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.     

Practical scheduling for call center operations

A practical spreadsheet-based scheduling method is developed to determine the optimal allocation of service agents to candidate tour types and start times in an inbound call center. A stationary Markovian queueing model with customer abandonment is employed to determine required staffing levels for a sequence of time intervals with varying call volumes, handling times, and relative agent availabilities. These staffing requirements populate a quadratic programming model for determining the distribution of agent tours that will maximize the fraction of offered calls beginning service within a target response time, subject to side constraints on tour type quantities. The optimal distribution is scaled to reflect the total number of scheduled agents, and a near-optimal integer solution is derived using rounding thresholds found by successive one-dimensional searches. This novel approach has been successfully implemented in large service centers at Qwest Communications and could easily be adapted to other operational environments.     

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.     

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).     

The Modern Call Center: A Multi‐Disciplinary Perspective on Operations Management Research

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.     

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.

     

Staff rostering in call centers providing employee transportation

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

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.     

动态需求下车辆路径问题的周期性优化模型及求解

针对客户点不断更新的动态需求车辆路径问题,依据滚动时域对配送中心工作时间进行划分,提出基于延迟服务的周期性客户点实时重置策略,策略中延迟服务机制能结合车辆启动延迟系数对照当前时域的时间进行检验,满足所有客户点的服务需求,保证车辆满足中心时间窗约束。设计多阶段求解的混合变邻域人工蜂群算法对各时间片内子问题进行连续迭代优化,算法中子路径动态转变的设计能较好平衡原有客户点和新客户点对路径更新和车辆实时信息匹配的要求。算例验证及对比分析表明本文策略和算法在求解动态问题时的有效性和可行性。     

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.     

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.     

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.     

求解带时间窗的客户需求可分条件下的车辆路径问题

物流运输中的车辆路径问题历来是一个重要的理论和实际问题,在同时考虑客户需求可分以及客户方和配送中心时间窗限制的前提下,重新构造了问题模型,并结合蚂蚁算法中转移概率的改进和最大-最小蚂蚁系统,设计了问题求解过程和分割点选取规则,计算结果显示出算法的可行性。另外还与客户需求不可分的情况进行了对比,从而说明在大规模物流运输需求下,可分能带来更好的效果。     

Analysis of Two‐Level Support Systems with Time‐Dependent Overflow—A Banking Application

In this paper, we analyze the performance of call centers of financial service providers with two levels of support and a time‐dependent overflow mechanism. Waiting calls from the front‐office queue flow over to the back office if a waiting‐time limit is reached and at least one back‐office agent is available. The analysis of such a system with time‐dependent overflow is reduced to the analysis of a continuous‐time Markov chain with state‐dependent overflow probabilities. To approximate the system with time‐dependent overflow, some waiting‐based performance measures are modified. Numerical results demonstrate the reliability of this Markovian performance approximation for different parameter settings. A sensitivity analysis shows the impact of the waiting‐time limit and the dependence of the performance measures on the arrival rate.     

基于模糊时间窗的车辆调度问题研究

基于现实生活中配送企业车辆资源有限和顾客对服务时间要求并非完全刚性的特征,通过时间窗模糊化处理将顾客服务的满意度量化为配送服务开始时间的模糊隶属度函数。在一定满意度下,构建了基于模糊时间窗的车辆调度模型,根据模型的特点,改进了基于客户的染色体编码方式,设定了一种新的约束处理方法,避免了惩罚策略中选取惩罚因子的困难。在算法中用模糊优化程序处理问题的模糊特征,通过对顾客服务时间的局部调整来确定最佳服务时间。最终通过实例验证与原结果比较发现,引用模糊时间窗函数不仅可以降低配送成本,而且有利于节省运力资源。     

A Model for Planning Resource Requirements in Health Care Organizations

In this paper we present a general model and solution methodology for planning resource requirements (i.e., capacity) in health care organizations. To illustrate the general model, we consider two specific applications: a blood bank and a health maintenance organization (HMO). The blood bank capacity planning problem involves determining the number of donor beds required and determining the size of the nursing and support staff necessary. Capacity must be sufficient to handle the expected number of blood donors without causing excessive donor waiting times. Similar staff, equipment, and service level decisions arise in the HMO capacity planning problem. To determine resource requirements, we develop an optimization/queueing network model that minimizes capacity costs while controlling customer service by enforcing a set of performance constraints, such as setting an upper limit on the expected time a patient spends in the system. The queueing network model allows us to capture the stochastic behavior of health care systems and to measure customer service levels within the optimization framework.     

