unity. The evlution of Eq. (40), insted, results in L )( k Y F Yk,.10 2 (41) which represents fuzzy number with the sme men vlue s L but with only hlf the rnge of uncertinty. The reson for the different results of L nd L must be seen in the light of uncertinties of different origin. Wheres the uncertinties in L re just nturl ones, rising from the initil ssumption of fuzzy prmeters, the numericl result L for the non-simplified expression in Eq. (37) includes dditionl, rtificil uncertinties induced by the specil solution technique itself. Explicitly, both frctions in Eq. (39) re numericlly treted the sme wy leding to fuzzy unity due to. Thus, in contrst to symboliclly preprocessed solution, purely numericl solution technique is only cpble of hndling the feture equlity of vlues, but not identity of vribles. n other words, the conventionl fuzzy rithmetic does only led to results free of rtificil uncertinties if unry rithmeticl opertion re crried out or for binry rithmeticl opertions if the opernds re completely independent, i.e. if they stem from fuzzy prmeters of different origin. n cse of completely dependent opernds, nother fuzzy rithmetic hs to be defined which is not bsed on Zdeh s extension principle. As for the fuzzy finite element problem, the results for the exct solution expressed by in Figure 7 re free of ny rtificil uncertinty since this

8 solution completely fulfills the requirement mentioned bove. Using the solution with inversion of the globl stiffness mtrix!, insted, the results re chrcterized by rtificil uncertinties tht re bout s four times s lrge s the nturl ones. These rtificil uncertinties cn effectively be reduced, lthough. they cnnot be voided, when using the T decomposition of! for numericlly solving the fuzzy finite element problem. MOFE SOLUTON PROCEURES / Since the T decomposition proves to be the type of numericl solution technique with the lowest degree of rtificil uncertinties, this method shll be used s the bsis for further improvements which finlly led to modified solution procedure. However, it is not possible to specify one globlly vlid solution technique for fuzzy finite element problems; there will insted be different modifictions depending on the prmeter configurtion of the problem. The following re two prcticl modifictions tht could be dopted: (1) n most cses the elsticity prmeters of different elements, i.e. the fuzzy vlued Young s moduli, cn be ssumed to be uncertin to the sme percentile extent. Thus, the fuzzy vlued nd Gussin shped prmeters & of the globl stiffness mtrix! cn be rewritten s &? * (42) where represents fuzzy zero with the men vlue? ) nd the stndrd devition G. Since hs the sme vlue for ll elements, it cn be fctored out of the stiffness mtrix. The T decomposition, including most of the numericl opertions to solve the problem, cn then be performed for crisp stiffness mtrix voiding ny production of rtificil uncertinties. y this, the criticl fuzzy rithmeticl opertions cn be reduced to smller number consisting of forwrd nd bck substitution ccording to Eq. (26). (2) f t lest some of the fuzzy prmeters show identicl percentile uncertinty, prtitioning of the globl stiffness mtrix! cn be performed, enbling prtilly crisp T decomposition of the mtrix s described bove. Using this block elimintion technique s modified solution procedure, the ocurrence of rtificil uncertinties cn lso be reduced. CONCLUSON The ppliction of fuzzy rithmetic to solve engineering problems with uncertinties in the prmeters is powerful but lso problemtic tool. Although the nturl uncertinties ssocited with mteril vribility cn be considered to be cceptble, the rtificil uncertinties tht rise from computtionl spects in the finite element procedure must be minimized. Thus, to chieve prcticl results, the different numericl techniques for solving finite element problems should not be pplied in their common form, but should be modified with respect to reduction of criticl fuzzy rithmeticl opertions tht cn cuse rtificil uncertinties. Presently, these modifictions of the solution techniques must be redefined for every specil prmeter configurtion by some symbolic preprocessing of the solution schemes. One concept to solve this limittion is the development of self-modifying solution procedures. Another pproch, insted, consists of introducing new fuzzy rithmetic which provides different reliztion of the opertions between dependent nd independent fuzzy numbers. Reserch ctivities in this field re currently in progress. REFERENCES [1 ALEFEL, G., HERZERGER, J. ntroduction to ntervl Computtions. Acdemic Press, New York, [2 RANKOV,., HELLENOORN, H., REN- FRANK, M. An ntroduction to Fuzzy Control. Springer-Verlg, erlin, 2. edition, [3 UOS,., PRAE, H. Fuzzy Sets nd Systems: Theory nd Applictions. Mthemtics in Science nd Engineering, Vol Acdemic Press, New York - London, [4 KAUFMANN, A., GUPTA, M. M. ntroduction to Fuzzy Arithmetic. Vn Nostrnd Reinhold, New York, [5 ZAEH, L. A. Fuzzy sets. nformtion nd Control, 8: , [6 ZAEH, L. A. The concept of linguistic vrible nd its ppliction to pproximte resoning. Memorndum ERL-M 411, erkeley, C., Oktober [7 ZMMERMANN, H.-J. Fuzzy Set Theory nd its Applictions. Kluwer Acdemic Publishers, ordrecht, 2. edition, 1991.