Effective Project Scheduling Under Workspace Congestion and Workflow Disturbance Factors

Transcription

1 Abstract Effective Project Schedulig Uder Workspace Cogestio ad Workflow Disturbace Factors Vitaly Semeov, Ato Aichki, Sergey Morozov, Oleg Tarlapa, Vladislav Zolotov (Istitute for System Programmig RAS, Moscow, Russia) Effective project maagemet implies the use of advaced plaig ad schedulig methods that allow to determie feasible sequeces of activities ad to complete a project o time ad o budget. Traditioal schedulig tools like fudametal Critical Path Method (CPM) ad various methods for Resource Costraied Project Schedulig Problem (RCPSP) ad Time Costraied Project Schedulig Problem (TCPSP) have may shortcomigs for costructio projects where spatial factor plays a critically importat role. Previous attempts to iterpret space as a specific resource were successful for particular problems of lie-of-balace schedulig, space schedulig, dyamic layout plaig, horizotal ad vertical logic schedulig, workspace cogestio mitigatig, schedulig multiple projects with movable resources, spatial schedulig of repeated ad grouped activities ad motio plaig. However, oe of these methods cosiders the spatiotemporal requiremets i a holistic framework of geeric RCPSP problem ad provides feasible results accoutig for workspace ad workflow factors. I this paper we start with the classical RCPSP statemet ad the preset mathematically strog formalisatio of the exteded geeralised problem, takig ito accout workspace cogestio ad workflow disturbace costraits specified i practically meaigful ad computatioally costructive ways. For the geeralised RCPSP problem a effective schedulig method is proposed. The method teds to miimise the project makespa while satisfyig timig costraits ad precedece relatios, ot exceedig resource utilisatio limits, avoidig workspace cogestios ad keepig workflows cotiuous. The method reuses so-called serial schedulig scheme ad provides for additioal computatioal routies ad heuristic priority rules to geerate feasible schedules satisfyig all the imposed requiremets. Advatages of the method ad prospects for its applicatio to idustrial eeds are outlied i the paper too. Keywords: Plaig ad schedulig, Resource-costraied project schedulig problem, Priority rules, 4D modellig, Workspace maagemet. Itroductio Effective project maagemet implies the use of advaced plaig ad schedulig methods that allow to determie feasible sequeces of activities ad to complete a project o time ad o budget. Critical Path Method (CPM) ad various methods for Resource Costraied Project Schedulig Problem (RCPSP) ad Time Costraied Project Schedulig Problem (TCPSP) are traditioal tools icorporated i most popular project maagemet systems like Microsoft Project, Oracle Primavera, Asta Powerproject. Developed i the 1950s, the CPM geerates useful iformatio about the project, such as the logest sequece of activities, the shortest project duratio, ad the total ad free floats of each activity. This iformatio is crucial to a project s success ad substatially importat for the project maager to pla ad cotrol it more actively ad efficietly. I the mai, critical activities havig zero floats should receive the maagemet attetio that might be uecessary o other activities. This maagemet by exceptio is a importat advatage of the CPM, especially o large, complex projects (Ahuja, 1976; Bowers, 1995). Later the origial CPM formulatio was geeralised to take ito accout resource limitatios withi the RCPSP ad TCPSP statemets. I most of real idustrial projects, schedulig without cosiderig these limitatios may lead to o-credible results, sice the executio of activities is strogly affected by resource availability. Various aalytical ad heuristic methods have bee developed to apply the resource availability ito the schedulig process

