Decision Science Letters

Transcription

1 Deco Scece etter 4 (5) Cotet lt avalable at GrowgScece Deco Scece etter homepage: Optmum hape deg of tructural model wth mprece coeffcet by parametrc geometrc programmg Samr Dey a ad Tapa Kumar oy b a Departmet of Mathematc, Aaol Egeerg College, Vvekaada Sara, Aaol-7335, Wet Begal, Ida b Departmet of Mathematc, Ida Ittute of Egeerg Scece ad Techology, Shbpur, P.O.-Botac Garde, Howrah-73, Wet Begal, Ida C H O N I C E A B S T A C T Artcle htory: eceved December, 4 eceved reved format: March, 5 Accepted March, 5 Avalable ole March 5 Keyword: Nolear Programmg Geometrc Programmg Structural Optmzato Fuzzy umber Iterval-valued fucto The artcle preet oluto procedure of geometrc programmg to olve the tructural model wth mprece coeffcet. We have codered a gle obectve tructural optmzato model wth weght a a obectve fucto. Geometrc programmg provde a powerful tool for olvg a varety of mprece optmzato problem. Here we ue earet terval approxmato method to covert a tragular fuzzy umber to a terval umber. I th paper, we traform th terval umber to a parametrc terval-valued fuctoal form ad the olve the parametrc problem by geometrc programmg techque. The advatage of th techque that we ca fd drectly optmal oluto of the obectve fucto wthout olvg two-level mathematcal program. Numercal example gve to llutrate the model through th approxmato method. 5 Growg Scece td. All rght reerved.. Itroducto Structural optmzato a crtcal actvty that ha receved coderable atteto the lat four decade. ually, tructural optmzato problem volve earchg for the mmum of the tructural weght. Th mmum weght deg varou cotrat o performace meaure, uch a tree ad dplacemet. Optmum hape deg of tructure oe of the challegg reearch area of the tructural optmzato feld. That why the applcato of dfferet optmzato techque to tructural problem ha attracted the teret of may reearcher. For example, artfcal bee coloy algorthm (Somez, ), partcle warm optmzato (uh et al., ), geetc algorthm (Dede et al., ), at coloy optmzato (Kaveh et al., ) etc. Correpodg author. Tel: Fax: /68 E-mal addre: amr_beu@redffmal.com (S. Dey) 5 Growg Scece td. All rght reerved. do:.567/.dl.5.3.

