OPTIMIZATION OF SKELETAL STRUCTURES USING A HYBRIDIZED ANT COLONY-HARMONY SEARCH- GENETIC ALGORITHM *

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1 IJST, Transactons of Cvl Engneerng, Vol. 38, No. C1, pp 1-20 Prnted n The Islamc Republc of Iran, 2014 Shraz Unversty OPTIMIZATION OF SKELETAL STRUCTURES USING A HYBRIDIZED ANT COLONY-HARMONY SEARCH- GENETIC ALGORITHM * M. H. TALEBPOUR 1, A. KAVEH 2** AND V. R. KALATJARI 3 1, 3 Dept. of Cvl Engneerng, Shahrood Unversty of Technology, Shahrood, I. R. of Iran 2 Centre of Excellence for Fundamental Studes n Cvl Engneerng, Iran Unversty of Scence and Technology, Tehran, I. R. of Iran Emal: alkaveh@ust.ac.r Abstract Ant Colony Optmzaton (ACO) has been used as one of the popular meta-heurstc algorthms n structural optmzaton. In ths algorthm, the selected cross sectons are chosen accordng to a parameter called probablty rato. Ths parameter and the way to choose the cross sectons from a lst of cross sectons, are the most mportant ponts n the optmzaton process. Though the Ant Colony algorthm has a specal ablty n achevng the optmal pont, n some cases n order to avod local optma, the utlzaton of specal technques s needed. In the present paper, the frst am s to use Harmony Search (HS) algorthm to ncrease the local search ablty of the ACO. In ths way a combned algorthm, denoted by HACOHS, s obtaned wth specal abltes to acheve a global optmum. For ths purpose, optmal desgn of skeletal structures such as trusses and steel frames s consdered usng the HACOHS. However, n the process of optmzaton by HACOHS method, several GA selectons are employed at the cross secton selecton stage. Utlzng the Tournament (HACOHS-T), Roulette wheel (HACOHS-Ro), and Rank (HACOHS-Ra) methods t s found that the HACOHS-T s the most effcent of these algorthms for optmal desgn of skeletal structures. Keywords Optmzaton, ant colony algorthm, harmony search, genetc algorthm, skeletal structures 1. INTRODUCTION In the last two decades, meta-heurstc algorthms have been used extensvely n optmal desgn of structures. These algorthms are ntellgent random search approaches, mostly based on some processes and rules from nature. The logc of these algorthms s such that they nvestgate and search the entre desgn space pont by pont and progress to the optmum pont through generatng mproved desgns n optmzaton process wthout any restrctons on desgn varable type and problem constrants [1, 2]. The man dea of meta-heurstc methods was frst exposed by Fogel through presentng an Evoluton Strategy n 1966 [3]. Thereafter, other researchers presented several other algorthms each havng ther advantages and dsadvantages. These algorthms whch were nspred by natural processes conssted of GA, ACO, HS [4-8]. In recent years, the performance of these algorthms has mproved and ther shortcomngs are reduced by combnng dfferent meta-heurstc approaches [9-12]. Ant colony algorthm s known as an effcent meta-heurstc method wth good performance [13, 14]. Ths method was frst ntroduced by Colorn et al. [6, 7] as Ant System (AS) to solve the travellng salesman problem. The man logc of the method was based on the nspraton of ants' behavor searchng for food. Ants as socal blnd nsects lve n a socety wth mutual cooperaton and use a chemcal substance called pheromone to dscover the shortest route towards the food source. Each nsect leaves a Receved by the edtors September 25, 2012; Accepted March 18, Correspondng author

2 2 M. H. Talebpour et al. small amount of pheromone from place to place to dentfy the way back and also facltate the route determnaton for the other ants and return to formcary from the prevous route. The more the pheromone of a route, the greater the chance for other ants to choose the same route. Consequently, the path to reach the food source may have a greater chance to renvest the pheromone and also be chosen by other ants. Pheromone rate of each path constantly changes proportonal to the passng rate of the other ants and also by the magntude of the evaporaton. Evaporaton process results n elmnatng the long and unsuccessful routes durng ants search acton so that the shortest path to the food source wll be detected by ants. Inspred by ths fact, structural optmzaton problem was nvestgated by several researchers and n some cases the standard algorthm s mproved through some enhancements [15-18]. In ths relaton, dfferent approaches based on Ant Colony algorthm prncples namely, Ant Colony System (ACS), Max-Mn Ant System (MMAS), Rank-Based Ant System (RBAS), Best and Worst Ant System (BWAS) were proposed by dfferent researchers [19-21]. In most cases, the ablty of the Ant Colony algorthm n obtanng the optmal pont (global search) has been mproved and even n some cases ts local search ablty s also enhanced. Harmony Search algorthm s best known for ts local search ablty wthn the range of optmum desgn. Lee and Jm (2001) proposed ths method, whch s nspred by the process that a muscan follows to search for an approprate status whle playng musc [8]. Accordng to ths algorthm, each muscan s replaced by desgn varable durng optmzaton process and entre muscans form the desgn varables vector. Beauty and qualty of the musc results n the objectve functon value for the vector of desgn varables. In structural optmzaton process based on the HS, a prmary populaton wthout constrant volatons s requred to detect optmum ponts n the vcnty of the present populaton and to replace them nstead of ts members based on parameters such as HMCR, PAR and bw [22, 23]. Therefore, several teratons are requred for ths method to generate a prmary populaton and also to acheve the global optmum pont, and t s consdered as an approprate method for local search wthn the optmum pont range. Thus combnng HS and ACO methods may result n a hgh performance algorthm wth global and local search capabltes to acheve a global optmum pont. On the other hand, n a structural optmzaton based on the ACO, cross sectons from the lst of sectons are selected accordng to a parameter called probablty rato whch s smlar to the roulette wheel selecton method of the Genetc algorthm. Accordngly, sectons wth hgher pheromone rate have a greater chance to be selected. Investgatons on GA reveal that n some cases other selecton methods such as tournament method produce better results and lead to a more sutable optmum pont [24]. The present research s concerned wth optmal desgn of frame and truss structures based on a combned algorthm of Ant Colony and Harmony Search. The performance of the HACOHS s mproved through usng dfferent approaches for the selecton process n GA for selectng approprate cross sectons from the secton. For ths purpose, the roulette wheel (HACOHS-RO), tournament (HACOHS-T) and rank (HACOHS-RA) methods are utlzed and the convergence path to attan an optmum pont s used as a crteron for comparson of the above mentoned methods. Results ndcate that the use of HACOHS-T method mproves the resultng responses of the optmum structures. 2. STRUCTURAL OPTIMIZATION BASED ON HACOHS Optmzaton process based on the HACOHS method s almost smlar to the ACO algorthm; however, the local search process s performed wth exertng a condton as local search condton of the HS method. Fgure 1 llustrates the flowchart of the HACOHS algorthm. IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014

