Tjdschrft voor Econome en Management Vol. LII, 3, 2007 Project Schedulng Optmzaton n Servce Centre Management By V. VALLS, D. GÓMEZ-CABRERO, M. ÁNGELES PÉREZ and S. QUINTANILLA Vcente Valls Dpto. de Estadístca e Investgacón Operatva, Facultad de Matemátcas, Unverstat de Valenca, Dr. Molner, 50, 46100 Burjassot, Valenca, Span. Vcente.Valls@uv.es. Davd Gómez-Cabrero Dpto. de Estadístca e Investgacón Operatva, Facultad de Matemátcas, Unverstat de Valenca, Dr. Molner, 50, 46100 Burjassot, Valenca, Span. Davd.Gomez@uv.es M. Ángeles Pérez Dpto. de Matemátcas para la Economía y la Empresa, Facultad de Economía, Unverstat de Valenca, Avda. de los Naranjos, s/n, Edfco Departamental Orental, 46022 Valenca, Span. Angeles.Perez@uv.es Sacramento Quntanlla Dpto. de Matemátcas para la Economía y la Empresa, Facultad de Economía, Unverstat de Valenca, Avda. de los Naranjos, s/n, Edfco Departamental Orental, 46022 Valenca, Span. Mara.Quntanlla@uv.es ABSTRACT Ths paper deals wth a real-lfe problem that comes up n the daly management of Servce Centres. It s a complex resource allocaton and task schedulng problem that has to be solved onlne n a dynamc and uncertan envronment. The uncertanty resdes both n the task duratons and the task arrval tmes. It s shown that ths problem can be formulated and solved usng project schedulng models, algorthms and concepts. To deal wth uncertanty, a b-crtera qualty robustness concept n dynamc envronments whch extends the usual concept of statc qualty robustness s ntroduced. Several proactve-reactve schedulng procedures and surrogate measures of robustness are proposed. The computatonal experments carred out on more than 700 randomly generated dynamc scenaros ndcate that proactve schedulng procedures are able to prepare the predctve schedule to antcpate future nfeasbltes. Keywords: Dynamc project schedulng, uncertanty, qualty robustness, proactve-reactve schedulng, Servce Centres. 341
I. INTRODUCTION Ths paper deals wth a complex problem of task schedulng and resource assgnments that comes up n the daly management of company Servce Centres (SC). SCs usually deal wth the requests reported by ether external customers/ctzens or nternal users of an organzaton/company. Internal as well as external servces must be provded wth a gven level of servce. A call from a customer or a vendor requestng nformaton about products or orders, or placng a clam or askng for techncal support, s an unexpected event that requres specalsed attenton from the organsaton. In opposton to unexpected events, programmed tasks are tasks, repettve or not, expected n advance, that can be scheduled wthn a tme wndow and must also be accomplshed by specalsed workers. A programmed task gves place to one or several events when planned for executon. Tools have been developed to automate and ratonalse the daly actvty of SCs. These tools can be termed Servce Centre Management Tools (SCMT). In partcular, once an event s dentfed by an SCMT, t s classfed accordng to a pre-defned herarchcal structure whch contans all events that can occur wthn the system. The database stores the herarchcal structure of possble events and the nformaton requred for ther resoluton. It also stores all nformaton relatve to the specfc resoluton of past events. We are aware of proposed SCMTs that can automate most of the operatons of an SC. In partcular, they usually nclude several software applcatons that allow us to record, track and report all the nformaton nvolved n the daly management of an SC. However, there are two mportant areas that have not been dealt wth, automatc task schedulng and automatc management of human resources. The SC workforce can be bascally dvded nto three levels. Level 1 or Call Centre s made up of telephonsts who follow prevously establshed procedures. Level 2 or Work Desk s made up of multdscplnary specalsts able to resolve the varous problems that were unresolved at level 1. Level 3 s made up of Management. The Level 1 workforce s homogeneous. Everyone s capable of dong the same tasks after recevng a basc tranng course wth the help of a knowledge database and establshed procedures. The Level 2 workforce s heterogeneous. It s generally scarce, hghly qualfed, and expensve. Each specalst s able to solve some, but not all events. Some members of ths workforce tend to specalse n certan tasks 342
and/or acqure greater experence wth certan clents. These members should then be preferentally selected to resolve approprate events. Ths paper s focused on the second level of the SC (SC-Level2). Once an SC event enters the SC-Level2, t gves place to one or several tasks to be performed by the SC-Level2 workforce. Tasks are assocated wth a great deal of nformaton, regardng: (1) a standard duraton, (2) a clent-company servce level agreement (SLA) that establshes maxmum response tmes taken nto account from the tme the events enter the SC, for the begnnng and the end of tasks wth penaltes for delays, (3) maxmal and mnmal generalzed precedence relatonshps between the tasks that reflect technologcal constrants and (4) a crtcalty level ndcatng the clent-prorty n processng the task as soon as possble, ndependently of ts servce level agreements. New events can be contnually ncorporated nto the system. The servce level agreed for each clent, project, and task s a fundamental aspect of the operaton of an SC. The servce level agreements and the crtcalty levels are negotated durng the contractual phase before the SC servces are provded. In practce, the maxmum response tmes are establshed for categores of events, not for each task