GENETIC ALGORITHM FOR PROJECT SCHEDULING AND RESOURCE ALLOCATION UNDER UNCERTAINTY

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1 Int. J. Mech. Eng. & Rob. Res. 03 Fady Safwat et al., 03 Research Paper ISS Vol., o. 3, July IJMERR. All Rghts Reserved GEETIC ALGORITHM FOR PROJECT SCHEDULIG AD RESOURCE ALLOCATIO UDER UCERTAITY Ahmed Farouk, haled El lany and Fady Safwat * *Correspondng Author: Fady Safwat, fadysafwat85@gmal.com Ths paper provde a soluton approach for plannng, schedulng and managng project efforts where there s sgnfcant uncertanty n the duraton, resource requrements and outcomes of ndvdual tasks. Our approach yelds a nonlnear (GA) optmzaton model for allocaton of resources and avalable tme to tasks. Ths formulaton represents a sgnfcantly dfferent vew of project plannng from the one mpled by tradtonal project schedulng, and focuses attenton on mportant resource allocaton decsons faced by project managers. The model can be used to maxmze any of several possble performance measures for the project as a whole. We nclude a small computatonal example that focuses on maxmzng the probablty of successful completon of a project whose tasks have uncertan outcomes. The resource allocaton problem formulated here has mportance and drect applcaton to the management of a wde varety of project-structured efforts where there s sgnfcant uncertanty. eywords: Project schedulng, Resource constrants projects, Genetcs algorthms, Uncertanty ITRODUCTIO In Engneerng and busness, projects are subjected to a multtude of uncontrollable factors that affect ther successful completon and are dffcult to be predcted precsely or wth certanty n the plannng phase. Weather varatons, workers productvty varaton, resources avalablty, falures of equpment, customer s acceptance or refusal at dfferent phases of a project, extent of maturty of adopted technology and budget uncertanty are few examples of these uncontrollable factors. Uncertanty manfests tself n the estmaton of actvtes duratons, n estmatng cost, n determnng resource requrements, n the precedence order of dfferent actvtes and even n the outcome of some actvtes: whether these actvtes wll be successfully Faculty of Engneerng, Arab Academy of Scence and Technology, Alexandra, Egypt. Faculty of Engneerng, Modern Scence and Arts Unversty, 6 th October Cty, Egypt.

2 accomplshed or they wll be falures entalng consderaton of dfferent courses of acton and/or rework. Problems of project plannng and management n case of lack of nformaton and uncertanty have attracted the attenton of researchers snce long tme ago. Project Evaluaton and Revew Technque, known as PERT was the frst model (Goldberg, 989) that consders the randomness of actvty duratons. The Threevalued estmaton of actvty duraton (optmstc, most lkely and pessmstc) s one of the powerful features of PERT, snce t s easy to collect them from experenced staffs. PERT evaluates the mean and varance of each actvty duraton n terms of the three values by assumng the Beta dstrbuton as the probablty dstrbuton for all dfferent actvtes. PERT uses the same technque of the Crtcal Path Method CPM (Forward and Backward Passes) n evaluatng the project completon tme, defnng the crtcal path(s) and the floats (total and free) for each actvty, all based on the mean actvty duraton. PERT apples the Central Lmt theorem and consders the project completon tme as a random varable dstrbuted accordng to normal dstrbuton. The probablty of successfully completng a project at that tme defned by PERT or less s nearly 50%, whch means that there s a rsk of falng to complete the project wth nearly 50% probablty. Havng a determnately defned crtcal path(s) contradcts the assumpton of the randomness of actvtes duratons. Ths and other facts, stated later, call for further nvestgatons for buldng models more realstc than PERT (Van Slyke, 963; Hller and Leberman, 995; and Feng et al., 997). It s worthwhle to state here the lmtatons of PERT that motvated the development of other more realstc models: Assumng Beta dstrbuton to model the duraton of all project actvtes wthout any regard to the dfferent natures of dfferent actvtes. Determnaton of the project completon tme usng averages of duratons of actvtes wth no account for ther varances. Gettng determnately defned crtcal path(s) s n contradcton wth the assumpton of randomness of duratons of actvtes. Under Uncertanty any path could be crtcal but of course wth dfferent probablty. The assumpton of ormalty of the project completon tme s approxmately vald only n case of havng large number of actvtes. The normal dstrbuton wth ts unlmted ends s not the proper model. PERT s not capable of modelng uncertantes n the order of precedence and n outcomes of actvtes. Tme-Cost tradeoffs under uncertanty n an attempt to enhance the probablty of project successful completon cannot be performed by usng PERT. The actvtes are assumed ndependent and hence the project varance s taken as the sum of varances of actvtes on the crtcal path. Correlatons between actvtes are gnored n PERT. The precedence relatonshps between actvtes are lmted only by one dscplnethe start to fnsh dscplne. Other precedence relatonshps such as: fnsh to

