Estimating projects duration in uncertain environments: Monte Carlo simulations strike back

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1 Estmatng proects duraton n uncertan envronments: Monte Carlo smulatons strke back Stefana Tatton 1, Massmlano M. Schrald ( Tor Vergata Unversty of Rome, Dept. of Enterprse Engneerng, Va del Poltecnco, Rome, Italy) Lug Laura ( Sapenza Unversty of Rome, Dept. of Computer Scence, Va Arosto, Rome, Italy) Abstract PERT (Program Evaluaton and Revew Technque), developed n the 1950 s, represented the frst attempt to ncorporate uncertanty n proect schedulng. Despte some weaknesses, t s stll wdely used n proect management mostly thanks to the smplcty of ts algorthm n operatng on actvty network dagrams. Today the ncreasng complexty of proects requres new technques and the ncreasng avalablty of computer power have not brought proect smulaton nto common usage as expected. Although several revews assert that smulatve approach has already superseded PERT when copng wth uncertan envronment; the reason why t s not dffused s that smulatons requre a too long computng tme. In ths paper we show through an algorthm and expermental results that the computatonal tme, hstorcally the maor drawback of Monte Carlo smulatons, s defntely mnmum thanks also to the computatonal power avalable nowadays. We present results of an effcent program made of few lnes of code and able to compute the completon tme of a network actvty dagram wth actvtes and about precedence constrants between them. Keywords Stochastc networks duraton, large scale dataset, PERT, Monte Carlo, smulaton, computng tme. 1. Introducton Nowadays the globalzaton and the nnovaton technology have changed the compettve envronment, makng the focus on clent fundamental, renderng the nternal flexblty the most mportant response to the new threats and opportuntes. These reasons have made proect management concepts spread to several ndustres together wth the mportance of multdscplnary. In ths context, where the turbulence and the shortest tme to market rule, manage a proect wthout proect management means to not respect a budget or due dates. Whle on a hand the growng adopton of proect management concepts by ndustres let ncrease the use of analytcal procedures, on the other hand the ncreasng complexty of proects and uncertanty n estmatng actvty duraton, makes dffcult to determne wth accuracy proects completon tme. PERT was the frst technque whch consdered uncertanty n proect networks, although t s currently appled for several knd of proects and t s the most dffused, some weaknesses are hdden n ts assumptons, Mac Crmmon & Ryavec (1964) consdered them on two separate levels; the level of the ndvdual actvtes (the mean, standard devaton, and dstrbuton of each actvty) and the level of the whole network (the calculaton of a network mean and standard devaton). The smulaton technques have proved to be a powerful addton to the proect analyss toolbox, because ther results are more accurate than ones gven by PERT. In lterature we have several revews of smulaton appled to stochastc networks and the most mportant are the followng: Monte Carlo method, PERT-path technque and the Condtonal Monte Carlo. Monte Carlo soluton can be used to solve PERT networks accurately and to gan estmates for quanttes not obtanable from the standard approach thanks to the use of the Central Lmt Theorem. Condtonal Monte Carlo technque add an ntermedate step n the Monte Carlo method whch conssts nto reducng networks to derve an equvalent network whch can be further analyzed by smulaton method. The PERT-path technque, usng the same nformaton of PERT allows to analyze the possble evoluton of the completon tme takng nto account all proects actvtes and not only those belongng to the crtcal path. Even f t has been demonstrated by more accurate estmates of tme completon that these methods are more effectve than PERT, do not exst any new consoldated standard. Am of ths paper wll be to show that a smple and careful mplementaton of Monte Carlo smulaton requres short computng tme. 2. Lterature revew An actvty network s a drected acyclc network where the arc lengths typcally represent tmes to complete tasks that are part of a larger proect. The completon tme of all the actvtes on a -th path s a random varable X, where X. (1) T P 1 Corrspondng author Emal addresses: laura@ds.unroma1.t (Laura L.), schrald@unroma2.t (Schrald M.), stefana.tatton@unroma2.t (Tatton S.) Tor Vergata Unversty of Rome, Dept. of Enterprse Engneerng, Va del Poltecnco, Rome (Italy), ph

