scispace - formally typeset

Journal ArticleDOI

A joint chance-constrained programming approach for call center workforce scheduling under uncertain call arrival forecasts

01 Jun 2016-Computers & Industrial Engineering (Pergamon)-Vol. 96, pp 16-30

AbstractWe study the call center shift scheduling problem under uncertain demand forecasts.Forecasting errors are seen as independent normally distributed random variables.The resulting stochastic problem is modeled as a joint chance-constrained program.A mixed-integer linear programming based solution approach is proposed.Numerical results based on a real case study and managerial insights are provided. We consider a workforce management problem arising in call centers, namely the shift-scheduling problem. It consists in determining the number of agents to be assigned to a set of predefined shifts so as to optimize the trade-off between manpower cost and customer quality of service. We focus on explicitly taking into account in the shift-scheduling problem the uncertainties in the future call arrival rates forecasts. We model them as independent random variables following a continuous probability distribution. The resulting stochastic optimization problem is handled as a joint chance-constrained program and is reformulated as an equivalent large-size mixed-integer linear program. One key point of the proposed solution approach is that this reformulation is achieved without resorting to a scenario generation procedure to discretize the continuous probability distributions. Our computational results show that the proposed approach can efficiently solve real-size instances of the problem, enabling us to draw some useful managerial insights on the underlying risk-cost trade-off.

Summary (4 min read)

1. Introduction

  • Call centers can be broadly defined as facilities designed to support the delivery of some interactive service via telephone communications ([9]).
  • Shortterm decisions (1-2 weeks ahead) involve the scheduling of an available pool of agents over an horizon typically spanning one week.
  • The present work is related to short-term workforce management decisions in call centers.
  • The input data of the shift scheduling problem are thus subject to uncertainty: not taking this into account while building the shift schedule might lead to significant discrepancies between the call center performance targeted at the time scheduling decisions are made and the one actually obtained in practice (see [10]).
  • The authors explain how, under the assumption of independence between the forecasting errors, it can be reformulated as a stochastic program involving a set of individual chance constraints.

2. Literature review

  • Given the size of the call center industry and the complexity associated with its operations, call centers have emerged as a fertile ground for Operations Research.
  • This amounts to using an Erlang C model to represent the call center in each period of the scheduling horizon (see [17] and [18]).
  • This might explain why, to the best of their knowledge, all previously published approaches for stochastic call center shift scheduling rely on the use of discrete probability distributions to represent the uncertainty on the call arrival rates and translate each corresponding call arrival rate scenario into an agent requirement scenario in a pre-optimization step.
  • Thus, [17] and [18] consider that the information on uncertainty is directly provided in the form of a discrete probability distribution.
  • In the present paper, the authors propose a one-stage stochastic programming approach using joint chance constraints.

3. Joint chance-constrained programming model

  • This section is devoted to the detailed presentation of the problem under study in the present paper: the stochastic shift scheduling problem in a single-class single-skill call center.
  • The authors then consider the stochastic variant of the problem and introduce the proposed joint chance-constrained programming model.

3.1. Deterministic formulation

  • The authors consider the shift scheduling problem for a single-class single-skill call center.
  • The call arrival process during a period t is thus modeled as a Poisson process with rate λt.
  • Finally, customers patience is limited, i.e. a customer placed in the queue might hang up before starting service.
  • Note that an analytical expression of the function φµ,γ,p∗ is not available.
  • Finally the authors introduce the integer decision variables xs defined as the number of agents assigned to shift s. 7.

3.2. Joint chance-constrained programming formulation

  • In terms of solution approaches, a variety of tractable approximations have been proposed to handle general joint chance-constrained problems.
  • Those methods require the generation of a subset of the p-efficient points of the probability distribution through an enumeration scheme (see e.g. [8], [3] and [16]).

3.3. Equivalent individual chance-constrained programming formulation

  • This method allows to take into account both the intra-day and the intra-week seasonality in the call arrivals and makes use of independent and identically normally distributed random variables with mean 0 to represent the forecasting residuals.
  • This leads to the following formulation which provides a feasible solution of problem JCCP.
  • A first step towards solving these two problems thus consists in building a numerical representation of F−1Nt by exploiting the relation.