2 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series (Ahuja, 1976; David ad Patterso, 1975; Hegazy, 1999). Aalytical methods attempt to fid the optimum solutio i terms of the miimum project duratio, but usually require very log computatioal time, makig them impractical. O the other had, heuristic approaches provide reasoable solutios for large-scale projects i practical time (Boctor, 1990; Hegazy, 1999). However, these methods igore diverget spatial factors ad caot guaratee the correctess of the prepared schedules i terms of lack of spatial coflicts commoly related to workspace cogestio ad workflow disturbace. Ideed, a activity ca be performed if oly all the eeded workspaces are reserved throughout its executio period ad if they are ot occupied by other competitive activities arraged at the same place at the same time. I some sese, workspaces ca be iterpreted as reewable resources shared amog cocurret project activities with predefied utilisatio rates. This observatio applies equally to spaces required to istall or to assemble product compoets, to store materials o logistics sites, spaces used as passageways to deliver resources to destiatio areas or reserved for parkig zoes or household rooms, ad spaces prevetig safety hazards. The workflow disturbace is aother factor prevetig prompt movemet of resources o a project site, icreasig idle time for labour ad equipmet, ad thereby, deterioratig their productivity. To allow for cost ad time efficiecies, it is ecessary to achieve workflow cotiuity by balacig the resource utilisatio ad replacemet. May researchers addressed these topics by meas of the itroduced cocepts of lie-ofbalace (LOB) schedulig (Pai et al, 2013), space schedulig (Choo ad Tommelei, 1999), dyamic layout plaig (Zouei ad Tommelei, 1999), horizotal ad vertical logic schedulig (Thabet ad Beliveau, 1994b), workspace cogestio mitigatig (Yeoh ad David, 2012), schedulig multiple projects with movable resources (Hegazy, 1999), spatial schedulig of repeated ad grouped activities (Thabet ad Beliveau, 1994a), ad motio plaig (Ellips ad Davoud, 2007). However, these attempts were successful oly for very particular statemets as well as did ot result i a holistic framework accoutig for workspace ad workflow factors ad extedig traditioal CPM, RCPSP ad TCPSP methods. I particular, LOB is a liear schedulig method that allows balacig of the operatios i the projects with repeated activities cotiuously performed i each cosecutive uit. Repeatig uits are commoly foud as typical floors i high rise buildigs, resideces i multi-housig developmets, statios i highways, meters i pipelie etwork, log bridges, tuels, railways, or water mais. Usig LOB, repetitive activities are scheduled i such a way to esure a smooth processio of resources from uit to uit with miimal coflicts. However, may researches idicated that this techique is suitable to model simple repetitive productio processes, but it is quite limited for the complex projects represeted by discrete activities with varied utilisatio ad productivity rates. I this paper, alterative schedulig formulatios are discussed to exted the classical CPM ad RCPSP statemets ad to accout for workspace ad workflow factors. I Sectio 2 we start with the classical statemets ad the provide mathematically strog formalisatio of the geeralised problem with the workspace cogestio ad workflow disturbace costraits specified i practically meaigful ad computatioally costructive ways. A effective schedulig method for the geeralised problem is preseted i Sectio 3. The method teds to miimise the project makespa while satisfyig timig costraits ad precedece relatios, ot exceedig resource utilisatio limits, avoidig workspace cogestio ad keepig workflows cotiuous. The method reuses so-called serial schedulig scheme ad provides for additioal computatioal routies ad heuristic rules to geerate feasible schedules satisfyig all the imposed requiremets. Sectio 4 is devoted to Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

3 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series prelimiary validatio of the method. I the coclusios sectio, advatages of the method are summarised ad prospects for its applicatio i idustrial practice are outlied. Geeralised Schedulig Problem Classical RCPSP formulatio The classical RCPSP problem ca be stated as follows. A sigle project ca be represeted by a etwork with N activities o the odes ad M liks o its arcs. Every activity a, = 1,..., N implies a uiterrupted process begiig at the time t ad havig the fixed duratio d 0. Every lik l m, m = 1,..., M reproduces the fiish-start precedece relatio betwee a predecessor activity a Pr(m) ad a successor activity a Sc(m) ad forces the successor activity ot to be started earlier tha the give lag τ m after its predecessor has bee fiished. A successor activity havig oly zero-lag liks caot start util all its predecessors have bee fiished. For the formalisatio uique dummy source ad sik activities a 1 ad a N of zero duratio d = 1 0, d = 0 N are itroduced ad they are liked with the project activities havig opeed starts ad opeed eds respectively. I order to be processed, a activity a may require u k uits of the reewable resource r k durig its executio. A costat availability of every resource r k, k = 1,..., K is assumed ad deoted as U k. I correctly scheduled pla it caot be exceeded at ay time poit t such that t1 t t throughout the whole project. I order to make the problem simple, activity splittig ad resource levellig are ot cosidered. The objective of the RCPSP is to schedule the activities such that the makespa of the project is miimised, all the precedece relatios are satisfied ad resource availability limits are ot exceeded. Let A (t) deotes a idex set of the activities beig i progress at the time t or formally A ( t) = { = 1,..., N, t t < t + d}, the the RCPSP problem ca be mathematically formulated for ukow variable X = { t } N = 1 as follows: mi t N subject to (1) tsc ( ) t ( ) + d ( ) + τ, m = 1,..., M (2) m u k A( t) Pr m k Pr m m U, k = 1,... K, t t t t (3) 1 The objective fuctio (1) miimises the completio time of the uique sik activity t N ad thereby the makespa of the whole project. Costraits (2) take ito cosideratio the liks betwee each pair of precedig ad succeedig activities. Fially, costraits (3) limit the total resource utilisatio at each time poit to the available amouts. To be correct from mathematical poit of view ad to guaratee the solutio existece, the RCPSP must avoid ay lik cycles ad exclude exceeded resource utilisatio for idividual activities so that uk U k, = 1,..., N k = 1,..., K. By relaxig the resource costraits (3), the RCPSP reduces to the CPM-case which ca be solved by forward recursio i polyomial time. But i geeral statemet the RCPSP belogs to the class of NP-hard problems (Kolisch et al, 1995; Lavalle, 2006). Existig dyamic programmig procedures as well as the brach ad boud techiques are too computatioally expesive to fid optimal solutios i most practical cases. Therefore, heuristic approaches, ad i particular, priority rule based schedulig methods, are usually employed withi commercial packages for such purposes. Geerally, such methods distiguish i a schedulig scheme (serial or parallel, sigle- or multi-pass) ad i a set of Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

