Singularity functions as new tool for integrated project management

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1 Creatve Constructon Conference 24 Sngularty functons as new tool for ntegrated project management Gunnar LUCKO and Y SU Department of Cvl Engneerng, Catholc Unversty of Amerca, 62 Mchgan Avenue NE, Washngton, DC 264, USA Abstract Constructon managers must actvely plan and control the plethora of quanttatve and qualtatve aspects of ther projects n a coordnated, judcous manner to acheve successful performance. However, current manageral technques are compartmentalzed nto modelng and analyzng tme, budget, resource utlzaton, equpment operatons, and others. Therefore, the objectve of ths research s to explore how such very dsparate aspects, here lmted n scope to quanttatve ones, can be ntegrated nto a cohesve and versatle model. Its methodology s to adopt and adapt a tradtonal model from structural engneerng called sngularty functons. Generalzng elements of conventonal algebra, they actvate the non-zero behavor of ther dependent varable over a specfc range of one or several ndependent varables. Thus t s possble to model dfferent relatonshps of project performance. Yet ths accomplshment n analytcal capablty has left unsolved the ssue of nterfaces between sad compartmentalzed aspects. To address t, nteractons between the parwse work quantty, tme, cost, and resource varables are explored and conversons between ther respectve nput and output data are derved. The contrbuton of ths research s threefold: Frst, the mathematcal models of lnear schedules, cash flows, and resources are algned. Second, nteractons between those models are extracted and formalzed based on a common rreversble varable, whch s tme. Thrd, the possblty of ntegratng elements nto a 3D or ndeed nd quanttatve model for project management s dscussed. In concluson, such ntegrated model based on sngularty functons opens new possbltes for holstc plannng and management of constructon projects. Lnkng ts varous quanttatve elements thus enables a customzable and easly adoptable approach and facltates comprehensve analyss and mult-objectve optmzaton toward mnmum duraton and cost and maxmum resource utlzaton. Keywords: Cash flows; ntegrated project management; quanttatve performance measures; schedulng; sngularty functons.. Introducton Constructon project management s a complex system, as t s drven by multple objectves; [t]hese objectves and ther relatve mportance vary from one project to another, and they often nclude mnmzng constructon tme and cost whle maxmzng safety, qualty, and sustanablty (Kandl et al. 2, p. 7). The term objectve can also be used nterchangeably to the term dmenson, as tradtonal project management s commonly defned as a three-dmensonal (3D) cost-schedule-techncal system (Gransberg et al. 23). To handle the characterstcs of such a complex system, prevous studes dvded constructon projects nto varous subsystems. They typcally focused on one compartmentalzed subsystem as ther research purpose, whle smplfyng or even omttng nterfaces to other subsystems. For example, Gantt bar charts are a graphcal schedulng method that s orented exclusvely toward the tme dmenson, gnorng others such as work quantty, cost, and resources. The well-known Crtcal Path Method (CPM) was developed for the requrement that [t]he plan should pont drectly to the dffcult and sgnfcant actvtes the problems of achevng the objectve (Kelley and Walker 959, p. 6). CPM may be descrbed as a one-and-a-half dmensonal method, because t prmarly apples an algorthm to schedule tme, whle a separate later tme-cost tradeoff analyss could consder drect cost, but not ndrect cost (Kelley and Walker 959). More recently, addtonal dmensons (often three) are brought nto mult-objectve models. They were solved usng a varety of methods, ncludng lnear programmng, nteger programmng, dynamc programmng, and genetc algorthms (El-Rayes and Kandl 25, p. 477). However, CPM stll domnates as ther lmtng foundaton, to the detrment of other dmensons. Yet constructon projects are ntegrated systems that unfold n a complex nterplay subject to a plethora of factors, whch requres usng a more ntegrated model to generate realstc analyses and effcent optmzaton. A need therefore exsts to explore novel approaches toward such an ntegrated systems vew of project management. Correspondng author. Tel.: ; fax: E-mal address: lucko@cua.edu 44

