Reliability Engineering & System Safety
Simultaneous Tasks Planning and Resources Assignment in Maintenance Scheduling
under Uncertainties
--Manuscript Draft-Manuscript Number:
JRESS-D-24-02521R1
Article Type:
VSI: OR&AI maintenance
Keywords:
Maintenance scheduling; Task planning; Limited space; Mixed Integer
Programming; Resource assignment
Corresponding Author:
Wenjin Zhu
Northwestern Polytechnical University
Xi'an, CHINA
First Author:
Bin Wu
Order of Authors:
Bin Wu
Wenjin Zhu
Xu Luo
Shubin Si
Abstract:
Effective maintenance scheduling and timely execution of maintenance tasks within the
given time duration are important to system safety and reliability. In practical
maintenance, the practicality of task planning is essential due to uncertainties arising
from the actual maintenance environment, limited maintenance operation space,
execution challenges, and equipment constraints. This study focuses on enhancing
maintenance planning by addressing uncertain factors, evaluating cost and risk,
constructing a decision model, and incorporating risk assessment for interactive
decisions. Unlike previous approaches assuming stable tasks, this study
acknowledges that unforeseen changes may occur, necessitating immediate repairs.
Thus, a rolling optimization approach is introduced and formulated as a mixed integer
programming problem, allowing priority adjustments and dynamic task planning when
maintenance resource uncertainty occurs. As one of the critical constraints, the space
conflict among maintenance tasks is considered. Numerical experiments are
conducted with 12 certain tasks and 7 potential tasks to show the optimal solutions with
different uncertain scenarios, and case studies verify the optimality of the solutions.
Suggested Reviewers:
Ben Niu
Shenzhen University
nb@szu.edu.cn
An expert in the field of optimization in task planning.
Jiawen Hu
University of Electronic Science and Technology of China
hdl@sjtu.edu.cn
An expert in maintenance planning and optimization.
Xian Zhao
Beijing Institute of Technology
zhaoxian@bit.edu.cn
An expert in reliability and maintenance optimization.
Gregory Levitin
levitin@iec.co.il
Response to Reviewers:
Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation
Cover Letter
July 22, 2024
Dear Editor,
We are pleased to submit a research paper for the potential publication in the special issue
“Operations research and artificial intelligent models for maintenance management of
engineering assets” in Reliability Engineering & System Safety. The title of this manuscript is
“Simultaneous Tasks Planning and Resources Assignment in Maintenance Scheduling under
Uncertainties,” authored by Bin Wu, Wenjin Zhu, Xu Luo and Shubin Si. I serve as the
corresponding author.
Effective maintenance scheduling and timely execution of maintenance tasks within the given
time duration are important to system safety and reliability. In practical maintenance, the
practicality of task planning is essential due to uncertainties arising from the actual maintenance
environment, limited maintenance operation space, execution challenges, and equipment
constraints. This study focuses on enhancing maintenance planning by addressing uncertain
factors, evaluating cost and risk, constructing a decision model, and incorporating risk
assessment for interactive decisions. Unlike previous approaches assuming stable tasks, this
study acknowledges that unforeseen changes may occur, necessitating immediate repairs. Thus, a
rolling optimization approach is introduced and formulated as a mixed integer programming
problem, allowing priority adjustments and dynamic task planning when maintenance resource
uncertainty occurs. As one of the critical constraints, the space conflict among maintenance tasks
is considered. Numerical experiments are conducted with 12 certain tasks and 7 potential tasks to
show the optimal solutions with different uncertain scenarios, and case studies verify the
optimality of the solutions.
At present, the manuscript is not being submitted to any other journals, conferences, or
workshops under review. We appreciate your time and efforts in processing our manuscript, and
look forward to hearing from you in the future. Please address all correspondence concerning
this manuscript to me and feel free to correspond with me by e-mail (wenjin.zhu@nwpu.edu.cn).
Sincerely,
Wenjin Zhu
Associate Professor
School of Mechanical Engineering
Northwestern Polytechnical University
Xi’an, Shaanxi
710072, China
Response to Reviewers
Response to Comments of the RESS Manuscript
Manuscript Number: JRESS-D-24-02521
Manuscript Title: Simultaneous Tasks Planning and Resources Assignment in Maintenance
Scheduling under Uncertainties
Review Date: 2024-Dec-24
1. General Responses from authors:
We would like to thank the editor and four anonymous referees for their time and efforts to review
the original manuscript. We are very appreciative of their constructive comments and suggestions
which result in an improved work after the revision.
Below we provide the point-to-point responses. For easy reference, reviewers’ original questions
or comments are reproduced followed by our responses. In the manuscript, revised areas also are
highlighted in blue for easy reference and assessment. If the title of the table or the caption of
figure is highlighted, meaning the content of the table or figure has been updated accordingly.
2. Response to Editor Comments
Associate Editor: The reviewers have provided several important comments on this work, which
are summarized below. While there are many other comments that require attention, it is crucial
to thoroughly revise this manuscript by carefully addressing each comment from the reviewer.
1) The novelty and practical contributions need to be clearly highlighted.
Response:
Thank you for your valuable suggestion regarding the need to clearly highlight the novelty and
practical contributions of our work. We have revised the introduction to explicitly outline our novel
approach, which integrates overlooked factors such as spatial conflicts in parallel maintenance
activities and the utilization of multiple types of reusable maintenance resources, particularly in
remote offshore environments. Additionally, we emphasize the practical implications of our
findings, including the importance of maintainability in equipment design and optimization, which
can significantly enhance equipment availability. Our dynamic integration of maintenance policies
with task scheduling addresses uncertainties in maintenance practices, while the introduction of a
rolling optimization approach ensures flexibility and robustness in scheduling. We have also
included illustrative examples to demonstrate the real-world applicability of our contributions. We
believe these revisions effectively clarify the novelty and practical significance of our research.
2) The literature review should be reorganized to clearly establish the connection between the
proposed work and previous studies.
Response:
Thank you for your suggestions. We have revised the literature review to clarify connections to
previous studies and added a new figure (Figure 1: The Link of Maintenance Policy and
Maintenance Engineering) to illustrate these relationships on Page 3. We believe these changes
enhance the clarity and relevance of our work.
Figure 1. The link of maintenance policy and maintenance engineering
3) The model's formulation requires improvement for clarity.
Response:
Thank you for your suggestion regarding the model's formulation. We have carefully revised this
section for clarity and highlighted the changes in red in the revised manuscript. We believe these
improvements enhance the overall understanding of the model.
4) The writing quality and consistency need enhancement.
Response:
Thank you for your suggestion regarding writing quality and consistency. We have made thorough
revisions to enhance these aspects and highlighted the changes in red in the revised manuscript.
3. Response to Reviewer 1:
The paper addresses an important topic in maintenance and industrial practice, which is related to
maintenance planning and resource assignment under uncertainties. The mathematical derivations
are detailed and rigorous. The linearized MILP model has been solved and optimized properly
with reasonable results in acceptable time. However, there are several areas that require
improvement to enhance the clarity and quality of the manuscript. Below are my detailed
comments and suggestions.
1) The problem of nomenclature consistency should be noticed. For example, the phrase "Set of
maintenance tasks…" should match the format of other definitions.
Response:
Thank you for your comments and suggestions. The authors have revised the phrase "Set of
maintenance tasks…" to ensure it matches the format of other definitions and have standardized
the formatting of all key terms throughout the document. The authors believe these changes
effectively address the issue of nomenclature consistency. The changes have been outlined by red
color.
2) There are some ambiguities in definitions and the nomenclature. For example, please clarify the
definition of DSR in the Nomenclature section. The current definition does not accurately reflect
the detailed relationship described in Section 4.2.
Response:
Thank you for your comments and suggestions regarding the ambiguities in definitions and
nomenclature. The authors have clarified the definition of DSR in the Nomenclature section to
ensure consistency with the description in Section 4.2. The authors have included the relevant
formulas to support this clarification on Page 6.
3) For the results analysis, the details of parameters for the method, such as the EA-DEAP, should
be included. Please provide detailed information of the parameters used in the EA-DEAP
algorithm, such as seed, pop_size, n_gen, and rate_elite, to enhance the understanding and
reproducibility of the experiments.
Response:
Thank you for your valuable suggestion regarding the parameters used in the EA-DEAP algorithm.
In the revised manuscript, we will include the following details: the population size was set to 50
individuals, the algorithm was run for a maximum of 1000 generations, the crossover probability
was 0.8, and the mutation probability was 0.05. We believe these additions will enhance the clarity
and reproducibility of our work. The authors have included the relevant formulas to support this
clarification on Pages 30-31.
4) The convergence analysis of the EA-DEAP algorithm to the exact solutions solved by the MILP
algorithm should be provided, which is important for the problem with larger scale.
Response:
Thank you for your valuable suggestion. Due to space limitations and the fact that this topic is not
directly related to the main subject of the paper, we have chosen not to include it. Additionally,
the convergence time of this method is relatively long, and its performance is inferior compared
to the approach we propose. We hope the reviewer can understand our decision to maintain the
overall coherence and integrity of the manuscript.
In this section, we include the iterative results of the EA-DEAP algorithm, as well as a comparison
of the results after convergence with the exact solution. It can be observed that when the number
of tasks is 12, the EA-DEAP algorithm can quickly or gradually converge to an approximate
optimal solution that is close to the exact solution, although it may not fully reach the exact solution.
As the number of tasks increases to 15 and 17, the algorithm may struggle to converge to the exact
solution and may even find it difficult to identify a feasible solution. In these cases, the iterative
curve generated by EA-DEAP appears as a straight line positioned in the upper part of the graph,
indicating issues with its convergence.
12 random selected tasks
15 random selected tasks
17 random selected tasks
Results of different selected tasks’ EA-DEAP iterations
5) The author could provide some more application scenarios or industrial cases related to the
proposed model. For example, when giving relevant assumptions, the authors can combine
engineering practice to illustrate the rationality of the assumptions.
Response:
Thank you for your comments and suggestions. The model proposed in this study is based on the
maintenance scheduling of a ship engine room. The original version of the example is as follows:
“Due to the ship's scale and operating mechanism, maintenance onboard usually consists of
periodic
or
scheduled
maintenance,
preventive/condition-based
maintenance,
corrective/emergency maintenance, and, recently, the so-called predictive maintenance. This
study presents a simplified and general model whose assumptions are abstracted based on the
practice experience for equipment maintenance and repair.
When conducting maintenance on critical equipment inside the ship engine room, as mentioned in
this paper, the surrounding areas of the equipment can be occupied due to ongoing maintenance
tasks. Therefore, when performing maintenance on densely distributed equipment or equipment
that occupies a large space, it is crucial to identify the maintenance-occupied space and avoid
spatial conflicts. This is essential for generating an efficient and safe maintenance task allocation
plan.
Implementing maintenance activities requires personnel equipped with tools and spare parts,
which can be referred to as maintenance sources. The time required for a maintenance task
depends to some extent on the resources allocated to it. However, ships' maintenance resources
are typically limited and not easily replenished or rescued in offshore areas. Therefore,
maintenance resources are one of the primary constraining factors affecting maintenance task
planning and completion.”
