An Algorithm for Solving Project Scheduling to Maximize Net Present Value

Transcription

1 World Applied Sciences Journal (): -, ISSN - IDOSI Publicaions, An Algorihm for Solving Projec Scheduling o Maximize Ne Presen Value Ahmad Ali Bozorgzad and A. Hadi-Vencheh Deparmen of Indusrial Engineering, Azad Universiy, Najafabad, Isfahan, Iran Deparmen of Mahemaics, Azad Universiy, Khorasgan, Isfahan, Iran Absrac: In his paper, we presen an innovaive and applicable algorihm for solving he projec scheduling o maximize ne presen value (NPV). Our algorihm has been coded in visual C++ and esed by some random problems. The obained resuls show ha he proposed algorihm can solve he projecs wih more han aciviies and wo complexiy nework coefficien (CNC) in less han wo seconds. Also i is able o improve he NPV of projec wih aciviies, CNC, discoun rae of % and days receiving periods, abou.%. Key words: Ne presen value, Criical pah mehod, Cash flow, Floa, Complexiy nework coefficien INTRODUCTION The res of his Paper is organized as follow: in he following secion we review he lieraure. In secion we One of he mos imporan goals of a projec is sae pre-hypoheses of our model. Secion devoed o profiabiliy and rying o increase he amoun of he proposed algorihm. In secion we es our proposed profiabiliy. The problem of projec ime-budge in order model. Secion 6 concludes. o maximize ne presen value (NPV) is a cerain case of problem in projec ime-budge ha addiion o ime- Lieraure Review: The problems of projec ime-budge, budge under differen condiions and limiaions, i henceforh PS-Max-NPV, have been analyzed and possible o maximize floas and ime opporuniies and o invesigaed from is raising in unil now in differen maximize he profiabiliy of he projec. Wih regard o aspec and saes. A general mehod ha has been long duraion of implemening he mos projecs in under similarly applied in all previous researches, is ime-budge developing counries, increased volume of cash flow of sandard problems (wihou considering NPV) by using during implemenaion of projecs, discoun rae, capial common echniques of projec managemen (PN, GEPT, limiaion and so on. We conclude ha he bes facor for PERT, CPM, ) and geing he ime needed for maximizing profiabiliy is NPV facor. compleing a projec, deerminaion of early and lae ime The main goal of his sudy is o achieve a mehod of saring and finishing of aciviies, compuaion of rae such ha in addiion o considering ime-limiaion of aciviy floa, deerminaion of criical pah, and pre-reques relaions of aciviies maximize NPV deerminaion of criical aciviies and so on. And hen and profiabiliy of a projec. In addiion o main goal, using differen mehods and algorihms (branch and he following goals are achievable oo: bound, abu search, simulaed annealing, innovaive and ) in order o use floas and displacemen capaciies By specifying manner and amoun of cash flow of projec aciviies for maximizing NPV. Generally, he during implemenaion of a projec and final field and range of problem diversiy of PS - MAX - NPV scheduling of projecs, i will be possible o provide and ypes of performed sudies can be summarized as a comprehensive cos planning. By cos planning follows: before saring a projec, i is possible o ge aware of ime and needed capial in differen ime secions. Resoluion of PS-MAX-NPV problems wih regard o By deermining ime and cos planning, i is possible considering or no considering resource resricion. o do an efficien cos conrol during implemening a Resoluion of PS-MAX-NPV problems in differen projec. And in addiion o examine deviaions in cos saes of cash flow (he ype of conrac masery on and income, i is possible o rack he effecs of real projec). progress on NPV and he siuaion of projec in erms Resoluion of PS-MAX-NPV problems in differen of profi or loss in differen ime secions. nework saes (AON-AOA). Corresponding Auhor: Ahmad Ali Bozorgzad, Deparmen of Indusrial Engineering, Azad Universiy, Najafabad, Isfahan, Iran

