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Research Objectives – Types, Examples and Writing Guide

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Research Objectives

Research objectives refer to the specific goals or aims of a research study. They provide a clear and concise description of what the researcher hopes to achieve by conducting the research . The objectives are typically based on the research questions and hypotheses formulated at the beginning of the study and are used to guide the research process.

Types of Research Objectives

Here are the different types of research objectives in research:

• Exploratory Objectives: These objectives are used to explore a topic, issue, or phenomenon that has not been studied in-depth before. The aim of exploratory research is to gain a better understanding of the subject matter and generate new ideas and hypotheses .
• Descriptive Objectives: These objectives aim to describe the characteristics, features, or attributes of a particular population, group, or phenomenon. Descriptive research answers the “what” questions and provides a snapshot of the subject matter.
• Explanatory Objectives : These objectives aim to explain the relationships between variables or factors. Explanatory research seeks to identify the cause-and-effect relationships between different phenomena.
• Predictive Objectives: These objectives aim to predict future events or outcomes based on existing data or trends. Predictive research uses statistical models to forecast future trends or outcomes.
• Evaluative Objectives : These objectives aim to evaluate the effectiveness or impact of a program, intervention, or policy. Evaluative research seeks to assess the outcomes or results of a particular intervention or program.
• Prescriptive Objectives: These objectives aim to provide recommendations or solutions to a particular problem or issue. Prescriptive research identifies the best course of action based on the results of the study.
• Diagnostic Objectives : These objectives aim to identify the causes or factors contributing to a particular problem or issue. Diagnostic research seeks to uncover the underlying reasons for a particular phenomenon.
• Comparative Objectives: These objectives aim to compare two or more groups, populations, or phenomena to identify similarities and differences. Comparative research is used to determine which group or approach is more effective or has better outcomes.
• Historical Objectives: These objectives aim to examine past events, trends, or phenomena to gain a better understanding of their significance and impact. Historical research uses archival data, documents, and records to study past events.
• Ethnographic Objectives : These objectives aim to understand the culture, beliefs, and practices of a particular group or community. Ethnographic research involves immersive fieldwork and observation to gain an insider’s perspective of the group being studied.
• Action-oriented Objectives: These objectives aim to bring about social or organizational change. Action-oriented research seeks to identify practical solutions to social problems and to promote positive change in society.
• Conceptual Objectives: These objectives aim to develop new theories, models, or frameworks to explain a particular phenomenon or set of phenomena. Conceptual research seeks to provide a deeper understanding of the subject matter by developing new theoretical perspectives.
• Methodological Objectives: These objectives aim to develop and improve research methods and techniques. Methodological research seeks to advance the field of research by improving the validity, reliability, and accuracy of research methods and tools.
• Theoretical Objectives : These objectives aim to test and refine existing theories or to develop new theoretical perspectives. Theoretical research seeks to advance the field of knowledge by testing and refining existing theories or by developing new theoretical frameworks.
• Measurement Objectives : These objectives aim to develop and validate measurement instruments, such as surveys, questionnaires, and tests. Measurement research seeks to improve the quality and reliability of data collection and analysis by developing and testing new measurement tools.
• Design Objectives : These objectives aim to develop and refine research designs, such as experimental, quasi-experimental, and observational designs. Design research seeks to improve the quality and validity of research by developing and testing new research designs.
• Sampling Objectives: These objectives aim to develop and refine sampling techniques, such as probability and non-probability sampling methods. Sampling research seeks to improve the representativeness and generalizability of research findings by developing and testing new sampling techniques.

How to Write Research Objectives

Writing clear and concise research objectives is an important part of any research project, as it helps to guide the study and ensure that it is focused and relevant. Here are some steps to follow when writing research objectives:

• Identify the research problem : Before you can write research objectives, you need to identify the research problem you are trying to address. This should be a clear and specific problem that can be addressed through research.
• Define the research questions : Based on the research problem, define the research questions you want to answer. These questions should be specific and should guide the research process.
• Identify the variables : Identify the key variables that you will be studying in your research. These are the factors that you will be measuring, manipulating, or analyzing to answer your research questions.
• Write specific objectives: Write specific, measurable objectives that will help you answer your research questions. These objectives should be clear and concise and should indicate what you hope to achieve through your research.
• Use the SMART criteria: To ensure that your research objectives are well-defined and achievable, use the SMART criteria. This means that your objectives should be Specific, Measurable, Achievable, Relevant, and Time-bound.
• Revise and refine: Once you have written your research objectives, revise and refine them to ensure that they are clear, concise, and achievable. Make sure that they align with your research questions and variables, and that they will help you answer your research problem.

Example of Research Objectives

Examples of research objectives Could be:

Research Objectives for the topic of “The Impact of Artificial Intelligence on Employment”:

• To investigate the effects of the adoption of AI on employment trends across various industries and occupations.
• To explore the potential for AI to create new job opportunities and transform existing roles in the workforce.
• To examine the social and economic implications of the widespread use of AI for employment, including issues such as income inequality and access to education and training.
• To identify the skills and competencies that will be required for individuals to thrive in an AI-driven workplace, and to explore the role of education and training in developing these skills.
• To evaluate the ethical and legal considerations surrounding the use of AI for employment, including issues such as bias, privacy, and the responsibility of employers and policymakers to protect workers’ rights.

When to Write Research Objectives

• At the beginning of a research project : Research objectives should be identified and written down before starting a research project. This helps to ensure that the project is focused and that data collection and analysis efforts are aligned with the intended purpose of the research.
• When refining research questions: Writing research objectives can help to clarify and refine research questions. Objectives provide a more concrete and specific framework for addressing research questions, which can improve the overall quality and direction of a research project.
• After conducting a literature review : Conducting a literature review can help to identify gaps in knowledge and areas that require further research. Writing research objectives can help to define and focus the research effort in these areas.
• When developing a research proposal: Research objectives are an important component of a research proposal. They help to articulate the purpose and scope of the research, and provide a clear and concise summary of the expected outcomes and contributions of the research.
• When seeking funding for research: Funding agencies often require a detailed description of research objectives as part of a funding proposal. Writing clear and specific research objectives can help to demonstrate the significance and potential impact of a research project, and increase the chances of securing funding.
• When designing a research study : Research objectives guide the design and implementation of a research study. They help to identify the appropriate research methods, sampling strategies, data collection and analysis techniques, and other relevant aspects of the study design.
• When communicating research findings: Research objectives provide a clear and concise summary of the main research questions and outcomes. They are often included in research reports and publications, and can help to ensure that the research findings are communicated effectively and accurately to a wide range of audiences.
• When evaluating research outcomes : Research objectives provide a basis for evaluating the success of a research project. They help to measure the degree to which research questions have been answered and the extent to which research outcomes have been achieved.
• When conducting research in a team : Writing research objectives can facilitate communication and collaboration within a research team. Objectives provide a shared understanding of the research purpose and goals, and can help to ensure that team members are working towards a common objective.

Purpose of Research Objectives

Some of the main purposes of research objectives include:

• To clarify the research question or problem : Research objectives help to define the specific aspects of the research question or problem that the study aims to address. This makes it easier to design a study that is focused and relevant.
• To guide the research design: Research objectives help to determine the research design, including the research methods, data collection techniques, and sampling strategy. This ensures that the study is structured and efficient.
• To measure progress : Research objectives provide a way to measure progress throughout the research process. They help the researcher to evaluate whether they are on track and meeting their goals.
• To communicate the research goals : Research objectives provide a clear and concise description of the research goals. This helps to communicate the purpose of the study to other researchers, stakeholders, and the general public.

Advantages of Research Objectives

Here are some advantages of having well-defined research objectives:

• Focus : Research objectives help to focus the research effort on specific areas of inquiry. By identifying clear research questions, the researcher can narrow down the scope of the study and avoid getting sidetracked by irrelevant information.
• Clarity : Clearly stated research objectives provide a roadmap for the research study. They provide a clear direction for the research, making it easier for the researcher to stay on track and achieve their goals.
• Measurability : Well-defined research objectives provide measurable outcomes that can be used to evaluate the success of the research project. This helps to ensure that the research is effective and that the research goals are achieved.
• Feasibility : Research objectives help to ensure that the research project is feasible. By clearly defining the research goals, the researcher can identify the resources required to achieve those goals and determine whether those resources are available.
• Relevance : Research objectives help to ensure that the research study is relevant and meaningful. By identifying specific research questions, the researcher can ensure that the study addresses important issues and contributes to the existing body of knowledge.

About the author

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Researcher, Academic Writer, Web developer

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• Aims and Objectives – A Guide for Academic Writing
• Doing a PhD

One of the most important aspects of a thesis, dissertation or research paper is the correct formulation of the aims and objectives. This is because your aims and objectives will establish the scope, depth and direction that your research will ultimately take. An effective set of aims and objectives will give your research focus and your reader clarity, with your aims indicating what is to be achieved, and your objectives indicating how it will be achieved.

Introduction

There is no getting away from the importance of the aims and objectives in determining the success of your research project. Unfortunately, however, it is an aspect that many students struggle with, and ultimately end up doing poorly. Given their importance, if you suspect that there is even the smallest possibility that you belong to this group of students, we strongly recommend you read this page in full.

This page describes what research aims and objectives are, how they differ from each other, how to write them correctly, and the common mistakes students make and how to avoid them. An example of a good aim and objectives from a past thesis has also been deconstructed to help your understanding.

What Are Aims and Objectives?

Research aims.

A research aim describes the main goal or the overarching purpose of your research project.

In doing so, it acts as a focal point for your research and provides your readers with clarity as to what your study is all about. Because of this, research aims are almost always located within its own subsection under the introduction section of a research document, regardless of whether it’s a thesis , a dissertation, or a research paper .

A research aim is usually formulated as a broad statement of the main goal of the research and can range in length from a single sentence to a short paragraph. Although the exact format may vary according to preference, they should all describe why your research is needed (i.e. the context), what it sets out to accomplish (the actual aim) and, briefly, how it intends to accomplish it (overview of your objectives).

To give an example, we have extracted the following research aim from a real PhD thesis:

Example of a Research Aim

The role of diametrical cup deformation as a factor to unsatisfactory implant performance has not been widely reported. The aim of this thesis was to gain an understanding of the diametrical deformation behaviour of acetabular cups and shells following impaction into the reamed acetabulum. The influence of a range of factors on deformation was investigated to ascertain if cup and shell deformation may be high enough to potentially contribute to early failure and high wear rates in metal-on-metal implants.

Note: Extracted with permission from thesis titled “T he Impact And Deformation Of Press-Fit Metal Acetabular Components ” produced by Dr H Hothi of previously Queen Mary University of London.

Research Objectives

Where a research aim specifies what your study will answer, research objectives specify how your study will answer it.

They divide your research aim into several smaller parts, each of which represents a key section of your research project. As a result, almost all research objectives take the form of a numbered list, with each item usually receiving its own chapter in a dissertation or thesis.

Following the example of the research aim shared above, here are it’s real research objectives as an example:

Example of a Research Objective

• Develop finite element models using explicit dynamics to mimic mallet blows during cup/shell insertion, initially using simplified experimentally validated foam models to represent the acetabulum.
• Investigate the number, velocity and position of impacts needed to insert a cup.
• Determine the relationship between the size of interference between the cup and cavity and deformation for different cup types.
• Investigate the influence of non-uniform cup support and varying the orientation of the component in the cavity on deformation.
• Examine the influence of errors during reaming of the acetabulum which introduce ovality to the cavity.
• Determine the relationship between changes in the geometry of the component and deformation for different cup designs.
• Develop three dimensional pelvis models with non-uniform bone material properties from a range of patients with varying bone quality.
• Use the key parameters that influence deformation, as identified in the foam models to determine the range of deformations that may occur clinically using the anatomic models and if these deformations are clinically significant.

It’s worth noting that researchers sometimes use research questions instead of research objectives, or in other cases both. From a high-level perspective, research questions and research objectives make the same statements, but just in different formats.

Taking the first three research objectives as an example, they can be restructured into research questions as follows:

Restructuring Research Objectives as Research Questions

• Can finite element models using simplified experimentally validated foam models to represent the acetabulum together with explicit dynamics be used to mimic mallet blows during cup/shell insertion?
• What is the number, velocity and position of impacts needed to insert a cup?
• What is the relationship between the size of interference between the cup and cavity and deformation for different cup types?

Difference Between Aims and Objectives

Hopefully the above explanations make clear the differences between aims and objectives, but to clarify:

• The research aim focus on what the research project is intended to achieve; research objectives focus on how the aim will be achieved.
• Research aims are relatively broad; research objectives are specific.
• Research aims focus on a project’s long-term outcomes; research objectives focus on its immediate, short-term outcomes.
• A research aim can be written in a single sentence or short paragraph; research objectives should be written as a numbered list.

How to Write Aims and Objectives

Before we discuss how to write a clear set of research aims and objectives, we should make it clear that there is no single way they must be written. Each researcher will approach their aims and objectives slightly differently, and often your supervisor will influence the formulation of yours on the basis of their own preferences.

Regardless, there are some basic principles that you should observe for good practice; these principles are described below.

Your aim should be made up of three parts that answer the below questions:

• Why is this research required?
• What is this research about?
• How are you going to do it?

The easiest way to achieve this would be to address each question in its own sentence, although it does not matter whether you combine them or write multiple sentences for each, the key is to address each one.

The first question, why , provides context to your research project, the second question, what , describes the aim of your research, and the last question, how , acts as an introduction to your objectives which will immediately follow.

Scroll through the image set below to see the ‘why, what and how’ associated with our research aim example.

Note: Your research aims need not be limited to one. Some individuals per to define one broad ‘overarching aim’ of a project and then adopt two or three specific research aims for their thesis or dissertation. Remember, however, that in order for your assessors to consider your research project complete, you will need to prove you have fulfilled all of the aims you set out to achieve. Therefore, while having more than one research aim is not necessarily disadvantageous, consider whether a single overarching one will do.

Research Objectives

Each of your research objectives should be SMART :

• Specific – is there any ambiguity in the action you are going to undertake, or is it focused and well-defined?
• Measurable – how will you measure progress and determine when you have achieved the action?
• Achievable – do you have the support, resources and facilities required to carry out the action?
• Relevant – is the action essential to the achievement of your research aim?
• Timebound – can you realistically complete the action in the available time alongside your other research tasks?

In addition to being SMART, your research objectives should start with a verb that helps communicate your intent. Common research verbs include:

Table of Research Verbs to Use in Aims and Objectives

Last, format your objectives into a numbered list. This is because when you write your thesis or dissertation, you will at times need to make reference to a specific research objective; structuring your research objectives in a numbered list will provide a clear way of doing this.

To bring all this together, let’s compare the first research objective in the previous example with the above guidance:

Checking Research Objective Example Against Recommended Approach

Research Objective:

1. Develop finite element models using explicit dynamics to mimic mallet blows during cup/shell insertion, initially using simplified experimentally validated foam models to represent the acetabulum.

Checking Against Recommended Approach:

Q: Is it specific? A: Yes, it is clear what the student intends to do (produce a finite element model), why they intend to do it (mimic cup/shell blows) and their parameters have been well-defined ( using simplified experimentally validated foam models to represent the acetabulum ).

Q: Is it measurable? A: Yes, it is clear that the research objective will be achieved once the finite element model is complete.

Q: Is it achievable? A: Yes, provided the student has access to a computer lab, modelling software and laboratory data.

Q: Is it relevant? A: Yes, mimicking impacts to a cup/shell is fundamental to the overall aim of understanding how they deform when impacted upon.

Q: Is it timebound? A: Yes, it is possible to create a limited-scope finite element model in a relatively short time, especially if you already have experience in modelling.

Q: Does it start with a verb? A: Yes, it starts with ‘develop’, which makes the intent of the objective immediately clear.

Q: Is it a numbered list? A: Yes, it is the first research objective in a list of eight.

Mistakes in Writing Research Aims and Objectives

1. making your research aim too broad.

Having a research aim too broad becomes very difficult to achieve. Normally, this occurs when a student develops their research aim before they have a good understanding of what they want to research. Remember that at the end of your project and during your viva defence , you will have to prove that you have achieved your research aims; if they are too broad, this will be an almost impossible task. In the early stages of your research project, your priority should be to narrow your study to a specific area. A good way to do this is to take the time to study existing literature, question their current approaches, findings and limitations, and consider whether there are any recurring gaps that could be investigated .

Note: Achieving a set of aims does not necessarily mean proving or disproving a theory or hypothesis, even if your research aim was to, but having done enough work to provide a useful and original insight into the principles that underlie your research aim.

2. Making Your Research Objectives Too Ambitious

Be realistic about what you can achieve in the time you have available. It is natural to want to set ambitious research objectives that require sophisticated data collection and analysis, but only completing this with six months before the end of your PhD registration period is not a worthwhile trade-off.

3. Formulating Repetitive Research Objectives

Each research objective should have its own purpose and distinct measurable outcome. To this effect, a common mistake is to form research objectives which have large amounts of overlap. This makes it difficult to determine when an objective is truly complete, and also presents challenges in estimating the duration of objectives when creating your project timeline. It also makes it difficult to structure your thesis into unique chapters, making it more challenging for you to write and for your audience to read.

Fortunately, this oversight can be easily avoided by using SMART objectives.

Hopefully, you now have a good idea of how to create an effective set of aims and objectives for your research project, whether it be a thesis, dissertation or research paper. While it may be tempting to dive directly into your research, spending time on getting your aims and objectives right will give your research clear direction. This won’t only reduce the likelihood of problems arising later down the line, but will also lead to a more thorough and coherent research project.

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Writing the Research Objectives: 5 Straightforward Examples

The research objective of a research proposal or scientific article defines the direction or content of a research investigation. Without the research objectives, the proposal or research paper is in disarray. It is like a fisherman riding on a boat without any purpose and with no destination in sight. Therefore, at the beginning of any research venture, the researcher must be clear about what he or she intends to do or achieve in conducting a study.

How do you define the objectives of a study? What are the uses of the research objective? How would a researcher write this essential part of the research? This article aims to provide answers to these questions.

Table of Contents

Definition of a research objective.

A research objective describes, in a few words, the result of the research project after its implementation. It answers the question,

“ What does the researcher want or hope to achieve at the end of the research project.”  

The research objective provides direction to the performance of the study.

What are the Uses of the Research Objective?

The uses of the research objective are enumerated below:

• serves as the researcher’s guide in identifying the appropriate research design,
• identifies the variables of the study, and
• specifies the data collection procedure and the corresponding analysis for the data generated.

The research design serves as the “blueprint” for the research investigation. The University of Southern California describes the different types of research design extensively. It details the data to be gathered, data collection procedure, data measurement, and statistical tests to use in the analysis.

The variables of the study include those factors that the researcher wants to evaluate in the study. These variables narrow down the research to several manageable components to see differences or correlations between them.

Specifying the data collection procedure ensures data accuracy and integrity . Thus, the probability of error is minimized. Generalizations or conclusions based on valid arguments founded on reliable data strengthens research findings on particular issues and problems.

In data mining activities where large data sets are involved, the research objective plays a crucial role. Without a clear objective to guide the machine learning process, the desired outcomes will not be met.

How is the Research Objective Written?

A research objective must be achievable, i.e., it must be framed keeping in mind the available time, infrastructure required for research, and other resources.

Before forming a research objective, you should read about all the developments in your area of research and find gaps in knowledge that need to be addressed. Readings will help you come up with suitable objectives for your research project.

5 Examples of Research Objectives

The following examples of research objectives based on several published studies on various topics demonstrate how the research objectives are written:

• This study aims to find out if there is a difference in quiz scores between students exposed to direct instruction and flipped classrooms (Webb and Doman, 2016).
• This study seeks to examine the extent, range, and method of coral reef rehabilitation projects in five shallow reef areas adjacent to popular tourist destinations in the Philippines (Yeemin et al ., 2006).
• This study aims to investigate species richness of mammal communities in five protected areas over the past 20 years (Evans et al ., 2006).
• This study aims to clarify the demographic, epidemiological, clinical, and radiological features of 2019-nCoV patients with other causes of pneumonia (Zhao et al ., 2020).
• This research aims to assess species extinction risks for sample regions that cover some 20% of the Earth’s terrestrial surface.

Finally, writing the research objectives requires constant practice, experience, and knowledge about the topic investigated. Clearly written objectives save time, money, and effort.

Once you have a clear idea of your research objectives, you can now develop your conceptual framework which is a crucial element of your research paper as it guides the flow of your research. The conceptual framework will help you develop your methodology and statistical tests.

I wrote a detailed, step-by-step guide on how to develop a conceptual framework with illustration in my post titled “ Conceptual Framework: A Step by Step Guide on How to Make One. “

Evans, K. L., Rodrigues, A. S., Chown, S. L., & Gaston, K. J. (2006). Protected areas and regional avian species richness in South Africa.  Biology letters ,  2 (2), 184-188.

Thomas, C. D., Cameron, A., Green, R. E., Bakkenes, M., Beaumont, L. J., Collingham, Y. C., … & Hughes, L. (2004). Extinction risk from climate change. Nature, 427(6970), 145-148.

Webb, M., & Doman, E. (2016). Does the Flipped Classroom Lead to Increased Gains on Learning Outcomes in ESL/EFL Contexts?. CATESOL Journal, 28(1), 39-67.

Yeemin, T., Sutthacheep, M., & Pettongma, R. (2006). Coral reef restoration projects in Thailand.  Ocean & Coastal Management ,  49 (9-10), 562-575.

Zhao, D., Yao, F., Wang, L., Zheng, L., Gao, Y., Ye, J., Guo, F., Zhao, H. & Gao, R. (2020). A comparative study on the clinical features of COVID-19 pneumonia to other pneumonias, Clinical Infectious Diseases , ciaa247, https://doi.org/10.1093/cid/ciaa247

© 2020 March 23 P. A. Regoniel Updated 17 November 2020 | Updated 18 January 2024

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About the author, patrick regoniel.

Dr. Regoniel, a faculty member of the graduate school, served as consultant to various environmental research and development projects covering issues and concerns on climate change, coral reef resources and management, economic valuation of environmental and natural resources, mining, and waste management and pollution. He has extensive experience on applied statistics, systems modelling and analysis, an avid practitioner of LaTeX, and a multidisciplinary web developer. He leverages pioneering AI-powered content creation tools to produce unique and comprehensive articles in this website.

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• A Research Guide
• Research Paper Guide

How to Write Research Objectives

• What are research objectives
• Step-by-step writing guide
• Helpful tips
• Research objectives examples

What are research objectives, and why are they important?

Step-by-step research objectives writing guide, step 1: provide the major background of your research, step 2: mention several objectives from the most to least important aspects, step 3: follow your resources and do not promise too much, step 4: keep your objectives and limitations mentioned, step 5: provide action verbs and tone, helpful tips for writing research objectives.

• Keep your content specific! It is necessary to narrow things down and leave no space for double meanings or confusion. If some idea cannot be supported with a piece of evidence, it’s better to avoid it in your objectives.
• Objectives must be measurable! It is crucial to make it possible to replicate your work in further research. Creating an outline as you strive for your goals and set the purpose is necessary.
• Keeping things relevant! Your research objectives should be related to your thesis statement and the subject that you have chosen to work with. It will help to avoid introducing ideas that are not related to your work.
• Temporal factor! Set deadlines to track your progress and provide a setting for your research if it is relevant. It will help your target audience to see your limitations and specifics.

Research objectives example

Research objective 1: The study aims to explore the origins and evolution of the youth movements in the Flemish provinces in Belgium, namely Chiro and KSA. This research evaluates the major differences during the post-WW2 period and the social factors that created differences between the movements. 

Research objective 2: This paper implements surveys and personal interviews to determine first-hand feedback from the youth members and the team leaders. 

Research objective 3: Aiming to compare and contrast, this study determines the positive outcomes of the unity project work between the branches of the youth movement in Belgium, aiming for statistical data to support it. 

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Research Aims, Objectives & Questions

The “Golden Thread” Explained Simply (+ Examples)

By: David Phair (PhD) and Alexandra Shaeffer (PhD) | June 2022

The research aims , objectives and research questions (collectively called the “golden thread”) are arguably the most important thing you need to get right when you’re crafting a research proposal , dissertation or thesis . We receive questions almost every day about this “holy trinity” of research and there’s certainly a lot of confusion out there, so we’ve crafted this post to help you navigate your way through the fog.

Overview: The Golden Thread

• What is the golden thread
• What are research aims ( examples )
• What are research objectives ( examples )
• What are research questions ( examples )
• The importance of alignment in the golden thread

What is the “golden thread”?  

The golden thread simply refers to the collective research aims , research objectives , and research questions for any given project (i.e., a dissertation, thesis, or research paper ). These three elements are bundled together because it’s extremely important that they align with each other, and that the entire research project aligns with them.

Importantly, the golden thread needs to weave its way through the entirety of any research project , from start to end. In other words, it needs to be very clearly defined right at the beginning of the project (the topic ideation and proposal stage) and it needs to inform almost every decision throughout the rest of the project. For example, your research design and methodology will be heavily influenced by the golden thread (we’ll explain this in more detail later), as well as your literature review.

The research aims, objectives and research questions (the golden thread) define the focus and scope ( the delimitations ) of your research project. In other words, they help ringfence your dissertation or thesis to a relatively narrow domain, so that you can “go deep” and really dig into a specific problem or opportunity. They also help keep you on track , as they act as a litmus test for relevance. In other words, if you’re ever unsure whether to include something in your document, simply ask yourself the question, “does this contribute toward my research aims, objectives or questions?”. If it doesn’t, chances are you can drop it.

Alright, enough of the fluffy, conceptual stuff. Let’s get down to business and look at what exactly the research aims, objectives and questions are and outline a few examples to bring these concepts to life.

