Advance Online Scheduling with Overtime : A Primal-Dual Approach

Introduction. We study a fundamental online resource allocation problem in the context of outpatient health-care scheduling in which a heterogeneous stream of patient (or customers interchangeably) requests arrive one at a time with a declared reward for receiving service from one of a finite number of heterogeneous resources. This is an online adversarial setting in which there is no knowledge about the future arrival process of customers. Our goal is to develop effective and efficient algorithms and methods that are robust to the arrival process. Upon arrival of each customer, the system chooses in real-time both a resource and a date for service over a planning horizon subject to capacity constraints without knowing any information of the subsequent incoming customers in the future. Customers have heterogeneous capacity requirements and rewards. Each resource has a finite regular capacity but can be expanded at the expense of overtime cost. For human servers, shifts and work schedules dictate the regular capacity. However, overtime decisions cannot be ignored in most service operations systems due to the uncertain arrival process and the fact that urgent and unpredictable events happen from time to time. We develop new online algorithms for making not only a resource-date allocation decision for each customer but also an overtime decision for each resource on each day within a specified planning horizon. The goal is to maximize the total reward less the total overtime cost. Our proposed online algorithms are (i) both easy-to-implement and extremely efficient to compute, and also (ii) admit a theoretical worst-case performance guarantee. To make them more valuable in applications (e.g., health-care operations), we extend our approach to a rolling horizon online scheduling setting. We investigate our proposed online policies under the paradigm of competitive analysis. In competitive analysis, no prior knowledge is given about the sequence of arrivals, nor are they assumed to follow any predictable pattern. Our online algorithms’ performance is stated as a fraction of an optimal performance achieved when knowing the entire arrival sequence in advance. For r ≤ 1, if an algorithm can guarantee that this fraction is at least r for every instance of customer sequence, then it is said to achieve a competitive ratio of r. The objective is to design robust online algorithms which achieve the optimal competitive ratio, i.e., a ratio r∗ such that no algorithm can do better without advance information on the arrival sequence. Our algorithms are very effective and efficient in devising robust online policies that can be implemented without any information about the evolution of future demands. Both simulation and case studies show that the proposed online policy performs much better than its theoretical worst-case performance guarantee. In addition to the health-care operations (our case study), our models and online algorithms can be tailored to other applications such as online advertising, revenue management (e.g., room/seat allocation in hotels/airlines), the airline industry (e.g., the online arrival process of customers who purchase tickets), online auction (e.g., eBay in which the arrival of bidding customers for an auction can be properly modeled using our online framework), online retailers (e.g., the real-time orders that Amazon receives and must fill quickly), make-to-order flexible manufacturing systems, and call-center operations.