3 Optmzaton of skeletal structures usng a 3 Start Defnton and formulaton of optmzaton problem Intalzaton of parameters Probablty value calculaton Generaton new populaton based on GA selecton methods Local pheromone updatng Global pheromone updatng Investgatng local search crteron Satsfyng local search crteron condton Not satsfyng local search crteron condton Local search based on Secton 2.7 Not satsfyng fnshng condton Investgatng convergence and controllng number generaton a) Formulaton of the optmzaton problem Satsfyng fnshng condton Determne and ntroduce the best structure as optmum desgn Fg. 1. Structural optmzaton algorthm by HACOHS Optmal desgn of skeletal structures s formulated as follows: Fnd the least value of the weght objectve functon under the constrants C1 and C2: End W Ne A l a (1) 1 C1 : σ j σ j all j = 1, 2,, Ne (2) C2 : k all k k = 1, 2,, Ndof (3) In Eq. (1), a vector of cross secton varables, the matrx [A], s defned as: [A]=[a 1, a 2,, a Nos ] ; a S ; =1,, Nos (4) In Eq. (1) to Eq. (4), ρ s the materals densty of the th member, l s the length of the th member, а s cross secton for the th member and Ne s number of structural members. S s the lst of avalable profles found for the numbers of Ns from whch the optmum desgns are chosen. Nos s the number of sectons for each desgn whch s determned accordng to the structural members groupng. σ j shows stress value of j th member and σ all s the value of allowable stress. Δ k ndcates dsplacement of k th degree of freedom and Δ all k s maxmum dsplacement of k th degree of freedom. NDOF s the number of actve degrees of freedom for actve jonts of the structure. Constrant C1: In an optmum structure, stress rased from load combnatons n all members must be n the allowable range whch s determned based on the code beng used. Accordngly, stress value of each member of the structure n the optmzaton process s controlled. Volaton of the stress constrant s February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

4 4 M. H. Talebpour et al. determned by Eq. (5). In nlc number of load combnatons status, values of the constrant volaton of all the members are added together. C3 C 3 C 3 0 f all 1 f all 1 0 all ; 1 0 1,, Ne ; 1,, Ne (5) Constrant C2: After structural analyss and calculatng the stresses, the dsplacements of the actve nodes n each desgn are calculated. If the th degree of freedom dsplacement s n the range, no penalty wll be consdered; otherwse, the desgn wll be penalzed proportonal to the volaton. The volaton of the dsplacement constrant s determned by Eq. (6). In the load combnatons status, the volatons of the nodal dsplacement constrants are also added together for nlc cases. C40 f C4 C4 1 all f all 10 ; all 10 1,, Ndof ; 1,, Ndof (6) b) Intalzaton of the parameters The present combned algorthm (HACOHS) at frst needs parameter ntalzaton smlar to the other meta-heurstc algorthms. In ths algorthm, n addton to the ntal parameters of the Ant Colony algorthm such as the number of members, α, β, evaporaton rate, and the parameters of the Harmony Search consstng of PAR, HMCR are also ntalzed. Moreover, the local search crteron whch provdes search terms through HS method s also determned. At ths stage, the amount of prmary pheromone for all possble status wll be ntalzed. Snce there s a choce for the number of sectons lsted for each member of structures, a matrx called T wth dmensons proportonal to the number of sectons from the avalable lst and the number of desgn varables (number of structural member groupng) wll be developed as Eq. (7) to determne the pheromone value. Each element of ths matrx (T j ) ndcates the amount of pheromone rate of the th state from the lst of sectons for the j th desgn varable. T T11 T21 T Ns1 T T T Ns 2 T T T 1Nos 2 Nos NsNos NsNos (7) The amount of the prmary pheromone n ths matrx s ntalzed accordng to Eq. (8). T j (8) W 0 1 Where W mn s the value of the objectve functon accounted for the frst state of the lst of sectons to all the desgn varables [25]. c) Probablty value calculaton Followng the ntalzaton of the parameters of the combned algorthm, selecton probablty of each current mode (proportonal to sectons lst) for each desgn varable s calculated as follows [17]: IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014 mn

