Omega 43 (2014) 41–53

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Staff rostering in call centers providing employee transportation

E. Lerzan Örmeci, F. Sibel Salman n, Eda YücelDepartment of Industrial Engineering, College of Engineering, Koç University, Istanbul, Turkey

a r t i c l e i n f o

Article history:Received 12 December 2011Accepted 12 June 2013Available online 25 June 2013

Keywords:Call center operationsWorkforce schedulingRosteringPick-and-drop servicesAgent satisfactionMixed integer programming

83/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.omega.2013.06.003

esponding author. Tel.: +90 212 338 1707; faxail address: ssalman@ku.edu.tr (F.S. Salman).

a b s t r a c t

We address the staff rostering problem in call centers with the goal of balancing operational cost, agentsatisfaction and customer service objectives. In metropolitan cities such as Istanbul and Mumbai, callcenters provide the transportation of their staff so that shuttle costs constitute a significant part of theoperational costs. We develop a mixed integer programming model that incorporates the shuttlerequirements at the beginning and end of the shifts into the agent-shift assignment decisions, whileconsidering the skill sets of the agents, and other constraints due to workforce regulations and agentpreferences. We analyze model solutions for a banking call center under various management prioritiesto understand the interactions among the conflicting objectives. We show that considering transporta-tion costs as well as agent preferences in agent-shift assignments provides significant benefits in terms ofboth cost savings and employee satisfaction.

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1. Introduction

The call center industry has secured a global place in customerrelationship management within the last 30 years. Workforcemanagement plays an essential role in the success of call centeroperations. Call centers face a dynamic demand pattern as callarrival rates vary by months, weeks, days and within differenthours throughout a day. The need to balance the variable workloadwith agents working in shifts, presence of different call types thatrequire various agent skills, constraints arising from agent shiftpreferences and work place rules, among other operational con-siderations, all complicate workforce scheduling at call centers.Furthermore, several conflicting objectives with respect to operat-ing costs, customer service and employee satisfaction come intopicture. In this study, our aim is to provide a mathematicalmodeling approach to generate viable and productive workforceschedules that balance the conflicting objectives according to thepriorities of call center managers.

This study has been initiated by our collaboration with two callcenters operating in Istanbul, which brought up their difficultieswith staff rostering. One of them is a banking call center, while theother provides global technical support. Both operate in a multi-skill environment with different skill structures. Operations ofthese call centers have a common interesting aspect: they providethe transportation of all employees in all shifts, which creates asignificant cost component. Like most call centers, they are subjectto meeting pre-specified service levels, but have difficulties in

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achieving these targets at certain time intervals. Furthermore, theyacknowledge the effect of work schedules on agent satisfaction,and aim to accommodate staff preferences to increase productivityand commitment.

Agent turnover rates in the call center industry are typicallyhigh, with an average of 33% per year and exceeding 50% per year inmany call centers (see [1,30,24]). If work schedules are generated tominimize costs only, then the working conditions of the agents mayshow high variability and the inherent unfairness which leads tohigher turnover rates. One way to reduce these rates is to increaseagent satisfaction by providing preferred, or at least acceptable,workforce schedules for all agents. This requires identifying theundesired features of an agent's schedule and limiting them asmuch as possible. Although it is hard to quantify agent preferences,experiences of call center managers allow them to point out majorcauses of discontent. In light of these causes, it is possible to defineappropriate discontent measures. Then, the agents’ individualpreferences for these measures can be collected through question-naires, or the call center managers may assess general importancescores for different agent discontent factors. Once the discontentscores are identified, it becomes possible to limit the total dis-content of each agent in a schedule, thereby targeting fairnessamong agent schedules.

The service level in call centers is generally measured by thepercentage of calls answered within a specified time limit, e.g. atleast 80% of all calls in less than 20 s. Call centers determine therequired number of agents for each 15- or 30-min period usingthis service level criterion. With the existing workforce and budgetlimitations, these requirements may not be achievable, as fre-quently observed in one of the call centers we work with. There-fore, considering the requirements as soft constraints may not only

E.L. Örmeci et al. / Omega 43 (2014) 41–5342

be necessary in some cases, but also provides flexibility in work-force planning, leading to improvements in other criteria.

As mentioned above, the call centers we worked with have amajor cost component due to providing the transportation of theirstaff by shuttles, a common practice among Turkish companiesoperating in large cities. Nevertheless, this type of pick-and-dropservices is not unique for Turkey. For example, most call centersoperating in Mumbai also provide such services in various forms,e.g., pick and drop from/to several central locations within the city,or pick and drop from/to home at night shifts. A typical call centeroperates for 24 h a day and 7 days a week. These continuous workhours, together with the non-stationary call arrivals, may require ahigh number of shifts per day, in some extreme cases as high as 50.As the shuttle running costs may become a considerable part ofthe total operational costs, some call centers even prefer hiringonly the agents residing close to the call center, at the expense oflosing more skilled labor force. On the other hand, the call centerswho provide this type of service need to consider the transporta-tion aspect in scheduling. Typically, they aim to increase thevehicle utilizations by considering the locations of the agentsand how they would be grouped into vehicles when assigningthem to shifts.

Workforce scheduling is generally carried out using standardindustry-specific software packages that do not have the capabilityto model agent satisfaction, or the transportation aspect of theproblem. In current practice, workforce schedules generated bythese packages are adjusted by planners to address these issues,which requires a considerable amount of time and effort. To avoidthese inefficiencies and to improve the schedules, we develop amathematical model for call center workforce scheduling thatoptimizes the three objectives of the problem simultaneously,namely total operating costs, employee satisfaction and under-staffing. The specifics of these objectives vary with the call centers.In this study, we model the problem as observed in the call centerswe worked with. Although the approach is general, some con-straints and performance metrics might be specific to these cases.

Using the data of the banking call center, we conduct a caseanalysis in which we analyze model solutions under varyingpriorities. The case study serves to: (1) reveal the tradeoffsbetween cost, customer service and employee satisfaction objec-tives, (2) quantify the benefits from incorporating the transporta-tion costs into the model, (3) demonstrate how model parameterscan be used to reflect management preferences, and hence howthe model can support the workforce scheduling process.

We present the literature review in the next section, anddescribe the problem in Section 3. We give the mixed integerprogramming model in Section 4. Section 5 describes the inputdata and analyzes the model solutions under different parametersettings and scenarios. Finally, we summarize our contributionsand discuss implementation issues in Section 6.

2. Literature review

Workforce scheduling problems have been studied extensivelysince Dantzig [19], as they arise in numerous organizations such ascall centers, airlines, hospitals, and postal services, to name a few.Ernst et al. [20] present a review of application areas, solutionmethods and models, whereas Ernst et al. [21] provide anextensive bibliography with a chronicle of over 700 papers. Areview of the subject on call centers, on the other hand, can befound in Aksin et al. [1,2].

In the literature, workforce scheduling is commonly conductedin three stages: (1) staffing, (2) shift scheduling and (3) rostering,that are addressed either individually or by combining two of thethree. In the former case, the outputs of the previous stages are

assumed to be known, whereas the latter studies, usually, iteratebetween the two stages. We solve a rostering problem taking thestaffing requirements and shift descriptions as inputs to our model.

Staffing refers to finding the minimal number of agents thatguarantees a service level during each time interval in a planninghorizon, which are typically called workforce requirements orstaffing requirements. The solution methodologies for staffingproblems mainly consist of simulation or queueing theory, or acombination of the two, which may be accompanied by anoptimization model in multi-skill settings.

In the context of call centers, the two complicating factors inthe staffing problem are the nature of call arrival rates (time-varying and/or random) and the multi-skill environment. Greenet al. [25] and Whitt [42] provide examples of papers that studythe staffing problem under time-varying and uncertain call arrivalrates, respectively. Most call centers are multi-skill environments,where each agent type can handle a different set of call types, i.e.the so-called skills. In such environments, three inter-relatedproblems arise: flexibility design, staffing and call routing.