4 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series rules to prioritise the cocurret activities which over-cosume the limited resources. Wellkow priority rules are most total successors (MTS), latest start time (LST), greatest rak positioal weight (GRPW), weighted resource utilisatio ratio ad precedece (WRUP), latest fiish time (LFF), miimum slack (MSLK). Beig combied ad implemeted withi multi-pass schemes, they show the best results obtaiable by heuristics today. For more details, please see (Kolisch et al, 1995). Workspace Cogestio Coditios As stated above, the RCPSP oly takes ito accout costraits for reewable resources. It ca be exteded by itroducig workspaces that allow explicit visual iterpretatio ad mathematically strog formalisatio. Let w i, i = 1,..., I are project workspaces geometrically represeted by solids beig coected, compact, orietable 3-dimesioal maifolds i Euclidea space. Typically they are the objects of simple shape: cuboids, cyliders, prisms, pyramids, spheres, coes, polyhedra. But they ca be compoud objects costructed from primitives by meas of Boolea operatios o sets: uio, itersectio ad differece. Beig adopted by the costructive solid geometry modellig (CSG), these operatios are traditioally deoted as, ad \ respectively. Workspaces ca overlap each other i differet dimesios ad across time ad therefore they caot be cosidered as idepedet resources. By cosumig u k uits of the resource r k with correspodig spatial rate v k ad operatioal time d k, the activity a utilises a workspace w i(, k ) with the factor vk d k uk if t t < t + d ρ t = v w d k ( ) ( i(, k ) ), (4) 0 otherwise where the fuctio v (w) returs the volume of the correspodig workspace. The itroduced factor ca be iterpreted as a averaged desity of the resource uits per uit volume per uit time. A spatial multiplier i the expressio gives a ratio of the space required by the resource uit to the total available space allocated to the activity. A temporal multiplier reflects the fact that workspaces may ot always be utilised throughout the activity's operatio time ad may be used to describe the itermittet ature of cotiuous activities. A otatio w i(, k ) is used here to emphasise that the workspace w i is associated with the activity a ad the related resource r k oly whe the activity performs. d 2,3 d 2, 4 Time A 1 A 2 A 3 A 4 v 2,4 W 1 W 2 W 1 W 2 W 1 W 2 W 3 W 4 W 3 W 4 W 3 W 4 Figure 1: Example of workspace competitio Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

5 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series Two workspaces w i(, k ), w i (, k ) are defied as iterferig oes if their origiatig activities overlap i a time iterval d 0 ad their solids itersect i a volume ( ) v i, i = v wi(, k) wi (, k ) >, > 0. The the workspace iterferece ca be quatified by multiplyig these factors. To be processed cocurretly the activities must avoid workspace competitio ad cogestio. By limitig the utilisatio ad cogestio factors we thus require that the workspace capacity must be large eough to allocate all the eeded amout of the resource uits, icludig uits allocated i other iterferig workspaces. I other words, the coflictig activities should have a opportuity to be rearraged so that the utilised resources ca be reallocated over free domais of the workspaces. Uder the suggestio that workspaces are cosumed by differet activities ad resources additively, this requiremet takes the followig form: for, ) i(, k) I( A U, t t t < t + d ρ d,(5) k ( t) v( wi (, k ) wi (, k ) ) d, v( wi (, k ) ) (, k ) I ( A, U ) where I ( A, U ) is a set of idex pairs for all the activities ad related resources so that u k 0. The costrait (5) is stated for every workspace w i(, k ) throughout the executio iterval of its origiatig activity t t < t + d. A summatio o the left side of the costrait is take over all the project workspaces, icludig the give workspace. It is essetial that the costraits (5) allow short-term itersectig or eve overlappig of workspaces o the coditio that their utilisatio ad cogestio factors are small eough. Cosider a sample schedule cosistig of four activities A 1 A 4, each of them utilises ow workspace W 1 W 4 correspodigly. The workspaces are represeted by solids havig simple box shape ad beig located as show by Figure 1. I spite of solids of the workspaces W 1, W 3 as well as W 2, W 4 itersect, the figure demostrates the oly case of competitio ad potetial cogestio of the workspaces W 2, W 4 itroduced by the activities A 2 ad A 4 overlappig i the time iterval d 2,4 = [ t4, t4 + d 4 ]. The workspaces W 1, W 3 ever iterfere each other as the associated activities A 1, A 3 are ot performed cocurretly accordig to the schedule. To detect if the activities A 2, A 4 should be rearraged the additioal aalysis of utilisatio ad cogestio factors for the workspaces W 2, W 4 is required. Ofte, the operatioal factor is removed from the cosideratio ad, thereby the resources are suggested to be utilised throughout the whole activity duratio. The, the costraits take the followig simplified form: v k for i(, k) I( A, U ) u k v( wi (, k ) wi (, k ) ) v( wi (, k ) ) (6) v (, k ) I (, k ) (, k ) A summatio o the left side of the costrait is take over all the workspaces I(, k) I( A, U ) iterferig with the give workspace w i(, k ). I order to guaratee the existece of a solutio, the costraits (5) ad (6) beig applied to workspaces occupied by every idividual activity must be satisfied. Noteworthy, differet models for quatifyig spatio-temporal iterferece betwee workspaces have bee proposed (Yeoh ad David, 2012). The formalized costrait (5) is quite similar to the model discussed i (Chua et al, 2010), while the form (6) is more close to the criteria preseted i (Chavada et al, 2012). Ideed, if three activities A 1, A 2, A 3 utilise the same resource r i the same workspace w throughout the same duratio with respective Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