5 Optmzaton of skeletal structures usng a T j Ns T kj k pj 1,, Ns ; j 1,, Nos k 1 (9) 5 Where P j s the selecton probablty of the th mode (path) for the j th desgn varable. v s the stablty coeffcent for the th mode from the lst of sectons whch s defned as [25]: 1 ; a S ; 1,, Ns a (10) As can be seen from Eq. (10) the lower the a value, the more v s and correspondngly p j ncrease wth the ncrease of v accordng to Eq. 9. In Eq. (9), α and β are two parameters that wegh the relatve mportance of the pheromone tral and the heurstc nformaton, respectvely. If α = 0, then p j wll be proportonal to v value and correspondngly proportonal to the selected cross secton value (a ). Therefore, the optmzaton process becomes randomzed. On the other hand, f β = 0, then only the pheromone mpact wll be effectve n the choce probablty functon whch can result n a rapd and early convergence and as a consequence, ncreases the probablty of obtanng a local optmum [25]. d) Generatng new populaton based on the GA selecton methods In the HACOHS algorthm, after calculatng the values of the selecton probablty, new populaton should be determned based on the p j values. Therefore, n the present paper, many dfferent selecton methods of GA [24] are nvestgated and utlzed. HACOHS-RO: Roulette Wheel procedure of GA s used n ths method [26]. Accordngly, the sum of p j values for the th desgn varable s equal to 1. Now, f the probablty obtaned for the j th desgn varable s depcted as a roulette wheel, p j values wll form ts sectors. Through generatng an addtonal number between 0 and 1, a cross secton from the sectons lst wth larger sector may have a better chance to be selected. In order to mplement ths method, the cumulatve probablty of P for the j th desgn varable s determned as follows: j P Pkj ; 1,... Ns k 1 (11) Now a random number s produced between 0 and 1. The selected cross secton s dentfed from the lst j of sectons by comparng the random number to P values. Ths procedure s performed for all the desgn varables to form the new desgn. The process s repeated for all the populaton to form the new populaton based on the p j value accordng to T j. HACOHS-Ra: Rankng technque of GA s used to select cross secton from the lst of sectons [27]. The p j values are ordered from smallest to largest values. For nstance, the worst cross secton wth the lowest probablty rato possesses the frst rank and lkewse contnues. Fnally, the best cross secton wth the hghest probablty rato wll have rankng equal to Ns. Then cross secton selecton s performed based on member rankng n the arranged lst and secondary probablty rato s estmated as follows: P j Where m ndcates the cross secton member n the arranged lst of sectons accordng to p j value. In other j words, m s equal to. The cumulatve values of, smlar to the roulette wheel method, are determned P Ns m j 1 j (12) February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

6 6 M. H. Talebpour et al. accordng to Eq. (11) and selected cross secton for the j th desgn varable s chosen by producng unform j numbers rangng from 0 to 1 and comparng to P. Ths procedure s performed for all the desgn varables to take shape for the new desgn. The process s repeated for the number of populaton members to construct the new populaton. HACOHS-T: In ths method, tournament selecton approach s used n GA to select cross secton for the th desgn varable [28]. Accordngly, some cross sectons, proportonal to tournament sze, are selected randomly for the th desgn varable and durng a competton, a cross secton from a lst of sectons s selected wth the hghest value of p j for the th desgn varable. Ths procedure s performed for all the desgn varables to form the new desgn. The process s repeated for the number of populaton members n order to form the new populaton based on p j value accordng to T j. e) Local updatng and ftness calculaton As llustrated n Eq. (13), followng new populaton formaton, pheromone rate correspondng to total selected cross sectons (passed routes) for each desgn varable s decreased wth a constant coeffcent whch prevents pheromone accumulaton on each path and unfavorable and faled decsons are also gnored [25]. new Tj 0 T (13) Where T j new and T j old are the new and old pheromone rates for passed routes, respectvely. ρ o s the local update coeffcent whch has a value rangng from 0 to 1. The value of the objectve functon s estmated usng Eq. (1), followng the local upgradng. Applyng the modfed objectve functon, constrant optmzaton problem s converted to an unconstrant optmzaton problem whch s descrbed by the followng equaton: old j A W A 1 K max 0, Gq nlc Q q 1 (14) Where W(A): The objectve functon Gq: The structural volaton rate related to each constrant [A]: The vector of desgn varables Q: The total constrants governng the problem nlc: The number of load combnatons K: The penalty constant As can be deduced from Eq. (14), for every desgn that volates the problem constrants more, the correspondng φ functon value wll be more as well and wll have lower ftness [29]. As a result, followng the estmaton of φ values correspondng to each desgn, the present populaton wll be ranked mert-based [16]. f) Global updatng and depostng pheromones After orderng the present populaton based on ftness, the pheromone rate of all modes n the lst of sectons for all the desgn varables at global upgradng stage are decreased wth a coeffcent called evaporaton rate. In the other words, all the pheromone matrx entres are reduced based on the followng equaton [15]. new T ( 1 e ) T (15) IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014 r old