The flexibility design problem determines the skill sets avail-able for the agents. Generally, the call center management decideson these sets and implements them through appropriate hiringand training policies. Once the flexibility structure is chosen, thestaffing problem determines the number of agents needed fromeach skill set. Note that the staffing problem can change theflexibility structure by assigning a zero level to certain skill sets.Call routing policies, on the other hand, assign calls to agents whocan serve them. The staffing levels and the flexibility design affectthe routing policies, but also different routing policies will call fordifferent staffing levels and flexibility designs. Gans et al. [24] andAksin et al. [1,2] review the literature on these topics as well as theinteractions between them. Below, we cite the more recent andrelated ones.

The literature on the staffing problem in multi-skill environ-ments generally take the flexibility structure as well as the routingpolicy as given and solve for the staffing problem, see e.g., Wallaceand Whitt [41], Cezik and L'Ecuyer [17], Pot et al. [36], Avramidiset al. [5], Feldman and Mandelbaum [23]. The literature thatcharacterizes optimal or near-optimal call routing policies, onthe other hand, assume that both the flexibility design and thestaffing levels are fixed, see e.g., Ormeci [35], Bhulai [10], Bhulaiand Roubos [11] and the references therein. Finally, some papersconsider both staffing and routing problems. However, they needto simplify the routing policies in order to find explicit solutions.Harrison and Zeevi [26] and Bassamboo et al. [9] formulate theproblem hierarchically under a time-varying arrival rate, so thatthe first stage determines the staffing levels, while the secondstage solves for the routing problem through a fluid approximationof the call center. The routing policies found in these papersprovide only the number of agents from each skill set who answera certain type of calls at any time, where these numbers are notnecessarily integers. Our approach is similar to these papers, asour model will output the proportion of time that each agent willdedicate to each of his/her skills at each shift.

Shift scheduling identifies which shifts are to be used and findsthe number of employees to be assigned to each shift in order tofulfill the workforce requirements. These problems, traditionally,rely on mixed integer programming (MIP) models. Thompson [39],Aykin [6], Bard et al. [8], and Bhulai et al. [12] provide someexamples for shift scheduling solved by integer programming,where the staffing levels are taken as input.

Rostering, which is also called tour scheduling, generates thecomplete schedule of all employees over the planning horizon byconsidering regulations and work place rules, such as the requirednumber of days off, in addition to the workforce requirements. Theproblem is generally formulated with MIP models. Ertogral and

E.L. Örmeci et al. / Omega 43 (2014) 41–53 43

Bamuqabel [22] generate weekly schedules for groups of agents ina call center. Thompson [40], on the other hand, evaluates differentshift and tour scheduling MIP models in terms of labor costs,service levels and agent utilization via a thorough simulationanalysis. Li et al. [32] present a multi-objective approach to anurse rostering problem by formulating a goal programmingmodel with hard and soft constraints. Robbins [37] discussesissues involved in scheduling and rostering in call centers andother service systems. Recently, Maenhout and Vanhoucke [34]address re-rostering under disruptions due to unplannedabsences, staff turnover, etc.

Studies that combine two of the three aforementioned stagesinclude the following. Brusco and Jacobs [14,15] consider rosteringand shift scheduling problems together using MIP models. Hen-derson and Mason [28], Atlason et al. [3] and Ingolfsson et al. [29]develop iterative methodologies to solve a combination of staffingand rostering problems. Atlason et al. [4] and Maenhout andVanhoucke [33] solve staffing and shift scheduling problemstogether for agents at call centers and nursing staff, respectively.Helber and Henken [27] develop a profit-oriented shift schedulingapproach to solve a combination of staffing and shift schedulingproblems.

We solve a rostering problem through a MIP model, whichtakes the staffing requirements and shift descriptions as inputs.We assign each agent individually to the shifts due to the presenceof pick-and-drop services, multi-skill agents and agent satisfactionin the model. More explicitly, agents are described by not onlytheir skills but also where they live, so that they cannot bemodeled as groups of homogeneous agents as seen commonly inthe literature. In addition, we trace and limit the total discontent ofeach agent over the planning horizon. A unique feature of our MIPmodel is that it addresses the transportation of the employees,who reside in different regions, while assigning them to the shifts.Thus, a transportation schedule is also generated along with thetour schedule. This provides an interesting new aspect to therostering problem that stems from operational needs.

3. Problem characteristics and the modeling approach

In this section, we describe the characteristics of the rosteringproblem we study with respect to workforce requirements, shiftdescriptions, multiple skill sets, agent pick-and-drop services andagent satisfaction. We discuss how these factors relate to eachother and how we model them, along with our assumptions. Thesection ends with an overview of our modeling approach.

3.1. Multi-skill workforce requirements and shift descriptions

The three inputs of our model are the current skill sets of theagents, the current shift descriptions of the call center we workwith and the targeted staffing levels for each skill, which aregenerated by the standard call-center software package used bythe call center.

The current skill sets of the call center reflect the managementstrategy on hiring and training policies. In general, both thenumber of skills and skill sets can be very high. In the call centerwe worked with, the number of skills is 8, while the number ofskill sets is 7. Their training program bundles a number of skills,which decreases the number of skill sets, as we will describe inmore detail in Section 5. We should note, however, that thenumber of skill sets does not affect the performance of the model,as the model considers each skill of an agent as a differentresource allowing flexible usage of each skill in his/her skill set.The standard call-center software package used by the call centersgenerally generate the targeted staffing levels for each skill in each

time interval. In a multi-skill call center, these levels shouldfurther be converted to the number of agents from different skillsets. As explained in the literature review, the call routing policieshave significant effects on this conversion. One output of ourmodel is the proportion of time that each agent will dedicate toeach of his/her skill in each shift. These proportions will provide aguideline for developing routing algorithms, which will aim toachieve these proportions in real time. The model can be modifiedeasily to let the proportions change at each time interval, whichincreases the computational complexity substantially. Hence, themodel decides the proportion of time that each agent will dedicateto each of his/her skill in each shift, rather than in each interval.This does not decrease the model performance, since the propor-tions of different call types do not vary much over time.

The shifts are defined by their start and end times over theplanning horizon. The breaks are not considered in this definition,but they can be represented by adjusting the operator require-ments accordingly. Moreover, we assume that personnel onvacation or training is deducted from the total workforce atcorresponding periods at the beginning of the planning horizon.In principle, better shift descriptions might be found by using thestudies on shift scheduling. We experiment with additional shiftswhen solving for the case of the call center we worked with (seeSection 5.5.2).

The staffing requirements in a call center are, usually, calcu-lated to ensure serving at least a certain portion of customerswithin a target waiting time in each segment of the day, e.g.serving at least 80 % of customers within 20 s in each 30-mininterval. These requirements are strict in certain industries, suchas health care providers, whereas in certain settings they can berelaxed to meet the service levels within a day, week, or month,instead of shorter time periods, such as 15- or 30-min intervals.Such a relaxation would allow the call center to stay understaffedduring the peak periods, but to make up for this in other periods tohave the average service level satisfactory enough. The exactservice level goals are defined precisely in the financial agree-ments when the call center activities are outsourced. The literatureon workforce scheduling problems follows the industry require-ments, so that most studies take the staffing requirements in everyperiod as a hard constraint, whereas some relax this constraint tomeet the service levels within a larger time segment (see e.g.,[31,40] and the references in [16]).

The call center we work with, on the other hand, is a bank callcenter operated by the bank itself, where the staffing requirementsare not fulfilled in a number of time periods, for various reasonsthat include low workforce level and limited seat capacity.Although the call center management plans to expand the callcenter capacity, this requires a longer time period. Consequently,we assume that the current workforce level is fixed in this study,although it is insufficient to meet the targeted service level. Toaccount for this, we define soft staffing requirement constraints,while including a penalty cost for understaffing in each period.Therefore, the number of operators working in a period can be lessthan the requirements with a specified penalty cost per operator.In our model, treating the requirements as soft constraints allowscost savings in transportation and overtime in addition to betteragent satisfaction. Furthermore, allowing a tolerable level ofunderage may prevent excessive overstaffing for long periods.

