A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION

Transcription

1 INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING Int. J. Optm. Cvl Eng., 06; 6(4): Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION A. Kaveh *, and A. Zolghadr Centre of Excellence for Fundamental Studes n Structural Engneerng, Iran Unversty of Scence and Technology, Narma, Tehran 6, Iran ABSTRACT Ths paper presents a novel populaton-based meta-heurstc algorthm nspred by the game of tug of war. Utlzng a sport metaphor the algorthm, denoted as Tug of War Optmzaton (TWO), consders each canddate soluton as a team partcpatng n a seres of rope pullng compettons. The teams exert pullng forces on each other based on the qualty of the solutons they represent. The competng teams move to ther new postons accordng to Newtonan laws of mechancs. Unle many other meta-heurstc methods, the algorthm s formulated n such a way that consders the qualtes of both of the nteractng solutons. TWO s applcable to global optmzaton of dscontnuous, multmodal, non-smooth, and non-convex functons. Vablty of the proposed method s examned usng some benchmar mathematcal functons and engneerng desgn problems. The numercal results ndcate the effcency of the proposed algorthm compared to some other methods avalable n lterature. Keywords: tug of war optmzaton; meta-heurstc algorthm; mathematcal functons; optmal desgn; truss structures. Receved: 9 December 05; Accepted: 0 February 06. INTRODUCTION Nature-nspred meta-heurstc algorthms have been celebrated as powerful global optmzaton technques n the last few decades. These methods do not requre any gradent nformaton of the nvolved functons and are generally ndependent of the startng ponts. Due to these characterstcs, meta-heurstc optmzers are favorable choces when dealng wth dscontnuous, multmodal, non-smooth, and non-convex functons. Ths s especally the case when nearglobal optmum solutons are sought and the ntended computatonal effort s lmted. * Correspondng author: Centre of Excellence for Fundamental Studes n Structural Engneerng, Iran Unversty of Scence and Technology, Narma, Tehran 6, Iran E-mal address: alaveh@ust.ac.r (A. Kaveh)

2 470 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Many dfferent meta-heurstc optmzaton technques have been presented and successfully appled to dfferent problems. Examples nclude Genetc Algorthms (GA) [], Partcle Swarm Optmzaton (PSO) [], Ant Colony Optmzaton (ACO) [], Harmony Search (HS) [4], Bg Bang-Bg Crunch (BB-BC) [5], Charged System Search (CSS) [6], Magnetc Charged System Search (MCSS) [7], Ray Optmzaton (RO) [8], Democratc PSO (DPSO) [9], Dolphn Echolocaton (DE) [0], Colldng Bodes Optmzaton (CBO) [], Water Cycle, Mne Blast and Improved Mne Blast algorthms (WC-MB-IMB) [], Search Group Algorthm (SGA) [] and the Ant Lon Optmzer (ALO) [4]. Some applcatons of meta-heurstcs on optmzaton of structural engneerng problems could be found n [5-9]. The am of the present paper s to ntroduce a new meta-heurstc algorthm based on the physcal phenomenon assocated wth the game of tug of war. The algorthm, whch s called Tug of War Optmzaton (TWO), s then compared to some of the documented optmzaton technques usng benchmar problems. The remander of the paper s organzed as follows. The new algorthm together wth ts physcal bacground s presented n Secton. In Secton some mathematcal and engneerng desgn benchmar problems are studed n order to demonstrate the effcency of the proposed algorthm. The concludng remars are presented n Secton 4.. Idealzed tug of war framewor. TUG OF WAR OPTIMIZATION Tug of war or rope pullng s a strength contest n whch two competng teams pull on the opposte ends of a rope n an attempt to brng the rope n ther drecton aganst the pullng force of the opposng team. The actvty dates bac to ancent tmes and has contnued to exst n dfferent forms ever snce. There has been a wde varety of rules and regulatons for the game but the essental part has remaned almost unaltered. Naturally, as far as both teams sustan ther grps of the rope, movement of the rope corresponds to the dsplacement of the losng team. Fg. shows one of the competng teams n a tug of war. Fgure. A competng team n a tug of war Trumph n a real game of tug of war generally depends on many factors and could be dffcult to analyze. However, an dealzed framewor s utlzed n ths paper where the two teams havng weghts W and W j are consdered as two objects lyng on a smooth surface as shown n Fg..

3 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 47 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 μ s : Statc coeffcent of frcton μ : Knematc coeffcent of frcton Fgure. An dealzed tug of war framewor As a result of pullng the rope, the teams experence two equal and opposte forces (F p ) accordng to Newton's thrd law. For object, as far as the pullng force s smaller than the maxmum statc frcton force W s the object rests n ts place. Otherwse the non-zero resultant force can be calculated as: F r F W () p As a result, the object accelerates towards the object j accordng to the Newton's second law: Fr a W g Snce the object starts from zero velocty, ts new poston can be determned as: X new at X old () (). Tug of war optmzaton algorthm TWO s a populaton-based meta-heurstc algorthm, whch consders each canddate soluton as a team engaged n a seres of tug of war compettons. The weght of X x, j the teams s determned based on the qualty of the correspondng solutons, and the amount of pullng force that a team can exert on the rope s assumed to be proportonal to ts weght. Naturally, the opposng team wll have to mantan at least the same amount of force n order to sustan ts grp of the rope. The lghter team accelerates toward the heaver team and ths forms the convergence operator of the TWO. The algorthm mproves the qualty of the solutons teratvely by mantanng a proper exploraton/explotaton balance usng the descrbed convergence operator. The steps of TWO can be stated as follows: Step : Intalzaton A populaton of N ntal solutons s generated randomly:

