# Optimal Crew Scheduling in an Intensive Care Unit: A Case Study in a University Hospital
## 1 Introduction  
Intensive Care Units (ICUs) play a crucial role in healthcare by providing continuous and specialized care for critically ill patients. Given the high complexity and unpredictability of patient conditions, ICUs must operate 24/7, requiring a well-balanced and efficient shift-scheduling system for medical personnel. However, workforce scheduling in healthcare is inherently challenging due to numerous constraints, including labor regulations, staff preferences, fluctuating patient demand, and institutional policies. Poorly managed schedules can result in staff burnout, reduced efficiency, and compromised patient care. Traditionally, hospital shift scheduling is performed manually by senior nurses and doctors on a weekly or monthly basis, often requiring significant time and effort. [PERSON] [1] reported that manually scheduling a team of 20 nurses for a two-week period could take between 2 and 8 h of a head nurse's time. Manual scheduling is not only time consuming but also prone to errors and subjective decision making, leading to unfair shift distribution and dissatisfaction among employees. To address these challenges, mathematical models have been increasingly adopted to optimize scheduling processes. Computer-based schedulingensures fairness, reduces administrative workload, and enhances efficiency by considering multiple constraints simultaneously. The literature review examines scheduling problems related to doctors, nurses, and other healthcare personnel, categorizing existing studies in this field. Research indicates that nurse and doctor scheduling challenges are predominantly addressed using mathematical modeling, heuristic approaches, or hybrid methods that combine both techniques. [PERSON] (2008) conducted a comprehensive review of scheduling and rostering in healthcare and other sectors, highlighting the complexity of workforce management in hospital settings. Moreover, numerous studies have explored nurse scheduling, operating room scheduling, surgery scheduling, and other hospital workforce optimization problems ([PERSON], 2010). Several studies specifically focus on nurse scheduling, employing various mathematical and heuristic techniques. [PERSON] (2008) introduced an integer linear programming model, while [PERSON] (2010) proposed a mixed-integer goal programming approach. [PERSON] (2010) developed a goal programming model, whereas [PERSON] (2010) and [PERSON] (2011) utilized integer and 0-1 goal programming, respectively. [PERSON] (2010) presented a deterministic mathematical model, [PERSON] (2010) applied optimization techniques, and [PERSON] (2010) implemented a branch-and-price algorithm. Heuristic-based approaches have also been widely explored: [PERSON] et al. [PERSON] (2011) developed a hybrid heuristic algorithm, while [PERSON] et al. [PERSON] (2012) proposed a metaheuristic algorithm. Additionally, [PERSON] (2013) addressed nurse rostering using the Monte Carlo Tree Search heuristic method. Some studies combine mathematical modeling with heuristic approaches, such as those by [PERSON] and [PERSON] [PERSON] and [PERSON] (2014) and [PERSON] and [PERSON] and [PERSON] (2015), which integrate both methodologies to enhance nurse scheduling efficiency. Although doctor scheduling shares similarities with nurse scheduling, it has received considerably less attention in the literature. The complexity of physician schedules, combined with the larger number of nurses in hospital departments, makes it more challenging to manage both roles simultaneously. Recent research has started addressing this gap, with studies by [PERSON] (2016), [PERSON] (2017), [PERSON] (2018), [PERSON] (2019), and [PERSON] (2020) proposing various solutions. [PERSON] (2017) introduced a mathematical optimization model, while [PERSON] (2019) applied a stochastic programming approach. [PERSON] (2016) utilized an adaptive large-neighborhood-search heuristic method, and [PERSON] (2018) developed a hybrid GRASP method, integrating mathematical and heuristic modeling.