what is in an introduction of a research paper

How to Write a Research Paper Introduction (with Examples)

The research paper introduction section, along with the Title and Abstract, can be considered the face of any research paper. The following article is intended to guide you in organizing and writing the research paper introduction for a quality academic article or dissertation.

The research paper introduction aims to present the topic to the reader. A study will only be accepted for publishing if you can ascertain that the available literature cannot answer your research question. So it is important to ensure that you have read important studies on that particular topic, especially those within the last five to ten years, and that they are properly referenced in this section. 1 What should be included in the research paper introduction is decided by what you want to tell readers about the reason behind the research and how you plan to fill the knowledge gap. The best research paper introduction provides a systemic review of existing work and demonstrates additional work that needs to be done. It needs to be brief, captivating, and well-referenced; a well-drafted research paper introduction will help the researcher win half the battle.

The introduction for a research paper is where you set up your topic and approach for the reader. It has several key goals:

• Present your research topic
• Capture reader interest
• Summarize existing research
• Position your own approach
• Define your specific research problem and problem statement
• Highlight the novelty and contributions of the study
• Give an overview of the paper’s structure

The research paper introduction can vary in size and structure depending on whether your paper presents the results of original empirical research or is a review paper. Some research paper introduction examples are only half a page while others are a few pages long. In many cases, the introduction will be shorter than all of the other sections of your paper; its length depends on the size of your paper as a whole.

Break through writer’s block. Write your research paper introduction with Paperpal’s Generative AI features

Table of Contents

What is the introduction for a research paper, what are the parts of introduction in the research, 1. introduce the research topic:, 2. determine a research niche:, 3. place your research within the research niche:, how to use paperpal to write the introduction section, frequently asked questions on research paper introduction, key points to remember.

The introduction in a research paper is placed at the beginning to guide the reader from a broad subject area to the specific topic that your research addresses. They present the following information to the reader

• Scope: The topic covered in the research paper
• Context: Background of your topic
• Importance: Why your research matters in that particular area of research and the industry problem that can be targeted

Why is the introduction important in a research paper?

The research paper introduction conveys a lot of information and can be considered an essential roadmap for the rest of your paper. A good introduction for a research paper is important for the following reasons:

• It stimulates your reader’s interest: A good introduction section can make your readers want to read your paper by capturing their interest. It informs the reader what they are going to learn and helps determine if the topic is of interest to them.
• It helps the reader understand the research background: Without a clear introduction, your readers may feel confused and even struggle when reading your paper. A good research paper introduction will prepare them for the in-depth research to come. It provides you the opportunity to engage with the readers and demonstrate your knowledge and authority on the specific topic.
• It explains why your research paper is worth reading: Your introduction can convey a lot of information to your readers. It introduces the topic, why the topic is important, and how you plan to proceed with your research.
• It helps guide the reader through the rest of the paper: The research paper introduction gives the reader a sense of the nature of the information that will support your arguments and the general organization of the paragraphs that will follow. It offers an overview of what to expect when reading the main body of your paper.

A good research paper introduction section should comprise three main elements: 2

• What is known: This sets the stage for your research. It informs the readers of what is known on the subject.
• What is lacking: This is aimed at justifying the reason for carrying out your research. This could involve investigating a new concept or method or building upon previous research.
• What you aim to do: This part briefly states the objectives of your research and its major contributions. Your detailed hypothesis will also form a part of this section.

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How to write a research paper introduction?

The first step in writing the research paper introduction is to inform the reader what your topic is and why it’s interesting or important. This is generally accomplished with a strong opening statement. The second step involves establishing the kinds of research that have been done and ending with limitations or gaps in the research that you intend to address. Finally, the research paper introduction clarifies how your own research fits in and what problem it addresses. If your research involved testing hypotheses, these should be stated along with your research question. The hypothesis should be presented in the past tense since it will have been tested by the time you are writing the research paper introduction.

The following key points, with examples, can guide you when writing the research paper introduction section:

• Highlight the importance of the research field or topic
• Describe the background of the topic
• Present an overview of current research on the topic

Example: The inclusion of experiential and competency-based learning has benefitted electronics engineering education. Industry partnerships provide an excellent alternative for students wanting to engage in solving real-world challenges. Industry-academia participation has grown in recent years due to the need for skilled engineers with practical training and specialized expertise. However, from the educational perspective, many activities are needed to incorporate sustainable development goals into the university curricula and consolidate learning innovation in universities.

• Reveal a gap in existing research or oppose an existing assumption
• Formulate the research question

Example: There have been plausible efforts to integrate educational activities in higher education electronics engineering programs. However, very few studies have considered using educational research methods for performance evaluation of competency-based higher engineering education, with a focus on technical and or transversal skills. To remedy the current need for evaluating competencies in STEM fields and providing sustainable development goals in engineering education, in this study, a comparison was drawn between study groups without and with industry partners.

• State the purpose of your study
• Highlight the key characteristics of your study
• Describe important results
• Highlight the novelty of the study.
• Offer a brief overview of the structure of the paper.

Example: The study evaluates the main competency needed in the applied electronics course, which is a fundamental core subject for many electronics engineering undergraduate programs. We compared two groups, without and with an industrial partner, that offered real-world projects to solve during the semester. This comparison can help determine significant differences in both groups in terms of developing subject competency and achieving sustainable development goals.

Write a Research Paper Introduction in Minutes with Paperpal

Paperpal is a generative AI-powered academic writing assistant. It’s trained on millions of published scholarly articles and over 20 years of STM experience. Paperpal helps authors write better and faster with:

• Real-time writing suggestions
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With Paperpal, create a research paper introduction effortlessly. In this step-by-step guide, we’ll walk you through how Paperpal transforms your initial ideas into a polished and publication-ready introduction.

Step 1: Sign up on Paperpal and click on the Copilot feature, under this choose Outlines > Research Article > Introduction

Step 2: Add your unstructured notes or initial draft, whether in English or another language, to Paperpal, which is to be used as the base for your content.

Step 3: Fill in the specifics, such as your field of study, brief description or details you want to include, which will help the AI generate the outline for your Introduction.

Step 4: Use this outline and sentence suggestions to develop your content, adding citations where needed and modifying it to align with your specific research focus.

Step 5: Turn to Paperpal’s granular language checks to refine your content, tailor it to reflect your personal writing style, and ensure it effectively conveys your message.

You can use the same process to develop each section of your article, and finally your research paper in half the time and without any of the stress.

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The purpose of the research paper introduction is to introduce the reader to the problem definition, justify the need for the study, and describe the main theme of the study. The aim is to gain the reader’s attention by providing them with necessary background information and establishing the main purpose and direction of the research.

The length of the research paper introduction can vary across journals and disciplines. While there are no strict word limits for writing the research paper introduction, an ideal length would be one page, with a maximum of 400 words over 1-4 paragraphs. Generally, it is one of the shorter sections of the paper as the reader is assumed to have at least a reasonable knowledge about the topic. 2 For example, for a study evaluating the role of building design in ensuring fire safety, there is no need to discuss definitions and nature of fire in the introduction; you could start by commenting upon the existing practices for fire safety and how your study will add to the existing knowledge and practice.

When deciding what to include in the research paper introduction, the rest of the paper should also be considered. The aim is to introduce the reader smoothly to the topic and facilitate an easy read without much dependency on external sources. 3 Below is a list of elements you can include to prepare a research paper introduction outline and follow it when you are writing the research paper introduction. Topic introduction: This can include key definitions and a brief history of the topic. Research context and background: Offer the readers some general information and then narrow it down to specific aspects. Details of the research you conducted: A brief literature review can be included to support your arguments or line of thought. Rationale for the study: This establishes the relevance of your study and establishes its importance. Importance of your research: The main contributions are highlighted to help establish the novelty of your study Research hypothesis: Introduce your research question and propose an expected outcome. Organization of the paper: Include a short paragraph of 3-4 sentences that highlights your plan for the entire paper

Cite only works that are most relevant to your topic; as a general rule, you can include one to three. Note that readers want to see evidence of original thinking. So it is better to avoid using too many references as it does not leave much room for your personal standpoint to shine through. Citations in your research paper introduction support the key points, and the number of citations depend on the subject matter and the point discussed. If the research paper introduction is too long or overflowing with citations, it is better to cite a few review articles rather than the individual articles summarized in the review. A good point to remember when citing research papers in the introduction section is to include at least one-third of the references in the introduction.

The literature review plays a significant role in the research paper introduction section. A good literature review accomplishes the following: Introduces the topic – Establishes the study’s significance – Provides an overview of the relevant literature – Provides context for the study using literature – Identifies knowledge gaps However, remember to avoid making the following mistakes when writing a research paper introduction: Do not use studies from the literature review to aggressively support your research Avoid direct quoting Do not allow literature review to be the focus of this section. Instead, the literature review should only aid in setting a foundation for the manuscript.

Remember the following key points for writing a good research paper introduction: 4

• Avoid stuffing too much general information: Avoid including what an average reader would know and include only that information related to the problem being addressed in the research paper introduction. For example, when describing a comparative study of non-traditional methods for mechanical design optimization, information related to the traditional methods and differences between traditional and non-traditional methods would not be relevant. In this case, the introduction for the research paper should begin with the state-of-the-art non-traditional methods and methods to evaluate the efficiency of newly developed algorithms.
• Avoid packing too many references: Cite only the required works in your research paper introduction. The other works can be included in the discussion section to strengthen your findings.
• Avoid extensive criticism of previous studies: Avoid being overly critical of earlier studies while setting the rationale for your study. A better place for this would be the Discussion section, where you can highlight the advantages of your method.
• Avoid describing conclusions of the study: When writing a research paper introduction remember not to include the findings of your study. The aim is to let the readers know what question is being answered. The actual answer should only be given in the Results and Discussion section.

To summarize, the research paper introduction section should be brief yet informative. It should convince the reader the need to conduct the study and motivate him to read further. If you’re feeling stuck or unsure, choose trusted AI academic writing assistants like Paperpal to effortlessly craft your research paper introduction and other sections of your research article.

1. Jawaid, S. A., & Jawaid, M. (2019). How to write introduction and discussion. Saudi Journal of Anaesthesia, 13(Suppl 1), S18.

2. Dewan, P., & Gupta, P. (2016). Writing the title, abstract and introduction: Looks matter!. Indian pediatrics, 53, 235-241.

3. Cetin, S., & Hackam, D. J. (2005). An approach to the writing of a scientific Manuscript1. Journal of Surgical Research, 128(2), 165-167.

4. Bavdekar, S. B. (2015). Writing introduction: Laying the foundations of a research paper. Journal of the Association of Physicians of India, 63(7), 44-6.

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• If you are writing in a new discipline, you should always make sure to ask about conventions and expectations for introductions, just as you would for any other aspect of the essay. For example, while it may be acceptable to write a two-paragraph (or longer) introduction for your papers in some courses, instructors in other disciplines, such as those in some Government courses, may expect a shorter introduction that includes a preview of the argument that will follow.  
• In some disciplines (Government, Economics, and others), it’s common to offer an overview in the introduction of what points you will make in your essay. In other disciplines, you will not be expected to provide this overview in your introduction.  
• Avoid writing a very general opening sentence. While it may be true that “Since the dawn of time, people have been telling love stories,” it won’t help you explain what’s interesting about your topic.  
• Avoid writing a “funnel” introduction in which you begin with a very broad statement about a topic and move to a narrow statement about that topic. Broad generalizations about a topic will not add to your readers’ understanding of your specific essay topic.  
• Avoid beginning with a dictionary definition of a term or concept you will be writing about. If the concept is complicated or unfamiliar to your readers, you will need to define it in detail later in your essay. If it’s not complicated, you can assume your readers already know the definition.  
• Avoid offering too much detail in your introduction that a reader could better understand later in the paper.
• picture_as_pdf Introductions

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How to write an effective introduction for your research paper

Last updated

20 January 2024

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However, the introduction is a vital element of your research paper . It helps the reader decide whether your paper is worth their time. As such, it's worth taking your time to get it right.

In this article, we'll tell you everything you need to know about writing an effective introduction for your research paper.

• The importance of an introduction in research papers

The primary purpose of an introduction is to provide an overview of your paper. This lets readers gauge whether they want to continue reading or not. The introduction should provide a meaningful roadmap of your research to help them make this decision. It should let readers know whether the information they're interested in is likely to be found in the pages that follow.

Aside from providing readers with information about the content of your paper, the introduction also sets the tone. It shows readers the style of language they can expect, which can further help them to decide how far to read.

When you take into account both of these roles that an introduction plays, it becomes clear that crafting an engaging introduction is the best way to get your paper read more widely. First impressions count, and the introduction provides that impression to readers.

• The optimum length for a research paper introduction

While there's no magic formula to determine exactly how long a research paper introduction should be, there are a few guidelines. Some variables that impact the ideal introduction length include:

Field of study

Complexity of the topic

Specific requirements of the course or publication

A commonly recommended length of a research paper introduction is around 10% of the total paper’s length. So, a ten-page paper has a one-page introduction. If the topic is complex, it may require more background to craft a compelling intro. Humanities papers tend to have longer introductions than those of the hard sciences.

The best way to craft an introduction of the right length is to focus on clarity and conciseness. Tell the reader only what is necessary to set up your research. An introduction edited down with this goal in mind should end up at an acceptable length.

• Evaluating successful research paper introductions

A good way to gauge how to create a great introduction is by looking at examples from across your field. The most influential and well-regarded papers should provide some insights into what makes a good introduction.

Dissecting examples: what works and why

We can make some general assumptions by looking at common elements of a good introduction, regardless of the field of research.

A common structure is to start with a broad context, and then narrow that down to specific research questions or hypotheses. This creates a funnel that establishes the scope and relevance.

The most effective introductions are careful about the assumptions they make regarding reader knowledge. By clearly defining key terms and concepts instead of assuming the reader is familiar with them, these introductions set a more solid foundation for understanding.

To pull in the reader and make that all-important good first impression, excellent research paper introductions will often incorporate a compelling narrative or some striking fact that grabs the reader's attention.

Finally, good introductions provide clear citations from past research to back up the claims they're making. In the case of argumentative papers or essays (those that take a stance on a topic or issue), a strong thesis statement compels the reader to continue reading.

Common pitfalls to avoid in research paper introductions

You can also learn what not to do by looking at other research papers. Many authors have made mistakes you can learn from.

We've talked about the need to be clear and concise. Many introductions fail at this; they're verbose, vague, or otherwise fail to convey the research problem or hypothesis efficiently. This often comes in the form of an overemphasis on background information, which obscures the main research focus.

Ensure your introduction provides the proper emphasis and excitement around your research and its significance. Otherwise, fewer people will want to read more about it.

• Crafting a compelling introduction for a research paper

Let’s take a look at the steps required to craft an introduction that pulls readers in and compels them to learn more about your research.

Step 1: Capturing interest and setting the scene

To capture the reader's interest immediately, begin your introduction with a compelling question, a surprising fact, a provocative quote, or some other mechanism that will hook readers and pull them further into the paper.

As they continue reading, the introduction should contextualize your research within the current field, showing readers its relevance and importance. Clarify any essential terms that will help them better understand what you're saying. This keeps the fundamentals of your research accessible to all readers from all backgrounds.

Step 2: Building a solid foundation with background information

Including background information in your introduction serves two major purposes:

It helps to clarify the topic for the reader

It establishes the depth of your research

The approach you take when conveying this information depends on the type of paper.

For argumentative papers, you'll want to develop engaging background narratives. These should provide context for the argument you'll be presenting.