The pattern of call arrival rates throughout a day, hence theworkforce requirements, is bimodal in most call centers, withsteep changes especially in the mornings and evenings. Usuallyshift start times follow these changes in order to satisfy customerservice levels efficiently. For example, Fig. 1 illustrates the unevenrequirement distribution for one of the skills (skill 5) over 24 h in atypical weekday of the banking call we worked with. The bank callcenter currently uses the four shifts given in the figure, i.e., shift

Fig. 1. The required number of agents for skill 5 at low demand and shiftdescriptions.

E.L. Örmeci et al. / Omega 43 (2014) 41–5344

1 covers 0:00–9:00, shift 2 9:00–18:00, shift 3 9:00–20:00, andshift 4 15:00–24:00. The requirements are quite low during thenight shift (shift 1), but they increase sharply starting with shifts2 and 3. The peak requirements are observed in 11:00–17:00 witharound 130 agents attending skill 5. If all the requirements are tobe satisfied, an excessive overage will occur during 9:00–11:00where 64–97 agents are required. Furthermore, we need to assign53 agents to shift 4 to satisfy the requirements in periods 20:00–22:00. These agents will start working at 15:00, causing anoverage during 15:00–18:00 with 184 agents working while only125 agents are needed. We can avoid such cases by relaxing theconstraints on the workforce levels. In order to maintain tolerableunderage levels, the underage penalty cost is used as a controlparameter. The solution output by the model for this example willbe analyzed later in Section 5.

3.2. Effects of pick-and-drop services on workforce scheduling

Agents are transported from and to their home at the beginningand end of each shift. The pick-and-drop services are generallyoutsourced to a third party. Thus, the effects of these services onthe workforce scheduling depend on the contract between the callcenter management and the service provider. In the call centers weworked with, the accounting is carried out by the number of vehiclessent to the zones in the city. More explicitly, the city is divided into anumber of zones. The workforce schedule determines how manyagents to be transported from/to each zone at each shift. According tothese numbers, the shuttle company provides the required numberof shuttles for each zone. They are also responsible for picking up theagents from their homes in a timely manner to bring them to the callcenter before the shift starts. The shuttle company has several typesof vehicles with varying capacity, and charges the call center for eachtype of vehicle used with a fixed cost per trip from/to each zone. Thisfixed cost reflects the proximity of the zone to the call centerlocation. The call centers would like to minimize the total costaccruing from the number of shuttles utilized. Note that whether thevehicles return empty or full does not have any cost implications. Inother words, the additional convenience or inconvenience is handledcompletely by the third party.

One possible approach considered for practicality is to groupthe agents according to where they live and to assign them to thesame shifts. However, this is not possible due to several reasons:(1) the skill distribution of the agents in a group may not matchwith the requirements during a shift, (2) the requirementsfor different shifts vary considerably (for example, compare the

requirements during shift 1 and shift 2 in Fig. 1), (3) legalregulations and/or agent discontent and fairness issues mayprohibit such assignments. Thus, this approach is not valid. Ourmodel assigns agents individually to shifts by considering bothwhere they live and their skills. The resulting schedules providehigh vehicle utilizations which, in turn, decreases the requirednumber of shuttles (see Section 5).

Finally, we note the interaction between the transportation costsand underage levels. For example, if the maximum shuttle capacityis 15, while 16 agents are required during the peak periods of a shift,then an underage of one person in one or more periods within theshift may be preferable to using one more vehicle. Thus, the vehiclecosts and capacities also affect the underage decisions.

3.3. Factors affecting agent satisfaction

Different types of constraints arise due to work laws andregulations. The number of consecutive work days, total numberof work days in the planning horizon, and the number of workhours in a day are limited by legal regulations. Furthermore, atleast a certain amount of off-time should be assigned between twoconsecutive shifts of an employee, even though they may be ondifferent days. There are also constraints on the start time and theduration of the breaks. Even after satisfying these regulations,employee satisfaction is not guaranteed. It is desirable to incorpo-rate employee preferences into the schedules.

The individual preferences of the agents can vary widely. Forexample, some agents hate overtime shifts, while others preferworking overtime in order to earn more; or similarly some woulddetest working on the weekends while some others would prefer it.To represent a wide range of possibilities, we define a discontentscore for each shift and for each day, similar to Wright and Mahar[43]. Exceeding the regular number of workdays in the planninghorizon is another source of discontent, for which we define anotherdiscontent score. Each agent chooses a discontent score for each ofthese discontent sources. Then, our model addresses agent prefer-ences by limiting the total discontent penalty associated with eachemployee over the planning horizon. We should note that theseindividually set discontent scores do not bring additional computa-tional complexity to the model, and we expect that they do not causeadditional runtime. Call centers generally have different approachesfor setting the values of these parameters. In some call centers,seniority ranking or performance score of agents would also affecthow the shifts are assigned. In such cases, the discontent scoreschosen by the agents may be modified by the call center manage-ment, e.g., by multiplying the scores with a constant that reflects theseniority ranking and/or performance of the agents.

Several studies on workforce scheduling have addressed individualpreferences and requests of employees, for instance Bard and Pur-nomo [7] and Chiaramonte and Chiaramonte [18] in nurse schedulingand Yura [44] in production. In these studies the main goal is toimprove the schedule in terms of overall preferences. However, wealso aim to achieve fairness, as much as possible, by distributing thediscontent evenly among the agents. We discuss how our model canbe used to provide fairness among the agents in Section 5.5.

3.4. Modeling approach

We identified three categories of objectives, as total operatingcosts for the firm, service quality for customers and satisfaction foragents. Our model uses the following performance measures foreach of these objectives:

�

Operating costs: We consider the transportation and overtimecosts only. We do not consider regular labor cost since the callcenters take the workforce level fixed.

E.L. Örmeci et al. / Omega 43 (2014) 41–53 45

�

Service quality: We indirectly measure the service quality. Weassume that the workforce requirements are generated tosatisfy a target customer service level for each skill. Wemeasure the total underage level with respect to these require-ments. The penalty cost for unit underage for each skill andtime period indicates the relative importance of service quality.

�

Agent satisfaction: We calculate the total discontent score ofeach employee over the planning horizon and limit it by lowerand upper bounds.

The model parameters, namely the underage penalty cost(relative to the other costs) and the minimal and maximal agentdiscontent levels, represent the preferences of the call centermanagement in terms of these objectives. For example, if the callcenter management prioritizes operational costs and employeesatisfaction, then both the underage penalty cost and the maximaldiscontent level will be set to a low value. Section 5.5 illustrates theeffects of different parameter settings through numerical examples.

The model generates the shift assignments for each operator overthe planning horizon and can be used on a rolling horizon basisrepeatedly with updated data. In addition to the operator schedule,the model outputs a transportation schedule as well, by determiningwhich operators are to be transported from and to each zone at thebeginning and end of each shift. The model optimizes the number ofvehicles of different types to be dispatched. The resulting transporta-tion requirements can be used to optimize the vehicle routes in eachzone afterwards. Finally, the model also decides on how much timeeach agent will dedicate to each of his/her skills during each shift.Fig. 2 illustrates the inputs and the outputs of the model.

4. Mathematical model

Under the setting described in Section 3, we define the inputdata in terms of sets and parameters, and formulate a mixedinteger program.

SetsA set of operators (agents)S set of skills

Fig. 2. Overview of the model.