Research Aims: What are they?

Simply put, the research aim(s) is a statement that reflects the broad overarching goal (s) of the research project. Research aims are fairly high-level (low resolution) as they outline the general direction of the research and what it’s trying to achieve .

Research Aims: Examples  

True to the name, research aims usually start with the wording “this research aims to…”, “this research seeks to…”, and so on. For example:

“This research aims to explore employee experiences of digital transformation in retail HR.”   “This study sets out to assess the interaction between student support and self-care on well-being in engineering graduate students”  

As you can see, these research aims provide a high-level description of what the study is about and what it seeks to achieve. They’re not hyper-specific or action-oriented, but they’re clear about what the study’s focus is and what is being investigated.

Need a helping hand?

Research Objectives: What are they?

The research objectives take the research aims and make them more practical and actionable . In other words, the research objectives showcase the steps that the researcher will take to achieve the research aims.

The research objectives need to be far more specific (higher resolution) and actionable than the research aims. In fact, it’s always a good idea to craft your research objectives using the “SMART” criteria. In other words, they should be specific, measurable, achievable, relevant and time-bound”.

Research Objectives: Examples  

Let’s look at two examples of research objectives. We’ll stick with the topic and research aims we mentioned previously.  

For the digital transformation topic:

To observe the retail HR employees throughout the digital transformation. To assess employee perceptions of digital transformation in retail HR. To identify the barriers and facilitators of digital transformation in retail HR.

And for the student wellness topic:

To determine whether student self-care predicts the well-being score of engineering graduate students. To determine whether student support predicts the well-being score of engineering students. To assess the interaction between student self-care and student support when predicting well-being in engineering graduate students.

  As you can see, these research objectives clearly align with the previously mentioned research aims and effectively translate the low-resolution aims into (comparatively) higher-resolution objectives and action points . They give the research project a clear focus and present something that resembles a research-based “to-do” list.

Research Questions: What are they?

Finally, we arrive at the all-important research questions. The research questions are, as the name suggests, the key questions that your study will seek to answer . Simply put, they are the core purpose of your dissertation, thesis, or research project. You’ll present them at the beginning of your document (either in the introduction chapter or literature review chapter) and you’ll answer them at the end of your document (typically in the discussion and conclusion chapters).  

The research questions will be the driving force throughout the research process. For example, in the literature review chapter, you’ll assess the relevance of any given resource based on whether it helps you move towards answering your research questions. Similarly, your methodology and research design will be heavily influenced by the nature of your research questions. For instance, research questions that are exploratory in nature will usually make use of a qualitative approach, whereas questions that relate to measurement or relationship testing will make use of a quantitative approach.  

Let’s look at some examples of research questions to make this more tangible.

Research Questions: Examples  

Again, we’ll stick with the research aims and research objectives we mentioned previously.  

For the digital transformation topic (which would be qualitative in nature):

How do employees perceive digital transformation in retail HR? What are the barriers and facilitators of digital transformation in retail HR?  

And for the student wellness topic (which would be quantitative in nature):

Does student self-care predict the well-being scores of engineering graduate students? Does student support predict the well-being scores of engineering students? Do student self-care and student support interact when predicting well-being in engineering graduate students?  

You’ll probably notice that there’s quite a formulaic approach to this. In other words, the research questions are basically the research objectives “converted” into question format. While that is true most of the time, it’s not always the case. For example, the first research objective for the digital transformation topic was more or less a step on the path toward the other objectives, and as such, it didn’t warrant its own research question.  

So, don’t rush your research questions and sloppily reword your objectives as questions. Carefully think about what exactly you’re trying to achieve (i.e. your research aim) and the objectives you’ve set out, then craft a set of well-aligned research questions . Also, keep in mind that this can be a somewhat iterative process , where you go back and tweak research objectives and aims to ensure tight alignment throughout the golden thread.

The importance of strong alignment 

Alignment is the keyword here and we have to stress its importance . Simply put, you need to make sure that there is a very tight alignment between all three pieces of the golden thread. If your research aims and research questions don’t align, for example, your project will be pulling in different directions and will lack focus . This is a common problem students face and can cause many headaches (and tears), so be warned.

Take the time to carefully craft your research aims, objectives and research questions before you run off down the research path. Ideally, get your research supervisor/advisor to review and comment on your golden thread before you invest significant time into your project, and certainly before you start collecting data .  

Recap: The golden thread

In this post, we unpacked the golden thread of research, consisting of the research aims , research objectives and research questions . You can jump back to any section using the links below.

As always, feel free to leave a comment below – we always love to hear from you. Also, if you’re interested in 1-on-1 support, take a look at our private coaching service here.

Psst… there’s more (for free)

This post is part of our dissertation mini-course, which covers everything you need to get started with your dissertation, thesis or research project. 

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37 Comments

Thank you very much for your great effort put. As an Undergraduate taking Demographic Research & Methodology, I’ve been trying so hard to understand clearly what is a Research Question, Research Aim and the Objectives in a research and the relationship between them etc. But as for now I’m thankful that you’ve solved my problem.

Well appreciated. This has helped me greatly in doing my dissertation.

An so delighted with this wonderful information thank you a lot.

so impressive i have benefited a lot looking forward to learn more on research.

I am very happy to have carefully gone through this well researched article.

Infact,I used to be phobia about anything research, because of my poor understanding of the concepts.

Now,I get to know that my research question is the same as my research objective(s) rephrased in question format.

I please I would need a follow up on the subject,as I intends to join the team of researchers. Thanks once again.

Thanks so much. This was really helpful.

i found this document so useful towards my study in research methods. thanks so much.

This is my 2nd read topic in your course and I should commend the simplified explanations of each part. I’m beginning to understand and absorb the use of each part of a dissertation/thesis. I’ll keep on reading your free course and might be able to avail the training course! Kudos!

Thank you! Better put that my lecture and helped to easily understand the basics which I feel often get brushed over when beginning dissertation work.

This is quite helpful. I like how the Golden thread has been explained and the needed alignment.

This is quite helpful. I really appreciate!

The article made it simple for researcher students to differentiate between three concepts.

Very innovative and educational in approach to conducting research.

I am very impressed with all these terminology, as I am a fresh student for post graduate, I am highly guided and I promised to continue making consultation when the need arise. Thanks a lot.

A very helpful piece. thanks, I really appreciate it .

Very well explained, and it might be helpful to many people like me.

Wish i had found this (and other) resource(s) at the beginning of my PhD journey… not in my writing up year… 😩 Anyways… just a quick question as i’m having some issues ordering my “golden thread”…. does it matter in what order you mention them? i.e., is it always first aims, then objectives, and finally the questions? or can you first mention the research questions and then the aims and objectives?

Thank you for a very simple explanation that builds upon the concepts in a very logical manner. Just prior to this, I read the research hypothesis article, which was equally very good. This met my primary objective.

My secondary objective was to understand the difference between research questions and research hypothesis, and in which context to use which one. However, I am still not clear on this. Can you kindly please guide?

In research, a research question is a clear and specific inquiry that the researcher wants to answer, while a research hypothesis is a tentative statement or prediction about the relationship between variables or the expected outcome of the study. Research questions are broader and guide the overall study, while hypotheses are specific and testable statements used in quantitative research. Research questions identify the problem, while hypotheses provide a focus for testing in the study.

Exactly what I need in this research journey, I look forward to more of your coaching videos.

This helped a lot. Thanks so much for the effort put into explaining it.

What data source in writing dissertation/Thesis requires?

What is data source covers when writing dessertation/thesis

This is quite useful thanks

I’m excited and thankful. I got so much value which will help me progress in my thesis.

where are the locations of the reserch statement, research objective and research question in a reserach paper? Can you write an ouline that defines their places in the researh paper?

Very helpful and important tips on Aims, Objectives and Questions.

Thank you so much for making research aim, research objectives and research question so clear. This will be helpful to me as i continue with my thesis.

Thanks much for this content. I learned a lot. And I am inspired to learn more. I am still struggling with my preparation for dissertation outline/proposal. But I consistently follow contents and tutorials and the new FB of GRAD Coach. Hope to really become confident in writing my dissertation and successfully defend it.

As a researcher and lecturer, I find splitting research goals into research aims, objectives, and questions is unnecessarily bureaucratic and confusing for students. For most biomedical research projects, including ‘real research’, 1-3 research questions will suffice (numbers may differ by discipline).

Awesome! Very important resources and presented in an informative way to easily understand the golden thread. Indeed, thank you so much.

Well explained

The blog article on research aims, objectives, and questions by Grad Coach is a clear and insightful guide that aligns with my experiences in academic research. The article effectively breaks down the often complex concepts of research aims and objectives, providing a straightforward and accessible explanation. Drawing from my own research endeavors, I appreciate the practical tips offered, such as the need for specificity and clarity when formulating research questions. The article serves as a valuable resource for students and researchers, offering a concise roadmap for crafting well-defined research goals and objectives. Whether you’re a novice or an experienced researcher, this article provides practical insights that contribute to the foundational aspects of a successful research endeavor.

A great thanks for you. it is really amazing explanation. I grasp a lot and one step up to research knowledge.

I really found these tips helpful. Thank you very much Grad Coach.

I found this article helpful. Thanks for sharing this.

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• Writing Tips

How to Write Research Objectives

3-minute read

• 22nd November 2021

Writing a research paper, thesis, or dissertation ? If so, you’ll want to state your research objectives in the introduction of your paper to make it clear to your readers what you’re trying to accomplish. But how do you write effective research objectives? In this post, we’ll look at two key topics to help you do this:

• How to use your research aims as a basis for developing objectives.
• How to use SMART criteria to refine your research objectives.

For more advice on how to write strong research objectives, see below.

Research Aims and Objectives

There is an important difference between research aims and research objectives:

• A research aim defines the main purpose of your research. As such, you can think of your research aim as answering the question “What are you doing?”
• Research objectives (as most studies will have more than one) are the steps you will take to fulfil your aims. As such, your objectives should answer the question “How are you conducting your research?”

For instance, an example research aim could be:

This study will investigate the link between dehydration and the incidence of urinary tract infections (UTIs) in intensive care patients in Australia.

To develop a set of research objectives, you would then break down the various steps involved in meeting said aim. For example:

This study will investigate the link between dehydration and the incidence of urinary tract infections (UTIs) in intensive care patients in Australia. To achieve this, the study objectives w ill include:

• Replicat ing a small Singaporean study into the role of dehydration in UTIs in hospital patients (Sepe, 2018) in a larger Australian cohort.
• Trialing the use of intravenous fluids for intensive care patients to prevent dehydration.
• Assessing the relationship between the age of patients and quantities of intravenous fluids needed to counter dehydration.

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Note that the objectives don’t go into any great detail here. The key is to briefly summarize each component of your study. You can save details for how you will conduct the research for the methodology section of your paper.

Make Your Research Objectives SMART

A great way to refine your research objectives is to use SMART criteria . Borrowed from the world of project management, there are many versions of this system. However, we’re going to focus on developing specific, measurable, achievable, relevant, and timebound objectives.

In other words, a good research objective should be all of the following:

• S pecific – Is the objective clear and well-defined?
• M easurable – How will you know when the objective has been achieved? Is there a way to measure the thing you’re seeking to do?
• A chievable – Do you have the support and resources necessary to undertake this action? Are you being overly ambitious with this objective?
• R elevant – Is this objective vital for fulfilling your research aim?
• T imebound – Can this action be realistically undertaken in the time you have?

If you follow this system, your research objectives will be much stronger.

Expert Research Proofreading

Whatever your research aims and objectives, make sure to have your academic writing proofread by the experts!

Our academic editors can help you with research papers and proposals , as well as any other scholarly document you need checking. And this will help to ensure that your academic writing is always clear, concise, and precise.

Submit a free sample document today to trial our services and find out more.

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Writing Effective Research Aims and Objectives

• By: Margaret-Anne Houston , Marissa McDonagh Edited by: Margaret-Anne Houston
• Product: Sage Research Methods: Business
• Publisher: SAGE Publications Ltd
• Publication year: 2023
• Online pub date: March 21, 2023
• Discipline: Business and Management
• Methods: Research questions , Writing research , Research design
• DOI: https:// doi. org/10.4135/9781529668216
• Keywords: fuel poverty , social media Show all Show less
• Academic Level: Advanced Undergraduate Online ISBN: 9781529668216 More information Less information

The writing of effective research aims and objectives can cause confusion and concern to new and experienced researchers and learners. This step in your research journey is usually the first written method used to convey your research idea to your tutor. Therefore, aims and objectives should clearly convey your topic, academic foundation, and research design. In order to write effective research aims and objectives, researchers should consider all aspects of their proposed work. For example, the sample(s) to be approached for participation in the primary data collection. Identifying research objectives that are SMART is key to ensuring key aspects of the work are considered prior to any data collection. This includes consideration of access to samples and the ethics of researching the topic and research design. Finally, seeing your work as others will read it, can be an effective evaluation tool to ensure your own research objectives adequately capture and reflect your intended study. Therefore, this guide encourages you to consider common issues with identifying and writing research aims and objectives through consideration of examples.

Learning Outcomes

By the end of this guide, readers should be able to:

• Identify the meaning and purpose of a research aim within business research
• Understand the link between an effective research aim and the wider topic and literature/secondary sources, where appropriate
• Understand how to identify and write Specific, Measurable, Achievable, Realistic and Timely (SMART) Research objectives, research questions, and consideration of research hypothesis
• Recognize the link between writing an effective research aim and the research design. Write own research aim and objectives

Introduction

The writing of effective research aims and objectives can cause confusion and concern to new (and experienced!) researchers and learners. Attempting to identify the scope and focus of a project within a few specific statements, can take time and consideration of all aspects of your research design. If you are still unsure of your approach to your topic, or even the boundaries of the topic itself, this uncertainty can make the framing of an effective research aim seem like an uphill task.

However, even if this is your first time trying to convey your research idea within a few concise and precise statements, there are steps to take to ensure your work clearly communicates your meaning to your audience. This how-to-guide draws on examples of business topic research aims and objectives and explores techniques for reviewing their meaning. This active learning approach will enable you to grow confidence in framing and communicating your own research.

The importance of ensuring the research aim and objectives are not only reflective of the topic choice but are also achievable can be a fluid process, which in itself, can result in anxious researchers. Seeing your work as others will read it, can be an effective evaluation tool to ensure your own research objectives adequately capture and reflect your intended study. Therefore, this guide encourages you to consider common issues with identifying and writing research aims and objectives through consideration of examples.

Identify the Meaning and Purpose of a Research Aim with Business Research

Writing an effective research aim is an integral part of the research process. A research aim is a statement of intent. It should communicate your research goal clearly and should provide a focus for your work from the offset. It is important to differentiate between a research aim and the objectives. If a research aim tells the reader what you plan to achieve, then the research objectives should state how you would reach that goal. Often the objectives will provide a road map of the steps you will take in order to meet the research aim. Therefore, a research aim in business-related topics is typically a single sentence or even two, which conveys the overall purpose of the research-the end goal!

The terminology you use when writing your research aim is important. Note the following example aims from Business related topics:

• 1. This research aims to evaluate the lasting effects of lockdown and ‘work from home’ initiatives on productiveness in the financial service industry.
• 2. This research aims to establish a link between innovations in Artificial Intelligence (AI) and recruitment processes for The Royal Bank of Scotland.
• 3. This research aims to investigate to what extent Corporate Social Responsibility (CSR) initiatives can influence consumer behavior. A case study of Aldi UK.
• 4. This research aims to assess the effectiveness of technology companies’ risk management of cyber and information risks measured on the basis of supply chain resilience.
• 5. This research aims to explore the impact of Government funded initiatives to encourage social entrepreneurship in Scotland.

As evidenced above all of the aims stated contain verbs, these highlight how the research will be undertaken. Words such as to assess, to establish, to explore or to evaluate all reflect research analysis. This conveys your intention clearly to the reader and whilst it may not fully demonstrate exactly how the project will be undertaken, the verbs show what the goal is.

The objectives, which follow the aim, can help to show the exact ways the aim will be achieved, highlighting the research methods. It is important to think carefully about whether you plan to or will be to, come to a clear conclusion. Often it is not possible and this can be due to many factors such as the time or scope of the issue. For example, in the aims stated above number 2 is the only one that states it will ‘establish a link.’ This is because the aim is specific and measurable. The objectives should identify the specific processes it will examine and link to effective recruitment practices that are more effective than prior to AI being used.

However, for the other aims it is more appropriate to explore or investigate the topics, as opposed to ‘establishing’ or to ‘evidence an impact.’

Abbreviations are a useful way of shortening words or phrases and they can give writing a more coherent flow. It is worth noting that all abbreviations like AI or CSR should only be used when they are spelled out initially and if they appear frequently throughout your writing.

It is important to always check with your supervisor or course Handbook but typically, you should have a research question, a research aim, and objectives. The research question should capture what the issue is, often it will help to explain your research aim by offering a critical perspective. For example, if your research is to evaluate the effect of something then your question may be to what extent is that something works?

Finally, it is important to remember that the wording of your research aim may change slightly as your research progresses. Often students will modify the words to reflect what they are undertaking as the process develops.

Section Summary

• An effective research aim should clearly set out the goal of a project.
• Carefully consider the terminology you use at this stage, and ensure it reflects the outcome of the study.
• Remember a research aim can be fluid and the exact wording is likely to change as you progress through your research journey.

Understand the Link Between an Effective Research Aim and the Wider Topic and Literature/secondary Sources, Where Appropriate

When developing the research aim it is important to be engaged with the wider topic and associated literature and secondary sources from the offset. These sources will be crucial in helping you to tackle the topic successfully.

Identifying an idea for a research project can sometimes be a relatively simple first step in the research process. It is often narrowing the idea down to a research aim, which can be more difficult. A good way to start is to brainstorm ideas, think about what interests you the most about your studies, and note down keywords which can then be used as search terms. Researchers, at all levels of research and study, should consider information-seeking as a process through which they engage with the primary literature and secondary sources concerning their topic area. This will develop self-confidence in your ability to define the terms of reference of your work and studies. An inquiring mind and openness to a degree of flexibility of approach in these early stages of research, can be key to ensuring initial topic ideas can be molded into achievable research aims and objectives.

Research could be considered to be cyclical, not a one-off process. Therefore, in order to ensure a definable and achievable research topic, many projects use a mixture of sources. This requires a degree of confidence on the part of the researcher; to identify the relevant resources they require, a strategy for how to find them and also, a process for information management.

Many researchers will start with an online search for both academic and non-academic sources. The short-term success of this first step can be dictated by the choice of keywords and phrases. That is, those terms that the researcher believes are most relevant for, and most likely to come up with links to their research topic. However, caution should be employed in this initial task of online searching - this is an important opportunity to consider how we identify these specific keywords. A limited understanding of the area will be enhanced through further reading. It can allow the researcher to access previous studies in the same topic area and identify effective research methods. An informed research aim should be underpinned by reading and evaluating sources in relation to the research idea.

Using the research aims below as examples, note the sources required and some issues to consider for each source. By strategically linking your research aim to the wider area you will ensure your research is robust from the start.

• Reading combined with ongoing critical appraisal of associated sources can help to refine and focus your research aim and objectives.
• Think of your research as an ongoing process. Reading associated sources should be embedded in every stage of your research journey.
• Ensure you are acknowledging the wider research area and associated sources from the offset as this will help to refine and focus your research aim and objectives.

Understanding How to Identify SMART Research Objectives, Research Questions, and Consideration of Research Hypothesis

First-time final year undergraduates are normally expected to identify a research topic and research design that are realistic and achievable. Not only should they be realistic as topics but also achievable within a short time period when most learners have never undertaken such work previously. A common pitfall of many initial research topics is identifying an area that is too wide in scope. A simple step is to consider how to express and convey the work within a series of research objectives. Careful consideration of the content of these statements can help narrow the topic focus, and ensure the research design is relevant to the work to be undertaken. Therefore, writing your objectives should be viewed as a process and not a one-off exercise. Remember, they convey your work to an audience and set out the initial boundaries of the research to be undertaken.

Therefore, research aims and objectives should provide focus and direction for the research topic. Many business research methods texts will introduce the writing of research aims and objectives as a specific skill required to ensure they are Specific, Measurable, Achievable, Realistic and Timely (SMART). By following the SMART guidelines and analyzing examples of common issues within aims and objectives, learners can build confidence and ensure their aims and objectives are strong. Together with these five criteria, the language used can convey the depth of the inquiry. By way of explanation, consider the following topic submitted for consideration as a final-year project:

The research aim is to evaluate consumer perceptions of the impact of social media advertising on their car purchasing decisions. The fieldwork will examine consumer attitudes toward social media advertising and the benefits of this approach. This will be explored through the following research objectives:

• 1. Examine relevant literature concerning advertising, and trends in social media within the car industry;
• 2. Identify the attitudes of key players and stakeholders within the advertising industry toward the use of social media;
• 3. Discuss the effects of new technology on social media and advertising trends;
• 4. Evaluate how consumers relate to new technology with a view to making recommendations for improvement in the use of social media within online advertising.

S pecific – the research objectives reflect the terminology also used within the research aim.

M easureable – this does not necessarily mean that the work will involve quantitative data. Consider that the objectives identify the issues and samples and so the target of the work.

A chievable – does the work appear to be a piece of research that could be undertaken and completed within the confines of the undergraduate program? It could be achievable on the basis that the work does not appear to require a long time period to complete and the samples should be accessible. Achievability is also a consideration of university ethical consideration processes. For example, although a researcher is able to identify a sample of participants who are experiencing fuel poverty, consideration must be given to the possible ethical issues that surround requesting their participation. It may be deemed that the research could in some manner cause harm to the participants, such as stress through talking about their lived experiences. This stress could also be felt by the researcher who may not be trained to deal with such emotional situations. In both of these examples, the university ethics process could decide this work is unachievable.

R ealistic – the issue of social media advertising is realistic within the stated industry. The samples identified also appear linked to the topic. Furthermore, the academic foundation of the work is also identified – advertising. The work also appears to be realistic in terms of the resources required to complete it. The ethical use of data gathered from social media could also be relevant to determining if this topic is realistic. As with Achievability above, issues such as how the data was originally gathered and how it will then be used by the researcher, would be scrutinized by the Ethics process. Again, the principle of ‘do no harm’ would be applied to determine if the work is realistic.

T imely – although the work does not offer a specific timeframe, the use of social media for advertising is evident within the car industry. Therefore, this could be said to be timely.

Furthermore, the terminology is important. If you choose words that are descriptive, they will convey work that is also descriptive. So, try to use words such as ‘describe,’ ‘understand,’ or ‘gain an insight into’ only where they adequately reflect that your research is not an in-depth study. Consider using terms to evidence how you will approach each objective including: evaluate; critique; critically discuss and examine. All infer the research will go beyond a surface inquiry.

Now, at this stage, consider if the research wished to study the possible relationships between variables such as the impact on consumers of exposure to social media advertising on car sales decision-making. As with the approach to the similar topic above, this could be explored using qualitative data by gathering the experiences of consumers and/or people within the car industry. However, research that specifically wishes to explore possible links between issues and/or specific variables, could sometimes be better framed using a research hypothesis. This is a statement that identifies possible c ause and effect ’ relationships between variables. Therefore, the focus of the above topic could be reconsidered to identify the impact of social media advertising within the car industry. The new research question and hypothesis could be thus:

The research question: Do consumers perceive the impact of social media advertising on their car purchasing decisions?

Null Hypothesis: There is no difference in car purchasing decisions between those consumers who are exposed to social media advertising of cars compared to those who are not.

Alternative Hypothesis: There is a relationship between whether or not a consumer has been exposed to social media advertising and their car purchasing decision.

In order to address the hypothesis, some form of statistical testing would be required which is not covered in this guide. However, as a researcher, you should always consider what it is specifically that you wish to research when framing your work. This topic consideration could identify specific issues and/or variables which you wish to explore further to test if there are statistical relationships. In this situation, you could consider including hypothesis testing within your research design. As can be viewed above, related topics may be presented in different ways, with the inclusion or exclusion of a research hypothesis. The existence of possible relationships may be explored through research that seeks perceptions of advertising. However, research which seeks statistical evidence would be best represented with hypothesis testing.

• Research aim and objectives convey to your audience the topic and possible boundaries of your work. Therefore, ensuring they are presented as SMART, allows others to assess your work in the way you intended.
• Research ethics should be considered when writing research aims and objectives, including the potential impact of participation on individuals. Research should do no harm to the individuals involved, including the sample and researchers themselves.
• Research does not always necessitate consideration of the research hypothesis. However, in some circumstances, a well-considered hypothesis could offer statistical weight to your findings.

Recognizing the Link Between Writing an Effective Research Aim and the Research Design

The research aim and objectives should be written in a way that conveys the specific area or problem to be researched. This should allow anyone reading your research aim to understand the main focus of the work. For example, your work may aim to examine the lived experiences of individuals living with fuel poverty within a specific geographic area or demographic. In this example, you can clearly identify the topic – lived experiences of fuel poverty – and the focus – individuals within the chosen geographical area/demographic . To a more experienced researcher, it can also offer insight into the research design which may be reasonably expected. So, studies of ‘lived experiences’ can involve the gathering and/or analysis of qualitative data from individuals/communities as the researcher seek to gather the first-hand experiences of participants (individuals).

Clearly written research aim and objectives should allow the reader to consider the following information:

• 1. Wider academic area(s) within which the topic falls (for example, accountancy; marketing; management);
• 2. The main areas of the literature identified within the aim and/or objectives;
• 3. The data which would be expected to be gathered to in order to meet/address the research objectives;
• 4. The data collection methods which could be deemed relevant to the research aim and,
• 5. Overall, if the research aim and objectives are SMART (see above).

