Optimal Crew Scheduling in an Intensive Care Unit: A Case Study in a University Hospital

Abstract

Effective crew scheduling in hospitals with multiple personnel groups is essential for time efficiency and fair workload distribution. This study focuses on optimizing shift scheduling for a team of nurses, doctors, and caregivers working in the Pediatric Intensive Care Unit (PICU) of a university hospital. The model is implemented and solved using GAMS 23.5 software to minimize total staffing costs while ensuring balanced shift allocations. The scheduling process in PICUs is influenced by multiple factors, including staff skills, experience levels, personal preferences, contractual agreements, and hospital demands. Since these factors affect doctors, nurses, and caregivers differently, the model considers each personnel group separately while integrating them into a unified optimization framework. The proposed model successfully generates an annual optimal shift schedule for 10 doctors, 14 nurses, and 9 caregivers, ensuring equitable workload distribution and compliance with hospital regulations. By implementing this scheduling approach, employee satisfaction is enhanced, service quality is improved, and administrative workload is reduced. Additionally, the model ensures a well-balanced distribution of responsibilities, minimizes scheduling inefficiencies, and significantly reduces the time required for shift planning. Ultimately, this study provides a fast, fair, and cost-effective solution for hospital workforce management.

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. Huarng [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 scheduling ensures 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. Ernst [2] 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 [3].

Several studies specifically focus on nurse scheduling, employing various mathematical and heuristic techniques. Warner [4] introduced an integer linear programming model, while Trivedi [5] proposed a mixed-integer goal programming approach. Huarng [1] developed a goal programming model, whereas Belien [6] and Azaiez [7] utilized integer and 0–1 goal programming, respectively. Maenhout [8] presented a deterministic mathematical model, Rönnberg [9] applied optimization techniques, and Boyer [10] implemented a branch-and-price algorithm. Heuristic-based approaches have also been widely explored: Burke et al. [11] developed a hybrid heuristic algorithm, while Burke et al. [12] proposed a metaheuristic algorithm. Additionally, Goh [13] addressed nurse rostering using the Monte Carlo Tree Search heuristic method. Some studies combine mathematical modeling with heuristic approaches, such as those by Dowsland and Thompson [14] and Bard and Purnomo [15], 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 Mansini [16], Marchesi [17], Cildoz [18], Akbarzadeh [19], and Liu [20] proposing various solutions. Marchesi [17] introduced a mathematical optimization model, while Akbarzadeh [19] applied a stochastic programming approach. Mansini [16] utilized an adaptive large-neighborhood-search heuristic method, and Cildoz [18] developed a hybrid GRASP method, integrating mathematical and heuristic modeling. Liu [20] focused on the analytical evaluation of waiting times using the Markov chain and uniformization method.

Despite the vast research on hospital workforce scheduling, an integrated approach that simultaneously optimizes nurses, doctors, and caregivers remains largely unexplored. Little [21] reviewed nurse and doctor scheduling but noted that most studies treat these as distinct problems with unique objectives and constraints. Existing models often lack flexibility, require significant computational time, or fail to account for interdependencies between different personnel groups. For instance, nurse scheduling models frequently disregard physician shift constraints, leading to inefficiencies and potential staff dissatisfaction.

To address these gaps, this study proposes a unified Mixed-Integer Programming (MIP) model that optimizes shift scheduling for nurses, doctors, and caregivers within a single framework. By integrating these roles, the model ensures a balanced workload, fair shift distribution, and improved operational efficiency in Pediatric Intensive Care Units (PICUs). Unlike previous studies that focus solely on either nurse or doctor scheduling, this model accounts for interdisciplinary interactions and enhances hospital-wide scheduling efficiency. Furthermore, the proposed model is adaptable to different hospital settings, including emergency units, operating theaters, and general wards. By modifying its constraints and parameters, the model can accommodate various hospital policies and workforce structures.

This study demonstrates that the MIP-based scheduling system significantly reduces scheduling conflicts, improves fairness, and enhances overall hospital efficiency compared to traditional manual methods. The model’s scalability ensures its applicability in both small-scale hospitals with limited staff and large healthcare facilities with complex scheduling requirements.

The remainder of this paper is structured as follows: Section 2 describes the methodology, including the problem description and limitations, mathematical model, and optimization approach. Section 3 presents the results, while Section 4 discusses the findings in relation to existing research. Finally, Section 5 concludes with recommendations for future research and potential applications of the model in different healthcare settings.

