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Research Problem – Definition, Steps & Tips

Published by Jamie Walker at August 12th, 2021 , Revised On October 3, 2023

Once you have chosen a research topic, the next stage is to explain the research problem: the detailed issue, ambiguity of the research, gap analysis, or gaps in knowledge and findings that you will discuss.

Here, in this article, we explore a research problem in a dissertation or an essay with some research problem examples to help you better understand how and when you should write a research problem.

“A research problem is a specific statement relating to an area of concern and is contingent on the type of research. Some research studies focus on theoretical and practical problems, while some focus on only one.”

The problem statement in the dissertation, essay, research paper, and other academic papers should be clearly stated and intended to expand information, knowledge, and contribution to change.

This article will assist in identifying and elaborating a research problem if you are unsure how to define your research problem. The most notable challenge in the research process is to formulate and identify a research problem. Formulating a problem statement and research questions while finalizing the research proposal or introduction for your dissertation or thesis is necessary.

Why is Research Problem Critical?

An interesting research topic is only the first step. The real challenge of the research process is to develop a well-rounded research problem.

A well-formulated research problem helps understand the research procedure; without it, your research will appear unforeseeable and awkward.

Research is a procedure based on a sequence and a research problem aids in following and completing the research in a sequence. Repetition of existing literature is something that should be avoided in research.

Therefore research problem in a dissertation or an essay needs to be well thought out and presented with a clear purpose. Hence, your research work contributes more value to existing knowledge. You need to be well aware of the problem so you can present logical solutions.

Formulating a research problem is the first step of conducting research, whether you are writing an essay, research paper,   dissertation , or  research proposal .

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Step 1: Identifying Problem Area – What is Research Problem

The most significant step in any research is to look for  unexplored areas, topics, and controversies . You aim to find gaps that your work will fill. Here are some research problem examples for you to better understand the concept.

Practical Research Problems

To conduct practical research, you will need practical research problems that are typically identified by analysing reports, previous research studies, and interactions with the experienced personals of pertinent disciplines. You might search for:

• Problems with performance or competence in an organization
• Institutional practices that could be enhanced
• Practitioners of relevant fields and their areas of concern
• Problems confronted by specific groups of people within your area of study

If your research work relates to an internship or a job, then it will be critical for you to identify a research problem that addresses certain issues faced by the firm the job or internship pertains to.

Examples of Practical Research Problems

Decreased voter participation in county A, as compared to the rest of the country.

The high employee turnover rate of department X of company Y influenced efficiency and team performance.

A charity institution, Y, suffers a lack of funding resulting in budget cuts for its programmes.

Theoretical Research Problems

Theoretical research relates to predicting, explaining, and understanding various phenomena. It also expands and challenges existing information and knowledge.

Identification of a research problem in theoretical research is achieved by analysing theories and fresh research literature relating to a broad area of research. This practice helps to find gaps in the research done by others and endorse the argument of your topic.

Here are some questions that you should bear in mind.

• A case or framework that has not been deeply analysed
• An ambiguity between more than one viewpoints
• An unstudied condition or relationships
• A problematic issue that needs to be addressed

Theoretical issues often contain practical implications, but immediate issues are often not resolved by these results. If that is the case, you might want to adopt a different research approach  to achieve the desired outcomes.

Examples of Theoretical Research Problems

Long-term Vitamin D deficiency affects cardiac patients are not well researched.

The relationship between races, sex, and income imbalances needs to be studied with reference to the economy of a specific country or region.

The disagreement among historians of Scottish nationalism regarding the contributions of Imperial Britain in the creation of the national identity for Scotland.

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Step 2: Understanding the Research Problem

The researcher further investigates the selected area of research to find knowledge and information relating to the research problem to address the findings in the research.

Background and Rationale

• Population influenced by the problem?
• Is it a persistent problem, or is it recently revealed?
• Research that has already been conducted on this problem?
• Any proposed solution to the problem?
• Recent arguments concerning the problem, what are the gaps in the problem?

How to Write a First Class Dissertation Proposal or Research Proposal

Particularity and Suitability

• What specific place, time, and/or people will be focused on?
• Any aspects of research that you may not be able to deal with?
• What will be the concerns if the problem remains unresolved?
• What are the benefices of the problem resolution (e.g. future researcher or organisation’s management)?

Example of a Specific Research Problem

A non-profit institution X has been examined on their existing support base retention, but the existing research does not incorporate an understanding of how to effectively target new donors. To continue their work, the institution needs more research and find strategies for effective fundraising.

Once the problem is narrowed down, the next stage is to propose a problem statement and hypothesis or research questions.

If you are unsure about what a research problem is and how to define the research problem, then you might want to take advantage of our dissertation proposal writing service. You may also want to take a look at our essay writing service if you need help with identifying a research problem for your essay.

Frequently Asked Questions

What is research problem with example.

A research problem is a specific challenge that requires investigation. Example: “What is the impact of social media on mental health among adolescents?” This problem drives research to analyse the relationship between social media use and mental well-being in young people.

How many types of research problems do we have?

• Descriptive: Describing phenomena as they exist.
• Explanatory: Understanding causes and effects.
• Exploratory: Investigating little-understood phenomena.
• Predictive: Forecasting future outcomes.
• Prescriptive: Recommending actions.
• Normative: Describing what ought to be.

What are the principles of the research problem?

• Relevance: Addresses a significant issue.
• Re searchability: Amenable to empirical investigation.
• Clarity: Clearly defined without ambiguity.
• Specificity: Narrowly framed, avoiding vagueness.
• Feasibility: Realistic to conduct with available resources.
• Novelty: Offers new insights or challenges existing knowledge.
• Ethical considerations: Respect rights, dignity, and safety.

Why is research problem important?

A research problem is crucial because it identifies knowledge gaps, directs the inquiry’s focus, and forms the foundation for generating hypotheses or questions. It drives the methodology and determination of study relevance, ensuring that research contributes meaningfully to academic discourse and potentially addresses real-world challenges.

How do you write a research problem?

To write a research problem, identify a knowledge gap or an unresolved issue in your field. Start with a broad topic, then narrow it down. Clearly articulate the problem in a concise statement, ensuring it’s researchable, significant, and relevant. Ground it in the existing literature to highlight its importance and context.

How can we solve research problem?

To solve a research problem, start by conducting a thorough literature review. Formulate hypotheses or research questions. Choose an appropriate research methodology. Collect and analyse data systematically. Interpret findings in the context of existing knowledge. Ensure validity and reliability, and discuss implications, limitations, and potential future research directions.

You May Also Like

How to write a hypothesis for dissertation,? A hypothesis is a statement that can be tested with the help of experimental or theoretical research.

This article is a step-by-step guide to how to write statement of a problem in research. The research problem will be half-solved by defining it correctly.

Make sure that your selected topic is intriguing, manageable, and relevant. Here are some guidelines to help understand how to find a good dissertation topic.

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Organizing Your Social Sciences Research Paper

• 1. Choosing a Research Problem
• Purpose of Guide
• Design Flaws to Avoid
• Independent and Dependent Variables
• Glossary of Research Terms
• Reading Research Effectively
• Narrowing a Topic Idea
• Broadening a Topic Idea
• Extending the Timeliness of a Topic Idea
• Academic Writing Style
• Applying Critical Thinking
• Choosing a Title
• Making an Outline
• Paragraph Development
• Research Process Video Series
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• The C.A.R.S. Model
• Background Information
• The Research Problem/Question
• Theoretical Framework
• Citation Tracking
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• Evaluating Sources
• Primary Sources
• Secondary Sources
• Tiertiary Sources
• Scholarly vs. Popular Publications
• Qualitative Methods
• Quantitative Methods
• Insiderness
• Using Non-Textual Elements
• Limitations of the Study
• Common Grammar Mistakes
• Writing Concisely
• Avoiding Plagiarism
• Footnotes or Endnotes?
• Further Readings
• Generative AI and Writing
• USC Libraries Tutorials and Other Guides
• Bibliography

In the social and behavioral sciences, the subject of analysis is most often framed as a problem that must be researched in order to obtain a greater understanding, formulate a set of solutions or recommended courses of action, and/or develop a better approach to practice. The research problem, therefore, is the main organizing principle guiding the analysis of your research. The problem under investigation establishes an occasion for writing and a focus that governs what you want to say. It represents the core subject matter of scholarly communication and the means by which scholars arrive at other topics of conversation and the discovery of new knowledge and understanding.

Alvesson, Mats and Jörgen Sandberg. Constructing Research Questions: Doing Interesting Research . London: Sage, 2013; Jacobs, Ronald L. “Developing a Dissertation Research Problem: A Guide for Doctoral Students in Human Resource Development and Adult Education.” New Horizons in Adult Education and Human Resource Development 25 (Summer 2013): 103-117; Chapter 1: Research and the Research Problem. Nicholas Walliman . Your Research Project: Designing and Planning Your Work . 3rd edition. Thousand Oaks, CA: Sage Publications, 2011.

Choosing a Research Problem / How to Begin

Do not assume that identifying a research problem to investigate will be a quick and easy task! You should be thinking about it during the beginning of the course. There are generally three ways you are asked to write about a research problem : 1) your professor provides you with a general topic from which you study a particular aspect; 2) your professor provides you with a list of possible topics to study and you choose a topic from that list; or, 3) your professor leaves it up to you to choose a topic and you only have to obtain permission to write about it before beginning your investigation. Here are some strategies for getting started for each scenario.

I.  How To Begin:  You are given the topic to write about

Step 1 : Identify concepts and terms that make up the topic statement . For example, your professor wants the class to focus on the following research problem: “Is the European Union a credible security actor with the capacity to contribute to confronting global terrorism?" The main concepts in this problem are: European Union, security, global terrorism, credibility [ hint : focus on identifying proper nouns, nouns or noun phrases, and action verbs in the assignment description]. Step 2 : Review related literature to help refine how you will approach examining the topic and finding a way to analyze it . You can begin by doing any or all of the following: reading through background information from materials listed in your course syllabus; searching the USC Libraries Catalog to find a recent book on the topic and, if appropriate, more specialized works about the topic; conducting a preliminary review of the research literature using multidisciplinary databases such as ProQuest or subject-specific databases from the " By Subject Area " drop down menu located above the list of databases.

Choose the advanced search option in the database and enter into each search box the main concept terms you developed in Step 1. Also consider using their synonyms to retrieve additional relevant records. This will help you refine and frame the scope of the research problem. You will likely need to do this several times before you can finalize how to approach writing about the topic. NOTE: Always review the references from your most relevant research results cited by the authors in footnotes, endnotes, or a bibliography to locate related research on your topic. This is a good strategy for identifying important prior research about the topic because titles that are repeatedly cited indicate their significance in laying a foundation for understanding the problem. However, if you’re having trouble at this point locating relevant research literature, ask a librarian for help!

ANOTHER NOTE:   If you find an article from a database that's particularly helpful, paste it into Google Scholar , placing the title of the article in quotes. If the article record appears, look for a "cited by" reference followed by a number [e.g., C ited by 37] just below the record. This link indicates how many times other scholars have subsequently cited that article in their own research since it was first published. This is an effective strategy for identifying more current, related research on your topic. Finding additional cited by references from your original list of cited by references helps you navigate through the literature and, by so doing, understand the evolution of thought around a particular research problem. Step 3 : Since social science research papers are generally designed to encourage you to develop your own ideas and arguments, look for sources that can help broaden, modify, or strengthen your initial thoughts and arguments. For example, if you decide to argue that the European Union is inadequately prepared to take on responsibilities for broader global security because of the debt crisis in many EU countries, then focus on identifying sources that support as well as refute this position. From the advanced search option in ProQuest , a sample search would use "European Union" in one search box, "global security" in the second search box, and adding a third search box to include "debt crisis."

There are least four appropriate roles your related literature plays in helping you formulate how to begin your analysis :

• Sources of criticism -- frequently, you'll find yourself reading materials that are relevant to your chosen topic, but you disagree with the author's position. Therefore, one way that you can use a source is to describe the counter-argument, provide evidence from your own review of the literature as to why the prevailing argument is unsatisfactory, and to discuss how your approach is more appropriate based upon your interpretation of the evidence.
• Sources of new ideas -- while a general goal in writing college research papers in the social sciences is to examine a research problem with some basic idea of what position you'd like to take and on what basis you'd like to defend your position, it is certainly acceptable [and often encouraged] to read the literature and extend, modify, and refine your own position in light of the ideas proposed by others. Just make sure that you cite the sources !
• Sources for historical context -- another role your related literature plays in formulating how to begin your analysis is to place issues and events in proper historical context. This can help to demonstrate familiarity with developments in relevant scholarship about your topic, provide a means of comparing historical versus contemporary issues and events, and identifying key people, places, and events that had an important role related to the research problem. Given its archival journal coverage, a good multidisciplnary database to use in this case is JSTOR .
• Sources of interdisciplinary insight -- an advantage of using databases like ProQuest to begin exploring your topic is that it covers publications from a variety of different disciplines. Another way to formulate how to study the topic is to look at it from different disciplinary perspectives. If the topic concerns immigration reform, for example, ask yourself, how do studies from sociological journals found by searching ProQuest vary in their analysis from those in political science journals. A goal in reviewing related literature is to provide a means of approaching a topic from multiple perspectives rather than the perspective offered from just one discipline.

NOTE: Remember to keep careful notes at every stage or utilize a citation management system like EndNotes or RefWorks . You may think you'll remember what you have searched and where you found things, but it’s easy to forget or get confused. Most databases have a search history feature that allows you to go back and see what searches you conducted previously as long as you haven't closed your session. If you start over, that history could be deleted.

Step 4 : Assuming you have done an effective job of synthesizing and thinking about the results of your initial search for related literature, you're ready to prepare a detailed outline for your paper that lays the foundation for a more in-depth and focused review of relevant research literature [after consulting with a librarian, if needed!]. How will you know you haven't done an effective job of synthesizing and thinking about the results of our initial search for related literature? A good indication is that you start composing the outline and gaps appear in how you want to approach the study. This indicates the need to gather further background information and analysis about the research problem.

II.  How To Begin:  You are provided a list of possible topics to choose from Step 1 : I know what you’re thinking--which topic on this list will be the easiest to find the most information on? An effective instructor would never include a topic that is so obscure or complex that no research is available to examine and from which to design an effective study. Therefore, don't approach a list of possible topics to study from the perspective of trying to identify the path of least resistance; choose a topic that you find interesting in some way, that is controversial and that you have a strong opinion about, that has some personal meaning for you, or relates to your major or a minor. You're going to be working on the topic for quite some time, so choose one that you find interesting and engaging or that motivates you to take a position. Embrace the opportunity to learn something new! Once you’ve settled on a topic of interest from the list provided by your professor, follow Steps 1 - 4 listed above to further develop it into a research paper.

NOTE: It’s ok to review related literature to help refine how you will approach analyzing a topic, and then discover that the topic isn’t all that interesting to you. In that case, choose a different topic from the list. Just don’t wait too long to make a switch and, of course, be sure to inform your professor that you are changing your topic.

III.  How To Begin:  Your professor leaves it up to you to choose a topic

Step 1 : Under this scenario, the key process is turning an idea or general thought into a topic that can be configured into a research problem. When given an assignment where you choose the topic, don't begin by thinking about what to write about, but rather, ask yourself the question, "What do I want to understand or learn about?" Treat an open-ended research assignment as an opportunity to gain new knowledge about something that's important or exciting to you in the context of the overall subject of the course.

Step 2 : If you lack ideas, or wish to gain focus, try any or all of the following strategies:

• Review your course readings, particularly the suggested readings, for topic ideas. Don't just review what you've already read, but jump ahead in the syllabus to readings that have not been covered yet.
• Search the USC Libraries Catalog for a recently published book and, if appropriate, more specialized works related to the discipline area of the course [e.g., for the course SOCI 335: Society and Population, search for books on "population and society" or "population and social impact"]. Reviewing the contents of a book about your area of interest can give you insight into what conversations scholars are having about the topic and, thus, how you might want to contribute your own ideas to these conversations through the research paper you write for the class.
• Browse through some current scholarly [a.k.a., academic, peer reviewed] journals in your subject discipline. Even if most of the articles are not relevant, you can skim through the contents quickly. You only need one to be the spark that begins the process of wanting to learn more about a topic. Consult with a librarian and/or your professor about what constitutes the core journals within the subject area of the writing assignment.
• Think about essays you have written for other courses you have taken or academic lectures and programs you have attended outside of class. Thinking back, ask yourself why did you want to take this class or attend this event? What interested you the most? What would you like to know more about? Place this question in the context of the current course assignment. Note that this strategy also applies to anything you've watched on TV or has been shared on social media.
• Search online news media sources, such as CNN , the Los Angeles Times , Huffington Post , MSNBC , Fox News , or Newsweek , to see if your idea has been covered by the media. Use this coverage to refine your idea into something that you'd like to investigate further, but in a more deliberate, scholarly way in relation to a particular problem that needs to be researched.

Step 3 : To build upon your initial idea, use the suggestions under this tab to help narrow , broaden , or increase the timeliness of your idea so you can write it out as a research problem.

Once you are comfortable with having turned your idea into a research problem, follow Steps 1 - 4 listed in Part I above to further develop it into an outline for a research paper.