2 48 I practce, the problem of tructural deg may be formed a a typcal o-lear programmg problem wth o-lear obectve fucto ad cotrat fucto fuzzy evromet. Zadeh (965) frt troduced the cocept of fuzzy et theory. The Zmmerma (978) appled the fuzzy et theory cocept wth ome utable memberhp fucto to olve lear programmg problem wth everal obectve fucto. Some reearcher appled the fuzzy et theory to tructural model. For example, Wag et al. (985) frt appled α -cut method to tructural deg where the o-lear problem were olved wth varou deg levelα, ad the a equece of oluto were obtaed by ettg dfferet level-cut value of α. ao (987) appled the ame α -cut method to deg a four bar mecham for fucto geeratg problem. Structural optmzato wth fuzzy parameter wa developed by Yeh et al. (99). Xu (989) ued two-phae method for fuzzy optmzato of tructure. Shh et al. (4) ued level-cut approach of the frt ad ecod kd for tructural deg optmzato problem wth fuzzy reource. Shh et al. (3) developed a alteratve α -level-cut method for optmum tructural deg wth fuzzy reource. Geometrc Programmg (GP) method a effectve method ued to olve a o-lear programmg problem lke tructural problem. It ha certa advatage over the other optmzato method. Here, the advatage that t uually much mpler to work wth the dual tha the prmal oe. Solvg a o-lear programmg problem by GP method wth degree of dffculty (DD) play eetal role. (It defed a DD = total umber of term obectve fucto ad cotrat total umber of deco varable ). Sce late 96, GP ha bee kow ad ued varou feld (lke O, Egeerg cece etc.). Duff et al. (967) ad Zeer (97) dcued the bac theore o GP wth egeerg applcato ther book. Aother famou book o GP ad t applcato appeared 976 (Beghtler et al., 976).The mot remarkable property of GP that a problem wth hghly olear cotrat ca be traformed equvaletly to a problem wth oly lear cotrat. I real lfe, there are may dvere tuato due to ucertaty udgmet, lack of evdece etc. Sometme t ot poble to get relevat prece data for the cot parameter. The dea of mprecee (fuzze) GP.e. fuzzy geometrc programmg wa propoed by Cao (987). Yag et al. () dcued about the bac ad t applcato of fuzzy geometrc programmg. Oha et al. () ued bary umber for plttg the cot coeffcet, cotrat coeffcet ad expoet ad the olved t by GP techque. A oluto method of poyomal geometrc programmg wth terval expoet ad coeffcet wa developed by u (8). Naer et al. (4) olved two bar tru olear problem by ug geometrc programmg techque to the form of two-level mathematcal programmg. I th paper, we traform terval umber to a parametrc terval-valued fuctoal form ad the tructural model become parametrc tructural model, whch olved by geometrc programmg techque. The propoed procedure more effectve ad eay to calculate the dfferet value of the obectve fucto for dfferet value of the parameter. The ma beeft of th approxmato procedure that t ot requred to create two-level mathematcal programmg.. Structural Optmzato Model I zg optmzato problem, the am to mmze a gle obectve fucto, uually the weght of the tructure, uder certa behavoral cotrat o tre ad dplacemet. The deg varable are mot frequetly choe to be dmeo of the cro-ectoal area of the member of the tructure. Due to fabrcato lmtato the deg varable are ot cotuou but dcrete ce croecto belog to a certa et. A dcrete tructural optmzato problem ca be formulated the followg form m f A () g A, =,,..., m. d A, =,,...,.

3 S. Dey ad T. K. oy / Deco Scece etter 4 (5) 49 where f ( A ) repreet obectve fucto, g( A ) the behavoral cotrat, m ad are the umber of cotrat ad deg varable, repectvely. A gve et of dcrete value expreed by deg varable A ca take value oly from th et. I th paper, obectve fucto take a m f ( A) = ρal = d ad ad cotrat are choe to be tre of tructure a follow, σ (3) g ( A) =, =,,..., m σ where ρ ad l are weght of ut volume ad legth of th elemet, repectvely, m the umber of the tructural elemet, σ ad σ are the th tre ad allowable tre, repectvely. 3. Mathematcal Aaly 3. Geometrc Programmg Geometrc program (GP) ca be codered a a ovatve modu operad to olve a olear problem comparo wth other olear techque. It wa orgally developed to deg egeerg problem. It ha become a very popular techque ce t cepto olvg olear problem. The advatage of th method that,th techque provde u wth a ytematc approach for olvg a cla of olear optmzato problem by fdg the optmal value of the obectve fucto ad the the optmal value of the deg varable are derved. Alo th method ofte reduce a complex olear optmzato problem to a et of multaeou equato ad th approach more ameable to the dgtal computer. GP a optmzato problem of the form: ( x ) m g (4), =,,..., g x m x > =,,..., g x ( =,,,..., m) are poyomal or gomal fucto, x deco varable vector x =,,...,. where of compoet 3.. Geometrc Programmg Problem () ( x ) m g δ, ( =,,..., ) >, ( =,,..., ) g x b m x N k where g ( x) = δ c x ( =,,,..., m) k k k = = α ( m) δ ( m k N ) δ =± =,,...,, =± =,,,..., ; =,,...,, x ( x, x,..., x ) k T (5)