2 The network represents the precedence relatons among the tasks so that the length of the longest (crtcal) path through the network s the tme to complete the proect whch can be descrbed also by the am to fnd the c.d.f. (cumulatve dstrbuton functon) of a random varable of the crtcal path, expressed by the equaton: max X. (2) T 1,2,..., m The soluton of ths problem has gven by solvng the followng multvarate ntegral: n F t... df t 1,2,..., m (3) T where x x. t P t 1 Of prmary nterest for researchers has been the estmatons of expected value, of dstrbuton of the proect completon tme and the probablty that a gven arc wll be on the crtcal path (arc crtcalty). In lterature the technques used for evaluatng these estmates followed three dfferent approaches: analytc, approxmated and smulatve. The analytc approach was based on the assumpton that t s easest to use smple forms of df t and to presuppose specal networks n whch (3) can be solved dvdng t n a seres of snglevarate ntegrals. Martn (1965), studed the case of polynomal arc densty functons. Other analytc approaches (Charnes, A., Cooper, W. W., Thompson, G. L., 1964 and Robllard, P., Trahan, M., 1977) focused ther attenton prmarly on the proect completon tme. The approxmated approach, where the duratons of ndvdual actvtes are normally and ndependently dstrbuted to nvolve the manpulaton of dstrbuton parameters (Scull, D., 1983), has the great dsadvantage to not evaluate the level of the estmates approxmatons of the sngle actvtes even f Clark C. E. (1960) has approxmated the functon (2). Van Slyke (1963) was one of the frst researchers to use Monte Carlo smulaton to evaluate the proect completon tme and arc crtcaltes. Smulatve approach s the most dffused n lterature and t s becomng the approved method (at least by researchers) for stochastc network analyss. The PRAM Gude (Smon et al, 1997) says that The PERT technque has been superseded by the more powerful Monte Carlo smulaton modelng supported by computer-based tools, and PERT s no longer consdered to be a sutable rsk analyss technque. Among these dfferent approaches the last one s what gves the best results and on whch the researchers have wrtten more. After straghtforward smulaton used by Van Slyke n 1963, recent research has been focused on mprovng the effcency (both statstcal and computatonal) of the smulaton estmates. Burt and Garman (1971) use the dea of condtonal samplng to reduce statstcal varaton when each path n the network has an arc unque on t. Dodn (1985) explots specal structure to smplfy networks, also Fatem Ghom, S. M. T. and Rabban, M., (2003) consder, from structural vewpont, a mechansm based on graph theory whch changes structure of network to seres parallel network wth contracton of some arcs. Nowadays a new method s gong to be used: the PERT-path technque, Mummolo, (1997) and Potrandolfo (2000), based on the dea of proect evoluton, defned as a sequence of states (path) wth the dates of transton between states and assumng that proect actvtes are characterzed by stochastc ndependence. In every paper based on smulaton approach, t s dscussed the estmate of completon tme and the results are always better than the ones gven by the tradtonal PERT technque; however untl now the weakness of the smulaton appled to the stochastc networks was the computng tme requred to generate random varables and statstcs tests on the networks (Van Slyke, 1963). Ths paper wants to show that smulatve technques can become a standard method used by proect managers and analysts because t s smple to use and requres few seconds. In order to demonstrate ths, results of run, realzed by a program able to do Monte Carlo smulatons on networks wth actvtes about edges, wll be gven. However, there s an mportant flaw whch cannot be consdered n ths work: the actons taken by the proect manager to recover any proect slppage are mssed out of the model entrely (Wllams, T. 2003). 3. Computng the smulatons In ths secton we descrbe a straghtforward mplementaton of the Monte Carlo smulatons; we wll consder an actvty on node (AON) network dagram; each actvty (.e., each node) has an assocated probablty dstrbuton functon. Note that f we want to deal wth an actvty on arc (AOA) network dagram the algorthm proposed, t can be easly adapted. The smulaton can be dvded nto three steps: 1. Computng a Topologcal Order of the Network Dagram. 2. Computng the duraton of each actvty from the assocated probablty dstrbuton functon. 3. Computng the overall completon tme. Before detalng each of the above steps, t s mportant to note that the frst one s a preprocessng step that needs to be computed once for each network dagram, whle the second and thrd steps are the ones to be repeated at wll (n

