steps of science problem solving

• Science Notes Posts
• Contact Science Notes
• Todd Helmenstine Biography
• Anne Helmenstine Biography
• Free Printable Periodic Tables (PDF and PNG)
• Periodic Table Wallpapers
• Interactive Periodic Table
• Periodic Table Posters
• Science Experiments for Kids
• How to Grow Crystals
• Chemistry Projects
• Fire and Flames Projects
• Holiday Science
• Chemistry Problems With Answers
• Physics Problems
• Unit Conversion Example Problems
• Chemistry Worksheets
• Biology Worksheets
• Periodic Table Worksheets
• Physical Science Worksheets
• Science Lab Worksheets
• My Amazon Books

Steps of the Scientific Method 2

The scientific method is a system scientists and other people use to ask and answer questions about the natural world. In a nutshell, the scientific method works by making observations, asking a question or identifying a problem, and then designing and analyzing an experiment to test a prediction of what you expect will happen. It’s a powerful analytical tool because once you draw conclusions, you may be able to answer a question and make predictions about future events.

These are the steps of the scientific method:

• Make observations.

Sometimes this step is omitted in the list, but you always make observations before asking a question, whether you recognize it or not. You always have some background information about a topic. However, it’s a good idea to be systematic about your observations and to record them in a lab book or another way. Often, these initial observations can help you identify a question. Later on, this information may help you decide on another area of investigation of a topic.

• Ask a question, identify a problem, or state an objective.

There are various forms of this step. Sometimes you may want to state an objective and a problem and then phrase it in the form of a question. The reason it’s good to state a question is because it’s easiest to design an experiment to answer a question. A question helps you form a hypothesis, which focuses your study.

• Research the topic.

You should conduct background research on your topic to learn as much as you can about it. This can occur both before and after you state an objective and form a hypothesis. In fact, you may find yourself researching the topic throughout the entire process.

• Formulate a hypothesis.

A hypothesis is a formal prediction. There are two forms of a hypothesis that are particularly easy to test. One is to state the hypothesis as an “if, then” statement. An example of an if-then hypothesis is: “If plants are grown under red light, then they will be taller than plants grown under white light.” Another good type of hypothesis is what is called a “ null hypothesis ” or “no difference” hypothesis. An example of a null hypothesis is: “There is no difference in the rate of growth of plants grown under red light compared with plants grown under white light.”

• Design and perform an experiment to test the hypothesis.

Once you have a hypothesis, you need to find a way to test it. This involves an experiment . There are many ways to set up an experiment. A basic experiment contains variables, which are factors you can measure. The two main variables are the independent variable (the one you control or change) and the dependent variable (the one you measure to see if it is affected when you change the independent variable).

• Record and analyze the data you obtain from the experiment.

It’s a good idea to record notes alongside your data, stating anything unusual or unexpected. Once you have the data, draw a chart, table, or graph to present your results. Next, analyze the results to understand what it all means.

• Determine whether you accept or reject the hypothesis.

Do the results support the hypothesis or not? Keep in mind, it’s okay if the hypothesis is not supported, especially if you are testing a null hypothesis. Sometimes excluding an explanation answers your question! There is no “right” or “wrong” here. However, if you obtain an unexpected result, you might want to perform another experiment.

• Draw a conclusion and report the results of the experiment.

What good is knowing something if you keep it to yourself? You should report the outcome of the experiment, even if it’s just in a notebook. What did you learn from the experiment?

How Many Steps Are There?

You may be asked to list the 5 steps of the scientific method or the 6 steps of the method or some other number. There are different ways of grouping together the steps outlined here, so it’s a good idea to learn the way an instructor wants you to list the steps. No matter how many steps there are, the order is always the same.

Related Posts

2 thoughts on “ steps of the scientific method ”.

You raise a valid point, but peer review has its limitations. Consider the case of Galileo, for example.

That’s a good point too. But that was a rare limitation due to religion, and scientific consensus prevailed in the end. It’s nowhere near a reason to doubt scientific consensus in general. I’m thinking about issues such as climate change where so many people are skeptical despite 97% consensus among climate scientists. I was just surprised to see that this is not included as an important part of the process.

Comments are closed.

What is the Scientific Method: How does it work and why is it important?

The scientific method is a systematic process involving steps like defining questions, forming hypotheses, conducting experiments, and analyzing data. It minimizes biases and enables replicable research, leading to groundbreaking discoveries like Einstein's theory of relativity, penicillin, and the structure of DNA. This ongoing approach promotes reason, evidence, and the pursuit of truth in science.

Updated on November 18, 2023

Beginning in elementary school, we are exposed to the scientific method and taught how to put it into practice. As a tool for learning, it prepares children to think logically and use reasoning when seeking answers to questions.

Rather than jumping to conclusions, the scientific method gives us a recipe for exploring the world through observation and trial and error. We use it regularly, sometimes knowingly in academics or research, and sometimes subconsciously in our daily lives.

In this article we will refresh our memories on the particulars of the scientific method, discussing where it comes from, which elements comprise it, and how it is put into practice. Then, we will consider the importance of the scientific method, who uses it and under what circumstances.

What is the scientific method?

The scientific method is a dynamic process that involves objectively investigating questions through observation and experimentation . Applicable to all scientific disciplines, this systematic approach to answering questions is more accurately described as a flexible set of principles than as a fixed series of steps.

The following representations of the scientific method illustrate how it can be both condensed into broad categories and also expanded to reveal more and more details of the process. These graphics capture the adaptability that makes this concept universally valuable as it is relevant and accessible not only across age groups and educational levels but also within various contexts.

Steps in the scientific method

While the scientific method is versatile in form and function, it encompasses a collection of principles that create a logical progression to the process of problem solving:

• Define a question : Constructing a clear and precise problem statement that identifies the main question or goal of the investigation is the first step. The wording must lend itself to experimentation by posing a question that is both testable and measurable.
• Gather information and resources : Researching the topic in question to find out what is already known and what types of related questions others are asking is the next step in this process. This background information is vital to gaining a full understanding of the subject and in determining the best design for experiments. 
• Form a hypothesis : Composing a concise statement that identifies specific variables and potential results, which can then be tested, is a crucial step that must be completed before any experimentation. An imperfection in the composition of a hypothesis can result in weaknesses to the entire design of an experiment.
• Perform the experiments : Testing the hypothesis by performing replicable experiments and collecting resultant data is another fundamental step of the scientific method. By controlling some elements of an experiment while purposely manipulating others, cause and effect relationships are established.
• Analyze the data : Interpreting the experimental process and results by recognizing trends in the data is a necessary step for comprehending its meaning and supporting the conclusions. Drawing inferences through this systematic process lends substantive evidence for either supporting or rejecting the hypothesis.
• Report the results : Sharing the outcomes of an experiment, through an essay, presentation, graphic, or journal article, is often regarded as a final step in this process. Detailing the project's design, methods, and results not only promotes transparency and replicability but also adds to the body of knowledge for future research.
• Retest the hypothesis : Repeating experiments to see if a hypothesis holds up in all cases is a step that is manifested through varying scenarios. Sometimes a researcher immediately checks their own work or replicates it at a future time, or another researcher will repeat the experiments to further test the hypothesis.

Where did the scientific method come from?

Oftentimes, ancient peoples attempted to answer questions about the unknown by:

• Making simple observations
• Discussing the possibilities with others deemed worthy of a debate
• Drawing conclusions based on dominant opinions and preexisting beliefs

For example, take Greek and Roman mythology. Myths were used to explain everything from the seasons and stars to the sun and death itself.

However, as societies began to grow through advancements in agriculture and language, ancient civilizations like Egypt and Babylonia shifted to a more rational analysis for understanding the natural world. They increasingly employed empirical methods of observation and experimentation that would one day evolve into the scientific method . 

In the 4th century, Aristotle, considered the Father of Science by many, suggested these elements , which closely resemble the contemporary scientific method, as part of his approach for conducting science:

• Study what others have written about the subject.
• Look for the general consensus about the subject.
• Perform a systematic study of everything even partially related to the topic.

By continuing to emphasize systematic observation and controlled experiments, scholars such as Al-Kindi and Ibn al-Haytham helped expand this concept throughout the Islamic Golden Age . 

In his 1620 treatise, Novum Organum , Sir Francis Bacon codified the scientific method, arguing not only that hypotheses must be tested through experiments but also that the results must be replicated to establish a truth. Coming at the height of the Scientific Revolution, this text made the scientific method accessible to European thinkers like Galileo and Isaac Newton who then put the method into practice.

As science modernized in the 19th century, the scientific method became more formalized, leading to significant breakthroughs in fields such as evolution and germ theory. Today, it continues to evolve, underpinning scientific progress in diverse areas like quantum mechanics, genetics, and artificial intelligence.

Why is the scientific method important?

The history of the scientific method illustrates how the concept developed out of a need to find objective answers to scientific questions by overcoming biases based on fear, religion, power, and cultural norms. This still holds true today.

By implementing this standardized approach to conducting experiments, the impacts of researchers’ personal opinions and preconceived notions are minimized. The organized manner of the scientific method prevents these and other mistakes while promoting the replicability and transparency necessary for solid scientific research.

The importance of the scientific method is best observed through its successes, for example: 

• “ Albert Einstein stands out among modern physicists as the scientist who not only formulated a theory of revolutionary significance but also had the genius to reflect in a conscious and technical way on the scientific method he was using.” Devising a hypothesis based on the prevailing understanding of Newtonian physics eventually led Einstein to devise the theory of general relativity .
• Howard Florey “Perhaps the most useful lesson which has come out of the work on penicillin has been the demonstration that success in this field depends on the development and coordinated use of technical methods.” After discovering a mold that prevented the growth of Staphylococcus bacteria, Dr. Alexander Flemimg designed experiments to identify and reproduce it in the lab, thus leading to the development of penicillin .
• James D. Watson “Every time you understand something, religion becomes less likely. Only with the discovery of the double helix and the ensuing genetic revolution have we had grounds for thinking that the powers held traditionally to be the exclusive property of the gods might one day be ours. . . .” By using wire models to conceive a structure for DNA, Watson and Crick crafted a hypothesis for testing combinations of amino acids, X-ray diffraction images, and the current research in atomic physics, resulting in the discovery of DNA’s double helix structure .

Final thoughts

As the cases exemplify, the scientific method is never truly completed, but rather started and restarted. It gave these researchers a structured process that was easily replicated, modified, and built upon. 

While the scientific method may “end” in one context, it never literally ends. When a hypothesis, design, methods, and experiments are revisited, the scientific method simply picks up where it left off. Each time a researcher builds upon previous knowledge, the scientific method is restored with the pieces of past efforts.

By guiding researchers towards objective results based on transparency and reproducibility, the scientific method acts as a defense against bias, superstition, and preconceived notions. As we embrace the scientific method's enduring principles, we ensure that our quest for knowledge remains firmly rooted in reason, evidence, and the pursuit of truth.

The AJE Team

See our "Privacy Policy"

Choose Your Test

• Search Blogs By Category
• College Admissions
• AP and IB Exams
• GPA and Coursework

The 6 Scientific Method Steps and How to Use Them

General Education

When you’re faced with a scientific problem, solving it can seem like an impossible prospect. There are so many possible explanations for everything we see and experience—how can you possibly make sense of them all? Science has a simple answer: the scientific method.

The scientific method is a method of asking and answering questions about the world. These guiding principles give scientists a model to work through when trying to understand the world, but where did that model come from, and how does it work?

In this article, we’ll define the scientific method, discuss its long history, and cover each of the scientific method steps in detail.

What Is the Scientific Method?

At its most basic, the scientific method is a procedure for conducting scientific experiments. It’s a set model that scientists in a variety of fields can follow, going from initial observation to conclusion in a loose but concrete format.

The number of steps varies, but the process begins with an observation, progresses through an experiment, and concludes with analysis and sharing data. One of the most important pieces to the scientific method is skepticism —the goal is to find truth, not to confirm a particular thought. That requires reevaluation and repeated experimentation, as well as examining your thinking through rigorous study.

There are in fact multiple scientific methods, as the basic structure can be easily modified.  The one we typically learn about in school is the basic method, based in logic and problem solving, typically used in “hard” science fields like biology, chemistry, and physics. It may vary in other fields, such as psychology, but the basic premise of making observations, testing, and continuing to improve a theory from the results remain the same.

The History of the Scientific Method

The scientific method as we know it today is based on thousands of years of scientific study. Its development goes all the way back to ancient Mesopotamia, Greece, and India.

The Ancient World

In ancient Greece, Aristotle devised an inductive-deductive process , which weighs broad generalizations from data against conclusions reached by narrowing down possibilities from a general statement. However, he favored deductive reasoning, as it identifies causes, which he saw as more important.

Aristotle wrote a great deal about logic and many of his ideas about reasoning echo those found in the modern scientific method, such as ignoring circular evidence and limiting the number of middle terms between the beginning of an experiment and the end. Though his model isn’t the one that we use today, the reliance on logic and thorough testing are still key parts of science today.

The Middle Ages

The next big step toward the development of the modern scientific method came in the Middle Ages, particularly in the Islamic world. Ibn al-Haytham, a physicist from what we now know as Iraq, developed a method of testing, observing, and deducing for his research on vision. al-Haytham was critical of Aristotle’s lack of inductive reasoning, which played an important role in his own research.

Other scientists, including Abū Rayhān al-Bīrūnī, Ibn Sina, and Robert Grosseteste also developed models of scientific reasoning to test their own theories. Though they frequently disagreed with one another and Aristotle, those disagreements and refinements of their methods led to the scientific method we have today.

Following those major developments, particularly Grosseteste’s work, Roger Bacon developed his own cycle of observation (seeing that something occurs), hypothesis (making a guess about why that thing occurs), experimentation (testing that the thing occurs), and verification (an outside person ensuring that the result of the experiment is consistent).

After joining the Franciscan Order, Bacon was granted a special commission to write about science; typically, Friars were not allowed to write books or pamphlets. With this commission, Bacon outlined important tenets of the scientific method, including causes of error, methods of knowledge, and the differences between speculative and experimental science. He also used his own principles to investigate the causes of a rainbow, demonstrating the method’s effectiveness.

Scientific Revolution

Throughout the Renaissance, more great thinkers became involved in devising a thorough, rigorous method of scientific study. Francis Bacon brought inductive reasoning further into the method, whereas Descartes argued that the laws of the universe meant that deductive reasoning was sufficient. Galileo’s research was also inductive reasoning-heavy, as he believed that researchers could not account for every possible variable; therefore, repetition was necessary to eliminate faulty hypotheses and experiments.

All of this led to the birth of the Scientific Revolution , which took place during the sixteenth and seventeenth centuries. In 1660, a group of philosophers and physicians joined together to work on scientific advancement. After approval from England’s crown , the group became known as the Royal Society, which helped create a thriving scientific community and an early academic journal to help introduce rigorous study and peer review.

Previous generations of scientists had touched on the importance of induction and deduction, but Sir Isaac Newton proposed that both were equally important. This contribution helped establish the importance of multiple kinds of reasoning, leading to more rigorous study.

As science began to splinter into separate areas of study, it became necessary to define different methods for different fields. Karl Popper was a leader in this area—he established that science could be subject to error, sometimes intentionally. This was particularly tricky for “soft” sciences like psychology and social sciences, which require different methods. Popper’s theories furthered the divide between sciences like psychology and “hard” sciences like chemistry or physics.

Paul Feyerabend argued that Popper’s methods were too restrictive for certain fields, and followed a less restrictive method hinged on “anything goes,” as great scientists had made discoveries without the Scientific Method. Feyerabend suggested that throughout history scientists had adapted their methods as necessary, and that sometimes it would be necessary to break the rules. This approach suited social and behavioral scientists particularly well, leading to a more diverse range of models for scientists in multiple fields to use.

The Scientific Method Steps

Though different fields may have variations on the model, the basic scientific method is as follows:

#1: Make Observations 

Notice something, such as the air temperature during the winter, what happens when ice cream melts, or how your plants behave when you forget to water them.

#2: Ask a Question

Turn your observation into a question. Why is the temperature lower during the winter? Why does my ice cream melt? Why does my toast always fall butter-side down?

This step can also include doing some research. You may be able to find answers to these questions already, but you can still test them!

#3: Make a Hypothesis

A hypothesis is an educated guess of the answer to your question. Why does your toast always fall butter-side down? Maybe it’s because the butter makes that side of the bread heavier.

A good hypothesis leads to a prediction that you can test, phrased as an if/then statement. In this case, we can pick something like, “If toast is buttered, then it will hit the ground butter-first.”

#4: Experiment

Your experiment is designed to test whether your predication about what will happen is true. A good experiment will test one variable at a time —for example, we’re trying to test whether butter weighs down one side of toast, making it more likely to hit the ground first.

The unbuttered toast is our control variable. If we determine the chance that a slice of unbuttered toast, marked with a dot, will hit the ground on a particular side, we can compare those results to our buttered toast to see if there’s a correlation between the presence of butter and which way the toast falls.

If we decided not to toast the bread, that would be introducing a new question—whether or not toasting the bread has any impact on how it falls. Since that’s not part of our test, we’ll stick with determining whether the presence of butter has any impact on which side hits the ground first.

#5: Analyze Data

After our experiment, we discover that both buttered toast and unbuttered toast have a 50/50 chance of hitting the ground on the buttered or marked side when dropped from a consistent height, straight down. It looks like our hypothesis was incorrect—it’s not the butter that makes the toast hit the ground in a particular way, so it must be something else.

Since we didn’t get the desired result, it’s back to the drawing board. Our hypothesis wasn’t correct, so we’ll need to start fresh. Now that you think about it, your toast seems to hit the ground butter-first when it slides off your plate, not when you drop it from a consistent height. That can be the basis for your new experiment.

#6: Communicate Your Results

Good science needs verification. Your experiment should be replicable by other people, so you can put together a report about how you ran your experiment to see if other peoples’ findings are consistent with yours.

