Balancing waiting times in sequential servers appointment system using distributionally robust optimization

• Main Author: Goh, You Hui
• Other Authors: Yan Zhenzhen
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2023
• Subjects: Science::Mathematics::Applied mathematics::Optimization; Science::Mathematics::Applied mathematics::Operational research
• Online Access: https://hdl.handle.net/10356/166911
• Institution: Nanyang Technological University

This thesis studies an appointment scheduling problem with multiple sequential servers from a distributionally robust optimization (DRO) perspective. Using a moment-based ambiguity set along with conic optimization, our DRO models can capture the correlation between the random service times. A cost minimization model assuming linear waiting cost is first used to formulate a DRO model to find an optimal appointment schedule that minimizes the total systems' cost. We find that the schedule obtained can lead to an imbalanced waiting time across the servers, concentrating on the downstream servers. To ensure a balanced waiting time across the sequential servers, two approaches are studied. The first approach adopts an idea in the queuing literature to strategically idle the upstream server. A DRO model is proposed to jointly optimize the strategic idling (SI) policy and the appointment schedule. Extensive numerical studies are conducted to thoroughly examine the role of SI in a system's performance and the effect of correlation information on the optimal schedule and SI policy. Using data from a clinic, we conduct a case study to demonstrate the performance advantage of our SI policy, over existing SI policies in the literature. The second approach contests the common usage of the linear waiting cost function by instead using a piecewise linear convex waiting cost function. Such a convex function will allow us to penalize long waits. We generalize our DRO model with a linear cost function to a DRO model with a piecewise linear convex cost function. To investigate the opposite effect of incentivizing long waits, we derive another DRO model that uses a piecewise linear concave waiting cost function. In the reformulation of a general concave cost DRO model, we encounter a separable concave minimization problem. The problem is challenging since it is nonconvex. We propose to approximate the problem using a piecewise linear approximation, which leads to a mixed integer linear program (MILP). To reduce its problem size, we investigate the problem of minimizing the number of pieces in the piecewise linear approximation of the separable concave minimization problem. We propose a greedy binary search algorithm to design the piecewise linear approximation while ensuring a tight approximation guarantee of 1 + Ɛ.