6 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series factors ρ 1, ρ 2, ρ 3, the workspace mitigatio requiremet takes the trivial form ρ 1 + ρ2 + ρ3 1. The geeral forms (5), (6) are iteded for more sophisticated cases whe cocurret activities partially overlap i time ad utilise differet resources allocated i differet, partially crossed workspaces. Mai disadvatage of the reduced represetatio (6) compared with the geeral form (5) is that eve a short time overlay of the workspaces with relatively high utilisatio factors may lead to a breach of the cogestio coditios although ulikely that such workspace coflicts could ot be resolved i practice. Nevertheless, i this paper the form (6) is used as more solid requiremet imposed upo the iterferig workspaces. Moreover, it requires less computatio which is especially attractable for schedulig of large projects. Workflow Disturbace Coditios Workflow cotiuity is aother importat factor affectig the schedule feasibility ad its practical value. Beig arraged i differet places (sometimes, i differet sites or eve i geographically remote regios) the project activities eed the resources to be reallocated ad replaced i proper workspaces. Regardless of how resources are moved, these factors add o project costs ad duratio ievitably. Igorig these factors, the methods usually geerate schedules with high resource traffic ad ureasoable discotiuous workflows what makes them useless for practical purposes. This is a serious shortcomig of classical CPM, RCPSP, TCPSP methods for properly modellig the real-world costraits. Ufortuately, the studies metioed above did ot result i commo visio o the workflow pheomeo ad did ot provide a solid basis to specify workflow costraits i a formal way. I this sectio ow model for workflow maagemet is preseted beig tightly coected with issues of the spatio-temporal allocatio of resources amog workspaces. The model assumes the permaet use of a global pool of resources ad local resource pools associated with separate workspaces. The global project pool stores the total amout of uits for each resource type available at the curret time. Local pools store similar amouts of uits assiged to every workspace idividually. At the iitial time momet all the resource uits are assiged to oe or more workspaces emulatig logistics sites, warehouses, parkig zoes or household rooms. Wheever a ew activity starts ad requests a fixed umber of resource uits, it should be take ito accout ot oly the availability of the required uits at the give time momet, but also their distributio over workspaces. If the requested uits are available, the key issue arise here is which workspaces the resources should be supplied from. Oce the decisio is made, global ad local pools are updated properly so that the total amout of the resource uits is decreased by the utilised amout. Whe the activity is complete, these uits are released ad placed i the same workspace where they have bee utilised by the activity before. As they become available for other activities, the global pool ad local pool of this workspace are updated so that the total amout of the available resource uits is icreased by the released amout. It is see the followig tight relatioship betwee resource flows ad workflows i the scope of the model above. By supplyig resource uits from the earest workspaces ad miimizig resource reallocatio time, the workflows become more regular. The reallocatio processes ca be simulated by additioal liks betwee the activities releasig ad cosumig the same resource uits. Lag of every such lik could be determied by meas k of a user-defied trasfer fuctio τ ( u, i, i ) that returs the time eeded to move u uits of the resource r k from the source workspace w i to the destiatio workspace w i. If the activity a requires u k uits of the reewable resource r k ad these uits ca be delivered from the workspaces of the fiished activities a, a 1,, a so that 2 M Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

