In China, emergency room residents (EMRs) generally face high working intensity. It is particularly important to arrange the working shifts of EMRs in a scientific way to balance their work and rest time. However, in existing studies, most of the scheduling models are based on the individual doctor or nurse as a unit, less considering the actuality of operation and management of emergency department (ED) in large public hospitals in China. Besides, the depiction of the hard and soft constraints of EMR scheduling in China is insufficient. So in order to obtain the scientific and reasonable scheduling shifts, this paper considers various management rules in a hospital, physicians’ personal preferences, and the time requirements of their personal learning and living and takes the minimum deviation variables from the soft constraints as the objective function to construct a mixed integer programming model with the doctor group as the scheduling unit. The analytic hierarchy process (AHP) is used to determine the weights of deviation variables. Then, IBM ILOG CPLEX 12.8 is used to solve the model. The feasibility and effectiveness of the scheduling method are verified by the actual case from West China Hospital of Sichuan University. The scheduling results can meet the EMRs’ flexible work plans and the preferences of the doctor teams for the shifts and rest days. Compared with the current manual scheduling, the proposed method can greatly improve the efficiency and rationality of shift scheduling. In addition, the proposed scheduling method also provides a reference for EMR scheduling in other China’s highgrade large public hospitals.
The emergency department (ED) is not only a rescue department for severely ill patients but also a window for a hospital. The medical technology level and service quality of the ED are important aspects of social evaluation of hospitals. As the key department to provide all kinds of critical life support for patients, the ED must be open for 24 hours a day, 365 days a year. So the emergency room residents (EMRs) must work more shifts, often working at night and on weekends. In China, the day shift or the night shift of emergency room doctors is for up to 12 hours. Heavy work intensity and long shift time result in EMRs prone to fatigue, anger, pain, resentment, and other bad emotions [
The scheduling problem of medical staff is a combinatorial optimization problem. The current research mainly focuses on general medical departments, especially the nurse scheduling problem in these departments [
However, the problem of emergency doctor scheduling has not received much attention, and only few scholars have made preliminary exploration [
To conclude, in the existing research, most of the scheduling models are based on the individual doctor or nurse as a unit, less considering the actuality of operation and management of large public hospitals in China. Besides, the depiction of the hard and soft constraints of EMR scheduling in China is insufficient. So in this paper, we consider various management rules in a hospital, physicians’ personal preferences, and the time requirements of their personal learning and living to arrange their work shifts. In addition, according to the current hospital management situations in China, the doctors are divided into groups, and the multiobjective programming model is constructed to schedule physicians by groups in the emergency room. Finally, the validity of the model is validated by taking WCH as an example. The research results are expected to provide some references for scheduling EMRs in other China’s highgrade large public hospitals.
This paper is organized as follows: Section
The EMRs in largest public hospitals in China are usually divided into some groups (no less than three groups). These EMRs fall into two categories: firstclass doctors and secondclass doctors. The work shifts include day shifts (8 a.m. to 8 p.m.), strengthen shifts (including two time intervals, 8 a.m. to 3 p.m. and 2 p.m. to 9 p.m.), and night shifts (8 p.m. on the first day to 8 a.m. on the second day). Only one doctor group works in the day shift or night shift or strengthen shift. So three doctor groups are scheduled everyday. The doctor group scheduling model is constrained by the hard constraints (national laws and hospital regulations) and soft constraints (doctors’ personal preferences and flexible work rules). So taking one month as a cycle (30 days), how to schedule the doctor groups into different work shifts is a challenging problem.
In addition, considering the rationality, fairness, and humanization of EMR scheduling, some assumptions are made as follows:
The constraints of the scheduling model are in line with labor laws in China and regulations
Each doctor corresponds to a seniority level. There is no difference in the quality of work between doctors at the same level
The difference of the workload of doctors at the same level in the same scheduling period is as small as possible
The number of night shifts must be scheduled fairly and reasonably
Doctors’ personal research and teaching hours, expected rest days, and preferences for different shifts should be satisfied as much as possible
In this study, a multiobjective programming model is proposed, and the model is divided into two stages. In the first stage, the doctors are assigned to some medical teams. The doctors who want the same rest days at most are assigned into a group as much as possible. And each group must include a firstlevel doctor for the needs of internal exchange and learning each other within a group. In addition, the number of doctors in each group is as equal as possible. The result of grouping is obtained by the genetic algorithm. In the second stage, the assigned teams are scheduled to meet the requirements of soft and hard constraints, especially to satisfy the soft constraints as far as possible. This paper will focus on the second stage to solve the medical group scheduling problem.
In order to construct the model, the parameters and variables shown in Tables
Notation and description of parameters.
Notation  Description 