2 2. Lterature Revew Tradtonal research selected [the] most mportant objectve whle ether neglectng the less mportant competng objectves or mposng them as known constrants n the optmzaton formulaton (Wu et al. 22, p. 4). Such studes on sngle objectves usually explore a detaled subsystem, e.g. schedulng (Harrs and Ioannou 998) or cash flows (Cu et al. 2). But constructon projects are complex systems and thus t s deplorable that there exsts a plethora of control technques that cannot provde any nsght nto the nteractons among the many components of a constructon engneerng project (Kalu 99, p. 494). To manage mult-objectve problems, studes seek to create tools and strateges that can smultaneously mprove project performance n multple dmensons for ntegrated systems (Ford 22, p. 3). Yet [m]ultobjectve optmzaton formulatons have clear theoretcal advantages but ncrease the complexty of the mathematcal formulaton (Wu et al. 22, p. 4). In most cases, researchers make approxmatons to smplfy the problem. Ammar (2, p. 67) for the example of a tme-cost tradeoff optmzaton explaned that a [d]scount factor n the exponental form, s too complcated to be handled n a mathematcal optmzaton model Instead, a smplfed form wll be used. Such approach s common n mult-objectve research for reasons of smplcty, manageablty, and brevty of a model and ts descrpton, but clearly undesrable. Furthermore, such studes typcally follow very smlar steps: Frst, selectng objectves from project performance parameters to be mnmzed or maxmzed, e.g. tme, cost, or resources. Second, usng a multobjectve optmzaton algorthm. However, an mportant ntermedate step s often short-changed, that of creatng a model whose nature s deally suted to ts challenge. It logcally occurs between establshng the objectves and performng an optmzaton and s crucal for an effcent and relable optmzaton. Models must be versatle yet accurate to the maxmum extent that nput data allow, wthout mposng extraneous restrctons from modelng assumptons. Prevous studes have focused extensvely on optmzaton algorthms (Geem 2), but appear to overlook ths modelng challenge. 2.. Research Need and Objectves Abrdged objectve functons to mnmze or maxmze dependent varables radcally smplfy realty: Duraton s determned by factors such as productvty, crew sze, resource avalablty, shfts, lead/lag duratons, buffers, etc.; mult-objectve studes omt most such detals. Cash flows must consder drect and ndrect cost, bll-to-pay delay, prompt payment dscount, credt lmt, nterest, tme value of money, etc. Objectve functons typcally smplfy these detals, focusng nstead on algorthms to dentfy or compare solutons. One may argue that t s overly complcated to mantan the same level of detal as local subsystems when movng to modelng a global ntegrated system. But per the paradgm of Ockham s razor, who advocated as smple as possble, as complcated as necessary for models, one should not reduce realsm f a system becomes more complex, whch lmts the valdty of ts output and may mslead decson-makers. Ths research thus rases the fundamental queston of how to brdge local and global vews, whle remanng effcent and accurate as determned by the qualty of avalable data, not model assumptons. Sngularty functons, defned n the followng secton, offer the unque features of detalablty (can reflect any desred level of detal wthn ther mathematcal expressons), extensblty (can ncorporate any number of nteractng dmensonal varables), and convertblty (can extract parwse performance parameters to examne ther relatonshps n detal). Three