3 start, start to start and fnsh to fnsh could not be ncluded n PERT model. Buldng mathematcal models that are capable of overcomng all lmtatons of PERT s an extremely dffcult, f not mpossble task. The advent and fast progress of buldng computatonal models enables researchers to ntroduce Smulaton Modelng as more relable models to tackle such problems. Monte Carlo Smulaton was frst ntroduced snce sxtes of the last century (Van Slyke, 963). Monte Carlo Smulaton enabled planners to use dfferent probablty dstrbutons for duratons of dfferent actvtes, to ntroduce what s called crtcalty ndex for dfferent paths and dfferent actvtes and to ft probablty dstrbutons to project completon tme based on statstcs obtaned from several smulaton runs. Monte Carlo Smulaton has the common drawback of all smulaton technques,.e., the statstcal nature of the results and moreover t s not capable to perform constraned resource-based schedulng and tme-cost optmzaton analyss. Dscrete-event Smulaton (Hong et al., 004) s more powerful technque and may allow the use of other models of uncertantes such as fuzzy models of actvty duraton. Because of non-lnearty, commonly notced, n Tme-Cost Optmzaton (TCO) and resource allocaton problems, lnear programmng cannot be used. Recently, Genetc Algorthms (GA) as search engnes are wdely used n solvng TCO problems, constraned resource allocaton and resource levelng problems Goldberg (989), Feng et al. (997) and Tarek (999). The present work s manly concerned wth the evaluaton of the probablty of success of projects under uncertanty. Ths problem was consdered before by Turnqust and Lnda (00). They proposed Webull dstrbuton to model actvty duraton because of the unversalty of ths dstrbuton. Also n Turnqust and Lnda (00), a modfcaton was ntroduced n Webull dstrbuton n order to account for the effect of ncreasng resources levels on the actvty duraton. The Modfed Webull dstrbuton takes the followng form Turnqust and Lnda (00): f d E d d E mn d d exp E mn...() and E are the two newly ntroduced parameters that defnng resource multpler and the elastcty of the actvty duraton to be compressed as more resources are appled to t respectvely. The effect of allocatng more resources ( tmes the normal resource level) to actvtes s shown n Fgure. Fgure : Webull Densty Functon f(d) wth Dfferent Resource Multpler The concept of elastcty (E < = ) as a measure of the capablty to compress the actvty duraton by allocatng more resources of a certan type s clearly llustrated n Fgure. 3

