Scheduling of Construction Projects with a Hybrid Evolutionary Algorithm s Application

Transcription

1 15 Schedulng of Constructon Projects wth a Hybrd Evolutonary Algorthm s Applcaton Wojcech Bożejko 1, Zdzsław Hejduck 1, Magdalena Rogalska 2 and Meczysław Wodeck 3 1 Wrocław Unversty of Technology, Wrocław, Wybrzeże Wyspańskego 27, 2 Lubln Unversty of Technology, Lubln, Nadbystrzycka 40, 3 Wrocław Unversty, Wrocław, Jolot-Cure 15, Poland 1. Introducton In the works [5,16] there s presented a possblty of applcaton of crtcal chan schedulng /buffer management methodology (CCS/BM) n nvestment enterprse and constructon plannng. The completon of a cvl structure or a buldng complex s an undertakng consstng of the followng factors: fulfllng the requrements (qualty), cost, tme of executon, range and resources [7]. In the artcle there are presented the outcomes of the tests concernng an mprovement of methods used for nvestment schedulng and constructon project wth the mplementaton of mementc algorthm (.e. hybrd evolutonary, HEA, [1]). The am of the work was fndng an optmal (for a taken goal functon) level of workers employment that s mnmzaton of a dvergence from the average level of employment wth the mplementaton of CCS/BM methodology [5]. Schedulng of nvestment and constructon project s connected to an optmzaton task. It s related to fndng the best soluton fulfllng the constranng condtons and takng nto consderaton the goal functon. There are many known methods of optmzaton appled n specfc cases. Among others, there can be mentoned, for nstance, for contnuous tasks methods of lnear smplex, tasks of global optmzaton when a goal functon n the feld of accepted solutons has more than one local mnmum, dscrete tasks wth a greater complexty of calculatons based most commonly on dvson or constrant methods, non determned methods usng random generatng of solutons, a smulated annealng method as a modfcaton of random walk wth the mprovement of qualty of goal functon, tabu search, that s wth a lst of revsed varants and others. There are also technques appled wth the use of bologcal systems evolutonary algorthms, genetc, evolutonary strateges, evolutonary programmng and genetc programmng ([3],[18]). The general scheme of evolutonary algorthm s operaton resdes n creatng a loop embracng reproducton, genetc operatons, evaluaton and successon. The classc scheme of operaton of evolutonary algorthm s presented below accordng to [3]. Ths paper s a contnuaton of the topc presented n work [15], concernng the applcaton of genetc algorthms [7] to steerng of a level of an employment n nvestment and constructon projects. Treatng the evolutonary algorthm as a typcal method of proceedng concernng

2 296 Evolutonary Algorthms searchng for better solutons, takng nto consderaton the closest envronment, there was a slghtly dfferent approach proposed than the one appled n modelng of evoluton wth a applcaton of a bnary genetc code. The new one conssts n replcatng of randomly chosen varants (ndvduals) wth a possblty of multple copyng of the same soluton, n whch the random choce takes nto account a better adaptaton of a varant (ndvdual). Solvng practcal ssues of operatonal tasks, ncludng schedulng of nvestment and constructon project requres a selecton of such an optmzaton method whch leads to the best soluton wth the mnmal cost. It s assumed [3], that one of the most effcent optmzaton algorthms s the evolutonary algorthm wth strategy, consttutng frameworks for an dea of a mementc algorthm HEA, [1]. 2. Defnng the problem An optmzaton task refers to plannng of regularty of workers employment level [8,17,14,11,10] durng realzaton of constructon undertakng wth regard towards CCM/BM methodology [5,16]. Tme buffers ntroduced to schedulng task ncrease space of accepted varatons of realzng the project and an extent of a task. In order to solve an optmzaton task a hybrd evoluton algorthm HEA created by Bożejko and Wodeck [1] was appled. The dea of a algorthm HEA conssts of creaton of a start-up populaton n whch for every ndvdual there s a permutaton appled n order to fnd a local mnmum. Then, there s a passng towards separate populatons wth defnng a number and poston n a set of ndvduals. Theory of Constrans (TOC) by Goldratt [2,3,4,5,6] and ts practcal applcaton n managng of projects known as Crtcal Chan Schedulng (CCS) and Buffer Management (BM), n short defned as CCS/BM Method [7], stays n the center of nterest of many scentfc goups. Precursors of ths methods were Gffler and Thomson [8] together wth West [17] who ntroduced the concept of crtcal sequences determned not only by technologcal sequencng and adjustng the tme of tasks completon but also by constrants of resources n creaton of schedules. Creatng and steerng of schedules s one of the ams of managng buldng projects. Hgh level of smplfcaton of the two systems: Crtcal Path Method (CPM) together wth Program Evaluaton and Revew Technque (PERT) led to ther popularzng and common use all over the world. Changng standards, growng expectatons, cost cuttng, mnmzng fnes and the extent of nvestment projects, clearness of expressons led to undertakng of many works amng at ncreasng the effectveness of schedulng treated n relaton to tradtonal CPM/PERT methods. 2.1 Theory of constrans framework Theory of Constrants (TOC) can be appled n all projects whch am at reachng revenue. TOC s based on fve basc steps: 1. Identfcaton system constrants. 2. Decson of maxmum utlzaton of constraned resources. 3. Subordnaton of all processes to above decson. 4. Increasng number of constrants created as a result of a lqudaton of a constrant defned n the frst step as a bottleneck of the system. 5. Identfcaton of new constrants of resources created as a result of bottleneck elmnaton n pont 4; f constrant n pont 4 s elmnated then return to step one n order not to allow any nternal factors to constrant the whole system.

