an integrated approach for shift scheduling and rostering...

Research ArticleAn Integrated Approach for Shift Scheduling and RosteringProblems with Break Times for Inbound Call Centers

Turgay Tuumlrker 1 and Ayhan Demiriz 12

1Department of Industrial Engineering Sakarya University 54187 Sakarya Turkey2Department of Industrial Engineering Gebze Technical University Kocaeli 41400 Turkey

Correspondence should be addressed to Ayhan Demiriz ademirizgmailcom

Received 25 April 2018 Revised 17 September 2018 Accepted 15 October 2018 Published 21 November 2018

Academic Editor Purushothaman Damodaran

Copyright copy 2018 Turgay Turker and Ayhan Demiriz This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

It may be very difficult to achieve the optimal shift schedule in call centerswhich have highly uncertain and peaked demand duringshort time periods Overlapping shift systems are usually designed for such cases This paper studies shift scheduling and rosteringproblems for inbound call centerswhere overlapping shift systems are used An integer programmingmodel that determines whichshifts to be opened and how many operators to be assigned to these shifts is proposed for the shift scheduling problem For therostering problem both integer programming and constraint programming models are developed to determine assignments ofoperators to all shifts weekly days-off and meal and relief break times of the operators The proposed models are tested on realdata supplied by an outsource call center and optimal results are found in an acceptable computation time An improvement of15 in the objective function compared to the current situation is observed with the proposed model for the shift schedulingproblemThe computational performances of the proposed integer and constraint programming models for the rostering problemare compared using real data observed at a call center and simulated test instances In addition benchmark instances are used tocompare our Constraint Programming (CP) approach with the existing models The results of the comprehensive computationalstudy indicate that the constraint programming model runs more efficiently than the integer programming model for the rosteringproblem The originality of this research can be attributed to two contributions (a) a model for shift scheduling problem and twomodels for rostering problem are presented in detail and compared using real data and (b) the rostering problem is considered asa task-resource allocation and considerably shorter computation times are obtained by modeling this new problem via CP

1 Introduction

Service industry is known for its labor-intensive structureTherefore any improvement on labor costs will result inconsiderable savings in total cost Moreover effective laborschedules can lead tomutual satisfaction among the manage-ment customers and the staff in the service sector In thisregard it is inevitable for such businesses to focus more onthe effective use of the labor which is the most significant andthe most expensive resource for them Indeed several studieson the problem of workforce scheduling also prove thatreducing the employee cost expenses is beneficial to the firms[1]

Considering the direct communicationwith the customerside of the service sector it is crucial to do the planningin the call centers where hundreds of operators work and

where thousands of calls are processed Since workforce costis a significant component of the operating costs of the callcenters effective use of the workforce is vital to the callcenters In the work of Gans et al [2] it is stated that laborcosts are almost 70 of the total cost According to Dietz[3] this ratio indicates that the economic efficiency of theoperations is determined by the quality of the workforceplanning process

Several studies on the workforce planning of the callcenters focus on generating workforce schedules meetingboth the necessary service levels constraints and the require-ments of some laws and regulations While generating theseschedules it is extremely hard to find good solutions Therearise conflicting objectives when considering preferencesof the employees minimizing the costs and meeting allthe restrictions of the workplace [4] As indicated in [5]

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7870849 19 pageshttpsdoiorg10115520187870849

2 Mathematical Problems in Engineering

Workload prediction

Staffing levels

Shifts RostersStaffing Shi

schedulingRostering

Figure 1 Subproblems of workforce scheduling

Phase 1 Shift scheduling Phase 2 Rostering

Shi information+

Number of operators neededper period

Model 1IP based sub-model

Model 2aIP based sub-model

Model 2bCP based sub-model

Rosters with break times


Shi schedules

Figure 2 Subproblems considered within this study

the workforce scheduling problem in a call center has fourdifferent subproblems which are shown in Figure 1

The first subproblem is about forecasting the work loadand number of calls received within certain periods (usually30minutes or 15minutes [6])The frequency and the numberof calls received during the planning horizon are the maintopic of this subproblem The second subproblem staffingis about finding the required minimum number of opera-tors depending on the incoming call rate In other wordsstaffing is converting the estimated number of incomingcalls to the operator requirements at periodic levels Thethird subproblem focuses on choosing the optimal shifts thatmeet the demands for operators With this subproblem theshifts to be activated (opened) are also determined insteadof simply assigning the operators to the available shifts Thefourth subproblem rostering deals with the assignment ofthe available operators to the shifts and determining the dailybreaks andor usually weekly days-off of the operators

In shift scheduling the organization of the shifts in theplanning horizon is arranged Schedules generally includenonoverlapping shifts Allocation problem is easy to solve inthis case However call centers may have overlapping shiftsTherefore a shift scheduling system is crucial in call centersIn the rostering problem an operator-shift matching is madebased on definite constraints Also the days-off and the breaktimes of each operator are determined Here wewant to focuson the issue of determining the break times As it is knownan operator usually works between seven and nine hours aday and (s)he needs breaks during this period In their studySungur et al [7] emphasized that operator-break assignmentsshould be handled during the scheduling process Besidesother studies in the literature state that only meal breaks areincluded in their problem formulations and that relief breaksshould be taken into consideration as well [7]

Our aim in this study is to provide aworkforce schedulingmodel including shift scheduling and rostering The general

framework of the study is illustrated in Figure 2 First wesolve the shift scheduling problem by taking shift informationand the staffing requirements as inputs into our integerprogramming (IP) model (Model 1) Staffing requirementsmeans the minimum number of operators ensuring the ser-vice quality level within each period of the planning horizonAfterwards a rostering problemwill be solvedwhen the shiftsare identified based on the solutions of the first problem

Two different models are proposed for the rosteringproblem traditional IP modeling approach (Model 2a) andconstraint programming (CP) approach (Model 2b)WithCPapproach the problem is handled as a task-source allocationand modeled with this perspective The main contribution ofthis paper is to propose a CP model to solve the rosteringproblem in call center as an alternative to the IP model

The remainder of this paper is organized as followsSection 2 introduces the related literature review on work-force scheduling problems In Section 3 IP based model forshift scheduling problem and then IP and CP based modelsfor rostering problem are introduced Section 4 describesexperimental study based on real data and test instancesderived from the real data as well as the staff schedulingproblem benchmark instances [8] The paper is concludedand the future directions are highlighted in Section 5

2 Literature Review

There exist a lot of models developed in the literature forworkforce scheduling problem [9] Edie [10] analyzed thetraffic delays in the tolls with an experimental study andformulized fixed queueing systems Later Dantzig [11] intro-duced the integer linear programming model with regard toEdiersquos [10] study However the workforce scheduling problemhad gone through many evolutions in time

The studies of Ernst et al [4 12] and the study ofBrucker et al [9] presented a detailed literature review on

Mathematical Problems in Engineering 3

the application areas of these type of problems A summaryof studies in call center workforce scheduling between 2003and 2018 is presented in Tables 1(a) and 1(b) which depictsome of the characteristics and modeling approaches of thecall center workforce scheduling studies in the literatureThe types of subproblems handled in these studies are alsopointed in Tables 1(a) and 1(b) There is a lack of study inhandling shift scheduling and rostering problems togetherdetermining break times (especially relief breaks) and usingthe CP model for rostering in the literature

21 Literature on Subproblems of Workforce Scheduling inCall Centers

211 Call Forecasting The first subproblem of workforcescheduling problem is to forecast incoming calls withinparticular periods [5] There are two different types of callsin the call centers outbound and inbound calls Planning theoutbound calls is not difficult Due to the uncertainty in thearrival rate and duration of the inbound calls planning thistype of calls is considered challenging [13]

Bianchi et al compared Holt-Winters (HW) exponentialweighted moving average smoothing model and Box-Jenkins(ARIMA) autoregressive integrated weighted moving modelfor call forecasting in [14] Although a higher accuracy ratein HW for simple systems is achieved it is observed thatARIMA gives more realistic results for more complicated sys-tems Taylor [15] compares the forecast accuracy of differenttime series including a version of Holt-Winters smoothingmethod and concludes that simple forecast techniques are notvery successful in large processes

212 Staffing The arrival rate of the inbound calls in callcenters varies in time The number of operators should beplanned carefully to be flexible enough to ensure the cus-tomer satisfaction There are many studies in the literature ondetermining the optimal number of operators based on thearrival rateThe existing studies reveal that call center systemcan be regarded as a queueing system [16] MMS in otherwords Erlang C model is the commonly used and the sim-plest call center queueing model due to obtaining closed formanalytical solution [17]Themodels developed to calculate thenumber of essential workforces for the call centers are usuallyalternative of Erlang C calculations [2 16 18]

Brown et al present some statistical analyses of theoperations of a call center using queueingmodels in [18]Theypropose practical and beneficial mathematical models Themain inputs of mathematical models are the statistics such asthe number of the operators volume of the calls service timeand holding time The empirical analysis in [18] supports thepreference of the Erlang models for forecasting the customerdelays regardless of whether basic assumptions are satisfiedor not for the Erlang models

213 Shi Scheduling When the work day requires morethan one shift and someof themare overlapping determiningthe number of workforce to be assigned to these shiftsand planning the starting and ending times of these shiftsbecome important problems The main objective is to meet

the workforce demand at each period with the minimum cost[19]

There are several studies conducted on different areasof the literature In [20] Thompson proposed two differentmodels for shift scheduling While one of them proposes anapproach based on the acceptable service level per period theother one reveals an approach considering the average servicelevel during the planning horizon The developed models areMIP based Atlason et al [21] suggested that the model ofThompson [20] can be modeled by a simulation basedapproach Although both works consider period-based mon-itoring and assignment like our approach their performancesare not studied in a multishift case found in call centersettings

Koole and Van der Sluis [19] mention in their study thatthe service level should be provided at the period level in thetraditional approach and this is actually a hard constraintInstead of this approach they propose a model to meet thetarget of the general service level over the planning horizonIn [5] an easy model is implemented and it has a shortcomputational time They propose a method for multiskilledagent case which is different from our approach In additionto these studies Henderson and Berry [22] andAykin [23 24]provide some basic examples The performances of thesemethods are not known inmultishift settings like call centersIndeed they are not studied for the overlapping shifts

214 Rostering Well-constituted rosters have some advan-tages such as low cost efficient use of the resources increasein the employee satisfaction and fair work load and allocationof the shifts [41] Aykin [23] proposes an integer model forthe rostering problems His study includes multiple relief andmeal break time windows and different break variables aredefined for each shift The main problem has two differentshifts Each shift lasts for nine hours including the breaktimes The employees have a 30-minute lunch break and two15-minute relief breaks The break approach presented in hispaper is crucial in terms of being a basis for the prospectivestudiesThe break policy in our approach is inspired from thismodel