Alderman, Jim. "Choosing a Research Topic." Beginning Library and Information Systems Strategies. Paper 17. Jacksonville, FL: University of North Florida Digital Commons, 2014; Alvesson, Mats and Jörgen Sandberg. Constructing Research Questions: Doing Interesting Research . London: Sage, 2013; Chapter 2: Choosing a Research Topic. Adrian R. Eley. Becoming a Successful Early Career Researcher . New York: Routledge, 2012; Answering the Question. Academic Skills Centre. University of Canberra; Brainstorming. Department of English Writing Guide. George Mason University; Brainstorming. The Writing Center. University of North Carolina; Chapter 1: Research and the Research Problem. Nicholas Walliman . Your Research Project: Designing and Planning Your Work . 3rd edition. Thousand Oaks, CA: Sage Publications, 2011; Choosing a Topic. The Writing Lab and The OWL. Purdue University;  Mullaney, Thomas S. and Christopher Rea. Where Research Begins: Choosing a Research Project That Matters to You (and the World) . Chicago, IL: University of Chicago Press, 2022; Coming Up With Your Topic. Institute for Writing Rhetoric. Dartmouth College; How To Write a Thesis Statement. Writing Tutorial Services, Center for Innovative Teaching and Learning. Indiana University; Identify Your Question. Start Your Research. University Library, University of California, Santa Cruz; The Process of Writing a Research Paper. Department of History. Trent University; Trochim, William M.K. Problem Formulation. Research Methods Knowledge Base. 2006.

Resources for Identifying a Topic

Resources for Identifying a Research Problem

If you are having difficulty identifying a topic to study or need basic background information, the following web resources and databases can be useful:

• CQ Researcher -- a collection of single-themed public policy reports that provide an overview of an issue. Each report includes background information, an assessment of the current policy situation, statistical tables and maps, pro/con statements from representatives of opposing positions, and a bibliography of key sources.
• New York Times Topics -- each topic page collects news articles, reference and archival information, photos, graphics, audio and video files. Content is available without charge on articles going back to 1981.
• Opposing Viewpoints In Context -- an online resource covering a wide range of social issues from a variety of perspectives. The database contains a media-rich collection of materials, including pro/con viewpoint essays, topic overviews, primary source materials, biographies of social activists and reformers, journal articles, statistical tables, charts and graphs, images, videos, and podcasts.
• Policy Commons -- platform for objective, fact-based research from the world’s leading policy experts, nonpartisan think tanks, and intergovernmental and non-governmental organizations. The database provides advanced searching across millions of pages of books, articles, working papers, reports, policy briefs, data sets, tables, charts, media, case studies, and statistical publications, including archived reports from more than 200 defunct think tanks. Coverage is international in scope.

Descriptions of resources are adapted or quoted from vendor websites.

Writing Tip

Not Finding Anything on Your Topic? Ask a Librarian!

Don't assume or jump to the conclusion that your topic is too narrowly defined or obscure just because your initial search has failed to identify relevant research. Librarians are experts in locating and critically assessing information and how it is organized. This knowledge will help you develop strategies for analyzing existing knowledge in new ways. Therefore, always consult with a librarian before you consider giving up on finding information about the topic you want to investigate. If there isn't a lot of information about your topic, a librarian can help you identify a closely related topic that you can study. Use the Ask-A-Librarian link above to identify a librarian in your subject area.

Another Writing Tip

Don't be a Martyr!

In thinking about what to study, don't adopt the mindset of pursuing an esoteric or overly complicated topic just to impress your professor but that, in reality, does not have any real interest to you. Choose a topic that is challenging but that has at least some interest to you or that you care about. Obviously, this is easier for courses within your major, but even for those nasty prerequisite classes that you must take in order to graduate [and that provide an additional tuition revenue for the university], try to apply issues associated with your major to the general topic given to you. For example, if you are an international relations major taking a GE philosophy class where the assignment asks you to apply the question of "what is truth" to some aspect of life, you could choose to study how government leaders attempt to shape truth through the use of nationalistic propaganda.

Still Another Writing Tip

A Research Problem is Not a Thesis Statement

A thesis statement and a research problem are two different parts of the introduction section of your paper. The thesis statement succinctly describes in one or two sentences, usually in the last paragraph of the introduction, what position you have reached about a topic. It includes an assertion that requires evidence and support along with your opinion or argument about what you are researching. There are three general types of thesis statements: analytical statements that break down and evaluate the topic; argumentative statements that make a claim about the topic and defend that claim; and, expository statements that present facts and research about the topic. Each are intended to set forth a claim that you will seek to validate through the research you describe in your paper.

Before the thesis statement, your introduction must include a description of a problem that describes either a key area of concern, a condition to be improved upon, a difficulty to be eliminated, or a troubling issue that exists . The research problem describes something that can be empirically verified and measured; it is often followed by a set of questions that underpin how you plan to approach investigating that problem. In short, the thesis statement states your opinion or argument about the research problem and summarizes how you plan to address it.

Tips and Examples for Writing Thesis Statements. The Writing Lab and The OWL. Purdue University; Write a Strong Thesis Statement! The Writing Center, University of Evansville; Thesis Statements. The Writing Center. University of North Carolina; Tutorial #26: Thesis Statements and Topic Sentences. Writing Center, College of San Mateo; Creswell,  John W. and J. David Creswell. Research Design: Qualitative, Quantitative, and Mixed Methods Approaches . 5th edition. Thousand Oaks, CA: SAGE Publications, 2017.

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How to identify and resolve research problems

Updated July 12, 2023

In this article, we’re going to take you through one of the most pertinent parts of conducting research: a research problem (also known as a research problem statement).

When trying to formulate a good research statement, and understand how to solve it for complex projects, it can be difficult to know where to start.

Not only are there multiple perspectives (from stakeholders to project marketers who want answers), you have to consider the particular context of the research topic: is it timely, is it relevant and most importantly of all, is it valuable?

In other words: are you looking at a research worthy problem?

The fact is, a well-defined, precise, and goal-centric research problem will keep your researchers, stakeholders, and business-focused and your results actionable.

And when it works well, it's a powerful tool to identify practical solutions that can drive change and secure buy-in from your workforce.

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What is a research problem?

In social research methodology and behavioral sciences , a research problem establishes the direction of research, often relating to a specific topic or opportunity for discussion.

For example: climate change and sustainability, analyzing moral dilemmas or wage disparity amongst classes could all be areas that the research problem focuses on.

As well as outlining the topic and/or opportunity, a research problem will explain:

• why the area/issue needs to be addressed,
• why the area/issue is of importance,
• the parameters of the research study
• the research objective
• the reporting framework for the results and
• what the overall benefit of doing so will provide (whether to society as a whole or other researchers and projects).

Having identified the main topic or opportunity for discussion, you can then narrow it down into one or several specific questions that can be scrutinized and answered through the research process.

What are research questions?

Generating research questions underpinning your study usually starts with problems that require further research and understanding while fulfilling the objectives of the study.

A good problem statement begins by asking deeper questions to gain insights about a specific topic.

For example, using the problems above, our questions could be:

"How will climate change policies influence sustainability standards across specific geographies?"

"What measures can be taken to address wage disparity without increasing inflation?"

Developing a research worthy problem is the first step - and one of the most important - in any kind of research.

It’s also a task that will come up again and again because any business research process is cyclical. New questions arise as you iterate and progress through discovering, refining, and improving your products and processes. A research question can also be referred to as a "problem statement".

Note: good research supports multiple perspectives through empirical data. It’s focused on key concepts rather than a broad area, providing readily actionable insight and areas for further research.

Research question or research problem?

As we've highlighted, the terms “research question” and “research problem” are often used interchangeably, becoming a vague or broad proposition for many.

The term "problem statement" is far more representative, but finds little use among academics.

Instead, some researchers think in terms of a single research problem and several research questions that arise from it.

As mentioned above, the questions are lines of inquiry to explore in trying to solve the overarching research problem.

Ultimately, this provides a more meaningful understanding of a topic area.

It may be useful to think of questions and problems as coming out of your business data – that’s the O-data (otherwise known as operational data) like sales figures and website metrics.

What's an example of a research problem?

Your overall research problem could be: "How do we improve sales across EMEA and reduce lost deals?"

This research problem then has a subset of questions, such as:

"Why do sales peak at certain times of the day?"

"Why are customers abandoning their online carts at the point of sale?"

As well as helping you to solve business problems, research problems (and associated questions) help you to think critically about topics and/or issues (business or otherwise). You can also use your old research to aid future research -- a good example is laying the foundation for comparative trend reports or a complex research project.

(Also, if you want to see the bigger picture when it comes to research problems, why not check out our ultimate guide to market research? In it you'll find out: what effective market research looks like, the use cases for market research, carrying out a research study, and how to examine and action research findings).

The research process: why are research problems important?

A research problem has two essential roles in setting your research project on a course for success.

1. They set the scope

The research problem defines what problem or opportunity you’re looking at and what your research goals are. It stops you from getting side-tracked or allowing the scope of research to creep off-course .

Without a strong research problem or problem statement, your team could end up spending resources unnecessarily, or coming up with results that aren’t actionable - or worse, harmful to your business - because the field of study is too broad.

2. They tie your work to business goals and actions

To formulate a research problem in terms of business decisions means you always have clarity on what’s needed to make those decisions. You can show the effects of what you’ve studied using real outcomes.

Then, by focusing your research problem statement on a series of questions tied to business objectives, you can reduce the risk of the research being unactionable or inaccurate.

It's also worth examining research or other scholarly literature (you’ll find plenty of similar, pertinent research online) to see how others have explored specific topics and noting implications that could have for your research.

Four steps to defining your research problem

Image credit: http://myfreeschooltanzania.blogspot.com/2014/11/defining-research-problem.html

1. Observe and identify

Businesses today have so much data that it can be difficult to know which problems to address first. Researchers also have business stakeholders who come to them with problems they would like to have explored. A researcher’s job is to sift through these inputs and discover exactly what higher-level trends and key concepts are worth investing in.

This often means asking questions and doing some initial investigation to decide which avenues to pursue. This could mean gathering interdisciplinary perspectives identifying additional expertise and contextual information.

Sometimes, a small-scale preliminary study might be worth doing to help get a more comprehensive understanding of the business context and needs, and to make sure your research problem addresses the most critical questions.

This could take the form of qualitative research using a few in-depth interviews , an environmental scan, or reviewing relevant literature.

The sales manager of a sportswear company has a problem: sales of trail running shoes are down year-on-year and she isn’t sure why. She approaches the company’s research team for input and they begin asking questions within the company and reviewing their knowledge of the wider market.

2. Review the key factors involved

As a marketing researcher, you must work closely with your team of researchers to define and test the influencing factors and the wider context involved in your study. These might include demographic and economic trends or the business environment affecting the question at hand. This is referred to as a relational research problem.

To do this, you have to identify the factors that will affect the research and begin formulating different methods to control them.

You also need to consider the relationships between factors and the degree of control you have over them. For example, you may be able to control the loading speed of your website but you can’t control the fluctuations of the stock market.

Doing this will help you determine whether the findings of your project will produce enough information to be worth the cost.

You need to determine:

• which factors affect the solution to the research proposal.
• which ones can be controlled and used for the purposes of the company, and to what extent.
• the functional relationships between the factors.
• which ones are critical to the solution of the research study.

The research team at the running shoe company is hard at work. They explore the factors involved and the context of why YoY sales are down for trail shoes, including things like what the company’s competitors are doing, what the weather has been like – affecting outdoor exercise – and the relative spend on marketing for the brand from year to year.

The final factor is within the company’s control, although the first two are not. They check the figures and determine marketing spend has a significant impact on the company.

3. Prioritize

Once you and your research team have a few observations, prioritize them based on their business impact and importance. It may be that you can answer more than one question with a single study, but don’t do it at the risk of losing focus on your overarching research problem.

Questions to ask:

• Who? Who are the people with the problem? Are they end-users, stakeholders, teams within your business? Have you validated the information to see what the scale of the problem is?
• What? What is its nature and what is the supporting evidence?
• Why? What is the business case for solving the problem? How will it help?
• Where? How does the problem manifest and where is it observed?

To help you understand all dimensions, you might want to consider focus groups or preliminary interviews with external (including consumers and existing customers) and internal (salespeople, managers, and other stakeholders) parties to provide what is sometimes much-needed insight into a particular set of questions or problems.

After observing and investigating, the running shoe researchers come up with a few candidate questions, including:

• What is the relationship between US average temperatures and sales of our products year on year?
• At present, how does our customer base rank Competitor X and Competitor Y’s trail running shoe compared to our brand?
• What is the relationship between marketing spend and trail shoe product sales over the last 12 months?

They opt for the final question, because the variables involved are fully within the company’s control, and based on their initial research and stakeholder input, seem the most likely cause of the dive in sales. The research question is specific enough to keep the work on course towards an actionable result, but it allows for a few different avenues to be explored, such as the different budget allocations of offline and online marketing and the kinds of messaging used.

Get feedback from the key teams within your business to make sure everyone is aligned and has the same understanding of the research problem and questions, and the actions you hope to take based on the results. Now is also a good time to demonstrate the ROI of your research and lay out its potential benefits to your stakeholders.

Different groups may have different goals and perspectives on the issue. This step is vital for getting the necessary buy-in and pushing the project forward.

The running shoe company researchers now have everything they need to begin. They call a meeting with the sales manager and consult with the product team, marketing team, and C-suite to make sure everyone is aligned and has bought into the direction of the research topic. They identify and agree that the likely course of action will be a rethink of how marketing resources are allocated, and potentially testing out some new channels and messaging strategies .

Can you explore a broad area and is it practical to do so?

A broader research problem or report can be a great way to bring attention to prevalent issues, societal or otherwise, but are often undertaken by those with the resources to do so.

Take a typical government cybersecurity breach survey, for example. Most of these reports raise awareness of cybercrime, from the day-to-day threats businesses face to what security measures some organizations are taking. What these reports don't do, however, is provide actionable advice - mostly because every organization is different.

The point here is that while some researchers will explore a very complex issue in detail, others will provide only a snapshot to maintain interest and encourage further investigation. The "value" of the data is wholly determined by the recipients of it - and what information you choose to include.

To summarize, it can be practical to undertake a broader research problem, certainly, but it may not be possible to cover everything or provide the detail your audience needs. Likewise, a more systematic investigation of an issue or topic will be more valuable, but you may also find that you cover far less ground.

It's important to think about your research objectives and expected findings before going ahead.

Ensuring your research project is a success

A complex research project can be made significantly easier with clear research objectives, a descriptive research problem, and a central focus. All of which we've outlined in this article.

If you have previous research, even better. Use it as a benchmark

Remember: what separates a good research paper from an average one is actually very simple: valuable, empirical data that explores a prevalent societal or business issue and provides actionable insights.

And we can help.

Sophisticated research made simple with Qualtrics

Trusted by the world's best brands, our platform enables researchers from academic to corporate to tackle the hardest challenges and deliver the results that matter.

Our CoreXM platform supports the methods that define superior research and delivers insights in real-time. It's easy to use (thanks to drag-and-drop functionality) and requires no coding, meaning you'll be capturing data and gleaning insights in no time.

It also excels in flexibility; you can track consumer behavior across segments , benchmark your company versus competitors , carry out complex academic research, and do much more, all from one system.

It's one platform with endless applications, so no matter your research problem, we've got the tools to help you solve it. And if you don't have a team of research experts in-house, our market research team has the practical knowledge and tools to help design the surveys and find the respondents you need.

Of course, you may want to know where to begin with your own market research . If you're struggling, make sure to download our ultimate guide using the link below.

It's got everything you need and there’s always information in our research methods knowledge base.

Scott Smith

Scott Smith, Ph.D. is a contributor to the Qualtrics blog.

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The Research Problem & Statement

What they are & how to write them (with examples)

By: Derek Jansen (MBA) | Expert Reviewed By: Eunice Rautenbach (DTech) | March 2023

If you’re new to academic research, you’re bound to encounter the concept of a “ research problem ” or “ problem statement ” fairly early in your learning journey. Having a good research problem is essential, as it provides a foundation for developing high-quality research, from relatively small research papers to a full-length PhD dissertations and theses.

In this post, we’ll unpack what a research problem is and how it’s related to a problem statement . We’ll also share some examples and provide a step-by-step process you can follow to identify and evaluate study-worthy research problems for your own project.

Overview: Research Problem 101

What is a research problem.

• What is a problem statement?

Where do research problems come from?

• How to find a suitable research problem
• Key takeaways

A research problem is, at the simplest level, the core issue that a study will try to solve or (at least) examine. In other words, it’s an explicit declaration about the problem that your dissertation, thesis or research paper will address. More technically, it identifies the research gap that the study will attempt to fill (more on that later).

Let’s look at an example to make the research problem a little more tangible.

To justify a hypothetical study, you might argue that there’s currently a lack of research regarding the challenges experienced by first-generation college students when writing their dissertations [ PROBLEM ] . As a result, these students struggle to successfully complete their dissertations, leading to higher-than-average dropout rates [ CONSEQUENCE ]. Therefore, your study will aim to address this lack of research – i.e., this research problem [ SOLUTION ].

A research problem can be theoretical in nature, focusing on an area of academic research that is lacking in some way. Alternatively, a research problem can be more applied in nature, focused on finding a practical solution to an established problem within an industry or an organisation. In other words, theoretical research problems are motivated by the desire to grow the overall body of knowledge , while applied research problems are motivated by the need to find practical solutions to current real-world problems (such as the one in the example above).

As you can probably see, the research problem acts as the driving force behind any study , as it directly shapes the research aims, objectives and research questions , as well as the research approach. Therefore, it’s really important to develop a very clearly articulated research problem before you even start your research proposal . A vague research problem will lead to unfocused, potentially conflicting research aims, objectives and research questions .

What is a research problem statement?

As the name suggests, a problem statement (within a research context, at least) is an explicit statement that clearly and concisely articulates the specific research problem your study will address. While your research problem can span over multiple paragraphs, your problem statement should be brief , ideally no longer than one paragraph . Importantly, it must clearly state what the problem is (whether theoretical or practical in nature) and how the study will address it.

Here’s an example of a statement of the problem in a research context:

Rural communities across Ghana lack access to clean water, leading to high rates of waterborne illnesses and infant mortality. Despite this, there is little research investigating the effectiveness of community-led water supply projects within the Ghanaian context. Therefore, this study aims to investigate the effectiveness of such projects in improving access to clean water and reducing rates of waterborne illnesses in these communities.