4 Fgure : Webull Densty Functon f(d) wth Two Dfferent Values of Elastcty Under the Same Resource Multpler : Resultant resource multpler appled to th actvty L q : Upper lmt of resource type q multpler to be appled to actvty m : Most lkely estmaton of duraton of th actvty : umber of actvtes pred j : Set of predecessors to j th actvty Q: umber of resources types R qu : Avalable amount of q th resource n the u th perod Actvtes wth zero elastcty to a specfed resource would not respond to any ncrease n level of that resource. Ths could be explaned by unavalablty of space needed by more resource to operate such as for example weldng n confned spaces n shpbuldng. It should be emphaszed that an actvty could have dfferent elastcty s wth dfferent types of resources. Later, ths pont wll be pursued analytcally. PROBLEM FORMULATIO otatons and Symbols a : Optmstc evaluaton of duraton of th actvty b : Pessmstc estmaton of duraton of th actvty d : Duraton of th actvty d o : Mnmum duraton of th actvty E q : Elastcty of th actvty to the applcaton of resource q q : Resource multpler appled to th actvty S : Start of th actvty T: Project completon tme U q : umber of contguous perods of resource q avalablty, : Webull parameters of duraton of th actvty uq : Start of u th perod of q th resource Gven a project wth actvtes ( =,,, ). The precedence order of the actvtes s gven. Actvty duraton d s random and dstrbuted accordng to Modfed Webull dstrbuton gven n (). There are Q types of resources (manpower, cash, equpment,, etc.) avalable to fnsh the project n a predefned completon tme T. Resources avalablty along the project tme s lmted by the allowable resource levels at dfferent perods. The length of these perods and assocated wth them allowable levels of resources dffer from resource to another. Therefore, the project tme s subdvded nto a number of equal or non-equal contguous perods (u q =,,, U q ) as regards to each resource type q. It s requred to determne the 4

5 tme wndow d granted to complete each actvty amng at maxmzng the probablty that the project could be completed n tme T or to mnmze the probablty of falure to complete the project n tme T and to determne start tmes of each actvty S and resource multplers q so that the consumpton of resources of dfferent types wll not exceed the allocated amounts of these resources. Decson Varables As already dscussed, decson varables to be determned n the formulated problem are: d : Allowable tme wndow to complete works on th actvty S : Start tme of th actvty q : q th Resource multpler representng the ntensty of resource allocaton necessary and suffcent to complete works on th actvty n the predetermned tme wndow d wth maxmum possble probablty of success. The resultant resources multpler s proposed, n the present work, to be the geometrc mean of all multplers q as follows: Q q Eq q Q...() Objectve Functon The Success Probablty PAS (d, ) of completng the th actvty n tme wndow d can be evaluated as follows: PAS PAS d d, f t, d 0 dt d d o, exp...(3) Snce all actvtes of the project should be completed successfully n order to succeed to fnsh the project n the target tme T, then the probablty of the project success PPS(T) s obtaned as the product of probabltes of success of all actvtes: PPS PAS d,...(4) It should be noted here that f there could be optonal precedence order as n case of GERT networks, the probablty of project success wll be determned n a dfferent way takng nto account the dfferent optons of project completon paths. Constrants There are three types of constrants mposed on the decson varables: Completon Constrant Let( + ) th a dummy actvty wth zero duraton and zero resource requrement to act as an end actvty, then S + < = T...(5) Precedence Order Constrants S j S d pred j ( =,,..., ) (j =, 3,..., + )...(6) Resources Avalablty Constrants Resources of dfferent types are necessary to complete project works n the predefned project tme T. As already stated before, resource of type q s to be made avalable wth predefned quantty R qu n the u th perod. The sum of quanttes of the q th resource requred by all actvtes n perod u should be equal to or less than R qu. In order to calculate the 5

6 amount of consumed resources n dfferent perods, a resource hstogram should be frstly constructed for each resource type. The level of resource q, denoted by r q s taken constant along the extenson of the th actvty duraton d,.e., the resource q s assumed to be unformly consumed by the th actvty. A resource hstogram s an arrangement of rectangles wth heghts r q and wth wdths d for all actvtes preservng the precedence order of the actvtes. A typcal resource hstogram s depcted n Fgure 3 subdvded as regards to the avalablty of q th resource nto four equal contguous ntervals. The resource level r q can be evaluated as follows: rq q Aq d S t S d r q = 0 Otherwse...(7) where, A q s the nomnal requrement of th actvty from resource type q. For example how many man days nomnally requred to complete th actvty? It s clear from Fgure 3 that wthn the lmts of a perod u, project actvtes may be classfed nto three categores as regards to Fgure 3: Resource q Hstogram ther contrbuton n the resource demand durng perod u. Category : Actvtes completely embedded nsde the perod,.e., ther starts s equal to or larger than perod start uq and ther fnsh s equal to or less than perod fnsh uq +. Ths category contrbutes wth the full value of the requred resource q A q. Category : Actvtes partally le nsde the perod u S q uq S d uq S and u S d u u q q Ths category contrbutes wth a part of the value of the requred resource, uq q A q 0 u q Category 3: Actvtes le completely outsde the perod, S Or S uq d u q 0 u q q Based on the above classfcaton, the resources constrants may be expressed n the followng form: uq q A q R qu 0 u...(8) q It should be noted that there s a gven upper lmt L q for the resource multpler q. 6