Thompson [20] compares different shift and rosteringmodels in terms of workforce cost service level and work-force utilization rate via simulation analyses Our modelsconsider only the cost minimization but a multicriteriaobjective function is taken in [20]

Ertogral and Bamuqabel [34] introduce two differentmodels for the shift scheduling problem by dividing theoperators as flexible and nonflexible ones However they donot consider break times Dietz [3] proposes a quadraticmodel for rostering problem and proves that this quadraticmodel yields better results compared to a classical IP modelOverlapping and multishift configurations are consideredbut decisions for the break times are not included in [3]

Nah and Kim [46] propose a model of the rosteringproblem for a hospital reservation center They found outthat the developed model decreases the expected cost andthe abandonment rate of the company by using an appliedcase study Considering various types of costs such as laborwaiting and abandonment costs increases the significance


Table 1(a) Literature summary some characteristics of workforce scheduling studies in call centers

Reviewed literature Year Considered subproblems Break(s)Call forecasting Staffing Shift scheduling Rostering M R

Alfares [25] 2007 Asgeirssonamp Sigurethardottir [26] 2014 Atlason et al [27] 2008 Atlason et al [21] 2004 Avradimis et al [28] 2008 Avramidis et al [29] 2009 Bhulai et al [5] 2008 Canon [13] 2007 Castillo et al [30] 2009 CezikampLrsquoEcuyer [31] 2008 Defraeye et al [32] 2016 Dietz [3] 2011 Dudin et al [33] 2013 ErtogralampBamuqabel [34] 2008 Excoffier [35] 2016 Gong et al [17] 2015 Green et al [36] 2007 HelberampHenken [37] 2010 Hojati [38] 2010 IbrahimampLEcuyer [39] 2013 Ingolfsson et al [40] 2010 Kooleampvan der Sluis [19] 2003 Kyngas et al [41] 2012 Liao et al [42] 2009 Liu et al [43] 2018 Mattia et al [44] 2017 Millan-RuizampHidalgo [45] 2013 NahampKim [46] 2013 Ormeci et al [47] 2014 Pot et al [48] 2008 Restrepo [49] 2017 RobbinsampHarrison [50] 2008 RobbinsampHarrison [51] 2010 TaskiranampZhang [52] 2017 WallaceampWhitt [53] 2005 Weinberg et al [54] 2007 defined otherwise undefined M meal break R relief break(s)

(b) Literature summary modeling approaches of workforce scheduling studies in call centers

Reviewed literature LP IP MIP Queuing Constructive heuristic Other heuristic Simulation CP OthersAlfares [25] Asgeirssonamp Sigurethardottir [26] Atlason et al [27] Atlason et al [21] Avradimis et al [28] Avramidis et al [29] Bhulai et al [5] Canon [13] Castillo et al [30] DEACezikampLrsquoEcuyer [31]


(b) Continued

Reviewed literature LP IP MIP Queuing Constructive heuristic Other heuristic Simulation CP OthersDefraeye et al [32] Dietz [3] Dudin et al [33] ErtogralampBamuqabel [34] AHPExcoffier [35] SPGong et al [17] Green et al [36] HelberampHenken [37] Hojati [38] IbrahimampLEcuyer [39] RMIngolfsson et al [40] Kooleampvan der Sluis [19] Kyngas et al [41] Liao et al [42] SPLiu et al [43] Mattia et al [44] Millan-RuizampHidalgo [45] NNNahampKim [46] Ormeci et al [47] Pot et al [48] Restrepo [49] SPRobbinsampHarrison [50] RobbinsampHarrison [51] SPTaskiranampZhang [52] WallaceampWhitt [53] Weinberg et al [54] BF defined otherwise undefined LP linear programming IP integer programming MIP mixed integer programming CP constraint programmingDEA data envelopment analysis AHP analytic hierarchy process SP-stochastic programming NN neural network RM regression models BF bayesianforecasting

of the work [46] Break times are also determined Howeverthere are only two shifts for the planning horizon in [46]Asgeirsson and Sigurethardottir [26] formulate a model forrostering problem and state that their model yields betterresults compared to a local search based algorithm Theyimplement their model for four different companies Highquality feasible staff schedules are achieved

Ormeci et al [47] examine the rostering problem inthe call centers with the aim of balancing the operationalcosts customer representative satisfaction and customerservice level objectives It is also emphasized that call centersoffer transportation services to its employees and since thisconstitutes a major part of the operational costs they includethe transportation requirement into the models EspeciallyCanonrsquos study [13] is very similar to our work from theperspective of usage of CP in workforce scheduling of callcenters The first three subproblems are studied in [13] bymodelingMIP CP and Tabu Search Unlike our study Canon[13] studies just the staffing subproblem by modeling CPHowever only meal breaks are considered for the breakscheduling and models are not tested on known benchmarksets in [13]

22 Constraint Programming Constraint programming isa powerful tool for solving combinatorial search problemsbased on techniques such as artificial intelligence operationsresearch and graph theoryThe basic idea in the CP is that theuser specifies constraints and solves these constraints throughgeneral-purpose solvers [55] It is generally used to solve real-life problems in several application areas such as rosteringscheduling and manufacturing [56]

Ernst et al [4] present a review of workforce schedulingproblems In their study workforce scheduling methods andtechniques are reviewed According to their study classifica-tion of the different approaches contains five groups andCP isone of them A review of the literature on workforce schedul-ing problems is presented in [1] too CP is incorporated intosolution techniques in their study There are many studieswith regard to CP in different application areas of workforcescheduling literature Examples of the application areas arenurse schedulingrostering [57ndash61] health care [62 63] bustransportation [64] general [65ndash68] retail [69]military [70]and call center [13] (Canonrsquos paper) We would like to drawattention to the lack of usage of CP in call center workforcescheduling studies


In general the main advantages of CP can be listedas follows (1) it deals with heterogeneous constraints andnonconvex solution sets (2) the domain size of the problemis independent and (3) it enables the usage of optimizationprogramming language (OPL) [71] Furthermore the CPmodels affordmore useful analyses for real cases by requiringless computational effort [72] Consequently the usage ofCP offers an ideal framework for various complicated andconstrained workforce scheduling problems [1]

3 Models and Formulations

As it is seen in Tables 1(a) and 1(b) there are many differentmethods for workforce scheduling problems in the literatureThemethods commonly used for call forecasting subproblemare ARIMA-HW [14 15] regression models [39] queueingtheory [16] simulation based algorithms [73] stochasticmodels [74 75] Bayes approach [54 76] and neural networks[45] Researchers generally use mathematical (queueing)models [2 18 77] for staffing subproblem Simulation basedapplications [21 31] are also considered as an alternative inrecent years At first MIP and IP models are considered forsolving shift scheduling subproblem in many studies but asthe problem sizes ie variable and constraint numbers areincreased and the problems become more complicated it isobserved that simulation based and heuristicmetaheuristictechniques are preferred [1] Likewise there are many studieswhich include IP MIP heuristics and hybrid models forrostering subproblem Some of these studies are listed inTables 1(a) and 1(b) Our proposed models are explained inthis section

31 Shi Scheduling Model (Model 1) The solution of shiftscheduling problemmainly involves determining the numberof the shifts to be chosen and the number of operators forthese shifts The result of the staffing problem indicates theassignment of the operators to fulfill the service level require-ment Minimizing the unused capacity means meeting thedemand optimally Besides the decision of whether openinga shift or not will have its own associated costs whereasunfulfillment of the demand might lead to the other costs Inthis study a shift scheduling model is proposed to minimizenumber of shifts the excess and unfulfilled demands Thedetails of the model are as follows

Sets 119863 Set of days in the planning horizon (forall119889 isin 119863 =1|119863|))119878 Set of shifts in a day (forall119904 isin 119878 = 1|119878|))119875 Set of periods in a work day (forall119901 isin 119875 = 1|119875|))Parameters119862119875 Working cost per period119862119878 Cost of opening a shift119862119880 Cost for unfulfilling the demand per period119864119904119901 1 if shift s covers period p 0 otherwise

120574 Maximum number of the operator to be assignedto a shift119873119889119901 Number of operators needed at period p of dayd

Decision Variables119886119889119904 Number of operators assigned to shift s of day d(forall119889 isin 119863 forall119904 isin 119878)119901119889119904 Binary variable indicating if shift s of day d ischosen (forall119889 isin 119863 forall119904 isin 119878)119906119889119901 Unfulfilled demand in period p of day d (forall119889 isin119863 forall119901 isin 119875)

Decision Expressions The number of the unnecessary opera-tors in period p of day d is calculated as follows119903119889119901 = sum

119904isinS(119864119904119901119886119889119904) minus 119873119889119901 (forall119889 isin 119863 forall119901 isin 119875) (1)

Mathematical Model The mathematical model can bedepicted as follows119872119894119899119894119898119894119911119890 sum

119889isin119863119901isin119875

119862119875119903119889119901 + sum119889isin119863119904isin119878

119862119878119901119889119904 + sum119889isin119863119901isin119875

119862119880119906119889119901 (2)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878

(119864119904119901 lowast 119886119889119904) + 119906119889119901 ge 119873119889119901(forall119889 isin 119863 forall119901 isin 119875) (3)

119901119889119904 le 119886119889119904 le 120574119901119889119904 (forall119889 isin 119863 forall119904 isin 119878) (4)119901119889119904 119864119904119901 isin 0 1(forall119889 isin 119863 forall119904 isin 119878 forall119901 isin 119875) (5)

119886119889119904 119906119889119901 119873119889119901 120574 ge 0 119886119899119889 119894119899119905119890119892119890119903(forall119889 isin 119863 forall119904 isin 119878 forall119901 isin 119875) (6)

The objective function of the proposed model Eq (2) iswritten based on the minimization of the operator costs shiftopening cost and unfulfilled demand per shift respectivelyThere are no differences among the operators in terms ofproductivity and skills (ability) The cost parameters are allthe same for each operator in all of the proposed modelsEquation (3) guarantees the assignment of the operatorsneeded for each period of each day Equation (4) enables theassignment of an operator if a shift is open ie it makes surethat each shift has enough operators Similarly the maximumnumber of operators per shift will not be exceeded and ldquo0rdquooperator assignment to the opened shifts will be avoided with(4) Finally (5) and (6) define the domain constraints

