SCHEDULING OF CONSTRUCTION PROJECTS BY MEANS OF EVOLUTIONARY ALGORITHMS

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1 SCHEDULING OF CONSTRUCTION PROJECTS BY MEANS OF EVOLUTIONARY ALGORITHMS Magdalena Rogalska 1, Wocech Bożeko 2,Zdzsław Heduck 3, 1 Lubln Unversty of Technology, 2- Lubln, Nadbystrzycka 4., Poland. E-mal:[email protected] 2 Wrocław Unversty of Technology,. Wybrzeże Wyspańskego 2.,-3 Wrocław, Poland. E-mal: [email protected] 3 Wrocław Unversty of Technology, Wybrzeże Wyspańskego 2,-3 Wrocław, Poland. E-mal: [email protected] Abstract. The artcle presents the results of a computaton experment n whch a genetc algorthm (GA) and a hybrd evolutonary algorthm (HEA) were used. The respectve results are compared for an obectve functon descrbng employment level regularty. Keywords : evolutonary algorthm, genetc algorthm, tme couplng method TCM 1. Introducton The fundamental papers on the applcaton of evolutonary algorthms are the ones by Lawrence Fogel [1], Ingo Rechenberg, Hans-Paula Schewefel [2,3] and John Holland [4]. The classcal genetc algorthms are wdely descrbed n the paper by Davd Goldberg [] whle the fundamentals and applcatons of evolutonary algorthms are presented n the paper by Zbgnew Mchalewcz []. The development of optmzaton technques was spurred by a search for better methods of schedulng producton (ncludng constructon proects). One such method s evolutonary computaton (computer systems explotng the prncples of the evoluton of lvng organsms) whch ncludes: genetc algorthms, evolutonary strateges, evolutonary programmng, evolutonary plannng, genetc programmng. The methods nvolve creatng a populaton of ndvduals (schedules) through a random search for feasble solutons. The methods dffer n ther approach to creatng new generatons of ndvduals and selectng and applyng operators. Schedulng of constructon proects nvolves, among other thngs, the optmzaton of demand for realzaton means [,1,,12,], e.g.: optmze the employment level for an assgned proect duraton, mnmze proect duraton for an assgned level of means. Ths paper concerns the optmzaton problem wth an obectve functon n the form of a measure of employment level average devaton dsperson for an assgned constructon proect lead tme. The am s to compare, through a computaton experment, the results obtaned by applyng a hybrd evolutonary algorthm (HEA) [] and a classcal genetc algorthm (GA) [] to the schedulng of a constructon proect wth regard to employment level optmzaton. The problem belongs to a class of NP dffcult problems. 2. Basc propertes of genetc algorthms Optmzaton by means of a genetc algorthm conssts of the followng steps: dentfy the optmzaton problem soluton structure, encode the soluton as sequences, defne an obectve functon to be optmzed, defne GA operators and boundary crtera, perform calculatons wthn the constrants, decode the calculated optmum soluton sequences [].

