Multiobjective fuzzy optimization method

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Edmund Jennings
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1 Buletnul Ştnţfc al nverstăţ "Poltehnca" dn Tmşoara Sera ELECTRONICĂ ş TELECOMNICAŢII TRANSACTIONS on ELECTRONICS and COMMNICATIONS Tom 49(63, Fasccola, 24 Multobjectve fuzzy optmzaton method Gabrel Oltean Abstract -The paper proposes a new multobjectve optmzaton method, based on fuzzy technques. The method performs a real multobjectve optmzaton, every parameter modfcaton tang nto account the unfulfllment degrees of all the requrements. It uses fuzzy sets to defne fuzzy objectves and fuzzy systems to compute new parameter values. The strategy to compute new parameter values uses local gradent nformaton and encapsulates human expert thnng. After ntroducng our optmzaton method, we optmze the desgn of a fnte response flter. Keywords: multobjectve, fuzzy objectve, fuzzy system, populaton of solutons I. INTRODCTION Mathematcal formulaton of general optmzaton problem (GP - General Problem s [],[2]: Fnd x that mnmze f ( x subject to: g j (x, j =,...,m h q (x =, q =,...,p ( xl x x u It s formulated an constraned optmzaton problem wth a sngle objectv (a scalar one. The rgdty of the mathematcal problem posed by the general optmzaton formulaton s often remote from that of a practcal desgn problem. Rarely does a sngle objectve wth several hard constrants adequately represent the problem beng faced. More often there s a vector of objectves f(x = {f (x, f 2 (x,, f o (x} that must be traded off n some way. Multobjectve optmzaton s concerned wth the mnmzaton of a vector of objectves f(x that may be subject of a number of constrans or bounds: Fnd x that mnmse {f (x, f 2 (x,, f o (x} subject to: g j (x, j =,...,m h q (x =, q =,...,p (2 xl x x u Because f(x s a vector, f any of ts components are competng, there s no unque soluton to ths problem. Instead, the concept of nonnferorty (also called Pareto optmalty must be used to characterze the objectves [2], [], [4]. A nonnferor soluton s one n whch an mprovement n one objectve requres a degradaton of another. The technques for multobjectve optmzaton are wde and vared. Goal attanment method of Gembc nvolves expresng a set of desgn goals f r = {f r, f r 2,, f r o} assocated wth the set of desgn objectves. The multobjectve optmzaton problem s then expressed as a standard optmzaton problem usng the formulaton [5]: mnmse γ, γ R r such that f (x-w(xγ f ; =,...,o (3 In order to transpose the real word problem n the mathematcal language for the optmzaton frst the objectve functons should be defned. The objectve functons are the measures between the actual values of the functons and the requred values. The objectve functons must be carefully selected so that they lead to the requrements achevement. In ths paper, we propose a new fuzzy multobjectve optmzaton method. Our method smultaneously optmzes all the objectves so t drectly solves the multobjectve problem, wthout any transformaton nto an one objectve problem. The algorthm relays on fuzzy sets to formulate the optmzaton objectves and on fuzzy systems to compute the new values for the varables n every optmzaton teraton. Because t s a gradent method, a populaton of soluton can be employed n order to dramatcally ncrease the chance to obtan the best possble soluton. II. THE FZZY OPTIMIZATION METHOD The optmzaton method should converge to a global optmal soluton n a reduced number of teratons. Ths s not a smple tas due to the complex relatons between varables and nonlnear mult-varables functons to be optmzed. A varable can affects qute dfferent more than one functon at a tme, so when t s modfed n order to mprove one objectve functon t can damage another. A. Formulaton of the optmzaton problem Consecrated formulaton of the optmzaton problem s somehow rgd and does not always reflect the realty. Such a formulaton, very restrctve, reduces the possblty to mae trade-offs, a very mportant Techncal nversty of Cluj-Napoca 22