4 47 A. Kaveh and A. Zolghadr 0 xj x j,mn rand( x j,max x j, mn ) j,,..., n (4) Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 0 where x j s the ntal value of the jth varable of the th canddate soluton; x j, max and x j, mn are the maxmum and mnmum permssble values for the jth varable, respectvely; rand s a random number from a unform dstrbuton n the nterval [0, ]; n s the number of optmzaton varables. Step : Evaluaton of canddate desgns and weght assgnment The objectve functon values for the canddate solutons are evaluated. All of the ntal solutons are sorted and recorded n a memory denoted as the league. Each soluton s consdered as a team wth the followng weght: ft ( ) ft worst W ( ),,..., N (5) ft ft best worst where ft() s the ftness value for the th partcle; The ftness value of the th team, evaluated as as the penalzed objectve functon value for constraned problems; ft best and ft worst are the ftness values for the best and worst canddate solutons of the current teraton. Accordng to Eq. (5) the weghts of the teams range between and. Step : Competton and dsplacement In TWO each of the teams of the league competes aganst all the others one at a tme to move to ts new poston n each teraton. The pullng force exerted by a team s assumed to be equal to ts statc frcton force W s. Hence the pullng force between teams and j (F p,j ) can be determned as max{ W s, Wjs }. Such a defnton eeps the poston of the heaver team unaltered. The resultant force affectng team due to ts nteracton wth heaver team j n the th teraton can then be calculated as follows: F r,j F W (6) p,j where F p, j s the pullng force between teams and j n the th teraton and s coeffcent of nematc frcton. Consequently, team accelerates towards team j: a j Fr,j ( ) gj (7) W where a j s the acceleraton of team towards team j n the th teraton; gravtatonal acceleraton constant defned as: g j s the

5 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 47 g j X X (8) j Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 where X j and X are the poston vectors for canddate solutons j and n the th teraton. Fnally, the dsplacement of the team after competng wth team j can be derved as: X j j a t ( X X ) randn (, n) (9) max The second term of Eq. (9) ntroduces randomness nto the algorthm. Ths term can be nterpreted as the random porton of the search space traveled by team before t stops after the appled force s removed. The role of s to gradually decrease the random porton of the team's movement. For most of the applcatons could be consdered as a constant chosen from the nterval [0.9, 0.99]; bgger values of decrease the convergence speed of the algorthm and help the canddate solutons explore the search space more thoroughly. s a scalng factor whch can be chosen from the nterval (0,]. Ths parameter controls the steps of the canddate solutons when movng n the search space. When the search space s supposed to be searched more accurately wth smaller steps, smaller values should be chosen for ths parameter. For our numercal examples values between 0.0 and 0.05 seem to be approprate for ths parameter; X max and X mn are the vectors contanng the upper and lower bounds of the permssble ranges of the desgn varables, respectvely; denotes element by element multplcaton; randn (, n) s a vector of random numbers drawn from a standard normal dstrbuton. It should be noted that when team j s lghter than team, the correspondng dsplacement of team wll be equal to zero (.e. X j ). Fnally, the total dsplacement of team n teraton s equal to ( not equal j): N j mn X X (0) j The new poston of the team at the end of the th teraton s then calculated as: X X X () Step 4: Updatng the league Once the teams of the league compete aganst each other for a complete round, the league should be updated. Ths s done by comparng the new canddate solutons (the new postons of the teams) to the current teams of the league. That s to say, f the new canddate soluton s better than the Nth team of the league n terms of objectve functon value, the Nth team s removed from the league and the new soluton taes ts place.

6 474 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Step 5: Handlng the sde constrants It s possble for the canddate solutons to leave the search space and t s mportant to deal wth such solutons properly. Ths s especally the case for the solutons correspondng to lghter teams for whch the values of X s usually bgger. Dfferent strateges mght be used n order to solve ths problem. For example, such canddate solutons can be smply brought bac to ther prevous feasble poston (flybac strategy) or they can be regenerated randomly. In ths paper a new strategy s ntroduced and ncorporated usng the global best soluton. The new value of the jth optmzaton varable of the th team that volated sde constrants n the th teraton s defned as: x j randn GBj ( )(GBj xj ) () where GB j s the jth varable of the global best soluton (.e. the best soluton so far); randn s a random number drawn form a standard normal dstrbuton. There s a very slght possblty for the newly generated varable to be stll outsde the search space. In such cases a flybac strategy s used. The abovementoned strategy s utlzed wth a certan probablty (0.5 n ths paper). For the rest of cases the volated lmt s taen as the new value of the jth optmzaton varable. Step 6: Termnaton Steps through 5 are repeated untl a termnaton crteron s satsfed. The pseudo code of TWO s presented n Table. Table : Pseudo-code of the TWO algorthm developed n ths study procedure Tug of War Optmzaton begn Intalze parameters; Intalze a populaton of N random canddate solutons; Intalze the league by recordng all random canddate soluton; whle (termnaton condton not met) do Evaluate the objectve functon values for the canddate solutons Sort the new solutons and update the league Defne the weghts of the teams of the league W based on ft(x ) for each team for each team j f (W < W j ) Move team towards team j usng Eq. (9); end f end for Determne the total dsplacement of team usng Eq. (0) Determne the fnal poston of team usng Eq.() Use the sde constrant handlng technque to regenerate volatng varables end for end whle end

