A Practical Staff Scheduling Strategy Considering Various Types of Employment in the Construction Industry
Round 1
Reviewer 1 Report
The paper presents a staff scheduling model in a construction company. The paper is easy to follow and has practical applications, however, there are several concerns that must be addressed to make the paper appropriate for publication:
1- The paper must be edited for English.
2-What do you mean by "irregular" worker? Does it mean part-time? There is no definition of irregular employee in the paper.
3-The major issue across the equations is that the authors did not pay attention to the notation. In modeling, the variables and parameters are case-sensitive. For example, there is a difference between Ad and ad, x and X, r and R, etc. You defined your decision variables in Table 1 using lower case, but the equations have upper case alphabets, which is wrong. The same issue exists for the majority of parameters in the paper. Therefore, Table 1 and all the equations must be updated for consistency.
4-What is the purpose of equation 3?
5- what is the unit of cost in tables 4 and 5? $/hour or $/day?
make it clear in the table.
6- There is no information about the solution procedure and timing. I understand the scope of the paper is not the solution approach, however, there must be some sort of information about it in the paper.
Author Response
Paper title: Practical staff scheduling strategy considering various types of employment in the construction industry
Paper ID: algorithms-1851780
Comments received: August 16, 2022
Comments responded to: August 30, 2022
General Response from the Authors
We appreciate the valuable comments provided by the three anonymous reviewers regarding the contribution, the proposed method, and format of our manuscript. We really appreciate the efforts of the three reviewers to improve our manuscript. Therefore, we have made a great effort to enhance the presentation of the paper. An additional explanation about the modifications undertaken are as follows:
- Contribution of the study is supplemented throughout the overall manuscript.
- Introduction and Literature Review are revised to support this paper’s purpose.
- Numerical experiment is supplemented to make the manuscript meaningful.
- The format of manuscript was revised to be clearer and more comprehensive.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Academic editor
Q1. the authors fail to position their work in the academic literature. As a result, it is not clear what the contribution of the paper is and how to advances the state of the art in staff scheduling.
Responses: Thank you for your adequate comments. First of all, I revise the introduction to emphasize on what this paper considers and do. The revised part is as follow:
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.
This paper considers the actual situation of the construction industry in Korea such as the workforce shortage due to an increase in labor cost and workforce availability reduction by the change of the labor policy. The main thing considered in the paper is ‘no-show’, which is a real problem of the actual company that employed irregular employees to deal with the workforce shortage problem. Despite the no-show of irregular employees, it is difficult for the company to control irregular employees since the authority of giving penalties normally belongs to their outsourcing company in Korea. Nevertheless, the reason a company uses irregular employees is because of lower labor costs than local regular employees. Therefore, the company needs to respond to the unexpected situation that irregular employees suddenly do not come to work. To do this, this research uses linear programming to derive a deterministic staff schedule considering the average percent of irregular employee absences. This can be one of the proactive measures to respond to the stochastic characteristics of an irregular employee’s absence. Moreover, the simulation of a deterministic staff schedule is conducted to figure out what expectedly happens when assuming the application to a company in practice. Consequently, the performance of how well a staff schedule works is evaluated, and how many irregular employees do not come to work is expected. This can play a role as basic data that helps managers come down with deciding on a staff schedule.”
Furthermore, the contributions part is revised and added in the conclusion section in details. The added part is as follow:
“6.1 Contributions
This study proposes a staff scheduling strategy to minimize the total labor cost considering the actual features of the Korean construction industry such as an increase in labor cost and the change in labor policy. The contributions of this paper are as follows. First, this paper suggests a mathematical model that can save the total labor cost and aid the management of human resources. As shown in the numerical experiment, a mathematical model decreases the unnecessary number of employees from about 46.6 to 45 a day on average. This can allow a company to have an opportunity to save labor costs and rethink and optimize the number of employees they should have. Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea. this paper allows the manager of a company to come down with decision-making for staff schedule by providing various proactive cases while considering the sudden absence of employees. Despite the difficulty to consider unpredictable employee absence due to the inherently uncertain nature of irregular employees, the additional numerical experiments, which are the simulation and sensitivity analysis are performed in the paper. The result of the simulation predicts what expectedly happens when assuming the application of a deterministic staff schedule in practice. Especially, the predictive information on how many irregular employees would not come to work is given. It is impossible to guarantee that they would follow staff schedules perfectly in the real world. Nevertheless, the reason to perform the simulation is to allow managers to have a systematic procedure to respond to the absence of irregular employees, not to provide a staff schedule that makes all irregular employees follow the staff schedule. Moreover, through this sensitivity analysis with the change of the average percent of absence, this paper observes the influence of irregular employee absence on the change of the value of the objective function and staff schedule. Through the additional numerical experiments, various proactive cases are provided by considering the absence of irregular employees. This can help managers prepare the precautious measures to respond to the stochastic irregular employee's absence. Thus, it is expected to contribute to helping the manager with decision-making for staff schedules.”
Q2. 'no-shows' or employee absences have been studied by different authors. As they are inherently of an uncertain nature, the established manner of handling such problem characteristics is through stochastic modelling. The authors fail to acknowledge this and do not provide a motivation for their proposed approach.
Responses: Thank you for your adequate comments. I would like to appreciate to let me know the parts that I did not consider well again. As you mentioned, I added some previous references on employee absence studied by different authors.
“To improve the practicality, the uncertainty of irregular employees should be reflected when developing a staff scheduling strategy. Several studies considered the uncertainty of irregular employees. Research on workforce scheduling that considers the unpredictable employee’s absence was done to guarantee sufficient coverage in medical facilities. Becker et al. [29] proposed integer programming that considers assigning ex-post duties. This paper proved the practicality of the developed models by applying them to local medical facilities in practice. As a result, staff scheduling complexity was reduced. Ingels and Maenhout [30] stated many organizations make staff schedule under a deterministic operating environment. However, the stochastic environment such as employee absence occurs. To manage this uncertainty, this paper proposed a proactive approach to respond to schedule disruptions. By comparing a diversity of the proactive strategies, a proposed preemptive programming approach in this paper was assessed. Steenweg et al. [31] mentioned that short-term uncertainty in the workforce causes gaps between the planned and real shift schedule. Therefore, the unpredictable absence of employees should be considered when deciding on shift scheduling. This paper suggested a framework that can assign heterogeneously skilled employees to jobs optimally to respond to sudden absence. For this framework, the workforce availability was modeled by the stochastic simulation. As a result, the practicality of the framework that can provide efficient shift scheduling was proved by applying it to the actual cases.”
In addition, the different points from other studies are presented to provide motivation for this paper as well. It is as follows:
“This paper also considers the uncertainty of employees, especially irregular employee absence in the construction industry. Many studies about the staff scheduling that handles the uncertain nature of employees suggested stochastic modeling. In addition, other studies proposed the strategies for assignment of skilled employees to shift to fill in the absence. However, this paper develops a mathematical model that derives a deterministic staff schedule by considering the average percent of irregular employees’ absences. This is because the expected average percent of irregular employees’ absences could be different from the actual average percent of their absences in practice. That is, even a staff schedule derived from the stochastic modeling cannot ensure that all irregular employees follow a staff schedule because it is just performed only relying on the probability of absence. Therefore, this paper focuses on providing various proactive cases by deriving a deterministic staff schedule and conducting a simulation about it to evaluate performance. Consequently, the simulation helps a company expect how many irregular employees would not come to work when applying a deterministic staff schedule to a company. However, this cannot also guarantee that all irregular employees follow a staff schedule well because the possibility of their absences still exists. However, deriving a deterministic staff schedule and simulating it can contribute to providing a systematic procedure to manage the sudden absences of irregular employees. Therefore, this procedure can help managers prepare the precautious measures to rapidly respond to irregular employees’ absences.”
Q3. the presentation and grammar are poor, making parts of the paper hard to understand.
Responses: Thank you for your valuable comments. I agree with the point that English used in the paper is poor. So, I received English manuscript proofreading services from professors who are native English speakers at my university. I have received the revised manuscript, and I believe that the problem with English expressions has now been solved. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #1
Q1. The paper must be edited for English.
Responses: Thank you for your valuable comments. I agree with the point that English used in the paper is poor. So, I received English manuscript proofreading services from professors who are native English speakers at my university. I have received the revised manuscript, and I believe that the problem with English expressions has now been solved. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q2. What do you mean by "irregular" worker? Does it mean part-time? There is no definition of irregular employee in the paper.
Responses: Thank you for your adequate comments. First of all, the definition of irregular employees is as below.
“Irregular employees that are often shown in this paper are defined as foreign and local irregular employees who do not belong to a company and are paid daily. They are not part-time workers”
I added the explanation on irregular employee to help to understand what irregular employees are in this paper.
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.”
Q3. The major issue across the equations is that the authors did not pay attention to the notation. In modeling, the variables and parameters are case-sensitive. For example, there is a difference between Ad and ad, x and X, r and R, etc. You defined your decision variables in Table 1 using lower case, but the equations have upper case alphabets, which is wrong. The same issue exists for the majority of parameters in the paper. Therefore, Table 1 and all the equations must be updated for consistency.
Responses: Thank you for your valuable comments. I revised the decision variables in Table 2 using upper case. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q4. What is the purpose of equation 3?
Responses: Thank you for adequate comments. I omitted the explanation on Equation (3). It is added in the part of explanation about the Equations.
“Equation (2) and (3) represents the relationship between decision variables Ri,j,p,d and Yi,j,p,d. The actual number of irregular employees is decided by equation (2) which multiplies the number of irregular workers following the schedules without absence by . Equation (3) determines value of decision variable Ri,j,p,d and Ri,j,p,d can be 1 when decision variable Yi,j,p,d is 1.”
Q5. what is the unit of cost in tables 4 and 5? $/hour or $/day?
Responses: Thank you for your adequate comments. I put the information on the unit of cost in Table 4 and 5. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q6. There is no information about the solution procedure and timing. I understand the scope of the paper is not the solution approach, however, there must be some sort of information about it in the paper.
Responses: Thank you for your adequate comments. I think many studies using linear programming focus on the numerical experiment than solution procedure when solving the developed mathematical model. However, I agree with the comments that there must be some sort of information. So, I added the simple process as follow:
“4. Solution Process
Linear programming used as a method in this paper can be solved by the simplex method. The simplex method can be utilized to solve linear programming [32]. The simplex method is based on forming the inverse of the basic matrix and updating the inverse [33]. That is, it is the repetition of searching for feasible solutions, calculating the value of the objective function, and comparing the derived value and the optimal value till finding the optimal solution. The process of the simplex is shown in Figure 2. First of all, all equations should be converted into standard equation form after inserting slack variables and create an initial simplex table. After this, the thing to do is find the entering variables. When solving the maximization problem, the variable with the largest absolute value of the coefficient becomes the entering variable. The next thing is to find leaving variable. To do this, the ration between entering variable and solution should be calculated. At this time, the variable with the lowest ration value becomes the leaving variable. After deciding on entering variable and leaving variable, the intersection point of the entering variable and leaving variable becomes the pivot element and all values of the row in the simplex table with the pivot element should be divided by the pivot element. The rest of the rows should be updated by using the new pivot row made very previously. Lastly, if there is no new leaving variable, iterations terminate. If not, go back to the step of Finding the entering variable. Various computer packages based on the simplex method were already developed. A developed mathematical model in the paper is also solved by one of the packages. The package used in the paper is CPLEX, which is the mathematical optimization software package. The version is 20.1.0. The darker parts in Figure 2 are processed by CPLEX.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #2
Q1. I recommend emphasizing the associated research question in the introduction section. Future research opportunities should also be better emphasized. It is possible to emphasize the boundaries of research and the conditions of use of the selected method.
Responses: Thank you for your adequate comments. First of all, the boundary of research is staff scheduling using linear programming. Furthermore, the linear programming is normally used to minimize or maximize the objective function by assigning constrained resource efficiently. The objective function of this paper is to minimize the total labor cost.
As you mentioned, I emphasized the associated research in the introduction. I revised related parts as follow:
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.
This paper considers the actual situation of the construction industry in Korea such as the workforce shortage due to an increase in labor cost and workforce availability reduction by the change of the labor policy. The main thing considered in the paper is ‘no-show’, which is a real problem of the actual company that employed irregular employees to deal with the workforce shortage problem. Despite the no-show of irregular employees, it is difficult for the company to control irregular employees since the authority of giving penalties normally belongs to their outsourcing company in Korea. Nevertheless, the reason a company uses irregular employees is because of lower labor costs than local regular employees. Therefore, the company needs to respond to the unexpected situation that irregular employees suddenly do not come to work. To do this, this research uses linear programming to derive a deterministic staff schedule considering the average percent of irregular employee absences. This can be one of the proactive measures to respond to the stochastic characteristics of an irregular employee’s absence. Moreover, the simulation of a deterministic staff schedule is conducted to figure out what expectedly happens when assuming the application to a company in practice. Consequently, the performance of how well a staff schedule works is evaluated, and how many irregular employees do not come to work is expected. This can play a role as basic data that helps managers come down with deciding on a staff schedule.”
Q2. Figure 1: at the end of the x-axis there are incomprehensible orders.
Responses: Thank you for your valuable comments. I revised the x-axis. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q3. For the mathematical formulas, I recommend processing a flowchart of the entire layout, which can then be referenced in the Results section.
Responses: Thank you for your valuable comments. I revised the mathematical model part for the better readability, and I added the solution process as below.
“4. Solution Process
Linear programming used as a method in this paper can be solved by the simplex method. The simplex method can be utilized to solve linear programming [32]. The simplex method is based on forming the inverse of the basic matrix and updating the inverse [33]. That is, it is the repetition of searching for feasible solutions, calculating the value of the objective function, and comparing the derived value and the optimal value till finding the optimal solution. The process of the simplex is shown in Figure 2. First of all, all equations should be converted into standard equation form after inserting slack variables and create an initial simplex table. After this, the thing to do is find the entering variables. When solving the maximization problem, the variable with the largest absolute value of the coefficient becomes the entering variable. The next thing is to find leaving variable. To do this, the ration between entering variable and solution should be calculated. At this time, the variable with the lowest ration value becomes the leaving variable. After deciding on entering variable and leaving variable, the intersection point of the entering variable and leaving variable becomes the pivot element and all values of the row in the simplex table with the pivot element should be divided by the pivot element. The rest of the rows should be updated by using the new pivot row made very previously. Lastly, if there is no new leaving variable, iterations terminate. If not, go back to the step of Finding the entering variable. Various computer packages based on the simplex method were already developed. A developed mathematical model in the paper is also solved by one of the packages. The package used in the paper is CPLEX, which is the mathematical optimization software package. The version is 20.1.0. The darker parts in Figure 2 are processed by CPLEX.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q4. Results chapter: I recommend not to start with a picture.
Responses: Thank you for your valuable comments. As you mentioned, I moved the position of a picture after paragraph.
“5.2 Result
5.2.1. Before applying a mathematical model
This company did not have a staff schedule before. Due to this, they suffered from managing human resources efficiently. A staff schedule in Figure 3 is made based on the record of employee working patterns. Before the explanation of Figure 3, how to see a staff schedule is explained. In Figure 3, the row number means employees and the column number represents days. The numbers that are the darkest gray in row number stands for managers. The number in each cell indicates the assigned workplace and the cells with the diagonal pattern are Job 2, and without the diagonal pattern are Job 1. When it comes to jobs, Job 1 means normal work and Job 2 is equipment supervisor who takes responsibility for everything regarding equipment. If looking at Figure 3, more employees sometimes came to work than the number of employees a company needs, and fewer employees sometimes came to work than the number of employees a company needs. An example of the former one is Day 3 and the latter is Day 1. Especially, when looking at Day 24, placing enough workforce on the 2nd floor failed. Originally, more than 15 employees should be assigned on each floor during the daytime. However, only 12 employees were assigned that day. This directly shows inefficient human resource management. Moreover, about 46.6 employees came to work on average, which means it was a bigger number than the daily number of employees, 45, a company needs. This causes larger labor costs than when 45 employees come to work on average.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q5. The conclusion could be extended to the possibility of a next specific application of the proposed solution in practice. Future research directions may also be highlighted. In the manuscript, please emphasize the novelty of your approach.
Responses: Thank you for your valuable comments. First of all, a mathematical model is applied to an actual company in numerical experiment. Furthermore, the possible application of the proposed solution in practice is added as one of main contributions for this paper.
“Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea.”
In addition, I revised the future research part and added the future research directions. Some parts I added are as follows:
“6.2 Future Research
This paper could be expanded to another staff scheduling research that considers shifts. This paper does not consider shifts due to no shifts in the construction industry in general. However, many studies about staff scheduling consider shifts. Thus, if the developed mathematical model in this paper is improved considering the characteristics of shift, it is expected that the developed mathematical model could be applied to relatively more various industries. Furthermore, many objective functions can be considered to gain an optimal solution for staff scheduling. The current objective function is to minimize labor cost. This means that this staff scheduling strategy considers a company more than employees despite considering the policy called the 52-hour workweek policy. However, it seems necessary to create the objective function for a mathematical model that considers employees as well as the company in the future because employees' satisfaction with job assignments plays a crucial role in an organization’s success [34]. Moreover, research on figuring out the factors that influence the sincerity of irregular employees is needed in terms of their welfare. Doing this research can help make a proper working environment. In conclusion, various factors regarding both of company and employees in various ways should be considered to derive a good solution when staff scheduling problem is handled.”
Lastly, I added different points from the previous research in section 2 to emphasize the novelty of this paper.
“This paper also considers the uncertainty of employees, especially irregular employee absence in the construction industry. Many studies about the staff scheduling that handles the uncertain nature of employees suggested stochastic modeling. In addition, other studies proposed the strategies for assignment of skilled employees to shift to fill in the absence. However, this paper develops a mathematical model that derives a deterministic staff schedule by considering the average percent of irregular employees’ absences. This is because the expected average percent of irregular employees’ absences could be different from the actual average percent of their absences in practice. That is, even a staff schedule derived from the stochastic modeling cannot ensure that all irregular employees follow a staff schedule because it is just performed only relying on the probability of absence. Therefore, this paper focuses on providing various proactive cases by deriving a deterministic staff schedule and conducting a simulation about it to evaluate performance. Consequently, the simulation helps a company expect how many irregular employees would not come to work when applying a deterministic staff schedule to a company. However, this cannot also guarantee that all irregular employees follow a staff schedule well because the possibility of their absences still exists. However, deriving a deterministic staff schedule and simulating it can contribute to providing a systematic procedure to manage the sudden absences of irregular employees. Therefore, this procedure can help managers prepare the precautious measures to rapidly respond to irregular employee absence.”
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #3
Q1. This paper attempts to optimize various types of staff scheduling problems using mathematical programming models. It is true that the application of mathematical programming to staff scheduling is thoughtful, but the paper does not have a significant contribution to make it suitable for publication in this highly respected journal.
Responses: Thank you for your adequate comments. Following your advice, I rewrote the contributions part in the conclusion section. The added contributions are as follow:
“6.1 Contributions
This study proposes a staff scheduling strategy to minimize the total labor cost considering the actual features of the Korean construction industry such as an increase in labor cost and the change in labor policy. The contributions of this paper are as follows. First, this paper suggests a mathematical model that can save the total labor cost and aid the management of human resources. As shown in the numerical experiment, a mathematical model decreases the unnecessary number of employees from about 46.6 to 45 a day on average. This can allow a company to have an opportunity to save labor costs and rethink and optimize the number of employees they should have. Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea. Lastly, this paper allows the manager of a company to come down with decision-making for staff schedule by providing various proactive cases while considering the sudden absence of employees. Despite the difficulty to consider unpredictable employee absence due to the inherently uncertain nature of irregular employees, the additional numerical experiments, which are the simulation and sensitivity analysis are performed in the paper. The result of the simulation predicts what expectedly happens when assuming the application of a deterministic staff schedule in practice. Especially, the predictive information on how many irregular employees would not come to work is given. It is impossible to guarantee that they would follow staff schedules perfectly in the real world. Nevertheless, the reason to perform the simulation is to allow managers to have a systematic procedure to respond to the absence of irregular employees, not to provide a staff schedule that makes all irregular employees follow the staff schedule. Moreover, through this sensitivity analysis with the change of the average percent of absence, this paper observes the influence of irregular employee absence on the change of the value of the objective function and staff schedule. Through the additional numerical experiments, various proactive cases are provided by considering the absence of irregular employees. This can help managers prepare the precautious measures to respond to the stochastic irregular employee's absence. Thus, it is expected to contribute to helping the manager with decision-making for staff schedules.”
Q2. The optimal schedule result (Figure 2 on page 9) is not a scheduling result. Is it possible for a regular worker to work without a stable working schedule?
Responses: Thank you for your valuable comments. First of all, the original purpose of this paper is to help a manager of a company with decision making to minimize the total labor costs by presenting various proactive cases depending on the change of the average percent of irregular employee absence.
Furthermore, I added the part of comparison of two staff schedules between before and after applying a mathematical model to make this paper meaningful. The added parts are as below. I added the related pictures in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
“5.2 Result
5.2.1. Before applying a mathematical model
This company did not have a staff schedule before. Due to this, they suffered from managing human resources efficiently. A staff schedule in Figure 3 is made based on the record of employee working patterns. Before the explanation of Figure 3, how to see a staff schedule is explained. In Figure 3, the row number means employees and the column number represents days. The numbers that are the darkest gray in row number stands for managers. The number in each cell indicates the assigned workplace and the cells with the diagonal pattern are Job 2, and without the diagonal pattern are Job 1. When it comes to jobs, Job 1 means normal work and Job 2 is equipment supervisor who takes responsibility for everything regarding equipment. If looking at Figure 3, more employees sometimes came to work than the number of employees a company needs, and fewer employees sometimes came to work than the number of employees a company needs. An example of the former one is Day 3 and the latter is Day 1. Especially, when looking at Day 24, placing enough workforce on the 2nd floor failed. Originally, more than 15 employees should be assigned on each floor during the daytime. However, only 12 employees were assigned that day. This directly shows inefficient human resource management. Moreover, about 46.6 employees came to work on average, which means it was a bigger number than the daily number of employees, 45, a company needs. This causes larger labor costs than when 45 employees come to work on average.
5.2.2. After applying a mathematical model
Unlike a staff schedule before applying a mathematical model, A derived staff schedule is more efficient. As shown in Figure 4, all constraints regarding work in the daytime are satisfied. For example, 45 employees are scheduled to work if looking at Day 1. In addition, job assignments are met as well. Continued looking at Day 1, more than 15 employees are assigned on each floor and two of the employees conduct job 2 as equipment supervisors. Furthermore, more than one manager is assigned on each floor to supervise other regular and irregular employees as well. Moreover, over two of the employees work at open storage and a manager works in an office. That is, the failure of employees assignment caused by a manager’s random employee assignment does not happen. Other days as well as Day 1 meet all constraints for job assignments. Along with this, the 52-hour workweek policy is well applied in the mathematical model as the result shows. There are no employees who come to work for more than five days a week. All employees have a day-off on Sunday.
Furthermore, the purpose of the developed mathematical model is to minimize the total labor cost of regular and irregular employees. This means that the result in the numerical experiment is for a company that has a priority of saving the total labor cost. Therefore, if looking at the result in Figure 4, as many irregular employees as possible are scheduled to be on each workday even though the average percent of irregular employee absence is considered. It is more advantageous for a company to minimize the total labor costs by using as many irregular employees as possible out of 22 irregular employees due to relatively low labor costs. However, it is seen that as many irregular employees as possible are not used. This is because there are rules on doing jobs of a company and there are jobs that can be conducted by regular employees. A company does overtime work by assigning only regular employees to jobs for consideration for regular employees as well. To do overtime work, regular employees should work during the daytime. Thus, the result in Figure 4 is derived even though a company has enough irregular employees that can cover more jobs.
Comparing a staff schedule before and after applying a mathematical model, the following things can be found. First, there is a difference in the number of employees who come to work per day on average. Consequently, about 1.6 employees come to work less than before after applying a mathematical model every day. Second, a developed mathematical model prevents situations where employees come to work more or less than the number of employees a company needs a day. This can aid in the improvement of human resource management. Third, a company can save total labor costs even though the company has the same number of employees as before. It was $143,525 before applying a mathematical model and it is $138,335 after applying a mathematical model. That is, a company can save a total labor cost of $5,190. Lastly, the different employee work pattern is found due to consideration of the change in labor policy. Some employees came to work more than 6 days a week before applying a developed model. However, it is constrained by a mathematical model at the moment because a company should follow the change of labor policy, which is each employee should work less than 5 days a week. The number of days regular employees come to work is constrained after applying a mathematical model due to the change in labor policy and high labor costs. Some regular employees come to work 4 days a week. This could be shown not to allow them to work more. However, it indirectly proved that a company that did not efficiently manage human resources has several employees more than the number they need to have. This can be an opportunity to rethink and optimize the number of employees a company needs to possess.
According to Figure 5, the rules related to overtime work are also well-considered. First, nobody is assigned to the office. Second, only one manager is assigned to supervise regular employees on one of the two floors. For instance, one manager of four managers is assigned on the first floor to play a role of supervisor during overtime work if you look at the first day. Third, an employee who performs equipment supervisor is assigned to one of two floors. Lastly, one employee is also allocated to open storage at least. To sum up, a staff schedule at daytime and nighttime that satisfies constraints is derived considering total labor cost.
In addition, I added the simulation part to consider the stochastic characteristics of irregular employees. The added part is as below. I added the related picture and table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
5.3 Evaluation for a derived staff schedule
“This paper provides a conservative and deterministic staff schedule by using a mathematical model that considers the average percent of irregular employees’ absences, which is an uncertain characteristic. Even though irregular employee absence is stochastic, the reason this paper derives a deterministic staff schedule is that it is impossible to predict if irregular employees come to work or do not come to work perfectly only based on the probability of each irregular employee’s absence. That is, even the result derived from the stochastic model cannot guarantee that each irregular employee follows a staff schedule without absences. However, the reason to perform the simulation after deriving a deterministic staff schedule is that it is meaningful in that it can give the result that can help managers respond to the irregular employee’s absence. Figure 6 is the result of the simulation. Row numbers from 41 to 62 mean irregular employees and column numbers mean days of work. The result shows whether each irregular employee follows their schedule when considering each employee’s absence percent.
If looking at Table 6 below, the actual percent of each irregular employee’s absence is different from the expected percent of each employee’s absence. In other words, even though the average percent of irregular employees’ absence is considered as the stochastic characteristic through the simulation, it cannot ensure that managers have a staff schedule that can predict the absences of irregular employees perfectly. Furthermore, As shown in Figure 6, the lacking number of irregular employees can be confirmed. If managers prepare a conservative measure, they will assign more irregular employees on the days when some irregular employees are expected not to come to work. However, preparing additional irregular employees makes extra labor cost for a construction company. Nevertheless, it is more important to finish work by assigning extra irregular employees within due time. Saving labor costs is the next thing to do. Moreover, according to the days of absence shown in Table 6, the number of days when each irregular employees do not come to work can be confirmed. For example, the 44th irregular employee does not come to work 7 days out of 13 days. This irregular employee has an actual percent of absence, of 53.8%. It is 23.8% higher than an expected percent of absence, 30%. This type of employee is the main cause to make a company fail to follow a staff schedule and assign enough workforce to jobs. This finding can be an opportunity that can help a manager rethink the recruitment of other irregular employees. To sum up, the reason this paper derives a conservative and deterministic staff schedule and performs the simulation about it is that it can contribute to presenting the systematic procedure for irregular employee absence. This procedure could aid managers to prepare precautious measures for a staff schedule while considering the stochastic characteristics of irregular employee absence.”
Q3. The technique section of this manuscript is poor. It is incorrect to call the parameters in Table 1 (page 5) indexes, and they are not even used in Equations (1) to (31). For examples, W1, W2, …, etc. Are they Sets or Values? There is no doubt that they are not indexes. The table numbers in section 4.1, from lines 280 to 294, are not reflected in the paragraph.
Responses: Thank you for your valuable comments. As you mentioned, I expressed parameters in Table 1 incorrectly. I appreciate to allow me to find wrong things. Everything the referee pointed is Set, not indexes. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.”
In addition, Equation (25) is revised by using the parameter that was not used, which is
Lastly, I revised the paragraph that referee mentioned by using the right number of Tables.
“The company has 40 regular employees who consist of 36 normal regular workers and 4 managers. Furthermore, they have 22 irregular employees. The minimum necessary number of employees is at least 45 during the daytime. In addition, the number of workplaces is 4 spaces such as the 1st floor, 2nd floor, open storage, and the office. Other known parameters are set and presented in Table 3 except for known parameters regarding costs. The daily labor cost of each employee for both daytime and overtime is presented in Table 4 and Table 5. The known parameters to create an optimal schedule for the numerical experiment are in Table 3.
The cost for all employees is as Table 4. Each employee is paid daily according to company policy. Regular employees are paid more than irregular employees. Managers are paid about $20 more than normal regular employees. The reason the different levels in paycheck between weekdays and weekends is because of the implementation of a 52-hour workweek policy that regards Saturday as overtime workday. Table 5 represents the paycheck for regular employees at nighttime. The reason the company uses only regular employees at night is because of the rules of the company considered in the paper. They want to allow regular employees to earn money in the middle of a situation where most work is gradually conducted by irregular employees due to the high labor cost of regular employees.”
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Dear authors,
I recommend emphasizing the associated research question in the introduction section. Future research opportunities should also be better emphasized. It is possible to emphasize the boundaries of research and the conditions of use of the selected method.
Figure 1: at the end of the x-axis there are incomprehensible orders. For the mathematical formulas, I recommend processing a flowchart of the entire layout, which can then be referenced in the Results section. Results chapter: I recommend not to start with a picture.
The conclusion could be extended to the possibility of a next specific application of the proposed solution in practice. Future research directions may also be highlighted. In the manuscript, please emphasize the novelty of your approach.
Overall the article is interesting. I wish you good luck in further research.
Kind regards,
Reviewer
Author Response
Paper title: Practical staff scheduling strategy considering various types of employment in the construction industry
Paper ID: algorithms-1851780
Comments received: August 16, 2022
Comments responded to: August 30, 2022
General Response from the Authors
We appreciate the valuable comments provided by the three anonymous reviewers regarding the contribution, the proposed method, and format of our manuscript. We really appreciate the efforts of the three reviewers to improve our manuscript. Therefore, we have made a great effort to enhance the presentation of the paper. An additional explanation about the modifications undertaken are as follows:
- Contribution of the study is supplemented throughout the overall manuscript.
- Introduction and Literature Review are revised to support this paper’s purpose.
- Numerical experiment is supplemented to make the manuscript meaningful.
- The format of manuscript was revised to be clearer and more comprehensive.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Academic editor
Q1. the authors fail to position their work in the academic literature. As a result, it is not clear what the contribution of the paper is and how to advances the state of the art in staff scheduling.
Responses: Thank you for your adequate comments. First of all, I revise the introduction to emphasize on what this paper considers and do. The revised part is as follow:
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.
This paper considers the actual situation of the construction industry in Korea such as the workforce shortage due to an increase in labor cost and workforce availability reduction by the change of the labor policy. The main thing considered in the paper is ‘no-show’, which is a real problem of the actual company that employed irregular employees to deal with the workforce shortage problem. Despite the no-show of irregular employees, it is difficult for the company to control irregular employees since the authority of giving penalties normally belongs to their outsourcing company in Korea. Nevertheless, the reason a company uses irregular employees is because of lower labor costs than local regular employees. Therefore, the company needs to respond to the unexpected situation that irregular employees suddenly do not come to work. To do this, this research uses linear programming to derive a deterministic staff schedule considering the average percent of irregular employee absences. This can be one of the proactive measures to respond to the stochastic characteristics of an irregular employee’s absence. Moreover, the simulation of a deterministic staff schedule is conducted to figure out what expectedly happens when assuming the application to a company in practice. Consequently, the performance of how well a staff schedule works is evaluated, and how many irregular employees do not come to work is expected. This can play a role as basic data that helps managers come down with deciding on a staff schedule.”
Furthermore, the contributions part is revised and added in the conclusion section in details. The added part is as follow:
“6.1 Contributions
This study proposes a staff scheduling strategy to minimize the total labor cost considering the actual features of the Korean construction industry such as an increase in labor cost and the change in labor policy. The contributions of this paper are as follows. First, this paper suggests a mathematical model that can save the total labor cost and aid the management of human resources. As shown in the numerical experiment, a mathematical model decreases the unnecessary number of employees from about 46.6 to 45 a day on average. This can allow a company to have an opportunity to save labor costs and rethink and optimize the number of employees they should have. Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea. this paper allows the manager of a company to come down with decision-making for staff schedule by providing various proactive cases while considering the sudden absence of employees. Despite the difficulty to consider unpredictable employee absence due to the inherently uncertain nature of irregular employees, the additional numerical experiments, which are the simulation and sensitivity analysis are performed in the paper. The result of the simulation predicts what expectedly happens when assuming the application of a deterministic staff schedule in practice. Especially, the predictive information on how many irregular employees would not come to work is given. It is impossible to guarantee that they would follow staff schedules perfectly in the real world. Nevertheless, the reason to perform the simulation is to allow managers to have a systematic procedure to respond to the absence of irregular employees, not to provide a staff schedule that makes all irregular employees follow the staff schedule. Moreover, through this sensitivity analysis with the change of the average percent of absence, this paper observes the influence of irregular employee absence on the change of the value of the objective function and staff schedule. Through the additional numerical experiments, various proactive cases are provided by considering the absence of irregular employees. This can help managers prepare the precautious measures to respond to the stochastic irregular employee's absence. Thus, it is expected to contribute to helping the manager with decision-making for staff schedules.”
Q2. 'no-shows' or employee absences have been studied by different authors. As they are inherently of an uncertain nature, the established manner of handling such problem characteristics is through stochastic modelling. The authors fail to acknowledge this and do not provide a motivation for their proposed approach.
Responses: Thank you for your adequate comments. I would like to appreciate to let me know the parts that I did not consider well again. As you mentioned, I added some previous references on employee absence studied by different authors.
“To improve the practicality, the uncertainty of irregular employees should be reflected when developing a staff scheduling strategy. Several studies considered the uncertainty of irregular employees. Research on workforce scheduling that considers the unpredictable employee’s absence was done to guarantee sufficient coverage in medical facilities. Becker et al. [29] proposed integer programming that considers assigning ex-post duties. This paper proved the practicality of the developed models by applying them to local medical facilities in practice. As a result, staff scheduling complexity was reduced. Ingels and Maenhout [30] stated many organizations make staff schedule under a deterministic operating environment. However, the stochastic environment such as employee absence occurs. To manage this uncertainty, this paper proposed a proactive approach to respond to schedule disruptions. By comparing a diversity of the proactive strategies, a proposed preemptive programming approach in this paper was assessed. Steenweg et al. [31] mentioned that short-term uncertainty in the workforce causes gaps between the planned and real shift schedule. Therefore, the unpredictable absence of employees should be considered when deciding on shift scheduling. This paper suggested a framework that can assign heterogeneously skilled employees to jobs optimally to respond to sudden absence. For this framework, the workforce availability was modeled by the stochastic simulation. As a result, the practicality of the framework that can provide efficient shift scheduling was proved by applying it to the actual cases.”
In addition, the different points from other studies are presented to provide motivation for this paper as well. It is as follows:
“This paper also considers the uncertainty of employees, especially irregular employee absence in the construction industry. Many studies about the staff scheduling that handles the uncertain nature of employees suggested stochastic modeling. In addition, other studies proposed the strategies for assignment of skilled employees to shift to fill in the absence. However, this paper develops a mathematical model that derives a deterministic staff schedule by considering the average percent of irregular employees’ absences. This is because the expected average percent of irregular employees’ absences could be different from the actual average percent of their absences in practice. That is, even a staff schedule derived from the stochastic modeling cannot ensure that all irregular employees follow a staff schedule because it is just performed only relying on the probability of absence. Therefore, this paper focuses on providing various proactive cases by deriving a deterministic staff schedule and conducting a simulation about it to evaluate performance. Consequently, the simulation helps a company expect how many irregular employees would not come to work when applying a deterministic staff schedule to a company. However, this cannot also guarantee that all irregular employees follow a staff schedule well because the possibility of their absences still exists. However, deriving a deterministic staff schedule and simulating it can contribute to providing a systematic procedure to manage the sudden absences of irregular employees. Therefore, this procedure can help managers prepare the precautious measures to rapidly respond to irregular employees’ absences.”
Q3. the presentation and grammar are poor, making parts of the paper hard to understand.
Responses: Thank you for your valuable comments. I agree with the point that English used in the paper is poor. So, I received English manuscript proofreading services from professors who are native English speakers at my university. I have received the revised manuscript, and I believe that the problem with English expressions has now been solved. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #1
Q1. The paper must be edited for English.
Responses: Thank you for your valuable comments. I agree with the point that English used in the paper is poor. So, I received English manuscript proofreading services from professors who are native English speakers at my university. I have received the revised manuscript, and I believe that the problem with English expressions has now been solved. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q2. What do you mean by "irregular" worker? Does it mean part-time? There is no definition of irregular employee in the paper.
Responses: Thank you for your adequate comments. First of all, the definition of irregular employees is as below.
“Irregular employees that are often shown in this paper are defined as foreign and local irregular employees who do not belong to a company and are paid daily. They are not part-time workers”
I added the explanation on irregular employee to help to understand what irregular employees are in this paper.
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.”
Q3. The major issue across the equations is that the authors did not pay attention to the notation. In modeling, the variables and parameters are case-sensitive. For example, there is a difference between Ad and ad, x and X, r and R, etc. You defined your decision variables in Table 1 using lower case, but the equations have upper case alphabets, which is wrong. The same issue exists for the majority of parameters in the paper. Therefore, Table 1 and all the equations must be updated for consistency.
Responses: Thank you for your valuable comments. I revised the decision variables in Table 2 using upper case. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q4. What is the purpose of equation 3?
Responses: Thank you for adequate comments. I omitted the explanation on Equation (3). It is added in the part of explanation about the Equations.
“Equation (2) and (3) represents the relationship between decision variables Ri,j,p,d and Yi,j,p,d. The actual number of irregular employees is decided by equation (2) which multiplies the number of irregular workers following the schedules without absence by . Equation (3) determines value of decision variable Ri,j,p,d and Ri,j,p,d can be 1 when decision variable Yi,j,p,d is 1.”
Q5. what is the unit of cost in tables 4 and 5? $/hour or $/day?
Responses: Thank you for your adequate comments. I put the information on the unit of cost in Table 4 and 5. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q6. There is no information about the solution procedure and timing. I understand the scope of the paper is not the solution approach, however, there must be some sort of information about it in the paper.
Responses: Thank you for your adequate comments. I think many studies using linear programming focus on the numerical experiment than solution procedure when solving the developed mathematical model. However, I agree with the comments that there must be some sort of information. So, I added the simple process as follow:
“4. Solution Process
Linear programming used as a method in this paper can be solved by the simplex method. The simplex method can be utilized to solve linear programming [32]. The simplex method is based on forming the inverse of the basic matrix and updating the inverse [33]. That is, it is the repetition of searching for feasible solutions, calculating the value of the objective function, and comparing the derived value and the optimal value till finding the optimal solution. The process of the simplex is shown in Figure 2. First of all, all equations should be converted into standard equation form after inserting slack variables and create an initial simplex table. After this, the thing to do is find the entering variables. When solving the maximization problem, the variable with the largest absolute value of the coefficient becomes the entering variable. The next thing is to find leaving variable. To do this, the ration between entering variable and solution should be calculated. At this time, the variable with the lowest ration value becomes the leaving variable. After deciding on entering variable and leaving variable, the intersection point of the entering variable and leaving variable becomes the pivot element and all values of the row in the simplex table with the pivot element should be divided by the pivot element. The rest of the rows should be updated by using the new pivot row made very previously. Lastly, if there is no new leaving variable, iterations terminate. If not, go back to the step of Finding the entering variable. Various computer packages based on the simplex method were already developed. A developed mathematical model in the paper is also solved by one of the packages. The package used in the paper is CPLEX, which is the mathematical optimization software package. The version is 20.1.0. The darker parts in Figure 2 are processed by CPLEX.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #2
Q1. I recommend emphasizing the associated research question in the introduction section. Future research opportunities should also be better emphasized. It is possible to emphasize the boundaries of research and the conditions of use of the selected method.
Responses: Thank you for your adequate comments. First of all, the boundary of research is staff scheduling using linear programming. Furthermore, the linear programming is normally used to minimize or maximize the objective function by assigning constrained resource efficiently. The objective function of this paper is to minimize the total labor cost.
As you mentioned, I emphasized the associated research in the introduction. I revised related parts as follow:
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.
This paper considers the actual situation of the construction industry in Korea such as the workforce shortage due to an increase in labor cost and workforce availability reduction by the change of the labor policy. The main thing considered in the paper is ‘no-show’, which is a real problem of the actual company that employed irregular employees to deal with the workforce shortage problem. Despite the no-show of irregular employees, it is difficult for the company to control irregular employees since the authority of giving penalties normally belongs to their outsourcing company in Korea. Nevertheless, the reason a company uses irregular employees is because of lower labor costs than local regular employees. Therefore, the company needs to respond to the unexpected situation that irregular employees suddenly do not come to work. To do this, this research uses linear programming to derive a deterministic staff schedule considering the average percent of irregular employee absences. This can be one of the proactive measures to respond to the stochastic characteristics of an irregular employee’s absence. Moreover, the simulation of a deterministic staff schedule is conducted to figure out what expectedly happens when assuming the application to a company in practice. Consequently, the performance of how well a staff schedule works is evaluated, and how many irregular employees do not come to work is expected. This can play a role as basic data that helps managers come down with deciding on a staff schedule.”
Q2. Figure 1: at the end of the x-axis there are incomprehensible orders.
Responses: Thank you for your valuable comments. I revised the x-axis. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q3. For the mathematical formulas, I recommend processing a flowchart of the entire layout, which can then be referenced in the Results section.
Responses: Thank you for your valuable comments. I revised the mathematical model part for the better readability, and I added the solution process as below.
“4. Solution Process
Linear programming used as a method in this paper can be solved by the simplex method. The simplex method can be utilized to solve linear programming [32]. The simplex method is based on forming the inverse of the basic matrix and updating the inverse [33]. That is, it is the repetition of searching for feasible solutions, calculating the value of the objective function, and comparing the derived value and the optimal value till finding the optimal solution. The process of the simplex is shown in Figure 2. First of all, all equations should be converted into standard equation form after inserting slack variables and create an initial simplex table. After this, the thing to do is find the entering variables. When solving the maximization problem, the variable with the largest absolute value of the coefficient becomes the entering variable. The next thing is to find leaving variable. To do this, the ration between entering variable and solution should be calculated. At this time, the variable with the lowest ration value becomes the leaving variable. After deciding on entering variable and leaving variable, the intersection point of the entering variable and leaving variable becomes the pivot element and all values of the row in the simplex table with the pivot element should be divided by the pivot element. The rest of the rows should be updated by using the new pivot row made very previously. Lastly, if there is no new leaving variable, iterations terminate. If not, go back to the step of Finding the entering variable. Various computer packages based on the simplex method were already developed. A developed mathematical model in the paper is also solved by one of the packages. The package used in the paper is CPLEX, which is the mathematical optimization software package. The version is 20.1.0. The darker parts in Figure 2 are processed by CPLEX.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q4. Results chapter: I recommend not to start with a picture.
Responses: Thank you for your valuable comments. As you mentioned, I moved the position of a picture after paragraph.
“5.2 Result
5.2.1. Before applying a mathematical model
This company did not have a staff schedule before. Due to this, they suffered from managing human resources efficiently. A staff schedule in Figure 3 is made based on the record of employee working patterns. Before the explanation of Figure 3, how to see a staff schedule is explained. In Figure 3, the row number means employees and the column number represents days. The numbers that are the darkest gray in row number stands for managers. The number in each cell indicates the assigned workplace and the cells with the diagonal pattern are Job 2, and without the diagonal pattern are Job 1. When it comes to jobs, Job 1 means normal work and Job 2 is equipment supervisor who takes responsibility for everything regarding equipment. If looking at Figure 3, more employees sometimes came to work than the number of employees a company needs, and fewer employees sometimes came to work than the number of employees a company needs. An example of the former one is Day 3 and the latter is Day 1. Especially, when looking at Day 24, placing enough workforce on the 2nd floor failed. Originally, more than 15 employees should be assigned on each floor during the daytime. However, only 12 employees were assigned that day. This directly shows inefficient human resource management. Moreover, about 46.6 employees came to work on average, which means it was a bigger number than the daily number of employees, 45, a company needs. This causes larger labor costs than when 45 employees come to work on average.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q5. The conclusion could be extended to the possibility of a next specific application of the proposed solution in practice. Future research directions may also be highlighted. In the manuscript, please emphasize the novelty of your approach.
Responses: Thank you for your valuable comments. First of all, a mathematical model is applied to an actual company in numerical experiment. Furthermore, the possible application of the proposed solution in practice is added as one of main contributions for this paper.
“Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea.”
In addition, I revised the future research part and added the future research directions. Some parts I added are as follows:
“6.2 Future Research
This paper could be expanded to another staff scheduling research that considers shifts. This paper does not consider shifts due to no shifts in the construction industry in general. However, many studies about staff scheduling consider shifts. Thus, if the developed mathematical model in this paper is improved considering the characteristics of shift, it is expected that the developed mathematical model could be applied to relatively more various industries. Furthermore, many objective functions can be considered to gain an optimal solution for staff scheduling. The current objective function is to minimize labor cost. This means that this staff scheduling strategy considers a company more than employees despite considering the policy called the 52-hour workweek policy. However, it seems necessary to create the objective function for a mathematical model that considers employees as well as the company in the future because employees' satisfaction with job assignments plays a crucial role in an organization’s success [34]. Moreover, research on figuring out the factors that influence the sincerity of irregular employees is needed in terms of their welfare. Doing this research can help make a proper working environment. In conclusion, various factors regarding both of company and employees in various ways should be considered to derive a good solution when staff scheduling problem is handled.”
Lastly, I added different points from the previous research in section 2 to emphasize the novelty of this paper.
“This paper also considers the uncertainty of employees, especially irregular employee absence in the construction industry. Many studies about the staff scheduling that handles the uncertain nature of employees suggested stochastic modeling. In addition, other studies proposed the strategies for assignment of skilled employees to shift to fill in the absence. However, this paper develops a mathematical model that derives a deterministic staff schedule by considering the average percent of irregular employees’ absences. This is because the expected average percent of irregular employees’ absences could be different from the actual average percent of their absences in practice. That is, even a staff schedule derived from the stochastic modeling cannot ensure that all irregular employees follow a staff schedule because it is just performed only relying on the probability of absence. Therefore, this paper focuses on providing various proactive cases by deriving a deterministic staff schedule and conducting a simulation about it to evaluate performance. Consequently, the simulation helps a company expect how many irregular employees would not come to work when applying a deterministic staff schedule to a company. However, this cannot also guarantee that all irregular employees follow a staff schedule well because the possibility of their absences still exists. However, deriving a deterministic staff schedule and simulating it can contribute to providing a systematic procedure to manage the sudden absences of irregular employees. Therefore, this procedure can help managers prepare the precautious measures to rapidly respond to irregular employee absence.”
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #3
Q1. This paper attempts to optimize various types of staff scheduling problems using mathematical programming models. It is true that the application of mathematical programming to staff scheduling is thoughtful, but the paper does not have a significant contribution to make it suitable for publication in this highly respected journal.
Responses: Thank you for your adequate comments. Following your advice, I rewrote the contributions part in the conclusion section. The added contributions are as follow:
“6.1 Contributions
This study proposes a staff scheduling strategy to minimize the total labor cost considering the actual features of the Korean construction industry such as an increase in labor cost and the change in labor policy. The contributions of this paper are as follows. First, this paper suggests a mathematical model that can save the total labor cost and aid the management of human resources. As shown in the numerical experiment, a mathematical model decreases the unnecessary number of employees from about 46.6 to 45 a day on average. This can allow a company to have an opportunity to save labor costs and rethink and optimize the number of employees they should have. Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea. Lastly, this paper allows the manager of a company to come down with decision-making for staff schedule by providing various proactive cases while considering the sudden absence of employees. Despite the difficulty to consider unpredictable employee absence due to the inherently uncertain nature of irregular employees, the additional numerical experiments, which are the simulation and sensitivity analysis are performed in the paper. The result of the simulation predicts what expectedly happens when assuming the application of a deterministic staff schedule in practice. Especially, the predictive information on how many irregular employees would not come to work is given. It is impossible to guarantee that they would follow staff schedules perfectly in the real world. Nevertheless, the reason to perform the simulation is to allow managers to have a systematic procedure to respond to the absence of irregular employees, not to provide a staff schedule that makes all irregular employees follow the staff schedule. Moreover, through this sensitivity analysis with the change of the average percent of absence, this paper observes the influence of irregular employee absence on the change of the value of the objective function and staff schedule. Through the additional numerical experiments, various proactive cases are provided by considering the absence of irregular employees. This can help managers prepare the precautious measures to respond to the stochastic irregular employee's absence. Thus, it is expected to contribute to helping the manager with decision-making for staff schedules.”
Q2. The optimal schedule result (Figure 2 on page 9) is not a scheduling result. Is it possible for a regular worker to work without a stable working schedule?
Responses: Thank you for your valuable comments. First of all, the original purpose of this paper is to help a manager of a company with decision making to minimize the total labor costs by presenting various proactive cases depending on the change of the average percent of irregular employee absence.
Furthermore, I added the part of comparison of two staff schedules between before and after applying a mathematical model to make this paper meaningful. The added parts are as below. I added the related pictures in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
“5.2 Result
5.2.1. Before applying a mathematical model
This company did not have a staff schedule before. Due to this, they suffered from managing human resources efficiently. A staff schedule in Figure 3 is made based on the record of employee working patterns. Before the explanation of Figure 3, how to see a staff schedule is explained. In Figure 3, the row number means employees and the column number represents days. The numbers that are the darkest gray in row number stands for managers. The number in each cell indicates the assigned workplace and the cells with the diagonal pattern are Job 2, and without the diagonal pattern are Job 1. When it comes to jobs, Job 1 means normal work and Job 2 is equipment supervisor who takes responsibility for everything regarding equipment. If looking at Figure 3, more employees sometimes came to work than the number of employees a company needs, and fewer employees sometimes came to work than the number of employees a company needs. An example of the former one is Day 3 and the latter is Day 1. Especially, when looking at Day 24, placing enough workforce on the 2nd floor failed. Originally, more than 15 employees should be assigned on each floor during the daytime. However, only 12 employees were assigned that day. This directly shows inefficient human resource management. Moreover, about 46.6 employees came to work on average, which means it was a bigger number than the daily number of employees, 45, a company needs. This causes larger labor costs than when 45 employees come to work on average.
5.2.2. After applying a mathematical model
Unlike a staff schedule before applying a mathematical model, A derived staff schedule is more efficient. As shown in Figure 4, all constraints regarding work in the daytime are satisfied. For example, 45 employees are scheduled to work if looking at Day 1. In addition, job assignments are met as well. Continued looking at Day 1, more than 15 employees are assigned on each floor and two of the employees conduct job 2 as equipment supervisors. Furthermore, more than one manager is assigned on each floor to supervise other regular and irregular employees as well. Moreover, over two of the employees work at open storage and a manager works in an office. That is, the failure of employees assignment caused by a manager’s random employee assignment does not happen. Other days as well as Day 1 meet all constraints for job assignments. Along with this, the 52-hour workweek policy is well applied in the mathematical model as the result shows. There are no employees who come to work for more than five days a week. All employees have a day-off on Sunday.
Furthermore, the purpose of the developed mathematical model is to minimize the total labor cost of regular and irregular employees. This means that the result in the numerical experiment is for a company that has a priority of saving the total labor cost. Therefore, if looking at the result in Figure 4, as many irregular employees as possible are scheduled to be on each workday even though the average percent of irregular employee absence is considered. It is more advantageous for a company to minimize the total labor costs by using as many irregular employees as possible out of 22 irregular employees due to relatively low labor costs. However, it is seen that as many irregular employees as possible are not used. This is because there are rules on doing jobs of a company and there are jobs that can be conducted by regular employees. A company does overtime work by assigning only regular employees to jobs for consideration for regular employees as well. To do overtime work, regular employees should work during the daytime. Thus, the result in Figure 4 is derived even though a company has enough irregular employees that can cover more jobs.
Comparing a staff schedule before and after applying a mathematical model, the following things can be found. First, there is a difference in the number of employees who come to work per day on average. Consequently, about 1.6 employees come to work less than before after applying a mathematical model every day. Second, a developed mathematical model prevents situations where employees come to work more or less than the number of employees a company needs a day. This can aid in the improvement of human resource management. Third, a company can save total labor costs even though the company has the same number of employees as before. It was $143,525 before applying a mathematical model and it is $138,335 after applying a mathematical model. That is, a company can save a total labor cost of $5,190. Lastly, the different employee work pattern is found due to consideration of the change in labor policy. Some employees came to work more than 6 days a week before applying a developed model. However, it is constrained by a mathematical model at the moment because a company should follow the change of labor policy, which is each employee should work less than 5 days a week. The number of days regular employees come to work is constrained after applying a mathematical model due to the change in labor policy and high labor costs. Some regular employees come to work 4 days a week. This could be shown not to allow them to work more. However, it indirectly proved that a company that did not efficiently manage human resources has several employees more than the number they need to have. This can be an opportunity to rethink and optimize the number of employees a company needs to possess.
According to Figure 5, the rules related to overtime work are also well-considered. First, nobody is assigned to the office. Second, only one manager is assigned to supervise regular employees on one of the two floors. For instance, one manager of four managers is assigned on the first floor to play a role of supervisor during overtime work if you look at the first day. Third, an employee who performs equipment supervisor is assigned to one of two floors. Lastly, one employee is also allocated to open storage at least. To sum up, a staff schedule at daytime and nighttime that satisfies constraints is derived considering total labor cost.
In addition, I added the simulation part to consider the stochastic characteristics of irregular employees. The added part is as below. I added the related picture and table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
5.3 Evaluation for a derived staff schedule
“This paper provides a conservative and deterministic staff schedule by using a mathematical model that considers the average percent of irregular employees’ absences, which is an uncertain characteristic. Even though irregular employee absence is stochastic, the reason this paper derives a deterministic staff schedule is that it is impossible to predict if irregular employees come to work or do not come to work perfectly only based on the probability of each irregular employee’s absence. That is, even the result derived from the stochastic model cannot guarantee that each irregular employee follows a staff schedule without absences. However, the reason to perform the simulation after deriving a deterministic staff schedule is that it is meaningful in that it can give the result that can help managers respond to the irregular employee’s absence. Figure 6 is the result of the simulation. Row numbers from 41 to 62 mean irregular employees and column numbers mean days of work. The result shows whether each irregular employee follows their schedule when considering each employee’s absence percent.
If looking at Table 6 below, the actual percent of each irregular employee’s absence is different from the expected percent of each employee’s absence. In other words, even though the average percent of irregular employees’ absence is considered as the stochastic characteristic through the simulation, it cannot ensure that managers have a staff schedule that can predict the absences of irregular employees perfectly. Furthermore, As shown in Figure 6, the lacking number of irregular employees can be confirmed. If managers prepare a conservative measure, they will assign more irregular employees on the days when some irregular employees are expected not to come to work. However, preparing additional irregular employees makes extra labor cost for a construction company. Nevertheless, it is more important to finish work by assigning extra irregular employees within due time. Saving labor costs is the next thing to do. Moreover, according to the days of absence shown in Table 6, the number of days when each irregular employees do not come to work can be confirmed. For example, the 44th irregular employee does not come to work 7 days out of 13 days. This irregular employee has an actual percent of absence, of 53.8%. It is 23.8% higher than an expected percent of absence, 30%. This type of employee is the main cause to make a company fail to follow a staff schedule and assign enough workforce to jobs. This finding can be an opportunity that can help a manager rethink the recruitment of other irregular employees. To sum up, the reason this paper derives a conservative and deterministic staff schedule and performs the simulation about it is that it can contribute to presenting the systematic procedure for irregular employee absence. This procedure could aid managers to prepare precautious measures for a staff schedule while considering the stochastic characteristics of irregular employee absence.”
Q3. The technique section of this manuscript is poor. It is incorrect to call the parameters in Table 1 (page 5) indexes, and they are not even used in Equations (1) to (31). For examples, W1, W2, …, etc. Are they Sets or Values? There is no doubt that they are not indexes. The table numbers in section 4.1, from lines 280 to 294, are not reflected in the paragraph.
Responses: Thank you for your valuable comments. As you mentioned, I expressed parameters in Table 1 incorrectly. I appreciate to allow me to find wrong things. Everything the referee pointed is Set, not indexes. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.”
In addition, Equation (25) is revised by using the parameter that was not used, which is
Lastly, I revised the paragraph that referee mentioned by using the right number of Tables.
“The company has 40 regular employees who consist of 36 normal regular workers and 4 managers. Furthermore, they have 22 irregular employees. The minimum necessary number of employees is at least 45 during the daytime. In addition, the number of workplaces is 4 spaces such as the 1st floor, 2nd floor, open storage, and the office. Other known parameters are set and presented in Table 3 except for known parameters regarding costs. The daily labor cost of each employee for both daytime and overtime is presented in Table 4 and Table 5. The known parameters to create an optimal schedule for the numerical experiment are in Table 3.
The cost for all employees is as Table 4. Each employee is paid daily according to company policy. Regular employees are paid more than irregular employees. Managers are paid about $20 more than normal regular employees. The reason the different levels in paycheck between weekdays and weekends is because of the implementation of a 52-hour workweek policy that regards Saturday as overtime workday. Table 5 represents the paycheck for regular employees at nighttime. The reason the company uses only regular employees at night is because of the rules of the company considered in the paper. They want to allow regular employees to earn money in the middle of a situation where most work is gradually conducted by irregular employees due to the high labor cost of regular employees.”
Author Response File: Author Response.pdf
Reviewer 3 Report
My first conclusion of the review is that this manuscript should be REJECTED.
This paper attempts to optimize various types of staff scheduling problems using mathematical programming models. It is true that the application of mathematical programming to staff scheduling is thoughtful, but the paper does not have a significant contribution to make it suitable for publication in this highly respected journal. Firstly, the optimal schedule result (Figure 2 on page 9) is not a scheduling result. Is it possible for a regular worker to work without a stable working schedule? As a result, the proposed mathematical model does not make any sense, and it may not be suitable for regular workers.
Furthermore, the technique section of this manuscript is poor. It is incorrect to call the parameters in Table 1 (page 5) indexes, and they are not even used in Equations (1) to (31). For examples, W1, W2, …, etc. Are they Sets or Values? There is no doubt that they are not indexes. The table numbers in section 4.1, from lines 280 to 294, are not reflected in the paragraph.
As a whole, the manuscript seems to be more concerned with the calculation of the number of manpower than scheduling.
Author Response
Paper title: Practical staff scheduling strategy considering various types of employment in the construction industry
Paper ID: algorithms-1851780
Comments received: August 16, 2022
Comments responded to: August 30, 2022
General Response from the Authors
We appreciate the valuable comments provided by the three anonymous reviewers regarding the contribution, the proposed method, and format of our manuscript. We really appreciate the efforts of the three reviewers to improve our manuscript. Therefore, we have made a great effort to enhance the presentation of the paper. An additional explanation about the modifications undertaken are as follows:
- Contribution of the study is supplemented throughout the overall manuscript.
- Introduction and Literature Review are revised to support this paper’s purpose.
- Numerical experiment is supplemented to make the manuscript meaningful.
- The format of manuscript was revised to be clearer and more comprehensive.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Academic editor
Q1. the authors fail to position their work in the academic literature. As a result, it is not clear what the contribution of the paper is and how to advances the state of the art in staff scheduling.
Responses: Thank you for your adequate comments. First of all, I revise the introduction to emphasize on what this paper considers and do. The revised part is as follow:
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.
This paper considers the actual situation of the construction industry in Korea such as the workforce shortage due to an increase in labor cost and workforce availability reduction by the change of the labor policy. The main thing considered in the paper is ‘no-show’, which is a real problem of the actual company that employed irregular employees to deal with the workforce shortage problem. Despite the no-show of irregular employees, it is difficult for the company to control irregular employees since the authority of giving penalties normally belongs to their outsourcing company in Korea. Nevertheless, the reason a company uses irregular employees is because of lower labor costs than local regular employees. Therefore, the company needs to respond to the unexpected situation that irregular employees suddenly do not come to work. To do this, this research uses linear programming to derive a deterministic staff schedule considering the average percent of irregular employee absences. This can be one of the proactive measures to respond to the stochastic characteristics of an irregular employee’s absence. Moreover, the simulation of a deterministic staff schedule is conducted to figure out what expectedly happens when assuming the application to a company in practice. Consequently, the performance of how well a staff schedule works is evaluated, and how many irregular employees do not come to work is expected. This can play a role as basic data that helps managers come down with deciding on a staff schedule.”
Furthermore, the contributions part is revised and added in the conclusion section in details. The added part is as follow:
“6.1 Contributions
This study proposes a staff scheduling strategy to minimize the total labor cost considering the actual features of the Korean construction industry such as an increase in labor cost and the change in labor policy. The contributions of this paper are as follows. First, this paper suggests a mathematical model that can save the total labor cost and aid the management of human resources. As shown in the numerical experiment, a mathematical model decreases the unnecessary number of employees from about 46.6 to 45 a day on average. This can allow a company to have an opportunity to save labor costs and rethink and optimize the number of employees they should have. Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea. this paper allows the manager of a company to come down with decision-making for staff schedule by providing various proactive cases while considering the sudden absence of employees. Despite the difficulty to consider unpredictable employee absence due to the inherently uncertain nature of irregular employees, the additional numerical experiments, which are the simulation and sensitivity analysis are performed in the paper. The result of the simulation predicts what expectedly happens when assuming the application of a deterministic staff schedule in practice. Especially, the predictive information on how many irregular employees would not come to work is given. It is impossible to guarantee that they would follow staff schedules perfectly in the real world. Nevertheless, the reason to perform the simulation is to allow managers to have a systematic procedure to respond to the absence of irregular employees, not to provide a staff schedule that makes all irregular employees follow the staff schedule. Moreover, through this sensitivity analysis with the change of the average percent of absence, this paper observes the influence of irregular employee absence on the change of the value of the objective function and staff schedule. Through the additional numerical experiments, various proactive cases are provided by considering the absence of irregular employees. This can help managers prepare the precautious measures to respond to the stochastic irregular employee's absence. Thus, it is expected to contribute to helping the manager with decision-making for staff schedules.”
Q2. 'no-shows' or employee absences have been studied by different authors. As they are inherently of an uncertain nature, the established manner of handling such problem characteristics is through stochastic modelling. The authors fail to acknowledge this and do not provide a motivation for their proposed approach.
Responses: Thank you for your adequate comments. I would like to appreciate to let me know the parts that I did not consider well again. As you mentioned, I added some previous references on employee absence studied by different authors.
“To improve the practicality, the uncertainty of irregular employees should be reflected when developing a staff scheduling strategy. Several studies considered the uncertainty of irregular employees. Research on workforce scheduling that considers the unpredictable employee’s absence was done to guarantee sufficient coverage in medical facilities. Becker et al. [29] proposed integer programming that considers assigning ex-post duties. This paper proved the practicality of the developed models by applying them to local medical facilities in practice. As a result, staff scheduling complexity was reduced. Ingels and Maenhout [30] stated many organizations make staff schedule under a deterministic operating environment. However, the stochastic environment such as employee absence occurs. To manage this uncertainty, this paper proposed a proactive approach to respond to schedule disruptions. By comparing a diversity of the proactive strategies, a proposed preemptive programming approach in this paper was assessed. Steenweg et al. [31] mentioned that short-term uncertainty in the workforce causes gaps between the planned and real shift schedule. Therefore, the unpredictable absence of employees should be considered when deciding on shift scheduling. This paper suggested a framework that can assign heterogeneously skilled employees to jobs optimally to respond to sudden absence. For this framework, the workforce availability was modeled by the stochastic simulation. As a result, the practicality of the framework that can provide efficient shift scheduling was proved by applying it to the actual cases.”
In addition, the different points from other studies are presented to provide motivation for this paper as well. It is as follows:
“This paper also considers the uncertainty of employees, especially irregular employee absence in the construction industry. Many studies about the staff scheduling that handles the uncertain nature of employees suggested stochastic modeling. In addition, other studies proposed the strategies for assignment of skilled employees to shift to fill in the absence. However, this paper develops a mathematical model that derives a deterministic staff schedule by considering the average percent of irregular employees’ absences. This is because the expected average percent of irregular employees’ absences could be different from the actual average percent of their absences in practice. That is, even a staff schedule derived from the stochastic modeling cannot ensure that all irregular employees follow a staff schedule because it is just performed only relying on the probability of absence. Therefore, this paper focuses on providing various proactive cases by deriving a deterministic staff schedule and conducting a simulation about it to evaluate performance. Consequently, the simulation helps a company expect how many irregular employees would not come to work when applying a deterministic staff schedule to a company. However, this cannot also guarantee that all irregular employees follow a staff schedule well because the possibility of their absences still exists. However, deriving a deterministic staff schedule and simulating it can contribute to providing a systematic procedure to manage the sudden absences of irregular employees. Therefore, this procedure can help managers prepare the precautious measures to rapidly respond to irregular employees’ absences.”
Q3. the presentation and grammar are poor, making parts of the paper hard to understand.
Responses: Thank you for your valuable comments. I agree with the point that English used in the paper is poor. So, I received English manuscript proofreading services from professors who are native English speakers at my university. I have received the revised manuscript, and I believe that the problem with English expressions has now been solved. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #1
Q1. The paper must be edited for English.
Responses: Thank you for your valuable comments. I agree with the point that English used in the paper is poor. So, I received English manuscript proofreading services from professors who are native English speakers at my university. I have received the revised manuscript, and I believe that the problem with English expressions has now been solved. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q2. What do you mean by "irregular" worker? Does it mean part-time? There is no definition of irregular employee in the paper.
Responses: Thank you for your adequate comments. First of all, the definition of irregular employees is as below.
“Irregular employees that are often shown in this paper are defined as foreign and local irregular employees who do not belong to a company and are paid daily. They are not part-time workers”
I added the explanation on irregular employee to help to understand what irregular employees are in this paper.
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.”
Q3. The major issue across the equations is that the authors did not pay attention to the notation. In modeling, the variables and parameters are case-sensitive. For example, there is a difference between Ad and ad, x and X, r and R, etc. You defined your decision variables in Table 1 using lower case, but the equations have upper case alphabets, which is wrong. The same issue exists for the majority of parameters in the paper. Therefore, Table 1 and all the equations must be updated for consistency.
Responses: Thank you for your valuable comments. I revised the decision variables in Table 2 using upper case. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q4. What is the purpose of equation 3?
Responses: Thank you for adequate comments. I omitted the explanation on Equation (3). It is added in the part of explanation about the Equations.
“Equation (2) and (3) represents the relationship between decision variables Ri,j,p,d and Yi,j,p,d. The actual number of irregular employees is decided by equation (2) which multiplies the number of irregular workers following the schedules without absence by . Equation (3) determines value of decision variable Ri,j,p,d and Ri,j,p,d can be 1 when decision variable Yi,j,p,d is 1.”
Q5. what is the unit of cost in tables 4 and 5? $/hour or $/day?
Responses: Thank you for your adequate comments. I put the information on the unit of cost in Table 4 and 5. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q6. There is no information about the solution procedure and timing. I understand the scope of the paper is not the solution approach, however, there must be some sort of information about it in the paper.
Responses: Thank you for your adequate comments. I think many studies using linear programming focus on the numerical experiment than solution procedure when solving the developed mathematical model. However, I agree with the comments that there must be some sort of information. So, I added the simple process as follow:
“4. Solution Process
Linear programming used as a method in this paper can be solved by the simplex method. The simplex method can be utilized to solve linear programming [32]. The simplex method is based on forming the inverse of the basic matrix and updating the inverse [33]. That is, it is the repetition of searching for feasible solutions, calculating the value of the objective function, and comparing the derived value and the optimal value till finding the optimal solution. The process of the simplex is shown in Figure 2. First of all, all equations should be converted into standard equation form after inserting slack variables and create an initial simplex table. After this, the thing to do is find the entering variables. When solving the maximization problem, the variable with the largest absolute value of the coefficient becomes the entering variable. The next thing is to find leaving variable. To do this, the ration between entering variable and solution should be calculated. At this time, the variable with the lowest ration value becomes the leaving variable. After deciding on entering variable and leaving variable, the intersection point of the entering variable and leaving variable becomes the pivot element and all values of the row in the simplex table with the pivot element should be divided by the pivot element. The rest of the rows should be updated by using the new pivot row made very previously. Lastly, if there is no new leaving variable, iterations terminate. If not, go back to the step of Finding the entering variable. Various computer packages based on the simplex method were already developed. A developed mathematical model in the paper is also solved by one of the packages. The package used in the paper is CPLEX, which is the mathematical optimization software package. The version is 20.1.0. The darker parts in Figure 2 are processed by CPLEX.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #2
Q1. I recommend emphasizing the associated research question in the introduction section. Future research opportunities should also be better emphasized. It is possible to emphasize the boundaries of research and the conditions of use of the selected method.
Responses: Thank you for your adequate comments. First of all, the boundary of research is staff scheduling using linear programming. Furthermore, the linear programming is normally used to minimize or maximize the objective function by assigning constrained resource efficiently. The objective function of this paper is to minimize the total labor cost.
As you mentioned, I emphasized the associated research in the introduction. I revised related parts as follow:
“About 50 percent are irregular employees out of the total workforce according to the statistics data by Korean bank. The reason the percent of irregular employees is high is that many people do not want to be employed in the construction industry due to the possible dangers that can lead to death or disability [6]. In addition, if they employ regular employees, they need to keep paying for them even when they do not have work because of the difficulty of firing them. Due to this, the construction company suffers from workforce shortage regarded as one of the most serious problems in all industries [7]. Moreover, companies want to employ irregular employees including foreign employees from an outsourcing company to reduce costs due to an increase in the labor cost [8]. According to the [9], foreign employees account for 19.5 percent of the whole workforce size in the construction industry. In addition, it is expected that irregular foreign employees continue to increase. The government presented that about 57 thousand are the proper number of foreign employees in 2019. However, the number of foreign employees was about 210 thousand in the same year [10]. That is, the Korean construction industry faces a situation that has no choice but to employ a greater number of irregular foreign employees.
Several countries already used irregular employees to deal with the workforce shortage. For instance, Malaysia has already been using the irregular foreign workforce actively to manage the workforce shortage problem [11]. Furthermore, a portion of temporary jobs is increasing in the UK labor market [12]. Employing irregular foreign employees can be a realistic way to respond to the workforce shortage. However, it is important to know and understand the Korean way to use irregular employees. Most irregular employees are used in the way of getting paid daily. It is thought that this feature of the way of using irregular employees would be different from other countries that normally contract irregular employees and use them during the contracted period. However, because of the way to use irregular employees, irregular employees do not have loyalty so that they also have low responsibility for working. In addition, they could have different irregular employees every day because they are normally contracted daily. Thus, they have difficulty keeping tracking irregular employees and giving them penalties even when they make trouble due to a daily contract. These make companies difficult to do human resource management efficiently.
This paper considers the actual situation of the construction industry in Korea such as the workforce shortage due to an increase in labor cost and workforce availability reduction by the change of the labor policy. The main thing considered in the paper is ‘no-show’, which is a real problem of the actual company that employed irregular employees to deal with the workforce shortage problem. Despite the no-show of irregular employees, it is difficult for the company to control irregular employees since the authority of giving penalties normally belongs to their outsourcing company in Korea. Nevertheless, the reason a company uses irregular employees is because of lower labor costs than local regular employees. Therefore, the company needs to respond to the unexpected situation that irregular employees suddenly do not come to work. To do this, this research uses linear programming to derive a deterministic staff schedule considering the average percent of irregular employee absences. This can be one of the proactive measures to respond to the stochastic characteristics of an irregular employee’s absence. Moreover, the simulation of a deterministic staff schedule is conducted to figure out what expectedly happens when assuming the application to a company in practice. Consequently, the performance of how well a staff schedule works is evaluated, and how many irregular employees do not come to work is expected. This can play a role as basic data that helps managers come down with deciding on a staff schedule.”
Q2. Figure 1: at the end of the x-axis there are incomprehensible orders.
Responses: Thank you for your valuable comments. I revised the x-axis. I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q3. For the mathematical formulas, I recommend processing a flowchart of the entire layout, which can then be referenced in the Results section.
Responses: Thank you for your valuable comments. I revised the mathematical model part for the better readability, and I added the solution process as below.
“4. Solution Process
Linear programming used as a method in this paper can be solved by the simplex method. The simplex method can be utilized to solve linear programming [32]. The simplex method is based on forming the inverse of the basic matrix and updating the inverse [33]. That is, it is the repetition of searching for feasible solutions, calculating the value of the objective function, and comparing the derived value and the optimal value till finding the optimal solution. The process of the simplex is shown in Figure 2. First of all, all equations should be converted into standard equation form after inserting slack variables and create an initial simplex table. After this, the thing to do is find the entering variables. When solving the maximization problem, the variable with the largest absolute value of the coefficient becomes the entering variable. The next thing is to find leaving variable. To do this, the ration between entering variable and solution should be calculated. At this time, the variable with the lowest ration value becomes the leaving variable. After deciding on entering variable and leaving variable, the intersection point of the entering variable and leaving variable becomes the pivot element and all values of the row in the simplex table with the pivot element should be divided by the pivot element. The rest of the rows should be updated by using the new pivot row made very previously. Lastly, if there is no new leaving variable, iterations terminate. If not, go back to the step of Finding the entering variable. Various computer packages based on the simplex method were already developed. A developed mathematical model in the paper is also solved by one of the packages. The package used in the paper is CPLEX, which is the mathematical optimization software package. The version is 20.1.0. The darker parts in Figure 2 are processed by CPLEX.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q4. Results chapter: I recommend not to start with a picture.
Responses: Thank you for your valuable comments. As you mentioned, I moved the position of a picture after paragraph.
“5.2 Result
5.2.1. Before applying a mathematical model
This company did not have a staff schedule before. Due to this, they suffered from managing human resources efficiently. A staff schedule in Figure 3 is made based on the record of employee working patterns. Before the explanation of Figure 3, how to see a staff schedule is explained. In Figure 3, the row number means employees and the column number represents days. The numbers that are the darkest gray in row number stands for managers. The number in each cell indicates the assigned workplace and the cells with the diagonal pattern are Job 2, and without the diagonal pattern are Job 1. When it comes to jobs, Job 1 means normal work and Job 2 is equipment supervisor who takes responsibility for everything regarding equipment. If looking at Figure 3, more employees sometimes came to work than the number of employees a company needs, and fewer employees sometimes came to work than the number of employees a company needs. An example of the former one is Day 3 and the latter is Day 1. Especially, when looking at Day 24, placing enough workforce on the 2nd floor failed. Originally, more than 15 employees should be assigned on each floor during the daytime. However, only 12 employees were assigned that day. This directly shows inefficient human resource management. Moreover, about 46.6 employees came to work on average, which means it was a bigger number than the daily number of employees, 45, a company needs. This causes larger labor costs than when 45 employees come to work on average.”
I added the related picture in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
Q5. The conclusion could be extended to the possibility of a next specific application of the proposed solution in practice. Future research directions may also be highlighted. In the manuscript, please emphasize the novelty of your approach.
Responses: Thank you for your valuable comments. First of all, a mathematical model is applied to an actual company in numerical experiment. Furthermore, the possible application of the proposed solution in practice is added as one of main contributions for this paper.
“Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea.”
In addition, I revised the future research part and added the future research directions. Some parts I added are as follows:
“6.2 Future Research
This paper could be expanded to another staff scheduling research that considers shifts. This paper does not consider shifts due to no shifts in the construction industry in general. However, many studies about staff scheduling consider shifts. Thus, if the developed mathematical model in this paper is improved considering the characteristics of shift, it is expected that the developed mathematical model could be applied to relatively more various industries. Furthermore, many objective functions can be considered to gain an optimal solution for staff scheduling. The current objective function is to minimize labor cost. This means that this staff scheduling strategy considers a company more than employees despite considering the policy called the 52-hour workweek policy. However, it seems necessary to create the objective function for a mathematical model that considers employees as well as the company in the future because employees' satisfaction with job assignments plays a crucial role in an organization’s success [34]. Moreover, research on figuring out the factors that influence the sincerity of irregular employees is needed in terms of their welfare. Doing this research can help make a proper working environment. In conclusion, various factors regarding both of company and employees in various ways should be considered to derive a good solution when staff scheduling problem is handled.”
Lastly, I added different points from the previous research in section 2 to emphasize the novelty of this paper.
“This paper also considers the uncertainty of employees, especially irregular employee absence in the construction industry. Many studies about the staff scheduling that handles the uncertain nature of employees suggested stochastic modeling. In addition, other studies proposed the strategies for assignment of skilled employees to shift to fill in the absence. However, this paper develops a mathematical model that derives a deterministic staff schedule by considering the average percent of irregular employees’ absences. This is because the expected average percent of irregular employees’ absences could be different from the actual average percent of their absences in practice. That is, even a staff schedule derived from the stochastic modeling cannot ensure that all irregular employees follow a staff schedule because it is just performed only relying on the probability of absence. Therefore, this paper focuses on providing various proactive cases by deriving a deterministic staff schedule and conducting a simulation about it to evaluate performance. Consequently, the simulation helps a company expect how many irregular employees would not come to work when applying a deterministic staff schedule to a company. However, this cannot also guarantee that all irregular employees follow a staff schedule well because the possibility of their absences still exists. However, deriving a deterministic staff schedule and simulating it can contribute to providing a systematic procedure to manage the sudden absences of irregular employees. Therefore, this procedure can help managers prepare the precautious measures to rapidly respond to irregular employee absence.”
Reviewer Comments and Author Responses
1st Round Revision, algorithms-1851780
I appreciate all the efforts and supports of referees on the evaluation and important comments for my manuscript. In the revised version, I tried to sincerely handle every concern and suggestion raised by referees. Please refer following responses.
Referee #3
Q1. This paper attempts to optimize various types of staff scheduling problems using mathematical programming models. It is true that the application of mathematical programming to staff scheduling is thoughtful, but the paper does not have a significant contribution to make it suitable for publication in this highly respected journal.
Responses: Thank you for your adequate comments. Following your advice, I rewrote the contributions part in the conclusion section. The added contributions are as follow:
“6.1 Contributions
This study proposes a staff scheduling strategy to minimize the total labor cost considering the actual features of the Korean construction industry such as an increase in labor cost and the change in labor policy. The contributions of this paper are as follows. First, this paper suggests a mathematical model that can save the total labor cost and aid the management of human resources. As shown in the numerical experiment, a mathematical model decreases the unnecessary number of employees from about 46.6 to 45 a day on average. This can allow a company to have an opportunity to save labor costs and rethink and optimize the number of employees they should have. Second, the practicality to apply the developed mathematical model to other industries is proved. This study considers an actual construction company suffering from the problem of, no-show, derived from the situation that the company increases irregular employees to handle workforce shortage due to the rise of labor costs and a new labor policy in Korea. Through the numerical experiment with system parameters derived from human resources, types of jobs, workplaces, and the rules of work of a company, the applicability of the developed mathematical model to a company is proved in practice. Therefore, this paper would help conduct research on staff scheduling for the actual company as one of the basic references. Third, the change in labor policy is considered when developing a mathematical model. The optimal solution for staff scheduling by using this mathematical model is derived in the situation of workforce availability reduction compared to the same number of employees due to the influence of changed labor policy. In other words, this paper could play a role as a guide for staff scheduling that must consider the change of labor policy in the future in Korea. Lastly, this paper allows the manager of a company to come down with decision-making for staff schedule by providing various proactive cases while considering the sudden absence of employees. Despite the difficulty to consider unpredictable employee absence due to the inherently uncertain nature of irregular employees, the additional numerical experiments, which are the simulation and sensitivity analysis are performed in the paper. The result of the simulation predicts what expectedly happens when assuming the application of a deterministic staff schedule in practice. Especially, the predictive information on how many irregular employees would not come to work is given. It is impossible to guarantee that they would follow staff schedules perfectly in the real world. Nevertheless, the reason to perform the simulation is to allow managers to have a systematic procedure to respond to the absence of irregular employees, not to provide a staff schedule that makes all irregular employees follow the staff schedule. Moreover, through this sensitivity analysis with the change of the average percent of absence, this paper observes the influence of irregular employee absence on the change of the value of the objective function and staff schedule. Through the additional numerical experiments, various proactive cases are provided by considering the absence of irregular employees. This can help managers prepare the precautious measures to respond to the stochastic irregular employee's absence. Thus, it is expected to contribute to helping the manager with decision-making for staff schedules.”
Q2. The optimal schedule result (Figure 2 on page 9) is not a scheduling result. Is it possible for a regular worker to work without a stable working schedule?
Responses: Thank you for your valuable comments. First of all, the original purpose of this paper is to help a manager of a company with decision making to minimize the total labor costs by presenting various proactive cases depending on the change of the average percent of irregular employee absence.
Furthermore, I added the part of comparison of two staff schedules between before and after applying a mathematical model to make this paper meaningful. The added parts are as below. I added the related pictures in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
“5.2 Result
5.2.1. Before applying a mathematical model
This company did not have a staff schedule before. Due to this, they suffered from managing human resources efficiently. A staff schedule in Figure 3 is made based on the record of employee working patterns. Before the explanation of Figure 3, how to see a staff schedule is explained. In Figure 3, the row number means employees and the column number represents days. The numbers that are the darkest gray in row number stands for managers. The number in each cell indicates the assigned workplace and the cells with the diagonal pattern are Job 2, and without the diagonal pattern are Job 1. When it comes to jobs, Job 1 means normal work and Job 2 is equipment supervisor who takes responsibility for everything regarding equipment. If looking at Figure 3, more employees sometimes came to work than the number of employees a company needs, and fewer employees sometimes came to work than the number of employees a company needs. An example of the former one is Day 3 and the latter is Day 1. Especially, when looking at Day 24, placing enough workforce on the 2nd floor failed. Originally, more than 15 employees should be assigned on each floor during the daytime. However, only 12 employees were assigned that day. This directly shows inefficient human resource management. Moreover, about 46.6 employees came to work on average, which means it was a bigger number than the daily number of employees, 45, a company needs. This causes larger labor costs than when 45 employees come to work on average.
5.2.2. After applying a mathematical model
Unlike a staff schedule before applying a mathematical model, A derived staff schedule is more efficient. As shown in Figure 4, all constraints regarding work in the daytime are satisfied. For example, 45 employees are scheduled to work if looking at Day 1. In addition, job assignments are met as well. Continued looking at Day 1, more than 15 employees are assigned on each floor and two of the employees conduct job 2 as equipment supervisors. Furthermore, more than one manager is assigned on each floor to supervise other regular and irregular employees as well. Moreover, over two of the employees work at open storage and a manager works in an office. That is, the failure of employees assignment caused by a manager’s random employee assignment does not happen. Other days as well as Day 1 meet all constraints for job assignments. Along with this, the 52-hour workweek policy is well applied in the mathematical model as the result shows. There are no employees who come to work for more than five days a week. All employees have a day-off on Sunday.
Furthermore, the purpose of the developed mathematical model is to minimize the total labor cost of regular and irregular employees. This means that the result in the numerical experiment is for a company that has a priority of saving the total labor cost. Therefore, if looking at the result in Figure 4, as many irregular employees as possible are scheduled to be on each workday even though the average percent of irregular employee absence is considered. It is more advantageous for a company to minimize the total labor costs by using as many irregular employees as possible out of 22 irregular employees due to relatively low labor costs. However, it is seen that as many irregular employees as possible are not used. This is because there are rules on doing jobs of a company and there are jobs that can be conducted by regular employees. A company does overtime work by assigning only regular employees to jobs for consideration for regular employees as well. To do overtime work, regular employees should work during the daytime. Thus, the result in Figure 4 is derived even though a company has enough irregular employees that can cover more jobs.
Comparing a staff schedule before and after applying a mathematical model, the following things can be found. First, there is a difference in the number of employees who come to work per day on average. Consequently, about 1.6 employees come to work less than before after applying a mathematical model every day. Second, a developed mathematical model prevents situations where employees come to work more or less than the number of employees a company needs a day. This can aid in the improvement of human resource management. Third, a company can save total labor costs even though the company has the same number of employees as before. It was $143,525 before applying a mathematical model and it is $138,335 after applying a mathematical model. That is, a company can save a total labor cost of $5,190. Lastly, the different employee work pattern is found due to consideration of the change in labor policy. Some employees came to work more than 6 days a week before applying a developed model. However, it is constrained by a mathematical model at the moment because a company should follow the change of labor policy, which is each employee should work less than 5 days a week. The number of days regular employees come to work is constrained after applying a mathematical model due to the change in labor policy and high labor costs. Some regular employees come to work 4 days a week. This could be shown not to allow them to work more. However, it indirectly proved that a company that did not efficiently manage human resources has several employees more than the number they need to have. This can be an opportunity to rethink and optimize the number of employees a company needs to possess.
According to Figure 5, the rules related to overtime work are also well-considered. First, nobody is assigned to the office. Second, only one manager is assigned to supervise regular employees on one of the two floors. For instance, one manager of four managers is assigned on the first floor to play a role of supervisor during overtime work if you look at the first day. Third, an employee who performs equipment supervisor is assigned to one of two floors. Lastly, one employee is also allocated to open storage at least. To sum up, a staff schedule at daytime and nighttime that satisfies constraints is derived considering total labor cost.
In addition, I added the simulation part to consider the stochastic characteristics of irregular employees. The added part is as below. I added the related picture and table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.
5.3 Evaluation for a derived staff schedule
“This paper provides a conservative and deterministic staff schedule by using a mathematical model that considers the average percent of irregular employees’ absences, which is an uncertain characteristic. Even though irregular employee absence is stochastic, the reason this paper derives a deterministic staff schedule is that it is impossible to predict if irregular employees come to work or do not come to work perfectly only based on the probability of each irregular employee’s absence. That is, even the result derived from the stochastic model cannot guarantee that each irregular employee follows a staff schedule without absences. However, the reason to perform the simulation after deriving a deterministic staff schedule is that it is meaningful in that it can give the result that can help managers respond to the irregular employee’s absence. Figure 6 is the result of the simulation. Row numbers from 41 to 62 mean irregular employees and column numbers mean days of work. The result shows whether each irregular employee follows their schedule when considering each employee’s absence percent.
If looking at Table 6 below, the actual percent of each irregular employee’s absence is different from the expected percent of each employee’s absence. In other words, even though the average percent of irregular employees’ absence is considered as the stochastic characteristic through the simulation, it cannot ensure that managers have a staff schedule that can predict the absences of irregular employees perfectly. Furthermore, As shown in Figure 6, the lacking number of irregular employees can be confirmed. If managers prepare a conservative measure, they will assign more irregular employees on the days when some irregular employees are expected not to come to work. However, preparing additional irregular employees makes extra labor cost for a construction company. Nevertheless, it is more important to finish work by assigning extra irregular employees within due time. Saving labor costs is the next thing to do. Moreover, according to the days of absence shown in Table 6, the number of days when each irregular employees do not come to work can be confirmed. For example, the 44th irregular employee does not come to work 7 days out of 13 days. This irregular employee has an actual percent of absence, of 53.8%. It is 23.8% higher than an expected percent of absence, 30%. This type of employee is the main cause to make a company fail to follow a staff schedule and assign enough workforce to jobs. This finding can be an opportunity that can help a manager rethink the recruitment of other irregular employees. To sum up, the reason this paper derives a conservative and deterministic staff schedule and performs the simulation about it is that it can contribute to presenting the systematic procedure for irregular employee absence. This procedure could aid managers to prepare precautious measures for a staff schedule while considering the stochastic characteristics of irregular employee absence.”
Q3. The technique section of this manuscript is poor. It is incorrect to call the parameters in Table 1 (page 5) indexes, and they are not even used in Equations (1) to (31). For examples, W1, W2, …, etc. Are they Sets or Values? There is no doubt that they are not indexes. The table numbers in section 4.1, from lines 280 to 294, are not reflected in the paragraph.
Responses: Thank you for your valuable comments. As you mentioned, I expressed parameters in Table 1 incorrectly. I appreciate to allow me to find wrong things. Everything the referee pointed is Set, not indexes. I added the related table in the file of ‘Cover letter_Author Responses_algorithms-18551780’. Please refer to that file.”
In addition, Equation (25) is revised by using the parameter that was not used, which is
Lastly, I revised the paragraph that referee mentioned by using the right number of Tables.
“The company has 40 regular employees who consist of 36 normal regular workers and 4 managers. Furthermore, they have 22 irregular employees. The minimum necessary number of employees is at least 45 during the daytime. In addition, the number of workplaces is 4 spaces such as the 1st floor, 2nd floor, open storage, and the office. Other known parameters are set and presented in Table 3 except for known parameters regarding costs. The daily labor cost of each employee for both daytime and overtime is presented in Table 4 and Table 5. The known parameters to create an optimal schedule for the numerical experiment are in Table 3.
The cost for all employees is as Table 4. Each employee is paid daily according to company policy. Regular employees are paid more than irregular employees. Managers are paid about $20 more than normal regular employees. The reason the different levels in paycheck between weekdays and weekends is because of the implementation of a 52-hour workweek policy that regards Saturday as overtime workday. Table 5 represents the paycheck for regular employees at nighttime. The reason the company uses only regular employees at night is because of the rules of the company considered in the paper. They want to allow regular employees to earn money in the middle of a situation where most work is gradually conducted by irregular employees due to the high labor cost of regular employees.”
Author Response File: Author Response.pdf