2 For solving PS-MAX-NPV problems in differen (MIP) and resolved random produced problems by using saes, resource limiaion, he ype of nework used Lindo sofware and argued ha presened model offers and he ype of cash flow, differen soluions have opimal soluion []. Sepil and Orac expanded Kazaz and been offered. For he firs ime, Russell proposed he Sepil model and by considering limiaion of renewable problems of projec ime-budge for maximizing NPV []. resources hey presened an innovaive mehod for The hypoheses of Russell s model prined in he journal resolving PS-MAX NPV problems []. Shub and Egar of managemen science were considered a nework fixed all of heir previous hypoheses and used simulaed AOA ype, fixed cash flow and independen of ime annealing insead of he branch and bound for resolving of evens and aciviies and prerequisie relaions of FS PS - MAX - NPV []. They produced 6 random problems ype wih zero delay raes. Russell proposed a non linear and esed branch and bound algorihm and concluded mahemaical model for maximizing NPV and hen solved ha he qualiy of branch and bound soluions is objecive funcion by using firs senence of Taylor series somewha beer han simulaed annealing (SA). Also (Russell,, ). hey concluded ha compuaion ime comparison o SA Grinold convered Russell s model o a linear mehod has been decreased a grea deal. Egar and Shub mahemaical model and proposed wo soluions for PS- [] expanded Elmaghraby and Herroelen mehod and for MAX-NPV in AOA nework and wihou resource he firs ime considered cash flow as non - increasing limiaion (Grinold,, ). linear. They explained heir algorihm wih an example, Doerch and Paerson proposed a planning bu did no presen any compuaion resuls. mahemaical model wih zero and one in he sae of Vanhoucke e al. [] by considering non - increasing budge limiaion (Doersh,, ). Russell in linear process raised by Egar and Shub () used compleing his previous work and adding resource recursive search algorihm for is resoluion. They esed limiaion o ha model esed six innovaive soluions for heir model by some random problems. PS-MAX-NPV problems in problems and concluded ha no mehod solely can be appropriae for problems Pre-Hypoheses of Model: Considering he fac ha he (Russell, 6, ). Smih-Daniel and Aquilano [] by main goal of his sudy is o use floas and ime deadline coninuing Russell s work and considering he ime of of projec aciviies, in order o maximize he NPV; so acualizing posiive and negaive cash flow, is acualized by considering projec deadline and oher possible in early sage of each aciviy and posiive cash flow resricions ime siuaion of projec aciviies change in solely is acualized a he end of projec and afer such a way ha maximize NPV and as a resul profiabiliy compleing projec. Then, hey wih regard o above of implemening projec reaches maximum. hypoheses and by using an example showed ha ime- In he presened model i has been aemped o budge of all non criical aciviies ha have negaive cash consider pre - hypoheses in such a way ha have flow will maximize NPV in laes possible ime and ime- maximum adapabiliy o realiies. budge aciviies wih posiive cash flow will maximize NPV in earlies possible ime. Elmaghraby and Herrolen [] Cash Flow Mode: Cash inflow and ouflow during presened a mehod known as Elmaghraby inerpolaion implemening projec has been considered separaely. for resolving PS-MAX-NPV problems wih a leas Cash ouflow or a negaive amoun ha is paymens of node and aciviies ha were produced main conracor o sub - conracors is done on nodes randomly.they argued ha produced programs by his afer compleing relaed aciviy. Cash inflow or posiive algorihm were resolved in less han seconds by PC amouns received from employer are acualized in fixed compuers. Sepil and Orac [] showed ha he mehod ime inerval (T, T, T ) for compleing aciviies presened by Elmaghraby and Herroelen only offer one during ha period. The reason for considering posiive beer soluion (parial opimal) and can no always offer cash flow is ha indeed cash inflow or amoun received final opimal soluion. Egar e al. enered non increasing by conracor is no done separaely a saring or ending cash flow in PS - Max - NPV for he firs ime []. They every aciviy separaely, bu is done in cerain siuaions raised dependency of cash flow aciviies o compleing of projec and wih regard o performed work unil he ime ime and argued ha he volume of cash flow of every and in definie ime inervals, for insance monhly or each aciviy changes non- increasing wih regard o is hree monhs. One of he main differences beween our compleion and hen offered proximiy mehod know as algorihm and he previous sudies which have increased simulaed annealing. Kazaz and Sepil presened a efficiency and realism of he model a grea deal is mahemaical mehod called mixed ineger programming considering posiive cash flow as above.