7 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 475. TEST PROBLEMS AND OPTIMIZATION RESULTS Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 In order to evaluate the effcency of the proposed algorthm, some benchmar test problems are consdered from the lterature. A set of un-modal and mult-modal mathematcal optmzaton problems are studed n Secton.. In addton, sx well-studed engneerng desgn problems are nvestgated n Secton.. Except for the last three examples a populaton of 0 agents and a maxmum number of permtted teratons of 00 are used for all test problems. For the fourth, ffth, and sxth structural desgn problems the numbers of agents are taen as 0, 40, and 40, respectvely whle 400 teratons are used for these examples. The coeffcent of statc frcton ( s ) s taen as unty, whle the coeffcent of nematc frcton ( ) vares lnearly from to 0.. Snce smaller values of let the teams slde more easly towards each other and vce versa, such a parameter selecton helps the algorthm's agents to explore the search space at early teratons wthout beng severely affected by each other. As the optmzaton process proceeds, the values of gradually decrease allowng for convergence. It was found that usng the same value of nematc frcton coeffcent for all teams yeld the best optmzaton results for TWO.. Mathematcal optmzaton problems In ths secton the effcency of the TWO s evaluated by solvng the mathematcal benchmar problems summarzed n Table. These benchmar problems are taen from Ref. [0], where some varants of GA were used as the optmzaton algorthm. The results obtaned by TWO are presented n Table along wth those of GA varants. Each objectve functon s optmzed 50 tmes ndependently startng from dfferent ntal populatons and the average number of functon evaluatons requred by each algorthm s presented. The numbers n the parentheses ndcate the rato of the successful runs n whch the algorthm has located the global mnmum wth predefned accuracy, whch s taen as f f. The absence of the parentheses means that the algorthm has been mn 0 4 fnal successful n all ndependent runs. Table : Detals of the benchmar mathematcal problems solved n ths study Sde Functon name Functon constrants Aluff-Pentny Bohachevsy Bohachevsy Becer and Lago Brann Camel X [ 0,0] X [ 00,00] X [ 50,50] X [ 0,0] 0 x 5 5 x 0 X [ 5,5] 4 f ( X ) x x x x x x cos(x ) cos(4x ) 0 0 x x cos(x )cos(4x ) 0 f ( X ) f ( X ) f ( X ) ( x 5 ) ( x 5 ) 5. 5 f ( X ) ( x x x ) 0( )cos( x ) f ( X ) 4x x x xx 4x 4x Global mnmum

8 476 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Cb Cosne mxture DeJoung Exponental Goldsten and prce Grewan Hartman Hartman 6 X [ 5,5] n 4, X [,] X [ 5.,5.] n n,4,8, X [,] X [,] X [ 00,00] X [0,] 6 X [0,] n f ( X ) f ( X f ( X ) [ ( x x x ) f ( X x x xx x n f ( X ) n x cos(5x ) 0 x x x ) exp 0.5 n [0 (x x ) (8 x x x ) (94x x 4x f ( X ) f ( X ) 4 00 x 48x 6x x 6x x x ] x cos( ) c exp aj ( x j pj ) j a. c , p f ( X ) a p c exp 6 j a ( x j j and. p j ) c ,. and x ] Table : Performance comparson of TWO and GA varants n the mathematcal optmzaton problems Functon name GEN GEN S GEN S M GEN S M LS TWO AP,60 (0.99),60,77,5 09 Bf,99,56, Bf 0,4,7, BL 9,596 4,49,46 6 Brann,44,48,404,57 89 Camel,58,58,6,00 Cb 9,77,045,6,8 99 CM,05,05,74,59 40

9 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 477 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 DeJoung 9,900,040,46,8 46 Exp Exp4,7,7,054, Exp8,7,7,054, Goldsten and Prce,478,478,408,5 690 Grewan 8,88 (0.9), (0.9),764,65 (0.99) 766 Hartman,50,50,,74 06 Harman6,56 (0.54),56 (0.54),50 (0.67),865 (0.68) 60 (0.68) As t can be seen from Table, TWO generally performs better than GA and ts varants n the mathematcal optmzaton problems consdered n ths study.. Structural desgn problems In order to further nvestgate the effcency of the TWO, three engneerng desgn problems are consdered n ths secton. These problems have been prevously studed usng dfferent optmzaton algorthms. These constraned optmzaton problems are turned nto unconstraned ones usng a penalty approach. If the constrants are satsfed, then the amount of penalty wll be zero; otherwse ts value can be calculated as the rato of volated constrant to the correspondng allowable lmt... Desgn of a tenson/compresson sprng Weght mnmzaton of the tenson/compresson sprng shown n Fg., subject to constrants on shear stress, surge frequency, and mnmum deflecton s consdered as the frst engneerng desgn example. Ths problem was frst descrbed by Belegundu [] and Arora []. The desgn varables are the mean col dameter D(x ), the wre dameter d(x ), and the number of actve cols N (x ). The cost functon can be stated as: Fgure. Schematc of the tenson/compresson sprng whle the constrants are: f cost ( X ) ( x )x x ()

