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Note: The APA Publication Manual, 7 th Edition specifies different formatting conventions for student  and  professional  papers (i.e., papers written for credit in a course and papers intended for scholarly publication). These differences mostly extend to the title page and running head. Crucially, citation practices do not differ between the two styles of paper.

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Note: For accessibility purposes, we have used "Track Changes" to make comments along the margins of these samples. Those authored by [AF] denote explanations of formatting and [AWC] denote directions for writing and citing in APA 7. 

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A Quick Guide to Harvard Referencing | Citation Examples

Published on 14 February 2020 by Jack Caulfield . Revised on 15 September 2023.

Referencing is an important part of academic writing. It tells your readers what sources you’ve used and how to find them.

Harvard is the most common referencing style used in UK universities. In Harvard style, the author and year are cited in-text, and full details of the source are given in a reference list .

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Table of contents

Harvard in-text citation, creating a harvard reference list, harvard referencing examples, referencing sources with no author or date, frequently asked questions about harvard referencing.

A Harvard in-text citation appears in brackets beside any quotation or paraphrase of a source. It gives the last name of the author(s) and the year of publication, as well as a page number or range locating the passage referenced, if applicable:

Note that ‘p.’ is used for a single page, ‘pp.’ for multiple pages (e.g. ‘pp. 1–5’).

An in-text citation usually appears immediately after the quotation or paraphrase in question. It may also appear at the end of the relevant sentence, as long as it’s clear what it refers to.

When your sentence already mentions the name of the author, it should not be repeated in the citation:

Sources with multiple authors

When you cite a source with up to three authors, cite all authors’ names. For four or more authors, list only the first name, followed by ‘ et al. ’:

Sources with no page numbers

Some sources, such as websites , often don’t have page numbers. If the source is a short text, you can simply leave out the page number. With longer sources, you can use an alternate locator such as a subheading or paragraph number if you need to specify where to find the quote:

Multiple citations at the same point

When you need multiple citations to appear at the same point in your text – for example, when you refer to several sources with one phrase – you can present them in the same set of brackets, separated by semicolons. List them in order of publication date:

Multiple sources with the same author and date

If you cite multiple sources by the same author which were published in the same year, it’s important to distinguish between them in your citations. To do this, insert an ‘a’ after the year in the first one you reference, a ‘b’ in the second, and so on:

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A bibliography or reference list appears at the end of your text. It lists all your sources in alphabetical order by the author’s last name, giving complete information so that the reader can look them up if necessary.

The reference entry starts with the author’s last name followed by initial(s). Only the first word of the title is capitalised (as well as any proper nouns).

Harvard reference list example

Sources with multiple authors in the reference list

As with in-text citations, up to three authors should be listed; when there are four or more, list only the first author followed by ‘ et al. ’:

Reference list entries vary according to source type, since different information is relevant for different sources. Formats and examples for the most commonly used source types are given below.

  • Entire book
  • Book chapter
  • Translated book
  • Edition of a book

Journal articles

  • Print journal
  • Online-only journal with DOI
  • Online-only journal with no DOI
  • General web page
  • Online article or blog
  • Social media post

Sometimes you won’t have all the information you need for a reference. This section covers what to do when a source lacks a publication date or named author.

No publication date

When a source doesn’t have a clear publication date – for example, a constantly updated reference source like Wikipedia or an obscure historical document which can’t be accurately dated – you can replace it with the words ‘no date’:

Note that when you do this with an online source, you should still include an access date, as in the example.

When a source lacks a clearly identified author, there’s often an appropriate corporate source – the organisation responsible for the source – whom you can credit as author instead, as in the Google and Wikipedia examples above.

When that’s not the case, you can just replace it with the title of the source in both the in-text citation and the reference list:

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Harvard referencing uses an author–date system. Sources are cited by the author’s last name and the publication year in brackets. Each Harvard in-text citation corresponds to an entry in the alphabetised reference list at the end of the paper.

Vancouver referencing uses a numerical system. Sources are cited by a number in parentheses or superscript. Each number corresponds to a full reference at the end of the paper.

A Harvard in-text citation should appear in brackets every time you quote, paraphrase, or refer to information from a source.

The citation can appear immediately after the quotation or paraphrase, or at the end of the sentence. If you’re quoting, place the citation outside of the quotation marks but before any other punctuation like a comma or full stop.

In Harvard referencing, up to three author names are included in an in-text citation or reference list entry. When there are four or more authors, include only the first, followed by ‘ et al. ’

Though the terms are sometimes used interchangeably, there is a difference in meaning:

  • A reference list only includes sources cited in the text – every entry corresponds to an in-text citation .
  • A bibliography also includes other sources which were consulted during the research but not cited.

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Caulfield, J. (2023, September 15). A Quick Guide to Harvard Referencing | Citation Examples. Scribbr. Retrieved 31 May 2024, from https://www.scribbr.co.uk/referencing/harvard-style/

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What’s Included: Research Paper Template

If you’re preparing to write an academic research paper, our free research paper template is the perfect starting point. In the template, we cover every section step by step, with clear, straightforward explanations and examples .

The template’s structure is based on the tried and trusted best-practice format for formal academic research papers. The template structure reflects the overall research process, ensuring your paper will have a smooth, logical flow from chapter to chapter.

The research paper template covers the following core sections:

  • The title page/cover page
  • Abstract (sometimes also called the executive summary)
  • Section 1: Introduction 
  • Section 2: Literature review 
  • Section 3: Methodology
  • Section 4: Findings /results
  • Section 5: Discussion
  • Section 6: Conclusion
  • Reference list

Each section is explained in plain, straightforward language , followed by an overview of the key elements that you need to cover within each section. We’ve also included links to free resources to help you understand how to write each section.

The cleanly formatted Google Doc can be downloaded as a fully editable MS Word Document (DOCX format), so you can use it as-is or convert it to LaTeX.

FAQs: Research Paper Template

What format is the template (doc, pdf, ppt, etc.).

The research paper template is provided as a Google Doc. You can download it in MS Word format or make a copy to your Google Drive. You’re also welcome to convert it to whatever format works best for you, such as LaTeX or PDF.

What types of research papers can this template be used for?

The template follows the standard best-practice structure for formal academic research papers, so it is suitable for the vast majority of degrees, particularly those within the sciences.

Some universities may have some additional requirements, but these are typically minor, with the core structure remaining the same. Therefore, it’s always a good idea to double-check your university’s requirements before you finalise your structure.

Is this template for an undergrad, Masters or PhD-level research paper?

This template can be used for a research paper at any level of study. It may be slight overkill for an undergraduate-level study, but it certainly won’t be missing anything.

How long should my research paper be?

This depends entirely on your university’s specific requirements, so it’s best to check with them. We include generic word count ranges for each section within the template, but these are purely indicative. 

What about the research proposal?

If you’re still working on your research proposal, we’ve got a template for that here .

We’ve also got loads of proposal-related guides and videos over on the Grad Coach blog .

How do I write a literature review?