G set of geographical regionsR set of regular shifts in a dayO set of overtime shifts in a dayT set of time intervals in a dayD set of days in the planning horizonV set of vehicles

Parameters

m number of days in the planning horizonn number of weeks in the planning horizono overtime cost per shift per operatorcvgtd transportation cost of vehicle type v for region g at time

interval t of day d ðv∈V; g∈G; t∈T ;d∈DÞustd penalty coefficient for unsatisfied requirement of skill s

at time interval t of day d ðs∈S; t∈T ; d∈DÞrstd requirement of skill s at time interval t of day d

ðs∈S; t∈T ; d∈DÞpas 1, if operator a has skill s; 0, otherwise ða∈A; s∈SÞlag 1, if operator a lives in region g; 0, otherwise ða∈A; g∈GÞκv capacity of vehicle type v ðv∈VÞη call center seat capacity, which limits the number of

operators that can work simultaneouslyθ number of days that an operator can work consecutivelyτmax maximum number of days that an agent is allowed to

work over the planning horizon (by law and regulations)τ maximum number of days that an agent is allowed to work

over the planning horizon without causing discontentβ maximum number of hours that an agent is allowed to

work in a week (by law and regulations)γmax maximum allowed discontent score for an operator over

the planning horizonγmin minimum allowed discontent score for an operator over

the planning horizonαdayd ðaÞ discontent score for working on a day d chosen by agent

a ða∈A; d¼ 1;2;…;7Þαshfw ðaÞ discontent score for working in shift w chosen by agent a

ða∈A;w∈ðR∪OÞÞαðaÞ discontent score for working more than τ days in a

planning horizon chosen by agent a ða∈AÞewt 1, if shift w contains time interval t; 0, otherwise

ðw∈ðR∪OÞ; t∈T Þbwt 1, if shift w starts at time interval t; 0, otherwise

ðw∈ðR∪OÞ; t∈T Þfwt 1, if shift w finishes at time interval t; 0, otherwise

ðw∈ðR∪OÞ; t∈T Þhw number of time intervals that shift w contains ðw∈ðR∪OÞÞyw1 ;w2

1, if an operator can be assigned to shift w1 on day d andto shift w2 on day dþ 1; 0, otherwise ðw1;w2∈ðR∪OÞÞ

Decision variables

Xawd binary variable indicating if operator a is assigned to shiftw on day d ða∈A;w∈ðR∪OÞ; d∈DÞ

Auxiliary variables

Paswd the proportion of time that operator a uses for skill s inshift w on day d ða∈A; s∈S;w∈ðR∪OÞ; d∈DÞ

Wa binary variable indicating if operator a works more thanτ days ða∈AÞ

Za total discontent of operator a over the planning horizonða∈AÞ

Ustd amount of unsatisfied requirement for skill s at timeinterval t of day d ðs∈S; t∈T ; d∈DÞ

E.L. Örmeci et al. / Omega 43 (2014) 41–5346

Ivgtd number of vehicles of type v arriving from region g at thebeginning of time interval t of day d ðv∈V; g∈G; t∈T ; d∈DÞ

Lvgtd number of vehicles of type v leaving for region g at thebeginning of time interval t of day d ðv∈V; g∈G; t∈T ; d∈DÞ

Minimize ∑a∈A

∑w∈O

∑d∈D

oXawd þ ∑v∈V

∑g∈G

∑t∈T

∑d∈D

cvgtdðIvgtd þ LvgtdÞ

þ∑s∈S

∑t∈T

∑d∈D

ustdUstd ð1Þ

subject to∑a∈A

∑w∈ðR∪OÞ

lagbwtXawd ≤ ∑v∈V

κvIvgtd; ∀g∈G; t∈T ; d∈D; ð2Þ

∑a∈A

∑w∈ðR∪OÞ

lagf wtXawd ≤ ∑v∈V

κvLvgtd; ∀g∈G; t∈T ; d∈D; ð3Þ

rstd− ∑a∈A

∑w∈ðR∪OÞ

ewtPaswd≤Ustd; ∀s∈S; t∈T ; d∈D; ð4Þ

Paswd≤pasXawd; ∀a∈A; s∈S; w∈ðR∪OÞ; d∈D; ð5Þ

∑s∈S

Paswd≤Xawd; ∀a∈A; w∈ðR∪OÞ; d∈D; ð6Þ

∑a∈A

∑w∈ðR∪OÞ

ewtXawd ≤η; ∀t∈T ; d∈D; ð7Þ

∑w∈ðR∪OÞ

Xawd ≤1; ∀a∈A; d∈D; ð8Þ

∑iþθ

d ¼ i∑

w∈ðR∪OÞXawd ≤θ; ∀a∈A; i¼ 1;2;…;m−θ; ð9Þ

∑iþ6

d ¼ i∑

w∈ðR∪OÞhwXawd ≤β; ∀a∈A; i¼ 1;2;…;m−6 ð10Þ

Xaw1d þ Xaw2dþ1≤yw1w2þ 1; ∀a∈A; w1;w2∈ðR∪OÞ; ð11Þ

∑w∈ðR∪OÞ

∑d∈D

Xawd−τ≤ðτmax−τÞWa; ∀a∈A; ð12Þ

Za ¼ ∑w∈ðR∪OÞ

∑n

d ¼ 1αdaymod7ðdÞðaÞXawd

þ ∑w∈ðR∪OÞ

∑n

d ¼ 1αshfw ðaÞXawd þ αðaÞWa; ∀a∈A; ð13Þ

Za≤γmax; ∀a∈A; ð14Þ

Za≥γmin; ∀a∈A; ð15Þ

Xawd;Wa∈f0;1g; ∀a∈A;w∈ðR∪OÞ; d∈D;

Ivgtd; Lvgtd≥0 and integer; ∀v∈V; g∈G; t∈T ; d∈D;

Za;Ustd≥0; ∀a∈A; s∈S; t∈T ; d∈D:

where mod7ðiÞ in (13) is the modulus of i with respect to (7).In the objective function, there are three cost terms: overtime

costs, transportation costs and unsatisfied demand penalty. Theconstraints ensure the following:

�

Constraints (2) and (3) find the required number of vehicles atthe start and end of the shifts.

�

Constraints (4)–(6) find the amount of unsatisfied skillrequirement in each time interval of each day in the planninghorizon.

�

Constraints (7) limits the number of operators that can worksimultaneously by the call center seat capacity, η.

�

Constraints (8) ensure that an operator can be assigned to atmost one shift per day.

�

Constraints (9) restrict the number of days an operator canwork consecutively. In our case, an operator cannot work morethan 6 days consecutively by law, i.e., θ¼ 6.

�

Constraints (10) impose weekly working hour restrictions. Inour case, the law prohibits an operator fromworking more than56 h in a week, i.e., β¼ 56.

�

Constraints (11) guarantee that an operator has enough restingtime between consecutive shifts through the parameter yw1 ;w2

,which indicates whether a shift w1 on a day can be followed byshift w2 on the following day. In our case, an operator shouldhave at least 11 h between two consecutive shifts and yw1 ;w2

should be constructed accordingly. For example, in Fig. 1 if anoperator works in shift 4 on a day, (s)he cannot work in shift 1,2 or 3 on the next day.

�

Constraints (12) limit the number of working days of anoperator during the planning horizon by τmax and meanwhile,ensures that if an operator works more than τ days, his/herdiscontent variable Wa is set to 1. In our case, τmax ¼ 21 andτ¼ 20.

�

Constraints (13) calculate the total discontent of an operatorover the planning horizon. The three discontent factors (work-ing more than τ days, working on weekends, and workingovertime) are multiplied by their penalty coefficients. We notethat the days ð7i−1Þ and ð7iÞ for i¼ 1;2;…;n correspond toSaturdays and Sundays of the planning horizon, respectively.

�

Constraints (14) and (15) restrict the total discontent score ofan operator.

�

The remaining constraints define the nonnegativity and inte-grality conditions.

5. Case analysis

We analyze the characteristics of the solutions output by theworkforce scheduling model through numerical experiments thatare based on the 4-week data obtained from the banking callcenter. Section 5.1 describes the data and how different instancesare generated. In Section 5.2, we discuss the model size andcomputational time under different settings. Section 5.3 definesadditional performance measures. Section 5.4 analyzes the modelsolutions, while Section 5.5 presents different scenarios and thesensitivity analysis of the model solutions with respect to inputparameters.