This may be useful for class or a science fair. Professional scientists may publish their findings in scientific journals, where other scientists can read and attempt their own versions of the same experiments. Being part of a scientific community helps your experiments be stronger because other people can see if there are flaws in your approach—such as if you tested with different kinds of bread, or sometimes used peanut butter instead of butter—that can lead you closer to a good answer.

A Scientific Method Example: Falling Toast

We’ve run through a quick recap of the scientific method steps, but let’s look a little deeper by trying again to figure out why toast so often falls butter side down.

#1: Make Observations

At the end of our last experiment, where we learned that butter doesn’t actually make toast more likely to hit the ground on that side, we remembered that the times when our toast hits the ground butter side first are usually when it’s falling off a plate.

The easiest question we can ask is, “Why is that?”

We can actually search this online and find a pretty detailed answer as to why this is true. But we’re budding scientists—we want to see it in action and verify it for ourselves! After all, good science should be replicable, and we have all the tools we need to test out what’s really going on.

Why do we think that buttered toast hits the ground butter-first? We know it’s not because it’s heavier, so we can strike that out. Maybe it’s because of the shape of our plate?

That’s something we can test. We’ll phrase our hypothesis as, “If my toast slides off my plate, then it will fall butter-side down.”

Just seeing that toast falls off a plate butter-side down isn’t enough for us. We want to know why, so we’re going to take things a step further—we’ll set up a slow-motion camera to capture what happens as the toast slides off the plate.

We’ll run the test ten times, each time tilting the same plate until the toast slides off. We’ll make note of each time the butter side lands first and see what’s happening on the video so we can see what’s going on.

When we review the footage, we’ll likely notice that the bread starts to flip when it slides off the edge, changing how it falls in a way that didn’t happen when we dropped it ourselves.

That answers our question, but it’s not the complete picture —how do other plates affect how often toast hits the ground butter-first? What if the toast is already butter-side down when it falls? These are things we can test in further experiments with new hypotheses!

Now that we have results, we can share them with others who can verify our results. As mentioned above, being part of the scientific community can lead to better results. If your results were wildly different from the established thinking about buttered toast, that might be cause for reevaluation. If they’re the same, they might lead others to make new discoveries about buttered toast. At the very least, you have a cool experiment you can share with your friends!

Key Scientific Method Tips

Though science can be complex, the benefit of the scientific method is that it gives you an easy-to-follow means of thinking about why and how things happen. To use it effectively, keep these things in mind!

Don’t Worry About Proving Your Hypothesis

One of the important things to remember about the scientific method is that it’s not necessarily meant to prove your hypothesis right. It’s great if you do manage to guess the reason for something right the first time, but the ultimate goal of an experiment is to find the true reason for your observation to occur, not to prove your hypothesis right.

Good science sometimes means that you’re wrong. That’s not a bad thing—a well-designed experiment with an unanticipated result can be just as revealing, if not more, than an experiment that confirms your hypothesis.

Be Prepared to Try Again

If the data from your experiment doesn’t match your hypothesis, that’s not a bad thing. You’ve eliminated one possible explanation, which brings you one step closer to discovering the truth.

The scientific method isn’t something you’re meant to do exactly once to prove a point. It’s meant to be repeated and adapted to bring you closer to a solution. Even if you can demonstrate truth in your hypothesis, a good scientist will run an experiment again to be sure that the results are replicable. You can even tweak a successful hypothesis to test another factor, such as if we redid our buttered toast experiment to find out whether different kinds of plates affect whether or not the toast falls butter-first. The more we test our hypothesis, the stronger it becomes!

What’s Next?

Want to learn more about the scientific method? These important high school science classes will no doubt cover it in a variety of different contexts.

Test your ability to follow the scientific method using these at-home science experiments for kids !

Need some proof that science is fun? Try making slime

Trending Now

How to Get Into Harvard and the Ivy League

How to Get a Perfect 4.0 GPA

How to Write an Amazing College Essay

What Exactly Are Colleges Looking For?

ACT vs. SAT: Which Test Should You Take?

When should you take the SAT or ACT?

Get Your Free

Find Your Target SAT Score

Free Complete Official SAT Practice Tests

How to Get a Perfect SAT Score, by an Expert Full Scorer

Score 800 on SAT Math

Score 800 on SAT Reading and Writing

How to Improve Your Low SAT Score

Score 600 on SAT Math

Score 600 on SAT Reading and Writing

Find Your Target ACT Score

Complete Official Free ACT Practice Tests

How to Get a Perfect ACT Score, by a 36 Full Scorer

Get a 36 on ACT English

Get a 36 on ACT Math

Get a 36 on ACT Reading

Get a 36 on ACT Science

How to Improve Your Low ACT Score

Get a 24 on ACT English

Get a 24 on ACT Math

Get a 24 on ACT Reading

Get a 24 on ACT Science

Stay Informed

Get the latest articles and test prep tips!

Melissa Brinks graduated from the University of Washington in 2014 with a Bachelor's in English with a creative writing emphasis. She has spent several years tutoring K-12 students in many subjects, including in SAT prep, to help them prepare for their college education.

Ask a Question Below

Have any questions about this article or other topics? Ask below and we'll reply!

Six Steps of the Scientific Method

Learn What Makes Each Stage Important

ThoughtCo. / Hugo Lin 

• Scientific Method
• Chemical Laws
• Periodic Table
• Projects & Experiments
• Biochemistry
• Physical Chemistry
• Medical Chemistry
• Chemistry In Everyday Life
• Famous Chemists
• Activities for Kids
• Abbreviations & Acronyms
• Weather & Climate
• Ph.D., Biomedical Sciences, University of Tennessee at Knoxville
• B.A., Physics and Mathematics, Hastings College

The scientific method is a systematic way of learning about the world around us. The key difference between the scientific method and other ways of acquiring knowledge is that, when using the scientific method, we make hypotheses and then test them with an experiment.

Anyone can use the scientific method to acquire knowledge by asking questions and then working to find the answers to those questions. Below are the six steps involved in the scientific method and variables you may encounter when working with this method.

The Six Steps

The number of steps in the scientific method can vary from one description to another (which mainly happens when data and analysis are separated into separate steps), however, below is a fairly standard list of the six steps you'll likely be expected to know for any science class:

• Purpose/Question Ask a question.
• Research Conduct background research. Write down your sources so you can cite your references. In the modern era, you might conduct much of your research online. As you read articles and papers online, ensure you scroll to the bottom of the text to check the author's references. Even if you can't access the full text of a published article, you can usually view the abstract to see the summary of other experiments . Interview experts on a topic. The more you know about a subject, the easier it'll be to conduct your investigation.
• Hypothesis Propose a hypothesis . This is a sort of educated guess about what you expect your research to reveal. A hypothesis is a statement used to predict the outcome of an experiment. Usually, a hypothesis is written in terms of cause and effect. Alternatively, it may describe the relationship between two phenomena. The null hypothesis or the no-difference hypothesis is one type of hypothesis that's easy to test because it assumes changing a variable will not affect the outcome. In reality, you probably expect a change, but rejecting a hypothesis may be more useful than accepting one.
• Experiment Design and experiment to test your hypothesis. An experiment has an independent and dependent variable. You change or control the independent variable and record the effect it has on the dependent variable . It's important to change only one variable for an experiment rather than try to combine the effects of variables in an experiment. For example, if you want to test the effects of light intensity and fertilizer concentration on the growth rate of a plant, you're looking at two separate experiments.
• Data/Analysis Record observations and analyze the meaning of the data. Often, you'll prepare a table or graph of the data. Don't throw out data points you think are bad or that don't support your predictions. Some of the most incredible discoveries in science were made because the data looked wrong! Once you have the data, you may need to perform a mathematical analysis to support or refute your hypothesis.
• Conclusion Conclude whether to accept or reject your hypothesis. There's no right or wrong outcome to an experiment, so either result is fine. Accepting a hypothesis doesn't necessarily mean it's correct! Sometimes repeating an experiment may give a different result. In other cases, a hypothesis may predict an outcome, yet you might draw an incorrect conclusion. Communicate your results. You can compile your results into a lab report or formally submit them as a paper . Whether you accept or reject the hypothesis, you likely learned something about the subject and may wish to revise the original hypothesis or form a new one for a future experiment.

scientific method

Our editors will review what you’ve submitted and determine whether to revise the article.

• University of Nevada, Reno - College of Agriculture, Biotechnology and Natural Resources Extension - The Scientific Method
• World History Encyclopedia - Scientific Method
• LiveScience - What Is Science?
• Verywell Mind - Scientific Method Steps in Psychology Research
• WebMD - What is the Scientific Method?
• Chemistry LibreTexts - The Scientific Method
• National Center for Biotechnology Information - PubMed Central - Redefining the scientific method: as the use of sophisticated scientific methods that extend our mind
• Khan Academy - The scientific method
• Simply Psychology - What are the steps in the Scientific Method?
• Stanford Encyclopedia of Philosophy - Scientific Method

scientific method , mathematical and experimental technique employed in the sciences . More specifically, it is the technique used in the construction and testing of a scientific hypothesis .

The process of observing, asking questions, and seeking answers through tests and experiments is not unique to any one field of science. In fact, the scientific method is applied broadly in science, across many different fields. Many empirical sciences, especially the social sciences , use mathematical tools borrowed from probability theory and statistics , together with outgrowths of these, such as decision theory , game theory , utility theory, and operations research . Philosophers of science have addressed general methodological problems, such as the nature of scientific explanation and the justification of induction .

The scientific method is critical to the development of scientific theories , which explain empirical (experiential) laws in a scientifically rational manner. In a typical application of the scientific method, a researcher develops a hypothesis , tests it through various means, and then modifies the hypothesis on the basis of the outcome of the tests and experiments. The modified hypothesis is then retested, further modified, and tested again, until it becomes consistent with observed phenomena and testing outcomes. In this way, hypotheses serve as tools by which scientists gather data. From that data and the many different scientific investigations undertaken to explore hypotheses, scientists are able to develop broad general explanations, or scientific theories.

See also Mill’s methods ; hypothetico-deductive method .

Lucidly exploring and applying philosophy

• Fun Quizzes
• Logic Course
• Ethics Course
• Philosophy Course

Chapter 6: Scientific Problem Solving

If you prefer a video, click this button:

Scientific Problem Solving Video

Science is a method to discover empirical truths and patterns. Roughly speaking, the scientific method consists of

1) Observing

2) Forming a hypothesis

3) Testing the hypothesis and

4) Interpreting the data to confirm or disconfirm the hypothesis.

The beauty of science is that any scientific claim can be tested if you have the proper knowledge and equipment.

You can also use the scientific method to solve everyday problems: 1) Observe and clearly define the problem, 2) Form a hypothesis, 3) Test it, and 4) Confirm the hypothesis... or disconfirm it and start over.

So, the next time you are cursing in traffic or emotionally reacting to a problem, take a few deep breaths and then use this rational and scientific approach. Slow down, observe, hypothesize, and test.

Explain how you would solve these problems using the four steps of the scientific process.

Example: The fire alarm is not working.

1) Observe/Define the problem: it does not beep when I push the button.

2) Hypothesis: it is caused by a dead battery.

3) Test: try a new battery.

4) Confirm/Disconfirm: the alarm now works. If it does not work, start over by testing another hypothesis like “it has a loose wire.”  

• My car will not start.
• My child is having problems reading.
• I owe $20,000, but only make $10 an hour.
• My boss is mean. I want him/her to stop using rude language towards me.
• My significant other is lazy. I want him/her to help out more.

6-8. Identify three problems where you can apply the scientific method.

*Answers will vary.

Application and Value

Science is more of a process than a body of knowledge. In our daily lives, we often emotionally react and jump to quick solutions when faced with problems, but following the four steps of the scientific process can help us slow down and discover more intelligent solutions.

In your study of philosophy, you will explore deeper questions about science. For example, are there any forms of knowledge that are nonscientific? Can science tell us what we ought to do? Can logical and mathematical truths be proven in a scientific way? Does introspection give knowledge even though I cannot scientifically observe your introspective thoughts? Is science truly objective?  These are challenging questions that should help you discover the scope of science without diminishing its awesome power.

But the first step in answering these questions is knowing what science is, and this chapter clarifies its essence. Again, Science is not so much a body of knowledge as it is a method of observing, hypothesizing, and testing. This method is what all the sciences have in common.

Perhaps too science should involve falsifiability, which is a concept explored in the next chapter.

Return to Logic Home                            Next (Chapter 7, Falsifiability)

Click on my affiliate link above (Logic Book Image) to explore the most popular introduction to logic. If you purchase it, I recommend buying a less expensive older edition.

• Skip to Page Content
• Skip to Navigation
• Skip to Search
• Skip to Footer
• COVID-19 Health and Safety
• Admissions and Ticketing
• Temporary Hall Closures
• Accessibility
• Field Trips
• Adult Group Visits
• Guided Tours
• Transportation
• Davis Family Butterfly Vivarium
• Invisible Worlds: Immersive Experience
• The Secret World of Elephants
• Turtle Odyssey
• Worlds Beyond Earth: Space Show
• Extinct and Endangered: Insects in Peril
• Grounded by Our Roots
• Ice Cold: An Exhibition of Hip-Hop Jewelry
• Opulent Oceans
• What's in a Name?
• Children & Family Programs
• Teen Programs
• Higher Education
• Adult Programs
• Educator Programs
• Evaluation, Research, & Policy
• Master of Arts in Teaching
• Online Courses for Educators
• Urban Advantage
• Viruses, Vaccines, and COVID-19
• The Science of COVID-19
• OLogy: The Science Website for Kids
• News & Blogs
• Science Topics
• Margaret Mead Festival
• Origami at the Museum
• Astrophysics
• Earth and Planetary Sciences
• Herpetology
• Ichthyology
• Ornithology
• Richard Gilder Graduate School
• Hayden Planetarium
• Center for Biodiversity and Conservation
• Institute for Comparative Genomics
• Southwestern Research Station
• Research Library
• Darwin Manuscripts Project
• Microscopy and Imaging Facility
• Science Conservation
• Computational Sciences
• Staff Directory
• Scientific Publications
• Ways to Donate
• Membership FAQ
• Benefit Events
• Corporate Support
• Planned Giving
• At the Museum
• Biodiversity
• Data Visualizations
• Dinosaurs and Fossils
• Earth and Climate
• In the Field
• Isaac Asimov Memorial Debate
• Kid Science
• Research and Collections
• SciCafe / Lectures

The Scientific Process

1. define a question to investigate.

As scientists conduct their research, they make observations and collect data. The observations and data often lead them to ask why something is the way it is. Scientists pursue answers to these questions in order to continue with their research. Once scientists have a good question to investigate, they begin to think of ways to answer it.

2. Make Predictions

Based on their research and observations, scientists will often come up with a hypothesis. A hypothesis is a possible answer to a question. It is based on: their own observations, existing theories, and information they gather from other sources. Scientists use their hypothesis to make a prediction, a testable statement that describes what they think the outcome of an investigation will be.

3. Gather Data

Evidence is needed to test the prediction. There are several strategies for collecting evidence, or data. Scientists can gather their data by observing the natural world, performing an experiment in a laboratory, or by running a model. Scientists decide what strategy to use, often combining strategies. Then they plan a procedure and gather their data. They make sure the procedure can be repeated, so that other scientists can evaluate their findings.

4. Analyze the Data

Scientists organize their data in tables, graphs, or diagrams. If possible, they include relevant data from other sources. They look for patterns that show connections between important variables in the hypothesis they are testing.

5. Draw Conclusions

Based on whether or not their prediction came true, scientists can then decide whether the evidence clearly supports or does not support the hypothesis. If the results are not clear, they must rethink their procedure. If the results are clear, scientists write up their findings and results to share with others. The conclusions they draw usually lead to new questions to pursue.

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Biology archive

Course: biology archive   >   unit 1, the scientific method.

• Controlled experiments
• The scientific method and experimental design

Introduction

• Make an observation.
• Ask a question.
• Form a hypothesis , or testable explanation.
• Make a prediction based on the hypothesis.
• Test the prediction.
• Iterate: use the results to make new hypotheses or predictions.

Scientific method example: Failure to toast

1. make an observation., 2. ask a question., 3. propose a hypothesis., 4. make predictions., 5. test the predictions..

• If the toaster does toast, then the hypothesis is supported—likely correct.
• If the toaster doesn't toast, then the hypothesis is not supported—likely wrong.

Logical possibility

Practical possibility, building a body of evidence, 6. iterate..

• If the hypothesis was supported, we might do additional tests to confirm it, or revise it to be more specific. For instance, we might investigate why the outlet is broken.
• If the hypothesis was not supported, we would come up with a new hypothesis. For instance, the next hypothesis might be that there's a broken wire in the toaster.

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

• Earth Science
• Physics & Engineering
• Science Kits
• Microscopes
• Science Curriculum and Kits
• About Home Science Tools

Science Projects > Science Fair Projects > Scientific Method for Grades K-12  

Scientific Method for Grades K-12

The scientific method is a problem-solving process used during experiments. It can be modified according to the age and ability of students and also to  develop particular skills .

Asking a question is the first step in the scientific method (e.g., Who, What, When, Where, Why, How). You’ll usually find an answer to a broad, simple question. Answers often lead to more questions. It’s here where the scientific method really begins.

In this article, the scientific method is laid out in four steps.

In practice, though, it’s usually not this neat. Scientists and students will often have to repeat steps and start over with the experiment, forming a new hypothesis, and repeating the series of steps. It’s part of the scientific process, the “ art of science ;” learning is not a sign of failure.

Once complete, the results of an experiment can be used as the starting point for a new experiment to answer new questions. This is called iteration.

Steps of the Scientific Method

Step 1: Start with a question.  What do you wonder about? What would you like to know? In the first step of the scientific method, you may need to do some background research to learn more. It can help you define your question and decide what you want to discover.