We have a wealth of free resources on the Grad Coach Blog that unpack how to write a literature review from scratch. You can check out the literature review section of the blog here.

How do I create a research methodology?

We have a wealth of free resources on the Grad Coach Blog that unpack research methodology, both qualitative and quantitative. You can check out the methodology section of the blog here.

Can I share this research paper template with my friends/colleagues?

Yes, you’re welcome to share this template. If you want to post about it on your blog or social media, all we ask is that you reference this page as your source.

Can Grad Coach help me with my research paper?

Within the template, you’ll find plain-language explanations of each section, which should give you a fair amount of guidance. However, you’re also welcome to consider our private coaching services .

Free Webinar: Literature Review 101


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Format Your Paper & Cite Your Sources

  • Harvard Style
  • Citing Sources
  • Avoid Plagiarism
  • MLA Style (8th/9th ed.)
  • APA Style, 7th Edition
  • Chicago Style

What is Harvard Style?

What you need to know, harvard style tutorial.

  • Other Styles
  • Annotated Bibliographies
  • How to Create an Attribution

Harvard Style

The Harvard referencing system is known as the Author-Date style . It emphasizes the name of the creator of a piece of information and the date of publication, with the list of references in alphabetical order at the end of your paper.

Unlike other citation styles, there is no single, definitive version of Harvard Style. Therefore, you may see a variation in features such as punctuation, capitalization, abbreviations, and the use of italics. 

Always check with your instructor and follow the rules he or she gives you.

  • Harvard Style Guidelines Your class handout
  • Harvard Referencing Quick Guide From Staffordshire University

Harvard Style will affect your paper in two places:

  • In-text citations in the body of your paper, and
  • The reference list at the end of your paper
  • All in-text citations should be listed in the reference list at the end of your paper.
  • Reference list entries need to contain all the information that someone reading your paper would need in order to find your source.
  • Reference lists in Harvard Style are arranged alphabetically by first author.
  • Begin your Reference list on a new page after your text and number it consecutively.

Sample References List:

Example of Harvard References List

Click on the Links Below to See Additional Examples:

  • Sample Paper Paper provided by Kurt Olson
  • Harvard Citation Examples Document created by The University of Western Australia

Click on the image below to launch this tutorial that was created by the University of Leeds. The section on Citing in Text is especially useful.

research paper reference model

  • << Previous: Chicago Style
  • Next: Other Styles >>
  • Last Updated: May 1, 2024 2:04 PM
  • URL: https://libguides.mjc.edu/citeyoursources

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Research Method

Home » Research Methodology – Types, Examples and writing Guide

Research Methodology – Types, Examples and writing Guide

Table of Contents

Research Methodology

Research Methodology


Research Methodology refers to the systematic and scientific approach used to conduct research, investigate problems, and gather data and information for a specific purpose. It involves the techniques and procedures used to identify, collect , analyze , and interpret data to answer research questions or solve research problems . Moreover, They are philosophical and theoretical frameworks that guide the research process.

Structure of Research Methodology

Research methodology formats can vary depending on the specific requirements of the research project, but the following is a basic example of a structure for a research methodology section:

I. Introduction

  • Provide an overview of the research problem and the need for a research methodology section
  • Outline the main research questions and objectives

II. Research Design

  • Explain the research design chosen and why it is appropriate for the research question(s) and objectives
  • Discuss any alternative research designs considered and why they were not chosen
  • Describe the research setting and participants (if applicable)

III. Data Collection Methods

  • Describe the methods used to collect data (e.g., surveys, interviews, observations)
  • Explain how the data collection methods were chosen and why they are appropriate for the research question(s) and objectives
  • Detail any procedures or instruments used for data collection

IV. Data Analysis Methods

  • Describe the methods used to analyze the data (e.g., statistical analysis, content analysis )
  • Explain how the data analysis methods were chosen and why they are appropriate for the research question(s) and objectives
  • Detail any procedures or software used for data analysis

V. Ethical Considerations

  • Discuss any ethical issues that may arise from the research and how they were addressed
  • Explain how informed consent was obtained (if applicable)
  • Detail any measures taken to ensure confidentiality and anonymity

VI. Limitations

  • Identify any potential limitations of the research methodology and how they may impact the results and conclusions

VII. Conclusion

  • Summarize the key aspects of the research methodology section
  • Explain how the research methodology addresses the research question(s) and objectives

Research Methodology Types

Types of Research Methodology are as follows:

Quantitative Research Methodology

This is a research methodology that involves the collection and analysis of numerical data using statistical methods. This type of research is often used to study cause-and-effect relationships and to make predictions.

Qualitative Research Methodology

This is a research methodology that involves the collection and analysis of non-numerical data such as words, images, and observations. This type of research is often used to explore complex phenomena, to gain an in-depth understanding of a particular topic, and to generate hypotheses.

Mixed-Methods Research Methodology

This is a research methodology that combines elements of both quantitative and qualitative research. This approach can be particularly useful for studies that aim to explore complex phenomena and to provide a more comprehensive understanding of a particular topic.

Case Study Research Methodology

This is a research methodology that involves in-depth examination of a single case or a small number of cases. Case studies are often used in psychology, sociology, and anthropology to gain a detailed understanding of a particular individual or group.

Action Research Methodology

This is a research methodology that involves a collaborative process between researchers and practitioners to identify and solve real-world problems. Action research is often used in education, healthcare, and social work.

Experimental Research Methodology

This is a research methodology that involves the manipulation of one or more independent variables to observe their effects on a dependent variable. Experimental research is often used to study cause-and-effect relationships and to make predictions.

Survey Research Methodology

This is a research methodology that involves the collection of data from a sample of individuals using questionnaires or interviews. Survey research is often used to study attitudes, opinions, and behaviors.

Grounded Theory Research Methodology

This is a research methodology that involves the development of theories based on the data collected during the research process. Grounded theory is often used in sociology and anthropology to generate theories about social phenomena.

Research Methodology Example

An Example of Research Methodology could be the following:

Research Methodology for Investigating the Effectiveness of Cognitive Behavioral Therapy in Reducing Symptoms of Depression in Adults


The aim of this research is to investigate the effectiveness of cognitive-behavioral therapy (CBT) in reducing symptoms of depression in adults. To achieve this objective, a randomized controlled trial (RCT) will be conducted using a mixed-methods approach.

Research Design:

The study will follow a pre-test and post-test design with two groups: an experimental group receiving CBT and a control group receiving no intervention. The study will also include a qualitative component, in which semi-structured interviews will be conducted with a subset of participants to explore their experiences of receiving CBT.


Participants will be recruited from community mental health clinics in the local area. The sample will consist of 100 adults aged 18-65 years old who meet the diagnostic criteria for major depressive disorder. Participants will be randomly assigned to either the experimental group or the control group.

Intervention :

The experimental group will receive 12 weekly sessions of CBT, each lasting 60 minutes. The intervention will be delivered by licensed mental health professionals who have been trained in CBT. The control group will receive no intervention during the study period.