5.1. Parameter settings

The call center data given to us includes the addresses and theskill sets of 250 operators on the payroll in a typical month, theworkforce requirements for that month and the realized servicelevels. The requirements are the output of the workforce manage-ment software used by the call center, which computes therequirements to provide a service level of answering at least 80%of all calls within 20 s in every 15-min time interval.

An analysis of the data for the realized service levels shows thatit is not infrequent for the service levels to drop to 0 during theday. In other words, in certain 15-min periods none of thecustomers was answered within 20 s. This can be attributed totwo main reasons: (1) the demand for the operators is higher thanthe number of currently employed agents, and (2) the number ofagents who can simultaneously be seated in the call center waslimited by 167. The call center management was well aware ofthese problems and had plans to both extend the seat capacity andto hire new agents. We should also note that the call center hadobserved an increasing trend in the call arrival rates. As thecapacity expansion of the call center takes longer time than the

E.L. Örmeci et al. / Omega 43 (2014) 41–53 47

increase in the call rates, the call center expected to operate at aneven heavier load in near future. In our experimental design, weset the medium demand level to represent the current conditions,and further investigate low and high demand levels, by taking 80%and 120% of the medium levels, respectively. As a result, our modelsolutions suffer from high underage levels, resulting in low servicelevels, especially for medium and high demand levels. However,this accurately reflects the situation of the call center at that time.

The call center has 250 employees, who answer 8 differenttypes of calls, where each type corresponds to a skill. We label theskills as 1;2;…;8: Skill 1 corresponds to bank operations (12% oftotal number of required agents over 28 days), skill 2 to Englishcalls (1%), skill 3 to more advanced banking operations (1.5%), skill4 to fund-related operations (2%), skill 5 to credit card operations(63%), skill 6 to the procedures related to credit card passwords(4%), skill 7 to outsourced banking operations (6%), and skill 8 toinitial set-up call for new customers (10.5%). The proportions ofrequired agents by skill type do not vary significantly over the timeperiods, except for the night time (time periods 1–8).

The call center management decides on the flexibility structureof the call center and designs their training program accordingly.As a result, there are 7 different skill sets, which are labeled asA;B;…;G. Table 1 presents the distribution of agents with respectto regions in which they reside and their skill sets. The definitionof the skill sets and the percentage of agents in each of them givenin the parenthesis are as follows: A¼ f6g (34%), B¼ f6;7g (0.4%),C ¼ f5;6;8g (32%), D¼ f5;6;7;8g (0.4%), E¼ f2;5;6;7g (2%),F ¼ f1;3;4;5;6;7;8g (29%), G¼ f1;2;3;4;5;6;7;8g (2%). Only threeof the skill sets have a significant number of agents, namely skillsets A, C, and F. Most of the agents start in skill set A, as their initialtraining is on answering calls for credit card passwords (skill 6).The mid-level training is on credit cards (skill 5) and the initial set-up procedures for new customers (skill 8). Finally, the last trainingbundles all other skills, except for English (skill 2). As a result, 34%of the agents have only one skill (skill 6), which constitutes 4% ofthe total required number of agents. Moreover, all agents in thecall center have skill 6. Consequently, the number of agents withskill 6 is unnecessarily high. Thus, an obvious suggestion for thecall center management is to offer the mid-level training earlier, sothat most of the agents can satisfy the calls for credit cards and theprocedures regarding the initial set-up of new customers, whichconstitute 73% of the calls.

The call center divides Istanbul to 6 regions, but all employeesof the call center reside in only 4 regions. The regions are labeledin the order of being closer to the call center, so that region I is theclosest and region IV is the farthest. Table 1 shows that 75% of allagents reside in the closest two regions, i.e., in regions I and II. Thisreflects the call center management's recruiting policy withrespect to the living zone of the agents.

We generated instances from this data set and designed param-eter settings with specific characteristics. In all the instances, 250operators, 8 skills, 28 days, 24 time intervals in a day, 4 shifts,2 types of vehicles and 4 regions exist. The following input

Table 1Distribution of the agents with respect to skill sets and regions.

Skill sets Region I Region II Region III Region IV Sum

A 23 47 4 10 84B 0 1 0 0 1C 26 35 8 12 81D 0 0 0 1 1E 0 5 0 0 5F 15 32 6 20 73G 2 2 0 1 5

Sum 66 122 18 44 250

parameters are set to several levels: Demand for operators(requirements) have three levels (Low, Medium, High); and seatcapacity for the call center has two levels (Tight seat capacity(167), Ample seat capacity (250)).

A detailed description of the remaining parameter values in thegenerated instances and related assumptions are given below:

�

Vehicles: The call center uses two types of vehicles withcapacities 12 and 4 people. We assume that a vehicle can beassigned to only one region.

�

Transportation cost: The call center works with a shuttle serviceprovider and pays a fixed cost per trip and per vehicle used. Theper trip transportation cost per vehicle changes with the typeof the vehicle (with a ratio of approximately 3) as well as withthe region. Moreover, the shuttles which correspond to theregular office hours, i.e., shift 2 in Fig. 1, cost much less than theothers, since the bank pays for a significant portion of thetransportation cost at the start and end times of this shift,rather than the bank call center.

�

Shifts: We use the four shifts that the call center uses (seeFig. 1). Three of them lasts 9 h, while the fourth is a so-calledovertime shift that lasts 11 h. The agents are assigned to theseshifts according to the workforce planning, including the over-time shift. Hence, the agents know when they are going towork overtime over the planning horizon. The overtime costincurred for the overtime shift accounts for only the additional2 h. We also note that any two successive shifts assigned to anagent should be at least 11 h apart.

�

Labor costs: The employees receive a fixed salary, and are paidfor each extra hour if they work overtime. In the model, weonly consider the overtime cost as 10 per overtime hour perperson, while ignoring the other labor costs since the callcenter asked us to use 250 as the fixed number of agents.

�

Planning period: We take 28 days (4 weeks) as the planninghorizon.

�

Seat capacity: The call center has 167 seats. However, we alsoconsider the case with ample seat capacity where all agents canwork at the same time, i.e., with 250 seats.

�

Successive working days: By law, the operators can work at most6 successive days.

�

Agent discontent: The managers have to set appropriate rules fordetermining the discontent coefficients and collect them fromall agents, or design appropriate questionnaires. In our case, thecall center management would neither support collecting theindividual preferences of the agents, nor would include differentweights due to seniority. Hence, we used the coefficients set bythe managers for all agents. Based upon their experiences, callcenter managers identify three major reasons and the corre-sponding discontent scores for employee dissatisfaction:(1) exceeding the regular number of workdays in the planninghorizon with a discontent score of 5 for working one extra dayover the planning horizon, (2) working on a weekend day with adiscontent score of 4, (3) working overtime, although paid, witha score of 2. Hence, we set all other discontent scores equal to 0.In our computations, we bound the total discontent over theplanning horizon for each employee by 0 and 40, but we providea sensitivity analysis for these values in Section 5.5.4.

�

Underage penalty: We take the underage penalty as 20 peragent per period, but Section 5.5.3 presents a sensitivityanalysis for this value.

�

Workforce requirements: We convert 15-min requirements to1 h requirements by taking the maximum staffing level in each15-min interval of an hour as the requirement of that hour. Theaverage of medium level corresponds to the average workforcedemand of the call center, whereas low and high levels haveaverages of 80% and 120% of that of medium level. In order to

Table 2Distribution of total costs with respect to different cost types.

Demand Tight seat capacity Ample seat capacity

Overtime Transport Underage Overtime Transport Underage

Low 15,440 5923 131,600 15,220 5994 131,640Medium 20,460 5891 275,520 20,190 6012 275,200High 21,350 7142 441,040 23,150 6501 437,860

Table 3Average performance measuresn for all instances.

Demand Tight seat capacity Ample seat capacity

UV% OS% UR% DL UV% OS% UR% DL

Low 9.96 41.16 16.27 22.34 10.43 40.09 16.27 22.09Medium 11.91 53.59 27.46 25.84 9.74 52.51 27.43 25.73High 13.99 56.11 36.84 26.47 13.96 59.16 36.57 25.85

n UV¼underutilized vehicle capacity, OS¼overtime shifts, UR¼unsatisfiedrequirements, DL¼discontent score.