For empirical papers, highlighting past research is the key. Often, there will be some questions that weren't answered in those past papers. If your paper is focused on those areas, those papers make ideal candidates for you to discuss and critique in your introduction.

Step 3: Pinpointing the research challenge

To capture the attention of the reader, you need to explain what research challenges you'll be discussing.

For argumentative papers, this involves articulating why the argument you'll be making is important. What is its relevance to current discussions or problems? What is the potential impact of people accepting or rejecting your argument?

For empirical papers, explain how your research is addressing a gap in existing knowledge. What new insights or contributions will your research bring to your field?

Step 4: Clarifying your research aims and objectives

We mentioned earlier that the introduction to a research paper can serve as a roadmap for what's within. We've also frequently discussed the need for clarity. This step addresses both of these.

When writing an argumentative paper, craft a thesis statement with impact. Clearly articulate what your position is and the main points you intend to present. This will map out for the reader exactly what they'll get from reading the rest.

For empirical papers, focus on formulating precise research questions and hypotheses. Directly link them to the gaps or issues you've identified in existing research to show the reader the precise direction your research paper will take.

Step 5: Sketching the blueprint of your study

Continue building a roadmap for your readers by designing a structured outline for the paper. Guide the reader through your research journey, explaining what the different sections will contain and their relationship to one another.

This outline should flow seamlessly as you move from section to section. Creating this outline early can also help guide the creation of the paper itself, resulting in a final product that's better organized. In doing so, you'll craft a paper where each section flows intuitively from the next.

Step 6: Integrating your research question

To avoid letting your research question get lost in background information or clarifications, craft your introduction in such a way that the research question resonates throughout. The research question should clearly address a gap in existing knowledge or offer a new perspective on an existing problem.

Tell users your research question explicitly but also remember to frequently come back to it. When providing context or clarification, point out how it relates to the research question. This keeps your focus where it needs to be and prevents the topic of the paper from becoming under-emphasized.

Step 7: Establishing the scope and limitations

So far, we've talked mostly about what's in the paper and how to convey that information to readers. The opposite is also important. Information that's outside the scope of your paper should be made clear to the reader in the introduction so their expectations for what is to follow are set appropriately.

Similarly, be honest and upfront about the limitations of the study. Any constraints in methodology, data, or how far your findings can be generalized should be fully communicated in the introduction.

Step 8: Concluding the introduction with a promise

The final few lines of the introduction are your last chance to convince people to continue reading the rest of the paper. Here is where you should make it very clear what benefit they'll get from doing so. What topics will be covered? What questions will be answered? Make it clear what they will get for continuing.

By providing a quick recap of the key points contained in the introduction in its final lines and properly setting the stage for what follows in the rest of the paper, you refocus the reader's attention on the topic of your research and guide them to read more.

• Research paper introduction best practices

Following the steps above will give you a compelling introduction that hits on all the key points an introduction should have. Some more tips and tricks can make an introduction even more polished.

As you follow the steps above, keep the following tips in mind.

Set the right tone and style

Like every piece of writing, a research paper should be written for the audience. That is to say, it should match the tone and style that your academic discipline and target audience expect. This is typically a formal and academic tone, though the degree of formality varies by field.

Kno w the audience

The perfect introduction balances clarity with conciseness. The amount of clarification required for a given topic depends greatly on the target audience. Knowing who will be reading your paper will guide you in determining how much background information is required.

Adopt the CARS (create a research space) model

The CARS model is a helpful tool for structuring introductions. This structure has three parts. The beginning of the introduction establishes the general research area. Next, relevant literature is reviewed and critiqued. The final section outlines the purpose of your study as it relates to the previous parts.

Master the art of funneling

The CARS method is one example of a well-funneled introduction. These start broadly and then slowly narrow down to your specific research problem. It provides a nice narrative flow that provides the right information at the right time. If you stray from the CARS model, try to retain this same type of funneling.

Incorporate narrative element

People read research papers largely to be informed. But to inform the reader, you have to hold their attention. A narrative style, particularly in the introduction, is a great way to do that. This can be a compelling story, an intriguing question, or a description of a real-world problem.

Write the introduction last

By writing the introduction after the rest of the paper, you'll have a better idea of what your research entails and how the paper is structured. This prevents the common problem of writing something in the introduction and then forgetting to include it in the paper. It also means anything particularly exciting in the paper isn’t neglected in the intro.

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Ertugrul Portakal

Apr 12, 2024

Writing a Research Paper Introduction (with 3 Examples)

Nail your research paper's introduction! Learn to captivate and inform readers from the start—our guide shows how!

TABLE OF CONTENTS

A catchy and informative introduction is essential in academic writing, especially if you want your readers to have background information about your paper. However, writing an interesting and informative introduction can sometimes be a time-consuming and tiring process. If you don't know where to start when crafting an introduction, no need to worry - we've got you covered!

In this article, we will explain step by step what an introduction is in academic writing and how to write it!

Ready? Let's start!

• An introduction is a paragraph that provides information about your entire paper and aims to attract and inform the reader.
• Before writing an introduction or even starting your paper, you need to research academic sources.
• The first one or two sentences of an introduction paragraph should be a hook to attract the reader's attention.
• Afterwards, you need to prepare the reader for your argument by giving background information about your topic.
• Finally, you should state your argument about your topic with a thesis statement.
• If you are writing a longer paper, you can inform your readers about the map of your paper.
• If you are looking for an AI assistant to support you throughout your writing process, TextCortex is designed for you with its advanced features.

What is an Introduction in a research paper?

In any academic writing, including essays and research papers, an introduction is the first paragraph that the reader will encounter. This paragraph should both attract the reader's attention and give them the necessary information about the paper. In any academic paper, the introduction paragraph constitutes 10% of the paper's total word count. For example, if you are preparing a 3,000-word paper, your introduction paragraph should consist of approximately 300 words. You should also write sentences within these 300 words that will attract the reader's attention and provide them with information about the paper.

Importance of an Introduction Paragraph

The biggest function of an introduction paragraph is to prepare the reader for the author's thesis statement. A traditional introduction paragraph begins with a few sentences or questions that will catch the reader's attention. After attracting the reader's attention, necessary background information on the subject is given. Finally, the author explains to the readers what the whole paper is about by stating the thesis. A thesis statement is the final sentence that summarizes the main points of your paper and conveys your claim.

First Things First: Preliminary Research

When working on any academic writing type, it is essential to start by researching your topic thoroughly before beginning to type. What sets academic writing apart from other writing types is the requirement for it to be written using accurate information from reliable sources.

Researching academic sources can be a time-consuming and unnecessary process. One has to read through hundreds of pages, review dozens of articles and verify the accuracy of each source. However, if you're looking to reduce your workload and maximize efficiency by automating repetitive tasks such as literature review, ZenoChat is the perfect solution for you. With its web search feature, ZenoChat can use the entire internet as a data source. Additionally, by activating the "scholar" option of the ZenoChat web search feature, you can ensure that it only uses academic sources when generating output.

How to Create an Introduction for Academic Writing?

Creating an introduction paragraph that is interesting, informative, and conveys your thesis is an easier process than it seems. As long as you have sufficient information about your topic and an outline , you can write engaging introductions by following a few simple steps. Let's take a closer look at how to write an introduction for academic writing.

1-) Start with a Catchy Hook

Your first sentence is one of the factors that most influence a reader's decision to read your paper. This sentence determines the tone of your paper and attracts the reader's attention. For this reason, we recommend that you start your introduction paragraph with a strong and catchy hook sentence.

• Avoid long and complex sentences
• Use clear and concise sentences
• Write a sentence that will spark the reader's curiosity
• You can ask questions that will encourage the reader to read the remaining paragraph
• Avoid fact or overly broad sentences
• Avoid using dictionary definitions as your hook

2-) Give Background Information

After writing a strong hook sentence, you need to provide basic information about your topic so that the reader can understand what they will learn about when they read your paper. In this section, you can benefit from opinions that support or oppose your argument. Additionally, this section should refer to the body paragraphs of your writing.

• You can write a background information sentence for each body paragraph.
• The information here should be concise and compact
• Avoid talking about your evidence and results unless necessary.

3-) State Your Thesis 

After attracting the reader's attention and providing background information, it is time to present your approach and argument towards the topic with a thesis statement. A thesis statement usually comprises one or two sentences and communicates the paper's argument to the reader. A well-written thesis statement should express your stance on the topic.

• Avoid merely stating a fact
• Claim your argument

4-) Tell Reader About Your Paper

Although you need to move on to body paragraphs after the thesis statement in short papers, it will be useful to add a few sentences that will guide the reader in your longer papers. This way, your readers can better understand which arguments they will encounter on which pages and the course of your paper. That leads the reader to clearly understand and follow your content.

Let’s Wrap it Up

Writing an interesting and informative introduction is usually a long process that requires a lot of rewriting. You may need to rewrite a sentence dozens of times so that your words and sentences clearly describe your paper and argument. Fortunately, you can generate state-of-the-art introductions using AI tools and use them with a little editing.

When it comes to text generation, paraphrasing, and grammar & spelling checking, TextCortex is the way to go with its advanced LLMs and customization options. With TextCortex, you can generate all writing types, including introduction, from scratch, rewrite your existing texts, change their tone of voice, or fix their grammar. TextCortex is available as a web application and browser extension. The TextCortex browser extension is integrated with 30,000+ websites and apps. So, you can complete your AI-driven writing tasks anywhere and anytime.

Let's examine a few sample introductions generated by TextCortex.

Example Introduction #1

“Should social media platforms be banned from collecting their users' data?”

Example Introduction #2

“Do electric vehicles decrease overall emissions?”

Example Introduction #3

“Is graffiti an act of vandalism or the creation of art?”

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Microsoft 365 Life Hacks > Writing > How to write an introduction for a research paper

How to write an introduction for a research paper

How do you start a research paper when the words are hard to find? Students and pros alike struggle with how to do this effectively. We’ve got you covered—read on to brush up on the purpose of a research paper introduction, what it should include, and how to write a good one.

What is an introduction in a research paper?

It may seem obvious, but introductions are the introductory (first) paragraph of a piece of writing, in this case, a research paper. Intros should give readers a “sneak peek” of what to expect and should include the following:

• A scope. This defines the topic you’ll be covering.
• Context. The background of your topic
• Importance. The “so what?” to your research—why it matters.

Your introduction will need to cover a lot of ground, but it will only be a half-page to a few pages long. The length depends on the size of your paper. In many cases, the introduction will be shorter than all the other sections of your paper.

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What is the point of an introduction?

The introduction to your research paper does far more than you may think. It’s the selling point to your readers. Your readers won’t know the full breadth of what your research paper is about from the title alone, so that’s the entire purpose of an introduction.

A good introduction:

• Helps your reader understand your topic’s background
• Explains why a research paper is worth reading
• Offers guidance for navigating the rest of the piece
• Piques your reader’s interest (contains a hook)

Without a clear introduction, your readers will struggle—if they don’t have the right context, they may feel confused when they start reading your paper. They might even give up entirely. Your introduction will ground them and prepare them for the in-depth research to come.

What should an introduction include?

All research paper introductions should strive to be engaging and informative, but your introduction will never be complete if it doesn’t contain these six essential items:

A clear overview of the topic.

Begin by providing a general overview of your topic. Gradually narrow your focus until you address the specific subject of your paper.

Prior research

Your introduction is the place to review what is currently known (and not known) about your research paper topic. This background information also establishes your credibility for readers. What are the relevant gaps in scholarship on your topic that readers should know about? Reference both older material and current scholars as appropriate.

The rationale for your research paper is the problem you are trying to solve or the question you aim to answer. Explain why your topic needs to be addressed right now. In other words, your introduction needs to answer the question, “So what?” Why should readers care about your topic?

Methodologies

Outline your processes to correctly document your methodology, whether it’s for a grade, peer review, or publication. Lay out your goal and the questions you will address. Reveal how you conducted research and describe how you measured results. Explain why you made key choices. For example, did your research project involve a control group? What methods did you use to gather your data? Examples could include a focus group, questionnaires, or a case study.

Thesis statement

Your main introduction should end with a thesis statement. This statement summarizes the ideas that will run through your entire research article. It’s what your research paper is trying to prove or argue.

Introductions often conclude with an outline of your article. Your layout should quickly review what you intend to cover in the following sections. Think of it as a roadmap that tells your reader what to expect throughout the paper.

How you use these components in your intro depends on your field. For example, a physics research paper might emphasize methodology since quantitative methods and labs are often used versus a humanities-geared paper.

Three tips for writing your introduction

Here are three tips to keep in mind as you write your intro, if you want it to shine:

Tip 1: Write your introduction last.

An introduction summarizes all the things you’ve learned from your research. While it can feel good to get your preface done quickly, you should write the rest of your paper first. Then, you’ll find it easy to create a clear overview.

Tip 2: Include a strong quotation or story upfront.

You want your paper to be full of substance. But that doesn’t mean it should feel boring or flat. Add a relevant quotation or surprising anecdote to the beginning of your introduction. This technique will pique the interest of your readers and leave them wanting more.

Tip 3: Be concise.

Research papers cover complex topics. To help your readers, try to write as clearly as possible. Use concise sentences. Check for confusing grammar or syntax . Read your introduction aloud to catch awkward phrases. Before you finish your paper, be sure to proofread, too.

Remember: the introduction is your reader’s first impression of your research paper, so you’ll want to dress your intro paragraphs as best as you can. Explore more tips and tools you can use to wow readers, like Microsoft Editor .

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How to write an introduction for a research paper

Writing an introduction for a research paper can be one of the hardest parts of the writing process. How do you get started? In this post, we discuss the components of an introduction and explore strategies for writing one successfully.

What is an introduction?

The introduction to a research paper provides background information or context on the topic. It also includes the thesis statement and signposts that let the reader know what you will cover in the rest of the paper.

Depending on the type of research paper that you’re writing, you may also include a brief state of the field in your introduction. You might also put that in a separate section, called a literature review. Before you tackle writing your introduction, be sure to consult the assignment guidelines for your paper.

How to write an introduction

An introduction provides an overview of your topic and any background information that your readers need to know in order to understand the context. It generally concludes with an explicit statement of your position on the topic, which is known as your thesis statement.

The opening section

Many papers begin with a hook: a short anecdote or scenario that draws the reader in and gives a hint of what the paper will cover. A hook allows you to capture your reader’s attention and provides an anchor for the context that you will provide in the bulk of the introduction.

Most of your introduction should be taken up with background information, but this doesn’t mean that you should fill your opening section with overly general statements. Instead, provide key pieces of information (like statistics) that a reader would need to know in order to understand your main argument.

The thesis statement

Towards the end of the introduction, you should state your thesis, preferably in the form of "I argue that..." or "This paper argues that..." or a similar phrase. Although it’s called a “thesis statement,” your thesis can be more than one sentence.

Finally, an introduction contains a brief outline or "signposts" of what the rest of the article will cover (also known as forecasting statements). You can use language like, “in what follows,” or “in the rest of the paper,” to signal that you are describing what you’ll do in the remainder of the paper.

Tips for writing an introduction

1. don’t rely on generalizations.

An introduction is not simply filler. It has a very specific function in a research paper: to provide context that leads up to a thesis statement.

You may be tempted to start your paper with generalizations like, “many people believe that...” or, “in our society...,” or a general dictionary definition, because you’re not sure what kind of context to provide. Instead, use specific facts like statistics or historical anecdotes to open your paper.