According to your suggestions, we put more practical details and modified it as follow:
An illustrative example of the research background
“The engine rooms of ships and other equipment are placed inside with limited and fixed space.
Large critical facilities such as energy systems, power systems, electrical systems, ventilation
systems, plumbing systems, hydraulic systems, control systems, etc. are organized in engine room.
Due to the ship's scale and operating mechanism, maintenance onboard usually consists of periodic
or scheduled maintenance, preventive/condition-based maintenance, corrective/emergency
maintenance and sometimes multiple maintenance activities are organized at the same period.
When conducting maintenance on critical equipment inside the ship engine room, the surrounding
areas of the equipment will be occupied by the technicians with the tools such as lifting equipment
and other auxiliary equipment due to disassembly. Therefore, when performing maintenance
activities on densely distributed equipment, it is crucial to identify the maintenance-occupied space
and avoid spatial conflicts among the ongoing maintenance activities for safety and human factors.
Besides, the technicians and tools are limited and not easily replenished or rescued in offshore
areas. The time required for a maintenance task depends to some extent on the technicians, the
necessary space and tools/spares. The medium or major repair period of a ship can range from
several days to several months.
Therefore, this study presents a simplified and general model whose assumptions are abstracted
based on the practice experience for equipment maintenance and repair. The technicians with tools
and spare parts are referred as maintenance sources. Thus, the efficient space and resources are
essential for maintenance task planning and completion.”
The above content is modified on Page 8.What’s more, some explanations corresponding to three
uncertainties on Page 16 are provided.
6) This assumption is not realistic. However, the condition of the part after maintenance is not
determined. In the conclusion, the authors should give more managerial implications to illustrate
the practical value of the paper.
Response:
Thank you for your comments and suggestions! The following content is added on Pages 35-36:
Industrial engineering requires close collaboration among various departments throughout
the entire production and operation process. However, in practice, due to conflicts between shortterm economic and various optimization objectives, the need for maintainability is always
overlooked. In certain industries, maintainability and maintenance engineering are important
factors in equipment support. In the equipment design and optimization stage, considering the
maintainability of each component will be beneficial for improving the availability of the
equipment. Meanwhile, when making decision of CBM and PdM, it is recommended to consider
whether the implementation conditions are met.
7) The writing skill should improve. Please check spelling errors and formatting issues. For
example, Integrated Optimization of Non-Permutation (1) and (2) should have punctuation marks.
Response:
Thank you for your comments and suggestions regarding the writing quality. We have checked
the document, and we have consistently used the format "Eqs. (i)" for referencing the equations,
so there are no issues in that regard. We appreciate your attention to detail.
4. Response to Reviewer 2:
This paper aims to plan and schedule maintenance tasks under both time and space constraints
with uncertain task addition. The planning problem is formulated as an integer optimization model
and solved with a rolling-horizon approach. The advantage of this modeling and solution approach
lies in the simultaneous maintenance implementation of multiple tasks over the time horizon given
the potential maintenance space conflict and added maintenance jobs. For maintenance job
scheduling, the nature of this type of problem is NP-hard. It is good that the authors have tested
the speed of the algorithm using different size of problems up to 50 instances, and the results are
clearly presented in the work.
Below are the comments arising during the review, and the authors can consider and appropriately
incorporate these suggestions into the revision provided these comments are in alignment with the
theme of the work.
Comments:
1) Section 1 shall be reorganized. For example, the authors reviewed the works of CBM, then the
question is what is the connection with your work? What makes your work differs from existing
CBM? The same question can be appliable to predictive maintenance, maintenance-production
planning models. Also, if want, the authors can use table to summarize the contributions of existing
works as well as to compare the novelty of the proposed new paper. In summary, a connection
shall be made between the papers being reviewed and the proposed maintenance task/job
scheduling problem because CBM, PdM and time-based maintenance policies are more related to
maintenance policy.
Response:
Thank you for your comments and suggestions! The maintenance task scheduling model proposed
in this study is more relevant to maintenance practice. The content in the introduction have been
modified carefully in the manuscript and outlined by red color. Besides, a figure will be added to
explain the logic of the research. The authors provide a short conclusion here:
(1) The proposed model is the successive stage of maintenance policy and maintenance
decision making. In maintenance practice it is usually carried out by different departments
or by different personnel within the same department. For the aim of equipment reliability
and availability, the procedure is approximately equivalent to a segmented optimal decision,
which may not necessarily be the global optimal decision. Making a tentative attempt to
bridge the research gap is one of the research objectives of this study.
(2) The part of maintenance production is not so relevant with the main objective of this study.
The authors have removed part of the descriptions.
(3) The study considers three different uncertainties in maintenance task scheduling, which
could provide some realistic scenarios for CBM and PdM. For example, for the
complicated maintenance scenarios where many maintenance tasks are waiting, then the
effectiveness of CBM and PdM will dependent on many factors.
Figure 1. The link of maintenance policy and maintenance engineering
2) Certain paragraphs of Section 1 are long and they can be appropriately divided into 2 or 3
paragraphs. That is, Section 1 needs to be reorganized in a way to achieve the following goals: (1)
stating clearly about the research motivation or background, (2) review or survey the state of the
art associated with your work, and (3) point out the research gap, and highlight the contributions
of your work.
Response:
Thank you very much for your comments and suggestions! The instruction is clearly and very
helpful and valuable. The authors sincerely appreciate it. The modification concerning the three
instructions is in the major part of the introduction (on Page 2-7). Hence the authors will not present
all the modifications here.
3) In Section 2.1.1, "Due to the ships scale and operating mechanism, maintenance onboard
usually consists of periodic or scheduled maintenance, preventive/condition-based maintenance,
corrective/emergency maintenance, and, recently, the so-called predictive maintenance. This study
presents a simplied and general model whose assumption s are abstracted based on the practice
experience for equipment maintenance and repair." This paragraph does not state clearly which
maintenance policy your work adopts? CBM, PdM or time-based PM? Also PM (preventive
maintenance) includes scheduled maintenance, CBM and even predictive maintenance by some
scholars in this domain. Therefore, a clarification is preferred in this work though there might be
no standard solution in maintenance community.
Response:
Thank you very much for your comments and suggestions! The authors did fail to explain well the
major difference between this study and the reviewed maintenance policies. The proposed model
puts more efforts on the maintenance task scheduling according to several constraints when facing
several types of uncertainties. In maintenance practice for large and complex system/equipment
with many critical components, it is probably that several maintenance policies/states occur
simultaneously. We can explain it by the following two cases:
(1) If it is an onshore maintenance, which means that a list of maintenance tasks will be
generated, then it is favorable to be a corrective maintenance, or a time-based maintenance
policy.
(2) If it is an offshore maintenance, then it is favorable to be a mixture of both CBM or PdM.
Based on the existing maintenance scheduling, whenever a new maintenance task is triggered,
either based on corrective or preventive maintenance, the rolling scheduling procedure will be
active.
4) In Section 2.1.2, "from an illustration" or "for an illustration"?
Response:
Thank you for your correction. In Section 2.1.2, it should indeed be "for an illustration." The
authors have modified it on the manuscript and removed the similar errors.
5) In Equation (3), the value of "0.1" shall be explained. For example, why not 0.05 or 0.2, etc.
Response:
Thank you for your question regarding the value of "0.1" in Equation (3). This value is chosen as
a parameter between 0 and 1 to assist in the linearization process. The specific choice of 0.1, rather
than alternatives like 0.05 or 0.2, was not previously discussed, as any number within this range
can serve the purpose of facilitating the linearization. The rationale for selecting this particular
value is provided in the red-highlighted section below Equation (3). The authors will ensure that a
detailed explanation of this term, along with its significance in the context of the linearization
process, is provided in the revised manuscript on Page 12.
6) If Model 1 is a mixed integer non-linear programming model, then use term "MINLP" might
be more accurate for Model 1. After this model is linearized, it can be called "MILP" if want. For
example Problem 2 can be called MILP as has been done in the paper.
Response:
Thank you for your suggestion. The authors have updated the document to replace the term "MIP"
with "MINLP" for Model 1, as it is indeed a mixed integer non-linear programming model. After
linearization, we can refer to it as "MILP," consistent with the terminology used in the paper for
Problem 2.
7) Non-linear constraints may or may not create computational challenges depending on the
characteristics of the constraints. If the non-linear constraint belongs to quadratic function or if the
constraints are convex, the global optimization generally can be guaranteed.
Response:
Thank you for your suggestion! The statement accurately reflects that non-linear constraints may
present computational challenges depending on their characteristics, and that global optimization
can generally be guaranteed for quadratic or convex constraints.
8) In Section 2.3.2, linearization method is discussed, and again term or value like "0.5" in the
formular shall be explained as well.
Response:
Thank you for your insightful comments. This concern is consistent with the earlier comment 6)
regarding Equation (3).
9) In Table 1, should Cij be Cji for the third column?
Response:
Thank you for your careful review. It should indeed be Cji instead of Cij, as the first column
already contains an analysis of Cij. I apologize for this oversight and will make the necessary
correction in the document on Page 15.
10) The line below equation (4), two Dij are shown.
Response:
Thank you for your comments regarding the line below equation (4). You are correct that the
second Dij should be modified to Dji. The authors have made the necessary correction on Page
15.
11) Change "And Constraints (2-2) to Constraints (2-7)" into "And Constraints (2-2) to (2-7)" if
want. Similar changes can be made in other sentences.
Response:
Thank you for your suggestion! The authors have changed "And Constraints (2-2) to Constraints
(2-7)" to "And Constraints (2-2) to (2-7)." Similar adjustments have been made in other
sentences as needed on Page 16.
12) In Equation (6), the second argument is either problematic or a typo?
Response:
Thank you for your comments! In Equation (6), the second argument was indeed a typo. I have
corrected it accordingly on Page 18.
13) In Section 3.3. "he set of tasks that will be updated in the next step", change "he" into "the".
Response:
Thank you for your suggestion! The authors have made the correction in Section 3.3 on Page 19,
changing "he set of tasks that will be updated in the next step" to "the set of tasks that will be
updated in the next step."
14) In Section 4.2, "The mixed integer quadratic programming problem Problem2: MILP is
solved based on Python and GUROBI Solver." This sentence is misleading. If the model
involves quadratic term, then it cannot be called "linear", though GUROBI Solver can handle
this type of problem.
Response:
Thank you for pointing out the ambiguity in the sentence! The authors have revised it to: "The
solutions to the two models, (MINLP) and (MILP), are based on Python and GUROBI Solver" on
Page 21.
15) In Figure 8(a), not sure the uncertain tasks "M7, M10, M12, M19" are displayed as well.
Response:
Thank you for your question regarding Figure 8(a). The uncertain tasks "M7, M10, M12, M19"
are represented across the figures, with M7 corresponding to Figure 8(a), M10 to Figure 8(b),
M12 to Figure 8(c), and M19 to Figure 8(d). In each subfigure, these uncertain tasks are involved
in the diffusion space conflicts caused by maintenance. The green areas indicate the diffusion
regions of the uncertain tasks, while the red areas represent the conflicts with other diffusion
spaces. (Attention: after modifications of adding a new figure, the Figure 8 moves to Figure 9!)
