Structural Design Optimization using Generalized Fuzzy number

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1 IOS Journal of Mathematcs (IOS-JM) e-issn: , p-issn: X. Volume, Issue 5 Ver. II (Sep. - Oct. 5), PP osrournals.org Structural Desgn Optmzaton usng Generalzed Fuzzy number Samr Dey, Tapan Kumar oy (Department of Mathematcs, sansol Engneerng College, Inda) (Department of Mathematcs, Indan Insttute of Engneerng Scence and Technology, Shbpur, Inda) bstract : Ths paper presents soluton technque of geometrc programmng th fuzzy parameters to solve structural model. Here e are consdered all fuzzy parameters as a generalzed fuzzy number.e. generalzed trangular fuzzy number and generalzed trapezodal fuzzy number. Here materal densty of the bar, permssble stress of each bar and appled load are fuzzy numbers. We use geometrc programmng technque to solve structural problem. The structural problem hose am s to mnmze the eght of truss system subected to the maxmum permssble stress of each member. Decson maker can take the rght decsons from the set of optmal solutons. Numercal examples are dsplayed to llustrate the model utlzng generalzed fuzzy numbers. Keyords - Structural optmzaton, Generalzed fuzzy number, Geometrc programmng I. Introducton Structural Optmzaton provdes a means to help the structural engneer to acheve such an am to fnd the best ay to mnmze the eght of structural desgn. Ths mnmum eght desgn s subected to varous constrants on performance measures, such as stresses and dsplacements. Optmum shape desgn of structures s one of the challengng research areas of the structural optmzaton feld. That s hy the applcaton of dfferent optmzaton technque to structural problems has attracted the nterest of many researchers. For example, artfcal bee colony algorthm (Sonmez, M., []), partcle sarm optmzaton (Luh et al., []), genetc algorthm (Dede et al., []), ant colony optmzaton (Kaveh et al., [9]) etc. In general, structural optmzaton problem s solved th the assumpton that the appled load, permssble stress of each members and thckness of the truss are specfed n an exact mode. In real lfe, due to hestaton n udgments, lack of confrmaton of otherse. Sometmes t s not possble to get sgnfcant exact data for the structural system. Ths type of mprecse data s alays ell represented by fuzzy number, so fuzzy structural optmzaton model s needed n real lfe problem. lso makng a decson, decson-makers have to reve the alternatves th fuzzy numbers. It can be seen that fuzzy numbers have a very mportant role to descrbe fuzzy parameters n several fuzzy structural optmzaton model from the dfferent ve-ponts of decson makers. Zadeh [] frst ntroduced the concept of fuzzy set theory. Then Zmmermann [3] appled the fuzzy set theory concept th some sutable membershp functons to solve lnear programmng problem th several obectve functons. Some researchers appled the fuzzy set theory to Structural model. For example Wang et al. [] frst appled -cut method to structural desgns here the non-lnear problems ere solved th varous desgn levels, and then a sequence of solutons ere obtaned by settng dfferent level-cut value of. ao [6] appled the same -cut method to desgn a four bar mechansm for functon generatng problem.structural optmzaton th fuzzy parameters as developed by Yeh et al. [5]. In 989, Xu [4] used to-phase method for fuzzy optmzaton of structures. In 4, Shh et al.[7] used level-cut approach of the frst and second knd for structural desgn optmzaton problems th fuzzy resources.shh et al.[8] developed an alternatve -level cuts methods for optmum structural desgn th fuzzy resources n 3.Dey and oy [] ntroduced fuzzy mult-obectve mathematcal programmng technque based on generalzed fuzzy set and they appled t n mult-obectve structural models. The non-lnear optmzaton problems have been solved by varous non-lnear optmzaton technques. Geometrc Programmng (GP) [4,6] s an effectve method among those to solve a partcular type of nonlnear programmng problem. Duffn, Peterson and Zener [6] lad the foundaton stone to solve de range of engneerng problems by developng basc theores of geometrc programmng and ts applcaton n ther text book. Chang [6] used geometrc programmng n Communcaton Systems. One of the remarkable propertes of Geometrc programmng s that a problem th hghly nonlnear constrants can be stated equvalently th a dual program. If a prmal problem s n posynomal form then a global mnmzng soluton of the problem can be obtaned by solvng ts correspondng dual maxmzaton problem because the dual constrants are lnear, and lnearly constraned programs are generally easer to solve than ones th nonlnear constrants. Cao [] dscussed fuzzy geometrc programmng (FGP) th zero degree of dffcult. In 987, Cao [4] frst ntroduced FGP. There s a good book dealng th FGP by Cao [3]. Islam and oy [7] used FGP to solve a fuzzy EOQ model th flexblty and relablty consderaton and demand dependent unt producton cost a space constrant. FGP method s rarely used to solve the structural optmzaton problem. But stll there are enormous DOI:.979/ osrournals.org 75 Page