2. State your thesis directly

Once you’ve provided the appropriate, and specific, background information on your topic, you can move on to stating your thesis. As a rule of thumb, state your thesis as directly as possible. Use phrases like “I argue that..” to indicate that you are laying out your main argument.

3. Include signposts

A strong introduction includes clear signposts that outline what you will cover in the rest of the paper. You can signal this by using words like, “in what follows,” and by describing the steps that you will take to build your argument.

4. Situate your argument within the scholarly conversation

Some types of research papers require a separate literature review in which you explore what others have written about your topic.

Even if you’re not required to have a formal literature review, you should still include at least a paragraph in which you engage with the scholarly debate on your chosen subject. Be sure to include direct quotes from your sources . You can use BibGuru’s citation generator to create accurate in-text citations for your quotes.

This section can come directly before your thesis statement or directly after it. In the former case, your state of the field will function as additional context for your thesis.

Frequently Asked Questions about how to write an introduction for a research paper

A good introduction provides specific background information on your topic, sets up your thesis statement, and includes signposts for what you’ll cover in the rest of the paper.

An introduction should include context, a thesis statement, and signposts.

Do not include generalizations, apologies for not being an expert, or dictionary definitions in your introduction.

The length of your introduction depends on the overall length of your paper. For instance, an introduction for an 8-10 page paper will likely be anywhere from 1-3 pages.

You can choose to start an introduction with a hook, an important statistic, an historical anecdote, or another specific piece of background information.

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Citing sources in the introduction, "introduction checklist" from: how to write a good scientific paper. chris a. mack. spie. 2018..

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This is where you describe briefly and clearly why you are writing the paper. The introduction supplies sufficient background information for the reader to understand and evaluate the experiment you did. It also supplies a rationale for the study.

• Present the problem and the proposed solution
• Presents nature and scope of the problem investigated
• Reviews the pertinent literature to orient the reader
• States the method of the experiment
• State the principle results of the experiment

It is important to cite sources in the introduction section of your paper as evidence of the claims you are making. There are ways of citing sources in the text so that the reader can find the full reference in the literature cited section at the end of the paper, yet the flow of the reading is not badly interrupted. Below are some example of how this can be done:     "Smith (1983) found that N-fixing plants could be infected by several different species of Rhizobium."     "Walnut trees are known to be allelopathic (Smith 1949,  Bond et al. 1955, Jones and Green 1963)."     "Although the presence of Rhizobium normally increases the growth of legumes (Nguyen 1987), the opposite effect has been observed (Washington 1999)." Note that articles by one or two authors are always cited in the text using their last names. However, if there are more than two authors, the last name of the 1st author is given followed by the abbreviation et al. which is Latin for "and others". 

From:  https://writingcenter.gmu.edu/guides/imrad-reports-introductions

• Indicate the field of the work, why this field is important, and what has already been done (with proper citations).
• Indicate a gap, raise a research question, or challenge prior work in this territory.
• Outline the purpose and announce the present research, clearly indicating what is novel and why it is significant.
• Avoid: repeating the abstract; providing unnecessary background information; exaggerating the importance of the work; claiming novelty without a proper literature search. 
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How to Write an Introduction for a Research Paper

How to write an introduction for a research paper? Eventually (and with practice) all writers will develop their own strategy for writing the perfect introduction for a research paper. Once you are comfortable with writing, you will probably find your own, but coming up with a good strategy can be tough for beginning writers.

The Purpose of an Introduction

Your opening paragraphs, phrases for introducing thesis statements, research paper introduction examples, using the introduction to map out your research paper.

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• First write your thesis.Your thesis should state the main idea in specific terms.
• After you have a working thesis, tackle the body of your paper before you write the rest of the introduction. Each paragraph in the body should explore one specific topic that proves, or summarizes your thesis. Writing is a thinking process. Once you have worked your way through that process by writing the body of the paper, you will have an intimate understanding of how you are supporting your thesis. After you have written the body paragraphs, go back and rewrite your thesis to make it more specific and to connect it to the topics you addressed in the body paragraph.
• Revise your introduction several times, saving each revision. Be sure your introduction previews the topics you are presenting in your paper. One way of doing this is to use keywords from the topic sentences in each paragraph to introduce, or preview, the topics in your introduction.This “preview” will give your reader a context for understanding how you will make your case.
• Experiment by taking different approaches to your thesis with every revision you make. Play with the language in the introduction. Strike a new tone. Go back and compare versions. Then pick the one that works most effectively with the body of your research paper.
• Do not try to pack everything you want to say into your introduction. Just as your introduction should not be too short, it should also not be too long. Your introduction should be about the same length as any other paragraph in your research paper. Let the content—what you have to say—dictate the length.

The first page of your research paper should draw the reader into the text. It is the paper’s most important page and, alas, often the worst written. There are two culprits here and effective ways to cope with both of them.

First, the writer is usually straining too hard to say something terribly BIG and IMPORTANT about the thesis topic. The goal is worthy, but the aim is unrealistically high. The result is often a muddle of vague platitudes rather than a crisp, compelling introduction to the thesis. Want a familiar example? Listen to most graduation speakers. Their goal couldn’t be loftier: to say what education means and to tell an entire football stadium how to live the rest of their lives. The results are usually an avalanche of clichés and sodden prose.

The second culprit is bad timing. The opening and concluding paragraphs are usually written late in the game, after the rest of the thesis is finished and polished. There’s nothing wrong with writing these sections last. It’s usually the right approach since you need to know exactly what you are saying in the substantive middle sections of the thesis before you can introduce them effectively or draw together your findings. But having waited to write the opening and closing sections, you need to review and edit them several times to catch up. Otherwise, you’ll putting the most jagged prose in the most tender spots. Edit and polish your opening paragraphs with extra care. They should draw readers into the paper.

After you’ve done some extra polishing, I suggest a simple test for the introductory section. As an experiment, chop off the first few paragraphs. Let the paper begin on, say, paragraph 2 or even page 2. If you don’t lose much, or actually gain in clarity and pace, then you’ve got a problem.

There are two solutions. One is to start at this new spot, further into the text. After all, that’s where you finally gain traction on your subject. That works best in some cases, and we occasionally suggest it. The alternative, of course, is to write a new opening that doesn’t flop around, saying nothing.

What makes a good opening? Actually, they come in several flavors. One is an intriguing story about your topic. Another is a brief, compelling quote. When you run across them during your reading, set them aside for later use. Don’t be deterred from using them because they “don’t seem academic enough.” They’re fine as long as the rest of the paper doesn’t sound like you did your research in People magazine. The third, and most common, way to begin is by stating your main questions, followed by a brief comment about why they matter.

Whichever opening you choose, it should engage your readers and coax them to continue. Having done that, you should give them a general overview of the project—the main issues you will cover, the material you will use, and your thesis statement (that is, your basic approach to the topic). Finally, at the end of the introductory section, give your readers a brief road map, showing how the paper will unfold. How you do that depends on your topic but here are some general suggestions for phrase choice that may help:

• This analysis will provide …
• This paper analyzes the relationship between …
• This paper presents an analysis of …
• This paper will argue that …
• This topic supports the argument that…
• Research supports the opinion that …
• This paper supports the opinion that …
• An interpretation of the facts indicates …
• The results of this experiment show …
• The results of this research show …

Comparisons/Contrasts

• A comparison will show that …
• By contrasting the results,we see that …
• This paper examines the advantages and disadvantages of …

Definitions/Classifications

• This paper will provide a guide for categorizing the following:…
• This paper provides a definition of …
• This paper explores the meaning of …
• This paper will discuss the implications of …
• A discussion of this topic reveals …
• The following discussion will focus on …

Description

• This report describes…
• This report will illustrate…
• This paper provides an illustration of …

Process/Experimentation

• This paper will identify the reasons behind…
• The results of the experiment show …
• The process revealed that …
• This paper theorizes…
• This paper presents the theory that …
• In theory, this indicates that …

Quotes, anecdotes, questions, examples, and broad statements—all of them can used successfully to write an introduction for a research paper. It’s instructive to see them in action, in the hands of skilled academic writers.

Let’s begin with David M. Kennedy’s superb history, Freedom from Fear: The American People in Depression and War, 1929–1945 . Kennedy begins each chapter with a quote, followed by his text. The quote above chapter 1 shows President Hoover speaking in 1928 about America’s golden future. The text below it begins with the stock market collapse of 1929. It is a riveting account of just how wrong Hoover was. The text about the Depression is stronger because it contrasts so starkly with the optimistic quotation.

“We in America today are nearer the final triumph over poverty than ever before in the history of any land.”—Herbert Hoover, August 11, 1928 Like an earthquake, the stock market crash of October 1929 cracked startlingly across the United States, the herald of a crisis that was to shake the American way of life to its foundations. The events of the ensuing decade opened a fissure across the landscape of American history no less gaping than that opened by the volley on Lexington Common in April 1775 or by the bombardment of Sumter on another April four score and six years later. The ratcheting ticker machines in the autumn of 1929 did not merely record avalanching stock prices. In time they came also to symbolize the end of an era. (David M. Kennedy, Freedom from Fear: The American People in Depression and War, 1929–1945 . New York: Oxford University Press, 1999, p. 10)

Kennedy has exciting, wrenching material to work with. John Mueller faces the exact opposite problem. In Retreat from Doomsday: The Obsolescence of Major War , he is trying to explain why Great Powers have suddenly stopped fighting each other. For centuries they made war on each other with devastating regularity, killing millions in the process. But now, Mueller thinks, they have not just paused; they have stopped permanently. He is literally trying to explain why “nothing is happening now.” That may be an exciting topic intellectually, it may have great practical significance, but “nothing happened” is not a very promising subject for an exciting opening paragraph. Mueller manages to make it exciting and, at the same time, shows why it matters so much. Here’s his opening, aptly entitled “History’s Greatest Nonevent”:

On May 15, 1984, the major countries of the developed world had managed to remain at peace with each other for the longest continuous stretch of time since the days of the Roman Empire. If a significant battle in a war had been fought on that day, the press would have bristled with it. As usual, however, a landmark crossing in the history of peace caused no stir: the most prominent story in the New York Times that day concerned the saga of a manicurist, a machinist, and a cleaning woman who had just won a big Lotto contest. This book seeks to develop an explanation for what is probably the greatest nonevent in human history. (John Mueller, Retreat from Doomsday: The Obsolescence of Major War . New York: Basic Books, 1989, p. 3)

In the space of a few sentences, Mueller sets up his puzzle and reveals its profound human significance. At the same time, he shows just how easy it is to miss this milestone in the buzz of daily events. Notice how concretely he does that. He doesn’t just say that the New York Times ignored this record setting peace. He offers telling details about what they covered instead: “a manicurist, a machinist, and a cleaning woman who had just won a big Lotto contest.” Likewise, David Kennedy immediately entangles us in concrete events: the stunning stock market crash of 1929. These are powerful openings that capture readers’ interests, establish puzzles, and launch narratives.

Sociologist James Coleman begins in a completely different way, by posing the basic questions he will study. His ambitious book, Foundations of Social Theory , develops a comprehensive theory of social life, so it is entirely appropriate for him to begin with some major questions. But he could just as easily have begun with a compelling story or anecdote. He includes many of them elsewhere in his book. His choice for the opening, though, is to state his major themes plainly and frame them as a paradox. Sociologists, he says, are interested in aggregate behavior—how people act in groups, organizations, or large numbers—yet they mostly examine individuals:

A central problem in social science is that of accounting for the function of some kind of social system. Yet in most social research, observations are not made on the system as a whole, but on some part of it. In fact, the natural unit of observation is the individual person…  This has led to a widening gap between theory and research… (James S. Coleman, Foundations of Social Theory . Cambridge, MA: Harvard University Press, 1990, pp. 1–2)

After expanding on this point, Coleman explains that he will not try to remedy the problem by looking solely at groups or aggregate-level data. That’s a false solution, he says, because aggregates don’t act; individuals do. So the real problem is to show the links between individual actions and aggregate outcomes, between the micro and the macro.

The major problem for explanations of system behavior based on actions and orientations at a level below that of the system [in this case, on individual-level actions] is that of moving from the lower level to the system level. This has been called the micro-to-macro problem, and it is pervasive throughout the social sciences. (Coleman, Foundations of Social Theory , p. 6)

Explaining how to deal with this “micro-to-macro problem” is the central issue of Coleman’s book, and he announces it at the beginning.

Coleman’s theory-driven opening stands at the opposite end of the spectrum from engaging stories or anecdotes, which are designed to lure the reader into the narrative and ease the path to a more analytic treatment later in the text. Take, for example, the opening sentences of Robert L. Herbert’s sweeping study Impressionism: Art, Leisure, and Parisian Society : “When Henry Tuckerman came to Paris in 1867, one of the thousands of Americans attracted there by the huge international exposition, he was bowled over by the extraordinary changes since his previous visit twenty years before.” (Robert L. Herbert, Impressionism: Art, Leisure, and Parisian Society . New Haven, CT: Yale University Press, 1988, p. 1.) Herbert fills in the evocative details to set the stage for his analysis of the emerging Impressionist art movement and its connection to Parisian society and leisure in this period.

David Bromwich writes about Wordsworth, a poet so familiar to students of English literature that it is hard to see him afresh, before his great achievements, when he was just a young outsider starting to write. To draw us into Wordsworth’s early work, Bromwich wants us to set aside our entrenched images of the famous mature poet and see him as he was in the 1790s, as a beginning writer on the margins of society. He accomplishes this ambitious task in the opening sentences of Disowned by Memory: Wordsworth’s Poetry of the 1790s :

Wordsworth turned to poetry after the revolution to remind himself that he was still a human being. It was a curious solution, to a difficulty many would not have felt. The whole interest of his predicament is that he did feel it. Yet Wordsworth is now so established an eminence—his name so firmly fixed with readers as a moralist of self-trust emanating from complete self-security—that it may seem perverse to imagine him as a criminal seeking expiation. Still, that is a picture we get from The Borderers and, at a longer distance, from “Tintern Abbey.” (David Bromwich, Disowned by Memory: Wordsworth’s Poetry of the 1790s . Chicago: University of Chicago Press, 1998, p. 1)

That’s a wonderful opening! Look at how much Bromwich accomplishes in just a few words. He not only prepares the way for analyzing Wordsworth’s early poetry; he juxtaposes the anguished young man who wrote it to the self-confident, distinguished figure he became—the eminent man we can’t help remembering as we read his early poetry.

Let us highlight a couple of other points in this passage because they illustrate some intelligent writing choices. First, look at the odd comma in this sentence: “It was a curious solution, to a difficulty many would not have felt.” Any standard grammar book would say that comma is wrong and should be omitted. Why did Bromwich insert it? Because he’s a fine writer, thinking of his sentence rhythm and the point he wants to make. The comma does exactly what it should. It makes us pause, breaking the sentence into two parts, each with an interesting point. One is that Wordsworth felt a difficulty others would not have; the other is that he solved it in a distinctive way. It would be easy for readers to glide over this double message, so Bromwich has inserted a speed bump to slow us down. Most of the time, you should follow grammatical rules, like those about commas, but you should bend them when it serves a good purpose. That’s what the writer does here.

The second small point is the phrase “after the revolution” in the first sentence: “Wordsworth turned to poetry after the revolution to remind himself that he was still a human being.” Why doesn’t Bromwich say “after the French Revolution”? Because he has judged his book’s audience. He is writing for specialists who already know which revolution is reverberating through English life in the 1790s. It is the French Revolution, not the earlier loss of the American colonies. If Bromwich were writing for a much broader audience—say, the New York Times Book Review—he would probably insert the extra word to avoid confusion.

