Examensarbete. Rotating Workforce Scheduling. Caroline Granfeldt

Transcription

1 Examensarbete Rotatng Workforce Schedulng Carolne Granfeldt LTH - MAT - EX / SE

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3 Rotatng Workforce Schedulng Optmerngslära, Lnköpngs Unverstet Carolne Granfeldt LTH - MAT - EX / SE Examensarbete: 30 hp Level: A Supervsor: Torbjörn Larsson, Optmerngslära, Lnköpngs Unverstet Supervsor: Ann Bertlsson, Schemag AB Examner: Elna Rönnberg, Optmerngslära, Lnköpngs Unverstet Lnköpng: November 2015

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5 Abstract Several ndustres use what s called rotatng workforce schedulng. Ths often means that employees are needed around the clock seven days a week, and that they have a schedule whch repeats tself after some weeks. Ths thess gves an ntroducton to ths knd of schedulng and presents a revew of prevous work done n the feld. Two dfferent optmzaton models for rotatng workforce schedulng are formulated and compared, and some examples are created to demonstrate how the addton of soft constrants to the models affects the schedulng outcome. Two large realstc cases, wth constrants commonly used n many ndustres, are then presented. The schedules are n these cases analyzed n depth and evaluated. One of the models excelled as t provdes good results wthn a short tme lmt and t appears to be a worthy canddate for rotatng workforce schedulng. Keywords: Optmzaton, Rotatng schedules, Cyclcal schedulng URL for electronc verson: Granfeldt, v

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7 Acknowledgements I would lke to express my deepest grattude to my supervsor, Torbjörn Larsson, at Lnköpng Unversty for all the support and pleasant conversatons throughout the work on ths thess. Your mentorshp s the foremost reason I am nspred to contnue workng n the feld of optmzaton. I would also lke to thank Elna Rönnberg and Nls-Hassan Quttneh at Lnköpng Unversty for all your encouragement. Furthermore, I would lke to dedcate apprecaton to my supervsor, Ann Bertlsson, at Schemag AB for all your thoughtful and valuable nput. I thank my opponents, Davd Larsson and Rebecka Håkansson, for your support and constructve thoughts whch greatly helped mprove my work. Lastly, I thank my wonderful fancé Oscar, whose love and support I would not have made t ths far wthout, and my frends and famly for always encouragng me. You have all helped brng out the best n me, through good and bad tmes. Granfeldt, v

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9 Contents 1 Introducton Background Ams and Goals Method Topcs Covered Prevous Work Models and Common Constrants Integer Model Days-Off Schedulng Network Model Soluton Approaches Constrant programmng Heurstc Approaches Polynomal Method Basc Models The Integer Model The Network Model Illustratve Examples 19 5 Case Shfts and Staffng Demand Constrants Results Case 1a - Longer Work Perods Case 1b - Shorter Work Perods Case 1c - Schedule wth 12 Weeks Case Shfts and Staffng Demand Constrants Results Results When Alterng a Constrant Dscusson Suggestons for Future Development Granfeldt, x

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11 Lst of Fgures 1.1 Example of a schedule wth day, evenng and nght shfts Example of a schedule wth day, evenng and nght shfts The matrx contanng the elements a j Network defnton Example of a network Resultng schedule when solvng the basc model Resultng schedule n Example Resultng schedule n Example Resultng schedule n Example Resultng schedule n Example Resultng schedule when solvng the ntal setup n Example Resultng schedule when solvng the modfed setup n Example The two dfferent four-week constrants The two possble layouts of weekends Resultng schedules for longer work perods wth one weekend off between work weekends Resultng schedules for longer work perods wth two weekends off between work weekends Resultng schedules for longer work perods wth two weekends off between work weekends Resultng schedules for longer work perods wth one weekend off between work weekends Resultng schedule when schedulng for 12 weeks n Case Resultng schedule when schedulng for 12 weeks n Case Resultng schedule n Case Resultng schedule when alterng a constrant n Case Granfeldt, x

12 x Lst of Tables 7.1 Example of a weekly schedule where sngle days off appear twce. 44 Lst of Tables 1.1 Example of a workload matrx Typcal constrants used n rotatng workforce schedulng The shfts used n ths chapter The workload matrx used n ths chapter The dfferent shfts occurrng n Case The workload matrx for Case The collaboraton dstrbuton found n the schedules n Fgure The collaboraton dstrbuton found n the schedules n Fgure The collaboraton dstrbuton found n the schedules n Fgure The collaboraton dstrbuton found n the schedules n Fgure The collaboraton dstrbuton when schedulng for 12 weeks The dfferent shfts occurrng n Case The workload matrx for Case

13 Chapter 1 Introducton Ths thess explores dfferent approaches to a specfc type of staff schedulng called rotatng workforce schedulng. The defnton of ths type of schedules s dscussed n Secton 1.1, along wth other terms used n ths thess. 1.1 Background In many ndustres, t s requred that work should be carred out 24 hours a day, 7 days a week. Typcal work schedules n such contexts consst of a cycle repeatng tself after a few weeks. The work s usually dvded nto dfferent shfts, typcally day (D), evenng (E) and nght (N) shfts. Fgure 1.2 below shows an example of how a rotatng workforce schedule can look lke, where the empty spaces ndcate the rest perods. The schedule repeats tself every sx weeks. It should be mentoned rght away that n every model and example n ths thess, the occurrence of more than one shft per day n each weekly schedule s prohbted. PersonMoTuWeTh Fr SaSuMoTuWeTh Fr SaSuMoTuWeTh Fr SaSuMoTuWeTh Fr SaSuMoTuWeTh Fr SaSuMoTuWeTh Fr SaSu A D D D D D D D E E N N N N E E E E E D D D D D D D D D D D D B D D E E N N N N E E E E E D D D D D D D D D D D D D D D D D C N N N N E E E E E D D D D D D D D D D D D D D D D D D D E E D E E E E E D D D D D D D D D D D D D D D D D D D E E N N N N E D D D D D D D D D D D D D D D D D D D E E N N N N E E E E E F D D D D D D D D D D D D D D E E N N N N E E E E E D D D D D Fgure 1.1: Example of a schedule wth day, evenng and nght shfts. The observant reader notces that all persons n the above schedule are on the same sx-week schedule, wth the only dfference beng an offset n the startng week. Hence, the schedule can be represented compactly by only showng one person s schedule. In the schedule shown n Fgure 1.2, person A begns the cycle on week 1, whle person B begns the cycle on week 2, person C begns the schedule on week 3, and so on. At the end of each week, each person moves down to the followng week and the last person moves up to the frst week. Hence, Fgures 1.1 and 1.2 llustrate the exact same schedule. Granfeldt,