16) Two additional reference that might be included in the literature review:
 "Simultaneous scheduling of maintenance crew and maintenance tasks in bus operating
companies: a case study," by Rodrigo Martins, Francisco Fernandes, Virginia Infante,
Antonio R. Andrade, published I Journal of Quality in Maintenance Engineering, 2021.
 "A two-stage optimization approach for aircraft hangar maintenance planning and staff
assignment problems under MRO outsourcing mode," by Yichen Qin et al, 2020, published
in Computers & Industrial Engineering, vol. 146.
Response:
Thank you for your valuable suggestions regarding additional references for the literature review.
I have included the following two references in the References section.
The first paper has been added into the item “Joint maintenance planning with other conditions”
of the Subsection “1.1. Maintenance policy” on page 4.
 Martins, R., Fernandes, F., Infante, V., & Andrade, A. R. (2022). Simultaneous
scheduling of maintenance crew and maintenance tasks in bus operating companies: A
case study. Journal of Quality in Maintenance Engineering, 28, 506–53
The second paper has been added into the Subsection “1.2. Maintenance Scheduling” on page 4.
 Qin, Y., Zhang, J. H., Chan, F. T. S., Chung, S. H., Niu, B., & Qu, T. (2020). A two-stage
optimization approach for aircraft hangar maintenance planning and staff assignment
problems under MRO outsourcing mode. Computers & Industrial Engineering, 146,
106607.
5. Response to Reviewer 3:
In this paper, the authors address simultaneous tasks planning and resources assignment in
maintenance scheduling under uncertainties. The authors first develop a mixed integer program
for simultaneous maintenance planning, then involves three types of uncertainties through rollinghorizon optimization. In general, this paper addresses an important maintenance task planning
problem and is well written. I have the following minor comment in spirit of improvement.
1) The contributions and novelty of this work should be highlighted in a more explicit manner.
Response:
Thank you very much for your comments and suggestion! The authors revise and rewrite the
contributions and novelty of this work carefully:
Thank you very much for your comments and suggestion! The authors revise and rewrite the
contributions and novelty of this work carefully:
“…
Thus, as the successive stage of maintenance policy, maintenance task scheduling is
important to realize high equipment availability. The segmented optimization of maintenance
decision and maintenance task scheduling separately and independently may compromise the
efforts on both stages.
Based on the goal of making a tentative attempt to bridge the research gap, this paper focuses
on a maintenance task scheduling problem from the following aspects:
(1) Integration of the overlooked factors in maintenance practice as a constraint: spatial
conflict in parallel maintenance activities;
(2) Integration multiple types of reusable maintenance resources as constraints to investigate
the resource utilization and efficiency, particularly in remote offshore environments;
(3) Characterization of uncertainties in maintenance practice such as operational delay, logistic
delay, and emergent task, providing a potential integration of dynamic maintenance policy
with maintenance task scheduling;
(4) Introduction of a rolling optimization approach, ensuring smooth task execution through
dynamic adjustments and real-time optimization, thereby enhancing scheduling flexibility
and system robustness.
”
2) In Introduction, the authors summarize three streams of maintenance research. I suggest the
authors add a brief introductory sentence for each stream before diving into the detailed literature
review. In addition, the following reference can be discussed in stream one (i.e., CBM).
J. Xu, B. Liu, X. Zhao, X.-L. Wang. (2024) Online reinforcement learning for condition-based
group maintenance using factored Markov decision processes, European Journal of Operational
Research, 315(1) 176-190.
Response:
Thank you for your valuable suggestion. We appreciate your suggestion to add a brief introductory
sentence for each stream of maintenance research in the Introduction section. We have
implemented this change to enhance clarity before delving into the detailed literature review.
Additionally, we have included the reference you provided in the discussion of stream one
(condition-based maintenance):
Xu, J., Liu, B., Zhao, X., & Wang, X.-L. (2024). Online reinforcement learning for conditionbased group maintenance using factored Markov decision processes. European Journal of
Operational Research, 315, 176–190.
3) Following the above comment, please make it clearer which type of maintenance is considered
in this work (periodic, condition-based, or predictive?).
Response:
Thank you for your comments. We recognize the critical importance of effective maintenance
strategies in industrial engineering and emphasize that maintenance types should be tailored to
specific contexts, particularly in maintenance scheduling. We acknowledge that maintainability
and maintenance engineering are often overlooked due to conflicts between short-term economic
goals and optimization objectives. Therefore, it is essential to consider maintainability during the
equipment design and optimization phases to enhance equipment availability. Additionally,
decisions regarding Condition-Based Maintenance (CBM) and Predictive Maintenance (PdM)
should assess whether the necessary implementation conditions are met to ensure their
effectiveness. Our research aims to provide a comprehensive framework for maintenance
scheduling applicable across various industrial scenarios.
4) The authors assume that maintenance resources are reusable and will be released for reuse as
soon as the maintenance tasks are completed. This assumption is certainly reasonable for
technicians and special devices, but may not be true for spare parts.
Response:
Thank you for highlighting important point regarding the assumption that maintenance resources
are reusable and will be released. This issue indeed touches on specific technical aspects of the
model. Firstly, the assumption is related to the maintenance strategy, where we generally assume
that spare parts are sufficient and always available. If spare parts are abundant, the constraints
related to their availability can be relaxed. However, if spare parts are limited, some maintenance
tasks may not be executable, which would shift the focus of the entire study to a different problem,
such as selective maintenance and maintenance with inventory control. They are very important
topic and we will consider them in the future. Additionally, we have considered three types of
unexpected situations, one of which involves the postponement of maintenance tasks due to a lack
of spare parts, necessitating a waiting time for the part logistic before maintenance can be
performed. We sincerely hope the explanation addresses your concerns under certain conditions.
5) Though the writing and presentation of this paper are fairly good, there are still some writing
and language issues. Please go through the paper carefully.
Response:
Thank you for your suggestion regarding the writing and presentation of the paper. The authors
appreciate your acknowledgment of the overall quality, and will certainly take your advice to
carefully review the paper for any writing and language issues. Ensuring clarity and precision in
our communication is important, and the authors will make the necessary revisions to enhance the
quality of the manuscript.
6) In the last paragraph in Page 7, some sentences are in the past tense. Please use the present tense
consistently.
Response:
Thank you for your suggestion regarding the tense consistency on Pages 8-9. I have revised the
section to ensure that all sentences are now in the present tense. This change enhances clarity and
maintains a consistent narrative throughout the text.
7) Please replace "It's" and "can't" with "It is" and "cannot", respectively.
Response:
Thank you for your suggestion regarding the use of contractions in the text. The authors have
replaced "It's" with "It is" and "can't" with "cannot," as well as expanded all other contractions
such as "doesn't" and "couldn't" to their full forms. This change enhances the formality and clarity
of the writing.
5. Response to Reviewer 4:
This work proposes a MIP model for maintenance task scheduling considering limited resources
and spatial constrain. Meanwhile, this work proposes to use rolling horizon method to meet the
challenge of dynamic task. In my opinion, this work is trying to study a practical problem. The
developed model and corresponding algorithm are useful. Please find comments below.
1) The authors are suggested to rewrite the introduction to emphasize the research gap and support
the necessity of this work. Currently, the classification of maintenance models is not correct.
Specifically, TBM and corrective maintenance are missing. Meanwhile, related studies on job
scheduling are also missing.
Response:
Thank you for your insightful comments regarding the introduction. We have made substantial
revisions to this section to better emphasize the research gap and the necessity of our work.
Specifically, we have corrected the classification of maintenance models to include TBM and
corrective maintenance, which were previously missing. Additionally, we have incorporated
relevant studies on job scheduling to provide a more comprehensive context for our research.
These enhancements aim to strengthen the rationale for our study and are highlighted in red in the
revised manuscript for your review on Pages 2-7.
2) The formulation of this problem should be revised. Currently, it is not easy to follow the work.
Response:
Thank you for your suggestion regarding the formulation of the problem. We recognize that clarity
is essential for understanding our work, and we have revised this section to improve its coherence
and accessibility. The revised formulation now clearly outlines the key components of the problem,
the objectives of our study, and the methodology employed. We believe these changes will
enhance the reader's ability to follow our work more easily.
3) This work assumes that the machines are all rectangular. Is this assumption general? More
explanations should be provided to validate this assumption.
Response:
Thank you for your question regarding the assumption that all machines are rectangular. In this
work, we approximate the planar projections of all other shapes of equipment as rectangles or use
the smallest enclosing rectangle to represent them. The analysis primarily focuses on the conflicts
related to the additional diffusion space required for maintenance, characterized by the diffusion
around the rectangles. We propose corresponding formulas and embed them as constraints in the
model. Analyzing other planar shapes or describing the additional diffusion space in different ways
would require approaches from fields such as topology optimization and computer graphics. This
can be considered a direction for future research but is not the main focus of this study on
maintenance scheduling optimization.
4) This work considers a dynamic scheduling problem. A simulation study is needed in the
numerical study to validate the robustness of developed model and algorithm.
Response:
Thank you for your valuable feedback. In response to your comment, we have added a subsection
on robustness simulation in the paper, specifically on pages 32-35. This subsection details a
simulation study that tests 1000 random new task time points generated from a Poisson distribution.
We validate the robustness of the developed model and algorithm by calculating the mean of the
optimal solutions corresponding to these different time points. This addition enhances the clarity
and thoroughness of our numerical study.
Highlights
Simultaneous Tasks Planning and Resources Assignment in
Maintenance Scheduling under Uncertainties
Highlights:

Quantifying and integration of spatial conflicts within the maintenance tasks
allocation model

Design of multiple types of reusable maintenance resources to quantify resource
utilization

Consider uncertainties in maintenance practice including operational delay and
emergent task

Design a rolling window optimization approach to reschedule maintenance tasks
facing uncertainties
Manuscript File(Editable format preferred with extension .docx,
.doc, or .tex.
Click here to view linked References
Simultaneous Tasks Planning and Resources Assignment in
Maintenance Scheduling under Uncertainties
Bin Wua , Wenjin Zhua,∗, Xu Luob , Shubin Sia
a
School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi,
710072, China
b
Science and Technology on Integrated Logistics Support Laboratory, National University of
Defense Technology, Changsha, Hunan, 410073, China
Abstract
Effective maintenance scheduling and timely execution of maintenance tasks within
the given time duration are important to system safety and reliability. In practical maintenance, the practicality of task planning is essential due to uncertainties
arising from the actual maintenance environment, limited maintenance operation
space, execution challenges, and equipment constraints. This study focuses on enhancing maintenance planning by addressing uncertain factors, evaluating cost and
risk, constructing a decision model, and incorporating risk assessment for interactive
decisions. It emphasizes the adaptability of maintenance planning through learning
and evolution based on historical planning. Unlike previous approaches assuming
stable tasks, this study acknowledges that unforeseen changes may occur, necessitating immediate repairs. Thus, a rolling optimization approach is introduced, allowing priority adjustments and dynamic task planning when maintenance resource
uncertainty occurs. As one of the critical constraints, the space conflict among maintenance tasks is considered. Numerical experiments are conducted with 12 certain
tasks and 7 potential tasks to show the optimal solutions with different uncertain
scenarios, and case studies verify the optimality of the solutions.