4 4 3.3 Dual Problem The dual problem of the prmal problem (5) a follow, α k w δ k m ckw max d( w; λ) = δ = w k N σkwk = δ (ormal codto) k = m N δ kα kwk = ( =,,..., ) (Orthogoalty codto). = k= where δ, (,,..., m), δ k (,,..., mk ;,,..., N ) =± = =± = = ad δ =+, ad oegatvty codto, w = δ δ kwk, wk, N (,,..., mk ;,,..., N ) k = = = ad w =. Cae I: For N +, the dual program preet a ytem of lear equato for the dual varable where the umber of lear equato ether le tha or equal to the umber of dual varable. A oluto vector ext for the dual varable (Beghtler et al., 976). Cae II: For N < +, the dual program preet a ytem of lear equato for the dual varable where the umber of lear equato greater tha the umber of dual varable. I th cae, geerally, o oluto vector ext for the dual varable. However, oe ca get a approxmate oluto vector for th ytem ug ether the leat quare or the lear programmg method. 3.4 Fuzzy umber ad t earet terval approxmato 3.4. Fuzzy umber A real umber A decrbed a fuzzy ubet o the real le whoe memberhp fucto ( x ) the followg charactertc wth < a a a3 < µ x f a x a µ ( x) = x f a A µ A x a3 otherwe A µ ha where µ :[, ] [,] cotuou ad trctly creag; :[, ] [,] x a a A µ x a A a3 cotuou ad trctly decreag. A α = x: µ x α, x X A α a o empty bouded cloed terval cotaed X ad t ca be deoted α level et: The α level of a fuzzy umber A defed a a crp et A where α [,]. by Aα = A( α), A( α). A ( α ) ad A,repectvely Iterval umber α are the lower ad upper boud of the cloed terval A terval umber A defed by a ordered par of real umber a follow A= [ a, a] = { x: a x a, x } where a ad a are the left ad rght boud of terval A, a a follow repectvely. The terval A, alo defed by ceter C a ad half-wdth W A

5 S. Dey ad T. K. oy / Deco Scece etter 4 (5) 4 { } A= a, a = x: a a x a + a, x where C W C W C W the half-wdth of A. a C a + a = the ceter ad a W a a = Nearet terval approxmato Here we wat to approxmate a fuzzy umber by a crp model. Suppoe A ad B are two fuzzy umber wth α cut are A( α), A( α) ad B, α B α repectvely. The the dtace betwee A ad B ( ) ( ) d AB, = A α B α dα + ( A( α) B( α) ) dα. Gve A a fuzzy umber. We have to fd a cloed tervalc ( d A ), whch the earet to A wth repect to metrc d. We ca do t ce each terval alo a fuzzy umber wth cotat α cut for all α [,]. Hece ( C ( A) ) α = [ C, C ]. Now we have to mmze d ( d( AC, d A) ) = ( A ) ( ). α C dα + A α C dα wth repect to C ad C. I order to mmze d AC A, t uffcet to mmze the fucto (, ), (, d ) ( ) D C C = d AC A. The frt partal dervatve are d ( ) ( α) D C, C = A dα + C C. ( D C, C ) = A α dα + C C ad. ( D C, C ) = ad ( D( C, C) ) =, we get C C Solvg = ( α) α ad ( α) C A d Aga ce D( C C), = > C C = A dα., = > ad C, D( C C) ( ) ( ) C C CC D C, C.e. d ( AC, ( d A) ) global mmum. Therefore, the terval ( ) ( α) α, ( α) α H C, C = D C, C. D C, C D C, C = 4>. So the earet terval approxmato of fuzzy umber A wth Cd A = A d A d repect to the metrc d. A= a, a, a be a tragular fuzzy umber. The α cut terval of A defed a et ( 3) Aα = A( α), A( α) where A ( α) = a+ α( a a) ad A ( α) a3 α( a3 a) =. By earet