14 2 Chapter 1. Introducton Week Mo Tu We Th Fr Sa Su 1 D D D D D 2 D D E E 3 N N N N 4 E E E E E 5 D D D D D 6 D D D D D D D Fgure 1.2: Example of a schedule wth day, evenng and nght shfts. To be able to construct rotatng workforce schedules, a workload matrx s requred. Smply put, the matrx shows the staffng demand on each specfc shft every week. Table 1.1 shows an example of a workload matrx, whch matches the schedule gven n Fgure 1.2. Table 1.1: Example of a workload matrx. Shft Mo Tu We Th Fr Sa Su N D E The process of constructng the above descrbed type of schedules s, not surprsngly, called rotatng workforce schedulng. In general, staff schedulng s a very complex problem. A dffculty wth rotatng workforce schedulng s that varous constrants make t dffcult to fnd an optmal soluton; health and safety regulatons need to be taken nto consderaton. For example, t s not allowed to work a day shft drectly after a nght shft, whle workng too many nght shfts n a row s tresome and thus causes a proneness to more accdents. Addtonally, most work places and ther employees have specfc requests, whch are unque compared to other employers. Ths could for example be that every employee should at some tme durng the cycle work wth every other employee, or that a small group of employees should work more frequently together. An advantage of these types of schedules s that they are far. The schedulng process can be problematc snce t s dffcult to desgn schedules that contans the same propertes, but are stll dfferent and adapted to ndvdual preferences. Wth rotatng workforce schedules, ths ssue s non-exstent snce everyone s on the same schedule. Another property s that t s the staffng demand together wth the desgn of shfts that decdes the number of schedulng weeks. For example, the workload matrx n Table 1.1 contans 30 shfts, and each shft s n ths case 8 hours long. A person works on average 40 hours per week, and thus 5 shfts. Hence, 30/5 = 6 persons are needed to cover the staffng demand, whch leads to 6 schedulng weeks. In another example, every shft mght be 10 hours long and the workload matrx requres 28 shfts. On average, a person works 4 shfts every week. In total, ths requres 28/4 = 7 persons and thus 7 schedulng weeks.

15 1.2. Ams and Goals Ams and Goals In short, ths thess explores the possbltes of how to perform rotatng workforce schedulng, takng nto consderaton dfferent demands from the customer as well as constrants followng from laws and regulatons. More specfcally, the man objectves of ths thess can be dvded nto three dfferent parts: 1. Research what has prevously been done n ths feld. 2. Construct basc optmzaton models: an nteger model and a network model. 3. Extend the models and see how they perform on two larger cases. 1.3 Method To begn wth, a good lterature base s researched and studed. The lterature below was used as startng pont: G. Laporte. The Art and Scence of Desgnng Rotatng Schedules. The Journal of the Operatonal Research Socety, 50: , 1999 N. Muslu. Heurstc Methods for Automatc Rotatng Workforce Schedulng. Internatonal Journal of Computatonal Intellgence Research, 2: , 2006 M. Rocha, J.F. Olvera and M.A. Carravlla. Cyclcal Staff Schedulng: Optmzaton Models for Some Real-Lfe Problems. Journal of Schedulng, 16: , 2013 Chapter 2 comples all the nformaton found n the lterature. Two basc mathematcal models were developed, both found n Chapter 3. These models are wrtten as nteger lnear programs and mplemented n the mathematcal programmng language AMPL, whch allows the optmzaton models to be expressed n a smple way. To solve the optmzaton problems wrtten n AMPL, the solver CPLEX s used. The nterested reader can fnd more nformaton about AMPL and CPLEX n [4]. A small test case was constructed by hand, whch makes t possble to see f the models perform adequately. Unfortunately, only one of the models s useful n practce wth the current software, somethng whch s further explaned at the end of Secton 3.2. Two larger cases, based on constrants that are farly common n real-lfe, were later produced, whch helped to develop and push the boundares of one of the models further. These cases, along wth the resultng schedules gven by the model, are dscussed n Chapters 5 and 6, respectvely. 1.4 Topcs Covered Ths thess conssts of seven chapters, ncludng ths ntroducton. A summary of prevous work done n ths feld s found n Chapter 2. Chapter 3 contans the two basc models explored n ths thess. The followng chapter, Chapter 4, gves a few examples of how dfferent constrants - when added to the basc model - affect the outcome. Case 1 s covered n Chapter 5. More specfcally, the constrants used n that case and the dfferent resultng schedules are showed. Chapter 6 s lke the prevous chapter, but nstead looks at Case 2. Last but

16 4 Chapter 1. Introducton not least, Chapter 7 contans a dscusson of the results and also suggests deas for future work n ths feld.

17 Chapter 2 Prevous Work The schedulng process n general s an ssue that has been studed n several contexts. If the reader wshes to elaborate further on the subject, an extensve bblography has been wrtten by Ernst et al. [5] n Work on rotatng workforce schedulng has been done before, wth many dfferent approaches. Ths chapter dscusses a few of them, by begnnng to gve the most common constrants appearng throughout the lterature. It should be mentoned that early solvers, especally for nteger problems whch are common n schedulng applcatons, were not capable of handlng large scale problems. Most lkely, ths s the reason why a lot of artcles contan a specal-purpose algorthms, nstead of smply usng a standard solver lke CPLEX. 2.1 Models and Common Constrants A couple of constrants seem to appear frequently n rotatng workforce schedulng, regardless of context. Some of the most common ones, n no partcular order, are gven n Table 2.1. Several of these constrants depend on the settng and could thus of course be modfed accordng to the stuaton at hand Integer Model The most common way to model the rotatng workforce schedulng problem, and probably the frst dea to come to mnd, s by an nteger model. Although ths s often used today, t was not the case n the earler days. The dgtal revoluton has greatly helped develop the schedulng feld durng the last decades. Ths secton gves two examples of nteger models. Laporte et al. [7] suggested an algorthm n 1980 whch uses an nteger lnear programmng model. The algorthm begns by creatng segments consstng of shfts followed by some days off whch vares n length and start on dfferent days of the week. Wth the help of an nteger model, t s then possble to select a set of work segments that cover all shfts. The model can be used (recall that the paper s from 1980) because of the small lengths of segments, thus avodng large scale optmzaton. The algorthm contnues by mergng the segments together usng an enumeraton process, resultng n a set of feasble schedules. The best schedule wth regard to desrable propertes not ncluded n the nteger model, Granfeldt,

