problem solving process as a scientific approach

How it works

For Business

Join Mind Tools

Article • 5 min read

Using the Scientific Method to Solve Problems

How the scientific method and reasoning can help simplify processes and solve problems.

By the Mind Tools Content Team

The processes of problem-solving and decision-making can be complicated and drawn out. In this article we look at how the scientific method, along with deductive and inductive reasoning can help simplify these processes.

‘It is a capital mistake to theorize before one has information. Insensibly one begins to twist facts to suit our theories, instead of theories to suit facts.’ Sherlock Holmes

The Scientific Method

The scientific method is a process used to explore observations and answer questions. Originally used by scientists looking to prove new theories, its use has spread into many other areas, including that of problem-solving and decision-making.

The scientific method is designed to eliminate the influences of bias, prejudice and personal beliefs when testing a hypothesis or theory. It has developed alongside science itself, with origins going back to the 13th century. The scientific method is generally described as a series of steps.

• observations/theory
• explanation/conclusion

The first step is to develop a theory about the particular area of interest. A theory, in the context of logic or problem-solving, is a conjecture or speculation about something that is not necessarily fact, often based on a series of observations.

Once a theory has been devised, it can be questioned and refined into more specific hypotheses that can be tested. The hypotheses are potential explanations for the theory.

The testing, and subsequent analysis, of these hypotheses will eventually lead to a conclus ion which can prove or disprove the original theory.

Applying the Scientific Method to Problem-Solving

How can the scientific method be used to solve a problem, such as the color printer is not working?

1. Use observations to develop a theory.

In order to solve the problem, it must first be clear what the problem is. Observations made about the problem should be used to develop a theory. In this particular problem the theory might be that the color printer has run out of ink. This theory is developed as the result of observing the increasingly faded output from the printer.

2. Form a hypothesis.

Note down all the possible reasons for the problem. In this situation they might include:

• The printer is set up as the default printer for all 40 people in the department and so is used more frequently than necessary.
• There has been increased usage of the printer due to non-work related printing.
• In an attempt to reduce costs, poor quality ink cartridges with limited amounts of ink in them have been purchased.
• The printer is faulty.

All these possible reasons are hypotheses.

3. Test the hypothesis.

Once as many hypotheses (or reasons) as possible have been thought of, then each one can be tested to discern if it is the cause of the problem. An appropriate test needs to be devised for each hypothesis. For example, it is fairly quick to ask everyone to check the default settings of the printer on each PC, or to check if the cartridge supplier has changed.

4. Analyze the test results.

Once all the hypotheses have been tested, the results can be analyzed. The type and depth of analysis will be dependant on each individual problem, and the tests appropriate to it. In many cases the analysis will be a very quick thought process. In others, where considerable information has been collated, a more structured approach, such as the use of graphs, tables or spreadsheets, may be required.

5. Draw a conclusion.

Based on the results of the tests, a conclusion can then be drawn about exactly what is causing the problem. The appropriate remedial action can then be taken, such as asking everyone to amend their default print settings, or changing the cartridge supplier.

Inductive and Deductive Reasoning

The scientific method involves the use of two basic types of reasoning, inductive and deductive.

Inductive reasoning makes a conclusion based on a set of empirical results. Empirical results are the product of the collection of evidence from observations. For example:

‘Every time it rains the pavement gets wet, therefore rain must be water’.

There has been no scientific determination in the hypothesis that rain is water, it is purely based on observation. The formation of a hypothesis in this manner is sometimes referred to as an educated guess. An educated guess, whilst not based on hard facts, must still be plausible, and consistent with what we already know, in order to present a reasonable argument.

Deductive reasoning can be thought of most simply in terms of ‘If A and B, then C’. For example:

• if the window is above the desk, and
• the desk is above the floor, then
• the window must be above the floor

It works by building on a series of conclusions, which results in one final answer.

Social Sciences and the Scientific Method

The scientific method can be used to address any situation or problem where a theory can be developed. Although more often associated with natural sciences, it can also be used to develop theories in social sciences (such as psychology, sociology and linguistics), using both quantitative and qualitative methods.

Quantitative information is information that can be measured, and tends to focus on numbers and frequencies. Typically quantitative information might be gathered by experiments, questionnaires or psychometric tests. Qualitative information, on the other hand, is based on information describing meaning, such as human behavior, and the reasons behind it. Qualitative information is gathered by way of interviews and case studies, which are possibly not as statistically accurate as quantitative methods, but provide a more in-depth and rich description.

The resultant information can then be used to prove, or disprove, a hypothesis. Using a mix of quantitative and qualitative information is more likely to produce a rounded result based on the factual, quantitative information enriched and backed up by actual experience and qualitative information.

In terms of problem-solving or decision-making, for example, the qualitative information is that gained by looking at the ‘how’ and ‘why’ , whereas quantitative information would come from the ‘where’, ‘what’ and ‘when’.

It may seem easy to come up with a brilliant idea, or to suspect what the cause of a problem may be. However things can get more complicated when the idea needs to be evaluated, or when there may be more than one potential cause of a problem. In these situations, the use of the scientific method, and its associated reasoning, can help the user come to a decision, or reach a solution, secure in the knowledge that all options have been considered.

Join Mind Tools and get access to exclusive content.

This resource is only available to Mind Tools members.

Already a member? Please Login here

Try Mind Tools for FREE

Get unlimited access to all our career-boosting content and member benefits with our 7-day free trial.

Sign-up to our newsletter

Subscribing to the Mind Tools newsletter will keep you up-to-date with our latest updates and newest resources.

Subscribe now

Business Skills

Personal Development

Leadership and Management

Member Extras

Most Popular

Newest Releases

What Is Gibbs' Reflective Cycle?

Team Briefings

Mind Tools Store

About Mind Tools Content

Discover something new today

Onboarding with steps.

Helping New Employees to Thrive

NEW! Pain Points Podcast - Perfectionism

Why Am I Such a Perfectionist?

How Emotionally Intelligent Are You?

Boosting Your People Skills

Self-Assessment

What's Your Leadership Style?

Learn About the Strengths and Weaknesses of the Way You Like to Lead

Recommended for you

The long game.

Dorie Clark

Expert Interviews

Business Operations and Process Management

Strategy Tools

Customer Service

Business Ethics and Values

Handling Information and Data

Project Management

Knowledge Management

Self-Development and Goal Setting

Time Management

Presentation Skills

Learning Skills

Career Skills

Communication Skills

Negotiation, Persuasion and Influence

Working With Others

Difficult Conversations

Creativity Tools

Self-Management

Work-Life Balance

Stress Management and Wellbeing

Coaching and Mentoring

Change Management

Team Management

Managing Conflict

Delegation and Empowerment

Performance Management

Leadership Skills

Developing Your Team

Talent Management

Problem Solving

Decision Making

Member Podcast

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"

• Table of Contents
• Random Entry
• Chronological
• Editorial Information
• About the SEP
• Editorial Board
• How to Cite the SEP
• Special Characters
• Advanced Tools
• Support the SEP
• PDFs for SEP Friends
• Make a Donation
• SEPIA for Libraries
• Entry Contents

Bibliography

Academic tools.

• Friends PDF Preview
• Author and Citation Info
• Back to Top

Scientific Method

Science is an enormously successful human enterprise. The study of scientific method is the attempt to discern the activities by which that success is achieved. Among the activities often identified as characteristic of science are systematic observation and experimentation, inductive and deductive reasoning, and the formation and testing of hypotheses and theories. How these are carried out in detail can vary greatly, but characteristics like these have been looked to as a way of demarcating scientific activity from non-science, where only enterprises which employ some canonical form of scientific method or methods should be considered science (see also the entry on science and pseudo-science ). Others have questioned whether there is anything like a fixed toolkit of methods which is common across science and only science. Some reject privileging one view of method as part of rejecting broader views about the nature of science, such as naturalism (Dupré 2004); some reject any restriction in principle (pluralism).

Scientific method should be distinguished from the aims and products of science, such as knowledge, predictions, or control. Methods are the means by which those goals are achieved. Scientific method should also be distinguished from meta-methodology, which includes the values and justifications behind a particular characterization of scientific method (i.e., a methodology) — values such as objectivity, reproducibility, simplicity, or past successes. Methodological rules are proposed to govern method and it is a meta-methodological question whether methods obeying those rules satisfy given values. Finally, method is distinct, to some degree, from the detailed and contextual practices through which methods are implemented. The latter might range over: specific laboratory techniques; mathematical formalisms or other specialized languages used in descriptions and reasoning; technological or other material means; ways of communicating and sharing results, whether with other scientists or with the public at large; or the conventions, habits, enforced customs, and institutional controls over how and what science is carried out.

While it is important to recognize these distinctions, their boundaries are fuzzy. Hence, accounts of method cannot be entirely divorced from their methodological and meta-methodological motivations or justifications, Moreover, each aspect plays a crucial role in identifying methods. Disputes about method have therefore played out at the detail, rule, and meta-rule levels. Changes in beliefs about the certainty or fallibility of scientific knowledge, for instance (which is a meta-methodological consideration of what we can hope for methods to deliver), have meant different emphases on deductive and inductive reasoning, or on the relative importance attached to reasoning over observation (i.e., differences over particular methods.) Beliefs about the role of science in society will affect the place one gives to values in scientific method.

The issue which has shaped debates over scientific method the most in the last half century is the question of how pluralist do we need to be about method? Unificationists continue to hold out for one method essential to science; nihilism is a form of radical pluralism, which considers the effectiveness of any methodological prescription to be so context sensitive as to render it not explanatory on its own. Some middle degree of pluralism regarding the methods embodied in scientific practice seems appropriate. But the details of scientific practice vary with time and place, from institution to institution, across scientists and their subjects of investigation. How significant are the variations for understanding science and its success? How much can method be abstracted from practice? This entry describes some of the attempts to characterize scientific method or methods, as well as arguments for a more context-sensitive approach to methods embedded in actual scientific practices.

1. Overview and organizing themes

2. historical review: aristotle to mill, 3.1 logical constructionism and operationalism, 3.2. h-d as a logic of confirmation, 3.3. popper and falsificationism, 3.4 meta-methodology and the end of method, 4. statistical methods for hypothesis testing, 5.1 creative and exploratory practices.

• 5.2 Computer methods and the ‘new ways’ of doing science

6.1 “The scientific method” in science education and as seen by scientists

6.2 privileged methods and ‘gold standards’, 6.3 scientific method in the court room, 6.4 deviating practices, 7. conclusion, other internet resources, related entries.

This entry could have been given the title Scientific Methods and gone on to fill volumes, or it could have been extremely short, consisting of a brief summary rejection of the idea that there is any such thing as a unique Scientific Method at all. Both unhappy prospects are due to the fact that scientific activity varies so much across disciplines, times, places, and scientists that any account which manages to unify it all will either consist of overwhelming descriptive detail, or trivial generalizations.

The choice of scope for the present entry is more optimistic, taking a cue from the recent movement in philosophy of science toward a greater attention to practice: to what scientists actually do. This “turn to practice” can be seen as the latest form of studies of methods in science, insofar as it represents an attempt at understanding scientific activity, but through accounts that are neither meant to be universal and unified, nor singular and narrowly descriptive. To some extent, different scientists at different times and places can be said to be using the same method even though, in practice, the details are different.

Whether the context in which methods are carried out is relevant, or to what extent, will depend largely on what one takes the aims of science to be and what one’s own aims are. For most of the history of scientific methodology the assumption has been that the most important output of science is knowledge and so the aim of methodology should be to discover those methods by which scientific knowledge is generated.

Science was seen to embody the most successful form of reasoning (but which form?) to the most certain knowledge claims (but how certain?) on the basis of systematically collected evidence (but what counts as evidence, and should the evidence of the senses take precedence, or rational insight?) Section 2 surveys some of the history, pointing to two major themes. One theme is seeking the right balance between observation and reasoning (and the attendant forms of reasoning which employ them); the other is how certain scientific knowledge is or can be.

Section 3 turns to 20 th century debates on scientific method. In the second half of the 20 th century the epistemic privilege of science faced several challenges and many philosophers of science abandoned the reconstruction of the logic of scientific method. Views changed significantly regarding which functions of science ought to be captured and why. For some, the success of science was better identified with social or cultural features. Historical and sociological turns in the philosophy of science were made, with a demand that greater attention be paid to the non-epistemic aspects of science, such as sociological, institutional, material, and political factors. Even outside of those movements there was an increased specialization in the philosophy of science, with more and more focus on specific fields within science. The combined upshot was very few philosophers arguing any longer for a grand unified methodology of science. Sections 3 and 4 surveys the main positions on scientific method in 20 th century philosophy of science, focusing on where they differ in their preference for confirmation or falsification or for waiving the idea of a special scientific method altogether.

In recent decades, attention has primarily been paid to scientific activities traditionally falling under the rubric of method, such as experimental design and general laboratory practice, the use of statistics, the construction and use of models and diagrams, interdisciplinary collaboration, and science communication. Sections 4–6 attempt to construct a map of the current domains of the study of methods in science.

As these sections illustrate, the question of method is still central to the discourse about science. Scientific method remains a topic for education, for science policy, and for scientists. It arises in the public domain where the demarcation or status of science is at issue. Some philosophers have recently returned, therefore, to the question of what it is that makes science a unique cultural product. This entry will close with some of these recent attempts at discerning and encapsulating the activities by which scientific knowledge is achieved.

Attempting a history of scientific method compounds the vast scope of the topic. This section briefly surveys the background to modern methodological debates. What can be called the classical view goes back to antiquity, and represents a point of departure for later divergences. [ 1 ]

We begin with a point made by Laudan (1968) in his historical survey of scientific method:

Perhaps the most serious inhibition to the emergence of the history of theories of scientific method as a respectable area of study has been the tendency to conflate it with the general history of epistemology, thereby assuming that the narrative categories and classificatory pigeon-holes applied to the latter are also basic to the former. (1968: 5)

To see knowledge about the natural world as falling under knowledge more generally is an understandable conflation. Histories of theories of method would naturally employ the same narrative categories and classificatory pigeon holes. An important theme of the history of epistemology, for example, is the unification of knowledge, a theme reflected in the question of the unification of method in science. Those who have identified differences in kinds of knowledge have often likewise identified different methods for achieving that kind of knowledge (see the entry on the unity of science ).

Different views on what is known, how it is known, and what can be known are connected. Plato distinguished the realms of things into the visible and the intelligible ( The Republic , 510a, in Cooper 1997). Only the latter, the Forms, could be objects of knowledge. The intelligible truths could be known with the certainty of geometry and deductive reasoning. What could be observed of the material world, however, was by definition imperfect and deceptive, not ideal. The Platonic way of knowledge therefore emphasized reasoning as a method, downplaying the importance of observation. Aristotle disagreed, locating the Forms in the natural world as the fundamental principles to be discovered through the inquiry into nature ( Metaphysics Z , in Barnes 1984).

Aristotle is recognized as giving the earliest systematic treatise on the nature of scientific inquiry in the western tradition, one which embraced observation and reasoning about the natural world. In the Prior and Posterior Analytics , Aristotle reflects first on the aims and then the methods of inquiry into nature. A number of features can be found which are still considered by most to be essential to science. For Aristotle, empiricism, careful observation (but passive observation, not controlled experiment), is the starting point. The aim is not merely recording of facts, though. For Aristotle, science ( epistêmê ) is a body of properly arranged knowledge or learning—the empirical facts, but also their ordering and display are of crucial importance. The aims of discovery, ordering, and display of facts partly determine the methods required of successful scientific inquiry. Also determinant is the nature of the knowledge being sought, and the explanatory causes proper to that kind of knowledge (see the discussion of the four causes in the entry on Aristotle on causality ).

In addition to careful observation, then, scientific method requires a logic as a system of reasoning for properly arranging, but also inferring beyond, what is known by observation. Methods of reasoning may include induction, prediction, or analogy, among others. Aristotle’s system (along with his catalogue of fallacious reasoning) was collected under the title the Organon . This title would be echoed in later works on scientific reasoning, such as Novum Organon by Francis Bacon, and Novum Organon Restorum by William Whewell (see below). In Aristotle’s Organon reasoning is divided primarily into two forms, a rough division which persists into modern times. The division, known most commonly today as deductive versus inductive method, appears in other eras and methodologies as analysis/​synthesis, non-ampliative/​ampliative, or even confirmation/​verification. The basic idea is there are two “directions” to proceed in our methods of inquiry: one away from what is observed, to the more fundamental, general, and encompassing principles; the other, from the fundamental and general to instances or implications of principles.

The basic aim and method of inquiry identified here can be seen as a theme running throughout the next two millennia of reflection on the correct way to seek after knowledge: carefully observe nature and then seek rules or principles which explain or predict its operation. The Aristotelian corpus provided the framework for a commentary tradition on scientific method independent of science itself (cosmos versus physics.) During the medieval period, figures such as Albertus Magnus (1206–1280), Thomas Aquinas (1225–1274), Robert Grosseteste (1175–1253), Roger Bacon (1214/1220–1292), William of Ockham (1287–1347), Andreas Vesalius (1514–1546), Giacomo Zabarella (1533–1589) all worked to clarify the kind of knowledge obtainable by observation and induction, the source of justification of induction, and best rules for its application. [ 2 ] Many of their contributions we now think of as essential to science (see also Laudan 1968). As Aristotle and Plato had employed a framework of reasoning either “to the forms” or “away from the forms”, medieval thinkers employed directions away from the phenomena or back to the phenomena. In analysis, a phenomena was examined to discover its basic explanatory principles; in synthesis, explanations of a phenomena were constructed from first principles.

During the Scientific Revolution these various strands of argument, experiment, and reason were forged into a dominant epistemic authority. The 16 th –18 th centuries were a period of not only dramatic advance in knowledge about the operation of the natural world—advances in mechanical, medical, biological, political, economic explanations—but also of self-awareness of the revolutionary changes taking place, and intense reflection on the source and legitimation of the method by which the advances were made. The struggle to establish the new authority included methodological moves. The Book of Nature, according to the metaphor of Galileo Galilei (1564–1642) or Francis Bacon (1561–1626), was written in the language of mathematics, of geometry and number. This motivated an emphasis on mathematical description and mechanical explanation as important aspects of scientific method. Through figures such as Henry More and Ralph Cudworth, a neo-Platonic emphasis on the importance of metaphysical reflection on nature behind appearances, particularly regarding the spiritual as a complement to the purely mechanical, remained an important methodological thread of the Scientific Revolution (see the entries on Cambridge platonists ; Boyle ; Henry More ; Galileo ).

In Novum Organum (1620), Bacon was critical of the Aristotelian method for leaping from particulars to universals too quickly. The syllogistic form of reasoning readily mixed those two types of propositions. Bacon aimed at the invention of new arts, principles, and directions. His method would be grounded in methodical collection of observations, coupled with correction of our senses (and particularly, directions for the avoidance of the Idols, as he called them, kinds of systematic errors to which naïve observers are prone.) The community of scientists could then climb, by a careful, gradual and unbroken ascent, to reliable general claims.

Bacon’s method has been criticized as impractical and too inflexible for the practicing scientist. Whewell would later criticize Bacon in his System of Logic for paying too little attention to the practices of scientists. It is hard to find convincing examples of Bacon’s method being put in to practice in the history of science, but there are a few who have been held up as real examples of 16 th century scientific, inductive method, even if not in the rigid Baconian mold: figures such as Robert Boyle (1627–1691) and William Harvey (1578–1657) (see the entry on Bacon ).

It is to Isaac Newton (1642–1727), however, that historians of science and methodologists have paid greatest attention. Given the enormous success of his Principia Mathematica and Opticks , this is understandable. The study of Newton’s method has had two main thrusts: the implicit method of the experiments and reasoning presented in the Opticks, and the explicit methodological rules given as the Rules for Philosophising (the Regulae) in Book III of the Principia . [ 3 ] Newton’s law of gravitation, the linchpin of his new cosmology, broke with explanatory conventions of natural philosophy, first for apparently proposing action at a distance, but more generally for not providing “true”, physical causes. The argument for his System of the World ( Principia , Book III) was based on phenomena, not reasoned first principles. This was viewed (mainly on the continent) as insufficient for proper natural philosophy. The Regulae counter this objection, re-defining the aims of natural philosophy by re-defining the method natural philosophers should follow. (See the entry on Newton’s philosophy .)

To his list of methodological prescriptions should be added Newton’s famous phrase “ hypotheses non fingo ” (commonly translated as “I frame no hypotheses”.) The scientist was not to invent systems but infer explanations from observations, as Bacon had advocated. This would come to be known as inductivism. In the century after Newton, significant clarifications of the Newtonian method were made. Colin Maclaurin (1698–1746), for instance, reconstructed the essential structure of the method as having complementary analysis and synthesis phases, one proceeding away from the phenomena in generalization, the other from the general propositions to derive explanations of new phenomena. Denis Diderot (1713–1784) and editors of the Encyclopédie did much to consolidate and popularize Newtonianism, as did Francesco Algarotti (1721–1764). The emphasis was often the same, as much on the character of the scientist as on their process, a character which is still commonly assumed. The scientist is humble in the face of nature, not beholden to dogma, obeys only his eyes, and follows the truth wherever it leads. It was certainly Voltaire (1694–1778) and du Chatelet (1706–1749) who were most influential in propagating the latter vision of the scientist and their craft, with Newton as hero. Scientific method became a revolutionary force of the Enlightenment. (See also the entries on Newton , Leibniz , Descartes , Boyle , Hume , enlightenment , as well as Shank 2008 for a historical overview.)

Not all 18 th century reflections on scientific method were so celebratory. Famous also are George Berkeley’s (1685–1753) attack on the mathematics of the new science, as well as the over-emphasis of Newtonians on observation; and David Hume’s (1711–1776) undermining of the warrant offered for scientific claims by inductive justification (see the entries on: George Berkeley ; David Hume ; Hume’s Newtonianism and Anti-Newtonianism ). Hume’s problem of induction motivated Immanuel Kant (1724–1804) to seek new foundations for empirical method, though as an epistemic reconstruction, not as any set of practical guidelines for scientists. Both Hume and Kant influenced the methodological reflections of the next century, such as the debate between Mill and Whewell over the certainty of inductive inferences in science.

The debate between John Stuart Mill (1806–1873) and William Whewell (1794–1866) has become the canonical methodological debate of the 19 th century. Although often characterized as a debate between inductivism and hypothetico-deductivism, the role of the two methods on each side is actually more complex. On the hypothetico-deductive account, scientists work to come up with hypotheses from which true observational consequences can be deduced—hence, hypothetico-deductive. Because Whewell emphasizes both hypotheses and deduction in his account of method, he can be seen as a convenient foil to the inductivism of Mill. However, equally if not more important to Whewell’s portrayal of scientific method is what he calls the “fundamental antithesis”. Knowledge is a product of the objective (what we see in the world around us) and subjective (the contributions of our mind to how we perceive and understand what we experience, which he called the Fundamental Ideas). Both elements are essential according to Whewell, and he was therefore critical of Kant for too much focus on the subjective, and John Locke (1632–1704) and Mill for too much focus on the senses. Whewell’s fundamental ideas can be discipline relative. An idea can be fundamental even if it is necessary for knowledge only within a given scientific discipline (e.g., chemical affinity for chemistry). This distinguishes fundamental ideas from the forms and categories of intuition of Kant. (See the entry on Whewell .)

Clarifying fundamental ideas would therefore be an essential part of scientific method and scientific progress. Whewell called this process “Discoverer’s Induction”. It was induction, following Bacon or Newton, but Whewell sought to revive Bacon’s account by emphasising the role of ideas in the clear and careful formulation of inductive hypotheses. Whewell’s induction is not merely the collecting of objective facts. The subjective plays a role through what Whewell calls the Colligation of Facts, a creative act of the scientist, the invention of a theory. A theory is then confirmed by testing, where more facts are brought under the theory, called the Consilience of Inductions. Whewell felt that this was the method by which the true laws of nature could be discovered: clarification of fundamental concepts, clever invention of explanations, and careful testing. Mill, in his critique of Whewell, and others who have cast Whewell as a fore-runner of the hypothetico-deductivist view, seem to have under-estimated the importance of this discovery phase in Whewell’s understanding of method (Snyder 1997a,b, 1999). Down-playing the discovery phase would come to characterize methodology of the early 20 th century (see section 3 ).

Mill, in his System of Logic , put forward a narrower view of induction as the essence of scientific method. For Mill, induction is the search first for regularities among events. Among those regularities, some will continue to hold for further observations, eventually gaining the status of laws. One can also look for regularities among the laws discovered in a domain, i.e., for a law of laws. Which “law law” will hold is time and discipline dependent and open to revision. One example is the Law of Universal Causation, and Mill put forward specific methods for identifying causes—now commonly known as Mill’s methods. These five methods look for circumstances which are common among the phenomena of interest, those which are absent when the phenomena are, or those for which both vary together. Mill’s methods are still seen as capturing basic intuitions about experimental methods for finding the relevant explanatory factors ( System of Logic (1843), see Mill entry). The methods advocated by Whewell and Mill, in the end, look similar. Both involve inductive generalization to covering laws. They differ dramatically, however, with respect to the necessity of the knowledge arrived at; that is, at the meta-methodological level (see the entries on Whewell and Mill entries).

3. Logic of method and critical responses

The quantum and relativistic revolutions in physics in the early 20 th century had a profound effect on methodology. Conceptual foundations of both theories were taken to show the defeasibility of even the most seemingly secure intuitions about space, time and bodies. Certainty of knowledge about the natural world was therefore recognized as unattainable. Instead a renewed empiricism was sought which rendered science fallible but still rationally justifiable.

Analyses of the reasoning of scientists emerged, according to which the aspects of scientific method which were of primary importance were the means of testing and confirming of theories. A distinction in methodology was made between the contexts of discovery and justification. The distinction could be used as a wedge between the particularities of where and how theories or hypotheses are arrived at, on the one hand, and the underlying reasoning scientists use (whether or not they are aware of it) when assessing theories and judging their adequacy on the basis of the available evidence. By and large, for most of the 20 th century, philosophy of science focused on the second context, although philosophers differed on whether to focus on confirmation or refutation as well as on the many details of how confirmation or refutation could or could not be brought about. By the mid-20 th century these attempts at defining the method of justification and the context distinction itself came under pressure. During the same period, philosophy of science developed rapidly, and from section 4 this entry will therefore shift from a primarily historical treatment of the scientific method towards a primarily thematic one.

