Optimization of Nurse Scheduling Problem with a Two-Stage
Mathematical Programming Model
Chang-Chun Tsaia,*, Cheng-Jung Leeb a
Department of Business Administration, Trans World University, Taiwan b
Department of Information Management, National Yunlin University of Science and Technology, Taiwan Accepted 6 June 2009
Abstract
This paper constructs a two-stage mathematical programming model to solve the nurse scheduling problem in order to assign nurses to shifts over a scheduling period so that certain constraints (organizational and personal) are satisfied. In the first stage, the nurse optimal vacation schedules are solved by a self-schedule programming that can check for any violation of government regulations, hospital management requirements, and scheduling fairness. In the second stage, the nurse roster schedule is arranged and a Genetic Algorithm (GA) is further adopted to derive the optimal schedule. An empirical case study is performed and the results show that the proposed approach can solve the nurse scheduling problem efficiently. In addition, it can also be easily modified to suit different cases encountered in hospitals.
Keywords: International mathematical programming model, genetic algorithm, self-schedule,
nurse scheduling
1. Introduction
Nursing staff scheduling is an essential task in manpower resource management. The scheduling quality directly influences the nursing quality and working moral. Nurse scheduling problems represent a subclass of scheduling problems (Ender, 2005). Typically, personnel scheduling problems are highly constrained and complex optimization problems (Ernst et al., 2004). The need to take into account individual preferences further complicates the process. In recent years, the emergence of larger and more constrained problems has presented a real challenge. Because obtaining good quality solutions can lead to a higher level of personnel satisfaction (Burke et al., 2006). Cheang et al. (2003) added nursing staff’s preferences into the factors to be considered when preparing work schedules. In addition, Bard and Purnomo (2007) considered other factors, such as nurse workforce, hospital work and hospital scheduling regulations, to establish a schedule-making decision tree. The constraint conditions for nurse scheduling are broad, and may differ from case to case. Some of the constraint conditions even conflict with each other. For instance, the shift preference of nursing staff may violate the requirement for shift fairness. In practice, the nurse chiefs arrange the schedule based on their subjective experience. To meet the complicated situations with ever-increasing patient demands and a limited nurse workforce, the chiefs may require more time and effort than ever to deal with the scheduling and still fail to be fair to all the staff. Consequently, the nurse scheduling issue remains challenging, and development of a more sophisticated approach to solve the problem deserves further exploration.
Rondeau (1990) and Beltzhoover (1994) maintained that self-scheduling reduces the ratio of shift-changes, increases opportunities for on-the-job training, and increases individual
autonomy and self-awareness, while also helping staff to better enjoy their social and family lives. Abbott (1995) also noted that self-scheduling reduces interference in the personal lives of nursing staff, reducing shift-changes situations and requests for leave, and ultimately boosting satisfaction towards work. Though self-scheduling has numerous advantages, communication and coordination problems are its major limitations. Consequently, this study employs a modified self-scheduling method to improve these weaknesses.
With regard to solving the nurse scheduling problem, Smith and Wiggins (1977) proposed three categories of approach: cyclical, heuristic, and mathematical programming. The cyclical scheduling approach has shift and vacation arrangements set up by the nurse chiefs based on the needs of the nurse unit, the regulations of hospital, and the number of nursing staff. The cyclical scheduling approach normally utilizes a cyclical scheduling pattern on a fixed time range. For the heuristic scheduling approach, the nurse chiefs often construct a decision tree with consideration of nursing staff workforce, nurse service patterns, hospital scheduling policy, and other factors, and then utilize the resulting schedule on a cyclical basis. The mathematical programming approach is a special mathematical model developed to respond to the scheduling problems for different cases. Normally, it is constructed with objective functions and constraint equations and then utilizes appropriate algorithms to search for the optimal solutions.