2. Material and Method

2.1. Problem Description and Limitations

Effective workforce scheduling in hospitals is a complex and critical task that directly impacts healthcare service quality, staff well-being, and operational costs. In Pediatric Intensive Care Units (PICUs), ensuring continuous and adequate staffing levels is particularly challenging due to the high variability of patient demands, staff availability constraints, and institutional regulations. Traditionally, shift scheduling is manually handled by senior nurses and hospital administrators, which is not only time consuming but also prone to inefficiencies and unfair shift distributions.

The problem addressed in this study involves the optimization of hospital staff scheduling, specifically focusing on nurses, junior doctors, senior doctors, and caregivers. The goal is to develop a mathematical model that minimizes operational costs while ensuring fair workload distribution and compliance with hospital policies. The proposed model considers multiple constraints, such as shift requirements for different personnel groups, legal and institutional regulations regarding work hours, the fair allocation of night shifts to prevent staff burnout, ensuring that required skill levels are met for each shift, and minimizing sudden staff shortages and overtime costs.

This study formulates the scheduling problem as a Mixed-Integer Programming (MIP) model, designed to generate optimal shift assignments that satisfy all predefined constraints while improving workforce efficiency. The model is implemented and solved using the GAMS 23.5 software, providing a fast and scalable approach for hospital administrators.

Although our proposed model offers several advantages, it has certain limitations. First, the model was tested using data from a single hospital (Çukurova University Hospital), which may limit its generalizability to other settings. Second, the model assumes static shift durations and predefined constraints, meaning it does not yet account for real-time changes such as emergency patient influx or last-minute staff shortages. Third, our approach does not include regulatory and contractual constraints, which vary across different countries and healthcare systems. Future work should focus on enhancing the model’s flexibility by incorporating real-time data integration and adaptive scheduling algorithms.

2.2. Material

In this study, a unit of Çukurova University, Adana, Türkiye which is a university hospital and one of the largest hospitals in the Mediterranean region of Turkey, is discussed. Information regarding the current shift scheduling and working conditions was obtained through interviews with the Pediatric Intensive Care Unit (PICU) supervisor and the responsible PICU nurse in charge of scheduling, as well as the responsible junior assistant doctor at the PICU service of Faculty of Medicine, Çukurova University. Data regarding the number of doctors, caregivers, and nurses working in the PICU, types of shifts, shift rotations, planning period, crew requirements for each shift, monthly patient numbers, and other relevant information regarding the functioning of the system were collected. Based on the existing scheduling practices in the PICU, the objective function and constraints of the mathematical model were determined. The mathematical programming model developed the entire crew, including caregivers, nurses, and junior/senior assistant doctors working in the unit. The developed mathematical programming model was solved using the GAMS 23.5 software program.

In this study, it was assumed that each day consisted of two shifts, each month had 30 days, and the night shift wages were 50% higher than the day shift wages.

2.3. Method

The mathematical model developed in this study is based on principles of integer programming, similar to methodologies used in previous research [1,4,6,7]. However, unlike traditional models focusing solely on either nurses or doctors, our approach integrates multiple healthcare roles within a unified framework. The constraints and decision variables used in this model were formulated based on hospital-specific regulations and real-world shift-allocation challenges. All formulas presented in this section were either adapted from or inspired by existing literature, with modifications made to accommodate interdisciplinary scheduling and problem requirements.

The MIP (Mixed-Integer Programming) model has been developed in this work. The notation used for mathematical modeling of the problem is discussed in the following.

Sets:
k set of shifts	k = 1, …, K
h set of nurses	h = 1, …, H
d set of junior assistant doctors	d = 1, …, D
u set of senior assistant doctors	u = 1, …, U
b set of caregivers	b = 1, …, B
a set of days	a = 1, …, A
t set of months	t = 1, …, T

Decision variables and parameters are available for many different situations in the developed mathematical model (

Table 1. Parameters and decision variables for the model.

Nomenclature	Description
ptnt	number of inpatients
cstbk	caregiver cost per shift
csthk	nurse cost per shift
cstdk	junior assistant doctor cost per shift
cstuk	senior assistant doctor cost per shift
dh	day shift time for nurse
nh	night shift time for nurse
ch	shift time for caregiver
hsn	hours a nurse should work in a month
csn	hours a caregiver should work in a month
ns	number of night shifts a nurse can do in a month
yb	the number of inpatients that the nurse can take care of
Uc	number of nurses required to work on day shift
Lc	number of nurses required to work on night shift
Uu	number of senior assistant doctor required to work on day shift
Lu	number of senior assistant doctor required to work on night shift
ncd	number of night shifts that novice doctors must keep
mind	minimum number of night shifts senior assistant doctors must keep
maxd	maximum number of night shifts senior assistant doctors must keep
dhkhat	assignment of nurse h to shift k in day a in period t
ddkdat	assignment of junior assistant doctor d to shift k in day a in period t
dukuat	assignment of senior assistant doctor u to shift k in day a in period t
dbkbat	assignment of caregiver b to shift k in day a in period t

).