4.1. Minimum number of agents required as a function of the call arrival rate

  • The first step of their solution approach consists in building a numerical representation of the inverse cumulative probability distribution F−1Nt of the random variable.
  • The authors defined φµ,γ,p∗ in subsection 3.2 as the function of the call arrival rate λ providing the minimum number of servers n needed to reach the target service level p∗ when the service and patience threshold rates are µ and γ, respectively.
  • This algorithm exploits previously published results on the performance evaluation of Erlang A systems (see e.g. [15] and [21]).
  • The authors thus use in what follows a numerical description of φµ,γ,p∗ over a finite interval [0;λmax] which is obtained by computing conservative estimations of the threshold values λ̃l.
  • The corresponding computation time is thus not included in the numerical results presented in Section 6.

4.2. Inverse cumulative probability distribution of random variables Nt

  • FNt is thus fully described by giving its values for the set of positive integer values of x.
  • The authors therefore focus on computing the value of FNt over the set N ∗. Let l ∈ N∗. Besides, they assume that the forecasting error ǫt follows a normal distribution N (0, σt).
  • Solving the resulting mixed-integer linear program then provides us with a feasible solution of problem JCCP.
  • This subsection is thus devoted to the study of the functions.
  • The authors denote νtm the integer value of Ψt(yt) over the interval [βt,m+1; βt,m[.

4.4. Reformulation of problem EDetF as a large-size MILP

  • Proposition 2 implies that the right hand side of constraints (22) is a nonincreasing piecewise constant function of yt.
  • The authors exploit this result to reformulate problem EDetF as a mixed-integer linear program (MILP) involving a large number of binary variables and constraints.
  • 18 and reformulate EDetF as: In constraints (37), the non-linear term F−1Nt (π yt) has been replaced by the linear expression νt,0 + νt,0 represents the minimum number of agents required in period t to ensure that the risk of not reaching the target quality of service is below 1−π.
  • This is the purpose of constraints (38)-(39) which impose that yt stays above a lower bound, the value of which depends on the values of the zt,m variables.

5. A small illustrative example

  • The authors introduce a small instance of the call center shift scheduling problem in order to illustrate the solution approach and compare between the two formulations EDetB and EDetF discussed in Section 4.
  • The solution approach proposed to solve problems EDetB and EDetF comprises four main steps.
  • Second, for each period t, the authors build a numerical representation of function F−1Nt with αmax = 0.999999 and use it to compute the right hand side value F−1Nt (1− π 1/T ) of the constraints (14) involved in problem EDetB.
  • These periods typically corresponds either to peak hours where the mean call arrival rate is large or to off-peak periods where only a few shifts are working.

6. Numerical results

  • The authors carried out some computational experiments on real data coming from an anonymous health insurance company in order to evaluate the solution approach presented in Section 4 and to compare it with a scenario-based approach.
  • The results of this computational study are then used to derive 22 some managerial insights on the risk-cost trade-off in stochastic call center shift-scheduling.

6.1. Instances

  • To carry out their computational experiments, the authors generated 400 instances based on real data coming from an anonymous health insurance company.
  • More precisely, the various instances tested have the following features.
  • The authors used these data to generate a larger set of S = 120 shifts: these shifts correspond to part-time and full-time positions similar to the ones used in their case study, but with more flexibility to place half-days and/or days off within the week.

6.2. Numerical assessment of the proposed solution approach

  • The authors use the solution approach presented in Section 4 (with the values of the parameters Kmax, λmax, ∆λ, αmax and ymin provided in Section 5) to solve problems EDetB an EDetF.
  • The numerical results obtained on the 400 studied instances with formulation EDetB are provided in Table 3 while those obtained with formulation EDetF are provided in Tables 4-7.
  • Results from Tables 4-7 show that, despite their size, these mixed-integer linear programs could be solved within a reasonable computation time for all the considered instances generated from their real-life 1All data related to their experiments (description of the instances, C++ source files and numerical results) are available upon request from the corresponding author.
  • Moreover, results from Tables 4-7 also show that two features seem to have a strong impact on the computation times, namely the forecast quality and the maximum acceptable risk level.

