CONSTRUCTION SCHEDULING WITH ARTIFICIAL AGENTS AND THE ANT COLONY OPTIMIZATION METAHEURISTIC

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1 CONSTRUCTION SCHEDULING WITH ARTIFICIAL AGENTS AND THE ANT COLONY OPTIMIZATION METAHEURISTIC Symeon Chrstodoulou 1 1 Department of Cvl and Envronmental Engneerng, Unversty of Cyprus, P.O. Box 20537, 1678 Ncosa, Cyprus The research outlned n ths paper ams the development of a methodology to arrve at crtcal path calculatons n constructon networks usng Ant Colony Optmzaton (ACO) algorthms. Ant Colony Optmzaton s a populaton-based, artfcal multagent, general-search technque for the soluton of dffcult combnatoral problems. The method s theoretcal roots are based on the behavour of real ant colones and the collectve tral-layng and tral-followng of ts members n searchng for optmal solutons n traversng multple paths. In essence, ACO s nspred by the foragng behavour of natural ant colones whch optmze ther path from an orgn (ant nest) to a destnaton (food source) by takng advantage of knowledge acqured by other ants that prevously traversed the possble paths. In computer mplementatons of the ACO algorthms, artfcal ants are both agents and soluton-constructon procedures that stochastcally buld solutons by consderng (1) artfcal pheromone trals whch change dynamcally at run tme to reflect the agents acqured search experence, and (2) heurstc nformaton on the problem/network beng solved. The paper outlnes the fundamental mathematcal background of the ACO method and a suggested possble mplementaton strategy for solvng for longest (crtcal) paths n constructon schedule networks. Keywords: ant colony optmzaton, constructon schedulng, crtcal path. INTRODUCTION Currently, the most wdely used method for constructon schedulng s the crtcal path method (CPM) whch, n essence, s based on forward and backward passes of the actvty network to arrve at longest path and total duraton calculatons. The method provdes planners and constructors wth a formalzed way of calculatng the earlest and latest dates each constructon actvty can start and fnsh at, subject to the envsoned network topology (actvty relatonshps) and the mposed tme, resource and logc constrants. Undoubtedly, over the years CPM has grown n functonalty, accessblty and acceptablty. Yet, some lmtatons of the method have recently gven rse to the search for complmentary methodologes that could address some of ether the perceved lmtatons or the desred addtonal features. Among the most notable lmtatons are: () the nablty of CPM-based tools to calculate longest (or shortest) paths from a node to any node, () the nablty of CPMbased tools to account for resource-drven actvty relatonshps ( AND / OR combnatons of resources), and () the computatonal neffcency of the crtcal path method. 1 schrsto@ucy.ac.cy Chrstodoulou, S (2005) Constructon schedulng wth artfcal agents and the ant colony optmzaton metaheurstc. In: Khosrowshah, F (Ed.), 21st Annual ARCOM Conference, 7-9 September 2005, SOAS, Unversty of London. Assocaton of Researchers n Constructon Management, Vol. 2,

