DEVELOPING NEW CHARGED SYSTEM SEARCH-BASED ALGORITHM: APPLICATION IN THE TIME-COST TRADE-OFF PROBLEMS

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1 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton DEVELOPING NEW CHARGED SYSTEM SEARCH-BASED ALGORITHM: APPLICATION IN THE TIME-COST TRADE-OFF PROBLEMS M.K. Sharbatdar Faculty of Cvl Engneerng, Semnan Unversty, Semnan, Iran msharbatdar@semnan.ac.r S. Talatahar Engneerng, Unversty of Tabrz, Tabrz, Iran Department of Cvl talatahar@tabrzu.ac.r M.R. Mousav Faculty of Cvl Engneerng, Semnan Unversty, Semnan, Iran rm_mousav@semnan.ac.r ABSTRACT The trade-off between the total cost and project duraton s one of the most mportant parameters of constructon project plannng. There are varous methods to optmze tme-cost trade-off problems. Mathematcal programg models as one type of them cannot solve large and complex networks effectvely. On the other hand, although the meta-heurstcs algorthms n many cases can fnd a complete set of solutons but to optmze the tme-cost trade-off problems n very massve constructon projects they need to spend a lot of tme, so exstence a powerful algorthm wth hgher convergence rate s necessary. In ths paper new procedures MAWA-CSS and SMOCSS are ntroduced to generalze the well-known CSS algorthm for solvng TCTP optmzaton problem and all mult-objectve optmzaton problems n dscrete and contnuous search space. The overall structure of SMOCSS algorthm s smlar to the MOPSO and to detere the Charge magntude of partcles a new smple method s ntroduced. The proposed method s exaed for dfferent test functons and the results are compared to the results of two well-known multobjectve algorthms (NSGA-II and MOPSO). In addton, two example of tme-cost optmzaton problem (Feng and Zheng network wth 8 and 7 actvtes respectvely) are used to evaluate the performance of the proposed algorthms. The results ndcate that the SMOCSS algorthm has the ablty to fnd out the optmal soluton and defne the Pareto front as well n reasonable tme. Hence the proposed approach n ths paper s much adaptve and sutable for tacklng TCTP, whch s useful and benefcal for decson-makng on the trade-off between project duraton and total cost. Keywords: Tme-Cost Trade-off Problems (TCTP), mult-objectve optmzaton, meta-heurstcs algorthms, Charged System Search (CSS) INTRODUCTION The complextes and dffcultes of a constructon project causes changes n costs and tg of the project n the mplementaton phase. Wth the rapd expanson of the use of varous systems project delvery, tme s a deterng factor n the evaluaton of tenders and the manufacturng process. Therefore, constructon managers not focus just on reducng costs but the project executon tme s also so mportant. Forasmuch the compresson of tme wll nevtably lead to a gradual ncrease n drect costs of the project, n project management, achevng optmal tme-cost of the project becomes very mportant [-]. Several mathematcal models such as lnear programg [, ], nteger programg, or dynamc programg [, 6, 7] and LP/IP hybrd models are used to solve TCTPs [8, 9]. These methods are not sutable for large scale projects and cannot fnd all possble solutons [0]. Because of the populaton-based Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 77