Consider the wording in the example below:

The research aim of this dissertation is to examine the lived experiences of people living with fuel poverty and their attitudes towards support services within a local council area. This research aim will be addressed through the following research objectives:

• 1. Critically review previous literature and evaluate the origins and purpose of different definitions of ‘fuel poverty.’
• 2. Explore the attitudes of individuals currently experiencing fuel poverty towards support agencies and other stakeholders.
• 3. Analyze the opportunities and barriers to support agencies and related stakeholders within a local council area with specific regard to supporting those experiencing fuel poverty.
• 4. Compare and contrast the lived experiences of individuals experiencing fuel poverty with those of the support agencies to identify potential service gaps.

Looking closely at the work above, it could be reasonable to make the following assumptions about the research:

The academic area(s) within which the topic falls (for example, accountancy; marketing; management; social sciences; economics). This can be researched and explored by keyword searching the research aim. In this example, there appear to be multiple academic roots to the work:

• ‘lived experiences of people living with fuel poverty’ – this could be viewed as a social science/economics topic or even an engineering area. Either would depend on the specific view taken to investigate fuel poverty, i.e., real-world examples of lived experiences, specially such as narratives about their daily life. Alternatively, this aim could encapsulate studies within engineering areas that seek to understand the impact of construction and design decisions on the daily life of individuals.
• ‘…and their attitudes towards support services within a local council area’ – by adding a focus for the study as being specific to support services, this work is now narrowed to more reflect the social sciences area.
• If the work was indeed to study any issues such as building construction, this would be expected to appear within the research aim to convey the topic clearly and precisely.

Therefore, it could be expected that if the research draws on wider academic areas, this should be evident from the terminology within the research objectives. A consistent use of terminology ensures the academic foundation of the work is identifiable throughout. It could also be reasonably presumed that the relevant issues of each sample (individuals within fuel poverty, the support services, and stakeholders), would be refined to include specific factors to ensure the work is focused on specific issues.

Next, consider the type of data you would expect to gather to in order to meet/address the research objectives. The following options appear to be linked to the wording of the objectives:

• Secondary Data: The objectives identify the need for literature in the first stages of work in order to address objectives 1, 2, and 3. As the research is based on lived experiences, this could include not only academic work but also charity and government reports. Given that this is a real-world issue, examples could also be identified from reputable news agencies. All of these sources could help identify possible issues that may be identified by research participants during the data gathering. If these issues are not identified by the participants, they could be used to form a critical discussion around opportunities or barriers (objective 3).
• Primary Data: Given the focus on lived experiences related to support services, the research may be presumed to include a qualitative study. A qualitative study would allow participants to use their own voices and language to explain their lived experiences. Whereas a quantitative study, by its nature, could explore the issues already known to the researcher when the instrument was written, e.g., survey. Qualitative data could perhaps encourage more personal issues to be identified by the individual participants, and also offer some context for their position.

Subsequently, consider which data collection methods you would expect to be used to address the research aim.

• Quantitative data gathering tools: Could quantitative data gathering explore the lived experiences of this sample? Many areas could be effectively explored however lived experiences tend to be personal to the individual and so qualitative could offer more depth and richness to the data.
• As both the research aim and objectives identify specific samples, the research could be considered to have a boundary around those to be invited to participate. Therefore, secondary data may identify the definitions of fuel poverty and offer reasons for any differences. It could also allow the identification of the roles and remits of support services and stakeholders. However, it will not offer specific lived experience details that can come from the sample of individuals.

If specific organizational sectors or companies were identified, the use of quantitative data-gathering tools, such as a survey, may allow more specific information to be gathered. Remember, the research aim identifies that the focus is the individuals who experience fuel poverty. Therefore, a survey could address issues such as knowledge and understanding of these service providers. However, it could then miss hearing about the informal networks used by individuals for support, which could come to light during a qualitative study.

Finally, if in doubt, show your research aim and objectives to a colleague and ask them to tell you , what they think your research is about. This simple exercise will enable you to realize what other people understand from your work and so, allow you to tweak where necessary. This should ensure your research is not only accessible to different audiences but ultimately, is a fair reflection of your topic choice.

• Clearly written research aims and objectives can effectively convey information about your work. This allows a reader to consider the key aspects of your topic and sets expectations about the contents of your report/dissertation/thesis.
• Always ensure that the language used to write a research aim and objectives, adequately convey the meaning and depth of your research. It should be specific to your topic but also accessible to the intended audience(s).
• SMART research objectives can convey your understanding of research design. This should be apparent from the layering of issues and identification of relevant samples.

In conclusion, this guide has offered practical steps through example-based exercises to help you format your idea into an effective research aim and objectives. Having progressed through the exercises, you will have considered issues such as the importance of understanding how a research aim can help you refine your idea. It is also the mechanism to convey your research intention to your audience. Through exploring the importance of linking your research aim to the wider research area this will give you the confidence to develop SMART objectives. Following this, your work will reflect key areas of your research design through the use of relevant research methods terminology.

Therefore, by following the steps in this guide you should now be confident to take your idea and form it into robust research aims and objectives.

Multiple-Choice Quiz Questions

1. The purpose of a research aim is to ______.

Incorrect Answer

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Correct Answer

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2. It is important to understand the link between the research aim and the wider topic because ______.

Feedback: This is not the correct answer. The correct answer is B.

3. How many research objectives are necessary to ensure a successful final-year project?

4. Research objectives reflect ______

5. Research design can be reflected in the research aim and objectives by ______.

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21 Research Objectives Examples (Copy and Paste)

Research objectives refer to the definitive statements made by researchers at the beginning of a research project detailing exactly what a research project aims to achieve.

These objectives are explicit goals clearly and concisely projected by the researcher to present a clear intention or course of action for his or her qualitative or quantitative study. 

Research objectives are typically nested under one overarching research aim. The objectives are the steps you’ll need to take in order to achieve the aim (see the examples below, for example, which demonstrate an aim followed by 3 objectives, which is what I recommend to my research students).

Research Objectives vs Research Aims

Research aim and research objectives are fundamental constituents of any study, fitting together like two pieces of the same puzzle.

The ‘research aim’ describes the overarching goal or purpose of the study (Kumar, 2019). This is usually a broad, high-level purpose statement, summing up the central question that the research intends to answer.

Example of an Overarching Research Aim:

“The aim of this study is to explore the impact of climate change on crop productivity.” 

Comparatively, ‘research objectives’ are concrete goals that underpin the research aim, providing stepwise actions to achieve the aim.

Objectives break the primary aim into manageable, focused pieces, and are usually characterized as being more specific, measurable, achievable, relevant, and time-bound (SMART).

Examples of Specific Research Objectives:

1. “To examine the effects of rising temperatures on the yield of rice crops during the upcoming growth season.” 2. “To assess changes in rainfall patterns in major agricultural regions over the first decade of the twenty-first century (2000-2010).” 3. “To analyze the impact of changing weather patterns on crop diseases within the same timeframe.”

The distinction between these two terms, though subtle, is significant for successfully conducting a study. The research aim provides the study with direction, while the research objectives set the path to achieving this aim, thereby ensuring the study’s efficiency and effectiveness.

How to Write Research Objectives

I usually recommend to my students that they use the SMART framework to create their research objectives.

SMART is an acronym standing for Specific, Measurable, Achievable, Relevant, and Time-bound. It provides a clear method of defining solid research objectives and helps students know where to start in writing their objectives (Locke & Latham, 2013).

Each element of this acronym adds a distinct dimension to the framework, aiding in the creation of comprehensive, well-delineated objectives.

Here is each step:

• Specific : We need to avoid ambiguity in our objectives. They need to be clear and precise (Doran, 1981). For instance, rather than stating the objective as “to study the effects of social media,” a more focused detail would be “to examine the effects of social media use (Facebook, Instagram, and Twitter) on the academic performance of college students.”
• Measurable: The measurable attribute provides a clear criterion to determine if the objective has been met (Locke & Latham, 2013). A quantifiable element, such as a percentage or a number, adds a measurable quality. For example, “to increase response rate to the annual customer survey by 10%,” makes it easier to ascertain achievement.
• Achievable: The achievable aspect encourages researchers to craft realistic objectives, resembling a self-check mechanism to ensure the objectives align with the scope and resources at disposal (Doran, 1981). For example, “to interview 25 participants selected randomly from a population of 100” is an attainable objective as long as the researcher has access to these participants.
• Relevance : Relevance, the fourth element, compels the researcher to tailor the objectives in alignment with overarching goals of the study (Locke & Latham, 2013). This is extremely important – each objective must help you meet your overall one-sentence ‘aim’ in your study.
• Time-Bound: Lastly, the time-bound element fosters a sense of urgency and prioritization, preventing procrastination and enhancing productivity (Doran, 1981). “To analyze the effect of laptop use in lectures on student engagement over the course of two semesters this year” expresses a clear deadline, thus serving as a motivator for timely completion.

You’re not expected to fit every single element of the SMART framework in one objective, but across your objectives, try to touch on each of the five components.

Research Objectives Examples

1. Field: Psychology

Aim: To explore the impact of sleep deprivation on cognitive performance in college students.

• Objective 1: To compare cognitive test scores of students with less than six hours of sleep and those with 8 or more hours of sleep.
• Objective 2: To investigate the relationship between class grades and reported sleep duration.
• Objective 3: To survey student perceptions and experiences on how sleep deprivation affects their cognitive capabilities.

2. Field: Environmental Science

Aim: To understand the effects of urban green spaces on human well-being in a metropolitan city.

• Objective 1: To assess the physical and mental health benefits of regular exposure to urban green spaces.
• Objective 2: To evaluate the social impacts of urban green spaces on community interactions.
• Objective 3: To examine patterns of use for different types of urban green spaces. 

3. Field: Technology

Aim: To investigate the influence of using social media on productivity in the workplace.

• Objective 1: To measure the amount of time spent on social media during work hours.
• Objective 2: To evaluate the perceived impact of social media use on task completion and work efficiency.
• Objective 3: To explore whether company policies on social media usage correlate with different patterns of productivity.

4. Field: Education

Aim: To examine the effectiveness of online vs traditional face-to-face learning on student engagement and achievement.

• Objective 1: To compare student grades between the groups exposed to online and traditional face-to-face learning.
• Objective 2: To assess student engagement levels in both learning environments.
• Objective 3: To collate student perceptions and preferences regarding both learning methods.

5. Field: Health

Aim: To determine the impact of a Mediterranean diet on cardiac health among adults over 50.

• Objective 1: To assess changes in cardiovascular health metrics after following a Mediterranean diet for six months.
• Objective 2: To compare these health metrics with a similar group who follow their regular diet.
• Objective 3: To document participants’ experiences and adherence to the Mediterranean diet.

6. Field: Environmental Science

Aim: To analyze the impact of urban farming on community sustainability.

• Objective 1: To document the types and quantity of food produced through urban farming initiatives.
• Objective 2: To assess the effect of urban farming on local communities’ access to fresh produce.
• Objective 3: To examine the social dynamics and cooperative relationships in the creating and maintaining of urban farms.

7. Field: Sociology

Aim: To investigate the influence of home offices on work-life balance during remote work.

• Objective 1: To survey remote workers on their perceptions of work-life balance since setting up home offices.
• Objective 2: To conduct an observational study of daily work routines and family interactions in a home office setting.
• Objective 3: To assess the correlation, if any, between physical boundaries of workspaces and mental boundaries for work in the home setting.

8. Field: Economics

Aim: To evaluate the effects of minimum wage increases on small businesses.

• Objective 1: To analyze cost structures, pricing changes, and profitability of small businesses before and after minimum wage increases.
• Objective 2: To survey small business owners on the strategies they employ to navigate minimum wage increases.
• Objective 3: To examine employment trends in small businesses in response to wage increase legislation.

9. Field: Education

Aim: To explore the role of extracurricular activities in promoting soft skills among high school students.

• Objective 1: To assess the variety of soft skills developed through different types of extracurricular activities.
• Objective 2: To compare self-reported soft skills between students who participate in extracurricular activities and those who do not.
• Objective 3: To investigate the teachers’ perspectives on the contribution of extracurricular activities to students’ skill development.

10. Field: Technology

Aim: To assess the impact of virtual reality (VR) technology on the tourism industry.

• Objective 1: To document the types and popularity of VR experiences available in the tourism market.
• Objective 2: To survey tourists on their interest levels and satisfaction rates with VR tourism experiences.
• Objective 3: To determine whether VR tourism experiences correlate with increased interest in real-life travel to the simulated destinations.

11. Field: Biochemistry

Aim: To examine the role of antioxidants in preventing cellular damage.

• Objective 1: To identify the types and quantities of antioxidants in common fruits and vegetables.
• Objective 2: To determine the effects of various antioxidants on free radical neutralization in controlled lab tests.
• Objective 3: To investigate potential beneficial impacts of antioxidant-rich diets on long-term cellular health.

12. Field: Linguistics

Aim: To determine the influence of early exposure to multiple languages on cognitive development in children.

• Objective 1: To assess cognitive development milestones in monolingual and multilingual children.
• Objective 2: To document the number and intensity of language exposures for each group in the study.
• Objective 3: To investigate the specific cognitive advantages, if any, enjoyed by multilingual children.

13. Field: Art History

Aim: To explore the impact of the Renaissance period on modern-day art trends.

• Objective 1: To identify key characteristics and styles of Renaissance art.
• Objective 2: To analyze modern art pieces for the influence of the Renaissance style.
• Objective 3: To survey modern-day artists for their inspirations and the influence of historical art movements on their work.

14. Field: Cybersecurity

Aim: To assess the effectiveness of two-factor authentication (2FA) in preventing unauthorized system access.

• Objective 1: To measure the frequency of unauthorized access attempts before and after the introduction of 2FA.
• Objective 2: To survey users about their experiences and challenges with 2FA implementation.
• Objective 3: To evaluate the efficacy of different types of 2FA (SMS-based, authenticator apps, biometrics, etc.).

15. Field: Cultural Studies

Aim: To analyze the role of music in cultural identity formation among ethnic minorities.

• Objective 1: To document the types and frequency of traditional music practices within selected ethnic minority communities.
• Objective 2: To survey community members on the role of music in their personal and communal identity.
• Objective 3: To explore the resilience and transmission of traditional music practices in contemporary society.

16. Field: Astronomy

Aim: To explore the impact of solar activity on satellite communication.

• Objective 1: To categorize different types of solar activities and their frequencies of occurrence.
• Objective 2: To ascertain how variations in solar activity may influence satellite communication.
• Objective 3: To investigate preventative and damage-control measures currently in place during periods of high solar activity.

17. Field: Literature

Aim: To examine narrative techniques in contemporary graphic novels.

• Objective 1: To identify a range of narrative techniques employed in this genre.
• Objective 2: To analyze the ways in which these narrative techniques engage readers and affect story interpretation.
• Objective 3: To compare narrative techniques in graphic novels to those found in traditional printed novels.

18. Field: Renewable Energy

Aim: To investigate the feasibility of solar energy as a primary renewable resource within urban areas.

• Objective 1: To quantify the average sunlight hours across urban areas in different climatic zones. 
• Objective 2: To calculate the potential solar energy that could be harnessed within these areas.
• Objective 3: To identify barriers or challenges to widespread solar energy implementation in urban settings and potential solutions.

19. Field: Sports Science

Aim: To evaluate the role of pre-game rituals in athlete performance.

• Objective 1: To identify the variety and frequency of pre-game rituals among professional athletes in several sports.
• Objective 2: To measure the impact of pre-game rituals on individual athletes’ performance metrics.
• Objective 3: To examine the psychological mechanisms that might explain the effects (if any) of pre-game ritual on performance.

20. Field: Ecology

Aim: To investigate the effects of urban noise pollution on bird populations.

• Objective 1: To record and quantify urban noise levels in various bird habitats.
• Objective 2: To measure bird population densities in relation to noise levels.
• Objective 3: To determine any changes in bird behavior or vocalization linked to noise levels.

21. Field: Food Science

Aim: To examine the influence of cooking methods on the nutritional value of vegetables.

• Objective 1: To identify the nutrient content of various vegetables both raw and after different cooking processes.
• Objective 2: To compare the effect of various cooking methods on the nutrient retention of these vegetables.
• Objective 3: To propose cooking strategies that optimize nutrient retention.

The Importance of Research Objectives

The importance of research objectives cannot be overstated. In essence, these guideposts articulate what the researcher aims to discover, understand, or examine (Kothari, 2014).

When drafting research objectives, it’s essential to make them simple and comprehensible, specific to the point of being quantifiable where possible, achievable in a practical sense, relevant to the chosen research question, and time-constrained to ensure efficient progress (Kumar, 2019). 

Remember that a good research objective is integral to the success of your project, offering a clear path forward for setting out a research design , and serving as the bedrock of your study plan. Each objective must distinctly address a different dimension of your research question or problem (Kothari, 2014). Always bear in mind that the ultimate purpose of your research objectives is to succinctly encapsulate your aims in the clearest way possible, facilitating a coherent, comprehensive and rational approach to your planned study, and furnishing a scientific roadmap for your journey into the depths of knowledge and research (Kumar, 2019). 

Kothari, C.R (2014). Research Methodology: Methods and Techniques . New Delhi: New Age International.

Kumar, R. (2019). Research Methodology: A Step-by-Step Guide for Beginners .New York: SAGE Publications.

Doran, G. T. (1981). There’s a S.M.A.R.T. way to write management’s goals and objectives. Management review, 70 (11), 35-36.

Locke, E. A., & Latham, G. P. (2013). New Developments in Goal Setting and Task Performance . New York: Routledge.

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Frequently asked questions

What is a research objective.

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 .

Frequently asked questions: Dissertation

The acknowledgements are generally included at the very beginning of your thesis or dissertation, directly after the title page and before the abstract .

If you only used a few abbreviations in your thesis or dissertation, you don’t necessarily need to include a list of abbreviations .

If your abbreviations are numerous, or if you think they won’t be known to your audience, it’s never a bad idea to add one. They can also improve readability, minimising confusion about abbreviations unfamiliar to your reader.

A list of figures and tables compiles all of the figures and tables that you used in your thesis or dissertation and displays them with the page number where they can be found.

A thesis or dissertation outline is one of the most critical first steps in your writing process. It helps you to lay out and organise your ideas and can provide you with a roadmap for deciding what kind of research you’d like to undertake.

Generally, an outline contains information on the different sections included in your thesis or dissertation, such as:

• Your anticipated title
• Your abstract
• Your chapters (sometimes subdivided into further topics like literature review, research methods, avenues for future research, etc.)

An abstract for a thesis or dissertation is usually around 150–300 words. There’s often a strict word limit, so make sure to check your university’s requirements.

The abstract appears on its own page, after the title page and acknowledgements but before the table of contents .

While it may be tempting to present new arguments or evidence in your thesis or disseration conclusion , especially if you have a particularly striking argument you’d like to finish your analysis with, you shouldn’t. Theses and dissertations follow a more formal structure than this.

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

For a stronger dissertation conclusion , avoid including:

• Generic concluding phrases (e.g. “In conclusion…”)
• Weak statements that undermine your argument (e.g. “There are good points on both sides of this issue.”)

Your conclusion should leave the reader with a strong, decisive impression of your work.

The conclusion of your thesis or dissertation shouldn’t take up more than 5-7% of your overall word count.

The conclusion of your thesis or dissertation should include the following:

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

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

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

In a thesis or dissertation, the discussion is an in-depth exploration of the results, going into detail about the meaning of your findings and citing relevant sources to put them in context.

The conclusion is more shorter and more general: it concisely answers your main research question and makes recommendations based on your overall findings.

A dissertation prospectus or proposal describes what or who you plan to research for your dissertation. It delves into why, when, where, and how you will do your research, as well as helps you choose a type of research to pursue. You should also determine whether you plan to pursue qualitative or quantitative methods and what your research design will look like.

It should outline all of the decisions you have taken about your project, from your dissertation topic to your hypotheses and research objectives , ready to be approved by your supervisor or committee.

Note that some departments require a defense component, where you present your prospectus to your committee orally.

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

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A Guide to Writing Research Objectives and Aims

• Post author: Research Zone
• Post published: September 6, 2021
• Post category: Academic Advice / Writing Thesis
• Post comments: 1 Comment

Introduction

In determining the success of your research project, it is crucial to understand your research objectives and aim. However, it is, unfortunately, an aspect that many students struggle with, resulting in poor performance. As a result of their importance, if you suspect even the slightest possibility that you belong to this group of students, we strongly recommend that you read this article in its entirety.

In this article, we describe what research aim and objectives are, what distinguishes them from each other, and how to write them correctly.

What is Research Aim?

Research aims describe the purpose or main goal of a project. It help your reader to understand the focus of your research and gives them an idea of what you are trying to accomplish. Research aims are found in their own subsection under the introduction section of all research documents, regardless of whether they are dissertations or research papers.

In most situations, a research aim is expressed as a broad statement of the main goal of the study and doesn’t need to be more than one sentence. The exact format of the outline will vary depending on your preference, but it should all explain the purpose (context), your objective (the actual aim), and how you plan to accomplish it (highlights of your objectives).

What are Research Objectives?

Research aim defines what your study is going to answer, but research objectives outline how it’s going to answer it.

Those objectives break down each component of your research project into smaller portions, each representing a key section of it. Thus, almost all dissertations and theses are organized into numbered lists, with each item getting its own chapter.

Understanding the difference between research objectives and aim

The above explanation should make clear the difference between aim and objectives, but to clarify:

• Aim focus on what a project proposes to achieve; objectives focus on how the project will achieve its goal.
• The research objectives are more specific than the research aims.
• Objectives focus on the short-term and immediate outcomes of a project while aim focus on its long-term outcomes.
• It would be best to write an objective as a numbered list; research aim can be written in one sentence or short paragraph.

How to Write the Aim and Objectives of Research?

It is important to note that there is no definitive way to write clear objectives and aim for research. Researchers typically formulate their goals and objectives in so many different ways, and your supervisor may often influence the formulation based on their preferences.

Nevertheless, there are a few basic principles you should observe to ensure good practice; these principles are listed below.

Research Aim

The aim should include three parts, which address the following questions:

• Why is this research necessary?
• What is the purpose of this study?
• How will you accomplish it?

It is easier to accomplish writing your research aim by addressing each question using its own sentence, although you can combine sentences for each or write multiple sentences for each question, the important thing is to address each one individually.

The first question, why, provides context to your research project, the second question, what describes the aim of your research, and the last question, how, acts as an introduction to your objectives which will immediately follow.

Research Objectives

Each of your research objectives should have the following:

• Specific: Is the action you intend to take unclear, or is it focused and well-defined?
• Assessable: What method will you use to measure your progress and determine when you have accomplished your goal?
• Obtainable: Do you have the necessary support, resources, and facilities to execute the project?
• Relevant: Does the action you propose support the achievement of your research aim?
• Timebound: Are you realistically able to complete the action within the given timeframe?

Properly formulating the aims and objectives of your thesis, dissertation, or research paper is an integral part of its success. Your goals and objectives will determine what your research ultimately looks like in terms of scope, depth, and direction. You will gain clarity in your research and in the minds of your readers if you establish clear aims and objectives, with your aim stating what you wish to achieve, and your objectives indicating how you will do that. Moreover, you will have a clearer direction for your research if you take the time to establish your research objective and aim. This will lead to fewer future issues, but also to a more thorough and cohesive research project.

Feel free to reach out to us here if you need assistance in writing your objectives or aim for your thesis or research. You can also click here to learn more about the services provided.

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How to Write Objectives in a Research Proposal

Last Updated: May 19, 2023 Fact Checked

This article was co-authored by Felipe Corredor . Felipe is a Senior College Admissions Consultant at American College Counselors with over seven years of experience. He specializes in helping clients from all around the world gain admission into America's top universities through private, one-on-one consulting. He helps guide clients through the entire college admissions process and perfect every aspect of their college applications. Felipe earned a Bachelor's Degree from the University of Chicago and recently received his MBA. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 124,085 times.

A research proposal is a detailed outline for a significant research project. They’re common for class assignments, capstone papers, grant applications, and even job applications in some fields, so it's possible you'll have to prepare one at some point. The objectives are a very important part of a research proposal because they outline where the project is headed and what it will accomplish. Developing objectives can be a little tricky, so take some time to consider them. Then work on wording them carefully so your readers understand your goals. With clear objectives, your research proposal will be much stronger.

Brainstorming Your Objectives

• For example, your research question might be “What is the effect of prolonged TV-watching on children?” You can then use that question to build your study around.
• Narrow down your research topic if it’s too broad. A broad research topic makes breaking the objectives down much more difficult. A research question like “How can we save the environment?” is a huge question. Something like “What safety measures would prevent ocean pollution?” is more specific and attainable. [2] X Research source

• Remember that in most cases, you shouldn’t state that your study will prove or disprove something exactly since you haven’t done the work yet. Don’t say “This study proves that honey is not an effective treatment for acne.” Instead, make it something like “This study will demonstrate whether or not honey is an effective treatment for acne.”

• If your research question was “What is the effect of prolonged TV-watching on children?” then there are a few categories you could look at. Objectives wrapped up within that question might be: 1) the incidence of eyestrain among children who watch a lot of TV, 2) their muscular development, 3) their level of socialization with other children. Design your objectives around answering these questions.

• You could always state in your research proposal that you plan to design future experiments or studies to answer additional questions. Most experiments leave unanswered questions and subsequent studies try to tackle them.

• A general objective might be "Establish the effect of diet on mental health." Some specific goals in that project could be 1) Determine if processed foods make depression worse, 2) Identify foods that improve mood, 3) Measure if portion sizes have an impact on mood.
• Not all research proposals want you to divide between general and specific goals. Remember to follow the instructions for the proposal you're writing.