ndvdually. A task nherts ts maxmum response tmes from the category to whch the event that gves place to t belongs to. The crtcalty level dstngushes between tasks that ntally have the same servce level agreement. It expresses specfc company prortes whose non-fulflment s not penalzed. For example, a task representng the fxng of a prnter may have a dfferent prorty dependng upon whether t s a server-prnter or a personal-prnter. The SC-Level2 workforce s made up of specalsts, wth specalsatons n one or more areas of knowledge. A knowledge area descrbes a set of tasks a specalst wth specalsaton n ths area can handle. For each area, the effcency level (junor, senor or standard) of a specalst descrbes how well he or she can handle the tasks n that area and can be quantfed by the average servce tme. Senor (Junor) ndcates that the person s one of the most (least) sutable for the area and mples a reducton (ncrease) of the standard duraton. Each specalst has hs or her own tmetable whch ndcates the workng hours every day. Pre-empton s not allowed, only nterruptons of the tasks due to breaks n the worker tmetables are permtted. To process a task just one specalst specalsed n the area to whch the task belongs s requred. In general terms, the man objectve for an SC-Level2 schedulng system s to obtan n real tme a feasble plan of acton (a task 343
schedule and an assgnment of workers to tasks) that satsfes the technologcal constrants and does not use more specalsts than are avalable n each perod of tme and that satsfes the servce level agreements. Furthermore, to mprove the effcency of an SC t s requred to reach some other secondary, although also mportant, objectves: attend the urgency levels mposed by the crtcalty task levels, to obtan well-balanced worker workloads and to assgn the most qualfed specalst to each task. Plans are made at certan ponts n tme. Durng executon, however, new events are contnuously ncorporated nto the system and resolved events are dropped from the system. The events arrve at uncertan tmes and ther servce tmes are also uncertan. Due to the full recordng capablty of SCMTs, we can realstcally assume that some advance knowledge about the probablty dstrbutons of both types of tmes s avalable. In ths type of rapdly changng envronment, plannng becomes a contnuous onlne process. SCMTs usually nclude so-called schedulng tools to help managers to dynamcally make allocaton and schedulng decsons at certan decson ponts. Wth the nformaton provded by these tools, managers apply ther own set of rules to gve response to the requests of new servces that have arsen. Ths s a case of pure reactve schedulng (Herroelen and Leus, 2005). Ths smple approach s amed at a quck generaton of a feasble plan. It s an approach that ams at quckly satsfyng the man constrants rather than an optmzng approach. At a gven decson pont, t only consders the tasks already n the system. Ths myopc approach mght and frequently does lead to nfeasble scenaros that could possbly be avoded by takng nto account the avalable statstcal knowledge on forthcomng tasks to generate robust plans whch preserve feature feasblty. If we restrct our attenton to those tasks already n the system up to a certan pont n tme and use the average estmaton of task duraton, ths manageral problem can be consdered statc and determnstc, and be modelled as a project schedulng problem (the Sklled Workforce Project Schedulng Problem, SWPSP) as wll be apparent n secton II. Ths paper explores the use of project schedulng technology n the desgn of a system management model and the correspondng soluton methodology to deal wth the SC-Level2 allocaton and schedulng problem. The am s not only to make the best allocaton and schedulng decson at certan decson ponts but to also learn from the past to antcpate the future and produce more robust and effectve solutons. 344
Intensve research efforts over the past decades have been made n the feld of project schedulng greatly expandng the varety of models and soluton procedures under study. Overvews of state-of-the-art models and algorthms can be found n Demeulemeester and Herroelen (2002) and Neumann et al. (2003). The noton of skll s well known n the feld of personnel assgnment (Jang et al. (2004)), but not often consdered n project schedulng. Néron (2002) and Bellenguez and Néron (2004) consder the Resource Constraned Project Schedulng problem (RCPSP) where resources are staff members that have one or more skll. To be processed, each task of the project needs a gven number of staff members for each skll. However, task duratons do not depend on the workers assgned to ther executon. A revew of the state of the art n project schedulng wth generalzed precedence relatonshps can be found n Neumann et al. (2003). The books of Bagch (1999) and T Kndt and Bllaut (2002) can be consdered as reference books for researchers workng on mult-crtera schedulng problems. Other nterestng mult-crtera papers studyng some elements of the SWPSP are the followng: Ca and L (2000) consdered the problem of schedulng staff wth mxed sklls, and Vana and Sousa (2000) appled multobjectve versons of smulated annealng and tabu search to the RCPSP. The nterest n schedulng consderng due dates has been wde n felds such as machne schedulng; a revew can be found n Gordon et al. (2002). In the area of project schedulng, the paper of Vanhoucke et al. (1999) s one of the frst works wth due dates. In ths paper, resources are not consdered and penalty costs for earlness or tardness are. In a more recent paper, Vanhoucke (2002) consdered the resource constrants. Valls et al. (2006) studed the RCPSP wth due dates. The papers of De Reyck and Herroelen (1999), Hartmann (2001), Helmann (2001) and Helmann (2003) looked at the multmode RCPSP, where the task duraton depends on the mx of resources assgned to t, the objectve s C max mnmzaton and no worker tmetables are consdered. Recent surveys about project schedulng wth varable avalablty resources, such as tmetables, can be found n Sanlavlle and Schmdt (1998) and Neumann et al. (2003). Artgues and Roubellat (2000) study the case where, n a multproject, mult-mode settng wth ready tmes and due dates, t s 345