2 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton nature of mult-objectve evolutonary algorthms and ther ablty to fnd multple optma smultaneously [, ], several dfferent algorthms have been suggested n recent years to solve TCTP whch can nclude Non-Doated Sortng Genetc Algorthm (NSGA_II) [, 0] Partcle Swarm Optmzaton (PSO) [], Adaptve Weghted Approach (AWA) and etc. [-6]. In ths study, the man contrbuton s on ntroducng dfferent mult-objectve CSS algorthms that, n addton to smplcty, they are capable on fndng global optmum solutons and have hgh convergence rate. The charged System Search (CSS) s one of the recently ntroduced sngle-objectve optmzaton algorthm whch have been used n scence and engneerng optmzaton problems successfully. Ths method mxes the governng moton from Newtonan mechancs and the governng Coulomb law from physcs. In the CSS algorthm, each possble soluton corresponds to a charged partcle (CP) whch can mpose an attractve electrc force on other CPs accordng to Coulomb s law. The next poston of each CP s detered by Calculatng the resultant forces actng on the CP and applyng the knematc equatons [7-9]. The results of ths algorthm compared wth each other and wth some other well-known mult-objectve optmzaton methods. Also we wll use the well-known these approaches to solve tme-cost problem. PRELIMINARIES BASIC CONCEPTS IN MULTI-OBJECTIVE OPTIMIZATION MOO For better understandng of the Mult-Objectve Problems (MOPs), acquantance wth the followng concepts are necessary [0]: General Mult-objectve Optmzaton Problem: The am n a mult-objectve optmzaton for mzaton problem s fndng a vector x =(x, x,..., x n ) whch satsfes k nequalty constrants as: q ( x ) 0 ( =,,...,k ) () and l equalty constrants: h ( x ) = 0 ( =,,...,l ) () and mzes the vector functon Mn x Ω F( x ) = { f( x ), f( x ),..., fm( x )} () Where m s the number of objectves and Ω s a set of the decson vector. Pareto Doance: A vector u = (u, u,...,u n ) doates vector v =(v, v,..., v n ) (denoted by u < v) f and only f u s partally less than v,.e., {,,...,n }, u v {,,...,n } : u v. () < Pareto Optmal: A soluton x Ω s Pareto Optmal wth respect to Ω f and only f there s no x ʹ Ω for whch v = (f (x ), f (x ),..., f n (x )) doates u = (f (x), f (x),..., f n (x)). Pareto Optmal Set: For a gven MOP, F(x), the Pareto Optmal Set, P, s defned as P = { x Ω xʹ Ω F( xʹ) < F( x )}. Pareto Optmal Front: For a gven MOP, F(x), and Pareto Optmal Set, P, the Pareto Front PF s defned as PF = {u = F(x) x P}. Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 77

3 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton A soluton s sad to be Pareto Optmal f t s not doated by any other solutons n the search space, also termed as non-doated soluton. In ths paper, we dstngush the real Pareto Optmal front, termed PF real, and the fnal set of non-doated solutons obtaned by a mult-objectve optmzaton algorthm, termed PF algorthm as defned by the am of the mult-objectve optmzaton algorthms s to fnd a well unformly dstrbuted PF algorthm that approxmates PF algorthm as close as possble. CHARGED SYSTEM SEARCH Charged System Search (CSS) algorthm, ntroduced by Kaveh and Talatahar [7], s based on electrostatc and Newtonan mechancs laws. In the CSS algorthm, charged partcles (CPs) are assumed to be the canddate solutons. Charged partcle are affected by the electrc feld that created by other partcles. The amount of force exerted on each CP, are obtaned usng the electrostatc rules. Also, the moton of each CP s detered by the rules of Newtonan mechancs and the charge magntude of each partcle wll be detered accordng to the value of the objectve functon [7-9]. Fg. shows the pseudo-code of the CSS algorthm. TCTP PROBLEM FORMULATION Fg. : pseudo-code of the CSS algorthm. In the trade-off between the total cost and project duraton problems (TCTP), there are twn objectves to be mzed: the project tme and ts cost. In a project, a two-objectve optmzaton problem should be solved as follows: T () C Where T s the total project tme and C s the total project cost that they are defned as: ( k ) ( k ) T = max t x L L L (6) k k ( k ) ( k ) ( k ) C = + dc x T c A Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 77