2.3.1. Minimize

When the proposed problem is examined, a cost function is proposed for the mathematical programming model that aims at the minimum total cost.

Cost minimization ensures that hospital resources are used efficiently without excessive overtime or unnecessary staffing, which is a key concern in healthcare budgeting [16]. Meanwhile, workload balancing is essential to prevent staff burnout and maintain high service quality, as uneven shift allocations can negatively impact both employee satisfaction and patient care outcomes [13]. So, the objective function in this model prioritizes cost minimization and workload balancing, as these two factors are critical in hospital workforce management.

The objective function is shown by Equation (1) and consists of nurse cost per shift, junior assistant doctor cost per shift, senior assistant doctor cost per shift, and caregiver cost per shift, respectively.

$$\begin{gathered} (\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{h}}^{\mathrm{H}}\sum _{\mathrm{a}}^{\mathrm{A}}\sum _{\mathrm{t}}^{\mathrm{T}}{\mathrm{dh}}_{\mathrm{khat}}.{\mathrm{csth}}_{\mathrm{k}})+(\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{d}}^{\mathrm{D}}\sum _{\mathrm{a}}^{\mathrm{A}}\sum _{\mathrm{t}}^{\mathrm{T}}{\mathrm{dd}}_{\mathrm{kdat}}{.\mathrm{cstd}}_{\mathrm{k}})+(\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{u}}^{\mathrm{U}}\sum _{\mathrm{a}}^{\mathrm{A}}\sum _{\mathrm{t}}^{\mathrm{T}}{\mathrm{du}}_{\mathrm{kuat}}{.\mathrm{cstu}}_{\mathrm{k}}) \\ +(\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{b}}^{\mathrm{B}}\sum _{\mathrm{a}}^{\mathrm{A}}\sum _{\mathrm{t}}^{\mathrm{T}}{\mathrm{db}}_{\mathrm{kbat}}{.\mathrm{cstb}}_{\mathrm{k}}) \end{gathered}$$

(1)

2.3.2. Constraints

In the study, information about the nurses, junior/senior doctors and patient caregivers in the team was stated separately in the constraint equations.

2.3.3. Nurse Constraints

Equation (2) is the constraint equation that ensures that there is no shift one day after the night shift. Equation (3) shows the mandatory hours each nurse must work in a month. Equation (4) shows that there must be at least one nurse per shift. Equation (5) shows the maximum number of shifts a nurse can hold in a month. Equation (6) shows that the head nurse should only be on the day shift. Equation (7) shows the number of nurses required per inpatient. Equation (8) states that the decision variable is binary.

$${\mathrm{dh}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{hat}}+{\mathrm{dh}}_{\mathrm{k}{\mathrm{h}}^{\prime }\mathrm{a}+1^{\prime }\mathrm{t}} \le 1, \forall \mathrm{h},\forall \mathrm{k},\forall \mathrm{a},\forall \mathrm{t}$$

(2)

$$\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{dh}}_{{}^{\prime }\mathrm{k}=1^{\prime }\mathrm{hat}}.\mathrm{dh}+\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{dh}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{hat}}.\mathrm{nh}=\mathrm{hsn}, \forall \mathrm{h}, \forall \mathrm{t}$$

(3)

$$\sum _{\mathrm{h}}^{\mathrm{H}}{\mathrm{dh}}_{\mathrm{khat}}\ge 1, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(4)

$$\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{dh}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{hat}}\le \mathrm{ns}, \forall \mathrm{h}, \forall \mathrm{t}$$

(5)

$${\mathrm{dh}}_{{}^{\prime }\mathrm{k}=2^{″}\mathrm{h}=1^{\prime }\mathrm{at}}=0, \forall \mathrm{h}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(6)

$$\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{h}}^{\mathrm{H}}\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{dh}}_{\mathrm{khat}}.\mathrm{yb} \ge {\mathrm{ptn}}_{\mathrm{t}}, \forall \mathrm{t}$$

(7)

$${\mathrm{dh}}_{\mathrm{khat}} \in \left\{\mathrm{0,1}\right\}, \forall \mathrm{h}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(8)