Social Optimal Location of Facilities with Fixed Servers,Stochastic Demand,and Congestion

We consider two capacity choice scenarios for the optimal location of facilities with fixed servers, stochastic demand, and congestion. Motivating applications include virtual call centers, consisting of geographically dispersed centers, walk‐in health clinics, motor vehicle inspection stations, automobile emissions testing stations, and internal service systems. The choice of locations for such facilities influences both the travel cost and waiting times of users. In contrast to most previous research, we explicitly embed both customer travel/connection and delay costs in the objective function and solve the location–allocation problem and choose facility capacities simultaneously. The choice of capacity for a facility that is viewed as a queueing system with Poisson arrivals and exponential service times could mean choosing a service rate for the servers (Scenario 1) or choosing the number of servers (Scenario 2). We express the optimal service rate in closed form in Scenario 1 and the (asymptotically) optimal number of servers in closed form in Scenario 2. This allows us to eliminate both the number of servers and the service rates from the optimization problems, leading to tractable mixed‐integer nonlinear programs. Our computational results show that both problems can be solved efficiently using a Lagrangian relaxation optimization procedure.     

物流配送车辆路径规划模型的知识表示研究

针对物流配送车辆路径规划的实时动态建模问题,以解决模型的目标函数和约束等符号化知识的知识表示及基于知识的求解机制为突破口,提出了以七元组M=(B,O,C,I,P,E,D)表示车辆路径规划模型的知识表示方法--BOCIPED表示法;并以沈阳昌达集团餐饮配送公司为应用背景,设计建立了相应的车辆路径规划问题的建模与求解系统,通过系统的实际应用,验证了系统中BOCIPED表示方法的可行性与有效性.本研究为车辆路径规划这一难题提供由计算机自动生成模型并求解的新方法,有利于建立高智能的物流配送实时调度系统.     

“How May I Help You?” Improving Telephone Customer Service in a Medical Clinic Setting

The current study was carried out per management request to improve the overall quality of telephone customer service among appointment coordinators in a medical clinic. Exceptional telephone customer service included (a) using a standard greeting, (b) speaking in the appropriate tone of voice throughout the call, and (c) answering every call received by the unit. A preintervention analysis suggested that performance deficiencies resulted from weak antecedents, poor knowledge and skills, and weak performance contingencies. Task clarification, goal setting, feedback, and performance-contingent consequences were combined to improve these customer service behaviors for 20 full-time appointment coordinators at the clinic. The study used an ABÁ reversal design with weekly maintenance and 5-month follow-up observations. Introduction of the multicomponent intervention produced visible improvements in greeting (38% increase) and friendly voice tone (22% increase) behaviors; performance was maintained above baseline levels at 5 months postmaintenance. Abandon rates (the percentage of calls not answered by a live voice) remained fairly stable, on average. Findings support the use of a multicomponent intervention to increase telephone customer service behavior in medical clinic settings.     

A New Approximation Algorithm for the Capacitated Vehicle Routing Problem on a Tree

This paper presents a new approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot. Customers are located on vertices of the tree, and each customer has a positive demand. Demands of customers are served by a fleet of identical vehicles with limited capacity. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. Each tour begins at the depot, visits a subset of the customers and returns to the depot without violating the capacity constraint. We propose a 1.35078-approximation algorithm for the problem (exactly, ), which is an improvement over the existing 1.5-approximation.