2 factor n the optmzaton. Consequently, the soluton space s confned and n many cases, an optmal soluton does not exst. One way to overcome these drawbacs s to use fuzzy sets to defne optmzaton objectves. We wll fuzzfy the requrements gettng ths way the possblty to consder dfferent degrees for requrement achevements and acceptablty degrees for a partcular soluton. We wll assocate wth each requrement one or two fuzzy sets whose membershp functons wll represent the correspondng fuzzy objectve functons. For example for the requrements greater or equal f (x f r, and equal f (x=f r the correspondng fuzzy objectve functons are presented n Fg. µ µ D D f (x * f r K f a b Fg.. Fuzzy objectve functons: a f K f r K ; b f K = f r K The fuzzy objectve functons are µ (f (x : D f [, ] (4 where D f s the range of possble values for f (x. µ (f (x ndcates the error degree n accomplshng the th requrement, so we wll call them unfulfllment degrees (D. A value µ = means a fully achevement of the fuzzy objectve, whle a value µ = means that fuzzy objectve s not acheve at all, ths occurs when f (x taes an unacceptable value. In Fg.. We can see, for the current value of the varables vector x * the correspondng value of the unfulfllment degree s D *. Our new multobjectve optmzaton problem formulaton s: Fnd x that mnmse {µ (f (x, µ 2 (f 2 (x,, µ o (f (x} (5 B. The dea of populaton of solutons Startng the optmzaton wth only one ntal soluton, we can reman bloced nto a local Pareto optmal pont, where an mprovement n one objectve requres a degradaton of another. If we can obtan a set of local Pareto optmal ponts, t s hghly possble to have the global Pareto optmal pont among them. So, nstead of usng one search path we suggest usng a parallel search dealng wth the dea of populaton of solutons consstng of canddate solutons. The optmzaton starts wth the ntal canddate solutons. In our mplementaton, these ntal canddate solutons can be obtaned n several ways: randomly f r K f (x * f generated, generated wth Latn Hypercube Technque, or user provded. In each teraton, for every canddate soluton the actual functon value, the Ds and new parameter values are computed. If the Ds for one canddate can not be decreased anymore, we have found a local Pareto optmal pont and the future teratons wll not vst ths canddate soluton, shortenng the entre optmzaton tme. The optmzaton algorthm stops n one of the followng stuatons: all the Ds become zero for one canddate soluton. Ths canddate soluton s consdered a global Pareto optmal pont and t s our fnal soluton. We wll not contnue to search other Pareto optmal pont on the remanng search paths. none of the canddate solutons can be further mproved, meanng that the set of local Pareto optmal ponts was obtaned. As the fnal optmal soluton we chose the one wth the mnmum value of the mean of unfulfllment degrees (MD, consdered as global optmal pont. Also the algorthm wll stop f the maxmum number of teratons s reached. C. New parameter values computng The method for computng the new values for the varables nvolves fuzzy technques and local gradent nformaton. Each varable can affect more or less each objectve functon. In our method the sgn and the value to modfy a certan varable taes nto account the Ds, the gradents and the relatve mportance of the nvolved varables n relatons wth the objectve functons. Our method acts as a human expert for a certan crcut performance: t s better to modfy more the parameter wth greater mportance, because t can really affect the performance, and the modfcaton also depends on the unfulfllment degrees of the correspondng requrements. the parameter wth lower mportance s modfed less or not at all, because ts nfluence on crcut performance s nsgnfcant. the fnal modfcaton of a parameter s a weghted sum of the partal modfcaton (mposed by every objectve functon. Such human expert nowledge s captured and ncorporated n our method by means of a fuzzy logc system. The algorthm to compute the new varable values follows: In each teraton: Compute the local gradents of the functons n relaton wth each varable, f ( x - the local gradent of f functon n relaton wth x varable. For every functon f we compute the mportance of the varables v f ( x that shows the relatve mportance of every x n modfyng the functon f. 22