3 Schedulng of Constructon Projects wth a Hybrd Evolutonary Algorthm s Applcaton 297 The proposed system ams at ongong producton mprovement n order to gan bgger proft n current and future undertakngs through dentfyng and elmnatng bottlenecks n producton. TOC can be used n realzaton of buldng projects. Let us assume that n the frst step a tower crane s dentfed as a constraned resource. Thus, n the second step we take a decson of maxmum utlzaton of crane s capacty shftng work schedule from 8 to 16 workng hours [14]. In the thrd step, as a consequence of the taken decson, we must subordnate all teams and machnes whch cooperate wth the crane. In the fourth step we analyze whether the resources pose a threat of becomng bottlenecks of the system. In case when nternal factors constran the whole system, we must return to step one and dentfy constrants one more tme. Whle mplementng a producton mprovement n accordance wth TOC we can expect an ncrease of profts. The example of practcal applcaton of Theory of Constrants s a project management method schedulng method usng crtcal chan and buffer management CCS/BM. 2.2 CCS/BM methodology Chronc problems appearng n realzaton of constructon tasks whch cannot be removed even wth the use of advanced technologes became a reason for development of CCS/BM methods [11]. Classc schedulng methods used n constructon projects CPM/PERT are characterzed by utlzaton of tme used for completng separate actvtes wth regard to tme lmts resultng from vald, approxmate, standard data. Goldratt perceves such reasonng as napproprate and n hs work [2] even calls ths process a thef of tme, whereas Turner [16] analyzes the nfluence of tme reserve dsperson on revenue ganed from undertakng. He assumes that a project conssts of n tasks and each of them s afflcted wth a contngency (randomness). Moreover, on the assumpton that there s an even schedulng of tme needed for task completon, t has a standard σ devaton. Thus, an overall contngent reserve equals nσ. The rghts to manage t are dspersed. Every partcpant of the project can manage ther tme reserve. From the pont of vew of a project manager an optmal stuaton would be f he could manage the tme hmself for the followng reasons: reserves not utlzed by separate partcpants of actvtes would not get wasted and total tme reserve could be smaller than n the frst case. From the calculatons presented by Turner t results that a total tme reserve should equal (n) 1/2 σ whch s a much smaller value than nσ. Implementaton of a method of tme reserve aggregaton results n a sgnfcant reducton of project costs. Moreover, lack of tme reserve for a project manager often leads to a falure n meetng the deadlnes. Goldratt recalls some known examples: a tunnel under Brtsh Channel or drllng towers n the North Sea. The sze of tme reserves and duraton of separate actvtes (not takng nto consderaton ts nternal reserves) s one of the basc problems of CCS/BM. Durng developng of CCS/BM method Goldratt was basng on the followng psychologcal assumptons: Student Syndrome do not begn work before all possbltes and tme lmts are used ; Parknson s Law every work wll be done n a assgned tme or longer ; roadrunner mentalty real race aganst tme, [7]; Conklng s Roadunner Geococcyx calfornanus s the quckest runner among ts speces reachng the speed of 30 km/h, t never moves slowly; Murphy s Law anythng that can go wrong, wll go wrong.

