Proposing a new method to solve line balancing bottleneck problem in the single-model line

Abstract

Many problems occur when assigning tasks to work centres, especially in determining the required number of workstations for line balancing which requires a minimum theoretical number of workstations. The most common problem is bottleneck. In this paper, a method is proposed to solve floating tasks problem in single-model line when the actual required number of workstations exceeds the minimum theoretical number, and the standard time of the floating task (work center) exceeds the cycle time. The floating task will represent a critical bottleneck activity in line. The proposed method depends on minimizing the standard time of critical bottleneck and non-critical activities by a minimum free-floating time depends on the average of slack times of the non-critical activities, and it will increase the line efficiency from (77%) to (88%), and balance delay is minimized from (23%) to (12%).

Citation

Alrawi, M.A. (2023), "Proposing a new method to solve line balancing bottleneck problem in the single-model line", Emerald Open Research, Vol. 1 No. 4. https://doi.org/10.1108/EOR-04-2023-0015

Introduction

Line balancing means assigning work to a smallest number of workstations in a line process to equalize workload between these stations and achieve the desired output rates ¹. There are many types of line models: single-model, multi-model, and mixed-model. For a single line model, two general groups depending on their configuration are commonly seen: traditional straight lines and U-shaped lines: traditional straight production line organizes the tasks sequentially in one direction to form stations, and U-shaped production line, however, is divided into two sub-lines, namely, the entrance sub-line and the exit sub-line, and, thus, an operator may perform tasks on either one of the two sub-lines or on both sub-lines simultaneously ². In single-model line, there is only one model and precedence diagram ³. The classification of a bottleneck may vary. Normally the definitions include but are not necessarily restricted to: physical constraints, economical characteristics, output limitations, capacity utilization, work-in-process limitations and capacity in relation to demand. So, a bottleneck may simply be defined as the production system stage that has the largest influence on limiting the throughput in the system ⁴. Line balancing is a necessary task, its importance appears when there are some changes in process, such as adding or deleting tasks, change of components, changes in processing time, and also in implementing of new processes ⁵. It is also a successful tool to reduce bottleneck by balancing the task time of each work station ⁶. Line balancing is a method to balance the assignment of some work elements from an assembly line balancing to the work station to minimize the number of workstations and minimize total of idle time on the whole work station on a certain level of output, therefore it is necessary to do the line balancing to reduce the bottleneck, increase line efficiency, and reduce balance delay ⁷. The bottlenecks could be viewed from two different perspectives: bottleneck workstations and bottleneck tasks. Bottleneck workstations can be easily identified within the assembly line as the station having the maximum cycle time (greater than the takt time). Bottleneck tasks are identified within each work stations, whereas tasks having the maximum activity time ⁸. Determining the bottleneck is an essential issue to control the process, throughput, and cycle time. Since task is the smallest work element in the line, then the cycle time cannot be smaller than the largest time of a task. The proposed method aims to minimize the standard time of bottleneck workstation or bottleneck task, and assign all tasks to workstations, to avoid floating tasks and solve the problem when the actual number of workstations exceeds the minimum theoretical.

When constructing a diagram to presents the work elements, especially in project management issues, and constructing a cost-duration graph, the most difficult task is to decide which activities to shorten and how far to carry the shortening process. Shortening an activity is called Crashing⁹. Crashing is a useful tool in project management, and when the information about normal and crash times and costs is available. It depends on minimizing the normal time (standard time) for critical activities which have the minimum slope. In line balancing, such information is not available always, and it is sometimes difficult to estimate it. So, crashing is not an efficient tool to minimize critical activities’ standard times. The proposed method in this work will presents an approach to minimize the standard time of critical activities depending on the average of slack times of the non-critical activities.

Problem statement and related works

The simple assembly line balancing problems (SALBP) are fundamental versions of the general assembly line balancing problems (ALBP), which has attracted the attention of practitioners and researchers of OR. With respect to the objective functions, the SALBP was classified into SALBP-1, SALBP-2 and SALBP-E. These deterministic problems are not always applicable for real assembly and production lines, since in practice the durations of the assembly operations and other parameters may depend on many factors and are not constant values throughout the lifecycle of the assembly and production lines ¹⁰.