The aforementioned three nurse scheduling approaches all have their advantages and disadvantages. The cyclical scheduling approach can be conveniently executed; however, a new schedule must be arranged when nurses need to change their job or shifts, which in is quite common in practice. The heuristic scheduling approach needs a decision-making tree to develop the scheduling rules, but because the interaction among nursing staff is very complicated, the tree required is usually huge. Thus, when the constraint conditions are numerous, it will generally be difficult for the heuristic scheduling approach to attain a reasonable solution, and the nurse scheduling activities cannot be easily processed (Harvey and Mona, 1998). The mathematical programming approach has a substantial level of dependency on the cases addressed, and when dealing with different cases requires further re-formulation, meaning additional time and effort need to be spent on the project.
A nurse scheduling system can be built up and executed in various ways. Ahuja and Sheppard (1975) developed a four-module four-stage interactive cyclical scheduling system, which used a computer to arrange schedules for different nursing staff. Their system consists of four modules: (a) work pattern selector - it identifies cyclical schedule patterns from the input information, and different case hospitals may have different work patterns; (b) work schedule assembler - it combines nursing staff with the work patterns generated by the first module; (c) work schedule projector - it displays the work schedules for both the individuals and the organization; (d) work prediction and allocation of staff - it designs a work load index according to the requirements of the nursing staff and then assigns work to staff based on this work load index. Smith and Wiggins (1977) developed a batched three-stage cyclical scheduling system to arrange the nurse scheduling, which includes (a) summarizing the requirements of staff for specified units on weekly bases; (b) generating preliminary schedules and evaluating the schedules with the constraint conditions to check if there is any conflict; (c) manually adjusting the preliminary schedule and creating the finalized schedule. Kostreva and Jenning (1991) indicated that a nurse scheduling system should include survey and scheduler modules.
Integer programming, mixed integer programming, goal programming, linear programming, network programming, and constraint programming have all been used to solve the nurse scheduling problem. For instance, Miller et al. (1976), Ozkarahan and Bailey (1988) utilized the integer programming to search for a schedule with the lowest aversion
deviations; whereas Warner (1976) employed goal programming with multiple-choices to solve nurse scheduling problems. In addition, Kostreva et al. (1978) and Bell et al. (1986) developed mixed integer programming models; while Arther and Ravindran (1981) used 0-1 goal programming to solve two-stage cyclical scheduling problems. Musa and Saxena (1984) adopted 0-1 goal programming and heuristic search, and Azaiez and Sharif (2005) also used 0-1 goal programming to solve nurse scheduling problem. Bailey (1985) developed a cyclical scheduling model with integer programming. Brigitte et al. (1998) utilized linear programming to obtain the solution that can simultaneously minimize the total payment, satisfy the staff preference, and level the nurse workforce. Harvey and Mona (1998) employed network programming to study the cyclical and non-cyclical nurse scheduling problems on the basis of 12-hour shifts. Finally, Meyer (2001) used constraint programming to solve nurse scheduling problems.
Other techniques have also been found, and Randhawa and Stiompul (1993) developed a heuristic scheduling decision-making support system to handle the multiple-goal programming problems with binary variables; whereas Aickelin and Dowsland (2000) used genetic algorithms to solve nurse scheduling problems. Dowsland and Thompson (2000) combined Tabu search and network programming to establish a non-cyclical scheduling system; while Dimitri (2002) used genetic algorithms to solve cyclical scheduling problems. Aickelin and Li (2006) used Bayesian optimization algorithm to solve nurse scheduling problems.
Goldberg (1989) and Sharif (2000) pointed out that the advantages of GA are numerous. GA can solve the optimization problem with multiple variables; it can perform parallel processing, which can effectively save processing time. GA does not need to calculate the differential value of the fitness function, because the natural selection process is determined by the fitness of chromosomes; it is a multiple points search instead of a single point search approach, which has a relatively high probability of finding the value that is quite close to the global optimum rather than being trapped in a local search. In addition, GA is suitable for use in a computerized environment. In light of these advantages, this study attempts to develop a different approach by integrating a mathematical programming approach with the GA to solve the nurse scheduling problem. When dealing with different cases, the solving algorithm does not need to be re-adjusted. Instead, we only need to set new objective functions and constraint conditions.