Keywords: Maintenance scheduling, Task planning, Limited space, Mixed Integer
Programming, Resource assignment
1. Introduction
∗
Corresponding author
Preprint submitted to Elsevier
December 25, 2024
In today’s industrial landscape, operational and maintenance costs constitute a
significant portion of overall life cycle expenses for various systems. The competitive
business environment has propelled companies to seek continuous cost reductions, underscoring the importance of efficient maintenance planning. Maintenance is pivotal
in maintaining reliability and reducing security risks in various sectors like aviation,
naval, and nuclear industries.
As illustrated in Fig. 1, the current mainstream and existing reliability modeling concerning maintenance models can be classified into several categories. These
classifications serve as a tentative bridge to connect maintenance policy with maintenance scheduling, highlighting the interplay between strategic decision-making and
the practical execution of maintenance tasks. By understanding these models, we
can better align maintenance policies with scheduling practices, ultimately enhancing
the effectiveness and efficiency of maintenance operations. The relevant literature
can be summarized as follows:
1.1. Maintenance policy
1) Condition-based Maintenance (CBM) is vital in modern practices, focusing on
maintenance basSed on equipment condition rather than fixed schedules. [1] introduced a condition-based inspection-maintenance policy for critical systems, while [2]
developed a group maintenance approach using Markov decision processes and reinforcement learning. [3] proposed a framework for optimizing maintenance planning
and technician routing, and [4] addressed delays caused by crew arrival. Additionally, [5] used a rolling horizon approach to adjust maintenance schedules based on the
stochastic arrival of tasks. These studies emphasize the need for adaptive strategies
to enhance reliability and efficiency, though CBM often neglects practical aspects like
resource availability and time constraints, introducing uncertainties in scheduling.
2) Predictive maintenance with uncertainty: [6] presents a dynamic reallocation
strategy for a 1-out-of-2 pairs balanced system to enhance performance and longevity.
[7] optimizes maintenance intervals for multi-state systems with performance sharing
through a reliability model. [8] formulates an opportunistic maintenance optimization problem for multi-component systems as an infinite-horizon Markov decision
process (MDP), proposing a multi-agent approach for scheduling and worker allocation. [9] integrates random-time component reallocation and system replacement
into a random maintenance policy. [10] develops a maintenance planning model
that incorporates probabilistic remaining useful life (RUL) prognostics and resource
availability. [11] focuses on predictive maintenance for multiple systems, integrating
prediction and scheduling under uncertainty using deep learning. [12] introduces
a margin-based approach for dynamic maintenance decisions, while [13] creates an
2
Fig. 1. The link of maintenance policy and maintenance engineering.
3
adaptive predictive maintenance policy that accounts for sensor degradation and
state estimation uncertainty. Despite advancements in reinforcement learning and
data-driven methods, challenges related to randomness and uncertainties in maintenance task modeling and scheduling remain.
3) Joint maintenance planning with other conditions: Several studies have explored this area, including [14], which integrates job scheduling and maintenance
planning, and [15], which proposes a two-stage optimization approach for aircraft
maintenance under outsourcing conditions, focusing on maintenance scheduling and
staff assignment to enhance operational efficiency. [16] minimizes makespan in parallel machine scheduling using a mixed-integer programming model, while [17] presents
a mixed-integer linear program for robust job-shop scheduling under machine unavailability. Additionally, [18] introduces a rolling-horizon approach with a mixed-integer
nonlinear model for maintenance selection and production scheduling. [19] addresses
resource-constrained project scheduling, and [20] and [21] improve scheduling for aircraft paint shops and flow-shop scheduling using genetic algorithms. [22] considers
uncertainties in maintenance durations, and [23] integrates shared due date information into scheduling decisions. Finally, [24] optimizes maintenance crew allocation
through a simulation algorithm, while [25] focuses on multiple unit maintenance in
task scheduling with preventive maintenance thresholds.
1.2. Maintenance scheduling
In addition to the aforementioned research, some more elaborate works considering the practical factors concerning the maintenance operation in realistic circumstances have contributed to the broader landscape of maintenance optimization. [26]
emphasize physical constraints, such as humans’ appropriate operation space in the
cooperative scenario during the process of group decision-making and risk assessment. In [27], the effort is made to improve the efficiency and quality of facility layout optimum design for maintainability of a ship cabin, where the factors concerning
the maintenance activities, such as maintenance operating space and distance requirement and personnel movement distance are all considered. In [28], the aviation
maintenance technician scheduling (AMTS) problem with a dynamic task disassembly mechanism (DTDM) is modeled to solve the problem of arranging maintenance
technicians across shifts under the horizon of short-term maintenance.
[29] developed a semi-Markov decision process model to optimize maintenance
under random production waits, minimizing long-run costs. The scheduling of maintenance tasks and resources is further emphasized by [30], who employed an integer
linear programming model to optimize the scheduling of maintenance crews and
tasks in bus operating companies, demonstrating significant cost reductions. [31] ad4
dresses uncertainties in mission time and operating conditions through a two-stage
stochastic programming approach, formulating a joint selective maintenance and repairperson assignment problem as a mixed-integer nonlinear program. [32] employs
the augmented epsilon constraints method for bi-objective optimization in maintenance planning, and [33] considers stochastic arrivals of corrective maintenance tasks
and resource availability in airline scheduling. Additionally, [34] studies maintenance
task allocation for aircraft fleets, and [35] develops a two-layer strategy for large-scale
maintenance in oil and gas fields. A review by [36] highlights the integration of maintenance and production scheduling, indicating a growing trend in research.
Thus, as the successive stage of maintenance policy, maintenance task scheduling
is important to realize high equipment availability. The segmented optimization of
maintenance decision and maintenance task scheduling separately and independently
may compromise the efforts on both stages.
Based on the goal of making a tentative attempt to bridge the research gap, this
paper focuses on a maintenance task scheduling problem from the following aspects:
• 1) Integration of the overlooked factors in maintenance practice as a constraint:
spatial conflict in parallel maintenance activities;
• 2) Integration multiple types of reusable maintenance resources as constraints
to investigate the resource utilization and efficiency, particularly in remote
offshore environments;
• 3) Characterization of uncertainties in maintenance practice such as operational
delay, logistic delay, and emergent task, providing a potential integration of
dynamic maintenance policy with maintenance task scheduling;
• 4) Introduction of a rolling optimization approach, ensuring smooth task execution through dynamic adjustments and real-time optimization, thereby enhancing scheduling flexibility and system robustness.
The remainder of the paper is organized as follows: Section 2 reviews related research work that published in recent years. Section 3 presents a detailed description
of the maintenance scheduling problem in the high-speed train depots. Section 4
proposes an integer programming model for the problem, followed by several linearization techniques and valid inequalities to improve the original mathematical
formulation. Section 5 reports the computational results of both the artificially generated instances and a real-world case study from the Shanghai South Depot. Finally,
conclusions are drawn, and future research directions are discussed in Section 6.
5
2. Model description
2.1. Assumptions and Definitions
2.1.1. Background of ship maintenance–An illustrative example of ship engine room
The engine rooms of ships and other equipment are placed inside with limited
and fixed space. Large critical facilities such as energy systems, power systems,
electrical systems, ventilation systems, plumbing systems, hydraulic systems, control
systems, etc. are organized in engine room. Due to the ship’s scale and operating
mechanism, maintenance onboard usually consists of periodic or scheduled maintenance, preventive/condition-based maintenance, corrective/emergency maintenance
and sometimes multiple maintenance activities are organized at the same period.
When conducting maintenance on critical equipment inside the ship engine room,
the surrounding areas of the equipment will be occupied by the technicians with the
tools such as lifting equipment and other auxiliary equipment due to disassembly.
Therefore, when performing maintenance activities on densely distributed equipment,
it is crucial to identify the maintenance-occupied space and avoid spatial conflicts
among the ongoing maintenance activities for safety and human factors. Besides,
the technicians and tools are limited and not easily replenished or rescued in offshore
areas. The time required for a maintenance task depends to some extent on the
technicians, the necessary space and tools/spares. The medium or major repair
period of a ship can range from several days to several months.
Therefore, this study presents a simplified and general model whose assumptions
are abstracted based on the practice experience for equipment maintenance and repair. The technicians with tools and spare parts are referred as maintenance sources.
Thus, the efficient space and resources are essential for maintenance task planning
and completion.
Uncertainties that change over time can impact the normal progress of projects,
potentially causing delays in some tasks and extending the overall project duration. During project execution, uncertainties such as operational difficulties and
insufficient tools due to the actual maintenance environment can affect maintenance
efficiency. In some cases, these uncertainties can even lead to interruptions in maintenance tasks. Therefore, there is a need to analyze maintenance plans and optimize
maintenance task planning to address these uncertainties and improve maintenance
efficiency.
Given the particularity of ship engine room maintenance tasks in this study, a
rolling optimization strategy is adopted to address uncertainties and dynamic changes
in task scheduling. This strategy allows for adjustments based on real-time conditions, ensuring that maintenance operations can respond effectively to various challenges.
6
The concept of a maintenance cycle is introduced and divided into multiple time
windows based on detection points. Each time window is planned in detail, allowing
for flexibility. Factors such as worker availability, equipment failures, or the sudden
urgency of maintenance tasks can lead to unexpected delays. Thus, the rolling optimization method is deemed a suitable countermeasure to enhance adaptability and
maintain operational efficiency.
At the end of each time window, the model is updated using the latest project
data and progress. This ensures that tasks for the upcoming period are re-optimized
to reflect the most accurate information and status. Through this strategy, the
scheduling model in this study ensures robust decision-making and enhances adaptability to uncertain environments, providing valuable support for the practicality and
rationality of ship maintenance task planning.
2.1.2. Required spaces and spatial conflicts
From this subsection, we introduce the assumptions, symbols, and variables for an
illustration. To simplify the model, the machines with different area occupancies in
the engine room are labeled as maintenance tasks Mi , i = 1, 2, ..., n with the meshed
layout of the size 10 × 10 units with n rectangular shadow area. The rule of the
numbering of the maintenance tasks is based on a coordinate system, which starts
from the two-dimensional origin (0, 0) and ends at (10, 10) as shown by Fig. 2. Each
maintenance task is identified according to its coordinates of the four corner points
of the rectangle shadow and labeled in ascending order according to the abscissa and
ordinate. For example, the machine located in the top left corner is labeled as M1
with coordinate {(3, 4), (6, 4), (3, 7), (6, 7)} and the machine located in the bottom
right corner is labeled as M2 with coordinate {(0, 4), (2, 4), (0, 7), (2, 7)} in Fig. 2(a).
The area of the maintenance task Mi is defined based on the coordinate and
precisely by si = (xri − xli )(yiu − yid ), where xli and xri are two coordinates of the Mi
along the horizontal direction, and yid and yiu are two coordinates of the Mi along
the vertical direction respectively. In this study, the time spent on the path to each
machine is negligible compared to the maintenance time. Implementing each maintenance task requires some specific work area that cannot overlap with other tasks.