2 The optmzaton calculaton procedure usng a genetc algorthm [, 1] s shown n fg. 1. Fg. 1. Optmzaton procedure usng genetc algorthms []. 3. Basc propertes of hybrd evolutonary algorthm The algorthm [] starts wth the (random) creaton of ntal populaton P. The best member of populaton P s adopted as suboptmal soluton π. Let be an algorthm teraton number. New populaton +1 (.e. set P + ) s generated as follows. For current populaton P a set of local mnma ( LM ) s fxed (by carryng out the LocalOpt( π ) procedure for each element π P ). Elements whch occur n the same postons at the local mnma are fxed (procedure FxeSet( LM, FS )) and a set of fxed elements and postons ( FS + ) s created. Each permutaton of new populaton P + has fxed elements (n fxed postons) from set FS +. Free elements are randomly assgned to the remanng (free) postons. If permutatons β LM and F( β ) < F( π ) exst, then β s adopted for permutaton π. The algorthm ends after a predetermned number of generatons have been generated. P START Randomly select populaton of characterstcs (sgns, detectors) Develop characterstcs usefulness for populaton Check boundary crtera SELECTION Select populatons and next generatons CROSSBREEDING Recombne characterstcs and select partners MUTATION Stochastcally verfy matched partners EVALUATION Evaluate new usefulness PROPAGATION Select next generatons on bass of usefulness Hybrd Evolutonary Algorthm (HEA) Intalzaton: randomly created populaton = { π, π,..., π }; 1 2 η π = the best member of populaton the number of teraton =; P ; FS = ; repeat Fx local mnma set LM = { ˆ π, ˆ π,..., ˆ π }, where π P ; F ˆ 1 2 ˆ ( π ) ( π ) η π = LocalOpt( π ), for :=1 to η do < F then π ˆ π ; Fx set FS + = FxSet( LM, FS ) and generate new populaton + 1 P : = NewPopulaton( FS ) ; =+1; untl not StopCrteron; Comparson of the algorthms propertes The genetc algorthm s prncple of operaton conssts n creatng a random populaton of n code combnatons consstng of z bnary symbols V={,1}, replcatng them, replacng parts of the combnatons and sometmes mutatng some of the bts. For ths purpose the schema (a standard descrbng a subset of smlar sequences wth good solutons []) theorem s used. As opposed to other methods, optmzaton computatons are performed for whole populatons n whch decson varable values are sought avodng the possblty that only a local maxmum wll be found. The evolutonary algorthm s prncple of operaton conssts n searchng for a global mnmum through a random choce of a populaton of ndvduals (e.g. schedules). Dfferent strateges, e.g. (l+1), based on mechansms of searchng sets of feasble solutons through the self-adaptaton of the mutaton range [] are used. The genetc algorthm s a specal case of the evolutonary algorthm. It has a predetermned constant populaton of ndvduals on whch replcatons are performed. The hybrd evolutonary algorthm (HEA) frst randomly creates an ntal populaton, fnds the best member and generates new populatons n whch local mnma are sought. A set of the latter s formed and for each permutaton of a new generaton a better soluton s adopted from the area between neghbourng local mnma. The above optmzaton IT tools can be used to carry out practcal constructon proect schedulng tasks, ncludng the optmzaton of demand for means. They do not guarantee that optmum solutons wll be found but for problems of practcal sze the results can be consderably mproved. Ths can be done usng classcal genetc algorthms or mproved evolutonary algorthms (such as HEA []). In order to mprove the optmzaton tools, faster and more accurate algorthms for searchng the space of solutons are developed [, 1, 1]. Ths paper presents the results of a computaton experment f

3 ndcatng that better optmzaton results can be obtaned through the use of a hybrd evolutonary algorthm. 4. Computaton experment case study The optmzaton problem s as follows: for an assgned constructon proect lead tme the best (accordng to the adopted optmzaton crteron) employment level must be found. One of the optmzaton crtera can be to nclude the average demand for resources n the constructed obectve functon. In order to search for a mnmum demand one can adopt the average devaton from the daly demand as a measure of the unevenness of the demand for workers. The obectve functon s: where: T 1 q ( x) ravg f( x) = (1) T r r = 1 avg avg 1 n T = 1 = d r (2) the average number of workers employed on each of the T days. n x R x=(x 1, x 2,, x n ) the vector of the tasks startng tmes, x [a, b ] a the earlest tme of startng task, b the latest tme of startng task, q (x) the number of persons employed on day, = 1,2,,T, T a tme horzon, d duraton of process r the number of persons employed to carry out process. In order to carry out optmzaton calculatons one must calculate the obectve functon for each teraton. The functon can be calculated as the average percentage devaton of the daly number of employed persons from the average employment durng all the T days,.e. f ( x) [,1] for a standard obectve functon. The numercal data for the optmzaton calculatons are for a complex of 12 buldng structures on whch constructon processes are to be carred out. The proect s represented by a matrx of constructon process (r = ) duratons for the structures (f = 12). T = The work duraton matrx elements were based on blls of quanttes for a housng complex. Tme couplng method TCM III [1] was used to determne the constructon proect cycle (T = 21 workng days). The method allows one to dentfy a sequence of crtcal processes and fx noncrtcal work commencement and completon dates. Owng to the changeable poston of the obs along the tme axs one can optmze the employment level accordng to the adopted obectve functon.. Concluson An optmzaton problem (12, T = 21 workng days) concernng employment level plannng for an assgned constructon proect lead tme was computed usng a genetc algorthm (GA) and a hybrd evolutonary algorthm (HEA) [, ]. After 1 teratons usng GA, obectve functon f=14. and the average percentage devaton of the daly number of employed persons from the average employment for all the T days amountng to.3123 were obtaned. After 1 teratons usng HEA, the obectve functon was f=1. and the average devaton amounted to.3. Thanks to the use of the evolutonary algorthm the result mproved by 3.%. It has been demonstrated that evolutonary algorthms can be used for optmzng demand for resources (workers) n tme couplng methods (TCM) [1]. It s also possble to employ metaheurstc (smulated annealng and taboo search) algorthms and evolutonary algorthms to solve task schedulng problems n TCM. The research nto ths possblty s underway.