Advances in logic and probability held out promise of the possibility of elaborate reconstructions of scientific theories and empirical method, the best example being Rudolf Carnap’s The Logical Structure of the World (1928). Carnap attempted to show that a scientific theory could be reconstructed as a formal axiomatic system—that is, a logic. That system could refer to the world because some of its basic sentences could be interpreted as observations or operations which one could perform to test them. The rest of the theoretical system, including sentences using theoretical or unobservable terms (like electron or force) would then either be meaningful because they could be reduced to observations, or they had purely logical meanings (called analytic, like mathematical identities). This has been referred to as the verifiability criterion of meaning. According to the criterion, any statement not either analytic or verifiable was strictly meaningless. Although the view was endorsed by Carnap in 1928, he would later come to see it as too restrictive (Carnap 1956). Another familiar version of this idea is operationalism of Percy William Bridgman. In The Logic of Modern Physics (1927) Bridgman asserted that every physical concept could be defined in terms of the operations one would perform to verify the application of that concept. Making good on the operationalisation of a concept even as simple as length, however, can easily become enormously complex (for measuring very small lengths, for instance) or impractical (measuring large distances like light years.)

Carl Hempel’s (1950, 1951) criticisms of the verifiability criterion of meaning had enormous influence. He pointed out that universal generalizations, such as most scientific laws, were not strictly meaningful on the criterion. Verifiability and operationalism both seemed too restrictive to capture standard scientific aims and practice. The tenuous connection between these reconstructions and actual scientific practice was criticized in another way. In both approaches, scientific methods are instead recast in methodological roles. Measurements, for example, were looked to as ways of giving meanings to terms. The aim of the philosopher of science was not to understand the methods per se , but to use them to reconstruct theories, their meanings, and their relation to the world. When scientists perform these operations, however, they will not report that they are doing them to give meaning to terms in a formal axiomatic system. This disconnect between methodology and the details of actual scientific practice would seem to violate the empiricism the Logical Positivists and Bridgman were committed to. The view that methodology should correspond to practice (to some extent) has been called historicism, or intuitionism. We turn to these criticisms and responses in section 3.4 . [ 4 ]

Positivism also had to contend with the recognition that a purely inductivist approach, along the lines of Bacon-Newton-Mill, was untenable. There was no pure observation, for starters. All observation was theory laden. Theory is required to make any observation, therefore not all theory can be derived from observation alone. (See the entry on theory and observation in science .) Even granting an observational basis, Hume had already pointed out that one could not deductively justify inductive conclusions without begging the question by presuming the success of the inductive method. Likewise, positivist attempts at analyzing how a generalization can be confirmed by observations of its instances were subject to a number of criticisms. Goodman (1965) and Hempel (1965) both point to paradoxes inherent in standard accounts of confirmation. Recent attempts at explaining how observations can serve to confirm a scientific theory are discussed in section 4 below.

The standard starting point for a non-inductive analysis of the logic of confirmation is known as the Hypothetico-Deductive (H-D) method. In its simplest form, a sentence of a theory which expresses some hypothesis is confirmed by its true consequences. As noted in section 2 , this method had been advanced by Whewell in the 19 th century, as well as Nicod (1924) and others in the 20 th century. Often, Hempel’s (1966) description of the H-D method, illustrated by the case of Semmelweiss’ inferential procedures in establishing the cause of childbed fever, has been presented as a key account of H-D as well as a foil for criticism of the H-D account of confirmation (see, for example, Lipton’s (2004) discussion of inference to the best explanation; also the entry on confirmation ). Hempel described Semmelsweiss’ procedure as examining various hypotheses explaining the cause of childbed fever. Some hypotheses conflicted with observable facts and could be rejected as false immediately. Others needed to be tested experimentally by deducing which observable events should follow if the hypothesis were true (what Hempel called the test implications of the hypothesis), then conducting an experiment and observing whether or not the test implications occurred. If the experiment showed the test implication to be false, the hypothesis could be rejected. If the experiment showed the test implications to be true, however, this did not prove the hypothesis true. The confirmation of a test implication does not verify a hypothesis, though Hempel did allow that “it provides at least some support, some corroboration or confirmation for it” (Hempel 1966: 8). The degree of this support then depends on the quantity, variety and precision of the supporting evidence.

Another approach that took off from the difficulties with inductive inference was Karl Popper’s critical rationalism or falsificationism (Popper 1959, 1963). Falsification is deductive and similar to H-D in that it involves scientists deducing observational consequences from the hypothesis under test. For Popper, however, the important point was not the degree of confirmation that successful prediction offered to a hypothesis. The crucial thing was the logical asymmetry between confirmation, based on inductive inference, and falsification, which can be based on a deductive inference. (This simple opposition was later questioned, by Lakatos, among others. See the entry on historicist theories of scientific rationality. )

Popper stressed that, regardless of the amount of confirming evidence, we can never be certain that a hypothesis is true without committing the fallacy of affirming the consequent. Instead, Popper introduced the notion of corroboration as a measure for how well a theory or hypothesis has survived previous testing—but without implying that this is also a measure for the probability that it is true.

Popper was also motivated by his doubts about the scientific status of theories like the Marxist theory of history or psycho-analysis, and so wanted to demarcate between science and pseudo-science. Popper saw this as an importantly different distinction than demarcating science from metaphysics. The latter demarcation was the primary concern of many logical empiricists. Popper used the idea of falsification to draw a line instead between pseudo and proper science. Science was science because its method involved subjecting theories to rigorous tests which offered a high probability of failing and thus refuting the theory.

A commitment to the risk of failure was important. Avoiding falsification could be done all too easily. If a consequence of a theory is inconsistent with observations, an exception can be added by introducing auxiliary hypotheses designed explicitly to save the theory, so-called ad hoc modifications. This Popper saw done in pseudo-science where ad hoc theories appeared capable of explaining anything in their field of application. In contrast, science is risky. If observations showed the predictions from a theory to be wrong, the theory would be refuted. Hence, scientific hypotheses must be falsifiable. Not only must there exist some possible observation statement which could falsify the hypothesis or theory, were it observed, (Popper called these the hypothesis’ potential falsifiers) it is crucial to the Popperian scientific method that such falsifications be sincerely attempted on a regular basis.

The more potential falsifiers of a hypothesis, the more falsifiable it would be, and the more the hypothesis claimed. Conversely, hypotheses without falsifiers claimed very little or nothing at all. Originally, Popper thought that this meant the introduction of ad hoc hypotheses only to save a theory should not be countenanced as good scientific method. These would undermine the falsifiabililty of a theory. However, Popper later came to recognize that the introduction of modifications (immunizations, he called them) was often an important part of scientific development. Responding to surprising or apparently falsifying observations often generated important new scientific insights. Popper’s own example was the observed motion of Uranus which originally did not agree with Newtonian predictions. The ad hoc hypothesis of an outer planet explained the disagreement and led to further falsifiable predictions. Popper sought to reconcile the view by blurring the distinction between falsifiable and not falsifiable, and speaking instead of degrees of testability (Popper 1985: 41f.).

From the 1960s on, sustained meta-methodological criticism emerged that drove philosophical focus away from scientific method. A brief look at those criticisms follows, with recommendations for further reading at the end of the entry.

Thomas Kuhn’s The Structure of Scientific Revolutions (1962) begins with a well-known shot across the bow for philosophers of science:

History, if viewed as a repository for more than anecdote or chronology, could produce a decisive transformation in the image of science by which we are now possessed. (1962: 1)

The image Kuhn thought needed transforming was the a-historical, rational reconstruction sought by many of the Logical Positivists, though Carnap and other positivists were actually quite sympathetic to Kuhn’s views. (See the entry on the Vienna Circle .) Kuhn shares with other of his contemporaries, such as Feyerabend and Lakatos, a commitment to a more empirical approach to philosophy of science. Namely, the history of science provides important data, and necessary checks, for philosophy of science, including any theory of scientific method.

The history of science reveals, according to Kuhn, that scientific development occurs in alternating phases. During normal science, the members of the scientific community adhere to the paradigm in place. Their commitment to the paradigm means a commitment to the puzzles to be solved and the acceptable ways of solving them. Confidence in the paradigm remains so long as steady progress is made in solving the shared puzzles. Method in this normal phase operates within a disciplinary matrix (Kuhn’s later concept of a paradigm) which includes standards for problem solving, and defines the range of problems to which the method should be applied. An important part of a disciplinary matrix is the set of values which provide the norms and aims for scientific method. The main values that Kuhn identifies are prediction, problem solving, simplicity, consistency, and plausibility.

An important by-product of normal science is the accumulation of puzzles which cannot be solved with resources of the current paradigm. Once accumulation of these anomalies has reached some critical mass, it can trigger a communal shift to a new paradigm and a new phase of normal science. Importantly, the values that provide the norms and aims for scientific method may have transformed in the meantime. Method may therefore be relative to discipline, time or place

Feyerabend also identified the aims of science as progress, but argued that any methodological prescription would only stifle that progress (Feyerabend 1988). His arguments are grounded in re-examining accepted “myths” about the history of science. Heroes of science, like Galileo, are shown to be just as reliant on rhetoric and persuasion as they are on reason and demonstration. Others, like Aristotle, are shown to be far more reasonable and far-reaching in their outlooks then they are given credit for. As a consequence, the only rule that could provide what he took to be sufficient freedom was the vacuous “anything goes”. More generally, even the methodological restriction that science is the best way to pursue knowledge, and to increase knowledge, is too restrictive. Feyerabend suggested instead that science might, in fact, be a threat to a free society, because it and its myth had become so dominant (Feyerabend 1978).

An even more fundamental kind of criticism was offered by several sociologists of science from the 1970s onwards who rejected the methodology of providing philosophical accounts for the rational development of science and sociological accounts of the irrational mistakes. Instead, they adhered to a symmetry thesis on which any causal explanation of how scientific knowledge is established needs to be symmetrical in explaining truth and falsity, rationality and irrationality, success and mistakes, by the same causal factors (see, e.g., Barnes and Bloor 1982, Bloor 1991). Movements in the Sociology of Science, like the Strong Programme, or in the social dimensions and causes of knowledge more generally led to extended and close examination of detailed case studies in contemporary science and its history. (See the entries on the social dimensions of scientific knowledge and social epistemology .) Well-known examinations by Latour and Woolgar (1979/1986), Knorr-Cetina (1981), Pickering (1984), Shapin and Schaffer (1985) seem to bear out that it was social ideologies (on a macro-scale) or individual interactions and circumstances (on a micro-scale) which were the primary causal factors in determining which beliefs gained the status of scientific knowledge. As they saw it therefore, explanatory appeals to scientific method were not empirically grounded.

A late, and largely unexpected, criticism of scientific method came from within science itself. Beginning in the early 2000s, a number of scientists attempting to replicate the results of published experiments could not do so. There may be close conceptual connection between reproducibility and method. For example, if reproducibility means that the same scientific methods ought to produce the same result, and all scientific results ought to be reproducible, then whatever it takes to reproduce a scientific result ought to be called scientific method. Space limits us to the observation that, insofar as reproducibility is a desired outcome of proper scientific method, it is not strictly a part of scientific method. (See the entry on reproducibility of scientific results .)

By the close of the 20 th century the search for the scientific method was flagging. Nola and Sankey (2000b) could introduce their volume on method by remarking that “For some, the whole idea of a theory of scientific method is yester-year’s debate …”.

Despite the many difficulties that philosophers encountered in trying to providing a clear methodology of conformation (or refutation), still important progress has been made on understanding how observation can provide evidence for a given theory. Work in statistics has been crucial for understanding how theories can be tested empirically, and in recent decades a huge literature has developed that attempts to recast confirmation in Bayesian terms. Here these developments can be covered only briefly, and we refer to the entry on confirmation for further details and references.

Statistics has come to play an increasingly important role in the methodology of the experimental sciences from the 19 th century onwards. At that time, statistics and probability theory took on a methodological role as an analysis of inductive inference, and attempts to ground the rationality of induction in the axioms of probability theory have continued throughout the 20 th century and in to the present. Developments in the theory of statistics itself, meanwhile, have had a direct and immense influence on the experimental method, including methods for measuring the uncertainty of observations such as the Method of Least Squares developed by Legendre and Gauss in the early 19 th century, criteria for the rejection of outliers proposed by Peirce by the mid-19 th century, and the significance tests developed by Gosset (a.k.a. “Student”), Fisher, Neyman & Pearson and others in the 1920s and 1930s (see, e.g., Swijtink 1987 for a brief historical overview; and also the entry on C.S. Peirce ).

These developments within statistics then in turn led to a reflective discussion among both statisticians and philosophers of science on how to perceive the process of hypothesis testing: whether it was a rigorous statistical inference that could provide a numerical expression of the degree of confidence in the tested hypothesis, or if it should be seen as a decision between different courses of actions that also involved a value component. This led to a major controversy among Fisher on the one side and Neyman and Pearson on the other (see especially Fisher 1955, Neyman 1956 and Pearson 1955, and for analyses of the controversy, e.g., Howie 2002, Marks 2000, Lenhard 2006). On Fisher’s view, hypothesis testing was a methodology for when to accept or reject a statistical hypothesis, namely that a hypothesis should be rejected by evidence if this evidence would be unlikely relative to other possible outcomes, given the hypothesis were true. In contrast, on Neyman and Pearson’s view, the consequence of error also had to play a role when deciding between hypotheses. Introducing the distinction between the error of rejecting a true hypothesis (type I error) and accepting a false hypothesis (type II error), they argued that it depends on the consequences of the error to decide whether it is more important to avoid rejecting a true hypothesis or accepting a false one. Hence, Fisher aimed for a theory of inductive inference that enabled a numerical expression of confidence in a hypothesis. To him, the important point was the search for truth, not utility. In contrast, the Neyman-Pearson approach provided a strategy of inductive behaviour for deciding between different courses of action. Here, the important point was not whether a hypothesis was true, but whether one should act as if it was.

Similar discussions are found in the philosophical literature. On the one side, Churchman (1948) and Rudner (1953) argued that because scientific hypotheses can never be completely verified, a complete analysis of the methods of scientific inference includes ethical judgments in which the scientists must decide whether the evidence is sufficiently strong or that the probability is sufficiently high to warrant the acceptance of the hypothesis, which again will depend on the importance of making a mistake in accepting or rejecting the hypothesis. Others, such as Jeffrey (1956) and Levi (1960) disagreed and instead defended a value-neutral view of science on which scientists should bracket their attitudes, preferences, temperament, and values when assessing the correctness of their inferences. For more details on this value-free ideal in the philosophy of science and its historical development, see Douglas (2009) and Howard (2003). For a broad set of case studies examining the role of values in science, see e.g. Elliott & Richards 2017.

In recent decades, philosophical discussions of the evaluation of probabilistic hypotheses by statistical inference have largely focused on Bayesianism that understands probability as a measure of a person’s degree of belief in an event, given the available information, and frequentism that instead understands probability as a long-run frequency of a repeatable event. Hence, for Bayesians probabilities refer to a state of knowledge, whereas for frequentists probabilities refer to frequencies of events (see, e.g., Sober 2008, chapter 1 for a detailed introduction to Bayesianism and frequentism as well as to likelihoodism). Bayesianism aims at providing a quantifiable, algorithmic representation of belief revision, where belief revision is a function of prior beliefs (i.e., background knowledge) and incoming evidence. Bayesianism employs a rule based on Bayes’ theorem, a theorem of the probability calculus which relates conditional probabilities. The probability that a particular hypothesis is true is interpreted as a degree of belief, or credence, of the scientist. There will also be a probability and a degree of belief that a hypothesis will be true conditional on a piece of evidence (an observation, say) being true. Bayesianism proscribes that it is rational for the scientist to update their belief in the hypothesis to that conditional probability should it turn out that the evidence is, in fact, observed (see, e.g., Sprenger & Hartmann 2019 for a comprehensive treatment of Bayesian philosophy of science). Originating in the work of Neyman and Person, frequentism aims at providing the tools for reducing long-run error rates, such as the error-statistical approach developed by Mayo (1996) that focuses on how experimenters can avoid both type I and type II errors by building up a repertoire of procedures that detect errors if and only if they are present. Both Bayesianism and frequentism have developed over time, they are interpreted in different ways by its various proponents, and their relations to previous criticism to attempts at defining scientific method are seen differently by proponents and critics. The literature, surveys, reviews and criticism in this area are vast and the reader is referred to the entries on Bayesian epistemology and confirmation .

5. Method in Practice

Attention to scientific practice, as we have seen, is not itself new. However, the turn to practice in the philosophy of science of late can be seen as a correction to the pessimism with respect to method in philosophy of science in later parts of the 20 th century, and as an attempted reconciliation between sociological and rationalist explanations of scientific knowledge. Much of this work sees method as detailed and context specific problem-solving procedures, and methodological analyses to be at the same time descriptive, critical and advisory (see Nickles 1987 for an exposition of this view). The following section contains a survey of some of the practice focuses. In this section we turn fully to topics rather than chronology.

A problem with the distinction between the contexts of discovery and justification that figured so prominently in philosophy of science in the first half of the 20 th century (see section 2 ) is that no such distinction can be clearly seen in scientific activity (see Arabatzis 2006). Thus, in recent decades, it has been recognized that study of conceptual innovation and change should not be confined to psychology and sociology of science, but are also important aspects of scientific practice which philosophy of science should address (see also the entry on scientific discovery ). Looking for the practices that drive conceptual innovation has led philosophers to examine both the reasoning practices of scientists and the wide realm of experimental practices that are not directed narrowly at testing hypotheses, that is, exploratory experimentation.

Examining the reasoning practices of historical and contemporary scientists, Nersessian (2008) has argued that new scientific concepts are constructed as solutions to specific problems by systematic reasoning, and that of analogy, visual representation and thought-experimentation are among the important reasoning practices employed. These ubiquitous forms of reasoning are reliable—but also fallible—methods of conceptual development and change. On her account, model-based reasoning consists of cycles of construction, simulation, evaluation and adaption of models that serve as interim interpretations of the target problem to be solved. Often, this process will lead to modifications or extensions, and a new cycle of simulation and evaluation. However, Nersessian also emphasizes that

creative model-based reasoning cannot be applied as a simple recipe, is not always productive of solutions, and even its most exemplary usages can lead to incorrect solutions. (Nersessian 2008: 11)

Thus, while on the one hand she agrees with many previous philosophers that there is no logic of discovery, discoveries can derive from reasoned processes, such that a large and integral part of scientific practice is

the creation of concepts through which to comprehend, structure, and communicate about physical phenomena …. (Nersessian 1987: 11)

Similarly, work on heuristics for discovery and theory construction by scholars such as Darden (1991) and Bechtel & Richardson (1993) present science as problem solving and investigate scientific problem solving as a special case of problem-solving in general. Drawing largely on cases from the biological sciences, much of their focus has been on reasoning strategies for the generation, evaluation, and revision of mechanistic explanations of complex systems.

Addressing another aspect of the context distinction, namely the traditional view that the primary role of experiments is to test theoretical hypotheses according to the H-D model, other philosophers of science have argued for additional roles that experiments can play. The notion of exploratory experimentation was introduced to describe experiments driven by the desire to obtain empirical regularities and to develop concepts and classifications in which these regularities can be described (Steinle 1997, 2002; Burian 1997; Waters 2007)). However the difference between theory driven experimentation and exploratory experimentation should not be seen as a sharp distinction. Theory driven experiments are not always directed at testing hypothesis, but may also be directed at various kinds of fact-gathering, such as determining numerical parameters. Vice versa , exploratory experiments are usually informed by theory in various ways and are therefore not theory-free. Instead, in exploratory experiments phenomena are investigated without first limiting the possible outcomes of the experiment on the basis of extant theory about the phenomena.

The development of high throughput instrumentation in molecular biology and neighbouring fields has given rise to a special type of exploratory experimentation that collects and analyses very large amounts of data, and these new ‘omics’ disciplines are often said to represent a break with the ideal of hypothesis-driven science (Burian 2007; Elliott 2007; Waters 2007; O’Malley 2007) and instead described as data-driven research (Leonelli 2012; Strasser 2012) or as a special kind of “convenience experimentation” in which many experiments are done simply because they are extraordinarily convenient to perform (Krohs 2012).

5.2 Computer methods and ‘new ways’ of doing science

The field of omics just described is possible because of the ability of computers to process, in a reasonable amount of time, the huge quantities of data required. Computers allow for more elaborate experimentation (higher speed, better filtering, more variables, sophisticated coordination and control), but also, through modelling and simulations, might constitute a form of experimentation themselves. Here, too, we can pose a version of the general question of method versus practice: does the practice of using computers fundamentally change scientific method, or merely provide a more efficient means of implementing standard methods?

Because computers can be used to automate measurements, quantifications, calculations, and statistical analyses where, for practical reasons, these operations cannot be otherwise carried out, many of the steps involved in reaching a conclusion on the basis of an experiment are now made inside a “black box”, without the direct involvement or awareness of a human. This has epistemological implications, regarding what we can know, and how we can know it. To have confidence in the results, computer methods are therefore subjected to tests of verification and validation.

The distinction between verification and validation is easiest to characterize in the case of computer simulations. In a typical computer simulation scenario computers are used to numerically integrate differential equations for which no analytic solution is available. The equations are part of the model the scientist uses to represent a phenomenon or system under investigation. Verifying a computer simulation means checking that the equations of the model are being correctly approximated. Validating a simulation means checking that the equations of the model are adequate for the inferences one wants to make on the basis of that model.

A number of issues related to computer simulations have been raised. The identification of validity and verification as the testing methods has been criticized. Oreskes et al. (1994) raise concerns that “validiation”, because it suggests deductive inference, might lead to over-confidence in the results of simulations. The distinction itself is probably too clean, since actual practice in the testing of simulations mixes and moves back and forth between the two (Weissart 1997; Parker 2008a; Winsberg 2010). Computer simulations do seem to have a non-inductive character, given that the principles by which they operate are built in by the programmers, and any results of the simulation follow from those in-built principles in such a way that those results could, in principle, be deduced from the program code and its inputs. The status of simulations as experiments has therefore been examined (Kaufmann and Smarr 1993; Humphreys 1995; Hughes 1999; Norton and Suppe 2001). This literature considers the epistemology of these experiments: what we can learn by simulation, and also the kinds of justifications which can be given in applying that knowledge to the “real” world. (Mayo 1996; Parker 2008b). As pointed out, part of the advantage of computer simulation derives from the fact that huge numbers of calculations can be carried out without requiring direct observation by the experimenter/​simulator. At the same time, many of these calculations are approximations to the calculations which would be performed first-hand in an ideal situation. Both factors introduce uncertainties into the inferences drawn from what is observed in the simulation.

For many of the reasons described above, computer simulations do not seem to belong clearly to either the experimental or theoretical domain. Rather, they seem to crucially involve aspects of both. This has led some authors, such as Fox Keller (2003: 200) to argue that we ought to consider computer simulation a “qualitatively different way of doing science”. The literature in general tends to follow Kaufmann and Smarr (1993) in referring to computer simulation as a “third way” for scientific methodology (theoretical reasoning and experimental practice are the first two ways.). It should also be noted that the debates around these issues have tended to focus on the form of computer simulation typical in the physical sciences, where models are based on dynamical equations. Other forms of simulation might not have the same problems, or have problems of their own (see the entry on computer simulations in science ).

In recent years, the rapid development of machine learning techniques has prompted some scholars to suggest that the scientific method has become “obsolete” (Anderson 2008, Carrol and Goodstein 2009). This has resulted in an intense debate on the relative merit of data-driven and hypothesis-driven research (for samples, see e.g. Mazzocchi 2015 or Succi and Coveney 2018). For a detailed treatment of this topic, we refer to the entry scientific research and big data .

6. Discourse on scientific method

Despite philosophical disagreements, the idea of the scientific method still figures prominently in contemporary discourse on many different topics, both within science and in society at large. Often, reference to scientific method is used in ways that convey either the legend of a single, universal method characteristic of all science, or grants to a particular method or set of methods privilege as a special ‘gold standard’, often with reference to particular philosophers to vindicate the claims. Discourse on scientific method also typically arises when there is a need to distinguish between science and other activities, or for justifying the special status conveyed to science. In these areas, the philosophical attempts at identifying a set of methods characteristic for scientific endeavors are closely related to the philosophy of science’s classical problem of demarcation (see the entry on science and pseudo-science ) and to the philosophical analysis of the social dimension of scientific knowledge and the role of science in democratic society.

One of the settings in which the legend of a single, universal scientific method has been particularly strong is science education (see, e.g., Bauer 1992; McComas 1996; Wivagg & Allchin 2002). [ 5 ] Often, ‘the scientific method’ is presented in textbooks and educational web pages as a fixed four or five step procedure starting from observations and description of a phenomenon and progressing over formulation of a hypothesis which explains the phenomenon, designing and conducting experiments to test the hypothesis, analyzing the results, and ending with drawing a conclusion. Such references to a universal scientific method can be found in educational material at all levels of science education (Blachowicz 2009), and numerous studies have shown that the idea of a general and universal scientific method often form part of both students’ and teachers’ conception of science (see, e.g., Aikenhead 1987; Osborne et al. 2003). In response, it has been argued that science education need to focus more on teaching about the nature of science, although views have differed on whether this is best done through student-led investigations, contemporary cases, or historical cases (Allchin, Andersen & Nielsen 2014)

Although occasionally phrased with reference to the H-D method, important historical roots of the legend in science education of a single, universal scientific method are the American philosopher and psychologist Dewey’s account of inquiry in How We Think (1910) and the British mathematician Karl Pearson’s account of science in Grammar of Science (1892). On Dewey’s account, inquiry is divided into the five steps of

(i) a felt difficulty, (ii) its location and definition, (iii) suggestion of a possible solution, (iv) development by reasoning of the bearing of the suggestions, (v) further observation and experiment leading to its acceptance or rejection. (Dewey 1910: 72)

Similarly, on Pearson’s account, scientific investigations start with measurement of data and observation of their correction and sequence from which scientific laws can be discovered with the aid of creative imagination. These laws have to be subject to criticism, and their final acceptance will have equal validity for “all normally constituted minds”. Both Dewey’s and Pearson’s accounts should be seen as generalized abstractions of inquiry and not restricted to the realm of science—although both Dewey and Pearson referred to their respective accounts as ‘the scientific method’.