10 478 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 ( 4 xx g X ) 7785x ( 4x xx g X ) ( x x x ) 508x 40.45x g ( X ) x x x x g4( X ) 0.5 The sde constrants are defned as follows: 0 0 (4) 0.05 x 0.5 x. (5) x 5 Ths problem has been solved by Belegundu [] usng eght dfferent mathematcal optmzaton technques. Arora [] utlzed a numercal optmzaton technque called a constrant correcton at the constant cost to nvestgate the problem. Coello [] and Coello and Montes [4] used a GA-based algorthm to solve the problem. He and Wang [5] used a co-evolutonary partcle swarm optmzaton (CPSO). Montes and Coello [6] used evoluton strateges. Kaveh and Talatahar used mproved ant colony optmzaton [7] and charged system search CSS [6]. Recently, the problem has been solved by Kaveh and Mahdav [] usng Colldng bodes optmzaton (CBO). The optmzaton results obtaned by TWO are presented n Table 4 along wth those of other methods. It can be seen that TWO found the best desgn overall. It should be noted that some constrants are slghtly volated by the desgns quoted by Kaveh & Talatahar [6, 6]. Table 5 shows the statstcal results obtaned for 0 ndependent optmzaton runs. Table 4: Comparson of the optmzaton results obtaned n the tenson/compresson sprng problem Optmal desgn varables Methods x ( x ( x ( N ) f cost Belegundu [] Arora [] Coello [] Coello & Montes [4] He and Wang [5] Montes and Coello [6] Kaveh & Talatahar [7] Kaveh & Talatahar [6] Kaveh & Mahdav [] Present wor

11 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 479 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Table 5: Comparson of statstcal optmzaton results obtaned n the tenson/compresson sprng desgn problem Methods Best Mean Worst Std dev Belegundu [] N/A N/A N/A Arora [] N/A N/A N/A Coello [] e-5 Coello & Montes [4] e-5 He and Wang [5] e-5 Montes and Coello [6] e-4 Kaveh & Talatahar [7] e-5 Kaveh & Talatahar [6] e-5 Kaveh & Mahdav [] e-5 Present wor e-4.. Desgn of a welded beam The second test problem regards the desgn optmzaton of the welded beam shown n Fg.4. Ths problem has been used for testng dfferent optmzaton methods. The am s to mnmze the total manufacturng cost subject to constrants on shear stress ( ), bendng stress ( ), buclng load ( P c ), and deflecton ( ). The four desgn varables, namely h (x ), l (x ), t (x ), and b (x 4 ), are also shown n the fgure. Fgure 4. Schematc of the welded beam The objectve functon can be mathematcally stated as: fcost 4 The optmzaton constrants are ( X ).047x x 0.048x x (4.0 x ) (6) g ( X ) ({ x }) 0 (7) max

12 480 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 where g( X ) ({ x }) max 0 ( X ) x x 0 g 4 g4( X ) 0.047x 0.048x x4(4.0 x ) 5.0 g5( X ) 0.5 x 0 g6 ( X ) ({ x }) max 0 g7( X ) P P c({ x }) 0 x R P MR ', " x x J ( X ) ( ') ' " ( x M P( L ), R J x 4 x x x x xx 4x ") x 6PL 4PL ( X), ( X) x Ex x 6 4 x x4 4.0E x E P 6 c ( X ) L L 4G P 6000lb, L 4n, 6 6 E 00 ps, G 0 ps 0 (8).6 0 ps, ps max max 00 max 0.5n The sde constrants can be stated as: 0. x, 0. x 0, 0. x 0, 0. x 4 (9) Radgsdell and Phllps [8] utlzed dfferent optmzaton methods manly based on mathematcal programmng to solve the problem and compared the results. GA-based methods are used by Deb [9], Coello [], Coello and Montes [4]. Ths test problem was also solved by He and Wang [5] usng CPSO and Montes and Coello [6] usng evoluton strateges. Kaveh and Talatahar employed ant colony optmzaton [7] and charged system search [6]. Kaveh and Mahdav [] solved the problem usng the colldng bodes

