A Stochastic Goal Program for Employee Scheduling*

Deterministic goal programs for employee scheduling decisions attempt to minimize expected operating costs by assigning the ideal number of employees to each feasible schedule. For each period in the planning horizon, managers must first determine the amount of labor that should be scheduled for duty. These requirements are often established with marginal analysis techniques, which use estimates for incremental labor costs and shortage expenses. Typically, each period in the planning horizon is evaluated as an independent epoch. An implicit assumption is that individual employees can be assigned to schedules with as little as a single period of work. If this assumption violates local work rules, the labor requirements parameters for the deterministic goal program may be suboptimal. As we show in this research, this well-known limitation can lead to costly staffing and scheduling errors. We propose an employee scheduling model that overcomes this limitation by integrating the labor requirements and scheduling decisions. Instead of a single, externally determined staffing goal for each period, the model uses a probability distribution for the quantity of labor required. The model is free to choose an appropriate staffing level for each period, eliminating the need for a separate goal-setting procedure. In most cases this results in better, less costly decisions. In addition, the proposed model easily accommodates both linear and nonlinear under- and overstaffing penalties. We use simple examples to demonstrate many of these advantages and to illustrate the key techniques necessary to implement our model. We also assess its performance in a study of more than 1,700 simulated stochastic employee scheduling problems.     

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.     

On‐Call Overtime for Service Workforce Scheduling when Demand Is Uncertain

This article builds on prior research to develop shift scheduling models that include on‐call overtime for service environments where demand is uncertain. The research is motivated by recent developments in nurse scheduling, such as laws prohibiting mandatory overtime and the popularity of self‐scheduling systems. For single‐period scenarios, models are developed, solution methods are described, and results are explored for a variety of environments. Results show that the use of on‐call overtime can reduce costs slightly, with the amount of savings dependent on characteristics of the scheduling environment. The factor that most significantly affects cost savings is the cost of outside agency workers relative to overtime workers. In addition to lowering costs, on‐call overtime greatly reduces reliance on outside agency workers, which can have important practical implications in terms of quality of service and workforce morale. Results based on single‐period models motivate multiperiod formulations for single‐ and multidepartment scenarios, and solution methods are outlined for those cases. The possibility of using multiperiod models within a rolling horizon framework with forecast updating is discussed. This goes along with an extension of the traditional workforce management hierarchy that separates overtime and regular‐time scheduling, as seen in practice with self‐scheduling and shift‐bidding systems.     

Service Completion Estimates for Cross‐trained Workforce Schedules under Uncertain Attendance and Demand

Although cross‐trained workers offer numerous operational advantages for extended‐hour service businesses, they must first be scheduled for duty. The outcome from those decisions, usually made a week or more in advance, varies with realized service demand, worker attendance, and the way available cross‐trained workers are deployed once the demands for service are known. By ignoring the joint variability of attendance and demand, we show that existing workforce scheduling models tend to overstate expected schedule performance and systematically undervalue the benefits of cross‐training. We propose a two‐stage stochastic program for profit‐oriented cross‐trained workforce scheduling and allocation decisions that is driven by service completion estimates obtained from the convolution of the employee attendance and service demand distributions. Those estimates, reflecting optimal worker allocation decisions over all plausible realizations of attendance and demand, provide the gradient information used to guide workforce scheduling decisions. Comparing the performance of workforce scheduling decisions for hundreds of different hypothetical service environments, we find that solutions based on convolution estimates are more profitable, favor proportionately more cross‐trained workers and fewer specialists, and tend to recommend significantly larger (smaller) staffing levels for services under high (low) contribution margins than workforce schedules developed with independent expectations of attendance and demand.     