7 Optmzaton of skeletal structures usng a 7 Where e r ndcates the pheromone global evaporaton rate. old and new transcrbers ndcate old and new pheromone matrces, respectvely. After performng the global pheromone evaporaton process, pheromone should be placed on the passed routes. In the present work, a small populaton, λ r, of the best present populaton (µ) s prmarly formed. λ r value s ntalzed at the frst stage. Afterwards, pheromone rate of the lst of sectons modes for desgn varables whch are selected n the selecton stage (passed routes) s ncreased as follows [25]: T j T j r T j r rk. Tj er. r. (16) best k k1 Where j ndcates the passed routes and r k shows each desgn number n µ populaton so that r k s always n the range of 1 to λ r. (ΔT j ) k represents the amount of pheromone needed to be placed n the j route whch depends on resulted response qualty rased from k th desgn and (ΔT j ) best corresponds to the best desgn. (ΔT j ) for k th desgn s calculated as follows: j k 1 Where φ(a) for the k th desgn based on Eq. (14) s among the best. Consderng Eq. (16), more pheromone s placed on the passed routes from µ populaton whch results n an ncrease of the convergence rate n the present algorthm [25]. g) Local search In the present paper, f no change s observed n successve generatons of the optmzaton process n ftness value of the best populaton desgn and small µ populaton possesses desgns wth no constrant volaton, then local search process nspred by HS method on the µ populaton s performed. To attan ths, µ populaton desgns called HM are placed n a matrx as follows: A k (17) HM 1 a1 2 a1 r a1 a a x r 2 a a x 1 Nos 2 Nos r Nos r Nos (18) Then, for the number of HM populaton, new desgn varable vector [A] = [a 1 ', a 2 ',., a Nos '] s formed, based on Fg. 2 accordng to the HS rules, HCMR and PAR parameters. If the new vector s better than the worst HM vector, t s replaced by the most ncompetent HM desgn; otherwse HM stays unchanged. Accordngly, every value of a' n the new vector [A'] can be based on the HMCR parameter or regenerated randomly, and/or can be determned based on a' - a λr as correspondng values n HM. Ths stage s performable through generatng a random number between 0 and 1 (Ran1) and comparng to the HMCR value. If the random number s greater than 1, a' wll be determned randomly from the lst of sectons; otherwse a' value s determned from HM. Determnaton of the a' from HM lmt s also performed based on the PAR parameter. For ths purpose, a' s determned by generatng a random number between 0 and 1 (Ran2) and comparng wth PAR value. If the random number s smaller than PAR, a' wll be selected from the correspondng values n HM; otherwse value of the a' s determned from a correspondng value n the vcnty of a' n lst of sectons based on bw value [23, 30]. Full descrpton of ths method s llustrated n Fg. 2, where FNs shows a functon whch determnes every cross secton's number from the lst of sectons, and FNs -1 determnes ts nverse functon. February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

8 8 M. H. Talebpour et al. = 0 = +1 > Nos No HCMR < Ran1 Yes Yes No = 0 and generate a new vector for the next HM member Select a random value for a from the profle lst 1 Select a random value between r a a and replace wth a No PAR > Ran2 Yes Yes 0.5 > Ran2 No D = FNs (a ) + nt (Ran2.bw) D = FNs (a ) nt (Ran2.bw) No D Nos No D Nos Yes a = FNs -1 (D) Fg. 2. Local search n the HACOHS Hereafter, the entres of the pheromone matrx correspond to a range of possble scenaros n the lst of sectons for the HM members whch are consdered to be equal to the ntal pheromone rate, and the remanng entres of the pheromone matrx are equated to zero. In other words, the pheromone correspondng to several cross sectons on the top and bottom of the HM desgns from lst of sectons n pheromone matrx s consdered equal to the prmary pheromone rate, and the remanng entres of the pheromone matrx are consdered as zero [16]. The desred range for the neghborhood of the HM members s determned proportonal to the value of the parameter bw. h) Termnaton crteron Several methods are avalable for termnaton condton n meta-heurstc algorthms [1]. In ths paper, HACOHS termnaton condton s satsfed wth controllng the number of teratons. After termnaton of the algorthm, the best desgn s obtaned as the optmum desgn, and the convergence curve s drawn and thus an accurate comparson among dfferent selected methods wth HACOHS for each example s acheved. 3. NUMERICAL EXAMPLES In ths secton, several examples of skeletal structures are optmzed for evaluatng the effcency of the HACOHS-RO, HACOHS-RA and HACOHS-T. Here, three steel trusses and two steel frames are desgned. To provde an accurate judgment and to avod the effect of random parameters, the convergence dagrams for each example are drawn usng an average of 30 dfferent runs. IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014