2 Structural Desgn Optmzaton usng Generalzed Fuzzy number scopes to develop a fuzzy structural optmzaton model through fuzzy geometrc programmng (FGP). The parameter used n the GP problem may not be fxed. It s more frutful to use fuzzy parameter nstead of crsp parameter. Yang et al. [3] dscussd about the basc and ts applcatons of fuzzy geometrc programmng. Oha et al.[8] used bnary number for splttng the cost coeffcents, constrants coeffcent and exponents and then solved t by GP technque. soluton method of posynomal geometrc programmng th nterval exponents and coeffcents as developed by Lu (Lu, S.T., [9]). Nasser et al.[] solved to bar truss nonlnear problem by usng geometrc programmng technque nto the form of to-level mathematcal programmng. In the present paper, e have consdered the coeffcents of the problem are generalzed fuzzy number and solve t by fuzzy geometrc programmng technque. The rest of ths paper s organzed n the follong ay. In secton II, e dscuss about structural optmzaton model. In secton III and IV, e dscuss about mathematcal analyss and fuzzy mathematcs prerequstes. In secton V, dfferent methods for defuzzfcaton of fuzzy number are dscussed. In secton VI, e dscuss about geometrc programmng technque th fuzzy coeffcent. In secton VII and VIII, crsp and fuzzy model of to bar truss are dscussed and fnally e apply geometrc programmng technque to solve to bar truss structural model respectvely. In secton IX, e dscuss about an llustratve example. Fnally e dra conclusons from the results n secton X. II. Structural Optmzaton Problem In szng optmzaton problems the am s to mnmze a sngle obectve functon, usually the eght of the structure, under certan behavoral constrants on stress and dsplacements. The desgn varables are most frequently chosen to be dmensons of the cross-sectonal areas of the members of the structure. Due to fabrcaton lmtatons the desgn varables are not contnuous but dscrete snce cross-sectons belong to a certan set. dscrete structural optmzaton problem can be formulated n the follong form M n m ze f,,,...,. su b ect to g m d,,,..., n. here f represents obectve functon, g s the behavoral constrant, m and n are the number of constrants and desgn varables, respectvely. gven set of dscrete values s expressed by varables can take values only from ths set. In ths paper, obectve functon s taken as m () d and desgn () f l and constrants are chosen to be stress of structures here g,,,..., m and l are eght of unt volume and length of structural elements, and are the (3) th element, respectvely, m s the number of the th stress and alloable stress, respectvely. III. Mathematcal nalyss 3. Geometrc Programmng Geometrc program (GP) can be consdered to be an nnovatve modus operand to solve a nonlnear problem n comparson th other nonlnear technque. It as orgnally developed to desgn engneerng problems. It has become a very popular technque snce ts ncepton n solvng nonlnear problems. The advantages of ths method s that,ths technque provdes us th a systematc approach for solvng a class of nonlnear optmzaton problems by fndng the optmal value of the obectve functon and then the optmal values of the desgn varables are derved. lso ths method often reduces a complex nonlnear optmzaton problem to a set of smultaneous equatons and ths approach s more amenable to the dgtal computers. GP s an optmzaton problem of the form: subect to M n m ze g x (4),,,..., g x m x,,..., n DOI:.979/ osrournals.org 76 Page