Step 2: Form a hypothesis.  A hypothesis is an educated guess or explanation for what you know. Forming a good hypothesis—a scientific hypothesis—is the starting point for the experiment (and further study). You can prove the hypothesis as  observably  correct or disprove it through experimentation. Observably, because scientific explanations for the results of an experiment evolve and change.

Step 3: Conduct an experiment, making observations, and tracking results . Set up a test experiment to see if your hypothesis is right or wrong. Make observations during your experiment and keep track of them by writing them down. Often replication of an experiment, in the exact same way, is necessary to be sure of your results.

Step 4: Come to a conclusion.  Decide whether your hypothesis was right or wrong.  What were the results of your experiment? Can you tell why it happened that way? Explain and communicate your results.

These principles can be used to study the natural world and  navigate life’s challenges . You can study anything from plants and  rocks  to biology or  chemical reactions  using these four steps. Even very young students can use a modified version of the scientific method to organize their thoughts.

Scientific Method for Younger Students

Younger students can study practical science using a simple version of the scientific method. You can use their natural curiosity to guide them and make it memorable. Try teaching the earliest grades the same steps, but making the language easier to understand.

• Wonder  — What do I want to know about the world around me?
• Think  – What do I think will happen?
• Act  – Test my idea. What happens?
• Say  – Am I right?

These students can conduct their own experiments to learn about the world around them. For example, young students can study the states of matter by melting ice in the sun and shade. Before beginning, ask a student to predict what will happen to ice placed in the sun vs. ice placed in the shade. Then test his or her idea, check on the ice cubes over time, and ask the student to explain what happened. Was the student right?

In another example, young students could study chemical reactions by adding soap and food coloring to milk. Again, before starting, ask a student to tell you what he or she thinks will happen when you add soap and food coloring to some milk. Test the experiment, watch for a reaction, and ask the student to explain what happened. Was the student right?

Spurred on by their natural curiosity, the youngest students can wonder, think, and observe. From the youngest ages, they can develop the ability to carefully observe and describe what they see in a simple scientific journal. They can begin to develop the critical thinking skills needed to determine whether an experiment turned out how they expected—the beginning of scientific reasoning!

Scientific Method for Middle School and High School Students

Older students can use the steps of the scientific method more independently to complete a science fair project or experiment on a topic in which they have an interest. Interest is key–without it, they’ll get bored.

Guide students’ learning with the following expansion on the last two steps of the scientific method, which require more advanced critical thinking skills.

Conduct an experiment, making observations, and tracking results.

Upper elementary, middle school, and high school students can design experiments to answer questions about the world. The complexity of an experiment will depend on the student’s abilities.

In designing their experiments, these students should pay close attention to:

• Repeating an experiment . To be sure of your results, an experiment will need to be repeated, always in the same way. The more times an experiment is repeated producing the same results, the more reliable it is said to be. Scientific progress depends on reliable experiments independent of the person conducting them.
• Controlling variables . A variable is a part of the experiment that can change. An experiment has an independent and dependent variable. You change or control the independent variable and record the effect it has on the dependent variable. It’s important to change only one variable at a time during an experiment rather than try to combine the effects of variables in an experiment. To ensure confidence in your results, whether proving or disproving your original hypothesis, nothing should change when an experiment is repeated. Everything that could vary, such as the amounts of a substance, the kind of a substance, the time of day, or the environment, should be “held constant” or “controlled.”

Design and perform an experiment to test your hypothesis. For example, if you want to test the effects of light intensity and fertilizer concentration on the growth rate of a plant, you’re really looking at two separate experiments.

• Changing only one variable at a time . All variables in an experiment affect the outcome. That’s why, when comparing experiments, it’s important to change only one variable at a time. This allows you to attribute differences in outcomes correctly. For example, if you want to find out how a plant’s growth rate is affected by water, you would control all variables (soil, light, air temperature) other than watering levels.
• Tracking results . What happened during your experiment? Identify all your variables and  keep track of your observations in a science notebook . Once you have all the information recorded (i.e., data), you can start analyzing.

Come to a Conclusion Using the Scientific Method Steps

What was the result of analyzing the results of all your observations? Did your experiment turn out as expected? Was your original hypothesis proven or falsifiable? If your results were surprising, you may not be able to come to a conclusion right away. You may want to reconsider all your variables, change a part of your design, and conduct another experiment, gathering more data. Arriving at a conclusion requires a thoughtful assessment of the results of your experiment. And the more data analysis you can provide is better because every piece of information in both your control group and your experimental group is critical to leading you to draw conclusions accurately.

The scientific method begins with inductive rather than deductive reasoning. Deductive reasoning moves from general concepts to more specific information. Inductive reasoning moves from specific facts or observations to a general conclusion. For example, dissecting a flower and examining its individual parts (e.g., ovary, petal, pistil) teaches us about flowers in general. By examining something up close, science uses the critical thinking skills of observing, comparing, contrasting, and analyzing to make a general conclusion.

The scientific method is a powerful tool to turn your questions into new discoveries!

Teaching Homeschool

Welcome! After you finish this article, we invite you to read other articles to assist you in teaching science at home on the Resource Center, which consists of hundreds of free science articles!

Shop for Science Supplies!

Home Science Tools offers a wide variety of science products and kits. Find affordable beakers, dissection supplies, chemicals, microscopes, and everything else you need to teach science for all ages!

Related Articles

29 Creative Ways to Use a Home Science Tools Beaker Mug

Infuse a dash of experimentation into your daily routine with a Home Science Tools Beaker Mug! As we gear up for our 29th Anniversary, we've compiled a list of 29 exciting ways to use your beaker mug in everyday life. From brewing up creative concoctions to unleashing...

Next Generation Science Standards (NGSS)

What are the Next Generation Science Standards (NGSS)?  These guidelines summarize what students “should” know and be able to do in different learning levels of science. The NGSS is based on research showing that students who are well-prepared for the future need...

The Beginners Guide to Choosing a Homeschool Science Curriculum

Get Started: Researching Homeschool Science   Curriculums  Teaching homeschool science is a great way for families to personalize their child's education while giving you the flexibility to teach it your way. There are many wonderful science curriculums...

Valentine’s Day Science Projects

Valentine’s Day is a great opportunity to inspire your student’s LOVE for science! Engage your kids with science concepts such as diffusion, density, and surfactants. These three, hands-on science projects include the Dancing Conversational Hearts, Rainbow Heart, and...

Synthetic Frog Dissection Guide Project

Frog dissections are a great way to learn about the human body, as frogs have many organs and tissues similar to those of humans. It is important to determine which type of dissection is best for your student or child. Some individuals do not enjoy performing...

JOIN OUR COMMUNITY

Get project ideas and special offers delivered to your inbox.

• Utility Menu

GA4 Tracking Code

fa51e2b1dc8cca8f7467da564e77b5ea

• Make a Gift
• Join Our Email List
• Problem Solving in STEM

Solving problems is a key component of many science, math, and engineering classes.  If a goal of a class is for students to emerge with the ability to solve new kinds of problems or to use new problem-solving techniques, then students need numerous opportunities to develop the skills necessary to approach and answer different types of problems.  Problem solving during section or class allows students to develop their confidence in these skills under your guidance, better preparing them to succeed on their homework and exams. This page offers advice about strategies for facilitating problem solving during class.

How do I decide which problems to cover in section or class?

In-class problem solving should reinforce the major concepts from the class and provide the opportunity for theoretical concepts to become more concrete. If students have a problem set for homework, then in-class problem solving should prepare students for the types of problems that they will see on their homework. You may wish to include some simpler problems both in the interest of time and to help students gain confidence, but it is ideal if the complexity of at least some of the in-class problems mirrors the level of difficulty of the homework. You may also want to ask your students ahead of time which skills or concepts they find confusing, and include some problems that are directly targeted to their concerns.

You have given your students a problem to solve in class. What are some strategies to work through it?

• Try to give your students a chance to grapple with the problems as much as possible.  Offering them the chance to do the problem themselves allows them to learn from their mistakes in the presence of your expertise as their teacher. (If time is limited, they may not be able to get all the way through multi-step problems, in which case it can help to prioritize giving them a chance to tackle the most challenging steps.)
• When you do want to teach by solving the problem yourself at the board, talk through the logic of how you choose to apply certain approaches to solve certain problems.  This way you can externalize the type of thinking you hope your students internalize when they solve similar problems themselves.
• Start by setting up the problem on the board (e.g you might write down key variables and equations; draw a figure illustrating the question).  Ask students to start solving the problem, either independently or in small groups.  As they are working on the problem, walk around to hear what they are saying and see what they are writing down. If several students seem stuck, it might be a good to collect the whole class again to clarify any confusion.  After students have made progress, bring the everyone back together and have students guide you as to what to write on the board.
• It can help to first ask students to work on the problem by themselves for a minute, and then get into small groups to work on the problem collaboratively.
• If you have ample board space, have students work in small groups at the board while solving the problem.  That way you can monitor their progress by standing back and watching what they put up on the board.
• If you have several problems you would like to have the students practice, but not enough time for everyone to do all of them, you can assign different groups of students to work on different – but related - problems.

When do you want students to work in groups to solve problems?

• Don’t ask students to work in groups for straightforward problems that most students could solve independently in a short amount of time.
• Do have students work in groups for thought-provoking problems, where students will benefit from meaningful collaboration.
• Even in cases where you plan to have students work in groups, it can be useful to give students some time to work on their own before collaborating with others.  This ensures that every student engages with the problem and is ready to contribute to a discussion.

What are some benefits of having students work in groups?

• Students bring different strengths, different knowledge, and different ideas for how to solve a problem; collaboration can help students work through problems that are more challenging than they might be able to tackle on their own.
• In working in a group, students might consider multiple ways to approach a problem, thus enriching their repertoire of strategies.
• Students who think they understand the material will gain a deeper understanding by explaining concepts to their peers.

What are some strategies for helping students to form groups?  

• Instruct students to work with the person (or people) sitting next to them.
• Count off.  (e.g. 1, 2, 3, 4; all the 1’s find each other and form a group, etc)
• Hand out playing cards; students need to find the person with the same number card. (There are many variants to this.  For example, you can print pictures of images that go together [rain and umbrella]; each person gets a card and needs to find their partner[s].)
• Based on what you know about the students, assign groups in advance. List the groups on the board.
• Note: Always have students take the time to introduce themselves to each other in a new group.

What should you do while your students are working on problems?

• Walk around and talk to students. Observing their work gives you a sense of what people understand and what they are struggling with. Answer students’ questions, and ask them questions that lead in a productive direction if they are stuck.
• If you discover that many people have the same question—or that someone has a misunderstanding that others might have—you might stop everyone and discuss a key idea with the entire class.

After students work on a problem during class, what are strategies to have them share their answers and their thinking?

• Ask for volunteers to share answers. Depending on the nature of the problem, student might provide answers verbally or by writing on the board. As a variant, for questions where a variety of answers are relevant, ask for at least three volunteers before anyone shares their ideas.
• Use online polling software for students to respond to a multiple-choice question anonymously.
• If students are working in groups, assign reporters ahead of time. For example, the person with the next birthday could be responsible for sharing their group’s work with the class.
• Cold call. To reduce student anxiety about cold calling, it can help to identify students who seem to have the correct answer as you were walking around the class and checking in on their progress solving the assigned problem. You may even want to warn the student ahead of time: "This is a great answer! Do you mind if I call on you when we come back together as a class?"
• Have students write an answer on a notecard that they turn in to you.  If your goal is to understand whether students in general solved a problem correctly, the notecards could be submitted anonymously; if you wish to assess individual students’ work, you would want to ask students to put their names on their notecard.  
• Use a jigsaw strategy, where you rearrange groups such that each new group is comprised of people who came from different initial groups and had solved different problems.  Students now are responsible for teaching the other students in their new group how to solve their problem.
• Have a representative from each group explain their problem to the class.
• Have a representative from each group draw or write the answer on the board.

What happens if a student gives a wrong answer?

• Ask for their reasoning so that you can understand where they went wrong.
• Ask if anyone else has other ideas. You can also ask this sometimes when an answer is right.
• Cultivate an environment where it’s okay to be wrong. Emphasize that you are all learning together, and that you learn through making mistakes.
• Do make sure that you clarify what the correct answer is before moving on.
• Once the correct answer is given, go through some answer-checking techniques that can distinguish between correct and incorrect answers. This can help prepare students to verify their future work.

How can you make your classroom inclusive?

• The goal is that everyone is thinking, talking, and sharing their ideas, and that everyone feels valued and respected. Use a variety of teaching strategies (independent work and group work; allow students to talk to each other before they talk to the class). Create an environment where it is normal to struggle and make mistakes.
• See Kimberly Tanner’s article on strategies to promoste student engagement and cultivate classroom equity. 

A few final notes…

• Make sure that you have worked all of the problems and also thought about alternative approaches to solving them.
• Board work matters. You should have a plan beforehand of what you will write on the board, where, when, what needs to be added, and what can be erased when. If students are going to write their answers on the board, you need to also have a plan for making sure that everyone gets to the correct answer. Students will copy what is on the board and use it as their notes for later study, so correct and logical information must be written there.

For more information...

Tipsheet: Problem Solving in STEM Sections

Tanner, K. D. (2013). Structure matters: twenty-one teaching strategies to promote student engagement and cultivate classroom equity . CBE-Life Sciences Education, 12(3), 322-331.

• Designing Your Course
• A Teaching Timeline: From Pre-Term Planning to the Final Exam
• The First Day of Class
• Group Agreements
• Classroom Debate
• Flipped Classrooms
• Leading Discussions
• Polling & Clickers
• Teaching with Cases
• Engaged Scholarship
• Devices in the Classroom
• Beyond the Classroom
• On Professionalism
• Getting Feedback
• Equitable & Inclusive Teaching
• Advising and Mentoring
• Teaching and Your Career
• Teaching Remotely
• Tools and Platforms
• The Science of Learning
• Bok Publications
• Other Resources Around Campus

What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

• Identify the issue : Recognize the problem that needs to be solved.
• Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
• Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
• Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
• Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
• Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
• Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

• Brainstorming with others
• Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
• Analyzing cause and effect
• Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

• Creating a list of potential ideas to solve the problem
• Grouping and categorizing similar solutions
• Prioritizing potential solutions based on feasibility, cost, and resources required
• Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

• SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
• Decision-making matrices
• Pros and cons lists
• Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

• Communicating the solution to relevant parties
• Setting timelines and milestones
• Assigning tasks and responsibilities
• Monitoring the solution and making adjustments as necessary
• Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

• Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
• Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
• SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
• Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

• Generate a diverse range of solutions
• Encourage all team members to participate
• Foster creative thinking

When brainstorming, remember to:

• Reserve judgment until the session is over
• Encourage wild ideas
• Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

• 5 Whys : Ask “why” five times to get to the underlying cause.
• Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
• Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

• List your problem’s strengths, such as relevant resources or strong partnerships.
• Identify its weaknesses, such as knowledge gaps or limited resources.
• Explore opportunities, like trends or new technologies, that could help solve the problem.
• Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

• Write the problem in the center of a blank page.
• Draw branches from the central problem to related sub-problems or contributing factors.
• Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

• Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
• Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
• Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

• Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
• Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
• Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

• Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
• Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
• Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
• How to Resolve Employee Conflict at Work [Steps, Tips, Examples]
• How to Write Inspiring Core Values? 5 Steps with Examples
• 30 Employee Feedback Examples (Positive & Negative)

Pardon Our Interruption

As you were browsing something about your browser made us think you were a bot. There are a few reasons this might happen:

• You've disabled JavaScript in your web browser.
• You're a power user moving through this website with super-human speed.
• You've disabled cookies in your web browser.
• A third-party browser plugin, such as Ghostery or NoScript, is preventing JavaScript from running. Additional information is available in this support article .

To regain access, please make sure that cookies and JavaScript are enabled before reloading the page.

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

• View all journals
• Explore content
• About the journal
• Publish with us
• Sign up for alerts
• Open access
• Published: 19 August 2024

A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem

• Jiale Zhao 1 , 4 ,
• Huijie Zhang 3 ,
• Huanhuan Yu 2 , 4 ,
• Hansheng Fei 2 , 4 ,
• Xiangdang Huang 2 , 4 &
• Qiuling Yang 2 , 4  

Scientific Reports volume  14 , Article number:  19140 ( 2024 ) Cite this article

Metrics details

• Computational models
• Computational neuroscience
• Computational science
• Computer science
• Machine learning

In recent years, researchers have taken the many-objective optimization algorithm, which can optimize 5, 8, 10, 15, 20 objective functions simultaneously, as a new research topic. However, the current research on many-objective optimization technology also encounters some challenges. For example: Pareto resistance phenomenon, difficult diversity maintenance. Based on the above problems, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, based on Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. Then, a repulsion field method is proposed. The diversity of the population in the objective space is maintained by the repulsion field, so that the population can be evenly distributed on the real Pareto front. Finally, a new concurrent algorithm framework is designed. In the algorithm framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can switch freely among these three states according to its own evolution. The MOEA/TS algorithm is compared with 7 advanced many-objective optimization algorithms. The experimental results show that the MOEA/TS algorithm is more competitive in many-objective optimization problems.

Introduction

In reality, many optimization problems involve multiple conflicting objectives, such as the design of urban public transport routes 1 , production scheduling 2 , securities portfolio management 3 and so on. These types of optimization problems are called multi-objective optimization problems (MOPs). This means that there is no one solution to make all the objectives reach the optimum simultaneously, that is, the optimization of one objective may lead to the deterioration of other objectives 4 , 5 . Consequently, the solutions of MOPs are usually a set of compromise solutions that weigh all objectives. The definition of MOPs is as follows:

Among them, \(f\left(x\right)\) is the m-dimensional objective vector, which contains m conflicting objective functions; \({f}_{i}\left(x\right)\) represents the i-th objective function; x represents the n-dimensional decision variable; \(\Omega \) represents decision space; R m represents the objective space.