2 Chrstodoulou Item () refers to the need to dentfy the longest (or shortest) path to a desred actvty from any other actvty n a project network. Ths feature would enable constructors dentfy the paths to start (or completon) of any desred actvty from any other actvty n the network and act accordngly to recover or accelerate the project and to redrect resources n the most effcent manner. Smlarly, ths added feature would enable planners and managers nterested at successful tmely completon of specfc actvtes dentfy the longest, or shortest, path to that actvty and medate assocated rsks as needed. Item () refers to the need to account for resource-based actvty relatonshps. In ths scenaro, tasks are performed and schedule advancement s made upon addtve or condtonal combnatons of resources. Even though CPM schedules can nowadays be resource-loaded, so that resources are accounted for durng the network calculaton phase, such schedules do not enable accountng for real-lfe stuatons where actvty sequencng s based on resource (rather than actvty) relatonshps. In other words, how can a planner account for the stuaton where an actvty can start only when a resource from a predecessor actvty s freed and then combned wth a resource from another predecessor actvty to enable the start of the successor actvty wthout completon of the two predecessor actvtes? Item () refers to the perceved computatonal neffcency of the tradtonal CPM algorthms. If one consders that soluton of the actvty network can be acheved by solvng the underlyng equatons whch represent the actvty nterrelatonshps, then possble soluton methodologes could nclude soluton of lnear equatons, lnear programmng, and forward/backward passes. The frst two approaches mply that the appled solver s capable of handlng a large number of equatons and constrants, as well as optmzaton. Despte that schedulng problems wthn certan categores n general have polynomal-type exact soluton algorthms and ther soluton s smple, n ts most general form, though, resource-constraned schedulng problems are NPhard, meanng that there are no known algorthms for fndng optmal solutons n polynomal tme. Ths class of problems (NP-class) requres computatonal power that ncreases exponentally wth the sze of the problem and has no exact soluton. In such cases a heurstc approach s warranted. The last approach mples exhaustve enumeraton of the possble paths n a project network, and calculaton through network-traversng. In essence, calculaton n ths case s acheved by startng from the frst actvty and exhaustvely dentfyng all possble paths (successve actvtes) untl reachng the last actvty, addng the duraton of each lnk to the total duraton of the dentfed path up to that pont (EarlyStart, EarlyFnsh dates) and then dentfyng the longest path (forward pass). A reverse pass (same exhaustve approach) s used to calculate the latest dates each actvty can start and fnsh subject to keepng the project end date fxed (as calculated durng the forward pass phase). The combnaton of the two network passes provdes the TotalFloat of each actvty (LateStart EarlyStart, or LateFnsh EarlyFnsh) and therefore the crtcal path (crtcal are the actvtes wth TotalFloat = 0). In summary, despte the methods relatve ease of applcaton they possess neffcences (.e. the exhaustve enumeraton of network paths) and computatonal constrants (.e. the maxmum number of equatons and nequaltes the appled solver can solve). 774

3 Constructon Schedulng wth Artfcal Agents and the ACO Metaheurstc ANT COLONY OPTIMIZATION Introducton Ant Colony Optmzaton (ACO) s a populaton-based general search technque nspred by the foragng behavour exhbted by real ant colones. The method was frst proposed by Dorgo (1991, 1992) for the soluton of dffcult combnatoral problems, and further expanded upon by several other researchers (Dorgo et al. 1996, Manezzo et al. 2004). The method s characterzed by the combnaton of a pror nformaton about the structure of a promsng soluton wth posteror nformaton about the structure of a prevously obtaned good soluton (Manezzo et al. 2004), an attrbute that s very sutable to network traversng and thus constructon schedulng. In essence, the method uses knowledge acqured by one artfcal agent durng ts path-searchng, when constructng the next feasble or optmal soluton of a gven path-traversng problem. The underlyng methodology s modelled after the behavour exhbted by real-lfe ant colones as they search for optmal solutons n node-to-node path traversng stuatons, durng whch a shortest path n a statc or dynamc topology s sought. The behavour exhbted by such ants s characterzed by a renforcement mechansm that helps steer succeedng ants to the most frequently prevously traversed path. In partcular, an ant randomly traversng possble paths can help fnd the shortest path between food sources and a nest and n dong so depost a chemcal substance called pheromone, formng pheromone trals whch can then be followed by other ants n the colony. When choosng ther way through the possble path routes, ants smell the deposted pheromone and tend to follow those paths marked by stronger pheromone concentratons. Therefore, whle an solated ant moves essentally at random, an ant encounterng a prevously traversed path and pheromone-lad tral can detect such, decde wth hgh probablty to follow t and subsequently renforce the tral wth ts own pheromone. The collectve behavour s therefore characterzed by a postve (renforcng) feedback loop where the probablty wth whch each ant chooses the path to follow ncreases wth the number of ants havng chosen the same path n the precedng steps. The pheromone tral s renforced wth each successve pass untl the ant populaton and path traversng converge to the shortest path between source and destnaton, and the fnal result s the relatvely quck convergence of the path-traversng to the shortest path. Theoretcal Framework of the ACO Metaheurstc A number of ACO algorthms, startng from the orgnal work by Dorgo (1991, 1992), have been developed and proposed over the years. The common framework for ACO applcatons was proposed posteror to be the ACO metaheurstc (Dorgo, Manezzo and Colorn 1999), wth artfcal ants seen as stochastc soluton procedures and actng as agents. The soluton constructon s based by the pheromone trals whch change at run-tme, the heurstc nformaton on the problem nstance and the ants prvate memory. The generc problem topology was outlned by Stützle and Dorgo (2002). It conssts (a) of a fnte set of components, C, (b) a set of problem states, x, defned n terms of sequences (relatonshps) over the elements of C, (c) a set of all possble sequences, denoted by X, and (d) a fnte set of constrants n the system, Ω, whch defnes the feasble states and the set of feasble solutons, S *, whch are a subset of the feasble 775