32 Rostering Models In this section the details of the IPand CP models developed for the rostering problems arediscussed


321 e IP Model (Model 2a) We define the input data ofan IP model for the rostering problem in this sectionSets 119860 Set of operators (agents) (forall119886 isin 119860 = 1|119860|)119884 Set of break types (forall119910 isin 119884 = 1|119884|)119861119910119904 Set of acceptable time intervals of break y in shift

s119878119874119901 Set of shifts that overlap with the time interval(period) p

Parameters119862119867 Cost of hourly work119866119889119904 Total number of operators assigned to shift s inday d119867 Number of work days in planning horizon119873119889119901 Number of operators needed at period p of dayd119871119910 Length of type y break in periods (size)119882119904 Total working hours of shift s

Decision Variables119909119886119889119904 Binary variable indicating if 119886th operator isassigned to shift s in day d (forall119886 isin 119860 forall119889 isin 119863 forall119904 isin 119878)119911119886119889119904119910119901 Binary variable indicating if 119886th operator isassigned to shift s in day d and the operator startshisher break y in period p (forall119886 isin 119860 forall119889 isin 119863 forall119904 isin119878 forall119910 isin 119884 forall119901 isin 119875)

Decision Expression Total number of operators who havetheir break y at period p of day d are computed as below

119882119889119910119901 = 119901sum119901=119901minus119871119910+1

( sum119886isin119860

119904isin119878119874119901

119911119886119889119904119910119901)(forall119889 isin 119863 forall119910 isin 119884 forall119901 isin 119875)

(7)

Mathematical Model Mathematical model can be depicted asfollows119898119894119899119894119898119894119911119890 sum

119886isin119860119889isin119863119904isin119878

(119862119867119882119904119909119886119889119904)(8)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878

119909119886119889119904 le 1 (forall119886 isin 119860 forall119889 isin 119863) (9)

sum119889isin119863119904isin119878

119909119886119889119904 le (119867 minus 1) (forall119886 isin 119860)(10)

119909119886119889119904 minus sum119901isin119861119910119904

119911119886119889119904119910119901 = 0(forall119886 isin 119860 forall119889 isin 119863 forall119904 isin 119878 forall119910 isin 119884) (11)

sum119886

119909119886119889119904 = 119866119889119904 (forall119889 isin 119863 forall119904 isin 119878) (12)sum119904isin119878119874119901

119866119889119904 minus sum119910

119882119889119910119901 ge 119873119889119901(forall119889 isin 119863 forall119901 isin 119875) (13)

119909119886119889119904 119911119886119889119904119910119901 isin 0 1(forall119886 isin 119860 forall119889 isin 119863 forall119904 isin 119878 forall119910 isin 119884 forall119901 isin 119875) (14)

119866119889119904119882119889119910119901119873119889119901119871119910119882119904 ge 0 and integer(forall119889 isin 119863 forall119904 isin 119878 forall119910 isin 119884 forall119901 isin 119875) (15)

119867 gt 1 and integer (16)

The objective function Eq (8) is written based on theminimization of the total workforce cost Equation (9) allowsmaximum one shift assignment to the operators each dayover the planning horizon Equation (10) enables taking aminimum of one day-off for the operators over the planninghorizon Equation (11) guarantees all the breaks of theoperators and determines the frequency of these breaksEquation (12) sets the number of the assigned operators equalto the number of operators in the shifts obtained as a resultof the shift scheduling problem That is one of the solutionvalues of the shift scheduling problem is used as a parameterof this rostering problem Equation (13) ensures the sufficientnumber of operators working in each period of each day overthe planning horizon Finally (14) through (16) defines thedomain constraints

322 e CP Model (Model 2b) The corner stones of anyscheduling problem modeled by CP are the tasks [72] Wesimplify the original scheduling problem by consideringeach break as a task and assigning workforce resources tothese tasks is the primary focus of our solution approachBy generating the solution of shift scheduling problem thenumber of operators assigned each day is determined Thenumber of breaks of an operator assigned to any shift givespractically the number of tasks that will be performed bythe operator For instance if operators have four breaksin a workday the total number of the tasks will be fourtimes the number of operators to be assigned during theplanning period Therefore we are able to reduce the size ofthe scheduling problem significantly In a way we solve anequivalent scheduling problem


ID Day Type Shi Size Start End

Figure 3 Data structure for tasks

In the CP model all the sets except the ldquoacceptable timeintervals of breaks in shiftsrdquo set which is explained in Model2a will be utilized In terms of parameters while the cost perhour (119862119867) per worker is considered in Model 2a the costper period (119862119875) per worker is considered in Model 2b Theutilization of interval length for the total cost calculation isthe reason of this difference For instance if 15-minute periodsystem is taken into consideration as there are four differentperiods in a one-hourworking period the periodical workingcost is calculated as 119862119875 = 1198621198784 The sets and the parametersof the proposed model are as follows

Sets 119869 Set of all tasks (forall119895 isin 119869 = 1120591|119884|)119872 Set of task dimension (forall119898 isin 119872 = 1|119872|)119877 Set of shifts (forall119903 isin 119877 = 1|119877|)119874 Task-resource allocation tuple119874 = ⟨119895 119886⟩ | forall119895 isin 119869 forall119886 isin 119860 (17)

119878119860 Set of active shifts(119878119860 = ⟨119903 119889 119904⟩ | forall119903 isin 119877 forall119889 isin 119863 forall119904 isin 119878) (18)

119878119875 The start and end time of shifts(119878119875 = ⟨119904 119894 119890⟩ | forall119904 isin 119878 forall119894 amp 119890 isin 119875) (19)

119879 Tuple of tasks (119879 = ⟨119895 119889 119910 119904 119898 119894 119890⟩ | forall119895 isin119869 forall119889 isin 119863 forall119910 isin 119884 forall119904 isin 119878 forall119898 isin 119872 forall119894 amp 119890 isin 119875)Parameters119862119875 Labor cost per period119866119889119904 Total number of operators assigned to shift s in

day d120591 Total number of assignments in planning horizon120575119910119904 Start time of break type y in shift s120576119910119904 End time of break type y in shift s

All possible tasks should be generated in the model beforerunning the CP model As it is seen in Figure 3 each task isidentified with 7-field tuple layout (119879) j is used for the taskindex number It expresses that type y task is done in day dshift s and between period i and period e with an intervalsize ofm Here the periods i and e which express the startingand ending periods of the related task type are derived fromparameters 120575119910119904 and 120576119910119904

The starting and ending periods (119878119875 ) and active shifts (119878119860 )are defined via 3-field tuple layout While the elements of 119878119875

Latest end timeEarliest start time

start time end timet

Task

Figure 4 An interval variable

tuple indicate that the shift s starts in the i119905ℎ period and endsin the e119905ℎ period elements of the 119878119860 tuple express that there isan assignment to 119889th day 119904th shift with 119903 index numberThe119874tuple generated for task-source sharing means that resourcea is used for the task with j index number

Decision Variables The time interval during which a task iscompleted in the CP can be represented with an intervaldecision variable In Figure 4 an interval decision variableincludes unique characteristics such as the earliest start timethe latest end time and duration

The position of a task between the earliest start timeand the latest end time is not known at the beginningThe positions of intervals are determined by the sequencevariables Besides the intervals can be considered optionallyby using different constraints The definitions of decisionvariables in our study are given as follows119909119896119886 Decision variable for 119886th operator in shift k

interval variable which is optional between timeperiods i and e and with a size of (e-i) forall119896 isin 119878119860 forall119886 isin119860 forall119894 amp 119890 isin 119878119875119908119905 Decision variable for task id t interval variablewhich is optional between time periods i and e andwith a size of m forall119905 isin 119879 (i e and m are given aselement of 119879)119911119900 Decision variable for resource allocation of task ointerval variable which is optional forall119900 isin 119874119902119896119886 Decision variable represents a total order over aset of 119911119900 sequence variable forall119896 isin 119878119860 forall119886 isin 119860

Cumulative Functions Ensuring that the number of activeoperators is equal to or less than the number of operatorsneeded for each period is a complex constraint consideringthat there are overlapping shifts in our problem With theproposed model this functionality can be achieved quiteeasily via cumulative functions In other words the tasksand sources (operators) of these tasks are represented asa function of time A cumulative function expression isrepresented by the OPL keyword cumulFunction and as


h

0

pulse (ah)

a

h

0

pulse (uvh)

u v

Figure 5 Example of pulse (a h) and pulse (u v h)

30

0

15

1 2 3 4 5

Figure 6 Example of cumulative function (stepwise 0997888rarr11997888rarr15 0997888rarr3 30997888rarr4 15997888rarr5)it is shown in Figure 5 when a is expressed with intervaldecision variable it is shown as 119891 = 119901119906119897119904119890(119886 ℎ) and whenit is expressed with any h parameter it is shown as 119891 =119901119906119897119904119890(119906 V ℎ) The value of function 119891 = 119901119906119897119904119890(119886 ℎ) changesto h at the start of an interval variable while the value offunction 119891 = 119901119906119897119904119890(119906 V ℎ) changes to h between times u andv

In short as it is shown in Figure 6 a cumulative functionexpresses a step function that has a decreasing or increasingtendency in related to the fixed or varying time intervals Thefunctions from (20) to (22) express the number of operatorsassigned number of the operators on a break and number ofthe operators needed per period respectively119891119882119889 = 119901119906119897119904119890 (119909119896119886 1)forall119889 isin 119863 forall119896 isin 119878119860 forall119886 isin 119860 119896119889 = 119889 (20)119891119861119889 = 119901119906119897119904119890 (119908119905 1) forall119889 isin 119863 forall119905 isin 119879 119905119889 = 119889 (21)119891119877119889 = 119901119906119897119904119890 (119901 minus 1 119901119873119889119901) forall119889 isin 119863 forall119901 isin 119875 (22)

eConstraint Programming Model Due to the differences inthe functional construction of available modeling languageson the market CP models are more dependent on theCP packages compared to the mathematical programmingmodels IBM ILOG CPLEX Optimization Studio 126 syntaxis used as the modeling language in this study Table 2illustrates the definitions of the syntax used in theOPLmodel(see Userrsquos Manual for IBM ILOG CPLEX)

The CP model for the rostering problem is formulated asfollows

Objective Function

119898119894119899119894119898119894119911119890 sum119896isin119878119860119886isin119860

(119862119875119897119890119899119892119905ℎ119874119891 (119909119896119886)) (23)

Equation (23) is written based on the minimization of work-force cost As it is underlined in Table 2 with the expressionof 119897119890119899119892119905ℎ119874119891 the period length of a119905ℎ operatorrsquos task in the kthassigned shift is found