Data Collection:

Quantitative data will be collected through the use of standardized measures such as the Beck Depression Inventory-II (BDI-II) and the Generalized Anxiety Disorder-7 (GAD-7). Data will be collected at baseline, immediately after the intervention, and at a 3-month follow-up. Qualitative data will be collected through semi-structured interviews with a subset of participants from the experimental group. The interviews will be conducted at the end of the intervention period, and will explore participants’ experiences of receiving CBT.

Data Analysis:

Quantitative data will be analyzed using descriptive statistics, t-tests, and mixed-model analyses of variance (ANOVA) to assess the effectiveness of the intervention. Qualitative data will be analyzed using thematic analysis to identify common themes and patterns in participants’ experiences of receiving CBT.

Ethical Considerations:

This study will comply with ethical guidelines for research involving human subjects. Participants will provide informed consent before participating in the study, and their privacy and confidentiality will be protected throughout the study. Any adverse events or reactions will be reported and managed appropriately.

Data Management:

All data collected will be kept confidential and stored securely using password-protected databases. Identifying information will be removed from qualitative data transcripts to ensure participants’ anonymity.


One potential limitation of this study is that it only focuses on one type of psychotherapy, CBT, and may not generalize to other types of therapy or interventions. Another limitation is that the study will only include participants from community mental health clinics, which may not be representative of the general population.


This research aims to investigate the effectiveness of CBT in reducing symptoms of depression in adults. By using a randomized controlled trial and a mixed-methods approach, the study will provide valuable insights into the mechanisms underlying the relationship between CBT and depression. The results of this study will have important implications for the development of effective treatments for depression in clinical settings.

How to Write Research Methodology

Writing a research methodology involves explaining the methods and techniques you used to conduct research, collect data, and analyze results. It’s an essential section of any research paper or thesis, as it helps readers understand the validity and reliability of your findings. Here are the steps to write a research methodology:

  • Start by explaining your research question: Begin the methodology section by restating your research question and explaining why it’s important. This helps readers understand the purpose of your research and the rationale behind your methods.
  • Describe your research design: Explain the overall approach you used to conduct research. This could be a qualitative or quantitative research design, experimental or non-experimental, case study or survey, etc. Discuss the advantages and limitations of the chosen design.
  • Discuss your sample: Describe the participants or subjects you included in your study. Include details such as their demographics, sampling method, sample size, and any exclusion criteria used.
  • Describe your data collection methods : Explain how you collected data from your participants. This could include surveys, interviews, observations, questionnaires, or experiments. Include details on how you obtained informed consent, how you administered the tools, and how you minimized the risk of bias.
  • Explain your data analysis techniques: Describe the methods you used to analyze the data you collected. This could include statistical analysis, content analysis, thematic analysis, or discourse analysis. Explain how you dealt with missing data, outliers, and any other issues that arose during the analysis.
  • Discuss the validity and reliability of your research : Explain how you ensured the validity and reliability of your study. This could include measures such as triangulation, member checking, peer review, or inter-coder reliability.
  • Acknowledge any limitations of your research: Discuss any limitations of your study, including any potential threats to validity or generalizability. This helps readers understand the scope of your findings and how they might apply to other contexts.
  • Provide a summary: End the methodology section by summarizing the methods and techniques you used to conduct your research. This provides a clear overview of your research methodology and helps readers understand the process you followed to arrive at your findings.

When to Write Research Methodology

Research methodology is typically written after the research proposal has been approved and before the actual research is conducted. It should be written prior to data collection and analysis, as it provides a clear roadmap for the research project.

The research methodology is an important section of any research paper or thesis, as it describes the methods and procedures that will be used to conduct the research. It should include details about the research design, data collection methods, data analysis techniques, and any ethical considerations.

The methodology should be written in a clear and concise manner, and it should be based on established research practices and standards. It is important to provide enough detail so that the reader can understand how the research was conducted and evaluate the validity of the results.

Applications of Research Methodology

Here are some of the applications of research methodology:

  • To identify the research problem: Research methodology is used to identify the research problem, which is the first step in conducting any research.
  • To design the research: Research methodology helps in designing the research by selecting the appropriate research method, research design, and sampling technique.
  • To collect data: Research methodology provides a systematic approach to collect data from primary and secondary sources.
  • To analyze data: Research methodology helps in analyzing the collected data using various statistical and non-statistical techniques.
  • To test hypotheses: Research methodology provides a framework for testing hypotheses and drawing conclusions based on the analysis of data.
  • To generalize findings: Research methodology helps in generalizing the findings of the research to the target population.
  • To develop theories : Research methodology is used to develop new theories and modify existing theories based on the findings of the research.
  • To evaluate programs and policies : Research methodology is used to evaluate the effectiveness of programs and policies by collecting data and analyzing it.
  • To improve decision-making: Research methodology helps in making informed decisions by providing reliable and valid data.

Purpose of Research Methodology

Research methodology serves several important purposes, including:

  • To guide the research process: Research methodology provides a systematic framework for conducting research. It helps researchers to plan their research, define their research questions, and select appropriate methods and techniques for collecting and analyzing data.
  • To ensure research quality: Research methodology helps researchers to ensure that their research is rigorous, reliable, and valid. It provides guidelines for minimizing bias and error in data collection and analysis, and for ensuring that research findings are accurate and trustworthy.
  • To replicate research: Research methodology provides a clear and detailed account of the research process, making it possible for other researchers to replicate the study and verify its findings.
  • To advance knowledge: Research methodology enables researchers to generate new knowledge and to contribute to the body of knowledge in their field. It provides a means for testing hypotheses, exploring new ideas, and discovering new insights.
  • To inform decision-making: Research methodology provides evidence-based information that can inform policy and decision-making in a variety of fields, including medicine, public health, education, and business.

Advantages of Research Methodology

Research methodology has several advantages that make it a valuable tool for conducting research in various fields. Here are some of the key advantages of research methodology:

  • Systematic and structured approach : Research methodology provides a systematic and structured approach to conducting research, which ensures that the research is conducted in a rigorous and comprehensive manner.
  • Objectivity : Research methodology aims to ensure objectivity in the research process, which means that the research findings are based on evidence and not influenced by personal bias or subjective opinions.
  • Replicability : Research methodology ensures that research can be replicated by other researchers, which is essential for validating research findings and ensuring their accuracy.
  • Reliability : Research methodology aims to ensure that the research findings are reliable, which means that they are consistent and can be depended upon.
  • Validity : Research methodology ensures that the research findings are valid, which means that they accurately reflect the research question or hypothesis being tested.
  • Efficiency : Research methodology provides a structured and efficient way of conducting research, which helps to save time and resources.
  • Flexibility : Research methodology allows researchers to choose the most appropriate research methods and techniques based on the research question, data availability, and other relevant factors.
  • Scope for innovation: Research methodology provides scope for innovation and creativity in designing research studies and developing new research techniques.