3 all the experments descrbed n ths paper, unless otherwse stated, we ran smulatons on each network dagram). Step 1: computng a Topologcal Order of the Network Dagram. In Fgure 1 t s possble to see a general drected acyclc graph (a) and ts topologcal order (b): we recall that a topologcal order of a graph s an orderng of the nodes n such a way that no node has an outgong edge to a node that precede t. For more detals on the computaton of a topologcal order see the book of Cormen et al. (2001); f we denote wth N the number of nodes (actvtes), and wth M the number of edges (precedence constrants), the computatonal cost of ths step s O(N+M). Fgure 1: a general drected acyclc graph (a) and ts topologcal order (b) Step 2: computng the duraton of each actvty from the assocated probablty dstrbuton functon. In ths step, for each actvty (node), we generate a random value accordng to the parameters of the assocated probablty dstrbuton functon. We consdered unform, trangular, beta, gaussan and exponental dstrbutons. That t s mportant to separate ths step from the next one n order to explot the localty prncple n the code. The computatonal cost of ths step s O(N (cost of random number generaton)). Step 3: computng the overall completon tme. From the prevous steps we acheved that 1) the nodes are sorted n such a way there s no edge from a node to one of ts predecessors, and 2) we computed the duraton (.e. we generated random values accordng to the assocated dstrbutons) of each actvty. Now, to compute the overall completon tme we smply need to compute, for each node (ordered by the topologcal order), whch node amongst ts predecessors s the latest endng one. The computatonal cost of ths step s O(N+M). In order to show that Monte Carlo smulatons are nowadays a practcal tool to estmate the overall completon tme of a proect, we mplemented the above algorthms. We wrote the program n the Java programmng language, whch has the advantage of beng easly portable between several operatng systems and computer archtectures. We used the Java Statstcal Classes (JSC, freely avalable from the web address to generate random numbers accordng to several probablty dstrbutons. We run all our experment on a normal notebook, an Apple MacBook Pro, wth a 2.4 Ghz Intel Core Duo processor and 4Gb RAM. The operatng system s MAC OS X Leopard (10.5.4). The code we developed s freely avalable, under the GPL lcense, from the web address 4. Expermental results In ths secton we descrbe the expermental results. Snce t s well known that the Monte Carlo smulaton approach outperforms the PERT, here we focus only on the computatonal tme that, hstorcally, snce the work of Van Sylke (1963) has been descrbed as the maor drawback of the Monte Carlo smulatons. More specfcally, we wants to emphasze that, wth the computatonal power avalable nowadays, the Monte Carlo smulatons are defntely a practcal approach: we were able to compute, wth a program made of few lnes of (carefully desgned) code, the latest completon tme of a network actvty dagram wth actvtes (nodes) and about precedence constrants (edges) between them n about half a second on a common notebook; we can run more than 100 teratons per mnute of a such huge network. If we move to more reasonably szed network dagrams, t took 15 seconds to compute dfferent smulatons on a network made of nodes and about edges. We dd not even explot the dual core of the processor (.e. we could halve the computaton tme by smply runnng two parallel experments, one on each core). It s clear that, wth ths computaton tmes, t s not even worth tryng to desgn sophstcated approaches, lke the one proposed by Burt and Garman (1971); note that also the PERT-path technque, of Mummolo (1997) and Potrandolfo (2000), s superseded by Monte Carlo smulatons both n the accuracy of the results and n the computatonal tme needed; furthermore the PERT-path technque scale wth a factor n!, that makes t unfeasble for medum to large network dagrams.