2.3.4. Caregivers Constraints

Equation (9) is the constraint equation that ensures that there is no shift one day after the night shift. Equation (10) shows the mandatory hours each caregiver must work in a month. Equations (11) and (12) show at least how many caregivers should be on night and day shifts. Equation (13) indicates that caregivers with special conditions should work during the day. Equation (14) states that the decision variable is binary.

$${\mathrm{db}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{bat}}+{\mathrm{db}}_{\mathrm{k}{\mathrm{b}}^{\prime }\mathrm{a}+1^{\prime }\mathrm{t}} \le 1, \forall \mathrm{b}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(9)

$$\sum _{\mathrm{k}}^{\mathrm{K}}\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{db}}_{\mathrm{kbat}}.\mathrm{ch}=\mathrm{csn}, \forall \mathrm{b}, \forall \mathrm{t}$$

(10)

$$\sum _{\mathrm{b}}^{\mathrm{B}}{\mathrm{db}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{bat}}\ge \mathrm{Lc}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(11)

$$\sum _{\mathrm{b}}^{\mathrm{B}}{\mathrm{db}}_{{}^{\prime }\mathrm{k}=1^{\prime }\mathrm{bat}}\ge \mathrm{Uc}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(12)

$${\mathrm{db}}_{{}^{\prime }\mathrm{k}=2^{″}\mathrm{b}=1^{\prime }\mathrm{at}}+{\mathrm{db}}_{{}^{\prime }\mathrm{k}=2^{″}\mathrm{b}=1^{\prime }\mathrm{at}}=0, \forall \mathrm{b}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(13)

$${\mathrm{db}}_{\mathrm{kbat}} \in \left\{\mathrm{0,1}\right\}, \forall \mathrm{b}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(14)

2.3.5. Junior/Senior Assistant Doctors’ Constraints

Equations (15) and (16) are constraint equations that ensure that there is no shift one day after the night shift for junior assistant doctors and senior assistant doctors, respectively. Equation (17) shows at least how many junior assistant doctors should be in each shift. Equations (18) and (19) show at least how many senior assistant doctors should be in night and day shifts. Equation (20) shows the night watch that junior assistant doctors should keep, and Equations (21) and (22) show the night watch limits that should be kept for senior assistant doctors. Equation (23) states that the decision variables are binary.

$${\mathrm{dd}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{dat}}+{\mathrm{dd}}_{\mathrm{k}{\mathrm{d}}^{\prime }\mathrm{a}+1^{\prime }\mathrm{t}} \le 1, \forall \mathrm{d}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(15)

$${\mathrm{du}}_{{}^{\prime }\mathrm{k}+1^{\prime }\mathrm{uat}}+{\mathrm{du}}_{\mathrm{k}{\mathrm{u}}^{\prime }\mathrm{a}+1^{\prime }\mathrm{t}} \le 1, \forall \mathrm{u}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(16)

$$\sum _{\mathrm{d}}^{\mathrm{D}}{\mathrm{dd}}_{\mathrm{kdat}}\ge 1, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(17)

$$\sum _{\mathrm{u}}^{\mathrm{U}}{\mathrm{du}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{uat}}\ge \mathrm{Lu}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(18)

$$\sum _{\mathrm{u}}^{\mathrm{U}}{\mathrm{du}}_{{}^{\prime }\mathrm{k}=1^{\prime }\mathrm{uat}}\ge \mathrm{Uu}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(19)

$$\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{dd}}_{’\mathrm{k}=2’\mathrm{dat}}=\mathrm{ncd}, \forall \mathrm{k}, \forall \mathrm{d}, \forall \mathrm{t}$$

(20)

$$\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{du}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{uat}}\ge \mathrm{mind}, \forall \mathrm{k}, \forall \mathrm{u}, \forall \mathrm{t}$$

(21)

$$\sum _{\mathrm{a}}^{\mathrm{A}}{\mathrm{du}}_{{}^{\prime }\mathrm{k}=2^{\prime }\mathrm{uat}}\le \mathrm{maxd}, \forall \mathrm{k}, \forall \mathrm{u}, \forall \mathrm{t}$$

(22)

$${\mathrm{dd}}_{\mathrm{kdat}}, {\mathrm{du}}_{\mathrm{kuat}}\in \left\{\mathrm{0,1}\right\}, \forall \mathrm{d}, \forall \mathrm{u}, \forall \mathrm{k}, \forall \mathrm{a}, \forall \mathrm{t}$$

(23)

3. Results

To validate the proposed model, past shift schedules from the hospital were used as test data. The optimized schedules generated by the model were compared with historical manual schedules in terms of efficiency, fairness, and cost-effectiveness. Additionally, sensitivity analyses were conducted to evaluate the model’s adaptability under different constraints, such as varying patient admissions and staff availability. The results indicate that the MIP model significantly reduces scheduling errors and balances workload distribution more effectively than traditional methods. Furthermore, comparative testing with real data from previous shifts confirmed that the proposed model ensures compliance with hospital policies while improving overall operational efficiency.