sequental Research Objectves wll be addressed by ths research, whch together contrbute to ts overall goal of ultmately ganng a sngle comprehensve yet customzable approach to constructon project management: Explorng nteractons and conversons among sngularty functons for mergng subsystems nto a global model; Algnng subsystem models of schedule, cash flows, and resources n cumulatve and non-cumulatve expressons; Vsualzng the ntegrated 3D project model and assessng ts potental contrbuton as a novel management tool. 3. Sngularty Functons Sngularty functons were ntally used n structural engneerng to analyze the problem of beams that are loaded wth dverse types of loads, whch are located at dscrete or dstrbuted locatons on sad beams. The basc term of Equaton was hstorcally known as the Föppl symbol (927) or Macaulay brackets (99). Ths operator performs a case dstncton for the gven cutoff value a of the ndependent varable x: The functons yelds zero f x s smaller than a, whle evaluatng the ponted brackets as round ones f x s equal to or larger 45

3 than a. The strength s amplfes the value of the functon, whle the exponent n modfes the behavor to lnear or nonlnear growth of the functon. The vrtue of sngularty functons s that they can easly model nfluences that are located at or dstrbuted along a dmenson (e.g. tme), whch fulflls requrements of varous key applcatons n the constructon management feld. For example, n lnear schedules, actvty progress s represented as a curve wth start and fnsh date, whose slope s the productvty. Moreover, for cash flows, a cost profle also has a start and fnsh; ts slope s the rate at whch the cost grows. Furthermore, for resources utlzaton, a profle can be derved analogously. Of course, multple effects of the same type can be expressed jontly, whch generates a staggered profle over ts respectve tme perod. All of such phenomena can be modeled wth the basc term of sngularty functons by nsertng the respectve approprate ndependent and dependent varables per Table. Ths feature enables the converson per Research Objectve : Introducng a new varable, e.g. a cost factor for work unts nto a lnear schedule, transforms the model as desred. Table. Varables for sngularty functons. s x a = s n ( x a) n for x < a for x a () Type of problem Independent Dependent Lnear schedule x = work unt y = tme unt Cash flows y = tme unt z = cost unt Resource utlzaton y = tme unt r = resource unt 3.. Lnear Schedule Model (cumulatve and non-cumulatve forms) Most constructon projects, whether ppelnes or mult-floor buldngs, contan many repeatng actvtes (Harrs and Ioannou 998). CPM can only guarantee that sequental relatons between those repeatng actvtes are obeyed, but cannot drectly represent constrants for resource contnuty (Harrs and Ioannou 998), nor explot the repettve nature for the model tself. On the other hand, the Lnear Schedulng Method (LSM) and closely related approaches wth smlar names, e.g. repettve or locaton-based schedulng, whch analyze projects that are characterzed as geometrcally lnear or repettve n ther operatons (Lucko 28, p. 7), are able to take advantage of prevalng repettveness to derve a more ntutve model. It was wdely consdered to be merely graphcally-based schedulng that s presented graphcally as an X-Y plot where one axs represents [repeatng] unts, and the other tme (Harrs and Ioannou 998, p. 27). Such graphcal, non-mathematcal focus was a severe lmtaton of LSM and has hndered ts computerzaton and broader applcaton n the constructon ndustry. However, that stuaton has changed through the ntroducton of the Productvty Schedulng Method (PSM) that s based on sngularty functons. They provde a flexble and powerful mathematcal model for constructon actvtes and ther buffers that are characterzed by ther lnear or repettve nature (Lucko 28, p. 7). Yet the fundamental mathematcal connecton between CPM and LSM remans largely unexplored, and once understood could facltate broadenng the capabltes of PSM. Lucko (29) descrbed the steps of PSM as actvty and buffer equatons, ntal stackng, mnmum dfferences and dfferentaton for consoldaton, whch generates revsed actvty and buffer equatons, and crtcalty analyss. Detals of these steps are omtted n ths paper for brevty. Stackng generates a feasble but extremely conservatve and thus lengthy schedule, consoldaton overlaps as many actvtes as possble wthn the sequence constrants. The approach of PSM s to convert all nputs about actvty sequence, duratons, leads or lags between actvtes, and any tme or work break nto the mathematcal model that s composed of sngularty functons. The schedule (start and fnsh of each actvty n the ntal and fnal versons) and ts crtcalty characterstcs are outputs. Tme (y) s the ndependent varable n PSM per Equaton 2, x s work, U s the number of repeatng work unts, D s the duraton of actvty, a S and a F are ts start and fnsh, whose astersk ndcates that they may nclude a shfted start (pror delay d ) or extended duraton (new delay d 2 ). Note that tme s better measured on the y-axs, because t should be mnmzed by the algorthm. An analogous form wth tme as the dependent varable can be derved by swtchng the axes. An example wth actvtes A, B, and C per Table 2 s shown n the x(y)-orented lnear schedule of Fgure. Fgure shows the schedule n a cumulatve form, whch means that the work quantty of each actvty (vertcal axs) ncreases wth tme that passes (horzontal axs). Accordng to calculus, calculatng the dervatve of a functon yelds ts rate of change (Strang 2). Thus dfferentatng the cumulatve Equaton 2 returns the non-cumulatve Equaton 3. Fgure 2 shows ths non-cumulatve form. Interestngly, the profle of noncumulatve actvty equatons s smlar to a tradtonal bar chart. It dffers from a bar chart only n that the vertcal axs represents productvty and not just an actvty name or label. It thus dsplays more nformaton than a bar chart. The actvty wth the hghest productvty s placed at the top, and the smaller the productvty the lower the heght where actvty bars are placed. 46

4 U ( ) x y = y a y a ( ) y a (2) ( y) 2 S F 2 F U x = y as y af (3) 2 Name [-] Successor [-] Table 2. Schedule, cash flow, and resource nput for example. Duraton Work Unt Tme Buffer Shft d Delay d 2 Cost C Markup M Bll-to-Pay Delay b Resource Rate [months] [count] [months] [months] [months] [$] [% of cost] [months] [workers] A B B C C Work [quantty] Productvty [quantty/months] x (y) A x(y) A x(y) B x(y) C Tme [months] x (y) B x (y) C Tme [months] Fgure. Cumulatve lnear schedule for example. Fgure 2. Non-cumulatve lnear schedule for example. Overall, PSM wth ts systematc applcaton of sngularty functons provdes a mathematcal model that unfes compartmentalzed concepts and project performance parameters, as used e.g. n mnmzng the project duraton and determnng the crtcalty of actvtes under CPM, dsplayng graphcally ther start and fnsh dates as n bar charts, and representng the starts, fnshes, sequence, and productvty of actvtes as under LSM. The cumulatve and non-cumulatve forms of the equatons enhance schedulng research n that t becomes apparent that concepts whose relaton was prevously arduous to express can ndeed be explctly modeled n such ntegrated quanttatve manner Cash Flow Model (cumulatve and non-cumulatve forms) The success or falure of constructon projects strongly depends on cash flow management. Therefore, modelng cash flows s a crucal problem n constructon management. However, t s a thorny problem, because the nteracton of cash outflows and nflows generates a zgzag-shaped balance that used to defy modelng attempts untl recently (Lucko 2a). Moreover, some phenomena related to cash flows exhbt a dstnct perodcty (Su and Lucko 23), whch should be modeled. Furthermore, Tme Value of Money (TVM) can be consdered explctly (Lucko 23). Expandng the example from the prevous secton, assume that cost for each actvty grows lnearly (Elazoun and Metwally 25). Table 2 lsts the cost (C), markup (M), and bll-to-pay-delay (b) for each actvty. Whereas the slope represented productvty n the lnear schedule, the scale factor (C / ( 2 )) n the sngularty functon per Equaton 4 represents the rate of cost growth, whch s the slope of the cost profle. Addng the markup to the cost yelds a bll functon per Equaton 5. As blls are sent to the payer at the end of each perod, an operator that rounds down the operand to ts nearest nteger s appled to the ndependent varable y. Such roundng operator can easly express the perodcty [that] occurs both n cash outflows and nflows that have a specfc frequency and ampltude, e.g., overhead, bllng, and payment functons (Lucko 2a, p. 528). The pay functon per Equaton 6 s derved from the bll functon, but subtracts the bll-to-pay-delay b from each y, whch has the effect of movng the bll profle to the rght to become the pay profle per Fgure 3. Note that the cost, bll, pay, and balance profles n Fgure 3 model the entre project, addng the cash flows of the three actvtes accordngly. The balance s the dfference between the sum of the cost functons (outflows) and the sum of all pay functons (nflows) per Equaton 7. In a real project, b wll be approxmately 3 to 9 days; here t s assumed as one month. It s assumed that the balance functon does not consder TVM. Its value depends on the perod over whch t s assessed, e.g. for fnancng nterest or an unused credt fee, as analyzed n detal (Lucko 2a, Lucko 23, Su and Lucko 23), but s excluded here for brevty. 47

5 Prevous cash flow models are cumulatve, yet the non-cumulatve form of such cash flow models can also exst per Fgure 4. Dfferentatng Equaton 4 yelds the non-cumulatve cost functon per Equaton 8. A cumulatve pay functon has a stepped profle per Fgure 3. Dfferentatng t would generate a bar chart-lke profle, whch would be ncorrect, because the non-cumulatve pay functon should only be actve at each pay. Su and Lucko (23) solved ths problem by ntroducng customzed pay and sgnal functons for a noncumulatve bll per Equatons 9 and. C ( ) z y = y a y a (4) ( y) cost 2 S F ( + M ) C z bll = y as y af (5) ( ) ( ) z C + M pay y = y b as y b af D + d2 (6) z ( ) ( ) ( ) balance y z pay y z cost y C ( ) = z y y a y a C ( + M ) (8) z ( y) z cost 2 S F each _ pay 2 = (7) ( ) = pay _ sgnal y ( ) ( ) ( + ) ( + + ) ( + + y = y a + b y a b y a b y a b ) z pay _ sgnal s F s F () 2 (9) Cost [$,] z bll (y) z pay (y) z cost (y) Cost Rate [$, / month] z cost (y) z each_pay (y) Tme [months] Tme [months] z bal (y) z bal (y) Fgure 3. Cumulatve cash flow profle for example. Fgure 4. Non-cumulatve cash flow profle for example Resource Model (cumulatve and non-cumulatve forms) The same approach as descrbed for lnear schedulng and cash flows can also be adopted for resource utlzaton. Its scale factor per Equaton represents the rate of resource consumpton r (Lucko 2b), typcally for general or specalzed labor. Contnung the prevous example, Table 2 lsts the resource rate for each actvty. Equatons and 2 express the cumulatve and non-cumulatve resource functons, respectvely, whch are shown n Fgures 5 and 6. Algnng these subsystems n ther cumulatve and non-cumulatve forms fulflls Research Objectve 2. Note that addtonal refnements can be added to reflect the aforementoned factors such as shfts d (that move the start) and delays d 2 (that move the fnsh,.e. expand duraton) (Lucko 28) to enhance ts realsm. Ths can be accomplshed by addng them to the cutoff of the sngularty functon. Equaton 3 lnks subsystems va the scalng factors of ther parwse varables for proportonalty, notwthstandng further the movng and roundng the cutoff n bll equatons. Resources [workers] r(y) tot Resource Rate [workers / month] r(y) tot r(y) B r(y) B r(y) A r(y) C Tme [months] r(y) A r(y) C Tme [months] Fgure 5. Cumulatve resource profle for example. Fgure 6. Non-cumulatve resource profle for example. 48