4 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton Where A s the number of actvtes, ( k ) t represents the duraton of actvty when perforg the kth opton, ( k ) x stands for the ndex varable of actvty when perforg the kth opton. If x = then the actvty perform the kth opton. The sum of ndex varables of all optons should be equal to. L k means the actvty sequence on the kth path, and L k = { k, k,, nk } where jk represents the sequence number of actvty j on the kth path. L stands for the set of all paths of a network, and L = {L k k=,,, m}, where m symbolzes ( k ) the number of all paths of a network. dc and c when perforg the kth opton, respectvely [,, 6]. ( k ) ( k ) represent the drect and ndrect cost of actvty NEW SIMPLE MULTI-OBJECTIVE CHARGED SYSTEM SEARCH (SMOCSS) Due to the hgh convergence and large capablty of CSS algorthm [7] we produced a new smple multobjectve optmzaton algorthm based on CSS algorthm (SMOCSS) that, n addton to smplcty, s capable of fndng global optmum solutons and have hgh convergence rate [-]. The flowchart of ths algorthm s shown schematcally n Fg.. Fg.: flowchart of SMOCSS algorthm CHARGED SYSTEM SEARCH FOR MODIFIED ADAPTIVE WEIGHT APPROACH (MAWA- CSS-TCTP) TCTP problem s a mult-objectve optmzaton problem. One postve trend to solve ths problem s Modfed Adaptve Weght Approach [6]. In ths paper based on Charged System Search algorthm and modfed adaptve weghtng method a model s developed and pareto front s acheved. The man steps of MAWA-CSS algorthm for TCTP are descrbed as follows: ) Randomly generaton of ntal solutons. ) Computng the project duraton and total cost from Eq.. Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/

5 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton ) Choosng the doant answers and addng to the Pareto soluton set ) Calculatng the alternatve ftness value as: Zt Zt + γ Zc Zc + γ f ( X ) = wt + w () max c max Z Z + γ Z Z + γ t t c c Whch Z t and Z t max as the mal value and maxmal value for the objectve of duraton respectvely, Z c and Z c max as the maxmal value and mal value for the objectve of cost respectvely, w t and w c are adaptve weght for the crteron of tme and cost respectvely that can be calculate by consderng ths condtons as: f Zt = Zt and Zc = Zc wt = wc = 0. else f Zt Zt and Zc = Zc wt = 0.9, wc = 0. else f Zt = Zt and Zc Zc wt = 0., wc = 0.9 else f Zt Zt and Zc Zc Zt Zc vt =, v = max c max Zt Zt Zc Zc vt vc wt =, wc = vt + vc vt + vc end ) Detere charge magntude for all the partcles n populaton as: ft( ) ftworst q =, =,,...,N () ftbest ftworst Where ftbest and ftworst are the so far best and the worst ftness of all partcles; ft() represents the objectve functon value or the ftness of the agent, and N s the total number of partcles. 6) Intalze the F vector (resultant force vector acted on each partcle) and detere the resultant force exerted to each partcle. 7) Compute the new poston and velocty of each partcle. 8) Mantan the partcles wthn the search space []. 9) Repeatng step to step 8 untl the generated Pareto solutons are repeated and there s no new soluton n the generated set. NUMERICAL EXAMPLES PERFORMANCE METRICS In order to provde a quanttatve assessment for the performance of an MO optmzer, three ssues are often taken nto consderaton []: a) The dstance of the resultng non-doated set to the Pareto-optmal front should be mzed. Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/