Solving actual assembly line balancing problems is difficult with the many real world constraints. So, line balancing bottlenecks are still problematic at present. Most of the literatures focus on defining and solving bottlenecks in scheduling job shops and flow shops, and different methods and heuristics were proposed to solve this problem such the Shifting Bottleneck Method which is considered as one of the most successful tools to identify bottlenecks, minimize total weighted tardiness, and control bottlenecks along the scheduling ¹¹. From the other hand, researches in determining the line balancing model and minimizing takt time have long occupied a prominent place in operations management literature. Bottlenecks in line balancing when its standard time is greater than the cycle time, and the actual required number of workstations exceeds the theoretical minimum had a little concern from researchers. This paper aims to address this issue by minimizing the standard time of critical bottleneck and non-critical activities by a minimum free floating time depends on the average of slack times of the non-critical activities.

Naveen Kumar and Dalgobind Mahto (2013) discussed the minimization of idle time of man and machine through distributing tasks over the workstation. Meby Mathew and D.Samuelraj (2013) studied the distributing of work load in an assembly line across successive workstations as an approach to reduce cycle time and wastes in resource and time. Varsha Narayan and Shriram Sane (2014) discussed waste identification and elimination, and de-bottlenecking to balance the line and optimize utilization of resources for improving the productivity. Shriram Sane, Varsha Karandikar, Rahul Pulkurte, and Subodh Patil (2014) discussed the effects of Lean Manufacturing tools such as cycle time study, line imbalance calculation, bottleneck identification, Kaizen, space utilization through layout change and workstations organization on the performance of assembly lines. Jaggi et al. (2015) studied the reduction in cycle time for single model assembly line when line is balanced which increase the efficiency by reducing non value added activities and other outcomes were that assembly line balanced by recommending new layout. Manaye (2019) balanced production line by using line balancing techniques which are Ranked Positional Weight and Largest Candidate method through work study method. As a result of analysis, the Ranked Positional Weight gives better results in the line efficiency and delay time minimization which compared to largest candidate technique. So that techniques minimize bottleneck operations and arranging the workload among work stations to increase line efficiency and minimize delay time ¹².

Problem formulation

In line balancing, a few requirements are needed for a set of tasks to be assigned to workstations. These requirements are: 1) a task that have been assigned to a workstation cannot be assigned to another workstation; 2) total time for each workstation to finish all their tasks must be less than or equal to cycle time; and 3) the precedence relationship must be followed rigidly by all tasks and cannot be disregard ¹³. A single-model line instance of (11) tasks (A to K) is considered as in Table 1 to assign tasks to workstations, and balance the line. Cycle Time (Ct) and theoretical minimum number of workstations (Ws) are calculated depending on Equation (1) and Equation (2) below. Precedence diagram is constructed for assembling the line as in Figure 1 as an initial state.

Ct = T R \ D ( 1 )

Ws = Jc \ Ct ( 2 )

Where: (T _R) is total required time of the working day, (D) is daily demand, and (Jc) is Job content which equals to (49) and represents the total of tasks standard times in Table 1.

According to Equation (1) and Equation (2), given the daily demand is (3000) unit and the available working time is (6) hours per day, then (Ct) is (7) seconds and (Ws) is (7) workstations. Table 2 presents the earliest and latest start and finish times, critical and non-critical tasks, and slack times for the tasks. Total completion time (C _max) is (30) seconds, and total slack time is (19) seconds.

The initial state indicated in Figure 1 shows that task C represents a problem in which it still floating, because its standard time is greater than cycle time, also the actual required number of workstations is (8) and exceeds the theoretical minimum. The floating task represents a critical bottleneck activity in the line, and its standard time must be reduced to meet minimum theoretical number of workstations and output rates.

The proposed method

Slack time is the difference between latest and earliest starting times of the non-critical activities. It means that there is a free float time for the work element. Decreasing total completion time of the (11) tasks instance presented in Figure 1 requires minimizing the standard time of the non-critical tasks by the minimum slack time of one of them.

The proposed approach must be run many times to meet the balancing requirements. Many experimental setups must be done to the system, and operations assumptions must be defined as below:

• - Determine the initial state.

• - Determine the minimum theoretical number of work stations.

• - Assigning work load to work stations, the duration time of each work center is pre-determined.

• - No handling operations, moving parts, and work in process are needed.

• - Assuming that the demand is stable on the line.

• - Assuming that the precedence relations are stable.

• - Determining the critical bottleneck and non-critical activities.

• - The approach will stop when meeting the balancing requirements, and the number of run times could not be pre-determined.