As you can see, this problem statement clearly and concisely identifies the issue that needs to be addressed (i.e., a lack of research regarding the effectiveness of community-led water supply projects) and the research question that the study aims to answer (i.e., are community-led water supply projects effective in reducing waterborne illnesses?), all within one short paragraph.

Need a helping hand?

Wherever there is a lack of well-established and agreed-upon academic literature , there is an opportunity for research problems to arise, since there is a paucity of (credible) knowledge. In other words, research problems are derived from research gaps . These gaps can arise from various sources, including the emergence of new frontiers or new contexts, as well as disagreements within the existing research.

Let’s look at each of these scenarios:

New frontiers – new technologies, discoveries or breakthroughs can open up entirely new frontiers where there is very little existing research, thereby creating fresh research gaps. For example, as generative AI technology became accessible to the general public in 2023, the full implications and knock-on effects of this were (or perhaps, still are) largely unknown and therefore present multiple avenues for researchers to explore.

New contexts – very often, existing research tends to be concentrated on specific contexts and geographies. Therefore, even within well-studied fields, there is often a lack of research within niche contexts. For example, just because a study finds certain results within a western context doesn’t mean that it would necessarily find the same within an eastern context. If there’s reason to believe that results may vary across these geographies, a potential research gap emerges.

Disagreements – within many areas of existing research, there are (quite naturally) conflicting views between researchers, where each side presents strong points that pull in opposing directions. In such cases, it’s still somewhat uncertain as to which viewpoint (if any) is more accurate. As a result, there is room for further research in an attempt to “settle” the debate.

Of course, many other potential scenarios can give rise to research gaps, and consequently, research problems, but these common ones are a useful starting point. If you’re interested in research gaps, you can learn more here .

How to find a research problem

Given that research problems flow from research gaps , finding a strong research problem for your research project means that you’ll need to first identify a clear research gap. Below, we’ll present a four-step process to help you find and evaluate potential research problems.

If you’ve read our other articles about finding a research topic , you’ll find the process below very familiar as the research problem is the foundation of any study . In other words, finding a research problem is much the same as finding a research topic.

Step 1 – Identify your area of interest

Naturally, the starting point is to first identify a general area of interest . Chances are you already have something in mind, but if not, have a look at past dissertations and theses within your institution to get some inspiration. These present a goldmine of information as they’ll not only give you ideas for your own research, but they’ll also help you see exactly what the norms and expectations are for these types of projects.

At this stage, you don’t need to get super specific. The objective is simply to identify a couple of potential research areas that interest you. For example, if you’re undertaking research as part of a business degree, you may be interested in social media marketing strategies for small businesses, leadership strategies for multinational companies, etc.

Depending on the type of project you’re undertaking, there may also be restrictions or requirements regarding what topic areas you’re allowed to investigate, what type of methodology you can utilise, etc. So, be sure to first familiarise yourself with your institution’s specific requirements and keep these front of mind as you explore potential research ideas.

Step 2 – Review the literature and develop a shortlist

Once you’ve decided on an area that interests you, it’s time to sink your teeth into the literature . In other words, you’ll need to familiarise yourself with the existing research regarding your interest area. Google Scholar is a good starting point for this, as you can simply enter a few keywords and quickly get a feel for what’s out there. Keep an eye out for recent literature reviews and systematic review-type journal articles, as these will provide a good overview of the current state of research.

At this stage, you don’t need to read every journal article from start to finish . A good strategy is to pay attention to the abstract, intro and conclusion , as together these provide a snapshot of the key takeaways. As you work your way through the literature, keep an eye out for what’s missing – in other words, what questions does the current research not answer adequately (or at all)? Importantly, pay attention to the section titled “ further research is needed ”, typically found towards the very end of each journal article. This section will specifically outline potential research gaps that you can explore, based on the current state of knowledge (provided the article you’re looking at is recent).

Take the time to engage with the literature and develop a big-picture understanding of the current state of knowledge. Reviewing the literature takes time and is an iterative process , but it’s an essential part of the research process, so don’t cut corners at this stage.

As you work through the review process, take note of any potential research gaps that are of interest to you. From there, develop a shortlist of potential research gaps (and resultant research problems) – ideally 3 – 5 options that interest you.

Step 3 – Evaluate your potential options

Once you’ve developed your shortlist, you’ll need to evaluate your options to identify a winner. There are many potential evaluation criteria that you can use, but we’ll outline three common ones here: value, practicality and personal appeal.

Value – a good research problem needs to create value when successfully addressed. Ask yourself:

• Who will this study benefit (e.g., practitioners, researchers, academia)?
• How will it benefit them specifically?
• How much will it benefit them?

Practicality – a good research problem needs to be manageable in light of your resources. Ask yourself:

• What data will I need access to?
• What knowledge and skills will I need to undertake the analysis?
• What equipment or software will I need to process and/or analyse the data?
• How much time will I need?
• What costs might I incur?

Personal appeal – a research project is a commitment, so the research problem that you choose needs to be genuinely attractive and interesting to you. Ask yourself:

• How appealing is the prospect of solving this research problem (on a scale of 1 – 10)?
• Why, specifically, is it attractive (or unattractive) to me?
• Does the research align with my longer-term goals (e.g., career goals, educational path, etc)?

Depending on how many potential options you have, you may want to consider creating a spreadsheet where you numerically rate each of the options in terms of these criteria. Remember to also include any criteria specified by your institution . From there, tally up the numbers and pick a winner.

Step 4 – Craft your problem statement

Once you’ve selected your research problem, the final step is to craft a problem statement. Remember, your problem statement needs to be a concise outline of what the core issue is and how your study will address it. Aim to fit this within one paragraph – don’t waffle on. Have a look at the problem statement example we mentioned earlier if you need some inspiration.

Key Takeaways

We’ve covered a lot of ground. Let’s do a quick recap of the key takeaways:

• A research problem is an explanation of the issue that your study will try to solve. This explanation needs to highlight the problem , the consequence and the solution or response.
• A problem statement is a clear and concise summary of the research problem , typically contained within one paragraph.
• Research problems emerge from research gaps , which themselves can emerge from multiple potential sources, including new frontiers, new contexts or disagreements within the existing literature.
• To find a research problem, you need to first identify your area of interest , then review the literature and develop a shortlist, after which you’ll evaluate your options, select a winner and craft a problem statement .

Psst... there’s more!

This post was based on one of our popular Research Bootcamps . If you're working on a research project, you'll definitely want to check this out ...

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How to formulate research problems?

June 16, 2023 4 min read

One of the most important steps in the research process is formulating a research problem. It establishes the framework for the whole study and directs the researcher in determining the research’s emphasis, scope, and goals. An effective research technique may be created with the support of a clearly defined research topic, which also aids in the generation of pertinent research questions.

This article will provide a general overview of the procedure involved in defining research problems, highlighting important considerations and steps researchers should take to formulate precise and insightful research problems.

What is a research problem?

It refers to a specific topic, problem, or knowledge gap that a researcher aims to study and address through a systematic inquiry. It establishes the foundation for a research project and guides the entire investigation.

When creating a research problem, researchers often start with a topic of interest before focusing on a particular issue or question. A substantial, relevant, and original challenge adds to the corpus of knowledge and has real-world applications.

A clearly stated research topic aids in the concentration of research resources and efforts, permits the development of an effective research technique, and directs the evaluation and interpretation of data acquired. It also helps in developing research goals and hypotheses by giving the investigation a distinct direction.

For instance, a research problem could be “What are the causes leading to the decline of bee populations in urban areas?” — This study challenge addresses a particular set of urban regions and draws attention to the problem of dwindling bee numbers. By focusing on this issue, researchers may analyze the various reasons for the loss, analyze how it affects the environment, and suggest conservation tactics.

Characteristics of an effective research problem

An effective research problem possesses several essential qualities that enhance its quality and suitability for examination. The key characteristics of a strong research problem are:

Significance

Should address an important issue or knowledge gap in the field of study, contributing to the existing body of knowledge.

Should be precisely stated, avoiding vague or overly general statements and providing a clear and concise description. This clarity enables the definition of research objectives and hypotheses and guides the research process.

Feasibility

Should be feasible in terms of the available time, resources, and skills. It can be realistically pursued, given the researcher’s capabilities and study circumstances. Sufficient data, research tools, and potential exploration paths should be reasonably accessible.

Should explore new facets, angles, or dimensions of the subject, offering fresh perspectives or approaches. This characteristic promotes intellectual progress and distinguishes the research from previous investigations.

Measurability

Should be formulated in a way that allows for empirical examination and the generation of quantifiable results. Data can be systematically collected and analyzed to answer the research questions or achieve the research goals, enhancing the objectivity and rigor of the research process.

Relevance and applicability

Should address relevant issues or help develop useful guidelines, regulations, or actions. It is more effective when it impacts multiple stakeholders and has the potential to produce practical results.

Interest and motivation

Should be intellectually engaging and interesting to the researcher and the academic community. It sparks curiosity and encourages further research, leading to high-quality research output.

Ethical consideration

Should adhere to ethical principles and rules, considering the welfare and rights of participants or subjects involved in the study.

ALSO READ: What is research design?

Types of research problems.

Research problems can be categorized into different types based on their nature and scope. The three most common types are:

Theoretical

It involves using theoretical frameworks, concepts, and models to investigate a subject or event. Theoretical research aims to extend existing knowledge, address unsolved disputes or gaps, or critique and evaluate preexisting theories.

It focuses on specific problems or challenges within a particular industry or sector and aims to provide practical solutions through systematic research. Applied research aims to bridge the gap between theory and practical application, optimizing existing processes, technologies, products, or services.

Action research combines research and action to address real-world issues. It encompasses problem-solving in various contexts, such as organizations, education, community development, policy implementation, and personal or professional development. Action research is flexible and can be tailored to different situations and issues.

Importance of research problems

Research problems play a vital role in shaping the direction and course of an investigation. They serve as the foundation for the entire research process, guiding researchers in their pursuit of knowledge and advancement in a specific field. The importance of research problems lies in the following:

Identifying knowledge gaps

Research problems help identify areas where knowledge is lacking or incomplete, highlighting the need for further investigation and addressing unanswered questions.

Providing direction

A well-defined research problem gives the research project focus and direction. It aids in the development of an effective research design, technique and the establishment of research objectives and questions.

Justifying the study’s significance

A clear research problem helps researchers justify the value and importance of their study by emphasizing its relevance, potential benefits, and contributions to the field.

Facilitating problem-solving and decision-making

Research problems often stem from real-world challenges or problems. By examining these problems, researchers can develop innovative ideas, methods, or strategies to solve practical issues or guide decision-making.

Advancing theory and knowledge

Research problems serve as a basis for developing new concepts, hypotheses, or models. By addressing research challenges, researchers contribute to understanding a subject, debunk preexisting beliefs, or propose new hypotheses.

Promoting intellectual curiosity and innovation

Research problems encourage intellectual curiosity and innovation by pushing researchers to explore fresh perspectives and methodologies. By encouraging critical thinking, generating original ideas, and developing unique research approaches, research problems foster innovation and creativity.

ALSO READ: The basics of market research

5 steps to formulate research problems.

Formulating research problems is a crucial initial step in conducting purposeful and targeted research. Here are five steps to follow:

Identify the broad research area

Determine the broad subject or field that interests you, considering discipline-specific topics or specific phenomena.

Conduct a literature review

Review existing literature and research in your chosen field to understand the current knowledge level and identify gaps or unsolved issues and areas requiring further research. Read relevant scholarly publications, books, and articles to gain a comprehensive understanding.

Narrow down the focus

Based on the literature review, select a specific component or subject within your chosen research field. Look for inconsistencies, contradictions, or open-ended questions in the existing literature that can present challenges for future research. Refine your research topic and focus it on a single problem or phenomenon.

Define clear objectives

Establish clear and concise research objectives that outline your investigation’s specific aims or outcomes. SMART (specific, measurable, attainable, relevant, and time-bound) objectives help maintain focus and guide the research process effectively.

Formulate research questions

Create distinct research questions or hypotheses that align with your research problem and objectives. Qualitative research often utilizes research questions, while quantitative research employs hypotheses. Ensure these inquiries or hypotheses are precise, concise, and aimed at addressing the stated research problem.

Remember that formulating research problems is an iterative process. As you learn more about the topic and develop new ideas, it can need several changes and improvements. You may establish a solid basis for your study and improve your chances of performing fruitful and influential research by adhering to these recommendations and continually improving your research problem.

Researchers can create precise and insightful research problems that add to the body of knowledge and progress in their particular fields of study by using the procedures described in this article. A research problem outlines the precise field of inquiry and knowledge gaps that the research attempts to address, defining the scope and objective of a study.

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Types of Research – Explained with Examples

• By DiscoverPhDs
• October 2, 2020

Types of Research

Research is about using established methods to investigate a problem or question in detail with the aim of generating new knowledge about it.

It is a vital tool for scientific advancement because it allows researchers to prove or refute hypotheses based on clearly defined parameters, environments and assumptions. Due to this, it enables us to confidently contribute to knowledge as it allows research to be verified and replicated.

Knowing the types of research and what each of them focuses on will allow you to better plan your project, utilises the most appropriate methodologies and techniques and better communicate your findings to other researchers and supervisors.

Classification of Types of Research

There are various types of research that are classified according to their objective, depth of study, analysed data, time required to study the phenomenon and other factors. It’s important to note that a research project will not be limited to one type of research, but will likely use several.

According to its Purpose

Theoretical research.

Theoretical research, also referred to as pure or basic research, focuses on generating knowledge , regardless of its practical application. Here, data collection is used to generate new general concepts for a better understanding of a particular field or to answer a theoretical research question.

Results of this kind are usually oriented towards the formulation of theories and are usually based on documentary analysis, the development of mathematical formulas and the reflection of high-level researchers.

Applied Research

Here, the goal is to find strategies that can be used to address a specific research problem. Applied research draws on theory to generate practical scientific knowledge, and its use is very common in STEM fields such as engineering, computer science and medicine.

This type of research is subdivided into two types:

• Technological applied research : looks towards improving efficiency in a particular productive sector through the improvement of processes or machinery related to said productive processes.
• Scientific applied research : has predictive purposes. Through this type of research design, we can measure certain variables to predict behaviours useful to the goods and services sector, such as consumption patterns and viability of commercial projects.

According to your Depth of Scope

Exploratory research.

Exploratory research is used for the preliminary investigation of a subject that is not yet well understood or sufficiently researched. It serves to establish a frame of reference and a hypothesis from which an in-depth study can be developed that will enable conclusive results to be generated.

Because exploratory research is based on the study of little-studied phenomena, it relies less on theory and more on the collection of data to identify patterns that explain these phenomena.

Descriptive Research

The primary objective of descriptive research is to define the characteristics of a particular phenomenon without necessarily investigating the causes that produce it.

In this type of research, the researcher must take particular care not to intervene in the observed object or phenomenon, as its behaviour may change if an external factor is involved.

Explanatory Research

Explanatory research is the most common type of research method and is responsible for establishing cause-and-effect relationships that allow generalisations to be extended to similar realities. It is closely related to descriptive research, although it provides additional information about the observed object and its interactions with the environment.

Correlational Research

The purpose of this type of scientific research is to identify the relationship between two or more variables. A correlational study aims to determine whether a variable changes, how much the other elements of the observed system change.

According to the Type of Data Used

Qualitative research.

Qualitative methods are often used in the social sciences to collect, compare and interpret information, has a linguistic-semiotic basis and is used in techniques such as discourse analysis, interviews, surveys, records and participant observations.

In order to use statistical methods to validate their results, the observations collected must be evaluated numerically. Qualitative research, however, tends to be subjective, since not all data can be fully controlled. Therefore, this type of research design is better suited to extracting meaning from an event or phenomenon (the ‘why’) than its cause (the ‘how’).

Quantitative Research

Quantitative research study delves into a phenomena through quantitative data collection and using mathematical, statistical and computer-aided tools to measure them . This allows generalised conclusions to be projected over time.

According to the Degree of Manipulation of Variables

Experimental research.

It is about designing or replicating a phenomenon whose variables are manipulated under strictly controlled conditions in order to identify or discover its effect on another independent variable or object. The phenomenon to be studied is measured through study and control groups, and according to the guidelines of the scientific method.

Non-Experimental Research

Also known as an observational study, it focuses on the analysis of a phenomenon in its natural context. As such, the researcher does not intervene directly, but limits their involvement to measuring the variables required for the study. Due to its observational nature, it is often used in descriptive research.

Quasi-Experimental Research

It controls only some variables of the phenomenon under investigation and is therefore not entirely experimental. In this case, the study and the focus group cannot be randomly selected, but are chosen from existing groups or populations . This is to ensure the collected data is relevant and that the knowledge, perspectives and opinions of the population can be incorporated into the study.

According to the Type of Inference

Deductive investigation.

In this type of research, reality is explained by general laws that point to certain conclusions; conclusions are expected to be part of the premise of the research problem and considered correct if the premise is valid and the inductive method is applied correctly.

Inductive Research

In this type of research, knowledge is generated from an observation to achieve a generalisation. It is based on the collection of specific data to develop new theories.

Hypothetical-Deductive Investigation

It is based on observing reality to make a hypothesis, then use deduction to obtain a conclusion and finally verify or reject it through experience.

According to the Time in Which it is Carried Out

Longitudinal study (also referred to as diachronic research).

It is the monitoring of the same event, individual or group over a defined period of time. It aims to track changes in a number of variables and see how they evolve over time. It is often used in medical, psychological and social areas .

Cross-Sectional Study (also referred to as Synchronous Research)

Cross-sectional research design is used to observe phenomena, an individual or a group of research subjects at a given time.

According to The Sources of Information

Primary research.

This fundamental research type is defined by the fact that the data is collected directly from the source, that is, it consists of primary, first-hand information.

Secondary research

Unlike primary research, secondary research is developed with information from secondary sources, which are generally based on scientific literature and other documents compiled by another researcher.