6 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton b) A good (n most cases unform) dstrbuton of the solutons found s desrable. The assessment of ths crteron mght be based on a certan dstance metrc. c) The extent of the obtaned non-doated front should be maxmzed,.e., for each objectve, a wde range of values should be covered by the non-doated solutons. Comparatve studes performed by researchers such as [-6] made use of a sute of unary performance metrcs pertnent to the optmzaton goals of proxmty, dstrbuton, and dversty. In ths paper, three dfferent qualtatve measures are utlzed. Generatonal dstance (GD) s a measure of the dstance between the true (PF real ) and generated Pareto front (PF algorthm ). Ths metrc of ndvdual dstance representng the dstance s gven by GD = n n pf d pf = () Where n pf s the number of members n PF algorthm and d s the Eucldean dstance between the th member n PF algorthm and ts nearest member n PF real. A smaller value of GD mples better convergence. The metrc of spacng (S) gves an ndcaton of how evenly the solutons are dstrbuted along the dscovered Pareto-front: npf n pf S = ( d d ), d = d () n = n = pf pf Where n pf s the number of members n PF algorthm and d s the Eucldean dstance (n the objectve space) between the th member n PF algorthm and ts nearest member n PF algorthm. A smaller value of S mples a more unform dstrbuton of solutons n PF algorthm. The metrc of maxmum spread (MS) measures how well the PF real s covered by the PF algorthm through hyper-boxes formed by the extreme functon values observed n the PF real and PF algorthm. It s defned as: 0. m max max ( f,f ) max( f,f ) MS = m max = F F () Where m s the number of objectves, f max and f are the maxmum and mum of the th objectve n PF algorthm, respectvely, and F max and F are the maxmum and mum of the th objectve n PF real, respectvely. A larger value of MS mples a better spread of solutons. TEST PROBLEMS Four benchmark problems ZDT, ZDT, FON and POL are selected to exae the performance of the proposed algorthm. The test problems are detaled n table [6-9]. Table.: Benchmark test problems ZDT, ZDT, FON and POL Test problem ZDT Mathematcal formulas Range of varables f( x ) = x x n = 9 0 x f( x ) g( x ), g( x ) = + x g( x ) n = Number of varables 0 Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/

7 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton ZDT FON POL f( x ) = x x x 0 x f ( x ) = g( x ) sn( 0π x ) g( x ) g( x ) f( x ) = exp ( x ) = x f = + ( x ) exp ( x ) = f ( x ) = + ( A B ) + ( A B ) f ( x ) = ( x + ) + ( x + ) A = 0. sn cos + sn.cos π x, x π A =. sn cos + sn.cos B = 0. sn( x ) cos( x ) + sn( x ). cos( x ) B =. sn( x ) cos( x ) + sn( x ) 0. cos( x ) 0 CASE STUDY IN TCTP In ths paper we wll use mult-objectve CSS algorthms mentoned n next sectons to solve two wellknown tme-cost optmzaton problems named as the Feng network actvtes and the Zheng network actvtes. The Feng network wth 8 actvtes s one of the most well-known examples for tme-cost optmzaton problem that has attracted the attenton of many researchers []. Confguraton and actvtes network optons of Feng network are shown n Fg. and Table respectvely. In addton to Feng actvtes network, another actvtes network wth 7 actvtes ntroduced by Zheng consdered as second example. Confguraton and actvtes network optons of Zheng network are shown n Fg. and Table respectvely [0]. Fg. : Confguraton of Feng network actvtes Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/

8 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton Table : Tme, Cost, Number of execute optons and Prerequstes of actvtes of Feng network No 6 Pr Op T Cost No Pr Op T Cost No Pr Op T Cost Fg. : Confguraton of Zheng network actvtes Table : Tme, Cost, Number of execute optons and Prerequstes of actvtes of Zheng network. No Pr - Op T 0 Cost Op T 6 0 Cost Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 No Pr 780

9 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton RESULTS In order to llustrate the effcency of our method, t s compared wth two of most well-known multobjectve optmzaton algorthms named as NSGA_II and MOPSO. Parameters for comparson are nclude: mentoned qualtatve measures (GD, S and MS) and the relatve average runnng tme of the algorthms (average runnng tme of the algorthms dvded on average runnng tme of the SMOCSS algorthm). A random ntal populaton s created for each of the 0 runs on each test problem. All of the algorthms are mplemented n Matlab. The results obtaned for test problems can be presented as follows: Test problem - ZDT The comparson of results between the Pareto fronts produced by NSGA-II, MOPSO and SMOCSS of ZDT and the true Pareto front are shown n Fg. (a) (c), respectvely. 7 6 Fg.: Pareto fronts produced by (a) NSGA-II, (b) MOPSO and (c) SMOCSS on test functon ZDT. Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 78