• The best goals align with each letter in the SMART acronym. The weaker ones are missing some letters. For example, you might come up with a topic that’s specific, measurable, and time-bound, but not realistic or attainable. This is a weak objective because you probably can’t achieve it.
• Think about the resources at your disposal. Some objectives might be doable with the right equipment, but if you don’t have that equipment, then you can’t achieve that goal. For example, you might want to map DNA structures, but you can’t view DNA without an electron microscope.
• Ask the same question for your entire project. Is it attainable overall? You don’t want to try to achieve too much and overwhelm yourself.
• The specific words in this acronym sometimes change, but the sentiment is the same. Your objectives should overall be clear and specific, measurable, feasible, and limited by time.

Using the Right Language

• Verbs like use, understand, or study is vague and weak. Instead, choose words like calculate, compare, and assess.
• Your objective list might read like this: 1) Compare the muscle development of children who play video games to children who don’t, 2) Assess whether or not video games cause eyestrain, 3) Determine if videogames inhibit a child’s socialization skills.
• Some proposals use the infinitive form of verbs, like “to measure” or “to determine.” This is also fine but refer to the proposal instructions to see if this is correct.

• You can further explain your objectives further in the research proposal. No need to elaborate a lot when you’re just listing them.
• If you’re having trouble shortening an objective to 1 sentence, then you probably need to split it into 2 objectives. It might also be too complicated for this project.

• For example, “Determine if sunlight is harmful” is too vague. Instead, state the objective as “Determine if prolonged sun exposure increases subjects’ risk of skin cancer.”
• It’s helpful to let someone else read your proposal and see if they understand the objectives. If they’re confused, then you need to be more specific.

• For example, don’t say “Measure the effect of radiation on living tissue.” Instead, say “Determine what level of radiation is dangerous to living tissue.”
• Remember, don’t state the objectives as you’ve already done the experiments. They’re still not answered.

Writing the Objectives

• This is a common format for research proposals, but not universal. Always follow the format that the instructions provided.
• Depending on how long your introduction has to be, you might also list the objectives there. This depends on whether or not you have room.

• At the very least, the abstract should list the general objective. This tells the readers what your study is working towards.

• In some research projects, the general objective is called a long-term goal instead. Adjust your language to the proposal requirements.
• Some proposals directions may just want the specific objectives rather than a division between the general and specific ones. Don’t divide them if the instructions tell you not to.

• Your introduction may be as follows: "My long-term objective with this project is determining whether or not prolonged video-game playing is harmful to children under 5. I will accomplish this aim by meeting the following objectives: 1) Compare the muscle development of children who play videogames to children who don’t 2) Assess whether or not videogames cause eyestrain 3) Determine if videogames inhibit a child’s socialization skills"
• The specific objectives are usually listed as a bullet or numbered points. However, follow the instructions given.

Research Proposal Templates

Expert Q&A

• It’s always a good idea to let someone else read your research proposals and make sure they’re clear. Thanks Helpful 0 Not Helpful 0
• Proofread! A great proposal could be ruined by typos and errors. Thanks Helpful 2 Not Helpful 0

• Some proposal instructions are very specific, and applicants that don’t follow the format are eliminated. Always follow the instructions given to stay within the requirements. Thanks Helpful 3 Not Helpful 0

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• ↑ https://uk.sagepub.com/sites/default/files/upm-assets/15490_book_item_15490.pdf
• ↑ https://research-methodology.net/research-methodology/research-aims-and-objectives/
• ↑ https://www.uh.edu/~lsong5/documents/A%20sample%20proposal%20with%20comment.pdf
• ↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3282423/
• ↑ https://www.cdc.gov/healthyyouth/evaluation/pdf/brief3b.pdf
• ↑ https://www.open.edu/openlearncreate/mod/oucontent/view.php?id=231&section=8.6.2
• ↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6398294/
• ↑ https://arxiv.org/pdf/physics/0601009.pdf
• ↑ https://www.bpcc.edu/institutional-advancement-grants/how-to-write-goals-and-objectives-for-grant-proposals
• ↑ https://guides.library.illinois.edu/c.php?g=504643&p=3454882

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• How to Write a Research Proposal | Examples & Templates

How to Write a Research Proposal | Examples & Templates

Published on October 12, 2022 by Shona McCombes and Tegan George. Revised on November 21, 2023.

A research proposal describes what you will investigate, why it’s important, and how you will conduct your research.

The format of a research proposal varies between fields, but most proposals will contain at least these elements:

Introduction

Literature review.

• Research design

Reference list

While the sections may vary, the overall objective is always the same. A research proposal serves as a blueprint and guide for your research plan, helping you get organized and feel confident in the path forward you choose to take.

Table of contents

Research proposal purpose, research proposal examples, research design and methods, contribution to knowledge, research schedule, other interesting articles, frequently asked questions about research proposals.

Academics often have to write research proposals to get funding for their projects. As a student, you might have to write a research proposal as part of a grad school application , or prior to starting your thesis or dissertation .

In addition to helping you figure out what your research can look like, a proposal can also serve to demonstrate why your project is worth pursuing to a funder, educational institution, or supervisor.

Research proposal length

The length of a research proposal can vary quite a bit. A bachelor’s or master’s thesis proposal can be just a few pages, while proposals for PhD dissertations or research funding are usually much longer and more detailed. Your supervisor can help you determine the best length for your work.

One trick to get started is to think of your proposal’s structure as a shorter version of your thesis or dissertation , only without the results , conclusion and discussion sections.

Download our research proposal template

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Writing a research proposal can be quite challenging, but a good starting point could be to look at some examples. We’ve included a few for you below.

• Example research proposal #1: “A Conceptual Framework for Scheduling Constraint Management”
• Example research proposal #2: “Medical Students as Mediators of Change in Tobacco Use”

Like your dissertation or thesis, the proposal will usually have a title page that includes:

• The proposed title of your project
• Your supervisor’s name
• Your institution and department

The first part of your proposal is the initial pitch for your project. Make sure it succinctly explains what you want to do and why.

Your introduction should:

• Introduce your topic
• Give necessary background and context
• Outline your  problem statement  and research questions

To guide your introduction , include information about:

• Who could have an interest in the topic (e.g., scientists, policymakers)
• How much is already known about the topic
• What is missing from this current knowledge
• What new insights your research will contribute
• Why you believe this research is worth doing

As you get started, it’s important to demonstrate that you’re familiar with the most important research on your topic. A strong literature review  shows your reader that your project has a solid foundation in existing knowledge or theory. It also shows that you’re not simply repeating what other people have already done or said, but rather using existing research as a jumping-off point for your own.

In this section, share exactly how your project will contribute to ongoing conversations in the field by:

• Comparing and contrasting the main theories, methods, and debates
• Examining the strengths and weaknesses of different approaches
• Explaining how will you build on, challenge, or synthesize prior scholarship

Following the literature review, restate your main  objectives . This brings the focus back to your own project. Next, your research design or methodology section will describe your overall approach, and the practical steps you will take to answer your research questions.

To finish your proposal on a strong note, explore the potential implications of your research for your field. Emphasize again what you aim to contribute and why it matters.

For example, your results might have implications for:

• Improving best practices
• Informing policymaking decisions
• Strengthening a theory or model
• Challenging popular or scientific beliefs
• Creating a basis for future research

Last but not least, your research proposal must include correct citations for every source you have used, compiled in a reference list . To create citations quickly and easily, you can use our free APA citation generator .

Some institutions or funders require a detailed timeline of the project, asking you to forecast what you will do at each stage and how long it may take. While not always required, be sure to check the requirements of your project.

Here’s an example schedule to help you get started. You can also download a template at the button below.

Download our research schedule template

If you are applying for research funding, chances are you will have to include a detailed budget. This shows your estimates of how much each part of your project will cost.

Make sure to check what type of costs the funding body will agree to cover. For each item, include:

• Cost : exactly how much money do you need?
• Justification : why is this cost necessary to complete the research?
• Source : how did you calculate the amount?

To determine your budget, think about:

• Travel costs : do you need to go somewhere to collect your data? How will you get there, and how much time will you need? What will you do there (e.g., interviews, archival research)?
• Materials : do you need access to any tools or technologies?
• Help : do you need to hire any research assistants for the project? What will they do, and how much will you pay them?

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

Methodology

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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 …

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

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

A PhD, which is short for philosophiae doctor (doctor of philosophy in Latin), is the highest university degree that can be obtained. In a PhD, students spend 3–5 years writing a dissertation , which aims to make a significant, original contribution to current knowledge.

A PhD is intended to prepare students for a career as a researcher, whether that be in academia, the public sector, or the private sector.

A master’s is a 1- or 2-year graduate degree that can prepare you for a variety of careers.

All master’s involve graduate-level coursework. Some are research-intensive and intend to prepare students for further study in a PhD; these usually require their students to write a master’s thesis . Others focus on professional training for a specific career.

Critical thinking refers to the ability to evaluate information and to be aware of biases or assumptions, including your own.

Like information literacy , it involves evaluating arguments, identifying and solving problems in an objective and systematic way, and clearly communicating your ideas.

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

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The Importance Of Research Objectives

Imagine you’re a student planning a vacation in a foreign country. You’re on a tight budget and need to draw…

Imagine you’re a student planning a vacation in a foreign country. You’re on a tight budget and need to draw up a pocket-friendly plan. Where do you begin? The first step is to do your research.

Before that, you make a mental list of your objectives—finding reasonably-priced hotels, traveling safely and finding ways of communicating with someone back home. These objectives help you focus sharply during your research and be aware of the finer details of your trip.

More often than not, research is a part of our daily lives. Whether it’s to pick a restaurant for your next birthday dinner or to prepare a presentation at work, good research is the foundation of effective learning. Read on to understand the meaning, importance and examples of research objectives.

Why Do We Need Research?

What are the objectives of research, what goes into a research plan.

Research is a careful and detailed study of a particular problem or concern, using scientific methods. An in-depth analysis of information creates space for generating new questions, concepts and understandings. The main objective of research is to explore the unknown and unlock new possibilities. It’s an essential component of success.

Over the years, businesses have started emphasizing the need for research. You’ve probably noticed organizations hiring research managers and analysts. The primary purpose of business research is to determine the goals and opportunities of an organization. It’s critical in making business decisions and appropriately allocating available resources.

Here are a few benefits of research that’ll explain why it is a vital aspect of our professional lives:

Expands Your Knowledge Base

One of the greatest benefits of research is to learn and gain a deeper understanding. The deeper you dig into a topic, the more well-versed you are. Furthermore, research has the power to help you build on any personal experience you have on the subject.

Keeps You Up To Date

Research encourages you to discover the most recent information available. Updated information prevents you from falling behind and helps you present accurate information. You’re better equipped to develop ideas or talk about a topic when you’re armed with the latest inputs.

Builds Your Credibility

Research provides you with a good foundation upon which you can develop your thoughts and ideas. People take you more seriously when your suggestions are backed by research. You can speak with greater confidence because you know that the information is accurate.

Sparks Connections

Take any leading nonprofit organization, you’ll see how they have a strong research arm supported by real-life stories. Research also becomes the base upon which real-life connections and impact can be made. It even helps you communicate better with others and conveys why you’re pursuing something.

Encourages Curiosity

As we’ve already established, research is mostly about using existing information to create new ideas and opinions. In the process, it sparks curiosity as you’re encouraged to explore and gain deeper insights into a subject. Curiosity leads to higher levels of positivity and lower levels of anxiety.

Well-defined objectives of research are an essential component of successful research engagement. If you want to drive all aspects of your research methodology such as data collection, design, analysis and recommendation, you need to lay down the objectives of research methodology. In other words, the objectives of research should address the underlying purpose of investigation and analysis. It should outline the steps you’d take to achieve desirable outcomes. Research objectives help you stay focused and adjust your expectations as you progress.

The objectives of research should be closely related to the problem statement, giving way to specific and achievable goals. Here are the four types of research objectives for you to explore:

General Objective

Also known as secondary objectives, general objectives provide a detailed view of the aim of a study. In other words, you get a general overview of what you want to achieve by the end of your study. For example, if you want to study an organization’s contribution to environmental sustainability, your general objective could be: a study of sustainable practices and the use of renewable energy by the organization.

Specific Objectives

Specific objectives define the primary aim of the study. Typically, general objectives provide the foundation for identifying specific objectives. In other words, when general objectives are broken down into smaller and logically connected objectives, they’re known as specific objectives. They help define the who, what, why, when and how aspects of your project. Once you identify the main objective of research, it’s easier to develop and pursue a plan of action.

Let’s take the example of ‘a study of an organization’s contribution to environmental sustainability’ again. The specific objectives will look like this:

To determine through history how the organization has changed its practices and adopted new solutions

To assess how the new practices, technology and strategies will contribute to the overall effectiveness

Once you’ve identified the objectives of research, it’s time to organize your thoughts and streamline your research goals. Here are a few effective tips to develop a powerful research plan and improve your business performance.

Set SMART Goals

Your research objectives should be SMART—Specific, Measurable, Achievable, Realistic and Time-constrained. When you focus on utilizing available resources and setting realistic timeframes and milestones, it’s easier to prioritize objectives. Continuously track your progress and check whether you need to revise your expectations or targets. This way, you’re in greater control over the process.

Create A Plan

Create a plan that’ll help you select appropriate methods to collect accurate information. A well-structured plan allows you to use logical and creative approaches towards problem-solving. The complexity of information and your skills are bound to influence your plan, which is why you need to make room for flexibility. The availability of resources will also play a big role in influencing your decisions.

Collect And Collate

After you’ve created a plan for the research process, make a list of the data you’re going to collect and the methods you’ll use. Not only will it help make sense of your insights but also keep track of your approach. The information you collect should be:

Logical, rigorous and objective

Can be reproduced by other people working on the same subject

Free of errors and highlighting necessary details

Current and updated

Includes everything required to support your argument/suggestions

Analyze And Keep Ready

Data analysis is the most crucial part of the process and there are many ways in which the information can be utilized. Four types of data analysis are often seen in a professional environment. While they may be divided into separate categories, they’re linked to each other.

Descriptive Analysis:

The most commonly used data analysis, descriptive analysis simply summarizes past data. For example, Key Performance Indicators (KPIs) use descriptive analysis. It establishes certain benchmarks after studying how someone has been performing in the past.

Diagnostic Analysis:

The next step is to identify why something happened. Diagnostic analysis uses the information gathered through descriptive analysis and helps find the underlying causes of an outcome. For example, if a marketing initiative was successful, you deep-dive into the strategies that worked.

Predictive Analysis:

It attempts to answer ‘what’s likely to happen’. Predictive analysis makes use of past data to predict future outcomes. However, the accuracy of predictions depends on the quality of the data provided. Risk assessment is an ideal example of using predictive analysis.

Prescriptive Analysis: 

The most sought-after type of data analysis, prescriptive analysis combines the insights of all of the previous analyses. It’s a huge organizational commitment as it requires plenty of effort and resources. A great example of prescriptive analysis is Artificial Intelligence (AI), which consumes large amounts of data. You need to be prepared to commit to this type of analysis.

Review And Interpret

Once you’ve collected and collated your data, it’s time to review it and draw accurate conclusions. Here are a few ways to improve the review process:

Identify the fundamental issues, opportunities and problems and make note of recurring trends if any

Make a list of your insights and check which is the most or the least common. In short, keep track of the frequency of each insight

Conduct a SWOT analysis and identify the strengths, weaknesses, opportunities and threats

Write down your conclusions and recommendations of the research

When we think about research, we often associate it with academicians and students. but the truth is research is for everybody who is willing to learn and enhance their knowledge. If you want to master the art of strategically upgrading your knowledge, Harappa Education’s Learning Expertly course has all the answers. Not only will it help you look at things from a fresh perspective but also show you how to acquire new information with greater efficiency. The Growth Mindset framework will teach you how to believe in your abilities to grow and improve. The Learning Transfer framework will help you apply your learnings from one context to another. Begin the journey of tactful learning and self-improvement today!

Explore Harappa Diaries to learn more about topics related to the THINK Habit such as  Learning From Experience ,  Critical Thinking  & What is  Brainstorming  to think clearly and rationally.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

11.1 The Purpose of Research Writing

Learning objectives.

• Identify reasons to research writing projects.
• Outline the steps of the research writing process.

Why was the Great Wall of China built? What have scientists learned about the possibility of life on Mars? What roles did women play in the American Revolution? How does the human brain create, store, and retrieve memories? Who invented the game of football, and how has it changed over the years?

You may know the answers to these questions off the top of your head. If you are like most people, however, you find answers to tough questions like these by searching the Internet, visiting the library, or asking others for information. To put it simply, you perform research.

Whether you are a scientist, an artist, a paralegal, or a parent, you probably perform research in your everyday life. When your boss, your instructor, or a family member asks you a question that you do not know the answer to, you locate relevant information, analyze your findings, and share your results. Locating, analyzing, and sharing information are key steps in the research process, and in this chapter, you will learn more about each step. By developing your research writing skills, you will prepare yourself to answer any question no matter how challenging.

Reasons for Research

When you perform research, you are essentially trying to solve a mystery—you want to know how something works or why something happened. In other words, you want to answer a question that you (and other people) have about the world. This is one of the most basic reasons for performing research.

But the research process does not end when you have solved your mystery. Imagine what would happen if a detective collected enough evidence to solve a criminal case, but she never shared her solution with the authorities. Presenting what you have learned from research can be just as important as performing the research. Research results can be presented in a variety of ways, but one of the most popular—and effective—presentation forms is the research paper . A research paper presents an original thesis, or purpose statement, about a topic and develops that thesis with information gathered from a variety of sources.

If you are curious about the possibility of life on Mars, for example, you might choose to research the topic. What will you do, though, when your research is complete? You will need a way to put your thoughts together in a logical, coherent manner. You may want to use the facts you have learned to create a narrative or to support an argument. And you may want to show the results of your research to your friends, your teachers, or even the editors of magazines and journals. Writing a research paper is an ideal way to organize thoughts, craft narratives or make arguments based on research, and share your newfound knowledge with the world.

Write a paragraph about a time when you used research in your everyday life. Did you look for the cheapest way to travel from Houston to Denver? Did you search for a way to remove gum from the bottom of your shoe? In your paragraph, explain what you wanted to research, how you performed the research, and what you learned as a result.

Research Writing and the Academic Paper

No matter what field of study you are interested in, you will most likely be asked to write a research paper during your academic career. For example, a student in an art history course might write a research paper about an artist’s work. Similarly, a student in a psychology course might write a research paper about current findings in childhood development.

Having to write a research paper may feel intimidating at first. After all, researching and writing a long paper requires a lot of time, effort, and organization. However, writing a research paper can also be a great opportunity to explore a topic that is particularly interesting to you. The research process allows you to gain expertise on a topic of your choice, and the writing process helps you remember what you have learned and understand it on a deeper level.

Research Writing at Work

Knowing how to write a good research paper is a valuable skill that will serve you well throughout your career. Whether you are developing a new product, studying the best way to perform a procedure, or learning about challenges and opportunities in your field of employment, you will use research techniques to guide your exploration. You may even need to create a written report of your findings. And because effective communication is essential to any company, employers seek to hire people who can write clearly and professionally.

Writing at Work

Take a few minutes to think about each of the following careers. How might each of these professionals use researching and research writing skills on the job?

• Medical laboratory technician
• Small business owner
• Information technology professional
• Freelance magazine writer

A medical laboratory technician or information technology professional might do research to learn about the latest technological developments in either of these fields. A small business owner might conduct research to learn about the latest trends in his or her industry. A freelance magazine writer may need to research a given topic to write an informed, up-to-date article.

Think about the job of your dreams. How might you use research writing skills to perform that job? Create a list of ways in which strong researching, organizing, writing, and critical thinking skills could help you succeed at your dream job. How might these skills help you obtain that job?

Steps of the Research Writing Process

How does a research paper grow from a folder of brainstormed notes to a polished final draft? No two projects are identical, but most projects follow a series of six basic steps.

These are the steps in the research writing process:

• Choose a topic.
• Plan and schedule time to research and write.
• Conduct research.
• Organize research and ideas.
• Draft your paper.
• Revise and edit your paper.

Each of these steps will be discussed in more detail later in this chapter. For now, though, we will take a brief look at what each step involves.

Step 1: Choosing a Topic

As you may recall from Chapter 8 “The Writing Process: How Do I Begin?” , to narrow the focus of your topic, you may try freewriting exercises, such as brainstorming. You may also need to ask a specific research question —a broad, open-ended question that will guide your research—as well as propose a possible answer, or a working thesis . You may use your research question and your working thesis to create a research proposal . In a research proposal, you present your main research question, any related subquestions you plan to explore, and your working thesis.

Step 2: Planning and Scheduling

Before you start researching your topic, take time to plan your researching and writing schedule. Research projects can take days, weeks, or even months to complete. Creating a schedule is a good way to ensure that you do not end up being overwhelmed by all the work you have to do as the deadline approaches.

During this step of the process, it is also a good idea to plan the resources and organizational tools you will use to keep yourself on track throughout the project. Flowcharts, calendars, and checklists can all help you stick to your schedule. See Chapter 11 “Writing from Research: What Will I Learn?” , Section 11.2 “Steps in Developing a Research Proposal” for an example of a research schedule.

Step 3: Conducting Research

When going about your research, you will likely use a variety of sources—anything from books and periodicals to video presentations and in-person interviews.

Your sources will include both primary sources and secondary sources . Primary sources provide firsthand information or raw data. For example, surveys, in-person interviews, and historical documents are primary sources. Secondary sources, such as biographies, literary reviews, or magazine articles, include some analysis or interpretation of the information presented. As you conduct research, you will take detailed, careful notes about your discoveries. You will also evaluate the reliability of each source you find.

Step 4: Organizing Research and the Writer’s Ideas

When your research is complete, you will organize your findings and decide which sources to cite in your paper. You will also have an opportunity to evaluate the evidence you have collected and determine whether it supports your thesis, or the focus of your paper. You may decide to adjust your thesis or conduct additional research to ensure that your thesis is well supported.

Remember, your working thesis is not set in stone. You can and should change your working thesis throughout the research writing process if the evidence you find does not support your original thesis. Never try to force evidence to fit your argument. For example, your working thesis is “Mars cannot support life-forms.” Yet, a week into researching your topic, you find an article in the New York Times detailing new findings of bacteria under the Martian surface. Instead of trying to argue that bacteria are not life forms, you might instead alter your thesis to “Mars cannot support complex life-forms.”

Step 5: Drafting Your Paper

Now you are ready to combine your research findings with your critical analysis of the results in a rough draft. You will incorporate source materials into your paper and discuss each source thoughtfully in relation to your thesis or purpose statement.

When you cite your reference sources, it is important to pay close attention to standard conventions for citing sources in order to avoid plagiarism , or the practice of using someone else’s words without acknowledging the source. Later in this chapter, you will learn how to incorporate sources in your paper and avoid some of the most common pitfalls of attributing information.

Step 6: Revising and Editing Your Paper

In the final step of the research writing process, you will revise and polish your paper. You might reorganize your paper’s structure or revise for unity and cohesion, ensuring that each element in your paper flows into the next logically and naturally. You will also make sure that your paper uses an appropriate and consistent tone.

Once you feel confident in the strength of your writing, you will edit your paper for proper spelling, grammar, punctuation, mechanics, and formatting. When you complete this final step, you will have transformed a simple idea or question into a thoroughly researched and well-written paper you can be proud of!

Review the steps of the research writing process. Then answer the questions on your own sheet of paper.

• In which steps of the research writing process are you allowed to change your thesis?
• In step 2, which types of information should you include in your project schedule?
• What might happen if you eliminated step 4 from the research writing process?

Key Takeaways

• People undertake research projects throughout their academic and professional careers in order to answer specific questions, share their findings with others, increase their understanding of challenging topics, and strengthen their researching, writing, and analytical skills.
• The research writing process generally comprises six steps: choosing a topic, scheduling and planning time for research and writing, conducting research, organizing research and ideas, drafting a paper, and revising and editing the paper.

Writing for Success Copyright © 2015 by University of Minnesota is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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• v.53(4); 2010 Aug

Research questions, hypotheses and objectives

Patricia farrugia.

* Michael G. DeGroote School of Medicine, the

Bradley A. Petrisor

† Division of Orthopaedic Surgery and the

Forough Farrokhyar

‡ Departments of Surgery and

§ Clinical Epidemiology and Biostatistics, McMaster University, Hamilton, Ont

Mohit Bhandari

There is an increasing familiarity with the principles of evidence-based medicine in the surgical community. As surgeons become more aware of the hierarchy of evidence, grades of recommendations and the principles of critical appraisal, they develop an increasing familiarity with research design. Surgeons and clinicians are looking more and more to the literature and clinical trials to guide their practice; as such, it is becoming a responsibility of the clinical research community to attempt to answer questions that are not only well thought out but also clinically relevant. The development of the research question, including a supportive hypothesis and objectives, is a necessary key step in producing clinically relevant results to be used in evidence-based practice. A well-defined and specific research question is more likely to help guide us in making decisions about study design and population and subsequently what data will be collected and analyzed. 1

Objectives of this article

In this article, we discuss important considerations in the development of a research question and hypothesis and in defining objectives for research. By the end of this article, the reader will be able to appreciate the significance of constructing a good research question and developing hypotheses and research objectives for the successful design of a research study. The following article is divided into 3 sections: research question, research hypothesis and research objectives.

Research question

Interest in a particular topic usually begins the research process, but it is the familiarity with the subject that helps define an appropriate research question for a study. 1 Questions then arise out of a perceived knowledge deficit within a subject area or field of study. 2 Indeed, Haynes suggests that it is important to know “where the boundary between current knowledge and ignorance lies.” 1 The challenge in developing an appropriate research question is in determining which clinical uncertainties could or should be studied and also rationalizing the need for their investigation.