Constraints

Presence (Logical Constraints)The presence of an optional interval can be determined

using the presenceOf constraint For instance given thatldquoardquo and ldquobrdquo are two optional intervals if ldquoardquo is present ldquobrdquo ispresent as well In a way this looks like Boolean expressions

sum119886isin119860

119901119903119890119904119890119899119888119890119874119891 (119909119896119886) = 119866119889119904 forall119896 isin 119878119860 forall119889 isin 119863 forall119904 isin 119878 (24)

sum119896isin119878119860

119901119903119890119904119890119899119888119890119874119891 (119909119896119886) le 1 forall119886 isin 119860 forall119889 isin 119863 119896119889 = 119889 (25)

sum119896isin119878119860

119901119903119890119904119890119899119888119890119874119891 (119909119896119886) le (119867 minus 1) forall119886 isin 119860 (26)

sum119905isin119879119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = (119901119903119890119904119890119899119888119890119874119891 (119909119896119886))forall119886 isin 119860 forall119896 isin 119878119860 forall119910 isin 119884 119905119895 = 119900119895 119900119886 = 119886 119905119904 = 119896119904 119905119889 = 119894119889 119905119910 = 119910 (27)

sum119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = 119901119903119890119904119890119899119888119890119874119891 (119908119905) forall119905 isin 119879 119905119895 = 119900119895 (28)


Table 2 Syntax definitions used in OPL expressions and constraints

Syntax Definition119897119890119899119892119905ℎ119874119891 Integer expression to access the length of an interval119899119900119874V119890119903119897119886119901 Constraint used to prevent intervals in a sequence from overlapping119901119903119890119904119890119899119888119890119874119891 Constraint used to enforce the presence of intervals119904119910119899119888ℎ119903119900119899119894119911119890 Constraint used to synchronize the start and end of intervals119904119905119890119901119908119894119904119890 Keyword for stepwise linear functions

Table 3 List of CP constraints used for benchmark instances

Constraint on [8] Definition of Constraint Our Constraint Structure1 An employee cannot be assigned more than one shift on a single day Cumulative function2 A minimum amount of rest is required after each shift Cumulative function3 The maximum numbers of shifts of each type that can be assigned to employees Cumulative function4 Minimum and maximum work time Presence (Logical constraints)5 Maximum consecutive shifts Count function6 Minimum consecutive shifts Count function7 Minimum consecutive days off Count function8 Maximum number of weekends Presence (Logical constraints)9 Days-off Presence (Logical constraints)10 Cover requirements Cumulative function

Equation (24) sets the number of operators to be assigned tothe shift during the day equal to the number of operators pershift obtained from the result of the shift scheduling problemEquation (25) enables the assignment of the operators tomaximum one shift in any day in the planning horizonEquation (26) allows the operators to have at least one day-off in the planning horizon Equation (27) guarantees thatthe operators get only one task (break) in each shift they areassigned Equation (28) enables that each task is done by onlyone operator

Synchronize Formation A synchronization constraint (key-word synchronize) between an interval decision variablea and a set of interval decision variables B makes all presentintervals in the setB start and end at the same times as intervala if it is present [78]With (29) the start and end times of theallocations (119911119900) are determined119904119910119899119888ℎ119903119900119899119894119911119890 (119908119905 1199111 119911119900 119900 isin 119874)forall119905 isin 119879 119905119895 = 119900119895 (29)

Overlap Prevention A noOverlap constraint (keywordnoOverlap) can be used to schedule the tasks that usespecific resources With (30) the assignment of an operatorto another task while (s)he is performing a task during thesame period is prevented119899119900119874V119890119903119897119886119901 (119902119896119886) forall119886 isin 119860 forall119896 isin 119878119860 (30)

Cumulative Functions Equation (31) guarantees that enoughoperators are to be assigned to each period of each day in theplanning horizon119891119882119889 minus 119891119861119889 ge 119891119877119889 forall119889 isin 119863 (31)

Domain Constraints Finally (32) defines the domain con-straints119866119889119904 120575119910119904 120576119910119904 120591 ge 0 and integerforall119889 isin 119863 forall119904 isin 119878 forall119910 isin 119884 (32)

In order to verify our CP model we used benchmark testinstances given in [8] For this purpose we rewrote themathematical model in [8] with CP constructs In otherwords we developed a CP model derived from our workbased on mathematical model proposed in [8] We are onlyable to compare our CP approach with known results in theliterature Themodel in [8] is not given in detail in our paperdue to space limitations We prefer giving CP equivalents ofthe constraints in [8] in Table 3

In addition to constraints given in Table 3 synchronizeand noOverlap constraints are used to find solutions to thebenchmark test instances We do not use count constraint inour original CP model for the rostering problem However itis used for benchmark test instances (see Section 422) Wereport our findings in the following section

4 Experimental Studies

41 A Real-Life Problem

411 Problem Definition Real data obtained from an out-source call center supplying services to various companies areused in this studyThe company provided us shift informationand the minimum number of operators needed at eachtime interval to meet the call demand The working dayis divided into 15-minute periods Therefore there are 96periods each day (ie |119875|=96) Planning horizon is settledas seven days (|119863|=7) Company provided data for a single


Table 4 Shift templates

Shiftnumber (s)

Time range(24h)

ExistingAdditional shifts

Shift periods Allotted periodsfor break type 1

Allotted periodsfor break type 2


Allotted periodsfor break type 4Start (i) End (e)

1 00-08 1 32 3-8 11-17 21-26 27-322 08-16 29 64 35-40 43-49 53-58 59-643 09-17 33 68 39-44 47-53 57-62 63-684 10-18 37 72 43-48 51-57 61-66 67-725 11-19 41 76 47-52 55-61 65-70 71-766 12-20 45 80 51-56 59-65 69-74 75-807 13-21 49 84 55-60 63-69 73-78 79-848 14-22 53 88 59-64 67-73 77-82 83-889 15-23 57 92 63-68 71-77 81-86 87-9210 16-00 61 96 67-72 75-81 85-90 91-96 existing shifts additional shifts in the analyses 119878119875 = ⟨119904 119894 119890⟩ | 119904 isin 119878 119894 amp 119890 isin 119875

planning horizon ie seven days The outline for the shiftsis displayed in Table 4 There are five shifts (119904 = 1 2 4 6 10)during the weekdays that is from Monday (119889 = 1) to Friday(119889 = 5) and three shifts (119904 = 1 2 10) during the weekends(119889 = 6 7) in this setup We add five more possible shifts tothis model and evaluate all of them regardless of weekday orweekend With the ones added to the analyses the possibleshift set has ten members (119878= ldquo00-08rdquo ldquo08-16rdquo ldquo09-17rdquo ldquo10-18rdquo ldquo11-19rdquo ldquo12-20rdquo ldquo13-21rdquo ldquo14-22 ldquo15-23rdquo ldquo16-00rdquo) in ourexperimental setting

Each operator has four break types during the shift (s)heis assigned (|119884|=4) in this firmThe break types of first thirdand fourth are relief break the type of second break is mealbreak The periods of all break types for each shift are showninTable 4There are one and half-hour break windows for therelief breaks and two- and a half-hour break windows for themeal break All shifts are 8-hour long (119882forall119904isin119878=8)

There are currently 76 operators working at this location(|119860|=76)These operators can only work in one shift in a dayand they have to take minimum one day-off within 7-dayplanning horizon At most twenty operators can be assignedto a shift due to the capacity constraints in the shift schedulingproblem (120574 = 20) Relief and meal breaks should be taken atbreak time windows given in Table 4 Each relief break maylast 15 minutes (1198711 1198713 and 1198714=1) and meal break lasts for30 minutes (1198712=2) for each operator The times of break maydiffer for each operator in the call center

The number of the workforce required for 15-minuteperiods for seven days (119873119889119901) in the planning horizon of thefirm is also known We illustrate the operator requirementsfor seven-day planning horizon in Figure 7

First the shift scheduling problem (Model 1) is solved forthis organization The solution of this problem will specifythe number of operators to be assigned to the 119904th shift in the119889th day (119866119889119904) However these numbers will be adjusted prorota in the rostering problem in order to meet the demandrequired per period during the breaks of the operators Theincrease (adjustment) rate can be at most 30 Besides atmost 26 operators can be assigned for a shift Skipping thisstep leads to an infeasible solution for the rostering problem

Table 5 Input data for Model 1 parameters

Parameters Value Unit119862119867 12 TLOperator per hour119862119875 3 TLOperator per period119862119878 300 TLShift119862119880 100 TLOperator per period119867 7 DaysTL=Turkish liras

The updated 119866119889119904 the tasks tuple (119879) and active shifts tuple(119878119860) will constitute the input for the rostering problem Therest of the input data used is summarized in Table 5

412 Implementation and Computational Results The solu-tions of the shift scheduling problem are presented and thenthe results obtained from both models (IPCP) of rosteringproblem are discussed in this section All of the proposedmodels in this study are implemented by using IBM ILOGCPLEXOptimization Studio which is available free of chargeat IBM Academic Initiative web site for the academic usersWe run our models on a laptop computer with Intel i5Processor and 8 GB memory The run time of CPLEX is setto 3600 seconds ie one hour Model 1 is solved both forthe existing shift system and for the shifts included in theanalyses For both cases an optimal solution is obtained andthe proposed case has shown an almost 15 improvementin the objective function compared to the existing case Thesolution details of the models are shown in Table 6

Day-shift assignment results for both cases are displayedin Table 7 The cells show the number of the requiredassignments for the related shift in the corresponding dayThecells including ldquo0rdquo indicate that the shift in the correspondingday will not be utilized and thus no assignments are required

While 31 shifts are available for the existing case in theplanning period 27 shifts (|119877| = 27) are utilized for theproposed case Hence four shifts are never used in planningin our case Since the firm did not prefer unmet demand allthe demands in the corresponding planning horizon are met


1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7

0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs

Figure 7 The number of operators needed at each period for 7-day planning horizon

Table 6 Details of the Model 1

Problem Objective Time (s) No of constraints No of variablesExisting 15501 486 983 1373Proposed 13437 477 952 1373

Table 7 The numbers of the operators assigned to the shifts according to the days (119866119889119904)Days

Shifts1 2 3 4 5 6 7 8 9 10

E P E P E P E P E P E P E P E P E P E P1 7 7 13 15 0 0 9 0 0 0 7 14 0 0 0 0 0 0 15 152 6 6 14 17 0 0 8 0 0 0 6 11 0 0 0 0 0 0 14 143 7 7 15 13 0 0 7 0 0 11 5 0 0 0 0 0 0 0 14 154 8 8 13 13 0 0 5 0 0 0 5 10 0 0 0 0 0 0 13 135 8 8 14 14 0 0 6 0 0 0 7 12 0 0 0 0 0 0 16 166 7 7 16 10 0 0 0 0 0 0 0 6 0 0 0 0 0 0 17 117 7 7 14 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 10E Existing Case P Proposed Case