9 Optmzaton of skeletal structures usng a 9 a) A 52-bar truss Optmal desgn of a 52-bar truss, shown n Fg. 3, s performed as the frst example. Here, E and ρ are assumed to be as MPa and 7860 kg/m 3, respectvely. Fg. 3. A 52-bar planar truss structure. In Fg. 3, the loads P x and P y are 100 kn and 200 kn, respectvely. Here, the truss members are categorzed nto 12 groups and the allowable stress constrants are consdered n the range of ±180 MPa. The sectons of the 52-bar truss are lsted n Table 1. Table 1. The avalable cross-secton areas of the AISC code No. n 2 mm 2 No. n 2 mm 2 No. n 2 mm 2 No. n 2 mm February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

10 10 M. H. Talebpour et al. Fgure 4 shows the convergence curves of the present truss as an average of 30 dfferent runs based on HACOHS-RO, HACOHS-RA and HACOHS-T methods. Ths confrms that the convergence rate of the HACOHS-T s hgher. Ths method leads to lghter weght than the other exstng approaches. Table 2 ncludes the results of the optmum desgn for the present algorthm and some other exstng approaches. Gr. Mem. Fg. 4. The convergence hstory of the proposed methods for the 52-bar truss structure Table 2. Comparson of the optmal desgns for the 52-bar planar truss structure (mm 2 ) Wu and Chow [31] Lee and Geem [30] L et al. [32] L et al. [32] L et al. [32] Kaveh and Talatahar [11] Caprles et al. [16] Kaveh and Talatahar [33] Ths Study GA HS PSO PSOPC HPSO DHPSACO RBAS LU,2 CSS HACOHS-T 1 A 1 -A A 5 -A A 11 -A A 14 -A A 18 -A A 24 -A A 27 -A A 31 -A A 37 -A A 40 -A A 44 -A A 50 -A Gq Weght-kg b) A 72-bar truss In ths example, the optmal desgn of a 72-bar truss, shown n Fg. 5, s performed. Here, E and ρ are consdered as ks ( MPa) and 0.1 lb/n 3 ( kg/m 3 ), respectvely. The stress range for truss members, and the maxmum nodal dsplacement are lmted to ±25 ks (± MPa) and ±0.25 n (0.635 Cm), respectvely. Present truss members are categorzed nto 16 groups. Table 1 contans the lst of sectons, and Table 3 shows the appled loads for 2 dfferent condtons. IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014

11 Optmzaton of skeletal structures usng a 11 Fg. 5. A 72-bar spatal truss structure Table 3. Loadng condtons for the 72-bar spatal truss structure Nodes Condton 1 Condton 2 P x kps (kn) P y kps (kn) P z kps (kn) P x kps (kn) P y kps (kn) P z kps (kn) (22.241) 5.0 (22.241) 5.0 (22.241) (22.241) (22.241) (22.241) (22.241) Fgure 6 llustrates the convergence curve for an average of 30 runs of the proposed methods. From ths fgure t can be deduced that HACOHS-T method s more successful and also possesses a hgher chance of obtanng lghter desgns than the other proposed methods. Fg. 6. The convergence hstory of the proposed methods for the 72-bar truss structure The best desgn for the 72-bar truss s also obtaned by HACOHS-T. Table 4 ncludes the results of the optmum desgn for the HACOHS-T n comparson to those of the other methods. It s obvous that the proposed method leads to lower weght desgn than the other algorthms. February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

12 12 M. H. Talebpour et al. Gr. Mem. Table 4. Comparson of the optmal desgns for the 72-bar spatal truss structure Wu and Chow [31] L et al. [32] L et al. [32] Optmal cross-sectonal area n 2 (mm 2 ) Kaveh and L et al. Talatahar [32] [11] [33] Kaveh and Talatahar Kalatjar and Talebpour [29] Ths Study GA PSO PSOPC HPSO DHPSACO CSS M.S.M HACOHS-T A 1 -A 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 5 -A 12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 13 -A (71.613) ( ) ( ) (71.613) (71.613) (90.968) (71.613) (71.613) A 17 -A (71.613) ( ) ( ) (71.613) (71.613) (71.613) (71.613) (71.613) A 19 -A 22 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 23 -A 30 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 31 -A (71.613) (90.968) ( ) ( ) (90.968) (71.613) (71.613) (71.613) A 35 -A (71.613) (71.613) ( ) (71.613) (71.613) (71.613) (71.613) (71.613) A 37 -A 40 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 41 -A 48 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 49 -A (71.613) (90.968) (71.613) (71.613) (71.613) (71.613) (71.613) (71.613) A 53 -A (71.613) ( ) ( ) (71.613) ( ) ( ) (71.613) (71.613) A 55 -A 58 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 59 -A 66 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 67 -A 70 (90.968) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A 71 -A 72 (71.613) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Weght-lb (kg) c) A 200-bar truss ( ) ( ) ( ) ( ) ( ) ( ) (177.63) ( ) Ths example deals wth optmzaton of a 200-bar truss, as llustrated n Fg. 7. Here, E and ρ are assumed to be Ks ( MPa) and lb/n 3 ( kg/cm 3 ), respectvely. The truss members are categorzed nto 29 groups, and the allowable stress s taken as ±10 ks (± MPa). The external loads are exerted to the truss n 3 dfferent condtons: (1) 1 kp (4.448 kn) load exerts to nodes 1, 6, 15, 20, 29, 34, 43, 48, 57, 62, 71 n x drecton. (2) 10 kp (44.48 kn) load exerts to nodes 1, 2,, 6, 8, 10, 12, 14, 15,, 20, 22, 24, 25,, 73, 74 and 75 n y drecton. (3) Combnaton of condtons 1 and 2. The avalable lst for optmzaton of the 200-bar truss s as follows: a S= {0.1 (64.516), ( ), 0.44 ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), 2.8 ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), 9.3 ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), 33.7 ( )} n 2 - (mm 2 ); = 1,, 29. IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014