10 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton From the Fg., t can be observed that all algorthms are able to fnd solutons near the global Pareto front. The values of the three metrcs for each algorthm are shown n Table. Consderng all of the metrcs from Table, t can be seen that new algorthm (SMOCSS) s the best among the three adopted algorthms. Table.: Results for ZDT: The mean values of the three metrcs and relatve run tme for each algorthm NSGA_II MOPSO SMOCSS GD (mean) S (mean) MS (mean) Relatve Run Tme.. Test problem - ZDT The comparson of results between the Pareto fronts produced by NSGA-II, MOPSO and SMOCSS of ZDT and the true Pareto front are shown n Fg. 6(a) (c), respectvely. Fg.6: Pareto fronts produced by (a) NSGA-II, (b) MOPSO and (c) SMOCSS on test functon ZDT. From the Fg.6 t can be revealed that the NSGA_II and the MOPSO algorthms are stuck n a local Pareto optmum, but the SMOCSS algorthm s able to evolve a dverse and well dstrbuted near-optmal Pareto front by spendng less run tme. The values of the three metrcs for each algorthm are shown n Table. Consderng all of the metrcs from Table, t can be seen that new algorthm (SMOCSS) s the best among the three adopted algorthms. Table.: Results for ZDT: The mean values of the three metrcs and relatve run tme for each algorthm NSGA_II MOPSO SMOCSS GD (mean) S (mean) MS (mean) Relatve Run Tme.89. Test problem - FON The comparson of results between the Pareto fronts produced by NSGA-II, MOPSO and SMOCSS of FON and the true Pareto front are shown n Fg. 7(a) (c), respectvely. Table.6: Results for FON: The mean values of the three metrcs and relatve run tme for each algorthm Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 78

11 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton NSGA_II MOPSO SMOCSS GD (mean) S (mean) MS (mean) Relatve Run Tme..09 From the Fg.7 t can be seen that all algorthms are able to evolve a dverse and well dstrbuted nearoptmal Pareto front for ths problem but the computaton tme requred for NSGA_II and MOPSO algorthms are more than the run tme requred for SMOCSS algorthm. So that even the NSGA_II run tme about. tmes more that the same for our algorthm. The values of the three metrcs for each algorthm are shown n Table 6. Consderng all of the metrcs from Table 6, t can be seen that all algorthms are usable to optmzng ths problem. Fg.7: Pareto fronts produced by (a) NSGA-II, (b) MOPSO and (c) SMOCSS on test functon FON. Test problem - POL The comparson of results between the Pareto fronts produced by NSGA-II, MOPSO and SMOCSS of POL and the true Pareto front are shown n Fg. 8(a) (c), respectvely. From the Fg.8 t can be seen that smlar to the prevous problem, all algorthms are able to evolve an almost dverse and well dstrbuted near-optmal Pareto front for ths problem but the computaton tme requred for NSGA_II and MOPSO algorthms are more than the run tme requred for SMOCSS algorthm. So that even the NSGA_II run tme about tmes more that the same for our algorthm. The values of the three metrcs for each algorthm are shown n Table Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 78

12 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton 7. Consderng all of the metrcs from Table 7, t can be seen that NSGA_II s better than MOPSO but t requres more run tme and both of ths two algorthms are not as good as SMOCSS. Fg.8: Pareto fronts produced by (a) NSGA-II, (b) MOPSO and (c) SMOCSS on test functon POL. Table.7: Results for POL: The mean values of the three metrcs and relatve run tme for each algorthm NSGA_II MOPSO SMOCSS GD (mean) S (mean) MS (mean) Relatve Run Tme.6.0 The Feng and the Zheng actvtes network, mentoned n secton, were solved by NSGA-II, MOPSO, MAWA-CSS and SMOCSS algorthms and the results were compared n the number of non-doated solutons and relatve run tme of algorthms. All of the algorthms are mplemented n Matlab and each problem was solved 0 tmes and mean of results was recorded. Table 8 shows the number of non-doated solutons and relatve run tme of algorthms. Table 8: The number of non-doated solutons and relatve run tme of algorthms. Network NSGA-II the number of nondoated solutons MOPSO MAWA- CSS SMOCSS RunTme RunTme of the SMOCSS NSGA-II MOPSO MAWA- CSS Feng... Zheng.8.9. SMOCSS It can be seen from table 8 that SMOCSS algorthm s faster than other algorthms and also s able to fnd all solutons n pareto set and MAWA-CSS s not as good as SMOCSS algorthm but s faster than MOPSO and NSGA-II. Furthermore, sometmes the MOPSO and the NSGA-II could not fnd all solutons of Feng Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 78