desred to nsert a new unexpected actvty nto a gven schedule such that the resultng mpact on maxmum lateness s mnmzed. As s ponted out n Van de Vonder et al. (2006), the vast majorty of the research efforts on project schedulng focuses on the development of exact and heurstc procedures for the generaton of a workable baselne schedule (predctve schedule), assumng complete nformaton and a statc and determnstc envronment. Durng executon, however, a project may be subject to consderable uncertanty, whch may lead to numerous schedule dsruptons that can lead to numerous types of costs. A baselne schedule wth express antcpaton of dsruptons, whch s protected aganst certan undesrable consequences of reschedulng, s called robust. Van de Vonder et al. (2005a) dstngush between two types of robustness: qualty robustness and soluton robustness. Qualty robustness s defned as the probablty that a project ends wthn the project deadlne. Soluton robustness, also referred to as stablty, s defned as a qualty of the schedulng envronment when there s lttle devaton between the baselne and the executed schedule. In general, there are two approaches to deal wth uncertanty n a schedulng envronment (Herroelen and Leus, 2005): proactve and reactve schedulng. Proactve schedulng constructs a baselne schedule that accounts for statstcal knowledge of uncertanty. The consderaton of uncertanty nformaton s used to make the predctve schedule more robust,.e. nsenstve to dsruptons. Reactve schedulng nvolves revsng or re-optmzng a schedule when an unexpected event occurs. Robust project schedulng s a growng feld of research. Recently, many proactve-reactve schedulng procedures (Herroelen and Leus, 2005; Van de Vonder et al., 2005ab; Van de Vonder et al., 2007abc; Lambrechts et al., 2007; Deblaere et al., 2007) have been proposed and computatonally tested for the case of a statc envronment n whch the uncertanty of the project resdes n the actvty duratons. However, the problem dealt wth n ths partcular paper s dynamc and consders both actvty duraton and actvty arrval tme uncertantes. It s nterestng to note that nether the ralway schedulng polcy nor the polcy of mantanng the baselne resource allocaton when developng reactve schedulng procedures are justfed n an SC-Level2 envronment. The management model we propose n ths paper ntegrates three man elements: a determnstc and statc optmzaton algorthm; a proactve procedure and a reactve procedure. At a gven tme t, the 346
determnstc and statc algorthm generates a predctve schedule, takng nto consderaton the unfnshed tasks already n the system and assumng the average actvty duratons as ther determnstc duratons. The predctve schedule mantans the current resource allocaton only for the tasks n process at tme t and not for the tasks prevously scheduled but not started yet. Then, a proactve schedulng procedure s appled to the predctve schedule to produce a baselne schedule that antcpates possble future nfeasbltes. Ths concatenaton of procedures s perodcally appled at subsequent tmes wth tme ntervals of length D (reschedulng tmes). Between tmes t and t + D, the problem s consdered statc: tasks arrvng wthn ths nterval are gnored untl tme t + D when the (re)schedulng determnstc and statc algorthm s called agan. Durng project executon, tasks may take longer or shorter than ntally expected (average duraton) causng devatons between the actually realzed and planned completon tmes. Each tme such a devaton occurs, a fast reactve schedulng procedure s appled to repar the schedule such that the protecton aganst future nfeasbltes s optmzed. The actual schedule that s obtaned after these modfcatons s called the realzed schedule. The rest of the paper s organsed as follows: In Secton 2 we state the SWPSP model and the essentals of the soluton approach proposed n Valls et al. (2007). In Secton III, we further elaborate on the dynamcs of the decson makng process, propose several surrogate measures of the b-crtera qualty robustness consdered n ths dynamc envronment, propose several proactve-reactve schedulng procedures and present the results of the computatonal experments carred out. Fnally, conclusons are gven n Secton IV. II. THE SKILLED WORKFORCE PROJECT SCHEDULING PROBLEM The Sklled Workforce Project Schedulng Problem les at the core of the proposed management model and s the project schedulng problem to be solved at each reschedulng tme. Valls et al. (2007) have formulated ths problem, shown ts NP-completeness and proposed an effcent heurstc soluton procedure for t. In ths research work, we use ths heurstc procedure to generate at each reschedulng tme a predctve schedule on whch the proactve schedulng procedure wll act. The types of nfeasbltes that such a predctve schedule may 347