Increasing one’s knowledge about the subject of interest can be accomplished in many ways. Appropriate methods include systematically searching the literature, in-depth interviews and focus groups with patients (and proxies) and interviews with experts in the field. In addition, awareness of current trends and technological advances can assist with the development of research questions. 2 It is imperative to understand what has been studied about a topic to date in order to further the knowledge that has been previously gathered on a topic. Indeed, some granting institutions (e.g., Canadian Institute for Health Research) encourage applicants to conduct a systematic review of the available evidence if a recent review does not already exist and preferably a pilot or feasibility study before applying for a grant for a full trial.

In-depth knowledge about a subject may generate a number of questions. It then becomes necessary to ask whether these questions can be answered through one study or if more than one study needed. 1 Additional research questions can be developed, but several basic principles should be taken into consideration. 1 All questions, primary and secondary, should be developed at the beginning and planning stages of a study. Any additional questions should never compromise the primary question because it is the primary research question that forms the basis of the hypothesis and study objectives. It must be kept in mind that within the scope of one study, the presence of a number of research questions will affect and potentially increase the complexity of both the study design and subsequent statistical analyses, not to mention the actual feasibility of answering every question. 1 A sensible strategy is to establish a single primary research question around which to focus the study plan. 3 In a study, the primary research question should be clearly stated at the end of the introduction of the grant proposal, and it usually specifies the population to be studied, the intervention to be implemented and other circumstantial factors. 4

Hulley and colleagues 2 have suggested the use of the FINER criteria in the development of a good research question ( Box 1 ). The FINER criteria highlight useful points that may increase the chances of developing a successful research project. A good research question should specify the population of interest, be of interest to the scientific community and potentially to the public, have clinical relevance and further current knowledge in the field (and of course be compliant with the standards of ethical boards and national research standards).

FINER criteria for a good research question

Adapted with permission from Wolters Kluwer Health. 2

Whereas the FINER criteria outline the important aspects of the question in general, a useful format to use in the development of a specific research question is the PICO format — consider the population (P) of interest, the intervention (I) being studied, the comparison (C) group (or to what is the intervention being compared) and the outcome of interest (O). 3 , 5 , 6 Often timing (T) is added to PICO ( Box 2 ) — that is, “Over what time frame will the study take place?” 1 The PICOT approach helps generate a question that aids in constructing the framework of the study and subsequently in protocol development by alluding to the inclusion and exclusion criteria and identifying the groups of patients to be included. Knowing the specific population of interest, intervention (and comparator) and outcome of interest may also help the researcher identify an appropriate outcome measurement tool. 7 The more defined the population of interest, and thus the more stringent the inclusion and exclusion criteria, the greater the effect on the interpretation and subsequent applicability and generalizability of the research findings. 1 , 2 A restricted study population (and exclusion criteria) may limit bias and increase the internal validity of the study; however, this approach will limit external validity of the study and, thus, the generalizability of the findings to the practical clinical setting. Conversely, a broadly defined study population and inclusion criteria may be representative of practical clinical practice but may increase bias and reduce the internal validity of the study.

PICOT criteria 1

A poorly devised research question may affect the choice of study design, potentially lead to futile situations and, thus, hamper the chance of determining anything of clinical significance, which will then affect the potential for publication. Without devoting appropriate resources to developing the research question, the quality of the study and subsequent results may be compromised. During the initial stages of any research study, it is therefore imperative to formulate a research question that is both clinically relevant and answerable.

Research hypothesis

The primary research question should be driven by the hypothesis rather than the data. 1 , 2 That is, the research question and hypothesis should be developed before the start of the study. This sounds intuitive; however, if we take, for example, a database of information, it is potentially possible to perform multiple statistical comparisons of groups within the database to find a statistically significant association. This could then lead one to work backward from the data and develop the “question.” This is counterintuitive to the process because the question is asked specifically to then find the answer, thus collecting data along the way (i.e., in a prospective manner). Multiple statistical testing of associations from data previously collected could potentially lead to spuriously positive findings of association through chance alone. 2 Therefore, a good hypothesis must be based on a good research question at the start of a trial and, indeed, drive data collection for the study.

The research or clinical hypothesis is developed from the research question and then the main elements of the study — sampling strategy, intervention (if applicable), comparison and outcome variables — are summarized in a form that establishes the basis for testing, statistical and ultimately clinical significance. 3 For example, in a research study comparing computer-assisted acetabular component insertion versus freehand acetabular component placement in patients in need of total hip arthroplasty, the experimental group would be computer-assisted insertion and the control/conventional group would be free-hand placement. The investigative team would first state a research hypothesis. This could be expressed as a single outcome (e.g., computer-assisted acetabular component placement leads to improved functional outcome) or potentially as a complex/composite outcome; that is, more than one outcome (e.g., computer-assisted acetabular component placement leads to both improved radiographic cup placement and improved functional outcome).

However, when formally testing statistical significance, the hypothesis should be stated as a “null” hypothesis. 2 The purpose of hypothesis testing is to make an inference about the population of interest on the basis of a random sample taken from that population. The null hypothesis for the preceding research hypothesis then would be that there is no difference in mean functional outcome between the computer-assisted insertion and free-hand placement techniques. After forming the null hypothesis, the researchers would form an alternate hypothesis stating the nature of the difference, if it should appear. The alternate hypothesis would be that there is a difference in mean functional outcome between these techniques. At the end of the study, the null hypothesis is then tested statistically. If the findings of the study are not statistically significant (i.e., there is no difference in functional outcome between the groups in a statistical sense), we cannot reject the null hypothesis, whereas if the findings were significant, we can reject the null hypothesis and accept the alternate hypothesis (i.e., there is a difference in mean functional outcome between the study groups), errors in testing notwithstanding. In other words, hypothesis testing confirms or refutes the statement that the observed findings did not occur by chance alone but rather occurred because there was a true difference in outcomes between these surgical procedures. The concept of statistical hypothesis testing is complex, and the details are beyond the scope of this article.

Another important concept inherent in hypothesis testing is whether the hypotheses will be 1-sided or 2-sided. A 2-sided hypothesis states that there is a difference between the experimental group and the control group, but it does not specify in advance the expected direction of the difference. For example, we asked whether there is there an improvement in outcomes with computer-assisted surgery or whether the outcomes worse with computer-assisted surgery. We presented a 2-sided test in the above example because we did not specify the direction of the difference. A 1-sided hypothesis states a specific direction (e.g., there is an improvement in outcomes with computer-assisted surgery). A 2-sided hypothesis should be used unless there is a good justification for using a 1-sided hypothesis. As Bland and Atlman 8 stated, “One-sided hypothesis testing should never be used as a device to make a conventionally nonsignificant difference significant.”

The research hypothesis should be stated at the beginning of the study to guide the objectives for research. Whereas the investigators may state the hypothesis as being 1-sided (there is an improvement with treatment), the study and investigators must adhere to the concept of clinical equipoise. According to this principle, a clinical (or surgical) trial is ethical only if the expert community is uncertain about the relative therapeutic merits of the experimental and control groups being evaluated. 9 It means there must exist an honest and professional disagreement among expert clinicians about the preferred treatment. 9

Designing a research hypothesis is supported by a good research question and will influence the type of research design for the study. Acting on the principles of appropriate hypothesis development, the study can then confidently proceed to the development of the research objective.

Research objective

The primary objective should be coupled with the hypothesis of the study. Study objectives define the specific aims of the study and should be clearly stated in the introduction of the research protocol. 7 From our previous example and using the investigative hypothesis that there is a difference in functional outcomes between computer-assisted acetabular component placement and free-hand placement, the primary objective can be stated as follows: this study will compare the functional outcomes of computer-assisted acetabular component insertion versus free-hand placement in patients undergoing total hip arthroplasty. Note that the study objective is an active statement about how the study is going to answer the specific research question. Objectives can (and often do) state exactly which outcome measures are going to be used within their statements. They are important because they not only help guide the development of the protocol and design of study but also play a role in sample size calculations and determining the power of the study. 7 These concepts will be discussed in other articles in this series.

From the surgeon’s point of view, it is important for the study objectives to be focused on outcomes that are important to patients and clinically relevant. For example, the most methodologically sound randomized controlled trial comparing 2 techniques of distal radial fixation would have little or no clinical impact if the primary objective was to determine the effect of treatment A as compared to treatment B on intraoperative fluoroscopy time. However, if the objective was to determine the effect of treatment A as compared to treatment B on patient functional outcome at 1 year, this would have a much more significant impact on clinical decision-making. Second, more meaningful surgeon–patient discussions could ensue, incorporating patient values and preferences with the results from this study. 6 , 7 It is the precise objective and what the investigator is trying to measure that is of clinical relevance in the practical setting.

The following is an example from the literature about the relation between the research question, hypothesis and study objectives:

Study: Warden SJ, Metcalf BR, Kiss ZS, et al. Low-intensity pulsed ultrasound for chronic patellar tendinopathy: a randomized, double-blind, placebo-controlled trial. Rheumatology 2008;47:467–71.

Research question: How does low-intensity pulsed ultrasound (LIPUS) compare with a placebo device in managing the symptoms of skeletally mature patients with patellar tendinopathy?

Research hypothesis: Pain levels are reduced in patients who receive daily active-LIPUS (treatment) for 12 weeks compared with individuals who receive inactive-LIPUS (placebo).

Objective: To investigate the clinical efficacy of LIPUS in the management of patellar tendinopathy symptoms.

The development of the research question is the most important aspect of a research project. A research project can fail if the objectives and hypothesis are poorly focused and underdeveloped. Useful tips for surgical researchers are provided in Box 3 . Designing and developing an appropriate and relevant research question, hypothesis and objectives can be a difficult task. The critical appraisal of the research question used in a study is vital to the application of the findings to clinical practice. Focusing resources, time and dedication to these 3 very important tasks will help to guide a successful research project, influence interpretation of the results and affect future publication efforts.

Tips for developing research questions, hypotheses and objectives for research studies

• Perform a systematic literature review (if one has not been done) to increase knowledge and familiarity with the topic and to assist with research development.
• Learn about current trends and technological advances on the topic.
• Seek careful input from experts, mentors, colleagues and collaborators to refine your research question as this will aid in developing the research question and guide the research study.
• Use the FINER criteria in the development of the research question.
• Ensure that the research question follows PICOT format.
• Develop a research hypothesis from the research question.
• Develop clear and well-defined primary and secondary (if needed) objectives.
• Ensure that the research question and objectives are answerable, feasible and clinically relevant.

FINER = feasible, interesting, novel, ethical, relevant; PICOT = population (patients), intervention (for intervention studies only), comparison group, outcome of interest, time.

Competing interests: No funding was received in preparation of this paper. Dr. Bhandari was funded, in part, by a Canada Research Chair, McMaster University.

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Title: m3: a multi-task mixed-objective learning framework for open-domain multi-hop dense sentence retrieval.

Abstract: In recent research, contrastive learning has proven to be a highly effective method for representation learning and is widely used for dense retrieval. However, we identify that relying solely on contrastive learning can lead to suboptimal retrieval performance. On the other hand, despite many retrieval datasets supporting various learning objectives beyond contrastive learning, combining them efficiently in multi-task learning scenarios can be challenging. In this paper, we introduce M3, an advanced recursive Multi-hop dense sentence retrieval system built upon a novel Multi-task Mixed-objective approach for dense text representation learning, addressing the aforementioned challenges. Our approach yields state-of-the-art performance on a large-scale open-domain fact verification benchmark dataset, FEVER. Code and data are available at: this https URL

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This paper is in the following e-collection/theme issue:

Published on 21.3.2024 in Vol 26 (2024)

Three-Day Monitoring of Adhesive Single-Lead Electrocardiogram Patch for Premature Ventricular Complex: Prospective Study for Diagnosis Validation and Evaluation of Burden Fluctuation

Authors of this article:

Original Paper

• Hyo-Jeong Ahn 1 , MD   ; 
• Eue-Keun Choi 1, 2 , MD, PhD   ; 
• So-Ryoung Lee 1, 2 , MD, PhD   ; 
• Soonil Kwon 1 , MD   ; 
• Hee-Seok Song 3 , MSc   ; 
• Young-Shin Lee 3 , MSc   ; 
• Seil Oh 1, 2 , MD, PhD  

1 Department of Internal Medicine, Seoul National University Hospital, Seoul, Republic of Korea

2 Department of Internal Medicine, Seoul National University College of Medicine, Seoul, Republic of Korea

3 Seers Technology Co, Ltd, Seongnam-si, Gyeonggi-do, Republic of Korea

Corresponding Author:

Eue-Keun Choi, MD, PhD

Department of Internal Medicine

Seoul National University Hospital

101 Daehak-ro

Seoul, 03080

Republic of Korea

Phone: 82 220720688

Email: [email protected]

Background: Wearable electrocardiogram (ECG) monitoring devices are used worldwide. However, data on the diagnostic yield of an adhesive single-lead ECG patch (SEP) to detect premature ventricular complex (PVC) and the optimal duration of wearing an SEP for PVC burden assessment are limited.

Objective: We aimed to validate the diagnostic yield of an SEP (mobiCARE MC-100, Seers Technology) for PVC detection and evaluate the PVC burden variation recorded by the SEP over a 3-day monitoring period.

Methods: This is a prospective study of patients with documented PVC on a 12-lead ECG. Patients underwent simultaneous ECG monitoring with the 24-hour Holter monitor and SEP on the first day. On the subsequent second and third days, ECG monitoring was continued using only SEP, and a 3-day extended monitoring was completed. The diagnostic yield of SEP for PVC detection was evaluated by comparison with the results obtained on the first day of Holter monitoring. The PVC burden monitored by SEP for 3 days was used to assess the daily and 6-hour PVC burden variations. The number of patients additionally identified to reach PVC thresholds of 10%, 15%, and 20% during the 3-day extended monitoring by SEP and the clinical factors associated with the higher PVC burden variations were explored.

Results: The recruited data of 134 monitored patients (mean age, 54.6 years; males, 45/134, 33.6%) were analyzed. The median daily PVC burden of these patients was 2.4% (IQR 0.2%-10.9%), as measured by the Holter monitor, and 3.3% (IQR 0.3%-11.7%), as measured in the 3-day monitoring by SEP. The daily PVC burden detected on the first day of SEP was in agreement with that of the Holter monitor: the mean difference was –0.07%, with 95% limits of agreement of –1.44% to 1.30%. A higher PVC burden on the first day was correlated with a higher daily ( R 2 =0.34) and 6-hour burden variation ( R 2 =0.48). Three-day monitoring by SEP identified 29% (12/42), 18% (10/56), and 7% (4/60) more patients reaching 10%, 15%, and 20% of daily PVC burden, respectively. Younger age was additionally associated with the identification of clinically significant PVC burden during the extended monitoring period ( P =.02).

Conclusions: We found that the mobiCARE MC-100 SEP accurately detects PVC with comparable diagnostic yield to the 24-hour Holter monitor. Performing 3-day PVC monitoring with SEP, especially among younger patients, may offer a pragmatic alternative for identifying more individuals exceeding the clinically significant PVC burden threshold.

Introduction

Premature ventricular complexes (PVCs) are commonly observed in individuals who have undergone long-term ambulatory monitoring [ 1 ]. Frequent PVCs can result in reversible cardiomyopathy, although patients with PVCs without underlying structural heart disease are usually expected to have a benign clinical prognosis [ 2 - 4 ].

Several risk factors are known to contribute to the overall risk of developing PVC-induced cardiomyopathy (PIC), and the burden of PVC is one of the most important predictors of clinical deterioration [ 1 , 5 , 6 ]. Although there is no absolute cutoff of PVC burden to identify patients at risk of developing PIC, 16%-26% of PVC burden is reported as a significant threshold to develop PIC [ 5 , 7 , 8 ]. Of note, at least 10% of the PVC burden is considered sufficient for developing PIC to warrant a regular assessment of structural and functional cardiac change [ 9 ].

For the determination of an appropriate treatment strategy for PVC, one of the essential pieces of information that should be accurately assessed is the PVC burden, together with the accompanying symptoms and the presence or absence of structural heart disease. However, defining the exact PVC burden remains a challenge because of its substantial hourly or daily variation, and recent evidence recommends that patients undergo ambulatory ECG monitoring throughout the day to observe its maximum burden [ 1 ]. The 24-hour Holter monitoring has been regarded as the gold standard for evaluating PVC burden, but the Holter monitor is uncomfortable to wear, and the 24-hour duration is insufficient to appreciate the substantial burden variation demonstrated among individuals [ 10 , 11 ].

With the advent of widespread use of wearable electrocardiogram (ECG) patch monitoring, studies have suggested that extended monitoring is required to estimate more accurate daily PVC burden [ 10 , 12 ]. Recently, an adhesive single-lead ECG patch (SEP) has been found to enable more convenient extended monitoring to capture fluctuations in the PVC burden, but the diagnostic yield of SEP has not been thoroughly validated. Moreover, the optimal duration of ambulatory ECG monitoring for PVC evaluation has not been determined, and the clinical significance of 3-day SEP monitoring (a shorter period than studied before) [ 13 ] is yet to be explored.

In this study, we investigated the (1) diagnostic yield of SEP for PVC detection through a direct comparison with the 24-hour Holter monitoring and (2) variations in the PVC burden recorded during the 3-day extended monitoring using SEP and the associated clinical factors. Ultimately, we aimed to explore the potential clinical utility of SEP as a more convenient diagnostic tool of PVC, with the goal of identifying more patients who have clinically significant PVC burden.

Ethics Approval

The institutional review board at Seoul National University Hospital authorized this study (E-2001-111-1096). All study participants provided written informed consent. This study is reported according to the STROBE (Strengthening the Reporting of Observational Studies in Epidemiology) guidelines ( Multimedia Appendix 1 ).

Study Design and Patient Selection

This is a single-center and prospective cohort study of patients with PVC, which was performed between May 2021 and June 2022. Patients (≥19 years old) with documented PVC on a 12-lead ECG and scheduled for evaluation of the PVC burden (percentage of number of PVCs over the total QRS complexes) by 24-hour Holter monitoring were included. Since SEP utilizes a smartphone app to transmit ECG data, those unable to operate and manipulate the app were excluded. Three electrophysiologists (Choi EK, Lee SR, and SO) screened and recruited eligible patients at the outpatient clinic. Baseline clinical features, including demographic data, symptoms regarding PVC, comorbidities, lifestyle behaviors, medication history, and echocardiographic parameters were investigated.

Validation of PVC Diagnosis by SEP Through Comparison With the Holter Monitor

The 24-hour Holter monitor (SEER Light, GE HealthCare) and SEP (mobiCARE MC-100, Seers Technology) record the electrical activity of the heart through 3 channels (leads I, V1, and V6) and a single channel (lead II), respectively. The process of validating the ECG algorithm for PVC detection by SEP and its performance are detailed in Multimedia Appendix 2 . Patients underwent simultaneous ECG monitoring with the Holter monitor and SEP on the first day of monitoring. After 24 hours of monitoring, the Holter device was detached, and ECG monitoring was continued for the following 48 hours with SEP; the assessment of PVC burden using the Holter monitor is a part of the guideline-adherent practice [ 14 ], with patients covering the associated costs through local health insurance coverage. The detailed specifications of the SEP and the attachment methods for both devices are described in our prior report [ 15 ]. Information on total wear time, proportion of signal noise, total QRS complexes, and PVC burden was collected. An example strip of PVC detection using SEP is shown in Multimedia Appendix 3 . For the evaluation of the diagnostic yield of PVC by SEP, the burdens of PVC assessed by the Holter monitor and SEP on the first day were compared with each other.

Variability of PVC Burden and Clinical Factors Associated With Greater Variability

The fluctuation of PVC burden was assessed by the changes in the daily and 6-hour PVC burden monitored over a 3-day period using SEP. The purpose of evaluating daily and 6-hour PVC burden variations was to verify the fluctuations in the values. This analysis enables us to address the challenge of identifying the accurate overall PVC burden, and consequently, determining the appropriate treatment strategy. We defined daily PVC burden variation as the difference between the maximum and the minimum daily PVC burden measured by the SEP. The 6-hour PVC burden was calculated as the burden of PVC from midnight to 6 AM, 6 AM to noon, noon to 6 PM, and 6 PM to midnight. Similarly, the 6-hour PVC burden variation was defined as the difference between the maximum and minimum 6-hour PVC burden. We estimated the proportion of patients who were additionally identified to reach a clinically significant daily PVC burden (10%, 15%, or 20%) [ 12 ] during the 3-day monitoring. Further, clinical factors associated with a greater variation in PVC burden and those additionally influencing the PVC burden to cross the clinically meaningful threshold during the extended monitoring were investigated.

Self-Reported Questionnaire About the Experience of the 3-Day Extended ECG Monitoring

The participants’ clinical experience of the 3-day extended ECG monitoring for PVC detection by SEP was evaluated using a self-reported questionnaire. The survey was conducted on a voluntary basis for all the study participants without offering any form of compensation on the day the SEP device was returned. Questions were based on the overall convenience of the Holter monitor and SEP, any instances of unexpected reattachment, and accessibility and user friendliness of mobile apps for SEP. Participants responded to the questions by using a scale ranging from 1 to 5 (1=no, 2=minimally, 3=sometimes, 4=likely, or 5=very likely).

Statistical Analysis

Continuous variables were reported as mean (SD) or median (IQR) and compared by Student 2-sided t test or Mann-Whitney test. Categorical variables were presented as n (%), and Pearson chi-square test or Fisher exact test were applied as required. The diagnostic yield of PVC by SEP was compared to that by the Holter monitor with Bland-Altman plots with limits of agreement and scatter plots. The degree of agreement for the burden of PVC monitored by each device was analyzed. The associations between individuals’ baseline characteristics and the additional identification of significant PVC thresholds (10%, 15%, or 20%) during the extended monitoring were evaluated by multivariable logistic regression analysis. All analyses were performed using Stata (version 17, StataCorp LLC).

Baseline Characteristics of the Patients

Of the data of 156 enrolled patients with documented PVC on a 12-lead ECG, the data of 13 patients could not be retrieved due to errors in their smartphones, unfamiliarity with app use, or missing telemetry data transmission of SEP. Nine patients withdrew from the examination due to discomfort and skin irritability: 5 patients due to frequent alarms and the requirement of keeping their smartphones in close proximity, 2 due to skin irritability, and 2 due to inexperienced operation of the app. Finally, data from 134 patients with enough wear time of the patch (≥4200 minutes) and whose data were transmitted completely were included in this analysis.

The baseline characteristics of the included patients are described in Table 1 . The mean age was 54.6 (SD 13.3) years, and 45 (33.6%) patients were males. The majority of the patients reported symptoms such as palpitation (96/134, 71.6%) or dizziness (15/134, 11.2%). Hypertension (46/134, 34.3%) and diabetes mellitus (21/134, 15.7%) were the common comorbidities. Congestive heart failure and cardiomyopathy were reported in 5.9% (8/134) and 2.9% (4/134) of the patients, respectively. Patients frequently took β-blockers (97/134, 72.4%) or calcium channel blockers (10/134, 7.5%) to relieve symptoms and PVC burden.

Diagnostic Performance of PVC Detection and Burden by SEP Compared to That by the Holter Monitor

An example of PVC detection by the adhesive SEP is presented in Multimedia Appendix 3 . Detailed parameters of monitoring and the overall burden of PVC were compared between the 2 monitoring methods (Holter monitor and SEP) and are summarized in Table 2 . The total mean wear time was 1395.1 (SD 33.5) minutes for the Holter monitor and 4318.0 (SD 12.4) minutes for SEP. The transmitted data were retrieved without significant noise for both devices (0.1% for the Holter monitor vs 0.9% for SEP). The median PVC burden of all the enrolled patients was 2.4% (IQR 0.2%-10.9%), as measured by the Holter monitor, and 3.3% (IQR 0.3%-11.7%), as measured by SEP during the 3-day extended monitoring, without significant difference ( P =.58). During the first 24-hour monitoring by SEP, the burden of PVC among all patients was 7.5% (SD 10%; median 2.8%, IQR 0.3%-11%) without a significant difference from the PVC burden measured by the Holter monitor ( P =.24). For those with PVC burden ≥5%, the overall PVC burden was detected as 13.3% (IQR 8%-24%) by the Holter monitor and 13.2% (IQR 9.7%-20.5%) by SEP during the 3-day extended monitoring ( P =.10). During the first 24-hour monitoring by SEP, the burden of PVC among patients with a burden ≥5% was 16.5% (SD 9.9%; median 13.7%, IQR 8.1%-24.4%); the P value compared to the PVC burden detected by the Holter monitor was .82 ( Table 2 ).