The message “Know your audience” applies to all writers. Don’t talk down to them by assuming they can’t get dressed in the morning. Don’t strut around showing off your book learnin’ by tossing in arcane facts and esoteric language for its own sake. Neither will win over readers.

Bromwich, Herbert, and Coleman open their works in different ways, but their choices work well for their different texts. Your task is to decide what kind of opening will work best for yours. Don’t let that happen by default, by grabbing the first idea you happen upon. Consider a couple of different ways of opening your thesis and then choose the one you prefer. Give yourself some options, think them over, then make an informed choice.

Whether you begin with a story, puzzle, or broad statement, the next part of the introduction should pose your main questions and establish your argument. This is your thesis statement—your viewpoint along with the supporting reasons and evidence. It should be articulated plainly so readers understand full well what your paper is about and what it will argue.

After that, give your readers a road map of what’s to come. That’s normally done at the end of the introductory section (or, in a book, at the end of the introductory chapter). Here’s John J. Mearsheimer presenting such a road map in The Tragedy of Great Power Politics . He not only tells us the order of upcoming chapters, he explains why he’s chosen that order and which chapters are most important:

The Plan of the Book The rest of the chapters in this book are concerned mainly with answering the six big questions about power which I identified earlier. Chapter 2, which is probably the most important chapter in the book, lays out my theory of why states compete for power and why they pursue hegemony. In Chapters 3 and 4, I define power and explain how to measure it. I do this in order to lay the groundwork for testing my theory… (John J. Mearsheimer, The Tragedy of Great Power Politics . New York: W. W. Norton, 2001, p. 27)

As this excerpt makes clear, Mearsheimer has already laid out his “six big questions” in the introduction. Now he’s showing us the path ahead, the path to answering those questions.

At the end of the introduction, give your readers a road map of what’s to come. Tell them what the upcoming sections will be and why they are arranged in this particular order.

After having written your introduction it’s time to move to the biggest part: body of a research paper.

Back to How To Write A Research Paper .

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How to Write a Research Introduction

Last Updated: December 6, 2023 Fact Checked

This article was co-authored by Megan Morgan, PhD . Megan Morgan is a Graduate Program Academic Advisor in the School of Public & International Affairs at the University of Georgia. She earned her PhD in English from the University of Georgia in 2015. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 2,657,632 times.

The introduction to a research paper can be the most challenging part of the paper to write. The length of the introduction will vary depending on the type of research paper you are writing. An introduction should announce your topic, provide context and a rationale for your work, before stating your research questions and hypothesis. Well-written introductions set the tone for the paper, catch the reader's interest, and communicate the hypothesis or thesis statement.

Introducing the Topic of the Paper

• In scientific papers this is sometimes known as an "inverted triangle", where you start with the broadest material at the start, before zooming in on the specifics. [2] X Research source
• The sentence "Throughout the 20th century, our views of life on other planets have drastically changed" introduces a topic, but does so in broad terms.
• It provides the reader with an indication of the content of the essay and encourages them to read on.

• For example, if you were writing a paper about the behaviour of mice when exposed to a particular substance, you would include the word "mice", and the scientific name of the relevant compound in the first sentences.
• If you were writing a history paper about the impact of the First World War on gender relations in Britain, you should mention those key words in your first few lines.

• This is especially important if you are attempting to develop a new conceptualization that uses language and terminology your readers may be unfamiliar with.

• If you use an anecdote ensure that is short and highly relevant for your research. It has to function in the same way as an alternative opening, namely to announce the topic of your research paper to your reader.
• For example, if you were writing a sociology paper about re-offending rates among young offenders, you could include a brief story of one person whose story reflects and introduces your topic.
• This kind of approach is generally not appropriate for the introduction to a natural or physical sciences research paper where the writing conventions are different.

Establishing the Context for Your Paper

• It is important to be concise in the introduction, so provide an overview on recent developments in the primary research rather than a lengthy discussion.
• You can follow the "inverted triangle" principle to focus in from the broader themes to those to which you are making a direct contribution with your paper.
• A strong literature review presents important background information to your own research and indicates the importance of the field.

• By making clear reference to existing work you can demonstrate explicitly the specific contribution you are making to move the field forward.
• You can identify a gap in the existing scholarship and explain how you are addressing it and moving understanding forward.

• For example, if you are writing a scientific paper you could stress the merits of the experimental approach or models you have used.
• Stress what is novel in your research and the significance of your new approach, but don't give too much detail in the introduction.
• A stated rationale could be something like: "the study evaluates the previously unknown anti-inflammatory effects of a topical compound in order to evaluate its potential clinical uses".

Specifying Your Research Questions and Hypothesis

• The research question or questions generally come towards the end of the introduction, and should be concise and closely focused.
• The research question might recall some of the key words established in the first few sentences and the title of your paper.
• An example of a research question could be "what were the consequences of the North American Free Trade Agreement on the Mexican export economy?"
• This could be honed further to be specific by referring to a particular element of the Free Trade Agreement and the impact on a particular industry in Mexico, such as clothing manufacture.
• A good research question should shape a problem into a testable hypothesis.

• If possible try to avoid using the word "hypothesis" and rather make this implicit in your writing. This can make your writing appear less formulaic.
• In a scientific paper, giving a clear one-sentence overview of your results and their relation to your hypothesis makes the information clear and accessible. [10] X Trustworthy Source PubMed Central Journal archive from the U.S. National Institutes of Health Go to source
• An example of a hypothesis could be "mice deprived of food for the duration of the study were expected to become more lethargic than those fed normally".

• This is not always necessary and you should pay attention to the writing conventions in your discipline.
• In a natural sciences paper, for example, there is a fairly rigid structure which you will be following.
• A humanities or social science paper will most likely present more opportunities to deviate in how you structure your paper.

Research Introduction Help

Community Q&A

• Use your research papers' outline to help you decide what information to include when writing an introduction. Thanks Helpful 0 Not Helpful 1
• Consider drafting your introduction after you have already completed the rest of your research paper. Writing introductions last can help ensure that you don't leave out any major points. Thanks Helpful 0 Not Helpful 0

• Avoid emotional or sensational introductions; these can create distrust in the reader. Thanks Helpful 51 Not Helpful 12
• Generally avoid using personal pronouns in your introduction, such as "I," "me," "we," "us," "my," "mine," or "our." Thanks Helpful 32 Not Helpful 7
• Don't overwhelm the reader with an over-abundance of information. Keep the introduction as concise as possible by saving specific details for the body of your paper. Thanks Helpful 25 Not Helpful 14

You Might Also Like

• ↑ https://library.sacredheart.edu/c.php?g=29803&p=185916
• ↑ https://www.aresearchguide.com/inverted-pyramid-structure-in-writing.html
• ↑ https://libguides.usc.edu/writingguide/introduction
• ↑ https://writing.wisc.edu/Handbook/PlanResearchPaper.html
• ↑ https://dept.writing.wisc.edu/wac/writing-an-introduction-for-a-scientific-paper/
• ↑ https://writing.wisc.edu/handbook/assignments/planresearchpaper/
• ↑ http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3178846/

About This Article

To introduce your research paper, use the first 1-2 sentences to describe your general topic, such as “women in World War I.” Include and define keywords, such as “gender relations,” to show your reader where you’re going. Mention previous research into the topic with a phrase like, “Others have studied…”, then transition into what your contribution will be and why it’s necessary. Finally, state the questions that your paper will address and propose your “answer” to them as your thesis statement. For more information from our English Ph.D. co-author about how to craft a strong hypothesis and thesis, keep reading! Did this summary help you? Yes No

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Frequently asked questions

What should i include in a research paper introduction.

The introduction of a research paper includes several key elements:

• A hook to catch the reader’s interest
• Relevant background on the topic
• Details of your research problem

and your problem statement

• A thesis statement or research question
• Sometimes an overview of the paper

Frequently asked questions: Writing a research paper

A research project is an academic, scientific, or professional undertaking to answer a research question . Research projects can take many forms, such as qualitative or quantitative , descriptive , longitudinal , experimental , or correlational . What kind of research approach you choose will depend on your topic.

The best way to remember the difference between a research plan and a research proposal is that they have fundamentally different audiences. A research plan helps you, the researcher, organize your thoughts. On the other hand, a dissertation proposal or research proposal aims to convince others (e.g., a supervisor, a funding body, or a dissertation committee) that your research topic is relevant and worthy of being conducted.

Formulating a main research question can be a difficult task. Overall, your question should contribute to solving the problem that you have defined in your problem statement .

However, it should also fulfill criteria in three main areas:

• Researchability
• Feasibility and specificity
• Relevance and originality

Research questions anchor your whole project, so it’s important to spend some time refining them.

In general, they should be:

• Focused and researchable
• Answerable using credible sources
• Complex and arguable
• Feasible and specific
• Relevant and original

All research questions should be:

• Focused on a single problem or issue
• Researchable using primary and/or secondary sources
• Feasible to answer within the timeframe and practical constraints
• Specific enough to answer thoroughly
• Complex enough to develop the answer over the space of a paper or thesis
• Relevant to your field of study and/or society more broadly

A research aim is a broad statement indicating the general purpose of your research project. It should appear in your introduction at the end of your problem statement , before your research objectives.

Research objectives are more specific than your research aim. They indicate the specific ways you’ll address the overarching aim.

Once you’ve decided on your research objectives , you need to explain them in your paper, at the end of your problem statement .

Keep your research objectives clear and concise, and use appropriate verbs to accurately convey the work that you will carry out for each one.

I will compare …

Your research objectives indicate how you’ll try to address your research problem and should be specific:

Research objectives describe what you intend your research project to accomplish.

They summarize the approach and purpose of the project and help to focus your research.

Your objectives should appear in the introduction of your research paper , at the end of your problem statement .

The main guidelines for formatting a paper in Chicago style are to:

• Use a standard font like 12 pt Times New Roman
• Use 1 inch margins or larger
• Apply double line spacing
• Indent every new paragraph ½ inch
• Include a title page
• Place page numbers in the top right or bottom center
• Cite your sources with author-date citations or Chicago footnotes
• Include a bibliography or reference list

To automatically generate accurate Chicago references, you can use Scribbr’s free Chicago reference generator .

The main guidelines for formatting a paper in MLA style are as follows:

• Use an easily readable font like 12 pt Times New Roman
• Set 1 inch page margins
• Include a four-line MLA heading on the first page
• Center the paper’s title
• Use title case capitalization for headings
• Cite your sources with MLA in-text citations
• List all sources cited on a Works Cited page at the end

To format a paper in APA Style , follow these guidelines:

• Use a standard font like 12 pt Times New Roman or 11 pt Arial
• If submitting for publication, insert a running head on every page
• Apply APA heading styles
• Cite your sources with APA in-text citations
• List all sources cited on a reference page at the end

No, it’s not appropriate to present new arguments or evidence in the conclusion . While you might be tempted to save a striking argument for last, research papers follow a more formal structure than this.

All your findings and arguments should be presented in the body of the text (more specifically in the results and discussion sections if you are following a scientific structure). The conclusion is meant to summarize and reflect on the evidence and arguments you have already presented, not introduce new ones.

The conclusion of a research paper has several key elements you should make sure to include:

• A restatement of the research problem
• A summary of your key arguments and/or findings
• A short discussion of the implications of your research

Don’t feel that you have to write the introduction first. The introduction is often one of the last parts of the research paper you’ll write, along with the conclusion.

This is because it can be easier to introduce your paper once you’ve already written the body ; you may not have the clearest idea of your arguments until you’ve written them, and things can change during the writing process .

The way you present your research problem in your introduction varies depending on the nature of your research paper . A research paper that presents a sustained argument will usually encapsulate this argument in a thesis statement .

A research paper designed to present the results of empirical research tends to present a research question that it seeks to answer. It may also include a hypothesis —a prediction that will be confirmed or disproved by your research.

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Starting Your Research Paper: Writing an Introductory Paragraph

• Choosing Your Topic
• Define Keywords
• Planning Your Paper
• Writing an Introductory Paragraph

The Dreaded Introductory Paragraph

Writing the introductory paragraph can be a frustrating and slow process -- but it doesn't have to be.  If you planned your paper out, then most of the introductory paragraph is already written.  Now you just need a beginning and an end.

Here's an introductory paragraph for a paper I wrote.  I started the paper with a factoid, then presented each main point of my paper and then ended with my thesis statement.

  Breakdown:

• << Previous: Planning Your Paper
• Last Updated: Feb 12, 2024 12:16 PM
• URL: https://libguides.astate.edu/papers

Organizing Academic Research Papers: 4. The Introduction

• Purpose of Guide
• Design Flaws to Avoid
• Glossary of Research Terms
• Narrowing a Topic Idea
• Broadening a Topic Idea
• Extending the Timeliness of a Topic Idea
• Academic Writing Style
• Choosing a Title
• Making an Outline
• Paragraph Development
• Executive Summary
• Background Information
• The Research Problem/Question
• Theoretical Framework
• Citation Tracking
• Content Alert Services
• Evaluating Sources
• Primary Sources
• Secondary Sources
• Tertiary Sources
• What Is Scholarly vs. Popular?
• Qualitative Methods
• Quantitative Methods
• Using Non-Textual Elements
• Limitations of the Study
• Common Grammar Mistakes
• Avoiding Plagiarism
• Footnotes or Endnotes?
• Further Readings
• Annotated Bibliography
• Dealing with Nervousness
• Using Visual Aids
• Grading Someone Else's Paper
• How to Manage Group Projects
• Multiple Book Review Essay
• Reviewing Collected Essays
• About Informed Consent
• Writing Field Notes
• Writing a Policy Memo
• Writing a Research Proposal
• Acknowledgements

The introduction serves the purpose of leading the reader from a general subject area to a particular field of research. It establishes the context of the research being conducted by summarizing current understanding and background information about the topic, stating the purpose of the work in the form of the hypothesis, question, or research problem, briefly explaining your rationale, methodological approach, highlighting the potential outcomes your study can reveal, and describing the remaining structure of the paper.

Key Elements of the Research Proposal. Prepared under the direction of the Superintendent and by the 2010 Curriculum Design and Writing Team. Baltimore County Public Schools.

Importance of a Good Introduction

Think of the introduction as a mental road map that must answer for the reader these four questions:

• What was I studying?
• Why was this topic important to investigate?
• What did we know about this topic before I did this study?
• How will this study advance our knowledge?

A well-written introduction is important because, quite simply, you never get a second chance to make a good first impression. The opening paragraph of your paper will provide your readers with their initial impressions about the logic of your argument, your writing style, the overall quality of your research, and, ultimately, the validity of your findings and conclusions. A vague, disorganized, or error-filled introduction will create a negative impression, whereas, a concise, engaging, and well-written introduction will start your readers off thinking highly of your analytical skills, your writing style, and your research approach.

Introductions . The Writing Center. University of North Carolina.

Structure and Writing Style

I. Structure and Approach

The introduction is the broad beginning of the paper that answers three important questions for the reader:

• What is this?
• Why am I reading it?
• What do you want me to think about / consider doing / react to?

Think of the structure of the introduction as an inverted triangle of information. Organize the information so as to present the more general aspects of the topic early in the introduction, then narrow toward the more specific topical information that provides context, finally arriving at your statement of purpose and rationale and, whenever possible, the potential outcomes your study can reveal.

These are general phases associated with writing an introduction:

• Highlighting the importance of the topic, and/or
• Making general statements about the topic, and/or
• Presenting an overview on current research on the subject.
• Opposing an existing assumption, and/or
• Revealing a gap in existing research, and/or
• Formulating a research question or problem, and/or
• Continuing a disciplinary tradition.
• Stating the intent of your study,
• Outlining the key characteristics of your study,
• Describing important results, and
• Giving a brief overview of the structure of the paper.