Occasionally, scientists make sweeping statements about a simple and distinct scientific method, as exemplified by Feynman’s simplified version of a conjectures and refutations method presented, for example, in the last of his 1964 Cornell Messenger lectures. [ 6 ] However, just as often scientists have come to the same conclusion as recent philosophy of science that there is not any unique, easily described scientific method. For example, the physicist and Nobel Laureate Weinberg described in the paper “The Methods of Science … And Those By Which We Live” (1995) how

The fact that the standards of scientific success shift with time does not only make the philosophy of science difficult; it also raises problems for the public understanding of science. We do not have a fixed scientific method to rally around and defend. (1995: 8)

Interview studies with scientists on their conception of method shows that scientists often find it hard to figure out whether available evidence confirms their hypothesis, and that there are no direct translations between general ideas about method and specific strategies to guide how research is conducted (Schickore & Hangel 2019, Hangel & Schickore 2017)

Reference to the scientific method has also often been used to argue for the scientific nature or special status of a particular activity. Philosophical positions that argue for a simple and unique scientific method as a criterion of demarcation, such as Popperian falsification, have often attracted practitioners who felt that they had a need to defend their domain of practice. For example, references to conjectures and refutation as the scientific method are abundant in much of the literature on complementary and alternative medicine (CAM)—alongside the competing position that CAM, as an alternative to conventional biomedicine, needs to develop its own methodology different from that of science.

Also within mainstream science, reference to the scientific method is used in arguments regarding the internal hierarchy of disciplines and domains. A frequently seen argument is that research based on the H-D method is superior to research based on induction from observations because in deductive inferences the conclusion follows necessarily from the premises. (See, e.g., Parascandola 1998 for an analysis of how this argument has been made to downgrade epidemiology compared to the laboratory sciences.) Similarly, based on an examination of the practices of major funding institutions such as the National Institutes of Health (NIH), the National Science Foundation (NSF) and the Biomedical Sciences Research Practices (BBSRC) in the UK, O’Malley et al. (2009) have argued that funding agencies seem to have a tendency to adhere to the view that the primary activity of science is to test hypotheses, while descriptive and exploratory research is seen as merely preparatory activities that are valuable only insofar as they fuel hypothesis-driven research.

In some areas of science, scholarly publications are structured in a way that may convey the impression of a neat and linear process of inquiry from stating a question, devising the methods by which to answer it, collecting the data, to drawing a conclusion from the analysis of data. For example, the codified format of publications in most biomedical journals known as the IMRAD format (Introduction, Method, Results, Analysis, Discussion) is explicitly described by the journal editors as “not an arbitrary publication format but rather a direct reflection of the process of scientific discovery” (see the so-called “Vancouver Recommendations”, ICMJE 2013: 11). However, scientific publications do not in general reflect the process by which the reported scientific results were produced. For example, under the provocative title “Is the scientific paper a fraud?”, Medawar argued that scientific papers generally misrepresent how the results have been produced (Medawar 1963/1996). Similar views have been advanced by philosophers, historians and sociologists of science (Gilbert 1976; Holmes 1987; Knorr-Cetina 1981; Schickore 2008; Suppe 1998) who have argued that scientists’ experimental practices are messy and often do not follow any recognizable pattern. Publications of research results, they argue, are retrospective reconstructions of these activities that often do not preserve the temporal order or the logic of these activities, but are instead often constructed in order to screen off potential criticism (see Schickore 2008 for a review of this work).

Philosophical positions on the scientific method have also made it into the court room, especially in the US where judges have drawn on philosophy of science in deciding when to confer special status to scientific expert testimony. A key case is Daubert vs Merrell Dow Pharmaceuticals (92–102, 509 U.S. 579, 1993). In this case, the Supreme Court argued in its 1993 ruling that trial judges must ensure that expert testimony is reliable, and that in doing this the court must look at the expert’s methodology to determine whether the proffered evidence is actually scientific knowledge. Further, referring to works of Popper and Hempel the court stated that

ordinarily, a key question to be answered in determining whether a theory or technique is scientific knowledge … is whether it can be (and has been) tested. (Justice Blackmun, Daubert v. Merrell Dow Pharmaceuticals; see Other Internet Resources for a link to the opinion)

But as argued by Haack (2005a,b, 2010) and by Foster & Hubner (1999), by equating the question of whether a piece of testimony is reliable with the question whether it is scientific as indicated by a special methodology, the court was producing an inconsistent mixture of Popper’s and Hempel’s philosophies, and this has later led to considerable confusion in subsequent case rulings that drew on the Daubert case (see Haack 2010 for a detailed exposition).

The difficulties around identifying the methods of science are also reflected in the difficulties of identifying scientific misconduct in the form of improper application of the method or methods of science. One of the first and most influential attempts at defining misconduct in science was the US definition from 1989 that defined misconduct as

fabrication, falsification, plagiarism, or other practices that seriously deviate from those that are commonly accepted within the scientific community . (Code of Federal Regulations, part 50, subpart A., August 8, 1989, italics added)

However, the “other practices that seriously deviate” clause was heavily criticized because it could be used to suppress creative or novel science. For example, the National Academy of Science stated in their report Responsible Science (1992) that it

wishes to discourage the possibility that a misconduct complaint could be lodged against scientists based solely on their use of novel or unorthodox research methods. (NAS: 27)

This clause was therefore later removed from the definition. For an entry into the key philosophical literature on conduct in science, see Shamoo & Resnick (2009).

The question of the source of the success of science has been at the core of philosophy since the beginning of modern science. If viewed as a matter of epistemology more generally, scientific method is a part of the entire history of philosophy. Over that time, science and whatever methods its practitioners may employ have changed dramatically. Today, many philosophers have taken up the banners of pluralism or of practice to focus on what are, in effect, fine-grained and contextually limited examinations of scientific method. Others hope to shift perspectives in order to provide a renewed general account of what characterizes the activity we call science.

One such perspective has been offered recently by Hoyningen-Huene (2008, 2013), who argues from the history of philosophy of science that after three lengthy phases of characterizing science by its method, we are now in a phase where the belief in the existence of a positive scientific method has eroded and what has been left to characterize science is only its fallibility. First was a phase from Plato and Aristotle up until the 17 th century where the specificity of scientific knowledge was seen in its absolute certainty established by proof from evident axioms; next was a phase up to the mid-19 th century in which the means to establish the certainty of scientific knowledge had been generalized to include inductive procedures as well. In the third phase, which lasted until the last decades of the 20 th century, it was recognized that empirical knowledge was fallible, but it was still granted a special status due to its distinctive mode of production. But now in the fourth phase, according to Hoyningen-Huene, historical and philosophical studies have shown how “scientific methods with the characteristics as posited in the second and third phase do not exist” (2008: 168) and there is no longer any consensus among philosophers and historians of science about the nature of science. For Hoyningen-Huene, this is too negative a stance, and he therefore urges the question about the nature of science anew. His own answer to this question is that “scientific knowledge differs from other kinds of knowledge, especially everyday knowledge, primarily by being more systematic” (Hoyningen-Huene 2013: 14). Systematicity can have several different dimensions: among them are more systematic descriptions, explanations, predictions, defense of knowledge claims, epistemic connectedness, ideal of completeness, knowledge generation, representation of knowledge and critical discourse. Hence, what characterizes science is the greater care in excluding possible alternative explanations, the more detailed elaboration with respect to data on which predictions are based, the greater care in detecting and eliminating sources of error, the more articulate connections to other pieces of knowledge, etc. On this position, what characterizes science is not that the methods employed are unique to science, but that the methods are more carefully employed.

Another, similar approach has been offered by Haack (2003). She sets off, similar to Hoyningen-Huene, from a dissatisfaction with the recent clash between what she calls Old Deferentialism and New Cynicism. The Old Deferentialist position is that science progressed inductively by accumulating true theories confirmed by empirical evidence or deductively by testing conjectures against basic statements; while the New Cynics position is that science has no epistemic authority and no uniquely rational method and is merely just politics. Haack insists that contrary to the views of the New Cynics, there are objective epistemic standards, and there is something epistemologically special about science, even though the Old Deferentialists pictured this in a wrong way. Instead, she offers a new Critical Commonsensist account on which standards of good, strong, supportive evidence and well-conducted, honest, thorough and imaginative inquiry are not exclusive to the sciences, but the standards by which we judge all inquirers. In this sense, science does not differ in kind from other kinds of inquiry, but it may differ in the degree to which it requires broad and detailed background knowledge and a familiarity with a technical vocabulary that only specialists may possess.

• Aikenhead, G.S., 1987, “High-school graduates’ beliefs about science-technology-society. III. Characteristics and limitations of scientific knowledge”, Science Education , 71(4): 459–487.
• Allchin, D., H.M. Andersen and K. Nielsen, 2014, “Complementary Approaches to Teaching Nature of Science: Integrating Student Inquiry, Historical Cases, and Contemporary Cases in Classroom Practice”, Science Education , 98: 461–486.
• Anderson, C., 2008, “The end of theory: The data deluge makes the scientific method obsolete”, Wired magazine , 16(7): 16–07
• Arabatzis, T., 2006, “On the inextricability of the context of discovery and the context of justification”, in Revisiting Discovery and Justification , J. Schickore and F. Steinle (eds.), Dordrecht: Springer, pp. 215–230.
• Barnes, J. (ed.), 1984, The Complete Works of Aristotle, Vols I and II , Princeton: Princeton University Press.
• Barnes, B. and D. Bloor, 1982, “Relativism, Rationalism, and the Sociology of Knowledge”, in Rationality and Relativism , M. Hollis and S. Lukes (eds.), Cambridge: MIT Press, pp. 1–20.
• Bauer, H.H., 1992, Scientific Literacy and the Myth of the Scientific Method , Urbana: University of Illinois Press.
• Bechtel, W. and R.C. Richardson, 1993, Discovering complexity , Princeton, NJ: Princeton University Press.
• Berkeley, G., 1734, The Analyst in De Motu and The Analyst: A Modern Edition with Introductions and Commentary , D. Jesseph (trans. and ed.), Dordrecht: Kluwer Academic Publishers, 1992.
• Blachowicz, J., 2009, “How science textbooks treat scientific method: A philosopher’s perspective”, The British Journal for the Philosophy of Science , 60(2): 303–344.
• Bloor, D., 1991, Knowledge and Social Imagery , Chicago: University of Chicago Press, 2 nd edition.
• Boyle, R., 1682, New experiments physico-mechanical, touching the air , Printed by Miles Flesher for Richard Davis, bookseller in Oxford.
• Bridgman, P.W., 1927, The Logic of Modern Physics , New York: Macmillan.
• –––, 1956, “The Methodological Character of Theoretical Concepts”, in The Foundations of Science and the Concepts of Science and Psychology , Herbert Feigl and Michael Scriven (eds.), Minnesota: University of Minneapolis Press, pp. 38–76.
• Burian, R., 1997, “Exploratory Experimentation and the Role of Histochemical Techniques in the Work of Jean Brachet, 1938–1952”, History and Philosophy of the Life Sciences , 19(1): 27–45.
• –––, 2007, “On microRNA and the need for exploratory experimentation in post-genomic molecular biology”, History and Philosophy of the Life Sciences , 29(3): 285–311.
• Carnap, R., 1928, Der logische Aufbau der Welt , Berlin: Bernary, transl. by R.A. George, The Logical Structure of the World , Berkeley: University of California Press, 1967.
• –––, 1956, “The methodological character of theoretical concepts”, Minnesota studies in the philosophy of science , 1: 38–76.
• Carrol, S., and D. Goodstein, 2009, “Defining the scientific method”, Nature Methods , 6: 237.
• Churchman, C.W., 1948, “Science, Pragmatics, Induction”, Philosophy of Science , 15(3): 249–268.
• Cooper, J. (ed.), 1997, Plato: Complete Works , Indianapolis: Hackett.
• Darden, L., 1991, Theory Change in Science: Strategies from Mendelian Genetics , Oxford: Oxford University Press
• Dewey, J., 1910, How we think , New York: Dover Publications (reprinted 1997).
• Douglas, H., 2009, Science, Policy, and the Value-Free Ideal , Pittsburgh: University of Pittsburgh Press.
• Dupré, J., 2004, “Miracle of Monism ”, in Naturalism in Question , Mario De Caro and David Macarthur (eds.), Cambridge, MA: Harvard University Press, pp. 36–58.
• Elliott, K.C., 2007, “Varieties of exploratory experimentation in nanotoxicology”, History and Philosophy of the Life Sciences , 29(3): 311–334.
• Elliott, K. C., and T. Richards (eds.), 2017, Exploring inductive risk: Case studies of values in science , Oxford: Oxford University Press.
• Falcon, Andrea, 2005, Aristotle and the science of nature: Unity without uniformity , Cambridge: Cambridge University Press.
• Feyerabend, P., 1978, Science in a Free Society , London: New Left Books
• –––, 1988, Against Method , London: Verso, 2 nd edition.
• Fisher, R.A., 1955, “Statistical Methods and Scientific Induction”, Journal of The Royal Statistical Society. Series B (Methodological) , 17(1): 69–78.
• Foster, K. and P.W. Huber, 1999, Judging Science. Scientific Knowledge and the Federal Courts , Cambridge: MIT Press.
• Fox Keller, E., 2003, “Models, Simulation, and ‘computer experiments’”, in The Philosophy of Scientific Experimentation , H. Radder (ed.), Pittsburgh: Pittsburgh University Press, 198–215.
• Gilbert, G., 1976, “The transformation of research findings into scientific knowledge”, Social Studies of Science , 6: 281–306.
• Gimbel, S., 2011, Exploring the Scientific Method , Chicago: University of Chicago Press.
• Goodman, N., 1965, Fact , Fiction, and Forecast , Indianapolis: Bobbs-Merrill.
• Haack, S., 1995, “Science is neither sacred nor a confidence trick”, Foundations of Science , 1(3): 323–335.
• –––, 2003, Defending science—within reason , Amherst: Prometheus.
• –––, 2005a, “Disentangling Daubert: an epistemological study in theory and practice”, Journal of Philosophy, Science and Law , 5, Haack 2005a available online . doi:10.5840/jpsl2005513
• –––, 2005b, “Trial and error: The Supreme Court’s philosophy of science”, American Journal of Public Health , 95: S66-S73.
• –––, 2010, “Federal Philosophy of Science: A Deconstruction-and a Reconstruction”, NYUJL & Liberty , 5: 394.
• Hangel, N. and J. Schickore, 2017, “Scientists’ conceptions of good research practice”, Perspectives on Science , 25(6): 766–791
• Harper, W.L., 2011, Isaac Newton’s Scientific Method: Turning Data into Evidence about Gravity and Cosmology , Oxford: Oxford University Press.
• Hempel, C., 1950, “Problems and Changes in the Empiricist Criterion of Meaning”, Revue Internationale de Philosophie , 41(11): 41–63.
• –––, 1951, “The Concept of Cognitive Significance: A Reconsideration”, Proceedings of the American Academy of Arts and Sciences , 80(1): 61–77.
• –––, 1965, Aspects of scientific explanation and other essays in the philosophy of science , New York–London: Free Press.
• –––, 1966, Philosophy of Natural Science , Englewood Cliffs: Prentice-Hall.
• Holmes, F.L., 1987, “Scientific writing and scientific discovery”, Isis , 78(2): 220–235.
• Howard, D., 2003, “Two left turns make a right: On the curious political career of North American philosophy of science at midcentury”, in Logical Empiricism in North America , G.L. Hardcastle & A.W. Richardson (eds.), Minneapolis: University of Minnesota Press, pp. 25–93.
• Hoyningen-Huene, P., 2008, “Systematicity: The nature of science”, Philosophia , 36(2): 167–180.
• –––, 2013, Systematicity. The Nature of Science , Oxford: Oxford University Press.
• Howie, D., 2002, Interpreting probability: Controversies and developments in the early twentieth century , Cambridge: Cambridge University Press.
• Hughes, R., 1999, “The Ising Model, Computer Simulation, and Universal Physics”, in Models as Mediators , M. Morgan and M. Morrison (eds.), Cambridge: Cambridge University Press, pp. 97–145
• Hume, D., 1739, A Treatise of Human Nature , D. Fate Norton and M.J. Norton (eds.), Oxford: Oxford University Press, 2000.
• Humphreys, P., 1995, “Computational science and scientific method”, Minds and Machines , 5(1): 499–512.
• ICMJE, 2013, “Recommendations for the Conduct, Reporting, Editing, and Publication of Scholarly Work in Medical Journals”, International Committee of Medical Journal Editors, available online , accessed August 13 2014
• Jeffrey, R.C., 1956, “Valuation and Acceptance of Scientific Hypotheses”, Philosophy of Science , 23(3): 237–246.
• Kaufmann, W.J., and L.L. Smarr, 1993, Supercomputing and the Transformation of Science , New York: Scientific American Library.
• Knorr-Cetina, K., 1981, The Manufacture of Knowledge , Oxford: Pergamon Press.
• Krohs, U., 2012, “Convenience experimentation”, Studies in History and Philosophy of Biological and BiomedicalSciences , 43: 52–57.
• Kuhn, T.S., 1962, The Structure of Scientific Revolutions , Chicago: University of Chicago Press
• Latour, B. and S. Woolgar, 1986, Laboratory Life: The Construction of Scientific Facts , Princeton: Princeton University Press, 2 nd edition.
• Laudan, L., 1968, “Theories of scientific method from Plato to Mach”, History of Science , 7(1): 1–63.
• Lenhard, J., 2006, “Models and statistical inference: The controversy between Fisher and Neyman-Pearson”, The British Journal for the Philosophy of Science , 57(1): 69–91.
• Leonelli, S., 2012, “Making Sense of Data-Driven Research in the Biological and the Biomedical Sciences”, Studies in the History and Philosophy of the Biological and Biomedical Sciences , 43(1): 1–3.
• Levi, I., 1960, “Must the scientist make value judgments?”, Philosophy of Science , 57(11): 345–357
• Lindley, D., 1991, Theory Change in Science: Strategies from Mendelian Genetics , Oxford: Oxford University Press.
• Lipton, P., 2004, Inference to the Best Explanation , London: Routledge, 2 nd edition.
• Marks, H.M., 2000, The progress of experiment: science and therapeutic reform in the United States, 1900–1990 , Cambridge: Cambridge University Press.
• Mazzochi, F., 2015, “Could Big Data be the end of theory in science?”, EMBO reports , 16: 1250–1255.
• Mayo, D.G., 1996, Error and the Growth of Experimental Knowledge , Chicago: University of Chicago Press.
• McComas, W.F., 1996, “Ten myths of science: Reexamining what we think we know about the nature of science”, School Science and Mathematics , 96(1): 10–16.
• Medawar, P.B., 1963/1996, “Is the scientific paper a fraud”, in The Strange Case of the Spotted Mouse and Other Classic Essays on Science , Oxford: Oxford University Press, 33–39.
• Mill, J.S., 1963, Collected Works of John Stuart Mill , J. M. Robson (ed.), Toronto: University of Toronto Press
• NAS, 1992, Responsible Science: Ensuring the integrity of the research process , Washington DC: National Academy Press.
• Nersessian, N.J., 1987, “A cognitive-historical approach to meaning in scientific theories”, in The process of science , N. Nersessian (ed.), Berlin: Springer, pp. 161–177.
• –––, 2008, Creating Scientific Concepts , Cambridge: MIT Press.
• Newton, I., 1726, Philosophiae naturalis Principia Mathematica (3 rd edition), in The Principia: Mathematical Principles of Natural Philosophy: A New Translation , I.B. Cohen and A. Whitman (trans.), Berkeley: University of California Press, 1999.
• –––, 1704, Opticks or A Treatise of the Reflections, Refractions, Inflections & Colors of Light , New York: Dover Publications, 1952.
• Neyman, J., 1956, “Note on an Article by Sir Ronald Fisher”, Journal of the Royal Statistical Society. Series B (Methodological) , 18: 288–294.
• Nickles, T., 1987, “Methodology, heuristics, and rationality”, in Rational changes in science: Essays on Scientific Reasoning , J.C. Pitt (ed.), Berlin: Springer, pp. 103–132.
• Nicod, J., 1924, Le problème logique de l’induction , Paris: Alcan. (Engl. transl. “The Logical Problem of Induction”, in Foundations of Geometry and Induction , London: Routledge, 2000.)
• Nola, R. and H. Sankey, 2000a, “A selective survey of theories of scientific method”, in Nola and Sankey 2000b: 1–65.
• –––, 2000b, After Popper, Kuhn and Feyerabend. Recent Issues in Theories of Scientific Method , London: Springer.
• –––, 2007, Theories of Scientific Method , Stocksfield: Acumen.
• Norton, S., and F. Suppe, 2001, “Why atmospheric modeling is good science”, in Changing the Atmosphere: Expert Knowledge and Environmental Governance , C. Miller and P. Edwards (eds.), Cambridge, MA: MIT Press, 88–133.
• O’Malley, M., 2007, “Exploratory experimentation and scientific practice: Metagenomics and the proteorhodopsin case”, History and Philosophy of the Life Sciences , 29(3): 337–360.
• O’Malley, M., C. Haufe, K. Elliot, and R. Burian, 2009, “Philosophies of Funding”, Cell , 138: 611–615.
• Oreskes, N., K. Shrader-Frechette, and K. Belitz, 1994, “Verification, Validation and Confirmation of Numerical Models in the Earth Sciences”, Science , 263(5147): 641–646.
• Osborne, J., S. Simon, and S. Collins, 2003, “Attitudes towards science: a review of the literature and its implications”, International Journal of Science Education , 25(9): 1049–1079.
• Parascandola, M., 1998, “Epidemiology—2 nd -Rate Science”, Public Health Reports , 113(4): 312–320.
• Parker, W., 2008a, “Franklin, Holmes and the Epistemology of Computer Simulation”, International Studies in the Philosophy of Science , 22(2): 165–83.
• –––, 2008b, “Computer Simulation through an Error-Statistical Lens”, Synthese , 163(3): 371–84.
• Pearson, K. 1892, The Grammar of Science , London: J.M. Dents and Sons, 1951
• Pearson, E.S., 1955, “Statistical Concepts in Their Relation to Reality”, Journal of the Royal Statistical Society , B, 17: 204–207.
• Pickering, A., 1984, Constructing Quarks: A Sociological History of Particle Physics , Edinburgh: Edinburgh University Press.
• Popper, K.R., 1959, The Logic of Scientific Discovery , London: Routledge, 2002
• –––, 1963, Conjectures and Refutations , London: Routledge, 2002.
• –––, 1985, Unended Quest: An Intellectual Autobiography , La Salle: Open Court Publishing Co..
• Rudner, R., 1953, “The Scientist Qua Scientist Making Value Judgments”, Philosophy of Science , 20(1): 1–6.
• Rudolph, J.L., 2005, “Epistemology for the masses: The origin of ‘The Scientific Method’ in American Schools”, History of Education Quarterly , 45(3): 341–376
• Schickore, J., 2008, “Doing science, writing science”, Philosophy of Science , 75: 323–343.
• Schickore, J. and N. Hangel, 2019, “‘It might be this, it should be that…’ uncertainty and doubt in day-to-day science practice”, European Journal for Philosophy of Science , 9(2): 31. doi:10.1007/s13194-019-0253-9
• Shamoo, A.E. and D.B. Resnik, 2009, Responsible Conduct of Research , Oxford: Oxford University Press.
• Shank, J.B., 2008, The Newton Wars and the Beginning of the French Enlightenment , Chicago: The University of Chicago Press.
• Shapin, S. and S. Schaffer, 1985, Leviathan and the air-pump , Princeton: Princeton University Press.
• Smith, G.E., 2002, “The Methodology of the Principia”, in The Cambridge Companion to Newton , I.B. Cohen and G.E. Smith (eds.), Cambridge: Cambridge University Press, 138–173.
• Snyder, L.J., 1997a, “Discoverers’ Induction”, Philosophy of Science , 64: 580–604.
• –––, 1997b, “The Mill-Whewell Debate: Much Ado About Induction”, Perspectives on Science , 5: 159–198.
• –––, 1999, “Renovating the Novum Organum: Bacon, Whewell and Induction”, Studies in History and Philosophy of Science , 30: 531–557.
• Sober, E., 2008, Evidence and Evolution. The logic behind the science , Cambridge: Cambridge University Press
• Sprenger, J. and S. Hartmann, 2019, Bayesian philosophy of science , Oxford: Oxford University Press.
• Steinle, F., 1997, “Entering New Fields: Exploratory Uses of Experimentation”, Philosophy of Science (Proceedings), 64: S65–S74.
• –––, 2002, “Experiments in History and Philosophy of Science”, Perspectives on Science , 10(4): 408–432.
• Strasser, B.J., 2012, “Data-driven sciences: From wonder cabinets to electronic databases”, Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences , 43(1): 85–87.
• Succi, S. and P.V. Coveney, 2018, “Big data: the end of the scientific method?”, Philosophical Transactions of the Royal Society A , 377: 20180145. doi:10.1098/rsta.2018.0145
• Suppe, F., 1998, “The Structure of a Scientific Paper”, Philosophy of Science , 65(3): 381–405.
• Swijtink, Z.G., 1987, “The objectification of observation: Measurement and statistical methods in the nineteenth century”, in The probabilistic revolution. Ideas in History, Vol. 1 , L. Kruger (ed.), Cambridge MA: MIT Press, pp. 261–285.
• Waters, C.K., 2007, “The nature and context of exploratory experimentation: An introduction to three case studies of exploratory research”, History and Philosophy of the Life Sciences , 29(3): 275–284.
• Weinberg, S., 1995, “The methods of science… and those by which we live”, Academic Questions , 8(2): 7–13.
• Weissert, T., 1997, The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem , New York: Springer Verlag.
• William H., 1628, Exercitatio Anatomica de Motu Cordis et Sanguinis in Animalibus , in On the Motion of the Heart and Blood in Animals , R. Willis (trans.), Buffalo: Prometheus Books, 1993.
• Winsberg, E., 2010, Science in the Age of Computer Simulation , Chicago: University of Chicago Press.
• Wivagg, D. & D. Allchin, 2002, “The Dogma of the Scientific Method”, The American Biology Teacher , 64(9): 645–646

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Blackmun opinion , in Daubert v. Merrell Dow Pharmaceuticals (92–102), 509 U.S. 579 (1993).
• Scientific Method at philpapers. Darrell Rowbottom (ed.).
• Recent Articles | Scientific Method | The Scientist Magazine

al-Kindi | Albert the Great [= Albertus magnus] | Aquinas, Thomas | Arabic and Islamic Philosophy, disciplines in: natural philosophy and natural science | Arabic and Islamic Philosophy, historical and methodological topics in: Greek sources | Arabic and Islamic Philosophy, historical and methodological topics in: influence of Arabic and Islamic Philosophy on the Latin West | Aristotle | Bacon, Francis | Bacon, Roger | Berkeley, George | biology: experiment in | Boyle, Robert | Cambridge Platonists | confirmation | Descartes, René | Enlightenment | epistemology | epistemology: Bayesian | epistemology: social | Feyerabend, Paul | Galileo Galilei | Grosseteste, Robert | Hempel, Carl | Hume, David | Hume, David: Newtonianism and Anti-Newtonianism | induction: problem of | Kant, Immanuel | Kuhn, Thomas | Leibniz, Gottfried Wilhelm | Locke, John | Mill, John Stuart | More, Henry | Neurath, Otto | Newton, Isaac | Newton, Isaac: philosophy | Ockham [Occam], William | operationalism | Peirce, Charles Sanders | Plato | Popper, Karl | rationality: historicist theories of | Reichenbach, Hans | reproducibility, scientific | Schlick, Moritz | science: and pseudo-science | science: theory and observation in | science: unity of | scientific discovery | scientific knowledge: social dimensions of | simulations in science | skepticism: medieval | space and time: absolute and relational space and motion, post-Newtonian theories | Vienna Circle | Whewell, William | Zabarella, Giacomo

Copyright © 2021 by Brian Hepburn < brian . hepburn @ wichita . edu > Hanne Andersen < hanne . andersen @ ind . ku . dk >

• Accessibility

Support SEP

Mirror sites.