This study will incorporate the hospital management requirements, government regulations, and nursing staff’s shift preferences into a two-stage mathematical model. In the first stage, the nurse optimal vacation schedules are solved by a self-schedule programming that can check for any violation of government regulations, hospital management requirements, and the scheduling fairness. In the second stage, the nurse roster schedule is arranged and a Genetic Algorithm (GA) is further adopted to solve the optimal schedule. The remaining parts of this paper are organized as follows. Section two develops the nurse work and vacation model and the nurse roster schedule model. Section three introduces the GAs for solving both models. In section four, an Obstetrics hospital in Kaohsiung, Taiwan is studied as our empirical case, and finally the concluding remarks are given.
2. Model development
Two mathematical programming models are proposed in this study. Model I is the holiday schedule model based on self-scheduling in order to allow the shift-table to apply nursing staff resources most efficiently. Model II is scheduled the entire shift-table to obtain the most appropriate shift-table.
A blank shift-table will be offered for the next month due to the holiday self-schedule method. All nursing staffs mark their own desired off days on the blank shift-table, the scheduler has to negotiate and set priority to solve the conflict problem. To ease the burden of the scheduler and ensure the fairness, Model I is designed to identify the optimal solution of a complete off-shift table scheduling using LINGO due to the decision-making variables are determined.
Model II is created to result the entire shift-table by arranging the shifts for each nursing staff in the second stage. Model II is a mixed integer non-linear programming problem
(MINLP) using GA to solve this particular issue.
2.1 Mathematical model for the holiday self-schedule (Model I)
The execution steps for the holiday self-schedule method are listed below:
Step 1: List the number of off days and holidays available for each nursing staff, then have all of the nursing staff mark their own desired off days.
Step 2: When too many staff select the same day, the scheduler has to negotiate and set priority, ranking the importance of the reasons of different individuals for wanting the day off. When negotiation is unsuccessful, the dispute is settled by a lucky draw.
Step 3: Calculate the shortage of off days on holidays and weekdays for each staff member.
Step 4: Schedule a complete off shift table.
To ease the burden of the scheduler, and ensure the fairness and quality of the off shifts table, this study designs a mathematical programming model to identify the optimal solution of Step 4.
This study uses the following notation for problem modeling: (1) Indexing Sets:
i: nursing staff index ; i = 1, 2,…, N t: date index；t = 1, 2,…, T
(2) Parameters：
N: total numbers of nursing staff T: schedule days
:
H i shortage of off days on holidays for the i-th nursing staff
:
C i shortage of off-days on weekdays for the i-th nursing staff
:
Rt maximum number of nurses available for t-th day. Tsun: sets of Sundays.
Tsat: sets of Saturdays.
Thot: sets of National Holidays.
T1: sets of Holidays.
[
T1 =(Tsun ∪Tsat ∪Thol)]
T2: sets of weekdays.[
T2 =(T-T1)]
t-th day.
) N (
E ：the average days of evening (overnight) shift on which flexible shift nursing staff are required to work in the entire shift table.