NOTE: Even though the introduction is the first main section of a research paper, it is often useful to finish the introduction very late in the writing process because the structure of the paper, the reporting and analysis of results, and the conclusion will have been completed and it ensures that your introduction matches the overall structure of your paper.

II.  Delimitations of the Study

Delimitations refer to those characteristics that limit the scope and define the conceptual boundaries of your study . This is determined by the conscious exclusionary and inclusionary decisions you make about how to investigate the research problem. In other words, not only should you tell the reader what it is you are studying and why, but you must also acknowledge why you rejected alternative approaches that could have been used to examine the research problem.

Obviously, the first limiting step was the choice of research problem itself. However, implicit are other, related problems that could have been chosen but were rejected. These should be noted in the conclusion of your introduction.

Examples of delimitating choices would be:

• The key aims and objectives of your study,
• The research questions that you address,
• The variables of interest [i.e., the various factors and features of the phenomenon being studied],
• The method(s) of investigation, and
• Any relevant alternative theoretical frameworks that could have been adopted.

Review each of these decisions. You need to not only clearly establish what you intend to accomplish, but to also include a declaration of what the study does not intend to cover. In the latter case, your exclusionary decisions should be based upon criteria stated as, "not interesting"; "not directly relevant"; “too problematic because..."; "not feasible," and the like. Make this reasoning explicit!

NOTE: Delimitations refer to the initial choices made about the broader, overall design of your study and should not be confused with documenting the limitations of your study discovered after the research has been completed.

III. The Narrative Flow

Issues to keep in mind that will help the narrative flow in your introduction :

• Your introduction should clearly identify the subject area of interest . A simple strategy to follow is to use key words from your title in the first few sentences of the introduction. This will help focus the introduction on the topic at the appropriate level and ensures that you get to the primary subject matter quickly without losing focus, or discussing information that is too general.
• Establish context by providing a brief and balanced review of the pertinent published literature that is available on the subject. The key is to summarize for the reader what is known about the specific research problem before you did your analysis. This part of your introduction should not represent a comprehensive literature review but consists of a general review of the important, foundational research literature (with citations) that lays a foundation for understanding key elements of the research problem. See the drop-down tab for "Background Information" for types of contexts.
• Clearly state the hypothesis that you investigated . When you are first learning to write in this format it is okay, and actually preferable, to use a past statement like, "The purpose of this study was to...." or "We investigated three possible mechanisms to explain the...."
• Why did you choose this kind of research study or design? Provide a clear statement of the rationale for your approach to the problem studied. This will usually follow your statement of purpose in the last paragraph of the introduction.

IV. Engaging the Reader

The overarching goal of your introduction is to make your readers want to read your paper. The introduction should grab your reader's attention. Strategies for doing this can be to:

• Open with a compelling story,
• Include a strong quotation or a vivid, perhaps unexpected anecdote,
• Pose a provocative or thought-provoking question,
• Describe a puzzling scenario or incongruity, or
• Cite a stirring example or case study that illustrates why the research problem is important.

NOTE:   Only choose one strategy for engaging your readers; avoid giving an impression that your paper is more flash than substance.

Freedman, Leora  and Jerry Plotnick. Introductions and Conclusions . University College Writing Centre. University of Toronto; Introduction . The Structure, Format, Content, and Style of a Journal-Style Scientific Paper. Department of Biology. Bates College; Introductions . The Writing Center. University of North Carolina; Introductions . The Writer’s Handbook. Writing Center. University of Wisconsin, Madison; Introductions, Body Paragraphs, and Conclusions for an Argument Paper. The Writing Lab and The OWL. Purdue University; Resources for Writers: Introduction Strategies . Program in Writing and Humanistic Studies. Massachusetts Institute of Technology; Sharpling, Gerald. Writing an Introduction . Centre for Applied Linguistics, University of Warwick; Writing Your Introduction. Department of English Writing Guide. George Mason University.

Writing Tip

Avoid the "Dictionary" Introduction

Giving the dictionary definition of words related to the research problem may appear appropriate because it is important to define specific words or phrases with which readers may be unfamiliar. However, anyone can look a word up in the dictionary and a general dictionary is not a particularly authoritative source. It doesn't take into account the context of your topic and doesn't offer particularly detailed information. Also, placed in the context of a particular discipline, a term may have a different meaning than what is found in a general dictionary. If you feel that you must seek out an authoritative definition, try to find one that is from subject specific dictionaries or encyclopedias [e.g., if you are a sociology student, search for dictionaries of sociology].

Saba, Robert. The College Research Paper . Florida International University; Introductions . The Writing Center. University of North Carolina.

Another Writing Tip

When Do I Begin?

A common question asked at the start of any paper is, "where should I begin?" An equally important question to ask yourself is, "When do I begin?" Research problems in the social sciences rarely rest in isolation from the history of the issue being investigated. It is, therefore, important to lay a foundation for understanding the historical context underpinning the research problem. However, this information should be brief and succinct and begin at a point in time that best informs the reader of study's overall importance. For example, a study about coffee cultivation and export in West Africa as a key stimulus for local economic growth needs to describe the beginning of exporting coffee in the region and establishing why economic growth is important. You do not need to give a long historical explanation about coffee exportation in Africa. If a research problem demands a substantial exploration of historical context, do this in the literature review section; note in the introduction as part of your "roadmap" [see below] that you covering this in the literature review.

Yet Another Writing Tip

Always End with a Roadmap

The final paragraph or sentences of your introduction should forecast your main arguments and conclusions and provide a description of the rest of the paper [a "roadmap"] that let's the reader know where you are going and what to expect.

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The Research Process

• What is the Research Process?

There are many ways to describe the process of researching a topic for a paper or a project.

This description of the research process consists of ten actions across five stages. This process is cyclical and creative, not linear. Continue to adjust your terms, tools, and questions throughout the process and move back and forth across stages and actions as you explore your topic and find out new information.

Understand your assignment and timeline, and think about and write down the questions you'll be asking while you do your research. You may be asked to turn in a topic proposal for class, and this stage helps you complete your proposal.

Brainstorm the concepts, topics, and phrases you'll use to search for source material, and think about and choose the databases, websites, or other places you'll search.

Explore and search in the databases or other places you've chosen while developing your plan. Look through other databases or sites as new questions arise. Collect more sources than are required for your paper or project. Evaluate and select your sources. Take your time in this stage, and search again for different sources, if needed.

Read your selected sources carefully, taking notes while you do so. Read critically. Answer your existing questions and come up with new questions. Search again, if needed, so you can fill any gaps in knowledge and satisfy your curiosity about your topic.

Create your product, whether it's a research paper, a poster, a presentation, or something else. Draft your citations, and check your product against your assignment requirements. Edit your draft, proofread, incorporate feedback and turn it in.

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• Last Updated: Sep 16, 2024 10:38 AM
• URL: https://libguides.unomaha.edu/researchprocess

Minimum Wages in the 21st Century

This chapter surveys the literature on the impact of minimum wages on low-wage labor markets. We describe and critically review the empirical methods in the new minimum wage literature, particularly those leveraging quasi-experimental variation. We provide a quantitative overview of the most recent evidence on the employment and wage effects of the policy, while also exploring emerging research on its impact on other margins, including amenities, other inputs (such as capital and high-skilled workers), firm entry and exit, output prices and demand, profits, and productivity. This approach allows us to present a comprehensive picture of how minimum wage policies affect firms, workers, and labor markets. We also review the evidence on the policy’s impact on wage inequality and income distribution. Finally, we discuss how these effects can vary depending on the economic context and the level of a country’s development.

We thank Akash Bhatt, Annie McGrew, Jon Piqueras, and Linda Wu for excellent research assistance. We are also grateful to David Card, Christian Dustmann, Larry Katz, Pat Kline, Thomas Lemieux, Alan Manning, Michael Reich, Liyang Sun and the participants of the Conference for the Handbook of Labor Economics, Volume 5 in Berlin for the valuable comments and insights. This research has been funded by the European Union’s Horizon 2020 research and innovation program (grant agreement number 949995). The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.

MARC RIS BibTeΧ

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An enhanced Moth-Flame optimizer with quality enhancement and directional crossover: optimizing classic engineering problems

• Open access
• Published: 16 September 2024
• Volume 57 , article number  295 , ( 2024 )

Cite this article

You have full access to this open access article

• Helong Yu 1 ,
• Jiale Quan 1 ,
• Yongqi Han 1 ,
• Ali Asghar Heidari 2 &
• Huiling Chen 3  

As a popular meta-heuristic algorithm, the Moth-Flame Optimization (MFO) algorithm has garnered significant interest owing to its high flexibility and straightforward implementation. However, when addressing engineering constraint problems with specific parameters, MFO also exhibits limitations such as fast convergence and a tendency to converge to local optima. In order to address these challenges, this paper introduces an enhanced version of the MFO, EQDXMFO. EQDXMFO integrates a Quality Enhancement (EQ) strategy and a Directional Crossover (DX) mechanism, fortifying the algorithm’s search dynamics. Specifically, the DX mechanism is designed to augment the population’s diversity, enhancing the algorithm’s exploratory potential. Concurrently, the EQ strategy is employed to elevate the solution quality, which in turn refines the convergence precision of the algorithm. To verify the effectiveness of EQDXMFO, experiments are carried out on the test set of the IEEE CEC2017. A total of 5 classical algorithms, five excellent MFO variants, and seven state-of-the-art algorithms are selected for comparison, which confirm the significant advantages of EQDXMFO. Next, EQDXMFO is applied to five complex engineering constraint problems, demonstrating that EQDXMFO can optimize realistic problems. The comprehensive analysis shows that EQDXMFO has strong optimization capabilities and provides methods for research on other complex real-world problems.

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• Artificial Intelligence

Avoid common mistakes on your manuscript.

1 Introduction

In recent years, with the rapid development of science and technology, optimization problems in industry have been widely concerned (Chen et al. 2020 ; Cao et al. 2020a ). Nevertheless, classic optimization techniques are becoming less capable of satisfying the demands needed for solving more complex problems (Cao et al. 2021 ). Therefore, Meta-heuristic Algorithms (MAs) have been favored since their birth (Yang et al. 2023 ). This stochastic optimization method with larger search space can be used as a black box. It is the stochastic nature of the algorithm itself that drives the optimization task to be more flexible and pervasive. The key of optimization is to efficiently find a globally optimal solution to a complex real-world problem under certain conditions (Cao et al. 2019 ). Evolutionary methods aim to tackle complex, discrete, and constrained problems while creating as much value as possible at the lowest cost (Cao et al. 2020b ). The problems involved may involve single, multi, or many unbiased, based on the number of cost functions to be optimized at simultaneously (Duan et al. 2023 ). However, this paper focuses on single objective cases, and related method, while the optimization concepts are not limited to this scenario.

The main classification methods of MAs ( Vinoth Chandra and Anand 2022 ) are shown in Fig.  1 . MAs are classified into five distinct groups: physical-based methods, chemical-based techniques, human-based strategies, evolutionary algorithms, and swarm intelligence systems. The classics include Simulated Annealing (SA) (Aarts et al. 2005 ) and Sine Cosine Algorithm (SCA) (Mirjalili 2016 ). Among the most influential and applied evolutionary algorithms are Genetic Algorithm (GA) (Huang et al. 2022 ), Simulated Annealing (SA) (Sun et al. 2019 ), Differential Evolution (DE) (Price 2013 ), and Evolution Strategy (ES) (Rechenberg 1989 ). At present, the most popular class of algorithms are Swarm Intelligence (SI) algorithms (Tang et al. 2023 ). This is a bionic optimization algorithm based on population behavior, which provides important support for solving complex optimization problems, such as combination optimization (Zhou et al. 2023 ; Shi et al. 2022 ), economic emission (Dong et al. 2021 ; Dong 2023 ), medical diagnosis (Li et al. 2018 ; Zhao et al. 2019 ; Huang et al. 2019 ), and engineering design (Liang 2020 ; Yu 2020 ; Nautiyal 2021 ), etc. Three typical representatives are listed in Fig.  1 : Particle Swarm Optimization (PSO) (Cao et al. 2020c ), Ant Colony Optimization (ACO) (Sun et al. 2018 ), and Moth-Flame Optimization (MFO) (Mirjalili 2015 ). In addition, it also includes some of the latest popular algorithms, such as the Rime Optimization Algorithm (RIME) (Su 2023 ), Whale Optimization Algorithm (WOA) (Mirjalili and Lewis 2016 ), Harris Hawks Optimizer (HHO) (Heidari et al. 2019 ), Slime Mould Algorithm (SMA) (Chen 2022 ; Li et al. 2020 ), weIghted meaN oF vectOrs (INFO) (Ahmadianfar et al. 2022 ), Colony Predation Algorithm (CPA) (Tu et al. 2021 ), Hunger Games Search (HGS) (Yang et al. 2021 ), Runge Kutta Optimization (RUN) (Ahmadianfar et al. 2021 ), Liver Cancer Algorithm (LCA) (Houssein et al. 2023 ), and Parrot Optimizer (PO) (Lian et al. 2024 ), etc. They are widely used for their outstanding feasibility, flexibility and robustness. In addition to these methods, Grey Wolf Optimization (GWO) (Yin et al. 2020 ), Artificial Bee Colony (ABC) (Wang and Zhang 2023 ), Harmony Search (HS) (Luo et al. 2024 ), and Biogeography-based Optimization (BBO) (Wang et al. 2017a ) are widespread and have been utilized for various case studies.

Categories of MAs

However, according to the famous “No Free Lunch” theorem (Wolpert and Macready 1997 ), no optimization algorithm performs well on every problem. To fully explore the algorithm characteristics of SI, researchers continue to propose many variants to provide efficient solutions to complex practical problems. For example, to avoid the local optimization problem, Chen et al. (Zhao et al. 2019 ) introduced an aging mechanism and proposed a competitive variant of PSO (ALCPSO), which could achieve higher optimization accuracy while maintaining convergence speed. In order to overcome the problem of different defects in the two algorithms simultaneously, Nenavath et al. (Nenavath and Jatoth 2018 ) proposed a new optimization algorithm (SCADE) that combined the advantages of SCA and DE and made a major breakthrough in the target tracking problem. Xu et al. ( 2022 ) developed a multi-objective robust model to tackle the refined oil distribution case with supply shortages and a multi-objective PSO. To solve the feature selection problem, Fan et al. ( 2021 ) proposed an enhanced Fruit Fly Optimization Algorithm (FOA) (BSSFOA) based on the Bat-inspired Algorithm (BA) and fully solved the discrete optimization problem. In order to optimize discrete engineering design problems, Zhao et al. (Yu et al. 2023 ) introduced high-quality operators, proposed a strengthened version of SCA (CCEQSCA), and reasonably optimized classical realistic problems. For the engineering constraint problem, Qiao et al. ( 2022 ) designed an enhanced WOA algorithm (WDNMWOA) based on two renewal strategies to increase the possibility of population individuals approaching the optimal infinitely. The aforementioned classical improved algorithms have demonstrated excellent performance across various scenarios. However, due to the structural and parameter-specific characteristics of the original SI, these algorithms often struggle to explore a broader range of candidate solutions during the optimization process. This limitation has motivated the current study to select the classical MFO algorithm for a novel attempt at enhancement.

