A Project Scheduling Method Based on Fuzzy Theory

Transcription

1 Journal of Industral and ystems Engneerng Vol. No. pp prng 007 Proect chedulng Method Based on Fuzzy Theory hmad oltan * Rasoul Ha harf Unversty of Technology and Engneerng Research Insttute Mnstry of grcultural Jahad P. O. Bo: 5-75 Tehran Iran (soltan@er.ac.r) Department of Industral Engneerng harf Unversty of Technology P. O Bo: 65-9 Tehran Iran (ha@sharf.edu) BTRCT In ths paper a new method based on fuzzy theory s developed to solve the proect schedulng problem under fuzzy envronment. ssumng that the duraton of actvtes are trapezodal fuzzy numbers (TFN) n ths method we compute several proect characterstcs such as earlest tmes latest tmes and slack tmes n term of TFN. In ths method we ntroduce a new approach whch we call modfed backward pass (MBP). Ths approach based on a lnear programmng (LP) problem removes negatve and nfeasble solutons whch can be generated by other methods n the backward pass calculaton. We drve the general form of the optmal soluton of the LP problem whch enables practtoners to obtan the optmal soluton by a smple recursve relaton wthout solvng any LP problem. Through a numercal eample calculaton steps n ths method and the results are llustrated. Keywords: Proect schedulng Fuzzy theory Modfed backward pass (MBP) Trapezodal fuzzy number (TFN) Lnear programmng (LP). INTRODUCTION: chedulng s deemed to be one of the most fundamental and essental bases of the proect management scence. There are several methods for proect schedulng such as CPM PERT and GERT. nce too many drawbacks are nvolved n methods estmatng the duraton of actvtes these methods lack the capablty of modelng practcal proects. In order to solve these problems a number of technques lke fuzzy logc genetc algorthm (G) and artfcal neural network can be consdered. fundamental approach to solve these problems s applyng fuzzy sets. Introducng the fuzzy set theory by Zadeh n 965 opened promsng new horzons to dfferent scentfc areas such as proect schedulng. Fuzzy theory wth presumng mprecson n decson parameters and utlzng mental models of eperts s an approach to adapt schedulng models nto realty. To ths end several methods have been developed durng the last three decades. The frst method called FPERT was proposed by Chanas and Kamburowsk (98). They presented the proect completon tme n the form of a fuzzy set n the tme space. Gazdk (98) developed a fuzzy network of an a pror unknown proect to estmate the actvty duraton and used fuzzy algebrac operators to calculate the duraton of the proect and ts crtcal path. Ths work s called FNET. n etenson of FNET * Correspondng uthor

2 Proect chedulng Method 7 was proposed by Nusuton (99) and Lorterapong and Moselh (996). Followng on ths McCahon (99) Chang et al. (995) and Ln and Yao (00) presented three methodologes to calculate the fuzzy completon proect tme. Other researchers such as Kuchta (00) Yao and Ln (000) Chanas and Zelnsk (00) and Olveros and Robnson (005) usng fuzzy numbers presented other methods to obtan fuzzy crtcal paths and crtcal actvtes and actvty delay. Prevous work on network schedulng usng fuzzy theory provdes methods for schedulng proects. These methods however do not support backward pass calculatons n drect manner smlar to that used n the forward pass. Ths s manly due to the fact that fuzzy subtracton s not proportonate to the nverse of fuzzy addton. Therefore these methods are ncapable to calculate proect characterstcs such as the latest tmes and slack tmes. In ths paper a new method s ntroduced for proect schedulng n fuzzy envronment. Ths method s developed based on a number of assumptons and defntons n the fuzzy set and proect schedulng. In the fuzzy proect network consdered n ths paper we assume that the duraton of actvtes are trapezodal fuzzy numbers (TFN). The proect characterstcs such as fuzzy earlest tmes and fuzzy proect completon tme are calculated as TFN by forward pass. s mentoned above backward pass n fuzzy envronment fals to compute the fuzzy latest tmes and fuzzy slack tmes. Therefore for computaton of these parameters we propose a new approach whch we call modfed backward pass (MBP). Ths approach s based on the proect schedulng fundamental concepts and lnear programmng. In MBP usng the proect schedulng concepts fuzzy latest tmes and fuzzy slack tmes relatons are transformed to lnear programmng (LP) problem. fter that we drve the general form of the optmal soluton of the LP problem whch enables practtoners to obtan the optmal soluton by a smple recursve relaton wthout solvng any LP problem. The advantage of MBP approach n comparson wth the prevous approaches s that t does not use the fuzzy subtracton operator n ts relatons. Due to these modfcatons the nherent defects dscussed before n the fuzzy envronment wll remove. Therefore the obtaned fuzzy latest tmes and fuzzy slack tmes n the MBP approach are correct and calculated as TFNs as well. Fnally through a numercal eample calculaton steps n ths approach and result are llustrated.. DEFINITION ND UMPTION In ths secton some basc notons of the area of fuzzy theory that have been defned by Kaufmann and Gupta (985) and Zmermann (996) are ntroduced. Then proect network s defned as a drected and acyclc graph n fuzzy envronment... Defntons Defnton: Let R be the space of real numbers. fuzzy set s a set of ordered pars {( μ ( ) ) R } where μ ( ) : R [ 0 ] and s upper sem contnuous. Functon μ ( ) s called membershp functon of the fuzzy set. Defnton: conve fuzzy set s a fuzzy set n whch: y R λ [0 ] μ ( λ + ( λ) y ) mn[ μ ( ) μ ( y) ]