This approach does not ensure solving the bottleneck problem of the line, neither allows minimizing the standard time of the critical bottleneck task, nor meeting the theoretical minimum number of workstations. The proposed method will minimize the standard time for both of critical bottleneck and non-critical activities by a minimum free floating time (I _t) depends on the average of slack times of non-critical activities as in the equations below:

SLt ave = ∑ SL t \ N ( 3 )

DFS t = S Bt − Ct ( 4 )

Where SLt _ave is the average of slack time for non-critical tasks, DFS _t is the difference in time between the standard time S _Bt of the bottleneck task and cycle time, and N represents the number of non-critical tasks.

T BK = S Bt − P Bt ( 5 )

Where T _BK represents the bottleneck time, and P _Bt represents the standard time of the previous task on the critical path.

Now , if SLt ave ≥ DFS t ( 6 )

And DFS t ≤ T BK then I t = DFS t ( 7 )

Otherwise , if SLt ave < DFS t and SLt ave < T BK < DFS t then I t = SLt ave ( 8 )

Else , I t = T BK ( 9 )

At first, for the non-critical tasks in Table 2 and Equation (3), the proposed method is applied to get SLt _ave equals to (4) seconds with a predefined Ws = 7 workstations and Ct = 7 seconds, DFS _t = 1 seconds, and T _BK = 2 seconds. Considering Equation (6) and Equation (7) for the initial state in Figure (1), then I _t = 1 second. The tasks involved in the reduction are the critical task C and non-critical (B, D, F, G, and I) in which their standard times will be (7, 1, 6, 4, and 2) respectively as displayed in Table 3 which presents balancing the line. Figure (2) presents assigning work to workstations. Total completion time (C _max) is (29) seconds, and total slack time is (29) seconds.

Line efficiency and balance delay

To measure whether the line is efficient in producing the output rate, line efficiency and balance delay are the most measures used. To optimize line balancing, efficiency should be maximized and balance delay should be minimized. These measures are calculated by the following equations:

Efficiency ( % ) = JC \ Ws ( Ct ) ( 10 )

Balance Delay ( % ) = 100 − Efficiency ( 11 )

Results

The minimum free floating time (I _t) ensures minimizing both bottleneck time and completion time (C _max) of the line from (30) to (29) seconds, and meets the theoretical (Ws).

Table 4 presents a comparison between efficiency and balance delay before and after applying the proposed method.

Conclusions and suggestions for future work

The bottleneck problem in assembly line balancing has become a frequently encountered problem today. The unequal and random workload assignment to the work stations, floating task (work center) with standard time exceeds the cycle time, and actual required number of workstations exceeds the minimum theoretical number cause the occurrence of floating tasks, non-optimal use of resources, and decrease the line efficiency. For this reason, if these problems are solved in the work shop floor, a significant increase in productivity, efficiency, and profitability will be achieved, contributing to a significant reduction in completion times, line balance delay, and costs at the same rate.

The results of the proposed method show that the floating tasks which represent critical bottleneck activities have a significant influence on the assembly line balancing problem when assigning tasks and work elements to workstations if at least one of the tasks considered as bottleneck when its standard time is greater than cycle time, and the actual required number of workstations exceeds the theoretical minimum. The proposed method depends on minimizing the standard time of critical bottleneck and non-critical tasks by a minimum free floating time depends on the average of slack times of the non-critical activities. This minimum free floating time ensures minimizing both bottleneck time and the completion time (C _max) of the line from (30) to (29) seconds, and meets the theoretical (Ws), increases the line efficiency from (77%) to (88%), and balance delay is minimized from (23%) to (12%). The proposed method could be applicable in different industrial problems where the assembly line balancing is one of them, and in job shop and flow shop process environments.

For future works, solving bottleneck problems in other line balancing models such parallel or mixed line model studies could be considered. Minimizing workstation takt time is another attractive field for study.

Data availability

No data.

Publisher’s note

The original DOI of the article was 10.35241/emeraldopenres.14983.2

This is Version 2 of the article. Version 1 is available as supplementary material.