ncur depend on the characterstcs of the SWPSP model and the soluton approach appled. In ths secton, we provde a formulaton of the SWPSP and the essentals of the soluton procedure. We refer the nterested reader to the aforementoned paper for further detals. A. Tasks The project conssts of a set of n tasks numbered 1 to n (J = {1, 2,..., n}), where each task has to be processed wthout pre-empton to complete the project. The dummy tasks 1 and n represent the begnnng and end of the project. The clent-company servce level agreement establshes for each task j a maxmum startng tme, ms j, and a maxmum fnshng tme, mf j. Both dates can be exceeded, however some costs are ncurred. We defne sc j (fc j ) as the cost assocated to the delay of one unt n the startng (fnshng) tme of task j. Each task j has assocated a standard duraton, d j, and should be processed wthout pre-empton by just one worker. Interruptons of the tasks due to breaks n the worker tmetables are permtted. The crtcalty level, c j, models the clent-prorty n processng the task j as soon as possble and s ndependent of ts maxmum startng and fnshng tmes. A larger value of the crtcalty level ndcates a larger prorty. The dummy tasks 1 and n requre no resources, do not have assocated any maxmum startng and fnshng dates and ther duratons and crtcalty levels are null. Duratons, costs, maxmum dates and crtcalty levels are assumed to be non-negatve ntegers. B. Workforce The SC workforce s made up of nw workers, W = {w 1,,w nw }, wth specalsatons n one or more knowledge areas A = {A 1,, A na }. A knowledge area descrbes a set of tasks a worker wth specalsaton n ths area can handle. A s a partton of the set of tasks. We wll denote by A(j) A the area assgned to task j J, by W(A h ) W the set of workers that can handle the tasks of the area A h A, by W(j) W the set of workers that can handle the task j and by A(w k ) A the areas a worker w k W can be assgned to. 348
Gven an area A h, for each w k W(A h ) an effcency level s defned, e kh, descrbng how well w k can handle the tasks n area A h. The effcency level s represented as an nteger value, e kh 1. The smaller the value of e kh, the greater the sutablty. We wll consder three effcency values: e kh = 1, 2 and 3, representng respectvely that w k s a senor, a standard or a junor worker for area A h. Assgnng a senor (junor) worker to a job mples a reducton (ncrease) of the standard duraton of the job. We wll consder a reducton percentage of RP % and an ncrease percentage of IP %. The company has defned RP = IP = 25%. We wll denote by d jk the duraton of task j f processed by worker w k. If w k s a standard worker d jk = d j. In other case, d jk s calculated by applyng the reducton or ncrease percentage to d j and roundng up the result to the nearest nteger number. Each worker has hs or her own tmetable whch ndcates the workng hours every day. The tmetable of worker w k s represented by a vector (av kt ) t = 0,,PDUR where av kt wll denote the avalablty of worker w k n nstant t. If nstant t s a tmetable workng nstant for w k then av kt = 1 or f t s not av kt = 0. PDUR s the schedulng horzon. In our model, tasks can only be nterrupted at tmetable breaks and have to be reassumed mmedately at the end of the break. Ths means that n nterval [s j, f j [ task j s n progress at tme t exactly f av kt = 1. Thus gven a start tme s j and assumng task j s processed by w k, the fnshng tme f j of task j s unquely determned by: f t* mn t* avkt = d jk t ( s, w ) = 1 f * exsts t= sj + otherwse j j k Obvously f j (s j, w k ) s j + d jk. C. Generalzed Precedence Relatonshps The model consders generalzed precedence relatonshps (GPRs) between the tasks,.e. mnmal and maxmal tme lags between tasks startng and/or fnshng tmes. We consder four types of GPRs: startstart (SS), start-fnsh (SF), fnsh-start (FS) and fnsh-fnsh (FF). A mnmal tme lag ndcates that a task cannot start/fnsh earler than a certan number of tme unts after the start/fnsh of another task. 349
A maxmal tme lag ndcates that a task cannot start/fnsh later than certan tme unts after the start/fnsh of another task. Tme lags are consdered to be postve, negatve or zero ntegers. Furthermore, tme lags may represent ether tme perods or percentage work contents. In the latter case, the length of the tme lag s a percentage of the duraton of one of the two tasks nvolved n the relatonshp. Only mnmal relatonshps wll be consdered n the model wthout any loss of generalty, negatve/postve maxmal tme lags wll be replaced by equvalent postve/negatve mnmal tme lags of opposte drecton. A generalzed precedence relatonshp between two tasks, and j, s represented by a three components vector, d,j. The frst component ndcates the relatonshp type (FF, FS, SF or SS) and the second the tme lag. The thrd shows, n case of a percentage tme lag, the task used to calculate that percentage, n a tme perod lag the value s fxed to zero. For example, d 2,3 = (SS, 5, 2) means that task 3 can start after 5% of task 2 has been executed. Note that not all combnatons of relatonshp type and task make sense n the case of percentage tme lags. However, all generalzed precedence relatonshps that appear n practce can be modelled n ths way. To ensure that the dummy start and fnsh tasks (1 and n) correspond to the begnnng and the end of the project we nsert the generalzed precedence relatons d 1,j = (FS,0,0) j J, j 1 and d j,n = (FS,0,0) j J, j n. Gven a d,j = (SS, p, a) we wll denote as SS j the length n tme perods of lag p. If d,j s a tme perod relatonshp (a = 0) then obvously SS j = p. If d,j s a percentage relatonshp (a = or a = j), then SS j can only be calculated when the startng tme and the worker assgned to task a have already been establshed. For example, gven the relatonshp d,j = (SS, p, ) wth p 0, beng executed by w k, and s beng the startng tme of task, we calculate: SS j t p d mn t avkt = = * * 1 t= s 100 + k s f t* exsts otherwse In the same way, we defne SF j, FS j and FF j. 350