3 7 oltan and Ha Defnton: fuzzy set s called postve f ts membershp functon s such that μ ( ) 0 0. Defnton: Trapezodal Fuzzy Number (TFN) s a conve fuzzy set whch s defned as: μ( ) where ( ) μ ( ) b < c () b 0 a a c 0 a a < b c < d < d For convenence TFN represented by four real parameters by a tetraplod ( a b c d) (Fg.). a b c d ( a b c d) wll be denoted μ ( ) a b c d Fg. Trapezodal Fuzzy Number (TFN) Defnton5: Trapezodal fuzzy number ( a b c d) s called postve TFN f: 0 a b c d... Operaton on TFNs Chen and Hwang (99) and Dubos and Prade (988) have been defned a number of operatons can be performed on TFNs. The followng are employed operatons n the development of the proposed method: Let ( a b c ) and B ( a b c ) be any two TFNs then: d d B ( a + a b + b c + c d + ) () d B ( a b c c b d ) () a MX ( B ) ( ma( a a )ma( b b )ma( c c )ma( d d ) ) () MIN ( B ) ( mn( a a )mn( b b )mn( c c )mn( d d ) ) (5) where fuzzy addton; fuzzy subtracton; and M X and I N mnmum respectvely. M are fuzzy mamum and

4 Proect chedulng Method 7.. Fuzzy Proect Network network N V D beng a fuzzy proect model s gven. V s a set of nodes (events) and V V s a set of arcs (actvtes). The network N s a drected and acyclc graph n the fuzzy envronment. The set V {... n} s labeled n such a way that the followng condton holds: ( ) <. In the fuzzy envronment the duraton of ths actvty (D) s a postve TFN: D ( d d d d ). Let us denote by P( ) { V ( ) } the set of predecessors and by { V ( } ( ) ) the set of successors of event V respectvely. tartng tme of the fuzzy proect model s a postve TFN: T ( t t t t ).. FUZZY PROJECT CHEDULING Fuzzy proect schedulng conssts of the forward pass and modfed backward pass (MBP) calculatons to obtan the substantal proect characterstcs. In ths secton for the fuzzy proect network these characterstcs such as earlest tmes latest tmes and slack tmes are obtaned by carryng out the calculatons as follows... Fuzzy Forward Pass Calculatons The earlest tmes and also proect completon tme n a proect network can be detected by forward pass. In ths case usng the relatons of CPM n the fuzzy envronment results n the followng fuzzy forward calculatons: { } MX E D P( ) φ P( ) E ( e e e e ) T t t t t P( ) φ ( es es es es ) E E ( ) (7) ( ef ef ef ef ) E D EF (8) ( tf tf tf tf ) MX E T F (9) V In the above E s the fuzzy earlest tme of event ; T s the fuzzy tme of startng the proect; E s the fuzzy earlest startng of actvty ( ) ; E F s the fuzzy earlest fnshng of actvty ( ) and T F s the fuzzy tme of proect completon. Based on the above equatons E E E and T F can be calculated as postve TFNs. F.. Fuzzy Modfed Backward Pass (MBP) Calculatons Backward pass calculatons are employed to calculate the latest tmes n the proect network. In ths case f the backward pass calculatons of CPM are entrely done n the fuzzy envronment the fuzzy latest tme of event L ) can be wrtten as: ( (6)

