Rostering from Staffing Levels

Transcription

1 Rosterng from Staffng Levels a Branch-and-Prce Approach Egbert van der Veen, Bart Veltman 2 ORTEC, Gouda (The Netherlands), Egbert.vanderVeen@ortec.com 2 ORTEC, Gouda (The Netherlands), Bart.Veltman@ortec.com Abstract: Many rosterng methods frst create shfts from some gven staffng levels, and after that create rosters from the set of created shfts. Although such a method has some nce propertes, t also has some bad ones. In ths paper we outlne a method that creates rosters drectly from staffng levels. We use a Branch-and-Prce (B&P) method to solve ths rosterng problem and compare t to an ILP formulaton of the subclass of rosterng problems studed n ths paper. The two methods perform almost equally well. Branch-and-Prce, though, turns out to be a far more flexble approach to solve rosterng problems. It s not too hard to extend the Branch-and-Prce model wth extra rosterng constrants. However, for ILP ths s much harder, f not mpossble. Next to ths, the Branch-and-Prce method s more open to mprovements and hence, combned wth the larger flexblty, we consder t better suted to create rosters drectly from staffng levels n practce. Keywords: Personnel rosterng, staffng levels, employee preferences, Branch-and-Prce, column generaton Introducton The effcent use of human resources s mportant n many ndustres. For sure ths holds for health care, where the man costs are the loans of the workforce. Ineffcent rosterng leads to neffcent use of scarce and expensve resources, and hence to unnecessary costs. In most lterature the rosterng process s decomposed nto three mportant subphases. Frst, there s the staffng phase. In ths phase staffng levels are created from a predcted workload. Staffng levels represent the number and skll level of requred resources wthn a gven tme slot. For example, between 7:30 AM and 9:00 AM, two nurses need two be avalable at the South ward, from 8:00 AM tll 0:30 AM, an anesthetst needs to be avalable at Operatng Theatre 8, etc. Next, n the shft schedulng phase, shfts are created from the staffng levels. These shfts are supposed to cover the staffng levels as effcent as possble. After that, n the shft rosterng phase, workers are assgned to the created shfts,.e. rosters are created. Ths rosterng process s llustrated by Fgure. Fgure : Rosterng from staffng levels: a two-step approach.

2 As llustrated by Fgure, workng tme regulatons and employee preferences constran the creaton of shfts and rosters. However, where workng tme regulatons constran the creaton of the shfts (e.g. shft length between 6 and 9 hours) as well as the creaton of rosters (e.g. an employee s not allowed to work more than 6 shfts a week), employee preferences are only accounted for when creatng rosters. A major downsde of ths way of rosterng s that the shfts that result from the shft schedulng phase mght not allow for the creaton of good or even vald rosters, snce the employee preferences further constran the feasble shft set. To solve ths, we propose a rosterng method that creates rosters drectly from staffng levels, as s graphcally llustrated by Fgure 2. Fgure 2: Rosterng from staffng levels: a one-step approach. Now, as s observed from Fgure 2, ths new schedulng method creates rosters drectly from staffng levels, and, whle dong ths, t keeps track of workng tme regulatons and employee preferences smultaneously. In ths way, employee preferences are accounted for drectly durng the creaton of the shfts. Takng employee preferences nto account s a hot topc n health care schedulng due to a general lack of care and cure professonals lke nurses and physcans. Now, more than before, there s a strong orentaton on matchng staffng levels and employees as good as possble. Ths mples, n some extent, that the two-step approach was adequate n the past, but t s expected to be too lmted n the (near) future. The man objectve of ths paper s to develop a method that creates rosters drectly from staffng levels, and whle dong ths takng workng tme regulatons and employee preferences nto account. Ths paper s structured as follows. The next secton frst dscusses some lterature related to ths rosterng problem. After that, n Secton 3, the model used to solve the rosterng problem s outlned and Secton 4 dscusses the results of our software mplementatons. Secton 5 draws conclusons from ths research. 2 Related lterature Some lterature already (partly) looks at rosterng drectly from staffng levels. Keth (979) outlnes a method that creates rosters from shft templates. Wth ths method, only shfts that come from a predefned set of (template) shfts may be used when creatng the rosters. However, ths method does not keep track of employee preferences, because the cost of assgnng shfts to employees s not employee specfc. Dowsland and Thompson (2000) outlne a straghtforward extenson of the model of Keth that makes the cost of shfts employee specfc. By makng the cost of shfts employee specfc, employee preferences are taken nto account (to some extend). Although the classc schedulng method has some downsdes, as outlned n Secton, t has one major postve aspect. In ths schedulng method the larger problem of creatng a roster s decomposed nto two smaller subproblems, the shft schedulng and shft rosterng subproblems. Caprara, Monac and Toth (200) ntroduce a method that solves the shft rosterng and shft schedulng phase teratvely. When the shft schedulng phase comes up wth a set of shfts that 2

3 does not match employee preferences very well, a recreaton of shfts can take place, such that the new set of shfts (hopefully) meets the employee preferences better. However, the method of Caprara et al. cannot drectly be appled to our problem, snce they apply t to a tran personnel rosterng problem. Whereas n our problem we are rosterng from staffng levels, the tran personnel rosterng has to roster tasks. Furthermore, the objectve of the tran personnel rosterng problem of Caprara et al. s to mnmze the amount of tme needed to work a complete schedule, whereas n our problem the tme horzon s fxed; all staffng levels need to be matched wthn the gven tmeframe. The major problem, however, wth teratvely solvng the shft schedulng and shft rosterng phase s that t s unclear what nformaton should be provded to the shft schedulng phase when shft rosterng cannot fnd a soluton. Moreover, when the shft rosterng phase s unable to fnd a soluton, t s also not clear whether ths s due to the set of created shfts or whether there s no soluton at all. 3 Branch-and-Prce model In ths study we restrct ourselves to nstances where employees are allowed to work at most one shft that conssts of one (large) tme slot wthout nterruptons. Although ths seems rather restrctve, these nstances are already NP-complete, snce we are dealng wth staffng levels for multple sklls, see Van der Veen (2009). We use a Branch-and-Prce (B&P) method to create the rosters. The Brand-and-Prce method presented here can be easly extended to rosterng problems wth more general shfts. For readers not famlar to Branch-and-Prce methods we outlne a short ntroducton to ths topc n Secton 3.. A more elaborate ntroducton to the concept of Branch-and-Prce can e.g. be found n Hans (200). Both Hans (200) and we assume that readers are famlar wth lnear programmng theory, and concepts lke lnear programmng, dual problems and reduced cost. A general ntroducton to lnear programmng theory can be found n e.g. Chvátal (985). 3. Branch-and-Prce: general ntroducton Large lnear programs are often solved by means of column generaton. Ths method was frst appled by Glmore and Gomory (96). Wth a column generaton approach ntally only a small set of columns s ncluded n the lnear program, whch often s referred to as the master problem. After the master problem s solved t s checked, va the so-called prcng problem, whether there s a column wth proftable reduced cost that s not yet ncluded n the master problem. If t exsts t s added to the master problem. After that, the procedure s repeated untl the prcng problem cannot fnd addtonal columns wth reduced cost. However, when ths procedure s appled for nteger lnear programs we mght not end up wth an nteger soluton. To fnd a soluton anyway, some knd of branchng s appled. After a branchng took place, often a subset of the generated columns s not vald anymore for the current node. These columns are removed from the master problem, and, after that, the prcng problem s called untl no proftable column s found anymore. Two thngs can happen now. Frst, the soluton can be nteger. If t s a better soluton than the best one so far, t s set as the current best nteger soluton, otherwse t s fathomed. Secondly, the soluton can be fractonal. If the objectve value of the fractonal soluton s worse than the current best nteger soluton t s also fathomed, otherwse addtonal branchng s needed. In Fgure 3 the Branch-and-Prce method s schematcally summarzed. Note that a drect ILP formulaton of our rosterng problem, where every possble shft s represented by a varable, contans a huge number of varables. Moreover, for reasonable problem dmensons ths number of possble shfts would be too large to be modelled and solved by computers. Hence, we chose to apply a Branch-and-Prce method to solve the rosterng problem. 3

4 Solve master problem (Secton 3.2) Solve prcng problem (Secton 3.3) Add column to master problem New column generated? Yes No Integral soluton? No Apply branchng (Secton 3.4) Yes Node(s) left that need to be consdered? Yes No Optmal soluton found Fgure 3: Branch-and-Prce: a schematc overvew. The most mportant parts of ths Branch-and-Prce method, the master problem, the prcng problem and the branchng method are explaned n more detal n respectvely Secton 3.2, Secton 3.3 and Secton Branch-and-Prce: master problem Gven a set of employees ( =,,n), sklls (j =,,m), tme slots (t =,,T), one may thnk of an hour each, and staffng level d per combnaton of skll and tme slot, we come to the followng master problem n mn c x (a) = k K k k s.t. n = k K k K a k x k d xk for =, K, m for j =, K, m; t =, K, T (b) (c) x 0, nteger, for =, K m; k (d) k K 4

5 In the above model, c k denotes the cost of employee workng shft k. We defne c k to be equal to the number of workng perods n shft k. Next to ths, x k equals when employee works shft k. The set K denotes the set of shfts that are created specfcally for employee. Furthermore, a k equals when accordng to shft k the related employee would work on skll j durng perod t. It equals 0 otherwse. Hence, the objectve functon (a) mnmzes total cost. Constrants (b) thus mply that the staffng levels are met. Constrants (c) ensure that an employee works at most one shft. Constrants (d) mply that the x k are non-negatve and ntegral. Note that n fact the x k are bnary varables, but we modeled them as general non-negatve ntegers. By dong ths the dual problem of the master problem s smaller, because t does not nclude varables correspondng to the upper bound constrants on the varables. Ths makes the prcng problem that s presented n Secton 3.3 easer. To get the Branch-and-Prce method started we have to create some ntal varables (.e. columns) for the master problem. Otherwse there s no dual problem and then no reduced cost can be calculated. An easy way to ntalze the master problem s to set all K such that they contan exactly one shft k for whch a k = d. When one of these shfts s worked, the total demand s covered. Note that such a shft s not vald, snce we assumed an employee cannot work on two sklls at the same tme. However, by settng the c k = M, for these ntal shfts, where M s a sgnfcantly large number, we make sure that after generatng a number of addtonal (vald) columns, va the prcng problem, these ntal (nvald) shfts do not appear n the soluton. Hence, all nformaton about the valdty of shfts s assumed to be ncorporated n the prcng problem. We have to admt that ths s probably not the smartest way to ntalze the master problem. However, snce all nformaton about valdty of shfts s contaned n the prcng problem, ths way of ntalzaton works ndependent of the constrants that are mpled on shfts. 3.3 Branch-and-Prce: prcng problem To determne the next column to be added to the master problem, the prcng problem needs to be solved. The prcng problem s solved when the column wth the least reduced cost s found. Note that negatve reduced costs are proftable for our problem, snce we have a mnmzaton problem. Hence, the column wth the most negatve reduced cost s the one we lke to add to the master problem. However, n order to calculate reduced costs the dual of the master problem needs to be solved frst. The dual of the LP relaxaton of the master problem s gven by max s.t. m T n d + j = t = = m T j = t = a jkt π ν (2a) + ν c k for =, K, n; k K π (2b) 0 for j =, K, m; t =, K T π (2c), ν 0 for =, K n (2d), The reduced cost of a shft k of employee s now gven by m T * k jk t j = t = c a π ν (3) * where π * and v * denote the optmal values of π and v respectvely. To determne the shft wth the least reduced cost t s not too neffcent to enumerate all possble shfts, calculate ther reduced costs, and select the one wth the least reduced cost. To do ths, let J denote the set of sklls employee has. Frst determne 5

6 mn j J { * π } (4) for all t. Denote the vector contanng all these π * by π. Now, let w represent a workng pattern: a sequence of 0s and s ndcatng n whch tme slots the employee should () or should not (0) work. By enumeratng all workng patterns w that employee s allowed to work, and calculatng w٠π, we fnd the workng pattern wth the least reduced cost. Va the workng pattern and π we then determne for every perod on whch skll the employee s workng,.e. the shft s determned. If w٠π < ν * the reduced cost of ths shft s negatve, and t s added to the master problem. Let the mnmal and maxmal number of consecutve tme slots employee needs to work (f called to work) be denoted by mndu and maxdu, respectvely. Then the number of workng patterns s bounded by t = T mndu + T t t = T maxdu + t = { π j J t ( T + ) T T 2 2 Determnng mn } for t =,,T has tme complexty O(mT), hence the tme complexty of the prcng problem s O(mT + T 2 ), whch s polynomal. Employees are smply selected by ncreasng ndex,.e. frst we solve the prcng problem for employee and see whether a shft wth negatve reduced cost exsts for employee. If t exsts t s added to the master problem, and the master problem s resolved. Otherwse, employee 2 s selected and the procedure repeats. If there s no employee for whom a shft wth negatve reduced exsts we need to apply branchng. The branchng procedure s descrbed n the next secton. (5) 3.4 Branch-and-Prce: branchng When the soluton obtaned to the LP relaxaton after column generaton s fractonal, branchng needs to be appled. The branchng needs to result n an nteger soluton. It s also mportant that branchng decsons can be ncorporated n the prcng problem,.e. t must be possble to adjust the prcng problem n such a way that the generated columns respect the branchng decsons. Wth the rosterng problem we apply branchng to the tme slots n whch employees start and stop workng. Ths s easly ncorporated n the prcng problem, and t leads to an ntegral soluton. The former s trval; smply do not nclude shfts that have workng perods before the start perod or after the stop perod n the enumeraton. The latter s less obvous. Moreover, when start and stop perods have been fxed for all employees there mght not even be a feasble nteger soluton amongst the generated columns. Ths s llustrated by Example. Example. Gven a rosterng problem wth 2 employees, 2 tme slots and 2 sklls. Demand (d ) s gven by the followng matrx where j {,2 } and t {,2 } d D = d 2 d d 2 22 = And the generated shfts are a = (,) and b = (2,2) for employee ; and p = (,2) and q = (2,) for employee 2. The frst number ndcates the skll that s worked on durng tme slot, the second number ndcates the skll that s worked on durng tme slot 2. Furthermore, the cost of the shfts equal the number of workng perods n t, hence c a = c b = c 2p = c 2q = 2. For both employees and 2 we fxed tme slot as start perod and tme slot 2 as stop perod. The optmal soluton clearly equals x a = x b = x 2p = x 2q = ½. Furthermore, ths s the unque optmal soluton, and there does not exst an nteger soluton amongst the generated shfts. Furthermore, there are no shfts 6

7 wth negatve reduced cost (snce ths soluton s optmal gven the demand D) and the start and stop perods are fxed for both employees. Hence, the example shows that after full branchng an nteger soluton does not necessarly exst. An nteger soluton s easly constructed when start and stop perods are fxed for all employees. We do ths va the constructon of a b-matchng problem. A b-matchng problem s smlar to regular matchng problems except that n b-matchng for every vertex v the matchng has to contan exactly b v edges that are connected to v. A more detaled descrpton of (b-)matchng problems s found n e.g. Schrjver (2003). Now, for every tme slot t, create vertces for all employees that are allowed to work durng t. Furthermore, for every tme slot t, create vertces v for every skll j where d > 0 wth b v = d. Now, for every tme slot t, connect an employee vertex to a demand vertex whenever the employee has the correspondng skll. Note that we now have t ndependent b-matchng problems. After these b-matchng problems are solved for all tme slots t, shfts are created from ther solutons and ths offers a soluton to the rosterng problem for the current node of the branchng tree. Note that b-matchng problems are polynomally solvable, see Schrjver (2003). We know that there are feasble solutons to these b-matchng problems snce the fractonal solutons for the rosterng problem offer fractonal solutons for the b-matchng problems, whch mples that there are ntegral solutons to the b-matchng problems. Snce b-matchng has a totally unmodular technology matrx, such a fractonal soluton s a convex combnaton of ntegral solutons. Solvng the b-matchng problems drectly returns ntegral solutons, and from these the shfts are created. Hence, the Branch-and-Prce method, of whch the branchng scheme s an mportant part, leads to an ntegral soluton. As a fnal note to ths chapter we want to remark that although we engneered the above Branchand-Prce approach n such a way that t solves our rosterng problem, t offers a lot of flexblty to solve other rosterng problems. In fact the prcng problem determnes the knd of constrants that are mpled on shfts, snce the master problem s nothng but a constrant set coverng problem, and only shfts generated by the prcng problem are used here. So even when there are lots of constrants mpled on shfts, and t s not very easy to fnd addtonal shfts, ths dffculty s solated n the prcng problem part of the soluton approach. Furthermore, t s also straghtforward to combne heurstc and optmal prcng problems to generate addtonal columns, such that t s well possble to prevent the prcng problem from consumng too much solvng tme. Of course, the branchng part also needs to be adjusted when other rosterng problems are modeled, but we beleve that ths wll not be too hard. Note that for these larger rosterng problems columns correspond to week or month rosters, nstead of sngle day shfts. Fnally, we want to stress that there are lots of possbltes to ncorporate practcal constrants n the Branchand-Prce formulaton. For example, n practce t s often preferred that employees only work shfts from a predefned set of template shfts. Ths s easly ncorporated n the Branch-and- Prce model by smply lettng the prcng problem check whch of the template shfts offers the best reduced cost. 4 Expermental results To assess the performance of our Branch-and-Prce (B&P) method we created an ILP formulaton of the rosterng problem. In the prevous chapter we mentoned that an ILP formulaton based on shfts has too many varables for reasonable problem dmensons. Hence, we created an ILP formulaton where shfts are mplctly defned. To do ths, we created varables ndcatng whether an employee works on a specfc skll durng some specfc tme slot or not. Ths mples that the number of varables s lnear n the problem dmensons, and wll not become too large when the problem dmensons get larger. For a complete overvew of the created ILP model we refer to Van der Veen (2009). However, we need to stress here that ILP s not a very good way to solve (health care) rosterng problems. ILP works here, due to our restrcton that employees are allowed to 7

8 work only one shft that conssts out of a sngle tme slot wthout nterruptons. However, for ths ILP model, we need artfcal bnary varables for every employee and every tme slot to model the rosterng problem correctly. When employees are allowed to work multple shfts, or when more elaborate constrants are mpled on shfts, ths number of artfcal varables certanly ncreases further, makng the rosterng problem both hard to model and hard to solve n an ILP settng. The ILP model s solved va the ILOG CPLEX.0 callable lbrary. The B&P model s mplemented n SCIP. For a general descrpton of SCIP the reader s referred to Achterberg (2007). Wthn B&P, we agan use CPLEX.0 to solve the LP relaxatons. To assess the qualty of the B&P method we used randomzed datasets. Lots of nstances are generated and solved, n order to get a good ndcaton of the overall qualty of the algorthm. To create randomzed demand the followng procedure s appled for every skll category j. Frst, create n ntegers N for whch d ~ n N U (, T ), such that n satsfes N < T N Next, generate n ntegers D U, d ~ = ( 0 m n ) n =, and create sets = { } = + = n τ : τ, K, N ; τ N p, K, N p for 2 n ; τ n N p= p= p= +, p K Now, for all tme slots t τ set demand to D., T Note that wth ths demand we have blocks of tme slots wth equal demand. We thnk ths knd of demand better fts realty than when a unform random number s drawn for every ndvdual tme slot. The latter mples very strong fluctuatons n demand, whereas the former mples a more controlled demand pattern, whch, we thnk, better fts realty. Next to random demand, we created random sklls. For every employee frst a unform random number m s generated from [,m]. After that, unform random numbers are drawn from [,m] untl m dfferent numbers are obtaned, ndcatng the sklls the employee has. Wth these randomzed demand and sklls there s no assurance that there are enough avalable (sklled) employees to cover demand. However, n Van der Veen (2009) t s ndcated that, wth these randomzed skll sets, t s lkely that demand can be met. Detaled results of the experments we performed can be found n Van der Veen (2009). Table summarzes some mportant results. The man focus s on the comparson of solvng tmes of our B&P approach versus the ILP model. We do not have to look at soluton qualty, snce both mplementatons return optmal solutons. Test case # n m T mndu maxdu t : t T 4 T Implementaton avg solvng tme (sec.) #unsolved nstances B&P ILP B&P ILP B&P ILP

9 t : t T 4 T 5 5 : : B&P ILP B&P ILP B&P ILP Table : Computatonal experment: results The fve leftmost columns of Table show nformaton on the test case solved. Columns, 2 and 3 show respectvely the number of employees, the number of sklls, and the number of tme slots that are part of the test case. Columns 4 and 5 ndcate the mnmal number of tme slots an employee should work when called to work, and the maxmal number of tme slots an employee s allowed to work, respectvely. The sxth, seventh and eghth column of Table ndcate the average tme needed to fnd a soluton (column 7) by the correspondng solver (column 6), and the number of unsolved nstances (column 8). Note that for all parameter sets (set of fxed values of n, m, T, mndu and maxdu) 20 randomzed test nstances are created and solved. Ths mples that the total number of unsolved nstances per test case can be sgnfcant. Note that for test cases and 2 the values of mndu and maxdu are vared. For test cases 3 and 4 the value of T s vared, and for test cases 5 and 6 the values of mndu and maxdu are vared. Test cases for whch mndu > maxdu are gnored, snce they are nfeasble. Summarzng from Table we observe that B&P performs relatvely bad for the test cases where T grows large. Furthermore, we observe that B&P has a stronger dependence on m than ILP has. Ths can be observed from results and 2; the average solvng tme of B&P ncreases faster than the average solvng tme of ILP when m ncreases. Ths s also observed from test cases 5 and 6. The relatve bad performance of B&P for larger T s caused by the fact that the tme needed to solve the prcng problem depends quadratcally on the value of T, see Secton 3.3. Furthermore, for larger T (and m) the number of columns that needs to be generated s also larger, whch mples that the total tme consumed by the prcng problem grows larger and larger. For most cases we see that ILP outperforms B&P. Even though, the performance of B&P s qute acceptable n most cases. However, t s not too dffcult to mprove the B&P method, but mprovng the ILP mplementaton s much harder. We lst some good possbltes for mprovng the B&P mplementaton. As outlned above, the tme needed by the prcng problem ncreases a lot when T gets larger. There also s a slght dependence on the value of m. Decreasng these dependences wll mprove both the tme needed by the prcng problem as well as the overall solvng tme needed. Next to the prcng problem the most tme of the B&P algorthm s consumed by solvng the LP relaxatons of the master problem. For larger m and T the master problem gets very large durng the solvng process, and hence solvng the LP relaxatons consumes more and more tme. Column management strateges, that try to keep the master problem from gettng (very) large, probably further decrease the total solvng tme. Fnally, we only generate one column at a tme when the prcng problem s called. Generatng multple columns at once mght also mprove the solvng speed. Fnally, we want to stress that n ths paper we studed a relatvely smple rosterng problem. As already ndcated n ths secton, t s expected that for more realstc and more elaborate models, t s (almost) mpossble to model these problems wth ILP, and hence for those problems B&P wll be the preferred solvng method. 9

10 5 Conclusons Lots of rosterng methods frst create shfts based on staffng levels, and after that create rosters from ths set of created shfts. In order to create rosters drectly from staffng levels, whch allows accountng for employee preferences when creatng shfts, Branch-and-Prce turns out to be a flexble approach to do so. On a small subclass of rosterng problems Branch-and-Prce and ILP perform almost equally well. Although ths subclass of rosterng problems s NP-complete, t contans only a small set of constrants that are mpled on shfts n practce. However, due to the structure of the Branch-and-Prce method t s not too hard to extend t n order to nclude the complex constrants from practce. Moreover, due to the structure of Branch-and-Prce methods there s a lot of flexblty to deal wth these dffcult constrants effcently. However, extendng the ILP model wth more and hard constrants s not at all an easy task, f not mpossble. Furthermore, extendng the ILP model wth such practcal constrants results n a sgnfcant ncrement n solvng tme. Consderng the fact that Branch-and-Prce offers far more flexblty to deal wth practcal constrants than the ILP model does, we expect Branch-and-Prce to perform relatvely better for rosterng problems where more constrants are mpled on the shfts, or where week or month rosters are created. Furthermore, there are some good and easy ways to ncorporate practcal concepts, lke template shfts, nto the Branch-and-Prce approach, whch also makes t of better practcal use. References Achterberg, T. (2007) Constrant Integer Programmng, Ph.D. Thess Technsche Unverstät Berln Caprara, A.; Monac, M.; Toth, P. (200) A Global Method for Crew Plannng n Ralway Applcatons, In: Voß, S.; Daduna, J.R. (eds.) Computer-Aded Schedulng of Publc Transport, Lecture Notes n Economcs and Mathematcal Systems, Sprnger-Verlag, Berln Hedelberg, pp Chvátal, V. (983) Lnear Programmng, W.H. Freeman and Company, New York Dowsland, K.A. (998) Nurse Schedulng wth Tabu Search and Strategc Oscllaton. European Journal of Operatonal Research 06 (998) 2-3, pp Dowsland, K.A.; Thompson, J.M. (2000) Solvng a Nurse Schedulng Problem wth Knapsacks, Networks and Tabu Search. Journal of the Operatonal Research Socety 5 (2000) 7, pp Glmore, P.C.; Gomory, R.E. (96) A Lnear Programmng Approach to the Cuttng-Stock Problem, Operatons Research 9 (96) 6, pp Hans, E.W. (200) Resource Loadng by Branch-and-Prce Technques, Twente Unversty Press, Enschede Keth, G.K. (979) Operator Schedulng, IIE Transactons (979), pp Schrjver, A. (2003) Combnatoral Optmzaton Polyhedra and Effcency, Sprnger-Verlag, Berln Van der Veen, E. (2009) Rosterng from Staffng Levels: a Branch-and-Prce Approach, M. Sc. Thess Rjksunverstet Gronngen 0