In the field of multi-objective optimization, problems with 2 or 3 optimization objectives are called general multi-objective optimization problems (GMOPs). Problems with more than 3 optimization objectives are called many-objective optimization problems (MaOPs) 6 , 7 , 8 . GMOPs aren’t the focus of our attention, as there have been many reports about GMOPs 9 , 10 . On the contrary, MaOPs are the focus of our attention, as there are still some challenges to be solved. The fundamental difference between GMOPs and MaOPs is the number of optimization objectives. Assuming that the number of optimization objectives is m, the probability that one individual dominates another is \(1/{2}^{m-1}\) in theory 11 , 12 . This means that with the increase of the number of optimization objectives, traditional Pareto dominance will fail, Pareto resistance will occur, and most multi-objective optimization algorithms will lose selection pressure in terms of convergence.

In recent years, with the research and exploration of MaOPs, many-objective optimization technology has been developed to a certain extent, and basically 4 mainstream many-objective optimization algorithms have been formed 13 . The first is many-objective optimization algorithm based on dominance. The algorithm modifies the definition of traditional Pareto domination by domination relaxation technique to enhance the selection pressure of the algorithm in terms of convergence. \(\alpha\) -Dominance, \(\upepsilon \) -Dominance and Cone \(\upepsilon \) -Dominance are all common domination relaxation techniques. Compared with traditional Pareto dominance, the effectiveness of dominance relaxation technology has been reported in many works. Therefore, dominance relaxation technology has been widely used to solve MaOPs. However, the current domination relaxation technique also faces two problems: (1) With the increase of the number of optimization objectives, the effect of the domination relaxation technique is getting worse and worse; (2) The domination relaxation technique tends to make the population converge to a certain sub-region of the real Pareto front (PF).

The second is many-objective optimization algorithm based on index. The algorithm guides the selection and evolution of the population by integrating convergence and diversity into one index (such as IGD, HV). Its representative work includes: HypE, MaOEA/IGD, SMS-EMOA. However, the algorithm faces some problems when it is used to solve MaOPs, such as complex index calculation, difficult selection of reference point or reference PF.

The third is many-objective optimization algorithm based on decomposition. The algorithm transforms MaOPs into several single-objective optimization sub-problems through an aggregation function, and then drives the individuals in the neighborhood to update by neighborhood strategy, finally realizes the evolution of the whole population. Its representative work includes: MOEA/D, MOEA/D-D, MOEA/D-DU. However, many-objective optimization algorithm based on decomposition is only suitable for MaOPs with regular PF (such as the DTLZ1 problem). When dealing with MaOPs with irregular PF, many-objective optimization algorithm based on decomposition often performs poorly.

The fourth is many-objective optimization algorithm based on hybrid strategy. The algorithm adopts different search strategies in different environments (different stages or different sub-populations), and uses the advantages of their respective search strategies to deal with complex MaOPs. Its representative work includes: AHM, eMOFEOA, CPSO. In many reports, many-objective optimization algorithm based on hybrid strategy is more suitable for solving MaOPs.

According to the above analysis, this paper considers using the many-objective optimization algorithm based on hybrid strategy, and further proposes the many-objective evolutionary algorithm based on three states (MOEA/TS). The innovations and contributions of this paper are as follows: (1) A feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. (2) Based on the Pareto front layer, the concept of “individual importance degree” is proposed. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. (3) A repulsion field method is proposed. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF. (4) Design a new concurrent algorithm framework. In the framework, the algorithm is divided into three states, and each state focuses on a specific task. The population can freely switch among these three states according to its own evolution.

The remainder of this paper is organized as follows: Sect. " Preparatory work " introduces the basic definition, related work and research motivation. Sect. " Basic definition " introduces each part of the MOEA/TS algorithm in detail. Sect. " Related work " introduces the test results of MOEA/TS algorithm and 7 advanced many-objective optimization algorithms on various test problems, and then analyzes and summarizes them according to the test results. Sect. " Many-objective optimization algorithm based on dominance "summarizes this article and looks forward to future work.

Preparatory work

Basic definition.

In this section, we will introduce some basic definitions related to many-objective optimization technology.

Definition of dominance: if solution x isn’t worse than solution y in all objectives and solution x is better than solution y in at least one objective, it is said that x dominates y . That is, if \(\forall i\in \left\{\text{1,2},3,...,m\right\}\) satisfies \({f}_{i}\left(x\right)\le {f}_{i}\left(y\right)\) and \(\exists j\in \left\{\text{1,2},3,...,m\right\}\) satisfies \({f}_{j}\left(x\right)<{f}_{j}\left(y\right)\) , it is said that x dominates y .

Definition of non-dominated solution: if there are no solutions that can dominate x in the decision space, then x is called a Pareto optimal solution or a non-dominated solution. That is, if \(\nexists {x}^{*}\in \Omega \) makes x* dominate x , then x is called a Pareto optimal solution or a non-dominated solution.

Definition of Pareto optimal solution set: the set composed of Pareto optimal solutions is called the Pareto optimal solution set (PS). The mathematical description of PS is as follows:

Definition of Pareto front: the mapping of PS in the objective space is called Pareto front (PF). The mathematical description of PF is as follows:

The goal of the many-objective optimization technology is to find a set of non-dominated solutions that are close to the real PF (convergence) and make them well distributed on the real PF (diversity).

Related work

In recent years, many scholars have conducted in-depth research and exploration in the many-objective optimization technology.

Many-objective optimization algorithm based on dominance

Considering the limitations of Pareto dominance relationship in high-dimensional objective space, Zhou et al 14 proposed a many-objective optimization algorithm based on dominance relation selection. Firstly, they introduced an angle domination relationship with higher selection pressure based on the traditional Pareto domination relationship, and designed a new dominance selection strategy. Additionally, they proposed an angle-based individual distribution method to ensure even population distribution in the objective space. The algorithm shows strong competitiveness in solving MaOPs. Wang et al 15 believed that as the number of objectives increased, the traditional dominance relationship would become invalid. Therefore, they proposed a modified dominance relation. That is, they used penalty-based adaptive matrix regions to assist the traditional dominance relationship. Further, for MaOPs with irregular Pareto fronts, they introduced a population-based adaptive adjustment method to replace the predefined weight vector. On this basis, for MaOPs, they developed a many-objective optimization algorithm based on modified dominance relation and adaptive adjustment method. Zhang et al 16 believed that the current many-objective optimization algorithms focused too much on convergence, which would cause the population to converge to a certain sub-region of the real Pareto front. In order to solve this problem, they proposed a many-objective optimization algorithm based on double distance domination. In this algorithm, double distance can not only measure the convergence of the algorithm to adapt to different Pareto fronts, but also combine angle-based niche technology to emphasize the diversity of the algorithm. In addition, they also designed a special mutation operator. This operator can generate high-quality individuals in sparse areas to improve the diversity of the algorithm.

Many-objective optimization algorithm based on index

Aiming at the high complexity problem of hypervolume computation, Shang et al 17 proposed a new multi-objective evolutionary algorithm (MOEA) based on R2 index, namely the R2HCA-EMOA algorithm. The core idea of this algorithm is to use R2 index variables to approximate the contribution of hypervolume. The basic framework of the proposed algorithm is similar to that of SMS-EMOA. In order to improve the calculation efficiency of the algorithm, the utility tensor structure is introduced to calculate R2 index variables. In addition, the normalization mechanism is incorporated into the R2HCA-EMOA algorithm to improve its performance. Zhang et al 18 believed that the loss of selection pressure was the core reason for the poor performance of the algorithm. In order to solve this problem, they proposed a many-objective optimization algorithm based on fitness evaluation and hierarchical grouping. The fitness evaluation method combined the convergence measure based on the cos function and the diversity measure based on angle to create the selection pressure of convergence and diversity. In order to further strengthen the selection pressure, they proposed a hierarchical grouping strategy. Firstly, individuals are divided into different layers by front index, and then individuals in the same layer are divided into different groups by R2 index. Although some indexes can approximate the contribution of HV, However, Nan et al 19 believed that the key of performance evaluation was to find the worst solution rather than accurately approaching the HV value of each solution. In order to improve the ability to identify the worst solution, they proposed a two-stage R2 index evaluation method. In the first stage, the R2 indexes of all individuals are roughly evaluated to select some candidate solutions. In the second stage, these candidate solutions are accurately evaluated. Finally, they proposed a many-objective optimization algorithm based on the two-stage R2 index.

Many-objective optimization algorithm based on decomposition

In order to balance the convergence and diversity of the decomposition-based algorithm and reduce its dependence on the real PF direction, Wu et al 20 developed a many-objective optimization algorithm based on antagonistic decomposition method. This method utilizes the complementary characteristics of different sub-problems in a single example. Specifically, two populations are co-evolved by two sub-problems with different contours and opposite search directions. In order to avoid allocating redundant computing resources to the same area of PF, two populations are matched into one-to-one pairing according to their working areas on PF. In mating selection, each solution pair can only contribute one parent at most. In order to improve the performance of decomposition-based algorithms, Fan et al 21 proposed a differential multi-objective optimization algorithm based on decomposition. Firstly, they designed a neighborhood intimacy factor to improve the diversity of the algorithm based on the characteristics of neighborhood search. Then, they introduced a Gaussian mutation operator with dynamic step size to enhance the algorithm’s ability to escape from local optimal regions and improve convergence. Finally, they combined a difference strategy with the decomposition-based multi-objective optimization algorithm to further strengthen its evolutionary ability. Peng et al 22 believed that data dimensionality reduction could be applied to the objective space. Based on this consideration, they proposed a many-objective optimization algorithm based on projection. Firstly, they used the idea of data dimensionality reduction and spatial decomposition to divide the objective space into projection plane and free dimension. Then, a double elite strategy was used to maintain the balance between convergence and diversity of the algorithm. Finally, the algorithm based on decomposition was used as the algorithm of free dimension to solve MaOPs.

Many-objective optimization algorithm based on hybrid strategy

Aiming at convergence problem and diversity problem of the algorithm, Sun et al 23 proposed a many-objective optimization algorithm based on two independent stages. The algorithm deals with convergence and diversity problems in two independent and successive stages. Firstly, they introduced a non-dominated dynamic weight aggregation method, which is capable of identifying the Pareto optimal solutions of MaOPs. Then, they used these solutions to learn the Pareto optimal subspace in order to solve the convergence problem. Finally, the diversity problem was solved by using reference lines in the Pareto optimal subspace. Considering the advantages of the multi-objective and multi-population (MPMO) framework in solving MaOPs, Yang et al 24 proposed an algorithm based on the MPMO framework. The algorithm adopts the deviation sorting (BS) method to solve MaOPs, so as to obtain good convergence and diversity. In terms of convergence, the BS method is applied to each population in the MPMO framework, and the effect of non-dominant sorting is enhanced by the optimization objectives of the corresponding population. In terms of diversity, the maintenance method based on reference vector is used to save the diversity solutions. Aiming at the five-objective job shop scheduling problem (JSSP), Liu et al 25 proposed a new genetic algorithm based on the MPMO framework. Firstly, five populations are used to optimize five objectives, respectively. Secondly, in order to prevent each population from focusing only on its corresponding single objective, an archive sharing technology (AST) is proposed to store the elite solutions collected from five populations, so that the population can obtain the optimization information of other objectives from the archive. Thirdly, the archive updating strategy (AUS) is proposed to further improve the quality of the solutions in the archive.

Research motivation

Based on the related work, we believe that there are still the following problems in the current many-objective optimization technology:

(1) The diversity and convergence of the algorithm are difficult to balance. Most algorithms can’t coordinate the balance between them well, and they either emphasize convergence or diversity too much, which leads to poor quality of the non-dominated solution set.

(2) It is difficult to maintain the convergence of the algorithm. When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail.

(3) It is difficult to maintain the diversity of the algorithm. Especially when the real PF is complex or the latitude of the objective space is high, individuals may have the clustering effect, and the population may not be evenly distributed on the real PF.

(4) The evolution efficiency of the algorithm is low. The traditional evolution operators have strong randomness and low evolution efficiency, and aren’t suitable for dealing with MaOPs.

Therefore, solving these problems and providing a good many-objective optimization algorithm constitute the research motivation of this paper.

For problem 1, some work attempts to separate the convergence optimization and diversity optimization of the algorithm, thus designing a concurrent algorithm architecture. Concurrent algorithm architecture means that only one of convergence or diversity is considered in one iteration instead of considering both convergence and diversity simultaneously. In order to solve GMOPs, Professor Ye Tian 26 tried to design a concurrent algorithm architecture and proposed the MSEA algorithm, and the experimental results were satisfactory. Therefore, it seems to be a feasible path to solve MaOPs by using concurrent algorithm architecture. However, recent research 23 shows that in MaOPs, the concurrent algorithm architecture seems to be unstable, and the experimental results fluctuate greatly (such as MaOEA/IT algorithm). Because when the algorithm only considers the convergence of the population, it often affects the diversity of the population; Similarly, when the algorithm only considers the diversity of the population, it often affects the convergence of the population. If a coordination intermediary can be added to the concurrent algorithm architecture to alleviate the contradiction between diversity and convergence, the concurrent algorithm architecture will become stable and its superiority will be truly reflected. Based on this motivation, this paper proposes a new concurrent algorithm framework. In the new algorithm framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the population convergence; Diversity maintenance state is responsible for improving population diversity; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution.

For problem 2, some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. If we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Obviously, compared with domination relaxation technique, individual importance degree has greater advantages.

For problem 3, the traditional diversity maintenance technology isn’t suitable for high-dimensional objective space. For instance: the niche method, the density evaluation method, and the weight vector method. In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination. Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

For problem 4, traditional evolution operators aren’t suitable for dealing with MaOPs. Because traditional evolution operators have strong randomness and low evolution efficiency. For instance: the binary crossover operator, the polynomial mutation operator, and the differential evolution operator. In principal component analysis, the decomposition of the covariance matrix and correlation matrix is a very important step. By decomposing the covariance matrix or the correlation matrix, we can obtain a set of orthogonal bases. These orthogonal bases are the most important features of the original data 27 . Therefore, this paper designs a feature extraction operator based on Cholesky decomposition 28 . The feature extraction operator can be understood as a feature extractor. It can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Obviously, compared with traditional evolution operators, the feature extraction operator has higher evolution efficiency.

MOEA/TS algorithm

Feature extraction operator.

The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. The workflow of the feature extraction operator is shown in Fig.  1 .

The workflow of feature extraction operator.

In the first step, W high-quality solutions \({x}^{1},{x}^{2},{x}^{3},...,{x}^{W}\) are selected from the population. These W solutions will form the high-quality solution set S.

In the second step, calculate the mean \(\overline{x}\) and covariance matrix A of the high-quality solution set S:

Among them, \({x}^{i}={\left({x}_{1}^{i},{x}_{2}^{i},{x}_{3}^{i},...,{x}_{n}^{i}\right)}^{T}, i\in (1,...,W); {x}_{j}={\left({x}_{j}^{1},{x}_{j}^{2},{x}_{j}^{3},...,{x}_{j}^{W}\right)}^{T}, j\in (1,...,n)\)

In the third step, Cholesky decomposition is performed on the covariance matrix A. That is, the covariance matrix A is decomposed into the product of the lower triangular matrix and the transposition of the lower triangular matrix. Assuming that the lower triangular matrix is L, there is

Through formula \(A=L*{L}^{T}\) , we can calculate \({a}_{11}={l}_{11}^{2}\) , that is, \({l}_{11}=\sqrt{{a}_{11}}\) . Then, according to \({a}_{i1}={l}_{i1}*{l}_{11}\) , we can get \({l}_{i1}={a}_{i1}/{l}_{11}\) , so we can get the first column element of matrix L .

Assuming that we have calculated the first k-1 column elements of the matrix L. Through

In this way, we can solve the k-th column element of matrix L through the first k-1 column elements of matrix L. Then, we can solve matrix L by recursion.

In the fourth step, the sampling vector \(s={\left({s}_{1},...,{s}_{n}\right)}^{T}\) is generated by Gaussian distribution \(N\left(\text{0,0.7}\right)\) . Then, a feature solution is generated.

Among them, \({x}^{feature}={\left({x}_{1}^{feature},...,{x}_{n}^{feature}\right)}^{T}\)

It should be noted that the standard deviation std is an important parameter of the Gaussian distribution. In this paper, the standard deviation std is set to 0.7. The parameter analysis verifies that 0.7 is a reasonable standard deviation. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

In the fifth step, assuming that the selected individual is \({x}^{i}({x}_{1}^{i},...,{x}_{n}^{i})\) . Based on binary crossover operator 29 and feature solution, the formula of generating offspring individual is as follows:

Among them, \({c({c}_{1},...,{c}_{n})}^{T}\) is the offspring individual. \({\beta }_{k}\) is dynamically determined by the feature factor \(\mu \) :

Among them, \(rand\) is used to generate a random number between 0 and 1; \(r and i(\text{0,1})\) is used to generate 0 or 1 randomly.

For the design principle of formula ( 15 ), please browse the Supplementary Information Document.

In the sixth step, the individual \(c\) is detected and repaired. When some components in individual \(c\) exceed the upper bound or lower bound, these components need to be repaired. The repair formula is as follows:

Among them, \({c}_{i}^{u}\) , \({c}_{i}^{l}\) represent the upper bound and lower bound of the i-th component of individual \(c\) , respectively. \({{c}{\prime}({c}_{1}{\prime},...,{c}_{n}{\prime})}^{T}\) represents the repaired individual.