Author roles

Alrawi MA: Writing - Review & Editing

Amendments from Version 1

The proposed method improves the performances of the assembly line when a bottleneck problem is occurred with many cases such the floating tasks, number of workstations exceeds the theoretical number, and the standard time of the floating task exceeds the cycle time. These cases were not discussed by researchers in literatures, for that, the proposed method was described as a new problem and was not supported by existing literature. Because of that, the results presented were not validated against any other existing works in the literature. There was a need to define the experimental setups of the experiment on the assembly line. So it is added to the manuscript. The proposed approach must be run many times to meet the balancing requirements. Many experimental setups must be done to the system, and operations assumptions must be defined as below: Determine the initial state. Determine the minimum theoretical number of work stations. Assigning work load to work stations, the duration time of each work center is pre-determined. No handling operations, moving parts, and work in process are needed. Assuming that the demand is stable on the line. Assuming that the precedence relations are stable. Determining the critical bottleneck and non-critical activities. The approach will stop when meeting the balancing requirements, and the number of run times could not be pre-determined The impact of the study as a significant increase in productivity, efficiency, and profitability was defined.

The peer reviews of the article are included below.

Funding statement

The author(s) declared that no grants were involved in supporting this work.

Competing interests

No competing interests were disclosed.

Reviewer response for version 2

Mahathir Mohammad Bappy, Department of Industrial and Systems Engineering, Mississippi State University, Mississippi State, United States

Competing interests: No competing interests were disclosed.

This review was published on 14 September 2023.

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation: approve-with-reservations

Review:

• Rewrite the abstract with solid problem statement, and by incorporating practical implications of the proposed method.

• Briefly incorporate technical contribution of the study in the introduction.

• Describe the research gaps.

• Reference for equations is missing, please incorporate.

• Add a flow chart for the proposed method.

• In table 3 what is meant by critical bath.

• Make the figure’s font and operators consistent.

• In conclusion, incorporate the contribution of the study.

Is the argument information presented in such a way that it can be understood by a non-academic audience?

Yes

Is the rationale for developing the new method (or application) clearly explained?

Partly

Could any solutions being offered be effectively implemented in practice?

No

Is the description of the method technically sound?

Partly

Is real-world evidence provided to support any conclusions made?

No

Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Partly

Does the piece present solutions to actual real world challenges?

Yes

If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

No

Are sufficient details provided to allow replication of the method development and its use by others?

Yes

Reviewer Expertise:

Manufacturing

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Reviewer response for version 2

Abdullah Hulusi Kökçam, Industrial Engineering Department, Sakarya University, Sakarya, Turkey

Competing interests: No competing interests were disclosed.

This review was published on 07 September 2023.

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation: approve-with-reservations

The paper introduces a method to address the floating tasks issue within a single-model production line, where the actual workstation requirements exceed the theoretical minimum, and the standard time for floating tasks surpasses the cycle time. The method involves minimizing the standard duration of both critical bottleneck and non-critical activities by introducing a minimum free-floating time based on the average slack times of non-critical activities.

The method put forward in the study should be cleared of technical errors and explained more robustly. In particular, the impact of activity duration reduction on cost should be elaborated upon. Additionally, the paper would benefit from improved paragraph structure for smoother transitions, clarification of abbreviations upon their first use, inclusion of more recent literature, clarification of elements in figures, and addressing inconsistencies in terminology. The following points should be taken into account regarding the study.

• The paper's paragraphs are lengthy, and some sentence transitions lack coherence. For instance, it abruptly shifts from discussing line balancing to the bottleneck issue in the introduction and later returns to line balancing without clear transitions. New paragraphs could be introduced to improve readability.

• The paper should provide the expansion of abbreviations upon their first use, such as “OR.”

• Additionally, some referenced sources in the text do not appear in the bibliography, like Naveen Kumar and Dalgobind Mahto (2013), Meby Mathew and D.Samuelraj (2013), and Jaggi et al. (2015).

• Since most of the reviewed literature is from 2015 and earlier, it has become outdated, and the paper should incorporate more recent research.

• The meaning of activities connected in the precedence diagram in Figure 1 (e.g., B-E-G) should be clarified.

• It's unclear why activity C is highlighted in a different color.

• Just because an activity has slack time doesn't mean it will always have free float time. This statement in the proposed method should be revised.

• The statement “Decreasing total completion time of the (11) tasks instance presented in Figure 1 requires minimizing the standard time of the non-critical tasks by the minimum slack time of one of them.” suggests that shortening the duration of non-critical activities will reduce the total completion time. However, on the contrary, when the duration of critical activities is reduced, the total completion time may decrease.

• It would be more appropriate to explain the proposed approach with a flowchart.