3 Structural Desgn Optmzaton usng Generalzed Fuzzy number here g x,,,..., m are posynomal or sgnomal functons, x s decson varable vector of n components x,,..., n. 3.. Geometrc Programmng Problem M n m ze g x (5) subect to g x b,,,..., m x,,,..., n N k here g x,,,..., c x m k k g k n,,..., m,,,,..., m ; k,,..., N, x x, x,..., x n k 3.3 Dual Problem The dual problem of the prmal problem (5) s M a x m ze d N ; m c k k k k Subect to k k (Normal condton) k m N k k k,,..., n (Orthogonalty condton). k here,,,..., m,,,..., m ; k,,..., N k and, and non-negatvty condtons, N,,..., m ; k,,..., N and,, k k k k T. Case I: For N n,the dual program presents a system of lnear equatons for the dual varables here the number of lnear equatons s ether less than or equal to the number of dual varables. soluton vector exsts for the dual varable (Beghtler et al.,[5]). Case II: For N n,the dual program presents a system of lnear equatons for the dual varables here the number of lnear equaton s greater than the number of dual varables. In ths case, generally, no soluton vector exsts for the dual varables. Hoever, one can get an approxmate soluton vector for ths system usng ether the least squares or the lnear programmng method. IV. Fuzzy Mathematcs Prerequstes Fuzzy sets frst ntroduced by Zadeh [] n 965 as a mathematcal ay of representng mprecseness or vagueness n everyday lfe. Defnton 4.. Fuzzy Set fuzzy set n a unverse of dscourse X s defned as the follong set of pars x, ( x ) / x X.Here : X [,] s a mappng called the membershp functon of the fuzzy set and s called the membershp value or degree of membershp of x X n the fuzzy set. The larger x s the stronger the grade of membershp form n. Defnton 4.. Fuzzy Number fuzzy number s a fuzzy set n the unverse of dscourse X. It s both convex and normal. Defnton cut of a Fuzzy Number The -level of a fuzzy number s defned as a crsp set : ( ),, [,]. s non-empty bounded closed nterval contaned n X and t can be x x x X DOI:.979/ osrournals.org 77 Page

4 Structural Desgn Optmzaton usng Generalzed Fuzzy number denoted by L,, L and are the loer and upper bounds of the closed nterval, respectvely. Fgure shos a fuzzy number th cuts L,, L,. It s seen that f then L L and. x L L x Fg.. Fuzzy number th -cut Defnton 4.4. Convex fuzzy set fuzzy set of the unverse of dscourse X s convex f and only f for all x, x n X, x x m n x, x hen. Defnton 4.5. Normal fuzzy set fuzzy set of the unverse of dscourse X s called a normal fuzzy set mplyng that there exst at least one x X such that x. Defnton 4.6. Generalzed Fuzzy Number (GFN) Generalzed fuzzy number as a, b, c, d ; here and a, b, cand d are real numbers. The generalzed fuzzy number s a fuzzy subset of real lne, hose membershp functon x satsfes the follong condtons: x s a contnuous mappng from to the closed nterval [,]. x here x a ; 3 x s strctly ncreasng th constant rate on [ a, b ] x here b x c 4 ; 5 x s strctly decreasng th constant rate on [, ] 6 x here d x. c d ; Note: s a convex fuzzy set and t s a non-normalzed fuzzy number tll. If a b c d and, then s called a real number a. Here x, x th membershp functon x f x a f x a If a b and c d, then s called crsp nterval [ a, b ]..It ll be normalzed for DOI:.979/ osrournals.org 78 Page

5 Structural Desgn Optmzaton usng Generalzed Fuzzy number Here x, x th membershp functon x f a x d o th e r se If b c, then s called a generalzed trangular fuzzy number (GTFN) as a, b, c ; v If b c, then t s called a trangular fuzzy number (TFN) as a, b, c. Here x, x th membershp functon x a fo r a x b b a d x x fo r b x d d b o th e r s e x TFN GTFN a b d Fg.. TFN and GTFN v If b c, then s called a generalzed trapezodal fuzzy number (GTrFN) as a, b, c, d ; v If b c, then t s called a trapezodal fuzzy number (TrFN) as a, b, c, d. Here x, x th membershp functon x x a b a fo r a x b fo r b x c d x fo r c x d d c o th e r s e x T rf N G T rf N O a b c d Fg. 3. TrFN and GTrFN DOI:.979/ osrournals.org 79 Page