6 4 terval approxmato method the lower lmt of the terval a+ a C = A( α) dα = a+ α( a a) dα = ad the upper lmt of the terval a C = A d = a a a d = + a 3 α α 3 α 3 α. a + a a + a =. I the cetre ad half a+ a + a3, a3 a. 4 4 Therefore, the terval umber correpodg A, 3 [ m, ] wdth form the terval umber of A defed a 4. Parametrc Iterval-valued fucto m be a terval, where m>, >. From aalytcal geometry pot of vew, ay real umber ca be repreeted o a le. Smlarly; we ca expre a terval by a fucto. The parametrc, g = m for,, whch et [, ] terval-valued fucto for the terval [ m] ca be take a [ ] trctly mootoe, cotuou fucto ad t vere ext. et ψ be the vere of g( ), the logψ log m =. log log m 5. Geometrc Programmg wth fuzzy coeffcet Whe all coeffcet of Eq. (6) are tragular fuzzy umber, the the geometrc programmg problem of the form m T δ k k k = = = g x c x α k N α k δ k k δ for =,,3,..., m. k = = g x = c x b x > for =,,..,., 3 c k = c, c, c k k k where ck = ( c 3 k, ck, ck) ad ( 3 b,, ) = b b b g earet terval approxmato method, we traform all tragular fuzzy umber to terval umber.e. c k, c k, c k, c k ad, b b. The geometrc programmg problem wth mprece parameter of the followg form (6) m T k g ( x) = δkc k x α k = = N α k δ k k δ for =,,3,..., m. k = = g x = c x b x > for =,,..,. (7)

7 S. Dey ad T. K. oy / Deco Scece etter 4 (5) 43 where c k, c k ad b deote the terval couterpart.e. c k ck, ck, c k ck, ck, b b, b, c >, c >, b > for all ad k. g parametrc terval-valued fuctoal form, the problem k k (7) reduce to N k m g( x ; ) = δk ( ck) ( ck) x α k = = N ( ) α k δ ( ; ) δ g x = c c x b b k k k k = = x > for =,,..,., =,,3,..., m. Th a parametrc geometrc programmg problem. We get dfferet oluto of th problem for dfferet value of the parameter. 6. Two Bar Tru Structural Model The ymmetrc two-bar tru (Naer et al., 4) how Fg. codered here. The obectve to mmze the weght of tru the tre σ cotrat of each bar. There are two deg varable- mea tube dameter (d) ad heght (h) of the tru. (8) Fg.. Two bar tru uder load The weght of the tructure ρ( dπ t b ) + h ad tre ( ) model ca be wrtte a = ρ( π + ) m WT d, h d t b h P b + h σ ( dh, ) σ dπth dh>, P b + h /( dπth). The tructural et b + h = y b + h = y. Hece the ew cotrat b+ h y by + hy. Hece the crp tructural model (9)

8 44 m WT dhy,, = ρtdπy Pyh () σ ( dhy,, ) σ dπt by + hy dhy>,, where P = appled load; t = thcke of the bar; d = mea dameter of the bar (deco varable); b= the dtace betwee two hge. Model () a tadard poyomal geometrc programmg problem.e. δ =, δ = ad t = t, t, t P P, P, P σ = σ, σ, σ are tragular δ =. Whe, = ad 3 fuzzy umber.the the fuzzy tructural model m WT dhy,, = ρtd πy 3 3 Pyh () σ ( dhy,, ) σ dπ t by + hy dhy>,, g earet terval approxmato method, the terval umber correpodg tragular umber t = ( t, t, t3) t + t, t3+ t = [ t, ] t.smlarly terval umber correpodg P ad σ are P+ P P3 + P σ + σ σ + σ, = [ P, P] ad, 3 = [ σ, σo ] repectvely. The problem () reduce to = ρ[ ] m WT dhy,, t, t dπy ( dhy) [ P, P] yh dπ [ t, t ] [ ] σ,, σ, σ by + hy dhy>,, whch equvalet to () m WT dhy,, = ρ tdπy Pyh σ ( dhy,, ) dπ t by + hy dhy>,, σ where t [ t, t ], P [ P, P ] ad σ [ σ, σ ]. o (3) 7. Parametrc Geometrc Programmg Techque o Two bar Tru Structural Model Accordg to ecto 4, the fuzzy two bar tru tructural model (3) reduce to a parametrc programmg by replacg t = t t, P= P P ad σ = σ σ where [,]

