CONSTRUCTION PROJECT SCHEDULING WITH IMPRECISELY DEFINED CONSTRAINTS

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1 Management an Innovaton fo a Sustanable Bult Envonment ISBN: June 2011, Amsteam, The Nethelans CONSTRUCTION PROJECT SCHEDULING WITH IMPRECISELY DEFINED CONSTRAINTS JANUSZ KULEJEWSKI Wasaw Unvesty of Technology Pl. Poltechn Wasaw, Polan j.ulejews@l.pw.eu.pl Abstact Ths pape egas to the scheulng of a constucton poject une ll-efne constants of tme an esouces fo the executon of wos. Fuzzy numbes ae use fo moellng the mpecson of constants. Two methos of the measuement of fuzzy constants satsfacton ae pesente. The fst metho uses the possblty measues base stctly on the assumptons of the fuzzy sets theoy. The secon metho uses the measue base upon the concept of the α-cuts of a fuzzy numbe an the pobablty theoy. The numecal examples ae gven fo the compason of both methos. The esults confm that the use of the pobablstc measue poves the neutalzaton of the assessment of the fuzzy constants meetng an mpoves the constucton scheule. Keywos: constucton scheule, mpecson, possblty measue, pobablty measue. INTRODUCTION The plannng ata use n the scheulng of constucton pojects ae often mpecsely efne wth ega to the eque poject completon tme an the avalablty of enewable esouces (ey pesonnel an constucton equpment eque fo the poject executon. Ths s cause by the vaous ccumstances, e.g.: the unqueness of the any gven constucton poject maes t ffcult o even mpossble to use the statstcal methos fo the assessment of the poject maespan; the acquston of contacts n the teneng poceue oes not allow fo the pecse plannng of the owne enewable esouces stbuton to the nvual pojects; contons of contact pove fo a tme nteval between the completon tme pefee by the clent (ue-to ate an the completon tme eque by the clent une the penalty of the contact temnaton of contacto s fault (ealne; the exstence of such a tme nteval allows the contacto fo the flexble plannng of the suffcent tme fo the completon of the wos. As a esult, the plannng ata fo the planne ae often efne mpecsely [3], wth the use of a natual language, e.g.: "about two wees", "about two to thee wees," a lttle ove two wees, "about ffteen to twenty woes" an ale [14]. In the lteatue ealng wth the poblems of poject scheulng on the bass of mpecsely efne plannng ata, thee s a common appoach to use the fuzzy sets theoy fo moellng the mpecson of poject ata, n conjuncton wth the vaous scheule optmzaton methos, as fo example the banch-an-boun metho, e.g. [16], poty heustcs, e.g. [4], [15], [18] an metaheustc methos, e.g. [5], [11], [15], [18]. Howeve, the most of the lteatue une

2 conseaton tae nto account only the mpecson of uatons of wos an the mpecson of tme avalable fo the executon of wos. The avalablty of any enewable esouce s teate as well-nown, whch n the case of a eal constucton poject aely hols tue. The assessment of the fulflment of mpecsely efne tme constants s one wth the use of possblty measue fo the compason of two fuzzy numbes o the eal numbe an the fuzzy numbe, one epesentng the planne poject maespan an the othe epesentng the poject maespan lmt. The compason s one wth the Huwcz cteon, e.g. [8], [17], [18] an the esult s hghly affecte by the specfc s atttue of the assesso. In esult, two o moe pesons may expess ffeent opnons about the egee of meetng the fuzzy lmt of the poject maespan. In ths pape, the pncples of the fuzzy moellng of mpecsely efne plannng constants an the pncples of the assessment of the fuzzy constants satsfacton ae pesente. The poblem of the neutalsaton of the assessment of meetng the fuzzy tme an esouce constants s esolve wth the use of the α -cuts of a fuzzy numbe an the pobablty theoy. The pape also pesents a numecal example showng the avantages of the use of pobablty measue fo the optmzaton of the constucton scheule wth ega to the mpecsely efne tme an esouce constants. THE MODELING OF THE PROJECT CONSTRAINTS USING FUZZY SETS To moel the mpecson of the avalablty of the -th esouce, a planne can use a tapezoal fuzzy numbe ( R (1 (2 (3 (4 n the fom of the oee fou (,,, R =, whee eal numbes ( = 1,, 4 shoul satsfy the conton: 0. Fg. 1 shows an example of a tapezoal fuzzy numbe moellng mpecsely efne (2 (3 (1 esouce constants, expesse as fom about to about, but not less than an not (4 moe than. (1 (2 (3 (4 1,00 µ ( R R (1 (2 (3 (4 Fgue 1: An example of usng a tapezoal fuzzy numbe fo moellng the mpecsely efne avalablty of the enewable esouce constants. Souce: Own In the moe specfc case, a planne s able to naow the aea of mpecson an to (2 (1 expess hs opnon about esouce constants as about, but not less than an not moe than (3. To expess ths mpecson mathematcally, a planne can use a tangula