3 These mportance of the varables are computed based on absolute values of the local gradents: v f (x ( x = ; =,...n;,..., o (6 n f (x f = = r For every requrements f compute the D as a membershp degree of the actual value of the correspondng functon f (x * to the assocated fuzzy objectves. Two examples are shown n Fg.. v For every varable x and every functon f we compute a partal coeffcent to modfy that parameter. Ths partal coeffcent coef x (f s computed by a frst order Taag-Sugeno fuzzy system (Fg. 2. Fg.2. Partal coeffcent computng The fuzzy sets for the nput lngustc varables D and mportance and for output lngustc varable coef-part are not presented here due to the lac of space. The fuzzy rules are presented n Table. where, for example the 4 th column and the 3 nd row gve the followng fuzzy rules: If DR s Medum and mportance s Small then coef-part s Small. Table D Importance Z S M L Z Z S VS S M M S M L L S L VL Z Zero VS Very Small S -Small M Medum L Large VL Very Large The control surface generated by ths fuzzy system s presented n Fg. 3 gradent and on the drecton (go up or go down n whch the functon must be modfed. So we obtaned partal coeffcents wth sgn: scoef x ( f. v For every varable x we compute the functon nfluence upon varable modfcaton p x ( f that shows the relatve mportance of every functon f to compute the modfcaton of x parameter. p f (x ( f = ; =,..., o;,..., n (7 o f (x x = = v The coeffcents used for modfyng each parameter are computed as weghted sum of the partal coeffcents, the weght beng the nfluences ( x f p. It means that the greater the nfluence s, the greater the contrbuton on the partal coeffcent. v = scoef x = x new o ( px ( f scoef x ( f o p x = ( f Compute new varable values: = x + scoef x abs (8 ( x' + xmn (9 where x I taes the value of x from 3 teratons bac f n all these 3 teratons we have the same modfcaton sgn for t as n the actual teraton. Otherwse x I taes the value of x from the current teraton. We found that usng x I nstead of smple x we can change the varable sgn and mprove the convergence of the algorthm. Also more help n the sgn changng, convergence and accuracy of fnal soluton can by obtaned usng the varable xmn. It taes a default value to the begnnng of the optmzaton, and that value s dynamcally decreased f some oscllatons appear n the mean of unfulfllment degree. Fnally, we should menton that the optmzaton method acts n an adaptve manner: when the Ds are large (towards we have large coeffcents to modfy the varables (see Table. For small Ds we have small coeffcents to modfy the varables, so we can focus our search so that the soluton converges to the exact local Pareto optmal pont. III. IMPLEMENTATION Fg.3. Control surface to compute partal coeffcent v The partal coeffcents coef (f receve a plus or mnus sgn dependng on the sgn of the local x In order to chec and valdate our multobjectve optmzaton algorthm we mplemented a prototype system n Matlab for Wndows. The prototype conssts on a man functon optfuzz and other secondary functon. The man functon should be 222

4 nvoed from Matlab worspace wth a seres of arguments : fun a strng contanng the name of the Matlab functon that computes the objectve functons; reqs vector of numercal values of the requrements; sgn vector wth +, or values, wth the same length as reqs vector. When the values s + optfuzz attempt to mae the objectve functon greater or equal to correspondng requrements; for optfuzz attempt to mae the objectve functon less than the correspondng requrements; for optfuzz attempt to mae the objectve equal wth the correspondng requrements nrvar number of varables weght vector wth weght for the objectve functons proc vector wth values n (, that control the fuzzy sets defnng fuzzy objectves lb a vector of lower bounds of the varables; ub a vector of upper bounds of the varables; nt_sol varable that set the method for generatng the ntal soluton optons vector wth some optons of the optmzaton algorthm (number of teratons, number of canddate solutons, ntal value for xmn The user can provde empty values for some of the above arguments; n ths case, the default values are used. The user should