Agent-based modeling and simulation of multiproject

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1 Aget-based modelg ad smulato of multproject schedulg José Alberto Araúzo, Javer Pajares, Adolfo Lopez- Paredes Socal Systems Egeerg Cetre (INSISOC) Uversty of Valladold Valladold (Spa) {arauzo,pajares,adolfo}ssoc.es Jua Pavó Facultad de Iformátca Uversdad Complutese de Madrd Madrd (Spa) Abstract There are o aalytcal solutos for the problem of dyamc schedulg of resources for multple projects realtme. Mathematcal approaches, lke teger programmg or etwork based techques, caot descrbe complexty of real problems (mult-projects evromets have may terrelated elemets), ad have dffcultes to adapt the aalyss to dyamcs chages. However, ths complex problem ca be modeled as a mult-aget system, where agets egotate resources through a aucto spred mechasm. Agets ca be used to represet projects ad resources. Projects demad resources for fulfllg ther scheduled plaed work, whereas resources offer ther capabltes ad workforce. A aucto spred mechasm s used to allocate resources to projects ad the prce of resources emerges ad chages over tme depedg o supply ad demad levels each tme slot. By meas of ths mult-aget system, t s possble to overcome most of the problems faced mult-project schedulg such as chages resources capabltes, allocato flexblty, chages project strategc mportace, etc. Keywords aget-based modellg; aget-based smulato; mult-project evromets; aucto based resources allocato; project schedulg. I. INTRODUCTION The problem of allocatg resources for multple cocurret projects appears large cases of servce ad maufacturg orgazatos. A paradgmatc example ca be a egeerg projects offce. Ths orgazato makes dfferet kds of projects that are proposed at ay tme, whch must be hadled a gve tme frame. Each project cossts of a umber of actvtes (calculatos, desg, checks, budgetg, etc.) that are performed by workers ad wth some precedece relatoshps. The workers ca perform oe or several actvtes accordg to ther sklls. Decso makers have to reject advsable projects ad decde whch resources wll be allocated to whch projects ad whe. Prevous decsos have hgh mpact the offce s proft. I order to acheve strategc goals t s mportat to gve prorty to projects, ad to allocate actvtes to the most effcet workers at the approprate tme. Because of ths, before executg projects t s advsable to make a schedule that optmzes the allocato of resources. Classcal methods are based o mathematcal programmg ad ca solve ths problem whe the complexty s low. Ad there are some heurstcs ad meta-heurstcs that are able to provde good schedules for more complex problems [9]. The tradtoal schedulg ad cotrol systems propose herarchcal ad cetralzed archtectures, where a classcal scheduler system that has a global model of the mult-project evromet makes schedules accordg to the curret state of the system. Has et al. [4] revew exstg lterature herarchcal approaches ad propose a geerc project plag ad cotrol framework for helpg maagemet to choose betwee plag methods, depedg o orgasatoal ssues. But these techques are ot flexble or robust eough, ad have dffcultes to cosder may real factors. I addto, real evromets udergo frequet chages (ew resources, ew techologes) that force to modfy the schedulg system. The tradtoal schedulg ad cotrol systems, whch are based o herarchcal ad cetralzed archtectures, have ot eough flexblty to adapt themselves to the dyamsm ad complexty of mult-project evromets. These ssues have motvated, last years, successve proposals are appearg to mprove the schedulg ad cotrol a mult-project evromet. The paradgm of Mult-aget Systems (MAS) ca help to fd solutos, especally cases where some socal behavour emerges. Ths paper shows a aget-based approach for ole dyamc schedulg ad cotrol mult-project evromets that takes advatage of the ablty of agets to egotate ad adapt to chagg codtos. The MAS has bascally two types of agets: projects maagers ad resources maagers. Projects have scheduled work to be doe by dfferet resources. Resources are edowed wth some capabltes (kowledge, work force, etc.) that are eeded to do the work. Projects demad resources over tme ad resources offer ther capabltes ad tme avalablty. There s a aucto process, ad the prce of resource-tme slots emerges edogeously as a result of supply ad demad. The desg of the aucto process uses a techque that has bee proposed for dstrbuted schedulg the lterature [8], [14], [11]. Ths aget-based approach has two dstctve aspects wth respect to other works: the tegrato of strategc decsos