13 Optmzaton of skeletal structures usng a 13 Fg. 7. A 200-bar planar truss structure Fgure 8 shows the convergence curves obtaned by dfferent proposed methods ndcatng a better convergence for the HACOHS-T. To avod the effect of random parameters on the proposed methods, convergence curves for each method are drawn as the average of 30 runs. Accordng to Table 5, the results of the HACOHS-T method confrm the better performance of ths method. Fg. 8. The convergence hstory of the proposed methods for the 200-bar truss structure February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

14 14 M. H. Talebpour et al. Gr. Table 5. Comparson of the optmal desgns for the 200-bar planar truss structure Members Coello and Chrstansen [34] Optmal cross-sectonal area n 2 (mm 2 ) Toğan and Daloğlu [35] Kalatjar and Talebpour [36] Ths study GA GA M.S.M HACOHS-T 1 1, 2, 3, ( ) ( ) 0.1 (64.516) 2 5, 8, 11, 14, ( ) ( ) ( ) 3 19, 20, 21, 22, 23, (64.516) 0.1 (64.516) ( ) 4 18, 25, 56, 63, 94, 101, 132, 139, 170, (64.516) 0.1 (64.516) 0.1 (64.516) 5 26, 29, 32, 35, ( ) ( ) ( ) 6 6, 7, 9, 10, 12, 13, 15, 16, 27, 28, 30, 31, 33, 34, 36, ( ) ( ) ( ) 7 39, 40, 41, (64.516) 0.1 (64.516) 0.1 (64.516) 8 43, 46, 49, 52, ( ) ( ) ( ) 9 57, 58, 59, 60, 61, ( ) 0.1 (64.516) 0.1 (64.516) 10 64, 67, 70, 73, ( ) ( ) ( ) 11 44, 45, 47, 48, 50, 51, 53, 54, 65, 66, 68, 69,71,72, 74, ( ) 0.44 ( ) 0.44 ( ) 12 77, 78, 79, ( ) ( ) 0.1 (64.516) 13 81, 84, 87, 90, ( ) ( ) ( ) 14 95, 96, 97, 98, 99, ( ) 0.1 (64.516) 0.1 (64.516) , 105, 108, 111, ( ) ( ) ( ) 82, 83, 85, 86, 88, 89, 91, 92, 103, 104, 106, , 109, 110, 112, ( ) ( ) ( ) , 116, 117, ( ) ( ) ( ) , 122, 125, 128, ( ) ( ) ( ) , 134, 135, 136, 137, (64.516) 0.1 (64.516) 0.1 (64.516) , 143, 146, 149, ( ) 9.3 ( ) 9.3 ( ) 120, 121, 123, 124, 126, 127, 129, 130, 141, , 144, 145, 147, 148, 150, ( ) ( ) ( ) , 154, 155, ( ) ( ) ( ) , 160, 163, 166, ( ) ( ) ( ) , 172, 173, 174, 175, ( ) ( ) ( ) , 181, 184, 187, ( ) ( ) ( ) 158, 159, 161, 162, 164, 165, 167, 168, 179, , 182, 183, 185, 186, 188, ( ) ( ) ( ) , 192, 193, ( ) ( ) ( ) , 197, 198, ( ) ( ) ( ) , ( ) ( ) ( ) Weght-lb (kg) ( ) ( ) ( ) ( ) d) An eght-story, one-bay frame In ths example, an eght-story frame wth one bay, as llustrated n Fg. 9, s optmzed. Here, E and ρ are assumed as 200 GPa and 76.8 kn/m 3, respectvely, and the lateral drft at the top of the structure s the only performance constrant (lmted to 5.08 cm). Effectve loads are consdered for one condton as shown n Fg. 9. Members of the mentoned frame are categorzed nto 8 groups selected from a lst of 268-sectons (Table 6). Table 6. The avalable cross-secton areas of the AISC W-Secton No. Secton A cm 2 (n 2 ) I x cm 4 (n 4 ) S x cm 3 (n 3 ) I y cm 4 (n 4 ) S y cm 3 (n 3 ) 1 W44 x (98.3) (31100) (1410) (1200) (150) 2 W44 x (85.8) (27100) (1240) (1050) (133) 267 W5 x (4.68) (21.3) (8.51) (7.51) (1.27) 268 W4 x (3.83) (11.3) (5.46) (3.86) (1) IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014

15 Optmzaton of skeletal structures usng a 15 Fg. 9. A one-bay eght-story frame structure Dfferent proposed methods were appled to the optmal desgn of ths frame, and the average of 30 dfferent runs s used for drawng the convergence curves, Fg. 10. As t s shown n ths fgure, the HACOHS-T method has a better average performance than the other methods n obtanng the optmum desgn. Fg. 10. The convergence hstory obtaned by the proposed methods for the one-bay, eght story frame February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