Individual importance degree based on the Pareto front layer

When the number of optimization objectives is large, the algorithm will produce Pareto resistance, and the traditional Pareto dominance may fail. Some scholars try to modify the definition of traditional Pareto dominance by using dominance relaxation technology to enhance the selection pressure of the algorithm in terms of convergence. However, with the increase of the number of optimization objectives, the effect of dominance relaxation technology is getting worse and worse. They only focus on the modification of Pareto domination definition, but ignore the difference between objective values. Figure  2 shows 4 non-dominant individuals. Among them, individual B is the closest to Origin O, individual C is second, individual A is third, and individual D is the farthest from Origin O. This means that in the population, individual B is the most important, individual C is the second most important, individual A is the third most important, and individual D is the least important. In addition, we can also find from Fig.  2 that there is a significant difference between the objective values of individual B and the objective values of other individuals, that is, \(\sum_{X\in \left\{A,C,D\right\}}\sum_{i=1}^{2}{f}_{i}(B)-{f}_{i}(X)\) is the smallest. This shows that there is a special relationship between the importance of individuals and the difference of the objective values. Based on this discovery, if we can distinguish the importance of different individuals by using the difference between the objective values, we can further create the selection pressure of the algorithm in terms of convergence, and finally Pareto resistance will be eliminated. Therefore, based on Pareto front layer, we propose the concept of “individual importance degree”.

Schematic diagram of individual importance.

Assuming that there are n solutions in a certain Pareto front layer, the objective function values of these solutions are normalized to [0,1] based on the maximum and minimum values of each objective function. \({f}_{k}{\prime}({x}^{i})\) represents the k-th normalized objective function value of individual \({x}^{i}\) .

Define the Pareto dominance function

The trend of the Pareto dominance function is shown in Fig.  3 .

The trend of the Pareto dominance function.

Pareto dominance function can be used to reflect the dominance degree among different individuals. For example, \(PDF({f}_{k}{\prime}({x}^{i})-{f}_{k}{\prime}({x}^{j}))\) represents the dominance degree of individual \({x}^{i}\) to individual \({x}^{j}\) on the k-th objective function; \(PDF({f}_{k}{\prime}({x}^{j})-{f}_{k}{\prime}({x}^{i}))\) represents the dominance degree of individual \({x}^{j}\) to individual \({x}^{i}\) on the k-th objective function; Obviously, the greater the dominance degree, the better one individual is than another on one objective function. Therefore, on one objective function, the dominance degree of one individual to another can be expressed as:

On this basis, the dominance degree of one individual to another can be expressed as:

Further, the importance degree of one individual to another can be expressed as:

Importance degree can indicate the importance of one individual to another. The greater the importance degree, the more important one individual is than another.

Since a certain Pareto front layer has n solutions, each solution needs to be compared with other n-1 solutions, so as to construct n-1 competing pairs. Assuming that an individual is \({x}^{i}\) , then the n-1 competing pairs are \(\left({x}^{i},{x}^{1}\right),\left({x}^{i},{x}^{2}\right),...,\left({x}^{i},{x}^{j}\right),...,\left({x}^{i},{x}^{n}\right)\) , respectively (note: \(i\ne j\) ). Thus, the importance degree of individual \({x}^{i}\) to the other n-1 individuals is \(Imp\left({x}^{i},{x}^{1}\right),Imp\left({x}^{i},{x}^{2}\right),...,Imp\left({x}^{i},{x}^{j}\right),...,Imp\left({x}^{i},{x}^{n}\right)\) , respectively (note: \(i\ne j\) ).

Finally, the importance degree of the individual \({x}^{i}\) can be expressed as:

The importance degree of one individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer. The greater the importance degree of one individual, the more important it is in the same Pareto front layer.

Figure 4 shows the use of individual importance degree. Firstly, based on a certain Pareto front layer, the competition pools and competition pairs are constructed. Then, the individual importance degree of different individuals is calculated by Formula ( 24 ). Finally, the importance of different individuals in the same Pareto front layer is obtained.

The use of individual importance degree.

Taking the two-objective optimization problem as an example, it is assumed that there are 4 non-dominant individuals. They are \(A\left(\text{17,5}\right)\) , \(B\left(\text{9,7}\right)\) , \(C\left(\text{7,15}\right)\) and \(D\left(\text{5,25}\right)\) , respectively. It means that these 4 individuals belong to the first non-dominant layer, and their advantages and disadvantages can’t be compared by the non-dominant rank. The distribution of 4 individuals in the objective space is shown in Fig.  5 (a).

The distribution of 4 individuals.

In order to better compare the advantages and disadvantages of these 4 individuals, we use the individual importance degree to deal with these 4 individuals. Firstly, the objective function values of these 4 individuals are normalized to [0,1]. After normalization, the coordinates of these 4 individuals are \(A\left(\text{1,0}\right)\) , \(B\left(\text{0.333,0.1}\right)\) , \(C\left(\text{0.167,0.5}\right)\) and \(D\left(\text{0,1}\right)\) , respectively. The distribution of 4 individuals in the normalized objective space is shown in Fig.  5 (b). Next, according to Fig.  4 , the competition pools and competition pairs are constructed. Then, according to formula ( 22 ) and formula ( 23 ), the \(P\left({x}^{i},{x}^{j}\right)\) and \(Imp\left({x}^{i},{x}^{j}\right)\) of each competition pair are calculated. The calculation results are shown in Table 1 . Finally, according to the formula ( 24 ), the importance degree of these 4 individuals is 0.1918, 0.9488, 0.6673 and 0.1921, respectively. The results show that individual B is the most important, individual C is the second most important, individual D is the third most important, and individual A is the least important. This result is consistent with the intuitive perception that we get from Fig.  5 (b). Based on the above example, we believe that the concept of individual importance degree and related process are effective and can achieve the desired goals.

Repulsion field method

In the field of microphysics, when the distance between particles is too close, repulsion will push the particles away from their neighbors. On the contrary, when the distance between particles is too great, the repulsion will decrease and the particles tend to be close to the neighboring particles. This way makes the distribution of particles present a state of mutual coordination (As shown in Fig.  6 ). Based on the characteristics of particle distribution, a repulsion field method is proposed in this paper. The repulsion field is used to maintain the diversity of the population in the objective space, so that the population can be evenly distributed on the real PF.

The uniform distribution of microscopic particles.

Firstly, according to the maximum and minimum values of each objective function, the objective function values of all solutions in the population are normalized to [0,1]. \({f}_{k}{\prime}({x}^{i})\) represents the k-th normalized objective function value of individual \({x}^{i}\) .

Then, a repulsion potential field with repulsion radius r is constructed around each individual. Assuming that a repulsion potential field has been constructed for individual \({x}^{i}\) , then all individuals within the repulsion potential field will be subject to the repulsion potential from individual \({x}^{i}\) . The magnitude of the repulsion potential depends on the distance between other individuals and individual \({x}^{i}\) . When other individuals are outside the repulsion potential field of individual \({x}^{i}\) , the repulsion potential is 0. When other individuals are within the repulsion potential field of individual \({x}^{i}\) , the closer the other individuals are to individual \({x}^{i}\) , the greater the repulsion potential that they obtain. Assuming that there is individual \({x}^{j}\) , then the repulsion potential that individual \({x}^{j}\) obtains is

Among them, \(\rho \) is the gain coefficient of the repulsion potential field, usually set to 1; \(r\) is the radius of the repulsion potential field; \(dis\left({x}^{j},{x}^{i}\right)\) represents the euclidean distance between individual \({x}^{j}\) and individual \({x}^{i}\) in the objective space. The formula is as follows:

Further, the repulsion \(Rep\left({x}^{j},{x}^{i}\right)\) that individual \({x}^{j}\) obtains is the negative gradient of the repulsion potential \(Repfield\left({x}^{j},{x}^{i}\right)\) . The formula is as follows:

It means that when \(dis\left({x}^{j},{x}^{i}\right)\le r\) , the smaller \(dis\left({x}^{j},{x}^{i}\right)\) is, the larger \(Rep\left({x}^{j},{x}^{i}\right)\) is. when \(dis\left({x}^{j},{x}^{i}\right)>r\) , \(Rep\left({x}^{j},{x}^{i}\right)=0\) .

Based on the repulsion potential field, the total repulsion potential that individual \({x}^{j}\) obtains is

Finally, the total repulsion that individual \({x}^{j}\) obtains is

It should be noted that the repulsion potential and repulsion proposed in this paper are both vectors. It means that repulsion potential and repulsion have both magnitude and direction. The addition of different repulsion is the vector synthesis of repulsion, rather than the pure numerical addition. This is also an obvious feature that the repulsion field method is different from other scalar function methods (such as niche method). Figure 7 shows the vector synthesis process of repulsion in a two-dimensional space environment. Among them, F SUM is the total repulsion that individual A obtains; F BA is the repulsion generated by individual B to individual A; F CA is the repulsion generated by individual C to individual A.

The vector synthesis process of repulsion.

In the repulsion field method, the individual with large repulsion usually means that the individual is located in the multiple repulsion potential field that other individuals construct. It indicates that the individual is located in a dense area in the objective space and is close to other individuals. Therefore, individuals with large repulsion aren’t conducive to maintaining population diversity. Naturally, we hope that individuals with large repulsion can move away from dense areas in the objective space along the direction of repulsion. Based on this idea, firstly, we need to find some individuals closest to the direction of the repulsion to construct a high-quality solution set. Then, the feature extraction operator is used to extract the location features of the high-quality solution set. Finally, based on these features, individuals with large repulsion can evolve along the direction of repulsion. As shown in Fig.  8 , individual D and individual E are the individuals closest to the direction of repulsion. The feature extraction operator is used to extract the position features of these two individuals. Based on these features, individual A evolves into individual A*, which is far away from the previous dense area.

Individual A is far away from the dense area.

It should be noted that the feature extraction operator has the randomness caused by Gaussian sampling. Therefore, the evolution of individuals also has a certain degree of randomness.

Framework of MOEA/TS algorithm

The framework of the MOEA/TS algorithm is shown in Fig.  9 . Firstly, the relevant parameters of the algorithm are initialized; secondly, judge which state the algorithm is in. If the algorithm is in the convergence maintenance state, the following steps are adopted to improve the convergence of the algorithm: (1) Randomly select the parent individual. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the diversity maintenance state, the following steps are adopted to improve the diversity of the algorithm: (1) Select the individual with the worst diversity in the population. (2) Use feature extraction operator to generate offspring individuals. (3) If the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual. If the algorithm is in the coordination state, the following steps are adopted to coordinate the convergence and diversity of the algorithm: (1) Randomly select the parent individual. (2) Use the Gaussian mutation operator to generate offspring individuals. (3) If the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual. Then, it is judged whether the algorithm has completed the i-th iteration. If the algorithm doesn’t complete the i-th iteration, the corresponding maintenance step or coordination step is re-executed. If the algorithm completes the i-th iteration, the current state of the algorithm is updated. Finally, it is judged whether the algorithm ends. If the algorithm doesn’t end, the corresponding maintenance step or coordination step is performed according to the current state of the algorithm. If the algorithm is finished, the population is output.

The framework of MOEA/TS algorithm.

Description of MOEA/TS algorithm

Main framework.

This section describes the main framework of the MOEA/TS algorithm. The pseudo-code of the main framework of the MOEA/TS algorithm is shown in Algorithm 1. The main steps include: in line (1), initializing population P, repulsion field radius r, and state value (state=1 means that the algorithm is in convergence maintenance state, state=2 means that the algorithm is in diversity maintenance state, and state=3 means that the algorithm is in coordination state.); In line (2), the Front value, Imp value and Rep value of each solution in population P are calculated (The Front value is calculated by the fast non-dominated sorting method.); In line (3), it is judged whether the algorithm meets the termination condition (The termination condition is usually the maximum iterations.); In line (4), the count value is initialized. The count value is used to count the number of updated solutions in the i-th iteration; in lines (5)-(11), according to the current state of the algorithm, the update way of the population is selected. When state=1, the convergence of the population is updated. When state=2, the diversity of the population is updated. When state=3, the convergence and diversity of the population are coordinated; in line (12), the state value of the algorithm is updated according to the current state of the algorithm and the count value.

Main framework.

Convergence maintenance

This section mainly describes the convergence maintenance of the population. The pseudo-code of convergence maintenance is shown in Algorithm 2. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In lines (3)–(4), based on Front, Imp, the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the parent individual; In line (6), the individual with the worst convergence in the population is found; In lines (7)–(13), if the offspring individual is superior to the individual with the worst convergence in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst convergence in the population, the flag is marked as 0. Among them, the flag value is used to indicate whether the population P has changed (flag=0 means that the population P hasn’t changed, flag=1 means that the population P has changed.); In lines (14)–(16), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Convergence maintenance.

Diversity maintenance

This section mainly describes the diversity maintenance of the population. The pseudo-code of diversity maintenance is shown in Algorithm 3. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), the individual with the worst diversity in the population is found; In line (3), according to the direction of total repulsion that the worst individual obtains, the distance \({dis}_{j}\) can be calculated; In line (4), based on \({dis}_{j}\) , the high-quality solution set S is constructed; In line (5), the feature extraction operator is used to extract the features of the high-quality solution set S, and then assist the evolution of the worst individual; In lines (6)–(12), if the offspring individual is superior to the individual with the worst diversity in the population, the worst individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the individual with the worst diversity in the population, the flag is marked as 0; In lines (13)–(15), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Diversity maintenance.

Coordination of convergence and diversity

This section mainly describes the coordination of convergence and diversity of the population. The pseudo-code of coordination of convergence and diversity is shown in Algorithm 4. The main steps include: in line (1), the algorithm enters the i-th iteration; In line (2), one parent individual is randomly selected from population P; In line (3), based on the parent individual, the Gaussian mutation operator is used to generate the offspring solution; In lines (4)–(10), if the offspring individual is superior to the parent individual in convergence and diversity, the parent individual is replaced by the offspring individual and flag is marked as 1. If the offspring individual is inferior to the parent individual in convergence and diversity, the flag is marked as 0; In lines (11)–(13), it is judged whether flag equals 1. If flag equals 1, the count value is updated, and the Front value, Imp value and Rep value of each solution in population P are updated.

Coordination.

This section mainly describes the feature extraction operator. The pseudo-code of the feature extraction operator is shown in Algorithm 5. The main steps include: in line (1), the features \(\overline{x},L\) of the high-quality solution set S are extracted by formula ( 4 ) and formula ( 13 ); In line (2), the sampling vector \(s={\left({s}_{1},...,{s}_{n}\right)}^{T}\) is generated by the Gaussian distribution \(N\left(\text{0,0.7}\right)\) ; In line (3), based on \(\text{s},\overline{x},L\) , the feature solution \({x}^{feature}\) is generated by formula ( 14 ); In line (4), based on parent, \({x}^{feature}\) , the offspring solution O is generated by formula ( 15 ).

Feature extraction.

Update of algorithm state

In this paper, the algorithm state is further updated according to the current state of the algorithm and the stability of the population. The pseudo-code of the update of the algorithm state is shown in Algorithm 6. When the algorithm is in the convergence maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of convergence, then the algorithm turns to the diversity maintenance state; When the algorithm is in the diversity maintenance state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of diversity, then the algorithm turns to the coordination state; When the algorithm is in the coordination state and the number of updated solutions in the i-th iteration is less than or equal to 5%*N, it is considered that the population tends to be stable in terms of coordination, then the algorithm turns to the convergence maintenance state. It should be noted that the threshold value T is a key parameter in measuring whether the population tends to be stable or not. In this paper, the threshold value T is set to 5%. The parameter analysis verifies that 5% is a reasonable threshold. For more details on parameter analysis, please browse the experiment chapter (Parameter sensitivity analysis section).

Determination state.

Computational complexity of one iteration of MOEA/TS algorithm

Assuming that the size of the population is N, the number of the objective function is m, the dimension of the decision variable is n, and the size of the high-quality solution set is W, then the computational complexity of Rep is O(mN2), the computational complexity of Front is O(mN2), and the computational complexity of Imp is O(mN2). The core steps of the feature extraction operator (Algorithm 5) include the construction of the covariance matrix and Cholesky decomposition. The computational complexity of covariance matrix construction is O(Wn2) and the computational complexity of Cholesky decomposition is O(n3/6). Therefore, the computational complexity of the feature extraction operator (Algorithm 5) is O(Wn2+n3/6). The core steps of convergence maintenance (Algorithm 2) include population ranking, feature extraction operator, selection of the worst individual, and updating of Front, Imp and Rep. Their computational complexity is O(N2), O(Wn2+n3/6), O(N), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of convergence maintenance (Algorithm 2) is O(N(N2+Wn2+n3/6+N+3mN2)). The core steps of diversity maintenance (Algorithm 3) include selection of the worst individual, distance calculation, population ranking, feature extraction operator, and updating of Front, Imp and Rep. Their computational complexity is O(N), O(nN), O(N2), O(Wn2+n3/6), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of diversity maintenance (Algorithm 3) is O(N(N+nN+N2+Wn2+n3/6+3mN2)). The core steps of coordination of convergence and diversity (Algorithm 4) include the Gaussian mutation operator, and updating of Front, Imp and Rep. Their computational complexity is O(n), O(mN2), O(mN2), O(mN2), respectively. Therefore, the computational complexity of coordination of convergence and diversity (Algorithm 4) is O(N(n+3mN2)). The computational complexity of Determination State (Algorithm 6) is O(1). Based on the above computational complexity analysis, the computational complexity of one iteration of the MOEA/TS algorithm is max{O(N(N2+Wn2+n3/6+N+3mN2)), O(N(N+nN+N2+Wn2+n3/6+3mN2)), O(N(n+3mN2))}+O(1)≈max{O(NWn2+Nn3+mN3), O(NWn2+Nn3+mN3), O(mN3)}= O(NWn2+Nn3+mN3). In this paper, N>>max{W, n, m}. Therefore, the computational complexity of the MOEA/TS algorithm is O(mN3). As a reference algorithm, the computational complexity of the NSGA-III algorithm is O(mN2). The computational complexity of the MOEA/TS algorithm is an order of magnitude higher than that of the NSGA-III algorithm. This shows that the MOEA/TS algorithm is an expensive many-objective optimization algorithm.