13 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 48 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 optmzaton method. Table 6 compares the best results obtaned by the dfferent optmzaton algorthms consdered n ths study. The statstcal results for 0 ndependent runs are provded n Table 7. It can be seen from Table 6 that optmum desgn found by TWO s about 0.0 percent heaver than that found by CBO whch was the best desgn quoted so far n lterature. Table 6: Comparson of the optmzaton results obtaned n the welded beam problem Optmal desgn varables Methods x ( x ( l ) x ( ) x 4( b t ) f cost Radgsdell and Phllps [8] Deb [9] Coello [] Coello and Montes [4] He and Wang [5] Montes and Coello [6] Kaveh & Talatahar [7] Kaveh & Talatahar [6] Kaveh & Mahdav [] Present wor Table 7: Comparson of statstcal optmzaton results obtaned n the welded beam problem Methods Best Mean Worst Std Dev Radgsdell and Phllps [8].85 N/A N/A N/A Deb [9].46 N/A N/A N/A Coello [] Coello and Montes [4] He and Wang [5] Montes and Coello [6] Kaveh & Talatahar [7] Kaveh & Talatahar [6] Kaveh & Mahdav [] Present wor Desgn of a planar 0-bar truss structure subject to frequency constrants The szng optmzaton of a planar 0-bar truss subject to frequency constrants s selected as the thrd test case. The confguraton of the structure s depcted n Fg. 5. Ths s a wellnown benchmar problem n the feld of frequency constrant structural optmzaton. Each of the members' cross-sectonal area s assumed to be an ndependent varable. A nonstructural mass of g s attached to all free nodes. Table 8 summarzes the materal propertes, varable bounds, and frequency constrants for ths example. Ths problem has been nvestgated by dfferent researchers: Grandh and Venayya [0] usng an optmalty algorthm, Sedaghat et al. [] usng a sequental quadratc programmng and fnte element force method, Wang et al. [] usng an evolutonary node shft method, Lngyun et al. [] utlzng a nche hybrd genetc algorthm, Gomes [4] employng the standard partcle

14 48 A. Kaveh and A. Zolghadr swarm optmzaton algorthm and Kaveh and Zolghadr utlzng democratc PSO [9], and a hybrdzed PSRO algorthm [5] among others. Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Fgure 5. Schematc of the planar 0-bar truss structure Table 8: Materal propertes, varable bounds and frequency constrants for the 0-bar truss structure Property/unt Value E (Modulus of elastcty)/ N/m ρ (Materal densty)/ g/m Added mass/g Desgn varable lower bound/m Desgn varable upper bound/m L (Man bar s dmenson)/m 9.44 Constrants on frst three frequences/hz ω 7, ω 5, ω 0 Table 9 summarzes the desgn vectors of the optmal structures found by dfferent methods. Table 9: Optmzed desgns (cm ) obtaned for the planar 0-bar truss problem (the optmzed weght does not nclude the added masses) Grandh & Lngyun Kaveh & Zolghadr Element Sedaghat Wang et Gomes Venayya et al. number et al. [] al. [] [4] DPSO PSRO Present [0] [] [9] [5] wor Weght (g)

15 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 48 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Accordng to Table 9, the weght of the optmal structure found by TWO s 5.7 g whch s slghtly better than that of DPSO, the best desgn quoted so far n the lterature. The mean value and the standard devaton of 0 ndependent runs of TWO on ths problem are 57.4 and.4 g, respectvely. Table 0 shows the natural frequences of the optmzed structures obtaned by dfferent methods. It can be seen that all of the constrants are satsfed...4 Desgn of a spatal 5-bar truss structure Weght mnmzaton of the spatal 5-bar truss schematzed n Fg. 6 s consdered as the fourth desgn example. The materal densty and modulus of elastcty are 0. lb/n and 0000 s, respectvely. Table shows the two ndependent loadng condtons appled to the structure. The twenty fve bars of the truss are classfed nto eght groups, as follows: () A, () A -A 5, () A 6 -A 9, (4) A 0 -A, (5) A -A, (6) A 4 -A 7, (7) A 8 -A, and (8) A - A 5. Table 0: Natural frequences (Hz) evaluated at the optmzed desgns for the planar 0-bar truss Grandh Lngyun Kaveh and Zolghadr Frequenc Sedaghat Wang et Gomes &Venay et al. y number et al. [] al. [] [4] DPSO PSRO Present ya [0] [] [9] [5] wor Table : Independent loadng condtons actng on the spatal 5-bar truss Node Case Case P x ps P y ps P z ps P x ps P y ps P z ps Maxmum dsplacement lmtatons of 0.5 n are mposed on all nodes n all drectons. The axal stress constrants, whch are dfferent for each group, are shown n Table. The cross-sectonal areas vary contnuously from 0.0 to.4 n for all members.