7 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series u k 1 m M (, k ), a structure { } = u + u u k 2 k M k u k uk m= 1 = is called the route for the resource r k i coformity to the activity a. The additioal trasfer liks should be created with the coditios below: for k m = 1,..., N, k = 1,..., K, m = 1,..., M (, k) t t + d + τ ( u, i(, k), i(, k)) (7) Ufortuately, such liks caot be defied by the plaer i advace i the problem statemet phase ad eed to be determied directly whe the project is beig scheduled. Every created trasfer lik may delay the successor activity ad therefore adds o overall project duratio. The schedulig methods for thus formalized problem should take ito accout these circumstaces whe decidig o the activity priority miimizig the total project makespa ad o the resource reallocatio policy ot disturbig atural workflows. m m k m Time A 1 A 2 A 3 A 4 Used 5 uits τ (4,1,2) Used 4 uits τ (1,1,4) τ (3,2,3) Used 3 uits τ (3,3,4) Used 4 uits 5 uits released Required 4 uits Required 3 uits Required 4 uits 1 uits released 4 uits released Required 3 uits Required 4 uits W 1 W 2 W 3 W 4 W 1 W 2 W 3 W 4 τ (4,1,2) τ (3,1,3) τ (4,1,4) τ (3,2,3) τ (4,2,4) 1 uits released 4 uits reserved 1 uits released 1 uits released 3 uits released Required 4 uits W 1 W 2 W 3 W 4 W 1 W 2 W 3 W 4 τ (1,1,4) τ (3,3,4) Figure 2: Example of a schedule takig ito accout workflow cotiuity factor. Figure 2 presets a example of a schedule takig ito accout workflow cotiuity factor. The schedule cosists of four activities A 1 A 4 utilizig o-itersectig workspaces of simple box shape W 1 W 4 correspodigly ad requirig for executio the give amout of uits of a labour resource "worker" as show over the activity symbols at the Gatt chart. Let a crew cosistig of five workers is available to perform activities o the schedule. The work starts with the activity A 1 usig all the crew (5 uits) that is located at the workspace W 1. After the activity A 1 has bee completed all the resource uits become available for other activities of the schedule. Aalysis of time values retured by trasfer fuctios τ ( 4,1,2), τ (3,1,3), τ (4,1,4 ) determies the miimum as τ (4,1,2 ) movig four workers from W 1 to the closest workspace W 2. The the precedece relatioship betwee the activities A 1 ad A 2 is established so that A 2 should be started just after A 1 has bee fiished. A 2 reserves 4 resource uits moved to the workspace W 2. 1 resource uit has bee Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

8 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series left released ad located i the workspace W 1. No more activities ca be started cocurretly with A 2 because of lack of sufficiet free resource uits to perform them. Similarly, the activity A 3 is scheduled after A 2 has bee completed. Three workers ecessary to perform it are reserved ad moved to the workspace W 3. Two workers (oe i W 1, aother oe i W 2 ) are released, but this is ot eough to start the activity A 4 cocurretly. Fially, after the activity A 3 has bee completed, A 4 requirig 4 resource uits ca be started. Three uits are moved from the closest workspace W 3, oe more uit ca be moved either from the workspace W 1 or W 2. I spite of the workspace W 1 is outermost to W 4, the resource uit located i it has bee released earlier, so the time value τ (1,1,4 ) is less tha τ (1,2,4 ). The two precedece relatioships betwee the activities A 1 ad A 4 as well as A 3 ad A 4 should be defied i the schedule. Schedulig Method As metioed above, the objective of the RCPSP is to schedule the activities such that the makespa of the project is miimised (1), all the precedece relatios are satisfied (2) ad resource availability limits are ot exceeded (3). The ukow variable of the problem is a vector of activity start times X = { t } N = 1. The geeralised Workspace ad Workflow Costraied Project Schedulig Problem (WWCPSP) ca be itroduced as a project makespa miimizatio problem (1), (2), (3) with the additioal costraits o workspaces ad workflows (6), (7). It is worth otig that the solutio of the geeralised problem icludes ot oly activity start times, but also resource reallocatio routes which would eable the activities to start o the scheduled times. Thus, the ukow variable of the WWCPSP K problem is a structure X = { t { } } N, u k k= 1 = 1. Geeralizig the RCPSP, the WWCPSP remais to be a NP-hard problem ad requires log computatio time eve for fidig suboptimal solutios. Let discuss the proposed schedulig method for the WWCPSP. The method adopts so-called serial schedulig scheme ad provides for additioal computatioal routies ad heuristic rules to geerate feasible schedules satisfyig all the imposed requiremets. Schedulig Scheme To resolve the WWCPSP problem the serial schedulig scheme metioed i may works (Kolisch, 1996b) was adopted ad advaced. It assumes a stage-wise algorithm extedig a partial schedule (i.e. a schedule where oly a subset of the activities has bee scheduled ad assiged a start time). Two disjoit activity-sets are associated with each stage, amely: the scheduled set S ad the decisio set D. The set S is formed by idices of the activities which were already scheduled ad thus belog to the partial schedule. The decisio set D cotais idices of the uscheduled activities with every predecessor beig i the scheduled set. I each stage oe activity from the decisio set is selected with a priority rule (i case of ties the ext priority rule or the smallest activity umber is applied to select the activity) ad scheduled at its earliest precedece, resource ad workspace feasible start time. Afterwards, the selected activity is removed from the decisio set ad put ito the scheduled set. This, i tur, may place a umber of activities ito the decisio set, sice all their predecessors are ow scheduled. The algorithm termiates at the stage umber j = N, whe all activities are i the scheduled set. The advaced serial scheme ca be formally specified as follows: Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