7 q < = L q...(9) u q s a newly factor ntroduced n our work. It wll be called Contrbuton factor of th actvty n the u th perod. Factors u are not amenable q to smple computatons because of the step functons expressng resources dstrbutons as seen n Fgure 3. In an attempt to crcumvent these dffcultes n Turnqust and Lnda (00), authors proposed a sophstcated approach to compute factors uq. In ths approach, the step functons expressng the dstrbuton of resources are converted nto contnuous dfferentable functons of tme wrtten here n a smpler expresson than that n the referred work as follows: r q q A d t S wd q tan h tan h w...(0) As we sad; the constant w s ntroduced n order to counteract the effect of the two sngular ponts at t = S and t = S + d and render the varaton of functons r q gradual rather than abrupt at these ponts. Fgures 4a and 4b llustrate the effect of the constant w. Fgure 4a: w = 0. Fgure 4b: w = 0.0 In Fgure 4, two plots of the functon 0.5 tan h tanh w for two values of w(w = 0. and w = 0.0) (S = 0, d = 0). It s notced that the functon 0.5 tan h tanh w has the value of unty as the tme t beng nsde the range s < = t < = s + d whle t drops to zero outsde ths range. The constant w determnes the nature of the change of the functon at the start and fnsh of the actvty. As w decreases (w < 0.03), the change tends to be rather sharp (4b) whle, for bgger values of w the change of the functon at start and fnsh of the actvty s rather gradual (4a). Ths explans the role of the constant w n formula (0). As resource consumpton rate r q s already expressed n the form of a contnuous dfferentable functon, the consumpton of resource q durng the u th nterval can be evaluated by ntegratng r q on tme over the u th nterval and then summng up for all project actvtes. Proceedng n ths way, the resource constrant n (8) wll take the form: t uq Aq d uq t S t S d tan h tan h dt R wd wd qu...() 7

8 u q, uq are the tmes of the start and end of u th nterval of resource q Evaluate the ntegral I n (), cos hv coshv I wd Ln coshv coshv where, v v v v 3 4 uq S wd uq uq wd uq S S d wd wd S d 3 4 Substtutng n () we fnd, 3 q Aq Ln cos hv cos hv 4 w cos hv cos hv R qu...() Comparng (8) and () we fnd for factors u q measurng the contrbuton of the actvty n the consumpton of the resources n the nterval U: uq w cos hv coshv Ln cos hv coshv 0 u...(3) q It should be noted that, rrespectve of the attractveness of the above approach proposed n Turnqust and Lnda (00), t suffers from computatonal dffcultes of havng overflow n calculatng hyperbolc and 3 4 exponental functons. Therefore, another approach s proposed n the present work dealng wth the step functons. The approach s clearly presented n the followng flow chart n Fgure 5. The flow chart may be mplemented by means of VBA under Excel. Formulaton Summary Maxmze PPS Subject to: S T S j S d uq q PAS d, A 0 u q q R qu q (, j =,,, ) (q =,,, Q) Pred j (U q =,,, U q )...(4) The problem, formulated n (4), s a nonlnear program. The non-lnearty s severe and clearly notced n the objectve functon and the resource constrants. Soluton of such problems wth these types of non-lnearty s far from beng amenable to standard packages of optmzaton software. Therefore n the present work, Genetc Algorthm approach, as one of the most powerful computatonal modelng technques, wll be adopted. A GEETIC ALGORITHM (GA) MODEL The soluton of the problem formulated above s obtaned by evaluatng the set of decson 8