Parameters  

Set of doctor groups, indexed by 

Set of days of the monthly planning period, indexed by 

Set of doctors, indexed by 

Index of the shift type: 

Index of the seniority levels of the doctors: 

Number of doctors belonging to each seniority level: 

Difference in amounts of night shifts for the doctors at the same level within the planning period 

Total number of shifts that a doctor group should be assigned 

Set of doctor groups who need rest during the planning period 

Set of rest days of a doctor group 

Upper limit of the number of doctor groups required for every shift 

Lower limit of the number of doctor groups required for every shift 

Upper limit of the number of doctor groups required for all shifts per day 

Lower limit of the number of doctor groups required for all shifts per day 

Set of all Saturday within the planning period 
Notation and description of variables.
Notation  Description 




1 if the doctor group 

1 if the doctor group 





1 if the doctor group 

1 if the doctor group 

1 if the doctor group 

1 if the doctor group 

1 if the doctor group 

1 if the doctor group 





1 if the doctor group 





B shifts can be assigned to every doctor group within the planning period 
According to the investigation of the ED of the largest public hospitals in China and the analysis of various factors that affect EMR scheduling, the constraints of EMR scheduling are obtained. According to whether or not the constraints must be satisfied, the constraints of EMR scheduling are divided into hard constraints and soft constraints. Hard constraints refer to the conditions that must be met in any scheduling environment; otherwise, the scheduling scheme is not feasible. Hard constraints mainly include the labor regulations and hospital management systems. Soft constraints refer to the preferences of doctors for work shifts and flexible working rules in the scheduling period.
The hard constraints of the model are as follows:
Hard constraints meet national laws, hospital regulations, work shifts, and working hours for doctor groups and restrictions on the number of doctor groups per shift per day. Formula (
The soft constraints meet the flexible working rules and the doctor groups’ preferences for the shifts and expected rest days. The formula (
The objective function of the model is as follows:
The objective of this model is trying to meet most of the soft constraints and minimizing the deviation variables from the soft constraints. The deviation variables have different weight values according to the importance of the soft constraints. In this paper, each deviation variable is given a weight value which is gained by the application of the analytic hierarchy process (AHP). Suppose the value of the weight is
Take the No. 1 resuscitation room in the ED of WCH as a background. The composition of the doctors is shown in Table
Parameters about doctors.
Total number of scheduled doctors  Number of doctors with seniority level 1  Number of doctors with seniority level 2 

25  5  20 
Twentyfive emergency room doctors are replaced by numbers 1∼25, of which 1∼5 and 6∼25, respectively, refer to the doctors with seniority level 1 and seniority level 2. The scheduling period is 30 days. The expected rest days of each doctor are shown in Table
Expected rest days of doctors.
Doctor’s number  Each doctor’s expected rest days’ number 

1, 2, 3, 4, 5  
6  15, 16, 17, 18 
7, 8, 9, 10, 11  
12  2, 3, 4, 5 
13  22, 23, 24 
14, 15, 16  
17  10, 11, 12 
18, 19, 20  
21  25, 26 
22, 23, 24, 25 
According to the above parameters, firstly, the doctors with the same expected rest days are assigned to a group. According to the expected rest days of each doctor in Table
Results of grouping and arrangements of expected rest days.
Doctor group  Doctors  Expected rest days 

A  1, 11, 
2, 3, 4, 5 
B  2, 9, 
22, 23, 24 
C  3, 7, 14, 
10, 11, 12 
D  4, 8, 15, 
25, 26 
E  5, 
15, 16, 17, 18 
Firstly, the AHP is applied to gain the weight values of the deviation variables. Then, the CPLEX solver is used to obtain the scheduling table.
The criterion layer
The comparisons between any two factors of the above eight factors are made by the EMRs. A total of 25 score tables are issued, and 21 of them are successfully recovered. These collected score tables include 5 copies from 5 doctors at high seniority levels (
Finally, the degree of importance
Pairwise comparison of the soft constraints from 21 score tables and factors’ relative importance values are shown in Table
Pairwise comparison of the soft constraints.