Di(Ei,Ni)：accumulated frequency of i-th nursing staff is working on day (evening,
overnight) shift in the entire shift table. (3) Decision Variables
⎩ ⎨ ⎧
= 10 otherwisearrange the i-th nurse staff tohave off-dayson t-th day in the initial shift table
' it R ⎩ ⎨ ⎧
= 10 otherwisearrange the i-th nurse staff tohave off-dayson t-th day in theentireshift table. it
R
Dumit = 0 or 1, Dummy variable ⎩ ⎨ ⎧ = otherwise 0 day on tth shift overnight) (evening, day prearrange staff nurse ith the 1 ) , ( ' ' ' it it it E N D ⎩ ⎨ ⎧ = otherwise 0 day. on tth shift day have to staff nurse ith the arrange 1 it D ⎩ ⎨ ⎧ = otherwise 0 day. on tth shift evening have to staff nurse ith the arrange 1 it E ⎩ ⎨ ⎧ = otherwise 0 day. on tth shift overnight have to staff nurse ith the arrange 1 it N Min Z =
[
Z1,Z2,Z3]
（1）( )
sun N i it N i it R t T R Z ⎥ ∀ ∈ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =∑
∑
∈ ∈ , N N 2 2 1 （2）( )
sat N i it N i it R t T R Z ⎥ ∀ ∈ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =∑
∑
∈ ∈ , N N 2 2 2 （3）( )
hol N i it N i it R t T R Z ⎥ ∀ ∈ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =∑
∑
∈ ∈ , N N 2 2 3 （4） Subject to T t N i D R Rit = it' + umit, ∀ ∈ , ∈ （5） N i H D i T t it = ∀ ∈∑
∈ , um 1 （6） N i C D i T t it = ∀ ∈∑
∈ , um 2 （7） T t R R t N i it ≤ ∀ ∈∑
∈ , （8） T t N i R R R R R R Rit it it it it it it ∈ ∈ ∀ ≥ + + + + + + − − − − − − , , 1 ) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 ( （9）Equation (1) is the objective function of the off-shift table in this study, and was derived from formulas (2), (3), and (4). This equation integrates the three functional indexes above into a single objective function through linear amalgamation. The weight of each function is determined by ranking the staff preferences. Equations (2), (3) and (4) represent the variance of off-days number on the Saturday, Sunday and national holidays.
Constraint (5) indicates the definition of variable and ensures the rationality of, while avoiding contradiction between the original and final off-shift tables. Constraints (6) and (7) are designed to satisfy all individual members of nursing staff with regard to their off days and off holidays. Constraint (8) exists to prevent too many people taking leave on the same day and leading to an inadequate number staff being on duty. Finally, constraint (9) is created to satisfy the unit policy of no more than six consecutive work days for any individual.
2.2 Mathematical model of scheduling for the entire shift-table (Model II)
After scheduling a complete off-shift table, the entire shift-table can be simply completed by arranging the shifts for each nursing staff on their work days. Similarly, this study proposes a MINLP model as following:
Min
Z
=
[
Z
1,
Z
2,
Z
3,
Z
4,
Z
5,
Z
6,
Z
7]
（10）∑∑
− × − × − × = Eit Dit Eit Dit Z1 ( ( 3) ( 2) ( 1) ) （11）∑∑
− × − × = Dit Nit Rit Z2 ( ( 2) ( 1) ) （12）∑∑
− × − × = Eit Nit Rit Z3 ( ( 2) ( 1) ) （13）∑∑
∈ ∈ − − × × = N i t T it t i t i R N N Z4 ( ( 2) ( 1) ) （14）∑∑
∈ ∈ − + − + − × + + = N i it it it it it it T t it it it E N D D E E N N D Z5 ( ' ' ') ( ' ' ' )/2 （15） E E Ai = i −[ ]
∑
= 6 AiZ
[ ]
represents the Gaussian Function （16）N N Ai = i −
[ ]
∑
= 7 AiZ
[ ]
represents the Gaussian Function （17）Subject to
∑
∈ ∈ ∀ ≥ N i t it D t T D , （18）∑
∈ ∈ ∀ = N i t it E t T E , （19）∑
∈ ∈ ∀ = N i t it N t T N , （20）T
t
N
i
R
N
E
D
it+
it+
it+
it=
1
,
∀
∈
,
∈
（21）T
t
N
i
D
N
i(t−1)+
it≤
1
,
∀
∈
,
∈
（22）Equation (10) represents the objective function of this mathematical model, which is derived from equations (11) through (17). This study integrates the above-mentioned seven-functions into a single target one. The weight of each individual function is ranked based on nursing staff preference, and the most favored is assigned the largest index value, for example
seven. Finally, the indexes are totaled and the weight index is calculated based on the percentage taken by each function.
Equation (11) expresses the total number of times the evening-day pattern occurred during any four-consecutive days. Moreover, equations (12), (13) and (14) represent the total occurrence of day-overnight-off shifts, evening-overnight-off shifts and overnight-off-overnight shifts in any three consecutive days, respectively. Equation (15) calculates the number of infringements between the final fixed shift-table and the former staff-preferred shift-table; meanwhile, equations (16) and (17) express the total exceed numbers over the required evening and overnight shifts for all nursing staff.