The overall distribution of the patients with PVC burden is presented in Figure 1 A. Most of the patients (97/134, 72.4%) had a PVC burden <10%. PVC burden was reported as 10%-19.9% and ≥20% in 18 (13.4%) and 19 (14.2%) patients, respectively. An individual comparison of patients’ PVC detection by the Holter monitor and SEP is shown in Figure 1 B. The PVC burden evaluated by the Holter monitor was repeated with almost the same value as the PVC burden estimated by SEP on the first day, whereas PVC burden variation was observed across days on SEP ( Figure 1 B). The validation of the diagnostic yield of SEP was performed by Bland-Altman analysis, and high agreement of PVC detection between the Holter monitor and SEP was confirmed with a mean difference of –0.07% and 95% limit of agreement of –1.44% to 1.30% ( Figure 2 A). PVC burdens detected by the Holter monitor and SEP on the first day of monitoring were highly correlated with each other ( Figure 2 B, R 2 =0.995), indicating the accurate detection of PVC by SEP.

a ECG: electrocardiogram.

b N/A: not applicable.

c During the first 24-hour monitoring with the single-lead electrocardiogram patch, the average, minimum, and maximum heart rates (bpm) were 72.2 (SD 8.3), 48.8 (SD 6.2), and 123.2 (SD 17.7), respectively. The P values with the comparison of each value with the Holter monitor were <.001, .04, and .002, respectively.

d PVC: premature ventricular complex.

e During the first 24-hour monitoring with the single-lead electrocardiogram patch, the burden of premature ventricular complex among all patients was 7.5% (SD 10%; median 2.8%, IQR 0.3%-11%). The P value with the comparison of premature ventricular complex burden detected on the Holter monitor was .24.

f During the first 24-hour monitoring with the single-lead ECG patch, the burden of premature ventricular complex among patients with burden ≥5% was 16.5% (SD 9.9%; median 13.7%, IQR 8.1%-24.4%). The P value with the comparison of premature ventricular complex burden detected on the Holter monitor was .82.

PVC burden varied widely from day to day and hour to hour, as shown in Figure 1 B. Greater variations in PVC burden were observed when evaluated every 6 hours. In the same patient, the maximum daily PVC burden was 1.68-fold (IQR 1.31-3.02) higher than the minimum daily burden and the maximum 6-hour burden was 12.08-fold (IQR 4.00-57.50) higher than the minimum 6-hour burden; for those with PVC burden ≥1%, the maximum daily and 6-hour PVC burden were 1.52-fold (IQR 1.28-2.07) and 12.17-fold (IQR 5.06-49.46) higher than each minimum burden, respectively. Moreover, the variation was dependent on the individuals’ burden value; patients with higher PVC burden exhibited greater daily or hourly burden variation. Daily and 6-hour burden variations according to the value of PVC burden are presented in Figure 3 , showing a moderate to high degree of correlation ( R 2 =0.337 for daily and 0.483 for 6-hour burden variation). An example of an ECG strip of an individual presenting a high variation in daily and 6-hour PVC burden monitored using SEP is shown in Multimedia Appendix 4 .

The number of patients with PVC burden lower than the clinically significant thresholds was 42 for 10%, 56 for 15%, and 60 for 20%. However, the 3-day extended monitoring additionally identified that 29% (12/42), 18% (10/56), and 7% (4/60) of the patients reached 10%, 15%, and 20% thresholds of the PVC burden, respectively ( Figure 4 , Table 3 ).

The clinical features of those crossing the PVC burden cutoff of 10%, 15%, or 20% during the 3-day extended monitoring were compared with those of the other patients with PVC burden lower than the clinically significant thresholds ( Multimedia Appendix 5 ). Patients who were additionally identified to reach clinically significant thresholds during the 3-day extended monitoring were younger ( P =.03), had congestive heart failure ( P =.01) or cardiomyopathy ( P =.01), and were using amiodarone ( P =.04). When multivariable logistic regression analysis was performed, young age was independently associated with possible additional detection of PVC burden with clinical significance ( Table 4 ; β=.95, 95% CI .92-.99; P =.02 for age).

a PVC: premature ventricular complex.

b SEP: single-lead electrocardiogram patch.

a N/A: not applicable.

Patients’ Experience of 3-Day Extended ECG Monitoring

A survey on patients’ experience with Holter monitoring and SEP was completed by 124 patients (response rate, 124/134, 92.5%), and the results are detailed in Multimedia Appendix 6 . Overall, patients experienced lesser discomfort and lesser skin irritability with SEP compared to that with the Holter monitor ( P <.001 and P =.02, respectively), but they wished to stop monitoring with SEP due to the longer duration of the examination. Indeed, since the patients wished to stop monitoring with SEP due to the longer duration of the examination, they detached the SEP and reattached the SEP later, thereby resulting in more instances of reattachment of the SEP (14/112, 12.5% for the Holter monitor and 66/118, 55.9% for SEP; P =.02). Thus, patients were more favorable to using SEP than the Holter monitor; 75.7% (84/111) of the patients responded to choosing a patch monitor for their next examination.

Principal Results

Our principal findings are as follows: (1) SEP can diagnose PVC with comparable accuracy to the 24-hour Holter monitor, (2) the wide variation in the PVC burden recorded by SEP was proportional to the overall PVC frequency, (3) 3-day extended monitoring could identify those who reach clinically significant cutoffs of PVC burden, and (4) young age is associated with additional detection of clinically significant burdens of PVC during extended monitoring.

Comparison to Prior Work

Ambulatory ECG monitoring is the principal diagnostic tool for evaluating PVC burden, and recently developed wearable ECG monitoring devices enable extended monitoring for more than 24 hours [ 1 , 16 , 17 ]. In particular, SEP is considered to be a sufficient and practical method to evaluate the PVC burden accurately and comprehend the wide variations daily [ 11 , 18 ]. Previous studies [ 19 , 20 ] have demonstrated variations in the PVC burden to up to 23% change across days, while a recent study [ 10 ] has reported a 2.5-fold median difference between the maximum and minimum 24-hour PVC burden in the same patient over a 14-day monitoring period. Regarding the optimal duration of PVC monitoring, the median time to detect one’s maximum PVC burden was reported as 6 (IQR 2-11) days [ 12 ], and the ideal duration for accurate PVC burden assessment was suggested as 7 days [ 13 ]. Although the monitoring period in our study (3 days) was shorter than the observation period in previous reports [ 12 , 13 ], we still observed a considerable intraindividual median daily PVC burden difference of 1.7-fold. Given that the optimal period for monitoring PVCs has not been established and a long recording length could be burdensome for both patients and clinicians, we suggest a compromise in the monitoring length (3 days), thereby providing an acceptable level of daily PVC burden, which may improve patients’ compliance and adherence. Despite the general comfort of SEP, our survey results revealed that patients felt more likely to discontinue monitoring as the monitoring time became longer. Furthermore, an extended monitoring duration could potentially exacerbate patient discomfort and fatigue because of the requirement of carrying their mobile phones for data transmission. The discomfort could also be intensified by the alarm systems and skin irritability. In our study, 9 patients withdrew from the examination due to these practical challenges; thus, optimizing the monitoring duration could potentially decrease the dropout rate associated with these issues.

A wide variation in the daily PVC burden detected during the 3-day monitoring enabled us to additionally identify 6.7%-28.6% of the patients who reach clinically significant thresholds (10%, 15%, or 20%). Compared to a previous study [ 12 ] that showed that extended monitoring nearly doubled the identification of those reaching the 10% threshold during 14-day monitoring, the low proportion of identification in our study could be due to the shorter duration of monitoring (3 days). However, this finding is consistent with the fact that extended monitoring brings incremental gain in identifying patients reaching the significant cutoffs of daily PVC burden. We observed that SEP could diagnose PVC with almost the same accuracy as the Holter monitor, and 3-day monitoring could detect patients who might benefit from an advanced treatment modality such as radiofrequency catheter ablation.

Implications for Practice

Several risk factors are known to be associated with the presence of PVC and higher PVC frequency. Increased age, taller height, higher blood pressure, and unhealthy lifestyle behaviors such as smoking and less physical activity are consistently shown to be associated with the presence or higher frequency of PVC [ 21 - 23 ]. However, no study has demonstrated the clinical factors related to the large variations in PVC burden, which might be necessary to stratify patients requiring extended monitoring for a more accurate assessment of PVC burden. We found that the PVC burden evaluated during the first 24-hour monitoring was correlated not only with the day-to-day but also with the hour-to-hour variation. The higher frequency of PVCs was linked to a wider variation in the overall burden. In addition, we found that younger age, presence of congestive heart failure or cardiomyopathy, and use of amiodarone are clinical factors that may increase the likelihood that PVC burden reaches clinically significant thresholds during the 3-day extended monitoring. Above all, multivariable analysis showed that younger age was consistently associated with the significant variations in the PVC burden ( P =.02). Although no explainable mechanism for this association has been reported yet, the fact that older age and a lack of diurnal variation of PVC frequency are related to a higher risk of PIC [ 24 , 25 ] imply that there might be a link between young age and a greater variation in the PVC burden. Our analyses revealed patients who required extended monitoring for a more precise evaluation of PVC burden, thus improving the determination of the treatment strategy. The variable circadian distribution of PVC burden and its associated clinical factors can provide additional value, as this information can be used to guide the pharmacologic induction of PVCs during radiofrequency catheter ablation and predict the outcome in patients [ 25 ].

Strengths and Limitations

Our study has several limitations. First, the diagnostic yield of PVC via SEP and the degree of variation monitored for 3 days should be examined in a larger population. Second, since most of the included patients had normal ventricular systolic function, the accurate diagnosis of PVC and changes in PVC frequency monitored by SEP should be further validated in patients with reduced ejection fraction or structural heart diseases. Third, more than two-thirds of the patients had low PVC burden (<10%); thus, the extrapolation of our findings in patients with higher PVC burden (10%, 15%, or 20%) could be limited. Nonetheless, the distribution of the PVC burden in our study may represent the characteristics of patients in real-world settings, suggesting that our findings can be readily applied in a clinical context. Lastly, the external replication of our findings by using other wearable ECG patches in a multicenter setting would be required for the confirmation of generalizability.

Conclusions

In a single-center prospective registry of patients with PVC, we validated that SEP can accurately diagnose PVC with almost the same yield as the 24-hour Holter monitor. We found that during the 3-day extended monitoring with SEP, the significant fluctuations in daily and 6-hour PVC burden were proportional to the overall PVC frequency. Further, 3-day extended monitoring of PVC by using SEP enabled the identification of more patients exceeding the clinically significant burden threshold; thus, SEP monitoring could be a practical method with acceptable patient adherence to enhance the decision on the optimal treatment strategy (ie, catheter ablation) for PVC. Several clinical factors, especially younger age, were associated with a higher variation of daily PVC burden, implying the necessity of extended monitoring for accurate assessments.

Acknowledgments

This work was supported in part by the Korea Medical Device Development Fund grant funded by the Korean government (Ministry of Science and Information and Communication Technology, Ministry of Trade, Industry and Energy, Ministry of Health and Welfare, and Ministry of Food and Drug Safety; project HI20C1662, 1711138358, KMDF_PR_20200901_0173) and a grant funded by Seers Technology.

Data Availability

All data generated or analyzed during this study are included in this paper and Multimedia Appendices 1 - 6 . The data of this study will be shared upon reasonable request to the corresponding author.

Authors' Contributions

HJA conceptualized the study design, conducted data curation, performed formal analysis, visualized the data, and was primarily responsible for writing this paper. EKC conceptualized and reviewed the study design, performed data collection, reviewed the manuscript, supervised the study, and performed project administration with funding acquisition. SK and SO reviewed the methodology, investigation, and the manuscript. SRL collected data and reviewed the methodology, investigation, and manuscript. HSS and YSL conducted data curation, performed formal analysis, prepared the resources and software, and reviewed the manuscript. EKC is the guarantor of this work and takes responsibility for the integrity and accuracy of the data analysis. All authors read and approved the final manuscript.

Conflicts of Interest

HSS and YSL are stockholders of Seers Technology Co, Ltd. EKC received the research grants or speaking fees from Abbott, Bayer, BMS/Pfizer, Biosense Webster, Chong Kun Dang, Daewoong Pharmaceutical Co, Daiichi-Sankyo, DeepQure, Dreamtech Co, Ltd, Jeil Pharmaceutical Co Ltd, Medtronic, Samjinpharm, Seers Technology, and Skylabs. Stock options were received from Seers Technology and Skylab.

STROBE (Strengthening the Reporting of Observational Studies in Epidemiology) guidelines.

Process of validating the electrocardiogram algorithm for premature ventricular complex detection by single-lead electrocardiogram patch and its performance.

An example strip of premature ventricular complex detection by the single-lead electrocardiogram patch.

An example of an electrocardiogram strip of an individual presenting a high variation in daily and 6-hour premature ventricular complex burden monitored using a single-lead electrocardiogram patch.

Comparison of the clinical features of patients crossing the premature ventricular complex burden cutoffs of 10%, 15%, or 20% during the 3-day extended monitoring.

Survey completed by 124 patients on their experience with the Holter monitor and single-lead electrocardiogram patch.

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Abbreviations

Edited by T Leung, G Eysenbach; submitted 30.01.23; peer-reviewed by Y Guo, M Mekhael; comments to author 16.06.23; revised version received 13.08.23; accepted 12.02.24; published 21.03.24.

©Hyo-Jeong Ahn, Eue-Keun Choi, So-Ryoung Lee, Soonil Kwon, Hee-Seok Song, Young-Shin Lee, Seil Oh. Originally published in the Journal of Medical Internet Research (https://www.jmir.org), 21.03.2024.

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A simple, efficient and versatile objective space algorithm for multiobjective integer programming

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• Published: 21 March 2024

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• Kerstin Dächert   ORCID: orcid.org/0000-0002-6458-6480 1 ,
• Tino Fleuren 2   na1 &
• Kathrin Klamroth 3   na1  

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In the last years a multitude of algorithms have been proposed to solve multiobjective integer programming problems. However, only few authors offer open-source implementations. On the other hand, new methods are typically compared to code that is publicly available, even if this code is known to be outperformed. In this paper, we aim to overcome this problem by proposing a new state-of-the-art algorithm with an open-source implementation in C++ . The underlying method falls into the class of objective space methods, i.e., it decomposes the overall problem into a series of scalarized subproblems that can be solved with efficient single-objective IP-solvers. It keeps the number of required subproblems small by avoiding redundancies, and it can be combined with different scalarizations that all lead to comparably simple subproblems. Our algorithm bases on previous results but combines them in a new way. Numerical experiments with up to ten objectives validate that the method is efficient and that it scales well to higher dimensional problems.

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1 Introduction

Integer programming (IP) is a classical field of Operations Research with a wide range of economical and industrial applications. Prominent examples are knapsack and capital budgeting problems, assignment problems, routing problems and problems occuring in supply chain management applications. The practical success of integer programming is supported by the fact that highly efficient solvers are readily available. This is not the case for multiobjective integer programming (MOIP). Indeed, while there is a growing need for the consideration of multiple conflicting goals, including economical, ecological and robustness criteria, the development of MOIP solvers lags behind.

A simple yet efficient way to overcome this shortcoming can be seen in the recent trend towards so-called objective space methods or scalarization-based algorithms . Different from “purely” multiobjective approaches like, for example, multiobjective branch and bound algorithms, such methods rely on the iterative solution of appropriately defined single-objective IPs, hence taking advantage of the efficiency of single-objective IP solvers. Besides the fact that such methods largely benefit from the strength of single-objective IP solvers, they are independent of the particular problem structure and are thus very generally applicable, without the hassle of problem specific fine-tuning.

Objective space methods can be distinguished w.r.t. the way the objective space is decomposed and w.r.t. the way the subproblems are formulated. Both aspects, decomposition and subproblem formulation , have a direct impact on the computational efficiency of the overall method: The decomposition influences the number of solver calls, and the complexity of the subproblems has a decisive effect on the computational time required by each individual solver call. Different objective space methods have been proposed, e.g., in Klein and Hannan ( 1982 ); Sylva and Crema ( 2004 ); Laumanns et al. ( 2006 ); Sylva and Crema ( 2008 ); Lokman and Köksalan ( 2013 ); Özlen et al. ( 2014 ); Kirlik and Sayin ( 2014 ); Dächert and Klamroth ( 2015 ); Klamroth et al. ( 2015 ); Boland et al. ( 2016 ); Dächert et al. ( 2017 ); Boland et al. ( 2017b ); Holzmann and Smith ( 2018 ); Tamby ( 2018 ); Turgut et al. ( 2019 ); Joswig and Loho ( 2020 ) and Tamby and Vanderpooten ( 2020 ). We review the most prominent approaches in the light of a generic algorithmic description in Sect.  4 .

In the following subsections we first discuss the intrinsic difficulty of MOIP problems (Sect.  1.1 ) and then give a formal problem formulation and introduce the notation (Sect.  1.2 ). We provide a prototypical formulation of a generic scalarization-based algorithm in Sect.  2 , discuss the geometric complexity of the associated decomposition, and briefly review scalarization methods that can be used in this context. Section  3 presents how the number of integer problems can be reduced when using the \(\varepsilon \) -constraint scalarization. In Sect.  4 we review the related literature and provide a summary of the leading objective space algorithms for MOIP. An extensive computational study comparing different approaches on instances of knapsack, assignment travelling salesman problems with up to ten objective functions is presented in Sect.  5 . While the approach presented in this paper does not necessarily require the fewest number of integer programs to be solved, it is, nevertheless, the best with respect to CPU time.

1.1 Challenges in multiobjective integer programming

While MOIP problems are relatively well understood in the biobjective case, they are principally difficult in three and more dimensions (see, for example, Figueira et al. 2017 ). One reason for the specific role of biobjective problems is that in this case Pareto optimal solutions can be sorted such that their objective values are increasing in one objective and decreasing in the other objective. In this way, a natural ordering within the nondominated set is induced which facilitates central operations like, for example, decomposition, bound computations, and filtering for dominated solutions. This is used in the context of objective space methods, for example, in Aneja and Nair ( 1979 ); Chalmet et al. ( 1986 ); Ralphs et al. ( 2006 ). Since there is not such a natural ordering in the case of three and more objectives, many approaches do not directly transfer from the biobjective to the multiobjective case. Figure  1 illustrates this situation with the help of a simple example with two nondominated outcome vectors in the biobjective case (Fig.  1 a) and in the three-objective case (Fig.  1 b).

Dominated region in the biobjective ( a ) and in the three-objective case ( b ) for two points \(z^1=(4,6)\) and \(z^2=(7,3)\) ( a ) and \(z^1=(4,6,6)\) and \(z^2=(7,3,4)\) ( b ), respectively. Note that a can be interpreted as a projection of b onto the \(f_1-f_2\) -plane

In addition, when increasing the number of objective functions then also the number of efficient solutions usually increases. Even though already biobjective problems are intractable in the sense that the size of the nondominated set may grow exponentially with the problem size (see, for example, Ehrgott 2005 ), in practice the percentage of nondominated outcome vectors largely depends on two model characteristics: The problem structure and coefficients, and the number of objective functions. Indeed, when extending an MOIP problem by an additional objective function, then all formerly efficient solutions remain efficient for the extended problem, and chances are high that further solutions become efficient if they perform well w.r.t. the additional objective. It is easy to see that when the solution set is finite it is always possible to define a finite number of objective functions so that all feasible solutions are efficient.

1.2 Problem formulation and notation

In the following, we give a brief introduction to multiobjective optimization and to the classical concept of Pareto dominance. For an extensive introduction into the field, we refer to the textbooks (Ehrgott 2005 ) and Miettinen ( 1999 ). Throughout this paper we use the following notation to compare two vectors \(x^1,x^2\in \mathbb {R}^{n}\) :

The symbols \(\geqq , \geqslant \) and > are used analogously.

We consider multiobjective integer programming problems (MOIP) with \(p\ge 2\) functions given by

where \(C\in \mathbb {Z}^{p\times n}\) is the objective matrix, \(A\in \mathbb {Z}^{m\times n}\) is the constraint matrix, \(b\in \mathbb {Z}^m\) is the right-hand-side vector, and \(X=\{x\in \mathbb {Z}^n:\, Ax\leqq b\}\) denotes the set of feasible solutions of  MOIP . We assume that all coefficients are integer and that the variables may only take integer values. Non-negativity constraints can be included in the constraint system \(Ax\leqq b\) when required. We note that multiobjective binary programming problems (MOBP) where the solution vectors are constrained to \(x\in \{0,1\}^n\) are an important special case that is included in this formulation.

The p objective functions are given as a p -dimensional vector \(f=(f_1,\dots ,f_{p})\,:\, X\rightarrow \mathbb {Z}^{p}\) , where \(p\ge 2\) denotes the number of objective functions and hence the dimension of the objective space. The i th objective function is given as a linear function with cost vector \(c^i=(c_{i1},\dots ,c_{in})\) , i.e., \(f_i(x)=c^i x = \sum _{j=1}^n c_{ij}x_j\) , \(i=1,\dots ,p\) . Since we assume that all coefficients are integer, the set of attainable (i.e., feasible) outcome vectors \(Z=f(X)\) satisfies \(Z\subseteq \mathbb {Z}^{p}\) .

We focus on the determination of Pareto optimal (or efficient ) solutions which are those solutions for which none of the objective values can be improved without deterioration of at least one other objective value. This concept is based on the partial ordering induced by the natural ordering cone \(\mathbb {R}^{p}_{\geqslant }=\{z\in \mathbb {R}^{p}\,:\,z\geqslant 0\}=\mathbb {R}^{p}_{\geqq }{\setminus }\{0\}\) , where \(0\in \mathbb {R}^p\) denotes the p -dimensional zero-vector. Then \(z^1\leqslant z^2\) (i.e., \(z^1\) dominates \(z^2\) ) if and only if \(z^1\in z^2 - \mathbb {R}^{p}_{\geqslant }\) , and \({\bar{x}}\in X\) is Pareto optimal if and only if there does not exist \(x\in X\) such that \(f(x)\in f({\bar{x}})-\mathbb {R}^{p}_{\geqslant }\) .

The image of a Pareto optimal solution in the objective space is called nondominated point or nondominated outcome vector . The set of all Pareto optimal solutions (nondominated points) is referred to as the Pareto set ( nondominated set ) and denoted by \(X_E\subseteq X\) and \(Z_N\subseteq Z\) , respectively. For a given nonempty set \(N\subseteq Z\) of feasible outcome vectors, we refer to the set \(N+\mathbb {R}^{p}_{\geqq }\) as the dominated region of the set N , see Fig.  1 for an illustration. Note that further nondominated outcome vectors of  MOIP can only be located in the complement of the dominated region of the set N , i.e., in the set \(S(N):=\mathbb {R}^{p}\setminus (N+\mathbb {R}^{p}_{\geqq })\) which we call the search region w.r.t. N . This is in fact true for every nonempty set \(N\subseteq Z\) , irrespective of the fact whether all points in N are nondominated or not. In the special case that \(N=Z_N\) we additionally have that \(N+\mathbb {R}^{p}_{\geqq } = Z + \mathbb {R}^{p}_{\geqq }\) and that \(S(N)\cap Z=\emptyset \) .

2 Generic scalarization-based algorithm

In this section, we provide a generic formulation of an objective space algorithm that is based on an appropriate decomposition of the search region and on the iterative solution of scalarized single-objective IPs using standard IP solvers. We emphasize that this method does not assume a specific (combinatorial) problem structure and that it takes advantage of the efficiency of readily available IP solvers.

Given an initial area of interest, i.e., an initial search region, each solver call either generates a new nondominated outcome vector (which, by removing the region it dominates, leads to a reduction of the search region) or returns the information that the corresponding subproblem is infeasible (and that the associated part of the search region does not contain any further nondominated outcome vectors and can be excluded from further evaluations).

Key operations are hence the description and update operation for the search region as the basis for its decomposition into search zones (Sect.  2.1 ) and the formulation of appropriate subproblems in the respective search zones which is usually realized through scalarizations (Sect.  2.2 ). A pseudocode formulation of the general algorithmic framework is given in Algorithm 1.

Generic Scalarization-Based Algorithm

As said before, the driving factors for the efficiency of Algorithm 1 are the number and the complexity of subproblems that are solved during the course of the algorithm. These two factors are interrelated: By formulating more complex subproblems, larger reductions of the search region can be achieved which can in turn be used to reduce the number of solver calls. However, this usually comes at the price of more expensive solver calls. This trade-off is analyzed in the numerical tests in Sect.  5 . We note that an additional criterion when implementing Algorithm 1 is the avoidance of infeasible subproblems since detecting infeasibility often comes at a (slightly) higher computational cost.

2.1 The search region and its complexity

In this section we first focus on the mathematical description of the search region and show how it suggests a natural decomposition into non-redundant search zones. Towards this end, consider an arbitrary iteration of Algorithm 1 and let \(N\ne \emptyset \) denote the set of nondominated points computed so far. Then the search region S ( N ) that potentially contains further nondominated points is formally defined as

A thorough analysis of the geometric structure of the search region and its concise description is given in Dächert and Klamroth ( 2015 ) for the three-objective case, and in Klamroth et al. ( 2015 ) and Dächert et al. ( 2017 ) for the general case. See also Joswig and Loho ( 2020 ) for an alternative yet equivalent interpretation based on monomial cones and tropical algebra.

We follow the exposition in Klamroth et al. ( 2015 ) and assume that the search region is restricted to a bounding box with lower bound \((m,\dots ,m)^T\in \mathbb {Z}^{p}\) and upper bound \((M,\dots ,M)^T\in \mathbb {Z}^{p}\) (with \(m<M\) ). These bounds may be given by the decision maker or they may be derived, for example, from global bounds on the attainable objective function values. For example, the ideal point can be used to determine a global lower bound. For an MOIP problem the ideal point \(z^I \in \mathbb {Z}^p\) is defined component-wise as

To simplify the notation, we assume wlog that \(m=0\) . Moreover, let p dummy points \(d^i=M\cdot e^i\in \mathbb {Z}^{p}\) , \(i=1,\dots ,p\) be included in the set N that delimit the bounding box, where \(e^i\) denotes the i th unit vector in \(\mathbb {R}^{p}\) . Then the search region S ( N ) can be unambiguously described by a finite set of local upper bounds U ( N ) such that

where \(C(u)=u-\mathbb {R}^{p}_>\) denotes the search zone induced by the local upper bound u . Figure  2 illustrates the situation for the example problems from Fig.  1 .