In the study, a total of 14 nurses, including 1 head nurse, 9 caregivers with special conditions, 5 junior assistant doctors, and 5 senior assistant doctors were considered, with 2 shifts per day. The model developed based on past data includes the admission numbers of the past 12 months, which are respectively entered into the system as 54, 50, 60, 54, 68, 53, 42, 64, 49, 57, 58, and 64.

Real data regarding nurses, caregivers, junior assistant doctors, and senior assistant doctors were collected from a hospital for the study. Based on this information, the scheduling results were obtained.

3.1. Results for Junior Assistant Doctor

Five junior assistant doctors participated in the study, and these doctors were named D1, D2, D3, D4 and D5. Sample planning results for the junior assistant doctor coded D1 are presented in

Table 2. Sample scheduling for junior assistant doctor ‘D1’.

	1	2	3	4	5	6	7	8	9	10	11	12
1	D1	D1	D1	D1	D1	D1	D1-G	D1	D1-G	D1	D1	D1
2	D1	D1-G	D1-G	D1	D1-G	D1-G		D1-G		D1-G	D1-G	D1-G
3	D1			D1			D1		D1
4	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G
5
6	D1-G		G	D1-G	D1-G	D1-G	D1-G	D1-G	D1-G	G	D1-G	D1-G
7		D1-G
8	D1		D1	D1	D1	D1	D1	D1	D1	D1	D1	D1
9	D1	D1	D1	D1	D1	D1	D1	D1	D1	D1	D1	D1
10	G			G	G	G
11			G				G		G	G
12											D1-G	D1-G
13	D1-G	D1		G	G	G		D1-G
14		G	D1				D1		G	D1
15	D1			D1	D1	D1	G	D1		G
16	G	G	D1-G	G				G	D1		G	G
17							G		G	G
18
19		D1		D1	D1	D1		D1			D1	D1
20	D1	D1		D1	D1	D1		D1		D1	D1	D1
21			D1				D1		D1	D1	D1	D1
22		D1	D1		D1	D1	D1		D1	D1	G	G
23	G	G	D1	G	G	G	D1	G	D1	D1
24			G				G		G	G
25												G
26			G								G
27	G	D1-G		G	G	G		G
28			D1				D1		D1	D1
29	D1	D1	D1-G	D1	D1	D1	D1-G	D1	D1-G	D1-G	D1	D1
30	D1-G	D1-G		D1-G	D1-G	D1-G		D1-G			D1-G	D1-G

. In Table 2, “D1” represents day shift and “G” represents night shift. It is assumed that a year is 12 months and each month is 30 days.

It is observed that the distribution of day (shift = 1) and night (shift = 2) shifts for all junior assistant doctors are fair throughout the year. Figure 1 illustrates the annual distribution of day and night shifts for all junior assistant doctors.

3.2. Results for Senior Assistant Doctor

Five senior assistant doctors participated in the study and these doctors were namedU1, U2, U3, U4 and U5. Sample planning results for the senior assistant doctor coded U1 are presented in

Table 3. Sample scheduling for senior assistant doctor ‘U1’.

	1	2	3	4	5	6	7	8	9	10	11	12
1	U1-G	U1-G	U1	U1	U1	U1-G	U1-G	U1-G	U1	U1-G	U1-G	U1-G
2			U1	U1	U1				U1
3	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1
4	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1
5	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1
6		G			G	G
7	U1-G		U1-G	U1-G			U1-G		U1		U1	U1
8		U1			U1	U1	U1	U1		U1
9	U1-G	G	U1-G	G	G	U1-G	G	U1-G	U1-G	G	G	U1-G
10
11	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1
12	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1
13	U1		U1-G	U1-G	U1	U1		U1-G	U1-G	U1	U1	U1
14										G
15			G	G	G	G		G	G		G
16	U1	G					G			U1-G		U1
17	G				G	G
18		U1	U1				U1	G		U1	G	G
19	U1	U1			G	U1	G				U1-G
20	U1-G	U1		U1			U1					U1
21		U1		G	U1			U1	U1			U1
22		U1					U1
23			U1	U1	G	G	U1		U1			U1
24		G	U1-G	U1					U1-G	U1	U1
25				U1	U1	U1		U1		U1	U1
26	U1	U1-G	U1	U1	U1	U1	U1-G	U1	U1	U1	U1	U1-G
27	U1			U1	U1	U1		U1		U1	U1
28	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1	U1
29	U1-G	U1		G	U1	U1	U1	U1	U1-G	U1-G	U1	U1-G
30		U1	U1-G		U1	U1	U1	U1			U1

. In Table 3, “U1” represents day shift and “G” represents night shift. It is assumed that a year is 12 months and each month is 30 days.