only wrte hs objectve functons and run the optfuzz wth the arguments show above. The optmzaton routne return the fnal values of objectve functons, the values of the varables, the D for each requrements and a curve wth the evoluton of MD durng the optmzaton for the canddate soluton that provde fnal soluton. some tolerance. We must use the dscretzaton of the frequency doman we are nterested n. The number of functon to be optmzed equals the number of dscrete frequency, and the number of varables equals the number of a coeffcents. Frst we use only 5 (unform dstrbuted frequency n each doman, so a number of functon to optmze and a number of 5 varables. The value of the requrements are for the fve frequency n [;.] range, and for the fve frequency n [.5;.5]. Because frequency between.hz and.5 Hz are not specfed, no requrements are needed here. We run the optmzaton algorthm for a populaton of 9 canddate solutons, for 5 maxmum number of teratons, wth randomly generated ntal solutons. In order to see how the nternal computatons deploy, we reproduced n Fg.4. the evoluton of 3 (from a total of Ds durng the optmzaton, for st canddate soluton. D s Iteratons 3.5 x -4 3 a IV. RESLTS In order to hghlght the behavor of our new fuzzy multobjectve optmzaton method, we use t to solve some multobjectve optmzaton problems. Consder desgnng a lnear-phase Fnte Impulse Response (FIR flter. The problem s to desgn a low pass flter wth magntude one at all frequency between and. Hz and magntude zero between.5 and.5 Hz. The frequency response H ( f for such a flter s defned by D s Iteratons Fg. 4. Evoluton of there Ds. a Full process; b Detals: fnal teratons b 2M H( f = h n= M A( f = a n= j2πfn ( n e = A( f ( n cos( 2πfn j2πfm e ( where A(f s the magntude of the frequency response. So the problem s to compute the magntude coeffcents a(n so that the magntude response matches the desred response (at each frequency wth From the Fg.4. a one can see that n the frst teratons ( up to around 4 all the Ds have (large varatons. Ths s because we are far from a good soluton and each functon ass for hgh modfcaton of the varables. Remember that each functon depends on each varable. After ths transent regme, all Ds falls towards zero and contnue to decrease, up to an magntude order of -4 n the fnal teratons (fg.4. b, to reach, as close as possble, a value for Ds. Fg.5. depcts the evoluton of the mean unfulfllment degree MD (arthmetc mean of all Ds, that globally characterze the 223

5 optmzaton process, also for the st canddate soluton. In the frst 5 teratons, large oscllatons (due to large changes n each D and then a rapd mprovement n the value of MD can be seen. The algorthm s very close to a good soluton (MD= , n teraton 5 After that, the algorthm try to mprove the soluton, contnung D D Iteratons x Iteratons Fg. 5. Evoluton of the mean D. up: full process; bottom: detals for fnal teratons to decrease the MD up to the value n teraton 5. The convergence s slower, because we are close to the deal soluton, so the algorthm should move carefully around. If one wll run the algorthm for a larger number of teraton a better soluton can be reach. Let s menton that the necessary tme to run the optmzaton (9 canddate soluton, functons, 5 varable, 5 teraton was 583s on a Pentum IV, GHz, 256 Mo RAM machne. The results obtaned for all 9 canddate soluton are presented n Table 2 and Table 3. Table 2 contans nformaton about the ntal MD (for ntal value of the varables and fnal MD (for value of varables after optmzaton. For all canddate solutons, there s a very good evoluton of unfulfllment degrees from approxmately to. or even less. So, ndeed, ths multobjectve problem has more than one good soluton. The best soluton s gven by the st canddate soluton wth a fnal value of MD almost zero, an acceptable soluton for a practcal problem. Table 2 CANDIDATE MD SOLTION ntal fnal o o o o o o o o o Table 3 present the ntal and fnal values for every functon to be optmzed together wth the correspondng requrements, for three canddate solutons ( o, 4 o and 5 o. We can see that even for the soluton 4 o that s the poorest from the ntal set, the fnal values of the functon are very close to the requrements (e.g..989 for ;.6 for ;.6 for, -.2 for. For the best soluton ( o the dfferences are smaller (e.g..9 for ;.9992 for ; -. for and so on. CAND. SOL. Table 3 RE CAND. SOL. CAND. SOL. Q 4 5 nt. fnal nt. fnal nt. fnal s M D Iteratons Fg. 6. Evoluton of MDs for the canddate solutons o, 4 o, 5 o Also the evoluton of the MDs for these three canddate solutons are presented n Fg. 6. Now, let us consder a more complex stuaton. For the same FIR applcaton tae 5 (unform dstrbuted frequency n each doman, so a number of 224

