Shane G. Henderson

Call Center Staffing with Simulation and Cutting Plane Methods

We present an iterative cutting plane method for minimizing staffing costs in a service system subject to satisfying acceptable service level requirements over multiple time periods. We assume that the service level cannot be easily computed, and instead is evaluated using simulation. The simulation uses the method of common random numbers, so that the same sequence of random phenomena is observed when evaluating different staffing plans. In other words, we solve a sample average approximation problem. We establish convergence of the cutting plane method on a given sample average approximation. We also establish both convergence, and the rate of convergence, of the solutions to the sample average approximation to solutions of the original problem as the sample size increases. The cutting plane method relies on the service level functions being concave in the number of servers. We show how to verify this requirement as our algorithm proceeds. A numerical example showcases the properties of our method, and sheds light on when the concavity requirement can be expected to hold.

Approximate Dynamic Programming for Ambulance Redeployment

We present an approximate dynamic programming approach for making ambulance redeployment decisions in an emergency medical service system. The primary decision is where we should redeploy idle ambulances so as to maximize the number of calls reached within a delay threshold. We begin by formulating this problem as a dynamic program. To deal with the high-dimensional and uncountable state space in the dynamic program, we construct approximations to the value function that are parameterized by a small number of parameters. We tune the parameters using simulated cost trajectories of the system. Computational experiments demonstrate the performance of the approach on emergency medical service systems in two metropolitan areas. We report practically significant improvements in performance relative to benchmark static policies.

Optimizing Call Center Staffing Using Simulation and Analytic Center Cutting-Plane Methods

We consider the problem of minimizing staffing costs in an inbound call center, while maintaining an acceptable level of service in multiple time periods. The problem is complicated by the fact that staffing level in one time period can affect the service levels in subsequent periods. Moreover, staff schedules typically take the form of shifts covering several periods. Interactions between staffing levels in different time periods, as well as the impact of shift requirements on the staffing levels and cost, should be considered in the planning. Traditional staffing methods based on stationary queueing formulas do not take this into account. We present a simulation-based analytic center cutting-plane method to solve a sample average approximation of the problem. We establish convergence of the method when the service-level functions are discrete pseudoconcave. An extensive numerical study of a moderately large call center shows that the method is robust and, in most of the test cases, outperforms traditional staffing heuristics that are based on analytical queueing methods.

Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty

The recent development of intensity modulated radiation therapy (IMRT) allows the dose distribution to be tailored to match the tumours shape and position, avoiding damage to healthy tissue to a greater extent than previously possible. Traditional treatment plans assume that the target structure remains in a fixed location throughout treatment. However, many studies have shown that because of organ motion, inconsistencies in patient positioning over the weeks of treatment, etc, the tumour location is not stationary. We present a probabilistic model for the IMRT inverse problem and show that it is identical to using robust optimization techniques, under certain assumptions. For a sample prostate case, our computational results show that this method is computationally feasible and promising-compared to traditional methods, our model has the potential to find treatment plans that are more adept at sparing healthy tissue while maintaining the prescribed dose to the target.

Ambulance Service Planning: Simulation and Data Visualisation

The ambulance-planning problem includes operational decisions such as choice of dispatching policy, strategic decisions such as where ambulances should be stationed and at what times they should operate, and tactical decisions such as station location selection. Any solution to this problem requires careful balancing of political, economic and medical objectives. Quantitative decision processes are becoming increasingly important in providing public accountability for the resource decisions that have to be made. This chapter discusses a simulation and analysis software tool ‘BartSim’ that was developed as a decision support tool for use within the St. John Ambulance Service (Auckland Region) in New Zealand (St. Johns). The novel features incorporated within this study include the use of a detailed time-varying travel model for modelling travel times in the simulation, methods for reducing the computational overhead associated with computing time-dependent shortest paths in the travel model, the direct reuse of real data as recorded in a database (trace-driven simulation), and the development of a geographic information sub-system (GIS) within BartSim that provides spatial visualisation of both historical data and the results of what-if simulations.

