Optimal Sharing of Surgical Costs in the Presence of Queues

Resumen

The widespread access to Public Health Care in western European countries is placing the system at a point in which optimal allocation of resources becomes a major management problem. Citizens are particularly sensitive to any phenomena related to the provision of health services and among these phenomena, one of the most relevant is the severe congestion suffered by public health services. These congestion problems have as one of their most direct consequences the enormous length of waiting lists for surgical treatment. The formation of waiting lists to get elective surgery, can be framed as a queueing system. Assuming that the agents are served respecting their arrival order, the only control variable is the capacity to install. Consequently, the theory can help us to take decisions concerning that capacity, taking into account that the higher the capacity the higher the associated costs, but the shorter the expected queues. The queueing system arising in surgical treatment has some specific characteristics: 1. There are two sources for the formation of waiting lists. On the one hand, the capacity of the operation theatre, and, on the other hand, the bed capacity of the hospital; 2. Several medical procedures share both servers, namely, customers from different treatments need to use both the operation theatre and the beds; 3. Each of those procedures have their own rate of arrival; 4. Not all medical procedures are considered as equally urgent, in the sense that the average waiting time politically considered as adequate differs among procedures. In the managing of such a situation, a cost allocation problem arises: Since different procedures share both the operation theatre and the hospital beds, we have to design a cost allocation rule in order to share the joint costs. This is the main purpose of this paper. In order to construct a cost allocation rule, we use a game theoretical perspective, designing a cost allocation game. In the first part of the paper we concentrate ourselves on the costs associated to the operating theatre. Then, we construct a game by confronting two situations: one in which each medical procedure has its own operating theatre, and another one in which there is a unique theatre that serves all the diseases. We show that sharing the use of the operating theatre to treat the patients of the different medical procedures, leads to a cost reduction. Then, we construct a cost-sharing game and, given the characteristics of the game, we suggest a costsharing rule that recommends the Shapley value allocation of the cost-sharing game. Thus, our optimal tariff has all the nice properties of the Shapley value. The fact that this cooperative solution can be computed easily, is certainly an important property in a practical environment. The cost-sharing game emerging among the treatments is the sum of an additive game plus an �airport game�, where the different landing track capacities are translated in our model to the capacity required by the operating theatre in order to satisfy its demand, according to the maximum average waiting time guarantee. Up to this point, only the direct costs derived from surgical interventions were considered. However, we have to take into account that an operation generates also other costs, more precisely the costs incurred during the patients� hospitalization time for recovering. Then, we introduce in the model the post-operative costs and we study how they are affected by the cooperation among medical procedures. Treating the beds as servers, we may model the hospitalization stage also as a queueing system. Then, the number of servers (beds) required to guarantee the service, can be computed in different scenarios. Nonetheless, there is no possibility of arriving at general results, due to the lack of analytical solvability of the model. In spite of that, something general can be said about the average number of beds. By so doing, we show that sharing the use of the operating theatre has an ambiguous effect on average post-operative costs. If the medical procedure with the highest priority level, has a higher recovering time than the average hospitalization time of the rest of the pathologies, we can ensure that in average terms cooperation leads to post-operative cost savings. Finally, a numerical example with real data is analyzed. In this example, we compute the distribution of surgical costs, applying the theoretical results obtained previously. As for the number of beds required, we also compute them, under different scenarios. Also, we estimate the distribution of bed costs among the procedures, provided that an upper bound of .1 is set on the probability of waiting after the intervention.