9 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series INITIALISATION: j := 1 S := ; j WHILE j < N BEGIN COMPUTE D * j FOR { = 1,..., N S, m = 1,..., M, = Sc( m) Pr( m S } : = ) : = mi j { θ ( ) mi ( θ ( )) } D = j k = 1,..., K D j j BEGIN u* k = { u τ ( u ) = mi Sj ( τ ( u ))} : u U * k CREATE LINKS BY END u * k : * = earliest start preservig (2),(3),(6),(7) for t t * t t * d * t + D j + 1 = D j \ : * STOP; S j+ : S 1 = j j : = j +1 END; * Withi the preseted scheme at every step j = 1,..., N a decisio set j D is formed ad updated accordig to precedece relatios. Usig a priority rule fuctio θ (), the priority values are computed for all activities from the decisio set ad the activity with the maximum priority value is selected. Differet priority rules are admitted withi this scheme. For the prioritised activity a * ad for each its cosumable resource r k a optimum route u * k is determied to miimise the trasfer time τ ( u *k ) at the set of all the possible routes Sj m M (, k) U k = { uk } m= 1 m S j, m = 1,..., M (, k) }, origiatig from the workspaces of the scheduled activities whose idices are already cotaied i the set S j : { u τ ( u) = mi Sj ( τ ( ))} u* k : = u (8) u U * k Havig got the resource route, the trasfer liks are created with the lags correspodig to k m * m the time delays τ ( u * k, i( m, k), i(, k)) for each portio of the delivered resource u * k. Fially, the prioritised activity a * is scheduled so that both precedece relatios (2), resource limits (3), workspace mitigatio costraits (6), ad iduced workflow liks coditios (7) are satisfied. The decisio ad scheduled sets are updated properly ad the method proceeds to the ext step. Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

10 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series Priority Schedulig Rules The umber of priority rules proposed is relatively high. The MTS, LST, GRPW, WRUP, LFF, MSLK rules metioed above well suit to the RCPSP problems, but fully igorig the spatial factors ulikely they would remai workable for the cosidered WWCPSP statemets. Oce the workspace ad workflow costraits have bee specified, effective priority rules for the WWCPSP problem ca be proposed. I order to prevet the workflow disturbace ad to keep resource traffic reasoable, the rule should miimise the resource movig time. It ca be reached if the resources are supplied from the earest workspaces with a miimal trasfer time accordig to the fuctio (8). If the uits placed i a earby workspace are ot eough for the scheduled activity a, the the search is propagated over distat workspaces with a expectatio that the requested uits ca be collected from several workspaces. The trasfer time ca be estimated for every sort of the requested resource ad for every reallocated portio. This time may have impact to earliest start of the scheduled activity. As a result, the activity may be delayed by d due to all the resource trasfers. The activities havig miimum delay with respect to the origial duratio d d should be prioritised to avoid high resource traffic. We call this priority rule by a Movig Delay Ratio (MDR) ad apply it as the first priority rule ivocated by the fuctio θ (). I case of ties, LFF or MSLK are applied as secodary rules for the fuctio θ (). Resource Reallocatio Model First of all, the method is based o a assumptio that the time trasfer fuctio ca be simplified ad its depedecy o the amout of reallocated resource uits ca be represeted by a separate multiplier so that 1 k τ k ( u, i, i ) = ρ ( i, i ) k s ( u), (9) where the value s k ( u) plays role of the speed of the resource trasfer ad the factor ρ k ( i, i ) meas the legth of a traversig path from the workspace w i to the workspace w i. k The secod factor ca be give i a tabular form { ρ i, i } providig the path legths amog all the workspace pairs. Ufortuately, the use of the tabular form looks urealistic for large projects as it would require maual iput of huge data. A aalytical form based o Euclidea, Mahatta or maximum orm for the 3-dimesioal vector coectig geometric ceters or corers of the workspaces is more coveiet for practical purposes. However, it may produce wrog estimates of the path legths ad prevet right choice of ext activities whe traversig the project space. As a example, the distace from oe room to a adjacet room alog a corridor may be the same as the distace from the room to a upper room located at the ext floor. But the path legths betwee the workspaces seem to be essetially differet. Moreover, if the activities are prioritised so that the resources to move i earest, quickly reachable workspaces, orm-based estimates may become error-proe. A more promisig way is to use so-called space-fillig curves ad to defie the spatial factor by meas of a distace fuctio o these curves. Figure 3 presets row-wise (a), prime rowwise (b), spiral-wise (f), U-wise (i) space-fillig curves as well as the curves based o the well-kow orderigs by Morto (c), Peao-Hilbert (d), Cator (e) ad Gray (g, h). Every space-fillig patter ca be geeralised for the 3-dimesioal case ad adapted to the project space by meas of altered orietatios of axes of the uderlyig coordiate system, thereby producig 48 particular curves. Figure 4 illustrates this variety by givig a few particular curves for 3-dimesioal row-wise ad Peao-Hilbert space-fillig patters. By choosig oe of the patters, adjustig its orietatio i the project space ad settig the cell Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