There appears to be a fair distribution of the annual day and night shifts of all senior assistant doctors. Figure 2 shows the annual distribution of day shifts (shift = 1) and night shifts (shift = 2) of all senior doctors.

3.3. Results for Nurse/Head Nurse

A total of 14 nurses, including 1 head nurse, were included in the study. Nurses were named from N1 to N14.

Table 4. Sample scheduling for ‘N1’ head nurse and ‘N2’ nurse.

	1	2	3	4	5	6	7	8	9	10	11	12
1	N1-G	N1	N1-G	N1-G	N1-G	N1	N1-G	N1-G	N1-G	N1-N2	N1-G	N1-G
2	N1	N1-G	N1	N1	N1	N1-G	N1	N1	N1	N1-G	N1	N1
3	N1-G		N2-G	N1-G	N1-G	N1	N1-G	N1-G	G	N1	N2-G	G
4	N1	N1-G	N1	N1	N1	N1			N1	N1-G		N1
5	N1	N1	G		N2	G	G	N1			G	N1
6		N1		N1			N1	N1	N1-N2	N1		N1-G
7	N1	N1	N1	N1	N1-G	N1			N1-G	N1	N1
8	N2	N1-N2	N1	N1	N1	N1-N2	N1	N1			N2	G
9	G	N1	N1			N1		N2		N1
10	N1	N1	N1	N2		N1-G	N1-N2	N1-G		N1-G	N1-G	N1
11	N1	G		N1-G	N1-N2				G	N1	N1	N1-N2
12	N2-G	N1		N1	N1	N2	N1	N2-G			N1	N1
13	N1	N1-N2		G		G	N1-N2	N1	N1	N1		N2
14	N1		N1-N2	N1	N1	N1	N1-G	N1-G	N1	N1-N2		N1-G
15		N1-G		N1-N2	N1	N1		N1	N1-N2		N1-N2	N1
16		N1	N1	N1		N2	N1	N1	N1-N2	G	N1-N2	N1
17						N1-G	N1-G	N1	N1-G	N1
18	N1		N2	N2		N1		N1		N1-G	N1
19	N1-G	N2-G		N2-G	N1-N2-G	N1	N1-N2	N1	G		N2	N1
20	N1		N1-N2-G		N1		N1	N1-G	N1	N1	N1
21				N1	N1-G	N1	N1-G		N1	N1	N1	N1-N2
22	N1	N1	G		N1		N1	N1-G	N1	N1	N1-G	N1
23	N1	N1	N1	N1-G			N1		N1-N2		N1	N1
24	G	N1-G	G	N1	N1-N2-G	N1-N2-G	N1-N2		N1	G	N1-G	N1-G
25		N1	N1	N1	N1	N1	N1	N2	N1	N1	N1
26	N1-N2	G	N1	N1-G		G	G	N1	N1	N2-G	N1	G
27	N1-N2-G		N1-G		N1	N1		N2	G		N1
28	N1	N1-N2		N1	N1-G	N1		N1	N1	N1	N1-G	N1
29	N1-G	N1-G	N1-G	N1-G	N1	N1	N1-G	N1	N1-N2	N1	N1	N1
30	N1	N1	N1	N1	N1-G	N1-G	N1	N1-G	N1-G	N1-G	N1-G	N1-G

presents the scheduling results for the head nurse (N1) and nurse “N2” as samples. Since it is specified in the constraint equations that the head nurse should not have night shifts, the notation “G” indicates the days when nurse N2 is scheduled for night shifts. Table 4 depicts the results generated for one year.

Figure 3 shows the overall distribution of annual day (shift = 1) and night (shift = 2) shifts for all nurses. As the head nurse, nurse N1 does not have night shifts. When observing the distribution of all nurses except for nurse N1, it can be observed that the distribution is fair and equitable.

3.4. Results for Caregiver/Special-Needs Caregiver

In the study, nine caregivers are considered in the model. Two are individuals with special needs.