Constraint (18) ensures that the actual scheduling result meets the staffing requirements for the daily day-shift, while constraints (19) and (20) ensure that it is met for the daily evening and overnight shifts. Furthermore, constraint (21) sets the limit that each nursing staff should not work more than eight hours a day, while constraint (22) prohibits overnight-day shifts.
3. Model solving algorithms and process NCS
This study divides nurse scheduling operation into two models. The solving algorithms and process for both models is briefly introduced below:
(1) Holiday self-schedule
During the execution steps for the holiday self-schedule method (section 2.1), most decision-making variables are determined (from steps 1~3). Hence, this study seeks the solution with the software package LINGO for the step 4.
(2) Overall schedule
Model II is MINLP that can’t be solved by the LINGO, a GA is further adopted to solve the optimal schedule and is written using MATLAB language. The solution-seeking steps are presented below.
Step 1. Producing an individual is decided by coding
When applying the heredity algorithm to solve the problem, the variable designed must be transformed, regardless of whether it is a measuring index or a numerical index, into code form. Different coding methods are adopted for different types of problems. Genes are used code, and recorded along with the hereditary features of individual creatures.
The size of each individual is set as a matrix of N x T, with the horizontal axis as the scheduling date, and the vertical axis as the labeled number of each individual. The byte value is an integer from 1 to 4. “1” represents day-shifts, “2” denotes evening shifts, “3” stands for overnight shifts, and “4” represents off-shifts. Three nurses work the evening shift on the first day and will have a day off on the second day, work the day shift on the third day, and the overnight shift on the fourth day, (please refer to figure 1 for details).
Because the off days for each nursing staff are determined by holiday self-scheduling, the location of byte value 4 is fixed for each individual. Furthermore, each byte has only four possibilities: 1, 2, 3 and 4. Satisfaction of the limited formula 22 for each nursing staff means that they should not work more than eight hours a day.
Figure 1. The codes are arranged in the form of a string representing chromosomes. Step 2. Defining appropriate functions
Before seeking a solution using GA, the functions of appropriateness must be defined to assess the efficacy of the solution. This study sets the appropriate function as the objective function (17).
Step 3. Reproduction
When selecting an individual for the purpose of reproduction, natural selection indicates individuals with higher adaptation capacity have a better chance of being selected. The method of selection is the simplest and the most commonly used ratio method. In this, the rate of reproduction is determined by using the percentage of total function value represented by the function value of appropriateness of an individual; namely, the chance of an individual being chosen depends on its appropriateness. This study selected the presentation method of choosing probability, where
{
Npop}
t t t t X ,X ,..., X 2 1 =
ψ represents the Npop solutions in
generation t, andNpop represents the number of individuals in a race. The chance of each
solution i t
X being selected as a parent string is P(Xti)：
[
]
[
]
∑
∈−
−
=
t i t X i t t M i t t M i tX
f
f
X
f
f
X
P
ψψ
ψ
2 2)
(
)
(
)
(
)
(
)
(
（23）In formula (23), fM (ψt)is defined as the poorest target function value in the t generation chromosome, while ( i)
t
X
f represents the target function value of every solution of Xti in
generation t.
Step 4. Crossover
The crossover procedure is designed according to the special features of the model, explained as follows. Before mating is conducted, two parent individuals are selected by using the reproduction algorithm, and two cutting points were determined randomly. Genes are reproduced from the cutting points of the parent generation in the same location as the child generation. Other genes of the child generation are replaced by the genes within the cutting points of another parent, as presented in Figure 2 (represented by the first row). This study puts forward the hypothesis that if two parent individuals proceed to a crossover
Figure 2. Parent individual before crossover.
Figure 3. Child individual after crossover.
algorithm, the two cutting points are 1 and 5, and Figure 3 illustrates the results of the crossover.