According to How the Data is Obtained

Documentary (cabinet).

Documentary research, or secondary sources, is based on a systematic review of existing sources of information on a particular subject. This type of scientific research is commonly used when undertaking literature reviews or producing a case study.

Field research study involves the direct collection of information at the location where the observed phenomenon occurs.

From Laboratory

Laboratory research is carried out in a controlled environment in order to isolate a dependent variable and establish its relationship with other variables through scientific methods.

Mixed-Method: Documentary, Field and/or Laboratory

Mixed research methodologies combine results from both secondary (documentary) sources and primary sources through field or laboratory research.

A science investigatory project is a science-based research project or study that is performed by school children in a classroom, exhibition or science fair.

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Research Problem Statement — Find out how to write an impactful one!

Table of Contents

What Is a Research Problem Statement?

A research problem statement is a clear, concise, and specific statement that describes the issue or problem that the research project addresses. It should be written in a way that is easily understandable to both experts and non-experts in the field.

To write a research problem statement, you should:

• Identify the general area of interest: Start by identifying the general area of research that interests you.
• Define the specific problem: Narrow down the general area of interest to a specific problem or issue.
• Explain the significance of the problem: Provide context for the problem by explaining why it is important to study and what gap in current knowledge or understanding it fills.
• Provide a clear and concise statement: State the problem in a clear and concise manner, making sure to use language that is easily understood by your intended audience.
• Use a scientific and objective tone: The problem statement should be written in a neutral and objective tone, avoiding any subjective language and personal bias .

An Example of a Research Problem Statement

“The increasing prevalence of obesity in children is a growing public health concern. Despite the availability of information on healthy eating and physical activity, many children are still not engaging in healthy lifestyle behaviors. The problem this study addresses is the lack of understanding of the barriers and facilitators to healthy lifestyle behaviors in children.”

When to Write a Problem Statement in Research?

A research problem statement should be written at the beginning of the research process, before any data collection or analysis takes place. This is because the statement sets the foundation for the entire research project by clearly defining the problem that the research is trying to address.

Writing a problem statement early in the research process helps to guide the research design and methodology , and ensures that the research is focused on addressing the specific problem at hand. It also helps to ensure that the research is relevant and addresses a gap in current knowledge or understanding.

In addition, a well-written problem statement effectively communicates the purpose and significance of the research to potential funders, collaborators, and other stakeholders. It also generates interest and support for the research project.

It’s also important to note that, during the research process, the statement can be refined or updated as new information is discovered or as the research progresses. This is normal and it’s a good idea to revise the statement as needed to ensure that it remains clear and concise and that it accurately reflects the current focus of the research project.

What Does a Research Problem Statement Include?

A research problem statement typically includes the following elements:

1. The research topic:

The general area of interest or field of study that the research project addresses.

2. The specific problem or issue:

A clear and concise statement of the problem or issue that the research project aims to address.

3. The significance of the problem:

A discussion of why the problem is important and what gap in current knowledge or understanding it fills.

4. The research questions:

A set of questions that the research project aims to answer, in order to address the problem or issue.

5. The research objectives:

A set of specific and measurable objectives that the research project aims to achieve.

6. The scope of the research:

A description of the specific population, setting, or context that the research project will focus on.

7. The theoretical framework:

A discussion of the theoretical concepts and principles that inform the research project.

8. The research design:

A description of the research methodologies that will be used to collect and analyze data in order to address the research questions and objectives.

It’s important to note that the problem statement is usually brief and concise, typically a few sentences or a short paragraph. But it should provide enough information to convey the main idea of the research project.

Important Features of Research Problem Statement

The problem statement should be clear and easy to understand. Write it in a way that is accessible to both experts and non-experts in the field.

2. Specificity

The statement should be specific and clearly define the problem or issue that the research project aims to address. It should be narrow enough to be manageable, but broad enough to be of interest to others in the field.

3. Significance

The statement should explain why the problem is important and what gap in current knowledge or understanding it fills. It should provide context for the research project and help to justify its importance.

4. Relevance

The statement should be relevant to the field of study and address an issue that is currently of concern to researchers.

5. Research questions

The statement should include a set of research questions that the research project aims to answer in order to address the problem or issue.

6. Research objectives

The statement should include a set of specific and measurable objectives that the research project aims to achieve.

The statement should define the specific population, setting, or context that the research project will focus on.

8. Theoretical framework

The statement should provide an overview of the theoretical concepts and principles that inform the research project.

9. Research design

The statement should provide an overview of the research methodologies. This will be useful collect and analyze data in order to address the research questions and objectives.

Difference Between a Thesis Statement and a Problem Statement

A thesis statement and a problem statement are related but distinct elements of a research project.

A thesis statement is a statement that summarizes the central argument or claim of a research paper or essay. It presents the main idea of the paper and sets the direction for the rest of the content. It’s usually located at the end of the introduction, and it’s often one sentence.

A problem statement, on the other hand, is a statement that describes a specific problem or issue that the research project aims to address. It sets the foundation for the entire research project by clearly defining the research problem. It is usually located at the beginning of a research paper or proposal, and is of one or a few paragraphs.

In summary, a thesis statement is a summary of the main point or key argument of the research paper. A problem statement describes the specific issue that the research project aims to address. A thesis statement is more focused on the final outcome of the research. While a problem statement is focused on the current state of knowledge and the gap in understanding that the research project aims to fill.

In Conclusion

A problem statement is a critical component of the research project, as it provides a clear and concise roadmap for the research, and helps to ensure that the research is well-designed and addresses a significant and relevant issue.

We hope this blog has clarified your doubts and confusion associated with research problem statement and helps you write an effective statement for your research project!

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Good and straightforward explanations with ease of understanding for all the students and teachers alike.

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Research Problems: Traits, Types, and Examples

• By Zain Ul Abadin
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Embark on a journey into the world of research problems, where the quest for knowledge and innovation unfolds. Let’s unravel the intricacies together, exploring the types, traits, and real-world examples that shape the landscape of academic inquiry. From the abstract realms of ideas to the tangible evidence of real-world observations, let’s delve into the art of crafting compelling research topics that make a lasting impact.

Introduction

Picture this: You’re standing at the crossroads of academic exploration, armed with curiosity and a compass called research problems. In this guide, we’re not just unraveling the mysteries; we’re offering a roadmap. Let’s understand the importance, types, and traits of research problems with real-world examples spanning diverse disciplines.

Why is the Research Problem Important?

Before we dive into the nitty-gritty, let’s talk about why the research problem holds such significance. Think of it as the North Star for scholars, guiding them through uncharted territories. This section explores why crafting a meaningful research problem is the crucial first step in impactful scholarly contributions.

Types of Research Problems

A. conceptual research problems.

Ever been lost in the labyrinth of ideas? That’s the realm of conceptual research problems. We’ll navigate through the abstract, shaping the intellectual landscape of our chosen discipline with theories and constructs.

B. Empirical Research Problems

Now, let’s ground ourselves. Empirical research problems demand more than just theories; they crave real-world data and observations. It’s about making sense of the tangible, contributing to our practical understanding of phenomena.

Traits of Effective Research Problems

A. specificity and clarity.

Imagine giving someone directions without a GPS. That’s what happens when a research problem lacks specificity and clarity. We’ll explore why being crystal clear is the secret sauce for impactful research.

B. Relevance to the Field

Ever had a conversation that felt out of place? Effective research problems don’t do that. They resonate with ongoing discussions in the field, addressing real issues and contributing meaningfully to the conversation.

C. Feasibility and Scope

Let’s be practical. Effective research problems aren’t just brilliant; they’re doable. We’ll discuss why feasibility and a defined scope are the unsung heroes of well-crafted inquiries.

Characteristics of a Research Problem

Now, let’s delve into the DNA of a research problem. What makes it tick? We’ll explore seven key characteristics that set the stage for impactful research:

Imagine trying to read a map in the dark. Clarity in a research problem is like turning on the lights. It ensures everyone understands the problem, setting the stage for focused inquiry.

2. Precision

Ever tried hitting a bullseye with foggy glasses? Precision narrows down the focus, preventing our research from becoming a blurry mess. It guides us in formulating specific questions and hypotheses.

3. Relevance

Let’s stay in the conversation, not hover on the outskirts. Relevance to the field ensures our research aligns with current trends, issues, and gaps. It’s the key to making our study meaningful.

Picture a library filled with the same book. Novelty distinguishes our research from the crowd, addressing unexplored areas or offering fresh perspectives on existing issues.

5. Significance

Ever painted a masterpiece only to hide it away? Significance ensures our research holds value, making a meaningful impact on the academic community or society at large.

6. Feasibility

Let’s not plan a trip to Mars without a spaceship. Feasibility considers the practical aspects of our research, evaluating resources, time constraints, and ethical considerations.

7. Delimitation

Imagine a recipe with no ingredients. Delimitation sets boundaries, specifying the scope of our study. It prevents our research from becoming overly ambitious or unfocused.

Examples of Research Problems

A. Social Sciences

Our journey takes us through the diverse landscape of social sciences . From exploring the impact of social media on relationships to investigating the dynamics of political ideologies, we’ll discover a rich tapestry of research avenues.

B. Natural Sciences

Now, let’s don our scientific hats. In natural sciences, our research problems may involve unraveling cosmic mysteries, understanding molecular intricacies, or tackling environmental challenges.

Crafting Research Topics for Students

Students, this one’s for you! Crafting research topics isn’t just about grades; it’s about enhancing learning and nurturing critical thinking. Let’s explore how well-defined topics shape your academic journey.

Understanding the Difference

Research problem vs. research question.

Ever been unsure whether to ask for directions or look it up yourself? We’ll clarify the difference between research problems (demanding solutions) and questions (seeking information).

Exploring Research Topics in Biology

Key areas of interest.

Biologists join us on a journey through key areas of interest. From genetics to ecology, we’ll explore research opportunities that contribute meaningfully to the field.

Investigating Research Topics in Chemistry

Emerging trends.

Chemists, buckle up! Chemistry is ever-evolving, and we’ll dive into emerging trends like green chemistry, nanotechnology, and pharmaceuticals that keep you at the forefront.

Defining Research Topics

Let’s delve into the intricate process of defining research topics, where the art lies not only in selection but in seamless alignment with the broader context of our respective disciplines. Crafting topics that resonate with the academic pulse, we navigate the intellectual landscape with precision and purpose, ensuring our inquiries leave a lasting imprint on scholarly discourse.

Navigating Research Problem Meanings

Ever read between the lines? Interpreting meanings in research problems requires finesse. We’ll navigate through nuanced language and subtle distinctions for accurate understanding.

Ensuring Clarity in Research Problem Statements

Common pitfalls.

Crafting clear statements is an art. We’ll highlight common pitfalls researchers often encounter, from vague language to overcomplication, hindering the clarity of their inquiries.

The Art of Formulating Research Problem Ideas

Creativity in problem definition.

Let’s embrace the creative side. Formulating ideas isn’t a checklist; it’s an art. We’ll explore how creativity fosters a vibrant and dynamic academic landscape.

Addressing Burstiness in Research Problems

Ever noticed how no two fingerprints are alike? Burstiness in research problems signifies this variation. We’ll embrace this diversity, bringing depth and diversity to our inquiries.

Perplexity in Research Problem Solving

Let’s talk about the elephant in the room: complexity. Perplexity in research problem solving is acknowledging the intricacies. We’ll embrace this complexity to effectively tackle challenging dilemmas.

Overcoming Challenges in Research Topic Definitions

Tips for Researchers

Challenges are inevitable, but so is resilience. This section offers practical tips to overcome obstacles, refine problem statements, and contribute meaningfully to respective fields.

Understanding types is like having a compass; it guides focused studies. Let’s explore:

1. Descriptive Research Problems

These problems offer a detailed snapshot of a phenomenon. What’s happening? Let’s find out.

2. Exploratory Research Problems

Ever wanted to explore the unknown? These problems uncover new insights and generate hypotheses, laying the groundwork for further investigation.

3. Explanatory Research Problems

Curious about cause and effect? These problems focus on understanding relationships between variables, explaining why certain phenomena occur.

4. Comparative Research Problems

Spot the differences and similarities. These problems involve analyzing entities or groups, valuable for discerning patterns and trends.

5. Intervention Research Problems

Ready for action? These problems address the impact of specific interventions, assessing effectiveness in bringing desired changes.

6. Predictive Research Problems

Ever wanted to predict the future? These problems involve forecasting trends or outcomes based on existing data and patterns.

How to Handle These Research Problems Using AI Tools

In the vast landscape of research, AI tools become companions. Handling problems efficiently is an art, and one noteworthy tool in the market is Blainy. Picture Blainy as your sidekick, offering sophisticated assistance from problem definition to data analysis, streamlining your research journey with efficiency.

Key Takeaways

• Research problems vary in types, from conceptual explorations to empirical investigations, each serving a unique purpose in academic inquiry.
• Effective research problems share common traits such as specificity, relevance, and feasibility.
• Crafting research topics for students is essential for fostering critical thinking and analytical skills in educational settings.
• The distinction between research problems and questions lies in their focus on solutions and information, respectively.
• Burstiness and perplexity in research problems contribute to the richness and complexity of academic exploration.
• Overcoming challenges in research topic definitions requires creativity, precision, and an awareness of common pitfalls.

Now, let’s talk about significance. The research problem isn’t just a puzzle; it’s the heartbeat of scholarly endeavours. This section explores why it shapes academic exploration’s trajectory.

In the ever-evolving realm of academic exploration, AI tools emerge as invaluable companions. Navigating through research problems becomes not just efficient but a journey enriched with insights. Among the noteworthy tools, envision Blainy as your research confidant, adding a touch of sophistication to streamline the entire process. It’s not just problem-solving; it’s a symphony of efficiency and intellect in your scholarly pursuit.

In conclusion, the journey through research problems is an adventure. Mastering the art of formulating potent inquiries is a key for scholars, contributing to the ever-evolving tapestry of human knowledge.

Q1: What is the significance of specificity in research problems?

      Specificity ensures clarity and precision, guiding researchers toward meaningful solutions.

Q2: How can students benefit from crafting well-defined research topics?

      Crafted topics enhance learning, nurturing critical thinking and analytical skills crucial for academic success.

Q3: What challenges do researchers commonly face in formulating research problem statements?

      Common pitfalls include vague language and overcomplication, hindering the clarity of statements.

Q4: How does burstiness contribute to the richness of research inquiries?

      Burstiness enriches the academic landscape, bringing depth and diversity to research inquiries.

Q5: Why is creativity essential in formulating research problem ideas?

      Creativity fosters dynamic perspectives, contributing to a vibrant academic landscape.

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5 Sources of a Research Problem: The Complete Guide

by  Antony W

February 20, 2022

In this guide, you will learn about the best sources of a research problem for your next project. 

The term research problem refers to a clear expression of an area of concern that requires a clear understanding and deliberate investigation. While it offers a broad proposition and a valuable question, a research problem doesn’t demonstrate how to do something.

It’s worth looking at a research problem for a number of reasons. It introduces a reader to the topic under investigation and orients to the importance of the study.

Besides allowing you to define the most important parameter to investigate in your paper, a research problem offers you a concise guide to come up with research questions , make relevant assumptions, and formulate a proposition .

More importantly, a research problem gives you a more comprehensive framework to conduct extensive studies and explain your findings.

Need help with your research paper, dissertation, or thesis but you have no idea where to start? Hire Help for Assessment for Assistance.

Type of Research Problems

There are four types of research problems that you need to know before we look at the sources of a research problem.

These are casuist, difference, descriptive and relational research problems.

1. Relational Research Problem

A relational research problem suggests the need to investigate the correlation between two or more variables.

It’s the researcher’s responsibility to investigate a number of precise characteristics and identify the relationship between them.

2. Casuist Research Problem

Casuist research problem has something to do with the determination of what’s right and what’s wrong.

It questions human conduct by looking closely at the moral dilemmas by means of careful differentiation of cases as well as the application of general rules.

3. Descriptive Research Problem

In this case, a researcher looks forward to investigating a “what is” kind of issue.

The goal of examining a descriptive research problem is to determine the underlying significance of an event or the existence of a situation.

It’s with the descriptive research problem that a researcher can discover understudied or hidden issues.

5. Difference Research Problem

A difference research problem focuses on the distinction between two or more groups.  More often than not, researchers use this type of problem to compare and contrast more than one phenomenon.

What are the Sources of Research Problems?

Now that you know the types of possible research problems that you can focus on in a term paper , let’s look at the sources that you can use to identify research problems.

From a research perspective, the kind of research problem that you wish to investigate should meet two conditions.

First, the problem has to be unique and not something other researchers have already looked into exhaustively. Second, the problem has to be concise enough to raise specific issues that you can address in a research paper .

With that said, below are five sources of a research problem:

1. Interviews

Interviews sessions can be significant sources of research problems. The method gives you an opportunity to have formal discussions and informal interactions with individuals who can provide useful insights into research and make findings more relevant to future research. 

Consider having discussions with experts in the field you wish to investigate. These professionals mat be healthcare service providers, business leaders, teachers, social workers, attorneys, and accountants to mention but a few examples.

By interacting with these experts, you’re able to identify real-world problems that researchers have either ignored or understudied in the academic space.

Moreover, interview sessions give you the opportunity to get some practical knowledge that can help you to design and conduct your studies.

2. Personal Experiences

Your everyday experiences are a good source of research problem.

You have to think critically about your personal experiences with an issue that affects your family, your personal life, or your community.

A research problem derived from personal experience can spring from any issue and from anywhere.

For example, you can construct a research problem from events that appear to be out of the ordinary or from community relationships that don’t have clear explanations.

3. Deductions from Theory

A deduction from theory refers to inferences a researcher makes from the generalizations of life in a society that a researcher knows very well.