3 fuzzy numbe fuzzy numbe R (1 (2 (3 n the fom of the oee thee (,, R = o a tapezoal R (1 (2 (2 (3 n the fom of the oee fou (,,, R =. Smlaly, to moel the mpecson of the poject maespan lmtaton one can use a (1 tapezoal fuzzy numbe T = ( t, t, t, t, whee eal numbes satsfy the conton: t t t oee fou, assumng fo example: (2 (3 (1 (2 (3 (4 0 t (1 (4 ( t (=1,, 4. One can etemne the components of ths as the eal numbe t : the shotest feasble poject maespan, etemne as the esult of the netwo moel analyss wthout the enewable esouce avalablty constants; (2 as the eal numbe t : the lowe lmt of the poject maespan, evaluate by the scheule as havng the geatest chance une the gven ccumstances; (3 as the eal numbe t : the uppe lmt of the poject maespan, evaluate by the planne as havng the geatest chance une the gven ccumstances; as the eal numbet (4 : the poject completon tme eque by the clent. ASSESSMENT OF THE PROJECT FUZZY CONSTRAINTS SATISFACTION Applcaton of the possblty measues If the mum consumpton of the -th enewable esouce must not excee the mpecsely efne esouce avalablty lmtaton, the scheule must satsfy the followng elaton: R (1 The satsfacton of the elaton (1 means that the mum consumpton of the -th enewable esouce, expesse by the eal numbe, shoul not excee the lmt, whch s the unnown (yet output of the fuzzy numbe R. Smlaly, f the wos must be complete wthn the mpecsely pescbe tme peo, the scheule must satsfy the followng elaton: T t. (2 The satsfacton of the elaton (2 means that the planne poject maespan expesse by the eal numbe t, wll not be longe than the tme lmt, whch s the unnown (yet output of the fuzzy numbe elaton T. Usng the theoy of possblty [1], one shoul assess the egee of fulflment of the R an evaluate of the veacty of the statement: "the eal numbe not be geate than the unnown (yet output of the fuzzy numbe R wll usng the necessty measue N( an the possblty measue Π (. The necessty measue s R use to assess how much the occuence of the elaton R R s obvous thoughout the

4 state of the nowlege of the planne of the ccumstances whch ae lmtng the avalablty of the -th enewable esouce. The possblty measue s use to assess how much the occuence of the elaton R emans n complance wth the state of nowlege of the planne of the ccumstances whch ae lmtng the avalablty of the -th enewable esouce. Accong to [1] an [8], the appopate fomulas ae as follows: N( Π( R = sup µ R Π( R = sup µ R (, (3 (, (4 R = 1 Π( R, (5 whee µ ( s the membeshp coeffcent of the fuzzy set R. R It shoul be note that the possblty measue Π ( oes not have the R popety of complementaty,.e. Π( oes not have to be equal to 1 Π(. R R Usng the necessty measue an the possblty measue fo assessng the ceblty of the statements gven above, one shoul conse the cases shown n Fg. 2. a α µ ( R 1,00 R b µ ( R 1,00 α R (1 (2 (3 (4 (1 (2 (3 (4 c µ ( 1,00 R R (1 (2 (3 (4 Fgue 2: The altenatve schemes of elatons between the fuzzy numbe numbe. Souce: Own R an the eal On the bass of the fomulas (3, (4 an (5, one can conclue that:

5 1 Fo the case shown n Fg. 2a: Π ( R = α, N( R = 0; the evaluate statement may be tue to a egee of α, but the obvous tuth of ths statement s zeo; 2 Fo the case shown n Fg. 2b: Π ( R = 1, N( R = 1 α ; the evaluate statement s possbly tue, but the obvous tuth of ths statement s 1 α, 3 Fo the case shown n Fg. 2c: Π ( R = 1, N( R = 0; the evaluate statement s possbly tue, but the obvous tuth of ths statement s zeo. The meanng of the assessment pesente above may be sometmes ffcult to unestan fo the planne. Theefoe, fo the assessment of the egee of fulflment of the elaton R, thee s a sought afte the synthetc measue (mae hee as ST, havng n lne wth the ntuton of the planne the popety of complementaty: ST( R = 1 ST( R. (6 Usng the appoach shown n [4], [17] an [18], one can mplement the Huwcz cteon fo the assessment of the egee of omnaton of the fuzzy numbe numbe : ST( R ove the eal R = β Π( R + (1 β N( R, (7 whee β (0.0; 1.0 s the coeffcent of optmsm, whch chaactezes the s atttue of the planne. Fo example, assumng the neutal s atttue of the planne (β = 0.5, one can obtan: (1 1,0 fo, (1 (2 0,5 + 0,5(1 α fo, (2 (3 ST( R = 0,5 fo, (8 (3 (4 0,5α fo, (4 0,0 fo. In a smla way one can assess the egee of fulflment of the elatont T. As can be seen fom the fomula (7, the esult of the assessment of the egee of fulflment of the elatons R an t T stongly epens on the value of coeffcent β, whch chaactezes the s atttue of the planne. Applcaton of the pobablstc measue It shoul be note, afte [2], that the necessty measue an the possblty measue etemnate the lowe boun an the uppe boun of the pobablty: N( R P( R Π( R, (9 N( t T P( t T Π( t T. (10

6 Ths ases the queston whethe t s feasble to neutalze the assessment of the fulflment of elatons R an t T, though the ect use of the pobablstc measue. The esultng poblem can be escbe as follows: thee ae two numbes gven: (1 a eal numbe m, epesentng the mum consumpton of some enewable esouce o the planne poject maespan, an (2 a fuzzy numbe maespan; N, moellng the lmt of esouce avalablty o the lmt of the poject assess the pobablty P( m N that a eal numbe m, esultng fom the constucton poject scheule, wll be not geate than the unnown (yet output of a fuzzy numbe N. The ea of the assessment of the pobablty P( m N pesente below s base upon the use of the α -cuts of a fuzzy numbe N fo a fnte numbe of levels of cetanty of the mpecse estmaton of the gven constant. Fo the any gven α -cut of a fuzzy numbe N l u, an nteval numbe N = [ n, n ] s obtane. Symbol s an nex of a sequent α α -cut. An example of an nteval α α N α s shown n Fg ,00 α µ (x N x l n α Fgue 3: An example of an nteval m N α u n α. Souce: Own. l u On the bass of Fg. 3 one can conclue that f nα < m < n α, then an nteval N α s ve futhe nto subntevals [ l u n α, m] an [ m ]. The pobablty that a eal numbe m wll be not geate than the unnown (yet output of an nteval numbe etemne geometcally as: u α u α,n α l α N α can be n m P( m N =. (11 α n n If m then P( m N = 1an f l n α α m then P( m N = 0. The aggegaton u n α α of pobabltes P( m N α fo the fnte numbe of α -cuts of a numbe N, leas to the followng fomula: α P( m N α P( m N =, (12 α

