A Proposed Formulation for Long-Term Medical Trainee Scheduling Problem

Samsul Amar1, I G. B. Budi Dharma, Andi Sudiarso

Department of Mechanical and Industrial Engineering, Universitas Gadjah MadaJl. Grafika 2 Yogyakarta, Indonesia, 55281

1email: sams_amar@yahoo.com

AbstractMedical trainee scheduling is a complicated problem due to the scale, the constraints and the solving method. Some heuristics and exact methods have been developed to solve the problem. Branch and Price is one .

Keyword: medical trainee scheduling, exact method, heuristic/meta-heuristic method.

IntroductionMedical trainee program is an important part of medical education system. A trainee should finish this program before getting the license as a medical doctor. The reason is that he/she needs to get exercise in real problems under supervision of senior physicians before he/she can treat patients independently. In other hand, medical trainee program also becomes a part of personnel fulfillment of the hospitals and the clinics, which are related to the program. Before starting a medical training program, a schedule needs to be created considering some rules to make the program effective. Although medical trainee scheduling problem is important, only a few studies published about this problem compared with other scheduling problems in hospital such as nurse scheduling problem (NSP) and operating room scheduling. Burke et.al (2004) succeeded to compile 140 papers from international journal and 14 doctoral dissertations about nurse rostering (a class of nurse scheduling problem) while Bekkan (2010) can only compile ten of medical trainee scheduling problems. Scheduling trainees to perform some activities in a certain period is complicated. The problem contains a huge number, often hundreds of thousands, of variables and constraints. Medical trainee scheduling problem is more complicated than NSP (Belien, 2007) because of some conditions. First, medical trainee scheduling problem is usually deal with longer period than NSP; consequently, the number of variable will grow dramatically. Second, trainees still have to complete their education, consequently, they have to perform much various activities/posts, and the capacity of each post is usually limited. These will increase the number of variables and constraints. Third, unlike NSP, an activity for a trainee cannot be separated by other activities or by off time (vacancy period). Trainee also cannot be replaced by other trainee if he/she is not able to perform an activity. These conditions will add some complicated constraints in the modeling. The main issue in solving this kind of problem is the computation time. Using heuristic/ meta-heuristic method can reduce the computation time but the solution may not be the optimal one. Using exact method such as branch & bound and branch & price can produce the optimal solution but the computation time may be not reasonable for most problems.Formulation is an important step before solving the problem. A problem may be formulated into two or more formulations. A good formulation will lead to a better performance in term of faster computation time. This paper proposes a binary integer programming formulation to solve a medical trainee scheduling problem and compares the performance of the proposed formulation with the typical formulation using standard branch and bound method. The objective of this paper is to propose a formulation and to show that a good formulation can lead to much more better performance than other formulation.

Research MethodAt first, the problem relating to hospital trainee scheduling is explored. The exploration includes theory and literature work, and real problem general observation. After define the problem, the next activities are reviewing the theories and literatures related to the problem, and observe the real problem of hospital trainee scheduling. Observation and interview are done to get a clear perspective of the real problem. The subjects of interview are the scheduler team, head of department, senior physicians, trainees and personnel of the posts. To build the model for the clinical rotation scheduling, some data need to be collected. The data includes: the scheme of clerkship program, rules and regulation, capacity of each post and the period in which the trainee should perform, how the scheduler make the schedule at present, and preferences of the policy maker.Two models of hospital trainee scheduling is then constructed, i.e. the ordinary type of model used by most researchers (Model 1) and the proposed model (model 2). To compare the models, 8 problem instances are generated. Total computation time is the main parameters to measure performance of the model. In order to solve the problem instances using Model 1 and Model 2, MATLAB and LP-Solve software are used in a . The reason of using MATLAB is its capability to perform matrix operation easily. The matrix that represents the model is then solved using LP-Solve, an open source-software to solve linear programming problem.

Medical Trainee Scheduling ProblemThe main issue of medical trainee scheduling is how to schedule the trainees to conduct the activities (to stay in the posts). The trainees should conduct each activity in a certain period, without violating some constraints. The constraints usually emerge from the decision maker, law and regulation from the governments, safety, fairness, and trainee satisfaction. The constraints may include: 1) coverage/capacity of the posts, 2) number of period that a trainee should perform an activity, 3) preference of the trainee, 4) setup constraints (each trainee can only start each activity once), 5) no more than one activity per period for every trainee, 6) previous schedule that has been made, 7) routing rule of the activities, 8) maximum number of consecutive holidays, 9) maximum number of consecutive activities, and 10) minimum inter-arrival period of two trainees in a post. Ideally, all constraints are satisfied. However, this condition is difficult to be met. Therefore, the objective function of the model is usually to minimize the violation of the constraints. A penalty is charge whenever a constraint is violated. The value of the penalty can be various based on the negative effect of the violation. The set of constraints that can be violated are called soft constraints while the strict constraints that cannot be violated are called hard constraints.

SummaryMedical trainee scheduling problem is a complicated problem. Some heuristic and exact approaches have been reported able to solve real life problems. Branch & Bound and Branch & Price are two exact methods that have been used to solve such problem. Heuristic/meta-heuristic approaches that have been applied to solve medical trainee scheduling are Rounding Heuristic and Genetic Algorithm. The exact methods can find the optimal solution while heuristic/meta-heuristic can find a good solution in relatively sorter computation time but cannot guarantee to get the optimal solution. Other exact and heuristic methods such as Column Generation, Dynamic Programming, Branch & Cut, Branch & Cut & Price, Tabu Search, Simulated Annealing, Fuzzy Logic and Artificial Intelligence approaches may be considered as solving methods for medical trainee scheduling. Designing a good formulation and preprocessing technique may also able to reduce the computation time.Combination of exact and heuristic approach has been described in some research. The research conclude that the combination has some benefit and the next research in this area is promising.

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