A researcher takes the deduction, places them in an empirical frame, and then, based on a theory, they come up with a research problem and a hypothesis that suggests some findings based on given empirical results.

The research accounts for the relationship to observe if a theory summarizes the state of an affair.

A systematic investigation, which evaluates if the empirical information affirms or rejects the hypothesis , comes next.

4. Interdisciplinary Perspective

If you consider interdisciplinary perspective to identify a problem for a research study, you’ll have to look at scholarship and academic movements from outside your main area of investigation.

It’s an intellectually involving process, one that requires reviewing pertinent literature to discover unique avenues of exploration an analysis.

The benefit of using this approach to identify a research problem for your research paper assignment is that it presents an opportunity for you to understand complex issues with ease.

5. Relevant Literature

To generate a research problem from relevant literature, you first have to review research related to your area of interest.

Doing so allows you to find gaps on the topic, making it easy for you to understand just how much understudied your area of interest is.

Data collected from relevant literature is relevant because it helps to:

• Fill existing gaps in knowledge based on a specific research
• Determine if current studies can have implications on further research on the same issue
• See if it’s possible to conduct a similar study in a different area or apply the same in a different context
• Determine if the methods used in previous studies can be effective in solving future problems

We can’t stress enough on the value of existing literature. The results should point you towards an outstanding issue, give suggestion for future gaps, and make it possible to delineate gaps in existing knowledge.

Research Paper Writing Help

Finding a research problem is just one part of the research paper assignment. You have to develop a research question, formulate a hypothesis, write a thesis statement,  and then write your research paper. It can be a lot of work, which demands a lot of attention and time.

If you need help to brainstorm, research, and write your research paper, click the button below to place your order. 

About the author 

Antony W is a professional writer and coach at Help for Assessment. He spends countless hours every day researching and writing great content filled with expert advice on how to write engaging essays, research papers, and assignments.

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• How to Define a Research Problem | Ideas & Examples

How to Define a Research Problem | Ideas & Examples

Published on 8 November 2022 by Shona McCombes and Tegan George.

A research problem is a specific issue or gap in existing knowledge that you aim to address in your research. You may choose to look for practical problems aimed at contributing to change, or theoretical problems aimed at expanding knowledge.

Some research will do both of these things, but usually the research problem focuses on one or the other. The type of research problem you choose depends on your broad topic of interest and the type of research you think will fit best.

This article helps you identify and refine a research problem. When writing your research proposal or introduction , formulate it as a problem statement and/or research questions .

Table of contents

Why is the research problem important, step 1: identify a broad problem area, step 2: learn more about the problem, frequently asked questions about research problems.

Having an interesting topic isn’t a strong enough basis for academic research. Without a well-defined research problem, you are likely to end up with an unfocused and unmanageable project.

You might end up repeating what other people have already said, trying to say too much, or doing research without a clear purpose and justification. You need a clear problem in order to do research that contributes new and relevant insights.

Whether you’re planning your thesis , starting a research paper , or writing a research proposal , the research problem is the first step towards knowing exactly what you’ll do and why.

Prevent plagiarism, run a free check.

As you read about your topic, look for under-explored aspects or areas of concern, conflict, or controversy. Your goal is to find a gap that your research project can fill.

Practical research problems

If you are doing practical research, you can identify a problem by reading reports, following up on previous research, or talking to people who work in the relevant field or organisation. You might look for:

• Issues with performance or efficiency
• Processes that could be improved
• Areas of concern among practitioners
• Difficulties faced by specific groups of people

Examples of practical research problems

Voter turnout in New England has been decreasing, in contrast to the rest of the country.

The HR department of a local chain of restaurants has a high staff turnover rate.

A non-profit organisation faces a funding gap that means some of its programs will have to be cut.

Theoretical research problems

If you are doing theoretical research, you can identify a research problem by reading existing research, theory, and debates on your topic to find a gap in what is currently known about it. You might look for:

• A phenomenon or context that has not been closely studied
• A contradiction between two or more perspectives
• A situation or relationship that is not well understood
• A troubling question that has yet to be resolved

Examples of theoretical research problems

The effects of long-term Vitamin D deficiency on cardiovascular health are not well understood.

The relationship between gender, race, and income inequality has yet to be closely studied in the context of the millennial gig economy.

Historians of Scottish nationalism disagree about the role of the British Empire in the development of Scotland’s national identity.

Next, you have to find out what is already known about the problem, and pinpoint the exact aspect that your research will address.

Context and background

• Who does the problem affect?
• Is it a newly-discovered problem, or a well-established one?
• What research has already been done?
• What, if any, solutions have been proposed?
• What are the current debates about the problem? What is missing from these debates?

Specificity and relevance

• What particular place, time, and/or group of people will you focus on?
• What aspects will you not be able to tackle?
• What will the consequences be if the problem is not resolved?

Example of a specific research problem

A local non-profit organisation focused on alleviating food insecurity has always fundraised from its existing support base. It lacks understanding of how best to target potential new donors. To be able to continue its work, the organisation requires research into more effective fundraising strategies.

Once you have narrowed down your research problem, the next step is to formulate a problem statement , as well as your research questions or hypotheses .

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 …

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.

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

They summarise 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 .

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Home » Research Methodology – Types, Examples and writing Guide

Research Methodology – Types, Examples and writing Guide

Table of Contents

Research Methodology

Definition:

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

Structure of Research Methodology

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

I. Introduction

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

II. Research Design

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

III. Data Collection Methods

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

IV. Data Analysis Methods

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

V. Ethical Considerations

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

VI. Limitations

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

VII. Conclusion

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

Research Methodology Types

Types of Research Methodology are as follows:

Quantitative Research Methodology

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

Qualitative Research Methodology

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

Mixed-Methods Research Methodology

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

Case Study Research Methodology

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

Action Research Methodology

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

Experimental Research Methodology

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

Survey Research Methodology

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

Grounded Theory Research Methodology

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

Research Methodology Example

An Example of Research Methodology could be the following:

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

Introduction:

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

Research Design:

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

Participants:

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

Intervention :

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

Data Collection:

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

Data Analysis:

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

Ethical Considerations:

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

Data Management:

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

Limitations:

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

Conclusion:

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

How to Write Research Methodology

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

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

When to Write Research Methodology

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

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

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

Applications of Research Methodology

Here are some of the applications of research methodology:

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

Purpose of Research Methodology

Research methodology serves several important purposes, including:

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

Advantages of Research Methodology

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

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

Research Methodology Vs Research Methods

About the author.

Muhammad Hassan

Researcher, Academic Writer, Web developer

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Types of Research Problem, Theoretical, Applied & Action

There are three types of research problem

• Theoretical research problem
• Applied research problem
• Action research problem

Theoretical Research Problem

It is the theoretical explanation of a research problem. It gives only theory and meaning of the problem. It defines the problem theoretically. This type of research has no need of hypothesis and verification.

• It is exploratory
• It is theoretical in nature
• It provides basic meaning

Applied Research Problem

An applied social research problem is a practical use of the theoretical knowledge. In that type the field work and visits to the problematic situation is necessary. It has the design of hypothesis and verification.

• It is practical in nature
• It has exploratory hypothesis
• It wants the verification

Action Research Problem

Action research problem is that one for which the immediate solution is required. It has millions of problems in the world that type of research is continuous and needs quick solution in emergency basis.

• It is welfare oriented.
• It is service oriented.
• It is sensitive and immediate decision required for solution.

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Housing crisis worsens as extreme conditions for renters described as 'needle in a haystack type stuff'

There is further evidence Australia's ongoing housing crisis is deepening.

The national vacancy rate has hit a record low of 0.7 per cent, according to Domain.

In Perth and Adelaide, it is as low as 0.3 per cent.

"It's challenging and I think 0.3 per cent (vacancy rate), that's needle in a haystack type stuff," chief of research and economics at Domain Nicola Powell said.

Domain's February Vacancy Rates Report shows it's hard going for renters right across the country.

"And we've got some of our capital cities with much more extreme conditions, versus other cities that are actually better conditions for tenants — still technically a landlords market — but a higher vacancy rate in Canberra and Darwin sitting at 1.3 per cent."

Ms Powell said record low vacancy rates are being driven by ongoing factors like rapid population growth and rising property prices, leaving more renters stuck in the rental market.

But perhaps the biggest problem is a shortage of supply, and that sluggishness has been brought into sharp focus by the latest Bureau of Statistics building approvals data.

Building approvals fell a further 1 per cent in January, after falling 10.1 per cent in December.

CoreLogic's head of research Tim Lawless said the problem is construction companies are reluctant to build because they can't turn a profit.

"The biggest one clearly is this significant surge in construction costs we've seen through the pandemic, and even though we aren't seeing construction costs rise anywhere near as quickly as they used to be ... they are rising," Mr Lawless said.

"They're not going backwards.

"And this is where some of the challenges come for the building sector, is actually having a feasible development in place that's going to deliver some level of a profit margin is notoriously difficult now."

The problem seems to be most acute in the construction of detached homes where building approvals slid 9.9 per cent in January to an 11.5-year low.

"Add to that we're still seeing very tight labour capacity restraints in the building sector," Mr Lawless said.

"Materials costs are still very high as well. So you can see where some of this pain is coming from. And it doesn't look like it's turning around at all at the moment.

"And it's hard to see it turning around any time soon to be honest," Mr Lawless said.

Housing crisis deepening

The government has made building more homes a policy priority, committing to build 1.2 million new homes over five years.

Ms Powell says speeding up the approvals process at the council level would help achieve this target.

"And I think you know that starts at the grassroots level around reducing red tape," she said.

"It's about providing shovel-ready land at affordable prices for developers to actually be able to — you know, for the sums to stack up for developers to be able to come and actually provide that level of supply."

Director of Sky Planning Melissa Neighbour said loosening "planning controls", or what's allowed to be built on any given site, at the local council level, is key to boosting the housing supply.

"And that's a reality of developing," Ms Neighbour said,

"We need to be able to have more planning approvals in the system to allow more sites to stack up. If we are turning off the tap at the very top of the funnel, which is the planning controls, then we are already at a loss."

And for those hopeful renters looking for at least something to cling onto, Domain data shows, on a national scale, the average views per rental listing continued to decrease in February.

While it may mean that Australians are sleeping rough, on the flip side, Ms Powell said others are choosing to stay at home to save up for a house of their own.

"You know it's challenging for a tenant wanting to become a first home buyer," Ms Powell said.

"You can understand why people are moving back in with mum and dad.

"So it showcases that biting of affordability constraints is certainly there.

"And people are cost-cutting to be able to have shelter or to be able to squirrel away and save to actually buy a home."

But it also means, despite the lack of supply, fewer people are inspecting properties which reduces the competition among renters for existing places.

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A novel geometric method based on conformal geometric algebra applied to the resection problem in two and three dimensions

• Original Article
• Open access
• Published: 27 May 2024
• Volume 98 , article number  47 , ( 2024 )

Cite this article

You have full access to this open access article

• Jorge Ventura 1 ,
• Fernando Martinez 1 ,
• Francisco Manzano-Agugliaro   ORCID: orcid.org/0000-0002-0085-030X 1 ,
• Aleš Návrat 2 ,
• Jaroslav Hrdina 2 ,
• Ahmad H. Eid 3 &
• Francisco G. Montoya 1  

This paper introduces a novel method for solving the resection problem in two and three dimensions based on conformal geometric algebra (CGA). Advantage is taken because of the characteristics of CGA, which enables the representation of points, lines, planes, and volumes in a unified mathematical framework and offers a more intuitive and geometric understanding of the problem, in contrast to existing purely algebraic methods. Several numerical examples are presented to demonstrate the efficacy of the proposed method and to compare its validity with established techniques in the field. Numerical simulations indicate that our vector geometric algebra implementation is faster than the best-known algorithms to date, suggesting that the proposed GA-based methods can provide a more efficient and comprehensible solution to the two- and three-dimensional resection problem, paving the way for further applications and advances in geodesy research. Furthermore, the method’s emphasis on graphical and geometric representation makes it particularly suitable for educational purposes, allowing the reader to grasp the concepts and principles of resection more effectively. The proposed method has potential applications in a wide range of other fields, including surveying, robotics, computer vision, or navigation.

Avoid common mistakes on your manuscript.

1 Introduction

The resection problem, also known in surveying as the Snellius–Pothenot (SP) or the inverse intersection problem, involves calculating the position of an unknown point P (also called a station) using the positions of three known points A , B , and C , and relative angular measurements from P . It is a relevant problem not only in geodesy and surveying, but also in other disciplines such as robot path planning (Masselli and Zell 2014 ), positioning (Pierlot and Van Droogenbroeck 2014 ), navigation (Pereira et al. 2018 ) or computer graphics (Mazaheri and Habib 2015 ), and can be solved both geometrically and algebraically. The solution of the two-dimensional problem has been known for several centuries and has numerous variants [almost 500! according to Bock ( 1956 )]. The three-dimensional problem is much more intricate and only complex and sophisticated algebraic solutions are known. The two- and three-dimensional configurations are illustrated in Fig.  1 .

Representation of the resection problem in 2D (left) and 3D (right)

1.1 Motivation

The determination of the position of an observer based on angular measurements from known points is of interest in several disciplines, such as surveying, computer graphics, optics, and robotics. Traditionally, solutions to this problem have relied on heavily algebraically loaded methods, which can be complex and challenging to comprehend. Furthermore, these methods do not always provide an intuitive understanding of the geometric relationships involved in the problem.

In both two-dimensional (2D) and three-dimensional (3D) problems, there is a need for a more geometric or graphical approach that simplifies the study of the resection problem. Although there are existing graphical methods for solving the problem in 2D that have been known for some time, their algebraic implementation can be quite cumbersome, hindering their widespread adoption. When the problem is approached from a geometric perspective, a better understanding of the underlying structures and relationships can be achieved, making the problem more accessible to a wider range of researchers and practitioners. This could potentially lead to novel applications and advancements in the related areas mentioned above.

In light of the exposed ideas, the main motivation behind this paper is to develop a novel geometric method based on conformal geometric algebra (CGA) to address the resection problem in two- and three-dimensional ways. By leveraging the unifying properties of CGA to represent geometric primitives within a single mathematical framework, we aim to provide a more intuitive and geometric understanding of the problem, simplifying its study, and paving the way for further applications and advancements in the field.

1.2 Background and literature overview

The resection problem has been extensively studied in the literature, with various algebraic and geometric or graphical methods proposed for its solution. It appears that ancient Greeks, such as Hipparchus or Ptolemy, already studied this problem in the context of astronomy, although the first person to solve the problem in the context of surveying was the Dutch mathematician Willebrord Snel van Royen (known as Snellius) in 1617 (Wreede 2007 ). He achieved the goal by using geometric and algebraic methods mainly based on trigonometry. This same problem was addressed in 1671 by John Collins in his work Philosophical Transactions . Collins contributed significantly to the topic by presenting a new and elegant geometrical solution, which involves the use of just one circle and an auxiliary point. In 1692, Laurent Pothenot, who was working on the definition of the meridian north of Paris, presented a paper on the subject. However, according to McCaw ( 1918 ) and others, Pothenot did not contribute anything new to the solution of the problem and all he did was publish the works of Snellius and Collins under his own name. Other authors who have studied this issue are shown in Table  1 .

In the context of geodesy and surveying, the resection problem has been a fundamental problem for centuries. The problem has been addressed from various perspectives, including algebraic (Awange et al. 2010 ), geometric/graphical (Masselli and Zell 2014 ), and numerical (Dekov 2012 ) methods. Algebraic solutions to the problem are well established and have been extensively studied. However, these methods often involve complex algebraic manipulations and do not provide any intuitive geometric understanding of the problem.

The two-dimensional problem has been solved using approaches such as graphical methods (Bock 1956 ), analytical geometry (Bil 1992 ), matrix methods (Bähr 1991 ), and algebraic methods based on Sylvester resultants (Awange and Grafarend 2002 ), Groebner bases (Awange 2002 ), and other elimination techniques (Sturmfels 2002 ). Awange and Grafarend ( 2005 ) provided a detailed overview of several algebraic techniques for solving 2D and 3D resection.

The three-dimensional version is more complex, with most solutions relying on intricate algebraic methods. Grunert originally formulated the problem in 1841 and derived quartic equations to determine the unknown distances (Grunert 1841 ). Since then, numerous procedures have been developed to optimise Grunert’s formulation and reduce the computational steps involved (Awange and Grafarend 2005 ; Fischler and Bolles 1981 ). Algebraic techniques such as Groebner bases (Awange and Grafarend 2003a ), polynomial resultants (Awange and Grafarend 2003b ), and linear homotopy (Paláncz et al. 2010 ) have been effectively applied to solve the 3D problem. However, geometric solutions to the resection problem have been less explored. The geometric or graphical approach provides a more intuitive understanding of the problem and can be more easily visualised.

In recent years, there has been a growing interest in the application of geometric algebra (GA) to solve several geometric problems. GA provides a unified mathematical framework for representing and manipulating geometric objects, making it a powerful tool for geometric computations (Dorst et al. 2010 ). The application of GA to the resection problem was first proposed by Smith ( 2023b , 2023a ).

However, the application of GA to the resection problem is still in its early stages, and there is still a long way to go. In particular, the application of conformal geometric algebra (CGA) to the resection problem has not been fully explored. CGA extends GA by incorporating the concept of conformal transformations, which can provide a more powerful and flexible framework for geometric computations (Hestenes and Sobczyk 2012 ; Dorst et al. 2010 ). CGA provides a unified framework for handling points, lines, planes, circles, spheres, and other geometric entities. Each entity has a unique representation, and the relationships between entities correspond to algebraic relationships between their CGA representations. This eliminates the need for coordinate transformations when transitioning between reference frames.