6.3. Comparison with a scenario-based solution approach

  • In order to further assess the proposed solution approach, the authors compare it with a scenario-based approach, namely the sample approximation approach presented in [19].
  • This approximation enables to reformulate the stochastic problem as a large-size mixed-integer linear program.
  • This means that the shift schedules x∗ obtained through this approach are not feasible with respect to the joint chance-constraint (5).
  • It seems that, for the problem under study here, the sample size required to get such a near-optimal solution of JCCP is too large 30 31 to allow a resolution of formulation SA within reasonable computation time, especially for the small values of π.

6.4. Discussion and managerial insights

  • The authors now seek to derive from the results of their computational study some useful insights for call center managers faced with the problem of scheduling workforce under uncertain call arrival forecasts.
  • The authors first compare the two variants of the proposed solution approach: the one based on problem EDetB and the one based on formulation EDetF.
  • Namely, for the 400 considered instances, the total number of worked hours is reduced on average by 19% thanks to the use of the optimal sharing out of the risk between the scheduling periods carried out in problem EDetF.
  • On the contrary, increasing the value of 1−π might lead to significant cost savings.
  • Providing call center managers with such a quantified representation of the risk-cost trade-off might help them decide upon the risk level that they are ready to accept.

7. Conclusion and research perspectives

  • The authors studied the shift scheduling problem for a single-class single-skill call center with impatient customers and focused on explicitly taking into account in the related optimization problem the impact of the uncertainties in the call arrival rates forecasts.
  • Staffing a call center with uncertain non-stationary arrival rate and flexibility.
  • Modeling and theory, MPS/SIAM Series on Optimization 9, Society for Industrial and Applied Mathematics, Philadelphia. [31], also known as Lectures on stochastic programming.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

HAL Id: hal-01294589
https://hal.archives-ouvertes.fr/hal-01294589
Submitted on 29 Mar 2016
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A joint chance-constrained programming approach for
call center workforce scheduling under uncertain call
arrival forecasts
M. Excoer, C. Gicquel, O. Jouini
To cite this version:
M. Excoer, C. Gicquel, O. Jouini. A joint chance-constrained programming approach for call center
workforce scheduling under uncertain call arrival forecasts. Computers & Industrial Engineering,
Elsevier, 2016, 96, pp.16-30. �10.1016/j.cie.2016.03.013�. �hal-01294589�

A joint chance-constrained programming approach for
call center workforce scheduling under uncertain call
arrival forecasts
Mathilde Excoffier
1
, Céline Gicquel
1
, Oualid Jouini
2
1
Laboratoire de Recherche en Informatique, Université Paris Sud, Orsay, France
2
Laboratoire Génie Industriel, Ecole Centrale Paris, Chatenay-Malabry, France
September 2015
Working paper submitted for publication in Computers & Industrial Engineering

A joint chance-constrained programming approach for
call center workforce scheduling under uncertain call
arrival forecasts
Abstract
We consider a workforce managem ent problem arising i n call centers, namely
the shift-scheduling problem. It consists in determining the number of a gents
to be assigned to a set of predefined shifts so as to optimize the trade-off
between manp ower cost a n d custom er qual i ty of service. We fo cus on explic-
itly taking into account in the shift-scheduling problem the uncertainties in
the future call arri val ra t es forecasts. We model them as independent ran-
dom variables following a continuous probabili ty di st ri b u t i o n . The result-
ing stochastic optimization problem is handled as a joint chance-constrained
program and is reformulated as an equivalent large- si ze mixed-integer linear
program. One key point of the pro posed solutio n approach is that this r e-
formulation is achieved without resorting to a scenario generation procedure
to discretize the continuous p r ob a b i l i ty d i s tr i b u t i o n s. Our computatio n al re-
sults show that the proposed approach can efficiently solve real-size instances
of the problem, enabling us to draw some useful manager i a l insi ghts on the
underlying risk-cost trade-off.
Keywords: Personnel planni n g , Call center shift scheduling, Customer
abandonment, S t ochastic programming, Pro b a b i l i st i c constraints,
Mixed-integer lin e ar programming
1. Introduction
Call centers can be broadly defined as facili ti e s designed to sup port t h e
delivery of some i nteractive service via tele p hon e communications ([9]). Ap-
plications include among othe rs telemarketing, customer service, help desk
support and emergency dispatch. In most cases, the primary function of a
call center is to receive phone calls that have been initiated by customers.
Preprint submitted to Computers & Industrial Engineering March 29, 2016