It should be noted that although MOEA/TS algorithm has a higher computational complexity. But compared with the NSGA-III algorithm, the MOEA/TS algorithm also has greater advantages. In terms of convergence optimization, the NSGA-III algorithm adopts the traditional definition of Pareto domination. Obviously, the traditional definition can’t solve the problem of Pareto resistance. MOEA/TS algorithm uses the concept of “individual importance degree”. Individual importance degree can solve the problem of Pareto resistance. In terms of diversity optimization, the NSGA-III algorithm uses predefined reference points. The predefined reference points can’t solve the problem that the population can't be evenly distributed on the real PF in the high-dimensional objective space. MOEA/TS algorithm uses the repulsion field method. The repulsion field method can solve the problem that the population can’t be evenly distributed on the real PF in the high-dimensional objective space. In terms of algorithm architecture, the NSGA-III algorithm adopts the serial algorithm architecture. The serial algorithm architecture is difficult to coordinate the convergence optimization and diversity optimization of the algorithm. MOEA/TS algorithm adopts the concurrent algorithm architecture. The concurrent algorithm architecture can coordinate the convergence optimization and diversity optimization of the algorithm. In terms of operators, the NSGA-III algorithm uses the traditional binary crossover operator and polynomial mutation operator. The evolutionary ability of these two operators is weak. MOEA/TS algorithm uses feature extraction operator. Feature extraction operator has strong evolutionary ability. Therefore, the MOEA/TS algorithm has better performance than the NSGA-III algorithm. The comparison results support our conclusion. For the comparison results of these two algorithms, please browse Supplementary Information Document.

Experimental results and analysis

Experimental settings, configuration of experimental software and hardware.

The hardware and software configurations of the experiment are shown in Table 2 . Among them, PlatEMO 30 is a professional many-objective optimization experiment platform. The platform includes multiple test function sets and many-objective optimization algorithms.

Test function

The test functions used in the experiment include: DTLZ test function set (DTLZ1-7), MAF test function set (MAF1-6) and WFG test function set (WFG1-9). Literature 31 describes the characteristics of related test functions. The parameter settings of the related test functions are shown in Table 3 .

Comparison algorithm

In order to verify the performance of MOEA/TS algorithm in the many-objective optimization field, this paper compares MOEA/TS algorithm with 7 advanced many-objective optimization algorithms. These 7 many-objective optimization algorithms include: VMEF 32 , BiGE-BEW 33 , MOEA/DG 34 , MOEA/D 35 , LSMaODE 36 , MaOEA/IT 23 and MaOEA/IGD 37 .

For all test cases, Wilcoxon rank sum test at 5% significance level 38 is used to compare the significance of the difference between the MOEA/TS algorithm and the comparison algorithms. The symbol “+” indicates that the comparison algorithms are significantly better than the MOEA/TS algorithm; the symbol “-“indicates that the comparison algorithms are significantly inferior to the MOEA/TS algorithm. The symbol “=” indicates that there is no significant difference between the MOEA/TS algorithm and the comparison algorithms.

Performance evaluation

In the aspect of performance evaluation, this paper uses inverted generational distance plus (IGD+) and hypervolume (HV) 39 to measure the performance of many-objective optimization algorithm. The smaller the IGD+ value that the algorithm obtains, the better the performance of the algorithm. The larger the HV value that the algorithm obtains, the better the performance of the algorithm.

In order to facilitate observation, we provide the normalized HV value of each algorithm relative to the best HV result. This normalization makes all the results lie in the range [0,1], and 1 represents the best value.

Considering the length of the paper, we only show the IGD+ values of different algorithms in the experiment chapter. For the HV values of different algorithms, please browse the Supplementary Information Document.

Parameter setting

In terms of algorithm parameters, according to some existing parameter research results 13 , 40 , the feature factor \(\mu \) is set to 20 in this paper. According to the parameter sensitivity analysis, the number of high-quality solutions W is set to 9 in this paper. The parameter sensitivity analysis of W is detailed in the subsequent chapters.

The algorithm parameters of the 7 comparison algorithms are determined according to the best parameters provided by the corresponding literature.

Performance comparison under benchmark test functions

Performance comparison under dtlz test function set.

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 4 . As can be seen from Table 4 , MOEA/TS algorithm wins the first place in 15 test cases; BiGE-BEW algorithm wins the first place in 5 test cases; MOEA/D algorithm wins the first place in 15 test cases. In the 35 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 21, 27, 25, 16, 32, 35 and 31, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 5, 5, 15, 1, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 8, 3, 5, 4, 2, 0 and 4, respectively. Therefore, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm have the best performance. The performance of VMEF algorithm, MOEA/DG algorithm, BiGE-BEW algorithm and LSMaODE algorithm decreases in turn. The performance of MaOEA/IGD algorithm and MaOEA/IT algorithm is similar and the worst.

Based on Table 4 , we further analyze the performance of these algorithms. In the DTLZ test function set, MOEA/TS algorithm performs poorly on DTLZ1, DTLZ5 and DTLZ6 test functions. The possible reasons are that the DTLZ1 test function has multiple local optima, and the DTLZ5 and DTLZ6 test functions have a narrow convergence curve. In the DTLZ1 test function, although the repulsion field method of the MOEA/TS algorithm makes the population widely distributed. However, its population distribution isn’t uniform and regular. The population distribution of some algorithms using predefined weight vectors is uniform and regular. In the DTLZ5 and DTLZ6 test functions, the coordination mechanism of MOEA/TS algorithm fails. The narrow convergence curve makes the population more concentrated, but the repulsion field method will disperse the population. The coordination mechanism is difficult to play a role.

The real Pareto front of DTLZ test function set is regular and the function complexity isn’t high. Therefore, algorithms with better diversity may be more popular. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. Therefore, it has good performance. VMEF algorithm uses different convergence ranking methods to deal with different test problems. Therefore, VMEF algorithm is good in convergence and poor in diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. In theory, the algorithm should perform well. However, there are defects in its convergence and diversity measurement formula. Finally, the experimental results of the algorithm aren’t as good as the expected results. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence and good in diversity. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because the real Pareto front of DTLZ test function set isn’t complex, the advantage of this multi-population algorithm architecture isn’t obvious. Therefore, compared with other algorithms, its performance is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm's performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under MAF test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 5 . As can be seen from Table 5 , MOEA/TS algorithm wins the first place in 10 test cases; BiGE-BEW algorithm wins the first place in 8 test cases; MOEA/DG algorithm wins the first place in 2 test cases; MOEA/D algorithm wins the first place in 5 test cases; LSMaODE algorithm wins the first place in 5 test cases. In the 30 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 18, 25, 21, 20, 27 and 30, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 6, 11, 2, 5, 9, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 2, 1, 3, 4, 1, 2 and 0, respectively. Therefore, in the MAF test function set, MOEA/TS algorithm has the best performance. The performance of BiGE-BEW algorithm, LSMaODE algorithm, VMEF algorithm, MOEA/D algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. The performance of MaOEA/IGD algorithm is the worst.

Based on Table 5 , we further analyze the performance of these algorithms. In the MAF test function set, MOEA/TS algorithm performs poorly on MAF2 and MAF3 test functions. The possible reasons are that the MAF2 test function greatly increases the difficulty of convergence on the basis of the DTLZ2 test function, and the MAF3 test function has a convex Pareto front and many local fronts. In the MAF2 test function, although the MOEA/TS algorithm can recognize the advantage and disadvantage of different individuals in the same front layer, the evolutionary efficiency of the MOEA/TS algorithm isn’t ideal. In other words, after the algorithm is finished, the population still has the large evolution potential in convergence. In the MAF3 test function, MOEA/TS algorithm can effectively deal with the convex Pareto front. However, MOEA/TS algorithm is difficult to deal with multiple local fronts because feature extraction operator of MOEA/TS algorithm is difficult to extract features of multiple local fronts.

MAF test function set is the variety of DTLZ test function set. It adds a lot of characteristics to the DTLZ test function set. For example, degenerate, convex, concave, partial, multimodal, deceptive, et al. Therefore, the MAF test function set is more difficult in terms of convergence and diversity. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. VMEF algorithm uses different convergence ranking methods to deal with different test problems. However, the complex Pareto fronts and diversified characteristics still pose a great challenge to VMEF algorithm. Therefore, the performance of VMEF algorithm is mediocre. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. MOEA/D algorithm can easily deal with the DTLZ test function set. However, its performance isn’t ideal when dealing with more complex MAF test function set. Surprisingly, LSMaODE algorithm shows good performance. We speculate that the possible reason is that the real Pareto front of the MAF test function set is complex, and then the advantages of multi-population algorithm architecture can be reflected. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Performance comparison under WFG test function set

In this paper, each algorithm is executed 30 times to get the average data as shown in Table 6 . As can be seen from Table 6 , MOEA/TS algorithm wins the first place in 27 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 6 test cases; MOEA/DG algorithm wins the first place in 1 test case; LSMaODE algorithm wins the first place in 3 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 26, 29, 42, 45, 39, 45 and 43, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 10, 9, 3, 0, 3, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 9, 7, 0, 0, 3, 0 and 2, respectively. Therefore, in the WFG test function set, MOEA/TS algorithm has the best performance. The performance of VMEF algorithm, BiGE-BEW algorithm, LSMaODE algorithm and MOEA/DG algorithm decreases in turn. The performance of MaOEA/IGD algorithm, MOEA/D algorithm and MaOEA/IT algorithm is similar and the worst.

Based on Table 6 , we further analyze the performance of these algorithms. MOEA/TS algorithm performs well in all WFG test functions. The possible reason is that the problem characteristics of the WFG test function set are bias, fraud and degradation. The WFG test function set is more difficult than the DTLZ test function set. However, the problem characteristics of the WFG test function set don’t include multiple local fronts (From the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with multiple local fronts.). MOEA/TS algorithm can deal with these problem characteristics. Therefore, MOEA/TS algorithm performs well in all WFG test functions. It should be noted that the WFG3 test function has a narrow convergence curve, but the performance of MOEA/TS algorithm is still the best. This is an interesting phenomenon. Because from the previous analysis, we know that MOEA/TS algorithm isn’t good at dealing with test functions with narrow convergence curves (such as DTLZ5 and DTLZ6 test functions). Based on the convergence difficulty of the WFG test function set, we speculate that the performance of the other 7 algorithms is worse, thus highlighting the performance of MOEA/TS algorithm.

Compared with the DTLZ test function set, the MAF test function set is more difficult in terms of convergence and diversity. VMEF algorithm uses different convergence ranking methods to deal with different test problems. This approach helps VMEF algorithm to deal with different problem characteristics. Therefore, the performance of VMEF algorithm is good. Based on the convergence measure and diversity measure, BiGE-BEW algorithm transforms the many-objective optimization problem into a two-objective optimization problem. Although there are some defects in its diversity and convergence measurement formula, BiGE-BEW algorithm shows good performance in convergence when dealing with more complex MaOPs. MOEA/DG algorithm still uses the traditional dominance relationship to maintain the convergence of external archives. Therefore, MOEA/DG algorithm is poor in convergence. MOEA/D algorithm uses predefined weight vectors to maintain diversity and aggregation functions to maintain convergence. This approach isn’t suitable for dealing with test functions with bias characteristic. Therefore, the performance of MOEA/D algorithm is the worst. LSMaODE algorithm divides the population into two subpopulations and uses different strategies to optimize them. Because most WFG test functions have bias characteristic, LSMaODE algorithm doesn’t consider the bias problem. Therefore, the performance of LSMaODE algorithm is mediocre. MaOEA/IT algorithm optimizes convergence and diversity through two independent phases. However, the algorithm’s performance is always poor because it doesn’t alleviate the contradiction between convergence and diversity. The reference Pareto front of MaOEA/IGD algorithm is poor. Therefore, the algorithm’s performance is always poor.

Comparison and analysis

By synthesizing Tables 4 , 5 , 6 , we can obtain the data shown in Table 7 . As can be seen from Tables 4 , 5 , 6 , MOEA/TS algorithm wins the first place in 52 test cases; VMEF algorithm wins the first place in 8 test cases; BiGE-BEW algorithm wins the first place in 19 test cases; MOEA/DG algorithm wins the first place in 3 test cases; MOEA/D algorithm wins the first place in 20 test cases; LSMaODE algorithm wins first place in 8 test cases. As can be seen from Table 7 , in the 110 test cases, the number of MOEA/TS algorithm is significantly superior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 69, 74, 92, 82, 91, 107 and 104, respectively. The number of MOEA/TS algorithm is significantly inferior to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 22, 25, 10, 20, 13, 1 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to VMEF algorithm, BiGE-BEW algorithm, MOEA/DG algorithm, MOEA/D algorithm, LSMaODE algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm is 19, 11, 8, 8, 6, 2 and 6, respectively. Based on the above data, we can get the following conclusions: MOEA/TS algorithm has the best performance; the performance of BiGE-BEW algorithm, VMEF algorithm, MOEA/D algorithm, LSMaODE algorithm, MOEA/DG algorithm and MaOEA/IT algorithm decreases in turn. MaOEA/IGD algorithm has the worst performance.

In addition to the above conclusions, we can also observe 3 interesting phenomena:

(1) In the MAF test function set and WFG test function set, MOEA/TS algorithm has no competitors. However, in the DTLZ test function set, MOEA/TS algorithm and MOEA/D algorithm are competitors, and they have similar performance. This is because most DTLZ test functions have regular PF, while most MAF test functions and WFG test functions have more complex PF. It can be seen from Sect. " Introduction " that MOEA/D algorithm is suitable for MaOPs with regular PF. Therefore, in the DTLZ test function set, MOEA/D algorithm can compete with MOEA/TS algorithm. In the MAF test functions and WFG test functions, only MOEA/TS algorithm shows excellent performance.

(2) The performance of MOEA/TS algorithm is better on the test cases with 10 objectives, 15 objectives and 20 objectives. The performance of MOEA/TS algorithm is relatively ordinary on the test cases with 5 objectives and 8 objectives. This is because when the number of optimization objectives is small, most many-objective optimization algorithms perform well. Compared with other many-objective optimization algorithms, the advantages of MOEA/TS algorithm aren’t obvious. However, with the increase of the number of optimization objectives, the performance of other many-objective optimization algorithms becomes worse and worse. In contrast, the performance of MOEA/TS algorithm isn’t significantly affected. Therefore, compared with other many-objective optimization algorithms, MOEA/TS algorithm has obvious advantages. This shows that MOEA/TS algorithm is more suitable for solving MaOPs with more than 10 objectives.

(3) Without considering MOEA/TS algorithm, MOEA/D algorithm has the best performance in the DTLZ test function set. BiGE-BEW algorithm has the best performance in the MAF test function set. VMEF algorithm has the best performance in the WFG test function set. This shows that different many-objective optimization algorithms are suitable for different test function sets. However, MOEA/TS algorithm can show excellent performance on three test function sets. This indicates that MOEA/TS algorithm has strong universality and applicability.

Distribution diagram of solutions in the objective space

In order to describe the distribution of solutions in the high-dimensional objective space more intuitively, this paper draws the distribution diagram of solutions in the objective space. Considering the length of the paper, it is unrealistic to show the distribution diagrams of all test functions. Therefore, this section only shows the distribution diagrams of 3 representative test cases. These 3 test cases are DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives, respectively.

Figure 10 shows the distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives. It can be seen from Fig.  10 that distribution diagrams of MOEA/TS algorithm, BiGE-BEW algorithm, MOEA/DG algorithm and MOEA/D algorithm are similar, which indicates that these 4 algorithms are excellent in convergence and diversity; VMEF algorithm and LSMaODE algorithm are good in diversity, but poor in convergence; MaOEA/IT algorithm and MaOEA/IGD algorithm are very poor in convergence and diversity.

Distribution diagrams of each algorithm on DTLZ2 test case with 20 objectives.

Figure 11 shows the distribution diagrams of each algorithm on MAF1 test case with 15 objectives. It can be seen from Fig.  11 that MOEA/TS algorithm and VMEF algorithm are good in convergence, but poor in diversity; BiGE-BEW algorithm and LSMaODE algorithm are good in diversity, but poor in convergence. MOEA/DG algorithm, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Distribution diagrams of each algorithm on MAF1 test case with 15 objectives.

Figure 12 shows the distribution diagrams of each algorithm on WFG3 test case with 10 objectives. It can be seen from Fig.  12 that MOEA/TS algorithm has the best convergence and diversity; LSMaODE algorithm is also excellent, only slightly worse than MOEA/TS algorithm in terms of diversity; BiGE-BEW algorithm and MOEA/DG algorithm are good in diversity, but poor in convergence. VMEF algorithm is good in convergence, but poor in diversity. MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm are very bad in convergence and diversity.

Distribution diagrams of each algorithm on WFG3 test case with 10 objectives.

Evolution curve analysis of the algorithm

This section takes DTLZ2 test case with 20 objectives, MAF1 test case with 15 objectives and WFG3 test case with 10 objectives as examples to display the evolution curves of 8 algorithms (as shown in Figs.  13 , 14 , 15 ).

Evolution curve of each algorithm on DTLZ2 test case with 20 objectives.

In Figure 13 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of MOEA/DG algorithm, BiGE-BEW algorithm, MOEA/D algorithm, LSMaODE algorithm, VMEF algorithm and MaOEA/IGD algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on DTLZ2 test case with 20 objectives, and is suitable for solving DTLZ2 test problem with 20 objectives.

In Figure 14 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of BiGE-BEW algorithm, VMEF algorithm, LSMaODE algorithm, MaOEA/IGD algorithm, MOEA/D algorithm and MOEA/DG algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of all algorithms are smaller than the initial IGD+ values. This shows that all algorithms have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MaOEA/IT algorithm fluctuates greatly. This shows that MaOEA/IT algorithm isn’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on MAF1 test case with 15 objectives, and is suitable for solving MAF1 test problem with 15 objectives.

Evolution curve of each algorithm on MAF1 test case with 15 objectives.