The required work area is defined based on a designed parameter with the form of
(Ali , Ari , Adi , Aui ), where li and ri denote the number of rows of meshes located
in the left and right of Mi and similarly ui and di denote the number of rows of
meshes located upper and down of Mi . The required work area is the surrounding
area defined by the intersection of the four coordinates of the four corners. Fig. 2(b)
gives the examples of M1 to M6 . For detail, the texture area surrounding the M4
is the required work area of M4 . If (Al2 , Ar2 , Ad2 , Au2 ) = (0, 1, 1, 2) for M1 given
7
(a) Occupied area for equipment
(b) Extra area for maintenance
Fig. 2. Layout instance for six maintenance tasks.
(Al1 , Ar1 , Ad1 , Au1 ) = (1, 1, 1, 1), then the work areas of M2 and M1 are overlapping
at the red area ’part A’, which leads to the conflict between M2 and M1 if they are
scheduled to be maintained in the same time. By similar observation and analysis,
we can find that M1 and M3 overlap in the ‘Part B’ area, and M4 and M5 overlap
in the ‘Part C’ area. Therefore, we can conclude that in this instance, the pairs of
conflicting tasks are {M1 , M2 }, {M1 , M3 }, {M4 , M5 }. In other words, no simultaneous maintenance can occur in any of the above three tasks in the subsequent
maintenance scheduling process.
2.1.3. Simultaneous maintenance planning
As mentioned previously, given (Al1 , Ar1 , Ad1 , Au1 ) = (1, 1, 1, 1) for M1 and
(Al2 , Ar2 , Ad2 , Au2 ) = (0, 1, 1, 2) for M2 the pairs of conflicting tasks are {M1 , M2 },
{M1 , M3 }, {M4 , M5 }. Fig. 3 gives an example of a maintenance task scheduling
with two batches {M2 , M3 , M4 } in Fig. 3(a) and {M1 , M5 } in Fig. 3(b), where the
maintenance tasks included in each batch are spatial conflict-free.
2.2. The mathematical model of simultaneous maintenance planning
This study focuses on the success of the implementation of scheduled maintenance. Due to the limited maintenance staff and limited space in the machine room
8
(a) Work area for M2 , M3 and M4
(b) Work area for M1 and M5
Fig. 3. Two layouts of simultaneous maintenance schedules.
onboard, sometimes different maintenance activities have to wait in the queue until
the necessary staff, resources, and space are in position. Without considering the specific and useful purpose of each maintenance activity corresponding to each machine,
we label each specific maintenance activity as maintenance task {M1 , M2 , ..., Mn }
and simplify it by the following two properties: 1)maintenance space, and 2)maintenance resource. The properties consist of the constraints of the scheduling problem
and will be further introduced in the following sections.
2.2.1. Spatial conflict evaluation and simultaneous maintenance
Recall the definitions of xli , xri , yiu , yid and A(li , ri , di , ui ) in 2.1.1, Iij is defined as
follows:
1
(yid − di − yjd + dj )(yjd − dj − yiu − ui ) > 0
y
Iij =
(1)
0
otherwise
where i, j ∈ N , i ̸= j. The expression (yid − di − yjd + dj )(yjd − dj − yiu − ui ) means
that Mi and Mj are conflicted with each other in vertical direction and hence Iijy = 1
given yid − di < yjd − dj . For the scenario of yid − di > yjd − dj , a similar function can
be deduced. Moreover, it is obvious to calculate Iijx by substituting x for y in Eq.
(1).
9
Iij = Iijx ∗ Iijy , i, j = 1, 2, ..., n, i ̸= j
(2)
For i = j, i, j = 1, 2, ..., n, Iij = 0. If and only if both Iijx = 1 and Iijy = 1, i.e., the
conflict occurs both with the direction of the horizon and vertical, thus, Mi and Mj
are conflicted with each other, which means that Mi and Mj can not be maintained
simultaneously.
After evaluating the spatial conflicts of two maintenance tasks, we use a matrix
B of size n × n to demonstrate the temporal relation between the two tasks. Let the
start time be si and maintenance duration be ti of maintenance task Mi , i = 1, 2, ..., j.
For the ith row of B, if task Mi and task Mj intersect at the start time point si of
Mi (that is, the start time of task Mi is not later than the start time of task Mj ),
then the jth column of the row is assigned by a value of 1, i.e., Bij = 1; otherwise,
it is assigned by a value of 0 for Bij . If i = j, i, j = 1, 2, ..., n,Bij = 1. Similar to the
spatial conflict evaluation by Eq. (1), for i, j = 1, 2, ..., n the specific rule of value
assignment for matrix B is summarized as follows:
Bij =
1,
0,
sj ≤ si < sj + tj ⇐⇒ (si + 0.1 − sj )(si + 0.1 − sj − tj ) < 0
otherwise ⇐⇒ (si + 0.1 − sj )(si + 0.1 − sj − tj ) > 0
(3)
That is based on the truth that proposition sj ≤ si < sj + tj firstly can be
equivalently converted into a conjunctive proposition (si − sj ≥ 0) ∧ (si − sj − tj <
0), and the equality of si − sj ≥ 0 and si + 0.1 − sj > 0, si − sj − tj < 0 and
si +0.1−sj −tj < 0 is based on the integrity of si , sj . We can also change the number
‘0.1’ to another positive ‘number’ < 1. The purpose is to transform the inequality
before the equivalence symbol into an equivalent strict inequality. The left side of
this inequality is the product of (si + number − sj ) and (si + number − sj − tj ). Based
on the previous analysis, when this product is strictly less than 0, it is equivalent to
Bij = 1; when it is strictly greater than 0, it is equivalent to Bij = 0. This approach
avoids confusion when (si + number − sj ) and (si + number − sj − tj ) = 0, as it is
unclear whether this is due to si + number − sj = 0 or si + number − sj − tj = 0.
Furthermore, it transforms the original logical constraint sj ≤ si < sj + tj into a sign
judgment of the product (si + number − sj ) and (si + number − sj − tj ). Through
this method, the constraints (1-2) and (1-3) in Model 1 can be naturally introduced.
Finally, we use the multiplication operation to substitute the conjunctive proposition
equally. Moreover, it is intuitively natural to calculate Bij and judge whether two
different tasks Mi and Mj are simultaneously maintained by whether it takes 1 or
not. By combining Bij and Iij , we can deduce that no more than one of Bij and Iij
10
is 1 for a task planning scheme that satisfies spatial constraints, which means that
spatial conflicts and simultaneous maintenance can not occur together.
2.2.2. Multiple resources of multiple tasks
In maintenance practice, resources such as technicians, spare parts, and special
devices are limited, and different maintenance tasks will consume or occupy some
resources during the maintenance operation. Assume that the resources are reusable
and will be released for reuse as soon as the maintenance tasks are completed. Three
types of resources, including personnel, materials, and devices, are considered, where
the unit of each type of resource is 1. For convenience, assume that each task requires
a toolbox consisting of specific types and amounts of resources.
2.3. Mathematical model formulation
2.3.1. Avoid indicator constraints to formulate
Model 1
min
(MINLP)
(1-0)
max(si + ti )
i∈N
Bij + Iij ≤ 1,
(1-1)
i, j = 1, 2, ..., n, i ̸= j
(si − sj + 0.1)(si + 0.1 − sj − tj ) ≥ −N Bij ,
i, j = 1, 2, ..., n, i ̸= j
(si − sj + 0.1)(si + 0.1 − sj − tj ) ≤ N (1 − Bij ),
n
X
rkj Bij + rii ≤ Rk ,
i, j = 1, 2, ..., n, i ̸= j
i, j = 1, 2, ..., n, k = 1, 2, 3
(1-2)
(1-3)
(1-4)
j=1,j̸=i
Constraints (1-2) and Constraints (1-3) are introduced to avoid simultaneous
maintenance of tasks with spatial conflicts from Eq. (3). In Eq. (3), we got the
logistical relationship between Bij and si , sj , and then it is easy to shrink the value
of (si + 0.1 − sj )(si + 0.1 − sj − tj ) into (−1, 0) ∪ (0, 1) by dividing a large number
N . And the interval between −Bij and 1 − Bij is either (0, 1) or (−1, 0), then we
can get −Bij ≤ (si + 0.1 − sj )(si + 0.1 − sj − tj ) ≤ 1 − Bij which is equal to Eq. (3).
Finally, we obtain Constraints (1-2) and Constraints (1-3) to substitute the indicator
expressions Eq. (3).
11
N is a sufficiently large number, allowing us to optimize the problem size by
evaluating conflicts between two tasks without considering each time point within
the time window. It also facilitates dynamic programming in case of unexpected
situations. Here, since the start and end points of tasks do not consume resources,
and the problem is an integer programming problem, we use N and add a small
perturbation of 0.1 to handle the critical points (task start and end times) as a
non-consuming resource and conflict-free situations.
Constraints (1-4) focuses on the cumulative maintenance resources formed by
each task batch in the batch maintenance scenario. It has been proven earlier that
when the cumulative resources of each batch satisfy the resource constraints, the
cumulative resources at every time point within the maintenance time window will
also satisfy the resource constraints. Conversely, suppose the cumulative resources
of each batch do not meet the resource constraints. In that case, the cumulative resources at every time point within the maintenance time window will also fail to meet
the resource constraints. Therefore, these two approaches for resource accumulation
are equivalent regarding resource constraints.
When dealing with spatiotemporal conflict, because spatial conflict judgment Iij
is a known variable calculated in advance, it is necessary to continuously update
decision variables si and sj along with model optimization.
2.3.2. Linearization by big-M method
Obviously, Iij is a nonlinear equation due to its expression, and the judgment
of its sign is also a generalized logic constraint, increasing the model’s complexity.
The powerful branch-cut algorithm in the GUROBI solver and other methods are
suitable for mixed integer linear programming models. Considering that there are
nonlinear constraints in the mixed integer non-linear program (MINLP) model by
(MINLP), some methods are needed to avoid the nonlinear constraints, and hence
the original nonlinear constraints are linearized.
[32] applied a heuristic linearization technique to reduce model complexity and
increase model tractability using a large number of ‘M’. Thus, we introduce a sufficiently large integer M together with decision variable Bij by B such that for all
i, j ∈ N , the inequality |si + 0.5 − si − tj | < M holds. Here, we introduce two new
matrices C and D with the same size as B. Hence, we can calculate the range of
Cij = −(si + 0.5 − si − tj )/(2M ), and the results are included in (0,1). We can use
the 0-1 index variable Dij to control Cij by the operation Dij − 0.5 < Cij < Dij + 0.5.
Then we can analyze different conditions with Dij , i, j = 1, 2, ..., n, and list the corresponding values of Dij and the simultaneous maintenance relationship between Mi
and Mj under different values of Cij in Table 1:
12
1. Dij = 0, in this situation, we get −0.5 < Cij < 0.5, which means that si +
0.5 − sj − tj > 0. And si − sj − tj is an integer, which is either greater than
−0.5 or less than −0.5; while it is greater than −0.5, we could induce that
si ≥ sj + tj , which means that Mi does not start later than Mj ’s finish. Hence,
this situation illustrates that Mi and Mj do not have a time conflict.