Such operations, k n own as ”inbound” cal l centers, are the topic of the present
paper.
Personnel plann i n g is a key issue in call center management. Namely, as
reported in [1], call centers are labor-i ntensive operations in which the cost o f
the staff members handling the phone calls (known as the agents) typically
accounts for 60% to 80% of all the operating expenses. An efficient workforce
management is thus crucial to achieve profita b i l i ty in a call center.
Call center workforce management involves three main levels of decis i on -
making (see e.g. [23] ) . Long-term planning deci s i on s (6-12 months ah ea d )
include the determination of how many agents to hire and train at what
times based on aggregate long-term forecasts of demand for services. Short-
term decisions (1-2 weeks ahead) involve the scheduling of an available pool
of agents over an horizon typically spann i n g one week. These decisions are
based on detail ed short-te rm forecasts of agent requirements. Finally, real-
time adjustment decisions, such as agent schedule updating and call routing,
have to be made on a intra-day basis.
The present work is related to short-term workforce management deci-
sions in call centers. Th ese decisions usually involve two main steps. First, a
range of possible shift patter n s is defined an d managers have to determine the
number of agents to be assigned to each shift. Second, a rostering procedure
combines shifts into rosters and assigns rosters to indivi d u a l employees.
In the present pap er , we focus on the first st ep of this process and consider
the shift scheduling problem. We thus seek to determine the number of agents
to be assigned to a set of predefined shifts so as to meet two potentially
conflicting objectives, namely minimizing the manpower cost and delivering
a high quality of serv i ce to the call center cus to m er s. To achieve this, we
will look for shift schedules where the supply of agent resources is matched
as closely as possible with the demand for services. Namely, understaffing
would lead to customer di ssa ti s fact i o n due to a poor q u a l i ty o f service while
overstaffing would result in useless over-service to customers and higher than
needed operating costs.
One of the major difficult i es to be tackled wh i l e try i n g to m a t ch supply
and demand in a call center is that the level of demand for services, i.e. the
workload, is highly variable. This is mainly due to the fact that the call ar-
rival rate (number of calls reaching the call center per unit of time) is subject
to strong fluctuatio n s over the co u rs e of a day or a week. This difficulty is
usually handled in practice by d i vi di n g the scheduling horizon into a number
of time periods of 15 to 6 0 minutes. In each period, the ca l l arrival pro-
2