Research Methodology Vs Research Methods

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  • Published: 28 May 2024

Quantum computing for several AGV scheduling models

  • Liang Tang 1 ,
  • Chao Yang 1 ,
  • Kai Wen 2 ,
  • Wei Wu 3 &
  • Yiyun Guo 4  

Scientific Reports volume  14 , Article number:  12205 ( 2024 ) Cite this article

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  • Applied mathematics
  • Computer science
  • Quantum physics

Due to the high degree of automation, automated guided vehicles (AGVs) have been widely used in many scenarios for transportation, and traditional computing power is stretched in large-scale AGV scheduling. In recent years, quantum computing has shown incomparable performance advantages in solving specific problems, especially Combinatorial optimization problem. In this paper, quantum computing technology is introduced into the study of the AGV scheduling problem. Additionally two types of quadratic unconstrained binary optimisation (QUBO) models suitable for different scheduling objectives are constructed, and the scheduling scheme is coded into the ground state of Hamiltonian operator, and the problem is solved by using optical coherent Ising machine (CIM). The experimental results show that compared with the traditional calculation method, the optical quantum computer can save 92% computation time on average. It has great application potential.

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The scale of the logistics industry has maintained a considerable growth rate, and many human and material resources have been invested in it, thus creating the labor-intensive industry. Improving the automatic and intelligent level of the logistics industry has become an important issue for industry and academia. In recent years, some of the industry’s leading enterprises have already carried out technological reforms. For example, the retail giant, Amazon, has established a huge logistics center, that uses intelligent sorting technology, delivery drones, automated guided vehicles (AGVs), etc. China’s e-commerce giant, Jingdong, has also set up an ‘Asia One’ warehouse, in which more than 100 AGVs are used for transportation operations at the same time 1 . In addition, technical innovation also occurred in ports. In 1993, the world’s first automated wharf was built in the Amsterdam Port in the Netherlands, and dozens of AGVs were used for container transshipment. With the introduction of technology and the accumulation of operation experience, the construction of automatic terminals has been tried all over the world, and the usage of AGVs has gradually increased. Currently, the use of AGVs has penetrated all aspects of logistics, transportation and production, which has greatly promoted the level of industrial automation and intelligence and improved efficiency.

To meet the needs of application scenarios, the amount of parallel work of AGVs is increasing, which brings great difficulty to the AGV scheduling. AGV scheduling problem is a difficult combinatorial optimization problem, and a large number of researchers have devoted themselves to this field and made some contributions. Singh et al. 2 considered AGV scheduling with battery constraints, developed a mixed-integer programming model with the objective of minimizing the combined task delay cost and AGV transportation cost, and designed a customized adaptive large-neighborhood search algorithm to solve the model. Zhang et al. 3 studied an AGV scheduling problem in matrix manufacturing plants, and proposed a mixed integer programming model to minimize the generalized transportation cost, based on which an improved iterative greedy algorithm was designed and compared with six other algorithms to show its superior solution performance. For the scheduling problem of AGVs in smart factories, Zhang et al. 4 proposed a self-organized dynamic scheduling method, that groups multiple AGVs to perform tasks among themselves and uses improved gene expression programming to learn dynamic scheduling rules. The numerical experimental results show that the method can considerably reduce system costs. Wang and Zeng 5 studied the port AGV scheduling and path planning problem under conflict-free paths, established a mixed integer model with the objective of minimizing task completion time, and proposed a customized branch and bound algorithm combined with a heuristic algorithm to solve the small-scale problem, and further developed a two-stage greedy heuristic algorithm to quickly obtain a satisfactory solution for the large-scale problem. Sagar and Jerald 6 proposed a real-time scheduling strategy for AGVs based on deep reinforcement learning technology, established a Markov decision model for real-time scheduling, and developed a Q-learning algorithm. The superiority of the method is shown through numerical experiments. Considering the scheduling and path planning model of shop floor AGVs, Saidi et al. 7 developed a discrete-time model and proposed a two-stage ant colony algorithm to solve the model. From the above literature, the research on scheduling problems of AGVs covers several scenarios such as workshops and terminals. Researchers have built mixed integer programming models, integer programming models, Markov decision process models, etc. The methods used are scheduling rules, exact algorithms, heuristic algorithms, reinforcement learning algorithms, etc. From the results, it is observed that the exact algorithm can generate optimal scheduling solutions, however, its computational time is prohibitively slow, rendering it impractical for large-scale problems. Inexact algorithms exhibit favorable efficiency but often converge to local optima. The provision of high-quality scheduling solutions within a short timeframe poses a significant challenge.

In recent years, significant advancements have been made in both theoretical understanding and practical applications of quantum computing. The fundamental distinction between quantum computers and traditional computers lies in their reliance on quantum mechanical principles. Quantum computers utilize quantum bits (qubits) as the fundamental units of information storage 8 , which can exist in superposition states of both 1 and 0, enabling them to hold exponentially more information compared to traditional computers. It is well recognized that quantum computers offer substantial advantages, particularly in addressing specific problems such as combinatorial optimization, often described as the superiority of quantum computing. Many combinatorial optimization problems are NP-hard, presenting significant challenges for traditional computers to solve. Combinatorial optimization problem can be mapped to the ground state search problem of Ising model. Hardware systems can be built in many different ways to simulate the process of Hamiltonian reduction, such as adiabatic quantum computing (AQC), quantum annealing (QA), etc. However, it is always a difficult problem to improve the connection density between qubits, which will affect the efficiency of problem solving 9 , 10 . Coherent Ising Machine (CIM) is a quantum computer developed according to the optical principle 11 , 12 , 13 , 14 , 15 , 16 , 17 , which can work at room temperature and deal with large-scale problems, such as compression sensor problems 18 and polyhedron problems 19 . CIM uses laser pulses in optical fiber as qubits for quantum calculation. The early prototype of CIM is injection synchronous laser Ising machine. The number of coupled lasers in this scheme is proportional to the square of qubits, which is quite difficult. On this basis, optical delay linear CIM and measurement feedback CIM using nonlinear optical crystal instead of laser are developed. The latter uses measurement feedback to avoid the challenge that the former needs to control a large number of optical delay lines accurately 14 . The machine used in this study is measurement feedback CIM.

AGV scheduling problem can be understood as a kind of routing problem. Most traditional solutions to routing problems require sacrificing large amounts of computational resources and Osaba et al. 20 indicated that quantum computing techniques have great potential in the area of solving routing and optimization problems. In the early days, Goswami et al. 21 developed a phase estimation technique to solve the traveling salesman problem (TSP), using IBM’s quantum simulator to provide results for four city cases. Then, researchers tried to solve more complex problems with quantum computing. Feld et al. 22 presented a quadratic unconstrained binary optimization (QUBO) formulation for solving the vehicle routing problem with capacity constraints, evaluated the solution quality and computation time and compared it with classical solution methods. Bao et al. 23 proposed a two-stage QUBO formulation of the vehicle routing problem with balanced pickup, mapping the first stage to a clustering problem and describing the second stage as a TSP problem, and evaluated it against traditional methods in terms of numerical experimental results. Harwood et al. 24 tried to establish a qubo model to describe the vehicle routing problem by using the modeling idea of node and arc, and evaluated the model by using analog quantum devices. Geitz et al. 25 built a QUBO model to solve the job-shop scheduling problem, using quantum computers or simulators, constrained programming and tabu search. The calculation results proved the effectiveness of quantum computing in small-scale situations. And the established QUBO model can be extended to AGV scheduling problem. Ohzeki et al. 26 formulated an Ising model for the collision-free scheduling problem of AGVs within a factory setting. They utilized a quantum annealing machine to solve the model, with results demonstrating the potential application of quantum annealing machines in addressing real-world industrial challenges. Based on the above cases, it can be seen that some researchers have begun to use quantum computing to solve practical problems in the field of optimization. However, the research on quantum computing related to AGV scheduling has just started, and many researchers used simulators to solve them, because the current physical real machine resources are scarce, and the scale of solving problems is still relatively small, and it is easy to make mistakes and lacks the running data of physical real machines.