7 whee = 1,, I s an nex of sequent α -cut of a numbe N. SCHEDULE OPTIMIZATION PROBLEMS FORMULATION In ths pape, an actvty on noe netwo moel wth fnsh to stat elatons between actvtes s aopte to epesent the constucton poject. The stat ate of the poject s set to zeo. Only the mpecson of the scheule constants s consee. Theefoe, the fomula fo calculatng the scheule poject maespan can be expesse as: t = { s + }, = 1, 2,..., n, whee s s the stat ate of actvty, s the uaton of actvty, an n s the total numbe of actvtes. The followng two altenatve optmzaton poblems can be fomulate, base upon the two altenatve measues of the complance wth the fuzzy lmt of the poject maespan: fn the stat ates of actvtes so as to mze the egee of complance wth the fuzzy lmt of the poject maespan: ST : ST = ST( t T, (13 fn the stat ates of actvtes so as to mze the pobablty of complance wth the fuzzy lmt of the poject maespan: P : P = P( t T, (14 whee t s the eal numbe, epesentng the planne poject maespan, an s the fuzzy numbe, moellng the mpecsely specfe constant fo the poject maespan. Tang nto account the elatons of type fnsh to stat between the actvtes, the soluton of the poblem (14 o of the poblem (15 must fulfl the followng conton: s s + { Pec( j}, (15 j whee Pec ( j s the set of peecessos of an actvty j n the poject netwo moel. The soluton of the poblem (14 o of the poblem (15 must also tae nto account the fuzzy constants of the enewable esouces avalablty. The mum consumpton of the -th esouce can be etemne as: = { }, (16 pτ p { A( τ } whee {A (τ} s the set of opeatons execute n a tme peo τ, τ = 1,...,t, pτ s the consumpton of the -th enewable esouce fo the executon of an actvty p n a tme peo τ, an t s the planne poject maespan. Accong to the two altenatve measues of the complance wth the fuzzy esouce constants, the planne shoul assess the eque egee ST of complance wth the fuzzy constant of -th esouce avalablty o the eque pobablty P of complance wth the fuzzy constant of -th esouce avalablty. Ths leas to the followng contons: fo the solutons of the poblem (14: the conton of meetng the eque egee of complance (ST wth the fuzzy lmt of avalablty of the th enewable esouce: ST( R < ST, (17 T

8 fo the solutons of the poblem (15: the conton of meetng the eque pobablty of complance (P wth the fuzzy lmt of avalablty of the th enewable esouce: P( R SCHEDULE OPTIMIZATION PROBLEMS SOLVING < P. (18 Despte the specfc measue aopte fo the assessment of the fuzzy constants satsfacton, optmzaton poblems pesente above ae the esouce constane poject scheulng poblems, whch belong to the class of NP-ha poblems [4]. Fo solvng such poblems, the use of heustc o metaheustc methos s well justfe. The etale escpton s omtte hee. The suveys appopate fo the constucton poject scheulng has been one by the othes, e.g. [6], [7], [9] [13], [16]. In ths pape, the consee scheule optmzaton poblems wee tanslate nto numecal optmzaton poblems solve wth the use of Genetc Algothm (GA. To peseve the eque technologcal peceence elatonshps among actvtes n the poject netwo moel, the geneal ea was to use the GA technque to establsh the atonal esouce elatonshps among some actvtes. Those atonal elatonshps wee base upon the selecte poty ules. On the bass of the soluton pesente by the m-th chomosome, the stat ates s j of each actvty wee calculate usng the followng fomula: s ( m = { s ( m + }. (19 j { Pec( j} The followng ftness functons wee use to assess the esultng constucton scheule: 1 fo the poblem of mzaton the egee of complance wth the fuzzy lmt of the poject maespan: K F = 1 f ( m = ST( m, (20 2 fo the poblem of mzaton the pobablty of complance wth the fuzzy lmt of the poject maespan: K F = 1 f ( m = P( m, (21 whee: f(m the value of ftness functon fo the soluton pesente by the m-th chomosome; ST(m the egee of complance wth the fuzzy lmt of the poject maespan, esultng fom the scheule awn upon the soluton pesente by the m-th chomosome: ST( m = ST( t( m T ; (22 P(m the pobablty of complance wth the fuzzy lmt of the poject maespan, esultng fom the scheule awn upon the soluton pesente by the m-th chomosome: P( m = P( t( m T ; (23 t(m the planne poject maespan, esultng fom the scheule awn upon the soluton pesente by the m-th chomosome; F the penalty (lage enough postve eal numbe fo the falue to meet the eque egee of complance wth the fuzzy lmt of avalablty of the -th enewable esouce; G the penalty (lage enough postve eal numbe fo the falue to meet the eque pobablty of complance wth the fuzzy lmt of avalablty of the -th enewable esouce; K the numbe of types of enewable esouces wth lmte avalablty.