2 (accept or reject ew projects) ad operatve aspects (resource allocato), ad the ablty to maage resource flexblty. Ths allows magers to study the advsablty of creasg the flexblty of resources. The ext secto troduces the role of aget-based modelg ad smulato project schedulg. Secto 3 presets the MAS for the real-tme schedulg problem, whch has bee specfed wth a aget-oreted modelg laguage, INGENIAS [10]. Ths has bee the bass for mplemetg a smulato, whch s descrbed secto 4, ad whose results are dscussed secto 5. Fally, secto 6 presets ma coclusos of usg ths aget-based modelg ad smulato approach. II. AGENT ORIENTED MODELING AND SIMULATION FOR REAL-TIME SCHEDULING OF MULTIPLE PROJECTS Mult-projects evromets are complex ad dyamc systems. They clude may compoets ad depedeces, ad may chages may occur the executo of projects. Moreover, projects are heretly dstrbuted; each task may be completed by dfferet resources or dfferet geographcal locatos ad each project maager may be dfferet places. MAS have bee show to deal wth problems of complexty, opeess (compoets of the system are ot kow advace, ca chage over tme, ad are hghly heterogeeous, dyamc project maagemet terms), wth dyamcal ad ukow evromets chagg over tme (ucertaty) ad ubquty (the actvty s dstrbuted over the complete structure) [5] [12]. I the partcular case of mult-project systems, the agets ca be abstracted as tasks, resources, project maagers, etc. Ths desg eables to dstrbute the maagemet system elemetal compoets drectly detfable the target system, ad hece gvg the opportuty to create systems easer to desg, to adapt ad to mata. Moreover, sce the system s dstrbuted accordg to ts structure, ay chage the structure ca be easly traslated to the maagemet system. Ths decetralzed approach facltates the desg of market mechasms to solve the schedulg problem by meas of dstrbuted approxmatos [2]. Recetly, Lee, Kumara ad Chatterjee [7] have proposed a aget-based dyamc resource schedulg for multple dstrbuted projects usg market mechasms. Followg the same research le, Cofessore et al. propose [3] aother teratve combatoral aucto mechasm. Other examples of aget-based approaches project maagemet ca be foud the works of Km ad colleagues [6], Wu ad Kotak [13], ad Cabac [1]. III. A MAS MODEL FOR MULTIPLE PROJECT SCHEDULING The system ca be modeled wth two types of agets represetg project ad resource maagers. Agets have the ablty to teract wth each other. I ths case, t s mportat to defe a aucto protocol for project agets to compete for the use of resources. Resource Maager Agets teract wth project agets to form o the status, capabltes ad cost at each specfc tme. A thrd type of aget s cluded the system to create ew agets ad motorg the global behavor. A. Project Maager Agets Each project s assocated to a Project Maager Aget. The system s cosdered dyamc: whle some projects are beg developed other projects ca be cluded or rejected realtme, whch mples the creato ad deleto of the correspodg agets. At ay stat t there are I projects the system, each oe deoted by. Each oe s characterzed by a value V, that ca be terpreted as the reveue obtaed for the project, a weght w represetg the strategc mportace gve to the specfc project, a desrable delvery date D, a lmt delvery date D *, whch caot be exceeded, a arrval date of the project to the system, B, ad a lmt aswer date R that represets the latest date to decde whether to accept or reject the project. Each project cossts of J actvtes, each oe deoted by j, where {1, 2,, I} ad j {1, 2,, J }. Every actvty j of a project s assocated wth a competece h(,j). Ay actvty j wth a gve competece h(,j) ca be performed by a resource m just f m s edowed wth the competece h(,j). The durato of the actvty j depeds o the resource assged to perform t. The durato of actvty j resource m s deoted as d jm. It s calculated accordg to d jm =d j /e m,h(j), where d j s the stadard durato of actvty j of project ad e m,h(j) s the effcecy of resource m to perform the competece h(,j). Ths frst smplfed model assumes that the actvtes of ay project should be performed sequetally the order defed by j ad oly oe resource ca be assged to a actvty. There s also the assumpto that oce some resource has begu a task, the actvty caot be terrupted; the resource eeds to fsh t to be assged to ay other actvty. Fgure 1. MAS orgazato model (wth INGENIAS otato [10]). The dagram shows a orgazato (Egeerg Compay), whch has several departmets (Projects Offce ad Producto Ut). The Projects Offce has oe Motor Aget ad several Project Maager Agets. The Producto Ut has Resource Maagers that take care of the use of Resources.