11 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series sizes alog differet dimesios, the user defies a simple automatic routie for estimatig resource reallocatio time. If the chose patter ad made adjustmets match to a real project eviromet, the estimates become realistic so they ca be applied whe decidig o activity priorities ad keepig workflows cotiuous. (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 3: Space-fillig curves for the trasfer time estimatio. Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

12 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series (a) (b) (c) (d) (e) (f) Figure 4: Space-fillig curves geeralised for 3D case. Computatioal Experimets A simple project o refurbishig a small hotel with partial replacemet of supply lies has bee used for computatioal experimets. The hotel is a two-storeyed buildig havig 34 rooms of three types varied i size: stadard, studio ad family. Time required to refurbish a room depeds o its size ad is 2 days for stadard, 4 days for studio ad 6 days for family room. Refurbishig is performed by a crew of workers room-by-room sequetially. The sequece of rooms selected for refurbishig has o matter. Replacemet of supply lies is accomplished by aother crew i parallel with refurbishig ad subdivided ito 5 stages, each of 10 days. The first stage icludes works related to all the hotel buildig, the other 4 stages ca be performed after the first oe has bee fiished ad cover oly oe buildig aisle (left ad right at the first ad secod storey correspodigly). Traditioal plaig systems usually cosider workspace as a spatial resource ad assig it to all the activities that should be performed i it to cotrol its availability. As applied to the cosidered project, 34 spatial resources (each represets a separate room) are created. All the resources are assiged to the activity represetig the first stage of supply lie replacemet. Resources represetig rooms located at the correspodig hotel aisle are assiged to other supply lie replacemet activities. Oly oe spatial resource is assiged to the activity o refurishig the correspodig room. The the activities are scheduled accordig to a predefied priority rule, for example, latest fiish time (LFT). Schedulig algorithm based o this rule is simple eough: activities with the largest duratio (or latest fiish time) should be scheduled first. A activity with the smallest idex is selected first for Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

13 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series schedulig amog oes with equal duratio. The prepared schedule is preseted i the Figure 5a. (a) (b) Figure 5: Hotel refurbishig schedule prepared usig the traditioal RCPSP method. Visualisatio of crew movemet routes (see Figure 5b) demostrates obvious shortcomig of the prepared schedule. Accordig to it the refurbishig crew is forced to move 5 times from oe aisle to aother at the secod storey, 3 times from oe storey to aother, 3 times from oe aisle to aother at the first storey. The supply lie replacemet crew moves 3 times from oe storey to aother. Such chaotic movemet is icoveiet for the workers ad may lead implicitly to overheads. Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

14 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series Coceptual differece of the proposed algorithm cosists i takig ito accout spatial factor durig schedulig procedure. I additio, the model of workspaces used i it is more accurate ad flexible tha reducig the workspaces to resources utilised i classical approaches. It allows loadig the workspaces as much as possible. Beig applied to the discussed project, it takes ito accout the fact that two crews ca share the same workspace at the same time if allowable by its utilisatio factor. Oe more differece of the algorithm is that the prioritizatio of activities depeds o resource trasfer time: activities with the miimum trasfer time should be performed first. The time is estimated whe bypassig possible routes of resource reallocatio betwee workspaces. Activities with equal priority are scheduled accordig to classical approaches. The prepared schedule is preseted i Figure 6a. It is essetial that the project makespa accordig to the schedule prepared usig the proposed method is 10 days smaller tha i the previous example. (a) (b) Figure 6: Hotel refurbishig schedule prepared usig the WWCPSP method. Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