Even if there are unfulfilled demands the cost of this wouldbe bore and any necessary changes in the number of operatorsassigned to the shifts will be adjusted while passing to therostering model

The active shifts (119878119860) to meet the demands are deter-mined The solutions for the operator assignment to activeshifts to their breaks and to their days-offwill be given Phase2 is only solved for the proposed case by determining theadjusted demand There are not any changes in the programparameters for Model 2a except adding a time constraint Inorder to get more effective results in CP some parameters canbe adjusted Most of the parameter settings are used in theirdefault values The parameters that are different from theirdefault values are given in Table 8

Since long computation time may be required for obtain-ing the optimized solution in CP optimizer fail limit is set to100000 Number of workers is chosen as 1 If the number of

Table 8 Parameter setting for the CP solver

Parameter SettingFail limit 100000Number of workers 1Search phase 119909119896119886Search type Restart

workers is set to n the CP optimizer engine creates nworkerseach in their own thread and thatwill work in parallel to solvethe problem Using multiple workers requires more memorythan using a single worker A search phase allows us to specifythe order of the search moves and the order in which thevalues must be tested To prioritize the operator assignmentsto the shifts we set the search phase as119909119896119886 ldquoRestartrdquo is chosen


Table 9 Model 2a and Model 2b details

Inc rate() 120591 Model 2a (IP) Model 2b (CP)

Obj Time (s) No of const No of variab Obj Time (s)(fail lim) No of const No of variab

15 365 Infeasiblea No solutionb



22lowast 386 37056 87723

677214 876709

37056 707 (26306) 264586 12299223lowast 386 37056 88641 37056 709 (26365) 264586 12299224 390 37440 89857 37440 616 (23189) 267066 12422425 390 37440 90635 37440 629 (22930) 267066 12422430 409 39264 91533 39264 675 (26041) 278846 13007646 456 43776 92812 43776 809 (20965) 307986 14455247 462 Infeasiblea No solutionb

The results in bold are proved to be optimallowastThe increase rate (Inc rate) corresponding to the lowest optimal value of the caseaNo integer solution foundbSearch terminated by limit no solution found

as ldquosearch strategyrdquo by considering the study of Regin [79]The solution details of the models are illustrated in Table 9

According to the results the increase rate in the numberof operators to be assigned to the 119904th shift in the 119889th day(119866119889119904) which arises as a result of shift scheduling stage willnot be feasible for rostering stage if it is less than 22 ormore than 46 All the results in this range provide optimalresults for both models ie Models 2a and 2b of rosteringproblem If the rate is 22 or 23 there will be a total of120591=386 assignments and the lowest optimal solution (386 x 12TLh x 8 h=37056TL) will be reached It is understood thatthe operators have approximately 192 dayweek days-off

Model 2b results in an impressively better performancethan Model 2a in terms of solution time In Table 9 while thevalues outside the parentheses in CP time column show thetime to reach the optimum result the values in parenthesesshow the time to reach the fail limit While Model 2a reachesthe minimum optimal result in 87723 seconds Model 2breaches the same result in 707 seconds When all casesare considered it can be seen that the time to reach theoptimal results in Model 2a is almost 130 times more thanthat in Model 2b While the total number of constraints andvariables does not depend on the increase rate in Model 2ait changes in Model 2b because the total number of tasks(|119869| = 120591|119884|) increases in proportion to the total number ofthe assignments

42 Experimental Trials In this section the proposedmodelsfor the rostering problem are tested on simulated instancesand benchmark test instances in [8]

421 Results from Simulated Test Instances We generatedeleven test instances by varying the demand within certainlimits per period from small to large to compare the per-formances of rostering models First we solved each testinstance via Model 1 to figure out what shifts should be usedTable 10 presents the input and output values and test results

of Model 1 Then by increasing the number of operatorsassigned to the shifts in certain rates we generated the inputsfor the rostering problem by balancing the total number ofoperators and the total number of assignments Hence thenumber of operators varied between 27 and 185 for the testinstances

Table 10 summarizes the results of the computations ofModel 1 An optimal result is found for all instances andobjective function values solution times the numbers ofthe constraints and the variables are shown in Table 10 Thenumbers of the constraints and the decision variables are thesame for all instances Average solution time is 443 seconds

Table 11 compares the computational results of the twomodels for the rostering problem The increase rate variesbetween 10 and 30 by 5 steps The optimal solutionvalues for all instances are shown in Table 11 For example theoptimal solution is reached with a 26 increase rate and 141operators are assigned to active shifts in the planning horizonResults obtained between 26 and 30 increase rates areoptimal and size of the operator set is equal to 27 for the firstinstance

Whenbothmodels are compared in terms of all instancesit can be inferred that Model 2b is much more efficient thanModel 2a The IP model contains a noticeably larger numberof constraints and decision variables in all problems thanthe CP model Moreover the number of variables is morethan the number of constraints in IP model On the otherhand the number of the constraints is more than that of thevariables in CP While the numbers of the constraints andthe variables are independent of the increase rate for eachinstance in IP model the variable and the constraint sizes arein direct relation to the total number of assignments in CPmodel

The proposed CP model can provide a feasible solutionwithin seconds When all cases are considered it is seen thatthe time to reach the optimal results in IP is almost 290 timesmore than that of CP In the biggest problem (the eleventh


Table 10 Information on test instances and details of Model 1

Ins Operator demand per period 120574 Total number ofthe opened shifts 120591 Obj Time (s) No of const No of variab

Min Ave Max1 0 354 9 10 23 100 9360 462 952 13732 1 551 13 10 26 147 10806 497 952 13733 1 758 18 10 27 196 11625 427 952 13734 3 1082 23 15 26 274 12282 397 952 13735 2 1140 26 15 28 299 14121 430 952 13736 2 1140 26 20 27 299 13821 468 952 13737 3 1407 32 30 27 367 14973 462 952 13738 3 1407 32 30 27 367 14973 458 952 13739 4 2099 47 40 27 546 18201 407 952 137310 4 2280 52 40 27 598 19542 407 952 137311 5 2666 60 40 27 701 21648 462 952 1373The results in bold are proved to be optimal

instance) the IP model cannot provide a solution within thespecified run time (3600 seconds) and CP model can givea solution only in 20 seconds A large memory usage of 63GB for this instance is seen in CPmodel It cannot be found asolution with CP by using current limits and settings in largermodelsHowever we can say that the feasible solutions can beobtained with IP by increasing the run time and with CP byusing a computer with a higher memory

422 Results of Benchmark Test Instances (from httpwwwschedulingbenchmarksorg) In order to test our CPapproach on publicly available benchmark test instanceswe modified our model accordingly Table 12 compares thecomputational results of models in [8] and our proposedCP model The instances are available for download athttpwwwcsnottacuksimtecNRP where all the requiredinformation on each instance best solutions and lowerbounds is also available Models studied in [8] are anEjection Chain metaheuristic a Branch and Price methodand an Integer Programming formulation coded on Gurobi563 As seen in Table 12 the instances range from very smallto very large

As it is seen from Table 12 the branch and price methodis very effective for small and medium size benchmark testinstances However it fails to return solutions for largerinstances instance 13 and beyond due to the lack of memoryissues On the other hand our CP approach is able to returnsolutions for Instances 14-19 Our approach also fails dueto the memory issues for instances between 20 and 24 Forthese instances Gurobi 563 is not able to find solutionseither Ejection Chain metaheuristics method yielded somesolutions for the larger instances though not necessarily goodsolutions

Our approach yielded the optimal solutions for thefirst four instances too within very reasonable run timescompared to the others For the larger instances near optimalor near lower bound solutions are found Nevertheless itachieves better results than Ejection Chain metaheuristicsfor the instances that feasible solutions can be found In

general Gurobi 563 found the best near optimal solutionsHowever our approach yielded the best results for instances15 18 and 19 Therefore we can conclude that our approachis competitive enough in general with methods discussed in[8]

5 Conclusion

Call centers are business units where workforce is an expen-sive resource and an extensively used oneThemain objectivein these business units is to ensure the maximum customerand employee satisfaction via the minimum operational andworkforce costs In order tomeet this objective the firms haveto utilize several techniques Workforce scheduling modelscan be regarded as one of the most useful tools in this regard

This paper addresses an integrated solution for the shiftscheduling and the rostering problems within call centersShift scheduling problem is formulated as an IP model Weintroduce an integer programming and a constraint program-ming model to solve the rostering problem Assignments ofoperators to proposed shifts weekly days-off and meal andrelief break times of the operators are determined with theserostering models It is shown that even a detailed rosteringproblem can be easily regarded as a task-resource assignmentproblem with CP approach

The proposed models are tested by using real-life datatest problems generated with a series of scenarios and bench-mark test instances Cross-comparison through experimentalresults is conductedWhile comparing these results the focusis mostly on the models developed for rostering problemsThe results indicate that CP modeling is more preferablethan the IP modeling in terms of facilitation flexibility andhigh speed in computation time Moreover it is observedby analyzing benchmark results that CP can be utilizedeffectively for solving general workforce scheduling problemstoo The advantage of CP over IP is related to the fact thatthe declarative language of logic has an explanatory powerThe growth in the size of the models does not affect CPin terms of finding solutions On the other hand temporal


Table 11 Comparison of Model 2a and Model 2b for test problems

Ins Incrate () 120591 |119860| Model 2a (IP) Model 2b (CP)

Obj Time (s) No of const No of variab Obj Time (s) (fail lim) No of const No of variab1 10 123 27 Infeasiblea No solutionb

15 127 27 Infeasiblea No solutionb



26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146

2 10 173 38 Infeasiblea No solutionb

15lowast 186 38 17856 22232 409314 541283 17856 075 (12884) 70969 309923 10 223 51 Infeasiblea No solutionb



23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()

120591 |119860| Model 2a (IP) Model 2b (CP)Obj Time (s) No of const No of variab Obj Time (s) (fail lim) No of const No of variab



20 852 185 Timeoutc No solutionb

25lowast 889 185 Timeoutc 85344 2067 (93082) 1387842 671406The results in bold are proved to be optimallowastThe increase rate (Inc rate) corresponding to the lowest optimal value of the caseaNo integer solution foundbSearch terminated by limit no solution foundcTimeout (3600 seconds)