16 484 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Table : Member stress lmts for the 5-bar spatal truss Element group Compressve stress lmts s (MPa) Tensle stress lmt s (MPa) 5.09 (4.96) 40.0 (75.80).590 (79.9) 40.0 (75.80) 7.05 (9.) 40.0 (75.80) (4.96) 40.0 (75.80) (4.96) 40.0 (75.80) (46.60) 40.0 (75.80) (47.98) 40.0 (75.80) 8.08 (76.40) 40.0 (75.80) Fgure 6. Schematc of the spatal 5-bar truss structure Table shows that the dfferent optmzaton methods converged almost to the same structural weght. It seems that the results obtaned by dfferent methods are very close together for ths example. The best result obtaned by TWO s g whch s only slghtly heaver than that of HS. However, t should be noted that accordng to our codes the optmum desgn of HS slghtly volates dsplacement constrants. Moreover TWO requres 000 analyses to complete the optmzaton process, whle HS has used 5000 analyses. The mean value and standard devaton of 0 ndependent optmzaton runs by TWO are and 0. lb, respectvely. Small dfferences n weght may orgnate from the level of precson n the mplementaton of the optmzaton problem. For example, the dsplacement lmt of 0.50n set for TWO resulted n feasble optmzed desgn quoted n Table yeldng a maxmum dsplacement of 0.504n whch can be rounded to the above mentoned lmt.

17 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 485 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th Desgn of a spatal 7-bar truss structure A spatal 7-bar truss structure shown n Fg. 7 s consdered as the ffth desgn example. The elements are grouped to form 6 desgn varables accordng to Table 5. The materal densty and the modulus of elastcty are taen as 0. lb/n and 0,000 s, respectvely. All members are subjected to a stress lmtatons of ±5 s. The dsplacements of the uppermost nodes along x and y axes are lmted to ±0.5 n. Cross-sectonal areas of bars can vary between 0.0 and 4.00 n. The two loadng condtons actng on the structure are summarzed n Table 4. Ths problem has been studed by Erbatur et al. [9] usng Genetc Algorthms, Camp and Bchon [40] usng Ant Colony Optmzaton, Perez and Behdnan [4] usng Partcle Swarm Optmzaton, Camp [4] usng Bg Bang-Bg Crunch algorthm, and Kaveh and Khayatazad [8] usng Ray Optmzaton among others. Table : Comparson of the optmzaton results obtaned n the spatal 5-bar truss problem Element group Optmal cross-sectonal areas (n ) Rajeev and Schutte and Lee and Geem, Present Krshnamoorthy, GA Groenwold, HS [8] wor [6] PSO [7] A A -A A 6 -A A 0 -A A -A A 4 -A A 8 -A A -A Best weght (lb) Average weght (lb) N/A N/A Std Dev (lb) N/A.478 N/A 0. No. of analyses N/A Table 4: Independent loadng condtons actng on the spatal 7-bar truss node Case Case P x ps (N) P y ps (N) P z ps (N) P x ps(n) P y ps(n) P z ps (N) _ -5 _ -5 4 _ -5

18 486 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Fgure 7. Schematc of the spatal 7-bar truss structure Table 5 compares the results obtaned by the present method to those prevously reported n the lterature. The weght of the best result obtaned by TWO s lb whch s the best among the compared methods. Moreover, the mean weght of the results for 0 ndependent optmzaton runs of TWO s 8.98 lb whch s less than all other methods. Also, TWO requres only 6000 structural analyses whle ACO, BB-BC and RO requre 8,500, 9,6, and 9,084, respectvely. Table 5: Comparson of the optmzaton results obtaned n the spatal 7-bar truss problem Optmal cross-sectonal areas (n ) Camp and Perez and Kaveh and Element group Erbatur et Present Bchon Behdnan Camp [4] Khayatazad al. [9] wor [40] [4] [8]

19 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 487 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th Best weght (lb) Mean weght (lb) N/A 8.6 N/A Standard devaton (lb) N/A.66 N/A.9..6 Number of analyses N/A 8,500 N/A 96 9, Desgn of a 0-bar dome truss The last test problem s the weght mnmzaton of a 0-bar dome truss shown n Fg. 8. Ths structure was consdered by Soh and Yang [4] as a confguraton optmzaton problem. It has been solved by Lee and Geem [8], Kaveh and Talatahar [44], Kaveh et al. [45], Kaveh and Khayatazad [8], and Kaveh and Mahdav [] as a szng optmzaton problem. The members of the structure are dvded nto seven groups as shown n Fg. 8. Fgure 8. Schematc of the 0-bar dome truss structure

20 488 A. Kaveh and A. Zolghadr The allowable tensle and compressve stresses are set accordng to the AISC-ASD [46] provsons as follows: Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th F y for for 0 0 where s the compressve allowable stress and depends on the elements slenderness rato Cc E F y 5 8Cc 8Cc for for C C where E s the modulus of elastcty, F y s the yeld stress of the materal, s the KL slenderness rato ( ), K s the effectve length factor, L s the member length and r r s the radus of gyraton. C c s the crtcal slenderness rato separatng elastc and nelastc buclng regons ( C E ). c F y The modulus of elastcty and the materal densty are taen as 0,450 s (0 GPa) and 0.88 lb/n, respectvely. The yeld stress s taen as 58.0 s (400 MPa). The radus of b gyraton s expressed n terms of cross-sectonal areas of the members as r aa [47]. Constants a and b depend on the types of sectons adopted for the members such as ppes, angles, etc. In ths example ppe sectons are used for the bars for whch a=0.499 and b= The dome s consdered to be subjected to vertcal loads at all unsupported nodes. These vertcal loads are taen as -.49 ps (60 N) at node, ps (0 N) at nodes through 4, and -.48 ps (0 N) at the other nodes. Four dfferent problem varants are consdered for ths structure: wth stress constrants and no dsplacement constrants (Case ), wth stress constrants and dsplacement lmtatons of ±0.969 n (5 mm) mposed on all nodes n x and y drectons (Case ), no stress constrants and dsplacement lmtatons of ±0.969 n (5 mm) mposed on all nodes n z drecton (Case ), and all the abovementoned constrants mposed together (Case 4). For cases and the maxmum cross-sectonal area s taen as 5.0 n (.6 cm ) whle for cases and 4 t s taen as 0 n (9.0 cm ). The mnmum cross-sectonal s taen as n (5 cm ) for all cases. Table 6 compares the results obtaned by dfferent optmzaton technques for ths example. It can be seen that the results found by TWO are comparable to those of other methods. In case the best result obtaned by TWO s the same as that of CBO whch s the best result so far. In cases and 4 the results obtaned by TWO are better than those of RO, IRO, and CBO and are only slghtly heaver than that of HPSACO (0.0 and percent n cases and 4, respectvely). The average number of structural analyses requred by the RO and IRO algorthms were reported as 9,900 and 8,00, respectvely, whle TWO uses c c (0) ()

