Improving operational effectiveness of tactical master plans for emergency and elective patients under stochastic demand and capacitated resources

Abstract

This paper develops a two-stage planning procedure for master planning of elective and emergency patients while allocating at best the available hospital resources. Four types of resources are considered: operating theatre, beds in the medium and in the intensive care units, and nursing hours in the intensive care unit. A tactical plan is obtained by minimizing the deviations of the resources consumption to the target levels of resources utilization, following a goal programming approach. The MIP formulation to get this tactical plan is specifically designed to account for emergency care since it allows for the reservation of some capacity for emergency patients and possible capacity excess. To deal with the deviation between actually arriving elective patients and the average number of patients on which the tactical plan is based, we consider the possibility of planning a higher number of patients than the average to create operating slots in the tactical plan (slack planning). These operating slots are then filled in the operational plan following several flexibility rules. We consider three options for slack planning that lead to three different tactical plans on which we apply three flexibility rules to get finally nine alternative weekly schedules of elective patients. We then develop an algorithm to modify this schedule on a daily basis so as to account for emergency patients’ arrivals. Scheduled elective patients may be cancelled and emergency patients may be sent to other hospitals. Cancellation rules for both types of patients rely on the possibility to exceed the available capacities. Several performance indicators are defined to assess patient service and hospital efficiency. Simulation results show a trade-off between hospital efficiency and patient service.

Introduction

Master surgical plans can contribute to a balanced use of resources such as beds, operating theatres and nursing staff. Belien and Demeulemeester (2007) propose a three-stage procedure for developing effective operating room (OR) schedules with a multi-resource perspective: allocation of OR time to surgical specialties at strategic level, development of a master surgery schedule at tactical level and scheduling individual patients at operational level. Van Oostrum et al. (2008) build their two-phase decomposition approach around surgical procedure types: defining the mix of procedures to be performed and scheduling the surgery types. Both studies report on improved performance on the use of resources by using a cyclic master surgical plan.

However, the operational performance can differ strongly from the expected performance of the tactical plan for several reasons. One reason is that the tactical master plan needs to be translated into an operational plan used for scheduling the patients; this operational plan may deviate from the tactical plan due to short-term availability of resources or due to the need to schedule more patients because of an unacceptable rise in waiting time. Another reason can be emergency admissions, because they can have priority to elective admissions. What happens when there are more arriving emergencies than expected? This may place more demand on intensive care (IC) beds. It may therefore be necessary to cancel elective patients that would need an IC bed after the surgical procedure. Thus, the balanced use of resources suggested by the master tactical plan may result in a much poorer performance on an operational level. In this study we consider the questions of how to develop more robust tactical plans and how to translate those plans into operational plans. First we summarize the literature tackling this issue. Then we introduce our approach to this problem.

Harper (2002) emphasizes the need for patient classification techniques for a better management of hospital resources. This author develops a simulation model designed to assess the impact of the flow of emergency and elective patients over the use of three major resources: beds, operating theatres and medical staff. Simulation models are used for their predictive value as in the work of Ridge et al. (1998) which evaluates in particular the bed occupancy level when setting to some value the transfer rate of emergency patients to other hospitals. Emergency transferral as a way to manage patient overflow also is analysed in the paper of Litvak et al. (2008). The authors consider several regional hospitals that jointly reserve a number of beds in their intensive care unit for emergency patients. The authors develop an analytical approach to compute the number of regional beds to be reserved for any acceptance rate and show that cooperation leads to a higher acceptance rate with a lower number of reserved beds. One way to deal with overflows of patients is to develop operational strategies to gain capacity. To this purpose, Cochran and Roche (2009) design a split patient flow model to treat lower acuity patients in a separate queue from higher acuity patients waiting for a bed.