D. SWPSP formulaton In ths secton, we provde a conceptual formulaton of SWPSP wth the am of unambguously establshng the constrants and objectves nvolved. A schedule or soluton S of an SWPSP nstance s represented by two vectors S = (s, w) where the vector s = (s 1, s 2,..., s n ) of non negatve ntegers, ndcates for each task j ts startng tme and w = (w(1), w(2),..., w(n)) assgns to each task a worker w() W(), w(1) = w(n) s a dummy worker. A schedule S = (s, w) s a feasble schedule f the startng tmes fxed by s satsfy the generalzed precedence relatonshps, the resource constrants and the maxmum startng and fnshng dates. Therefore, S s a feasble schedule f S satsfes the followng constrants: (1) w() W() V (2) s j s SS j (,j) E SS (3) f j s SF j (,j) E SF (4) s j f FS j (,j) E FS (5) f j f FF j (,j) E FF (6) s ms V (7) f mf V { k } (8) V w( ) = w and s t < f 1 w k W, 0 t < PDUR (9) av ws () = 1 V (10) 0 s PDUR V (11) s 1 = 0 where f = f (s,w()) V Equatons (1) ensure that each task s assgned to a unque worker of W(). Equatons (2)-(5) denote GPRs mnmal tme lags, where SS j, SF j, FS j, FF j are calculated as ndcated n secton 2.3. Equatons (6) and (7) represent the date constrants. Equatons (8) guarantee that each worker does not execute more than one task smultaneously. Equatons (9) force tasks to start n a tmetable workng nstant. Equatons (10) and (11) fx the task startng tmes wthn the schedulng horzon. 351
Gven the nherent complexty of the problem we are dealng wth, the man objectve of an SC schedulng system s to obtan a feasble soluton, S, satsfyng all the prevous constrants. However, to mprove the effcency of the SC, t s requred to reach some other secondary, although also mportant, objectves. These objectves, ranked n order of mportance, are the followng: 1. To attend to the urgency levels mposed by crtcalty levels. To acheve ths objectve, we have defned the followng functon that measures f the more crtcal tasks are carred out before the least crtcal ones. 2. To assgn the best worker to each task, modelled by: 3. To obtan well-balanced worker workloads levels, represented by Z S Loads S 3 Z S Crtcalty S c s 1 n ( ) = ( ) = n ( ) = ( ) = () () = 1 Z2 S AD S e wa ns ( ) = ( ) = k = 1 j= 1 loadk ave load, ave load j j where dk J w k ns loadk = ()= and ave load = 1 1 load f n 1 ns k = 1 av t= 0 kt k The optmum s found accordng to a lexcographcal order defned by the prevous crtera ndces, Z 1 Z 2 Z 3, and denoted by mn- Lex(Z). Determnng an optmal soluton of mn Lex (Z) s equvalent to fndng a soluton S* S 3 wth: 0 { } () 1 S = S S s feasble 1 ( 2) S = S S Z( S ) = mn Z S ( 1, 2, 3 1 ( )) = S S 352
One of the man objectves of an SC schedulng system s to get rd of nfeasble plans and to manage only feasble ones. Therefore, specal attenton has been pad to the search of feasble solutons and that s why feasblty-objectves appear n the frst postons of the rankng. The SWPSP can be seen as a case of a mult-mode resource constraned project schedulng problem. However, the SWPSP sgnfcantly dffers from the mult-mode constraned project schedulng problem (Helmann, 2001) n two man aspects: the objectve of the optmzaton problem and the consderaton of worker tmetables. E. Soluton approach The proposed soluton approach combnes local searches wth typcal populaton management technques of genetc algorthms. These types of algorthms are commonly referred to as Hybrd Genetc Algorthms or Genetc Local Searches (Gendreu and Potvn, 2005), and some of them are categorzed as Memetc Algorthms (Moscato, 1989). For an ntroducton nto GAs, we refer to Goldberg (1989). As a feasble soluton may not exst for certan nstances of the SWPSP and even the problem of knowng f t exsts s an NP-complete problem, the soluton methodology s prepared to handle nfeasble solutons where some constrants are relaxed. The constrants that can be unfulflled and the functons to measure the gaps of these constrants are the followng: Cycles caused by GPRs represent an essental element of an SWPSP nstance. The problem of testng f there s a worker assgnment that makes all generalzed precedence relatonshps n strong components feasble s NP-complete. Therefore generalzed precedence constrants wthn strong components wll be consdered as soft constrants and may not be satsfed. To measure the cycle nfeasbltes we wll use: + + ( ) = + + + + GPR S SSj sj s SFj f j s (, j) ESS / C()= C( j) (, j) ESF / C()= C( j) + FSj s j + f + FF f + f (, j) EFS / C()= C( j) (, j) EFF / C()= C( j) j j + where C() denotes the strong component contanng, V and a + = max(a,0) a. 353