9 Fgure 5: The Flow Chart varables d, S, q ( =,,, ), (q =,,, Q) that maxmze the Objectve functon (4) and satsfyng the constrants (5), (6), (8). GA approach s a search technque searches for an optmum or near optmum soluton(s) n a space of solutons. The space of solutons s 9

10 ntally bult of a populaton of chromosomes representng solutons to the problem randomly generated. The space of solutons s evolved by means of the three Genetc Operators namely Crossover Operator, Copyng Operator and Mutaton Operator. The evoluton of the search space and ts consttuents Chromosomes s governed by a law smlar to the Law of atural Selecton n bology,.e., survval of only the fttest (Goldberg, 989). A ftness functon s appled n order to dscard solutons (chromosomes) havng lower ftness and keepng only chromosomes wth hgher ftness. The process of evoluton contnues untl no further mprovements could be attaned. Chromosome Structure and Intal Populaton Each chromosome conssts of (Q + ) genes such that the frst genes carry values responsble for fndng actvty duratons, the second genes carry values responsble for fndng startng tmes of actvtes and the rest Q genes carry values q responsble for fndng the resource multplers. The quanttes, and q are random numbers rangng from 0 to and unformly dstrbuted. ext, a method wll be developed n order to transfer these random numbers, and q nto decson varables d, S, k q respectvely. The Inverse Problem of Project Schedulng Tradtonally, the drect problem of project schedulng s to fnd a project completon tme T havng duratons of all project actvtes and actvtes precedence order. On the contrary, n the formulated n ths work problem, the project completon tme T s gven wth the precedence order of the actvtes and requred to determne the allowable duratons of the actvtes. Ths Inverse problem wll be solved n the followng steps: Snce the mnmum possble duraton of actvtes d o are gven, the mnmum possble completon tme T mn can be obtaned by the drect approach by Crtcal Path Method (CPM). The random quanttes occupyng the frst genes n a chromosome are used as transtonal actvty duratons n a CPM to evaluate a transtonal project completon tme Temp. ote that the precedence order s preserved n evaluatng Temp. The nverse problem s now ready to be solved to fnd actvty duratons that result n a completon tme T gven n the formulated problem. The soluton s proposed n the followng expresson: d do T Tmn...(5) Temp Startng Tmes of Actvtes In order to fnd the values of the second set of decson varables, we proceed as follows: Havng already determned duratons d apply CPM to determne the earlest start ES and free float FF of each actvty. The random quanttes occupyng the second set of genes n a chromosome are used to determne the startng tme of actvtes by the followng expresson: S ES FF...(6) Resource Multplers The random quanttes q occupyng the last Q genes are used to evaluate the resource multplers as follows: 0

11 q L...(7) q q The Ftness Functon The optmum or near optmum soluton wll be found as one of the feasble soluton that has the maxmum value of, a desgned for the purpose, Ftness Functon. The feasblty of a soluton s realzed by satsfyng all the constrants n (4). In the framework of GA approach, the nfeasble solutons should be penalzed by ntroducng a bg negatve value dependng on the amount of devaton from the rght hand sde of the unsatsfed constrant. Therefore, the Ftness Functon FIT for every chromosome wll take the form: FIT Q u q d d exp mn R qu q q uq o uq q A q, 0...(8) Determnaton of Webull Parameters, Selecton of Webull dstrbuton as the law of probablty dstrbuton of the actvty duratons as random varable s justfed by ts unversalty as several well-known dstrbutons could be derved from Webull dstrbuton as specal cases by changng the value of the shape parameter. Two methods are proposed here to calculate and dependng on the nput data. Gven Mean and Standard Devaton of Actvty Duraton d o...(9)...(0) d o s the mnmum duraton. Dvdng (0) by the square of (9) and performng smple manpulaton we get: do...() Equaton n () s n one unknown and can be solved easly usng one of the tools of Excel The Goal Seek. Substtutng n (9), we fnd. In practce, mean and specally varance of dfferent actvtes duratons are mostly unavalable because of lack of statstcs and also unqueness of projects. The three valued estmatons a, m; b of an actvty duraton whch s commonly used n PERT could be collected more easly than mean and varance n specal sessons wth experenced personnel as optmstc, most lkely and pessmstc estmatons. Gven a, m, b Estmatons of Actvty Duraton The optmstc estmaton a wll be taken as the mnmum duraton d o. The most lkely estmaton m s that duraton at whch the Webull pdf attans ts maxmum value. The pessmstc estmaton b wll be equated to a (-R) percentle (0 < = R < = ). Therefore, from the dfferentaton of Equaton (), we get: m a...()