9 NUMERICAL EXAMPLES The scope of the exemplay constucton poject coves moenzaton of an exstng housng estate. Ths nclues the enovaton of exstng bulngs A an B, the enovaton of the exstng estate oa, ca pang, an the constucton of new bulngs C an D wth the ancllay facltes. The netwo of the poject actvtes s shown n Fg.4. The netwo ata ae gven n Table Fgue 4: An example of a constucton poject netwo moel. Souce: Own Neglectng the lmt of wofoce avalablty, the shotest feasble poject maespan s t s = 37 wees, wth the mum employment of = 49 woes pe wee. The contacto assumes that the numbe of avalable woes wll be pobably lmte to woes. In any case t wll be not less than 25 an not moe than 40 woes. The mpecsely specfe lmt of wofoce avalablty can be moele by the fuzzy tapezoal numbe R = (25, 30, 35, 40. The clent eques that the poject has to be complete wthn a mum peo of 50 wees fom the ate of commencement. On the bass of hs past expeence, the contacto assumes that he shoul be techncally able to execute the wos wthn about wees. Due to the commtments of the contacto to the othe clents, the poject maespan shoul not excee 45 wees, an the owne of ths poject wll absolutely not accept the poject maespan exceeng the peo of 50 wees. The mpecsely specfe lmt of tme avalable fo the executon of the wos can be moele by the tapezoal fuzzy numbe T = (37, 40, 45, 50. Table 1. Data fo the poject netwo moel shown n Fg. 4. Souce: Own

10 Actvty No Descpton Constucton ste pepaaton Eathwos fo bulngs C an D Renovaton of founatons of bulng A Renovaton of an exstng estate oa Renovaton of the oof of bulng A Renovaton of ntenal sevces n bulng B Duaton (wees Reque numbe of woes Ealest feasble stat ate fnsh ate Founaton of bulng C Founaton of bulng D Renovaton of the exstng pang Renovaton of ntenal sevces n bulng A Reecoaton of bulng B Supestuctue of bulng C Supestuctue of bulng D Reecoaton of bulng A Intenal sevces n bulng C Intenal sevces n bulng D Repa of auxlay facltes Fnshng wos n bulng C Fnshng wos n bulng D Constucton ste emoval In the fst example, the planne s oblge to scheule the poject to the hghest egee of complance wth the mpecsely specfe tme lmt fo the executon of the wos.

11 The planne s s neutal (β = 0,5. Moeove, the scheule shoul guaantee the egee of complance wth the fuzzy lmt of wofoce avalablty not less than ST w = 0,50. The poblem gven above s escbe by the fomula (13: ST : ST = ST( t T, wth the conton (15, concenng the fnsh to stat elatons among the actvtes n the poject netwo moel: s s + { Pec( j}, j an wth the conton (17, concenng the eque egee of complance wth the fuzzy lmt of wofoce avalablty: ST( R 0,50. The esultng constucton scheule s pesente n Fg. 5. The planne poject maespan s t = 44 wees an the egee of complance wth the fuzzy lmt of tme avalable fo the executon of the wos s ST( t T = 0,50. The mum wofoce employment s 35 woes pe wee an the esultng egee of complance wth the fuzzy lmt of wofoce avalablty s ST( R = 0,50. It shoul be note that the esultng egee of complance wth the fuzzy lmt of tme avalable fo the executon of the wos s ate sgnfcantly hghe by the moe optmstc planne (see Fg. 7. It shoul be also note, that the shotenng of the planne poject maespan to the level of 40 wees oes not change the esultng egee of complance wth the fuzzy lmt of tme avalable fo the executon of the wos (see Fg. 7. Smlaly, the eucton of the mum wofoce employment to the level of 30 woes pe wee oes not change the esultng egee of complance wth the fuzzy lmt of wofoce avalablty Fgue 5: Constucton scheule ensung the hghest egee of complance wth the mpecsely specfe tme lmt fo the executon of the wos. Souce: Own In the secon example, the planne s oblge to scheule the poject to the hghest pobablty of complance wth the mpecsely specfe lmt of tme avalable fo the