cess is modeled as a Poisson process with a constant and deterministically
known arrival rate and the call center i s treated as an independent queu i n g
system in stationary state (see e.g. [11 ]). This allows to determine the re-
quired staffing level for each period which is set to be the minimum number
of agents for which the target quality of service (expressed e.g. as the ma x -
imum allowed proportion of custome rs hanging up before being answered)
is reached. A deterministic optimization problem has then to be solved to
find the m i n i mum-cost shift schedule ensuring that the required number of
agents is staffed in each period of the hori zon .
However, as mentioned e.g. by [1], at the time when decision on shift
schedules is made, call arrival rates a r e most often not deterministically
known. We only have estimation s obtained via a forecast i n g procedure whose
outcome is a point forecast and some probabilistic representation of the fore-
casting errors. The input data of the shift scheduling problem are thus
subject to uncertainty: not taking this into account while building the shift
schedule might lead to significant discrepancies between the call center per-
formance targeted at the time scheduling decisions are made and the one
actually obt ai n e d in practice (see [10]).
In the present paper, we propose a stochastic programming based ap-
proach to explicitly take into account the uncertainty on the call arrival rates
in t h e shift scheduling problem. Our contributions are threefold. Fi r st we
model the forecasting error on the call arrival rate in each period as a random
variable following a continuous probability distribution. This is i n contrast
with most previously published approaches which rely on dis cr et e probability
distributions through the u s e of a finite set of scenarios to represent th e uncer-
tainty. Secon d , we propose to model the stochastic shift scheduling problem
as a one-stage stochastic program involving a joint chance constraint. Such
a model is particularly relevant when the call center is evaluated based on
its ability at reaching, on all periods of the scheduling horizo n , the target
quality of service and when the call center management focuses on the risk
of not meeting this objective. Another advantage is that it does not re-
quire introducing a penalty cost to be incurred when the target quality of
service is not reached, the valu e of which might be difficul t to estimate in
practice. Third we present an efficient solution approach based on the refo r -
mulation of the stochastic program as an equival ent deterministic large-size
mixed-integer linear program. This approach relies on the assu m p ti o n that
the r an do m variables modeling the f or eca st i n g error s are independent from
one another and comprises two mai n steps: (1) the reformulation of the joint
3

Citations
More filters

Journal ArticleDOI
TL;DR: Compared to joint-probabilistic chance-constrained programming (JCP), the CFSP method is more effective for handling multiple random parameters associated with different probability distributions in which their correlations are unknown.
Abstract: In this study, a copula-based flexible-stochastic programming (CFSP) method is developed for planning regional energy system (RES). CFSP can deal with multiple uncertainties expressed as interval values, random variables and fuzzy sets as well as their combinations employed to objective function and soft constraints. It can also reflect uncertain interactions among random variables through using copula functions even having different probability distributions and previously unknown correlations. Then, based on the developed CFSP approach, a CFSP-RES model is formulated for planning RES of the urban agglomeration of Beijing and Tianjin (China). Results disclose that uncertainties existed in the system components have significant effects on the outputs of decision variables and system cost, and the variation of system cost is reached 16.3%. Results also reveal that air pollutant emissions can be mitigated if the urban agglomeration can co-implement renewable energy development plans (REDP) over the planning horizon, with the reductive rates of [3.3, 7.6] % of sulfur dioxide (SO2), [2.7, 4.1] % of nitrogen oxides (NOx) and [7.0, 11.5] % of particulate matter (PM10). Compared to joint-probabilistic chance-constrained programming (JCP), the CFSP method is more effective for handling multiple random parameters associated with different probability distributions in which their correlations are unknown. Thus, it is not limited to some unjustified assumptions and can be applied to a wider range of problems than previous studies. The findings are helpful to explore the influence of interaction among random variables on modeling outputs and provide in-depth analysis for identifying desired decision schemes for planning RES.

38 citations


Cites background from "A joint chance-constrained programm..."

  • ...Besides, the conventional joint-probabilistic chance-constrained programming (JCP) methods for reflecting interactive relationships among a set of probabilistic constraints are based on assumptions that all of random variables employed to probabilistic constraints are normally and independently distributed [12, 13]....

    [...]


Journal ArticleDOI
TL;DR: A copula-based interval two-level programming (CITP) method is applied to planning the energy-water nexus system (EWNS) of Henan Province (China), where various decision-making levels and diverse risk-interaction scenarios are analyzed and results can provide decision supports for the coordinated development of regional-scale EWNS management.
Abstract: The management of water resources system and energy system belongs to different decision-making departments, and there is a certain hierarchical relationship between them. Optimizing the configuration of regional-scale water and energy systems from a global perspective, and considering the correlations between water resources shortage risk and energy shortage risk as well as their joint-risk interaction, can improve the accuracy and efficiency of management decisions. This study aims to propose a copula-based interval two-level programming (CITP) method by integrating a copula-based interval stochastic programming (CISP) method and two-level programming (TP) method. CITP cannot only balance the goals and preferences among different decision-making levels but also analyze the risk interactions between water resources availability and electricity demand. The CITP method is then applied to planning the energy-water nexus system (EWNS) of Henan Province (China), where various decision-making levels and diverse risk-interaction scenarios are analyzed. Results reveal that: during the planning horizon, a) the total electricity-generation amounts can change by 7.31 × 103 GWh from S1 to S5; b) the future electricity-supply structure will toward a more sustainable aspect, and the electricity generated from gas-fired, hydro and wind power can increase by 6.2 × 103 GWh, 3.7 × 103 GWh and 5.8 × 103 GWh, respectively. Results can provide decision supports for the coordinated development of regional-scale EWNS management among water, energy, economy and society as well as environment.