4 Chrstodoulou states. Furthermore, a cost functon ( s t) f, can be assocated wth each canddate soluton, s, and n some cases a separate cost functon s defned and assocated to states other than solutons. The generc behavor of the artfcal ants was also outlned by Stützle and Dorgo (2002) as follows: Ants buld solutons by movng on the constructon graph G=(C, R), where C s the set of components n the network, and R s the set of relatonshps (connectons) fully connectng the components. Even though both feasble and nfeasble solutons can be bult, artfcal ants, n general, try to buld feasble solutons. The problem constrants, Ω, are mplemented durng the networktraversng and the polcy followed by the artfcal ants. The components, c C, and connectons, rj R, can have a pheromone tral, τ, assocated wth them whch allows for the mplementaton of a long-term memory polcy about the ant search process. Smlarly, the components, c C, and connectons, rj R, can have a heurstc value, η, whch allows ncorporaton of problem-specfc nformaton. The path that each artfcal ant, k, follows can be stored n the ant s memory M k. k Each artfcal ant, k, can be assgned a start state, xs, and one or more k termnaton condtons, e. The constructon procedure of ant k stops when at k least one of the termnaton condtons e s satsfed. When n state x ( x ) r = r 1, an ant attempts to move to any node j n the feasble solutons subset (mmedate successors) N. If ths s not possble, t mght be allowed to move to any other node that t s not part of the mmedate-successors subset. The move to a successor node s determned by a stochastc decson rule and t s subject to a functon of the locally pheromone and the connecton s heurstc, the ant s memory, and the problem constrants. Each addton of a component c j to the current soluton updates the pheromone tral assocated wth t. Once a soluton s bult, the ant retraces the same path backwards and updates the pheromone trals of the used components or connectons. CONSTRUCTION SCHEDULING USING ANT COLONY OPTIMIZATION Introducton Constructon schedulng exhbts many smlartes to the ACO metaheurstc, snce the underlyng network topologes and path-searchng approach to longest (or shortest) path calculatons are comparable. If one substtutes the search for shortest path (ACO) to the search for longest path (CPM) and treats ACO ants, states, connectons and cost functon to CPM s resources, actvtes, relatonshps and duratons respectvely then the ACO metaheurstc can be employed n solvng for the longest path n connected, acyclc graphs (such as constructon actvty networks). k 776

5 Constructon Schedulng wth Artfcal Agents and the ACO Metaheurstc Furthermore, by consderng dfferent nodal states and ant types then the search for longest path can be accompaned wth calculatons of longest path from a node to any node, as well as wth resource optmzaton. ACO-Based Algorthm For a gven constructon network topology (project schedule) defned by a graph G=(N,A) wth N beng the set of nodes (actvtes) and A beng the set of arcs (actvty relatonshps) connectng the subject nodes, the proposed ACO-based procedure for fndng the crtcal path(s) between chosen nodes N 1 and N 2 can be summarzed by the followng steps: 1. Intalze all arcs wth small amount of pheromone, τ 0. Ths value can be an nverse lne-dstance between the nodes N 1 and N 2, or the nverse lne-dstance of the subject arc. 2. An artfcal ant s launched from node N 1 (the start node) pseudo-randomly walkng from a node to a successor node va the connectng arcs untl t reaches ether the end-node ( N 2 ) or a dead end. When at a gven node, the artfcal ant s selecton of an arc to follow s probablstc, based on a stochastc assgnment of each th arc s lkelhood of selecton, as defned by p = τ η β β τ η In the above equaton, τ s the pheromone concentraton on the th arc, η s an a pror avalable heurstc value for the th arc and β s a parameter determnng the relatve nfluence of the heurstc nformaton. The value of η can be defned ether as the nverse of the length of the arc, or the nverse of the length of the arc plus the lne-dstance between the subject node and N 2. It should be noted that prevously vsted arcs are excluded from the selecton (to enable complete tree spannng and avod memorzaton ). 3. The selecton s further asssted by the consderaton of a randomly generated number, 0 q 1, whch s compared to a predefned value, q 0, specfc to the network topology. If q q0 then the arc wth the hghest value p s selected. Otherwse, a random selecton of an arc s used based on the dstrbuton defned by the equaton for p. 4. Upon crossng each th arc durng the aforementoned soluton-constructng phase a local pheromone update rule s appled to update the level of pheromone concentraton at the gven arc. The updated pheromone level s defned by ( 1 ρ) τ ρτ 0 τ = + (2) where ρ s another network topology parameter ( 0 ρ 1). As already noted, the goal of the local updatng s to enable exploraton of more path/route varatons by makng already traversed arcs less lkely to be chosen agan durng the randomzaton of the arc selecton process. (1) 777