Table 12 Comparison of our model and other models in [8] for benchmark instances

Ejection Chain Branch and Price Gurobi 563 Our Proposed CP ModelIns W St Sh LB 10 min 60 min Sol Time (s) Sol Time (s) Sol Time (s) First Time (s)1 2 8 1 607 607 607 607 027 607 162 607 075 0752 2 14 2 828 923 837 828 013 828 522 828 6269 62693 2 20 3 1001 1003 1003 1001 045 1001 1354 1001 15833 158334 4 10 2 1716 1719 1718 1716 150 1716 15899 1716 51848 518485 4 16 2 1143 1439 1358 1160 2561 1143 152024 1180 360015 722626 4 18 3 1950 2344 2258 1952 1046 1950 44093 2014 360059 3588837 4 20 3 1056 1284 1269 1058 9373 1056 215248 1091 360050 3196738 4 30 4 1281 2529 2260 1308 360000 1323 359983 1522 360176 3581429 4 36 4 247 474 463 439 7699 439 359985 452 360285 31640210 4 40 5 4631 4999 4797 4631 11344 4631 24420 4666 360451 35720511 4 50 6 3443 3967 3661 3443 1911 3443 10992 3455 361079 35079712 4 60 10 4040 5611 5211 4046 13364 4040 230384 4409 367693 29980313 4 120 18 1346 8707 3037 Out of Memory 3109 360055 Out of Memory14 6 32 4 1277 2542 1847 Out of Memory 1280 360013 1345 360670 35318915 6 45 6 3806 6049 5935 Out of Memory 4964 360000 4548 362562 33175916 8 20 3 3211 4343 4048 3323 26502 3233 359999 3332 360162 35804817 8 32 4 5726 7835 7835 Out of Memory 5851 360000 5911 360798 35692118 12 22 3 4351 6404 6404 Out of Memory 4760 359999 4668 360473 35876019 12 40 5 2945 6522 5531 Out of Memory 5420 360590 4900 362612 34045820 26 50 6 4743 23531 9750 Out of Memory - 360005 Out of Memory21 26 100 8 20868 38294 36688 Out of Memory - 360021 Out of Memory22 52 50 10 - - 516686 Out of Memory - 360019 Out of Memory23 52 100 16 - - 54384 Out of Memory - 360043 Out of Memory24 52 150 32 - 156858 Out of Memory Out of Memory Out of MemoryKnown optimal solutions are in boldW= Number of Weeks St= Number of Staff Sh= Number of Shift Type LB=Best Known Lower Bound

constraints prevent IP models for finding a feasible solutionOne of the limitations of CP is exceeding thememory size dueto the increase in size of model In such occasions CPmodelscannot provide any solutions because of the inadequaciesin resources Even though a modest laptop with 8 GB ofRAM has been used in this study some CP models arerun on computers with larger memory like 48 GB in theliterature We expect that if our CP model is run on sucha system it will find feasible even optimal solutions withinacceptable run time limits for larger benchmark instan-ces

Additional real-life constraints eg an operator whoworks in a specified shift of a related day may not work ina prespecified shift in the following day or if the operator hasa break at a certain period the next break time should be afterprespecified periods etc can be added to the model in futurestudies Multiskill rostering problem also is left to be studiedin a follow-up work

Data Availability

The data used in our experiments are available upon requestfrom the corresponding author


Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] J van den Bergh J Belien P de Bruecker E Demeulemeesterand L de Boeck ldquoPersonnel scheduling a literature reviewrdquoEuropean Journal of Operational Research vol 226 no 3 pp367ndash385 2013

[2] N Gans G Koole and A Mandelbaum ldquoTelephone call cen-ters tutorial review and research prospectsrdquoManufacturing ampService Operations Management vol 5 no 2 pp 79ndash141 2003

[3] D C Dietz ldquoPractical scheduling for call center operationsrdquoOmega vol 39 no 5 pp 550ndash557 2011

[4] A T Ernst H Jiang M Krishnamoorthy and D Sier ldquoStaffscheduling and rostering a review of applications methods andmodelsrdquo European Journal of Operational Research vol 153 no1 pp 3ndash27 2004

[5] S Bhulai G Koole and A Pot ldquoSimple methods for shiftscheduling in multiskill call centersrdquoManufacturing amp ServiceOperations Management vol 10 no 3 pp 411ndash420 2008

[6] Z Aksin M Armony and V Mehrotra ldquoThe modern call cen-ter amulti-disciplinary perspective on operationsmanagementresearchrdquo Production Engineering Research and Developmentvol 16 no 6 pp 665ndash688 2007

[7] B Sungur C Ozguven and Y Kariper ldquoShift scheduling withbreak windows ideal break periods and ideal waiting timesrdquoFlexible Services and Manufacturing Journal vol 29 no 2 pp203ndash222 2017

[8] T Curtois and R Qu ldquoTechnical report Computational resultson new staff scheduling benchmark instancesrdquo Tech Rep 2017

[9] P Brucker R Qu and E Burke ldquoPersonnel scheduling modelsand complexityrdquo European Journal of Operational Research vol210 no 3 pp 467ndash473 2011

[10] L C Edie ldquoTraffic Delays at Toll Boothsrdquo Journal of the Opera-tions Research Society of America vol 2 no 2 pp 107ndash138 1954

[11] G B Dantzig ldquoLetter to the EditormdashA Comment on EdiersquosldquoTraffic Delays at Toll Boothsrdquordquo Journal of the OperationsResearch Society of America vol 2 no 3 pp 339ndash341 1954

[12] A T Ernst H Jiang M Krishnamoorthy B Owens and DSier ldquoAn annotated bibliography of personnel scheduling androsteringrdquoAnnals of Operations Research vol 127 pp 1ndash4 2004

[13] C Canon ldquoPersonnel scheduling in the call center industryrdquo4OR vol 5 no 1 pp 89ndash92 2007

[14] L Bianchi J Jarrett and R C Hanumara ldquoImproving fore-casting for telemarketing centers by ARIMA modeling withinterventionrdquo International Journal of Forecasting vol 14 no 4pp 497ndash504 1998

[15] J W Taylor ldquoA comparison of univariate time series methodsfor forecasting intraday arrivals at a call centerrdquo ManagementScience vol 54 no 2 pp 253ndash265 2008

[16] G Koole and A Mandelbaum ldquoQueueing models of call cen-ters an introductionrdquo Annals of Operations Research vol 113pp 41ndash59 2002

[17] J Gong M Yu J Tang and M Li ldquoStaffing to maximize profitfor call centers with impatient and repeat-calling customersrdquoMathematical Problems in Engineering vol 2015 Article ID926504 10 pages 2015

[18] L Brown N Gans AMandelbaumet al ldquoStatistical analysis ofa telephone call center a queueing-science perspectiverdquo Journalof the American Statistical Association vol 100 no 469 pp 36ndash50 2005

[19] G Koole and E van der Sluis ldquoOptimal shift scheduling with aglobal service level constraintrdquo Institute of Industrial Engineers(IIE) IIE Transactions vol 35 no 11 pp 1049ndash1055 2003

[20] G M Thompson ldquoLabor staffing and scheduling models forcontrolling service levelsrdquo Naval Research Logistics (NRL) vol44 no 8 pp 719ndash740 1997

[21] J Atlason M A Epelman and S G Henderson ldquoCall centerstaffing with simulation and cutting plane methodsrdquo Annals ofOperations Research vol 127 no 1 pp 333ndash358 2004

[22] W B Henderson and W L Berry ldquoDetermining Optimal ShiftSchedules for Telephone Traffic Exchange Operatorsrdquo DecisionSciences vol 8 no 1 pp 239ndash255 1977

[23] T Aykin ldquoOptimal shift scheduling with multiple break win-dowsrdquoManagement Science vol 42 no 4 pp 591ndash602 1996

[24] T Aykin ldquoComparative evaluation of modeling approaches tothe labor shift scheduling problemrdquo European Journal of Oper-ational Research vol 125 no 2 pp 381ndash397 2000

[25] H K Alfares ldquoOperator staffing and scheduling for an IT-helpcall centrerdquo European Journal of Industrial Engineering vol 1no 4 pp 414ndash430 2007

[26] E I Asgeirsson and G L Sigurethardottir ldquoNear-optimal MIPsolutions for preference based self-schedulingrdquoAnnals of Oper-ations Research vol 239 no 1 pp 273ndash293 2016

[27] J Atlason M A Epelman and S G Henderson ldquoOptimizingcall center staffing using simulation and analytic center cutting-plane methodsrdquo Management Science vol 54 no 2 pp 295ndash309 2008

[28] A N Avramidis M Gendreau P LrsquoEcuyer and O PisacaneldquoSimulation-based optimization of agent scheduling in mul-tiskill call centersrdquo in Proceedings of the 2007 5th Interna-tional Industrial Simulation Conference ISC 2007 pp 255ndash263Netherlands June 2007

[29] A N Avramidis W Chan and P LrsquoEcuyer ldquoStaffing multi-skillcall centers via search methods and a performance approxima-tionrdquo Institute of Industrial Engineers (IIE) IIE Transactions vol41 no 6 pp 483ndash497 2009

[30] I Castillo T Joro and Y Y Li ldquoWorkforce scheduling withmultiple objectivesrdquo European Journal of Operational Researchvol 196 no 1 pp 162ndash170 2009

[31] M T Cezik and P LrsquoEcuyer ldquoStaffing multiskill call centers vialinear programming and simulationrdquoManagement Science vol54 no 2 pp 310ndash323 2008

[32] M Defraeye and I Van Nieuwenhuyse ldquoA branch-and-boundalgorithm for shift scheduling with stochastic nonstationarydemandrdquo Computers amp Operations Research vol 65 pp 149ndash162 2016

[33] Sergey Dudin Chesoong Kim Olga Dudina and JanghyunBaek ldquoQueueing System with Heterogeneous Customers as aModel of a Call Center with a Call-Back for Lost CustomersrdquoMathematical Problems in Engineering vol 2013 Article ID983723 13 pages 2013

[34] K Ertogral and B Bamuqabel ldquoDeveloping staff schedules for abilingual telecommunication call center with flexible workersrdquoComputers amp Industrial Engineering vol 54 no 1 pp 118ndash1272008

[35] M Excoffier C Gicquel and O Jouini ldquoA joint chance-constrained programming approach for call center workforce


scheduling under uncertain call arrival forecastsrdquo Computers ampIndustrial Engineering vol 96 pp 16ndash30 2016

[36] L V Green P J Kolesar and W Whitt ldquoCoping with time-varyingdemandwhen setting staffing requirements for a servicesystemrdquo Production Engineering Research andDevelopment vol16 no 1 pp 13ndash39 2007

[37] S Helber and K Henken ldquoProfit-oriented shift scheduling ofinbound contact centers with skills-based routing impatientcustomers and retrialsrdquo OR Spectrum vol 32 no 1 pp 109ndash134 2010