15 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series Figure 6b demostrates that the crews should move from oe storey to aother oly oce. This is more reasoable ad coveiet comparig to the previous schedule. Thus, the proposed schedulig algorithm ad the correspodig priority rule does ot icrease the total project duratio ad improve quality of workflow comparig to the classical approaches. Coclusios Thus, the ew WWCPSP problem geeralizig the classical RCPSP by takig ito accout spatial factor has bee stated ad formalized. The effective schedulig method for the problem has bee proposed, ivestigated ad approved. Like the classical approaches it teds to miimise the project makespa while satisfyig timig costrait, precedece relatios ad ot exceedig resource utilisatio limits. I additio, the method takes ito accout workspace cogestio ad workflow disturbace factors ad allows ot oly to determie availability of resources to perform the project activities, but also to cotrol overloadig of workspaces ad to miimise time overheads required for reallocatio of resources. The coducted computatioal experimets showed that the method geerates feasible schedules ear to optimal solutio at least for the low-scale bechmark problems. The reached advatages allow employig the method for the idustrial eeds. Such activities are plaed for the ext research phase. Refereces Ahuja H. N. 1976, Costructio Performace Cotrol by Networks, Joh Wiley & Sos, New York. Boctor F. F. 1990, Some efficiet multi-heuristic procedures for resource costraied project schedulig, Europea Joural of Operatioal Research, 49, Bowers J. A. 1995, Criticality i resource-costraied etworks, Joural of the Operatioal Research Society, 46(1), Chavada R., Dawood N & Kassem M. 2012, Costructio workspace maagemet: the developmet ad applicatio of a ovel D plaig approach ad tool, Joural of Iformatio Techology i Costructio, 17, Choo H. J. & Tommelei I. D. 1999, Space Schedulig Usig Flow Aalysis, Proceedigs IGLC-7, Chua D., Yeoh K & Sog Y. 2010, Quatificatio of spatial temporal cogestio i Four-dimesioal computer-aided desig, Joural of Costructio Egieerig ad Maagemet, 136 (6), David E. W & Patterso J. H. 1975, A compariso of heuristic ad optimum solutios i resourcecostraied project schedulig, Maagemet Sciece, 21(8), Ellips M & Davoud S. 2007, Classic ad Heuristic Approaches i Robot Motio Plaig - A Chroological Review, Proceedigs of world academy of sciece, egieerig ad techology 23, Hegazy T. 1999, Optimizatio of resource allocatio ad levelig usig geetic algorithms, Joural of Costructio Egieerig ad Maagemet, 125(3), Kolisch R., Sprechrr A & Drexl A. 1995, Characterizatio ad geeratio of a geeral class of resource-costraied project schedulig problems, Maagemet Sciece, 41(10), Kolisch R. 1996a, Efficiet priority rules for the resource-costraied project schedulig problem, Joural of Operatios Maagemet, 14, Kolisch R. 1996b, Serial ad parallel resource-costraied project schedulig methods revisited: Theory ad computatio, Joural of Operatios Maagemet, 90, Lavalle S. M. 2006, Plaig algorithms. Cambridge Uiversity Press, UK. Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),

16 Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series Semeov V. A., Kazakov K. A & Zolotov V.A. 2011, Virtual Costructio: 4D Plaig ad Validatio, Proceedigs of the XI Iteratioal Coferece o Costructio Applicatios of Virtual Reality, Pai S. K., Verguese, P & Rai, S. 2013, Applicatio of Lie of Balace Schedulig Techique (LOBST) for a Real estate sector, Iteratioal Joural of Sciece, Egieerig ad Techology Research (IJSETR), 2(1), Thabet W. Y. & Beliveau Y. J. 1994a, Modelig Work Space to Schedule Repetitive Floors i Multistory Buildigs, Joural of Costructio Egieerig ad Maagemet, 120(1), Thabet W. Y & Beliveau Y. J. 1994b, HVLS: Horizotal ad Vertical Logic Schedulig for Multistory Projects, Joural of Costructio Egieerig ad Maagemet, 120(4), Yeoh K. W & Chua D.K.H. 2012, Mitigatig Workspace Cogestio: A Geetic Algorithm Approach, EPPM 2012 Coferece, Zouei P. P & Tommelei I. D. 1999, Dyamic Layout Plaig Usig a Hybrid Icremetal Solutio Method, Joural of Costructio Egieerig ad Maagemet, 125(6), Semeov V, Ato Aichki A, Morozov S, Tarlapa O & Zolotov V. 2014, Effective Project Schedulig Uder Workspace Cogestio Ad Workflow Disturbace Factors, Australasia Joural of Costructio Ecoomics ad Buildig Coferece Series, 2(1),