21 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 489 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 Element group 40 canddate solutons and 400 teratons le CBO resultng n a maxmum number of analyses of HPSACO and HS, respectvely, performed 0000 and 5000 structural analyses to obtan ther optmal results. It should be noted that HPSACO s a hybrd method whch combnes good features of PSO, ACO and HS. Table 6: Comparson of the optmzaton results obtaned n the 0-bar dome problem Optmal cross-sectonal areas (n ) Case Case HS [8] HPSACO CBO Present HPSACO CBO Present RO [8] HS [8] RO [8] [44] [] wor [44] [] wor Best weght (Ib) Average weght (Ib) Std (Ib) Case Case 4 HPSACO Present HPSACO CBO Present RO [8] CBO [] RO [8] IRO [45] [44] wor [44] [] wor Best weght (Ib) Average weght (Ib) Std (Ib) CONCLUSIONS In ths paper a novel populaton-based meta-heurstc algorthm based on a game of tug of war s ntroduced. The algorthm, denoted as Tug of War Optmzaton (TWO), consders each canddate soluton as a team partcpatng n a seres of tug of war compettons. Le other meta-heurstc optmzaton algorthms TWO uses a combnaton of randomness and explotaton of prevously obtaned favorable results to perform global optmzaton. The qualty of the canddate solutons mproves teratvely as the optmzaton process proceeds. Unle many other meta-heurstc methods, the algorthm s formulated n such a way that consders the qualtes of both of the nteractng teams. In other words, snce the weghts of the teams are nvolved n the convergence operator, two losng teams are not treated the same by a partcular wnnng team. The team wth less weght would probably move more drastcally.

22 490 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th 08 As a meta-heurstc optmzaton algorthm, TWO does not requre nformaton on the dervatves of the objectve functon and the constrants. Ths maes the algorthm applcable to a wde range of optmzaton problems from dfferent felds. Some classcal mathematcal and structural optmzaton problems are solved to evaluate the effcency of the proposed algorthm. Results ndcate the superorty of TWO compared to some other state-of-the-art optmzaton algorthms. REFERENCES. Holland JH. Adaptaton n Natural and Artfcal Systems, Ann Arbor: Unversty of Mchgan Press, Eberhart RC, Kennedy J. A new optmzer usng partcle swarm theory, In: Proceedngs of the Sxth Internatonal Symposum on Mcro Machne and Human Scence, Nagoya, Japan, Dorgo M, Manezzo V, Colorn A. The ant system: optmzaton by a colony of cooperatng agents, IEEE Trans Syst Man Cybern B 996; 6: Geem ZW, Km JH, Loganathan GV. A new heurstc optmzaton algorthm: harmony search, Smulat 00;76(): Erol OK, Esn I. New optmzaton method: bg bang-bg crunch, Adv Eng Softw 006; 7: Kaveh A, Talatahar S. A novel heurstc optmzaton method: charged system search, Acta Mech, 00; : Kaveh A, Mote Share MA, Mosleh M. A new meta-heurstc algorthm for optmzaton: magnetc charged system search, Acta Mech 0; 4: Kaveh A, Khayatazad M. A novel meta-heurstc method: ray optmzaton, Comput Struct 0; -: Kaveh A, Zolghadr A. Democratc PSO for truss layout and sze optmzaton wth frequency constrants, Comput Struct 04; 0: Kaveh A, Farhoud N. A new optmzaton method: dolphn echolocaton, Adv Eng Softw 0; 59: Kaveh A, Mahdav VR. Colldng bodes optmzaton: A novel meta-heurstc method, Comput Struct 04; 9: Sadollah A, Esandar H, Bahrennejad A, Km JH. Water cycle, mne blast and mproved mne blast algorthms for dscrete szng optmzaton of truss structures, Comput Struct 05; 49: -6.. Gonçalves MS, Lopez RH, Mguel LFF. Search group algorthm: A new metaheurstc method for the optmzaton of truss structures, Comput Struct 05; 5: Mrjall S. The ant lon optmzer, Adv Eng Softw 05; 8: Kaveh A, Zolghadr A. Comparson of nne meta-heurstc algorthms for optmal desgn of truss structures wth frequency constrants, Adv Eng Softw 04; 76: Kaveh A, Zolghadr A. Magnetc charged system search for structural optmzaton, Perod Polytech Cvl 04; 58(): Kaveh A, Zolghadr A. Topology optmzaton of trusses consderng statc and dynamc constrants usng the CSS, Appl Soft Comput 0; : 77-4.