Tour Scheduling and Task Assignment of a Heterogeneous Work Force: A Heuristic Approach

The dual problem of work tour scheduling and task assignment involving workers who differ in their times of availability and task qualifications is examined in this paper. The problem is presented in the context of a fast food restaurant, but applies equally well to a diverse set of service operations. Developing a week-long labor schedule is a nontrivial problem, in terms of complexity and importance, which a manager spends as much as a full workday solving. The primary scheduling objective (the manager's concern) is the minimization of overstaffing in the face of significant hourly and daily fluctuations in minimum staffing requirements. The secondary objective (the workers’ concern) is the minimization of the sum of the squared differences between the number of work hours scheduled and the number targeted for each employee. Contributing to scheduling complexity are constraints on the structure of work tours, including minimum and maximum shift lengths and a maximum number of workdays. A goal programming formulation of a representative problem is shown to be too large, for all practical purposes, to be solved optimally. Existing heuristic procedures related to this research possess inherent limitations which render them inadequate for our purposes. Subsequently, we propose and demonstrate a computerized heuristic procedure capable of producing a labor schedule requiring at most minor refinement by a manager.     

A METHODOLOGY FOR LABOR SCHEDULING IN A SERVICE OPERATING SYSTEM

In the increasingly competitive services sector, utilization of the labor force can make the difference between profits or losses. Until recently, service operations managers had a limited set of tools, most of them computer-based, for scheduling labor. This paper offers a manual heuristic for labor scheduling that outperforms traditional algorithmic solution approaches. Specifically, this study examines the problem of scheduling employees in service delivery system subject to demand variability. The manual heuristic proposed asigns full-time empolyees to weekly work schedules with the objective of minimizing the total number of labor hours scheduled. The performance of the manual heuristic is compared to the classical algorithmic solution and to a lower bound for a variety of demand distributions and system operating conditions. The heuristic is shown to produce a smaller work force than the classical approach in 106 of the 108 demand-operating condition patterns examined.     

Investigating Unpaid Overtime Working among the Part‐time Workforce

This paper considers whether unpaid overtime working relates to contracted working hours (i.e. whether an employee works part‐time or full‐time) in Britain. It uses the authoritative 2004 British data set Workplace Employment Relations Survey to derive a sample of 4,530 workers, from 735 workplace establishments, who worked unpaid overtime. It tests hypotheses linking contracted working hours to unpaid overtime, and whether this link is moderated by gender, occupational group and the availability of flexible working arrangements. Part‐time workers were found to work significantly more unpaid overtime hours compared with their full‐time counterparts. Gender, occupation and flexible working practices moderated this relationship, where the extent to which part‐timers work more unpaid overtime than their full‐time counterparts was greater for men than for women, was greater for professional/managerial part‐time workers compared with other occupations, and was more evident in establishments less likely to offer flexible working arrangements. The findings raise concerns about the exploitation of part‐time workers.     

Staffing a Multiskilled Workforce with Varying Levels of Productivity: An Analysis of Cross-training Policies*

Service operations that utilize cross-trained employees face complex workforce staffing decisions that have important implications for both cost and productivity. These decisions are further complicated when cross-trained employees have different productivity levels in multiple work activity categories. A method for policy analysis in such environments can be beneficial in determining low-cost staffing plans with appropriate cross-training configurations. In this paper, we present an integer linear programming model for evaluating cross-training configurations at the policy level. The objective of the model is to minimize workforce staffing costs subject to the satisfaction of minimum labor requirements across a planning horizon of a single work shift. The model was used to evaluate eight cross-training structures (consisting of 36 unique cross-training configurations) across 512 labor requirement patterns. These structures, as well as the labor requirement patterns, were established based on data collected from maintenance operations at a large paper mill in the United States. The results indicate that asymmetric cross-training structures that permit chaining of employee skill classes across work activity categories are particularly useful.     

Reexamining the Nurse Scheduling Problem: Staffing Ratios and Nursing Shortages*

Legislators at the state and national levels are addressing renewed concerns over the adequacy of hospital nurse staffing to provide quality care and ensure patient safety. At the same time, the well‐known nursing shortage remains an ongoing problem. To address these issues, we reexamine the nurse scheduling problem and consider how recent health care legislation impacts nursing workforce management decisions. Specifically, we develop a scheduling model and perform computational experiments to evaluate how mandatory nurse‐to‐patient ratios and other policies impact schedule cost and schedule desirability (from the nurses' perspective). Our primary findings include the following: (i) nurse wage costs can be highly nonlinear with respect to changes in mandatory nurse‐to‐patient ratios of the type being considered by legislators; (ii) the number of undesirable shifts can be substantially reduced without incurring additional wage cost; (iii) more desirable scheduling policies, such as assigning fewer weekends to each nurse, have only a small impact on wage cost; and (iv) complex policy statements involving both single‐period and multiperiod service levels can sometimes be relaxed while still obtaining good schedules that satisfy the nurse‐to‐patient ratio requirements. The findings in this article suggest that new directions for future nurse scheduling models, as it is likely that nurse‐to‐patient ratios and nursing shortages will remain a challenge for health care organizations for some time.     