4 298 Evolutonary Algorthms Goldratt proposes tme reducton for a completon of an ndvdual processes together wth nformng tasks executors only about due dates. A schedule ncludng tme buffers s accessble exclusvely for a project manager. Partcpants, not possessng any tme reserves, try to complete ther tasks as quck as possble (roadrunner mentalty), start ther job wth a full capacty (Student Syndrome), n case of threat of fallng behnd the deadlne (Murphy s Law & Parknson s Law) the project manager has tme reserves to modfy and steer the course of works. Goldratt solves the problem of tme reserves n an arbtrary manner. Namely, he ntroduces reducton of an actvty duraton by half (50%) and creates tme buffers of a 50% value of a new, shortened actvty duraton. In reference to sub-crtcal chans Goldratt calls t a feedng buffer FB, whereas behnd a crtcal chan he places a project buffer PB. Accordng to the followng assumptons: Project buffer PB s a tme reserve placed at the end of crtcal chan, stayng to project managers dsposal, ntroduced n order to protect completon of a project, calculated on the bass of crtcal chan tme duraton (accordng to Goldratt a project buffer s 25% of crtcal chan duraton), there s only one project buffer n schedulng. Feedng Buffer FB s a tme reserve placed at the end of non-crtcal chan (feedng chan) stayng to managers dsposal n order to protect tasks placed n a crtcal chan, ntroduced as a protecton of deadlne of crtcal tasks, calculated on the bass of noncrtcal chan duraton (n Goldratt s method a feedng buffer equals 25% of non-crtcal chan duraton), there are as many feedng buffers as the number of non-crtcal chans (FB appears always when a non-crtcal chan lnks wth a crtcal chan). Resource buffer RB s a tme reserve placed n schedule before enterng of a new resource nto a crtcal chan, calculated on the bass of logstc dependences and possbltes, ntroduced n order to ensure ntaton of a process n a crtcal chan, n a planned due date. Crtcal chan CC s a set of processes appearng n front of a project buffer, determned by: actvtes duraton tme,, ther techncal sequence of completon, accessblty and resource requrements; processes n a crtcal chan are deprved of tme reserves taken nto consderaton n CPM/PERT (accordng to Goldratt the length of a crtcal chan s calculated as 50% of crtcal path CP); n a crtcal chan cannot appear any processes not havng a drect nfluence on due task tme (owng to tme, techncal or resource constrants); n case of appearng of more than one crtcal path one must choose one of them and transform a crtcal chan (there can be only one crtcal chan). Fgure 1 presents a graph of actvtes lastng adequately 16,4,8,8,4 tme unts. There are dependences between actvtes shown and ther sequence has been marked n fgure 1. Accordng to CCS/BM methodology and applyng Goldratt s assumpton concernng a duraton of tme of processes and length of feedng (FB) and project buffer (PB) we obtan a graph depcted n fgure 2. An ndvdual assessment, based on system analyss of shortenng separate processes, creaton of crtcal chan and sze of feedng and project buffers allows for creaton of ratonal schedule ncludng not only techncal but also resources, organzatonal, fnancal constrants. Dependences between process duraton tme, crtcal chan buffer and optmal utlzaton of resources wth applcaton of mementc algorthm (HEA) to calculatons have been thoroughly examned. Expanson of mplemented n fnancal calculatons method of contngency onto area of schedulng of constructon projects wth applcaton of TOC and CCS/BM methods accompaned by HEA mght lead to shortenng of tme and costs of a

5 Schedulng of Constructon Projects wth a Hybrd Evolutonary Algorthm s Applcaton 299 buldng process. Works concernng ths problem wll be contnued n the future by the authors Fg. 1. Graph of actvtes and tmes of ther duraton n a classc formulaton PB FB Fg. 2. Graph of actvtes and tme of ther duraton accordng to CCS/BM ncludng project (PB) and feedng buffers (FB). 3. Hybrd Evolutonary Algorthm Hybrd evolutonary algorthm was proposed by Bożejko and Wodeck [1] and t s a general method of solvng dscreet optmzaton problems. Therefore some elements of the algorthm have to be addressed n detal to use t for solvng automaton problems n constructon, especally the method of the problem s code for HEA, determnng of set of fxed elements and local optmzaton approach. 0 The algorthm starts by formng resdual populaton P (whch can be randomly formed). The best element of populaton P 0 s adopted as suboptmum soluton π *. Let be the 1 algorthm teraton number. New populaton +1 (.e. set ) s generated as follows. For P +

