Trainee scheduling at Hospital: a Paper ReviewSamsul Amar1, I G. B. Budi Dharma2

1Department of Industrial Engineering, Trunojoyo UniversityPO BOX 2, Kamal, Bangkalan, East Java, IndonesiaE-mail: sams_amar@yahoo.com2Department of Mechanical & Industrial Engineering, Gadjah Mada UniversityJl. Grafika 2 Yogyakarta, Indonesia, 55281

AbstractTrainee program (for prevocational doctors) is an important part of medical education system. It also becomes a part of personnel fulfillment of the hospital. Yet, only a few studies have been conducted to solve trainee scheduling problem. Belien and Demeulemeester (2004b) give one of excellent model. Therefore, this paper will explore and develop this model.The problem of trainee scheduling is how to schedule trainees to perform some activities during a planning horizon. Belien and Demeulemeester (2004b) use Integer Linear Programming (ILP) model and solve the model using branch and price approach. The model has several constraints including coverage constraints, formation requirements, trainees non-availability constraints, and setup constraints. In the real problem, these constraints are hard to be satisfied overall. Therefore, the constraints may be violated with some penalty costs and the objective of the model is to minimize total penalty costs. Some development can be made to make the model more suitable to real condition such as considering the previous schedule, routing rule, and revising the definition of setup variable.Keyword: Trainee scheduling, Integer Linear Programming (ILP), constraint, penalty cost.

IntroductionMany hospitals, especially university hospital, have a trainee program for prevocational doctors. The trainee program is an important part of medical education system. This program is conducted to train graduate students of medical to make them ready to be doctors. Typically, a prevocational doctor (next called trainee in this paper) has to perform a number of activities, each in a certain period. Trainee program also become a part of personnel fulfillment of the hospital (Belien and Demeulemeester, 2004). Therefore, a number of trainees should be assigned to each activity in each period to fulfill personnel requirement. Even though trainee scheduling is an important field, only a few studies have been conducted in this field, contradict with other scheduling problem in hospital like nurse rostering and scheduling or operating room scheduling which got a lot of attention. The writers can only find four articles in trainee scheduling at hospital, which are Belien and Demeulemeester (2004a), Belien and Demeulemeester (2004b), Belien (2006), and Dexter et al. (2010). The last paper (Dexter et al, 2010) specialize their study on trainee scheduling of operating room, and cannot be generalized to other trainee problem and therefore, will not explored in this paper review.

SEMINAR NASIONAL TEKNIK INDUSTRI UNIVERSITAS GADJAH MADA 2011Yogyakarta, 26 Juli 2011

Program Studi Teknik Industri Jurusan Teknik Mesin dan Industri FT UGMISBN XXX-XXXXX-X-X

Belien and Demeulemeester (2004a) describe a trainee scheduling problem at the ophthalmology department of the university hospital Gasthuisberg, Leuven, Belgium. They use Integer Linear Programming (ILP) to model the problem and solve the problem using branch and price method. In the model, two types of constraint are used, those are hard and soft constraints. Hard constraints are the constraints that are not allowed to be broken. These constraints consist of work coverage constraints and formation requirements. Soft constraints including trainees preference and setup restriction, in opposite, may be violated with some penalty cost. The objective is to minimize total penalty cost. However, in their model, only trainees preference constraints are set to be violate-able. Belien (2006) in his thesis report writes back the model of Belien and Demeulemeester (2004a) with the addition of other model for scheduling in hospital including nurse scheduling and operating room scheduling and describes some exact and heuristic methodologies to solve the problems. Belien and Demeulemeester (2004b) give a comprehensive model of trainee scheduling. They expand the soft constraints and make coverage constraints as soft constraints. The model is more applicable to general trainee scheduling problem, not only for prevocational doctor trainee program, but also for other trainee program scheduling. Therefore, in this paper, the model of Belien and Demeulemeester (2004b) will be reviewed. However, this paper will only concern about the model and does not cover the techniques to solve the model.

Problem Statement and the Mathematic ModelingTo complete the education, trainees have to follow some trainee posts (called activities in this paper) each in a certain period. The trainees should complete to perform all activities in a certain period. Each activity has a certain capacity and personnel need. In the following, the model of Belien and Demeulemeester (2004) will be presented.If, i is period index, j is trainee index, and k is activity index, the decision variable of the model is defined as:= 1, if during period i, trainee j is scheduled to perform activity k.= 0, otherwise.and a dummy variable to define starting time for each trainee to conduct each activity.= 1, if during period i, trainee j starts to perform activity k.= 0, otherwise.The model of trainee scheduling problem contains four set of constrains, those are coverage constraints, formation requirements constraint, non-availability constraints and setup constrains. All constraints can be violated with some penalty costs. Ideally, the constraints are satisfied as much as possible; however, in the real problem, there is no even a feasible solution for the problem. Therefore, the objective of the model is to minimize the total penalty costs.The coverage constraints means that each activity has to be performed by a certain number of trainees who have the appropriate skills. Each coverage constraint is associated with an activity but an activity can have some coverage constraints. Let C represent the set of coverage constrains, and , , , , and respectively be the activity, the trainee set having the appropriate skills, the number of trainees required, and the start and end period of constraint coverage time horizon. Then, let and be the number of trainees scheduled too many and too few in period i for coverage c, and and their respective associated penalty cost. The coverage constraint can be stated as follows:

The second constraints are formation requirements. These constraints ensure that each trainee has to perform each activity for some periods. Let define Fjk as target number of period trainee j has to perform activity k, n as the number of periods in the scheduling horizon, m the number of trainees and p the number of activities. Then, let and as number of periods too many and too few trainee j is scheduled to perform activity k (compared with formation requirement) and and their associated penalty cost. The constraints can be stated as the following:

In the real problem, even though the number of period for each trainee to perform each activity is flexible, it should be in the range of minimum and maximum number of period allowed. In addition to the second constraints, let Ljk and Ujk be the strict minimum and maximum number of period trainee j has to perform activity k. Two constraint sets can be added as follows:

The third constraints is non-availability constraints. Let define Nj as all non-available periods for trainee j, and dij as dummy variable set to be 1 if trainee j is scheduled to perform activity k during his/ her non-available period and wij is respective penalty cost. This penalty cost is trainee preference. Therefore:

Another fact is that a trainee cannot perform more than one activity at a certain period. This constraints below is not allowed to be violated.

Setup constraint means that each trainee can only restart each activity once. If the trainee j restarts the activity k more than once, say hjk times, a gjk penalty cost is charge as the reduction of efficiency of learning time. A set of constraints can be stated as follows.

Finally, an integer constraints are stated:

The objective function is then defined as minimizing total penalty costs because of violating some constraints:

Review and Model DevelopmentThe model of Belien and Demeulemeester (2004b) as presented before is an exelent model. It is applicable for real problem, even though it needs to be modified for specific problem. However, some modification can be made to make the model more suitable for wider application.First of all, the schedule is usually made for a long time horizon, say 1 or 2 year to give a certainty to the trainees. When the new trainees entry, the schedule is adjusted. Changing the schedule will lead to dissatisfaction to the trainees. It also needs administrative adjustment and may decrease customer satisfaction. For example, when a trainee treat a patient during some period and then replaced by other trainee, a patient could be disappointed because (s)he has built good relationship with the previous trainee. Therefore, changing the schedule should lead to a penalty. Let oldijk as old schedule for trainee j in period i to perform activity k, then:oldijk = 1, if trainee j was scheduled to perform activity k during period i.= 0, otherwise.By defining changeijk as the change of schedule from being scheduled to be not scheduled, a new constraint set can be built: oldijk - xijk changeijk If a changing has a penalty cost pchange, then we can add a new element of objective function as follows:

Secondly, a routing rule may appear in the real problem. It means that some activities can only be performed after a trainee finished performing some other activities. Let Priork as a set of activities that should be done before perform activity k where (the set of activities which have prior activities). A new set of constraint can be added:

Finally, an activity cannot be said to be restarted when it separated by off period (periods which is not scheduled for any activity). Constraint (8) shows that an activities will be stated as restarting if separated by off periods. Therefore, the setup constraints may be improved in the future.

ConclusionThe ILP model of Belien and Demeulemeester (2004b) has given a comprehensive view of trainee scheduling. The objective of the model is minimizing total penalty costs as a result of violating some constraints. The model has included coverage constraints, formation requirements, trainees non-availability constraints, and setup constraints. However, some development can be proposed to make the model more suitable for real condition. The development can include considering the previous schedule, routing rule, and revising the definition of setup variable in the constrain set.

RefferencesBelien, J., 2006, Exact and Heuristic Methodologies for Scheduling in Hospitals: Problems, Formulations and Algorithms, Ph.D. dissertation, Faculteit Economische en Toegepaste Economische Wetenschappen, Katholieke Universiteit Leuven, Belgium.Belien, J. and Demeulemeester, E., 2004a, Scheduling trainees at a hospital department using a branch-and-price approach, Research Report OR 0403, Katholieke Universiteit Leuven, Department of Applied Economics.Belien, J. and Demeulemeester, E., 2004b, Heuristic Branch-and-Price for Building Long Term Trainee Schedules, Research Report OR 0402, Katholieke Universiteit Leuven, Department of Applied Economics.Dexter, F., Wachtel, R. T., Epstein, R. H., Ledolter, J., Todd. M. M., 2010, Analysis of Operating Room Allocations to Optimize Scheduling of Specialty Rotations for Anesthesia Trainees, Anesthesia Analgesia, Vol 111:2, p520-524.