12 citations


Dissertation
25 Mar 2020
TL;DR: This thesis shows that under some assumptions that hold in call center examples, one can obtain the optimal solutions of the original problem by solving its SAA with large enough sample sizes, and indicates the viability of the SAA approach in this context, in both theoretical and practical aspects.
Abstract: In this thesis, we study the staffing optimization problem in multiskill call centers, in which we aim at minimizing the operating cost while delivering a high quality of service (QoS) to customers. We also introduce the use of chance constraints which require that the QoSs are met with a given probability. These constraints are adequate in the case when the performance is measured over a short time interval as QoS measures are random variables in a given time period. The proposed staffing problems are challenging in the sense that the stochastic constraints have no-closed forms and need to be approximated by simulation. In addition, the QoS functions are typically non-linear and non-convex. We consider staffing optimization problems in different settings and study the proposed models in both theoretical and practical aspects. The methodologies developed are general, in the sense that they can be adapted and applied to other staffing/scheduling problems in queuing-based systems. The thesis consists of three articles dealing with different challenges in modeling and solving staffing optimization problems in multiskill call centers. The first and second articles concern a two-stage staffing optimization problem under uncertainty. While in the first one, we study a general two-stage discrete stochastic programming model to provide a theoretical guarantee for the consistency of the sample average approximation (SAA) when the sample sizes go to infinity, the second one applies the SAA approach to solve the two-stage staffing optimization problem under arrival rate uncertainty. Both papers indicate the viability of the SAA approach in our context, in both theoretical and practical aspects. To be more precise, in the first article, we consider a general two-stage discrete stochastic problem with expected value constraints. We formulate its SAA with nested sampling. We show that under some assumptions that hold in call center examples, one can obtain the optimal solutions of the original problem by solving its SAA with large enough sample sizes. Moreover, we show that the probability that the optimal solution of the sample problem is an optimal solution of the original problem, approaches one exponentially fast as we increase the sample sizes. These theoretical findings are important, not only for call center applications, but also for other decision-making problems with discrete decision variables. The second article concerns solution methods to solve a two-stage staffing problem under arrival rate uncertainty. It is motivated by the fact that the SAA version of the two-stage staffing problem becomes expensive to solve with a large number of scenarios, as for each scenario, one needs to use simulation to approximate the QoS constraints. We develop an algorithm that combines simulation, cut generation, cut strengthening and Benders decomposition to solve the SAA problem. We show the efficiency of the approach, especially when the number of scenarios is large.

9 citations


Cites background from "A joint chance-constrained programm..."

  • ...Research studies have focused traditionally on single-skill call centers, see Green et al. (2003) for example, but there exist a few studies on the optimization of multiskill call centers....

    [...]

  • ...In another study, Excoffier et al. (2015a) consider the case where the call arrival rates are subject to uncertainty and follow unknown continuous probability distributions....

    [...]

  • ...Excoffier et al. (2014) consider the multi-period shift-scheduling problem for single-call type, single-agent group call centers with uncertainties in the future call arrival rates....

    [...]