5 7 F { } MIN L D ( ) φ () L T ( ) φ oltan and Ha (0) s mentoned above the fundamental manner of the backward pass s based on an nverson between addton and subtracton. In a crsp envronment the equaton of +B B s always correct but the addton and subtracton are not always nverse n the fuzzy theory. It means that and B do not satsfy the relaton B B. Therefore the fuzzy backward pass n the equaton above faces serous problems. For eample for a typcal proect data n whch ( ) {} L (000) and (5050) D usng (0) t s found that: L ( 055 ) It s clear that L s a trapezodal fuzzy number wth a negatve part. It depcts that the latest tme of event may happen n a negatve tme. But the negatve tme s not feasble snce t s not defned n the proect schedulng. To avod ths problem we propose a new approach whch we call modfed backward pass (MBP). Therefore accordng to the concept of L and usng (0) the fuzzy latest tme of event L ) can be defned as: ( { } MIN X X D L ( ) φ () L T F ( ) φ () Usng ths relaton leads to a full postve tetraplod ( l l l l ). Therefore the problem due to the appearance of the negatve tme s removed. But n some cases the calculated L may not satsfy the defnton of TFN. For eample for a typcal proect data n whch ( ) {} L (000) and D (5050) usng () s found that: L (5070). It can be seen that L s not a trapezodal fuzzy number because t volates the conve condtons ( ). Therefore the calculaton should be done n such a way that L becomes a trapezodal fuzzy number. By addng the trapezodal condton to the above relatons the followng relaton s obtaned: L X D L MIN MX X ( ) φ T F ( ) φ () X s Postve TFN () In the above relaton we defne the relaton for any two TFNs such as ( a a a a ) and B ( b b b b ) as: B a b a b a b a b where () φ. Relaton () results n the followng fuzzy mathematcal programmng problem:

6 Proect chedulng Method 75 { ( )} L MIN MX X () ubect to X D L () () 0 Ths problem can be rearranged n a more convenent form as followng: L MIN Y ( y y y y ) s. t. y l l y y y l l 0 ) ) ) ) ) ) ) ) By replacng the obectve functon wth MIN y + y + y + y the problem above converts to a lnear programmng problem. It can be easly observed that the optmal soluton of ths problem L s obtaned as a postve TFN usng a smple recursve relaton: L ( l l l l ) : l ma ( 0 mn ( l d ) ) () () () () l ma ( 0 mn ( l mn ( l d ) ) ) l ma ( 0 mn ( l mn ( l d ) ) ) l ma( 0 mn( l mn( l d In the MBP followng the calculaton of L the fuzzy latest fnshng of actvtes ( L F ) s calculated as follows: LF ( lf lf lf lf ) L (6) Based on the equatons above L can be calculated as a postve TFN. F () (5)

7 76 oltan and Ha nother mportant characterstc of the backward pass s the fuzzy latest startng of actvtes ( L ). In order to calculate L when the backward pass calculatons of CPM are appled drectly to fuzzy envronment the followng relaton s obtaned: L LF D (7) In the relaton above the use of fuzzy subtracton s requred. Due to the presence of fuzzy subtracton smlar to that used n calculaton of L n some cases the calculatng of L may face some obstacles. Then by addng postve and trapezodal condtons and also usng a lnear programmng problem smlar to () and (5) the followng relatons would be obtaned: L ls ls ls ls ls ls ls ls ( ): ma ( 0 ( lf ma ( 0 mn ma ( 0 mn ma ( 0 mn ( lf ( lf ( lf ) ) ( ( ( lf lf lf ) ) ) ) ) ) ) ) ) (8).. Fuzzy lack Tmes One of the man characterstcs n proect control and plannng s the slack tme. There are three types of slacks for any actvty.e. fuzzy total slack ( T F ) fuzzy free slack ( F F ) and fuzzy ndependent slack I F ). If classcal relatons of CPM are appled for the calculaton of these ( characterstcs n the fuzzy envronment we can wrte the followng relatons: TF LF EF (9) FF E EF (0) IF E L D () By usng the relatons above the slack tmes may be out of postve TFNs defnton. Therefore smlar to the calculaton of L n modfed backward pass the followng relatons are proposed for slack tmes: TF ( tf tf tf tf tf tf tf ma (0 ( tf lf ma (0 mn ma (0 mn ); ( tf ( tf ma (0 mn ( tf ( lf )) ( lf ( lf ()