3 B. Resource Maager Agets A resource s modelled as a Resource Maager Aget. There are M resources, whch ca be assged smultaeously to oe actvty. Each resource s edowed wth a gve cost rate per ut of tme, c m (m {1, 2, 3 M}), ad a subset H m of competeces that ca be performed (H={h 1, h 2,... h K } s the set of competeces that are ecessary to complete the projects). Each resource has a certa grade or ablty to perform a competece. Therefore, the work capacty of resources ca be symbolzed by meas of a vector of abltes per resource e m =(e m1, e m2,,e mk ), where e mf 0 shows the ablty degree of resource m to perform the competece h f. If e mf = 0 the the resource m has ot the competece h f, f 0 <e mf < 1 the resource s able to perform effcetly the competece h f, f e mf = 1 t has stadard effcecy to perform the competece, ad f e mf > 1 t wll do t effcetly. C. Motorg Aget A Motorg Aget has the resposblty to vsualze the curret state of the system to the user. Moreover, ths aget allows the user to create ew Project Maager Agetsm, as show Fgure 1. IV. AGENT WORKFLOWS AND INTERACTIONS The aget workflows ad teractos must be desged order to maxmze the global effcecy of the system, whch wll be evaluated by the average beeft obtaed a certa tme terval T accordg to: Effcecy B T T = = ( V Cost( )) for all projects that are fshed T, Cost() s the cost to complete the project. Ths cost has two compoets, the drect resource cost ad the delay cost: d Cost( ) = C + w ( D F ) T (1) j 2 m(j) (2) j ejm The frst added correspods to the drect resource cost to fsh each actvty j. m( j ) deotes the resource selected to comply wth actvty j. The secod added s the delay cost assocated wth the project, where F s the real delvery date. The problem cosders the decso to reject projects. Ths could happe ay of the followg cases: The reveue obtaed from the project does ot compesate the costs. The schedulg exceeds the D * of the project. The mpact o the schedulg of the rest of the projects s ot acceptable. Ths may happe for two causes. Frst, f the ew project oblges to delay a commtted project beyod D *, t wll be rejected. If ot, but the cluso of the ew project creases the delay costs of the other projects more tha the drect beeft obtaed for the project, t wll also be rejected. A. Aucto Iteractos At ay tme, the system has as may Project Maager Agets as projects are ordered. Each oe represets a partcular project characterzed by ts tasks, precedece relatoshps, due date, value, local programs ad ther executo state. Ther goal s to look for cotracts wth resources that ca perform the requred actvtes ad hece completg successfully the project. I order to acheve ther goal, Project Maager Agets make plas that take to accout oly ther ow actvtes (local schedule). The decso-makg process s decetralzed as t emerges from teractos amog the agets a aucto process. Each project maager creates ts ow schedule (local schedule) by takg to accout ts ow project goals ad ts ow kowledge. Ths procedure ca brg compatble local schedules (several projects try to use the same resource at the same momet). Moreover, the local schedules ca be globally effcet (proftable projects are rejected; most mportat projects have delays; etc). These dffcultes that arse from the autoomy of each aget are solved wth a market mechasm that esures that local schedules are early compatble ad globally effcet accordg to the expresso (1). Ths aucto based mult-project schedulg approach s fouded o Lagraga Relaxato [8][11][14], a decomposto techque for mathematcal programmg problems. I order to apply the market metaphor, the perods whe resources are avalable are subdvded a set of small tme tervals or tme slots. Each tme slot o each resource s modelled as a good that ca be sold a aucto, where each resource acts as a seller. Thus, a local schedule wll be a budle of tme slots that has bee allocated to a project. The umber of sellers s equal to the umber of resources the system. Each resource proposes a prce for the tme slots from the curret tme to the ed of the schedulg horzo. The schedulg horzo chages dyamcally by cocdg wth the latest tme slot that some project has asked at ay momet. Each project aget plays the role of a bdder that partcpates auctos by askg the Resource Maager Agets for the set tme slots that t requres to execute ts pedg tasks at the curret tme. It wll try to fd a set of tme slots (Z ) through the resource pool whle currg the mmum possble local cost (LC ). Ths cost has two compoets, the sum of the prce of the selected tme slots ad the delay cost (expresso 3): LC = p Z + w 2 ( D F ) where p s the prce of the tme slot (t) of the resource (m). (3)