E.L. Örmeci et al. / Omega 43 (2014) 41–5348

introduce random fluctuations in the workforce demand, weadded random terms of small variance to the averages.

5.2. Model size and run times

For the described instances with 250 agents, 28 day planninghorizon with 24 time intervals per day, 4 shifts, 8 skills, 2 types ofvehicles and 4 regions, the model contains a total of 268,628variables. Among those, 28,250 are binary, and 10,752 take integervalues. The number of assignment variables increases linearly inthe number of agents, the number of shifts and the length of theplanning horizon. Integer variables correspond to the number ofvehicles at the start and end of the shifts. The number ofconstraints is 281,674. Note that the requirement constraints aredefined for each period on each day and each skill type, whileagent satisfaction constraints are defined for each agent. As aresult, the model size increases linearly with the input data size.

All instances are solved using the CPLEX solver 12.2 on aworkstation with Xeon E5520 @ 2.27 GHz processor and 48 GBof memory, with a time limit of 24 h and gap tolerance of 1%. Thatis, the optimization model is terminated when a feasible solutionhas been proved to be within 1% of optimality without reaching24 h time limit of run time. In all cases, the solver outputs afeasible solution within 1% gap from the best lower bound within24 h.

In terms of the hardness of instances with differing data andparameter values, we observed the following: Typically, instanceswith low demand are easier and the problem gets harder formoderate values of the demand. Also, instances with a tight seatcapacity constraint are harder to solve than those in which thisconstraint is relaxed. Within the different values of the upperbound for agent discontent, the hardest instances correspond tothe ones with larger values of the bound. The model that mini-mizes the sum of total discontents of all agents is much easier tosolve, as a feasible solution with less than 1% gap is found within1 h in most of the instances.

5.3. Additional performance measures

To evaluate the model solutions, we define new performancemeasures in addition to the costs. To observe the effect ofaccounting for the transportation costs, we use the percentage ofunderutilized vehicle capacity:

UV%¼ 1−2∑a∈A∑w∈R∪O∑d∈DXawd

∑v∈Vκv∑g∈G∑t∈T∑d∈DðIvgtd þ LvgtdÞ

� �100;

where the nominator in the second term gives the total number ofagents to be transported to/from the call center, and its denomi-nator is the total capacity of all vehicles used.

Another measure is the percentage of overtime shifts in thetotal number of shifts used in the planning horizon:

OS%¼ ∑a∈A∑w∈O∑d∈DXawd

∑a∈A∑w∈O∪R∑d∈DXawd100:

The effect of operator underage on the customer service levelscan be understood by computing the percentage of unsatisfiedrequirements over all skills and time periods:

UR%¼ ∑s∈S∑t∈T∑d∈DUstd

∑s∈S∑t∈T∑d∈Drstd100:

Finally, we measure the overall discontent level of the agentsby the average discontent score:

DL¼ ∑a∈ATDa

jAj :

5.4. Analysis of the model solutions

This section presents the model solutions for different settingscharacterized by different demand levels and seat capacities(Section 5.4.1). We observe how the underage/overage levelschange through a typical weekday in Section 5.4.2. We alsopresent the shift assignments with respect to regions and shiftdescriptions (Section 5.4.3), and the average percentage of timethat agents from each skill set allocate to his/her skills in themodel solutions (Section 5.4.4). We set the underage penalty to 20,the bounds for the total discontent to 0 and 40 (γmin ¼ 0 andγmax ¼ 40), and the seat capacity to 167 through out this section,unless it is specified otherwise.

5.4.1. Effect of demand levels and seat capacityWe first analyze how the objective function values are dis-

tributed among the different cost components. Table 2 presentsthe cost distribution for all instances. We can see that the under-age cost dominates the other cost types by accounting for 86–94%of total costs with an average of 92% and they increase signifi-cantly as the demand level increases. Note that the underagepenalty indicates the importance of customer service and the totalunderage cost is not part of the operating costs incurred. Hence,we concentrate on the transportation and overtime costs. Theoverall proportion of the transportation costs in all instances is24%, while overtime costs account for the rest. The overtime costsincrease with the demand level, whereas the transportation costsare high when the demand level is high, but do not varysignificantly for low and medium demand levels. Having enoughseat capacity decreases the total costs, but only marginally with amaximum of 0.4%. Its effect on different types of costs varies, withno monotonicity and no significant impact.

Table 3 presents how the other performance measures change.The percent of underutilized vehicle capacity is low, as it rangesbetween 9% and 14%. Hence, the model is effective in terms ofgrouping the operators with respect to the zones. The averagevehicle utilizations are not affected by the parameters signifi-cantly, except for the high demand level case. All the otherperformance measures, i.e., the proportion of overtime shifts andof unsatisfied requirements as well as the discontent levels,increase by the average demand level. The seat capacity does notaffect these measures significantly.

Fig. 3. The number of agents required in each hour and the number of agents assigned in each shift at medium demand with tight seat capacity for (a) skill 5, (b) skill 1.

Table 4Distribution of the agents with respect to shifts and regions.

Shifts Region I Region II Region III Region IV Sum

1 86 34 3 0 1232 15 515 243 779 15523 398 1072 68 6 15444 505 27 0 0 532

Table 5Average percentage of time allocated to each skill from each skill set.

Skill sets Skills # of agents

1 2 3 4 5 6 7 8

A – – – – – 100 – – 84B – – – – – 0 100 – 1C – – – – 99 0 – 1 81D – – – – 71 0 19 10 1E – 10 – – 58 0 – 32 5F 27 – 3 6 32 0 14 18 73G 20 16 2 3 34 0 10 15 5

% requirements 12 1 1 2 63 4 6 10

E.L. Örmeci et al. / Omega 43 (2014) 41–53 49

5.4.2. Underage and overage levelsWe observe the changes in the underage/overage levels during

a typical workday. We plot the workforce demand levels for skills1 and 5, which correspond to 12% and 63% of the total require-ments, respectively, and the number of operators allocated tothese skills on a Tuesday when the demand level is medium. Fig. 3shows the demands for operators as bars and the shift assign-ments as shaded boxes. Note that the number of agents assignedto skill s in shift w on day d is found by ∑a∈APaswd from the modelresults, which is constant for each shift w.

We observe that the requirements are rarely satisfied through-out the day for both skills. The number of agents allocated to skill1 is close to the requirements during 9:00–20:00, as they are shortby at most 4 agents (a required number of 25 versus an assignednumber of 21 in interval 15). However, as no agents are assigned toskill 1 during shift 4, none of skill 1 calls will be answered during20:00–24:00. The underage level for skill 5, on the other hand, is ata level of around 30 during 11:00–15:00, and it is even higherduring 20:00–24:00, reaching a maximum of 48 at 20:00. Themain reason for the high levels of underage during 20:00–24:00 isthe low number of agents assigned to shift 4 (16:00–24:00).

5.4.3. Shift assignments with respect to regions and shift descriptionsTable 4 presents the distribution of agents with respect to the

regions in which they reside and the shifts they are assigned toduring the 28-day period. Shift 1 is a night shift, which is used theleast. We observe that most agents are assigned to shifts 2 and 3,whereas shift 4 is used sparingly, which provides a reason for thehigh shortages during 20:00–24:00 (see Fig. 3). The agents fromfarther regions (Regions III and IV) are mostly assigned to theregular working hours, i.e., shift 2. Moreover, they never work atshift 4, while working overtime (shift 3) for a limited numberof times.