6 300 Evolutonary Algorthms current populaton P a set of local mnma LM s fxed (by carryng out procedure LocalOpt( π ) for each element π P ). Elements occupyng the same postons n the local mnma are fxed (procedure ( FxeSet LM, FS )), formng a set of fxed elements and 1 postons FS + 1. Each permutaton of new populaton P + has fxed elements (n fxed 1 postons) from set FS +. Free elements are randomly assgned to the remanng * (unoccuped) postons. If there s a permutaton β LM and F( β ) < F( π ), then β s * adopted as permutaton π. The algorthm stops when t has generated a predetermned number of generatons. We apply followng notaton: LocalOpt( π ) : * π : sub-optmal soluton determned by the algorthm, η : number of elements n populaton (the same n each generaton), P : populaton n the teraton of algorthm, P = { π 1, π2,..., πη }, LM : local optmzaton algorthm to determnng local mnmum, where π s a startng soluton of the algorthm, a set of local mnma n teraton, LM = { ˆ π 1, ˆ π2,..., ˆ πη } where ˆ π j = LocalOpt( π j), π j P, j = 1,2,..., η. a set of fxed elements and poston n permutatons of FS : populaton P, a procedure whch determnes a set of fxed elements and FxSet( LM, FS ) : postons n next teraton of evolutonary algorthm,, a procedure whch generates a new populaton n next teraton NewPopulaton( FS ) : 1 of the algorthm, P + = NewPopulaton( FS ). The code of the proposed hybrd evolutonary algorthm s gven below and fgure 3. Algorthm (HEA) Intalzaton: 0 randomly formed populaton P = { π 1, π2,..., πη }; * 0 π = the best element of populaton P ; Iteraton number =0; 0 FS = ; repeat Determne a set of local mnma LM = { ˆ π 1, ˆ π2,..., ˆ πη }, where ˆ j π = LocalOpt( π j ), π j P ; for j:=1 to η do * * f F( ˆ π j ) < F( π ) then π ˆ π j ; 1 FS + = FxSet( LM, FS ); {fx set} + 1 P : = NewPopulaton( FS ); {generate new populaton} =+1; not untl Stop Crteron (exceedng tme or a number of teratons).

7 Schedulng of Constructon Projects wth a Hybrd Evolutonary Algorthm s Applcaton 301 ter:= ter+1 Start ter := 0 Determnng free elements and postons ter > Max_ter No Yes End Insertng the new elements Random populaton Deletng the oldest elements Local optmzaton Descent Search or Tabu Search Changng of the element s age Auto-tune of the parameters Fg. 3. Hybrd evolutonary algorthm. 4. Problem codng and notaton The problem can be defned as follows. There are: a set of n jobs J={1,2,,n}, a set of m machnes M={1,2,,m}. Job j J, conssts of a sequence of m operatons O j1, O j2,, O jm. Operaton O jk corresponds to the processng of job j on machne k durng an unnterrupted processng tme p jk. We want to fnd a schedule such that the maxmum completon tmes s mnmal. Such a problem s known as a flow shop problem n lterature. Let π =(π(1), π(1),,π(n)) be a permutaton of jobs {1,2,,n} and Π be the set of all permutatons. Each permutaton π Π defnes a processng order of jobs on each machne. Completon tme of job π(j) on machne k can be found usng the recursve formula: C π(j)k =max{c π(j-1)k, C π(j)k-1 }+p π(j)k }, where π(0)=0, C 0k =0, k=1,2,...,m and C 0j =0, j=1,2,...,n. Local optmzaton (procedure LocalOpt) The local search (LS) method s a metaheurstc approach desgned to fnd a near-optmal soluton of combnatoral optmzaton problems. The basc verson of LS starts from an 0 ntal soluton x. The elementary step of the method performs, for a gven soluton x, a search through the neghborhood Nx ( ) of x. The neghborhood Nx ( ) s defne by move performed from x. A move transforms a soluton nto another soluton. The am of ths 1 elementary search s to fnd n Nx ( ) a soluton x + wth the lowest cost functons. Then the search repeats from the best found, as a new startng soluton.