6 Chrstodoulou 5. Steps (2) - (4) are repeated for all ants n the ant colony and the most successful ant (.e. the one whose path defnes the soluton) s used to globally update the network s pheromone trals. The global update rule s defned by ( α ) τ ατ L τ = 1 + (3) whereα s yet another network topology parameter ( 0 α 1) whose value determnes the level of evaporaton of pheromone concentratons. The factor τ L s a value nversely proportonal to the path length of the best soluton n case of an arc vsted by the best ant or zero for all other ants. 6. The global update rule can be appled by ether the global-best or the teratonbest ant. In the frst case, the ant to perform the update s the one that obtaned the best soluton (found the longest path n the network) durng the entre optmzaton process. In the second case the update s performed by the ant reachng the best soluton durng each teraton of the algorthm. 7. Steps (2) - (6) are repeated for ether a fxed number of teratons or untl a predefned condton s met, and upon termnaton of the algorthm the pheromone tral n the graph G=(N,A) s used to determne the soluton (the arcs wth hghest pheromone concentraton form the longest path of the network). Case Study The ACO algorthm was mplemented by means of custom software (Chrstodoulou 2005). The software was desgned and developed so as to be able not only to emulate tradtonal (legacy) constructon schedulng applcatons and ntegrate wth ther underlyng databases but to also be able to complement them wth the ACO metaheurstc features. Furthermore, the developed software applcaton can be executed n ether test mode (randomzed network topologes) or project mode (actual constructon schedules to be solved) and be ntegrated wth external database management systems to account for addtonal common constructon features such as actvty costs, resource types, etc. The ablty to ntegrate wth legacy schedulng software (such as Prmavera s Prmavera Project Planner, or Mcrosoft s Project), furnshes users wth the ablty to get the most out of CPM and ACO applcatons. Fgure 1: Topology of case study project network 778

7 Constructon Schedulng wth Artfcal Agents and the ACO Metaheurstc The developed software was tested on several case studes, one of whch s presented heren. The case study presented (Fgure 1) s based on a randomzed topology of 10 nodes, 18 node connectons (a duplcate connecton generated by the program was subsequently gnored), and assumed topology parameters of q = , β = 1. 0, ρ = 0.5, α = 0. 5, and C = 50. Of the 10 network nodes ncluded n the assumed network topology, 4 are ant nests (.e. nodes wth no predecessor nodes), 5 are regular nodes and 1 s a food source (.e. a node wth no successor nodes). It s also noted that the generated network was based on undrectonal nodal connectons (an acyclc graph) so as to emulate real-lfe constructon networks, and that the network constructon was based on an assumed maxmum number of three successor connectons per node, to smplfy the manual calculatons and subsequent verfcaton by standard CPM procedures. The crtcal path calculatons on the case study topology, and the resultng EarlyStart, EarlyFnsh, LateStart, LateFnsh and TotalFloat values obtaned by applyng tradtonal CPM procedures are tabulated n Table 1. As shown, the CPM calculatons result n dentfyng actvtes 0-4, 4-6, 6-7 and 7-9 as crtcal (TotalFloat = 0), thus ndcatng a total project duraton of = 110 tme-unts. Table 1: Soluton of the case-study network topology usng the Crtcal Path Method (CPM) Start Node End Node Duraton Successor Nodes Early Start Early Fnsh Late Start Late Fnsh Total Float Crtcal? , Yes , , , , , , Yes Yes Yes Applcaton of the ACO methodology generates dfferent topology states at the end of each teraton (as shown n Fgure 2). Snce the method s ntellgently teratve (each successve teraton s selectvely dependant on prevously acqured knowledge about the topology) the crtcal path may change several tmes before convergng and stablzng to the correct soluton (Fgure 2). At each teraton, though, the algorthm generates the pheromone concentraton levels (acqured knowledge on the crtcalty of each connecton and node) and the resultng longest path (thus the crtcal actvtes). 779