16 16 M. H. Talebpour et al. The results ndcate that the proposed HACOHS-T method s more effcent than the methods of the other researchers (Table 7). Gr. Khot et al [37] Table 7. Comparson of optmal desgns for the one-bay, eght story frame Kaveh et al [12] Camp et al [38] Kaveh et al [9] Kaveh et al [15] Kaveh et al [12] Kaveh et al [12] Kaveh et al [17] Kaveh et al [9] Ths Study GA PSO FEAPGEN PSOPC+ACO ACO GA HGAPSO ACO DPSACO HACOHS-T 1 W14 x 34 W21 x 44 W18 x 46 W18 x 35 W21 x 50 W21 x 44 W18 x 35 W18 x 40 W18 x 35 W 18 x 35 2 W10 x 39 W16 x 26 W16 x 31 W16 x 31 W16 x 26 W18 x 35 W18 x 35 W16 x 26 W16 x 31 W 16 x 31 3 W10 x 33 W21 x 44 W16 x 26 W14 x 22 W16 x 26 W14 x 22 W14 x 22 W16 x 26 W16 x 26 W 16 x 26 4 W8 x 18 W12 x 16 W12 x 16 W12 x 16 W12 x 14 W12 x 14 W12 x 16 W12 x 14 W14 x 22 W 12 x 16 5 W21 x 68 W14 x 30 W18 x 35 W21 x 48 W16 x 26 W16 x 26 W16 x 31 W21 x 44 W16 x 31 W 18 x 35 6 W24 x 55 W21 x 44 W18 x 35 W18 x 40 W18 x 40 W18 x 40 W21 x 44 W18 x 35 W18 x 40 W 18 x 35 7 W21 x 50 W14 x 22 W18 x 35 W16 x 31 W18 x 35 W18 x 35 W18 x 35 W18 x 35 W16 x 26 W 18 x 35 8 W12 x 40 W16 x 26 W16 x 26 W16 x 36 W14 x 22 W12 x 22 W16 x 26 W12 x 22 W14 x 22 W 16 x 26 w-kn e) A fve-story, two-bay frame In ths example a fve-story frame wth two bays s studed, Fg. 11. For all structural members, E and ρ are assumed to be GPa and 78 kn/m 3, respectvely. Accordng to references [39,40], the allowable stress for all the structural members s equal to ±166.6 MPa and the allowable dsplacement for nodes of the last story s 1/500 of the frame heght. Fg. 11. A fve-story, two-bay unbraced frame Dead, lve, and wnd loads are appled to the structure as the followng three cases (the magntude and drectons of the loads are defned n Table 8). ) DL+0.9(LL+WL) ) DL+WL ) DL+ LL IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014

17 Optmzaton of skeletal structures usng a 17 Loadng type Dead load Table 8. Appled loads to the fve-story, two-bay unbraced frame Magntude and drecton W y = kn/m on members P y = 19.6 kn at nodes 1 and 3 P y = 40.2 kn at nodes 4, 6, 7, 9, 10, 12, 13 and 15 Lve load W y = kn/m on members P x =5.684 kn at node 1 P x =7.252 kn at node 4 Wnd load P x =6.664 kn at node 7 P x =5.978 kn at node 10 P x =6.272 kn at node 13 Columns must have a constant cross secton n each story for the desgned frame. On the other hand, snce all the structural beams are desgned ndependently, a total of 15 desgn varables are selected and consdered from the lst of sectons of Table 9. Table 9. Avalable cross-secton for the fve-story, two-bay unbraced frame Secton A cm 2 I number y cm 4 S y cm 3 I x cm 4 S x cm 3 Secton number A cm 2 I y cm 4 S y cm 3 I x cm 4 S x cm In Fg. 12, the convergence curves for ths example are llustrated for the proposed methods. Each curve s obtaned usng the average of 30 dfferent runs. Therefore, the effect of random parameters on the proposed methods s reduced and more accurate judgment can be made. Results show that the HACOHS- T method has better performance than the other methods n obtanng the optmum desgn. Fnal desgn obtaned from HACOHS-T method has lower weght compared to the other proposed methods and references. Fg. 12. The convergence hstory obtaned wth the proposed methods for the present frame after 30 runs February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