8 302 Evolutonary Algorthms Local search algorthm Select a startng pont: x ; xbest : = x; repeat Select a pont y Nx ( ) accordng to the gven crteron based on the value of the goal functon Fy ( ); x : = y; f Fy ( ) > Fx ( best ) then xbest : = y; untl some termnaton condton s satsfed. A fundamental element of the algorthm, whch has a crucal nfluence on qualty and tme of computaton, s a neghborhood. A neghborhood s generated by the nsert moves n the best local search algorthms wth the permutaton representaton of the soluton. A set of fxed elements and poston (procedure FxSet) A set of fxed elements and postons FS (n -th teraton of the algorthm) conssts of quads ( alα,,, ϕ ), where a s an element of the set N (a N ), l s a postons n a soluton (1 l n ) and α, ϕ are attrbutes of a par ( al, ). Parameter α means ftness and decdes on belongng to the set, ϕ s an age element s removed from the set after exceedng some number of teratons (here: 3 teratons). In every teraton of the algorthm, after determnng the local mnma (procedure LocalOpt), a new set FS +1 = FS s establshed. Next, a FxSet(LM, FS ) procedure s called, n whch there are executed the followng operatons : a. changng of the age of each element, b. deletng the oldest elements, c. nsertng the new elements. Insertng elements Let P = {π 1, π 2,..., π η } be a populaton of η elements n teraton. For each permutaton π j from P, applyng the local search algorthm (LocalOpt(π j ) procedure), a set of local mnma LM = { ˆ π, ˆ π,..., ˆ π } s determned. Each permutaton Let 1 2 η ˆ π = ( ˆ π (1), ˆ π (2),..., ˆ π ( )), j = 1,2,... η. j j j j n nr( a, l) { ˆ π LM : ˆ π ( l) = a}. j It s a number of permutatons from the set LM, n whch there s an element a n the poston l. If a N s a free element and nr( a, l) α = Φ () η then the element a s fxed n the poston l. j

9 Schedulng of Constructon Projects wth a Hybrd Evolutonary Algorthm s Applcaton 303 A new populaton (procedure NewPopul) To generate a new populaton P +1, randomly drawn free elements are nserted n remanng free postons of the elements of populaton P. NewPopulaton( FL ) 1 P + ; Determne a set of free elements: + 1 FE = { a N : ( a, l, αϕ, ) FS } and a set of free postons + 1 FP = { l : ( a, l, αϕ, ) FS } ; for j:=1 to η do {Insertng of fxed elements} + 1 for each ( al,, αϕ, ) FS do π j( l) : = a; W FE; for s:=1 to n do {Insertng of free elements} f s FP then π j( s) = w, where w= random( W) and W W \{ w} ; P+ 1 P+ 1 { π j}. Functon random generates from a unform dstrbuton an element of the set W. Computatonal complexty of the algorthm s O(η n). 5. Case study The subject matter of the analyss s a network model of an nvestment and constructon project (ICP) accordng to [9,13], contanng of n = 16 of lnked buldng processes. There has been a computatonal test carred out n order to prove the possblty of tme buffers nfluence on regularty of workers employment n an enterprse schedule. Wth regard to taken constrants the values of a objectve functon F were adopted as follows: T 1 n f ( x) = q j( x) dr j= 1 T = 1 where: n x R, x=(x 1, x 2,, x n ) vector of moments of ntatng tasks executon, x [a, b ], a earlest moment of task begnnng, b latest moment ntatng tasks executon, q j (x) number of workers employed on a j day, j = 1,2,,T, T tme horzon, d tme of process duraton, r number of workers employed n order to carry out the process. (1)

10 304 Evolutonary Algorthms Below there s presented a network model of an exemplary enterprse takng nto consderaton tme buffers; feedng FB [12] and project buffer PB. There s nformaton concernng a number of actvtes, tme of actvtes accordng to Goldratt method [4] and forecasted resources (number of employees). Fg. 4. The graph of ICP Process number Duraton tme Earlest term of ntalzng Earlest term of fnshng Latest term of beggnnng Latest term of fnshng Number of workers PB FB FB FB Table 1. The data of actvtes

11 Schedulng of Constructon Projects wth a Hybrd Evolutonary Algorthm s Applcaton 305 Table 1 ncludes basc nformaton concernng a constructon project modeled by network. Basc tme parameters forecasted to carry out constructon works were counted. Numercal data necessary n order to carry out optmzaton calculatons are presented below: Fg. 5. The Gantt Chart, value of objectve functon for earlest terms f= for latest terms f= Fgure 5 depcts The Gantt Chart together wth a graph of workers employment for earlest and latest terms of works begnnng. For these extreme terms the values of goal functon were calculated,.e. f= and f= adequately. Fgure 6 shows Gantt s lnear graph presentng an optmal schedule of a constructon enterprse, for a taken goal functon. Placement of a crtcal path does not change, whereas non-crtcal actvtes were put on a tme scale takng nto account tme buffers (FB), n an optmal way wth regard to a mnmal value of a goal functon. Below Gantt s lnear graph there s a graph of workers employment correspondng to planned tasks. It shows the best soluton concernng a mnmal average dvergence from an