9 The model (4) take the reduce form a follow = ρ π ( ) m WT dhy,, d yt t P P yh ( ) ( )( ) dπ t t σ σ by + hy dhy>,, S. Dey ad T. K. oy / Deco Scece etter 4 (5) 45 Applyg Geometrc Programmg Techque, the dual programmg of the problem (4) w ( ) ( ) π ( t t)( σσ ) w w πρ t t P P b max gw = ( w w w w + w ) w = (Normalty codto) w ( w+ w ) For prmal varable y :. w + w + ( ). w + ( ). w = (orthogoal codto) (5) For prmal varable h :. w + ( ). w +. w +. w = (orthogoal codto) For prmal varable d :. w + ( ). w +. w +. w = (orthogoal codto) w, w, w, w > Th a ytem of four lear equato wth four ukow. Solvg we get the optmal value a follow w =, w =, w =.5 ad w =.5 From prmal dual relato we get ( ) = ( ) ( t t)( ) ρdπy t t w g ( w) P P w yd h = π σ σ w w by = w + w w hy = w + w The optmal oluto of the model (4) through parametrc approach gve by w ( ) ( ) π ( t t)( σσ ) w w w πρ t t P P b g ( w) = w + w w w w y b ( w + w ) =, w h bw =, d w ( ) ( t t)( ) ( w + w ) P P b w w w ( + ) = π σ σ w bw Note that the optmal oluto of GP techque parametrc approach deped o. (4)

10 46 8. A llutratve example The put data for the tructural optmzato problem () gve a follow: Support dtace Materal dety Permble tre oad (P) lb Thcke (t) 3 (b) ( ρ ) lb / ( σ ) p 33, , Formulato of the ad model preeted a follow m WT dhy,, =.88yd.75 yd h (6) 9y + hy dhy>,, Th a Poyomal Geometrc Programmg Problem wth degree of dffculty (DD) = 4 (3 + ) =. Applyg Geometrc Programmg Techque, the dual programmg of the problem (6) w w w.88 w 9 ( w+ w ) max gw = (.75) ( w + w ) w w w (Normalty codto) w = For prmal varable y :. w + w + ( ). w + ( ). w = (orthogoal codto) For prmal varable h :. w + ( ). w +. w +. w = (orthogoal codto) For prmal varable d :. w + ( ). w +. w +. w = (orthogoal codto) w, w, w, w > Th a ytem of four lear equato wth four ukow. Solvg we get the optmal value a follow w =, w =, w =.5 ad w =.5 From prmal dual relato we get w.88 yd = wg ( w),.75yd h w =, 9y w = ad hy = w w + w w + w Solvg th we get the optmum oluto of the problem (6) by Geometrc Programmg (GP) Techque preeted Table Table Optmal oluto of Two Bar Tru Structural Model Method Weght WT (lb) Dameter Heght d () h () y () GP NP Schmt (98)