On a more tactical level, the available capacity has to be balanced between elective patients and emergencies. Dealing with emergency patients may amount to determining the level of capacity required to operate on a pattern of elective patients so as to keep the deferral rate as low as possible. For instance, Utley et al. (2003) use generating functions to estimate the bed needs depending on overall levels of elective admissions and emergency arrivals. Lamiri et al. (2008) develop a stochastic model for operating room planning to deal with both an elective and emergency demand for surgery. The objective is to assign elective patients to periods so as to minimize the sum of elective patient related costs and overtime costs of operating rooms.

The purpose of this paper is twofold. First we design an optimisation approach to determine a tactical plan with reserved capacity for emergency patients. Then we develop operational strategies to deal with the actual flow of elective and emergency patients on a weekly and daily basis. We will illustrate our approach with data related to thoracic surgery, using a setting from a Dutch Thorax Centre described hereafter. We consider several groups of patients according to their demand profile on resources; each of these groups being homogeneous in terms of the use of four main resources: operating theatres, beds in intensive care unit, beds in medium care unit and nursing staff. The groups can be linked to pathology categories or procedure types.

In the medium-run, operating theatre management requires a plan for all elective patients to be operated on over a given horizon, generally four weeks, in order to allocate at best the major resources while not exceeding the available capacity. Part of this capacity is reserved for emergency patients. The problem is modelled as a mixed integer program the objective function of which consists in minimizing a weighted sum of deviations of resources consumptions to the target levels of resources utilization in order to obtain a smooth allocation of patients over the planning horizon. The optimization programme is therefore a goal programming approach. The solution is a tactical plan which is used to derive a weekly operational plan for elective patients (see Fig. 1). Emergency arrivals and actual lengths of stay lead the operational plan to be altered and, in this planning phase, the only remaining control option is to cancel some operations. This results in a final daily plan described as the executed plan.

Since the actual flow of elective patients may differ from the average flow on the basis of which the tactical plan is determined, we use several strategies to schedule the actual elective patients while obtaining a better spread of resources usage over time. We consider the possibility of planning a higher number of patients than the average to create operating slots in the tactical plan (slack planning). These operating slots are then filled in the operational plan following several flexibility rules. Slack planning therefore consists in increasing the number of slots in the tactical plan, and flexibility deals with the use of the slots in the weekly schedule of patients (operational plan). Flexibility leaves on each day the possibility of changing patient groups initially planned for operation. For instance, if no patients of a patient group are available on the waiting list, an elective patient of another group can be scheduled. Slack planning and flexibility appeared to be the most promising strategies amongst several strategies developed in Dellaert and Jeunet (2010) who examined a situation in which only elective patients were considered. In the present contribution, slack planning and flexibility are applied in a situation where we also have emergency patients.

We design several rules to deal with emergency patients on a daily basis. Some of the emergency patients may be sent to other hospitals or elective patients may be postponed to avoid capacity excess. Handling emergency patients therefore leads to modifying the operational plan of elective patients. These emergency scheduling rules and the flexibility strategies are assessed by several indicators: the waiting time, the number of cancelled operations and the number of schedule changes between the tactical plan, the operational plan and the executed plan. The performance will ultimately be expressed in terms of two indicators that we both want to minimize: patient dissatisfaction and hospital inefficiency.

The central question we want to answer in this paper is which combinations of actions on the three planning levels lead to the most efficient performance and what recommendations can be given to make the final plan.

The remainder of the paper is organized as follows. Section 2 provides a general description of the problem and the approach we follow. In Section 3, we present the mathematical model used to obtain the planning of patients on a tactical level and we shortly discuss its connection with goal programming. Section 4 begins with a brief description of the scheduling rules initially designed by Dellaert and Jeunet (2010) to obtain an operational plan of elective patients on a weekly basis. We then turn to a presentation of the daily scheduling algorithm we developed in this paper to deal with emergency patients. We finally present several indicators we defined to assess the performance of the developed strategies and algorithm. Numerical experiments are discussed in Section 5. Section 6 draws some conclusions and formulates recommendations for further work.