6 Structural Desgn Optmzaton usng Generalzed Fuzzy number Fg. 3. shos GTrFNs a, b, c, d ; and TrFN a, b, c, d hch ndcate dfferent decson maker s opnons for dfferent values of,.the values of represents the degree of confdence of the opnon of the decson maker. V. Dfferent Methods for Defuzzfcaton of Fuzzy Number In real lfe, bulk of the nformaton s assmlated as fuzzy numbers but there ll be a need to defuzzfy the fuzzy number. ctually defuzzfcaton s the converson of the fuzzy number to precse or crsp number. Several processes are used for such converson. Here e have dscussed four types of defuzzfcaton method; 5.. Type-I: Center of Mass (COM) Method a u Let be a fuzzy number then the defuzzfcaton of s gven by DOI:.979/ osrournals.org 8 Page a u a l ˆ a u a l x x d x x d x here a and l are the loer and upper lmts of the support of.the value  represents the centrod of the fuzzy number. 5..a. Defuzzfcaton of G T F N a, b, c ; by COM method ˆ a b c 5..b. Defuzzfcaton of G T rf N a, b, c, d ; by COM method 3 ˆ d c b a d c b a 3 d c b a Note: 5.. For COM method, defuzzfcaton of GTFN and GTrFN does not depend on. In ths case, defuzzfcaton of generalzed fuzzy number and normalzed fuzzy number ll be same. 5.. Type-II: Mean of -Cut (MC) Method Let be a fuzzy number then the defuzzfcaton of s gven by ˆ m, d m a x L here s the heght of m ax, L, a an cut,.e. L L, here L and are the left and rght lmts of the cut of m the a fuzzy number. 5..a. Defuzzfcaton of G T F N a, b, c ; by MC method ˆ a b c.here a b a and c c b 5..b. Defuzzfcaton of G T rf N a, b, c, d ; by MC method ˆ a b c d.here and d d c L a b a Note: 5.. For MC method, defuzzfcaton of TFN and TrFN (normalzed fuzzy number obtaned by puttng n the defuzzfcaton rule of GTFN (5..a) and GTrFN (4..b) respectvely Type III: emoval area () method ccordng to Kaufmann and Gupta [5], let us consder an ordnary number k and a fuzzy number. The left sde removal of th respect to,, the left sde of the fuzzy number. Smlarly, the rght sde removal l 4 k k,s defned as the area bounded by x k and fuzzy number th respect to x k s defned as the mean of Thus, k l, k r, k. r, 4 L k s defned. The removal of the, k and, k. l r

7 Structural Desgn Optmzaton usng Generalzed Fuzzy number 5.3.a. Defuzzfcaton of G T F N a, b, c ; by method, the removal number of th respect to orgn s defned as the mean of to areas, b c r,, ˆ, a b c. 4 a b l, and 5.3.b. Defuzzfcaton of G T rf N a, b, c, d ; by method, the removal number of th respect to orgn s defned as the mean of to areas, ˆ, a b c d. 4 a b l, and c d r,, Note:5.3. For method, defuzzfcaton of TFN and TrFN are obtaned by puttng n the defuzzfcaton rule of GTFN (4.3.a), GTrFN (5.3.b) respectvely. Note: 5.4. Defuzzfcaton of GTFN and GTrFN by type-ii and type-iii method are same but these are dfferent th type-i Mean of Expected Interval (MEI) Method The level set of s defned as, nterval of fuzz number,denoted as L E I s value of s gven by d d M E I L. ccordng to Helpern [8], the expected E I d d L,. The approxmated. 5.4.a. Defuzzfcaton of G T F N a, b, c ; by MEI method ˆ a c a b c and c c b L a b a 5.4.b. Defuzzfcaton of G T rf N a, b, c, d ; by MEI method ˆ a d a b c d and d d c L a b a Note:4.6. For MEI method, defuzzfcaton of TFN and TrFN are obtaned by puttng n the defuzzfcaton rule of GTFN (4.4.a), GTrFN (5.4.b) respectvely. VI. Geometrc Programmng Wth Fuzzy Coeffcent When all coeffcents of (5) are generalzed fuzzy number, then the geometrc programmng problem s of the form M n m z e g x k c k x subect to N T k k n n k k k g x c x b k for,, 3,..., m. x for,,.., n. here c k, c k and b are generalzed fuzzy number. Usng dfferent dfuzzfcaton methods, e transform all generalzed fuzzy number nto crsp number.e. c k, c and k b. The geometrc programmng problem th mprecse parameters s of the follong form M n m z e g x k c k x T k n k.here.here (6) DOI:.979/ osrournals.org 8 Page