13 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton actvtes network. As a result of solvng the Zheng and Feng actvtes networks wth SMOCSS Followng solutons acheved (table 9 and table 0). Table 9: Results of solvng the Zheng actvtes networks wth SMOCSS Tme Cost Tme Cost Tme Cost Tme Cost Table 0: Results of solvng the Feng actvtes networks wth SMOCSS Tme Cost Tme Cost Tme Cost Tme Cost Fg. 9 and Fg. 0 shows all results of solvng the Zheng and Feng actvtes networks by SMOCSS respectvely. Fg.9: All results of solvng the Zheng actvtes networks by SMOCSS. Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/060 78

14 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton Fg.0: All results of solvng the Feng actvtes networks by SMOCSS. CONCLUSION In ths paper, a new mult-objectve optmzaton algorthm, named as SMOCSS, s proposed. The proposed algorthm s based on the recently developed algorthm, charge system search (CSS). In ths approach, each partcle creates an electrcal feld around tself and consequently affects other partcles by a force whch s a result of ths attracton feld. By ths pattern, better partcles attract worse partcles. The overall structure of ths new algorthm s smlar to the MOPSO and to detere the Charge magntude of partcles a new smple method s ntroduced. A comparatve study of SMOCSS and two state-of-the-art MO algorthms on four benchmark test problems s presented. The results of applyng three performance metrcs clearly ndcate that SMOCSS s compettve and even outperforms most of the selected MO algorthms. The comparson of the computatonal tme shows that SMOCSS requres less computer tme than the selected MO algorthms on some of the selected benchmark test problems. Constructon tme-cost trade-off problems are large scale two-objectve optmzaton problems and fndng out all solutons n Pareto set wth spendng reasonable tme s very mportant. By generalzng the wellknown Charged Search System (CSS), wth usng of a new method (SMOCSS), to optmze mult-objectve problems, not only the optmal soluton could be found n reasonable tme, but also the Pareto front could be defned that wll provde more effectve nformaton and bass for further and correct decson-makng. The result of proposed model s compared wth MAWA-CSS algorthm, as another mult-objectve optmzer, and also NSGA-II and t s concluded that current procedure of modelng TCTP has several advantages especally n terms of generatng better solutons and consug less tme for solvng the problem. REFRENCES T.J Hndelang, J.F Muth, A dynamc programg algorthm for decson CPM networks. Operat Res, pp. -, 979. Elmaghraby, S. E. Resource allocaton va dynamc programg n actvty networks, Eur. J. Operatonal Res., Vol. 6, pp Feng, C.W., Lu, L. and Burns, S.A. Usng genetc algorthms to solve constructon tme-cost trade-off problems, Journal of Computng n Cvl Engneerng, ASCE, Vol., No., pp Kelly, James, E. Jr. Crtcal path plannng and schedulng: mathematcal bass, Oper. Res., Vol. 9 no., pp Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/