8 Chrstodoulou Fgure 2: Network states at dfferent ACO teratons The fnal (soluton) results obtaned by the ACO algorthm are shown n Fgure 3, tabulated as FnalPheromone values. The algorthm consders these values durng the soluton phase and decdes whch actvtes on a contnuous path are crtcal, also outputtng the Crtcal Path (Nodes), Crtcal Path (Connectons) and Duraton values (Fgure 2). As seen, for the partcular case study, the ACO algorthm dentfes 780

Constructon Schedulng wth Artfcal Agents and the ACO Metaheurstc actvtes 0-4, 4-6, 6-7 and 7-9 (connectons 1, 12, 16 and 18 respectvely) as crtcal and calculates the crtcal path to be of 110 tme-unts

9 Constructon Schedulng wth Artfcal Agents and the ACO Metaheurstc actvtes 0-4, 4-6, 6-7 and 7-9 (connectons 1, 12, 16 and 18 respectvely) as crtcal and calculates the crtcal path to be of 110 tme-unts total duraton, whch agrees wth the results gven by tradtonal CPM-based software. Fgure 3: Soluton of sample network usng the ACO algorthm Varatons and Extensons of Appled Method Even though ths paper presents longest-path calculatons n constructon networks and dentfcaton of longest path actvtes and project total duraton, the appled ACO methodology can be modfed and extended to account for resource and cash flows, and node-to-node longest path calculatons. The former can be acheved by consderng the respectve actvty assgnments and ncludng them n the cost functon of each arc, used n the optmzaton process. The latter can be acheved by settng the start node of the node-to-node sequence of nterest as ant nest and the end node as food source and reconstructng the soluton path (longest path) whle all other nodes are set as plan network nodes. CONCLUSIONS AND FUTURE WORK The ACO metaheurstc provdes users wth an alternatve way of constructng longest-path solutons n acyclc (undrectonal) network topologes. Despte the seemngly teratve approach of the ACO method, the method utlzes ntellgent selecton procedures to perform the optmzaton and arrve at the longest paths n a prescrbed network topology. Past sample case studes ndcate quck convergence to the fnal soluton and thus low computatonal tme, and the performance of the appled optmzaton model can be further ncreased by varyng the topology parameters. 781

10 Chrstodoulou Ongong work on the subject matter addresses the ncluson of resource-based schedulng technques to account for AND/OR resource-combnaton requrements at the network nodes, and ways to generate total-float values for each actvty (the current ACO method does not generate these values). REFERENCES Chrstodoulou, S. (2005) Ant Colony Optmzaton n Constructon Schedulng. In: ASCE s 2005 Internatonal Conference On Computng n Cvl Engneerng, July 2005, Cancun, Mexco. Amercan Socety of Cvl Engneers (ASCE). Dorgo, M. (1991) Ant Colony Optmzaton. In: G. C. Onwubolu and B. V. Babu (eds.) New Optmzaton Technques n Engneerng. Sprnger-Verlag Berln Hedelberg, Dorgo, M. (1992) Optmzaton, Learnng and Natural Algorthms, Ph.D.Thess, Poltecnco d Mlano, Italy. Dorgo, M., Manezzo, V., and Colorn, A. (1996) The Ant System: Optmzaton by a Colony of Cooperatng Agents. IEEE Transactons on Systems, Man, and Cybernetcs Part B, 26(2), Dorgo, M., D Caro, G. and Gambardella, L. M. (1999) Ant Algorthms for Dscrete Optmzaton. Artfcal Lfe, 5(2), Manezzo V, Gambardella L.M., De Lug F. (2004). In: G. C. Onwubolu and B. V. Babu (eds.) New Optmzaton Technques n Engneerng. Sprnger-Verlag Berln Hedelberg, Stützle, T. and Dorgo, M. (2002). The Ant Colony Optmzaton Metaheurstc: Algorthms, Applcatons, and Advances. In: F. Glover and G. Kochenberger (eds.) Handbook of Metaheurstcs. Kluwer Academc Publshers, Norwell, MA. 782

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