4 Fg. 2. Process of optmzaton by means of evolutonary algorthms [] Fg.3. Gantt chart after genetc algorthm applcaton, mnmum obectve functon f=14..

5 Fg. 4. Gantt chart after evolutonary algorthm applcaton, mnmum obectve functon f=1.. References 1. Fogel L., Owens A., Walsh M., Artfcal ntellgence through smulated evoluton. Chchester, Wley, Rechenberg I., Evolutonsstratege: Optmerung technscher Systeme nach Prnzpen der bologschen Evoluton. Stuttgart, Frohman-Holzboog,. 3. Schwefel H-P., Numercal optmzaton of computer models. Chchester, Wley,. 4. Holland J. H., Adaptaton n natural and artfcal systems. Ann Arbor, Unversty of Mchgan Press, 1.. Goldberg D., Algorytmy genetyczne w zastosowanach. Warsaw, WNT, 1.. Mchalewcz Z., Genetc algorthms + data structures = evolutonary programs (n Polsh). Warsaw, WNT, 1.. Bożeko W., Wodeck M., A hybrd evolutonary algorthm for some dscrete optmzaton problems. IEEE Computer Socety, P22 ISBN --22-, 2.. Rogalska M., Bożeko W. Heduck Z., Employment level control usng genetc algorthms (n Polsh). 1st KILW PAN and KN PZITB Scentfc Conference Gdańsk- Krynca 2, pp LEU, S.S., YANG, C.H., HUANG J.C. (2) Resource levellng n constructon by genetc algorthm-based optmzaton and ts decson support system applcaton. Automaton n Constructon 1, Nassar K., Evolutonary optmzaton of resource allocaton n repettve constructon schedules, ITcon., Vol. 1 (2) p.2.. Borne P., Evolutonary algorthms for ob-shop schedulng. Int. J. Appl. Math. Comput. Sc., 24, Vol., No 1, Toklu Y.C., Applcaton of genetc algorthms to constructon schedulng wth or wthout resource constrants. Can. J. Cv. Eng. 2, (22).. Ugwu O.O., Tah J.H.M., Development and applcaton of a hybrd genetc algorthm for resource optmzaton and management. ECAM 22 /4, Otterloo van Suwert, Evolutonary algorthms and Schedulng Problems. Unverstet Utrecht, Reeves C.R. Yamada T., Genetc Algorthms Path Relnkng on the Flowshop Sequencng Problem. Evolutonary Computaton Journal, MIT Press, Vol.., No 1, pp , Yamada T., Solvng the Csum Permutaton Flowshop Schedulng Problem by Genetc Local Search ICEC (Internatonal Conference on Evolutonary Computaton), pp Heduck Z., Rogalska M., Tme Couplng Methods (TCM), Przegląd Budowlany 2/2, 3-4.