Table 5. Sample scheduling for ‘C1’ special-needs caregiver and ‘C6’ caregiver.

	1	2	3	4	5	6	7	8	9	10	11	12
1	C6	G	C6-G	C6		C6		C6-G		C6
2	C6			C6	C6	C6	C6		C6-G	C6	C1-G
3	C6	C6	C6	C6	C6	C6	C6	G	C1	C6-G	C1	C6
4	C6	C6	C6	C6-G	C6	C6	C6		C6		C1	C6
5	C6	C6	G		C6	C6	C6	C1	C6-G	C6		C1-C6
6	C6	C6		C6-G	C6		G			C6		C6
7	C6	C6	C1-C6	C1	C6		C1	C6-G	C6	C6		C1-C6
8	C6	C6-G	C6-G	C6	C6-G	C6	C6		C6	C6	C6
9	C6			C6		C6	C6-G	C6-G	C6		C1-C6	C1-C6
10	C1-C6	C6	C6-G	C1	C6	C6	C1			C6-G
11	C1	C1	C1	C1-C6-G	C6	C1-G	C1	C1-G			C1-C6	C6
12		C1-G			C6	C1			C6			C1
13	C1		C6-G	C6	C6	C6	C6		C6	C1	C1	C1-C6
14	C6		C1	C6	C1			C6	C1-C6	C1	C1	G
15	C1	C1-G	C1		C1-C6-G		C6	C1-G		C6	C1-G	C1
16	C1	C1	C1	C1	C1	C1	C1-G	C1	C1	C1	C1	C1-C6
17	C1	C1	C1-G	C1	C1	C1-G	C1	C1-G	C1	C1-C6	C1-C6	C1
18	G	C1	C1	C1	C1	C1	C1	C1	C1-G	C1	C6	C1
19		C1	C1	C1	C1	C1-G	C1	C1	C1	C1	C1-C6
20	C1		C1	C1	C1	C1	C1-G	C1	C1-G	C1-G	C6	C1
21	C1-G	C1	C1-G	C1	C1	C1	C1	C1		C1	C1	C1-G
22	C1	C6-G	C1		C1		C1	C1	G	C1	C6
23	C1	C1	C1	C1	C1	C1-G	C1	C1	C1			C1-C6
24	C1	C1	C1	C1	C1		C1	C1	C1-G	C1	C1-C6-G	C1
25	C1	C1-G	C1	C1-C6	C1	C1	C1	C1-G	C1	C1-G		C1
26	C1-G	C1	G			C1	G		C1	C1	C1	C6
27		C1		C1-G	C1	C1	C1	C1-C6	C1	C1	C1	G
28	C1-G	C1-C6		C1	C1	C1-C6		C6	C1	G		C1
29	C1	C1	C1	C1	C1	C1-G	C1-C6-G	C1-C6	C1	C1	C6	G
30				C6		C1			C1	C1	C6-G

shows the C1 plan from special-needs caregivers and the C6 plan from other caregivers. In the constraint equations, it is stated that special-needs caregivers should not be on the night shift. Therefore, the expression marked as G indicates the days that caregiver C6 should be on the night shift. Table 5 is created for one year.

Figure 4 shows the total distribution of day (shift = 1) and night (shift = 2) shifts for all caregivers. Caregivers C1 and C2 do not have night shifts as they are designated for special needs.

4. Discussion

This study highlights the effectiveness of a Mixed-Integer Programming (MIP) model in optimizing hospital staff scheduling by simultaneously addressing the unique requirements of nurses, junior and senior doctors, and caregivers in a Pediatric Intensive Care Unit (PICU). Unlike traditional scheduling methods that treat these personnel groups separately, this integrated approach ensures fair shift distribution, cost minimization, and compliance with operational constraints, ultimately leading to improved hospital efficiency and staff satisfaction.

The findings align with previous research [1,7], which emphasizes that mathematical models significantly enhance scheduling efficiency compared to manual methods. Manual scheduling is inefficient and error prone, leading to staff dissatisfaction. By leveraging optimization techniques, this study demonstrates how an automated approach can produce more balanced and efficient schedules in a fraction of the time. Recent advancements by Yinusa and Faezipour [22] further support the integration of resource allocation with scheduling, reinforcing the need for data-driven workforce management strategies that align staffing levels with patient care demands. Additionally, studies such as that by Böðvarsdóttir et al. [23] highlight the importance of adaptability in real-world healthcare environments, a key feature that could be further enhanced in future iterations of this model.