Cameron and Lasenby ( 2008 ) showed that CGA subsumes projective geometry and is more computationally efficient than matrix methods. CGA has found applications in computer vision (Wareham et al. 2004 ), robotics (Zaplana et al. 2022 ), and other geometric computing problems (Hitzer et al. 2022 ). However, its potential for solving resection-type problems in geodesy and surveying remains relatively unexplored. This paper seeks to fill the void by introducing a new geometric solution based on CGA for the 2D and 3D resection problem.

1.3 Contributions

In this paper, a new method for solving the resection problem using the mathematical framework geometric algebra in 2D and 3D dimensions is proposed. The method provides a simple solution based on purely geometric and graphical principles. Specifically, the 3D version of the problem is thoroughly analysed in detail. Specific novel contributions are the following:

A novel conformal geometric algebra (CGA)-based method is presented to solve the two- and three-dimensional resection problem, providing a more intuitive and unified geometric approach.

The proposed method is compared with established techniques, and its advantages and efficacy are demonstrated through numerical examples.

The potential applications of the method in several fields such as computer graphics, optics, and robotics are highlighted, emphasising its versatility, and paving the way for future advancements in geometric research.

Several algorithms have been developed that provide better results than the best-known algorithms to date from a computational perspective.

1.4 Outline

The remainder of the article is organised as follows. Section  2 provides an introduction to geometric algebra and conformal geometric algebra. Section  3 revisits the resection problem and reviews traditional methods. Section  4 presents existing and new GA-based methods. Section  4 presents the proposed method based on CGA to solve the resection problem. Section  5 provides several application examples to demonstrate the effectiveness of the proposed method, along with benchmarks for computational efficiency. Section  6 provides an error and uncertainty analysis of the proposed methods. Finally, Sect.  7 concludes the article with a summary and some suggestions for future work.

2 Basics concepts in geometric algebra

Geometric algebra is a mathematical framework for representing geometric objects and transformations in a unified way (Hestenes and Sobczyk 2012 ). GA extends the algebra of vectors to include other geometric objects such as points, lines, planes, and volumes. GA provides a powerful tool to solve geometric problems and has applications in a wide range of fields, including electrical engineering (Montoya et al. 2019 , 2021 ), computer vision (Hrdina and Návrat 2017 ), robotics (Hrdina et al. 2017 ), and other engineering fields [see Hitzer et al. ( 2022 ) and references there in].

GAs are used mainly in situations where Euclidean transformations play a significant role. A simple way to introduce GAs is to understand them through more familiar tools such as complex numbers or quaternions (indeed, they are subalgebras of GA). For example, implementing Euclidean rotations using quaternions is essential to improve computational capabilities and adopt an object-oriented approach. We can see quaternions \({\mathbb {H}}\) as a natural extension of the complex numbers \({\mathbb {C}}\) in the form

where \(a,b,c,d \in \mathbb R\) and the products of the basis elements \(\varvec{i},\varvec{j},\varvec{k}\) satisfy the multiplications rules \(\varvec{i}^2=\varvec{j}^2=\varvec{k}^2=-1\) and \(\varvec{ij}=-\varvec{ji}=\varvec{k}\) . The quaternion \(\text {Im} (\varvec{z})=b \varvec{i} + c \varvec{j} +d\varvec{k}\) is called the imaginary part of \(\varvec{z}\) , and \(\text {Re} (\varvec{z})=a\) is called the real part of \(\varvec{z}\) . Note the use of bold letters for quaternions and regular font for real numbers.

The operation of rotating an object by an angle \(\theta \) around the axis \(\varvec{n}= n_x \varvec{i}+ n_y \varvec{j}+ n_z \varvec{k} \) can be represented by the quaternion

acting on the vector \(\varvec{q}= x\varvec{i} +y\varvec{j} +z\varvec{k}\) . Note that \(||\varvec{n}||=\sqrt{\varvec{n}\bar{\varvec{n}}}=1\) , where the bar decoration stands for quaternionic conjugation. The representation is done in a similar way as in the case of complex numbers. The difference is that quaternions are not commutative and thus they act via the so-called sandwich product

So, we are working in a four-dimensional linear space. If we want to realise also the translations, we have to extend the algebra by another dimension using an element \(\varvec{\epsilon }\) such as \(\varvec{\epsilon }^2=0\) and \(\varvec{\epsilon } \varvec{i}=\varvec{\epsilon } \varvec{j}=\varvec{\epsilon } \varvec{k}=0\) . This algebra is usually called dual quaternions. For more on the use of quaternions in general engineering topics, see Selig ( 2005 ).

On the other hand, the wedge operation on a vector space allows us to work with linear subspaces. A line can be characterised by a vector \(\varvec{\ell }\) , so x belongs to a line if and only if \(x \wedge \varvec{\ell }=0\) . The wedge of two vectors then characterises the plane in the same way. A vector space closed in the wedge operation is called a Grassmannian algebra. The combination of these two concepts leads to the notion of GA.

2.1 Euclidean vector GA

Vector geometric algebra (VGA) is perhaps one of the simplest GAs. For the two-dimensional case, the VGA ( \(\mathbb G_2\) ) is a Grassmannian algebra based on two orthonormal basis vectors ( \(\varvec{\sigma }_1\) , \(\varvec{\sigma }_2\) ) together with a bilinear operation known as geometric product satisfying the following identities:

The term \(\varvec{\sigma }_1 \wedge \varvec{\sigma }_2\) is known as bivector ( \(\varvec{\sigma }_{12}\) for short). Geometrically, a bivector represents an oriented plane segment spanned by the two vectors. We can use bivectors to represent rotations in the following way. The collection of bivectors \(\varvec{\sigma }_1 \wedge \varvec{\sigma }_2\) forms a one-dimensional vector space that is closed under multiplication. We can then generate rotations by applying the exponential map to bivectors. For example, the exponential of the bivector

is used as in ( 3 ) to perform a rotation by an angle \(\theta \) in the plane spanned by \(\varvec{\sigma }_1\) and \(\varvec{\sigma }_2\) . In this way, the group of rotations generated by the bivectors is mathematically equivalent to the group of unitary complex numbers \(e^{i\theta }\) , which also represent rotations in the complex plane (note that \((\varvec{\sigma }_1 \wedge \varvec{\sigma }_2 )^2 =-1\) ). However, bivectors provide a more intuitive geometric representation of rotations directly in vector space.

Similarly, we can introduce the \(\mathbb G_3\) algebra for the purposes of reasoning in the 3D space. The algebra \(\mathbb G_3\) is based on three generators \(\varvec{\sigma }_1, \varvec{\sigma }_2\) and \(\varvec{\sigma }_3\) together with a geometric product defined by the following identities:

In Sect.  4.1 , it will be shown how to use the VGA-based method to solve a 2D version of the resection problem. This procedure is, in fact, a use of GA \(\mathbb G_2\) .

2.2 Conformal geometric algebra

The goal here is to create a model of Euclidean geometry. Specifically, geometry whose symmetry group contains the Euclidean symmetries (rotations, translations, etc.). For that purpose, a nondegenerate quadratic form will be chosen, thus obtaining the conformal geometric algebra (CGA). In the case of CGA for a two-dimensional Euclidean space, a GA of signature (3, 1) is obtained [also known as Compass Rule Algebra CRA, see Hildenbrand ( 2018 )], with basis vectors \(\varvec{\sigma }_1,\varvec{\sigma }_2,\varvec{\sigma }_+\) and \(\varvec{\sigma }_-\) such that

For mathematical convenience, it is advisable to define two new basis vectors

with properties

Vector \(\varvec{\sigma }_0\) is known to represent the Euclidean point at the origin of the coordinate system, and vector \(\varvec{\sigma }_{\infty }\) represents the point at infinity. Using the rules above, it can be proved that any Euclidean point X can be mapped to the CGA vector space as

In “Appendix A.1”, explicit calculations that justify our choices can be found. As in Sect.  2.1 , the extension to higher dimensions is straightforward by adding the element \(\varvec{\sigma }_3\) . This will make \(\mathbb G_3\) appear instead of \(\mathbb G_2\) , and quaternions will appear in the bivectors.

In the notation used throughout this paper to ensure clarity, regular font indicates real numbers, boldface denotes vectors in the CGA space, boldface with an overhead arrow (e.g. \(\vec {\varvec{x}}\) ) represents vectors in the Euclidean space, and uppercase letters in boldface indicate multivectors in CGA.

The usefulness of CGA and CRA is based on the fact that the distance between Euclidean points is encoded in the scalar product. The points X and Y are represented by the Euclidean vectors \(\vec {\varvec{x}}\) and \(\vec {\varvec{y}}\) , respectively. The mapping defined in ( 12 ) maps these Euclidean points (vectors) to vector elements \(\varvec{{x}}\) and \(\varvec{{y}}\) in the CGA space.

As shown in “Appendix A.2”, the scalar product between two CGA points codes the distance:

Thus, the Euclidean point X (represented by the Euclidean vector \(\vec {\varvec{x}}\) ) lies on the sphere (circle) S with centre in point C and radius r if and only if it satisfies the identity \(\varvec{{x}} \cdot \varvec{{c}} = -\frac{1}{2} r^2 \) in the conformal space. Since \(\varvec{x}\cdot \varvec{\sigma }_{\infty }=-1\) for each point, this identity may be written as

Thus, the vector element \(\varvec{{c}} - \frac{1}{2} r^2 \varvec{\sigma }_{\infty } \) represents the sphere S in the CGA space. It should be emphasised that within CGA, there exist two distinctive methods to reference the identical geometric entity: IPNS and OPNS (see “Appendix A.3”). IPNS is superior for transformations and intersections, whereas OPNS is advantageous for blending and morphing tasks. It is crucial to be aware of the representation we are operating in; however, transitioning between representations can be achieved seamlessly using the dual operator ( \(^\star \) ), as described in “Appendix A.3”.

3 Revisiting the resection problem

In the context of geodesy and surveying, the resection problem plays an important role in determining the position of an observer based on angular measurements from three known reference points. Over the years, various approaches have been developed to address this problem, ranging from graphical or geometrical methods to algebraic ones. However, many of these methods can be complex or tedious, particularly when addressing the problem in three dimensions. With the growing importance of accurate positioning in modern applications, it is essential to revisit the resection problem and explore innovative approaches that offer more intuitive solutions.

3.1 Traditional methods

In this section, traditional methods for solving the resection problem are reviewed. They can be classified into four basic groups: trigonometric, geometric/graphical, iterative (numerical) and others (see Table  2 ). A comprehensive list of methods for 2D resection problems is already presented in Table  1 . Note that sometimes the frontier between trigonometric and geometric methods is not as clear because both solutions exist at the same time (e.g. the Cassini method).

Trigonometric methods are some of the oldest and most famous procedures for solving the 2D resection problem. They are based on the use of trigonometric functions to compute the position of the observer using the angles between the known points and the observer.

The Pothenot–Snellius method, also known as the Kästner–Burkhardt method, is one of the most known and oldest procedures in this group. The Cassini method is another example of a trigonometric solution (with a graphical solution, too), which is similar to the one described by Esteves et al. and Cohen and Koss. The Easton and Cameron method is also a trigonometric approach, similar to the Cassini method.

Geometric or graphical methods, on the other hand, use geometrical constructions and properties of the known points to calculate the position of the observer. These methods are based on the use of geometric and graphic principles to solve the problem. Esteves et al. proposed a geometric method that uses the intersection of circles to calculate the position of the observer. Cohen and Koss also proposed a geometric method that uses the intersection of circles, but with a different approach from Esteves et al. Font-Llagunes proposed a method that uses the intersection of lines and similarity between triangles to calculate the position of the observer. Pierlot et al. and Ligas proposed a method that uses the intersection of power lines to calculate the position of the observer, although the solution is given in algebraic form.

Iterative methods use iterative algorithms to converge to the observer position. These methods are based on the use of an initial estimate of the observer position, which is refined iteratively until convergence is achieved. Iterative search is an example of an iterative method that uses a search algorithm to find the robot position. Sanchiz et al. proposed a method that uses an iterative search algorithm to find the observer position.

Finally, there are other methods that do not fit the previous groups. The Tienstra method is one such example, which is a completely different approach based on barycentric coordinates. Another method like Willerding is based on the use of complex numbers to compute rotations in the Argand plane to find the observer position.

4 Resection using geometric algebra

Nowadays, methods based on geometric algebra (GA) have been developed, providing a new solution to the resection problem while maintaining the focus on its geometrical roots. GA offers a versatile framework that can be adapted to different contexts based on the selection of specific metrics and the number of dimensions. For example, when all elements of the base \(\varvec{\sigma }_i\) square to \(+1\) , the resulting algebra is known as vector geometric algebra (VGA) (see Sect.  2.1 ). By extending the ability of the basis elements to square to \(-1\) or 0 or by accommodating more dimensions, it becomes possible to explore alternative forms of GA. One such example is CGA, which incorporates two additional dimensions: one dimension squaring to \(+1\) and another squaring to \(-1\) . This flexibility enables GA to solve a wide array of applications and problem domains, seamlessly scaling the number of dimensions in a straightforward way.

4.1 Vector GA method

The 2D VGA-based method has been recently proposed by Smith ( 2023a ) and published as disseminative material. The process is mainly geometric and results in obtaining a vector \(\varvec{p}\) that describes the position of the point P when choosing the middle point ( B ) as the origin. In this case, we start with a vector basis consisting of two elements \(\varvec{\sigma } =\{\varvec{\sigma }_1, \varvec{\sigma }_2\}\) . Figure  2 illustrates a representation of the problem, as well as a detailed sequence of steps carried out.

Vector GA-based method steps to solve the 2D resection problem

First, using the known data ( A , B , C , \(\alpha \) , and \(\beta \) ), circles \(c_1\) and \(c_2\) are drawn using points A , B , P and B , C , P , respectively (see Fig.  2 a). These circles serve as an auxiliary element for better understanding the solution, but are not required as such. With the help of the central angle theorem, the vectors \(\varvec{d}_1\) and \(\varvec{d}_2\) are obtained

where \(\varvec{v}_1=A-B\) and \(\varvec{v}_2=C-B\) . Note that B was chosen as the origin, but any other point can also be selected under the condition that \(\alpha \) or \(\beta \) is not null. This situation occurs when P is collinear with two of the three known points. In such a case, other points can be selected as the origin. Equation ( 15 ) indicates that vectors \(\varvec{d}_i\) are the result of rotating and scaling vectors \(\varvec{v}_i\) , as shown in Fig.  2 b. Note that the rotation angle is given by \((90-\alpha )\) and \((90-\beta )\) , respectively. The next step involves determining the vector \(\varvec{d}\) as \(\varvec{d}_2-\varvec{d}_1\) . Finally, the desired vector \(\varvec{p}\) is the rejection of \(\varvec{d}_1\) or \(\varvec{d}_2\) on \(\varvec{d}\) (see Fig.  2 c). In VGA, the above steps are summarised in the following equation

It should be noted that the proposed solution is remarkably simple and does not involve the use of any type of coordinates. The result is obtained by simple geometric operations, such as rotation, scaling, and rejection, applied to the inherent primitives of VGA, such as vectors in this specific case.

The proposed method has advantages over the existing ones. It avoids some limitations as in Tienstra’s method where no solution can be found if the points A , B , and C are collinear. Furthermore, it is feasible to obtain an indicator of how close P is to the forbidden circle (defined by points A , B , and C ) by means of the length of vector \(\varvec{d}\) . If the point P is on this circle, then it can be easily checked that \(d=\Vert \varvec{d}\Vert = 0\) . Consequently, small values of d suggest that we should relocate the station to another site to ensure a reduced source of error (see Sect.  6 for a detailed error analysis).

4.2 Conformal GA method

The methodology employing VGA, as outlined in Sect.  4.1 , primarily utilised the GA \(\mathbb {G}_2\) . However, given that the resection problem primarily deals with circles, utilising its conformal extension, specifically the compass ruler algebra (CRA), appears to be more suitable (see Hildenbrand 2018 ). When dealing with geometric problems, it is highly advantageous to have a tool that can express graphical methods algebraically.

On the basis of the postulates presented in Sect.  2.2 , two traditional and well-known graphical methods are proposed to solve the resection problem using CRA: Cassini and Collins. By leveraging CRA, both methods can receive clear algebraic interpretations, as explained in the following sections.

For a more in-depth example of CRA using the Clifford library in Python, see the GitHub repository.

4.2.1 Cassini construction

The Cassini method provides a solution to the resection problem by leveraging the inscribed angle theorem. The solution is obtained by determining the intersection of two circles: one passing through points A , B , and P , and the other through points B , C , and P as shown in Fig.  3 . To determine the centres of the circles, two lines must be intersected. The step-by-step graphical approach underlying the Cassini method can be elucidated, along with the equivalent steps, using the CRA algebra (see Fig.  4 ).

2D graphical resection procedure using Cassini method

Graphical solution of Cassini method step by step. The point P is found by intersecting the two circles \(c_1\) and \(c_2\) where all the points see AB with angle \(\alpha \) and BC with angle \(\beta \) , respectively

CRA version of Cassini’s method step by step

CRA Mapping : The problem starts by mapping three known Euclidean points, A , B , and C , to the CRA domain. For example, the point A with coordinates \((a_1,a_2)\) is mapped as

For the rotated lines \(\varvec{\ell }_{AO_{1}}\) and \(\varvec{\ell }_{CO_{2}}\) , the following rotors and translators must be first defined

, and thus,

Note that rotations are always relative to the origin. To rotate around an arbitrary point, it must first be translated to the origin. The rotation is then applied, followed by another translation that returns the point back to its original position.