3 NCF i NPV i NCFi NPV i Fig. : Posiive and negaive cash flow In Fig., separae posiive and negaive cash flow has been showed. Numbers on each node represen oal coss of aciviies ended o ha node and relaion shown a he end of T period represens ne calculaion posiive Fig. : Ne cash flow and NPV for node i cash flow in T ime. In he menioned relaion a he end of T period, is Shub and Egar by considering AOA nework expanded projec profi coefficien ha is esimaed experimenally branch and bound algorihm and like Vanhouke e al. on he basis of previous similar work or is deermined by esed heir algorihm by random problems and presened employer on he basis of conrac beween employer and heir resuls. The proposed algorihm in his sudy is conracor. shows ha conracor for undergoing cos derived from recursive search algorihm, bu major C will receive amoun of C profi. differences in he pre-hypoheses, differen ours from Vanhouke model. By comparing ne funcion of cash flow Type of Nework: By considering he srucure of cash in Egar and Shub wih ne funcion of cash flow in he flow and in order o simplify he model and o coordinae proposed model (Fig. ) and also he ype of proposed nework wih using algorihm ha is a recursive search nework in wo neworks, AOA has been used; I can be algorihm, AOA neworks has been applied. In his seen ha he presened innovaive algorihm is a nework, wih regard o cash flow mode, every node has combinaion of branch and bound mehod and recursive a negaive cash flow (ouflow) ha represens oal coss search of Vanhouke e al. of implemening aciviies o ha node. The proposed recursive search algorihm in his sudy hereafer is called "sep by sep recursive search Type of Cash Flow: In he presened model wha is algorihm" due o is specific orienaion. imporan is considering projec deadline or in he In order o simplify work and beer undersanding oher words, ime of acualizing end node. Also ype of of mehod he hypoheses of acualizing posiive cash cash flow and performing aciviies one by one are very flow in fixed period is omied and i is supposed ha imporan. ha is, fixed cash flow ype. posiive cash flow similar o negaive cash flow is acualized on nodes. Hence for every node a cash flow Type of Limiaions: The only limiaion of model is ha can be posiive, negaive or zero is compuable. So limiaion of prerequisie relaion in ype of end o sar his ype of cash flow is viewed posiive. In our mehod (FS) wih delay rae of zero. firsly, projec nework is drawn and he earlies and he laes ime of all nodes are compued. Then, all nodes are The Algorihm: According o wha menioned, he mos scheduled in he earlies ime and iniial NPV is compued complee and he bes mehod in resolving projec and his NPV is considered as low bound. Then search scheduling problems in order o maximize NPV in he sars from end of projec and firs i is examined for every case of non-limiaion of resources are [] and []. node ha wheher his is allowed o change ime or no Vanhouke e al. presened an innovaive algorihm called (because all nodes are scheduled in he earlies ime. By recursive search algorihm for resolving PS-MAX-NPV changing ime nodes we mean movemen of ime node problems and esed heir model by producing random oward projec deadline). Allowable nodes firs, have ime problem in differen saes, nodes numbers, complexiy deadline, second change heir ime because increasing coefficien and projec deadline and showed he resuls. projec NPV.

4 According o Fig., ransfer of ime of every node wih negaive cash flow oward he laes ime of node, NPV increases NPV of node, conrary, change of nodes ime posiive cash flow from he earlies ime cause decreasing NPV of ha node. So, for every node i is examined ha weaher is has negaive or posiive cash flow. If i has.6 6. posiive cash flow is no allowable o change ime and. may cause NPV and projec decreasing. Thus, hese. nodes are relieved and search wihin previous nodes is coninued. Bu if hese nodes have negaive cash flow are allowable o change ime. I is clear from Fig. ha he bes siuaion for his node is scheduling in he laes Fig. : NPV for node i in Example- possible ime. So, his node emporary is scheduled a he laes ime. Because delaying his node may cause delaying pos requisie nodes, ha par of nework afer expeced node is rescheduled and new NPV is compued on his basis. If compued NPV is larger han low bound of problem, his scheduling is acceped and search wih previous nodes is coninued. If NPV has no been beer, scheduling of nework is convered o previous sae (because ransfer forward