Step5. Mutation
The heuristic mutation of this study is self-designed based on the special features of model, explained in the constraints, above, and each individual attempts to satisfy constraints (19), (20), and (21), ensuring that the schedule result satisfies the daily requirement for day-shifts and maintains the minimum nursing manpower for evening and overnight day-shifts. The mutation rules are thus as follows: (a) each individual is capable of mutational calculation and each satisfies constraints (19), (20) and (21). (b) The unit of individual mutation is the line, and the mutational algorithm treats each line one by one.
4. Empirical case analysis
4.1 Case overview and problem description
This study adopted nursing staff scheduling in the Dept. of Obstetrics and Genecology (Ob-Gyn) of a hospital in Kaohsiung as a case study. This case study involved three shifts a day, namely day-shift (AM 8:00~PM 4:00), evening shift (PM 4:00~AM 0:00) and overnight shift (AM 0:00~AM 8:00). A one-month cycle time was used. The unit contained 14 nursing staff, with required manpower of five staff for the day-shift, and three and two staff for evening and overnight shifts, respectively.
The constants in this study’s scheduling system are set as follows: 1. Shifts during annual leave are scheduled through lottery. Moreover, the head nurse gives the privilege of off shifts on New Year’s Day, Tomb-Sweeping Day, Dragon-boat Festival and Mid-autumn Festival to those who did not take annual leave; 2. Leave on New Year’s Day, Tomb-Sweeping Day, Dragon-boat Festival and Mid-autumn Festival should not be registered in advance; 3. The shift-table marks the number of off days and holidays available per staff member per month; 4. Holidays include Saturdays, Sundays and National Holidays, but not New Year’s Day, Tomb-Sweeping Day, Dragon-boat Festival and Mid-autumn Festival; 5. Unless necessary, the scheduling of regular off days on Saturdays, Sundays, or National Holidays is avoided, with attempts being made to evenly distribute off days at these times; 6. The maximum number of consecutive working days scheduled for a given individual is six; 7. Prior scheduling is permitted for special reasons (births, weddings, engagements, overseas travels, or special long leave.)
For acquiring the optimal result of Model I and Model II, two software (LINGO and MATLAB) were run on a personal computer with a Pentium ΙΙI 450MHz CPU. With regarding to the size of the Model II is listed as following: the number of variables is equal to “4 * N * T + 3 * T”; the number of constraints is “3 * T + 2 * N * T”. The information on off days for each nursing staff in the Ob-GYN should be entered. This study is conducted in March 2005; the analysis and comparison between the shift-table. The original manual schedule is as follows.
4.2 Analysis of the holiday self-schedule in the OB-GYN Dept.
Based on the execution steps and the mathematical model (model I) of the holiday self-schedule method, the off day shift table is shown in the Table 1. During the self-schedule rotation, nursing staff scheduled off days as they desired (113 days in total). However, owing to surplus manpower on specific days, the scheduler had to make some coordination and adjustment. One hundred and two days of the actual off day shift table were the same as the original schedule from the preferred off day table. Therefore, most of off days (90.3%) are identical to the original schedule chosen by the staff. Table 2 gives the results of the survey of the nursing staff regarding the holiday self-schedule method.
4.3 System parameters and the solving result
In the second stage, GA is adopted to optimize the nurse schedule. This study established parameters such as population size, probability of mutation and so on for executing GA. Table 3 lists the selected factors and the level values that were selected to increase the efficiency of problem solving. A total of eight (2 x 2 x 2) combinations of different controlling factors were used. Thirty tests were conducted for each combination. From Table 4, the termination condition is set at the 50th generation, and the chance of obtaining the best solution is set at the 30th generation. Therefore, this study set the termination condition at the 50th generation. From test numbers 5~8 in Table 5, under the termination condition, population size and the increase in the probability of mating provided no clear assistance in obtaining the optimal solution, and significantly increased the system calculation time. Therefore, this study sets the population size at 40 and the probability of mating at 0.7.
Table 1. The off-day shift table.
Table 2. Survey on holiday self-scheduling in the OB-GYN Dept. Items Strongly Agree Agree No Comment Disagree Strongly Disagree
1. The practice of holiday self-scheduling increases self-scheduling autonomy and makes it easier for me to decide my own holidays.
92.86％ 7.14％ 0 0 0 2. The system improves my chances
of getting days off when I want them, and enhances my
satisfaction regarding scheduling.
85.6％ 14.4％ 0 0 0 3. My chances of getting the days
off that I wanted were increased, and with a few exceptions most of the off days were self-scheduled.
85.6％ 14.4％ 0 0 0
In comparison with the entire shift-table, Table 5 is derived from using the GA and the original manual shift-table. The results acquired from the model can be easily obtained, and the approach presented in this study is better than the manually scheduled method, as it can determine the most appropriate shift-table in just five minutes. Therefore, the model presented here serves as a good reference for any medical unit seeking to solve its nursing shift schedule problem.
Table 3. Selected controlling factors and their level value.
Alias Controlling Factors Level Value
A Termination Condition 2nd Level (30，50) B Population Size 2nd Level (40，80) C Probability of Mutation 2nd Level (0.3，0.7)
Table 4. Parameters: Combination and related solution qualities.
Test No. Termination condition (A) Size of population (B) Probability of mutation (C) Best solution- seeking results Probability of getting the best solution Average solving time (sec.) 1 30 40 0.7 0 0.73 (22/30) 219.68 2 30 40 0.3 0 0.73 (22/30) 230.52 3 30 80 0.7 0 0.77 (23/30) 243.26 4 30 80 0.3 0 0.83 (25/30) 259.75 5 50 40 0.7 0 0.90 (27/30) 285.36 6 50 40 0.3 0 0.87 (26/30) 314.82 7 50 80 0.7 0 0.93 (28/30) 335.15 8 50 80 0.3 0 0.90 (27/30) 352.23 5. Concluding remarks
This study presents two efficient mathematical programming models and applies GA to determine the optimal nursing staff shift schedule. Unlike most existing approaches, the new approach has the ability to build schedules. This study demonstrates a two-stage efficient and
flexible mathematical programming model and arrived with solid result rather than traditional scheduling. Upon the needs of the nursing staff, this approach is able to adjust the objective function, constraints and their weighted value to increase the flexibility and generality of the scheduling model. In particular, this study proposed two significant issues highly valued: one is the staff fairness principle, and the other is the self-determined scheduling principle. Hence, the ultimate schedule truly reflects the preferences of nurse staff as well as the management requirements. The experimental results demonstrated the strength of this approach.
The contributions are as follows. Firstly, this study applied GA to obtain the entire shift-table. This model copes with different scheduling cases with the proposed solid core algorithm. It is highly adaptive to other applications. Secondly, this study proposes a heuristic mutation method to significantly reduce the scheduling setup time. Thirdly, this study developed a scheduling system to handle the scheduling activities. This approach can reduce the nurse chief’s scheduling workload and also increase the nurse staff satisfaction by providing the staff vacation fairness and more respect on self-determination schedule.
Future research could be extended to reflect the effect on employee satisfaction and financial performance improvement. This research gives some preliminary answers of how to include human-like learning into scheduling algorithms and this approach might be interested to practitioners and researchers in areas of scheduling and evolutionary computation.
Table 5. The comparison between the manual and entire shift-tables.
Evaluation Index Manual
Shift-Table
Entire Shift-Table No evening-day-evening-day shifts on any four consecutive
days (number of violations). 0 0
No day-overnight-off shifts on any three consecutive days
(number of violations). 4 0
No evening-overnight-off shifts on any three consecutive
days (number of violations). 3 0
No overnight-off-overnight shifts on any three consecutive
days (number of violations). 0 0
Matching the preferred tentative shift-table
(number of violations). 2 0
The total exceed numbers over the required evening
shifts for all nursing staff. 6 0
The total exceed numbers over the required overnight
shifts for all nursing staff. 6 0
Satisfying day-shift needs (number of violations). 0 0
Maintaining minimum labor requirements regarding evening
shifts (number of violations). 0 0
Maintaining minimum labor requirements regarding
overnight shifts (number of violations). 0 0
Ensuring there are no overnight-day shifts on any two
consecutive days (number of violations). 0 0
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