8 Proect chedulng Method 77 FF IF ( ff ff ff ff ff f f f f ( f ma (0 mn( ff ff ff ma (0 ( ff e ma (0 mn( ma (0 mn( f f f ma (0 ( e ma (0 mn( f ); ff ff l ma (0 mn( f ma (0 mn( f ); ( e ( e ( e ( e ( e ( e l l l () (). NUMERICL EXMPLE The network representng a structure of proect s gven n Fg Fg. Proect network n the numercal eample The duraton of actvtes are postve TFNs (Table). The fuzzy start tme of ths eample s. Table. D of the numercal eample ctvty() Duraton D ) ( () (585) () ( ) () (78) () (0550) (5) (585) (6) (55560) (7) ( ) (57) ( ) (67) (586)

9 78 oltan and Ha Usng the prevously descrbed relatons the man fuzzy characterstcs for the numercal eample are obtaned. These values are postve TFNs. Table represents fuzzy earlest and latest tmes of events by usng (6) and (5). Table. Calculated values of E and Event () E (585) ( ) ( ) ( ) (80000) (557595) L for the numercal eample L (5860) ( ) ( ) ( ) (07569) (557595) For eample: P() φ E ( e e e e ) T P() {} E ( e e e e ) E D (585) () {5} L ( l l l l ) (5860) : l ma(0mn(( l5 d5)( l ma(0mn(606)) 60 l ma(0mn( lmn(( l5 d5)( l ) ma(0mn(60mn(857 8 l ma(0mn( lmn(( l5 d5)( l ) ma(0mn(8mn( l ma(0mn( lmn(( l5 d5)( l ) ma(0mn(mn(5 5 The fuzzy tme of proect completon T F s calculated usng (9) as: (557595). E E F L and L F are obtaned usng (7) (8) (6) and (8) respectvely. The results have been presented n Table. s an eample for actvty (-) we obtan: Table. Calculated values of E E F () () () () () (5) (6) (7) (57) (67) (585) ( ) (585) ( ) ( ) ( ) (80000) E E F (585) ( ) ( ) ( ) ( ) (80000) (057595) (55970) (97856) L and L (065) L F for the numercal eample (576) ( ) (5860) ( ) ( ) ( ) (07569) L F (5860) ( ) ( ) ( ) ( ) (07569) (557595) (557595) (557595)

10 Proect chedulng Method 79 E ( es es es es ) E (585) EF ( ef ef ef ef ) E D ( ) LF ( lf lf lf lf ) L ( ) L ( ls ls ls ls ) (576) : ls ma ( 0 ( lf ) ) ma ( 0 6 ) 6 ls ma ( 0 mn ( lf ( lf ) ) ) ma ( 0 mn ( 6 57 ) ) 57 ls ma ( 0 mn ( lf ( lf ) ) ) ma ( 0 mn ( 57 ) ) ls ma ( 0 mn ( lf ( lf ) ) ) ma ( 0 mn ( )) Usng ()-() and the results presented n tables and the fuzzy slack tmes are calculated (Table ). These values can be used for calculaton of the crtcalty of the actvtes as well as determnaton of the crtcal paths. Table. Calculated values of () T F T F F F and F F I F for the numercal eample I F () (065) () () (8557) (557) () () (5) (065) (6) (779) (7) (57) (065) (065) (67) (779) (779) 5. CONCLUION Prevous works on network schedulng usng fuzzy sets theory provdes methods for schedulng proects. These methods however do not support the backward pass calculatons n drect manner smlar to that used n the forward pass. In ths paper a new method based on the fuzzy theory has been developed to solve the proect schedulng problem under the fuzzy envronment. In ths method duraton of actvtes are consdered as postve trapezodal fuzzy numbers. Then proect characterstcs such as earlest tmes latest tmes and slack tmes are calculated as trapezodal fuzzy numbers (TFNs). maor advantage of ths method s to employ drect arthmetc fuzzy operatons n obtanng meanngful computable results. In ths method we ntroduced a new approach whch we called Modfed Backward Pass (MBP). Ths approach based on a lnear programmng (LP) problem removes negatve and nfeasble solutons whch can be generated by other methods n the backward pass calculaton. We drove the general form of the optmal soluton of the LP problem whch enables practtoners to obtan the optmal soluton by smple recursve relaton wthout

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