Our contributions

In a word, most of the existing AGV scheduling research adopts traditional models and methods, which can not effectively meet the actual needs of large-scale scheduling. Quantum computing has great application potential in solving specific problems that traditional computers cannot solve, and researchers have tried in optimization fields. However, as far as we know, there are few literatures about AGV scheduling using quantum computing technology. Based on these facts, the idea of carrying out this research came into being. The main contributions of this paper are summarized as follows.

In traditional research on the AGV scheduling problem, the computation time increases greatly with an increase in the number of AGVs and tasks. We introduce quantum computing technology into the research of the AGV scheduling problem and construct new QUBO models of AGV scheduling. In real scenarios, dispatchers often set different scheduling objectives according to the nature of the work, among which minimizing the total AGV travel time and minimizing the task completion time (makespan) are the two most common objectives. According to the different objectives, we have deduced different QUBO models, and given the model solutions and related theoretical basis under two different objectives.

We use traditional computer and CIM to carry out numerical experiments on the traditional model and QUBO model proposed by us respectively. The experimental results show that the computation speed of CIM is much faster than that of traditional computer, and the average calculation time is saved by 92%, which proves that CIM has great application potential in solving AGV scheduling problem and similar combinatorial optimization problems.

AGV scheduling model

AGV scheduling problems have many classifications according to different scenarios and considerations. For example, consider the time window of the task, joint optimization of scheduling and path, cooperation with other devices, charging strategy and so on. Due to the limitation of quantum bits of CIM, it is impossible to solve the AGV scheduling problem in complex scenes 27 , 28 , 29 . Therefore, we simplify the problem and keep the essence of AGV scheduling problem. On this basis, we construct the AGV scheduling model. In this section, we present the classical AGV scheduling model based on mixed integer programming (MIP), and propose two new models, which we call the node and arc models.

Problem description

figure 1

The AGV scheduling problem and a feasible solution. All AGVs start from a fixed start node, perform transportation tasks, and reach the end node after performing all tasks. ’S’ represents the starting point of a transportation task, and ’E’ represents the end point of a transportation task. Different colors represent different AGV’s mission routes.

We consider an AGV scheduling problem (to make the problem more general, we do not set up a working scenario), as shown in Fig.  1 . Given an AGV set, all AGVs have a unified starting node and ending node, and all AGVs start to accept tasks from the unified starting node until all transportation tasks are completed, and then return to the unified ending node. In the AGV scheduling problem, we are given a set of transportation tasks, each with a starting point and an ending point. For an AGV, the process of completing the transportation task can be described as first arriving at the starting point of the task to load the transported goods, then transporting them to the ending point of the task, and then driving to the starting point of the next task to perform the next task. The travel time of the AGV between any two task points is known. We consider two optimization objectives, the first is to minimize the total AGV travel time and the second is to minimize the maximum task completion time (makespan). The first objective is generally used when the task is not urgent to achieve a reduction in total system energy consumption, while the second objective is set to complete transportation tasks quickly.

Next, we elaborate on the symbolic settings in the problem as follows.

\(V=\{1,\ldots ,k,\ldots ,K \}\) set of AGVs,

\( R=\{1,\ldots ,r,{{r}^{'}},{{r}^{''}},\ldots ,n-1 \}\) set of actual tasks,

\({{R}_{1}}=\{0,1,\ldots ,r,{{r}^{'}},{{r}^{''}},\ldots ,n-1\}\) set of actual tasks and virtual start task,

\({{R}_{2}}=\{1,\ldots ,r,{{r}^{'}},{{r}^{''}},\ldots ,n\}\) set of actual tasks and virtual end task,

\({{R}^{'}}=\{0,1,\ldots ,r,{{r}^{'}},{{r}^{''}},\ldots ,n\}\)  set of all tasks,

\(A=\{(r,{{r}^{'}})\mid r,{{r}^{'}}\in {{R}^{'}}\}\) arc set that consists of all valid task pairs that can be conducted adjacently,

\(\pi =\{1,\ldots ,t,\ldots ,N \}\) set of the sequence of tasks performed by an AGV

a        a single task arc

\(\theta _{r}^{+}\)      an arc with task r as the left node

\(\theta _{r}^{-}\)      an arc with task r as the right node

\(r_{\text{s}}\)       starting point of task r ,

\(r_{\text{d}}\)       ending point of task r .

\({{e}_{{{r}_\text{s}}{{r}_\text{d}}}}\)     indicates the travel time of the AGV from two points \({{r}_\text{s}}\) and \({{r}_\text{d}}\) ,

\({{e}_{{{r}_\text{d}}r_\text{s}^{'}}}\)     indicates the travel time of the AGV from two points \({{r}_\text{d}}\) and \(r_\text{s}^{'}\) ,

\({{c}_{r{{r}^{'}}}}\)      contains two parts of time, the first part is the time from the start of task r to the end of task r , and the second part is the time from the end of task r to the start of task \({r}^{'}\) , \({{c}_{r{{r}^{'}}}}={{e}_{{{r}_{s}}{{r}_{d}}}}+{{e}_{{{r}_{d}}r_{s}^{'}}}\) .

Mixed integer programming model

In this subsection, we introduce the classical model of AGV scheduling. The classical model is formed as a mixed integer programming, and it has the following variables.

\({{y}_{r,{{r}^{'}},k}}\)       binary variable, equal to 1 if task r is performed directly prior to \({{r}^{'}}\) by AGV k , 0 otherwise;

\(f^\text{s}_{rk}\)          arrival time of AGV k at the start node of task r ;

\(f^\text{d}_{rk}\)          arrival time of AGV k at the end node of task r ;

T           the makespan for AGVs to perform transportation tasks.

The first optimization objective of the MIP model is to minimize the total AGV travel time, and its model is presented as follows.