These assignments are a direct consequence of the transporta-tion cost structure. As mentioned earlier, the bank pays for asignificant portion of the transportation cost of the agents who

start working at 9:00 and finish working at 18:00. Hence, the callcenter contributes to the transportation cost of shift 2 onlymarginally. As a result, shift 2 covers all the peak demands witha very low cost, which increases its marginal benefit and makes itthe most valuable shift. The transportation cost of shift 3 isrelatively low as well, since the cost at the beginning of this shiftis also compensated by the bank. Consequently, the marginalbenefit of the other shifts is much lower, which explains the lownumber of assignments to shift 4, especially from the fartherregions.

5.4.4. Percentage of time allocated to each skillTable 5 presents the average percentage of time that an agent

from a skill set uses each of his/her skills. Note that the average istaken over all shifts, days and all agents in a skill set. The dashesmean that a skill set does not contain the indicated skill. Forconvenience, the table also includes the number of agents fromeach skill set, as well as the percentage of required agents for eachskill. The percentage of time allocated to this skill is generallydecreasing by the number of skills in a skill set. In particular,agents from skill set F (with 7 skills) answer calls requiring skill5 for 32% of their time, while those from skill set C (3 skills)answer this type of calls 99% of their time. This table also confirmsour earlier observation that there are too many agents serving skill6. This imbalance contributes to the high levels of underageas well.

5.5. Scenario and sensitivity analysis

In this section, we investigate several what-if scenarios toevaluate the benefits of representing the transportation in themodel and additional shifts, and conduct sensitivity analysis of the

E.L. Örmeci et al. / Omega 43 (2014) 41–5350

model parameters: the underage penalty and the bounds on theagent discontent score. We also develop a new model whichminimizes the total discontent scores of all agents, and evaluateits performance. The fairness issue is also discussed in the contextof agent discontent scores.

5.5.1. Benefits from including transportation in the modelTo analyze the benefits from including transportation costs in

the model, we solved our model with zero transportation costs.We take the shift assignments from the solution of this new modeland compute the transportation costs that would accrue by findingthe optimal number and type of vehicles required.

Table 6 presents the solutions of the two models when the seatcapacity is tight. The columns labeled by T refers to the solution ofthe model with the actual values of transportation costs, NT to thesolution of the model with no transportation costs and Δ% givesthe percentage of the relative change between the two cases(computed as the ratio of the difference in the performancemeasures of the two solutions to the performance measure ofthe solution with no transportation).

Accounting for the transportation costs brings significantimprovements in the transportation costs in a range of 25–34%and underutilized vehicle capacity in a range of 34–56%, while itdoes not affect the other performance measures in a substantialway as all the relative changes are below 1%. This is very positiveas it shows that we can find better solutions in terms oftransportation costs without worsening the other two criteria,namely customer service quality and the agent satisfaction.

5.5.2. Benefits of additional shiftsIn this section, we consider two additional shifts with start and

finish times 16:00–1:00 and 6:00–15:00. Both of these shifts covera significant portion of high demand periods. The shifts during6:00–15:00 and 15:00–24:00 (shift 4) together cover a periodwhich has higher demand than the rest of the day.

Solving the model with the additional shifts to optimalityrequires considerable amount of computation time and hencewe set a run time limit of 60 h to achieve a gap of at most 1%. Weexperiment with these shifts for all demand levels and both tightand ample seat capacities. In all these cases, the objective functionvalues do not improve those of the original solutions by morethan 1%.

We observe that the new shifts are not used much. This can beexplained by the transportation cost structure that subsidizes shift2 significantly and shift 3 partially (see Section 5.4.3). Hence, weconclude that the call center's current shifts provide sufficient flex-ibility in shift timing for the given structure of transportation cost.

5.5.3. Effects of underage penalties on solution characteristicsWe set the underage penalty as 20 in our numerical experi-

ments. The underage penalty should be higher than the overtimecost (which is 10 per hour), since otherwise overtime shifts willnever be used. Moreover, to obtain a reasonable trade-off betweenthe underage levels and transportation activities, the underage

Table 6Comparing the performance measuresa for the model with transportation costs (T) and

Demand Transportation costs UV%

T NT Δ% T NT Δ%

Low 5923 8718 32.06 9.96 22.57 55Medium 5891 8903 33.83 11.91 21.29 44High 7142 9536 25.11 13.99 21.40 34

a UV¼underutilized vehicle capacity, UR¼unsatisfied requirements, DL¼discontent

penalty is set to be between the minimum and maximumtransportation costs over all regions and vehicle types. In thissection, we experiment with different values of underage penaltyto demonstrate how customer service levels may be controlledthrough this parameter.

In Section 5.4.2, we analyzed the shift assignments at themedium demand level, where we observe that the requirementsat the medium demand level are very tight with the currentnumber of agents and seat capacity as illustrated in Fig. 3. Hence,in this section we restrict our experiments to systems with lowdemand level and ample seat capacity to see the effect of underagepenalty levels.

Regarding the effects of the underage penalty, we first note itsrole as a leverage for setting the relative importance of customerservice, compared to employee satisfaction and operating costobjectives. Table 7 summarizes the performance measures as theunderage penalty level varies in the range 5–40. We first observethat the performance measures do not change much for theunderage penalties higher than 15. This is because the currentnumber of agents is not sufficient to satisfy all the requirementseven at the low demand level. When we restrict our observationsfor the underage penalty in the range 5–15, we see that a higherunderage cost increases the proportion of overtime shifts anddiscontent level, while decreasing the underage levels. We expectthat the effect of the underage penalty will be more pronouncedwhen the number of agents is sufficient to satisfy the existingworkforce demand.

5.5.4. Analysis of agent discontent and fairnessThe first part of this section conducts sensitivity analysis on the

discontent limits and compares the effect of modeling agentdiscontent in two alternative ways. Then, we address the fairnessissue, as the lack of fairness is another source of discontent amongthe agents.

Our model allows limiting the total discontent of each agentfrom below and above. In our case analysis, we bound thisquantity by 40 from above, to ensure that the model does notassign a particularly unpleasant work schedule to any employee.For example, the model will not assign an operator a total of 21working days (a discontent of 5), all weekends (a discontent of8�4) and two overtime shifts (a discontent of 2�2) at the sametime over a planning horizon of 28 days, since the total discontentof an agent with such a schedule amounts to 41.

In order to understand the effect of this bound on theperformance measures, including the costs, we set the upperbound to 100, which practically corresponds to the case with noupper bound. Table 8 presents the percentage of the relativechange in the related performance measures when the upperbound is increased from 40 to 100. More explicitly, Δ% iscomputed as the ratio of the difference in the performancemeasures of the two solutions to the performance measure ofthe solutions corresponding to the upper bound 100.

In all demand levels, the upper bound of 40 decreases both theaverage and the standard deviation of the discontent levels

the model with no transportation cost (NT).

UR% DL

T NT Δ% T NT Δ%

.87 16.27 16.20 −0.43 22.34 22.40 0.27

.06 27.43 27.40 −0.11 25.84 25.97 0.5

.63 36.57 36.91 0.92 26.47 26.63 0.6

score.

E.L. Örmeci et al. / Omega 43 (2014) 41–53 51

significantly, especially for the high demand. When the discontentis not limited, the minimum discontent level for all demand levelsis 5, while the maximum discontent increases to above 60.Naturally, the use of overtime shifts is also higher for this case,since it is one of the discontent sources. On the other hand, boththe underutilized vehicle capacities (UV) and the performancemeasures related to the unsatisfied agent requirements (UR)improve when the upper bound on the discontent is relaxed.However, the improvements are less than 5%, except for UV for thehigh demand case. As a result, we can conclude that the modelchooses a better solution in terms of discontent from a large poolof good feasible solutions when the total discontent of each agentis limited.

Next, we evaluate further the effect of the upper bound valueson the optimal solutions when the demand level is low. FromTable 9, we see that as the upper bound value ðγmaxÞ increases, theaverage and standard deviation of the discontent levels of theagents increase until γmax ¼ 45. When γmax≥40, all performancemeasures are somewhat constant.

Table 7Effect of varying underage penalties on the performance measuresa and costdistribution for low demand level and ample seat capacity.