12 executon of the wos. Moeove, the scheule shoul guaantee the pobablty of complance wth the fuzzy lmt of wofoce avalablty not less than P w = 0,50. Ths poblem s escbe by the fomula (14: P : P = P( t T, wth the conton (15, concenng the fnsh to stat elatons among the actvtes n the poject netwo moel: s s + { Pec( j}, j an wth the conton (18, concenng the eque pobablty of complance wth the fuzzy lmt of wofoce avalablty: P( R 0,50. The esultng constucton scheule s pesente n Fg. 6. The planne poject maespan s t = 41 wees an the pobablty of complance wth the fuzzy lmt of tme avalable fo the executon of the wos s P( t T = 0,70. The mum wofoce employment s 32 woes pe wee an the esultng pobablty of complance wth the fuzzy lmt of wofoce avalablty s P( R = 0,56. It shoul be note that any shotenng of the planne poject maespan mpoves the pobablty of complance wth the fuzzy lmt of tme avalable fo the executon of the wos (Fg. 7, egaless the s atttue of the planne. Smlaly, any eucton of the mum wofoce employment mpoves the pobablty of complance wth the fuzzy lmt of wofoce avalablty Fgue 6: Constucton scheule ensung the hghest pobablty of complance wth the mpecsely specfe tme lmt fo the executon of the wos. Souce: Own

13 1,0 P( t T,ST( t T P( t T ST( t T fo β = 0,8 ST( t T fo β = 0,5 T Fgue 7: Compason of the functons ST( t T an P( t T. Souce: Own Suppose now that the wofoce s ve nto two man taes: constucton woes (Tae I fo tems 1 5, 7 9, 11 14, an nstalles (Tae II fo tems 6, 10, 15, 16. The mpecsely specfe lmt of constucton woes avalablty s about 30, but not less than 28 an not moe than 32 woes. Ths lmt can be moelle by the fuzzy tapezoal numbe R I = (28, 30, 30, 32. Futhemoe, the mpecsely specfe lmt of nstalles avalablty s about 20, but not less than 18 an not moe than 22 woes. Ths lmt can be moelle by the fuzzy tapezoal numbe R II = (18, 20, 20, 22. The mpecsely specfe lmt of tme avalable fo the executon of the wos emans as befoe: 40, 45, 50. T = (37, The planne s oblge to scheule the poject to the hghest pobablty of complance wth the mpecsely specfe lmt of tme avalable fo the executon of the wos. Moeove, the scheule shoul guaantee the pobablty of complance wth the fuzzy lmt of constucton woes avalablty not less than P I = 0,50 an the pobablty of complance wth the fuzzy lmt of nstalles avalablty not less than P II = 0,50. Ths poblem s escbe by the fomula (14: P : P = P( t T, wth the conton (15, concenng the fnsh to stat elatons among the actvtes n the poject netwo moel: s s + { Pec( j}, j an wth the contons (18, concenng the eque pobablty of complance wth the fuzzy lmts of constucton woes an nstalles avalablty: P( RI 0,50, I