In Fig.  15 , in terms of the final IGD+ value of the algorithm, MOEA/TS algorithm has the smallest IGD+ value, while the IGD+ values of LSMaODE algorithm, MOEA/DG algorithm, VMEF algorithm, BiGE-BEW algorithm, MaOEA/IGD algorithm and MOEA/D algorithm successively increase, and MaOEA/IT algorithm has the largest IGD+ value. This shows that MOEA/TS algorithm has the best convergence and diversity within the specified number of iterations. In terms of the evolution of the algorithm, the final IGD+ values of the MaOEA/IT algorithm, VMEF algorithm, MaOEA/IGD algorithm, BiGE-BEW algorithm and VMEF algorithm are all greater than the initial IGD+ values. This shows that the performance of these 5 algorithms deteriorates during evolution, and they aren’t suitable for dealing with WFG3 test problem with 10 objectives. The initial IGD+ value of MOEA/DG algorithm is close to the final IGD+ value, and the IGD+ value of MOEA/DG algorithm fluctuates little during the evolution. This shows that MOEA/DG algorithm is insensitive to evolution. Only the final IGD+ values of LSMaODE algorithm and MOEA/TS algorithm are less than the initial IGD+ values. This shows that LSMaODE algorithm and MOEA/TS algorithm have strong evolution ability, especially MOEA/TS algorithm has the strongest evolution ability. In terms of algorithm fluctuation, MOEA/D algorithm, MaOEA/IT algorithm and MaOEA/IGD algorithm have greater fluctuation. This shows that these 3 algorithms aren’t stable. Based on the above analysis, we believe that MOEA/TS algorithm has the best comprehensive performance on WFG3 test case with 10 objectives, and is suitable for solving WFG3 test problem with 10 objectives.

Evolution curve of each algorithm on WFG3 test case with 10 objectives.

In addition, we can also observe an interesting phenomenon from Fig.  13 to Fig.  15 : the IGD+ values of some algorithms sometimes increase significantly with the increase of iterations. That is, the performance of some algorithms sometimes deteriorates seriously with the increase of iterations. The reasons for this phenomenon may include three aspects: (1) The algorithm doesn’t adopt the elite preservation strategy. Some high-quality solutions may gradually disappear; (2) Due to the complexity of the optimization problems, the evolutionary direction of the population may be misled by some pseudo-elite individuals; (3) The convergence optimization and diversity optimization of the algorithm aren’t coordinated. The optimization of convergence may affect the optimization of diversity or the optimization of diversity may affect the optimization of convergence. It can be seen from the pseudo-code of the algorithm in Section 3.5 that the MOEA/TS algorithm proposed in this paper considers the above three aspects. Therefore, MOEA/TS algorithm can effectively alleviate this phenomenon.

Effectiveness verification of innovation part

In order to verify the effectiveness of the innovative parts, 4 variants are designed in this section. As follows:

MOEA/TS-1 algorithm: The feature extraction operator in MOEA/TS algorithm is changed to the binary crossover operator and polynomial mutation operator;

MOEA/TS-2 algorithm: The repulsion field method in MOEA/TS algorithm is removed;

MOEA/TS-3 algorithm: The concurrent architecture in MOEA/TS algorithm is changed to serial architecture;

MOEA/TS-4 algorithm: The individual importance degree in MOEA/TS algorithm is removed.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 5 algorithms. In this paper, 5 algorithms are executed 30 times to get the average data as shown in Table 8 . As can be seen from Table 8 , MOEA/TS algorithm wins the first place in 24 test cases; MOEA/TS-1 algorithm wins the first place in 13 test cases; MOEA/TS-2 algorithm wins the first place in 7 test cases; MOEA/TS-3 algorithm wins the first place in 1 test case. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 21, 30, 40 and 45, respectively. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 11, 6, 0 and 0, respectively. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-1 algorithm, MOEA/TS-2 algorithm, MOEA/TS-3 algorithm and MOEA/TS-4 algorithm is 13, 9, 5 and 0, respectively. The average ranking of MOEA/TS algorithm is about 1.64; the average ranking of MOEA/TS-1 algorithm is about 2.02; the average ranking of MOEA/TS-2 algorithm is about 2.62; the average ranking of MOEA/TS-3 algorithm is about 3.71; the average ranking of MOEA/TS-4 algorithm is 5.

Therefore, we think that the 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm. This shows that our innovations are effective. In addition, based on the above data, we can also find that “individual importance degree” has the greatest influence on the algorithm; the algorithm architecture ranks second; the repulsion field method ranks third; the feature extraction operator ranks fourth.

Ablation experiment of selection approach

In the feature extraction operator, we select W high-quality solutions. To prove the effectiveness of this selection approach over random selection, the ablation experiment will be performed in this sect. " Introduction " variant is designed in this section. As follows:

MOEA/TS-5 algorithm: W solutions are randomly selected in the feature extraction operator.

This paper takes WFG test function set (45 test cases) as samples, and then verifies the performance of 2 algorithms. In this paper, 2 algorithms are executed 30 times to get the average data as shown in Table 9 . As can be seen from Table 9 , MOEA/TS algorithm wins the first place in 45 test cases. In the 45 test cases, the number of MOEA/TS algorithm is significantly superior to MOEA/TS-5 algorithm is 42. The number of MOEA/TS algorithm is significantly inferior to MOEA/TS-5 algorithm is 0. Statistically, the number of MOEA/TS algorithm is similar to MOEA/TS-5 algorithm is 3. Therefore, we believe that the performance of MOEA/TS algorithm is better than MOEA/TS-5 algorithm in the WFG test function set. It proves that the selection approach that we use is better than random selection in the feature extraction operator.

In addition, the performance of MOEA/TS-5 algorithm isn’t as good as that of MOEA/TS-1 algorithm. It means that the performance of the feature extraction operator based on random selection is even worse than that of some classical operators. The possible reason is that the randomly selected solution set will cause the feature extraction operator to extract many bad features. These bad features hinder individual evolution, which makes the convergence maintenance state and diversity maintenance state of MOEA/TS algorithm fail for a long time, and only the coordination state can play some role. The architecture of the MOEA/TS algorithm is undermined by some bad features.

Parameter sensitivity analysis.

The algorithm parameters analyzed in this paper are mainly the number of high-quality solutions W, threshold value T, standard deviation std. Due to the high complexity of the WFG3 test case with 10 objectives, it is difficult for the population of each algorithm to cover the real Pareto front, so this paper considers the WFG3 test case with 10 objectives as the main function of parameter analysis.

The initial value and value range of each parameter are shown in Table 10 .

As shown in Fig.  16 , when \(W<9\) , the IGD + value of the algorithm decreases significantly with the increase of W . It means that when \(W<9\) , the performance of the feature extraction operator is greatly improved with the increase of W . This is because the features extracted by the feature extraction operator are closer to the ideal situation. When \(W=9\) , the IGD + value of the algorithm is minimum. This shows that when \(W=9\) , the feature extraction operator performs best. When \(W>9\) , the IGD + value of the algorithm increases slowly. It means that when \(W>9\) , the performance of the feature extraction operator deteriorates gradually with the increase of W . This is because some features are over-extracted by feature extraction operators. Therefore, for WFG3 test case with 10 objectives, \(W=9\) is the best parameter selection.

The corresponding relationship between IGD + value and W.

As shown in Fig.  17 , when \(T<5\%\) , the IGD + value of the algorithm decreases significantly with the increase of T . This is because if the threshold value T is too small, the algorithm will remain in the same state for a long time, and it is difficult to be adjusted to other states. Convergence and diversity of algorithm will also be difficult to balance. This situation will be improved with the increase of T . When \(T=5\%\) , the IGD + value of the algorithm is minimum. This shows that when \(T=5\%\) , the algorithm has the best performance. When \(T>5\%\) , the IGD + value of the algorithm increases gradually with the increase of T . This is because if the threshold value T is too large, the algorithm’s state will be adjusted frequently. Even if the population isn’t stable in one state (convergence, diversity, coordination), the algorithm will also be adjusted to other states. This isn’t conducive to improving the convergence and the diversity of the algorithm. The efficiency of the algorithm will also be affected. Therefore, for WFG3 test case with 10 objectives, \(T=5\%\) is the best parameter selection.

The corresponding relationship between IGD + value and T.

As shown in Fig.  18 , when \(std<0.7\) , the IGD + value of the algorithm decreases significantly with the increase of std . This is because if std is too small, the results of Gaussian sampling are too concentrated in the middle region, and the randomness of the sampling vector is weak, which isn’t conducive to the use of features and generation of diversified feature solutions. When \(std=0.7\) , the IGD + value of the algorithm is minimum. This shows that when \(std=0.7\) , the feature extraction operator performs best. When \(std>0.7\) , the IGD + value of the algorithm increases significantly with the increase of std . This is because if the std is too large, the result of Gaussian sampling is too scattered, the randomness of the sampling vector is strong, some components are easy to exceed the upper bound or lower bound, and some features are easy to be eliminated by the repair operator. Therefore, for WFG3 test case with 10 objectives, \(std=0.7\) is the best parameter selection.

The corresponding relationship between IGD + value and std.

Based on the above analysis of algorithm parameters, we think \(W=9, T=5\%, std=0.7\) are the best parameter combinations in WFG3 test case with 10 objectives. Further, we test the performance of the above parameter combinations in more test cases. The experimental results show that the above parameter combinations perform well in most test cases. Therefore, this paper sets the number of high-quality solutions \(W\) , the threshold value \(T\) and the standard deviation \(std\) to 9, 5% and 0.7, respectively.

Practical problem testing

This section mainly explores the performance of MOEA/TS algorithm in practical problems. The practical problem selected in this section is the industrial internet optimization problem based on the blockchain provided in reference 40 .

The industrial internet can support effective control of the physical world through a large amount of industrial data, but data security has always been a challenge due to various interconnections and accesses. Blockchain technology supports the security and privacy protection of industrial internet data with its trusted and reliable security mechanism. Fragmentation technology can help improve the overall throughput and scalability of the blockchain network. However, due to the uneven distribution of malicious nodes, the effectiveness of fragmentation is still challenging. In addition, the conflict between multiple industrial network indicators is also a problem we have to consider. Therefore, the industrial internet optimization problem based on blockchain is an important research problem.

In this section, the industrial internet optimization problem based on blockchain has the following 4 optimization objectives:

(1) Minimizing the shard invalidation probability (SIP);

(2) Minimizing the transmission delay (TD);

(3) Maximizing the throughput (TP);

(4) Minimizing the load of Malicious Nodes (LMN).

The research background of the industrial internet based on blockchain and the calculation formulas of these 4 objectives are detailed in reference 40 .

In this section, we set the population size to 220, the number of iterations to 300, and the number of function evaluations to 66000. We still use inverted generational distance plus (IGD+) to measure the performance of many-objective optimization algorithms. However, the real PF of the practical problem is unknown. Therefore, we run these algorithms many times to obtain the different non-dominated solution sets. The non-dominated union set of the different non-dominated solution sets is considered as the real PF. The relevant parameters of these algorithms are shown in Section 4.1.

In this section, each algorithm is executed 30 times to get the data as shown in Table 11 . As can be seen from Table 11 , MOEA/TS algorithm has absolute advantages. The performance of BiGE-BEW algorithm and MOEA/DG algorithm is good and similar. The performance of VMEF algorithm and MOEA/D algorithm in practical problems is obviously not as good as that in benchmark test functions. This is because the real PF of the practical problem is more complex. The performance of LSMaODE algorithm is close to that of MOEA/D algorithm. The performance of MaOEA/IT algorithm and MaOEA/IGD algorithm is the worst. Based on the above observations and analysis, we believe that MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Considering that the solutions obtained by the many-objective optimization algorithms are the population, it is unrealistic to compare different network indicators of different algorithms intuitively. However, in practical applications, we only need to make choices according to the specific needs or preferences of users or enterprises. In this section, we first select the individuals with the largest throughput in each algorithm, and then compare the MOEA/TS algorithm with other algorithms on the basis of ensuring the maximum throughput. The network indicators obtained by these 8 algorithms are shown in Table 12 . As can be seen from Table 12 , in terms of SIP and TP, MOEA/TS algorithm has the best performance; In terms of TD, MOEA/TS algorithm ranks second; In terms of LMN, MOEA/TS algorithm ranks third. Therefore, we believe that the MOEA/TS algorithm has the best comprehensive performance in the industrial internet optimization problem based on blockchain, and various network indicators are at the forefront.

Based on the experimental analysis from Section 4.2 to Section 4.8, we can obtain the following conclusions:

(1) In the benchmark test cases, MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms.

(2) MOEA/TS algorithm is more suitable for dealing with the MaOPs with more than 10 objectives.

(3) MOEA/TS algorithm can show excellent performance in different test function sets, and has strong universality and applicability.

(4) MOEA/TS algorithm has the best convergence and diversity, the strongest evolution ability and the fastest convergence speed.

(5) The 4 innovative parts of MOEA/TS algorithm are necessary and indispensable. The lack of any innovative parts will seriously affect the performance of MOEA/TS algorithm.

(6) MOEA/TS algorithm still has excellent performance and strong applicability in practical problems.

Summary and future work

Aiming at some difficulties in the many-objective optimization field, this paper proposes a many-objective evolutionary algorithm based on three states (MOEA/TS). Firstly, a feature extraction operator is proposed. The feature extraction operator is a feature extractor, which can extract the features of the high-quality solution set, and then assist the evolution of the current individual. Secondly, in terms of convergence maintenance, this paper doesn’t consider using domination relaxation technique, because the current domination relaxation technique still faces some problems. Based on Pareto front layer, this paper proposes the concept of “individual importance degree”. The importance degree of an individual can reflect the importance of the individual in the same Pareto front layer, so as to further distinguish the advantages and disadvantages of different individuals in the same front layer, and effectively solve the phenomenon of Pareto resistance. Then, in terms of diversity maintenance, this paper considers maintaining the diversity of the population in the objective space by repulsion field, so that the population can be evenly distributed on the real PF. Finally, a new concurrent algorithm framework is designed. In the framework, the algorithm is divided into three states, namely, convergence maintenance state, diversity maintenance state and coordination state. Each state focuses on a specific task. That is, the convergence maintenance state is responsible for improving the convergence of population; Diversity maintenance state is responsible for improving the diversity of population; the coordination state is responsible for coordinating the contradiction between diversity and convergence. The population can freely switch among these three states according to its own evolution. The experimental results show that MOEA/TS algorithm is superior to the other 7 advanced many-objective optimization algorithms. In addition, the effectiveness of the innovation parts is further verified.

However, MOEA/TS algorithm also has obvious defects: MOEA/TS algorithm isn’t good at dealing with test problems with narrow convergence curves or multiple local fronts. Therefore, in the future, we will further improve MOEA/TS algorithm, so that MOEA/TS algorithm can deal with test problems with narrow convergence curve or multiple local fronts. In addition, constrained MOPs and high-dimensional MOPs are also the focus of our future research.

Data availability

All data generated or analysed during this study are included in this published article.

Lin, H. F. & Tang, C. P. Analysis and optimization of urban public transport lines based on multiobjective adaptive particle swarm optimization. IEEE Trans. Intell. Transp. Syst. 23 , 16786–16798. https://doi.org/10.1109/TITS.2021.3086808 (2022).

Article   Google Scholar  

Qin, X., Fang, Z. H. & Zhang, Z. X. Multi-objective optimization for production scheduling ofprecast components considering resource constraints. Comput. Int. Manuf. Syst. 27 , 2248–2259. https://doi.org/10.13196/j.cims.2021.08.008 (2021).

Saric, F., Begusic, S., Mercep, A. & Kostanjcar, Z. Statistical arbitrage portfolio construction based on preference relations. Expert Syst. Appl. 238 , 1–12. https://doi.org/10.1016/j.eswa.2023.121906 (2023).

Chen, Y., Zhong, J., Feng, L. & Zhang, J. An adaptive archive-based evolutionary framework for many-task optimization. IEEE Trans. Emerg. Top. Comput. Intell. 4 , 369–384. https://doi.org/10.1109/TETCI.2019.2916051 (2020).

Article   CAS   Google Scholar  

Pradhan, D., Wang, S., Ali, S., Yue, T. & Liaaen, M. CBGA-ES+: A cluster-based genetic algorithm with non-dominated elitist selection for supporting multi-objective test optimization. IEEE Trans. Softw. Eng. 47 , 86–107. https://doi.org/10.1109/TETCI.2019.2916051 (2021).

Bian, H. L., Tian, J., Yu, J. L. & Yu, H. Bayesian co-evolutionary optimization based entropy search for high-dimensional many-objective optimization. Knowl. Based Syst. 274 , 1–13. https://doi.org/10.1016/j.knosys.2023.110630 (2023).

Li, W., Chen, Y. T., Dong, Y. H. & Huang, Y. A solution potential-based adaptation reference vector evolutionary algorithm for many-objective optimization. Swarm Evol. Comput. 84 , 1–15. https://doi.org/10.1016/j.swevo.2023.101451 (2023).

Wang, Y. J., Gao, P. & Chen, Y. An improved farmland fertility algorithm for many-objective optimization problems. Sci. Rep. 12 , 1–24. https://doi.org/10.1038/s41598-022-06329-x (2022).

Khurana, D., Yadav, A. & Sadollah, A. A non-dominated sorting based multi-objective neural network algorithm. MethodsX 10 , 1–16. https://doi.org/10.1016/j.mex.2023.102152 (2023).

Huang, H. J., Zheng, B. F., Wei, X. X., Zhou, Y. Q. & Zhang, Y. D. NSCSO: A novel multi-objective non-dominated sorting chicken swarm optimization algorithm. Sci. Rep. 14 , 1–38. https://doi.org/10.1038/s41598-024-54991-0 (2024).

Li, M. Q., Yang, S. X. & Liu, X. H. Bi-goal evolution for many-objective optimization problems. Artif. Intell. 228 , 45–65. https://doi.org/10.1016/j.artint.2015.06.007 (2015).

Article   MathSciNet   Google Scholar  

Wei, L. S. & Li, E. C. A many-objective evolutionary algorithm with population preprocessing and projection distance-assisted elimination mechanism. J. Comput. Design Eng. 10 , 1988–2018. https://doi.org/10.1093/jcde/qwad088 (2023).