View this site from another server:

• Info about mirror sites

The Stanford Encyclopedia of Philosophy is copyright © 2023 by The Metaphysics Research Lab , Department of Philosophy, Stanford University

Library of Congress Catalog Data: ISSN 1095-5054

Choose Your Test

Sat / act prep online guides and tips, 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

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.

Student and Parent Forum

Our new student and parent forum, at ExpertHub.PrepScholar.com , allow you to interact with your peers and the PrepScholar staff. See how other students and parents are navigating high school, college, and the college admissions process. Ask questions; get answers.

Ask a Question Below

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

Improve With Our Famous Guides

• For All Students

The 5 Strategies You Must Be Using to Improve 160+ SAT Points

How to Get a Perfect 1600, by a Perfect Scorer

Series: How to Get 800 on Each SAT Section:

Score 800 on SAT Math

Score 800 on SAT Reading

Score 800 on SAT Writing

Series: How to Get to 600 on Each SAT Section:

Score 600 on SAT Math

Score 600 on SAT Reading

Score 600 on SAT Writing

Free Complete Official SAT Practice Tests

What SAT Target Score Should You Be Aiming For?

15 Strategies to Improve Your SAT Essay

The 5 Strategies You Must Be Using to Improve 4+ ACT Points

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

Series: How to Get 36 on Each ACT Section:

36 on ACT English

36 on ACT Math

36 on ACT Reading

36 on ACT Science

Series: How to Get to 24 on Each ACT Section:

24 on ACT English

24 on ACT Math

24 on ACT Reading

24 on ACT Science

What ACT target score should you be aiming for?

ACT Vocabulary You Must Know

ACT Writing: 15 Tips to Raise Your Essay Score

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?

Is the ACT easier than the SAT? A Comprehensive Guide

Should you retake your SAT or ACT?

When should you take the SAT or ACT?

Stay Informed

Get the latest articles and test prep tips!

Looking for Graduate School Test Prep?

Check out our top-rated graduate blogs here:

GRE Online Prep Blog

GMAT Online Prep Blog

TOEFL Online Prep Blog

Holly R. "I am absolutely overjoyed and cannot thank you enough for helping me!”

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.1: The Scientific Method

• Last updated
• Save as PDF
• Page ID 123904

• Teresa Friedrich Finnern
• Norco College

Learning Objectives

• Identify the shared characteristics of the natural sciences.
• Summarize the steps of the scientific method.
• Compare inductive reasoning with deductive reasoning.
• Describe the goals of basic science and applied science.

The Process of Science

Science includes such diverse fields as astronomy, biology, computer sciences, geology, logic, physics, chemistry, and mathematics (Figure \(\PageIndex{1}\)). However, those fields of science related to the physical world and its phenomena and processes are considered natural sciences . Natural sciences could be categorized as astronomy, biology, chemistry, earth science, and physics. One can divide natural sciences further into life sciences, which study living things and include biology, and physical sciences, which study nonliving matter and include astronomy, geology, physics, and chemistry. Some disciplines such as biophysics and biochemistry build on both life and physical sciences and are interdisciplinary. Natural sciences are sometimes referred to as “hard science” because they rely on the use of quantitative data; social sciences that study society and human behavior are more likely to use qualitative assessments to drive investigations and findings.

Not surprisingly, the natural science of biology has many branches or subdisciplines. Cell biologists study cell structure and function, while biologists who study anatomy investigate the structure of an entire organism. Those biologists studying physiology, however, focus on the internal functioning of an organism. Some areas of biology focus on only particular types of living things. For example, botanists explore plants, while zoologists specialize in animals.

Scientific Reasoning

One thing is common to all forms of science: an ultimate goal “to know.” Curiosity and inquiry are the driving forces for the development of science. Scientists seek to understand the world and the way it operates. To do this, they use two methods of logical thinking: inductive reasoning and deductive reasoning.

Inductive reasoning is a form of logical thinking that uses related observations to arrive at a general conclusion. This type of reasoning is common in descriptive science. A life scientist such as a biologist makes observations and records them. These data can be qualitative (descriptive) or quantitative (numeric), and the raw data can be supplemented with drawings, pictures, photos, or videos. From many observations, the scientist can infer conclusions (inductions) based on evidence. Inductive reasoning involves formulating generalizations inferred from careful observation and the analysis of a large amount of data.

Deductive reasoning ,   or deduction, is the type of logic used in hypothesis-based science. In deductive reason, the pattern of thinking moves in the opposite direction as compared to inductive reasoning; that is, specific results are predicted from a general premise. Deductive reasoning is a form of logical thinking that uses a general principle or law to forecast specific results. From those general principles, a scientist can extrapolate and predict the specific results that would be valid as long as the general principles are valid. Studies in climate change can illustrate this type of reasoning. For example, scientists may predict that if the climate becomes warmer in a particular region, then the distribution of plants and animals should change. These predictions have been made and tested, and many such changes have been found, such as the modification of arable areas for agriculture, with change based on temperature averages. 

Inductive and deductive reasoning are often used in tandem to advance scientific knowledge (Example \(\PageIndex{1}\)) . Both types of logical thinking are related to the two main pathways of scientific study: descriptive science and hypothesis-based science. Descriptive (or discovery) science , which is usually inductive, aims to observe, explore, and discover, while hypothesis-based science , which is usually deductive, begins with a specific question or problem and a potential answer or solution that one can test. The boundary between these two forms of study is often blurred, and most scientific endeavors combine both approaches.

Example \(\PageIndex{1}\)

Here is an example of how the two types of reasoning might be used in similar situations.

In inductive reasoning, where a conclusion is drawn from a number of observations, one might observe that members of a species are not all the same, individuals compete for resources, and species are generally adapted to their environment. This observation could then lead to the conclusion that individuals most adapted to their environment are more likely to survive and pass their traits to the next generation.

In deductive reasoning, which uses a general premise to predict a specific result, one might start with that conclusion as a general premise, then predict the results. For example, from that premise, one might predict that if the average temperature in an ecosystem increases due to climate change, individuals better adapted to warmer temperatures will outcompete those that are not. A scientist could then design a study to test this prediction.

The Scientific Method

Biologists study the living world by posing questions about it and seeking science-based responses. The scientific method is a method of research with defined steps that include experiments and careful observation. The scientific method was used even in ancient times, but it was first documented by England’s Sir Francis Bacon (1561–1626; Figure \(\PageIndex{2}\)), who set up inductive methods for scientific inquiry. The scientific method is not exclusively used by biologists but can be applied to almost all fields of study as a logical, rational problem-solving method.

It is important to note that even though there are specific steps to the scientific method, the process of science is often more fluid, with scientists going back and forth between steps until they reach their conclusions.

Observation and Question

Scientists are good observers. In the field of biology, naturalists will often will make an observation that leads to a question. A naturalist is a person who studies nature. Naturalists often describe structures, processes, and behavior, either with their eyes or with the use of a tool such as a microscope. A naturalist may not conduct experiments, but they may ask many good questions that can lead to experimentation. Scientists are also very curious. They will research for known answers to their questions or run experiments to learn the answer to their questions.

Let’s think about a simple problem that starts with an observation and apply the scientific method to solve the problem. One Monday morning, a student arrives at class and quickly discovers that the classroom is too warm. That is an observation that also describes a problem: the classroom is too warm. The student then asks a question: “Why is the classroom so warm?”

Proposing a Hypothesis

A hypothesis is an educated guess or a suggested explanation for an event, which can be tested. Sometimes, more than one hypothesis may be proposed. Once a hypothesis has been selected, the student can make a prediction. A prediction is similar to a hypothesis but it typically has the format “If . . . then . . . .”.

For example, one hypothesis might be, “The classroom is warm because no one turned on the air conditioning.” However, there could be other responses to the question, and therefore one may propose other hypotheses. A second hypothesis might be, “The classroom is warm because there is a power failure, and so the air conditioning doesn’t work.” In this case, you would have to test both hypotheses to see if either one could be supported with data.

A hypothesis may become a verified theory . This can happen if it has been repeatedly tested and confirmed, is general, and has inspired many other hypotheses, facts, and experimentations. Not all hypotheses will become theories.

Testing a Hypothesis

A valid hypothesis must be testable. It should also be falsifiable , meaning that it can be disproven by experimental results. Importantly, science does not claim to “prove” anything because scientific understandings are always subject to modification with further information. This step—openness to disproving ideas—is what distinguishes sciences from non-sciences. The presence of the supernatural, for instance, is neither testable nor falsifiable. To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. Each experiment will have one or more variables and one or more controls. A variable is any part of the experiment that can vary or change during the experiment. The control group contains every feature of the experimental group except that it was not manipulated. Therefore, if the results of the experimental group differ from the control group, the difference must be due to the hypothesized manipulation, rather than some outside factor. Look for the variables and controls in the examples that follow. To test the first hypothesis, the student would find out if the air conditioning is on. If the air conditioning is turned on but does not work, there should be another reason, and this hypothesis should be rejected. To test the second hypothesis, the student could check if the lights in the classroom are functional. If so, there is no power failure, and this hypothesis should be rejected. Each hypothesis should be tested by carrying out appropriate experiments. Be aware that rejecting one hypothesis does not determine whether or not the other hypotheses can be accepted; it simply eliminates one hypothesis that is not valid (Figure \(\PageIndex{3}\)). Using the scientific method, the hypotheses that are inconsistent with experimental data are rejected.

While this “warm classroom” example is based on observational results, other hypotheses and experiments might have clearer controls. For instance, a student might attend class on Monday and realize she had difficulty concentrating on the lecture. One observation to explain this occurrence might be, “When I eat breakfast before class, I am better able to pay attention.” The student could then design an experiment with a control to test this hypothesis.

Visual Connection

The scientific method may seem too rigid and structured. It is important to keep in mind that, although scientists often follow this sequence, there is flexibility. Sometimes an experiment leads to conclusions that favor a change in approach; often, an experiment brings entirely new scientific questions to the puzzle. Many times, science does not operate in a linear fashion; instead, scientists continually draw inferences and make generalizations, finding patterns as their research proceeds. Scientific reasoning is more complex than the scientific method alone suggests. Notice, too, that the scientific method can be applied to solving problems that aren’t necessarily scientific in nature (Example \(\PageIndex{2}\)).

Example \(\PageIndex{2}\)

In the example below, the scientific method is used to solve an everyday problem. Match the scientific method steps (numbered items) with the process of solving the everyday problem (lettered items). Based on the results of the experiment, is the hypothesis correct? If it is incorrect, propose some alternative hypotheses.

Steps of the Scientific Method

• Observation
• Hypothesis (answer)

Process of Solving an Everyday Problem

• There is something wrong with the electrical outlet.
• If something is wrong with the outlet, my coffee maker also won’t work when plugged into it.
• My toaster doesn’t toast my bread.
• I plug my coffee maker into the outlet.
• My coffee maker works.
• Why doesn’t my toaster work?

Two Types of Science: Basic Science and Applied Science

The scientific community has been debating for the last few decades about the value of different types of science. Is it valuable to pursue science for the sake of simply gaining knowledge, or does scientific knowledge only have worth if we can apply it to solving a specific problem or to bettering our lives? This question focuses on the differences between two types of science: basic science and applied science.

Basic science or “pure” science seeks to expand knowledge regardless of the short-term application of that knowledge. It is not focused on developing a product or a service of immediate public or commercial value. The immediate goal of basic science is knowledge for knowledge’s sake, though this does not mean that, in the end, it may not result in a practical application.

In contrast, applied science or “technology,” aims to use science to solve real-world problems, making it possible, for example, to improve a crop yield or find a cure for a particular disease. In applied science, the problem is usually defined for the researcher.

Some individuals may perceive applied science as “useful” and basic science as “useless.” A question these people might pose to a scientist advocating knowledge acquisition would be, “What for?” A careful look at the history of science, however, reveals that basic knowledge has resulted in many remarkable applications of great value. Many scientists think that a basic understanding of science is necessary before an application is developed; therefore, applied science relies on the results generated through basic science. Other scientists think that it is time to move on from basic science and instead to find solutions to actual problems. Both approaches are valid. It is true that there are problems that demand immediate attention; however, few solutions would be found without the help of the wide knowledge foundation generated through basic science.

One example of how basic and applied science can work together to solve practical problems occurred after the discovery of DNA structure led to an understanding of the molecular mechanisms governing DNA replication. Strands of DNA, unique in every human, are found in our cells, where they provide the instructions necessary for life. During DNA replication, DNA makes new copies of itself, shortly before a cell divides. Understanding the mechanisms of DNA replication enabled scientists to develop laboratory techniques that are now used to identify genetic diseases, pinpoint individuals who were at a crime scene, and determine paternity. Without basic science, it is unlikely that applied science would exist.

Another example of the link between basic and applied research is the Human Genome Project, a study in which each human chromosome was analyzed and mapped to determine the precise sequence of DNA subunits and the exact location of each gene. (The gene is the basic unit of heredity; an individual’s complete collection of genes is their genome.) Other less complex organisms have also been studied as part of this project in order to gain a better understanding of human chromosomes. The Human Genome Project (Figure \(\PageIndex{4}\)) relied on basic research carried out with simple organisms and, later, with the human genome. An important end goal eventually became using the data for applied research, seeking cures and early diagnoses for genetically related diseases.

While research efforts in both basic science and applied science are usually carefully planned, it is important to note that some discoveries are made by serendipity , that is, by means of a fortunate accident or a lucky surprise. Penicillin was discovered when biologist Alexander Fleming accidentally left a petri dish of Staphylococcus bacteria open. An unwanted mold grew on the dish, killing the bacteria. The mold turned out to be Penicillium , and a new antibiotic was discovered. Even in the highly organized world of science, luck—when combined with an observant, curious mind—can lead to unexpected breakthroughs.

Reporting Scientific Work

Whether scientific research is basic science or applied science, scientists must share their findings in order for other researchers to expand and build upon their discoveries. Collaboration with other scientists—when planning, conducting, and analyzing results—are all important for scientific research. For this reason, important aspects of a scientist’s work are communicating with peers and disseminating results to peers. Scientists can share results by presenting them at a scientific meeting or conference (Figure \(\PageIndex{5}\)), but this approach can reach only the select few who are present. Instead, most scientists present their results in peer-reviewed manuscripts that are published in scientific journals. Peer-reviewed manuscripts are scientific papers that are reviewed by a scientist’s colleagues, or peers. These colleagues are qualified individuals, often experts in the same research area, who judge whether or not the scientist’s work is suitable for publication. The process of peer review helps to ensure that the research described in a scientific paper or grant proposal is original, significant, logical, and thorough. Grant proposals, which are requests for research funding, are also subject to peer review. Scientists publish their work so other scientists can reproduce their experiments under similar or different conditions to expand on the findings. The experimental results must be consistent with the findings of other scientists.

A scientific paper is very different from creative writing. Although creativity is required to design experiments, there are fixed guidelines when it comes to presenting scientific results. First, scientific writing must be brief, concise, and accurate. A scientific paper needs to be succinct but detailed enough to allow peers to reproduce the experiments.

The scientific paper consists of several specific sections—introduction, materials and methods, results, and discussion. This structure is sometimes called the “IMRaD” format, an acronym for Introduction, Method, Results, and Discussion. There are usually acknowledgment and reference sections as well as an abstract (a concise summary) at the beginning of the paper. There might be additional sections depending on the type of paper and the journal where it will be published; for example, some review papers require an outline.

The introduction starts with brief, but broad, background information about what is known in the field. A good introduction also gives the rationale of the work; it justifies the work carried out and also briefly mentions the end of the paper, where the hypothesis or research question driving the research will be presented. The introduction refers to the published scientific work of others and therefore requires citations following the style of the journal. Using the work or ideas of others without proper citation is considered plagiarism .

The materials and methods section includes a complete and accurate description of the substances used, and the method and techniques used by the researchers to gather data. The description should be thorough enough to allow another researcher to repeat the experiment and obtain similar results, but it does not have to be verbose. This section will also include information on how measurements were made and what types of calculations and statistical analyses were used to examine raw data. Although the materials and methods section gives an accurate description of the experiments, it does not discuss them.

Some journals require a results section followed by a discussion section, but it is more common to combine both. If the journal does not allow the combination of both sections, the results section simply narrates the findings without any further interpretation. The results are presented by means of tables or graphs, but no duplicate information should be presented. In the discussion section, the researcher will interpret the results, describe how variables may be related, and attempt to explain the observations. It is indispensable to conduct an extensive literature search to put the results in the context of previously published scientific research. Therefore, proper citations are included in this section as well.

Finally, the conclusion section summarizes the importance of the experimental findings. While the scientific paper almost certainly answered one or more scientific questions that were stated, any good research should lead to more questions. Therefore, a well-done scientific paper leaves doors open for the researcher and others to continue and expand on the findings.

Review articles do not follow the IMRaD format because they do not present original scientific findings (they are not primary literature); instead, they summarize and comment on findings that were published as primary literature and typically include extensive reference sections.

Attributions

Curated and authored by Kammy Algiers using  1.2 (The Process of Science)  from Biology 2e  by OpenStax (licensed CC-BY ).

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.

Mechanics (Essentials) - Class 11th

Course: mechanics (essentials) - class 11th   >   unit 2.

• Introduction to physics
• What is physics?

The scientific method

• Models and Approximations in Physics

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..

• Observation: the toaster won't toast.

2. Ask a question.

• Question: Why won't my toaster toast?

3. Propose a hypothesis.

• Hypothesis: Maybe the outlet is broken.

4. Make predictions.

• Prediction: If I plug the toaster into a different outlet, then it will toast the bread.

5. Test the predictions.

• Test of prediction: Plug the toaster into a different outlet and try again.
• 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..

• Iteration time!
• 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?

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.

• Bipolar Disorder
• Therapy Center
• When To See a Therapist
• Types of Therapy
• Best Online Therapy
• Best Couples Therapy
• Best Family Therapy
• Managing Stress
• Sleep and Dreaming
• Understanding Emotions
• Self-Improvement
• Healthy Relationships
• Student Resources
• Personality Types
• Guided Meditations
• Verywell Mind Insights
• 2023 Verywell Mind 25
• Mental Health in the Classroom
• Editorial Process
• Meet Our Review Board
• Crisis Support

Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

Follow Now : Apple Podcasts / Spotify / Google Podcasts

You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.2: The Scientific Approach to Knowledge

• Last updated
• Save as PDF
• Page ID 37664

Learning Objectives

• To identify the components of the scientific method

Scientists search for answers to questions and solutions to problems by using a procedure called the scientific method . This procedure consists of making observations, formulating hypotheses, and designing experiments, which in turn lead to additional observations, hypotheses, and experiments in repeated cycles (Figure \(\PageIndex{1}\)).

Observations can be qualitative or quantitative. Qualitative observations describe properties or occurrences in ways that do not rely on numbers. Examples of qualitative observations include the following: the outside air temperature is cooler during the winter season, table salt is a crystalline solid, sulfur crystals are yellow, and dissolving a penny in dilute nitric acid forms a blue solution and a brown gas. Quantitative observations are measurements, which by definition consist of both a number and a unit. Examples of quantitative observations include the following: the melting point of crystalline sulfur is 115.21 °C, and 35.9 grams of table salt—whose chemical name is sodium chloride—dissolve in 100 grams of water at 20 °C. An example of a quantitative observation was the initial observation leading to the modern theory of the dinosaurs’ extinction: iridium concentrations in sediments dating to 66 million years ago were found to be 20–160 times higher than normal. The development of this theory is a good exemplar of the scientific method in action (see Figure \(\PageIndex{2}\) below).

After deciding to learn more about an observation or a set of observations, scientists generally begin an investigation by forming a hypothesis , a tentative explanation for the observation(s). The hypothesis may not be correct, but it puts the scientist’s understanding of the system being studied into a form that can be tested. For example, the observation that we experience alternating periods of light and darkness corresponding to observed movements of the sun, moon, clouds, and shadows is consistent with either of two hypotheses:

• Earth rotates on its axis every 24 hours, alternately exposing one side to the sun, or
• The sun revolves around Earth every 24 hours.

Suitable experiments can be designed to choose between these two alternatives. For the disappearance of the dinosaurs, the hypothesis was that the impact of a large extraterrestrial object caused their extinction. Unfortunately (or perhaps fortunately), this hypothesis does not lend itself to direct testing by any obvious experiment, but scientists collected additional data that either support or refute it.

After a hypothesis has been formed, scientists conduct experiments to test its validity. Experiments are systematic observations or measurements, preferably made under controlled conditions—that is, under conditions in which a single variable changes. For example, in the dinosaur extinction scenario, iridium concentrations were measured worldwide and compared. A properly designed and executed experiment enables a scientist to determine whether the original hypothesis is valid. Experiments often demonstrate that the hypothesis is incorrect or that it must be modified. More experimental data are then collected and analyzed, at which point a scientist may begin to think that the results are sufficiently reproducible (i.e., dependable) to merit being summarized in a law , a verbal or mathematical description of a phenomenon that allows for general predictions. A law simply says what happens; it does not address the question of why.

One example of a law, the Law of Definite Proportions , which was discovered by the French scientist Joseph Proust (1754–1826), states that a chemical substance always contains the same proportions of elements by mass. Thus sodium chloride (table salt) always contains the same proportion by mass of sodium to chlorine, in this case 39.34% sodium and 60.66% chlorine by mass, and sucrose (table sugar) is always 42.11% carbon, 6.48% hydrogen, and 51.41% oxygen by mass. Some solid compounds do not strictly obey the law of definite proportions. The law of definite proportions should seem obvious—we would expect the composition of sodium chloride to be consistent—but the head of the US Patent Office did not accept it as a fact until the early 20th century.

Whereas a law states only what happens, a theory attempts to explain why nature behaves as it does. Laws are unlikely to change greatly over time unless a major experimental error is discovered. In contrast, a theory, by definition, is incomplete and imperfect, evolving with time to explain new facts as they are discovered. The theory developed to explain the extinction of the dinosaurs, for example, is that Earth occasionally encounters small- to medium-sized asteroids, and these encounters may have unfortunate implications for the continued existence of most species. This theory is by no means proven, but it is consistent with the bulk of evidence amassed to date. Figure \(\PageIndex{2}\) summarizes the application of the scientific method in this case.

Example \(\PageIndex{1}\)

Classify each statement as a law, a theory, an experiment, a hypothesis, a qualitative observation, or a quantitative observation.

• Ice always floats on liquid water.
• Birds evolved from dinosaurs.
• Hot air is less dense than cold air, probably because the components of hot air are moving more rapidly.
• When 10 g of ice were added to 100 mL of water at 25 °C, the temperature of the water decreased to 15.5 °C after the ice melted.
• The ingredients of Ivory soap were analyzed to see whether it really is 99.44% pure, as advertised.

Given : components of the scientific method

Asked for : statement classification

Strategy: Refer to the definitions in this section to determine which category best describes each statement.

• This is a general statement of a relationship between the properties of liquid and solid water, so it is a law.
• This is a possible explanation for the origin of birds, so it is a hypothesis.
• This is a statement that tries to explain the relationship between the temperature and the density of air based on fundamental principles, so it is a theory.
• The temperature is measured before and after a change is made in a system, so these are quantitative observations.
• This is an analysis designed to test a hypothesis (in this case, the manufacturer’s claim of purity), so it is an experiment.

Exercise \(\PageIndex{1}\)

• Measured amounts of acid were added to a Rolaids tablet to see whether it really “consumes 47 times its weight in excess stomach acid.”
• Heat always flows from hot objects to cooler ones, not in the opposite direction.
• The universe was formed by a massive explosion that propelled matter into a vacuum.
• Michael Jordan is the greatest pure shooter ever to play professional basketball.
• Limestone is relatively insoluble in water but dissolves readily in dilute acid with the evolution of a gas.
• Gas mixtures that contain more than 4% hydrogen in air are potentially explosive.

qualitative observation

quantitative observation

Because scientists can enter the cycle shown in Figure \(\PageIndex{1}\) at any point, the actual application of the scientific method to different topics can take many different forms. For example, a scientist may start with a hypothesis formed by reading about work done by others in the field, rather than by making direct observations.

It is important to remember that scientists have a tendency to formulate hypotheses in familiar terms simply because it is difficult to propose something that has never been encountered or imagined before. As a result, scientists sometimes discount or overlook unexpected findings that disagree with the basic assumptions behind the hypothesis or theory being tested. Fortunately, truly important findings are immediately subject to independent verification by scientists in other laboratories, so science is a self-correcting discipline. When the Alvarezes originally suggested that an extraterrestrial impact caused the extinction of the dinosaurs, the response was almost universal skepticism and scorn. In only 20 years, however, the persuasive nature of the evidence overcame the skepticism of many scientists, and their initial hypothesis has now evolved into a theory that has revolutionized paleontology and geology.

Chemists expand their knowledge by making observations, carrying out experiments, and testing hypotheses to develop laws to summarize their results and theories to explain them. In doing so, they are using the scientific method.

Fundamental Definitions in Chemistry: https://youtu.be/SBwjbkFNkdw

The Robots That Will Change the World Are Already Among Us

Climate change more like ecosocial crisis, openmind books, scientific anniversaries, kary mullis, the genius of a scientist, the eccentricity of a celebrity, featured author, latest book, solving problems visually.