Local upper bounds shown by filled circles in the biobjective ( a ) and in the three-objective case (b) for the two points introduced in Fig.  1 . The empty circles in a show the projections \({\hat{u}}^{31}\) and \({\hat{u}}^{33}\) of the local upper bounds \(u^{31}\) and \(u^{33}\) that are only present in the three-objective case, see b

Intuitively, local upper bounds are the maximal points in the objective space that are not dominated by any of the points in N . As a consequence, all components \(u_i\) , \(i=1,\dots ,p\) , of a local upper bound \(u\in U(N)\) are induced by a corresponding component \(z_i^i\) of a point \(z^i\in N\) that satisfies \(z_i^i=u_i\) and \(z^i_j < u_j\) for all \(j\in \{1,\dots ,p\}{\setminus }\{i\}\) . We call such a point a defining point for (component i of) the local upper bound u . Note that local upper bounds are thus also integer points, i.e., \(U(N)\subset \mathbb {Z}^{p}\) .

Moreover, as soon as at least one nondominated point z in the interior of the bounding box is found (this is usually the case after solving the first subproblem), then the global upper bound \((M,\dots ,M)^T\) is dominated and every local upper bound has at least one defining point that is not a dummy point.

More generally, whenever a new nondominated point \({\bar{z}}\) is found in U ( N ), then the set U ( N ) needs to be updated: For all \(u\in U(N)\) with \({\bar{z}}<u\) , u is replaced by p smaller local upper bounds \(u^1, \dots , u^p\) , where

I.e., to compute \(u^i\) , the i -th component of u is replaced by the value of the nondominated point \({\bar{z}}\) in component i while all other components remain unchanged. Hence, \(u^i\leqslant u\) for all \(i=1,\dots ,p\) , that is, the search region is reduced. For convenience we call the resulting local upper bound \(u^i\) an i - child of u . Note that \({\bar{z}}<u\) may be satisfied for more than one local upper bound in U ( N ). In this case, some of the generated i -children may be redundant for the description of \(U(N\cup \{{\bar{z}}\})\) and can be removed. The detection of redundant local upper bounds is described in detail in Dächert and Klamroth ( 2015 ), Klamroth et al. ( 2015 ) and Dächert et al. ( 2017 ).

In the biobjective case (see Fig.  2 a for an illustration), the set U ( N ) is immediately obtained from the points in N by first sorting these points, for example, w.r.t. \(f_1\) , and then combining their respective components in increasing order of \(f_1\) (in Fig.  2 a, this yields \(U(N)=\{u^1,u^2,u^3\}\) ). Figure  2 b illustrates the set \(U(N)=\{u^1,u^{21},u^{22},u^{31},u^{33}\}\) for an example problem with three objectives that was computed by recursively introducing the outcome vectors \(z^1\) and \(z^2\) . Efficient operations to iteratively update the set U ( N ) when further nondominated points are added to N are described in Klamroth et al. ( 2015 ) and Dächert et al. ( 2017 ).

Given the set U ( N ), a single-objective subproblem can be solved for each of the associated search zones \(C(u)=u-\mathbb {R}^{p}_{>}\) to either find a new (mutually nondominated) point, or to show that the corresponding search zone does not contain any further nondominated points. The cardinality of the set U ( N ) is thus decisive for the number of subproblems that need to be solved in Algorithm 1. A worst case bound on | U ( N )| can be derived from results of Boissonnat et al. ( 1998 ) and Kaplan et al. ( 2008 ) from the field of algorithmic geometry: The number of local upper bounds is bounded by

This bound is tight in the sense that there are instances where this bound is attained. However, in practice the value of | U ( N )| is often much smaller, often even by orders of magnitude. In the special case of two objectives ( \(p=2\) ) the number of local upper bounds is exactly \(|U(N)|=|N|+1\) , irrespective of the considered problem instance. Moreover, Dächert and Klamroth ( 2015 ) showed that for three objectives ( \(p=3\) ) it holds that \(|U(N)|=2|N|+1\) (under the assumption that no two points in N share the same value in one of their components). Hence, for \(p=3\) the number of search zones still grows only linearly with the number of nondominated points.

We argue that the decomposition of the search region S ( N ) into | U ( N )| pairwise non-redundant search zones C ( u ), \(u\in U(N)\) , is a natural decomposition that immediately leads to the formulation of associated pairwise non-redundant single-objective subproblems. The efficiency of this method is validated in the numerical tests presented in Sect.  5 . While other decompositions of the search region are of course possible and have other pros and cons, the structure of the set U ( N ) and particularly its cardinality remain decisive for the computational complexity of the respective methods.

Indeed, when the number of objective functions p is fixed and when the natural decomposition is combined with a suitable scalarization method, then ( 3 ) implies a polynomial bound on the overall number of iterations of Algorithm 1. In this situation we call a scalarization method suitable if and only if either a new nondominated point is found or the respective search zone is identified as an empty zone (which can be excluded from further consideration) when solving an appropriate scalarization in a search zone. This property is, e.g., satisfied by the frequently applied \(\varepsilon \) -constraint scalarization and by the weighted Tchebychev scalarization, see Sect.  2.2 below for further details.

When the natural decomposition of the search region into non-redundant search zones is combined with a suitable scalarization method, then the overall number of iterations of Algorithm  1 can be bounded by

which is polynomial in the size of the nondominated set \(|Z_N|\) when p is fixed.

First consider the case where the \(|Z_N|\) nondominated points are found within the first \(|Z_N|\) iterations, and after that the remaining search zones are investigated which are empty. In this case, the number of subproblems solved by Algorithm 1 is bounded by

i.e., \(|Z_N|\) iterations in each of which a nondominated point is determined, and at most \(O(|Z_N|^{\lfloor \frac{p}{2}\rfloor })\) iterations to detect that all search zones induced by these points are empty. C.f. ( 3 ) for the corresponding bound on the number of search zones induced by \(Z_N\) .

Now consider the general case where empty search zones may be detected at any time. In this case the above argumentation still holds since an empty search zone is never changed in later iterations, irrespective from the fact whether it is detected earlier or later during the procedure. Indeed, only those search zones are updated and split that contain a newly detected nondominated point, and this situation can not occur when the considered search zone is empty. Therefore, the iteration at which an empty search zone is detected does not matter. \(\square \)

Note that Theorem  1 relies on two central properties: (a) The complexity of the decomposition used in Algorithm 1 grows only polynomially with the number of nondominated points (which is guaranteed in the case of the natural decomposition), and (b) the employed scalarization must be tailored for the respective search zones in the sense that with each solver call, either a new nondominated point is found or the search zone is excluded. We will focus on this property when discussing suitable scalarizations, tailored for the natural decomposition, in the following section.

To the best of our knowledge the above approach is the only objective space method that has a provably polynomial bound on the number of required solver calls while keeping these solver calls simple, i.e., without relying on disjunctive constraints and/or additional integer variables.

2.2 Scalarization methods

We now assume that an intermediate set N of already computed nondominated points and a decomposition of the associated search region into a finite set of search zones is given. We exemplify the following discussion at the natural decomposition described in Sect.  2.1 above, where each search zone C ( u ), \(u\in U(N)\) , corresponds to a hyperrectangle with lower bound 0 and upper bound u .

Algorithm 1 then iteratively selects a search zone in order to decide whether it is feasible and hence contains further nondominated points–in this case, such a nondominated point has to be computed–or whether it is infeasible and can be excluded from further consideration. This is realized by formulating an appropriate single-objective subproblem. For search zones with a simple structure as in the case of the natural decomposition, this can be realized by appropriate scalarizations , see, for example, Ehrgott ( 2006 ) for a survey in the context of multiobjective integer programming problems. In the following, we focus on \(\varepsilon \) -constraint scalarizations and weighted Tchebychev scalarizations since they are particularly well suited in this setting.

2.2.1 \(\varepsilon \) -constraint scalarization

The most commonly used scalarization in this context is the \(\varepsilon \) - constraint scalarization : Given a search zone C ( u ) with local upper bound u , the associated \(\varepsilon \) -constraint subproblem w.r.t. the i th objective, \(i\in \{1,\dots ,p\}\) , is given by

where \(J_i:=\{1,\dots ,p\}\setminus \{i\}\) denotes the index set of all non-selected objective functions. Figure  4 a illustrates the additional constraints together with an exemplary level curve of the objective function. The following lemma illuminates the role of the defining points of the local upper bound u for the associated \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_i(u)\) ). We refer to Tamby and Vanderpooten ( 2020 ) for a similar result.

Let C ( u ) be a search zone from the natural decomposition of the search region, let \(i\in \{1,\dots ,p\}\) be a selected objective function such that \(u_i\) is not defined by a dummy point, and let \(z^i\) be a defining point for the i -th component of u with pre-image \(x^i\in X\) . Then it holds:

\(x^i\) is a feasible solution for the \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_i(u)\) ).

If \(x^i\) is an optimal solution of ( \(P^{\varepsilon }_i(u)\) ), then the search zone C ( u ) does not contain any further nondominated points.

Since \(z^i\in N\setminus \{d^1,\dots ,d^{p}\}\) is a defining point of u we have that \(z^i_j< u_j\) for all \(j\in J_i\) . Since in addition \(z^i=Cx^i\) with \(x^i\in X\) , it follows that \(z^i\in \mathbb {Z}^{p}\) and hence \(z^i_j=c^jx^i\le u_j-1\) for all \(j\in J_i\) . Thus, \(x^i\) is feasible for problem ( \(P^{\varepsilon }_i(u)\) ).

Now let \(x^i\) be an optimal solution of problem ( \(P^{\varepsilon }_i(u)\) ). Then there is no \({\bar{x}}\in X\) with \(c^j{\bar{x}} < u_j\) , \(j\in J_i\) (recall that \(c^j{\bar{x}}\) is integral for all \(j\in \{1,\dots ,p\}\) ) and \(c^i{\bar{x}}<z^i_i=u_i\) , and hence \(Z\cap C(u)=\emptyset \) . \(\square \)

If the \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_i(u)\) ) is solved by an IP-solver that can be sped up when providing a feasible starting solution, then Lemma  1 suggests to select a minimization objective \(f_i\) , \(i\in \{1,\dots ,p\}\) , such that the corresponding defining point \(z^i\) is not a dummy point since then a corresponding pre-image \(x^i\in X\) with \(Cx^i=z^i\) is already known and can be used as a feasible starting solution when solving problem ( \(P^{\varepsilon }_i(u)\) ). Moreover, problem ( \(P^{\varepsilon }_i(u)\) ) is then feasible which usually has a positive effect on the computational time (as compared to infeasible subproblems).

The following properties are well-known; we refer to the textbook (Ehrgott 2005 ) for further details.

Every optimal solution \({\bar{x}}\in X\) of an \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_i(u)\) ) is weakly efficient for MOIP , and at least one of the optimal solutions of ( \(P^{\varepsilon }_i(u)\) ) is efficient for  MOIP .

When combining the results of Lemmas  1 and 2 we can conclude that if an \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_i(u)\) ) returns an optimal solution \({\bar{x}}\in X\) with \(c^i{\bar{x}}<z^i_i\) , then this solution is in the search zone C ( u ) and it is at least weakly efficient for ( \(P^{\varepsilon }_i(u)\) ). In this case, \(Z_N\cap C(u)\ne \emptyset \) and a nondominated point in C ( u ) can be found by, for example, performing a lexicographic optimization over the set of optimal solutions of problem ( \(P^{\varepsilon }_i(u)\) ). The latter can be realized, e.g., by solving a second-stage problem

over the set of optimal solutions of problem ( \(P^{\varepsilon }_i(u)\) ). Note that a feasible starting solution is readily available by the optimal solution from the first stage problem.

Note that this two-stage approach requires two solver calls whenever a search zone contains further nondominated points. This can be avoided by adding an appropriate augmentation term to problem ( \(P^{\varepsilon }_i(u)\) ), leading to an augmented \(\varepsilon \) - constraint scalarization

which returns an efficient solution of  MOIP . Here, \(\rho >0\) is an appropriately chosen small augmentation parameter . We refer to Ehrgott and Ruzika ( 2008 ) for a detailed discussion on augmented \(\varepsilon \) -constraint scalarizations and appropriate choices of \(\rho \) .

Since an optimal solution \({\bar{x}}\) of the \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_i(u)\) ) minimizes the selected objective function \(f_i\) in the search zone C ( u ), there can not be any further nondominated points in C ( u ) that have an even better value in objective \(f_i\) . As a consequence, when updating the decomposition w.r.t. the new point \({\bar{z}}=C{\bar{x}}\) , search zones \(C({\bar{u}})\) with \({\bar{u}}\le u\) and \({\bar{u}}_i\le {\bar{z}}_i\) are known to be empty and can be removed from the search region without further consideration. Depending on the applied decomposition, this fact usually results in a reduction of solver calls and is thus an advantage of using \(\varepsilon \) -constraint scalarizations. In the example depicted in Fig.  3 , \(u^{11} \le u^1\) and \(u^{11}_1 \le z^3_1\) , hence, \(u^{11}\) can be removed without further inspection. We discuss the realization of such a reduction in detail in Sect.  3 below.

a The 3d situation from Fig.  2 b before solving the next subproblem; b 2-dimensional projection of the \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_1(u^1)\) ) in \(C(u^1)\) minimizing \(f_1\) with feasible set constrained by \(c^jx\le u^1_j-1\) , \(j=2,3\) (shaded) and level curve associated to an optimal point \(z^3=(3,4,8)\) (dashed) for the problem shown in a ; c 3d situation with updated local upper bounds for the same problem after adding the newly generated point \(z^3=(3,4,8)\) to the set N

2.2.2 Weighted Tchebychev scalarization

The weighted Tchebychev scalarization minimizes a weighted Tchebychev distance \(l_{\infty }^w(z,z^r)=\max _{j=1,\dots ,p} |w_j(z_j-z_j^r )|\) between a reference point \(z^r\) and the closest feasible point \(z\in Z\) in the objective space.

In order to determine whether a given search zone C ( u ) with \(u\in U(N)\) contains further nondominated points, a natural choice for the reference point is the origin, i.e., \(z^r=0\) . The weights \(w_j>0\) , \(j=1,\dots ,p\) of the weighted Tchebychev distance are then set such that the level set of level 1 passes through all defining points of u and has its upper right corner in u , see Fig.  4 b for an illustration. This yields \(w_j=\frac{1}{u_j}\) , \(j=1,\dots ,p\) . Note that we assume wlog that \(u>0\) for all \(u\in U(N)\) so that these weights are well-defined. Since we assume that the bounding box is contained in \(\mathbb {R}^{p}_\geqq \) we can omit all absolute values and formulate the weighted Tchebychev scalarization w.r.t. the search zone C ( u ) as

Note that the second, constrained reformulation can be immediately passed to a standard IP-solver.

Let C ( u ) be a search zone from the natural decomposition of the search region. Then it holds:

Every pre-image \(x^i\in X\) of a defining point \(z^i\) of u , \(i\in \{1,\dots ,p\}\) , that is not a dummy point is a feasible solution for the weighted Tchebychev scalarization ( \(P^{T}(u)\) ) with objective value 1.

If the optimal objective value of problem ( \(P^{T}(u)\) ) is 1, then the search zone C ( u ) does not contain any further nondominated points.

Since every solution in X is feasible for problem ( \(P^{T}(u)\) ) this is also true for every pre-image \(x^i\) of a defining point \(z^i\) of u , \(i\in \{1,\dots ,p\}\) . Since \(z^i_i=c^ix^i=u_i\) and \(z^i_j=c^jx^i<u_j\) for all \(j\in J_i\) it follows that \(\max _{j=1,\dots ,p} w_j\, c^jx^i = \max _{j=1,\dots ,p} \frac{c^jx^i}{u_j}=\frac{c^ix^i}{u_i}=1\) .

Now suppose that the optimal objective value of problem ( \(P^{T}(u)\) ) is 1. Then there is no solution \({\bar{x}}\in X\) that satisfies \(c^jx<u_j\) for all \(j=1,\dots ,p\) , and hence \(Z\cap C(u)=\emptyset \) . \(\square \)

(a) \(\varepsilon \) -constraint scalarization in \(C(u^1)\) minimizing \(f_1\) with feasible set constrained by \(f_2(x)=c^2x\le u^1_2-1\) (shaded) and level curve (dashed) through \(z^1\) , which can be used as feasible starting solution; (b) weighted Tchebychev scalarization in \(C(u^1)\) with feasible set (shaded) and level curve (dashed) passing through \(u^1\) , using also \(z^1\) as feasible starting solution

Note that different from the \(\varepsilon \) -constraint scalarization ( \(P^{\varepsilon }_i(u)\) ), all previously determined solutions, and hence in particular all defining points of the considered search zone or its upper bound, respectively, that are not dummy points, can be used as (relatively good) feasible starting solutions for the weighted Tchebychev scalarization ( \(P^{T}(u)\) ).

In general, the theoretical properties of the weighted Tchebychev scalarization and of the \(\varepsilon \) -constraint scalarization are very similar. We refer again to the textbook (Ehrgott 2005 ) for the following result.

Every optimal solution \({\bar{x}}\in X\) of a weighted Tchebychev scalarization ( \(P^{T}(u)\) ) is weakly efficient for ( MOIP ), and at least one of the optimal solutions of ( \(P^{T}(u)\) ) is efficient for ( MOIP ).

Similar to the case of the \(\varepsilon \) -constraint scalarization, weakly efficient solutions can be avoided by solving a second-stage problem

over the set of optimal solutions of problem ( \(P^{T}(u)\) ), where \({\bar{\alpha }}\) denotes the optimal objective value of problem ( \(P^{T}(u)\) ). Again, a feasible starting solution for this second stage problem is known from solving the first stage problem.

Rather than solving two IP-problems, the original problem can also be extended by an augmentation term to the objective, yielding an augmented weighted Tchebychev scalarization

We emphasize that the augmentation parameter \(\rho >0\) has to be chosen carefully. Indeed, while \(\rho \) has to be chosen small enough such that no efficient solution is missed, an overly small choice of \(\rho \) may lead to numerical problems. A detailed analysis with a concrete suggestion for an “optimal” choice of the augmentation parameter when \(p=2\) is given in Dächert et al. ( 2012 ).

Figure  4 illustrates the additional constraints added to the problem in the respective scalarizations ( \(\varepsilon \) -constraint versus weighted Tchebychev). In addition, exemplary level curves of the respective objective functions are shown.

3 Reduction of the search region when using the \(\varepsilon \) -constraint method

If we use the \(\varepsilon \) -constraint scalarization in Line  3 of Algorithm 1, we can reduce the search region further. This is due to a specific property of the the \(\varepsilon \) -constraint scalarization which allows to remove one newly created search zone directly in every iteration in which a new nondominated point is found. However, to make use of this additional reduction, we cannot select the zone in which we search for a new nondominated point arbitrarily but have to select it according to some criterion. Note that the idea itself is not new but has already been elaborated in Dächert and Klamroth ( 2015 ) for the tricriteria case.

In the following we present a selection criterion that is valid for any number of criteria. We prove that this selection criterion allows to reduce the total number of evaluated search zones by the number of nondominated points. In the proof we make use of certain results from Dächert et al. ( 2017 ), namely Lemma 3.8 and Proposition 4.3, which we recall below before stating and proving the selection criterion. They use both the notion of a neighbor from Definition 3.6 in Dächert et al. ( 2017 ) which we present in a shorter form below.

Definition 1

[Neighbor] Two local upper bounds \(u,u' \in U(N)\) are neighbors if they share \(p-1\) defining points, among which exactly one changes its position from some \(j \in \{1,\dots ,p\}\) to some \(k \in \{1,\dots ,p\}, k \ne j\) . That is, there are two indices j ,  k with \(j \ne k\) such that \(z^j(u) = z^k(u')\) while \(z^i(u) = z^i(u'\) ) for all \(i \in \{1,..., p\}\backslash \{j,k\) }. We then say that u is a j -neighbor of \(u'\) , and that \(u'\) is a k -neighbor of u .

(Dächert et al. 2017 ) Let \(u, u' \in U(N)\) be local upper bounds such that \(u'\) is a k -neighbor of u , and u is a j -neighbor of \(u'\) . Then, we have \(u_j < u'_j\) , \(u_k > u'_k\) and \(u_i=u'_i\) for \(i \ne j,k\) .

Proposition 1

(Dächert et al. 2017 ) Let \(u \in U_{{\bar{z}}} = \{ u' \in U(N): {\bar{z}} < u' \}\) . Then, for every \(k=1,\dots , p\) , one of the following two assertions holds exclusively:

\(u^k \in U(N \cup \{ {\bar{z}} \} )\) , where \(u^k\) denotes the k -child of u

\(u' \in U_{{\bar{z}}}\) , where \(u'\) is the k -neighbor of u .

(Min-component selection criterion) If, in every iteration of Algorithm  1 , we select a search zone \({\bar{u}}\) which is minimal in some component \(i \in \{1,\dots ,p\}\) , i.e., for which

for some \(i \in \{1,\dots ,p\}\) holds, and if we then solve an \(\varepsilon \) -constraint problem of the form \(P^{\varepsilon ,2}_i({\bar{u}})\) or \(P^{\varepsilon ,\rho }_i({\bar{u}})\) , then, whenever there is a solution \({\bar{z}} \in Z_N\) with \({\bar{z}} < {\bar{u}}\) , the i -child of \({\bar{u}}\) exists, i.e., \({\bar{u}}^i \in U(N \cup \{ {\bar{z}} \})\) . Moreover, \({\bar{u}}^i\) can not contain further nondominated points, i.e., \(\{ z \in Z_N: z < {\bar{u}}^i \}= \emptyset \) .

We first show that \({\bar{u}}^i \in U(N \cup \{ {\bar{z}} \})\) . Since \({\bar{z}} < {\bar{u}}\) , using the notation of Proposition  1 , \({\bar{u}} \in U_{{\bar{z}}}\) holds.

Assume that \({\bar{u}}^i \notin U(N \cup \{ {\bar{z}} \})\) . Then, due to Proposition  1 , the i -neighbor \(u'\) of \({\bar{u}}\) contains \({\bar{z}}\) . According to Lemma  5 , \(u'_i < {\bar{u}}_i\) holds for the i -neighbor \(u'\) of \({\bar{u}}\) . However, this is a contradiction to \({\bar{u}}\) being minimal with respect to component  i . Hence, \({\bar{u}}^i \in U(N \cup \{ {\bar{z}} \})\) must hold.

It remains to show that \({\bar{u}}^i\) does not contain further nondominated points. Assume that there is a \(z' \in Z_N\) with \(z' < {\bar{u}}^i\) . Then, in particular, \(z'_i < {\bar{u}}^i_i = {\bar{z}}_i\) would hold. This, however, is a contradiction to \({\bar{z}}\) being optimal for the \(\varepsilon \) -constraint problem \(P^{\varepsilon ,2}_i({\bar{u}})\) or \(P^{\varepsilon ,\rho }_i({\bar{u}})\) . \(\square \)

Thanks to Theorem  2 we can remove one search zone per iteration, in which a new nondominated point is detected, when using the \(\varepsilon \) -constraint scalarization. The presented selection criterion is simple but not the only possible one. Another one is presented as a heuristic in Tamby and Vanderpooten ( 2020 ) (see their Section 3.3). In the following we formally prove that also their criterion yields the desired result of avoiding to solve unnecessary scalarizations.

Tamby and Vanderpooten ( 2020 ) select a search zone \(u^*\) by computing

and \(z^I\) denotes the ideal point, see ( 1 ). The index \(i^*\) serves to make use of starting solutions. If we ignore this additional feature, their selection criterion for a fixed index i reads

for a certain \(i \in \{1,\dots ,p\}\) . Note that, except in the first iteration, a product consisting of \(p-1\) terms has to be computed for every search zone with \(u_i \ne M\) to find the search zone with the largest projected hypervolume.

[Max-projected-volume selection criterion] If, in every iteration of Algorithm 1, we select a search zone \({\bar{u}}\) according to ( 8 ), and if we then solve an \(\varepsilon \) -constraint problem of the form \(P^{\varepsilon ,2}_i({\bar{u}})\) or \(P^{\varepsilon ,\rho }_i({\bar{u}})\) , then, whenever there is a solution \({\bar{z}} \in Z_N\) with \({\bar{z}} < {\bar{u}}\) , the i -child of \({\bar{u}}\) exists, i.e., \({\bar{u}}^i \in U(N \cup \{ {\bar{z}} \})\) . Moreover, \({\bar{u}}^i\) can not contain further nondominated points, i.e., \(\{ z \in Z_N: z < {\bar{u}}^i \}= \emptyset \) .

The proof is similar to the proof of Theorem  2 . Assume that \({\bar{u}}^i \notin U(N \cup \{ {\bar{z}} \})\) . Then, due to Proposition  1 , the i -neighbor \(u'\) of \({\bar{u}}\) contains \({\bar{z}}\) . According to Lemma  5 , we then have \(u'_i < {\bar{u}}_i\) , \(u'_k > {\bar{u}}_k\) for some \(k \ne i\) and \(u'_j={\bar{u}}_j\) for \(j \ne i,k\) . However, then

would hold, a contradiction to \({\bar{u}}\) satisfying ( 8 ). Hence, \({\bar{u}}^i \in U(N \cup \{ {\bar{z}} \})\) must hold.

The rest of the proof is the same as for Theorem  2 . \(\square \)

Theorem  3 demonstrates that the heuristic selection criterion of Tamby and Vanderpooten ( 2020 ) has indeed the same beneficial property as our selection criterion ( 6 ). However, ours is much simpler, since it is solely based on one component value of a search zone, an information that is available anyway. It does not require any computation of additional figures as, e.g., the projected hypervolumes computed in Tamby and Vanderpooten ( 2020 ).

As done in ( 7 ), we could additionally choose the index  i which then determines the index to be optimized in the \(\varepsilon \) -constraint problem. Then, previously generated nondominated points can be used as starting solutions as discussed in Lemma  1 .