2. Dij = 1, in this situation, we get 0.5 < Cij < 1, which means that si + 0.5 −
si − tj < 0. And si − sj − tj is an integer, which is either greater than -0.5 or
less than -0.5; while it is less than -0.5, we could induce that si < sj + tj , which
means that Mj finish later than Mi ’s start, and we can not induce whether
there is a time conflict between the two. Hence, we should need the extra
information of Dji , if Dji = 1, then we can deduce a time conflict between Mi
and Mj .
Table 1: Simultaneous maintenance analysis with different values of Dij corresponding to Cij .
Dij + Dji − 1 Simultaneous or not
Cij
Dij
Cji
Dji
0.5 < Cij < 1
1
0.5 < Cji < 1
1
1
Yes
0 < Cij < 0.5
0.5 < Cij < 1
0
1
0.5 < Cji < 1
0 < Cji < 0.5
1
0
0
No
0 < Cij < 0.5
0
0 < Cji < 0.5
0
Maintenance order contradiction
As we can see from Eq. (3), Bij is not only related to the value of si +0.5−sj −tj ,
but also the value of si + 0.5 − sj . Hence we do the similar way as Cij to set
Eij = (Si + 0.5 − sj + M )/2M , and the same way as Dij to control 0-1 index variable
Fij by the equality Fij −0.5 ≤ Eij ≤ Fij +0.5. In conclusion, we obtain the equivalent
relationships:
Dij = 1 ⇐⇒ si + 0.5 − sj − tj < 0 Fij = 1 ⇐⇒ si + 0.5 − sj > 0
(4)
Returning to Eq. (3) for Bij , and combining it with Eq. (4) for Dij and Dji ,
we obtain the equivalence relation between Bij and Dij , Fij :
Bij = 1 ⇐⇒ Dij = Fij = 1
(5)
To get the linear constraints, we consider the sum Dij + Fij ; it could be 0, 1, 2.
And when the sum is 2, that is equivalent to Dij = Fij = 1. Hence, it is necessary
for us to divide the sum into two conditions: 0, 1 and 2. We consider the expression
(Dij + Fij − 1.5)/3, then we can find that it will lie in (−1, 0) if Dij + Fij = 0, 1,
13
it will lie in (0, 1) if Dij + Fij = 2. Hence, we could use inequality Bij − 1 <
(Dij + Fij − 1.5)/3 < Bij to control Bij by Dij , Fij .
Hence, we use linear Constraints to ensure that time and spatial conflicts can
not be met simultaneously. And Constraints (2-2) to (2-7) play a role in the same
as Constraints (1-2) and Constraints (1-3), it is for linerization the defining of Bij .
Constraints (2-2) and Constraints (2-3) are used for control Dij , Constraints (2-4)
and Constraints (2-5) are used for control Fij , and those are based on Eq. (4).
THen, Constraints (2-7) are used for contron Bij based on Eq. (5). Finally, we get
the linearization of (MINLP), which is displayed by (MILP).
Model 2
min
(MILP)
(2-0)
max(si + ti )
i∈N
Bij + Iij ≤ 1,
(2-1)
i, j = 1, 2, ..., n, i ̸= j
sj + tj − si − 0.5 + M
≥ Dij − 0.5,
2M
i, j = 1, 2, ..., n, i ̸= j
(2-2)
sj + tj − si − 0.5 + M
≤ Dij + 0.5,
2M
i, j = 1, 2, ..., n, i ̸= j
(2-3)
si + 0.5 − sj + M
≥ Fij − 0.5,
2M
i, j = 1, 2, ..., n, i ̸= j
(2-4)
si + 0.5 − sj + M
≤ Fij + 0.5,
2M
i, j = 1, 2, ..., n, i ̸= j
(2-5)
Dij + Fij − 1.5 ≤ 3Bij ,
i, j = 1, 2, ..., n, i ̸= j
i, j = 1, 2, ..., n, i ̸= j
(2-7)
i = 1, 2, ..., n, k = 1, 2, 3
(2-8)
Dij + Fij − 1.5 ≥ 3(Bij − 1),
n
X
rkj Bij + rki ≤ Rk ,
(2-6)
j=1,j̸=i
14
3. Uncertainties in Maintenance Task Scheduling
3.1. Occurrences of Unexpected events during maintenance planning
During the implementation of maintenance tasks in the ship cabin, the actual
maintenance environment often presents uncertainties, such as operational difficulties and insufficient resource planning. These uncertainties necessitate a reevaluation
and optimization of existing maintenance planning. To address this, scientific methods for characterizing uncertainty factors are researched, analyzing the costs and
risks of adjusting maintenance plans. In this subsection, a maintenance planning
decision model is constructed, incorporating interactive decision-making based on
risk assessment and leveraging knowledge and historical data to evolve continuously
and improve the robustness and rationality of maintenance planning decisions. This
is crucial for enhancing the practical applicability of maintenance task planning.
For the previous model by (MILP), it was assumed that no unforeseen changes
would occur during the maintenance process, allowing for a globally optimal plan for
all maintenance tasks initially. However, when a series of uncertainties occur, they
may damage the existing optimal scheduling and thus bring the risk of significant
losses if they are not promptly addressed. Hence, the maintenance tasks of the
existing scheduling changes should be prioritized at specific time points. Notice that
in this study, the concept of "priority" is equivalent to the order of each maintenance
task in the optimal schedule, which means that the task planned to be maintained
earlier has higher priority than the others. At these points, it is necessary to re-plan
the maintenance tasks that are scheduled yet waiting for maintenance operation in
the queue. The process of progressively planning the remaining tasks is known as
rolling optimization. The time points at which task priorities change are referred to
as detection points. The dynamic maintenance task planning process is dependent
on the priority changes occurring at these detection points.
The paper will delve into the following three types of uncertainties:
1) During the maintenance process, sudden additions to the list of tasks may arise.
Initially, a list of critical equipment requiring maintenance is compiled, along with
a corresponding maintenance task plan, through engineer analysis. As maintenance
progresses on various components of the list, steps such as disassembly, detection,
and in-depth testing are executed to acquire more comprehensive information regarding the health status of equipment. Through this procedural sequence, potential
underlying faults may be diagnosed. Consequently, the initial list of maintenance
tasks undergoes update due to the emergence of new maintenance tasks during the
maintenance process.
2) During the maintenance process, there may be instances where the repair
time of a specific task experiences unexpected delay. Initially, engineers formulate
15
the initial maintenance task list based on engineering practice experience and the
repair procedures and processes specified in maintenance manuals, thereby estimating the required repair time for each task. However, during the actual execution
of maintenance operations, measures such as disassembly and thorough inspections
serve to refine and update the actually required repair duration. As a result, certain
maintenance tasks may encounter unexpected increases and delays in repair time.
3) During the maintenance process, delays in repairs can occur due to insufficient
resources. The successful implementation of maintenance operations requires specific
repair tools, equipment, spare parts, and technical engineers. Maintenance tasks can
be executed according to plan only when all necessary conditions are met. However,
in the actual execution of maintenance tasks, factors such as delays in updating
tools and spare parts inventory or unexpected situations involving technical engineers
can lead to a lack of the aforementioned repair resources, resulting in delays in
maintenance tasks.
3.2. Maintenance tasks are planned on a rolling optimization in batches
The uncertainty over time in the planning process is introduced previously, which
leads to changes in the state of each task (changes in priority, different occupied
resources due to different degradation levels, etc.) over time. So for an existing
planning scheme, it is necessary to set the detection time point τ and then detect
the two types of task sets at this moment in time, respectively, in the planning
process: the set of tasks of which maintenance tasks have already started before τ :
−
1) Ω+
τ = {i ∈ Ω|si < τ }, and 2) Ωτ = {i ∈ Ω|si ≥ τ }.
Based on these, once an incremental sequence of detection time points of finite
length τ0 , τ1 , τ2 ...τn is selected, then for each detection time point τi , two sets of tasks
can be decided according to the detection time τi :
−
+
−
Ω+
τi ∪ Ωτi = Ω and Ωτi ∩ Ωτi = ∅
(6)
where Ω+
τi contains all the maintenance tasks that have been started before τi and
−
Ωτi contains all the maintenance tasks that will start after τi . Hence, Ω contains all
the tasks. For a complete dynamic maintenance planning scheme by rolling optimization, the detection time points are set as follows: τ0 is the global planning scheme
−
start point, τn is the global planning scheme endpoint, which means Ω+
τ0 = Ω τn = ∅
+
and Ω−
τ0 = Ωτn = Ω. Moreover, there is the following relationship between the sets
of tasks given τi < τj , i < j:
+
−
−
∅ ⊂ Ω+
τi ⊂ Ωτj ⊂ Ω and ∅ ⊂ Ωτj ⊂ Ωτi ⊂ Ω
16
(7)
Therefore, each step of the rolling optimization (for a certain detection time) is
actually an update of the tasks in the planning to be maintained. With step-by-step
update planning, the tasks are gradually updated until empty so that each step of
the rolling dynamic planning problem size will be gradually reduced, and ultimately
can generate a global planning program to consider the uncertainty of the state of
the task. The condition encompasses both the uncertainty by generating detection
time points and uncertain scenarios for maintenance tasks.
3.3. Generating specific detection time point series
This study considers the detection points determined by the latest completion
time in the first batch of tasks. Before the global planning starts, there is limited information about the detection points other than the starting point, and it is supposed
to obtain the detection time points one by one in an iterative way:
i) Initially, carry out the task planning at the time τ0 of all the tasks contained in
Ω in the process of planning; and then take the set of the first batch of tasks
with parallel maintenance in the optimization solution as Ω+
τ1 .
ii) Then find out the set of tasks for which the updated planning is to be carried
+
out next as Ω−
τ1 = Ω \ Ωτ1 , i.e., the set of tasks that will be updated in the next
step.
iii) Then the earliest time point of the maintenance tasks in the first batch of machines to be repaired is denoted as τ1 , and then update the tasks of Ω−
τ1 during
planning, and take the set composed of the first batch of machines to be repaired
at the same time in the optimization result as Ω+
τ2 ...
iv) Continue this way, we can obtain the detection point data series and carry out
the dynamic maintenance planning through iteration until the whole planning
is finished.
By this way, dynamic maintenance planning can be carried out while obtaining
detection point time through continuous iteration until the end of the whole global
planning. The planning scale at each step of dynamic maintenance planning is decreasing gradually.
3.4. Rolling horizon: flowchart and algorithm
4. Solutions and Experimental Results
4.1. Parameters setting
A case of a two-dimensional layout with 19 maintenance tasks is considered as
shown in Fig. 5. By meshing densely, the 19 maintenance tasks are characterized as
17
Algorithm 1 Rolling Approach for Dynamic Maintenance Planning.