18 6 Chapter 2. Prevous Work Table 2.1: Typcal constrants used n rotatng workforce schedulng. The last column shows examples of artcles whch nclude the constrant. Nr Constrant Artcles 1 Shft change, lke gong from a day shft to an evenng shft, can only occur after at least one day off. 2 The number of consecutve work days must not exceed a preset bound, typcally 6 or 7, and must not be less than 2 or 3. 3 The number of consecutve rest days must not exceed a preset bound, typcally 6 or 7, and must not be less than 2 or 3. 4 Long work perods are normally followed by long rest perods and short work perods should be followed by short rest perods. 5 Schedules should contan as many full weekends off as possble. [1],[6],[7], [8] [1],[2],[6], [7],[8],[9], [10],[11], [12],[13] [1],[6], [7], [8],[10] [1],[7] [1],[6],[7], [8] 6 Weekends off should be well spaced out n the cycle. [1], [7], [8] 7 In some contexts, teams of employees must reman ntact, [1] that s, they cannot be dvded and recombned dfferently. 8 Forward rotatons (day, evenng, nght) should be used nstead of backward rotatons (day, nght, evenng). [1],[14] 9 Staffng demand should always be satsfed. Alternatvely, a maxmum number of shortage or excess workers mght be acceptable. [1],[3],[6], [13],[15], [16] 10 Exactly one shft should be used each day. [1],[15],[16] 11 Every week should contan 2 consecutve days off. [15],[16], [17] for example how weekends off are spread out, s then selected by the user. The algorthm was successful at ts tme and was used to create schedules for at least two polce statons and one fre staton. In 2013, Rocha et al. [3] proposed a mxed nteger problem to solve the rotatng workforce schedulng problem. In a mxed nteger problem, some of the varables are constraned to be ntegers whle the others are allowed to be fractonal. The paper begns by gvng a general mathematcal model, whch the authors adapt to two dfferent real-lfe case studes: a glass ndustry and a contnuous care unt. In the glass ndustry the workforce s homogeneous concernng sklls, wth everyone workng full-tme. The contnuous care unt s heterogeneous, wth dfferent demands for dfferent shfts and both full-tme as well as part-tme workers. Results are presented for both adjusted models, whch culmnates n the concluson that the models are able to produce optmal soluton n an adequate amount of tme.

19 2.1. Models and Common Constrants Days-Off Schedulng A common type of problems n earler days of rotatng workforce schedulng s the so called days-off schedulng problem, where we have a 5-day work week but a 7-day operatng week. It s often also requred that every employee s enttled to 2 consecutve days off each week. The problem conssts of determnng the off-work days for each worker, thus the name of the problem, whle mantanng the mnmum requred workforce sze. Durng the mddle of the seventes, Baker [15, 16] suggested methods for solvng the cyclc days-off schedulng problem. In 1974, Baker [15] developed a smple algorthm for the objectve to fnd a mnmum staff sze capable of meetng the requrements. The approach s heurstc and conssts of two dfferent steps. In the same year, Baker [16] tackles the problem of schedulng wth both full-tme and part-tme staff. Just lke n [15], he develops an algorthm though wth the objectve ths tme beng to mnmze the number of part-tme employees. Unlke the procedure n ths thess, the approaches used by Baker n both papers allow fndng the solutons by hand calculatons. In 1977, Baker and Magazne [18] contnued to work on the problem. In ths paper, they examne a number of methods to solve dfferent day-off polces. For each case, a formula for optmal workforce sze together wth an algorthm for constructng a feasble schedule s ncluded. Other contrbutons n the area are gven by Bennett and Potts [17] n 1968, Bechtold [19] n 1981 and by Alfares [20] n The days-off schedulng problem s a lttle dfferent from what s tred to acheve n ths thess. Its type of constrants s usually ncluded n the rotatng workforce schedulng problem. Hence, the days-off schedulng constrants can be vewed upon as a subset of the constrants found n the later problem Network Model Another possble approach to the rotatng workforce schedulng problem s to use a network flow model. It s usually dffcult to ncorporate all the constrants n the network, whch makes t necessary to have sde constrants. Hence, the resultng models are usually only network flow-based, as s the case n the paper by Balakrshnan and Wong [21] from Ther model deals wth ssues lke rest-perod dentfcaton and work/rest perod sequencng. All the constrants are ncluded n the network tself, wth the excepton of the constrants descrbng the staffng demand whch are thus treated as sde constrants. An optmal soluton corresponds to a path n the network. The unqueness wth ther approach s that, unlke prevous suggested algorthms durng that tme, t handled all requrements smultaneously. The common problems of nfeasblty and non-optmalty, whch can arse n a sequental approach, are therefore avoded. For example, the work perods mght be constructed n the frst step whle sequenced n the next, ergo a sequental approach. Balakrshnan and Wong use nodes that correspond to the days wthn the plannng horzon, where the source node represents the frst day and the snk node represents the last day. Each arc n the network corresponds to a work perod allocated to a specfc shft, or alternatvely to a rest perod. Each path from the source node to the snk node wll consequently represent a sequence of work and rest perods, thus a rotatng workforce schedule. The shortest path, where the length of the path s modeled as the cost of that sequence, gves the

20 8 Chapter 2. Prevous Work best resultng schedule wth regard to the cost. The number of paths n the network wll of course be very large for large scale problems, whch made a complete enumeraton non-vable at that tme. The authors therefore decded to use a soluton technque based on Lagrangan dualty theory followed by a K-shortest path enumeraton process. In ths thess, although dfferent from the model suggested by Balakrshnan and Wong, a network flow-based model s also constructed. 2.2 Soluton Approaches Before dscussng dfferent soluton methods for the rotatng workforce schedulng problem, a manual approach suggested by Laporte [1] n 1999 s looked upon. In hs paper, Laporte argues that fndng an optmal schedule can be more of an art than a scence. He states that desgn by hand, where t s possble to relax certan constrants, can sometmes yeld very reasonable schedules. Desgnng schedules wth the constrants mentoned n Table 2.1 s often relatvely easy and can be done by hand wthout dffculty. Dffcultes arse however when, for example, the number of workng hours per week must fall wthn an acceptable range. Stll, by bendng the rules a lttle, workable solutons can be found. The followng are a few suggestons, gven by Laporte n hs paper, of how ths can be done. If we look at Constrant 1, t states that a shft change can only occur after at least one day off, whle Constrant 2 states that breaks should be at least two days long. One day breaks are often avoded snce they are usually not favoured by the staff. However, breakng the frst rule mght be acceptable f some other unappealng property s elmnated, for example a very long work stretch. Another aspect s that of extendng shfts to satsfy workng hours per week. If the number of work hours per week s too low, t can be adjusted by extendng some shfts so that they overlap each other. The problem wth ths adjustment mght nonetheless be that t causes redundances wth too many workers on duty when the shfts overlap, but also addng extra stran on the workers these extended shfts. If the number of workers s large, such that they work n teams, t s possble to ntroduce an occasonal relef team. The relef team, consstng of fewer workers than an ordnary team, works the same schedule every week. Ths schedule concde wth the schedule of one of the weeks from the ordnary cyclcal schedule, say the frst week for smplcty. The regular team then take turns between themselves to take the frst week off, whle beng replaced by the relef team. Although ths breaks Constrant 7, t ntroduces the possblty of havng an occasonal week off. Asde from the suggestons above, the artcle by Laporte also ncludes an example of manpower schedulng at Quebec Mnstry of Transportaton. In the presented case, a couple of other constrants besdes those n Secton 2.1 are added. Solvng ths schedulng case farly easy, Laporte contnued by argung that schedulng by hand combned wth the technques suggested by hm and descrbed above gave a fast and satsfactory result, and that ths s also the case n many other examples. In ths thess, we do not schedule by hand but rather ntroduce models to solve the problem for us. However, the models nclude soft constrants, whch

21 2.2. Soluton Approaches 9 are constrants that are relaxed and penalzed n the objectve functon (see the begnnng of Chapter 3 for a more detaled defnton). Thus, even though we do not schedule by hand, we can stll bend the rules to obtan schedules wth desred propertes smply be usng dfferent values of penalty parameters. Exhaustve Enumeraton Early on, smple exhaustve enumeraton procedures were commonly used. An example of ths s the paper by Smth [22] from 1976, whch ncludes an algorthm that attempts to mnmze the staffng demand subject to constrants regardng rest perods. A form of feedback loop s often used between steps to correct any nfeasble or less desrable schedules, lke for example adjustng the staffng demand and consequently ncreasng or decreasng the workforce sze. Ths decomposton approach mght seem good n theory, but the algorthms are rarely able to manage large scale problems even wth today s computer technology. Commercal Software The rotatng workforce schedulng problem s common n several ndustres. Thus, t s only natural that a need for commercal software to smplfy the process has arsen. An example of ths s the Frst Class Scheduler, developed by Gärtner et al. [23] n The purpose of ths software s to fnd hghqualty schedules that are at least as good as solutons found by professonal planners, and to generate these schedules n a reasonably small amount of tme. Furthermore, hard constrants should be easy to control by the user and also ndvdual preferences should be taken nto account. In 2002, Muslu et al. [9] extended the software by addng a four-step algorthm whch focuses on splttng the schedulng process nto smaller problems. In each of these problems, a human decson-maker s nvolved n order to ncorporate preferences regardng soft constrants. Combned wth problem-orented ntellgent backtrackng algorthms, the method delvers good solutons for real-lfe problems. Another software for rotatng software schedulng s CP-Rota, developed by Trska and Muslu [12] n The software uses constrant programmng, further explaned n Secton 2.2.1, to solve real-lfe problems from the ndustry. The purpose of the software s to complement prevously mentoned Frst Class Scheduler Constrant programmng Constrant programmng can be counted as both a model and a method. The technque orgnates from artfcal ntellgence, where certan problems can not be solved wthn polynomal tme. Just lke the name mples, the programmng paradgm s founded on relatons between varables as constrants. The use of constrant programmng for rotatng workforce schedulng was frst ntroduced by Chan and Wel [24] n Laporte and Pesant [14] further appled the concept n ther paper from 2004, when they developed an algorthm whch s able to handle a large varety of constrants and has been appled to several real-lfe nstances and produced good-qualty solutons n each case.

22 10 Chapter 2. Prevous Work Heurstc Approaches Heurstcs s a class of methods often used to solve dffcult optmzaton problems such as nteger problems (see Lundgren et al. [25]). In short, a heurstc s a method whch generates a good soluton wthn a lmted soluton tme but wthout a guarantee of the soluton s qualty. The soluton found s however n most cases near-optmal, but t s often troublesome to decde how close to the optmum t s. These methods are usually developed for a partcular class of optmzaton problems and adapted to take advantage of that problem s specfc structure. In ths thess, heurstcs are not used to fnd feasble schedules but ths subsecton gves some examples of how heurstcs could be utlzed. In 1987, Burns and Koop [6] ntroduced a heurstc approach for fndng feasble schedules. These schedules use no more than the mnmum number of workers necessary whle stll satsfyng the constrants gven n Table 2.1. Ths heurstc algorthm s among the frst to nclude mult-shfts (dfferent types of shfts), as prevous algorthms only focused on sngle-shft (one type of shft) manpower schedulng. Montoya and Mejía [13] used n 2006 an teratve procedure wth local search. Local search s a method whch teratvely mproves a feasble soluton untl a termnaton crteron s met. As the name suggests, the resultng objectve value s a local optmum. In ther artcle, the authors avod local optma by guaranteeng that the algorthm never returns to a prevously vsted soluton. The algorthm takes several constrants nto account, among them mnmum and maxmum number of workng hours and rest perods. The results appear promsng as optmal solutons are found on all test nstances, and the procedure also shows a good behavour n terms of executon tmes. In , Muslu [11, 26, 2] ntroduced the concept of combnng tabu-search wth mn-conflct heurstcs to solve the schedulng problem. Tabu-search s a metaheurstc whch combnes local search wth the ablty to move towards solutons wth an nferor objectve value. Ths property gves a global dmenson to the local-search snce t enables more local optma to be found. Mn-conflct heurstcs are often used to solve constrant satsfacton problems. The algorthm selects at each teraton a random conflcted varable, whch thus volates one or more constrants. A new value s then assgned for the selected varable such that t mnmzes the number of conflcts. Ths process s terated untl a soluton s found, or a pre-selected maxmum number of teratons s reached. Returnng to the paper, the suggested algorthms appear to perform well and produce good results. However, a feasble soluton s not always guaranteed. Nevertheless, the methods work on all the test nstances (gven by lterature and real-lfe problems from a broad range of organzatons) ncluded n the paper. Muslu and Mörz [10] presented n 2004 a genetc algorthm. A genetc algorthm s a search heurstcs whch mmcs the process of natural selecton, where solutons are generated by technques nspred by evoluton prncples such as nhertance, mutaton, selecton and crossover. The results were promsng and the algorthm succeeded n generatng feasble solutons, but was nevertheless outperformed by earler methods (especally the one used n [9]). The authors do however leave suggestons for further mprovement.

23 2.2. Soluton Approaches Polynomal Method A more algebrac computatonal approach s a Boolean polynomal method, whch was recently suggested by Falcón et al. [8]. The man dea s to count and enumerate all feasble rotatng schedules. The polynomal structure s mproved by the use of Gröbner bases, whch reduces the computatonal tme sgnfcantly. The method produces good results on all nstances tested. Another advantage wth ths method s that problems whch are nfeasble are easly detected, makng t possble to analyze constrants and how they affect the feasblty of the problem.

24 12 Chapter 2. Prevous Work

25 Chapter 3 Basc Models In ths chapter, two basc models are ntroduced. Secton 3.1 presents an nteger model, whle Secton 3.2 presents a network model. In general, two dfferent types of constrants are used when modelng: Hard constrants are always satsfed and wrtten as explct constrants n the model. An example of ths s that exactly one schedule should be chosen each week. All constrants n ths chapter are hard unless otherwse stated. Soft constrants are constrants whch we would lke to be satsfed, although t s not necessary. For nstance, we may wsh that our schedule contans fve consecutve days off somewhere n the cycle. The way ths s accomplshed s by addng a penalty parameter and an auxlary varable n the objectve functon, where the varable becomes actve f the constrant s not satsfed. The penalty parameter, usually a very large number when mnmzng, wll then cause an ncrease n the objectve functon value. The advantage of soft constrants s that t s possble to prortze dfferent constrants to dfferent needs wth the help of the penalty parameters. Chapter 4 gves a few examples of hard constrants as well as soft constrants, and also how they affect the outcome. It should be mentoned that possble weekly schedules are generated by enumeraton. Infeasble schedules, wth regard to some exstng hard constrants, are then fltered out. An example of such a constrant s the maxmum number of consecutve work days that schedule should contan. Thus, several hard constrants are hdden n the generaton of possble weekly schedules. Ths method can unfortunately not cover all the hard constrants. Instead, explct hard constrants are added to the models. Granfeldt,

26 14 Chapter 3. Basc Models 3.1 The Integer Model Our frst mathematcal descrpton of the problem to construct rotatng workforce schedules s henceforth denoted as the nteger model. The schedulng s done over a cycle of W weeks, wth N possble weekly schedules to choose from every week. The man varable s defned as { 1 f weekly schedule s chosen for week v x v = 0 otherwse. Exactly one weekly schedule should be used each week, thus Satsfyng the staffng demand s somethng of hgh prorty. parameters x v = 1, v. To model ths, H pd = the staffng demand for shft type p on day d, and { 1 f weekly schedule contans shft type p on day d b pd = 0 otherwse and auxlary varables y u pd = the number of personnel lackng for shft type p on day d, y u pd 0, y o pd = the number of extra personnel for shft type p on day d, y o pd 0, are ntroduced. The staffng demand s modeled as two soft constrants, one for ypd u and one for yo pd, and the penalty parameter M n the objectve functon s assgned a large value. Ths guarantees that the staffng demand s always satsfed, f ths s at all possble. The constrants are wrtten as H pd b pd x v ypd, u p, d, v b pd x v H pd ypd, o p, d. v Another useful constrant s to fx a shft to an arbtrary week and day. Suggestvely, ths shft should concde wth the staffng demand. For nstance, f the frst week s Monday should contan a day shft the constrant below s obtaned: b 2,,1 x 1, = 1 There are two reasons for ths constrant: 1. It s easer to read and compare schedules f they all begn alke. 2. Every schedule has W equvalent solutons snce the full schedule s cyclcal wth W weeks, wth the only dfference beng whch week begns the schedule. By addng ths constrant, symmetry s avoded as t removes W 1 of the equvalent solutons. Observe that ths does not remove any unque solutons snce ths shft has to be ncluded n some week. The symmetry constrant only decdes whch week that s.

27 3.1. The Integer Model 15 The dffculty wth rotatng workforce schedulng les n mergng the possble weekly schedules together n the best possble way wthout volatng any constrants. To help wth ths, the followng parameter s ntroduced: a j = 1 f weekly schedule week v can be followed by weekly schedule j week v otherwse where a j can capture dfferent desred constrants. The a j -parameters are created when enumeratng possble weekly schedules. Lookng at the problem of mergng week v and v +1 wth each other, the parameter a j can be expressed n a matrx structure. Fgure 3.1 gves a graphcal descrpton of ths. Fgure 3.1: The matrx contanng the elements a j. Two constrants are needed for the mergng: a j x (v mod W )+1,j x v, j a j x v x (v mod W )+1,j, v, v, j where W s the number of weeks the full schedule conssts of. The modulus operaton s used because the full schedule s cyclcal, meanng that (W + 1) mod W = 1. If x v = 1 n the top constrant, then at least one of the elements n the sum must be used. In other words, f a possble weekly schedule s chosen a certan week, then t must be able to be followed by a possble weekly schedule j. The second constrant work the other way around: f a weekly schedule j s chosen a specfc week, then t has to be preceded by a weekly schedule. To summarze, the ntroduced constrants say that exactly one weekly schedule should be used every week and that the staffng demand should be satsfed. Addtonally, the constrants also consders whch weekly schedules that are allowed to follow a week v. Hence, wth the nformaton above, a very smple optmzaton model s now formed:

28 16 Chapter 3. Basc Models subject to mn z = M p (ypd u + ypd) o x v = 1, v H pd b pd x v ypd, u v p, d b pd x v H pd ypd, o p, d v d b 2,,1 x 1, = 1 a j x (v mod W )+1,j x v, v, j a j x v x (v mod W )+1,j, v, j x v = 0/1, v, M s redundant here snce the objectve only penalzes unfulflled staffng demand and not volaton the of any other soft constrants. 3.2 The Network Model Our second mathematcal descrpton of the problem s a network model wth addtonal sde constrants, where each node represents a weekly schedule. Hence, t s called the network model. In Fgure 3.2 below, f the arc s used, then schedule s used week v whle schedule j s used week v + 1. The arcs work the same way as the parameter a j does n the nteger model, thus the arcs only exst f the constrants allow schedule to be followed by schedule j. v x vj v + 1 j Fgure 3.2: Network defnton An example of a network can be seen Fgure 3.3. The rows conssts of all possble weekly schedules, whle the columns corresponds to weeks. The last week, the arc connects back to the weekly schedule chosen the frst week.

29 3.2. The Network Model 17 possble weekly schedules weeks Fgure 3.3: Example of a network. Lke wth the frst model, a few varables and parameters need to be ntroduced. The man varable s defned as follows: 1 f arc between node and node j s used from week v to week x vj = v otherwse. The network model uses the same parameters as the nteger model, wth the only dfference beng the parameters a j. Instead the set A s used, whch contans all arcs from week v to week v + 1. Observe that the set A s the same for all weeks snce every week has the same possble schedules. The arcs however represent a j (they are actually created from that matrx) whch bascally means that a j s stll used, but represented by a set of all feasble arcs. The relaton between the prevous varables x v and the new x vj s x v x vj. In some (,j) A constrants, the frst sum can smply be replaced by the latter. The frst constrant ntroduced s the followng flow conservaton constrant: x vk x (v mod W )+1,k,j = 0, v, k (,k) A (k,j) A Ths constrant says that everythng that flows nto a node must also flow out of t. Just lke before, exactly one weekly schedule should be used each week. Ths s enforced by x 1,j = 1, (,j) A together wth the flow conservaton constrant. To avod symmetry, the constrant below s added: (,j) A r x 1,j = 1,

30 18 Chapter 3. Basc Models where A r s a subset of A restrcted such that A r contans all the possble weekly schedules that have a day shft on the Monday. The staffng demand s, just lke n the prevous model, gven by the followng two soft constrants: H pd b pd x vj ypd, u p, d, v v (,j) A (,j) A b pd x vj H pd y o pd, p, d. The only dfference s that x v s replaced wth x vj. The entre network (,j) A model s therefore the followng: mn z = M (ypd u + ypd) o p subject to (,k) A v (,j) A (,j) A d x 1,j = 1 x 1,j = 1 (,j) A r H pd b pd x vj ypd, u v (,j) A b pd x vj H pd ypd, o x vk (k,j) A x (v mod W )+1,k,j = 0, p, d p, d v, k x vj = 0/1, v,, j Unfortunately, ths model resulted n very long soluton tmes, whch made t unusable n practce. The most probable reason for ths behavour s that although the problem has a network structure wth addtonal sde constrants, the varables used are ntegers. The current CPLEX-solver s of lmted applcablty snce t does not have features and optons that enables solvng sde-constraned network problems wth nteger varables by explotng the network structure. The consequence of ths s ncreased soluton tmes, as seen when runnng the network model. Hence, the results n ths report wll only cover the nteger model, whch from now on wll be shortened to the model.

31 Chapter 4 Illustratve Examples In ths chapter, a few examples of hard and soft constrants and the resultng schedules that the nteger model produces are gven. The dfferent shfts used can be seen n Table 4.1, whle the staffng demand s shown n Table 4.2. The tables make t clear that the schedulng s done wth a demand for nght, day and evenng shfts. The staffng demand s for 30 shfts, and, wth the gven shft lengths, a person works on average 5 shfts a week. Thus, the staffng demand requres 6 persons, whch leads to 6 schedulng weeks. In every example, nformaton s also gven about the sze of the problem and the soluton tmes. Table 4.1: The shfts used n ths chapter. Shft Tme Break N mn D mn E mn Table 4.2: The workload matrx used n ths chapter. Shft Mo Tu We Th Fr Sa Su N D E The followng hard constrants are taken nto consderaton n the enumeraton process: It s prohbted to work more than 6 consecutve days. After the last nght shft n a workng perod, there should be at least a perod of 50 hours off untl the next shft starts. The possble weekly schedules are enumerated, and then the nfeasble schedules wth regard to the above constrants are fltered out. The a j -parameters captures these restrctons and assst n mergng of weekly schedules. Granfeldt,

32 20 Chapter 4. Illustratve Examples Recall the basc model from Chapter 3.1: subject to mn z = M p (ypd u + ypd) o d x v = 1, v H pd b pd x v ypd, u v b pd x v H pd ypd, o v b 2,,1 x 1, = 1 a j x (v mod W )+1,j x v, j a j x v x (v mod W )+1,j, x v = 0/1, v, p, d p, d v, v, j If ths model s solved, usng the two restrctons mentoned above, the schedule seen n Fgure 4.1 s produced. Week Mo Tu We Th Fr Sa Su 1 D D D D D D 2 D E E E E D 3 N N N N 4 E D D D D D 5 D D 6 D D D D D D Fgure 4.1: Resultng schedule when solvng the basc model. Soluton tme: 7 s # possble weekly schedules: 563 # varables: 3420 # constrants: 6805 Ths schedule mght not look very pleasant to use, but t satsfes all of the requrements. The hard constrants are clearly met snce only one schedule each week s used, and the frst week s Monday contans a day shft. In addton the constrants descrbed by the a j -parameters are also fulflled snce the resultng schedule has a maxmum of 6 consecutve work days, and after the last nght shft there s 84 hours off. The staffng demand s obvously also met snce the

33 21 numbers from the workload matrx, Table 4.2, match the shfts n the resultng schedule. In the followng examples, and addton of some constrants to the basc model s done to see what happens. Thus, the resultng schedules are always compared to the schedule n Fgure 4.1. Example 4.1. In ths example, the model s extended by addng a constrant sayng that f you work anythng durng the weekend, you must also work the entre weekend. Ths mplementaton s ncorporated n the enumeraton of the possble weekly schedules and n the values of the a j -parameters. Solvng the basc model now nstead gves the schedule n Fgure 4.2. Week Mo Tu We Th Fr Sa Su 1 D D D D D 2 E E E E E 3 D D D D D D 4 D D D D D D 5 D D D D 6 N N N N Fgure 4.2: Resultng schedule n Example 4.1, where the a-parameters have an added constrant. Soluton tme: 1 s # possble weekly schedules: 298 # varables: 1830 # constrants: 3625 Lookng at the resultng schedule, t s apparent that the only sgnfcant dfference between ths schedule and the schedule n Fgure 4.1 s how the weekends appear. The constrants from the basc model are stll satsfed, but wth the added condton of havng the weekends clustered together. If you work Saturday the condton forces you to also work Sunday and vce versa. Example 4.2. In ths example, sngle days off are unfavourable and thus to be penalzed. Hence, soft constrants whch have the penalty parameter Q n the objectve functon are added. To help wth ths, the followng parameters are added: { 1 f weekly schedule has a sngle day off n the mddle of the week e = 0 otherwse, { e s 1 f weekly schedule ends wth exactly n days off, n = 0, 1 n = 0 otherwse, { e b 1 f weekly schedule begns wth exactly n days off, n = 0, 1 n = 0 otherwse.

34 22 Chapter 4. Illustratve Examples The auxlary varable { 1 f a sngle day off occurs n the mergng of week v and v + 1 s v = 0 otherwse s used n the objectve functon. A sngle day off occurs f a week ends wth a sngle day off and the next week begns wth work, or f a week ends wth work and the next week begns wth a sngle day off. The dea s to capture ths phenomenon wth the help of e s n,eb n and s v. If the descrbed stuaton arses, s v wll be actvated and thus gvng an ncrease to the objectve functon value. Usng the parameters and auxlary varable above, the basc model can be expanded: mn z = M p (ypd u + ypd) o + Q v d ( e x v + s v ) subject to x v = 1, v H pd v b pd x v ypd, u p, d b pd x v H pd ypd, o v p, d b 2,,1 x 1, = 1 a j x (v mod W )+1,j x v, j a j x v x (v mod W )+1,j, v, v, j e s,1x v + j e s,0x v + j e b j,0x (v mod W )+1,j s v + 1, e b j,1x (v mod W )+1,j s v + 1, v v x v = 0/1, v, Solvng the above model yelds the schedule seen n Fgure 4.3.

35 23 Week Mo Tu We Th Fr Sa Su 1 D D E E D 2 N N N N 3 E E D D D 4 D D D D D D 5 D D D E 6 D D D D D D Fgure 4.3: Resultng schedule n Example 4.2. Soluton tme: 6 s # possble weekly schedules: 563 # varables: 3426 # constrants: 6817 Ths schedule has 3 sngle days off, whle the schedule n Fgure 4.1 has 6 sngle days off. Hence, the added constrant cuts the sngle days off n half wthout loosng any other qualty (wth regard to the constrants). Example 4.3. Ths example examnes the possblty of penalzng a certan number of consecutve work days. To help us wth ths, we ntroduce the penalty parameter Q n, where n s the number of consecutve work days. The followng parameters are used: 1 f weekly schedule contans n consecutve work days n the f n = mddle of the week 0 otherwse, { fn s 1 f weekly schedule ends wth exactly n consecutve work days = 0 otherwse, 1 f weekly schedule begns wth exactly n fn b = consecutve work days 0 otherwse. An auxlary varable s also needed: 1 f the jont between week v and week v + 1 accumulates n y nv = consecutve work days 0 otherwse. The constrant used n ths example s smlar to the constrant for sngle days off n Example 4.2. If a weekly schedule has n consecutve work days n the mddle of the week, t s penalzed n the objectve functon by the parameter Q n. If some work perod length s feasble, the penalty parameter s smply zero for that length n. Consecutve work days can also appear when weekly schedules are merged together. The parameters fn s and f n b keep track of how many work days a weekly schedule ends wth and begns wth respectvely. If

36 24 Chapter 4. Illustratve Examples an unwanted work perod length occurs n the mergng, the auxlary varable y nv s actvated. For example, f a week v ends wth 2 work days and schedule v + 1 begns wth 3 work days, then the mergng accumulates a work perod of 5 days. The auxlary varable y 5,v s then actvated, whch leads to a penalty n the objectve functon value accordng to Q 5. The resultng model s gven below. mn z = M (ypd u + ypd) o + Q n (y nv + f n x v ) p n v subject to d x v = 1, v H pd b pd x v ypd, u v b pd x v H pd ypd, o v b 2,,1 x 1, = 1 a j x (v mod W )+1,j x v, j a j x v x (v mod W )+1,j, p, d p, d v, v, j fdx s v + j f b j,c dx (v mod W )+1,j y cv + 1, v, c = 1,.., 6, d = 0,.., c x v = 0/1, v, The resultng schedule becomes as n Fgure 4.4, where sngle work days have been penalzed. Week Mo Tu We Th Fr Sa Su 1 D D E E E D 2 N N N N 3 D D D D 4 E E D D D D 5 D D D D 6 D D D D D D Fgure 4.4: Resultng schedule n Example 4.3. Soluton tme: 7 s # possble weekly schedules: 563 # varables: 6841 # constrants: 7057

37 25 The schedule above clearly has no sngle work days, whle the schedule n Fgure 4.1 has 3 of them. Example 4.4. A restrcton s added to the basc model about how many weekends off there should be n the full schedule. Specfcally, the schedule should have at least L = 3 weekends off. A consequence of ths constrant together wth the staffng demand used n ths chapter s that f a work day occurs durng the weekend, both Saturday and Sunday should be worked. Thus, the resultng schedule should be somewhat smlar to the result n Example 4.1. We ntroduce the parameter { 1 f weekly schedule has the weekend off l = 0 otherwse, and the auxlary varable t = the number of weekends mssng to satsfy L = 4 weekends off, t 0 to model ths soft constrant. Lke n Example 4.2, Q s used as the penalty parameter. If weekends off are mssng to satsfy the requrement, the auxlary varable t wll be actvated and a penalty n objectve functon value s receved. The resultng mathematcal model can be seen below. subject to mn z = M p (ypd u + ypd) o + Qt d x v = 1, v H pd b pd x v ypd, u p, d v b pd x v H pd ypd, o p, d v b 2,,1 x 1, = 1 a j x (v mod W )+1,j x v, v, j a j x v x (v mod W )+1,j, v, j L v l x v t x v = 0/1, v, If ths model s solved, the schedule n Fgure 4.5 s receved.