4 Search strategies in objective space methods

Objective space methods can be distinguished w.r.t. the way in which they search for nondominated outcome vectors in the objective space and, in particular, in the search region. While the previous sections focused on a formal description of the search region and its decomposition into (rectangular) search zones, which naturally implies decomposition-based methods, we will also review alternative search strategies and provide a brief summary of some of the most successful objective space methods. We classify these methods into three categories: Decomposition, recursive dimension reduction and disjunctive constraints. However, some methods can not be assigned clearly to one of these categories, wherefore we also discuss hybrid approaches in the end.

4.1 Decomposition

The concept of decomposition relies on the idea that the search region can be described by the union of rectangular sets which in turn can be described based on already generated nondominated points. In every iteration, one of these rectangular sets can be investigated which results in a rather simple IP to be solved. A natural and efficient decomposition of the search region was described in Sect.  2.1 above. However, there are different ways of decomposing the search region which will be briefly reviewed in the following.

One way consists in subdividing the search region into disjoint cells each of which is defined by a lower and upper bound vector. When a new nondominated point is generated, its components are inserted as axis-parallel “hyper-cuts” to all (concerned) cells. Dominated cells can be deleted immediately. Nevertheless, the number of remaining cells grows quickly. The union of all disjoint cells represents the search region. This idea is used in Laumanns et al. ( 2006 , 2005 ), and Kirlik and Sayin ( 2014 ). The performance of an algorithm based on such a disjoint decomposition highly depends on the order in which the cells are investigated. Kirlik and Sayin ( 2014 ) select the box with the largest volume, where the volume is defined by the upper bound of the cell and the ideal point. Thereby they can avoid the generation of dominated points. However, without further enhancement strategies, it might happen that a nondominated point is computed more than once.

Other algorithms use a non-disjoint decomposition which leads to a smaller number of boxes to be saved and investigated. In this case, the boxes are usually only defined by upper bound vectors. Przybylski et al. ( 2010 ) use the idea of upper bounds in the second phase of their algorithm. They remove redundant upper bounds by a filtering step in every iteration. Lokman and Köksalan ( 2013 ) also implicitly use a sort of a non-disjoint decomposition by varying the right-hand side vectors of the constraints on the objective vectors in a systematic way based on all already generated nondominated points. Dächert and Klamroth ( 2015 ) propose a non-disjoint decomposition for tricriteria optimization problems which results in a number of upper bound vectors that depends linearly on the number of nondominated points. Klamroth et al. ( 2015 ) present a constructive algorithm that generates the set of local upper bound vectors based on a stable set of points, i.e., mutually nondominated points. The main idea is to use this stable set of points as defining points and to formulate a criterion which guarantees that the number of local upper bounds is minimal. This method will be referred to as the defining point algorithm (DPA) in Sect.  5 below. Another non-disjoint decomposition is proposed in Dächert et al. ( 2017 ) that is based on the inherent neighborhood structure between nondominated points and local upper bounds. Different to most other recursive and decomposition approaches, Dächert and Klamroth ( 2015 ), Klamroth et al. ( 2015 ) and Dächert et al. ( 2017 ) can guarantee that each nondominated point is only generated once during the algorithm which implies that no strategy based on the examination of all previously generated nondominated points is needed. Moreover, the DPA approach is not restricted to the application of \(\varepsilon \) -constraint scalarizations for the solution of subproblems. A specific implementation of the DPA approach is presented in Tamby and Vanderpooten ( 2020 ). It uses the particular properties of \(\varepsilon \) -constraint scalarizations in combination with a strategy for choosing the next search zone, which allows to use feasible starting solutions for the next IP to be solved. Holzmann and Smith ( 2018 ) suggest to use modified augmented weighted Tchebychev scalarizations to solve the subproblems, both in combination with simple non-disjoint decompositions including redundancies, and in combination with the DPA approach. Numerical tests confirm the clear superiority of the non-redundant decomposition from DPA.

4.2 Recursive dimension reduction

The idea of recursion is to reduce the dimension of the objective space (and hence the search region) until single criterion or bicriteria (sub-)problems are obtained. The results of lower level formulations are then used as bounds in the next higher level problem in a systematic way.

There are different ways how to organize the recursion. One way consists in selecting two objectives (typically the first and the second) and computing all points that are nondominated with respect to these two objectives. All other objectives are bounded from above, where the upper bounds are narrowed iteratively in every level of recursion. This is the basic idea of Chalmet et al. ( 1986 ), Özlen and Azizoğlu ( 2009 ) and Özlen et al. ( 2014 ).

Others also compute all possible combinations of recursions. Given a multiobjective optimization problem with \({p}\) objectives they consider all \({p}\) corresponding \(({p}-1)\) -dimensional optimization problems and so on. Therefore, a tree is created with bicriteria optimization problems in its leaf nodes. It is shown in Ehrgott and Tenfelde-Podehl ( 2003 ) that the points obtained by solving all \(({p}-1)\) -criteria problems represent a subset of the nondominated set of the original problem. This way of recursion has been used, e.g., in Tenfelde-Podehl ( 2003 ) and as a subprocedure in Dhaenens et al. ( 2010 ) and Przybylski et al. ( 2010 ).

A drawback of recursive algorithms is the fact that nondominated points are typically computed several times since they are often optimal for multiple recursive subproblems. Since the solution of IPs is costly, it is critical for the performance to avoid the repeated generation of already known nondominated points. In order to deal with this issue the information from all problems solved before can be stored. More precisely, this requires to save the right-hand side values of the bounded objectives as well as the result of the corresponding optimization run, i.e., the optimal solution of the problem or the information that it is infeasible. Before solving a new problem, the list of already solved problems is scanned. This idea is implemented in Özlen et al. ( 2014 ) and improved the number of solved IPs drastically.

4.3 Disjunctive constraints

Using disjunctive constraints is a general concept in optimization that is tailored for the case that the feasible set is non-convex. Typically, artificial binary variables are used to activate or deactivate parts of the feasible set. We refer, e.g., to the textbook Nemhauser and Wolsey ( 1999 ) for the general concept.

In the context of multiobjective optimization problems this concept is useful to describe the search region, which is a non-convex set. It is then possible to consider all remaining parts of the search region (that are not already known to be empty) simultaneously, i.e., to search for a new nondominated point by solving one single IP formulated over the complete (non-convex) search region. In this case, every nondominated point is computed exactly once. However, with every new nondominated point, the number of constraints and artificial binary variables increases which makes this approach computationally demanding and hardly ever competitive. Disjunctive constraints are used in Klein and Hannan ( 1982 ), Sylva and Crema ( 2004 ) and Lokman and Köksalan ( 2013 ).

4.4 Hybrid approaches

There are also hybrid methods that combine ideas from the three categories mentioned above. Boland et al. ( 2016 ) propose a method called L -shape search method for tricriteria problems that is a hybrid of disjunctive constraints and decomposition. The idea is to use disjunctive constraints only with respect to the lastly generated nondominated point which results in an L -shape element that is investigated with priority, i.e., shrunk quickly towards the ideal point. During the procedure, unexplored rectangular sets are saved and investigated in the later phase of the algorithm. Boland et al. ( 2017a ) extend the L -shape search method to any number of objectives. The approach of Boland et al. ( 2017b ) for tricriteria optimization problems can be seen as a hybrid of recursion and decomposition. It deals with upper bound vectors similar to other decomposition approaches. However, for a certain sequence of problems to be solved it keeps the bound on one of the objectives fixed as in a recursive method.

Bektaş ( 2018 ) presents a hybrid approach for multiobjective IPs that combines disjunctive constraints, inspired by Klein and Hannan ( 1982 ) and Sylva and Crema ( 2004 ), with decomposition. The disjunctive single-objective IPs use fewer variables than the original approaches, and the number of solver-calls (i.e., single-objective IPs solved) is reduced as compared to pure decomposition methods. However, the higher complexity of the individual solver calls still counteracts the speed-up obtained from solving fewer IPs.

5 Numerical study

For our numerical study we provide an implementation of the defining point algorithm (DPA) in C++ . The source files as well as our own test files can be downloaded from https://github.com/kerstindaechert/DefiningPointAlgorithm . For the test files taken from other authors we refer to the respective urls. We compare our implementation to three state-of-the-art algorithms for which open source code is available and which are also implemented in C++ . Moreover, all three methods invoke the commercial solver CPLEX to solve the single-objective scalarizations. Note that the respective implementations are of different types according to the classification shown in Sect.  4 : disjoint decomposition (method “Epsilon” by Kirlik and Sayin 2014 ), recursive dimension reduction (method “AIRA” by Özlen et al. 2014 ) and a hybrid of disjunctive constraints and decomposition (method “DCM” by Boland et al. 2017a ), supplemented by our “DPA ” implementation which is a pure non-disjoint and non-redundant decomposition method.

We acknowledge that there are several other objective space methods that could be used for further comparisons, such as, for example, Bektaş ( 2018 ), Holzmann and Smith ( 2018 ), Tamby and Vanderpooten ( 2020 ). Our comparison are focused on the above methods Epsilon, AIRA and DCM for the following reasons: (1) C++ implementations are available for these methods, and they can be combined with the same versions of single-objective solvers ( CPLEX in our case). This leads to a fair comparison and avoids bias resulting from translations from other programming languages and/or re-implementations of existing code in C++ . (2) The selected methods are representative in the sense that they cover different variants of objective space methods as discussed in Sect.  4 above. (3) In particular, Epsilon, the disjoint decomposition method of Kirlik and Sayin ( 2014 ), is used in a majority of all comparative studies on objective space methods. It can thus be seen as a general reference and may provide information on the relative performance w.r.t. other methods that are not included in this study.

In Sect.  5.1 we present our implementation in more detail. In Sect.  5.2 we state the benchmark algorithms with their current sources. We present the test instances in Sect.  5.3 , followed by the discussion of the numerical results in Sect.  5.4 .

5.1 Implementation

We implement the defining point algorithm (DPA) in C++ . It is based on the original implementation of Klamroth et al. ( 2015 ) in C which has been designed to find and insert discrete sets of (nondominated) points. Hence, our main adjustment besides the translation to C++ is to make it applicable to solve multiobjective integer programming problems. To do so we add an interface to the commercial solver CPLEX to solve \(\varepsilon \) -constraint scalarizations. Moreover, we implement the enhancement discussed in Sect.  3 .

Algorithm 2 shows the pseudocode of our implementation. Note that it has the same structure as Algorithm 1, the Generic Scalarization-Based Algorithm. We read the MOIP problem from a textfile and transform it into minimization format if needed. In ObtainBounds we compute the ideal point \(z^I\) as well as a global upper bound \(z^M\) by minimizing or maximizing each objective function individually, respectively. The values of \(z^M\) incremented by 1 are used to define the upper bound u of the initial box. We initialize the list of boxes U by appending the initial box. As long as U is not empty, we select a box in SelectBoxToRefine . While it is possible to select any box here, in order to make use of Theorem  2 , we select a box with minimal value \(u_i\) according to ( 6 ), where the index  i has been specified by the user and is part of the Input of Algorithm 2.

Defining Point Algorithm with \(\varepsilon \) -constraint scalarization

We then solve a scalarization, more precisely a two-stage or augmented \(\varepsilon \) -constraint method as presented in Sect.  2.2 . We call the resulting variants DPA-TS and DPA-A, respectively. Note that we do not make use of the option of feeding the solver with a feasible starting solution. However, the defining point of the selected local upper bound \({\bar{u}}\) in component i is typically a feasible solution for ( \(P^{\varepsilon }_i(u)\) ) and ( \(P^{\varepsilon ,\rho }_i(u)\) ) and might be the result of the scalarization when the box is empty. Hence, in case of feasibility, we additionally check whether \(z^s_i < {\bar{u}}_i\) . We append \(z^s\) only to the list N if the inequality is satisfied, otherwise we remove \({\bar{u}}\) from U .

In case a new nondominated point has been found, the list of local upper bounds is updated in UpdateList \((U, z^s)\) which is basically a re-implementation of Algorithm 5 of Klamroth et al. ( 2015 ), however, without generating the i -child of \({\bar{u}}\) which yields an empty search zone according to Theorem  2 . For the pseudocode and justification of this routine we refer to Klamroth et al. ( 2015 ).

5.2 Benchmark algorithms

In our numerical study we take into account recently proposed algorithms that offer an open-source implementation in C++ and invoke the commercial solver CPLEX to solve scalarizations. To the best of our knowledge the following algorithms satisfy these requirements:

Epsilon (Kirlik and Sayin 2014 ): http://home.ku.edu.tr/~moolibrary/

AIRA (Özlen et al. 2014 , Pettersson and Ozlen 2019 ): An implementation in C is available at https://bitbucket.org/melihozlen/moip_aira (last update 2017). A parallelized version of this code in C++ is available at https://github.com/WPettersson/moip_aira . It can also be used in a non-parallel fashion, hence, we used this implementation for our comparison.

DCM ( Boland et al. ( 2017a ): http://www.eng.usf.edu/~hcharkhgard/

5.3 Instances

For problems with three to five objectives we use publicly available benchmark instances of knapsack, assignment and travelling salesman instances. For our tests with up to ten objectives we create new instances of knapsack and assignment problems.

5.3.1 Knapsack problem

We use knapsack instances from http://home.ku.edu.tr/~moolibrary/ which have been used as a benchmark in Kirlik and Sayin ( 2014 ) and are still used for comparison in recent publications. These instances include problems with three objectives and \(10, 20, \dots , 100\) items, problems with four objectives and 10, 20, 30, 40 items as well as problems with five objectives and 10 and 20 items. For each problem size there are 10 instances, so in total this set consists of 160 test files.

In order to create instances with six to ten criteria we used the test problem creator from https://github.com/wpettersson/ProblemGenerator .

5.3.2 Assignment problem

Assignment instances are also available at http://home.ku.edu.tr/~moolibrary/ . These instances include problems with three objectives and \(5, 10, \dots , 50\) items. Again, for each problem size there are 10 instances.

We also created instances with more criteria with the help of the test problem creator from https://github.com/wpettersson/ProblemGenerator .

5.3.3 Travelling salesman problem

We use instances with three and four objectives from Pettersson and Ozlen ( 2019 ). Thereby, the problems with three objectives are available under the direct link https://figshare.com/articles/3_objective_problem_test_cases/4814695 .   The problems with four objectives are available here: https://figshare.com/articles/4_objective_problem_test_cases/4814698 .

5.4 Numerical results

All algorithms are implemented in C++ and compiled for the Microsoft Windows 64 Bit operation system. For solving the optimization problems, the IBM CPLEX Optimizer 12.10 solver is used. The comparisons were executed on an Intel Core i7–7500U @2.90 GHz with 16GB of RAM, which is a rather slow computer in comparison to most popular processors. However, it demonstrates that the described algorithms also run on personal laptops and do not require high-performance compute clusters. All algorithms generate a textfile as output that contains the generated nondominated points. We compare these output files to make sure that all algorithms work correctly.

Some of the compared algorithms measure the runtime when executing, others do not. Since we do not want to alter the algorithms, we measure runtime as the time between calling the main routine until its termination. In order to avoid excessive runtimes we use the following timeout rule for all knapsack and assignment instances: We measure the runtime of DPA-TS and use it as a reference. The maximum runtime for all other algorithms is at least 30 minutes and at most ten times the reference runtime. For the TSP instances we reduce the upper bound of the timeout to three (instead of ten) times the runtime of DPA-TS to obtain shorter overall running times. In order to obtain stable run times, we run every benchmark instance twice and compute the average runtime.

5.4.1 Knapsack instances from Kirlik and Sayin ( 2014 )

We first study the knapsack instances with three to five objectives in Tables  1 , 2 , 3 and 4 . Thereby, Table  1 shows CPU times, Table  2 the number of calls to CPLEX, thus, the number of solved IPs. For better comparison, the following two Tables  3 and  4 show the same results but this time relative w.r.t. the best performing algorithm per problem, which means relative CPU time with respect to the fastest and the relative number of IP calls with respect to the one with the smallest number of IPs. Note that since we are showing average figures, it might happen that none of the algorithms reaches a value of 1 which means that it is not superior in all of the 10 problems.

Considering CPU times we note that DPA outperforms the three benchmark algorithms Epsilon, DCM and AIRA considerably, especially for large instance sizes. With growing problem size, the superiority of DPA becomes more pronounced. For example, for \(p=3\) and \(n=100\) , AIRA requires a relative runtime of 2.17, EPS a relative runtime of 2.64 and DCM a relative runtime of even 7.22 compared to DPA-A. Expressed in absolute terms, this means a runtime of 758.72 s for DPA-A, 1613.51 s for AIRA, 2220.52 s for EPS and 5753.29 s for DCM.

Note that EPS ran out of the given time limit of ten times the runtime of DPA-TS in 13 of the 160 problems, all of them from instances 4–30, 4–40 and 5–20. To some extent this matches the results presented in Kirlik and Sayin ( 2014 ), where for 4–40 only five of the ten problems could be solved within a time limit of 25000 s of CPU time. AIRA and DCM solve all problems reliably.

Comparing DPA within its two variants, DPA-A is fastest in most of the instances. The two-stage variant DPA-TS requires between 1.12 up to 1.43 times longer CPU times. This goes in line with the fact that the two-stage variants solve one IP more per nondominated point compared to the augmented variant.

Comparing the number of CPLEX calls in Tables  2 and  4 , DCM solves less IPs than all the other algorithms for the small instances. This is not surprising since DCM uses disjunctive programs to search multiple rectangular boxes at once. This reduces the number of CPLEX calls on the one hand, but yields more complicated models involving additional binary variables on the other hand. As can be seen from the CPU times, the strategy of solving less but more complicated problems does not pay off. Even if DPA-A solves much more IPs in some instances, it requires considerably less CPU time.

5.4.2 Assignment instances from Kirlik and Sayin ( 2014 )

The results obtained for the benchmark assignment instances with three objectives are given in Tables  5 , 6 , 7 , and  8 . Again, we depict absolute CPU times and number of solved IPs as well as their relative conterparts. While the average number of nondominated points of these instances is similar to those of the knapsack instances, it is common knowledge that assignment problems are much harder to solve.

With respect to CPU time, the DPA algorithms perform best again. For instance 3–50 EPS requires 1.69 more runtime than DPA-A, followed by AIRA with a factor of 1.97 and DCM with a factor of 10.26. While DCM does not only consume much more CPU time than all other algorithms for this instance, it also does not find all nondominated points reliably. More precisely, for the ten problems contained in 3–50, it misses on average around \(10\%\) of the nondominated set.

With respect to the number of IPs, DPA-A solves the smallest number in all instances, followed by DCM, DPA-TS, EPS and AIRA, respectively.

5.4.3 Travelling salesman instances from Pettersson and Ozlen ( 2019 )

As a third test bed we use travelling salesman instances with three and four objectives. Note that each problem size consists of five instances. Moreover, we set the timeout here to at most three times the runtime of DPA-TS (but again at least 30 min). The results are shown in Tables  9 , 10 , 11 , and 12 . The results are in line with the results for knapsack and assignment instances, and, thus, confirm the general behavior for three and four objectives: DPA-A is fastest in nearly all instances, followed by DPA-TS. The behavior of the three remaining algorithms depends on the problem size. For smaller problems, EPS is faster than DCM, which in turn is faster than AIRA. At some problem size, AIRA gets better than DCM. For large problems, AIRA outperforms DCM and EPS.

5.4.4 High-dimensional instances

Finally we investigate how the algorithms perform when applied to instances with up to 10 objectives. To the best of our knowledge, we are the first to present results for problems with more than six objectives by exact algorithms. Since the computational effort increases considerably with an increasing number of objectives, we restrict the number of knapsack items (variables) to at most 15. For the assignment instances with six to ten objectives we present results for 5 items, i.e., 25 variables. Moreover, we evaluate only five or three problems per instance.

The results of the knapsack instances are shown in Tables  13 and  14 . First we state that both variants, DPA-TS and DPA-A, solve all instances reliably. Moreover, we observe that the main difference between both variants, the saving of \(|Z_N|\) integer programs when using the augmented variant, looses importance with increasing number of objectives. This also translates to the CPU times. Considering the results of EPS we observe that it can only solve the smaller instances within the given time frame. While, e.g., DPA-TS solves instance 7–10 with on average 81.8 nondominated points within 34.67 s, EPS can not provide the nondominated set within half an hour. The smaller instances 6–5, 6–10, 7–5, 8–5 and 9–5 can be solved by EPS, but in all but one instance EPS is the slowest algorithm. AIRA also shows difficulties with the larger instances. In general, it performs better than EPS (except for instance 6–5) but worse than the DPA variants and DCM. DCM outperforms the DPA variants with respect to CPU times for the smaller instances 6–5, 7–5, 8–5, 9–5 and 10–5, for which the solution time is less than 1 s or slightly above. However, the DPA variants show their superiority in all larger instances. DCM can only partially solve these instances. The average figures shown in Table  13 only take the solved problems into account. Even for these, the average CPU times are, in general, higher than those of the DPA variants. Looking at the number of IPs, DCM solves significantly less IPs than both DPA variants in all high-dimensional instances. This underlines again that the number of IPs is not the main figure to look at but that the structure of these IPs, in particular the number of the involved integer variables, is decisive for the CPU time.

The results of the assignment instances are given in Tables  15 and  16 . In general, the results are similar to the knapsack results. However, the DPA variants perform even better. Again, DPA-TS and DPA-A solve all instances reliably and are, apart from instances 4–5 and 5–5, clearly the fastest algorithms compared to the other three algorithms. However, different to the knapsack results, DCM is outperformed by the DPA variants also for most of the smaller instances. Besides, AIRA performs better for the assignment instances. It outperforms DCM in the larger instances 4–15, 4–20 and 5–10 with respect to CPU time, but is still inferior to the DPA variants.

6 Summary and further ideas

In this paper we combine the defining point algorithm (DPA) introduced in Klamroth et al. ( 2015 ) with suitable scalarization methods to obtain a new versatile approach to compute the nondominated set of multiobjective integer programming problems. Our theoretical and numerical analysis provide evidence that DPA finds a competitive balance between the number of required solver calls on the one hand, and the numerical complexity of each individual solver call on the other hand. Indeed, we show that the number of solver calls can be bounded by a polynomial in the number of nondominated points in the worst case. At the same time, subproblems are kept simple by using basic \(\varepsilon \) -constraint or weighted Tchebychev scalarizations. Two variants of DPA are implemented in C++ and compared to available state-of-the-art open-source solvers. We demonstrate the clear superiority of DPA with respect to CPU time on common benchmark instances as well as newly generated instances with up to ten objectives. Setting a time limit of at most ten times the solution time of the fastest algorithm, respectively, only DPA solves all instances reliably.

We present our algorithm as versatile and modular in the sense that it can be combined with any scalarization. Our numerical study concentrates on the \(\varepsilon \) -constraint scalarization due to the advantages described in Sect.  3 when the goal is to find every efficient solution. A numerical comparison using an \(\varepsilon \) -constraint scalarization and additionally a weighted Tchebychev scalarization has been presented in Dächert and Klamroth ( 2015 ). The algorithm therein is structurally similar to the defining point approach used in this paper, so the findings can be transferred directly. It is shown that when the goal is to generate the entire nondominated set, the (augmented) \(\varepsilon \) -constraint scalarization is always superior to the weighted Tchebychev scalarization due to its reduced number of IPs. However, when the goal is to generate only a subset of the nondominated set, a so-called incomplete representation, then other scalarizations as, e.g., the weighted Tchebychev scalarization promise to be more useful. We refer to Doğan et al. ( 2022 ) for an implementation in the context of multi-objective mixed-integer linear programming (MOMILP) problems. An adaptation of DPA to other types of representations is a promising direction for future research. This topic is, however, beyond the scope of this paper and left for further research.

Our study also shows that with an increasing number of objectives, the computational time increases as well. Therefore, in the future, a parallel variant should be developed. Parallel variants based on other algorithms have already been proposed in Pettersson and Ozlen ( 2019 ) and Turgut et al. ( 2019 ). It remains to be studied how a parallel defining point algorithm competes with these approaches.

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Acknowledgements

We thank two anonymous referees for their valuable comments which helped to improve the paper. Kathrin Klamroth acknowledges financial support by the Deutsche Forschungsgemeinschaft, project number KL 1076/11-1.

Open Access funding enabled and organized by Projekt DEAL.

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Tino Fleuren and Kathrin Klamroth have been contributed equally to this work.

Authors and Affiliations

Faculty of Informatics/Mathematics, Hochschule für Technik und Wirtschaft Dresden - University of Applied Sciences, Friedrich-List-Platz 1, 01069, Dresden, Germany

Kerstin Dächert

Fraunhofer Institute for Industrial Mathematics, Fraunhofer-Platz 1, 67663, Kaiserslautern, Germany

Tino Fleuren

Department of Mathematics and Computer Science, University of Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany

Kathrin Klamroth

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Correspondence to Kerstin Dächert .

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We assure that there is no conflict of interest concerning the publication of this manuscript in MMOR. Kathrin Klamroth declares financial support by the Deutsche Forschungsgemeinschaft, project number KL 1076/11-1. The other two authors have not received specific funding. All sources of data that have been used for this work are properly declared and referenced.

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Dächert, K., Fleuren, T. & Klamroth, K. A simple, efficient and versatile objective space algorithm for multiobjective integer programming. Math Meth Oper Res (2024). https://doi.org/10.1007/s00186-023-00841-0

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Received : 26 February 2023

Revised : 02 August 2023

Accepted : 31 October 2023

Published : 21 March 2024

DOI : https://doi.org/10.1007/s00186-023-00841-0

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