6 functon to optmze and a number of 5 varables. The problem s two-fold complcated. Frst, the number of functon to be optmzed s hgher ( nstead of. Second, we have only 5 varables to set the requred value for each functon. After runnng the optmzaton we reached our best fnal soluton after 57 teraton, wth a fnal MD of The evoluton of the MD durng the optmzaton s presented n Fg. 7. Further teratons can not mprove the soluton. havng smaller oscllatons than the other one (bottom..8.6 M D Iteraton Fg.7. Evoluton of MD for optmzaton functons To see the power of our method we compared the prevous result wth the result obtan wth the Goal Attanment method. The Goal Attanment method s also a multobjectve optmzaton method, mplemented n the Optmzaton Toolbox from Matlab [5]. For the same problem, wth the same ntal pont ( for all 5 ntal varables, the results obtaned wth both method are presented n Tabel 4. In order to have the same measure for both method we computed the absolute error between the functon value after optmzaton and the requred value, for every functon. The mean absolute error over all functons shows that our method (Fuzzy multobjectve optmzaton provded a more accurate soluton than the Goal attanment method,.952 beng les than The prce pad s a larger number of teratons and accordngly more tme to complete the optmzaton. Anyway, the tme for our method remans small enough for practcal applcatons. Table 4 METHOD MEAN ABS. ERROR MAX. ABS. ERROR ITER. TIME Fuzzy s Goal attan s Fg. 8. presents a comparson of the magntude response computed wth the varables (the a coeffcents provded by the optmzatons method (up fuzzy multobjectve optmzaton and bottom Goal attanment optmzaton wth the deal magntude response. We can easly see that both optmzaton methods ensure nce frequency characterstcs. The characterstc provded by our optmzaton method (up s closer to the deal one, Fg. 8. Magntude response wth varable values provded by Multobjectve optmzaton up; Goal Attanment - bottom V. CONCLSION In ths paper a new multobjectve optmzaton method usng fuzzy logc has been ntroduced. The method really allows optmzaton of several objectves smultaneously because the modfcaton of each parameters s a functon of the unfulfllment degrees of all the requrements. The results obtaned after optmzng the coeffcents of a FIR flter show that our method wors very well. Due to the populaton of solutons, we can fnd a set of optmal ponts. The method has a very large chance to fnd the global optmal soluton due to ts multple search paths. Also n the proxmty of the fnal solutons, the method wors well to contnue decrease MD up to the local optmal ponts. The qualty of each fnal soluton s very hgh. Ths s possble because the method uses local gradent nformaton and wors n an adaptve manner: whle the Ds decrease, the step n the parameter modfcaton also decreases. Compared wth other multobjectve optmzaton method (Goal Attanment our method assures a better accuracy of the fnal solutons. REFERENCES [] Boyd, S., Vandeberghe, L., Introducton to Convex Optmzaton wth Engneerng Applcaton, Stanford nversty, 999. [2] Branch, Mary Ann, Grace, A., Optmzaton Toolbox. For se wth Matlab, The MathWor Inc., 996 [3] Grmbleby, J., B., Computer-Aded Analyss and Desgn of Electronc Networs, Ptman Publshng 99, pp.57-9; [4] Maul, P.C, Comments on FPAD: A Fuzzy Nonlnear Programmng Approach to Analog Crcut Desgn, IEEE Trans. on Computer Aded Desgn of Integrated Crcuts and Systems, No.6, June 997, pp.656; [5] ****, Optmzaton Toolbox Help for Matlab R3, The MathWor Inc

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