Maximizing the nurses’ preferences in nurse scheduling problem: mathematical modeling and a meta-heuristic algorithm

Maximizing the nurses' preferences in nurse scheduling problem: mathematical modeling and a meta-heuristic algorithm Hamed Jafari 0 Nasser Salmasi 0 0 Department of Industrial Engineering, Sharif University of Technology , Tehran , Iran The nurse scheduling problem (NSP) has received a great amount of attention in recent years. In the NSP, the goal is to assign shifts to the nurses in order to satisfy the hospital's demand during the planning horizon by considering different objective functions. In this research, we focus on maximizing the nurses' preferences for working shifts and weekends off by considering several important factors such as hospital's policies, labor laws, governmental regulations, and the status of nurses at the end of the previous planning horizon in one of the largest hospitals in Iran i.e., Milad Hospital. Due to the shortage of available nurses, at first, the minimum total number of required nurses is determined. Then, a mathematical programming model is proposed to solve the problem optimally. Since the proposed research problem is NP-hard, a meta-heuristic algorithm based on simulated annealing (SA) is applied to heuristically solve the problem in a reasonable time. An initial feasible solution generator and several novel neighborhood structures are applied to enhance performance of the SA algorithm. Inspired from our observations in Milad hospital, random test problems are generated to evaluate the performance of the SA algorithm. The results of computational experiments indicate that the applied SA algorithm provides solutions with average percentage gap of 5.49 % compared to the upper bounds obtained from the mathematical model. Moreover, the applied SA algorithm provides significantly better solutions Health systems; Nurse scheduling problem; Preference scheduling; Mathematical programming; Neighborhood structure; Meta-heuristic algorithms - & Nasser Salmasi in a reasonable time than the schedules provided by the head nurses. Introduction Healthcare services consume a considerable share of the budget in each country. Hospitals are the largest organizations in providing health care services. Nurses, as one of the major portion of hospitals human resources, account for a considerable part of a hospitals annual budget. Thus, the hospitals policy makers have to efficiently arrange the available nurses. This problem is worsened by the shortage of available nurses in many countries. For instance, it is expected a shortage of 400,000 registered nurses in the United States of America by 2020 (Janiszewski 2003). The major reasons for nursing shortage are changing work climate in hospitals, low salary paid to nurses, decline in enrollment at nursing schools, and reduction of nurses job satisfaction (Murray 2002). Lu et al. (2002) study the relationships among professional commitment and job satisfaction for registered nurses. They distribute a structured self-administered questionnaire, including the professional commitment scale, job satisfaction, and demographic data to 2197 registered female nurses with an average age of 28.56 years that 72 % of them had an associates degree. They found a positive correlation between job satisfaction and professional commitment to leave the profession. The discriminate analysis indicated low job satisfaction is the major reason of 30.5 % of nurses who leave their profession. Thus, factors that increase nurses job satisfaction are very important for policy makers. An effective way to increase the job satisfaction rate is assigning the desirable working shifts to nurses. The assignment of nurses to the shifts is called nurse scheduling problem (NSP) (De Causmaecker and Vanden Berghe 2011). In the NSP, the goal is to assign shifts to the nurses in order to satisfy the hospitals demand during the planning horizon. The NSP has been studied with several objective functions and different assumption sets. Several mathematical models, heuristic and meta-heuristic algorithms, and hybrid methods are proposed to solve the problem so far which are discussed in the following paragraphs. There are several proposed mathematical models to solve the NSP. Miller et al. (1976) develop a two-stage mathematical model to balance the trade-off between staffing coverage and schedule preferences of individual nurses. A feasible solution is generated in the first stage, and then the generated solution is improved at the second stage. Arthur and Ravindran (1981) propose a two-stage multi-objective mathematical model to solve the research problem optimally. In their approach, working days of each nurse are specified using the goal programming method at the first stage, and working shifts are assigned to nurses at the second stage. Azaiez and Al-Sharif (2005) propose a binary goal programming model to solve a multi-objective NSP. The proposed model is used for problems with at most 22 nurses. Al-Yakoob and Sherali (2007) propose a mixed integer programming model to achieve fairness in the generated employee schedules by minimizing the total sum of absolute differences between employee preference indices and central preference values. Valouxis et al. (2012) apply a two-stage mathematical programming model where at the first stage, the workload for each nurse is determined, while at the second stage, the daily shifts are assigned to the nurses. They consider only two constraints in their model: the schedule should provide a specific number of personnel for each scheduling period and a nurse can start only one shift per day. Wright and Mahar (2013) propose a centralized model for the NSP by considering minimization of costs and overtime, simultaneously. MHallah and Alkhabbaz (2013) apply a simple Operations Research tools to a common and sensitive problem. They investigate the problem of designing timetables for the nurses working in Kuwaiti health care units. In details the constraints of the problem, they propose a mixed integer linear programming model and solve the mathematical model for the case of a specific health care unit using an off-the-shelf optimizer. Moreover, Guo et al. (2014) study assigning a set of nurses to surgeries scheduled on each workday in an operating room suite. Due to significant uncertainty in surgery durations, designing schedules that obtain high nurse efficiency is complicated by the competing objective of ensuring ontime start of surgeries. For trading off between the two performance objectives, they formulate the problem as a mixed integer programming model with explicit probability modeling of uncertainty. Bard and Purnomo (2007) propose a Lagrangian-based algorithm for the cyclic NSP. The objective is to strike a balance between satisfying individual preferences and minimizing personnel costs. Belien and Demeulemeester (2008) use branch-and-price algorithm to solve the NSP problem. They present a model that integrates the scheduling process of nurses and operating rooms, simultaneously. For ease o (...truncated)