Fairness and efficiency in web server protocols

We consider the problem of designing a preemptive protocol that is both fair and efficient when one is only concerned with the sojourn time of the job and not intermediate results. Our Fair Sojourn Protocol (FSP) is both efficient, in a strong sense (similar to the shortest remaining processing time protocol: SRPT), and fair, in the sense of guaranteeing that it weakly outperforms processor sharing (PS) for every job on any sample path.Our primary motivation is web serving in which the standard protocol is PS, while recent work proposes using SRPT or variants. Our work suggests both a framework in which to evaluate proposed protocols and an attractive new protocol, FSP.

Forecasting emergency medical service call arrival rates

We introduce a new method for forecasting emergency call arrival rates that combines integer-valued time series models with a dynamic latent factor structure. Covariate information is captured via simple constraints on the factor loadings. We directly model the count-valued arrivals per hour, rather than using an artificial assumption of normality. This is crucial for the emergency medical service context, in which the volume of calls may be very low. Smoothing splines are used in estimating the factor levels and loadings to improve long-term forecasts. We impose time series structure at the hourly level, rather than at the daily level, capturing the fine-scale dependence in addition to the long-term structure. Our analysis considers all emergency priority calls received by Toronto EMS between January 2007 and December 2008 for which an ambulance was dispatched. Empirical results demonstrate significantly reduced error in forecasting call arrival volume. To quantify the impact of reduced forecast errors, we design a queueing model simulation that approximates the dynamics of an ambulance system. The results show better performance as the forecasting method improves. This notion of quantifying the operational impact of improved statistical procedures may be of independent interest.

Chessboard Distributions and Random Vectors with Specified Marginals and Covariance Matrix

There is a growing need for the ability to specify and generate correlated random variables as primitive inputs to stochastic models.Moti vated by this need, several authors have explored the generation of random vectors with specified marginals, together with a specified covariance matrix, through the use of a transformation of a multivariate normal random vector (the NORTA method).A covariance matrix is said to be feasible for a given set of marginal distributions if a random vector exists with these characteristics. We develop a computational approach for establishing whether a given covariance matrix is feasible for a given set of marginals. The approach is used to rigorously establish that there are sets of marginals with feasible covariance matrix that the NORTA method cannot match. In such cases, we show how to modify the initialization phase of NORTA so that it will exactly match the marginals, and approximately match the desired covariance matrix.An important feature of our analysis is that we show that for almost any covariance matrix (in a certain precise sense), our computational procedure either explicitly provides a construction of a random vector with the required properties, or establishes that no such random vector exists.

Travel time estimation for ambulances using Bayesian data augmentation

We introduce a Bayesian model for estimating the distribution of ambulance travel times on each road segment in a city, using Global Positioning System (GPS) data. Due to sparseness and error in the GPS data, the exact ambulance paths and travel times on each road segment are unknown. We simultaneously estimate the paths, travel times, and parameters of each road segment travel time distribution using Bayesian data augmentation. To draw ambulance path samples, we use a novel reversible jump Metropolis-Hastings step. We also introduce two simpler estimation methods based on GPS speed data. We compare these methods to a recently published travel time estimation method, using simulated data and data from Toronto EMS. In both cases, out-of-sample point and interval estimates of ambulance trip times from the Bayesian method outperform estimates from the alternative methods. We also construct probability-of-coverage maps for ambulances. The Bayesian method gives more realistic maps than the recently published method. Finally, path estimates from the Bayesian method interpolate well between sparsely recorded GPS readings and are robust to GPS location errors.

Two Issues in Setting Call Centre Staffing Levels

Motivated by a problem facing the Police Communication Centre in Auckland, New Zealand, we consider the setting of staffing levels in a call centre with priority customers. The choice of staffing level over any particular time period (e.g., Monday from 8 am–9 am) relies on accurate arrival rate information. The usual method for identifying the arrival rate based on historical data can, in some cases, lead to considerable errors in performance estimates for a given staffing level. We explain why, identify three potential causes of the difficulty, and describe a method for detecting and addressing such a problem.