Mirjalili et al. ( 2015 ) proposed an MFO algorithm inspired by moths’ lateral localization navigation mechanism in 2015. Subsequently, MFO gained popularity due to its simple parameters, ease of implementation, and better scalability (Hussien et al. 2020 ). The algorithm performs well on statistical indicators and has become an effective tool for improving NP-hard problems such as shop floor scheduling (Cui et al. 2020 ), power system planning (Reddy et al. 2018 ), and feature selection (Abd Elaziz et al. 2020 ). However, it has been observed that the MFO algorithm exhibits certain limitations, notably an imbalance between exploration and exploitation capabilities. Therefore, to improve the algorithm’s balance ability and apply to different fields, many excellent variants of MFO have been derived. For example, Xu et al. ( 2019a ) introduced Gaussian mutation and chaotic local search strategies and proposed an enhanced algorithm to avoid MFO falling into local optima. A hybrid kernel extreme learning model is built for financial forecasting, thus providing an effective forecasting tool for financial scenarios.

Xu et al. ( 2019b ) introduced a series of enhancement operators to enhance the randomness of the population during the optimization process. They combined the exploration performance of the strategy to derive a strong algorithm, LGCMFO, which can prove that LGCMFO is significantly stronger than other algorithms even though it consumes more wall-clock time. Shan et al. ( 2021 ) have enhanced the MFO algorithm by incorporating a double adaptive weighting mechanism, denoted as WEMFO. This modification has led to a significant enhancement in the convergence accuracy of the original MFO algorithm and has bolstered its adaptive regulation capabilities. The WEMFO algorithm has been successfully applied to a range of practical problems, including those in engineering and medical treatment, where it has yielded remarkable results. Yu et al. ( 2020 ) proposed a novel fusion algorithm, QSMFO, based on SA and quantum revolving gate, which increased the diversity of moths while increasing the search ability of the algorithm, providing auxiliary means for feature selection and engineering constraint problems. Wang et al. ( 2017b ) introduced a Kernel Extreme Learning Machine (KELM) scheme based on Chaotic MFO (CMFO) to make the algorithm have regular ergodicity, which can be used in the field of medical diagnosis and significantly improve the prediction accuracy. Despite the numerous enhancements and successful applications of MFO variants across various operational domains, yielding satisfactory results, ongoing research has revealed opportunities for further refinement. Subsequent studies are thus directed towards optimizing the convergence rate of the algorithm and elevating the quality of the derived optimal solutions.

The improved MFO algorithm has achieved a notable degree of breakthrough in its performance. Nevertheless, to effectively address discrete optimization problems, the algorithm must exhibit higher stability. Therefore, this paper proposes a more stable MFO variant, EQDXMFO. In order to solve the problem of premature convergence of MFO, the Directional Crossover (DX) mechanism is introduced into MFO for the first time in this study. Based on the idea of a random selection of parents in GA, this method can guide population individuals to continuously search for the best scheme, accelerate the search efficiency of the algorithm, and achieve the purpose of enhancing its exploitation ability. Then, a Quality Enhancement (EQ) strategy is added to the MFO. The aim is to explore the candidate solutions of a larger region, improve the quality of the solutions by searching the optimal solutions of three paths, and finally improve the convergence accuracy. In order to verify the validity of EQDXMFO, the IEEE CEC2017 test set is used for evaluation and validation. EQDXMFO has been compared with 17 well-known algorithms to highlight the advantages of this algorithm in the optimization process. Finally, applying EQDXMFO to discrete optimization problems proves that EQDXMFO can be used as a valuable optimization tool in practical constraint problems. The main contributions of this study are as follows:

DX is introduced into MFO to guide the algorithm toward the position of the optimal solution and enhance the exploitation capability of EQDXMFO while preserving the population’s diversity.

Second, EQ is introduced to increase the level of quality of candidate solutions, while keeping the capability of exploring prospective candidate solutions in different ranges to boost the convergence reliability of EQDXMFO.

Through a large number of experiments on the IEEE CEC2017 and compared with 17 algorithms, including the champion algorithm, the robustness and competitiveness of EQDXMFO are fully demonstrated.

EQDXMFO is used for five typical discrete engineering design problems and is proven to excel in solving real-world and complex problems.

The paper is organized as follows: Sect.  2 offers a comprehensive overview of the MFO algorithm, establishing a foundation for subsequent discussions. Section  3 introduces two distinct optimization strategies that are pertinent to the enhancement of the MFO algorithm. Section  4 presents a detailed discussion of the comparative experimental results, highlighting the performance of the proposed EQDXMFO. Section  5 delves into the practical application of the EQDXMFO to engineering problems, illustrating its real-world utility. Finally, Sect.  6 makes the paper’s findings and proposes directions for future research.

2 Overview of MFO

The MFO algorithm (Mirjalili 2015 ) is a novel optimizer inspired by the lateral localization navigation mechanism of moths during night flight. The moths mistook the light of the flame for the light of the moon, thus deriving the spiral mechanism that laid the foundation for the emergence of MFO.

The MFO has two main components: moths and flames. In the population initialization phase, the moths are candidate solutions, and the position of the moths in the solution space represents the problem variables. The matrix of the moth population \(M\) in \(d\) -dimensional space is described as:

where \(n\) is the number of moths. The fitness values of individual moths generate the \({F}_{X}\) array as follows:

The moth spawns a new flame location with each iteration. As shown below, \(A\) is the flame position matrix and its fitness values are stored in \({F}_{A}\) :

After initialization, the laterally positioned flight path of the moth is simulated and modeled. Each moth updates its position according to the following formula:

where \({X}_{i}\) denotes the the \(i\) -th moth and \({A}_{j}\) denotes the \(j\) -th flame in Eq. ( 5 ). The unfolded formula is shown in Eq. ( 6 ), using a logarithmic spiral as the main update mechanism. Within the search space, the updating formula starts with the moth and ends with the flame. Where \(\left|{A}_{j}-{X}_{i}\right|\) denotes the distance between the \(i\) -th moth and the \(j\) -th flame. \(b\) denotes the shape constant of the logarithmic spiral function and \(t\) represents a random number within the range \([-1, 1]\) , which determines the distance between the next position of the moth and the position of the flame.

However, the update of the moth’s position is only related to the optimal flames in the last steps of the iteration. To maximize the application of the best solution, an adaptive framework is presented to limit the number of flames and create an equilibrium among exploration and exploitation trends. This important formula is as follows:

where \(FEs\) denotes the current number of iterations, \(MaxFEs\) denotes the maximum number of objective function evaluations and \(N\) represents the maximum number of flames, which also represents the maximum population size.

The flowchart of MFO is shown in Fig.  2 .

Flowchart of MFO

3 MFO-based improvement approach

The EQDXMFO algorithm introduces the EQ and the DX. Specifically, MFO tends to fall into local optima. As a result, EQ is implemented to increase the superiority and convergence degree of the best solution for the population. The problem of local stagnation is likely to occur while finding the optimal solution. To improve the method's capability to escape local optimal conditions, the diversity of the population can be increased to some extent. DX is introduced to enhance the balance between exploration and exploitation. This section describes the fundamentals of improvement strategies in EQDXMFO.

3.1 Quality enhancement strategy

The Enhanced Solution Quality (ESQ) strategy, introduced by Ahmadianfar et al. ( 2021 ), is an optimization technique designed to elevate solution quality, initially implemented within the RUN algorithm. In this paper, ESQ is slightly modified and applied to the MFO algorithm, named EQ. Each solution is guaranteed to continuously move to a better position, improving the search accuracy and global exploration capability. The mathematical model is shown below:

where \({X}^{best}\) denotes the global optimal position. \({X}_{avg}\) is the average of three random solutions ( \({X}_{a}, {X}_{b}\text{ and }{X}_{c}\) ). Combining the optimal solution ( \({X}^{best}\) ) produces the new position ( \({X}_{new1}\) ), where \({\beta }_{1}\) is a random number in the range \(\left[\text{0,1}\right]\) . \({X}_{new2}\) then denotes the new position resulting from the addition of the EQ. Where \({r}_{0}\) denotes an integer with a value of \(\pm 1\) or \(0\) and \(\omega\) denotes a dynamic factor that is negatively correlated with the number of evaluations. When \(\omega <1\) , the population enters a local optimal-seeking phase with increased exploitation capacity. When \(\omega \ge 1\) , the population enters the global exploration phase. Where a stochastic parameter \(\mu\) is added, which mainly improves the ability of \({X}_{new1}\) to jump out of the local optimal in order to increase the population diversity. Here \(c\) is also a random number in the range \(\left[\text{0,5}\right]\) .

In this process, it is likely that \({X}_{new2}\) is worse than the current solution. To avoid this phenomenon, a new solution \({X}_{new3}\) will be proposed as a way to improve the quality of the solution. The new location \({X}_{new3}\) that will be generated is defined as shown below:

where \(v\) is a random number in the range \(\left[\text{0,2}\right]\) that determines the quality of the optimal solution. The position update formula is shown below (listing the important parameters):

\(SF\) denotes the self-adaptive factor, which ensures that a dynamic balance between exploration and exploitation is maintained. Both \(a\) and \(b\) are constants with default values of 20 and 12, respectively. \(FEs\) is the current evaluation values, \(MaxFEs\) is the maximum objective function evaluation count and \({X}_{RK}\) is determined by four coefficients ( \({k}_{1},{k}_{2},{k}_{3},{k}_{4}\) ) which expressed in Eqs. ( 16 – 20 ):

where \(rand\) , \({rand}_{1}\) and \({rand}_{2}\) are three random numbers in the range \(\left[\text{0,1}\right]\) . \(u\) is also an important stochastic parameter that enhances the optimal solution during searching process ( \({X}^{best}\) ), \(u\) is defined in Eq. ( 16 ). The individual with the lowest fitness value in the current population is denoted as \({X}_{w}\) , and \(\Delta x\) represents the distance increment which calculated in Eqs. ( 21 and 22 ):

where \(ub\) and \(lb\) form a solution space and \(\gamma\) is the scale factor determined by the solution space.

3.2 Directional crossover mechanism

DX utilizes the gradient information of the algorithm to guide the individuals. It is based on the evolution of the crossover and mutation mechanism of the GA algorithm, known as DX. In this paper, the DX mechanism is introduced into the MFO algorithm, marking the first instance of its kind in the literature, which is designed to bolster the algorithm’s exploratory and exploitative capabilities and give it the ability to jump out of the local optima. The mathematical model of the DX to find the optimal solution is as follows:

There are four important parameters involved here: crossover rate ( \({p}_{c}\) ), variable crossover probability ( \({p}_{cv}\) ), directional probability ( \({p}_{d}\) ) and multiplication factor ( \(\alpha\) ). The first case is when \({p}_{best}^{j}\) ≥ \({p}_{mean}^{j}\) :

where \({c}_{1}\) and \({c}_{2}\) are individuals generated after directed crossover. The parental individuals are determined by the populations \({p}_{1}^{j}\) , \({p}_{2}^{j}\) in the space \(j\in \left[1,d\right]\) . \({p}_{mean}^{j}\) and \({p}_{best}^{j}\) are the mean and the optimal value of the parent individual in the \(j\) -th dimension, respectively. The second case is when \({p}_{best}^{j}<{p}_{mean}^{j}\) :

The third case is when \({p}_{best}^{j}\ne {p}_{mean}^{j}\) under the condition that the parental individuals have the same value:

where \({r}_{3}\) and \({r}_{4}\) are both two random numbers in the range \([0, 1]\) , which are determined by the two parameters \(val\) and \(\beta\) during each iteration. \({y}_{u}^{j}\) and \({y}_{l}^{j}\) together form the upper and lower bounds of the \(j\) -th dimension.

3.3 Overview of EQDXMFO

This subsection describes the implementation of EQDXMFO in detail. Firstly, the DX is placed in the position after the population update of the MFO. This favors the ability of the algorithm to jump out of local optima after the first update. It also addresses the propensity of the MFO algorithm to prematurely converge on local optima. Secondly, EQ is placed at the end of the calculation. EQ acts as an iterative filter to strengthen the algorithmic competitiveness of the global search by continuously improving the quality of candidate solutions. The flowchart of EQDXMFO is shown in Fig.  3 .

Flowchart of EQDXMFO

The following shows the specific implementation flow of EQDXMFO:

Step 1: Initialize the parameters involved in EQDXMFO, the moth population \(X\) and the flame population \(A\) .

Step 2: Calculate the fitness value for each individual of the moth population.

Step 3: Determine whether the number of iterations is less than the maximum number of objective function evaluations \((FEs<MaxFEs)\) . If “Yes”, the number of flames is updated by Eq. ( 7 ) to balance exploitation and exploration. Conversely, the optimal flame is output.

Step 4: Determine whether \(i<=n\) . If “Yes”, the position of the moth population is updated by Eq. ( 5 ). Meanwhile, calculate the current optimal fitness of the moth population.

Step 5: Introduce the DX and generate two random numbers \(num1\) and \(num2\) .

Step 6: \(new{X}_{1}\) and \(new{X}_{2}\) are generated, increasing the exploitation ability of the algorithm.

Step 7: The best one is selected to be updated as the current optimal individual \({X}^{best}\) .

Step 8: Introduce the EQ, which first determines whether \(rand\) is less than 0.5, provided that \(rand\) is less than \(w\) . If “Yes”, the position of the resulting candidate solution \({X}_{new2}\) is calculated by Eq. ( 8 ). Otherwise, return to step 3.

Step 9: Denote the fitness value of \({X}_{new2}\) by \(\text{f }\left({X}_{new2}\right)\) . Denote the fitness value of \(X\) by \(\text{f }(X)\) . If \(\text{f }(X)<\text{f }({X}_{new2})\) , updating the position of the best solution.

Step 10: Determine if \(rand\) is less than \(w\) . If “Yes”, calculate the position of the newly generated candidate solution \({X}_{new3}\) by Eq. ( 12 ). Further enhance the quality of the candidate solution by introducing \({X}_{new3}\) . Otherwise, return to step 3.

Step 11: Calculate the fitness value \(\text{f }({X}_{new3})\) and compare it with the current solution. If \(\text{f }(X)>\text{f }({X}_{new3})\) , the current optimal solution is updated to \({X}_{new3}\) . Otherwise, return to step 3.

Step 12: Output the global optimal position and optimal fitness.

The pseudo-code of EQDXMFO is shown below:

In addition, the time complexity determines the running time of the algorithm. There are three main influences here: the maximum number of objective function evaluations ( \(MaxFEs\) ), the population size ( \(N\) ) and the population dimension ( \(d\) ). The overall time complexity is calculated by \(O(EQDXMFO)=MaxFEs\times [O(MFO)+O(DX)+O(EQ)]\) . The original time complexity of MFO is \(O(MFO)=MaxFEs\times [O({N}^{2}+N\times d)]\) . The time complexity of DX is \(O(DX)=MaxFEs\times [O(N\times d)]\) . The time complexity of EQ is \(O(EQ)=MaxFEs\times [O(n\times d)]\) . Thus, the final time complexity of EQDXMFO is \(O(EQDXMFO)=MaxFEs\times [O({N}^{2})+3O(N\times d)]\) .