A schedule S that satsfes all the temporal constrants, GPR(S) = 0, s called tme-feasble. The opposte wll be called tme-nfeasble. Maxmum date s constrants, (6) and (7) n the model, are also consdered to be soft constrants and therefore can be unfulflled. The tardness n the dates wll mply some economc penaltes for the company (sc j and fc j, prevously defned). The followng functon measures, n soluton S = (s, w), the weghted total delay, wth respect to the date agreements: A schedule S that satsfes the maxmum date constrants, TARD(S) = 0, s called date-feasble. The opposte wll be called date-nfeasble. The resource avalablty model constrants, (8), are also consdered to be soft constrants. The use of more resource unts than the ones avalable can be made up wth extra workng hours or contractng more workers. We defne wc k as the cost assocated wth the excess of one unt n the tmetable of worker w k. The followng functon measures the cost of the worker avalablty excess n soluton S = (s, w): Resource S ns TARD S sc s ms fc f mf PDUR k = 1 t= 0 n + n ( ) = j j j + j j j= 1 j= 1 ( ) = ( ) wc k V s t < f and w( ) = k av max 0, 1 kt A schedule S that satsfes the resource constrants, Resource(S) = 0, s called resource-feasble. The opposte wll be called resourcenfeasble. Model constrants (1), (2)-(5) for relatonshps between tasks from dfferent strong components, (9), (10) and (11) wll always be satsfed. The algorthm was tested on over 720 randomly generated nstances. The nstances had 50, 100 or 500 tasks. The computatonal experments ndcated that the procedure s effectve and that the computng tmes requred by the algorthm are small enough to be of use n the onlne management of a Servce Centre. j + 354
Ths algorthm can be easly modfed to work only wth resourcefeasble schedules. In what follows, we wll refer to ths modfed verson as algorthm SWPSP. III. DYNAMIC PROACTIVE-REACTIVE SCHEDULING PROCEDURES When an event occurs at a gven tme, t gves place to a task. If the event occurs agan, t gves place to another task whch s a copy of the prevous task. Tasks orgnated by the occurrence of the same event have the same characterstcs and we wll say that they belong to the same type of task. To ease the notaton n ths secton, we wll consder that all workers are avalable at any tme nstant. We wll also say that the workers that can handle the same types of tasks wth the same effcency levels belong to the same type of worker. Gven a schedule S and a worker type, we wll denote by TASSIGN() the set of tasks assgned to any worker of type n S. At each re-schedulng tme, the algorthm SWPSP s appled to generate a predctve schedule. Two types of schedule nfeasbltes can be consdered: tme-nfeasblty whch s measured by the functon GPR() and date nfeasblty whch s measured by the functon TARD(). In ths context, the am of a proactve schedulng procedure s to prepare the predctve schedule n such a way that forthcomng tasks can be feasbly scheduled as much as possble. In other words, the goal s to mnmze both the values of GPR(realzed schedule) and that of TARD(realzed schedule). Ths s a case of b-crtera qualty robustness n a dynamc envronment. In ths context, qualty robustness refers to the nsenstvty of the values of some determnstc objectve functons of the baselne schedule not necessarly the makespan to dstortons. One way to make the problem workable s to develop proactve schedulng procedures amed at optmzng some easy and quck to calculate surrogate measures of the qualty robustness consdered n ths paper that take nto account the current predctve schedule and the avalable statstcal knowledge on forthcomng tasks. In ths secton, we ntroduce three schemes for generatng surrogate measures of ths type. We also present fve proactve schedulng algorthmc schemes, a reactve schedulng procedure and the results of the computatonal experments carred out. 355
A. Surrogate measures based on expected workload Let us suppose that S s the predctve schedule generated at re-schedulng tme t. The surrogate measures based on expected workload combne two knds of nformaton: probablstc nformaton about tasks possbly comng after tme t and worker type workloads assgned by S. Gven the nherent rapdly changng nature of the problem, we are nterested only n the tasks that mght arrve up to a certan tme t 1 > t. If t s expected that many tasks of type j wll arrve nto the system n the tme perod [t, t 1 ], then t would be a sensble polcy to modfy the predctve schedule n such a way that as many workers able to process type j tasks as possble would be unoccuped n the tme perod [t 2, t 3 ] where t s expected that the comng type j tasks wll be processed. In subsecton III.E, we wll further elaborate on t 1, t 2 and t 3. We assume that for each type of task the probablty dstrbutons of the tme between successve task arrvals and of the number of task arrvals n any tme perod of a gven length can be approxmated from hstorcal data. Then, the followng expected value and probabltes can easly be computed: W W W p j l = p j b = probablty that a type j task arrves nto the system between tmes last j and t where last j s the arrval tme of the last type j task arrved at pror to t. probablty that a type j task arrves nto the system before tme t 1 gven that t has arrved after tme t. ENT j = expected number of type j tasks to arrve n tme perod [t, t 1 ]. Then, the followng weghtng factors can be computed: l b l 1 p j 2 p j 3 ENTj 4 p j * j = W W W l j = b j = j = l p p ENT p * b l b 5 p j * ENTj 6 pj * d j 7 pj * d j j = W W b j = l j = b p * ENT p * d p * d l b 8 ENTj * d j 9 pj * ENTj * d j 10 pj j = W j = W l j = b ENT* d p * ENT* d p where d j s the standard duraton of type j tasks. ENT j ENT * ENT * d j * ENT * d j 356