14 P( RI I 0,50. The esultng constucton scheule s pesente n Fg. 8. I I Fgue 8: Constucton scheule ensung the hghest pobablty of complance wth the mpecsely specfe tme lmt fo the executon of the wos n the case of two taes wth lmte avalablty. Souce: Own In ths case, the planne poject maespan s t = 40 wees an the pobablty of complance wth the fuzzy lmt of tme avalable fo the executon of the wos s P( t T = 0,84. The mum employment of constucton woes s 30 woes pe wee an the esultng pobablty of complance wth the fuzzy lmt of wofoce avalablty s P( I RI = 0,50. The mum employment of nstalles s 19 woes pe wee an the esultng pobablty of complance wth the fuzzy lmt of wofoce avalablty s P( RI I = 0,97. I I CONCLUSIONS The theoy of possblty allows fo the moellng of mpecsely efne plannng constans by means of tapezoal o tangula fuzzy numbes. If tapezoal fuzzy numbes ae use, some ffcultes ase n etemnng the pope numecal value of coeffcent β chaactezng the s atttue of the assesso. Ths may cause the ffeent assessments of the egee of satsfacton of fuzzy plannng constants, fomulate by planne an by the ecson mae. In aton, the esult of evaluaton emans constant when the planne poject maespan o the planne esouce consumpton taes the value fom the coe of the tapezoal fuzzy numbe, moellng the gven plannng constant. Ths avesely affects the optmalty of solutons to scheulng poblems wth mpecsely efne constants. If tangula fuzzy numbes ae use, thee stll eman ffcultes n the numecal

15 chaactestcs of the atttue towas s. The appoach pesente n ths pape combnes the elements of the theoy of possblty an the elements of theoy of pobablty. The mpecson of poject constants s moelle by fuzzy numbes, whle the level of satsfacton of fuzzy plannng constants s assesse by the use of pobablty measue. It has been emonstate that the use of pobablty measue neutalze the assessment of complance wth the fuzzy constans. Moeove, the esults of the optmzaton of constucton scheule ae mpove. The appoach pesente n ths pape can be aopte when also the uncetantes of poject actvty uatons ae moelle by fuzzy numbes. LITERATURE [1] Dubos D., Pae H., Possblty theoy, Plenum Pess, New Yo [2] Dubos D., Pae H., When uppe pobabltes ae possblty measues, Fuzzy Sets an Systems, 49 (1, 1992, pp [3] Dubos, D., Fage, H., Fotemps, F., Fuzzy scheulng: Moellng flexble constants vs. copng wth ncomplete nowlege, Euopean Jounal of Opeatonal Reseach, 147(2, 2003, pp [4] Hape M., Słowńs R., Fuzzy poty heustcs fo poject scheulng, Fuzzy Sets an Systems 83, 1996, pp [5] Hape M., Słowńs R., Fuzzy set appoach to mult-objectve an mult-moe poject scheulng une uncetanty n: Hape M., Słowńs R. (e., Scheulng Une Fuzness, Physca-Velag, Heelbeg, 2000, pp [6] Hegazy T., Optmzaton of esouce allocaton an levelng usng genetc algothms, Jounal of Constucton Engneeng an Management, 125 (3, 1999, pp [7] Jaśows P., Sobota A., Scheulng constucton pojects usng evolutonay algothm, Jounal of Constucton Engneeng an Management, 132 (8, 2006, pp [8] Kuchta D., Soft mathematcs n management. The applcaton of nteval an fuzzy numbes n management accountng (n Polsh. Ofcyna Wyawncza Poltechn Wocławsej, Wocław [9] Lee Z.J., Su S.F., Lee C.Y., Hung Y.S., A heustc genetc algothm fo solvng esouce allocaton poblems, Knowlege an Infomaton Systems, 5, 2003, pp [10] Leu S.S.,Yang C.H: GA-base multctea optmal moel fo constucton scheulng, Jounal of Constucton Engneeng an Management, 125 (6, 1999, pp [11] Leu S.S., Chen A.T.,Yang C.H., A GA-base fuzzy optmal moel fo constucton tme-cost tae-off, Intenatonal Jounal of Poject Management, 19, 2001, pp [12] L H., Love P.E.D., Usng mpove genetc algothms to facltate tme cost optmzaton, Jounal of Constucton Engneeng an Management, 123 (3, 1997, pp [13] L H., Cao J.N., Love P.E.D., Usng machne leanng an GA to solve tme cost tae off poblems, Jounal of Constucton Engneeng an Management, 125 (5, 1999, pp [14] Loteapong P., Moselh O., Poject netwo analyss usng fuzzy sets theoy, Jounal of Constucton Engneeng an Management, ASCE, 122 (4, 1996, pp [15] Pan H., Yeh C.H., Fuzzy poject scheulng, Poceengs of the IEEE Intenatonal Confeence on Fuzzy Systems, 2003, pp

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