Coping with Time‐Varying Demand When Setting Staffing Requirements for a Service System

We review queueing‐theory methods for setting staffing requirements in service systems where customer demand varies in a predictable pattern over the day. Analyzing these systems is not straightforward, because standard queueing theory focuses on the long‐run steady‐state behavior of stationary models. We show how to adapt stationary queueing models for use in nonstationary environments so that time‐dependent performance is captured and staffing requirements can be set. Relatively little modification of straightforward stationary analysis applies in systems where service times are short and the targeted quality of service is high. When service times are moderate and the targeted quality of service is still high, time‐lag refinements can improve traditional stationary independent period‐by‐period and peak‐hour approximations. Time‐varying infinite‐server models help develop refinements, because closed‐form expressions exist for their time‐dependent behavior. More difficult cases with very long service times and other complicated features, such as end‐of‐day effects, can often be treated by a modified‐offered‐load approximation, which is based on an associated infinite‐server model. Numerical algorithms and deterministic fluid models are useful when the system is overloaded for an extensive period of time. Our discussion focuses on telephone call centers, but applications to police patrol, banking, and hospital emergency rooms are also mentioned.     

A Queueing Theoretic Approach to Set Staffing Levels in Time‐Dependent Dual‐Class Service Systems*

This article addresses the optimal staffing problem for a nonpreemptive priority queue with two customer classes and a time‐dependent arrival rate. The problem is related to several important service settings such as call centers and emergency departments where the customers are grouped into two classes of “high priority” and “low priority,” and the services are typically evaluated according to the proportion of customers who are responded to within targeted response times. To date, only approximation methods have been explored to generate staffing requirements for time‐dependent dual‐class services, but we propose a tractable numerical approach to evaluate system behavior and generate safe minimum staffing levels using mixed discrete‐continuous time Markov chains (MDCTMCs). Our approach is delicate in that it accounts for the behavior of the system under a number of different rules that may be imposed on staff if they are busy when due to leave and involves explicitly calculating delay distributions for two customer classes. Ultimately, we embed our methodology in a proposed extension of the Euler method, coined Euler Pri, that can cope with two customer classes, and use it to recommend staffing levels for the Welsh Ambulance Service Trust (WAST).     

Value-Based Routing and Preference-Based Routing in Customer Contact Centers

Telephone call centers and their generalizations—customer contact centers—usually handle several types of customer service requests (calls). Since customer service representatives (agents) have different call-handling abilities and are typically cross-trained in multiple skills, contact centers exploit skill-based routing (SBR) to assign calls to appropriate agents, aiming to respond properly as well as promptly. Established agent-staffing and SBR algorithms ensure that agents have the required call-handling skills and that call routing is performed so that constraints are met for standard congestion measures, such as the percentage of calls of each type that abandon before starting service and the percentage of answered calls of each type that are delayed more than a specified number of seconds. We propose going beyond traditional congestion measures to focus on the expected value derived from having particular agents handle various calls. Expected value might represent expected revenue or the likelihood of first-call resolution. Value might also reflect agent call-handling preferences. We show how value-based routing (VBR) and preference-based routing (PBR) can be introduced in the context of an existing SBR framework, based on static-priority routing using a highly-structured priority matrix, so that constraints are still met for standard congestion measures. Since an existing SBR framework is used to implement VBR and PBR, it is not necessary to replace the automatic call distributor (ACD). We show how mathematical programming can be used, with established staffing requirements, to find a desirable priority matrix. We select the priority matrix to use during a specified time interval (e.g., 30-minute period) by maximizing the total expected value over that time interval, subject to staffing constraints.     

Emergency Department Staff Planning to Improve Patient Care and Reduce Costs

In the face of high staffing costs, uncertain patient arrivals, and patients unsatisfied with long wait times, staffing of medical emergency departments (EDs) is a vexing problem. Using empirical data collected from three active EDs, we develop an analytic model to provide an effective staffing plan for EDs. Patient demand is aggregated into discrete time buckets and used to model the stochastic distribution of patient demand within these buckets, which considerably improves model tractability. This model is capable of scheduling providers with different skill profiles who work either individually or in teams, and with patients of varying acuity levels. We show how our model helps to balance staffing costs and patient service levels, and how it facilitates examination of important ED staffing policies.     