Circle building : The centres of the circles can now be found as the intersection of the lines \(\varvec{\ell }_{AB}\) and \(\varvec{\ell }_{AO_{1}}\) and \(\varvec{\ell }_{BC}\) and \(\varvec{\ell }_{CO_{2}}\)

The result is a flat point, representing the wedge of the sought point and the point at infinity. Footnote 1 The extraction of the point of interest is straightforward by factoring out the point at infinity

The radius of the circles can be computed as

, and the circles themselves are determined as

Intersection of circles : The desired result can be obtained from the intersection of the two circles \(\varvec{c}_1\) and \(\varvec{c}_2\) as

Two intersecting circles in CGA yield a couple of points (1D sphere), also known as pair-point \(\varvec{P}\) . Finally, the Euclidean point P is recovered by the classic formula (see Hildenbrand 2018 )

One of the points \( P_{\pm }\) is exactly the point B , and the other is the sought point P (see Fig.  3 ).

Figure  5 shows a concise and condensed summary of the key steps involved and discussed above. It provides valuable visual depictions that enhance the geometric intuition underlying the method, offering an algebraic interpretation of the graphical approach. It also helps to reinforce the strong connection between the graphical Cassini method and its algebraic translation using CRA.

The GitHub repository shows several examples derived from the code developed by the authors. The computational procedures in CGA are found to be straightforward. The methodology involves the manipulation and combination of geometric objects, thus justifying the occasional reference to GA as an object-oriented approach.

Collin’s method for solving the resection problem

4.2.2 Collins construction

The graphical method of Collins provides a solution to the resection problem using the intersection of the line passing through the point B and the so-called Collins auxiliary point E with the circle containing the points A , C , and E . The step-by-step procedure for applying Collins’ method graphically is depicted in Fig.  6 .

CRA version of Collins’ Method step by step

CRA Mapping : Similarly to the Cassini method, the points A , B , and C must be mapped to CGA as

Auxiliary Point : To determine the auxiliary point E , it is necessary to define the lines \(\ell _{AE}\) and \(\ell _{CE}\) as shown in Fig.  6 b by first rotating the line through A and C \((\ell _{AC})\) an angle \(\beta \) around the point A in a counterclockwise direction

and then rotate the line \(\ell _{AC}\) around C by an angle \(\alpha \) clockwise

The intersection of the two lines determines the position of the Collins auxiliary point E

We get the Collins point as a flat point and use the standard projection to receive E .

Circle and Line intersection : Using the points A , C , and E , the circle c that passes through them is built

The end point P we are looking for is found by intersecting the circle c and the line through E and B (see Fig.  6 d),

The intersection of a line and a circle results in a pair-point \(\varvec{P}\) . As in the previous method, Eq. ( 26 ) is used to extract the two distinct points, one of them being the point E and the other the sought point P .

Figure  7 shows a visual summary of the key steps of the Collins method.

Step-by-step construction of the graphical solution for the resection problem in 3D

4.3 Resection method in 3D

The construction procedure for the 3D case is carried out in a series of detailed steps, as in the 2D sections and as illustrated in Fig.  8 . The main idea is to project all points into a plane parallel to the reference plane \(\varvec{\sigma }_{12}\) (the plane xy ). It can be any, but we chose the one that passes through one of the known points (for example, point A ). Then, we apply one of the methods presented in the previous sections to this simplified 2D version of the original problem, and finally, we solve for the \(\varvec{\sigma }_3\) ( z ) coordinate of the projected point. Section  5 provides specific examples.

CRA Mapping : Represent Euclidean points \(A,B,C \in \mathbb R^3\) in VGA as vectors \(\vec {\varvec{x}}=x_1 \varvec{\sigma }_1 +x_2 \varvec{\sigma }_2 + x_3 \varvec{\sigma }_3\) , project them into the xy plane ( \(\varvec{\sigma }_1 \wedge \varvec{\sigma }_2 = \varvec{\sigma }_{12}\) ) by formulas

Note that, for computational purposes, it is sufficient to remove component \(x_3\) for this specific case. The points are now mapped to the CRA domain

2D Solution : Use one of the methods described in Sect.  4.2 to solve the problem restricted to the plane xy , obtaining the points \(P_1\) and \(P_2\) . The choice point will be designated as \(P_p\) for subsequent steps.

3D Reconstruction : Find the line \(\ell \) perpendicular to the xy plane that passes through the point \(P_p\)

where \(\varvec{u}_3=\varvec{\sigma }_3 \wedge \varvec{\sigma }_{\infty }\) is the CGA representation of a free vector in direction of \(\varvec{\sigma }_3\) . Now, find the plane \(\varvec{\pi }_A\) perpendicular to the line \(\ell \) that passes through the point A

The intersection of the line \(\ell \) and the plane \(\varvec{\pi }_A\) determines the point G (obtained as a flat point)

, and thus, the point at infinity must be removed using the standard formula, recovering the intersecting point G in CGA

Using the inverse map from CGA to VGA, we end up with \(\vec {\varvec{g}}\) and the point of interest \(\vec {\varvec{p}}\) ( P ) can be obtained as

To obtain the point P without performing the last trigonometric operation, the line AG can be rotated around A by the angle \(\gamma _1\) in the plane perpendicular to the plane xy containing the line AG .

Cassini’s method when P is collinear with two points

Collins’ method when P is collinear with two points

4.4 Singularities of the methods

All three methods presented so far, i.e. VGA, Cassini CRA, and Collins CRA, encounter the same intrinsic limitation known as the dangerous circle. This issue is unavoidable with the Snellius–Pothenot formulation, where the unknown point P is part of the circle determined by the three known points. However, the graphical methods of Cassini and Collins offer distinct advantages over the VGA approach by avoiding the use of trigonometry, regardless of the challenges posed by singularities. This simplifies certain computational aspects and potentially enhances the geometric intuition behind the solutions, providing a more straightforward path to address the resection problem in specific scenarios.

For example, the angles \(\alpha \) and \(\beta \) in the denominators of Eq. ( 15 ) play a crucial role in the VGA method. If either angle is zero, then the result is infinity (meaning that point P lies on line segment AB or BC ). This is a common issue in all traditional methods where some trigonometric formulation is used [see Pierlot and Van Droogenbroeck ( 2014 ) for details]. To mitigate this effect in the VGA method, different reference stations can be selected to avoid zero values for \(\alpha \) or \(\beta \) .

In contrast, the Cassini CRA method can elegantly solve this issue graphically. Moreover, the CRA formulation also manages to capture the simplicity of the geometric solution. Figure  9 a shows the specific situation where the point P is collinear with AB (a similar case occurs for collinearity with BC , see Fig.  9 b). In either case, one of the circles degenerates into a line, and the solution simply consists of intersecting this line with the remaining circle. This is a clear example of how to deal with the singularities inherent in the trigonometric formulation by leveraging the basic principles of planar geometry. This provides a more intuitive approach to handling special alignments compared to ad hoc fixes needed for traditional resection techniques. It should be noted that no singularity occurs if P is collinear with the segment AC .

Interestingly, the Collins method presents an additional singularity beyond that of the Cassini method, occurring when the point P is collinear with segment AC .

The singularity due to the alignment of P with the sides AB or BC is solved as shown in Fig.  10 . Essentially, the point P is obtained from the intersection of the line passing through A and B and another line built by rotating line AC around C by an angle equal to \(\gamma _1=\theta -\beta \) (or rotating AC around A by \(\gamma _2=\varphi -\alpha \) if alignment occurs with BC ). This elegant geometric construction circumvents the infinity values that arise in the trigonometric formulation when P aligns with any side of the triangle ABC . The Collins CRA approach thereby extends the advantages of the Cassini method to fully address all possible singular configurations through basic planar geometry principles.

Finally, the singularity caused by the collinearity between P and AC is solved as follows. Note that in this situation, the rotated lines \(\ell _{AE}\) and \(\ell _{CE}\) are parallel as in Fig.  11 . We can recognise this by the fact that the projection of the flat point ( 38 ) is not a point, but a direction at infinity. Footnote 2 However, the search line is then characterised by the point B and the directional vector \(\varvec{u}=(\varvec{\sigma }_4 \wedge \varvec{\sigma }_3) \cdot \varvec{L}_{AE}\) , that is, \(\varvec{L}_{BE}=\varvec{b} \wedge \varvec{u} \wedge \varvec{\sigma }_{\infty }\) . The point we are looking for is \(\varvec{P}= \varvec{L}_{BE} \vee \varvec{L}_{AC}\) , which is a flat point originating from the wedge product of our sought point P and the point at infinity \(\varvec{P}=\varvec{p}\wedge \varvec{\sigma }_{\infty }\) .

5 Numerical simulations and benchmarks

5.1 two-dimensional example.

Both the VGA and CGA approaches detailed in Sects.  4.1 and 4.2 , respectively, have been implemented using specific GA software to address the 2D resection problem. Clifford (Hadfield et al. 2021 ) and GA-FuL (Eid [36]) packages were used to facilitate the work with GA. The algorithms we devised have been uploaded to GitHub, accessible via the following link: https://github.com/jorgeven98/Resection-Problem-3D . These algorithms were rigorously tested using various configurations, as illustrated in Fig.  12 . The locations of the known points, as well as the angles \(\alpha \) and \(\beta \) , can be retrieved from the code shared on GitHub in every instance. Remarkably, GA-based methods enable us to precisely solve any initial point arrangement.

Singular position. A , P , and C aligned

Tested point configurations using triangle ABC

In Table  3 , numerical data regarding the positions of known points A , B , and C , as well as angles \(\alpha \) and \(\beta \) , are presented for the four scenarios. To address these cases, the coordinates of all points (including the unknown point P ) are initially established. An auxiliary mathematical function is then employed, in conjunction with these coordinates, to ascertain the angles with precision. Subsequently, this derived angular information, together with the locations of the known points, is incorporated into our computational algorithms. The primary objective is to confirm that the calculated results are consistent with the coordinates initially designated for the point P .

5.2 Three-dimensional example

To evaluate the newly developed 3D resolution method, several examples have been proposed and solved, each featuring a distinct configuration of 3D points (see Table  4 ). For the purpose of this study, we will focus on configuration a), while the remaining examples are analysed similarly.

In the first step, we convert points A , B , and C into VGA vectors

Assuming that the position of the unknown point \(\vec {\varvec{p}}\) is \((6.8103,-6.6307,14.5388)\) , the values of the horizontal angles are \(\alpha =-39.7471^{\circ }\) , \(\beta =305.0165^{\circ }\) and the vertical angle is \(\gamma _1=31.1521^{\circ }\) . It should be noted that \(\alpha \) and \(\beta \) are the angles between points \(A_p\) and \(B_p\) , and \(B_p\) and \(C_p\) , respectively, as seen from \(P_p\) . The subscript p indicates the projection of that point in the xy plane. On the other hand, \(\gamma _1\) is the angle through which A sees P measured in the plane perpendicular to the \(\varvec{\sigma }_3\) axis and taking as a reference the line parallel to the xy plane. In this case, the point A has been selected, but any other point could be used.

3D visualisation for configurations b) and c) in Table  4 . a Points A , B , and C aligned. b The projections of A , B , and C in the plane xy form a triangle with P projected inside

The procedure proposed in Sect.  4.3 is followed. By projecting onto the xy plane and solving the 2D version of the problem, we obtain the following solution

The intermediate geometric objects are also computed, with the following results:

As a result, our desired point is \(P=(6.8103,-6.6307, 14.5388)\) , matching the original position vector \(\vec {\varvec{p}}\) .

Figure  13 shows the point configurations for examples b) and c) in Table  4 . By applying the same method as in example a), we can determine the precise location of point P , regardless of the positions of points A , B , and C with no limitations, but the unsolvable forbidden circle (all points lying in the same plane and P belonging to the circle defined by points A , B , and C ).

5.3 Benchmarks

To evaluate the computational efficiency of our GA-based algorithms, extensive benchmarking tests were performed and the results were compared with state-of-the-art methods. Our implementation, which exploits the power of code generation, has achieved superior performance, outperforming the best-known algorithms to date, such as the ToTal algorithm and the method proposed by Ligas. Moreover, sophisticated CGA-based approaches have shown excellent results and are among the most efficient in terms of execution time.

A comprehensive set of benchmarks was conducted, comparing our algorithms against top-efficient algorithms operating in the whole plane and accommodating any ordering for the known points. Each algorithm was executed \(10^6\) times at random locations within the same square-shaped area used for error analysis (see next section). The tests were performed on an Intel Core i7-9700K CPU 3.60GHz (Coffee Lake) with 8 logical and 8 physical cores (8 GB RAM, Windows 11, C#,.NET SDK 8.0.102).

Our findings reveal that our VGA-based algorithm outperforms state-of-the-art methods (see Table  5 ), executing approximately 13.3% and 30% faster than the previously best-known algorithms by Pierlot with Total #1 and Total #2.

The primary advantage of our methodology and implementation is the utilisation of GA-FuL’s comprehensive code generation capabilities. These capabilities range from generating code for individual multivector operations to creating full software libraries with proper software architecture and nested folder/file structure, enabling efficient and optimised geometric algebra computations.

6 Uncertainty analysis

This section investigates the impact of measurement uncertainties on the efficacy of the proposed GA methods. Given the intrinsic presence of noise in practical measurements, it is crucial to assess the sensitivity and resilience of the method to such perturbations (Font-Llagunes and Batlle 2009a ; Pierlot and Van Droogenbroeck 2014 ).

Previous discussions have highlighted that the resection problem faces an intractable challenge within the forbidden circle defined by points A , B , and C . This limitation is inherent in the nature of the problem rather than a reflection of the methods used to solve it. Novel methodological approaches should be considered with caution as they may inadvertently introduce undesirable scenarios and singularities not covered by the original formulation of the problem. In this regard, the proposed methods handle degenerate or limiting cases in a convenient and elegant graphical way, as shown in Fig.  11 .

This study presents three approaches to address the problem of 2D and 3D resection. In order to improve the ability to evaluate the accuracy of the algorithms and determine the station position error, a new metric has been developed. Consequently, we propose three formulations that are essentially different variations of the same underlying approach, each defining the metric D based on the square of a distance to denote the proximity of the station P to the forbidden region.

VGA method : The vector \(\varvec{d}\) is used as the difference between the vectors \(\varvec{d}_1\) and \(\varvec{d}_2\) (refer to Eq. ( 15 )). When the station P is located on the forbidden circle, the vectors \(\varvec{d}_1\) and \(\varvec{d}_2\) converge, resulting in a null vector \(\varvec{d}\) . Therefore, minimal values of D indicate close proximity to the forbidden circle

Cassini method : The determination of D using the CGA Cassini method requires the calculation of the squared distance between the centres of the circles \({c_1}\) and \({c_2}\) (as shown in Fig.  4 ), marked by \(O_1\) and \(O_2\) , respectively. In situations where the station is positioned on the forbidden circle, the two circles converge, causing their centres to overlap and the intercentre distance to become 0. Thus, according to Eq. ( A3 ), D is defined as

Collins method : Similarly to the previous method, D is determined as the squared distance between two specific points. The auxiliary Collins point E and the point B are used for this purpose. They coincide when the station P is situated on the forbidden circle. The formula used to compute this distance is as follows

Error analysis for two point configurations using a square window of \(2\times 2\) m.. Config #1 is an equilateral triangle, while in config #2 the points are collinear. Both configurations have been tested with standard deviation \(\sigma =0.01\) and \(\sigma =0.1\) . The three GA methods (VGA, CollinsCGA, and CassiniCGA) are compared based on their position error (first row) and metric 1/ D (second row), respectively

To validate the sensitivity of the proposed algorithms, a series of simulations have been proposed. The simulation framework is designed within a square area measuring 4 by 4 square metres, incorporating two unique configurations for three known points. The first configuration is an equilateral triangle, with the points located at the positions \(A = (0,1)\) , \(B = (-0.866, -0.5)\) , and \(C = (0.866, -0.5)\) . The points in the second configuration are linearly arranged at \(A = (-0.866,0)\) , \(B = (0, 0)\) , and \(C = (0.866, 0)\) . A spacing of 2 cm is used in each direction across the grid. At each point on the grid, the angles \(\alpha \) and \(\beta \) seen from P are calculated. Gaussian noise is introduced into these angles, characterised by a zero mean and two distinct standard deviations ( \(\sigma \) = 0.01 degrees and \(\sigma \) = 0.1 degrees). The algorithms use these modified angles as input to determine the estimated position of the unknown point. The discrepancy in position ( \(\Delta d\) ) is quantified by the Euclidean distance between the exact and estimated location of the point P .

The study performs 1000 iterations for each position to determine the standard deviation of the position error. The resulting standard deviations are shown in Fig.  14 . The results for the equilateral triangle configuration, with standard deviations of \(\sigma \) = 0.01 degrees and \(\sigma \) = 0.1 degrees, are presented in the first and second columns, respectively. Similarly, the results for the second configuration (collinear points) are detailed in the third and fourth columns, respectively, using the same standard deviations. The figure shows the standard deviation of the position error and the mean error measure 1/ D in the first and second rows, in that order. It is important to note that the scales used in the graphic representation are not linear. To enhance the visual clarity of these images and emphasise the correlation between the position errors and our new error metric, we applied histogram equalisation to the images.

6.1 Discussion of the results

The simulations performed are in agreement with those reported in the literature and support the case studies of each of the three methods presented. Figure  14 shows the different point configurations, where the forbidden circle is clearly identifiable due to the increase in standard deviation of the position error as it is approached. Minimal errors are observed inside the circle, while errors increase with distance outside the circle. It is remarkable that the three methods provide almost identical results for the error position.

The 1/ D metric plots maintain a shape similar to the position error plots near the forbidden circle. High red values are observed as D becomes smaller, while minimum values are observed in the lines \(\ell _{AB}\) and \(\ell _{BC}\) in Cassini and \(\ell _{AC}\) in Collins due to an increase of D (see Figs.  9 and 10 ).

Configuration 2 transforms the forbidden circle into a straight line. In this region, the error distribution is high and increases progressively with distance from the line. Because this configuration represents a degenerate situation, the case where D becomes zero does not occur. The different methods used to define D lead to overlapping points at infinity, causing D to take high values near the critical line and resulting in a minimal value for 1/ D . The metric D was chosen for configuration 2 due to this property.