The workload of the workers of type n tme nterval [t 2,t 3 ] s defned as: Occupancy ( ) = Workload t, t ( ) ( 2, 3) t t * NWT 3 2 ( ) = ( ) ( h) where Workload t, 2, t3 mn t3, fh max t2, s h TASSING() AT( t2, t3) where NWT s the number of workers of type and TASSIGN() s the set of tasks assgned to workers of type and AT(t 2,t 3 ) = { h J fh t2 and sh t3}. Combnng both knds of nformaton, we propose the followng scheme for generatng surrogate measures of robustness: EXWORK1 = W * j * Occupancy( j) where W j * denotes any of the W j k, k = 1,.., 10, defned above. The applcaton of ths measure may result n unbalanced worker type occupances: occupances of worker types wth large weghts tend to decrease to zero and, therefore, occupances of worker types wth small weghts tend to ncrease to one. A way of smoothng ths effect s to use the exponental functon and defne: EXWORK2 = W j j Occupancy j where q s an nput parameter. Note that each scheme gves rse to ten surrogate measures whose relatve effcency has to be expermentally tested. B. Surrogate measure based on GPRmax slack The surrogate measure presented n ths subsecton s amed at measurng how well the predctve schedule s prepared to absorb forthcomng tasks wthout deteroratng the value of the tme nfeasblty functon GPR. The tme nfeasblty of a schedule results from the non-fulflment of maxmal GPRs. Gven a schedule S and a task j belongng to a strong component we defne the GPRmax slack of task j, GPRmax-slack(j), as the maxmum number of tme unts that * j * q [ ] ( ) 357
task j can be delayed wthout volatng any maxmal GPR. If a task j does not belong to any strong component, then GPRmax-slack(j) = 0. It seems apparent that the greater the total GPRmax slack of a schedule the more preferable the schedule s. Also, t seems more preferable to use a schedule where the total GPRmax slack s evenly dstrbuted over all the tasks than a schedule where an actvty has a hgh GPRmax slack and the others have a low GPRmax slack. For ths reason, the contrbuton of the GPRmax slack of a task to the surrogate measure s lmted to a maxmum value of maxslack, where maxslack s an nput parameter. As before, we are only nterested n the GPRmax slack of the tasks scheduled up to a certan tme t 4 > t. Therefore, gven a task j, we defne: flexblty(j) = 0, f j does not belong to a strong component or s(j) > t 4 ; and flexblty(j) = mn{gprmaxslack(j), maxslack}, otherwse. Then, the thrd surrogate measure s defned as FLEX( S) = flexblty( j). j C. Proactve schedulng procedures The frst three proactve schedulng procedures we propose are Local Search Algorthms focused on optmzng the surrogate measures EXWORK1, EXWORK2 or FLEX of any gven schedule. The proactve schedulng procedures wll be denoted wth the name of the surrogate measure they optmze. They are manly based on two operators, Inserton and Swap. The Inserton operator selects a task assgned to a type of worker and nserts t nto the lst of tasks assgned to another type of worker. The Swap Operator conssts n selectng two tasks assgned to two dfferent worker types and exchangng ther postons n the lsts of tasks assgned to the worker types. To mprove the surrogate FLEX measure, the Swap operator swaps frst those tasks wth lower GPRmax-slack value. To mprove the surrogate EXWORK measures, tasks assgned to worker types whch contrbuton to the EXWORK measure s large are ether reassgned to worker types wth low contrbutons, or swapped wth lower duraton tasks assgned to worker types wth low contrbutons. We also propose two other proactve schedulng procedures, FLEX- EXWORK1 and FLEX-EXWORK2, that consst of the consecutve applcaton of the procedure FLEX and the procedures EXWORK1 or EXWORK2, respectvely. None of the developed Local Search Algorthms are allowed to ncrease the GPR value of the schedule they start wth. However, they 358
are allowed to ncrease the TARD value up to a percentage of the ntal value TARD(predctve schedule). In the two cases where the FLEX procedure s frst appled, the value of the FLEX measure of the resultng schedule cannot be decreased by the posteror applcaton of an EXWORK procedure. D. Reactve schedulng procedure The reactve schedulng procedure has the role of reactng when a dsrupton occurs. To be effcent n an onlne envronment t needs to be fast and to avod ncreasng the GPR() value, f possble. In our model, reactve schedulng s used when the actual completon tme of a task j s dfferent from that whch was planned. If the actual completon tme of task j s smaller than expected, then the next task, next_task, assgned to the same worker that has processed task j, w(j), s scheduled to start as early as possble. If the actual completon tme of task j s bgger than expected, then two dfferent cases can be consdered. In the frst case, the delay n the process of task j does not force next_task to start later than planned; then, next_task s scheduled to start as early as possble. In the second case, the startng tme of next_task must be delayed. Before delayng any task, the contrbuton of each mnmal generalzed precedence relatonshp to the GPR value of the current schedule s computed and called the ntal contrbuton of ths GPR. The reactve schedulng procedure starts by schedulng next_task at the new completon tme of task j. Then, the mnmal GPRs wth orgn n task next_task are consdered n turn. If the current contrbuton of a gven mnmal GPR s greater than ts ntal contrbuton, then the start of ts fnal task s delayed untl the ntal contrbuton s restored. Ths procedure s appled agan to all delayed tasks untl no more tasks are delayed. In case that next_task has been delayed, then the whole process starts agan wth j = next_task. E. Computatonal experments The computatonal experments are based on stochastc smulaton analyss. Smulaton analyss s based on generatng a heterogeneous set of possble scenaros and evaluatng the qualty of the procedures at each scenaro. Each scenaro has a fxed set of types of workers and a fxed set of types of tasks. Each type of task has assgned two exponental dstrbutons: one for the tmes between arrvals and one 359