The objective function ( 1 ) is to minimize the total travel time of the AGV. Constraints ( 2 ) and ( 3 ) ensure that all AGVs need to complete the virtual start task and the virtual end task, and constraints ( 4 ) guarantee that all actual tasks are uniquely assigned to a particular AGV. Constraints ( 5 ) ensure that each AGV completes its task satisfying the flow balance. Then, constraints ( 6 ) guarantee that the virtual start task starts and ends at moment 0. Constraints ( 7 ) states that the time to reach the end of a task is equal to the time to reach the start of that task plus the transport time from the start to the end. Constraints ( 8 ) indicates that the time to reach the start of a task is later than the time to reach the end of the previous task plus the transportation time required to reach the start of that task from the end of the previous task, where M is a sufficiently large value. The last constraints ( 9 ) eliminates task self-citation. Constraints ( 10 ) and ( 11 ) represent range limits of variables.

A slight modification of the above model can be used as a model for minimizing the makespan for AGVs to perform transportation tasks, which is given as follows, Eqs. ( 2 )–( 11 ).

where T denotes the makespan. The objective function is to minimize T , and constraints ( 13 ) denotes that T must be no less than the time required by the last AGV to complete the task.

QUBO and Ising model

This section mainly describes the concepts of QUBO model and Ising model and their relationship. QUBO is an expression of optimization problem, and its goal is to find binary variables that minimize quadratic polynomials. Ising model was first put forward and applied in statistical physics. It describes a system composed of interacting units, in which each spin particle must have two possible random states (such as + 1 and − 1), and then it was introduced into the field of mathematics as a model to describe a series of optimization problems. Many combinatorial optimization problems can be expressed in the form of quadratic unconstrained binary optimization or ising model, and they can be transformed into each other 30 , 31 , 32 . The general expression of QUBO model is shown in Eq. ( 14 ).

where x is a z dimensional vector of binary variables, Q is the quadratic coefficient matrix, and \({{c}^\texttt{T}}\) is the coefficient matrix of the primary term.

The above model in the form of QUBO can be easily transformed into an Ising model, and the variable range of the Ising model is \(\left\{ -1,1 \right\} \) . Specifically, it can be realized by variable substitution \({{\sigma }_{i}}=2{{x}_{i}}-1\) . Then, the optimization function can be expressed in the following form.

where the \({{\sigma }_{i}}\) is spin variable, \({{J}_{ij}}\) and \({{h}_{i}}\) are the quadratic and linear coefficients. The solution of Ising problem is to find the ground state of Hamiltonian. CIM solves the Ising problem according to the principle of minimum gain, and can find the ground state or low energy state of Ising Hamiltonian. The method is to map the QUBO problem into a fully connected Ising Hamiltonian with programmable parameters, and obtain the solution of the problem through controllable quantum phase transition 33 , 34 .

In this section, we will describe the node model. The core idea of node model is to regard tasks as nodes and the order of task execution as the order of vehicles passing through nodes.The node model has the QUBO form and is suitable for quantum computing, the variables of the model are described as below.

\({{x}_{r,t,k}}\) binary variable, equal to 1 if task r is assigned to AGV k as it is t -th task, 0 otherwise.

The first optimization objective of the node model is to minimize the total AGV travel time, and the model is shown below.

\({{\partial }_{i}}\) \((i=1,\ldots ,6)\) are weights correspond to each objective function. The objective function ( 17 ) is to minimize the total travel time of all the AGVs. The minimization function ( 18 ) and ( 19 ) ensures that for each AGV, the virtual start task and the virtual end task must be the first task and the last task, respectively. For each actual task, we want it to be assigned exactly to one AGV, so we add the minimization function ( 20 ). We also consider the minimization function ( 21 ) in order to make each AGV perform at most one task on each task sequence. For a particular AGV, the order of its tasks must be continuous, based on which we set the minimization function ( 22 ).

The model described below is modified from the above model to accommodate the goal of minimizing the makespan.

The least time-consuming task model requires finding the AGV with the longest task execution time and minimizing its execution task time, which leads to inequality constraints, as follows.

The objective function ( 24 ) is to minimize the makespan T .

Then we add slack variables to transform the above inequality constraint into an equation, as follows.

\({{T}_{k}}\) \((k\in V)\) is the slack variable, and both T and \({{T}_{k}}\) \((k\in V)\) need to be represented by binary variables. \({{\delta }_{i}}\) \( (i=1,2,\ldots ,m)\) and \({{\delta }_{ik}}\) \((i=1,2,\ldots ,{m}^{'}, {k}\in {V})\) are the discretized auxiliary variables we introduce, whose number is related to the size of the arithmetic case and needs to be estimated. A large number of auxiliary variables will make the difficulty of solving soar. In general, to represent an integer between 0 and \(\varsigma \) , \(\left[ {{\log }_{2}} \varsigma \right] +1\) discretized auxiliary variables need to be introduced, where \([\varsigma ]\) denotes the largest integer that does not exceed \(\varsigma \) . Of course, if there are non-integer values introduced in the calculation example, then it is necessary to introduce an approximate representation. First, we introduce the precision matrix as follows:

Then the real numbers \({{\varpi }_{i}}\) \((i=1,\ldots ,L)\) in some interval can be approximated as follows.

where \({{b}_{i}}\in {{\{0,1\}}^{L}}\) \((i=1,\ldots ,L)\) . If the error is expressed in terms of \(\phi \) , the error satisfies the following relation:

In this way, we can rewrite ( 24 ) and ( 25 ) as follows.

The positive real numbers can be approximated using integer auxiliary variables. \({{\sigma }_{j}}\) \((j=1,2\ldots ,L)\) and \({{\sigma }_{jk}}\) \((j=1,2,\ldots ,{{L}^{'}},k\in V)\) are used to approximate the decimal part. The number of binary variables used is related to the required approximate accuracy, as shown in Formula ( 28 ).

Thus, the model under this objective can be represented as follows, Eqs. ( 18 )–( 22 ), ( 24 )–( 25 ), ( 31 )–( 32 ).

In ( 33 ), \({{\varepsilon }_{i}}\) \((i=1\ldots ,7)\) are weights for each objectives. The model does not satisfy the QUBO form, because \({{H}_{Z}}\) is a quadrinomial binary polynomial, which needs to be degenerated. Next, we provide some analysis of the \({{H}_{Z}}\) . First, to make it easier to show our results, we perform a variable substitution, as follows:

The total number of tasks is N , and \(\eta \) contains \(({{N}^{2}}-3N+3)(N-1)\) monomials. Then \({{H}_{Z}}\) can be expanded as follows.

In the \({{H}_{Z}}\) , \({{\eta }^{2}}\) is a quadrinomial binary polynomial, \(-2\eta T\) and \(2\eta {{T}_{k}}\) are cubic binary polynomials, and \({{T}^{2}}\) , \(T_{k}^{2}\) and \(-2T{{T}_{k}}\) are all quadratic binary polynomials, so we need to reduce \({{\eta }^{2}}\) , \(-2\eta T\) and \(2\eta {{T}_{k}}\) . The number of quardrinomial binary monomials in \({{\eta }^{2}}\) is represented by \({{\tau }_{4}}\) , and \({{\tau }_{4}}\) is as follows:

where \({{\mathbb {Z}}^{+}}\) represents the set of all positive integers. Equation ( 37 ) indicates that any power of the binary variable itself is equal to itself, and thus, \(\tau _{4}^{'}\) quardrinomial monomial can be directly reduced, and its specific number is as follows.