of one or several nodes wih posiive NPV has been considered due o ransfer Fig. : NPV for node i when [Ei, Li] =[, ] forward). In he nex sep, by considering he fac ha every generalizaion o his sae and an appropriae mehod is ransfer o forward of his node wih negaive cash flow needed for selecing hese ypes of nodes. The following cause increasing NPV, minimum inerval of his node ha example shows how o selec allowable nodes o change is scheduling of i, if his rae is larger han zero, his ime and compuaion of opimal value of his change ime. node in ime [minimum inerval of node wih laer nodes + Example. Consider node i wih he earlies ime of and he earlies ime of node], is scheduled and new NPV is he laes ime of uni and cash ouflow of -. compued, by making sure of he fac ha new NPV is Suppose acualizing posiive cash flow, ineres rae and relaed o low bound of problem i is sored as low bound coefficien profiabiliy for his node are ime uni and search wih previous nodes is coninued. (T=),. and., respecively. Fig. shows NPV for Minimum inerval of every node wih laer node is he his node. period ha can shif he ime of one node forward, wihou According o Fig. opimal sae for maximizing NPV effecing on laer nodes. The rae of his inerval is of his node is in ime or firs receive of posiive cash minimum free floa of all aciviies ha is sared. flow is in inerval beween he earlies and he laes ime Considering concep of minimum inerval of every node of acualizing his node. On he hypoheses hree cases wih laer nodes and considering Fig. one can conclude can be considered for every node, say i, in erms of ime ha shif of one node wih negaive cash flow cerainly acualizaion: will improve NPV of he node projec. Afer clarifying he mehod, he hypohesis of If ineger coefficien like f (f =,,. m, where m is acualizing posiive cash flow in fixed ime period ha he number of aciviies) is found, such ha Ei = ft increases model realism and considered indicaor of (Ei is he earlies ime of node i). Then, opimal ime of model, is added o model. By adding his, selecion of his node is he earlies ime and any ime variaion allowable nodes needs change and modificaion. oward deadline of projec cause decreasing NPV. According o menioned, hose nodes were allowable o If ineger coefficiens like f is found such ha change ime ha had negaive cash flow bu by adding (f=,,, m) Ei <ft Li, Li is he laes ime of node his hypohesis o model cash flow of each node is i, hen according o example, he opimal siuaion calculaed by (-ci+ci(+ )) and i is posiive for all nodes. of his node for maximizing NPV is scheduling in he Thus, idenifying nodes allowable o change ime is no firs ime of receive beween inerval [Ei, Li ]. 6

5 NPV And i is sored as new low bound of he problem and search is coninued wih previous nodes. Bu if here is a need o more ime change from minimum inerval of node i wih laer nodes, emporarily his node is scheduled in ime of firs period of receive and all he oher nodes are scheduled afer his node and NPV of he projec is compued based on he new scheduling and compared o low bound of he problem. In he case of improving, Fig. : NPV for node i when T= and [Ei, Li ] = [, ] obained NPV is sored as low bound and search is coninued wih previous nodes. In he case of no ineger coefficien like f (f=,,, m) can be improving, node in ime [he earlies node ime + minimum found such ha Ei ft Li, hus according o inerval of node wih laer nodes ], is scheduled and Figure he bes siuaion for his node is is resuled NPV wihou comparison o lower bound is scheduling in he laes possible ime. sored as low bound. In he same way search is coninued from one node o las o he firs node. According o wha menioned above, he general According o wha menioned he proposed scheme of our algorihm is as follow: firs nework of algorihm is as follows: projec is drawn and he earlies and he laes ime and negaive and posiive cash flow is compued for every Sep. : The nework of he projec is drawn and by node. Then, all nodes are scheduled in he earlies helping criical pah mehod he earlies and he laes ime possible ime and NPV resuled from his scheduling is of each node is compued. sored as low bound of problem. Afer compuaion of Since he cos of every aciviy is known, hence he iniial NPV, search is sared from he end node o he compue he cash flow of each node and by having firs one. As posiive cash flow of he las node is coefficiens and -ci compue posiive cash flow of each acualized in he ime of is acualizaion, he opimal sae node by ci(+ ). In his sep