Home healthcare integrated staffing and scheduling

Workforce planning for home healthcare represents an important and challenging task involving complex factors associated with labor regulations, caregivers’ preferences, and demand uncertainties. This task is done manually by most home care agencies, resulting in long planning times and suboptimal decisions that usually fail to meet the health needs of the population, to minimize operating costs, and to retain current caregivers. Motivated by these challenges, we present a two-stage stochastic programming model for employee staffing and scheduling in home healthcare. In this model, first-stage decisions correspond to the staffing and scheduling of caregivers in geographic districts. Second-stage decisions are related to the temporary reallocation of caregivers to neighboring districts, to contact caregivers to work on a day-off, and to allow under- and over-covering of demand. The proposed model is tested on real-world instances, where we evaluate the impact on costs, caregiver utilization, and service level by using different recourse actions. Results show that when compared with a deterministic model, the two-stage stochastic model leads to significant cost savings as staff dimensioning and scheduling decisions are more robust to accommodate changes in demand. Moreover, these results suggest that flexibility in terms of use of recourse actions is highly valuable as it helps to further improve costs, service level, and caregiver utilization.     

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.     

Determining overtime policies for a repair shop

Problems arise when a repair shop with a fixed capacity has to cope with fluctuations in the arrivals of failed assemblies. The effective use of priority scheduling rules and labor assignment rules will enable the repair shop to work efficiently but only to the extent of the repair capacity of the shop. Increasing the number of spares in stock to accommodate these short-term fluctuations would be costly in the long run. In order to maintain a certain level of service, the repair shop often needs to resort to overtime. This paper examines various overtime policies, namely, proactive and reactive overtime policies, under various failure distributions and budget constraints for spares. The findings show that reactive overtime policies generally perform better than proactive overtime policies under the conditions tested. Benefit-cost ratio curves are developed for these overtime policies which illustrate the stochastic interaction effects of the priority scheduling rule, spares budget, and failure arrival distribution.     

Reserving Capacity for Urgent Patients in Primary Care

This paper examines the effect of the common practice of reserving slots for urgent patients in a primary health care practice on two service quality measures: the average number of urgent patients that are not handled during normal hours (either handled as overtime, referred to other physicians, or referred to the emergency room) and the average queue of non‐urgent or routine patients. We formulate a stochastic model of appointment scheduling in a primary care practice. We conduct numerical experiments to optimize the performance of this system accounting for revenue and these two service quality measures as a function of the number of reserved slots for urgent patients. We compare traditional methods with the advanced‐access system advocated by some physicians, in which urgent slots are not reserved, and evaluate the conditions under which alternative appointment scheduling mechanisms are optimal. Finally, we demonstrate the importance of patient arrival dynamics to their relative performance finding that encouraging routine patients to call for same‐day appointments is a key ingredient for the success of advanced‐access.     

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.     

A MARKET UTILITY‐BASED MODEL FOR CAPACITY SCHEDULING IN MASS SERVICES

Only a small set of employee scheduling articles have considered an objective of profit or contribution maximization, as opposed to the traditional objective of cost (including opportunity costs) minimization. In this article, we present one such formulation that is a market utility‐based model for planning and scheduling in mass services (MUMS). MUMS is a holistic approach to market‐based service capacity scheduling. The MUMS framework provides the structure for modeling the consequences of aligning competitive priorities and service attributes with an element of the firm's service infrastructure. We developed a new linear programming formulation for the shift‐scheduling problem that uses market share information generated by customer preferences for service attributes. The shift‐scheduling formulation within the framework of MUMS provides a business‐level model that predicts the economic impact of the employee schedule. We illustrated the shift‐scheduling model with empirical data, and then compared its results with models using service standard and productivity standard approaches. The result of the empirical analysis provides further justification for the development of the market‐based approach. Last, we discuss implications of this methodology for future research.     