8 Structural Desgn Optmzaton usng Generalzed Fuzzy number subect to N n k k k for,, 3,..., m. g x c x b k x for,,.., n. Ths s a geometrc programmng problem. (7) VII. To Bar Truss Structural Model To bar truss model s developed and ork out under the follong notatons. 7.. Notaton We defne the follong varables and parameters; P = appled load; t = thckness of the bar; d = mean dameter of the bar (decson varable); b= the dstance beteen to hnged pont. WT= eght of the structure; h = the perpendcular dstance from appled load pont to the base lne (decson varable); y = depends on b and h (decson varable); 7.. Crsp Structural Model The symmetrc to-bar truss shon n Fgure 4 has been studed by several researchers lke [7,]. Here e consder same model. The obectve s to mnmze the eght of truss system subect to the maxmum permssble stress n each member s. There are to desgn varables- mean tube dameter (d) and heght (h) of the truss. Fg. 4.To bar truss under load The eght of the structure s d t b h P b h and stress s The structural model can be rtten as, M n m z e W T d h d t b h d th. P b h S u b e c t to d, h ; d th d, h ; (8) Let b h y b h y. Hence the ne constrant s Hence the structural model s b h y b y h y. DOI:.979/ osrournals.org 8 Page

9 M n m z e W T d, h, y td y Structural Desgn Optmzaton usng Generalzed Fuzzy number P y h S u b e c t to d, h, y ; d t b y h y ; (9) d, h, y ; The above problem (9) can be treated as a Posynomal Geometrc Programmng problem th zero Degree of Dffculty Fuzzy Structural Model The obectve as ell as constrant goal can nvolve many uncertan factors n a structural optmzaton problem. Therefore the structural optmzaton model can be represented n fuzzy envronment to make the model more flexble and adoptable to the human decson process. If the coeffcent of obectve functon and constrant goal of (9) are fuzzy n nature.then the crsp model (9) s transformed nto fuzzy model as follos M n m z e W T d, h, y t d y S u b e c t to P y h d t ; () b y h y ; here P, a n d are fuzzy n nature. d, h, y ; VIII. Soluton Procedure of Fuzzy Structural Model throgh Geometrc Programmng fter defuzzfcatonof the fuzzy parameters, the fuzzy to bar truss structural model () reduces as M n m z e W T d, h, y ˆ d y t S u b e c t to P y h d t ; b y h y ; () d, h, y ; pplyng Geometrc Programmng Technque, the dual programmng of the problem () s t P b m ax g ( ) t ( ) su b ect to (Normalty condton) For prmal varable y :. ( ). ( ). (orthogonal condton) For prmal varable h :. ( )... (orthogonal condton) For prmal varable d :. ( )... (orthogonal condton),,, Ths s a system of four lnear equaton th four unknons. Solvng e get the optmal values as follos From prmal dual relaton e get,,.5 a n d.5 b ˆ d yt g ( ) P t y d y h So the dual obectve value s gven by and h ˆ t P b g ( ) t y ( ) () DOI:.979/ osrournals.org 83 Page