for the duratons. For each type of tasks there s a set of tasks: each task has an arrval tme and a duraton, both values beng computed from the related exponental dstrbutons. At each scenaro, the GPR graph structure s generated usng the methodology presented n Schwndt (1996). We defned a set of generator parameters (under study) and for each combnaton we generate several scenaros. Near to 100 000 scenaros were generated. From ths ntal set, the hardest scenaros for each combnaton of a subset of generator parameters were selected. Ths prelmnary selecton was based on the computatonal results of an ntal verson of the algorthm; a total number of 729 benchmark scenaros were selected. Wth the am of fnetunng the proposed proactve schedulng procedures, we use the FRace procedure. FRace s a heurstc method for fnetunng metaheurstcs. Ths procedure was frst used n Brattar (2002). At each teraton, t emprcally selects from a set of confguratons a set of canddates by dscardng bad ones as soon as statstcally suffcent evdence s gathered aganst them. FRace s based on the Fredman test, a statstcal method for hypothess testng also known as Fredman two-way analyss of varance by ranks. Ths procedure allows comparng many confguratons wth lower computatonal tme than a brute-force (or complete) comparson method. There are fve proactve schedulng procedures (EXWORK1, EXWORK2, FLEX, FLEX-EXWORK1 and FLEX-EXWORK2) and for each one of them there are several optons for weghtng factors, t 1, t 2 and t 3 for EXWORK measures; and several optons of t 4 and maxslack for FLEX measure. There s a consderable number of possble combnatons, from all of them FRace selected (non-dscarded) a reduced number after runnng 300 nstances, addng 50 nstances at each FRace teraton. From the fnal set of confguratons we have selected three promsng optons, C2, C3 and C4, respectvely the best confguratons for FLEX, FLEX-EXWORK1 and FLEX-EXWORK2 not dscarded by FRace. FRace dscarded all the combnatons that consdered only EXWORK measures. C1 denotes the algorthm that does not use any proactve schedulng procedure. Tables 1 and 2 show computatonal results for GPR(realzed schedule) and TARD(realzed schedule), respectvely. The results show that proactve schedulng procedures are able to reduce average GPR(realzed schedule). Among them, EXWORK procedures (C3 and C4) decrease most the average GPR(realzed schedule), but the medan s better for C2, so a further study s needed. Consderng TARD(real- 360
zed schedule), t s clear that EXWORK procedures mprove the qualty of the algorthm. TABLE 1 Computatonal Results: GPR Average Standard Mnmum Maxmum Percentle Devaton 25 50 75 C1 443.48 401.22 8 3,030 202.00 334.50 537.25 C2 422.06 401.83 8 3,626 184.75 310.50 497.25 C3 420.03 314.55 7 3,420 222.00 346.50 530.75 C4 420.03 298.25 30 3,420 232.00 355.50 527.00 TABLE 2 Computatonal Results: TARD Average Standard Mnmum Maxmum Percentle Devaton 25 50 75 C1 217,624.56 167,659.86 2,531 950,049 92,083.00 173,028.50 302,834.50 C2 217,422.60 167,290.18 2,467 951,213 92,947.25 174,569.50 300,616.50 C3 163,938.38 137,538.36 2,147 928,826 64,332.25 124,371.00 230,066.00 C4 157,545.84 131,280.03 2,080 948,754 63,889.75 119,604.50 219,445.00 Statstcal tests are necessary n order to confrm computatonal dfferences between confguratons. The Kolmogorov-Smrnov test rejects normal dstrbuton hypothess for the TARD() measure and Q-Q plots reject the normal dstrbuton hypothess for the GPR() measure; consequently, we use the non-parametrc Wlcoxon sgned-rank test. The Wlcoxon test does not requre assumptons about the form of the dstrbuton of the data. We perform the Wlcoxon test on the two qualty measures GPR() and TARD(). Wth regard to GPR(), C2 obtans the best sgned rank value and tests show that the dfferences between C2 and the rest of confguratons are sgnfcant. Among the rest of the confguratons, there are no sgnfcant dfferences (see Table 3). Regardng TARD(), dfferences are sgnfcant for each par wse comparson, (see Table 4). C4 s the confguraton that obtans the best rank-sgned results followed by C3, C2 and C1, n ths order. 361
TABLE 3 Sgnfcance of Wlcoxon tests: GPR C3 vs C1 C2 vs C1 C4 vs C1 C2 vs C3 C4 vs C2 C3 vs C4 Sgnfcance (two-taled) 0.00 0.00 0.000 0.52 0.56 0.59 TABLE 4 Sgnfcance of Wlcoxon tests: TARD C3 vs C1 C2 vs C1 C4 vs C1 C2 vs C3 C4 vs C2 C3 vs C4 Sgnfcance (two-taled) 0.00 0.01 0.000 0.00 0.00 0.00 IV. CONCLUSIONS In ths paper, we have dealt wth a real-lfe problem that comes up n the daly management of Servce Centres. It s a complex resource allocaton and task schedulng problem that has to be solved onlne n a dynamc and uncertan envronment. The uncertanty resdes both n the task duratons and the task arrval tmes. We have shown that ths problem can be formulated and solved usng project schedulng models, algorthms and concepts. At certan re-schedulng tmes, a predctve schedule s constructed by solvng an nstance of the Sklled Workforce Project Schedulng Problem. Problem SWPSP s a statc and determnstc project schedulng problem that ncorporates some real features not prevously consdered together n the lterature, whch greatly ncreases the computatonal dffculty of the problem. Three of the authors of ths paper have developed a rapd and effcent enough algorthm for the onlne resoluton of ths complex mult-objectve and mult-mode project schedulng problem. The man objectve of an SC plannng system s to generate feasble solutons. Therefore, we have proposed to generate robust schedules that preserve future feasblty. To ths end, we have ntroduced a b-crtera qualty robustness concept n dynamc envronments whch extends the usual concept of statc qualty robustness. We have also 362
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