[38] M Hojati ldquoNear-optimal solution to an employee assignmentproblem with seniorityrdquo Annals of Operations Research vol 181pp 539ndash557 2010

[39] R Ibrahim and P LrsquoEcuyer ldquoForecasting call center arrivalsFixed-effectsmixed-effects andbivariatemodelsrdquoManufactur-ing and Service OperationsManagement vol 15 no 1 pp 72ndash852013

[40] A Ingolfsson F Campello X Wu and E Cabral ldquoCombininginteger programming and the randomization method to sched-ule employeesrdquo European Journal of Operational Research vol202 no 1 pp 153ndash163 2010

[41] N Kyngas J Kyngas and K Nurmi ldquoOptimizing Large-ScaleStaff Rostering Instancesrdquo in Proceedings of the InternationalMulti Conference of Engineers and Computer Scientists vol IIpp 1524ndash1531 2012

[42] S Liao C Van Delftrsquo G Koole Y Dallery and O JouinildquoCall center capacity allocation with random workloadrdquo inProceedings of the 2009 International Conference on Computersand Industrial Engineering CIE 2009 pp 851ndash856 France July2009

[43] Z Liu Z Liu Z Zhu Y Shen and J Dong ldquoSimulated anneal-ing for a multi-level nurse rostering problem in hemodialysisservicerdquo Applied So Computing vol 64 pp 148ndash160 2018

[44] S Mattia F Rossi M Servilio and S Smriglio ldquoStaffing andscheduling flexible call centers by two-stage robust optimiza-tionrdquo OMEGA - e International Journal of Management Sci-ence vol 72 pp 25ndash37 2017

[45] D Millan-Ruiz and J I Hidalgo ldquoForecasting call centrearrivalsrdquo Journal of Forecasting vol 32 no 7 pp 628ndash638 2013

[46] J E Nah and S Kim ldquoWorkforce planning and deployment fora hospital reservation call center with abandonment cost andmultiple tasksrdquo Computers amp Industrial Engineering vol 65 no2 pp 297ndash309 2013

[47] E L Ormeci F S Salman and E Yucel ldquoStaff rostering in callcenters providing employee transportationrdquo OMEGA - eInternational Journal of Management Science vol 43 pp 41ndash532014

[48] A Pot S Bhulai and G Koole ldquoA simple staffing method formultiskill call centersrdquo Manufacturing and Service OperationsManagement vol 10 no 3 pp 421ndash428 2008

[49] M I Restrepo B Gendron and L-M Rousseau ldquoA two-stage stochastic programming approach for multi-activity tourschedulingrdquo European Journal of Operational Research vol 262no 2 pp 620ndash635 2017

[50] TRRobbins andT PHarrison ldquoA simulation based schedulingmodel for call centers with uncertain arrival ratesrdquo in Proceed-ings of the 2008 Winter Simulation Conference WSC 2008 pp2884ndash2890 USA December 2008

[51] T R Robbins and T P Harrison ldquoA stochastic programmingmodel for scheduling call centers with global service levelagreementsrdquoEuropean Journal of Operational Research vol 207no 3 pp 1608ndash1619 2010

[52] G Kilincli Taskiran and X Zhang ldquoMathematical models andsolution approach for cross-training staff scheduling at callcentersrdquoComputers ampOperations Research vol 87 pp 258ndash2692017

[53] R BWallace andWWhitt ldquoA staffing algorithm for call centerswith skill-based routingrdquo Manufacturing amp Service OperationsManagement vol 7 no 4 pp 276ndash294 2005

[54] J Weinberg L D Brown and J Stroud ldquoBayesian forecastingof an inhomogeneous Poisson process with applications to callcenter datardquo Journal of the American Statistical Association vol102 no 480 pp 1185ndash1198 2007

[55] F Rossi P Van Beek and T Walsh Handbook of ConstraintProgramming Elsevier Amsterdam Netherland 2006

[56] R Soto B Crawford W Palma et al ldquoTop- k Based AdaptiveEnumeration in Constraint ProgrammingrdquoMathematical Prob-lems in Engineering vol 2015 Article ID 580785 12 pages 2015

[57] L Trilling A Guinet and D Le Magny ldquoNurse schedulingusing integer linear programming and constraint program-mingrdquo IFAC Proceedings Volumes vol 39 no 3 pp 671ndash6762006

[58] J-P Metivier P Boizumault and S Loudni ldquoSolving nurse ros-tering problems using soft global constraintsrdquo Lecture Notes inComputer Science (including subseries Lecture Notes in ArtificialIntelligence and Lecture Notes in Bioinformatics) Preface vol5732 pp 73ndash87 2009

[59] R Qu and F He ldquoAHybrid Constraint Programming Approachfor Nurse Rostering Problemsrdquo in Applications and Innovationsin Intelligent Systems XVI pp 211ndash224 Springer London UK2009

[60] F He and R Qu ldquoA constraint programming based columngeneration approach to nurse rostering problemsrdquo Computersamp Operations Research vol 39 no 12 pp 3331ndash3343 2012

[61] E Rahimian K Akartunalı and J Levine ldquoA hybrid integerand constraint programming approach to solve nurse rosteringproblemsrdquoComputersampOperations Research vol 82 pp 83ndash942017

[62] R Cipriano L Di Gaspero and A Dovier ldquoHybrid approachesfor rostering A case study in the integration of constraintprogramming and local searchrdquo Lecture Notes in ComputerScience (including subseries LectureNotes in Artificial Intelligenceand Lecture Notes in Bioinformatics) Preface vol 4030 pp 110ndash123 Springer Berlin Germany 2006

[63] S Bertels and T Fahle ldquoA hybrid setup for a hybrid scenarioCombining heuristics for the home health care problemrdquoComputers amp Operations Research vol 33 no 10 pp 2866ndash2890 2006

[64] T H Yunes A V Moura and C C de Souza ldquoHybrid columngeneration approaches for urban transit crew managementproblemsrdquo Transportation Science vol 39 no 2 pp 273ndash2882005

[65] G Laporte and G Pesant ldquoA general multi-shift schedulingsystemrdquo Journal of the Operational Research Society vol 55 no11 pp 1208ndash1217 2004

[66] S Demassey G Pesant and L-M Rousseau ldquoA Cost-RegularBased Hybrid Column Generation Approachrdquo Constraints AnInternational Journal vol 11 no 4 pp 315ndash333 2006

[67] M-C Cote B Gendron and L-M Rousseau ldquoModeling theRegular Constraint with Integer Programmingrdquo in LectureNotes in Computer Science vol 4510 pp 29ndash43 Springer BerlinGermany 2007


[68] M-C Cote B Gendron C-G Quimper and L-M RousseauldquoFormal languages for integer programming modeling of shiftscheduling problemsrdquo Constraints An International Journalvol 16 no 1 pp 54ndash76 2011

[69] N Chapados M Joliveau and L-M Rousseau ldquoRetail storeworkforce scheduling by expected operating incomemaximiza-tionrdquo Lecture Notes in Computer Science (including subseriesLecture Notes in Artificial Intelligence and Lecture Notes inBioinformatics) Preface vol 6697 pp 53ndash58 2011

[70] H Li and KWomer ldquoA decomposition approach for shipboardmanpower schedulingrdquo Military Operations Research vol 14no 3 pp 67ndash90 2009

[71] R Sahraeian and M Namakshenas ldquoOn the optimal modelingand evaluation of job shops with a total weighted tardinessobjective constraint programming vs mixed integer program-mingrdquo Applied Mathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 39 no2 pp 955ndash964 2015

[72] C Sel B Bilgen JM Bloemhof-Ruwaard and J G A J van derVorst ldquoMulti-bucket optimization for integrated planning andscheduling in the perishable dairy supply chainrdquo Computers ampChemical Engineering vol 77 pp 59ndash73 2015

[73] A N Avramidis W Chan M Gendreau P LrsquoEcuyer and OPisacane ldquoOptimizing daily agent scheduling in amultiskill callcenterrdquo European Journal of Operational Research vol 200 no3 pp 822ndash832 2010

[74] A N Avramidis A Deslauriers and P LrsquoEcuyer ldquoModelingdaily arrivals to a telephone call centerrdquo Management Sciencevol 50 no 7 pp 896ndash908 2004

[75] S Bhulai ldquoDynamic routing policies for multiskill call centersrdquoProbability in the Engineering and Informational Sciences vol23 no 1 pp 101ndash119 2009

[76] T Aktekin ldquoCall center service process analysis Bayesianparametric and semi-parametric mixture modelingrdquo EuropeanJournal of Operational Research vol 234 no 3 pp 709ndash7192014

[77] A Sencer and B Basarir Ozel ldquoA simulation-based decisionsupport system for workforce management in call centersrdquoSimulation vol 89 no 4 pp 481ndash497 2013

[78] ldquoDetailed Scheduling in IBM ILOG CPLEX OptimizationStudio with IBM ILOG CPLEX CP Optimizerrdquo Ibm 2010Accessed 11-Jan-2016 Available httpswwwresearchgatenetpublication328530729 Detailed Scheduling in IBM ILOGCPLEX Optimization Studio

[79] J-C Regin ldquoSolving Problems with CP Four Common Pitfallsto Avoidrdquo in Proceedings of the 17th International Conference onPrinciples and Practice of Constraint Programming - CP 2011 pp3ndash11 Springer Berlin Germany 2011

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Submit your manuscripts atwwwhindawicom


Workload prediction

Staffing levels

Shifts RostersStaffing Shi

schedulingRostering

Figure 1 Subproblems of workforce scheduling

Phase 1 Shift scheduling Phase 2 Rostering

Shi information+

Number of operators neededper period

Model 1IP based sub-model

Model 2aIP based sub-model

Model 2bCP based sub-model



Shi schedules

Figure 2 Subproblems considered within this study

the workforce scheduling problem in a call center has fourdifferent subproblems which are shown in Figure 1

The first subproblem is about forecasting the work loadand number of calls received within certain periods (usually30minutes or 15minutes [6])The frequency and the numberof calls received during the planning horizon are the maintopic of this subproblem The second subproblem staffingis about finding the required minimum number of opera-tors depending on the incoming call rate In other wordsstaffing is converting the estimated number of incomingcalls to the operator requirements at periodic levels Thethird subproblem focuses on choosing the optimal shifts thatmeet the demands for operators With this subproblem theshifts to be activated (opened) are also determined insteadof simply assigning the operators to the available shifts Thefourth subproblem rostering deals with the assignment ofthe available operators to the shifts and determining the dailybreaks andor usually weekly days-off of the operators