Underagepenalty

UV% OS% UR% DL Costs

Overtime Transport Underage

5 15.37 0 22.21 10.10 0 3654 44,98010 6.10 30.78 17.16 19.22 11,600 4962 69,55015 13.60 40.40 16.40 22.03 15,190 5977 99,57020 10.43 40.09 16.27 22.09 15,220 5994 131,64025 11.81 40.28 16.27 22.01 15190 5884 164,62530 14.16 40.37 16.27 22.12 15,220 6025 197,43035 9.81 40.53 16.17 22.12 15,310 5966 229,21540 12.68 40.68 16.27 22.20 15,410 6042 263,480

a UV¼underutilized vehicle capacity, OS¼overtime shifts, UR¼unsatisfiedrequirements, DL¼discontent score.

Table 8Percentage change in performance measuresa when total discontent upper boundis relaxed from 40 to 100 for tight seat capacity.

Demand Δ% Discontent level Δ% Δ% Δ% Costs

DL sDL Min Max UV OS UR Over Trans Under

Low 8.5 7.0 100 34.4 −4.4 −0.7 0.2 −0.7 −3.6 0.3Medium 16.4 7.7 100 34.4 −1.4 14.3 −2.6 14.4 10.9 −2.6High 23.5 17.2 100 38.5 −10.1 21.1 −3.3 22.5 7.0 −3.3

a UV¼underutilized vehicle capacity, OS¼overtime shifts, UR¼unsatisfiedrequirements, DL¼discontent score.

Table 9Effect of varying discontent upper bounds on the performance measuresa for low dema

Upper bound Discontent level UV %

DL sDL Min Max

20 12.96 8.46 0 20 18.2925 16.72 11.60 0 25 15.8130 19.37 13.43 0 30 13.4535 22.25 15.78 0 35 12.7040 22.09 17.24 0 40 9.9645 24.18 17.53 5 45 10.2050 24.23 17.58 5 49 7.44

a UV¼underutilized vehicle capacity, OS¼overtime shifts, UR¼unsatisfied requirem

We also experiment with the model by incorporating the totaldiscontent of all agents in the objective function, thus minimizingit instead of limiting the total discontent of each agent. We run thisnewmodel with different coefficient values for penalizing the totaldiscontent in the objective. Table 10 summarizes the correspond-ing results. We first observe that the solutions for 20 (with amaximal discontent of 16) and 50 (with a maximal discontent of 0)are not acceptable due to the very high underage levels. In theacceptable cases, we evaluate the performances of the two models(one limiting and the other minimizing the discontent), bycomparing the performance measures when the unsatisfiedrequirements of the two models are close to each other. Accord-ingly, the solution for 0 discontent coefficient is coupled with thecase when the discontent is limited by 40, the solution for 5 iscoupled with 30, and 10 with 25. In all these three cases, theperformance measures related to discontent of agents are betterwhen total discontent is limited, while the other performancemeasures do not vary significantly. Hence, we can conclude thatlimiting the total discontent of each agent generates bettersolutions in terms of employee satisfaction.

Finally, we consider the fairness issue. Blochliger [13] definesfairness as distributing unpopular shifts evenly among all employ-ees. Similarly, we define fairness as distributing the factors causingdiscontent in the schedules evenly among all agents. In accordancewith Blochliger's proposal, we measure fairness by the range or thestandard deviation of the agent discontent values. An investigationof these values in the above settings reveals a high variation. Thestandard deviation of the discontent scores drops below 10 onlywhen the upper bound is 20 or for the unacceptable cases whenthe discontent coefficient is 20 or 50 in the model which mini-mizes the total discontent. Hence, these models are not sufficientto ensure fairness among the agents.

One way to handle fairness is to implement Blochliger's followingsuggestion: compensate for the high discontent of an agent in thecurrent planning horizon by limiting his/her discontent by a lowervalue in the next planning horizon. Another way is to set tighterlower and upper bounds on the total discontent of each agent. InTable 11 we see that as the allowed range of total discontentdecreases, the standard deviation decreases significantly in thesolution. However, this improvement comes at an increased under-age cost, although both overtime and transportation costs decrease.Hence, setting tight bounds for total discontent may be a viableoption to achieve better fairness among operator schedules, but itintroduces unnecessary discontent for some of the agents.

6. Conclusions

We study the staff rostering problem for call centers thatoperate in a multi-skill environment and provide pick-and-dropservices for their employees with three categories of objectives, as

nd level and tight seat capacity.

OS % UR % Costs

Overtime Transport Underage

15.05 19.89 5600 5539 161,06024.82 17.76 9320 6317 143,74033.35 16.96 12,510 5669 137,44041.24 16.30 15,480 6121 131,88041.16 16.27 15,440 5923 131,60041.44 16.23 15,540 5860 131,54041.38 16.30 15,490 5714 131,900

ents, DL¼discontent score.

Table 10The performance measuresa for the solution of the model that minimizes the sum of the total discontents for low demand level and tight seat capacity.

Discontent Discontent level UV % OS % UR % Costs

coefficient DL sDL Min Max Overtime Transport Underage

0 24.15 18.53 5 61 9.54 40.86 16.27 15,340 5718 132,0005 18.52 16.48 0 51 11.85 31.52 17.23 11,820 5216 139,640

10 15.46 14.09 0 47 13.62 23.70 18.26 8880 5731 147,88020 2.56 3.63 0 16 16.81 0 26.19 0 3209 212,16050 0 0 0 0 13.01 0 28.25 0 3568 228,800

a UV¼underutilized vehicle capacity, OS¼overtime shifts, UR¼unsatisfied requirements, DL¼discontent score.

Table 11Effect of varying discontent lower and upper bounds on the performance measuresa for low demand level and tight seat capacity.

Lower bound Upper Discontent level UV % OS % UR % Costs

bound DL sDL Min Max Overtime Transport Underage

0 40 22.09 17.24 0 40 9.96 41.16 16.27 15,440 5923 131,60012 32 25.04 8.34 13 32 12.74 36.05 16.63 14,100 5874 134,54017 27 23.7 4.61 17 27 11.34 27.58 17.40 10,980 5593 140,780

a UV¼underutilized vehicle capacity, OS¼overtime shifts, UR¼unsatisfied requirements, DL¼discontent score.

E.L. Örmeci et al. / Omega 43 (2014) 41–5352

total operating costs for the firm, service quality for customers andsatisfaction for agents. We develop a mixed integer programmingmodel that considers the transportation costs of agents from/totheir homes as well as the agent satisfaction while allowingunderstaffing with a corresponding penalty cost. The modelprovides solutions that tradeoff underage costs with overtimeand transportation costs, while providing agent satisfaction viaconstraints. The model is effective in identifying the solutions withminimal transportation costs, while keeping the other perfor-mance measures at a constant level.

The model parameters, namely the underage penalty coeffi-cient and the bounds on the total discontent of each agent, allowthe managers to select solutions which fit their needs. In ournumerical analysis, we provide a sensitivity analysis with respectto these parameters. We should note that the parameter settingsprovide a desired level of control only when the system has someflexibility, as observed in our numerical experiments. If the work-force demands are too high for the current workforce level, theeffect of these parameters on the solutions is limited.

The model we developed can be implemented into daily opera-tions by integrating it with the information system at the call center,as it can be solved to near-optimality within reasonable time. It canalso be used as a scheduling tool for planning purposes. For instance,it may guide the hiring and staff training decisions by consideringskills and regions of the agents. The underage and overage valuesmay point out weaknesses of the current shift descriptions, whichmay lead to improvements in the shift structures.

The schedule can also be revealed to the agents to obtainfeedback, which can then be incorporated to the model. The shiftassignments output by the model can be used either directly orwith adjustments by the managers. For instance, the performanceof the schedule can be evaluated via simulation to observe theeffects of underage on service levels as in Sencer and Basarir [38].Moreover, the routes of the vehicles can be optimized according tothe transportation schedule.

Acknowledgments

This project was financially supported by TUBITAK. We thankFinansbank and Siemens call centers for providing data and guidance.

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