4 Experimentation and analysis of results

In this section, the EQDXMFO is tested. The algorithm’s effectiveness in a series of experiments is verified using 30 functions of IEEE CEC2017 test suite, which shows that the algorithm has a strong performance. To validate the advantages of the EQDXMFO, two main methods have been adopted. One of them is the nonparametric test of the Wilcoxon Signed-rank Test (WSRT) (Derrac et al. 2011 ). In the WSRT test, “ + ”, “ − ” and “ = ” symbols are used to indicate that the performance of EQDXMFO is better, worse than and equal to other comparison algorithms, respectively. Furthermore, it is crucial to note that the WSRT includes a significant metric known as the p-value. When \(p\) is less than the significant level 0.05, then it presents a significant difference. Additionally, we employed the Friedman Test (FT) (Alcalá-Fdez et al. 2009 ), which is a nonparametric method for ordinal analysis of variance. The averages and rankings further illustrate that the current optimization algorithm outperforms competitors and wins on the IEEE CEC 2017.

Sect.  4.1  fully describes the environment setup and algorithm setup to assure experiment fairness Sections  4.2 , 4.3 , 4.4 , 4.5 , 4.6 , 4.7 are a series of experiments based on the two main validation methods described above to confirm the validity of EQDXMFO. First of all, a comparative experiment of strategy combination is carried out. Through combinatorial experiments, it is verified that the EQDXMFO has a significant capacity to move away from the local optimal, achieve global convergence, and finally find the optimal solution. Secondly, a balance and diversity analysis of EQDXMFO and MFO will be conducted. The advantages and significance of EQDXMFO are compared by analyzing the development trend of the two algorithms in the process of exploration and exploitation. Next, the stability of EQDXMFO in different dimensions is analyzed and verified by stability experiments. Then, the EQDXMFO is compared with 5 original algorithms to verify that the EQDXMFO has excellent performance. Finally, EQDXMFO is compared with 5 variants of MFO and 7 advanced algorithms to thoroughly verify EQDXMFO’s effectiveness. In addition, the running time of all algorithms in this chapter is compared and analyzed so as to provide a more complete evaluation of EQDXMFO.

4.1 Experimental settings

In order to experiment pretty and effectively, this section is divided into two sections. Section  4.1.1 introduce the configuration parameters of the experimental environment. Section  4.1.2 provides detailed configuration information for any algorithm employed during the study.

4.1.1 Experimental environment settings

This section uses a unified standard server to conduct a series of experiments to ensure the fairness of the experiments. At the same time, it can provide reference value for another research. The model parameters of the experimental equipment are as follows:

CPU: Intel (R) Xeon (R) W-2245 CPU @ 3.90 GHz

RAM: 128.0 GB

OS: Windows 10 Professional

Software: MATLAB 2018a

4.1.2 Set up for the algorithms

In this section, the EQDXMFO is compared with the classical algorithms by testing on 30 benchmark functions of IEEE CEC2017, such as WOA (Mirjalili and Lewis 2016 ), HHO (Heidari et al. 2019 ), PSO (Kennedy and Eberhart 1995 ), SMA (Li et al. 2020 ) and RUN (Ahmadianfar et al. 2021 ). The EQDXMFO also is compared with excellent variants of the MFO algorithm, which included CLSGMFO (Xu et al. 2019a ), LGCMFO (Xu et al. 2019b ), WEMFO (Shan et al. 2021 ), QSMFO (Yu et al. 2020 ) and CMFO (Wang et al. 2017b ). Finally, the EQDXMFO is compared with state-of-the-art algorithms, including ALCPSO (Chen et al. 2012 ), SCADE (Nenavath and Jatoth 2018 ), BSSFOA (Fan et al. 2021 ), CCEQSCA (Yu et al. 2023 ), WDNMWOA (Qiao et al. 2022 ), LSHADE (Tanabe and Fukunaga 2014 ), and SHADE (Tanabe and Fukunaga 2013 ).

In this study, we selected 30 benchmark functions from the IEEE CEC2017 suite to assess the algorithm’s performance. A detailed description of these functions can be found in Tables A1–A2of the supplementary material. To comprehensively validate the efficacy of the novel EQDXMFO, we uniformly set the number of objective function evaluations to 300,000. This decision is made to provide a robust statistical basis for the performance analysis. Additional parameter configurations for the involved algorithms are elaborated in Table  1 .

4.2 Combinatorial experiments with strategies

This subsection introduces the EQ and the DX for the first time to improve the optimized performance of the MFO. During the introduction of the strategy, different MFO variants are constructed through four sets of permutation experiments as shown in Table  2 . The EQDXMFO represents an augmentation of the original MFO algorithm with both EQ strategy and the DX mechanism. In contrast, the EQMFO algorithm integrates only the EQ strategy into the MFO framework, while the sole inclusion of the DX mechanism characterizes the DXMFO algorithm. It is pertinent to highlight that the application of a given strategy to the basic MFO is denoted by “1”, signifying the presence of the strategy. Conversely, the absence of any strategy application is denoted by “0”, indicating a pure MFO approach.

The strategy combination experiments are executed on a comprehensive set of 30 benchmark functions of IEEE CEC2017. Furthermore, to augment the persuasiveness of our experimental findings, both the WSRT and the FT are employed to rigorously assess the statistical significance of the performance outcomes. The results and rankings of the four combined algorithms in the WSRT are given in Table  3 . Based on the description of the “ ± / = ” notation at the beginning of this chapter, the analysis concludes that EQDXMFO outperforms MFO by 30 IEEE CEC2017 benchmark functions and ranks first with a mean value of 1.63. In addition, this section also gives the FT ranking results as shown in Table  4 , where EQDXMFO competed for the victory with a running score of 1.87. Both WSRT and FT are ranked first in the combined rankings for both metrics, which is very convincing for the optimization performance of EQDXMFO. The final results show that the optimal searching ability of the algorithm is the most remarkable only when EQ and DX are introduced simultaneously.

4.3 Balance and diversity analysis of EQDXMFO and MFO

To further validate the enhancement of MFO capability by EQ and DX strategies, this subsection analyzes the balance and diversity of EQDXMFO and MFO. The experiment selected six representative sets of functions: F1, F3, F10, F18, F27 and F30, as shown in Fig.  4 .

a Balance plots of EQDXMFO b Balance plots of MFO c Diversity plots of population. (Color figure online)

As shown in Fig.  4 a and b, it can be clearly seen that the equilibrium value of EQDXMFO is effectively improved by almost 20% compared to MFO. The figure mainly introduces a green curve that effectively contrasts the change in balance between development phases, called the incremental–decremental curve. The balance and comparison analysis diagram of the two algorithms can be roughly divided into the following three trends. For example, F1 and F18 indicate that the exploitation and exploration capabilities of EQDXMFO are steadily improving compared with MFO. F3 shows that EQDXMFO maintains a thorough global search in the early stages, where turning points represent the algorithm’s escape from local traps. F10, F27, and F30 appear as obvious incremental peaks, indicating that the pre-search exploration occupies the main position, and the accuracy is improved after reaching the balance. The rest of the images also show all three trends and a qualitative leap in exploration capabilities. Overall, EQDXMFO has a more consistent optimization capability.

Figure  4 c represents the comparative diversity analysis plot of EQDXMFO and MFO. The average distance between searching individuals represents the diversity of a population. It can be seen from the figure that at the beginning of the algorithm, both algorithms are in the stage of high population diversity. However, MFO rapidly converges, and EQDXMFO slowly descends. This is due to the introduction of DX, which maintains the population diversity of EQDXMFO in the early stage. With the gradual narrowing of the search range, the algorithm is forced to jump out of the trap in reverse, and the rate is increased to achieve the optimization result in the global scope. EQ is used to increase the convergence accuracy. The comprehensive description ensures good momentum and increases the global search range of EQDXMFO.

4.4 Analysis of the stability experiments

In this section, we test the scalability of different algorithms by setting different dimensions to verify the stability of the improved optimization algorithm. Here, we select four dimensions, 10, 30, 50 and 100, set up 300, 000 evaluations and test EQDXMFO and MFO on 30 benchmark functions. Comparison results are shown in Table  5 . The mean of EQDXMFO and MFO is expressed in Avg and the standard deviation is expressed in Std. The primary results are highlighted by employing bold formatting to distinguish them. EQDXMFO outperformed MFO in almost all the mean and the standard deviation functions. To facilitate review of the comparison results, the average comparison results for EQDXMFO and MFO are shown in Table B1 of the supplementary material. The table shows that the p-value of EQDXMFO is less than 0.05 for most of the functions, indicating that EQDXMFO is significantly more stable than MFO, making the experiment more meaningful for research. In addition, the WSRT and FT test results of the EQDXMFO and MFO algorithms in the four dimensions are presented in Table  6 . The use of the “ + / − / = ” symbols given initially in this section yielded that EQDXMFO has 25 functions higher than MFO, and 5 functions equal to MFO in the 10 dimensions. When the dimensions are 30, 50 and 100, EQDXMFO performs better than MFO and ranks first on all 30 functions. Finally, the WSRT and the FT results show that EQDXMFO ranks first in the results of all the dimensions listed. This suggests that the EQDXMFO outperforms the MFO algorithm in terms of optimization performance and demonstrates more excellent stability across varying dimensions.

4.5 Comparative experiments with the original algorithms

Global optimization experiments are conducted from subsect.  4.5 until subsect.  4.7 to validate the convergence performance of EQDXMFO further. The first step is to compare EQDXMFO with the five classical algorithms. Three classical high-performance optimization algorithms (WOA, HHO, and PSO) and two newly proposed popular optimization algorithms (SMA and RUN) are selected. Six algorithms are simultaneously put on the 30 benchmark functions from the IEEE CEC2017 for comparative validation.

Table 7 shows the means and standard deviations of the algorithms. EQDXMFO has the most minimum means and standard deviations, with only minor differences from other algorithms. It shows that the EQDXMFO algorithm is the most stable algorithm in the experiments and has a strong optimization performance. Next, the optimized performance of EQDXMFO is verified analytically by significant differences between the algorithms, placing the results in Table B2 of the supplementary material.  The improved algorithm's capacity to identify whether the p-value is less than 0.05 demonstrates EQDXMFO's superior optimization capability.

Table 8 shows the results of six algorithms, including EQDXMFO on WSRT and FT. “ + / − / = ” indicates the number of algorithms for which EQDXMFO is stronger, weaker or equal to the other algorithms, respectively. Firstly, the advantages and disadvantages of the algorithm and these algorithms are compared. It can be seen that EQDXMFO is excellent, including 30 functions better than WOA, 29 functions better than HHO, 25 functions better than PSO, and only one function weaker than PSO. EQDXMFO is compared to two popular new algorithms (SMA, RUN), in which more than half of the functions are excellent in both cases. This is followed by comparing the mean score and rank of EQDXMFO and other algorithms on WSRT and FT, which are ranked first with mean scores of 1.53 and 1.77, respectively. Therefore, it can be shown that the EQDXMFO has excellent optimized performance.

In addition, to compare the EQDXMFO with other peers, the convergence plots of EQDXMFO, WOA, HHO, PSO, SMA, and RUN are generated on the set of 30 functions from IEEE CEC2017, as shown in Fig.  5 . The ability of the EQDXMFO to escape local optima is exemplified by its performance on the benchmark function F1, where a collapse in the optimization landscape indicates a successful exploration of global optimum. As can be seen from other graphs, EQDXMFO has the fastest convergence speed among the six algorithms, which shows that it has strong development ability and can find the optimal solution quickly. The optimal solution quality of EQDXMFO performs far more effectively than other methods, especially as shown in F9. The analysis demonstrates that EQDXMFO converges with excellent accuracy and can identify the best solution rapidly and correctly.

Convergence curves of EQDXMFO compared to 5 original algorithms. (Color figure online)

4.6 Comparative experiments with variants of MFO

This subsection compares the EQDXMFO with other excellent MFO variants to further validate its performance. EQDXMFO is compared to five excellent variants of MFO: CLSGMFO, LGCMFO, WEMFO, QSMFO, and CMFO.

The comparison results are recorded in Table  9 . EQDXMFO has an advantage in most functions for mean and standard deviation comparisons. More than half of the mean and standard deviation are at the minimum, which indicates that EQDXMFO can obtain a more stable optimal solution. To make the experimental results more convincing, the WSRT test is performed, and the results are placed in Table B3of the supplementary material. The statistical analysis of the data indicates that for the majority of the benchmark functions, the p-values are below the threshold of 0.05, suggesting that the observed differences are statistically significant. The results further demonstrate that the EQDXMFO exhibits a significant divergence from other algorithms in terms of performance. Specifically, EQDXMFO has outperformed its counterparts in the majority of the functions, indicating that its superior performance is not an isolated incident but a consistent trend across various functions. This consistency underscores the robustness and reliability of EQDXMFO, confirming that its advantages are statistically significant.

Secondly, the results of WSRT and FT of the five MFO variants compared with EQDXMFO are recorded in Table  10 . The following verification results are obtained based on the judgment criterion of “ + / − / = ” symbols. EQDXMFO is better than the other five algorithms in more than half of the functions; 26 functions are especially better than WEMFO, and 30 functions are better than CMFO. This indicates that EQDXMFO is highly optimized and competitive. Next, the scores and rankings of the WSRT and FT tests for EQDXMFO and the other compared algorithms are listed. EQDXMFO has a mean score of 1.60 for the WSRT and a mean score of 1.90 for the FT, which is in first place in the combined rankings for both tests. Therefore, it can be confirmed that the proposed EQDXMFO in this paper has a strong advantage in the variant algorithms of the same type of MFO, which is of research significance.

To further verify the convergence performance of EQDXMFO, the convergence curves are shown in Fig.  6 . On the 30 functions of the IEEE CEC 2017, the objective function is evaluated 300,000 times with the red curve representing EQDXMFO. As can be seen from the graph, both F1 and F2 tend to jump out of the local optimal solution. This means that MFO has enhanced global exploration capabilities with an additional strategic mechanism. The remaining function convergence graphs reveal that the EQDXMFO algorithm exhibits the swiftest convergence rate, with a pronounced convergence effect. Finally, global convergence is attained, and the most optimal solution is identified, particularly on the F9 function. In summary, the EQDXMFO demonstrates superior adaptability in comparison to other algorithms and possesses commendable exploitation and exploration capabilities.

Convergence curves of EQDXMFO compared to the 5 MFO variants

4.7 Comparative experiments with advanced algorithms

However, comparing the original classical algorithm with the MFO variant algorithms is not enough. Therefore, this section compares EQDXMFO with five advanced optimization algorithms, CCEQSCA, WDNMWOA, ALCPSO, BSSFOA, and SCADE, as well as two champion algorithms, LSHADE and SHADE. Further, verify the optimized performance of EQDXMFO on the IEEE CEC2017.

Table 11 represents the results of comparing EQDXMFO and the other seven advanced algorithms. In 30 independent experiments, half of the mean and standard deviation of EQDXMFO are minimum values, and the excellent and good ratio is good in these experiments, which is comparable to the two champion algorithms. It is enough to show that the MFO optimization algorithm, with the addition of EQ and DX strategies, has strong stability and excellent optimization performance. The experiments continued to compare the advantages and disadvantages of the EQDXMFO algorithm with other algorithms using the WSRT test at the p = 0.05 significance level to characterize. The results of the supplementary material are presented in Table B4. EQDXMFO is significantly different from other algorithms for most functions, indicating that EQDXMFO has significant advantages over other algorithms.

Next, the results of the comparison of WSRT and FT for these eight algorithms are placed in Table  12 . The first row of each test in the table is the number of algorithmic strengths and weaknesses indicated by the “ + / − / = ” symbols, and it can be seen that the comparison results are very significant and strong. EQDXMFO is stronger than LSHADE and SHADE on more than half of the functions, and is more excellent than ALCPSO on 27 functions. It is worth noting that the EQDXMFO completely outperforms the remaining four advanced algorithms on 30 functions. Secondly, the performance of EQDXMFO is verified by the score and ranking in each validation method. As can be seen from the data in the table, EQDXMFO ranks second only to SHADE in WSRT evaluation, while it ranks first in FT evaluation with a score of 1.95. Combining the above data, it can be concluded that EQDXMFO performs much better than other advanced variants and has a great advantage in optimizing performance.