In shift scheduling the organization of the shifts in theplanning horizon is arranged Schedules generally includenonoverlapping shifts Allocation problem is easy to solve inthis case However call centers may have overlapping shiftsTherefore a shift scheduling system is crucial in call centersIn the rostering problem an operator-shift matching is madebased on definite constraints Also the days-off and the breaktimes of each operator are determined Here wewant to focuson the issue of determining the break times As it is knownan operator usually works between seven and nine hours aday and (s)he needs breaks during this period In their studySungur et al [7] emphasized that operator-break assignmentsshould be handled during the scheduling process Besidesother studies in the literature state that only meal breaks areincluded in their problem formulations and that relief breaksshould be taken into consideration as well [7]

Our aim in this study is to provide aworkforce schedulingmodel including shift scheduling and rostering The general

framework of the study is illustrated in Figure 2 First wesolve the shift scheduling problem by taking shift informationand the staffing requirements as inputs into our integerprogramming (IP) model (Model 1) Staffing requirementsmeans the minimum number of operators ensuring the ser-vice quality level within each period of the planning horizonAfterwards a rostering problemwill be solvedwhen the shiftsare identified based on the solutions of the first problem

Two different models are proposed for the rosteringproblem traditional IP modeling approach (Model 2a) andconstraint programming (CP) approach (Model 2b)WithCPapproach the problem is handled as a task-source allocationand modeled with this perspective The main contribution ofthis paper is to propose a CP model to solve the rosteringproblem in call center as an alternative to the IP model

The remainder of this paper is organized as followsSection 2 introduces the related literature review on work-force scheduling problems In Section 3 IP based model forshift scheduling problem and then IP and CP based modelsfor rostering problem are introduced Section 4 describesexperimental study based on real data and test instancesderived from the real data as well as the staff schedulingproblem benchmark instances [8] The paper is concludedand the future directions are highlighted in Section 5

2 Literature Review

There exist a lot of models developed in the literature forworkforce scheduling problem [9] Edie [10] analyzed thetraffic delays in the tolls with an experimental study andformulized fixed queueing systems Later Dantzig [11] intro-duced the integer linear programming model with regard toEdiersquos [10] study However the workforce scheduling problemhad gone through many evolutions in time

The studies of Ernst et al [4 12] and the study ofBrucker et al [9] presented a detailed literature review on























(b) Continued

















119889isin119863119901isin119875

119862119875119903119889119901 + sum119889isin119863119904isin119878

119862119878119901119889119904 + sum119889isin119863119901isin119875

119862119880119906119889119901 (2)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878












119882119889119910119901 = 119901sum119901=119901minus119871119910+1

( sum119886isin119860

119904isin119878119874119901


(7)


119886isin119860119889isin119863119904isin119878

(119862119867119882119904119909119886119889119904)(8)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878


sum119889isin119863119904isin119878


119909119886119889119904 minus sum119901isin119861119910119904


sum119886


119866119889119904 minus sum119910




















Task








h

0

pulse (ah)

a

h

0

pulse (uvh)

u v


30

0

15

1 2 3 4 5





Objective Function

119898119894119899119894119898119894119911119890 sum119896isin119878119860119886isin119860

(119862119875119897119890119899119892119905ℎ119874119891 (119909119896119886)) (23)


Constraints



sum119886isin119860


sum119896isin119878119860


sum119896isin119878119860


sum119905isin119879119900isin119874


sum119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = 119901119903119890119904119890119899119888119890119874119891 (119908119905) forall119905 isin 119879 119905119895 = 119900119895 (28)


















Shiftnumber (s)

Time range(24h)



















1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018






























(b) Continued

















119889isin119863119901isin119875

119862119875119903119889119901 + sum119889isin119863119904isin119878

119862119878119901119889119904 + sum119889isin119863119901isin119875

119862119880119906119889119901 (2)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878












119882119889119910119901 = 119901sum119901=119901minus119871119910+1

( sum119886isin119860

119904isin119878119874119901


(7)


119886isin119860119889isin119863119904isin119878

(119862119867119882119904119909119886119889119904)(8)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878


sum119889isin119863119904isin119878


119909119886119889119904 minus sum119901isin119861119910119904


sum119886


119866119889119904 minus sum119910




















Task








h

0

pulse (ah)

a

h

0

pulse (uvh)

u v


30

0

15

1 2 3 4 5





Objective Function

119898119894119899119894119898119894119911119890 sum119896isin119878119860119886isin119860

(119862119875119897119890119899119892119905ℎ119874119891 (119909119896119886)) (23)


Constraints



sum119886isin119860


sum119896isin119878119860


sum119896isin119878119860


sum119905isin119879119900isin119874


sum119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = 119901119903119890119904119890119899119888119890119874119891 (119908119905) forall119905 isin 119879 119905119895 = 119900119895 (28)


















Shiftnumber (s)

Time range(24h)



















1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018



















119889isin119863119901isin119875

119862119875119903119889119901 + sum119889isin119863119904isin119878

119862119878119901119889119904 + sum119889isin119863119901isin119875

119862119880119906119889119901 (2)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878












119882119889119910119901 = 119901sum119901=119901minus119871119910+1

( sum119886isin119860

119904isin119878119874119901


(7)


119886isin119860119889isin119863119904isin119878

(119862119867119882119904119909119886119889119904)(8)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878


sum119889isin119863119904isin119878


119909119886119889119904 minus sum119901isin119861119910119904


sum119886


119866119889119904 minus sum119910




















Task








h

0

pulse (ah)

a

h

0

pulse (uvh)

u v


30

0

15

1 2 3 4 5





Objective Function

119898119894119899119894119898119894119911119890 sum119896isin119878119860119886isin119860

(119862119875119897119890119899119892119905ℎ119874119891 (119909119896119886)) (23)


Constraints



sum119886isin119860


sum119896isin119878119860


sum119896isin119878119860


sum119905isin119879119900isin119874


sum119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = 119901119903119890119904119890119899119888119890119874119891 (119908119905) forall119905 isin 119879 119905119895 = 119900119895 (28)


















Shiftnumber (s)

Time range(24h)



















1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018














119882119889119910119901 = 119901sum119901=119901minus119871119910+1

( sum119886isin119860

119904isin119878119874119901


(7)


119886isin119860119889isin119863119904isin119878

(119862119867119882119904119909119886119889119904)(8)

119904119906119887119895119890119888119905 119905119900 sum119904isin119878


sum119889isin119863119904isin119878


119909119886119889119904 minus sum119901isin119861119910119904


sum119886


119866119889119904 minus sum119910




















Task








h

0

pulse (ah)

a

h

0

pulse (uvh)

u v


30

0

15

1 2 3 4 5





Objective Function

119898119894119899119894119898119894119911119890 sum119896isin119878119860119886isin119860

(119862119875119897119890119899119892119905ℎ119874119891 (119909119896119886)) (23)


Constraints



sum119886isin119860


sum119896isin119878119860


sum119896isin119878119860


sum119905isin119879119900isin119874


sum119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = 119901119903119890119904119890119899119888119890119874119891 (119908119905) forall119905 isin 119879 119905119895 = 119900119895 (28)


















Shiftnumber (s)

Time range(24h)



















1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018





















Task








h

0

pulse (ah)

a

h

0

pulse (uvh)

u v


30

0

15

1 2 3 4 5





Objective Function

119898119894119899119894119898119894119911119890 sum119896isin119878119860119886isin119860

(119862119875119897119890119899119892119905ℎ119874119891 (119909119896119886)) (23)


Constraints



sum119886isin119860


sum119896isin119878119860


sum119896isin119878119860


sum119905isin119879119900isin119874


sum119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = 119901119903119890119904119890119899119888119890119874119891 (119908119905) forall119905 isin 119879 119905119895 = 119900119895 (28)


















Shiftnumber (s)

Time range(24h)



















1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018









h

0

pulse (ah)

a

h

0

pulse (uvh)

u v


30

0

15

1 2 3 4 5





Objective Function

119898119894119899119894119898119894119911119890 sum119896isin119878119860119886isin119860

(119862119875119897119890119899119892119905ℎ119874119891 (119909119896119886)) (23)


Constraints



sum119886isin119860


sum119896isin119878119860


sum119896isin119878119860


sum119905isin119879119900isin119874


sum119900isin119874

119901119903119890119904119890119899119888119890119874119891 (119911119900) = 119901119903119890119904119890119899119888119890119874119891 (119908119905) forall119905 isin 119879 119905119895 = 119900119895 (28)


















Shiftnumber (s)

Time range(24h)



















1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018

























Shiftnumber (s)

Time range(24h)



















1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018









1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96periods in a day


0

5

10

15

20

25

30

35

num

ber o

f ope

rato

rs





Shifts1 2 3 4 5 6 7 8 9 10















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018















22lowast 386 37056 87723

677214 876709






















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018

















5 Conclusion











26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018















26lowast 141 27 13536 13966 331764 444186 13536 034 (6531) 39846 1703430 142 27 13632 13167 331764 444186 13632 039 (7213) 40074 17146





23lowast 257 51 24672 36777 500964 656034 24672 096 (19444) 124906 5621025lowast 257 51 24672 36265 500964 656034 24672 104 (19879) 124906 56210





26lowast 357 66 34272 56691 606714 788439 34272 120 (25900) 213933 9910830 372 66 35712 55943 606714 788439 35712 126 (18805) 222033 103128



19lowast 371 78 35616 77832 691314 894363 35616 265 (30549) 262823 12160420lowast 371 78 35616 83424 691314 894363 35616 249 (30174) 262823 121604



19lowast 367 78 35232 85623 691314 894363 35232 267 (31792) 259342 12018420lowast 367 78 35232 86155 691314 894363 35232 255 (31113) 259342 120184




22lowast 459 90 44064 107475 775914 1000287 44064 379 (31365) 365902 17193625 472 90 45312 113363 775914 1000287 45312 405 (29734) 375418 176668




22lowast 459 109 44064 144044 909864 1168000 44064 538 (41609) 441978 20784625 472 109 45312 161664 909864 1168000 45312 459 (38822) 453470 213566







26lowast 768 160 73728 296778 1269414 1618177 73728 1244 (83295) 1045410 50323230 789 160 75744 322878 1269414 1618177 75744 1289 (82612) 1072542 516756


Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018









Table 11 Continued

Ins Incrate ()










Data Availability





References


























































































Journal of












Journal of







Volume 2018






Volume 2018











References


























































































Journal of












Journal of







Volume 2018






Volume 2018






























































Journal of












Journal of







Volume 2018






Volume 2018








an integrated approach for shift scheduling and rostering...

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