Staffing Call Centers with Uncertain Arrival Rates and Co‐sourcing

In a call center, staffing decisions must be made before the call arrival rate is known with certainty. Once the arrival rate becomes known, the call center may be over‐staffed, in which case staff are being paid to be idle, or under‐staffed, in which case many callers hang‐up in the face of long wait times. Firms that have chosen to keep their call center operations in‐house can mitigate this problem by co‐sourcing; that is, by sometimes outsourcing calls. Then, the required staffing N depends on how the firm chooses which calls to outsource in real time, after the arrival rate realizes and the call center operates as a M/M/N + M queue with an outsourcing option. Our objective is to find a joint policy for staffing and call outsourcing that minimizes the long‐run average cost of this two‐stage stochastic program when there is a linear staffing cost per unit time and linear costs associated with abandonments and outsourcing. We propose a policy that uses a square‐root safety staffing rule, and outsources calls in accordance with a threshold rule that characterizes when the system is “too crowded.” Analytically, we establish that our proposed policy is asymptotically optimal, as the mean arrival rate becomes large, when the level of uncertainty in the arrival rate is of the same order as the inherent system fluctuations in the number of waiting customers for a known arrival rate. Through an extensive numerical study, we establish that our policy is extremely robust. In particular, our policy performs remarkably well over a wide range of parameters, and far beyond where it is proved to be asymptotically optimal.     

Optimizing the Staffing and Routing of Small‐Size Hierarchical Call Centers

Multiple‐skill call centers propagate rapidly with the development of telecommunications. An abundance of literature has already been published on call centers. Here, we want to focus on centers that would typically occur in business‐to‐business environments; these are call centers that handle many types of calls but where the arrival rate for each type is low. To find an optimal configuration, the integrality of the decision variables is a much more important issue than for larger call centers. The present paper proposes an approach that uses elements of combinatorial optimization to find optimal configurations. We develop an approximation method for the evaluation of the service performance. Next, we search for the minimum‐cost configuration subject to service‐level constraints using a branch‐and‐bound algorithm. What is at stake is to find the right balance between gains resulting from the economies of scale of pooling and the higher cost or cross‐trained agents. The article shows that in most cases this method significantly decreases the staffing cost compared with configurations with only cross‐trained or dedicated operators.     

Schedule Recovery: Unplanned Absences in Service Operations*

The U.S. service sector loses 2.3% of all scheduled labor hours to unplanned absences, but in some industries, the total cost of unplanned absences approaches 20% of payroll expense. The principal reasons for unscheduled absences (personal illness and family issues) are unlikely to abate anytime soon. Despite this, most labor scheduling systems continue to assume perfect attendance. This oversight masks an important but rarely addressed issue in services management: how to recover from short‐notice, short‐term reductions in planned capacity. In this article, we model optimal responses to unplanned employee absences in multi‐server queueing systems that provide discrete, pay‐per‐use services for impatient customers. Our goal is to assess the performance of alternate absence recovery strategies under various staffing and scheduling regimes. We accomplish this by first developing optimal labor schedules for hypothetical service environments with unreliable workers. We then simulate unplanned employee absences, apply an absence recovery model, and compute system profits. Our absence recovery model utilizes recovery strategies such as holdover overtime, call‐ins, and temporary workers. We find that holdover overtime is an effective absence recovery strategy provided sufficient reserve capacity (maximum allowable work hours minus scheduled hours) exists. Otherwise, less precise and more costly absence recovery methods such as call‐ins and temporary help service workers may be needed. We also find that choices for initial staffing and scheduling policies, such as planned overtime and absence anticipation, significantly influence the likelihood of successful absence recovery. To predict the effectiveness of absence recovery policies under alternate staffing/scheduling strategies and operating environments, we propose an index based on initial capacity reserves.     