 


1  1/9  1/3  1  1/3  1  1/8  1/8 

9  1  8  8  9  9  1  9 

3  1/8  1  1/9  1  1/9  1/8  1/8 

1  1/8  9  1  9  1  1/7  1/7 

3  1/9  1  1/9  1  1/9  1/8  1/8 

1  1/9  9  1  9  1  1/7  1/7 

8  1  8  7  8  7  1  8 

8  1/9  8  7  8  7  1/8  1 
According to Table
The root method is applied to calculate the weight value of each criterion element as follows:
The maximum eigenvalue of matrix
The consistency index
The random index
Values of the random index

1  2  3  4  5  6  7  8  9  10 


0  0  0.58  0.89  1.12  1.24  1.32  1.41  1.45  1.49 
As the consistency evaluation index, the consistency ratio
Referring to Table
After 30 corrections by the induced matrix modification method, the new judgment matrix
The new weight value of each criterion element is obtained as follows:
The maximum eigenvalue of matrix
Considering that the mathematical model is the MILP model, the numbers of variables and constraints are small, and the data scale in this case is also small, it is appropriate to use the CPLEX solver to solve the problem. The multiobjective scheduling model was solved using IBM ILOG CPLEX 12.8 on a computer with 2.30 GHz Intel i5 processor and 64bit 8.0 GB RAM. The solved model includes 1,310 binary variables, 10 integer variables, and 1,475 constraints. The computational time is 2.26 s.
The scheduling table is shown in Table
Scheduling table of doctor groups in the emergency room.
Weekday  Weekend  Weekday  Weekend  Weekday  Weekend  Weekday  Weekend  Weekday  BC  YB  

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

1  3  1  2  1  3  1  3  1  3  3  2  1  1  2  1  2  3  18  6  
B  1  1  3  3  2  2  3  2  1  2  3  2  3  1 

2  1  3  1  18  6  
C  2  2  1  1  2  3  2 


3  1  3  3  3  2  3  2  1  1  2  18  6  
D  3  3  2  1  2  1  1  3  2  3  2  2  1  2  2  2  2  1  1  3 

3  18  6  
E  3  2  1  2  1  1  3  2  3  2 

2  1  1  2  3  3  2  3  18  6 
Some results can be obtained by comparison of the above scheduling table with the soft constraints of the model (see Table
Comparison of the soft constraints with scheduling results.
Soft constraints  Results of the scheduling table 

Two consecutive night shifts  0 
Three consecutive night shifts  0 
Three consecutive day shifts  0 
Four consecutive day shifts  0 
Three consecutive upper half of strengthen shifts  0 
Four consecutive upper half of strengthen shifts  0 
Three consecutive lower half of strengthen shifts  0 
Four consecutive lower half of strengthen shifts  0 
Doctors who do not have rest days at weekends  0 
Doctors whose expected rest days do not meet  0 
The doctor group whose total monthly work shifts are not equal to 18  0 
In the current situation in the No. 1 resuscitation room in the emergency department of West China Hospital of Sichuan University, it takes 12 days to construct a onemonth schedule manually by trial and error. Using the proposed method in this paper, a highquality schedule is generated in reasonable time. Besides, there may be some limitations through manual scheduling that some soft constraints cannot be satisfied. On the contrary, the scheduling result by the proposed model can be obtained in reasonable time, which can better meet various management rules in a hospital, physicians’ personal preferences, and the time requirements of their personal learning and living. Therefore, the proposed model can greatly improve the efficiency and rationality of shift scheduling for China’s highgrade large public hospitals.
Doctors are the most important medical resources in a hospital. EMRs undertake the long and intensive work. Scientific and reasonable scheduling shifts are of great significance for relieving work pressure and improving the quality of medical service. Based on the actual situation of the ED in China’s highgrade large public hospitals and the fact that most of the scheduling models are based on the individual doctor as a unit and that the depiction of the hard and soft constraints of EMR scheduling in China is insufficient in the existing research, a multiobjective programming model with the doctor group as the scheduling unit is proposed aimed at satisfying the doctors’ personal preferences as the soft constraint under the national laws and hospital rules. The mathematical model of the scheduling problem is to satisfy more soft constraints as far as possible. The CPLEX solver is used to obtain the scheduling table. The scheduling result satisfies the doctors’ personal preferences. The feasibility and effectiveness of the method are verified by the actual case from West China Hospital of Sichuan University. The methods and ideas for scheduling EMRs can be applied to other hospitals all over the world.
Nonetheless, there are some limitations in this paper, and future research may expand further. In this paper, the AHP is applied to compute deviation variable weight values. However, the AHP is a decisionmaking method simulating the human brain, and it has strong subjectivity. Future research may seek more objective methods to obtain the weight values. Besides, the doctors’ preferences for different shifts as the soft constraint may be considered further. Moreover, because the scheduling of the EMRs is influenced by various factors, exploring the factors that affect EMR scheduling and the index of the doctor’s satisfaction further is also the direction of future research.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Natural Science Foundation of China (grant numbers 71532007, 71131006, and 71172197) and Key Research and Development Program, Science & Technology Department of Sichuan Province (grant numbers 2017SZ0007 and 2019YFS0385). We would like to thank the emergency physicians of West China Hospital of Sichuan University for their assistance in the investigation.