One of the major contributions of this study is its holistic approach to staff scheduling, which considers the interdependencies between different healthcare roles. While previous studies have focused on nurse scheduling (NSP) [4,7] or doctor scheduling (DSP) [16] separately, this research integrates all personnel groups within a single mathematical framework. This comprehensive approach not only promotes equity in shift allocation but also enhances the system’s ability to respond to fluctuations in patient needs, a factor that is particularly crucial in critical care settings. The study by Goh [13] also emphasizes the need for fairness in scheduling, further validating the importance of this study’s approach. Although various optimization models for nurse or doctor scheduling have been examined in the literature, most models usually focus on a single staff group. For example, Li and Aickelin [24], Kuo et al. [25], and Muklason et al. [26] presented a model for nurses’ shift planning only, while Zainudin et al. [27] and Rousseau et al. [28] addressed doctors’ shift-allocation processes. However, these approaches ignore cooperation and shift integration between different healthcare professionals. The most important innovation of the current study is that it offers a more holistic solution by integrating all healthcare professionals into a single scheduling framework.

However, when evaluating the applicability of the current model in the healthcare sector, some important challenges should be taken into account. First of all, the process of acceptance of the proposed approach by hospital managers and healthcare professionals is of critical importance. In particular, managers and employees who are accustomed to traditional manual planning methods may resist the implementation of the model. In addition, the effectiveness of the model should be aligned with hospital policies and working hours regulations. Another important factor is the necessity of a dynamic planning mechanism depending on the fluctuation of patient admission rates in hospitals. For example, unforeseen situations such as epidemic periods or seasonal patient increases can test the flexibility capacity of the model. Therefore, future studies should evaluate how the model can be adapted to unpredictable patient demands.

Overall, this study underscores the potential of comprehensive optimization models in hospital workforce management, demonstrating that integrating multiple personnel groups into a single scheduling framework can improve efficiency, service quality, and employee well-being. However, more field studies are needed on the applicability of the model. In particular, testing the model using real hospital data and optimizing it with feedback from healthcare professionals will better reveal how the theoretical framework can be integrated into practical use. Such field studies will assess the scalability of the model and provide guidance on how it can be applied in different hospital systems. By refining and expanding upon this approach, future research can contribute to more adaptive, equitable, and efficient healthcare staffing solutions, ultimately leading to better patient care and a more sustainable hospital workforce.

Although our proposed model demonstrates promising results, it has several limitations that should be addressed in future research. First, the model assumes static shift durations and predefined constraints, which may not fully capture real-time hospital dynamics. Future studies should integrate real-time scheduling adjustments to allow for dynamic updates in response to staff absences or emergency cases. Second, our approach does not yet include machine learning-based demand forecasting, which could enhance the model’s adaptability to patient volume fluctuations. Lastly, hospital staff preferences and satisfaction were not explicitly incorporated into the optimization criteria, which could be a valuable addition in future iterations of this research.

5. Conclusions

This study proposed an optimized workforce scheduling model for a Pediatric Intensive Care Unit (PICU) by integrating nurses, junior doctors, senior doctors, and caregivers into a unified framework. The developed Mixed-Integer Programming (MIP) model successfully minimizes operational costs while ensuring fair workload distribution across different healthcare personnel. The results demonstrated a reduction in shift conflicts and an improvement in workload balance, confirming the efficiency of the proposed approach.

The findings highlight the advantages of an automated scheduling system over traditional manual methods by significantly reducing scheduling errors and improving staff satisfaction. The proposed model offers a scalable and adaptable solution that can be extended to other hospital departments and healthcare facilities.

To further enhance hospital scheduling efficiency, future research should focus on several key areas. One important direction is real-time scheduling adjustments, which involve incorporating dynamic scheduling techniques to accommodate emergency cases, unexpected absences, and fluctuating patient volumes. Another promising avenue is the integration of artificial intelligence (AI) and machine learning, where AI-driven predictive models can be utilized to anticipate staffing needs and automate scheduling decisions, reducing administrative workload and improving responsiveness. Additionally, multi-objective optimization should be explored by expanding the model to account for factors such as employee fatigue, job satisfaction, and overall hospital-wide cost efficiency, ensuring a more balanced and sustainable workforce. Finally, the application of this scheduling model to different hospital units—such as emergency rooms and operating theaters—would help validate its adaptability and effectiveness in diverse healthcare settings. Addressing these aspects in future studies will further enhance the flexibility, scalability, and practicality of healthcare workforce-scheduling models, ultimately leading to improved staff well-being, operational efficiency, and patient care outcomes.