15 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton Hendrckson, C. and Au,T. Project Management for Constructon, Prentce-Hall, Inc, Englewood Clffs, N.J Butcher, W.S. Dynamc programg for project cost-tme curve, Journal of Constructon Dvson, ASCE, Vol. 9 (C0), pp Robnson, D. R. A dynamc programg soluton to cost-tme tradeoff for CPM, Mgmt. Sc., Vol., No., pp Lu, L., Burns,S.A. and Feng, C. Constructon tme-cost trade-off analyss usng LP/IP, Journal of Constructon Engneerng and Management, ASCE, Vol. (), pp Burns, S.A., Lu, L. and Feng, C.W. The LP/IP hybrd method for constructon tme-cost trade off analyss, Constr. Manag. Econ., Vol., No., pp Zheng, D. X.M., Ng, S.T. and Kumaraswamy, M.M. Applyng a genetc algorthm-based mult-objectve approach for tme-cost optmzaton, Journal of Constructon Engneerng and Management, Vol. 0, No., pp Carlos A. Coello Coello, Davd A. Van Veldhuzen, Gary B. Lamont. Evolutonary Algorthms for Solvng Mult-Objectve Problems. Kluwer Academc Publshers, Boston. ISBN Davd A. Van Veldhuzen, Gary B. Lamont. Mult-Objectve Evolutonary Algorthms: Analyzng the Stateof-the-Art. Evolutonary Computaton, 7():-6, 000. Yang, I.T. Usng eltst partcle swarm optmzaton to facltate bcrteron tme-cost trade-off analyss, J. Constr. Eng. Manage., Vol., No. 7, pp Geem, Z. W. Multobjectve optmzaton of tme-cost trade-off usng harmony search, Journal of Constructon Engneerng and Management, ASCE, Vol. 6, No. 6, pp Ng, S. T. and Zhang, Y. Optmzng constructon tme and cost usng ant colony optmzaton approach, Journal of Constructon Engneerng and Management, ASCE, Vol., No. 9, pp Dasy X. M. Zheng, S. Thomas Ng, and Mohan M. Kumaraswamy. Applyng Pareto Rankng and Nche Formaton to Genetc Algorthm-Based Multobjectve Tme Cost Optmzaton. J. Constructon. Eng. Manage. (), Kaveh A, Talatahar S. A novel heurstc optmzaton method: charged system search. Acta Mech; : Kaveh, A., & Talatahar, S. Optmal desgn of skeletal structures va the charged system search algorthm. Structural and Multdscplnary Optmzaton (7), Kaveh, A., & Talatahar, S. Charged system search for optmum grllage systems desgn usng the LRFD- AISC code. Journal of Constructonal Steel Research (66), Coello Coello, C. A., Van Veldhuzen, D. A., & Lamont, G. B. Evolutonary algorthms for solvng mult objectve problems. New York: Kluwer Academc Publshers. 00. Q, K., Le, W., & Qd, W. A novel self-organzng partcle swarm optmzaton based on gravtaton feld model. In Proceedng of the Amercan Control Conference, New York (pp. 8 ) Coello Coello, C. A., Puldo, G. T., & Lechuga, M. S. handlng multple objectves wth partcle swarm optmzaton. IEEE Transacton on Evolutonary Computaton, 8(), Ztzler, E., Laumanns, M., & Thele, L. SPEA: Improvng the strength Pareto evolutonary algorthm. Zurch, Swtzerland: Swss Federal Insttute Technology. 00. Ztzler, E., Deb, K., & Thele, L. Comparson of mult-objectve evolutonary algorthms: Emprcal results. Evolutonary Computaton, 8(), Deb, K., Pratap, A., Agarwal, S., & Meyarvan, T. A fast and eltst mult-objectve genetc algorthm: NSGA-II. IEEE Transacton on Evolutonary Computaton, 6(), Ztzler, E., & Thele, L. Mult objectve evolutonary algorthms: A comparatve case study and the strength Pareto approach. IEEE Transactons on Evolutonary Computaton, (), Tsa, S. J., Sun, T. Y., Lu, C. C., Hseh, S. T., Wu, W. C., & Chu, S. Y. An mproved mult-objectve partcle swarm optmzer for mult-objectve problems. Expert Systems wth Applcatons, 7(8), Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/

16 The Turksh Onlne Journal of Desgn, Art and Communcaton - TOJDAC August 06 Specal Edton Chen, J., Ln, Q., & J, Z. A hybrd mmune mult objectve optmzaton algorthm. European Journal of Operatonal Research (0), Deb, K. Mult objectve optmzaton usng evolutonary algorthms. Wley, U.K: Chchester. 00. Submt Date:.06.06, Acceptance Date:.07.06, DOI NO: 0.76/060AGSE/