Journal ArticleDOI
TL;DR: The results of the comprehensive computational study indicate that the constraint programming model runs more efficiently than the integer programming model for the rostering problem.
Abstract: It may be very difficult to achieve the optimal shift schedule in call centers which have highly uncertain and peaked demand during short time periods. Overlapping shift systems are usually designed for such cases. This paper studies shift scheduling and rostering problems for inbound call centers where overlapping shift systems are used. An integer programming model that determines which shifts to be opened and how many operators to be assigned to these shifts is proposed for the shift scheduling problem. For the rostering problem both integer programming and constraint programming models are developed to determine assignments of operators to all shifts, weekly days-off, and meal and relief break times of the operators. The proposed models are tested on real data supplied by an outsource call center and optimal results are found in an acceptable computation time. An improvement of 15% in the objective function compared to the current situation is observed with the proposed model for the shift scheduling problem. The computational performances of the proposed integer and constraint programming models for the rostering problem are compared using real data observed at a call center and simulated test instances. In addition, benchmark instances are used to compare our Constraint Programming (CP) approach with the existing models. The results of the comprehensive computational study indicate that the constraint programming model runs more efficiently than the integer programming model for the rostering problem. The originality of this research can be attributed to two contributions: (a) a model for shift scheduling problem and two models for rostering problem are presented in detail and compared using real data and (b) the rostering problem is considered as a task-resource allocation and considerably shorter computation times are obtained by modeling this new problem via CP.

7 citations


22 Aug 2016
TL;DR: A sample average approximation (SAA) version of this staffing problem with probabilistic constraints in an emergency call center whose solution converges to that of the exact problem when the sample size increases.
Abstract: We consider a staffing problem with probabilistic constraints in an emergency call center. The aim is to minimize the total cost of agents while satisfying chance constraints defined over the service level and the average waiting time, in a given set of time periods. We provide a mathematical formulation of the problem in terms of probabilities and expectations. We define a sample average approximation (SAA) version of this problem whose solution converges to that of the exact problem when the sample size increases. We also propose a quick and simple simulation-based (heuristic) algorithm to compute a good (nearly optimal) staffing solution for the SAA problem. We illustrate and validate our algorithm with a simulation model based on real data from the 911 emergency call center of Montreal, Canada.

7 citations


References
More filters

Journal ArticleDOI
Abstract: A new approach to optimizing or hedging a portfolio of nancial instruments to reduce risk is presented and tested on applications. It focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value-at-Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is anyway considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage rms, mutual funds, and any business that evaluates risks. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can be applied also to the optimization of percentiles in contexts outside of nance.

4,862 citations


Book
24 Sep 2009
TL;DR: The authors dedicate this book to Julia, Benjamin, Daniel, Natan and Yael; to Tsonka, Konstatin and Marek; and to the Memory of Feliks, Maria, and Dentcho.
Abstract: List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

2,163 citations


Journal ArticleDOI
TL;DR: This work begins with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations, which identifies important problems that have not been addressed and identifies promising directions for future research.
Abstract: Telephone call centers are an integral part of many businesses, and their economic role is significant and growing. They are also fascinating sociotechnical systems in which the behavior of customers and employees is closely intertwined with physical performance measures. In these environments traditional operational models are of great value--and at the same time fundamentally limited--in their ability to characterize system performance.We review the state of research on telephone call centers. We begin with a tutorial on how call centers function and proceed to survey academic research devoted to the management of their operations. We then outline important problems that have not been addressed and identify promising directions for future research.

1,330 citations


Journal ArticleDOI
TL;DR: A large deviation-type approximation, referred to as “Bernstein approximation,” of the chance constrained problem is built that is convex and efficiently solvable and extended to the case of ambiguous chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set.
Abstract: We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its computationally tractable approximation, i.e., an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem. We construct a general class of such convex conservative approximations of the corresponding chance constrained problem. Moreover, under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independent-of-each-other random variables, we build a large deviation-type approximation, referred to as “Bernstein approximation,” of the chance constrained problem. This approximation is convex and efficiently solvable. We propose a simulation-based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and well-known scenario approximation approaches. Finally, we extend our construction to the case of ambiguous chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly, while the chance constraint should be satisfied for every distribution given by this set.