What’s the best way to approach (and successfully solve) a mathematical problem statement? Perhaps by drawing a picture? That, at least, is the conclusion of a striking new study by researchers at the Universities of Geneva and Burgundy.

It’s not a trivial assumption. It is thought that when we face a mathematical problem that contains both mathematical information (numbers and arithmetic operations) and non-mathematical information (the context of the problem and the characteristics of the entities involved), our brains process this combination of verbal and numerical information and convert it into a mental representation in order to identify the best strategy for solving it. On the other hand, more and more studies suggest that the schematic drawings that are usually made to solve this type of problem are a reflection of these mental representations.

Game 1: Not a game, an experiment

In the study, participants were asked to solve 12 simple arithmetic problems in as few steps as possible and to draw a picture that would help them understand and solve the problem.

Here are two of these problems, and we invite you to solve them in the same way: in as few steps as possible, and with a drawing to help you understand the problem.

Problem 1 : Paul has five red marbles and also has some blue marbles. In total he has eleven marbles. Julie’s marbles are green and blue. Julie has as many blue marbles as Paul and also has two fewer green marbles than Paul has red marbles. How many marbles does Julie have?

Problem 2 : Lisa takes the train during the day, travels for 5 hours and arrives at her destination at 11am. Fred got on the train at the same time as Lisa and his trip took 2 hours less. What time did Fred arrive at his destination?

Independently of the above, many studies postulate that relying on drawings, diagrams or other types of graphical representations when processing information has numerous benefits: it improves our ability to learn and remember, it helps us to understand complex concepts, it reinforces critical and scientific thinking, and it fosters a transversal and interdisciplinary interpretation. And from a mathematical point of view, using these representations makes it easier to establish the relationships between different data, to visualise the information implicit in the statement and to identify the most direct and simplest solution strategy.

Use the drawings to answer these complex and hieroglyphic questions.

A recent study goes a step further by suggesting that the verbal information in the problem statement influences the type of diagram shown and also the strategy chosen to solve the problem. More specifically, the study has found that the type of diagram preferentially chosen depends on whether the statement is cardinal or ordinal in nature.

Thus, when the context alludes to the cardinal properties of the quantities involved—the number of elements in a set—a drawing based on groupings of entities (crosses, circles, etc.) that sometimes overlap (or intersect) is usually chosen. This in turn leads to a three-step arithmetic strategy. On the other hand, when the statement of the problem focuses on the ordinal properties of numbers—the position they occupy in a set—we usually opt for drawings based on axes, graduations or intervals, which lead to a more direct and simpler one-step solution strategy.

And this is observed even when the problems are analogous from a mathematical point of view: they have the same structure, the same numerical values and can be solved with the same strategy (as in the case of the two problems in Game 1).

But perhaps the most interesting reflection is that, knowing this, it is possible to guide and train the student to apply this second type of diagram, thereby facilitating the identification of the best way to solve it.

Game 3:  A high-flying challenge

Sara wants to travel from Madrid to Tokyo. To do so, she flies first to New York, from where she takes a plane to London and from there to Tokyo.

Paul also wants to go from Madrid to Tokyo, but in his case he flies directly from Madrid to London and then takes a flight to Tokyo.

If Sara flies for a total of 27hrs 15min and Paul for 14hrs 30min, and given that the flight from New York to London takes 4hrs 45min longer than the flight from Madrid to London, and the flight from London to Tokyo takes 12 hours, how long is the flight from Madrid to New York?

And if both Sara and Paul lose only one hour at each stopover, what will the local time be when they each arrive in Tokyo if they both depart Madrid at 2pm?

            M                                 NY    M-L + 4:45   L                    12:00 h                   T

                                                   M           L                                       12:00 h                    T 

14 hrs 30 min

The Madrid-London flight takes 2hrs 30min. New York to London is 2hrs 30min + 4hrs 45min = 7hrs 15min. And the Madrid to New York flight is 27hrs 15min – 12hrs – 7hrs 15min = 8 hours.

With this, and bearing in mind that each stopover only takes one hour:

If Sara leaves at 2pm from Madrid then: 2pm + 8hrs – 6hrs (time difference) + 1hr (at NY airport) + 7hrs 15min + 5hrs (time difference) + 1hr + 12hrs + 8hrs (time difference) = 2:15am on day 3.

In Paul’s case: 2pm + 2hrs 30min – 1hr (time difference) + 1hr + 12hrs + 8hrs (time difference) = 12:30pm on day 2.

Miguel Barral

Related publications.

• What Purpose Do the Great Mathematical Problems Serve?
• David Hilbert: The Architect of Modern Mathematics
• Magic Squares: When Art is Squared With Mathematics

More about Science

Environment, leading figures, mathematics, scientific insights, more publications about ventana al conocimiento (knowledge window), comments on this publication.

Morbi facilisis elit non mi lacinia lacinia. Nunc eleifend aliquet ipsum, nec blandit augue tincidunt nec. Donec scelerisque feugiat lectus nec congue. Quisque tristique tortor vitae turpis euismod, vitae aliquam dolor pretium. Donec luctus posuere ex sit amet scelerisque. Etiam sed neque magna. Mauris non scelerisque lectus. Ut rutrum ex porta, tristique mi vitae, volutpat urna.

Sed in semper tellus, eu efficitur ante. Quisque felis orci, fermentum quis arcu nec, elementum malesuada magna. Nulla vitae finibus ipsum. Aenean vel sapien a magna faucibus tristique ac et ligula. Sed auctor orci metus, vitae egestas libero lacinia quis. Nulla lacus sapien, efficitur mollis nisi tempor, gravida tincidunt sapien. In massa dui, varius vitae iaculis a, dignissim non felis. Ut sagittis pulvinar nisi, at tincidunt metus venenatis a. Ut aliquam scelerisque interdum. Mauris iaculis purus in nulla consequat, sed fermentum sapien condimentum. Aliquam rutrum erat lectus, nec placerat nisl mollis id. Lorem ipsum dolor sit amet, consectetur adipiscing elit.

Nam nisl nisi, efficitur et sem in, molestie vulputate libero. Quisque quis mattis lorem. Nunc quis convallis diam, id tincidunt risus. Donec nisl odio, convallis vel porttitor sit amet, lobortis a ante. Cras dapibus porta nulla, at laoreet quam euismod vitae. Fusce sollicitudin massa magna, eu dignissim magna cursus id. Quisque vel nisl tempus, lobortis nisl a, ornare lacus. Donec ac interdum massa. Curabitur id diam luctus, mollis augue vel, interdum risus. Nam vitae tortor erat. Proin quis tincidunt lorem.

Do you want to stay up to date with our new publications?

Receive the OpenMind newsletter with all the latest contents published on our website

OpenMind Books

• The Search for Alternatives to Fossil Fuels
• View all books

About OpenMind

Connect with us.

• Keep up to date with our newsletter

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
• My Account Login
• Explore content
• About the journal
• Publish with us
• Sign up for alerts
• Open access
• Published: 10 April 2024

A hybrid particle swarm optimization algorithm for solving engineering problem

• Jinwei Qiao 1 , 2 ,
• Guangyuan Wang 1 , 2 ,
• Zhi Yang 1 , 2 ,
• Xiaochuan Luo 3 ,
• Jun Chen 1 , 2 ,
• Kan Li 4 &
• Pengbo Liu 1 , 2  

Scientific Reports volume  14 , Article number:  8357 ( 2024 ) Cite this article

391 Accesses

Metrics details

• Computational science
• Mechanical engineering

To overcome the disadvantages of premature convergence and easy trapping into local optimum solutions, this paper proposes an improved particle swarm optimization algorithm (named NDWPSO algorithm) based on multiple hybrid strategies. Firstly, the elite opposition-based learning method is utilized to initialize the particle position matrix. Secondly, the dynamic inertial weight parameters are given to improve the global search speed in the early iterative phase. Thirdly, a new local optimal jump-out strategy is proposed to overcome the "premature" problem. Finally, the algorithm applies the spiral shrinkage search strategy from the whale optimization algorithm (WOA) and the Differential Evolution (DE) mutation strategy in the later iteration to accelerate the convergence speed. The NDWPSO is further compared with other 8 well-known nature-inspired algorithms (3 PSO variants and 5 other intelligent algorithms) on 23 benchmark test functions and three practical engineering problems. Simulation results prove that the NDWPSO algorithm obtains better results for all 49 sets of data than the other 3 PSO variants. Compared with 5 other intelligent algorithms, the NDWPSO obtains 69.2%, 84.6%, and 84.6% of the best results for the benchmark function ( \({f}_{1}-{f}_{13}\) ) with 3 kinds of dimensional spaces (Dim = 30,50,100) and 80% of the best optimal solutions for 10 fixed-multimodal benchmark functions. Also, the best design solutions are obtained by NDWPSO for all 3 classical practical engineering problems.

Similar content being viewed by others

Neural operators for accelerating scientific simulations and design

Physics-informed machine learning

Principal component analysis

Introduction.

In the ever-changing society, new optimization problems arise every moment, and they are distributed in various fields, such as automation control 1 , statistical physics 2 , security prevention and temperature prediction 3 , artificial intelligence 4 , and telecommunication technology 5 . Faced with a constant stream of practical engineering optimization problems, traditional solution methods gradually lose their efficiency and convenience, making it more and more expensive to solve the problems. Therefore, researchers have developed many metaheuristic algorithms and successfully applied them to the solution of optimization problems. Among them, Particle swarm optimization (PSO) algorithm 6 is one of the most widely used swarm intelligence algorithms.

However, the basic PSO has a simple operating principle and solves problems with high efficiency and good computational performance, but it suffers from the disadvantages of easily trapping in local optima and premature convergence. To improve the overall performance of the particle swarm algorithm, an improved particle swarm optimization algorithm is proposed by the multiple hybrid strategy in this paper. The improved PSO incorporates the search ideas of other intelligent algorithms (DE, WOA), so the improved algorithm proposed in this paper is named NDWPSO. The main improvement schemes are divided into the following 4 points: Firstly, a strategy of elite opposition-based learning is introduced into the particle population position initialization. A high-quality initialization matrix of population position can improve the convergence speed of the algorithm. Secondly, a dynamic weight methodology is adopted for the acceleration coefficients by combining the iterative map and linearly transformed method. This method utilizes the chaotic nature of the mapping function, the fast convergence capability of the dynamic weighting scheme, and the time-varying property of the acceleration coefficients. Thus, the global search and local search of the algorithm are balanced and the global search speed of the population is improved. Thirdly, a determination mechanism is set up to detect whether the algorithm falls into a local optimum. When the algorithm is “premature”, the population resets 40% of the position information to overcome the local optimum. Finally, the spiral shrinking mechanism combined with the DE/best/2 position mutation is used in the later iteration, which further improves the solution accuracy.

The structure of the paper is given as follows: Sect. “ Particle swarm optimization (PSO) ” describes the principle of the particle swarm algorithm. Section “ Improved particle swarm optimization algorithm ” shows the detailed improvement strategy and a comparison experiment of inertia weight is set up for the proposed NDWPSO. Section “ Experiment and discussion ” includes the experimental and result discussion sections on the performance of the improved algorithm. Section “ Conclusions and future works ” summarizes the main findings of this study.

Literature review

This section reviews some metaheuristic algorithms and other improved PSO algorithms. A simple discussion about recently proposed research studies is given.

Metaheuristic algorithms

A series of metaheuristic algorithms have been proposed in recent years by using various innovative approaches. For instance, Lin et al. 7 proposed a novel artificial bee colony algorithm (ABCLGII) in 2018 and compared ABCLGII with other outstanding ABC variants on 52 frequently used test functions. Abed-alguni et al. 8 proposed an exploratory cuckoo search (ECS) algorithm in 2021 and carried out several experiments to investigate the performance of ECS by 14 benchmark functions. Brajević 9 presented a novel shuffle-based artificial bee colony (SB-ABC) algorithm for solving integer programming and minimax problems in 2021. The experiments are tested on 7 integer programming problems and 10 minimax problems. In 2022, Khan et al. 10 proposed a non-deterministic meta-heuristic algorithm called Non-linear Activated Beetle Antennae Search (NABAS) for a non-convex tax-aware portfolio selection problem. Brajević et al. 11 proposed a hybridization of the sine cosine algorithm (HSCA) in 2022 to solve 15 complex structural and mechanical engineering design optimization problems. Abed-Alguni et al. 12 proposed an improved Salp Swarm Algorithm (ISSA) in 2022 for single-objective continuous optimization problems. A set of 14 standard benchmark functions was used to evaluate the performance of ISSA. In 2023, Nadimi et al. 13 proposed a binary starling murmuration optimization (BSMO) to select the effective features from different important diseases. In the same year, Nadimi et al. 14 systematically reviewed the last 5 years' developments of WOA and made a critical analysis of those WOA variants. In 2024, Fatahi et al. 15 proposed an Improved Binary Quantum-based Avian Navigation Optimizer Algorithm (IBQANA) for the Feature Subset Selection problem in the medical area. Experimental evaluation on 12 medical datasets demonstrates that IBQANA outperforms 7 established algorithms. Abed-alguni et al. 16 proposed an Improved Binary DJaya Algorithm (IBJA) to solve the Feature Selection problem in 2024. The IBJA’s performance was compared against 4 ML classifiers and 10 efficient optimization algorithms.

Improved PSO algorithms

Many researchers have constantly proposed some improved PSO algorithms to solve engineering problems in different fields. For instance, Yeh 17 proposed an improved particle swarm algorithm, which combines a new self-boundary search and a bivariate update mechanism, to solve the reliability redundancy allocation problem (RRAP) problem. Solomon et al. 18 designed a collaborative multi-group particle swarm algorithm with high parallelism that was used to test the adaptability of Graphics Processing Units (GPUs) in distributed computing environments. Mukhopadhyay and Banerjee 19 proposed a chaotic multi-group particle swarm optimization (CMS-PSO) to estimate the unknown parameters of an autonomous chaotic laser system. Duan et al. 20 designed an improved particle swarm algorithm with nonlinear adjustment of inertia weights to improve the coupling accuracy between laser diodes and single-mode fibers. Sun et al. 21 proposed a particle swarm optimization algorithm combined with non-Gaussian stochastic distribution for the optimal design of wind turbine blades. Based on a multiple swarm scheme, Liu et al. 22 proposed an improved particle swarm optimization algorithm to predict the temperatures of steel billets for the reheating furnace. In 2022, Gad 23 analyzed the existing 2140 papers on Swarm Intelligence between 2017 and 2019 and pointed out that the PSO algorithm still needs further research. In general, the improved methods can be classified into four categories:

Adjusting the distribution of algorithm parameters. Feng et al. 24 used a nonlinear adaptive method on inertia weights to balance local and global search and introduced asynchronously varying acceleration coefficients.

Changing the updating formula of the particle swarm position. Both papers 25 and 26 used chaotic mapping functions to update the inertia weight parameters and combined them with a dynamic weighting strategy to update the particle swarm positions. This improved approach enables the particle swarm algorithm to be equipped with fast convergence of performance.

The initialization of the swarm. Alsaidy and Abbood proposed 27 a hybrid task scheduling algorithm that replaced the random initialization of the meta-heuristic algorithm with the heuristic algorithms MCT-PSO and LJFP-PSO.

Combining with other intelligent algorithms: Liu et al. 28 introduced the differential evolution (DE) algorithm into PSO to increase the particle swarm as diversity and reduce the probability of the population falling into local optimum.

Particle swarm optimization (PSO)

The particle swarm optimization algorithm is a population intelligence algorithm for solving continuous and discrete optimization problems. It originated from the social behavior of individuals in bird and fish flocks 6 . The core of the PSO algorithm is that an individual particle identifies potential solutions by flight in a defined constraint space adjusts its exploration direction to approach the global optimal solution based on the shared information among the group, and finally solves the optimization problem. Each particle \(i\) includes two attributes: velocity vector \({V}_{i}=\left[{v}_{i1},{v}_{i2},{v}_{i3},{...,v}_{ij},{...,v}_{iD},\right]\) and position vector \({X}_{i}=[{x}_{i1},{x}_{i2},{x}_{i3},...,{x}_{ij},...,{x}_{iD}]\) . The velocity vector is used to modify the motion path of the swarm; the position vector represents a potential solution for the optimization problem. Here, \(j=\mathrm{1,2},\dots ,D\) , \(D\) represents the dimension of the constraint space. The equations for updating the velocity and position of the particle swarm are shown in Eqs. ( 1 ) and ( 2 ).

Here \({Pbest}_{i}^{k}\) represents the previous optimal position of the particle \(i\) , and \({Gbest}\) is the optimal position discovered by the whole population. \(i=\mathrm{1,2},\dots ,n\) , \(n\) denotes the size of the particle swarm. \({c}_{1}\) and \({c}_{2}\) are the acceleration constants, which are used to adjust the search step of the particle 29 . \({r}_{1}\) and \({r}_{2}\) are two random uniform values distributed in the range \([\mathrm{0,1}]\) , which are used to improve the randomness of the particle search. \(\omega\) inertia weight parameter, which is used to adjust the scale of the search range of the particle swarm 30 . The basic PSO sets the inertia weight parameter as a time-varying parameter to balance global exploration and local seeking. The updated equation of the inertia weight parameter is given as follows:

where \({\omega }_{max}\) and \({\omega }_{min}\) represent the upper and lower limits of the range of inertia weight parameter. \(k\) and \(Mk\) are the current iteration and maximum iteration.

Improved particle swarm optimization algorithm

According to the no free lunch theory 31 , it is known that no algorithm can solve every practical problem with high quality and efficiency for increasingly complex and diverse optimization problems. In this section, several improvement strategies are proposed to improve the search efficiency and overcome this shortcoming of the basic PSO algorithm.

Improvement strategies

The optimization strategies of the improved PSO algorithm are shown as follows:

The inertia weight parameter is updated by an improved chaotic variables method instead of a linear decreasing strategy. Chaotic mapping performs the whole search at a higher speed and is more resistant to falling into local optimal than the probability-dependent random search 32 . However, the population may result in that particles can easily fly out of the global optimum boundary. To ensure that the population can converge to the global optimum, an improved Iterative mapping is adopted and shown as follows:

Here \({\omega }_{k}\) is the inertia weight parameter in the iteration \(k\) , \(b\) is the control parameter in the range \([\mathrm{0,1}]\) .

The acceleration coefficients are updated by the linear transformation. \({c}_{1}\) and \({c}_{2}\) represent the influential coefficients of the particles by their own and population information, respectively. To improve the search performance of the population, \({c}_{1}\) and \({c}_{2}\) are changed from fixed values to time-varying parameter parameters, that are updated by linear transformation with the number of iterations:

where \({c}_{max}\) and \({c}_{min}\) are the maximum and minimum values of acceleration coefficients, respectively.

The initialization scheme is determined by elite opposition-based learning . The high-quality initial population will accelerate the solution speed of the algorithm and improve the accuracy of the optimal solution. Thus, the elite backward learning strategy 33 is introduced to generate the position matrix of the initial population. Suppose the elite individual of the population is \({X}=[{x}_{1},{x}_{2},{x}_{3},...,{x}_{j},...,{x}_{D}]\) , and the elite opposition-based solution of \(X\) is \({X}_{o}=[{x}_{{\text{o}}1},{x}_{{\text{o}}2},{x}_{{\text{o}}3},...,{x}_{oj},...,{x}_{oD}]\) . The formula for the elite opposition-based solution is as follows:

where \({k}_{r}\) is the random value in the range \((\mathrm{0,1})\) . \({ux}_{oij}\) and \({lx}_{oij}\) are dynamic boundaries of the elite opposition-based solution in \(j\) dimensional variables. The advantage of dynamic boundary is to reduce the exploration space of particles, which is beneficial to the convergence of the algorithm. When the elite opposition-based solution is out of bounds, the out-of-bounds processing is performed. The equation is given as follows:

After calculating the fitness function values of the elite solution and the elite opposition-based solution, respectively, \(n\) high quality solutions were selected to form a new initial population position matrix.

The position updating Eq. ( 2 ) is modified based on the strategy of dynamic weight. To improve the speed of the global search of the population, the strategy of dynamic weight from the artificial bee colony algorithm 34 is introduced to enhance the computational performance. The new position updating equation is shown as follows:

Here \(\rho\) is the random value in the range \((\mathrm{0,1})\) . \(\psi\) represents the acceleration coefficient and \({\omega }{\prime}\) is the dynamic weight coefficient. The updated equations of the above parameters are as follows:

where \(f(i)\) denotes the fitness function value of individual particle \(i\) and u is the average of the population fitness function values in the current iteration. The Eqs. ( 11 , 12 ) are introduced into the position updating equation. And they can attract the particle towards positions of the best-so-far solution in the search space.

New local optimal jump-out strategy is added for escaping from the local optimal. When the value of the fitness function for the population optimal particles does not change in M iterations, the algorithm determines that the population falls into a local optimal. The scheme in which the population jumps out of the local optimum is to reset the position information of the 40% of individuals within the population, in other words, to randomly generate the position vector in the search space. M is set to 5% of the maximum number of iterations.

New spiral update search strategy is added after the local optimal jump-out strategy. Since the whale optimization algorithm (WOA) was good at exploring the local search space 35 , the spiral update search strategy in the WOA 36 is introduced to update the position of the particles after the swarm jumps out of local optimal. The equation for the spiral update is as follows:

Here \(D=\left|{x}_{i}\left(k\right)-Gbest\right|\) denotes the distance between the particle itself and the global optimal solution so far. \(B\) is the constant that defines the shape of the logarithmic spiral. \(l\) is the random value in \([-\mathrm{1,1}]\) . \(l\) represents the distance between the newly generated particle and the global optimal position, \(l=-1\) means the closest distance, while \(l=1\) means the farthest distance, and the meaning of this parameter can be directly observed by Fig.  1 .

Spiral updating position.

The DE/best/2 mutation strategy is introduced to form the mutant particle. 4 individuals in the population are randomly selected that differ from the current particle, then the vector difference between them is rescaled, and the difference vector is combined with the global optimal position to form the mutant particle. The equation for mutation of particle position is shown as follows:

where \({x}^{*}\) is the mutated particle, \(F\) is the scale factor of mutation, \({r}_{1}\) , \({r}_{2}\) , \({r}_{3}\) , \({r}_{4}\) are random integer values in \((0,n]\) and not equal to \(i\) , respectively. Specific particles are selected for mutation with the screening conditions as follows:

where \(Cr\) represents the probability of mutation, \(rand\left(\mathrm{0,1}\right)\) is a random number in \(\left(\mathrm{0,1}\right)\) , and \({i}_{rand}\) is a random integer value in \((0,n]\) .

The improved PSO incorporates the search ideas of other intelligent algorithms (DE, WOA), so the improved algorithm proposed in this paper is named NDWPSO. The pseudo-code for the NDWPSO algorithm is given as follows:

The main procedure of NDWPSO.

Comparing the distribution of inertia weight parameters

There are several improved PSO algorithms (such as CDWPSO 25 , and SDWPSO 26 ) that adopt the dynamic weighted particle position update strategy as their improvement strategy. The updated equations of the CDWPSO and the SDWPSO algorithm for the inertia weight parameters are given as follows:

where \({\text{A}}\) is a value in \((\mathrm{0,1}]\) . \({r}_{max}\) and \({r}_{min}\) are the upper and lower limits of the fluctuation range of the inertia weight parameters, \(k\) is the current number of algorithm iterations, and \(Mk\) denotes the maximum number of iterations.

Considering that the update method of inertia weight parameters by our proposed NDWPSO is comparable to the CDWPSO, and SDWPSO, a comparison experiment for the distribution of inertia weight parameters is set up in this section. The maximum number of iterations in the experiment is \(Mk=500\) . The distributions of CDWPSO, SDWPSO, and NDWPSO inertia weights are shown sequentially in Fig.  2 .

The inertial weight distribution of CDWPSO, SDWPSO, and NDWPSO.

In Fig.  2 , the inertia weight value of CDWPSO is a random value in (0,1]. It may make individual particles fly out of the range in the late iteration of the algorithm. Similarly, the inertia weight value of SDWPSO is a value that tends to zero infinitely, so that the swarm no longer can fly in the search space, making the algorithm extremely easy to fall into the local optimal value. On the other hand, the distribution of the inertia weights of the NDWPSO forms a gentle slope by two curves. Thus, the swarm can faster lock the global optimum range in the early iterations and locate the global optimal more precisely in the late iterations. The reason is that the inertia weight values between two adjacent iterations are inversely proportional to each other. Besides, the time-varying part of the inertial weight within NDWPSO is designed to reduce the chaos characteristic of the parameters. The inertia weight value of NDWPSO avoids the disadvantages of the above two schemes, so its design is more reasonable.

Experiment and discussion

In this section, three experiments are set up to evaluate the performance of NDWPSO: (1) the experiment of 23 classical functions 37 between NDWPSO and three particle swarm algorithms (PSO 6 , CDWPSO 25 , SDWPSO 26 ); (2) the experiment of benchmark test functions between NDWPSO and other intelligent algorithms (Whale Optimization Algorithm (WOA) 36 , Harris Hawk Algorithm (HHO) 38 , Gray Wolf Optimization Algorithm (GWO) 39 , Archimedes Algorithm (AOA) 40 , Equilibrium Optimizer (EO) 41 and Differential Evolution (DE) 42 ); (3) the experiment for solving three real engineering problems (welded beam design 43 , pressure vessel design 44 , and three-bar truss design 38 ). All experiments are run on a computer with Intel i5-11400F GPU, 2.60 GHz, 16 GB RAM, and the code is written with MATLAB R2017b.

The benchmark test functions are 23 classical functions, which consist of indefinite unimodal (F1–F7), indefinite dimensional multimodal functions (F8–F13), and fixed-dimensional multimodal functions (F14–F23). The unimodal benchmark function is used to evaluate the global search performance of different algorithms, while the multimodal benchmark function reflects the ability of the algorithm to escape from the local optimal. The mathematical equations of the benchmark functions are shown and found as Supplementary Tables S1 – S3 online.

Experiments on benchmark functions between NDWPSO, and other PSO variants

The purpose of the experiment is to show the performance advantages of the NDWPSO algorithm. Here, the dimensions and corresponding population sizes of 13 benchmark functions (7 unimodal and 6 multimodal) are set to (30, 40), (50, 70), and (100, 130). The population size of 10 fixed multimodal functions is set to 40. Each algorithm is repeated 30 times independently, and the maximum number of iterations is 200. The performance of the algorithm is measured by the mean and the standard deviation (SD) of the results for different benchmark functions. The parameters of the NDWPSO are set as: \({[{\omega }_{min},\omega }_{max}]=[\mathrm{0.4,0.9}]\) , \(\left[{c}_{max},{c}_{min}\right]=\left[\mathrm{2.5,1.5}\right],{V}_{max}=0.1,b={e}^{-50}, M=0.05\times Mk, B=1,F=0.7, Cr=0.9.\) And, \(A={\omega }_{max}\) for CDWPSO; \({[r}_{max},{r}_{min}]=[\mathrm{4,0}]\) for SDWPSO.