Require: Set of tasks to be scheduled Ω, initial time τ0
Ensure: Detection point series, dynamic maintenance plan
+
1: Initialize: τ0 ← 0, Ω−
τ0 ← Ω, Ωτ0 ← ∅, i ← 1
2: Solve the pre-planned scheme for tasks in Ω
3: Generate detection point τi
4: Determine the set of tasks already started, Ω+
τi ; and record their certain information
5: Check for uncertain scenarios at τi and find immediate tasks
6: Determine the set of tasks for the next updated planning, Ω−
τi
7: while Ω−
is
not
empty
do
τi
8:
Incorporate certain information into the dynamic maintenance planning and
treat tasks that are in progress but not completed at τi as virtual tasks
9:
Solve task planning at τ for tasks in Ω−
τ with certain information and virtual
tasks
10:
i←i+1
11:
Generate detection point τi and Check for uncertain scenarios
−
12:
Update Ω+
τi and Ωτi
13: end while
14: Get all of the dynamic maintenance planning schemes at each generated detection
point τi
the following Table 2 where xli and yid denote the coordinates of the left lower corner
of Mi , and xri − xli and yiu − yid denote the length and the width of the location of Mi .
li ,ri ,di and ui denote the required work area surrounding the maintenance task Mi
in the four directions. The required area for maintenance of Mi , where Mi is colored
grey, t is colored green; and the conflict of the required area is highlighted in red,
and the spare area is colored blue.
In this study, we consider constraints on three types of maintenance resources,
where the available amount of each type of resource is limited and is greater than
1. For convenience of record, assume that each task has a demand for each type
of resource, i.e., a toolbox with a specific amount of different resources denoted by
a ternary array representation. For each original task, the occupation maintenance
resources during maintenance operation are summarized by Table 3. Notice that the
tasks in {M4 , M7 , M10 , M12 , M14 , M16 , M19 } are deliberately not shown in Table 3 as
they are designed as the uncertainties in the maintenance scheduling process. To
highlight the impact of resource constraints on the optimal solution, this subsection
18
Fig. 4. Dynamic maintenance planning in rolling horizon.
selects a batch of tasks with less space conflict. In the following subsection, the research will be conducted by combining both spatial conflicts and resource constraints.
4.2. 12 tasks with different resources limit
The solutions to the two models, (MINLP) and (MILP) are based on Python
and GUROBI Solver. First, we will design the following experiments to verify the
optimality of the solutions.
Fig. 6 presents the optimal maintenance P
scheduling results under a demand to
supply ratio DSR = 0.3, which makes Ri = ⌊( k∈Ω rki )×DSR⌋. The use of the floor
function ‘⌊, ⌋’ is due to the fact that excess decimal parts do not increase the resource
utilization of the task. Fig. 6(a) is a Gantt chart showing the optimal makespan of
31. Fig. 6(b), Fig. 6(c), and Fig. 6(d) are resource histograms illustrating resource
19
Table 2: Parameters setting of 19 maintenance tasks (M1 ∼ M19 ).
xi
yi
li
wi
Ai (l)
Ai (r)
Ai (d)
Ai (u)
xi
yi
li
wi
Ai (l)
Ai (r)
Ai (d)
Ai (u)
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
54
109
67
133
16
36
19
58
222
58
108
56
13
25
8
50
167
177
55
74
28
55
52
60
91
317
76
36
12
50
54
33
289
198
40
29
57
54
10
8
73
392
97
64
12
7
31
10
394
148
45
85
57
44
36
23
422
36
88
71
6
44
7
28
235
353
55
86
36
47
17
20
495
220
83
106
47
9
18
53
M11
M12
M13
M14
M15
M16
M17
M18
M19
355
417
55
79
49
34
31
13
560
75
85
78
45
8
11
34
468
385
55
74
43
40
42
7
665
171
44
62
56
38
11
50
729
70
118
60
10
10
9
54
602
422
5
74
49
34
33
15
697
298
55
74
31
54
50
38
840
200
63
146
15
19
60
43
772
424
105
42
15
18
61
20
Table 3: Required resources for maintenance tasks.
Task
Resource
M1
M2
M3
M5
M6
M8
M9
M11
M13
M15
M17
M18
I
II
III
2
0
2
0
2
2
0
3
3
0
2
2
1
3
2
2
3
0
3
1
3
3
1
0
3
0
2
3
1
3
1
0
1
3
1
0
utilization under these periods: L1 : 0 ∼ 8, L2 : 8 ∼ 10, L3 : 10 ∼ 11, L4 : 11 ∼
11, L5 : 11 ∼ 15, L6 : 15 ∼ 16, L7 : 16 ∼ 17, L8 : 17 ∼ 22, L8 : 22 ∼ 24, L9 : 24 ∼
29, L10 : 29 ∼ 31.
Fig. 7 illustrates the optimal maintenance scheduling results under a demand
to supply ratio DSR = 0.5. The Gantt chart in Fig. 7(a) displays the schedul20
Fig. 5. A two-dimensional layout with 19 maintenance tasks(12 determined tasks and 7 uncertain
tasks).
ing of maintenance tasks, ensuring all tasks are completed in the shortest possible time of 20. Fig. 7(b) shows the utilization of Type I resources with availability R1 = 10. Fig. 7(c) shows the utilization of Type II resources with availability
R2 = 8. Fig. 7(d) shows the utilization of Type III resources with availability R3 = 10
under these periods: L1 : 0 ∼ 4, L2 : 4 ∼ 5, L3 : 5 ∼ 7, L4 : 7 ∼ 9, L5 : 9 ∼ 10, L6 :
10 ∼ 11, L7 : 11 ∼ 12, L8 : 12 ∼ 13, L9 : 13 ∼ 20.
Fig. 8 illustrates the optimal maintenance scheduling results under a demand
to supply ratio DSR = 0.8. The Gantt chart in Fig. 8(a) displays the scheduling of maintenance tasks, ensuring all tasks are completed in the shortest possible time. Fig. 8(b) shows the utilization of Type I resources with availability
R1 = 16. Fig. 8(c) shows the utilization of Type II resources with availability
R2 = 13. Fig. 8(d) shows the utilization of Type III resources with availability
21
(a) Gantt chart with the optimal makespan 31
(b) Resource Histogram with (c) Resource Histogram with (d) Resource Histogram with
availability R1 = 6
availability R2 = 5
availability R3 = 6
Fig. 6. The optimal maintenance scheduling under demand to supply ratio DSR = 0.3.
R3 = 16 under these periods: L1 : 0 ∼ 1, L2 : 1 ∼ 5, L3 : 5 ∼ 7, L4 : 7 ∼ 8, L5 : 8 ∼
9, L6 : 9 ∼ 10, L7 : 10 ∼ 12, L8 : 12 ∼ 16, L9 : 16 ∼ 17.
These charts clearly demonstrate the resource utilization under different availability conditions and how optimal scheduling can achieve the shortest makespan.
As the demand-to-supply ratio (DSR) gradually increases, the availability of resources becomes larger, allowing more simultaneous maintenance tasks. This results
in a more compact maintenance scheduling scheme, thereby reducing the optimal
makespan. Subsequent experiments revealed that when DSR reaches 1, the optimal
22
(a) Gantt chart with the optimal makespan 20
(b) Resource Histogram with (c) Resource Histogram with (d) Resource Histogram with
availability R1 = 10
availability R2 = 8
availability R3 = 10
Fig. 7. The optimal maintenance scheduling under demand to supply ratio DSR = 0.5.
makespan remains 17. This indicates that once DSR increases to a certain level,
the impact of resource constraints on the makespan diminishes, and spatial conflicts
become the primary factor affecting the makespan.
4.3. 13 tasks including an unexpected task with fixed resource
Add the uncertain tasks M7 , M10 , M12 , M19 to the original 12-task-maintenance
determined model, and the newly added tasks layout instances are as Fig. 9(a)
to Fig. 9(d). It can be found that when the added tasks carry out maintenance, poten23
(a) Gantt chart with the optimal makespan 17
(b) Resource Histogram with (c) Resource Histogram with (d) Resource Histogram with
availability R1 = 16
availability R2 = 13
availability R3 = 16
Fig. 8. The optimal maintenance scheduling under demand to supply ratio DSR = 0.8.
tial spatial conflicts will occur with the neighboring {M2 , M3 , M4 , M6 }, {M7 , M13 },
{M8 , M14 }, {M17 , M18 } (red parts), so maintenance cannot be operated at the same
time with these four tasks. It is obvious that those simultaneous tasks at each time
period Li do not face spatial conflicts as referred to in Fig. 9. Next, we verify the
satisfaction of each type of resource constraint by plotting.
The specific experiment process is as follows: when the detection point τ is
determined, the additional task should be scheduled to start at τ , and the decision
variables array S(task start time) and decision variables array E(task end time) in the
24
(a) The uncertain task M7
(b) The uncertain task M10
(c) The uncertain task M12
(d) The uncertain task M19
Fig. 9. Layout instances of 12 determined tasks and four uncertain additional tasks.
program have to be increased by one length accordingly, while the auxiliary variable
matrices have to be increased by one row and one column. Certain information about
tasks already started should be considered; the remaining duration of tasks already
started but not finished should be considered as virtual tasks with starts of τ .
In this case, the tasks are added at two different time points with respect to
each different uncertain additional task, and the following results are obtained using
dynamic rolling optimization at each point to be detected:
When the maintenance task process needs to add a new task uncertainly, suppose that under the pre-planned scheme as Fig. 7, the sudden situation of adding
a new task M7 occurs at the time point τ = 1, and M7 needs to be added to
25
the task set to be planned to form Ω− = {M1 , M2 , M5 , M6 , M7 , M8 , M11 , M13 , M15 }.
Considering the virtual task formed by the task {M3 , M9 , M17 , M18 } being executed
is still necessary. The optimal completion after the rolling approach is 21. And
While M7 is additionally started at the another time point τ = 5, we can see
Ω− = {M1 , M2 , M5 , M6 , M7 , M8 , M11 , M15 }, M3 and M17 meet their ending time.
The optimal makespan is 24, which is more than 4 than the pre-planned. As we can
see from the two pairs of charts in two columns, the maintenance schedule before
τ is the same as the pre-planned one. Taking into account the urgency of the new
task, when there is a task such as M6 to be started at the time τ such as τ = 5,
and the task cannot be started at the same time as M7 due to space conflicts or
resource constraints, then M7 will be prioritized to start at the tau time, resulting
in the postponement of the original task M6 .
Fig. 10. The Gantt chart of the optimal scheduling with uncertain addition M7 .
Fig. 11 illustrates the impact of adding a new task M10 at different time points
on the maintenance task scheduling. The left two columns of charts correspond to
the scenario where the new task M10 is added at time point τ = 4, while the right
two columns correspond to the scenario where the new task M10 is added at time
point τ = 12.
When the new task M10 is added at time point τ = 4, then Ω− = {M1 , M2 , M5 , M6 ,
M8 , M10 , M11 , M13 , M15 } needs to be re-planned. The optimal completion time after
the rolling approach is 27, resulting in a delay of 7 units compared to the pre-planned
26
schedule.
When the new task M10 is added at time point τ = 12, the task set Ω− =
{M8 , M11 , M15 } also needs to be re-planned. The optimal completion time after the
rolling approach is 28, resulting in a delay of 8 units compared to the pre-planned
schedule.
