Multimodularity in the Stochastic Appointment Scheduling Problem with Discrete Arrival Epochs

79 Pages Posted: 18 Nov 2018

See all articles by Christos Zacharias

Christos Zacharias

University of Miami Business School

Tallys Yunes

University of Miami - Department of Management Science

Date Written: October 26, 2018

Abstract

We address the problem of designing appointment scheduling strategies in a stochastic environment accounting for patient no-shows, non-punctuality, general stochastic service times, and unscheduled emergency walk-ins. A good appointment schedule seeks to help outpatient clinics to utilize their resources efficiently while containing patients' waiting times. The task of identifying an optimal schedule is modeled as a nonlinear integer program, where the objective function is the outcome of stochastic analysis in transient state. We maintain the discrete nature of the appointment scheduling problem by considering arrival epochs with discrete supports. By looking at discrete-time snapshots of the random evolution of a single-server queueing model, we characterize probabilistically the system's workload over time as a function of an appointment schedule and we derive recursive expressions for the performance measures of interest. Subsequently, we unfold discrete convexity properties of the optimization problem. We prove that under general conditions the objective function is supermodular and componentwise convex. Under assumptions on patient punctuality, we prove that the optimal scheduling strategy minimizes a multimodular function; a property which guarantees that a locally optimal schedule is also globally optimal. The size of the local neighborhood, however, grows exponentially with the dimension of the problem. To the best of our knowledge, this study is the first to develop and implement an algorithm for minimizing locally a multimodular function over nonnegative integer vectors via submodular set-function minimization over ring families; a task that can be performed in polynomial time.

Keywords: transient queues, appointment scheduling, discrete optimization, multimodularity

Suggested Citation

Zacharias, Christos and Yunes, Tallys, Multimodularity in the Stochastic Appointment Scheduling Problem with Discrete Arrival Epochs (October 26, 2018). University of Miami Business School Research Paper No. 18-14. Available at SSRN: https://ssrn.com/abstract=3273649 or http://dx.doi.org/10.2139/ssrn.3273649

Christos Zacharias (Contact Author)

University of Miami Business School ( email )

United States

HOME PAGE: http://moya.bus.miami.edu/~czacharias/

Tallys Yunes

University of Miami - Department of Management Science ( email )

United States

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