Besides, the experimental data are retained to two decimal places, but some experimental data will increase the number of retained data to pursue more accuracy in comparison. The best results in each group of experiments will be displayed in bold font. The experimental data is set to 0 if the value is below 10 –323 . The experimental parameter settings in this paper are different from the references (PSO 6 , CDWPSO 25 , SDWPSO 26 , so the final experimental data differ from the ones within the reference.

As shown in Tables 1 and 2 , the NDWPSO algorithm obtains better results for all 49 sets of data than other PSO variants, which include not only 13 indefinite-dimensional benchmark functions and 10 fixed-multimodal benchmark functions. Remarkably, the SDWPSO algorithm obtains the same accuracy of calculation as NDWPSO for both unimodal functions f 1 –f 4 and multimodal functions f 9 –f 11 . The solution accuracy of NDWPSO is higher than that of other PSO variants for fixed-multimodal benchmark functions f 14 -f 23 . The conclusion can be drawn that the NDWPSO has excellent global search capability, local search capability, and the capability for escaping the local optimal.

In addition, the convergence curves of the 23 benchmark functions are shown in Figs. 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 and 19 . The NDWPSO algorithm has a faster convergence speed in the early stage of the search for processing functions f1-f6, f8-f14, f16, f17, and finds the global optimal solution with a smaller number of iterations. In the remaining benchmark function experiments, the NDWPSO algorithm shows no outstanding performance for convergence speed in the early iterations. There are two reasons of no outstanding performance in the early iterations. On one hand, the fixed-multimodal benchmark function has many disturbances and local optimal solutions in the whole search space. on the other hand, the initialization scheme based on elite opposition-based learning is still stochastic, which leads to the initial position far from the global optimal solution. The inertia weight based on chaotic mapping and the strategy of spiral updating can significantly improve the convergence speed and computational accuracy of the algorithm in the late search stage. Finally, the NDWPSO algorithm can find better solutions than other algorithms in the middle and late stages of the search.

Evolution curve of NDWPSO and other PSO algorithms for f1 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f2 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f3 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f4 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f5 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f6 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f7 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f8 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f9 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f10 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f11(Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f12 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f13 (Dim = 30,50,100).

Evolution curve of NDWPSO and other PSO algorithms for f14, f15, f16.

Evolution curve of NDWPSO and other PSO algorithms for f17, f18, f19.

Evolution curve of NDWPSO and other PSO algorithms for f20, f21, f22.

Evolution curve of NDWPSO and other PSO algorithms for f23.

To evaluate the performance of different PSO algorithms, a statistical test is conducted. Due to the stochastic nature of the meta-heuristics, it is not enough to compare algorithms based on only the mean and standard deviation values. The optimization results cannot be assumed to obey the normal distribution; thus, it is necessary to judge whether the results of the algorithms differ from each other in a statistically significant way. Here, the Wilcoxon non-parametric statistical test 45 is used to obtain a parameter called p -value to verify whether two sets of solutions are different to a statistically significant extent or not. Generally, it is considered that p  ≤ 0.5 can be considered as a statistically significant superiority of the results. The p -values calculated in Wilcoxon’s rank-sum test comparing NDWPSO and other PSO algorithms are listed in Table  3 for all benchmark functions. The p -values in Table  3 additionally present the superiority of the NDWPSO because all of the p -values are much smaller than 0.5.

In general, the NDWPSO has the fastest convergence rate when finding the global optimum from Figs. 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 and 19 , and thus we can conclude that the NDWPSO is superior to the other PSO variants during the process of optimization.

Comparison experiments between NDWPSO and other intelligent algorithms

Experiments are conducted to compare NDWPSO with several other intelligent algorithms (WOA, HHO, GWO, AOA, EO and DE). The experimental object is 23 benchmark functions, and the experimental parameters of the NDWPSO algorithm are set the same as in Experiment 4.1. The maximum number of iterations of the experiment is increased to 2000 to fully demonstrate the performance of each algorithm. Each algorithm is repeated 30 times individually. The parameters of the relevant intelligent algorithms in the experiments are set as shown in Table 4 . To ensure the fairness of the algorithm comparison, all parameters are concerning the original parameters in the relevant algorithm literature. The experimental results are shown in Tables 5 , 6 , 7 and 8 and Figs. 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 and 36 .

Evolution curve of NDWPSO and other algorithms for f1 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f2 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f3(Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f4 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f5 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f6 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f7 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f8 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f9(Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f10 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f11 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f12 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f13 (Dim = 30,50,100).

Evolution curve of NDWPSO and other algorithms for f14, f15, f16.

Evolution curve of NDWPSO and other algorithms for f17, f18, f19.

Evolution curve of NDWPSO and other algorithms for f20, f21, f22.

Evolution curve of NDWPSO and other algorithms for f23.

The experimental data of NDWPSO and other intelligent algorithms for handling 30, 50, and 100-dimensional benchmark functions ( \({f}_{1}-{f}_{13}\) ) are recorded in Tables 8 , 9 and 10 , respectively. The comparison data of fixed-multimodal benchmark tests ( \({f}_{14}-{f}_{23}\) ) are recorded in Table 11 . According to the data in Tables 5 , 6 and 7 , the NDWPSO algorithm obtains 69.2%, 84.6%, and 84.6% of the best results for the benchmark function ( \({f}_{1}-{f}_{13}\) ) in the search space of three dimensions (Dim = 30, 50, 100), respectively. In Table 8 , the NDWPSO algorithm obtains 80% of the optimal solutions in 10 fixed-multimodal benchmark functions.

The convergence curves of each algorithm are shown in Figs. 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 and 36 . The NDWPSO algorithm demonstrates two convergence behaviors when calculating the benchmark functions in 30, 50, and 100-dimensional search spaces. The first behavior is the fast convergence of NDWPSO with a small number of iterations at the beginning of the search. The reason is that the Iterative-mapping strategy and the position update scheme of dynamic weighting are used in the NDWPSO algorithm. This scheme can quickly target the region in the search space where the global optimum is located, and then precisely lock the optimal solution. When NDWPSO processes the functions \({f}_{1}-{f}_{4}\) , and \({f}_{9}-{f}_{11}\) , the behavior can be reflected in the convergence trend of their corresponding curves. The second behavior is that NDWPSO gradually improves the convergence accuracy and rapidly approaches the global optimal in the middle and late stages of the iteration. The NDWPSO algorithm fails to converge quickly in the early iterations, which is possible to prevent the swarm from falling into a local optimal. The behavior can be demonstrated by the convergence trend of the curves when NDWPSO handles the functions \({f}_{6}\) , \({f}_{12}\) , and \({f}_{13}\) , and it also shows that the NDWPSO algorithm has an excellent ability of local search.

Combining the experimental data with the convergence curves, it is concluded that the NDWPSO algorithm has a faster convergence speed, so the effectiveness and global convergence of the NDWPSO algorithm are more outstanding than other intelligent algorithms.

Experiments on classical engineering problems

Three constrained classical engineering design problems (welded beam design, pressure vessel design 43 , and three-bar truss design 38 ) are used to evaluate the NDWPSO algorithm. The experiments are the NDWPSO algorithm and 5 other intelligent algorithms (WOA 36 , HHO, GWO, AOA, EO 41 ). Each algorithm is provided with the maximum number of iterations and population size ( \({\text{Mk}}=500,\mathrm{ n}=40\) ), and then repeats 30 times, independently. The parameters of the algorithms are set the same as in Table 4 . The experimental results of three engineering design problems are recorded in Tables 9 , 10 and 11 in turn. The result data is the average value of the solved data.

Welded beam design

The target of the welded beam design problem is to find the optimal manufacturing cost for the welded beam with the constraints, as shown in Fig.  37 . The constraints are the thickness of the weld seam ( \({\text{h}}\) ), the length of the clamped bar ( \({\text{l}}\) ), the height of the bar ( \({\text{t}}\) ) and the thickness of the bar ( \({\text{b}}\) ). The mathematical formulation of the optimization problem is given as follows:

Welded beam design.

In Table 9 , the NDWPSO, GWO, and EO algorithms obtain the best optimal cost. Besides, the standard deviation (SD) of t NDWPSO is the lowest, which means it has very good results in solving the welded beam design problem.

Pressure vessel design

Kannan and Kramer 43 proposed the pressure vessel design problem as shown in Fig.  38 to minimize the total cost, including the cost of material, forming, and welding. There are four design optimized objects: the thickness of the shell \({T}_{s}\) ; the thickness of the head \({T}_{h}\) ; the inner radius \({\text{R}}\) ; the length of the cylindrical section without considering the head \({\text{L}}\) . The problem includes the objective function and constraints as follows:

Pressure vessel design.

The results in Table 10 show that the NDWPSO algorithm obtains the lowest optimal cost with the same constraints and has the lowest standard deviation compared with other algorithms, which again proves the good performance of NDWPSO in terms of solution accuracy.

Three-bar truss design

This structural design problem 44 is one of the most widely-used case studies as shown in Fig.  39 . There are two main design parameters: the area of the bar1 and 3 ( \({A}_{1}={A}_{3}\) ) and area of bar 2 ( \({A}_{2}\) ). The objective is to minimize the weight of the truss. This problem is subject to several constraints as well: stress, deflection, and buckling constraints. The problem is formulated as follows:

Three-bar truss design.

From Table 11 , NDWPSO obtains the best design solution in this engineering problem and has the smallest standard deviation of the result data. In summary, the NDWPSO can reveal very competitive results compared to other intelligent algorithms.

Conclusions and future works

An improved algorithm named NDWPSO is proposed to enhance the solving speed and improve the computational accuracy at the same time. The improved NDWPSO algorithm incorporates the search ideas of other intelligent algorithms (DE, WOA). Besides, we also proposed some new hybrid strategies to adjust the distribution of algorithm parameters (such as the inertia weight parameter, the acceleration coefficients, the initialization scheme, the position updating equation, and so on).

23 classical benchmark functions: indefinite unimodal (f1-f7), indefinite multimodal (f8-f13), and fixed-dimensional multimodal(f14-f23) are applied to evaluate the effective line and feasibility of the NDWPSO algorithm. Firstly, NDWPSO is compared with PSO, CDWPSO, and SDWPSO. The simulation results can prove the exploitative, exploratory, and local optima avoidance of NDWPSO. Secondly, the NDWPSO algorithm is compared with 5 other intelligent algorithms (WOA, HHO, GWO, AOA, EO). The NDWPSO algorithm also has better performance than other intelligent algorithms. Finally, 3 classical engineering problems are applied to prove that the NDWPSO algorithm shows superior results compared to other algorithms for the constrained engineering optimization problems.

Although the proposed NDWPSO is superior in many computation aspects, there are still some limitations and further improvements are needed. The NDWPSO performs a limit initialize on each particle by the strategy of “elite opposition-based learning”, it takes more computation time before speed update. Besides, the” local optimal jump-out” strategy also brings some random process. How to reduce the random process and how to improve the limit initialize efficiency are the issues that need to be further discussed. In addition, in future work, researchers will try to apply the NDWPSO algorithm to wider fields to solve more complex and diverse optimization problems.

Data availability

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

Sami, F. Optimize electric automation control using artificial intelligence (AI). Optik 271 , 170085 (2022).

Article   ADS   Google Scholar  

Li, X. et al. Prediction of electricity consumption during epidemic period based on improved particle swarm optimization algorithm. Energy Rep. 8 , 437–446 (2022).

Article   Google Scholar  

Sun, B. Adaptive modified ant colony optimization algorithm for global temperature perception of the underground tunnel fire. Case Stud. Therm. Eng. 40 , 102500 (2022).

Bartsch, G. et al. Use of artificial intelligence and machine learning algorithms with gene expression profiling to predict recurrent nonmuscle invasive urothelial carcinoma of the bladder. J. Urol. 195 (2), 493–498 (2016).

Article   PubMed   Google Scholar  

Bao, Z. Secure clustering strategy based on improved particle swarm optimization algorithm in internet of things. Comput. Intell. Neurosci. 2022 , 1–9 (2022).

Google Scholar  

Kennedy, J. & Eberhart, R. Particle swarm optimization. In: Proceedings of ICNN'95-International Conference on Neural Networks . IEEE, 1942–1948 (1995).

Lin, Q. et al. A novel artificial bee colony algorithm with local and global information interaction. Appl. Soft Comput. 62 , 702–735 (2018).

Abed-alguni, B. H. et al. Exploratory cuckoo search for solving single-objective optimization problems. Soft Comput. 25 (15), 10167–10180 (2021).

Brajević, I. A shuffle-based artificial bee colony algorithm for solving integer programming and minimax problems. Mathematics 9 (11), 1211 (2021).

Khan, A. T. et al. Non-linear activated beetle antennae search: A novel technique for non-convex tax-aware portfolio optimization problem. Expert Syst. Appl. 197 , 116631 (2022).

Brajević, I. et al. Hybrid sine cosine algorithm for solving engineering optimization problems. Mathematics 10 (23), 4555 (2022).

Abed-Alguni, B. H., Paul, D. & Hammad, R. Improved Salp swarm algorithm for solving single-objective continuous optimization problems. Appl. Intell. 52 (15), 17217–17236 (2022).

Nadimi-Shahraki, M. H. et al. Binary starling murmuration optimizer algorithm to select effective features from medical data. Appl. Sci. 13 (1), 564 (2022).

Nadimi-Shahraki, M. H. et al. A systematic review of the whale optimization algorithm: Theoretical foundation, improvements, and hybridizations. Archiv. Comput. Methods Eng. 30 (7), 4113–4159 (2023).

Fatahi, A., Nadimi-Shahraki, M. H. & Zamani, H. An improved binary quantum-based avian navigation optimizer algorithm to select effective feature subset from medical data: A COVID-19 case study. J. Bionic Eng. 21 (1), 426–446 (2024).

Abed-alguni, B. H. & AL-Jarah, S. H. IBJA: An improved binary DJaya algorithm for feature selection. J. Comput. Sci. 75 , 102201 (2024).

Yeh, W.-C. A novel boundary swarm optimization method for reliability redundancy allocation problems. Reliab. Eng. Syst. Saf. 192 , 106060 (2019).

Solomon, S., Thulasiraman, P. & Thulasiram, R. Collaborative multi-swarm PSO for task matching using graphics processing units. In: Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation 1563–1570 (2011).

Mukhopadhyay, S. & Banerjee, S. Global optimization of an optical chaotic system by chaotic multi swarm particle swarm optimization. Expert Syst. Appl. 39 (1), 917–924 (2012).

Duan, L. et al. Improved particle swarm optimization algorithm for enhanced coupling of coaxial optical communication laser. Opt. Fiber Technol. 64 , 102559 (2021).

Sun, F., Xu, Z. & Zhang, D. Optimization design of wind turbine blade based on an improved particle swarm optimization algorithm combined with non-gaussian distribution. Adv. Civ. Eng. 2021 , 1–9 (2021).

Liu, M. et al. An improved particle-swarm-optimization algorithm for a prediction model of steel slab temperature. Appl. Sci. 12 (22), 11550 (2022).

Article   MathSciNet   CAS   Google Scholar  

Gad, A. G. Particle swarm optimization algorithm and its applications: A systematic review. Archiv. Comput. Methods Eng. 29 (5), 2531–2561 (2022).

Article   MathSciNet   Google Scholar  

Feng, H. et al. Trajectory control of electro-hydraulic position servo system using improved PSO-PID controller. Autom. Constr. 127 , 103722 (2021).

Chen, Ke., Zhou, F. & Liu, A. Chaotic dynamic weight particle swarm optimization for numerical function optimization. Knowl. Based Syst. 139 , 23–40 (2018).

Bai, B. et al. Reliability prediction-based improved dynamic weight particle swarm optimization and back propagation neural network in engineering systems. Expert Syst. Appl. 177 , 114952 (2021).

Alsaidy, S. A., Abbood, A. D. & Sahib, M. A. Heuristic initialization of PSO task scheduling algorithm in cloud computing. J. King Saud Univ. –Comput. Inf. Sci. 34 (6), 2370–2382 (2022).

Liu, H., Cai, Z. & Wang, Y. Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl. Soft Comput. 10 (2), 629–640 (2010).

Deng, W. et al. A novel intelligent diagnosis method using optimal LS-SVM with improved PSO algorithm. Soft Comput. 23 , 2445–2462 (2019).

Huang, M. & Zhen, L. Research on mechanical fault prediction method based on multifeature fusion of vibration sensing data. Sensors 20 (1), 6 (2019).

Article   ADS   PubMed   PubMed Central   Google Scholar  

Wolpert, D. H. & Macready, W. G. No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1 (1), 67–82 (1997).

Gandomi, A. H. et al. Firefly algorithm with chaos. Commun. Nonlinear Sci. Numer. Simul. 18 (1), 89–98 (2013).

Article   ADS   MathSciNet   Google Scholar  

Zhou, Y., Wang, R. & Luo, Q. Elite opposition-based flower pollination algorithm. Neurocomputing 188 , 294–310 (2016).

Li, G., Niu, P. & Xiao, X. Development and investigation of efficient artificial bee colony algorithm for numerical function optimization. Appl. Soft Comput. 12 (1), 320–332 (2012).

Xiong, G. et al. Parameter extraction of solar photovoltaic models by means of a hybrid differential evolution with whale optimization algorithm. Solar Energy 176 , 742–761 (2018).

Mirjalili, S. & Lewis, A. The whale optimization algorithm. Adv. Eng. Softw. 95 , 51–67 (2016).

Yao, X., Liu, Y. & Lin, G. Evolutionary programming made faster. IEEE Trans. Evol. Comput. 3 (2), 82–102 (1999).

Heidari, A. A. et al. Harris hawks optimization: Algorithm and applications. Fut. Gener. Comput. Syst. 97 , 849–872 (2019).

Mirjalili, S., Mirjalili, S. M. & Lewis, A. Grey wolf optimizer. Adv. Eng. Softw. 69 , 46–61 (2014).

Hashim, F. A. et al. Archimedes optimization algorithm: A new metaheuristic algorithm for solving optimization problems. Appl. Intell. 51 , 1531–1551 (2021).

Faramarzi, A. et al. Equilibrium optimizer: A novel optimization algorithm. Knowl. -Based Syst. 191 , 105190 (2020).

Pant, M. et al. Differential evolution: A review of more than two decades of research. Eng. Appl. Artif. Intell. 90 , 103479 (2020).

Coello, C. A. C. Use of a self-adaptive penalty approach for engineering optimization problems. Comput. Ind. 41 (2), 113–127 (2000).

Kannan, B. K. & Kramer, S. N. An augmented lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J. Mech. Des. 116 , 405–411 (1994).

Derrac, J. et al. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput. 1 (1), 3–18 (2011).

Download references

Acknowledgements

This work was supported by Key R&D plan of Shandong Province, China (2021CXGC010207, 2023CXGC01020); First batch of talent research projects of Qilu University of Technology in 2023 (2023RCKY116); Introduction of urgently needed talent projects in Key Supported Regions of Shandong Province; Key Projects of Natural Science Foundation of Shandong Province (ZR2020ME116); the Innovation Ability Improvement Project for Technology-based Small- and Medium-sized Enterprises of Shandong Province (2022TSGC2051, 2023TSGC0024, 2023TSGC0931); National Key R&D Program of China (2019YFB1705002), LiaoNing Revitalization Talents Program (XLYC2002041) and Young Innovative Talents Introduction & Cultivation Program for Colleges and Universities of Shandong Province (Granted by Department of Education of Shandong Province, Sub-Title: Innovative Research Team of High Performance Integrated Device).

Author information

Authors and affiliations.

School of Mechanical and Automotive Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

Jinwei Qiao, Guangyuan Wang, Zhi Yang, Jun Chen & Pengbo Liu

Shandong Institute of Mechanical Design and Research, Jinan, 250353, China

School of Information Science and Engineering, Northeastern University, Shenyang, 110819, China

Xiaochuan Luo

Fushun Supervision Inspection Institute for Special Equipment, Fushun, 113000, China

You can also search for this author in PubMed   Google Scholar

Contributions

Z.Y., J.Q., and G.W. wrote the main manuscript text and prepared all figures and tables. J.C., P.L., K.L., and X.L. were responsible for the data curation and software. All authors reviewed the manuscript.

Corresponding author

Correspondence to Zhi 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 4.0 International License, which permits use, sharing, adaptation, 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 changes were made. 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/4.0/ .

Reprints and permissions

About this article

Cite this article.

Qiao, J., Wang, G., Yang, Z. et al. A hybrid particle swarm optimization algorithm for solving engineering problem. Sci Rep 14 , 8357 (2024). https://doi.org/10.1038/s41598-024-59034-2

Download citation

Received : 11 January 2024

Accepted : 05 April 2024

Published : 10 April 2024

DOI : https://doi.org/10.1038/s41598-024-59034-2

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

• Particle swarm optimization
• Elite opposition-based learning
• Iterative mapping
• Convergence analysis

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.

Solving the uncapacitated facility location problem under uncertainty: a hybrid tabu search with path-relinking simheuristic approach

• Open access
• Published: 24 April 2024

Cite this article

You have full access to this open access article

• David Peidro   ORCID: orcid.org/0000-0001-8678-6881 1 ,
• Xabier A. Martin 1 ,
• Javier Panadero 2 &
• Angel A. Juan 1  

The uncapacitated facility location problem (UFLP) is a well-known combinatorial optimization problem that finds practical applications in several fields, such as logistics and telecommunication networks. While the existing literature primarily focuses on the deterministic version of the problem, real-life scenarios often involve uncertainties like fluctuating customer demands or service costs. This paper presents a novel algorithm for addressing the UFLP under uncertainty. Our approach combines a tabu search metaheuristic with path-relinking to obtain near-optimal solutions in short computational times for the determinisitic version of the problem. The algorithm is further enhanced by integrating it with simulation techniques to solve the UFLP with random service costs. A set of computational experiments is run to illustrate the effectiveness of the solving method.

Avoid common mistakes on your manuscript.

1 Introduction

The facility location problem (FLP) was initially addressed by Stollsteimer [ 1 ], Kuehn and Hamburger [ 2 ], Manne [ 3 ] and Balinski [ 4 ] to determine a set of facilities that minimizes the aggregation of two inversely correlated costs: (i) the cost of opening facilities; (ii) the cost related to servicing customers from the opened facilities. In most formulations of the problem, a set of customers and a set of potential facility locations are known in advance. Likewise, the opening costs associated with each facility and the costs of servicing each customer from every potential facility are also known. Hence all inputs are deterministic insofar as they are static inputs that are given from the very beginning. So the FLP is a frequent optimization problem used in very diverse application fields, from logistics and inventory planning ( e.g. , where to allocate distribution or retailing centers in a supply chain) to telecommunication and computing networks ( e.g. , where to allocate cloud service servers in a distributed network, cabinets in optical fiber networks, etc.).

Illustrative example of a solution for the UFLP under uncertainty

As one of the most frequent optimization problems in the logistics and supply chain management area, several versions of the FLP have been analyzed in the scientific literature. The uncapacitated FLP (UFLP) assumes that each facility’s capacity is virtually unlimited or is, at least, far beyond expected customer demands. Despite being known as the simple facility location problem, the simple warehouse location problem, or the simple plant location problem for its apparent simplicity [ 5 ], the UFLP has been proved to be NP-hard [ 6 ]. Therefore, heuristic and metaheuristic approaches become a natural choice for solving large-scale instances of the UFLP in reasonably short computing times. This is because exact and approximate methods are unable to accurately solve large instances during short periods of time. Another variant is the capacitated FLP (CFLP), where each open facility has a limited servicing capacity; i.e., there is limit to customers’ demand that can be served by any single facility. According to Silva and de la Figuera [ 7 ], both Lagrangian-based heuristics and metaheuristics have been demonstrated as effective methods for solving the CFLP. The single-source CFLP (SSCFLP) also requires each customer having to be supplied by exactly one facility. As stated by Klose and Drexl [ 8 ], the SSCFLP is generally more difficult to solve than the multiple-source CFLP. In fact the SSCFLP also belongs to the class of NP-hard problems [ 9 ]. Therefore, heuristic approaches become a natural choice for solving large-scale instances of the SSCFLP in reasonably short computing times.

Uncertainty permeates all real-world systems, including supply chain management, logistics and facility locations. The inherent unpredictability of various variables poses a significant challenge in designing efficient and cost-effective solutions. Correia and Saldanha-da Gama [ 10 ] reviewed the FLP with stochastic components by exploring different methods proposed in the recent literature to optimize the FLP under uncertainty. These stochastic components arise when inputs, such as customer demands or service costs, are random variables instead of deterministic values. Effectively recognizing and accounting for this uncertainty become crucial in providing optimal solutions for real-life combinatorial optimization problems. Simulation-based optimization approaches have been proposed to tackle such problems [ 11 , 12 ]. These approaches encompass diverse optimization methods, including mathematical programming, metaheuristics, and even machine learning. In recent decades, the hybridization of metaheuristics with simulation has emerged as a popular and effective approach for solving stochastic optimization problems [ 12 ]. Simheuristics, a simulation-optimization method that combines simulation with metaheuristics, has been widely used to address various combinatorial optimization problems with stochastic elements [ 13 , 14 ]. This work proposes a novel simheuristic algorithm to address the UFLP under uncertainty by specifically considering stochastic components in the form of random service costs. These service costs can be modeled using probability distributions that can be either theoretical or empirical. Figure  1 provides an illustrative example of a solution for the UFLP under uncertainty. The example depicts the locations where facilities have been opened (indicated by red squares) and the closed facilities (represented by white squares). Customers, depicted by blue circles, are served by the open facilities to which they are actively connected. Each facility has a fixed opening cost, and servicing a customer throughout an open facility has an associated service cost, which is uncertain and, therefore, modeled as a random variable.