all nodes are scheduled a of his node is scheduling in he earlies possible ime so, he earlies ime and by considering negaive and posiive he las node is relieved and search is coninued from one cash flow (for each node) and ineres rae and ime period node o las. of posiive cash flow he NPV is compued and In he algorihm for every node i is examined ha considered as low bound of problem. wheher here is a receive ime ha be beween he earlies and laes ime of acualizaion of node or no (wheher Se: i= n- Ei ft Li). If no, hen node i is scheduled in he laes ime and if his ime change is greaer han minimum Sep. : For node I, examine if here is any ineger like f inerval wih laer nodes, hen all of he laer nodes are (f=,.m), such ha Ei ft Li. If so, go o he nex rescheduled and NPV resuled from his rescheduling is sep oherwise coninue from sep 6. compued and compared o low bound of problem. In he case of improving obained NPV is sored as low bound Sep. : Calculae fi such ha Ei fit Li. If here is more and search is coninued wih previous nodes; oherwise han one fi, hen consider he smalles one. Compue his node in ime [he earlies ime of he node + minimum minimum inerval of node i wih laer nodes. inerval wih laer node] is scheduled and new NPV is If fit Ei+mfi, hen consider fit as he ime for compued. This new NPV is sored as low bound. node i and compue new NPV and wihou comparison If a coefficien like f is found (f:,, m) such ha sore i as low bound and go o sep, oherwise go o he Ei ft Li, i is examined ha if scheduling of he node in nex sep. ime of he firs period of receive beween he earlies and he laes ime does no need more ime change from Sep. : Schedule node i in fit (i = fit ) and reschedule all minimum inerval beween he node wih laer node. nodes afer node i. Compue NPV obained from new Hence his node is scheduled in ime of firs receive scheduling. If NPV is beer han low bound accep new beween he earlies and he laes ime of node and new scheduling and sore resuled NPV as low bound and go NPV is compued, wihou comparison wih low bound. o sep, oherwise go o he nex sep.

6 Sep. : Compue minimum inerval of node i wih laer firs and final resuls are shown. Afer providing nodes. If i is greaer han zero coninue he process, sofware program efficiency coefficien and capaciy oherwise go o sep. of he algorihm is examined by several randomly Schedule node i in Ei+ mfi ime, m is he number of produced problems. In order o examine efficiency aciviies and compue new NPV and by ensuring ha coefficien of presened algorihm, sensiiviy rae of wo NPV is less han lower bound and wihou comparison, variable of model indicaor i.e. compuaion ime and sore i as low bound and coninue from sep. NPV improvemen percen as a resul of performing algorihm relaed o changes of four effecive parameers, Sep. 6: Temporarily schedule node i in he laes ime and m (number of aciviies), q (projec deadline) CNC (projec reschedule all nodes afer he node i. If obained NPV is complexiy coefficien) and T (inerval beween receive greaer han low bound accep his scheduling and sore periods) are examined. See Figs.6-. NPV as low bound, oherwise go o nex sep. Fig. 6 shows compuaion ime relaed o increasing aciviies. CPU compuaion ime is also increased Sep. : If minimum inerval of node i wih laer nodes is gradually; his increase of ime afer 6 aciviies has greaer han zero hen se he ime of node i equal o Ei+ more severiy. Increasing ime of compuaion is due o mfi and compue NPV resuled from his ime change and logical aciviies, because by fixing CNC and increasing sore i as low bound, oherwise go o nex sep. number of aciviies, number of nodes increase oo and considering node o node search, by increasing nodes Sep. : Replace i = i-. If i > = go o sep, oherwise compuaion ime is also increased. As i has been accep presened scheduling as opimal scheduling sore increasing ime of compuaion is due o increasing i and end. number of aciviies, our algorihm can resolve projecs As we see he oupu of algorihm is he opimal wih acivaes and complexiy coefficien of in less scheduling of nodes of a projec and value of NPV. Bu, hen second. Anoher parameer ha is sensiiviy has he main goal of his sudy is opimal scheduling of been examined is NPV improvemen percen. According o projec aciviies in order o maximize projec NPV, hence Fig., NPV improvemen percen relaed o number of he algorihm needs a final sep in order o urn nodes aciviies is also mildly increased wih regard o very lile scheduling o scheduling of projec acivaes. increase in NPV