23 A NOVEL META-HEURISTIC ALGORITHM: TUG OF WAR OPTIMIZATION 49 Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th Kaveh A, Zolghadr A. Truss optmzaton wth natural frequency constrants usng a hybrdzed CSS-BBBC algorthm wth trap recognton capablty, Comput Struct 0; 0 0: Kaveh A, Zolghadr A. Shape and sze optmzaton of truss structures wth frequency constrants usng enhanced charged system search algorthm, Asan J Cvl Eng 0; (4): Tsoulos IG. Modfcatons of real code genetc algorthm for global optmzaton, Appl Math Comput 008; 0: Belegundu AD. A study of mathematcal programmng methods for structural optmzaton, Ph.D. thess, Department of Cvl and Envronmental Engneerng, Unversty of Iowa, Iowa, USA, 98.. Arora JS. Introducton to Optmum Desgn, New Yor, McGraw-Hll, Coello CAC. Use of a self-adaptve penalty approach for engneerng optmzaton problems, Comput Ind 000; 4: Coello CAC, Montes EM. Constrant-handlng n genetc algorthms through the use of domnance-based tournament, IEEE Trans Relablty 99; 4: He Q, Wang L. An effectve co-evolutonary partcle swarm optmzaton for constraned engneerng desgn problem, Eng Appl Artf Intell 007; 0: Montes EM, Coello CAC. An emprcal study about the usefulness of evoluton strateges to solve constraned optmzaton problems, Int J Gen Syst 008; 7: Kaveh A, Talatahar S. An mproved ant colony optmzaton for constraned engneerng desgn problems, Eng Comput 00b; 7: Ragsdell KM, Phllps DT. Optmal desgn of a class of welded structures usng geometrc programmng, ASME J Eng Ind Ser B 976; 98: Deb K. Optmal desgn of a welded beam va genetc algorthms, AIAA J 99; 9: Grandh RV, Venayya VB. Structural optmzaton wth frequency constrants, AIAA J 988; 6: Sedaghat R, Suleman A, Tabarro B. Structural optmzaton wth frequency constrants usng fnte element force method, AIAA J 00; 40: Wang D, Zha WH, Jang JS. Truss optmzaton on shape and szng wth frequency constrants, AIAA J 004; 4: Lngyun W, Me Z, Guangmng W, Guang M. Truss optmzaton on shape and szng wth frequency constrants based on genetc algorthm, Comput Mech 005; 5: Gomes MH. Truss optmzaton wth dynamc constrants usng a partcle swarm algorthm, Expert Syst Appl 0; 8: Kaveh A, Zolghadr A. A new PSRO algorthm for frequency constrant truss shape and sze optmzaton, Struct Eng Mech 04b; 5: Rajeev S, Krshnamoorthy CS. Dscrete optmzaton of structures usng genetc algorthms, ASCE, J Struct Eng 99; 8: Schutte JJ, Groenwold AA. Szng desgn of truss structures usng partcle swarms, Struct Multdscp Optm 00; 5: Lee KS, Geem ZW. A new structural optmzaton method based on the harmony search algorthm, Comput Struct 004; 8:

24 49 A. Kaveh and A. Zolghadr Downloaded from joce.ust.ac.r at :4 IRDT on Tuesday September th Erbatur F, Hasançeb O, Tütüncü I, Klç H. Optmal desgn of planar and space structures wth genetc algorthms, Comput Struct 000; 75: Camp CV, Bchon J. Desgn of space trusses usng ant colony optmzaton, ASCE, J Struct Eng 004; 0; Perez RE, Behdnan K. Partcle swarm approach for structural desgn optmzaton, Comput Struct 007; 85: Camp CV. Desgn of space trusses usng Bg Bang Bg Crunch optmzaton, ASCE, J Struct Eng 007; : Soh CK, Yang J. Fuzzy controlled genetc algorthm search for shape optmzaton, J Comput Cvl Eng, ASCE 996; 0: Kaveh A, Talatahar S. Partcle swarm optmzer, ant colony strategy and harmony search scheme hybrdzed for optmzaton of truss structures, Comput Struct 009; 87: Kaveh A, Ilch Ghazaan M, Bahshpoor T. An mproved ray optmzaton algorthm for desgn of truss structures, Perod Polytech Cvl Eng 0; 57: Amercan Insttute of Steel Constructon (AISC), Manual of Steel Constructon Allowable Stress Desgn, 9th ed, Chcago, Illnos, Saa MP. Optmum desgn of pn-joned steel structures wth practcal applcaton, ASCE, J Struct Eng 990; 6: