Post, Gerhard

Veen, Egbert van der

Veldhoven, Sophie van

English

We discuss Days off scheduling as intermediate phase to personnel rostering

van Veldhoven, Sophie

Post, Gerhard F.

van der Veen, Egbert

University of Twente Research Information

Noname manuscript No.(will be inserted by the editor)Days off schedulingA 2-phase approach to personnel rosteringSophie van Veldhoven · Gerhard Post · Egbertvan der VeenReceived: date / Accepted: date1 IntroductionThe personnel rostering problem (or nurse rostering problem) is a well-known com-binatorial optimization problem, with a rich literature, see [Ernst et al(2004)] and[Burke et al(2004)]. In many practical situations the large number of working timeregulations and preferences make it difficult and time consuming to construct a goodschedule.One way to reduce the complexity of the personnel rostering problem is to decom-pose the problem into subproblems that are easier to solve. Though the subproblemscan possibly be solved to optimality in a reasonable time, the combination of the sub-problems does not necessarily lead to an optimal solution to the original problem. Inthis work we present a 2-phase decomposition model. In the first phase we constructa days off schedule, that is a schedule that specifies for each employee the workingdays and the days off. We also consider a refinement in which we consider day off,day shift or night shift. In the second phase we specify which shifts are actually as-signed to the employees on their working days, which means that we solve a shiftrostering problem respecting the days off schedule found in the first phase.In practice, the construction of a schedule with working days and days off can bea separate step in the process of assigning staff. For individual employees, it may bepleasant to know the working days a long time ahead, so that they can plan their freeSophie van VeldhovenAviv, Langestraat 11, 7511 HA Enschede, The Netherlands,E-mail: sophievveldhoven@gmail.comGerhard Post (corresponding author)Department of Applied Mathematics, University of Twente, P.O. Box 217, Enschede, The Netherlands,and ORTEC, Groningenweg 6k, 2803 PV Gouda, The Netherlands,E-mail: g.f.post@utwente.nlEgbert van der VeenDepartment of Applied Mathematics, University of Twente, P.O. Box 217, Enschede, The Netherlands,and ORTEC, Groningenweg 6k, 2803 PV Gouda, The Netherlands,E-mail: Egbert.vanderVeen@ortec.com2 Sophie van Veldhoven et al.time. The exact hours they have to work, are not essential to know in the long term.In addition, requests for vacation can be taken into account long before the actualplanning, which can alert the planner for possible capacity problems. The assignmentof the various shifts to the employees can be done in a later stage, for example, amonth before the start of the new planning period.The aim of this paper is two-fold: investigate the computation times for a classof personnel rostering problems with and without decomposition, and, secondly, in-vestigate the quality of the schedules in the 2-phase approach compared to solutionsobtained in one run. We will use integer linear programming (ILP) in our research.Hence we will present a mathematical program for the construction of days off sched-ules, based on the constraints of the original problem.2 ApproachAs data sets we use the Employee Scheduling Benchmark Data Sets [Curtois(2007)].We formulate formulate an ILP for these data sets, which are solved using CPLEX12.2, with the time limit set to 1 hour. These results serve as the benchmark for ourtests. Next we formulate the days off scheduling problems, which we derive from theinstances. We consider two variants:(On, Off): Distinguish working days and days off.(Day, Night, Off): Distinguish day shifts, night shifts, and days off.We include the refinement (Day, Night, Off) in our study, because in many cases therequirements before and after night shifts are very different from (all) other shifts.In fact we reduce the original shift scheduling problem to a shift scheduling problemwith 2 shift types (on, off), respectively 3 shift types (day, night, off). Hence in thefirst and second phase we can use similar ILP models, with the addition that in thesecond phase the assignment of shifts to employees should obey the decisions ofthe first phase. Though the ILP models are similar, the reduction in the first phaseis not straightforward, and the next subsections describe how this is done. For this,we consider the different types of constraints present in the benchmark, and explainshortly how we handle them.Requests and fixed assignmentsEmployees can have shifts preassigned, can have requests for shifts on or can haverequests for a day off. Those can be transferred to the days off scheduling problemdirectly: work requests or shift on requests result in days on. Shift off requests areignored, since the employee might work on the same day in another shift.Cover requirementsPer day the instance describes a minimum, maximum, or preferred cover per shifttype or per time period. These have to be aggregated to be useful in the days offschedule. In case the cover requirements are in (overlapping) shift types we formulatean ILP model to determine the minimum number of employees needed, and use thisamount for the days off schedule. In case that time periods are used, we proceed inthe same way.Days off scheduling 3Pattern requirementsThe pattern requirements can express a wide variety of constraints for individualschedules. It can contain constraints like the maximum number of shifts per planninginterval or week, shift sequences, shifts in weekends, etc. If a pattern concerns allshift types, we can use it in the days off schedule. In the case of (day, night, off), wecan incorporate more pattern requirements, potentially leading to better results.Workload requirementsWorkload requirements describe the number of hours an employee should work inthe planning period. Clearly, these are difficult to use in the days off schedule ifdifferent shift types have different lengths. Since we use only necessary conditions,the best we can do is to calculate upper and lower bounds on the working days, andadd those conditions as constraints to the ILP. In the shift scheduling phase we havefull information, so that we can use the correct workload requirements.3 ResultsWe tested the 2-phase decomposition on 16 instances present at [Curtois(2007)] with3 to 12 shift types. For almost all of these instances optimal solutions are known.Running CPLEX 12.2 for 1 hour on a standard dual core PC yields an optimal solu-tion in 9 cases. For these cases we find that the (on, off) decomposition gives onlygood results for 4 of these instances, where using the (day, night, off) decompositionleads to good results for 6 instances. For the other 7 cases we compare the decompo-sition results with the results without decomposition. Then we find good results in 3out of 7 cases if we use the (on, off) decomposition, and in 5 cases if we use the (day,night, off) decomposition. In most cases the calculation times reduce by more than98%. Here ‘good results’ is meant in the sense that the cost is of the same order ofmagnitude; sometimes the results are better, but usually slightly worse.Several improvements can be made to the model and the solution method. Forthe model we can improve the way that the pattern constraints are dealt with; nowin several instances we use only a minor portion of these in the days off schedule,which in some instances leads to high costs in phase 2. For the solution part, we canapply the decomposition several times, say 10 times, with a constraint added thatmakes sure we do not generate the same solution. On some instances, this approachsignificantly improves the results.References[Burke et al(2004)] Burke EK, De Causmaecker P, Vanden Berghe G, Van Landeghem H (2004) The stateof the art of nurse rostering. Journal of Scheduling 7(6): pp. 441–499.[Curtois(2007)] Curtois T (2007) http://www.cs.nott.ac.uk/∼tec/NRP, Employee SchedulingBenchmark Data Sets.[Ernst et al(2004)] Ernst AT, Jiang H, Krishnamoorthy M, Owens B, Sier D (2004) An annotated bibliog-raphy of personnel scheduling and rostering. Annals of Operations Research 127: pp. 21–144.

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