percen. One can argue ha NPV Considering definiion of AOA nework and also improvemen percen of projecs is no sensiive o he based on he pre- hypoheses (acualizaion of cash ou number of projec aciviy and uilizing proposed flow in finishing every aciviy and acualizaion of cash algorihm for every projec, regardless of number of in flow afer compleing aciviy and a he end of fixed aciviies is appropriae. According o Fig., wih ime period), he bes ime for doing every aciviy for increasing CNC, compuaion ime is decreased. I is due increasing NPV is is scheduling in he laes possible o he fac ha CNC is raio of number of aciviies o ime. Hence, a sep is added as follows: number of nodes, by fixing number of aciviies and increasing CNC, number of nodes is decreased and by Sep. : Replace j =. decreasing number of nodes, ime consumpion (algorihm.. Place ime of compleing all acivaes ended o is based on node o node search process), is decreased. node j equal o j. Also, according o Fig., by increasing CNC, NPV Se si= fi - di for all aciviies ha was seleced in he improvemen percen is mildly decreased, as menioned, previous sage. by increasing complexiy coefficien and fixing number of If j n go o., oherwise sore scheduling of aciviies, projec nodes relaed o number of aciviies is projec aciviies as opimal scheduling. decreased. Considering he proposed algorihm, by decreasing number of nodes wih respec o number of Tes he Model: The proposed algorihm has been aciviies, number of saes ha can increase NPV, is ++ programmed by C sofware and is performance accuracy decreased and as a resul by increasing CNC, followed by has been verified by previous manual solved problems. decreasing nodes of projec, NPV improvemen percen is The inpus of he program are: program ineres rae, decreased. Figures and show ime change and projec profiabiliy, esimaed cos and ime for every improvemen of NPV wih respec o increasing of aciviy and finally pre-requisie. And wo oupu as he projec deadline. From hes figures we see ha

7 CPU imes (/ s ) The char of changing he CPU ime o increasing in aciviies quaniy aciviies quaniy Fig. 6: Changing he CPU ime wih respec o increasing in aciviies quaniy NPV improvmen precenage 6 The char of NPV improvmen precenage o he change aciviies quani aciviies quaniy Fig. : NPV improvemen percenage wih respec o he change aciviies quaniy CPU ime (/ s). The char of changing he CPU ime o increasing he CNC CNC Fig. : Changing he CPU ime wih respec o increasing he CNC NPV improvmen precenage The char of changing NPV improvmen precenage o increasing CNC CNC Fig. : NPV improvemen percenage o increasing he CNC

8 CPU ime (/ s) The char of changing CPU ime o increasing in projec deadline deadline. Fig. : Changing he CPU ime o increasing in projec deadline NPV improvmen precenage 6 The char of changing NPV improvmen precenage o increasing in projec deadline deadline Fig. : Changing NPV improvemen percenage o increasing he in projec deadline (/ s) CPU ime 6 6. The char of changing CPU ime o increasing in receiving ime periods Receiving ime periods Fig. : Changing he CPU ime o increasing in receiving ime periods NPV improvmen precenage 6 The char of NPV improvmen precenage o increasing in receiving ime period Receiving ime periods Fig. : NPV improvemen percenage o increasing receiving ime periods

9 improvemen of NPV and ime consumpion of he of projec deadline, bu is severely sensiive o change of proposed algorihm are no sensiive o change deadline. inerval of received periods. And by doubling his According o Fig., improvemen of NPV percen is inerval, percen of NPV improvemen is also double. severely sensiive relaed o inerval change, beween Hence, by increasing inerval beween received amouns, received periods. And by increasing his period NPV is effeciveness rae of he algorihm is increased oward almos increased linearly. I should be noed ha NPV improvemen. For example, for a projec wih monhly increasing NPV percen due o increasing T rae does no received period and CNC=, q=, m= and á = % mean ha increasing ime inerval of received period is a (annual), NPV improvemen is.%, bu for his projec benefi for conacor and increase NPV of projec, bu by wih hree monh received inerval his percen is.%. increasing inerval beween received periods early NPV is decreased. Because, conacor should wai longer for REFERENCES receiving spen coss. Hence, increasing T is no benefi for conracor and decreases NPV of projec. The rend of. Doersch, R.H. and J.H. Paerson,. Scheduling a NPV improvemen percen due o increasing inerval projec o maximize is presen value: A zero-one beween received represen increasing effec of using programming approach. Managemen Science, recursive search algorihm in order o maximize NPV and : -. by increasing T he necessiy uilizing his algorihm is. Elmaghraby, S.E. and W. Herroelen,. The increased. Also wih aenion o Fig., one can scheduling of aciviies o maximize he ne presen conclude ha if ime of he projec is greaer, hen value of projec. European Journal of Operaional effeciveness of performing he algorihm o obain more Research, : -. profi will be greaer.. Egar, R., A. Shub and L.J. LeBlanc, 6. Scheduling projecs o maximize ne presen value- CONCLUSION he case of ime-dependen, coningen cash flows. European Journal of Operaional Research, 6: -6. According o obained resuls, one can conclude Egar, R. and A. Shub,. Scheduling projec ha he proposed algorihm has a high efficiency and is aciviies o maximize ne presen value- he case able o solve more han aciviies and complexiy of linear ime-dependen, coningen cash flows. coefficien of in less han seconds. Also he rae of Inernaional Journal of Producion Research, : algorihm effeciveness is ousanding, such ha percen -. of NPV improvemen by using our algorihm for projecs. Grinold, R.C.,. The paymen scheduling problem. wih aciviies, CNC =, á = % and received period Naval Research Logisics Quarerly, : -6. of days is moderaely. %. 6. Herroelen, W. and E. Gallens,. Compuaional By examining sensiiviy analysis i is clear ha he experience wih an opimal procedure for he ime consumpion of he algorihm wih respec o projec scheduling of aciviies o maximize he ne presen deadline and received inerval period is indifference; and value of projecs. European Journal of operaional by generaing more severe changes in hese wo Research, 6: -. parameers, angible change is no produced in he. Kazaz, B. and C.B. Sepil, 6. Projec scheduling compuaion ime. Bu, compuaional ime is sensiive o wih discouned cash flows and progress paymens. number of aciviies and complexiy coefficien of projec Journal of he Operaional Research Sociey, and has direc relaion wih number of aciviies and : 6-. conrariwise wih CNC change (his is due o he naure. Russell, A.H.,. Cash flows in neworks. of node o node search and aciviy o aciviy rend of Managemen Science, 6: -. his mehod). Anoher imporan parameer ha is. Russell, R.A., 6. A comparision of heurisics for sensiiviy o four menioned parameers was examined scheduling projecs wih cash flows and resource is NPV improvemen percen due o using he resricions. Managemen Science, : -. proposed algorihm. According o obained resuls, NPV. Sepil, C. and N. Orac,. Performance of he improvemen percen wih respec o wo parameers, heurisic procedures for consrained projecs wih number of aciviies and complexiy coefficien of projec, progress paymen. Journal of he Operaional has very low sensiiviy and is indifference o parameer Research Sociey, : -.

10 . Shub, A. and R. Egar,. A branch-and bound. Vanhouke, M., E. Demeulemeeser and W. Herrielen, algorihm for scheduling projecs o maximize he. Scheduling projecs wih linearly imene presen value: The case of ime independen, dependen cash flows o maximize he ne presen coningen cash flows. Inernaional Journal of value. Inernaional Journal of Producion Research, Producion Research, : 6-. : 6-.. Smih-Daniels, D.E. and N.J. Aquilano,. Using a lae-sar resource-consrained projec schedule o improve projec ne presen value. Decision Sciences, : 6-6. Appendix : In his appendix we give flowchar of he proposed algorihm. Parameers are: A: ineres rae N: number of nodes n: projec deadline fii: ne inflow of node i pi: pre-requisie node of node i h: projec profiabiliy coefficien m: number of aciviies T: recived inerval foi: ne ouflow of node i di: duraion of aciviy i lb: lower bound sra i =f i T f i T<=E i +mf i?, a, n, m,t,d, -fo i? p i? d i Compue new NPV fi i = fo i (+?) Change : Call CPM funcion and compue E i and L i for each node. i = E i Call NVP funcion and compue iniial NVP. lb = NPV and i = n- i =f i T i >= Call CPM funcion and reschedule all nodes afer node i Compue New NVP Compue mf i. NPV > lb Is here any ineger, say f, such ha E i <=f i T<=L i lb = NPV i = i- mf i > i = E i +mf i

11 i = L i j = Call CPM f uncion and reschedule all nodes afer node i Compue New NVP Aciviies eplacemen for which end o node i: F i = i NPV > lb Aciviies replacemen for which end o node i : S i = F i d i mf i > j = j+ i = E i +mf i j<=n Compue New NVP END lb = NPV i = i-