中小呼叫中心月度排班优化模型与算法

国内中小呼叫中心制定坐席人员月度排班表时,通常考虑劳动法规合同约束和体现企业自身用工管理诉求。构建坐席人员月度排班优化问题的二次整数规划模型。鉴于问题模型难解性,依据调研企业需求和模型逻辑结构分析,把问题分解成三个子问题。通过构建整数规划模型和提出启发式算法来求出子问题解,从而生成排班问题优化解。问题实例计算表明,模型算法能够有效控制人力成本和兼顾员工同班次管理目标。与周排班方法比较,该方法能够充分体现月度排班人力灵活性来实现人力优化配置。     

AN IMPROVED HEURISTIC FOR STAFFING TELEPHONE CALL CENTERS WITH LIMITED OPERATING HOURS

Many telephone call centers that experience cyclic and random customer demand adjust their staffing over the day in an attempt to provide a consistent target level of customer service. The standard and widely used staffing method, which we call the stationary independent period by period (SIPP) approach, divides the workday into planning periods and uses a series of stationary independent Erlang‐c queuing models—one for each planning period—to estimate minimum staffing needs. Our research evaluates and improves upon this commonly used heuristic for those telephone call centers with limited hours of operation during the workday. We show that the SIPP approach often suggests staffing that is substantially too low to achieve the targeted customer service levels (probability of customer delay) during critical periods. The major reasons for SIPP‘ s shortfall are as follows: (1) SIPP's failure to account for the time lag between the peak in customer demand and when system congestion actually peaks; and (2) SIPP’ s use of the planning period average arrival rate, thereby assuming that the arrival rate is constant during the period. We identify specific domains for which SIPP tends to suggest inadequate staffing. Based on an analysis of the factors that influence the magnitude of the lag in infinite server systems that start empty and idle, we propose and test two simple “lagged” SIPP modifications that, in most situations, consistently achieve the service target with only modest increases in staffing.     

Joint chance-constrained staffing optimization in multi-skill call centers

This paper concerns the staffing optimization problem in multi-skill call centers. The objective is to find a minimal cost staffing solution while meeting a target level for the quality of service (QoS) to customers. We consider a staffing problem in which joint chance constraints are imposed on the QoS of the day. Our joint chance-constrained formulation is more rational capturing the correlation between different call types, as compared to separate chance-constrained versions considered in previous studies. We show that, in general, the probability functions in the joint-chance constraints display S-shaped curves, and the optimal solutions should belong to the concave regions of the curves. Thus, we propose an approach combining a heuristic phase to identify solutions lying in the concave part and a simulation-based cut generation phase to create outer-approximations of the probability functions. This allows us to find good staffing solutions satisfying the joint-chance constraints by simulation and linear programming. We test our formulation and algorithm using call center examples of up to 65 call types and 89 agent groups, which shows the benefits of our joint-chance constrained formulation and the advantage of our algorithm over standard ones.

     

Time‐Staged Overtime Staffing for Services with Updated Forecasts and Availabilities

This article develops a framework for staffing in a service environment when multiple opportunities exist for prescheduling overtime prior to the start of a shift. Demand forecasts improve as the shift approaches, while the availability of workers to be scheduled for overtime decreases. First, a single‐shift model is developed and used in computational studies to evaluate the benefits of time‐staged overtime staffing, which include slightly lower costs and significant reductions in unscheduled overtime and outside agents. A multishift model is then developed to consider constraints on consecutive hours worked and minimum rest intervals between shifts. A multishift computational study shows how the benefits of time‐staged overtime staffing depend on problem characteristics when interactions between shifts are considered. The article discusses how single‐shift and multishift models relate to each other and alternative ways the models may be used in practice, including decentralized open shift management and centralized overtime scheduling.     

Revenue Management Through Dynamic Cross Selling in Call Centers

This paper models the cross‐selling problem of a call center as a dynamic service rate control problem. The question of when and to whom to cross sell is explored using this model. The analysis shows that, under the optimal policies, cross‐selling targets may be a function of the operational system state. Sufficient conditions are established for the existence of preferred calls, i.e., calls that will always generate a cross‐sell attempt. These provide guidelines in segment formation for marketing managers, and lead to a static heuristic policy. Numerical analysis establishes the value of different types of information, and different types of automation available for cross selling. Increased staffing for the same call volume is shown to have a positive and increasing return on revenue generation via cross selling, suggesting the need to staff for lower loads in call centers that aim to be revenue generators. The proposed heuristic leads to near optimal performance in a wide range of settings.     