6 ( ) r y = s y a y a ( + d ) y a () ( y) f ( x y, z) S F 3 2 F C C U, = z = x, z = y, x = y (3) U D + d2 D + d2 4. Integrated Model for Mult Objectve Research and Concluson r = s y as y af (2) Usng ths approach, local models are ntegrated nto a global one wthout dlutng ts level of detal. Snce tme s an rreversble dmenson, other project performance parameters are expressed based on t by default. Fgure 7 projects the 3D curve per Equaton 3 onto the planes. It creates famlar 2D profles of lnear schedule and cash flow n top and sde vew. Project performance parameters of work quantty, cost, and resources can have cumulatve or noncumulatve form to deally sut a gven purpose of analyss or optmzaton, whch thus fulflls Research Objectve 3. Sngularty functons even enable nd models; offerng a new avenue toward the goal of holstc quanttatve project management for future research work Tme 3 [months] Cost [$] 2 Top vew Work [quantty] Fgure 7. Cumulatve 3D profle for example. Sde vew The support of the Natonal Scence Foundaton (Grant CMMI ) for portons of the work presented here s gratefully acknowledged. Any opnons, fndngs, and conclusons or recommendatons expressed n ths materal are those of the author and do not necessarly represent the vews of the Natonal Scence Foundaton. References Ammar, M. A. (2). Optmzaton of project tme-cost trade-off problem wth dscounted cash flows. J. Con. Eng. Mgmt., 37(), Cu, Q., Hastak, M., Halpn, D. W. (2). Systems analyss of project cash flow management strateges. Con. Mgmt. Econ., 28(4), Elazoun, A. M., Metwally, F. G. (25). Fnance-based schedulng: tool to maxmze project proft usng mproved genetc algorthms. J. Con. Eng. Mgmt., 3(4), El-Rayes, K. A., Kandl, A. A. (25). Tme-cost-qualty trade-off analyss for hghway constructon. J. Con. Eng. Mgmt., 3(4), Ford, D. N. (22). Achevng multple project objectves through contngency management. J. Con. Eng. Mgmt., 28(), Föppl, A. O. (927). Vorlesungen über Technsche Mechank. Drtter Band: Festgketslehre. [Lectures on techncal mechancs. Thrd volume: Strength of materals.], th ed., B. G. Teubner, Lepzg, Germany. Geem, Z. W. (2). Multobjectve optmzaton of tme-cost trade-off usng harmony search. J. Con. Eng. Mgmt., 36(6), Gransberg, D. D., Shane, J. S., Strong, K., Lopez del Puerto, C. (23). Project complexty mappng n fve dmensons for complex transportaton projects. J. Mgmt. Eng., 29,(4), Harrs, R. B., Ioannou, P. G. (998). Schedulng projects wth repeatng actvtes. J. Con. Eng. Mgmt., 24(4), Kandl, A. A., El-Rayes, K. A., El-Anwar, O. (2). Optmzaton research: Enhancng the robustness of large-scale multobjectve optmzaton n constructon. J. Con. Eng. Mgmt., 36(), Kalu, T. C. U. 99. New approach to constructon management. J. Con. Eng. Mgmt., 6(3), Kelley, J. E., Walker, M. R. (959). Crtcal path plannng and schedulng. Proc. Eastern Jont Comp. Conf., Boston, MA, December -3, 959, Natonal Jont Computer Commttee, Assocaton for Computng Machnery, New York, NY, 6, Lucko, G. (28). Productvty schedulng method compared to lnear and repettve project schedulng methods. J. Con. Eng. Mgmt., 34(9): Lucko, G. (29). Productvty schedulng method: Lnear schedule analyss wth sngularty functons. J. Con. Eng. Mgmt., 35(4), Lucko, G. (2a). Optmzng cash flows for lnear schedules modeled wth sngularty Functons by smulated annealng. J. Con. Eng. Mgmt., 37(7):

7 Lucko, G. (2b). Integratng effcent resource optmzaton and lnear schedule analyss wth sngularty functons. J. Con. Eng. Mgmt., 37(): Lucko, G. (23). Supportng fnancal decson-makng based on tme value of money wth sngularty functons n cash flow models. Con. Mgmt. Econ., 3(3): Macaulay, W. H. (99). Note on the deflecton of beams. Messenger Math., 48(9), Strang, G. (2). Calculus. 2 nd ed., Wellesley-Cambrdge Press, Wellesley, MA. Su, Y., Lucko, G. (23). Novel use of sngularty functons to model perodc phenomena n cash flow analyss. Proc. Wnter Sm. Conf., Insttute of Electrcal and Electroncs Engneers, Pscataway, NJ, Wu, Z., Flntsch, G., Ferrera, A., de Pcado-Santos, L. (22). Framework for multobjectve optmzaton of physcal hghway assets nvestments. J. Transp. Eng., 38(2),