Accordingly, the main contributions of this paper can be summarized as follows: (i) a tabu search metaheuristic [ 15 ] to efficiently solve the UFLP in short computational times; (ii) a path-relinking approach to obtain near-optimal solutions by exploring paths that connect good quality solutions; (iii) a simheuristic algorithm that integrates the tabu search metaheuristic and path-relinking approach with simulation techniques to efficiently solve the aforementioned problem. Note that the tabu search and path-relinking algorithms are shown to be highly effective in practice for finding near-optimal or optimal solutions to large-scale optimization instances in short computational times [ 16 , 17 ]. However, without the support of simulation-based extensions like that introduced in this paper, these techniques do not account for the inherent unpredictability found in real-life systems when solving the UFLP under uncertainty.

The remainder of this paper is structured as follows: Section  2 formulates the mathematical model for the UFLP with random service costs. Section  3 reviews the literature on the UFLP under uncertainty. Section  4 presents a tabu search metaheuristic with path-relinking as a solving method for the UFLP, and how the algorithm can be extended into a simheuristic one to solve the UFLP under uncertainty. Section  5 describes the computational study performed to test the proposed solving method. Finally, Section  6 discusses the main conclusions of this work and open research lines.

2 Model formulation

By following the mixed-integer linear programming model for the deterministic UFLP proposed by Erlenkotter [ 18 ], the stochastic version of the problem can be modeled using the following set of parameters and variables

I - Set of m alternative facility locations, indexed by i .

J - Set of n customer zones, indexed by j .

\(f_i\) - Fixed cost of establishing a facility at location i .

\(C_{ij}\) - Random variable that models the cost of servicing customer j

from facility i .

\(x_{ij}\) - It takes the value 1 if the demand of customer j is supplied from

facility i , and 0 otherwise.

\(y_{i}\) - It takes the value 1 if facility i is open, and 0 otherwise.

Then the stochastic UFLP considered in this work is formulated as follows:

The objective function ( 1 ) represents the minimization of the sum of the expected total cost of servicing customers and the total fixed costs of facility establishment.

Constraint ( 2 ) ensures that the demand for each customer zone is met. Constraint ( 3 ) makes sures that customer demand can only be produced and shipped where the facility is established. Constraint ( 4 ) features probabilistic constraints. These constraints add a probabilistic aspect to the model by ensuring that the cost of serving each customer from each facility falls within a certain threshold \(c_{max}\) with a specified level of confidence \(p_0\) . This allows the model to account for the uncertainty in the cost values and helps more robust decisions to be made that consider the risk associated with the random variable \(C_{ij}\) . As discussed later, this can be considered a soft constraint, which will generate a penalty cost every time it is violated. Finally, Constraints ( 5 ) and ( 6 ) define the binary decision variables. The deterministic version of this model (considering constant costs \(c_{ij}\) instead of variable ones \(C_{ij}\) ) was implemented using Python and solved with commercial solver Gurobi to calculate the optimal values for different instances of the deterministic UFLP. In this way, the performance of the metaheuristic proposed in this work can be compared to the mathematical approach. Additionally, a simheuristic was developed to solve the stochastic version of the problem at different uncertainty levels, which is a very common situation in real-life problems.

3 Related work on the UFLP under uncertainty

The FLP was initially introduced as the plant location problem by Stollsteimer [ 1 ] and Balinski [ 4 ]. Traditionally, the FLP has been approached from several perspectives, including worst-case analysis, probabilistic analysis and empirical heuristics. Although exact algorithms for the problem can be found in the existing literature, the NP-hard nature of the FLP makes heuristics a more practical approach for quickly obtaining solutions, especially for larger and more realistic instances. One of the first works on the FLP was carried out by Efroymson and Ray [ 19 ], who developed a branch-and-bound algorithm. They utilize a compact formulation of the FLP by leveraging the fact that its linear programming relaxation could be solved through inspection. However, this linear programming relaxation is known to be weak and does not, therefore, provide tight lower bounds. Another early approach was proposed by Spielberg [ 20 ], which employs a direct search or implicit enumeration method. The authors present two different algorithms based on the same directed search: one considering the facilities to be initially open and another one contemplating initially closed ones. Later Erlenkotter [ 18 ] proposed a dual-based exact approach that differed from previous approaches by considering a dual objective function. An improved version of this work was presented by Körkel [ 21 ]. Although exact approaches provide optimal solutions for small or medium instances, they are unsuitable for solving large-scale real-world problems in reasonable computational times. Therefore, employing approximate methods is advisable. One of the earliest approximation methods was that by Hochbaum [ 22 ], and it consists of a simple and fast greedy heuristic. More recently, Ghosh [ 23 ] put forward a neighborhood search heuristic by incorporating a tabu search as the local search component. This approach yields competitive solutions in significantly shorter computational times compared to exact algorithms. A similar approach with a tabu search was proposed by Michel and Van Hentenryck [ 24 ]. This algorithm shows competitive performance compared to previous literature results. The algorithm utilizes a linear neighborhood approach, where a single facility is flipped at each iteration. Resende and Werneck [ 25 ] introduced an algorithm based on the GRASP metaheuristic. This algorithm incorporates a greedy construction phase combined with a local search operator and path-relinking. It obtains results that come close to the lower bound values for a wide range of different instances. Recently, Martins et al. [ 26 ] introduced an ‘agile optimization’ framework for the UFLP. This framework combines a biased-randomized algorithm with parallel programming techniques to offer real-time solutions. The key feature of this approach is its ability to react and swiftly adapt to rapidly changing customer demands. It achieves this by re-optimizing the system whenever new information is incorporated into the model.

The FLP was initially introduced as the plant location problem by Stollsteimer [ 1 ] and Balinski [ 4 ]. Traditionally, the FLP has been approached through various perspectives, including worst-case analysis, probabilistic analysis, and empirical heuristics. While exact algorithms for the problem can be found in the existing literature, the NP-hard nature of the FLP makes heuristics a more practical approach for obtaining solutions quickly, especially for larger and more realistic instances. One of the first works on the FLP was carried out by Efroymson and Ray [ 19 ] who developed a branch-and-bound algorithm. They utilized a compact formulation of the FLP, leveraging the fact that its linear programming relaxation could be solved through inspection. However, this linear programming relaxation is known to be weak and therefore does not provide tight lower bounds. Another early approach was proposed by Spielberg [ 20 ], which employed a direct search or implicit enumeration method. The authors present two different algorithms based on the same directed search, one considering the facilities initially open and another one considering the facilities initially closed. Later, Erlenkotter [ 18 ] propose a dual-based exact approach, differing from previous approaches by considering a dual objective function. An improved version of this work was presented by Körke [ 21 ]. Although exact approaches provide optimal solutions for small or medium instances, they are unsuitable for solving large-scale real-world problems in reasonable computational time. Therefore, it is advisable to employ approximate methods. One of the earliest approximation methods was proposed by Hochbaum [ 22 ], consisting of a simple and fast greedy heuristic. More recently, Ghosh [ 23 ] proposed a neighborhood search heuristic, incorporating tabu search as the local search component. This approach yields competitive solutions within significantly reduced computational times compared to exact algorithms. A similar approach using a tabu search was proposed by Michel and Van Hentenryck [ 24 ]. This algorithm has shown competitive performance compared to previous literature results. The algorithm utilizes a linear neighborhood approach, where a single facility is flipped at each iteration. Resende and Werneck [ 25 ] introduced an algorithm based on the GRASP metaheuristic. This algorithm incorporates a greedy construction phase combined with a local search operator, and a path-relinking. It achieved results close to the lower bound values for a wide range of different instances. Recently, Martins et al. [ 27 ] introduced an Agile Optimization framework [ 26 ] for the UFLP. This framework combines a biased-randomized algorithm with parallel programming techniques to offer real-time solutions. The key feature of this approach is its ability to react and adapt swiftly to rapidly changing customer demands. It achieves this by re-optimizing the system whenever new information is incorporated

A wide range of variants of the FLP have been extensively addressed in the literature, regardless of the employed solution method. One of the most studied variants is the SSCFLP. Similarly to the original version of the FLP, several exact methods have also been proposed to solve small instances of this problem. For example, Holmberg et al. [ 28 ] described a matching algorithm incorporated into a Lagrangian heuristic. In another study, Díaz and Fernández [ 29 ] developed an exact algorithm that integrates a column generation procedure for finding upper and lower bounds within a branchand-price framework. Similarly, Yang et al. [ 9 ] introduced an exact algorithm based on a cut-and-solve framework designed explicitly for the SSCFLP. Regarding the use of approximate methods, Chen and Ting [ 30 ] proposed a hybrid algorithm that combines a Lagrangian-based heuristic with an ant colony method. This approach aims to leverage the strengths of both techniques to solve the problem. Ahuja et al. [ 31 ] presented a large-scale neighborhood search algorithm, which focuses on efficiently exploring the solution space of the problem. This approach allows large-scale instances to be effectively handled in a reasonable computational time. Filho and Galvão [ 32 ] proposed a tabu search metaheuristic that is also able to handle large instances, while Delmaire et al. [ 33 ] put forward a more sophisticated algorithm also based on a tabu search, which combines a reactive GRASP algorithm with a tabu search. This algorithm incorporates elements of both techniques to enhance search capabilities and to find high-quality solutions. Finally, Estrada-Moreno et al. [ 34 ] presented a biased-randomized iterated local search metaheuristic to solve the SSCFLP with soft capacity constraints. This variant of the SSCFLP assumes that the maximum capacity at each facility can be potentially exceeded by incurring a penalty cost, which increases with the constraint-violation gap. For a more comprehensive literature review on the FLP and its variants, readers are referred to [ 8 , 35 ], and the book edited by Eiselt and Marianov [ 36 ] covers many relevant works on the FLP.

In real life, the inputs of combinatorial optimization problems are typically nondeterministic. This means that they are subject to random events; i.e.., random failures of some components, stockouts due to random demands, etc. Therefore, Simulation-based optimization approaches are required. These methods aim to find a solution that performs well for any possible realization of the random variables, i.e., a robust solution that can handle variations and fluctuations in the problem parameters that may occur in real-world scenarios s [ 37 ]. Although the stochastic FLP has not been paid significant attention in the literature, we can find several works, some of which have been published only a few years after the problem’s definition. For example, Balachandran and Jain [ 38 ] presented a stochastic FLP model with piece-wise linear production. This model takes demands to be random and continuous variables. Later Correia and Saldanha-da Gama [ 10 ] examined distinct modeling frameworks for facility location under uncertain conditions, which specifically distinguish among robust optimization, stochastic programming and chance-constrained models. Another simulation-optimization approach to address this problem is the simheuristics [ 39 ], which is a promising approach based on the combination of simulation with metaheuristics, used for solving efficiency different combinatorial optimization problems with stochastic elements. Indeed simheuristics has been used by different authors to solve the UFLP under uncertainty. De Armas et al. [ 40 ] proposed a simheuristic approach to address the UFLP with random service costs. This simheuristics combines an ILS metaheuristic with Monte Carlo simulation to deal with uncertainty by providing flexibility to consider diverse optimization objectives beyond minimizing the expected cost. A similar approach is that proposed in Quintero-Araujo et al. [ 41 ], which proposes a SimILS framework, a simheuristic algorithm that combines Monte Carlo simulation with a biased-randomized metaheuristic algorithm to solve the capacitated location routing problem (CLRP) with stochastic demands. Another interesting approach i that reported Bayliss and Panadero [ 42 ], where a learnheuristic algorithm [ 43 ] is presented to solve the temporary-facility location and queuing problem. It integrates a biased randomization algorithm with simulation and a machine-learning component to tackle not only uncertain components, but also dynamics components of problems.

To further extend the landscape of facility location problems under uncertainty, other methodological approaches have emerged in recent years. Marques and Dias [ 44 ] introduced a dynamic UFLP by considering uncertainty in fixed and assignment costs, customer and facility locations. They aimed to minimize the expected total cost, while explicitly considering regret. Regret, as a measure of loss for not choosing an optimal scenario solution, is upper-bounded. Their mixed integer programming model and solution approach demonstrates potential through illustrative examples and computational results. Another exciting approach was that proposed by Zhang et al. [ 45 ], which tackles another variant, the squared metric two-stage stochastic FLP by focusing on uncertainty in client sets and facility costs with squared metric connection costs. They proposed a new integer linear programming formulation and evaluated two algorithms’ performance by analyzing approximation ratios and per-scenario bounds. Moreover, Ramshani et al. [ 46 ] explored disruptions in two-level UFLPs by including additional facilities between customers and the main facilities, and by acknowledging disruptions’ impact on facility reliability for meeting customer demands. They developed mathematical formulations and algorithms, such as a tabu search and a problem-specific heuristic, to address disruptions in a two-level distribution chain scenario. By extending this research, Koca et al. [ 47 ] addressed two-stage stochastic UFLPs by emphasizing system nervousness. They introduced models to consider uncertain location and allocation decisions adaptable to realizations of uncertain parameters. Their proposed models incorporated restricted recourse to control deviations between first-stage and scenario-dependent allocation decisions by showcasing Benders-type decomposition algorithms and computational enhancements.

4 Proposed methodology

Simulation methods are frequently employed by experts to address stochastic uncertainty because they enable several scenarios to be analyzed to support decision-making processes. However, it is important to note that simulation itself is not an optimization tool. Therefore, hybrid simulation-optimization methodologies have been proposed to efficiently cope with large-scale optimization problems under uncertainty. One such simulation-optimization method is simheuristics, which combines metaheuristics with simulation [ 48 ]. Its efficiency as a method for solving different combinatorial optimization problems with stochastic elements has been shown in several studies [ 43 ]. This success can be attributed to the method’s ability to evaluate solutions using simulation and problem-specific analytical expressions. Simheuristic algorithms are ‘white-box’ approaches designed specifically to solve large-scale and NP-hard combinatorial optimization problems with stochastic elements, which can come in the form of stochastic objective functions or probabilistic constraints [ 49 ]. The proposed simheuristic method combines a tabu search metaheuristic with path-relinking, which is then integrated with Monte Carlo simulation (MCS) to solve the UFLP with random service costs. Using the tabu search metaheuristic with path-relinking combination allows us to obtain near-optimal solutions to the deterministic version of the problem in short computational times. This combination is successfully employed to solve NP-hard optimization problems in vehicle routing and scheduling domains [ 50 , 51 ]. The algorithm is further combined with simulation techniques to guide the algorithm during the search for near-optimal solutions to the stochastic version of the problem.

4.1 Tabu search with path-relinking

The algorithm for a tabu search with path-relinking is now described. We represent a state in the tabu search by vector \(y_i\) as defined in the model formulation. It is a natural choice seeing that the facility locations are the only combinatorial component. To solve the UFLP, it is enough to know the set of open facilities because customer zones are connected to the cheapest open facilities.

As shown in Algorithm 1, three sections can be identified. The first section applies the tabu search metaheuristic N times to the same problem instance, but uses different seeds. The best solutions are stored in a pool of elite solutions of size L . Subsequently, the path-relinking technique is applied for each pair of elements in the elite solution pool. The new solutions produced to create the next pool of elite solutions are employed (new generation pool). Only those solutions that improve the two used in the path-relinking process are considered the candidates to be inserted into the new generation pool of size L . Note that the best solution found so far is inserted directly into the new generation of elite solutions (1-elitism). The criterion for inserting a new solution into the elite solutions pool is the lowest total cost. The creation of new pools is repeated as long as new generation pools of at least two solutions are produced. After the path-relinking phase, the algorithm begins a local search process around \(sol_{best}\) . This procedure consists of applying a well-known 2-opt local search to every possible combination of two facility locations in the solution.

Tabu search with path-relinking

• Tabu search

The tabu search is presented in Algorithm 2. First, a random solution is generated to provide each facility location with an opening probability. The neighborhood consists in flipping the state of a facility from open to closed, or vice versa. After scanning the neighborhood, if there is an improvement is calculated. Then the facility location involved in the improvement is added to the tabu list to prevent it from being chosen for a defined number of iterations (tenure). The tabu list is implemented by associating a simple counter tabu [ i ] for each facility i . When a facility is inserted on the tabu list, the counter tabu [ i ] is updated by adding the actual iteration count, plus the tenure ( \(tabu[i] = iteration+tenure\) ). This expression states that the facility will remain in the tabu search memory for as many iterations as the value of the tenure parameter. The algorithm can quickly check if a facility is on the tabu list if \(tabu[i] > iteration\) . If there is no improvement in the solution, the algorithm randomly closes a previously open facility location. If only one site remains to be closed, a new one is randomly opened. Tenure is adjusted during the search process. If the algorithm finds improvements in the neighborhood, it decreases tenure to improve exploitation. On the contrary, if it does not find improvements when changing the state of any of the facility locations of a solution, tenure is increased to improve search diversification (never without exceeding a given maximum value). The process is repeated until the stopping criterion is met, which consists in exceeding a number M of iterations without finding an improvement in the solution. The tabu search algorithm uses three different pieces of information for each customer zone j : the number of open facilities that offer the cheapest connection to j, the cost of that connection, and the cost of the second cheapest connection to an open facility. These details allow the gains from opening and closing a facility to be incrementally updated by eliminating the need to compute the entire objective function for a new solution (line 10). A similar approach is used in Michel and Van Hentenryck [ 24 ].

• Path-relinking

The path-relinking subroutine is comprehensively described in Algorithm 3. Given the origin and reference solutions, we first calculate which one gives the best cost and store it in \(sol_{best}\) . Then cardinality is calculated as the number of differences between the origin and reference solutions. Next, the algorithm starts from \(sol_O\) and gradually transforms into \(sol_R\) by flipping the facilities that are different between the two solutions. During this transformation process, intermediate solutions are evaluated to see if they improve the overall result. Finally, the best found solution is returned.

The algorithm’s computational complexity is next reported. For the tabu search component, the complexity per iteration is \(O(n \cdot log(m))\) , where n represents the number of clients and m denotes the number of facilities. This complexity accounts for the calculation of the cost of flipping each facility in a solution, while tracking the best and second-best facilities to assign clients to. These operations are repeated until a number of failed runs is reached, which contributes to the constant factor of the computational complexity. In other words, as the number of iterations is constant (not dependent on the size of the problem), the overall computational complexity of the tabu search component is expressed as \(O(n \cdot log(m))\) . The path-relinking component has a similar complexity per iteration of \(O(n \cdot log(m))\) . This also involves updates to the best and second-best facilities assigned to clients, because intermediate solutions have to be evaluated to see if they improve current solutions. Similar to the tabu search component, the actual number of iterations performed by the path-relinking component depends on the cardinality of solutions along with the number of elite generations, which are not dependent on the size of the instance. Thus, the overall computational complexity of the path-relinking component is \(O(n \cdot log(m))\) . The local search component, the computational complexity per iteration is associated with the process of exploring the neighborhood of the current solution. This involves flipping each facility in the solution and checking if the new solution is improved, whose computational complexity is \(O(n \cdot log(m))\) . This process is repeated until no further improvements are found, so the overall computational complexity of the local search is \(O(n \cdot log(m))\) . Finally, as the three components discussed are executed in a serial manner, we can conclude the proposed algorithm has a worst-case computational complexity of \(O(n \cdot log(m))\) .

Flowchart of the simheuristic methodology

4.2 The simheuristic framework

Figure  2 illustrates a flow chart of our simheuristic methodology for dealing with the UFLP under stochastic uncertainty. Every time a new solution is generated by the tabu search, path-relinking or a local search, it is simulated with a few runs to obtain an estimate of the solution’s average stochastic cost. On the one hand, regarding the tabu search, all the N times when the algorithm reaches the stopping criterion and finds the best solution (Algorithm 1), the fast simulation is run. On the other hand, the simulations of the path-relinking solutions are the most important in the proposed simheuristic. Path-relinking as an intensive process done on a set of elite solutions allows a sufficiently large set of potential solutions to be explored that could perform well in a stochastic environment. For this reason, every time a new solution is generated during the gradual transformation of \(sol_O\) into \(sol_R\) , it is simulated regardless of whether it improves the deterministic solution. In addition, whenever the local search procedure reaches a new improved solution, this is also simulated. The solutions with the best-estimated cost are stored in a pool of elite stochastic solutions. Once an overall stopping criterion is met, a limited set of elite stochastic solutions is sent to a more intensive simulation stage to obtain accurate estimates on their behavior in a stochastic scenario. The general stopping criterion consists of satisfying the stopping criteria of each section of the proposed metaheuristic ( M failed tabu search runs, a new pool generation with < than 2 solutions during path-relinking and no new improvements during the local search).

Algorithm 4 depicts the main characteristics of our simheuristic algorithm. The code is similar to that shown in Algorithm 1. The solutions obtained by the tabu search are quickly simulated, while those with the lowest expected cost are included in the pool of elite stochastic solutions (lines 13-17). Then when the path-relinking ( PR-SIMH ) and local search ( LS-SIMH ) functions are called, the pool is passed as an argument to be updated according to the simulations of the newly generated solutions (as explained above). In lines 34-38, intensive simulation is performed to obtain better estimates of the elite solutions. The best solutions are selected according to the lowest expected cost. However, other estimates, such as risk or reliability, may also be used as discussed in Chica et al. [ 52 ].

Simheuristic for the UFLP

5 Computational study

Several computational experiments were carried out to evaluate and assess the performance of our solving method. To comprehensively illustrate the experiments conducted in this section, a detailed presentation of the utilized benchmark instances, that outlines their characteristics, is first provided. Next the presented benchmark instances are extended to consider random service costs, which model the uncertainty that is inherent in practical scenarios, and penalty costs, by addressing several real-world situations. Then Section  5.4 identifies the significantly affected parameters and decides the optimal parameter settings for the algorithm through the Design of Experiments (DoE). Last, the obtained results and the discussion are presented in Section 5.5. All the numerical experiments were implemented using the C++ programming language. The code was compiled on a Manjaro Linux machine with the GCC version 12.3 compiler and executed on a computer with an Intel Core i5-9600K 3.7 GHz and 16 GB RAM.

5.1 Description of benchmark instances

In order to evaluate the performance of the proposed simheuristic algorithm for the UFLP with random service costs, we used the classic instances originally proposed for the p -median problem by Ahn et al. [ 53 ], later employed in the UFLP context by Barahona and Chudak [ 54 ]. This set of large instances is called MED, which is the most used set of instances in the UFLP literature for being the most challenging ones to solve. To the best of our knowledge, the results of these instances have not been improved from 2006 [ 25 ]. This makes them the perfect benchmarks to test the quality of our algorithm. Each instance is composed of a set of n points picked uniformly at random in the unit square. A point represents both a user and facility, and the corresponding Euclidean distance determines service costs. Additionally, each instance is characterized by the following nomenclature \(x - y\) , where x represents the number of facilities and customers, while y refers to the opening cost scheme. The set of instances consists of six different subsets, each with a different number of facilities and customers (500, 1000, 1500, 2000, 2500 and 3000), and three different opening cost schemes per subset ( \(\sqrt{n}/10\) , \(\sqrt{n}/100\) , and \(\sqrt{n}/1000\) corresponding to 10, 100, and 1000 instance suffixes, respectively). The larger instances and those with suffix 1000 are the most difficult to solve. As they have lower opening costs, the number of open facilities in the solution is bigger, which increases the number of possible combinations and, thus, the complexity of finding the optimal solution.

5.2 Extending benchmark to address uncertainty

To the best of our knowledge, there are no stochastic FLP instances that employ random service costs to be used as a benchmark. So instead of assuming constant service costs \(c_{ij}\) ( \(\forall i\in I, \forall j\in J\) ), we consider a more realistic scenario in which service costs are modeled as random variables \(C_{ij}\) . The service cost represents the cost required to service a customer zone \(j \in J\) throughout a facility location \(i \in I\) under perfect conditions. Accordingly, we extend the previously described set of instances called MED to assess both the performance and quality of the proposed simheuristic algorithm. The dataset construction process involves transforming the deterministic service costs into random service costs following a probability distribution function. Specifically, the deterministic service costs are transformed into stochastic ones when solutions are sent to the simulation phase. Hence the service costs \(c_{ij}\) found in instances are used as the expected values of the random service costs \(C_{ij}\) , which are modeled to follow a log-normal probability distribution. The log-normal distribution is a natural choice for describing nonnegative random variables, such as service costs [ 55 ]. The log-normal distribution has two parameters: the location parameter, \(\mu _{ij}\) , and the scale parameter,  \(\sigma _{ij}\) , which relate to the expected value \(E[C_{ij}]\) and to variance \(Var[C_{ij}]\) , respectively. Equations  7 - 8 define how these parameters have been modeled. We assume service cost \(C_{ij} \sim LogNormal(\mu _{ij}, \sigma _{ij})\) with \(E[C_{ij}] = c_{ij}\) , where \(c_{ij}\) is the deterministic service delay found in instances, and variance \(Var[C_{ij}] = k \cdot E[C_{ij}]\) . Parameter k is a design parameter that allows us to set up the uncertainty level. It determines how much stochastic costs deviate from their expected values ( \(c_{ij}\) derived from deterministic costs) and, therefore, influences the degree of randomness or uncertainty in the simulated costs. So as k converges to zero, the more the results will move toward the deterministic state, and by increasing the value of k , the results will converge to the stochastic state. The selection of k values depends on the uncertainty level desired in stochastic costs. We consider three different uncertainty levels: low ( \(k = 5\) ), medium ( \(k = 10\) ) and high ( \(k = 20\) ). The main objective of setting different k values is to simulate a range of real-world scenarios for the variability and unpredictability of service costs. By employing multiple k values, the algorithm’s performance can be evaluated under diverse conditions, which allows for a comprehensive assessment of its robustness and adaptability to varying cost uncertainty levels.

This approach enables us to test the effectiveness of our simheuristic algorithm in handling UFLP instances with varying uncertainty levels, and to showcase its adaptability at different degrees of random service costs without having to create brand-new datasets.

5.3 Extend benchmark with penalty costs

In order to address real-world situations where certain facilities’ costs need to be controlled or limited, a nonlinear penalty cost is added. This approach aligns the optimization process with practical considerations and allows decision makers to make more informed choices regarding facility selection and cost management. Note in the basic UFLP under uncertainty with a linear objective function that the solution which minimizes the total expected cost will be the same as the optimal solution for the deterministic UFLP. This property will not hold if, for instance, a nonlinear penalty cost is added in the objective function to account for facilities with a total higher service cost than a threshold, etc. Thus we introduce a non linear penalty cost that is twice the cost of opening a facility if the instance maximum service cost of a deterministic solution is exceeded. This parameter, denoted as \(c_{\text {max}} = \max {c_{ij}}\) , represents the maximum \(c_{ij}\) value between all the facilities opened in the best-found deterministic solution. It is worth noting that when this nonlinear penalty is incorporated into the objective function of the UFLP under uncertainty, the solution that minimizes the total expected cost may no longer be the same as the optimal solution for the deterministic UFLP. This deviation from linearity arises because the penalty cost is included. In other words, the best-known deterministic solution, which will still be optimal in a deterministic scenario for not being affected by the \(c_{\text {max}}\) threshold, will perform quite poorly in a stochastic scenario because in many simulation runs there will be service costs that will exceed the \(c_{\text {max}}\) threshold, which will result in a severe penalty. Furthermore, the simheuristic algorithm tends to open some more facilities than those in the deterministic solution to avoid the number of times that the threshold is exceeded. Therefore, the simheuristic approach aims to search for solutions that mitigate the impact of penalties associated with exceeding the \(c_{\text {max}}\) threshold in multiple simulation runs.