38 26 Chapter 4. Illustratve Examples Week Mo Tu We Th Fr Sa Su 1 D D D D D D 2 E E D D D 3 D E D D D D 4 D D E E 5 N N N N 6 D D D D D Fgure 4.5: Resultng schedule n Example 4.4. Soluton tme: 7 s # possble weekly schedules: 563 # varables: 3421 # constrants: 6806 The schedule above has 3 full weekends off, as requested, wthout breakng any of the other constrants. If compared to Fgure 4.2 n Example 4.1, t s seen that they are very smlar. Besdes the obvous result of both havng 3 work weekends, they both also have entre work perods of nght shfts and evenng shfts. The structure of the day shfts are a lttle dssmlar, but the dfference s neglgble snce we could easly move the shfts n ether schedule to receve the other wthout any notable negatve effect. Example 4.5. In ths fnal example, the model from Example 4.4 s used but nstead wth L = 4. In other words, 4 weekends off s requred. However, ths s mpossble wthout falng to satsfy the staffng demand. Id the staffng demand s penalzed harder than the weekend constrants, the schedule seen n Fgure 4.6 s receved. Week Mo Tu We Th Fr Sa Su 1 D D D D D D 2 D D D D D D 3 D D D D D 4 D D D D 5 N N N N 6 E E E E E Fgure 4.6: Resultng schedule when solvng the ntal setup n Example 4.5. Soluton tme: 6 s # possble weekly schedules: 563 # varables: 3421 # constrants: 6806 The resultng schedule, as expected, does not gve 4 weekends off. Instead, just as t should be, t has full demand coverage. However, t does have 3 full weekends off snce ths accommodates to the constrant n the best possble way. As the reader mght recall, we penalze harder the more number of weekends

39 27 that are mssng to satsfy the constrant. If the staffng demand s penalzed softer than the weekends off, the schedule n Fgure 4.7 s receved. Week Mo Tu We Th Fr Sa Su 1 D D D D D D 2 D D D D D D 3 D D 4 D D D D D 5 N N N N 6 E E E E E Fgure 4.7: Resultng schedule when solvng the modfed setup n Example 4.5. Soluton tme: 7 s # possble weekly schedules: 563 # varables: 3421 # constrants: 6806 In ths schedule, we see that there are 4 weekends off, and thus personnel s mssng one weekend. We have now gone through the bascs of the mathematcal modellng and how soft constrants work. Thus, we are ready to move on to the case studes. The model used n the case studes are not shown n detal snce the prncple already has been shown n ths chapter. Hence, the constrants used are mentoned but not how they are mathematcally modeled.

40 28 Chapter 4. Illustratve Examples

41 Chapter 5 Case 1 Case 1 follows the standard of a sx week schedule for sx persons, just as n prevous examples. However the types of shfts are a lttle dfferent from what has been seen before n ths thess, wth two nght shfts and two day shfts. At the end of ths chapter, the expanson of schedulng for twelve weeks s explored to see f ths results n dfferent solutons. Every schedule contans nformaton about the sze of the problem and the soluton tmes, just as n Chapter Shfts and Staffng Demand In ths case, we use four dfferent shfts as seen n Table 5.1. Table 5.2 shows the staffng demand. Table 5.1: The dfferent shfts occurrng n Case 1. Shft Tme Break NL mn NS mn DL mn DS mn Table 5.2: The workload matrx for Case 1. Shft Mo Tu We Th Fr Sa Su NL NS DL DS 1 1 Granfeldt,

42 30 Chapter 5. Case Constrants Case 1 has several hard and soft constrants besdes the most basc ones gven n Chapter 3: Every day should nclude at least 11 consecutve hours off. The begnnng of the day cycle can be set at a chosen tme and does not have to start at mdnght. Every week should contan at least 36 consecutve hours off. For every four week nterval, from a chosen startng pont, the average workng week should not exceed 40 hours. If the startng pont s set to the begnnng of the frst week, three dfferent ntervals wll appear untl the pattern repeats tself. Rememberng that the full schedule s cyclcal, Fgure 5.1a shows these three ntervals, wth the braces representng the sx week schedule. If we nstead schedule for 12 weeks, ths hard constrant wll cover the ntervals seen n Fgure 5.1b. (a) The four-week ntervals when schedulng for 6 weeks. (b) The four-week ntervals when schedulng for 12 weeks. Fgure 5.1: The two dfferent four-week constrants. To avod symmetry, the frst week s Monday must contan shft DL. After the last nght shft n a work perod, there should be a perod of at least 50 hours off untl the next shft starts. A schedule should contan a maxmum of 5 consecutve work days. We wsh to fnd a schedule where everyone works wth everyone else as much as possble. In other words, person A should never encounter person B only once every cycle, but everyone else several more tmes per cycle. However, how these encounters are dstrbuted durng the cycle does not matter, as we are only lookng on how many tmes a person works wth another person. It should be clarfed that shfts NL and NS count as a meet, as well as DL and DS. If someone works a shft durng the weekend, then both Saturday and Sunday should be workdays. Wth the staffng demand gven for ths case, ths means that every cycle contans two work weekends and four weekends off. It s prohbted to work two weekends n a row. Ths means that n a schedule wth a sx week cycle, there are two dfferent possbltes of weekend layouts. There s ether one weekend off between the work weekends, or two weekends off between the work weekends. Fgure 5.2 clarfes ths. We are nterested n both possbltes.