• The statement “ P _Bt represents the standard time of the previous task on the critical path” is mentioned, but the specific activity to which it refers is unclear.

• The term “I _t” is used without prior definition.

• There is a typographical error in the title of Table 3, where “bath” should be corrected to “path.”

• Do “Jc” and “JC” refer to the same term in Equations 2 and 10?

• The choice of the “\” symbol in equations instead of “/” should be clarified.

• The paper focuses on reducing activity durations in the proposed method. Still, it doesn't address whether activities can be shortened, the extent to which they can be shortened, and the associated cost increase.

• The term “floating task (work center)” implies that “floating task” and “work center” are the same, which should be clarified.

Is the argument information presented in such a way that it can be understood by a non-academic audience?

No

Is the rationale for developing the new method (or application) clearly explained?

Partly

Could any solutions being offered be effectively implemented in practice?

No

Is the description of the method technically sound?

Partly

Is real-world evidence provided to support any conclusions made?

No

Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Partly

Does the piece present solutions to actual real world challenges?

No

If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Yes

Are sufficient details provided to allow replication of the method development and its use by others?

Yes

Reviewer Expertise:

Scheduling, Optimization, Artificial Intelligence, Machine Learning

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Reviewer response for version 1

mohd nor akmal khalid, School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Japan; School of Computer Sciences, Universiti Sains Malaysia, Pulau Pinang, Malaysia

Competing interests: No competing interests were disclosed.

This review was published on 27 April 2023.

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation: approve-with-reservations

The author proposed a line balancing bottleneck in a single-model line, which claimed that some performances of the assembly line were improved to some degree.

Although this is well and good, the manuscript fell short on many fronts.

Firstly, the manuscript refers to many rather outdated manuscripts, and the only manuscript that is worth considering at this time is by Manaye (2019). However, I failed to find such a paper in the reference section. Rather the reference section listed many unrelated references, some even not being cited in the main text. As such, the current contributions to the body of knowledge seem rather questionable.

Secondly, the proposed method was fully described and was not supported by existing literature. In addition, only performance measures were formulated where neither the approach's process nor any model was present. Therefore, it is challenging to determine the extent of the manuscript's contribution.

Thirdly, the experimental setups were missing entirely. Basically, what are the system settings of the experiment done on the assembly line? How many runs/simulations were conducted? How the best results were determined? Is there a parameter involved in the experiments?

Fourthly, the results presented, although they showed some improvement, it is rather arbitrary and were not validated against any other existing works in the literature. In addition, it was unclear where or how the data came to be. Also, how significant is the improvement to real-world settings (statistical analysis is needed)?

Finally, no limitations and managerial implications were discussed. Basically, the results should be discussed from an industrial standpoint, which needs to consider other/many moving parts of the manufacturing process. As such, the impact of the study is also unclear.

Due to these facts, the manuscript will be no more than a paper suited as a conference paper instead of a journal submission.

Is the argument information presented in such a way that it can be understood by a non-academic audience?

No

Is the rationale for developing the new method (or application) clearly explained?

Partly

Could any solutions being offered be effectively implemented in practice?

Not applicable

Is the description of the method technically sound?

Partly

Is real-world evidence provided to support any conclusions made?

Not applicable

Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Partly

Does the piece present solutions to actual real world challenges?

Not applicable

If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

Partly

Are sufficient details provided to allow replication of the method development and its use by others?

Partly

Reviewer Expertise:

artificial intelligence, planning and scheduling, evolutionary algorithms, game analytics, data sciences

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

Maha Alrawi, University of Technology-Iraq, Iraq

Competing interests: No competing interests were disclosed.

This review was published on 29 Jun 2023.

Many thanks to the reviewer for his significant comments.

The proposed method improves the performances of the assembly line when a bottleneck problem is occurred with many cases such the floating tasks, number of workstations exceeds the theoretical number, and the standard time of the floating task exceeds the cycle time. These cases were not discussed by researchers in literatures, for that, the proposed method was described as a new problem and was not supported by existing literature. Because of that, the results presented were not validated against any other existing works in the literature.

I agree with the reviewer in that there is a need to define the experimental setups of the experiment on the assembly line. So it is added to the manuscript. The data is an experimentally instance. No need at this stage for statistical analysis because there is no comparison between two proposed methods.

The impact of the study as a significant increase in productivity, efficiency, and profitability was defined.

A couple of updated manuscripts were added.

Maha Alrawi, University of Technology-Iraq, Iraq

Competing interests: No competing interests were disclosed.

