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

TL;DR: In this article, a mixed-integer linear programming based solution approach is proposed to solve the shift scheduling problem under uncertain demand forecasts, where forecasting errors are seen as independent normally distributed random variables.

Abstract: We 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

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.

Figures

Table 2: Illustrative example: optimal solutions of problems EDetB and EDetF

Table 6: Resolution of problem EDetF - instances with σt λt = 1.5

Table 7: Resolution of problem EDetF - instances with σt λt = 2

Table 8: Comparison with the sample approximation approach - instances with σt λt = 2

Figure 2: General risk-cost trade-off analysis

Table 1: Illustrative example: input data

Figure 1: Illustrative example: function Ψ1

Table 4: Resolution of problem EDetF - instances with σt λt = 0.5

Table 3: Resolution of problem EDetB - instances with σt λt ∈ {0.5; 1; 1.5; 2}

Table 5: Resolution of problem EDetF - instances with σt λt = 1

Figure 3: Impact of forecast quality on the risk-cost trade-off