12 306 Evolutonary Algorthms average level of workers employment. A strped lne shows tme buffers whch are n fact a tme reserve enablng ts utlzaton for an optmal placement of tasks on a tme axs of constructon works, so that the possble level of workers employment can be kept regular. Fg. 6. The Gantt Chart after optmzaton, mnmal value of objectve functon f= In an analyzed example there has been a genetc algorthm (GA) appled ensurng after th teraton the result of f= , whereas after applcaton of hybrd evolutonary algorthm (HEA) the result of f= just after 100 teratons. The calculatons were carred out on a Pentum IV computer wth a clock of 3 GHz. The tme of calculatons n the frst case was 2 seconds whereas n the second below 1 mllsecond,. e. over 2000 tmes faster. 6. Summary The calculatons have been based on an elaborated optmzaton programs wth applcaton of a genetc algorthm GA a hybrd evolutonary algorthm. For a presented optmzaton task of n = 16 sze, acheved values of a goal functon equal: n case of applcaton of a

13 Schedulng of Constructon Projects wth a Hybrd Evolutonary Algorthm s Applcaton 307 genetc algorthm (GA), f m = whereas for a hybrd evolutonary algorthm (HEA) also f mn = Whle analyzng the results n case a genetc algorthm (GA) from an example [13], hybrd evolutonary algorthm (HEA) and after ntroducton of tme buffers accordng to CCS/BM methodology one can state that applcaton of tme buffers of a zero load of resources (team workers) ncreases the extend of a optmzaton task and ensures the smallest value of a total dvergence from an average level of employment. 7. References [1] Bożejko W., Wodeck M., A hybrd evolutonary algorthm for some dscrete optmzaton problems, IEEE Computer Socety, 2005, [2] Fox R., Goldratt E.M., The Race. [Croton-on-NY]: North Rver Press, [3] Goldratt E.M., The Goal. [Great Barrngton, MA]: North Rver Press, 1 st ed. 1984, 2 nd revsed ed, [4] Goldratt E.M.,. Crtcal chan. [Great Barrngton, MA]: North Rver Press, [5] Goldratt E.M., The Haystack Syndrome: sftng nformaton out of the data ocean, New York, North Rver Press, [6] Goldratt E.M.. It s no luck. [Great Barrngton, MA]: North Rver Press, [7] Herroelen W., Leus R., On the merts and ptfalls of crtcal chan schedulng. Journal of Operatons Management, 19, 2001, [8] Gffler,B., Thompson, G.L., Algorthms for solvng producton-schedulng problems. Operatons Research, 8, 1960, [9] Leu S.S., Yang C.H., Huang J.C., Resource levelng n constructon by genetc algorthmbased optmzaton and ts decson support system applcaton. Automaton n Constructon, 10, 2000, [10] Radovlsky Z.D., A quanttatve approach to estmate the sze of the tme buffer n the theory of constrants. Internatonal Journal of Producton Economcs 55, 1998, [11] Rand G.K., Crtcal Chan: theory of constrants appled to project management. Internatonal Journal of Project Management, 18, 200, [12] Rogalska M., Hejduck Z., Shortenng the realsaton tme of buldng project wth applcaton of theory of constrants and crtcal chan schedulng.journal of Cvl Engneerng and Management, vol. X, Suppl 2, 2004, [13] Rogalska M., Bożejko W., Hejduck Z., Employment levelcontrol usng genetc algorthms (n Polsh). 51st KILW PAN and KN PZITB Scentfc Conference Gdańsk-Krynca, 2005, [14] Scavno N.J., Effect of Multple Calendars on Total Float and Crtcal Path. Cost Engneerng, vol.45, No.6, [15] Steyn.H., Project management applcatons of the theory of constrans beyond crtcal chan schedulng. Internatonal Journal of Project Management, 20, 2002, [16] Turner JR., Controllng progress wth planned cost or budgeted cost. Internatonal Journal of Project Management, 18(3), 2000, [17] West, J.D., Some propertes of schedules for large projects wth lmted resources. Operatons Research, 12, 1964,

14 308 Evolutonary Algorthms [18] Yang J.B., Applyng the theory of constrants to constructon schedulng. Proceedngs of Second Internatonal Structural Engneerng and Constructon Conference (ISEC 02), Vol. 1, 2003,