936 citations


Journal ArticleDOI
Abstract: Call centers are an increasingly important part of today's business world, employing millions of agents across the globe and serving as a primary customer-facing channel for firms in many different industries. Call centers have been a fertile area for operations management researchers in several domains, including forecasting, capacity planning, queueing, and personnel scheduling. In addition, as telecommunications and information technology have advanced over the past several years, the operational challenges faced by call center managers have become more complicated. Issues associated with human resources management, sales, and marketing have also become increasingly relevant to call center operations and associated academic research. In this paper, we provide a survey of the recent literature on call center operations management. Along with traditional research areas, we pay special attention to new management challenges that have been caused by emerging technologies, to behavioral issues associated with both call center agents and customers, and to the interface between call center operations and sales and marketing. We identify a handful of broad themes for future investigation while also pointing out several very specific research opportunities.

723 citations


"A joint chance-constrained programm..." refers background or methods in this paper

  • ...by Aksin et al. (2007), at the time when decision on shift schedules is made, call arrival rates are most often not deterministically known....

    [...]

  • ...We refer the reader to Aksin et al. (2007) and Gans et al. (2003) for a general introduction to this field and focus in what follows on the recently emerged research stream on stochastic call center shift scheduling. We distinguish three main features to classify the related papers: the call center setting, the representation of the uncertainty and the risk management measures. In terms of call center architecture, the simplest case consists in a setting where a single pool of homogeneous agents handles a single class of infinitely patient calls. This amounts to using an Erlang C model to represent the call center in each period of the scheduling horizon (see Liao, Koole, van Delft, & Jouini, 2012; Liao, van Delft, & Vial, 2012). However, the importance of modeling customer impatience and abandonment in call centers has been underlined in several papers such as Gans et al. (2003) and Mandelbaum and Zeltyn (2009b)....

    [...]

  • ...We refer the reader to Aksin et al. (2007) and Gans et al. (2003) for a general introduction to this field and focus in what follows on the recently emerged research stream on stochastic call center shift scheduling. We distinguish three main features to classify the related papers: the call center setting, the representation of the uncertainty and the risk management measures. In terms of call center architecture, the simplest case consists in a setting where a single pool of homogeneous agents handles a single class of infinitely patient calls. This amounts to using an Erlang C model to represent the call center in each period of the scheduling horizon (see Liao, Koole, van Delft, & Jouini, 2012; Liao, van Delft, & Vial, 2012). However, the importance of modeling customer impatience and abandonment in call centers has been underlined in several papers such as Gans et al. (2003) and Mandelbaum and Zeltyn (2009b). Thus, similarly to Gans et al....

    [...]

  • ...We refer the reader to Aksin et al. (2007) and Gans et al. (2003) for a general introduction to this field and focus in what follows on the recently emerged research stream on stochastic call center shift scheduling. We distinguish three main features to classify the related papers: the call center setting, the representation of the uncertainty and the risk management measures. In terms of call center architecture, the simplest case consists in a setting where a single pool of homogeneous agents handles a single class of infinitely patient calls. This amounts to using an Erlang C model to represent the call center in each period of the scheduling horizon (see Liao, Koole, van Delft, & Jouini, 2012; Liao, van Delft, & Vial, 2012). However, the importance of modeling customer impatience and abandonment in call centers has been underlined in several papers such as Gans et al. (2003) and Mandelbaum and Zeltyn (2009b). Thus, similarly to Gans et al. (in press) and Robbins and Harrison (2010), we use in the present paper a representation of the call center as an Erlang A model. For both the Erlang C and the Erlang A models, the performance evaluation of the call center can be done by exploiting analytical results available in the queuing theory literature. A more complicated setting corresponds to skill-based routing call centers. In this case, the performance evaluation of the call center has to be made by relying either on simulation or on approximations under various asymptotic regimes. Stochastic shift scheduling for skillbased routing call centers has been studied by Bodur and Luedtke (2014), Gurvich, Luedtke, and Tezcan (2010), Helber and Henken (2010) and Ye et al....

    [...]

  • ...We refer the reader to Aksin et al. (2007) and Gans et al....

    [...]