All methods produce identical position error plots, indicating consistency and conformity with the results obtained by most algorithms to solve the resection problem. This confirms that the sensitivity to calculate the position of the point P , even with noisy measured angles, is independent of the method used and unique, as discussed in Pierlot and Van Droogenbroeck ( 2014 ) and Font-Llagunes and Batlle ( 2009b ). The metric 1/ D can be used as an indicator of proximity to the forbidden circle. When dealing with aligned beacons, the value of D should be used directly. In other cases, the similarity between this metric and the position error suggests that the former can approximate the latter if a function of the other problem parameters is applied.

7 Conclusions

This article presents a novel approach to solving the resection problem in two and three dimensions using conformal geometric algebra (CGA). The CGA framework allowed representing points, lines, planes, circles, and spheres in a unified mathematical structure and offered a more intuitive understanding and efficient solution to the resection problem compared to existing algebraic techniques.

The proposed method leveraged the ability of CGA to transition between different reference frames without requiring coordinate transformations. This eliminated the need for multiple calculation steps and complex algebraic manipulations that are characteristic of traditional algebraic solutions. Through extensive numerical simulations, we have demonstrated the validity and efficacy of our GA-based approach, achieving accuracy comparable to that of established algebraic techniques, while significantly improving computational efficiency and providing valuable geometric insights.

Our findings suggest that the geometric algebra framework has strong potential for solving resection-type problems not only in surveying and geodesy but also in computer graphics, robotics, computer vision, and navigation. By exploiting geometric relationships between entities, CGA paves the way for more intuitive solutions that unify computations involving different geometric primitives.

Future research can build upon the ideas presented here to address more complex variants of the resection problem that involve additional constraints. The CGA method can also be extended to address intersection problems and other spatial geometric computations across diverse disciplines. By harnessing the power of geometric algebra and the versatility of conformal geometric methods, this work opens up new possibilities for advancing geometric research and computational techniques.

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Jorge Ventura, Fernando Martinez, Francisco Manzano-Agugliaro & Francisco G. Montoya

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Appendix A Some calculations over CGA

CGA has been briefly introduced in Sect.  2.2 . In this Appendix, some fundamental calculations are presented to illustrate the computational efficiency inherent in this algebra. For further information and a detailed understanding, the reader is referred to Hrdina et al. ( 2021 ), Dorst et al. ( 2010 ), Hestenes and Sobczyk ( 2012 ), Hildenbrand ( 2018 ).

1.1 A.1 Conformal inclusion

We’ll show how the translations and rotations are connected and related to the insertion of the Euclidean space. If we choose vector \(\varvec{\sigma }_0\) as the origin of the coordinate system and use the element

, then it can act as a translation, allowing us to define an inclusion as a translation of the origin \(\varvec{\sigma }_0\) in the direction of the vector \(\vec {\varvec{x}}=(x \varvec{\sigma }_1 + y \varvec{\sigma }_2 +z \varvec{\sigma }_3)\) by the sandwich product as

So, we see that we can identify points in \(\mathbb R^3\) with vectors in CGA through the inclusion

where the element \(\varvec{T}\) acts as a translation. Using a similar reasoning, the element \(\varvec{R}=e^{n_1\varvec{\sigma }_2\varvec{\sigma }_3 +n_2\varvec{\sigma }_1\varvec{\sigma }_3+n_3\varvec{\sigma }_1\varvec{\sigma }_2}\) acts as a rotation due to identification \(\text {Im } {\mathbb {H}} =\langle \varvec{\sigma }_2\varvec{\sigma }_3,\varvec{\sigma }_1\varvec{\sigma }_3, \varvec{\sigma }_1\varvec{\sigma }_2 \rangle \) and the following computation

1.2 A.2 Distance between points

One of the key distinctions between CGA and VGA lies in the interpretation of the inner product. In CGA, the inner product represents the distance between points, which is fundamentally a quadratic concept in VGA. By incorporating two additional dimensions, CGA linearises this quadratic function. The equation below elucidates this:

This calculation explicitly demonstrates how CGA linearises the quadratic nature of the distance representation found in VGA, offering nuanced insights into geometric relations and interactions between points.

Furthermore, this characteristic is utilised to depict the midline \(\varvec{M}_{}\) as illustrated in expression ( 18 ). A midline is established by two control points, \(\varvec{a}\) and \(\varvec{b}\) , and encompasses points that maintain equal distances to these control points. This means that \( (\varvec{x}-\varvec{a})^2 = (\varvec{x}-\varvec{b})^2 \) . In the context of CGA, this condition can be articulated as follows:

i.e. \(\varvec{x} \in \varvec{M}_{AB} \leftrightarrow \varvec{x} \cdot (\varvec{a} -\varvec{b}) =0\) , so \((\varvec{a} - \varvec{b})\) is the IPNS representation of the midline defined by control points \(\varvec{a}\) and \(\varvec{b}\) .

1.3 A.3 IPNS vs OPNS representation of objects

In GA, we have several types of product, so it is possible to represent objects in different ways. With the help of the inner product, we can define the object \({\mathcal {C}}\) as follows.

In CGA, for example, the object \({\mathcal {C}} = n_1 \varvec{\sigma }_1+n_2 \varvec{\sigma }_2 +n_3 \varvec{\sigma }_3 + d \varvec{\sigma }_{\infty }\) can be tested and verify that it represents a plane

where \(\vec {\varvec{n}} = n_1 \varvec{\sigma }_1+n_2 \varvec{\sigma }_2+n_3 \varvec{\sigma }_3\) . Equation ( A5 ) describes a plane with normal vector \(\vec {\varvec{n}}\) and distance from the origin d . On the other hand, the wedge product defines an object \({\mathcal {D}}\) as follows

Again, in CGA, for example, the object

defines the line goes through the points \(\varvec{a}\) and \(\varvec{b}\) :

which represented the line based on the points \(\varvec{a}\) and \(\varvec{b}\) .

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Ventura, J., Martinez, F., Manzano-Agugliaro, F. et al. A novel geometric method based on conformal geometric algebra applied to the resection problem in two and three dimensions. J Geod 98 , 47 (2024). https://doi.org/10.1007/s00190-024-01854-1

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Money blog: UK's best pub chef shares amazing cheap pasta recipe; beach-goers face £1,000 fine for taking pebbles

For the latest instalment of our Cheap Eats series, we speak to Dave Wall, head chef at the UK's number one gastropub, The Unruly Pig in Suffolk. Read this and the rest of today's consumer and personal finance news in the Money blog below, and leave your thoughts in the comments box.

Wednesday 29 May 2024 06:48, UK

• Beach-goers warned they face £1,000 fine if they take pebbles
• Average amount spent on holiday revealed - and it's higher than two years ago
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• Raheem Sterling to pay for 14 people to go to university and applications close this Thursday

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Every Wednesday we ask Michelin chefs to pick their favourite Cheap Eats where they live and when they cook at home. This week we speak to Dave Wall, head chef at the UK's number one ranked gastropub, The Unruly Pig in Suffolk.

Hi Dave , c an you tell us your favourite places in Suffolk  where you can get a meal for two for less than £40?

Honey + Harvey . A cracking spot for breakfast, brunch or lunch. They have the most delicious coffee and a cracking full English, the vibe is super-chilled and laidback and I always feel so relaxed there.

Lark . A beautiful little independent restaurant in Bury St Edmunds with the most incredible selection of small plates and top-drawer cooking. Admittedly, I find myself spending a fair bit more than £40 at Lark because I love James Carn's cooking so much that I end up going way over the top and ordering far too many dishes.

What's your go-to cheap meal at home?

Anchovy pasta is one. I get that anchovy is often considered a Marmite ingredient. I love them, but if you are in the "hate" camp, then please bear with me, as I want to persuade you to give these versatile little wonders a second look (and perhaps not tar all anchovies with the same brush).

My recipe below uses both brown and brined anchovies. It is an easier but still utterly delicious version of the dish I've served at The Unruly Pig (which also comes with an oyster velouté). This is comfort food at its best. Buon appetito!

• 250g butter
• 70g brown anchovies (ideally Cantabrian)
• 1 clove garlic
• 30g double cream
• 25g of brined anchovies

Add all the ingredients to a pan. Bring to a slow simmer on a low heat. Once the mixture starts to boil, remove, and transfer to blender. Blend for two minutes until the mixture is well emulsified. Set aside.

Pangrattato

Three bread slices, crusts removed (staler the better)

• 1 garlic clove
• 1 lemon zest
• Pinch salt & pepper

Blend all the ingredients in food processor, making sure the crumb is fine. On a low heat, gently toast the crumbs until they become golden.

• 125 g of fresh spaghetti per person
• Grated Parmesan, brined anchovy, celery leaf to garnish 

Gently the cook the pasta in simmering boiling water, add plenty of salt to the pasta water so it tastes like sea water. Cook for 1-2 minutes - or to instructions if using dried.

Bring it all together

Meanwhile, gently heat the anchovy pasta sauce in a large pan so it becomes warm. Be careful not to boil. Once the pasta is cooked, gently remove and put it straight in to the warmed anchovy sauce. Add a splash of the pasta water to retain some of the starch (as this will help thicken your sauce).

Gently cook the pasta in the anchovy sauce until it becomes thick and creamy, and the sauce coats the pasta. Serve into a bowl and add the Parmesan, fresh anchovies and celery leaf on top.

Generously sprinkle the pasta with the golden pangrattato to add a wonderful texture and crunch.

We've spoken to lots of top chefs and bloggers - check out their cheap eats from around the country here...

Beach-goers in Cumbria have been warned they could face a fine of up to £1,000 if they remove pebbles or shells across the area.

Cumberland Council has told visitors it is unlawful to take natural materials such as sand, shells and pebbles from the beach under the Coast Protection Act.

Cumberland councillor Bob Kelly said it was important to "ensure that our beaches remain vibrant and intact for future generations".

"I understand people's reluctance to follow this guidance, as I have been a collector of shells myself. But taking a pebble or a shell from a beach can in fact damage the environment," he said.

"Pebbles and other natural matter act as a natural sea defence against coastal erosion, natural flood defences and wildlife habitats, which many experts warn has become even more of an issue due to climate change."

People are spending more on holiday than they were two years ago, the latest data from ABTA Travel Money has shown. 

On average, UK travellers are spending £369 each during a short break abroad - up more than £59 since 2022. 

For a longer break, the typical amount rises to £660, which is up £231 a person since 2022. 

Families with children over five are likely to spend the most while on a short holiday, totalling £431.

But the highest spend comes from travellers aged 55-64, who spend an average of £721.  

"People are spending more while on holiday overseas and that can't just be put down to inflation," Graeme Buck, director of communications for ABTA Travel Money, said. 

"Over the past two years, UK prices have risen by a total of 9.3% whereas overseas holiday spend is up by 54% for a longer holiday.

"Add in more favourable exchange rates for many holiday destinations, we see over the last few years that there has been a clear shift towards people spending the spare money they may have on holidays and creating memories that will last a lifetime." 

Visitors to all Euro currency destinations this summer will see a little more for their money, as the pound has increased against the Euro (up 2.1%). 

The UK has the highest diesel prices in Europe, according to new analysis.

The RAC, which carried out the research, found the average price of a litre of diesel at UK forecourts is 155p - 5p more than Ireland and Belgium.

Although duty on both petrol and diesel was cut from 57.95p to 52.95p in spring 2022, the UK still has the highest rate of duty on diesel in Europe alongside Italy, but Italy's average pump price is 7p per litre cheaper at 148p.

France's duty rate is the equivalent of just 1p per litre lower than in the UK, but its average price for diesel is 9p per litre cheaper at 146p.

The analysis is based on figures from the European Commission and the UK's Competition and Markets Authority.

Simon Williams, fuel spokesman for the RAC, said: "Having the most expensive diesel in Europe despite the current 5p duty cut is a very dubious honour."

Despite the RAC bringing the issue to the attention of energy secretary Claire Coutinho in a letter just over a week ago, he said, "the price of diesel at the pump has barely fallen".

"We can see no good reason why retailers in Great Britain aren't cutting their prices at the pumps," he added.

Thieves are targeting electric car charging cables in the latest spate of car crimes.

Data from Instavolt, the UK's largest operator of rapid chargers, found gangs had targeted 27 sites in Yorkshire and the Midlands since last November and stolen 174 cables.

With each cable costing at least £1,000, the operator, which runs Osprey Charging and BP Pulse, said this was affecting electric vehicle drivers.

It also risked deterring prospective drivers who wanted to make the move to electric cars, they said.

The company is now introducing a range of measures at charging stations to deter thieves, including installing extra CCTV, security patrols, using SmartWater to tag property and tracking devices.

Instavolt CEO Delvin Lane told Autocar : "These thefts are extremely frustrating for our customers and for us."

He also noted that it was a "misconception" that the copper in chargers brought real financial gain. 

"The value of any metal stolen is insignificant. The thefts just cause disruption to EV drivers - including those in the emergency services - looking to charge their vehicles," he said.

By Sarah Taaffe-Maguire , business reporter

A company that makes microchips for artificial intelligence and became the first chipmaker to be worth first $1trn then $2trn has today reached another record high.

Nvidia shares are now going for a record $1,132.19 after it posted higher-than-expected quarterly profits and made strong forecasts. Its value is now $2.62trn (£2.05trn)

The US-based, New York-listed company is in the ranks of tech giants worth the eye-watering trillion sum, including Amazon, Apple, Microsoft, and Google parent company Alphabet, as investors expect the company will benefit from the AI revolution.

Also making headlines was UK company Boohoo, the Manchester-based fast fashion retailer, as it cancelled annual bonuses worth £3m. 

A pay proposal for bosses was also ditched after talks with shareholders as the company has experienced losses after the pandemic-era online shopping boom faded and a cost of living crisis eroded consumer spending power.

Revolution Bars has rejected a proposed offer from rival Nightcap, warning it is "incapable of being delivered".

The hospitality group launched a sale process and restructuring plans last month amid efforts to stay afloat. The company's restructuring plans include £12.5m in fundraising and the closure of 18 venues.

But Revolution has said the non-binding proposal from Nightcap  did not include the proposed fundraising and would not work as it was "highly conditional".

Read the full story here ...

The rate of price rises in UK shops has returned to "normal levels", according to new industry figures.

Overall annual shop inflation eased to 0.6% in May, down from 0.8% in April, the British Retail Consortium (BRC) and NielsenIQ said.

The figure is the lowest since November 2021.

More than 120 business leaders have written an open letter giving their backing to Labour in the general election.

The letter printed in The Times has been signed by figures including the founders of Wikipedia Jimmy Wales, chef Tom Kerridge and former CEOs of Heathrow, JP Morgan and Aston Martin.

Openreach plans to build full fibre broadband in over 500 more locations across the UK, it has been announced.

The new locations include 400,000 sites in the hardest to reach, most rural parts of the country, including Tobermory in Argyll and Bute, Haworth in West Yorkshire, Saundersfoot in South Wales, Pinxton in Derbyshire, Harlow in Essex and Roborough in Devon.

The work, which is part of Openreach's £15bn project to upgrade the UK's broadband infrastructure, will cover a further 2.7 million homes and businesses by the end of 2026.

Clive Selley, chief executive of Openreach, said the plan was to build right across the UK, "from cities and towns to far-flung farms and island communities".

"Over time, we've learnt to deliver predictably, consistently and at a rapid pace - despite this being a hugely complex national engineering project," he said.

Check your full fibre availability here .

London is officially the world's cleverest city, according to an annual study.

Oxford Economics , an independent economic advisory firm, found the capital topped the list when it came to "human capital" - this encompasses the collective knowledge and skills of a city's population.

In its report, the firm found London came out on top in part due to the number of higher education institutions in the city, "which helps it achieve one of the highest rates of educational attainment in the world".

"London also attracts many highly educated people from abroad and several global corporations are headquartered in the city to take advantage of this world-class talent pool," the report added.

In second place was Tokyo and in third place was Riyadh.

Top cities by human capital score: 

1. London, UK

2. Tokyo, Japan

3. Riyadh, Saudi Arabia

4. New York, US

5. Seoul, South Korea

6. Paris, France

7. Washington, DC, US

8. Abu Dhabi, United Arab Emirates

9. Sydney, Australia

10. Boston, US

Concerns are being raised as fast fashion retailer Shein has its sights set on a London stock market listing.

Our City editor Mark Kleinman previously reported the fast fashion giant had held talks with the London Stock Exchange about staging a blockbuster public listing in the UK. 

Shein has quickly become one of the world's biggest online clothing retailers and the talks came at a time of crisis for the City as a listing venue for large multinationals.

But senior MPs are calling for more scrutiny of the Chinese company, which sells ultra-cheap clothing and has faced allegations of labour malpractices. 

Alicia Kearns, the Conservative chair of the Commons Foreign Affairs Committee, wrote on X: "With Shein's prices so low, the London Stock Exchange needs to ask itself, whose suffering is subsiding those prices?

"A company which has failed to make full disclosures about its supply chains as required by UK law, and where there are grave concerns about its factory working conditions, has no place in London."

Sarah Champion, the Labour chair of the International Development Committee, told The Guardian: "Transparency in supply chains is vital and something all governments should be demanding. Serious concerns have been raised about the use of modern slavery by Shein which need investigating."

A Shein spokesperson told the Money blog that it acknowledged its role in "creating a more sustainable and responsible fashion industry" and "believed it was healthy to attract scrutiny and transparency" and "wanted to be held to the highest standards".

"Shein has a zero-tolerance policy for forced labour and we are committed to respecting human rights," they said in a statement.

"We take visibility across our entire supply chain seriously and we require our contract manufacturers to only source cotton from approved regions."

The company also said it was "investing millions of pounds in strengthening governance and compliance across its supply chain".

"Our regular supplier audits are showing a consistent improvement in performance and compliance by our supplier partners. This includes improvements in ensuring that workers are compensated fairly for what they do," they added.

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