4 To select the set of tme slots (Z ) that mmzes ther local cost, Project Maager Agets use a dyamcal programmg algorthm where all possble combatos of tme slots ad resources are cosdered [13]. I ther decso, they take to accout that oly those resources edowed wth the ecessary competeces ca carry out a certa actvty. Moreover, the umber of tme slots ecessary to complete a task (durato) are determed accordg to the ablty degree of the resource the competece. Each project aget wll regard as schedulg horzo the tme slot that goes from the curret tme to the lmt delvery date (D * ). If some project aget caot fd a set of tme slots such a maer that t allows to schedule tasks before D *, wth a smaller cost tha ts value (V ), the t wll ot ask for ay set of tme slots. Ths mples that the project s uproftable at the correspodet roud of bddg ad must be rejected. Each Resource Maager Aget determes the prce charged for the tme slots wth the purpose of reducg resource coflcts ad maxmzg ther reveue. I order to get ths goal a subgradet optmzato algorthm s used to adjust prces at each roud of bddg. By meas of ths algorthm the Resource Maager Agets crease the prce of the tme slots where there s coflct (more tha oe project maager has asked for ths tme slot) ad reduce the prce of the tme slots that have ot bee demaded. The process of prce adjustmet ad bd calculato cotues deftely. At each roud of bddg the resource coflcts wll be reduced. At the frst roud of bddg, the tme slots prces for the resource (m) are equal to the resource cost rate (c m ). At the rest of bddg roud, the prces wll be updated by meas of the expresso 4. α s calculated accordg to [8]. p + 1 = max { c, p + α g } m Where: +1 p s the prce of the tme slot (t) of resource (m) at the roud (+1) p s the prce of the tme slot (t) of resource (m) at the roud () α s the step at the roud (). It decreases whe () creases. Ad ( g a 1) s the subgradet, where = s the demad of slot (t) of resource (m) (4) a B. Cotract Iteractos By meas of the aucto mechasm descrbed above, project agets buld compatble ad globally effcet local schedules for ther pedg actvtes. Moreover, at the same tme, agets teract through a complemetary process to make frm agreemets based o the local schedules that have bee created by meas of the aucto process. These agreemets determe fxed programs for earlest scheduled tasks. Whe these agreemets are obtaed, project agets wll ever cosder the tasks cluded as frm cotracts as pedg. The global effcecy ad the compatblty of local schedules deped o the degree of covergece of market prces to the equlbrum prces. If the prces get closer to the equlbrum prce, they wll be represetatve of the system state; they wll have formato about ay system feature ad local schedules wll be compatble ad globally effcet. If agets are makg frm cotracts whe prces are ot represetatve of the system state, the compatbltes could take part. I these cases, the agets resolve compatbltes by meas of local schedule based heurstcs rules. More exactly, whe several actvtes use the same resource at the same momet, the actvty that has bee earlest programmed local schedule wll have prorty to be cotracted frm agreemets. Although ths heurstc does ot esure global effcecy, t wll acheve perfect compatblty fal decsos. V. SIMULATION AND RESULTS The system has bee mplemeted ad smulated wth dfferet scearos. Here the aalyss focuses o the role of resource capabltes ad the opto of project rejecto. The frst scearo shows a smple