10 Structural Desgn Optmzaton usng Generalzed Fuzzy number y b ( ), h b d P b ( ) t b IX. Numercal Expose We assume that materal densty of the bar, permssble stress of each bar and appled load are fuzzy n nature. We take to types of fuzzy generalzed, GTFN, GTrFN as nput data nstead of crsp coeffcent. Table-: Input data for fuzzy model () as TFN P 3, 3 3, :.,.3,.5; 6, 6, ; Table-: Input data for fuzzy model () as TrFN P 3, 3 6, 3 3 5, :.,.,.4,.5 5; 5 8 5, 5 9, 6 7 5, 6 5 ; For COM defuzzfcaton rule s not consdered for dfferent values of (only s consdered) for numercal result of dfferent types of generalzed fuzzy number hch are exhbted n table-3, table-4, table-5 and table-6. Table 3. Optmal soluton of To Bar Truss Structural Model () by GP method hen nput data are GTFN Weghts W T (lb s ) Dameter d (n ) DOI:.979/ osrournals.org 84 Page Heght h (n ) y (n ) Defuzzfczton Type Type-I Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV The table (3) gves the result of optmum eght for to bar truss usng generalzed trangular fuzzy number by defuzzfcaton rule of COM method, MC method, method and MEI method. For MC method and method outcome are same. Table 4. Optmal soluton of To Bar Truss Structural Model () by NLP method hen nput data are GTFN Weghts W T (lb s ) Dameter d (n ) Heght h (n ) y (n ) Defuzzfczton Type Type-I Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV

11 Structural Desgn Optmzaton usng Generalzed Fuzzy number Table (4) dsplayed the result of to bar truss model by non-lnear programmng by Lngo softare takng GTFN. It s notce that GP method gves batter result for some case otherse almost same. Table 5. Optmal soluton of To Bar Truss Structural Model () by GP method hen nput data are GTrFN Weghts W T (lb s ) Dameter d (n ) DOI:.979/ osrournals.org 85 Page Heght h (n ) y (n ) Defuzzfczton Type Type-I Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV The table (5) gves the result of optmum eght for to bar truss usng generalzed trapezodal fuzzy number by defuzzfcaton rule of COM method, MC method, method and MEI method. For MC method and method outcome are same. Table 6. Optmal soluton of To Bar Truss Structural Model () by NLP method hen nput data are GTFN Weghts W T (lb s ) Dameter d (n ) Heght h (n ) y (n ) Defuzzfczton Type Type-I Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV Type-II&III Type-IV Table (6) dsplayed the result of to bar truss model by non-lnear programmng by Lngo softare takng GTrFN. It s notce that GP method gves batter result for some case otherse almost same. X. Concluson We have consdered to bar truss structural model hose am s to mnmze the eght of truss system subected to the maxmum permssble stress of each member. We use geometrc programmng technque to solve structural problem th fuzzy coeffcents. Here materal densty of the bar, permssble stress of each bar and appled load are generalzed fuzzy numbers. In many stuatons, problem parameters are more competent to take as GFN for real lfe examples. Hence ths ork gves more sgnfcant for structural engneer for decsonmakng. Ths technque can be appled to solve the dfferent decson makng problems n other engneerng and management scences th dfferent types of fuzzy number. cknoledgements The authors also deeply acknoledge the poneers lke S.S.ao and C.J.Shh and all other authors ho played pvotal role n dealng th non-lnear structural model. Because of ther beautful contrbutons e could enter nto ths area. eferences [] Wang, G.Y. & Wang, W.Q., Fuzzy optmum desgn of structure. Engneerng Optmzaton, 8, 985, 9-3. [] Zadeh, L.., Fuzzy set. Informaton and Control, 8(3),985, [3] Zmmermann, H.J., Fuzzy lnear programmng th several obectve functon.fuzzy sets and systems,,978, [4] Xu, C., Fuzzy optmzaton of structures by the to-phase method. Computer and Structure, 3(4),989, [5] Yeh, Y.C. & Hsu, D.S., Structural optmzaton th parameters. Computer and Structure,37(6),99, [6] ao, S.S., Descrpton and optmum desgn of fuzzy mathematcal systems.journal of mechancal desgn,9(,)987, 6-3. [7] Shh,C. J. & Lee, H. W., Level-cut pproaches of Frst and Second Knd for Unque Soluton Desgn n Fuzzy Engneerng Optmzaton Problems. Tamkang Journal of Scence and Engneerng, 7(3),4, [8] Shh,C.J., Ch,C.C. & Hsao,J.H., lternatve -level-cuts methods for optmum structural desgn th fuzzy resources, Computers and Structures, 8,3,

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