Equivalent Alternate Solutions for the Tour Scheduling Problem*

Achieving minimum staffing costs, maximum employee satisfaction with their assigned schedules, and acceptable levels of service are important but potentially conflicting objectives when scheduling service employees. Existing employee scheduling models, such as tour scheduling or general employee scheduling, address at most two of these criteria. This paper describes a heuristic to improve tour scheduling solutions provided by other procedures, and generate a set of equivalent cost feasible alternatives. These alternatives allow managers to identify solutions with attractive secondary characteristics, such as overall employee satisfaction with their assigned tours or consistent employee workloads and customer response times. Tests with both full-time and mixed work force problems reveal the method improves most nonoptimal initial heuristic solutions. Many of the alternatives generated had more even distributions of surplus staff than the initial solutions, yielding more consistent customer response times and employee workloads. The likelihood of satisfying employee scheduling preferences may also be increased since each alternative provides a different deployment of employees among the available schedules.     

Managing appointments with waiting time targets and random walk-ins

The random arrivals of walk-in patients significantly affect the daily operations of healthcare facilities. To improve the performance of outpatient departments, this paper attempts to make an appointment schedule by considering walk-ins and the waiting time target (WTT) for appointment patients. A stochastic programming model is proposed to solve this problem with the objective of minimizing the weighted patient waiting and makespan cost. A non-decreasing waiting cost function is used to capture the WTT fulfillment of appointment patients, whereas walk-ins incur a linear waiting cost. A finite-horizon Markov Decision Process model is formulated to establish the optimal real-time scheduling policy under a given appointment schedule. The appointment schedule is determined by a two-stage stochastic programming approximation and a local search improvement. Structural properties of the optimal appointment scheduling and real-time scheduling policies are established. In particular, it is shown that appointment overbooking is allowed only at the end of the regular session, and the optimal real-time scheduling policy is an easy-to-implement threshold policy with bounded sensitivity. Numerical experiments based on real data are performed to investigate the influence of different parameters and to compare different schedules. The optimal schedule demonstrates superior performance by allowing reasonable waiting times for appointment patients depending on their WTTs. Managerial insights are also provided to hospital managers. Finally, the basic model is extended by incorporating random service times and random arrivals of appointment patients. The latter includes the random number of patients that show up for service or call for appointments, and the random arrival time (unpunctuality). Appointment overbooking strategies are shown to have different structures under some stochastic factors.     

A non-queueing model to predict teller requirements in retail bank branches

Two models are developed in this paper to determine optimum teller scheduling at bank branches. The first is a parametric model consisting of three phases: predicting the aggregate number of transactions for the bank based on the factors influencing customer demand and arrival rates; determining the percent of the aggregate transactions volume performed at each retail branch based on customer profiles and branch locations; formulating a transaction sensitive teller staffing model for each retail branch based upon predetermined customer service time and personnel utilization criteria. The second model is an integer programming model to determine the optimum teller staffing requirements based on the results of the first model and constrained by traditional current banking practices.     

Staffing and scheduling flexible call centers by two-stage robust optimization

We study the shift scheduling problem in a multi-shift, flexible call center. Differently from previous approaches, the staffing levels ensuring the desired quality of service are considered uncertain, leading to a two-stage robust integer program with right-hand-side uncertainty. We show that, in our setting, modeling the correlation of the demands in consecutive time slots is easier than in other staffing approaches. The complexity issues of a Benders type reformulation are investigated and a branch-and-cut algorithm is devised. The approach can efficiently solve real-world problems from an Italian call center and effectively support managers decisions. In fact, we show that robust shifts have very similar costs to those evaluated by the traditional (deterministic) method while ensuring a higher level of protection against uncertainty.     