12 If generally we consder (-R) percentle, then b a exp R b a Ln R Dvdng () by (3), we get Ln R m a b a...(3)...(4) Equaton (4) s n a sngle unknown and can be solved by Goal Seek of Excel. Substtutng n () or (3) we may fnd. ILLUSTRATIVE EXAMPLES As a more llustraton of the modelng approach outlned before, we apply our approach on a new constructon project. For analyss of ths example project, we focus on the probablty of successfully reachng the end node n 8 months (588 workng days), and our formulaton of the objectve functon n 6 s Z F n F n. The dependence of each F (d, k ) term on d and k has been suppressed to smplfy the notaton. Table summarzes the nput data for the 6 tasks. The optmstc duraton (a) s the value of d o for each task. The most lkely (m) and the pessmstc tme (b) to complete each tasks successfully are the bass for specfyng the two parameters of the Webull dstrbuton for each task; gven those two values, they solved Table : Input Data Task a m b P B E for and for each task. The elastcty values (E ) n Table defne the percentage reducton n the scale parameter of the dstrbuton of tme to successful completon for each task, resultng from a one percent ncrease n resources appled to the task. The two columns labeled omnal Person-Days and omnal Budget (B) specfy the Aq values for the two resources for each task. We appled our approach on ths data, as to fnd

13 Probablty of Success Project Avalable Tme 588 Table : Result of Optmzaton from our Approach Actvty umber Actvty Duraton Actvty Start Actvty Total Float Actvty Resource Multpler Actvty Success Probablty E E E E E E E the probablty of completng the project durng the avalable perod and resources. We transfer our formulaton nto code for the Genetc Algorthm. The Genetc Algorthm has been mplemented on Vsual Basc under Excel. The optmzed results for task start tmes, allowable duratons, and resource multplers, as well as the resultng probabltes for successful completon of each task, are shown n Table. Ths set of values results n a probablty of success for the project as a whole of

14 Table 3: The Dstrbuton of Resources Over the Perods Perod Avalable Manpower Consumed Manpower Avalable Budget Consumed Budget Also the dstrbutons of resources over each perod are shown n Table 3. REFERECES. Feng S, Lu L and Burns S (997), Usng Genetc Algorthm to Solve Constructon Tme and Cost Trade Off Problems, Journal of Constructon Engneerng and Management, Vol. 3, o. 3.. Goldberg D E (989), Genetc Algorthm n Search Optmzaton and Machne Learnng, Addson Wesley. 3. Hller F S and Leberman G J (995), Introducton to Operatons Research, McGraw-Hll Inc., ew York. 4. Hong Zhang, Heng L and Tam C M (004), Fuzzy Dscrete-Event Smulaton for Modelng Uncertan Actvty Duraton, Constructon and Archtectural Management, Vol., o. 6, pp Ioannou P G and Martnez J C (998), Project Schedulng Usng State-Based Probablstc Decson etwork, Proceedng of the 998 Wnter Smulaton Conference. 6. Tarek Hegazy (999), Optmzng Resource Allocaton and Levelng Usng Genetc Algorthm, Journal of Constructon Engneerng and Management, Vol. 5, o Turnqust M A and Lnda ozck (00), Allocatng Tme and Resources n Project Management Under Uncertanty, Proceedng of the 36 th Hawa Internatonal Conference on System Scences. 8. Van Slyke R M (963), Monte Carlo Methods and the PERT Problem, Operatons Research, Vol., o. 5, pp