18 18 M. H. Talebpour et al. Table 10 ncludes the results of the optmum desgn of other references compared to those of the HACOHS-T method. Table 10. Comparson of the optmal desgns for the fve-story, two-bay unbraced frame Group Member Cha and Sun [39] Juang and Chang [40] Ths Study RDQA DLM HACOHS-T Weght-kN CONCLUDING REMARKS In ths paper, a new combned method s proposed for optmal desgn of skeletal structures such as frames and trusses through combnng the ACO, HS and GA selecton methods. In ths algorthm, the global and local search processes are smultaneously performed based on the above mentoned methods. The ACO method deals wth global search process whereas HS method and ACO perform local search. On the other hand, dfferent selecton approaches consstng of Roulette Wheel (HACOHS-RO), Rank (HACOHS-RA) and Tournament (HACOHS-T) methods are used n the combned HACOHS algorthm. It s found that the HACOHS-T method has a better performance than the other methods. Ths can be easly observed from Fgs. 4, 6, 8, 10 and 12. It should be mentoned that the HACOHS-RO method also has approprate convergence property n small search spaces wth low number of cross lst of sectons, whle n large search spaces ths s not as successful as HACOHS-T method. Thus t s not advsable for problems wth large search spaces. On the other hand, havng slow convergence rate, the HACOHS-Ra does not provde a proper process for the present problems to obtan optmum pont. Therefore, applyng ths method s not recommended. Consderng the qualty of results, the proposed HACOHS-T method s superor to the other proposed methods and other exstng algorthms. Ths pont can also be observed form the results of the nvestgated examples consstng of the 8-story frame 1- bay, 5-story 2- bays and the 200-bar, 72-bar and 52-bar truss examples. In all the optmzed structures, the algorthms of some other researchers are also mproved, and the HACOHS-T acheved a desgn of smaller weght. The advantages of GA, HS and ACO algorthms are used n HACOHS-T algorthm so that the search space s searched more accurately and the probablty of beng trapped n local optmum s reduced. Furthermore, the convergence of the HACOHS-T method ndcates that the algorthm does not need the tunng of parameters of the consttutng algorthms. Accordng to HACOHS-T method, sutable cross sectons are selected for the members based on the IJST, Transactons of Cvl Engneerng, Volume 38, Number C1 February 2014

19 Optmzaton of skeletal structures usng a 19 tournament method of Genetc algorthm, and global and local search abltes of the ant colony algorthm and Harmony search are smultaneously utlzed. Acknowledgement: The second author s grateful to the Iran Natonal Scence Foundaton for support. REFERENCES 1. Glover, F. & Kochenberger, G. A. (2003). Handbook of metaheurstcs. Kluwer Academc Publshers, Boston, USA. 2. Dreo, J., Petrowsk, A., Sarry, P. & Tallard, E. (2006). Metaheurstcs for hard optmzaton. Sprnger-Verlag, Berln, Hedelberg, Germany. 3. Fogel, L. J, Qwens, A. J. & Walsh, M. J. (1966). Artfcal ntellgence through smulated evoluton. John Wley & Sons, Chchester, UK. 4. Holland, J. H. (1975). Adaptaton n natural and artfcal system. Unversty of Mchgan Press, Ann Arbor, MI, USA. 5. Goldberg, D. E. (1989). Genetc algorthm n search optmzaton and machne learnng. Addson-Wesley, Boston, USA. 6. Colorn, A., Dorgo, M. & Manezzo, V. (1991). Dstrbuted optmzaton by ant colony. In: Proceedng of the frst European conference on artfcal lfe, USA, pp Dorgo, M. (1992). Optmzaton learnng and natural algorthm (n Italan). PhD. Thess, Dparmento d, Elettronca, Poltecnco d Mlano, Mlan, Italy. 8. Geem, Z. W., Km, J. H. & Loganathan, G. V. (2001). A new heurstc optmzaton algorthm: harmony search. Int J Model Smulat, Vol. 76, No. 2, pp Kaveh, A. & Talatahar, S. (2007). A dscrete partcle swarm ant colony optmzaton for desgn of steel frames. Asan J Cvl Eng, Vol. 9, No. 6, pp Kao, Y. T. & Zahra, E. (2008). A hybrd genetc algorthm and partcle swarm optmzaton for multmodal functon. Appl Soft Comput, Vol. 8, No. 2, pp Kaveh, A. & Talatahr, S. (2009). A partcle swarm ant colony optmzaton for truss structures wth dscrete varables. J Construct Steel Res, Vol. 65, Nos. (8-9), pp Kaveh, A. & Malakoutrad, S. (2010). Hybrd genetc algorthm and partcle swarm optmzaton for the force method-based smultaneous analyss and desgn. Iranan J Sc Tech, Vol. 34, pp Camp, C. V. & Bchon, J. B. (2004). Desgn of space trusses usng ant colony optmzaton. J Struct Eng ASCE, Vol. 130, No. 5, pp Camp, C. V., Bchon, J. B. & Stovall, S. P. (2004). Desgn of steel frames usng ant colony optmzaton, J Struct Eng ASCE, Vol. 131, No. 3, pp Kaveh, A. & Shojaee, S. (2007). Optmal desgn of skeletal structures usng ant colony optmzaton. Int J Numer Methods Eng, Vol. 70, No. 5, pp Caprles, V. S. Z., Fonseca, L. G., Barbosa, H. J. C. & Lemonge, A.C.C. (2007). Rank-based ant colony algorthms for truss weght mnmzaton wth dscrete varables. Commun Numer Methods Eng, Vol. 23, No. 6, pp Kaveh, A. & Talatahar, S. (2010). An mproved ant colony optmzaton for the desgn of planar steel desgn frames. Eng Struct, Vol. 32, No. 3, pp Kaveh A. & Talatahar S. (2010). An mproved ant colony optmzaton for constraned engneerng desgn problems. Eng Comput, Vol. 27, No. 1, pp Stützle, T. & Hoos, H. (1997). Improvement on the ant system: Introducng MAX-MIN ant system. Proceedng of the nternatonal conference on artfcal neural network and genetc algorthm, Wen, pp February 2014 IJST, Transactons of Cvl Engneerng, Volume 38, Number C1

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