Sun, Y. F., Bian, K., Liu, Z., Sun, X. & Yao, R. X. Adaptive strategies based on differential evolutionary algorithm for many-objective optimization. Discret. Dyn. Nat. Soc. 2021 , 1–17. https://doi.org/10.1155/2021/2491796 (2021).

Zhou, S. Q., Dai, Y. R. & Chen, Z. H. Dominance relation selection and angle-based distribution evaluation for many-objective evolutionary algorithm. Swarm Evol. Comput. 86 , 1–19. https://doi.org/10.1016/j.swevo.2024.101515 (2024).

Wang, X. W., Xie, Z. H., Zhou, X. & Gu, X. S. A two-stage adaptive reference direction guided evolutionary algorithm with modified dominance relation for many-objective optimization. Swarm Evol. Comput. 78 , 1–14. https://doi.org/10.1016/j.swevo.2023.101272 (2023).

Zhang, W., Liu, J. C., Liu, J. H., Liu, Y. C. & Tan, S. B. A dual distance dominance based evolutionary algorithm with selection-replacement operator for many-objective optimization. Expert Syst. Appl. 237 , 1–25. https://doi.org/10.1016/j.eswa.2023.121244 (2023).

Shang, K. & Ishibuchi, H. A new hypervolume-based evolutionary algorithm for many-objective optimization. IEEE Trans. Evol. Comput. 24 , 839–852. https://doi.org/10.1109/TEVC.2020.2964705 (2020).

Zhang, W., Liu, J. C., Liu, J. H., Liu, Y. C. & Wang, H. H. A many-objective evolutionary algorithm based on novel fitness estimation and grouping layering. Neural Comput. Appl. 35 , 24283–24314. https://doi.org/10.1007/s00521-023-08950-x (2023).

Nan, Y., Shang, K., Ishibuchi, H. & He, L. J. A Two-stage Hypervolume Contribution Approximation Method Based on R2 Indicator. In: 2021 IEEE Congress on Evolutionary Computation (IEEE CEC 2021) . 2468-2475. https://doi.org/10.1109/CEC45853.2021.9504726 (2021).

Wu, M., Li, K., Kwong, S. & Zhang, Q. Evolutionary many-objective optimization based on adversarial decomposition. IEEE Trans. Cybern. 50 , 753–764. https://doi.org/10.1109/TCYB.2018.2872803 (2020).

Article   PubMed   Google Scholar  

Fan, M. W. et al. Improved multi-objective differential evolution algorithm based on a decomposition strategy for multi-objective optimization problems. Sci. Rep. 12 , 1–14. https://doi.org/10.1038/s41598-022-25440-7 (2022).

Peng, F. A., Lv, L., Chen, W. R. & Wang, J. A projection-based evolutionary algorithm for multi-objective and many-objective optimization. Processes 11 , 1–22. https://doi.org/10.3390/pr11051564 (2023).

Sun, Y., Xue, B., Zhang, M. & Yen, G. G. A new two-stage evolutionary algorithm for many-objective optimization. IEEE Trans. Evol. Comput. 23 , 748–761. https://doi.org/10.1109/TEVC.2018.2882166 (2019).

Yang, Q. T., Zhan, Z. H., Kwong, S. & Zhang, J. Multiple populations for multiple objectives framework with bias sorting for many-objective optimization. IEEE Trans. Evol. Comput. 27 , 1340–1354. https://doi.org/10.1109/TEVC.2022.3212058 (2023).

Liu, S. C. et al. Many-objective job-shop scheduling: A multiple populations for multiple objectives-based genetic algorithm approach. IEEE Trans. Cybern. 53 , 1460–1474. https://doi.org/10.1109/TCYB.2021.3102642 (2023).

Tian, Y., He, C., Cheng, R. & Zhang, X. A multistage evolutionary algorithm for better diversity preservation in multiobjective optimization. IEEE Trans. Syst. Man Cybern. Syst. 51 , 5880–5894. https://doi.org/10.1109/TSMC.2019.2956288 (2021).

Sun, C. H., Wang, Y. H., Wan, P. & Du, Y. A cooperative spectrum sensing algorithm based on principal component analysis and K-medoids clustering. In: 2018 33rd Youth Academic Annual Conference of Chinese Association of Automation (YAC). 835-839. https://doi.org/10.1109/YAC.2018.8406487 (2018).

Osinsky, A., Bychkov, R., Trefilov, M., Lyashev, V. & Ivanov, A. Regularization for cholesky decomposition in massive MIMO detection. IEEE Wirel. Commun. Lett. 12 , 1603–1607. https://doi.org/10.1109/LWC.2023.3284349 (2023).

Gu, Q. H., Gao, S., Li, X. X., Xiong, N. N. & Liu, R. R. An adaptive adjacent maximum distance crossover operator for multi-objective algorithms. Soft Comput. 27 , 7419–7438. https://doi.org/10.1007/s00500-023-07978-4 (2023).

Tian, Y., Cheng, R., Zhang, X. Y. & Jin, Y. PlatEMO: A MATLAB platform for evolutionary multi-objective optimization [educational forum]. IEEE Comput. Intell. Mag. 12 , 73–87. https://doi.org/10.1109/MCI.2017.2742868 (2017).

He, Z., Yen, G. G. & Zhang, J. Fuzzy-based Pareto optimality for many-objective evolutionary algorithms. IEEE Trans. Evol. Comput. 18 , 269–285. https://doi.org/10.1109/TEVC.2013.2258025 (2014).

Qiu, W. et al. Ensemble many-objective optimization algorithm based on voting mechanism. IEEE Trans. Syst. Man Cybern. Syst. 52 , 1716–1730. https://doi.org/10.1109/TSMC.2020.3034180 (2020).

Wang, J. & Chen, H. A Weight Vector Bi-Objective Evolutionary Algorithm with Bi-criterion Evolution for Many-Objective Optimization. In: 2022 IEEE 4th International Conference on Power, Intelligent Computing and Systems (ICPICS). 273-279. https://doi.org/10.1109/ICPICS55264.2022.9873807 (2022).

He, X. & Dai, C. An Improvement Evolutionary Algorithm Based on Decomposition and Grid-based Pareto Dominance for Many-objective Optimization. In: 2022 Global Conference on Robotics, Artificial Intelligence and Information Technology (GCRAIT). 145-149. https://doi.org/10.1109/GCRAIT55928.2022.00039 (2022).

Zhang, Q., Liu, W. & Li, H. The performance of a new version of MOEA/D on CEC09 unconstrained MOP test instances. In: 2009 IEEE Congress on Evolutionary Computation (CEC). 203–208. https://doi.org/10.1109/CEC . 2009.4982949 (2009).

Zhang, K., Shen, C. & Yen, G. G. Multipopulation-based differential evolution for large-scale many-objective optimization. IEEE Trans. Cybern. 53 , 7596–7608. https://doi.org/10.1109/TCYB.2022.3178929 (2023).

Sun, Y., Yen, G. G. & Yi, Z. IGD indicator-based evolutionary algorithm for many-objective optimization problems. IEEE Trans. Evol. Comput. 23 , 173–187. https://doi.org/10.1109/TEVC.2018.2791283 (2019).

Wilcoxon, F. Individual comparisons by ranking methods. Breakthr. Stat. 1 , 196–202. https://doi.org/10.1007/978-1-4612-4380-9_16 (1992).

Ishibuchi, H., Imada, R., Masuyama, N. & Nojima, Y. Comparison of Hypervolume, IGD and IGD+ from the Viewpoint of Optimal Distributions of Solutions. In: 2019 International Conference on Evolutionary Multi-Criterion Optimization (EMO). 332–345. https://doi.org/10.1007/978-3-030-12598-1_27 (2019).

Cai, X. J. et al. A sharding scheme-based many-objective optimization algorithm for enhancing security in blockchain-enabled industrial internet of things. IEEE Trans. Ind. Inform. 17 , 7650–7658. https://doi.org/10.1109/TII.2021.3051607 (2021).

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 62362026 and 62162021; in part by the specific research fund of The Innovation Platform for Academicians of Hainan Province under grant YSPTZX202314; in part by the Key Project of Hainan Province under Grant ZDYF2023GXJS158.

National Natural Science Foundation of China, 62362026, 62362026, Specific research fund of The Innovation Platform for Academicians of Hainan Province, YSPTZX202314, YSPTZX202314, Key Project of Hainan Province, ZDYF2023GXJS158, ZDYF2023GXJS158.

Author information

Authors and affiliations.

School of Information and Communication Engineering, Hainan University, Haikou, 570228, China

School of Computer Science and Technology, Hainan University, Haikou, 570228, China

Huanhuan Yu, Hansheng Fei, Xiangdang Huang & Qiuling Yang

Academic Affairs Office, Shandong Agricultural University, Taian, 271018, China

Huijie Zhang

Innovation Platform for Academicians of Hainan Province, Hainan University, Haikou, 570228, China

Jiale Zhao, Huanhuan Yu, Hansheng Fei, Xiangdang Huang & Qiuling Yang

You can also search for this author in PubMed   Google Scholar

Contributions

Conceptualization, J.L.Z., H.J.Z. and X.D.H.; Methodology, J.L.Z., H.J.Z. and X.D.H.; Software, J.L.Z., H.J.Z., H.H.Y. and H.S.F.; Validation, H.H.Y. and H.S.F.; Formal analysis, J.L.Z. and H.J.Z.; Investigation, H.H.Y. and H.S.F.; Resources, Q.L.Y.; Data curation, H.H.Y. and H.S.F.; Writing-original draft, J.L.Z.; Writing-review&editing, J.L.Z., H.J.Z., H.H.Y. and H.S.F.; Visualization, J.L.Z. and H.J.Z.; Supervision, H.H.Y. and H.S.F.; Project administration, J.L.Z. and Q.L.Y.; Funding acquisition, Q.L.Y.; All authors have read and agreed to the published version of the manuscript.

Corresponding author

Correspondence to Qiuling Yang .

Ethics declarations

Competing interests.

The authors declare no competing interests.

Additional information

Publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Supplementary information., rights and permissions.

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ .

Reprints and permissions

About this article

Cite this article.

Zhao, J., Zhang, H., Yu, H. et al. A many-objective evolutionary algorithm based on three states for solving many-objective optimization problem. Sci Rep 14 , 19140 (2024). https://doi.org/10.1038/s41598-024-70145-8

Download citation

Received : 26 January 2024

Accepted : 13 August 2024

Published : 19 August 2024

DOI : https://doi.org/10.1038/s41598-024-70145-8

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

By submitting a comment you agree to abide by our Terms and Community Guidelines . If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Quick links

• Explore articles by subject
• Guide to authors
• Editorial policies

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

share this!

August 19, 2024

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

fact-checked

trusted source

written by researcher(s)

Prison is a disaster for women—here's what could work instead

by April Smith, The Conversation

The early days of the new Labor government have given prison reform advocates reason to be hopeful. Two announcements—the appointment of James Timpson as prisons minister and an expanded early release program—are promising steps towards repairing a criminal justice system that has been devastated by cuts and neglect.

Prison overcrowding is the most pressing issue, with the current population standing at more than 87,000. This significantly exceeds the Ministry of Justice's "safe and decent" capacity of 79,695 inmates . England and Wales have one of the highest rates of imprisonment in western Europe, almost twice as high as Germany.

Approximately 4% of people in prison in England and Wales are women, and in 2020, 72% of female prison admissions were for non-violent crimes . As someone who studies women's prisons, I hope that plans for change will include revising how we punish and rehabilitate women.

Timpson has said that prisons, especially for women, are a "disaster." He has noted correctly that prisons often lead women back into the cycle of offending . Women released from prison reoffend more frequently and more quickly than those who serve community sentences .

Women frequently experience a "revolving door" of short prison sentences—too short to engage in meaningful education, training or work opportunities. In 2017, 77% of custodial sentences for women were 12 months or less, with a 71% reoffending rate following such short sentences.

Incarcerated women face complex and severe challenges. They are disproportionately affected by self-harm , accounting for 29% of all incidents (despite making up less that 5% of the prison population). Many are mothers and primary caregivers, impacting an estimated 17,000 children annually.

Violence in women's prisons is another serious concern, with 469 incidents per 1,000 prisoners in the 12 months to June 2023. The assault rate is more than three and a half times higher than a decade ago, and significantly exceeds that in men's prisons. This increase in violence is often linked to mental health issues and substance abuse, exacerbated by overcrowding.

More than half of incarcerated women have experienced severe trauma, including domestic violence and childhood abuse . Women in the justice system need tailored support systems that offer trauma-informed care, mental health services and community-based support.

The 2007 Corston report was commissioned in response to the deaths of six women at HMP Styal between August 2002 and August 2003. It offered 43 recommendations for improving the treatment of vulnerable women in the criminal justice system . At the time, the government accepted 41 of these recommendations, including limiting imprisonment to serious offenses and enhancing health and support services.

In 2013, the House of Commons justice committee reviewed the progress since the Corston report, and found not much progress had been made. The report criticized prisons for treating women like children, and contrasted this to women's centers, which it argued empower women to take control of their lives.

There are around 40 women's centers in England and Wales, but there has been a lack of funding and political commitment to open more.

Gender-specific programs and support are critical to addressing the circumstances and needs of female offenders, including victimization, trauma, mental and physical health, pregnancy and parenting.

Other approaches—what works for women?

Prison often creates more problems than it solves, especially for women with complex issues, which short sentences fail to address. Rather than building more prisons, investing in comprehensive community support such as women's centers could be an effective alternative.

Timpson has advocated for community-based alternatives to imprisonment . One example is Hope Street in Southampton, a residential community where women live with their children.

Hope Street offers a community alternative for women on community orders, remand, or post-release. Several evaluations and research studies have documented the positive outcomes for women who engage with women's centers, including decreasing reoffending ).

The program will be evaluated over the next few years. If successful, a similar model could be implemented nationwide, reducing the number of women who end up in prison.

Scotland's Bella Center also represents a more progressive approach, focusing on community-based custody. This 16-space unit, free from prison bars, helps women reintegrate into society through community contact and access to local services. It is part of a £600 million plan to improve Scotland's custodial estate .

The government's focus on addressing the prison crisis presents a promising opportunity to explore more of these innovative, community-based alternatives to prison , and create meaningful change for female offenders.

Provided by The Conversation

Explore further

Feedback to editors

Small molecule organic eutectics show potential for replacing plastics

3 minutes ago

Animals with higher body temperatures are more likely to evolve into herbivores, study finds

32 minutes ago

Study explains why laws are written in an incomprehensible style

Understanding of early life ecosystems sheds light on evolution of life on Earth

Sea anemone study identifies potentially regenerative stem cells linked to conserved genes

3 hours ago

Ultrathin quantum light sources: Scientists show excitonic interactions boost efficiency of entangled photon generation

Study finds gun violence in rural America rises as deer hunting season begins

Mapping Martian meteorites: Tracing origins on the red planet

Scientists investigate triggers of explosive volcanic eruptions in lab simulation study

4 hours ago

Scientists uncover exciton behavior in van der Waals magnets

Relevant physicsforums posts, cover songs versus the original track, which ones are better, why are abba so popular.

Aug 17, 2024

Today's Fusion Music: T Square, Cassiopeia, Rei & Kanade Sato

Favorite songs (cont.).

Aug 16, 2024

Talent Worthy of Wider Recognition

History of railroad safety - spotlight on current derailments.

More from Art, Music, History, and Linguistics

Related Stories

British jails are at a breaking point—here's how the Dutch halved their prison population

Jul 10, 2024

How going from care to prison compounds women's trauma

Jan 18, 2022

Women's prisons: Why they do more harm than good

Jul 30, 2019

New-born deaths lead academics to call for law changes to prevent the imprisonment of pregnant women

The UK's prisons aren't just overcrowded—they need to be better designed

Jul 17, 2024

How to stop released prisoners reoffending: What the evidence says

Jul 21, 2024

Recommended for you

Spread the love (online): Study reveals in-party positivity drives online engagement more than out-party hostility

How some states help residents avoid costly debt during hard times

Singing from memory unlocks a surprisingly common musical superpower

Aug 15, 2024

Renewable energy policies provide benefits across state lines, study shows

Let us know if there is a problem with our content.

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

• Data Science
• Trending Now
• Data Structures
• System Design
• Foundational Courses
• Practice Problem
• Machine Learning
• Data Science Using Python
• Web Development
• Web Browser
• Design Patterns
• Software Development
• Product Management
• Programming

In this video, we will understand the Majority Element problem! 🧠 Perfect for coding enthusiasts, students gearing up for interviews, or professionals sharpening their problem-solving skills, this video offers a comprehensive guide to understanding and solving one of the most essential challenges in array processing.

What You'll Learn:

• Defining the Majority Element: We start by explaining what constitutes a majority element in an array and why it’s crucial in the realm of algorithms.
• Examples with Detailed Explanations: Follow along as we explore various examples to identify the majority element in different arrays. Understand how certain elements qualify and why some arrays lack a majority element altogether.
• Naive Approach: Learn the basic, yet fundamental, method of solving the Majority Element problem using a nested loop approach. We’ll break down the logic, show you how to implement it, and discuss its time complexity (O(n²)) and space complexity (O(1)).
• Efficient Approach (Moore’s Voting Algorithm): Step up your game with this optimized method that reduces the time complexity to O(n). We’ll guide you through the two-step process of finding the majority candidate and validating it, ensuring you grasp the algorithm's power and efficiency.

Key Points Covered:

• Introduction to the Majority Element problem.
• An in-depth explanation of the Naive Approach.
• Detailed walkthrough of Moore’s Voting Algorithm.
• Comparison of time and space complexity for both methods.
• Practical coding examples to solidify your understanding.
• Enhance your problem-solving toolkit with essential algorithms.
• Prepare for coding interviews with clear and concise explanations.
• Access practical examples in various programming languages to practice and perfect your skills.