As shown in Fig.  7 , the convergence results of EQDXMFO and seven other advanced algorithms are plotted. From this, we can visually compare the optimization characteristics and effects of different algorithms. EQDXMFO is represented by a bold red curve so that the algorithm can be clearly compared in the early, middle, and late stages of evolution. Observation shows that F3 and F22 have the highest rate of the algorithm, which also represents a strong exploitation. It can be seen from the convergence graphs of other functions that the red curve has the characteristics of fast speed and high convergence accuracy. EQDXMFO is basically on par with the champion algorithm and significantly better than others. The comprehensive results show that EQDXMFO is an excellent variant of the MFO algorithm, and its competitiveness can be confirmed.

Convergence curves of EQDXMFO compared to the 7 advanced algorithms

4.8 Comparative results of running time

The algorithm’s running time is also one of the indexes used to evaluate the algorithm’s performance. Therefore, this section evaluates the EQDXMFO from an algorithm runtime perspective. It is important to note that IEEE CEC2017 of 30 functions evaluates this experiment. The dimension is set to 30, the number of evaluations is 300,000 experiments are performed to complete the overall evaluation. Among them, the running time of the algorithm on a single function interval is selected for analysis and explanation, as shown in Fig.  8 .

Comparison of running time

Figure  8 shows that the running time of EQDXMFO is in the lower moderate range, specifically 3.67 s. EQDXMFO is only 1.50 s longer than the WOA for the shortest time from 2.17 s. However, this is not to say that EQDXMFO has no exploitable value. By combining the WSRT and FT standards in Sect.  4.5 , we can conclude that EQDXMFO’s performance is far superior to WOA’s. This is precisely because the EQ policy is added to increase the traversal time. In order to improve the quality of the solution, thorough exploration is carried out, resulting in a relatively large running time, which is also evident from the comparative results of the RUN. However, what is of concern is that SMA, LSHADE and SHADE all have significantly longer running times, which fully demonstrates that not all algorithms have higher performance with shorter running times. In addition, it can also be seen from the figure that the running time of the MFO algorithm is short, but the convergence accuracy is not high. Therefore, EQDXMFO benefits from the structural advantages of the MFO algorithm, which can give it strong optimization performance under relatively low running time. Finally, it is concluded that although the running time of EQDXMFO is not very prominent, its comprehensive performance is still worthy of recognition.

5 EQDXMFO in real-world problems

The global optimization capability of EQDXMFO has been fully verified by Sect.  4 , but its real performance cannot be explained. Therefore, this chapter applies EQDXMFO to five classical discrete engineering constraint problems to prove the advantages of the EQDXMFO algorithm. Including welded beam design (WBD), pressure vessel design (PVD), multiple disc clutch brake design (MDCBD), three-bar truss design (TBTD), and tension/compression spring design (TCSD). It is worth noting that the experiments uniformly set maximum evaluations of 2, 000 times and the best results of the problem are indicated in bold.

5.1 Welded beam design (WBD)

The EQDXMFO is applied to the classical WBD problems (Deb 1991 ). The model diagram is shown in Fig.  9 . The optimization objective is to pay the least welding cost under four constraints (shear stress ( \(\uptau\) ), bending stress ( \(\upsigma\) ), buckling load ( \({P}_{c}\) ) and deflection ( \(\updelta\) )). It also explores four parameters, including the thickness of wed ( \(h\) ), length of the champed bar ( \(l\) ), the height of the bar ( \(t\) ) and the thickness of the bar ( \(b\) ). The specific mathematical formula is shown below:

Model diagram of WBD

Subject to:

Variable range:

\(0.1\le {x}_{1}\le 2\) , \(0.1\le {x}_{2}\le 10\) , \(0.1\le {x}_{3}\le 1\) 0, \(0.1\le {x}_{4}\le 2\) ,

Variable ranges: \(P=6000lb, L=14in, E=30\times {10}^{6}psi, G=12\times {10}^{6}psi,\)

In this problem, five popular algorithms, including EO (Faramarzi et al. 2020 ), SASSA (Song et al. 2022 ), CLSGMFO (Xu et al. 2019a ), WEMFO (Shan et al. 2021 ) and ARSCA (Yang et al. 2023 ) are selected to compare with EQDXMFO. The remarkable effects of EQDXMFO are shown in Table  13 . The results show that EQDXMFO has the lowest payment cost and stands out among many optimization algorithms, and its payment cost is 1.724200. To make the experiment more convincing, the maximum minimum, mean and variance of EQDXMFO are compared with the other five algorithms as shown in Table  14 . EQDXMFO wins with superior conditions. Therefore, EQDXMFO is more suitable to solve the WBD issue.

5.2 Pressure vessel design (PVD)

PVD is the problem of designing a cylindrical pressure vessel at minimum cost, which is a common complex engineering type problem (Mohammed and Rashid 2020 ). This problem as shown in Fig.  10 contains four parametric conditions: the thickness of the shell ( \({T}_{s}\) ), the thickness of the head ( \({T}_{h}\) ), inner radius ( \(R\) ), length of the cylindrical section of the vessel ( \(L\) ). The mathematical model is shown below:

Model diagram of PVD

Variable ranges: \(0\le {x}_{1}\le 99, 0\le {x}_{2}\le 99\) , \(10\le {x}_{3}\le 200\) , \(10\le {x}_{4}\le 200\) .

The results of EQDXMFO are compared with HMCMMFO (Yu et al. 2022 ), LXMWOA (Qi et al. 2022a ), CSS (Kaveh and Talatahari 2010 ), AGOA (Wang et al. 2021 ), SDSMA (Qi et al. 2022b ) algorithms are shown in Table  15 . It can be clearly compared that EQDXMFO outperforms the rest of the algorithms with the lowest cost of 5967.832. This suggests that the method has high optimization ability in PVD problems and can successfully enhance the design parameters. Combined with the combined statistics in Table  16 , it effectively proves that EQDXMFO meets the optimality criteria in all criteria.

5.3 Multiple disc clutch brake design (MDCBD)

MDCBD can be considered one of the classic complex reality problems (Rao and Waghmare 2017 ). The aim is to design a multi-disc clutch brake with minimum mass ( \({f}_{1}\) ) and minimum stopping time ( \({f}_{2}\) ) as shown in Fig.  11 . There are five other parameters involved: internal radius ( \({r}_{i}\) ), external radius ( \({r}_{0}\) ), thickness of the disc ( \(t\) ), driving force ( \(F\) ), and the number of friction surfaces ( \(Z\) ). The specific mathematical model constructed is shown below:

Model diagram of MDCBD

The above parameters are as follows:

Table 17 shows the comparison results between EQDXMFO and the five algorithms: GSA (Rashedi et al. 2009 ), BareFOA (Yu 2020 ), DTSMA (Yin et al. 2022 ), hHHO-SCA (Kamboj et al. 2020 ) and WEMFO (Shan et al. 2021 ). EQDXMFO ranks first among these algorithms with a margin of 0.235242. It can be seen that EQDXMFO has obvious advantages in the MDCBD problem, and can play a role in optimizing the utilization of engineering materials under the premise of meeting constraints. In order to fully analyze the performance of EQDXMFO on the MDCBD problem, comparative means and variances etc. are shown in Table  18 . In conclusion, EQDXMFO can effectively reduce the material usage in this problem.

5.4 Three-bar truss design (TBTD)

TBTD is a classical engineering problem in the field of civil engineering (Fauzi and Batool 2019 ). The intention is to reduce the structural weight of the object while determining the area of the cross-section. The problem contains only two parameters ( \({\text{x}}_{1}\) , \({\text{x}}_{2}\) ), which satisfy three constraints: the stress, buckling, and deflection constraints of each steel bar. The structure is shown in Fig.  12 . The specific design of this model is described as follows:

Model diagram of TBTD

Objective function:

where \(0\le {x}_{i}\le 1; i=1, 2;\) l  = 100 cm, P  = 2 kN /cm 2 ; σ  = 2 kN /cm. 2

As shown in Table  19 , the results of EQDXMFO are compared with LSA-SM (Lu et al. 2017 ), PSO-DE (Liu et al. 2010 ), CDESSA (Zhang et al. 2023 ), SHO (Zhao et al. 2023 ), SSA (Jain et al. 2019 ), and GWO (Mirjalili et al. 2014 ). EQDXMFO wins first with optimum cost of 263.895738. The results of the comparison of mean, variance, and maximum and minimum values are shown in Table  20 . EQDXMFO performs superior in all the metrics. This shows that EQDXMFO has certain advantages in solving TBTD problems and can effectively reduce the cost of engineering problems.

5.5 Tension/compression spring design (TCSD)

TCSD is one of the common problems in optimization literature (Tzanetos and Blondin 2023 ). The goal is to minimize the mass of the spring while satisfying a certain load. This problem contains four constraints: minimum deflection, shear stress, oscillation frequency, and outer diameter limit. Three effective parameters need to be satisfied: the spring wire diameter (d), the average coil diameter (D), and the limited number of springs (N), as shown in Fig.  13 . To this end, the following mathematical model is established to solve the problem:

Model diagram of TCSD

Variable range: \(0.05\le {x}_{1}\le 2\) , \(0.25\le {x}_{2}\le 1.3\) , \(2\le {x}_{3}\le 150\) .

In this paper, the following representative algorithms and criteria are selected to compare with the performance of EQDXMFO on this problem: CCEQSCA (Yu et al. 2023 ), EO (Faramarzi et al. 2020 ), MSCA (Chen et al. 2020 ), RO (Kaveh and Khayatazad 2012 ), SCSO (Seyyedabbasi and Kiani 2023 ), and AVOA (Abdollahzadeh et al. 2021 ). Table 21 illustrates the capability performance of different algorithms when solving this problem. By comparison, EQDXMFO wins in this problem with the minimum cost of 0.012665. This shows that EQDXMFO can traverse the optimal solution to help researchers obtain the lowest cost in solving the minimum quality problem in engineering problems. Meanwhile, Table  22 shows the statistical results of EQDXMFO, the results show that EQDXMFO has low mean and variance, which again indicates that EQDXMFO has good adaptability and robustness. Taken together, it shows that EQDXMFO can contribute an effective solution to the problem of TCSD.

5.6 Analysis of the results of engineering design problems

With the complexity of engineering problems, it is necessary to use SI algorithm to give the optimal value of the problem. Through the comparison and optimization results of the above five practical engineering design problems, several reasons for EQDXMFO to stand out in constraint problems are analyzed as follows:

EQDXMFO is based on the basic characteristics of SI algorithm and relies on cooperative optimization between populations, which can thoroughly search for the optimal solution within the constraints. Especially for solving discrete engineering constraint problems in the real world, EQDXMFO can and efficiently find suitable solutions through strong robustness and flexibility.

EQDXMFO is based on the two policy mechanisms of DX and EQ to complete the optimization. DX can help the algorithm expand the space of the solution in the engineering problem and improve the probability of searching for the optimal global solution. Under the action of EQ, the convergence accuracy of the algorithm can be stabilized, and the quality of the best scheme can be effectively improved. Based on this, EQDXMFO can be among the best in engineering problems, which has practical significance.

In the real world, many complex problems with constraints are usually solved by the penalty function method (Oztas and Erdem 2023 ). The penalty function implies imposing a penalty on values that do not satisfy the solution criteria of the problem. In general, SI algorithms rely on internal punishment. The framework of the structural model is shown in the classical engineering problems listed above. The purpose is to transform the constrained problem into the unconstrained problem by introducing the penalty function so that the optimal solution of the problem can be obtained quickly.

6 Conclusion and future perspectives

In this paper, we propose a variant algorithm of MFO based on the EQ and DX mechanism, called EQDXMFO. Compared with the traditional MFO, EQDXMFO has faster convergence speed, more traversal time, and better-quality solutions. The algorithm introduces the DX mechanism in the middle and late optimization to increase the diversity of the population, escape from the local trap, and improve the ability to explore globally. At the same time, the EQ strategy is introduced to improve the quality of local solutions and increase the probability of selecting potential solutions to improve the convergence accuracy of EQDXMFO. With the cooperation of these two strategy mechanisms, the flexibility and robustness of EQDXMFO are comprehensively improved.

This study demonstrates the advantages of EQDXMFO through a series of validation experiments at IEEE CEC2017 in Sect.  4 . Firstly, the enhancement effects of EQ and DX on MFO are compared through ablation experiments. Only when both strategies are introduced at the same time is the optimization most significant and achieves the purpose of higher accuracy. Secondly, the balance and diversity experiments are carried out to find the appropriate balance point. Then, four common and representative dimensions (10, 30, 50, 100) are selected to comprehensively verify the stability of EQDXMFO, which lays a foundation for solving complex practical problems. In addition, EQDXMFO is compared with five original algorithms, five variants of MFO, and seven advanced algorithms in 30 dimensions. EQDXMFO achieves excellent results, indicating that the algorithm is outstandingly competitive.

Finally, EQDXMFO is applied to five complex discrete engineering constraint problems to explore the optimization ability of EQDXMFO further. Compared with other algorithms, EQDXMFO is very competitive on typical engineering problems, which has effective value for researching and applying complex practical problems.

In summary, EQDXMFO has better optimization performance and significantly improves complex practical problems. However, due to the randomness of the swarm intelligence algorithm, the balance between exploitation and exploration can still be further strengthened. Therefore, we hope that the algorithm structure can be further optimized in future research. In addition, EQDXMFO can be extended to other application fields, such as optimization problems in complex fields such as feature selection (Ma et al. 2023 ) and image classification (Chen 2024 ), to explore the research value in different scenarios.

Data availability

Data is provided within the manuscript or supplementary information files.

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Acknowledgements

This study was funded by the Key R&D Project of Jilin Province Science and Technology Development Plan (20220202032NC). This study was also supported in part by the Natural Science Foundation of Zhejiang Province (LZ22F020005), National Natural Science Foundation of China (62076185, 62301367).

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College of Information Technology, Jilin Agricultural University, Changchun, 130118, China

Helong Yu, Jiale Quan & Yongqi Han

School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran, Iran

Ali Asghar Heidari

Key Laboratory of Intelligent Informatics for Safety & Emergency of Zhejiang Province, Wenzhou University, Wenzhou, 325035, China

Huiling Chen

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Author 1: Helong Yu Contributions: writing—original draft, writing—review & editing, software, visualization, investigation. Author 2: Jiale Quan Contributions: writing—original draft, writing—review & editing, software, visualization, investigation. Author 3: Yongqi Han Contributions: conceptualization, methodology, formal analysis, investigation, writing—review & editing, funding acquisition, supervision. Author 4: Ali Asghar Heidari Contributions: writing—original draft, writing—review & editing, software, visualization, investigation. Author 5: Huiling Chen Contributions: conceptualization, methodology, formal analysis, investigation, writing—review & editing, funding acquisition, supervision.

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Yu, H., Quan, J., Han, Y. et al. An enhanced Moth-Flame optimizer with quality enhancement and directional crossover: optimizing classic engineering problems. Artif Intell Rev 57 , 295 (2024). https://doi.org/10.1007/s10462-024-10923-y

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Accepted : 22 August 2024

Published : 16 September 2024

DOI : https://doi.org/10.1007/s10462-024-10923-y

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