Health Care Management Science 3 (2000) 193–200 193 A mathematical programming approach for scheduling physicians in the emergency room Huguette Beaulieu a , Jacques A. Ferland b , Bernard Gendron b,* and Philippe Michelon c a Imperial Oil Limited, 111 St. Clair Ave. W., Toronto, Ontario, Canada M5W 1K3 b D´ epartement d’informatique et de recherche op´ erationnelle, Universit´ e de Montr´ eal, P.O. Box 6128, Succursale Centre-Ville, Montr´ eal, Qu´ ebec, Canada H3C 3J7 c Laboratoire d’informatique d’Avignon, Universit´ e d’Avignon et des Pays de Vaucluse, B.P. 1228, F-84911 Avignon, France Received 27 July 1999; accepted 4 February 2000 Preparing a schedule for physicians in the emergency room is a complex task, which requires taking into account a large number of (often conflicting) rules, related to various aspects: limits on the number of consecutive shifts or weekly hours, special rules for night shifts and weekends, seniority rules, vacation periods, individual preferences, ... In this paper, we present a mathematical programming approach to facilitate this task. The approach models the situation in a major hospital of the Montr´ eal region (approximately 20 physicians are members of the working staff). We show that the approach can significantly reduce the time and the effort required to construct a six-month schedule. A human expert, member of the working staff, typically requires a whole dedicated week to perform this task, with the help of a spreadsheet. With our approach, a schedule can be completed in less than one day. Our approach also generates better schedules than those produced by the expert, because it can take into account simultaneously more rules than any human expert can do. Keywords: health administration, emergency physician scheduling, mathematical programming 1. Introduction Preparing a schedule for physicians in the emergency room is a complex task, which requires taking into account a large number of (often conflicting) rules, related to var- ious aspects: limits on the number of consecutive shifts or weekly hours, special rules for night shifts and week- ends, seniority rules, vacation periods, individual prefer- ences, ... In this paper, we present a mathematical pro- gramming approach to facilitate this task. The approach models the situation in a major hospital of the Montr´ eal re- gion (Sacr´ e-Coeur Hospital, where approximately 20 physi- cians are members of the working staff at the emergency room). The paper is organized as follows. First, our contribu- tions are highlighted in section 2, which provides a review of the literature on methods for scheduling health care per- sonnel. Then, section 3 gives an overview of the problem faced by the planner in our case study. The mathematical programming model is presented in section 4. In section 5, we describe the solution method based on the model, we give a brief outline of its implementation (the interested reader is referred to [2] for further details), and we com- pare the schedules it produces with those proposed by a human expert (a member of the working staff who has been in charge of the scheduling task for many years). In the conclusion, we summarize our work and propose exten- sions. * Corresponding author. E-mail: gendron@iro.umontreal.ca. 2. Literature review Methods for generating workforce schedules are typi- cally divided into cyclic and non-cyclic approaches. Cyclic techniques proceed by defining fixed sequences of shifts, which are then assigned equally (or almost) among work- ers. Such techniques are well-adapted to situations where the same schedules can accommodate all workers. Even then, they must account for vacations and days-off, and in the presence of many seniority rules and individual prefer- ences, as in our case (see the next section), they are usually of no help. Non-cyclic methods must then be considered, which generally fall into two categories: (1) those requiring human expertise and the use of a spreadsheet (which helps the planner to balance the schedules among categories of workers); (2) optimization approaches. These latter meth- ods have two main advantages over the former ones: first, they require very little human intervention and can there- fore be (almost) fully computerized; second, when properly designed, they can handle many more rules simultaneously than any human expert can do, even with the help of a spreadsheet. However, mastering them is by no means a trivial task, and their application to a specific case might require several years of development. Much of the research on scheduling health care person- nel has been devoted to the case of hospital nurses [18]. A recent account of the nurse scheduling literature is given in [9]. An example of a non-cyclic method based on human expertise and the use of a spreadsheet is given in [20], while classical optimization approaches are described in [14,23]. Progress in computer technology and software tools has  Baltzer Science Publishers BV