5.4 Parameter analysis by design of experiments

In this section, we use DoE to identify the significantly affected parameters and decide the best combination of each parameter for the TS+PR algorithm. As shown in Table  1 , a two-level full factorial design with three replications [ 56 ] is adopted to investigate the significant effect of four parameters of the TS+PR algorithm on both the objective value and the calculation time. The following parameters are considered: N (maximum runs of the algorithm with different seeds), M (maximum number of the tabu iterations without improvement), L (elite pool length) and Tenure (number of iterations a move is considered to be tabu, expressed as a percentage of the number of facility locations of the problem). The high and low levels of each parameter are displayed in Fig.  4 .

The results are shown in Table  1 for a randomly generated test problem of size ( \(|I| = |J|=2000\) ). The problem was created following the same structure as the large instances called MED. The selected opening cost scheme was \(\sqrt{n}/1000\) because, as previously mentioned, it is the most challenging. The response variable AVG GAP (%) is the average gap of 30 different independent runs compared to the optimal solution obtained with Gurobi for the generated test problem. The response variable AVG TIME is the average execution time for the 30 independent runs. In all, 3 * \(2^{4}\) * 30 = 1440 trials were conducted in this experiment. Table  1 shows the average (best) values for the 30 runs and displays 48 trials.

Pareto chart for AVG GAP(%)

Main effect plot for AVG GAP(%)

According to the ANOVA results (Table  2 ) and the Pareto chart (Fig.  3 ) for the response AVG GAP (%), only parameters M and L are statistically significant for the proposed problem. The values RSquare = 92.21%, RSquare(Adj) = 88.19%, and Square(Pred) = 81.33% obtained in Statgraphics mean that this model fits the data well and can be used to accordingly determine the suitable parameters (Fig.  4 ).

For the response AVG TIME (see Fig.  5 ), all the parameters except Tenure are the algorithm’s most significant and critical parameters at \(\alpha = 0.05\) . The model fits the data very well (RSquare = 99.25%, RSquare(Adj) = 98.86%, and Square(Pred) = 98.20%).

Pareto chart for AVG TIME(s)

Main effect plot for AVG TIME(s)

To find the best parameters for the algorithm, we focus mainly on the response AVG GAP (%). The best parameter settings on N , M , L and Tenure , according to AVG GAP(%), are 64, 500, 10 and 10%, respectively (see Fig.  4 ). Although M and L prolong the computation time (see Fig.  6 ), they are statistically significant and their higher values are necessary. Coosing a value of 64 vs. 32 for N negatively affects the execution time (it increases by more than 40%) and is not statistically significant. Furthermore, the best gap for \(N=64\) is 0.02% and 0.03% for \(N=32\) , a minimal difference for the longer computational time required. Finally, Tenure does not statistically affect either AVG GAP or AVG TIME. As explained previously, the reason lies in the algorithm’s capacity to adjust the tenure value as the search progresses. So it does not affect starting with a slightly higher or lower value.

The main objective of this paper is not to obtain the best possible solutions in the deterministic environment. So we explore the possibility of considering \(N=64\) and of even analyzing the results of larger values for the most significant factors, such as M and L . This paper focuses on uncertain conditions using a simulation optimization approach, such as the simheuristics concept. For this reason, having a good deterministic search algorithm (i.e. 0.03% gap) with reasonable computation times is sufficient. In this way, the algorithm can be efficiently combined with Monte Carlo simulations to search for the best solutions under uncertainty. For all these reasons, the values chosen for the adjustment of the TS+PR algorithm, used later within the simheuristic framework, are \(N = 32, M = 1000, L = 10\) and \(Tenure = 10\%\) .

Three more parameters are defined by applying the simheuristic approach described in Algorithm 2, which are longSim , shortSim and L 2. The first parameter is defined as being large enough to have a full, meaningful simulation of a solution under stochastic uncertainty. The second parameter attempts to quickly evaluate a solution in an uncertain environment to guide the search for better ones. The last parameter consists of a pool of elite solutions (quickly evaluated) to be finally validated in the complete simulation. This pool is ordered from the best to the worst result of the fast simulation. A solution in a lower position may obtain a better result in the long simulation than another in a higher initial position. For this reason, pool size must be large enough to capture the best possible solution in the global simulation. The exploratory studies indicate that the best final solution after the full simulation appears before the first five positions of the elite pool. To ensure that the best solution is obtained in the complete simulation, a value higher than five is finally chosen.

Table  3 summarizes all the parameters and the values selected for the following computational experiments applied to the benchmark instances described in the previous points (Table 4 ).

5.5 Results and discussion

Table  5 includes the results obtained for four different approaches to solve the UFLP with the MED benchmark instances. The first column displays instances, whose names are combinations of the number of demand points and the opening cost scheme. The next two columns show the solutions obtained using the Gurobi Optimizer solver, along with the corresponding computational time required to reach those solutions. The subsequent three columns present the best-found solutions obtained from our tabu search with the path-relinking approach, along with the computational time needed to obtain them. Each instance is run 30 times using a different seed for the random number generator, and the best and average results are reported. Furthermore, the three columns that follow depict the best-found solutions achieved using the multi-start heuristic proposed by Resende and Werneck [ 25 ], along with the computational time required for each solution. The next three columns illustrate the best-found solutions obtained with the ILS-based approach proposed by De Armas et al. [ 40 ], along with the computational time taken to achieve those results. Finally, the last three columns display the best-found solutions obtained using the multi-wave algorithm approach by Glover et al. [ 57 ], plus the computational time taken to reach these solutions. To deal with the different CPU performance times for each previous method found in the literature, the results can be refactored in time using the performance characteristics of the employed CPUs. We used the tables published on specialized websites; i.e., http://www.techarp.com and https://www.userbenchmark.com , which evaluate CPU performance, to complete the conversion and normalisation for fair results. Based on the PLUS column, in Table  4 we include the number of times that our computer was faster than each computer used in the literature. It should be noted that it was not possible to normalize the times reported in Resende and Werneck [ 25 ] due to lack of comparative information and the difficulty of the conversion process when employing a CPU with a different architecture (RISC). However, it is reasonable to assume that the normalized times of their work would be very competitive.

Table  6 presents the results obtained for the uncertainty levels considered using the MED benchmark instances. Each instance was run 10 times using our simheuristic algorithm with different seeds for the random number generator, and the best results are reported. The first column identifies the instances, while the following three display the results obtained by our approach for the deterministic UFLP. First, the column labeled BKS reports the best-known solutions provided by the Gurobi Optimizer solver, along with our best-found deterministic solutions labeled OBD . We also calculate the percentage gaps of the best-found deterministic solutions compared to the best-known solutions. The remaining columns showcase the results obtained for three different uncertainty levels. In the second section of the table, we report the results obtained for the low uncertainty level. The OBD-S column displays the expected cost obtained when evaluating the best deterministic solution ( OBD ) in a stochastic scenario with a low uncertainty level. To compute the expected cost, an intensive simulation process is applied to the OBD . This process aims to assess the quality of our best-found deterministic solutions at varying uncertainty levels. The next column (OBS) shows the expected cost obtained using our simheuristic approach for the stochastic version of the problem. This approach considers the random service costs during the solution search. The subsequent section of the table presents the results for the medium uncertainty level. In the OBD-S column, we report the expected cost obtained when evaluating the best deterministic solution ( OBD ) in a stochastic scenario with medium uncertainty. Similarly, the next column ( OBS ) displays the expected cost obtained using our simheuristic algorithm. Finally, the last section of the table exhibits the results for the high uncertainty level. The OBD-S column shows the expected cost obtained when evaluating the best deterministic solution ( OBD ) in a stochastic scenario with high uncertainty. The last column ( OBS ) displays the expected cost obtained with our simheuristic approach.

For the sake of completeness, Table  7 covers the additional details of the computational results omitted in Table  6 . The first column identifies the instances, while the following four columns display the facilities open cost, the related nonlinear penalty cost, the threshold defined as the instance maximum service cost and the number of open facilities of the best-found solution, respectively. The remaining columns showcase the percentage gaps obtained for three different uncertainty levels, along with the number of open facilities of the best-found stochastic solutions. In the second section of the table, we report the results obtained for the low uncertainty level. The OBD-S GAP (%) column displays the percentage gap obtained when comparing the best deterministic solutions simulated in a low uncertainty scenario with the best-known solutions ( BKS ). The next column ( OBS GAP (%) ) shows the percentage gap obtained when comparing our simheuristic approach. The subsequent column ( OPEN ) presents the number of open facilities of the best-found stochastic solution. The subsequent section of the table offers the results for the medium uncertainty level. In the OBD-S GAP (%) column, we report the percentage gap obtained when comparing the best deterministic solution at a medium uncertainty level to the best-known solutions ( BKS ). The next column ( OBS (%) ) displays the percentage gap obtained using our simheuristic algorithm. Moreover, the subsequent column ( OPEN ) presents the number of open facilities of the best-found stochastic solution. Finally, the last section of the table exhibits the results for the high uncertainty level. The OBD-S GAP (%) column depicts the percentage gap obtained when comparing the best deterministic solution at a high uncertainty level. The next column ( OBS GAP (%) ) displays the percentage gap obtained using our simheuristic algorithm. The last column (OPEN) displays the number of open facilities of the best-found stochastic solution.

Gaps of \(\sqrt{n}/10\) opening cost scheme instances w.r.t the BKS

Gaps of \(\sqrt{n}/100\) opening cost scheme instances w.r.t the BKS

Gaps of \(\sqrt{n}/1000\) opening cost scheme instances w.r.t the BKS

Figures  7 ,  8 and  9 depict an overview of Table  6 by showing our algorithm’s performance for all the considered uncertainty levels. In these box plots, the horizontal and vertical axes represent the three uncertainty levels and the percentage gap obtained in relation to the BKS reported by the Gurobi Optimization solver, respectively. Note for the deterministic version of the UFLP, that our tabu search with the path re-linking approach nearly reaches the BKS for the \(\sqrt{n}/10\) , \(\sqrt{n}/100\) , and \(\sqrt{n}/1000\) opening cost schemes, with a gap of approximately \(0.00\%\) , \(0.03\%\) , and \(0.04\%\) , respectively. These results highlight the quality of our algorithm because our approach provides highly competitive solutions. Regarding the stochastic version of the UFLP, which is the main contribution of this paper, the obtained results show that the solutions provided by our heuristic approach for the three different uncertainty levels clearly outperform the solutions for the deterministic UFLP when they are simulated at the corresponding uncertainty level. In other words, our best-found solutions for the deterministic version of the problem ( OBD ) might be suboptimal when uncertainty is considered. Hence the importance of integrating simulation methods when dealing with optimization problems with uncertainty. Note also that the OBD can be seen as a reference lower bound in an ideal scenario with perfect information (i.e., without uncertainty) for the expected cost under uncertainty conditions. Similarly, OBD-S can be seen as an upper bound for the expected cost at the different uncertainty levels. As expected, the gaps for all three opening cost schemes worsen as the uncertainty level of k increases. For the specific case of the \(\sqrt{n}/10\) opening cost scheme, gaps increase more than for others. As shown in Table  7 , this is because the penalty cost (opening cost) is higher in this case. The penalty amounts to 3-5% of the total costs, but is less than 1.5% for the \(\sqrt{n}/100\) opening cost scheme and less than 0.5% for the remaining one. In addition, the number of open facilities is much smaller. These are cases in which a few facilities with high capacity serve many customers. The average number of open facilities for this opening cost scheme is around 18 in the best-found deterministic solutions, but rises to over 430 for the instances with suffix 1000. In this high opening cost scheme, where there are few viable facilities to open and with a high cost penalty, the OBD-S result is that which obtains the worst values on average, with a gap of 8.99%, 15.42% and 30.56% for all three uncertainty levels. In this environment, our simheuristic approach is able to achieve a 3 to 5 times better improvement (1.81%, 3.80%, and 10.71%). For the remaining cases (suffixes 100 and 1000) where the penalty costs are lower and the number of open installations is bigger, and the gaps shown in Table  7 and Figs.  7 ,  8 and  9 are smaller. It is not surprising that the OBD solutions better perform under uncertainty when the penalty cost is lower. Our OBS solutions improve the deterministic model’s performance for all the different k uncertainty levels, and also for all the opening cost schemes, which demonstrates the validity of the proposed approach.

When taking everything into account, the findings validate the significance of factoring in uncertainty while searching for a solution because it can greatly affect the quality of our best-found solutions. For instance, when our best deterministic solution is simulated in a stochastic scenario, it yields a much higher expected cost. Conversely, our best-found stochastic solution ( OBS ) provides a lower expected cost. Therefore, incorporating uncertainty aspects during the search process produces better results than when simulating uncertainty elements after finding a solution for the deterministic version of the problem.

6 Conclusions

In this paper, we analyze the UFLP under uncertainty, which occurs in several real-life systems. This uncertainty arises when inputs, such as customer demands or service costs, are random variables instead of deterministic values. To solve this optimization problem, we propose a novel solving methodology that combines a tabu search metaheuristic with path-relinking. The proposed algorithm is tested on the largest instances from the literature because they are the most challenging ones. The results show that our approach is capable of obtaining near-optimal solutions in short computational times. Additionally, the results are compared to the BKS from the literature, along with other competitive approaches, to validate our algorithm’s effectiveness in the deterministic scenario. Then the algorithm is converted into a simheuristic by integrating it with a simulation component to solve the UFLP with random service costs. The log-normal probability distribution is employed to model the random service costs. The obtained results show our simheuristic approach outperforms the simulated deterministic solutions when uncertainty is considered. Thus our simheuristic approach constitutes a general methodology that can be employed to efficiently solve several FLP variants under uncertainty.

One of the limitations of the present work is that it considers only uncertainty of a stochastic nature. Thus extending the simheuristic algorithm to address the UFLP in a more general setting, including stochastic and nonstochastic uncertainty elements like type-2 fuzzy systems [ 58 ] and rough sets [ 59 ], is a future research line. Another limitation is that simulations are performed on a single CPU core. As a future research line, it would be interesting to consider a parallel version of the simheuristic algorithm to obtain optimal solutions in an even shorter time, which may allow fast decision making in time-sensitive scenarios. Finally, another possible extension for this paper is to develop a similar combination of a tabu search metaheuristic with a path-relinking simheuristic to obtain near-optimal solutions in short computational times for other optimization problems under uncertainty, for example, vehicle routing problems, arc routing problems and scheduling problems.

Data availability statement

Data sharing is not applicable to this article because no datasets were generated or analyzed during the present study.

Stollsteimer JF (1961) The Effect of Technical Change and Output Expansion on the Optimum Number, Size, and Location of Pear Marketing Facilities in a California Pear Producing Region. University of California, Berkeley

Google Scholar  

Kuehn AA, Hamburger MJ (1963) A heuristic program for locating warehouses. Manage Sci 9(4):643–666

Article   Google Scholar  

Manne AS (1964) Plant location under economies-of-scale-decentralization and computation. Manage Sci 11(2):213–235

Balinski ML (1966) On finding integer solutions to linear programs. In: Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems, pp. 225–248. Yortown Heights, NY

Verter V (2011) Uncapacitated and capacitated facility location problems. Foundations of Location Analysis. 155:25–37

Cornuejol G, Nemhauser GL, Wolsey LA (1990) The uncapacitated facility location problem. In: Mirchandani, P.B., Francis, R.L. (eds.) Discrete Location Theory. Wiley Series in Discrete Mathematics and Optimization, pp. 119–171. John Wiley & Sons, New York

Silva FJF, Figuera DS (2007) A capacitated facility location problem with constrained backlogging probabilities. Int J Prod Res 45(21):5117–5134

Klose A, Drexl A (2005) Facility location models for distribution system design. Eur J Oper Res 162(1):4–29

Article   MathSciNet   Google Scholar  

Yang Z, Chu F, Chen H (2012) A cut-and-solve based algorithm for the single-source capacitated facility location problem. Eur J Oper Res 221(3):521–532

Correia I, Saldanha-da-Gama F (2019) Facility location under uncertainty. Location Science, 185–213

Figueira G, Almada-Lobo B (2014) Hybrid simulation-optimization methods: A taxonomy and discussion. Simul Model Pract Theory 46:118–134

Amaran S, Sahinidis NV, Sharda B, Bury SJ (2016) Simulation optimization: a review of algorithms and applications. Ann Oper Res 240(1):351–380

Castaneda J, Martin XA, Ammouriova M, Panadero J, Juan AA (2022) A fuzzy simheuristic for the permutation flow shop problem under stochastic and fuzzy uncertainty. Mathematics 10(10)

Hatami S, Calvet L, Fernandez-Viagas V, Framinan JM, Juan AA (2018) A simheuristic algorithm to set up starting times in the stochastic parallel flowshop problem. Simul Model Pract Theory 86:55–71

Glover F (1986) Future paths for integer programming and links to artificial intelligence. Computers & Operations Research. 13(5):533–549

Al-Sultan KS, Al-Fawzan MA (1999) A tabu search approach to the uncapacitated facility location problem. Ann Oper Res 86:91–103

Sun M (2006) Solving the uncapacitated facility location problem using tabu search. Computers & Operations Research. 33(9):2563–2589

Erlenkotter D (1978) A dual-based procedure for uncapacitated facility location. Oper Res 26(6):992–1009

Efroymson M, Ray T (1966) A branch-bound algorithm for plant location. Oper Res 14(3):361–368

Spielberg K (1969) Algorithms for the simple plant-location problem with some side conditions. Oper Res 17(1):85–111

Körkel M (1989) On the exact solution of large-scale simple plant location problems. Eur J Oper Res 39(2):157–173

Hochbaum DS (1982) Approximation algorithms for the set covering and vertex cover problems. SIAM J Comput 11(3):555–556

Ghosh D (2003) Neighborhood search heuristics for the uncapacitated facility location problem. Eur J Oper Res 150(1):150–162

Michel L, Van Hentenryck P (2004) A simple tabu search for warehouse location. Eur J Oper Res 157(3):576–591

Resende MG, Werneck RF (2006) A hybrid multistart heuristic for the uncapacitated facility location problem. Eur J Oper Res 174(1):54–68

C Martins L Hirsch P, Juan AA (2021) Agile optimization of a two-echelon vehicle routing problem with pickup and delivery. Int Trans Oper Res 28(1):201–221

Martins LDC, Tarchi D, Juan AA, Fusco A (2022) Agile optimization for a real-time facility location problem in internet of vehicles networks. Networks 79(4):501–514

Holmberg K, Rönnqvist M, Yuan D (1999) An exact algorithm for the capacitated facility location problems with single sourcing. Eur J Oper Res 113(3):544–559

Díaz JA, Fernández E (2002) A branch-and-price algorithm for the single source capacitated plant location problem. Journal of the Operational Research Society. 53(7):728–740

Chen C-H, Ting C-J (2008) Combining Lagrangian heuristic and ant colony system to solve the single source capacitated facility location problem. Transportation Research Part E: Logistics and Transportation Review. 44(6):1099–1122

Ahuja RK, Orlin JB, Pallottino S, Scaparra MP, Scutellà MG (2004) A multi-exchange heuristic for the single-source capacitated facility location problem. Manage Sci 50(6):749–760

Filho VJMF, Galvão RD (1998) A tabu search heuristic for the concentrator location problem. Locat Sci 6(1):189–209

Delmaire H, Díaz JA, Fernández E, Ortega M (1999) Reactive GRASP and tabu search based heuristics for the single source capacitated plant location problem. INFOR: Information Systems and Operational Research 37(3):194–225

Estrada-Moreno A, Ferrer A, Juan AA, Bagirov A, Panadero J (2020) A biased-randomised algorithm for the capacitated facility location problem with soft constraints. J Oper Res Soc 71(11):1799–1815

Fotakis D (2011) Online and incremental algorithms for facility location. ACM SIGACT News 42(1):97–131

Eiselt HA, Marianov V (eds) (2011) Foundations of Location Analysis. International Series in Operations Research & Management Science. Springer, Boston, MA

Snyder LV, Daskin MS (2006) Stochastic p-robust location problems. IIE Trans 38(11):971–985

Balachandran V, Jain S (1976) Optimal facility location under random demand with general cost structure. Naval Research Logistics Quarterly. 23(3):421–436

Reyes-Rubiano L, Ferone D, Juan AA, Faulin J (2019) A simheuristic for routing electric vehicles with limited driving ranges and stochastic travel times. SORT-Statistics and Operations Research Transactions. 43(1):0003–0024

De Armas J, Juan AA, Marquès JM, Pedroso JP (2017) Solving the deterministic and stochastic uncapacitated facility location problem: From a heuristic to a simheuristic. Journal of the Operational Research Society. 68(10):1161–1176

Quintero-Araujo CL, Guimarans D, Juan AA (2021) A simheuristic algorithm for the capacitated location routing problem with stochastic demands. Journal of Simulation. 15(3):217–234

Bayliss C, Panadero J (2023) Simheuristic and learnheuristic algorithms for the temporary-facility location and queuing problem during population treatment or testing events. J Simul 0(0):1–20

Juan AA, Keenan P, Martí R, McGarraghy S, Panadero J, Carroll P, Oliva D (2023) A review of the role of heuristics in stochastic optimisation: From metaheuristics to learnheuristics. Ann Oper Res 320(2):831–861

Marques MDC, Dias JM (2018) Dynamic location problem under uncertainty with a regret-based measure of robustness. Int Trans Oper Res 25(4):1361–1381

Zhang J, Li M, Wang Y, Wu C, Xu D (2019) Approximation algorithm for squared metric two-stage stochastic facility location problem. J Comb Optim 38:618–634

Ramshani M, Ostrowski J, Zhang K, Li X (2019) Two level uncapacitated facility location problem with disruptions. Computers & Industrial Engineering. 137:106089

Koca E, Noyan N, Yaman H (2021) Two-stage facility location problems with restricted recourse. IISE Transactions. 53(12):1369–1381

Gruler A, Quintero-Araújo CL, Calvet L, Juan AA (2017) Waste collection under uncertainty: A simheuristic based on variable neighbourhood search. European Journal of Industrial Engineering. 11(2):228–255

Fu MC (2003) Feature article: Optimization for simulation: Theory vs. practice. INFORMS Journal on Computing 14(3):192–215

Ho SC, Gendreau M (2006) Path relinking for the vehicle routing problem. Journal of Heuristics. 12:55–72

Peng B, Lü Z, Cheng TCE (2015) A tabu search/path relinking algorithm to solve the job shop scheduling problem. Computers & Operations Research. 53:154–164

Chica M, Juan AA, Bayliss C, Cordón O, Kelton WD (2020) Why simheuristics? benefits, limitations, and best practices when combining metaheuristics with simulation. SORT-Statistics and Operations Research Transactions. 44(2):311–334

MathSciNet   Google Scholar  

Ahn S, Cooper C, Cornuejols G, Frieze A (1988) Probabilistic analysis of a relaxation for the k-median problem. Math Oper Res 13(1):1–31

Barahona F, Chudak FA (2000) In: Pardalos, P.M. (ed.) Solving Large Scale Uncapacitated Facility Location Problems, pp. 48–62. Springer, Boston, MA

Kim JS, Yum B-J (2008) Selection between weibull and lognormal distributions: A comparative simulation study. Computational Statistics & Data Analysis. 53(2):477–485

Montgomery DC (2017) Design and Analysis of Experiments. John wiley & Sons, Hoboken, New Jersey

Glover F, Hanafi S, Guemri O, Crevits I (2018) A simple multi-wave algorithm for the uncapacitated facility location problem. Frontiers of Engineering Management. 5(4):451–465

Castillo O, Melin P, Kacprzyk J, Pedrycz W (2007) Type-2 fuzzy logic: theory and applications. In: 2007 IEEE International Conference on Granular Computing (GRC 2007), pp. 145–145. IEEE

Pawlak Z (1982) Rough sets. International Journal of Computer & Information Sciences. 11(5):341–356

Download references

Acknowledgements

This work has been partially supported by the Spanish Ministry of Science (PDC2022-133957-I00, PID2022-138860NB-I00 and RED2022-134703-T) and by the Generalitat Valenciana (PROMETEO/2021/065).

Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.

Author information

Authors and affiliations.

Research Center on Production Management and Engineering, Universitat Politécnica de Valéncia, Plaza Ferrandiz Carbonell, Alcoy, 03801, Spain

David Peidro, Xabier A. Martin & Angel A. Juan

Department of Computer Architecture and Operating Systems, Universitat Autónoma de Barcelona, Carrer de les sitges, Bellaterra, 08193, Spain

Javier Panadero

You can also search for this author in PubMed   Google Scholar

Contributions

Conceptualization: Angel A. Juan, David Peidro; Methodology: Angel A. Juan, Xabier A. Martin, Javier Panadero; Software: David Peidro; Validation: Angel A. Juan, Xabier A. Martin, David Peidro, Javier Panadero; Software: David Peidro; Writing-original draft preparation: Xabier A. Martin, David Peidro, Javier Panadero; Writing-review and editing: Angel A. Juan, David Peidro. All authors have read and agreed to the published version of the manuscript.

Corresponding author

Correspondence to David Peidro .

Ethics declarations

Ethics approval.

This article does not contain any studies with human participants or animals performed by any of the authors

Conflict of interest

The authors declare no conflict of interest

Additional information

Publisher's note.

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

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, 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 changes were made. 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/4.0/ .

Reprints and permissions

About this article

Peidro, D., Martin, X.A., Panadero, J. et al. Solving the uncapacitated facility location problem under uncertainty: a hybrid tabu search with path-relinking simheuristic approach. Appl Intell (2024). https://doi.org/10.1007/s10489-024-05441-x

Download citation

Accepted : 31 March 2024

Published : 24 April 2024

DOI : https://doi.org/10.1007/s10489-024-05441-x

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

• Facility location problem
• Uncertainty
• Simheuristics
• Find a journal
• Publish with us
• Track your research