Intelligent Procedures for Intra‐Day Updating of Call Center Agent Schedules

For nearly all call centers, agent schedules are typically created several days or weeks before the time that agents report to work. After schedules are created, call center resource managers receive additional information that can affect forecasted workload and resource availability. In particular, there is significant evidence, both among practitioners and in the research literature, suggesting that actual call arrival volumes early in a scheduling period (typically an individual day or week) can provide valuable information about the call arrival pattern later in the same scheduling period. In this paper, we develop a flexible and powerful heuristic framework for managers to make intra‐day resource adjustment decisions that take into account updated call forecasts, updated agent requirements, existing agent schedules, agents' schedule flexibility, and associated incremental labor costs. We demonstrate the value of this methodology in managing the trade‐off between labor costs and service levels to best meet variable rates of demand for service, using data from an actual call center.     

An appointment scheduling policy for healthcare systems with parallel servers and pre-determined quality of service

The appointment scheduling problem is well-known in the literature. The use of appointment systems has been adopted widely in many different fields, including service industries and especially healthcare.This research focuses on healthcare systems where patients arrive according to pre-assigned appointments. We consider healthcare systems with several parallel servers, where a given sequence of patients, with randomly distributed service durations and a possibility of no-shows, is to be scheduled. The aim is to minimize the end of day and increase resource utilization while a minimal probability of each appointment starting on time (quality of service) is required.We formulated the problem using mathematical programing and developed a multi-server numerical-based (MSN) algorithm to solve it. We conducted some experimental runs and checked the impact of the problem parameters on the end of day, customers’ average waiting time and the percentage of customers that waited for service. We also show how server pooling improves the above system measures. Finally, once the appointments are set, we develop a methodology to determine the shift length so as to balance overtime costs (costs of overtime hours) against undertime costs (costs of regular, unused hours).     

Appointment Overbooking in Health Care Clinics to Improve Patient Service and Clinic Performance

The problem of no‐shows (patients who do not arrive for scheduled appointments) is particularly significant for health care clinics, with reported no‐show rates varying widely from 3% to 80%. No‐shows reduce revenues and provider productivity, increase costs, and limit patient access by reducing effective clinic capacity. In this article, we construct a flexible appointment scheduling model to mitigate the detrimental effects of patient no‐shows, and develop a fast and effective solution procedure that constructs near‐optimal overbooked appointment schedules that balance the benefits of serving additional patients with the potential costs of patient waiting and clinic overtime. Computational results demonstrate the efficacy of our model and solution procedure, and connect our work to prior research in health care appointment scheduling.     

Strategies for Addressing the Nursing Shortage: Coordinated Decision Making and Workforce Flexibility

In this article, we present strategies to help combat the U.S. nursing shortage. Key considerations include providing an attractive work schedule and work environment—critical issues for retaining existing nurses and attracting new nurses to the profession—while at the same time using the set of available nurses as effectively as possible. Based on these ideas, we develop a model that takes advantage of coordinated decision making when managing a flexible workforce. The model coordinates scheduling, schedule adjustment, and agency nurse decisions across various nurse labor pools, each of differing flexibility levels, capabilities, and costs, allowing a much more desirable schedule to be constructed. Our primary findings regarding coordinated decision making and how it can be used to help address the nursing shortage include (i) labor costs can be reduced substantially because, without coordination, labor costs on average are 16.3% higher based on an actual hospital setting, leading to the availability of additional funds for retaining and attracting nurses, (ii) simultaneous to this reduction in costs, more attractive schedules can be provided to the nurses in terms of less overtime and fewer undesirable shifts, and (iii) the use of agency nurses can help avoid overtime for permanent staff with only a 0.7% increase in staffing costs. In addition, we estimate the cost of the shortage for a typical U.S. hospital from a labor cost perspective and show how that cost can be reduced when managers coordinate.     

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.