This review was published on 27 Jul 2023.

Many thanks to the reviewer for their significant comments.

The proposed method improves the performances of the assembly line when a bottleneck problem is occurred with many cases such the floating tasks, number of workstations exceeds the theoretical number, and the standard time of the floating task exceeds the cycle time. These cases were not discussed by researchers in literature, for that, the proposed method was described as a new problem and was not supported by existing literature. Because of that, the results presented were not validated against any other existing works in the literature.

I agree with the reviewer in that there is a need to define the experimental setups of the experiment on the assembly line. So it is added to the manuscript. The data is an experimentally instance. No need at this stage for statistical analysis because there is no comparison between two proposed methods.

The impact of the study as a significant increase in productivity, efficiency, and profitability was defined.

A couple of updated manuscripts were added.

Reviewer response for version 1

Osama Al Meanazel, Industrial Engineering Department, The Hashemite University, Zarqa, Jordan

Competing interests: No competing interests were disclosed.

This review was published on 11 April 2023.

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation: approve

The author provides new methodology for solving line balancing bottleneck problem in the single-model line. The article is clear of mistakes and no further modification is required. The introduction shows the related studies, and the methodology are shown. I would suggest that if the author provide an example of how to apply this methodology in real case study.

Is the argument information presented in such a way that it can be understood by a non-academic audience?

Yes

Is the rationale for developing the new method (or application) clearly explained?

Yes

Could any solutions being offered be effectively implemented in practice?

Yes

Is the description of the method technically sound?

Yes

Is real-world evidence provided to support any conclusions made?

Yes

Are the conclusions about the method and its performance adequately supported by the findings presented in the article?

Yes

Does the piece present solutions to actual real world challenges?

Yes

If any results are presented, are all the source data underlying the results available to ensure full reproducibility?

No source data required

Are sufficient details provided to allow replication of the method development and its use by others?

Yes

Reviewer Expertise:

Ergonomics and safety engineer

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Maha Alrawi, University of Technology-Iraq, Iraq

Competing interests: No competing interests were disclosed.

This review was published on 15 Apr 2023.

Dear sir

Thank you for the reviewer comments, they were scientific.

As for the suggestion of applying on real data, I would like to point out that the manuscript is not applied research, and a hypothetical example was tested for the proposed method.

Figures

Figure 1.

Precedence diagram of the initial state.

Figure 2.

Assigning work to workstations using the proposed method.

Table 1.

A single-model line instance.

Task	Precedence	Standard time (sec)
A	---	6
B	A	2
C	A	8
D	B	7
E	C	2
F	D	5
G	C	2
H	E	5
I	G	3
J	H,I	4
K	F,J	5

Table 2.

The critical bath calculations of the initial state.

Activity Name	On Critical Path	Activity Time	Earliest Start	Earliest finish	Latest start	Latest finish	Slack (LS-ES)
A	yes	6	0	6	0	6	0
B	No	2	6	8	11	13	5
C	yes	8	6	14	6	14	0
D	No	7	8	15	13	20	5
E	yes	2	14	16	14	16	0
F	No	5	15	20	20	25	5
G	No	2	14	16	16	18	2
H	yes	5	16	21	16	21	0
I	No	3	16	19	18	21	2
J	yes	4	21	25	21	25	0
K	yes	5	25	30	25	30	0
Project	Completion	Time	=	30	Seconds

Table 3.

Critical bath calculations of the proposed method.

Activity Name	On Critical Path	Activity Time	Earliest Start	Earliest finish	Latest start	Latest finish	Slack (LS-ES)
A	yes	6	0	6	0	6	0
B	No	1	6	7	13	14	7
C	yes	7	6	13	6	13	0
D	No	6	7	13	14	20	7
E	yes	2	13	15	13	15	0
F	No	4	13	17	20	24	7
G	No	1	13	14	17	18	4
H	yes	5	15	20	15	20	0
I	No	2	14	16	18	20	4
J	yes	4	20	24	20	24	0
K	yes	5	24	29	24	29	0
Project	Completion	Time	=	29	Seconds

Table 4.

Efficiency and balance delay before and after applying the proposed method.

Criteria	Before the propose method	After the propose method
C max(Seconds)	30	29
Job Content (Seconds)	49	43
Total Slack Time(Seconds)	19	29
Efficiency	77%	88%
Balance Delay	23%	12%