Minimizing the expected waiting time of emergency jobs

Journal of Scheduling https://doi.org/10.1007/s10951-022-00767-1 Minimizing the expected waiting time of emergency jobs Arne Schulz1 · Malte Fliedner1 Accepted: 28 October 2022 © The Author(s) 2022 Abstract We consider a scheduling problem where a set of known jobs needs to be assigned to a set of given parallel resources such that the expected waiting time for a set of uncertain emergency jobs is kept as small as possible. On the basis of structural insights from queuing theory, we develop deterministic scheduling policies that reserve resource capacity in order to increase the likelihood of resource availability whenever an emergency job arrives. Applications of this particular scheduling problem are, for instance, found in the field of surgical operations scheduling in hospitals, where high-priority but uncertain emergencies compete for scarce operating room capacity with elective surgeries of lower priority. We compare our approaches with other policies from the literature in a comprehensive simulation study of a surgical operations unit. Keywords Machine scheduling · Operating room scheduling · Non-elective surgery · Queueing theory 1 Introduction In typical scheduling problems, a set of jobs I has to be scheduled on a set of given resources O such that some (often time-oriented) objective is optimized. In many planning environments, not all jobs or their characteristics are known at the time of a scheduling decision, so that there is considerable uncertainty about future demand for resources and thus about ideal resource utilization. At the heart of many scheduling problems therefore lies a form of risk management that needs to trade off the consequences of scheduling a known job on some resource against future demand induced by uncertain jobs. The importance of the scheduling decision is exacerbated if jobs cannot be interrupted once started and in particular whenever the uncertain jobs foreseeably have an emergency character, so that the success of the total system crucially depends on whether emergency jobs can be served in a timely fashion. In this work, we study the decision problem of scheduling a known set of non-pre-emptive jobs on homogeneous resources in such a way that the waiting time of a set of uncertain emergency jobs which will be released to the system at B Malte Fliedner Arne Schulz 1 Institute of Operations Management, Universität Hamburg, Moorweidenstraße 18, 20148 Hamburg, Germany a later stage is kept low. Applications for this kind of planning problem have been extensively studied in the context of operating room scheduling (see the survey of Cardoen et al., 2010), where all surgeries that are carried out can typically be split up into a subset of known elective surgeries, which can be planned upfront on a daily or weekly basis, and a subset of non-elective surgeries, which most often constitute emergency surgeries that are uncertain with respect to the points in time of their occurrence and resource utilization. Due to the nature of emergencies, these latter non-elective surgeries often need to be serviced as fast as possible, so that a successful scheduling policy of elective surgeries should leave sufficient free capacity to service emergencies. Related problems have also been studied in the field of machine scheduling and maintenance in order to accommodate rush orders or maintenance operations (see Sect. 2). While it is possible to estimate the effects of stochastic resource demand and incorporate the estimates in a stochastic optimization model, in scheduling practice often simpler policies dominate that reserve a certain set of resources or a share of their productive time to the uncertain emergency jobs. In operating room scheduling, for instance, a standard practice is to reserve one or more operating rooms exclusively for emergency surgeries in order to always have available capacity for the emergency surgeries (provided that these resources have not already been seized by another emergency). This also reduces disruptions of the planned schedule comprised of known jobs, since they are effectively not com- 123 Journal of Scheduling peting for the same resources. However, this will typically come at the expense of a lower resource utilization whenever no or only few emergencies need to be processed and might be too inflexible whenever several emergencies occur one after the other. In this work, we will investigate deterministic scheduling policies that try to schedule known jobs in such a way that the expected waiting time of uncertain emergency jobs is reduced without directly having to anticipate future demand in the analysis. For this purpose, the paper is structured as follows: in Sect. 2, we provide a literature overview with a focus on operating room scheduling. In Sect. 3, the problem setting is described. We also analyse a simplified queueing model in order to derive theoretical insights into the problem structure (Sect. 3.1) and motivate a deterministic scheduling policy that arranges jobs without emergencies in a particular fashion (Sect. 3.2). We then develop two scheduling approaches in Sects. 4 (mixed-integer programme) and 5 (heuristic), which are tested in a computational study in Sect. 6. The paper closes with a conclusion in Sect. 7. 2 Literature review Although the investigated problem can be found in different areas of application in principle (e.g. production and maintenance scheduling), the presented literature review focuses on operating room scheduling since it is in this field of enquiry that the trade-offs between the interests of elective and non-elective patients have been studied most intensely (cf. Cardoen et al., 2010 and Samudra et al., 2016). Furthermore, we will contrast our policy with two approaches that have been developed explicitly in the context of operating room scheduling (Wullink et al., 2007 and van Essen et al., 2012). Nevertheless, we give some brief references to other areas of application at the end of this section. Operating room scheduling needs to consider a fundamental conflict between elective surgeries (jobs of normal priority), whose characteristics are known at the time of the scheduling decision, and non-elective surgeries, which typically arrive as emergency surgeries and thus have a very high priority, but are otherwise uncertain with respect to their characteristics and times of arrival (see van Riet and Demeulemeester 2015 for an in-depth discussion of the conflict of interest between elective and non-elective patients). Although non-elective surgeries are divided into urgent surgeries, which need care within a fixed time horizon of a couple of hours, and emergencies, which need care immediately, we focus only on emergencies in this paper. Both types of surgeries, electives and emergencies, have to be scheduled in a given number of operating rooms (OR) with a fixed time capacity. In order to accommodate emergency surgeries, two scheduling strategies are po (...truncated)