Modeling and optimizing the evacuation of hospitals based on the MRCPSP with resource transfers

Abstract

In this paper, we consider the problem of hospital evacuation and model it as a multi-mode resource-constrained project scheduling problem (MRCPSP) with additional resource transfers and blockings. Based on this model two heuristic decomposition approaches are proposed. The first uses a tabu search algorithm for the mode assignment problem, the scheduling subproblem is solved by adapted serial and parallel schedule generation schemes using priority rules. The second is based on a decomposition into an evacuation and a routing subproblem where solutions are represented by resource flows. Computational experiments were performed for both approaches using randomly generated instances based on real-world scenarios.

Introduction

Evacuation planning as a scientific research topic has received an increasing amount of attention in recent years. The field of evacuation planning covers a broad range of situations where evacuations might be necessary, ranging from building evacuations (e.g., due to fire) to evacuations of complete cities and regions (e.g., due to natural disasters). Generally, evacuations can be classified as either precautionary or life-saving evacuations (cf. Hamacher and Tjandra 2002). On the one hand, for precautionary evacuations, the hazard is known in advance, i.e., the evacuation time can be estimated and the evacuation can be planned in detail. Such problems are often modeled as network flow problems where people or vehicles are routed through a network modeling building sections or streets. On the other hand, life-saving evacuations are usually necessary if hazards occur with insufficient warning and the affected people have to be evacuated without pre-emergency planning. In this case, the problems that arise are more direct (e.g., the rescue of injured people, clearing routes for the evacuation, fire-fighting, etc.) and often have to be dealt with in real time.

In this paper, we concentrate on the evacuation of hospitals. The most important difference between the evacuation of hospitals and other evacuation scenarios is that the patients within hospitals often cannot help themselves in order to reach a safety zone but rely on the help of other people and aids (like stretchers, wheelchairs, hospital beds, etc.) for the transport. In the survey by Childers and Taaffe (2010) it is stated that the problem of evacuation planning for healthcare facilities has received less attention than other evacuation scenarios despite the additional difficulties encountered when evacuating these facilities. Various research opportunities to improve this situation are described, e.g., estimating the time required for the evacuation as well as planning the transport of patients both inside and outside the healthcare facility. An evaluation of 257 reported hospital evacuations in the United States between 1971 and 1999 (cf. Sternberg et al. 2004) revealed that the preparedness of hospitals is often insufficient due to a lack of exercises as well as a disregard of evacuation plans in the case of an actual emergency. A similar study of 522 hospitals in Germany (Lipp et al. 1998) showed that only $83.5\%$ of these hospitals had any emergency plan and $51.2\%$ of these hospitals have never staged an emergency exercise.

Most papers dealing with evacuation planning for healthcare facilities focus on decision making procedures as well as emergency preparedness training. For example, an emergency management system that can be used for decision making in the case of a hospital evacuation is the Hospital Incident Command System (HICS, cf. [4]). Furthermore, some papers report experiences from past evacuations. For example, Gray and Hebert (2007) deal with the evacuation of multiple hospitals in New Orleans due to the Hurricane Katrina in august 2005, while Katter et al. (2008) describe the evacuation of a hospital in Germany after a bomb from World War II was discovered in the vicinity of the hospital.

Only few papers deal with evacuation planning for healthcare facilities using optimization or simulation techniques. A simulation approach for estimating the time required to evacuate all patients from a hospital to sheltering facilities (e.g., other hospitals) is described in Taaffe et al. (2006). This model focuses on the transport of the patients from the affected hospital to sheltering facilities and incorporates a limited number of transporting vehicles as well as support staff for the transport. Additionally, only a limited amount of space in the staging area (the area where the patients enter the transporting vehicles) and a limited number of beds in the sheltering facilities might be available. In this approach, the transport of the patients within the hospital is modeled as a stochastic delay. An extension of this approach can be found in Tayfur and Taaffe (2007) where the influence of traffic on the transporting vehicles is added to the model. An integer linear programming formulation for this problem is described in Tayfur and Taaffe (2009). Similarly, Bish et al. (2014) use an integer linear program for the problem of transporting patients from a hospital to other receiving hospitals while minimizing the risk to the patients.

A prediction model for estimating the time required to evacuate all patients from one or multiple floors of a hospital is introduced in Golmohammadi and Shimshak (2011). Here, the time required to evacuate all patients from their initial locations within the hospital to a safe location inside or outside the hospital is calculated while staff members that are available to assist in the evacuation as well as elevators are regarded as bottleneck resources. A simulation approach for estimating the time required to evacuate patients during an internal danger (e.g., a fire) is presented in Wolf (2001). In this approach, the section of the hospital that has to be evacuated is modeled as a network and assistants as well as aids are preassigned to evacuate specific patients. A particular emphasis is put on representing real-world parameters such as the floor plan of the building as well as evacuation times of the patients using various aids.

In this paper, the problem of hospital evacuation is modeled as a multi-mode resource-constrained project scheduling problem (MRCPSP) with additional resource transfers and blockings. The RCPSP was chosen since it is able to model activities (the evacuation of patients) that require multiple types of resources (assistants, aids, and building sections). The activities have to be scheduled such that a given objective function is optimized (e.g., minimizing the time required to evacuate all patients). The multi-mode situation reflects the fact that the evacuation of a patient can be done in different modes (using different equipment or different routes through the building). Additionally, resource transfers with associated transfer times have to be taken into account for the transport of assistants or aids between different locations. Finally, blocking constraints arise due to limited capacities of building sections.

Resource-constrained project scheduling problems have been extensively studied in scientific literature since the early 1960s and several books have been published dealing with the RCPSP and extensions (e.g., Brucker and Knust 2011, Demeulemeester and Herroelen 2002). The RCPSP with additional transfer times has been considered in the context of multi-project scheduling in Krüger and Scholl (2009), Krüger (2009), Krüger and Scholl (2010). Besides a mixed-integer linear programming formulation of this problem, Krüger (2009) introduced adaptions of the serial as well as the parallel schedule generation scheme (cf. Kolisch 1996) that select resource transfers based on priority rules. Additionally, she presented a genetic algorithm where solutions are represented by activity lists. An alternative solution representation where solutions are represented by resource flows was suggested in Poppenborg and Knust (2015) (cf. also Poppenborg 2014).

Blocking constraints were mainly studied in shop scheduling problems where blockings occur if no intermediate storage between machines exists and a job remains on a machine until the next machine is available. Hall and Sriskandarajah (1996) give an overview of blocking shop scheduling problems with a focus on flow-shop problems, while Mascis and Pacciarelli (2002) consider the blocking job-shop problem. To the best of our knowledge, the RCPSP with blocking constraints has not been studied before.

The remainder of the paper is organized as follows. In Sect. 2 the studied problem of hospital evacuation is described more precisely. Afterwards, in Sect. 3 the problem is modeled as MRCPSP. In Sect. 4 a two-stage decomposition algorithm is presented which uses adapted serial and parallel schedule generation schemes based on priority rules. An alternative decomposition approach based on resource flows is described in Sect. 5. Computational experiments for both approaches are discussed in Sect. 6. Finally, conclusions can be found in Sect. 7.