4 Fgure 2: computatonal tme (seconds) versus the number of nodes (x-axs). In Fgure 2 we plot the computatonal tme as a functon of the number of nodes, whch vares from to ; the average outdegree (.e. the number of outgong edges) was 50, and therefore the number of edges ranged from to We recall that each pont represents the tme of teratons on the same network dagram. It s nterestng to notce that the shape of the plot s defntely lnear. Ths s not the case of the plot shown n Fgure 3, where the number of edges ranges from to wth a constant number of nodes (2000); ths s obtaned by varyng the average outdegree (from 10 to 100). Here the computatonal cost s almost constant. Together, these plots show that, n these experments, the domnant cost s the random number generaton, that s lnear n the number of the nodes, and ths explan the lnear shape of Fgure 2 and the (almost) constant aspect of Fgure 3, where the number of nodes s fxed. Fgure 3: computatonal tme (seconds) versus the number of edges (x-axs). 5. Conclusons In ths paper we showed that the Monte Carlo smulaton approach, that s well known to be more accurate than PERT snce the work of Van Sylke (1963), s nowadays no more a computatonally uncomfortable technque: we showed that, wth a smple and carefully wrtten program, we are able to deal wth massve scale networks (100k nodes, 50M edges) n less than a second on a normal notebook, and few seconds are suffcent to smulate 10 thousand tmes a large scale network (1k nodes, 50k edges). Ths confrms that Monte Carlo smulatons now supersede other approaches lke PERT-path technque, ntroduced by Mummolo (1997) and Potrandolfo (2000), and needs no sophstcated mprovements, lke the one proposed by Burt and Garman (1971), to be a practcal and useful tool n proect management.

5 References Burt, J. M., Garman, M. B., (1971), Condtonal Monte Carlo: a smulaton technque for stochastc network analyss, Management Scence, Vol.18, No.8, pp Charnes, A., Cooper, W. W., Thompson, G. L., (1964), "Crtcal Path Analyses Va Chance Constraned and Stochastc Programmng", Operatons Research, Vol.12, pp Clark, C. E., (1961), "The Greatest of a Fnte Set of Random Varables", Operatons Research, Vol.9, No.2, pp Cormen, T. H., Leserson, C.E., Rvest, R.L., and Sten, C., (2001), Introducton to Algorthms (second edton), MIT press. Dodn, B. M., Elmaghraby, S. E., (1985), Approxmatng the crtcalty ndces of the actvtes n the PERT network, Management Scence, Vol.31, pp Fatem Ghom, S. M. T., Rabban, M., (2003), A new structural mechansm for reducblty of stochastc PERT networks, European Journal of Operatonal Research, Vol.145, pp Mac Crmmon, C. A., Ryavec, K. R., (1964), "An Analytcal Study of the PERT Assumptons, Operatons Research, Vol.12, No.1, pp Martn, J. J., (1965), "Dstrbuton of the Tme Through a Drected Acyclc Network", Operatons Research, Vol.13, pp Mummolo, G., (1997), Measurng uncertanty and crtcalty n network plannng by PERT-path technque, Internatonal Journal of Proect Management, Vol.15, No.6, pp Pontrandolfo, P., (2000), Proect duraton n stochastc networks by PERT-path technque, Internatonal Journal of Proect Management, Vol.18, pp Robllard, P., Trahan, M., (1977), The Completon Tme of PERT Networks, Operatons Research, Vol.25, No.1, pp Scull, D., (1983), The Completon Tme of PERT Networks, Journal of Operatonal Research Socety, Vol.34, No.2, pp Van Slyke, R. M., (1963), Monte Carlo Methods and PERT Problem, Operatons Research, Vol.11, No.5, pp Wllams, T., (2003), Why Monte-Carlo Smulatons of Proect Networks are Wrong, Strathclyde Busness School, Scotland, Glasgow.