case to llustrate the ma features of the system, the ext subsecto. Ths s followed by a dyamc scearo order to evaluate the system performace evolvg complex evromets. A. Smple Case Study Cosder three dfferet resources (R1, R2 ad R3), edowed wth the competeces C1, C2 ad C3 respectvely. TABLE I. shows a portfolo of fve projects, ad the tasks eeded to complete each project. Each task s defed by meas of the pertag competece ad expected stadard tme to be completed. Proj. Task 1 Tasks Task 2 TABLE I. Task 3 SIMPLE CASE STUDY Arrval date Startg Date DD1 DD2 Value P1 C1 50 C2 25 C P2 C3 40 C1 45 C P3 C2 35 C1 40 C P4 C3 30 C1 50 C P5 C1 45 C3 20 C The arrval date s the date whe the project s cluded the system. Projects ca start-up the startg date; otherwse, they should have bee rejected before ths date. Due Date 1 (DD1) s the most desrable durato whereas Due Date 2 (DD2) s the maxmum allowed. All the projects have a weght of 1. Fgures 2 ad 3 show the system state at a gve tme (curret tme). I the upper area of the fgures the relatve dualty gap evoluto s preseted. The prces of tme slots are the soluto of the dual problem ad the dualty gap s a

5 measure of the dfferece betwee the prmal ad dual objectve fucto, so t quatfes the qualty of the soluto [8]. The relatve dualty gap s calculated as the dualty gap dvded by the dual soluto. A small relatve dualty gap meas that the prces are represetatve of the system state, thus, a good soluto s acheved. The lower part of the fgures preset charts of resources. These charts show the tasks that each resource has performed utl the curret tme (lower area of the resource charts) ad the tme slot prces (upper area of the resource chart). The tme slots prces prevous to curret tme are the prces whe agets were dog frm agreemets for those tme slots. The prces later tha curret tme are the estmated prces the curret roud of the aucto. Fgure 2. shows the system state at the momet 45, just before projects P4 ad P5 arrve at the system. Whe the frst projects (P1, P2 ad P3) are cluded the system, the dualty gap s hgh, dcato of a bad soluto. But the, the prce formato mechasm makes the prces to stablse, ad the gap becomes smaller. Ths meas that prces are close to equlbrum. Fgure 2. System state at the momet 45 Fgure 3. shows the evoluto of the tasks performed by each resource ad the prces of the tme slots after fshg the smulato. Ths fgure shows how the dualty gap creases whe the projects P4 ad P5 arrve to the system. At ths momet prevous prces dd ot reflect the ew system state (ew projects are the system). After some tme, the prces chage to adapt themselves to the ew system codtos, ad the gap decreases aga. It ca be observed that the ew prces are very dfferet from prevous prces. Ths happes especally resource R1 where prces are very hgh. Note that project P5 has bee rejected although t has a hgh value, because ts value was ot avalable at tme 0, whe projects P1, P2, P3 were watg to start-up. The calculus of the paymet that projects have doe for tme slots (TABLE II. ) shows that the same projects do a paymet hgher tha ther values. Whe projects P4 ad P5 arrve at the system the prces of tme slots of the resource R1 grow because P5 s able to pay hgher prces to be performed. Although P5 accepts hgher prces tha other projects, P1, P2 ad P3 caot be rejected ad fally they must pay the market prces. The fal total value (B T =total values of performed projects mus total delay cost) s TABLE II. PAYMENT FOR TIME SLOTS PER PROJECT Project Value Total paymet P P P P P5 0 0 The smulato ot oly gves the dyamc schedule ad the refused projects, but the value of each resource as well. For stace, Fgure 3. the prces of resource R1 are very hgh durg all tme slots. Ths meas that the resource competece s very valuable (bottleeck), so f the frm s gog to be egaged smlar projects the earby future, t would be useful to clude more resources wth the same competeces. O the other had, prces of resources R2 ad R3 are small, although they are workg o dfferet tasks durg the smulato. So, the possblty of ehacg the