Therefore, the number of quadrinomial polynomials that truly need to be reduced is \({{\tau }_{4}}-\tau _{4}^{'}\) terms. Looking at the part of cubic polynomial, the number of cubic monomials in \({{H}_{Z}}\) is represented by \({{\tau }_{3}}\) , and then \({{\tau }_{3}}\) is as follows:

Then, the number of all polynomials in \({{H}_{Z}}\) that need to be descended \(\tau \) is as follows:

At least \(\tau \) binary auxiliary variables must be introduced to complete the descending order according to a paper 35 . Due to the number of auxiliary variables introduced, the processing is more complex and it is difficult to calculate using existing quantum computers. So this model will not be introduced so far in this paper.

In this section we will describe the arc model. The core idea of arc model is that the sequence of tasks before and after execution is regarded as an arc connected between nodes, and building the model with arc as the basic unit can reduce the dimension. The arc model also has a QUBO form with the same parameter settings as the node model, and the decision variables are shown below.

\({{\upsilon }_{a,t,k}}\) binary variable, equal to 1 if arc a is assigned to AGV k in sequence t , 0 otherwise.

As with the node model, we first explore the model with the optimization objective of minimizing the total travel time.

\({{\beta }_{i}}\) \((i=1,\ldots ,8)\) are weights for the corresponding objective. The objective function ( 42 ) is to minimize the total travel time of the AGV. We want all AGVs to be executed in the first order a task arc that starts with a virtual start task, so we add the minimization function ( 43 ). The minimization function ( 44 ) means that the last task completed by each AGV must be a virtual end task. We want to complete at most one task arc in a certain order of an AGV, so we add the minimization function ( 45 ). The minimization function ( 46 ) indicates that for each actual task, we want a certain task arc with it as the starting node to be assigned to an AGV in a certain order of completion, and we want the virtual start task to be completed only once for each AGV, so we add the minimization function ( 47 ). Similarly, for each AGV, its virtual end task can only be completed once, so we add the minimization function ( 48 ). The minimization function ( 49 ) ensures that for each AGV, it is must to satisfy the flow balance when performing the task arc.

Next, we show a model with the goal of minimizing the makespan as follows, Eqs. ( 42 )–( 49 ).

Numerical experiments

In this section, we provide rich numerical experimental results, and research data can be obtained on public databases 36 . In the first subsection we use Gurobi 37 solver to solve the MIP model proposed above on a traditional computer, and show its computing performance under different problem scales. In the second subsection we use optical quantum computer to solve the problem cases of node model and arc model at different scales. And the computation performance is compared with that of traditional computers.

The CIM we used is provided by Beijing Qboson Quantum Technology Co.Ltd, and its structure and principle diagram are shown in Fig.  2 . The components of this CIM are mainly composed of optical parts and electrical parts. The optical part of the machine is composed of pulsed laser, erbium-doped fiber amplifier(EDFA), fiber rings and periodically poled lithium niobate (PPLN) crystals, while the electrical part is mainly composed of optical balanced homodyne detectors (BHD), analog-to-digital/digital-to-analog (AD/DA) converter and field-programmable gate array (FPGA). The laser emits laser with a repetition frequency of 100mhz, which is amplified by EDFA, and then the amplified laser frequency is doubled by PPLN crystal to generate 780 nm laser, which is used as the pump source to synchronously pump the phase sensitive amplifier, forming degenerate optical parametric oscillation(DOPO). There are 211 oscillation pulses in the fiber ring, and the time interval between adjacent pulses is 10 ns, so the transmission time of optical pulses in the ring is 2.11 µs. Then, the laser output in the fiber ring and the laser with the fundamental frequency of 1560 nm are determined by BHD, and the FPGA obtains the feedback signal of the next round trip according to the interaction intensity between spins in Ising Hamiltonian, which is used as the control signal of the intensity modulator (IM), and its sign defines the phase shift (0 or \(\pi \) ) of the phase modulator (PM) 9 , 14 , 34 , 38 .

To compare the performance between CIM and traditional computer, we also run our experiments with Gurobi 9.5.1 on a Mechrevo computer with 2.8 GHz Intel Core i7 CPU and 8GB memory, using up to four threads. The task points used in this experiment are randomly selected on the two-dimensional axis, ranging from 10 to 90, and then Euclidean distance is used as the length between two points, and we design the speed of each AGV to be constant, the time passing through the unit distance is the unit time.

figure 2

Schematic diagram of coherent Ising mechanism construction and principle.

Computing on a traditional PC

In this section, we use Gurobi to solve the mixed integer programming model of AGV scheduling for two optimization objectives. In “ Number of tasks ” we show experiments on the variation in computation time with the number of tasks, while in “ Number of AGVs ” we show experiments on the variation in computation time with the number of AGVs. We set a time limit to 1800 seconds for each run.

Number of tasks

In general, an increase in the number of tasks leads to a slower generation of AGV scheduling solutions. In this subsection, we investigate the effect of task number variation on the computational speed of the three models proposed in this paper. To achieve this goal, we generate instances of 4 tasks to 12 tasks with a fixed number of AGVs of 2 and obtain the computational time graphs shown in Fig.  3 , where the left figure takes the minimum total travel time as the objective function, and the right figure takes the minimum makespan as the objective function. The legend section represents the model number, which corresponds to the previous section number.

figure 3

Computation time versus number of tasks for MIP model. ( a ) Represents the change of MIP model computation time with the number of tasks under the goal of minimizing the total travel time. ( b ) Represents the change of MIP model computation time with the number of tasks under the goal of minimizing the makespan.

In Fig.  3 , we find that the computing speed of mixed integer programming model gradually slows down with the increase of the number of AGV tasks, and the computation time increases sharply when the number of tasks reaches a certain critical value, which is a common property reflected by two different objective functions. Especially when the number of tasks increases to 12, the computing time has exceeded 1800s, which reflects the weakness of traditional models in the face of large-scale problems.

Number of AGVs

In this subsection, we hope to explore the influence of the change of AGV number on the computing time of mixed integer programming model, so we fixed the number of tasks as 10 and 11, and set the number of AGVs in the range from 2 to 8. We show the results in Tables  1 and  2 . Notation “–” in the table implies that the corresponding model failed to obtain an optimal solution within the time limit. An instance with a tasks and b AGVs are denoted by “a–b”.

From the results obtained in Tables  1 and  2 , we can conclude that there is no strict correlation between the number of AGV and the computing time. Table  1 shows the computational performance of the three models under the objective of minimize the total travel time, and we can observe that the computational performance of the MIP model is very poor, and many groups of experiments failed to obtain an optimal solution within the limited time. Table   2 shows the computational performance of the MIP model under the objective of minimize the makespan, and the model performs much better, the optimal solution is obtained in the limited time in all groups of experiments. However, its computation time is still at a great disadvantage.