11 S. Dey ad T. K. oy / Deco Scece etter 4 (5) 47 Whe the put data of two bar tru tructural model () take a tragular fuzzy umber.e. 8,3,34 t =.6,.8,. σ = 55,59, 6.g earet P =, ad terval approxmato method, we get the correpodg terval umber ad terval-valued fucto.e. P = [ 3,33 ], P = ( 3) ( 33) [ 3,33], = [ ] t = (.7) (.) [.7,.] ad σ [ ] = 57, 6 σ = ( 57) ( 6) [ 57, 6] where [,] t.7,., The optmal oluto of the fuzzy model by terval-valued parametrc geometrc programmg preeted Table. Table Optmal oluto of Two Bar Tru Structural Model Weght Dameter WT (lb) d () Heght h () y () For =, the lower boud of the terval value of the parameter ued to fd the optmal oluto. For =, the upper boud of terval value of the parameter ued for the optmal oluto. Thee reult yeld the lower ad upper boud of the optmal oluto. The ma advatage of the propoed techque that oe ca get the termedate optmal reult ug proper value of. 9. Cocluo The advatage of th techque that we ca fd drectly optmal oluto of the obectve fucto wthout olvg two-level mathematcal program. Th method mple ad take mmal tme. Here deco maker (egeer) may obta the optmum reult a per h/her requremet.the methodology preeted th paper ca be appled other feld of egeerg optmzato. Ackowledgemet The author would lke to thak the aoymou referee for cotructve commet o earler vero of th paper. The great upport of my famly member make me poble to do t. eferece Wag, G.Y. & Wag, W.Q. (985). Fuzzy optmum deg of tructure. Egeerg Optmzato, 8, 9-3. Zadeh,.A. (965). Fuzzy et. Iformato ad Cotrol, 8(3), Zmmerma, H.J. (978).Fuzzy lear programmg wth everal obectve fucto.fuzzy et ad ytem,,

12 48 Xu, C. (989).Fuzzy optmzato of tructure by the two-phae method. Computer ad Structure, 3(4), Yeh, Y.C. & Hu, D.S. (99).Structural optmzato wth fuzzy parameter. Computer ad Structure,37(6), ao, S.S. (987). Decrpto ad optmum deg of fuzzy mathematcal ytem. Joural of mechacal deg,9(), 6-3. Shh,C. J. & ee, H. W.(4).evel-cut Approache of Frt ad Secod Kd for que Soluto Deg Fuzzy Egeerg Optmzato Problem. Tamkag Joural of Scece ad Egeerg, 7(3), Shh,C.J., Ch,C.C. & Hao,J.H.(3).Alteratve α -level-cut method for optmum tructural deg wth fuzzy reource, Computer ad Structure, 8, Kaveh, A. & Talatahar, S. ().A mproved at coloy optmzato for the deg of plaar teel frame. Egeerg Structure, 3(3), Dede, T., Bekroğlu, S. & Ayvaz, Y. ().Weght mmzato of true wth geetc algorthm. Appled Soft Computg,(), uh, G.C. &, C.Y.(). Optmal deg of tru-tructure ug partcle warm optmzato. Computer ad Structure, 89(34), 3. Somez, M. ().Dcrete optmum deg of tru tructure ug artfcal bee coloy algorthm. Structural ad multdcplary optmzato, 43(), Yag, H. & Cao, B. Y.(). Fuzzy geometrc programmg ad t applcato. Fuzzy formato ad egeerg, (), -. Zeer, C. (97). Egeerg deg by geometrc programmg, Wley, New York. Beghtler,C.S.& Phllp, D.T. (976).Appled geometrc programmg. Wley, New York. Duff,.J., Petero,E.. & Zeer,C.M. (967).Geometrc programmg- theory ad applcato. Wley, New York. Schmt,.A. (98). Structural ythe- t gee ad developmet. AIAA Joural, 9(), Oha, A. K. & Da, A.K. ().Geometrc programmg problem wth coeffcet ad expoet aocated wth bary umber. Iteratoal Joural of Computer Scece Iue, 7(), u, S. T. (8). Poyomal geometrc programmg wth terval expoet ad coeffcet. Europea Joural of Operatoal eearch, 86(), 7-7. Cao, B.Y. (987). Soluto ad theory of queto for a kd of fuzzy potve geometrc program. Proceedg of the ecod IFSA Cogre, Tokyo,, 5-8. Naer, S. H. & Alzadeh, Z. (4). Optmzed oluto of a two-bar tru olear problem ug fuzzy geometrc programmg. Joural of Nolear Aaly ad Applcato, -9. do:.5899/4/aa-3