rage of capabltes of resources R2 ad R3 should be cosdered; for stace, the case of huma resources, ths ca be doe by meas of trag. Fgure 4. Tasks performed by resources (competeces of resource R2 creased). Fgure 3. Tasks performed by resources. Tj deotes Task j of project P. Fgure 4. shows the evoluto of the system whe the resource R2 s also edowed wth the competece C1 wth effcecy 0.8. Compared wth the prevous case, ow the prce rage s lower for resource R1 ad hgher for R2. Although the durato of task T22 of project P2 s smaller resource R1 tha R2, ow the system have to reallocate real-tme ths actvty to R2. So, R1 ca perform tme the task T51 of the project P5. I ths expermet, the project P5 s accepted ad executed, ad the total value has bee creased from to

6 Ths shows that the system s capable to use the flexblty of resource R2 to mprove real-tme the global performace. B. Complex Dyamc Scearo I order to check the system performace very dyamc evromets, cosder 12 projects (table 3) that arrve at the system every 20 uts of tme (frst P1, secod P2,, ad fally P12). I TABLE III. DD1 ad DD2 are relatg to startg date. Resources ad competeces are smlar to the prevous case study. TABLE III. DYNAMIC PORTFOLIO OF PROJECTS (COMPLEX SCENARIO) Fgure 5. Total value complex expermets Proj. Tasks Task 1 Task 2 Task 3 DD1 DD2 Value P1 C1 50 C2 25 C P2 C3 40 C1 45 C P3 C2 35 C1 40 C P4 C3 30 C1 50 C P5 C1 45 C3 20 C P6 C3 10 C2 45 C P7 C1 20 C2 10 C P8 C3 40 C1 45 C P9 C2 35 C1 10 C P10 C3 30 C1 15 C P11 C1 15 C3 50 C P12 C3 35 C2 50 C We have doe several smulatos by chagg two types of parameters: the respose perod ad the set of competeces of resources. The respose perod s the tme terval betwee the arrval date ad the startg date. Durg ths perod, projects wat the system for rejecto or acceptace decso. If ths perod s log, more projects are watg for decso smultaeously, so decsos wll be more effcet. We have smulated three competece dstrbuto cases: case A (R1 has the competece C1, R2 the C2, R3 the C3), case B (R1 C1, R2 C1 ad C2, R3 C3), ad case C (R1 C1, R2 C1 ad C2, R3 C2 ad C3). Fgure 5. shows the total values obtaed dfferet expermets. Each curve represets the value varato for cases (A, B ad C) whe the respose perod creases. Note that the system effcecy s hgher whe the respose perod creases ad whe the resources are more flexble (they have more competeces). Ths shows that ths scearo the system performace s sutable; the software s able to maage complexty to mprove the global effcecy. VI. CONCLUSIONS Although project maagemet lterature has bee maly cocered wth maagg dvdual projects, practce frms usually work dyamc ad complex mult-project evromets. We propose a mult-aget system ad a aucto mechasm for ole dyamc schedulg mult-project evromets. Projects have tasks to be completed, so they compete for the resources edowed wth the capabltes requred to do some peces of work. The prces of resources emerge edogeously by meas of a aucto process. We show some of the possbltes of ths mult-aget approach to deal wth some of the decsos that maagers eed to take wth mult-project evromets. The system allocates dyamcally resources to projects, ad decdes what projects to accept or reject takg to accout project value, proftablty ad (feedback) operatoal formato. We also show how t s possble to dscover whch resources are the most valuable, so they should be added to the frm. Ths approach cotrbutes to fll the gap betwee the lterature portfolo project maagemet (usually focused o corporate strategy ad face) wth the work mult-project maagemet (maly cocered wth operatoal ssues, schedulg ad resource allocato). ACKNOWLEDGMENTS Ths work has bee doe the cotext of the followg projects: (1) Aget-based Modellg ad Smulato of Complex Socal Systems (SCoSSys), supported by Spash Coucl for Scece ad Iovato, wth grats TIN C03-01 ad TIN C03-02; (2) ABACO VA006A09, (3) the Programa de Creacó y Cosoldacó de Grupos de Ivestgacó UCM-BSCH GR58-08, ad (4) GR251/09 supported by the Juta de Castlla y Leo.

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