Computational experiment on CIM

In this subsection, we use the CIM to solve the QUBO model and compare its performance with the MIP solver on traditional PC. Since the maximum number of Quantum bits of the CIM used in this research is 100, all the comparative examples in this section limit the number of variables to 100. Based on this, we completed six groups of computation experiments with a quantum computer. In all the experiments, the number of AGV was limited to two. For the numerical experiment of node model, we set the number of tasks to 4 to 7, while for the numerical experiment of arc model, we fixed the number of tasks to 4.

figure 4

Evolution diagram of Hamiltonian with time under the objective functions of minimizing the total travel time in node model. ( a ) Represents the evolution diagram of Hamiltonian with time under the example of 4 tasks. ( b ) Represents the evolution diagram of Hamiltonian with time under the example of 5 tasks. ( c ) Represents the evolution diagram of Hamiltonian with time under the example of 6 tasks. ( d ) Represents the evolution diagram of Hamiltonian with time under the example of 7 tasks.

figure 5

Evolution diagram of Hamiltonian with time under two objective functions of arc mode. ( a ) represents the evolution diagram of Hamiltonian with time under the objective function of minimizing the total travel time. ( b ) Represents the evolution diagram of Hamiltonian with time under the objective function of minimizing the makespan.

In Figs.  4 and   5 , we plot the evolution of Ising Hamiltonian with time under node model and arc model, respectively. According to the above explanation of CIM principle construction, we can know that the time interval between every two adjacent data points in the figure is 2.11 microseconds. The Hamiltonian decreases with the passage of time, and the phase transition occurs as the power of the pump light gradually increases to the oscillation threshold. The solution obtained when reaching the lowest energy state is the result of CIM solution, and the corresponding time at this time is the computation time.

figure 6

Schematic diagram of quantum computing solutions for node model. ( a ) represents solution under 4 tasks. ( b ) represents solution under 5 tasks. ( c ) represents solution under 6 tasks. ( d ) represents solution under 7 tasks.

figure 7

Schematic diagram of quantum computing solutions for arc model. ( a ) represents the solution of 4 task under the objective function of minimizing the total time. ( b ) represents the solution of 4 task under the objective function of minimizing the makespan.

Figures  6 and 7 show the schematic diagrams of CIM’s solution under node model and arc model. Among them, Fig.  6 contains four parts, which respectively represent the schematic diagrams under 4-7 tasks, while Fig.  7 contains two parts, which respectively represent the schematic diagrams of solving two objective functions under 4 tasks. In fact, Ising model can be transformed into the corresponding representation of maximum cut problem 15 . The maximum cut problem is usually used as a measure and demonstration basis for the complexity of quantum computing problems and the distribution of solutions. The figure shows the solution of our problem expressed by the representation method of maximum cut problem. Points with different colors indicate that they are in different groups, and the connecting lines of points in the same group are gray, while those of points in different groups are red. In these figures, we can clearly perceive the complexity of each model at different scales.

Here, we compare the performance of node model and arc model on quantum computer with that of mixed integer programming model on traditional computer, and the comparison results are shown in Table  3 . An instance with a tasks and b AGVs are denoted by “a–b”.

Due to the limitation of hardware, the comparison of large-scale examples cannot be carried out. However, from Table  3 , we can see that the solutions obtained by CIM are all optimal solutions. And the CIM is much faster than the traditional computer in small-scale examples. We can observe that CIM has obvious performance advantages over traditional computers in small-scale examples. In particular, when the scale increases, the time required for CIM does not increase significantly as that of traditional computers. This shows that CIM has great development and application potential. In addition, there is little difference in computing performance between node model and arc model on quantum computer. Node model is slightly faster than arc model, but arc model is more universal than node model. In order to measure the improvement of CIM’s computing efficiency compared with the traditional computer in the given example, we propose the following computation formula.

IMP represents the calculation speed improvement rate, \(Q_{TRA}\) and \(Q_{CIM}\) respectively represent the computation time of traditional computer and CIM on the same example, and both node model and arc model participate in the comparison. After calculating the IMP of all examples, we find that the computation efficiency of CIM is \(92\%\) faster than that of traditional computers.

Conclusion and future research

We applied quantum computing technology to the research on AGV scheduling, and proposed QUBO models that adapts to solve the problem under two different criteria, minimizing total AGV travel time and makespan. Compared with the traditional MIP model, numerical experiments were carried out on traditional computers and CIM. The experimental results proved the superiority and great potential of quantum computing in this field. Of course, due to the limitation of hardware, there are still some shortcomings in this study, which can not show the advantages of quantum computing in large-scale situations. It is believed that with the continuous development of quantum computing technology, the outstanding performance of quantum computing will be demonstrated in solving large-scale problems in the future. In addition, we also summarized the situation that this study can expand, as follows.

First, the model can be improved and expanded. The model we considered above applies to AGVs with a uniform start node and a uniform end node. Realistic scenarios exist where AGVs have different start and end nodes, and our model is easily expandable for these types of cases. The solution is to involve the number of virtual start and end tasks based on the number of AGVs. Due to the difficulty of mapping large-scale optimization problems to the QUBO form with the small number of bits currently available for quantum computing, our proposed model is a pure scheduling problem that does not consider path optimization. Of course, subsequent researchers can consider this possibility in the case of mature technology. We believe that a two-layer planning model can be built on the basis of the existing model, where the scheduling and path planning problems are computationally solved alternately in two sub-models, which reduces both the modeling difficulty and the number of bits used in the solution of a single model.

Second, quantum computer and traditional computer can be combined to study AGV scheduling problem, and their respective characteristics can be better used to improve the efficiency of solving this problem.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.


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The authors would like to thank the team of Bose Quantum Technology Co., Ltd. for helping to obtain experimental data on a quantum computer, the members of this team include Kai Wen, Congyu Cao, Yin Ma and Hai Wei.

This work was partially supported by Grants (72372015, 71871038)from the National Natural Science Foundation of China, Liaoning Provincial Natural Science Foundation (2023-MS-125), and the General support project of China Postdoctoral Science Foundation (2019M661085), and the Humanities and Social Science Foundation of the Ministry of Education(21YJAZH070). Wei Wu was partially supported by JSPS KAKENHI [Grant No.~21K14367] and Industrial Technology Development Organization (NEDO) project (JPNP23003).

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L.T. has carried out research work division and personnel deployment. C.Y. and W.W are responsible for building the model, and C.Y. is also responsible for the numerical experiment of the classic computer. K.W and his team are responsible for numerical experiments and technical support related to quantum computing. Y.Y.G is responsible for providing guidance on relevant business background. Everyone participated in the writing and revision of the paper. Submission of a paper implies that the work described has not been published previously, that it is not under consideration for publication elsewhere and that its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out. Submission also implies that, if accepted, it will not be published elsewhere in the same form, in English or in any other language, without the written consent of the publisher.

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Tang, L., Yang, C., Wen, K. et al. Quantum computing for several AGV scheduling models. Sci Rep 14 , 12205 (2024). https://doi.org/10.1038/s41598-024-62821-6

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