Peakedness‐Based Staffing for Call Center Outsourcing

This study considers the staffing problem of a vendor call center in a co‐sourcing setting. The aim is to take short‐term variability and correlations in time for call arrivals at such a vendor call center into account. To do so, peakedness is proposed as a useful measure of the burstiness in the arrival stream. The study empirically demonstrates the presence of bursty arrivals at a call center and proposes an approach to the measurement of the peakedness of the arrival stream making use of standard call center data. The problematic nature of bursty arrivals in the context of call center co‐sourcing is demonstrated along with an asymptotic result establishing that the problem persists in large call centers. The study then analyzes two peakedness‐based staffing methods: one which is a well known extension of the square root staffing rule and another which makes use of the Hayward approximation principles. Both approaches are simple and enable the vendor to improve its staffing procedure with good accuracy.     

Optimal Configuration of a Service Delivery Network: An Application to a Financial Services Provider

Driven by market pressures, financial service firms are increasingly partnering with independent vendors to create service networks that deliver greater profits while ensuring high service quality. In the management of call center networks, these partnerships are common and form an integral part of the customer care and marketing strategies in the financial services industry. For a financial services firm, configuring such a call center service network entails determining which partners to select and how to distribute service requests among vendors, while incorporating their capabilities, costs, and revenue‐generating abilities. Motivated by a problem facing a Fortune 500 financial services provider, we develop and apply a novel mixed integer programming model for the service network configuration problem. Our tactical decision support model effectively accounts for the firm's costs by capturing the impact of service requirements on vendor staffing levels and seat requirements, and permits imposing call routing preferences and auxiliary service costs. We implemented the model and applied it to data from an industry partner. Results suggest that our approach can generate considerable cost savings and substantial additional revenues, while ensuring high service quality. Results based on test instances demonstrate similar savings and outperform two rule‐based methods for vendor assignment.     

Optimal Sequencing of Unpunctual Patients in High‐Service‐Level Clinics

Even though patients often arrive early and out of turn for scheduled appointments in outpatient clinics, no research has been undertaken to establish whether an available provider should see an early patient right away (preempt) or wait for the patient scheduled next. This problem, which we call the “Wait‐Preempt Dilemma,” is particularly relevant for “high‐service‐level” clinics (such as psychotherapy, chiropractic, acupuncture), where preempting may cause the missing patient to wait for an excessively long time, should she show up soon. Typically, the dilemma is resolved by preemption, where the provider starts serving the patient who has already arrived to avoid staying idle. By contrast, we analytically determine the time intervals where it is optimal to preempt and those where it is optimal to wait, and find that in some cases the provider should in fact stay idle, even in the presence of waiting patients. Our results suggest that the proposed analytical method outperforms the always‐preempt policy in clinics that do not overbook and have service times longer than 30 minutes. In these cases, the analytical method dramatically reduces patient waiting times at the cost of a modest increase in overtime. By contrast, in clinics that overbook or have short service times, the two policies perform similarly, and hence the always‐preempt policy is preferable due to its simplicity. A software application is provided that clinics can readily use to solve the wait‐preempt dilemma.     

Scheduling of Multi‐skilled Staff Across Multiple Locations

We address the problem of assigning airline customer service agents (CSAs) to tasks related to departing flights, such as selling tickets and collecting boarding cards, at an international terminal of a large airport. The airline specifies minimum and target levels of staff and required (or desired) types and levels of skills for each location in each time period. The assignment problem is complicated by staff heterogeneity, time required for moves between locations, and lunch and rest‐break requirements. We present a mixed‐integer formulation that considers both staffing shortages and skills mismatches and show that the problem is NP‐hard. We derive valid inequalities that tighten the bounds within a branch‐and‐cut procedure, enabling us to obtain near‐optimal solutions for problems of realistic size very quickly. We also present a generalization to simultaneously optimize shift starting times and task assignments, which can aid in longer term workforce planning. Finally, we utilize our procedure to obtain managerial insights regarding the benefits of flexibility derived from more highly skilled staff, allowing more frequent moves, and choices of shift starting times. We also demonstrate the benefits of our procedure vs. a heuristic that mimics what an experienced scheduler might choose.