Fig. 11. The Gantt chart of the optimal scheduling with uncertain additional M10 .
Fig. 12 demonstrates the impact of adding a new task M12 at different time points
on the maintenance task scheduling. The left two columns of charts correspond to
the scenario where the new task M12 is added at time point τ = 5, while the right
two columns correspond to the scenario where the new task M12 is added at time
point τ = 9.
When the new task M12 is added at time point τ = 5, the task set Ω− =
{M1 , M2 , M5 , M6 , M8 , M9 , M11 , M12 , M15 } needs to be re-planned. The optimal completion time after the rolling approach is 24, resulting in a delay of 4 units compared
to the pre-planned schedule.
When the new task M12 is added at time point τ = 9, the task set Ω− =
{M1 , M2 , M8 , M11 , M12 , M15 } also needs to be re-planned. The optimal completion
time after the rolling approach is 23, resulting in a delay of 3 units compared to the
pre-planned schedule.
Fig. 13 presents the consequences of introducing a new task M19 at different
time points on the maintenance task scheduling. The left two columns of the charts
illustrate the scenario where the new task M19 is added at time point τ = 11, while
27
Fig. 12. The Gantt chart of the optimal scheduling with uncertain additional M12 .
the right two columns depict the scenario where the new task M19 is added at time
point τ = 12.
When the new task M19 is introduced at time point τ = 11, the task set Ω− =
{M 2, M8 , M11 , M15 , M19 } requires rescheduling. The optimal completion time after
applying the rolling approach is 26, resulting in a delay of 6 units compared to the
pre-planned schedule.
Conversely, when the new task M19 is added at time point τ = 12, the task set
−
Ω = {M8 , M11 , M15 , M19 } also necessitates rescheduling. The optimal completion
time after the rolling approach is 27, leading to a delay of 7 units compared to the
pre-planned schedule.
5. Experimental results of performance with different methods
We randomly selected three scenarios of 50 instances. We adopt the GUROBI
10.0.2 solver to run the (MINLP) and (MILP) on PyCharm 2022.1 (Professional
Edition). Besides, we applied the evolutionary algorithm to solve the problem using
EA-DEAP, implemented by the Distributed Evolutionary Algorithms in Python,
a novel evolutionary computation framework for rapid prototyping and testing of
ideas. The termination runtimes of MINLP-GUROBI and EA-DEAP are set to 3600s
and 7200s, respectively. EA-DEAP used a population of 50 for 1000 generations,
28
Fig. 13. The Gantt chart of the optimal scheduling with uncertain additional M19 .
with crossover and mutation probabilities of 0.8 and 0.05. These experiments are
implemented on a desktop computer with an eight-core processor, specifically the
Intel(R) Core(TM) i7-9700 CPU @ 3.00GHz and 32 GB of RAM.
The gap for MINLP-GUROBI and EA-DEAP is defined by Eq. (8):
Gap =
ObjM ILP −GU ROBI − Objmodel
∗ 100%,
ObjM ILP −GU ROBI
(8)
where model includes MINLP-GUROBI and EA-DEAP. Through compared experiments, it is found that the MILP-GUROBI model can obtain the optimal solution
quickly, while the MINLP-GUROBI model finds it difficult to obtain the optimal
solution quickly. When faced with a large task scale, DEAP makes obtaining a suboptimal or feasible solution even more difficult. To facilitate the comparison of the
performance of solutions obtained by different models through data, when the model
does not obtain a feasible solution within the set terminal time, the solution obtained
is defined as the worst feasible solution: That is, from time 0, all tasks are executed
end to end, so that no two or more tasks are executed at the same time, and the
resulting makespan is the sum of the respective execution times of all tasks.
After obtaining the global solution of a specific instance by MILP-GRUOBI, we
recorded the runtime and made it the maximum runtime for MINLP-GUROBI and
EA-DEAP. Then, we compared their solutions within the same maximum runtime.
The available usage of each resource type is set to 80% of the total demand for
29
that type of resource. Firstly, as can be seen from the three figures: from Fig. 14
to Fig. 16, while the MILP-GUROBI gets the optimal precise solution within a maximum runtime, the other two models could hardly get the precise optimal solution.
While the scale of the problem increases as the number of tasks rises, the performance
gap between MILP-GUROBI and the other two models grows larger. Especially for
the results shown by Fig. 16, the MIL-GRUOBI and EA-DEAP could not find an
appropriate initial feasible solution so that those scatters corresponded to their solutions lie on the near top of the figure, which represents the worst feasible solution
far away from the MILP-GUROBI’s.
To confirm the correctness and effectiveness of the proposed MILP model, we
randomly selected three types of 50 instances. We adopt the GUROBI 10.0.2 solver
to run the MILP on PyCharm 2022.1 (Professional Edition). The termination runtimes of MINLP-GUROBI and EA-DEAP are set to 3600s and 7200s, respectively.
Moreover, as can be seen from the results in Table 4, after MINLP-GUROBI runs for
more time than MILP-GUROBI, it can also get the same precise optimal solution
as MILP-GUROBI, but the genetic algorithm under the framework of EA-DEAP
still cannot get the precise optimal solution after running to the maximum time of
7200s, but the suboptimal solution to a certain extent. And it can be seen that the
stability of MILP-GUROBI is better. For the same number of task instances, the
optimal solution can be achieved in a relatively stable runtime, but the stability of
MINLP-GUROBI will be slightly worse, especially for the larger 17-tasks-instance,
which took at least 17 seconds of the N.O.2, but the longest time took 1700 seconds
of the N.O.1 and EA-DEAP are set to 3600s and 7200s, respectively.
5.1. Simulation for robustness verification
To address the potential issue of duplicating random new task time points, we
generated 1,000 random numbers following a Poisson distribution with a mean of 5.
This approach allowed us to capture a realistic range of task initiation points. We
extracted unique values from this dataset and recorded their frequencies, as shown
in Table 5 and Fig. 17. We then focused on 14 specific new task time points ranging
from 0 to 13, selecting task M12 for analysis to evaluate how varying initiation times
affect overall completion durations.
After conducting simulations for each selected time point, we obtained optimal task completion durations, which are presented in the subsequent table. The
weighted average of these durations was calculated to be 20.066, close to the optimal
duration of 20, corresponding to the Poisson mean of 5. This result underscores
the effectiveness of our simulation approach in accurately modeling task completion
dynamics under different initiation conditions.
30
Table 4: Computational results for 3 scenarios of instances.
Instance
MILP-GUROBI
MINLP-GUROBI
N.O.
Obj
CPU(s)
Obj
CPU(s)
Gap(%)
Obj
CPU(s)
Gap(%)
12 tasks
1
2
3
4
5
19
24
22
20
21
0.062
0.084
0.052
0.064
0.025
19
24
22
20
21
1.1
2.9
84
1.8
11
0
0
0
0
0
26
29
30
25
27
7200
7200
7200
7200
7200
37
21
26
25
29
15 tasks
1
2
3
4
5
20
19
20
22
20
0.17
0.34
0.11
0.31
0.1
20
19
20
22
20
12
4.8
15
97
16
0
0
0
0
0
27
26
29
29
27
7200
7200
7200
7200
7200
35
37
45
32
35
17 tasks
1
2
3
4
5
22
19
20
21
21
1.6
0.38
0.23
0.28
0.23
22
19
20
21
21
1700
17
52
530
72
0
0
0
0
0
28
29
30
28
-
7200
7200
7200
7200
-
27
53
50
33
-
Scenario
EA-DEAP
N.O.: the instance number; Obj: the objective function value; CPU: the runtime.
−: the feasible solutions of MINLP-GUROBI(EA-DEAP) cannot be obtained within 3600(7200) s.
Fig. 14. Performance of instances containing 12 randomly selected tasks.
31
Fig. 15. Performance of instances containing 15 randomly selected tasks.
Table 5: Random time points and frequencies to add task M12 .
points
frequencies
optimals
0
1
2
3
4
5
6
7
8
9
10
11
12
13
11 34 99 129
17 17 18 19
171
20
184
20
129
20
100 64 40 24 9
21 22 23 24 25
3
26
3
27
6. Conclusion and future research
6.1. Conclusion
This study addresses the challenges posed by limited resources and conflicts in
spatial allocation for maintenance tasks, with the goal of minimizing the makespan.
We proposed a MINLP model, which was subsequently linearized into a MILP model
using the big-M method. The visualization of these scheduling schemes with increasing resource availability, along with the accumulation of resources over time, provided
a clear observation of how the resource constraints were being satisfied.
To address uncertainties due to unpredictable emergencies and variable maintenance environments. Tasks were categorized into pre-planned and to-be-planned
sets, with dynamic adjustment strategies ensuring smooth maintenance processes
32
Fig. 16. Performance of instances containing 17 randomly selected tasks.
and timely project completion. Our method, based on rolling time windows and
detection points, adapts intelligently to uncertainly additional tasks, maintaining
scheduling continuity and ensuring on-time project completion.
The performance comparison results demonstrated that the linearized MILP
model provides precise solutions within significantly shorter runtimes compared to
the MINLP model. Additionally, it exhibited superior stability when compared to
evolutionary algorithms implemented within the EA-DEAP framework.
6.2. Future research
Future research should examine the trade-offs between losses from delayed maintenance and extended makespan due to priority changes. Developing a multi-objective
ship maintenance task planning model that considers spatial constraints, work areas,
and cabin access will enhance efficiency. Additionally, focusing on dynamic adjustment strategies and resource optimization will improve the robustness of maintenance
scheduling in unforeseen situations. Industrial engineering requires close collaboration among various departments throughout the entire production and operation
process. However, in practice, due to conflicts between short-term economic and
various optimization objectives, the need for maintainability is always overlooked.
In certain industries, maintainability and maintenance engineering are important
33
Fig. 17. Different task adding time points for M12 .
34
factors in equipment support. In the equipment design and optimization stage, considering the maintainability of each component will be beneficial for improving the
availability of the equipment. Meanwhile, when making decision of CBM and PdM,
it is recommended to consider whether the implementation conditions are met.
Acknowledgement
This work is supported by the National Natural Science Foundation of China
(grant No.72231008 and grant No.72471188) and the Science and Technology Innovation Team of Shaanxi Provincial (Grant No.2024RS-CXTD-28).
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Declaration of Interest Statement
Conflict of Interest
Declaration of interests
 The authors declare that they have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may
be considered as potential competing interests:
Author Statement
Declaration of Generative AI and AI-assisted technologies in the
writing process
Paper title: Simultaneous Tasks Planning and Resources Assignment in
Maintenance Scheduling under Uncertainties
Author information:
Bin Wua , Wenjin Zhua,∗, Xu Luob, Shubin Sia
aMinistry of Industry and Information Technology Key Laboratory of Industrial Engineering and
Intelligent Manufacturing, Northwestern Polytechnical University, Xi’an 710072, China
bLaboratory of Science and Technology on Integrated Logistics Support, College of Intelligent
Sciences and Technology, National University of Defense Technology, Changsha, PR China
Statement: During the preparation of this work the authors did not use
Generative AI and AI-assisted technologies in the writing process.